Physical Properties of Polymers Handbook, 2nd edition

Physical Properties of Polymers Handbook Second Edition

Physical Properties of Polymers Handbook Second Edition

Edited by

James E. Mark Polymer Research Center and Department of Chemistry University of Cincinnati Cincinnati, Ohio

Editor: James E. Mark Distinguished Research Professor Department of Chemistry Crosley Tower, Martin Luther King Drive University of Cincinnati Cincinnati, OH 45221-0172 [email protected]

Library of Congress Control Number: 2005938500 ISBN-13: 978-0-387-31235-4 ISBN-10: 0-387-31235-8

eISBN-13: 978-0-387-69002-5 eISBN-10: 0-387-69002-6

Printed on acid-free paper. ß2007 Springer ScienceþBusiness Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer ScienceþBusiness Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 10 9 8 7 6 5 4 3 2 1 springer.com

Contents Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

PART I. STRUCTURE 1.

Chain Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. R. Sundararajan

3

2.

Names, Acronyms, Classes, and Structures of Some Important Polymers . . . . . . . . . . . . . . . . . . . . . . . Chandima Kumudinie Jayasuriya and Jagath K. Premachandra

25

PART II. THEORY 3.

The Rotational Isomeric State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carin A. Helfer and Wayne L. Mattice

43

4.

Computational Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joel R. Fried

59

5.

Theoretical Models and Simulations of Polymer Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrzej Kloczkowski and Andrzej Kolinski

67

6.

Scaling, Exponents, and Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohamed Daoud, H. Eugene Stanley, and Dietrich Stauffer

83

PART III. THERMODYNAMIC PROPERTIES 7.

Densities, Coefficients of Thermal Expansion, and Compressibilities of Amorphous Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert A. Orwoll

93

8.

Thermodynamic Properties of Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . George I. Makhatadze

103

9.

Heat Capacities of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianye Wen

145

10.

Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yong Yang

155

v

vi / CONTENTS 11.

Thermodynamic Quantities Governing Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Mandelkern and R. G. Alamo

165

12.

The Glass Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Donald J. Plazek and Kia L. Ngai

187

13.

Sub-Tg Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joel R. Fried

217

14.

Polymer—Solvent Interaction Parameter x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert A. Orwoll and Pamela A. Arnold

233

15.

Theta Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. R. Sundararajan

259

16.

Solubility Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Zeng, Y. Du, Y. Xue, and H. L. Frisch

289

17.

Mark—Houwink—Staudinger—Sakurada Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Zeng, Y. Du, Y. Xue, and H. L. Frisch

305

18.

Polymers and Supercritical Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annette D. Shine

319

19.

Thermodynamics of Polymer Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hany B. Eitouni and Nitash P. Balsara

339

PART IV. SPECTROSCOPY 20.

NMR Spectroscopy of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alan E. Tonelli and Jeffery L. White

21.

Broadband Dielectric Spectroscopy to Study the Molecular Dynamics of Polymers Having Different Molecular Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Kremer

385

Group Frequency Assignments for Major Infrared Bands Observed in Common Synthetic Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Noda, A. E. Dowrey, J. L. Haynes, and C. Marcott

395

22.

23.

Small Angle Neutron and X-Ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . George D. Wignall

359

407

PART V. MECHANICAL PROPERTIES 24.

Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Witold Brostow

423

25.

Chain Dimensions and Entanglement Spacings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. J. Fetters, D. J. Lohse, and R. H. Colby

447

26.

Temperature Dependences of the Viscoelastic Response of Polymer Systems . . . . . . . . . . . . . . . . . . . K. L. Ngai and D. J. Plazek

455

27.

Adhesives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alphonsus V. Pocius

479

CONTENTS

/ vii

28.

Some Mechanical Properties of Typical Polymer-Based Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianye Wen

487

29.

Polymer Networks and Gels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferenc Horkay and Gregory B. McKenna

497

30.

Force Spectroscopy of Polymers: Beyond Single Chain Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xi Zhang, Chuanjun Liu, and Weiqing Shi

525

PART VI. REINFORCING PHASES 31.

Carbon Black . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manfred Klu¨ppel, Andreas Schro¨der and Gert Heinrich

539

32.

Properties of Polymers Reinforced with Silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chandima Kumudinie Jayasuriya and Jagath K. Premachandra

551

33.

Physical Properties of Polymer/Clay Nanocomposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clois E. Powell and Gary W. Beall

561

34.

Polyhedral Oligomeric Silsesquioxane (POSS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guirong Pan

577

35.

Carbon Nanotube Polymer Composites: Recent Developments in Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. C. Weisenberger, R. Andrews, and T. Rantell

36.

Reinforcement Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gert Heinrich, Manfred Klu¨ppel, and Thomas Vilgis

585 599

PART VII. CRYSTALLINITY AND MORPHOLOGY 37.

Densities of Amorphous and Crystalline Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladyslav Kholodovych and William J. Welsh

611

38.

Unit Cell Information on Some Important Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Edward S. Clark

619

39.

Crystallization Kinetics of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rahul Patki, Khaled Mezghani, and Paul J. Phillips

625

40.

Block Copolymer Melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Castelletto and I. W. Hamley

641

41.

Polymer Liquid Crystals and Their Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Witold Brostow

653

42.

The Emergence of a New Macromolecular Architecture: ‘‘The Dendritic State’’ . . . . . . . . . . . . . . . . . Donald A. Tomalia

671

43.

Polyrotaxanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feihe Huang, Adam M.-P. Pederson, and Harry W. Gibson

693

44.

Foldamers: Nanoscale Shape Control at the Interface Between Small Molecules and High Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Morris M. Slutsky, Richard A. Blatchly, and Gregory N. Tew

699

viii / 45.

CONTENTS Recent Advances in Supramolecular Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Varun Gauba and Jeffrey D. Hartgerink

715

PART VIII. ELECTRICAL, OPTICAL AND MAGNETIC PROPERTIES 46.

Conducting Polymers: Electrical Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arthur J. Epstein

725

47.

Electroluminescent Polymer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leni Akcelrud

757

48.

Magnetic, Piezoelectric, Pyroelectric, and Ferroelectric Properties of Synthetic and Biological Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrzej Kloczkowski and Taner Z. Sen

49.

Nonlinear Optical Properties of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. M. K. P. Wijekoon, K.-S. Lee, and P. N. Prasad

50.

Refractive Index, Stress-Optical Coefficient, and Optical Configuration Parameter of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vassilios Galiatsatos

787 795

823

PART IX. RESPONSES TO RADIATION, HEAT, AND CHEMICAL AGENTS 51.

Ultraviolet Radiation and Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anthony L. Andrady

857

52.

The Effects of Electron Beam and g-Irradiation on Polymeric Materials . . . . . . . . . . . . . . . . . . . . . . . . K. Dawes, L. C. Glover, and D. A. Vroom

867

53.

Flammability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Archibald Tewarson

889

54.

Thermal-Oxidative Stability and Degradation of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladyslav Kholodovych and William J. Welsh

927

55.

Synthetic Biodegradable Polymers for Medical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laura J. Suggs, Sheila A. Moore, and Antonios G. Mikos

939

56.

Biodegradability of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anthony L. Andrady

951

57.

Properties of Photoresist Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qinghuang Lin

965

58.

Pyrolyzability of Preceramic Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yi Pang, Ke Feng, and Yitbarek H. Mariam

981

PART X. OTHER PROPERTIES 59.

Surface and Interfacial Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Afshin Falsafi, Subu Mangipudi, and Michael J. Owen

1011

60.

Acoustic Properties of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moitreyee Sinha and Donald J. Buckley

1021

61.

CONTENTS

/ ix

Permeability of Polymers to Gases and Vapors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. A. Stern and J. R. Fried

1033

PART XI. MISCELLANEOUS 62.

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ping Xu

1051

63.

Units and Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuhong Wang

1057

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1063

Contributors R. G. Alamo Department of Chemical and Biomedical Engineering, Florida Agricultural and Mechanical University, and Florida State University College of Engineering, Tallahassee, FL 32310-6046, [email protected] Anthony L. Andrady Engineering and Technology Division, RTI International, Research Triangle Park, NC 27709, [email protected] Pamela A. Arnold Chemistry Department, Gettysburg College, Gettysburg, PA 17325, [email protected] Rodney Andrews University of Kentucky Center for Applied Energy Research, 2540 Research Park Dr, Lexington, KY 40511, [email protected] Nitash P. Balsara Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA 94720, [email protected] Gary W. Beall Center for Nanophase Research, Southwest Texas State University, San Marcos, TX 78666, gb11@ txstate.edu Richard A. Blatchly Chemistry Department, Keene State College, Keene NH 03435, [email protected] Witold Brostow Department of Materials Science and Engineering and Department of Physics, University of North Texas, PO Box 305310, Denton, TX 76203-5310, [email protected] Donald J. Buckley General Electric Global Research Center, One Research Circle, Niskayuna, NY 12309, buckley@crd. ge.com V. Castelletto Department of Chemistry, University of Reading, Reading, RG6 6AD, UK. Edward S. Clark Department of Materials Science and Engineering, The University of Tennessee, Knoxville, TN 37996, [email protected] R. H. Colby Materials Science and Engineering, Penn State University, University Park, PA 16802, [email protected] Mohamed Daoud Laboratoire Leon Brillouin (CEA-CNRS), CE Saclay, Gif-sur-Yvette, Cedex, France, daoud@llb. saclay.cea.fr K. Dawes Department of Materials Science and Engineering, North Carolina State University, Campus Box 7907, Raleigh, NC 27695, [email protected] A. E. Dowrey Miami Valley Innovation Center, 11810 E. Miami River Rd., Cincinnati, OH 45242, [email protected] Hany B. Eitouni Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA 94720, [email protected] Arthur J. Epstein Department of Physics and Department of Chemistry, The Ohio State University, Columbus, OH 43210-1117, [email protected] xi

xii / CONTRIBUTORS Afshin Falsafi 3 M Company, 3M Center, 260-2B-12, St. Paul, MN 55144, [email protected] Lewis J. Fetters School of Chemical Engineering, Cornell University, Ithaca, NY 14853, [email protected] Joel R. Fried Department of Chemical and Materials Engineering, Mail Location #0012, The University of Cincinnati, [email protected] Richard H. Friend, F. R. S., Optoelectronics Group, Cavendish Laboratory, Madingly Road, Cambridge, CB3 OHE, UK, [email protected] Harry L. Frisch Department of Chemistry, State University of New York at Albany, Albany, NY 12222, hlf04@albany. Edu Vassilios Galiatsatos Equistar Chemicals, LP, 11530 Northlake Dr., Cincinnati, OH 45249, [email protected], [email protected] Varun Gauba Department of Chemistry and Bioengineering, 6100 Main Street, Rice University, Houston, TX 77005, [email protected] Harry W. Gibson Department of Chemistry, Virginia Polytechnic & State University, Blacksburg, VA 24061, hwgibson@ vt.edu L. C. Glover Tyco Electronics, 305 Constitution Dr, Menlo Park, CA 94025, [email protected] N. C. Greenham Optoelectronics Group, Cavendish Laboratory, Madingly Road, Cambridge, CB3 OHE, UK, email address not available Ian W. Hamley Department of Chemistry, University of Reading, Reading, RG6 6AD, UK. [email protected] Jeffrey D. Hartgerink Department of Chemistry and Bioengineering, 6100 Main Street, Rice University, Houston, TX 77005, [email protected] J. L. Haynes The Procter & Gamble Company, Beckett Ridge Technical Center, 8611 Beckett Road, West Chester, OH 45069, [email protected] Gert Heinrich Leibniz Institut fu¨r Polymerforschung Dresden e. V., Hohe Strasse 6, D-01069 Dresden, Germany, [email protected] Carin A. Helfer Institute of Polymer Science, The University of Akron, Akron, OH 44325-3909, [email protected] Ferenc Horkay National Institutes of Health, National Institute of Child Health and Human Development, Laboratory of Integrative and Medical Biophysics, Section on Tissue Biophysics and Biomimetics,, Bethesda, Maryland 20892, horkay@ helix.nih.gov Feihe Huang Department of Chemistry, Virginia Polytechnic & State University, Blacksburg, VA 24061, fhuang@chem. utah.edu Vladyslav Kholodovych Department of Pharmacology, University of Medicine & Dentistry of New Jersey (UMDNJ), Robert Wood Johnson Medical School and the UMDNJ Informatics Institute, Piscataway, NJ 08854, [email protected] Andrzej Kloczkowski L.H. Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA 50011, [email protected] Manfred Klu¨ppel Deutsches Institut fu¨r Kautschuktechnologie e. V., Eupener Straße 33, D-30519 Hannover, Germany, [email protected]

CONTRIBUTORS

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xiii

Andrzej Kolinski Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland, kolinski@chem. uw.edu.pl F. Kremer Universitat Leipzig, Fakultat f. Physik u. Geowissenschaften, Leipzig, Germany, [email protected] Chandima Kumudinie Jayasuriya Department of Chemistry, University of Kelaniya, Sri Lanka, [email protected] Kwang-Sup Lee Department of Polymer Science and Engineering, Hannam University, Daejeon 306-791, Korea, [email protected] Qinghuang Lin IBM Thomas J. Watson Research Center, 1101 Kitchawan Rd, Route 134/PO Box 218, Yorktown Heights, NY 10598, [email protected] Chuanjun Liu Department of Chemistry, Tsinghua University, Beijing 100084, P. R. China, [email protected] D. J. Lohse ExxonMobil Research and Engineering Company, Annandale NJ 08801-0998, [email protected] George Makhatadze Department of Biochemistry and Molecular Biology, Penn State University College of Medicine, Hershey, PA 17033, [email protected] L. Mandelkern Department of Chemistry and Biochemistry, Florida State University, Tallahassee, FL 32306-3015, [email protected] Subu Mangipudi Medtronic Corporation, 6800 Shingle Creek Parkway, Brooklyn Center, MN 55430, subu.mangipudi@ medtronic.com C. Marcott Miami Valley Innovation Center, 11810 E. Miami River Rd., Cincinnati, OH 45242, [email protected] J. E. Mark Department of Chemistry, Crosley Tower, Martin Luther King Drive, The University of Cincinnati, Cincinnati, OH 45221-0172, [email protected] Wayne L. Mattice Institute of Polymer Science, The University of Akron, Akron, OH 44325-3909, wlm@polymer. uakron.edu Gregory B. McKenna Department of Chemical Engineering, Texas Tech University, Lubbock, TX 79409-3121, greg. [email protected] Khaled Mezghani Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Box 169, Dhahran 31261, Saudi Arabia, [email protected] Antonios G. Mikos Department of Bioengineering, PO Box 1892, MS-142, Rice University, Houston, TX 77005-1892, [email protected] Sheila A. Moore Department of Bioengineering, PO Box 1892, MS-142, Rice University, Houston, TX 77005-1892, [email protected] Kia L. Ngai Code 6807, Naval Research Laboratory, Washington, DC 20375-5320, [email protected] Isao Noda The Procter & Gamble Company, Beckett Ridge Technical Center, 8611 Beckett Road, West Chester, OH 45069, [email protected] Robert A. Orwoll Department of Chemistry, College of William and Mary, Williamsburg, VA 23187-8795, [email protected] Michael J. Owen Dow Corning Corporation, Midland, MI 48686-0994, [email protected]

xiv /

CONTRIBUTORS

Guirong Pan Department of Chemical and Materials Engineering, The University of Cincinnati, Cincinnati, OH 45221-0012, [email protected] Yi Pang Department of Chemistry, University of Akron, Akron, OH 44325-3601, [email protected] Rahul Patki Department of Chemical and Materials Engineering, The University of Cincinnati, Cincinnati, OH 45221-0012, [email protected] Adam M. -P. Pederson Department of Chemistry, Virginia Polytechnic & State University, Blacksburg, VA 24061, adamp@ vt.edu Paul J. Phillips Department of Chemical and Materials Engineering, The University of Cincinnati, Cincinnati, OH 452210012, [email protected] Donald J. Plazek Department of Materials Science and Engineering, University of Pittsburgh, Pittsburgh, PA 15261, [email protected] Aphonsus V. Pocius 3 M Corporate Research Materials Laboratory, St. Paul, MN 55144-1000, [email protected] Clois E. Powell Center for Nanophase Research, Southwest Texas State University, San Marcos, TX 78666, cp21@ txstate.edu P. N. Prasad Department of Chemistry, The State University of New York at Buffalo, Buffalo, NY 14260-3000, pnprasad@ acsu.buffalo.edu Jagath K. Premachandra Department of Chemical and Process Engineering, University of Moratuwa, Sri Lanka, [email protected] T. Rantell University of Kentucky Center for Applied Energy Research, 2540 Research Park Dr, Lexington, KY 40511, [email protected] Andreas Schro¨der Rheinchemie Rheinan GmbH, Du¨sseldorfer str. 23–27, D-68219 Mannheim, Germany Taner Z. Sen Department of Biochemistry, Biophysics, and Molecular Biology, Iowa State University, Ames, IA 50011, [email protected] Weiqing Shi Department of Chemistry, Tsinghua University, Beijing 100084, P. R. China, [email protected] Annette D. Shine Department of Chemical Engineering, University of Delaware, Newark, DE 19716, shine@donald. che.udel.edu Moitreyee Sinha General Electric Global Research Center, One Research Circle, Niskayuna, NY 12309, [email protected] Morris Slutsky Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA 01003, [email protected] H. Eugene Stanley Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, [email protected] Dietrich Stauffer Institute of Theoretical Physics, Cologne University, D-50923 Koln, Euroland, stauffer@thp. Uni-Koeln.DE S. Alexander Stern Department of Biomedical and Chemical Engineering, Syracuse University, Syracuse, NY 13244, USA, [email protected]

CONTRIBUTORS

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Laura J. Suggs Department of Biomedical Engineering, University of Texas at Austin, Austin, TX 78712, Laura.Suggs@ engr.utexas.edu P. R. Sundararajan Department of Chemistry, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada K1S 5B6, [email protected] Gregory N. Tew Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA 01003, [email protected] Archibald Tewarson FM Global, Research, 1151 Boston Providence Turnpike, Norwood, MA 02062, archibald. [email protected] Donald A. Tomalia Dendritic Nanotechnologies Inc./Central Michigan University, 2625 Denison Drive, Mt. Pleasant, MI 48858, [email protected] Alan E. Tonelli Fiber & Polymer Science Program, North Carolina State University, Raleigh, NC 27695, [email protected] Thomas Vilgis Max Planck Institut fur Polymerforschung, Postfach 3148, D-6500, Mainz, Germany 55021, vilgis@ mpip-mainz.mpg.de, D. A. Vroom Tyco Electronics, 305 Constitution Dr, Menlo Park, CA 94025, [email protected] Shuhong Wang DuPont Performance Elastomers L.L.C., DuPont Experimental Station, P.O. Box 80293, Wilmington, DE 19880, [email protected] M. C. Weisenberger University of Kentucky Center for Applied Energy Research, 2540 Research Park Dr, Lexington, KY 40511, [email protected] William J. Welsh Department of Pharmacology, University of Medicine & Dentistry of New Jersey (UMDNJ), Robert Wood Johnson Medical School and the UMDNJ Informatics Institute, Piscataway, NJ 08854, [email protected] Jianye Wen ALZA Corp., 1900 Charleston Rd., Mountain View, CA 94039, [email protected], [email protected] Jeffery L. White Department of Chemistry, Department of Chemistry, Oklahoma State University, [email protected] George D. Wignall Center for Neutron Scattering, Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6393, [email protected], [email protected] W. M. K. P. Wijekoon Applied Materials, 3303 Scott Blvd; M/S 10852, Santa Clara, CA 95054, kapila_wijekoon@ amat.com Ping Xu W.L. Gore & Associates, Inc., Cherry Hill Division, 2401 Singerly Road, P.O. Box 1220, Elkton, MD 21922-1220, [email protected], [email protected] Yong Yang Benjamin Moore and Co., Flanders, NJ 07836, [email protected] Wanxue Zeng Albany NanoTech, CESTM Building, 251 Fuller Road, Albany, NY 12203, [email protected] Xi Zhang Department of Chemistry, Tsinghua University, Beijing 100084, P. R. China, [email protected]

Preface to the Second Edition As before, the goal of this handbook is to provide concise information on the properties of polymeric materials, particularly those most relevant to the areas of physical chemistry and chemical physics. The hope is that it will simplify some of the problems of finding useful information on polymer properties. All of the chapters of the first edition were updated and 11 entirely new chapters added. Four of them focus on novel polymeric structures, specifically dendrimers, polyrotaxanes, foldamers, and supramolecular polymers in general. Another group of chapters covers reinforcing phases in polymers, including carbon black, silica, clays, polyhedral oligomeric silsesquioxanes (POSS), carbon nanotubes, and relevant theories. The final new chapter describes experiments on single polymer chains. It is a pleasure to acknowledge with gratitude the encouragement, support, and technical assistance provided by Springer, particularly David Packer, Lee Lubarsky, Felix Portnoy, and, earlier, Hans Koelsch. The editor also wishes to thank his wife Helen for the type of understanding and support that helps get one through book projects of this complexity.

James E. Mark Cincinnati, Ohio December 2006

xvii

Preface to the First Edition This handbook offers concise information on the properties of polymeric materials, particularly those most relevant to the areas of physical chemistry and chemical physics. It thus emphasizes those properties of greatest utility to polymer chemists, physicists, and engineers interested in characterizing such materials. With this emphasis, the more synthetic–organic topics such as the polymerization process and the chemical modification of polymers were considered beyond its scope. The contributors to this handbook have endeavored to be highly selective, choosing and documenting those results considered to have the highest relevance and reliability. There was thus no attempt to be exhaustive and comprehensive. The careful selection of the results included, however, suggests it should nonetheless provide the great majority of topics and data on polymer properties likely to be sought by members of the polymer community. Extensive indexing should facilitate locating the desired information, and it is hoped that the modest size of the handbook will give it considerable portability and wide availability. Every attempt has been made to include modern topics not covered in a convenient handbook format elsewhere, such as scaling and fractal dimensions, computational parameters, rotational isomeric state models, liquid–crystalline polymers, medical applications, biodegradability, surface and interfacial properties, microlithography, supercritical fluids, pyrolyzability, electrical conductivity, nonlinear optical properties, and electroluminescence. All contributions to this volume were extensively reviewed by a minimum of two referees, to insure articles of the highest quality and relevance. Many of the reviewers were chosen from the Editorial Board of the AIP Series in Polymers and Complex Materials, of which this handbook is a part. Their important contributions are gratefully acknowledged, as are those of the Editors-in-Chief of the Series, Ronald Larson and Philip A. Pincus. One Editorial Board member, Robert E. Cohen, deserves special acknowledgment and sincere thanks. He not only originated the idea of doing a handbook of this type, but also contributed tremendously to its realization. Charles H. Doering and Maria Taylor (and earlier, Zvi Ruder) also provided unfailing support and encouragement in this project. It has been a distinct pleasure working with them and other members of the AIP Press: K. Okun, K. S. Kleinstiver, M. Star, and C. Blaut. The editor also wishes to thank his wife Helen for the type of understanding and support that is not always easy to put into words. Both the editor and contributors to this volume would feel well rewarded if this handbook helps relieve some of the problems of finding useful information on polymer properties in the ever-growing scientific literature.

James E. Mark Cincinnati, Ohio November 1995

xix

CHAPTER 1

Chain Structures P. R. Sundararajan Department of Chemistry, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada K1S 5B6

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymers with Macrocyclic and Other Photoactive Groups . . . . . . . . . . . . . . . . . . . . . Polymers with Fullerene and Carbon Nanotube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotaxanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dendrimers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supramolecular Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 9 11 11 13 13 20 20 20

and molecular shuttle have been the focus of several research groups (see below). The intent is to acquire the ability to control the material at the atomic/molecular level, i.e., on the nano scale [1–5]. This chapter gives an overview of the literature on microstructures, "photonic" polymers, fullerence-based polymers, cyclics, rotaxanes, and dendrimers. The properties of polymers with other architectures and morphologies are discussed in various other chapters of this handbook. Please note that in this chapter, in the previous edition of this handbook, we had listed examples from published articles in Tables 1.1–1.8. Most of the topics discussed at that time were new and emerging. Since that time, publications in each of these topics have been numerous and cannot be accommodated within the scope and size of this chapter. The original tables are kept, however, since these include the initial work in these areas.

1.1 INTRODUCTION It is known that the physical properties of a polymer depend not only on the type of monomer(s) comprising it, but also on the secondary and tertiary structures, i.e., the stereochemistry of the linkage, the chain length and its distribution, its ability to crystallize or remain amorphous under various conditions, and the shape or distribution of the shapes of the chain in the crystalline and amorphous states. Through advances in polymer chemistry, in most cases polymers can be designed with specific properties. Control of the microstructure, e.g., the tacticity and molecular weight distribution of vinyl polymers, has been the focus of a number of papers in the last two decades. In most applications, a polymer, once designed as a product, has to be stable and maintain its structure and morphology under various temperatures and other environmental conditions during the lifetime of the product. However, the recent interest is also in changing the shape or morphology of the molecule instantaneously and reversibly, without any memory or hysteresis effects, with electrical, optical or mechanical stimulus. These "smart" materials are aimed towards such applications as information processing, storage, and retrieval, and molecular recognition similar to the biological systems. Synthetic efforts on in situ devices such as the photonic molecular wire, electronic molecular wire,

1.2 MICROSTRUCTURE Since the stereospecific polymerization of polyolefins pioneered by Natta, an extensive literature has developed in the synthesis, characterization, and utilization of polymers of defined microstructure. Although x ray diffraction could confirm the existence or absence of regular 3

4 / CHAPTER 1 diad is racemic (Fig. 1.1b) and regular alternation of the d and l centers along the chain defines a syndiotactic polymer. Random occurrence of d and l centers along the chain leads to an atactic polymer, as shown schematically in Fig. 1.1c. The convenience of defining the tacticity of a vinyl polymer in this manner and its application to developing the matrix methods for calculating the configurational average properties of these chains have been discussed by Flory [6]. The effect of tacticity on the properties of polymers has long been recognized, with such basic differences as in the glass transition temperature. Lemieux et al. [7] studied the effect of the tacticity of poly(methyl methacrylate) (PMMA) on its miscibility with poly(vinyl chloride) (PVC), chlorinated PVC and Saran. In a series of papers, Beaucage and Stein [8] and Beaucage et al. [9] examined the effect of the tacticity of poly(vinyl methyl ether) on its blend characteristics with polystyrene. Many of the ‘‘regular’’ or isotactic polymers have been studied in terms of the crystalline structure, crystal growth, and morphology [10,11]. These studies also prompted development of theories on chain folding, nucleation, and growth, etc., to model the experimental observations, as well as to predict the properties of these

microstructure, and infrared spectroscopy could be used to estimate the isotactic or syndiotactic content of a polymer, it was not until the development of NMR spectroscopy for microstructure analysis that the isotactic, syndiotactic, or atactic perpetuation extending to pentads and hexads could be determined quantitatively and accurately. This is dealt with in detail in the chapter by Tonelli in this handbook. A schematic for defining the tacticity of vinyl polymers of the type [(CH2)—(CHR)]n is shown in Fig. 1.1. If, as shown in Fig. 1.1(a), the skeletal bonds are in the trans conformation and lie in the plane of the paper, the R groups on successive asymmetric carbons projecting on the same side (up in this figure) defines a meso diad and perpetuation of this configuration leads to an isotactic polymer. Assignment of a configuration d to the asymmetric carbons in this figure is arbitrary. If, by a 1808 rotation of the chain, all the R groups are rendered to lie below the plane of the paper, the carbon centers are assigned an l configuration. The stereochemistry of the chain would not differ, however, if the chain ends are indistinguishable. Thus, an ‘‘all d’’ or ‘‘all l’’ chain is isotactic in character. If one of the asymmetric carbons of the diad is in the d configuration and the other is in l, the

H

H C

H

H

C

C (d)

(d)

C

C

C

H

R

(a)

H

R

H

C

C

H

H

C

H

C (d)

(d)

C

C

C

H

R

H

H

C

H

H

C

C

(l)

(d)

(l)

(d)

C

C

C

C

C

R

H

H

R

R

racemic diad H

H

H

C

C (l)

(l)

C

C

C

R

H

H

H

H

(l)

H

H

H

C (l) C

R

R H

syndiotactic polymer

H

R

R

H

H

(d)

R

H

R

isotactic polymer

H

H

H

H

(d)

H

R

H

H

C

meso diad H

(c)

H

H

(d)

H

(b)

H

H

C

R

H

H C

R

(l)

(d)

C

C

H

H

R

atactic polymer

FIGURE 1.1. Schematic of the definition of tacticity of an asymmetric chain of the type [(CH2 )(CHR)]n .

CHAIN STRUCTURES polymers. Solution properties of these isotactic chains could in most cases be interpreted in terms of the local conformation of the chain segments using the rotational isomeric state schemes. However, the rationalization of these properties for stereoirregular or syndiotactic chains was impeded to some extent by the lack of experimental results on polymer samples with precisely tailored microstructure. In a highly isotactic chain, the stereo defects can be not only an isolated r diad, but a short perpetuation of it. Zhu et al. [12], from 13C NMR analysis of highly isotactic polypropylene, concluded that isolated racemic units can occur up to a pentad (rrrr) sequence. Whereas most of the early work on crystallization, etc., were concerned with predominantly isotactic chains, the recent developments in synthetic methodologies have enabled the preparation of highly syndiotactic polymers [13,14]. Since the high stereoregularity of these syndiotactic polymers facilitates their crystallization, several papers have been published on the x-ray crystal structure and polymorphism of syndiotactic polystyrene [15–18]. The chain conformation in the crystalline state has also been analyzed using NMR [19]. Similarly, the crystal structure of syndiotactic polypropylene has also been studied by a number of authors [20–22]. Liquori et al. [23] first discovered that isotactic and syndiotactic PMMA chains form a crystalline stereocomplex. A number of authors have since studied this phenomenon [24]. Buter et al. [25,26] reported the formation of an ‘‘in situ’’ complex during stereospecific replica polymerization of methyl methacrylate in the presence of preformed isotactic or syndiotactic PMMA. Hatada et al. [24] reported a detailed study of the complex formation, using highly stereoregular PMMA polymers with narrow molecular weight distribution. The effect of tacticity on the characteristics of Langmuir-Blodgett films of PMMA and the stereocomplex between isotactic and syndiotactic PMMA in such monolayers at the air-water interface have been reported in a series of papers by Brinkhuis and Schouten [27,27a]. Similar to this system, Hatada et al. [28] reported stereocomplex formation in solution and in the bulk between isotactic polymers of R-(þ)- and S-(
)-a-methylbenzyl methacrylates.

1.3 ARCHITECTURE In addition to the tacticity, the molecular weight and its distribution are also major factors which influence the ultimate properties of these chains. Whereas a wide molecular weight distribution can even be a merit for some commodity resin applications, consistent control of the distribution is obviously a requirement for commercial applications. With a wide molecular weight distribution, factors of concern are the internal plasticization of the high molecular weight component by the low molecular weight fraction and the resultant effects on properties such as the Tg. Recent syn-

/

5

thetic efforts focus on controlling not only the tacticity but the molecular weight distribution as well. Anionic living polymerization was used by Hatada et al. [29,30] to prepare narrow molecular weight, highly stereoregular poly(methyl methacrylate). These authors also discussed isolation of stereoregular oligomers of PMMA using a preparative supercritical fluid chromatography method [31]. Preparation of heterotactic-rich poly(methyl methacrylate) and other alkyl methacrylates has also been described [32,33]. The living anionic polymerization of methacrylic esters and block copolymers with low dispersity has been discussed by Teyssie´ et al. [34,35], Bayard et al. [36], and Baskaran [36a]. Diblock copolymers of styrene and t-Bu acrylate with Mw/Mn = 1.05 have been obtained. Wang et al. [37] presented an extensive set of results on the effect of various types of ligands and different solvents and solvent mixtures on the stereochemistry of anionically polymerized poly (methyl methacrylate). Predominantly isotactic or syndiotactic polymers, with narrow polydispersity or bimodal or multimodal distribution of molecular weights were obtained depending on the synthetic conditions. Using different types of catalysts, Asanuma et al. [38] prepared iso- and syndiotactic poly(1-butene), poly(1-pentene), poly(1-hexene), and poly(1-octene) with narrow molecular weight distribution. Whereas the authors cited above employed anioinic polymerization to control the molecular weight distribution, Georges et al. [39–42] developed a living, stable-free radical polymerization process that can be performed in solution, bulk, or suspension. This was also extended to emulsion polymerization of block copolymers [43a]. Since then, there has been a burst of activity on several polymerization methods such as atom transfer radical polymerization (ATRP) [43b–e], living metal catalyzed radical polymerization [43f], and living cationic polymerization [43g]. Designing novel polymer topologies using living ROMP methods has also been developed [43h]. Table 1.1 summarizes some of the work on the control of tacticity and molecular weight distribution with common polymers such as the PMMA and polystyrene. In addition to the occurrence of defects in a stereoregular vinyl polymer in terms of a diad of alternate tacticity, the head-to-head/tail-to-tail (H-H/T-T) defect is also of interest [44]. This type of defect is shown schematically in Fig. 1.2. Different types of polymerization conditions which would introduce these defects have been summarized by Vogl and Grossman [45]. The H-H content has been known to vary from about 4% in PVC to 30% in polychlorotrifluoroethylene. Such a linkage would no doubt affect the properties of the chain to different extents. Indirect synthetic methods (e.g., hydrogenation of polydienes) have been developed to specifically prepare H-H polymers and compare their properties with regular head-to-tail (HT) counterparts. For example, Fo¨ldes et al. [46] have developed a synthetic route to prepare H-H polystyrene, with molecular weights ranging from 240 000 to 1 200 000, and close to

6 / CHAPTER 1 TABLE 1.1. Microstructure. Polymer Atactic poly(alkyl methacrylate)s

Heterotactic poly(aklyl methacrylate)s

Isotactic poly(alkyl methacrylate)s

Syndiotactic poly(alkyl methacrylate)s

Syndiotactic poly(alkyl methacrylate)s

Syndiotactic poly(alkyl methacrylate)s

Isotactic poly(2-Ncarbazolylethyl acrylate) Poly(cyclobutyl methacrylate) Poly(cyclodecyl methacrylate) Poly(cyclododecyl methacrylate) Poly(cycloheptadecyl methacrylate) Poly(cyclooctyl methacrylate) Poly(cyclopentyl methacrylate) Isotactic poly(ethyl methacrylate) Isotactic oligo(methyl methacrylate)

Syndiotactic oligo(methyl methacrylate) Atactic poly(methyl methacrylate) Atactic poly(methyl methacrylate)-d2 Heterotactic poly(methyl methacrylate) Isotactic poly(methyl methacrylate)

Tacticity (%) and remarks

Reference

Methyl methacrylate: rr ¼ 64%, mr ¼ 31%, mm ¼ 5%; Mn ¼ 33; 000; Mw =Mn ¼ 1:14 Ethyl methacrylate: rr ¼ 66%, mr ¼ 28%, mm ¼ 6% Isobutyl methacrylate: rr ¼ 66%, mr ¼ 28%, mm ¼ 6%; Mn ¼ 30; 000; Mw =Mn ¼ 1:31 Langmuir Blodgett monolayer behavior with tacticity discussed. Ester group: -CH2 CH3 : mr ¼ 87:2%; Mn ¼ 7010; Mw =Mn ¼ 1:08 -CH2 CH2 CH2 CH3 : mr ¼ 87:1%; Mn ¼ 9300; Mw =Mn ¼ 1:07 -CH2 CH(CH3 )2 : mr ¼ 78:4%; Mn ¼ 6350; Mw =Mn ¼ 1:07 -CH(CH3 )2 : mr ¼ 69:2%; Mn ¼ 4730; Mw =Mn ¼ 1:07 Tacticity variation of poly(ethyl methacrylate) with synthetic conditions discussed in detail. Methyl methacrylate: mm>97%; Mn ¼ 36; 000; Mw =Mn ¼ 1:17 Ethyl methacrylate: mm ¼ 95%; Mv ¼ 115; 000 Isobutyl methacrylate: mm ¼ 95%; Mn ¼ 3200; Mw =Mn ¼ 4:8 Langmuir Blodgett monolayer behavior with tacticity discussed. rr content with side group: C2 H5 : 90% CH(CH3 )2 : 92% (CH2 )3 CH3 : 92% CH2 -CH(CH3 )2 : 93% C(CH3 )3 : rr ¼ 57%, mr ¼ 33% Mn : 6000
8690; Mw =Mn : 1:06
1:64 Various types of side chain ester groups. rr : 82–92% DP 31–421; Mw =Mn ¼ 1:07
1:43 Stereocomplex with iso-PMMA discussed. Methyl methacrylate: rr ¼ 85%, mr ¼ 14%; Mn ¼ 46; 000; Mw =Mn ¼ 1:2 Ethyl methacrylate: rr ¼ 88%, mr ¼ 9%; Mv ¼ 93,000 Isobutyl methacrylate: rr ¼ 97%, mr ¼ 3%; Mn ¼ 16,000; Mw =Mn ¼ 1:09 Langmuir Blodgett monolayer behavior with tacticity discussed. m ¼ 87–97%; Mn ¼ 0:56  104 to 5:104 ; Mw =Mn ¼ 4:0
4:8 Hole mobility is discussed. rr : 65%, (mr þ rm): 32%; Mw ¼ 13:9:104 ; Mw =Mn ¼ 1:3; Tg ¼ 78  C rr : 67%, (mr þ rm): 30%; Mw =Mn ¼ 1:7; Tg ¼ 58  C rr : 63%, (mr þ rm): 34%; Mw ¼ 9:8  104 ; Mw =Mn ¼ 1:4; Tg ¼ 56  C rr ¼ 67%, (mr þ rm): 31%; Mw =Mn ¼ 1:6; Tg ¼ 56  C

[27]

rr : 63%, (mr þ rm): 34%; Mw ¼ 12:1  104 , Mw =Mn ¼ 1:4; Tg ¼ 73  C rr : 66%, (mr þ rm): 32%; Mw ¼ 11: 0  104 , Mw =Mn ¼ 1:2; Tg ¼ 75  C mm ¼ 97% mm :mr :rr ¼ 96.1:3.9:0 19–29-mer isolated by preparative supercritical fluid chromatography; DP ¼ 28:6; Mw =M Mn ¼ 1:15; Tg of 28mer ¼ 34.5 8C; Stereocomplex with syndiotactic oligo(methyl methacrylate) discussed. mm :mr :rr ¼ 0.3:7.6:92.1 isolated by preparative supercritical fluid chromatography; DP ¼ 26:8; Mw =M Mn ¼ 1:09; stereocomplex with isotactic oligo(methyl methacrylate) discussed. mm ¼ 6%, mr ¼ 36%, rr ¼ 58%; Mw ¼ 124,000; Mw =Mn ¼ 2:8; FTIR spectroscopic analysis of the conformational energy differences between rotational isomeric states is presented. mr ¼ 67.8%, rr ¼ 20.6%, mm ¼ 11.6%; Mn ¼ 11640; Mw =Mn ¼ 1:09
1:14; Tg ¼ 102:2  C; Various mr and rr contents result depending on the synthetic conditions. mm>98%; Mw ¼ 115 000; Mw =Mn ¼ 2:8

[33]

[27a]

[139]

[24]

[27a]

[140] [141] [141] [141] [141] [141] [141] [142] [31]

[31]

[143] [32] [32]

[143]

CHAIN STRUCTURES

/

TABLE 1.1. Continued. Polymer Isotactic poly(methyl methacrylate)-d2 Isotactic poly(methyl methacrylate) Isotactic poly(methyl methacrylate) Isotactic poly(methyl methacrylate)-b-poly (ethylmethacrylate) Isotatic poly(methyl methacrylate)-b-poly (ethylmethacrylate)-b-poly (methyl methacrylate)-bPoly(methylmethacrylate-bethylmethacrylate) Isotactic poly(methyl methacrylate)-copoly(ethylmethacrylate) Poly(iso-MMA-b-syndio-MMA)

Syndiotactic poly(methyl methacrylate) Syndiotactic poly(methyl methacrylate)-d2 Syndiotactic poly(methyl methacrylate) Syndiotactic poly(methyl methacrylate)

Syndiotactic poly(methyl methacrylate)

Syndiotactic poly-1,2(4-methyl-1,3-pentadiene)

Isotactic poly(1-pentene) Syndiotactic poly(1-pentene) Syndiotactic polypropylene Syndiotactic polypropylene

Tacticity (%) and remarks FTIR spectroscopic analysis of the conformational energy differences between rotational isomeric states is presented. mm ¼ 96%; Mw =Mn ¼ 1:1 mm ¼ 97%, mr ¼ 2%; rr ¼ 1%; Mn ¼ 33030, Mw =Mn ¼ 1:25 Stereocomplexation with syndiotactic methacrylates discussed. DP : 59/59, 97/151, 64/182; mm ¼ 95–97%; Mw =Mn ¼ 1:29
2:11

DP : 35/50/200, 25/28/25; mm 95–97%; Mw =Mn ¼ 1:17, 1:42 rr : 89–91% (mm ¼ 0, mr ¼ 9–11%); Mn 8900–12,300; Mw =Mn ¼ 1:07
1:25

Reference

[30] [24] [142,144]

[142,144] [139]

mm ¼ 96–97%; MMA/EMA 78/22 to 26/74; Mw =Mn ¼ 1:53
3:57

[142]

Stereoblock polymer with isotactic and syndiotactic blocks. Mw =Mn ¼ 1:27; Isotactic block: mm ¼ 97%, mr ¼ 2%, rr ¼ 1% Syndiotactic block: mm ¼ 7%, mr ¼ 17%, rr ¼ 76% Stereocomplex between the block polymer and iso or syndiotactic PMMA discussed. rr ¼ 76%, mr ¼ 22%, mm ¼ 2%; Mw ¼ 152,000; Mw =Mn ¼ 2:0; FTIR

[24]

Spectroscopic analysis of the conformational energy differences between rotational isomeric states is presented. Two samples with rr : 89.5 and 91.5%. Mw ¼ 2:6  105 to 5:5  105 ; Mw =Mn ¼ 1:3
1:4 Aggregation process in n-butyl acetate discussed. Two samples (i) mm ¼ 2%, mr ¼ 8.5%, rr ¼ 89.5% (ii) mm ¼ 3%, mr ¼ 31%, rr ¼ 66% (i) Mn ¼ 145,000; Mw =Mn ¼ 1:6 (ii) Mn ¼ 45,000 NMR, IR studies of aggregation in solution; IR and x-ray studies of crystallinity. Sample (i) crystallinity 27–32%, crystallite size 46–57 A˚ rr up to 96%. Anionic living polymerization; Mn ¼ 2000
14500; Mw =Mn ¼ 1:13
5:5; effect of synthetic variables on tacticity, molecular weight and distribution and yield discussed. More than 88% 1,2 content; amorphous; hydrogenation produced crystalline syndiotactic poly(4-methyl-1-pentene) with Tm ¼ 186  C.

mmmm (pentad) ¼ 90%; Mw ¼ 17 000; Mw =Mn ¼ 2:3; Tm ¼ 64  C; x ray and NMR data. rrrr (pentad) ¼ 85%; Mw ¼ 65000; Mw =Mn ¼ 3:0; Tm ¼ 42  C; Tg ¼
22:7  C; crystallinity 30%; x ray and NMR data. rrrr 74–86%; Mw ¼ 52  103 to 777  103 ,Mw =Mn ¼ 1:8
2:4 rrrr ¼ 91.5%; Mw ¼ 1:5  105 ; Mw =Mn ¼ 1:9; crystalline structure of the zig-zag form is reported.

[143]

[145]

[146]

[139,147]

[148] [for the crystal structure of the hydrogenated polymer, poly (4-methyl1-pentene), see 149] [150] [150] [151] [152]

7

8 / CHAPTER 1 TABLE 1.1. Continued. Polymer

Tacticity (%) and remarks

Reference

Syndiotactic polypropylene

rrrr pentads 81.4–94.5% Mw ¼ 9:6  104
17:3  104 ; Tm : 135
186  C; thermal behavior discussed.

Polypropylene

Five samples, mm ¼ 92.2–94.9% (NMR); Mw ¼ 22 000
947 000; fractionation according to stereoregularity; crystallization and melting behavior with IR tacticity studied. rr>98%, rrrrrr>94%; NMR, IR and x ray diffraction results discussed; Tm  270  C. rr>98% rrrr>96%; Tm ¼ 260
270  C; IR and Raman spectroscopic studies of local chain conformation in glass and gels. m>95%; Tm ¼ 230  C; IR and Raman spectroscopic studies of local chain conformation in glass and gels. mr ¼ 67%; mm ¼ 18%; rr ¼ 15% mm>98% Cystallization and melt behavior discussed. Mv ¼ 400000; maximum spherulite growth rate at Tc ¼ 165  C; Tm  ¼ 212:5  C.

Syndiotactic polystyrene Syndiotactic polystyrene Syndiotactic polystyrene Isotatic polystyrene Heterotactic poly(vinyl alcohol) Isotactic poly(2-vinyl pyridine)

100% conversion. The synthesis and properties of H-H polymers have been reviewed by Vogl [47] and Vogl and Grossman [45]. The chain flexibility of a H-H polymer either increases or decreases as compared to the H-T chain, depending on the nature of the side group. A comparison [45,47] of the glass transition temperatures of some of the polymers is given in Table 1.2. Arichi et al. [48] found

H

H

H

H

C

C

H

R

H

R

C

H

H

C

R

H

H

FIGURE 1.2. Schematic of the (a) head-to-tail and (b) headto-head/tail-to-tail placements. Note the sequence of the "markers", ****** in (a) versus ****** in (b).

[13,155] [14] [156,157] [157] [158] [159]

that the theta temperature of H-H polypropylene in isoamylacetate was about 98 higher than that of atactic H-T polypropylene (34 8C). On the other hand, a study of dilute solution properties of H-H polystyrene by Strazielle et al. [49] showed that the theta temperature in cyclohexane was 19 8C, which is lower by 168 than the theta temperature of H-T polystyrene in the same solvent. Hattam et al. [50] studied the solution properties of H-H polypropylene (see chapter on ‘‘Theta Temperatures’’). Another type of specificity that can occur is the chirality. Isotactic poly(triphenylmethyl methacrylate) is the first known case in which the helicity of the polymer leads to chirality and optical activity [51,52]. A conformational analysis of this polymer has been reported by Cavallo et al. [53]. TABLE 1.2. Glass transition temperatures of some head-tohead and head-to-tail polymers. Polymer

C

C

H

R

H

C

C

R

H R

H

C

H

C

H R

H

C

(b)

C

H R

H

C

C

H

H

H

C

C

(a)

H

H

[153] [for the crystal structure using one of these samples, see 22] [154]

Poly(isobutylene) Poly(methyl acrylate) Poly(methyl crotonate) Poly(methyl cinnamate) Poly(methyl methacrylate) Poly(propylene) Poly(styrene) Poly(vinyl cyclohexane) Poly(vinyl chloride) Taken in part from Ref. 45.

Head-to-Head (8C)

Head-to-Tail (8C)

87 40 107 210 160–170
39 97 88 91


61 12 80 190 100
17 98 138 83

CHAIN STRUCTURES

/

9

Apart from the carbon chain polymers discussed above, the silicon chain polymers have also been investigated extensively in terms of microstructure. The stereochemistry of polysilanes has been studied using 29 Si-NMR spectroscopy [54,55]. Wolff et al. [56] concluded that for a poly (phenylmethyl silane), the ratio of mm:rr:mr(rm) to be 3:3:4 and that the spectra of poly(1,2,2-trimethyl -1-phenyldisilane) are consistent with approximately equal amounts of head-to-head and head-to-tail sequences and an atactic configuration.

1.4 POLYMERS WITH MACROCYCLIC AND OTHER PHOTOACTIVE GROUPS Synthetic efforts in designing polymers with functional moieties in the main chain or the side chain to impart photoconductivity, eletro-optic, nonlinear optical properties, etc., has been an active area in recent years [57,58]. Covalent tagging of chromophores to polymers in order to study the conformational dynamics and to study charge transfer complexes has been reported by a number of authors [59–61]. A summary of the work on p and s conjugated oligomeric tetrathiafulvalenes for increasing the dimensionality of electrical conduction was presented by Adam and Mu¨llen [62]. In searching for polymers with photorefractivity, photoconductivity and optical nonlinearity, metalloporphyrins, and metallophthalocyanines have been candidate materials for inclusion in the main chain or the side chain. A brief overview of this area was discussed by Allcock [63]. For example, initial designs on the molecular electronic wires, with backbone-linked porphyrins have been reported by Crossley and Burn [64]. Following this analogy, a molecular photonic wire was announced by Wagner and Lindsey [65]. In the latter, a boron-dipyrromethene dye provides an optical input at one end of the chain, a linear array of three zinc porphyrins serves as a signal transmission element and a free base porphyrin provides an optical output at the other end of the chain. In the case of main chain porphyrin or phthalocyanine polymers, (1) the central metal atoms are covalently linked by a single atom such as O such that the porphyrin (porph) or phthalocyanine (Pc) macrocyclic rings are cofacial, as shown in Fig. 1.3a or (2) the central metal atoms are linked by a flexible or rigid spacer. In the case of side chain polymers, polymerization is performed via side chains attached to the macrocyclic using an acrylic or methacrylic polymer as the backbone. This is illustrated in Fig. 1.3b. (The designs of Crossley and Burn [64], and Wagner and Linsey [65] are different from these general classes.) Intraor inter-molecular p overlap of these macrocyclics dictate the ultimate properties. It is known from the work on small molecule analogues that the extent of p overlap of these macrocyclics influence the photoconductivity, absorption wavelength, etc. [66]. The flexible spacers as well as the side groups attached to these macrocyclics improve their

FIGURE 1.3. The main chain and side chain polymers incorporating metallophthalocyanines are shown schematically. (a) The main chain formed by linking the metallo-PC units, with an oxygen atom, leading to a cofacial arrangement of the macrocyclic rings. Flexible spacers can also be used instead of a single oxygen atom. (b) The metallo-Pc is attached to a side group of a chain such as PMMA. Although two adjacent Pc’s are shown here in the cofacial arrangement, such an intramolecular overlap would depend on the tacticity and the conformation of the chain. The metal M can be Cu, Al, Si, Ge, etc.

solubility and processibility. It is well known that phthalocyanines, without any flexible side groups, are notoriously insoluble in any convenient solvent. A summary of some of these activities are presented in Table 1.3. Phthalocyaninecontaining polymers [66a] and conjugated polymer-based chemical sensors [66b] have been discussed. The asymmetrically substituted porphyrins or phthalocyanines exhibit isomerism. A theoretical treatment of this aspect was published by Knothe [67]. With the emergence of photonics for telecommunication applications, there have been extensive activities related to the development of polymeric materials to this end. Several reviews are available on the synthesis and fabrication of polymer-based molecular wires and switches [67a–g]. In addition, a number of studies on azobenzene-containing

10 / CHAPTER 1 TABLE 1.3. Polymers with macrocyclic photoactive groups. Polymer Poly(5-[4-(acryloyloxy)phenyl]-10,15,20triphenylporphyrin) Poly(5-[4-(methacryloyloxy)phenyl]-10,15,20triphenylporphyrin) Poly(methylmethacrylate-co-5-[4-(methacryloyloxy) phenyl]10,15,20-triphenylporphyrin) Poly[porph-O-CH2 -](I) Poly[porph-O-(CH2 )6 -](II) Poly[porph-O-(CH2 )8 -](III) Poly[porph-O-(CH2 )8 -](IV) Poly[(O-porph-O-CH2 )x -co-(O-bisph-O-CH2 )y ] Poly[(porph-O-CH2 -O)x -co-(bisph-O-CH2 -O)y ] (Porph-Ph-porph)3 1,4-phenylene bridged porphyrins Poly(porph-phenylenevinylene)

Tetrakisporphyrin and oligomeric porphyrin

Boron-dipyrromethene dye-(ZnPorph)3 -Porph Poly(N-vinyl-2-pyrrolidone-Porph) Poly[(AIPc)–F] Poly[(GaPc)-F] Poly[(CrPc)-F] Poly[(SiPc)-O] Poly[(GePc)-O] Poly[(SnPc)-O] Poly[(SiPc)-O] Poly[(GePc)-O] Poly[(SnPc)-O] Poly(CuPc) Poly(CoPc) Poly(NiPc) Poly(AlPc-F) Poly(SiPc-O) Poly[octakis(decyloxy)SiPc-O]

Poly(CH2 -CHCOOROPc)R: C8 H17 or C12 H25

Poly[(SiPc)-O] Poly[(SiPc)-O-(CH2 )4 -Si(C6 H5 )2 -(CH2 )4 -O] Poly[(SiPc)-O-(CH2 )4 -Si(CH3 )2 -C6 H4 -Si(CH3 )2 (CH2 )4 -O] Poly[(SiPc)-O-(CH2 )4 -Si(CH3 )2 -(O-Si(CH3 )2 )4 (CH2 )4 -O] Poly[(SiPc)-(Si(CH3 )(C6 H5 ) )n ] (n  6.6)

Remarks

Reference

Polyacrylate or methacrylate polymers with pendant porphyrin units; hypochromism and hyperchromism discussed.

[160]

Main chain porphyrin polymers; meso-tetraarylporphyrins used; I, II, III with CH3 substituted Porph; IV with CH2 CH3 substitution; III and IV also with Co, Mn, and Zn transition metals. Main chain porphyrin polymers; various copolymer compositions; Mw up to 125 000; Co and Cu transition metal inclusion with some copolymers. Metal free and Zn; several types of substitutions on the porphyrin. Main chain porphyrin polymers; 68% yield; metallized with Zn, Cu, or Ni; NMR, FTIR and cyclic voltammetric results. An approach to molecular electronic wire; Tetrakisporphyrin (C322 H370 N28 ) is about 65 A˚ long. The tert-butyl groups along the backbone provide an insulating sheath around the conjugated core and enables solubility. An approach to photonic molecular wire. Absorption and emission spectra discussed. Poly(N-vinyl-2-pyrrolidone) with metal free or Mg porphyrin side group; spectroscopic behavior discussed. Co-facial packing of Pc rings; spectroscopy, electrical conductivity, electron microscopy, effect of doping discussed.

[161]

DP ¼ 120 with SiPc (density r ¼ 1:432), 70 with GePc (r ¼ 1:512) and 100 with SnPc (r ¼ 1:719). Cofacial structure of Pc rings. Crystal structure discussed based on powder x ray diffraction data and modeling. Sheet polymers of metal phthalocyanines; insoluble; electronic spectra, magnetic susceptibility, electrical conductivity, x ray diffraction discussed. Main chain metallized phthalocyanine (Pc) polymers; electron microscopy and electron diffraction; both AIPc and SiPc are cofacial. Substituted SiPc main chain, soluble polymer; Mw ¼ 118 000 and 250 000 by SAXS of heptane solutions; rod length 219 and 472 A for the two samples. Side chain phthalocyanine polymers; Pc substituted with various groups; metal free, Cu and Ni Pc’s; Mw up to 47 000; liquid crystalline; absorption and fluorescence spectroscopy. Main chain Pc polymers; symmetric and unsymmetric alkyl substituted Si phthalocyanines; absorption and emission spectroscopy, optical properties, cyclicvoltammetry are discussed in terms of packing and interactions of phthalocyanines.

[161a]

[162] [163]

[64]

[65] [164] [165]

[166,167]

[168]

[169]

[170]

[171]

[172]

CHAIN STRUCTURES

/ 11

TABLE 1.3. Continued. Polymer Tetra(methoxy)-tetra(octyloxy)-phthalocyaninatopolysiloxane Poly[(SiPc)-O] Poly[acrylamide-CuPc(NO)2 ] Poly[vinylcarbazole-CuPc(NO)2 ]

Poly[2-[[11-(methacryloyloxy)undecyl]oxy]-3methoxy9,10,16,17,23,24, hexakis (dodecyloxy)phthalocyanine] (polyundecyloxy methacrylate with side chain metal-free phthalocyanine) Poly(perylene imide): Poly(perylene-R) subtituted perylenes; R : Linkage : (CH2 )9 or Ph-O-Ph or Ph-CH2 -Ph Poly(4’-dialkylamino-4-nitrostilbene acrylate-b-methyl methacrylate) Poly(4’-dialkylamino-4-nitrostilbene methacrylate-b-methyl methacrylate) Poly(4’-dialkylamino-4-nitroazobenzene methacrylate-b-methyl methacrylate) Poly[(R,R)-dibenzo-19-crown-6] Poly[(S,S)-dibenzo-19-crown-6]

Remarks

Reference

Langmuir-Blodgett film properties studied.

[173]

SAXS from dilute solutions. Side chain phthalocyanine polymer; water soluble; also doped with iodine; photoconductivity discussed. Copolymer with vinyl carbazole and dinitro CuPc covalently attached to the carbazole moiety. 21 mol % CuPc(NO)2 bonded to PVK. The polymer shows better photoconductivity than monomeric CuPc or CuPc(NO2 )4 . Langmuir-Blodgett monolayer formation studied with IR, ellipsometry, electron diffraction; effect of adding 1-arachidic acid discussed.

[174] [175] [176]

[177]

Main chain perylene polyimide; Mw up to 64 100; soluble in various solvents; absorption, fluorescence spectra discussed. Polyacrylates with NLO active side chains; wide range of Mw up to 186 000; many soluble in methylene chloride or THF; two samples show liquid crystallinity; microscopy and thermal analysis discussed.

[178]

Polymeric chiral crown ethers; hostguest complexation discussed.

[180]

polymers have been reported [67h,i], including the fabrication of light driven organized layered materials [67j]. Advances in polymerization methods have played a key role in recent efforts to design materials with specific properties. As an example, the ATRP technique mentioned in Sect. 1.3 has recently been used to tailor the photochromic performance of polymer-dye conjugates [67k]. 1.5 POLYMERS WITH FULLERENE AND CARBON NANOTUBE Summarizing the research activities on fullerenes, Baum [68] wrote ‘‘ . . . the question most commonly asked of fullerene` researchers has been very simple: What is it good for? The answer to that question has generally gone something like this: We don’t yet know what applications will be discovered for C60 and the other fullerenes. However, the remarkable properties of these new forms of carbon will inevitably lead to many new products that will range from new types of polymers . . . ’’ Fullerenes have been incorporated into polymeric backbones as ‘‘pearl necklace’’ or as side chains (‘‘charm bracelet’’). The synthesis of dendrimers with C60 has also been reported [69]. A brief summary of polymer related fullerene work was given by Hirch [70]. Chemical derivitization of C60 has been described by Petrie

[179]

et al. [71]. A summary of the initial work on polymers incorporating C60 is given in Table 1.4. Various forms of polymeric fullerenes have been prepared in the past decade: side chain polymers, main chain polymers, dendritic fullerenes, star-shaped polymers, fullerene endcapped polymers, etc. [71a–d]. With the invention of the carbon nanotubes [71e,f] and the development of methods to functionalize them [71g–i], their applications in the area of polymers range from opto-electronic devices to biosensors [71j–m]. 1.6 CYCLIC POLYMERS The cyclic polymers or oligomers are distinct in their physical properties from the corresponding linear chains. There has been considerable interest in the synthesis, isolation, characterization, and utilization of polymeric cyclics, although in a number of cases, they are at best oligomeric. A collection of reviews on various aspects of cyclic polymers has been published and an introduction to this area has been given [72–74a–c]. Chromatographic methods are normally used experimentally to determine the population of the cyclics which may coexist with the corresponding linear chains. Three methods have been reviewed by Semlyen [73] to

12 / CHAPTER 1 TABLE 1.4. Polymers with fullerenes. Polymer Polyarenefullerenes C60 -p-xylylene copolymer Poly(4,4’-diphenyl-C61 sebacate) Poly(bisphenol A hexamethyleneurethane-C61 ) Poly(ethylene imine-C60 ) Poly(ethylene propylene)terpolymerC60 (amine functionalized) Poly[4-[[(2-aminoethyl)imino]methyl] styrene-C60 ] C60 ( ¼ C ¼ C ¼ C58 )n Poly[(styrene)-co-(styrene/C60 )]

Poly(RbC60 ) and poly(KC60 ) Polyether dendrimer-C60

Remarks Reaction with benzene and toluene led to C60 -(C6 H6 )12 and C60 (C6 H5 -CH3 )12 , respectively. Cross-linked; insoluble; xylylene/C60 rato: 3.4:1.0 Solid state NMR spectra discussed. Side chain C60 polymer; 61% yield; soluble in nitrobenzene, benzonitrile; NMR, IR, cyclic voltammetry are discussed.

Reference [181] [182] [183] [183]

Side chain C60 polymer; molar mass 35 000 gmol
1 ; molar ratio polymer/C60 : 18/1. Side chain C60 polymer. Side chain C60 polymer; molar mass 20 000 gmol
1 ; molar ratio polymer/C60 : 19/1; soluble in toluene, carbon disulfide. Solid state photopolymerization; soluble in boiling isodurene. Side chain C60 polymer; polymers made with 5.5% (Tg ¼ 112  C), 21% (Tg ¼ 142  C) and 29% (Tg ¼ 166  C) (w/w) of C60 ; Mw ¼ 38500; single Tg ; soluble in methylenechloride, THF. Lattice parameters and x ray data. A deuteriated fourth generation azide dendrimer138 with C60 as the core; 68% yield; extremely soluble; Tg ¼ 52  C (138 higher thanstarting dendrimer).

theoretically calculate the population of the cyclics in cyclic-chain equilibrium [75–80]. The molar cyclization equilibrium constants have been determined both experimentally and by calculations for a number of cases such as dihydrogen siloxanes [81], dimethyl siloxanes [76,81–86], and sodium metaphosphates [87], cyclic nylon 6 [88], poly(ethylene terephthalate) [89], and liquid sulphur [90]. Cyclics offer a wide range of opportunities for polymer synthesis, processing, and modification. Ring opening polymerization of cyclics leads to high molecular weight polymers. This was demonstrated using octamethylcyclotetrasiloxane to synthesize long chain polysiloxanes [91]. This process of involving the cyclics has also been used to control the block length of polysiloxane in the preparation of siloxane-styrene-siloxane or siloxane-isoprene-siloxane triblock copolymers [92] and styrene-dimethylsiloxane diblock copolymers [93]. The cyclics offer the advantage of the ease of processing due to their low viscosity at the product fabrication temperature. Hence, in applications such as injection molding, the cyclics can be used for postpolymerization to achieve high molecular weight polymer end products [94]. Macrocyclic oligmers of bisphenol A polycarbonate have been used to prepare polycarbonates of very high molecular weight (Mw ¼ 200 000–400 000), by ring-opening polymerization which avoids the creation of byproducts. Cyclics with 2–21 monomer units have been prepared and the cyclics yield can be varied from 0% to over 85% by manipulating the synthetic conditions [95–97]. Synthesis and polymerization of cyclic oligomeric arylates [98] and cyclic ether

[184] [185] [184] [186] [187]

[188] [69]

ketones, ether sulfones, and ether imides have also been reported [99]. Mark and Semlyen, in a series of papers, have studied the mechanism and the effect of trapping cyclics in end-linked elatomeric networks [100–103]. Sharp fractions of cyclics of poly(dimethylsiloxane) (PDMS), varying in size from 31 to 517 skeletal atoms, were mixed with linear chains for different periods of time and the linear chains were then end-linked using a tetrafunctional silane. The untrapped cyclics were extracted to determine the amount trapped. It was found that while cyclics with less than 38 skeletal atoms were not at all trapped, for n>38, the percentage of cyclics trapped increased with size, with 94% trapped in the case of the cyclic with 517 skeletal atoms. In effect, the system of trapped cyclics in the end linked PDMS network is a polymeric catenane. It is thus possible to control the elastomeric properties of the network by incorporating the appropriate sized cyclics. This study has been extended to cyclic PDMS in poly(2,6-dimethyl-1,4-phenylene oxide) [104,105] and cyclic polyesters in PDMS [106]. Percec and coworkers [107–110] have synthesized liquid crystalline cyclic oligomeric polyethers based on 1-(4-hydroxy-4’-biphenyl)-2-(4hydroxyphenyl)butane with dibromoalkanes. Rings varying from 2 to 5 monomer units were prepared and show isotropic-nematic transition. The nematic order is modeled to arise from the collapse of the rings in the form of a ‘‘folded chain’’ structure, as shown schematically in Fig. 1.4. This is similar to the case of chain folded crystallization of cyclic alkanes (with 34–288 CH2 groups) and cyclic urethanes [111–114].

CHAIN STRUCTURES Isotropic

/ 13

Nematic T

T

(a)

FIGURE 1.4. A model of the isotropic ! nematic transition in cyclic oligomeric polyethers via intramolecular collapse of the cyclic.

T

T

(b)

Table 1.5 summarizes the studies on cyclics of poly(dimethyl siloxane) and derivatives, and Table 1.6, those of other polymers. In these tables, Kx refers to the molar cyclization equilibrium constant and RIS, to the rotational isomeric state scheme to analyze chain conformations. 1.7 ROTAXANES Polyrotaxanes (the name derived from Latin words for wheel and axle) are essentially in situ molecular composites consisting of a linear chain threaded through a cyclic molecule. The interior diameter of the cyclic must be large enough to accommodate the linear chain. Large end groups might be necessary to prevent the unthreading of the chain from the cyclic. Two principal approaches have been used in the synthesis of polyrotaxanes. In the statistical method, no specific interaction exists between the linear and the cyclic species. The equilibrium for threading is driven by entropic factors. This hence provides a wide choice of pairs of cyclics and linear chains. However, the resulting yield is often low. In the template or directed method, specific attractive interaction (such as metal chelation, charge transfer interactions, etc.) between the cyclic and the linear species is taken advantage of. The polyrotaxanes can be of the ‘‘main chain’’ or the ‘‘side chain’’ type [115], as illustrated schematically in Fig. 1.5, along with a bulky terminal group to prevent dethreading. In the former, the cyclic is threaded through a linear chain and is free to glide along the chain as the steric interactions would permit. In the case of side chain rotaxanes, the cyclic is threaded through a long side chain of a polymer. Thus, a wide range of options and architectures are possible. Rotaxanes have also been part of dendrimers, i.e., dendritic molecules containing rotaxane-like bonds to link

FIGURE 1.5. A schematic representation of the (a) main chain and (b) side chain rotaxanes. The T represents a large terminal group which may be used to prevent dethreading of the cyclic from the chain.

their components, either at the core, termini, or branches [115a]. Comprehensive reviews of the history, chemistry, and physical chemistry of rotaxanes and the related architecture, the catenanes, have been published [115–120]. Joyce et al. [121] reported a molecular modeling study of cyclics of poly(dimethylsiloxane) to understand the energetics of the threading process of linear chains with particular reference to rotaxanes. Of relevance is also the exhaustive review by Wenz [122] on the role of cyclodextrins in the building of supramolecular structures. Cyclodextrins are cyclics of D-glucose, with a-1,4’ linkages. The common ones are a,b,g-cyclodextrins, with 6, 7, and 8 D-glucose units, respectively. The rotaxanes offer a plethora of possibilities in terms of supramolecular architecture [115, 116, 122a]. Applications toward molecular machines, motors, and switches have been extensively explored [122b–e]. Table 1.7 lists the examples of different types of rotaxanes reported in the literature. Some of the early work on the chemistry is omitted but can be found in the reviews cited above. In naming the rotaxanes in this table, we follow the nomenclature of Gibson and Marand [115]: polymerrotaxa-cyclic. 1.8 DENDRIMERS Dendrimers (Greek) or arborols (arbor: treeþalcohol ¼ arborol) are tree-like macromolecular structures topologically controlled during the synthesis. Starting from

14 / CHAPTER 1 TABLE 1.5. Cyclics of poly(dimethyl siloxane) and its derivatives. Number of monomers in cyclics

Polymer Poly(n-butyl methylsiloxane) Poly(dimethyl siloxane) Poly(dimethyl siloxane)

4–200 4–40

Poly(dimethyl siloxane) Poly(dimethyl siloxane)

4–200

Poly(dimethyl siloxane) Poly(dimethyl siloxane)

Poly(dimethyl siloxane) Poly(dimethyl siloxane)

3–6 up to approx. 650 (number average number of skeletal bonds 1300) 65–275 16–259

Poly(dimethyl siloxane)

92

Poly(dimethyl siloxane)

33–122

Poly(dimethyl siloxane-bstyrene-b-dimethylsiloxane) Poly(ethyl methyl siloxane)

4–8

Poly(ethyl methyl siloxane) Poly(hydrogen methyl siloxane) [-H(CH3 )SiO-] Imide-disiloxane Paraffin-siloxanes Paraffin siloxane [-(CH3 )2 Si-(CH2 )4 -(CH3 )2 Si-O-] Poly(phenyl methylsiloxane)

4–20

4–15

Remarks K4 : 0.37; K5 : 0.19 Kx determined. Kx determined with bulk and solution equilibrates; characteristic ratio measured. Calculations with RIS models. K4 : 0:19; K5 : 0:09 Kx measured; dilute solution behavior and RIS treatment discussed. [h]-Mw relationships. Construction of a preparative GPC for isolating sharp fractions of cyclics; Mw =Mn ¼ 1:05; viscosity of fractions measured.

Neutron scattering measurements of chain dimensions. Used in trapping experiments with linear chains; effect of ring size on trapping and elastic properties of the network studied. Cyclics trapped to form catenanes with poly(2,6-dimethyl-1,4-phenylene oxide). 26% w/w cyclics permanently captured in network. 1m phase domains seen after extraction; mechanical and thermal properties discussed. Trapped to form catenanes with poly(2,6-dimethyl-1,4-phenylene oxide); weight fraction of cyclics trapped increases with their DP. Two Tg ’s observed; phase separated domains form. Solution equilibrates; short styrene and DMS blocks, Mw approx 1:1  104 . Equilibrated in bulk and in toluene; 25.8% w/w cyclics; Kx measured K4 : 0:25; K5 : 0:16. Equilibrated in bulk and in toluene; 12.5% w/w cyclics; Kx measured.

2–10

Kx measured for x ¼ 1
6.

3–50

30% w/w undiluted, 90% with toluene; K4 ¼ 0:25; K5 : 0:16; populations of configurational isomers for x ¼ 3
5. Equilibrated in bulk and in toluene; 31% w/w cyclics; Kx measured. >80% (w/w) population.

Poly(n-propylmethyl siloxane)

4–8

3,3,3-trifluoropropyl methyl siloxane Poly(3,3,3-trifluoropropylmethyl siloxane) Poly(vinylmethyl siloxane)

3–13 4–20 4–23

Equilibrated in bulk and in cyclohexanone; 82.7% w/w cyclics; Kx measured. Kx , dilute solution viscosity, melt viscosity, Tg measured.

a single branch cell, repeat units or branch cells are iteratively added, to produce ‘‘star-burst’’ structures, one generation after another (STARBURST is a registered

Reference [189] [82] [83]

[189] [84] [190] [191]

[192] [100–103, 106]

[104]

[105]

[193] [194] [189] [194] [98] [195] [81] [189]

[194] [196] [194] [197]

trademark of Dow Chemical Company) [123]. During such multiplicative growth, the polymer adopts a spherical shape, free of chain entanglements, as shown schematically

CHAIN STRUCTURES

/ 15

TABLE 1.6. Cyclics of other polymers.

Polymer

Number of monomers in cyclic

Cycloalkane Cycloalkane

34 12–84

Cycloalkane Cycloalkane Cyclic amides

36 24–288

Amide-imides Bisphenol-A-co-4,5-bis(phydroxyphenyl)-2-(p- nitrophenyl)-oxazole Polycarbonate 2–21

Polycarbonate Poly(decamethylene adipate) Poly(decamethylene adipate)

2,4 1–5

Poly(1,3-dioxolane) 2,2’-dithiobis(2-methyl propionaldehyde Polyester [-(CH2 )10 COO-]

2–9 ester tetramer 5–15

Polyethers with 1-(4-hydroxy4’-biphenyl)-2-(4- hydroxyphenyl) butane and dibromoalkanes Ether imides Ether ketones Ether sulphones Ethylene terephthalate Tris(ethylene terephthalate) Tris(ethylene terephthalate) 2-Norbornene Nylon-6 Oligo(cyclooctene)s 1,3,6-trioxacyclooctane

2–5

1,3,6,9-tetra oxacycloundecane

Reference

X ray crystal structure. Thermal behavior studied; entropy of fusion/unit increases with chain length; melting temperatures discussed. X ray crystal structure Tm , LAM frequencies. Discussion of chain folding. Prepared with dodecanedioyl dichloride and pure isomers of 4,4’-methylenedicyclohexylamine; NMR, IR, x ray diffraction discussed.

[111] [198]

Macrocycles for NLO copolycarbonates; Tg increases with 2nd component.

Polycarbonate Polycarbonate

(2R,5R,8R,11R)-2,5,8,11tetra-tertbutyl-1,4,7,10- tetraoxa cyclododecane 1,3,6,9,12,15-hexa oxacycloheptadecane 1,3,6,9,12-penta oxacyclotetradecane

Remarks

3–9 3 3 2–7 1–6 2–10 1–9

4

Up to 95% yield of cycllcs; large cyclics with Mw more than 100,000. Cyclics content (0–85%) depends on catalyst. Dimer 5%; trimer 18%; tetramer 16%; pentamer 12%; hexamer 9%; higher oligomers 25%. X ray crystal structure. Kx measured and compared with theory; RIS treatment discussed. Kx measured; RIS treatment discussed. 95% yield, Tm ¼ 182  C Trapping experiments with linear PDMS chains; molecular modeling of the phenomenon. Exhibit nematic mesophase; chiral cyclics display cholesteric phase.

25–75% yield. 40–52% yield. 40–52% yield. Extracted from PET; Kx given. Synthesized. Synthesized; 46% yield. 2% w/w cyclics; melt equilibrates; Kx measured. Tm for x ¼ 2: 60  C x ¼ 4: 35  C x ¼ 6: 40  C Kx given. Tm ¼ 168  C NMR, conformation discussed.

1–5 1–7

2–8

[113] [114] [199]

[98] [200]

[201] [95, 97] [202] [203] [75a] [204] [205] [206] [106] [107,109,110]

[99] [99] [99] [207] [208] [209] [210] [88,211] [212,212a] [213]

[214]

[213] Tm for x ¼ 2: 44  C x ¼ 4: 61  C

[213]

[215]

16 / CHAPTER 1 TABLE 1.6. Continued.

Polymer

Number of monomers in cyclic

1,3,6,9-tetra oxacycloundecane

1–8

Poly(phenyl methyl silane)

6

N-methylated oligo(pphenyleneterephthal amide) Polyphosphate [NaPO3 ]x Polystyrene

3–6 3–7 Mw : 4 500
20 000

Polystyrene Polystyrene

Polystyrene

Polystyrene

Polystyrene

Poly(trimethylene succinate)

1–7

Poly(sulphur) Poly(sulphur) Cyclo(oligourethane)s

>8 6 to 26 1–7

Poly(tetra hydrofuran)

4–6

Remarks Tm for x ¼ 2: 88  C x ¼ 4: 54  C x ¼ 6: 39  C Kx given. Two isomers characterized by NMR; crystal structure determined for one.

Reference [213]

[216] [217]

10 w/w cyclics Yields from 10 to 45%

[218] [219]

Synthesis and fractionation described; Mw of rings 5  103 to 4:5  105 ; yield decreases with Mw from 55% to 18% Mw 11 100 to 181 500, Mw =Mn < 1:2; quasi elastic light scattering experiments to determine translational diffusion coefficients and compare with linear chains. Mw 12 000
22 000 SANS experiments to determine mean square radii of gyration and second virial coefficient for cyclic and linear chains. Macrocyclic fractions with 11,500  Mw  181 000; thermodynamic and hydrodynamic properties in dilute solution and melt viscosity studied and compared with corresponding linear fractions. Macrocyclic fractions with 1:9  104  Mw  3:9  105 viscoelastic properties measured and compared with theory. Kx measured, compared with theory; RIS treatment discussed.

[220]

Melting temperatures, long periods, x ray diffraction and structure discussed.

in Fig. 1.6. However, theory predicts [124] a limit of 10 generations beyond which the reaction rates decrease significantly and defects begin to predominate. Extensive reviews of advances in this field have been published [123,125–128]. The synthesis of ‘‘cascade molecules’’ or arborols as spherical micelles has been described by Newkome et al. [129,130] who also proposed a nomenclature for such structures. A designation [m]-[n]-[p] would refer to a case in which m and p represent the number of surface groups and n denotes the bridge size. The conventional synthesis of dendrimers involves the ‘‘divergent’’ method, in which branch cells are constructed in situ around the initiator core or preformed branch cells are attached to the core. Successive generations are then built. On the other hand, in the ‘‘convergent’’ method [131–133] the dendritic fragments are prepared by starting from frag-

[221]

[222]

[223, 224]

[225]

[204] [90] [73] [112] [226]

ments which would ultimately comprise the periphery and progressing inward. The resulting dendritic wedges, after several generations of growth, are coupled to a polyfunctional core. A double-stage convergent growth approach has also been described by Wooley et al. [134], which enables synthesis of a dendrimer with a ‘‘hypercore’’ made of flexible segments and a rigid outer layer or vice versa. Dendrimers have also been used as macroinitiators for forming hybrid linear-globular AB block copolymers [135]. A summary of all known synthetic strategies to dendrimers has been given by Tomalia [127]. Self-aggregation of certain dendrimers into lyotropic and thermotropic mesophases has been reported [108, 136, 137]. In particular, the review by Tomalia et al. [124] traces the similarities and scope of the dendrimer designs to biological systems tailored by Mother Nature and the prospects and applications of supramolecular mimetics via man-made

CHAIN STRUCTURES

/ 17

TABLE 1.7. Poly(rotaxanes). Poly(rotaxane) Poly(amide)-rotaxa-b-cyclodextrin Poly(ethylene glycol)-rotaxa-acyclodextrin

Poly(ethylene glycol) bisaminerotaxa-a-cyclodextrin

Poly(iminoundecamethylene)rotaxa-a-cyclodextrin Poly(iminoundecamethylene)rotaxa-heptakis(2,6-di-O-methyl)b-cyclodextrin Poly(iminotrimethylene-iminodecamethylene)rotaxa-a-cyclodextrin Poly(iminotrimethylene-iminodecamethylene)rotaxa-heptakis(2,6-di-O-methyl)b-cyclodextrin Poly(isobutylene)-rotaxa-b-cyclodextrin Poly(isobutylene)-rotaxa-g-cyclodextrin

Poly(propylene glycol)-rotaxa-b cyclodextrin Poly(propylene glycol)-rotaxa-a cyclodextrin Polyvinylidene chloride-rotaxa-b cyclodextrin Poly(azomethine)-rotaxa-42-crown-14

Poly(butylene sebacate)rotaxa-(30-crown-10) Poly(butylene sebacate)rotaxa-(60-crown-20) Poly(decamethylene sebacate)rotaxa-(30-crown-10) Polystyrene-rotaxa-bisparaphenylene34-crown-10 Polystyrene-rotaxa-30-crown-10 Poly(triethyleneoxy sebacate)rotaxa-(30-crown-10) Poly(triethyleneoxy sebacate)rotaxa-(60-crown-20) Poly(urethane-rotaxa-bis(p-phenylene)34-crown-10 Poly(urethane)-rotaxa-(60-crown-20) Poly(urethane)-rotaxa-(36-crown-12)

Remarks

Reference

Rotaxane becomes insoluble; no melting observed in DSC before decomposition. PEG Mw from 400 to 10 000 studied; complexation in aqueous solution at room temperature; maximum complexation with Mw ¼ 1000; two ethylene glycol units/a-CD cavity; x ray study leads to an extended columnar structure. PEG end capped with 2,4-dinitrofluorobenzene after complexation in aqueous solution; 20–23 alpha-CD molecules entrapped on each chain; rotaxane is insoluble in water; hydrogen bonds between entrapped alpha-CD’s suggest alternating face-toface, back-to-back arrangement of the CD’s. A nicotinoyl group was used for endcapping to prevent dethreading.

[227]

Complex formed in water although PIB is insoluble in it; complexation with b-and g-CD’s show opposite dependence on Mw of PIB. More than 90% yield with g-CD and PIB Mw between 800 and 1350. 96% yield with PPG Mw ¼ 1000 and beta-CD and decreases with increasing Mw . Two PPG repeat units per b-CD. More than 70% yield with PPG Mw 400–1000 and g-CD. Complexation in aqueous solution at room temperature. In situ polymerization; x ray diffraction of the inclusion compound discussed. p-tri(p-t-butylphenyl) derivatives used as blocking groups; liquid crystalline; Tm is 67 8C, and smectic between 67 and 73 8C. Ti ¼ 123  C. Parent polymer is insoluble; polyrotaxane is soluble in chloroform, acetone, etc. With and without a triarylmethyl derivative as blocking group; efficiency of threading depends on size of the macrocycle; blocking group is not always needed for stability.

[228]

[229,230]

[231]

[232]

[233]

[234,235] [236]

[237]

[238] With a triphenyl blocking group; free-radical and anionic polymerizations. Two Tg ’s observed. With and without a triarylmethyl derivative as blocking group; efficiency of threading depends on size of the macrocycle; blocking groups is not always needed for stability.

[239]

Segmented polyurethane; host-guest complexation approach; 80% threading efficiency. Statistical threading approach; threading efficiency 57% with 60Cr20 and 16% with 36Cr12. Tg of rotaxanes (
28 8C for 60Cr20 and 19 8C for 36Cr12) lower than the parent polymer (51 8C).

[240]

[237]

[241]

m-crown-n refers to a crown ether containing m backbone atoms, with n oxygen atoms. a,b,g- cyclodextrins: cyclics with 6, 7, and 8 a-1, 4’-D-glucose units, respectively.

18 / CHAPTER 1

Core

from the core in a concentric radial fashion in the case of starbranched systems. On the other hand, in the case of starburst dendrimers with symmetrical branch cells and topologies, the density increases with generations and it remains constant for the asymmetrically branched starburst dendrimers. Other physical properties which contrasts them with linear polymers are [126] (1) the radius of gyration of a dendrimer is larger than that of a linear chain with the same Mw ; (2) the Tg depends on the terminal groups as well as on the nature of the repeat unit blocks; (3) the high degree of branching prevents any interchain entanglements; and (4) the Mark-Houwink type relationship between Mw and intrinsic viscosity does not apply. The hydrodynamic radius increases more rapidly with generations than the radius of gyration. Wooley et al. [138] made a systematic study of the glass transition temperatures of several types of dendrimers. They also derived an equation to relate the Tg of a dendrimer to Tg1 corresponding to infinite molecular weight:

FIGURE 1.6. A schematic representation of a symmetric dendrimer.

Tg ¼ Tg1
K 0 (ne =M),

dendrimer structures. Tomalia et al. [123] and Fre´chet [128] also discuss the similarities and differences between starburst dendrimers and other starbranched or hyperbranched systems. While they are similar in terms of a core, radial branches, and perhaps telechelic functionality, the structure of hyperbranched polymers is neither regular nor highly symmetrical. The branch-segment densities decrease

(1:1)

here ne is the number of chain ends per molecule, M is the molecular weight, and K 0 includes several parameters such as the free volume per chain end, etc. It was found that the nature of the chain ends dramatically affects the Tg and that the latter increases with the polarity of the chain ends. The internal composition, as in the case of block copolymer dendrimers, also influences the Tg . A summary of various published dendritic structures is given in Table 1.8.

TABLE 1.8. Polymeric dendrimers. Dendrimer Acid-terminated dendrimers

Ammonium ion dendrimers Aramid dendrimers

Aryl ester dendrimers Carbosilane dendrimers ‘‘Comb burst’’ dendrimer

Crown ether dendrimers Dendrimer-b-polyethyleneglycolb-dendrimer

Remarks

Reference

Z- cascade:methane[4]:(3-oxo-6-oxa-2-azaheptylidyne):(3-oxo2-azapentylidyne):propanoic acids. Five generations; pH dependence of hydrodynamic radii discussed.

[242]

Fully aromatic amide dendrimers; 1,3,5-benzenetricarboxamide core; 3,5-dicarboxamidophenyl repeat units; phenyl or 3,5-ditbutyl isophthalate termini. Molecular modeling showed that the conformations of the isophthalate segment lead to the either open or conjested structure. Based on symmetrically substituted benzenetricarboxylic acid esters; convergent synthesis; four generations. Tetravinylsilane as the core; dichloromethylsilane as the propagating unit; four generations; Mw ,[h] discussed. Backbone: styrene/divinylbenzene copolymer; teeth: triethanolamine; ion exchange material (anionic: core and branch points, cationic: termini). Based on N-benzyloxycarbonyl-1,4,10,13-tetraoxa-7,16 diazacyclooctadecane and with trichloroformyl mesitylene as core. Third and fourth generation polyether dendrimers used; DP of PEG from 24 to 447: Mw of copolymers 3600–20 300; Mw =Mn #1:1; size and shape of the block copolymers discussed.

[243] [244]

[245] [246] [247]

[248] [249, see also 250 for further characterization]

CHAIN STRUCTURES

/ 19

TABLE 1.8. Continued. Dendrimer Dendrimer-b-polyethyleneoxide-bdendrimer Dendrimer-b-polyethyleneglycol-bdendrimer Phosphonium dendrimers Phenylene dendrimers

Phenyl acetylene dendrimer Propylene imine dendrimers

Quaternary ammonium ion dendrimer Trimethylene imine dendrimers Aromatic polyamide dendrimers Polyamide dendrimers

Polyamido amine dendrimers

Polyarylamines Polyester dendrimers

Polyester dendrimers

Polyether dendrimer

Polyether dendrimer

Polyether dendrimer (thermotropic)

Polyether dendrimers (thermotropic)

Remarks Polyether dendrimers from Hawker and Fre´chet [132] used; reactivities in the melt and solution discussed.

Based on tris(p-methoxymethylphenyl)phosphine. 1,3,5-phenylene based hydrocarbon dendrimers; Mw =Mn ¼ 1:08; soluble in common organic solvents; spectroscopy, thermal analysis described. 94 monomer units (C1134 H1146 ); 3,5-di-tert-butylphenyl peripheral group; Mw ¼ 14776; Mw =Mn ¼ 1:03; soluble in pentane. Method for large scale (several Kg) synthesis presented; diaminobutane as the core; NH2 or CN end groups; five generations; Tg , viscosity discussed. 36 terminal trimethylammonium groups, catalysis applications discussed. Ammonia as the initiator core; nitrile or amine end groups; five generations. Lyotropic. Three polyamides with the same internal hierarchical architectures but with either acidic, neutral or basic terminal functionality; 5 generations, 972 terminal groups in 5th gen.; the polymers shrink or swell upon pH change (‘‘smart behavior’’). Ethylenediamine or ammonia core, N-(2-aminoethyl)acrylamide repeat units, seven generations; Mw ¼ 43 451 with ammonia core, 57 972 with ethylenediamine core. With CO2 Me or NH2 head groups; hydrodynamic radii ([h]-Mw measurement), and surface area with generations discussed. Based on 2,4-dinitrofluorobenzene and anilines. Complexation with iodine; Tm and cyclic voltammetry discussed. Based on 3,5-bis(trimethylsiloxy) benzoyl chloride; Mw from 30 000 to 200 000; Mw and dispersity depend on temperature; 55 to 60% branching. Hyperbranched aromatic polyesters based on 5-acetoxyisophthalic acid (Tg ¼ 239  C) and 5-(2-hydroxyethoxy)isophthalic acid (Tg ¼ 190  C); Carboxylic acid terminal groups; degree of branching: 50%; [h], polyelectrolyte properties discussed. 3,5-dihydroxybenzyl alcohol monomer unit reacted with benzylic bromide; 1, 1, 1-tris(4’ hydroxyphenyl)ethane core; six generations; Mw ¼ 154 000; Mw =Mn ¼ 1:02. Size exclusion chromatography, [h]-Mw relationship; linear dependence of hydrodynamic radii on generation; a characteristic maximum in [h] observed; samples from Hawker and Fre´chet [131, 132] used. Dibromalkanes with 1-(4-hydroxy-4’-biphenylyl-2-(4hydroxyphenyl); 6–10 methylene units as flexible spacers; those with 6, 8, and 10 CH2 units or with benzyl chain end exhibit nematic mesophase. Monomers: 6-bromo-1-(4-hydroxy-4’-biphenylyl)-2-(4-hydroxyphenyl)hexane (TPH); 13-bromo-1-(4-hydroxyphenyl)-2- [4-(6-hydroxy-2naphthalenylyl)-phenyl]tridecane (BPNT); 13- bromo-1-(4-hydroxyphenyl)-2-(4-hydro-4’’-p- terphenylyl)tridecane (TPT); Chain ends: benzyl or allyl or alkyl. TPH and BPNTshow narrow nematic meosphase; TPT nematic mesophase extends over 82 8C. Degree of branching for TPT with allyl end is 0.82.

Reference [2, 49, 250]

[250a, see also 250] [253]

[254,255] [256]

[257] [258] [136] [259]

[260, 261]

[262–264] [265] [266]

[267]

[131, 132]

[268]

[108]

[137]

20 / CHAPTER 1 TABLE 1.8. Continued. Dendrimer Polyether-b-polyester dendrimer

Polyether dendrimer-styrene copolymer Polyether dendrimer-C60

Polyphenylene dendrimer Polyphenolic dendrimers Silane dendrimers Silicone dendrimers Polysiloxysilane dendrimers Polysiloxane dendrimers

Siloxane starburst dendrons and dendrimers

Remarks

Reference

3,5-dihydroxybenzyl alcohol as monomer for ether-linked fragments and 2,2,2-trichloroethyl 3,5-dihydroxybenzoate for ester linked fragments; radially alternating the dendritic segments produced segmented-block polymer and concentric alternation gave layer-block polymer; spectroscopy, thermal characterization described. Copolymer of the type poly[(styrene)y -co-(styrene-G-4)x ] where the fourth generation dendrimer is p-linked to styrene. A deuteriated fourth generation azide dendrimer [138], with C60 as the core; 68% yield; extremely soluble: Tg ¼ 52  C (138 higher than starting dendrimer). Carboxylate form is water soluble. complexation with p-toluidine reported. A double stage convergent synthesis is described; a ‘hypercore’ based on 4,4- bis(4’-hydroxyphenyl)pentanol. Tetraallylsilane as the zeroth generation; five generations synthesized (C4368 H7764 Si485 ).

[269]

Tris-[(phenyldimethylsiloxy) dimethylsiloxy]methylsilane core; bis[(phenylimethylsiloxy)methyl siloxy] dimethylsilanol as the building block; Three generations; [h]-Mw relationship/MarkHouwink constants, NMR, molecular diameter discussed; Mw =Mn #1:1. Allylbis[4-(hydroxydimethylsily)phenyl]-methylsilane as the building block; Four generations; [h]-Mw relationship and Tg reported.

1.9 SUPRAMOLECULAR POLYMERS Using molecular recognition and self-assembly to construct macromolecules and to design devices using macromolecules is emerging as a fertile area of research. Both covalent and noncovalent bonding are used to this end. Along with interactions such as hydrogen bonding, charge transfer complex, ionic bonding, etc. the secondary and tertiary structures can be designed in a controlled and reversible manner [278–285]. Chain folding as a precursor to self-assembly has also been realized. Sequences of rigid hydrophobic chromophores, linked by flexible hydrophilic segments, fold and unfold [286]. While the intramolecular interaction between the chromophores leads to chain folding, intermolecular attractions favor self-assembly. Incorporating a specific "foldamer" [287] to cause, for example, a U turn in the polymer structure has been accomplished [288,289]. The possibilities for molecular architectures are thus endless. ACKNOWLEDGMENTS Financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.

[270] [69]

[271] [134] [272, 273] [274] [275] [276]

[277]

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52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 66a. 66b. 67. 67a. 67b. 67c. 67d. 67e. 67f. 67g. 67h. 67i. 67j. 67k. 68. 69. 70. 71. 71a. 71b. 71c.

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CHAPTER 2

Names, Acronyms, Classes, and Structures of Some Important Polymers Chandima Kumudinie Jayasuriya* and Jagath K. Premachandray *Department of Chemistry, University of Kelaniya, Dalugama, Kelaniya, Sri Lanka; yDepartment of Chemical and Process Engineering, University of Moratuwa, Katubedda, Moratuwa, Sri Lanka

Acronym, alternate name

Common name Amylose

Class

Structure of repeat unit

Polysaccharide OH O

OH O

HO OH

Bisphenol A polysulfone

O O

HO

OH

n

Polysulfone O

O

SO2

C(CH3)2 n

Cellulose

Cellulose acetate

Rayon, cellophane, regenerated cellulose

CA

Polysaccharide

HO

OH

OH

O O

OH

Cellulose ester

O

OR

O HO

OH

n

OR

OR

O O

OR

O

O OR

OR

n R = −COCH3

Cellulose nitrate

CN

Cellulose ester OR

OR

OR

O O

O O OR

OR

OR

n R = −NO2

25

26 / CHAPTER 2 Continued.

Common name Hydroxypropyl cellulose

Acronym, alternate name HPC

Class

Structure of repeat unit

Cellulose ester OR

OR

OR

O O

OR

O

O OR

OR n

R = −(CH2)3−OH

Ladder polymer

Double-strand polymer n

Polyacetal

Polyether

H C

O n

R

Polyacetaldehyde

Polyether

H C

O n

CH3

Polyacetylene

Polyalkyne CH

CH n

Polyacrylamide

Vinyl polymer, acrylic polymer

CH

CH2 n

C

NH2

O

Poly(acrylic acid)

Vinyl polymer, acrylic polymer

CH

CH2 n

C O

OH

NAMES, ACRONYMS, CLASSES, AND STRUCTURES OF SOME IMPORTANT POLYMERS

/ 27

Continued.

Common name

Acronym, alternate name

Polyacrylonitrile

PAN

Class Vinyl polymer, acrylic polymer

Structure of repeat unit CH2

CH

n CN

Poly(L-alanine)

Polypeptide

O NH

C

CH

n CH3

Polyamide

Nylon

Polyamide

O NH

R

NH

O

C

R'

C n

Polyamide imide

PAI

O C NH

N

C C O

R n

O

Polyaniline

Polyamine NH n

Polybenzimidazole

PBI

Polyheteroaromatic

N

N

NH

NH

Polybenzobisoxazole

PBO

Polyheteroaromatic

N

O

Polybenzobisthiazole

PBT

Polyheteroaromatic

n

N

O

N

N

S

S

n

n

28 / CHAPTER 2 Continued.

Common name Poly(g-benzyl-Lglutamate)

Acronym, alternate name PBLG

Class

Structure of repeat unit

Polypeptide

O NH

CH

C n

(CH2)2 O

1,2-Polybutadiene

PBD

C

CH2

O

Diene polymer CH2

CH

n CH

cis-1,4Polybutadiene

PBD

CH2

Diene polymer

H

trans-1,4Polybutadiene

PBD

H

Diene polymer H n

H

Poly(1-butene)

n

PB-1

Polyolefin CH

CH2 n

CH2 CH3

Polybutylene terephthalate

PBT

Polyester (CH2)4

O

O

O

C

C n

Poly(e-caprolactam)

Nylon 6

Polyamide

O NH

C

(CH2)5 n

Poly(e-caprolactone)

Polyester

O O

C

(CH2)5 n

NAMES, ACRONYMS, CLASSES, AND STRUCTURES OF SOME IMPORTANT POLYMERS

/ 29

Continued.

Common name

Acronym, alternate name

Polycarbonate

PC

Class

Structure of repeat unit

Polyester

CH3 O

O

C

O

CH3

cis-1,4Polychloroprene

Neoprene

n

Diene polymer n H

Cl

trans-1,4Polychloroprene

Neoprene

Diene polymer H

Cl

Polychlorotrifluoroethylene

PCTFE

Vinyl polymer, Fluoro polymer

Cl

F

C

C

F

F

n

n

Poly(10-decanoate)

Polyester

O (CH2)9

O

C n

Poly(diethyl siloxane)

PDES

Polysiloxane

CH2 CH3 Si

O n

CH2 CH3

Poly(dimethyl siloxane)

PDMS

Polysiloxane

CH3 Si

O n

CH3

C

30 / CHAPTER 2 Continued.

Common name Poly(diphenyl siloxane)

Acronym, alternate name PDPS

Class

Structure of repeat unit

Polysiloxane

Si

O n

Polyester

O

O O

R

O

C

R'

C n

Polyether ketone

PEK

Polyketone

O O

C n

Polyether etherketone

PEEK

Polyketone

O O

C

O n

Polyether sulfone

PES SO2

O

n

Polyethylene

PE

Polyolefin CH2

CH2 n

Poly(ethylene imine)

Polyamine CH2

CH2

NH n

Poly(ethylene oxide) [Poly(ethylene glycol)]

PEO [PEG]

Polyether CH2

CH2

O n

NAMES, ACRONYMS, CLASSES, AND STRUCTURES OF SOME IMPORTANT POLYMERS

/ 31

Continued.

Common name Poly(ethylene terephthalate)

Acronym, alternate name PET

Class

Structure of repeat unit

Polyester (CH2)2

O

O

O

C

C n

Polyglycine

Polypeptide

O NH

CH2

C n

Poly(hexamethylene adipamide)

Nylon 6,6

Polyamide

O (CH2)6

NH

O (CH2)4

C

NH

C n

Poly(hexamethylene sebacamide)

Nylon 6,10

Polyamide

O (CH2)6

NH

O (CH2)8

C

NH

C n

Polyhydroxybutyrate

PHB

Polyester

CH3 O

O CH2

CH

C n

Polyimide

PI

Polyimide

O

O

C

C

N

Polyisobutylene

Butyl rubber

Vinylidine polymer

N C

C

O

O

CH3 C

CH2 n

CH3

Polyisocyanate

PIC

Polyamide

O N

C n

R

n

32 / CHAPTER 2 Continued.

Common name

Acronym, alternate name

Polyisocyanide

Class Polyisocyanide

Structure of repeat unit N

R

C n

cis-1,4-Polyisoprene

cis-PIP, natural rubber

Diene polymer n H

trans-1,4Polyisoprene

trans-PIP Gutta percha

CH3

Diene polymer CH3

H

Polylactam

n

Polyamide

O (CH2)m

NH

C n

Polylactone

Polyester

O O

(CH2)m

C

n

Poly(methacrylic acid)

Vinyl polymer, acrylic polymer

COOH CH2

C CH3

Poly(methyl acrylate)

PMA

Vinyl polymer, acrylic polymer

CH2

CH C O

n

O

n CH3

NAMES, ACRONYMS, CLASSES, AND STRUCTURES OF SOME IMPORTANT POLYMERS

/ 33

Continued.

Common name Poly(methyl methacrylate)

Acronym, alternate name PMMA

Class Vinylidene polymer, acrylic polymer

Structure of repeat unit CH3 CH2

C C

n CH3

O

O

Poly(4-methyl pentene)

Polyolefin

CH2

CH n CH2 CH(CH3)2

Poly(a-methyl stryrene)

Vinylidene polymer

CH3 CH2

C

n

Poly(methylene oxide)

PMO Polyformaldehyde

Polyether CH2

O n

Poly(methyl phenyl siloxane)

PMPS

Polysiloxane

CH3 Si

O n

Poly(m-phenylene terephthalamide)

Nomex

Polyaramid

O HN

NH

C

O C n

34 / CHAPTER 2 Continued.

Common name Poly(p-phenylene terephthalamide)

Acronym, alternate name Kevlar

Class

Structure of repeat unit

Polyaramid HN

NH

O

O

C

C n

Polynitrile

Polyimine

C

N n

R

Polynucleotide

Polynucleotide

base phosphate

sugar n

Poly(n-pentene-2)

Poly(a-olefin)

CH

CH

CH3

CH2 CH3

CH

CH2

n

Poly(n-pentene-1)

Poly (1-pentene)

Poly(a-olefin)

n

CH2 CH2 CH3

Polypeptides [Poly(a-amino acid)]

Polypeptide

O NH

CH

C n

R

Poly(p-methyl styrene)

Vinyl polymer CH

CH2 n

CH3

Poly(p-phenylene)

PP

Polyaromatic

n

NAMES, ACRONYMS, CLASSES, AND STRUCTURES OF SOME IMPORTANT POLYMERS

/ 35

Continued.

Common name Poly(p-phenylene oxide)

Acronym, alternate name PPO

Class

Structure of repeat unit

Polyether O n

Poly(p-phenylene sulfide)

PPS

Polysulfide S n

Poly(p-phenylene vinylene)

Polyaromatic CH

CH n

Polyphenylsulfone

Polysulfone

SO2

O

O n

Polyphosphate

Inorganic polymer

O P

O

R

O

OR'

Polyphosphazene

Inorganic polymer

n

R P

N n

R'

Polyphosphonate

Inorganic polymer

O P

O

R

O n

R'

Poly(3-propionate)

Poly(bpropiolactone)

Polyester

O O

CH2 CH2

C n

Polypropylene

PP

Poly(a-olefin)

CH CH3

CH2 n

36 / CHAPTER 2 Continued.

Common name Poly(propylene glycol)

Acronym, alternate name PPG

Class

Structure of repeat unit

Polyether HOCHCH2

CH2

O

CH

OH n

CH3

Poly(propylene oxide)

PPO

CH3

Polyether CH2

CH

O n

CH3

Polypyrazine

Heterocyclic polymer

N Ar n

N

Polypyrazole

Heterocyclic polymer

R

R

N

N

N

N R

Poly(pyromellitimide-1,4diphenyl ether)

Kapton

Polyimide

O

O

C

C

C

C

O

O

N

Polypyrrole

n

O

N

n

Heterocyclic polymer

N

Polyquinoxaline

R

n

Heterocyclic polymer N

N

N

N n

Polysilane

Inorganic polymer

R Si R'

n

NAMES, ACRONYMS, CLASSES, AND STRUCTURES OF SOME IMPORTANT POLYMERS

/ 37

Continued.

Common name

Acronym, alternate name

Polysilazane

Polysiloxane

Class Inorganic polymer

Silicone

Inorganic polymer

Structure of repeat unit R Si

N

R'

R"

n

R Si

O n

R'

Polystyrene

PS, styrofoam

Vinyl polymer CH2

CH

n

Polysulfide

Thiokol

Polysulfide

R

Sm n

Polysulfur

Polysulfur

S n

Polytetrafluoroethylene

Poly(tetramethylene oxide)

PTFE, Teflon

PTMO

Poly(a-olefin)

F

F

C

C

F

F

n

Polyether CH2

CH2

CH2

CH2

O n

Polythiazyl S

N n

Polythienyl vinylene CH=CH S

n

38 / CHAPTER 2 Continued.

Common name

Acronym, alternate name

Polythiopene

Class

Structure of repeat unit

Polyheterocyclic

S

Poly(trimethylene ethylene urethane)

n

CH2CH2CH2 OCONH CH2CH2

O

NHCO n

Polyurea

Polyurea

O NH

R

NH

O

C

NH

R'

NH

C n

Polyurethane

Adiprene

Polyurethane

O O

R

O

O

C

NH

R'

NH

C n

Poly(L-valine)

Polypeptide

O NH

CH

C n

CH(CH3)2

Poly(vinyl acetate)

PVAc

Vinyl polymer CH2

CH

n O

CH3

C O

Poly(vinyl alcohol)

PVA

Vinyl polymer

CH

CH2 n

OH

Poly(vinyl butyral)

PVB

Vinyl polymer

CH2 CH2

CH

CH

O

O CH (CH2)2 CH3

n

NAMES, ACRONYMS, CLASSES, AND STRUCTURES OF SOME IMPORTANT POLYMERS

/ 39

Continued.

Common name

Acronym, alternate name

Class

Poly(vinyl carbazole)

Structure of repeat unit CH2

CH n N

Poly(vinyl chloride)

PVC

Vinyl polymer

CH2

CH

n Cl

Poly(vinyl fluoride)

PVF

Vinyl polymer, fluoro polymer

CH2

CH

n F

Poly(vinyl formal)

Vinyl polymer

CH2 CH2

CH

CH

O

O CH2

Poly(2-vinyl pyridine)

PVP

Vinyl polymer

CH2

CH

n N

Poly(N-vinyl pyrrolidone)

Vinyl polymer

CH

CH2 n

N

Poly(vinylidene chloride)

PVDC

Vinylidene polymer

O

Cl C

CH2 n

Cl

n

40 / CHAPTER 2 Continued.

Common name Poly(vinylidene fluoride)

Acronym, alternate name PVDF

Class Vinylidene polymer

Structure of repeat unit F CH2

C

n F

Poly(p-xylylene) CH2

CH2

n

Vinyl polymer

Vinyl polymer

R

R"

C

C

R'

R" "

n

ACKNOWLEDGEMENT The authors wish to acknowledge the contribution of W. Zhao, formerly of SRI International, to this chapter.

CHAPTER 3

The Rotational Isomeric State Model Carin A. Helfer and Wayne L. Mattice Institute of Polymer Science, The University of Akron, Akron, OH 44235-3909

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History and Noteworthy Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship to Simpler Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rotational Isomeric State Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Statistical Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Conformational Partition Function, Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Stereochemical Sequence in Vinyl Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extraction of Useful Information From Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Virtual Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Expression for the Dimensions of a Specified Conformation . . . . . . . . . . . . Averaging the Dimensions over all the Conformations in Zn . . . . . . . . . . . . . . . . . . . Use of Cn for Calculation of C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Applications of the RIS Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why are some chains described with more than one RIS Model? . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 44 44 45 45 46 48 50 51 51 52 53 53 55 56 56

ences in energy for conformations produced by rotation about a bond or pair of bonds) as most chemists are likely to desire. Figure 3.1 defines the values of u and f that are associated with bond i. This detailed description of the local chain structure is presented in a mathematical framework that often permits extremely fast calculation of the average values of many conformation-dependent physical properties of individual polymer chains in the amorphous bulk state and in dilute solution in a solvent chosen so that the excluded volume effect is negligible. The speed of the calculation follows from the formulation of the problem as the serial product of n matrices. Computers are easily trained to rapidly, and accurately, compute this serial product. The most commonly calculated property is the mean square unperturbed dimension. It is usually represented by the mean square unperturbed end-to-end distance, hr 2 i0 , obtained by averaging the square of the length of the end-to-end vector, r, over all conformations under conditions where the chain is unperturbed by long-range interactions.

3.1 INTRODUCTION Flexible macromolecules populate an enormous number of conformations at ordinary temperature, T. If n stable conformations are available to each internal bond in a chain of n bonds, the chain can access nn
2 conformations. When n ¼ 3, as is appropriate for many simple polymers, the number of distinguishable conformations exceeds 10100 when n > 211. This number is achieved by polyethylene at the relatively low molecular weight of 2,984. Scientists and engineers need information about the average properties of specific chains with much larger values of n, where the conformation-dependent physical properties of interest depend on an appropriate average over a truly enormous number of conformations. Among the several models that have been proposed for averaging over this ensemble of conformations, the rotational isomeric state (RIS) model is unique in its combination of structural detail with computational efficiency. It incorporates as much structural detail (bond lengths, l, bond angles, u, torsion angles, f, differ43

44 / CHAPTER 3

qi

fi

speed with structural detail. Numerous other conformationdependent physical properties are accessible also from this model. This chapter will describe the RIS method, present a few illustrative results, and cite many of the RIS models for specific polymers that have been presented in the literature.

li+1

li li⫺1

3.2 HISTORY AND NOTEWORTHY REVIEWS

FIGURE 3.1. The definitions of ui and fi associated with bond i.

r¼

n X

li ,

(3:1)

i¼1 n n
1 X n
X X
2  r 0 ¼ hr  ri0 ¼ l2i þ 2 li  lj 0 : i¼1

(3:2)

i¼1 j¼iþ1

The double sum in Eq. (3.2) can be handled efficiently by matrix methods when the chain is in its unperturbed, or Q, state. The result for hr 2 i0 is usually presented as the dimensionless characteristic ratio, Cn .
2 r 0 : (3:3) Cn ¼ nl2 As defined in Eq. (3.3), Cn is the ratio of the mean square unperturbed end-to-end distance to the value expected for the freely jointed chain with the same number of bonds, of the same length. If the bonds in the chain are of different lengths, as in polyoxyethylene, l2 in the denominator is replaced by the mean square bond length. For any flexible unperturbed chain, Cn approaches a limit, C1 , as n ! 1. The final approach to this limit is usually from below, but it can sometimes be from above. The latter situation can be encountered when Cn passes through a maximum at finite n [1]. The mean square unperturbed radius of gyration, hs2 i0 , is accessible by similar methods that rapidly evaluate and sum the mean square end-to-end distances for all of the subchains, denoted by hrij2 i0 , where i and j identify the chain atoms at the ends of the subchain. If all n þ 1 chain atoms, indexed from 0 to n, can be taken to have the same mass, hs2 i0 is given by the expression in Eq. (3.4).
2 s 0¼

n
1 X n D E X 1 r2 : (n þ 1)2 i¼0 j¼iþ1 ij 0

(3:4)

If the chain is flexible, the RIS calculations produce hs2 i0 ¼ hr 2 i0 =6 in the limit as n ! 1, as expected [2]. The ratio hr 2 i0 =hs2 i0 may differ from 6 at finite n because hr 2 i0 and hs2 i0 do not have the same approach to their limiting behavior [1]. For many real chains, hr 2 i0 =hs2 i0 > 6 at finite n, although hr 2 i0 =hs2 i0 ! 6 as n ! 1. The RIS model has no peer for the calculation of hs2 i0 or hr 2 i0 as a function on n because it combines

The speed of calculations using the RIS model arises from its formulation of the problem as a serial product of matrices. The generator matrix technique, which lies at the heart of the calculation, predates the appearance of the RIS model by 10 years [3]. The application of the RIS technique to polymers is now over five decades old [4], although its appearance in the polymer literature did not begin to mushroom until a decade after its first appearance [5–9]. Because computers at that time did not have nearly the speed and widespread availability that is seen today, there was a strong motivation for formulation of the problem in a manner that allowed efficient calculation. With today’s computers, the most popular calculations require no more than a few seconds of cpu time. The first important general work on the RIS model is Flory’s classic book, which first appeared in 1969 [10]. His book was followed 5 years later by an excellent review in Macromolecules that presented a more general and concise formulation of the RIS method [11]. Another book on the RIS model appeared during the year of the 25th anniversary of the first publication of Flory’s original text [12]. It was soon followed by an exhaustive compilation, in a standardized format, of the RIS models presented in the literature over the four decades that ended in the mid1990s [13]. 3.3 RELATIONSHIP TO SIMPLER MODELS The information incorporated in the RIS model and several simpler models is summarized in Table 3.1. The freely jointed chain has n bonds of length l, with no correlation whatsoever in the orientations of any pair of bonds. The contribution of the double sum in Eq. (3.2) is nil, and Cn ¼ 1 at all values of n. Fixing the bond angle, but allowing free rotation about all bonds, produces the freely rotating chain. If u 6¼ 90 , Cn will depend on n. The asymptotic limit as n ! 1 is (1
cos u)=(1 þ cos u). If u > 90 , as is usually the case with real polymers, the freely rotating chain model yields C1 > 1. Energetic information, in the form of a torsional potential about the internal bonds, E(f), is incorporated in the third model in Table 3.1. If the same symmetric torsion potential is applied independently to all internal bonds, the characteristic ratio depends on the average value of the cosine of the torsion angle. The term from the freely rotating chain is retained, and it is multiplied by another term that arises from the symmetric hindered rotation, C1 ¼ [(1
cos u)=(1 þ cos u)] [(1
hcos fi)=(1 þ hcos fi)].

THE ROTATIONAL ISOMERIC STATE MODEL

/ 45

TABLE 3.1. Information incorporated in the RIS model and in several simpler models. Model Freely jointed chain Freely rotating chain Simple chain with symmetric hindered rotation RIS model

Geometric informationa

Energetic information

n, l n, l, u n, l, u,f

None None First-order interactions (independent bonds, symmetric torsion) First- and higher-order interactions (interdependent bonds, torsion need not be symmetric)

n, l, u,f

a

All bonds are assumed to be identical in the usual implementations of the first three models. The assumption of identical bonds is easily discarded in the RIS model.

Ð Since hcos fi depends on T, hcos fi ¼ { exp [
E(f)=kT] Ð df}
1 cos f exp [
E(f) =kT] df, this model is the only one in this paragraph that explicitly says the mean square unperturbed dimensions are temperature dependent. The Boltzmann constant is denoted by k. All of the results in this paragraph assume that the bonds are identical. In general, it is difficult or impossible to write the results for C1 with such simple closed-form expressions when the torsions become interdependent and the bonds are not all identical. However, Nature asks that we take account of the interdependence of the torsions, because nearly all of the real-world polymers have bonds that are subject to interdependent torsions. And many important polymers are made up of bonds with different lengths. The closest one can come to a general and simple expression is something of the form given in Eq. (3.5). C1 ¼ Lim

n!1

G1 G2    Gn 1 ¼ Lim G1 G2    Gn : n!1 Zn U1 U2    Un

(3:5)

As we shall see below, the denominator in Eq. (3.5) is the conformational partition function, Zn , for the RIS model of the chain. It is constructed as a sum of Boltzmann factors that depend on T and the energies of the first- and higherorder interactions present in all of the conformations of the chain. Structural information does not appear explicitly in Zn . However, a wealth of structural information (l, u, f) can appear in the numerator of Eq. (3.5). The numerator also contains all of the thermal and energetic information from Zn . The combination of this information allows a rapid estimation of Cn , even at large n, because computers can rapidly calculate the serial matrix products that appear in the numerator and denominator of Eq. (3.5). U1 . . . Un is a simpler serial product than G1 . . . Gn , because it does not include structural information explicitly. For this reason, the easiest introduction to the RIS model is to focus first on Zn , rather than G1 . . . Gn .

3.4

THE ROTATIONAL ISOMERIC STATE APPROXIMATION

The basis for the RIS model is most easily seen if we consider a chain where the torsion angles at internal bonds

are restricted to a small set of values. For many simple polymers, the RIS models use n ¼ 3, but the model is sufficiently robust so that it can be used with other choices also. The number of conformations of a chain of n bonds is nn
2 , which becomes enormous when n is large enough so that the molecule becomes of interest to polymer scientists. A pair of two consecutive bonds, bonds i
1 and i, has n2 conformations. The n2 conformations can be presented in tabular form, where the columns represent the n conformations at bond i, and the rows represent the n conformations at bond i
1. Each entry in the table corresponds to a specific choice of the conformations at these two bonds. In the RIS model, this table becomes a matrix. The elements in the matrix represent contributions to the statistical weights for the conformation adopted at bond i (which depends on the column in the matrix), for a specific choice of the conformation at the preceding bond (which depends on the row in the matrix). 3.5 THE STATISTICAL WEIGHT MATRIX The statistical weight matrix for bond i, denoted Ui , is usually formulated as the product of two matrices. Ui ¼ Vi Di :

(3:6)

Interaction energies that depend only on the torsion at bond i are responsible for the statistical weights that appear along the main diagonal in Di . These interactions are termed firstorder interactions because they depend on a single degree of freedom, fi . For the example of a polyethylene-like chain with a symmetric three-fold torsion potential, the rotational isomeric states are t, gþ , g
(trans, gaucheþ , gauche
). In the approximation that all bonds are of the same length, all bond angles are tetrahedral, and the torsion angles for the t and g states are 1808 and + 608, the separation of the terminal atoms in a chain of three bonds is (19=3)1=2 l in the t state, but this separation falls to (11=3)1=2 l in the g states. This change in separation usually produces different energies in the t and g states. The influence of these energies on the conformation of the chain is taken into account in D. Often a statistical weight is calculated from the corresponding energy as a Boltzmann factor, w ¼ exp (
E=RT). The t state is usually taken as the reference point, with Et ¼ 0

46 / CHAPTER 3 and a statistical weight of one, and the g states have a statistical weight of s ¼ exp [
(Eg
Et )=RT] if the torsion is symmetric, with Egþ ¼ Eg
. A pre-exponential factor may also be necessary if the t and g wells have significantly different shapes. When the order of indexing of the rows and columns is t, gþ , g
, this diagonal matrix takes the form shown in Eq. (3.7). 2 3 1 0 0 Di ¼ diag(1,s,s) ¼ 4 0 s 0 5, 1 < i < n: (3:7) 0 0 s Real chains often have s < 1, as in polyethylene [14], but a few chains, such as polyoxymethylene, have s > 1 [15]. The second-order interactions depend jointly on fi
1 and fi . For a simple chain with n ¼ 3 and symmetric torsion about its internal bonds, the second-order interactions in the tgþ , tg
, gþ t, and g
t states are identical, as are the interactions in the gþ gþ and g
g
states, and the interactions in the gþ g
and g
gþ states. The general form of Vi under these conditions appears in Eq. (3.8), where the order of indexing is t, gþ , g
for both rows and columns, and the reference point for the second-order interactions is any of the four conformations where one bond is t and the other is g. 2 3 t 1 1 Vi ¼ 4 1 c v 5, 2 < i < n: (3:8) 1 v c The state at bond i
1 indexes the rows, and the state at bond i indexes the columns. Figure 3.2 depicts the four possible separations of the terminal atoms in a chain of four bonds when all bonds are of the same length, bond angles are tetrahedral, and the torsion angles for the t and g states are 1808 and 608. By far the shortest separation is seen when the two internal bonds adopt g states of opposite sign. This short distance

causes most real chains to have severely repulsive secondorder interactions in the gþ g
and g
gþ conformations, producing v < 1, as in both polyethylene and polyoxyethylene. Often the other second-order interactions are weak enough so that little error is introduced if they are ignored, which frequently leads to the approximation t ¼ c ¼ 1. The statistical weight matrix incorporates the first- and second-order interactions, according to Eq. (3.6). For the chain with a symmetric three-fold torsion potential and pairwise interdependent bonds, Ui adopts the form in Eq. (3.9). 2 3 t s s Ui ¼ 4 1 sc sv 5, 2 < i < n: (3:9) 1 sv sc The first rotatable bond in the chain, with i ¼ 2, is a special case because there is no preceding rotatable bond for use in defining the statistical weights to be incorporated in V2 . This situation is handled by formulating U2 using only the statistical weights for first-order interactions, which is achieved by using a value of 1 for every element in V2 . For the simple chain considered in the examples presented in Eqs. (3.7)–(3.9), all of the Ui are square and identical [14]. In polyoxymethylene, all of the Ui are square, but there are two square Ui with distinctly different numerical values of the elements [15]. Half of the statistical weight matrices for polyoxymethylene incorporate second-order interactions between pairs of oxygen atoms, and the other half incorporate second-order interactions between pairs of methylene groups. For other polymers, such as the polycarbonate of bisphenol A [16], some of the Ui may be rectangular but not square, because there is a different number of rotational isomeric states at bonds i
1 and i. The RIS model does not require that all bonds adopt the same value for n. 3.6 THE CONFORMATIONAL PARTITION FUNCTION, Zn

4l tt

Distance

3l

tg g+ g+

2l g+ g⫺ l

0

FIGURE 3.2. The four distinguishable separations of the terminal atoms in a chain of four bonds when all bonds are of the same length, l, all bond angles are tetrahedral, and the three states at each internal bond have torsion angles of 1808 (t) or + 608 (g þ and g
).

The conformational partition function, in the RIS approximation, is the sum of the statistical weights for the nn
2 conformations in the RIS model. The terminal row and column vectors must be formulated so that they will extract the desired sum of the statistical weights of all conformations from V2 D2 . . . Vn
1 Dn
1 . Zn ¼ J V2 D2 V3 D3 . . . Vn
1 Dn
1 J ¼ J U2 U3 . . . Un
1 J:

(3:10)

J denotes a row of n elements in which the first element is 1 and all following elements are 0, and J denotes a column of n elements in which every element is 1. The statistical weight matrix Ui , 1 < i < n, is the product Vi Di . Often J and J are written instead as U1 and Un [11]. Zn ¼ U1 U2 U3 . . . Un
1 Un ¼

n Y i¼1

Ui :

(3:11)

THE ROTATIONAL ISOMERIC STATE MODEL

/ 47

As a specific example, Zn for a polyethylene chain can be calculated by the combination of Eqs. (3.9) and (3.11), in the approximation where t ¼ c ¼ 1 [14]. 2

Zn ¼ ½ 1

0

32 s 1 s 54 1 s 1

1 s 0 4 1 s 1 s

s s sv

For ethyl terminated polyoxyethylene, the main portion of the calculation employs a repetition of three distinct statis-

2 3n
3 2 3 1 1 s sv 5 4 1 5 ¼ J 4 1 1 1 s

s s sv

3n
2 s sv 5 J: s

(3:12)

tical weight matrices, containing two distinct s’s and two distinct v’s [15].

2

3 02 32 32 31(n
4)=3 2 3 1 sa 1 sb 1 sa 1 sa s a sa sb sa 1 sa sa 4 1 sa sa va 5J: Zn ¼ J 4 1 sa sa 5@4 1 sa sa va 54 1 sb sb vb 54 1 sa sa vb 5A 1 sa s a 1 s a va s a 1 sb vb sb 1 sa vb sa 1 sa va sa

Here sa and sb denote the statistical weights for the firstorder interaction of two methylene groups and two oxygen atoms, respectively, in g states. The statistical weights for the second-order interactions of two methylene groups and a methylene group with an oxygen atom in g  g  states are denoted by va and vb , respectively. Table 3.2 summarizes RIS models for several chains with pair-wise interdependent bonds subject to a symmetric three-fold torsion potential, such that U is given by Eq. (3.9). The torsion angles are 1808 and (608 þ Df). Every entry has Ev < 0. This energy is listed as being infinite when the population of the gþ g
and g
gþ states is so small that it can be ignored. In contrast with Ev , the table contains entries for Es that are of either sign.

(3:13)

Table 3.3 summarizes selected literature citations for RIS models for homopolymers with 1–7 bonds per repeat unit, with all bonds subject to symmetric torsions. The list in Table 3.3 terminates with poly(6-aminocaproamide), nylon 6, although RIS models for chains with much longer repeat units have been reported in the literature [13]. The selection of entries in Table 3.3 is based in part on recognizing contributions of historical interest, and in part on more recent models that exploit computational methods and experiments that were not readily available during the early days of the development of RIS models. It does not include all applications of the RIS model to a given polymer. In some cases this number would be huge. For example, there are well over 100 applications of RIS models for polyethylene in the literature.

TABLE 3.2. RIS models for several chains with pair-wise interdependent bonds subject to a symmetric threefold torsion potential and Et ¼ Ec ¼ 0. Lengths in nm, angles in degrees, energies in kJ/mol. Polymer

Bond

l

u

Df

Es

Ev

Reference

Polymethylene Polymethylene Polyoxymethylene

C–C C–C C–O O–C Si–C C–Si Si–O O–Si C–O O–C C–C O–C C–C C–C C–O

0.153 0.153 0.142 0.142 0.190 0.190 0.164 0.164 0.143 0.143 0.153 0.143 0.153 0.153 0.143

112 112 112 112 115 109.5 143 110 111.5 111.5 111.5 111.5 111.5 111.5 111.5

7.33 03 5 5 0 0 0 0 10 10 10 10 0 0 10

1.1–1.9 1.8–2.5
5.9
5.9 0 0 3.6 3.6
1.7 to
2.1
1.7 to
2.1 3.8 3.8
1.7
1.7 3.8

5.4–6.7 7.1–8.0 1 6.3 1 0.80 1 4.2 1.7 1 1.7 1 1 2.5 1

[14] [14] [15]

Polydimethylsilmethylene Polydimethylsiloxane Polyoxyethylene

Poly(trimethylene oxide)

[17] [18] [15]

[15]

48 / CHAPTER 3 TABLE 3.3. Selected literature citations for RIS models of polymers without rings in the backbone and with bonds subject to symmetric torsions. x denotes the number of chain atoms in the repeat unit. For a given value of x, the models are listed in the order of increasing molecular weight of the repeat unit specified in the third column. x

Polymer

1

Polymethylene, polyethylene Polysilane Polymeric sulfur Polytetrafluoroethylene Polydimethylsilylene Polymeric selenium Polyoxymethylene Polysilylenemethylene Polydihydrogensiloxane Polyisobutylene Polyvinylidene fluoride Polydimethylsilylenemethylene Polydimethylsiloxane Polyphosphate Polyvinylidene chloride Polydichlorophosphazene Polyvinylidene bromide Polydiphenylsiloxane Polyoxyethylene Polyglycine Polythiaethylene Poly(oxy-1,1-dimethylethylene) Poly(1,4-cis-butadiene) Poly(1,4-trans-butadiene) Poly(trimethylene oxide) Poly(1,4-cis-isoprene) Poly(1,4-trans-isoprene) Poly(trimethylene sulfide) Poly(3,3-dimethyloxetane) Poly(3,3-dimethylthietane) Poly(tetramethylene oxide) Poly(1,3-dioxolane) Poly(pentamethylene sulfide) Poly(thiodiethylene glycol) Poly(hexamethylene oxide) Poly(6-aminocaproamide)

2

3

4

5 6 7

Repeat unit

References

---CH2 -----SiH2 --–S– ---CF2 -----Si(CH3 )2 --–Se– ---CH2 ---O--CH2 SiH2 OSiH2 ---CH2 ---C(CH3 )2 -----CH2 ---CF2 -----CH2 ---Si(CH3 )2 -----O---Si(CH3 )2 -----O---PO2 -----CH2 ---CCl2 -----N---PCl2 -----CH2 ---CBr2 -----O---Si(C6 H5 )2 -----O---CH2 ---CH2 -----NH---CH2 ---CO-----S---CH2 ---CH2 -----O---CH2 ---CH(CH3 )2 -----CH2 ---CH==CH---CH2 -----CH2 ---CH==CH---CH2 -----CH2 ---CH2 ---CH2 ---O-----CH2 ---CH==C(CH3 )---CH2 -----CH2 ---CH==C(CH3 )---CH2 -----CH2 ---CH2 ---CH2 ---S-----O---CH2 ---C(CH3 )2 ---CH2 -----S---CH2 ---C(CH3 )2 ---CH2 -----CH2 ---CH2 ---CH2 ---CH2 ---O-----CH2 ---CH2 ---O---CH2 ---O-----S---CH2 ---CH2 ---CH2 ---CH2 ---CH2 -----O---CH2 ---CH2 ---S---CH2 ---CH2 -----O---CH2 ---CH2 ---CH2 ---CH2 ---CH2 ---CH2 -----NH---CH2 ---CH2 ---CH2 ---CH2 ---CH2 ---CO---

[14,19] [20] [21–23] [24–26] [20] [21–23, 27] [15,28,29] [30] [31] [32–35] [36,37] [17,30,38] [18,39] [40] [41] [42] [36] [43] [15,44–46] [47] [48,49] [50] [51–54] [53–56] [15] [51,52,57] [53,55,56] [58] [59,60] [61] [15] [62] [63] [64] [65] [66]

3.7 THE STEREOCHEMICAL SEQUENCE IN VINYL POLYMERS Vinyl polymers, for which polypropylene serves as a prototype, present some additional issues not encountered in chains with symmetric torsions. The physical properties of these chains depend on the stereochemical composition and stereochemical sequence of the chain, and this dependence must be reflected in Z. Two equivalent methods have been used for description of the stereochemistry of vinyl polymers. One approach uses pseudoasymmetric centers [67]. Although the fragment denoted by ---CH2 ---CHR--CH2 --- does not contain a chiral center, it can be treated as though it were chiral if one CH2 group is distinguished from the other. This distinction is drawn when the bonds in the

chain are indexed from one end to the other, because then the CH2 ---CHR bond preceding the pseudoasymmetric center bears a different index from the following CHR---CH2 bond. The –CHR-group is defined here to be in the d (l) configuration if the z component of the nonhydrogen substituent is positive (negative) in a local coordinate system for the CHR---CH2 bond. This local coordinate system is defined as follows: The x-axis for the CHR---CH2 bond is parallel with the bond and oriented from CHR to CH2 . The y-axis is in the plane of the chain atoms in ---CH2 ---CHR---CH2 ---, and oriented with a positive projection on the x-axis for the preceding CH2 ---CHR bond, which points from CH2 to CHR. The z-axis completes a righthanded Cartesian coordinate system. Alternatively (and equivalently), the stereochemical sequence can be described

THE ROTATIONAL ISOMERIC STATE MODEL as sequences of meso diads (two successive identical pseudoasymmetric centers) and/or racemo diads (two successive nonidentical pseudoasymmetric centers) [67]. Statistical weight matrices that include all first- and second-order interactions can be formulated using Eq. (3.6) and the additional matrix defined in Eq. (3.14). 2 3 1 0 0 Q ¼ 4 0 0 1 5: (3:14) 0 1 0

/ 49

The double subscript on v shows whether the second-order interaction is between two groups in the backbone, vCC , two side chains, vRR , or a side chain and a group in the backbone, vCR . If the two pseudoasymmetric centers have opposite chirality, the two possibilities are given in Eq. (3.20). 2 3 h vCR tvRR 1 tvCC 5: Udl ¼ QUld Q ¼ 4 hvCR (3:20) hvRR vCC tv2CR

Q has the useful property that Q ¼ E, where E denotes the identity matrix. For a vinyl polymer with a nonarticulated side chain (such as a halogen atom), the ---CHR---CH2 --- bond immediately following a pseudoasymmetric center has a D matrix that is either

When pseudoasymmetric centers are used for the description of the stereochemical sequence, six distinct statistical weight matrices, Eqs. (3.18)–(3.20), are required. They can be replaced by a total of three statistical weight matrices, denoted by Up ,Um , and Ur , if the stereochemical sequence is described instead as a sequence of meso and racemo diads.

Dd ¼ diag(h,1,t)

Up ¼ QUd ¼ Ul Q,

(3:21)

Um ¼ Udd Q ¼ QUll ,

(3:22)

Ur ¼ Udl ¼ QUld Q:

(3:23)

2

(3:15)

or Dl ¼ diag(h,t,1) ¼ QDd Q,

(3:16)

depending on whether the CHR is a d or l pseudoasymmetric center. The three-rotational isomeric states are t,gþ , and g
, in that order. The conformation weighted by t has two firstorder interactions, which occur between the underlined pairs of atoms in CH2 ---CHR---CH2 ---C and CH2 ---CHR---CH2 ---C (this t is different from the one used in Eq. (3.9)). The conformation with only the second of these first-order interactions is weighted by h, and the conformation with only the first of these first-order interaction is the reference point, with a statistical weight of 1. The most important secondorder interactions are independent of the configuration of the side chain because they only involve atoms in the main chain. 2 3 1 1 1 Vd ¼ Vl ¼ 4 1 1 v 5: (3:17) 1 v 1 The complete statistical weight matrices for this bond in the two stereochemical configurations are obtained as VD, from Eq. (3.6). 2 3 h 1 t Ud ¼ QUl Q ¼ 4 h 1 tv 5: (3:18) h v t Proceeding in the same manner, there are four possible statistical weight matrices for the CH2 ---CHR bond immediately before a pseudoasymmetric center, depending on the stereochemistry at this center and the preceding pseudoasymmetric center. If both pseudoasymmetric centers have the same chirality, the two possibilities, Udd and Ull , can be interconverted using Q. 2 3 tvCR 1 hvRR Udd ¼ QUll Q ¼ 4 h tvCR (3:19) vCC 5: hvCR tvCC vRR vCR

The conformational partition function for a vinyl polymer of specified stereochemical sequence is formulated as a string of matrices where every second matrix is Up , and the intervening matrices are either Um or Ur , depending on the sequence of the diads in the chain. The definitions of the rotational isomeric states must change also when Eqs. (3.21)–(3.23) are used. Instead of using the t,gþ , and g
states that are appropriate for d and l pseudoasymmetric centers, one uses t, g, and g
states for meso and racemo diads. The gauche state that has t in its statistical weight is denoted by g
, and the other gauche state is denoted simply by g. Indexing of rows and columns in Up ,Um , and Ur is in the sequence t, g, g
. Several vinyl polymers, such as polystyrene, have statistical weights such that t and all of the v’s are much smaller than 1. Under these circumstances little error may be introduced if the g
state is ignored because its statistical weight always includes tv. This simplification allows construction of Z with 2  2 matrices where the rows and columns are indexed t, g [68].   1 1 Up ¼ , (3:24) 1 v  2  h vRR h Um ¼ , (3:25) h vCC  2  h hvCR Ur ¼ : (3:26) hvCR 1 In these three equations, all of the statistical weights for first-order interactions in the diad have been placed in Um and Ur . The Up Um sequence shows that an isotactic chain can avoid conformations weighted by v (a very small number in most polymers) if it adopts an ordered sequence that is either tg or gt, which is consistent with the commonly observed chain conformation in crystalline isotactic

50 / CHAPTER 3 polymers. In contrast, the Up Ur sequence shows that a syndiotactic chain is more likely to crystallize in either the tt or gg conformation, because these conformations do not require weighing with v. Literature citations for RIS models for several vinyl polymers, as well as related polymers for which stereochemical compositions and stereochemical sequences are issues, are summarized in Table 3.4. 3.8 EXTRACTION OF USEFUL INFORMATION FROM Zn The conformational partition function is subject to the same types of manipulations as are other partition functions encountered in statistical mechanics. Thus the average conformational energy of the chain is obtained from the temperature dependence of Zn .   @ ln Zn hEi
E0 ¼ kT 2 : (3:27) @T

Zn depends on T because the elements of the statistical weight matrices are Boltzmann factors. The conformational entropy is obtained from this result and ln Zn . S¼

hEi
E0 þ k ln Zn : T

The same approach can be used to deduce average conformations of local portions of the chain. The probability that bond i is in a particular rotational isomeric state, h, is obtained by dividing Zn into the sum of the statistical weights of all conformations where this bond is in the desired state. ph;i ¼ Z
1 U1 U2    Ui
1 U0h;i Uiþ1    Un :

Polypropylene Poly(vinyl alcohol) Poly(vinyl fluoride) Poly(1-butene)a Polysilapropylene Poly(propylene oxide) Poly(vinyl methyl ether)a Poly(vinyl chloride) Poly(methyl vinyl ketone) Poly(propylene sulfide) Poly(trifluoroethylene) Head-to-head, tail-to-tail polypropyleneb Poly(vinyl acetate) Poly(methyl acrylate) Poly(tert-butyl vinyl ketone) Poly(methyl methacrylate) Poly(methyl phenyl siloxane) Polystyrene Poly(2-vinylpyrridine) Poly(vinyl bromide) Poly(N-vinyl pyrrolidone) Poly(a-methylstyrene) Polysilastyrene Poly(p-chlorostyrene) Poly(phenyl acrylate) Poly(N-vinyl carbazole) Poly(methylphenylsilylene) Asymmetrically substituted polysilylenemethylenea a

(3:29)

The modified statistical weight matrix denoted by U0h;i is obtained by zeroing all columns of Ui except the column that indexes the desired state, h. This operation has the effect of ignoring the statistical weights of all conformations of the chain where bond i is not in the desired state, while keeping intact the statistical weights of all chain

Table 3.4. RIS models for selected vinyl polymers and related polymers for which stereochemical composition and stereochemical sequence are issues. Chains are listed in the order of the molecular weight of their repeat unit. Polymer

(3:28)

Repeat unit

References

---CH2 ---CH(CH3 )-----CH2 ---CH(OH)-----CH2 ---CHF-----CH2 ---CH(C2 H5 )-----CH2 ---SiH(CH3 )-----O---CH2 ---CH(CH3 )-----CH2 ---CH(OCH3 )-----CH2 ---CHCl-----CH2 ---CH(COCH3 )-----S---CH2 ---CH(CH3 )-----CF2 ---CHF-----CH2 ---CH(CH3 )---CH(CH3 )---CH2 -----CH2 ---CH(OCOCH3 )-----CH2 ---CH(COOCH3 )-----CH2 ---CH[COC(CH3 )3 ]-----CH2 ---C(CH3 )(COOCH3 )-----O---Si(CH3 )(C6 H5 )-----CH2 ---CH(C6 H5 )-----CH2 ---CH(C5 NH4 )-----CH2 ---CHBr-----CH2 ---CH(C4 NOH6 )-----CH2 ---C(CH3 )(C6 H5 )-----SiH2 ---SiH(C6 H5 )-----CH2 ---CH(C6 H4 Cl)-----CH2 ---CH(COOC6 H5 )-----CH2 ---CH(C12 NH8 )-----Si(CH3 )(C6 H5 )-----CH2 ---Si(CH3 )[O(CH2 )3 OC6 H4 C6 H5 ]---

[69–72] [73,74] [75] [76] [77] [78,79] [80] [81,82] [83] [79,84,85] [75] [86] [87] [88–90] [91] [92–94] [95] [68,96,97] [98] [99] [100] [101] [102,103] [104] [105] [106,107] [102,108] [109]

These articulated side chains require more elaborate methods that are extensions of the simpler ones described in this Chapter. b Hydrogenated poly(2,3-dimethylbutadiene).

THE ROTATIONAL ISOMERIC STATE MODEL conformations where this bond is in state h. The ph;i are useful in the interpretation of conformation-dependent properties of polymers that are highly local in origin. Examples are the coupling constants in NMR spectra [110], as evaluated by a Karplus relationship [111–114], and the optical activity of chiral vinyl polymers [115]. An extension of this approach yields the probability that bonds i
1 and i are simultaneously in states j and h, respectively. pjh;i ¼ Z


1

U1 U2    Ui
1 U0jh;i Uiþ1

   Un :

(3:30)

The matrix U0jh;i is obtained from Ui by zeroing every element except the one in the row and column indexed by j and h, respectively. A useful method for calculating the probabilities for longer sequences of bonds, in the approximation where there is interdependence of nearest-neighbor pairs of bonds, makes use of another probability that can be calculated from the results of Eqs. (3.29) and (3.30). qjh;i ¼ pjh;i =pj;i
1 :

(3:31)

This term is the probability that bond i is in state h, given that bond i
1 is in state j. (This restriction on the state at bond i
1 was absent in the definition of pjh;i .) The differences in qjh;i and pjh;i are apparent from examination of the types of summations that must be performed in order to achieve unit probability. n X n X

pjh;i ¼ 1,

(3:32)

j¼1 h¼1 n X

qjh;i ¼ 1:

(3:33)

h¼1

The normalization is achieved differently for pjh;i and qjh;i . If s ¼ 0:543 and v ¼ 0:087, a C–C bond in the middle of a long polyethylene chain has the following values for these probabilities, where each set of probabilities is presented in the form of a 3  3 matrix with rows and columns indexed in the order t, gþ , g
. 2 3 0:321 0:138 0:138 pjh;i ¼ 4 0:138 0:0591 0:00516 5, (3:34) 0:138 0:00516 0:0591 2

qjh;i

0:538 ¼ 4 0:682 0:682

0:231 0:292 0:026

3 0:231 0:026 5: 0:292

/ 51

The probability that bonds i
2, i
1, and i are in states z, j, and h, respectively, is given by pz;i
2 qzj;i
1 qjh;i , which results from the logical extension of Eq. (3.30). This approach can be extended to the probabilities for observations of longer sequences of bonds in specified states. 3.9 VIRTUAL BONDS Many important chains contain bonds that are locked into a single conformation due to restrictions imposed by ring formation, as in the benzene ring of poly(ethylene terephthalate), or electronic structures (as in the amide unit of nylon-6, which strongly prefers the planar trans conformation). These rigid units are often treated with virtual bonds, where a single virtual bond spans the rigid unit. Several instances where virtual bonds have been used are summarized in Table 3.5. 3.10 MATRIX EXPRESSION FOR THE DIMENSIONS OF A SPECIFIED CONFORMATION The geometry for a specified conformation of a chain of n bonds is formulated in a manner that will facilitate averaging of the result with the aid of the information contained in Zn . We will defer the averaging process until the next section, and focus here on a single conformation. The statistical weight of this conformation is irrelevant in the present section, but it will become highly relevant in the next section. The local Cartesian coordinate system depicted in Fig. 3.3 is affixed to each bond in the chain. Bond i runs from chain atom i
1 to chain atom i. The x-axis for this bond is parallel with the bond, and oriented from chain atom i
1 to chain atom i. The y-axis is in the plane of bonds i and i
1, and oriented with a positive projection on bond i
1. The z-axis completes a right-handed Cartesian coordinate system. Since the first bond does not have a previous bond for use in defining the y-axis, an imaginary zeroth bond is used. This bond is oriented such that it produces a trans state TABLE 3.5. Examples of the use of virtual bonds in the construction of RIS models. Rigid unit

Examples

References

Polybenzoxazine Polycarbonates Polyesters Polypyrrole Polysaccharides Nucleic acids Poly(amino acids) Poly(lactic acid) Polybutadiene

[116] [16,117] [118–120] [121] [122–124] [125] [126–128] [129] [51]

(3:35)

The elements in Eq. (3.34) illustrate the importance of the interdependence of the bonds in polyethylene. The probability for a pair of bonds in g states depends strongly on whether they are of the same or opposite sign. The interdependence of the bonds is also apparent in Eq. (3.35). If the bonds were independent, all rows of qjh;i would be identical.

Aromatic ring

Aliphatic ring Amide group Ester group CH2 CH¼CHCH2 unit

52 / CHAPTER 3 The matrix that transforms a vector from its representation in the coordinate system of bond i þ 1 into its representation in bond i is denoted Ti . It depends on the angle made by these two bonds and the torsion at bond i. 2 3
cos u sin u 0 Ti ¼ 4
sin u cos f
cos u cos f
sin f 5: (3:37)
sin u sin f
cos u sin f cos f

yi li+1

li

xi

li⫺1

FIGURE 3.3. Local coordinate system for bond i. The x and y axes are drawn on the figure. The z-axis (not drawn) completes a right-handed Cartesian coordinate system.

at the first bond. With these definitions of the local coordinate systems, a bond vector in its own coordinate system can be quickly written. 2 3 l li ¼ 4 0 5 (3:36) 0 2

r2 ¼ [ 1

2lT1 T1

1 l21 ] 4 0 0

2lT2 T2 T2 0

3 2 lT2 1 l2 5    4 0 1 0

The expression for s2 is written in the approximation where all of the chain atoms can be taken to be of the same mass. s2 ¼ (n þ 1)
2 H1 H2 . . . Hn
1 Hn :

(3:40)

Matrices H1 and Hn are written as the first row and last column, respectively, of the general expression for Hi;1 < i < n, as was true also for G [11]. 3 2 1 1 2lTi Ti l2i l2i 6 0 1 2lT Ti l2 l2 7 i i 7 i 6 (3:41) Hi ¼ 6 Ti li li 7 7, 1 < i < n: 60 0 40 0 0 1 15 0 0 0 0 1 The squared dipole moment, m2 , of polar chains such as polyoxymethylene can be treated using a simple modification of Eq. (3.39). The bond vector, li , for bond i is replaced by the dipole moment vector, mi , for the same bond [15]. For polyoxymethylene, the bond vectors are connected in a head-to-tail fashion, but the bond dipole moment vectors are connected in a head-to-head, tail-to-tail fashion, with the oxygen atom being at the negative end of mi . Extension to polyoxyethylene requires that some of the mi be null vectors, as would be the case for the CH2 ---CH2 bond [15]. In some chains, such as poly(vinyl chloride), the important mi are not aligned with the li [130].

The expression for liþ1 in the coordinate system of bond i is Ti liþ1 . With this notation, the end-to-end vector in a specified conformation can be written as a serial product of n matrices constructed from T and l [11].      T2 l2 Tn
1 ln
1 ln r ¼ [ T1 l1 ]  0 1 0 1 1 ¼ A1 A2    An
1 An :

(3:38)

All of the geometric information (li , ui , fi ) pertinent to bond i appears in Ai . The squared end-to-end distance and squared radius of gyration can be calculated using exactly the same information, but with the information presented in larger matrices. 2lTn
1 Tn
1 Tn
1 0

32 2 3 ln l2n
1 ln
1 54 ln 5 ¼ G1 G2    Gn
1 Gn : 1 1

(3:39)

3.11 AVERAGING THE DIMENSIONS OVER ALL THE CONFORMATIONS IN Zn The matrices denoted by Ai , Gi , and Hi (1 < i < n) in Eqs. (3.38)–(3.40) depend on the rotational isomeric state assigned to bond i through the appearance of fi in Ti . The states at this bond also index the columns of Ui . We now seek a pairing of the appropriate statistical weight from Ui with the geometry in Ti . This objective is achieved by expansion of each element in Ui through multiplication of each of its elements by the appropriate Ai , Gi , or Hi , depending on whether the target of the calculation is hri0 , hr 2 i0 , or hs2 i0 . As an example, the calculation of hr 2 i0 can be written as a serial product of n G matrices [11]. hr 2 i0 ¼ Z
1 G1 G2 . . . Gn
1 Gn :

(3:42)

The internal Gi are constructed by expansion of each element in Ui , denoted ujh , by Gh , such that Gi becomes a 5n  5n matrix, whereas Ui was a n  n matrix. The terminal Gi are either a row or column of 5n elements. When n ¼ 3, the G matrices take the forms shown in Eqs. (3.43)– (3.45).   G1 ¼ 1 2lT1 T1 l21 0    0 , (3:43)

THE ROTATIONAL ISOMERIC STATE MODEL 2

u11 Gt Gi ¼ 4 u21 Gt u31 Gt

u12 Ggþ u22 Ggþ u32 Ggþ

3 u13 Gg
u23 Gg
5, u33 Gg
2

l2n

1 < i < n,

9

(3:44)

8 7

3

6 ln 7 6 7 617 6 27 6l 7 6 n7 7 Gn ¼ 6 6 ln 7: 617 6 7 6 l2 7 6 n7 4 ln 5 1

/ 53

6 Chain a Chain b Chain c

5 Cn

(3:45)

4 3 2 1 0 0

0.2

0.4

0.6

0.8

1

1/n

3.12 USE OF Cn FOR CALCULATION OF C1 The results of an illustrative calculation are depicted in Fig. 3.4. The chain has u ¼ 112 , n ¼ 3, and f ¼ 180 and 60 . The statistical weight matrix for all internal bonds is given by Eq. (3.9) with t ¼ c ¼ 1. When s and v are also 1, the chain has the same Cn as the freely rotating chain with the same bond angle. Imposition of a symmetric torsional potential that penalizes the g states, with s ¼ 0:4, increases the Cn . Introduction of a pair-wise interdependence, via s ¼ 0:4 and v ¼ 0:1, produces a further increase in Cn . Obviously, the interdependence of the bonds can have a strong effect on the unperturbed dimensions of the chain. The value of C1 can be reliably determined from the results depicted in Fig. 3.4 if s ¼ v ¼ 1, but the limiting value is less reliably defined when s ¼ 0:4 and v ¼ 0:1. This problem is alleviated by plotting the same data in another manner, as shown in Fig. 3.5. The linear extrapolation of the data to 1=n ¼ 0 leads unambiguously to the value for C1 . This linear extrapolation is theoretically justified both for hr 2 i0 =nl2 and for hs2 i0 =nl2 [1] and also for the

corresponding ratio constructed from the mean square unperturbed dipole moment, hm2 i0 =nm2 [131]. 3.13 OTHER APPLICATIONS OF THE RIS MODEL The previous portion of this chapter has focused primarily on the use of the RIS model for the computation of the mean square unperturbed dimensions, because that is the most frequent application of the model. This section describes briefly many other applications of the RIS model. All of these applications employ the conformational partition function, but the additional information incorporated in the calculation, and the manipulation of Z, depend on the application. 3.13.1 Applications That Depend Only on the Energetic Information Contained in Z

8 7 6 5 Cn

FIGURE 3.5. The data in Fig.3. 4 plotted as Cn vs. 1/n, along with linear extrapolations to 1=n ¼ 0. The values of C1 , which are (a) 2.198, (b) 4.396, and (c) 7.869, can be estimated with an error no larger than 1% by linear extrapolation of the data for n no larger than 128. The linear extrapolation shown, which uses C64 and C128 , leads to estimates for C1 of (a) 2.198, (b) 4.393, and (c) 7.858.

Chain a

4

Chain b Chain c

3 2 1 0 0

10 20 30 40 50 60 70 80 90 100 110 120 130 n

FIGURE 3.4. Cn vs. n for chains with u ¼ 112 , n ¼ 3, and f ¼ 180 and 608. U for all internal bonds is given by Eq. (3.9) with t ¼ c ¼ 1. The other statistical weights are (a) s ¼ v ¼ 1, (b) s ¼ 0:4, v ¼ 1, and (c) s ¼ 0:4, v ¼ 0:1.

The calculation of ph;i as a function of i provides an estimate of the distance that end effects penetrate into a long unperturbed chain. This calculation shows that end effects for polyethylene are confined to the first few bonds at the end of the chain [10]. The end effects can extend much further into the chain when the second-order interactions become more severe, as is frequently the case for the probability of a helical conformation, ph;i , in a long homopolypeptide near the midpoint of its helix-coil transition [132]. In proton NMR, the values of ph;i are helpful in understanding the values of the spin–spin coupling constants, using the Karplus relationship [111–114], and in understanding the g effect on the chemical shift in 13 C NMR spectra [110,133]. The values of ph;i have also been used to interpret the optical activity exhibited by chiral poly(a-olefins) [115] and other polymers [134,135].

54 / CHAPTER 3 The combination of ph and pjh gives the number of bonds in a run of state h. An illustrative use is in the determination of the average number of residues in a helical segment in a homopolypeptide as ph =pc h, where h denotes the helical state, and c denotes any other state [10,132]. The stereochemical composition of vinyl polymers after epimerization to stereochemical composition can be determined from the information contained in a more elaborate form of Z that takes account of the conformations of all stereochemical sequences, with all sequences weighted with respect to the same definition for the zero point of the conformational energy [136,137].

3.13.2 Applications That Use Properties Accessible From Z and the Geometry of the Chains The higher even moments of the unperturbed dimensions, hr 2p i0 and hs2p i0 , p > 1, are accessible through an appropriate expansion in the dimensions of the generator matrices used for the simpler cases where p ¼ 1 [11]. Dimensionless ratios formed from appropriate combinations of these even moments provide information about the shape of the distribution functions. Thus hr 2 i0 measures the average value of r 2 ,hr 4 i0 =hr 2 i20 measures the width of the distribution for r 2 , and hr 6 i0 =hr 2 i30 measures the skewness of this distribution function. All flexible homopolymers will approach the Gaussian limit of hr 4 i0 =hr 2 i20 ¼ 5=3 as n ! 1, but narrower distributions (smaller hr 4 i0 =hr 2 i20 ) are typical at finite n [138]. Macrocyclization equilibria can be understood in terms of these dimensionless ratios, via an elaboration of the Jacobson–Stockmayer approach [139,140]. More accurate results, particularly for rather short chains, are obtained when the hr 2p i0 =hr 2 ip0 are supplemented by additional terms, calculated from the RIS model, that monitor the angular correlation between bonds 1 and n in the unperturbed chain as r 2 ! 0 [141,142]. Although the averages of many conformation-dependent physical properties of interest can be extracted rapidly from the RIS model by a matrix multiplication scheme of the type shown in Eq. (3.42), with Gi defined as appropriate for the specific property of interest, there are numerous other properties that cannot be evaluated by this simple device. For these other properties, an efficient Monte Carlo (MC) simulation can often be constructed, using the information in the RIS model. The information in Z allows rapid computation of the ph;2 and qjh;i , 2 < i < n. These normalized probabilities and a random number generator allow rapid generation of a representative sample of unperturbed chains. If the sample is sufficiently large, the simple average of r 2 over all chains in the sample will approach the value of hr 2 i0 specified by Eq. (3.42). The MC simulation is less efficient than Eq. (3.42) in the calculation of hr 2 i0 , but it offers the opportunity for the calculation of other physical properties that cannot be formulated as a serial matrix product. An example is provided by the angular scattering function,

P(q), where q is related to the scattering angle, u, and wavelength of the radiation, l, by Eq. (3.47).

n X n X sin (qrij ) 1 , (3:46) P(q) ¼ qrij (n þ 1)2 i¼0 j¼0   4p u sin q¼ : l 2

(3:47)

An illustrative example is provided by the use of this approach to determine how the scattering function of unperturbed poly(methyl methacrylate) depends on the stereochemical composition of the chains [143]. The method can also be employed to generate accurate distribution functions for the end-to-end distance in unperturbed chains that are sufficiently short so that there are strong departures from the Gaussian distribution which would be achieved in the limit as n ! 1 [144]. The representative sample of unperturbed chains can be edited to generate other useful ensembles. One of the most common examples is to discard chains with r larger than a specified cutoff. As this cutoff becomes smaller and smaller, the ensemble of surviving chains approaches the ensemble for the unperturbed macrocycle [145]. This ensemble can be used to evaluate properties of the ensemble directly, or to determine how easily polydimethylsiloxane macrocycles of a given n can be threaded [146]. The unperturbed ensemble can also be edited to discard chains that attempt placement of their atoms in regions of space that are deemed to be inaccessible. This approach generates ensembles of chains that are tethered by one end to an impenetrable surface [147], or chains in a melt that contains impenetrable spherical filler particles [148]. The complete ensemble of unperturbed chains can also be perturbed by the introduction of new interactions, not considered explicitly in the RIS model, as in the re-weighting of the ensemble to investigate the properties of bolaform electrolytes (polymer with ionic groups at their ends) [149]. The ph;i and qjh;i can be used to cause coarse-grained chains to mimic the conformational properties of specific real chains, because these probabilities enforce the proper distribution function for r for the entire coarse-grained chain, as well as all of its subchains [150,151]. This feature facilitates the recovery of atomistically detailed models from equilibrated ensembles of coarse-grained chains [152]. It also causes the coarse-grained chains to be sensitive to subtleties such as the dependence of the miscibility of polypropylene chains in the melt on their stereochemical composition [153,154]. 3.13.3 Applications That Depend on Properties in Addition to Z and the Geometry of the Chains Generator matrices are easily formulated for the computation of the mean square dipole moment, hm2 i0 , using the analogy between formulating r as a sum of bond vectors and

THE ROTATIONAL ISOMERIC STATE MODEL

/ 55

formulating m as a sum of bond dipole moment vectors [10,15,130]. This analogy can be extended, via the valence optical scheme, to conformation-dependent properties that depend on the anisotropy of the polarizability of a bond [10,12]. This analogy leads to generator matrices for the optical anisotropy, stress-optical coefficient, electrical birefringence, i.e., molar Kerr constant, and magnetic birefringence, i.e., molar Cotton-Mouton constant.

TABLE 3.6. Representative polymers that have been described by RIS models with different n’s

3.14 WHY ARE SOME CHAINS DESCRIBED WITH MORE THAN ONE RIS MODEL?

Polyisobutylene

The appearance in the literature of several RIS models for a single polymer may initially be confusing, but it is not at all surprising when one considers the objectives of the RIS approach. The exact description of the physical properties of a polymer would start from the Schro¨dinger equation, in a manner similar to one appropriate for small molecules. That approach is not practical. Therefore we resort to practical models that contain sufficient detailed information to let us account in a satisfactory manner for the physical properties that are of interest to us. Sometimes this objective can be obtained to a similar degree of accuracy with somewhat different values for the parameters in the model. This situation is illustrated by the first two entries in Table 3.2. The two RIS models for polyethylene place the gauche states at slightly different displacements, (120
Df), from the trans state. In order to maintain the proper values for Cn , a change in the geometry of the chain, produced by a change in the value of Df, requires compensating changes in the weighting of the chains, which is achieved by adjustments in the values of Es and Ev [14]. If the value of Df were increased from 0 to 7.58 without any other changes in the model, Cn would increase because the g states would have been moved closer to the geometry of the t state. This increase in Cn can be avoided by increasing slightly the probability for g state, and that objective is achieved by the changes in Es and Ev . The literature contains many similar examples where a given chain is described by various RIS models that have the same form, but slightly different values of the parameters. There are also numerous examples where the RIS models have more substantial differences, because they use statistical weight matrices of different dimensions. Several examples are presented in Table 3.6. An obvious origin of the differences in dimension of the U’s is a difference in the number of rotational isomeric states assigned to individual bonds. Thus polyethylene has been described with RIS models that assign three [14], five [14], or seven [155] states to each internal bond. An increase in n should lead to a more accurate model, because it permits the incorporation of more detail into the calculation. Of course, it also introduces more parameters into the model, with the added burden on the user of assigning values to these parameters. The most popular RIS models for polyethylene use n ¼ 3 because

Polymer

n

References

Polyethylene

3 5 7 3 4 6 3 4 6 3 6 3 4 5 6 2 3 6 2 4

[14] [14] [155] [24] [25] [25] [32] [34,35] [33,35] [156] [41] [69,71] [155] [72,157] [70] [92] [94] [93] [117,158] [16]

Polytetrafluoroethylene

Poly(vinylidene chloride) Polypropylene

Poly(methyl methacrylate)

Polycarbonate

the increased accuracy accessible with a larger value of n usually is not justified because it makes a trivial improvement in the agreement between the calculated values and experiment. The principle here is to incorporate into the RIS model as much detail as is necessary . . . but no more detail than necessary. The necessary amount of detail required in the RIS model may depend on the physical property that is calculated from the model. For example, the dependence of C1 on stereochemical composition in poly(methyl methacrylate) is described nearly as well by a relative simple three-state model [94] and by a much more complex six state model that contains many more parameters [93]. However, the sixstate model is superior to its simpler relative in the description of the scattering function, P(m), which is sensitive to the precise description of the conformations of relatively short subchains [159]. In the case of polypropylene, the stereochemical composition achieved after epimerization to stereochemical equilibrium is captured correctly by a three-state model [71], but accurate description of the behavior of C1 with changes in stereochemical composition is better achieved with a five-state model [72]. The stereochemical composition at stereochemical equilibration does not depend explicitly on the geometry (l, u, f) when it is calculated with the RIS model [71], but C1 is obviously sensitive to this geometry [72]. In particular, the manner in which C1 depends on the probability of a meso dyad, pm , as pm ! 1 can be improved by going from a three-state to a five-state model. The dimensions of U change, at constant n, if higher order interactions are incorporated in the RIS model. Thus

56 / CHAPTER 3 polyethylene has been treated using a 9  9 representation of U. The calculation retains n ¼ 3, but the increase in dimensions of U was necessary to test the potential importance of third-order interactions [160]. In order to introduce into U a statistical weight that depends on a thirdorder interaction, the rows are indexed by the states at bonds i
2 and i
1, and the columns are indexed by the states at bonds i – 1 and i, leading to a n2  n2 representation for U. The only nonzero elements in this U are those where the row and column agree on the state at bond i – 1. For this reason, 2/3 of the elements are zero. Any nonzero element corresponds to a unique combination of rotational isomeric states at bond i – 2, i – 1, and i. Third-order interactions have also been included in U for polyoxyethylene, requiring an expansion in the dimension of U, even though n ¼ 3 [45]. Interactions of higher than third order are sometimes important, as illustrated by the transition from a random coil to an intramolecular antiparallel sheet with tight bends [161]. Under these circumstances, each U becomes a sparse matrix. The sparse character of the matrix can be exploited in writing the computer code required for numerical evaluation of the model [161]. ACKNOWLEDGMENT Over his career, much of WLM’s research using the RIS model has been supported by various sources. Current financial support for this research is from NSF DMR 0098321 and from the Collaborative Center in Polymer Photonics, funded jointly by the Air Force Office of Scientific Research, Wright-Patterson Air Force Base, and The University of Akron. Some items coauthored by WLM in the reference list were supported by FAA, NIH, and other NSF grants. REFERENCES 1. W. L. Mattice, C. A. Helfer, and A. P. Sokolov, Macromolecules 37, 4711 (2004). 2. P. Debye, J. Chem. Phys. 14, 636 (1946). 3. H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252 (1941). 4. M. V. Volkenstein, Dokl. Acad. Nauk SSSR 78, 879 (1951). 5. Yu. Ya. Gotlib, Zh. Tekhn. Fiz. 29, 523 (1959). 6. T. M. Birshtein and O. B. Ptitsyn, Zh. Tekhn. Fiz. 29, 1048 (1959). 7. S. Lifson, J. Chem. Phys. 30, 964 (1959). 8. K. Nagai, J. Chem. Phys. 31, 1169 (1959). 9. C. A. J. Hoeve, J. Chem. Phys. 32, 888 (1960). 10. P. J. Flory, Statistical Mechanics of Chain Molecules (Wiley, New York, 1969). Reprinted with the same title by Hanser, Mu¨nchen, in 1989. 11. P. J. Flory, Macromolecules 7, 381 (1974). 12. W. L. Mattice and U. W. Suter, Conformational Theory of Large Molecules. The Rotational Isomeric State Model in Macromolecular Systems (Wiley, New York, 1994). 13. M. Rehahn, W. L. Mattice, and U. W. Suter, Adv. Polym. Sci. 131/ 132 (1997). 14. A. Abe, R. L. Jernigan, and P. J. Flory, J. Am. Chem. Soc. 88, 631 (1966). 15. A. Abe and J. E. Mark, J. Am. Chem. Soc. 98, 6468 (1976). 16. M. Hutnik, A. S. Argon, and U. W. Suter, Macromolecules 24, 5956 (1991).

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THE ROTATIONAL ISOMERIC STATE MODEL 70. R. H. Boyd and S. M. Breitling, Macromolecules 5, 279 (1972). 71. U. W. Suter, S. Pucci, and P. Pino, J. Am. Chem. Soc. 97, 1018 (1975). 72. U. W. Suter and P. J. Flory, Macromolecules 8, 765 (1975). 73. R. M. Wolf and U. W. Suter, Macromolecules 17, 669 (1984). 74. A. E. Tonelli, Macromolecules 18, 1086 (1985). 75. A. E. Tonelli, Macromolecules 13, 734 (1980). 76. W. L. Mattice, Macromolecules 10, 1171 (1977). 77. P. R. Sundararajan, Macromolecules 23, 3178 (1990). 78. A. Abe, T. Hirano, and T. Tsurura, Macromolecules 12, 1092 (1979). 79. Y. Sasanuma, Macromolecules 28, 8629 (1995). 80. A. Abe, Macromolecules 10, 34 (1977). 81. J. E. Mark, J. Chem. Phys. 56, 451 (1972). 82. G. D. Smith, P. J. Ludovice, R. L. Jaffe, and D. Y. Yoon, J. Phys. Chem. 99, 164 (1995). 83. U. W. Suter, J. Am. Chem. Soc. 101, 6481 (1979). 84. A. Abe, Macromolecules 13, 541 (1980). 85. Y. Sasanuma, Y. Hayashi, H. Matoba, I. Touma, H. Ohta, M. Sawanobori, and A. Kaito, Macromolecules 24, 8216 (2002). 86. E. D. Akten and W. L. Mattice, Macromolecules 34, 3389 (2001). 87. P. R. Sundararajan, Macromolecules 11, 256 (1978). 88. P. J. Flory, J. Am. Chem. Soc. 89, 1798 (1967). 89. D. Y. Yoon, U. W. Suter, P. R. Sundararajan, and P. J. Flory, Macromolecules 8, 784 (1975). 90. E. A. Ovalvo, E. Saiz, R. M. Masegosa, and I. Herna´ndez-Fuentes, Macromolecules 12, 865 (1979). 91. J. A. Guest, K. Matsuo, W. H. Stockmayer, and U. W. Suter, Macromolecules 13, 560 (1980). 92. P. R. Sundararajan and P. J. Flory, J. Am. Chem. Soc. 96, 5025 (1974). 93. M. Vacatello and P. J. Flory, Macromolecules 19, 405 (1986). 94. P. R. Sundararajan, Macromolecules 19, 415 (1986). 95. J. E. Mark and J. H. Ko, J. Polym. Sci., Polym. Phys. Ed. 13, 2221 (1975). 96. A. D. Williams and P. J. Flory, J. Am. Chem. Soc. 91, 3111 (1969). 97. R. F. Rapold and U. W. Suter, Macromol. Theory Simul. 3, 1 (1994). 98. A. E. Tonelli, Macromolecules 18, 2579 (1985). 99. E. Saiz, E. Riande, M. P. Delgado, and J. M. Barrales-Rienda, Macromolecules 15, 1152 (1982). 100. A. E. Tonelli, Polymer 23, 676 (1982). 101. P. R. Sundararajan, Macromolecules 10, 623 (1977). 102. W. J. Welsh, J. R. Damewood, Jr., and R. C. West, Macromolecules 22, 2947 (1989). 103. P. R. Sundararajan, Macromolecules 24, 1420 (1991). 104. J. E. Mark, J. Chem. Phys. 56, 458 (1972). 105. J. S. Saiz, E. Riande, J. San Roma´n, and E. L. Madruga, Macromolecules 23, 786 (1990). 106. P. R. Sundararajan, Macromolecles 13, 512 (1980). 107. A. Abe, H. Kobayashi, T. Kawamura, M. Date, T. Uryu, and K. Matsuzaki, Macromolecules 21, 3414 (1988). 108. P. R. Sundararajan, Macromolecules 21, 1256 (1988). 109. C. A. Helfer, W. L. Mattice, and D. Chen, Polymer 45, 1297 (2004). 110. A. E. Tonelli, NMR Spectroscopy and Polymer Microstructures. The Conformational Connection. (VCH, New York, 1989). 111. M. Karplus, J. Chem. Phys. 30, 11 (1959). 112. M. Karplus, J. Chem. Phys. 33, 1842 (1960). 113. M. Karplus, J. Am. Chem. Soc. 35, 2870 (1963). 114. F. A. Bovey, A. I. Brewster, D. J. Patel, A. E. Tonelli, and D. A. Torchia, Acct. Chem. Res. 5, 193 (1972). 115. A. Abe, J. Am. Chem. Soc. 90, 2205 (1968). 116. W. K. Kim and W. L. Mattice, Comput. Theor. Polym. Sci. 8, 339 (1998).

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CHAPTER 4

Computational Parameters Joel R. Fried Department of Chemical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221-0012

4.1 4.2

Molecular Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Force Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 61 65

This chapter discusses the form and parameterization of the potential energy terms that are used for the atomistic simulation of polymers. The sum of potential terms constitutes a molecular force field that can be used in molecular mechanics, molecular dynamics, and Monte Carlo simulations of polymeric systems. Molecular simulation methods can be used to determine such properties as PVT data, selfdiffusion coefficients, modulus, phase equilibrium, x-ray and neutron diffraction spectra, small molecule solubility, and glass transition temperatures with considerable accuracy and reliability using current force fields. Included in the coverage of Chapter 4 is a review of the fundamentals of molecular mechanics and a survey of the most widely used force fields for the simulation of polymer systems. In addition, references to the use of specific force fields in the study of important polymer groups are given.

in the system. Torsional contributions also may include improper torsion and out-of-plane bending terms in some force fields as discussed in Section 4.2. Nonbonded terms typically include steric (e.g., van der Waals) and electrostatic (e.g., Coulombic) terms but may also include polarization contributions. Force field parameters for each bonded or nonbonded term are obtained by fitting potential energy terms to ab initio (e.g., HF/6-31G*) or DFT calculations of small molecules or by fitting to experimental data such as crystal structure and the heat of vaporization1 (DHV ) for low-molecular-weight compounds. The form of specific terms used by different commercial, public domain, and customized force fields for polymer simulations are given in the sections that follow.

4.1 MOLECULAR MECHANICS

Bond Stretching

Traditional molecular mechanics methods developed by Allinger [1] and others view a molecule as a series of beads (i.e., the nuclei) joined together by springs (i.e., the bonds). The total potential energy, V, of the system is the sum of all bonded and nonbonded terms as

Bond-stretching terms can have several different forms including the simple harmonic function 2 1 X bond
V bond (rij ) ¼ kij rij
rij0 , (4:3) 2 bonds

V(r) ¼ V B (r) þ V NB (r):

where kijbond is the bond-stretching parameter and rij0 is the equilibrium bond distance (for which the potential energy contribution is zero). The summation is taken over all bonds in the system. Alternately, additional higher order (i.e.,

4.1.1 Bonded Terms

(4:1)

The bonded terms include bond stretching, angle bending, and dihedral (i.e., torsional) contributions as X X X V B (r) ¼ V bond (rij ) þ V bend (uijk ) þ V tors (fijkl ), bonds

bends

dihedrals

1 The heat of vaporization is related to the cohesive energy density (CED), the total intermolecular energy, through the expression r ECED ¼ (DHV
RT) M where M is the molecular weight and r is the density of the low-molecularweight compound.

(4:2) where the summations are made over all contiguous atoms constituting bonds (i.e., two-body interactions), angles (threebody interactions), and torsions (four-body interactions) 59

60 / CHAPTER 4 anharmonic) terms may be included as a polynomial such as the quartic expression 
2
3 1 X k2 rij
rij0 þk3 rij
rij0 V bond (rij ) ¼ 2 bonds
4  0 þ k4 rij
rij : (4:4) A Morse exponential potential [2] can also be used for the bond-stretching term in the form i o2 X n h
Dij exp
a rij
rij0
1 , (4:5) V bond (rij ) ¼ bonds

V tors (fijkl ) ¼

5 X X

an cosn fijkl :

(4:11)

dihedrals n¼0

Improper (out-of-plane bending) torsion potentials appear in some force fields. These are used to represent potential energy required to maintain the configuration of four contiguous atoms within certain geometric limits. The form of this potential term can be written as 2 1 X oop
V oop (vijkl ) ¼ kijkl vijkl
v0ijkl , (4:12) 2 improper torsions

v0ijkl

where Dij is the bond dissociation energy and   kij 1=2 a¼ : 2Dij

where represents the equilibrium (i.e., minimum energy) improper torsion angle. (4:6)

The Morse function is an accurate representation of the bond-stretching potential since the exponential term in Eq. (4.5) implicitly includes anharmonic terms. Angle Bending The harmonic term for (valence) angle bending can be written as 2 1 X bend
V bend (uijk ) ¼ kijk uijk
u0ijk , (4:7) 2 bends u0ijk

where is the equilibrium (i.e., minimum energy) valence angle. The quartic form may be written as 
2
3 1 X bend k2 uijk
u0ijk þk3 uijk
u0ijk V (uijk ) ¼ 2 bends
4  þ k4 uijk
u0ijk : (4:8) An alternative to the harmonic expression Eq. (4.7) is the Urey–Bradley expression X VUB ¼ KUB ðS
S0 Þ2 : (4:9) UB

where S is the Urey–Bradley 1,3 distance (i.e., the A–C distance in bond angle ABC).

Cross-Coupling Terms Cross-coupling terms have been used in several force fields as a means to represent the effect of one type deformation on another such as the interrelationship between bond stretching and angle bending which can be expressed in a bond–bend potential term as 2
2 1 X X b,b
V b,b (rij ,uijk ) ¼ k rij
rij0 uijk
u0ijk : 2 bonds bends (4:13) Other cross-coupling terms include bond–torsion and bend– bend–torsion. Cross-coupling terms are important for accurate modeling of normal mode vibrational frequencies and to better model the potential at large deformation (i.e., positions far from the potential minimum).

4.1.2

Nonbonded Terms

Nonbonded terms include intramolecular interactions between pairs of atoms separated by three or more bonds and those belonging to different molecules (i.e., intermolecular interactions). Interactions between pairs of atoms separated by one or two bonds are contained in the bonded energy terms of the bond-stretch and angle-bending terms, respectively. All interactions in a simulation system may be included (i.e., Ewald summation) or distance cutoffs, typically ˚ , may be used. in the range from 8 to 12 A

Torsion Torsional terms can have several different forms such as V tors (fijkl ) ¼

X 1 X ktors (n)[1
cos (nfijkl )], 2 dihedrals n¼1,2,... ijkl (4:10)

where n is the periodicity of the torsional motion. Another torsional form that has been used is the Ryckaert–Bellemans potential [3,4]

Steric Terms Steric interactions are typically represented by some form of a Lennard-Jones (LJ) potential such as the LJ 6–12 potential or the LJ 6–9 potential as illustrated below 2 ! ! 3 0 9 0 6 X r r ij ij 5: V LJ ¼ (4:14) «ij 42
3 r r ij ij i6¼j

COMPUTATIONAL PARAMETERS

/ 61

The 6th order term in the LJ expression represents dispersion (long-range) interactions while the 9th (or 12th) order term represents short-range repulsion. Sometimes an exponential potential may be used as the short-range term in combination with a 6th order dispersion term in the form " # X Cij exp
6 V ¼ Aij exp (
Bij rij )
6 : (4:15) rij i6¼j

hydrogen bonding. Incorporation of multiple nonbonded terms, including polarization terms as discussed in the next section, significantly adds to the computational time since nonbonded interactions must be calculated between thousands of atoms at each timestep, typically 1 fs.

The exponential form is a better representation of repulsive interactions than the LJ inverse-12 form. The combination of an exponential and a 6th order term has been called the Buckingham potential function, the exponential-6 equation, or the modified Hill equation. The LJ parameters «ij and rij0 appearing in Eq. (4.14) are obtained by a combination rule using individual atomic parameters. These combination rules include the Lorentz and Berthelot rule and the 6th order combination law given as [5] " # 0 6 0 6 1=6 (r ) þ (r ) i j rij0 ¼ (4:16) 2

Some force fields also include a polarization term, V pol , along with steric (i.e., LJ or Buckingham) and electrostatic terms in the nonbonded potential expression as

and «ij ¼

2(«i «j )1=2 (ri0 rj0 )3 (ri0 )6 þ (rj0 )6

:

(4:17)

Electrostatic Terms The electrostatic terms include the simple Coulombic expression in the general form X fqi qj V es ¼ , (4:18) rij i6¼j where qi represents the charge on atom i of the atom pair i, j, rij is the separation between atoms i and j, and f ¼ 1=p«0 where «0 is the dielectric constant.

Hydrogen Bonding Contributions An additional nonbonded term sometimes appearing in force fields for biological systems (especially older force field versions) is used to model the interaction between hydrogen donor and acceptor atoms involved in hydrogen bonding. An example is the potential energy term ! X Cij Dij HB V ¼
10 cos4 uDHA , (4:19) 12 r rij ij i6¼j where uDHA is the angle between the donor (D), hydrogen (H), and acceptor (A) atoms. Current force field versions do not explicitly treat hydrogen bonding since extensive parameterization of nonbonded terms ideally should include

Polarization

V NB (r) ¼ V steric (r) þ V es (r) þ V pol (r):

(4:20)

An example of the form of a polarization term is [6] 1X V pol ¼ m Ei , (4:21) 2 i i where mi is the dipole moment associated with atom i and Ei is the electrostatic field experienced at atom i. A detailed discussion of polarization contributions is given by Smith and Borodin [7]. Polarizable force fields allow the charge distribution to respond to the dielectric environment [8] and are particularly important in the atomistic simulation of water and the detailed simulation of biological systems in general. A problem associated with inclusion of a polarization potential term is the additional computational cost incurred by including another nonbonded term. In the case of polymers, polarizable force fields are particularly important in the treatment of polymer electrolytes including those of poly(ethylene oxide)/Liþ as discussed by Smith and Borodin [7] and in the atomistic simulation of systems in which chemical reactions can occur as in the case of proton transfer (e.g., fuel cell applications) or the simulation of combustion events. Force fields that can treat bond formation or breaking include ReaxxFF [9] as discussed briefly in the next section. 4.2 FORCE FIELDS Many different force fields are now available from commercial and other sources. Some force fields like the MM series2 of force fields developed by Allinger and the Merck MM [10] have been parameterized primarily for molecular mechanics and dynamics of small molecules. Due to their limited importance for polymer simulations, they will not be covered in this section; however, they have been used to study conformational properties of model compounds for some aromatic polymers. In some cases, force fields primarily developed for biomolecules such as AMBER, CHARMM, and GROMOS have been used in the molecular simulation of polymeric systems. Force fields having particular importance for polymers include simple but versatile 2

The most recent version is MM4.

62 / CHAPTER 4 generic Class I force fields like DREIDING [11]. At the upper end are ab initio parameterized Class II force fields such as the consistent force field (CFF) family to which the force field COMPASS3 [12] belongs. COMPASS has been extensively parameterized using physical property data and includes anharmonic and cross-coupling contributions in the bonded interactions (Section 4.1.1). In the discussion that follows, force fields are grouped into the categories of generic force fields, biological force fields, and Class II force fields. As shown by references given in Table 4.1 that surveys the literature from 1990 to 2005, all the force fields discussed in this section have been used for the atomistic simulation of polymeric systems. Many of these articles provide information on parameterization. References prior to 1990 were included in the previous review by Roe [13].

TABLE 4.1. Literature citations (1990–2005) for force fields used in the atomistic simulations of polymers. Polymer

Force field

Poly(aryl ether ether ketone)

TRIPOS DREIDING CHARMM TRIPOS DREIDING CHARMM CFF93 DREIDING TRIPOS TRIPOS ReaxxFF DREIDING custom custom COMPASS CFF93 custom CVFF PCFF DREIDING CFF93þ Custom DREIDING CFF93 DREIDING TRIPOS Custom PCFF AMBER PCFF PCFF DREIDING DREIDING UFF AMBER Custom COMPASS AMBER COMPASS AMBER CFF91 Custom GROMOS Tripos5.2 CFF93 CHARMM AMBER Custom DREIDING DREIDING custom CVFF, CFF91 PCFF2 Custom

Polyarylates Poly(2,5-benzimidazole) Polybenzoxazoles trans-1,4-Polybutadiene Polycarbonate

Polydimethylsiloxane Polyethersulfone Polyethylene

4.2.1 Generic Force Fields Universal. The parameters in the Universal force field (UFF) [14–16] are calculated using general rules based only upon the element, its hybridization, and its connectivity. For this reason, the UFF has broad applicability but is inherently less accurate than extensively parameterized force fields such as COMPASS. Bond-stretching terms in the UFF are either harmonic or Morse functions. The anglebending and torsion terms are described by a small cosine Fourier expansion. For nonbonded terms, the LJ 6–12 potential and Coulombic terms are used for steric and electrostatic terms, respectively. DREIDING. DREIDING is another general-purpose force field that uses generalized force constants and geometry parameters. Parameterization of DREIDING is biased toward the first row elements (and carbon); however, DREIDING can be custom parameterized from ab initio or semiempircal data from calculations of model compounds with very good success in the atomistic simulation of polymers as shown by Fried and Goyal [17] and others. The default form of DREIDING uses the harmonic term, Eq. (4.3), for bond stretching and a harmonic cosine form of the angle-bend term given as 2 1 X bend
V bend (uijk ) ¼ kijk cos uijk
cos u0ijk (4:22) 2 bends The torsion term has the form h io 1 X tor n V tor (fijk ) ¼ kjk 1
cos njk (fjk
f0jk ) (4:23) 2 dihedrals where f is the dihedral or torsional angle between the ijk and jkl planes formed by two consecutive bonds ij and kl. In addition, DREIDING includes an inversion term that has 3

Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies.

Poly(ethylene oxide)

Poly(ethylene terephthalate)

Poly(p-hydroxybenzoic acid) Polyimides Polyisobutylene Polyisoprene Polymethacrylates Poly(methyl methacrylate) Poly(naphthalic anhydride) Poly(p-phenylene) Poly(p-phenylene isophthalate) Poly(p-phenylene sulfide) Poly(p-phenylene terephthalate) Polyphosphazenes Polypropylene Poly(propylene oxide) Polypyrrole Polyrotaxanes Polysilanes Polystyrene syndiotactic-polystyrene Poly[1-(trimethylsilyl)-1-propyne] Polyurethanes Poly(vinyl chloride) Poly(vinyl methyl ether) Poly(vinylene fluoride)

Reference [41] [42] [43] [44] [45] [46] [36] [47,48] [49] [50] [9] [51] [52] [53] [54] [39] [55] [56] [57] [58] [37] [59] [60] [37] [61–66] [67] [53] [68] [69] [70] [70,71] [72] [73] [74] [75] [76] [77] [75] [40] [78] [79] [80] [81] [82] [38] [83,84] [85] [86] [17] [87] [88] [89] [90] [91]

COMPUTATIONAL PARAMETERS importance for atoms that are bonded to three other atoms (e.g., N in NH3 and P in PH3 ). The inversion term represents the difficulty of forcing all three bonds for atom i bonded to exactly three other atoms j, k, l, into the same plane. For nonbonded interactions, DREIDING uses a LJ 6–12 potential, a Coulombic expression for electrostatic interactions, and a term to accommodate hydrogen bonding.

V(r) ¼

4.2.2 Biological Force Fields Empirical force fields for biological macromolecules have been reviewed by Mackerell [6] and by Ponder and Case [18]. These include CHARMM, AMBER, OPLS, and GROMOS. All may be classified as a Class I force field of the general form given by Eg. (4.24)

X 1 X bond 1 X bend 1 X oop kij (rij
rij0 )2 þ kijk (uijk
u0ijk )2 þ kf [1 þ cos (nf
d)] þ kijkl (vijkl
v0ijkl )2 2 bonds 2 bends 2 improper torsions torsions 2 !12 !6 3 0 0 X qi qj X rij rij 5þ þ «ij 42
3 , rij rij «rij i6¼j i6¼j

X i

kb,i (ri
ri0 )2 þ

X

ku,i (ui
u0i )2 þ

i

X V0,i þ V1,i (1 þ cos fi )=2 i

 X X  

(qi qj e2 =rij ) þ 4«ij (sij =rij )12
(sij =rij )6 : þ V2,i (1
cos 2fi )=2 þ V3,i (1 þ cos 3fi )=2 þ i

TRIPOS. A force parameterized for biomolecules and small organic molecules, but sometimes used for polymers, is the TRIPOS force field [26] in the Sybyl molecular modeling package. The TRIPOS 5.2 force field includes harmonic bond stretching and angle bending with a torsional function consisting of a single cosine term. Nonbonded terms include a LJ 6–12 potential and a Coulombic term with either a constant or distance dependent dielectric function. GROMACS. Another force field originally targeted for the molecular simulations of biomolecules, but also useful for polymers, is GROMACS6 that runs molecular dynamics in a message-passing parallel mode. GROMACS [27] is a new implementation of GROMOS7 developed by van Gunsteren and Berendsen at the University of Groningen in the late 1980s [28,29]. Provisions are available in GROMACS for conversion between GROMACS

4 5 6 7

Chemistry at HARvard Macromolecular Mechanics). Optimized Potentials for Liquid Simulations. GROningen MAchine for Chemical Simulation. GROningen Molecular Simulation.

(4:24)

AMBER. AMBER [20,21] has been extensively used in the simulation of proteins and nucleic acids but recently has been generalized with parameters for most organic acid and pharmaceutical molecules [22]. OPLS. The OPLS5 force field [23–25] was introduced in the early 1980s to simulate liquid-state properties of water and more than 40 organic liquids. The form of the OPLSAA force field is given as [25]

where the bonded terms are all harmonic and there are no cross-terms. CHARMM. The CHARMM4 [19] force field includes harmonic terms for bond stretching and angle bending. Both proper and improper torsion terms are included in CHARMM as are LJ 6–12 and Coulombic nonbonded contributions.

V(r) ¼

/ 63

(4:25)

j

and GROMOS formats including the GROMOS87 and GROMOS96 force fields that are provided in GROMACS. Current features of GROMACS 3.0 have been reviewed by Lindahl et al. [30]. For bonded terms, GROMACS uses either a two-body harmonic potential (Eq. (4.3)) or Morse function for bond stretching and a three-body harmonic potential for angle bending (Eq. (4.7)). For both bond stretching and angle bending, a constraint can be used in place of the potential term. Four-body potentials include proper torsions and improper torsion potentials in the form of Eqs. (4.11) and (4.12), respectively. GROMACS uses a LJ 6–12 potential (an exponential short-range term, Eq. (4.15), is optional) and a Coulombic term (Eq. (4.18)) for nonbonded interactions. Force fields options include GROMOS, OPLS, and AMBER. Any united atom (UA) or all-atom force fields based on the general types of potential functions implemented in the GROMOS code can be used. GROMACS also permits the use of arbitrary forms of interactions with spline-interpolated tables as well as external potential terms for position-restraining forces and external acceleration (for nonequilibrium molecular dynamics).

64 / CHAPTER 4 CVFF. The consistent valence force field (CVFF) originally applied to biological systems [31] is a forerunner of the consistent force field (CFF) and its later derivatives (the polymer consistent force field PCFF and COMPASS) as discussed in Section 4.24. Terms in CVFF included a Morse potential for bond stretching, a harmonic term for angle bending, cosine torsional and out-of-plane torsional terms, four cross-coupling terms (bond–bond, angle–angle, bond–angle, and angle–torsion), and LJ 6–12 and Coulombic terms for nonbonded interactions. CVFF has been reported to perform less favorably than an early version of CFF (CFF91) for predicting the conformational energies of small molecules [32]. 4.2.3 Specialized Force Field and MD Codes ReaxFF. ReaxFF allows for bond breaking and bond formation in MD simulation so that thermal decomposition can be modeled as has been shown recently for polydimethylsiloxane [9]. ReaxxFF includes terms for traditional bonded potentials as well as nonbonded potentials (i.e., van der Waals and Coulombic). Bond breaking and bond formation are handled through a bond order/bond distance relationship. Parameterization is through high-level DFT calculations (B3LYP/6-311þþG**). DL_POLY. DL_POLY8 is a parallel molecular dynamics simulation package originally developed at the Daresbury Laboratory in England. Parameters for the current DL_POLY_3 force field may be obtained from the GROMOS, AMBER, and DREIDING force fields that share functional forms. LAMMPS. Another message-passing MD code is LAMMPS9[33] used for high-performance parallelized molecular dynamics calculations. The current version (version 17) is compatible with both AMBER and CHARMM. 4.2.4 Class II Force Fields Class II force fields make extensive use of both anharmonic and cross-coupling terms to adequately represent the ab initio potential energy surface (PES). These include the original consistent force field (CFF) that developed out of CVFF (Section 4.2.2) and subsequent variations, the most recent being the COMPASS force field. CFF. The consistent force field (CCF) [34] developed by Biosym10 is a descendent of CVFF but differs in the specific types of potential terms. The nonbonded terms of CFF include a quartic bond-stretch term (Eq. (4.4)), a quartic 8

http://www.cse.clrc.ac.uk/msi/software/DL_POLY/. Large-scale Atomic/Molecular Massively Parallel Simulator; http:// www.cs.sandia.gov/~sjplimp/lammps.html. 9

10 Biosym was merged with Molecular Simulations into the current company Accelrys.

angle-bending term (Eq. (4.8)), a three-term Fourier expansion term for torsion (Eq. (4.10)), and an out-of-plane torsion term (Eq. (4.12)). CFF includes several different versions (CFF91, CFF93 [35], CFF95) and the polymer consistent force field (PCFF). CFF93 has been parameterized for polycarbonates [36], aromatic polyesters [37], polysilanes [38], and poly(ethylene oxide) [39]. COMPASS. COMPASS is an example of a Class II force field parameterized by using an analytic representation of the ab initio (e.g., HF/6-31G*) potential energy surface. The functional form of the COMPASS force field is the same as CFF93 and includes an out-of-plane potential term (angle w), a LJ 6–9 potential as well as nonharmonic terms for bond stretching and angle bending, a Fourier cosine series for torsion, and a number of cross-coupled terms for the bonded interactions. The form of the COMPASS force field described in detail by Sun [12] is V(r) ¼ X

X

 k2 (b
b0 )2 þ k3 (b
b0 )3 þ k4 (b
b0 )4 þ

b

 k2 (u
u0 )2 þ k3 (u
u0 )3 þ k4 (u
u0 )4 þ

u

X

½k1 (1
cos f) þ k2 (1
cos 2f) þ k3 (1
cos 3f)þ

f

X w

X

k2 w2 þ


 X 0 k(b
b0 ) b0
b0 þ k(b
b0 )(u
u0 )þ b ,u b ,b 0

X

(b
b0 )½k1 (1
cos f) þ k2 (1
cos 2f) þ k3 (1
cos 3f)þ

b ,f


 X X
0 0 k u0
u0 (u
u0 ) þ k(u
u0 ) u0
u0 cos fþ b ,u u , u ,f 2 !9 !6 3 X qi qj X rij0 rij0 5: (4:26) þ «ij 42
3 rij rij rij i,j i,j

The partial charge for atom i in COMPASS is the sum of all charge bond increments, dij , as X dij : (4:27) qi ¼ j

The LJ parameters are obtained from the 6th order combination law (Eqs. (4.16) and (4.17)). The parameterization of COMPASS for bonded potential terms includes a fitting of the total energies as well their first derivatives (gradients) and second derivatives (Hessians) to ab initio (HF/6–31G*) calculations of lowmolecular-weight analogs. Examples of such analogs in the parameterization of COMPASS terms for polycarbonate are diphenyl carbonate, dimethyl carbonate, and 2,2diphenylpropane [36]. Nonbonded parameters are obtained from ab initio calculations and by parameter fitting to crystal structures. Valence parameters and charges are further scaled to fit experimental data. Full descriptions of the parameterization procedures and a tabulation of force constants for COMPASS have been given in several sources [12,40].*

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CHAPTER 5

Theoretical Models and Simulations of Polymer Chains Andrzej Kloczkowski* and Andrzej Kolinskiy L .H. Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA 50011, USA*; Faculty of Chemistry, Warsaw University, Pasteura 1, 02-093 Warsaw, Polandy

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Freely Jointed Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Freely Rotating Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chains with Fixed Bond Angles and Independent Potentials for Internal Bond Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chains with Interdependent Rotational Potentials. The Rotational Isomeric State Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theories of Polymer Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Theories of Real Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering from Polymer Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulations of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68 69 69 71 72 74 75 75 81

Theoretical models for other systems, such as star, branched, and ring polymers, random and alternating copolymers, graft and block copolymers are discussed in the book by Mattice and Suter [1]. Block copolymers are discussed in Chap. 32 of this Handbook [2]. Theories of branched and ring polymers are presented in the book by Yamakawa [3]. Liquid–crystalline polymers are discussed in the book by Grosberg and Khokhlov [4], and liquid crystalline elastomers in the recent book of Warner and Terentjev [5]. Bimodal networks are discussed by Mark and Erman [6,7]. Molecular theories of filled polymer networks are presented by Kloczkowski, Sharaf and Mark [8] and recently by Sharaf and Mark [9]. This first part of this article deals only with treatment of ‘‘bonded’’ interactions of polymer chains, appropriate only for modeling chains under Q-point conditions. Problems connected with effects of excluded volume are presented at the end of this chapter. The excluded volume effect for chains in good solvents are also presented in Chaps. IIB [10] and IIID [11] of this handbook and in books by Freed [12], de Gennes [13], des Cloizeaux and Jannink [14], and

5.1 INTRODUCTION In the first part of this article the review of various theoretical models for polymer chains is given. The models of freely jointed chains, freely rotating chains (including wormlike chains), and chains with fixed bond angles and independent rotational potentials and with interdependent potentials, including rotational isomeric state approximation, are presented. In the second part various theories of polymer networks are presented. The affine network model, phantom network, and theories of real networks are discussed. Scattering from polymer chains is also briefly presented. The third part of this article covers computer simulations of polymer chains. Methods of simulation of chains on lattices are presented and the equivalence between lattice chains and off-lattice chain models is discussed. The simulation of excluded volume effect is examined. The polymer chain collapse from random coil to dense globular state, and simulations of dense polymer systems are discussed. This article describes models for linear chains of homopolymers and for unimodal, unfilled polymer networks. 67

68 / CHAPTER 5 P

Forsman [15]. More information about computer modeling of polymers is provided by Binder [16,17], Baumgartner [18], Kolinski and Skolnick [19], and most recently by Kotelyanskii and Therodorou [20].

hs2 i0 

0 # I< j # n

lj )
(

i¼1

n X

lj )i0 ¼ nl 2

hs2 i0 1 ¼ : hr 2 i0 6

where T is the absolute temperature, k is the Boltzmann constant, u(lj ) is the potential energy of two segments connected by the j-th bond lj , and d denotes Dirac delta function. For the freely jointed chain model we have
 u(lj ) 1 d(jlj j  ‘): (5:7) exp ¼ 4pl 2 kT

(5:1)

By using the Fourier representation of the d function we obtain  n Z 1 ik
r sin (kl) P(r, n) ¼ 3 dk e 8p kl   Z 1 1 sin (kl) n ¼ 2 sin (kr) kdk: (5:8) 2p r 0 kl

(5:2)

It is convenient to compare real polymer chains with freely jointed chain by using the concept of the characteristic ratio defined as the ratio of the mean-square end-to-end vectors of a real chain and freely jointed chain with the same number of bonds

The solution of Eq. (5.8) is

hr 2 i0 : (5:3) nl 2 The characteristic ratio is a measure of chain flexibility. Flexible chains have Cn close to unity, while semiflexible and rigid polymers have usually much larger values of Cn . The mean-square radius of gyration for freely jointed chain is: Cn ¼

(2)

(5:5)

(5:6)

j¼1

for i 6¼ j:

P(r, n) ¼

1 nþ1 2 2 pl r(n  2)!

i#(nr=l)=2 X i¼0

n! (n  2i  r=l )n2 : (  1) i!(n  i)! i

q2 f3

(4)

I3

q1 I2

f2

(3)

I4 (n −2)

I5

(1) I1

f n −1 (0)

(5:4)

The freely jointed chain model has an exact analytical solution for the distribution function of the end-to-end vector. The probability that the chain of n bonds has the end-toend vector r is
 Z n X n u(lj ) P(r,n) ¼ dl1 dl2 . . . dln d[( li )  r]P exp , kT j¼1 i¼1

because hli
lj i0 ¼ 0

(n þ 2)nl 2 6 (n þ 1):

For longer chains (in the limit n ! 1) we have

The freely jointed chain model (known also as random flight model) was proposed for polymers by Kuhn in 1936. The chain is assumed to consist of n bonds of equal length l, jointed in linear succession, where the directions (u, f) of bond vectors may assume all values (0 # u # p; 0 # f # 2p) with equal probability (see Fig. 5.1). This means that directions of neighboring bonds are completely uncorrelated. The freely jointed chain model corresponds to a chain with fixed bond lengths and with unconstrained, free to adjust valence angles and with free torsional rotations. The mean square end-to-end vector hr 2 i0 in the unperturbed state (denoted by subscript 0) for the freely jointed chain is n X

¼

(n þ 1)2

5.2 THE FREELY JOINTED CHAIN

hr 2 i0 ¼ h(

hrij2 i0

In −1 (5) (n −1)

In

(n)

q n −1

FIGURE 5.1. Polymer chain composed of n bonds. Angles u are defined as complementary angles.

(5:9)

THEORETICAL MODELS AND SIMULATIONS OF POLYMER CHAINS In the limit n ! 1 the distribution function of the endto-end vector for freely jointed chain asymptotically approaches a Gaussian function
P(r, n) ¼

3 2pnl 2

3=2 exp


 3r 2 : 2nl 2

5.3.1 Worm-like Chain Model The projection of the end-to-end vector of a chain r on the direction of the first bond l1 =l for the freely rotating chain is

(5:10)

h

5.3 THE FREELY ROTATING CHAIN

2

hliþ1
lj i ¼ l cos u:

(5:11)

Similarly for two bonds i and iþk we have hliþk
li i ¼ l 2 ( cos u)k :

(5:12)

This result follows from the fact that the projection of a given bond on the preceding bond is cos u, while projections in two transverse directions averaged over free rotations are zero. This means that the projection of the kþi bond on the kþi1 bond is cos u, the projection of this projection on the kþi2 bond is (cos u)2 , etc., which finally leads to the Eq. (5.12). The mean square end-to-end vector for freely rotating chain is hr 2 i ¼

n X i¼1

hli i2 þ 2

n X ni X

n n1 X r
l1 1X i¼ hl1
li i ¼ l ( cos u)i l i¼1 l i¼0

¼l

The freely rotating chain model is a freely-jointed chain with fixed bond angles. It is assumed that all bond have equal length l and all bond angles are equal. The angle ui is defined as a supplementary angle of the skeletal bond angle at segment i as seen in Fig. 5.1, and therefore

hli
ljþk i

/ 69

1  ( cos u)n , 1  cos u

(5:16)

where u is the angle between bonds. In the limit n ! 1 this converges to r
l 1  a: (5:17) lim h i ¼ n!1 l 1  cos u The quantity a is called the persistence length and is a measure of chain stiffness. The wormlike chain model (sometimes called the Porod-Kratky chain) is a special continuous curvature limit of the freely rotating chain, such that the bond length l goes to zero and the number of bonds n goes to infinity, but the contour length of the chain L ¼ nl and the persistance length a are kept constant. In this limit r
l1 i ¼ a(1  eL=a ) l

(5:18)

hr 2 i0 a ¼ 2a[1  (1  eL=a )]: L L

(5:19)

h and

When the chain length L is much larger than the persistance length a, the effect of chain stiffness becomes negligible. In the limit L ! 1 we have hr 2 i0 =L ! 2a and the wormlike chain reduces to a freely-jointed chain.

i¼1 k¼1

  1 þ cos u 2 cos u[1  ( cos u)n  ¼ nl : 1  cos u n(1  cos u)2 2

(5:13)

For infinitely long chains the second term in Eq. (5.13) may be neglected and the characteristic ratio defined by Eq. (5.3) becomes: C1 ¼

1 þ cos u : 1  cos u

(5:14)

The mean square radius of gyration (defined by Eq. (5.4)) for freely rotating chain is hs2 i0 (n þ 2)(1 þ cos u) cos u  ¼ 6(n þ 1)(1  cos u) (n þ 1)(1  cos u)2 nl 2 þ

2( cos u)2 (n þ 1)2 (1  cos u)3



2( cos u)3 [1  ( cos u)n ] : n(n þ 1)2 (1  cos u)4

(5:15)

For very long chains the last three terms in Eq. (5.15) become negligible and hs2 i0 ¼ hr 2 i0 =6.

5.4 CHAINS WITH FIXED BOND ANGLES AND INDEPENDENT POTENTIALS FOR INTERNAL BOND ROTATION The more realistic model than freely rotating chain is a chain with fixed bond angles and hindered internal rotations. For simplicity it is assumed that the total configurational energy of the chain is a sum of configurational energies of chain bonds, and the energy of a given bond is independent on the configurational states of other bonds in the chain including the neighboring bonds. We should note that this is an approximation and for real polymer chains because of the steric interactions the energy of a given bond depends on the energy of its neighbors. We define a local Cartesian coordinate system for each of the bonds. We assume that the axis xi is directed along the bond i, and the yi axis lies in the plane formed by bonds i and i1, while the zi axis is directed to make the coordinate system right-handed. The components of the (iþ1)th bond liþ1 can be expressed in the coordinate system of the preceding bond i l0iþ1 ¼ Ti liþ1 ,

(5:20)

70 / CHAPTER 5 where Ti is the orthogonal matrix of the rotational transformation 2 3 sin ui 0 cos ui Ti ¼ 4 sin ui cos fi  cos ui cos fi sin fi 5 (5:21) sin ui sin
i  cos ui sin fi  cos fi Here ui is the supplementary bond angle (see Fig. 5.1) and fi is a dihedral angle between two planes defined by two pairs of bonds: bonds i1 and i, and i and i þ1. The scalar product of two bonds li
lj written in the matrix notation is lTi lj where lj is the column vector and lTi is the transpose of li (i.e., the row vector) 2 3 1 lj ¼ Ij 4 0 5 lTi ¼ lj [100], (5:22) 0 where li and lj are lenghts of bonds i and j (li ¼ lj ¼ l in our model but for polymers with different types of bonds in the backbone they may differ). Transforming successively over the intervening bonds the vector representation of the bond j to the coordinate system of bond i (j > i) we have hli
lj i ¼ hlTi Ti Tiþ1


Tj1 Ij i ¼ li lj hTi Tiþ1


Tj i11 : (5:23) Here hTi Tiþ1 . . . Tj1 i11 denotes configurational average of the (1–1) element of the matrix product Ti Tiþ1 . . . Tj1 . The configurational average of the product of rotational transformation matrices is generally given by hTi Tiþ1


Tj1 i ¼   R R l2 ,


,ln ) dl1 dl2


dln


(Ti Tiþ1


Tj1 ) exp E(l1 ,kT E(l1 ,l2 ,


,ln ) R R dl1 dl2


dln ,


exp kT (5:24) where k is the Boltzmann constant, T is the absolute temperature and E(l1 ,l2 , . . . ,ln ) is the conformational energy of the whole chain of n bonds. For a chain with fixed bond lenghts this energy depends only on the orientations of bonds described by bond angles ui and rotational angles fi , where 1 # i # n1, since the orientation of the last n-th bond is fully determined by the orientation of preceding bonds. For a chain with fixed bond angles the conformational energy is only a function of rotational angles fi , with 2 # i # n1, because f1 is undefined. For chains with independent potentials for internal bond rotation the conformational energy of the chain is a sum of bond energies Ei (fi ) E(f2 ,f3 ,


,fn1 ) ¼

n1 X

Ei (fi )

(5:25)

i¼2

and j1

hTi Tiþ1


Tj1 i ¼ P hTk i, k¼1

(5:26)

where for symmetric rotational potentials with ui (fi ) ¼ ui (  fi ) we have

2

sin uk cos uk hTk i ¼ 4 sin uk hcos fk i  cos uk hcos fk i 0 0

3 0 5: 0 hcos fk i (5:27)

Here hcos fk i is

h i k (fk ) cos fk exp EkT dfk h i : R 2p Ek (fk ) exp df k 0 kT

R 2p hcos fk i ¼

0

(5:28)

Using Eqs. (5.23) and (5.27) we may calculate the meansquare end-to-end vector for fixed bond angles and independent potentials for internal bond rotation " # n X ni X k 2 2 2 hr i0 ¼ nl þ 2l hTi i¼1 k¼1

11

  (E þ hTin ) 2 E þ hTi  2hTi ¼ nl , E  hTi n(E  hTi)2 11

(5:29)

where E is the unit matrix, and the subscript 11 denotes the (1–1) element of the matrix in square parenthesis. Equation (5.29) resembles Eq. (5.13) for freely rotating chain with cos u replaced by . Similarly to Eq. (5.15) the mean square radius of gyration is  hs2 i0 (n þ 2)(E þ hTi) hTi  ¼ 6(n þ 1)(E  hTi) (n þ 1)(E  hTi)2 nl 2 # 2hTi2 2hTi3 [1  hTin ]  : þ (n þ 1)2 (1  hTi)3 n(n þ 1)2 (1  hTi)4 11

(5:30) The general solution of eqs. (5.27) and (5.28) is possible by diagonaliziation of the matrix defined by Eq. (5.27). The eigenvalues of are l1,2 ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cos u(1  hcos fi)  cos2 u(1  hcos fi)2 þ 4hcos fi 2 :

l3 ¼ hcos fi

(5:31) 2

For example, the expression for hr i0 in terms of eigenvalues of is hr 2 i0 (1 þ cos u)(1 þ hcos fi) ¼ (1  cos u)(1  hcos fi) nl 2 

2l1 ( cos uhcos fi þ l1 )(1  ln1 ) n(l1  l2 )(1  l1 )2

þ

2l2 ( cos uhcos fi þ l2 )(1  ln2 ) : n(l1  l2 )(1  l2 )2

(5:32)

For very long chains only first terms in Eqs. (5.29) and (5.30) are important and we have   hr 2 i0 E þ hTi (1 þ cos u)(1 þ hcos fi) C1 ¼ lim ¼ ¼ n!1 nl 2 E  hTi 11 (1  cos u)(1  hcos fi) (5:33)

THEORETICAL MODELS AND SIMULATIONS OF POLYMER CHAINS and 



hs2 i0 nþ2 E þ hTi ¼ ¼ 6(n þ 1) E  hTi 11 nl 2 (n þ 2)(1 þ cos u)(1 þ hcos fi) : ¼ 6(n þ 1)(1  cos u)(1  hcos fi)

(5:34

5.5 CHAINS WITH INTERDEPENDENT ROTATIONAL POTENTIALS. THE ROTATIONAL ISOMERIC STATE APPROXIMATION In real polymer chains the rotational potentials depend on the steric interactions between pendant groups of neighboring bonds, and are generally not mutually independent. In the simplest case of hydrocarbons the bond rotational potential has three minima as shown in Fig. 5.2. The global minimum at the torsional angle 08 corresponds to the trans two other minima with the same energies at torsional angle around þ1208 and 1208 correspond to the gaucheþ and the gauche states (gþ and g ). The energy difference between the trans the gauche states for n-alkanes is about 500 cal/ (mole). We may use the rotational isomeric state approximation that each bond in the chain occur in one of these rotational states. This assumption enables us to replace all integrals over rotational angles in the partition function and statistical averages by summations over bonds rotational states. Additionally steric interactions between pendant groups of neighboring bonds become important, e.g., the sequence g g becomes energetically very unfavorable. We may neglect the longer range interactions and assume that the configurational energy is a sum of energies of nearest-neighbor pairs E(f2 ,f3 ,


,fn1 ) ¼

n1 X

Ei (fi1 ,fi ):

(5:35)

i¼2

The configurational partition function becomes   Z Z 1 Z ¼


exp  E(f2 ,


,fn1 ) df2


dfn1 kT   n 1 XY 1 ¼ exp  Ei (fi1 ,fi ) , (5:36) kT {f} i¼2 where {f} denotes the set of all available states (t,gþ ,g ) for all bonds in the chain. We define the statistical weight corresponding to bond i being in the h state while bond i  1 being in the z state (where h and z are sampled from the t,gþ ,g set)
 Ezh,i uzh,i ¼ exp (5:37) kT and the statistical weight matrix 2 utt,i utgþ ,i Ui ¼ 4 ugþ t,i ugþ gþ ,i ug t,i ug gþ ,i

Energy (kJ moI)

(5:38)

where sc and sv denote ugþ gþ and ugþ g , respectively. By using the statistical weight matrices we may express the configuration partition function as " # n1 n1 Y XY  Z¼ uhz,i ¼ J Ui J, (5:40) i¼2



where J and J are row and column vectors, respectively 2 3 1 J ¼ [ 1 0 0] J ¼ 4 1 5: (5:41) 1

20 15

For very long chains (in the limit n ! 1) the partition function is determined by the largest eigenvalue l1 of the statistical weight matrix U

10 5

−180

3 utg ,i ugþ g ,i 5: ug g ,i

It is convenient to express the energy of a given single bond relative to the energy of the trans state. The energy of a pair of bonds Ehz,i is defined relative to the state where the bond i is in the trans state, and all subsequent bonds j > i are also in the trans states. From this definition follows Eht ¼ 0 for z ¼ t,gþ ,g . Additionally Etgþ ¼ Etg ¼ Egþ gþ ¼ Eg g ( 500 cal/mol for n-alkanes), and Egþ g ¼ Eg gþ ( 3,000 cal/mole for n-alkanes). This means that the statistical weight matrix may be written as 2 3 1 s s U ¼ 4 1 sc sv 5, (5:39) 1 sv sc

{f} i¼2

25

/ 71

Z ffi ln2 1 : −120

−60

0

60

120

180

Torsional angle q (degrees)

FIGURE. 5.2. The dependence of the conformational energy on the torsional angle in n-alkanes.

(5:42)

The largest eigenvalue of the matrix U defined by Eq. (5.39) is  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 l1 ¼ 1 þ s(c þ v) þ [1  s(c þ v)]2 þ 8s : 2 (5:43)

72 / CHAPTER 5 The probability that bonds i  1 and i occur in states h and z, respectively is " # " # i1 n1 Y 1  Y @Ui Uk Uk J phz,i ¼ J Z @ ln uhz,i k¼iþ1 k¼2 ffi

@ ln l1 : @ ln uhz

(5:44)

The probability that bond i is in the state z, irrespective of the state of bond i1 is X X @ ln l1 phz,i ffi : (5:45) p z ,i ¼ @ ln uhz n¼t,gþ ,g h¼t,gþ ,g The conditional probability that bond i is in z state, given that bond i  1 is in state h is phz,i qhz,i ¼ : (5:46) ph,i1 In order to calculate the mean square end-to-end vector or a radius of gyration we have to calculate averages hTi Tiþ1 . . . Tj1 i (Eq. (5.24)). For bonds with independent rotational potentials this average is a product of averages hTi for single bonds. For chains with interactions between neighboring bonds we define for each bond i the supermatrix k Ti k of the order 99 2 3 T(f1 ) 5, k Ti k¼ 4 T(f2 ) (5:47) T(f3 ) i where T is rotation matrix given by Eq. (5.21), and the f1 ¼ 0 ,f2 ¼ 120 ,f3 ¼ 120 , are the torsional angles corresponding to the trans, gaucheþ and gauche states. We define also a direct product Ui E3 of the statistical weight matrix Ui (defined by Eq. (5.38)) and the unit matrix of order three E3 . 2 3 utt E3 utgþ E3 utg E3 Ui E3 ¼ 4 ugþ t E3 ugþ gþ E3 ugþ g E3 5: (5:48) ug t E3 ug gþ E3 ug g E3 The statistical average of the product of rotation matrices may then be written as  1 hTi Tiþ1 . . . Tj1 i ¼ hT(ji) i ¼ (J U(i2) ) E3  i 2 Z h i (U(nj) J) E3 ] [(U E3 ) k T k(ji) i j

(5:49) Then the mean-square end-to-end vector and the mean square radius of gyration are hr 2 i0 ¼ nl 2 þ 2

n1 X n X

lTi hT(ji) ilj i

(5:50)

i given by Eq. (5.49). Both hr 2 i0 and hs2 i0 may with hT(ji) i be written in a more compact form in terms of proper supermatrices. The details are given in Flory’s monograph [21]. Additional information is given in the Handbook chapter by Honeycutt [22]. 5.6 THEORIES OF POLYMER NETWORKS 5.6.1 The Affine Network The theory of affine networks was developed by Kuhn and improved by Treloar, and is based on the assumption that the network consists of n freely-jointed Gaussian chains and the mean-square end-to-end vector of network chains in the undeformed network is the same as of chains in the uncross-linked state. This assumption is supported by experimental data. It is also assumed that there is no change in volume on deformation and the junctions displace affinily with macroscopic deformation. The intermolecular interactions in the model are neglected, i.e., the system is similar to the ideal gas. The elastic free energy of a chain is related to the distribution function of the end-to-end vector P(r) 3 hr 2 i Ael ¼ c(T)  kT ln P(r) ¼ A (T) þ kT 2 2 hr i0

(5:52)

for the Gaussian distribution given by Eq. (5.10). Here c(T) and A (T) are constants dependent only on the temperature T, k is a Boltzmann constant, and hr 2 i0 is the average of the mean-square end-to-end vector in the undeformed state. The elastic free energy of the network DAel relative to the undeformed state is a sum of free energies of individual chains
2  3kT X 2 3 hr i 2 DAel ¼ (r hr i0 ) ¼ vkT 1 2hr 2 i0 v 2 hr 2 i0 (5:53) 2

Here hr i is the end-to-end vector in the deformed state averaged over the ensemble of chains hr 2 i ¼ hx2 i þ hy2 i þ hz2 i:

(5:54)

In the affine model of the network it is assumed all junction points are imbedded in the network, and each Cartesian component of the chain end-to-end vector transforms linearly with macroscopic deformation x ¼ lx x0 ,

y ¼ ly y0 , z ¼ lz z0

hx2 i ¼ l2x hr 2 i0 , hy2 i ¼ l2y hy2 i0 , hz2 i ¼ l2z hz2 i0

(5:55) (5:56)

and therefore

i¼1 j¼iþ1

1 DAel ¼ vkT(l2x þ l2y þ l2z  3): 2

and hs2 i0 ¼

1 (n þ 1)2

X

k k X X

0 # h # k # n i¼hþ1 j¼hþ1

lTi hT(ji) ilj (5:51)

(5:57)

Here, lx , ly , and lz are the components of the deformation tensor l, defined as the ratios of the final length of the

THEORETICAL MODELS AND SIMULATIONS OF POLYMER CHAINS sample Lt to the initial length Lt,0 in t ¼ x, y, and z direction, respectively. (The more rigorous statistical mechanical analysis by Flory [23] has shown that Eq. (5.57) should contain additional logarithmic term –mkT ln (V=V0 ), where m is the number of junctions, V is the volume of the network, and V0 is volume of the network at the state of formation). The force f under uniaxial tension in direction z is obtained from the thermodynamic expression:

 @DAel @DAel f ¼ ¼ L1 , (5:58) 0 @L T ,V @l T ,V where l ¼ lz ¼ Lz =Lz,0 . Because the volume of the sample is constant during deformation the x and y components of the deformation are lx ¼ ly ¼ l1=2 . Performing the differentiation in Eq. (5.58) leads to the elastic equation of state
 vkT (5:59) f ¼ (l  1=l2 ): L0

5.6.2 The Phantom Network Theory The theory of phantom network was formulated by James and Guth [24] in the forties. They assumed that chains are Gaussian with the distribution P(r) of the end-to-end vector  g 3=2 P(r) ¼ exp(  gr 2 ), (5:60) p where g¼

3 2hr 2 i0

(5:61)

and interact only at junction points. This means that chains may pass freely through one another, i.e., are ‘‘phantom’’, the excluded volume effects and chain entanglements are neglected in the theory. They assumed also that all junctions at the surface of the network are fixed and deform affinely with macroscopic strain, while all junctions and chains inside the bulk of the network fluctuate around their mean positions. The idea of the phantom network is very similar to the concept of the ideal gas. The theory based on these simple assumptions leads to significant improvements in the understanding of the properties of networks, such as microscopic fluctuations and neutron scattering behavior. The configurational partition function ZN of the phantom network is the product of the configurational partition functions of its individual chains. junctions i and j: Y ZN ¼ C exp(  3rij2 =2hrij2 i0 ) i<j

¼C

Y i<j

! 1XX  2 exp  gij jRi  Rj j : 2 i j

(5:62)

/ 73

Here, Ri and Rj are positions of junctions i and j, g ij ¼ 3=2hrij2 i0 if junctions i and j are connected by a chain, and zero otherwise, and C is a normalization constant. The position vectors Ri with i ranging from 1 to m, where m is a number of junctions, may be arranged in column form, represented as {R}. Equation (5.62) may then be written ZN ¼ Cexp(  {R}T G{R}),

(5:63)

where the superscript T denotes the transpose. The symmetric matrix G known as the Kirchhoff valency-adjacency matrix in the graph theory describes the connectivity of the network and its elements gij are ( g ij ¼Pg ij , P i 6¼ j G ¼ g ii ¼ g ij ¼ g ij : (5:64) j

j

James and Guth assumed that all m junctions in the network may be divided into two sets of junctions: (i) ms fixed junctions at the bounding surface of the polymer and (ii) mt free junctions fluctuating about their mean positions
t } inside the polymer. The partition function of the net{R work due to fluctuating junctions is ZN ¼ Cexp(  {DRt }T Gt {DRt }),

(5:65)

where {DRt } denotes fluctuations of free junctions {DRt } ¼ {Rt }  {Rt }:

(5:66)

The product of the fluctuations of two junctions i and j averaged over the network may be obtained from Eq. (5.65) as R DRi
DRj exp [  {DRt }T Gt {DRt }]d{DRt } R hDRi
DRj i ¼ exp [  {DRt }T Gt {DRt }]d{DRt } ¼

@ ln Zt , @gij

(5:67)

where d{DRt }  dDR1t dDR2t . . . dDRmt and
m 3=2 Z p t : Zt ¼ exp [  {DRt }T Gt {DRt ]d{DRt } ¼ detGt (5:68) This leads to the expression

.

hDRi DRj i ¼

3 @ 3 ln jdetGt j ¼ (G1 ) , 2 @gij 2 t ij

(5:69)

where (G1 t )ij denotes the (ij)-th element of the inverse matrix G1 t . Fluctuations of junctions from their mean positions in a phantom network depend on the network’s functionality f and are independent of macroscopic deformation. For example for the infinitely large network with the symmetrical tree-like topology (such as shown in Fig. 5.3) the mean-square fluctuations of junctions h(DR)2 i and

74 / CHAPTER 5 9 26 10

25

3

8 24 1

11 4

2

5

7

23

i.e.,

21 20

13 14

15

17

16

r ij ¼ r ij þ Dr ij

22

6

12

since instantaneous fluctuations and mean values are uncorrelated. From Eqs. (5.72) and (5.74) follows: 2 (5:75) h
r 2 i ¼ (1  )hr 2 i0 : f According to the theory the mean positions of junctions transform affinely with macroscopic strain while the fluctuations are strain independent:

18

19

correlations between fluctuations of two junctions i and j hDRi DRj i separated by m other junctions are: " #   1 3 (G1 h(DRi )2 i hDRi Rj i t )ii (Gt )ij ¼ 1 DRj DRi i h(DRj )2 i 2 (G1 t )ji (Gt )jj " # f1 1 3 f(f2)(f1)m f(f2) ¼ : (5:70) f1 1 2g f(f2)(f1) m f(f2)

.

The mean-square fluctuations of the distance rij ¼ jRi  Rj j between junctions i and j are h(Drij )2 i ¼ h(DRi  DRi )2 i 3 1 ) þ (G1 ¼ [(G1 t )jj  2(Gt )ij ] 2 t ii 2[(f  1)mþ1  1] 2 ¼ hr i0: (5:71) f(f  2)(f  1)m For a special case of mean-square fluctuations of the endto-end vector (m ¼ 0) we have 2 2 hr i0 : f

(5:72)

Equations (5.70) and (5.71) may be easily generalized for fluctuations of points along the chains in the network, since each point along the chain may be considered as bi-functional junction. As a consequence the valenceadjacency matrix in this generalized case contains additional elements describing the connectivity of bi-functional junctions. More details is provided in the review article by Kloczkowski, Mark, and Erman [25]. The vector rij between junctions i and j is r ij ¼ r ij þ Dr ij ,

(5:73)

where Drij is the instantaneous fluctuation of rij and rij is the time average of rij . Squaring both sides of the above equation and taking the ensemble average leads to hrij2 i

¼

hr 2ij i

2

þ h(Dr ij ) i

(5:74)

(5:77)

Using Eq. (5.53) for the elastic free energy, we obtain the following expression for the free energy of the phantom network 1 2 DAel ¼ (1  )nkT(l2x þ l2y þ l2z  3): 2 f

.

h(Dr)2 i ¼

"

2

FIGURE 5.3. First three tiers of a unimodal, symmetrically grown, tetrafunctional network (f ¼ 4) with tree-like topology.

.

# 2 2 2 2 lx þ ly þ lz 2 2 þ hr i0 : hr i ¼ (1  ) f f 3

(5:76)

(5:78)

Equation (5.78) is very similar to Eq. (5.57) for the affine network. The only difference is that the so called front factor (equal n=2 for affine network model) is replaced by j=2 for the phantom network model where j ¼ (1 

2 )n: f

(5:79)

The equation for the elastic force is similar to Eq. (5.59) for the affine network with n replaced by j. 5.7 STATISTICAL THEORIES OF REAL NETWORKS In real polymer network the effects of excluded volume and chain entanglements should be taken into account. In 1977 Flory [26] formulated the constrained junction model of real networks. According to this theory fluctuations of junctions are affected by chains interpenetration, and as the result the elastic free energy is a sum of the elastic free energy of the phantom network DAph (given by Eq. (5.78)) and the free energy of constraints DAc DAel ¼ DAph þ DAc

(5:80)

with DAc given by the formula X 1 DAc ¼ mkT ½Bt þ Dt  ln (1 þ Bt )  ln (1 þ Dt ), 2 t¼x,y,z (5:81) where Bt ¼

k2 (l2t  1) (l2t þ k2 )2

(5:82)

l2t Bt : k

(5:83)

and Dt ¼

THEORETICAL MODELS AND SIMULATIONS OF POLYMER CHAINS Here k is a parameter which measures the strength of the constraints. For k ¼ 0 we obtain the phantom network limit, and for infinitely strong constraints (k ¼ 1) the affine limit is obtained. Erman and Monnerie [27] developed the constrained chain model, where constraints effect fluctuations of the centers of the mass of chains in the network. Kloczkowski, Mark, and Erman [28] proposed a diffusedconstraint theory with continuous placement of constraints along the network chains. A different statistical–mechanical approach based on so called replica formalism was developed by Edwards and coworkers [29,30]. They studied the effect of topological entanglements between chains on the elastic free energy of the network and formulated the slip-link model. The elastic energy of constraints in the slip-link theory is  X (l2  1) 1 1 þ hl2t t DAc ¼ Ns kT ] , (5:84) þ ln [ 2 2 1þh t¼x,y,z 1 þ hlt where Ns is the number of slip-links and h is the slipage parameter. Equation (5.84) is very similar to Eq. (5.81) for the constrained junction model. Vilgis and Erman [31] showed that for small deformations both equations have the same form (except minor volume term) with k ¼ 1=h.

/ 75

where qx , qy , and qz are the components of the scattering vector q. The vector rij between two scattering centers may be written for a phantom network as rij ¼ rij þ Drij where rij is the time average of rij , and Drij is the instantaneous fluctuation of rij from its mean time-averaged value. Assuming that mean-square fluctuations are strain independent and that mean positions transform affinely with macroscopic strain and applying Eqs. (5.74)–(5.77) leads to " # hDx2ij i0 2 2 2 2 hxij i ¼ lx þ (1  lx ) 2 (5:88) hxij i0 , hxij i0 where lx is the x component of the principal deformation gradient tensor l, with similar expressions for the y and z components. For a freely jointed chain hx2ij i0 ¼ hrij2 i0 =3 ¼ hhr 2 i0 =3, where h ¼ ji  jj=N is the fractional distance, and hr 2 i0 is the mean-square end-to-end vector for the undeformed chain. Substituting these results to Eq. (5.87) leads to S(q) ¼


 N 1 X ji  jj 2 (f  2) ji  jj exp y ) 1  (1  l : N 2 i,j¼1 N f N

(5:89) In this equation

5.8 SCATTERING FROM POLYMER CHAINS The scattering form factor S(q) from a labeled chain in the network is given by the Fourier transform of the distribution function V(rij ) of the vector rij between two scattering centers i and j averaged over all pairs of scattering centers along the chain: N Z 1 X S(q) ¼ 2 exp (iq
rij )V(rij )drij : (5:85) N i, j¼1 Here q is the scattering vector representing the difference between the incident and scattered wave vectors k0 and k, respectively, and N is the total number of scattering centers along the chain. The distribution function V(rij ) of the vector rij between scattering centers in the undeformed state is assumed to be Gaussian. The distribution function V(rij ) in the deformed state is V(rij ) ¼ [(2p)3 hx2ij ihy2ij ihz2ij i]1=2 exp (  x2ij =2hx2ij i  y2ij =2hy2ij i  z2ij =2hz2ij i),

(5:86)

where hx2ij i, hy2ij i, and hz2ij i are the mean-square components of the vector rij in the deformed state. Substituting the expression for V(rij ) given by Eq. (5.86) into Eq. (5.85) leads to S(q) ¼

N 1 X exp (  q2x hx2ij i=2  q2y hy2ij i=2  q2z hz2ij i=2), N 2 i, j¼1 (5:87)

y ¼ q2 hr 2 i0 =6

(5:90)

l ¼ lq=q:

(5:91)

and the vector l is

For scattering parallel to the direction of extension l ¼ lk and for scattering p perpendicular to the direction of extension l ¼ l? ¼ 1= lk . Replacing the double summation by integration and evaluating one of the integrals leads to    Z 1 f2 S(q) ¼ 2 dh(1  h) exp yh 1  h(1  l2 ) f 0 (5:92) the result obtained by Pearson [32]. As the strain goes to zero Eq. (5.92) has the limiting form 2 lim S(q) ¼ (ey þ y  1) y

l!1

(5:93)

derived by Debye [33], corresponding to the scattering from an unperturbed Gaussian coil. Readers interested in scattering from labeled cross-linked paths in unimodal and bimodal networks should consult the review article by Kloczkowski, Mark, and Erman [25]. 5.9 SIMULATIONS OF POLYMERS System composed of polymers or containing polymers immersed in low molecular media are extremely complex

76 / CHAPTER 5 for many reasons. First, polymer chains (linear, branched, or cyclic) have often a huge molecular mass. Large fraction of single covalent bonds in the main chain imply at least a limited internal rotational freedom for each such bond, and consequently lead to an enormous number of available conformational isomers. Second, due to the excluded volume effect polymer chains are non-Markovian, i.e., conformational space accessible to a selected portion of the chain depends on the actual conformation of the remaining fragments. Consequently, a rigorous analytical treatment of polymer conformational statistics and dynamics is essentially impossible; although various aspects of polymer physics could be quite successfully addressed within framework of approximate theories (see the previous sections). Third, the chain connectivity imposes a complex network of topological obstacles. A moving chain cannot cross its own contour or the paths of the other chains present in the system. This has pronounced consequences for polymer dynamics in solutions and polymeric melts, where motion of polymers has to be extremely correlated and the correlation distances are several orders of magnitude larger than it is observed in typical disordered low molecular systems. The nature of these correlations could be extremely complex. For the above reasons computer simulations are very important components of methodology of theoretical polymer physics. Properly designed computational experiments expand our understanding of these complex systems, provide excellent test of the existing theories and stimulate development of new theoretical approaches. Due to the large size, time scales involved, and complexity of polymeric systems numerous new simulation techniques have been developed to meet these extreme computational demands. This way theoretical physics of polymers had significant influence on progress in computational physics in general. Simulations of polymers could be designed on various levels of molecular details treated in an explicit way [16– 20, 34–36]. Molecular Dynamics (or Brownian Dynamics) of all-atom systems are limited to short chains or/and to studies of local and fast relaxation processes. It is rather impractical, and often nonfeasible, to do MD simulations of long polymer collapse or a self diffusion of polymer chain in a melt, to give just a couple of typical examples. Monte Carlo simulations of the all-atom systems have a bit less limitations, but still large scale rearrangements are difficult to study. For these reasons frequently reduced representations of polymer conformational space are employed. These range from united atom models, where groups of atoms are treated as single interaction units, to lattice models where entire mers (or large united atoms) are restricted to a lattice, thereby enormously reducing the number of available states and simplifying energy calculations. While simple lattice models are of very limited utility in the physics of low molecular mass system, for polymers

they provided general solutions to very fundamental problems. This qualitative difference is strictly related to the difference in the correlation length scales in the two types of systems. In polymers the local details become usually irrelevant at large distances. Because of their importance for general physics of polymers and educational values we start from a discussion of simple lattice models of polymers and polymer dynamics.

5.9.1 Ideal Lattice Chains are Equivalent to Off-lattice Models Let us consider a chain restricted to a simple cubic lattice, with the lattice spacing equal to 1. The chain is a string of vectors with the six allowed orientations belonging to the following set {j1,0,0j, j  1,0,0j, j0,1,0j, j0,  1,0j, 0,0,1, j0,0,  1j}. A chain vector could be followed by any of the vectors from the set. Thus, there is no any average orientational correlation between the chain vectors, in spite of the lattice restrictions. Note, that for this ideal model a lattice site can be occupied by more than one bead of the chain. It could be immediately seen that the Eqs. (5.1) and (5.2) written for the freely joined chain are true as well for the ideal lattice chain. The models are equivalent, and an exact analytical theory of their conformational statistics exists. Such analogy goes much further. Let us now consider a chain restricted to the diamond lattice with a constant tetrahedral value of the valence angle and three discrete values of the torsional angle corresponding to the trans and two gauche states. Again, it is easy to note that this model is equivalent (in respect to its global properties) to the ideal, freely rotating chain with the tetrahedral value of the planar angle. It is also easy to show that such chain can mimic the chain with restricted rotations and interdependent rotations, provided Boltzmann weights are assigned to the trans and gauche conformations and proper correlations between the weights are taken into account. Equivalence of the ideal continuous and the lattice models extends also on the dynamic properties of a single chain. The Rouse model [37,38] , (or the bead and spring model) consists of a string of points (or beads) of equal mass connected by harmonic springs of equal length and equal strength of the harmonic potentials, although without any angular interactions. An exact analytical solution for the relaxation spectrum of this model is relatively easy to derive. For the ideal (without excluded volume limitations) lattice chain a simple model of dynamics, simulated by a long random sequence of small local conformational changes, could be formalized in a stochastic Master Equation of motion. It has been shown by Verdier and Stockmayer [39], that such model is equivalent to the Rouse model [37,38] in almost entire relaxation spectrum, except the fastest local oscillations involving a couple of chain segments.

THEORETICAL MODELS AND SIMULATIONS OF POLYMER CHAINS 5.9.2 Simulation of the Excluded Volume Effect in a Single Chain The ideal models described in the previous sections ignored a very important fact, that a polymer has its own volume, i.e., two segments cannot occupy the same place in space. Using a series of approximations Flory has shown that the exclude volume leads to a significant increase of the average random coil dimensions and changes the number of accessible conformers. Flory, has also shown that in a thermodynamically ‘‘poor’’ solvent the proper volume of the chain segments could be balanced by their mutual attractions, leading to a pseudoideal state, very similar to the Boyle point for the real gases. Typical, however, is the situation of a ‘‘good’’ solvent, where the effect of excluded volume is large. Exact analytical solution to the excluded volume problem does not exist. It is unknown how to calculate partition function of a single chain, since the probability of a given conformation of the (nþ1)th bond added to a chain depends on the conformation of the preceding n bonds. The process of virtual growth of a ‘‘real’’ (with excluded volume, in contrast to the ideal, lacking volume chains) chain is non-Markovian. This is exactly a situation where the data from computer simulations are needed for estimations of true (in silico) experimental properties of the model system and for subsequent evaluation of the assumptions and predictions of various approximate theories. In the context of a simple lattice model the problem could be formulated as follows. Compute the number of non-selfintersecting random walks on the lattice and the distribution of the segment density, size, shape, etc., of the resulting random coils as a function of the chain length. The first thought is to use computer for an exact enumeration of all possible conformations of a n-segment chain. Unfortu-

A

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nately, the number of possible random walks grows exponentially with the chain length. Exact enumeration is possible only for n range of few tens of segments. In this range the finite length effects are still large and an extrapolation of the obtained (exact) data to higher values of n is uncertain. Another approach is to employ a stochastic sampling (Monte Carlo method) to get a ‘‘representative’’ ensemble of non-self-intersecting random walks of the assumed length n. There the result is not exact, however avoids any systematic errors. The magnitude of the statistical error could be always reduced by the increase of the sample size. The algorithm is very simple. 1. Start from the first bond. 2. Add the next bond in a randomly selected direction (the simple ‘‘back’’ step could be a priori prohibited and the resulting bias easily removed from the results). 3. Check for non-self-intersection and repeat from (2) if a double occupancy of a lattice site is not detected, otherwise erase the chain and start from (1). 4. Stop the chain growth when the requested length n is reached and add the chain to the statistical ensemble. 5. Repeat the entire process starting from (1) until the required number of chain in the sample is collected. 6. Perform statistical analysis of the collected ensemble. The process of the MC chain growth is illustrated in Fig. 5.4. Situations, as that schematically depicted in Fig. 5.4B happen quite frequently. Therefore, the algorithm outlined above has a huge sample attrition rate; only a small fraction of the staring chains are finally accepted in the statistical

B

FIGURE 5.4. Two dimensional illustration of the MC growth of non-self-intersecting walks (see the text for details). On the left side (A) an example of the successful structure composed of n ¼ 15 segments is shown. On the right side (B) an intersection has been detected before reaching n ¼ 15, the final chain length, and the chain has to be removed from the statistical ensemble.

78 / CHAPTER 5 pool. To overcome this problem Rosenbluth and Rosenbluth [40] proposed a modified approach. The segments are selected only from the set of orientations which do not cause the intermediate chain clash. In the case shown in Fig. 5.4B the segment number 11 would be selected only from the following two possibilities; to the left, and to the top of the plane. Obviously, this introduces a bias to the sample. This bias could be easily removed with a proper weighting of the particular conformation with a factorials depending on the number of the allowed continuations at each step. The R&R method allows for generation of much longer chains. Their length is limited by the ‘‘cull-the-sack’’ effect, where the growing chain end is surrounded by the chain segments, blocking all the possibilities for the further continuation of the growth process. A number of extensions of the original R&R method have been proposed since then. In general, these methods look into possibility of a continuation in a larger perspective than just one segment ahead. There are several qualitatively different ways of sampling the polymer conformational space. One may start from a chain of a given length and successively modify its conformation. Two examples of such types of algorithms are illustrated in Fig. 5.5. One of them is the ‘‘pivot’’ algorithm, where a single step consist of a random selection of a bond and a rotation (in two dimensional case it is reduced just to a flip; vertical or horizontal) of the selected end of the chain. Advantage of this algorithm is that in a single step a large modification of the chain conformation is attempted. However the acceptance rate for longer chain could be rather small. A number of different global rearrangements of the chain conformations were designed aiming on a more efficient sampling. An example is the ‘‘reptation’’ algorithm, where a bond (or a small number of bonds) is cut-off from one end of the chain and added in a random direction on the

A

opposite end. The acceptance ratio for this type of global update algorithms could be quite high. Yet another example is a technique that could be viewed as a complex ‘‘pivotlike’’ algorithm, where a part of the chain on one end is erased and then re-grown in a random or semirandom fashion. Of course, the statistical sample is collected in along series of attempts (sometimes successful) to successive modifications of subsequently generated conformations. In the second type of algorithms (Fig. 5.5A) local micromodiffications of the chain conformation are randomly selected at random position of the chain. Marginally, let us note that the local move algorithm could be interpreted as a simulated Brownian motion of a polymer chain. This is a ‘‘real’’ chain version of the before mentioned Verdier–Stockmayer model [39] of polymer dynamics. Again, it should be stressed out, that an accurate analytical theory for the real chain dynamics does not exist. The local move algorithms are powerful tools for study of long-time (and large scale) polymer dynamics. There are however several problems with the models employing a limited set of local moves and low coordination number lattices. The algorithms could be nonergodic, or rather ergodic in a subset of its full conformational space. This is explained in Fig. 5.6. There is no path to- and no path from the conformation shown in the drawing. The problem may be cured using a higher coordination lattices and/or a larger set of ‘‘less-local’’ micromodiffications. An example of such larger scale move is shown in Fig. 5.6B. The backfire of such update of the local move algorithms is a less clear relation with the model of the Brownian motion. Perhaps, the ‘‘wave-like’’ move, when attempted rarely could be interpreted as a particular coincidence of a series of local moves, which somehow were able to pass the local conformational barriers. An additional flaw of the low coordination lattice models (beside the ergodicity

B

FIGURE 5.5. The idea of the pivot algorithm (A), and the local moves algorithm (B). The black contours indicate the initial structures, the lighter bonds show the accepted modifications. The local moves include (from top to the bottom of B): random chain end modification, a crankshaft move and a corner move.

THEORETICAL MODELS AND SIMULATIONS OF POLYMER CHAINS

B

FIGURE 5.6. A trapped conformation for the algorithm with only local moves for a chain on the simple square lattice (A). A longer distance move that guarantees the ergodicity of the algorithm, where a U shaped fragment at one part of the chain is cut-off and attached somewhere else (B).

problems) of polymer dynamics is the difficulty of controlling the effects of the lattice anisotropy on the observed motion. Obviously, simple models of polymer conformations and dynamics, very similar to those described above could be design in the continuous space. Such models could be sampled using MD, MC, or via various hybrid sampling techniques based on a combination of genetic algorithms (GA) and molecular mechanics (usually MC dynamics). The results from simplified lattice and off-lattice models are essentially equivalent. For instance, the average chain dimensions of the real chain models scale as hS2 i ng . Interestingly, the value of the universal constant for the 3-dimensional chains is close (but not identical) to the value resulting from the mean-field analytical theory of Flory. More qualitative differences are observed between the results of the ‘‘real’’ chain simulation of the polymer dynamics and the ideal chain theory of Rouse [37]. 5.9.3 Simulations of Polymer Chain Collapse Polymer chains in solution can undergo a collapse transition from an expanded random coil state to a dense globular state. The transition could be induced by decrease of temperature or by adding a ‘‘poor’’ solvent to the solution [41]. This process is difficult to describe analytically, but could be studied in details via computer simulations. Let us again consider a very simple lattice model. Representation of protein conformational state could be done using any kind of simple lattice. In such context it is easy to design a very simple potential mimicking the balance between the volume of the chain segments and their mutual attractions in the solution. The simplest form of such potential is given below:

(5:94)

In the formula above rij is the distance between two beads of the chain, 1 is the lattice spacing, and « is a negative constant. With « ¼ 0 the model reduces to the model of a ‘‘real’’ chain in a good solvent, where mutual attractions of the chain segments could be ignored. Energy of the entire chain is a sum of the binary contributions SEij . With decreasing temperature (or with increasing strength of the long-range interactions «) the mean dimensions of the chain decrease (the solid curve in Fig. 5.7). The curves become steeper with increasing chain length; nevertheless the collapse transition remains continuous. The dashed horizontal line corresponds to the dimensions of an ideal chain of the same local geometry. The vertical dashed line denotes the collapse transition temperature. Slightly higher than ideal dimensions of the real chain at the transition midpoint are due to a bit higher prefactor – the scaling of the mean dimension with the chain length is at this point the same as for an ideal chain i.e., hS2 i n. At very low temperatures, the globular state is a dense droplet with hS2 i n2=3 . Obviously, at very high temperatures the chain behaves as the thermal ‘‘real’’ chain discussed in the previous section, i.e., hS2 i ng . Very interesting are the models where on top of the long range interactions a local stiffness of the model chain is superimposed. Let us assume that we are dealing now with a simple chain restricted to the diamond lattice (although any other lattice or off-lattice model can include the short-range interactions that simulate the polymer limited flexibility). Then let us assume that the trans conformation is favored energetically in respect to the two gauche conformations. At some critical ratio of the potential energy

<S 2>/n

A

8 < 1, for rij < 1 Eij ¼ «, for rij ¼ 1 : 0, for r > 1: ij

/ 79

T*= –kBT/e

FIGURE 5.7. Collapse transition of a flexible polymer chain (solid line) and a semiflexible chain (dashed line) of a limited length (see the text for an explanation).

80 / CHAPTER 5 of these two types of local geometries the behavior of a chain of a limited length changes dramatically. In the range of high temperatures with decreasing temperature the chain dimensions increase due to increasing effect of the stiffness. Relatively long expanded segments could be seen at this range. At a critical temperature, these ‘‘rods’’ of fluctuating length coalesce due to a huge decrease of the potential energy of the long-range interactions for a small entropic expense. The transition is abrupt, highly cooperative (the average length of the expanded sequences jumps up at the transition), and has all features of the first-order phase transition, including easily detected metastable region, an ‘‘almost’’ singularity of the heat capacity and an extremely low population of the intermediate states. At the transition midpoint the simulated molecules adopt essentially only two types of conformations; swollen random coils with a short sequences of expanded states and a densely packed, highly ordered globular state, with much longer sequences of the expanded local conformations. This behavior of the semiflexible model has a number of essential properties of globular proteins. First, the collapse transition is pseudo first-order (all-or-none in the language of protein biophysics). Second, it is cooperative and the collapse induces a sudden increase of the length of the regular expanded fragments, very much as the formation of secondary structure during the protein folding transition. Third, the collapsed structure is highly ordered with relatively well defined (however not unique) number of ‘‘secondary structure’’ expanded elements. Note, that these striking similarities are observed in the homopolymer model where all polymer units are the same. This leads to the conclusion that one of the most important general aspects of protein folding is a competition between the long-range and the short-range (stiffness) interactions. In this picture, differentiation of the interactions along the polypeptide chains (sequence of amino acids) plays a ‘‘fine-tuning’’ role, selecting the structural detail of the globular state. This analogy to protein folding extends even further. As the length of the semiflexible chain increases the ordering of the globular state becomes modular – domains are formed upon the collapse. Each domain can form at slightly different temperature, within the range of the metastable states shown in Fig. 5.7. When the number of domains becomes large the collapse transition becomes continuous, as it should be for any infinitely long flexible (or semiflexible) polymer chain. Such detailed insight into the collapse transition of semiflexible polymers could be gain only from computer simulations, although a very approximate theories for a single globule collapse of semiflexible polymers were published in past. A single polymer simulations could address also the issues of chain topology, including the effect of polymer branching and macrocycles on the thermodynamics of the collapse transition and the dynamics in a diluted media. This can be addressed on various levels of details, from a large scale conformational sampling within a framework of reduced models to a detailed molecular mechanics study of

local conformational transitions. For instance, a very interesting simulation of DNA collapse has been recently performed using the bead and string model with a short range bending potential and the Brownian Dynamics as a sampling technique. These simulations led to a very plausible and nontrivial picture of the DNA collapse pathway. It is also possible to employ a multiscale sampling, where the large scale relaxations are modeled on a low resolution level and the details are studied with the all-atom representation. Finally, it is worth to mention a very broad class of approaches to a specific problem of polymer collapse transition, the protein folding transition. This field attracts a lot of researches due to its importance for molecular biology, and biotechnology, genetics, and molecular medicine (including new drugs design in particular). In the case of the protein folding problem, the details of physics and the pathway description of the collapse transition are (at least by now) of a lesser importance. The mean goal is to predict the unique structure of protein globular state. The task is nontrivial, since the copolymers of interest are composed of twenty different mers (amino acids) and the sequence of these mers dictates a vast variety of three-dimensional globular structures, with a very specific local conformations and their well defined mutual packing in the globule. Two types of algorithms are now the most successful. The first one uses a large set of ‘‘prefabricated’’ protein fragments, extracted from a collection of known three-dimensional structures, and the sampling scheme are based on an iterative shuffling of these fragments within the simulated chain. Another approach is more in spirit of the classic polymer algorithms. It employs a local move schemes, however with a complex representation of the polypeptide conformational space and elaborated set of mean field potentials, derived either from the physical properties of the small molecules or from statistical analysis of the structural regularities seen in known structures of globular proteins. An amazing progress was achieved in this field during the last few years. The second approach is probably somewhat more general; it opens a possibility of a qualitative study of protein folding pathways and molecular mechanisms, not only the predictions of the globular structure. The predictive power of the both type of approaches are similar. Nevertheless, the second one seems to be a bit more open for a wider range of applications. These applications include the bootstrapped (resolution- and time-vise multiscale) implementations of the polypeptide representation and dynamics. Coupling of the various levels of resolution enables for a quite detailed study of protein dynamics and thermodynamics. The simulation techniques and models developed specifically for proteins are easily adaptable for more general applications in polymer computational physics. [43] 5.9.4 Simulations of Dense Polymeric Systems Dense polymeric systems include polymer solutions, polymer networks, polymer melts, polymer liquid crystals

THEORETICAL MODELS AND SIMULATIONS OF POLYMER CHAINS and solids, and many more. There is a vast body of literature on each of these subjects [42]. The modeling approaches are also of great variety, from a simple reduced models (lattice and continuous) to the detailed molecular mechanics and even a quantum mechanics. It is beyond scope of this chapter to go through the detail of various applications. Let us just outline some of problems that could be addressed in computer simulations, increasing our understanding of complex systems and providing important stimuli for theoretical studies and practical applications in material science and biotechnology. Typical dense polymer solutions and melts are globally disordered; however the level of local ordering could be relatively high. This is a very complex phenomenon that involves long-range correlations that are the results of specific local interactions. A general insight could be gain from the low resolution models that allow for study of the large scale conformational rearrangements; although specific details could be very sensitive to the atomic structure and require extensive molecular mechanic study of carefully selected starting conformations. The same could be said about the phase transitions in bulk polymers. The rate polymer diffusion in polymer media spans orders of magnitude. The mechanism of the process is unclear. It is very difficult to provide even a qualitative mechanistic picture how a long chain can move throughout a complex network of entanglements superimposed by the other macromolecules. The reptation theory of DeGennes [13] is probably only qualitatively true and only for very specific conditions. Simulations could be extremely helpful in at lest qualitative understanding of this process. Another challenging (however not really macromolecular) polymeric system are biological membranes. It is known from various experiments that the spectrum of relaxation processes in membranes is extremely wide; from local cooperative motion of phospholipide chain and occasional jumping of molecules from one side of a membrane to the other one to a global flexing of the membrane and formation of vesicles. Simulations are done on various levels of generalization. There are mesoscopic model which treat the membrane as a kind of elastic network, but also a very detailed all-atom study of membrane structure and local dynamics. Bootstrapped, multiscale simulations could be a very promising way to attack this problem. REFERENCES 1. W.L. Mattice and U.W. Suter, Conformational Theory of Chain Molecules, Wiley, New York, 1994. 2. M.F. Schulz and F.S. Bates, this volume, Chap. 32

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3. H. Yamakawa, Modern Theory of Polymer Solutions, Harper & Row, New York, 1971. 4. A.Y. Grosberg and A.R. Khokhlov, Statistical Physics of Macromolecules, AIP, New York, 1994. 5. M. Warner and E.M. Terentjev, Liquid Crystal Elastomers, Oxford University Press, Oxford, 2003. 6. J.E. Mark and B. Erman, Rubberlike Elasticity. A Molecular Primer, Wiley, New York, 1988. 7. B. Erman and J.E. Mark, Structures and Properties of Rubberlike Networks, Oxford University Press, Oxford, 1997. 8. A. Kloczkowski, M.A. Sharaf and J.E. Mark, Chem. Eng. Sci. 49, 2889 (1994). 9. M.A. Sharaf and J.E. Mark, Polymer, 45, 3943 (2004). 10. W. Zhao and J.E. Mark, this volume, Chap. IIB. 11. J.D. Honeycutt, this volume, Chap. IIID. 12. K.F. Fried, Renormalization Group Theory of Macromolecules, Wiley, New York, 1987. 13. P.G. DeGennes, Scaling Concepts in Polymer Physics, Cornell University Press, New York, 1979. 14. J. des Cloizeaux and C. Jannink, Polymers in Solutions: Their Modeling and Structure, Clarendon, Oxford, 1990. 15. W.C. Forsman, Ed., Polymers in Solution, Plenum, New York, 1986. 16. K. Binder, Ed., Monte Carlo Methods in Statistical Physics, SpringerVerlag, Berlin Heidelberg New York, 1986. 17. K. Binder, Monte Carlo and Molecular Dynamics Simulations in Polymer Sciences, Oxford University Press, Oxford, 1995. 18. A. Baumgaertner, Simulation of polymer motion, Ann. Rev. Phys. Chem. 35, 419 (1984). 19. Kolinski and J. Skolnick, Lattice Models of Protein Folding, Dynamics and Thermodynamics. R.G. Landes, Austin, TX, 1996. 20. M. Kotelyanskii and D.N. Therodorou, Ed., Simulation Methods for Polymers, Marcel Dekker, New York, 2004. 21. P.J. Flory,Statistical Mechanics of Chain Molecules, Interscience, New York, 1969. 22. J.D. Honeycutt, this volume, Chap. XX. 23. P.J. Flory, Proc. Roy. Soc. London, Ser. A 351, 351 (1974). 24. H.M. James and E. Guth, J. Chem. Phys. 15, 669 (1947). 25. A. Kloczkowski, J.E. Mark and B. Erman, Comput. Polym. Sci. 2, 8 (1992). 26. P.J. Flory, J. Chem. Phys. 66, 5720 (1977). 27. B. Erman and L. Monnerie, Macromolecules, 22, 3342 (1989), 25, 4456 (1992). 28. A. Kloczkowski, J.E. Mark and B. Erman, Macromolecules, 28, 5089 (1995). 29. R.T. Deam and S.F. Edwards, Phil. Trans. R. Soc, A, 280, 317 (1976). 30. R.C. Ball, M. Doi and S.F. Edwards, Polymer, 22, 1010 (1981). 31. T. Vilgis and B. Erman, Macromolecules, 26, 6657 (1993). 32. D.S. Pearson, Macromolecules, 10, 696 (1977). 33. P. Debye, J. Phys. Colloid. Chem. 51, 18 (1947). 34. R.H. Boyd and P.J. Philips, The Science of Polymer Molecules, Cambridge, New York, 1993. 35. P.R. Schleyer, Ed. Encyclopedia of Computational Chemistry, Wiley, New York, 1998. 36. W.F. van Gunsteren and P.K. Weiner, Computer Simulations of Biomolecular Systems. Theoretical and Experimental Applications. Escom, Leiden, 1989. 37. P.E. Rouse, J. Chem. Phys. 21, 1272 (1953). 38. M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon, Oxford, 1986. 39. P.H. Verdier and W.H. Stockmayer, J. Chem. Phys. 36, 227 (1962). 40. M. Rosenbluth and N. Rosenbluth, J. Chem. Phys. 23, 356 (1955). 41. A. Montesi, M. Pasquali and M.C. MacKintosh, Phys. Rev. E 69, 021916 (2004). 42. J.E. Mark, K. Ngai, W. Graessley, L. Mandelkern, E. Samulski, J. Koenig and G. Wignall, Physical Properties of Polymers, Cambridge University Press, Cambridge 2004. 43. A. Kolinski, Acta Biochim. Polonica 51, 349 (2004).

CHAPTER 6

Scaling, Exponents, and Fractal Dimensions Mohamed Daoud,* H. Eugene Stanley,y and Dietrich Staufferz y

* Laboratorie Le´on Brillouin (CEA-CNRS), CE Saclay; Gif-sur-Yvette, Cedex, France Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 z Institute of Theoretical Physics, Cologne University, D-50923 Ko¨ln, Euroland

6.1 6.2

Linear Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gelation for Branched Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 86 89 89

P(r, N) ¼ (3=2pN‘2 )3=2 exp {
3r 2 =2N‘2 }:

6.1 LINEAR POLYMERS

(6:1)

From the second moment we define the fractal dimension df of the walk by hr 2 idf =2  N. For any spatial dimension d, the second moment R20  hr 2 i of P(r, N) is

Textiles, much of living matter, plastics, and many other materials consist of linear or branched polymers. Each polymer usually is a carbon chain consisting of many monomers like -CH2 -. We emphasize here the modeling of such polymers and compare the theoretical results with experiments. First, we consider the conformation of a random linear chain, which is a model for a dilute solution of a polymer in a solvent [1–6]. Typical examples are polystyrene in benzene or polydimethylsiloxane in toluene or cyclohexane. We assume that the macromolecules are made of N statistical units which are randomly oriented with respect to each other. Because the actual monomers have to respect chemical bond angles, independent units can be regarded as made of several monomers. It is possible to define such independent units which will be used in all cases. This procedure was first presented by Kuhn, who defined the concept of local rigidity of a polymer [1]. Here, we consider the chains as completely flexible, and we do not distinguish between actual monomers and statistically independent units.

R20  N‘2 ,

(6:2a)

Thus the fractal dimension [6] is df ¼ 2

(6:2b)

for any d. It is important to stress that any definition of a characteristic length for the random walk leads to this result. What is a fractal and its dimension? A long spaghetti is one dimensional since its mass increases linearly with the length. A pizza has a mass proportional to the square of the radius, if its thickness is constant, and thus is two dimensional. A glass of red wine has a volume and mass proportional to the third power of the length, and is three dimensional. Thus an object with mass proportional to (radius)df has a dimension df . It is called a fractal with the fractal dimension df if df differs from the Euclidean dimension (usually 3) of the space into which the object is embedded. The apple-like Mandelbrot set is perhaps the most famous deterministic fractal, whereas the random walks of Eq. (6.2a) are random fractals with mass / N / (radius)2 . In deterministic fractals, small parts are mathematically similar to suitably chosen large parts; in random fractals this ‘‘self-similarity’’ (a large branch of a tree looks similar to a small twig on it) is often described but seldom defined in any precise way. For a polymer chain, it is possible to use the mean square end-to-end distance, as we did above. It is also possible to

6.1.1 The Random Walk The simplest model to describe the structure of a linear chain made of N units of length l each is the random walk. This is an ideal chain where no interactions are present between monomers. The distribution function P(r,N), which is the probability that a chain made of N steps starts at the origin and ends at point r, is a Gaussian. In threedimensional space, 83

84 / CHAPTER 6 define the average radius of gyration. One finds that these lengths are proportional to each other if both are long, and that the fractal dimension is 2 (for a discussion, see Chapter 1 in [7]). This is the reason for using the sign , which denotes asymptotic proportionality, and scaling laws are assumed to be valid only asymptotically. It is important that the precise way the length is defined will change the prefactor, but not the exponents. In this sense, we can say that there is only one characteristic length, and we will not be interested in the differences between the prefactors. The fractal dimension may be observed experimentally by light or neutron scattering [8]. The scattered intensity S(~ q) is the Fourier transform of the pair correlation function S(~ q) ¼

N X

hexp [i~ q  (~ ri
~ rj )]i,

(6:3)

i,j¼1

where the brackets h  i represent an average over all configurations, and ~ q is the momentum transfer in the scattering experiment: for a neutron with wavelength l elastically scattered with an angle u, we have q  j~ qj ¼

4p u sin : l 2

(6:4)

Because Eq. (6.1) is valid for any pair of units in a random walk, Eq. (6.3) may be calculated exactly. This was done by Debye [1] some years ago. He found S(q) ¼

2
x (e
1 þ X), X2

(6:5)

with X ¼ q2 R20 =3:

(6:6)

Here, R0 is the radius of gyration of the ideal chain. In the intermediate range, l
1  q  R
1 0 , where the fractal nature of the walk appears, relation (6.5) may be approximated by S(q)  q
2 :

(6:7)

This relation provides a convenient way to measure the fractal dimension of a single polymer, whenever the intermediate range may be reached experimentally. Neutron scattering is an excellent technique for this: The available wave vector range is particularly well suited for polymers; since the typical unit size is around 10A, and the radius of ˚ ngstro¨ms. Linear chains begyration is several hundred A have actually as random walks in two cases: in a melt, when no solvent is present, and in a theta solvent [9]. The latter is introduced in Section 6.1.2 when we discuss the actual interactions between monomers. 6.1.2

The Self-Avoiding Walk

Random walks are ideal chains in the sense that there is no interaction between monomers. For actual polymers,

there is an interaction between any two monomers. The interaction consist of an attractive part for large distances, goes through a minimum at intermediate distances, and becomes a repulsive core at short distances. Because of this ‘‘steric’’ constraint, two monomers cannot be in the same location. At high temperatures, the repulsive core is dominant, and the local minimum may be neglected completely. This is the excluded volume effect, and corresponds to what is called a good solvent [10,11]. There exists a critical temperature called the Flory theta temperature, where the excluded volume effect and the attractive part compensate each other. Such solutions are said to be in a theta solvent [12–14]. For still lower temperatures, the attractive part of the potential becomes dominant, and although two monomers are not allowed to be in the same location, they tend to be in the vicinity of each other. As a consequence, the chain tends to collapse on itself [15–17]. Solvents in which this happens are known as poor solvents. As mentioned above, at the theta temperature, because of the compensation between attractive and repulsive parts of the potential, the random walk model gives an adequate description of a chain in three-dimensional space [1–6]. Actually, there are still logarithmic corrections, but they may be neglected. In two dimensions, a chain at theta temperature is still not equivalent to a random walk [18]. In what follows, we will be concerned with solutions in a good solvent. It was realized by Edwards [10] that the exact shape of the potential is not important, and that it could be described by a parameter y(T), where T is the temperature, called the excluded volume parameter, defined as ð y(T) ¼ {1
eV(r)=kT }dr, (6:8) like the classical second virial coefficient, where V(r) is the effective monomer–monomer potential. This parameter is positive in a good solvent, vanishes at the theta temperature and becomes negative in a poor solvent. In the good solvent, steric interactions are dominant, as mentioned above, and the polymer is swollen compared to the ideal chain. This swelling corresponds to a change in the fractal dimension of the chain, which now becomes smaller than 2. The fractal dimension was calculated by various renormalization group techniques and by computer simulations [19,20]. Here, we describe the Flory approximation which, although being wrong [1–6], gives the fractal dimension within a very good accuracy for all dimensions. In this approximation one assumes that the free energy can be written as F=kT ¼

R2 N2 þ y : Rd R20

(6:9)

The first term is the elastic energy, in which one considers the chain as a spring with spring constant 1=R20 , where R0 is

SCALING, EXPONENTS, AND FRACTAL DIMENSIONS the ideal radius from Eq. (6.2). The radius R is the actual radius of the chain, to be determined. The second term is the interaction energy which can be estimated as follows. In a unit volume, the number of monomers is N=Rd , the number of pair interactions scales as (N=Rd )2 , and the interaction energy is therefore y(N=Rd )2 . Thus, the total interaction energy in the volume Rd scales as Rd y(N=Rd )2 , which is the second term in Eq. (6.9). Minimizing F with respect to R gives the fractal dimension df of a linear chain in the Flory approximation, N  Rd f , df ¼

dþ2 : 3

(6:10a) (6:10b)

This prediction df ¼ 5=3 in three dimensions is close to the actual value near 1.7; df ¼ 1 and 4/3 for d ¼ 1 and 2, respectively, is even exact. Note that we recover the ideal chain dimension for d ¼ 4. This is the upper critical dimension above which the excluded volume interaction becomes irrelevant, and the chain is ideal. For higher dimensions, the interaction with itself is negligible for the exponents, because space is sufficiently large that the polymer almost does not cross itself. Therefore, for d  dc chains with or without interactions are equivalent. Equation (6.10) was checked directly, using polymers with different masses. It was also tested using scattering experiments, by measuring the Fourier transform S(q) of the pair correlation function. As above Eq. (6.7), one can show that the scattered intensity is related to the wave vector q by the fractal dimension. In d ¼ 3 one finds, using Eq. (6.10), that 5

S(q)  q
3 (‘
1 4q4R
1 0 ):

(6:11)

Relations (6.10) and (6.11) were tested experimentally by small angle neutron scattering. Let us mention that starshaped polymers are in this same universality class: the mass dependence of their radius of gyration also follows relations (6.10). However, it also depends on the number f of branches, indicating the special geometry of the object. For more details, the reader is referred to Refs. [21–25]. 6.1.3 Dilute Solutions So far, we have considered only a single polymer chain. Actual solutions contain many chains! We expect the above results to hold as long as the various polymers are far from each other. This is the case for dilute solutions, where we expect the concentration effects to be only perturbations to the various laws that we found. Let C be the monomer concentration. It is common to define the overlap concentration C where the distance between centers of masses of the chains is of the order of the radius of the macromolecules. Assuming the polymers

/ 85

are randomly distributed, the average distance between their centers of masses is d  (C=N)
1=d :

(6:12)

Equating Eq. (6.12) to the radius of gyration, and using Eq. (6.10), we get C  N 1
d=df  N
4=5

(d ¼ 3):

(6:13)

Relation (6.13) exhibits the fractal character of the chains; because they are fractals, their volume grows faster than their mass. Therefore, the overlap concentration decreases as the polymers become larger. As mentioned above, we expect two concentration regimes, with C=C smaller or larger than unity. Therefore, we do not expect N and C to act as independent variables for all the properties, but to appear only through the ratio C=C . This scaling behavior occurs in many properties, but we will consider here only the scaling behavior of the radius of gyration R and of the osmotic pressure p. In both cases, one may write a scaling relation deduced from the definition of the fractal dimension, Eq. (6.10): R(N,C)  N 1=df f (C=C ),

(6:14a)

and p(N,C) ¼

C g(C=C ): N

(6:14b)

Here the prefactor Cp  C=N in Eq. (6.14b) is merely the pressure of an ideal gas that is obtained for very low concentrations when the chains are very far from each other. The unknown functions f(x) and g(x) may be expanded for small x in the dilute regime but have singular behavior for large x in the semidilute regime. Therefore, in the dilute concentration regime, one expects corrections both for the radius and the osmotic pressure. In the latter case, we may write p(N,C) ¼

C {1 þ aC=C þ b(C=C )2 þ . . . }, N

(6:15a)

where a, b, . . . are constants. This may be identified with a virial expansion, p(N,C) ¼

C þ A2 C2 þ . . . N

(6:15b)

Comparing Eqs. (6.15a) and (6.15b) and using Eqs. (6.13) and (6.10) leads to the following expression for the second virial coefficient: A2  (NC )
1  N 3=df :

(6:16)

A similar expansion can be obtained for scattering intensity S(q,C) / 1=(1 þ q2 R2 þ . . . ). For very low q, in the Guinier regime qR51, this expansion is the basis for the so-called Zimm plots that are commonly used to determine the radius of a chain and the second virial coefficient of a solution.

86 / CHAPTER 6 6.1.4 Semidilute Solutions When the concentration C is increased above the overlap concentration C , one reaches a different regime where the macromolecules interpenetrate each other, and we expect the concentration effects to become dramatic. In dilute solutions, the concentration effects are represented by corrections to the power laws. Because the chains are fractals, the volume they occupy grows much faster than their mass. As indicated by Eq. (6.13), the larger the macromolecule, the smaller is C . For a typical polymer of 105 units, C is of the order of 10
2 g=cm3 . For infinite chains, the overlap concentration vanishes, and one is left only with the semidilute range. In this range, because the chains are flexible, they overlap each other, and we expect the simple laws we discussed above to break down. Still, the scaling laws (6.14) are valid, but one has to look for other limits. The basic idea to understand the behavior of a polymer in this regime was given in the limit of a melt by Flory [1] and was later generalized to semidilute solutions by Edwards [26]. In a melt, the average interaction should cancel, since each monomer is surrounded by other monomers, and therefore the polymer should behave as an ideal chain. We will see below that although this argument is valid for linear chains, it turns out to be wrong for branched polymers. The reason for this is related to the interpenetration of the various chains. For linear chains, this interpenetration effect was studied by Edwards [10], who introduced the concept of a screening length j. The idea here is that if we consider two monomers on a given chain, their total interaction is the sum of the direct excluded volume interaction and all the contributions coming from indirect interactions between them via other monomers belonging to other chains. This is equivalent to Debye–Hu¨ckel screening in an electrolyte solution. Because of this, the notion of ‘‘blob’’ was introduced. It corresponds to a part of the chain, made of g units, with radius j. Thus one may consider a polymer in a semidilute solution as an ideal chain if the blob is chosen as a statistical unit. Inside the blob, excluded volume interactions are still present. Note that at the overlap concentration, the blob is identical to the whole chain. Using these ideas, it was shown that j  C
3=4

(6:17)

and R


1=2 N j  N 1=2 C
1=8 : g

(6:18)

Note that as concentration increases, the sizes of the blob and of the chain decrease. In the bulk, we recover Flory’s results: the interaction completely screened, the size of the blob is the step length, and the chain is ideal. Thus the present model ensures a gradual cross-over from

the swollen to the ideal behaviors for increasing concentrations. The next quantity we will consider is the osmotic pressure [27]. We may use the same arguments as above to determine its dependence on C, starting with relation (6.14b). In the semidilute regime, we do not expect the expansion (6.15) to be valid, since the variable x ¼ C=C is larger than unity. Instead, we assume that g(x) behaves as a power law. Its exponent is determined by the following condition. In this concentration range, we expect the osmotic pressure to be given by the density of contacts between polymers. This is again a collective property of the solution that should depend only on concentration and not on the mass of the individual chains. Using this condition, we find p  C
d=(df
d)  C9=4 ,

(6:19)

a relation that was found first by des Cloizeaux [28]. Equation (6.19) was tested experimentally by Noda et al. [29]. Many points are remarkable in Eq. (6.19). The first is that these results differ strongly from what one would expect in a mean field approach. The second, and most remarkable result is that the fractal dimension controls the thermodynamic properties of the solution. This is extremely interesting because the fractal dimension was introduced to describe the properties of a single chain, where only small concentrations and distances in the order of several hun˚ ngstro¨ms were considered. We are now discussdreds of A ing thermodynamic, macroscopic, properties of a solution that is semidilute, and where the polymers strongly interact. Thus what was introduced to describe a local property of a single chain controls a solution that may be rather concentrated: even a 20% solution may be in this concentration range. 6.2 GELATION FOR BRANCHED POLYMERS So far we have considered polymers made of bifunctional units. These may react by two ends, or functionalities. When the monomers are more than bifunctional, polymerization leads to branched structures, and eventually to a solid called a gel [48]. In this section we will consider this case. As we will see, every polymer has still a fractal behavior. In addition to this, there is a very broad distribution of molecular weights, called polydispersity. Because of this, what is observed is an effective dimension that depends also on the dimension of the distribution. This holds for many polydisperse systems, with restrictions that will be discussed below. We will first present the distribution of molecular weights that is naturally found in the reaction bath. We will turn to dilute solutions, where the fractal dimension is smaller because of swelling. We will discuss the effective dimension that is observable. Then we will turn to the semidilute solutions and to the swollen gels. Finally, we will discuss the dynamics of these systems in the reaction bath.

SCALING, EXPONENTS, AND FRACTAL DIMENSIONS

Using relations (6.22) and (6.23), we find the fractal dimension df for percolation clusters,

6.2.1 The Sol–Gel Transition Let us consider a vessel with multifunctional monomers. Each monomer may react by one or more of its f functional groups. As time proceeds, there is a formation of dimers, trimers, . . . , polymers; this is the sol. This process makes the solution more and more viscous, because of the presence of large macromolecules. The viscosity diverges, and this defines a threshold time tc . For t > tc , in addition to the sol, there is an infinite molecule, the gel. Thus, there appears an elastic modulus due to the presence of a solid-like phase. Although there are probably other universality classes, this transition was successfully modeled by bond percolation [6]. Generally, bond percolation on a lattice has each bond (line connecting two neighboring lattice sites) present randomly with probability p and absent with probability 1– p. Clusters are groups of sites connected by present bonds. For p > pc an infinite cluster is formed. Percolation theory (in a Bethe lattice approximation) was invented by Flory (1941) to describe gelation for three-functional polymers. Sites are the unreacted monomers, bonds are the reacted functionalities. Clusters are the polymers, and the infinite cluster is the gel. We recall very briefly some results of percolation. The main result concerns the distribution of cluster sizes. This corresponds to what we called polydispersity. The distribution is very broad. If we call p the probability that a bond is reacted, pc is its value at the gelation threshold, the average probability P(N, «) that a randomly selected monomer belongs to a large polymer with N monomers each at a small ‘‘distance’’ «  p
pc from the threshold is P(N,«)  N 1
t f («N s ),

and (6:22)

Higher order moments defined the same way as above are proportional to Nz . The number of polymers with N units each is proportional to P(N,«)=N The exponent g is the susceptibility exponent in percolation. Similarly, one may also define a characteristic length, corresponding to the size of the typical polymers in the sol, dominating in diverging moments like Nz . This length diverges as j  «
n :

1=df ¼ sn:

(6:23)

(6:24)

Let us stress that this is the fractal dimension of the polymers in the reaction bath. We assume that all polymers that constitute the sol have this same fractal dimension. This was calculated by renormalization group techniques and computer simulations [31,32,33,34]. We will give a simple Flory derivation [35] that is close to the former results for all space dimensions. The polydispersity exponent t can be shown to be related to the fractal dimension, t ¼ 1 þ d=df :

(6:25)

This hyperscaling relation is valid for space dimensions d 6. Equation (6.25) implies that the distribution is very special; if one considers polymers with a given mass, they are in a C situation, i.e., they are in a space-filling configuration. Since they are fractals, however, voids are left in the structure. These voids are filled by polymers with smaller masses, with the same requirement for every mass: each one is in a C situation. Therefore, if one looks at the distribution, for any size considered, one always observes polymers at C*. In this sense, the distribution is fractal [36,37]. Note that it is possible to relate the ‘‘masses’’ Nz and Nw (in units of the monomer mass) defined above by eliminating «, Nz  Nwdf =(2df
d : (6:26a) Using relation (6.25), we get Nz  Nw1=(3
t) :

(6:20)

where t and s are percolation exponents [30] to be discussed below. The moments of the distribution have several interesting properties. The first moment is normalized below pc . The higher moments diverge with different exponents R NP(N,«)dN Nw  R (6:21)  «
g P(N,«)dN R 2 N P(N,«)dN  «
1=s : Nz  R NP(N,«)dN

/ 87

(6:26b)

Note that both relations (6.26a) and (6.26b) hold only if df < d, or equivalently if t > 2. If df ¼ d, or t ¼ 2, both masses become proportional to each other, and in our definitions, there is only one mass present in the problem. This will prove to be important in the discussion for the scattered intensity for dilute solutions below. 6.2.2

The Flory Approximation

Let us consider the large polymers, with mass Nz and radius j in the distribution. In the Flory approximation, one writes down a free energy made of two contributions F¼

j2 v Nz2 þ : j20 Nw jd

(6:27)

The first one is an entropic term where we assume that the polymer behaves like a spring with constant j20 , where j0 is the radius of an ideal chain when no interactions are present. The second term is the interaction energy in which y is the excluded volume interaction, discussed for linear chains. Except for the presence of Nw , this is very similar to what we considered for chains. The presence of this factor is due

88 / CHAPTER 6 to the fact that the large polymers are penetrated by the small ones. Because of this, there is a screening of the interactions, as in the semidilute case for linear chains. The precise form for the energy was evaluated by Edwards [26] and de Gennes [38] in a Debye–Huckel approximation. The ideal chain radius j0 was calculated on a Cayley tree [39] and was shown to be Nz 

j40 :

(6:28)

In the Flory approximation, all quantities, except the radius which is to be calculated, are assumed to have a mean-field behavior. Therefore there is a relation between Nz and Nw , Nz  Nw2 :

(6:29)

Minimizing the free energy with respect to j and using relations (6.23) and (6.24), we get the fractal dimension df of the large percolation clusters, 1 df ¼ (d þ 2): 2

(6:30)

This was tested indirectly by measurements of the mass distribution and the exponent t (6.25). 6.2.3

Dilute Solutions

Once the distribution of polymer sizes is known, it is possible to dilute the sol, and to consider dilute solutions. Let us stress that the growth of the polymers is quenched before dilution and that the distribution function is given. Because of the excluded volume interactions the polymers swell and their fractal dimension changes from df to dfa . The new fractal dimension dfa may be obtained within a Flory approximation by considering a free energy similar to that in Eq. (6.27). The difference between a dilute solution and the reaction bath which was considered above is in the interaction term. We expect that the excluded volume interactions are present in the dilute case whereas they are fully screened in the previous case [40]. Therefore, this contribution has the same form as in relation (6.9) for linear chains. It is straightforward to minimize the free energy with respect to the radius, which yields, 2 dfa ¼ (d þ 2): 5

(6:31)

The observation of this fractal dimension, however, is not easy, as we discuss now. Any experiment provides the average of the observed quantity over the whole distribution of masses. This averaging procedure leads to an effective dimension [41–44] that is different from the actual one. In order to see this, let us consider the scattered intensity. For a single mass, we have S1 (q, N) ¼ Ng[qR(N)]:

(6:32a)

where the function g(x) behaves as a power law in the fractal range, qR > 1 and qa 1, in such a way that the mass

dependence disappears. For a distribution of masses, and in the dilute regime where one may neglect correlations between monomers belonging to different polymers, the total scattered intensity [45] is X P(N,«)S1 (q, N): (6:32b) Stotal (q) ¼ Using relations (6.32), (6.20), and (6.21), we get Stotal (q) ¼ CNw g(qRz ),

(6:32c)

where C is the monomer concentration and Rz is the radius of the largest polymers, Ð NR2 (N)P(N,«)dN : (6:33) Rz  Ð NP(N,«)dN The radius Rz can be related to the largest masses Nz , Eq. (6.22), through the fractal dimension da

Rz  Nz f  Nw5=8

(d ¼ 3):

(6:34)

This relation was tested by light scattering measurements and found to be in good agreement with experimental results. In the intermediate scattering range, l
1 4q4R
1 z , the function g(x) in Eq. (6.32) behaves as a power law. The exponent of Stotal (q) is determined by the requirement that we are now in the fractal regime where no explicit mass dependence should appear. Using relations of Eq. (6.26), we get a

Stotal (q)  q
df (3
t)  q
8=5 (l
1 4q4R
1 z ) (d ¼ 3):

(6:35)

Therefore, an effective fractal dimension appears, that describes the behavior of the polydisperse system. As can be seen from Eq. (6.32), this effective dimension is related to the actual one, but also to the exponent t characterizing the distribution of masses. Note that this holds for percolation and for other distributions, as long as t > 2, as discussed above. The polydispersity effect disappears when t ¼ 2. In this sense, we will say that such systems are not polydisperse. This has an important implication. Measuring an exponent in a scattering experiment does not necessarily imply that one gets the fractal dimension directly. First one has to check the polydispersity by independent measurements, either with viscosity, or with second virial coefficient experiments. The latter may be calculated following the same steps as above and taking into account the interactions between the centers of masses of different polymers. Finally, we define two more exponents related to the viscosity h and the elastic modulus G h  «
k G  «f

(6:36)

for which the theory is less clear [46–51]. Note that for crosslinks between very long linear chains one expects the Bethe lattice or Cayley tree exponents to be valid except in an unmeasurably small interval at the transition. We end by

SCALING, EXPONENTS, AND FRACTAL DIMENSIONS noting that recently, very similar ideas were successfully applied to the diffusion of cancer cells in tissues modelled by a gel [52,53]. ACKNOWLEDGMENTS The authors are much indebted to M. Adam, E. Bouchaud, W. Burchard, R. Colby, M. Delsanti, D. Durand, B. Farnoux, P.G. de Gennes, O. Guiselin, G. Jannink, L.T. Lee, L. Leibler, J. E. Martin, and M. Rubinstein for many discussions. REFERENCES 1. P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953). 2. P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979). 3. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Science Publications, London, 1986). 4. J. des Cloizeaux and G. Jannink, Les polymers en solutions (ed. de Physique, Paris, 1987) [Engl. Transl: Polymers in Solution, Oxford University Press, 1990]. 5. A. Yu. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules, translated by Y. A. Atanov (AIP Press, New York, 1994). 6. D. Stauffer and H. E. Stanley, From Newton to Mandelbrot: A Primer in Theoretical Physics (Springer-Verlag, Heidelberg & New York, 1990). 7. A. Bunde and S. Havlin, eds., Fractals and Disordered Systems (Springer, Berlin, 1991). 8. J. Teixeira, in On Growth and Form, edited by H. E. Stanley and N. Ostrowsky (Martinus, Nijhoff, 1985). 9. J. P. Cotton, D. Decker, H. Benoit, B. Farnoux, J. Higgins, G. Jannink, R. Ober, C. Picot, and J. des Cloizeaux, Macromolecules 7, 863 (1974). 10. S. F. Edwards, Proc. Phys. Soc. 65, 613 (1965). 11. P. G. de Gennes, Phys. Lett. 38A, 339 (1972). 12. M. J. Stephen, Phys Lett. 53A, 363 (1975). 13. M. Daoud and G. Jannink, J. Phys. Lett. 37, 973 (1976). 14. M. Moore, J. Phys. A 10, 305 (1977). 15. I. M. Lifshitz, A. Grosberg, and A. Khoklov, Rev. Mod. Phys. 50, 685 (1978). 16. M. Nierlich, J. P. Cotton, and B. Farnoux, J. Chem. Phys. 69, 1379 (1978). 17. C. Williams, F. Brochard, and H. L. Frisch, Ann. Rev. Phys. Chem. 32, 433 (1981). 18. B. Duplantier, Phys. Rev. Lett. 59, 539 (1987). 19. B. Derrida, J. Phys. A 14, L5 (1981). 20. H. J. Hilhorst, Phys. Rev. B 16, 1253 (1977).

/ 89

21. M. Daoud and J. P. Cotton, J. Phys. 43, 531 (1982). 22. T. Birstein and E. Zhulina, Polymer 25, 1453 (1984). 23. G. S. Grest, K. Kremer, and T. A. Witten, Macromolecules 20, 1376 (1987). 24. J. Roovers, N. Hadjichristidis, and L. J. Fetters, Macromolecules 16, 214 (1983). 25. A. Halperin, M. Tirrell, and T. P. Lodge, Adv. Polym. Sci. 100, 31 (1991). 26. S. F. Edwards, Proc. Phys. Soc. 88, 265 (1966). 27. M. Daoud, J. P. Cotton, B. Farnoux, G. Jannink, G. Sarma, H. Benoit, R. Duplessix, C. Picot, and P. G. de Gennes, Macromolecules 8, 804 (1975). 28. J. des Cloizeaux, J. Phys. 36, 281 (1975). 29. I. Noda, N. Kato, T. Kitano, and M. Nagasawa, Macromolecules 14, 668 (1981). 30. H. Nakanishi and H. E. Stanley, Phys. Rev. B 22, 2466 (1980). 31. T. C. Lubensky and J. Isaacson, Phys. Rev. Lett. 41, 829 (1978); Phys. Rev. A 20, 2130 (1979). 32. G. Parisi and N. Sourlas, Phys. Rev. Lett. 46, 891 (1981). 33. P. J. Reynolds, W. Klein, and H. E. Stanley, J. Phys. C 10, L167 (1977). 34. B. Derrida and J. Vannimenus, J. Phys. Lett. 41, 473 (1980). 35. J. Isaacson and T. C. Lubensky, J. Phys. 42, 175 (1981). 36. M. E. Cates, J. de Phys. Lett. 38, 2957 (1985). 37. M. Daoud and J. E. Martin, in The Fractal Approach to Heterogeneous Chemistry, edited by D. Avnir (John Wiley, New York, 1990). 38. P. G. de Gennes, J. Polym. Sci. Polym., Symp. 61, 313 (1977). 39. B. H. Zimm and W. H. Stockmayer, J. Chem. Phys. 17, 1301 (1949). 40. M. Daoud and J. F. Joanny, J. de Phys. 42, 1359 (1981). 41. M. Daoud, F. Family, and G. Jannink, J. de Phys. Lett. 45, 119 (1984). 42. S. J. Candau, M. Ankrim, J. P. Munch, P. Rempp, G. Hild, and R. Osaka, in Physical Optics of Dynamical Phenomena in Macromolecular Systems (W. De Gruyter, Berlin, 1985), p. 145. 43. F. Schosseler and L. Leibler, J. Phys. Lett. 45, 501 (1984). 44. F. Schosseler and L. Leibler, Macromolecules 18, 398 (1985). 45. J. E. Martin and B. J. Ackerson, Phys. Rev. A 31, 1180 (1985). 46. L. de Arcangelis, Comput. Sci. Eng. 5, 78 (2004). 47. B. Ratajska-Gadomska and W. Gadomski, J. Phys. Cond. Matter 16, 9191 (2004). 48. J. P. Cohen, Physical Properties of Polymeric Gels (Addad, Wiley and Sons, 1996). 49. F. Prochazka, T. Nicolai and D. Durand, Macromolecules 29, 2260 (1996). 50. T. Nicolai, H. Randrianantoandro, F. Prochazka and D. Durand, Macromolecules 30, 5897 (1997). 51. D. Durand, and T. Nicolai, in Encyclopaedia of Materials Science and Technology, edited by K. H. J. Bushow, R. W. Cahn, M. C. Fleming, B. Ilschner, E. J. Kramer and S. Mahajan (Elsevier, 6, 6116, 2001). 52. G. C. Fadda, D. Lairez, B. Arrio, J. P. Carton and V. Larreta-Garde, Biophys. 85, 2808 (2003). 53. T. Abete, A. de Candia, D. Lairez and A. Coniglio, Phys. Rev. Lett. 93, 228302 (2004).

CHAPTER 7

Densities, Coefficients of Thermal Expansion, and Compressibilities of Amorphous Polymers Robert A. Orwoll Department of Chemistry, College of William and Mary, Williamsburg, VA 23187-8795

7.1 7.2

Densities as a Function of Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Densities as a Function of Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r(t) ¼ r(0)e
a0 t :

Tables in this chapter contain published pressure-volumetemperature data for amorphous homopolymers. Measurements below the melting temperatures for semi-crystalline materials are not included because of the potentially large variance among samples with differing degrees of crystallinity. Rogers [1] and Zoller [2] have also compiled equation-of-state data for amorphous polymers.

(7:2)

This latter form is for measurements in which the thermal expansion coefficient a¼

    1 @vsp 1 @r ¼
, vsp @t P r @t P

(7:3)

was found to be independent of temperature; i.e., a ¼ a0 . Additional density data can be found in Chapter 29. Tables 7.3 and 7.4 are compilations of densities and thermal expansion coefficients at one atmosphere tabulated at 20 8C intervals. Measurements below Tg are indicated by a g preceding the value. Some of the entries in these tables are taken directly from the published reports; consequently, there are occasional small variations between these values and those computed from the smoothing expressions in Tables 7.1 and 7.2. Densities judged to be more accurate than +0:001g=cm3 are recorded in Table 7.3 with an additional digit. Thermal expansion coefficients can also be determined from the power series expressions in Tables 7.1 and 7.2 using

7.1 DENSITIES AS A FUNCTION OF TEMPERATURE The temperature dependence of polymer densities at atmospheric pressure is given in Tables 7.1 and 7.2. Table 7.1 gives densities measured above the glass transition temperature Tg and, for semi-crystalline polymers, above the melting temperature. Table 7.2 lists densities of amorphous polymers below Tg . Volumetric data are presented here in terms of the density r rather than the more commonly reported specific volume nsp ¼ 1=r because, for most systems, r is a more linear function of temperature than is nsp . For most entries, the densities have been fitted to a power series r(t) ¼ a0 þ a1 t þ a2 t2 þ . . . ,

93 95 101

(7:1) a¼


with t representing the centigrade temperature. In the other cases, the polymer density is written as an exponential function of the centigrade temperature t

a1 þ 2a2 t þ . . . , a0 þ a1 t þ a2 t2 þ . . .

which follows from Eqs. (7.1) and (7.3).

93

(7:4)

94 / CHAPTER 7 TABLE 7.1. Densities, measured above Tg , as a function of temperature. Polymer Natural Rubber, unvulcanized Natural Rubber, cured Polyamide, Nylon 6 Polyamide, Nylon 6,6 Poly (butene-1), isotactic Poly(n-butyl methacrylate) Poly(«-caprolactone) Polycarbonate, (with Bisphenol A) Poly(cyclohexyl methacrylate) Poly(2,6-dimethylphenylene ether) Poly(dimethyl siloxane)

Poly(1,3-dioxepane) Poly(1,3-dioxolane) Polyetheretherketone Polyethylene, branched Polyethylene, linear Poly(ethylene terephthalate) Poly(ethyl methacrylate) Polyisobutylene Poly(methyl methacrylate)

Poly(methyl methacrylate), isotactic Poly(4-methyl pentene-1) Poly(o-methyl styrene) Polyoxybutylene Polyoxyethylene

Polyoxymethylene Polypropylene, atactic Polypropylene, isotactic

Polystyrene

Polysulfone, (with Bisphenol A) Polytetrafluoroethylene Polytetrahydrofuran Poly(vinyl acetate) Poly(vinyl chloride) Poly(vinylidene fluoride) Poly(vinyl methyl ether)

r, g=cm3 , (t in 8C)

Temp. range, 8C

Ref.

0:9283
6:10  10
4 t 0:9210
5:86  10
4 t 1:316 exp (
4:70  10
4 t) 1:306 exp (
6:60  10
4 t) 1:145
6:47  10
4 t 0:876 exp (
6:75  10
4 t) 1:0695
5:82  10
4 t
0:98  10
6 t 2 þ 0:241  10
8 t 3 1:070
6:95  10
4 t þ 0:40  10
6 t 2 1:110
7:81  10
4 t þ 0:519  10
6 t 2 1:254
6:35  10
4 t
0:116  10
6 t 2 1:2739 exp (
6:21  10
4 t) 1:1394
5:90  10
4 t
0:163  10
6 t 2 1:168
6:95  10
4 t
0:070  10
6 t 2 0:9919
8:925  10
4 t þ 0:265  10
6 t 2
0:0030  10
8 t 3 0:994
9:76  10
4 t þ 0:904  10
6 t 2 0:990
8:59  10
4 t þ 0:23  10
6 t 2 1:064
3:5  10
4 t 1:254
10:9  10
4 t 1:397 exp (
6:69  10
4 t) 0:868 exp (
6:73  10
4 t) 0:882
7:97  10
4 t þ 0:74  10
6 t 2 0:8674
6:313  10
4 t þ 0:367  10
6 t 2
0:055  10
8 t 3 0:863
4:73  10
4 t
0:38  10
6 t 2 1:390
7:82  10
4 t 1:156
6:59  10
4 t 0:9297
5:123  10
4 t þ 0:0615  10
6 t 2 1:223
5:29  10
4 t
0:507  10
6 t 2 1:2135
4:64  10
4 t
0:648  10
6 t 2 1:228 exp (
5:23  10
4 t) 1:211
5:96  10
4 t 1:229
7:12  10
4 t 1:252
8:40  10
4 t þ 0:56  10
6 t 2 0:843
5:11  10
4 t 1:064
5:58  10
4 t þ 0:125  10
6 t 2 0:985
6:82  10
4 t 1:142 exp (
7:09  10
4 t) 1:140
8:08  10
4 t 1:139
7:31  10
4 t 1:336 exp (
6:77  10
4 t) 0:848
0:19  10
4 t
3:05  10
6 t 2 0:859 exp (
6:60  10
4 t) 0:862 exp (
6:70  10
4 t) 0:8414
3:79  10
4 t
0:316  10
6 t 2 1:0865
6:19  10
4 t þ 0:136  10
6 t 2 1:077
5:49  10
4 t þ 0:124  10
6 t 2 1:067
5:02  10
4 t
0:135  10
6 t 2 1:314
6:49  10
4 t
0:018  10
6 t 2 2:340
24:33  10
4 t þ 0:309  10
6 t 2 0:996 exp (
6:691  10
4 t) 1:2124
8:62  10
4 t þ 0:223  10
6 t 2 1:394
2:03  10
4 t
2:19  10
6 t 2 1:816
26:77  10
4 t þ 4:75  10
6 t 2 1:0725
7:259  10
4 t þ 0:116  10
6 t 2

0–25 0–25 236–295 245–297 270–285 133–246 34–200 20–120 101–148 151–340 171–330 110–199 203–320 20–207 25–70 16–145 6–16 25–80 338–400 112–225 135–198 130–207 142–200 274–342 65–95 0–150 120–270 114–159 115–230 105–150 110–194 55–190 240–319 139–198 30–90 88–224 30–90 75–136 183–220 80–120 175–230 170–300 180–300 100–222 115–196 79–320 185–371 330–372 62–166 35–100 100–150 180–220 25–120

[3] [3] [4] [4] [5] [6] [7] [8] [1] [9] [38] [7] [10] [11] [12] [49] [43] [44] [13] [14] [7] [15] [7] [16] [8] [17] [18] [7] [4] [8] [48] [19] [45] [20] [42] [1,21] [42] [47] [22] [1] [4] [6] [38] [23] [20] [10] [24] [25] [1,21] [26] [1] [40] [27]

DENSITIES, THERMAL EXPANSIONS, AND COMPRESSIBILITIES

/ 95

TABLE 7.2. Densities of polymer glasses as a function of temperature. r, g=cm3 , (t in 8C)

Tg , 8C

Glassy polymer Poly(n-butyl methacrylate) Polycarbonate, (with Bisphenol A)

20 151 142

Temp. range, 8C

1:063
4:01  10
4 t


30–20


4

1:204
3:05  10 t
4

1:2044 exp (
2:21  10 t)
4

Ref. [8]

30–151

[9]

40–121

[38]

Poly(cyclohexyl methacrylate)

107

1:106
2:69  10 t

19–85

[7]

Poly(2,6-dimethylphenylene ether)

203

1:070 exp (
2:09  10
4 t)

30–203

[10]

Poly(ethyl methacrylate)

65

Poly(methyl methacrylate)

Poly(methyl methacrylate), isotactic Poly(o-methyl styrene)

100

Polysulfone, (with Bisphenol A) Poly(vinyl acetate)

1:131
3:06  10 t

[18]


6 2

17–80

[7]


30–105

[8]

1:175
2:47  10
4 t

105

1:187
2:35  10
4 t
0:67  10
6 t 2

20–95

[48]


4

9–26

[19]


4

29–82

[20]

8–75

[20]

30–79

[10]

30–186

[24]

1:225
2:65  10 t 1:027
2:64  10 t

92

1:048
1:88  10
4 t
0:608  10
6 t 2

79

1:052 exp (
2:86  10
4 t)

186


4

1:242
2:59  10 t
4

1:196
3:37  10 t

7.2 DENSITIES AS A FUNCTION OF PRESSURE Pressure-volume data have been reported for many polymers. In most cases the Tait equation r(0,t)  , P 1
C ln 1 þ B(t)

(7:5)

accurately represents the relationship between r and the pressure P. In it, the unitless parameter C is usually equated to 0.0894, while the temperature-dependent parameter B(t) is written as a function of two other empirical parameters b0 and b1 : B(t) ¼ b0 e
b1 t :

30–100


4

1:188
1:34  10 t
0:91  10 t

105

47

(7:6)

As above, t is the centigrade temperature. Literature values for C, b0 , and b1 are collected in Tables 7.5 (above Tg ) and 7.6 (below Tg ). The zero-pressure density r(0,t), which is virtually the same as the density at atmospheric pressure, can be computed from expressions in Tables 7.1 and 7.2. Table 7.7 lists equations for the temperature dependence of thermal pressure coefficients g ¼ (@p=@t)y that have been directly determined.

[8]


6 2

1:188
1:72  10 t
1:04  10 t

31

r(P,t) ¼


35–65


4

105

131

Polystyrene


4


30–20

[26]

Isothermal compressibilities     1 @vsp 1 @r ¼ , kT ¼
vsp @P t r @P t

(7:7)

at atmospheric pressure are summarized in Table 7.8. Most are derived from the Tait equation with parameters from Tables 7.5 and 7.6:

kT ¼

C b1 t e (at zero pressure): b0

(7:8)

Others are obtained from thermal pressure and expansion coefficients through the relation kT ¼

a : g

(7:9)

In most instances the latter values of kT have smaller uncertainties because of the greater accuracy that usually accompanies direct measurements of g and a. Values of kT for glasses are preceded with a g in Table 7.8. Related information can be found in Chapter 29.

g ¼ glass.

a

Polysulfone, (with Bisphenol A) Polytetrafluoroethylene Poly(vinyl acetate) Poly(vinyl chloride) Poly(vinylidene fluoride) Poly(vinyl methyl ether)

Polystyrene

Polypropylene, isotactic

Polyoxymethylene Polypropylene, atactic

Polyoxyethylene

Poly(methyl methacrylate), isotactic Poly(4-methyl pentene-1) Poly(o-methyl styrene) Polyoxybutylene

Poly(methyl methacrylate)

Poly(ethylene terephthalate) Poly(ethyl methacrylate) Poly(hydroxybutyrate) Polyisobutylene

Polyethylene, linear

Poly(1,3-dioxolane) Polyetheretherketone Polyethylene, branched

Poly(cyclohexyl methacrylate) Poly (2,6-dimethylphenylene ether) Poly(dimethyl siloxane)

Poly(«-caprolactone) Polycarbonate, (with Bisphenol A)

Poly(butene-1), isotactic Poly(n-butyl methacrylate)

Natural rubber, unvulcanized Natural rubber, cured Polyamide, Nylon 6 Polyamide, Nylon 6,6

Temperature (deg C)

g1.196

g1.175

0.9297

g1.131

g1.063a

0.9283 0.9211

0

TABLE 7.3. Densities (g=cm 3 ).

1.0580

g1.189

g1.044

g1.170 g1.182 g1.220

g1.184

g1.125 1.177 0.9195 0.912

0.974 0.973

0.9742

g1.101

1.057

0.9162 0.9093

20

1.0436

1.1783

g1.040 g1.040 g1.232

1.0294

1.1615

g1.035 g1.034 g1.226

1.0152

1.1449

1.026 g1.221

0.827

1.081

g1.006 0.931 0.944 1.086 1.075

g1.011 0.944

g1.016 0.958

1.091

g1.155 g1.164 1.189

g1.160 g1.171 1.204

g1.165 g1.177

1.107

0.8891 0.880 g1.171 g1.168

1.103

0.922 0.923 1.167

g1.180 g1.183 g1.084 g1.052 0.9222

1.018 1.017

80

0.8992 0.890 g1.177 g1.174

g1.113

g1.186 g1.188 g1.090 g1.057 0.9393 0.9389 0.939 0.939 1.188

1.032 1.030

60

0.9093 0.901 g1.181 g1.179

g1.119

g1.192 g1.194 g1.095 g1.061 0.9566 0.9566 0.956 0.956 1.210

1.045 1.043

40

1.0011

1.1285 1.352

1.016 g1.216

1.0260

0.816

1.063 1.066

0.931 1.070

1.174

g1.150

0.8791 0.870 g1.166

0.905 0.906

g1.048 0.9053

1.004 1.005 1.037 g1.174 g1.178

100

0.9871

1.338

1.0142 1.0125 1.005 g1.211

0.802

1.048 1.051

0.919 1.053

0.8691 0.861 1.153 1.148 1.153 1.140 1.144 1.160

0.801

0.889 0.890

0.990 0.993 1.023 g1.167 g1.173 1.066 g1.043 0.8887

120

1.322

1.0025 1.0021 0.994 g1.206

1.033

0.907 1.037

0.9881

0.8592 0.851 1.139 1.136 1.141 1.128 1.129 1.146

0.790 0.785 0.7847 0.789 0.784

0.873 0.874

1.054 g1.039 0.8722

1.010 g1.161

0.797 0.975

140

0.9909 0.9919 0.984 g1.201

0.773

1.018

0.895 1.022

0.9777

1.115 1.132

0.842 1.126 1.123 1.129

0.780 0.774 0.7735 0.778 0.774

0.857

1.041 g1.035 0.8560

1.150

0.786 0.961

160

1.488

0.762 0.764 0.763 0.763 0.9795 0.9818 0.973 g1.195

1.004

1.007

0.9674

1.101 1.119

1.117

1.112

0.769 0.763 0.7624 0.766 0.763

1.136 1.139 1.028 g1.030 0.8400

0.776 0.947

180

1.471

0.752 0.754 0.753 0.753 0.9681 0.9717 0.961 1.183

1.167

0.990

0.992

0.9571

1.106

1.097

0.759 0.752 0.7514 0.753

1.123 1.125 1.015 g1.026 0.8242

0.765 0.933

200

1.457

0.950 1.170

0.744 0.743 0.743 0.9569

1.151

0.976

1.094

1.082

0.749

1.012

1.109 1.111

0.755

220

0.939 1.157

0.734 0.732

0.720

1.067

0.997

1.095 1.097

0.745

1.176

240

0.928 1.144

0.724 0.721

0.710

1.052

0.983

1.081 1.084

1.100

1.165

260

0.916 1.130

0.714 0.711

0.700

1.172

0.968

1.067 1.070

1.154 0.963 1.086

280

0.905 1.117

0.705 0.699

0.690

1.156

0.953

1.053 1.057

1.071

1.143

300

0.893 1.104

0.679

1.140

0.939

1.039 1.044

320

1.091 1.548

1.125

1.113

1.025

340

1.078 1.504

1.098

360

1.084

380

[3] [3] [4] [5] [4] [6] [7] [8] [1] [9] [38] [7] [10] [11] [12] [41] [49] [44] [13] [14] [7] [15] [7] [41] [16] [8] [46] [17] [41] [18] [7] [4] [8] [48] [19] [45] [20] [42] [1,21] [41] [42] [1,21] [47] [22] [1] [41] [6] [38] [4] [23] [20] [10] [24] [25] [26] [1] [40] [27]

Ref.

7.0 6.1

7.2 6.1 6.4 g2.6 g2.2 5.9 g2.1 9.35

120

6.7 7.3

140

6.7 7.4

160

6.7 7.4

180

a

g ¼ glass.

Polysulfone, (with Bisphenol A) Polytetrafluoroethylene Poly(vinyl acetate) Poly(vinyl chloride) Poly(vinylidene fluoride) Poly(vinyl methyl ether)

Polystyrene g2.0

g2.8 7.17

6.92

6.87

g2.5 g2.9 g2.1

7.13

g2.3 g2.9 g2.1

6.96

7.20

5.0 g2.1

7.01

7.23 4.7

5.2 g2.1

5.78

7.7

6.8

7.06

5.5

5.79 5.1 5.3 g2.1

9.3

7.0

6.7 7.1

6.1

5.72 5.5 5.4 5.2 5.2

6.7 6.7

5.81 5.1 5.4 g2.1

6.7 7.1

5.3

5.9

5.75 5.8 5.7 5.2 5.2

7.14

6.7 6.7 7.5

6.0 g2.1 9.41

6.3 g2.6

7.1

6.7 7.1

5.82 5.1 5.6 g2.2

5.3

5.7

5.2

6.4

6.7 6.7 6.9 7.7 7.24 7.4 7.0 7.9

5.9 6.2 6.3 g2.1 9.53

5.3

5.8

6.1 6.0 5.2

7.6

6.7 6.7 7.2 7.5 7.18 7.3

6.2 g2.1 9.47

5.8

6.2

6.7 6.7 5.84 5.1 5.7 g2.2

6.6

6.1

7.5 6.7

6.7 7.1

6.2

g2.1

5.68 g2.7

g2.1 9.29

g2.6 g2.2

Polypropylene, atactic Polypropylene, isotactic

7.4

g2.6 7.3 6.7

g2.6 7.2

g2.6 7.1

g2.2

7.3

g2.1 g2.9 6.3

g2.1 g2.7 6.4

g2.1 g2.5

g2.1

g1.8

6.0 5.65 g2.4 g2.9

g2.7 5.61 g2.1 g2.5

g2.7 5.58 g1.8 g2.2

9.2 9.2

9.4 9.0

g2.6 g2.2 g2.5 g2.1 9.23

g2.7 5.54

g2.6 g2.2 g2.5 g2.1 9.17

g2.6 g2.2 g2.5 g2.1 9.11

7.9

g2.8

g2.1

g2.7 5.51

9.06 9.0

g2.4

6.4

6.8 6.2

100

Polyoxymethylene

Polyoxyethylene

Poly(methyl methacrylate), isotactic Poly(4-methyl pentene-1) Poly(o-methyl styrene) Polyoxybutylene

Poly(ethylene terephthalate) Poly(ethyl methacrylate) Polyisobutylene Poly(methyl methacrylate)

Polyethylene, linear

Poly(1,3-dioxolane) Polyetheretherketone Polyethylene, branched

Poly(cyclohexyl methacrylate) Poly(2,6-dimethylphenylene ether) Poly(dimethyl siloxane)

Poly(«-caprolactone) Polycarbonate, (with Bisphenol A)

g3.8a

6.5 6.3

80

6.7 7.4

200

6.7

220 240

5

6.6 6.9 6.7 6.7 5.85 5.1 5.8 5.5

6.8

7.1

5.2

6.7

8.0

7.8 7.32 7.5 7.0

5.9 5.6

5.87

6.6 7.1 6.7

6.8

7.1

7.1

5.2

7.0

7.6

6.7 6.7

7.1

6.2 6.2

6.7 6.7

6.1 6.2 6.4 g2.1 9.59

6.0 5.7

7.3 6.7

7.2

7.2

7.3

6.3 6.2

6.7

6.2 6.4

60

Poly(butene-1), isotactic Poly(n-butyl methacrylate)

6.6 6.4 6.7

40

4.7

6.6 6.5

Natural Rubber, unvulcanized Natural Rubber, cured

20

Polyamide, Nylon 6 Polyamide, Nylon 6.6

0

Temperature (deg C)

TABLE 7.4. Thermal expansion coefficients (10
4 K
1 ).

6.2 5.8

7.5 6.7

7.3

7.5

7.4

6.4 6.2

4.7 6.6

260

6.3 5.9

7.7 6.7

7.4

6.8

7.6

6.6 6.2

4.7 6.6 6.8

280

6.4 6.0

8.0 6.7

7.5

6.8

7.7

6.7 6.2

300

6.5 6.1

6.8

7.8

6.8

320

6.1 14.4

6.8

6.7

6.9

340

14.7

6.7

360

6.7

380 [3] [3] [28] [4] [4] [5] [6] [7] [8] [1] [9] [38] [7] [10] [11] [28] [12] [44] [13] [14] [7] [39] [15] [39] [29] [7] [16] [8] [17] [18] [7] [4] [8] [48] [19] [45] [20] [42] [1,21] [1,21] [42] [47] [29] [22] [1] [4] [38] [6] [29] [23] [20] [10] [24] [25] [26] [1] [40] [27]

Ref.

98 / CHAPTER 7 TABLE 7.5. Parameters for the Tait equation, Eqs. (7.5) and (7.6), for amorphous polymers above Tg . Polymer Natural Rubber, cured Natural Rubber, unvulcanized Polyamide, Nylon 6 Polyamide, Nylon 6,6 Poly(butene-1), isotactic Poly(n-butyl methacrylate) Poly(«-caprolactone) Polycarbonate, (with Bisphenol A) Poly(cyclohexyl methacrylate) Poly(2,6-dimethylphenylene ether) Poly(dimethyl siloxane)

Polyetheretherketone Polyethylene, branched

Polyethylene, linear Poly(ethylene terephthalate) Polyisobutylene Poly(methyl methacrylate)

Poly(methyl methacrylate), isotactic Poly(o-methyl styrene) Polyoxybutylene Polyoxyethylene Polyoxymethylene Polypropylene, atactic Polypropylene, isotactic

Polystyrene

Polysulfone, (with Bisphenol A) Polytetrafluoroethylene Poly(vinyl acetate) Poly(vinyl chloride) Poly(vinylidene fluoride)

C

b0 , bar

b1 , deg C
1

Temp. range, deg C

Pressure range, bar

Ref.

0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.1009 0.0988 0.0894 0.0894 0.0894 0.0894 0.0970 0.0894 0.0894 0.0894 0.0894 0.0894 0.0871 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.0894 0.1046 0.0894 0.0894 0.0894 0.0894

1916 1937 3767 3538 3164 1675 2267 1890 3100 2803 2952 2278 1041 960 885 915 3880 1867 1987 1771 1884 1767 1683 3697 1907 3000 2875 3850 4278 3006 2992 2619 1786 2077 2870 3058 1621 1491 1475 1705 2169 2521 2435 3659 4252 2231 2035 3522 2942 1066

0.00425 0.00517 0.00466 0.00376 0.00504 0.00453 0.00534 0.00393 0.00408 0.00387 0.00522 0.00429 0.00585 0.00604 0.00610 0.00609 0.00412 0.00439 0.00510 0.00470 0.00488 0.00466 0.00429 0.00415 0.00415 0.00508 0.00415 0.00672 0.00369 0.00426 0.00456 0.00411 0.00422 0.00395 0.00473 0.00433 0.00660 0.00418 0.00413 0.00498 0.00332 0.00408 0.00414 0.00376 0.00938 0.00343 0.00426 0.00565 0.00532 0.00042

0–25 0–25 236–295 270–285 245–297 133–246 34–200 100–148 151–340 171–330 123–200 203–320 25–70 29–60 25–70 18–150 338–400 112–225 140–200 130–200 140–203 142–200 140–200 274–342 53–110 105–255 114–160 120–139 115–230 110–194 55–190 139–198 62–166 88–224 75–136 183–220 80–120 170–300 175–230 180–300 115–196 79–320 115–249 186–370 343–372 64–120 35–100 82–97 100–150 180–220

0–500 0–500 0–2000 0–600 0–2000 0–2000 0–2000 0–2000 0–1800 100–2000 0–2000 0–1800 0–1000 0–800 0–900 0–2200 0–2000 0–2000 0–1000 0–2000 0–2000 0–2000 0–2000 0–2000 0–1000 0–1800 0–2000 0–2000 0–2000 0–2000 0–2000 0–2000 0–700 0–700 0–2000 0–2000 0–1000 0–2000 0–2000 100–2000 0–2000 0–1800 0–2000 0–2000 0–2000 0–1000 0–800 0–2000 0–2000 0–1200

[3] [3] [4] [5] [4] [6] [7] [1] [9] [38] [7] [10] [30] [31] [12] [49] [13] [14] [30] [7] [32,33] [7] [7] [16] [30] [18] [7] [33,34] [4] [48] [19] [20] [1,21] [1,21] [47] [22] [1] [6] [4] [38] [20] [10] [33,34] [24] [25] [30] [26,35] [33,34] [1] [40]

DENSITIES, THERMAL EXPANSIONS, AND COMPRESSIBILITIES

/ 99

TABLE 7.6. Parameters for the Tait equation, Eqs. (7.5) and (7.6), for polymer glasses. C

b0 , bar

b1 ,deg C
1

Tg , deg C

0.0894

3878

0.00261

151

30–151

0–1800

0.0894

3251

0.00133

142

40–141

100–1500

Poly(cyclohexyl methacrylate)

0.0894

3762

0.00298

107

19–74

0–2000

[7]

Poly(2,6-dimethylphenylene ether)

0.0894

3314

0.00200

203

30–203

0–1800

[10]

Poly(methyl methacrylate)

0.0894

3767

0.00470

100

30–100

0–1800

[18]

0.0894

3564

0.00323

105

17–91

0–2000

[7]

0.0894

3717

0.00396

111

20–100

0–2000

[33]

0.0894

4296

0.00416

105

20–95

0–2000

[48]

Poly(methyl methacrylate), isotactic

0.0894

4968

0.00670

47

9–26

0–2000

[19]

Poly(o-methyl styrene)

0.0894

3596

0.00355

131

29–82

0–2000

[20]

Polystyrene

0.0894

3449

0.00271

92

8–75

0–2000

[20]

0.0894

3917

0.00431

79

30–79

0–1800

[10]

0.0894

3337

0.00330

88

20–61

0–2000

[33,34]

Polysulfone, (with Bisphenol A)

0.0894

4323

0.00154

186

30–186

0–2000

[24]

Poly(vinyl acetate)

0.0894

3079

0.00097

31


30–20

0–800

[26,35]

Poly(vinyl chloride)

0.0894

3751

0.00241

75

20–51

0–2000

[33,34]

Glassy polymer Polycarbonate, (with Bisphenol A)

Temp. range, deg C

Pressure range, bar

Ref. [9] [38]

TABLE 7.7. Thermal pressure coefficients as a function of temperature. (@P=@t)v , bar/K, (t in 8C)

Temp. range, 8C

Ref.

Natural Rubber, cured

13:75
0:0545t
0:481  10
4 t 2

20–50

[28]

Poly(dimethyl siloxane)

8:71
0:0474t þ 0:93  10
4 t 2

25–162

[11]

Polymer

9:00
0:0538t þ 1:54  10
4 t 2 Polyethylene, high density


20–50

[28]

13:22
0:0558t þ 0:896  10 t

139–192

[15]


4 2


4 2

Polyisobutylene

12:66
0:0545t þ 1:02  10 t

10–172

[17]

Polyoxyethylene

15:79
0:0250t þ 1:532  10
4 t 2

50–103

[36]

Polypropylene, isotactic

23:19
0:1757t þ 4:22  10
4 t 2

158–195

[37]


4 2

27–100

[23]

20–120

[27]

Polystyrene Poly(vinyl methyl ether)

15:81
0:0859t þ 2:70  10 t


4 2

13:81
0:0514t þ 0:005  10 t

g ¼ glass.

a

Poly(vinylidene fluoride) Poly(vinyl methyl ether)

Poly(vinyl chloride)

Polysulfone, (with Bisphenol A) Polytetrafluoroethylene Polytetrahydrofuran Poly(vinyl acetate)

Polystyrene

Polypropylene, atactic Polypropylene, isotactic

Poly(methyl methacrylate), isotactic Poly(4-methyl pentene-1) Poly(o-methyl styrene) Polyoxymethylene Polyoxyethylene

Poly(methyl methacrylate)

Poly(ethylene terephthalate) Polyisobutylene

Polyethylene, linear

Polyetheretherketone Polyethylene, branched

Poly(cyclohexyl methacrylate) Poly(2,6-dimethylphenylene ether) Poly(dimethyl siloxane)

Poly(butene-1), isotactic Poly(n-butyl methacrylate) Poly(«-caprolactone) Polycarbonate, (with Bisphenol A)

Natural Rubber, unvulcanized Polyamide, Nylon 6 Polyamide, Nylon 6,6

Natural Rubber, cured

Temperature (deg C)

g 2.9

10.0

4.6 4.6

0

5.3

g 3.0 g 2.5

g 2.9

g 2.7

g 2.7 g 2.6 g 2.3 g 2.1

4.8

11.0

11.3

g 2.5

5.3 5.0 5.0

20

5.8

5.2 g 2.6

4.5 g 2.9 g 2.7 g 3.1 g 2.2

g 2.9

g 2.9 g 2.9 g 2.8 g 2.5

5.2

g 2.7 g 3.0 g 2.8 g 3.0 14.8

g 2.6a g 2.9 g 2.7 g 2.9 13.1 12.8 12.3 13.1 12.9 12.5

5.5 4.5 9.4

5.2

6.4

5.7

8.1

7.6 6.6 6.7 4.5 5.2

7.0 6.2 6.2 4.0

7.2

g 2.4

5.3

6.0

6.4 6.0 5.0 10.7

4.7

g 3.6

6.9 6.9 g 3.8

18.0

g 3.3 19.0

6.7 7.0 g 3.0 g 3.1

100

g 2.3

4.9

5.5

g 3.3

g 3.1

5.0 g 3.1 g 3.0 g 3.3 g 2.3

6.3 6.4 g 3.5 g 3.2 g 3.3 g 2.9 4.3

15.9

g 3.2 16.7

g 2.8 g 3.1

6.0

80

5.8 5.9 g 3.1 g 3.0 g 3.1 g 2.7 3.9

13.8 14.8 14.6 14.1

5.4

60

4.9

40

9.2

5.8

8.3 7.1

6.1 5.8 6.0 g 2.5

5.5 12.2

6.9

5.5 5.1 5.2 5.0 5.2

7.5

8.1

20.3

g 3.4 21.4

7.5 7.6 g 3.2 g 3.2

120

6.4

9.0

6.6 6.3 6.6 g 2.6

7.5

6.1

6.1 5.6 5.9 5.4 5.7

8.2

9.7

8.9 10.0 9.7 9.4 10.0

22.9

g 3.6 24.1

10.1 8.3 8.2 g 3.3 g 3.3

140

TABLE 7.8. Isothermal compressibilities (  10
5 bar
1 ) at atmospheric pressure.

9.8

7.0 6.8 7.1 g 2.6

11.2

8.1

6.6

5.9 6.2

6.7 6.0

9.7 11.0 10.7 10.4 11.0 10.7 10.6

6.3 g 3.7 27.0

5.5

11.0 9.3

160

8.9

7.5 7.4 7.7 g 2.7

12.7 12.7 12.8 13

8.8

7.2

6.4 6.8

7.4

10.6 12.2 11.8 11.4 12.1 11.7 11.5

6.0 6.4 7.0 g 3.9

12.1 10.3

180

9.0

8.0 8.4 5.2

14

13.8 13.8

6.9 9.5

8.2

12.9 12.5

11.5 13.5 12.9 12.6

6.5 6.9 7.7 g 4.0

13.2 11.5

200

9.0

8.7 9.1 5.6

16

15.0 15.0

7.6 10.3

9.1

12.6

7.1 7.5 8.6 10.1

14.5

220

9.4 9.9 6.0

17

16.3

19

10.1

11.0

7.7 8.1

15.8

7.3

240

6.5

10.2

19

17.8

20

12.0

8.3 8.7

10.5

8.0

260

7.0

11.1

21

19.3

22

7.7

13.0

9.0 9.4

8.8 7.2 11.6

280

7.5

12.1

23

21.0

24

8.4

14.2

9.8 10.2

300

8.1

13.1

27

9.1

15.5

10.6 11.0

320

8.8

9.9

9.4

11.5

340

9.4 61.6

10.2

360

11.0

380

[28] [3] [3] [4] [5] [4] [6] [7] [1] [9] [38] [7] [10] [11] [28] [30] [31] [12] [49] [13] [14] [30] [7] [32,33] [15] [7] [7] [16] [17] [30] [18] [7] [33,34] [48] [19] [45] [20] [22] [1,21] [36] [47] [1] [6] [4] [4,6,37] [38] [23] [20] [10] [33,34] [24] [25] [1,21] [30] [26,35] [33,34] [1] [40] [27]

Ref.

DENSITIES, THERMAL EXPANSIONS, AND COMPRESSIBILITIES REFERENCES 1. P. A. Rogers, J. Appl. Polym. Sci. 48, 1061 (1993). 2. P. Zoller, in Polymer Handbook, edited by J. Brandrup and E. H. Immergut (Wiley, New York, 1989), pp. VI/475–483. 3. L. A. Wood and G. M. Martin, J. Res. Nat. Bur. Stand. 68A, 259 (1964). 4. Y. Z. Wang, W. J. Chia, K. H. Hsieh, et al. J. Appl. Polym. Sci. 44, 1731 (1992). 5. H. W. Starkweather, Jr., P. Zoller, and G. A. Jones, J. Polym. Sci., Polym. Phys. Ed. 22, 1615 (1984). 6. P. Zoller, J. Appl. Polym. Sci. 23, 1057 (1979). 7. O. Olabisi and R. Simha, Macromolecules 8, 206 (1975). 8. S. S. Rogers and L. Mandelkern, J. Phys. Chem. 61, 985 (1957). 9. P. Zoller, J. Polym. Sci., Polym. Phys. Ed. 20, 1453 (1982). 10. P. Zoller and H. H. Hoehn, J. Polym. Sci., Polym. Phys. Ed. 20, 1385 (1982). 11. H. Shih and P. J. Flory, Macromolecules 5, 758 (1972). 12. R. N. Lichtenthaler, D. D. Liu, and J. M. Prausnitz, Macromolecules 11, 192 (1978). 13. P. Zoller, T. A. Kehl, H. W. Starkweather, Jr., et al. J. Polym. Sci. B. Polym. Phys. 27, 993 (1989). 14. P. Zoller, J. Appl. Polym. Sci. 23, 1051 (1979). 15. R. A. Orwoll and P. J. Flory, J. Am. Chem. Soc. 89, 6814 (1967). 16. P. Zoller and P. Bolli, J. Macromol. Sci.-Phys. B18, 555 (1980). 17. B. E. Eichinger and P. J. Flory, Macromolecules 1, 285 (1968). 18. C. K. Kim and D. R. Paul, Polymer 33, 2089 (1992). 19. A. Quach, P. S. Wilson, and R. Simha, J. Macromol. Sci.-Phys. B9, 533 (1974). 20. A. Quach and R. Simha, J. Appl. Phys. 42, 4592 (1971). 21. R. K. Jain and R. Simha, J. Polym. Sci., Polym. Phys. Ed. 17, 1929 (1979). 22. H. W. Starkweather, Jr., G. A. Jones, and P. Zoller, J. Polym. Sci. B, Polym. Phys. 26, 257 (1988). 23. H. Ho¨cker, G. J. Blake, and P. J. Flory, Trans. Faraday Soc. 67, 2251 (1971).

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101

24. P. Zoller, J. Polym. Sci., Polym. Phys. Ed. 16, 1261 (1978). 25. P. Zoller, J. Appl. Polym. Sci. 22, 633 (1978). 26. J. E. McKinney and M. Goldstein, J. Res. Nat. Bur. Stand. 78A, 331 (1974). 27. T. Shiomi, F. Hamada, T. Nasako, et al. Macromolecules 23, 229 (1990). 28. G. Allen, G. Gee, D. Mangaraj, et al. Polymer 1, 467 (1960). 29. V.-H. Karl, F. Asmussen, and K. Ueberreiter, Makromol. Chem. 178, 2037 (1977). 30. S. Beret and J. M. Prausnitz, Macromolecules 8, 536 (1975). 31. K. Kubota and K. Ogino, Macromolecules 11, 514 (1978). 32. D. P. Maloney and J. M. Prausnitz, J. Appl. Polym. Sci. 18, 2703 (1974). 33. K.-H. Hellwege, W. Knappe, and P. Lehmann, Kolloid-Z. Z. Polym. 183, 110 (1962). 34. R. Simha, P. S. Wilson, and O. Olabisi, Kolloid-Z. Z. Polym. 251, 402 (1973). 35. J. E. McKinney and R. Simha, Macromolecules 7, 894 (1974). 36. G. N. Malcolm and G. L. D. Ritchie, J. Phys. Chem. 66, 852 (1962). 37. G. C. Fortune and G. N. Malcolm, J. Phys. Chem. 71, 876 (1967). 38. Y. Sato, Y. Yamasaki, S. Takishima, et al. J. Appl. Polym. Sci. 66, 141 (1997). 39. L. Zhao, L. Capt, M. R., Kamal, et al., Polym. Eng. Sci. 44, 853 (2004). 40. N. Mekhilef, J. Appl. Polym. Sci. 80, 230 (2001). 41. R.-J. Roe, J. Phys. Chem. 72, 2013 (1968). 42. S.-M. Mai, C. Booth, and V. M. Nace, Eur. Polym. J. 33, 991 (1997). 43. J. Garza, C. Marco, J. G. Fatou, et al. Polymer 22, 477 (1981). 44. P. Archambault and R. E. Prud’homme, J. Polym. Sci., Polym. Phys. Ed. 18, 35 (1980). 45. P. Zoller, J. Appl. Polym Sci. 21, 3129 (1977). 46. P. J. Barham, A. Keller, E. L. Otun, et al. J. Mater. Sci. 19, 2781 (1988). 47. M. Schmidt and F. H. J. Maurer, J. Polym. Sci. B, Polym. Phys. 36, 1061 (1998). 48. M. Schmidt and F. H. J. Maurer, Macromolecules 33, 3879 (2000). 49. V. K. Sachdev, U. Yahsi, and R. K. Jain, J. Polym. Sci. B, Polym. Phys. 36, 841 (1998).

CHAPTER 8

Thermodynamic Properties of Proteins George I. Makhatadze Department of Biochemistry and Molecular Biology Penn State University College of Medicine, Hershey, PA 17033

8.1 8.2 8.3 8.4

Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydration Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Proteins are heteropolymers consisting of 20 different type of amino acid residues. They all share a common backbone

103 111 111 112 143

the increase of temperature with a slope from 0.005 to 0:008 J K
2 g
1 for small globular proteins depending on the protein [6]. The absolute value of CN p (T) at 25 8C ranges between 1.25 and 1:80 J K
1 g
1 , depending on the proteins [6]. The heat capacity of the denatured state, CD p (T), for small globular proteins is always larger than the heat capacity of the native state, and depends on temperature in a more complicated way. The temperature dependence is nonlinear at low temperatures (0---75  C). At temperatures above 75 8C the heat capacity function appears to be almost independent of temperature. At 25 8C, CD p (T) values for different proteins range from 1.85 to 2:20 J K
1 g
1 , while at 100 8C they are higher, 2:1---2:4 J K
1 g
1 . A nonlinear dependence of the heat capacity of denatured protein is in agreement (within 5%) with what one can expect for the heat capacity of an unfolded polypeptide chain, CU p (T), of the same amino acid composition [7–10]. The latter can be calculated as a sum of contributions from side chains and from the peptide unit:

H2 N-----CH-----COOH j R and differ by the side chain radical, R. Amino acid residues are arranged into a polypeptide chain through formation of peptide bonds 0 1 -----HN-----CH-----COOH HN-----CH-----CO A H2 N-----CH----- @ j j R R1 N N
2 The side chains, RN , vary in size and chemical nature (Table 8.1), allowing a great variety of properties. Naturally occurring polypeptides adopt a unique three dimensional conformation in solution. The structures of many of them have been solved by X-ray crystallography and 2D-NMR. 8.1 HEAT CAPACITY

CU p,2 ¼ (N
1)  Cp,2 (-----CHCONH-----) þ

N X

Cp,2 (
Ri ),

i¼1

Tables 8.2 and 8.3 present molar heat capacities of solid amino acids and polyamino acids. Table 8.4 presents specific heat capacities of anhydrous and hydrated proteins. All of the measurements were done by using adiabatic absolute calorimetry and their accuracy is better than 1–2%. Heat capacity of anhydrous proteins can be predicted using empirical approach developed by Wunderlich (see e.g., [1]). In the presence of the solvent water, the heat capacity of proteins increases. The partial molar heat capacity of proteins in the native state, CN p (T), appears to be a linear function of temperature [2–5]. The CN p (T) increases with

(8:1) where Cp,2 (-----CHCONH-----) is the heat capacity of the peptide unit, Cp,2 (
Ri ) is the heat capacity contribution of the side-chain of the ith amino acid residue, and N is the number of amino acid residues in the polypeptide chain. These contributions, for the temperature range 5–125 8C are listed in Table 8.5. The agreement is observed not only in the temperature dependence of these two functions (CD p (T) and CU (T) ) but in majority of cases in the absolute values as well p [7–10]. The close correspondence between the measured 103

His Ile Leu Lys Met Phe Pro Ser Thr Trp Tyr Val

Histidine

Isoleucine Leucine Lysine Methionine

Phenylalanine

Proline Serine Threonine

Tryptophane

Tyrosine

Valine

V

---CH---(CH3 )2 a-amino a-carboxyl

99.14 6.8–7.9 3.5–4.3

140

193.6

163.18

Y

25.1

227.8

186.21

W

9.7

122.7 89.0 116.1

* ---CH2 ---OH ---CH2 ---(CH3 )---OH

97.12 87.08 101.11

166.7 166.7 168.6 162.9

153.2

88.6 173.4 117.7 111.1 108.5 143.9 138.4 60.1

P S T

53.6

43.6

1.6

4.6 36.0

44.9

Volume ˚3 A

189.9

10.4

6.2

4.6

4.5 9.1–9.5

12

pKa

DHion kJ mol
1

147.18

113.17 113.17 128.18 131.21

137.15

71.08 156.20 114.11 115.09 103.14 128.14 129.12 57.06

Mol Mass Da

F

I L K M

---CH(CH3 )---C2 H5 ---CH(CH3 )2 ---CH2 ---(CH2 )4 ---NH2 ---(CH2 )2 ---S---CH3

---CH3 ---(CH2 )3 ---CNH(==NH)NH3 ---CH2 ---CONH2 ---CH2 ---COOH ---CH2 ---SH ---(CH2 )2 ---CONH2 ---(CH2 )2 ---COOH ---H

A R N D C Q E G H

Side Chains (–R) R---CH(NH2 )COOH

One letter code

43

42

42

38 42 44

43

42 43 44 44

43

46 45 45 45 36 45 45 85

ASAmc A˚ 2

117

144

190

105 44 74

175

140 137 119 117

102

67 89 44 48 35 53 61

ASAnpl sc ˚2 A

43

27

36 28

48 43

49

107 69 58 69 91 77

ASApol sc A˚ 2

*Proline is an imono acid; Enthalpies of ionization of side chains at 25 8C, DHion , are from [19]; van der Waals volume from [20]; ASAmc , surface area of the backbone, pol ASAnpl sc , nonpolar surface area of the side chains, and, ASAsc , polar surface area of the side chains are from [14].

Ala Arg Asn Asp Cys Gln Glu Gly

Alanine Arginine Asparagine Aspartic acid Cystein Glutamine Glutamic acid Glycine

Name

Three letter code

TABLE 8.1a. Properties of 20 naturally occurring amino acids.

104 / CHAPTER 8

THERMODYNAMIC PROPERTIES OF PROTEINS

/

105

TABLE 8.1b. Heat capacities (J K
1 mol
1 ) of solid amino acids. T/K L-Alanine

10 15 20 25 30 35 40 45 50 55 60 70 80

Cp (89.1 Da) [21] 0.494 1.674 3.849 6.724 10.09 13.75 17.37 20.91 24.40 27.78 31.02 37.03 42.64

L-Aspartic

10 15 20 25 30 35 40 45 50 55 60 70 80

Cp

T/K

Cp

90 100 110 120 130 140 150 160 170 180 190 200 210

47.87 52.55 56.94 61.09 65.10 68.99 72.72 76.32 79.83 83.22 86.53 89.79 93.01

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

96.23 99.58 102.93 106.32 111.55 113.05 116.40 119.66 122.84 125.98 114.10 122.26 126.02

(210.7 Da) [22] 2.326 6.481 12.25 19.21 26.91 34.91 42.93 50.71 58.16 65.27 72.05 84.43 95.48

90 100 110 120 130 140 150 160 170 180 190 200 210

105.52 114.68 123.26 131.46 139.37 147.15 154.72 162.09 169.41 176.65 183.76 190.87 197.90

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

205.10 212.21 219.37 226.52 233.72 241.00 248.20 255.27 262.25 269.16 243.26 260.96 269.28

acid (134.1 Da) [23] 0.732 2.686 6.033 10.31 15.14 20.09 25.18 30.06 34.70 39.10 43.30 50.88 57.70

90 100 110 120 130 140 150 160 170 180 190 200 210

64.02 69.45 74.64 79.54 84.27 88.83 93.26 97.57 101.80 106.06 110.21 114.31 118.41

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

122.51 126.69 130.88 134.98 139.03 143.14 147.32 151.63 155.98 160.37 144.47 155.19 160.42

L-ArginineHCl

10 15 20 25 30 35 40 45 50 55 60 70 80

T/K

106 / CHAPTER 8 TABLE 8.1b. Continued. T/K

Cp

L-Asparagine  H2 O

10 15 20 25 30 35 40 45 50 55 60 70 80 L-Cystine

10 15 20 25 30 35 40 45 50 55 60 70 80

(240.3 Da) [24] 1.644 5.284 15.25 18.03 25.49 33.14 40.63 47.87 54.85 61.51 67.91 79.87 91.21

L-Glutamic

10 15 20 25 30 35 40 45 50 55 60 70 80

(150.1 Da) [23] 0.686 2.343 5.376 9.531 14.63 20.29 26.18 32.13 37.95 43.56 48.95 58.99 68.12

acid (147.1 Da) [23] 0.276 2.782 6.259 10.86 16.12 21.69 27.18 32.59 37.80 42.68 47.28 55.65 63.18

T/K

Cp

T/K

Cp

90 100 110 120 130 140 150 160 170 180 190 200 210

76.48 83.93 91.13 98.07 104.85 111.46 117.95 124.35 130.71 136.94 143.09 149.12 155.19

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

161.21 167.28 173.34 179.41 185.44 191.38 197.28 203.18 208.99 214.81 193.26 207.90 214.89

90 100 110 120 130 140 150 160 170 180 190 200 210

102.13 111.13 120.00 128.66 137.24 145.52 153.64 161.59 169.54 177.23 184.81 192.30 199.70

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

206.98 214.18 221.33 228.49 235.60 242.59 249.53 256.40 263.17 269.91 244.81 261.92 269.99

90 100 110 120 130 140 150 160 170 180 190 200 210

70.17 76.53 82.34 87.95 93.22 98.37 103.39 108.41 113.39 118.24 122.93 127.70 132.47

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

137.24 142.01 146.78 151.54 156.31 161.17 166.11 171.08 175.98 180.79 162.72 175.06 180.87

THERMODYNAMIC PROPERTIES OF PROTEINS

/

107

TABLE 8.1b. Continued. T/K

T/K

Cp

T/K

Cp

(146.2 Da) [23] 0.728 2.460 5.791 10.48 15.91 21.80 27.72 33.46 38.93 44.06 48.87 57.66 65.48

90 100 110 120 130 140 150 160 170 180 190 200 210

72.55 78.99 84.94 90.67 196.23 101.76 107.28 112.63 117.95 123.18 128.37 133.60 138.74

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

144.06 149.45 154.72 159.91 165.02 170.04 175.10 180.12 185.10 189.83 171.63 184.18 189.91

Glycine (75.1 Da) [21] 10 0.255 15 0.967 20 2.393 25 4.464 30 7.037 35 9.933 40 13.00 45 16.12 50 19.25 55 22.26 60 25.15 70 30.49 80 35.26

90 100 110 120 130 140 150 160 170 180 190 200 210

39.53 43.26 46.69 50.00 53.09 56.07 58.91 61.67 64.39 67.03 69.66 72.30 74.89

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

77.49 80.08 82.84 85.60 88.45 91.25 94.06 96.90 99.75 102.63 92.13 99.20 102.68

Glycylglycine (132.1 Da) [25] 10 0.707 15 2.607 20 6.071 25 10.70 30 15.87 35 21.23 40 26.56 45 31.65 50 36.58 55 41.21 60 45.61 70 53.81 80 61.21

90 100 110 120 130 140 150 160 170 180 190 200 210

67.86 73.85 79.50 84.89 89.87 94.81 99.58 104.27 108.74 113.09 117.40 121.71 125.90

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

130.12 134.35 138.62 142.84 147.19 151.50 155.85 160.29 164.81 169.45 152.88 163.97 169.54

L-Glutamine

10 15 20 25 30 35 40 45 50 55 60 70 80

Cp

108 / CHAPTER 8 TABLE 8.1b. Continued. T/K

Cp

T/K

Cp

T/K

Cp

90 100 110 120 130 140 150 160 170 180 190 200 210

102.63 111.09 119.04 126.57 133.85 140.96 147.90 154.81 161.63 168.36 175.14 181.92 188.66

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

195.39 202.13 208.91 215.81 222.80 229.87 236.90 243.89 250.79 257.65 232.09 249.53 257.73

(131.2 Da) [26] 2.741 6.343 10.80 15.82 20.93 26.12 31.22 36.22 41.08 45.82 50.38 58.95 67.15

90 100 110 120 130 140 150 160 170 180 190 200 210

74.89 81.80 88.41 94.68 100.63 106.36 111.88 117.24 122.55 127.70 132.72 137.65 142.59

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

147.53 152.59 157.74 162.97 168.16 173.26 178.53 183.76 189.28 194.89 174.93 188.28 194.97

(131.2 Da) [26] 2.452 6.468 11.50 16.93 22.53 28.08 33.32 38.40 43.26 47.82 52.17 60.42 68.20

90 100 110 120 130 140 150 160 170 180 190 200 210

75.44 82.22 88.53 94.64 100.50 106.15 111.63 117.03 122.38 127.70 132.93 138.11 143.43

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

148.91 154.60 160.46 166.52 172.93 179.70 186.77 194.31 202.51 211.38 181.88 200.96 211.50

L-HistidineHCl

10 15 20 25 30 35 40 45 50 55 60 70 80 L-Isoleucine

10 15 20 25 30 35 40 45 50 55 60 70 80 L-Leucine

10 15 20 25 30 35 40 45 50 55 60 70 80

(191.6 Da) [22] 1.711 5.690 11.93 19.33 27.29 35.38 43.26 50.88 58.12 64.81 71.17 82.80 93.22

THERMODYNAMIC PROPERTIES OF PROTEINS

/

109

TABLE 8.1b. Continued. T/K L-LysineHCl

10 15 20 25 30 35 40 45 50 55 60 70 80 L-Methionine

10 15 20 25 30 35 40 45 50 55 60 70 80

Cp

T/K

Cp

T/K

Cp

(182.7 Da) [22] 2.460 6.594 12.23 18.82 25.89 33.08 40.11 46.94 53.64 60.12 66.32 77.70 88.12

90 100 110 120 130 140 150 160 170 180 190 200 210

97.74 106.36 114.47 122.22 129.54 136.40 143.05 149.58 155.94 162.09 168.16 174.26 180.41

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

186.61 192.92 199.33 205.81 212.42 219.16 225.94 232.97 240.25 247.86 221.25 238.91 247.99

(149.2 Da) [24] 2.017 5.883 11.48 17.79 24.23 30.65 36.70 42.43 47.87 53.01 57.91 66.99 75.19

90 100 110 120 130 140 150 160 170 180 190 200 210

82.68 89.50 95.90 101.96 107.70 113.26 118.66 124.01 129.24 134.47 139.70 145.19 151.04

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

157.28 164.14 172.05 181.13 192.67 208.28 229.79 259.20 298.28 298.19 214.30 290.04 296.85

90 100 110 120 130 140 150 160 170 180 190 200 210

74.73 80.96 86.94 92.72 98.49 104.31 110.17 116.11 122.01 127.95 133.93 139.87 145.85

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

152.05 158.49 165.02 171.54 178.07 184.60 191.13 197.69 204.22 210.71 186.65 203.01 210.79

L-Phenylalanine

10 15 20 25 30 35 40 45 50 55 60 70 80

(165.2 Da) [27] 2.594 6.682 11.87 17.36 23.16 28.93 34.36 39.44 44.23 48.74 53.01 60.79 67.99

110 / CHAPTER 8 TABLE 8.1b. Continued. T/K

Cp

T/K

Cp

T/K

Cp

(115.1 Da) [27] 1.301 4.105 8.314 13.01 17.85 22.65 27.17 31.42 35.38 39.02 42.43 48.62 54.10

90 100 110 120 130 140 150 160 170 180 190 200 210

59.08 63.60 67.91 72.17 76.36 80.63 84.98 89.45 93.93 98.41 102.84 107.24 111.63

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

116.06 120.50 124.93 129.33 133.85 138.16 142.67 147.28 152.09 156.98 141.00 151.17 157.07

(105.1 Da) [28] 0.816 2.703 5.556 9.063 13.03 17.21 21.46 25.63 29.69 33.59 37.31 44.18 50.33

90 100 110 120 130 140 150 160 170 180 190 200 210

55.90 61.09 65.73 70.12 74.35 78.41 82.34 86.11 89.83 93.47 97.11 100.67 104.22

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

107.78 111.25 114.77 118.28 121.84 125.39 128.99 132.59 136.23 139.91 126.52 135.56 140.00

90 100 110 120 130 140 150 160 170 180 190 200 210

84.56 91.92 99.04 106.06 113.18 120.33 127.45 134.64 141.80 148.99 156.27 163.59 170.88

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

178.20 185.69 193.34 201.08 208.87 216.52 224.22 231.92 239.58 247.32 218.95 238.15 247.44

L-Proline

10 15 20 25 30 35 40 45 50 55 60 70 80 L-Serine

10 15 20 25 30 35 40 45 50 55 60 70 80

L-Tryptophane

10 15 20 25 30 35 40 45 50 55 60 70 80

(204.2 Da) [27] 4.393 10.02 16.51 23.04 29.35 35.28 40.78 45.61 51.00 55.73 60.17 68.70 76.86

THERMODYNAMIC PROPERTIES OF PROTEINS

/

111

TABLE 8.1b. Continued. T/K

Cp

T/K

Cp

T/K

Cp

(181.2 Da) [27] 1.389 3.908 7.724 12.57 18.07 23.91 29.62 35.24 40.59 45.56 50.29 59.20 67.49

90 100 110 120 130 140 150 160 170 180 190 200 210

75.06 82.05 88.83 95.56 102.26 108.87 115.52 122.26 128.99 135.77 142.38 149.12 155.81

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

162.72 169.45 176.23 183.05 189.91 196.86 203.84 210.79 217.74 224.60 199.08 216.44 224.72

(117.2 Da) [24] 1.100 3.192 6.615 10.69 15.18 19.86 24.46 29.04 33.50 37.87 42.09 50.04 57.61

90 100 110 120 130 140 150 160 170 180 190 200 210

64.73 71.34 77.57 83.55 89.16 94.56 99.83 104.98 109.91 114.73 119.37 123.89 128.41

220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

133.01 137.57 142.05 146.61 151.17 155.85 160.50 165.06 169.70 174.31 157.28 168.82 174.39

L-Tyrosine

10 15 20 25 30 35 40 45 50 55 60 70 80 L-Valine

10 15 20 25 30 35 40 45 50 55 60 70 80

heat capacity of the denatured protein and the calculated one for the unfolded polypeptide chain is a strong argument that heat denatured proteins are indistinguishable thermodynamically from fully unfolded (D  U).

more details, compilation of the experimental data on the partial volume of proteins is recommended [12].

8.2 PARTIAL VOLUME

Because the absolute value of the heat capacity of the native state is always larger than the heat capacity of the unfolded state, and because they have different temperature dependencies, the heat capacity change upon unfolding is positive and temperature dependent itself:

The partial specific volume of proteins can be measured experimentally [11,12] or can be estimated from the amino acid composition of the protein using equation V2 ¼ (N
1)  V2 (-----CHCONH-----) þ

N X

8.3 THERMODYNAMIC FUNCTIONS

U N DU N Cp (T) ¼ Cp (T)
Cp (T),

V2 (-----Ri ), (8:2)

i¼1

where N is number of amino acid residues in the protein sequence, V2 (-----CHCONH-----) is the contribution of a peptide unit, and V2 (-----Ri ) is the contribution of the ith amino acid side chain. These contributions are listed in Table 8.6. For

(8:3)

where DU N Cp (T) is a nonlinear function of temperature. It has maximum at around 50 8C, decreases at high temperatures and appears to reach zero at 140 8C). The heat capacities of proteins in the native and unfolded states and the heat capacity change upon unfolding are presented in Table 8.7.

112 / CHAPTER 8 TABLE 8.2. Temperature dependence of the partial molar volumes (cm3 mol
1 ) of the peptide unit and of the side chains of amino acid residues for the temperature range 5–125 8C [11]. Temperature (8C)

–CHCONH– ---Ri Ala Arg Asn Asp Cys Gln Glu Gly His Ile Leu Lys Met Phe Pro Ser Thr Trp Tyr Val

5

25

50

75

100

125

28.3

28.0

28.6

28.9

29.0

29.4

24.8 81.2 44.1 39.0 40.5 58.4 54.9 10.9 65.1 64.5 75.0 72.4 73.3 82.0 36.0 27.1 44.2 110.9 88.2 60.7

27.2 85.7 45.7 41.5 40.5 61.0 57.5 10.3 64.0 66.1 77.6 73.5 72.1 86.3 35.8 27.8 44.7 110.9 90.0 63.0

30.1 88.4 47.8 44.5 40.5 63.2 60.7 9.6 63.1 67.7 80.2 75.8 70.2 92.0 34.1 28.8 45.9 110.9 92.2 65.7

33.1 90.4 49.3 47.3 40.5 66.0 63.6 9.2 62.4 69.8 82.9 78.5 69.0 97.9 33.7 29.7 47.6 110.9 94.1 68.5

36.3 92.1 50.8 49.9 40.5 68.5 66.3 8.9 61.8 71.8 86.5 81.3 67.9 102.6 33.5 30.7 50.9 110.9 95.9 71.7

39.4 93.5 52.4 52.6 40.5 69.7 69.3 8.5 61.1 74.0 89.4 84.8 66.7 107.9 33.2 32.0 54.6 110.9 97.8 74.5

The non-zero heat capacity change upon protein unfolding means that all other thermodynamic functions are also temperature dependent: DU N H(T)

¼

DU N H(T0 )

þ

ZT

DU N Cp (T)dT

(8:4)

T0

NMR structure of the native protein by computing the surface formed by the center of a spherical probe rolling on the surface of the protein (see e.g., [13,14]). As was shown earlier, the hydration effects of protein groups exposed to water upon unfolding [15–18] can be expressed as follows: X ^ hyd DChyd DU (8:7) N ASAk,i  DCp,k,i (T) p,k (T) ¼ i

U DU N S(T) ¼ DN S(T0 ) þ

ZT

DU N Cp (T)d ln T

(8:5)

DHkhyd (T) ¼

¼

DU NH




TDU NS

(8:6)

These functions are presented in Table 8.7 for a selected set of 20 proteins, for which the thermodynamics of unfolding have been measured with highest accuracy.

8.4 HYDRATION EFFECTS The contribution of solvent (water) to the observed changes of the thermodynamic parameters of proteins can be assumed to be proportional to the changes in water accessible surface area of the protein groups. The water accessible surface area can be calculated from the X-ray or

^ hyd DU N ASAk,i  DH k,i (T)

(8:8)

^hyd DU N ASAk,i  DSk,i (T)

(8:9)

i

T0

DU NG

X

DShyd k (T) ¼

X i

DGhyd k (T) ¼

X

^ hyd DU N ASAk,i  DGk,i (T),

(8:10)

i

where DU N ASAk,i is the change of water accessible surface area of protein group i of type k upon unfolding and ^ hyd (T), DS^hyd (T), DG ^ hyd (T), and DH ^ hyd (T) are the hyDC p,k,i k ,i k ,i k ,i dration heat capacity change, enthalpy, entropy, and Gibbs energy change of this type of group the normalized per square angstrom. These normalized hydration effects were determined by the transfer of model compounds from the gaseous phase to water and their values are listed in Table 8.8.

THERMODYNAMIC PROPERTIES OF PROTEINS

/

113

TABLE 8.3. Heat capacities (J K
1 mol
1 ) of solid poly (amino acids). Poly-L-Alanine (m.w 15,000 Da) [29] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

42.65

590

120.06

0.2

0.00

200

44.96

600

121.57

0.3

0.00

210

47.19

610

123.07

0.4

0.00

220

49.46

620

124.55

0.5

0.00

230

51.71

630

126.02

0.6

0.00

240

53.93

640

127.47

0.7

0.00

250

56.18

650

128.91

0.8

0.00

260

58.39

660

130.34

0.9

0.00

270

60.61

670

131.76

1.0

0.00

273.15

61.30

680

133.16

1.2

0.00

280

62.79

690

134.56

1.4

0.00

290

64.97

700

135.95

1.6

0.00

298.15

66.76

710

137.32

1.8

0.00

300

67.17

720

138.69

2

0.01

310

69.30

730

140.05

3

0.02

320

71.40

740

141.40

4

0.05

330

73.54

750

142.75

5

0.10

340

75.60

760

144.08

6

0.17

350

77.64

770

145.41

7

0.27

360

79.65

780

146.74

8

0.40

370

81.64

790

148.06

9

0.56

380

83.72

800

149.38

10

0.75

390

85.66

810

150.70

15

1.95

400

87.58

820

152.01

20

3.31

410

89.47

830

153.32

25

4.66

420

91.35

840

154.63

30

5.95

430

93.31

850

155.94

40

8.40

440

95.13

860

157.25

50

10.75

450

96.93

870

158.55

60

13.06

460

98.72

880

159.86

70

15.35

470

100.48

890

161.16

80

17.63

480

102.22

900

162.47

90

19.89

490

103.94

910

163.78

100

22.15

500

105.64

920

165.10

110

24.43

510

107.32

930

166.41

120

26.70

520

108.98

940

167.74

130

28.99

530

110.62

950

169.06

140

31.25

540

112.24

960

170.39

150

33.52

550

113.84

970

171.73

160

35.78

560

115.42

980

173.09

170

38.05

570

116.98

990

174.44

180

40.37

580

118.53

1000

175.80

114 / CHAPTER 8 TABLE 8.3. Continued. Poly-L-ArginineHCl (m.w 132,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

168.39

590

410.48

0.2

0.00

200

174.72

600

415.75

0.3

0.00

210

180.99

610

420.90

0.4

0.00

220

187.37

620

426.02

0.5

0.00

230

193.59

630

431.08

0.6

0.00

240

199.90

640

436.12

0.7

0.00

250

206.23

650

441.09

0.8

0.00

260

212.56

660

446.07

0.9

0.00

270

218.86

670

451.00

1.0

0.01

273.15

220.85

680

455.91

1.2

0.01

280

225.21

690

460.79

1.4

0.01

290

231.62

700

465.91

1.6

0.02

298.15

236.88

710

470.48

1.8

0.03

300

237.99

720

475.28

2

0.04

310

244.49

730

480.05

3

0.14

320

250.83

740

484.81

4

0.34

330

257.21

750

489.54

5

0.66

340

263.60

760

494.23

6

1.13

350

269.96

770

498.98

7

1.77

360

276.31

780

503.65

8

2.57

370

282.71

790

508.31

9

3.51

380

289.10

800

512.93

10

4.58

390

295.25

810

517.55

15

10.76

400

301.53

820

522.18

20

17.16

410

307.69

830

526.80

25

23.29

420

313.80

840

531.38

30

29.14

430

319.89

850

535.98

40

40.34

440

325.97

860

540.57

50

51.21

450

332.02

870

545.16

60

61.91

460

338.03

880

549.75

70

72.42

470

343.99

890

554.32

80

82.61

480

349.82

900

558.90

90

92.43

490

355.58

910

563.48

100

101.85

500

361.27

920

568.07

110

110.73

510

366.89

930

572.67

120

119.11

520

372.45

940

577.26

130

127.01

530

377.96

950

581.87

140

134.57

540

383.44

960

586.49

150

141.75

550

388.86

970

591.12

160

148.67

560

394.31

980

595.77

170

155.41

570

399.73

990

600.42

180

161.97

580

405.15

1000

605.07

THERMODYNAMIC PROPERTIES OF PROTEINS

/

115

TABLE 8.3. Continued. Poly-L-Aspartic acidNa (m.w 42,500 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

105.03

590

249.25

0.2

0.00

200

109.57

600

252.01

0.3

0.00

210

114.02

610

254.74

0.4

0.00

220

118.55

620

257.41

0.5

0.00

230

122.86

630

260.03

0.6

0.00

240

127.19

640

262.63

0.7

0.00

250

131.48

650

265.19

0.8

0.00

260

135.71

660

267.74

0.9

0.00

270

139.84

670

270.26

1.0

0.00

273.15

141.14

680

272.77

1.2

0.01

280

143.96

690

275.25

1.4

0.01

290

148.05

700

278.00

1.6

0.01

298.15

151.40

710

280.19

1.8

0.02

300

152.07

720

282.63

2

0.02

310

156.17

730

285.05

3

0.08

320

160.08

740

287.47

4

0.19

330

164.01

750

289.86

5

0.38

340

167.90

760

292.22

6

0.65

350

171.71

770

294.65

7

1.01

360

175.46

780

297.02

8

1.47

370

179.27

790

299.37

9

2.01

380

182.83

800

301.65

10

2.62

390

186.42

810

303.98

15

6.16

400

190.09

820

306.30

20

9.83

410

193.59

830

308.62

25

13.34

420

197.03

840

310.94

30

16.69

430

200.44

850

313.25

40

23.10

440

203.81

860

315.55

50

29.33

450

207.18

870

317.85

60

35.48

460

210.50

880

320.15

70

41.58

470

213.78

890

322.43

80

47.59

480

216.97

900

324.72

90

53.48

490

220.11

910

327.02

100

59.27

500

223.19

920

329.32

110

64.90

510

226.23

930

331.62

120

70.40

520

229.22

940

333.93

130

75.70

530

232.18

950

336.24

140

80.90

540

235.10

960

338.55

150

85.89

550

237.99

970

340.87

160

90.88

560

240.94

980

343.21

170

95.65

570

243.74

990

345.55

180

100.39

580

246.51

1000

347.89

116 / CHAPTER 8 TABLE 8.3. Continued. Poly-L-Asparagine (m.w 10,400 Da) [30] T/K

Cp

T/K

Cp 97

T/K

Cp

0.1

0.00

190

590

235.78

0.2

0.00

200

100.96

600

238.61

0.3

0.00

210

104.88

610

241.40

0.4

0.00

220

108.90

620

244.14

0.5

0.00

230

112.73

630

246.84

0.6

0.00

240

116.63

640

249.51

0.7

0.00

250

120.52

650

252.16

0.8

0.00

260

124.40

660

254.80

0.9

0.00

270

128.19

670

257.41

1.0

0.00

273.15

129.40

680

260.00

1.2

0.01

280

132.02

690

262.58

1.4

0.01

290

135.84

700

265.41

1.6

0.01

298.15

138.99

710

267.70

1.8

0.02

300

139.62

720

270.23

2

0.03

310

143.51

730

272.75

3

0.09

320

147.23

740

275.27

4

0.22

330

150.98

750

277.77

5

0.43

340

154.71

760

280.23

6

0.74

350

158.39

770

282.77

7

1.14

360

162.02

780

285.24

8

1.64

370

165.73

790

287.71

9

2.22

380

169.20

800

290.09

10

2.85

390

172.73

810

292.54

15

6.39

400

176.35

820

294.98

20

9.92

410

179.79

830

297.41

25

13.29

420

183.19

840

299.85

30

16.50

430

186.56

850

302.27

40

22.69

440

189.91

860

304.69

50

28.72

450

193.27

870

307.11

60

34.66

460

196.59

880

309.52

70

40.49

470

199.87

890

311.93

80

46.16

480

203.07

900

314.34

90

51.66

490

206.22

910

316.76

100

56.96

500

209.32

920

319.18

110

62.02

510

212.39

930

321.61

120

66.88

520

215.42

940

324.04

130

71.54

530

218.41

950

326.47

140

76.09

540

221.37

960

328.91

150

80.41

550

224.31

970

331.36

160

84.70

560

227.30

980

333.82

170

88.84

570

230.15

990

336.28

180

92.96

580

232.98

1000

338.75

THERMODYNAMIC PROPERTIES OF PROTEINS

/

117

TABLE 8.3. Continued. Poly-L-Glutamic AcidNa (m.w 74,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

100.57

590

282.92

0.2

0.00

200

106.20

600

286.38

0.3

0.00

210

111.79

610

289.78

0.4

0.00

220

117.49

620

293.13

0.5

0.00

230

122.98

630

296.42

0.6

0.00

240

128.45

640

299.68

0.7

0.00

250

133.90

650

302.91

0.8

0.00

260

139.28

660

306.11

0.9

0.00

270

144.57

670

309.28

1.0

0.00

273.15

146.23

680

312.42

1.2

0.00

280

149.83

690

315.54

1.4

0.01

290

155.07

700

318.90

1.6

0.01

298.15

159.33

710

321.72

1.8

0.01

300

160.21

720

324.77

2

0.02

310

165.42

730

327.80

3

0.07

320

170.42

740

330.83

4

0.16

330

175.41

750

333.82

5

0.31

340

180.36

760

336.78

6

0.53

350

185.21

770

339.80

7

0.83

360

189.98

780

342.76

8

1.20

370

194.80

790

345.70

9

1.64

380

199.34

800

348.55

10

2.14

390

203.89

810

351.47

15

5.03

400

208.50

820

354.37

20

8.02

410

212.91

830

357.26

25

10.89

420

217.26

840

360.15

30

13.62

430

221.56

850

363.03

40

18.86

440

225.83

860

365.90

50

23.96

450

230.06

870

368.76

60

28.99

460

234.25

880

371.62

70

34.05

470

238.38

890

374.46

80

39.17

480

242.39

900

377.32

90

44.38

490

246.33

910

380.15

100

49.71

500

250.21

920

383.01

110

55.12

510

254.02

930

385.88

120

60.64

520

257.79

940

388.74

130

66.24

530

261.50

950

391.60

140

71.96

540

265.17

960

394.47

150

77.65

550

268.80

970

397.35

160

83.41

560

272.47

980

400.25

170

89.13

570

276.00

990

403.15

180

94.87

580

279.48

1000

406.04

118 / CHAPTER 8 TABLE 8.3. Continued. Poly-Glycine II (m.w 15,000 Da) [29] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

64.83

590

163.11

0.2

0.00

200

67.40

600

165.32

0.3

0.00

210

69.93

610

167.40

0.4

0.00

220

72.48

620

169.46

0.5

0.00

230

75.06

630

171.51

0.6

0.00

240

77.62

640

173.53

0.7

0.00

250

80.17

650

175.55

0.8

0.00

260

82.75

660

177.56

0.9

0.00

270

85.25

670

179.55

1.0

0.00

273.15

86.07

680

181.53

1.2

0.00

280

87.91

690

183.50

1.4

0.01

290

90.45

700

185.46

1.6

0.01

298.15

92.53

710

187.40

1.8

0.02

300

93.01

720

189.34

2

0.02

310

95.58

730

191.26

3

0.07

320

98.42

740

193.17

4

0.17

330

100.85

750

195.07

5

0.34

340

103.43

760

196.97

6

0.58

350

105.99

770

198.85

7

0.88

360

108.61

780

200.73

8

1.25

370

112.07

790

202.60

9

1.67

380

113.75

800

204.47

10

2.11

390

116.28

810

206.33

15

4.52

400

118.87

820

208.18

20

6.87

410

121.42

830

210.03

25

9.09

420

123.91

840

211.88

30

11.22

430

126.39

850

213.72

40

15.34

440

128.84

860

215.57

50

19.38

450

131.26

870

217.41

60

23.38

460

133.69

880

219.26

70

27.32

470

136.15

890

221.10

80

31.17

480

138.47

900

222.93

90

34.86

490

140.93

910

224.74

100

38.46

500

143.23

920

226.58

110

41.89

510

145.51

930

228.42

120

45.16

520

147.77

940

230.26

130

48.25

530

150.01

950

232.13

140

51.22

540

152.23

960

233.99

150

54.11

550

154.43

970

235.84

160

56.91

560

156.72

980

237.71

170

59.56

570

158.86

990

239.56

180

62.21

580

161.00

1000

241.44

THERMODYNAMIC PROPERTIES OF PROTEINS

/

119

TABLE 8.3. Continued. Poly-L-Histidine (m.w 49,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

113.43

590

322.88

0.2

0.00

200

119.22

600

327.05

0.3

0.00

210

124.97

610

331.18

0.4

0.00

220

131.00

620

335.25

0.5

0.00

230

136.77

630

339.26

0.6

0.00

240

142.71

640

343.24

0.7

0.00

250

148.55

650

347.16

0.8

0.00

260

154.44

660

351.08

0.9

0.00

270

160.16

670

354.96

1.0

0.00

273.15

161.98

680

358.80

1.2

0.01

280

165.99

690

362.62

1.4

0.01

290

171.78

700

366.68

1.6

0.01

298.15

176.53

710

370.18

1.8

0.02

300

177.51

720

373.92

2

0.02

310

183.37

730

377.62

3

0.08

320

188.85

740

381.32

4

0.19

330

194.70

750

385.35

5

0.38

340

200.39

760

388.63

6

0.65

350

206.06

770

392.33

7

1.01

360

211.37

780

395.95

8

1.46

370

217.02

790

399.55

9

2.00

380

222.34

800

403.07

10

2.61

390

227.64

810

406.63

15

6.13

400

233.12

820

410.19

20

9.77

410

238.35

830

413.74

25

13.26

420

243.62

840

417.29

30

16.60

430

248.73

850

420.81

40

22.98

440

253.79

860

424.33

50

29.18

450

258.82

870

427.83

60

35.30

460

263.82

880

431.33

70

41.45

470

268.78

890

434.81

80

47.55

480

273.62

900

438.34

90

53.66

490

278.34

910

441.82

100

59.79

500

282.98

920

445.31

110

65.84

510

287.56

930

448.80

120

71.95

520

292.09

940

452.29

130

77.90

530

296.63

950

455.74

140

83.91

540

301.19

960

459.24

150

89.79

550

305.55

970

462.75

160

95.72

560

309.97

980

466.26

170

101.66

570

314.36

990

469.77

180

107.57

580

318.66

1000

473.28

120 / CHAPTER 8 TABLE 8.3. Continued. Poly-L-Leucine (m.w 150,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

116.64

590

300.37

0.2

0.00

200

121.12

600

304.48

0.3

0.00

210

125.58

610

308.56

0.4

0.00

220

130.13

620

312.57

0.5

0.00

230

134.68

630

316.52

0.6

0.00

240

139.08

640

320.46

0.7

0.00

250

143.73

650

324.36

0.8

0.00

260

148.34

660

328.25

0.9

0.00

270

152.93

670

332.10

1.0

0.00

273.15

154.40

680

335.93

1.2

0.01

280

157.73

690

339.73

1.4

0.01

290

162.40

700

343.79

1.6

0.01

298.15

166.35

710

347.30

1.8

0.02

300

167.14

720

351.05

2

0.03

310

172.11

730

354.77

3

0.10

320

176.90

740

358.49

4

0.23

330

181.86

750

362.18

5

0.45

340

186.62

760

365.84

6

0.77

350

191.58

770

369.57

7

1.20

360

196.37

780

373.22

8

1.74

370

201.29

790

376.87

9

2.39

380

206.03

800

380.43

10

3.11

390

210.82

810

384.04

15

7.31

400

215.77

820

387.65

20

11.65

410

220.55

830

391.24

25

15.81

420

225.28

840

394.84

30

19.78

430

230.00

850

398.43

40

27.39

440

234.67

860

402.00

50

34.77

450

239.32

870

405.57

60

42.05

460

243.94

880

409.14

70

49.23

470

248.57

890

412.70

80

56.23

480

253.07

900

416.26

90

63.02

490

257.58

910

419.83

100

69.61

500

262.12

920

423.39

110

75.80

510

266.50

930

426.97

120

81.73

520

270.85

940

430.55

130

87.31

530

275.16

950

434.13

140

92.66

540

279.43

960

437.72

150

97.73

550

283.68

970

441.32

160

102.69

560

287.99

980

444.93

170

107.40

570

292.15

990

448.55

180

112.07

580

296.28

1000

452.17

THERMODYNAMIC PROPERTIES OF PROTEINS

/

121

TABLE 8.3. Continued. Poly-L-LysineHBr (m.w 560,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

145.15

590

369.70

0.2

0.00

200

150.98

600

374.53

0.3

0.00

210

156.76

610

379.32

0.4

0.00

220

162.67

620

384.05

0.5

0.00

230

168.43

630

388.73

0.6

0.00

240

174.25

640

393.34

0.7

0.00

250

180.11

650

397.95

0.8

0.00

260

185.97

660

402.53

0.9

0.00

270

191.81

670

407.07

1.0

0.00

273.15

193.66

680

411.59

1.2

0.01

280

197.69

690

416.07

1.4

0.01

290

203.67

700

420.81

1.6

0.02

298.15

208.57

710

424.99

1.8

0.03

300

209.60

720

429.41

2

0.04

310

215.67

730

433.88

3

0.12

320

221.57

740

438.27

4

0.28

330

227.54

750

442.64

5

0.55

340

233.53

760

446.93

6

0.95

350

239.48

770

451.33

7

1.47

360

245.45

780

455.65

8

2.14

370

251.45

790

459.97

9

2.93

380

257.16

800

464.20

10

3.81

390

262.98

810

468.44

15

8.97

400

268.86

820

472.72

20

14.30

410

274.58

830

476.98

25

19.40

420

280.24

840

481.25

30

24.27

430

285.88

850

485.49

40

33.60

440

291.53

860

489.73

50

42.66

450

297.16

870

493.97

60

51.59

460

302.74

880

498.20

70

60.41

470

308.28

890

502.42

80

69.03

480

313.68

900

506.65

90

77.39

490

319.02

910

510.88

100

85.48

500

324.29

920

515.12

110

93.23

510

329.49

930

519.36

120

100.61

520

334.64

940

523.61

130

107.63

530

339.75

950

527.86

140

114.40

540

344.82

960

532.12

150

120.83

550

349.88

970

536.39

160

127.20

560

354.99

980

540.68

170

133.25

570

359.90

990

544.98

180

139.25

580

364.88

1000

549.28

122 / CHAPTER 8 TABLE 8.3. Continued. Poly-L-Methionine (m.w 160,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

108.51

590

281.22

0.2

0.00

200

113.39

600

284.80

0.3

0.00

210

118.22

610

288.35

0.4

0.00

220

123.15

620

291.84

0.5

0.00

230

127.96

630

295.28

0.6

0.00

240

132.69

640

298.70

0.7

0.00

250

137.51

650

302.09

0.8

0.00

260

142.24

660

305.45

0.9

0.00

270

146.91

670

308.80

1.0

0.00

273.15

148.39

680

312.12

1.2

0.01

280

151.59

690

315.42

1.4

0.01

290

156.30

700

318.96

1.6

0.01

298.15

160.16

710

321.96

1.8

0.02

300

160.95

720

325.21

2

0.02

310

165.71

730

328.44

3

0.08

320

170.29

740

331.66

4

0.19

330

174.90

750

334.86

5

0.38

340

179.52

760

338.03

6

0.65

350

184.14

770

341.27

7

1.02

360

188.63

780

344.44

8

1.48

370

193.22

790

347.60

9

2.02

380

197.58

800

350.67

10

2.63

390

201.95

810

353.81

15

6.19

400

206.41

820

356.94

20

9.87

410

210.69

830

360.07

25

13.39

420

214.94

840

363.19

30

16.76

430

219.15

850

366.30

40

23.20

440

223.35

860

369.41

50

29.47

450

227.55

870

372.51

60

35.63

460

231.71

880

375.61

70

41.81

470

235.82

890

378.69

80

47.91

480

239.84

900

381.79

90

53.95

490

243.80

910

384.89

100

59.95

500

247.75

920

387.99

110

65.83

510

251.60

930

391.10

120

71.58

520

255.42

940

394.21

130

77.17

530

259.20

950

397.33

140

82.68

540

262.94

960

400.45

150

87.98

550

266.66

970

403.59

160

93.28

560

270.43

980

406.73

170

98.43

570

274.06

990

409.89

180

103.53

580

277.65

1000

413.04

THERMODYNAMIC PROPERTIES OF PROTEINS

/

123

TABLE 8.3. Continued. Poly-L-Phenylalanine (m.w 23,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

105.52

590

320.05

0.2

0.00

200

111.58

600

324.19

0.3

0.00

210

117.56

610

328.26

0.4

0.00

220

123.77

620

332.27

0.5

0.00

230

129.77

630

336.22

0.6

0.00

240

135.84

640

340.12

0.7

0.00

250

141.94

650

343.98

0.8

0.00

260

148.03

660

347.80

0.9

0.00

270

154.06

670

351.58

1.0

0.00

273.15

155.97

680

355.33

1.2

0.00

280

160.10

690

359.04

1.4

0.01

290

166.22

700

362.99

1.6

0.01

298.15

171.17

710

366.38

1.8

0.01

300

172.21

720

370.01

2

0.02

310

178.30

730

373.60

3

0.07

320

184.21

740

377.19

4

0.16

330

190.14

750

380.73

5

0.32

340

196.03

760

384.24

6

0.54

350

201.84

770

387.81

7

0.85

360

207.56

780

391.32

8

1.23

370

213.32

790

394.79

9

1.68

380

218.82

800

398.15

10

2.19

390

224.34

810

401.59

15

5.15

400

229.91

820

405.01

20

8.21

410

235.28

830

408.42

25

11.14

420

240.58

840

411.82

30

13.93

430

245.81

850

415.23

40

19.30

440

250.98

860

418.69

50

24.54

450

256.11

870

421.97

60

29.82

460

261.18

880

425.33

70

35.20

470

266.17

890

428.68

80

40.69

480

271.04

900

432.03

90

46.37

490

275.83

910

435.38

100

52.17

500

280.53

920

438.73

110

57.97

510

285.16

930

442.10

120

63.85

520

289.73

940

445.46

130

69.77

530

294.23

950

448.82

140

75.67

540

298.67

960

452.19

150

81.61

550

303.05

970

455.57

160

87.59

560

307.46

980

458.96

170

93.53

570

311.71

990

462.36

180

99.53

580

315.91

1000

465.75

124 / CHAPTER 8 TABLE 8.3. Continued. Poly-L-Proline (m.w 44,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

75.68

590

224.54

0.2

0.00

200

79.12

600

227.75

0.3

0.00

210

82.65

610

230.92

0.4

0.00

220

86.17

620

234.04

0.5

0.00

230

89.84

630

237.11

0.6

0.00

240

93.57

640

240.14

0.7

0.00

250

97.35

650

243.15

0.8

0.00

260

101.24

660

246.14

0.9

0.00

270

105.05

670

249.09

1.0

0.00

273.15

106.28

680

252.02

1.2

0.00

280

108.98

690

254.93

1.4

0.01

290

112.99

700

258.08

1.6

0.01

298.15

116.23

710

260.69

1.8

0.01

300

116.90

720

263.54

2

0.02

310

120.99

730

266.37

3

0.06

320

125.05

740

269.19

4

0.14

330

129.05

750

271.98

5

0.28

340

133.07

760

274.76

6

0.48

350

137.11

770

277.52

7

0.75

360

141.16

780

280.27

8

1.08

370

145.18

790

283.00

9

1.48

380

149.17

800

285.72

10

1.93

390

153.17

810

288.43

15

4.54

400

157.15

820

291.13

20

7.25

410

160.94

830

293.81

25

9.83

420

164.77

840

296.49

30

12.30

430

168.58

850

299.16

40

17.03

440

172.40

860

301.82

50

21.63

450

176.17

870

304.48

60

26.15

460

179.91

880

307.13

70

30.64

470

183.62

890

309.78

80

35.04

480

187.24

900

312.42

90

39.31

490

190.88

910

315.06

100

43.46

500

194.46

920

317.70

110

47.45

510

197.95

930

320.34

120

51.27

520

201.40

940

322.98

130

54.95

530

204.81

950

325.63

140

58.51

540

208.19

960

328.28

150

62.00

550

211.54

970

330.93

160

65.43

560

214.84

980

333.59

170

68.84

570

218.11

990

336.26

180

72.27

580

221.35

1000

338.94

THERMODYNAMIC PROPERTIES OF PROTEINS

/

125

TABLE 8.3. Continued. Poly-L-Serine (m.w 6,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

73.88

590

188.29

0.2

0.00

200

77.23

600

190.62

0.3

0.00

210

80.53

610

192.92

0.4

0.00

220

83.94

620

195.17

0.5

0.00

230

87.14

630

197.39

0.6

0.00

240

90.38

640

199.59

0.7

0.00

250

93.62

650

201.77

0.8

0.00

260

96.82

660

203.93

0.9

0.00

270

99.94

670

206.07

1.0

0.00

273.15

100.94

680

208.20

1.2

0.00

280

103.08

690

210.31

1.4

0.01

290

106.22

700

212.68

1.6

0.01

298.15

108.82

710

214.51

1.8

0.01

300

109.32

720

216.60

2

0.02

310

112.52

730

218.67

3

0.06

320

115.57

740

220.74

4

0.13

330

118.65

750

222.79

5

0.26

340

121.72

760

224.81

6

0.44

350

124.74

770

226.90

7

0.68

360

127.71

780

228.93

8

0.99

370

130.76

790

230.95

9

1.36

380

133.58

800

232.90

10

1.77

390

136.46

810

234.90

15

4.16

400

139.44

820

236.90

20

6.64

410

142.26

830

238.89

25

9.01

420

145.06

840

240.89

30

11.27

430

147.83

850

242.88

40

15.60

440

150.58

860

244.86

50

19.82

450

153.34

870

246.83

60

23.96

460

156.08

880

248.81

70

28.13

470

158.79

890

250.77

80

32.26

480

161.41

900

252.74

90

36.36

490

164.00

910

254.72

100

40.44

500

166.55

920

256.69

110

44.46

510

169.06

930

258.68

120

48.42

520

171.54

940

260.66

130

52.29

530

174.00

950

262.64

140

56.10

540

176.43

960

264.63

150

59.75

550

178.84

970

266.62

160

63.42

560

181.32

980

268.63

170

66.95

570

183.67

990

270.64

180

70.46

580

185.99

1000

272.64

126 / CHAPTER 8 TABLE 8.3. Continued. Poly-L-Tryptophane (m.w 160,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

110.46

590

392.95

0.2

0.00

200

117.96

600

398.53

0.3

0.00

210

125.38

610

404.03

0.4

0.00

220

133.19

620

409.44

0.5

0.00

230

140.76

630

414.76

0.6

0.00

240

148.48

640

420.03

0.7

0.00

250

156.27

650

425.24

0.8

0.00

260

164.15

660

430.40

0.9

0.00

270

171.89

670

435.50

1.0

0.00

273.15

174.37

680

440.57

1.2

0.00

280

179.76

690

445.58

1.4

0.01

290

187.75

700

450.81

1.6

0.01

298.15

194.19

710

455.48

1.8

0.01

300

195.57

720

460.36

2

0.02

310

203.53

730

465.21

3

0.06

320

211.25

740

470.03

4

0.15

330

219.15

750

474.98

5

0.29

340

226.94

760

479.53

6

0.49

350

234.59

770

484.32

7

0.77

360

242.15

780

489.05

8

1.12

370

249.75

790

493.72

9

1.53

380

257.06

800

498.26

10

1.99

390

264.45

810

502.88

15

4.67

400

271.85

820

507.48

20

7.45

410

279.00

830

512.06

25

10.12

420

286.06

840

516.63

30

12.66

430

293.07

850

521.19

40

17.53

440

300.03

860

525.81

50

22.31

450

306.93

870

530.23

60

27.14

460

313.70

880

534.73

70

32.15

470

320.37

890

539.22

80

37.44

480

326.92

900

543.70

90

43.07

490

333.34

910

548.18

100

48.96

500

339.67

920

552.66

110

55.06

510

345.91

930

557.14

120

61.47

520

352.06

940

561.62

130

68.11

530

358.13

950

566.10

140

74.83

540

364.12

960

570.59

150

81.71

550

370.03

970

575.08

160

88.75

560

375.95

980

579.59

170

95.85

570

381.69

990

584.11

180

103.12

580

387.36

1000

588.62

THERMODYNAMIC PROPERTIES OF PROTEINS

/

127

TABLE 8.3. Continued. Poly-L-Tyrosine (m.w 125,000 Da) [30] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

112.30

590

351.96

0.2

0.00

200

119.12

600

356.61

0.3

0.00

210

125.88

610

361.20

0.4

0.00

220

132.86

620

365.71

0.5

0.00

230

139.62

630

370.15

0.6

0.00

240

146.44

640

374.54

0.7

0.00

250

153.27

650

378.89

0.8

0.00

260

160.08

660

383.20

0.9

0.00

270

166.81

670

387.46

1.0

0.00

273.15

168.94

680

391.68

1.2

0.00

280

173.55

690

395.86

1.4

0.01

290

180.35

700

400.28

1.6

0.01

298.15

185.86

710

404.13

1.8

0.01

300

187.02

720

408.22

2

0.02

310

193.78

730

412.26

3

0.07

320

200.35

740

416.30

4

0.16

330

206.93

750

420.29

5

0.31

340

213.47

760

424.24

6

0.54

350

219.92

770

428.26

7

0.84

360

226.28

780

432.22

8

1.22

370

232.68

790

436.13

9

1.66

380

238.80

800

439.92

10

2.16

390

244.94

810

443.79

15

5.09

400

251.13

820

447.65

20

8.12

410

257.11

830

451.49

25

11.01

420

263.01

840

455.32

30

13.78

430

268.84

850

459.15

40

19.09

440

274.61

860

463.04

50

24.28

450

280.34

870

466.74

60

29.52

460

285.98

880

470.53

70

34.92

470

291.55

890

474.29

80

40.53

480

297.00

900

478.07

90

46.43

490

302.35

910

481.84

100

52.53

500

307.62

920

485.61

110

58.80

510

312.81

930

489.39

120

65.28

520

317.93

940

493.17

130

71.86

530

322.97

950

496.95

140

78.48

540

327.95

960

500.74

150

85.16

550

332.87

970

504.54

160

91.94

560

337.81

980

508.35

170

98.69

570

342.59

990

512.17

180

105.51

580

347.30

1000

515.98

128 / CHAPTER 8 TABLE 8.3. Continued. Poly-L-Valine (m.w 7,230 Da) [29] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

100.26

590

255.89

0.2

0.00

200

104.36

600

259.29

0.3

0.00

210

108.39

610

262.65

0.4

0.00

220

112.55

620

265.96

0.5

0.00

230

116.53

630

269.23

0.6

0.00

240

120.58

640

272.47

0.7

0.00

250

124.67

650

275.70

0.8

0.00

260

128.75

660

278.90

0.9

0.00

270

132.79

670

282.06

1.0

0.00

273.15

134.07

680

285.22

1.2

0.01

280

136.83

690

288.36

1.4

0.01

290

140.90

700

291.75

1.6

0.01

298.15

144.29

710

294.60

1.8

0.02

300

144.98

720

297.70

2

0.03

310

149.19

730

300.78

3

0.09

320

153.30

740

303.85

4

0.21

330

157.63

750

306.90

5

0.41

340

161.52

760

309.92

6

0.70

350

165.61

770

313.02

7

1.09

360

169.67

780

316.04

8

1.57

370

173.83

790

319.05

9

2.13

380

177.77

800

321.99

10

2.75

390

181.78

810

324.98

15

6.26

400

185.90

820

327.97

20

9.82

410

189.87

830

330.94

25

13.20

420

193.83

840

333.93

30

16.43

430

197.71

850

336.89

40

22.65

440

201.57

860

339.86

50

28.70

450

205.43

870

342.82

60

34.68

460

209.28

880

345.77

70

40.64

470

213.09

890

348.72

80

46.51

480

216.82

900

351.67

90

52.26

490

220.56

910

354.63

100

57.88

500

224.34

920

357.59

110

63.32

510

227.95

930

360.56

120

68.54

520

231.54

940

363.52

130

73.53

530

235.10

950

366.49

140

78.37

540

238.62

960

369.47

150

82.98

550

242.13

970

372.46

160

87.52

560

245.69

980

375.46

170

91.86

570

249.12

990

378.47

180

96.11

580

252.52

1000

381.47

THERMODYNAMIC PROPERTIES OF PROTEINS

/

129

TABLE 8.3. Continued. Poly(L-LysineHBr-Alanine) (molar ratio 46 / 54, m.w 37,000 Da) [31] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

206.38

590

506.35

0.2

0.00

200

214.69

600

512.75

0.3

0.00

210

222.88

610

518.97

0.4

0.00

220

231.18

620

525.12

0.5

0.00

230

239.30

630

531.21

0.6

0.00

240

247.45

640

537.26

0.7

0.00

250

255.56

650

543.26

0.8

0.00

260

263.67

660

549.20

0.9

0.01

270

271.64

670

555.12

1.0

0.01

273.15

274.17

680

561.01

1.2

0.01

280

279.76

690

566.87

1.4

0.02

290

287.85

700

572.96

1.6

0.03

298.15

294.45

710

578.50

1.8

0.04

300

295.86

720

584.27

2

0.06

310

304

730

590.01

3

0.20

320

312.22

740

595.73

4

0.48

330

320.06

750

601.42

5

0.93

340

328.04

760

607.07

6

1.58

350

335.95

770

612.79

7

2.45

360

343.82

780

618.42

8

3.51

370

352.64

790

624.04

9

4.73

380

359.42

800

629.57

10

6.07

390

367.07

810

635.16

15

13.49

400

374.86

820

640.74

20

20.89

410

382.42

830

646.30

25

27.91

420

389.84

840

651.87

30

34.62

430

397.23

850

657.42

40

47.57

440

404.59

860

662.97

50

60.21

450

411.91

870

668.51

60

72.70

460

419.19

880

674.07

70

85.10

470

426.46

890

679.59

80

97.26

480

433.46

900

685.14

90

109.08

490

440.54

910

690.66

100

120.58

500

447.40

920

696.22

110

131.63

510

454.17

930

701.79

120

142.27

520

460.88

940

707.36

130

152.37

530

467.53

950

712.98

140

162.12

540

474.12

960

718.58

150

171.43

550

480.65

970

724.21

160

180.55

560

487.33

980

729.85

170

189.29

570

493.72

990

735.51

180

197.91

580

500.06

1000

741.19

130 / CHAPTER 8 TABLE 8.3. Continued. Poly (L-LysineHBr-Phenylalanine) (molar ratio 51/49, m.w 49,700 Da) [31] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

231.52

590

695.94

0.2

0.00

200

244.11

600

705.37

0.3

0.00

210

256.66

610

714.69

0.4

0.00

220

269.59

620

723.87

0.5

0.00

230

282.20

630

732.92

0.6

0.00

240

294.93

640

741.90

0.7

0.00

250

307.72

650

750.80

0.8

0.00

260

320.48

660

759.62

0.9

0.00

270

333.16

670

768.37

1.0

0.01

273.15

337.16

680

777.05

1.2

0.01

280

345.88

690

785.66

1.4

0.02

290

358.75

700

794.76

1.6

0.02

298.15

369.22

710

802.74

1.8

0.03

300

371.42

720

811.20

2

0.05

310

384.33

730

819.67

3

0.16

320

396.86

740

828.05

4

0.37

330

409.47

750

836.36

5

0.72

340

422.04

760

844.52

6

1.23

350

434.47

770

852.88

7

1.92

360

446.82

780

861.11

8

2.79

370

459.25

790

869.28

9

3.81

380

471.09

800

877.26

10

4.97

390

483.04

810

885.32

15

11.68

400

495.11

820

893.40

20

18.63

410

506.80

830

901.45

25

25.28

420

518.34

840

909.50

30

31.63

430

529.79

850

917.53

40

43.79

440

541.18

860

925.61

50

55.66

450

552.50

870

933.49

60

67.48

460

563.70

880

941.46

70

79.47

470

574.76

890

949.39

80

91.68

480

585.56

900

957.34

90

104.15

490

596.21

910

965.29

100

116.76

500

606.70

920

973.24

110

129.47

510

617.04

930

981.21

120

142.32

520

627.25

940

989.18

130

155.16

530

637.35

950

997.16

140

168.01

540

647.34

960

1005.15

150

180.69

550

657.26

970

1013.16

160

193.48

560

667.26

980

1021.20

170

206.10

570

676.89

990

1029.26

180

218.82

580

686.52

1000

1037.32

THERMODYNAMIC PROPERTIES OF PROTEINS

/

131

TABLE 8.3. Continued. Poly(L-Glutamic AcidNa-Tyrosine) (molar ratio 50/50, m.w 30,000 Da) [31] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

223.26

590

637.55

0.2

0.00

200

235.44

600

645.58

0.3

0.00

210

247.51

610

653.52

0.4

0.00

220

259.88

620

661.31

0.5

0.00

230

271.82

630

668.98

0.6

0.00

240

283.82

640

676.57

0.7

0.00

250

295.79

650

684.09

0.8

0.00

260

307.68

660

691.54

0.9

0.00

270

319.39

670

698.92

1.0

0.01

273.15

323.08

680

706.23

1.2

0.01

280

331.08

690

713.48

1.4

0.01

290

342.82

700

721.21

1.6

0.02

298.15

352.36

710

727.84

1.8

0.03

300

354.34

720

734.93

2

0.04

310

366.04

730

741.98

3

0.15

320

377.35

740

748.99

4

0.35

330

388.67

750

755.91

5

0.68

340

399.92

760

762.78

6

1.17

350

410.98

770

769.79

7

1.83

360

421.89

780

776.66

8

2.65

370

432.89

790

783.48

9

3.63

380

443.36

800

790.10

10

4.73

390

453.86

810

796.85

15

11.12

400

464.48

820

803.57

20

17.73

410

474.70

830

810.27

25

24.06

420

484.79

840

816.97

30

30.10

430

494.76

850

823.65

40

41.68

440

504.65

860

830.37

50

52.98

450

514.47

870

836.91

60

64.23

460

524.18

880

843.53

70

75.67

470

533.74

890

850.11

80

87.33

480

543.08

900

856.72

90

99.27

490

552.26

910

863.31

100

111.42

500

561.30

920

869.92

110

123.70

510

570.19

930

876.55

120

136.19

520

578.97

940

883.16

130

148.71

530

587.64

950

889.79

140

161.26

540

596.19

960

896.43

150

173.67

550

604.65

970

903.08

160

186.19

560

613.19

980

909.77

170

198.54

570

621.40

990

916.47

180

210.96

580

629.53

1000

923.16

132 / CHAPTER 8 TABLE 8.3. Continued. Poly(L-Proline-Glycine-Proline) (molar ratio 66/34, m.w 5,300 Da) [31] T/K

Cp

T/K

Cp

T/K

Cp

0.1

0.00

190

194.56

590

568.71

0.2

0.00

200

203.74

600

576.61

0.3

0.00

210

213

610

584.43

0.4

0.00

220

222.29

620

592.12

0.5

0.00

230

231.85

630

599.69

0.6

0.00

240

241.51

640

607.20

0.7

0.00

250

251.27

650

614.63

0.8

0.00

260

261.23

660

621.99

0.9

0.00

270

271.04

670

629.29

1.0

0.01

273.15

274.20

680

636.53

1.2

0.01

280

281.05

690

643.71

1.4

0.02

290

291.24

700

651.36

1.6

0.03

298.15

299.50

710

657.92

1.8

0.04

300

301.22

720

664.96

2

0.05

310

311.51

730

671.95

3

0.18

320

321.71

740

678.90

4

0.43

330

331.81

750

685.80

5

0.84

340

341.90

760

692.66

6

1.44

350

351.99

770

699.48

7

2.22

360

362.07

780

706.27

8

3.19

370

372.08

790

713.02

9

4.29

380

382.11

800

719.74

10

5.50

390

392.03

810

726.43

15

12.24

400

401.90

820

733.10

20

18.95

410

411.35

830

739.74

25

25.31

420

420.84

840

746.36

30

31.40

430

430.41

850

752.96

40

43.14

440

439.84

860

759.54

50

54.63

450

449.18

870

766.11

60

65.95

460

458.42

880

772.67

70

77.21

470

467.57

890

779.22

80

88.32

480

476.53

900

785.77

90

99.14

490

485.51

910

792.31

100

109.70

500

494.35

920

798.85

110

119.99

510

502.98

930

805.39

120

129.90

520

511.52

940

811.94

130

139.55

530

519.96

950

818.49

140

148.95

540

528.32

960

825.06

150

158.18

550

536.58

970

831.64

160

167.27

560

544.75

980

838.25

170

176.34

570

552.83

990

844.87

180

185.52

580

560.81

1000

851.51

THERMODYNAMIC PROPERTIES OF PROTEINS

/

133

TABLE 8.4. Heat capacities (J K
1 g
1 ) of anhydrous and hydrated proteins. Anhydrous bovine zinc insulin (5,665 Da) [32] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102 1.987 4.427 7.305 10.42 13.64 16.86 19.95 22.90 25.74 28.48 31.14 36.21 41.00

T/K 90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102 45.56 49.87 53.93 57.78 61.51 65.27 69.04 72.80 76.53 80.21 83.81 87.40 91.04

T/K 220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

Cp  102 94.73 98.58 102.42 106.32 110.21 114.14 118.03 122.05 126.11 130.25 115.35 125.35 130.29

Hydrated (4.0% H2 O) bovine zinc insulin (5,665 Da) [32] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102 1.933 4.339 7.284 10.49 13.89 17.23 20.46 23.56 26.55 29.43 32.20 37.46 42.38

T/K 90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102 46.94 51.25 55.40 59.41 63.39 67.32 71.30 75.23 79.20 83.14 87.03 90.96 94.94

T/K 220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

Cp  102 98.95 103.01 107.07 111.21 115.44 119.75 124.10 128.45 132.84 137.24 121.13 132.01 137.28

Anhydrous bovine chymotrypsinogen (25,666 Da) [32] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102 1.874 4.343 7.297 10.53 13.87 17.15 20.33 23.38 26.39 29.34 32.19 37.48 42.26

T/K 90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102 46.86 51.09 55.31 59.33 63.26 67.15 70.96 74.77 78.58 82.43 86.23 90.08 93.97

T/K 220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

Cp  102 97.82 101.67 105.56 109.50 113.55 117.57 121.67 125.86 130.08 134.27 118.87 129.29 134.35

134 / CHAPTER 8 TABLE 8.4. Continued. Hydrated (10.7% H2 O) bovine chymotrypsinogen (25,666 Da) [32] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102

T/K

1.778 4.184 7.309 10.84 14.43 18.06 21.61 25.03 28.28 31.45 34.55 40.45 45.94

90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102 51.13 55.98 60.71 65.44 70.12 74.81 79.50 84.22 88.95 93.76 98.70 103.85 109.12

T/K 220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

Cp  102 114.56 120.16 125.86 131.59 137.40 143.30 149.24 155.23 161.34 167.11 145.35 160.42 167.40

Anhydrous native bovine serum albumin (66,433 Da) [33] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102

T/K

1.854 4.234 7.058 10.14 13.36 16.60 19.72 22.71 25.71 28.61 31.37 36.62 41.59

90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102

T/K

46.23 50.54 54.69 58.74 62.63 66.44 70.25 74.06 77.82 81.59 85.31 89.08 92.80

220 230 240 250 260 270 280 290 300 310

Cp  102 96.48 100.25 104.01 107.91 111.88 115.90 119.96 124.14 128.32 132.55

Anhydrous denatured bovine serum albumin (66,433 Da) [33] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102 1.866 4.272 7.176 10.36 13.62 16.87 20.00 23.05 26.08 29.06 31.92 37.10 41.88

T/K 90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102 46.44 50.79 54.94 58.99 62.93 66.78 70.63 74.43 78.20 81.92 85.69 89.50 93.30

T/K 220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

Cp  102 97.15 101.04 105.02 109.08 113.22 117.45 121.71 126.02 130.37 134.77 118.78 129.54 134.81

THERMODYNAMIC PROPERTIES OF PROTEINS

/

135

TABLE 8.4. Continued. Hydrated (2.14% H2 O) bovine serum albumin (66,433 Da) [33] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102

T/K

1.678 4.038 7.079 10.54 13.70 16.96 20.11 23.18 26.20 29.09 31.90 37.29 42.34

Cp  102

90 100 110 120 130 140 150 160 170 180 190 200 210

47.07 51.46 55.65 59.75 63.68 67.66 71.67 75.73 79.75 84.31 87.70 91.71 95.73

T/K 220 230 240 250 260 270 280 290 300 310

Cp  102 99.75 103.81 107.86 111.92 116.06 120.33 124.52 128.78 133.05 137.28

Anhydrous bovine serosal collagen (280,950 Da) [33] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102

T/K

1.774 4.105 6.891 10.00 13.18 16.33 19.42 22.37 25.28 28.07 30.77 35.94 40.71

90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102 45.10 49.20 53.22 57.11 60.88 64.56 68.16 71.71 75.23 78.83 82.43 86.02 89.58

T/K 220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

Cp  102 93.09 96.65 100.25 103.93 107.70 111.46 115.31 119.12 122.93 126.73 112.68 122.22 126.78

Hydrated (13.53% H2 O) bovine serosal collagen (280,950 Da) [33] T/K 10 15 20 25 30 35 40 45 50 55 60 70 80

Cp  102 1.536 3.895 6.945 10.41 14.00 17.68 21.26 24.53 27.86 31.10 34.16 39.95 45.44

T/K 90 100 110 120 130 140 150 160 170 180 190 200 210

Cp  102 50.63 55.52 60.25 64.94 69.58 74.35 78.87 83.55 88.28 93.09 98.07 103.14 108.32

T/K 220 230 240 250 260 270 280 290 300 310 273.15 298.15 310.15

Cp  102 113.60 118.95 124.52 130.29 136.23 142.34 148.57 155.02 161.63 168.41 144.31 160.42 168.49

136 / CHAPTER 8 TABLE 8.4. Continued. Rat tail tendon collagen at different water content, R (grams of water per gram of protein) [34] Temperature (8C) R (g/g)


60


40


20

0

20

3.033 2.816 2.812 2.669 2.481 2.339 2.259 2.176 2.000

3.117 2.908 2.900 2.812 2.586 2.435 2.335 2.247 2.155

3.293 3.138 2.966 2.862 2.715 2.607 2.552 2.481 2.393

3.548 3.439 3.314 3.264 3.130 2.853 2.778 2.653 2.569

Before denaturation 1.44 1.01 0.93 0.79 0.65 0.527 0.468 0.428 0.376

1.582 1.548 1.598 1.582 1.556 1.577 1.498 1.494 1.381

1.778 1.849 1.891 1.862 1.841 1.883 1.799 1.757 1.715

2.343 2.469 2.510 2.594 2.803 2.761 2.803 2.469 1.925 After denaturation

1.44 1.02 0.93 0.79 0.651 0.524 0.468 0.428 0.375

1.674 1.686 1.695 1.715 1.715 1.766 1.674 1.582 1.464

1.904 1.946 1.950 2.029 2.050 2.121 2.033 1.941 1.799

2.929 3.347 3.766 3.347 3.347 3.347 3.347 2.510 2.929

Keratin (merino wool) [35] Water content (% of weight of keratin) T / 8C








70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100

0

2.8

7.6

0.887 0.929 0.971 1.008 1.046 1.088 1.125 1.167 1.205 1.243 1.284 1.322 1.364 1.402 1.443 1.481 1.523 1.561

0.925 0.971 1.013 1.054 1.096 1.142 1.184 1.226 1.272 1.314 1.356 1.402 1.443 1.494 1.540 1.590 1.640 1.690

1.004 1.059 1.109 1.159 1.213 1.264 1.318 1.368 1.423 1.473 1.527 1.582 1.636 1.695 1.757 1.820 1.887 1.950

10.4 1.013 1.067 1.121 1.176 1.230 1.280 1.335 1.393 1.452 1.523 1.594 1.665 1.732 1.803 1.879 1.962 2.054 2.151

16.9 1.088 1.159 1.234 1.305 1.377 1.448 1.519 1.590 1.665 1.745 1.837 1.933 2.025 2.105 2.180 2.238 2.293 2.330

20.9 1.176 1.255 1.335 1.414 1.494 1.577 1.657 1.745 1.837 1.933 2.050 2.142 2.201 2.255 2.314 2.360 2.406 2.448

24.8 1.230 1.322 1.414 1.506 1.607 1.715 1.787 1.883 1.975 2.067 2.197 2.255 2.310 2.356 2.402 2.448 2.494 2.536

28.4 1.247 1.351 1.456 1.561 1.665 – – 1.992 2.079 2.167 2.255 2.330 2.372 2.402 2.439 2.494 2.540 2.586

30.7 1.264 1.364 1.464 1.565 1.682 – – 2.063 2.146 2.230 2.310 – – 2.460 2.494 2.540 – –

33.9 1.230 1.343 1.460 1.573 1.674 – – 2.079 2.159 2.234 2.318 2.356 2.393 2.439 2.481 2.519 2.556 2.602

THERMODYNAMIC PROPERTIES OF PROTEINS

/

137

TABLE 8.5. Temperature dependence of the partial molar heat capacities (J K
1 mol
1 ) of the peptide unit and of the side chains of amino acid residues for the temperature range 5–125 8C [18]. Temperature (8C)

–CHCONH– ---Ri Ala Arg Asn Asp Cys Gln Glu Gly His Ile Leu Lys Met Phe Pro Ser Thr Trp Tyr Val

5

25

50

75

100

125

3.7

15.2

26.2

29.8

33.7

33.7

175.7 204.6 72.9 72.8 225.4 168 168.3 82.3 205.7 406.8 385.9 215.1 197.1 395.7 214.6 75.6 194.2 471.2 310.6 324.6

166.7 273.4 88.8 89 237.6 180.2 179 78 179.6 402.3 381.7 249.8 175.9 383 177.7 81.2 184.5 458.5 301.7 314.4

156.2 305.8 109.8 106.2 250.8 193.4 192 71.7 177.2 397.1 377.8 266.9 158.1 370.3 152.3 85.7 182.2 445.8 295.2 305

144.7 315.1 125.2 124.5 260.7 203.3 203.7 66.4 179.6 390.8 372.9 274.4 150.3 358.4 142.8 91.4 186.5 433.9 294.5 294.7

134.6 318.7 140.5 140.7 268.2 210.8 211.4 59.7 187.1 386 369.4 278.1 148.1 348.3 135.6 97.3 199 423.8 300.1 285.7

124.1 318.5 154.2 154.3 276.1 218.7 217.8 53.9 196.8 380.8 365.5 274.4 143.9 339.6 130.1 102.1 216.2 415.1 304 269.6

TABLE 8.6. Temperature dependence of the partial molar volumes (cm 3 mol
1 ) of the peptide unit and of the side chains of amino acid residues for the temperature range 5–125 8C [11]. Temperature (8C)

–CHCONH– ---Ri Ala Arg Asn Asp Cys Gln Glu Gly His Ile Leu Lys Met Phe Pro Ser Thr Trp Tyr Val

5

25

50

75

100

125

28.3

28.0

28.6

28.9

29.0

29.4

24.8 81.2 44.1 39.0 40.5 58.4 54.9 10.9 65.1 64.5 75.0 72.4 73.3 82.0 36.0 27.1 44.2 110.9 88.2 60.7

27.2 85.7 45.7 41.5 40.5 61.0 57.5 10.3 64.0 66.1 77.6 73.5 72.1 86.3 35.8 27.8 44.7 110.9 90.0 63.0

30.1 88.4 47.8 44.5 40.5 63.2 60.7 9.6 63.1 67.7 80.2 75.8 70.2 92.0 34.1 28.8 45.9 110.9 92.2 65.7

33.1 90.4 49.3 47.3 40.5 66.0 63.6 9.2 62.4 69.8 82.9 78.5 69.0 97.9 33.7 29.7 47.6 110.9 94.1 68.5

36.3 92.1 50.8 49.9 40.5 68.5 66.3 8.9 61.8 71.8 86.5 81.3 67.9 102.6 33.5 30.7 50.9 110.9 95.9 71.7

39.4 93.5 52.4 52.6 40.5 69.7 69.3 8.5 61.1 74.0 89.4 84.8 66.7 107.9 33.2 32.0 54.6 110.9 97.8 74.5

138 / CHAPTER 8 TABLE 8.7. Thermodynamics characteristics of the proteins (partial molar heat capacity of the native, CpN (kJ K
1 mol
1 ),
1
1 and unfolded state, CpU (kJ K
1 mol
1 ), and the heat capacity, DU mol
1 ), enthalpy, DU ), entropy, N Cp (kJ K N H(kJ mol
1
1 U
1 DU S(J K mol ), and Gibbs energy changes upon unfolding, D G(kJ  mol ))*. N N Temperature (8C) Protein ROP1

2

SH3 domain

25

50

75

CpN CpU DU N Cp

7.9 11.9 7.9

9.0 12.9 7.9

10.4 13.9 7.0

11.7 14.2 5.0

13.1 14.5 2.9

14.4 14.6 0.4

DU NH CpN CpU DU N Cp

107 9.7 14.7 5.0

265 10.6 15.4 4.8

451 11.7 16 4.3

580 12.8 16.4 3.6

601 13.9 16.6 2.7

700 15.0 16.6 1.6


32
166 14.1 8.7

52 126 14.5 9.5

146 428 7.8 10.5

222 653
5.2 11.4

274 797
23.3 12.4

296 856
44.7 13.4

CpU DU N Cp DU NH DU NS

11.5 2.8 72 87

12.5 3.0 130 288

13.1 2.6 200 514

13.5 2.1 259 690

13.7 1.3 303 809

13.7 0.3 323 864

DU NG CpN CpU DU N Cp

47.8 12.4 15.8 3.4

44.2 13.3 16.9 3.6

34.0 14.5 17.8 3.3

18.9 15.6 18.0 2.4

1.2 16.7 18.2 1.5


20.9 17.8 18.0 0.2

DU NH

66

135

221

292

341

362

DU NS DU NG

119

360

636

850

986

1041

DU NH DU NS DU NG CpN

BPTI3

4

CI2

Eglin-c

5

5

Protein G

27.7

15.6


3.8


26.8


52.3

CpN

11.9

12.9

14.1

15.4

16.6

17.9

CpU

16.1

16.8

17.6

17.9

18.2

18.1

4.2

3.9

3.5

2.5

1.6

0.2

DU NH

33

115

208

283

335

358

DU NS DU NG


22

262

561

786

929

989

39.1

36.9

26.8

9.5


11.5


35.6

CpN

8.1

8.9

9.9

10.9

11.9

12.9

CpU

11.7

12.5

13.2

13.5

13.9

13.9

3.6

3.6

3.3

2.6

2.0

1.0

DU N Cp

Tendamistat

7

125

32.9

DU N Cp

6

100

DU NH


4

67

153

227

283

320

DU NS DU NG


103

145

422

643

801

897

24.6

23.8

16.7

3.2


15.8


37.0

CpN

11.4

12.1

13.0

13.9

14.8

15.7

CpU

14.9

15.7

16.6

16.8

17.1

17.1

3.5

3.6

3.6

2.9

2.3

1.4

DU N Cp DU NH


22

70

176

262

321

351

DU NS DU NG


213

109

452

711

877

955


6.1


29.1

37.2

37.5

30.0

14.6

THERMODYNAMIC PROPERTIES OF PROTEINS

/

139

TABLE 8.7. Continued. Temperature (8C) Protein Ubiquitin

8

5

25

50

75

CpN

11.2

12.6

14.3

16.0

17.7

19.4

CpU

17.0

18.3

19.4

19.8

20.2

20.2

5.8

5.7

5.1

3.8

2.5

0.8

DU N Cp DU NH DU NS DU NG RNase T1

9

Met-J

11

Cytochrome c

393


444


44

393

727

959

1068

CpN

14.6

16.1

17.8

19.7

21.9

24.3

CpU

20.1

21.4

22.8

24.0

24.9

25.1

5.5

5.3

5.0

4.3

3.0

0.8

173

281

410

528

621

672

444

817

1233

1584

1845

1976

49.6

37.5

11.7


23.2


67.2


114.4

CpN

37.8

40.2

43.2

46.2

49.2

52.2

CpU DU N Cp DU NH

45.8

49.2

52.1

53.2

54.3

54.3

8.0

9.0

8.9

7.0

5.1

2.1

CpN

15.8

17.5

19.7

21.9

24.0

26.2

CpU

22.7

24.3

25.8

26.3

26.8

26.8

6.9

6.8

6.1

4.4

2.8

0.6

92

270

498

692

832

902


53

89

268

421

532

593


319

174

752

1210

1520

1681

35.7

37.1

25.1


1.1


35.0


76.0

CpN

16.5

18.5

21.0

23.5

26.0

28.5

CpU

23.6

25.4

26.9

27.4

28.0

28.1

7.1

6.9

5.9

3.9

2.0

0.1

167

307

467

590

664

690

379

866

1384

1752

1959

2029

61.6

48.9

20.0


19.7


66.7


117.5

CpN

19.5

20.8

22.5

24.2

25.8

27.5

CpU

24.1

26.0

27.8

28.5

29.3

29.5

4.6

5.2

5.3

4.3

3.5

2.0

DU NH DU NS DU NG Lysozyme

351


32.1

DU N Cp

11

273


6.7

DU NH DU NS DU NG RNase A

162

20.0

DU N Cp

11

27

35.1

DU NH DU NS DU NG Barnase


88

40.1

DU N Cp

12

125

35.4

DU N Cp DU NH DU NS DU NG 10

100

220

294

405

512

603

664

641

896

1254

1574

1826

1989

41.8

27.0


0.0


35.8


78.1


127.6

CpN

18.2

20.0

22.2

24.4

26.7

28.9

CpU

26.7

29.1

31.1

31.8

32.4

32.5

8.5

9.1

8.9

7.4

5.7

3.6

DU N Cp DU NH

111

242

408

562

683

753

140 / CHAPTER 8 TABLE 8.7. Continued. Temperature (8C) Protein DU NS 13

Interleukin 1b

T4-Lyz

14

15

Papain

Chymotrypsin

15

15

Pepsinogen

1

2

164

618

50

75

100

125

1153

1615

1954

2138

65.4

57.8

35.6


0.0


45.8


97.9

CpN

26.4

28.1

30.1

32.2

34.3

36.4

CpU

33.3

35.6

37.6

38.4

39.2

39.2

6.9

7.5

7.5

6.2

4.9

2.8

DU NS Myoglobin

25

DU NG

DU N Cp DU NH

11

5

7

151

330

501

640

736


99

401

1006

1516

1903

2155

DU NG

34.5

31.5

5.1


26.6


69.8


121.7

CpN

21.8

24.2

27.3

30.3

33.4

36.5

CpU

35.6

37.6

39.5

40.0

40.5

40.3

DU N Cp DU NH

14.4

14.0

12.8

10.3

7.7

4.4


231

6

291

555

774

920

DU NS


919


116

805

1595

2207

2588

DU NG

24.5

40.6

31.0


0.1


49.2


110.0

CpN

25.2

28.1

31.8

35.5

39.1

42.8

CpU DU N Cp

36.2

39.1

41.5

42.2

43.1

43.0

11.0

11.0

9.7

6.7

4.0

0.2

DU NH

20

240

499

671

805

856

DU NS


190

576

1413

1928

2302

2439

DU NG

72.8

68.4

42.6

0.1


53.6


114.7

CpN

28.1

32.0

36.8

41.7

46.5

51.4

CpU DU N Cp

45.0

48.0

50.5

51.3

52.1

51.9

16.9

16.0

13.7

9.6

5.6

0.5

DU NH


166

164

535

826

1015

1091

DU NS


911

236

1438

2312

2840

3042

CpN

34.3

37.7

42.0

46.3

50.6

54.8

CpU DU N Cp DU NH

49.2

51.8

54.3

55.1

56.0

55.8

14.9

14.1

12.3

8.8

5.4

1.0


21

268

598

862

1039

1119

DU NS


260

746

1813

2602

3099

3312

DU NG

51.3

45.7

12.4


43.5


116.9


199.2

CpN

44.2

51.7

61.1

70.4

79.8

89.2

CpU DU N Cp DU NH

78.4

82.3

86.3

87.7

89.6

89.7

34.2

30.6

25.2

17.3

9.8

0.5

DU NS 3


577
2279 4

5

72

770

1301

1639

1767


19

2242

3739

4687

5030

6

7

8

9

ROP[36]; SH3[37]; BPTI[5]; CI-2[38]; Eglin-c[39]; protein G[40]; Tendamistat[41]; Ubiquitin[42]; RNase T1[43]; 10MetJ[44]; 11Cytochrome c, RNase A, Lysozyme, Myoglobin[45,46]; 12Barnase[47]; 13Interleukin-1b[48]; 14T4-Lyz[49]; 15Papain, Chymotrypsin, Pepsinogen[50]. *Reported entropies and Gibbs energies are for the conditions of maximal stability [17]. The entropies and Gibbs energy of unfolding for rop and met-J are not available since the unfolding represents a bimolecular two-state process.

THERMODYNAMIC PROPERTIES OF PROTEINS

/

141

TABLE 8.8. Normalized values of the heat capacities (DC^phyd in J K
1 mol
1 A˚
2 ), enthalpies (DH^ hyd in J mol
1 A˚
2 ), ^ hyd in J K
1 mol
1 A˚
2 ) of hydration of various surfaces [17]. entropies (DS^hyd in 10
3 J K
1 mol
1 A˚
2 ), and Gibbs energies (DG p Temperature (8C) 5 Aliphatic

^ hyd DC p

2.14

50 Nonpolar surfaces 2.03

75 1.91

100 1.80

125 1.66

^ hyd


166


122


70


21

26

69

^ hyd


730


578


409


263


134


22

^ hyd DG

37

50

62

71

75

77

DH DS

Aromatic

2.24

25

^ hyd DC p

1.65

1.55

1.41

1.29

1.19

1.09

^ hyd DH


180


148


111


77


46


18

^ hyd


430


319


199


98


12

62

^ hyd DG


61


53


47


43


42


43

DS

Polar surfaces of: Arg

Asn

Asp

Cys

Gln

^ hyd DC p


0.20


0.12


0.04

0.01

0.08

^ hyd DH


821


827


831


833


834


833

^ hyd DS


458


478


492


497


498


495

^ hyd DG


694


685


672


660


647


635

^ hyd DC p


1.27


1.01


0.67


0.41


0.16

0.09

^ hyd

DH


871


894


915


928


936


936

^ hyd DS


575


654


723


763


783


785

^ hyd DG


711


699


681


663


643


623

^ hyd DC p


1.72


1.40


1.07


0.71


0.40


0.11

^ hyd

DH


684


715


746


768


782


788

^ hyd DS


360


469


569


636


675


691

^ hyd DG


584


575


562


547


530


513

^ hyd DC p

1.80

2.01

2.23

2.42

2.54

2.70

^ hyd

DH


309


271


218


160


98


32

^ hyd DS


535


402


232


59

113

283

^ hyd DG


160


151


143


139


140


145

^ hyd DC p


0.38


0.22


0.06

0.07

0.17

0.30

^ hyd


697


703


706


706


703


697

^ hyd

DS


571


591


604


603


594


579

^ hyd DG


538


527


511


497


481


467

DH

Glu


0.38

^ hyd DC p


0.71


0.55


0.35


0.17


0.05

0.09

^ hyd


549


562


573


580


583


582

^ hyd

DS


392


436


473


492


500


498

^ hyd DG


440


432


420


409


396


383

DH

142 / CHAPTER 8 TABLE 8.8. Continued. Temperature (8C)

His

^ hyd DC p

Met

Ser

Thr

CONH

75

100

125


1.96


2.43


2.38


2.26


2.07


1.82


1,084


1,128


1,188


1,247


1,301


1,349


542


693


888


1060


1211


1337

^ hyd


933


922


901


878


848


816

^ hyd DC p


1.31


1.53


1.59


1.36


1.15


0.94

^ hyd DH


685


714


753


789


821


847

^ hyd DS


384


482


609


716


804


870

^ hyd DG


578


570


556


540


519


498

^ hyd DC p


3.51


3.83


4.07


4.04


3.91


3.75

^ hyd DH


399


473


572


672


774


869

^ hyd DS


158


412


732


1031


1308


1555

^ hyd DG


356


350


335


315


283


247

^ hyd DC p


1.62


1.40


1.20


0.96


0.72


0.48

^ hyd DH


1015


1045


1078


1104


1126


1140

^ hyd DS


878


983


1089


1168


1227


1265

^ hyd DG


771


752


726


698


667


636

^ hyd DC p


1.09


1.29


1.22


0.89


0.29

0.55

^ hyd


1262


1287


1318


1343


1359


1356

^ hyd DS


971


1053


1156


1232


1274


1265

^ hyd DG


992


972


944


916


881


850

^ hyd DC p

1.05

0.96

1.07

1.08

1.03

1.05

^ hyd


1,181


1,161


1,135


1,110


1,084


1,055

^ hyd DS


766


693


615


534


460


392

^ hyd DG


968


954


936


924


912


899

DH

Tyr

50

^ hyd DS

DH

Trp

25

^ hyd DH

DG Lys

5

^ hyd DC p


1.46


1.48


1.36


1.15


0.86


0.59

^ hyd

DH


824


854


889


921


946


963

^ hyd DS


314


415


531


625


695


742

^ hyd DG


735


730


717


703


686


667

^ hyd DC p


2.08


1.81


1.56


1.53


1.49


1.55

^ hyd


1662


1702


1745


1785


1823


1862

^ hyd


890


1026


1162


1278


1383


1481

^ hyd DG


1415


1396


1370


1340


1307


1272

DH DS

THERMODYNAMIC PROPERTIES OF PROTEINS REFERENCES 1. Zhang G, Lebedev BV, Wunderlich B, Zhang JY J. P.S.P. B-P. P. 1995; 33: 2449–2455. 2. Privalov PL, Khechinashvili NN J. Mol. Biol. 1974; 86: 665–684. 3. Privalov PL, Tiktopulo EI, Venyaminov SY, Griko YV, Makhatadze GI, Khechinashvili NN. J. Mol. Biol. 1989; 205: 737. 4. Sturtevant JM. Ann. Rev. Phys. Chem. 1987; 38: 463–488. 5. Makhatadze GI, Kim KS, Woodward C, Privalov PL. Prot. Sci. 1993; 2: 2028. 6. Makhatadze GI. Heat capacities of amino acids, peptides and proteins. Biophys. Chem. 1998; 71: 133–156. 7. Privalov PL, Makhatadze GI. J. Mol. Biol. 1990; 213: 385. 8. Privalov PL, Makhatadze GI. J. Mol. Biol. 1992; 224: 715–723. 9. Richardson JM, McMahon KW, MacDonald CC, Makhatadze GI. Biochemistry 1999; 38: 12869–12875. 10. Richardson JM, Makhatadze GI. Temperature dependence of the thermodynamics of helix-coil transition. J. Mol. Biol. 2004; 335: 1029–1037. 11. Makhatadze GI, Medvedkin VN, Privalov PL. Biopolymers 1990; 30: 1001–1010. 12. Durchschlag H. In: Hinz H-J, editor. Thermodynamic Data for Biochemistry and Biotechnology. New York: Springer-Verlag; 1986. p 45–128. 13. Lee BK, Richards FM. J. Mol. Biol. 1971; 55: 379–400. 14. Miller S, Janin J, Lesk AM, Chothia C. J. Mol. Biol. 1987; 196: 641–656. 15. Wesson L, Eisenberg D. Protein Sci. 1992; 1: 227–235. 16. Oobatake M, Ooi T. Prog. Biophys. Mol. Biol. 1993; 59: 237–284. 17. Makhatadze GI, Privalov PL. Adv. Protein Chem. 1995; 47: 307. 18. Makhatadze GI, Privalov PL. J. Mol. Biol. 1990; 213: 375. 19. Izatt RM, Christensen JJ. In: Fasman GD, editor. The CRC Handbook of Biochemistry and Molecular Biology, Physical and Chemical Data Boca Raton, FL: CRC Press 1976. p 151–269. 20. Zamyatnin AA. Prog. Biophys. Mol. Biol. 1972; 24: 107–123. 21. Hutchens JO, Cole AG, Stout JW. J. Am. Chem. Soc. 1960; 82: 4813. 22. Cole AG, Hutchens JO, Stout JW. J. Phys. Chem. 1963; 67: 2245. 23. Hutchens JO, Cole AG, Robie RA, Stout JW. J. Biol. Chem. 1963; 238: 2407.

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CHAPTER 9

Heat Capacities of Polymers Jianye Wen ALZA Corporation, 1900 Charleston Road, Mountain View, CA 94039

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Heat capacity is a fundamental property of any material. It can be measured by adiabatic calorimetry (0–100 K), differential scanning calorimetry (DSC) (above 100 K), and some other techniques [2–4]. Theoretical interpretation of heat capacity has been carried out by various researchers through analysis based on the separation of the vibrational spectrum into group and skeletal vibrations as well as normal mode and their dispersion analysis [5–8]. The table in this chapter contains the heat capacity data in the literature for selected polymers. The heat capacity for any given polymer is a temperature-dependent quantity. Due to the space limitations, the heat capacity values at selected temperatures are listed in this table for the selected polymers. Temperatures are chosen such that at least two data points are included in both the glassy and rubbery regions. A crude extrapolation could be used to compare experimental data at other temperatures. The specific heat increment at Tg , DCp , is also given for the selected polymers. Readers can find more detailed information about DCp from the original references. Since heat capacity also depends on the state of the polymer, the state of the polymer is specified by the following abbreviations whenever possible:

9.1 INTRODUCTION The heat capacity of a substance can be defined as the amount of heat required to change its temperature by one degree. A more useful quantity is specific heat capacity, which is the amount of heat required to change the temperature of one unit mass of a material by one degree. Heat capacity is a fundamental property of any material. It is a macroscopic parameter that can be linked to molecular structure and vibrational motions at microscopic level [1]. Heat capacity under constant pressure (Cp ) is defined as the heat quantity which is required to increase the temperature of the unit mass of a material by 1 K or 18C under constant pressure. It is given by the following equation: Cp ¼ DQ=mDT

(unit: J=(g K) ),

(9:1)

where DQ is the required heat quantity (in Joules), m is the mass of the sample (in grams or kilograms), DT is the temperature increase from T1 to T2 (in degrees Celsius or Kelvins). The molar heat capacity under constant pressure is defined as that heat quantity which is required to heat the unit mole of a material through 1 K or 18C under constant pressure (in J/(mol K)). The experimental heat capacities are measured at constant pressure. However, in order to link heat capacity with the vibrational spectrum, Cp must first be converted to the heat capacity at constant volume. At constant volume, heat capacity can be defined as follows; Cv ¼ DQ=mDT

(unit: J=(g K) ):

a ¼ amorphous; c ¼ crystalline; s ¼ solid; m ¼ melt; sc ¼ semicrystalline; g ¼ glassy: The description of the theory of heat capacity and the application of heat capacity measurements have been given by Wunderlich and other researchers [2,3–17]. The most comprehensive and updated heat capacity data are collected in the ATHAS data bank (Advanced THermal AnalysiS) which has been developed over the last 25 years by Wunderlich (Chemistry Department, The University of Tennessee), and coworkers.

(9:2)

The relationship between two heat capacities is Cp
Cv ¼ Va2 T=KT

145 154

(9:3)

where a is the thermal expansion coefficient and KT is the isothermal bulk modulus.

145

146 / CHAPTER 9 TABLE 9.1. Heat capacities of selected polymers. Cp b Polymer

Moleculara Temperature Abbreviations weight (g/mol) Tg (K) (K) (kJ/(kg K))

(J/(mol K))

DCp c (J/(mol K))

Ref.

1. Main-chain carbon polymers Poly(iso-butyl acrylate)

PiBA

128.17

Poly(n-butyl acrylate)

PnBA

128.17

Poly(ethyl acrylate)

PEA

100.12

Poly(methyl acrylate)

PMA

86.09

1,4-Poly(butadiene) cis-

PBD

54.09

Poly(acrylics) 249 220 240 300 500 218 80 180 300 440 249 90 200 300 500 279 100 200 300 500 Poly(dienes)

1.2156 1.3365 1.8108 2.3388 0.5598 1.0632 1.8201 2.1803 0.5792 1.0301 1.7867 2.2189 0.6154 0.9816 1.765 2.143

155.80 171.30 232.09 299.77 71.75 136.27 233.28 279.45 57.99 103.13 178.88 222.16 52.98 84.51 151.99 184.49

50 150 300 350 50 150 300 500 100 200 300 600

0.3694 0.8967 1.960 2.214 0.3465 0.9057 NA 2.616 0.6733 1.2190 2.086 3.071

19.98 48.50 106.00 114.90 18.74 48.99 NA 141.50 37.78 68.40 117.02 172.31

30 130 300 450 190 30 130 300 450 Poly(alkenes) 252 100 200 300

0.2140 0.7775 1.924 2.409 0.1761 0.7898 1.924 2.409 0.674 1.110 1.555 2.202 3.127 0.7020 1.3319 1.903 2.079

171

trans-

180

Poly(1-butene)

PB

56.11

Poly(1-butenylene) cis-

PBUT

55.10

249

171

trans-

Poly(ethylene)

PE

14.03

Poly(1-hexene)

PHE

84.16

223

600 100 200 250 290

36.60

[18]

45.40

[18]

45.60

[18]

42.30

[18]

29.10

[19,20]

28.20

[19,20]

23.06

[19]

11.79 42.838 106.03 132.73 9.704 43.516 106.03 132.73

28.91

[3,19,20]

26.48

[3,19,20]

9.45 (c) 15.57 21.81 (s) 30.89 (m) 43.87 59.08 (a) 112.09 160.18 (a) 174.98 (a)

10.1

[21]

25.1

[19]

HEAT CAPACITIES OF POLYMERS

/

147

DCp c (J/(mol K))

Ref.

TABLE 9.1. Continued. Cp b Polymer

Abbreviations

Moleculara weight (g/mol)

Tg (K)

Temperature (K)

(kJ/(kg K))

(J/(mol K))

200

50 150 300 380

0.2440 0.8660 1.962 2.311

13.69 (a) 48.59 110.09 (a) 129.66

22.29

[19]

200

50 150 300 360 80 180 250 300 200 220 300 470 100 200 300

0.3573 0.9025 1.911 2.216 0.5610 1.090 1.4449 1.728 1.253 1.338 2.058 2.770 0.6238 1.132 1.622 2.099 3.178

24.34 61.48 130.20 144.80 47.21 91.75 121.60 145.40 87.90 93.82 144.34 194.32 26.25 (c) 47.63 (c) 68.24 (s) 88.34 (m) 133.73 (a)

30.87 (a)

[19]

33.7 (a)

[19]

27.03 (a)

[19]

17.37

[22]

0.5472 1.1557 1.8524 2.3673 1.2229 1.5710 2.0190 2.1127 0.5155 1.4666 1.9489 2.0462 1.8264 1.9091 2.2396 0.5248 0.9456 1.307 0.5904 1.032 1.395 0.5742 1.3755 2.0766 2.4323

77.81 164.34 263.41 336.63 173.90 223.40 287.10 300.43 58.84 167.42 222.47 233.57 310.77 324.83 381.06 45.18 81.41 112.50 50.25 87.81 118.70 57.49 137.72 207.91 243.52

29.70

[18]

39.00

[18]

31.70

[18]

—

[18]

—

[18]

—

[18]

0.4548 1.2230 1.2730

47.37 (g) 127.38 132.58

Poly(isobutene)

PiB

56.11

Poly(2-methylbutadiene) cis-

PMBD

68.12

Poly(4-methyl-1-pentene)

P4MPE

84.16

303

Poly(1-pentene)

PPE

70.14

233

Poly(propylene)

PP

42.08

260

Poly(n-butyl methacrylate)

PnBMA

Poly(i-butyl methacrylate)

PiBMA

Poly(ethyl methacrylate)

PEMA

Poly(hexyl methacrylate)

PHMA

Poly(methacrylic acid)

PMAA

Poly(methacrylamide)

PMAM

Poly(methyl methacrylate)

PMMA

Poly(styrene)

PS

600 Poly(methacrylics) 142.20 293 80 200 300 450 142.20 326 230 300 350 400 114.15 338 80 300 350 380 170.25 268 270 300 420 86.09 — 100 200 300 85.11 — 100 200 300 100.12 378 100 300 400 550 Poly(styrenes) 104.15 373 100 300

33.5

[18]

30.7 (a)

[23] [4]

148 / CHAPTER 9 TABLE 9.1. Continued. Cp b Polymer

—, a-methyl

—, p-bromo-

—, p-chloro-

—, p-fluoro-

—, p-iodo-

—, p-methyl-

Poly(acrylonitrile)

Poly(chlorotrifluoroethylene)

Poly(tetrafluoroethylene)

Poly(trifluoroethylene)

Poly(vinyl chloride)

Poly(vinylidene chloride)

Poly(vinylidene fluoride)

Moleculara Temperature DCp c Abbreviations weight (g/mol) Tg (K) (K) (kJ/(kg K)) (J/(mol K)) (J/(mol K)) 400 1.9322 600 2.4417 PaMS 118.18 441 100 0.4712 300 1.2752 460 2.1868 490 2.3331 PBS 183.05 410 300 0.79650 350 0.92349 420 1.2651 550 1.4641 PCS 138.60 406 300 1.0229 350 1.19848 410 1.6331 550 1.9134 PFS 122.14 384 130 0.47611 200 0.62048 300 0.93079 380 1.2672 PIS 230.05 424 300 0.67607 400 0.89102 430 1.1145 550 1.2570 PMS 118.18 380 300 1.2743 350 1.4917 390 1.9449 500 2.2766 Poly(vinyl halides) and poly(vinyl nitriles) PAN 53.06 378 100 0.5695 200 0.9286 300 1.297 370 1.624 PC3FE 116.47 325 80 0.2787 200 0.6257 300 0.85945 320 0.90667 PTFE 50.01 240 100 0.3873 200 0.6893 300 0.9016 1.028 700 1.454 P3FE 82.02 304 100 0.4049 200 0.7128 300 1.078 PVC 62.50 354 100 0.4291 300 0.9496 360 1.457 380 1.569 PVC2 96.95 255 100 0.3745 200 0.5932 250 0.7115 300 NA PVF2 64.03 233 100 0.4435

201.24 254.30 55.69 150.70 (g) 258.44 275.72 145.800 169.045 231.582 267.995 141.780 166.110 226.345 265.195 58.152 75.786 113.687 154.773 155.53 204.980 256.41 289.17 150.600 176.290 229.846 269.05 30.22 49.27 68.83 86.16 32.46 72.87 100.10 105.60 19.37 34.47 45.09 (s) 51.42 (m) 72.69 33.21 58.46 88.40 26.82 (g) 59.35 (g) 91.08 98.05 36.31 57.51 68.98 NA 28.40

Ref.

25.3

[19]

31.9

[3,24]

31.1

[3,24]

33.3

[3,24]

37.9

[3,24]

34.6

[3,24]

—

[18]

—

[19]

7.82

[3,19,25]

21.00

[19]

19.37 (a)

[19]

70.26

[19]

22.80

[19]

HEAT CAPACITIES OF POLYMERS

/

149

TABLE 9.1. Continued. Cp b Polymer

Moleculara Temperature Abbreviations weight (g/mol) Tg (K) (K) (kJ/(kg K))

Poly(vinyl fluoride)

PVF

46.04

314

Poly(p-phenylene)

PPP

76.10

Others —

Poly(vinyl acetate)

PVAc

Poly(vinyl alcohol)

PVA

Poly(vinyl benzoate)

PVBZ

Poly(p-xylylene)

PPX

Poly(iminoadipoyliminododecamethylene)

Nylon 612

Poly(iminoadipoyliminohexamethylene)

Poly(iminohexamethyleneiminoazelaoyl)

Poly(iminohexamethyleneiminosebacoyl)

Nylon 66

Nylon 69

Nylon 610

150 230 250 300 100 200 300 310

0.6185 0.8918 0.7856 NA 0.5204 0.8692 1.301 1.353

80 0.3708 150 0.58135 250 0.92926 300 1.117 86.09 304 80 0.3230 300 1.183 320 1.8409 370 1.898 44.05 358 60 0.2674 150 0.7187 250 1.185 300 1.546 148.16 347 190 0.71808 300 1.1025 400 1.8390 500 2.0333 104.15 286 220 0.91445 250 1.0576 300 1.3022 410 1.8686 2. Main-chain heteroatom polymers

(J/(mol K)) 39.60 57.10 50.30 NA 23.96 40.02 59.91 62.29 28.22 (sc) 44.241 (sc) 70.717 (sc) 85.040 (sc) 27.81 101.86 158.48 163.37 11.78 31.66 52.21 68.11 106.39 163.35 272.47 301.25 95.241 (sc) 110.149 (sc) 135.622 (sc) 194.619 (sc)

DCp c (J/(mol K))

17.80 (a)

—

53.7

—

Ref.

[19]

[26,27]

[19]

[19]

69.5

[19]

37.6 (a)

[3,28]

310.48

Poly(amides) 319 230

1.2296

381.78

214.8 (a)

[3,29,30]

226.32

323

300 400 600 230

1.5926 2.4842 3.1596 1.1139

494.48 771.30 980.986 252.10

145.0 (a)

[31]

331

300 400 600 230

1.4638 2.3794 2.793 1.1980

331.30 538.50 632.1 321.53

—

[3,29,30]

323

300 400 600 230

1.5204 2.3840 3.0720 1.2069

408.080 639.874 824.534 340.870

—

[3,29,30]

300 400 600

1.5644 2.3975 3.1041

441.820 677.125 876.685

268.40

282.43

150 / CHAPTER 9 TABLE 9.1. Continued. Cp b Polymer Poly(imino-(1oxohexamethylene))

Poly(imino-1oxododecamethylene)

Poly(imino-1oxoundecamethylene)

Moleculara Temperature DCp c Abbreviations weight (g/mol) Tg (K) (K) (kJ/(kg K)) (J/(mol K)) (J/(mol K)) Nylon 6

Nylon 12

Nylon 11

113.16

197.32

183.30

Poly(methacrylamide)

PMAM

85.11

Poly(L-alanine)

PALA

71.08

Poly(L-asparagine)

PASN

114.10

Polyglycine

PGLY

57.05

Poly(L-methionine)

PMET

131.19

Poly(L-phenylalanine)

PPHE

147.18

Poly(L-serine)

PSER

87.08

Poly(L-valine)

PVAL

99.13

Poly(butylene adipate)

PBAD

200.24

Ref.

313

70

0.4400

49.78

93.6 (a)

[31]

314

300 400 600 230

1.5023 2.5186 2.7881 1.2874

170.00 285.00 315.50 254.020

—

[3,29,30]

316

300 400 600 230

1.6952 2.4709 3.2786 1.2996

334.49 487.565 646.945 238.21

—

[3,29,30]

300 400 600 — 100 200 250 300 Poly(amino acids) — 230 300 350 390 — 230 300 350 390 — 230 300 350 390 — 220 300 350 390 — 220 300 350 390 — 220 300 350 390 — 230 300 350 390 Poly(esters) 199 80 150 300 450

1.7507 2.4567 3.2449 0.5904 1.032 1.214 1.395

320.91 450.314 594.794 50.25 87.81 103.30 118.70

—

[18]

1.102 1.315 1.498 1.622 0.958 1.218 1.397 1.537 0.929 1.170 1.356 1.516 0.936 1.347 1.595 1.768 0.830 1.153 1.382 1.548 0.959 1.297 1.541 1.747 1.213 1.455 1.647 1.802

78.33 93.47 106.5 115.3 109.3 139.0 159.4 175.4 53.00 66.75 77.36 86.49 122.8 176.7 209.3 232.0 122.1 169.7 203.4 227.8 83.50 112.9 134.2 152.1 120.2 144.2 163.3 178.6

—

[32]

—

[33]

—

[32]

—

[33]

—

[33]

—

[33]

—

[32]

0.54302 0.87449 1.9706 2.2147

108.734 175.107 394.595 443.470

140.046

[26,27]

HEAT CAPACITIES OF POLYMERS

/

151

TABLE 9.1. Continued. Cp b Polymer Poly(butylene terephthalate)

Poly(ethylene terephthalate)

Moleculara Temperature Abbreviations weight (g/mol) Tg (K) (K) (kJ/(kg K)) PBT

PET

220.23

192.16

(J/(mol K))

DCp c (J/(mol K))

248

150

0.61075

134.505 (sc)

320

200 300 400 570 100

0.82262 1.6134 1.8187 2.1678 0.4393

181.166 355.311 400.532 477.407 84.42

1.172 1.8203 2.1136 0.95 1.45 1.79 2.15 NA

225.2 349.80 406.15 202 308 380 457 NA

—

[35]

—

[3,36]

342

106.77

Ref. [3]

77.812

77.8 (a)

[3,34,31]

Poly(tridecanolactone)

PTDL

212.34

237

Poly(trimethylene adipate)

PTMA

186.21

—

300 400 600 185 260 300 395 300

331

310 330 360 180

1.8710 1.9137 1.9776 0.73

348.401 356.341 368.252 150.53

—

[37]

—

250 350 400 300

0.97 1.54 1.79 NA

201.29 318.41 368.75 NA

—

[3,36]

310 330 360 100 210 300 350 100 200 300

1.8401 1.8721 1.9201 0.6012 1.024 1.810 1.870 0.62322 1.0243 1.4229 1.8138 1.9415 0.5250 1.127 1.999 2.098 0.5568 1.044 1.878 2.081 0.49910 1.1175 1.6395 1.7012

291.014 296.074 303.664 51.760 88.170 155.858 (m) 161.031 (m) 71.140 116.923 162.42 207.04 (s) 221.62 (m) 30.470 65.42 116.039 (m) 121.75 (m) 40.130 75.220 135.354 (m) 149.994 (m) 57.930 129.705 190.295 (m) 197.456 (m)

Poly(trimethylene terephthalate)

Poly(trimethylene succinate)

PTT

PTMS

206.2

158.15

Poly(g-butyrolactone)

PBL

86.09

214

Poly(e-caprolactone)

PCL

114.15

209

Poly(glycolide)

PGL

58.04

318

Poly(b-propiolactone)

PPL

72.07

249

Poly(ethylene oxalate)

PEOL

116.07

306

350 100 300 400 550 100 240 300 400 100 300 320 360

57.4

[3,31,38]

59.5

[3,31,38,39]

44.4

[3,31,38,39]

50.4

[3,31,38,39]

56.23

[3,40,41]

152 / CHAPTER 9 TABLE 9.1. Continued. Cp b Polymer

Moleculara Temperature Abbreviations weight (g/mol) Tg (K) (K) (kJ/(kg K))

Poly(ethylene sebacate)

PES

228.29

Poly(oxy-2,6-dimethyl-1, 4-phenylene)

PPO

120.15

245

120 200 300 410

Poly(oxides) 482 80

Poly(oxyethylene)

POE

44.05

206

Polyoxymethylene

POM

30.03

190

Poly(oxy-1,4-phenylene)

POPh

92.10

358

Poly(oxypropylene)

POPP

58.08

198

Poly(oxytetramethylene)

PO4M

72.11

189

Poly(oxytrimethylene)

PO3M

58.08

195

Poly(diethyl siloxane)

PDES

102.21

Others 135

Poly(dimethyl itaconate)

PDMI

158.16

377

Poly(dimethyl siloxane)

PDMS

74.15

146

Poly(4-hydroxybenzoic acid)

PHBA

120.11

434

300 500 570 100 200 300

0.66292 0.95269 1.9245 2.1923 0.4418

(J/(mol K)) 151.338 (s) 217.490 (sc) 439.34 (m) 500.500 (m) 53.08

330

1.2459 2.1232 2.2555 0.6114 0.9507 1.257 1.995 2.223 0.5554 0.7266 1.283 1.920 2.292 1.185 1.367 1.694 2.003 0.537 1.014 1.915 2.105 0.5465 1.033 1.985 2.081 0.5095 0.9464 1.373 2.055 2.107

149.70 255.10 271.00 26.93 (s) 41.88 (s) 55.36 (s) 87.89 (m) 97.91 16.68 (s) 21.82 (s) 38.52 (s) 57.67 (m) 68.83 109.10 (s) 125.90 (s) 156.00 (m) 184.50 (m) 31.21 (s) 58.89 (s) 111.23 (m) 122.27 (m) 39.41 (s) 74.52 (s) 143.15 (m) 150.04 (m) 29.59 (s) 54.97 (s) 79.73 (s) 119.34 (m) 122.37

50 100 300 360 110 300 400 450 50 100 300 340 170 300 400 434

0.38820 0.73995 1.6184 1.7525 0.59700 1.3183 1.9282 2.0009 0.3672 0.7131 1.591 1.657 0.58914 1.0207 1.3662 1.4686

39.678 (sc) 75.630 (sc) 165.417 (m) 179.125 (m) 94.419 (a) 208.507 (a) 304.968 (m) 316.463 (m) 27.23 52.88 118.0 122.9 70.762 122.60 164.091 176.399

450 100 150 300 600 300 350 400 600 80 180 300 370 80 180 300 340 80 180 300

DCp c (J/(mol K)) 154.059

Ref. [3,31]

31.9 (a)

[42]

38.96

[42,43]

27.47

[42,43]

21.4 (a)

[42]

32.15

[42]

46.49

[42]

50.73

[42]

30.189

[3,44–47]

54.23

[48]

27.7 (a)

[44,49,50]

34

[51]

HEAT CAPACITIES OF POLYMERS

/

153

Moleculara Temperature DCp c Abbreviations weight (g/mol) Tg (K) (K) (kJ/(kg K)) (J/(mol K)) (J/(mol K))

Ref.

TABLE 9.1. Continued. Cp b Polymer Poly(lactic acid)

PLA

72.07

Poly(4,4’-isopropylidene diphenylenecarbonate)

PC

254.27

Poly(oxy-1,4-phenylene-oxy-1, 4-phenylene-carbonyl-1, 4-phenylene)

Poly(oxy-1,4-phenylenesulphonyl-1, 4-phenyleneoxy-1,4-phenylene(1-methylidene)-1, 4-phenylene)

Poly(p-phenylenebenzobisoxazole)

Poly(1,4-phenylene sulphonyl)

PEEK

PBISP

PBO

PAS

Poly(1-propene sulphone)

P1PS

Trigonal selenium

SEt

288.30

442.54

234.21

140.16

332.5

0.97 1.32 2.09 2.16 0.43143

69.75 95.30 150.56 155.77 109.70 (s)

—

[52]

418

200 300 400 470 100

48.5

[44]

419

300 450 560 300

1.207 1.9570 2.207 NA

306.8 (s) 497.60 (m) 561.3 (m) NA

78.1

[53–55]

458.5

419 500 750 200

1.789 1.928 2.358 0.75870

515.8 555.9 679.8 335.754

102.482

[3,56,57]

—

300 500 540 10

1.1161 1.9436 2.0251 0.01

493.934 860.132 896.19 2.69

—

[58]

492.6

100 200 300 150

0.32 0.64 0.97 0.597

76.00 148.82 226.84 83.7

—

[57]

300 500 620 0.01580 30 100 300 400

1.009 1.571 1.642 1.677 1.165 0.2304 0.318 0.3338 0.4777 0.4343

141.4 220.2 230.1 — 123.7 18.19 (s) 25.11 26.36 (s) 37.72 (m) 34.29

106.14

10

78.96

303.4

600 a

[59] 13.29

[60]

This is the molecular weight of the repeat unit of the polymer. Except the data for PTDL and P1PS, Cp data reported in the unit of kJ/(kg K) were converted from the Cp data in J/(mol K) which were directly cited from the literature, using the molecular weight of the repeat unit. c Specific heat increment at Tg . b

154 / CHAPTER 9 REFERENCES 1. A. Einstein, Ann. Physik, 22, 180, 800, 1907. 2. B. Wunderlich, ‘‘Thermal Analysis’’, Academic Press, Inc., San Diego, CA, 1990. 3. M. Varma-Nair and B. Wunderlich, J. Phys. Chem. Ref. Data, 20, 349, 1991. 4. Y. Agari, A. Ueda, and S. Nagai, J. Polym. Sci., Polym. Phys. Ed., 33, 33, 1995. 5. B. Wunderlich, Thermochimica, 300, 43, 1997. 6. B. Wunderlich, Pure Appl. Chem., 67, 1019, 1995. 7. P. Tandon, V. D. Gupta, O. Prasad, S. Rastogi, and V. P. Gupta, J. Polym. Sci.: Part B: Polym. Phys., 35, 2281, 1997. 8. R. Pan, M. Y. Cao, and B. Wunderlich, ‘‘Polymer Handbook’’, 3rd ed., J. Brandrup and E. H. Immergut (eds.), Wiley-Interscience, New York, 1989, VI 371. 9. R. Agarwal, R. Misra, P. Tandon, and V. D. Gupta, Polymer, 45, 5307, 2004. 10. B. Wunderlich, S. Z. D. Cheng, and K. Loufakis, ‘‘Encyclopedia of Polymer Science and Engineering’’, Vol. 16, 2nd ed., Wiley & Sons Limited, New York, 1989, p. 767. 11. W. W. Wundlandt, ‘‘Thermal Analysis’’, 3rd ed., Wiley & Sons, Inc., New York, 1985. 12. W. Reese, J. Macromol. Sci. Chem., A3, 1257, 1969. 13. H. E. Bair, ‘‘ASTM STP 1249’’, R. J. Seyler (ed.), American Society for Testing and Materials, Philadephia, 1994, pp. 50–74. 14. H. Baur and B. Wunderlich, Adv. Polym. Sci., 7, 151, 1970. 15. T. Hatakeyama, H. Kanetsuna, and S. Ichihara, Thermochim. Acta, 146, 311, 1989. 16. Y. Jin and B. Wunderlich, Thermochim. Acta, 226, 155, 1993. 17. A. Boller, Y. Jin, and B. Wunderlich, J. Therm. Anal., 42, 307, 1994. 18. U. Gaur, S. F. Lau, B. B. Wunderlich, and B. Wunderlich, J. Phys. Chem. Ref. Data, 11, 1065, 1982. 19. U. Gaur, S. F. Lau, B. B. Wunderlich, and B. Wunderlich, J. Phys. Chem. Ref. Data, 12, 29, 1983. 20. J. Grebowicz, W. Avcock, and B. Wunderlich, Polymer, 27, 575, 1986. 21. U. Gaur and B. Wunderlich, J. Phys. Chem. Ref. Data, 10, 119, 1981. 22. U. Gaur and B. Wunderlich, J. Phys. Chem. Ref. Data, 10, 1051, 1981. 23. U. Gaur and B. Wunderlich, J. Phys. Chem. Ref. Data, 11, 313, 1982. 24. L. H. Judovits, R. C. Bopp, U. Gaur, and B. Wunderlich, J. Polym. Sci., Polym. Phys. Ed., 24, 2725, 1986. 25. S. F. Lau, H. Suzuki, and B. Wunderlich, J. Polym. Sci., Polym. Phys. Ed., 22, 379, 1984. 26. I. B. Rabinnovich, V. P. Nistratov, A. G. Babinkov, K. G. Shvetsova, and V. N. Larina, Vysokomol. Soedin, A26, 743, 1984. 27. I. B. Rabinnovich, V. P. Nistratov, A. G. Babinkov, K. G. Shvetsova, and V. N. Larina, Polym. Sci., U.S.S.R., 26, 826, 1984. 28. D. E. Kirkpatrick, L. Judovits, and B. Wunderlich, J. Polym. Sci., Polym. Phys. Ed., 24, 45, 1986. 29. A. Xenopoulos and B. Wunderlich, Polymer, 31, 1260, 1990. 30. A. Xenopoulos and B. Wunderlich, J. Polym. Sci., Polym. Phys. Ed., 28, 2271, 1990. 31. U. Gaur, S. F. Lau, B. B. Wunderlich, and B. Wunderlich, J. Phys. Chem. Ref. Data, 12, 65, 1983.

32. K. A. Roles, A. Xenopoulos, and B. Wunderlich, Biopolymers, 31, 477, 1992. 33. K. A. Roles, A. Xenopoulos, and B. Wunderlich, Biopolymers, 31, 753, 1992. 34. S. J. Collocot, J. Phys. Chem., Solid State Phys., 20, 2995, 1987. ˚ . Fransson, J. Polym. Sci., Polym. Phys. Ed., 32, 35. P. Skoglund and A 1999, 1994. 36. R. Pan, ‘‘Heat Capacities of Linear Macromolecules’’, PhD Thesis, Dept. of Chem., Rensselaer Polytechnic Inst., Troy, NY, 1987. 37. M. Pyda, A. Boller, J. Grebowicz, H. Chuah, B. V. Lebedev, B. Wunderlich, J. Polym. Sci.: Part B: Polym. Phys., Vol. 36, 2499– 2511, 1998. 38. B. Lebedev and A. Yevstropov, Makromol. Chem., 185, 1235, 1984. 39. R. Pan, Master Thesis, Dept. of Chem., Rensselaer Polytechnic Inst., Troy, NY, 1986. 40. B. V. Lebedev, T. G. Kulajina, Y. B. Lyudvig, and T. N. Ovchinnkova, Vysokomol. Soedin, A24, 1490, 1982. 41. B. V. Lebedev, T. G. Kulajina, Y. B. Lyudvig, and T. N. Ovchinnkova, Polym. Sci. USSR, 24, 1695, 1982 42. U. Gaur and B. Wunderlich, J. Phys. Chem. Ref. Data, 10, 1001, 1981. 43. H. Suzuki and B. Wunderlich, J. Polym. Sci., Polym. Phys. Ed., 23, 1671, 1985. 44. U. Gaur, S. F. Lau, and B. Wunderlich, J. Phys. Chem. Ref. Data, 12, 91, 1983. 45. B. V. Lebedev, T. G. Kulagina, U. S. Svistunov, V. S. Papkov, and A. A. Zhdanov, Vysokomol. Soedin, A26, 2476, 1984. 46. B. V. Lebedev, T. G. Kulagina, U. S. Svistunov, V. S. Papkov, and A. A. Zhdanov, Polym. Sci. USSR, 26, 2773, 1984. 47. J. P. Wesson, ‘‘Mesophase Transitions in Poly(diethyl siloxane)’’, PhD Thesis, Dept. of Chem., Rensselaer Polytechnic Inst., Troy, NY, 1988. 48. J. M. G. Cowie, I. J. McEwen, and B. Wunderlich, Macromolecules, 16, 1151, 1983. 49. V. A. Lebedev, N. N. Mukhina, and T. G. Kulagina, K. Vysokomol. Soedin. Ser., A20, 1297, 1978. 50. V. A. Turdakin, V. V. Tarasov, and A. K. Mal’tesv, Zh. Fiz. Khim., 50, 1980, 1976. 51. M. Y. Cao and B. Wunderlich, J. Polym. Sci., Polym. Phys. Ed., 23, 521, 1985. 52. M. Pyda, R. C. Bopp, and B. Wunderlich, J. Chem. Thermodyn., 36, 731, 2004. 53. D. J. Kemmish and J. N. Hay, Polymer, 26, 905, 1985. 54. S. Z. D. Cheng, M.-Y Cao, and B. Wunderlich, Macromolecules, 19, 1868, 1986. 55. S. Z. D. Cheng and B. Wunderlich, J. Polym. Sci., Polym. Phys. Ed., 24, 1755, 1986. 56. N. V. Novoselova, L. Y. Tsvetkova, I. B. Rabinovich, E. M. Moseeva, and L. A. Faninskaya, J. Phys. Chem., 59, 350, 1985. 57. M. Varma-Nair, J. Yimin, and B. Wunderlich, Polymer, 33, 5272, 1992. 58. K. Saito, Y. Takahashi, M. Sorai, J. Polym. Sci., Part B: Polym. Phys., Vol. 38, 1584–1588, 2000. 59. F. S. Dainton, D. M. Evans, F. E. Hoare, and T. P. Melia, Polymer, 3, 310, 1962. 60. U. Gaur and B. Wunderlich, J. Phys. Chem. Ref. Data, 10, 89, 1981.

CHAPTER 10

Thermal Conductivity Yong Yang Benjamin Moore and Company, Flanders, NJ 07836

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The units for thermal conductivity k are expressed as W/(m K) in SI units, Btu in./(ft2 h 8F) in English units, and cal=(cm s 8C) in cgs units. The corresponding units for heat flux are expressed as W/(m2 ), Btu/(ft2 h), and cal/(cm2 s), respectively. The conversion factors of units transform for k in these systems are listed in Part XI, Chapter 63. Polymers and foamed polymers. Temperature, pressure, density of polymer, orientation of chain segments, crystal structures, degree of crystallinity, and many other factors may significantly affect thermal conductivity of polymers [9–18]. Therefore the thermal conductivity values can be varied in literatures for same polymers. In addition, discrepancies also occur for thermal conductivity values obtained using different test methods [19]. The data in the tables in this chapter may be the representative values of thermal conductivity and are not necessarily the absolute ones. Generally, pure polymers have low thermal conductivities, ranging from 0:1 to 0:6 W=(m K), as listed in Table 10.1. Foaming polymers may further enhance this low thermal conductivity. Polymer foams with lower density have more air and thus have lower thermal conductivity. The cell size of foamed polymers may also have an effect on thermal conductivity. Smaller foam cell size tends to yield lower thermal conductivity. Most foamed polymers have thermal conductivity values in the order of 10
2 W=(m K), which is about 10 times less than the same polymers. Table 10.2 is the list of thermal conductivities for common commercial foamed polymers. The effect of temperature on polymers is of practical importance because most polymers are processed at relatively high temperature and have applications in a wide temperature range. The temperature dependence of thermal conductivity of polymers has been studied from extremely low temperatures at 0.1 K to above melting point [11,13– 18,20]. Generally, with increase in temperature, thermal conductivity for amorphous polymers increases gradually

Thermal conductivity of polymers is an important thermal property for both polymer applications and processing. Polymers typically have intrinsic thermal conductivity much lower than those for metals or ceramic materials, and therefore are good thermal insulators. Further enhancement of this thermal insulating quality can be achieved by foaming polymers. In other applications which require higher thermal conductivity, such as in electronic packaging and encapsulations, satellite devices, and in areas where good heat dissipation, low thermal expansion and light weight are needed, polymers reinforced with fillers, organic or inorganic, are becoming more and more common in producing advanced polymer composites for these applications [1–8]. Most polymeric materials are processed and fabricated at elevated temperatures, often above their melting temperatures. This process may be long and expensive because of the low thermal conductivity of polymers. Subsequently, the cooling process or annealing may also be controlled by heat transport properties of polymers, which eventually affect the physical properties of the materials. One example is crystalline polymers, for which the structural and morphological features may be significantly changed with the speed of cooling. Careful consideration in designing polymer processing is vital to achieve desired properties. Definition. For one-dimensional and rectilinear heat flow, the steady-state heat transfer in polymeric materials can be described by the Fourier’s law of heat conduction: q ¼
k

dT , dx

162

(10:1)

where q is the heat flux (i.e., the heat transfer rate per unit area normal to the direction of flow), x is the thickness of the material, dT/dx is the temperature gradient per unit length, and the proportionality constant k is known as the thermal conductivity.

155

156 / CHAPTER 10 TABLE 10.1. Thermal conductivity k of polymers. Polymer

Temperature (K)

k (W/m K)

Reference

Polyamides Polyauryllactam (nylon-12) Polycaprolactam (nylon-6) Moldings Crystalline Amorphous Melt Poly(hexamethylene adipamide) (nylon-6,6) Moldings Crystalline Amorphous Melt Poly(hexamethylene dodecanediamide)(nylon-6,12) Poly(hexamethylene sebacamide) (nylon-6,10) Polyundecanolactam (nylon-11)

0.25 0.19

[21] [22]

293 303 303 523

0.24 0.43 0.36 0.21

[19] [9] [9] [9]

239 303 303 523

0.24 0.43 0.36 0.15 0.22

[19] [9] [9] [23] [24]

0.22

[19]

0.23

[21]

Polycarbonates, polyesters, polyethers, and polyketones Polyacetal 0.3 0.23 Polyaryletherketone 293 0.30 Poly(butylene terephthalate) (PBT) 293 0.29 0.16 Polycarbonate (Bisphenol A) 293 0.20 Temperature dependence 300–573 a 150–400 a Poly(dially carbonate) 0.21 Poly(2,6-dimethyl-1,4-phenylene ether) 0.12 Polyester Cast, rigid 0.17 Chlorinated 0.33 Polyetheresteramide 303 0.24–0.34 353 0.20–0.26 Polyetheretherketone (PEEK) 0.25 Poly(ethylene terephthalate) (PET) 293 0.15 Temperature dependence 200–350 a Poly(oxymethylene) 293 0.292 293 0.44 Temperature dependence 100–400 a Poly(phenylene oxide) Molding grade 0.23

[21] [19,24] [24] [19] [9] [9,19] [13] [14] [19] [22] [19] [19] [25] [25] [22] [19] [26,27] [23] [26] [26] [21]

Epoxides Epoxy resin Casting grade Temperature dependence Polychlorotrifluoroethylene Poly(ethylene–tetrafluoroethylene) copolymer Polytetrafluoroethylene

Low temperature dependence Poly(tetrafluoroethylene–hexafluoropropylene)

293 300–500 Halogenated olefin polymers 293 311–460

293 298 345 5–20.8

0.19 0.19–0.34

[19] [28]

0.29 0.146–0.248 0.238

[19] [20] [22]

0.25 0.25 0.34 a

[19] [29] [29] [20]

THERMAL CONDUCTIVITY /

157

TABLE 10.1. Continued. Polymer Copolymer(Teflon EEP) Poly(vinyl chloride) Rigid Flexible Chlorinated Temperature dependence

Poly(vinylidene chloride) Poly(vinylidene fluoride)

Temperature (K)

293 293 293 103 273 373 293 293 298–433

Hydrocarbon polymers Polybutene Polybutadiene Extrusion grade 293 Poly(butadiene–styrene) copolymer (SBR) 23.5% Styrene content Pure gum vulcanizate Carbon black vulcanizate Polychloroprene (Neoprene) Unvulcanized 293 Pure gun vculcanizate Carbon black vulcanizate Polycyclooctene 80% trans content Poly(1,3-cyclopentylenevinylene) [poly(2-norbornene)] Polyethylene Low density Medium density High density Temperature dependence 20–573 Molecular weight dependence Poly(ethylene–propylene) copolymer Polyisobutylene Polyisoprene (natural rubber) Unvulcanized Pure gum vulcanizate Carbon black vulcanizate Poly(4-methyl-1-pentene) 293 Polypropylene 293 Temperature dependence Polystyrene

273 373 473 573 673

Poly(p -xylylene) (PPX)

k (W/m K)

Reference

0.202

[23]

0.21 0.17 0.14 0.129 0.158 0.165 0.13 0.13 0.17–0.19

[19] [19] [19] [23] [23] [23] [23] [19,21] [22]

0.22

[22]

0.22

[19]

0.190–0.250 0.300

[9] [9]

0.19 0.192 0.210

[9,19] [9] [9]

0.27 0.29

[4] [9]

0.33 0.42 0.52 a a 0.355 0.13

[19] [19] [19] [9,13,20] [20] [9] [29]

0.13 0.15 0.28 0.167 0.12 0.2 a 0.105 0.128 0.13 0.14 0.160 0.12

[9,19] [9] [9] [22] [19] [9] [15–17] [9] [9] [9] [9] [22] [22]

0.07

[24]

0.11 0.23–0.50 a

[19,24] [24] [14]

Polyimides Polyetherimide Polyimide Thermoplastic Thermoset Temperature dependence

293 300–500

158 / CHAPTER 10 TABLE 10.1. Continued. Polymer

Temperature (K)

k (W/m K)

Reference

Phenolic resins Poly(phenol–formaldehyde) resin Casting grade Molding grade Poly(phenol–furfural) resin Molding grade

293

0.15 0.25

[19] [19]

0.25

[19]

Polysaccharides Cellulose Cotton Rayon Sulphite pulp, wet Sulphite pulp, dry Laminated Kraft Paper Alkali cellulose Different papers Cellulose acetate Cellulose acetate butyrate Cellulose nitriate Cellulose propionate Ethylcellulose

[23] [19] [19] [24] [21] [30]

0.25 0.20 0.20 0.17

[31] [31] [31] [31]

0.158 0.150 0.144 0.143 0.136 0.127 0.141 0.137 0.132

[32] [32] [32] [32] [32] [32] [32] [32] [32]

0.18 0.18 0.29 0.288 0.18 0.26

[9] [9] [21] [9] [22] [9]

293 293

0.21 0.31

[19] [19]

Vinyl polymers 293

0.26

[19,24]

333 413

0.251 0.184

[9] [9]

293

0.33 0.18

[19] [9]

0.13 a

[18] [18]

303–333 293 293

Polysiloxanes 230 290 340 410

Poly(dimethylsiloxane)

Poly(methylphenylsiloxance) 9.5% Phenyl, d ¼ 1,110 kg=m3

273 323 373 273 323 373 273 323 373

48% Phenyl, d ¼ 1,070 kg=m3

62% Phenyl, d ¼ 1,110 kg=m3

[23] [23] [23] [23] [23] [23]

0.071 0.054–0.07 0.8 0.067 0.13 0.046–0.067 0.029–0.17 0.20 0.33 0.23 0.20 0.21

Polysulfide and polysulfones Polyarylsulfone Polyethersulfone Poly(phenylene sulfide)

293 240–310

Poly(phenylene sulfone) Udel polysulfone Polyurethanes Polyurethane Casting resin Elastomer Polyacrylonitrile Poly(acrylonitrile–butadiene) copolymer (NBR) 35% Acrylonitrile Poly(acrylonitrile–butadiene–styrene) copolymer (ABS) Injection molding grade Poly(acrylonitrile–styrene) copolymer Poly(i-butyl methacrylate) At 0.82 atm Pressure and temperature dependence

THERMAL CONDUCTIVITY /

159

TABLE 10.1. Continued. Polymer Poly(n-butyl methacrylate) At 0.82 atm Pressure and temperature dependence Poly(butyl methacrylate–triethylene glycol dimethacrylate) copolymer Poly(chloroethylene–vinyl acetate) copolymer

Poly(dially phthalate) Poly(ethyl acrylate)

Poly(ethyl methacrylate)m At 0.82 atm Pressure and temperature dependence Poly(ethylene vinyl acetate) Poly(methyl methacrylate) Temperature dependence Poly(methyl methacrylate–acrylonitrile) copolymer Polymer(methyl methacrylate–styrene) copolymer Poly(vinyl acetate) Poly(vinyl acetate–vinyl chloride) copolymer Poly(vinyl alcohol) Poly(N-vinyl carbozole) Poly(vinyl fluoride) Poly(vinyl formal) Molding grade a

k (W/m K)

Reference

0.45 a 0.15

[18] [18] [29]

0.134 0.146 0.218 0.21 0.213 0.230 0.213

[20] [20] [20] [19] [20] [20] [20]

0.175 a 0.34 0.21 a 0.18

[18] [18] [33] [19] [20,34–37] [21]

0.12–0.21

[24]

293 443 243 333

0.159 0.167 0.2 0.126 0.168 0.14 0.17

[9] [30] [9] [22] [22] [9] [9]

293

0.27

[19]

Temperature (K)

293 325 375 310.9 422.1 533.2 273

293

See references citied in column 4.

in the glassy region and decreases slowly or remains constant in the rubbery region. For crystalline polymers, thermal conductivity decreases steadily with the increase in temperature below the melting point. At temperature above the melting point, it behaves in a similar way as amorphous polymers. Some of the values of polymer thermal conductivity measured at various temperatures are listed in Table 10.1. More results on temperature dependence can be found in the references cited. Thermal conductivity of polymers is highly dependent on polymer chain segment orientation [10,12]. This is because thermal energy transports more efficiently along the polymer chain. Crystalline polymers have highly ordered chain segments, and therefore have higher thermal conductivity than amorphous polymers. Amorphous polymers may exhibit anisotropic thermal transport properties if polymer chains are partially oriented, with thermal conductivity along the chains higher than that perpendicular to the chains. Reinforced polymers and thermally conductive polymer composites. Polymers are often reinforced with fillers to

improve their mechanical, electrical, and thermal properties. The thermal conductivity of filled polymers is primarily determined by the type and amount of fillers used. The thermal properties of the filler, the size, shape, and orientation of filler particles or fibers in polymer matrix, and the percentage of fillers are all important factors that determine the thermal conductivity of reinforced polymers. As shown in Table 10. 3, polymers reinforced with inorganic fillers usually increase their thermal conductivities from a few percent to a few times. The filler percentages listed in Table 10.3 are weight percentages, unless otherwise indicated. Highly thermal conductive polymer composites can be obtained by using fillers with high thermal conductivity and at high percentages. Thermally conductive polymer composites find wide applications in semiconductor industry such as electronic components encapsulation, in which the heat dissipation of circuit boards becomes more and more important as silicon chips get denser and faster. Other applications of thermally conductive polymer composites

160 / CHAPTER 10 TABLE 10.2. Thermal conductivity k of foamed polymers. Polymer Poly(acrylonitrile–butadiene) copolymer d ¼ 160---400 kg=m3 Cellulose acetate d ¼ 96---128 kg=m3 Polychloroprene (neoprene) d ¼ 112 kg=m3 192 kg=m3 Poly(dimethylsiloxane) Sheet, d ¼ 160 kg=m3 Epoxy d ¼ 32---48 kg=m3 80---128 kg=m3 Polyethylene Extruded plank d ¼ 35 kg=m3 64 kg=m3 96 kg=m3 144 kg=m3 Sheet, extruded, d ¼ 43 kg=m3 Sheet, crosslinked, d ¼ 26---38 kg=m3 Polyisocyanurate d ¼ 24---56 kg=m3 Polyisoprene (natural rubber) d ¼ 56 kg=m3 320 kg=m3 Phenolic resin d ¼ 32---64 kg=m3 112---160 kg=m3 Polypropylene d ¼ 54---96 kg=m3 Polystyrene d ¼ 16 kg=m3 32 kg=m3 64 kg=m3 96 kg=m3 160 kg=m3 Poly(styrene–butadiene) copolymer (SBR) d ¼ 72 kg=m3 Poly(urea–formaldehyde) resin d ¼ 13---19 kg=m3 Polyurethane Air blown, d ¼ 20---70 kg=m3 At 08C At 208C At 708C CO2 blown, d ¼ 64 kg=m3 , at 208C 20% closed cells, at 208C 90% closed cells, at 208C 500 mm cell size, at 208C 100 mm cell size, at 208C Poly(vinyl chloride) d ¼ 56 kg=m3 d ¼ 112 kg=m3

k (W/m K)

Reference

0.036–0.043

[9]

0.0450–0.45

[9]

0.040 0.065

[9] [9]

0.086

[9]

0.016–0.022 0.035–0.040

[9] [9]

0.053 0.058 0.058 0.058 0.040–0.049 0.036–0.040

[9] [9] [9] [9] [9] [9]

0.012–0.02

[9]

0.036 0.043

[9] [9]

0.029–0.032 0.035–0.040

[9] [9]

0.039

[9]

0.040 0.036 0.033 0.036 0.039

[19] [19] [19] [19] [19]

0.30

[9]

0.026–0.030

[9]

0.033 0.036 0.040 0.016 0.033 0.016 0.024 0.016

[38] [38] [38] [19] [19] [19] [19] [19]

0.035 0.040

[9] [9]

THERMAL CONDUCTIVITY /

161

TABLE 10.3. Thermal conductivity k of reinforced polymers and thermally conductive polymer composites. Polymer Polyacetal 5–20% PTFE Poly(acrylonitrile–butadiene–styrene) copolymer (ABS) 20% Glass fiber Polyaryletherketone 40% Glass fiber Polybenzoxazine (bisphenol A/methylamine/formaldehyde 93% Aluminum nitride 94.1% Aluminum oxide Polybenzoxazine (bisphenol A/methylamine/formaldehyde 50–90% Boron nitride Polybenzoxazine (bisphenol F/methylamine/formaldehyde) 85% Boron nitride Polybenzoxazine (bisphenol A/aniline) 85% Boron nitride Polybenzoxazine (bisphenol F/aniline) 85% Boron nitride Poly(butylenes terephthalate) (PBT) 30% Glass 40–50% Glass fiber Polycarbonate 10% Glass fiber 30% Glass fiber Polychloroprene (neoprene) 33% Carbon black Polycyclooctene 80% trans content, 20% boron oxide Poly(dially phthalate) Glass fiber Epoxy 50% Aluminum 25% Al2 O3 50% Al2 O3 75% Al2 O3 30% Mica 50% Mica Silica Epoxy (cresol–novolak) 70% Boron nitride Polyetheretherketone (PEEK) 30% Glass fiber 30% Carbon fiber Polyethylene 5–25% (vol.) Al2 O3 30% Glass fiber Poly(ethylene terephthalate) (PET) 30% Glass fiber 45% Glass fiber 30% Graphite fiber 40% PAN carbon fiber Polyimide Thermoplastic, 15% graphite 40% graphite Thermoset, 50% glass fiber

k (W/m K)

Reference

0.20

[24]

0.20

[19]

0.44

[24]

7.4 3.4

[5] [5]

1.7–37.5

[5]

20.9

[5]

19.8

[5]

10.6

[5]

0.29 0.20 0.42

[19] [9] [24]

0.22 0.32

[19] [19]

0.210

[9]

0.41

[4]

0.21–0.62

[22]

1.7–3.4 0.35–0.52 0.52–0.69 1.4–1.7 0.24 0.39 0.42–0.84

[39] [39] [39] [39] [19] [19] [19,39]

6.07–11.6

[6]

0.21 0.21

[24] [24]

1–1.6 0.36–0.46

[2] [24]

0.29 0.31 0.71 0.72

[9,19] [9] [24] [24]

0.87 1.73 0.41

[39] [24,39] [24]

162 / CHAPTER 10 TABLE 10.3. Continued. Polymer Polyisoprene (natural rubber) 33% carbon black Poly(melamine–formaldehyde) resin Asbestor Cellulose fiber Glass fiber Macerated fabric Wood flour/cellulose Poly(melamine–phenolic) resin Cellulose fiber Wood flour Nylon-6 (polycaprolactam) 30–35% Glass fiber Nylon-6,6 [poly(hexamethylene adipamide)] 30–33% Glass fiber 40% Glass fiber and mineral 30% Graphite or PAN carbon fiber Nylon-6,12 [poly(hexamethylene dodecanediamide)] 30–35% Glass fiber Poly(phenylene oxide) 30% Glass fiber Poly(phenylene sulfide) 40% Glass fiber 30% Carbon fiber Polypropylene 40% Talc 40% CaCO3 40% Glass fiber Polyurethane 49% Boron nitride,1% silicon dioxide Polystyrene 20% Glass fiber Poly(styrene–acrylonitrile) copolymer 20% Glass fiber Poly(styrene–butadiene) copolymer (SBR) 33% Carbon black Polytetrafluoroethylene 25% Glass fiber Poly(urea–formaldehyde) resin 33% Alpha cellulose

include communication satellite device, structural components for spacecraft, computer cases, and many others that require light weight and good thermal conduction. Fillers used in thermally conductive polymers include metal and metal oxides, boron oxide, and graphitic carbon fibers. High thermalconductive polymer composites may have thermal conductivities 10 to over a 100 times higher than those of pure polymers, equal to or higher than thermal conductivities for some metals. Table 10.3 also includes some of those high thermal conductivity polymer composites.

k (W/m K)

Reference

0.28

[9]

0.544–0.73 0.27–0.42 0.42–0.48 0.443 0.17–0.48

[9] [24] [24] [9] [24]

0.17–0.29 0.17–0.29

[39] [39]

0.24–0.28

[24]

0.21–0.49 0.46 1.0

[24] [24] [24]

0.427

[24]

0.16

[24]

0.288 0.28–0.75

[9] [24]

0.32 0.29 0.37

[19] [19] [19]

0.55

[40]

0.25

[26]

0.28

[24]

0.300

[9]

0.33–0.41

[24]

0.423

[9]

REFERENCES 1. M. J. Hodgin and R. H. Estes, Proceedings of the Technical Programs, NEPCO WEST Conference, Anaheim, CA (1999), pp. 359–366. 2. I. H. Tavman, in Nanoengineered Nanofibrous Materials, NATO Science Series II, Mathematics, Physics and Chemistry, edited by S. I. Guceri, Y. Gogotsi, and V. Kuzentsov (Kluwer Academic Book Pub., Dordrecht, Netherlands, 2004), Vol. 169, p. 449. 3. Ho-Sung Lee and S-Won Eun, in Composites 2004 Convention and Trade Show, (American Composites Manufacturers Association, Tampa, Florida, 2004). 4. C. Liu and T. Mather, ANTEC 2004 (Society of Plastic Engineers, 2004), pp. 3080–3084.

THERMAL CONDUCTIVITY / 5. H. Ishida and S. Heights, Composition for Forming High Thermal Conductivity Polybenzoxazine-Based Material and Method, US Patent 5,900,447 (1999). 6. H. R. Frank and D. S. Phillip, Enhanced Boron Nitride Composition and Polymer Based High Thermal Conductivity Molding Compound, EP 0 794 227 B1 (2002). 7. R. D. Hermansen, Room-Temperature Stable, One-Component, Thermally-Conductive, Flexible Epoxy Adhesives, EP 0 754 741 B1 (2001). 8. H. Ishida, Surface Treated Boron Nitride for Forming A Low Viscosity High Thermal Conductivity Polymer Based on Boron Nitride Composition and Method, US Patent, 6,160,042 (2000). 9. Encyclopedia of Chemical Technology, 3rd ed., edited by H. F. Mark, D. F. Othmer, C. G. Overberger, and G. T. Seaborg (WileyInterscience, New York, 1989). 10. K. Kurabayashi, Int. J. Thermophys. 22(1), 277 (2001). 11. D. M. Finlayson, P. Mason, J. Phys. C: Solid State Phys. 18, 1777 (1985). 12. K. Kurabayashi, M. Asheghi, M. Touzelbaev, and K. E. Goodson, IEEE J. Microelectromech. Syst. 8(2), 1057 (1999). 13. S. Pattnaik and E. V. Thompson, Polym. Prepr. 22(1), 299 (1981). 14. L. C. Choy, W. P. Leung, and Y. K. Ng, J. Polym. Sci. Polym. Phys. Ed. 25, 1779 (1987). 15. K. Eiermann, Kolloid Z. Z. Polym. 180, 163 (1962). 16. T. R. Fuller and A. L. Fricke, J. Polym. Sci. 15, 1729 (1971). 17. J. C. Ramsey III, A. L. Fricke, and J. A. Caskey, J. Polym. Sci. 17, 1597 (1973). 18. R. S. Frost, R. Y. S. Chen, and R. E. Barker, Jr., in Thermal Conductivity 14, Proceedings of the 14th International Thermal Conference, edited by P. G. Klemens and T. K. Chu (Plenum, New York, 1975). 19. E. V. Thompson, in Encyclopedia of Polymer Science and Engineering, edited by H. F. Mark, N. M. Bikales, C. G. Overberger, G. Menges, and J. I. Kroschwitz (Wiley-Interscience, New York, 1985), Vol. 16, pp. 711–737. 20. Thermal Conductivity, Nonmetallic Solid, Vol. 2 of Thermophysical Properties of Matter, edited by Y. S. Touloukian, R. W. Powell, C. Y. Ho, and P. G. Klemens (IFI/Plenum, New York, 1970). 21. International Plastics Handbook, 2nd ed., edited by H. Saechtling (Hanser, 1987)

163

22. Encyclopedia of Polymer Science and Engineering, edited by H. F. Mark, N. M. Bikales, C. G. Overberger, G. Menges, and J. I. Kroschwitz (Wiley-Interscience, New York, 1985), Vols. 2, 4, 9, 13, 16, and 19. 23. Polymer Handbook, edited by J. Brandrup and E. H. Immergut (WileyInterscience, New York, 1989). 24. Handbook of Plastics, Elastomers, and Composites, edited by C. A. Harper (McGraw-Hill, New York, 1992). 25. Handbook of Thermoplastic Elastomers, 2nd ed., edited by B. M. Walker and C. P. Rader (Van Nostrand Reinhold, New York, 1978). 26. F. C. Chen, Y. M. Poon, and L. L. Choy, Polymer 18, 135 (1977). 27. K. Eiermann and K. H. Kellwege, J. Polym. Sci. 57, 99 (1962). 28. B. C. Chern et al., in Thermal Conductivity 14, Proceedings of the 14th International Thermal Conference, edited by P. G. Klemens and T. K. Chu (Plenum, New York, 1975). 29. Handbook of Thermoproperties of Solid Materials, edited by A. Gold-Smith, T. E. Waterman, and J. Hirschborn (MacMillan, New York, 1961), Vol. IV. 30. Plastic Mold Engineering Handbook, 4th ed., edited by J. H. Bubois and W. I. Pribble (Van Nostrand Reinhold, New York, 1987). 31. A. Sugawarea and T. Takehashi, in Thermal Conductivity 14, Proceedings of the 14th International Thermal Conference, edited by P. G. Klemens and T. K. Chu (Plenum, New York, 1975). 32. R. T. Jamieson and J. B. Irving, in Thermal Conductivity 14, Proceedings of the 14th International Thermal Conference, edited by P. G. Klemens and T. K. Chu (Plenum, New York, 1975). 33. D. A. T. A. Digest, 11th ed., edited by J. A. Morgon (Morgan, San Diego, 1990), Vol. 1. 34. K Ueberreiter, Kolloid Z. Z. Polym. 216, 217 (1967). 35. J. S. Fox and M. Imber, J. Appl. Polym. Sci. 12, 571 (1968). 36. P. Lohe, Z. Z., Kolloid Polym. 203, 115 (1965). 37. P. Anderson and B. Sundgvist, J. Polym. Sci. Polym. Phys. Ed. 13, 243 (1975). 38. Polyurethane Handbook, edited by G. Oertel (Janser, New York, 1985). 39. Handbook of Thermoset Plastics, edited by I. Goodman and H. Sidney (Noyes, Park Ridge, NJ, 1986). 40. M. V. Huynh, Low Dielectric Constant Polyurethane Filleting Composition, US Patent: 5, 221, 783 (1993).

CHAPTER 11

Thermodynamic Quantities Governing Melting L. Mandelkern* and R. G. Alamoy *Department of Chemistry and Biochemistry, Florida State University, Tallahassee, FL 32306-3015; yDepartment of Chemical and Biomedical Engineering, Florida Agricultural and Mechanical University and Florida State University College of Engineering, 2525 Pottsdamer St., Tallahassee, FL 32310–6046

11.1 11.2

Prefatory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Fusion Per Repeating Unit, DHu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 168 184

conducted at temperatures well below the equilibrium melting temperature. Consequently, a polycrystalline, partially ordered, system is the one that actually develops. This crystalline system is well removed from equilibrium and can be considered to be in a metastable state. A structural and morphological complex system is thus developed that governs properties, including thermodynamic ones. The kinetic restraints that are placed on the crystallization of polymers make it difficult, if not impossible to directly determine their equilibrium melting temperatures. The directly observed melting temperatures are primarily a reflection of the structure and morphology of the actual crystalline systems. The primary factors involved are the crystallite thickness, the interfacial free energy, and the influence, if any, of the noncrystalline region. There are, however, indirect methods by which to estimate the value of T 0m . One of these is a theoretical method. The others are based on extrapolative procedures. To properly use the T 0m values that are tabulated, and to understand their limitations, the basic assumption involved and the problems in execution need to be recognized. The theoretical method is based on the work of Flory and Vrij [2]. Here, the equilibrium melting temperatures of a series of oligomers (low molecular weight chains of uniform length) are analyzed. In this treatment an additional entropic term is added to the expression for the molar entropy of fusion of long-chain molecules in an ordered crystalline array. This term, which takes into account the disruption on fusion of the terminal groups of neighboring molecules, exerts an important influence in analyzing the experimental melting temperatures for n-paraffins and their convergence to the limit, T 0m for the infinite length chain. By analyzing the fusion of the n-alkanes, up to C100 H202 , the T 0m value for linear polyethylene was found to be 418:7
1 K. This is the only polymer to

11.1 PREFATORY REMARKS In this section a compilation and tabulation of the key thermodynamic parameters that govern the fusion of crystalline homopolymers is presented. The key parameters of interest are the equilibrium melting temperature, T 0m , defined as the melting temperature of a large perfect crystal comprising infinite molecular weight chains, the enthalpy of fusion per chain repeating unit, DHu , which is independent of the level of crystallinity, and the entropy of fusion, DSu , which is obtained from the above quantities. The data listed in Tables 11.1–11.3 are limited to those homopolymers for which a complete set of thermodynamic parameters are available. Copolymers will not be included since this would involve specifying the sequence distribution of the comonomer. An exception will be made, however, for stereoregular polymers which, from the point of view of crystallization behavior, should be treated as copolymers [1]. Most polymers in this category are not completely stereoregular and cognizance must be taken of this fact when using the data assigned. We shall not be tabulating melting temperatures by themselves, without the auxiliary thermodynamic parameters. Before examining and using the data listed it is important to understand the theoretical and experimental foundations for the quantities that are given and the limitations that are imposed. Although a collection of regularly structured, flexible chains will crystallize, they never do so completely. Depending on the molecular constitution, the chemical nature of the chain and the crystallization conditions the level of crystallinity attained can range from 90% to just a few percentage. In order for the crystallization of polymers from the melt to be carried out at finite rates, it must be 165

166 / CHAPTER 11 which this theoretical method has been applied. There are not enough data available for oligomers with other type repeating units to apply this method to other polymers. It is important in utilizing this method that all the molecules be of uniform chain length so that molecular crystals can be formed. Both of the extrapolative methods are based on the Gibbs–Thomson equation for the melting of crystallites of finite size [3]. For a lamellar crystallite, whose length is very much greater than its thickness, this equation can be expressed as:   2sec 0 Tm ¼ T m 1  : (11:1) DHu l Here Tm is the experimentally observed melting temperature of a crystallite of thickness l and sec is the interfacial free energy associated with the basal plane of the lamellar crystallite. Eq. (11.1) was developed to describe the melting of small crystals in equilibrium with the melt. Hence only two phases (or regions) are involved. The implicit assumption is made that the boundary between them is a sharp one. In general, long-chain molecules do not fulfill this condition. It is now well established, by theory and experi˚ ment, that there is a diffuse interfacial region, 10–30 A thick, which connects the crystalline with the liquid-like region [4]. The analog to Eq. (11.1) to account for the melting of small crystallites, when equilibrium between three phases is involved, has yet to be developed. Despite the problem described above, Eq. (11.1) has been used to determine T 0m by measuring the observed melting temperature, Tm , as a function of the crystallite thickness. Accordingly, a plot of Tm against 1/l should be linear and extrapolate to T 0m . An example of this procedure is illustrated in Fig. 11.1 for poly(tetrafluoroethylene) [5]. A straight line clearly results in the plot of Tm against 1/l. An extrapolated value of 335 8C is obtained for T 0m . In this example the largest crystallite thickness that was generated ˚ . Thus, the extrapolation was not was about 5,000 A unduly long. This method appears to be quite satisfactory for poly(tetrafluoroethylene). However, for many of the flexible chain-type polymers, such as linear polyethylene,

Melting temperature (⬚C)

340

335

330

325

0

2

4

6

8

10

12

Reciprocal fold length x 104(Å–1)

FIGURE 11.1. Melting temperature, in degrees Celsius, as a function of reciprocal crystallite thickness in A˚ngstroms for poly(tetrafluoroethylene) [5].

poly(propylene), poly(hexamethylene adipamide), poly (pivalolactone), among others, the maximum thickness ˚ . In used in the extrapolation is usually less than 200 A ˚ many cases it is less than 100 A. Hence, most commonly there is a very long and treacherous extrapolation to l ¼ 1. There are also other matters of concern that need to be taken into account when using this method. It is very rare to develop a uniform crystallite thickness distribution. In particular, the distribution becomes very broad after hightemperature crystallization. The question then arises as to what portion of the distribution curve corresponds to the observed melting temperature. Usually some average crystallite thickness is measured. The use of this method with polymer systems where a large proportion of the crystallinity develops during the quenching process, after isothermal crystallization, would introduce major uncertainties if the crystallite thickness (l) is measured at room temperature. When linearly extrapolating Eq. (11.1) it is tacitly assumed that sec is independent of the crystallite thickness. Crystallite thicknesses are usually controlled by varying the crystallization temperature. Theory has shown that, among other factors, sec depends on the flux of chains emanating from the basal plane of the lamellar crystallite [4]. This flux will in term depend on the tilt angle of the chain, i.e., the angle between the chain axis and the normal to the basal plane [6]. For many polymers the tilt angle depends on the crystallization temperature. Consequently, a variation in sec with crystallite thickness can be expected, for many systems. Whether using Eq. (11.1), or the appropriate equivalent equation (when available), the functionality between sec and l is needed to properly estimate T 0m . Consideration should also be given to the meaning of the extrapolated value, obtained by measuring melting of folded chain crystallites, when the equilibrium state requires an almost completely extended state [7,8]. The interfacial free energies will be different in the two cases. Small-angle x-ray scattering is one of the standard methods that are used to determine crystallite thickness. A crystalline system comprising a set of stacked lamellae gives a characteristic long period. The long period must then be corrected for the level of crystallinity to obtain the crystallite thickness. In many studies this correction has not been made. Consequently, in these cases there is an indeterminate error in the extrapolation to T 0m . For the reasons that have been cited above, it is not surprising that there are serious discrepancies in the T 0m values determined by different studies of the same polymer using this method. For example, linear polyethylene, T 0m values, which were obtained by this extrapolative method, vary from 411.3 to 419.0 K among different investigators [9]. Although the difference is not a large range on an absolute basis, it is crucially important in analyzing crystal
lization kinetics because the undercooling, T 0m  Tc is involved. The other extrapolative method that is used was also developed from the Gibbs–Thomson equation [10,11].

THERMODYNAMIC QUANTITIES GOVERNING MELTING

For a three-dimensional homogeneous nucleus, or for certain types of heterogeneous nuclei [13–l5], under the same assumptions   b b 0 Tc : Tm ¼ T m 1  (11:3) þ 2m 2m The further assumption is commonly made that b ¼ 1. In this case Eq. (11.2) becomes (m  1) 0 Tc Tmþ 2m 2m

(11:4)

(2m  1) 0 Tc Tmþ : 2m 2m

(11:5)

Tm ¼ and Eq. (11.3) becomes Tm ¼

For this type of extrapolation we need not concern ourselves with which type of nucleation is controlling the crystallization since the only interest is evaluating T 0m . In either case, assuming that m is constant over the complete interval of crystallization temperatures, there is a linear relation between Tm and Tc . Thus, a linear extrapolation to the Tm ¼ Tc straight line should in principle lead to T 0m . The concept, as outlined above, requires low levels of crystallinity so that the mature crystallite will resemble the critical nucleus as closely as possible. However, independent experiments have made clear that the quantity m, a measure of crystallite thickening, is strongly dependent on the crystallization temperature [16,17]. It is also dependent on molecular weight. In addition it is reasonable to expect that sec > sen so that b will not be constant. With this theoretical background we can examine some of the experimental results typical of this popular extrapolative method that is used to obtain T 0m . In brief summary, the results depend on the nature of the polymer, the level of crystallinity that is developed, and the range of crystallization temperatures that is used. A set of Tm  Tc data for poly(ethylene oxide), Mw ¼ 6:1  105 , are given in Fig. 11.2 [18]. At the low levels of crystallinity the data are represented by two intersecting straight lines. The line with the steeper slope extrapolates to a T 0m value of 349.2 K. When the crystallinity level is increased, the observed melting points are represented by a single straight line. The extrapolated T 0m value of 341.2 K is now much lower. The value obtained at the lower crystallinity level is preferred by

167

Tm (⬚C)

75

70

Tm = Tc

65 50

55

60

65

70

75

Tc (⬚C) FIGURE 11.2. Plot of melting temperature against crystallization temperature for poly(ethylene oxide), (.) high crystallinity;  low crystallinity. Temperature in degrees Celsius [18].

theory. However, this quantity still does not represent the highest T 0m value that has been deduced for this polymer (see Table 11.1). It is important to note that the data obtained at the low crystallinity level can also be represented by a continuous curve. The question arises as to what happens if the Tc range is extended to higher temperatures. An example is given in Fig. 11.3 for a linear polyethylene sample having a most probable molecular weight distribution, Mw ¼ 351; 000, that was crystallized to a crystallinity level of less than 10% [12]. For the range in crystallization temperatures shown the data are best represented by a curve. If, however, the measurements are restricted to 126 8C, or less, the data would be represented by two intersecting straight lines. Within this range the extrapolated T 0m ¼ 417 K  418 is in agreement with theoretical expectations [2]. However, if the range of crystallization temperatures is increased to 131 8C (a high Tc for this polymer), then, the data points deviate from the straight line toward high values and a linear extrapolation is no longer operationally feasible. This behavior 150 PE, Mw = 351,000

140 Tm(⬚C)

Here, l and sec are related to the corresponding quantities that define a nucleus of critical size, namely, l* and sec . For a Gibbs type two-dimensional coherent nucleus l ¼ 2sen =DGu , where DGu ¼ DHu DT=T 0m [12] and DT ¼ T 0m  Tc . The interfacial free energy sen now represents that of the critical size nucleus. If it is assumed that l=l ¼ m and b ¼ sec =sen over the complete range of crystallization temperatures then   b b Tm ¼ T 0m 1  (11:2) þ Tc : m m

/

Tm = Tc 130

120 110

(1–λ) < 10% ∆H

120

130

140

150

TC(⬚C)

FIGURE 11.3. Plot of observed melting temperature, Tm , against crystallization temperature, Tm , for a linear polyethylene having a most probable molecular weight distribution, Mw ¼ 351,000. Degree of crystallinity less than 10%. Temperature in degrees Celsius [12].

168 / CHAPTER 11 is not unique to linear polyethylene, it is also shown by many other polymers. There is, therefore, a serious dilemma in using this method. If Tc is too high then the extrapolation cannot be made. In order to carry out the extrapolation an arbitrary decision must be made as to the highest Tc value to be used. Thus the value obtained for T 0m will also involve an element of arbitrariness. The fact that the parameter m varies with the crystallization temperature is a major reason for the curvature that is observed. Several theoretical proposals have been made to improve the data analysis in order to obtain a more reliable extrapolated value of T 0m [19,20]. The validity and reliability of the proposed methods awaits further study. There are obvious difficulties in obtaining T 0m by either of these extrapolative methods. Therefore, caution must be used in accepting, and using, the values so obtained. Equilibrium melting temperatures listed in Tables 11.1 and 11.3 have been obtained by one or the other of these methods, except for the theoretical value for linear polyethylene.

11.2 ENTHALPY OF FUSION PER REPEATING UNIT, DHu There are several experimental methods that allow DHu to be determined. Two of these are based on direct thermodynamic methods. They do not involve knowledge of the crystallinity level of the polymer. A third method is an indirect one that does require such knowledge. One of the thermodynamic methods involves the adaptation of the classical freezing point-composition relation to the melting of polymer-diluent mixtures. The most common situation encountered when a low molecular weight diluent is admixed with a polymer, is that the crystalline phase remains pure. Under these conditions, for polymers of high molecular weight, the melting temperature of a polymerdiluent mixture, Tm , can be expressed as [7,21]  1 1 R Vu   0 ¼ (1  v2 )  w1 (1  v2 )2 : Tm T m DHu V1

(11:6)

In deriving this equation, the chemical potentials in the melt of the two components are given by the Flory–Huggins theory [22]. In Eq. (11.6), Vu and V1 are the molar volumes of the chain repeating unit and diluent, respectively; v2 is the volume fraction of polymer in the mixture and w1 is the Flory–Huggins interaction parameter [22]. The implicit assumption is made in deriving Eq. (11.6) that sec is independent of composition. The similarity of Eq. (11.6) to the

classical expression for the freezing point depression is readily apparent. It has been found that, for a given polymer, the same value of DHu is obtained irrespective of the diluent used, giving support to the validity of Eq. (11.6) [23]. Measurement of the melting point depression is a powerful method for determining DHu . Enthalpies of fusion per repeating unit that were determined by this method are listed in Table 11.1. The other thermodynamic method that can be used to determine DHu involves the variation of the equilibrium melting temperature with applied hydrostatic pressure, p. The Clapeyron equation dT 0m DVu ¼ T 0m dp DHu

(11:7)

has been shown to apply to polymers [24], as it does to all substances. In this equation DVu is the difference between the volume, per unit, of the pure melt and of the pure crystal. It is important, therefore, that these two quantities be known as a function of both temperature and pressure in order to properly apply Eq. (11.7). The DHu values that have been obtained by this method are given in Table 11.2. The DHu values, obtained by indirect methods, of polymers that, except for minor exceptions, are not included in Table 11.1 or 11.2, are listed in Table 11.3. In general, the enthalpy of fusion is measured for a given sample and the level of crystallinity of the same specimen is determined, by one of the many methods that are available. The DHu value can then be calculated from these measurements. One problem associated with this procedure is that not all methods of measuring the level of crystallinity give exactly the same value. Small but significant differences are found between different techniques that reflect different sensitivities to the elements of phase structure [25]. A corollary to this procedure is to determine the enthalpy of fusion as a function of density for a sample that is crystallized in different ways. The results are then extrapolated to the density of the unit cell in order to obtain DHu . The density of the unit cell needs to be well established, and a long extrapolation is usually involved. The purpose of these prefatory remarks has been to explain the underlying theoretical basis for the data that appear in the tables that follow and the experimental difficulties that are involved. Such tabulations must always be taken cautiously and critically. However, with proper care, and understanding, the data should be very useful. Related information can be found in Chapters 30 and 31.

THERMODYNAMIC QUANTITIES GOVERNING MELTING

/

169

TABLE 11.1. Thermodynamic quantities determined by use of diluent equation [Eq. (11.6)]. T 0m (K)

DHu (J=mol)

DHu =M0 (J=g)

DSu (J=K mol)

References

418.7

4,142

295.8

9.9

[26–28]

a b

485.2 465.2

8,786 8,201

208.8 194.9

18.1 17.6

[29–36]

(I) (II) (III)

408.7 397.2 379.7

6,318 6,276 6,485

112.5 111.9 115.6

15.5 15.8 17.1

[37–38]

523.2

5,297

63.7

10.1

[39]

268.2

10,857

86.2

40.5

[40]

Polymer Ethylene [CH2]n Isot.-propylenea [CH2 −CH]n CH3 Isot.-butene-1 [CH2 −CH]n CH2 CH3 4-Methyl pentene-1 [CH − CH2]n CH2 CH CH3 CH3 1-Methyl octamer [CH − (CH2)6 − CH2]n CH3 Isot.-styrene

>516.2b

8,682c

83.4

16.8

[41–44]

>560.5d

8,577

82.4

15.3

[45–47]

523.2

6,862

156.1

13.1

[48–50]

593.2

5,021

94.7

8.5

[51,52]

450.2

5,857

51.4

13.0

[53]

369.2 421.2

13,807 4,602

255.7 85.2

37.4 10.9

[54]

273.2

9,205

170.4

33.7

[55]

⎯[ CH2 − CH ⎯ ]n Synd.-styrene ⎯[ CH2 − CH ⎯ ]n Vinyl alcohol [CH2 − CH]n OH Acrylonitrile [CH2 − CH]n C≡N Isot.-iso -propyl acrylate [CH2 − CH]n C

O O

CH3 CH CH3

trans-1,4 Butadiene H

(I) (II)

[CH2 − C=C− CH2]n H cis-1,4 Butadiene H H [CH2 − C=C − CH2]n

170 / CHAPTER 11 TABLE 11.1. Continued. Polymer trans-1,4 Isoprene CH3

(a) (b)

T 0m (K)

DHu (J=mol)

DHu =M0 (J=g)

DSu (J=K mol)

360.2 354.2

12,719 10,544

187.0 155.1

35.3 29.8

[56–58]

308.7

4,393

64.6

14.2

[59,60]

380.2

8,368e

94.6

22.0

[61–63]

References

[CH2− C=C − CH2]n H cis-1,4 Isoprene H3C H [CH2− C=C − CH2]n trans-1,4 Chloroprene CI [CH2− C=C − CH2]n H trans-Pentenamer [CH=CH − CH2 − CH2 − CH2]n

307.2

12,008

176.3

39.1

[64,65]

trans-Octenamere [CH=CH − (CH2)5 − CH2]n

350.2

23,765

215.7

67.9

[66]

cis-Octenamer [ CH=CH − (CH2)5 − CH2]n

311.2

21,000

190.9

67.5

[67]

trans-Decenamere [CH=CH − (CH2)7 − CH2]n

353.2

32,844

237.6

92.9

[66]

trans-Dodecenamer [CH=CH − (CH2)9 − CH2]n

357.2

41,171

247.6

115.3

[66,68]

Methylene oxide [CH2 − O)n

479.2

7,012

233.7

14.6

[69–72]

Ethylene oxide [CH2 − CH2 − O]n

353.2

8,703

197.8

24.6

[73–75]

Isot.-propylene oxide [CH2 − CH − O]n

355.2

7,531 8,368f

129.8

21.2

[76,77]

323.2

8,786

151.5

27.2

[78]

Tetramethylene oxide [(CH2)4 − O]n

330.2

15,899

220.8

48.2

[79]

Hexamethylene oxide [(CH2)6 − O]n

346.7

23,640

236.4

68.2

[80]

1,3 Dioxolane [O − CH2 − O − (CH2)2]n

366.2

15,481

209.2

42.3

[81]

1,3 Dioxepane [ O− CH2 − O − (CH2)4]n

303.0

14,454

141.7

47.7

[82]

1,3 Dioxocane [O − CH2 − O − (CH2)5]n

319.2

7,740

75.9

24.3

[83]

349.2 329.2

9,205 7,448

107.0 86.6

26.3 22.6

[84]

CH3 Trimethylene oxide [(CH2)3 − O]n

3,3 Dimethyl oxetane CH3 [O − CH2 − C − CH2]n CH3

II III

THERMODYNAMIC QUANTITIES GOVERNING MELTING

/

171

TABLE 11.1. Continued. T 0m (K)

Polymer

DHu (J=mol)

DHu =M0 (J=g)

DSu (J=K mol)

References

334.2

6,276

62.8

18.8

[85]

373.2 353.2

10,460 10,042

91.8 88.1

28.0 28.4

[86] [86]

398.2

9,414

54.1

23.6

[87]

401.2

53,555

318.8

133.5

[87]

2,6 Dimethyl,1,4 phenylene oxide CH3 [ O ]n CH3

548.2

5,230

43.6

9.5

[88,89]

2,6 Dimethoxy,1,4 phenylene oxide

560.2

3,184

20.9

5.7

[90]

363.2

10,460

141.4

28.8

[91]

>528.2

12,600

225.0

513:2

22,348

74.0

< 43.5

DSC-x ray

[189]

496 470

26,051

101.8

52.5

DSC-x ray

[190]

567

32,940

122

58.1

DSC-x ray

[191,192]

151.2

DSC-x ray

[193]

93.9

DSC-density

[194]

O O

O

C n

CH3

β α

Trimethylene 2,6 naphthalate

=

O

−CH2CH2CH2O

=

O−

O

n

=

Butylene 2,6 naphthalate

−CH2CH2CH2CH2O

O

=

O−

O

n

LARC-CPI

663.2

O

O

N

O

100,294

127.6

O

N O O

O

O

O

New-TPI O

O

N

N

O

O

O

n

679.2

63,800

116

235.2

4,619k

62.4

19.6

CalorimetryDSC

[195–197]

306.2

8,380

71.0

27.4

DSC-x ray

[198]

440.2 434.2 453.2 428.2 438.2 419.2

41,547 41,840 48,534 41,463 51,882 58,534

180.6 171.5 188.1 152.4 181.4 161.8

94.4 96.4 107.1 96.8 118.4 115.8

DSC-x ray DSC-x ray DSC-x ray DSC-x ray DSC-x ray DSC-x ray

[111] [111] [111] [111] [111] [111]

O n

Dimethylsiloxane j CH3 [Si − O]n CH3 Dichlorophosphazene Cl −P

N− n

Cl

Urethanel n¼2 n¼3 n¼4 n¼5 n¼6 n¼7

THERMODYNAMIC QUANTITIES GOVERNING MELTING

/

183

TABLE 11.3. Continued. T 0m (K)

DHu (J=mol)

DHu =M0 (J=g)

DSu (J=K mol)

Method

References

n¼8 n¼9 n ¼ 10 Urethanem n¼2 n¼3 n¼4 n¼5 n¼6 n¼7 n¼8 n¼9 n ¼ 10

430.2 420.2 427.2

55,229 52,300 56,484

175.9 159.5 165.2

128.4 124.5 132.2

DSC-x ray DSC-x ray DSC-x ray

[111] [111] [111]

510.2 500.2 505.2 462.2 470.2 464.2 469.2 463.2 465.2

48,534 47,279 52,718 51,045 51,882 50,626 59,413 58,576 69,036

155.6 145.0 155.1 144.2 141.0 132.5 150.0 142.9 162.8

95.1 94.5 104.4 110.4 110.3 109.1 126.6 126.4 148.4

DSC-x ray DSC-x ray DSC-x ray DSC-x ray DSC-x ray DSC-x ray DSC-x ray DSC-x ray DSC-x ray

[111] [111] [111] [111] [111] [111] [111] [111] [111]

Cellulose Tributyrate

465

12,851

34.5

27.6

DSC-heat capacity

[199]

Polymer

CH2X

O

O O

H X

X = − O − C − (CH2)2CH3

H H

H

X

a

For a sample 94% syndiotactic content (diads based analysis by 13C NMR), Tm ¼ 160 8C, DHu ¼ 1920 cal/mol. Data from S. & D. Cheng et al [205]. b For 64% syndiotactic polymer. c Calculated in Reference 145 for 100% syndiotactic material by modifying the data of D. Koekott, [206]. d Samples do not have most probable molecular weight distribution. e Cited R. P. Pearce and R. H. Marchessault, [207]. f Average of literature values. g The observed heat of fusion of a highly regular poly L-lactide (>99.9%) is 98.4 J/g (see E.J. Munson et al. [208]). h Taking Vc a ¼ 0:814 cm3 /g from D. R. Holmes, et al. [209]. If all literature values for Vc a are considered, DHu ranges from 6215 cal/mol to 7250 cal/mol (26000 J/mol–30334 J/mol). i

O O−(CH2CH2O)n

O

C N C

C

O

N C

O j

O

Higher alkyl siloxanes are not included due to their liquid-crystal characteristics Average value.

k l

O H

H O

− C − N − (CH2)6 N − C − O(CH2)n − O

x

m

H O

O H −C−N

CH2

N−C−O−(CH2)n− O− x

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202. K. Mezghani, R.A. Campbell, and P.J. Phillips, Macromolecules 27, 997 (1994). 203. R-M. Ho, C-P. Lin, H-Y. Tsai, and E-M. Woo, Macromolecules 33, 6517 (2000). 204. M.I. Aranguren, Polymer, 39, 4897 (1998). 205. J. Rodriguez-Arnold, A. Zhang, S.Z.D. Cheng, A.J. Lovinger, E.T. Hsieh, P. Chu, T.W. Johnson, K.G. Honnell, R.G. Geerts, S.J. Palackal, G.R. Hawley, and M.B. Welch, Polymer, 35, 1884 (1994). 206. D. Kockott, Kolloid-Z.Z. Polym. 198, 17 (1964). 207. R.P. Pearce and R.H. Marchessault, Macromolecules 27, 3869 (1994). 208. K.A.M. Thakur, R.T. Kean, J.M. Zupfer, N.U. Buehler, M.A. Doscotch, and E.J. Munson, Macromolecules 29, 8844 (1996). 209. D.R. Holmes, C.W. Bunn, and D.J. Smith, J. Polym. Sci. 17, 159 (1955).

CHAPTER 12

The Glass Temperature Donald J. Plazek* and Kia L. Ngaiy *Department of Materials Science and Engineering, University of Pittsburgh, Pittsburgh, PA 15261; y Naval Research Laboratory, Washington, DC 20375

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dependence of Tg on the Rate of Cooling q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume and Enthalpy Variations and the Fictive Temperature Tf . . . . . . . . . . . . . . Isothermal Contraction Near and Below Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Concentration Dependence of the Glass Temperature, Tg (f2 ) . . . . . . . . . . . . . Dependence of Tg on Molecular Weight and Crosslinking . . . . . . . . . . . . . . . . . . . . Dependence of Tg on the Degree of Crystallinity and Morphology . . . . . . . . . . . . Dependence of Tg on Intermolecular Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Effect of Pressure on Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Molecular Structure on Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differences of Opinion Concerning Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Corresponding Properties at Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscoelastic Behavior at Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Universal Behavior at Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of Tg from the Compiance Functions . . . . . . . . . . . . . . . . . . . . . . . . . Tg of Polymer Thin Films and Polymer Confined in Nanometer Scale Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.17 When do Volume and Entropy First Enter into Determining Molecular Mobility? . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15 12.16

187 188 189 190 190 192 196 197 197 197 199 200 202 204 204 207 212 213

so-called Kauzman Paradox or ‘‘catastrophe’’ drew attention to the precipitous decrease of the entropy of disordered glasses toward values which were less than that of the ordered crystalline state [1]. A second order thermodynamic transition appeared to be necessary to avert the ‘‘catastrophe.’’ Kinetic phenomena were encountered that led to decades of controversy about the nature of the glass ‘‘transition’’ temperature. Nearly all of the manifestations of glassy behavior have been determined to be kinetic in nature. In fact, it will be seen below that glass formation and the associated time and rate dependent changes in properties are examples of volume viscoelasticity. Any liquid which does not crystallize upon cooling is destined to become a glass. When a liquid is cooled continuously, the rate of diffusion decreases while the viscosity increases, reflecting a diminishing molecular mobility.

12.1 INTRODUCTION The equilibrium liquid state certainly is not as well understood as the crystalline state, and glasses which are nonequilibrium liquids are less well understood, but their unusual time-dependent properties have fascinated investigators for many years. There is no intention here to completely cover all the known properties of glasses or the ground covered by the numerous exhaustive reviews [1–10] and pertinent monographs, [11–34] but an attempt will be made to direct the reader to many of the significant observations, papers, theories, monographs, and review articles dealing with many of their interesting facets. Many of the kinetic phenomena exhibited by glasses were described in the early monograph by G. O. Jones [12]. The 187

188 / CHAPTER 12 The enhanced sluggishness of molecular response is due to the increasing molecular crowding and the attendant cooperativity of the molecular motions. At relatively high temperatures the mobility of the molecules of a liquid is great enough to maintain an equilibrium density, reflecting the more efficient packing that occurs during cooling. Eventually the mobility decreases to the point where the molecular rearrangements, necessary to alter the liquid structure, cannot keep up with a given fixed rate of cooling. Subsequently the liquid’s specific volume becomes increasingly greater than its equilibrium value at each lower temperature. At the same time, the liquid exhibits glassy properties, an extremely high viscosity, greater than 1012 poise (1011 P sec), and a low compliance, about 10
10 cm2 =dyne (10
9 Pa
1 ). Most significantly, at temperatures near the departure from an equilibrium density during cooling, any and all liquids are measurably, if not markedly, viscoelastic. The departure from equilibrium during cooling signals the glass temperature Tg . Note, we do not call it a glass ‘‘transition’’ temperature since no transition occurs in the liquid. It is still a liquid below Tg , albeit with an enormous, but measurable, viscosity. For Tg to be a material characterizing function, it must be defined in a cooling experiment, whether it be a quench or a constant rate of cooling, so that it emanates from a unique equilibrium condition. When determined from a cooling experiment, Tg is found to be a unique function of the cooling rate, q. This is because it is a manifestation of viscoelastic behavior. Viscoelastic behavior is a time-, rate, and frequency-dependent behavior. Tg reflects a characteristic molecular mobility, virtually by the definition given above. When the rate of molecular rearrangement cannot keep up with the rate of cooling, equilibrium is lost. While polymer glasses are of principal interest here, it must be kept in mind that qualitatively all glasses behave similarly whether they are organic, inorganic, or metallic. The glass temperature Tg of a given polymer depends on the rate of cooling, q, the pressure, P, the number average molecular weight, Mn , and if in solution, its volume fraction, f2 . In short, Tg (q,P,Mn ,f2 ). For a polycrystalline polymer, changes of Tg occur with a variation of the degree of crystallinity and the nature of the morphology of the material. The effect of each of these experimental variables will be discussed below with minimal reference to model-dependent analyses. The emphasis will be on the phenomenology that has been observed. The free volume [36–38], entropy [39–42], coupling [43], and fictive temperature [44–46] models have all been used in analyzing the phenomena with conflicting results in many cases. In addition to the original presentations, the models have been outlined in the many review papers referred to above. 12.2 DEPENDENCE OF Tg ON THE RATE OF COOLING q Although the rate dependence of Tg is constantly acknowledged, its direct observation has rarely been systematically

0,980 v,cm3/g

–q, K/h

PS

120 0 6 1, 2 0,042

0,975

0,970

0,965 T, ⬚C

60

70

80

90

100

110

FIGURE 12.1. Volume-temperature curves of PS extending through the glass temperature under various rates of cooling, as indicated. (From Greiner and Schwarzl, by permission [5b].)

observed. Fig. 12.1 shows the dilatometric results of Greiner and Schwarzl [5b] on a polystyrene covering 3 1/2 decades of cooling rate, q. The Tg , determined from the intersection of the equilibrium line with the varying glassy lines varies from 96 8C at the highest q of 2.0 8C/min down to 86 8C at the lowest rate of cooling employed which was 7  10
4 8C=min. Illustrative values of the thermal contraction coefficient, a, curves calculated from the curves of Fig. 12.1 are seen in Fig. 12.2
 1 @y a¼ y @T P A slightly lower glassy a is found with decreasing q but the temperature span between the limiting equilibrium and glassy lines is found to significantly diminish with decreas-

α,⫺10⫺4K⫺1 5 ⫺q = 120 K/h 4

⫺q = 6 K/h ⫺q = 0,042 K/h

3

T,⬚C 60

70

80

90

100

110

FIGURE 12.2 Thermal expansion coefficient, calculated from the data of Fig. 12.1, for PS under various rates of cooling. (From Greiner and Schwarzl, by permission [5b].)

THE GLASS TEMPERATURE

189

7

RATE

6 ~ ⌬V x 104 (cm3/g)

ing q. This observation is in accord with that reported by Moynihan et al. [46,47] on several inorganic glasses. Using the Macedo-Litovitz hybrid equation, Rekhson and Scherer rationalized this broadening [48]. Bero and Plazek observed the same broadening on a fully cured epoxy resin which is a viscoelastic solid since it is comprised of a molecular network which precludes flow [49]. This is in contrast with the polystyrene of Greiner and Schwarzl which is a viscoelastic liquid, because it is constituted of linear molecules and it does flow. The specific volume-temperature cooling curves for the epoxy resin are shown in Fig. 12.3. The extent of the temperature range between equilibrium liquid-like and glassy contractions is shown in Fig. 12.4. The rate dependence of Tg is also presented by Greiner and Schwarzl [5b] for polymethyl-methacrylate, PMMA, polyvinylchloride, PVC, and polycarbonate, PC.

/

⫺0.90 ⬚C/min ⫺0.250 ⫺0.050 ⫺0.003

5 4 3 2 1 0 110

120

130

140

TEMP.(⬚C)

12.3 VOLUME AND ENTHALPY VARIATIONS AND THE FICTIVE TEMPERATURE Tf With great care and effort, Richardson and Savill [50] have succeeded in measuring the Tg of polystyrene in cooling as a function of rate in a differential scanning calorimeter,

0.855

EPON IOOIF/DDS

0.854

0.852 0.851

DSC, and have compared the results obtained dilatometrically. Their study covered the molecular weight range from 580 up to 2:0  106 . The checks agreed within about 1 8C. The difficulty of the study was made clear. Most investigators do not have the opportunity to exercise the effort needed to obtain accurate Tg s from DSC measurements. Most DSC measurements are carried out as heating scans [51] which result in yielding something close to the fictive temperature Tf of Tool [44]. Investigators of inorganic glasses have long appreciated the distinction between Tg and Tf while polymer scientists in general are not aware of or ignore it. Figure 12.5 shows the glass temperature as the intersection point of the volume, y, or enthalpy, H, temperature lines of the equilibrium (or metastable equilibrium) liquid and the glass

0.850 0.849 0.848 0.847

RATE (⬚C/min) Tg (⬚C) ⫺0.90 ⫺0.250 ⫺0.050 ⫺0.003

0.846

Enthalpy or Volume

SPECIFIC VOLUME (CM3/g)

0.853

FIGURE 12.4. Deviation of measured specific volume points from the equilibrium and glass lines of Fig. 12.1 plotted as a function of temperature, showing the extent of the transformation range and its change with the rate of cooling.

131.7 130.5 129.9 126.1

0.845

110

115

120

125

130

135

140

145

TEMPERATURE (⬚C)

FIGURE 12.3 The specific volume V
(cm3=g) of EPON 1001F fully cured with a stoichiometric amount of 4,4’–diamino diphenyl sulfone (DDS) shown as a function of temperature at four different rates of cooling 0.90, 0.25, 0.050, and 0.003 8C/min. Glass temperatures identified by the intersection point of the equilibrium and glass lines are listed.

Tf

Tg

Temperature

FIGURE 12.5. Schematic plot of the enthalpy or the volume as a function of temperature for glass-forming liquids. The fictive, Tf and glass, Tg , temperatures are indicated.

190 / CHAPTER 12 eous contraction is monitored as a function of time. The results of such quenches and annealing are illustrated in Fig. 12.6. A commercial polystyrene (Dylene 8; Arco Polymers, Mn ¼ 0:93  105 , Mw ¼ 2:2  105 ) was studied [55b]. The fractional excess (above equilibrium) specific volume (y(t)=y inf )
1 is plotted as a function of the logarithmic annealing or aging time. y
(t) is the time-dependent specific volume and y
inf ¼ y(1) is the equilibrium value for the temperature at which the densification is occurring. Measurements such as these can be used to define a Tg which would be a material-characterizing function of the time of annealing (or physical aging [31]). In spite of the acknowledged intrinsic nonlinearity of the contractions linear parallel segments of the response are observed which can be easily extrapolated to zero excess specific volume [3]. In Fig. 12.6, lines tangent at the points of inflection can be extrapolated to zero. This intersection with the logarithmic time axis yields the time-temperature relationship for this Tg : Tg would then be a function of the aging time. The advantage of this kind of Tg is the greater resolution that is possible relative to that available from a cooling curve. The intersection point of the equilibrium and glass lines yields a Tg value which is probably valid to within a degree. The rate of time-scale shifts with temperature in the quench experiments is equivalent to that of viscoelastic processes [49], which are large. An order of magnitude change in rate of transport processes near Tg requires a temperature change from 1.58 to 6 8C depending on the material. 12.5 THE CONCENTRATION DEPENDENCE OF THE GLASS TEMPERATURE, Tg (f2 ) Usually a diluent decreases the Tg of a polymer severely. Early measurements [56] indicated that solvents with lower Tg s of their own decreased the Tg of a polymer to a greater 88.0 ⬚C

3.5

Polystyrene Dylene 8 (Mw=2.20x105) Physical Aging after quencing from 104.0 ⬚C

3.0 103[V(t)/Vinf)-1]

obtained in cooling. The intersection point is usually chosen as Tg. [3,23,34,49,50]. For a given substance this definition is a material characterizing function of the rate of cooling since it is the measure of the departure from a unique equilibrium. Figure 12.5 also shows how Tf which identifies the state of a specimen. Tf is the result of a simple geometric operation. The specific volume or enthalpy of a specimen must be known and a line having the slope of a glass line is drawn through it. The intersection of this glass line with the equilibrium line is Tf . Note that Tf is not a function of the heating rate. A fictive temperature measured by heating at the same rate as that of an immediately preceding cooling from above Tg approximates Tg and is called Tf ,g . In an actual heating curve, if it is slow enough, an appreciable spontaneous contraction or decrease in H can occur during the heating, which would yield an intersection point slightly below Tf . For equal cooling and heating rates, Tf is measurably lower than Tg . Without corrections for thermal lags, actual scans often show indicated temperatures where Tf > Tg . This is a clear indication of the error incurred. DSC measurements in the past have almost universally been carried out in the heating mode [51,52] because thermal lags can be corrected with melting point standards. Since freezing is a nucleated process, super cooling always occurs which prevents an accurate calibration during cooling. However, it has been observed that some liquid-crystal meso-phase transitions do not show super-cooling, thus making accurate calibrations during cooling a convenient possibility [53]. In addition, it should be noted that, while it is generally recognized that the dynamic loss tangent peaks are 158– 20 8C higher than Tg (q ¼ 1 8=min), [54,55a] the temperatures of these maxima continue to be reported as Tg s. Twenty degrees above Tg , the rate of molecular motion in many polymers is about a million times greater than that found at Tg . Therefore, predictions for rate processes based on such incorrect Tg s can be in error by six orders of magnitude. The uncertainty of many of the reported values, therefore, should not be taken lightly for practical purposes. In addition, serious differences of opinion exist concerning the molecular mobility at Tg . Such differences cannot be resolved until a better collection of Tg s is available. Even then, different analyses and models appear to lead to differing conclusions. Some discussion of these will be found below.

91.0 ⬚C

2.5 2.0

94.0 ⬚C

1.5

12.4 ISOTHERMAL CONTRACTION NEAR AND BELOW Tg

1.0

The kinetics of spontaneous time-dependent contraction of a glass following a quench from equilibrium above Tg is frequently studied. This kind of measurement is one of the experiments carried out by Kovacs [3] in his classical studies on the time-dependent variation of the specific volume of glasses. The most common study involves quenching from a fixed temperature above Tg down to different temperatures below Tg . At each chosen lower temperature, the spontan-

0.0

0.5

97.0 ⬚C 100.0 ⬚C

1

2

3 4 Log (t-100)

5

6

FIGURE 12.6. The fractional excess specific volume [v
(t)
v
(1)]=v
(1) for a polystyrene at different temperatures shown as a function of the logarithmic time after quenching from T ¼ 104.0 8C. (100 sec is subtracted as the approximate time to reach the temperature indicated.)

THE GLASS TEMPERATURE extent. This can be seen in Fig. 12.7 where the compositional variation of the Tg of polystyrene in a number of solvents is shown. Solutions of a polymer in solvents with Tg s higher than its own will usually have a greater value than that of the neat polymer [57]. The impression of a simple relationship given by Fig. 12.7 is misleading since a continuously decreasing negative slope is indicated. For polystyrene, PS, solutions in toluene [58] and m-tricresyl phosphate [59,60] this has been shown not to be the case. In 1964 Braun and Kovacs reported results on the polystyrene/toluene system which showed two descending curves which came together in a cusp [58] (see Fig. 12.8). The results were rationalized by fitting the Kelley–Bueche equation [61], Tg ¼

Tg,2⫺Tg,x , ⬚C

50

f2 a2 Tg2 þ f1 a1 Tg1 , f2 a2 þ f1 a1

100

150

200

cc 250 0

60° 40°

0°

Tg

−20° −40° −60° b −Naphthyl salicylate Phenyl salicylate Tricresyl phosphate Methyl salicylate Nitrobenzene

−140°

Chloroform

Benzene

Methyl acetate

Toluene

Ethyl acetate

Amyl butyrate

Carbon disulfide

−160°

0

0,50

0,75

c

1

iso-free volume temperature.a f is the volume faction; a is the cubical thermal expansion coefficient of the fractional free volume, f ¼ y f =y; subscripts 1 and 2 represent the solvent and polymer, respectively. y f is the free volume and y the measured volume. Below Tc , Braun and Kovacs used the equation [58],
 fg2 f2 Tg ¼ Tg1 þ , a1 f1

20°

−120°

0,25

FIGURE 12.8. The depression of Tg of a polystyrene by dissolution in toluene. x is the weight fraction of toluene. Filled circles represent dilatometric determinations and unfilled circles were obtained by means of differential thermal analysis, DTA. The crosses represent the results of Jenckel and Heusch [56]. (From Braun and Kovacs by permission, [58].)

80°

−100°

191

0

to the data above the temperature of the cusp, Tc . This equation was derived assuming that the free volumes of the polymer and the solvent were additive and that Tg is an

−80°

/

0.20

0.40

w1

0.60

0.80

1.00

FIGURE 12.7. The compositional variation of the glass temperature Tg of polystyrene in 12 different solvents. W1 is the weight fraction of solvent. (From Jenckel and Heusch, by permission, [56].)

where fg2 is the fractional free volume of the polymer at its Tg . Pezzin et al. found the same kind of behavior exhibited by polyvinyl chloride PVC in two different plasticizers, dibutyl phthalate, DBP, and dicyclohexylphthalate, DCHP [62,63]. The same two equations provided excellent fits to the PVC solution data (see Fig. 12.9). For polystyrene dissolved in m-tricresyl phosphate, TCP, additional factors were noted. Differential thermal analysis measurements indicated double Tg s for solutions with lower polymer concentrations beyond the depicted cusp in Fig. 12.10 [59]. Creep recovery measurements on this system showed that the solvent molecules in the solutions have higher mobilities than the polymer chain segments. A lower temperature and greater crowding is therefore necessary to force the solvent molecules from an equilibrium response. Therefore, it was a

Other treatments are now available [137,138].

192 / CHAPTER 12

350

Tg, °K

300

250

200 0

0.2

0.4

w1

0.8

0.6

1.0

FIGURE 12.9. The Tg of a polyvinlychloride as a function of diluent concentration in two different solvents. Open circlesdicyclohexylplthate and filled circles-dibutylphthalate. W1 is the weight fraction of diluent. (From Pezzin, Omacini, and Zilio-Grandi by permission [62].)

concluded that the higher Tg s reflected those of the polymer chain segments in a solvent-altered environment and the lower Tg s reflected the solvents mobility in the presence of the polymer chain segments [60]. For different bulk polymers, Tg local mode or short-range molecular motions are found within experimental uncertainty at the same place on the time or frequency scales of response [64–66]. For the solutions of PS in TCP, the Tg s from the lower ‘‘altered solvent’’ curve were necessary to bring the solvent contribution to the recoverable compliance into correspondence with other solutions. The higher Tg s were also needed to

100

Polystyrene-Tri Cresyl Phosphate (TCP)

80 60

Tg °C

40 20 0

bring the contribution from polymer local modes into correspondence [60]. The mobility of solvent molecules and how it is influenced by the presence of polymer solute molecules has been investigated and documented by measurements of pulsed field-gradient NMR [67], oscillatory electric bire-fringence [68,69], 13 C NMR relaxation [70], photon correlation spectroscopy [71–73a], and dielectric dispersion [73b]. Solvent molecules generally have a decreased mobility in the presence of polymer molecules with greater Tg s. In most cases, free volume concepts at least qualitatively predict the shift of solvent and polymer chain segment mobilities towards one another. The most frequently encountered case, where a polymer with a higher Tg than that of the solvent, brings to the solution a smaller contribution of free volume which decreases the solvent’s mobility. At the same time the solvent, being far above its Tg , brings a large contribution of free volume to the solution which accelerates polymer segment motions. However, there are solutions where the solvent’s mobility is increased by the presence of a polymer whose undiluted Tg is higher than that of the solvent. In this case, obviously, free volume concepts are inadequate. In these cases, the coupling model of Ngai has been able to attribute the unexpected solvent acceleration to a ‘‘primitive’’ or uncoupled relaxation time of the polymer which is smaller than that of the solvent [73].

12.6 DEPENDENCE OF Tg ON MOLECULAR WEIGHT AND CROSSLINKING During the transformation of a monomer into a polymer, ˚) many atoms separated by van der Waals distances (5 A ˚ ). participate in the formation of covalent bonds (1–3 A Therefore during polymerization, an increase in the macroscopic density ensues, while on the molecular level a decrease in free volume and entropy occurs while the cooperativity of motions increase. Concomitantly the glass temperature can increase by more than 100 8C. Several important adhesive systems are based on this increase. The cyanoacrylate ‘‘Super Glue’’ starts as a monomer with a Tg < 0 8C and it polymerizes to a linear soluble polymer with a Tg which is in the neighborhood of 100 8C upon application under anaerobic conditions. For linear polymers the simple equation [74] Tg ¼ Tg (1)
K=Mn ,

−20 −40 −60 −80 0

20

40

60

80

100

% TCP

FIGURE 12.10. The Tg s of a polystyrene as a function of the weight percent of the solvent m-tricresyl phosphate.

which reflects the linear decrease in Tg with the increase in concentration of polymer chain ends does an adequate job of describing most existing data in the literature. Deviations may occur at very low molecular weight. The more elaborate Gibbs-Dimarzio theory [40,41] which concentrates on the conformational entropy can fit data to lower molecular weights. Uncatenated cyclic polydimethylsiloxane, PDMS, shows [75] a slight increase in Tg with decreasing molecular

THE GLASS TEMPERATURE

CH2

CH3 CH

CH3

O

C

CH2

CH3

where (nþ1) is the average number of repeat units in the epoxy resin molecule. Diamines are most often used in curing these resins. Two examples are: 4,4’-methylene dianiline, MDA, H

H2N

C

NH2 ,

H

whose molecular weight is 198.3 g/mol, melting point range 90–93 8C and density, r(23 8C) ¼ 1:16 g=cm3 ; and 4,4’ diamino diphenyl sulfone, DDS O

H2N

S

NH2 ,

O

M W 248.3 g/mol, Mp 175–178 8C, r(23 8C) ¼ 1:38 g=cm3 . The effect of the developing network during curing on Tg and the mechanical behavior is illustrated with an epoxy resin Eponb 1001F (n ¼ 2:3) cured with DDS [81]. The degree of network development was determined by reacting the epoxy resins stoichiometrically with varying ratios of the tetrafunctional crosslinker DDS and the chain-stopping monofunctional methyl aniline. Some properties of the resin at different stages of network development are given in Table 12.1. Fictive temperatures Tf ,g [82–84] which approximate Tg are presented along with closely related temperatures. The Tf ,g (108/min) for the neat unreacted epoxy b

Shell Corp. trademark.

Tg ’ Tg (1)
K=Mn þ Kx r, where r is the number of crosslinks/gram. Epoxy resins based on the diglycidyl ether of bisphenol A, DGEBA, are the most widely produced and studied. Their chemical structure is represented by

CH

O

CH3

OH O

193

Crosslinking can increase the Tg above that of the infinite molecular weight linear polymer. This increase can be accounted for with the equation of Fox and Loshaek [74].

weight. This trend has been rationalized by Gutman and Dimarzio [76]. However, it was observed that the rate of creep at 120 8C for cyclic polystyrene, PS, with Mw ¼ 1:11  104 in the softening dispersion was the same within experimental uncertainty as a cyclic sample with a Mw ¼ 1:85  105 . This indicates that the Tg s of the two samples had to be within one or two tenths of a degree of one another [77]. For their linear counterparts, the rate of creep of the lower molecular weights is about 100 times faster where the difference in Tg s is about 10 8C.

O

/

CH2

O n

C

O

CH2

CH

CH2 ,

CH3

resin was 31 8C. The short-term (0:1---102 sec) behavior at Tg can be described by a viscoelastic recoverable compliance which appears to be the same for all amorphous materials, without secondary viscoelastic dispersions contributions in this timescale range. A recoverable creep compliance is found that can be fitted to the Andrade equation [85–87]. Jr (t) ¼ Jg þ bt1=3 , where Jg is the glassy compliance which has a value in the neighborhood of 1  10
9 Pa
1 (1  10
10 cm2 =dyne); t is the time; and b is a constant which depends on the choice of Tg (q). For the choice of q ¼ 10 8=min it can be seen in Table 12.1 that b(Tf ,g ) ’ 2  10
9 (Pa sec1=3 )
1 . The first three levels of crosslinking yielded viscoelastic liquids (gel fraction¼0) that flow and exhibit steady-state recoverable compliances [18]. A material which was extremely close to the incipient point of gelation was yielded by 45% DDS, where a macroscopic molecular network just appears. At this point the recoverable compliance at long times is immeasurably high; i.e., Je0 is operationally infinite. At higher crosslinker ratios, the equilibrium compliance Je of the molecular network is readily measurable. At 50% crosslinking agent Je ’ 1:0  10
4 Pa
1 (see Fig. 12.11). Tg s of fully cured bisphenol-A-based epoxy resins with varying crosslink densities were measured during cooling (58/min) in a pressurized bellows dilatometer (5 MPa). The crosslinking agent was DDS. The results are shown in Fig. 12.12. The molecular weights per crosslinked unit, Mx , were 420 for Epon 828; 910 for 1001F; 1520 for 1004F; and 2870 for 1007F. The Tg s obtained were 2048, 1278, 1128, and 101 8C, respectively. The loosest network epoxy, 1007F, has the highest specific volumes and the lowest Tg . The volume changes during the curing of Epon 1001F are shown in Figs. 12.13a and 12.13b. In Fig. 12.13a the temperature history of the cure is shown along with the volume changes due to heating, curing, and cooling as a

194 / CHAPTER 12 TABLE 12.1. Characterizing parameters. DDS

Tf ,g a

T0,J b

T0,a c

Jg  1010 d (cm2 =dyne)

b  1010 d

r(25 8C)e (g=cm3 )

log Je f (cm2 =dyne)

Mx g

0.25 0.35 0.40 0.45 0.50 0.60 0.70 1.0

63.0 65.0 70.6 72.5 73.4 80.0 86.4 132.0

64.1 66.0 69.9 72.5 75.4 84.9 89.2 135.6

66.3 65.0 66.4 69.9 73.7 82.0 88.7 135.5

1.13 0.685 0.65 1.45 1.06 1.15 1.24 1.36

1.46 1.43 1.95 2.06 1.19 2.63 1.78 1.95

— 1.188 — 1.189 1.191 1.194 1.218 1.205


5.53
4.10
3.60 —
5.00
5.98
6.43
7.04

— — — — 2:95  105 3:10  104 1:12  104 680

a

Measured using DSC; all heating rates were 10 8C/min, following cooling at a rate of 10 8C/min (20 8C/min rate of cooling for 0.35 and 0.45 DDS; and 80 8C/min for 1.0 DDS). b Reference temperatures that match the softening region on the creep compliance reduced time scale. Reference system was 0.45 DDS. c Reference temperatures obtained from the temperature shift factor, aT , analysis (see Fig. 12 in Ref. 81). d Andrade equation parameters for T ¼ T0,J . e Density of fully cured samples measured by flotation; followed by pycnometry. f For the viscoelastic liquids Je is the steady-state recoverable compliance and for the viscoelastic solids, beyond the gel point it is the equilibrium compliance. g Average molecular weight per crosslinked unit calculated from Je : Mx ¼ rRTJe .

function of time [82]. The volume changes are presented in Fig. 12.13b as a function of temperature, where the initial fictive temperature, Tf , of the reactant mixture and the glass temperature, Tg , of the fully cured resin can be seen.

T0 °C

−3 −4 −5

64.1

0.35 DDS

66.0

0.40 DDS

69.9

0.45 DDS

72.5

0.50 DDS

75.4

0.60 DDS

84.9

0.70 DDS

89.2

1.0 DDS

135.6

0

0.90

−1 −2 0.40 DDS

−3 0.35 DDS 0.50 DDS 0.25 DDS

−6

−4

0.60 DDS −5 0.70 DDS

−7

−6 1.0 DDS

−8

−7

1007/DDS

0.89

0.45 DDS

1004/DDS

0.88 SPECIFIC VOLUME (ml/g)

Log [J(t)− Jg,0] cm2/dyne

−2

0.25 DDS

Log [J(t)− Jg,0] Pa−1

−1

The increase in Tf ,g during curing has been seen to track the degree of cure. This can be seen in Fig. 12.14 where five variables have been monitored during the curing of Epon 1001F at 142 8C which is about 10 8C above the Tg of the fully cured material. The degree of cure was monitored by the increase in density. The gel fraction became measurable after several hours signaling the presence of a macroscopic

1001/DDS

0.87 0.86 0.85

828/DDS

0.84 0.83

−9

−8

−10 −2

−9

0

2

4

6

8

10

12

14

Log t/aT (sec)

FIGURE 12.11. Comparison of all of the time-dependent reduced compliance [J(t)
Jg ,0 ] as logarithmic functions of log t=aT . Comparison temperatures are indicated above and in Table 12.1. Jr (t)
Jg ,0 is shown for the viscoelastic liquids (dashed lines); i.e., specimens with no gel fraction.

PRESSURE = 5 MPa COOLING RATE = 5°/min

0.82 0.81

0

50

100 150 TEMP (°C)

200

250

FIGURE 12.12. Specific volume - temperature curves for four epoxy resins with increasing crosslink density from 1007/DPS to 828/DPS. Measurements were made during cooling at 58/min under a pressure of 5 MPa.

THE GLASS TEMPERATURE

/

195

SAMPLE TEMP

SPECIFIC VOLUME (ml/g)

0.88

150

100

0.87 0.86

50

SPECIFIC VOLUME

SPECIFIC VOLUME

SAMPLE TEMP.(°C)

200

0

0.85 0.84

EPON 1001F/DDS PRESSURE = 5 MPa

0.83 0

2

4

6

(a)

8 48 50 TIME (HOURS)

52

54

56

0.88 EPON 1001F/DDS PRESSURE = 5 MPa

SPECIFIC VOLUME (ml/g)

0.87

HEATING COOLING RATE = 5°/min

0.86

0.85

Tf

0.84

0.83 (b)

Tg

0

50

100

- HEATING & SOAKING - COOLING

150

200

SAMPLE TEMPERATURE (°C)

FIGURE 12.13 (a) Specific volume - temperature history of 1001F/DDS epoxy resin during curing under a pressure of 5 MPa. (b) Specific volume data from Fig. 12.13a plotted as a function of the temperature.

molecular network. Before the point of incipient gelation, the viscosity can be seen to climb toward infinity before gelation. After gelation, the precipitous drop in the equilibrium compliance of the developing network continues until a complete cure is achieved in about two days. Using the starting and final Tf ,g s as limits it can be seen that they follow the degree of cure. When an epoxy resin is cured at a temperature which is below its ultimate Tg (1) at full cure, the Tf ,g will

increase rapidly as the reaction proceeds until the Tg becomes equal and surpasses the temperature of cure Tc . The reaction then becomes diffusion-controlled and the rate of reaction and increase in Tf ,g decelerates to a much lower but still perceptible value as can be seen in Fig. 12.15 [82]. The curing reaction continues significantly even when Tc is 50 8C below Tg . Room temperature cures of epoxy resins can therefore be expected to continue for years.

196 / CHAPTER 12 −5.5

1.0 5.0

120

−6.0

100

−6.5

80

−7.0

0.4 3.5

60

−7.5

0.2 3.0

40 0

3

4 Log tc (sec)

5

Log Je (cm2/dyne)

EPON 1001 F/DDS

Tf (°C)

0.6 4.0

Logh (poise)

Degree of cure & Gel Fraction

0.8 4.5

Degree of cure Gel Fraction Log h Tf Log Je

6

−8.0

FIGURE 12.14. Comparison of changing parameters during the curing of 1001F/DDS at 142 8C. The logarithm of the viscosity h(p) and the equilibrium compliance Je (cm2 =dyn) as well as the fictive temperature Tf , the degree of cure, and the gel fraction are shown as functions of the logarithm of the time of curing Tc (s).

140

EPON 828/MDA

120

T1 (°C)

100 80 60 40

TCURE°C

20

100 80 60 40

0 −20 2

3

4 5 Log tCURE (sec)

6

FIGURE 12.15 Fictive temperatures, Tf , observed during the curing of an epoxy resin at temperatures below the Tg (1) of the fully cured state.

12.7 DEPENDENCE OF Tg ON THE DEGREE OF CRYSTALLINITY AND MORPHOLOGY Nearly all crystalline polymers contain chain segments that do not reside in a crystalline lattice. Usually these noncrystalline segments can be considered to constitute an amorphous phase which therefore can become glassy. The Tg of this amorphous phase depends on the degree of crystallinity. Tg increases and decreases with the presence of

crystallinity. It can increase or decrease with the degree of crystallinity depending on the relative density of the amorphous and crystalline states. Most often the more orderly crystalline state has the higher density at Tg and the noncrystalline molecular chains are constrained by being anchored to the immobile crystallites and Tg increases. On rare occasions the crystalline state has a lower density than the amorphous material [88]. In this case, less constraint on the noncrystalline chain segments increases the entropy causing Tg to decrease. Tg is not a unique function of the degree of crystallinity. At least in the usual case where the density of the crystalline state at Tg is higher than that of the amorphous state, the temperature at which the crystallites are formed plays a dominating role. At temperatures above the maximum in the crystal growth rate [89], and of course, below the melting temperature Tm , the rate of nucleation is low and therefore relatively few spherulites are formed and the tie molecules in between crystallites are relatively unconstrained. Hence the increase in Tg with increased crystallinity is relatively slight [89] as seen in Fig. 12.16. However, at temperatures below the maximum in crystal growth rate and near Tg , the rate of nucleation can be profuse. Many crystallites are formed and tie molecules are therefore shorter and more constrained. Therefore a given degree of crystallinity is more effective in raising Tg (see Fig. 12.17). The above conclusions are drawn from the mechanical results reported by Groeninckx et al. [89] where the onset of the steep decrease in log Er (10) with temperature can be considered as a rough estimate of Tg . Direct confirmation with proper Tg measurements is desirable.

THE GLASS TEMPERATURE

197

50 % (30 min)

40 % (27 min)

34 % (17u)

Er(10)(dynes/cm2)

Er(10)(dynes/cm2)

1010

/

1010

28.5 % (22 min)

109

14.5 % (11 min)

29.5 % (5 min 1/2)

Amorphous 108 60

80

100

23.5 % (4 min 1/2) 120 Temperature(°C)

FIGURE 12.16. Ten-second stress relaxation moduli Er (10) of PET crystallized at 2278C to different degrees presented as a function of temperature. Crystallization times are also shown as well as the estimated degree of crystallinity from density measurements. (From Groeninckx, Berghmans, and Smets, by permission, [89].)

12.8 DEPENDENCE OF Tg ON INTERMOLECULAR FORCES The greater the intermolecular interaction, all other things being equal, the higher Tg will be. It has been proposed that Tg is a linear function of the cohesive energy density CED [90]. CED ¼ 0:5MRTg
25M, where R is the gas constant and M is a parameter analogous to the number of degrees of freedom of a molecule. Eisenberg has shown how Tg increases dramatically in a phosphate glass with the decrease in size of incorporated anions and with increases in their charge [91]. He has also shown how in ionomers (specifically ethyl acrylate-acrylic acid copolymers neutralized with various cations) Tg is a common smoothly increasing function of c q/a as shown in Fig. 12.18. c is the cation concentration, q is its charge, and a is the distance between centers of charge, as shown in Fig. 12.18. The onset of the observed sigmoid coincides with the domination of a wide range of properties by ionic clusters. 12.9 THE EFFECT OF PRESSURE ON Tg An increase in pressure on an amorphous material increases molecular crowding and interactions along with decreasing the entropy. Regardless of the variable which one looks upon as significant, an increase in Tg is expected.

Amorphous 109 60

80

100

120

Temperature(°C)

FIGURE 12.17. Ten second stress relaxation moduli Er (10) of PET crystallized at 1208C to different degrees presented as a function temperature. (From Groeninckx, Berghmans, and Smets, by permission, [89].)

If a rapid pressure increase is incurred, a time-dependent decrease in volume will follow if the temperature is near or below Tg . This is indeed a simple voluminal viscoelastic creep process. The resultant behavior is quite analogous to that seen following a temperature quench as discussed above. The similar response to pressure jumps has been shown by Goldbach and Rehage [92]. The expected increase in Tg with increases in pressure has been documented by McKinney and M. Goldstein [93]. Discussions in the literature are confounded by not restricting the definition of Tg to cooling from an equilibrium state.

12.10 EFFECT OF MOLECULAR STRUCTURE ON Tg 12.10.1 Internal Plasticization and Chain Stiffness If side chains on the polymer backbone are increased in length only, it is generally observed that Tg decreases, ostensively because the linear side chains increase the fractional free volume between the chains and any structural change which increases the diameter of the side chains reverses the tendency; i.e., Tg increases. This is a reflection of the following general principle. Any structural feature which increases the size of the jumping unit of the molecular chain will increase Tg . An

198 / CHAPTER 12

T, (⬚C)

200

100

0

0

0.1

0.2

0.3

0.4

cq/a

FIGURE 12.18. Tg of ethyl acrylate - acrylic acid copolymers neutralized with various cations shown as a function of cq/a, where c is the cation concentration, q is its charge, and a is the distance between centers of charge. (From Matsuura and Eisenberg, by permission, [91b].) Different symbols represent different cations.

increase in chain stiffness resulting from longer rigid units in the chain backbone or more bulky side groups which drastically increase the potential barriers to rotation, cause substantial increases in Tg . Steric barriers to rotation are raised appreciably if a second side group is introduced on alternate chain backbone atoms. The pairs, polymethylmethacrylate PMMA (Tg ’ 115c)-polymethylacrylate PMA (14 8C) and poly(a-methylstyrene) (168 8C)polystyrene (100 8C), illustrate the effect. Introducing paraphenyl rings into a chain backbone increases Tg since a longer portion of the chain has to be involved in a molecular segmental displacement. To the contrary introducing additional methylene (-----CH2 -----) groups or ether oxygens into the polymer chain backbone lowers Tg because of the increased chain flexibility. With no side atoms or groups and a wide open bond angle (> 1118) the ether oxygen is considered the premier flexibilizing unit.

Tg of polyacrylonitrile (  103 8C) relative to that of polypropylene (
14 8C) must be due to the large electron affinity of the nitrile group (-----C  N). Enhancing the flexibility of a polymer chain by introducing an ether oxygen into its backbone will lower Tg and the melting temperature, Tm . Increasing the polarity or the opportunity for hydrogen bonding between neighboring chain segments increases Tg and Tm . The effects of such variations are illustrated in Table 12.2 where the coupling units are varied in a family of otherwise similar polymers. TABLE 12.2. Effect of polarity and hydrogen bonding. Low Tg s and TM s

O

R⬘

polyethers

O

12.10.2 Effect of Polarity The stronger the interactions between neighboring polymer chain segments, the greater the thermal kinetic energy must be to create holes of sufficient size to allow a diffusive jump of a chain segment to occur. Therefore, the greater the polarity of a polymer, the higher the Tg will be. The greatest effect will be due to resultant components of dipole units which are perpendicular to the chain backbone. The greater

R

High Tg s and TM s

polyesters

C

O

H

O

N

C

H

O

N

C

H

O

H

N

C

N

O

polyurethanes

polyamides

polyureas

THE GLASS TEMPERATURE 12.10.3 Influence of Symmetry On the basis of some of the above comments it would be expected that the Tg of polyisobutylene PIB [CH2 -----C (CH3 )2 ]n would be higher than that of polypropylene PP [CH2 -----C(H)CH3 ]n . The polarity of PIB is lower than that of PP because the opposing dipoles tend to cancel one another. However, the polarity is low in both cases and would not be expected to be a dominating factor. The two side groups on alternate carbon backbone atoms of the PIB certainly should present overall higher barriers to rotation than that present in the PP. Yet the Tg of PIB at
73 8C is some 59 8 lower than that of PP. Likewise, the Tg of polyvinylidene chloride [CH2 -----CCl2 ]n at
17 8C is 638 lower than that of polyvinylchloride [CH2 -----CClH]n . It was discovered by Boyd and Breitling [95] that whereas there are indeed higher barriers to rotation in the vinylidene polymers, there are adjacent potential energy wells with an extremely low barrier in between them, thus allowing for very free rotation over a limited angle of about 208 which permits liquid structures to adjust far more rapidly than expected.

12.10.4 Effects of Tacticity It is surprising that stereochemical variations in tacticity [96] have no measurable effect on the Tg of polymethylacrylate PMA and polystyrene PS, but they have a substantial effect on that of polymethylmethacrylate PMMA and poly(a-methylstyrene) PaMS [97]. The explanation appears to lie in the added steric repulsion to rotation due to the presence of the asymmetric double side groups on alternate chain backbone atoms. Extended planar zig-zag configurations of the chains are not possible and it is clear that different helical forms of highly isotactic and syndiotactic chains obtain. The stiffness of the helices are obviously significantly different. A reflection of the differences in the helical character of the chain conformation is seen in the variation of the dielectric permittivity as a function of the tacticity [98]. Highly syndiotactic samples of PMMA have a dominant b loss peak reflecting independent motion of the carbonyl dipole in the ester side group. However, as the degree of istacticity is increased, the dipole activity is shifted to the a backbone loss peak. As the limit of total isotacticity is approached, the b loss peak virtually vanishes, indicating that the carbonyl dipole motion is locked in with the chain backbone motion. A consequence of the lack of independent side chain motion (b mechanism) in iso-PMMA is a much smaller glassy compliance, Jg [99]. Secondary (sub-tg) loss mechanisms are listed in this handbook in the chapter authored by Fried.

/

199

12.11 DIFFERENCES OF OPINION CONCERNING Tg Qualitatively free volume concepts usually provide a rationale for observed behavior [18], but clearly they do not provide a comprehensive understanding. Interactions and coupling also play a role. Occasionally, as mentioned above, a polymer with a Tg , which is higher than that of its solvent will increase the mobility of the solvent molecules. This is contrary to what is expected according to free volume concepts. Such acceleration can be understood if the uncoupled mobility of the polymer is greater than that of the solvent [73]. In addition to the free volume [36,37] and coupling [43] models, the Gibbs–Adams–DiMarzo [39–42], (GAD), entropy model and the Tool–Narayanaswamy–Moynihan [44–47], (TNM), model are used to analyze the history and time-dependent phenomena displayed by glassy supercooled liquids. Havlicek, Ilavsky, and Hrouz have successfully applied the GAD model to fit the concentration dependence of the viscoelastic response of amorphous polymers and the normal depression of Tg by dilution [100]. They have also used the model to describe the compositional variation of the viscoelastic shift factors and Tg of random Copolymers [101]. With Vojta they have calculated the model molecular parameters for 15 different polymers [102]. They furthermore fitted the effect of pressure on kinetic processes with this thermodynamic model [103]. Scherer has also applied the GAD model to the kinetics of structural relaxation of glasses [104]. The GAD model is based on the decrease of the conformational entropy of polymeric chains with a decrease in temperature. How or why it applies to nonpolymeric systems remains a question. The TNM model has been used to describe structural relaxation during the heating and cooling of amorphous crowded liquids by O’Reilly [8] and by Hodge [10]. A disturbing result of the application of the TNM model is that the effective relaxation time, t, is not constant at Tg but varies almost eight orders of magnitude when comparing values for different materials [8]. This variation is in serious conflict with the nearly constant rate of creep at Tg observed on a wide variety of amorphous materials [60,64–66]. The TNM model employs the stretched time-scale of the Kohlrausch [105]-William-Watt [106] function: i.e., relaxation which is proportional to exp [
(t=t)m ]. The Kovacs, Aklonis, Hutchinson, and Ramos, KAHR, model employs a distribution function of retardation times to describe volume memory and other viscoelastic effects. In the KAHR model it is assumed that the retardation function shifts to longer times with decreasing temperature while its shape is conserved. This requires thermorheological simplicityc which does not always hold near and below Tg [78,107]. c

See the discussion on thermorheological simplicity in Chapter 26 on Viscoelastic Behavior.

200 / CHAPTER 12 The influence of rates on Tg and glass formation has recently been treated by Shi [108]. His results should be examined and tested. He predicts upper and lower bounds for Tg based on thermodynamic and kinetic factors.

12.12 SOME CORRESPONDING PROPERTIES AT Tg In Table 12.3 a number of properties determined in the neighborhood of Tg are listed [109]. The Tg s were determined at or reduced to a rate of cooling of 0.28/min. A wide variety of amorphous materials are represented. The first five materials tri-a-naphthylbenzene, Tri cresyl phosphate Aroclor 1248, 6 phenyl ether, and 1,2 diphenylbenzene (o-terphenyl) are nonpolymeric compounds; Tl2 SeAs2 Te3 is an inorganic glass-former; selenium is a polymeric element; polystyrene, amorphous polypropylene, polyvinylacetate, and polyisobutylene are linear organic polymers; Epon 1004/DDS, Epon 1007/DDS, and Viton 10A are crosslinked organic polymers; and 20 PB/A1248 is a solution of a linear organic polymer, 1,2 polybutadiene in an organic solvent, Aroclor 1248. The T1 is the hypothetical temperature in the Vogel, Fulcher, Tamman, and Hesse (VFTH) equation [18] and the Williams, Landel, and Ferry (WLF) equation [18] where the viscosity becomes infinite. It is hypothetical

because it is implicitly assumed that an equilibrium density persists. At an infinitely slow rate of cooling it appears by extrapolation that Tg (q ¼
1) ¼ T1 [49] which, in this limit, makes it a candidate for the second order thermodynamic transition of the GAD model. According to the ‘‘universal’’ form of the WLF equation Tg
T1 ’ 52. In Table 12.3, the values vary from 298 to 175 8C. Logarithmic values of the glassy shear compliance Jg (cm2 =dyne) at Tg are given. Most values are close to
10.0 as is often observed. Values as high as
9.4 have been reported [110] and the chalcogenide glass Tl2 SeAs2 Te3 is the lowest at
11.1. Logarithmic values for the steady-state recoverable shear compliance Je0 (cm2 =dyne) are also shown. For the nonpolymer Je0 values are 2.5 to 3.0 times larger than Jg . For the linear polymers, since Je0 is a function of the molecular weight up to value 5 times that of the entanglement value, Me , and its distribution, much larger ratios are seen. Of course, the network polymers exhibit equilibrium compliances that are determined by the level of crosslinking and are not steadystate values. Steepness indices, STg , ¼
Tg [d log aT =dT]Tg which indicate the temperature sensitivity of viscoelastic processes at Tg are also listed. STg correlates with the breadth of the effective viscoelastic spectrum; i.e., the higher STg is, the broader the viscoelastic response is. Thus, the tempera-

TABLE 12.3. Properties of amorphous materials near and below Tg .

Material Tri-a-naphthylbenzene Tri-cresyl phosphate Aroclor 1248a 6-Phenyl etherb 1,2 Diphenylbenzene Tl2 SeAs2 Te3 Se Polystyrenec Polypropylened Polyvinylacetatee Polyisobutylenef EPON 1004/DDSg EPON 1007/DDSg Viton 10Aj 20PB/A1248k a

T1

Jg

LogJe0 (Tg )

Tg

STg

Logh(Tg )

bTg  101 2

Tg1‘

DTg

8C
73.2
116.7
181.1
99.8
101.3
113.2 10.5 69.0
46.4 6.0
135 60.8 50.7
54.0
163.6


10.022
10.110
10.372
10.140
10.347
11.102
10.482
10.022
9.934
10.046
10.460
9.879
9.928
9.969
10.495


9.548
9.586
9.987
9.766
9.926
10.64
8.480
4.660
4.820
4.650
5.715
7.390
7.100
5.940
4.750

8C 63.8i
73.0i
50.0i
25.0i
32.4i 67.4 32.0 97.5i
14.0 34.8i
76.2i 107.7 97.3
26.1
50.0i

74.8 80.5 62.0 84.6 87.6 60.5 155 137.4 135.4 94.2 45.1 147.8 144.0 136.3 61.7

poise 12.445 12.905 12.746 12.677 12.367 12.246 12.67 13.74h 13.85h 17.05h 15.23h — — — 17.0h

cm2 =dyne-s1=3 16.67 10.09 6.146 7.700 9.746 3.980 7.590 20.88 15.80 16.95 14.62 9.816 7.590 14.56 9.900

8C 60.8
73.0
47.7
23.9
31.3 69.9 32.8 95.7
15.0 36.4
77.4 107.5 98.1
26.9
49.8

8C 3.0 —
2.3
1.1
1.1
2.5
0.8 1.8 1.0
1.6 1.2 0.2
0.8 0.8
0.2

Aroclor 1248 is a chlorinated biphenyl. 6-Phenyl ether is an abbreviation for bis(m-(m-phenoxy phenoxy)phenyl)ether. c Commercial Polystyrene (Mw ¼ 2:20  105 ) d Amorphous Polypropylene (Mw ¼ 2:05  105 ) e Polyvinylacetate (Mw ¼ 6:50  105 ) f Polyisobutylene (Mw ¼ 7:80  104 ) g EPONs are epoxy resins consisting of a diglycidyl ether of bisphenol A cross-linked using diaminodiphenyl sulfone(DDS) as the curing agent. They are network polymers that do not flow. h Viscosities were calculated using the Vogel, Fulcher, Tamman, and Hesse (VFTH) equation. i Glass temperatures were measured at a cooling rate of 0.2 8C/min otherwise they were calculated using the shift factors. j Viton 10A is a fluoroelastomer. k 20PB/A1248 is a 20-wt % solution of a Polybutadiene (Mw ¼ 1:34  105 ) in Aroclor 1248. l 0 Tg is the temperature at which b ¼ 10  10
12 cm2 =dyne-sec1=3 . b

THE GLASS TEMPERATURE

0.5

COOLED FROM 134 °C ti =100 sec

T = 131.5 °C

0.0

HEATED FROM 129 °C ti =200 sec

−0.5 0

1

2 3 log(t−ti) (sec)

4

5

FIGURE 12.19. Fractional volume deviation from equilibrium as a function of time at 131.5 8C after cooling from 134 8C and after heating from 129 8C plotted versus the logarithm of the corrected aging time t
ti where ti is the estimated time for the specimen to reach a uniform temperature. Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789–5802 with permission from Institute of Physics Publishing.

201

incurred after equilibrium was achieved. The results on an epoxy resin derived from a diglycidyl ether of bisphenyl A (Epon 1001), which was fully cured with DDS (4,4’-diamino diphenyl sulphone), are shown in Fig. 12.20, where V(t) is the specific volume after the temperature step, and V(1) is the equilibrium volume at the final temperature. Since the thermal driving force was virtually the same with each step, simple temperature reduction could be attempted with horizontal time-scale shifts. The successful reduction is presented in Fig. 12.21. It was assumed that the results were reasonably close to linear behavior and the more extensive curve obtained from the cooling steps was analyzed to obtain a normalized retardation spectrum L(t). ð L0 (t)dlnt ¼ 1 then V(0þ )
V(t) ¼ V(0þ )
V(1)

ð þ1   t L0 (t)(1
e
t )dlnt :
1

The L0 (t) that was obtained is shown in Fig. 12.22 with the retardation spectrum obtained from the shear creep compliance function, J(t). The levels of the short time behavior are matched. Three features should be noted. The functionality at short times is the same within experimental uncertainty. The 1/3 slope of log L(t) at short time indicates that the response is dominated by motions that contribute to Andrade creep. [66,85–87,141,142] JA (t) ¼ JA þ b t1=3 , where t is the time of creep, and JA and b are characterizing constants. The positive curvature and the following maximum of the creep compliance L(t) indicate the contribution of polymeric molecular modes of motion to the recoverable compliance. Since no positive curvature is seen in the volume contraction L0 (t), no polymeric modes are present and no long range coordinated motions of any kind are detected. ⎥(V(t)/V∞)−I)⎥ ⫻103

(V (t )/V (∞)-!)⫻103

ture dependence of dissipative processes is related to their time and frequency dependence [139]. Although the viscosities of nonpolymeric liquids have close to the same viscosity at Tg of  3  1012 poise ( 3  1011 Pa sec), this value is smaller than the classically referred to value of 1:0  1013 poise in spite of the fact that most common Tg s have been determined at a rate of cooling of 18/min. At the cooling rate of 0.28/min the Tg s reported here would be several degrees lower and thus higher viscosities would be expected. The widely accepted value for h(Tg ) ’ 1013 poise does not hold at all for linear polymers, as noted above, because of the dependence of h on the molecular weight [140], and for crosslinked polymers, which do not flow. The local segmental mobility is obviously the determining factor for the packing density of amorphous polymers, whether or not chain-like molecules are entangled or crosslinked. Local molecular packing and motions are believed to be the determining factors for the liquid structure. Long range motions cannot be influential in defining the density. Molecular packing beyond several nearest neighbors in an amorphous system such as a supercooled liquid cannot be coordinated because of the absence of long-range order. Evidence for this assertion can be seen by comparing the retardation spectra obtained from volume contraction with that obtained from shear creep. The kinetics of isothermal volume contraction and expansion of a cured epoxy resin, which is completely amorphous, below its Tg has been followed [49]. Since the free volume determines the molecular mobility, contraction following a decrease in temperature is going to be faster than the expansion following an increase to the same temperature since its starting specific volume is larger. This is the reason for the well known asymmetry of approach toward an equilibrium density. To minimize the asymmetry a series of small temperature jumps of 2.5 8C were utilized in following the contraction and expansion kinetics. Figure 12.19 shows that the asymmetry is minimal with such temperature changes. This study of physical aging was different from conventional studies in that each temperature increment was

/

0.5 COOLING

0.0 0.5

FINAL TEMP. 124.0 °C 126.5 129.0 131.5 134.0 136.5

HEATING

0.0 0

1

2

3 4 log(t −ti ) (sec)

5

6

FIGURE 12.20. Absolute fractional volume deviation from equilibrium as a function of the logarithm of the corrected aging time for five temperature after cooling 2.5 8C from equilibrium and four temperatures after heating. Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789–5802 with permission from Institute of Physics Publishing.

⎥(V (t )/V (∞)−I )⎥ ⫻103

202 / CHAPTER 12

134.0 ˚C 131.5 129.0 126.5 124.0

0.5

COOLING

0.0

log ov 0.0 1.0 1.8 2.6 4.2

log ov 136.5 ˚C −0.4 0.0 134.0 0.7 131.5 1.4 129.0

T0=134 °C

0.5

HEATING

0.0 −4

−3

−2

−1

0

1 log(t /aT)

2

3

4

5

FIGURE 12.21. Volume contraction and expansion from Fig. 12.3 reduced by time-scale shifts to the chosen reference temperature To ¼ 134 8C. The time t=aT is the corrected reduced value. Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789–5802 with permission from Institute of Physics Publishing. −8

0 T-JUMPON Jr(t )

−9 log L(t )

−1 T0 =130 °C

−10

−2

polystyrene (M ¼ 1:64  104 ) and a high molecular weight entangled polystyrene (M ¼ 3:8  106 ) has been shown in the form of many of the commonly presented functions. They are shown as logarithmic functions of the reduced time t=aT (sec) and/or frequency vaT (rad/sec), where aT is the temperature shift factor [18]. The viscoelastic functions presented include: 1. The shear creep & recoverable (dashed line) compliance J(t) & Jr (t)cm2 =dyne or Pa
1 ; 2. The stress relaxation modulus G(t), dynes=cm2 or Pa;

−11

1

2

3

4 5 log(t/aT) (sec)

6

7

8

−3

FIGURE 12.22. Comparison of retardation spectra for voluminal L0 and shear deformation Ls . In this double logarithmic plot of the distribution functions of retardation times t the ordinate scales have been adjusted to superpose the shorttime results. t=aT is the reduced retardation time.Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789– 5802 with permission from Institute of Physics Publishing.

Only local mode molecular motions contribute to changes in the local packing structure of a liquid and the mobility of those modes decrease rapidly with decreasing temperature. At some temperature, because of the diminishing rate of possible molecular rearrangements, the equilibrium liquid structure can no longer be maintained. Below this temperature the liquid is a glass. The average relaxation time for these motions should be same for all amorphous materials, since the local molecular rearrangement rate that is insufficient to keep up with a given rate of cooling is only a function of that rate of cooling. 12.13 VISCOELASTIC BEHAVIOR AT Tg The viscoelastic behavior of an amorphous nonpolymeric dehydroabietic acid DHAA as seen at the glass temperature along with that of a low molecular weight unentangled

3. The dynamic storage J’(v) and loss J’’(v) compliances; 4. The dynamic storage G’(v) and loss G’’(v) moduli; 5. The loss tangent tan d ¼ J 00 (v)=J 0 (v) ¼ G00 (v)=G0 (v); 6. The real component of the dynamic viscosity h0 (v) and the imaginary component h00 (v); 7. The retardation function L(t)cm2 =dyne or Pa
1 ; and 8. The relaxation spectrum H(t)dyne=cm2 or Pa. The systematic variation of the viscoelastic functions with molecular structure can clearly be seen. Because of the additivity of strains arising from different molecular mechanisms [143,144] it should be noted that the clearest picture is seen in the development of L( log t) with changes in molecular weight. Effects of branching and molecular weight distribution are not considered here. The simplest behavior is exhibited by the nonpolymeric DHAA (See Fig. 12.23 [115]). The logarithm of the retardation spectrum exhibits a linear increase with the logarithmic reduced time scale time scale at relatively short times up to a rather abrupt maximum value. The slope of the linear portion has a value of 1/3 within experimental uncertainty. This slope reflects the fact that the recoverable creep compliance appears to display Andrade creep with a proportionality to the cube root of time before the long-time limiting steady-state recoverable compliance is approached. Although the amorphous materials that have been examined can be fit to the Andrade t1=3 linearity at short times near Tg there is the

THE GLASS TEMPERATURE

/

203

−9.0

DHAA

Log G ( t ) (dyne/cm2)

Log J ( t ) (cm2/dyne)

10 J (t )

43 °C

−9.5

J r (t )

9

8

7

−10.0 0

1

2

3

6 −3

4

−2

−1

Log t /aT (sec)

0

1

2

3

4

5

Log t /aT (sec)

−9

Log G (dynes/cm2)

Log J (cm2/dyne)

10 J'

−10 J−1/w h

J"

−11

−5

−4

−3

−2

−1

8

7

1

0

G" 9

G'

−5

−4

Log waT (sec−1)

−3 −2 −1 Log w aT (sec−1)

0

1

2.0 13 Log h (dyne-sec/cm2)

1.5

Log tan δ

1.0 0.5 −0.0 −0.5 −1.0

−5

−4

−3 Log w aT

−10.0

−2

h"

h' 10

−1

0

−5

1

−4

−3

−2

−1

1

0

Log w aT (sec−1)

(sec−1) 10

−10.5

Log H (t) (dynes/cm2)

Log L(τ) (cm2/dyne)

11

9

−1.5

−11.0 −11.5 −12.0 −12.5 −13.0

12

−3

−2

−1

0

1

2

Log(τ/aT) (sec)

3

4

5

9 8 7 6 5 −3

−2

−1

0 1 2 Log (t /aT) (sec)

3

4

5

FIGURE 12.23. Viscoelastic functions of a nonpolymeric glass former (DHAA). Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789–5802 with permission from Institute of Physics Publishing.

204 / CHAPTER 12 possibility that such fits are an approximation to a functionality that varies from material to material. The functionality is the generalized Andrade creep t1
n [145–155], where (1
n) is the fractional exponent of the Kohlrausch relaxation function, exp [
(t=t)1
n ] [156]. The complement of the Kohlrausch exponent, n, is the coupling parameter of the Coupling Model [145–155]. To be able to utilize an operationally effective means to determine corresponding glass temperature we will assume that the Andrade creep observed is real. In any case the contributions to the recoverable deformation in this regime are identified as local mode intermolecular motions which are also seen in permittivity measurements and are referred to as the alpha mechanism. The same behavior can be seen in low molecular weight polymers as is illustrated with the polystyrene PS A61 [3]. It’s molecular weight is just below the molecular weight between entanglements, Me ¼ 17,000 for polystyrene. Hence there is no entanglement network. In fact the presence of entanglements is not seen until M > Mc ¼ 35,000, for polystyrene. Above Mc the viscosity is proportional to M3:4 and the entanglement plateau appears in the compliance and modulus functions. The additional feature seen in the log [L(t)] for this polystyrene in Fig. 12.24 as a function of the logarithmic reduced time is the pronounced peak seen at long-times [115]. The molecular motions contributing to the recoverable deformation at Logarithmic reduced times between 4 and 7 are believed to be the polymeric normal Rouse modes [18]. The most complicated viscoelastic response is exhibited by high molecular weight polymers with entangled linear molecular chains (Fig. 12.25 [115]). The data shown were obtained on a narrow distribution polystyrene PS F380 with a molecular weight of 3:8  106 . At moderate long-times beyond the local mode and the Rouse normal mode motions a second Andrade t1=3 region is seen in the entanglement rubbery region where polymeric chains with varying chain lengths are sequentially being partially oriented. At still longer times where the polymeric chains are disentangling permanent viscous flow deformation is accumulating linearly with time and the maximum orientation per unit stress is being approached, involving the molecular contortions with retardation times between Log(t=aT ) ¼ 11 and 14.

amorphous materials at their respective Tg ’s we find that they are all close to one another as seen in Fig. 12.26. Fourteen amorphous materials including organic and inorganic polymers and nonpolymers are represented. They are identified in Table 12.3 [109]. In producing Fig. 12.26 the best Tg ’s that were available for a cooling rate Q ¼ 0:2 8C/min were used to fix the time-scale. The variability relative to the average position is about one decade on the time-scale. Tricresyl phosphate (TCP) was chosen as the reference material; i.e., the position of it’s log L(t) was assumed to be correct for its Tg (0.2 8C/min cooling). The Tg ’s of the other glass-formers were adjusted to match the position of the glassy Andrade region to that of TCP. The required changes of the Tg ’s were about 2 8C or less. The Tg adjustments are listed in Table 12.3. Since glass temperatures are often in doubt by more than several degrees it is felt that the Andrade line seen in Fig. 12.27 represents all of the materials within experimental uncertainty and therefore the common curve can be used to determine relative Tg ’s with a precision that is not possible by any other means.

12.14 UNIVERSAL BEHAVIOR AT Tg

and

As mentioned above, if it is assumed that Andrade creep is the true functional form exhibited by amorphous materials at short times at Tg , a common behavior is operationally observed that can be used to determine corresponding Tg ’s for these materials. Since the mobility of the local mode motions determines the Tg , a kinetic characterizing variable, such as the Andrade coefficient b, which is determined by these motions, can be associated with Tg . If we examine the glassy Andrade regions of the retardation spectra for many

12.15 DETERMINATION OF Tg FROM THE COMPIANCE FUNCTIONS From the common line in the retardation functions shown at Tg in Fig. 12.27 the Andrade coefficient bTg can be calculated. Smith showed [157] that when Andrade creep is observed L(t) ¼ 0:246bt 1=3 : Therefore since at Tg , L(t) ¼ 2:24  10
12 at t ¼ 1 sec bTg ¼ 9:11  10
12 (cm2 =dyne sec1=3 Þ or

 bTg ¼ 9:11  10
11 m2 =N sec1=3 ),

one simply has to determine b as a function of temperature to find out where b has this value to determine Tg . The Tg defining bTg can be obtained from dynamic mechanical properties as well as from the recoverable creep compliance Jr (t), since we showed [87] that J 0 (v) ¼ JA þ 0:773bv
1=3

J 00 (v) ¼ 0:446bv
1=3 when J 0 (v) and J 00 (v) are the storage and loss components of the complex dynamic compliance J  (v) ¼ J 0 (v)
iJ 00 (v) and b is the same Andrade coefficient seen in Jr (t). At present, bTg seems to be the best indicator of the mobility at Tg . Some Tg s with the experimental conditions, where available, are given in Table 12.4. They appear to be

THE GLASS TEMPERATURE

J r (t )

−8

205

10

J (t)

PS A61[3] T0 = 93 °C

Log G (t )(dyne/cm2)

Log J (t )(cm2/dyne)

−7

/

−10

−10

8 6 4 2

−2

0

2

4

6

−4

8

−2

0

Log t /aT (sec)

2

4

6

8

Log t /aT (sec)

−4 −6 −7

10

J" Log G (dynes/cm2)

Log J (cm2/dyne)

−5 J'

−8 −9 −10

J "−1/w h

8 G"

6 4

G'

2

−11 0

−12 −10

−8

−6

−4

−2

Log w aT

0

2

−10

4

12 Log h (dyne-sec/cm2)

Log tand

2 1 0 −1

−2

0

2

4

0

2

4

h'

11 10 9

h"

8 7 6

−8

−6

−4

−2

Log w aT

−7

0

2

5 −10

4

−8

(sec−1)

−2

−4

−6

Log w aT

(sec−1)

10

−8

8 Log H(t )(dyne/cm2)

Log L(t )(cm2/dyne)

−4

13

3

−9 −10 −11 −12 −13

−6

Log w aT (sec−1)

4

−2 −10

−8

(sec−1)

6 4 2 0

−2

0

2

Log t /aT (sec)

4

6

−2 −4

−2

0

2

4

6

8

Log t /aT (sec)

FIGURE 12.24. Viscoelastic functions of a low molecular weight polystyrene (16,400). Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789–5802 with permission from Institute of Physics Publishing.

206 / CHAPTER 12 −4

10 PS F380 T0 = 98 ⬚C

j (t )

−6

Log G (t) (dynes/cm2)

Log J (t) (cm2/dyne)

−5

jr (t )

−7 −8 −9

9 8 7 6 5 4 3 2

−10

−1 −2

0

2

4

6

8

−1 −2

10 12 14

0

Log t /aT (sec) −4

Log J (t) (cm2/dyne)

9

J ⬙

8

−6

7

J ⬙⫺1/(wh)

−7

G⬘

6

−8

5

−9

4

G⬙

3

−10

−14 −12 −10

−4 −2

−8 −6

0

−14 −12 −10 −8

2

Log t /aT (sec)

−4

−2

0

2

−2

0

2

(sec−1)

18 16 Log h (dynes/cm2)

1 Log tane

−6

Log w aT

2

0 −1 −2

h⬙

14 12

h⬘

10 8

−14 −12 −10 −8

−6

−4

−2

0

−14 −12 −10 −8

2

Log w /aT (sec−1) −7

9

−8

8

−9 −10 −11 −12 −4 −2

0

2

4

6

8

Log t /aT (sec)

−6

Log w aT

Log H (t) (dynes/cm2)

Log L(t) (cm2/dyne)

10 12 14

10

J⬘

−5

2 4 6 8 Log t /aT (sec)

10 12 14

−4

(sec−1)

7 6 5 4 3 2

−4 −2

0

2

4

6

8

10 12 14

Log t /aT (sec)

FIGURE 12.25. Viscoelastic functions of a high molecular weight polystyrene, 3:8  106 . Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789–5802 with permission from Institute of Physics Publishing.

THE GLASS TEMPERATURE

12 14

T0 = T0(0.2 ⬚C/min)

⫺7 8

Log L(t)(cm2/dyne)

⫺8 11

3

⫺9 10

9

⫺10 ⫺11

4

⫺12

7

6

⫺13

5

2 13

⫺14 ⫺15

⫺5

0

5 10 Log t (sec)

15

20

FIGURE 12.26. Comparison of the retardation spectra at T0 ¼ Tg (0:2 8C=min) for (1) 6PE, (2) Aroclor 1248, (3) polypropylene, (4) TCP, (5) OTP, (6) Tl2 SeAs2 Te3 , (7) PS Dylene 8, (8) PIB, (9) Viton 10A, (10)Epon 1004/DDS, (11) Epon 1007/DDS, (12)PB/Aroclor 1248 soln., (13) Se, (14) PVAc. Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789–5802 with permission from Institute of Physics Publishing. ⫺5 12

⫺6

14 ⫺7

Log L(t)(cm2/dyne)

⫺8

8 11

⫺9

3

10

9

⫺10 ⫺11

4

⫺12

1 6 2 5

⫺13

7 13

⫺14 ⫺15

⫺5

0

5 10 Log t (sec)

15

207

12.16 Tg OF POLYMER THIN FILMS AND POLYMER CONFINED IN NANOMETER SCALE DIMENSIONS

⫺5 ⫺6

/

20

FIGURE 12.27. Superposition of the retardation spectra at short times for the glass-formers of Fig. 12.26. Reproduced from Donald J. Plazek and Craig A. Bero, Precise Glass Temperature, J. Phys.: Condens. Matter. 15 (2003) 5789– 5802 with permission from Institute of Physics Publishing.

among the best published values. All of the above mentioned caveats should be noted when utilizing these literature values.

The change of the Tg of bulk polymers when reducing one or more of its dimensions to nanometer scale is of interest to workers in fundamental research as well as in applications to technology. The experimental activities in the last decade are mostly on measurements of Tg of nanoscale polymer thin films, and only in recent years some studies of polymers confined inside nanoporous host systems have been reported [158]. The results reported so far are confusing. The glass temperature of polymers subject to nanometer-scale confinement have been reported to increase, decrease, or not changed, depending on the polymer, the geometry and nature of the confinement, and the technique of measurement [159]. For example, the Tg of free-standing thin polystyrene films was reported to decrease continuously with thickness h by about 60 K when h is near 20 nm as measured by ellipsometry and Brillouin scattering [160]. On the other hand, much smaller reductions of Tg were reported for thin polymer films on substrates, and free-standing atactic poly(methyl methacrylate) thin films of comparable molecular weight and thickness as polystyrene [158]. While Tg of isotactic poly (methyl methacrylate) films on aluminum are lower than that of the bulk and decrease with film thickness, the opposite is found if the films are sandwiched between two polystyrene films [161]. The change of Tg thus depends on the nature of the interfaces of the polymer thin film. This dependence resembles that found by molecular dynamics simulations of thin films of binary Lennard-Jones particles nanoconfined by walls defined by a smooth repulsive potential or by frozen binary Lennard-Jones particles [162]. A plausible explanation of the observed dependence of change of Tg on interface was given by the Coupling Model [163]. Experimental techniques and computer simulations that probe the viscoelastic response from global chain dynamics do not find the reduction of Tg in nanometer thin films as deduced by the studies of the local segmental dynamics [164]. For studies of polymers confined inside nanoporous host systems, the most comprehensive study was reported for poly(dimethyl siloxane) and poly(methylphenyl siloxane) down to 5 nm by neutron scattering and dielectric and calorimetric measurements [165]. An increase of molecular mobility, implying a decrease of Tg , was observed on decreasing the pore size. The increment of the specific heat capacity at the glass transition normalized by the mass of confined polymers also decreases with pore size, indicating a concomitant decrease of the cooperative length scale with a decrease of Tg . An explanation has been offered [163]. Excluding some experimental results which may turn out to be artifacts, the majority of the data in the literature are real and worth consideration. The variability of the results, arising from dependence on polymer, nature and geometry of confinement, and experimental techniques, does not necessarily mean that the situation is unmanageable. It makes

208 / CHAPTER 12 TABLE 12.4. Selected Tg s for some common polymers. Name

Tg (qc ) 8C(deg= min :)

Mn

Repeat unit

Tf ,g (qc ,qh ) 8C

Methoda

Reference


26(
,2)
120

Lin. dil. Lin. dil. Dil.

[112] [122] [135]


10(10,10)

DSC

[113]

Dil. Dil.

[114] [115]

Lin. dil.

[122]


58

Dil.

[132]


105
114

Dil. Dil.

[121] [129]


102

Dil.

[129]


7

Dil.

[129]

Nonaromatic hydrocarbon backbone polymers

Polyethylene

Polypropylene (amorphous)

H

H

(C

C)

H

H

H

H

2  105
20,
130

6  10

C)

(C

4

H CH3

H CH3

Polyisobutylene

3

4:9  10 6:6  105

C)

(C


76(0.017)
76(0.2)

H CH3

Cis 1,4 polyisoprene

Trans 1,4 polyisoprene

H

H

(C

C

C

H

H

CH3 H

H

H

(C

C

C)

H C

H

H

H

H

(C

C

C

C)

H

Trans 1,4 polybutadiene

H

H

H

H

H

(C

C

C

C)

H

H

H

1,2 Polybutadiene

C)

CH3 H

H

Cis 1,4 polybutadiene


70

H

H

(C

C)

H

H

C

CH2

THE GLASS TEMPERATURE

/

209

TABLE 12.4. Continued. Name Cis neoprene

Trans neoprene

Mn

Repeat unit H

H (C

C

C

C)

H

H

CI

H

H

H

H

C

H

Polyacrylates Polylacrylic acid

H (C H

C

C)

CI

H

Tf ,g (qc ,qh ) 8C

Methoda

Reference

DTA

[130]

DTA

[130]

Dil.

[132]

103

Lin. dil.

[127]

14(20) 8

DSC Dil.

[125] [123]

Dil.

[123]

117(0.017) 43(0.5)

Dil. DTA DTA Dil. DTA DTA Dil. Dil.

[5b] [116] [116] [116] [116] [116] [116] [136]

66 65

Dil. Dil.

[123] [126]


20

 105 (C

Tg (qc ) 8C(deg= min :)

H


45
45 Lin. dil.
40

131

C) C

OH

O

Polymethylacrylate

H (C H

H

9  105

C) C

O

CH3

O

Polybutylacrylate

H

H

(C

C)

H

C


24

O

C4H9

O

Polymethyl-methacrylate

H (C H

CH3 C) C

O

CH3

O

Polyethyl-methacrylate

H (C H

 105(b) 6:0  104(c) 6:0  104(c) 6:0  104(c)  3  105b  3  105b  3  105b  3  105d

CH3

 105

C) C O

O

C2H5

102.8(1) 108(1) 110(1)e 106(0.017) 120(
1) 122(
,1)e

210 / CHAPTER 12 TABLE 12.4. Continued. Name Poly-i-propylmethacrylate

H (C H

CH3 CH3

C) C

O

C

O

Poly-n-propylmethacrylate

Mn

Repeat unit

H (C H

Tf ,g (qc ,qh ) 8C

Methoda

Reference

81 88

Dil. Dil.

[123] [126]

35 43

Dil. Dil.

[124] [126]

19 20 23

Dil. Dil. Dil.

[123] [124] [126]


5

Dil.

[124]


20

Dil.

[124]


65

Dil.

[124]

 105

H

CH3

CH3

 105

C) C

Tg (qc ) 8C(deg= min :)

O

C3H7

O

Poly-n-butylmethacrylate

H

CH3

(C

C)

H

C

 105 O

C4H9

O

Poly-n-hexylmethacrylate

H (C H

CH3 C) C

O

C6H13

O

Poly-n-octylmethacrylate

H (C H

CH3 C) C

O

C8H17

O

Poly-n-dodecylmethacrylate

H (C H

CH3 C) C O

O

C12H25

THE GLASS TEMPERATURE

/

211

TABLE 12.4. Continued. Name Poly-i-butylmethacrylate

H (C H

Poly-cyclo-hexylmethacrylate

Mn

Repeat unit

H

CH3 C)

H

C

O

O

CH3

H

Tf ,g (qc ,qh ) 8C

Methoda

Reference

53

Dil.

[124]

66 62

Dil. Dil.

[123] [126]

Dil. Dil. Dil.

[5b] [5b] [115]

C2H5

CH3

 105 (C

Tg (qc ) 8C(deg= min :)

C) C

O

O

Vinyl polymers Polystyrene

Polyvinylacetate

H

H

2:1  105

(C

C)

9:2  104

94.8(1.0) 92.1(0.1) 97.5(0.2)

H

O

H

H

 3  105

34.8(0.2)

Dil.

[115]

2:8  103

71.0(1.0)

Dil.

[5b]

DTA

[128]

Dil.

[117]

(C H

C) O

C

CH3

O

Polyvinylchloride

Polyvinylalcohol

H

H

(C

C)

H

CI

H

H

(C

C)

H

OH

95

Heterogeneous backbone polymers Polydimethylsiloxane

CH3 ( Si CH3


123(0.3) O)

212 / CHAPTER 12 TABLE 12.4. Continued. Name

Mn

Repeat unit

Bisphenol A polycarbonate

CH3 (

144

O

C

O

C

O

Tg (qc ) Tf ,g (qc ,qh ) 8C(deg= min :) 8C Methoda Reference Lin. dil.

[118]

DSC

[119]

50(
,1) 52

DTA DTA

[133] [134]

40

DTA

[134]

DSC

[120]

O)

CH3

Polyethylene-terephthalate

Nylon 66

H

H

(C

C

H

H

O

O C

O

O

O

H

( C (CH2 )4C

–
3  104

O N

O

H

O)

O O

O)

N( CH2 )6N )

( N ( CH2 )5 C

Polyetherimide

C

70(10,10)

O

H

Nylon 6

O

O

N

O

CH3 O

O

O

213(80,10)

C CH3

a

Dil.  volume dialtometry; Lin dil. linear dilatometry; DSCdifferential scanning calorimetry; DTAdifferential thermal analysis. b 75% syndiotactic. c Ideally atactic. d 99% isotactic. e Dried at 170 8C in vacuum for 68 h.

the problem more difficult, but more studies in the future should help to improve the understanding of the diverse experimental results.

12.17 WHEN DO VOLUME AND ENTROPY FIRST ENTER INTO DETERMINING MOLECULAR MOBILITY? Conventional models or theories consider only the segmental relaxation of amorphous polymers and the primary relaxation of nonpolymeric glass-formers in the change of molecular mobility with temperature and pressure (and concomitant changes in free volume and/or configurational entropy) leading to vitrification. Here we wish to recognise two different kinds of secondary relaxation processes. There

is one that precedes and leads to the primary relaxation and others that are not related such as side group motion which are independent of the chain backbone motion. The same can be said for secondary relaxations in nonpolymeric glassformers that involve isolated intramolecular motion of a part of the basic unit. However, there are secondary relaxations which must involve the polymer backbone like polybutadiene (PB) and even polyisoprene (PI). The existence of a secondary relaxation in PB is well known [166], but in PI it was found only recently [167]. Equally intriguing is the appearance of secondary relaxations in rigid small molecule glass-formers such as toluene and chlorobenzene [168,169], where there are no internal degrees of freedom. Therefore these secondary relaxations must originate from some local intermolecular motion of the entire molecule. Such secondary relaxations are called the Johari–Goldstein (JG)

THE GLASS TEMPERATURE b-relaxation to honor their discovery of secondary relaxations even in totally rigid molecules. They are supposedly universal, existing in all glass-formers, and are considered to be the precursor of the primary structural relaxation. Some criteria for distinguishing JG b-relaxation from other garden variety of secondary relaxations have been established based on their properties that mimic the primary relaxation [170]. The relaxation time t b of JG b-relaxation has Arrhenius temperature dependence in the glassy state, but the actual temperature dependence of t b at temperatures above Tg is not a continuation of the Arrhenius temperature dependence below Tg . It is more like another VFTH temperature dependence that is weaker than that of t a [170–172]. Also t b is pressure dependent in the equilibrium liquid state (T > Tg ) [170], and increases on physical aging in the glassy state [173]. Polymerization and cross-linking experiments on Epon 828 have shown that t b increases with the increase of covalent bonds formed during the process, which follows the trend of the primary relaxation [174–176]. In binary miscible mixtures of two glassformers, the relation between t b and the primary relaxation time of a component changes systematically with the composition [177]. Although all the properties are less spectacular than that of the primary relaxation, they indicate that t b depends on volume and entropy. There is also good correspondence between t b and the primitive relaxation time t 0 of the Coupling Model [170–171,173,178], the latter is definitely a precursor of the primary relaxation and is volume and entropy dependent. The relaxation strength, D«b , of the JG relaxation in all these glass-formers is found to change on heating through the glass temperature in a similar manner as the changes observed in the enthalpy H, entropy S, and volume V [179,180]. The derivative of D«b with respect to temperature, dD«b =dT, increases from lower values at temperatures below Tg to higher values at temperatures above Tg , a mimicry of the same behavior of the specific heat Cp and the expansion coefficient, which are the derivatives dH=dT and dV=dT, respectively. Thus, volume and entropy have already entered into the determination of molecular mobility of the JG b-relaxation and t b , at times long before the emergence of the arelaxation and t a in the equilibrium liquid state. Since time is the natural variable, the dependence of molecular mobility on temperature, pressure, volume, and entropy originate in t b or t 0 . The stronger dependences of the primary relaxation time on the same variables are the consequence of the many-molecule (cooperative and dynamically heterogeneous) dynamics that increase the magnitude of t b naturally. It is the involvement of an increasing number of molecules (proportional to the cooperative length-scale and the width of the dispersion) in the primary relaxation than that involved in the local JG b-relaxation. The results suggest models and theories of vitrification that address only the a-relaxation need a new paradigm [181]. Related information can be found in Chapter 13.

/

213

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156. R. Kohlrausch, Pogg. Ann. Phys. 12(3), 393 (1847). 157. T. L. Smith, Personal communication (1959). 158. For collection of papers, see (i) B. Frick, R. Zorn, and H. Bu¨ttner (eds.) Proceedings in International Workshop on Dynamics in Confinement, J. Phys. IV 10, Pr7 (2000). (ii) B. Frick, M. Koza, and R. Zorn (eds.) Proceedings of 2nd International Workshop on Dynamics in Confinement, Eur. Phys. J. E 12, 5–194 (2003). (iii) Special issue on Properties of Thin Polymer Films G. Reiter and J. A. Forrest (eds) Eur. Phys. J. E 8, 101–266 (2002). 159. A summary of the confusing results can be found in G. B. McKenna, Eur. Phys. J. E 12, 191 (2003). 160. J. A. Forrest and K. Dalnoki-Veress, Adv. Colloid Interf. Sci. 94, 167 (2001). 161. M. R. Wu¨bbenhorst, C. A. Murray, and J. R. Dutcher, Eur. Phys. J. E 12: S109–S112 Suppl. (2003). 162. P. Scheidler, W. Kob, and K. Binder, Eur. Phys. J. E 12, 5 (2003). 163. K. L. Ngai, Phil. Mag. B. 82, 291 (2002). 164. For references to experimental works and computer simulations, see K. L. Ngai, Eur. Phys. J. E 8, 225 (2002). A plausible explanation is given therein. 165. A. Scho¨nhals, H. Goering, Ch. Schick, B. Frick, and R. Zorn, Eur. Phys. J. E 12, 173 (2003), and to be published. 166. R. Casalini, K. L. Ngai, C. G. Robertson, and C. M. Roland, J. Polym. Sci. Polym. Phys. Ed. 38, 1841 (2001). 167. C. M. Roland, M. J. Schroeder, J. J. Fontanella, and K. L. Ngai, Macromolecules 37, 2630 (2004). 168. G. P. Johari and M. Goldstein, J. Chem. Phys. 53, 2372 (1970). 169. G. P. Johari, Ann. N. Y. Acad. Sci. 279, 117 (1976). 170. K. L. Ngai and M. Paluch, J. Chem. Phys. 120, 2857 (2004). 171. M. Paluch, C. M. Roland, S. Pawlus, J. Ziolo, and K. L. Ngai, Phys. Rev. Lett. 91, 115701 (2003). 172. See Figure 9 in S. C. Kuebler, D. J. Schaefer, C. Boeffel, U. Pawelzik, and H. W. Spiess, Macromolecules 30, 6597 (1997). The secondary relaxation of poly(ethyl metacrylate) involves some motion of the main chain and is hence a JG relaxation according to [170]. 173. D. Prevosto, S. Capaccioli, M. Lucchesi, P. A. Rolla, and K. L. Ngai, J. Chem. Phys. 120, 4808 (2004). 174. M. G. Parthun and G. P. Johari, J. Chem. Phys. 103, 7611 (1995); 103, 440 (1995). 175. D. A. Wasylyshyn and G. P. Johari, J. Chem. Phys. 104, 5683 (1996). 176. M. Beiner and K. L. Ngai, Macromolecules, 38, 7033–7042 (2005). 177. K. L. Ngai and C. M. Roland, Rubber Chem. Tech. Rubber Rev. 77, 579 (2004). 178. K. L. Ngai, J. Phys.: Condens. Matter 15 (2003) S1107. K. L. Ngai, in AIP Conference Proceedings, 708, p. 515 (2004), Am. Inst. Phys. Melville NY. 179. G. P. Johari, G. Power, and J. K. Vij, J. Chem. Phys. 116, 5908 (2002); 117, 1714 (2002). 180. G. Power, G. P. Johari, and J. K. Vij, J. Chem. Phys. 119, 435 (2003). 181. K. L. Ngai, J. Non-Cryst. Solids, 351, 2635–2642 (2005).

CHAPTER 13

Sub-Tg Transitions Joel R. Fried Department of Chemical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221–0012

13.1 13.2

Amorphous Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semicrystalline Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

There have been a number of good general reviews of relaxation processes that occur at temperatures below the glass-transition temperature (Tg ) [1–6]. Early dynamicmechanical studies were surveyed by Woodward and Sauer [7]. Although now outdated, the seminal work reviewing both dynamic-mechanical and dielectric data is still the excellent 1967 monograph by McCrum, Read, and Williams [8]. It is not the purpose of this review to approach the comprehensive coverage provided by McCrum et al. but to summarize important results for the major polymer groups and to include more recent studies, especially for engineering thermoplastics not available prior to 1967. Where appropriate, mention is made of recent efforts at molecular modeling that aid in understanding the nature of molecular processes that operate below Tg . A variety of techniques can be used to detect relaxational processes occurring below Tg . These include dynamicmechanical, dielectric, NMR (e.g., 1 H line width and pulsed 13 C NMR relaxation times), and thermally stimulated discharge current (TSC) measurements. Of these, dynamicmechanical methods [9] have been the most widely used to study secondary-relaxation processes in polymers. These include free vibration methods, principally torsional pendulum [10,11] and torsional braid [12], and forced oscillation (FO) methods utilizing mechanically driven tensile, torsional, and flexural strains provided by a number of commercial instruments. The ability to vary oscillation frequency over a wide range makes FO (ca. 0.016–160 Hz) and dielectric techniques (ca. 10---106 Hz) especially useful for the determination of activation energies. Early dynamic methods also included resonance electrostatic methods that provide dynamic data over a higher (acoustic) frequency range (ca. 103 ---104 Hz) than is possible using FO dynamic mechanical methods. Basic principles and instrumentation

217 225 230 230

for resonance electrostatic and other dynamic measurements are given by Ferry [13]. In the following sections, the results of dynamic-mechanical and dielectric measurements of important amorphous and semicrystalline polymers are summarized and conclusions regarding the origin of sub-Tg molecular motions are offered. As illustrated by Fig. 13.1, temperature assignments for principal relaxations are the temperatures at the maximum of the dynamic-mechanical or dielectric tan d or loss modulus peaks at the reported frequencies. Values determined from tan g data are slightly higher than those determined from loss modulus values, and temperature assignments increase with increasing frequency. Typically, peak assignments for Tg are slightly higher (ca. 15–20 8C) than obtained by dilatometry at low cooling rates. Where available, data for only dry, unconditioned samples are reported in this review.

13.1 AMORPHOUS POLYMERS The prevailing view is that the glass transition (a relaxation in amorphous polymers) is associated with the coordinated motion of 50–100 carbon atoms and associated substituent groups about the chain axis, while secondary relaxations reflect the motions of smaller numbers (e.g., 4–8) of carbon atoms about the chain axis (e.g., crankshafttype motions) or motions of substituent groups [1]. Sub-Tg relaxations occurring in the amorphous glass are labeled in order of decreasing temperature assignment as b, g, and d and reflect motions of progressively smaller molecular units with correspondingly lower activation energies. The b relaxation in amorphous glasses has been assigned to the onset of motions which are precursors to 217

218 / CHAPTER 13

Log tan δ or loss modulus

a

g

b

Tg

Tb

Ta

Temperature

FIGURE 13.1. Idealized representation of the dynamic mechanical spectrum for an amorphous polymer illustrating temperature assignments for the a(Tg ), b, and g relaxations.

the long-range segmental motions occurring at Tg and, therefore, may be considered to be a general phenomenon of glassy materials, polymeric or otherwise [14–16]; however, the b relaxation is not always detectable and remains a somewhat controversial subject. In the case of a number of polymers such as polycarbonate [17], polysulfone [18,19], and polyarylates [20], the b relaxation has been attributed to defects in the glass and as such is affected by thermal treatment sometimes evident when the sample has been quenched from the melt but reduced or eliminated by annealing. Several relationships have been proposed to relate the temperature assignment for the b transition to Tg . For example, Boyer [3] has suggested that (at 100 Hz) Tb
0:75Tg

(13:1)

for both amorphous and semicrystalline polymers, where temperature is in degrees Kelvin. Van Krevelen [21] has proposed that Tb þ Tg
635

(13:2)

(b transition) but may have little affect on lower-temperature relaxations (i.e., g peak) [22,23]. Secondary relaxation processes have been correlated with a number of physical and mechanical properties. For example, there is a good correlation between impact strength and the occurrence of main-chain secondary loss processes [24–26], especially for amorphous polymers [21]. There is also reasonable correlation between secondary loss processes and gas permeability [27–29]. Energies required for main-chain and side-group motions can be obtained by determining the effect of frequency on the maximum temperatures of the loss or tan d peaks. The temperature at the peak maxima, Tmax , increases with increasing frequency and the activation energy, Ea , of the relaxational process may be determined from the slope of a semilog plot of frequency ( f ) versus reciprocal peaktemperature (1=Tmax ) as   Ea 1 ln f ¼  þ ln f0 , R Tmax

(13:4)

where f0 is a constant obtained from the intercept. Typically, activation energies for low-temperature (i.e., g) relaxations are small (ca. 10---80 kJ mol1 ). Heijboer [30] has suggested that for sub-Tg relaxations other than local main-chain motions Ea ¼ 0:060Tmax (at1 Hz),

(13:5)

where Ea is in kcal mol1 and Tmax is in degrees K. Activation energies are about 20% higher than given by Eq. (13.5) for main-chain motions. Corresponding activation energies for the glass transition, reflecting longer-range cooperative motions, are about an order of magnitude greater than those for sub-Tg relaxations.y High activation energies mean that the temperature location of the glass transition (i.e., the a relaxation in amorphous glasses) is relatively insensitive to a change in frequency compared to secondary-relaxation processes.

for amorphous polymers and Tb
0:8Tg  40 ¼ 0:5Tm  25

(13:3)

for semicrystalline polymers, where Tm is the crystallinemelting temperature. In general, secondary-relaxation processes are affected by sample history and the presence of diluents. For example, the method of film preparation (e.g., molding or casting), thermal history (annealed or quenched), and moisture absorption can affect the temperature range, activation energy, and magnitude of some sub-Tg transitions. The dynamicmechanical and dielectric spectra of polyamides and related polymers such as poly(amide–imides) that are highly water absorbent are particularly sensitive to the presence of absorbed water. Thermal treatment (annealing or quenching) will affect the tail (onset) of the glass transition peak

13.1.1 Poly(Alkyl Acrylates) and Poly(Alkyl Methacrylates) Dynamic-mechanical and dielectric data for several poly(alkyl acrylates) including poly(methyl acrylate) (PMA) (1), poly(ethyl acrylate) (PEA) (2), poly(n-butyl acrylate) (PBA) (3), and poly(cyclohexyl acrylate) (PCA) (4)

y In the case of the glass transition, the relationship between Tm and frequency is given by the WLF equation and not the Arrhenius relationship given by Eq. (13.4). Therefore, a semilog plot of f versus 1=Tmax will appear curved over a sufficiently wide frequency range. An apparent activation energy for the glass (aa ) transition may be calculated from use of Eq. (13.4) over a more limited frequency range as is typical for most dynamic-mechanical measurements.

SUB-Tg TRANSITIONS CH2 CH2

CH2

CH

C

O

CH2

CH C

O

C

O

O

CH3

C2H5

C 4H 9

1

2

3

CH2

CH3

C C

CH2 O

O CH3 CH3 C O

O H3C

C CH3

10

O

H3

CH3

C C

CH2 O

CH2 O

O

O

C3H7 8

C C

O

C2H4OH 7

CH3

C C

O

C2H5 6

C

CH2

O

5

CH2

CH3

C C

4

case of PMMA, the intensity of the b relaxation of amorphous samples increases with increasing syndiotacticity but decreases with increasing isotacticity [34].13 C NMR studies have indicated that the nature of the b relaxation in PMMA may be a p (1808) flip of the OCO plane of the side group coupled to a random rotation (208 amplitude) of the main chain [35]. In the case of poly(methacrylates) with longer alkyl groups such as poly(isobutyl methacrylate) (PiBMA) and poly(n-butyl methacrylate) (PnBMA), a low-temperature (g) relaxation is reported near 125–133 K due to side-chain motions involving the four-atom sequence -----O-----C-----C-----C or -----C-----C-----C-----C----- [36]. Poly(isopropyl methacrylate) (PiPMA) which does not have this sequence does not exhibit a relaxation at 120 K but does show one at 50 K (1,000 Hz) similar to that reported for PEMA (see Table 13.1). In those two cases, the low-temperature (d) relaxation is attributed to rotation of the isopropyl or ethyl group that is attached to the COO group [37]. Esteve-Marco et al. [38] have reported results of dielectric measurements and molecular mechanics calculations of poly(chloroethyl methacrylate) and poly(chloropropyl methacrylate). Molecular-mechanics simulation has been used extensively to study molecular motions in poly(alkyl methacrylates). As an example, Cowie and Ferguson [33,39] have

are given in Table 13.1. As shown by this data, the predominant (b) relaxation process in poly(alkyl acrylates) occurs at low temperatures (145–197 K) with an activation energy of ca. 30---60 kJ mol1 . Both Tg and the b-relaxation temperature decrease with increasing length of the side chain in the case of linear alkane substitution (i.e., PMA>PEA>PBA) [31]. From 13 C NMR (spin-lattice relaxation time) measurements and molecular-dynamics simulations, Kikuchi et al. [32] have concluded that molecular motions in PEA and PBA consist of internal rotation or torsional oscillation of each functional group and a slower motion induced by backbone motion. Also included in Table 13.1 are data for several poly(alkyl methacrylates) including poly(methyl methacrylate) (PMMA) (5), poly(ethyl methacrylate) (PEMA) (6), poly(2-hydroxyethyl methacrylate) (PHEMA) (7), poly(npropyl methacrylate) (PnPMA) (8), poly(n-butyl methacrylate) (PnBMA) (9), and poly(isobutyl methacrylate) (PiBMA) (10). The predominant sub-Tg process in poly(alkyl methacrylates) is the b relaxation (activation energy of 80---121 kJ mol1 ) appearing around 285–336 K and nearly independent of the length of the alkyl group (e.g., methyl, ethyl, propyl, and butyl) while Tg decreases with increasing side chain length (i.e., internal plasticization) [33]. In the

CH3

O

O

O

O

219

CH C

CH

/

C4H9 9

220 / CHAPTER 13 TABLE 13.1. Glass-transition and secondary-relaxation temperatures of poly(alkyl acrylates) and poly(alkyl methacrylates). Polymera

Techniqueb

PMA PEA PBA PCA

D D D Dilatometry FO D TP FO TBA TBA TBA D FO

PMMA PMMA a-PMMA s-PMMA i-PMMA PMMA i-PMMA amorph. crystal. PEMA PEMA PHEMA PnPMA PnBMA PnBMA PnBMA PiBMA

D D D VR VR D

f c Hz 1,000 1,000 1,000 1 1 1 1 1.24 1.25 1.4

Tg (K)

Ea d kJ mol 1

307 278 250 290

386 388 403 336

955

Tb (K)

Ea kJ mol 1

195 177 145

43 50 29

197 192 299 281 297 300 285

60

Tg (K)

Ea kJ mol 1

Td (K)

Ea kJ mol 1

[31] [31] [31] [118]

[24] [119] [34]

81

80 3 3 110 10 0.02 40–600 40–600 30

336 311

105

310 323

80 121

[120] [121]

50

123 115 133

23

125

25

9

[122] [120] [120] [110] [110] [37] [110] [37]

80 D

Ref.

10

a

PMA, poly(methyl acrylate); PEA, poly(ethyl acrylate); PBA, poly(n-butyl acrylate); PCA, poly(cyclohexyl acrylate); PMMA, poly(methyl methacrylate); PEMA, poly(ethyl methacrylate); PHEMA, poly(2-hydroxyethyl methacrylate); PnPMA, poly(n-propyl methacrylate); PnBMA, poly(n-butyl methacrylate); PiBMA, poly(isobutyl methacrylate). b ES, resonance electrostatic method; FO, forced oscillation dynamic-mechanical analysis; FV, free vibration; TP, torsion pendulum; TSC, thermally stimulated discharge current measurement; D, dielectric; VR, vibrating reed. c v ¼ 2pf where v is the angular frequency (rad s1 ) and f is frequency in units of Hz; 10 rad1 ¼ 1:5915 Hz. d Apparent activation energy calculated from Eq. (13.4); 1 kJ mol1 ¼ 0:2387 kcal mol1 ¼ 0:0104 eV=molecule.

concluded that the b relaxation of PMMA is due to the rotation of the oxycarbonyl unit with an activation energy of 70---90 kJ mol1 . Studies by Heijboer et al. [40] of (syndiotactic) PMMA indicate that the calculated barrier to alkoxycarbonyl group rotation is lower than experimentally observed for the b relaxation unless main-chain torsion angles are constrained. In the case of PnPMA, molecularmechanics calculations suggest that the g relaxation observed at about 90 K at 1 Hz (activation energy of 22 kJ mol1 ) could be attributed to hindered rotation around the O-----CH2 -----CH2 bond of the propyl group; similar loss peaks have been observed in the case of poly(nalkyl methacrylates) with longer alkyl groups (e.g., butyl, pentyl, and hexyl) [41]. Similar correlation of the d relaxations of syndiotactic PEMA, PIPMA, and poly(cyclohexyl methacrylate) with limited rotations around the O-alkyl bond was made more recently by Heijboer et al. [42] through the use of molecular-mechanics calculations. 13.1.2 Polycarbonates The most extensive studies of the dynamic-mechanical properties of polycarbonates have been reported by Yee et al.

[17,43,44] and by Vardarajan and Boyer [45]. Some results of dynamic-mechanical and dielectric measurements of bisphenol-A polycarbonate (PC) and two tetrasubstituted bisphenol-A polycarbonates, X

X CH3

O O

C

C

O

CH3 X

X

tetramethylbisphenol-A polycarbonate (TMPC, X ¼ CH3 ) and tetrachlorobisphenol-A polycarbonate (TCPC, X ¼ Cl), are summarized in Table 13.2. Results for PC reveal a b relaxation in the range from 320 to 370 K and a g relaxation in the range from 150 to 230 K. In addition, Varadarajan and Boyer [45] report a very low-temperature transition (d) near 53 K. As appears to be the case of for several other thermoplastics such as poly(2,6-dimethyl-1, 4-phenylene oxide) and polysulfone, the b relaxation is affected by thermal history (i.e., observable in samples

SUB-Tg TRANSITIONS

/

221

TABLE 13.2. Glass-transition and secondary-relaxation temperatures of polycarbonates. Polymera

Techniqueb

f c Hz

Tg (K) 423 423

PC PC PC PC PC PC PC

TP D

0.5–1.2 100

TP TP TP TSC

1 1 1.24 0.032

PC PC PC PC

FO D FO D

1 10 1 120

PC PC PC PC PC

FO TP FO D D FO FO FO FO NA TO FO FO FO NA TP

10–50 1 1 106 100 90 1 11 1

PC PC PC PC PC PC TMPC TMPC TMPC TCPC

110 15.9 1 11

Ea d kJ mol1

Tb (K)

Ea kJ mol1

Ea kJ mol1

164 153 373 340

426 423 423 420

Tg (K)

800 800

>343 shoulder 353

838 343

411

343

431

321

193

183 173 165 140 219 163 173 173 160 220 187 173 178 230 188 222 188 188

36 44 27 50 52 10–45 54

59 18 48

56

436 423 476

0.5–1.2

363

168 193 176 323 347 318 351

45 80 103

Ref. [123] [124] [125] [24] [49] [73] [46] [119] [48] [17] [47] [45] [126] [127] [128] [129] [130] [43] [131] [106] [78] [132] [133] [43] [106] [123]

a

Polymer abbreviations: PC, bisphenol-A polycarbonate; TMPC, tetramethylbisphenol-A polycarbonate; TCPC, tetrachlorobisphenol-A polycarbonate. b–d Legend in Table 13.1.

quenched from the melt but absent in annealed samples) [46,47]. There is evidence that the g relaxation may consist of two [46–48] and possibly three [45] overlapping relaxations. The intensity of the g relaxation peak has been reported to increase with increasing water content [49]. The molecular basis for sub-Tg molecular relaxations in the case of PC may include segmental motion and rotations of phenyl and methyl groups. The nature of these motions have been studied in detail by 13 C NMR spectroscopy and explored by semiempirical molecular-orbital (MO) calculations. Results of 13 C NMR measurements (spin–lattice relaxation times) by Jones and Bisceglia [50] indicate that several molecular processes may be coupled or synchronous. NMR studies by Schaefer et al. [51] have shown that the dominant motion in PC may be p flips of the phenylene ring about the main chain extending over a broad frequency range and superimposed on 308 ring oscillations; chlorine substitution of the rings eliminates both ring and main-chain motions. Activation energies of 37---50 kJ mol1 for phenylene group motion have been obtained from NMR measurement [52,53]. Methyl-group substitution at the orthopositions (e.g., TMPC) shifts the

onset of fast ring flips by about 180 K [54]. As indicated by the dynamic-mechanical data given in Table 13.2, the low-temperature relaxation (comparable to the g relaxation in PC) reported for TMPC and TCPC occurs at substantially higher temperatures (ca. >320 K) than for PC in agreement with the NMR results. Comparison of the dynamic mechanical spectra of bisphenol-A PC with trimethylcyclohexylbisphenol PC and spirobisindane-PC for which phenylene ring motion is greatly restricted has led Wimberger-Friedl and Schoo [55] to conclude that the g relaxation originates from motion of the carbonate group while phenylene group motion contributes as a separate mechanism to the hightemperature side of this relaxation. Semiempirical MO calculations of model compounds suggest that the g relaxation of PC may result from phenylene-ring flips (calculated activation energies of about 41 kJ mol1 ) as well as methyl-group rotation while the d relaxation mentioned earlier may be due to oscillations of the phenylene ring and the methyl group as well as rocking motions of carbonyl groups [56,57]. In agreement with results from NMR studies, semiempirical MO calculations of TMPC indicate that phenylene rotation is restricted due to

222 / CHAPTER 13 repulsion between the aromatic methyl group and the carbonyl oxygen atom [58]. 13.1.3 Polyimides and other Imide Polymers Polyimides (PIs) represent a broad class of high-Tg polymers derived from the polycondensation of an aromatic dianhydride and diamine. The most widely investigated polyimide is polypyromellitimide or poly(4,4’-oxydiphenylenepyromellitimide) (Kapton1) whose repeat unit structure is given below

O

O

N

N

O

O

O

n

Very recently, Wang et al. [29] have reported dynamic mechanical data for a number of polyimides derived from 1,4-bis(4-aminophenoxy) 2-tert-butylbenzene (BATB) and 3,3’,5,5’-tetramethyl-bis[4-(4-aminophenoxy)phenyl]sulfone (TMBPS). Temperatures for the g relaxation (DMA, 1 Hz) ranged from 152 to 185 K for the BATB-based polyimides and 150–161 K for the TMBPS-based polyimides. In general, bulky groups in the dianhydride segment, such as the hexafluoroisopropylidene group of 6FDA, reduces polymer packing, increases fractional free volume, and consequently causes the g relaxation to occur at lower temperature. Other polymers that contain imide groups include polyetherimide (PEI) (e.g., Ultem1) O

O C

N

CH3

N C

C

O

O

O O

CH3

general, an important sub-Tg relaxation for PIs is the b relaxation observed in the temperature range between 338 and 405 K and having an activation energy of about 84---117 kJ mol1 [59]. In addition, PIs exhibit a g relaxation in the range between 160 and 250 K that has been attributed to water absorption. For example, early dynamic-mechanical measurements of Kapton PI revealed two sub-Tg relaxations at 15,000 Hz—one at 400 K (b) attributed to torsional oscillations of the phenylene ring and another at 250 K identified here at the g relaxation which was observed to increase in intensity with sorbed water [60]. Computer modeling suggests that the b relaxation is probably associated with the relatively noncooperative motion of the diamine unit which is suppressed by crystallinity or orientation [59]. Other molecular-dynamics simulations of Kapton PI reveal near out-of-phase torsional motions about the nitrogen– phenyl bonds that involve the whole chain and is not localized in one small region [61]. Molecular dynamics studies of a semicrystalline PI (PTDA–DMDA) by Natarajan and Mattice [62] suggest that p-flips of phenoxy rings in the amorphous phase covers a broad range of activation energies. Dynamic-mechanical data for PEI given in Table 13.3 indicates a b relaxation at about 340–380 K and a g relaxation at about 160–186 K. These relaxations are comparable to those cited above for Kapton although they occur at slightly lower temperatures. As in the case of Kapton, the g-relaxational peak of PEI is reported to increase in intensity with sorbed water [63]. From comparison of the dielectric spectra of PEI, poly(ether sulfone) (PES) (see Section 13.1.6), and polyarylates with their corresponding low-molecular-weight compounds, Schartel and Wendorff [64] have concluded that both intrachain and interchain interactions contribute to the g relaxation in these polymers. Results of dynamic-mechanical measurements of a sample of PAI dried at 1908C are summarized in Table 13.3. The locations of the b- and g-relaxational peaks at 338 and 204 K (at 1 Hz) are comparable to that of Kapton and PEI. As in the previous cases, sorbed water has been observed to increase the intensity and decrease slightly the temperature of the g relaxation while the temperature and activation energy of the b relaxation increases with increasing water content [65].

poly(amide-imide) (PAI) (e.g., Torlon1) 13.1.4 Poly(phenylene oxides) O C

O

Dynamic-mechanical and dielectric properties of three poly(phenylene oxides)

C N

O

NH

C O

R

O

Representative dynamic-mechanical and dielectric data for Kapton PI, PEI, and PAI are given in Table 13.3. In

R

SUB-Tg TRANSITIONS

/

223

TABLE 13.3. Glass-transition and secondary-relaxation temperatures of imide polymers. Polymera PI PI PI PEI PEI PEI PEI PEI PAI

Techniqueb

f c Hz

ES ES TP TP FO FO D FO D FO

15,000 14,000 1 (1) 1 1 1,000 35

Tg (K)

485 492 501 513

Ea d kJ mol1

Tb (K)

Ea Kj mol1

Ea kJ mol1

Ref.

400 405

84–105

250

66

185 168 160 (shoulder)

44

343 355

[60] [134] [69] [63] [130] [135]

379

186

Tg (K)

330–1,250

43 1

549

338

117

204

[136] [64] [65]

a

PI, polypyromellitimide (Kapton polyimide) or poly(4,4’-oxydiphenylenepyromellitimide); PEI, poly(ether-imide); PAI, poly(amide-imide). b–d Legend given in Table 13.1.

poly(p-phenylene oxide) (R ¼ H), poly(2,6-dimethyl-1, 4-phenylene oxide) (R ¼ CH3 ), and poly(2,6-diphenyl-1,4phenylene oxide) (R ¼ C6 H5 ), are summarized in Table 13.4. Dynamic measurements of poly(p-phenylene oxide) (H2 PPO) reveal a g relaxation in the region of 120–160 K (1 Hz) having an activation energy of about 50 kJ mol1 . The majority of dynamic-mechanical studies for poly(2,6-dimethyl-1,4-phenylene oxide) (PPO) provide evidence for only a weak shoulder (g relaxation) in the vicinity of 125–160 K; however, a distinct peak has been observed by dielectric measurements [66,67]. In addition, there is evidence for a broad, low-intensity b peak in the range from 240 to 370 K. Sample preparation and impurities appear to have a significant effect on the appearance of the

weak sub-Tg relaxational processes in PPO [68,69]. By comparison, dynamic-mechanical data for poly(2,6-diphenyl-1,4-phenylene oxide) (P2 PPO), a semicrystalline polymer (Tm ¼ 753 K), suggests as many as three distinct sub-Tg relaxations [68,70]. In terms of intramolecular flexibility, the poly(2,6-disubstituted-1,4-phenylene oxides) are freely rotating chains [71]; however, intermolecular steric effects may limit phenylene rotation in the solid state and perhaps account for the absence of detectable sub-Tg relaxational processes. For example, results of 13 C NMR measurements indicate that the phenylene rings of PPO can execute only small amplitude motions due to the relative stiffness and dense packing of the PPO chain and blockage from rings on adjacent chains.

TABLE 13.4. Glass-transition and secondary-relaxation temperatures of poly(phenylene oxides). Polymera

Techniqueb

f c Hz

Tg (K)

H2 PPO H2 PPO H2 PPO PPO PPO PPO PPO PPO PPO PPO PPO PPO PPO P2 PPO

ES TP FO ES TP ES FO D D TP TP TP ES TP

7,000 1 110 7,040 1 7,000 110 100 100 1 1.3 1 9,640 1

423

P2 PPO

FO

110

a

Ea d kJ mol1

Tb (K)

Ea kJ mol1

363 370 273 370 240 512 517

84

628

277 286 373 348 502

84

363

78 67 96

Tg (K) 121 (shoulder) 160 155 140 (shoulder) 158 140 (shoulder) 158 157 205 125 (shoulder) 135 (shoulder) 126 g 238 d 83 143

Ea kJ mol1 50

36 44 40 42 29–34 50 17

Ref. [134] [68] [70] [137] [24] [134] [138] [66] [67] [49] [73] [68] [69] [68] [70]

H2 PPO, poly(p-phenylene oxide); PPO, poly(2,6-dimethyl-1,4-phenylene oxide); P2 PPO, poly(2,6-diphenyl-1,4-phenylene oxide). b–d Legend given in Table 13.1.

224 / CHAPTER 13 and polyethersulfone (PES)

13.1.5 Polystyrenes Molecular motions in polystyrene (PS) have been extensively reviewed by Boyer [72]. Results of dynamicmechanical studies of polystyrene, poly(4-methylstyrene) (P4MS), poly(4-chlorostyrene) (P4CS), and poly(a-methylstyrene) (PAMS) are summarized in Table 13.5. These and other studies show evidence for three transitions for PS below Tg . These include b (ca. 325 K), g (ca. 130–180 K), and d (ca. 30–40 K) transitions with activation energies of about 147, 42, and 8---13 kJ mol1 , respectively. The d relaxation has been associated with hindered partial rotation and wagging of the phenyl group [73]. It decreases in intensity with crystallinity in isotactic PS [74]. The origin of the g transition is less certain and may be due to motion of end groups. Results of molecular-dynamics simulations suggest that sub-Tg relaxations may include crankshaft-type motions of the PS backbone and librational motions of the pendant phenyl rings that depend upon the local environment [61]. NMR measurements indicate that the most prevalent molecular motion is restricted phenyl-group rotation with an average total displacement of ranging from 408 for orthosubstituted polystyrene to 708 for para-substituted polystyrenes having bulky nonpolar substituent groups [75]. Restrictions are due to intramolecular steric interactions and interchain packing (for unsubstituted PS). These conclusions are consistent with molecular mechanics studies reported by Khare and Paulaitis [76]. 13.1.6 Polysulfones Extensive studies of the dynamic-mechanical properties of a number of different polysulfones has been reported by Robeson et al. [77] and by Aitken et al. [78] Most of studies reported in the literature have focused on the two commercially important polysulfones — bisphenol-A polysulfone (PSF) O

CH3 O

O

C

S O

CH3

O O

S O

Results of dynamic-mechanical and dielectric studies of PSF and PES are summarized in Table 13.6. Results for PSF indicate a well-defined g relaxation located near 162–229 K. There is substantial evidence that the intensity of the g-relaxational peak increases with sorbed water content [49,77,79]. Substitutions that hinder phenylene mobility increase the temperature of the g relaxation [78]. There is controversy concerning the existence of a b relaxation located around 330–360 K that is sensitive to thermal history as has been reported for polycarbonate [18,19,80]. The dynamic-mechanical behavior of PES, which has a slightly lower Tg , is similar to that of PSF with a prominent g relaxation that is also water sensitive [49] and is located in the region from 163 to 265 K. Results of semiempirical molecular orbital (CNDO/2) calculations suggest that the g relaxation is due to phenylene-group rotation of the isopropylidene moiety with a calculated activation energy of 42 kJ mol1 , rotation of the methyl groups in the isopropylene group with an activation energy of 41 kJ mol1 , and possibly the diphenyl sulfone rotation with an activation energy of 42 kJ mol1 while the b relaxation is attributed to diphenyl ether rotation with an activation energy of 167 kJ mol1 [18,19]. Molecular simulations have shown that the rotational barriers for C-----O or C-----C bonds are higher than those for C–S bonds in PSF and that the mechanism for relaxation in the bulk state may be due to cooperative ring-flip motions associated with rotations about the C-----S linkage [81]. NMR studies have indicated that the b-relaxation is due to p flips of the aromatic rings that are unaffected by sorbed water but decrease in frequency in the presence of antiplasticizers [82]. Dynamic mechanical studies of PES by Shi et al. [83] indicated that the low-temperature (g) transition (ca. 193 K) is associated

TABLE 13.5. Glass-transition and secondary-relaxation temperatures of polystyrenes. Polymera PS PS PS PS PS P4MS P4CS PAMS a

Techniqueb

f c Hz

ES FO TP D TP ES ES ES

5,590 110 1 100 1.7 9,700 8,330 7,850

Ta (K)

Tb (K)

Tg (K)

Td (K)

Ea kJ mol1

Ref.

8.4

325

185 133

38

379

[134] [138] [74] [67] [73] [134] [134] [134]

38 394

154 33 92 95 126

8.8 9.2 16

PS, polystyrene; P4MS, poly(4-methylstyrene); P4CS, poly(4-chlorostyrene); PAMS, poly(a-methylstyrene). Legend given in Table 13.1.

b–d

SUB-Tg TRANSITIONS

/

225

TABLE 13.6. Glass-transition and secondary-relaxation temperatures of bisphenol-A polysulfone and polyethersulfone. Polymera PSF PSF PSF PSF PSF PSF PSF PSF PSF PES PES PES PES PES PES PES PES PES a

Techniqueb

f c Hz

ES TP TP D TP TP FO FO FO FO TP TP TP D TP FO

6,088 1 1 1,000 0.67 1 11 1.59 1 110 1 1 1 2:1  106 1 35

FO D

110

Tg (K)

Ea d kJ mol1

Tb (K)

Ea kJ mol1

468

448 333 920 470 459

282 358 273

476

498

Tg (K)

Ea kJ mol1

Ref.

229 163 206 207 162 183 173 173 177 193 163 226 183 265 178 170 175 193

50

[79] [139] [49]

70

45

55 21

45

[73] [140] [80] [19] [130] [78] [139] [49] [140] [128] [141] [136] [106] [78] [64]

PSF, bisphenol-A polysulfone; PES, polyethersulfone. Legend given in Table 13.1.

b–d

with the rotation of phenylene rings. Recent quasielastic neutron scattering of PES [84] and PSF [85] have suggested that the g and d relaxations may be associated with p-flips and oscillations of phenyl rings, respectively. 13.1.7 Poly(vinyl chloride) The dynamic spectrum of poly(vinyl chloride) (PVC) has been widely reported and reveals a major b relaxation located in the range from 195 to 273 K. Havriliak and Shortridge [86,87] have suggested that the molecular nature of the b relaxation in PVC is a hindered rotation of a segment about its main-chain axis. Results of dynamicmechanical and dielectric studies of PVC are summarized in Table 13.7. PVC is weakly crystalline due to syndiotactic sequences of repeating units. Harrell and Chartoff [88] have shown that crystallinity shifts the a relaxation to slightly higher temperature, shifts the b relaxation to lower temperature, and reduces peak intensity. Kakutani et al. [89] have suggested that the b relaxation may be composed of two overlapping relaxational processes at 223 and 273 K due to motions in the amorphous and crystalline regions, respectively. Chlorination of PVC shifts the a relaxation to higher temperatures and broadens the b relaxation peak while plasticization decreases the b relaxation and shifts it to lower temperature (i.e., the high-temperature portion of the b peak is reduced) [90]. Molecular dynamics studies of torsional angles changes by Meier and Struik [91] suggest that a localized five-bond transition may result in activation energies responsible for the b relaxation and higher activation energies associated with the glass transition.

13.2 SEMICRYSTALLINE POLYMERS Boyd has provided a detailed review of relaxational processes that occur in semicrystalline polymers [92] as well as a discussion of their molecular origins [93]. In general, the dynamic-mechanical and dielectric spectra of semicrystalline polymers are more complex than those of amorphous ones. This complexity results from the presence of additional transitions resulting from crystalline regions, varying degrees of crystallinity in different samples, and the possibility of different crystalline forms. While discussing the dynamic-mechanical and dielectric properties of semicrystalline polymers in this section, the usual convention will be used where the a peak is now associated with the crystalline–melting transition, the b peak is commonly identified with the glass-transition of the amorphous region, and subTg relaxations are indicated as g and d [92]. 13.2.1 Polyamides Dynamic-mechanical and dielectric data have been widely reported for most aliphatic polyamides, especially poly(«-caprolactam) (nylon-6 or PA-6; Tg
313 K) and poly(hexamethylene adipamide) (nylon-6,6 or PA-6,6; Tg
323 K). Results of dynamic-mechanical and dielectric measurements of PA-6 and PA-6,6 (Table 13.8) provide evidence for three relaxations (b, g, and d) in these polymers at temperatures below their crystalline–melting temperature Tm (487–506 K for PA-6 and 523–545 K for PA6,6) [8]. The b relaxation (located at above 310–347 K for PA-6,6 and 357–370 K for PA-6,6) is associated with high

226 / CHAPTER 13 TABLE 13.7. Glass-transition and secondary-relaxation temperatures of PVC. Techniquea

f b Hz

Tg (K)

2

358

D TP D TP FO (D)

Ea c kJ mol1

Tb (K)

364 223 524

D TP FO FO FO TSC

1 110

208–218 223 (b2 ) 273 (b1 )

2 3.5

233 239 226 235 195

360

754

11

Ea kJ mol1

Ref.

63 63 63 42–54 67 45 50–60

[96] [90] [142] [143] [89] [144] [145] [88] [119] [146] [147]

65 54

a

ES, resonance electrostatic method; FO, forced oscillation dynamic-mechanical analysis; FV, free vibration; TP, torsion pendulum; TSC, thermally stimulated discharge current measurement; D, dielectric; VR, vibrating reed. b v ¼ 2pf where v is the angular frequency (rad s1 ) and f is frequency in units of Hz; 10 rad1 ¼ 1:5915 Hz. c Apparent activation energy calculated from Eq. (4); 1 kJ mol1 ¼ 0:2387 kcal mol1 ¼ 0:0104 eV=molecule. TABLE 13.8. Secondary-relaxation temperatures of polyamides. Polymer

a

Technique

b

c

f Hz

Tb (K)

Ea d kJ mol 1

PA-6 PA-6 PA-6 PA-6 PA-6 PA-66 PA-66 PA-66 PA-66 Nomex

D ES TP FO FO VR FO FO FO ES

Nomex Kevlar

TP ES

1 10,000

550

665

Kevlar Kevlar

FO TP

110 1

733 816

767 813

Tg (K)

Ea kJ mol 1

Td (K)

Ea kJ mol 1

61 10,000 1 1 1 40–600 11 1 10,000

310 330 347 335 363 357 370

241 203

156 123

220 249

147 156

245

186

291 (g) 442 (g ) 352 291 (g) 417 (g ) 333 235 (g) 440 (g )

83 63 92 204 54 83

Ref. [96] [97] [94] [131] [148] [110] [149] [95] [131] [97]

120

24

[98] [97]

243 115

52 21

[99] [98]

a

Polymer abbreviations: PA-6,6, nylon-66; PA-6, nylon-6; Nomex, poly(m-phenylene isophthalamide); Kevlar, poly(p-phenylene terephthalamide). b–d Legend given in Table 13.1.

activation-energy molecular motions occurring in the amorphous phase (i.e., glass transition) that are affected by the overall degree of crystallinity and crystalline morphology [94] as well as plasticization by sorbed water [95]. The g relaxation (ca. 200–240 K for PA-6) is comparatively weak with an activation energy approximately 61 kJ mol1 [96]. The g relaxation increases in intensity with sorbed water suggesting motions involving (amide) carbonyl groups that are hydrogen bonded to water molecules. The intensity of the g relaxation increases with the relative concentration of

amide groups and shifts to lower temperatures in the presence of sorbed water [97]. The d relaxation (activation energy of ca. 42 kJ mol1 in the case of PA-6) occurs at ca. 120–160 K and is believed to be due to motions of the methylene groups [94]. Frosini and Butta [97] have suggested that the d relaxation occurs at about the same temperature (160–180 K at  104 Hz) for all aliphatic polyamides but increases in intensity with increasing number of methylene groups. Two important aromatic polyamides (aramids) are poly(m-phenylene isophthalamide) (Nomex) (Tg > 503 K)

SUB-Tg TRANSITIONS O

O

C

C

NH

NH

and poly(p-phenylene terephthalamide) (Kevlar) (Tg  618 K) O

O

C

C

NH

/

227

(oxymethylene) (POM), poly(ethylene oxide) (PEO), poly(propylene oxide) (PPO), and poly(tetramethylene oxide) (PTMO) are summarized in Table 13.9. The sub-Tg g-relaxation in PEO has been attributed to local twisting motion of main chains in both amorphous region and defective regions of the crystalline regions [100]. The g relaxation (amorphous) has been reported to increase as the proportion of oxygen in the main chain increases (i.e., POM > PEO > PPO > PTMO). 13.2.3 Poly(aryl ether ether)

NH

Unfortunately, the number of good dynamic mechanical and dielectric studies of well-characterized samples is limited. Some dynamic mechanical data for these two aramids are given in Table 13.8. A study by Badayev et al. [98] (Table 13.7) indicates that although the b and g peaks of these aramids are located at temperatures higher than those for the aliphatic polyamides, their activation energies are comparable with those of the two aliphatic polyamides. Both aliphatic and aromatic polyamides display a lowtemperature relaxation (d) at ca. 115–190 K; however, a molecular mechanism other than methylene group motion suggested for aliphatic polyamides must exist for the aramids. Kunugi et al. [99] report three principle relaxations (b, g, and d) for Kevlar fiber where the d relaxation was observed to be more prominent in the presence of sorbed water. The results of Kunugi et al. for annealed samples and those of several other investigations (Table 13.8) indicate that the g peak of aramids may appear as two weak peaks – one in the region above and below room temperature (291–333 K) and one at 417–440 K (g ) with a slightly higher activation energy.

13.2.2 Poly(alkylene oxides) Results of dielectric and dynamic-mechanical measurements of several poly(alkylene oxides) including poly-

Results of dynamic-mechanical measurements of poly(aryl ether ether ketone) (PEEK) (Tm ¼ 613---673 K) O O

O

C

are summarized in Table 13.10. An b peak corresponding to the glass transition occurs at about 420 K and is sensitive to the crystalline morphology of the sample [101]. Candia et al. [102] report two b peaks at 423 and 488 K where the higher temperature peak represents reorganization following the crystallization process. In addition, there is evidence for a broad g relaxation in the region between 170 and 213 K which may be due to contributions from two [103,104] or three [105] overlapping relaxations. This low-temperature sub-Tg relaxation has been attributed to localized wagging of the polar bridges within the amorphous regions while the high-temperature relaxation is attributed to a combination of wagging motions and phenylene flips and is affected by intermolecular interactions [101]. Sommer et al. [106] have looked at the effect of substituent group structure of the bisphenol groups on the Tg and the g-relaxation temperature of the following four polyetherketones (PEKs):

TABLE 13.9. Glass-transition and secondary-relaxation temperatures of poly(alkylene oxides). Polymera POM POM PEO PEO PEO PPO PPO PPO PTMO PTMO a

Techniqueb

f c Hz

TP D TP D D VR TP D VR D

1 2:1  106 1 12,800 20 1 2,000 40–600 2,000

Tb (K)

Ea d kJ mol1

Tg (K) 203 247

206 236 228 211 208 221 188–198

Ea kJ mol1 19

126–147

130–155 155–163

198 140 164

38 33

150 164

25–30 13–21

POM, polyoxymethylene; PEO, poly(ethylene oxide); PPO, poly(propylene oxide); PTMO, poly(tetramethylene oxide). Legend given in Table 13.1.

b–d

Ref. [24] [128] [150] [100] [151] [110] [150] [152] [110] [152]

228 / CHAPTER 13 O CH3 C O

O

C CH3

10

O CH3 C O

C

O

CH3 11 O C O

O

12 O C O

O

13

As shown by the data in Table 13.10, the temperature for the g relaxation increases with increasing steric hindrance to rotation of the bisphenol group in the order PEK(10) >TMBPA-PEK(11) > BPZ-PEZ(12). Methyl substitution of the cyclohexylidene ring in the case of TMC-PEK(13) lowers the g temperature compared to PPZ-PEZ.

13.2.4 Polyesters Dielectric and dynamic mechanical data for poly(ethylene terephthalate) (PET) (Tm
538 K), poly(butylene terephthalate) (PBT) (Tm
493 K), and several fully aromatic polyesters or polyarylates having the general structure shown below

TABLE 13.10. Glass-transition and secondary-relaxation temperatures of poly(aryl ether ether). Polymera PEEK

PEK TMBPA-PEK BPZ-PEK TMC-PEK a

Techniqueb

f c Hz

Tb (K)

TP TP TP FO FO FO NA

1 1 1 35 5 110

423 416

Ea d kJ mol1

amorph. 1,250–1,900 423

Tg (K) 193 176 183 170 213 208 180 198 209 191

Ea kJ mol1

30–100 80 31

Ref. [105] [63] [103] [136] [104] [102] [106] [106] [106] [106]

PEEK, poly(aryl ether ether ketone); PEK, bisphenol-A polyetherketone; TMBPA-PEK, tetramethylbisphenol-A PEK; BPZPEK, PEK from cyclohexylidene; TMC-PEZ, PEK from trimethyl-cyclohexylidene PEK. b–d Legend given in Table 13.1.

SUB-Tg TRANSITIONS R1

O

C

O

O

C

C

R4 R2

229

substitution of the bisphenol moiety. From molecular-mechanics calculation of conformational energies of three polyarylates derived from terephthalic acid, Charati et al. [108] have concluded that low-energy p phenyl-ring flips are possible through cooperative motions of both rings. From dielectric measurements of several polyarylates and related polymers, Schartel and Wendorff [64] concluded that the d relaxation (activation energy of  46 kJ mol1 ) must involve both intrachain and interchain contributions with a correlation length of no greater than a single repeat unit. del Campo et al. [109] have reported the dielectric spectra of a series of nematic polyesters following the form of structure 14 where R is a methylene chain of 4, 6, 8, 10, or 12 units. They attributed the b and g relaxations to local reorganizations of the mesogenic units and the methylene units of the spacer groups, respectively. The characteristics of the molecular motions associated with the b relaxation are influenced by the conformation arrangement of chains in the nematic phase.

O

R5 R3

/

R6

are summarized in Table 13.11. Results for both PET and PBT are comparable with a b(Tg ) relaxation observable in crystalline samples in the region from 344 to 366 K and a sub-Tg g-relaxation in the region of 188–237 K. The b-relaxation temperature increases with crystallinity while the g relaxation is relatively unaffected [107]. Charati et al. [20] have reported dynamic-mechanical data for a number of different polyarylates. They concluded that the g relaxation originates from defects of the glass and is reduced through thermal annealing. A d relaxation (water sensitive) was attributed to phenylene motion in the bisphenol moiety and is shifted to high temperatures with O

O

C

O

R

O

C

O

C(CH3)3

14

13.2.5 Polyolefins The presence of a varying numbers of side branches having different lengths and varying levels of crystallinity in different grades of polyethylene (PE), including low-density (LDPE), linear low-density (LLDPE), and high-density polyethylene (HDPE), complicate the interpretation of the dynamic-mechanical spectrum of this polymer. In addition, the nonpolar nature of polyethylene makes it unsuitable for dielectric analysis in its unmodified form although electrical

O

properties can be enhanced by irradiation in air. In general, the relaxational processes in polyethylene may be characterized as a, b, and g in order of decreasing temperature. The a process has been associated with the melting of PE crystallites of different sizes and decreases in intensity with decreasing crystallinity as may be achieved through irradiation [110] or chlorination as examples. The a-peak temperature is higher for high-density samples. There is evidence that the overall a process may result from two and possibly three different mechanisms [111]. Alberola et al. [112] report

TABLE 13.11. Glass-transition and secondary-relaxation temperatures of polyesters. Polymera

Techniqueb

PET

TP

PET amorph. Crystal. PET PET PBT PBT

D

PBT PBT PAR a

f c Hz 1

Tb (K) 292 (amorph.) 365 (cryst.)

Ea d kJ mol1 770

Tg (K) 208

Td (K)

52 (wet) 71 (dry)

Ref. [153]

1

366

10,000 10 1 1,600 1.6

344 334 353 355 423–587

63 NA 50

281

237 188

483 1,060–1,144

198 353–493

50 55

[154] [131] [154] [155]

27 369

[131] [107] [20]

172–383

PET, poly(ethylene terephthalate); PBT, poly(butylene terephthalate); PAR, various polyarylates. Legend given in Table 13.1.

b–d

Ea kJ mol1

[142] 829 387

D FO D D FV FO D FO

Ea kJ mol1

230 / CHAPTER 13 TABLE 13.12. Glass-transition and secondary-relaxation temperatures of polyolefins. Polymera

Techniqueb

f c Hz

LDPE LDPE

VR D

40–600 1,000

HDPE HDPE LLDPE i-PP i-PP

TP TP FV FV TP

1 1 1.59 1 1

Ta (K)

Ea d kJ mol1

Tb (K)

Ea kJ mol1

265

297–303 325–400 343–359

256 275 279

110–170

Tg (K) 159 153 175 153 152 149

380

Ea kJ mol1

Ref.

48

[110] [113] [24] [114] [156] [115] [116]

a

LDPE, low-density polyethylene; HDPE, high-density polyethylene; LLDPE, linear low-density polyethylene; i-PP, isotactic polypropylene. b–d Legend given in Table 13.1.

two high-temperature (a) processes in the temperature range from 303 to 393 K with activation energies of 80---210 kJ mol1 . The weak b relaxation, particularly evident in low-density samples, is typically observed between 220 and 280 K and is sometimes identified with motions in the interlaminar region (amorphous–crystalline interphase) and is observed to decrease in intensity upon annealing [113]. The g relaxation typically lies in the temperature range from 123 to 153 K. Willbourn [110] has suggested that this transition is characteristic of coordinated motions of a minimum of three or four methylene groups (e.g., crankshaft-type mechanism). The g relaxation is observed to increase in intensity as sample crystallinity increases and the a peak decreases. In the case of LDPE, as many as three lowtemperature (g) relaxations have been detected by TSC measurements [113]. Both the b and g relaxations have characteristics that may be associated with the glass transition such as high activation energy (ca. 150---200 kJ mol1 ) and WLF dependence [112] although the b relaxation may be associated with side chain motions involving CH3 groups [114]. It is also noted that polyethylene can achieve structural recovery by annealing at temperatures below the g-relaxation temperature [112]. Some representative dynamic-mechanical data for isotactic polypropylene (i-PP) are given in Table 13.12. It is noted that the activation energy of the a relaxation is smaller than that of the b relaxation (glass transition of amorphous fraction) suggesting a less cooperative process attributed to the diffusion of defects in the crystalline phase [115]. Jawad and Alhaj-Mohammad [116] associated the b relaxation with the glass transition and report that the intensity of the b peak decreases with drawing suggesting that b relaxation depends on mobility of chains in the amorphous regions.

13.2.6 Poly(phenylene sulfide) Poly(p-phenylene sulfide) (PPS) (Tm
558 K) exhibits only weak sub-Tg relaxational processes. Dynamicmechanical data [68,70] suggest a low-temperature (d)

relaxation in the region of 165–233 K with an activation energy of about 46 kJ mol1 [68]. There may be a highertemperature relaxation at 313 K (g) and one at 361 K (b) [70]. Deuterium NMR studies of deuterated amorphous PPS by Henrichs et al. [117] suggest that PPS can undergo rapid p flips with a distribution of activation energies centered about 46 kJ mol1 . ACKNOWLEDGEMENT The author would like to acknowledge the contribution of R.-J. Roe to the version of this chapter appearing in the first edition of this handbook. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

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CHAPTER 14

Polymer–Solvent Interaction Parameter w Robert A. Orwoll* and Pamela A. Arnoldy * Department of Chemistry, College of William and Mary, Williamsburg, VA 23187-8795 y Chemistry Department, Gettysburg College, Gettysburg, PA 17325

14.1 14.2 14.3

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Features and Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

¼
R(n1 ln f1 þ n2 ln f2 ): DS(comb) mix

Many thermodynamic properties of polymer solutions such as solubilities, swelling equilibria, and the colligative properties can be expressed in terms of the polymer-solvent interaction parameter w. This unitless quantity was originally introduced by P. J. Flory [1] and M. L. Huggins [2] as an exchange interaction parameter in their lattice model of polymer solutions. In their definition, the quantity kTw (k is the Boltzmann constant; T, the absolute temperature) is the average change in energy when a solvent molecule is transferred from pure solvent to pure, amorphous polymer. The reader is referred to Flory [3] for details. However, as explained in the following section, for this compilation w is defined empirically, independent of the Flory–Huggins or any other model. Values of w have been collected in the table below for binary mixtures of homopolymers and low molecular weight liquids. Interaction parameters for systems with two polymeric components, i.e., polymer blends, can be found in Chapter 19. Other tabulations of w are available [119,120,136].

(14:2)

Here, R is the gas constant and f1 and f2 are the volume fractions of solvent and polymer, respectively, in the resulting solution. The volume fraction of polymer can be expressed in terms of the weight fraction w2 of polymer and densities r1 and r2 of the pure components: w2 r1 (14:3) f2 ¼ 1
f1 ¼ r2 þ w2 (r1
r2 ): That part of DGmix that exceeds the contribution from the combinatorial entropy, namely, DGRmix ¼ DGmix
(
TDS(comb) mix ),

(14:4)

is the residual free energy [4]. Many thermodynamic properties of interest can be directly related to the change that the chemical potential of the solvent undergoes on mixing,   @DGmix m1
m01 ¼ (14:5) @n1 T ,P,n2 Differentiation of Eq. (14.4) with respect to n1 yields the residual chemical potential,

14.1 DEFINITION

(m1
m01 )R ¼ (m1
m01 )
RT[ ln (1
f2 )þ f2 (1
1=x)],

The change in the Gibbs free energy for mixing two components at constant temperature T and pressure P depends on the heat DHmix and entropy DSmix of mixing through the general thermodynamic relation DGmix ¼ DHmix
TDSmix :

233 234 234 256

(14:6)

where x is the ratio of the molar volume of polymer to that of solvent:

(14:1)

x¼

In the case of n2 moles of an amorphous polymer dissolving in n1 moles of solvent, the combinatorial contribution to the entropy of mixing [1–3] is

r1 M2 : r2 M1

(14:7)

M1 and M2 are the (number-average) molecular weights. The unitless interaction parameter w is defined for this 233

234 / CHAPTER 14 compilation as a reduced residual chemical potential using Eq. (14.6): w¼ ¼

(m1
m01 )R f22 RT (m1
m01 ) ln (1
f2 ) þ f2 (1
1=x) :
f22 RT f22

(14:8)

14.2 METHODS OF MEASUREMENT Most of the entries in the table below were obtained from osmotic pressure, vapor sorption, or inverse gas chromatography measurements [5]. Osmotic pressure measurements can be used to evaluate w at small volume fractions of polymer. The osmotic pressure P of a solution relative to pure solvent is related to the chemical potential and, with Eq. (14.8), to w through the thermodynamic expression m1
m01 ¼
PV1 ,

(14:9)

where V1 is the molar volume of solvent. The interaction parameter in the limit of infinite dilution can also be determined from the second virial coefficient A2 , i.e., the slope of a plot of P=RTc2 versus the concentration c2 ¼ r2 f2 (i.e., mass of polymer per volume of solution) at c2 ¼ 0: w¼

1
A2 V1 r22 : 2

(14:10)

Vapor sorption studies yield values of w for solutions at intermediate-to-high polymer concentrations. The vapor pressure P1 of solvent above a polymer solution relative to that of pure solvent P01 at the same temperature is (m1
m01 ) ¼ RT ln

P1 : P01

(14:11)

Substitution for (m1
m01 ) in Eq. 14.8 from Eq. 14.11 yields w. Inverse gas chromatography can be used to obtain the polymer-solvent interaction parameter in the limit of f2 ¼ 1. Here w is found from the retention volume of the low molecular-weight component in the vapor phase as it is eluted over the polymer which is the stationary component in a gas-phase chromatography experiment. The Flory Q-temperature affords another means of determining the interaction parameter. The Q-temperature is defined [3] such that at T ¼ Q and f2 ¼ 0, w ¼ 1=2. Qtemperatures are tabulated in Chapter 15. 14.3 GENERAL FEATURES AND SIGNIFICANCE For many systems, w has been found to increase with polymer concentration and decrease with temperature with a dependence that is approximately linear with, but in general not proportional to, 1/T. According to Eqs. 14.5 and

14.8, for a given volume fraction f2 of polymer, the smaller the value of w, the greater the rate at which the free energy of the solution decreases with the addition of solvent. Consequently, liquids with the smallest w’s are usually the best solvents for a polymer. Negative values of w often indicate strong polar attractions between polymer and solvent. Solutions for which w increases with increasing temperature at constant f2 have a negative partial molar heat of mixing, (@DHmix =@n1 )P,T ,n2 0:95

8–160 13.6–1200

2–6.8

Poly(4-methylpentene-1), isotatic

Poly(2-methylpentene-1 sulphone)

Mn ¼ 4:7---56:3

Poly(octene-1) Poly(pentene-1), isotactic syndiotactic isotactic isotactic isotactic, atactic

25–400 Mn ¼ 10:8 Mn ¼ 2:8 Mn ¼ 13:2---42:7 6.1–306 0.2–23.1 (isotactic) 0.2–55 (atactic)

Poly(1-phenyl-1-propyne) Poly(propylene) syndiotactic Poly(propylene), isotactic

1.5–145 11.7

2.8–56.4

1.06–1.09

Solvent

Theta temperature 8C

Ku :104 [dl:g
1 (g mol wt)
1=2 ]

methylcylohexane/noctanol (56/44) methylcyclohexane/npentanol (65.2/34.8) methylcyclohexane/npropanol (74.2/25.8) benzene n-pentane dibutyl ether i-amyl isovalerate i-amyl isovalerate i-amyl benzyl ether i-amyl n-butyrate n-amyl n-butyrate benzene n-butyl n-butyrate 3-methyl 5-heptanone benzene i-amyl isovalerate i-amyl isovalerate a-chloronaphthalene

24.5 u1 :76.0 u1 :204.0 22.1 21.0 23.7 28.0 22.0 22.8 46.2 55.5 25.0 25.0 27.0 165.0

o-dichlorobenzene biphenyl

133.0 194.6

diphenyl ether diphenyl methane MEK/isopropanol (39.5/60.5) MEK/n-hexane (35.4/64.6) phenetole phenetole phenetole isoamyl acetate 2-pentanol anisole

210.0 176.6 22.5 11.5 50.4 56.0 48.5 31.0 64.0 85.0

9.1 6.55

i-butylacetate diphenylmethane phenetole phenylether cyclohexane isoamyl acetate phenyl ether isoamyl acetate p-tert-amylphenol benzyl phenyl ether benzyl propionate n-butanol p-tert-butyl phenol dibenzyl ether biphenyl diphenyl ether

32.5 121.0 64.0 149.0 36.0 42.0 145.4 70.0 140.8 181.8 157.5 147.2 166.0 183.2 125.1 142.8

i:12.0;a:10.0 i:9.8 i:11.3;a:9.8 i:9.8;a:9.4

Refs.

[46]

11.0

[47] [48]

10.8 11.4 11.5 10.9 [49]

8.88

[50] [51]

11.5 [52]

9.1

12.1 i:10.6;a:9.9

17.2

10.6 15.2 13.7

[41]

[53] [54] [55] [56] [57]

[58] [59] [60] [61] [62,63]

264 / CHAPTER 15 TABLE 15.1. Continued.

Polymer

Dispersity (Mw =Mn )

Solvent

12.8 12.5 15.8 16.8 11.5 13.2

[64]

1.02–1.35 1.04–1.12

125.0 143.0 153.5 74.0 92.0 129.0 146.0 58.0 34.0 77.0 37.6

14.1 13.0

2.3–42.0 3.91–37.1

diphenyl phenylether phenyl ether 1-chloronaphthalene cyclohexanone biphenyl diphenyl ether isobutyl acetate isoamylacetate 1-octanol 2-octanol

Mn ¼ 0:22---4:19 1.64–43.3

1.0–1.07

isoamylacetate isoamylacetate

43.0 56.8

10.3 12.6

[69] [70]

Mw  10
4 Mh ¼ 5:0---53:1

Poly(propylene), atactic

Mh ¼ 4:3---70:8

Poly(propylene), atactic [hydrogenated poly (2-methyl-1,3-pentadiene)] fm ¼ 0:502 Polypropylene (head-to-head)

Ku :104 [dl:g
1 (g mol wt)
1=2 ]

Theta temperature 8C

Refs. [64] [60] [65,66]

[67] [68]

TABLE 15.2. Poly(alkenes).

Polymer Poly(butadiene), (cis: 0.98) Poly(butadiene), cis (cis: 0.95; trans: 0.01; 1,2: 0.04)

Mw  10
4

Dispersity (Mw =Mn )

Mn ¼ 5:3---48:9

Poly(butadiene), cis

5.0–110

Poly(butadiene), cis (cis: 0.93; trans: 0.04; 1,2: 0.03)

Mh ¼ 10:8---69:2

Solvent isobutyl acetate diethyl ketone methylisoamyl ketone methyl n-propyl ketone methylisoamyl ketone/ methyl n-propyl ketone (3/1) (1/1) (1/3) diethyl ketone/methyl n-propyl ketone (3/2) n-heptane/n-hexane (50/50) diethyl ketone

Poly(butadiene), cis, cyclized Deg. Cycl.

Mh ¼ 4:7---19:3

#1.1

20.5 10.3 12.6 59.7

18.5 18.1 17.8 15.7

22.3 32.7 46.2 30.0

17.5 17.1 16.7 17.4

Refs. [71] [72]

[73] [4]

dioxane

uu :14.0 ui :208.0 uu :
22.0 ui :237.0 uu :35.0 ui :141.0 26.5

diethyl ketone

213.0

ethyl propyl ketone propylene oxide cyclohexane/dioxane

240.0 146.0

propylene oxide 1.08–57.1

Ku :104 [dl:g
1 (g mol wt)
1=2 ]

5.0

ethyl propyl ketone

Poly(butadiene) (cis:0.36; trans: 0.57; 1,2 vinyl: 0.07) Poly(butadiene), trans (trans: 0.94; 1,2: 0.06)

Theta temperature 8C

17.8

[74] [4]

Ku as a function of cyclization is given in Figure 6 of this ref.

[75]

THETA TEMPERATURES /

265

TABLE 15.2. Continued.

Polymer 9% 31% 46% 63% 81% 1,4-poly(2,3-dimethyl butadiene) (>85% trans-1,4; 3% 1,2 vinyl) 1,4-1,2-poly(dimethylbutadiene) (trans 1,4: 23%; cis 1,4: 32%; 3,4: 45%) Poly(chloroprene)

Poly(isoprene) (43% brominated) Poly(isoprene) Poly(isoprene), cis Poly(isoprene), cis (96%) Poly(isoprene), cis (94%) 3 branches 11 branches 22 branches Poly(isoprene), trans (96%)

Poly(isoprene), cis: 0.70; trans: 0.23; 3,4: 0.07, star linear 4-arm star 6-arm star

8-arm star 12-am star Poly(isoprene) (1,2: 0.35; 3,4: 0.65) Poly(2-methyl-1,3-pentadiene) (cis/trans: 64/36) Poly(methylbutylene) (hydrogenated 1,4 polyisoprene) 1,4 polymyrcene (1,4: 90%; 3,4: 10%)

Mw  10
4

Ku :104 [dl:g
1 (g mol wt)
1=2 ]

(12/88) (17/83) (21/79) (30/70) (39/61) cyclohexane/n-propanol (81.3/18.7) isoamylacetate

30.0 30.0 30.0 30.0 30.0 25.0

22.7

[76]

30.0

12.6

[70]

MEK cyclohexane cylopentane MEK trans decalin cyclohexane methyl isobutyl ketone n-hexane/isopropanol (50/50) dioxane methyl isobutyl ketone methyl propyl ketone methyl propyl ketone methyl propyl ketone methyl propyl ketone methyl isobutyl ketone dioxane toluene/n-propanol (68.4/31.6) (67.6/32.4) (66.5/33.5) (65.8/34.2) (64.5/35.5) (63.8/36.2)

25.0 45.5 56.3 25.0 2.0 20.0 13.0 21.0

11.6 10.7 11.3

[77]

Dispersity (Mw =Mn )

10.8–51.8 6.2–116 4.8–34.6 3.6–110 5.5–86 Mn ¼ 5:01 and 1:81 2.51–23.8

1.03–1.19

5.6–79.4

1.22–1.5

15.1–304 58.7–171 380 5–100 6.9–75 linear 9.4

Theta temperature 8C

1.29–1.98

5.7; Br: 1.75 18; Br: 1.6 34.2; Br: 1.6 Mh ¼ 9:2---27

Solvent

31.2 16.5 33.0 33.0 27.8 23.5 15.0 47.7 25.0 30.0 35.0 40.0 45.0 50.0

16.6

Refs.

[78] [79] [80] [81] [73]

13.4

[82] [83]

19.1 22.2 21.9 21.7 21.4 21.3 21.1

[82]

[74,84, 85] 0.5–222.5 3.8–195 4.5–144.6

1.06–1.17 1.0–1.1 1.01–1.04

arm:0.51–75; star: 4.1–590 arm:0.35–44.5; star: 4.1–526

4.45–37.7

dioxane dioxane dioxane methyl isobutyl ketone methyl n-propyl ketone dioxane

34.1 33.4 33.5 12.2 25.4 32.8

dioxane

32.9

benzene/isopropanol (55/45) 2-octanol

20.0 28.9

11.2

[87]

18.9

[88]

0.84–60.0

1.02–1.13

n-hexyl acetate

60.9

6.66–58.3

#1.07

2-octanol

35.9

11.5 8.9 7.2

86

6.28

[70]

266 / CHAPTER 15 TABLE 15.2. Continued.

Polymer

Mw  10
4

Dispersity (Mw =Mn )

Solvent

Theta temperature 8C

Poly(pentenamer) (80% trans) Poly(vinylethylene) Poly(vinylethylene) (1,2: 98%; 1,4: 2%)

3.6–63.5 1.22–52.1 1.12–48.9

1.0–1.22 1.02–1.13 #1.07

isoamyl acetate 2-octanol 1-hexanol

38.0 32.8 66.0

Ku :104 [dl:g
1 (g mol wt)
1=2 ] 23.4 12.0 11.8

Refs. [89] [90] [70]

TABLE 15.3. Poly(vinyl)s.

Polymer

Mw  10
4

Dispersity (Mw =Mn )

Poly(ethylethylene)

2.66–55 5.2–54.5

1.03–1.11 1.03–1.11

Poly(vinyl acetate)

87–346 2.7–126.8

2.0

4.1–83 0.35–150 3.0–32

0.8–130 Branched Poly(9-vinyladenine) Poly(vinyl alcohol)

Poly(vinyl alcohol), urethanized 4.9%; 8.1% or 11.5%

Poly (vinyl carbanilate)

Mn ¼ 9:4, 13, and 51 1.35, 3.44, and 7.41

DP ¼ 770– 2040

1.02–1.05

Solvent 2-octanol 2-octanol 1-octanol methyl isopropyl ketone/n-heptane (73.2/26.8) n-heptane/methyl isopropyl ketone (27.3/72.7) ethyl n-butyl ketone ethyl isoamyl ketone methanol ethyl n-butyl ketone carbon tetrachloride ethanol 3-heptanone cetyl alcohol ethanol 3-heptanone sodium cacodylate/water (0.1 m/L)

Theta temperature 8C

Ku :104 [dl:g
1 (g mol wt)
1=2 ]

Refs.

23.5 21.0 53.0 25.0

7.4 8.22 7.49

30.0

9.2

[92]

9.29 8.20 10.1 9.55

[93]

29.0 66.0 6.0 29.0 46.4 19.0 29.0 123.0 12–15 26.0 40.0

[90] [70] [91]

[94] [95] [83]

5.37

[96] [83]

2.92

[[97]

water

97.0

[98]

t-butanol/water (32/68/w/w) ethanol/water (41.5/58.5 w/w) methanol/water (41.7/58.3 w/w) i-propanol/water (39.4/60.6 w/w) n-propanol/water (35.1/6.49 w/w) n-propanol/water(30/40)

25.0 25.0 25.0 25.0 25.0 30.0

[99]

n-propanol/water(40/50) toluence cyclohexanol

60.0 37.0 55

diethyl ketone

35

[100]

7.62

[101] [101a]

THETA TEMPERATURES /

267

TABLE 15.3. Poly(vinyl)s.

Polymer Poly(N-vinylcarbazole)

Mw  10
4 34.6–229 7.6–56.4

Dispersity (Mw =Mn ) 1.2–1.3

7.2–4.9 Poly(vinyl chloride) 4.3–48.7

fr ¼ 0:53---0:57 Poly(3-vinylpyrene) Poly(2-methyl-5-vinyl pyridine)

Mn ¼ 1:89---10:2 6.8 and 11.8 3.5–48.7

1.1

10.4–99

6.7–120

Poly(2-vinylpyridine)

42–445

Poly(N-vinylpyrrolidone)

3.5–23.3 1.15 Mn ¼ 9:9---45:7

Poly(vinyl sulfonic acid)

Poly(vinyltriazole)

DP¼366–3640

197

#1.2 #1.4

Solvent chlorobenzene nitrobenzene chlorobenzene/methanol (85.9/14.1) 1,3-dichlorobenzene/ methanol(85/15) benzyl alcohol THF/water (100/11.9); (100/9.5) n-butanol/cyclohexanone (15.8/100) n-butanol/cyclohexanone (41.5/100) cyclohexane/DMF (100/12.8) cyclohexanone DMF THF/water (100/5) methanol/THF (42/58) THF/water (91/9) chloroform methanol/THF (8/92) isoamyl acetate n-amyl acetate isobutyl acetate n-butyl acetate ethyl n-butyrate ethyl propionate methyl isobutyl ketone n-propyl acetate n-propyl propionate propionitrile tetrahydronaphthalene n-amyl acetate n-butyl acetate methyl isobutyl ketone n-heptane/n-propanol (59.6/40.4 w/w) benzene acetone/water (66.8/33.2) acetone/water (66.8/33.2) Na2 SO4 =water (0.55 m/l) KBr/water (0.347M) KCl/water (0.349M) KCl/water (0.650M) KCl/water (1.001M) NaBr/water (0.346M) NaBr/water (1.008M) NaCl/water (1.003M) dimethyl formamide/dioxane (76.6/23.4)

Theta temperature 8C

Ku :104 [dl:g
1 (g mol wt)
1=2 ]


36.5
20.4 25.0

7.38

25.0

8.77

155.4 30.0 59.0

Refs. [102]

15.6

[103]

[104] [105] [106]

72.0 40.5 51.0 36.5 17.0 22.0 25.0 25.0 25.0 53.2 48.2 49.0 21.8 50.0 25.4 37.4 19.3 58.0
3.6 49.5 48.2 21.8 37.4 25.0 15.0 25.0 25.0 28.0 5.7 5.5 26.0 44.5
0.6 40.1 32.4 25.0

5.2

[107] [108] [109]

8.4 (average)

[110]

8.3 8.4 8.0 12.0 7.2 7.4 5.8 7.72

[111]

[112] [113] [114] [115] [116]

7.65 8.91 9.01 10.72 10.6 10.78 [117]

268 / CHAPTER 15 TABLE 15.4. Poly(styrenes).

Polymer Poly (p-tert-butylstyrene) (fi ¼ 0:15, fs ¼ 0:55, fh ¼ 0:30) Poly (p-tert-butylstyrene)

Poly(p-chlorostyrene)

Mw  10
4

Solvent

Ku :104 [dl:g
1 (g mol wt)
1=2 ]

Refs.

1.8–640

1.02–1.22

1-nitropropane

31.0

6.1

[118,119]

2.74–45.5 and 5.1–748

#1.11 and 1.05–1.17

3-nonanol 2-octanol 1-hexanol 1-nitropropane ethyl benzene ethyl carbitol ethyl chloroacetate methyl chloroacetate i-propyl acetate i-propyl benzene i-propyl chloroacetate benzene

10.9 32.7 65.0 31.0
14.7 27.8
1.8 64.6 75.7 59.0
8.2 8.0

5.95 5.57 5.11 6.00 6.4 6.4 5.9 6.4 6.4 6.5 5.6

[120–122]

20.1–125

Mh ¼34.1– 1.79.9

Poly(p-decylstyrene) Poly(3,4-dichlorostyrene)

21.3–57 35–540

Poly(p-hexylstyrene) Poly(p-isopropylstyrene) Poly(p-methoxystyrene)

60–90

Poly(p-methoxystyrene)

Mh ¼ 7.6
63.1

Poly(a-methyl styrene)

29.4; 30.6; and 38.0

Poly(a-methyl styrene) fmm : 0.08; fmr : 0.48; frr : 0.44 fmr : 0.10; frr : 0.90 fmr : 0.19; frr : 0.81 fmm : 0.03; fmr : 0.29; frr : 0.68 fmm : 0.06; fmr : 0.40; frr : 0.55 fmm : 0.11; fmr : 0.45; frr : 0.44 Poly(a-methyl styrene)

Dispersity (Mw =Mn )

Theta temperature 8C

1.37–1.79

8–150

[125]

benzene/methanol (5.5/1) (5/1) (4.5/1) MEK/butylethylketone (1/1.24) butyl acetate/butyl alcohol (13/1 w/w) MEK dioxane/isopropanol (35/65) t-butyl benzene isoamylacetate dichlorodecane methyl isobutylketone benzene/methanol (73.7/26.3) chloroform/methanol (66.5/33.5) chloroform/cyclohexane (36.1/63.9) MEK/n-heptane (80/20) MEK/n-propanol (80.6/16.4) toluene/cyclohexanol (62.8/37.2) cyclohexane

30.2 20.0 52.2 75.0 92.6 23.4 25.0 25.0 25.0 25.0 25.0 25.0 36.2

cyclohexane cyclohexane cyclohexane cyclohexane cyclohexane cyclohexane cyclohexane trans-decalin

37.0 32.5 33.3 34.5 35.6 37.0 34.5 9.5

7.89 7.3 6.7

1-chloro-n-hexane cyclohexane 1-chloro-n-octane

10.0 34.5 53.0

6.51 6.88 6.31

26.7 32.4 41.6 21.2 32.9

[123,124]

5.61 5.6

7.4 6.9

[126] [127] [126] [86] [128]

6.4 [129]

[81] [130,131]

9–400 2.7–367 2.6–17.5

20.4–747; also two samples with Mn ¼ 3.93 and 8.01 2.01–90.3

1.8–2.8 1.11–1.60

1.01–1.12

7.8 7.39 7.52

[132]

[133]

THETA TEMPERATURES /

269

TABLE 15.4. Continued.

Mw  10
4

Polymer Poly (a-methyl styrene) 53% syndio; 41% hetero; 5% iso Poly (a-methyl styrene) fr ¼ 0:72 Poly(p-methylstyrene) Polystyrene

Dispersity (Mw =Mn )

Solvent

Theta temperature 8C

Ku :104 [dl:g
1 (g mol wt)
1=2 ]

Refs.

5.9–341

1.03–1.15

cyclohexane

36.2

[133a]

0.053–322

1.01–1.05

cyclohexane

30.5

[133b]

diethyl succinate cyclohexane ethylcyclohexane cyclohexane MEK/isopropanol (6/1) toluene/methanol (76.9/23.1) (75.2/24.8) (72.8/27.2) decalin (23.1% cis) cyclohexane 1-chloro-n-undecane diethyl malonate cyclohexane n-butyl formate cyclohexane cyclohexanol decalin diethyl malonate diethyl oxalate hexyl-m-xylene methylcyclohexane benzene/cyclohexanol (38.4/61.6) benzene/n-hexane (34.7/65.3) benzene/methanol (77.8/22.3) benzene/isopropanol (64.2/35.8) butanone/methanol (88.7/11.3) carbontetrachloride/methanol (81.7/28.3) chlorobenzene/di-isopropyl ether (32/68) chloroform/methanol (75.2/24.8) dioxane/methanol (71.4/28.6) methanol/tetrahydrofuran (28.7/71.3) 1-chloro-n-decane 1-chloro-n-undecane 1-chloro-n-dodecane benzene/isopropanol (66/34) dioxane/isopropanol (55/45) cyclohexane n-hexane/3-methylbutanone (48/52) cyclohexane methyl cyclohexane diethyl malonate diethyl oxalate cyclohexane/ methylcyclohexane (2/1) (1/1) (1/2)

16.4 34.0 70.0 34.0 23.0

7.0 7.9 7.3 8.2 7.30

25.0 34.0 45.0 19.3 35.0 32.8 35.9 34.8
9.0 34.0 83.5 29.5 31.0 51.5 12.5 68.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 6.6 32.8 58.6 20.0 20.0 34.5 20.0 34.5 70.5 34.2 55.8

9.2 8.9 8.8

43.0 48.0 54.0

7.8 7.5 7.3

16.3–197 Mh ¼ 7.0–127 Mh ¼ 0.4–697 4–146 Mh ¼ 10.2–53.2

Mh ¼ 1.8–271.6 8.5–395 40.6

1.05

17

33.5

40.6

Mh = 1.04– 123 Mn = 0.054– 0.72

#1.1

8.0 7.86 7.69 8.66

[134] [135] [43] [136] [137]

[138] [139] [140]

[141]

[142]

[143]

[86] 8.4 7.8 7.0 7.2 7.3

[144] [145] [72]

270 / CHAPTER 15 TABLE 15.4. Continued.

Polymer

Mw  10
4

Dispersity (Mw =Mn )

Solvent

9–56 7.0; 16, and 800

½ z = 87 Å MW(SANS) = 96,500;MW (Osmometry) = 97,200

2

MW = 693 ⫻ 103 (SANS) MW = 60 ⫻ 103 (GPC) Rg = 368 Å

1

0.05

0

3

0 0

10

20

30

40

50

60

70

80

90

100

Q 2⫻105[Å−2]
1

2 FIGURE 23.2. d
d (Q) VS Q for sample containing 5 wt % labeled PSD molecules in polystyrene (PSH).

scattering lengths of the labeled (C8 D8 ) and unlabeled (C8 H8 ) are 2:328 10
12 cm and 10:66  10
12 cm, respectively, [via Eq. (23.10) and Table 23.1]. Given a density, r ’ 1:05 gm cm
3 , the segment volume is 164:5  10
24 cm3 and the extrapolated cross section [dS=dV(0) ¼ 17:4  0:5 cm
1 ], leads to a polymerization index of the labeled chains of ND ¼ 928  30 or a molecular weight of (96:5  3)  103 , in reasonable agreement with independent determinations via osmometry [32]. The radius of gyration is close to that measured in ideal Q-solvents [8,21] and this supports the unperturbed Gaussian coil as a good approximation to the molecular configuration in amorphous polymers [8,13,32]. 23.3.2 SANS and SAXS from Melt-Crystallized Polyethylene Figure 23.3 shows a Zimm plot of the SANS differential scattering cross section for 6.0 wt% (wD ¼ 0:053) of deuterated polyethylene (PED) in a matrix of unlabeled PEH after rapidly quenching from the melt. The coherent scattering lengths of C2 H4 and C2 D4 are
0:166  10
12 and 4:00  10
12 cm
1 , respectively, [via Eq. (23.10) and Table 23.1], and based on an average density of r ’ 0:94 gm cm
3 , the segment volume is 49:5 10
24 cm3 . Thus the extrapolated

0

1

2 Q 2 ⫻ 10−4 (Å−2)

3

4

FIGURE 23.4. Typical Zimm plot for 6 wt% PED molecules in PEH matrix slow cooled (18C
1 ) from the melt.

cross section [dS=dV(0) ¼ 28:0 2 cm
1 ] leads to a polymerization index (N) of 1,600, which is of the same order as the value from gel permeation chromatography [33]. However, when the same sample is slow cooled from the melt [Fig. 23.4], the extrapolated cross section increases by over an order of magnitude. It is clear that these data do not originate in the scattering from single molecules, and it has been shown that the excess intensity is caused by aggregation or clustering of the labeled molecules [19], though this would not be apparent if the data were in arbitrary units. This behavior illustrates the point referred to above that the intensity is extremely sensitive to the particle or molecular dimensions and even an approximate (25%) absolute calibration is sufficient to reveal the presence of such artifacts. Figure 23.5 shows SAXS data from the polyethylene sample described above. Because PED and PEH have the same electron density, there is no contrast between the different isotopes and PEH, PED and partially labeled

dS −1 (Q) ⫻ 102(cm) dW

Quenched 10

MW = 46 ⫻ 103 (SANS) MW = 60 ⫻ 103 (GPC) Rg = 132 Å

5

0 0

1

2 3 Q 2⫻10−4(Å−2)

4

5

FIGURE 23.3. Typical Zimm plot for 6 WT % PED molecules in PEH matrix quenched from the melt.

dS FIGURE 23.5. dV (Q) vs Q for SAXS and SANS data from melt crystallized polyethylenes.

SMALL ANGLE NEUTRON AND X-RAY SCATTERING /

where rT ¼ 0:282  10
12 cm is the Thompson scattering factor of one electron, and 4:00  10
12 cm is the neutron scattering length of a C2 D4 monomer, which contains 16 electrons. Thus the measured (1.31 + 0.1) and theoretical ratios are in good agreement [34]. 23.3.3 Application of Contrast Variation Methods to Core–Shell Latex Structures Contrast variation methods can sometimes be used to remove a component of the scattering by matching its scattering power with that of the medium in which it is dispersed. This principle can be used in SANS experiments via isotopic solvent mixtures (e.g. H2 O=D2 O) to adjust the scattering power of the medium, as for example in studies of polymer latexes. Grancio and Williams [35] postulated a polymer-rich spherical core surrounded by a monomer-rich shell which serves as the major locus of polymerization, thus giving rise to core–shell morphology. Thus, the first formed

where rnp and rns are the neutron scattering length densities of the particle and solvent, respectively, Np is the number of particles per unit volume, Vp is the particle volume, and P (Q) is the particle form factor [P(0) ¼ 1]. According to Grancio and Williams [35], polymerization takes place in a surface shell and thus if the monomer feed is changed from protonated to deuterated material, this will result in a predominantly D-labeled shell. When examined by SANS in an H2 O---D2 O mixture which matches the scattering length density of the protonated core, the scattering will arise from a hollow sphere with a particle form factor [41] given by [ sin (QR)
sin (QR1)
QR cos (QR) þQR1 cos (QR1)]2 P(Q) ¼ 9 , (23:16) Q6 R6 (1
l)6 where R and a are the outer and inner radii, respectively, (l ¼ a=R). Figure 23.7 shows SANS data from a 4.6 vol% latexes ˚) with a fully deuterated PMMA-D shell, (thickness 30 A ˚ ), after polymerized on a PMMA-H core (radius, a ¼ 498 A desmearing corrections for the finite instrumental resolution [37,42]. The absolute intensity at zero scattering angle is given by Eq. (23.15) with P(0) ¼ 1 and Vp equal to the volume of the D-labeled polymer in the shell with SLD4, rnp ¼ 6:97  1010 cm
2 . The SLD of the solvent is close to that of the PMMA-H core (rns ¼ 1:06  1010 cm2 ) and thus 104 102 103 101 SAXS

102

100 101 10−1 SANS

100

10−2

(Q) for SAXS data (cm−1)

(23:14)

(23:15)

dS dW

(0:282  10
12  16)2 ¼ 1:27, (4:00  10
12 )2

dS (Q) ¼ (rnp
rns )2 Np Vp2 P(Q), dV

(Q) for SANS data (cm−1)

R¼

polymer constitutes the core and the second formed polymer makes up the shell, and neutron scattering has been used to test this hypothesis by isotopically labeling chains generated at specific points in the polymerization process [36–40]. For a homogeneous particle, suspended in a solvent, the neutron scattering cross section is given by

dS dW

samples all have the same SAXS profile. The background due to Compton scattering is virtually zero in this Q-range [4] and the signal arises from density fluctuations [13]. The ˚
1 is proportional to the interlamellar peak at Q  0:025 A square of electron density difference between the amorphous and crystalline regions (lamellae). The upturn as Q ! 0 probably arises from voids and other large scale structures such as spherulites. Figure 23.5 also shows the SANS data for PEH (solid circles), where the coherent signal is superimposed on a flat (incoherent) background  1 cm
1 . The open circles show the extra (coherent) cross section produced by adding 2% deuterated molecules (PED), which is proportional to the contrast difference (aH
aD )2 between deuterated and protonated segments. Departures from the flat incoherent background of the PEH sample (solid circles) are caused by density fluctuations in the sample and it is just possible to see the peak ˚
1, due to the periodic stacking of crystalat Q  0:025 A line lamellae alternating with amorphous regions. The SANS coherent signal in PEH is very weak, however, due to the cancellation between the scattering lengths of carbon and hydrogen (Table 23.1), which makes the SLD very small for PEH (Table 23.3). In the case of PED, there is no cancellation between the coherent scattering lengths of carbon and deuterium (Table 23.1), and the incoherent background is very much smaller than for PEH. Thus, PED should have virtually identical SAXS and SANS cross sections apart from a scale factor. Figure 23.6 shows absolute SAXS and SANS data for the same sample of PED, which should scale as the ratio of the electron density to the scattering length density. As the number of segments per unit volume is the same for SAXS and SANS, this term cancels and the ratio (R) reduces to

413

10−1 0

0.02

0.04

0.06 Q (Å−1)

0.08

0.10

0.12

dS FIGURE 23.6. dV (Q) vs Q for deuterated polyethylene sample after subtraction of incoherent background.

4 This value is slightly different to that quoted in Table 23.3, based on a density of 1:2 gm cm
3 , which is an average over the atactic, isotactic and syndiotactic homopolymers. For atactic PMMA, r ’ 1:18 gm cm
3 .

414 / CHAPTER 23 4.00

103

dS dW

(O)=1.7⫻103

cm−1

(Experiment)

(O)=2.3⫻103 cm−1 (Calculated)

102

dS (Q) (cm−1) dW

a = 67.3 Å dS (0) = 24.5 x 103 cm−1 dW = 28.2 x 103 cm−1 calculated for 2-phase structure of pure PED and PEH

x10-2 dS −½ (Q) (cm½) dW

dS dW

0.00 0.00

101

1.0

2.0

3.0

4.0

104Q2 (Å−2)

FIGURE 23.8. Debye–Bueche plot for phase separated blend of deuterated HDPE and protonated LDPE slow cooled from the melt AT 0:75 8C
1. Reprinted with permission from R. G. Alamo, J. D. Londono, L. Mandelkern, F. C. Stehling and G. D. Wignall, Macromolecules, 27, 411 (1994). Copyright (1994) American Chemical Society.

100

10−1 0.005 0.010 0.015 0.020 0.025 Q = 4π sin q [Å−1] λ

FIGURE 23.7. Desmeared SANS data (O) for PMMA Latex CI (4.6 vol%) with 30 A˚ D-DMMA shell on surface (core contrast matched) compared with theoretical hollow shell scattering. Reprinted from the Journal of Colloid and Interface Science, 123, L. W. Fisher, S. Melpolder, J. M. O’reilly, V. Ramakrishnan, and G. D. Wignall, ‘‘Neutron Scattering from Interfacially polymerized Latexes’’, 29–33, Copyright (1988) with permission of Elsevier.

the core contrast is negligible compared to the PMMA-D shell. The extrapolated cross section, dS=dV(0) ¼ 1:7 103 cm
1 , is in good agreement with the calculated value of (2:3  103 cm
1 ), in view of the extreme sensitivity of the calculations to slight variations in shell thickness, mismatches in SLD, surface roughness etc. [37,42]. Similarly the particle dimensions from SANS are in excellent agreement with independent techniques (e.g., LS).

where a1 is a length characterizing the structure, w1 and w2 are the volume fractions, and rn1 and rn2 are the neutron scattering length densities of the two phases [43,44]. Figure 23.8 shows a DB plot [d S=d V(Q)
1=2 vsQ2 ] of the data for a 50/50 blend after cooling from the melt at 0:75 C min
1 . The extrapolated cross section [d S=d V(0) ¼ 24:5 103 cm
1 ] is well over an order of magnitude higher than in the melt, indicating that the components have phaseseparated on cooling. The plot is reasonably linear and the (Q ¼ 0) cross section is given by Eq. (23.17) where the correlation length (a) is derived from the ratio of slope/ intercept of the DB plot [44–47]. Assuming complete separation of the H- and D-labeled components, the SLDs in the solid state can be scaled (via the density) from the melt values shown in Table 23.3, to give a calculated cross section of 28:2  103 cm
1 . In view of the fact that the experiments are independently calibrated with no arbitrary fitting factors in the intensity scale, the agreement with the absolute cross sections calculated from the DB theory is excellent. Interpenetrating Polymer Networks Figure 23.9 shows DB plots of SANS data from polystyrene–polybutadiene interpenetrating polymer networks [47].

23.3.4 SAXS and SANS from 2-Phase Systems Blends of High- and Low-Density Polyethylenes

6

dS 8pa3 w1 w2 [rn1
rn2 ]2 (Q) ¼ , dV (1 þ Q2 a21 )2

(23:17)

−½ dS (Q) (cm½) dW

7.0

SANS experiments have indicated that blends of high density (linear) and long-chain branched low density polyethylenes (HDPE/LDPE) are homogeneous in the melt, though the components may separate on slow cooling due to the difference in crystallization mechanisms [43]. The semicrystalline blends form effectively two-phase systems in the solid state, and it was shown [43, 44] that the Debye– Bueche (DB) [45,46] model was appropriate to describe the morphology, with a SANS cross section of the form

a = 106 Å

a = 53 Å

3.5

3 PB(0.2% DICUP)/PS semi-I IPN Fully deuterated PS

PB(0.1 % DICUP)/PS semi-I iPN Fully deuterated PS

0

0 0

2.5

5

0

2.5

5

Q2x104 Å−2


dS 
1=2 FIGURE 23.9. dV (Q) vs: Q2 for two PB/PS semi-I-IPN systems with different amounts of cross-linker (DICUP).

SMALL ANGLE NEUTRON AND X-RAY SCATTERING / Assuming complete segregation of the components, d S=d V(0) may be calculated from Eq. (23.17), via the measured correlation lengths (Fig. 23.9) and the SLDs given in Table 23.3. For the data from the two samples shown in Fig. 23.9, this leads to calculated values of 17:2  103 cm
1 and 2:7  103 cm
1 , compared to experimental determinations of 21:6  103 cm
1 and 2:0 103 cm
1 . The discrepancies are not unreasonable in view of the strong dependence of the cross section on the domain dimensions, which is a general feature of absolute intensity comparisons. However, this illustrates the point made earlier that even an approximate ( + 25%) absolute calibration is sufficient to test the assumption of complete phase separation of the blend components. In the limit Qa41, Eq. (23.17) reduces to d S=dV  PQ
4 P ¼ 2P(rn1
rn2 )2 S=V,

(23:18)

where P is the Porod constant, which contains information on the specific surface of the material, i.e., the total interphase surface (S) area per unit volume (V). By comparison of Eqs. (23.17) and (23.18) S=V ¼ 4w1 w2 =a:

(23:19)

For the data shown in Fig. 23.9, this leads to specific surface values in the range (58---150)  104 cm
1 or (58---150)m2 gm
1 (r ’ 1:0 gm cm
3 ). Void Content of Hompolymers via SAXS Invariant Analysis Figure 23.10 shows a Kratky plot [(Q2 d S=d V(Q) vs Q] for a polyimide sample made from the condensation of pyromellitic-dianhydride and oxydianiline (PMDA-ODA). The integrated area under this curve is the invariant which for a 2-phase system is given by

Q0 ¼

Z

415

1

Q2 d S=dV(Q)dQ

0

¼ 2p2 w1 w2 rT2 [re1
re2 ]2 ,

(23:20)

where w1 , w2 , and re1 , re2 are the volume fractions and electron densities of the two phases, respectively. PMDAODA may be regarded as a 2-phase system consisting of polymer and voids [48], with w1 51 and (1
w1 ) ’ 1. The polymer has a density of 1:4 gm cm
3 and the repeat unit (mass 382) contains 196 electrons, so re2 ¼ 0:43  1024 electrons cm
3 and re1 ¼ 0. From Fig. 23.10 the invariant, ˚
3 , or 0:25  1020 cm
4 , giving a Q0 ¼ 0:25  10
4 cm
1 A void fraction, w1 ’ 8:7  10
5 , which is typical for such materials [48]. An alternative estimate for w may be obtained via Guinier analysis if the voids are reasonably monodisperse, as indicated in Fig. 23.11. Assuming that the voids are spherical, the radius (R) may be obtained from the measured Rg via ˚ . The extrapolated cross section R ¼ (5=3)0:5 Rg ’ 348 A
1 dS=dV(0) ’ 142 cm is given by dS (0) ¼ NP V 2 rT2 w1 w2 [re1
re2 ]2 , (23:21) dV ˚ 3 (or 176  10
18 cm3 ) is where V ¼ 4=3pR3 ¼ 176  106 A the particle (void) volume and Np is the number of particles per unit volume. For w1 51, Np V ’ w1 and Eq. (23.21) gives w1 ’ 5:6  10
5 . The two estimates from the invariant and Guinier analysis are of the same order, and the difference probably results from the Guinier assumption of a relatively monodisperse void distribution. Departures from nonlinearity in the Guinier plot observed at the higher Q-values in Fig. 23.11 may reflect polydispersity effects, and thus the estimate via invariant analysis (which is independent of such assumptions) is probably more accurate.

7 6 0.0040

Rg = 270 Å

5 4

Q2 dS (Q) (cm)−1 Å−2) dW

dS In dW (Q)

0.0035

0.0020

3 2 1 0

0.0015 −1 0

−2 0

0.005

0.010 0.015 Q (Å−1)

0.020

0.025

FIGURE 23.10. Integrated scattering curve for PMDA–ODA imidized at 350 8C.

0

0.4

0.8 1.2 Q 2 x 104 (Å−2)

1.6

2.0

FIGURE 23.11. SAXS Guinier plot for PMDA–ODA Imidized at 350 8C.

416 / CHAPTER 23 23.3.5 Characterization of Multiphase Systems by SANS and SAXS

compared with calculations for two possible morphologies suggested by DSC and TEM analysis:

Blends of High-Density and Linear Low-Density Polyethylenes

A. A fraction of the HDPE-D component segregates during isothermal crystallization and the remainder cocrystallizes with LLDPE-H on cooling.

Equations (23.17)–(23.20) are strictly valid only for two-phase morphologies, and for multiphase systems, an extension of the SAXS invariant analysis may be employed by generalizing Eq. (23.20) and summing over the number of phases involved X Q0 ¼ 2p2 rT2 wi wj [re1
re2 ]2 : (23:22)

B. HDPE-D partially segregates from the LLDPE-H during isothermal crystallization and the remainder also segregates on cooling. A compositionally mixed homogeneous amorphous phase was assumed to surround the crystals in both cases.

i6¼j

Such an analysis has been applied to semicrystalline blends of polycaprolactone and polycarbonate via SAXS [49] and complementary SANS experiments [50] have employed the DB analysis described above [see Sections ‘‘Blends of High-and Low-Density Polyethylene’’ and ‘‘Interpenetrating Polymer Networks’’]. For semicrystalline blends of high density and short-chain branched linear low density polyethylenes (LLDPE), complementary DSC and TEM techniques indicate that the compositions of the various crystals and surrounding amorphous regions are such that the system cannot be described by a two-phase model [51]. The (Q ¼ 0) cross section [dS=dV(0)] cannot be calculated for multiphase systems, though it may be estimated via a ‘‘pseudo two-phase’’ model to a good approximation. For example, with deuterated HDPE-D and protonated LLDPE-H (to provide SANS contrast), the SLD of the HDPE-D crystal is 8:57  1010 cm
2 , whereas the SLDs of the mixed (HDPE-D/LLDPE-H) crystals and amorphous are 0.44 and 0.46 (1010 cm
2 ). Thus, the SANS cross section [dS=dV(0)] can be calculated to a good approximation by grouping the mixed phases into an average background (rav ¼ 0:45  1010 cm
2 ) surrounding pure HDPE-D crystal in a pseudo two-phase model. The SANS invariant may be calculated for a multiphase morphology [Eq. (23.22)] by substituting the neutron scattering length densities (rn ) for the x-ray scattering length density (rT re1 ) and summing over the various phases [51]. The (Q ¼ 0) cross section may be estimated via Eq. (23.17) for the pseudo two phase model. For series of HDPE-D/LDPE-H samples isothermally crystallized from the met at 117 8C, the experimental data are

The experimental and calculated Q0 values are listed in Table 23.4, and for the 18/82 (vol%) blend the calculated Q0 and the experimental data are identical for morphology type A. Similarly, the value of d S=dV(0) calculated from the pseudo two-phase model and Eq. (23.1) is 41:4  103 cm
1 for morphology type A, which agrees closely with the experimental value of 39:3  103 cm
1 . When morphology type B is assumed, the calculated values do not agree with the experimental data for this blend. Thus, SANS supports the idea that predominantly LLDPE-rich blends crystallize isothermally with morphology A, where a fraction of the HDPE-D component segregates during isothermal crystallization and the remainder co-crystallizes with the LLDPE-H on cooling. For the linear-rich, 78/22 blend the agreement between the experimental and calculated Q0 and d S=dV (0) values is closer for morphology type B. SANS indicates that for this blend the intensity and invariant conforms a more segregated morphology of the linear and branched components than for the LLDPE-rich blend. For 50/50 blends, the measured and calculated values of Q0 and d S=dV(0) indicate an intermediate between the A and B types, where part of the HDPE component that crystallizes on cooling is cocrystallized with the branched LLDPE and part crystallizes as pure HDPE [51]. In view of the fact that the experiments are independently calibrated with no arbitrary fitting factors in the intensity scale and that the crystal/amorphous compositions are obtained from DSC, the general agreement with the SANS data is excellent. Thus, the two-phase approximation is able to reproduce not only the SANS invariant, but also the (Q ¼ 0) cross section with good accuracy.

TABLE 23.4. Measured and calculated cross sections and invariants for HDPE-D/LDPE-H blends isothermally crystallized at 117 8C. dS=dV(Q ¼ 0) 10
3 (cm
1 ) expt.

˚
3 Q0 cm
1 A expt.

Proposed morphology

dS=dV(Q ¼ 0) 10
3 (cm
1 ) calc.

Q0 cm
1 A˚
3 calc.

18/82

41.4

0.009

78/22

36.1

0.0158

A B A B

39.2 58.1 18.4 33.1

0.009 0.013 0.007 0.013

Composition (% volume) HDPE-D/LLDPE-H

SMALL ANGLE NEUTRON AND X-RAY SCATTERING / Carbon-filled polyethylenes Other multiphase systems involving polymers include composite materials produced by mixing with filler particles to modify their mechanical properties or conductivity. For example, carbon black has been extensively used as a reinforcing filler in a number of applications such as automotive tires and can also be blended with insulators such as semicrystalline polyethylene (PE) to produce conductive composites used in electrical products. When the concentration of carbon black at room temperature is above the percolation threshold, the composite is conducting. However, at higher current loading, the system heats and expands the polyethylene (PE) matrix, and when this approaches the percolation threshold it becomes highly resistive [52]. This results in a lower current and the device cools to its original state, so a mixture of carbon black and polyethylene acts as a resettable fuse [53]. For materials with particle sizes in the range ˚ , both SANS and SAXS may be used to ex 10---1,000 A

plore the morphology and a combination of these techniques can provide greater insight than either technique in isolation. For example, combined SAXS/SANS studies of carbon-PE composites [52] suggested the presence of a third phase (voids) and subsequent experiments using the contrast options available from deuterium-labeling of the PE-matrix were designed to quantify the void fraction and its variation with temperature [53]. Figure 23.12 illustrates schematically the contrast options available from the combination of SAXS/SANS and deuterium labeling in the study of the three-phase system (polymer, carbon black, and voids), and makes it clear that one cannot resolve void morphology solely with SAXS. However, if one examines a normal composite (with protonated or H-labeled polymer) via SANS, the sample is essentially two-phase because the neutron scattering length densities of polyethylene and voids are virtually identical (see Table 23.5). For such a two-phase system, it has been shown that the morphology may be described by an extension of the DB theory [45,46] and Eq. (23.17) is modified to

dS 8pa31 w1 w2 f [rn1
rn2 ]2
Q2 a22 3=2 3 2 (Q) ¼ , þ p a w w (1
f )[r
r ] exp 1 2 n1 n2 2 2 2 dV 4 (1 þ Q2 a1 )

where a2 is a second correlation length characterizing long range structural features. (1
f ) and f are the fractional ˚ ) and contribution of long ranged (a2  500---860 A ˚ ) components of the strucshort ranged (a1  130---290 A tural model, respectively, [46,52,53]. As before, w1 and w2 are the volume fractions and rn1 and rn2 are the neutron scattering length densities of the effectively two-phase system of carbon (SLD ¼ 6:4  1010 cm
2 ) and polyethylene/ voids (SLD  0); see Table 23.5. Typically f is in the range 0:82 < f < 0:97 for 0:27 < w < 45:5 vol% and for f ¼ 1, Eq. (23.23) reduces to Eq. (23.17). Thus, a ‘‘pseudo twophase model’’ may be again applied to this three-phase composite material, as in section ‘‘Blends of High-Density and Linear Low-Density Polyethylenes’’, and it was shown [52] that Eq. (23.23) gives excellent fits to the SANS data over a wide range of carbon black compositions. Table 23.6 compares the measured and calculated cross sections at (Q ¼ 0) and it may be seen that the discrepancies for any given concentration are in the range + 25%. Such fluctuations are not unexpected in view of the extreme sensitivity of the cross section to the fitted correlation lengths, both of which are cubed to calculate dS=dV(0). However, the overall agreement is excellent, as there is no systematic distortion and the deviations are both positive and negative in virtually equal proportions. If one blends carbon black with deuterated polyethylene, it may be seen from Fig. 23.12, that presence of voids is highlighted within the carbon black/d-polyethylene matrix. Through a combination of SAXS/SANS experiments, one can extract information about void size and quantity [53]

417

(23:23)

using the theoretical formalism developed by Wu [54] to model microvoids in composite materials. Typical ˚ in size were measvoid concentrations  2 vol%, 400---500 A ured at room temperature in composite materials containing 30–40 vol% carbon. These voids decrease significantly in concentration during the melt transition however, dropping by an order of magnitude to  0:2 vol%. This decrease might be expected and suggests that the polyethylene domains

Contrast Options for SAXS and SANS Studies of Carbon-Polythylene Composite Materials

SAXS contains contributions from all three phases

SANS from carbon in H-PE gives structure of carbon alone

SANS from carbon in D-PE gives structure of voids alone

FIGURE 23.12. Contrast options for SAXS and SANS studies of carbon–polyethylene composite materials.

418 / CHAPTER 23 TABLE 23.5. Neutron (rn ) and X-ray (rx ¼ rT re1 ) scattering length densities of components of carbon-polyethylene composite materials. Species

Density, r(gm cm
3 )

X-Ray scattering length density, rx (1010 cm
2 )

Neutron scattering length density, rn (1010 cm
2 )

1.92 0.0 0.95 1.08

16.2 0.0 9.12 9.12

6.4 0.0
0.34 8.13

Carbon black Voids Polyethylene Polyethylene-d4

grow at the expense of the voids as the temperature is brought above the melting point. 23.3.6 Isotope Effects in SANS

where S (Q) is the structure factor, which contains information regarding both molecular architecture and thermodynamic interactions. In the mean field random phase approximation [59], S (Q) is given by S
1 (Q) ¼[wA NA PA (QRgA )]
1 þ

SANS studies of deuterium labeled polymers were based initially on the assumptions that the molecular configurations and interactions are independent of deuteration, and the interaction parameter between D-labeled and unlabeled segments of the same species wHD is zero. Nevertheless, there have been several experimental observations which suggested that isotopic substitution does influence polymer thermodynamics and Buckingham and Hentschell [55] suggested that this might arise from a finite interaction parameter (wHD  10
4
10
3 ) between H- and D-labeled segments. Subsequently, SANS was used to measure wHD for a range of isotopic mixtures [56–58], to delineate the circumstances under which demixing can occur. For a blend of two polymer species (A and S), one of which (A) is deuterium labeled, the coherent cross section (after subtracting the incoherent background) is given [8,13] by dS (Q) ¼ V
1 (aH
aD )2 S(Q), dV

(23:24)

TABLE 23.6. Comparison of measured and calculated values of the absolute SANS cross section at Q ¼ 0 for carbon–polyethylene composite materials. Vol % carbon black 44.5 39.5 34.8 26.3 16.5 12.4 5.6 1.08 0.53 0.27

dS=dV(Q ¼ 0) 10
3 (cm
1 ) measured

dS=dV(Q ¼ 0) 10
3 (cm
1 ) calculated

98 159 177 237 317 247 178 49 32 8

77 118 166 200 380 345 182 50 24 11

(23:25)

[(1
wA )NS PS (QRgS )]
1
2w ,

where wA ¼ wD is the volume fraction of the A species and RgA , RgS , NA and NS are the radii of gyration and polymerization indices of the two species. The intra-chain functions PA (Q) and PS (Q) are represented by Debye functions [60], based on the assumption of a Gaussian distribution of chain elements. P(Q) ¼ 2[R2g Q2 þ exp (
R2g Q2 )
1]=(R4g Q4 ): (23:26) By regarding the H- and D- molecules as different species with volume fractions wH and wD , the random phase approximation [Eq. (23.19)] may be fitted to the data with xHD as the only adjustable parameter [56–58]. Complementary experiments on polystyrene [61] and poly (dimethyl siloxane) [62] confirm the existence of a universal isotope effect, arising from the small differences in volume and polarizability between C–H and C–D bonds [63]. Table (23.7) lists typical values of the isotopic interaction parameter for various polymers in the concentration range 0:2 < wD < 0:8,

TABLE 23.7. Isotopic interaction parameter for various polymers. wD

T 8C

104 xHD

Polystyrene

0.5

160

1,4 Polybutadiene 1,2 polybutadiene (polyvinyl ethylene) 1,2 Polybutene (Polyethyl ethylene) Polydimethylsiloxane Polyethylene

0.31 0.5

50 47

1.8a 2.3b 7.2c 6.8d

47 296 160

8.8d 17e 4.0f

Polymer

a

Reference [56]. Reference [61]. c Reference [56]. d Reference [58]. e Reference [62]. f Reference [65]. b

0.5 0.5

SMALL ANGLE NEUTRON AND X-RAY SCATTERING / 1,400 1,200

361K 274 248

1,000 dS (Q) (cm−1) dW

where wHD has been shown to be relatively independent of concentration [58,61]. The above results raise the important question of how SANS studies are influenced by isotope effects. As explained earlier, initial SANS experiments on polymers relied on analogies with LS, where the limit of zero concentration was required to eliminate inter-chain scattering. Under such conditions, the isotope effect contributes almost insignificantly to the intensity, and this may be illustrated by calculating d S=dV(0) via Eqs. (23.24) and (23.25) for the sample of 5.0 wt% PSD in PSH as in Section 23.3.1. The inclusion of an isotopic interaction parameter wHD ¼ 1:8  10
4 changes d S=dV(0) to 17:5 cm
1 compared to 17:4 cm
1 calculated from Eq. (23.11) in the absence of isotope effects. Upon recognizing that information on chain statistics could equally well be obtained from concentrated isotopic mixtures, many experiments were conducted under such conditions in order to enhance the intensity. It is under these conditions that isotope-induced segregation effects are manifested. In the bulk state many of the systems studied are solids at room temperature and have been exposed for only a limited time in the liquid state, as for example during melt pressing. For polybutadiene, with a glass transition temperature below
90 8C, isotopic blends are liquid at room temperature, and this facilitates the attainment of equilibrium. Hence, isotope effects can be particularly dramatic in this system and Fig. (23.13) shows the scattering cross section of mixtures of deuterated (ND ¼ 4,600) and protonated (NH ¼ 960) as a function of temperature. It can be seen that the extrapolated zero-Q cross section exceeds by large factors the value it would have ( 100 cm
1 ) if the H–D interactions were negligible. For sufficiently high molecular weight, this system will even phase separate [64], as will other isotopic mixtures (e.g., polyethylene [65]). Thus, it is prudent to evaluate future experiments, based on measured values of xHD (Table 23.7), and to check for excess scattering. This is best accomplished by calibrating data on an absolute

419

800

ND = 4,600 NH = 960

600 400 200 x=0 0

0

0.004

0.008 0.012 Q (Å−1)

0.016

dS FIGURE 23.13. dV (Q) VS Q for blend of 69 vol% protonated and 31% deuterated 1, 4-Polybutadiene at the critical composition. The curves were obtained from the homogeneous mixture scattering function by adjusting xHD (one adjustable parameter). Reprinted with permission from F. S. Bates, G. D. Wignall and W. C. Koehler, Phys Rev. Lett., 55, 2425 (1985). Copyright (1985) American Physical Society.

scale and comparing the measured and theoretical intensities [66]. Table 23.8 summarizes the formulae for the scattering parameters defined in the above examples. Scattering techniques have been one of the main sources of structural information since the beginnings of polymer science. Over the past two decades, SANS has been extensively applied to complement existing scattering methods (SAXS, WAXS, LS etc.) and the above examples illustrate the new information which it has provided. The complementary aspects of neutron, light, and X-ray scattering, as applied to polymers and colloids, have been surveyed in a current volume edited by Lindner and Zemb [12]. A wide range of applications of neutron scattering to study polymer structure have been described by Higgins and Benoit [29], and Gabrys [67].

TABLE 23.8. Scattering parameters and formulae. Scattering parameter Forward (Q ¼ 0) cross section, dS=dV(0) Forward (Q ¼ 0) cross section, dS=dV(0) R1 Invariant, Q0 0 Q 2 dS=dV(Q)dQ Porod constant P ¼ Q 4 dS=dV(Q) a

Particular assumptions of Modela Guinier model (relatively monodisperse particles) Debye–Bueche model (randomly intermixed phases)

Formula

Equation

dS dV (0)

¼ Np Vp2 (rn1
rn2 )2 (SANSb )

(23.15)

dS dV (0)

¼ 8pa3 w1 w2 (rn1
rn2 )2 (SANSb )

(23.17)

Q0 ¼ 2p2 w1 w2 rT2 [re1
re2 ]2 (SAXSb ) P ¼ 2p(rn1
rn2 )2 S=V (SANSb )

(23.20) (23.18)

The Guinier, Debye–Bueche, Invariant and Porod analyses are all based on the assumption of well defined phases with sharp interfacial boundaries. In addition, the Guinier approach is based on the assumption that the length distribution function (23.15), or probability P00 (r ) that a randomly placed rod (length, r) can have both ends in the same scattering particle (phase) is zero beyond a well defined limit. For example, for monodisperse spheres, diameter D, P00 ¼ 0, for r > D. In the Debye–Bueche model, P00 has no cut off and approaches zero via an exponential correlation function only in the limit r ! 1 [45,46]. b For SAXS, the neutron scattering length density (rn ) is replaced by the product of the electron density (re ) and the Thompson scattering length (rT ¼ 0:282  10
12 cm), and vice-versa.

420 / CHAPTER 23 ACKNOWLEDGMENTS The author wishes to thank W. L. Wu and P. A. Egelstaff for permission to include Table 23.2 and Fig. 23.1, respectively. This research was supported by the Division Materials Science, US Department of Energy under contract DEAC05-00OR22725 with the Oak Ridge National Laboratory, managed by UT-Battelle, LLC.

REFERENCES 1. P. J. Flory, Principles of Polymer Chemistry, Cornell Press, Ithaca, New York p. 22 (1953). 2. L. E. Alexander, X-Ray Diffraction Methods in Polymer Science, Krieger Publishing Co. Inc., London (1969). 3. Chapter IV, Polymer Handbook, ed. J. Brandrup and E. H. Immergut, Wiley Interscience, New York (1975). 4. Chapter 3, International Tables for X-ray Crystallography, ed. C. H. Macgillavry and G. D. Rieck, Kynoch Press, Birmingham (1968). 5. A. Avitabile, R. Napolitano, B. Pirrozzi, K. D. Rouse, M. W. Thomas and B. T. M. Willis, J. Polym. Sci., Polym. Lett. Ed., 13, 351 (1975). 6. M. Stamm, J. Polym. Sci., Polym. Phys. Ed., 20, 235 (1982); M. Stamm, E. W. Fischer, M. Dettenmaier, and P. Convert, Discuss. Faraday Soc. 68, 263 (1979). 7. G. D. Wignall, Chapter 4 in Applied Fiber Science ed. F. W. Happey, Academic Press, London (1978). 8. G. D. Wignall, Encyclopedia of Polymer Science and Engineering, ed. M. Grayson and J. Kroschwitz, Wiley and Sons, New York, 10, 112 (1978). 9. G. Allen and J. S. Higgins, Rep. Prog. Phys. 36, 1073 (1973). 10. K. Nicholson, Contemp. Phys. 22, 451 (1981). 11. J. S. Higgins and A. Maconnachie, Chapter 22 in Neutron Scattering, eds. K. Skold and D. L. Price, Academic Press, New York, p. 287 (1987). 12. Neutron, X-Ray and Light Scattering, ed. P. Lindner and T. Zemb, North-Holland Delta Series, Elsevier Publishers, New York, (1991). 13. G. D. Wignall, Chapter 7, p. 313 in The Physical Properties of Polymers, ed. J. E. Mark, ACS Books (1993). 14. A. Guinier and G. Fourne`t, Small-Angle Scattering of X-Rays, John Wiley, New York (1955). 15. O. Glatter and O. Kratky, Small-Angle X-Ray Scattering, Academic Press, London (1982). 16. M. G. Huglin, Light Scattering from Polymer Solutions, Academic Press, London (1982). 17. G. Hadziioannou and R. S. Stein, Macromolecules, 17, 567 (1984); Macromolecules, 17, 1059 (1984). 18. R. G. Kirste, W. A. Kruse and K. Ibel, Polymer, 16, 120 (1975). 19. J. Schelten, G. D. Wignall, D. G. H. Ballard and G. W. Longman, Polymer, 18, 1111 (1977). 20. J. B. Hayter and J. Penfold, Colloid Polym. Sci. 261, 1022 (1983). 21. J. S. Higgins and R. S. Stein, J. Appl. Cryst. 11, 346 (1978). 22. V. E. Turchin, Slow Neutrons, Israel Program for Scientific Translations, Jerusalem (1965). 23. G. E. Bacon, Neutron Diffraction, Clarendon Press, Oxford, England (1971). 24. R. W. James, The Optical Principles of the Diffraction of X-rays, Bell, London (1958). 25. A. Maconnachie, Polymer, 25, 1068 (1984). 26. L. D. Coyne and W. Wu, Polym. Commun. 30, 312 (1989). 27. J. P. Cotton, B. Farnoux, G. Jannink, J. Mons and C. Picot, C. R. Acad. Sci. (Paris), 275, 3C, 175 (1972). 28. J. S. King, W. Boyer, G. D. Wignall and R. Ullman, Macromolecules, 18, 709 (1985). 29. J. S. Higgins and H. Benoit, Neutron Scattering from Polymers, Clarendon Press, Oxford (1994). 30. B. H. Zimm, J. Chem. Phys. 16, 157 (1948).

31. J. B. Hayter in V. Degiorgio and M. Corti, eds. Proceedings of Enrico Fermi School of Physics Course XC, North Holland, Amsterdam (1985), p. 59. 32. G. D. Wignall, D.G.H. Ballard and J. Schelten, Eur. Polym. J., 10, 861 (1974). 33. J. Schelten, D.G.H. Ballard, G. D. Wignall, G. Longman, and W. Schmatz, Polymer, 27, 751 (1976). 34. T. P. Russell, J. S. Lin, S. Spooner and G. D. Wignall, J. Appl. Cryst. 21, 629 (1988). 35. M. P. Grancio and D. J. Williams, J. Polym. Sci. (Al), 8, 2617 (1970). 36. M. P. Wai, R. A. Gelman, M. G. Fatica, R. H. Hoerl and G. D. Wignall, Polymer, 28, 918 (1987). 37. L. W. Fisher, S. M. Melpolder, J. M. O’Reilly, V. R. Ramakrishnan and G. D. Wignall, J. Colloid Interface Sci. 123, 24 (1988). 38. J. W. Goodwin, R. H. Ottewill, N. M. Harris and J. Tabony, J. Colloid and Polym. Sci. 78, 253 (1980); J. Colloid Interface Sci. 78, 253 (1980). 39. K. Alexander, D. J. Cebula, J. W. Goodwin, R. H. Ottewill and A. Parentich, Colloids Surf. 7, 233 (1983). 40. D. J. Cebula, J. W. Goodwin, R. H. Ottewill, G. Jenkin and J. Tabony, Colloid and Polym. Sci. 261, 555 (1983); Discuss. Faraday Soc. 76, 37 (1983). 41. Lord Rayleigh, Proc. R. Soc. London A. 84, 24 (1911). 42. G. D. Wignall, V. R. Ramakrishnan, M. A. Linne, A. Klein, L. H. Sperling, M. P. Wai, R. A. Gelman, M. G. Fatica, R. H. Hoerl, L. W. Fisher, S. M. Melpolder and J. M. O’Reilly, Mol. Cryst. Liq. Cryst. 180A, 25 (1990). 43. R. G. Alamo, J. D. Londono, L. Mandelkern, F. C. Stehling and G. D. Wignall, Macromolecules, 27, 411 (1994). 44. G. D. Wignall, J. D. Londono, J. S. Lin, R. G. Alamo and L. Mandelkern, Macromolecules, 28, 3156 (1995). 45. P. Debye and A. M. Bueche, J. Appl. Phys. 20, 518 (1949). 46. P. Debye, H. R. Anderson and H. Brumberger, J. Appl. Phys. 28, 679 (1957). 47. A. M. Fernandez, L. H. Sperling, and G. D. Wignall, Multicomponent Polymer Materials, ACS Advances in Chemistry Series 211 (1985). 48. T. P. Russell, Polym. Eng. Sci. 24, 345 (1984). 49. Y. W. Cheung, R. S. Stein, J. S. Lin and G. D. Wignall, Macromolecules, 27, 2520 (1994). 50. Y. W. Cheung, R. S. Stein, G. D. Wignall and H. E. Yang, Macromolecules, 26, 5365 (1993). 51. G. D. Wignall, R. G. Alamo, J. D. Londono, J. S. Lin, L. Mandelkern, M. H. Kim and G. M. Brown, Macromolecules, 33, 551–561 (2000). 52. G. D. Wignall, N. R. Farrar, and S. Morris, J. Mater. Sci. 25, 69–77 (1990). 53. D. W. Marr et al., Macromolecules, 30, 2120–2130 (1997). 54. W.-L. Wu, Polymer, 23, 1907–1912 (1982). 55. A. B. Buckingham and H. G. E. Hentschel, J. Polym. Sci. Polym. Phys. Ed. 18, 853 (1984). 56. F. S. Bates, G. D. Wignall and W. C. Koehler, Phys. Rev. Lett. 55, 2425 (1985). 57. F. S. Bates and G. D. Wignall, Macromolecules 19, 932 (1986). 58. F. S. Bates, M. Muthukumar, G. D. Wignall and L. J. Fetters, Macromolecules, 89, 535 (1988); Macromolecules, 21, 1086 (1988). 59. P. G. deGennes, Chapter 5 in Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca (1979). 60. P. Debye, J. Appl. Phys. 15, 338 (1944). 61. D. Schwahn, K. Hahn, J. Streib and T. Springer, J. Chem Phys. 93, 8383 (1989). 62. A. Lapp, C. Picot and H. Benoit, Macromolecules, 18, 2437 (1985). 63. F. S. Bates and G. D. Wignall, Phys. Rev. Lett. 57, 1429 (1986). 64. F. S. Bates, S. B. Dierker and G. D. Wignall, Macromolecules, 19, 1938 (1986). 65. J. D. Londono, A. H. Narten, G. D. Wignall, K. G. Honnell, E. T. Hsieh, T. W. Johnson and F. S. Bates, Macromolecules, 27, 2864 (1994). 66. G. D. Wignall and F. S. Bates, J. Appl. Cryst. 20, 28 (1987). 67. Applications of Neutron Scattering to Soft Condensed Matter (Edited by Barbara J. Gabrys), Gordon and Breach Science Publishers (2000). 68. J. M. O’Reilly, D. M. Teegarden and G. D. Wignall, Macromolecules 18, 2747 (1985). 69. B. Crist and G. D. Wignall, J. Appl. Cryst. 21, 701 (1988).

CHAPTER 24

Mechanical Properties Witold Brostow Department of Materials Science and Engineering and Department of Physics, University of North Texas, POBox 305310, Denton, TX 76203-5310, USA; http://www.unt.edu/LAPOM/; [email protected]

24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8

Relaxational and Destructive Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fracture Mechanics for Polymeric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasistatic Testing and Transient Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impact Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscoelasticity and Dynamic Mechanical Testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables of Selected Mechanical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423 426 429 436 438 440 442 442 442 444

2. Can we get a material or component with better properties?

24.1 RELAXATIONAL AND DESTRUCTIVE PROCESSES

While both questions are often asked simultaneously, the second question deals with development of new materials and will not be considered per se in this Chapter; some answers are provided in Chapter 41 on polymer liquid crystals. The first question shows that failure is related to prediction of performance under given service conditions, and this is the way we are going to tackle this problem. More specifically, we need prediction of long-term performance from short-term tests, and this will be one of the leitmotivs of the present chapter. The subject of this chapter is a vast one. There exist entire books devoted to it, including classical books by Ferry [1] and Aklonis and McKnight [2] as well as more recent ones [3,4].

24.1.1 Introduction; Service Performance and Reliability Service performance and reliability constitute the bottom line of the entire polymer science and engineering. Since this statement might appear an exaggeration, let me immediately explain why. Synthesis of macromolecules is of interest primarily to synthetic chemists; polymer rheology is of interest to polymer rheologists; rotational injection molding is of interest to rotational injection molders; and so on. There is, however, an exception: reliability of polymeric materials and components is of interest to everybody—polymer scientists, polymer engineers, and all laymen including those who do not even know what the word ‘‘polymer’’ means. A very good example provides a little girl playing with a plastic doll. If the doll will break into pieces, the girl will certainly cry first. Somewhat later, however, some captains of industry might cry also. Given this situation, let us formulate two highly pertinent and often asked questions:

24.1.2 The Chain Relaxation Capability (CRC) Polymeric materials are all viscoelastic. The ‘‘face’’ each polymer shows to the observer—elastic, viscous flow, a combination of both—depends on the rate and duration of force application as well as on the nature of the material and external conditions including the temperature T. We discuss the nature of viscoelasticity below and additionally in Section 5. In general, properties of viscoelastics depend on time, in contrast to metals and ceramics.

1. Will a given polymeric material or component serve for a reasonable amount of time, or will it fail prematurely? 423

424 / CHAPTER 24 To get a clear picture of the problem we are about to tackle, let us return to the girl with her plastic doll. Playing with the doll, the girl applies forces with various duration, direction(s) and application rate(s). For instance, the girl applied a tensile force to the head and both legs of the doll. The doll is a physical system which thus received energy U0 from outside. Important for the girl—and for us—is the question number 1 formulated above. Will the energy U0 be spent on destruction and eventual fracture of the doll, or will it get somehow dissipated and the doll will ‘‘live long’’? We can write a general equation [5–7] U ¼ U0
Ub
Ur ,

(24:1)

here U is the energy furnished from outside which at a given time has not yet been spent one way or the other; Ub (b for bond breaking) at the same time has been spent on destructive processes (such as crack formation or propagation); Ur at the given time has been dissipated, that is spent on nondestructive processes. Dissipation in a viscoelastic material is largely related to relaxational processes; the subscript r stands for relaxation. The quantities in Eq. (24.1) may refer to the material as a whole, but it is usually convenient to take them per unit weight of the polymer such as 1 g. Ur is quite important. It will be related soon to the chain relaxation capability (CRC) which has been defined [5–7] as follows: CRC is the amount of external energy dissipated by relaxation in a unit of time per unit weight of polymer. In the following we shall use the abbreviation CRC for the concept and the symbol UCRC for the, respective, amount of energy. Thus, at a given time t ðt Ur ¼ UCRC dt: (24:2) 0

The main reason why the concept of CRC is so useful is the following fact: it takes approximately 1,000 times more energy to break a primary chemical bond such as a carbon–carbon bond in a carbonic chain (what contributes to Ub and to crack propagation) than to execute a conformational rearrangement around the same bond. This is the basis of the following key statement [5–7]: Relaxational processes have priority in the utilization of external energy. The excess energy which cannot be dissipated by such processes goes into destructive processes. Nature is very kind to us! A viscoelastic material will relax rather than fracture—as long as it can go on relaxing. Unless there is a high concentration of external energy at a particular location, and as a consequence a number of primary bonds will break starting a crack, that energy will be dissipated. In contrast to nonchain materials, when we pull at a polymeric chain we gradually engage all segments of it; this by itself lowers the probability of local concentration of external energy and of destruction. Of course, there exist local energy concentrators and we shall discuss them below. There exist a number of constituents of CRC; we have just named one of them, but let us list them together: 1. Transmission of energy across the chain producing intensified vibrations of the segments.

2. Transmission—mainly by entanglements but also by segment motions—of energy from the chain to its neighbors. 3. Conformational rearrangements (such as cis into trans in carbonic macromolecules) executed by the chains. 4. Elastic energy storage resulting from bond stretching and angle changes. 5. Phase transformation toughening first observed by Kim and Robertson [8] and also studied by Karger-Kocsis [9]. Incidentally, fairly often the penultimate factor is excluded—with bad consequences for models based on such an assumption. 24.1.3 Correspondence Principles Given the conclusions from the previous section, we naturally ask: when will a given polymeric material or component have high CRC—so that we can expect a reasonable service time? We need to answer this question before dealing with specific properties and specific classes of materials. Paul Flory has shown how free volume vf is important for thermophysical properties of materials—and not only polymeric ones [10,11]; see also a chapter by Orwoll in this Handbook [12]. There are also seminal papers by Litt and Tobolsky [13] and Tschoegl [14–16] showing importance of vf for mechanical properties of viscoelastic materials. Consider now our CRC from this point of view. It is easy to envisage that the larger vf is, the larger is the maneuvering ability of the chains—what means the higher is CRC. Using specific quantities (typically per 1 g), we write v ¼ v þ vf :

(24:3)

Here v is the total specific volume and v is the characteristic (hard-core, incompressible) volume. The last two names are based on the concept of ‘‘squeezing out’’ the whole free volume by applying a very high pressure so that only v remains. Instead of free volume, some people work with the reduced volume v~ ¼ v=v ¼ 1 þ vf =v :

(24:4)

Equations such as (3) or (4) are not usable until a specific equation of state of the general form v~ ¼ v~(P~, T~) or P~ ¼ P~(~ v, T~) is assumed. Here P is the pressure and we need two more reduced quantities: P~ ¼ P=P and T~ ¼ T=T  :

(24:5)

The idea of reduced quantities goes all the way back to Johannes D. van der Waals in the eighteenth century. Thus, an equation of state requires three reducing quantities, v ,P , and T  . We have found repetitively good results using the Hartmann equation of state [17–19] P~v~5 ¼ T~3=2
ln v~:

(24:6)

MECHANICAL PROPERTIES Since experiments are often conducted at the atmospheric pressure P  0:1 J cm
3 , then the term containing P~ in Eq. (24.6) is negligible, and we have simply v~ ¼ exp [T~3=2 ]:

(24:7)


3

The pressure unit of J cm has been used for instance by Flory [11] and in contrast to Pa saves our time in calculations. Fortunately 1 J cm
3 ¼ 1 MPa ¼ 1 MN m
2 ¼ 107 erg cm
3 ¼ 107 dyne cm
2 ¼ 10 bar ¼ 145:04 psi ¼ 9:86923 atm.; the last number depends on the geographic location. Given Eqs. (24.6) and (24.7), we need to evaluate the characteristic parameters v , T  and if we deal not only with the atmospheric pressure also P . One can use the thermomechanical analysis (TMA) in the expansion mode to determine at the atmospheric pressure the dependence of specific volume v on temperature T. By fitting the experimental results to Eq. (24.7) one obtains the characteristic parameters v and T  . Zoller and coworkers have long ago developed a so-called Gnomix apparatus which performs full P–V–T determination [20]. There are several machines around the world based on the Zoller invention. We have used a Gnomix to advantage for organic polymers [21,22] as well as for inorganic ones [23]. One then represents experimental results by Eq. (24.6) and one calculates by a least-squares procedure the parameters P , v , and T  . To connect free volume to mechanical properties, we now need the classical Doolittle equation ln h ¼ A0 þ Bv =vf ,

(24:8)

where h is the viscosity. The connection can be made through correspondence principles which now we are going to discuss. Consider first a conformational rearrangement in a polymeric chain so fast that one cannot record it at room temperature. Clearly the total volume decreases when the temperature decreases, and along with it the free volume becomes smaller too. Thus, we can reach a temperature low enough to ‘‘catch’’ the process under investigation. This idea works also in the opposite direction. Instead of conducting experiments for 100 years at the ambient temperature, we can go to a higher temperature, thus produce higher free volume vf in the material, and ‘‘catch’’ within, say, 10 hours the same series of events. This is the basis for the time–temperature correspondence. Clearly we now have what we have been looking for: the capability to predict long-term behavior from short term tests. One performs experiments at a series of temperatures. There exists a temperature of particular interest, for instance 20 8C. There is also at least one parameter of particular interest, such as the tensile compliance D(t). In elastic materials we simply have D(t) ¼ «(t)=s ¼ 1=E, where E is the tensile modulus. However, our strain depends on time t; at constant T and s we have generally D(t) ¼ «(t)=s ¼ 1=E(t):

(24:9)

/

425

We now create a large diagram of D ¼ D(t) (or more often of log D ¼ log d(log t). We begin with results for 20 8C and also include isothermal results for all other temperatures. Then, without moving the curve for 20 8C, we shift results for all other temperatures so that they would form a single curve. We shall show below examples of such diagrams, often called master curves, an approach advocated for a long time by Ferry and his coworkers [27,1]. Each D(t) isotherm is moved left or right by a distance aT called the shift factor; clearly aT is different for each temperature. The whole procedure is also known as the method of reduced variables and means that D(t, T; s ¼ const:) ¼ D(t=aT , Tref ; s ¼ const:): (24:10) Here Tref (often also denoted by T0 ) is the temperature to which the master curve pertains. Thus, in our case Tref ¼ 20  C while in general aT (Tref ) ¼ 1. Changing the temperature is not the only option. By varying stress we can also change the free volume. Time– stress correspondence has been demonstrated experimentally already in 1948 by O’Shaughnessy [24]. Little attention has been paid to it, except for work in Latvia summarized by Goldman [25]. Only in 2000 an equation which makes possible quantitative predictions has been developed [26]. We can also apply an oscillating (typically sinusoidal) force to a polymeric component. If the frequency n of the oscillations is low, the chains will be able to adjust better to the externally imposed field, just as they do at higher temperatures. The inverse is true as well: high frequencies will give little opportunity for such rearrangements—as if the free volume and the temperature were low. Thus, we have time–frequency correspondence. We can write a series of approximate proportionalities [7] CRC  vf  T
n  r
1

(24:11)


1

Here r ¼ n is the mass density. The correspondence principles allow us to achieve our goal: prediction of long-term mechanical properties—and thus performance and reliability—from short-term tests. It is possible to predict behavior for, say, 16 decades of time from experiments each of which was made over four decades only; examples will be given below. It is easy to see that, when using the time–temperature correspondence, essential is the capability to predict the temperature shift factor aT (T). Similarly, when using the time–stress correspondence one needs the stress shift factor as (s). Starting from the Doolittle equation (24.8), it was possible to obtain a general equation [26] ln aT ,s ¼ AT ,s þ ln Tref =T þ ln [v(T, s)=vref ] þ B=(~ v
1) þ C(s
sref ):

(24:12)

Here vref pertains to the stress level of interest and is thus similar to Tref . If we assume a constant stress level, we obtain an equation which allows us to apply the time– temperature correspondence:

426 / CHAPTER 24 ln aT ¼ AT þ B=(~ v
1):

(24:13)

Similarly, if we assume a constant temperature and perform experiments at several stress levels, from Eq. (24.12) we obtain ln as ¼ As þ ln Tref =T þ ln [v(s)=vref ]þ B=(~ v
1) þ C(s
sref ):

(24:14)

Later in this Chapter we shall show applications of these concepts. Before doing so, however, we need to deal with the essential concepts of fracture mechanics.

incision with a pair of scissors on one side of the sheet led to success. The incision was in fact a notch—and created stress concentration defined by Eq. (24.15). Equation (24.15) corresponds to our intuitive notions about the deterioration produced by a crack. The deeper the crack is (h larger) the more ‘‘evil’’ it can produce. The more blunt the crack is (less sharp, larger r), the more ‘‘benign’’ it will turn out to be when external forces ‘‘attack’’ the component. 24.2.2 Stress Intensity Factor

24.2 FRACTURE MECHANICS FOR POLYMERIC MATERIALS 24.2.1 Stress Concentrators and Stress Concentration Factor As noted in the beginning of this Chapter, fracture is the bottom line of polymer science and engineering, and indeed of the entire materials science and engineering. As a result of the processing procedures used, plus handling in transport, etc., polymeric materials and components exhibit structure imperfections at various levels. Thus, there exist knit lines: areas in injection-molded parts of thermoplastics in which separate polymer melt flows arise, meet, and then to some extent—but not quite—combine together during manufacturing. Consequences of the presence of knit lines on mechanical properties are discussed by Criens and Mosle´ [28]. Due to the presence of crazes, scratches, cracks and other imperfections, mechanical properties of real polymeric materials are not as good as they theoretically could be. In this section we shall deal particularly with stress concentrators such as cracks (which appear although we did not want them) and notches (which are well-defined cracks introduced deliberately). The deteriorating effects of cracks and notches on material properties are represented by the stress concentration factor Kt ¼ 1 þ 2(h=r)1=2 :

(24:15)

Here h is the depth (length) of the crack or notch, or one-half of the length of the major axis in an elliptical hole; r is the radius of curvature at the tip of the notch, or at each end of the major axis of an elliptical crack. The name stress concentration factor is very appropriate. Consider again a tensile test with the stress s applied to the ends of the specimen (for details see below Section 3). The lines of force applied to these ends cannot go through the air; they must go through the material, and thus around the crack. As a consequence, when the lines meet (or separate, depending on the direction) at the crack tip, that tip is subjected not to the stress s, but to the stress s  Kt . The phenomenon is well known to anybody who wanted to make two smaller sheets from a plastic sheet and found that his or her own hands are not strong enough for this operation. However, a small

To account for differences on the loading modes (tensile, shear or tearing), a somewhat different measure of the ‘‘evil’’ produced by a crack or notch called the stress intensity factor is used Kl ¼ a p1=2 sh1=2 :

(24:16)

Kl characterizes the stress distribution field near the crack tip; the subscript Roman one, I, refers to the opening or tensile mode of crack extension; a is a geometric factor appropriate to a particular crack and component shape; the remaining symbols are the same as in Eq. (24.15). Unfortunately, Kt and Kl have similar symbols, similar names, and are expressed in terms of the same quantities. However, our effort to change this situation would largely be wasted. For an infinite plate in plane stress, the geometric factor a ¼ 1. Plane stress means that the stress sz along the z axis perpendicular to the plane surface is equal to zero; in practice this is not exactly true, but represents a reasonable approximation. For other geometries there exist tabulations of a values [29]. 24.2.3 Griffith’s Theory of Fracture Entire books have been written on fracture of polymers, so here we shall quote the most important results. We go back to the story of the girl with her plastic doll. Griffith [30,31] considered for elastic bodies the question: when will a crack propagate? His answer was: this will happen if the crack growth will lower the overall energy. He considered three contributions: (1) the potential energy of the external forces which are doing work on the body deforming it, (2) the stored elastic strain energy, and (3) the work done against the cohesive forces as new crack surfaces are formed. He thus derived an equation which we can write as scr ¼ (2GE=ph)1=2 :

(24:17)

Here scr is the stress level at and above which the crack will propagate; G is the surface energy per unit area (corresponds to the last of the three factors): E is the elastic modulus (also often called the Young modulus); h is the same as before. Thus, if the actual stress imposed is s < scr , the material will sustain the stress without the crack growing. The

MECHANICAL PROPERTIES equation is the same for both constant load and constant displacement conditions, hence it should work also for any intermediate conditions. Equation (24.17) has been the inspiration for much further work—some pertinent and some just rewriting it introducing new symbols and new names. One of these reformulations is Gr ¼ phs2 =E,

(24:18)

where Gr is known as the elastic energy release rate. Another such quantity is R ¼ 2G

(24:19)

Which is called the crack resistance. Substituting into Eq. (24.19) the value of 2G from Eq. (24.17), we get R ¼ phscr =E,

(24:20)

where the right hand sides of Eqs. (24.18) and (24.20) are similar. This leads to a new concept of Gcr ¼ phs2cr =E,

(24:21)

where Gcr is called the critical energy release rate. This is followed by a statement such as: when the elastic energy release rate Gr given by Eq. (24.18) becomes equal to the crack resistance R, then Gr acquires the critical value Gcr and a crack will propagate. It is amazing how many people are investing their efforts into rewording knowledge created by others! The whole story from Eq. (24.18) to (24.21) is nothing new beyond what we have learned already from the Griffith Eq. (24.17). We are mentioning this only because quantities such as the energy release rate are in use. For the same reason we still need to mention connections resulting from Eqs. (24.18), and (24.21) and the definition (16) of Kl . Making pairwise comparisons, we immediately find Kl ¼ a (Gr E)1=2

(24:22)

Klc ¼ a (Gcr E)1=2 ,

(24:23)

and

where, as expected, Klc is called the critical stress intensity factor; it is also known as fracture toughness. Important, however, is the following generalization of Eq. (24.17): scr ¼ [2[G þ G p )E=ph]1=2 :

(24:24)

Recall that the whole theory of Griffith has been developed for elastic bodies—what applies to metals within a certain range of imposed stresses. Thus, Eqs. (24.17)–(24.23) form the essence of linear elastic fracture mechanics (LEFM). In Eq. (24.24) a ‘‘plastic’’ term G p has been added to the elastic term G; metals exhibit also plasticity, hence the improvement displayed in Eq. (24.24). If we make a further step and assume that G p includes all nonelastic contributions, we shall have an equation usable also for viscoelastic materials. We, therefore, have to use Eq. (24.24) instead of (24.17) while in Eqs. (24.18)–(24.23) we need to put G þ G p instead

/

427

TABLE 24.1. Fracture toughness Klc values for selected polymers. Polymer Epoxy Polyester thermoset Polystyrenes high-impact polystyrenes Poly(methyl methacrylate)s Poly(ether sulfone) Acrylonitrile-butadiene-styrene Polycarbonate Poly(vinyl chloride)s Polyamide (nylon 6,6) Polyethylenes Polypropylenes Polyoxymethylene Poly(ethylene terephthalate)

Klc =(J cm
3 m

1=2

)

0.6 0.6 0.7–1.1 1–2 0.7–1.6 1.2 2.0 2.2 2–4 2.5–3 1–6 3–4.5 4 5

Note: J cm
3 m1=2 ¼ MPa m1=2 ¼ 0:9100 ksi in:1=2

of just G. Values of Klc for a number of polymers are listed in Table 24.1. The impact strength values listed at the end of this chapter are also pertinent since they represent a different measure of fracture toughness. 24.2.4 Crazes and Shear Yielding We need to consider the problem of the origin of the cracks. Crazes constitute one source of cracks. They are observed in glassy thermoplastics. Originally, crazes were thought to be just tiny cracks, but this turned out not to be true. We now recognize three kinds of these structures: surface crazes, internal crazes, and crazes at the crack tip. All three kinds consist of elongated voids and fibrils. The fibrils consist of highly oriented chains while each fibril is oriented at approximately 908 to the craze axis. The fibrils span the craze top-to-bottom, resulting in an internal sponge-like structure. Extensive studies of crazes and their behavior under loads have been conducted by Kramer and his school [32–42] and have been reviewed by Donald [43]. We know from their work that there are two unique regions within a craze: (1) the craze/bulk interface, a thin (10– 25 nm) strain-softened polymer layer in in which the fibrillation (and thus craze widening) takes place; and (2) the craze midrib, a somewhat thicker (50–100 nm wide) layer in the craze center which forms immediately behind the advancing craze. The relative position of the midrib does not change as the craze widens. By contrast, as the phase boundaries advance, new locally strain-softened regions are continuously generated, while strain-hardened craze fibrils are left behind. We already know that cracks are more dangerous than crazes. The latter are capable of bearing significant loads thanks to the fibrils. Therefore, we need to know under what conditions can crazes transform into cracks? Kramer,

428 / CHAPTER 24 Donald, and coworkers have established that the craze fibril stability depends on the average number of effectively entangled strands ne that survive the formation of fibril surfaces. Equations for calculating the original number of strands n0 as well as the number ne have been developed by Kramer and Berger [38]. It turns out that polymers with ne > 11:0  1025 strands m
3 and concomitantly a short entanglement length le are ductile and deform by shear yielding. Such materials exhibit engineering strains up to « ¼ 0:25 or even more prior to macroscopic fracture. Polymers with ne < 11:0  1025 strands m
3 and thus with large le are brittle and deform by crazing only. For polymers with intermediate values of ne and le there is a competition between shear deformation and crazing. In Fig. 24.1 we show a part of a craze. The parameter D is the (mean) craze fibril diameter while D0 is the craze fibril spacing. Both D and D0 increase somewhat with increasing ne . Berger [42] traced the craze fibril breakdowns to the formation of small pear-shaped voids at the craze/bulk interface. The results in [42] confirm the microscopic model of Kramer and Berger [38] which we see in Fig. 24.1. In general, providing from outside energy in excess of CRC may result in crazing, shear yielding, or cracking. In shear yielding oriented regions are formed at 458 angles to the stress. The shear bands are birefringent; in contrast to crazes, no void spaces are produced. Thus, crazing—created by tensile fields—is accompanied by volume dilation while shear yielding—created by compressive fields—is not. Combined fields result in mixed responses. The presence of liquids or vapors in the environment of a polymeric component affects the response to external mechanical forces. Thus, for instance polyarylate (Par) under uniaxial extension exhibits exclusively shear yielding without crazing. However, exposure to organic vapor (methylethyl ketone) results in crystallization, embrittlement, and conversion of the response to deformation from shear yielding to crazing [42]. Finally, let us mention that crack healing is possible. This phenomenon has been investigated by Kausch and also by Wool and reviewed by these authors [44,45].

24.2.5 Rapid Crack Propagation and Its Prevention The general definition of CRC in Subsection 24.1.2 does not specify a quantitative measure. Such a measure has to be defined for each specific problem. As an example, we shall now consider rapid crack propagation (RCP). RCP is a dangerous process. Velocities of 100---400 m s
1 (that is 300---1,400 feet s
1 ) have been observed in polyethylene (PE) pipes. Since such pipes are being used for fuel gas distribution within localities; RCP might be accompanied by an explosion of the gas pressurized inside. Given the importance of the problem, studies were made with the objective of connecting the crack length L with a variety of parameters: fuel pressure inside, pipe fatigue, tensile behavior of the piping material, and so on. L was determined by a standard procedure of Greig and Smith [46] such that a knife is pushed through a pressurized pipe by falling weight; given the rate at which RCP takes place, the length L is achieved almost instantaneously. However, no such connections were found—until Gaube and Mu¨ller [47] found a correspondence between the notch impact energy Ul (see Section 24.4.2) and L. An analysis of the problem [48] led to the following equation: L ¼ L0 þ L1 =Ul ,

(24:25)

where L0 is a material constant with the dimensions of length, L1 is another constant with the dimensions of length and energy, and Ul is the notch impact energy. What is required here is a criterion showing when RCP will not occur. Since the notch impact energy Ul is the independent variable in Eq. (24.25), it constitutes the appropriate measure of CRC for the problem under consideration. Ul can be determined by an independent and fairly widely available experimental procedure. L can be measured in an outdoor 14 m long stand at Hoechst AG in Frankfurt-on-the-Main (although such facilities are not widely available). Therefore, CRC will be represented here by a limiting impact energy Ul-lim defined as L(Ul > Ul-lim ) ¼ 0:

(24:26)

Now we simply substitute the definition of Ul-lim from Eq. (24.26) into Eq. (24.25), with the result Ul-lim ¼
L1 =L0 :

(24:27)

Polymer glass

Active zone

D

Do

Fibril

Void

FIGURE 24.1. A schematic of a fraction of one side of a craze.

An example of the application of the criterion just defined is shown in Fig. 24.2. The coordinates are L and Ul , as defined by Eq. (24.25). The criterion applies to all classes of materials; if all plastic pipes were identical, we would have only one point on the diagram, so here again differences in processing, handling, transport etc. appear. Each pipe is slightly different, and there is a certain scatter due to the limited accuracy of the two kinds of experiments, but it is clear that Eq. (24.25) is obeyed. Therefore, the criterion Eq. (24.27) derived from (24.25) is valid. For the data shown in Fig. 24.2 we have L0 ¼
896 mm, L1 ¼ 269 J mm,

L/mm

MECHANICAL PROPERTIES

429

Since notches with h < hcr do not cause crack propagation, it was only natural to assume

800

dh ¼ b(h
hcr ) for h $ hcr , dt

700 600

(24:29)

where b is a time-independent proportionality factor characteristic for the material since it depends on CRC. We do not have space here to provide details of the derivation; the final result [49] is

500 400

log Kl ¼ (1=2) log (a2 2GE)þ

300

(1=2) log [1 þ (1=bhcr )]dh=dt:

200 100 0 2

/

3

4

5

6

J/U FIGURE 24.2. Length L/mm of the cracks in PE pipes determined by the Greig-Smith test 50 vs. the reciprocal Charpy impact energy (U=J)
1 ; after [48].

therefore Ul-lim ¼ 0:300 J. If the impact energy determined in the Charpy test (see the section on impact behavior) is higher than this value, rapid crack propagation will not occur. Since the criterion is defined for a class of polyethylenes, a safety factor somewhat larger than unity may be introduced. 24.2.6 Slow Crack Propagation and Its Prediction The slow crack propagation (SCP) is vastly different from RCP, not at all spectacular but in fact ‘‘quiet’’ and insidious. The crack propagation rate dh/dt might be only, say, 1 mm per month; an observation for instance two weeks after installing a polymeric component might reveal nothing. Experimentalists customarily present the dh/dt rates as a function of the logarithmic stress intensity factor Kl as defined by Eq. (24.16); we now use h as the crack length (as we did before) to differentiate it from the length L which pertained to RCP. The problem clearly consisted in relating dh/dt to Kl . It was solved [49] by using the CRC approach in conjunction with the Eq. (24.17) of fracture mechanics. The problem was different than that of Griffith. He needed the critical stress scr above which crack propagation occurs for a given crack length h. In our problem we need to know whether the crack length is below a certain value, call it hcr , so that the crack will not propagate [49]. We therefore reformulate the Griffith Eq. (24.17) as hcr ¼ 2GE=ps3 :

(24:28)

By definition, the crack will propagate only when h > hcr . This is not only a consequence of the CRC concept but also supported by the molecular dynamics computer simulations [50,51] showing that a crossover exists from the force field region dominated by chain relaxation to one in which crack propagation occurs.

(24:30)

Equation (24.30) provides the desired connection between Kl and dh/dt. In the derivation both the stress level s and the original crack length h0 were used but both canceled out, with the unexpected result that the crack propagation rate is independent of both! The experimental results support Eq. (24.30) as shown for instance in Fig. 24.3 for Hoechst PEs studied under uniaxial tension in water medium at 60 8C. Each symbol pertains to a different stress level and a different original notch length. It is clear that all polyethylenes with the molecular mass MA form a common curve, and the same is true for the other molecular masses. Moreover, we see that a higher M results in a lower crack propagation rate; this result is related to the constituents of CRC listed at the end of Section 24.1.3, particularly the first two of them. In the beginning we have called SCP ‘‘insidious’’. The lowest experimental crack propagation rate value in Fig. 24.3 is dh=dt ¼ 10
8 cm s
1 ; this is only 0.315 cm per year, but the crack does grow. This fact gives us an idea on the utility of Eq. (24.30). 24.3 QUASISTATIC TESTING AND TRANSIENT TESTING 24.3.1 Types of Testing Procedures We have already referred to various kinds of data on mechanical behavior of polymers. We are now going to consider methods of acquisition of such information. The most frequently used are the so-called quasistatic methods which involve relatively slow loading. Tension, compression, and flexure belong here. The quasistatic methods have to be distinguished from so-called transient tests which include stress relaxation and creep. There are also impact tests and dynamic mechanical procedures which will be defined later. Specimens for testing may be produced by processing operations such as injection molding, compression molding, or machining from sheets. Machined surfaces have to be smoothed in their long axis direction with abrasive paper. Any flash on molded specimens shall be removed; the crosssectional area has to be uniform along the whole length subjected to testing. Consequences of any nonuniformity would show up as stress concentrators discussed above.

430 / CHAPTER 24 10 −5 2

Stress intensity factor K1

Jcm

5 4

B

A

C

3 2

1 T = 60°C water 0.5

10−8

5

10−7

5

5 10−5 cm/s 10−6 Crack propagation rate dh/dt

5

10−4

FIGURE 24.3. Crack propagation rate vs. the stress intensity factor for Hoechst polyethylenes. Each PE class such as A has the same molecular mass, with MA < MB < MC ; after [49].

The recommended number of tests on each sample is at least five, 10 or more are preferred. If producing design data for a particular application is the objective, the samples must be prepared by the same method as the part in question. Testing of materials is governed by standards. We shall often refer below to those of the American Society for Testing and Materials (ASTM), West Conshohocken, PA. However, as national economies become more and more connected into a global economy, the use of standards defined by the International Standards Organization (ISO) is on the increase. In Table 24.2 we list several ISO and ASTM tests. 24.3.2 Tensile Properties Tensile testing is the most frequently used method to characterize the material strength. The machine used is presented schematically in Fig. 24.4. It should be of the constant-rate-of-crosshead-movement type, consisting of one fixed and one movable member, both carrying self-aligning grips. The movable member shall move with a uniform, controlled velocity with respect to the stationary one. An extensometer is used to determine the distance between two designated points within the gage length of the test specimen as this is stretched. Speed of testing is defined as the relative rate of motion of the grips or test fixtures. It is specified for different types of specimens, varying typically from 1 to 500 mm/min (0:2---20 in: min
1 ). The lowest speed that produces rupture in the time range 0.5–5 min for the specimen geometry used is to be selected. One tests dumbbell-shaped or straight-sided specimens under defined conditions of pretreatment, temperature, humidity, and deformation rate. The former specimens are shown in Fig. 24.5.

There are two essential properties determined each time. The first is the engineering stress s ¼ F=A0 ,

(24:31)

where F is the applied force and A0 is the initial crosssectional area. Determination of the true stress based on the actual cross-sectional area A which changes during the

TABLE 24.2. ISO and ASTM tests for important mechanical properties. Property Tensile modulus Yield stress Yield strain Nominal strain at break Elongation at break Stress at 50% strain Stress at break Strain at break Flexural modulus Flexural strength Charpy impact strength at
30 8C Charpy impact strength at þ23 8C Charpy notched impact strength at
30 8C Tensile impact Izod impact strength at
30 8C Izod impact strength at þ23 8C Izod notched impact strength at
30 8C Izod notched impact strength at þ23 8C

ISO standard

ASTM standard

527-1 & 2 527-1 & 2 527-1 & 2 527-1 & 2 527 527-1 & 2 527-1 & 2 527-1 & 2 178 178 179 179 179

D 638 D 638 D 638 — D 638 — D 638 D 638 D 790 D 790 D 256 D 256 D 256

8256 180 180 180

D 1822 D 4812 D 4812 D 256

180

D 256

MECHANICAL PROPERTIES

/

431

there is no clear linearity of the initial portion of the stress– strain curve, the modulus is calculated by dividing the nominal (¼ engineering) stress value by the corresponding designated strain (secant modulus). In Fig. 24.6 we show several types of behavior seen in tensile testing of polymers. For performing a specific test, consult one of the standards listed in Table 24.2. 24.3.3 Compressive Properties

FIGURE 24.4. The machine for quasistatic testing—including tension, compression, 3-point bending and/or 4-point bending.

experiment is possible but more difficult. The other key property is the engineering strain (also known as the nominal tensile strain) « ¼ (l
l0 )=l0 ¼ Dl=l0 :

(24:32)

Here l is the current length of the specimen while l0 is the original length. The quantities obtained most often from tensile testing are: Tensile strength: The maximum load divided by A0 . Percent elongation: If the specimen gives a yield load larger than the load at break, calculate percent elongation at yield. Otherwise, percent elongation at break is reported. Modulus of elasticity: It is the proportionality factor E appearing in Hooke’s law: s ¼ E«

(24:33)

Of course, in compressive testing the strain defined by Eq. (24.32) is negative, but the definitions (31)–(33) are applicable. Basically two different testing methods are available here. In the first one the sample is deformed at a constant rate under simultaneous recording of the stress and deformation. This method, in essence a mirror image of the tensile test, is defined in ASTM D 695M. According to the second method, a constant load is applied to the specimen, the deformation of which is recorded after a given period of time with additional reading of the recovery of the specimen following unloading. This method, basically a compressive creep recovery test, is the subject of ASTM D 621. Compression is an important mode of load application. An example of compressive loading is assemblies of conductors and insulators held together by suitable fastening devices. However, the compressive strength as such has a rather limited design value, since this type of loading apart from exceptions, such as collapsing foams or shatter of brittle plastics, seldom results in failure. Testing of flexible materials, like rubbers, may involve complications due to their deformability. For instance, one finds that compressive stiffness is markedly dependent on contact surface constraints and specimen shape.

Stress A

and is also often called Young’s modulus. It is calculated from the initial linear portion of the load vs. extension curve giving us the stress vs. strain curve. For materials where

C B D E

w

F

WC

Strain WC

W

FIGURE 24.5. The dumbbell (‘‘dogbone’’) specimens for tensile testing.

FIGURE 24.6. Typical engineering tensile stress vs. engineering strain curves. Points A, C, E, and F correspond to the tensile strength and elongation at break, D and B at yield. The curve ending at A represents a brittle material, those with C and E tough materials each with a yield point, while the curve ending at F shows a tough material without a yield point.

432 / CHAPTER 24 24.3.4 Flexure and Bending We already mentioned that the machine shown in Fig. 24.4 serves also for bending. Most popular are two kinds, 3 point and 4 point, shown in Fig. 24.7, and described in standards D 790, D 790 M (¼metric) and ISO 178. There are also less used but more specific standards: ASTM D 747 for apparent bending modulus of plastics by means of a cantilever beam and D 648 for deflection temperature of plastics under flexural load. For brittle materials, flexure testing is believed to yield more reliable strength, modulus, and other data than the tensile method, this primarily by reducing the pronounced effects of misalignment in tension. For sheet materials (except laminated thermosets, high-strength reinforced composites) the dimensions of the specimens depend on whether tested flatwise or edgewise; the thickness of the sheet is the depth, or width, respectively. The depth shall not exceed the width in the latter case. ASTM standards specify also that, for sheets less than 1.5 mm in thickness, a specimen 50 mm long by 10 mm wide shall be tested flatwise on a 25 mm support span. Molding materials shall be 80 by 10 by 4 mm tested flatwise on a 64 mm support span. Special rules apply to laminated thermosets and highly anisotropic composites, which shall be tested with a larger span-tothickness ratio (up to 60:1). Anisotropic materials require four different specimens, tested edgewise and flatwise, and cut in lengthwise and crosswise directions.

24.3.5 Stress Relaxation Stress relaxation is typically determined in the uniaxial mode in a specimen or part kept at constant deformation.

This pertains to parts in service such as fasteners, seals, or screws. An example of results of such a test are shown in Fig. 24.8. The relaxing stress could conceivably fall to zero (curve a in the bottom part of Fig. 24.8) but in practice the behavior displayed as curve b is observed, so that a certain level of internal stress si is established. The concept of internal stress is very useful for bringing out common features of stress relaxation behavior of different kinds of materials. Instead of plotting stress vs. time t, let us plot (s
si )=(s0
si ) ¼ si =s0 vs: t. Here s0 pertains to the time of strain imposition. Such a plot was proposed by Kuba´t already in 1965 [52]. An example is shown in Fig. 24.9. We see that curves for ostensibly very different materials have similar shapes. A large central part of each curve has almost the same slope s as the other curves, so that s ¼ (
ds=d ln t)max ¼ (0:1  0:01)(s0
si ):

(24:34)

To explain the situation displayed in Fig. 24.8, Kuba´t has proposed a cooperative theory of stress relaxation [53,54]. He assumed that single units (metal atoms, polymer chain segments) do not relax individually but clusters of such units relax together. Thus, the Kuba´t theory is quite general an explains the observed behavior of metals and polymers alike. Molecular dynamics computer simulations have confirmed that indeed cluster relaxations prevail over individual relaxation, and this both for metals [55] and for polymers [56,57]. In Section 24.1.3 we have discussed among others the time–temperature correspondence principle. An example of application of that principle is shown in Fig. 24.10. The results pertain to high density polyethylene (HDPE) subjected to different levels of predrawing [58]. The draw ratio is defined as

Strain

L

Time Stress s0

b s∞ a

L /4

L /2

L /4

FIGURE 24.7. 3-point and 4-point loading modes in bending.

t0

Time

FIGURE 24.8. Stress relaxation represented by strain vs. time and stress vs. time curves. Explanation in text.

s ∗/s ∗0

MECHANICAL PROPERTIES

/

433

1.0

24.3.6 Creep

0.8

Creep denotes the time-dependent elongation of a specimen or part subjected to a constant stress. Normally, the deformation range is relatively limited; the stress provided by a dead-weight can thus be considered as fairly constant and the change in the cross-section during the process neglected. Such a loading mode emulates the loading situations normally encountered in engineering practice. The pertinent standards include ASTM D 2990. Figure 24.11 shows a schematic picture of a creep curve plotted as strain vs. time. There is an initial elastic deformation which at higher stress levels may also include a plastic component. This is followed by the primary creep stage characterized by a decreasing creep rate—stabilizing at a level corresponding to the secondary or stationary creep stage. In the end phase of the process, called tertiary creep, the rate becomes higher again, eventually resulting in creep rupture. It is to be noted that long-term failure may occur at significantly lower stresses than those determined in normal tensile testing. The logarithm of the time to rupture is often found to decrease linearly with the applied load. Primary (transient) creep can be considered as a consolidation process during which the structure of the material adjusts itself to the following steady-state creep stage. In some instances, like in cross-linked elastomers at low stresses, the steady state is absent, with the creep rate decreasing to zero, and the total creep strain remaining constant. In this case, primary creep is a delayed response of the material to the applied stress. At higher stress levels, chain scission, oxidation effects etc. may influence this simple behavior.

0.6 1

0.4

2

3

4 5

6

7

0.2

1

10

10

2

10 3

10 4

10 5

Time (s)

FIGURE 24.9. Stress relaxation curves—as explained in the text—for polyisoprene (natural rubber, 1), oriented low density polyethylene (LDPE) with the draw ratio l ¼ 1.8 (curve 2), indium (3), unoriented LDPE (4), cadmium (5), polyisobutylene (6), and lead (7).

l ¼ « þ 1,

(24:35)

where « is the engineering strain defined by Eq. (24.32). The curves in Fig. 24.10 have the same shape as those in Fig. 24.9. The final horizontal parts are fairly long in Fig. 24.10, a consequence of prediction over 16 decades of time. The necessary shift factor values have been calculated from ln aT ¼ 1=(a þ cl) þ B=(~ v
1):

(24:36)

Equation (24.36) reduces to Eq. (24.13) for l ¼ 1. Equation (24.7) has been also used along with a representation of T  as a quadratic function of l. We see that indeed predict of long-term behavior from short-term tests can be accomplished. 1

0.25

0.15

2 D

0.10

Strain

s /(103 Jcm−3)

0.20

3

0.05

C

0.00 −8

−6

−4

−4

0

2

4

6

8

B

10

log t /aT

FIGURE 24.10. Master stress relaxation curves for HDPE at the reference temperature T ¼ 313.2 K (¼ 40 8C), the constant tensile strain « ¼ 0:025 and at different values of the draw ratio: l ¼ 12:2 in the top (1) curve; l ¼ 5:5 in the middle (2) curve; and the material without predeformation (l ¼ 1) in the bottom (3) curve. The symbols pertaining to the experiment temperatures are the same in all three curves: & for
50 8C; & for
30 8C; ~ for
10 8C; ~ for 0 8C; * for þ 20 8C; * for þ 40 8C; ^ for þ 60 8C; ^ for þ 80 8C; and  for þ 100 8C. The vertical coordinate is the tensile stress s, the horizontal is log t=aT ; after [58].

A

Time

FIGURE 24.11. A schematic of a creep curve. A ¼ instantaneous initial deformation which may contain a plastic component; B ¼ primary, C ¼ secondary and D ¼ tertiary creep stage.

434 / CHAPTER 24 Important here of course is whether the shift factor aT values calculated from Eq. (24.13) agree with the experimental ones. These results are displayed in Fig. 24.15. The continuous line is calculated from our Eq. (24.13). The dotted line is from an equation proposed in 1955 by Williams, Landel, and Ferry (WLF) [27], a pioneering aT (T) formula at that time. We see that the WLF equation works well in a certain temperature range—this seems the reason it is still in use—but fails miserably outside of that range. Nobody else but Ferry [1] stated that range of application of WLF amounts to 50 K or so, not more. If one makes a primitive and unfounded assumption in our Eq. (24.13), one gets from it the WLF equation as a special case [6]. The problem is when people use the WLF equation blindly in wide temperature ranges, obtain bad results, and draw a false conclusion that the time—temperature correspondence principle does not work. As already mentioned, stress relaxation was also determined for PET/0.6PHB [59]. We do not present the results here, although also in this case one obtains a master curve which covers 16 decades of time. Important, however, is the comparison of aT (T) values from creep and stress relaxation. This is made in Fig. 24.16. The continuous line is again obtained from Eq. (24.13). We see that the aT values obtained from these two kinds of experiments practically coincide. Thus, Eq. (24.13) serves to predict a true material property rather than a property related to just one kinds of experiments. The time—stress correspondence principle as embodied by Eq. (24.14) has also been used successfully [61]. We do not include such results for brevity. One could argue that the use of equations discussed in Section 24.1 requires fairly large amounts of experimentation. This impression might be

During the steady-state stage the material flows in a viscous (plastic) manner. In some instances, this stage may not be clearly discernible, constituting only a transition between the primary and tertiary portions of the creep curve. It may be noted that the acceleration of the creep rate in the latter part is not due entirely to a decrease in the cross-section of the specimen and thus to an increase in the stress level in tests where the specimen is loaded with a dead-weight. We have already mentioned creep recovery. An example including the recovery stage is shown in Fig. 24.12. We observe that the recovery curve is almost a mirror image of the primary creep stage. In Section 24.1 we have defined ways of prediction of long-term behavior from short-term tests. Let us now provide more examples of application of these concepts. Creep and stress relaxation have been determined for PET/ 0.6PHB, where PET is the poly(ethylene terephthalate), PHB, the p-hydroxybenzoic acid, and 0.6 is the mole fraction of the latter in the copolymer [58]. PET/0.6PHB is a polymer liquid crystal, see chapter 41 on PLCs in this Handbook. In temperature ranges of interest it forms 4 coexisting phases [60]. Conventional wisdom said that prediction methods work only for so-called rheologically simple materials, practically for one-phase polymers. Therefore, we have decided to apply as severe a test as possible to our prediction methods and a multiphase PLC is a good choice. In Fig. 24.13 we show several isotherms of tensile creep compliance (see Eq. (24.9)) for PET/0.6PHB [58]. In Fig. 24.14 we show a master curve for Tref ¼ 62  C (the glass transition temperature of PET, the nonliquid crystalline component of the PLC) based on the curves from Fig. 24.13. We see a successful prediction over 16 decades of time.

2.8 2.4

Deformation mm

2.0 1.6 1.2 0.8 0.4 0

0 4 8 Time in (min)

12

16

20

24

28

32

36

40

44

48

52

56

60

FIGURE 24.12. Creep and creep recovery of an oriented polypropylene monofilament with 0.35 mm in radius at 60.7 8C and stress level s ¼ 36 J cm
3 unloaded at 35.5 and 45.5 min. Deformation in mm relates to a specimen length l0 ¼ 100 mm.

MECHANICAL PROPERTIES

/

435

−1.50

log [D /(J cm−3)]

−2.00

20 40 50 60 70 80 90 100 110 120

−2.50

−3.00

−3.50

−4.00 1.00

2.50

2.00

1.50

3.50

3.00

log (t/s)

FIGURE 24.13. Experimental tensile creep compliance for PET/0.6PHB in logarithmic coordinates at 20 8C (the bottom curve) and other temperatures indicated in the insert; after [59].

−2.00

−2.20 −2.40

log [D /(J cm−3)]

−2.60

20 40

−2.80

50 60 70 80 90 100

−3.00 −3.20

110 120

−3.40 −3.60 −3.80 −4.00 0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

log (t /s)

FIGURE 24.14. Tensile creep compliance for PET/0.6PHB in logarithmic coordinates as the master curve for 62 8C; after [59].

436 / CHAPTER 24 slow-loading process, such as a part (an early stage) of quasistatic loading by compression. Then UCRC featured in Eq. (24.2) might be relatively low; as a consequence Ur will be low too, but still Ur > U0 , and the material or component will ‘‘survive an attack’’. However, if the loading occurs at a fast rate, the same external energy U0 will exceed Ur because relaxational processes take time, and fracture will occur. We shall now consider impact testing with this situation in mind.

15

log (a T)

10

5

0 −5

24.4.2 Impact Testing

−10 0

20

40

60

T /°C

80

100

120

140

FIGURE 24.15. The temperature shift factor aT (T ) for PET/ 0.6PHB for 62 8C. Circles are experimental values, the dotted line from the WLF equation and the continuous line from Eq. (24.13) in conjunction with Eq. (24.7); after [59].

confirmed for instance by our Fig. 24.13 which contains 10 isotherms. Therefore, methods of prediction of long-term behavior from short-term tests based on our Eqs. (24.12)– (24.14) have been developed [62, 63] such that one uses two or three experimental isotherms or results for two or three stress levels. Again, we are not going to discuss these results here for brevity. 24.4 IMPACT BEHAVIOR 24.4.1 Rates of Force Application

The most frequently applied impact tests are shown in Fig. 24.17 A and B. A pendulum (shown as a filled arrow) falls from a certain height; the loss in the potential energy of the pendulum is assumed equal (with a correction for losses such as friction) to the energy U0 absorbed by the specimen; see Eq. (24.1). The Charpy test is described by the ASTM D 256 standard method B, the lzod test by the same standard method A. We see (Fig. 24.17 A) that in the Charpy test there is a symmetry with respect to the center of the specimen. By contrast, in the lzod test (Fig. 24.17 B) the bottom half of the specimen remains ‘‘untouched’’ while the top part is broken off. We—and more and more laboratories around the world—perform now both tests with a sensor installed on the pendulum and connected to a computer. Thus, not only a single value of the energy but a whole curve is obtained. For convenience single values of impact strength (IS) for a number of polymers are listed in Tables at the end of this chapter. There is also a combination of tension with impact shown schematically in Fig. 24.17 C. This test is also symmetric with respect to the center, just as the Charpy procedure.

We have noted in Subsection 24.2.5 that a measure of CRC has to be defined for each specific problem. Imagine a 24.4.3 Impact Transition Temperature: Determination and Prediction 5

Traditionally—and that started with metals—one distinguishes two types of mechanical behavior of polymers: brittle and ductile. It will be clear to us after discussion of the free volume concept in Subsection 24.1.3 that brittle behavior will dominate at low temperatures when the free

0 −5 In (a T)

−10 −15 −20 −25 −30 −35 20

a

40

60

80 T /°C

100

120

140

FIGURE 24.16. Experimental shift factors aT (T ) from creep (full circles) and from stress relaxation (empty circles). The continuous line is from Eq. (24.13) in conjunction with Eq. (24.7); after [59].

w

(A)

Radius r

(B)

(C)

FIGURE 24.17. Schematics of impact tests showing geometry, loading mechanisms, and clamping modes.

MECHANICAL PROPERTIES volume is low. Therefore, there is a transition temperature Tl above which the material will be ductile. We shall discuss the CRC connections and a way to predict Tl in the next Subsection. Now we shall define a procedure of experimental determination of Tl . It should be noted immediately that the index I refers to impact; determination of brittle-to-ductile transition by loading at a rate slower than impact will result in finding not a single temperature, but a temperature range; the range might be as large as 10 K [64]. In view of this, we define Tl as the temperature at which the response of the material changes from brittle to ductile under high-impact conditions. The Charpy test described above can be used to achieve those conditions [6]. As discussed in Subsection 24.2.5, two specimens are hardly ever identical. At Tl we have, therefore, 50% failing in the brittle way and the other half in the ductile way. The difference between the two kinds of failure are easily visible when one compares fracture surfaces, macroscopically as well as in micrographs obtained by scanning electron microscopy (SEM). Macroscopically, the fracture surface of a brittle failure appears smooth. SEM micrographs show in this case a ‘‘flaky’’ surface. By contrast, ductile failure is characterized by ‘‘hills and valleys’’ with deformed strands coming out from the surface, as well as holes in the surface left by strands which at break time have ‘‘joined’’ the other surface. Examples of the two types of micrographs are shown, respectively, in Figs. 24.18 and 24.19. There is a whole book by Michler [65] on polymer micromechanics which contains many instructive SEM micrographs of fracture surfaces as well as crazes, shear yielding, and also combinations such as crazes crossing shear bands. Using the concepts discussed in Sections 24.1 and 24.2, the following equation [6] was derived: Kt ¼ F  e
B=(vl
1)

/

437

100µm

10µm

FIGURE 24.18. SEMicrograph of a brittle fracture surface; after [65].

tionships which work well, it was tempting to see whether the relationships can be used also in the opposite direction: going from mechanical properties toward volumetric ones. Thus, Eq. (24.37) was used in this opposite direction [66]:

(24:37)

here Kt is the stress concentration factor as defined by Eq. (24.15); B is the Doolittle constant from Eq. (24.8); and the reduced volume v~l is that at the impact-transition temperature Tl . Thus, we have an implicit formula for Tl which can be related to v~l by an equation of state such as Eq. (24.6) or (24.7); there is a Tl value corresponding to each stress concentration factor. Equation (24.37) was tested for LDPE for which sufficient data were available. The results are shown in Fig. 24.20. We see that the equation is obeyed within the limits of the experimental accuracy. Thus, two pairs of Tl and Kt values are sufficient for the calculation of the parameters F and B and for subsequent prediction of the entire diagram.

24.4.4 Prediction of Volumetric Properties from Impact Data We have used above free volume to explain mechanical properties. Since we have at our disposal quantitative rela-

100µm

5µm

FIGURE 24.19. SEMicrograph of a ductile fracture surface; after [65].

438 / CHAPTER 24 TI 300 280 260 240 220 200 180 160

0

4

8

12

16

20

24

28

Kt

FIGURE 24.20. Relation between the stress concentration factor Kt and the impact transition temperature Tl in K for LDPE. Circles represent experimental values obtained by the Charpy method and crosses those calculated from Eq. (24.37).

specific volume v was obtained for the first time from mechanical parameters—the impact transition data—via an equation of state. The result was prediction of v over a temperature range of 100 K. The average difference between calculated and experimental specific volume values was only 0.092%. This constitutes one more confirmation— and of a different type—of the physical significance of the CRC concept and of the relations based on that concept. 24.5 VISCOELASTICITY AND DYNAMIC MECHANICAL TESTING

primary technique for the study of dissipation mechanisms, and thus of CRC. Clearly DMA data are of importance in designing products to be used in, for instance, vibration isolation, where the mechanical damping properties are used to convert mechanical vibrations into heat. Methods of this type are also highly useful in studies of phase separation in multicomponent systems, effects of fillers and other additives, different processing variables, degree of crystallinity, molecular orientation, internal stresses, etc. Consider a material subjected to an oscillating load of small amplitude that is in the linear viscoelastic range. The angular frequency of the sinusoidal oscillation is v. A sinusoidal stress s will produce a sinusoidal strain «, and vice versa. However, because of the viscous component of the deformation, there will be a phase shift between stress and strain. The pertinent quantities can be represented as follows: « ¼ «0 sin vt s ¼ s0 sin (vt þ d) ¼ s0 sin vt cos d þ s0 cos vt sin d:

As noted in Subsection 24.1.2, viscoelasticity of polymers represents a combination of elastic and viscous flow material responses. Dynamic mechanical analysis (DMA, also called dynamic mechanical thermal analysis, DMTA) enables simultaneous study of both elastic (symbol ’) and viscous flow (symbol ’’) types of behavior. One determines the response of a specimen to periodic deformations or stresses. Normally, the specimen is loaded in a sinusoidal fashion in shear, tension, flexion, or torsion. If, say, the experiment is performed in tension, one determines the elastic tensile modulus E’ called storage modulus and the corresponding viscous flow quantity E’’ called the loss modulus. Diagrams showing the temperature or frequency dependence of storage and loss modulae can be used to locate the thermal transition regions such as the glass transition— although other methods such as differential scanning calorimetry (DSC) can be used for that purpose as well. At the same time, the dynamic mechanical methods constitute the

(24:39)

Here s0 and «0 denote, respectively, the amplitudes of stress and strain, t the time, and d the phase shift between stress and strain. An illustration is provided in Fig. 24.21. As already mentioned, the description of the response of a viscoelastic material to a sinusoidal tensile strain requires the introduction of two modulae; they are defined as s0 E0 ¼ cos d ¼ Ed cos d (24:40) «0 s0 sin d ¼ Ed sin d, (24:41) E00 ¼ «0 Ed is named the absolute value of the dynamic modulus. Obviously, Ed ¼ [(E0 )2 þ (E00 )2 ]1=2 :

24.5.1 Objectives and Definitions

(24:38)

(24:42)

The introduction of E’ and E’’ enables us to write Eq. (24.39) as s ¼ «0 E0 sin vt þ «0 E00 cos vt:

(24:43)

s
s0 sin(wt + d ) e
e 0 sin(wt)

d

FIGURE 24.21. The phase lag of the strain « resulting from an applied sinusoidal stress s.

MECHANICAL PROPERTIES The ratio

D0 ¼

E00 ¼ tan d E0

(24:44)

is the mechanical loss factor. It is a measure of the energy dissipated during a loading cycle relative the energy stored elastically in the material. Sometimes the term internal friction is used instead. Another way of describing this type of response is to use the similarity between Eq. (24.42) and the decomposition of a number in the complex plane into its real and imaginary components. We can thus define a complex dynamic modulus E in the following way E ¼ E0 þ iE00 ¼ Ed eid ,

(24:45)

where Ed is the absolute value of the dynamic modulus introduced in Eqs. (24.40) and (24.41) and equal to s0 =«0 . Figure 24.22 illustrates the decomposition of E into its components according to Eq. (24.45). As can be seen, the complex representation is equivalent to that introduced above; see Eqs. (24.40) and (24.41). The modulae relating to dynamic shear and hydrostatic compression, that is G and K, respectively, are defined in the same way as E in the above equations. In some cases, the inverse values of the complex modulae named compliances are used; these are similar to the transient modulae and compliances such as seen in Eq. (24.9). The complex tensile compliance D is thus defined as D ¼

1 , E

(24:46)

and the complex shear compliance J  as J ¼

1 : G

(24:47)

The following equations relate the components of D and E: E0 ¼

D0 D00 , E00 ¼ 2 , 2 Dd Dd

(24:48)

ImD*

ImE*

E D E

ReE *

439

(24:49)

where Dd is given by Dd Ed ¼ 1. Similar relations apply to the other moduli and the corresponding compliances. The graphical visualization of the compliance components using the complex plane is shown also in Fig. 24.22. It should be remembered that the moduli and compliances under discussion are functions of frequency. The quantities E’, D’ etc. should thus be written E0 (v), D0 (v), and so forth. The frequency dependence of these quantities is governed by the same distribution of relaxation or retardation times as is stress relaxation, creep or other time-dependent mechanical phenomena. Single relaxation or retardation times cannot depict the frequency dependence of the dynamic mechanical behavior of polymers. There is just one book in the world literature on the subject of dynamic mechanical analysis (DMA) which discusses the quantities briefly defined above, namely by Menard [67]. A summary is provided also by Menard in a book chapter [68]. 24.5.2 Experimental Procedures Dynamic mechanical testing allows the use of a variety of instrument types and a wide range of experimental conditions. The temperature may range from practically obtainable subambient up to levels where thermal degradation occurs, the frequencies typically from 0.01 to 1,000 Hz. The results should be examined for possible self-resonances. The elastic modulus of the material to be examined may range from 0:1 J cm
3 to 100 J cm
3 depending on type of polymer, temperature, and frequency. The different techniques available for the determination of dynamic mechanical properties include several modes of load application and a number of dependent variables (temperature, frequency, and time). ASTM D 4092 provides a collection of definitions and terms, the most important of them described in Section 24.5.1. ASTM D 4065 describes standard practice in determining dynamic mechanical properties according to a variety of experimental methods; see Fig. 24.23. 24.5.3 Fatigue Determination

Ed δ

E0 E0 0 , D ¼ , E2d E2d

/

δ

ReD * D Dd

FIGURE 24.22. Graphical representation of the storage and loss moduli E ’ and E ’’ as components of a vector Ed in the complex plane. Ed is the absolute value of the dynamic modulus. The corresponding compliances are shown in the right hand part of the figure.

Plastics parts subjected to repeated loading may undergo failure by so-called dynamic fatigue. The term dynamic intends to distinguish this type of failure from that mentioned in static loading—as for instance in creep where the term static fatigue is sometimes used; see Section 24.3.6. The stress levels leading to failure are in both cases lower than those recorded in short-term tests. In dynamic fatigue, it is often observed that no failure occurs when the stress amplitude is lower than a certain value, the so-called fatigue or endurance limit, often characteristic of the material being studied.

440 / CHAPTER 24 25

Tensile

Compression

Torsion

Shear

log 2 tan d

20 15 10 5 Flexure

Dual cantilever

FIGURE 24.23. Schematic picture of various loading modes used in dynamic mechanical testing.

0 −4

−3

−2

0 −1 log aT /w

1

2

3

FIGURE 24.24. The master curve for HDPE of tan d vs. log aT =v for 40 C and l ¼ 1; after [58].

Fatigue testing of polymers cannot be accelerated by simply increasing the loading frequency. The reason is the relatively high level of mechanical damping (internal friction) in common polymers which would produce an excessive heating of the specimen. Fatigue tests provide data on the number of loading cycles producing certain types of deterioration of the material (crack initiation and propagation, fatigue failure, softening due to energy dissipation). The ASTM test D 671, based on a constant force amplitude, allows these effects to be studied at varying stress levels and environmental conditions. When used for design purposes, the testing and end-use conditions are to be similar. Differences in the fatigue behavior may also be noted when employing testing equipment different from that described in the standard. There exists a related but different German Standard DIN 53 442 which uses dumb-bell-shaped specimens differing from those used for tensile testing by a rounded middle section. Another difference in comparison with the above ASTM method is the use of constant deformation amplitude of the vibrations. This results in a stress amplitude decreasing with time due to stress relaxation. Apart from this, the stress amplitude diminishes also due to the heating of the specimen. The results are reported in a similar manner as required by the ASTM standard with the stress amplitude relating to the first cycle. 24.5.4 Application of Time–Frequency Correspondence Principle We have explained the correspondence principles in Section 24.1.3, including the time—frequency correspondence. We were not able to apply this particular principle before becoming familiar with dynamic mechanical experiments. We need to provide at least an example of the application of the correspondence in the frequency domain. In Fig. 24.24 we show results from [58] pertaining to HDPE. The shift factors used to obtain that diagram have been calculated from equations in Section 24.1.3. More examples can be found for instance in the same paper [58].

24.6 ELASTOMERS 24.6.1 Mechanical Behavior as a Function of Temperature The most amazing thing about elastomeric polymers is the fact that they can be stretched by several hundreds of percent and still behave elastically; that is the engineering stress s (Eq. (24.31) will still be directly proportional to the engineering strain « (Eq. (24.32)). This in contrast to other polymers, and in an even sharper contrast to metals and ceramics in which the elastic region ends at one percent elongation or even less. As a result, the elastic tensile modulus E (see Eq. (24.33)) is 1:1  105 J cm
3 for copper, 7:2  104 J cm
3 for clear fused quartz, 2  103 J cm
3 for nylon (that is a nonelastomeric polymer) and only about 1 J cm
3 for gum rubber. The explanation of the behavior which is ordinarily called rubbery lies in the huge number of possible conformations in elastomeric chains. When a copper wire is drawn, we soon come to weakening and eventual destruction of primary chemical bonds between Cu atoms. When a rubber band is drawn, rotations and other changes results in new conformations, but the primary bonds are preserved. This can be described as unkinking and straightening out of kinked and ‘‘mixed up spaghetti-like’’ elastomeric chains. It is essential to note that elastomers do not always behave in the manner known from stretching a rubber band at room temperature. Some of us might have seen an experiment when such a rubber band was put into liquid nitrogen, became brittle, and when stretching was attempted the band broke into little fragments. Thus, in general the type of behavior of an elastomer depends on the temperature. This is shown in Fig. 24.25: the elastic modulus E (for a certain fixed time after the imposition of a force) as a function of temperature T. At low temperatures we have the brittle behavior—as the rubber band in liquid nitrogen;

MECHANICAL PROPERTIES log E

/

441

of one segment. We can easily verify that the last result is true: when we put a stretched piece of rubber between our lips, under given tension the specimen shrinks when warmed. In other words, since F, N, and h2 are all constant, an increase in T must produce a decrease in l. Other cases such as biaxial extension and shear are treated in the already quoted book of Mark and Erman [69].

Glass

Leather

Rubber

Crosslinked Uncrosslinked Flowing liquid T

Tg

FIGURE 24.25. Dependence of the tensile modulus E (for a fixed time t since the imposition of a force) on temperature T for an elastomer.

the modulus E is relatively high. Then from the glass transition temperature Tg up to approximately Tg þ 30 K we have the leathery state—with retarded high elasticity. Then comes the rubbery behavior known to us from stretching the elastomeric band at room temperature: instantaneous high elasticity. Finally, if the elastomer is not cross-linked, we have melting and liquid flow. If the elastomer is crosslinked, the rubbery plateau persists. We conclude that an elastomer might exhibit glassy, leathery, rubbery, or liquid flow behavior. 24.6.2 Thermodynamic and Molecular Behavior We have already referred to the fact that the explanation for the instantaneous high elasticity lies at the molecular level. This is a vast area of active research, and we do not have space to discuss details, but we can recommend to the reader a book by Mark and Erman [69] which covers precisely that field. Here we shall mention only two facts. First, the behavior at the molecular level can be related to the macroscopic thermodynamic description. For the simple uniaxial tension we have dU ¼ TdS
PdV þ Fdl

(24:50)

dA ¼
SdT
PdV þ Fdl,

(24:51)

where the symbols have the same meaning as before: U is energy; S, entropy; F, force; and l is the length while A is the Helmholtz function. Second, Eqs. (24.50) and (24.51) can be used in conjunction with the analysis of a memoryless system (also known as the story of the drunkard walk) to obtain the following relation: F ¼ kTl=Nh2 ,

(24:52)

where N is a constant proportional to the degree of polymerization, k is the Boltzmann constant while h is the length

24.6.3 Swelling of Networks Some polymeric materials are water-repellent and thus used for instance as impregnation of overcoats, but some elastomeric and other networks absorb liquid penetrants avidly and swell—until an equilibrium degree of swelling is reached. Since this is only one chapter in the Handbook, of limited length, again we shall take the same short-cut as in the preceding Subsection: we recommend to the reader the book by Mark and Erman [69]. Since the behavior of elastomers is characterized in terms of energy and Helmholtz function, as in our Eqs. (24.50) and (24.51), we need a relation for the calculation of change of A caused by swelling [70]: DAswelling ¼ DAel þ DAmix :

(24:53)

That is, the change in the Helmholtz function on swelling consists of an elastic (‘‘mechanical’’) contribution DAel resulting from the change of dimensions of the network caused by the solvent penetration and also from the ‘‘thermodynamic’’ contribution DAmix caused by polymer þ solvent interactions upon mixing. The latter can be calculated for the swelling process by similar procedures as for polymer þ polymer or liquid þ liquid systems. Equation (24.53) is thus the starting point for dealing with mechanical, thermodynamic, and molecular behavior of swollen networks. The assumption that there is no mixed term, that is mechanical effects do not affect thermodynamic ones nor vice versa, has been supported by results for several systems [70]. 24.6.4 Filled Elastomers Natural rubber crystallizes on elongation—a phenomenon called strain-induced crystallization—what enhances mechanical properties. However, a filler in the form of carbon black is typically added to natural rubber to additionally modify the mechanical properties. Elastomers which cannot undergo strain-induced crystallization contain even more fillers. Carbon black is used in such cases also, but silicone rubbers are filled with silica. Automotive tires constitute the classic example of carbonblack reinforced elastomers. The elastomer can be either natural rubber—as typically is the case of truck and aircraft tires, or else a synthetic rubber—as is typical for automobile tires. However, reinforcing fillers constitute only one of many additives. There are also antioxidants, light stabilizers,

442 / CHAPTER 24 plasticizers, antiplasticizers, impact modifiers, processing aids, colorants, flame retardants, crosslinking agents etc. There exists a thorough collective book edited by Zweifel [71] on polymer additives, which discusses filers and reinforcements in some detail. 24.7 OTHER ISSUES 24.7.1 Brittleness and Aging There is still a number of topics related to mechanical properties of polymers which we did not cover. One of them is aging in the glassy state: tending toward equilibrium, the material increases its density, and thus lowers its free volume. We do not have space for it, but aging is understandable in terms of CRC as explained in Section 24.1, and is discussed in some detail by Robertson and Kim [72]. An important result of aging is brittleness. Of course, there are also materials which are brittle even without aging. Brittleness is not a simple inverse of ductility (for which there is more than one definition) nor of toughness. Brittleness has been defined [73] as B ¼ 1=(eb  E0 )

(24:54)

where eb is the elongation at break in tensile testing (along with the stress at break sb and other quantities, see Tables below) while E’ is known to us From Section 24.5.1. Thus, the first term in the denominator comes from quasi-static tensile testing and the second from DMA. Application of Eq. (24.54) shows that polystyrene is highly brittle, what explains odd behavior of PS in a variety of circumstances [73].

surfaces in relative motion [78]. Rabinowicz [78] describes vividly huge annual losses to industry caused by wear. Some tribologists claim that their discipline is not a part of mechanics but independent and comparable to mechanics in its importance. A review of polymer tribology which includes fundamental definitions is available [79]. Similarly as mechanical properties, tribological properties of polymers can be varied by using additives; thus, carbon black can be used for the purpose [80]. By contrast, using external liquid lubricants—which work so well for metal surfaces—is in many cases dangerous because of swelling described above. Another option is application of magnetic fields which cause polymer orientation and thus can improve scratch resistance [81].

24.8 TABLES OF SELECTED MECHANICAL DATA Following are selected data for the most often used polymers. They have been divided (partly arbitrarily, because of the overlap in definitions) into four tables, numbered from 24.3 to 24.6 respectively, for general purpose polymers, engineering polymers, thermosets, and elastomers. The third column in each of these tables shows the values of density, the fourth of the tensile modulus, the fifth the stress at break, the sixth the elongation at break; IS denotes the Izod impact strength for notched specimens. The letters A and C in the last column in Tables 24.3 and 24.4 pertain respectively to amorphous and crystalline thermoplastic polymers.

24.7.2 Nanoindentation

ACKNOWLEDGMENTS

Nanoindentation is a technique gaining increasing popularity [74–76]. Actually, the technique is sometimes abused by attempts to calculate the elastic modulus E on the basis of a model valid for fully elastic materials only [74]. While such attempts fail, a connection has been found by Fujisawa and Swain between E and the unloading strain rate [75]. As shown by Tweedie and Van Vliet [76], spherical indentation provides lower contact strains and more reliable results than conical indentation. A modification providing repetitive indenter hits perpendicular to the specimen surface at the same spot and thus nanoindentation fatigue testing (NIFT) exists also [77].

Professor Josef Kuba´t of the Chalmers University of Technology in Gothenburg and Dr. Michael J. Kuba´t of the Royal Institute of Technology in Stockholm have participated in writing this chapter for the first Handbook edition. Thanks are due to Professor G.H. Michler of the Institute of Materials Science of the Martin Luther University in Merseburg for providing us with the micrographs used in Section 24.4. I appreciate discussions with: Professor Michael Bratychak, Lvivska Politechnika National University; Dr. Georg Broza, Technical University of Hamburg; Dr. Rimantas Levinskas, Lithuanian Energy Institute, Kaunas; Professor Robert Maksimov, Institute of Polymer Mechanics, the Latvian National University, Riga; Professor Moshe Narkis, Technion, Haifa, Dr. Dorota Pietkiewicz, LAPOM, University of North Texas, Denton; and Professor Anuvat Sirivat, Chulalongkorn University, Bangkok.

24.7.3 Tribology Another important area is tribology which includes friction, scratch resistance, wear and design of interactive

MECHANICAL PROPERTIES

/

443

TABLE 24.3. Mechanical properties of thermoplastics: commodity (general purpose) plastics. Polymer PE LD, LDPE Polyethylene, low density PE HD, HDPE Polyethylene, high density PE UHMW, UHMW PE Polyethylene, ultra-high molecular weight PP polypropylene PP polypropylene PP PVC Poly(vinyl chloride) PVC

Grade

Homopolymer
40% glass fiber filled Copolymer Rigid (RPVC) Flexible (FPVC, plasticized)

PS Polystyrene SB Styrene-butadiene

Rubber-modified PS High-impact PS, HIPS ABS Acrylonitrile-butadiene-styrene Medium IS ABS Acrylonitrile-butadiene-styrene High IS SAN Styrene-acrylonitrile ASA Acrylate-styrene-acrylonitrile

r=(g=cm3 )

E/(GPa)

sb /(MPa)

«b /(%)

0.915–0.93

0.14–0.3

7–17

200–900

NB

C

0.94–0.97

0.7–1.4

20–40

100–1000

30–200

C

0.93–0.94

0.1–0.7

20–40

200–500

NB

C

0.90–0.91 1.22–1.23

1.1–2 6.8–7.2

30–40 60–110

100–600 1.5–4

20–75 75–110

C

0.89–0.905 0.9–1.2 1.32–1.58 1–3.5 1.16–1.70 0.05–0.15

28–40 40–75 6–25

200–500 30–80 150–400

60–750 20–1000 —

C A A

1.04–1.05 0.98–1.10

2.4–3.2 1.5–2.5

30–60 15–40

1–4 15–60

13–25 50–400

A A

1.03–1.06 1.01–1.04 1.07–1.09 1.05–1.07

2–2.8 1.6–2.5 3.4–3.7 2.2–2.4

30–50 30–40 55–75 30–50

15–30 5–70 2–5 20–40

130–320 350–600 15–30 450–600

A A A A

r=(g=cm3 ) E/(GPa)

sb /(MPa)

«b /(%)

IS/(J/m)

Structure

1.13

80 50 170–180 110 75–90

IS/(J/m)

Structure

TABLE 24.4. Mechanical properties of thermoplastics: engineering plastics. Polymer PA 6 Polyamide 6 (Polycaprolactam) PA 6 Polyamide 6 (Polycaprolactam) PA 6 Polyamide 6 (Polycaprolactam) PA 6 Polyamide 6 (Polycaprolactam) PA 66 Polyamide 66 [Poly(hexamethyleneadipamide)] PA 66 Polyamide 66 [Poly(hexamethyleneadipamide)] PA 11 Polyamide 11 [Poly(11-aminoundecanoic acid)] POM Polyacetal Polyoxymethylene POM Polyacetal Polyoxymethylene PET Poly(ethylene terephthalate) PBT Poly(butylene terephthalate) PBT Poly(butylene terephthalate) PC Polycarbonate CA Cellulose acetate CAB Cellulose acetate butyrate PMMA Poly(methyl methacrylate) PTFE Polytetrafluoroethylene PSU (PSO) Polysulfone PES Polyethersulfone PPS Poly(phenylene sulfide) PPO (PPE) Poly(phenylene oxide) or -ether PEEK Polytetheretherketone PEEK Polyetheretherketone a

dry as molded. at 50% relative humidity. c thickness 3.2 mm. b

Grade a b a

30–35% glass fiber 1.35–1.42 30–35% glass fiber a 1.14 b

b

3 1.5 8–10 5.5 3.4 17–2

50

a

1.04

Homopolymer Copolymer

Modified with PS

1.42 3.1 65–70 1.41 2.8 65–72?? 1.29–1.40 3 50 1.31 2.3–2.5 50–60 1.52 10 100–140 1.2 2.1–2.4 70–90 1.27–1.32 1.5–2.5 25–45 1.18 1.4–1.8 30–35 1.17–1.20 2.5–3.3 55–75 2.15–2.20 0.41 7–30 1.25 2.5–2.6 70 1.37 2.5 80–90 1.35 3.6 65–75 1.06–1.08 2.2–2.7 50–60

30% glass fiber

1.32 1.49

þ 30% glass fiber

1.5

3.6 10

45–50

90–200 100

50–120 30–120 160–200 160 2–4 50 95 20 30–55 80

C C C

50–110

C

400–500 100–NB

C

25–75 40–75 50–300 120–200 2–4 100–120 10–70 30–100 3–5 200–400 50–100 40–80 1–2 200–350

C C C C

50 2

60–120 50–80 12–40 40–55 80–130 650–1000c 100–450 50–500 10–20 150 65–70 75–120 70 200–370

A A A A C A A C A

80 100

C C

444 / CHAPTER 24 TABLE 24.5. Mechanical properties: thermosets. Grade

r=(g=cm3 )

E/(GPa)

sb /(MPa)

«b /(%)

IS/(J/m)

Wood-flour (ca. 50%) filled molding compound Impact modified, cellulose filled (ca. 50%) Cellulose filled (ca. 50%) Cellulose filled (ca. 50%) Cast, rigid Premix, chopped glass Woven glass cloth Unfilled Glass fiber reinforced SMC

1.37–1.46

5.5–12

30–60

0.4–0.8

10–30

25–45

1–2

20–60

35–100 40–60 30–40 20–60 200–350 30–90 140–250

0.5–1 0.4–0.8 1.5–2.5 1 1–2 1–2 0.5–2

10–20 10–20 10–20 80–320 300–1600 10–50 1600–2100

Polymer PF Phenol-formaldehyde resin PF Phenol-formaldehyde resin MF Melamine-formaldehyde resin UF Urea-formaldehyde resin Polyester thermosetting resin Polyester thermosetting resin Polyester thermosetting resin Epoxy resin SMC Sheet molding compound

1.38–1.42 1.47–1.52 1.46–1.48 1.04–1.46 1.65–2.30 1.5–2.1 1.2–1.3 1.6–2

8–10 7–9 2–4.5 7–17 10–30 3–5 15–30

TABLE 24.6. Mechanical properties: elastomers. Polymer

Grade

NR (Natural rubber) Cis-polyisoprene

Unfilled vulcanisate 50 pph CB, vulc.a SBR Styrene-butadiene rubber Unfilled, vulc. (23–25% styrene) 50 pph CB, vulc.a IIR (Butyl rubber) Isobutylene-isoprene rubber Unfilled, vulc. 50 pph CB, vulc.a NBR (Nitrile rubber) Acrylonitrile-butadiene rubber Unfilled, vulc. (AN content 26–27%) 50 pph CB, vulc.a CR (Chloroprene rubber) Poly(2-chloro-1,3-butadiene) Unfilled, vulc. 50 pph CB, vulc.a EPDM Ethylene-propylene rubber Unfilled, vulc. 50 pph CB, vulc.a a

r=(g=cm3 ) E/(MPa) sb /(MPa) 0.93 0.93–1.0

0.91–0.98 0.92

1.2–1.25 0.85–0.87

«b

1–2 3.5–6 1–2

17–30 14–28 1.4–2.8

650–900 450–600 450–600

14–19 — 4–10 —

14–27 17–21 9–21 4–7

400–650 750–950 300–700 350–800

8–18 1–3 3–5 — 5–10

10–30 13–22 23–25 1.2 10–16

350–800 800–1000 200–450 400 250–750

pph¼parts per hundred; CB¼carbon black.

REFERENCES 1. J.D. Ferry, Viscoelastic Properties of Polymers, 3rd edition (Wiley, New York 1980). 2. J.J. Aklonis and W.J. MacKnight, Introduction to Polymer Viscoelasticity, 2nd edition (Wiley, New York 1983). 3. Failure of Plastics, edited by W. Brostow and R.D. Corneliussen (Hanser, Munich, Vienna, New York 1986, 1989, 1992). 4. Performance of Plastics, edited by W. Brostow (Hanser, Munich, Cincinnati 2000). 5. W. Brostow, Mater. Chem. Phys. 13, 47 (1985). 6. W. Brostow, Chapter 10 in Ref. 3. 7. W. Brostow, Chapter 5 in Ref. 4. 8. J. Kim and R.E. Robertson, J. Mater. Sci. 27, 300 (1992). 9. J. Karger-Kocsis, Polym. Eng. Sci. 36, 203 (1996). 10. P.J. Flory, Statistical Mechanics of Chain Molecules (Wiley, New York 1969). 11. P.J. Flory, Faraday Soc. Disc. 49, 7 (1970). 12. R.A. Orwoll, Chapter 7 in this Handbook. 13. M.H. Litt and A.V. Tobolsky, J. Macromol. Sci. Phys. 1, 433 (1967). 14. D.G. Fesko and N.W. Tschoegl, J. Polymer Sci. C 35, 51 (1971). 15. R.W. Fillers and N.W. Tschoegl, Trans. Soc. Rheol. 21, 51 (1977). 16. W.K. Moonan and N.W. Tschoegl, Internat. J. Polym. Mater. 10, 199 (1984).

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C.S. Henkee and E.J. Kramer, J. Mater. Sci. 21, 1398 (1986). L.L. Berger and E.J. Kramer, J. Mater. Sci. 22, 2739 (1987). L.L. Berger and E.J. Kramer, J. Mater. Sci. 23, 3536 (1988). E.J. Kramer and L.L. Berger, Adv. Polym. Sci. 91/92, 1 (1990). L.L. Berger, Macromolecules 22, 3162 (1989). L.L. Berger, Macromolecules 23, 2926 (1990). C.J.G. Plummer and A.M. Donald, Polymer 32, 409 (1991). L.L. Berger, J. Polym. Sci. Phys. 27, 1629 (1989). A.M. Donald, Chapter 13 in Ref. 4. H.H. Kausch, Chapter 5 in Ref. 3. R.P. Wool, Chapter 15 in Ref. 4. J.M. Greig and T.R. Smith, Paper 3 at the Conference on Designing to Avoid Mechanical Failure, Plastics Institute of London, Cranfield Institute of Technology, 8–10 January 1973. E. Gaube and W.F. Mu¨ller, Kunststoffe 70, 72 (1980). W. Brostow and W.F. Mu¨ller, Polymer 27, 76 (1986). W. Brostow, M. Fleissner and W.F. Mu¨ller, Polymer 32, 419 (1991). R. Simoes, A.M. Cunha and W. Brostow, e-Polymers 2004, no. 067. R. Simoes, A.M. Cunha and W. Brostow, Model. & Simul. Mater. Sci. & Eng. 14, 157 (2006); R. Simoes, A.M. Cunha and W. Brostow, Comput. Mater. Sci. 36, 319 (2006). J. Kuba´t, Nature 204, 378 (1965). J. Kuba´t, Phys. Status Solidi B 111, 599 (1982). J. Kuba´t and M. Rigdahl, Chapter 4 in Ref. 3. W. Brostow and J. Kuba´t, Phys. Rev. B 47, 7659 (1993). W. Brostow, J. Kuba´t and M.J. Kuba´t, Mater. Res. Soc. Symp. 321, 99 (1994). S. Blonski, W. Brostow and J. Kuba´t, Phys. Rev. B 49, 6494 (1994). Yu. M. Boyko, W. Brostow, A. Ya. Goldman and A.C. Ramamurthy, Polymer 36, 1383 (1995). W. Brostow, N.A. D’Souza, J. Kuba´t and R. Maksimov, J. Chem. Phys. 110, 9706 (1999). W. Brostow, M. Hess and B.L. Lo´pez, Macromolecules 27, 2262 (1994).

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61. A.E. Akinay, W. Brostow and R. Maksimov, Polym. Eng. Sci. 41, 977 (2001). 62. A.E. Akinay and W. Brostow, Polymer 42, 4527 (2001). 63. A.E. Akinay, W. Brostow, V.M. Castan˜o, R. Maksimov and P. Olszynski, Polymer 43, 3593 (2002). 64. E.H. Andrews, Fracture in Polymers (American Elsevier, New York 1968). 65. G.H. Michler, Kunststoff-Mikromechanik (Carl Hanser Verlag, Mu¨nchen, Wien 1992). 66. W. Brostow and M.A. Macip, Macromolecules 22, 2761 (1989). 67. K.P. Menard, Dynamic Mechanical Analysis – An Introduction (CRC Press, Boca Raton – London, 1999). 68. K.P. Menard, Chapter 8 in Ref. 4. 69. J.E. Mark and B. Erman, Rubberlike Elasticity - A Molecular Primer 2nd edn. (Cambridge University Press, 2000). 70. W. Brostow, Macromolecules 4, 742 (1971). 71. Plastics Additives Handbook, edited by H. Zweifel (Hanser, Munich, Cincinnati 2000). 72. R.E. Robertson and J.-H. Kim, Chapter 14 in Ref. 4. 73. W. Brostow, H.E. Hagg Lobland and M. Narkis, J. Mater. Res. 21, 2422 (2006). 74. W.C. Oliver and G.M. Pharr, J. Mater. Res. 7, 1564 (1992); W.C. Oliver and G.M. Pharr, J. Mater. Res. 19, 3 (2004). 75. N. Fujisawa and M.V. Swain, J. Mater. Res. 21, 708 (2006). 76. C.A. Tweedie and K.J. Van Vliet, J. Mater. Res. 21, 1576 (2006). 77. B.D. Beake, B. Bilyeu, W. Brostow and W. Chonkaew, Polymer Internat. 56 (2007) to be published. 78. E. Rabinowicz, Friction and Wear of Materials, 2nd edn. (Wiley, New York, 1995). 79. W. Brostow, J.-L. Deborde, M. Jaklewicz and P. Olszynski, J. Mater. Ed. 24, 119 (2003). 80. W. Brostow, M. Keselman, I. Mironi-Harpaz, M. Narkis and R. Peirce, Polymer 46, 5058 (2005). 81. W. Brostow and M. Jaklewicz, J. Mater. Res. 19, 1038 (2004).

CHAPTER 25

Chain Dimensions and Entanglement Spacings L. J. Fetters*, D. J. Lohsey, and R. H. Colbyz *Chemical and Biomedical Engineering, Cornell University, Ithaca, NY 14853-5201; yExxonMobil Research and Engineering Company, Annandale, NJ 08801-0998; zMaterials Science and Engineering, Penn State University, University Park, PA 16802

25.1 25.2 25.3 25.4

Chain Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chain Entanglement and Tube Diameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Molecular Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Dependence of Chain Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

447 448 448 451 453

end-to-end distance 0 and their molar mass M is a constant, for large M, that characterizes their chain dimensions. In practice, the ratio 0 =M depends weakly on temperature in the melt and the specific choice of u-solvent, imparting a weak temperature dependence to various quantities calculated from that ratio. The Kuhn length b of a polymer is the ratio of the meansquare end-to-end distance 0 and the fully extended size Rmax

This chapter summarizes data on chain dimensions and entanglement spacings for a number of linear flexible polymers. The polymers are listed in the Appendix along with their abbreviations used in the Tables. The equations relating various important parameters are from the literature [1–3]. While polymer chain entanglement is far from being understood [4–6], one natural idea based on overlap [7] appears useful for thinking about entanglement effects in polymer melts. This concept leads to the entanglement criterion: a fixed number of entanglement strands (Pe ) share a volume equal to the cube of the tube diameter (a3 ) [8–12]. One of the main purposes of this chapter is to test this criterion using literature data on flexible polymer melts and evaluate this universal number. Empirical relations useful for estimating the plateau modulus and entanglement molar mass of polymer melts emerge from this analysis. Chain entanglement is important, not merely for melt rheology, but also for mechanical properties of glassy [13] and semicrystalline polymers [14]. This chapter first discusses chain dimensions of polymers and then discusses chain entanglement and the tube diameter. The critical molar mass for entanglement effects in melt viscosity is then discussed, followed by the temperature dependence of chain dimensions.

b


hR2 i0 : Rmax

(25:1)

Aliphatic backbone polymers have n backbone bonds, with a well-defined average backbone bond length l, and known backbone bond angle u, making Rmax ¼ nl cos (u=2) in the all-trans conformation. Flory defined [15] the characteristic ratio C1 as the ratio of the actual unperturbed mean-square end-to-end distance 0 and that of a freely jointed chain nl2 , which is a polymer-specific constant at large M C1


hR2 i0 mb hR2 i0 : ¼ 2 nl2 lM

(25:2)

The second equality uses the equation n ¼ M=mb , where mb is the average molar mass per backbone bond. Using this definition of C1 , the Kuhn length can be rewritten as:

25.1 CHAIN DIMENSIONS b¼ In either the melt state or in a u-solvent solution, linear flexible polymers adopt Gaussian statistics, and their average conformation is described as a random walk. Consequently, the ratio of their unperturbed mean-square

C1 nl2 C1 l : ¼ nl cos (u=2) cos (u=2)

(25:3)

It is common to assume the fully extended conformation is a 0 linear chain (ignoring bond angles) and hence Rmax ¼ nl, 0 and b ¼ C1 l. Since the bond angle of a polyethylene chain 447

448 / CHAPTER 25 is u ¼ 68 , b0 =b ¼ cos (u=2) ¼ 0:83. In principle, either convention may be utilized; here we use Eq. (25.3) to calculate the Kuhn length. The Kuhn length is the effective monomer size for the equivalent freely jointed chain (N Kuhn monomers of length b instead of n backbone bonds of length l ) hR2 i0 ¼ C1 nl2 ¼ Nb2 ,

Rmax ¼ Nb:

(25:4)

The molar mass of a Kuhn monomer is M0 ¼ M=N and the volume occupied by the Kuhn monomer is n0 ¼ M0 =rNAv , where r is the density and NAv is Avogadro’s number. This description of chain dimensions can be used to calculate many quantities. For example, combining Eqs. (25.3) and (25.4) yields the number of main chain bonds in a Kuhn monomer n=N ¼ C1 = cos2 (u=2). Witten et al. [16] define the packing length p as the ratio of the occupied volume of a chain M=rNAv and the meansquare end-to-end distance p


M M0 n0 ¼ 2 ¼ 2: 2 hR i0 rNAv b rNAv b

(25:5)

volume and entanglement strand volume (an overlap parameter [3] for entanglement) Pe


a3 a ¼ : Ve p

This number appears to be constant for flexible polymers, with the average value Pe ¼ 20:6(  8%). Table 25.1 shows data for polyolefin melts listing density r, plateau modulus Ge , melt chain dimensions from SANS 0 =M, entanglement molar mass Me calculated from Eq. (25.6), Kuhn length b, packing length p, tube diameter a, and the overlap parameter for entanglement Pe , all at temperature T. Since Pe is apparently a polymer-independent constant, Eq. (25.10) suggests that the tube diameter and packing length are proportional, and the constant of proportionality thus has the empirical temperature dependence [1,2]: a ¼ 14:0 exp (T=1270) p:

Ge ¼

The plateau modulus Ge defines the entanglement spacing of a polymer melt, and the entanglement molar mass Me [17] Me


rRT ¼ Ve rNAv , Ge

(25:6)

where R ¼ kNAv is the ideal gas constant (k is the Boltzmann constant), T is the absolute temperature, and Ve
kT=Ge ¼ Me =( rNAv ) is the entanglement volume. The length scale associated with the entanglement spacing is the tube diameter a [3]. Since a chain in the melt is Gaussian on all scales larger than the Kuhn length, and for flexible chains a  b, the tube diameter is related to the entanglement molar mass through the chain dimensions rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi hR2 i0 Me a¼ ¼ b Ne , (25:7) M where Ne is the number of Kuhn monomers in an entanglement strand (of molar mass Me ). The occupied volume of an entanglement strand is Ve a2 Ve ¼ n0 Ne ¼ n0 ¼ a2 p: (25:8) b In analogy to polymer networks (where the equilibrium modulus is kT per network strand) the plateau modulus is kT per entanglement strand kT b2 kT kT Ge ¼ ¼ ¼ 2 : 2 Ve n 0 a a p

(25:11)

Using Eqs. (25.6), and (25.9)–(25.11), we can obtain useful empirical equations [2] for the entanglement molar mass and the plateau modulus Me ¼ P2e p3 rNAv ¼ 200 exp (T=635) rp3 NAv ,

25.2 CHAIN ENTANGLEMENT AND TUBE DIAMETER

(25:10)

exp ( T=635) kT : 200 p3

(25:12) (25:13)

Table 25.1 lists these quantities for polyolefins. Table 25.2 lists these quantities for polydienes while polyacrylics and polymethacrylics are listed in Table 25.3. Table 25.4 lists these quantities for various other flexible linear polymers. 25.3 CRITICAL MOLECULAR WEIGHT The critical molar mass (Mc ) parameter [18,19] denotes the transition in the melt viscosity/molar mass relation as the exponents change from 1 to 3.4. Table 25.5 presents the polymers for which the Mc values are known while Table 25.6 lists the polymers for which, seemingly, Mc =Me is one. The ratio of Mc =Me was long taken to be 2 [18] and thus to be species independent. However, a recent empirical compilation [19] has shown that the ratio is p dependent and varies from 3.5 (PE; p ¼ 1.69) to 1.4 (a-PCHE; p ¼ 5.59). Based upon the data of Table 25.5 this ratio is empirically expressed as:   0:534 Mc p ¼ 3:42p0:534 ¼ : (25:14) Me p Mc hence follows the empirical expression: Mc ¼ Me

  0:534 p , p

(25:15)

(25:9)

The number of entanglement strands Pe within the confinement volume a3 is determined as the ratio of confinement

˚ With Mc expressed in where Mc ¼ Me at p ¼ p  10 A. ˚ this fashion, Me overtakes Mc as p approaches p in the 10 A ˚ range. At least four polymers exist with p ffi 10 A (see Table

CHAIN DIMENSIONS AND ENTANGLEMENT SPACINGS

/

449

TABLE 25.1. Molecular characteristics of olefinic polymers and copolymers. Polymer

T (K)

r ( g cm3 )

Ge (MPa)

o =M (A˚ 2 )

C1

b (A˚)

no (A˚ 3 )

Mo

p (A˚)

Me

a (A˚)

Ne

Pe

PE PE PEB-2 PEB-2 PEB-5 PEB-7 PEB-10 PEB-12 PEB-12 alt-PEP alt-PEP alt-PEP PEB-18 PEB-18 HPI-16 HPI-20 PEB-25 PEB-25 a-PP a-PP a-PP a-PP i-PP s-PP HHPP HHPP HPI-34 alt-PEB alt-PEB PEB-32 PEB-32 HPI-50 PEB-40 PEB-40 PIB PIB a-PEE a-PEE HPI-75 HPMYRC a-PHEX HPMYRC-64 a-PCHE

298 413 389 413 413 413 413 298 413 298 373 413 298 413 373 373 298 413 298 348 413 463 463 463 298 413 373 298 413 298 413 373 298 413 298 413 298 413 300 324 273 308 433

0.851 0.785 0.802 0.785 0.788 0.789 0.791 0.860 0.793 0.856 0.812 0.790 0.860 0.797 0.812 0.812 0.864 0.799 0.852 0.825 0.791 0.765 0.766 0.766 0.878 0.810 0.812 0.861 0.800 0.863 0.802 0.812 0.864 0.805 0.918 0.849 0.866 0.807 0.855 0.832 0.871 0.871 0.920

3.5 2.6 2.5 2.2 1.90 1.55 1.35 1.50 1.20 1.10 1.03 0.97 1.12 0.90 0.88 0.79 0.69 0.67 0.48 0.48 0.47 0.42 0.43 1.35 0.52 0.52 0.50 0.58 0.52 0.44 0.43 0.35 0.24 0.30 0.34 0.30 0.18 0.20 0.12 0.12 0.14 0.10 0.068

1.40 1.25 1.25 1.22 1.15 1.05 1.05 1.04 0.952 0.924 0.871 0.834 0.926 0.913 0.813 0.788 0.800 0.799 0.678 0.678 0.678 0.678 0.694 1.03 0.691 0.691 0.703 0.725 0.692 0.641 0.692 0.632 0.570 0.595 0.570 0.557 0.480 0.508 0.452 0.434 0.542 0.409 0.323

8.26 7.38 7.70 7.51 7.47 7.08 7.53 7.72 7.06 7.21 6.80 6.51 7.42 7.31 6.51 6.45 7.08 7.07 6.00 6.00 6.00 6.00 6.15 9.12 6.12 6.12 6.25 6.88 6.57 6.22 6.71 6.21 6.06 6.32 6.73 6.58 5.67 6.00 6.67 6.73 9.60 7.36 7.49

15.4 13.7 14.3 14.0 13.9 13.2 14.0 14.3 13.1 13.4 12.6 12.1 13.8 13.6 12.1 12.0 13.2 13.1 11.2 11.2 11.2 11.2 11.4 16.9 11.4 11.4 11.6 12.8 12.2 11.5 12.5 11.5 11.3 11.7 12.5 12.2 10.5 11.1 12.4 12.5 17.8 13.7 13.9

329 318 339 338 353 347 391 382 379 376 374 368 396 421 368 372 416 449 358 369 385 398 407 604 353 383 392 434 446 400 465 430 427 478 496 524 443 503 660 720 1119 895 1082

168.3 150.4 163.5 159.5 167.3 164.9 186.2 197.6 180.9 194.0 182.9 175.1 205.1 202.2 180.1 181.9 216.4 216.2 183.4 183.4 183.4 183.4 187.8 278.7 187.0 187.0 192.0 225.2 214.9 208.0 224.6 210.5 222.1 231.8 274.2 267.9 230.9 244.3 339.7 360.6 586.6 457.5 599.4

1.39 1.69 1.66 1.73 1.83 1.90 2.00 1.86 2.20 2.10 2.35 2.52 2.09 2.28 2.52 2.60 2.40 2.60 2.88 2.97 3.10 3.20 3.12 2.10 2.74 2.97 2.91 2.66 3.00 3.00 2.99 3.24 3.37 3.47 3.18 3.51 4.00 4.05 4.30 4.60 3.52 4.78 5.59

602 1,040 1,040 1,220 1,420 1,750 2,010 1,420 2,270 1,930 2,440 2,790 1,900 3,040 2,860 3,190 3,100 4,090 4,390 4,970 5,780 7,010 6,850 2,180 4,180 5,350 5,030 3,680 5,280 4,860 6,400 7,190 8,910 9,210 6,690 9,710 11,900 13,800 17,800 18,700 14,100 21,700 48,700

29.0 36.0 36.0 38.7 40.5 42.8 46.0 38.4 46.5 42.2 46.1 48.3 42.0 52.7 48.2 50.1 49.8 57.2 54.6 58.1 62.6 68.9 69.0 47.4 53.8 60.8 59.5 51.6 60.4 55.8 66.6 67.4 71.3 74.0 61.7 73.6 75.6 83.9 89.6 90.0 87.5 94.3 125.4

3.58 6.89 6.34 7.68 8.51 10.6 10.8 7.18 12.5 9.93 13.4 16.0 9.27 15.0 15.9 17.5 14.3 18.9 23.9 27.1 31.5 38.2 36.5 7.83 22.4 28.6 26.2 16.3 24.6 23.3 28.5 34.2 40.1 39.7 24.4 36.3 51.6 56.7 52.3 51.8 24.1 47.5 81.2

20.8 21.3 21.7 22.3 22.1 23.0 23.0 20.7 21.1 20.1 19.6 19.2 20.1 23.1 19.2 19.3 20.7 22.0 19.0 19.5 20.2 21.5 22.1 22.5 19.6 20.5 20.4 19.4 20.1 18.6 22.2 20.8 21.1 21.3 19.4 20.9 18.9 20.7 20.8 19.6 24.9 19.7 22.4

TABLE 25.2. Molecular characteristics of polydiene polymers and copolymers. Polymer

T (K)

r ( g cm3 )

Ge (MPa)

o =M ˚ 2) (A

C1

b (A˚)

no ˚ 3) (A

Mo

p (A˚)

Me

a (A˚)

Ne

Pe

cis-PI PI-7 PI-16 PI-20 PI-29

298 298 298 298 298

0.910 0.900 0.899 0.898 0.896

0.58 0.35 0.35 0.35 0.35

0.679 0.596 0.593 0.591 0.587

5.20 4.70 4.88 4.96 5.20

9.34 8.44 8.82 8.98 9.39

235 221 243 253 279

128.6 119.6 131.2 136.5 150.3

2.69 3.10 3.12 3.13 3.16

3,890 6,370 6,360 6,350 6,340

51.4 61.6 61.4 61.3 61.0

30.2 53.2 48.5 46.5 42.2

19.1 19.9 19.8 19.6 19.3

450 / CHAPTER 25 TABLE 25.2. Continued. Polymer

T (K)

r ( g cm3 )

Ge (MPa)

o =M ˚ 2) (A

C1

b (A˚)

no (A˚ 3 )

Mo

p (A˚)

Me

a (A˚)

Ne

Pe

PI-34 PI-50 PI-75 cis-PBd PBd-7 PBd-15 PBd-18 PBd-20 PBd-23 PBd-26 PBd-30 PBd-62 PBd-98 SBR** PEBd 55-DMBD PMYRC-0 PMYRC-64

298 298 298 298 298 298 298 298 298 298 298 298 300 298 298 348 298 298

0.895 0.893 0.890 0.900 0.895 0.896 0.895 0.895 0.895 0.895 0.894 0.890 0.890 0.913 0.891 0.861 0.892 0.891

0.35 0.41 0.37 0.76 1.15 1.10 1.05 1.07 1.05 1.00 0.98 0.81 0.57 0.78 0.29 0.33 0.10 0.071

0.585 0.528 0.563 0.758 0.876 0.854 0.846 0.841 0.832 0.824 0.813 0.727 0.661 0.818 0.543 0.640 0.398 0.374

5.26 4.80 8.07 4.61 5.52 5.54 5.56 5.61 5.66 5.68 5.67 6.17 7.39 6.41 4.85 7.31 5.30 5.87

9.58 8.80 15.0 8.28 9.93 10.0 10.1 10.1 10.2 10.3 10.3 11.3 13.7 11.9 9.02 13.6 9.85 10.9

291 273 745 167 209 218 222 227 234 238 244 328 532 316 279 556 454 592

156.9 146.6 399.3 90.5 112.5 117.7 119.8 122.4 125.9 128.0 131.2 175.9 284.8 173.6 149.7 288.4 243.8 317.5

3.17 3.52 3.32 2.44 2.12 2.17 2.19 2.21 2.23 2.25 2.28 2.57 2.82 2.22 3.43 3.01 4.68 4.98

6,330 5,390 5,960 2,930 1,930 2,020 2,110 2,070 2,110 2,220 2,260 2,720 3,890 2,900 7,610 7,550 22,100 31,100

60.9 53.4 57.9 47.1 41.1 41.5 42.3 41.7 41.9 42.7 42.9 44.5 50.7 48.7 64.3 69.5 93.8 107.8

40.4 36.8 14.9 32.4 17.1 17.1 17.6 16.9 16.8 17.3 17.2 15.5 13.7 16.7 50.8 26.2 90.6 97.9

19.2 15.1* 17.5 19.4 19.4 19.1 19.3 18.9 18.8 19.0 18.8 17.4 18.0 22.0 18.7 23.1 20.0 21.7

*The low value of Pe likely indicates the real plateau modulus is lower. **Styrene content 25 wt%.

TABLE 25.3. Molecular characteristics of poly(acrylics) and poly(methacrylics). Polymer

T (K)

r ( g cm3 )

a-PMA a-PEA a-POA a-PMMA a-PEBMA a-PHMA a-POMA a-PDDMA a-PAPHMA a-PBPHMA

298 298 298 413 373 373 373 298 393 393

1.11 1.13 0.98 1.13 0.988 0.960 0.923 0.929 1.00 1.00

Ge (MPa)

o =M (A˚ 2 )

C1

b (A˚)

no ˚ 3) (A

Mo

p (A˚)

Me

a (A˚)

Ne

Pe

0.25 0.36 0.16 0.31 0.15 0.090 0.033 0.016 0.012 0.0092

0.436 0.463 0.442 0.390 0.315 0.366 0.272 0.254 0.167 0.154

7.91 9.76 17.1 8.22 11.3 13.1 11.4 13.6 14.5 15.2

14.7 18.1 31.9 15.3 21.0 24.4 21.1 25.3 26.9 28.2

740 1,040 3,890 880 2,350 2,800 2,950 4,500 7,220 8,600

494.6 710.1 2,295 598 1,396 1,622 1,635 2,513 4,348 5,173

3.43 3.17 3.83 3.77 5.34 4.73 6.61 7.04 9.95 10.8

11,000 7,770 15,200 12,500 20,400 33,100 86,700 144,000 272,000 355,000

69.2 60.0 81.9 69.9 80.2 110.0 153.6 191.1 213.2 233.8

22.2 10.9 6.61 20.9 14.6 20.4 53.0 57.2 62.6 68.6

20.2 18.9 21.4 18.5 15.0* 23.3 23.2 27.1* 21.4 21.7

*These two samples yield Pe values at odds with the value of 21. This indicates the potential presence of pronounced errors in the chain dimension and/or plateau modulus values. From the trend shown in the chain dimension column the primary error seems to exist with this parameter.

TABLE 25.4. Molecular characteristics of miscellaneous polymers. Polymer

T (K)

r(g cm3 )

Ge (MPa)

o =M (A˚ 2 )

p (A˚)

Me

a (A˚)

Pe

a-PaMS a-PS i-PS a-PtBS a-PVA a-PVME m-AEK Me-PEEK PC

473 413 413 473 333 303 473 463 473

1.04 0.969 0.969 0.957 1.08 1.05 1.20 1.16 1.14

0.32 0.20 0.19 0.10 0.35 0.41 2.2 3.3 2.7

0.442 0.437 0.420 0.361 0.490 0.580 0.775 0.834 0.864

3.61 3.92 4.08 4.81 3.14 2.73 1.79 1.72 1.69

12,800 16,600 17,500 37,600 8,540 6,450 2,140 1,350 1,660

75.1 85.2 85.7 116.5 64.7 61.2 40.8 33.6 37.9

20.8 21.7 21.0 24.2 20.6 22.4 22.8 19.6 22.5

CHAIN DIMENSIONS AND ENTANGLEMENT SPACINGS

/

451

TABLE 25.4. Continued. Polymer

T (K)

r(g cm3 )

Ge (MPa)

˚ 2) o =M (A

p (A˚)

Me

a (A˚)

Pe

PDMS PET PN6 PEO POM PPO PSF PTFE RADEL-R

298 548 543 353 473 505 523 653 555

0.970 0.989 0.985 1.06 1.14 0.998 1.15 1.46 1.22

0.20 3.1 1.8 1.8 1.7 1.2 2.1 1.7 3.6

0.422 0.845 0.853 0.805 0.763 0.741 0.756 0.598 0.821

4.06 1.99 1.98 1.95 1.91 2.24 1.91 1.90 1.66

12,000 1,450 2,470 1,730 2,640 3,500 2,380 4,660 1,560

71.2 35.0 45.9 37.3 44.8 50.9 42.4 52.8 35.8

17.5 17.6 23.2 19.2 23.5 22.7 22.2 27.7 21.6

TABLE 25.5. Entanglement and critical molecular weights of miscellaneous polymers. Polymer

T (K)

r(g cm3 )

˚ 2) o =M (A

p (A˚)

Me

Mc

Mc =Me

PE PBd-7 PI-7 PEO SBR a-PVA alt-PEP PI-7 a-PMMA PBd-98 a-PaMS PDMS PIB a-PS PIB a-PCHE

443 298 243 353 298 428 373 298 490 300 459 298 298 490 490 453

0.768 0.895 0.919 1.081 0.930 1.08 0.812 0.900 1.09 0.889 1.04 0.970 0.918 0.959 0.817 0.920

1.21 0.876 0.618 0.805 0.708 0.490 0.871 0.625 0.425 0.720 0.460 0.422 0.570 0.434 0.570 0.323

1.79 2.12 2.92 1.91 2.52 3.14 2.40 2.95 3.58 2.59 3.47 4.06 3.17 3.39 3.57 5.59

980** 2,000 3,250* 2,000 2,960 9,100 3,100 6,025 13,600 3,850 13,300 12,000 6,900 18,100 10,500 48,750

3,480 6,380 10,000 5,870 8,210 24,500 8,100 13,100 29,500 8,200 28,000 24,500 13,100 31,200 17,000 80,000

3.5 3.2 3.0 2.9 2.8 2.7 2.6 2.2 2.2 2.1 2.1 2.0 1.9 1.7 1.6 1.6

*Calculated value is 6,000. **Measured value at 413 K. The calculated value (via Eq. (25.12)) at 443 K is 1,150. TABLE 25.6. Polymers with large packing lengths. Polymer

T (K)

r(g cm3 )

o =M (A˚ 2 )

p (A˚)

Me

[Mc =Me ]

a-PHDEC a-PAPHMA a-PBPHMA PMA-CH3

418 393 393 363

0.796 1.0 1.0 1.17

0.213 0.167 0.154 0.123

9.79 9.94 10.8 11.5

173,000* 268,000 355,000 485,000*

1.01** 1.00 0.96 0.93

*Via Eq. (25.12). **Via Eq. (25.15).

25.6). Since it seems improbable that Me >Mc for any polymer, we expect the limiting value for Mc =Me of 1, inde˚ An unexplained facet of these pendent of p for p 10 A. empirical observations is that as p increases, fewer entanglement events are seemingly required to reach the regime where the melt viscosity becomes proportional to the 3.4 power of molar mass. This is displayed in Table 25.5 where the Me and Mc data for various flexible polymers are listed. Note that while the relation between Me and packing length is understood [1], the corresponding state of play between p and Mc remains purely empirical [19].

25.4 TEMPERATURE DEPENDENCE OF CHAIN DIMENSIONS A feature of chain dimensions is their temperature dependence that is expressed in terms of k


d ln hR2 i0 fe =f ¼ T dT

(25:16)

where fe =f denotes the energetic fraction of the temperaturedependent force in a polymer network at constant volume.

452 / CHAPTER 25 TABLE 25.7. Melt state values of k ¼ d ln 0 =dT . k  103 (K1 )

Polymer samples alt-PEP (PEP)* HPI-50 a-PCHE a-PEE a-PMMA a-PP a-PS a-PPEN alt-PEB HHPP i-PP PBd-7 PDMS PEB-2(PE) PEB-5 PEB-7 PEB-10 PEB-12 PEB-18 PEB-25 PEB-32 PEB-40 PEO PIB HPI-75 PI-7 (cis-PI)

From SANS 1.1 [25] 0.2 [21] 0 [21] þ0.40 [28,29] þ0.10 [31] 0.1 [32] 0 [31] — 0 [21] 0 [21] 0 [35] — — 1.2 [38] 1.3 [29] 0.65 [29] 0.44 [29] 0.44 [29] 0.1 [29] 0 [29] þ0.63 [29] þ0.55 [29] 0.30 [40] — þ1.2 [21] þ0.40 [21]

From fe =f 1.5 [26,27] — — þ0.30 [30] 0.10 [24] — þ0.17 [33,34] þ0.33 [30] — — — þ0.16 [36] þ0.78 [37] 1.2 [39] — — — — — — — — þ0.03 [41] 0.28 [42] — þ0.41 [43]

The empirical sign of k can be þ, 0, or – (see Table 25.7). The modes of measurement have included theta condition measurements utilizing a family of theta solvents or melt state measurements. The latter include thermoelastic measurements on networks [22] and small angle neutron scattering (SANS) measurements on labeled chains in a polymer melt [23]. Generally, the theta condition approach (multiple theta solvents over a wide temperature range) is recognized to be unreliable [21,24]. An example of this [21] is a-PEE, where extensive theta condition work (over the temperature range of 200 K) led to a negative value of k ¼ 1:2  103 K1 , as opposed to the two positive values found for the melt state by SANS and thermoelastic measurements; see Table 25.7. The role of k on the packing length, plateau modulus and entanglement volume can be significant, particularly for cases where the melt rheology is studied over a wide temperature range. A 200 K range is common for amorphous polymers having low glass transition temperatures. The packing length has two sources of temperature dependence (0 and r, see Eq. (25.5)). For typical temperatures (350 K) the change in density for polymer liquids as a function of temperature is d ln r=dT  6 104 K1 :0 is the more interesting parameter, in that it can increase, decrease, or remain constant as temperature is changed.

*Ethylene–propylene random copolymer.

APPENDIX Alphabetical Listing of Polymers. Name

References

Description

alt-PEB alt-PEP a-PaMS a-PAPHMA a-PBPHMA a-PCHE a-PDDMA a-PEA a-PEBMA a-PEE

[44,45] [25,46,47] [1,17,18,48,49] [50,51] [50,51] [1,19,21,52] [53,54] [53,55] [17,53,56] [28,57]

a-PHDEC a-PHEX a-PHMA a-PMA a-PMMA a-POA a-POMA a-PP a-PPEN a-PS a-PtBS

[58] [59,60] [53,61] [55,62] [17,18,21,31,63] [55,59] [17,53,54,64] [32,65,66] [59] [17–19,31,67] [1,68]

essentially alternating poly(ethylene-co-1-butene); hydrogenated PEBd essentially alternating poly(ethylene-co-propylene); hydrogenated PI-7 atactic poly(a-methyl styrene) atactic poly[6-{4(anisyloxycarbonyl)phenoxy}-hexyl methacrylate] atactic poly[6-{4(butoxycarbonyl)phenoxy}-hexyl methacrylate] atactic poly(cyclohexyl)ethylene or poly(vinyl cyclohexane) atactic poly(dodecyl)methacrylate atactic poly(ethyl)acrylate atactic poly(ethyl butyl)methacrylate atactic poly(ethyl ethylene); also called poly(butene-1);may be made via the hydrogenation of poly(vinyl ethylene) atactic poly(hexadecene-1) atactic poly(hexene-1) atactic poly(hexyl)methacrylate atactic poly(methyl)acrylate atactic poly(methyl)methacrylate atactic poly(octyl)acrylate atactic poly(octylmethyl)methacrylate atactic polypropylene; hydrogenated poly(2-methyl 1,3-pentadiene) atactic poly(pentene-1) atactic polystyrene atactic poly(t-butyl styrene)

CHAIN DIMENSIONS AND ENTANGLEMENT SPACINGS

/

453

APPENDIX Continued. Name

References

Description

a-PVA a-PVME cis-PBd cis-PI 55-DMBD HHPP

[17,18,54] [69,70] [17,71] [17,72,73] [1] [1,21,45]

HPI-x HPMYRC-x i-PMMA i-PP i-PS m-AEK Me-PEEK

[1,26,74] [1,45] [54,63] [35,66] [54,75] [76] [77,78]

atactic poly(vinyl acetate) atactic poly(vinyl-methylether) 1,4-polybutadiene  96% cis content. 1,4-polyisoprene  100% cis content; natural rubber poly-2,3(dimethyl butadiene) 55% 1,4; 45% 3,4 content. hydrogenated poly(2,3 dimethyl)butadiene:head-to-head polypropylene (alternating copolymer of ethylene and butene-2). hydrogenated polyisoprene where x ¼ 3,4 content of parent polyisoprene hydrogenated poly(myrcene) with x% 3,4 isotactic-poly(methylmethacrylate) isotactic polypropylene isotactic polystyrene poly(m-arylene–ether–ketone) methyl-poly(aryl–ether–ether–ketone); prepared from methyl hydroquinone and 4,4’difluorobenzophenone. polybutadiene, x ¼ vinyl percent; for 100% vinyl content the material is identified as poly(vinyl ethylene) or 1,2-polybutadiene. polycarbonate of bisphenol A(4, 4’-isopropylidenediphenol) poly(dimethylsiloxane) polyethylene poly(ethyl butadiene)  75/20/5 cis/trans/3,4 poly(ethylene–butene) random copolymer; x denotes number of ethyl branches per 100 backbone carbons poly(ethylene oxide) poly(ethylene terephthalate) polyisobutylene 1,4-polyisoprene where x ¼ 3,4 content; PI-75 is 75% 3,4, and 25% 1,2 (with essentially no 1,4 addition) main-chain liquid crystal polyester poly(myrcene) with x% 3,4 [myrcene ¼ 1,6-octadiene-7-methyl-3-methylene] polycaprolactam-nylon 6 poly(oxymethylene) poly(phenylene oxide) alternating copolymer of bisphenol A and dichlorodiphenyl sulfone (UDEL) poly(tetrafluoro)ethylene alternating copolymer of 4,4’-biphenol and dichlorodiphenyl sulfone solution prepared copolymer (anionic polymerization) styrene–butadiene (34% vinyl; 19% cis and 47% trans) 25 wt% styrene syndiotactic polypropylene

PBd-x

[57,79–82]

PC PDMS PE PEBd PEB-x

[1,54,76,83,84] [17,85–87] [17,19,38,88–90] [1,44] [29,57]

PEO PET PIB PI-x

[18,19,40] [54,91,92] [17,18,93–95] [21,47,96–98]

PMA-CH3 PMYRC-x PN6 POM PPO PSF PTFE RADEL-R SBR

[99] [1,44,45] [54,100,101] [28,54] [76,102] [76,103] [104,105] [76,103] [17,106]

s-PP

[66,107,108]

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66. Eckstein, A.; Suhm, J.; Friedrich, C.; Maier, R.-D.; Sassmannshausen, J.; Bochmann, M.; Mu¨lhaupt, R. Macromolecules 1998, 31, 1335. 67. Onogi, S.; Masuda, T.; Kitagawa, K. Macromolecules 1970, 3, 109. 68. Mays, J. W.; Ferry, W. M.; Hadjichristidis, N.; Funk, W. G.; Fetters, L. J. Polymer 1986, 27, 129. 69. Kannan, R. M.; Lodge, T. P. Macromolecules 1997, 30, 3694. 70. Choi, S.; Liu, X.; Briber, R. M. J. Polym. Sci., Part B: Polym. Phys. 1998, 36, 1. 71. Brandrup, J.; Immergut, E. H., Eds. Polymer Handbook, 3rd Ed., Wiley, New York (1989). 72. Ansorena, F. J.; Revuelta, L. M.; Guzma´n, G. M.; Iruin, J. J. Eur. Polym. J. 1982, 18, 19. 73. Sanders, J. F.; Ferry, J. D.; Valentine, R. H. J. Polym. Sci. A-2 1968, 6, 967. 74. Mays, J. W.; Hadjichristidis, N.; Fetters, L. J. Macromolecules 1984, 17, 2723. 75. Guenet, J. M.; Picot, P.; Benoit, H. Macromolecules 1979, 12, 86. 76. Roovers, J. Toporowski, P. M.; Ethier, R. High Perform. Polym. 1990, 2, 165. 77. Hermann-Scho¨nherr, O.; Schneller, A.; Seifert, A. M.; Soliman, M.; Wendorff, J. H. Makromol. Chem. 1992, 193, 1955. 78. Wang, F.; Roovers, J.; Toporowski, P. M. Macromolecules 1993, 26, 3826. 79. Fetters, L. J.; Lohse, D. J.; Graessley, W. W. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 1023. 80. Roovers, J. Polym. J. 1986, 18, 153. 81. Roovers, J.; Toporowski, P. M. Rubber Chem. Tech. 1990, 63, 734. 82. Colby, R. H.; Fetters, L. J.; Graessley, W. W. Macromolecules 1987, 20, 2226. 83. Hutnik, M.; Argon, A. S.; Suter, U. W. Macromolecules 1991, 24, 5956. 84. Aloisio, C. J.; Boehm, V. W. in Rheology Vol.2 (G. Astarita and G. Marrucci, L. Nicholais, Eds.) Plenum Press, New York (1980), p. 513. 85. Kirste, R. G.; Lehnen, B. R. Makromol. Chem. 1976, 177, 1137. 86. Beltzung, M.; Picot, C.; Rempp, P.; Herz, J. Macromolecules 1982, 15, 1594. 87. Plazek, D. J.; Dannhauser, W.; Ferry, J. D. J. Colloid Sci. 1961, 16, 101. 88. Lieser, G.; Fischer, E. W.; Ibel, K. J. Polym. Sci., Polym. Lett. Ed., 1975, 13, 39. 89. Pearson, D. S.; Fetters, L. J.; Graessley, W. W.; Ver Strate, G.; von Meerwall, E. Macromolecules 1994, 27, 711. 90. Han, J.; Jaffe, R. L.; Yoon, D. Y. Macromolecules 1997, 30, 7245. 91. McAlea, K. P.; Schultz, J. M.; Gardner, K. H.; Wignall, G. D. Macromolecules 1985, 18, 447. 92. Wallach, M. L. Makromol. Chem. 1967, 103, 19. 93. Hayashi, H.; Flory, P. J.; Wignall, G. D. Macromolecules 1983, 16, 1328. 94. Fetters, L. J.; Graessley, W. W.; Kiss, A. D. Macromolecules 1991, 24, 3136. 95. Pyckhout-Hinzen, W.; Fetters, L. J. unpublished data (PIB at 298K). 96. Nemoto, N.; Moriwaki, M.; Odani, H.; Kurata, M. Macromolecules 1971, 4, 215. 97. Nemoto, N.; Odani, H.; Kurata, M. Macromolecules 1972, 5, 531. 98. Abdel-Goad, M.; Pyckhout-Hintzen, W.; Kahle, S.; Allgaier, J.; Richter, D.; Fetters, L. J. Macromolecules 2004, 37, 8135. 99. Fourmaux-Demange, V.; Boue´, F.; Brulet, A.; Keller, P.; Cotton, J. P. Macromolecules 1998, 31, 801. 100. Mattiussi, A.; Gechele, G. B.; Francesconi, R. J. Polym. Sci. A 1969, 7, 411. 101. Flory, P. J.; Williams, A. D. J. Polym. Sci. A-2 1967, 5, 399. 102. Cai, H.; Ait-Kadi, A.; Brisson, J. Polymer 2003, 44, 1481. 103. Roovers, J.; Toporowski, P. M.; Ethier, R. High Perform. Polym. 1990, 2, 151. 104. Tuminello, W. H.; Treat, T. A.; English, A. D. Macromolecules 1988, 21, 2606. 105. Chu, B.; Wu, C. Macromolecules 1987, 20, 93. 106. Kraus, G. in The Stereo Rubbers (Saltman, W. M. Ed.) Wiley, New York, (1977); p. 613. 107. Wheat, W. R. ANTEC Proc. 1995, p. 2275. 108. Jones, T. D.; Chaffin, K. A.; Bates, F. S.; Annis, B. K.; Hagaman, E. W.; Kim, M.-H.; Wignall, G. D.; Fan, W.; Waymouth, R. Macromolecules 2002, 35, 5061.

CHAPTER 26

Temperature Dependences of the Viscoelastic Response of Polymer Systems K. L. Ngai* and D. J. Plazeky *Naval Research Laboratory, Washington, D.C. 20375-5320; yDepartment of Materials Science and Engineering, University of Pittsburgh, Pittsburgh, PA 15216

26.1 26.2 26.3 26.4 26.5

The WLF Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation of WLF Equation to Free Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermorheological Complexities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Dependences Under Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Important Secondary Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

g

measured at Tg , it was initially noted that the constants C1 g and C2 assume values close to 17.448 and 51.68, respectively, for 17 polymers [12]. Individual treatment of the data on a wide range of polymers indicates that Cg1 may take values between 158 and 268 and Cg2 between 208 and 1308. The fit extends to temperatures below Tg if the polymer is at its equilibrium density. The WLF expression has been shown [12] to be related to the Vogel–Fulcher–Tammann–Hesse equation [14–16],

26.1 THE WLF EQUATION The most successful temperature dependence for the viscous flow [1,2], viscoelastic response [1], dielectric dispersion [3–5], nuclear magnetic resonance response [6–8] and dynamic light scattering [9–10] of polymers and supercooled liquids with various chemical structures is the Williams, Landel, and Ferry (WLF) equation [11,12] log

455 455 456 456 475 476

Js0 (T)h(T) t(T) C1 (T  T0 )
log ¼ log aT ¼  , Js0 (T0 )h(T0 ) t(T0 ) C2 þ T  T0 (26:1)

log t i ¼ log A þ

(C=2:303) , T  T1

(26:2)

for h or t where A, C and T1 are empirical constants. It follows that

where Js0 is the steady state recoverable compliance; h is the shear viscosity; t is a retardation or relaxation time; aT is the time-scale shift factor; T0 is the chosen reference temperature; and C1 and C2 are characterizing constants. Js0 (T) is a very weak function of the temperature. In fact in the temperature range where T=Tg varies from 1.2 to 2.0, Js0 has been found to be independent of temperature [13]. Therefore its variation is often ignored. It will be ignored in this chapter. Some authors identify magnitude variations with temperature which are reported as bT ¼ Js0 (T)=Js0 (T0 ). Williams, Landel, and Ferry [12] reported that such an expression is valid for polymers over the temperature range Tg < T < Tg þ 100 . When Tg is chosen as the reference temperature, i.e., when the response curves measured at different temperatures are shifted primarily along the time or frequency scales to superimpose upon the response curve

log aT ¼

C=2:303 C=2:303  : T  T0 T0  T1

(26:3)

This is identical to the WLF expression provided the Vogel parameters and the WLF parameters are related as C ¼ 2:303C1 C2

(26:4)

T0  T1 ¼ C2 :

(26:5)

and

26.2 RELATION OF WLF EQUATION TO FREE VOLUME The WLF equation for aT has been rationalized in terms of Doolittle’s free volume theory [17]. According to this theory 455

456 / CHAPTER 26 that portion of the volume which is accessible to the kinetic process of interest is considered to be the free volume vf ¼ v  v0 , where v is the measured volume and the inaccessible volume v0 is called the occupied volume. The Doolittle equation states that the viscosity is an exponential function of the reciprocal of the relative free volume f  vf =v0 . h ¼ Aeb=f ,

(26:6)

where A and b are characterizing constants. Williams, Landel, and Ferry chose to use the fractional free volume f ¼ vf =v in place of f. This substitution made no difference in their derivation of the equation for the temperature shift factor aT   b 1 1  , log aT ¼ (26:7) 2:303 f fg since [1=f  1=fg ] ¼ [1=f  1=fg ]. With the assumption that the fractional free volume is a linear function of temperature f ¼ f0 þ af (T  T0 ):

(26:8)

Substituting of Eq. 26.8 in Eq. 26.7 yields log aT ¼ 

(B=2:303f0 )(T  T0 ) f0 =af þ T  T0

(26:9)

which is identical in form with the WLF equation.

to a single-loss tangent maximum; however mechanisms contributing to different loss peaks inevitably have different temperature dependences. This is widely recognized for the sub-Tg loss peaks, identified by the Greek letters b, g, and d. However, it is not as widely recognized that the so-called a mechanism, which is normally seen above Tg involves contributions from possibly three groups of molecular mechanisms with specifically different sensitivities to the variation of temperature [19–21]. In spite of the fact that a single loss peak is generally observed, a growing body of knowledge shows that local mode, sub-Rouse, and Rouse normal modes of chain backbone motions have different temperature dependences [19–24] which are most often different from that of the mechanisms of the terminal zone [19,25–32] which leads to steady-state behavior. Consequently, the temperature dependence of the viscosity is usually different from that of the glass to rubber-softening dispersion. Therefore, it is important to know from which region or viscoelastic zone the WLF constants C1 and C2 were determined. Table 26.1 presents WLF constants and contains such information, when possible. It should be noted that the shift factors that can be fitted to the WLF equation show positive curvature when plotted logarithmically against the temperature. Quite often, log aT values obtained near and below Tg show negative curvature at low temperatures simply is an indication that the lower temperature measurements were made before the density of the material reached its equilibrium value.

26.3 THERMORHEOLOGICAL COMPLEXITIES In the frame work of the free volume theory, the molecular mobility at any temperature is assumed to depend primarily on the free volume remaining. It is generally further assumed in this approach that this molecular mobility determines the temperature dependence of the shift factors of all different kinds of molecular motions involving various length scales in the polymer melt. Hence free volume approach usually purports that the temperature shift factors of different viscoelastic mechanisms are the same. This result coming from the free volume theory of molecular mobility is perhaps the justification of the practice of obtaining the (meaning that there is only one) shift factor curve, aT , which is usually derived by superposing curves of viscoelastic functions measured at different temperatures within the time or frequency range of the instrumentation. So long as the curve of the stress relaxation modulus, the creep and recoverable compliance, dynamic moduli or compliance do not change their shape in logarithmic plots, unique reduced curves with extended time or frequency range can be obtained. This will be the case, principally, if all of the molecular mechanisms contributing to the time- and frequency-dependent modulus and compliance functions, have the same temperature dependence. When this is so, the polymer is identified as being thermorheologically simple [18]. This appears to be true, in general, for closely related mechanisms, i.e., those within a group contributing

26.4 TEMPERATURE DEPENDENCES UNDER PRESSURE The loss of molecular mobility on approaching the glassy state by decreasing temperature may be due to increased molecular crowding (decrease in free volume) as well as a decrease in thermal energy (decrease in entropy). The relative importance of these two factors, volume (or free volume) and thermal energy (or entropy) has been a controversial issue for many years. It cannot be resolved by temperature variations alone in experimental studies, since the volume, entropy, and thermal energy all depend on temperature. The introduction of pressure, P, as an additional experimental variable makes a difference, because the specific volume, V, can be altered while maintaining temperature, T, constant. By combining the dielectric or light scattering results for polymeric and nonpolymeric glass-formers with the corresponding equation of state (PVT data), the volume and temperature dependence of the primary (local segmental for polymers) relaxation times t a can be obtained [181–185]. The results indicate that in general neither T (or entropy) nor V is exclusively the appropriate thermodynamic variable for describing the dynamics of glass formers, but rather t a is a function of the product variable, T 1 V g , where g is material-dependent, reflecting the nature of the intermolecular potential. Number polymers have been studied thus far, yielding the following

TEMPERATURE DEPENDENCES OF THE VISCOELASTIC RESPONSE OF POLYMER SYSTEMS

/

457

TABLE 26.1. WLF parameters characterizing temperature dependencies of shift factors for relaxation and retardation times in various polymer systems. T0 K

C1  K

C2  K

Tg  K

Ref.

Poly(acetaldehyde)

243

14.5

24

243

[33]

.

Polyethylene (solution chlorinated) Cl content ¼ 56.6 w/w, amorphous Poly(hexene-1)

312.3

12.7

63.3

317

[180]

.

218

17.4

51.6

218

[34]

.



Polymer

Comments

Nonaromatic hydrocarbon backbone polymers

Polyisobutylene PIB (the NBS sample with M ¼ 1:3  106 distributed by R.S. Marvin [35–37] on which the most comprehensive studies of viscoelastic properties were carried out in many laboratories)

298

8.61

200.4

201

[38,39]

.

197

6.14

56

201

[40–42], [19–20], [43]

.

228.5

13.18

130.9

201

[20,43]

.

245.3

21.24

147.7

[43,20]

.

215.3

20.36

117.7

[43,20]

.

aT ,a of local segmental motion from dielectric relaxation data. aT ,a of local segmental motion from a combination of dielectric relaxation and dynamic mechanical relaxation data for log t a ¼ 12:7 þ 804= (T  249 K). The shift factors, aT ,S , of the softening dispersion with G 0 ranging from about 106 < G 0 < 109:2 dyne=cm2 and temperature T from 27.5 8C to 70 8C. aT ,S of the entire softening dispersion, from dynamic mechanical measurement of J *(f ), 10 3  104 s from 74.2 to 35.8 8C. t Rouse (66.9 8C) ¼ 2.74. aT ,a of local segmental mode from G(t ) and Jr (t) data in the softening dispersion for viscoelastic response with Jr (t) < Jea (where Jea is the relaxed compliance of the local segmental motion and has the value of approximately 5 times the glassy compliance Jg ) and combined with data from the resolved local segmental motion obtained by photon correlation spectroscopy. Log t a (T ¼ 66:9C) ¼ 0:5. aT ,sub-Rouse the sub-Rouse modes in the softening dispersion located in the compliance range of Jea < Jr (t) < 108 cm2 dyne and resolved as a tan d peak situated at a higher frequency than the Rouse tan d peak by a combination of isothermal creep and dynamic mechanical measurements in the real time range of 106 < t < 3  104 s from 74.2 to 35.8 8C. log tsub-Rouse (66.9 8C) ¼ 1.63.

458 / CHAPTER 26 TABLE 26.1. Continued. Polymer PIB

PIB (E-19) M ¼ 78 500 PIB

PIB 2:7  104 < Mw < 7  105 PIB (M ¼ 4900) Polypropylene PP (atactic )

T0 K

C1  K

C2  K

Tg  K

Ref.

205

14.3

72.5

201

[44]

.

205

13.7

64.8

201

[44]

.

198

15.99

62.99 200.4

[45]

.

202

16.96

80

202

[47]

.

298.2

7.49 192

205

[48]

.

298.2

7.60 184

[49]

.

298

7.53

85

262

[29]

.

298

6.86

65

262

[29]

.



Comments

Same sample as above

267.74

12.9

34.74 262

[50]

.

Same sample as above

262.65

13.14

21.7

[51]

.

PP (atactic )

253

18.2

47.6

[7a]

.

258

14.5

30

[7b]

.

262

258

aT ,Rouse resolved by a combination of stress relaxation and dynamic birefringence measurements in the range 100:2 < t < 103:8 s. Log t Rouse (66.9 8C) ¼ 2.13. This result is in fair though not perfect agreement with that obtained by the tan d peak (see above). aT ,G of the ‘‘stress that relaxes through monomer rotation around the chain axis’’ resolved by a combination of stress relaxation and dynamic birefringence measurements in the range 100:2 < t < 103:8 s. Log t G (  66:9 C) ¼ 0:1. This technique has not resolved the sub-Rouse modes possibly because time temperature superposition was used. Hence the result is a compromise between the local segmental mode and the sub-Rouse modes. Log t G (  68:2 C) ¼ 0:77 which is close to the average between ta and t sub-Rouse at 66.9 8C obtained from isothermal mechanical data taken over 9 decades of real time (see above). aT ,J(t) of the entire viscoelastic spectrum from Jr (t). Its T-dependence is similar to that of viscous flow, h. aT ,h of viscosity h. When extrapolated down to lower temperatures, its temperature dependence remarkably (in the sense that this does not happen in most other polymers) is nearly the same as that of aT ,G(t) discussed above [Tobolsky and coworkers, Refs. 40–42]. aT ,h of the terminal dispersion measured from 243 to 473 K. aT ,h of viscosity from 50 to 170 8C. Its temperature dependence similar to that given above. aT ,h of viscosity measured up to 1012:6 poise at 266 K. It has a weaker temperature dependence than that of aT ,S of the softening dispersion in the temperature range where the viscoelastic response has Jr (t) principally less than 108 cm2 =dyne. aT ,S of the softening dispersion from Jr (t). It has a stronger temperature dependence than that of aT ,h . aT ,f of local segmental motion from correlation functions measured by photon correlation spectroscopy (PCS) carried out in the temperature range of 268 200 S=cm show Drude behavior for a small fraction of the conduction electrons which essentially percolate through the film while the remaining conduction electrons are more localized.

doping routes, and processing are improved, further advances in materials properties can be expected. Estimates of the ultimate conductivity [122] of the quasi-onedimensional polymer systems assuming the primary momentum relaxations are from 2kF phonons which have modest population at room temperature suggests the ultimate conductivity for (CH)x is
2  106 S=cm (compared to 5:5  105 S=cm for copper). From the experimental data which exist for current systems, estimates of the intrinsic conductivity also can be made [197]. The intrinsic Drude nature of metallic carriers has been identified using both microwave and optical techniques. Both of these techniques have identified the presence of a group of carriers which demonstrate Drude behavior with a long scattering time (t
1011 s). The Drude conductivity for traditional metals is given by

46.9 ULTIMATE CONDUCTIVITY 46.9.1 Drude Model Analysis The intrinsic conductivity of conducting polymers is of interest for fundamental science and future materials applications. Though the materials currently have sDC less than common metals, the conducting polymers are not fully crystalline. Past progress suggests that as the synthesis,

OPtical Conductivity (S/cm)

1200

900 A

600 B 300 C 0 0.01

0.10

1.0

Energy (eV)

FIGURE 46.32. Room temperature optical conductivity versus frequency for PPy(PF6 ) (A), PPy(TsO) (B), and PPy(S-PHE) (C) (from Refs. [74, 192, and 224]).

744 / CHAPTER 46

CONDUCTIVITY / 103 Scm−1

2.0

1.0

1 2 0 101

102

3 103 104 −1 WAVE NUMBER / cm

105

FIGURE 46.33. Optical conductivity versus frequency for poly-(methylthiophene) (PMT) doped with PF6 as a function of doping level (from Ref. [190]). The spectra are for different doping levels [with (1) being highest and (3) the lowest doping level].

s ¼ ne2 t=m . In the present systems, only a small fraction (
0.1%) of the conduction electrons show this Drude behavior. If all of the conduction electrons (determined by doping percentage) have a scattering time equivalent to t
1011 s, then sultimate
107 S=cm. 46.9.2 Resonance Quantum Transport in Doped Conducting Polymers Metallic doped polymers (polyaniline and polypyrrole) have an electromagnetic response that, when analyzed within the standard theory of metals, is provided by an extremely small fraction of the total number of available electrons
0.1% (in contrast to
100% for common metals) but with anomalous long scattering time t $ 1013 s (more then 100 times longer than for common metals). Prigodin and Epstein have shown [220] that a network of metallic grains (polymer’s crystalline domains) connected by resonance quantum tunneling through localized states in surrounding disordered medium produces this behavior. The small fraction of electrons is assigned to the low density of resonance states and the long scattering time is related to the narrow width of energy levels in resonance. This differs from the general consensus [221–224] that the difference in doped conducting polymers between metal and insulator at low temperatures is caused by disorder driven localization of the conduction electrons (Anderson insulator–metal transition (IMT)) [153].

Figure 46.34 reviews the two conventional single particle transport mechanisms. For band transport (Drude theory) electrons behave as ‘‘free particles’’. They are accelerated by the applied electric field and loose their momentum through scattering by impurities and phonons. As a result, an electron’s motion may be described as quantum diffusion. At low temperature the phonon scattering becomes weak and the conductivity increases with decreasing temperature to its residual value. For hopping the zero temperature electrical conductivity is zero because the charge carriers are localized. At finite temperatures electrons hop from one localized state to another by absorbing or emitting phonons. In contrast to band transport, hopping conductivity increases with temperature because of increased availability of phonons (see Fig. 46.34). The dielectric constant « can be used to identify the mechanism of charge transport [222,224]. For band transport, « is negative because of inertia of ‘‘free electrons’’; for hopping « is positive and proportional to the square of the size of the localized state. Hopping often is expected to be the basic mechanism of transport for polymers because of their irregular structure and because even weak disorder localizes electrons in single isolated chains. This type of conductivity is observed in the dielectric polymers (doped polymers that become insulators at low temperatures) and their behavior qualitatively follows the above dependencies. However, the behavior of metallic polymers can be understood within neither the above hopping model nor the model of conventional metal with band electrons. The metallic conductivity of doped polymers decreases with decreasing temperature and a finite residual conductivity is within a decade of the room temperature value. Though the decrease of the dc conductivity for metallic polymers with decreasing temperature can be accounted for within the band model by effects of localization caused by disorder, the experimental optical and low frequency conductivity and dielectric constant in the metallic state of doped polymers in principle cannot be accounted for by band models with homogeneous disorder. Experiments [221–224] show that the high frequency ($ 0.1 eV) conductivity and dielectric constant generally follows a Drude law with the number of electrons
1021 cm3 corresponding to the total density of conduction electrons and conventional scattering time
1015 s in both the metallic and dielectric phases (Figs. 46.26–46.28). At decreasing frequency the polymers in the dielectric phase progressively display insulator properties and « becomes positive for frequency # 0.1 eV signaling that charge carriers are now localized. Microwave frequency (
6.6 GHz) « experiments [222] yield localization lengths
5 nm, depending on sample. A surprising and remarkable feature of the metallic phase in polymers is that «(v) is similar to that of dielectric samples in magnitude and sign with decreasing frequency, also changing sign from negative to positive at approximately the same frequency
0.1 eV. However, for metallic

CONDUCTING POLYMERS: ELECTRICAL CONDUCTIVITY /

Band Transport

Hopping Transport

1. Extended states: s (T=0) = s0

1. Localized states: s (T=0) = 0

2. Polarization is out-of-phase with an applied electric field due to electron inertia:

2. Polarization is in-phase with an applied electric field because the electron is bound within x :

e(w ~0) = −e 2 N(e F) l 2 sB > sC > sD (Inset, sDC (T)). (after Ref. [222]).

These experimental data contrast with the Anderson IMT [153] in which electronic behavior is controlled by homogeneous disorder. In the dielectric phase electrons are bound by fluctuations of the random potential. On the metallic side of the transition, free carriers have short scattering times. In the metallic phase near the transition, « is positive because the disorder causes dynamic polarization due to slowing diffusion by localization effects. When approaching the IMT the localization effects increase and « diverges (‘‘dielectric catastrophe’’ [200]). The small plasma frequency and very long t of the metallic state in doped polymers can be explained [222, 224,225] assuming that the conductivity is provided by a small fraction
0.1% of the total carriers with long scattering time > 1013 s. However, it is difficult to reconcile this conclusion with the behavior for high frequencies which supports that the scattering time is usual
1015 s and all available electrons participate in conduction. To account for these anomalies the possible presence of a collective mode, as in a charge density wave conductor, or superconductor, was suggested [199]. Prigodin and Epstein proposed [220] that in highly conducting polymers there is a new mechanism of charge transport, resonance quantum tunneling among metallic domains. These materials are strongly inhomogeneous [18,62,127,133] with ‘‘crystalline’’ regions within which polymer chains are well ordered (Fig. 46.36). Electrons are

746 / CHAPTER 46 simplicity one may assume that the two nearest grains are electrically connected by
N? =z chains, where z is the number of nearest neighboring grains. In the metallic phase the intergrain coupling leads to broadening of quantized levels in the grains, dE ¼ 2N? gDE, where g is the transmission coefficient between grains through a single chain. The IMT occurs when dE
DE and the critical chain-link coupling gc satisfies

LII

2N? gc ¼ 1:

R

FIGURE 46.36. Schematic view of the structure of polyaniline and polypyrrole. The lines represent polymer chains. The dashed squares represent regions where polymer chains have crystalline order.

considered to be three-dimensionally delocalized over grains due to good overlap between chains within the grains. At least, when the IMT is approached, an electron’s delocalization first occurs inside these regions. Outside the crystalline regions the chain order is poor and the electronic wave functions are strongly localized within single chains. Therefore, the crystalline domains can be considered as nanoscale metallic dots embedded in a disordered poorly conducting medium. The metallic grains remain always spatially separated by disordered regions, and, therefore, direct tunneling between grains is exponentially suppressed. The intergrain tunneling is possible through intermediate localized states in the disordered portion with strong contribution from resonance states whose energy is close to the Fermi level (Fig. 46.37). The dynamics of resonance tunneling can account for the frequency-dependent anomalies in the conductivity and dielectric constant of the metallic phase of these doped polymers. Reference [220] provides a detailed description and quantitative analysis of this model. Within the Prigodin and Epstein model each grain is coupled to other grains by 2N? independent chains. For

(46:12)

For PAN(HCl) and similar PAN(CSA) this yields gc
102 . If g < gc the system is a dielectric and the behavior (46.4) is retained for all vt T  1. However, on the metallic side (g > gc ) electrons are delocalized and their low-frequency motion is a random walk among the grains. Introducing the mean transition rate, W, for hopping between the grains and the mean distance between the centers of neighboring grains, R(b
(Lk =R)3 ), the corresponding diffusion coefficient D3 and the macroscopic conductivity are D3 ¼ R2 W, s(v
0) ¼ be2 N(«F )D3 :

(46:13)

When approaching the IMT from the metallic side W tends to 0 as [12]: W ¼ (DE=(2z) ) exp [  2p(gc =(g  gc ) )1=2 ]. In the metallic phase the hopping frequency W is related to the above model parameters as W ¼ dE=(2z) ¼ (N? =z)gDE,

(46:14)

and the whole system can be represented as a network of random conductors. The nodes represent the grains where randomization of electronic motion happens. Further details of analysis and its application to specific polymers are in Ref. [220]. A principal difference between direct and resonance tunneling is the time for tunneling. Direct tunneling that occurs in a conventional granular metal is an almost instantaneous process, i.e., its characteristic time is the scattering time t. Resonance tunneling that is anticipated to be in the metallic polymers shows a delay determined by the level width g.

EF

EF

δE

∆E

Dot 1

Dot 2

FIGURE 46.37. Schematic illustration of the electrical coupling of the metallic grains being provided by resonance tunneling of between the quantized states of the metallic grains through localized states in the amorphous region.

CONDUCTING POLYMERS: ELECTRICAL CONDUCTIVITY /

The use of plastics and their composites is rapidly increasing in numerous areas. However, the final assembly of products is often limited by the capability of existing joining techniques. The ability of ICPs, especially polyanilines, to absorb electromagnetic radiation and convert it into heat introduces another application in the welding of thermoplastics and thermosets [210]. With the rapid advances and broad implementation of computer and telecommunication technologies there is an increased need to shield EMI, especially in the radio and microwave frequency ranges. Intrinsically conducting polymers are promising materials for shielding of EMI because of their relatively high conductivity and dielectric constant and the ease of control of their conductivity and dielectric constant through chemical processing [207]. Also, they are relatively lightweight compared to standard metals, flexible, and do not corrode as common metals. The microwave conductivity and dielectric constant of polyanilines are controllable through chemical processing (e.g., stretch ratio, molecular weight, doping level, counter ion, solvent, etc.). Figure 46.38 compares the total shielding efficiency of PAN–CSA (m-cresol) materials with that of copper on the base of mass/area, while Fig. 46.39 compares the shielding efficiency of several different conducting polymer systems. The ability to disperse conducting polymers into insulating hosts such as poly(3-octylthiophene) in polyethylene [217] and PAN–CSA in polymethylmethacrylate [218], or nylon [219], and achieve percolation at less than 1%, increases opportunities for applications. Active electronic devices fabricated from semiconducting organic and polymer materials have become of increasing interest. In particularly, regioregular poly(3-hexylthiophene) (P3-HT) has been of interest because its relative

The temperature dependence of the dc conductivity in the metallic phase follows from that expected for resonant tunneling through the strongly localized states in the amorphous regions of the metallic polymer. With increasing temperature phonons increase the localization length of the states in the disordered regions thereby increasing the resonant transmission rate between grains and increasing the conductivity. As a result the low-frequency part of electromagnetic response is shifted with increasing temperature to a range of higher frequencies as it experimentally is observed [222]. 46.10 APPLICATIONS Intrinsically conducting polymers (ICPs) also are of interest for a wide range of applications [204]. The ICPs have been proposed for use as conducting wires, in batteries [205], as electromagnetic interference (EMI) shielding materials [206–209], joining (welding) of plastic materials [210], light emitting diodes (LEDs) [211], sensors [212], anticorrosive coatings [213], etc. In LED studies, doped polyaniline and its blends have been used for the hole injecting layer [214], and undoped poly(p-phenylene vinylene) and other materials have been utilized for the light emitting layer [211]. More recently, a symmetrically configured alternating voltage light emitting (SCALE) device based on electronic polymers has been demonstrated [215,225,226,227]. The advantages of ICPs for light emitting devices include flexibility, mechanical strength, and relatively easy control of the color of light emission. Transparent conducting polymers may be incorporated as electrodes in efficient LEDs [216].

10000

747

SET (DB)

500 Cu 100

1000

Cu

polyaniline

SET (DB)

0

0

2 4 6 8 mass / area (mg/cm2)

10

500

polyaniline 0 0

20

40

mass / area

60

80

100

(mg/cm2)

FIGURE 46.38. Comparison of total shielding efficiency (SET ) of PAN–CSA (m-cresol) samples and copper (Cu) as a function of mass/area. Inset: magnification below 10 mg=cm2 (from Ref. [207]).

748 / CHAPTER 46 on Fig. 46.40 the ID current decreases with gate voltage VG similarly to that normally observed for conventional semiconductors. The ratio ION =IOFF reaches up to 104 in some devices [232–236]. Epstein et al. suggest [232,235] that the observed field effect in conducting polymers based ‘‘transistor’’ is closely related to the mesoscopic inhomogeneity of conducting polymers. They emphasize that the field effect cannot be observed if the conducting polymer is a conventional conductor. For conducting polymers with conductivity of
30 S/cm, the screening radius for an electric field produced by the FET gate electrode is expected to be less then 2 nm [232]. Epstein et al. propose that the field effect in conducting polymers is due to the ionic component of their charge conductivity. The inhomogeneous structure leaves enough free space for mobile ions. The ions inside the polymers produce the additional screening of the external field, but the crucial feature of ions is the ability of ions to migrate between the conducting polymer film and external dielectric layer interface. As a result the concentration of ions inside of the polymer is controlled by the gate potential. Due to electroneutrality the internal ionic density determines the concentration of primary charge carriers in the polymer and therefore it is anticipated that the polymer conductivity is governed by the gate potential. The critical concentration of ions for disrupting the conductivity along the polymer chains within the disordered regions was estimated within the following simple picture. If 50% crystallinity and size grain
10 nm were assumed, then intergrain hopping includes
ten intermediate sites along the chain linking two ordered regions. To interrupt this connection it is enough to introduce only one ion, i.e., approximately 5% ‘‘dedoping’’ would produce an appreciable effect. Figure 46.41 schematically illustrates the ability of a small concentration of excess ions to control the electrical conductivity of a doped polymers.

200 sample σmw(S/cm) A B C D E

SET (dB)

150

4700 940 560 290 110

10−5 εp ⏐tanδ⏐ −17 0.77 −3.9 0.67 −0.70 2.2 −1.0 0.80 0.99 0.32

A

B

100 C

D

50

E highly conducting polymers 0 20

0

40 Thickness (m m)

60

80

FIGURE 46.39. Comparison of total shielding efficiency (SET ) of highly conducting polymers versus sample thickness. Sample A: stretched heavily iodine doped Tsukamoto polyacetylene; sample B: unstretched heavily iodine doped Tsukamoto polyacetylene; sample C: PAN–CSA (m-cresol); sample D: PPy(PF6 ); sample E: PPy(TsO) (from Ref. [207]). The inset shows the microwave transport parameters smw , «t , and tan d for each of the materials.

structural order leads to a high mobility, in the range of 0:2 cm2 =(Vs) [228–231]. Recently a new interesting phenomenon, an electric field effect, was reported for the doped highly conducting polymers [232–235]. Figure 46.40 shows I–V characteristics of such a transistor [236] in which conducting polymer PEDOT:PSS [poly(ethylene dioxythiophene): poly(styrenesulfonic acid)] with sRT
30 S=cm is used as active electronic element and PVP [poly(vinylphenol)] is used as a dielectric separating the gate and source–drain channel. As it is shown

2.0µ

PEDOT:PSS(60 nm)/PVP(25 nm)/Al(40 nm)

1.0µ

lD(A)

VGS=+3.0V 0.0 Source electrode (S)

+1.5V −1.0µ

Gate electrode (G) +1.0V −2.0µ

+0.0V −5

−4

−4

Drain electrode (D) −4

−4

0 VD (V)

1

2

3

4

5

FIGURE 46.40. The drain–source current as the function of drain voltage of a thin PEDOT:PSS/PVP film. The insert shows the polymer-based transistor configuration (after Park et al. [236]).

CONDUCTING POLYMERS: ELECTRICAL CONDUCTIVITY /

⫹

FIGURE 46.41. Schematic illustration of ionic suppression of intergrain hopping. Compensation by cations of charge of acceptors removes the nearby localized states on the polymer backbone that provide for holes easy hopping of electrons between grains. Reprinted from Ref. [235] ß (2005) with permission from Elsevier.

46.11 NANOSTRUCTURING OF CONDUCTING POLYMERS The conventional chemical polymerization of polyaniline only produces nonfibrous or irregular shaped morphologies [237,238]. In the past several years, a variety of chemical methods were reported that yield polyaniline nanofibers, such as use of hard templates [239,240], soft templates [241], electrospinning [242], interfacial polymerization [237], and seeding polymerization [238]. Recently, Chiou and Epstein discovered that polyaniline nanofibers can be directly synthesized in dilute chemical polymerization

749

without aid of specific templates or techniques [243] (Fig. 46.42). The nanofibrous structures prepared from dilute polymerization have no significant change when dedoped and redoped multiple times by base and acid solutions, respectively. The dispersion in deionized water of nanofibers is stable for several minutes without aggregation and precipitation [239]. Furthermore, a highly porous film of nanofibers is obtained as the dispersion is cast and dried on the glass or silicon wafer substrates. The UV/vis absorption patterns of polyaniline nanofibers obtained are consistent with previously reported results for conventionally synthesized polyaniline. 46.12 SUMMARY The electrical transport properties of conducting polymers span the behaviors associated with semiconductors through to metals. Their properties depend critically upon the history of chemical synthesis, processing, and resulting structural order. Highly conducting doped polyacetylene, doped polypyrrole, and protonated polyaniline have similar dielectric responses, though on somewhat different scales. For each of these systems a positive dielectric constant is recorded at microwave frequencies for less conducting samples. A complex ‘‘metallic Drude response,’’ involving both an all conduction band electron response and a delocalized electron response, is detected for well-processed, highly conducting samples. The ability to engineer the electrical and dielectric properties using chemistry opens the opportunity for a wide range of applications from electrostatic dissipation to sensors and field effect transistors.

FIGURE 46.42. Nanofiber network of polyaniline prepared by dilute polymerization. Reprinted from [243] ß (2005) with permission from Wiley-VCH.

750 / CHAPTER 46 Development of nanofibers and other morphologies will lead to new fundamental science and new opportunities for applications.

ACKNOWLEDGMENTS This work was supported in part by the Office of Naval Research and National Science Foundation. The author thanks R. Kohlman, J. Joo, Vladimir Prigodin, Fang-Chi Hsu, June Hyoung Park, and Nan-Rong Chiou for discussions.

GLOSSARY OF TERMS Anderson Localization: spatial localization of electronic wavefunctions due to randomness of the electronic potential which causes a metal–insulator transition in sufficiently disordered materials. Antisoliton: solitons are present in materials with two degenerate phases A and B. If a soliton is a kink between A and B phase, then an antisoliton is a kink between B and A phase. Bipolaron: a bipolaron is similar to a polaron except that it is doubly charged, spinless, and both of its energy states in the band gap are totally filled or empty. Bloch Waves: delocalized electronic wavefunctions which have the form ck (~ r ) ¼ uk (~ r ) exp (i~ k ~ r ), where uk (~ r) is a function with the periodicity of the lattice unit cell and exp (ik~ ~ r ) is a wave of wavelength l ¼ 2p=k. Commensurate Charge Density Wave: a static modulation of the charge density in the system with a periodicity equal to a rational number multiplied by the underlying periodicity of the lattice. Due to the charge density wave, a gap is opened at the Fermi level which lowers the total energy of the system. Crosslinked Polymers: polymers with greater interaction between chains either through regions of greater crystallinity (physical crosslinks between the polymer chains), or through chemical bonding between chains. Crystalline Coherence Length: a length which characterizes the spatial correlations for the polymer chain, indicating the length over which the local order randomizes. This length is determined from the width of x-ray scattering peaks from the Scherrer formula. Curie Susceptibility: Paramagnetic susceptibility due to uncoupled spins free to align in a magnetic field and subject only to thermal fluctuations. The Curie susceptibility is given by wCurie ¼ C=T, where C is the Curie constant (0.375 emu K/mol) and T is the temperature. Degenerate Ground State: for a degenerate ground state, the conjugation path is such that reversal of the single and double bonds results in a phase of the system with an equivalent energy.

Doping: a process whereby charges are removed or added to the polymer chain, altering the electronic structure and response. Dru¨de Model: this model of the electrons in a conductor treats the electrons as free, subject only to dissipative, inertial, and electromagnetic forces. In this model, the conductivity s(v) and the dielectric function «(v) are given as s(v) ¼ (V2p t=4p)=(1  ivt) and «(v) ¼ «B  V2p =(v(v þ i=t) ), where Vp is the plasma frequency and t is the mean scattering for transport. Electron–Electron Interactions: a broad term referring to the electromagnetic interaction between electrons as well as some of the effects of the Pauli exclusion principle. Exciton: an electron–hole pair bound by Coulombic forces capable of transferring energy but not charge because it is electrically neutral. Hole: a vacant orbital in an energy band which acts as a positive charge in an applied electric or magnetic field. Hopping Transport: a form of charge transport which involves electron motion from one spatially localized state to another accompanied by the absorption or emission of a phonon. Incommensurate Charge Density Wave: similar to a commensurate charge density wave except that the periodicity of the charge density modulation does not equal a rational number multiplied by the periodicity of the underlying lattice. Inhomogeneous Disorder: Structural configuration for a polymer solid which consists of a mixture of ordered (crystalline) and disordered regions of the polymer. Kramers–Kronig Analysis: a set of mathematical relations due to causality which relate the real (dispersive) and imaginary (absorptive) parts of a physical quantity. These relations can be used to determine the imaginary part of a quantity given information about the real part and vice versa. Localization Modified Drude Model: a model for conduction electrons which includes suppression of the Drude conductivity at low frequencies due to finite localization lengths for the electrons. Localized States: electronic states which are not extended over the entire solid as Bloch waves are localized states. The spatial dependence of the wavefunctions of a localized state is usually assumed to vary as jc(~ r )j
exp (  j~ r ~ r0 j=j), decaying exponentially in a characteristic length j, the localization length, away from ~ r0 . Charge transport by electrons in these states is due to hopping. Lorentz Model: this model treats electrons as bound strongly to an atom, subject to dissipative, inertial, electromagnetic, as well as restoring forces. In this model, the dielectric function «(v) is given by «(v) ¼ «B þ V2p = (v20  v2  iv=t), where «B is the background dielectric function due to everything else, Vp is the plasma frequency, v0 is the binding energy, and t is the mean scattering time.

CONDUCTING POLYMERS: ELECTRICAL CONDUCTIVITY / Mesoscopic Metallic State: a metallic state in an inhomogeneous system in which conduction electrons are delocalized over a number of crystalline regions (with disordered polymer regions between them). The size of the localization length is
102 ---103 A, smaller than macroscopic dimensions (104 A or greater). Mobility Edge: the critical energy which separates electronic states which are spatially localized due to disorder and thus have zero contribution to the electrical conductivity at very low temperature from those which are delocalized and therefore have nonzero contribution to the electrical conductivity at low temperature. Mott Variable Range Hopping: a form of hopping transport which results when the electron may hop to a distant site instead of just a neighboring site if the energy difference between its current site and the distant site is smaller than the difference between its current site and the neighboring sites. Mott variable range conductivity has a form given by s(T) ffi s0 exp [  (T0 =T)1=(dþ1) ], where d is the dimensionality of the hops and T0 is a reduced activation energy. Nondegenerate Ground State: for a nondegenerate ground state, the conjugation path is such that reversal of the single and double bonds results in a distinctly different energy. One-Dimensional Chain: a linear system for which the interactions along the chain direction are much stronger than the interactions perpendicular to the chain. pz Orbitals: electron wavefunctions with atomic principal quantum number p character which have a node in the x–y plane (the x–y plane is usually taken as the plane of the polymer sp2 bonds.) Electrons in these orbitals usually pair to form double bonds and provide the conjugation responsible for the interesting electronic properties of conducting polymers. p Conjugation: alternating single and double bonds in a single plane due to the overlap of atomic pz orbitals along the polymer backbone. p conjugation leads to the electronic bands responsible for the interesting electronic properties of conducting polymers. Pauli Susceptibility: paramagnetic, approximately temperature independent magnetic susceptibility due to conduction electrons. The Pauli susceptibility, wPauli ¼ 2m2B N(EF ), where mB is a Bohr magneton and N(EF ) is the density of states at the Fermi level. Peierls Instability: an instability prominent in quasi-onedimensional systems with strong electron–phonon interactions due to which the lattice spontaneously distorts with a 2kF (kF is the Fermi wavevector) periodicity, forming a gap at the Fermi level which lowers the total energy of the system. Percolation: in a solid made up of more than one component (a composite system), the volume fraction of the different components can be varied. Percolation refers to the transitions which occur when the volume fraction of a component is such that there are connected paths of that

751

component across the material. For example, in a composite of a metal and an insulator, the metal particles percolate when they form a connected path across the material and finite dc conductivity becomes possible. Phonon: a quantum of lattice vibrational energy which reflects the normal vibrational modes of the lattice allowed by symmetry. Phonon Induced Delocalization: in disordered solids, localization can result when a wavefunction interferes with itself due to elastic scattering and forms a standing wave. Phonon scattering can destroy this interference effect and cause the wavefunctions to be more extended. Photoexcitation: the use of light (photons) to cause transitions of electrons from the ground state to excited states of the system. Plasma Frequency: defined as Vp ¼ 4pne2 =m , where n is the volume density of conduction electrons, e is the charge of an electron, and m is the effective mass renormalized from the free electron mass by lattice and interaction effects. Plasma Response: an excitation of a solid for which the negative charge in the solid is displaced uniformly with respect to the ions. Plasma oscillations occur when the dielectric function is equal to zero. Polaron: most generally a localized electronic state accompanied by a surrounding lattice distortion. In conduction polymers, it has been discussed as a bound state of a soliton and an antisoliton. This excitation can occur in degenerate and nondegenerate ground state polymers. A polaron possesses a single charge with normal spin–charge relations (i.e., with a single charge, it also has spin 1/2) and usually two states in the bandgap of the neutral polymer, one of which is half filled. Polaron Lattice: a uniform periodic array of ‘‘polarons’’ assumed stabilized against a Peierls distortion by interchain interaction. The band structure for a polaron lattice is metallic. Screened Plasma Frequency: the plasma frequency normalized or screened by a background dielectric constant. pffiffiffiffiffi The screened plasma frequency, vp ¼ Vp = «B , where Vp is the plasma frequency and «B is the background dielectric constant due to all excitations except the conduction electrons. Soliton: a low energy excitation of the electronic system which is localized in space and maintains its identity in the presence of other excitations. In conducting polymers, a soliton takes the form of a kink or misfit between two distinct energetically equivalent phases (in degenerate ground state systems). Its properties include a reversed spin–charge relation (i.e., when it is charge neutral, it has spin 1/2), the introduction of a single energy level within the band gap of the polymer, and a lattice distortion surrounding the soliton. Time Reversal Symmetry: a symmetry of a system characterized by replacing t (time) with –t without changing the physics of the system.

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CHAPTER 47

Electroluminescent Polymer Systems Leni Akcelrud Departamento de Quimica, Centro Politecnico da UFPR, Universidade Federal do Parana, CP 19081, CEP 81531-990 Curitiba, Parana, Brazil

47.1 47.2

47.3 47.4 47.5 47.6 47.7 47.8 47.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light Emitting Devices and Mechanism for Light Emission . . . . . . . . . . . . . . . . . . . 47.2.1 Interchain Excitons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.2.2 Transport Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PPV and PPV-Type Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugation Confinement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polythiophenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyano polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poly(p-phenylene)s (PPP) and Polyfluorenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silicon-containing polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nitrogen-Containing Conjugated Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

757 758 758 759 759 763 765 767 768 775 777 782

broad range of colors can be achieved with one polymer. It may be noted that both low and high molecular weight organic EL materials are now under wide study. The perceived advantages of the former include the possibility of a definitive chemical structure, chemical purification at high purity levels by sublimation, and facile manufacture of complex 3D architectures, while polymers are favored for their mechanical properties, principally the ready formation of robust films and their processibility using easily accessible technology for simple device architectures. The two classes of materials are not infrequently used together in multicomponent chromophoric layers. A typical design of a polymer LED is shown in Fig. 47.1. More elaborate architectures may vary from this basic scheme, and may involve the use of a multicomponent chromophore and one or more transport layers (see below). Recent advances include microfabrication of diode-pixel arrays [19], patterned light emission with sizes of the order of 0.8 mm [20], polarized EL based on stretch or, rub aligned Langmuir-Blodgett deposited polymers, or specifically synthesized liquid crystal polymers [21–27]. The application of ultrathin and selfassembling films is an important development in LED technology. In this case the film in the device is not cast using a traditional processing technique such as spin coating, but using electrostatic layer-by-

47.1 INTRODUCTION Organic polymers that emit light on the imposition of an electric field have commanded increasing attention in the last decade both for their scientific interest and as potential materials for electrooptical and optoelectronic applications. A number of reviews on electroluminescent polymers focusing the basic physics [1–5], synthesis and properties [6,7], device operation and materials [8–11], design and synthesis [12] blue emitting structures [13] have been published. Some books are also out on the subject [14–18]. Apart from intrinsic electronic features, the color emitted by small organic molecules depends on microenvironment characteristics, such as their location in the device and medium polarity. When attached to a polymer chain, the mobility of the chromophore is restricted in all directions, and the emission becomes dependent on the structural features of the macromolecule (including the molecule’s architecture, such as regioregularity, location and distribution of chromophores, etc.). This restriction opened a new avenue in the development of electroluminescent polymers: color tuning could be achieved by introducing variations in the polymeric structure, since in doing so the energy gap of the p---p
transition responsible for the color emitted is changed, and a 757

758 / CHAPTER 47 Metal Cathode

Electron Transport Layer (ETL) E L Polymer Hole Transport Layer (HTL)

Glass Substract ITO Anode

Light Output

FIGURE 47.1. Design of a polymer LED showing optional HTL and ETL.

layer selfassembly methodologies [28–31]. This means that recent advances in the molecular level processing of conducting polymers have made it possible to fabricate thin film multilayer heterostructures with a high degree of control over the structural features and thickness of the deposited layers. As the dimensions of the individual layers approach molecular scales it may be possible to approach quantum effects in these multilayer contacts. Further progress is represented by the development of surface light-emitting devices (SLEDS), in which both anode and cathode lay underneath the electroluminescent layer, so that no transparent materials are required in the LED construction. These SLEDS were microfabricated using conventional silicon processing [32]. The patterning of light-emitting layers is the most important step in the manufacturing of multicolor organic electroluminescent devices, and should combine large area coatings with device patterning. One very promising methodology employs an ink jet patterning process [33–35]. Light emission of poly [3-(2-benzotriazolo)ethylthiophene] has been investigated by the cathodoluminescence (CL) spectroscopy. An electron beam was used to inject directly electrons and holes in the polymer. The introduction of benzotriazole, an electronwithdrawing moiety, to the thiophene was done to enhance electronic affinity [36]. 47.2 LIGHT EMITTING DEVICES AND MECHANISM FOR LIGHT EMISSION Light-emitting devices as shown in Fig. 47.1 can be operated in a continuous DC or AC mode. They behave like a rectifier, the forward bias corresponding to a positive voltage on the ITO electrode and are also called light-emitting diodes in analogy to p–n junction devices. Light emission is transmitted from the transparent side normal to the plane of the device. The polymer layer is usually deposited by spin coating, but dipping techniques can also be used. The cathode injects electrons in the conduction band of the polymer (p
state), which corresponds to the lowest unoccupied molecular orbital (LUMO), and the anode injects holes in the valence band (p state), which corresponds to the highest occupied molecular orbital (HOMO). The injected charges (polarons) can travel from one electrode to the other, be

annihilated by any specific process such as multiphonon emission, Auger processes, or surface recombination. These concepts have been extensively studied in inorganic systems and may also apply to polymer systems. A simplified description involves the formation of a neutral species, called an exciton, through the combination of electrons and holes. The exciton can be in the singlet or triplet state according to spin statistics. Because only singlets can decay radiatively, and there is only one singlet for each three triplet states, the maximum quantum efficiency (photons emitted per electron injected) attainable with fluorescent polymers is theoretically 25%. This limit can be overcome by using phosphorescent materials that can generate emission from both singlet and triplet excitons [37]. As a result, the internal quantum efficiency can reach 100%. Forrest et al. [38] reported highly efficient phosphorescent LEDs by doping an organic matrix with heavy atoms containing phosphorus. Polymer devices were also fabricated by using polyfluorene [39] or poly(vinyl carbazole) [40] as the host for the phosphorescent dye. The singlet exciton decay time is typically in the ns timescale, whereas the triplet survives for up to 1 ms at low temperatures [41]. A major fact determining the internal quantum yield for luminescence (ratio of radiative to nonradiative processes) is the competition between radiative and nonradiative decays of the electron–hole pairs created within the polymer layer. These pairs can migrate along the chains and are therefore susceptible to trapping at quenching sites where nonradiative processes may occur. The sequence of charge processes leading to exciton formation is charge injection, transport, and recombination. These processes are difficult to separate on the basis of the device electrical characteristics, and the transport mechanism affects the other two. Two modes of injection mechanisms have been discussed for the operation of LEDs: thermoionic emission over a Schottky-like contact, and tunneling into the transport bands. Theoretical modeling of charge injection has been attempted by several approaches [42–48]. 47.2.1 Interchain Excitons In LEDs the polymers are thin films, leading to the possibility of electronic interactions between neighboring

ELECTROLUMINESCENT POLYMER SYSTEMS chains and the creation of new excited state species. This topic has been the subject of many recent investigations [49–52] mainly related to PPV [53,54], PPV derivatives such as poly(2-methoxy-1,4-PPV), MEH-PPV [55–60], CN-PPV [51,55,61], poly(p-pyridyl vinylene) [59,62], acetoxy PPV [63], and other light-emitting polymers [49,64–68], showing good evidence to suggest that interchain excitations play a significant role. The importance of these interchain excitations continues to be one of debate: if they are nonemissive, then they are detrimental to device operation, but if they are emissive, they can be used effectively [69]. The final morphology has a direct effect on the performance of MEH-PPV based LEDs. Higher degrees of interchain interactions enhance the mobility of charge carriers at the expense of lower quantum efficiencies for EL. The reduction in efficiency in well-packed regions is attributed to rapid formation of nonemissive interchain species without the involvement of ground state dimers or aggregates [60,70].

47.2.2 Transport Layers Single layer device architecture is typically employed and is appropriate for evaluation of new polymer chromophores and for measurements of their EL and PL spectra. In the simplest cases the two spectra are quantitatively identical, although in numerous cases their noncoincidence reveals a more complicated exciton formation and decay related to the differing modes of energy input. In devices which are intended to maximize photonic output and efficiency, however, it is established practice to employ additional layers of organic material (polymeric or low molecular weight) interspersed between chromophore film and the electrodes using materials chosen for their functional ability to facilitate charge transport and block (localize) carriers, avoiding their crossing the device without recombination. Usually the carriers do not form junctions with identical (or zero) barrier heights and therefore one carrier will be preferentially injected. If the two junction barriers are not identical, higher electric fields would be required near the junction with the greater barrier energy in order to equalize the injected current density from each contact [71]. The LED efficiency is also reduced if the excitons are formed at the interface of the polymer and the electrode, lowering the carrier injection. This location is also where the greatest number of defects is expected and can act as quenching sites [72,73]. The transport layer also decreases exciton quenching near the metal electrode by acting as spacer separating the metallic contact from the active luminescent layer. To confine holes in the emissive layer an electronconducting-hole blocking layer should be used (electron transport layer, ETL). Its valence band should be lower in energy than the EL layer and its electron affinity should be equal to or greater than the EL layer. In this way holes are confined between the emissive layer and the ETL, and the space charge formed provides a higher electric field across

/

759

the interface with a more uniform distribution of charge, thus improving the balance between carriers. The same reasoning is valid for the use of hole transport layers (HTL). The use of transport materials represents an improvement in device stability since it makes it possible in certain cases to change from a lower work function fw metal electrode as calcium (fw ¼ 2.9 eV) which is unstable in atmospheric conditions to a higher work function material as aluminum (fw ¼ 4.3 eV), and adds it directly to the emitting polymer. Materials for ETL are electron deficient and the most used are oxadiazole compounds in the ‘‘free’’ form as PBD or grafted to a polymer main chain. Apart from PPV, a variety of electron-accepting polymers such as poly(vinyl carbazole) (PVK) [74,75], poly(pyridine-2,5-diyl), poly(1,10-phenanthroline-3-diyl), or poly(4,40-disubstituted-2,20-bithiazole5,50-diyl) have been used as HTL materials. The incorporation of the transport/blocking material can also be made directly by blending them with the emissive material [76], as in the case of the green emitter poly(2-cholestanoxy-5-hexyldimethylsylyl-1,4-phenylene vinylene) (CCSPPV). With the combination of PVK as HTL and PBD as ETL it was possible to achieve internal quantum efficiency in excess of 4%. Several p-doped conjugated polymers have been used as hole injecting electrodes, like polypyrrole, polythiophene derivatives, and polyaniline [77–82], which have high work functions, providing low barriers for hole injection. There are reports of stable operation over long times for devices using polymeric dopants, which are expected to be relatively immobile. These include polystyrenesulfonic acid used to dope poly(dioxyethylene thienylene) (PEDOT) [3].

47.3 PPV AND PPV-TYPE STRUCTURES 47.3.1 Precursor Routes 47.3.1.1

The Wessling method

Like most highly conjugated materials, semiconducting polymers show poor solubility in organic solvents. Structural changes have been made to overcome this difficulty. The first highly structured electroluminescent polymer, PPV, a green-yellow emitter, was prepared via a precursor route because its insolubility in poly reactions resulted in only oligomeric materials. The precursor route involves the preparation of a soluble polymer intermediate that is cast in the appropriate substrate and after thermal treatment is converted to the final product in situ. This involves producing a polymer in which the arylene units are connected by ethylene units. The saturated units in the precursor contain a group which not only solubilizes the macromolecule and allows for processing, but also acts as a leaving group, thus affording the unsaturated vinylene units of a fully conjugated polymer. One of the most important soluble precursor routes to PPV was developed by Wessling and co-workers in the 1960s [83,84] based upon aqueous solvent synthesis

760 / CHAPTER 47

CIH2C

CH3 − CH2 − S⊕ Cl

CH3 Cl

− ⊕S

50°C

CH2Cl + H3C—S—CH3

CH2

CH3

MeOH / H2O (8:2)

OH − /0°C Cast Film (In vacuum)

CH3 Cl− ⊕ H3C S

CH3

CH2-CH

elimination T = 200°C

n

Precursor

n

PPV

FIGURE 47.2. The sulfonium precursor route (Wessling) to PPV.

of poly(p-xylylene-a-dialkylsulfonium halides) from a-a0 bis(dialkylsulfonium salts), followed by thermolytic formation of the final conjugated polymer, as shown in Fig. 47.2. The charged sulfonium groups solubilize the polymer and are removed during the conversion step. The mechanism is believed to proceed according to a chain growth polymerization via the in situ generation of the monomer, a pquinomethane-like intermediate [85,86], based on the facts that high molecular weight is formed very quickly, within the first minutes of the reaction and also that various radical inhibitors limit or prevent formation of long polyelectrolyte chains. However, the initiation process was not unequivocally identified [87–91]. Figure 47.3 depicts the radical chain mechanism for PPV synthesis [92]. A modified Wessling route where the solubilizing and leaving group is an alkoxy group has been developed and gives methoxyprecursor polymers, which are soluble in polar aprotic solvents such as chloroform, dichloromethane, and tetrahydrofurane [93,94]. The generation of precursor copolymers containing randomly placed methoxy and acetate groups (which are expected to be more labile to elimination) was an approach

A−

A− + S R3 R4

R1

A−

+ S R3 R4

used to prepare poly(2,5-dimethoxy-p-phenylene vinylene) (DMPPV) of controlled conjugation length [95,96]. A soluble PPV derivative which could be used directly without a second step treatment was a natural development since it would simplify device fabrication and at the same time allow for less imperfections in the final structure since the conversion process inevitably introduces defects (chemical, morphological, etc.) into the chain with the result that there is a distribution of effective conjugation lengths and these are far shorter than the nominal degree of polymerization. In fact, the different precursor polymers discussed above all give PPV, but the structural and hence electronic properties can vary quite dramatically depending on which precursor polymer was utilized [97]. Derivatization of PPV with long alkyl and/or alkoxy ramifications (RPPV, ROPPV) was the first approach for the obtainment of soluble electroluminescent polymers. The solubility by derivatization is due to the lowering of the interchain interactions, which should not in principle change the rigid rod-like character of the main chains. A variety of PPV derivatives can be obtained from p-xylylenes by analogous routes used to obtain PPV. A wide

OH−

+ S R3 R4

R1 A− + S R3 R4

R2

R2

R2(R3) poly(p-phenylene vinylene)

R R2(R1) S 3 R4 R1 R2(R3)

+

n

n R2(R3) precursor

A−

+

R2

A−+ R3 R1(R2) S R4

R1(R2)

R1(R2)

A

−

R2(R3)

R1(R2)R1(R2) − + S R3 A − S R3 R4 R4 +

FIGURE 47.3. Mechanistic processes for the sulfonium precursor synthesis of poly(phenylene vinylene)s, showing the ylide, the xylylene, and the poly(p-xylylene-a-dialkylsulfonium halide) (PXD). Substituents X and Y can be alkyl, alkoxy, and aryl groups. Reprinted with permission from Synthesis, properties of poly(phenylene vinylene)s, related poly (arylene vinylene)s. 1998. p. 61. chapter 3. ß 1998 Marcel Dekker, Inc.

ELECTROLUMINESCENT POLYMER SYSTEMS variety of substituents are tolerated by the soluble sulfonium precursor route affording alkoxy [98–102], alkyl [103,104], alkyl and aryl [105] substituted PPVs. Attachment of alkoxy side chains to the polymer backbone lowers the optical bandgap of most polymers, thereby playing an important role in the color tuning of the polymeric materials. The solid-state properties (color, absorption, emission, fluorescence quantum yield, photoconductivity, etc.) of these polymers were found to be greatly dependent on the number, length position, and geometry of the grafted alkoxy side chains [106]. Substituted PPVs are soluble in organic solvents, which is a very useful feature in the preparation of polymer LEDs. One of the first PPV soluble derivatives, prepared by the Santa Barbara group in California via the precursor route, was methoxy-ethylhexyloxy PPV (MEH-PPV), which emitted a red-orange color [107,108]. Subsequently another PPV derivative with the bulky cholestanoxy group was prepared, namely the poly [2,5-bis(3a-5b cholestanoxy)-1,4-phenylene vinylene] (BCHAPPV). A blue shift was observed in relation to MEH-PPV; BCHA-PPV emitted in the yellow region [109,110].

47.3.1.2.

The chlorine precursor route

An important soluble precursor route for PPV and related polymers involves the polymerization of 1,4-bis(chloromethyl or bromomethyl) arenes by treatment with potassiumt-butoxide in nonhydroxylic solvents like tetrahydrofuran. This methodology was first used by Gilch and Wheelright [111] as one of the most successful early PPV synthesis.

ClCH2

ClCH2

(1)

t-BuOK

S(Me)2

/

761

Hoerhold and co-workers elaborated fairly extensively on this method and recently applied it to synthesis of PPVs with large solubilizing groups on the aryl ring such as cholestanoxy (Fig. 47.4(a)), high molecular weight, highly phenylated PPVs [112–118] such as the diphenyl-4-biphenyl ring substituted PPV which showed green EL [119], poly(2,3-diphenyl-1,4-phenylene vinylene) (DP-PPV) [115, 120–122] (Fig. 47.4(b)). Without the presence of solubilizing side chains on the arene ring, premature precipitation can occur, but otherwise the method has the advantage of producing a precursor that is soluble in organic nonhydroxylic solvents, and therefore useful for electronic applications that require processing. The chlorine precursor route has been also applied to the synthesis of the extensively explored poly(2-methoxy 5-(2-ethyl hexyloxy polyphenylene vinylene) (MEH-PPV). A plausible source of defects in the precursor routes resides in the remaining saturated linkages between aromatic links, which can lead to localized traps via hydrogen abstraction, causing premature device decay. The regiochemical randomness associated with the Wessling based routes [123–126] is an important point, since it affects the solid-state morphology and electronic states connected with molecular architecture. In another variation of the Wessling route, nonionic sulfinyl groups in the ethylene moiety are the leaving groups [127,128]. In addition to the substituted aryl rings that can be incorporated in PPVs by the soluble precursor routes, it is also possible to use other aryl rings, like condensed ones, as long as they are derivable from p-xylylenes or their monocyclic analogs. The obtainment of PPVs with aryl or alkyl substituents at the phenylene or vinylene group of PPV can also be accomplished by the Cl − ⊕ (Me)2SCH2

Cl

−

⊕ CH2S(Me)

(2)

NaOH

Cl − ⊕ S(Me)2

(DP-PPV)

FIGURE 47.4. The dehydrohalogenation route to PPV derivatives illustrated in the synthesis of poly(2,3-diphenyl-1,4-phenylene vinylene) (DP-PPV) poly(1-methoxy-4-(2-ethylhexyloxy)-p-phenylene vinylene) (MEH-PPV).

762 / CHAPTER 47 of ITO/PEDOT/polymer /Ca/Al exhibited a low turn-on voltage (4.0 V), a very high external quantum efficiency (3.39 cd/A), and the highest brightness found in this survey (16,910 cd=m2 ) [133]. Recently, a new synthetic route toward PPV and its derivatives has been reported in which the monomer is polymerized toward a dithiocarbamate precursor polymer by the addition of a strong base. The corresponding conjugated polymer is obtained via a heart treatment of the precursor polymer. This dithiocarbamate precursor route represents a compromise between several straightforward but sometimes troublesome precursor routes and the more complex sulfinyl precursor route [134]. A natural development was the introduction of hole and/ or electron transporting groups in EL polymers aiming the improvement of injection/transporting properties as in the case of the incorporation of triphenyl amine and cyano groups in MEH-PPV [135,136]. Recent advances in PPVrelated structures include soluble PPV derivatives containing pyrene [137] or perylene [138] dyes in the main and C60 grafted units [139]. Energy migration from a large bandgap polymer to another with lower bandgap is possible when the absorption of the latter overlaps with the emission of the former to a certain

palladium catalyzed coupling of dihalogenoarenes and ethylene [129]. It is a general trend that when electron donating alkoxy groups are attached to phenylene rings of PPV the bandgap is reduced and the wavelength of the emitted light shifts to red from the green region [104,107,110,130,131]. RO-PPVs where the alkoxy RO– length varied from C5 to C12 showed increasing EL intensity with increasing side chain length. This was attributed to the reductions of nonradiative decay processes due to preventing migration of excitons to traps. Apart from electronic effects, intermacromolecular packing is a major factor in determining emission color and photoluminescence efficiency (PLeff). Since this quantity is a key factor in LED efficiency (along with balanced charge injection and carrier mobility as seen above), steric effects are important in the design of EL polymers. Figure 47.5 shows the influence of side groups on the emission characteristics of some important PPV derivatives [132]. As a general trend close packing as in BEH-PPV (due to its lateral symmetry) results in reduced PLeff, whereas polymers bearing bulky side groups show increased PLeff, as BCHAPPV, despite the symmetry which gives higher order in the polymer films. Devices using poly(2,3-diphenyl-1,4-phenylenevinylene) derivatives containing long branched alkoxy chains or fluorenyl substituents with the configuration

O O

O

n

O

n

BEH-PPV 583,626nm (Orange)

MEH-PPV 587,631nm (Orange)

O O n O

n O

M3O-PPV 566,602nm (Yellow)

n

BUEH-PPV 524,554nm (Green)

FIGURE 47.5. Influence of side groups on the emission properties of some important PPV derivatives. Reprinted with permission from Synth Met 1997;85(1–3):1275. ß 1997 Elsevier Science.

ELECTROLUMINESCENT POLYMER SYSTEMS extent, and the result is an enhancement of the lower bandgap emission. The dynamics of the excitation transfer process, measured in the ps timescale using an ultrafast Ti/sapphire laser, indicate that the energy transfer was completed in 10 ps when m-EHOP-PPV (poly [2-(m-2’-ethylhexoxyphenyl)-1,4phenylene vinylene]) was used as the host with BCHA-PPV (poly[2,5-bis(cholestanoxy)-1,4-phenylene vinylene]) and BEH-PPV (poly[2,5-bis(2’-phenylene vinylene]) as the guests [140]. Mixtures of poly(2-methoxy-5-(2-ethylhexyloxy-1,4-phenylenevinylene) (MEH-PPV) which emits at 600 nm (yellow-orange) with poly [1,3-propanedioxy1,4-phenylene-1,2-ethenylene(2,5-bis(bimethylsylyl)-1, 4-phenylene) -1,2-ethenylene-1,4-phenylene]) a conjugated– nonconjugated block copolymer, DSiPV) which emits at 450 nm (blue), yielded only the large wavelength emission. By varying the ratio DSiPV/MEH-PPV from 9:1 to 1:15 the relative quantum efficiency increased by a factor of 500. This was attributed not only to energy migration of the excitons from DSiPV to MEH-PPV but also to a dilution factor. As the EL active MEH-PPV is diluted by DSiPV, the intermolecular nonradiative decay is diminished by blocking of the charge carriers [141].

47.4 CONJUGATION CONFINEMENT 47.4.1 Conjugated–Nonconjugated Block Copolymers So far we have seen that introducing substituents in the PPV molecule leads to various EL polymers, emitting in various regions of the visible spectrum according to their chemical structures. A theoretical study of the effects of derivatization can be found in reference [142]. From the red shift of the peaks in PL found with increasing chain length, the effective conjugation length for long chain precursor route samples of PPV are theoretically estimated to be 10–17 repeat units [143]. However, experimental work with oligomeric models led to the conclusion that the effective conjugation length of the solid polymer is not larger than 7–10 units [144]. Thus fully conjugated polymers may have chromophores with different energy gaps because the effective length of conjugation is statistically distributed. However, in the mixture, the chromophores with lower energy gaps will be the emitting species because of energy transfer. To solve this problem several approaches have been developed. The confinement of the conjugation into a well-defined length of the chain is one of the most successful strategies developed so far. Illustrative examples of EL structures exploring the concept of conjugation confinement are shown in Fig. 47.6 [145–147]. Copolymers in which a well-defined emitting unit is intercalated with nonemitting blocks have demonstrated that the emitted color was not affected by the length of the inert spacers but the EL efficiency of the single layer LEDs fabricated with the copolymers was a function of the length of the nonconjugated blocks; copolymers with longer spacers yielded higher efficiency devices [148]. Those conjugated–nonconjugated co-

/

763

polymers (CNCPs) are soluble, homogeneous in terms of conjugation length, and can be designed to emit in any of the visible spectrum [148–151]. In such structures energy transfer from high bandgap to lower bandgap sequences in which excitons may be partially confined will provide higher luminescence efficiency when compared to similar structures of uniform conjugation [152]. In soluble poly(dialkoxy-pphenylene vinylene)s, the systematic variation on the degree of conjugation showed that PL and EL increased with the fraction of nonconjugated units. At the same time, confinement of the effective conjugation has proved to be an efficient means for blue shifting the spectrum because the conjugated emitters can allow charge carriers to form but not to diffuse along the chain, thus limiting the transport to quenching sites [153,154]. This electronic localization results in a large p---p* bandgap which decreases with conjugation length [155]. A widely used route to CNCPs involves the Wittig type coupling of dialdehydes with bis(phosphoranylidene)s [156,157]. A series of CNCPs was prepared, varying the O(CH2)n O spacer length, and chromophore’s structure (PPV type) and length allowing to correlate conjugation length with emission color and device efficiency [148–151]. Changing the aromatic ring from a p-phenylene to a p-thienylene residue caused bandgap shifts in which the emission changed from blue to yellow [150,158, 159]. The introduction of nonconjugated segments not only confines the p electrons in the conjugated part but also imparts solubility and improves the homogeneity of the films. The Wittig route (as with other condensation routes) does not lead to high molecular weight polymers because these become insoluble after a certain degree of polymerization is reached. In CNCPs the solubility provided by the spacer permits the obtainment of high molecular weight materials. Conjugation confinement can also be achieved by tailoring the polymer structure in other ways, like inserting kink (ortho and meta) linkages or imposing steric distortions. Alkoxy substituted PPVs usually carry the alkoxy groups at the 2,5-positions in the ring and are red shifted in relation to unsubstituted PPV. By placing these substituents at the 2,3-positions and the ring in a meta-configuration it was possible to obtain blue emitting alkoxy PPVs [160]. Efficient blue-green polymer light-emitting diodes were prepared with block copolymers composed of the fluorescent segments, 1,4-di[2-(1-naphthyl)vinyl] benzene or 2, 5-dimethyloxy-1,4-di[2-(1-naphthyl)vinyl] benzene and the flexible segments, tri(ethylene oxide) [161]. A new type of cyclolinear polymer, poly(phenylene vinylene-alt-cyclotriphosphazene), was synthesized through Heck-type coupling reaction. Apart from controlling the conjugation length and solubility, the nonemitting cyclophosphazene rings were capable of accommodating a wide variety of substituents with minimal effect on the electronic properties [162]. 47.4.2 Chromophores as Side Groups An extension of the concept of CNCPs is the attachment of the fluorofore as a pendant group to a nonemitting

764 / CHAPTER 47 a) CH2O

b)

CH3 OCH3 O

OCH3

CH3 OC3H2O

CH2O CH3O

n OCH3 n

d) O

c)

O

N OAc

O

F2C

OMe x

CF2

N

MeO

1-x

MeO

n

PPx-0

f) e)

O

O

n H2CO

O n

g)

R

R R

R

R

R R

R

FIGURE 47.6. Examples of EL polymers exploring the concept of conjugation confinement. (a) An aliphatic spacer separating PPV type blocks; (b) dimethylsilane groups separating PPV type blocks; (c) partially eliminated MEH-PPV; (d) hexafluoroisopropylidene nonconjugated segment separating polyquinoline emitting units; (e) kink (meta-) linkages in MEH-PPV; (f) adamantane moiety as spacer; (g) the planarity is interrupted by the twisted p-phenylene groups, schematically illustrated with the wiggled lines.

random coil polymer. This idea should in principle present several advantages: the synthetic route would be simpler than that used for main-chain polymers, the solubility would be dominated by the nature of the backbone, the emission wavelength would be predetermined, and crystallization of the chromophore with concomitant degradation of the diode (in comparison with small molecular weight sublimable systems) would be prevented. In addition, an electroluminescent group could be placed on every repeat unit or in a controlled frequency along the backbone. Some

representative EL structures with emitting pendant groups are shown in Fig. 47.7. Using polystyrene as the main chain, stilbene groups were attached to every repeat unit, in every other repeating unit, or in every third repeat unit (Fig. 47.7(a)) [64,163,164], resulting in blue emitting polymers. Grafting anthracene derivatives (2,3,7,8-tetramethoxy-9,10dibutyl anthracene) (Fig. 47.7(b)) and N-methyl naphthalimide (Fig. 47.7(c)) gave blue and green PMMA based light emitting materials. Charge transfer and emission from associated forms (ground state dimers or excimers) are common

ELECTROLUMINESCENT POLYMER SYSTEMS (CH2-CH)2

CH2-CH

/

765

n CH3 CH3

CH2 C

n

CH2 C

C

O O

CH2

n

C

O

O

O

(CH2)2 N H3C CH3O

O

OCH3 N OCH3

CH3O a)

b)

C3H7

c)

O

C4H9

FIGURE 47.7. Examples of EL polymers with emitting pendant groups. (a) Stilbene chromophores linked to a polystyrene backbone; (b) anthracene derivatives linked to a poly(methyl methacrylate) backbone; (c) naphthalimide based chromophore as side chain in poly(methyl methacrylate).

events in pendant chromophore structures. Examples include the pyrene excimer emission only of polysiloxanes bearing a pyrene group in each mer and the suppression of carbazole emission of copolymers containing carbazole and pyrene attached to a polysiloxane backbone or carbazole and fluoranthene attached to a PMMA main chain [165]. PPV has also been used as a backbone for grafting of lumophores, giving rise to a structure with more than one simultaneously emitting center, like PPV containing the electron accepting trifluoromethyl stilbene moiety. This group emits in the violet, but the substituted PPV showed only the PPV characteristic emission, due to energy transfer [166]. The concept of pendant chromophores has been also explored to afford better transport properties, by covalently attaching charge transport groups to the emitting polymer. The hole transporting carbazole and the electron transporting 2-(4-biphenyl)-5-(4-t-butylphenyl)-1,3,4-oxadiazole) (PBD) were placed as side groups in each mer of PPV. A slight interaction between the p-electrons of the PPV backbone and those of the pendant groups was detected. Also, blue-shifted absorptions indicated that steric effects partially disrupted the conjugation in PPV; the copolymers showed overlapped emissions of the main chain and the side groups. The direct PBD attachment to PPV improved the EL efficiency to a great extent, but the carbazole insertion resulted in an increased imbalance in carrier transport, since PPV itself accepts and transports holes more readily than electrons [167,168]. Apart from designing a molecule capable of emitting light in a defined region of the visible spectrum, a very interesting approach is to design structures that can emit light over a broad spectral range so that the color emitted is white or close to it. With this objective polymers carrying more than one chromophore were prepared like a ring anthracenyl substituted PPV [169]. An

interesting blend was prepared using a side chain copolymer with pendant perylene groups and carrier transporting copolymers, in which hole and electron transporting units were incorporated in the same chain [170].

47.5 POLYTHIOPHENES Among various polymers for LED fabrication poly(3alkylthiophene) (PAT) [171] has stimulated much interest because it was the first soluble and even fusible conducting polymer, and it demonstrated novel characteristics such thermochromism [172] and solvatochromism [173]. EL in these materials was first reported by Ohmori [174,175] and it is now possible to tune the emission of substituted polythiophenes from ultraviolet to IR by changing the substituent [176]. LEDs made with PAT emitted a red orange color [177] peaking at 640 nm. For the series in which the side chain is an aliphatic branch of 12, 18, or 22 carbons the EL intensity increased linearly, the latter (22 carbons) being five times brighter than the former (12 carbons). This was explained in terms of confinement of carriers on the main chain where longer substitutions accounted for greater interchain distance decreasing the probability for quenching [165,178]. The emission intensity of PAT-based LEDs increases with increasing temperature (20–808C) contrary to inorganic GaAs and InGaP semiconductor diodes [179]. This was explained in terms of changes in effective conjugation length with temperature due to changes in the main chain conformation which decreased the nonradiative recombination probability. Some representative polythiophene structures are shown in Fig. 47.8. Polythiophene and substituted polythiophenes can be prepared by chemical or electrochemical routes [180]. The eletrochemical method

766 / CHAPTER 47

CH3

S

S

n

POPT

S n PDCHT

n

PCHT

S n PMOT

S S

n

PTOPT

n

S

S n PCHMT

PDOPT

PTOPT PCHT

PDPT

POPT

Electroluminescence intensity (a.u.)

PCHMT

300

400

500

600

700

800

900

Wavenumber (nm)

FIGURE 47.8. Effect of substitution on the emitting properties of polythiophenes. POPT* and POPT** are different forms of the same polymer, due to thermal treatment.

gives crosslinked materials, and chemical synthesis is most straightforward, in the iron chloride oxidative polymerization route. A particular point in this aspect is that of obtaining regioregular polymers, since regioregularity strongly influences the optical and transport properties of polythiophenes [181]. The dihedral angle and thus the p-orbital overlap between adjacent thiophene rings along the polymer backbone determine the conjugation length along the polymer chain. Short conjugation gives a blue-shifted emission and long conjugation gives a red-shifted emission. Three main strategies have been used for controlling the conjugation length and bandgap in polythiophenes. In the first the conjugation length is modified by adding different substituents on the repeating unit, imposing continuous steric tor-

sions of the main chain [182]. In Fig. 47.8 polythiophenes bearing substituents at positions 3 and 4 in the ring are shown and illustrate the shifts in emission resulting from different degrees of torsion. The larger substituents give a large dihedral angle between the rings, and short conjugation along the polymer backbone is achieved, resulting in blue-shifted emission. This way emission from the blue (PCHMT), green (PCHT), orange (PTOPT) to red (and NIR) (POPT) were observed [32,183]. With mixtures of these polymers it was also possible to obtain voltage controlled EL and white light emitters. For poly(3-(2,5-octyldiphenyl)thiophene) (PDOPT) the bulky side chains efficiently separate the backbones giving the polymer a high PL yield (0.37 in. solution and 0.24 in. film). PTOPT

ELECTROLUMINESCENT POLYMER SYSTEMS has a lower density of side chains, and the PL yield reduces from 0.27 to 0.05 going from solution to thin film. PMOT is twisted out of planarity by sterical hindrance and shows blue-shifted absorption and emission [184]. Substituted polythiophene-containing electron transporting groups such as benzotriazole, chlorobenzotriazole, and fluorene have also been reported [185,186]. Poly(3-octyl thiophene), which can be obtained as a 95% regioregular material, offers an example of how super structure can affect the electronic properties of an emissive polymer. Changing from poly(3hexylthiophene) to poly(3-dodecylthiophene) increased the maximum efficiency from 0.05 to 0.2% with calcium electrodes [187]. The phase structure in blends of one or more polythiophenes with a PMMA matrix allowed the fabrication of nano-LEDs giving white light emission. The thiophene backbone has been functionalized with a wide variety of organic moieties including alkyl, fluoroalkyl, alkylthio, alkoxy, alcohol/thiol, amino, cyano, ester, carboxylic acid, and sulfonate side chains. Nitrogen-derivatized polythiophenes permit further modification of the polymers [188]. Other approaches to tune the emission color of polythiophene LEDs are the preparation of completely coplanar systems with controlled inclusion of head-to-tail dyads or the preparation of alternating block copolymer. The insertion of p-phenylene ring to head-to-head thiophene dyad linked [189], with different substituents on both thiophene and phenylene enhanced by 29% the PL efficiency, in comparison with other polythiophenes, and by changing the substitution on both the phenylene and thiophene rings, the electronic spectrum of the polymers could be tuned, emitting blue to green light. Photophysical and electrooptical properties of regioregular polythiophenes functionalized with tetrahydropyran moieties tethered to the main chain by alkyl spacers were prepared to access structure–property relationships of regioregular THP-bearing poly(3-alkylthiophene)s. In particular, aggregation phenomena were addressed by investigating the influence of the alkyl chain length with respect to their photophysical and electrooptical properties [190]. The emission of a series of p–n diblock copolymers with good electron transporting properties where oligothiophenes were linked with oxadiazolyl-dialkoxybenzene units could be tuned from blue to green to orange by increasing the number of thiophene rings from 1 to 3 [191,192]. In a recent study [193] of the transport properties of a polythiophene derivative, poly(3-(2’-methoxy-5’-octylphenyl)thiophene) (POMeOPT) the current–voltage characteristics of single layer devices were measured in two regimes: contact limited current and bulk-limited current. The passage from one regime to another was done upon insertion of a conducting polymer poly(3,4 ethylenedioxythiophene) doped with poly(4-styrenesulfonate) (PEDOT-PSS) between the metallic electrode and the POMeOPT. The measured mobility was seven times higher than that for MEH-PPV in the same conditions, illustrating the good transport properties and high mobility that can be attained with regioregular

/

767

substituted polythiophenes. An interesting property of polthiophenes is phosphorescence emission which can be obtained by doping the polymer with a phosphorescent heavy metal as iridium, platinum, and others as in the case of poly(3-methyl-4-octylthiophene) as host and the phosphorescent compounds bis(2-phenylbenzothiazole) iridium acetylacetonate (BTIr) or platinum(II) 2,8,12,17-tetraethyl3,7,13,18-tramethyl porphyrin as guest [194,195]. Introduction of the electron withdrawing groups as bithithiophene, pyridinyl, dipyridyl, and phenanthroline can modify their optical and electrical properties. These structures are low bandgap conjugated polymers with higher conductivity (carrier mobility), and may be transparent in visible light. Therefore, they have a great potential application in transistor, transparent conductor, nonlinear optical devices, and smart windows [196,197]. An alternating structure in which an unsubstituted thiophene ring was linked to a 3-alkyl-substituted thiophene, the two repeating units being alternated and a bulky group in the side chain showed an interesting peculiarity of combining high conjugation length with large interchain distances. Differently from the regioregular PATs, the copolymer showed high PL efficiency both in the solid form and in partially aggregated solutions [198].

47.6 CYANO POLYMERS Most of the electroluminescent polymers are suitable as hole-injecting and transporting materials. To set an adequate balance in the injection flows coming from each side of the device it has been necessary to use electron transporting layers and/or low work function metals at the cathode, like calcium, which are unstable at atmospheric conditions. The synthesis of polymers with high electron affinity as the solution processable poly(cyanoterephthalydene)s which are derivatives of PPV with cyano groups attached to the vinylic carbons has provided the material necessary to complement the existing hole transport PPVs [57,142,199–205]. Poly(arylene vinylene)s bearing electron withdrawing groups are not easily available by application of the Wessling and related procedures and thus these cyano derivatives of PPV were synthesized via a Knoevenagel condensation route between an aromatic diacetonitrile and the corresponding aromatic dialdehyde [206–208] as exemplified in Fig. 47.9(a) or by copolymerization of dibromoarenes in basic medium. This approach permits adjustment of the bandgap by varying the proportion of the two comonomers [209]. The synthesis of fully conjugated PPV type structures containing cyano groups attached to the ring afforded a more perfect structure when a Wittig type condensation was followed, as in Fig. 47.9(b) in relation to the Knoevenagel route, emitting orange light (3000 cd=m2 at 20 V) in a double layer device with PPV as HTL [210]. A variety of monomers with different substituents in the ring as alkyl or alkoxy solubilizing groups (as hexyloxy or

768 / CHAPTER 47 a) OC8H13 CN

OC8H13

OC8H13CN

CHO

(Bu4N)+(OH)−

+

n

OC8H13

THF, BuOH OHC CN

OC8H13

OC8H13 CN

OC8H13

OC8H13 O CH2Br

CH2P(OEt)2 OC16H33

OC16H33 HBr(CH2O)n H33C16O

H33C16O

OC16H33

P(OEt)3 ∆ H33C16O

CH2P(OEt)2

CH2Br

O 1) OHC

CH(OEt)2

(CH3)3COK/THF

H33C16O OHC

2) H3O+

CHO H33C16O CN

Al3PCH2

CH2PAl3

OC16H33

NC

CN

DBU H33C16O b)

n

NC

FIGURE 47.9. Synthetic routes to CN-substituted EL polymers. (a) Knoevenagel route leading to CN placement in the double bond; (b) Wittig route used to place the CN group in the aromatic ring in a conjugated–nonconjugated block copolymer.

methoxy-ethyl-hexyoxy as in MEH-PPV) were used to prepare cyano PPV like polymers emitting in the full-visible spectrum. The inclusion of thiophene units in the main chain lowers the bandgap and shifts the emission to the infrared [211]. Examples EL polymers bearing cyano groups are given in Fig. 47.10. The electron withdrawing effect of the cyano group is calculated to increase the binding energies of both occupied p and unoccupied p---p
states, while at the same time keeping a similar p---p
gap [212]. The photophysical behavior of these polymers indicated that aggregates or excimers were probably the emitting associated form [213–216].

47.7 POLY(P-PHENYLENE)S (PPP) AND POLYFLUORENES 47.7.1 Polyphenylenes Poly(p-phenylene) (PPP) is an interesting material for electrooptical applications as its bandgap is in the blue region of the visible spectrum and its thermal stability is combined with high PL. However, it is insoluble and infusible making it difficult to fabricate thin films. In the early stages of the search for PPP synthesis the limitations were related to the difficulties in the preparation of polymers possessing a defined architecture. Since only a few ‘‘classical’’

ELECTROLUMINESCENT POLYMER SYSTEMS b)

/

769

CN

a)

n O

O

CN

CN

n

CN O

N

O

c)

OC16H33

CN

H33C16O

n NC

FIGURE 47.10. Examples of various EL polymers bearing the CN group in the double bond or in the aromatic ring.

organic reactions are known to generate a direct link between aromatic units, metal-catalyzed coupling reactions are commonly used for this purpose. The most successful routes are the Yamamoto route and the Suzuki crosscoupling reactions (SCC). The Yamamoto route involves the Ni mediated coupling of arenes by the reaction of the correspondent dibromo-substituted compounds [217]; the SCC involves the palladium-catalyzed crosscoupling reaction between organoboron compounds and organic halides. When applied to polymer synthesis, it proved to be a powerful tool to prepare poly(arylene)s and related polymers. In this case, SCC is a step-growth polymerization (Suzuki crosscoupling polymerization, SCP) of bifunctional aromatic monomers. The general method has been reviewed [218] and a wide variety of polymer structures prepared through this method [219]. In Fig. 47.11, a schematic representation of the step growth SCP is shown. Alkylated, soluble PPPs prepared via

(RO)2B—Ar—B(RO)2

+

X—Ar'—X

coupling reactions using the Yamamoto [219] or Suzuki [220] routes yielded significant torsion angles. The interring twisting significantly changes the electronic structure as well as the conjugation length [221]. Copolymers consisting of oligo p-phenylene sequences linked by ethylene, vinylene, or units have been reported. By the combination of different AA/BB type monomers in various concentrations in a Suzuki coupling as polymerization route, a variety of well-defined structures were prepared with high quantum yields in solution [222] as shown in Fig. 47.12. Matrix-assisted laser desorption ionization time of flight mass spectrometry (MALDI-TOF-MS) and HPLC analyses of monodisperse-substituted PPP fractions indicated that the effective conjugation length was around 11 phenylene units [223]. One way of obtaining a planar conjugated backbone was to incorporate the phenyl rings into a ladder-type structure where four C-atoms of each phenyl

Pd(0) Ar—Ar' Na2CO3 H2O/Toluene

R= H, alkyl

n

X= Br, l, OTf

Pd(0) (RO)2B—Ar—X

Ar Na2CO3 H2O/Toluene

n

FIGURE 47.11. Schematic representation of the SCP —Ar— represent aromatic units, typically benzene derivatives.

770 / CHAPTER 47

O

Px:

x O

Px Px

Px

n

x = 3,7,10,20

k

n

Px n

x=3

x = 2,3,5

k=4

Px Px n

Px

Px k

n

x=3

k

Px n

x = 2; k = 4 x = 2; k = 4

x = 3; k = 5

x = 3; k = 5

FIGURE 47.12. Polymers containing oligo-p-phenylene sequences linked by ethylene (E), vinylene (V) or ethynylene (A). The numbers correspond to the degree of polymerization of the phenylene sequences. The monomers were connected through the Suzuki coupling method.

ring are connected with neighboring rings (LPPP) in combination with an additional attachment of solubilizing side groups, thus creating a solution processable structure [224– 229]. The forced planarity of the molecule led to a high degree of intrachain order, with a conjugation length of about eight phenyl rings [230,231]. The EL spectrum of the structures showed two emissions: a blue (461 nm) and a yellow (600 nm) which was attributed to the formation of excimers. A blue emitting PPP copolymer was reported in which tri-(p-phenylene) (LPP) and oligo(phenylene vinylene) segments were linked in an orthogonal arrangement to decrease quenching processes. Analogous structures with oligo (p-phenylene) units orthogonally and periodically tethered to a polyalkylene main chain have been prepared by the polymerization of oligomeric fluoreneacenes via an SN2 type of mechanism [232] Another class of PPP-type polymer is exemplified by the poly(benzoyl-1,4-phenylene) in a head-to-tail configuration [233]. The introduction of the carbazole unit in the ladder type tetraphenylene, blue emitting polymers brought about a slight bathochromic shift of the emission. The polymers exhibited good EL properties in initial PLED tests with high luminance values typically over

700–900 cd=m2 at a bias of 10 but a definitive suppression of the excimer was not demonstrated [234].

47.7.2 Polyfluorenes Recently, polyfluorenes were introduced as a prospective emitting layer for polymer LEDs. These materials are thermally stable and display high PL efficiencies both in solution and in solid films [235–239] with emission wavelengths primarily in the blue spectral region. Their photostability and thermal stability are also found to be better than those of the poly(phenylene vinylene)s. Polyfluorenes contain a rigidly planarized biphenyl structure in the fluorene repeating unit, while the remote substitution at C-9 produces less steric interaction in the conjugated backbone itself than in comparison with PPP, in which this interaction can lead to significant twisting of the main chain since the substituents used to control solubility are ortho to the aryl chain linkage, as is the case for the monocyclic monomers [240,241] discussed in Section 47.7.1. In this regard polyfluorenes can be considered as another version of PPP with pairs of phenylene

ELECTROLUMINESCENT POLYMER SYSTEMS

771

to the synthesis of a wide number of polyfluorenes and related structures. In the case of alternating copolymers obtained by SCP the optical and electronic properties of the polymers were tailored through selective incorporation of different aromatic units into the system. A variety of chromophores intercalated with fluorene has been reported, such as phenylene, naphthalene, anthracene, stilbene, cyanovinylene, thiophene, bithiophene [245] pyrazoline, quinoxaline, 1,2-cyanostilbene, pyridine, and carbazole [220, 246, 247].

rings locked into a coplanar arrangement by the presence of the C-9 atom. Liquid crystallinity was observed in poly (dioctyl fluorene), which is important for the obtainment of polarized EL [26,242]. A representative number of polyfluorenes and related structures are shown in Fig. 47.13. The nickel-mediated coupling of arylene dihalides, the Yamamoto route, has been used to prepare a variety of fluorene and substituted fluorene homo- and copolymers [235–244]. As with PPPs, the SCP has been recently applied

a)

/

b)

n

O x R

PFE

N

R

c)

N

n

R

R

N

N

n

TPD-PFE

S

S d) S C8H17

O

O

e)

S

n C8H17

C8H17

O

C8H17

n

O

PBTF Ph

Ph Ph

Ph

Ph

Ph

Ph

Ph

f)

g)

Ph

x H17C8

C8H17

y

Ph

n

FIGURE 47.13. Examples of various fluorene based polymers. (a) Fluorene copolymer with triple bonds, poly(2,7-9,9-di2-ethylhexylfluorenylene ethynylene); (b) alternating copolymers of 9,9-dioctylfluorene and oxadiazole; (c) copolymer containing the electron-accepting moiety 2,7-diethynylfluorene and the electron-donating moiety tetraphenyl diaminobiphenyl (TPD); (d) poly[2,20-(5,50-bithienylene)-2,7-(9,9-dioctylfluorene)] (PBTF); (e) poly[2,20-(5,50-di(3,4-ethylenedioxythienylene))-2,7-(9, 9-dioctylfluorene)] (PdiEDOTF); (f) polyfluorenes with perylene groups in the main chain; (g) a dendronized polyfluorene.

772 / CHAPTER 47 lene)s, demonstrated strong EL, and their effective conjugation length was calculated to be around 10 fluorene units (Fig. 47.13(a)) [250]. Devices with fluorene polymers appear to have electrons as the majority carriers and their performance is notably improved when modified with an appropriate HTL. Hole transporting moieties such as tertiary amines and TPD [251] have been incorporated to polyfluorenes in attempts to optimize LED performance. The HOMO levels of fluorene-based poly(iminoarylene)s (5.1 eV) were close to the work function of ITO, and their use as buffer layers has been suggested (buffer layers are inserted

Figures 47.14 and 15 show the Yamamoto and the SCP routes to synthesize fluorene-based copolymers, respectively. Well-defined monodisperse oligomers were prepared via SCP to access the effect of conjugation length on photoluminescent properties of polyfluorenes [248]. The Yamamoto route was also used to prepare 9-di-hexyl substituted oligofluorenes, containing 3–10 repeating units. The effective conjugation length was estimated to be 12 bonded fluorene units, by extrapolation of spectral data [249]. Substituted oligofluorenes in which the fluorene units alternate with triple bonds, namely oligo(9,9-dihexyl-2,7-fluorene ethyny-

Br

+

Br

H13C6

C6H13

Br

Ar

Br

Ni(COD)2, Dipyridyl, COD DMF / Toluene

Ar

n H13C6

m

C6H13

H Ar:

C6H13

Ph

Ph H

H13C6

H F3C

CF3

H

FIGURE 47.14. The Yamamoto route to polyfluorene based copolymers: nickel mediated coupling of 2,7-dibromo-9,9-dialkyl fluorene and various dibromoarenes.

ELECTROLUMINESCENT POLYMER SYSTEMS

O

O B

+

Br

Ar

Br

C6H13 C6H13

Ar:

H13C6

C6H13

H13C6

O

H13C6

COOEt

H13C6

n

C6H13

C6H13

O

N CN

Ar

2]toluene K2CO3 reflux

O H13C6

773

1)[(PPh3)4Pd(0)

B

O

/

N

H21C10

C10H21

S EtOOC

FIGURE 47.15. The SCP route to fluorene-based alternating copolymers: tetrakis(triphenylphosphine)palladium mediated condensation 9,9-dialkylfluorene-2,7-bis(trimethylene boronate) and various dibromoarenes.

between ITO anode and HTLs, as TPD). On the other hand, the incorporation of the electron withdrawing 1,3,4-oxadiazole units brought the electron affinity of the copolymers close to the work functions of Ca. These structures, shown in Fig. 47.13(b), prepared via the SCP, contained the oxadiazole evenly dispersed in the main chain, at every one, three, or four 9,9-dioctyl fluorene mers. All copolymers fluoresced in the blue range with quantum yields of about 70% in solution [252]. The combination of donor and accepting moieties in fluorene-based structures has been accomplished by alternating TPD (electron donating) with 2,7-diethylhexyl fluorene or diethynylfluorene units (electron donors), as shown in Fig. 47.13(c). A fluorinated copolymer formed by alternating mers of [2,3,5,6 tetrafluoro-1,4 phenylene] and [9,9’-dihexyl-2,7 fluorene] emitting blue light with low turn on voltages, showed a superior performance to that of the nonfluorinated analog copolymer and of the corresponding poly(9,9’dihexyl-2,7 polyfluorene) homopolymer [253]. An alternating polyfluorene with low bandgap segments has been designed and synthesized aiming to tune the emission. The low bandgap segment consists of an electron acceptor (thiophene, A), fenced by electron donors (benzodiathyazole, D). This D–A–D configuration lead to a partial charge transfer in the polymer backbone, and thereby a low bandgap (1.3 eV) [254]. The same approach was used by incorporating an analog of the red emitting dye DCM [(4-phenylamino)vinyl) pyran-4-ylidene-malononitrile] as a comonomer into the polyfluorene backbone. The emission was in the range 573–620 nm (greenish-yellow to red). DCM dye has

an electron-deficient 2-pyran-4-ylidene malononitrile (PM) group and an electron-rich aromatic amine group, so both the absorption and emission show a red region because of the effect of charge transfer from triphenylene (TPA) to the PM group [255]. Another fluorene copolymer containing the luminescent dye [4-dicyanomethylene-2-methyl-6-4H-pyran (DCM) as acceptor compound was irradiatiated with UV light in the presence of gaseous trialkylsilanes. This reagent selectively saturates the C ¼ C bonds in the DCM comonomer units while leaving the fluorene units essentially unaffected. As a result of the photochemical process, the red electroluminescence of the acceptor compound vanishes, and the bluegreen electroluminescence from the polyfluorene units is recovered. Compared with previous processes based on polymer blends, this copolymer approach avoids problems associated with phase-separation phenomena in the active layer of OLEDs [256]. Orange-red emission was also seen in single layer devices of a series of conjugated copolymers of fluorene and 2- [2, 6-bis(2-arylvinyl)pyridine-4-ylidene]-malononitrile [257]. Another kind of red-emitting polyfluorenes with high electron affinity was reported, namely 9,9-dihexylfluorene and diketopyrrolopyrrole [258]. The Foerster-type energy transfer was efficiently used to tune the solid-state emission color of fluorene based copolymers bearing perylene dyes as end groups or side chains, as shown in Fig. 47.13(f). The emission coming almost exclusively from the perylene dyes could be tuned from yellow-green (558 nm) to red (675 nm)

774 / CHAPTER 47 [259]. Color tuning to the deep-red and NIR region was achieved by incorporating a selen-containing heterocycle, benzoselenadiazole, a selenium analog of benzothiadiazole in different compositions, resulting in a significant red shift in comparison with its sulfur analogue [260]. The abundant literature in polyfluorene and derivatives show that nowadays it is possible to tune the emission of polyfluorene derivatives from bluish-violet to deep red and near infrared [261]. One problem with polyfluorenes is the occurrence of an undesired low-energy band at 500–600 nm in the photo- and electroluminescence spectra of the pristine polymer or after annealing or the passage of current. The low-energy green band limits the emission efficiency and damages the blue color purity and stability as well. Two opposite points of view on the origin of this green emission have been reported. According to the first, the green emission is attributable to the interchain aggregates and/or excimers. Consequently, dendronization, introduction of spiro- or crosslinks, substitution with bulky side groups such as tetraphenylthiophene, blending, and the introduction of disorder units such as carbazole, pyridine, and thiophene have been applied to suppress intermolecular interaction [262]. The second point of view states that the green emission band is caused by keto defects of polyfluorenes, which are generated during the handling of the materials in air, or by a reaction with residual oxygen over the course of photophysical experimentation. Certain authors have proposed that the origin of the green-emission band stands on the fluorenone moiety and contradicted experimentally the assumption that intermolecular aggregates or excimers are involved. A series of well-defined 9,9’-dihexylfluorene-co-fluorenone copolymers with various fluorenone contents and a set of monodisperse oligofluorenes in the chain center have been prepared to elucidate the exact origin of the low-energy emission in polyfluorenes. On the basis of the steady-state photoluminescence (PL) and PL decay dynamics of the fluorenone-containing oligomers and copolymers both in dilute solutions and in thin films, the origin of the controversial low-energy emission band was attributed to the interaction between intrachain fluorenone moieties instead of the intermolecular aggregates or excimers. It was also proposed that a fluorene pentamer with a central fluorenone unit would be more appropriate to represent the actual chromophore responsible for the green emission in the copolymers [263]. Nevertheless the question remains still controversial. The introduction of 9-hexylcarbazole and 9-dimethylaminopropylcarbazole moieties into polyfluorene chain was claimed to effectively prevent excimer formation in the polymers [264], With the same idea 9,9dihexylfluorenyl was inserted as a pendant group in a chain of poly(biphenylene vinylene). The insertion brought about steric interactions between adjacent rings, reducing conjugation length, but at the same time inhibited the formation of excimers. The polymer showed bright and stable blue emission [265].

Miller and co-workers at IBM have managed to overcome the low-energy emission by incorporating anthracene units which show stable blue emission even after annealing at 2008C for 3 days [266]. Mullen et al. [267] at the MaxPlanck Institute in Germany produced nonaggregating polyfluorenes by the insertion of dendron side chains, as shown in Fig. 47.13(g) [268], giving a polymer with pure blue emission, as the bulky side chains do not cause distortion between the fluorene units. Recently, dendritic structures were attached to polyfluorenes with further addition of a low percentage of surface-modified semiconductor nanoparticles [269]. Starlike materials tethered to polyfluorene derivatives emitting blue, green, or red light were developed. Polyhedral oligomeric silsesquioxanes were incorporated into the center core of the derivatives to enhance thermal stability and reduce linear aggregation [270]. The optical properties of a series of light-emitting hyperbranched polyfluorenes through 1,3,5-substituted benzene crosspoints were investigated. With increase in crosspoint density, the emission color of the PEDOT-containing LEDs shift from green to violet and showed higher EL efficiency due to the effective exciton confinement and the reduction of intrachain or interchain exciton annihilation [271]. A series of electron-deficient, oxadiazole-, quinoline-, quinoxaline- and phenylenecyanovinylene-containing copolymers bearing ethyl hexyl in the fluorene unit was developed. These materials possess low-lying LUMO energy levels (3.01 to 3.37 eV) and low-lying HOMO energy levels (6.13 to 6.38 eV), with sharp blue emission, and may be promising candidates for electron transport-holeblocking materials in LED fabrication. The film emissions were only 7–11 nm red shifted in comparison with the solution emissions, indicating that excimer formation was suppressed. This was explained in terms of the prevention of molecular stacking by the presence of sterically demanding ethyl-hexyl substitutions at the fluorene unit [272]. The formation of a network is a useful strategy in the obtainment of various performance improvements in polyfluorenes. For example, the attachment of styryl end groups, via reaction of the bromo-terminated polymer with bromostyrene in a Yamamoto coupling, allowed the deposition of a crosslinkable layer through the thermal polymerization of the terminal styrene groups. Apart from the added advantage of further casting other layers, the immobilization of the chains leads to suppression of intermolecular excited state interactions, hampering the ability to p stack [273,274]. Another kind of fluorene-containing structure consisted of conjugated polyfluorene/poly(p-phenylenevinylene) copolymer containing the pendant bis(4-alkoxyphenyl) groups in the C-9 position of every alternating fluorene unit. The main advantage of the use of an extended 9,9-bis(4-hydroxyphenyl)fluorenyl core in the polymerization reaction is that the insertion of a rigid phenylene spacer between the large side chain and the polymer backbone may lead to a more efficient shielding effect on the polyfluorene main chain, which would suppress the formation of aggregates/excimers while

ELECTROLUMINESCENT POLYMER SYSTEMS not blocking the reaction sites of the macromonomer from the palladium catalyzed polymerization reaction. The chain stacking, however, was not completely avoided, and a green electroluminescence was observed [275]. Energy migration has been explored in polyfluorenes to enhance emission intensity. For example, devices of poly (9,9-dioctylfluorene) mixed with the amine-substituted copolyfluorene poly(9,9-dioctyl-fluorene-co-bis-N,N’-phenyl1,4-phenylenediamine), showed a blue emission with a luminance of 1550 cd=m2 and a maximum external quantum efficiency of 0.4%, much larger than the original homopolymer. White-light-emitting devices have been demonstrated with new single-component fluorene-acceptor copolymers with three emitting units: blue-emitting 9,9-dihexyl-fluorene, green emitting quinoxaline (or yellow-emitting 2,1,3benzothiadiazole), and red-emitting (thieno [3,4-b]-pyrazine) units in the same chain. The energy-transfer between the emitting moieties suggests the white-light emission could be obtained by a relatively small fraction of the acceptor moieties. The EL devices typically had a luminance of 1,870 cd=m2 at 10 V. The CIE coordinates of this device are (0.33, 0.34), which are almost identical to the standard white emission, and they exhibit insignificant changes in driving voltages. The results suggest that very bright and highly stable white-emission devices could be achieved by single-component fluorene-acceptor copolymers with three emitting moieties as an emissive layer [276]. A recent aspect of the research in polyfluorenes is related to supramolecular ordering of these conjugated polymers by making rod-coil block copolymers. The rod-like conjugated polyfluorene was end capped on one or both ends with polyethylene oxide, forming di- or triblock copolymers. The solid-state fluorescence spectra of these materials had better resolution than the homopolymer, indicating an enhanced number of well-ordered rods in the films and an additional increase in long wave emission. Multilayer fluorene-based LEDs were reported by a Japanese group [277] where a three layer device having the structure ITO/N,N’-bis (2,5-ditertbutylphenyl)-3,4,9,10-perylene dicarboxamide (BPPC)/ N,N’-diphenyl-N,N’-(3-methylphenyl)-1,10-biphenyl-4,40diamine (TPD)/poly(9,9-dihexylfluorene) (PDHF) was able to emit either red or blue by changing the polarity of the applied voltage. TPD is a material mainly used for hole transport, BPPC is a red emitter, and the polymer emits in the blue region. The particular set of gap conditions in this system allowed the emission of blue light under positive bias conditions (ITO anode, AI cathode) and emission of red light under negative conditions. Furthermore, the device can be driven with an AC field and the emission color can be gradually modulated by changing the frequency of the applied AC field. Placing a small amount of surface-tailored CdS nanoparticles into the dendritic structure of copolyfluorene substantially improves the efficiency of the polymer’s light emission, as well as the purity of the emitted light. One possible explanation for the enhancements in PL and EL may be the reduction in the concentration of interpolymer

/

775

excimers, i.e., the CdS nanoparticles caused an increase in the interpolymer chain distance. An intermediate structure between PPP and polyfluorene has been developed, the poly(2,8-indenofluorene). This blue emitting polymer is stable up to 3808C, and shows thermotropic LC behavior at high temperatures (250–3008C) making it a good candidate as the active material in polarized LEDs [278]. Models of spin statistics predict that the electron–hole recombination event should produce three times as many triplets as singlets, and this has been confirmed experimentally for electroluminescent devices. Considerable effort has been devoted nowadays to attach phosphors covalently to a conjugated polymer backbone so as to allow efficient energy transfer between polymer and phosphor. Electrofosforencesce seems to be the new trend to maximize LED performance. One exemple is the red Electrofosforescent Light Emiiting Diode based on iridium complexes with the [lr(btp)2(acac)] fragment (where btp is 2-(2’-benzo [b]thienyl)pyridinato and acac is acetylacetonate). The fragment was attached directly or through a ---(CH2)8 –spacer chain at the 9-position of a 9-octylfluorene host. The dibromofunctionalized spacerless or octamethylene-tethered fluorene monomers were chain extended by Suzuki polycondensations using the bis(boronate)-terminated fluorene macromonomers in the presence of end-capping chlorobenzene solvent to produce the statistical spacerless and octamethylene-tethered copolymers containing an even dispersion of the pendant phosphorescent fragments [279].

47.8 SILICON-CONTAINING POLYMERS The interest in silicon-based polymers resides in the delocalization of the s electrons over a Si backbone providing electronically analogous properties to the p-conjugated polymers. Polysilanes are s-conjugated polymers with a one-dimensional (1D) Si chain backbone and organic side chain substituents. Progress in understanding their electronic structure derived from both theoretical and experimental studies has revealed that they are quasi-1D semiconductors with a direct and wide bandgap (4 eV), and that the sconjugated electronic structure typically observed in silane high polymers appears in Si chains with more than 20–25 Si units [280]. Polysilanes exhibit photoconductivity, intense near-UV absorption, and strong PL of small Stokes shift and high hole mobility (on the order of 104 cm2 s1 ) [281]. Near-UV or UV emitting LEDs of diaryl, dialkyl, monoalkyl-aryl polysilanes have been reported [282–285], and the bandgap energies tend to shift to lower values based on the size of substituents with aromatic side groups [286]. The emissions in polysilanes have been attributed to the s---s
transitions of 1D excitons in the Si backbone. Nevertheless, PL studies of poly(methylphenylsilane) demonstrated the existence of another emission due to a charge transfer state from the intrachain s to pendant p
groups which appear in

776 / CHAPTER 47 Silicon-containing PPV derivatives have been developed in which the silicon unit acts a spacer to improve solubility, film forming characteristics, and confine conjugation, in analogy of the conjugated–nonconjugated block copolymers with an aliphatic spacer. While the aliphatic segments as spacers can act as a barrier to injection and mobility of the charge carriers, resulting in higher threshold voltages, the silicon units with an aromatic or flexible group are able to produce the same spacer effects with low operating voltages [290]. It has been argued that the participation of the d-orbital of the Si atom could be assisting to increase the

larger wavelengths (400–500 nm) [287]. Being typical ptype semiconductors, polysilanes cannot transport electrons, and for this reason the incorporation of electron transporting groups with emitting properties seemed to be an interesting way of combining good properties [288]. Polymethyl phenyl silane (PMPS), poly [bis(p-n-butylphenylphenyl)silane] (PBPS), and poly(2-naphthyl phenyl silane (PNPS) are examples of polysilanes, shown in Fig. 47.16(a). A blue emission (480 nm) with a PL of 87% was achieved with a poly(methylphenylsilane) containing anthracene units in the polymer backbone [289].

C4H3

CH3 (a)

Si

Si

n

Si

n

n

C4H3 PNPS

PMPS

PBPS

Me (b) Cl

Me

Me

Si

Si

nPr

nPr

Cl Cl +

Si

Cl

Toluene Reflux

Si

Me

Me

Si

Si

Si

nPr

nPr x

R E1OHNa CH2PPh3 + OHC—X—CHO

1-BuOH / CHCl3

R’

(c)

R,R’ : alkyl, phenyl X: aromatic residue

y z

R Ph3PH2C

Me Na

X

Si R’

n

(1) Wittig reaction

NC R

R NOCH2

Si R’

CH2CN + OHC—X—CHO

Bu4NOH THF,1-BuOH 20 min, 50°C

Si R’

X CN n

(2) Knoevenagel reaction

FIGURE 47.16. Examples of (a) polysilanes, where the main chain is made up of Si–Si bonds: poly(phenyl methyl silane) (PMPS), poly(naphthylmethyl silane) (PNPS), poly[bis(p-n-butylphenyl)silane] (PBPS); (b) synthetic route to poly(methyl phenyl silane) containing anthracene units; (c) copolymers with Si inserted between p-conjugated blocks: Wittig (top) and Knoevenagel (bottom) routes.

ELECTROLUMINESCENT POLYMER SYSTEMS effective conjugation length, thus facilitating charge mobility [291], although previous theoretical work has demonstrated that Si bonds break effectively the p-conjugation [292]. A variety of PPV-related structures such as copolymers of diphenyl/dibutylsilane [293], dibutyl, butyl/methyl, diphenyl silanes [294] with PPV and alkoxy PPV [295] have been reported, in which the organosilicon groups are used as spacers [296]. Some representative examples are shown in Fig. 47.16(b). The EL spectrum of the diphenyl-substituted copolymer (SiPhPPV) gave the highest peak (450 nm) when the operating voltage of 9 V was applied. With 12 V applied bias a strong white color was emitted due to additional emissive bands. This threshold voltage was further decreased to 7 V by the introduction of a CN group into the double bond of PPV (Fig. 47.16(c)) [291]. Similar results were obtained in alternating copolymers of silane and carbazolyl or fluorenyl derivatives, peaking around 440–476 nm with operating voltages of 6–12 [297]V. In contrast to alkoxy groups, the lack of electron donor capacity (and consequent red shifting of the emission) of alkylsilicon groups has suggested the introduction of these groups as ramifications in EL polymers, improving processing characteristics [298,299]. One of the unique properties of polysilanes is the SiSi bond scission of the backbone chain under UV radiation. In the presence of oxygen it is accompanied by a SiOSi bond formation that leads to the conversion into an insulator, with no hole transport ability. Using this property, a patterningimage-display electroluminescent device was built. Before turning on the voltage, the anthracene-containing polysilane LED was irradiated in order to pattern an image onto the emission area, from the glass substrate side. Blue patterned light was obtained, corresponding to the negative photomask used [289,300].

47.9 NITROGEN-CONTAINING CONJUGATED POLYMERS 47.9.1 Pyridine-Containing Conjugated Polymers Due to their strong electron-acceptor character, nitrogencontaining groups of various kinds have been incorporated to conjugated polymeric structures. The most extensively studied structures carry the pyridine moiety, in homopolymers (poly(2,5-pyridine), poly(3,5 pyridine)) [301,302], in PPVtype structures (poly(p-pyridylene vinylene))s [302,303], in copolymers with PPV (poly(phenylene vinylene pyridylene vinylene)s) or in p-phenylene derivatives [304,305]. Alternating pyridine-based backbone copolymers with substituted phenylene and fluorene units have been reported for tunability of electronic properties with enhanced stability [306]. As compared to phenylene-based analogues, one of the most important features of pyridine-based polymers is the higher electronic affinity. As a consequence, the polymer is more resistant to oxidation and shows better electron transport properties. The higher electron affinity enables the use

/

777

of relatively stable metals such as Al, Cu or Au, or doped polyaniline as electrodes [302,307]. The pyridine-containing conjugated polymers are highly luminescent, especially the copolymers with phenylene vinylene. The solubility of polypyridines in organic solvents represents another advantage as compared to PPV for device fabrication. The ability to protonate and quaternize the nitrogen makes it possible to manipulate the electronic structure and thereby the emission wavelength [305,308]. The synthesis of poly(2,5-pyridine) and poly(3,5-pyridine) is straightforward: one step coupling polymerization of the 2,5 (or 3,5) dibromopyridine using a metal catalyst [309]. It was proposed that two blue electroluminescent devices emitting at 420 and 520 nm can be constructed by varying the degree of protonation. Another example illustrating the possibility of tuning spectroscopic properties by protonation of the lone pair of electrons of the pyridine ring is the red shift observed in the fluorescence and EL emission of poly(2,5-pyridylene-co-1,4-(2,5bis(ethylhexyloxy)phenylene)[310]. Excitation profiles show that emission arises from both protonated and nonprotonated sites in the polymer chain. Protonation is also accompanied by intramolecular hydrogen bonding to the oxygen of the adjacent solubilizing alkoxy group, providing a new mechanism for driving the polymer into a near planar conformation, extending the conjugation and tuning the emission profiles. The PL and EL spectra of copolymers of 1,4-phenylene vinylene and 2,6-pyridylene vinylene(co(2,6PyV–PV)) could be tuned, respectively, in function of the excitation wavelength of the light and the external voltage applied in LED devices. The incorporation of the pyridine moiety increased the EL efficiency of the devices by a factor of 5 in relation to PPV [302,308,311]. A series of poly(2,5dialkoxy-1,4-phenylene-alt-2,5-pyridine)s in which the [alkoxy phenylenepyridyilene] structural unit acts as donor–acceptor pair was synthesized via a SCP [304]. The electron affinity of these polymers is ca. 2.5 eV, comparable to that of copolymers containing oxadiazole moieties. The electron-withdrawing pyridinylene groups were able to lower the LUMO energy in such a way that these polymers may have similar electron injection properties as typical oxadiazole-containing electron transport polymers, when they are used as active materials in LEDs. The Wittig and Wittig–Horner reactions have been employed to prepare copolymers containing bipyridine and silicon units. The organosilicon moiety improves the solubility and limits the conjugation length [312]. Recently, the bipyridine moiety linked to metals as iridium was used to fabricate blue phosphorescent LEDs with high emission intensity [313].

47.9.2 Oxadiazole-Containing Conjugated Polymers One of the best electron transport structures is the oxadiazole group, as noted above. The covalent attachment of the PBD moiety to an emitting polymer (Fig. 47.4(b)), was a natural development in LED technology, avoiding the

778 / CHAPTER 47 N—N a) O

n

OC12H25 N—N

O

b)

O(CH2CH2O)3 O

n

N—N

C12H25O

FIGURE 47.17. Oxadiazole-containing EL polymers. (a) Poly(2,5-diphenylene-1,3,4-oxadiazole)-4,40-vinylene; (b) alternating oxadiazole-alt-alkoxyphenylene and a aliphatic spacer.

problems with the deposition of an additional layer in diode construction and the anticipated phase separation after film preparation or under operating conditions. Some examples of oxadiazole-containing EL polymers are given in Fig. 47.17. Among the various examples found in the literature one can cite the use of oxadiazole incorporated in the main chain [168,314–316] or as a pendant group [168,214,317–320]. In the first case the oxadiazole moiety was used as a comonomer in chromophoric imides [168,321], copolymers containing thiophene sequences [322], or in a PPV type main chain [168,267,323]. Synthetic approaches for the oxadiazole-containing polymers were the Heck reaction or via the formation of a polyhydrazide precursor [324]. Oxadiazole moieties linked to PPV [168,325], alkyl-PPV, and polythiophene chains as pendant groups with improved EL efficiencies due to higher electron affinity, better injection, and transport properties have also been reported [326]. The insertion of electron-transporting groups in p-type polymers brings about bipolar carrier transport ability. The combination of donor–acceptor groups in the same chain in attempts to achieve balanced electron–hole injection and transport avoiding the use of intermediate transport layers has been widely explored, such as the combination in the same chain. Due to their strong electron-acceptor character, oxadiazole and PPV [158,326] or oxadiazole and oligothiophenes [327] have been reported, for example for PPV type chains [169,326,328] or other chromophores like naphthalimide [329], fluorene [330], anthracene, triphenyl amine.

47.9.3 Polyquinolines and Polyquinoxalines Polyquinolines and polyquinoxalines are n-type (electron transporting) polymers and therefore offer alternative EL device engineering in conjunction with the extensively studied p-type (hole transporting) polymers such as poly(p-phenylene vinylene)s, polyphenylenes, and polythiophenes. A variety of polyquinolines [331,332] and polyquinoxalines [333] has been reported in the literature, acting as the active

emitting layer [331,334,335] or as the transport layer [336– 338] in LEDs. Figure 47.18 shows some representative examples of these emitting and electron-transporting polymers. The increased electron deficiency of the quinoxaline ring due to additional imine nitrogen, compared to the quinoline ring, enhances the electron-accepting ability of conjugated polyquinoxalines, thus improving electron transport protonated by means of acidic solvents, intense blue emission was observed at 450 nm. When the positive charge on the nitrogen atom of the quinolines reached a critical value, intermolecular electrostatic repulsion prevented the aggregation, and the emission spectra were those of the isolated chain. The nonprotonated forms formed LC structures, which formed excimers with emissions in the 550– 600 nm range. pH-Tunable PL was also demonstrated in poly(vinyl diphenylquinoline) with emission maxima varying from 486 to 529 nm (blue to green). Intramolecular excimer emission was observed in acidic solutions but not in neutral solutions or thin films of the polymer. The polymer, which was obtained from a modification of polystyrene, was introduced as a prospective high Tg , electron transporting counterpart of the hole transporting side chain PVK. The electron transporting properties of copolymers bearing fluorene and quinoline units with conjugation confinement varied with the chain rigidity and conjugation length and proved to be useful in double and triple layer devices [339]. A series of polyquinolines containing 9,9’spirobifluorene was recently reported. The two fluorene rings were orthogonally arranged and were connected via a common tetracoordinated carbon. The incorporation of the piro moiety provided good solubility due to a decrease in the degree of molecular packing and crystallinity while imparting a significant increase in both Tg and thermal stability, by restricting segmental mobility. Polymers incorporating bis(phenylquinoline) and regioregular dialkylbithiophene in the backbone showed substantial enhancement in device performance under ambient air conditions (1.4% external quantum efficiency, 2,170 cd=m2 ) in bilayer MEH-PPV LEDs [340] properties [334,335]. Polyquinolines are

ELECTROLUMINESCENT POLYMER SYSTEMS

a)

/

779

b) R’ N N N n

n

R’ R’= H, CH3, nC9H19

PPQ

FIGURE 47.18. Examples of EL polyquinolines.

usually prepared by the acid-catalyzed Friedlandler condensation of bis(o-aminoketone)s and bis(ketomethylene) monomers, have good mechanical properties and high thermal stability, and can be processed into high quality thin films. A new synthetic route to organic solvent-soluble conjugated polyquinolines incorporating bis(4-alkylquinoline) units through a new A–A monomer was used to prepare 3,3’-dinonanoylbenzidine copolymers, poly(2,2¢-arylene4,4’-bis(4-alkylquinoline)) [341]. Variations in the polyquinoline backbone linkage (R) and pendant side groups provided a means to regulate the intramolecular and supramolecular structures which in turn enabled tuning of the light emission from blue to red. Studies of supramolecular photophysics of selfassembled block copolymers bearing styrene–polyquinoline sequences demonstrated evidence of J-aggregation in well-defined, ordered structures such as micelles and vesicles. The polymers represent a novel class of functional luminescent materials [342]. In a recent contribution, Jenekhe reported a detailed study on voltage-tunable multicolor emissionbilayer LEDs combining PPV (as typical p-type layer) with a series of polyquinolines, polyanthrazolines, polybenzothiazoles, and a poly(benzimidazobenzophenanthroline) ladder, addressing the influence of the polymer–polymer interface in the diode efficiency and luminance and showing that its electronic structure plays a more important role than the injection barrier at the cathode/polymer interface [343].

47.9.4 Carbazole-Containing Conjugated Polymers Due to its electro- and photoproperties, the carbazole molecule has been used in various technological applications [344,345]. Polymers based on this compound have been used to enhance LED emission and also for color

tuning. In Partridge’s study of EL of dye doped PVK systems [346–348], the hole transport properties of this polymer were demonstrated. Devices in which the emitting layer was formed by PVK blended with other polymeric systems have shown remarkable increases in luminescence efficiency, as compared to those in which PVK was not incorporated. As an example, the blue emitting device with the ITO/polymer blend/Ca configuration made of poly(pphenyl phenylene vinylene) (PPPV) blended with PVK showed a quantum efficiency of 0.16% which is a good result for a blue emitter [349]. As in all cases of PVK blends, it was observed that there is an optimum molar ratio between the emitting polymer and PVK for the increase in EL intensity. If the diode is doped excessively with PVK the conductivity can be reduced, when the other polymer is more conductive than PVK itself. Apart from the homopolymer, the carbazole moiety can be incorporated in polymer chains as part of the main chain, like poly(2,5-dihexylphenylene-alt-N-ethyl-3,6-carbazolevinylene), which was combined with an electron transporting oxadiazole-containing structure [350]. Alternating structures containing units of 2,5-bis-trimethylsilyl-p-phenylene vinylene, or 2-methoxy5-(2-ethylhexyloxy)-p-phenylene vinylene with N-ethyl hexyl-3,6-carbazolevinylene or 9,9-n-hexyl fluorenevinylene were prepared via Wittig polycondensation. Among the four combinations the silyl-substituted carbazole copolymers presented the most interesting properties, for example, its EL quantum efficiency was 32 times higher than the MEH-PPV analog (film quantum efficiency was 0.81, one of the highest reported). The participation of the silyl group in the PL enhancement and also to the blue shift observed was attributed to its sterical hindrance and lack of electron donating ability as compared to alkoxy substituents [350]. Well defined carbazol-3,9-diyl based oligomer homologues were prepared by the Ullmann condensation and used for multilayer LEDs [351].

780 / CHAPTER 47 The hole conduction of the carbazole unit was studied by comparing polydiphenylacetylene derivatives without and with a carrier transport moiety as poly[1-(p-n-butylphenyl)2-phenylacetylene] and poly[1(p-n-carbazolylphenyl)2-phenylacetylene], respectively. It was shown that hole mobility enhancement by the attachment of the carbazolyl side groups brings about a remarkable improvement in the EL devices. Recently carbazole-imide moieties combined with fluorene were used to prepare LEDs with a brightness of 14,228 cd=m2 , maximum luminous efficiency of 4.53 cd/ A and maximum power efficiency of 1.57 lm/W [352]. A write-once read-many-times (WORM) memory device based on an acrylate polymer containing electron donating carbazole pendant groups, or poly(2-(9H-carbazol-9yl)ethyl methacrylate) (PCz), was demonstrated [353]. Carbazole–pyrene-based compounds when blended to PVK afforded EL devices with green emission with luminance up to 1,000 cd=m2 [354]. Semiladder poly(p-phenylene)s containing carbazole and fluorene moieties exhibited maximum luminescence of 5,500 cd=m2 and maximum luminance efficiency of 0.556 cd/A in single-layer light emitting devices with pure-blue emission (lmax 447 nm) [355]. A new type of electron-transporting polymer was reported as the first stereoregular polymerization of (N-noctyl-3-carbazoyl)acetylene, initiated with a Rh norbornadiene catalyst. The resulting polymers were composed of amorphous cis-transoid isomers, called a columnar, with wide range of emitting colors [356]. Light-emitting alternating copolymers of 9,9-dialkylfluorene and N-hexylcarbazole with conjugated and d-Si interrupted structures have been synthesized as an approach or the synthesis of nonaggregating optoelectronic polymers [357].

47.9.5 Other Nitrogen-Containing Conjugated Polymers Other electron affinity enhancer nitrogen-containing heterocyclic groups include the triazole [358], bithiazole [359] and nonconjugated amino groups as substituents, such as ¼¼N(CH3,C6H13) attached to the ring in (poly(2,5-bis(Nmethyl-N-alkylamino)phenylene vinylenes) [360]. In this case the nitrogen atom does not participate in the conjugation system, and its electron donor character provides a stronger donor effect to the amino substituent than the corresponding alkoxy substituents. As mentioned earlier (under transport layers), triamines have been used as hole transport materials. In a recent report, this class of compounds have shown emissive capacity as well, when inserted as pendant groups in conjugated backbones. Single layer devices fabricated from these copolymers can emit light ranging from yellow to bright red, depending on the aromatic units incorporated [361]. The attachment of carbazolyl groups as pendant groups grafted to a backbone

as PMMA, which was blended with MEH-PPV, enhanced the emission four times when compared to that of the pure polymer [362]. Apart from hole transport ability, PVK can interact with low molecular weight compounds or polymers to form new emitting species as exciplexes [363], and consequently bring about a shift in the emission wavelength. In the case of PPPV/PVK [364] mentioned above, where PVK was used both as matrix and hole transport material, the EL spectra of the system tended to blue shift as compared to the respective pure PPP. It was speculated that this was caused by a change in molecular conformation or aggregation of the PPP in the PVK matrix. In blends of the conjugated–nonconjugated multiblock copolymer PPV derivative (CNMBC) with PVK, the EL blue shifted according to PVK increments in the blend. The new emission was attributed to an exciplex formed by the two polymers. As the composition of the blend changed from a PVK-poor to a PVK-rich ratio the emitted light changed from green to blue. Further blends of composition 97/3 (wt%) PVK/block copolymer yielded an EL spectrum with a single emission peak in the blue region different in location from either single component, whereas for a certain range of composition the new peak coexisted with those of the pure components. An exciplex where an excited PVK combines with ground state copolymer was proposed.

47.10 POLYACETYLENES AND POLYMERS WITH TRIPLE BONDS IN THE MAIN CHAIN 47.10.1 Polyacetylenes Polyacetylene is the first conjugated polymer to exhibit a metallic conductivity. However, polyacetylene shows a very low PL efficiency [365]. Recently, in contrast, a number of good light emitting polyacetylene derivatives have been produced, covering the visible spectrum. The poor solubility of these rod-like structures with concomitant color tuning was addressed in three ways: by substituting the hydrogen atoms with alkyl or aryl groups as done in PPV, by means of copolymerization, or by a combination of both methods. In the first approach many of substituents were tried [366,367]. Monosubstituted polyacetylenes were often referred to as nonluminescent polymers, but the insertion of mesogenic pendants afforded intense blue emitting materials [368] as shown in Fig. 47.19(a). The emission observed from a series of alkyl and phenyl disubstituted polyacetylenes changed from blue-green to pure blue and the luminescence intensity was enhanced when the length of the alkyl side chain increased [369]. This effect was also observed in poly(3alkylthiophene) and in PPV derivatives [370], indicating that the p---p
interband transition increases with the length of the side chain, and at the same time the diffusion rate of the excitons to quenching sites is reduced by the longer interchain distances. Examples of EL disubstituted polyace-

ELECTROLUMINESCENT POLYMER SYSTEMS

CH

C

(CH2CH2)mDCO

R:

n

/

781

OC3H15

R m 2, 3, 4, 9

C

CH

n

H C

H

H C

C

n

CH2

C

n

C

n

CH2

C

N

C

C

C

N

n

Si

C

CH3

S a)

PIMA

PTEPA

PDPAA

PCINA

PDMPSiPA

C C

C

n

C

CH3 C

C

C

PDPA-Ad

C

n

n

n O

b)

CH3

PMeNA

PDPA-CPh

Si

PDPA-SiPh3

FIGURE 47.19. Examples of EL polyacetylenes. (a) Monosubstituted; (b) disubstituted. Reprinted with permission from Synth Met 1997;91:283. ß 1997 Elsevier Science.

tylenes are given in Fig. 47.19(b). Aryl-substituted polyacetylenes were stable to 2008C in either air or nitrogen, according to thermogravimetric analysis. With the interest in combining the electrooptical properties of PPA and PCz in a CPN film, we have synthesized a series of substituted poly(phenylacetylene)s containing carbazole unit as side groups, which subsequently through. Electropolymerization or chemical oxidation resulted in a conjugated polymer network having both inter- and intramolecular crosslinkages between the pendant monomer units [371].

47.10.2 Poly(phenylene ethynylene)s The HOMO–LUMO energy gap of some alkoxy substituted poly(phenylene ethynylene)s (ROPPE) is higher than that of the corresponding ROPPVs, indicating that introdu-

cing triple bonds in the main chain shortens the effective conjugation length. For example, poly(3,4-dialkyl-1,6-phenylene ethynylene) has a bandgap of 3.1 eV [372]. Poly(pphenylene ethynylene)s showed a lower energy barrier for electron injection than for hole injection, in contrast with the PPV analogs. This is an important feature for cathode stability, since more stable metals can be used [373]. A high quantum efficiency was obtained with poly(2,5 dialkoxy1,4-phenylene ethynylene) (ROPPE) in which the triple bond is equivalent to the double bond in poly(2,5-dialkoxy1,4-phenylene vinylene) (ROPPV) [374]. Bright blue-green EL was observed from an LED made with the copolymer ROPPE and pyridine (Py) with AI/ROPPE–Py/ITO. In comparison with PPV it was suggested that the triple bonds in the chain were responsible for the blue shift and enhancement of the EL, due to the shortening of the effective conjugation length and effective confinement of the

782 / CHAPTER 47 excitons. However, the effects of the triple bonds were suppressed by the introduction of electron-rich moieties in the chain, such as in poly(2,5-dialkoxy-1,4-phenylenediethylene-co-9,10-anthracenylene) (ROPPE–An). In this case the insertion of the 9,10-anthracenyl group caused a delocalization of the p-electrons and enhancement of interchain interactions [372,375]. Linear copolymers containing phenylene ethynylene linkages have been synthesized using precursor routes [376], inserting m-linkages [377] or aliphatic spacers [378,379]. In the precursor route, regular building blocks of p-phenylene, norbonadiene, and diethynyl benzene made up the soluble precursor, which was converted to a polymer with enyne units upon the release of cyclopentadiene through a retroDiels–Alder pathway. The copolymer with phenylene with m-linkages made through a Heck-type route, using Pd as catalyst, displayed a PL four times stronger than that of the corresponding p-substituted analog. This synthetic method is very often employed in a general way in poly(arylene ethynylene)s chemistry [380]. PPE analogs obtained by copolymerization with di(2-octyldecyl)anthracene yielded stable blue and green emission without chain stacking [381]. REFERENCES 1. Bradley DDC. Synth Met 1993;54(1–3):401–15. 2. Kalinowski J. Electronic processes in organic electroluminescence. In: Miyata S, Nalwa S, editors. Organic Electroluminescent Materials and Devices. Japan: Gordon & Breach; 1997. p. 1. 3. Friend RH, Gymer RW, Holmes AB, Burroughes JH, Marks RN, Taliani C, Bradley DDC, dos Santos DA, Bre¨ das JL, Lo¨ glund M, Salaneck WR. Nature 1999;397:121–8. 4. Cacialli F. Phil Trans R Soc Lond Ser A— Math Phys Eng Sci 2000; 358(1765):173–92. 5. Cacialli F. Curr Opin Colloid Interf Sci 1999;4(2):159–64. 6. Segura JL. Acta Polym 1998;49(7):319–44. 7. Kraft A, Grimsdale A, Holmes AB. Angew Chem Int 1998;37(4):402–28. 8. Mori Y. Single layer organic electroluminescent devices. In: Miyata S, Nalwa S, editors. Organic Electroluminescent Materials and Devices. Japan: Gordon & Breach; 1997. p. 391. 9. Leventis N, Huang L-Y. Polym News 1995;20(10):307–13. 10. Rothberg LJ, Lovinger AJ. J Mater Res 1996;11(12):3174–87. 11. Sheats JR, Chang YL, Roitman DB, Socking A. Acc Chem Res 1999;32(3):193–200. 12. Greiner A. Polym Adv Technol 1998;9(7):371–89. 13. Kim DY, Cho HN, Kim CY. Prog Polym Sci 2000;25(8):1089–139. 14. Conwell EM, Stolka M, Miller MR. Electroluminescent materials, devices and large- screen displays. International Society for Optical Engineering (SPIE) Proceedings, San Jose, CA, 1993. 15. Miyata S, Nalwa HS. Organic Electroluminescent Materials and Devices. Amsterdam: Gordon & Breach; 1997. 16. Hsieh BR, Wei Y. Semiconducting polymers: properties and synthesis. ACS Symp Series 735, American Chemical Society; 1999. 17. Kippelen B, Bradley D. Polymer photonic devices. IV. International Society for Optical Engineering (SPIE) Proceedings, vol. 3281, San Jose, CA, 1998. 18. Akcelrud L. Prog Polym Sci 2003;28:875–962. 19. Faraggi EZ, Davidov D, Cohen G, Noah S, Golosovsky M, Avny Y, Neumann R, Lewis A. Synth Met 1997;85(1–3):1187–90. 20. Rogers JA, Bao Z, Dhar L. Appl Phys Lett 1998;73(3):294–6. 21. Li AK, Yang SS, Jean WY, Hsu CS, Hsieh BR. Chem Mater 2000;12(9):2741–4.

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CHAPTER 48

Magnetic, Piezoelectric, Pyroelectric, and Ferroelectric Properties of Synthetic and Biological Polymers Andrzej Kloczkowski* and Taner Z. Sen*,y *L .H. Baker Center for Bioinformatics and Biological Statistics, Iowa State University, Ames, IA 50011, USA; Department of Biochemistry, Biophysics, and Molecular Biology, Iowa State University, Ames, IA 50011, USA

y

48.1 48.2 48.3 48.4 48.5

Magnetic Polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piezoelectric Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pyroelectric Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ferroelectric Polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piezoelectric, Pyroelectric, and Ferroelectric Properties of Biopolymers. . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

787 789 790 791 792 793

uted throughout the volume of the material, and the total magnetic moment of the material is zero. Paramagnetics magnetize parallel to the direction of the external magnetic field, the magnetization of the paramagnetic material vanishes when the external magnetic field is switched off. Ferromagnetism is a property exhibited by compounds of certain metals: iron group or rare-earth elements. The atomic magnetic moments of these materials, below the so called Currie temperature, orient in one direction. The resulting magnetic susceptibility is very high and shows a saturation effect. The large magnetization of the material is preserved when the external field is switched off, i.e., ferromagnetics may have spontaneous magnetization below the Curie point. Above the Curie temperature ferromagnetics behave as ordinary paramagnetics. Ferromagnetic materials are composed of small, spontaneously magnetized regions, so called domains. The directions of magnetization of different domains do not coincide and the net magnetic moment of the substance in the absence of the external magnetic field may be zero. The application of even weak external magnetic field aligns magnetic moments of domains of the ferromagnet and gives rise to the high magnetic moment of the sample. Antiferromagnetism is a property of certain crystals containing transition elements which show spontaneous antiparallel magnetic ordering below Ne´el temperature. Above the Ne´el temperature the antiferromagnetic

48.1 MAGNETIC POLYMERS Depending on the behavior in the presence of the external magnetic field all materials may be classified as diamagnetic, paramagnetic, ferromagnetic (or antiferromagnetic). Diamagnetism is exhibited by materials with a negative magnetic susceptibility, i.e., they magnetize in a direction opposite to the direction of the applied magnetic field, and are repelled by a magnet. All materials with zero net electronic orbital moment and zero net spin magnetic moment are diamagnetics. Most of the organic molecules and polymers have a closed shell electronic structure with a singlet ground state with all electron spins paired, and therefore have diamagnetic properties. The diamagnetism follows from the Lenz’s law, that the induced field opposes the applied magnetic field. All substances including paramagnets (or ferromagnets) have diamagnetic susceptibility components, but these components are very small, usually much smaller in comparison to paramagnetic susceptibilities. Substances with nonzero net electronic orbital moments or with unpaired electron spins are paramagnetics. Paramagnetic materials have open-shell electronic structure, with some unpaired electrons. Organic molecules with one unpaired electron in doublet ground state show paramagnetic properties and are called free radicals. The magnetic moments of molecules of paramagnetic substances are randomly distrib787

788 / CHAPTER 48 ordering vanishes. The quantum theory of ferromagnetism was developed by Heisenberg. According to the Heisenberg theory there is an effective interaction between the electron spins, which arises from to the interplay between electrostatic Coulomb forces and the effect of the Pauli exclusion principle. The exchange energy of this interaction is Eij ¼
2Jij si  sj ,

(48:1)

where si and sj are spin angular momenta for electrons i and j, and the coupling constant Jij is called the exchange integral. If the coupling constant J is positive and large enough the ferromagnetic ordering in the system is favorable. For large negative values of J the antiparallel antiferromagnetic alignment of spins is favored. If the exchange integral is small, the ferromagnetic (or antiferromagnetic) state is not possible. Ferromagnetism of the transition and rare-earth metal salts and complexes is due to high-spin states of those atoms. The electronic structure of these metal atoms is open-shell, with d-orbitals and f-orbitals containing several unpaired electrons [1–3]. Until recently the only known ferromagnetic polymers were metallo-polymers containing transition metals or rareearth elements. The presence of the metal, particularly Fe, Ni, Co may lead to ferromagnetic behavior due to the formation of extrinsic separate magnetic phase of the metal. Another possible mechanism of ferromagnetism in metallopolymers, of the intrinsic nature, is the strong coupling of the spins of metal ions with organic polymeric ligand. The problem if the ferromagnetic behavior of the metallopolymer has intrinsic or extrinsic origin is very complicated [4]. There are several known polymers containing iron atoms which show ferromagnetic properties. An example is complex of the PPH polymer with FeSO4  7H2 O shown below, containing each iron ion surrounded by six nitrogen atoms

HC

N

N

CH N

(CH2)6

N

(CH2)6

Fe N HC

N

CH

existence of organic ferromagnetic compounds has been theoretically postulated in 1963 by McConnell for high spin charge transfer complexes [5]. The proposed systems contained alternating linear sequences of donor–acceptor pairs in triplet and singlet ground states. A parallel ordering of all spins in those systems was predicted [6]. In 1968 Mataga proposed theoretically several magnetic polymer structures containing high spin blocks, as shown below with dots above carbon atoms denoting free radicals [7].

.

.

.

.

C

C

C

C

.

.

.

.

C

C

C

C

.

.

.

.

C

C

C

C

.

.

C

.

.

C

.

C

.

C

.

C

.

C

.

C

.

C

C

.

C

C

Mataga predicted that according to Hund’s rule unpaired electrons in these polymers may have parallel spins, leading to ferromagnetic behavior. In 1987 Ovchinnikov et al. announced the discovery of the first ferromagnetic organic polymer based on polydiacetylene chain with dangling nitroxyl radicals [8,9]. The compound was obtained by bulk polymerization of stable paramagnetic biradical BIPO initiated by light. The chemical structure of this compound is shown below. O

HO

N C OH

The observed ferromagnetic behavior of this polymer was first attributed to the intrinsic mechanism, but later studies have shown that the ferromagnetism is of the extrinsic nature, due to the formation of the ferromagnetic phase containing Fe2 O3 [4]. Discoveries of purely-organic ferromagnetic materials, i.e., materials which do not contain transition metals or rare-earth elements, have been reported. The possibility of

.

C

C

C C n

N O

The saturation magnetization of this material was very low (0.1% of the theoretical prediction), and the fraction of crystallites showing ferromagnetic properties was also extremely small. The Curie temperature was reported to be 440  20 K. Those results, however, were unverified by other laboratories. Also Torrance reported the synthesis of

PROPERTIES OF SYNTHETIC AND BIOLOGICAL POLYMERS an organic ferromagnet by iodine oxidation of 1,3,5-triaminobenzene [10]. Iwamura obtained fully characterized ferromagnetically coupled oligomers of polycarbenes [11,12]. These announcements have been followed by many other reports on discoveries of ferromagnetic polymers, by various authors [4]. However, all those results were unconvincing because of the poor characterization of those polymers and problems with reproducibility of the results. There is a growing list of well-characterized high spin polymers, mostly based on polyacetylene chain with pendant groups containing radical centers, but the only observed magnetic behavior of these compounds is a weak antiferromagnetism. Generally, the fraction of the bulk of the polymers exhibiting ferromagnetic behavior is very small, usually less than 5%. It is very probable that the reported ferromagnetic properties of organic polymers are due to such accidental factors, as ferrometalic impurities [4]. Several liquid– crystalline ferromagnets have also been reported [13]. Another approach to magnetic polymers are so-called polaronic ferromagnets, with spins introduced to the polymer molecules by doping, which form radical ions named polarons. With proper molecular topology ferromagnetic coupling among polarons may be possible [2]. The field of magnetic polymers are reviewed by Miller and Epstein [14] and by Kamachi [15].

48.2 PIEZOELECTRIC POLYMERS Piezoelectricity is an effect of the generation of electric polarity in certain dielectric materials (i.e., electric insulators) in response to the applied mechanical pressure. The effect was discovered by Pierre and Jacques Curie in 1880. The polarization of the material is proportional to the applied stress. The piezoelectric effect has very important practical applications in transducers, i.e., devices which convert electric signals to mechanical signals, or the mechanical signals to electric signals, such as microphones, loudspeakers, gramophone needles, ultrasound generators, quartz clocks, etc [16,17]. The most known piezoelectric materials are some inorganic crystals, such as: quartz, barium titanate BaTiO3 , ammonium dihydrogen phosphate NH4 H2 PO4 , Rochelle salt (potassium sodium tartrate tetrahydrate NaKC4 H4 O6  4H2 O), or zinc blende ZnS. The electric polarization generates the electric potential between opposite surfaces of the crystal, and gives rise to the electric current if these two sides of the crystal are connected by a wire. The converse piezoelectric effect, i.e., the generation of the mechanical stress in the piezoelectric material by applying the electric voltage to opposite sides of the crystal is also observed. The generated piezoelectric strain is proportional to the applied electric field. (There is an important difference between converse piezoelectric effect and electrostriction, where the deformation is proportional to the square of the field). On the molecular level, piezoelectricity occurs

/

789

when elastic deformations of the body are accompanied by unequal vectorial displacements of centers of gravity of positive and negative charges (ions) leading to the net polarization of the sample. The unequal vectorial displacements in the crystal require the lack of the crystallographic center of symmetry. There is a correlation between the symmetry of the crystal and the type of stresses which generate the piezoelectric polarization. The lower the symmetry of the crystal, the more different types of stresses produce piezoelectric polarization. Generally, each of the three components of the polarization DPi is linearly related to each of the nine components of the stress tensor sjk DPi ¼ dijk sjk ,

(48:2)

where dijk is the third-rank tensor having 27 components called piezoelectric coefficients. In the absence of body torques the stress tensor is symmetric, i.e., sjk ¼ skj and has only 6 components so the maximum number of piezoelectric coefficients is 18 [18]. It is therefore convenient to represent the stress tensor as a six component vector sm , and the piezoelectric coefficients as a second-rank tensor dim , where m ¼ 1–6, so that DPi ¼ dim sm :

(48:3)

Similarly in the converse piezoelectric phenomena each of the 6 components of the strain tensor em is related to each of the 3 components of the electric field em ¼ dcim Ei ,

(48:4)

where dcim are converse piezoelectric coefficients. The stress and strain tensors are related through the relations em ¼ smn sn

(48:5)

sm ¼ cmn en ,

(48:6)

and

where smn and cmn (with m, n ¼ 1–6) are the elastic compliance tensor and the elastic stiffness tensor, respectively [17]. The most known polymer exhibiting piezoelectric behavior is poly(vinylidene fluoride) (PVF2 ) which has the structure (CF2 CH2 )n [19]. The piezoelectric polymer sample is obtained by orienting dipole moments in the polymer by poling. The symmetry of the highly oriented poled polymer film reduces the number of possible piezoelectric coefficients from 18 to 5. Using the convention where 1 is the direction of drawing, 2 is in the direction of the film, and 3 is normal to the film surface the only nonzero piezoelectric coefficients for uniaxially oriented film are d31 ,d32 ,d33 ,d15 , and d24 [18]. For biaxially oriented films d31 ¼ d32 and d15 ¼ d24 . The coefficient d33 is very difficult to measure, so sometimes it is calculated from the hydrostatic piezoelectric coefficient dh ¼ d31 þ d32 þ d33 :

(48:7)

790 / CHAPTER 48 Table 48.1 shows some piezoelectric coefficients for thin PVF2 films obtained by Kepler and Anderson from static measurements [18]. It should be noted that piezoelectric coefficients, especially d33 are very difficult to measure and stress-induced changes in polymer crystallinity contribute significantly to the piezoelectric coefficients [20]. Because of the viscoelastic properties of polymers the piezoelectric charge signal is not in phase with the stress or strain. Piezoelectric coefficients obtained from dynamic measurements depend on the frequency. It has been found that experimental values of the piezoelectric coefficients also depend on the method of processing of polymer films. For example, Wang found that in drawn and rolled films, d31 is about 25% higher than in films only drawn [21]. It has been shown that annealing has a large effect on the magnitude of the piezoelectric coefficients. For example Takase et al. found that annealing the stretched PVF2 films in temperatures between 160 and 180 8C increased d31 from 20 to 28 pCN
1 [22]. Piezoelectric coefficients depend also on temperature. It has been found that piezoelectric coefficients increase with increasing temperature, especially rapidly above the glass transition temperature (
50 8C for PVF2 ) [23]. Table 48.2 shows piezoelectric coefficients of PVF2 obtained from dynamic measurements at frequencies 10 Hz and 25 Hz [20]. Other most known piezoelectric polymers are: poly(vinylchloride) (PVC), poly(vinylfluoride) (PVF), and Nylon 11. The piezoelectric coefficients of these polymers are much lower than piezoelectric coefficient of PVF2 . Table 48.4 shows piezoelectric coefficients of these polymers and compares them with those of some piezoelectric inorganic crystals.

TABLE 48.1. Piezoelectric coefficients of thin PVF2 films (in pCN
1 ) [18]. Coefficient d31 d32 d33 dh d33 (calculated from dh )

Biaxially oriented film

Uniaxially oriented film

4.34 4.36
12.4
4.8
13.5

21.4 2.3
31.5
9.6
33.3

TABLE 48.2. Piezoelectric coefficients of PVF2 (in pCN
1 ) obtained from dynamic measurements [20]. Coefficient d31 d32 d33 dh

Frequency 10 Hz

Frequency 25 Hz

28 4
35
3

17.5 3.2

48.3 PYROELECTRIC POLYMERS Pyroelectricity is the effect of the generation of electric polarization in certain dielectrics by the change of temperature. The change of the electric polarization DPi (for each component of the polarization vector) in the pyroelectric material is proportional to the change of the temperature DT DPi ¼ pi DT,

(48:8)

where pi (with i ¼ 1–3) are pyroelectric coefficients. Pyroelectricity is related to piezoelectricity, since it also occurs in crystals with crystallographic cells without a center of symmetry. Additional requirement for pyroelectric crystal is the occurrence of a polar axis, which reduces the number of possible crystal classes in comparison with piezoelectrics. Out of the total 32 possible crystallographic point groups, 20 groups fulfill requirements for piezoelectricity, but only 10 groups exhibit pyroelectricity. The most known pyroelectric crystals are: barium titanate BaTiO3 , cane sugar, and tourmaline. Because pyroelectric materials have polar axis they have permanent electric polarization, which is usually compensated by free charges on the surface of the material. The change in temperature alters the electric polarization, and leads to the pyroelectricity. The converse pyroelectric effect, the so called electrocalorimetric effect in pyroelectrics is also observed. The external electric field applied to a pyroelectric material changes its temperature. In order to eliminate the effect of the thermal expansion, measurements of the pyroelectric effect are often performed for mechanically constrained (e.g., by clamps) materials. We may also define two types of pyroelectricity: the primary pyroelectricity, which is measured when the dimensions of the sample are fixed (by clamps), and secondary pyroelectricity containing additional contribution to the polarization resulting from the thermal expansion. The secondary pyroelectricity coefficients psi are related to piezoelectric coefficients dim , elastic stiffness tensor cmn , and thermal expansion coefficients an [18]. psi ¼ dim cmn an :

(48:9)

The pyroelectric effect is widely used for the detection of infrared radiation, and for very precise measurements of the temperature [16,17]. The most known polymer showing the pyroelectric effect is poly(vinylidene fluoride) (CF2 CH2 )n (PVF2 ) [19]. Polymer samples exhibiting pyroelectric properties are prepared by poling. Pyroelectric coefficients for polymers depend on the electric poling field. With the increasing poling field pyroelectric coefficients increase. Table 48.3 shows primary and calculated secondary pyroelectric coefficients of PVF2 films, together with thermal expansion coefficients, and elastic stiffness coefficients, obtained by Kepler and Anderson [18]. The corresponding piezoelectric coefficients were earlier shown in Table 48.1.

PROPERTIES OF SYNTHETIC AND BIOLOGICAL POLYMERS Other popular pyroelectric polymers are: poly(vinylchloride) (PVC), poly(vinylfluoride) (PVF), and Nylon 11, but their pyroelectric coefficients are smaller than pyroelectric coefficient of PVF2 . On the other hand the pyroelectric coefficient of PVF2 is much smaller than pyroelectric coefficients of inorganic crystals. Table 48.4 shows the comparison of piezoelectric and pyroelectric coefficients of some inorganic crystals and polymers [24]. An extensive review of pyroelectric materials including single crystals, ceramics, liquid crystals, and composites has been recently published by Lang and Das-Gupta [25]. 48.4 FERROELECTRIC POLYMERS Ferroelectrics are materials with built-in spontaneous, permanent electric polarization. Similarly as ferromagnetics possess the spontaneous, permanent magnetization, ferroelectrics have spontaneous polarization, which can be reversed by applying external electric field. The most known ferroelectrics are inorganic monocrystals, such as Rochelle salt (potassium sodium tartrate tetrahydrate

TABLE 48.3. Pyroelectric coefficients pi and psi , thermal expansion coefficients an , and elastic stiffness coefficients cmn of PVF2 films [18]. Coefficient

Biaxially ordered film

Uniaxially ordered film


1.25
0.44 1.24 1.00 5.04 3.25


2.74
1.48 0.13 1.45 5.04 3.25

p3 (10
5 Cm
2 K
1 ) ps3 (10
5 Cm
2 K
1 ) a1 (10
4 K
1 ) a2 (10
4 K
1 ) c11 (109 Pa) c12 (109 Pa)

TABLE 48.4. Piezoelectric coeffiecients d and pyroelectric coefficients p of several inorganic crystals and several polymers with references to original papers.
1

Material

d(in pCN )

Ref.

Quartz PZT-4 (lead zirconate titanate) BaTiO3 Rochelle Salt Triglycine sulfate Sr0:5 Ba0:5 Nb2 O6 PVC PVF Nylon 11 PVF2

2.3 (d11 ) 289 (d33 )

[26] [27]

190 (d33 ) 53 (d25 ) 50

[27] [26] [29]

95 0.7 1 0.26 28 (d31 )

[29] [30] [19] [30] [23]

p(in10
5 Cm
2 K
1 )

Ref.

27

[28]

20

[28]

30

[28]

60 0.1 1.0 0.5 4

[29] [30] [31] [28] [32]

/

791

NaKC4 H4 O6  4H2 O), monopotassium dihydrophosphate (KH2 PO4 ), or barium titanate (BaTiO3 ). At sufficiently high temperatures ferroelectrics show normal dielectric behavior. However, below a certain critical temperature (so called, Curie temperature), even a small electric field causes a large polarization, which is preserved even if the external field is switched off. This means that below the Curie point ferroelectric materials show spontaneous polarization. The phase transition at the Curie temperature is related to the change of the lattice symmetry of the sample. The spontaneously polarized crystal is anisotropic and has lower symmetry than the nonpolarized one. Ferroelectric materials below the Curie temperature are also always piezoelectric, because the polarized sample has no center of symmetry. If the nonpolarized crystal has the center of symmetry, the piezoelectricity of the sample vanishes above the Curie temperature. All ferroelectrics below the Curie temperature also always show pyroelectric behavior. Until the late sixties the only known ferroelectrics, piezoelectrics, and pyroelectrics were certain inorganic monocrystals, or polycrystalline ceramics like lead titanate zirconate perovskites. Other known materials with macroscopic polarization were electrets, (for example mixtures of beeswax and rosin) in which the polarization was produced by application of the electric field in the melted state and then by cooling and the solidification of the polarized material. In 1969 Kawai discovered that the poly(vinylidene fluoride) polymer (known as PVF2 or PVDF) is ferroelectric [19]. PVF2 has a simple molecular structure (CF2 CH2 )n with the degree of polarization n usually greater than 10,000. PVF2 is a crystalline polymer, with about the half of the polymer molecules in the amorphous noncrystalline state and another half in crystalline lamella state [33]. The crystallographic studies have shown that PVF2 may form four different phases. One of the phases called the a-phase (or phase II) obtained by quenching from the melt is nonpolar, while other three phases called the b,g, and the d-phase (or phases I, III and IV), respectively, are ferroelectric [18]. The most important is the b-phase, in which the polymer has the all-trans conformation (and the largest dipole moment) and two chains in the unit cell are aligned in the same direction. The phases g and d have the electric polarization per unit cell smaller then the b-phase. The phase transformations between different phases of PVF2 are possible by means of annealing, drawing, or poling. The ferroelectric phase of PVF2 was discovered by poling PVF2 film, i.e., by heating a film above temperature 100 8C and applying high electric field (above 50 MVm
1 ) and then cooling the sample to the room temperature. The resulting material had very large piezoelectric coefficient (6:7 pCN
1 ), about ten times larger than for any other polymer, and possessed also pyroelectric properties, at temperatures well above the glass transition temperature (
40 8C) of the polymer [16]. The magnitude of the piezoelectric coefficient and the induced polarization increase with the increasing

792 / CHAPTER 48 voltage and the increasing poling temperature. Poling can take place even at room temperature and below, if sufficiently high electric field is applied. The PVF2 b-phase melts in temperature range 175–180 8C and no ferroelectric phase transition below the melting temperature is observed. The ferroelectric phase transition is observed in copolymers of PVF2 with poly(trifluoroethylene) (PF3 E) or with poly (tetrafluoroethylene) (PF4 E). For example the study of the 51%mol PVF2 – 49%mol PF3 E copolymer shows the ferroelectric-to-paraelectric first order phase transition at the temperature 65 8C. The addition of PF4 E or PF3 E to PVF2 increases the number of defects in the PVF2 chains and lowers the Curie temperature of the copolymer in comparison to the pure PVF2 . The extrapolation of the plot of the Curie temperature as a function of the PVF2 content in the copolymer to the pure PVF2 gives the estimation of the Curie point of PVF2 at 190 8C. This temperature is above the melting point temperature of PVF2 and therefore the Curie point for pure PVF2 is not observed [18]. Ferroelectric properties (much weaker than PVF2 ) are shown also by other polymers like poly(vinylchloride) (PVC) or poly(vinylfluoride) (PVF). (See Table 48.3 for

O –

–

OC–O–(CH2)10 – C – O –

CH3

–

O –

(–CH2–C(CH3)–)n

comparison of piezoelectric and pyroelectric coefficients of PVC and PVF with those for PVF2 and for some ferroelectric inorganic crystals). In the mid-seventies it was shown that because of the symmetry requirements, chiral smectic C liquid crystals are ferroelectric [34]. Liquid crystals are formed by anisotropic rod-like molecules. The most known liquid crystals are nematics, with orientational ordering of molecular axis of mesogenic molecules around the preferred direction, but no spatial ordering. The smectic A liquid crystals have (in addition to the orientational ordering) spatial alignment of molecules in layers, with the preferred direction perpendicular to the plane of the layers. In the smectic C phase the direction of orientational ordering is tilted relative to the normal to the plane of the layers. Chiral molecules have no center of symmetry, i.e., the mirror image of the molecule differs from the molecule itself. If a chiral mesogenic molecule has permanent dipole moment, then the unit cell of the smectic C phase has no center of symmetry and the material should be ferroelectric. The first ferroelectric liquid crystalline polymer synthesized by Shibaev and Plate [35] is shown below

– OCO –

– C – O – (CH2)2 – CH C2H5

The molecule is a liquid crystalline polymer with chiral smectic C phase forming parts attached as side chains. The field required to switch the direction of polarization of the polymer is very low (0:3 MVm
1 ). There is a lot of interest in liquid crystalline ferroelectric polymers, because of their possible use for fast-switching electro-optical devices. More information about ferroelectric liquid crystals can be found in references [36,37]. Another group of ferroelectric polymers are odd nylons. The most known is Nylon 11 O k (
NH
C
(CH2 )10
)n In crystal phase Nylon 11 molecules are packed in sheets with hydrogen bonds between oxygen atoms and NH groups of neighboring chains, and dipole moments of Nylon 11 chains are aligned. The piezoelectric and pyroelectric coefficients of Nylon 11 are smaller than those for PVF2 (see Table 48.4 for comparison). A comprehensive bibliography for ferroelectric materials (including polymers) has been recently published by Toyoda [38].

48.5 PIEZOELECTRIC, PYROELECTRIC, AND FERROELECTRIC PROPERTIES OF BIOPOLYMERS Piezoelectricity, pyroelectricity, and ferroelectricity is hardly confined to synthetic polymers. Some biopolymers also possess these properties, and scientists study them to understand how nature exploits these properties. The earliest studies of biopolymer piezoelectricity, for example, go back to 1960s when Morris Shamos and Leroy Lavine (with Michael Morris) studied bone piezoelectricity [39] and later postulated piezoelectricity as a ‘‘fundamental property’’ of tissues of biological origins [40]. In 1968, RNA ferroelectricity was demonstrated by Stanford and Lorey [41]. However, the scientific interest in these properties of biological molecules was dwarfed by the interest in other materials. In 1999 Sidney Lang [42] indicated that compared to ‘‘thousands’’ of publications on piezoelectric, pyroelectric, and ferroelectric materials, only less than 100 of them were biologically related. Piezoelectricity in biological molecules can be found both in soft and hard tissues as shown in Table 48.5. In this table, dij represent piezoelectric coefficient components, which

PROPERTIES OF SYNTHETIC AND BIOLOGICAL POLYMERS TABLE 48.5. Piezoelectric coefficients (in 10
12 m=V) in biomaterials (taken from Lemanov [43]). Material Bovine achiles tendon Horse femur Silk

d14

d15

d31

d33


2.7
0.2
1.1

1.4 0.04 0.25

0.09 0.003 0.02

0.07 0.003 0.023

TABLE 48.6. Pyroelectric coefficients (in 10
5 Cm
2 K) in biomaterials (taken from Lemanov [43]). Material

Coefficient

Hoof tendon Insect thorax Wheat Plant leaves

0.00004 0.35 0.46 0.015

are previously explained under piezoelectric polymers section. Pyroelectric coefficients for some biological materials are similarly given in Table 48.6. As explained under pyroelectric polymers section, pyroelectric coefficients are related to piezoelectric coefficients. Compared to those of PVF2 films in Table 48.3, piezoelectric coefficients of biopolymers shown in Table 48.6 are very small and have positive signs. Piezoelectricity and pyroelectricity in biopolymers originate from their building blocks: biopolymers are copolymers with 20 different monomers. These monomers are called amino acids and consist of a chiral carbon (except glycine) surrounded by an amino group, hydrogen atom, carboxyl group, and a side chain R as shown below. H NH2 – C – COOH

/

793

TABLE 48.7. Piezoelectricity and symmetry groups of amino acids [45] (L and D denote left-sided and right-sided enantiomers, and DL is the equimolar mixture of enantiomers). Amino acid a-glycene g-glycene L-alanine L-valine L-isoleucine L-glutamic acid L-cysteine DL-alanine DL-valine DL-serine DL-aspartic acid DL-lysine DL-tyrosine DL-trptophan

Piezoelectricity þ þ þ þ
þ
þ



þ


Symmetry C2h C3 D2 C2 C2 ,D2 C2 ,D2 C4 ,D6 C2v C1 C2h C2h D2 C2v C1

in cell division and to transport motor proteins, the orientation of tubulin proteins becomes crucial for cell fitness. This orientation may be controlled by electric fields [46]. Microtubules alignment was reviewed by Cyr [47]. Another interesting biological example that shows ferroelectric properties is voltage-dependent ion channels. These channels are glycoproteins located in the cell membrane, and they are found in either open or closed conformations. In the open conformation, these proteins are ion conductors. In the closed conformation, they become non-conducting, yet they still retain their ferroelectricity, even showing liquid crystalline properties [36,48]. There is a growing interest in studying the piezoelectricity, pyroelectricity, and ferroelectricity in biological systems to understand what roles they play in cellular functions, and this interest is expected to increase in the future.

R

Different side chain groups determine the type of a given amino acid. These side chains can be classified as neutral, acidic, or basic; aliphatic or aromatic; hydrophobic or polar, etc. As discussed previously, the piezoelectricity and pyroelectricity requires restrictions on crystal symmetry (e.g., 20 crystallographic groups can be piezoelectric, and only 10 groups demonstrates pyroelectricity). The piezoelectric response of amino acids to the high-frequency electric pulses is shown in Table 48.7; the pyroelectric properties of amino acids were also measured, but found not significant [44]. Ferroelectricity is a requirement for some biopolymers for their biological function. Microtubules which hold the cell structurally intact are a good example. These biostructures consist of identical a and b tubulin proteins which have permanent dipole moments. Since cells utilize microtubules

REFERENCES 1. Kahn O. Molecular magnetism, New York: VCH publishers, 1993. 2. CJ O’Connor, editor, Research frontiers in magnetochemistry, Singapore: World scientific, 1993. 3. D. Gatteschi et al., editors, Magnetic molecular materials, Dordrecht: Kluwer, 1991. 4. Miller JS. Adv. Mater. 1992; 4:298. 5. McConnell HM. J. Chem. Phys. 1963;39:1910. 6. Miller JS, Epstein AJ, Rief WM. Chem.Rev. 1988; 88:201. 7. Mataga N. Theor. Chim. Acta 1968; 10:372. 8. Korshak YV, Medvedeva TV, Ovchinnikov AA. Nature 1987;326:370. 9. Ovchinnikov AA, Spector VN. Synth. Met. 1988;27:B615. 10. Torrance JB, Oostra S, Nazzal A. Synth. Met. 1987;19:709. 11. Iwamura H. Pure Appl.Chem. 1986;58:187. 12. Iwamura H. Adv. Phys. Org. Chem. 1990;26:179. 13. Palacio F, Ramos J, Castro C. Mol. Cryst. Liq. Cryst. Technol. Sect. A 1993;232:173. 14. Miller JS, Epstein AJ. Chem. Eng. News 1995;(October 2):30. 15. Kamachi M. J. Macromol. Sci. Part C Polym. Rev. 2002;42(4):541–561.

794 / CHAPTER 48 16. Wang TT, Herbert JM, Glass AM. editors, The application of ferroelectric polymers, Glasgow: Blackie and Son, 1988. 17. Wong CP, editor, Polymers for electronic and photonic applications, San Diego: Academic Press, 1993. 18. Kepler RG, Anderson RA. J. Appl. Phys. 1978;49:4490. 19. Kawai H. Jpn. J. Appl. Phys. 1969;8:975. 20. Schewe H, McAroy BR, editors. Ultrasonics Symposium Proceedings. New York:IEEE, 1982. pp. 519. 21. Wang TT. J. Appl. Phys. 1979;50:6091. 22. Takase Y, Scheinbeim JI, Newman BA. J. Polym. Sci., Part B: Polym. Phys. 1989;27:2347. 23. Ohigashi H. J. Appl. Phys. 1976;47:949. 24. GM Sessler, editor, Electrets, 2nd. ed. Berlin: Springer-Verlag, 1987. 25. Lang SB, Das-Gupta DK. Ferroelectrics Rev. 2000;2:217–354. 26. W.G. Caddy, Piezoelectricity, New York: McGraw-Hill, 1946. 27. Berlincourt DA, Curran DR, Jaffe H. Physical Acoustics. New York: Academic Press, 1964. pp. 1. 28. Garn LE, Sharp EJ. IEEE Trans. PHP 1974;10:28. 29. Liu ST, Long D. Proc. IEEE 1978;66:14. 30. Litt MH, Hsu C, Basu P. 2005;No: 5 on Office of Naval Research Contract No. N00014–75-c-0842 31. Phelan RJ Jr., Mahler RJ, Cook AR. Appl. Phys. Lett. 1971;19:337. 32. Day GW, Hamilton RL, Phelan RJ Jr., Mullen LO. Appl. Phys. Lett. 1974;24:456.

33. Lovinger AJ. Science 1983;220:1115. 34. Meyer RB, Liebert L, Strzelecki L, Keller P. J. Phys. Lett. 1975;36:L69. 35. Shibaev V, Plate N. Pure Appl. Chem. 1985;57:1589. 36. JW Goodby et al., Ferroelectric liquid crystals: principles, properties, and applications, Philadelphia: Gordon and Breach, 1991. 37. Barny PL, Dubois JC, McArdle CB, editors. Side chain liquid crystal polymers. Glaskow:Blackie and Son, 1989. 38. Toyoda K. Ferroelectrics 2003;282:57–216. 39. Shamos MH, Shamos MI, Lavine LS. Nature 1963;197:81. 40. Shamos MH, Lavine LS. Nature 1967;213:267–269. 41. Stanford AL, Lorey RA. Nature 1968;219:1250–1251. 42. Lang SB. 10th Inter. Symp. Electrets 1999. 43. Lemanov VV. Piezo-, Pyro-, and Ferroelectricity in biological materials. In:Galassi C, Dinescu M, Uchino K, Sayer M, editors. Piezoelectric materials: advances in science, technology and applications. Kluwer, 2000. 44. Lemanov VV, Popov SN, Pankova GA. Ferroelectrics 2003;285: 581–590. 45. Lemanov VV. Ferroelectrics 2000;238:211–218. 46. White RG, Hyde GJ, Overall RL. Protoplasma 1990;158:73–85. 47. Cyr RJ. Annu. Rev. Cell Biol. 1994;10:153–180. 48. Helluin O, Beyermann M, Leuchtag HR, Duclohier H. IEEE Trans. Diec. Electr. Insul. 2001;8(4):637–643.

CHAPTER 49

Nonlinear Optical Properties of Polymers W. M. K. P. Wijekoon, K.-S. Lee, and P. N. Prasad The Institute for Lasers, Photonics and Biophotonics, Department of Chemistry, The State University of New York at Buffalo, Buffalo, NY 14260-3000

49.1 49.2 49.3 49.4 49.5 49.6 49.7 49.8 49.9 49.10 49.11 49.12 49.13

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurements of b of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurements of w(2) of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b and w(2) Values of Polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Third-Order NLO Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Third-Harmonic Generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degenerate Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Kerr Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Focusing and Defocusing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g and w(3) Values of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TPA Cross-Section Values of Organics and Polymers . . . . . . . . . . . . . . . . . . . . . . . . Variation in the
(2) ,
(3) , and s2 value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

In general, the polarization induced in an optical medium by incident radiation can be written as [2,3]

49.1 INTRODUCTION Nonlinear optical (NLO) properties of organic polymers can be viewed as dielectric phenomena, and their origins can conveniently be explained by considering a planar wave propagation through a nonlinear dielectric medium [1–4]. In a dielectric medium the polarization P induced by the incident field E can be written as a power series of the field strength as follows: . . P ¼ w(1)
E þ w(2) ..EE þ w(3) ..EEE þ


,

795 796 797 798 804 805 805 806 806 807 808 811 815 818

.. v1 v2 (2) v PI (v) ¼ w(1) IJ (  v)
EJ þ wIJK (  v;v1 ,v2 ).EJ EK .. v1 v2 v3 þ w(3) IJKL (  v;v1 ,v2 v3 ).EJ EK EL þ


: (49:2) (3) Obviously the larger the values of w(2) IJK and wIJKL , the higher the second- and third-order polarization created in the sample, respectively. In polymeric materials, the origin of the optical nonlinearity can be traced to the molecular constituents, and therefore, one can identify polarization due to molecular units. Accordingly, the interaction of radiation with materials can be expressed in terms of the induced molecular polarization (induced dipole moment) as follows [3]:

(49:1)

where P and E are vectors and w(1) , w(2) , and w(3) , which relate P and E, are tensors. The linear polarizability tensor w(1) is a second-rank tensor. w(2) is the hyperpolarizability tensor and is a third-rank tensor and w(3) , the second hyperpolarizability tensor, is a fourth-rank tensor. The first-order term w(1) describes linear processes such as absorption, refraction, and scattering while the higher-order terms w(2) and w(3) describe the second- and third-order NLO processes, respectively.

pi (v) ¼ aij (  v)
Fvj þ bijk (  v;v1 ,v2 ): Fvj 1 Fvk 2 . þ g ijkl (  v;v1 ,v2 ,v3 )..Fvj 1 Fvk 2 Fvl 3 þ


, (49:3) where the molecular susceptibilities b and g are analogous to the bulk susceptibilities w(2) and w(3) given in Eq. (49.2) 795

796 / CHAPTER 49 and the local fields Fqv p take into account the difference between the actual field seen by the molecules and the applied field Eqv p . There are several experimental techniques that can be employed to evaluate NLO coefficients b, g, w(2) , and w(3) of polymeric materials. At present, polymers that exhibit large nonresonant optical nonlinearities are of considerable interest in scientific and industrial circles. Organic polymers offer significant tailoring flexibility in that their chemical structures can be modified to optimize the microscopic NLO response at the molecular level. Furthermore, at the bulk and microstructure level polymers can be processed as fibers, thin films in oriented or unoriented forms, glasses, and gels. NLO processes provide important functions of frequency conversion (for example, frequency doubling to increase the density of data storage), light controlled by electric field and even optical processes such as light controlled by light, the manifestation of which can be utilized to build photonic devices. A photonic device utilizes photons instead of electrons to acquire, process, transmit, and store information [3]. In order to be successfully applied in light-wave technology and optical circuitry a material should satisfy many criteria: easy processing, high transparency, high physical, mechanical, thermal, electrical, and chemical stability, compatibility with the other materials used for microelectronics, high optical power damage threshold, high optical nonlinearity, and reasonably low cost. Polymeric materials may be, perhaps, the first to combine most of these properties, avoiding stringent tradeoffs. Concepts of optical computing and optical signal and image processing have been developed utilizing NLO processes to perform the functions of frequency conversions, light modulation, optical switching, optical logic, optical memory storage, and optical limiter functions. Devices performing these functions utilize two important manifestations: frequency conversion and refractive-index modulation. In the case of the second-order NLO effects refractive-index modulation is produced by application of an external electric field. Using second-order effects polymeric materials may find applications such as in secondharmonic generation, high-density data storage, and electrooptic spatial light modulation in the near future. As far as the third-order NLO effects are concerned, the refractive index is modulated by controlling the intensity of the optical field, and it provides the mechanism for optical switching and optical bistability. The main advantage of using all-optical processing is the gain of speed and gain in connectivity. Polymeric materials posses fast NLO response and also can be fabricated in the form of fibers and channel waveguides to satisfy the necessary requirements. In addition, an important third-order NLO phenomenon is two-photon absorption (TPA) in which a molecule can simultaneously absorb two photons, when irradiated by intense laser pulses. Since the availability of femtosecond (fs) laser in the 1990s, a great deal of work has been done for developing the efficient two-photon absorbing materials. Such TPA materials can be

employed for various photonic applications including 3-D optical data storage, 3-D microfabrication, photodynamic therapy, and optical power limiting. In this chapter, we discuss the topics related to certain theoretical aspects, concepts of material design, and evaluation techniques for second- and third-order NLO polymers as well as for two-photon absorbing organic and polymeric materials. 49.2 MEASUREMENTS OF b OF POLYMERS Organic polymers, being amorphous, do not exhibit any second-order NLO effect, even though the molecules themselves are acentric. Therefore, in order to observe any second-order NLO effect, the isotropy of the medium has to be perturbed. This is usually accomplished by application of a strong dc electric filed. In the liquid phase, measurements are made by the technique known as electric-field-induced second-harmonic (EFISH) generation [5]. In this technique the solution is contained in a wedge-shape fused silica cell. The cell is sandwiched between two electrodes. A large dc voltage pulse is applied to the electrodes to disturb the average molecular orientation. At the same time the probing laser beam also is incident on the cell. The emerging second-harmonic generation (SHG) signal at 2v from the cell is recorded, as the cell is translated across the laser beam (Fig. 49.1). The NLO polarization responsible for the EFISH process can be expressed as [3] v v 0 Pi (2v) ¼ w(3) ijkl (  2v;v,v,0)Ej Ek El :

(49:4)

As the cell is moved across the incident laser beam, the optical path inside the cell varies linearly with the translated distance, so the resulting SHG signal exhibits an oscillatory behavior. The intensity of the EFISH signal can be approximated by
 8p(2v)2 sin2 [Dk(l=2)] (3) 2 2 2 I2v ¼ jw j l Iv , (49:5) n2v n22v c2 «20 [Dk(l=2)]2 Laser EFISH Cell Beam Splitter PMT Monochromator Pulsed HV Power Supply

Monochromator PMT Reference

BOXCAR

FIGURE 49.1. Schematics of experimental arrangement of electric-field-induced second-harmonic generation.

NONLINEAR OPTICAL PROPERTIES OF POLYMERS where nvi are the refractive indices at corresponding frequencies, c is the velocity of light in vacuum, l is the thickness of the cell, «0 is the permittivity of free space, and Dk is the phase mismatch. Because of the dispersion of the refractive index, the fundamental and the SHG waves propagate with two different phase velocities (vi =Ki ). Therefore, in general there exists a phase mismatch between the fundamental and the SHG waves. As shown in Eq. (49.5) when the cell is moved across the incident laser beam optical path inside the cell varies, so the emerging SHG signal generates fringes. These fringes are numerically analyzed to obtain coherence length and amplitude, which contain the necessary information to deduce b. In fact, in an EFISH experiment what one measures is an effective third-order nonlinearity w(3) EFISH , which can be given by  X
mb (3) (0) (v) (v) (2v) þ gel , N wEFISH ¼ f f f f (49:6) 5KT c where N is the number density, kT is the thermal energy, and the summation is over all components of the solution (the solute and the solvent). gel is the effective third-order hyperpolarizability for the pure four-wave mixing process (2v ¼ v þ v þ 0) and it can be determined by examining the temperature behavior of w(3) EFISH or can be approximated from results of a third-harmonic generation or degenerate four-wave mixing experiment. However, the magnitude of g el is generally an order of magnitude smaller than the mb=5kT contribution and, therefore, can be neglected. In a centrosymmetric medium (m ¼ 0) the EFISH signal is generated only by the third-order polarizability. The terms f describe the local field factor, which relates the applied field E(v) to the local field F(v).

v v Pi (2v) ¼ w(2) ijk (  2v;v,v)EJ Ek :

/

797 (49:8)

In the Maker fringe method the sample is rotated in a plane perpendicular to the plane of the probing laser beam (Fig. 49.2), giving rise to a fringe pattern. The magnitude of the second-order polarization created in the sample depends on a number of parameters as shown in Eq. (49.9), where w(2) is the NLO coefficient, l is the thickness of the sample, niv is the refractive index at the relevant frequency, Pv is the intensity of the fundamental beam, and Dk is the phase mismatch between the fundamental and the SHG beams:
 2 1 sin [Dk(l=2)] (2) 2 2 P2v / jw j l jpv j2 : (49:9) 2 nv n2v [Dk(l=2)]2 As shown in Eq. (49.9) the SHG polarization generated in the sample shows an oscillatory behavior as a function of the sample thickness. Therefore, if the sample is rotated the SHG intensity shows a fringe pattern as a function of the angle of rotation. These fringes under the condition Dk 6¼ 0 are known as Maker fringes and their period is related to the coherent length lc . The fringes are originated due to the phase mismatch (Dk) between the forced and harmonic waves. The SHG intensity oscillates with the angle as shown in Eq. (49.10), where l is the sample thickness, Im (u) is the envelope function, and lc (u) is the coherence length. Experimentally, the coherent length is determined by fitting the Maker fringes to the appropriate theoretical expression. Once the coherent length is determined by a fitting procedure, with the proper choice of input and output polarization combinations, one can evaluate the NLO coefficients with respect to a known reference (for example, quartz) by using the simplified (nv  n2v  n) relationship given in Eq (49.11) [2]:

49.3 MEASUREMENTS OF x(2) OF POLYMERS In polymeric materials the second-order bulk susceptibility w(2) can be related to the hyperpolarizability b through the relationship [3] 2v v v w(2) IJK (  2v;v,v) ¼ NbIJK fI fJ fK hO(u,f)i,

(49:7)

Laser Sample Beam Spilitter PMT Monochromator

fpqv

where N is the molecular number density, is the localfield factor, and hO(u,f)i is the orientation factor that projects bijk on to macroscopic coordinates IJK. It should be noted that in the past the majority of the SHG data have been presented in terms of a dijk coefficient, which is related to w(2) by w(2) ijk ¼ 2dijk . The number of nonzero components in the w(2) tensor depends on the point-group symmetry of the ijk material. There are several techniques, both absolute and relative, which can be used to evaluate w(2) ijk components of a material [3]. However, in the case of polymeric materials the most useful method is Maker fringe method [6–8]. The general form of the second-order polarization created in a sample by the incoming fundamental beam can be approximated as [3]

Monochromator PMT Reference PMT

BOXCAR

FIGURE 49.2. Schematics of experimental arrangement for measuring x(2) values of oriented polymers by the secondharmonic generation. The same arrangement can be used for the measurements ox x (3) via third-harmonic generation. In the case of the third-harmonic generation the measurement is usually carried out in a vacuum.

798 / CHAPTER 49

w(2) s


 2 pl Iv ¼ Im2 (u) sin2 , 2lc (u)

(49:10)


1=2
2 1=2 p n3s lc I2v, S (2)  w : 2 l2s n3r I2v, r r

(49:11)

which can be attributed to the electron donating strength of the amino group [10]. The early studies on electric field poling were performed mainly on the guest-host systems, for instance, DANS dissolved in thermotropic nematic liquid-crystalline polymers [12]. In this guest-host system polar order decays rapidly. The w(2) zzz value was about 1 pm/V. Therefore, the later investigations on guest-host systems were focused on both increasing the temporal stability as well as the w(2) zzz value. Among guest-host systems, the most studied system is Disperse Red-doped polymethylmethacrylate (PMMA) (Fig. 49.3) [13]. The early poling experiments on this system were accomplished by using parallel-plate electrodes and different doping levels. The maximum w(2) zzz value obtained with parallel-plate poling was 5 pm/V. Subsequent experiments employed corona poling, and it was possible to achieve a w(2) zzz value of 13.4 pm/V in thus poled samples [14]. Table 49.2 shows d33 ( ¼ 0:5w(2) zzz ) values of various NLO chromophores in a PMMA polymer host [13–17]. One efficient way to increase the number density of NLO chromophores in a polymeric host, without crystallization, phase separation, and any concentration gradient, is to attach them as side chains of a polymer [15,18]. Also, the temporal stability of the poled structures of side-chain NLO polymers has been proven to be much better than that of the same guest–host system due to the higher glass transition temperature of the polymer. A number of poled side-chain NLO polymers (Fig 49.4) have employed for SHG measurements. Table 49.3 exhibits d33 values of some side-chain NLO polymers [16–28]. Many of the side-chain NLO polymers, both copolymers and homopolymers, have been designed using a PMMA polymer skeleton as the backbone. This includes novel methacrylate polymers which contain a molecular-ionic NLO chromophore, N-akkylpyridinium salt, in the side chain [25,26]. The corona-poled polymer films of such polymers showed a larger w(2) value of the homopolymer

However, organic polymers, being amorphous, do not show second-order NLO effects. In order to employ them in second-order NLO measurements a polar order is induced by an external means such as electric field poling process or the Langmuir–Blodgett (LB) technique. Poled polymers and the majority of the LB films possess C1y symmetry. Therefore, the second-order optical susceptibility tensor for SHG has only three independent nonzero elements, namely, (2) (2) w(2) zzz , wzxx and wxxz . Under Klienman symmetry [9] conjecture (2) this reduces to two independent elements w(2) zzz and wzxx . These two elements can be evaluated by measuring the p-polarized SHG intensity emerging from the sample by incident p- and s-polarized fundamental beams, respectively. 49.4 b AND w(2) VALUES OF POLYMERS The b and w(2) values have been measured for a large number of chromophores in solution and in polymeric films in attempts to identify efficient second-order NLO polymers. Table 49.1 shows mb values that were derived from EFISH measurements of several organic polymers. All polymers in Table 49.1 have basically the same hyperpolarizability as their analogous monomers [10]. Also the copolymer and the homopolymers exhibit the same mb values in solution, indicating no enhancement of NLO properties due to cooperative effects. In the case of 4-methoxy4’-carbomethoxy-a-amino-a0 -cyanostilbene polymers, again the mb values are approximately the same [11]. However, in the case of NSPMAn ,NSVn , and NBSBMAn polymers, NSPMAn has a noticeably larger hyperpolarizability

TABLE 49.1. The product (mb) obtained from EFISH measurements of several second-order NLO polymers. Polymer NSPMAn NSVn NSME-BP6-5 NBSBMAn 4-Methoxy-4’-carbomethoxy-a-amino-a0 cyanostilbene: R ¼ 4  CH2 CH2 O 4-Methoxy-4’-carbomethoxy-a-amino-a0 cyanostilbene: R ¼ 4  CH2 O 4-Methoxy-4’-carbomethoxy-a-amino-a0 cyanostilbene: R ¼ 4  O 4-Methoxy-4’-carbomethoxy-a-amino-a0 cyanostilbene: copolymer

mb(1048 esu)

lm

References

93.1 315 305 561 93 265 74

1.9074 1.064 1.907 1.064 1.907 1.064 1.907

[9] [9] [9]

61

1.907

[10]

63

1.907

[10]

79

1.907

[10]

[9] [10]

NONLINEAR OPTICAL PROPERTIES OF POLYMERS CH3 CH2 C

CH3 CH2 C n C O

O

C

O (CH2)5 C O O

CH3 CH2 C n C O

n

/

799

C O CH

O

O

C O

(CH2)6

(CH2)6

O

O

O

O

(CH2)6 O

N N

NO2 NO2

O

NO2

(CH2)5 O NO2

n

NSME-BP6-5

NBSBMAn

NSVn

NSPMAn

C O R

N CN n

CH3O

4-methoxy-4’-carbomethoxy-α-amino-α’-cyanostilbene C O CH2

CH2

C O

O

CH2

N

O

CN

N CN

n

CH3O

CH3O

4-methoxy-4’-carbomethoxy-α-amino-α’-cyanostilbene: copolymer

FIGURE 49.3. Molecular structures of the NLO polymers described in Table 49.1.

TABLE 49.2. The NLO coefficient d33 ( ¼ 0:5x(2) 333 ) of various NLO chromophores doped in a PMMA polymer matrix. NLO chromophores have been oriented by the poling process. Polymer PMMA:Disperse Red (plate poling) PMMA:Disperse Red (corona poling) PMMA:4-(dicyanovinyl)-4-(dialkylamino) azobenzene PMMA:Disperse Orange 3 PMMA:4-(tricyanovinyl)-N, N’-dimethylaniline PMMA:Disperse Red PMMA:3-(dicanomethylene)-5,5dimethyl-1[[5-(dimethylamino)-2-thienyl]vinyl]cyclohexene PMMA:3-(dicyanomethylene)-5,5-dimethyl-1-[p(dimethylamino)styryl]cyclohexene PMMA:2-[[4-[-(dimethylamino)styryl]phenyl] methylene]propanediritrile

d33 (pm/V)

l(mm)

References

2.5 6.7 74 5.8 16 5.8 38

1.58 1.58 1.58 1.064 1.064 1.064 1.3

[12] [13] [14] [15]

26

1.3

[16]

27

1.3

[16]

[16]

800 / CHAPTER 49 CH3

CH3 CH2 C C O

C

CH2 C

C O

O

O

OCH3

CH3

CH3 CH2 C

CH2 C

C

O

O

OCH3

R N

DCV-MMA

N

N

R

CH2

CH2

CH2

N

DCV-Bis-Azo

Bis-Azo

DR1

N

R

R

N N N

N N

N N

CN

H

NO2

CN

N

H3C N

NO2

CH2

CN

H

CH3

CH3

C

CH

CH2 C

CH2 C

C O

O NH

N

OCH3

C O O (CH2)6 O

OH

HNPP-PHS

N

CN

CH3 CH2 CH

CH3 N

HO

NO2

NO2

MONS:MMA

PNA-polyethylene CH3 CH2 C C

O

NO2

O (CH2)n O

HPnl

N CH3

FIGURE 49.4. Molecular structures of the NLO polymers described in Table 49.3.

(w(2) zzz ¼ 15:9 pm=V) compared to that of the copolymer (2) (w(2) zzz ¼ 10:0 pm=V). The lower wzzz value of the homopolymer has been attributed to the lower chromophore concentration in the copolymer. The temporal stability as well as the poling-induced alignment of these side-chain molecularionic polymers can be largely improved by incorporation of bulky counterions. For example, the SHG intensity in a bulky counterion, tetraphenylborate, -containing polymer is about five times larger than that of the analogous iodidecontaining polymer [25]. Somewhat larger d33 values (40 pm/V at 1:064 mm) have been obtained in several poly(styrene-co-acrylic acid esters)

side-chain copolymers that were synthesized by attaching hydroxy-functionalized azobenzene, benzylidene aniline, and coumarin chromophores. The stability of the poled structures of these polymers was found to be better than that of NLO guest-loaded PMMA films [29]. Temporal stability of poled structures can be improved if NLO chromophores are attached to the polymer backbone forming a main-chain polymer (Fig. 49.5) (Table 49.4). One such example is poly(urea) [30]. This polymer can be prepared from vapor deposition polymerization. Some main-chain NLO polymers in which the dipole moment of the NLO chromophore is perpendicular to the polymer backbone

NONLINEAR OPTICAL PROPERTIES OF POLYMERS

/

801

TABLE 49.3. The NLO coefficient d33 ( ¼ 0:5x(2) 333 ) of poled side-chain NLO polymers. d33 (pm/V)

l(mm)

References

>50 43 69 150 >30 7.5 3 27 15.6 (108 esu) 36.2 (108 esu) 7.6 (108 esu) 21.8 (108 esu) 11.8 10.8 20 7.3 (108 esu) 7.0 (108 esu) 1 (108 esu) 20 (108 esu) 0.04–0.4

1.58 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064 1.064

[15] [18] [18] [18] [19] [20] [21] [22,23] [24] [24] [24] [24] [25] [25] [25] [9] [9] [9] [26] [27]

Polymer DCV-MMA DR1-tethered PMMA bis-Azoanalog of DR1-tethered PMMA Dicyanovinyl bis-Azoanalog of DR1-tethered PMMA PNA attached polyethylene HNPP-PHS Poly(N-MNA acrylamide) NPP-PPO HPOB HP6B HP6I HP10B HPBR15 HPBR21 CP6I-HEMA NSPMAn NBSBMAn NSME-BP6-5 MONS:MMA EPO-TEK 310-2 and DANS

OH CH2

OH CH2

CH

N

CH2

CH2 O

CH

O

BISA-NAT

NO2 OH CH2

OH

CH

CH2

N

CH2

CH

CH2

OH

O

BISA-NA

NO2

CH2

O

OH

CH

CH2

N

CH2

CH

O

CH2

O

OH

NO2

CH2 CH N CH2 CH CH2

O

CH2 O

OH

BISA-NPDA

H

H

H

H

N

N

N

N

O

O

Polyurea

FIGURE 49.5. Molecular structures of the NLO polymers described in Table 49.4.

802 / CHAPTER 49

FIGURE 49.5. Continued

have also been used for NLO measurements. An example of such a polymer is bisA-ANT [31]. This type of main-chain polymer is expected to be easier to pole than the head-to-tail main-chain polymers due to their enhanced flexibility. The bulky tolane unit in bisA-ANT is supposed to improve the temporal stability of the poled films. In fact, the decay of polar order of bisA-ANT was found to be much better than that of the analogous pNA polymer. The temporal stability of poled NLO materials is considerably improved by incorporating them in cross-linked systems (Fig. 49.6). Interesting results have also been obtained recently by Dalton and co-workers in cross-linked poly(urethanes). In their study the NLO chromophore was polymerized with bifunctional isocyanate to obtain the corresponding urethane polymer [32,33]. Both cross-linked and non-cross-linked polymers show no thermal transitions below 2708C. Although cross-linking can remarkably improve the temporal stability of poled structures, thermal and chemical cross-linking require processing of materials at elevated temperatures. These temperatures sometimes tend to degrade NLO chromophores. Therefore, an alternative cross-linking procedure such as photochemical cross-linking has been studied [34,35]. One such system is poly-

R O

O F F

O F O

F F

O

O

F O O

O O

O

O

O

O NC

O R

CN NC

O

O

O

R⫽

S

CN

N

Chromophore density: 33 wt % Mw: 4664 r = 60 pm/V at 1.55 µm 33

R

FIGURE 49.6. Chemical structure of a multiarm crosslinkable Nlo dendrimer.

NONLINEAR OPTICAL PROPERTIES OF POLYMERS (vinylcinnamate) doped with a NLO chromophore. In this guest–host system it was possible to achieve a 20% loading level of guest molecules without phase separation. Although the Tg of the system is found to be dependent on the chromophore concentration, plasticization of the chromophore was remarkably small. Absorption measurements indicate that the polar order in the poled film persists for a long period of time (many months) without any change [35]. Table 49.4 shows d33 values obtained from poled films of some side-chain polymers [36–43]. Also, the Langmuir–Blodgett (LB) technique has been employed to prepare oriented thin films of organic polymers for NLO investigations. An LB film containing 20 bilayers of alternating monolayer of two different polymers yield a 16 thickness of 60 nm and a w(2) esu cm per zzz of 11:2  10 bilayer at 1:047 mm [44]. Molecular nonlinearity in this LB film is supposed be arising from charge-transfer excitation in the polymer with the sulfonyl group as the acceptor and amino group as the donor. Optical waveguides capable of guiding blue light with low loss (2---6 dB cm1 ) over several centimeters have been fabricated from NLO polymers using the LB technique. However, it was necessary to use another NLO inactive polymer (t-butyl methacrylate polymer) as buffer layer in order to prevent the formation of inversion center in the film [45–48]. An LB film containing 168 bilayers (thickness  0:44 mm) showed waveguide attenuation at 457.9 nm of 5.8 dB/cm for TE polarization and 2.6 dB/cm for TM polarization. The SHG measurements on a 0.88-mm-thick LB film results in a w(2) zzz value of 9 8  109 esu. Similar w(2) esu) zzz values (d33 ¼ 5:3  10 have been obtained for Langmuir–Schaefer films of mesogenic moieties containing polysilane copolymers [49]. Current strategies for developing highly w(2) activities of NLO polymers basically involve molecular design concepts, same as presented in the previous sections. The key steps are: (i) design of high mb chromophores; (ii) improvement of temporal stability of NLO response in polymer matrices; and (iii) poling efficiency of the polymeric systems [50]. Among these, the most important factor might be to design highly efficient NLO chromophores. Therefore, to under-

/

803

stand the structure–property relationship of chromophores with high w(2) values, quantum mechanical analysis based on a simple two-level model description and bond-order alternation principle have been investigated [51,52]. Table 49.5 shows values of mb at the off-resonance wavelength of 1907 nm for some representative chromophores, with improved optical nonlinearity [53,54]. A large enhancement of optical nonlinearity can be obtained by extending the length of p-bridged chains with strong electron acceptor such as the cyano group. It was shown that very large electro-optic (E-O) coefficients could be achieved by introducing highly efficient chromophores into amorphous polymer matrix such as poly(methyl methaacrylate)(PMMA), polycarbonate, polyquinoline, etc. [55–59]. In the case of PMMA added with 17.5% of chromophore 10, the E-O coefficient was reported to be r33 ¼ 105 pmV1 at 1:33 mm [60]. This is more than three times higher than that of lithium niobate (LiNbO3 ; r 33 ¼ 31 pmV1 ), which has been used as a material for E-O modulators. To improve the temporal and the thermal stabilities of poled NLO dipoles, polyurethanes can be considered as one of the best matrix materials, because of their extensive formation of H-bonding between the urethane linkages which would increase the rigidity of the polymer chain. Several researchers succeeded in making greatly enhanced NLO polymeric systems by using polyurethane backbone [61–68]. Also, polyimides [69–80], polyetherimides [81,82], polyamides [83–89], and polyesters [90–94] as matrix polymers can also provide an enhanced thermal stability of aligned dipoles due to high glass transition temperature (Tg ) characteristics which also result in the chain stiffness. By incorporating the NLO chromophores into these polymer backbones, much improved long-term thermal stability of NLO activity were obtained. Another promising approach for making efficient NLO systems is the employment of dendritic polymers [95–111]. Compared to common polymers, dendritic NLO polymers can provide many advantages from the viewpoint of structural variation, optimization of E-O values, thermal stability, optical loss, etc. Jen’s group prepared a multiarm

TABLE 49.4. The NLO coefficient d33 ( ¼ 0:5x(2) 333 ) of poled main-chain NLO polymers. Polymer

d33 (pm/V)

l(mm)

Polyurea BisA-NAT BisA-NAT BisA-NPDA

6.6 90 30 13.5 d31 ¼ 3 40 120 4.6–5.5 60 28,000 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 13,650 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 162 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 17,250 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 2,000 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 237 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 28,000 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 340 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 3,780 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 410 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 550 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 5,970

Refractive index n

Temperature (8C)

1.511 1.6298 1.6298 1.5592 1.5682 1.5059 1.5178 1.4631 1.50 1.5390 1.586 1.6056 1.6290 1.554–1.558 1.3900 1.542 1.4969 1.5570 1.4990 1.572 1.544 1.4035 1.403 1.395 1.403 1.4035 1.399 1.4031 1.4034 1.388 1.4015 1.3935 1.4025 1.398 1.6539 1.4740 1.5300 1.60 1.537–1.55 1.53–1.57 1.523–1.54 1.50–1.57 1.53–1.58 1.658 1.4685 1.5020 1.4903 1.485 1.502 + 0.001 1.555 1.5063 1.455 1.455 1.459 1.570 + 0.003

20 20 20 20 20 20 20 30 20 23 25 25 25 20 25 20 25

25 20 25 23 23 23 23 20 25 20 20–25 25 20 20 20 20 20 20–22

Reference [4,14] [5] [4,14] [4,14] [4,14] [5] [9,10] [5] [1] [5] [13,18] [4,14] [8,17] [5] [8,17] [5] [5] [8,17] [5] [4,5] [5] [8,17] [11] [11] [11] [11] [11] [11] [11] [11] [11] [11] [11] [11] [8,17] [5] [8,17] [1] [6] [6] [5] [13,18] [13,18] [1] [5] [5] [5] [5] [4,14] [4,14] [5] [9] [9] [9] [4,14]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

829

TABLE 50.1. Continued. Polymer Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 770 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 9,430 Poly(dimethylsiloxane) trimethylsiloxy terminated M.W. ¼ 950 Poly[diphenylmethane bis(4-phenyl) carbonate] Poly(diphenylmethyl methacrylate) Poly(dodecyl methacrylate) Poly(e-caprolactam) Poly(ester carbonate) (polyphthalate) Poly(ester) cast resin, rigid (see Ftn. 31) Poly(etherimide) Poly(ethersulfone) Poly(ethyl acrylate) Poly(ethyl a-chloroacrylate) (see Ftn. 32) Poly(ethyl glycolate methacrylate) Poly(ethyl methacrylate) Poly(ethyl sulfide methacrylate) Poly(ethylene chlorohydrin methacrylate) Poly(ethylene dimethacrylate) Poly(ethylene glycol benzoate methacrylate) Poly(ethylene glycol dimethacrylate) Poly(ethylene glycol fumarate) Poly(ethylene glycol methyl ether) M.W. ¼ 350 Poly(ethylene glycol methyl ether) M.W. ¼ 550 Poly(ethylene glycol methyl ether) M.W. ¼ 750 Poly(ethylene glycol monomethacrylate) Poly(ethylene glycol phthalate) (see Ftn. 33) Poly(ethylene glycol) (see Ftn. 34) Poly(ethylene glycol) (Carbowax) Poly(cyclohexyl-a -chloroacrylate) Poly(cyclohexyl-a -ethoxyacetate) Poly(ethylene glycol) M.W. ¼ 300 Poly(ethylene glycol) M.W. ¼ 400 Poly(ethylene glycol) methyl ester Poly(ethylene glycol) tetrahydrofurfuryl ester Poly(ethylene maleate) Poly(ethylene oxide) Poly(ethylene succinate) Poly(ethylene terephthalate) (PET) Poly(ethylene terephthalate) grounded (PETG) Poly(ethylene) (see Ftn. 35) Poly(ethylene) (density 0:914 g=cm3 ) Poly(ethylene) (density 0:94---0:945 g=cm3 ) Poly(ethylene) (density 0:965 g=cm3 ) Poly(ethylene); type 1—lower density, melt index 0.3–3.6 Poly(ethylene); type 1—lower density, melt index 200 Poly(ethylene); type 1—lower density, melt index 6–26 Poly(ethylene); type 2—med. density, melt index 1.0–1.9 Poly(ethylene); type 2—med. density, melt index 20 Poly(ethylene); type 3—higher density, melt index 0.1–12.0 Poly(ethylene); type 3—higher density, melt index 0.2–0.9 Poly(ethylene); type 3—higher density, melt index 1.5–15 Poly(ethyleneglycol) 200 monononylether Poly(ethyleneglycol) 300 monononylether Poly(ethyleneglycol) 300 monononylether (fractionated, fraction 2) Poly(ethyleneglycol) 300 monononylether (fractionated, fraction 4)

Refractive index n

Temperature (8C)

1.459 1.459 1.463 1.465 1.467 1.455 1.462 1.56–1.57 1.4840 1.52–1.53 1.545 1.54 1.54 1.51 1.51 1.51 1.54 1.54 1.448 1.452 1.451 1.453 1.454 1.455 1.459 1.5300 1.547 1.51 1.40 1.3825 1.5381 1.575 + 0.003 1.4750 1.5300 1.481 1.505–1.51 1.477 1.5045 1.5050 1.521 1.5191 1.4728 1.645 1.6006 1.52 1.472–1.48 1.5672 1.517 1.52 1.4893 1.49 1.383 1.381 1.382 1.379

20 20 20 20 20 20 20 25 20 20 23 23 23 23 23 23 20 20 20 20 20 20 20 25 20

20 30 25 20 20 20 25 20 20 20 20 20

20 20 20 23 20

Reference [10] [9] [9] [9] [9] [10] [10] [4,14] [8,17] [5] [5] [1] [13,18] [13,18] [13,18] [13,18] [13,18] [13,18] [9] [9] [9] [9] [9] [9] [9] [8,17] [5] [5] [11] [11] [5] [4,14] [5] [8,17] [10] [5] [5] [10] [8,17] [5] [9] [5] [4,14] [4,14] [5] [5] [4,5] [5] [5] [5] [9] [11] [11] [11] [11]

830 / CHAPTER 50 TABLE 50.1. Continued. Polymer Poly(ethyleneglycol) 300 monononylether (fractionated, fraction 5) ˚) Poly(ethyleneglycol) 400 monononylether (at 6,563 A ˚) Poly(ethyleneglycol) 600 monononylether (at 4,861 A Poly(ethylidene dimethacrylate) (see Ftn. 36) Poly(ethylmercaptyl methacrylate) Poly(eugenol methacrylate) Poly(fluorenyl methacrylate) Poly(furfuryl methacrylate) Poly(glycerol phthalate) Poly(glycerol rosin–maleate) Poly(glycol maleate) Poly(glycol succinate) Poly(heptafluorobutyl acrylate) Poly(hexadecyl methacrylate) Poly(hexamethylene adipamide) Poly(hexamethylene glycol dimethacrylate) (see Ftn. 37) Poly(hexamethylene sebacamide) Poly(hexyl methacrylate) Poly(isobutene) Poly(isobutyl methacrylate) (see Ftn. 38) Poly(isobutylene) (see Ftn. 39) Poly(isoprene) (see Ftn. 40) Poly(isopropyl methacrylate) (see Ftn. 41) Poly(lead dimethacrylate) Poly(m-cresyl methacrylate) Poly(m-nitrobenzyl methacrylate) Poly(methacrylate methyl salicylate) Poly(methacryl phenyl salicylate) Poly(methacrylic anhydride) Poly(methacrylonitrile) (see Ftn. 42) Poly(methyl acrylate) (see Ftn. 43) Poly(methyl a-bromoacrylate) Poly(methyl a-chloroacrylate) Poly(methyl isospropenyl ketone) Poly(methyl isopropenyl ketone) Poly(methyl methacrylate) Poly(methyl methacrylate) Poly(methyl methacrylate) Poly(methyl methacrylate), beads Poly(methyl phenyl siloxane), trimethylsiloxy terminated Poly(methyl-3,3,3-trifluoropropylsiloxane), M.W. ¼ 14,000 Poly(methyl-3,3,3-trifluoropropylsiloxane), M.W. ¼ 2,350 Poly(methyl-3,3,3,-trifluoropropylisoxane), M.W. ¼ 4,600 Poly(methyl-3,3,3-trifluoropropylsiloxane), silanol terminated Poly(methyl-a-chloroacrylate) (see Ftn. 44) Poly(methyl-a-methylene butyrolactone) Poly(methylene-a-valerolactone) Poly(methylhexadecylsiloxane) Poly(methylhydrosilane) Poly(methylhydrosiloxane) Poly(methylhydrosiloxane) trimethylsilyl terminated M.W. ¼ 1,500 Poly(methylhydrosiloxane) trimethylsillyl terminated M.W. ¼ 2,270 Poly(methylhydrosiloxane) trimethylsilyl terminated M.W. ¼ 360–420 Poly(methylhydrosiloxane) trimethylsilyl terminated M.W. ¼ 4,500–5,000 Poly(methyloctadecylsiloxane)

Refractive index n

Temperature (8C)

1.5172 1.5155 1.5248 1.5118 1.451 1.3979 1.398 1.396 1.382 1.397 1.443 1.445 1.463 1.455 1.43 1.5857 1.5476 1.5965 1.5246 1.4830 1.4813 1.4840 1.360 1.5823 1.6040 1.6098 1.5707 1.507 1.5705 1.5932 1.5750 1.6400 1.4630 1.4800 1.4900 1.5500 1.5600 1.4430 1.4450 1.4510 1.4550 1.3830 1.3970 1.465 1.5671 1.5745 1.5602 1.5827 1.5792 1.5900 1.6539 1.5937 1.5702 1.5850 1.6056

20

20 20 20

23 20 20 20 20 25 20 25 25 20 20 20 20 20 20 25 25 25 25 25 25 25 25 25 25 25 25 25 50

Reference [4,14] [4,14] [4,14] [4, 14] [11] [9] [10] [11] [11] [11] [11] [11] [1] [11] [6] [5] [4,5] [4,5] [4,14] [8,17] [4,5] [4,8] [5] [4,5] [5] [4,5] [5] [4,14] [4,5] [4,5] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [8,17] [5] [5] [5] [5] [5] [5] [5] [5] [5] [5] [5] [5]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

831

TABLE 50.1. Continued. Polymer

Refractive index n

Poly(methyloctylsiloxane) 1.43 Poly(methylpentene) 1.4563 Poly(methyltetradecylsiloxane) 1.51–1.54 Poly(monochlorotrifluoroethylene) 1.555 Poly(monofluorethyl methacrylate) 1.4840 Poly[N(-2-methoxythyl)methacrylamide] 1.4744 Poly[N-(2-phenylethyl)methacrylate] 1.5750 Poly(N-allyl methacrylamide) 1.48 Poly(N-benzyl methacrylamide) 1.4690 Poly(N-b-methoxyethyl methacrylamide) 1.5840 Poly(N-b-phenylethyl methacrylamide) 1.4495 Poly(n-butyl methacrylate) (see Ftn. 45) 1.5559 Poly(N-butyl-methacrylamide) 1.5575 Poly(n-hexyl methacrylate) 1.6150 Poly(N-methyl-methacrylamide) 1.554 Poly(n-propyl methacrylate) 1.552 Poly(N-vinylphthalimide) 1.5967 Poly(nonafluoropentyl acrylate) 1.6690 Poly(o-chlorobenzhydryl methacrylate) 1.71 Poly(o-chlorobenzyl methacrylate) 1.608 Poly(o-chlorodiphenylmethyl methacrylate) 1.339 Poly(o-chlorostyrene) 1.584 + 0.003 Poly(o-cresyl methacrylate) (see Ftn. 46) 1.385 Poly(o-methoxyphenyl methacrylate) 1.612 Poly(o-methoxystyrene) 1.5624 Poly(o-methoxystyrene) 1.5706 Poly(octafluoropentyl acrylate) 1.6300 Poly(octylmethylsilane) 1.447 Poly[oxy(2,6-dimethyl-1,4-phenylene)] 1.4570 Poly[oxy(2,6-diphenyl-1,4-phenylene)] 1.5960 Poly[oxy(acryloxypropylmethylsilylene)] 1.49 Poly[oxy(dicyanopropylsilylene)] 1.49 Poly[oxy(mercaptopropylmethylsilylene)] 1.4735 Poly[oxy(methyl m-chlorophenylethylsilylene)] 1.5030 Poly[oxy(methyl m-chlorophenylsilylene)] 1.541 Poly[oxy(methyl n-hexadecylsilylene)] 1.542 Poly[oxy(methyl n-hexylsilylene)] 1.500 Poly[oxy(methyl n-octadecylsilylene)] 1.6568 Poly[oxy(methyl n-octylsilylene)] 1.59–1.592 Poly[oxy(methyl n-tetradecyl silylene)] 1.59–1.60 Poly[oxy(methyl-t-trifluoropropylsilylene)] 1.5901 Poly[oxy(methylhydrosilylene)] 1.5858 Poly[oxy-1-oxopentamethylene] 1.5916 Poly(oxy-2,6-dimethylphenylene) 1.59–1.60 Poly[oxycarbonyloxy-1,4(2,6-dichloro)phenylne-isopropylidene-1,4(2,6-dichloro)phenylene] 1.57–1.60 Poly(oxycarbonyloxy-1,4-phenylene-1,3-dimethyl-butylidene-1,4-phenylene) 1.59–1.60 Poly(oxycarbonyloxy-1,4-phenylene-1-methyl-butylidene-1,4-phenylene) 1.6 Poly(oxycarbonyloxy-1,4-phenylene-1-propyl-butylidene-1,4-phenylene) 1.6–1.7 Poly(oxycarbonyloxy-1,4-phenylene-sec-butylidene-1,4-phenylene) 1.633 Poly(oxycarbonyloxy-1,4-phenylenebutylidene-1,4-phenylene) 1.4638 Poly(oxycarbonyloxy-1,4-phenylenecyclohexylidene-1,4-phenylene) 1.4638 Poly(oxycarbonyloxy-1,4,phenylenediphenyl-methylene-1,4-phenylene) 1.4746 Poly(oxycarbonyloxy-1,4-phenyleneethylidene-1,4-phenylene) 1.346 Poly(oxycarbonyloxy-1,4-phenyleneisobutylidene-1,4-phenylene) 1.348 Poly(oxycarbonyloxy-1,4-phenyleneisopropylidene-1,4-phenylene) 1.360

Temperature (8C) Reference 30 20 25 25 20 20 25 20 20 20 20 20 20 20 25 20 20 25 20 25 25 20 20 25 20 25 25 20 20 20 20 25 25 25 20 25 20

23 23

25 20 30 25 25 25

[5] [5] [5] [5] [5] [5] [5] [5] [8,17] [5] [5] [4,5] [4,5] [4,5] [4,5] [4,5] [4,5] [8,17] [5] [4,5] [5] [4, 14] [5] [4,5] [4,5] [4,5] [8,17] [10] [8,17] [8,17] [10] [1] [5] [5] [10] [5] [4,5] [8,17] [5] [4,14] [4,14] [4,14] [10] [1] [1] [6] [13,18] [4,5] [1,5] [8,17] [10] [5] [5] [5] [5]

832 / CHAPTER 50 TABLE 50.1. Continued. Polymer

Refractive index n

Temperature (8C)

Poly(oxycarbonyloxybis[1,4(3,5-dichlorophenylene)]) Poly(oxydimethylsilylene) Poly(oxyethylene) (see Ftn. 47) Poly[(oxyethylene)-a-benzoate-v-methacrylate] Poly(oxyethyleneoxymaleoyl) [poly(ethylene maleate)] Poly(oxyethyleneoxysuccinoyl) [poly(ethylene succinate)] Poly(oxyethyleneoxyterephthaloyl) [poly(ethylene terephthalate)] Poly(oxymethylene) Poly(oxyoctamethylene) Poly(oxypentaerythritoloxyphthaloyl) Poly(oxypropylene) Poly(p,p’-xylyenyl dimethacrylate) Poly(p-bromophenyl methacrylate) Poly(p-cyclohexylphenyl methacrylate) Poly(p-divinylbenzene) Poly(p-isopropylstyrene) Poly(p-methoxybenzyl methacrylate) Poly(p-methoxystyrene) Poly(p-xylylene) Poly(pentabromophenyl methacrylate) Poly(pentachlorophenyl methacrylate) Poly(pentadecafluorooctyl acrylate) Poly(pentaerythritol phthalate) Poly(pentaerythritol tetramethacrylate) Poly(pentafluoropropylacrylate) Poly(pentafluorovinyl propionate) Poly(phenyl a-bromoacrylate) Poly(phenyl cellosolve methacrylate) Poly(phenyl methacrylate) Poly(phenylmethylsilane) Poly(propylene glycol) Poly(propylene oxide) Poly(propylene sulfide) Poly(propylene) (see Ftn. 48) Poly(propylene) (density 0:9075 g=cm3 ) Poly(propylene), atactic (density 0:8575 g=cm3 ) Poly(propylene), chlorinated Poly(sec-butyl a-bromoacrylate) Poly(sec-butyl a-chloroacrylate) Poly(styrene sulfide) Poly(styrene) (see Ftn. 49) Poly(sulfides) (Thiokol) Poly(sulfone) Poly(t-butyl methacrylate) (see Ftn. 50) Poly(terpineyl methacrylate) Poly(tetradecyl methacrylate) Poly[tetrafluoro-3-(heptafluoropropoxy)propyl acrylate] Poly[tetrafluoro-3(pentafluoroethoxy)propyl acrylate] Poly[tetrafluoro-3(trifluoromethoxy)propyl acrylate] Poly(tetrafluoroethylene) (PTFE) (see Ftn. 51) Poly(tetrahydrofuran) M.W. ¼ 250 Poly(tetrahydrofuran) M.W. ¼ 650 Poly(tetrahydrofuran), with oxirane Poly(tetrahydrofurfuryl methacrylate) Poly(triethoxyl silicol methacrylate)

1.35 1.35 1.35 1.35 1.46 1.465 1.466 1.5096 1.4889 1.43 1.407 1.437 1.4177 1.375 1.356 1.5–1.6 1.4628 1.48–1.50 1.4665 1.45–1.47 1.49–1.53 1.49–1.53 1.5775 1.4563 1.485 1.47–1.49 1.48–1.49 1.52–1.53 1.54–1.55 1.52–1.55 1.52 1.55 1.54–1.55 1.512 1.502 1.454 1.50 1.4750 1.4591 1.4507 1.5129 1.466 1.474 1.683 1.55 1.60–1.63 1.60–1.63 1.42 1.42 1.6818 1.52 1.5196 1.54 1.5059 1.4631

20 23

20 20 20 20 20 23 25 20 25

30 20 23 23 20 20 20 23 23 23

23 25 20 20 23 20 30 30 20 20 30 20 20 23 23 25 23 20 20 20 30

Reference [9] [13,18] [12] [1] [10] [10] [10] [5] [5] [3,13] [5] [5] [5] [5] [5] [5] [5] [5] [5] [6] [5] [6] [4,5] [5] [9,10] [6] [6] [6] [5] [12] [5] [5] [6] [5] [5,10] [10] [6] [5] [5] [5] [5] [10] [5] [4,5] [5] [3,13] [6] [5] [13,18] [5] [4,14] [4,14] [4,14] [4,14] [4,14]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

833

TABLE 50.1. Continued. Polymer Poly(triethylcarbinyl methacrylate) (see Ftn. 52) Poly(trifluorochloroethylene) (PTFCE) Poly(trifluoroethyl acrylate) Poly(trifluoroethyl methacrylate) Poly(trifluoroisopropyl methacrylate) (see Ftn. 53) Poly(trifluorovinyl acetate) Poly(undecafluorohexyl acrylate) Poly(urethane), rigid (see Ftn. 54) Poly(vinyl acetal) (see Ftn. 55) ˚ ) (see Ftn. 56) Poly(vinyl acetate) (at 6,563 A Poly(vinyl alcohol) (see Ftn. 57) Poly(vinyl benzoate) Poly(vinyl butyl ether) Poly(vinyl butyral) (see Ftn. 58) Poly(vinyl chloride acetate) molding compound—rigid Poly(vinyl chloride) (PVC) rigid (see Ftn. 59) Poly(vinyl chloride–acetate) Poly(vinyl chloroacetate) (see Ftn. 60) Poly(vinyl cychlohexene dioxide) Poly(vinyl decyl ether) Poly(vinyl dodecyl ether) Poly(vinyl ethyl ether) (see Ftn. 61) Poly(vinyl formal) molding compound (see Ftn. 62) Poly(vinyl formate) (see Ftn. 63) Poly(vinyl hexyl ether) Poly(vinyl isobutyl ether) (see Ftn. 64) Poly(vinyl methacrylate) (see Ftn. 65) Poly(vinyl methyl ether) (see Ftn. 66) Poly(vinyl methyl ketone) Poly(vinyl naphthalene) Poly(vinyl octyl ether) Poly(vinyl pentyl ether) Poly(vinyl phenyl sulfide) Poly(vinyl propionate) (see Ftn. 67) Poly(vinyl sec-butyl ether) (isotactic) Poly(vinyl triophene) Poly(vinyl-2-ethylhexyl ether) Poly(vinylcarbazole) Poly(vinylfuran) (see Ftn. 68) Poly(vinylidene chloride) molding compound (see Ftn. 69) Poly(vinylidene-fluoride) (PVDF) (see Ftn. 70) Poly(vinylnaphthalene) Poly(vinylpyrrolidone) Poly(vinylsulfonic acid) (sodium salt) Propylmethyl homopolymer, vinyldimethyl terminated Pyralin Polyimide Film-Pl-2540 Pyralin Polyimide Film-Pl-2550 Rosin (grade M, wood) Rosin ester Rubber (hard, 32% S) Rubber hydrochloride (Pilofilm) Shellac, bleached, dry Shellac, orange Silicone, Dow-Corning 2102 Sulfonamide resins

Refractive index n

Temperature (8C)

1.4634 1.554 1.39–1.43 1.394 1.5067 1.5066 1.5099 1.65 1.4855 1.5013 1.483–1.485 1.51–1.52 1.517 1.5063 1.5119 1.52–1.57 1.477 1.4728 1.5228 1.499 1.52 1.4725 1.4893 1.4768 1.5117 1.514 1.376 1.5096 1.4177 1.48–1.50 1.47–1.49 1.4667 1.545–1.555 1.51–1.55 1.47–1.49 1.54–1.56 1.565 1.525–1.529 1.5303 1.4757 1.5129 1.467 1.55 1.53 1.43 1.78 1.70 1.525 + 0.003 1.496 + 0.003 1.60 1.540–1.550 1.534 + 0.003 1.516 + 0.003 1.419 1.56–1.596

25 20 20 20 20 23 20 20 25 20 20 20 25 20 20 20 20 23 20 20 25 20 20

24 25

40 21 20 20 20 20 20

20 21–23 25 25 20–22 20–22

Reference [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [1] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [1] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [11] [7,16] [7,16] [4,14] [4,14] [4,5] [4,14] [4,14] [4,14] [4,14] [4,14]

834 / CHAPTER 50 TABLE 50.1. Continued. Polymer Terpene resin (Piccolyte S-10) Terpene resin (Piccolyte S-40) Toluenesulfonamide formaldehyde Urea formaldehyde Urea–thiourea formaldehyde Footnote 1 CP-41/l/ gen. purpose, lowest heat resistance CP-51/lE/ gen. purpose, medium heat resistance CP-61/lE/ gen. purpose, medium heat resistance CP-71/l/ gen. purpose, maximum heat resistance CP-75/l/ gen. purpose, maximum heat resistance, medium flow CP-80/E/ gen. purpose, unlubed CP-81/E/ gen. purpose, high heat resistance CP-82/l/ gen. purpose, maximum heat resistance CP-924/l/ good impact str. exc. light trans. CP-927/I/ better impact str. exc. surface gloss Footnote 2 H-11/IE/ highest heat resistance water white H-12/IE/ improved flow properties water white H-15-011/IE/ UV absorbing automotive applications H-15-012/I/ UV transmitting automotive applications H-15/IE/ increased toughness maximum service temp. white L-40/IE/ improved flow easy molding water white M-30/IE/ improved flow easy molding water white Footnote 3 6/I/ thin walled and complicated parts 6E/EI/ good flow, tough technical parts 7/I/ good flow dimen. stability 7E/EI/ high melt viscosity 8/I/ good surface hardness 8E/E/ high melt viscosity, good stab. Footnote 4 DR G/I/ high-impact g radiation DR M/I/ high-impact, low reflectance DR/IE/ high-impact acrylic molding G-UVT// gen. purpose optical quality G// gen. purpose cellcast unshrunk nat. HFI-10/I/ high-impact high-flow pellets HFI-7/I/ medium-impact high-flow pellets II-UVA// gen. purpose preshrunk II-UVT// gen. purpose increases uv stab K// superior thermoformability MC// plastic sheet economical material MI-7/I/ medium-impact high flow MI-7G/IE/ medium-impact light transmission UF-3// ultraviolet radiation V-044/E/ high heat resistance V-045/IE/ high heat resistance V-052/I/ high heat resistance V-811/IE/ maximum heat resistance V-825/IE/ maximum heat resistance V-920/IE/ high heat resistance VM/IE/ medium heat resistance VS/I/ maximum flow lowest heat resistance

Refractive index n 1.506 1.515 1.596 1.54–1.56 1.66

Temperature (8C) 25 25 25

Reference [4,14] [4,14] [4,14] [1,14] [4,14]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

835

TABLE 50.1. Continued. Polymer Footnote 5 300// cross-linked commercial continuous cast Acrysteel I-GP// continuous cast impact clear GPA// gen purpose cast acrylic sheet clear Footnote 6 Cellulose acetate (partly saponified) Cellulose acetate-molding Cellulose acetate-sheet Footnote 7 ASTM Grade: H2-1 ASTM Grade: H4-1 ASTM Grade: H6-1 ASTM Grade: MH-1, MH-2 ASTM Grade: MS-1, MS-2 ASTM Grade: S2-1 Footnote 8 ASTM Grade: H4 ASTM Grade: MH ASTM Grade: S2 Cellulose acetate butyrate Cellulose acetate butyrate Footnote 9 Cellulose acetate propionate; ASTM Grade: 1 Footnote 10 Cellulose nitrate Footnote 11 Congo copal Congo ester Footnote 12 Damar No. 1 (Singapore) Footnote 13 Epoxy resins Epoxy, diglycidyl ether of bisphenol A, cast flexible or molded Footnote 14 Gutta percha, alpha Gutta percha, beta Footnote 15 Nylon 6 (Poly[imino(1-oxohenamethylene)]) Nylon 6,10 [Poly(iminoadipoyliminotetramethylene)] Nylon 6,6 [Poly(iminoadipoylaminohexamethylene)] Nylon molding compound Nylons (polyamide) type 11 Nylons (polyamide) type 6,6 Footnote 16 Phenol–formaldehyde Footnote 17 Poly(4-methyl-1-pentene) Footnote 18 Poly(abietic acid) Footnote 19 Poly(acrylic acid) Footnote 20 Poly(acrylonitrile) Poly(acrylonitrile) Footnote 21 Poly(allyl methacrylate)

Refractive index n

Temperature (8C)

Reference

1.54 1.46–1.50 1.49–1.50

23 23

[4,14] [6] [6]

1.46–1.49 1.46–1.49 1.46–1.49 1.46–1.50 1.494

21

[13,18] [13,18] [13,18] [4,14] [4,14]

1.46–1.49

23

[13,18]

1.501

21

[4,14]

1.540–1.541 1.506 + 0.003

25 20

[4,14] [4,14]

1.515 + 0.003

20

[4,14]

1.57–1.61 1.61

23

[4,14] [13,18]

1.514 1.509

50 50

[4,14] [4,5]

1.53 1.53 1.53 1.53–1.55 1.52 1.53

[5] [5] [5] [4,14] [1] [1]

1.5–1.7

[4,14]

1.4975

20

1.529

[4,5] [5]

1.5196

20

[5]

1.4917 1.63

20

[4,14] [4,5]

1.5624

20

[4,14]

836 / CHAPTER 50 TABLE 50.1. Continued. Polymer Footnote 22 Poly(a-naphthyl methacrylate) Poly(a-naphthyl methacrylate) Footnote 23 Poly(b-naphthyl methacrylate) Footnote 24 Poly(bornyl methacrylate) Footnote 25 Poly(butadiene) Footnote 26 Poly(butyl acrylate) Poly(butyl acrylate) Footnote 27 Poly(carbonate) Poly(carbonate) of bisphenol-A Poly(carbonate) of bisphenol-A Footnote 28 Poly(chlorotrifluoroethylene) Poly(chlorotrifluoroethylene) Footnote 29 Poly(cyclohexyl methacrylate) Footnote 30 Poly(diallyl phthalate) Footnote 31 Poly(ester) cast resins—flexible Poly(ester) cast resins—rigid Poly(ester) cast, flexible Poly(ester) cast, rigid Poly(ester) resin, rigid (ca. 50% styrene) Footnote 32 Poly(ethyl a-chloroacrylate) Footnote 33 Poly(ethylene glycol phthalate) Footnote 34 Poly(ethylene glycol) Footnote 35 Poly(ethylene) Poly(ethylene) lonomer Poly(ethylene) molding cmpd. Footnote 36 Poly(ethylidene dimethacrylate) Footnote 37 Poly(hexamethylene glycol dimethacrylate) Footnote 38 Poly(isobutyl methacrylate) Footnote 39 Poly(isobutylene) Footnote 40 Poly(isoprene) Poly(isoprene) cis Poly(isoprene) Footnote 41 Poly(isopropyl methacrylate)

Refractive index n

Temperature (8C)

Reference

1.5487 1.5933

20 20

[4,14] [4,14]

1.466

20

[5]

1.467

20

[10]

1.585

[4,14]

1.5951 1.5067

20 20

[4,14] [5]

1.532 + 0.001 1.4969 1.5857

25 20 25

[4,14] [4,14] [8,17]

1.571–1.572 1.62–1.64

20

[4,14] [4,14]

1.375

[11]

1.5933

20

[5]

1.4539 1.4744 1.64 1.51 1.567

20 25

[10] [8,17] [1] [5] [1]

20

1.51

23

[6]

1.4831

20

[5]

1.5714

20

[4,14]

1.5191 1.533 1.5705

20 20

[10] [11] [4,14]

1.5431

20

[4,14]

1.5246

20

[5]

1.5857

20

[4,14]

1.4831

25

[4,14]

1.5398 1.6200 1.5219

20 20 20

[4,5] [4,5] [4,14]

1.604

20

[4,14]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

837

TABLE 50.1. Continued. Polymer Footnote 42 Poly(methacrylonitrile) Footnote 43 Poly(methyl acrylate) Poly(methyl acrylate) Footnote 44 Poly(methyl-a-chloroacrylate) Poly(methyl-a-chloroacrylate Footnote 45 Poly(n-butyl methacrylate) Footnote 46 Poly(o-cresyl methacrylate) Footnote 47 Poly(oxyethylene) (high molecular weight) Footnote 48 Poly(propylene) Poly(propylene) Footnote 49 Poly(styrene) Poly(styrene) Poly(styrene) Poly(styrene) Poly(styrene) (general purpose) Poly(styrene) (general purpose) Poly(styrene) (heat and chemical) Poly(styrene) modified molding compound—heat and chemical resistance type Poly(styrene) molding compound—unfilled Footnote 50 Poly(t-butyl methacrylate Poly(t-butyl methacrylate) Footnote 51 Poly(tetrafluoroethylene) (PTFE) Poly(tetrafluoroethylene) (PTFE) Poly(tetrafluoroethylene) (PTFE) Poly(tetrafluoroethylene) (PTFE) Poly(tetrafluoroethylene) (PTFE) Poly(tetrafluoroethylene) (PTFE) molding compound Footnote 52 Poly(triethylcarbinyl methacrylate) Footnote 53 Poly(trifluoroisopropyl methacrylate) Footnote 54 Poly(urethanes) Footnote 55 Poly(vinyl acetal) Footnote 56 ˚) Poly(vinyl acetate) (at 4,861 A Poly(vinyl acetate) Poly(vinyl acetate) (medium acetate type) Poly(vinyl acetate) (low acetate type) Poly(vinyl acetate) molding compound

Refractive index n

Temperature (8C)

1.5874

20

[5]

1.380 1.4780

25 25

[5] [8,17]

1.575 1.6056

Reference

[5] [5]

1.5964

20

[4,5]

1.364

25

[5]

1.35

23

[6]

1.4640 1.4540

30 30

[5] [5]

1.467 1.4700 1.5000 1.6818 1.4613 1.4665 1.4581 1.57–1.60

20 30 20 20 30 20 30 23

[5,10] [5] [8,10] [4,14] [5] [5] [5] [6]

1.6568

20

[4,5]

1.6376 1.4626

30

[4,14] [5]

1.389 1.356 1.4638 1.546 1.527 1.529

20

1.539

20

[4,14]

1.547 + 0.001

25

[4,14]

1.525

20

[4,14]

1.5001

20

[4,14]

1.5100 1.5174 1.4903 1.523–1.57 1.4685

20 20 20

[4,14] [4,14] [4,14] [4,14] [4,14]

20 25 25

20

[10] [4,14] [4,14] [4,14] [4,14] [4,14]

838 / CHAPTER 50 TABLE 50.1. Continued. Polymer Footnote 57 Poly(vinyl alcohol) molding compound) Poly(vinyl alcohol) molding compound) Footnote 58 Poly(vinyl butyral) Poly(vinyl butyral) molding compound—flexible, unfilled Poly(vinyl butyral) molding compound—rigid Footnote 59 Poly(vinyl chloride) (PVC) Poly(vinyl chloride) (PVC) Poly(vinyl chloride) (PVC) Poly(vinyl chloride) (PVC) rigid Poly(vinyl chloride) (PVC) þ 40% dioctyl phthalate Poly(vinyl chloride) (PVC) þ 40% tricrexyl phosphate Poly(vinyl chloride) (PVC) molding compound—rigid Poly(vinyl chloride) (PVC) rigid Footnote 60 Poly(vinyl chloroacetate) Footnote 61 Poly(vinyl ethyl ether) Poly(vinyl ethyl ether), low molecular weight Footnote 62 Poly(vinyl formal) Footnote 63 Poly(vinyl formate) Footnote 64 Poly(vinyl isobutyl ether) Footnote 65 Poly(vinyl methacrylate) Footnote 66 Poly(vinyl methyl ether) Poly(vinyl methyl ether) (isotactic) Footnote 67 Poly(vinyl propionate) Footnote 68 Poly(vinylfuran) Footnote 69 Poly(vinylidene chloride) Footnote 70 Poly(vinylidene fluoride) (PVDF)

Refractive index n

Temperature (8C)

Reference

1.547 1.512–1.519

20 25

[4,14] [4,14]

1.4563 1.51–1.54 1.514

30 25 25

[4,14] [4,14] [4,14]

1.4831 1.5381 1.541 1.484 1.4744 1.5048 1.5066 1.52–1.55

20 20 25 25 25 23 20

[4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [4,14] [1]

1.505–1.51 1.48 1.52

[4,14] 25 20

1.485–1.49 1.49

[4,14] [4,14] [4,14]

20

[4,14]

1.49

[4,14]

1.35–1.38

[4,14]

1.436 1.4889

20 20

1.49–1.53 1.512

[4,14] [4,14] [4,14]

25

[4,14]

1.454

[4,14]

1.452

[4,14]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

839

TABLE 50.2. Refractive indices of copolymers. All values reported are at a wavelength of 5,893 A˚. Copolymer Acrylics multipolymer Acrylonitrile butadiene styrene copolymer (ABS), transparent Arylef U100 (equimolar random copolymer of bisphenol-A terephthalate and bisphenol-A isophthalate) Dimethyl (48–52%)–phenylmethyl siloxane (48–52%) copolymer, trimethylsiloxy terminated Dimethyl (70%)–dodecyl (15%)–tetradecyl siloxane (15%), terpolymer Dimethyl (70%)–methyloctadecyl siloxane (30%), copolymer Dimethyl (79–82%)–diphenylsiloxane (18–21%) copolymer, silanol terminated Dimethyl (79–82%)–diphenylsiloxane (18–21%) copolymer, trimethylsiloxy terminated Dimethyl (82–88%)–diphenylsiloxane (12–18%) copolymer, silanol terminated Dimethyl (88–92%)–phenylmethyl siloxane (18–12%) copolymer, trimethylsiloxy terminated Dimethyl (94–96%)–diphenylsiloxane (4–6%) copolymer, trimethylsiloxy terminated Dimethyl (97–98%)–diphenylsiloxane (2–3%) copolymer, silanol terminated Dimethyl–tetrachlorophenyl siloxane, copolymer, branched, trimethylsiloxy terminated Diphenyl (76.5%)–dimethyl (23.5%) copolymer, vinyl terminated, M.W. ¼ 13, 200 Diphenyl (84%)–dimethyl (16%) copolymer, vinyl terminated, M.W. ¼ 9,300;18,600; 18,900;19,500;35,300;54,900 Diphenyl (95%)–dimethyl (5%) copolymer, vinyl terminated, M.W. ¼ 17,300; 49,000;80,400 Diphenyl (97%)–dimethyl (3%) copolymer, vinyl terminated, M.W. ¼ 15,600;62,300 Diphenyl (97%)–dimethyl (3%) copolymer, vinyl terminated, M.W. ¼ 78,000 Methyl hydro (3–4%)–cyanopropylmethylsiloxane copolymer Methyl hydro (0.5–1.0%)–dimethylsiloxane copolymer, trimethylsilyl terminated, M.W. ¼ 10,000,13,000 Methyl hydro (30–35%)–dimethylsiloxane copolymer, trimethylsilyl terminated, M.W. ¼ 2,000–2,100 Methyl hydro (15–18%)–dimethylsiloxane copolymer, trimethylsilyl terminated, M.W. ¼ 25–35 Methyl hydro (50–55%)–dimethylsiloxane copolymer, trimethylsilyl terminated, M.W. ¼ 900–1,000 Methyl hydro (25–30%)–methyloctyl siloxane copolymer Methyl hydro (40–60%)–methyloctyl siloxane copolymer Methyl hydro (45–50%)–phenylmethylsiloxane copolymer, dimethylsiloxy terminated Methyloctyl (35–40%)–vinylmethyl (3–4%)–dimethylsiloxane (56–64%), terpolymer Methylphenyl (45–55%)–diphenylsiloxane (45–55%) copolymer, trimethylsiloxy terminated Poly(butadiene-co-acrylonitrile) Poly(butadiene-co-styrene) (30% styrene) block copolymer Poly(butadiene-co-styrene) (75%–25%) Poly(ethylene-co-propylene) (EPR-rubber) Poly(ethylene–co-vinyl acetate) Poly(methyl-3,3,3-trifluoropropylsiloxane–50% dimethyl siloxane) copolymer Poly(oxyethyleneoxymaleoyl) [poly(ethylene maleate)] Poly(oxyethyleneoxysuccinoyl) [poly(ethylene succinate)] Poly(oxyethyleneoxyterephthaloyl) [poly(ethylene terephthalate)] Poly(styrene-co-acrylonitrile) (75%–25%) Poly(styrene-co-maleic anhydride) Poly(tetrafluoroethylene-co-hexafluoropropylene)

Refractive index n 1.52 1.536 1.61

Temperature (8C)

25

Reference [12] [1] [8]

1.5

[11]

1.43 1.44 1.485

[11] [11] [11]

1.488

[11]

1.473

[11]

1.425

[11]

1.422

[11]

1.42 1.428

[11] [11]

1.493 1.465

22–25 22–25

[11] [11]

1.432

22–25

[11]

1.420 1.421 1.446 1.404

22–25 22–25

[11] [11] [11] [11]

1.399

[11]

1.4

[11]

1.394

[11]

1.44 1.435 1.5 1.437 1.582

[11] [11] [11] [11] [11]

1.52 1.53 1.535 1.4748–1.48 1.47–1.5 1.387 1.4840 1.4744 1.5750 1.57 1.564 1.338

[5] [5] [5] [5] [5] [11] [5] [5] [5] [5] [5] [5]

25 25 20 21

840 / CHAPTER 50 TABLE 50.2. Continued. Refractive index n

Copolymer Poly(vinyl chloride þ 40% dioctyl phthalate Poly(vinyl chloride) þ 40% tricrexyl phosphate Poly(vinyl chloride-co-vinyl acetate) (95/5-90/10) Styrene acrylonitrile (SAN) Styrene acrylonitrile copolymer (unfilled) Styrene butadiene Styrene butadiene thermoplastic elastomers Styrene maleic anhydride (SMA) Styrene methylmethacrylate (SMMA)

Temperature (8C)

1.52 1.55 1.525–1.535 1.565–1.569 1.56–1.57 1.571 1.52–1.55 1.6 1.56

Reference [5] [5] [5] [3] [1] [1] [1] [1] [1]

23

TABLE 50.3. Optical configuration parameters (Da) and stress-optical coefficients (C) of polymers. All values reported are at a wavelength of 6,328 A˚, D.S. denotes degree of substitution, and v 2 is the polymer volume fraction.

Polymer Benzyl cellulose, D.S. ¼ 2.5 Cellulose acetate D.S. ¼ 2.4 Cellulose acetate D.S. ¼ 3.0 Cellulose acetate D.S. ¼ 3.0 Cellulose acetate D.S. ¼ 3.0 Cellulose acetate D.S. ¼ 3.0 Cellulose benzoate D.S. ¼ 3.0 Cellulose benzoate D.S. ¼ 3.0 Cellulose benzoate D.S. ¼ 3.0 Cellulose benzoate D.S. ¼ 3.0 Cellulose benzoate D.S. ¼ 3.0 Cellulosedimethylphosphonocarbamate D.S. ¼ 2.0 Cellulose dimethylphosphonocarbamate D.S. ¼ 2.0 Cellulose diphenylacetate Cellulose diphenylacetate Cellulose diphenylphosphonocarbamate D.S. ¼ 2.0 Cellulose monophenylacetate D.S. ¼ 2.8 Cellulose monophenylacetate D.S. ¼ 2.8 Cellulose nitrate D.S. ¼ 1.9 Cellulose nitrate D.S. ¼ 1.9 Cellulose nitrate D.S. ¼ 2.3 Cellulose nitrate D.S. ¼ 2.7 Cellulose nitrate D.S. ¼ 2.7 Cellulose nitrate D.S. ¼ 2.7 Cellulose nitrate D.S. ¼ 2.7 Cellulose nitrate D.S. ¼ 2.7 Cellulose nitrate D.S. ¼ 2.8 Cellulose phenylcarbamate Cellulose phenylcarbamate D.S. ¼ 2.2 Cellulose phenylcarbamate D.S. ¼ 3.0 Cellulose phenylcarbamate D.S. ¼ 3.0 Cellulose phenylcarbamate D.S. ¼ 3.0 Cellulose phenylcarbamate D.S. ¼ 3.0 Cellulose phenylcarbamate D.S. ¼ 3.0

Da (A˚3 )

Diluent

Temp (8C)

v2

Stress optical coefficient C (109 Pa
1 )

Reference

294 0
34 35 61 144
617
763
914
830
447 710

Dioxane Pyridene Tetrachloroethane Bromoform Dioxane Methyl ethyl ketone Dimethylformamide Chloroform Bromobenzene Dimethyl phthalate Dioxane 0.1 M NaCl

[65] [65] [65] [65] [100] [65] [65] [65] [65] [65] [100] [101]

640

0.2 M NaCl

[101]

1,360 1,030 626–640

Acetophenone Dioxane Dioxane

[102] [100] [65]

600 478
62 149
330
540
320
115
300
140
820
1,100
1,880
742
572
1,830
872
560

Bromobenzene Bromoform Cyclohexanone Dioxane Cyclohexanone Cyclohexanone Amyl acetate Acetone Butyl acetate Ethyl acetate Cyclohexanone Acetophenone Dioxane Benzophenone Benzophenone Dioxane Dioxane Ethyl acetate

[65] [65] [65] [65] [65] [65] [65] [65] [65] [65] [65] [102] [65] [103] [103] [65] [65] [65]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

841

TABLE 50.3. Continued.

Da (A3 )

Polymer Cellulose stearate D.S. ¼ 2.0 Cyanoethyl cellulose D.S. ¼ 26 Cyanoethylacetyl cellulose Cyanoethylacetyl cellulose Cyanoethylacetyl cellulose Cyanoethyltrityl cellulose DNA Ethyl cellulose Ethyl cellulose D.S. ¼ 2.5 Poly(1,2,3-trimethyl-2,3-dihydro-1,6indendiyl-1,4-phenylene-ethylene) Poly(1,2,3-trimethyl-2,3-dihydro-1, 6indenedinyl) Poly(1-butene) atactic Poly(1-butene) isotactic Poly(1-decene) isotactic Poly(1-dodecene) isotactic Poly(1-heptene) isotactic Poly(1-hexadecene) Poly(1-hexene) atactic Poly(1-hexene) isotatic Poly(1-octadecene) isotactic Poly(1-octene) isotactic Poly(1-pentene) isotactic Poly(1-pentene) isotactic Poly(1-tetradecene) isotactic Poly(1-tetradecene) isotactic Poly(2.5-dichlorostyrene) Poly(2.5-dimethylstyrene) Poly(2-methyl-5-vinyl-N-butylpyridinium bromide) Poly(2-methyl-5-vinyl-N-butypyridinium bromide) Poly(2-methyl-5-vinyl-pyridine) Poly(2-methyl-5-vinyl-pyridinium chloride) Poly[2-propenoic acid-4-(phenylazoxy)phenyl ester] Poly(2-propenoic acid-4-[(4-butylphenyl) azoxy]-phenyl ester) Poly(3,4-dichlorostyrene) Poly(3-methyl tetrahydrofuran); isotactic, planar, trans config Poly(3-methyl tetrahydrofuran); isotactic, planar, trans config Poly(3-methyl tetrahydrofuran); isotactic, planar, trans config. Poly(3-methyl tetrahydrofuran); isotactic, planar, trans config. Poly(3-methyl tetrahydrofuran); isotactic, planar, trans config. Poly(3-methyl-1-butene-silsesquioxane) Poly(3-methyl-1-butene-silsesquioxane) Poly[4-(4-nonyloxy-benzamido) styrene]

Diluent

Temp (8C) v 2

Stress optical coefficient C (109 Pa
1 ) Reference


500 900 15 390 90 220
30,000 430 512 142

Tetrachloroethane Cyclohenanone Cyclohexanone Acetone Dimethylformamide Cyclohexanone Aqueous 0.2 M NaCl Carbon tetrachloride Dioxane Bromoform

[104] [65] [105] [65] [65] [105] [65] [106]] [65] [85]

78

Bromoform

[85]

33.4 25.2
82.5
120
24.5 (
205)–(
213) 12.1
6.5
257
39 8.0 9.3
176
171
265
180
900

Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Toluene Bromoform Bromoform 0.01 M NaCl

[30] [30] [30] [30] [30] [30] [30] [30] [30] [30] [30] [30] [30] [30] [67] [67] [72]


270

0.1 M NaCl

[72]


260
300
450

Bromoform 0.1 M HCl Tetrachloroethane

[73] [74] [65]


510

Tetrachloroethane

[65]


300 2.473

Tetrabromoethane 20

1.25

[69] [21]

2.563

30

1.20

[21]

2.653

40

1.16

[21]

2.750

40

1.13

[21]

2.835

60

1.10

[21]


570
400
2,500

Benzene Butyl acetate Benzene

[65] [65] [65]

842 / CHAPTER 50 TABLE 50.3. Continued.

Polymer Poly(4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly (4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly(4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly(4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly(4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly(4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy] phenyl ester) Poly(4-(hexadecyloxy)-benzoic acid-4-[(4-[(2-methyl-1-oxo-2-propenyl) oxy]benzoyl)oxy]phenyl ester) Poly(4-(hexadecyloxy)-benzoic acid-4-[(4-[(2-methyl-1-oxo-2-propenyl) oxy]benzoyl)oxy]phenyl ester) Poly(4-(hexyloxy)-3-nitro-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly(4-(hexyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly(4-(nonyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2-propenyl) oxy]phenyl ester) Poly(4-[(1-oxo-2-propenyl)oxy]-benzoic acid-4-butoxyphenyl ester) Poly(4-[(1-oxo-2-propenyl)oxy]-benzoic acid-4-ethoxyphenyl ester) Poly(4-[(1-oxo-2-propenyl)oxy]-benzoic acid-4-methoxyphenyl ester) Poly(4-[(1-oxo-2-propenyl)oxy]-benzoic acid-4-propoxyphenyl ester) Poly(4-[(1-oxo-2-propenyl)oxy]-benzoic acid-phenyl ester) Poly(4-[(2-methyl-1-oxo-2-propenyl)oxy]benzoic acid-4-(dodecyloxy) phenyl ester) Poly(4-[(2-methyl-1-oxo-2-propenyl)oxy]benzoic acid-4-(nonyloxy)phenyl ester) Poly(4-[(2-methyl-1-oxo-2-propenyl) oxy]benzoic acid-4-cyanophenyl ester) Poly(4-[(2-methyl-1-oxo-2-propenyl) oxy]benzoic acid-4-methoxyphenyl ester) Poly(4-[(2-methyl-1-oxo-2-propenyl) oxy]benzoic acid-hexadecyl ester)

Da (A3 )

Diluent

Temp (8C) v 2

Stress optical coefficient C (109 Pa
1 ) Reference


1,000 Bromoform

[66]


1,600 Benzene

[66]


1,400 Chloroform

[66]


2,700 Carbon tetrachloride

[65]


890

[66]

Tetrahydrofuran


4,200 Benzene/heptene, 52/48, 66/34

[67]


4,900 Chloroform

[65]


3,000 Benzene

[65]


1,200 Dioxane

[65]


370

Benzene

[65]


600

Carbon tetrachloride

[65]


800

Tetrachloroethane

[65]


630

Tetrachloroethane

[65]


520

Tetrachloroethane

[65]


810

Tetrachloroethane

[65]


520

Tetrachloroethane

[65]


2,350 Carbon tetrachloride

[65]


2,700 Carbon tetrachloride

[65]


240

Dimethylformamide

[65]


500

Tetrachloromethane

[65]


445

Carbon tetrachloride

[65]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

843

TABLE 50.3. Continued.

Polymer Poly(4-[(11-[(2-methyl-1-oxo-2propenyl)oxy]-1-oxoundecyl)oxy]benzoic acid-4-butoxyphenyl ester) Poly(4-[2-(2-methyl-1-methylene-allyloxy)ethoxy]-benzoic acid-4-methoxyphenyl ester] Poly(4-propoxy-benzoic acid-4-[(2-methyl1-oxo- 2-propeny)oxy]phenyl ester) Poly(4-vinylpyridene) Poly(4-vinylpyridinium chloride) Poly(4-vinylpyridinium chloride) Poly(acenaphthylene) helix 4/1 Poly(acenaphthylene) trans Poly(acrylic acid) Poly(acrylic acid), sodium salt Poly(acrylic acid), sodium salt Poly(acrylonitrile) Poly(a-methylstyrene); glass Poly(a-methylstyrene) Poly(benzimidizole-2,6-diyl-1,4phenylenebenzimidazol6,2-diyl-iminoterephthaloylimino) Poly(benzimidizole-2,6-diyliminoterephthaloylimino-1, 4-phenylene) iminoterephthaloylimino-1,3-(4hydroxyphenylene) Poly(benzoxazole-2,6-diyl-iminoterephthaloylimino-1, 4-(3-hydroxyphenylene) Poly(benzoxazole-2,6-diyl-iminoterephthaloylimino-1, 4-phenylene) Poly(benzyl methacrylate); glass Poly(b-naphthyl methacrylate) Poly(b-vinylnapthalene) Poly(b-vinylnapthalene) Poly(butadiene) 1,4-cis Poly(butadiene) 1,4-cis Poly(butadiene) 1,4-cis Poly(butadiene) 1,4-cis Poly(butadiene) 1,4-cis Poly(butadiene) 1,4-trans Poly(butadiene) 1,4-trans Poly(butadiene) 1,4-trans Poly(butadiene) 1,4-trans Poly(butadiene) 1,4-trans Poly(butadiene); (isotactic) melt Poly(butyl acrylate); fraction of meso dyads, Wm ¼ 0.3 Poly(carbonate) of bisphenol-A Poly(carbonate) of bisphenol-A, benzyl substituted Poly(carbonate) of bisphenol-A, phenyl substituted Poly(carbonate) of bisphenol-A; glass

Da (A3 )

Diluent

Temp (8C)

v2

Stress optical coefficient C (109 Pa
1 ) Reference


90

Dioxane

[65]


117

Carbon tetrachloride

[65]


320

Dimethylformamide/toluene

[65]


240
260
440
300
300
0.5
20 (
4)–(
10)
23

Bromoform 0.1 M HCl 0.05 M HCl Bromoform Bromoform Dioxide 0.0012 M NaCl, pH 7 water, pH 6.1 Dimethylformamide


133 490

Tetrabromoethane Sulfuric acid

[75] [75] [75] [83] [83] [42] [43] [43] [64] [17,68] [70] [65]

5,080

Sulfuric acid

[65]

2,340

Sulfuric acid

[65]

5,940

Sulfuric acid

[65]

5,940

Sulfuric acid

[65]


60
440
4.4 61.3–63 53.5 57.3 72 86.9 71 61.1 57.3 81.6 101

Tetrabromoethane Bromoform Benzene Benzene Carbon tetrachloride Cyclohexane Toluene p-Xylene Benzene Carbon tetrachloride Cyclohexane Toluene p-Xylene

0.002

0.025–0.045


0.95

22 50

3.3 1.0

[17,50] [64] [83] [27] [24] [24] [24] [24] [24] [25] [24] [24] [24] [24] [17,26] [20,49]

170–230 220

3.5–3.7 40

[17,29] [17,29]

250

1.8–2.1

[17,29]

0.111

[17,50]

844 / CHAPTER 50 TABLE 50.3. Continued.

Polymer Poly(carbonate) of bisphenol-A; melt Poly(cetyl acrylate) Poly(cetyl acrylate) Poly(cetyl methacrylate) Poly(chloroethyl methacrylate); glass Poly(chlorohexyliminocarbonyl) Poly(chloroprene) Poly(chloroprene) Poly(chloroprene) Poly(chloroprene) Poly(chloroprene) Poly(chloroprene) Poly(chloroprene) Poly(chloroprene) Poly(cholestryl acrylate) Poly(cis-1,4-cyclohexane dimethanol sebacate) Poly(cis-1,4-cyclohexane dimethanol sebacate) Poly(cis-1,4-cyclohexane dimethanol sebacate) Poly(cis-1,4-cyclohexane dimethanol sebacate) Poly(cis-1,4-cyclohexane dimethanol sebacate) Poly(cis-1,4-cyclohexane dimethanol sebacate) Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 4,700 g=mol; trans content ¼ 10% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 4,700 g=mol; trans content ¼ 10% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 4,700 g=mol; trans content ¼ 10% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 4,700 g=mol; trans content ¼ 10% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 4,700 g=mol; trans content ¼ 10% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 4,700 g=mol; trans content ¼ 10% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 4,700 g=mol; trans content ¼ 10% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 6,000 g=mol; trans content ¼ 70% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 6,000 g=mol; trans content ¼ 70% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 6,000 g/mol; trans content ¼ 70% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 6,000 g/mol; trans content ¼ 70%

Da (A3 )

Diluent

Temp (8C) 200


141
164
160

Decalin Toluene Benzene

3,000 110 33 64 39 99 46 67 88
360 9.15 8.76 8.5 8.37 8.28 7.5 7.344

Carbon tetrachloride a-Bromonaphthalene Carbon tetrachloride Chlorobenzene Dichloromethane a-Methylnaphthalene Tetrachloromethane Toluene p-Xylene Benzene

v2 1.0

Stress optical coefficient C (109 Pa
1 ) 3.5

2.560

[22] [46] [45] [56] [17,57] [92] [27] [27] [27] [27] [27] [27] [27] [27] [47] [20,111] [20,111] [20,111] [20,111] [20,111] [20,111] [23]


0.0056

1,3,5-Triethylbenzene

40 50 60 70 80 40 70

1.0 1.0 1.0 1.0 1.0 0.336

Reference

0.352 0.326 0.306 0.292 0.280

7.300

60

2.626

[23]

7.310

50

2.719

[23]

7.234

40

2.783

[23]

7.238

30

2.884

[23]

7.237

20

2.989

[23]

7.229

10

3.099

[23]

9.978

70

3.414

[23]

9.819

60

5.553

[23]

9.969

50

3.727

[23]

9.849

40

3.8.8

[23]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

845

TABLE 50.3. Continued.

Polymer Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 6,000 g/mol; trans content ¼ 70% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 6,000 g/mol; trans content ¼ 70% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde) network; Mn ¼ 6,000 g/mol; trans content ¼ 70% Poly(cis/trans-1,4-cyclohexane dimethanol-alt-formaldehyde); fraction of rings in trans configuration ¼ 0.1 Poly(cis/trans-1,4-cyclohexane dimethanolalt-formaldehyde); fraction of rings in trans configuration ¼ 0.7 Poly(cyclohexyl methacrylate); glass Poly(decyl acrylate) Poly(decyl acrylate) Poly(dichlorophenyl-silsesquioxane) Poly(dichlorophenyl-silsesquioxane) Poly(diethylene glycol terephthalate) Poly(diethylene glycol terephthalate) Poly(diethylene glycol trans-1,4cyclohexane dicarboxylate) Poly(dimethyl siloxane) Poly(dimethyl siloxane) Poly(dimethyl siloxane) Poly(dimethyl siloxane) Poly(dimethyl siloxane) Poly(dimethyl siloxane) Poly(dimethyl siloxane) Poly(dimethyl siloxane) Poly(dimethyl sil-methylene) Poly(dimethyl sil-methylene) Poly(diphenylpropylene) Poly(epsilon, N-carbobenzoxy-Llysine) helix Poly(ethyl acrylate) Poly(ethyl acrylate) Poly(ethyl acrylate) Poly(ethyl acrylate) Poly(ethyl acrylate) Poly(ethyl acrylate); fraction of meso dyads, Wm ¼ 0.3 Poly(ethylene) Poly(ethylene) Poly(ethylene) Poly(ethylene) Poly(ethylene) Poly(ethylene) Poly(ethylene) Poly(ethylene) Poly(ethylene), cross-linked Poly(ethylene), linear melt Poly(gamma-benzyl-L-glutamate) coil

Da (A3 )

Diluent

Temp (8C)

v2

Stress optical coefficient C (109 Pa
1 ) Reference

9.988

30

3.957

[23]

9.714

20

4.029

[23]

9.435

10

4.060

[23]

7.24

30

[20,23]

9.99

30

[20,23]


0.059
74
95
4,450
4,700 19.8 26.5 7.4 0.81 0.68 0.3–0.7 0.51 0.38 0.18 0.15–0.20 4.27 2.5 80 3,600

Decalin Toluene Bromoform Tetrabromoethane Tricresylphosphate

70 70 70 Decalin 70 Cyclohexane 70 Carbon tetrachloride 70 Carbon tetrachloride 70 20–190 50 1,3,5-Triethylbenzene 50 Bromoform Dimethylformamide

3.0 10
37
14
11
0.82

Benzene Bromobenzene Bromoform Dibromoethane Dimethylformamide

60 30 7.8 8.4 6.5 5.1 40 3.9–4.0

Tetralin, xylene Decalin

230

70 70

50

n-C22H46 n-C12H26 Decalin Decalin

Dichloroacetic acid

150 150 150 150 150 150 130–180 150–190

1.0 0.58 1.0 1.0 1.0 1.0 0.25 0.16 0.16 0.35–0.71 0.135–0.260 1.0

[17,57] [45] [45] [65] [65] [20,111] [20,111] [20,111] [20,32] [20,105] [20,108] [20,32] [20,32] [20,32] [20,109] [17,29] [20,110] [20,110] [39] [93] [48] [48] [48] [48] [48] [20,49]

1.0

1.0 1.0 0.41 0.44 0.33 0.33 1.5–2.2 1.8–2.4

[31] [31] [20,35] [20,34] [20,32] [20,32] [20,32] [20,33] [17,29] [17,29] [65]

846 / CHAPTER 50 TABLE 50.3. Continued.

Polymer

Da (A3 )

Diluent

Temp (8C)

v2

Stress optical coefficient C (109 Pa
1 )

Reference

Poly(gamma-benzyl-L-glutamate) helix Poly(glycol methacrylate) Poly(glycol methacrylate) Poly(glycol methacrylate) Poly(hexyl methacrylate) Poly(hexyl methacrylate) Poly(hydrazocarbonyl-1,4-phenyleneiminoterephthaloyl) Poly(imino-1,3-phenyleneimino-sophthaloyl) Poly(imino-1,4-phenyleneimino-terephthaloyl) Poly(imino-1,4-terephthaloylimino-1, 4-phenylenediphenylmethyl-1,4-phenylene) Poly(iminocarbonyl,1,4-phenylene) Poly(iminocarbonyl-cyclohexylene) Poly(isobutene)

25,000 1.0
6.0
6.0
40
9.7 3,630

Dichloroethane Dimethylformamide Ethyl alcohol Water Benzene Carbon tetrachloride Dimethylsulfoxide

[65] [58] [58] [58] [59] [59] [65]

360 5,250 130

Sulfuric acid Sulfuric acid Sulfuric acid

[65] [65] [65]

10,500 390 45–59

[65] [65] [27]

Poly(isobutene) Poly(isobutene) Poly(isobutene) Poly(isobutylene) Poly(isobutylene) Poly(isobutylene) Poly(isobutylene) Poly(isobutylene) Poly(isobutylene) Vistanex B-100 Poly(isobutylphenyl-silsesquioxane) (1:1) Poly(isohexylphenyl-silsesquioxane) (1:1) Poly(isoprene) cis Poly(isoprene) cis Poly(isoprene) trans Poly(isoprene); melt Poly(isoprene); natural rubber (cross-linked) Poly(isoprene); trans, gutta percha (cross-linked) Poly(L-glutamic acid) coil

35 30 69 4.10 3.5 3.36 3.79 2.84

Sulfuric acid Sulfuric acid Tetrachloroethylene, m-Xylene Carbon tetrachloride Decalin p-Xylene

Poly(L-glutamic acid) helix

1,900

Poly(m,p-aromatic ester) m/p ¼ 1/3 Poly(m-chlorophenyl-silsesquioxane) Poly(methacrylic acid) Poly(methacrylic acid) Poly(methacrylic acid), sodium salt Poly(methacrylic acid), sodium salt Poly(methacrylic acid), sodium salt Poly(methyl acrylate) Poly(methyl acrylate) Poly(methyl acrylate) Poly(methyl acrylate); fraction of meso dyads, Wm ¼ 0.3 Poly(methyl acrylate); fraction of meso dyads, Wm ¼ 0.5 Poly(methyl methacrylate) atactic

750
4,700 50 150 150 400 56–300 17 16 26
0.34


840
980 48 53.1 49

136

Carbon tetrachloride Cyclohexane Bromotrichloromethane Decalin Benzene Benzene Benzene Squalene Benzene

1.9 1.80–2.05

85–250

3.0

[17,29]

30 30 30 30 30 25

1.0 1.0 0.16 0.17 0.19 0.03

1.7

Phosphate buffer, 0.1 M, pH ¼ 12.5 Phosphate buffer, 0.1 M, pH ¼ 4.2 Dichloroacetic acid Carbon tetrachloride Methanol 0.002 M HCl 0.012 M NaCl, pH ¼ 7 0.0012 M NaCl, pH ¼ 7 Water Benzene Toluene Toluene


0.84 2.0

22 20–100

[27] [31] [27] [20,37] [20,38] [20,37] [20,37] [20,37] [36] [65] [65] [25] [28] [25] [17,26] [17,29]

Benzene

[93] [93]

50

1.0

[91] [65] [61] [61] [61] [61] [62] [44] [45] [44] [20,49]

50

1.0

[20] [63]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

847

TABLE 50.3. Continued.

Polymer

Da (A3 )

Poly(methyl methacrylate) isotactic

25

Diluent

Temp (8C)

v2

Benzene


12.1

Reference [63]

(
0.0033) – (0.0045)

Poly(methyl methacrylate), glass Poly(methyl phenyl siloxane); fraction of meso dyads, Wm ¼ 0.5 Poly(methyl phenyl siloxane); fraction of meso dyads, Wm ¼ 0.5 Poly(methyl phenyl siloxane); fraction of meso dyads, Wm ¼ 0.5 Poly(methyl phenyl siloxane); fraction of meso dyads, Wm ¼ 0.5 Poly(methyl phenyl siloxane); fraction of meso dyads, Wm ¼ 0.5 Poly(methyl phenyl siloxane); fraction of meso dyads, Wm ¼ 0.5 Poly(N,N-piperazindiyl-2,5-diketo1,3-pyrrolidindiylhexamethylene2,5-diketo-1,3-pyrrolidindiyl) Poly(N,N-piperazindiyl-2,5-diketo1,3-pyrrolidindiylhexamethylene2,5-diketo- 1,3-pyrrolidindiyl) Poly(N-2,4-dimethylphenylmaleimide) Poly(n-butyl acrylate) Poly(n-butyl acrylate) Poly(n-butyl acrylate) Poly(n-butyl acrylate) Poly(n-butyl methacrylate) atactic Poly(n-butyl methacrylate) isotactic Poly(n-butyliminocarbonyl) a.k.a. poly(butyl isocyanate) Poly(n-butyliminocarbonyl) a.k.a. poly(butyl isocryanate) Poly(N-isobutylmaleimide) Poly(N-methylcitraconimide) Poly(N-phenylmathacrylamide) Poly(n-tolyliminocarbonyl) Poly(ocatdecyl acrylate) Poly(ocatdecyl acrylate) Poly(octyl acrylate) Poly(octyl acrylate) Poly(octyl acrylate; fraction of meso dyads, Wm ¼ 0.3) Poly(octyl methacrylate) Poly(octyl methacrylate) Poly(oxadiazole-2,5-diyl1,4-phenylene) Poly(oxycarbonyloxy1,4-phenylenecyclcohexylidene1,4-phenylene) Poly(oxydecamethyleneoxycarbonylphenyleneoxyterephthaloyloxy phenylenecarbonyl) Poly(oxydimethysilyene) Poly(oxyethyleneoxyterephthaloyl) Poly(oxyethyleneoxyterephthaloyl)

Stress optical coefficient C (109 Pa
1 )

[17,50]

25

1.000

[20,113]


11.5

Decalin

25

0.673

[20,113]


9.12

Decalin

25

0.488

[20,113]


8.32

Decalin

25

0.350

[20,113]


8.10

Decalin

25

0.211

[20,113]


8.50

Decalin

25

0.204

[20,113]

56

Benzyl alcohol

[98]

30

Bromoform/cyclohexanol, 60/40

[98]


200
11
17.8
10.1
6.5
14
2.0 4,100

Bromoform Benzene Decalin Toluene Toluene Benzene Benzene Carbon tetrachloride

[65] [44] [45] [45] [44] [51] [52] [65,107]

800

Pentafluorophenol 0.9/0.1

[65]

160 150
103
39
190
232
57.4
47.9
1.1

Chlorobenzene Bromoform o-Toluidine Bromoform Decalin Toluene Decalin Toluene

[65] [97] [69] [65] [45] [45] [45] [45] [20,49]


47
12.5 1,750

Benzene Carbon tetrachloride Sulfuric acid

[59] [59] [65,112]


114

Bromoform

[88]

200

Dichloroacetic acid

[90]

4.7 70 48.7

Petroleum ether Dichloroethane/phenol 1/1 Dichloroacetic acid

[94] [89] [90]

50

1.0

848 / CHAPTER 50 TABLE 50.3. Continued.

Da (A3 )

Polymer Poly(oxyhexamethyleneoxycarbonylphenyleneoxyterephthaloyloxyphenylenecarbonyl) Poly(oxymethylphenylsilylene) (50–62.5)/ (50–37.5) Poly(oxymethylphenylsilylene) 50/50 Poly(oxymethylphenylsilylene) 75/25 Poly(oxymethylphenylsilylene) 75/25 1 Poly(oxymethylphenylsilylene) 87.5/12.5 Poly(oxymethylphenylsilylene) 90/10 Poly(oxymethylphenylsilylene) atactic Poly(oxymethylphenylsilylene) atactic Poly(oxymethylphenylsilylene) isotactic Poly(oxyphenysilylene) Poly(oxypropylene) Poly(oxypropylene); Mc ¼ 2,000 Poly(oxypropylene); Mc ¼ 3,000 Poly(oxypropylene); Mc ¼ 3,000 Poly(oxypropylene); Mc ¼ 3,000 Poly(oxytetramethyleneoxycarbonylphenyleneoxyterephthaloyloxyphenyleneoxycarbonyl) Poly(p-carboxyphenyl methacrylate) Poly(p-carboxyphenyl methacrylate) Poly(p-carbethoxy-Nphenylmethacrylamide) Poly(p-chloro-N-phenylmethacrylamide) Poly(p-chlorostyrene) Poly(p-chlorostyrene); glass Poly(p-methylcarboxy–phenyl methacrylate) Poly(p-methylstyrene) atactic Poly(p-methylstyrene) isotactic Poly(p-t-butyl styrene); glass Poly(p-t-butylphenyl methacrylate) Poly(p-tolymaleiimide) Poly(phenyl methacrylate) Poly(phenyl methacrylate); glass Poly(phenyl-silsesquioxane) Poly(propylene) Poly(propylene) atactic Poly(propylene) atactic Poly(propylene) atactic Poly(propylene) atactic Poly(propylene) isotactic Poly(propylene); isotactic (melt) Poly(propylene); M.W. ¼ 43,000–48,000 Poly(S-carbobenzoxymethyl-L-cystein) Poly(styrene) Poly(styrene) Poly(styrene) atactic Poly(styrene) isotactic Poly(styrene); atactic, M.W. ¼ 9,000,000 Poly(styrene); glass

Diluent

Temp (8C)

v2

Stress optical coefficient C (109 Pa
1 ) Reference

250

Dichloroacetic acid

[90]


36

Decalin

[95]


36
13.6
2
10
2.3
25.5
82
82
85
18 4.660.04 4.330.09 3.05 3.87 280

Decalin Benzene Decalin Decalin Petroleum ether Benzene Decalin, tetralin Decalin, tetralin Benzene Cyclohexane

Decalin POP oligomers Dichloroacetic acid

[95] [96] [95] [95] [94] [96] [95] [95] [96] [86] [20,87] [20,87] [20,87] [20,87] [90]

180 370
230

Dioxane 0.1 M NaCl o-Toluidine

[55] [55] [54]


160.0
230

o-Toluidine Bromoform


77

Dibromoethane

[54] [67] [17,68] [55]


147
140

Bromoform Bromoform


90
160
10.5

Bromobenzene Bromoform Bromobenzene

(
1,060)–(
1,800) 0.3 45 30 30 55 30

Bromoform

25 25 25 25

1.0 1.0

0.024

0.011

0.040

Benzene, xylene Carbon tetrachloride Decalin Toluene Carbon tetrachloride 210 210

22
1.7

Dichloroacetic acid


145
224

Bromoform Bromoform Bromoform

0.9 0.9

(
4.1)–5.2

25

0.01


6.9 0.010

[71] [71] [17,68] [53] [99] [64] [17,57] [65] [20,41] [31] [40] [31] [30] [31] [17,29] [19,36] [93] [20,41] [17,29] [67] [76] [19,36] [17,50]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

849

TABLE 50.3. Continued.

Polymer Poly(styrene); M.W. ¼ 50,000– 860,000 Poly(t-butyl methacrylate) atactic Poly(t-butyl methacrylate) isotactic Poly(triethylene glycol terephthalate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl acetate) Poly(vinyl butyral) Poly(vinyl butyral) Poly(vinyl butyrate) Poly(vinyl butyrate) Poly(vinyl butyrate) Poly(vinyl chloride) Poly(vinyl chloride) Poly(vinyl chloride); M.W. ¼ 14,000 Poly(vinyl cinnamate) Poly(vinyl propionate) Poly(vinyl propionate) Poly(vinyl propionate) Poly(vinyl stearate) Poly(vinyl toluene); glass Poly(vinylpyrrolidone) Poly([1,3,5,7-tetraoxo-2,3,6,7-tetrahydro-1H, 5H-benzo-(1,2-c:4,5-c’) dipyrrol- 2,6-diyl]1,4-phenylene) Poly[(2,2’-diphenyl(6,6’-biquinoxaline)-3,3’diyl)(1,1’-biphenyl)-4,4’-diyl] Poly[(3-b)-cholest-5-en-3-ol-2-propenoate] Poly[(ar,ar’-diphenyl(biquinoxaline)-ar, ar’-diyl]-1, 4-phenyleneoxy (1,1’-biphenyl)- 4,4’diyloxy1,4-phenylene) Poly[(imino-4,6-dicarboxyisophthaloylimino) biphenyl-4-4’-diyl]

Da (A3 )

2.1 19.3 13.3
20 4.0–5.9 9.4
20
16
26 14
34.9
24
23
36
39
25
33 10 13.5 19 2.0 81
2.68
1.90 173
80
36
48 40
420

Diluent

Temp (8C) 153–215

v2

Stress optical coefficient C (109 Pa
1 )

Reference

1.0

(
4)–(
5)

[19,36]

Benzene Benzene 1.0 Acetone Benzene Bromobenzene Bromoform Carbon tetrachloride Carbon tetrachloride Chlorobenzene Chloroform Chloroform Cyclohexane Dichloroethane Dichloroethane Tetrabromoethane Tetrabromoethane Toluene Toluene Toluene o-Xylene Chloroform 1,3,5-Triethylbenzene Toluene–phenol, 79–21 Benzene Carbon tetrachloride Chloroform Tetrahydrofuran Bromoform


430
31
40 1.3
130

Tetrabromoethane Carbon tetrachloride Chloroform Toluene Carbon tetrachloride


75 250–440

Benzyl alcohol Sulfuric acid

175

Tetrachloroethane, chloroform Benzene Tetrachloroethane, chloroform

50 50

1.0 0.6

210

1.0


0.5
0.5

0.015


300 106

75

Dimethylacetamide

[44] [44] [20,111] [77] [27] [78] [77] [27] [79] [27] [79] [27] [27] [27] [27] [27] [77] [27] [44] [79] [27] [27] [20,80] [20,80] [27] [27] [27] [27] [81] [82] [17,29] [19,36] [64] [27] [27] [27] [84] [17,68] [64] [65]

[65] [65] [65]

[65]

850 / CHAPTER 50 TABLE 50.3. Continued.

Polymer Poly[(phenylquinoxalinediyl) oxy (phenylquixalinediyl)-1, 4phenyleneoxy (1,1’-biphenyl)-4,4’-diyloxy1,4-phenylene] Poly[imino(1-oxohexamethylene)] a.k.a. [Poly(epsilon-caprolactam)] Poly[imino-1,4-phenyleneimino-(4,6dicarboxy-isophthaloylimino)–1,4, phenylene-terephthaloyl] Sulfate cellulose, sodium salt D.S. ¼ 0.4 Sulfate cellulose, sodium salt D.S. ¼ 0.4 Sulfate cellulose, sodium salt D.S. ¼ 0.4 Sulfate cellulose, sodium salt D.S. ¼ 0.4 Sulfate cellulose, sodium salt D.S. ¼ 0.4 Sulfate cellulose, sodium salt D.S. ¼ 0.4

Da (A3 )

Temp (8C)

Diluent

v2

Stress optical coefficient C (109 Pa
1 )

Reference

65

Tetrachloroethane, chloroform

[65]

63

Sulfuric acid

[65]

88

Dimethylacetamide

[65]

634 645 680 980 1,300 1,750

Aqueous NaCl 0.2 M Aqueous NaCl 0.15 M Aqueous NaCl 0.10 M Aqueous NaCl 0.01 M Aqueous NaCl 0.005 M Aqueous NaCl 0.001 M

[65] [65] [65] [65] [65] [65]

TABLE 50.4. Optical configuration parameters of copolymers. All values reported are at a wavelength of 6,328 A˚. Copolymer Poly(methyl methacrylate-co-p-tert-butylphenyl methacrylate) 25/75 mol % Poly(methyl methacrylate-co-p-tert-butylphenyl methacrylate) 50/50 mol % Poly(methyl methacrylate-co-p-tert-butylphenyl methacrylate) 80/20 mol % Poly(methyl methacrylate-co-p-tert-butylphenyl methacrylate) 91/9 mol % Poly(methyl methacrylate-graft-styrene) (70–90)/(30–10) mol % Poly(methyl methacrylate-graft-styrene) 10/90 mol % Poly(methyl methacrylate-graft-styrene) 87/13 mol % Poly(n-butyl methacrylate-graft-styrene) 8/92 mol % Poly[p-(4-cetoxybenzoxy)-phenyl methacrylate-co-cetyl methacrylate] 15/85 mol % Poly[p-(4-cetoxybenzoxy)-phenyl methacrylate-co-cetyl methacrylate] 22/78 mol % Poly[p-(4-cetoxybenzoxy)-phenyl methacrylate-co-cetyl methacrylate] 39/41 mol % Poly[p-(4-cetoxybenzoxy)-phenyl methacrylate-co-cetyl methacrylate] 4/96 mol % Poly[p-(4-cetoxybenzoxy)-phenyl methacrylate-co-cetyl methacrylate] 60/40 mol % Poly[p-(4-cetoxybenzoxy)-phenyl methacrylate-co-cetyl methacrylate] 8/92 mol % Poly[p-(4-cetoxybenzoxy)-phenyl methacrylate-co-cetyl methacrylate] 81/19 mol % Poly(phenylbutyl isocyanate-co-choral) 50/50 mol % Poly(propylene-graft-atactic styrene) 30/70 mol % Poly(styrene-block-propylene), atactic, 64/36 mol %

Da(A3 )

Diluent

Reference


44

Chlorobenzene

[5]


30.4

Chlorobenzene

[5]


7.4

Chlorobenzene

[5]

1.5

Chlorobenzene

[5]

100–1,100 700–7,000 30 1,190
277

Bromoform Bromoform Bromoform Bromoform Carbon tetrachloride

[39] [114] [115] [115] [116]


400

Carbon tetrachloride

[116]


540

Carbon tetrachloride

[116]


16

Carbon tetrachloride

[116]


920

Carbon tetrachloride

[116]


180

Carbon tetrachloride

[116]


2,000

Carbon tetrachloride

[116]

12 22
8

Carbon tetrachloride Chlorobenzene Chlorobenzene

[92] [117] [118]

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION

/

851

TABLE 50.4. Continued. Copolymer

Da(A3 )

Poly(styrene-co-chlorostyrene) 39.5/60.5 mol % Poly(styrene-co-chlorostyrene) 55/45 mol % Poly(styrene-co-chlorostyrene) 68.7/31.3 mol % Poly(styrene-co-chlorostyrene) 83.2/16.8 mol % Poly(styrene-co-chlorostyrene) 89.7/10.3 mol % Poly(styrene-co-methyl methacrylate) 30/70 mol % Poly(styrene-co-methyl methacrylate) 50/50 mol % Poly(styrene-co-methyl methacrylate) 70/30 mol % Poly(styrene-co-N-methylcitraconimide) 48/52 mol % Poly(styrene-co-N-methylcitraconimide) 54/46 mol % Poly(tert-butyl methacrylate-graft-styrene) 8/92 mol % Poly(vinyl chloride-graft-styrene) 12.1/87.9 wt% Poly(vinyl chloride-graft-styrene) 30.7/69.3 wt% Poly(vinyl chloride-graft-styrene) 5.2/94.8 wt% Poly(vinyl chloride-graft-styrene) 5.6/94.4 wt% Poly[(imino-1,4-phenyleneimino-terephthaloyl)co-(iminocarbonyl-1,4-phenylene)] (ratio 1/9) Poly{[(4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1oxo-2-propenyl)oxy]phenyl ester)]-co-cetylmethacrylate} 50/50 mol % Poly{[(4-(hexadecyloxy)-benzoic acid-4-[(2-methyl-1-oxo-2propenyl)oxy]phenyl ester)]-co-cetyl methacrylate}-70/30 mol %


226
202
198
172
165
34
68
88
34
26 540 155 110 330 180 4,380

Bromoform Bromoform Bromoform Bromoform Bromoform Chlorobenzene Bromobenzene Bromobenzene Bromoform Bromoform Bromoform Benzene Benzene Benzene Benzene Sulfuric acid

[119] [119] [119] [119] [119] [120] [120] [120] [97] [97] [121] [81] [81] [81] [81] [65]


680

Carbon tetrachloride

[65]


1,050

Carbon tetrachloride

[65]

REFERENCES 1. D. V. Rosato, Plastics Encyclopedia and Dictionary, Hanser, New York, 1993, p. 781. 2. Nunez, Nunez, and Mann eds. CenBASE/Materials in Print, Wiley, New York, 1990. 3. J. F. Shackelford and W. Alexander, eds. CRC Materials Science and Engineering Handbook, CRC, Boca Raton, 1992, p. 776. 4. H. F. Mark, N. G. Gaylord, and N. M. Bikales, eds. Encyclopedia of Polymer Science and Technology: Plastics, Resins, Rubbers, Fibers, Interscience, New York, 1964. 5. J. Brandrup and E. H. Immergut, eds. Polymer Handbook, 3rd ed. Wiley, New York, 1989, pp. 451–461. 6. C. L. Mantell, ed. Engineering Materials Handbook (McGraw-Hill, New York, 1958), Sec. 33. 7. S. H. Goodman, Handbook of Thermoset Plastics, Noyes Park Ridge, NJ, 1986, p. 281. 8. J. Bicerano, Prediction of Polymer Properties, Dekker, New York, 1993, pp. 179–197. 9. Janssen Chimica Catalog, Janssen, Belgium, 1990. 10. Aldrich Catalog, Aldrich Chemical, Milwaukee, WI, 1990. 11. R. Anderson, G. L. Larson, and C. Smith, eds., Huls Silicon Compounds Register and Review, Huls, Piscataway, N.J., 1991. 12. C.A. Harper, ed. Handbook of Plastics and Elastomers, McGrawHill, New York, 1975. 13. C. T. Lynch, ed., CRC Handbook of Materials Science, CRC Press, Boca Raton, FL, 1975, Vol. 3. 14. G. M. Kline, Analytical Chemistry of Polymers, Interscience, New York, 1962, Part III. 15. E. I. duPont de Nemours and Co., Kapton, Summary of Property Report No. E-50533, 1982. 16. E. I. duPont de Nemours and Co., Preliminary Process Bulletin No. PC-1. 17. D. W. van Krevelen, Properties of Polymers, 2nd ed. Elsevier, Amsterdam, 1976. 18. Engineering Plastics, Vol.2 of Engineering Materials Handbook, ASM International, Metals Park, OH, 1988.

Diluent

Reference

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78. V. N. Tsvetkov, A. E. Grishchenko, L. E. De-Millo, and E. N. Rostovskii, Vysokomol. Soedin. 6, 384, 1964. 79. E. V. Frisman and V. A. Andreichenko, Vysokomol. Soedin. 4, 1559, 1962. 80. E. Riande, E. Saiz, and J. E. Mark, J. Polym. Sci. Polym. Phys. Ed. 22, 863, 1984. 81. S. P. Micengendler, K. I. Sokolova, G. A. Andreeva, A. A. Korotkov, T. Kadirov, S. I. Klenin, and S. Ja. Magarik, Vysokomol. Soedin. 9A, 1133, 1967. 82. V. N. Tsvetkov, E. N. Zakharova, G. A. Fomin, and P. N. Lavrenko, Vysokomol. Soedin. 14A, 1956, 1972. 83. V. N. Tsvetkov, M. G. Vitovskaja, P. N. Lavrenko, E. N. Zakharova, I. F. Gavrilenko, and N. N. Stefanovskaja, Vysokomol. Soedin. 13A, 2532, 1971. 84. V. S. Skazka, G. A. Fomin, G. V. Tarasova, I. G. Kirillova, V. M. Jamshchikov, A. E. Gryshchenko, and I. A. Alekseeva, Vysokomol. Soedin. 15A, 2561, 1973. 85. V. N. Tsvetkov and S. Ja. Magarik, Dokl. Acad. Nauk. SSSR 115, 911, 1957. 86. V. N. Tsvetkov, T. I. Garmonova, and R. P. Stankevitch, Vysokomol. Soedin. 8, 980, 1966. 87. A. L. Andrady, M. A. Llorente, and E. Saiz, J. Polym. Sci. Polym. Phys. Ed. 25, 1935, 1987. 88. T. I. Garmonova, M. G. Vitovskaja, P. N. Lavrenko, V. N. Tsvetkov, and E. V. Korovina, Vysokomol. Soedin. 13, 884, 1971. 89. S. M. Savvon and K. K. Turoverov, Vysokomol. Soedin. 6, 205, 1964. 90. V. N. Tsvetkov, L. N. Andreeva, P. N. Lavrenko, E. V. Baleiva, O. V. Okatova, A. Yu. Bilibin, and S. S. Skorokhodov, Eur. Polym. J. 20, 817, 1984. 91. V. N. Tsvetkov, L. N. Andreeva, P. N. Lavrenko, O. V. Okatova, E. V. Baleiva, A. Yu. Bilibin, and S. S. Skorokhodov, Eur. Polym. J. 21, 933, 1985. 92. V. N. Tsvetkov, I. N. Shtennikova, E. I. Rjumtsev, and Ju. P. Getmanchuk, Eur. Polym. J. 1, 767, 1971. 93. V. N. Tsvetkov, I. N. Shtennikova, V. S. Skazka, and E. I. Rjumtsev, J. Polym. Sci. C 16, 3205, 1968. 94. V. N. Tsvetkov, E. V. Frisman, and N. N. Boitsova, Vysokomol. Soedin. 2, 1001, 1960. 95. V. N. Tsvetkov, K. A. Andrianov, E. L. Vinogradov, S. E. Yakushkina, and Ts. V. Vardasanidse, Vysokomol. Soedin. 9B, 893, 1967. 96. V. N. Tsvetkov, K. A. Andrianov, E. L. Vinogradov, V. I. Pakhomov, and S. E. Yakushkina, Vysokomol. Soedin. 9A, 1967. 97. M. G. Vitovskaja, V. N. Tsvetkov, L. I. Godunova, and T. V. Sheremeteva, Vysokomol. Soedin. 9A, 1682, 1967. 98. T. I. Garmonova, M. G. Vitoskaja, S. V. Bushin, and T. V. Sheremeteva, Vysomokol. Soedin. 16A, 265, 1974. 99. V. N. Tsvetkov, N. N. Kupriyanova, G. V. Tarasova, P. N. Lavrenko, and I. I. Migunova, Vysomokol. Soedin. 12A, 1974, 1970. 100. V. N. Tsvetkov, E. I. Rjumtsev, I. N. Shtennikova, T. V. Peker, and N. V. Tsetkova, Dokl. Akad. Nauk. SSSR 207, 1173, 1972. 101. E. N. Zakharova, L. I. Kutsenko, V. N. Tsvetkov, V. S. Skakza, G. V. Tarasova, and V. M. Jamshchikov, Leningrad Univ. Vestnik, Ser. Fiz. Khim. N16, 55, 1970. 102. G. I. Okhmenko, Thesis, Inst. Macromolecular Compounds Acad. Sci. SSSR, Leningrad, 1969. 103. H. Janetchitz-Kriegl and W. Burhard, Adv. Polym. Sci. 6, 170, 1969. 104. V. N. Tsvetkov, S. Ja. Ljubina, I. A. Strelina, S. I. Klenin, and V. I. Kurljankina, Vysomokol. Soedin. 15A, 691, 1973. 105. V. N. Tsvetkov and A. E. Grishchenko, J. Polym. Sci. C 6, 3195, 1968. 106. V. N. Tsvetkov and I. N. Shtennikova, Vysomokol. Soedin. 2, 808, 1960. 107. V. N. Tsvetkov and A. V. Grishchenko, Polym. Sci. USSR 7, 902, 1965; J. Polym. Sci. C 16, 3195, 1968. 108. N. J. Mills and D. W. Saunders, J. Macromol. Sci. B. 2, 369, 1968. 109. N. J. Mills, Polymer 12, 658, 1971. 110. M. A. Llorente, J. E. Mark, and E. Saiz, J. Polym. Sci. Polym. Phys. Ed. 21, 1173, 1983. 111. E. Riande, J. Guzman, and J. G. de la Campa, Macromolecules 21, 2128, 1988. 112. S. Ja. Magarik and V. N. Tsvetkov, Zh. Fiz. Khim. 33, 835, 1959. 113. M. Llorente, I. Fernandez de Pierola, and E. Saiz, Macromolecules 18, 2663, 1985.

REFRACTIVE INDEX, STRESS-OPTICAL COEFFICIENT, AND OPTICAL CONFIGURATION 114. V. N. Tsvetkov, S. Ja. Magarik, T. Kadirov, and G. A. Andreeva, Vysokomol. Soedin. 10A, 943, 1968. 115. V. N. Tsvetkov, G. A. Andreeva, I. A. Branovskaja, V. E. Eskin, S. I. Klenin, and S. Ja. Magarik, J. Polym. Sci. C 6, 239, 1967. 116. V. N. Tsvetkov, E. I. Rjumtsev, I. N. Shtennikova, E. V. Korneeva, Okhrimenko, N. A. Mikhailova, A. A. Baturin, Ju. A. Amerik, and B. A. Krentsel, Vysokomol. Soedin. 15A, 2570, 1973. 117. V. N. Tsvetkov, E. I. Rjumtsev, I. N. Shtennikova, E. V. Korneeva, Okhrimenko, N. A. Mikhailova, A. A. Baturin, Ju. A. Amerik, and B. A. Krentsel, Vysokomol. Soedin. 15A, 2570, 1973.

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118. A. Romanov, S. Ja. Magarik, and M. Lazar, Vysokomol. Soedin. 9B, 292, 1967. 119. T. M. Birshtein, V. P. Budtov, E. V. Frisman, and N. K. Janoskaja, Vysokomol. Soedin. 4, 455, 1962. 120. E. V. Frisman and N. N. Boitsova, Leningrad Univ. Vestnik. N4, 26, 1959. 121. V. N. Tsvetkov, G. A. Andreeva, I. A. Branovskaja, V. E. Eskin, S. I. Klenin, and S. Ja. Magarik, J. Polym. Sci. C 16, 239, 1967.

CHAPTER 51

Ultraviolet Radiation and Polymers Anthony L. Andrady Camille Dreyfus Laboratory, Research Triangle Institute, Research Triangle Park, NC 27709

51.1 51.2

Mechanisms of Photodegradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Solar UV Induced Degradation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

859 861 865

Absorption of electromagnetic radiation is a necessary prerequisite for photodegradation; Table 51.1 summarizes the ultraviolet cut-off wavelength and relative stability of commodity polymers and biopolymers. Polymers such as polyolefins, which theoretically should be transparent to ultraviolet (UV) light, nevertheless do absorb UV radiation due to the presence of impurities from several sources. In most practical applications, the compounding ingredients in the formulation and the processing operation itself yield sufficient chromophores to allow these polymers to absorb UV radiation and therefore to undergo photodegradation. Also included in Table 51.1 is data on spectral sensitivity of common polymers. Wavelength sensitivity of a given photodegradation process can be measured experimentally using either monochromatic radiation or a specific source such as a white light source. In the former technique, the change in a specific property (such as yellowness or absorbance at a selected wavelength) per unit available photon is obtained. When a specific light source is used, samples of the polymer are exposed behind a series of cut-on filters. Each cut-on filter allows only the fraction of light having a wavelength longer than the cut-on value to reach the sample. Samples exposed to the same source behind different cut-on filters therefore photodegrade at different rates and sometimes even via different mechanisms. A comparison of the changes in a selected property of the samples exposed behind different filters allows the identification of the approximate spectral region which causes the most damage. This range is both source-dependent and damage-dependent. The data in Table 51.1 pertain to white light spectrum from borosilicate-filtered xenon source radiation, which is similar to direct solar radiation at unit air mass. However, the spectral sensitivity depends on the property or mode of damage used in its determination and also on compounding

The energies associated with near-ultraviolet radiation quanta are about 3.0–4.3 eV which correspond to 72–97 kcal/mol. Common covalent bonds encountered in polymers have bond dissociation energies which for the most part are either lower or within this energy range. Provided the ultraviolet radiation is absorbed by the polymer and suitable pathways are available for the photoexcited singlet (S) and triplet (T) species to transfer the absorbed energy to cause photochemical reactions, light-induced damage to the polymer can take place. In most systems a variety of competing photophysical processes, such as phosphorescence (from (T1 ! S0 ) transition) or fluorescence (from (S1 ! S0 ) transition), may preclude chemical reaction. When photochemical reactions of polymers do take place, they tend to involve the triplet excited states of molecules rather than the ground or singlet stage species because of the relatively longer lifetime of the former. The lowest excited triplet state, T1 , is formed by radiationless intersystem crossing from the lowest excited singlet state, S1 . Higher triplet states can form only from a triplet-triplet absorption where a molecule in the T1 state absorbs a photon. When photochemical pathways are available, they often involve sequences of chemical reactions with specific energy requirements. The number of photochemical degradative pathways available to polymers is quite extensive. As the photon energy is a function of the wavelength of radiation, it is reasonable to expect the high energy, short wavelength ultraviolet radiation to be more effective than visible light in promoting a wider range of these reactions. This is found to be the case; for instance, solar UV-B range (extending from about 290 nm to 315 nm) is well known to be the most deleterious wavelengths to polymers exposed to sunlight. 857

858 / CHAPTER 51 TABLE 51.1. Absorption of UV-visible radiation by common synthetic and natural polymers. Spectral sensitivity Polymer

Cut-off [nm]a

Polyethylene Polypropylene

Kaptone (PI) > PEEK > PES > Upilexe-S(PI) {4} U-PS > U-Polymer The order of resistance for electron beam irradiation is slightly different [187]: Upilexe-R(PI) ¼ Upilexe-S(PI) > Kaptone (PI) > PEEK > PES {4} U-PS > U-Polymer The polyimides and PEEK show high radiation resistance to attenuation of physical properties. The major component gases are: H2 and N2 for polyimides; CO2 and CO for PEEK; CO2 , CO, and SO2 for polysulphones; and CO2 and CO for U-Polymer. The yields for gaseous evolution are very low and are given in Table 52.26 for electron beam and Table 52.27 for g-irradiation. An increase in the glass transition temperature Tg occurs when PEEK, either in the crystalline form PEEK-c or the amorphous form PEEK-a, is irradiated with g-irradiation. This is indicative that a crosslinking process is occurring [188]. The irradiation of both amorphous and semicrystalline poly(phenylene sulfide) (PPS) with an electron beam in the presence of nitrogen shows no noticeable change in the

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883

mechanical or thermal properties at least to 104 kGy [189]. On the other hand, irradiation in air instead of nitrogen showed a change in both mechanical and thermal properties. At very high doses, 4
104 kGy, the amorphous PPS loses about 62% of its original tensile strength while the semicrystalline PPS loses about 57%. The Tm also changes, decreasing by about 10–2718C. 52.3.16 Other Polymers Table 52.28 gives the G(X) and the G(S) values for a list of different polymers which will not be discussed in detail. The polyoxymethylene, cellulose, and polyisobutylene are all readily degraded upon irradiation. The irradiation of some composite materials such as epoxy/graphite, polyimide/graphite, and polysulfone/graphite fibers have shown that the effects for irradiation up to 5
104 kGy for electron radiation and up to 3,500 kGy for g-radiation are negligible provided the irradiation is carried out in the absence of oxygen [196,197]. Polycarbonates, although they tend to strongly discolor for unstabilized grades, are relatively resistant to irradiation showing retention of elongation at yield and tensile modulus after irradiation up to 1,000 kGy [198].

Polymer Blends The effect of electron beam irradiation on the miscible poly(styrene) and poly(vinyl methyl ether) (PVME) blend has been studied. The poly(styrene), being much more resistant to effects of irradiation, does not offer any protection to the poly(vinyl methyl ether). Gel content studies indicated significant crosslinking [199]. Further studies of this

TABLE 52.26. Yields for gas evolution G(Gas)(104 ) for electron beam irradiation. Polymer

G(H2 )

KaptonE PEEK-c PEEK-a UpilexE-R UpilexE-S

4.8 7.5 12 1.3 2.3

G(N2 )

G(CO)

0.15 – – 0.10 2.9

3.5 3.4 5.2 2.1 1.9

G(CO2 )

G(CH4 )

Dose (MGy)

11 11.3 16 3.4 8.2

0.89 0.16 0.22 0.07 0.27

6.0 5.8 6.0 5.0 5.0

TABLE 52.27. Yields for gas evolution G(Gas)(104 ) for g- irradiation under vacuum. Polymer KaptonE PEEK-c PEEK-a UpilexE-R UpilexE-S

G(H2 )

G(N2 )

G(CO)

G(CO2 )

G(CH4 )

Dose (MGy)

2.1 6.3 12 0.38 8.4

3.6 – – 9.8 13

3.9 12 6.5 2.5 1.8

7.4 5.5 12 5.2 15

0.89 0.14 0.20 0.08 0.30

7.4 8.1 7.4 5.7 8.1

884 / CHAPTER 52 TABLE 52.28. Yields of crosslinking G(X) and scission G(S) for polymeric materials. Polymer

G(X)

G(S)

Reference

Polyoxymethylene Polyisobutylene Cellulose Polyvinylacetate (O2 ) Polyvinylacetate (N2 ) Poly(vinyl ether) Polypropylene oxide (atactic) Polypropylene oxide (isotactic)

6.5 – – 0.3 0.15 5.8 0.15 0.31

11.1 5 11 0.07 0.06 – 0.22 0.51

[190] [191] [192] [26] [193] [194] [195] [195]

polymer blend with gamma irradiation and deuterated PS showed that a significant amount of grafting between the blend components occurred [200]. The gamma irradiation of a PS and PMMA blends showed that the polystyrene did not offer radiation protection for the PMMA. However, in the copolymer, poly(styrene- co-methylmethacrylate), a protective effect from the polystyrene was observed [201]. Some radiation(electron beam and gamma) crosslinking in PS/PMMA has also been reported [202]. A more recent study has shown the effect of gamma irradiation on the glass transition temperature (Tg ) of the miscible blend [203]. Gamma irradiation of the highly miscible poly(vinyl alcohol)/polyacrylamide blends up to 100 kGy has been show to increase the thermal stability of the blend [204]. Recent irradiation studies with blends of PVC and modifiers such as flexible polymers (EVA [205] or ENR –epoxidized natural rubber [206]) or PFMs (polyfunctional monomers) have shown that the irradiation achieves more crosslinking and less degradation (chlorine loss) at lower doses. Seven PFMs, used at 10 parts per hundred rubber (phr), were compared for effectiveness for increasing softening temperature, gel yield and swelling ratio in PVC wire formulations [207]. EVA blends with PE (usually LDPE) have been studied and found to be more sensitive in achieving property improvements at lower doses [208,209]. In one case, a thermoplastic elastomer (TPE) with lower set was formed at < 50 kGy [210,211].

52.4 ADDITIVES The above review of the effects of high energy irradiation on polymeric materials has covered the effects on the ‘‘pure’’ polymer, that is, the materials without the addition of additives except the ones added by the manufacturer, such as antioxidants. With many of the materials discussed above, the effect of high energy irradiation can be dramatically changed by the addition of additives. For example, more efficient crosslink-

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207. H. A. Youssef, Z. I. Ali and A. H. Zahran, Polym. Degrad. Stab. 74, 213, (2001). 208. J. Sharif, S. H. S. A. Aziz and K. Hashim , Radiat. Phys. Chem. 58, 191 (2000). 209. S. M. A. Salehi, G. Mirjalili and J. Amrollahi, J. Appl. Polym. Sci. 92, 1049, (2004). 210. S. Chattopadhyay, T. K. Chaki and A. K. Bhowmick, J. Appl. Polym. Sci. 79, 1877, (2001). 211. S. Chattopadhyay, T. K. Chaki and A. K. Bhowmick, J. Appl. Polym. Sci. 81, 1936, (2001). 212. K. Posselt, Kolloid-Z. Z. Polym. 223, 104, (1968). 213. Y. Okada, Adv. Chem. Ser. 66, 44, (1967). 214. B. J. Lyons, Nature, 185, 604, (1960). 215. A. A. Miller, J. Appl. Polym. Sci. 5, 388, (1961). 216. B. J. Lyons and P. E. Cross, Trans. F. Soc. 59, 2350, (1963). 217. R. G. Bauman, J. Appl. Polym. Sci. 2, 328, (1959). 218. R. G. Bauman and J. W. Born, J. Appl. Polym. Sci. 1, 351, (1959). 219. A. Chapiro, ‘‘Radiation Chemistry of Polymeric Systems,’’ Chapter 12, p. 596, Interscience, New York, 1962.

CHAPTER 53

Flammability Archibald Tewarson FM Global Research, 1151 Boston-Providence Turnpike, Norwood, MA 02062

53.1 53.2 53.3 53.4 53.5 53.6 53.7 53.8 53.9 53.10 53.11 53.12

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fire properties Associated with the Pyrolysis of the Polymer: Heat of Gasification and Surface Re-radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fire Properties Associated with Ignition of the Polymer: Critical Heat Flux and Thermal Response Parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fire Properties Associated with Combustion of the Polymer: Flame Heat Flux, Heat of Gasification, and Surface Re-Radiation Loss . . . . . . . . . . . . . . . . . . . . . . . . Fire Properties Associated with Flame Propagation: Limiting Oxygen Index and Fire Propagation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing Methods for Flame Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fire Properties Associated with the Generation of Products: Yields of Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fire Properties Associated with the Generation of Heat . . . . . . . . . . . . . . . . . . . . . . Fire Properties Associated with Corrosion and Smoke Damage . . . . . . . . . . . . . . . Fire Properties Associated with Fire Suppression/Extinguishment . . . . . . . . . . . . . Standards and Testing of Polymer Products and Materials . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

889 890 892 893 897 898 902 907 909 910 913 923 924

Combustion is a process in which the solid surface of the polymer or the gasified polymer reacts with the oxygen from air with a visible flame (flaming combustion) or without a visible flame (nonflaming combustion). Flame propagation is a process in which the pyrolysis front accompanied by flaming- or nonflaming combustion moves with time beyond the point of origin. Flame extinction is a process where the pyrolysis, ignition, combustion, and fire propagation processes are interrupted by applying agents such as water, inert or chemically active gases, liquids or solids, or reducing the oxygen concentration. Heat and products are generated in pyrolysis, ignition, combustion, and flame propagation processes, presenting hazards to life and property. Hazard due to release of heat (high temperature and radiation) is defined as thermal hazard [1]. Hazard due to release of products is defined as

53.1 INTRODUCTION The flammability of a polymer is an interaction of pyrolysis, ignition, combustion, flame propagation, and flame extinction processes. The processes are brought about by the heat exposure of the polymer. Pyrolysis is an endothermic process and involves softening, melting, discoloration, cracking, decomposition, vaporization, etc. of the polymer and release of pyrolysis products. The boundary of the pyrolysis process on the surface of the polymer is defined as the pyrolysis front. Pyrolysis process is also defined as the gasification of the polymer. Ignition is a process in which the gasified polymer mixes with air, forms a combustible mixture and the mixture ignites by itself (auto-ignition) or is ignited by a flame, a hot object, an electrical spark, etc., (pilotedignition). 889

890 / CHAPTER 53 nonthermal hazard [2]. Nonthermal hazard is due to toxic and corrosive products which interfere in light transmission (reducing visibility) and in electrical operations of delicate electrical components and equipment, and impart discolor and malodor. For the assessment of thermal and nonthermal hazards, fire prevention and protection, several types of models have been developed. All these models use fire properties of polymers associated with pyrolysis, ignition, combustion, and flame propagation as inputs [2,3]. These properties are listed in Table 53.1 and are discussed in this chapter.

53.2 FIRE PROPERTIES ASSOCIATED WITH THE PYROLYSIS OF THE POLYMER: HEAT OF GASIFICATION AND SURFACE RE-RADIATION LOSS The steady state polymer gasification rate is expressed as [2,3]: 00

00

q_
q_ rr m_ f ¼ e , DHg 00

(53:1)

00

where m_ f is the polymer gasification rate or the mass loss 00 rate in pyrolysis (kg=m2
s), q_ e is the external heat flux

TABLE 53.1. Fire properties associated with pyrolysis, ignition, combustion, and fire propagation processes. Fire property

Fire property description

Pyrolysis process Heat of gasification (DHg ) in MJ/kg 00

Surface re-radiation Loss (q_ rr ) in kW=m2 Yield of a product (yj ) in kg/kg Product generation parameter (PGP) Ignition process 00 Critical Heat Flux, CHF, (q_ cr ) in kW=m2 Thermal Response Parameter (TRP) in kJ=m2 for thermally thin polymer and in kW-s1=2 m2 for thermally thick polymer Combustion process 00 Flame heat flux (q_ f ) in kW=m2 Net Heat of Complete Combustion (DHT ) in MJ/kg Chemical Heat of Combustion (DHch ) in MJ/kg Convective heat of combustion (DHcon ) in MJ/kg Radiative heat of combustion (DHrad ) in MJ/kg Yield of a product (yj ) in kg/kg Heat release parameter (HRP) Product generation parameter (PGP) Fire propagation Limiting oxygen index (LOI) Flame propagation rate Fire propagation index (FPI)

Energy required to vaporize a unit mass of the polymer originally at ambient temperature. Heat lost to the environment from the hot surface of the polymer. Amount of a product generated per unit mass of the polymer gasified. Defines product generation rate in non-flaming combustion for a specified heat flux exposure. Minimum heat flux at or below which a flammable vapor-air mixture is not created. It is related to the fire point or ignition temperature. Resistance to ignition and fire propagation.

Heat flux transferred from the flame back to the surface of the burning polymer. Amount of energy released in the complete combustion of a unit mass of the polymer with water as a gas. Amount of energy actually released in the flaming combustion of a unit mass of the polymer. Component of the chemical heat of combustion carried away from the flame by combustion product-air mixture. Component of the chemical heat of combustion transmitted away from the flame by radiation. Amount of a product generated per unit mass of the polymer gasified. Defines heat release rate in the combustion process for a specified heat flux exposure. Defines product generation rate in the combustion process for a specified heat flux exposure. Defines propagating and non-propagating fire behavior.

FLAMMABILITY 00

(kW=m2 ), q_ rr is the surface re-radiation loss (kW=m2 ), and DHg is the heat of gasification (MJ/kg). Pyrolysis experiments are performed where the polymer is heated in an inert environment and various measurements are made. The most widely used techniques are: (1) differential scanning calorimetry, and (2) mass pyrolysis technique. In the differential scanning calorimetry, measurements are made for the specific heat, heats of melting, vaporization, decomposition, etc., in a differential scanning calorimeter and the data are used in the following equation to calculate the heat of gasification, such as for a melting polymer [2]: Z Tm Z Ty DHg ¼ cp,a dT þ DHm þ cp,l dT þ DHy , (53:2) Ta

Tm

/

891

where DHm and DHy are the heats of melting and vaporization at the respective melting and vaporization temperatures in MJ/kg, cp, s and cp, l are the specific heats of the polymer in the solid and molten states in MJ/kg respectively, and Ta , Tm , and Ty are the ambient, melting, and vaporization temperatures in K, respectively. For polymers which do not melt, but sublime, decompose or char, Eq. (53.2) is modified accordingly. The values of the heat of gasification calculated from the differential scanning calorimetry in our laboratory are listed in Table 53.2 [4]. In the mass pyrolysis technique, the mass loss rate is measured as a function of external heat flux in the presence of co-flowing nitrogen or air with an oxygen concentration of 10% by volume, and the data are used in Eq. (53.1). The heat of gasification is determined from the linear regression

TABLE 53.2. Surface re-radiation loss and heat of gasification of polymers.a Heat of gasification (MJ/kg) Polymerb

Surface reradiation loss (kW/m2)

ASTM E 2058 FPA

DSC

Natural polymers Filter paper Corrugated paper Wood (Douglas fir) Plywood/FR Particle board

10 10 10 10 —

3.6 2.2 1.8 1.0 3.9

— — — — —

Synthetic polymers Epoxy resin Polypropylene Polyethylene (low density) Polyethylene (high density) Polyethylene foams Polyethylene/25% Chlorine Polyethylene/36% Chlorine Polyethylene/48% Chlorine Rigid polyvinylchloride (PVC) PVC/plasticizer Plasticized PVC, LOI ¼ 0.20 Plasticized PVC, LOI ¼ 0.25 Plasticized PVC, LOI ¼ 0.30 Plasticized PVC, LOI ¼ 0.35 Rigid PVC, LOI ¼ 0.50 Polyisoprene PVC panel Nylon 6/6 Polyoxymethylene Polymethylmethacrylate Polycarbonate Polycarbonate panel Isophthalic polyester Polyvinyl ester Acrylonitrile-Butadiene-Styrene Styrene-Butadiene Expanded Polystyrene Polystyrene (granular)

— 15 15 15 12 12 12 10 15 10 10 — — — — 10 17 15 13 11 11 16 — — 10 10 10–13 13

2.4 2.0 1.8 2.3 1.4–1.7 2.1 3.0 3.1 2.5 1.7 2.5 2.4 2.1 2.4 2.3 2.0 3.1 2.4 2.4 1.6 2.1 2.3 3.4 1.7 3.2 2.7 1.3–1.9 1.7

— 2.0 1.9 2.2 — — — — — — — — 2.1 2.4 2.3 — — — 2.4 1.6 — — — — — — — 1.8

892 / CHAPTER 53 TABLE 53.2. Continued. Heat of gasification (MJ/kg) Polymerb

Surface reradiation loss (kW/m2)

Expanded polyurethane (flexible) Expanded polyurethane (Rigid) Expanded polyisocyanurate Expanded phenolic Expanded phenolic/FR TefzelT (ETFE) TeflonT (FEP) TeflonT (TFE) TeflonT (PFA) PEEK-30% fiber glass Polyethersulfone-30% fiber glass Polyester1-fiber glass Polyester2-fiber glass Polyester3-fiber glass Polyester4-fiber glass Polyester5-fiber glass Phenolic-fiber glass (thick sheet) Phenolic-Kevlar (thick sheet)

16–19 14–22 14–37 20 20 27 38 48 37 — — — 10 10 15 10 20 15

ASTM E 2058 FPA 1.2–2.7 1.2–5.3 1.2–6.4 1.6 3.7 0.9 2.4 0.8;1.8 1.0 7.9 1.8 2.5 1.4 6.4 5.1 2.9 7.3 7.8

DSC 1.4 — — — — — — — — — — — — — — — — —

a

Data are from the Flammability Laboratory of the FM Global using the ASTM E 2058 FPA shown in Fig. 53.1 and a differential scanning calorimeter. b Abbreviations listed in the nomenclature.

analysis of the data. An external heat flux value at which there is no measurable mass loss rate for 15 min of heat flux exposure is taken as the value for the surface re-radiation loss. The mass pyrolysis technique is used in the ASTM E 2058 FPA, shown in Fig. 53.1, originally designed in our laboratory for this application [4]. Table 53.2 lists the values of the heat of gasification and surface re-radiation loss using the mass pyrolysis technique in our laboratory [4]. The interruption of pyrolysis by passive and/or active fire protection techniques would prevent fires to propagate beyond the ignition zone resulting in reduced fire hazards. The passive fire protection technique involves changes in the polymer to increase the values of the surface re-radiation loss and heat of gasification.

53.3 FIRE PROPERTIES ASSOCIATED WITH IGNITION OF THE POLYMER: CRITICAL HEAT FLUX AND THERMAL RESPONSE PARAMETER Ignition of a polymer involves formation of a flammable vapor air mixture and initiation of combustion on its own (auto-ignition) or assisted by a small heat source (piloted ignition). Minimum heat flux at or below which there is no ignition is defined as the critical heat flux (CHF). The surface of a polymer exposed to heat flux is at a higher temperature than the interior. A polymer with a steep temperature gradient between the surface and the interior is defined as thermally thick. If there is no temperature gradi-

ent between the surface and the interior, the polymer is defined as thermally thin. Thermally thick and thin conditions depend on the actual thickness of the polymer, heating rates, chemical structures of the polymers and additives. The time to ignition and external heat flux satisfy the following relationships [2,3]: 00

1 p q_ e ¼ , tig 4 Kthin

(53:3)

for thermally thin polymers where Kthin ¼ rcp dDTig is defined as the thermal response parameter (TRP) for thermally thin polymers (kJ=m2 ), r is the density of the polymer (kg=m3 ), cp is the specific heat of the polymer (MJ/kg-K), d is the actual thickness of the polymer (m), and Tig is the ignition temperature above ambient (K); and sffiffiffiffiffi rffiffiffiffi 00 1 p q_ e ¼ , (53:4) tig 4 Kthick pffiffiffiffiffiffiffiffiffi for thermally thick polymers where Kthick ¼ DTig krcp is the defined as the TRP for thermally thick polymers (kW
s=1=2 m2 ), and k is the thermal conductivity of the polymer (kW.m-K). The TRP represents resistance of a polymer to generate flammable vapor-air mixture. The CHF and TRP values of the polymers are obtained by using the ignition technique [3,4]. External heat flux at which there is no ignition for 15 min is taken as the CHF. The CHF value is generally close to the value for the surface re-radiation loss. In the ignition technique, time to ignition is measured at various external

FLAMMABILITY

2.07m UPPER SECTION

TWO PORTS-180° APART, ONE & TWO-OPTICAL TRANSMISSION MEASUREMENT THREE PORTS-120° APART, ONE-GAS TEMPERATURE MEASUREMENT, TWO-PRODUCT SAMPLING, THREE-NOT IN USE

0.04

0.14 0.12

0.03 Thermally-Thick

0.10 0.08

0.02

0.06

Thermally-Thin

0.04 0.02

0.01 CHF

0.00 0

3

152mm ID TEFLON COATED STAINLESS STEEL DUCT

20

40

60

80

0.00 100

External Heat Flux (kW/m2)

FIGURE 53.2. Time to ignition versus external heat flux for a 100100 mm10 mm thick silicone based polymer. Data were measured in the ASTM E 2058 FPA. Data satisfy the thermally-thick behavior away from the critical heat flux value.

FOUR INFRA-RED HEATERS

METAL SCREEN PLATFORM ALUMINIUM CYLINDER AIR & OXYGEN

1.46m LOWER SECTION

VERTICAL SLAB 162mm ID, 260mm LONG EXTENSION 162mm ID, 432mm LONG QUARTZ TUBE

ALUMINIUM AIR DISTRIBUTION BOX

893

Time-to-lgnition)−1 (s)−1

THREE PORTS-120° APART, ONE-PARTICULATE SAMPLING, TWO-PRESSURE MEASUREMENT, THREE-CORROSION MEASUREMENT

(Time-to-lgnition)−1/2 (s)−1/2

0.16 BLAST GATE & LINEAR ACTUATOR CONTROL

/

LOAD CELL UNISTRUT STEEL FRAME

FIGURE 53.1. The ASTM E 2058 Fire Propagation Apparatus (FPA).

53.4 FIRE PROPERTIES ASSOCIATED WITH COMBUSTION OF THE POLYMER: FLAME HEAT FLUX, HEAT OF GASIFICATION, AND SURFACE RE-RADIATION LOSS The steady state relationship for polymer gasification rate or mass loss rate is similar to the relationship for the pyrolysis condition [Eq. (53.1)], except for an additional term for the flame heat flux [2,3]: 00

00

heat flux values and the data are used in Eqs. 53.3 and 53.4. A linear relationship between the time to ignition or square root of time to ignition and external heat flux, away from the CHF value [Eqs. (53.3) or (53.4), respectively], is indicative of the thermally thin or thick behavior, such as shown in Fig. 53.2 for a silicone polymer, which behaves as a thermally thick polymer. The TRP value is obtained from the linear regression analysis of the data in the linear portion of the curve, away from the CHF value. The ASTM E 2058 FPA, shown in Fig. 53.1, has been designed to use this technique. The TRP values from the ignition technique are listed in Table 53.3. The values of the ignition temperature, thermal conductivity, and specific heat, which are individual components of TRP, taken from Tewason et al. [7,8] are listed in Table 53.4. The CHF and TRP values depend on the physical and chemical characteristics of the polymers. Increasing the CHF and TRP values of the polymers by various passive protection techniques would delay initiation of combustion and flame would propagate at lower rate or there would be no fire propagation beyond the ignition zone.

m_ f ¼ 00

00

00

q_ e þ q_ f
q_ rr , DHg

(53:5)

where m_ f is the mass loss rate in combustion (kg=m2
s) 00 and q_ f is the flame heat flux transferred back. The values of the heat of gasification and surface reradiation loss determined in pyrolysis are used. The flame heat flux is determined from the flame radiation scaling technique [2,3,13]. This technique utilizes the knowledge that in small-scale fires, flame radiative heat flux increases with increase in the oxygen mass fraction (Y0 ) [2,3,13]. For Y0 $0:30, the flame radiative heat flux reaches an asymptotic limit comparable to the limit for large-scale fires burning in the open [2,3,13]. In the flame radiation scaling technique, mass loss rate is measured with co-flowing air having various oxygen mass fractions. Flame heat flux is calculated by using the mass loss rate data in Eq. (53.5), along with the values of the heat of gasification and surface radiation loss measured in pyrolysis [13]. The convective component of the flame heat flux is determined from the combustion of methanol dominated by convective heat transfer [13]. The flammability

894 / CHAPTER 53 TABLE 53.3. Critical heat flux and thermal response parameter.a Thermal response parameter Critical heat flux (kW=m2 )

Thermally thick (kW-s1=2 =m2 )

Thermally thin (kJ=m2 )

Natural polymers 100% cellulose Tissue paper News paper Wood (red oak) Wood (Douglas fir) Wood (Douglas fir/FR) Wood (hemlock) Corrugated paper Wool 100%

13 13 11 10 10 10 — 13 —

— — — 134 138 251 175 — 252

159 130 175 — — — — 385 —

Nonhalogenated synthetic polymers Epoxy resin Polystyrene Polypropylene Styrene-butadiene Crosslinked polyethylenes Polyvinyl ester Polyoxymethylene Nylon Polymethylmethacrylate Isophthalic polyester Acrylonitrile-butadiene -styrene Polyethylene (high density) Polyethylene/NH- FR Polycarbonate

15 13 15 10 15 — 13 15 11 — 13 15 15 15

457 162 193 198 224–301 263 269 270 274 296 317 321 652–705 331

— — — — — — — — — — — — — —

Halogenated synthetic polymers Isoprene Plasticized PVC, LOI ¼ 0.20 Plasticized PVC, LOI ¼ 0.25 Plasticized PVC, LOI ¼ 0.30 Plasticized PVC, LOI ¼ 0.35 Rigid PVC, LOI ¼ 0.50 Rigic PVC TefzelT (ETFE) TeflonT (FEP)

10 10 10 10 10 10 15 27 38

174 285 401 397 345 388 406 356 682

— — — — — — — — —

Composite systems Polyester-0% fiber glass Polyester1-30% fiber glass Polyester2-70% fiber glass Polyester3-70% fiber glass Polyester4-70% fiber glass Polyester5-70% fiber glass Polyester-77% fiber glass Epoxy-0% fiber glass Epoxy-fiber glass (thin sheet) Epoxy-65% fiber glass Epoxy-76% fiber glass Vinyl ester-0% fiber glass Vinyl ester-62% fiber glass Vinyl ester-69% fiber glass Polyimide-fiber glass PPS-fiber glass PPS-84% fiber glass Bismaleimide-fiber glass

— — 10 10 15 10 — — 10 10 15 — — — — — 20 —

296 256 275 382 406 338 426 257 156 420 667 263 312 444 833 588 909 625

— — — — — — — — — — — — — — — — — —

Polymer

FLAMMABILITY

/

895

TABLE 53.3. Continued. Thermal response parameter Critical heat flux (kW=m2 )

Thermally thick (kW-s1=2 =m2 )

Thermally thin (kJ=m2 )

Phenolic-fiber glass (thin sheet) Phenolic-80% fiber glass Phenolic-fiber glass Epoxy/phenolic-fiber glass PEEK-30% fiber glass Polyethersulfone-30% FG Phenolic-kevlar (thin sheet) Phenolic-84% kevlar Cyanate-73% graphite Epoxy-71% graphite Epoxy-graphite Bismaleimide-graphite PPS-graphite PEEK-graphite Phenolic-graphite

33 20 — 20 — — 20 15 20 24 — — — — —

105 610 345–769 1250 301 256 185 403 1000 667 476–667 526–588 333 526 400–714

— — — — — — — — — — — — — — —

Expanded synthetic polymers Polyurethanes Polystyrenes Phenolics Neoprenes

13–40 10–15 20 16

55–221 111–317 610 113–172

— — — —

Polymers with fiberweb, net-like and multiplex structures Polypropylenes Polyester-polypropylene Wood pulp-polypropylene Polyesters Rayon Polyester-rayon Wool-nylon Nylon Cellulose Cellulose/polyester

8–15 10 8 8–18 14–17 13–17 15 15 13 13–16

— — — — — — — — — —

278–385 139 130 161–303 161–227 119–286 293 264 159 149–217

Polymers as electrical power cable insulation and jackets PVC/PVC PE/PVC Silicone/PVC Silicone/XLPO EPR/EPR EPR,FR/EPR,FR XLPE/XLPE XLPE/EVA XLPE/Neoprene XLPO/XLPO XLPO,PVF/XLPO EPR/CLS-PE

13–25 15 19 25–30 20–23 14–28 20–25 12–22 15 16–25 14–17 14–19

156–341 221–244 212 435–457 467–567 289–448 273–386 442–503 291 461–535 413–639 283–416

— — — — — — — — — — — —

Polymers as communications cable insulation and jackets PVC/PVC PE/PVC XLPE/XLPO Si/XLPO EPR-FR Chlorinated PE ETFE/EVA PVC/PVF

15 20 20 20 19 12 22 30

131 183 461–535 457 295 217 454 264

— — — — — — — —

Polymer

a

Data from the ASTM E 2058 FPA at FMRC [2,3,5–8] or calculated from the data reported in Refs. 9 and 10.

896 / CHAPTER 53 TABLE 53.4. Ignition temperature, thermal conductivity, and specific heats of polymers.a

Polymers Natural polymers Cotton News paper White pine, shavings

Ignition temperature (K)b

Specific heat (kJ/kg-K)

Thermal conductivity (kW=m-K)  104

527 503 533

— — —

— — —

Nonhalogenated synthetic polymers ABS Acetal homopolymer Acrylics Cellulose acetate Epoxy Epoxy/silica Nylon 6/6 Nylon 6/6/33% glass Nylon 6 Nylon 6/30–35% glass Polymethylmethacrylate Polyethylene Low density Medium density High density Polypropylene Polystyrene Polycarbonate Polyester Polyester/premix chopped glass Polyaryl ether Polyether sulfone Phenol-formaldehyde Polyphenylene oxide Polyurethane Styrene-acrylonitrile (SAN) Styrene-butadiene (SB)

527 — — — — — 785 — — — 651

1.26–1.67 1.46 1.46 1.26–2.09 1.05 0.84–1.13 1.67 1.26 1.67 2.09 2.09

1.88–3.35 2.30 1.67–2.51 1.67–3.35 1.67–2.09 4.18–8.37 2.43 2.13 2.43 2.43 2.68

622 — — 736 675 651 — — — — — — — — 645

2.30 2.30 2.30 1.92 1.34 1.17–1.26 1.17–2.30 1.05 1.46 1.09 1.59–1.76 1.34 1.67–1.88 1.34–1.42 1.88–2.09

3.35 3.35–4.18 4.60–5.19 1.17 1.00–1.38 1.92 1.76–2.89 4.18–6.69 2.98 1.34–1.84 1.26–2.51 1.88 0.63–3.10 1.21–1.26 1.51

Halogenated synthetic polymers PVC PVC2 PTFE FEP PVF2 PCTFE

675 — 767 900 — —

1.34 1.34 1.05 1.17 1.38 0.92

1.25–2.93 1.26 2.51 2.51 1.26 1.97–2.22

— — — — — —

— — — — — —

2.00 10.5 17.2 50.2 240 850

Inert fibers for composite systems Kevlar Glass Quartz Graphite Sapphire (aluminum oxide) Silicone carbide a b

Data taken from Refs. 11 and 12. Estimated from the CHF value in Table 53.3.

apparatus, shown in Fig. 53.1, has been designed to use this technique. The asymptotic values for the mass loss rate in combustion and flame heat flux determined from the radiation

scaling technique in the ASTM E 2058 FPA are listed in Table 53.5. The measured asymptotic values of the mass loss rate in combustion in large-scale fires reported in the literature are also listed in Table 53.5. The asymptotic flame

FLAMMABILITY

/

897

TABLE 53.5. Asymptotic mass loss rate and flame heat flux. Mass loss rate (kg=m2 -s)  103

Flame heat flux (kW=m2 )

Flame rad. Scaling techb

Largescale

Flame rad. scaling techb

Largescale

Aliphatic carbon-hydrogen atoms Polyethylene Polypropylene Heavy fuel oil (2.6–23 m) Kerosene (30–80 m) Crude oil (6.5–31 m) n-Dodecane (0.94 m) Gasoline (1.5–223 m) JP-4 (1.0–5.3 m) JP-5 (0.60–17 m) n-Heptane (1.2–10 m) n-Hexane (0.75–10 m) Transformer fluids (2.37 m)

26 24 — — — — — — — 66 — 27–30

— — 36c 65c 56c 36c 62c 67c 55c 75c 77c 25–29

61 67 — — — — — — — 32 — 23–25

— — 29 29 44 30 30 40 39 37 37 22–25

Aromatic carbon-hydrogen atoms Polystyrene (0.93 m) Xylene (1.22 m) Benzene (0.75–6.0 m)

36 — —

34 67c 81c

75 — —

71 37 44

Aliphatic carbon-hydrogen-oxygen atoms Polyoxymethylene Polymethylmethacrylate (2.37 m) Methanol (1.2–2.4 m) Acetone (1.52 m)

16 28 20 —

— 30 25 38c

50 57 22 —

— 60 27 24

21–27 22–25

— —

64–76 49–53

—

16 14 7

— — —

50 50 52

Polymers/Liquidsa

Aliphatic carbon-hydrogen-oxygen-nitrogen atoms Expanded Polyurethanes (flexible) Expanded Polyurethanes Rigid Aliphatic carbon-hydrogen-halogen atoms Polyvinylchloride TefzelT (ETFE) TeflonT (FEP) a

Numbers in parentheses are the pool diameters in meters. Flame radiation scaling technique: pool diameter fixed at 0.10 m, Y0 $0:30. ASTM E 2058 FPA. c Taken from various references in the literature. b

heat flux values, determined in the ASTM E 2058 FPA are in good agreement with the values derived from the mass loss rate in large-scale fires. The asymptotic flame heat flux values vary from 22 to 77 kW=m2 , dependent primarily on the pyrolysis mode rather than on the chemical structures. For examples, for the liquids, which vaporize primarily as monomers or as very low molecular weight oligomer, the asymptotic flame heat flux values are in the range of 22---44 kW=m2 , irrespective of their chemical structures. For polymers, which vaporize as high molecular weight oligomer, the asymptotic flame heat flux values increase substantially to the range of 49 to 71 kW=m2 , irrespective of their chemical structures. The independence of the asymptotic flame heat value from the chemical structure is consistent with the dependence of

the flame radiation on optical thickness, soot concentration and flame temperature. Decrease in the flame heat flux and increase in the heat of gasification and surface re-radiation loss through various passive fire protection techniques would prevent the fire to grow and propagate beyond the ignition zone and the thermal and nonthermal hazards would be reduced and/or eliminated.

53.5 FIRE PROPERTIES ASSOCIATED WITH FLAME PROPAGATION: LIMITING OXYGEN INDEX AND FIRE PROPAGATION INDEX Flame propagation is a process where the pyrolysis front moves beyond the ignition zone over the polymer surface,

898 / CHAPTER 53 accompanied by the sustained combustion process. The rate of the movement of the pyrolysis front, accompanied by the sustained combustion process, is defined as the flame propagation rate. For a sustained fire propagation process, flame or external heat sources need to transfer heat flux ahead of the pyrolysis front to satisfy the CHF and TRP values. Flame propagation can occur in the downward, upward, and horizontal directions. Three test apparatuses and methods have been developed to determine the fire properties associated with flame propagation: (1) the ASTM D 2863 oxygen index test method for downward flame propagation for small samples [14]; (2) the ASTM E 1321-90 lateral ignition and flame spread (LIFT) test method for horizontal and lateral flame propagation [15,16]; and (3) the fire propagation index (FPI) test method for vertical flame propagation [2,3,17,18]. In the LIFT and FPI test methods, the following definition of the flame propagation velocity for thermally thick polymers is utilized [2,3,7,15]: 00

u¼

funct(q_ f ) [DTig2 krcP ]

,

crease in the gas temperature as indicated by the LOI values of the composite systems in Table 53.6. The oxygen index test utilizes the flame radiation scaling technique for small samples and indirectly assesses heat flux from the flame through LOI. At or below the LOI value of a polymer, the heat flux requirements for CHF and TRP values for fire propagation are not satisfied. The higher is the LOI of a polymer, higher are its CHF and TRP values and/or lower is the heat flux provided by its flame, and the polymer is considered as fire hardened. The oxygen index test is used for molded polymers, fabrics, expanded polymers, thin films, polymers which form char, drip, or soften, and for liquids. The data are reproducible. The test is used to study polymer combustion chemistry, fire retardant treatment of the polymers and for screening the polymers. No relationships have been established between LOI and the flame heat flux, CHF, TRP, and fire propagation rate. The application of the oxygen index test data to predict the fire propagation behavior of polymers expected in actual fires is thus uncertain.

(53:6) 00

where u is the flame propagation rate in m/s, funct (q_ r ) is a function representing the flame heat flux transferred to the surface of the polymer ahead of the pyrolysis front (kW2 =m3 ), r is the density of the polymer (kg=m3 ), cp is the specific heat of the polymer (MJ/kg-K), k is the thermal conductivity of the polymer (kW/m-K) and DTig is the ignition temperature above ambient (K) [see the definition of the TRP for thermally thick polymer in Eq. (53.4)].

53.6 TESTING METHODS FOR FLAME PROPAGATION 53.6.1 The ASTM D 2863 Oxygen Index Test In this test, downward flame propagation for small vertical sheets (6.5-mm wide, 70–150-mm long, 3-mm thick) is examined, in air flowing in the opposite direction with variable oxygen concentration [14]. Minimum oxygen concentration (volume percent) at or below which the downward flame propagation cannot be sustained, defined as the limiting oxygen index (LOI), is determined [14]. The LOI values reported in the literature [8,19] are compiled in Table 53.6. For PMMA, LOI¼17.3 in Table 53.6 which is higher than the oxygen concentration of 16.0% required for flame extinction for larger PMMA slabs [6]. The difference is probably due to differences in the flame radiation and flow characteristics. For example, for larger PMMA slabs exposed to external heat flux values of 40, 60, and 65 kW=m2 in the ASTM E 2058 FPA, flame extinction occurs at oxygen concentrations of 13.0%, 12.0%, and 11.5%, respectively [6]. The LOI value decreases with in-

53.6.2 The ASTM E 1321-90 Lateral Ignition and Flame Spread (LIFT) Test Equation (53.6) is expressed as [15]: u¼

C , TRP2

(53:7)

where C is defined as the flame heating parameter (kW2 =m3 ). The ignition and flame spread tests are performed in normal air at various external heat flux values [15,16]. In the ignition tests, 155-155-mm samples are exposed to various external heat flux values and times to flame attachment are measured [15,16]. The values of k, r, cp , DTig are determined from the relationship between the time to flame attachment and external heat flux [15,16]. These values can be used to calculate the TRP value [Eq. (53.4)]. In the flame spread tests, 155-mm wide and 800-mm long horizontal samples in a lateral configuration are used [15,16]. The samples are exposed to an external heat flux which is 5 kW=m2 higher than the CHF value in the ignition zone [15,16]. Beyond the ignition zone, the external heat flux decreases gradually and is significantly lower than the CHF value at the end of the sample [15,16]. The sample is preheated to thermal equilibrium and ignited with a pilot flame in the ignition zone. The pyrolysis front is tracked as a function of time and is used to determine the flame heating parameter and used in Eq. (53.7) along with the TRP value to calculate the flame propagation rate. The flame propagation rate calculated from the data reported in Refs. 15 and 19 are listed in Table 53.7. The relative flame propagation rate is also listed in Table 53.7. In the LIFT Apparatus, most of the common polymers and carpets have faster lateral flame propagation than

FLAMMABILITY

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TABLE 53.6. Limiting oxygen indices for polymers.a Polymer Cotton Cotton (loosely woven) Filter paper Wood (birch) Wood (red oak) Wood (plywood) Cellulose Cellulose acetate (dry) Cellulose acetate (4.9% water) Cellulose butyrate (0.06% water) Cellulose butyrate (2.8% water) Cellulose acetate-butyrate Rayon Wool (loosely woven) Wool fiber (dry cleaned) Leather (chrome based) Natural rubber foam Polyacetal (CelconT) Polyacetal-30% FG Polyformaldehyde Poly(ethylene oxide) Polyoxymethylene (DelrinT) Polyphenylene oxide Polyurethane foam Polyvinyl alcohol Polysulfone PVC fiber PVC (rigid) PVC (chlorinated) Poly(vinyl fluoride) (TedlarT) Polyethylene (20% chlorine) Neoprene Neoprene rubber Polyisoprene Poly(vinylidene chloride) (SaranT) Polytrichlorofluoroethylene TeflonT (TFE) NomexT Polyester fabric Polyester Polyester-70% fiber glass; (heated to 8C) 25 100 200 300 Epoxy3–65% fiber glass, (heated to 8C) 25 100 200 300 Phenolic resin Phenol-formaldehyde resin Phenolic-80% fiber glass, (heated to 8C) 25 100 a

Data taken from Refs. 8 and 19.

LOI

Polymer

LOI

16–17 18.5 18.2 20.5 23.0 23.0 19.0 16.8 18.1 18.8 19.9 19.6 18.7–18.9 23.8 25.2 34.8 17.2 14.9 15.6 15.0 15.0 14.9 29.9 16.5 22.5 30.0–32.0 37.1 45.0–49.0 45.0–60.0 22.6 24.5 40.0 26.3 18.5 60.0 95.0 95.0 28.5 20.6 41.5

Polyethylene1 Polyethylene2 Polyethylene-50% Al2 O3 Polypropylene Polypropylene-30% FG Polystyrene1 Polystyrene2 Polymethylmethacrylate(PlexiglasT) Polycarbonate1 Polycarbonate2 Polycarbonate3 ABS-1 ABS-2 ABS-20% FG SBR foam Nylon fiber Nylon-6,6 Nylon-6,6 Nylon-6,12 Nylon-6 Polyacrylonitrile Polyimide (KaptonT) Silicon rubber Polyester2–70% fiber glass, 8C 25 100 200 300 Polyester3–70% fiber glass, 8C 25 100 200 300 Epoxy resin Epoxy1–65% fiber glass, 8C 25 100 200 300 Epoxy2–65% fiber glass, 8C 25 100 200 300

17.4 17.4 19.6 17.4 18.5 17.8 17.6–18.3 17.3 22.5 24.9 26.0–28.0 18.3–18.8 18.8 21.6 16.9 20.1 24.3 24.0–29.0 25.0 25.0–26.0 18.0 36.5 30.0

23.0 23.0 nylon (car bon-hydrogen-oxygen-nitrogen atom aliphatic bonds)>PE and PP (carbon-hydrogen atom aliphatic bonds)>wood (carbon-hydrogen-oxygen atom aliphatic bonds)>PMMA (carbon-hydrogen-oxygen atom aliphatic bonds). This order is opposite to the order for CO, but is expected on the basis of the fundamental understanding of the smoke formation in the combustion of the polymeric materials. The concentration predictions can be used to define the experimental conditions in various toxicity, corrosion, and smoke damage evaluation tests. The correlations for the concentration predictions can be combined with various toxicity, corrosion, and smoke damage relationships, as inputs to the hazard assessment models. 53.8 FIRE PROPERTIES ASSOCIATED WITH THE GENERATION OF HEAT The heat release rate is directly proportional to the mass loss rate and the proportionality constant is defined as the heat of combustion: 00 00 Q_ i ¼ DHi m_ f , 00

(53:14)

where G_ i is the heat release rate (kW=m2 ), DHi is the heat 00 of combustion (kJ/kg), and m_ f is the mass loss rate (kg=m2 -s). The rate of heat release in a combustion process, within the flame, is defined as the chemical heat release rate. The chemical heat released within the flame is carried away from the flame by flowing product-air mixture and is emitted to the environment as radiation. The component of the chemical heat release rate carried away by the flowing products-air mixture is defined as the convective heat release rate. The component of the chemical heat release rate emitted to the environment is defined as the radiative heat release rate. The heat of combustion is defined respectively as the chemical, convective, and radiative heat of combustion. The chemical heat release rate is determined from the carbon dioxide generation (CDG) and oxygen consumption (OC) calorimetries [2,3]. In the CDG calorimetry, the chemical heat release rate is determined from the mass generation rate of CO2 corrected for CO [2,3]. In the OC calorimetry, the chemical heat release rate is determined from the mass consumption rate of O2 [2,3,24]. The convective heat release rate is determined from the gas temperature rise

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907

(GTR) calorimetry [2,3,25]. The radiative heat release rate is determined from the difference between the chemical and convective heat release rates [2,3]. 53.8.1 The CDG Calorimetry The chemical heat release rate is determined from the following relationships: 00  _ 00  _ 00 Gco2 þ DHco Gco , Q_ ch ¼ DHco 2

 DHco ¼ 2

 DHco

DHT , Cco2

 DHT
DHco Cco ¼ , Cco

(53:15) (53:16)



(53:17)

00

 where Q_ ch is the chemical heat release rate (kW=m2 ), DHco2 is the net heat of complete combustion per unit mass of CO2  generated (MJ/kg), DHco is the net heat of complete combustion per unit mass of CO generated (MJ/kg), DHT is the net heat of complete combustion per unit mass of fuel consumed (kJ/g), Cco2 is the stoichiometric yield of CO2 00 (kg/kg), Cco is the stoichiometric yield of CO (kg/kg), G_ co2 00 is the generation rate of CO2 (kg=m2 -s) and G_ co is the generation rate of CO (kg=m2 -s).   The values of DHco2 and DHco for over 200 fuels are tabulated in Tewarson [2]. The values depend on the chemical structures of the fuels. With some exceptions, the values remain approximately constant within each generic group of fuels. For approximate calculations, the average values can  be used, which are: DHco2 ¼ 13:3 kJ=g  11%, and  DHco ¼ 11:1 kJ=g  18%. In the CDG calorimetry, the CO correction for wellventilated combustion is very small, because of the small amounts of the CO generated. The variations of 11% and   18% in the DHco2 and DHco values, respectively, would reduce significantly if values for low molecular weight hydrocarbons with small amounts of O, N, and halogen were used in averaging. For the determination of the chemical heat release rate, mass generation rates of CO2 and CO are measured and   actual values of DHco2 and DHco are used for accuracy or the average values for approximate results. The CO2 and CO measurement details are described in Tewarson [2].

53.8.2 The OC Calorimetry The chemical heat release rate is determined from the following relationship: 00 00 Q_ ch ¼ DH0 C_ 0 ,

DH0 ¼

DHT , C0

(53:18) (53:19)

908 / CHAPTER 53 where DH0 is the net heat of complete combustion per unit 00 mass of oxygen consumed (MJ/kg), C_ 0 is the mass consumption rate of oxygen (kg=m2 -s) and C0 is the stoichiometric mass-oxygen-to-fuel ratio (kg/kg). The values of DH0 for over 200 fuels are tabulated in Tewarson [2]. The values depend on the chemical structures of the fuels. With some exceptions, the values remain approximately constant within each generic group of fuels. For approximate calculations, the average value can be used, which is: DH0 ¼ 12:8 kJ=g  7%. The variation of + 7% would reduce significantly if values for low molecular weight hydrocarbons with small amounts of O, N, and halogen were used in averaging. For the determination of the chemical heat release rate, mass consumption rate of O2 is measured and the actual value of DH0 is used for accuracy or the average value for approximate results. The O2 measurement details are described in Tewarson [2]. The chemical heat release rates determined from the CDG and OC calorimetries are very similar.

of combustion obtained in this fashion are listed in Table 53.9. The radiative heat of combustion is obtained from the difference between the chemical and convective heats of combustion, as heat losses are negligibly small in the ASTM E 2058 FPA. 53.8.5 Heat Release Rate at Various Heat Fluxes The heat release rate can be predicted at various heat flux values from the following relationship obtained from Eqs. (53.5) and (53.14):   DHi 00 00 00 00 (53:22) (q_ e þ q_ f
q_ rr ), Q_ i ¼ DHg

53.8.3 The GTR Calorimetry

where DHi =DHg is defined as the heat release parameter (HRP) (kJ/kJ). HRP values are independent of the fire size, but depend on the ventilation and can be calculated from the data such as listed in Tables 53.2 and 53.9 or from the slopes of the lines obtained by plotting the heat release rates against the external heat flux. The heat release rate can be calculated 00 from the HRP and q_ rr values for the fire scenario for specified external and flame heat flux values.

The convective heat release rate is determined from the following relationship:

53.8.6 Generation of Heat and Ventilation

W_ cp (Tg
Ta ) 00 Q_ con ¼ , A

(53:20)

00

where Q_ con is the convective heat release rate (kW=m2 ), cp is the specific heat of the combustion product-air mixture at the gas temperature (MJ/kg-K), Tg is the gas temperature _ is the total mass flow (K), Tg is ambient temperature (K), W rate of the fire product-air mixture (kg/s), and A is the total exposed surface (m2 ). For the determination of the convective heat release rate, temperature and total mass flow rate of the fire-products air mixture are measured. The literature value of the specific heat of air at the gas temperature is used as the fire products are diluted by fresh air by about 20 times their volume. The temperature and mass flow rate measurement details are described in Tewarson [2]. 53.8.4 Heat of Combustion The average heat of combustion is determined from the ratio of the energy, Ei , obtained from the summation of the chemical, convective, and radiative heat release rates and the total mass of gasified polymer, Wf , obtained from the summation of the mass loss rate: DHi ¼

Ej , Wf

(53:21)

where DHi is the average chemical, convective, or radiative heat of combustion (MJ/kg). The values of the average heat

The relationship between heat release rate or heat of combustion and the equivalence ratio is expressed as [21]:   a 00 00 _ _ Qi,y ¼ Qi,1 1
, (53:23) exp bF
j  DHi,y ¼ DHi,1

 a 1
, exp bF
j

 wy ¼ w1 1
00

 a , exp bF
j

(53:24)

(53:25)

where Q_ i,y is the heat release rate, DHi,y is the heat of combustion, and wy is the combustion efficiency for the 00 ventilation-controlled combustion (kW=m2 ), Q_ i,1 is the heat release, DHi,1 is the heat of combustion, and w1 is the combustion efficiency for the well-ventilated combustion (kW=m2 ), and a, b and j are the ventilation correlation coefficients. Combustion efficiency is the ratio of the chemical heat release rate or chemical heat of combustion to the heat release rate for complete combustion or net heat of complete combustion. For the nonhalogenated polymers, a ¼ 0:97, b ¼ 2:5, and j ¼ 1:2 for the chemical heat release rate or the chemical heat of combustion and a ¼ 1:0, b ¼ 2:5, and j ¼ 2:8 for the convective heat release rate or the convective heat of combustion. For the halogenated polymers, a ¼ 0:30, b ¼ 0:001, and j ¼ 11 for the chemical heat release rate or the chemical heat of combustion. Chemical heat release rate, chemical heat

FLAMMABILITY of combustion, and combustion efficiency decrease with ventilation restriction or increase in the equivalence ratio. Figure 53.5 shows the combustion efficiency calculated from Eq. (53.25). As expected, combustion efficiency decreases with increase in the equivalence ratio due to limitation in the availability of air. For the nonhalogenated polymers, flame extinction occurs for combustion efficiency between about 0.20 and 0.40. The halogenated polymer burns with a low combustion efficiency; a slight decrease in the combustion efficiency (below about 0.30) results in flame extinction, although combustion remains well ventilated. The combustion efficiency decreases rapidly with increase in the equivalence ratio for the low molecular fuel (natural gas, methane) compared to the polymers, which gasify as higher molecular weight oligomer. 53.9 FIRE PROPERTIES ASSOCIATED WITH CORROSION AND SMOKE DAMAGE

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909

Corrosion is measured by exposing metal surfaces to the flowing fire products in the sampling duct of the ASTM E 2058 FPA. The change in resistance due to corrosion is measured as a function of time such as shown in Fig. 53.6. ˚ /min). In The slopes of the lines represent corrosion rates, (A Fig. 53.6, corrosion rate is faster for the polyester-PVC-fiber glass sample than it is for the fire-retarded (FR) polypropylene (PP) sample. PP polymer sample shows negligible corrosion as it does not contain halogen atoms. The total mass loss of the polymer lost (g), total volumetric flow rate of the mixture of fire products and air mixture (m3 =s) and combustion duration (s) are measured in the experiments. The CI values for various polymers have been determined in the flammability apparatus; Table 53.12 lists values for some selected polymers, as examples. The CI values for nonhalogenated polypropylene and wood are negligibly small. For the highly halogenated polymers, the CI value is

53.9.1 Corrosion Damage 500

CI ¼ {dloss =Dtexposure }={Wf =V_T Dttest },

(53:26)

˚ /min)/(mg/g), dloss is the metal loss due to where CI is in (A ˚ corrosion (A), Dtexposure is the time the corrosive product deposit is left on the surface of the metal (min), Wf is the total mass of the polymer lost in the experiment (g), V_T is the total volumetric flow rate of the mixture of fire products and air (m3 =s) and Dttest is the combustion duration (s).

400 Plyester-PVC-fiber glass

Corrosion (A)

The fire property associated with corrosion damage is defined as corrosion index (CI), which is the corrosion rate ˚ /min)/(g of per unit mass concentration of the products [(A 3 polymer gasified products/m of air flow)] [2,3]:

300

200

FR polypropylene-inert PVC-plasticizers

100

0

Polypropylene polymer

0

300

600

900

1200

1500

Time ( minutes) 1.0 Non–Flaming

0.9 PMMA

Combustion Efficienty

0.8 0.7

PP,PE,Nylon

TABLE 53.12. Corrosion index for selected polymers.a

0.6 0.5 0.4 0.3

PS, Wood

Natural Gas, NIST

Materials

0.0 10–1

Corrosion index [(A˚/min)/(g polymer gasified/m3 of air)]

PVC

0.2 0.1

FIGURE 53.6. Corrosion of thin copper strip on a fiber glass polyester plate by the flowing combustion products -air mixture in the sampling duct of the ASTM E 2058 FPA.

Flaming

Transition

100

101

Equivalence Ratio

FIGURE 53.5. Calculated combustion efficiency versus the equivalence ratio. For the calculations, Eq. (53.25) and data from Table 53.9 and [2] were used. For natural gas, the data measured at the National Institute of Standards and Technology (NIST) as reported in [22] were used.

Polyvinylchloride (PVC)b-1 PVC-2 PVC-3 PVC-4 Polypropylene Polypropylene/fire retardant TeflonT (TFE) Wood a

1.8 0.78 0.60 0.36 0.074 1.7 0.28 0.088

Determined in the ASTM E 2058 FPA at the Factory Mutual Research Corporation. b Amount of nonhalogenated additive increasing from 1 to 4.

910 / CHAPTER 53 high if hydrogen atoms are present in the structure (PVC) or halogenated fire-retardant additive is present (PP/FR), and is low if there are no hydrogen atoms in the structure (Teflon1 TFE). The difference in the CI values indicate the importance of water as a combustion product to generation acids for PVC and PP/FR. Teflon1 (TFE) does not generate water as a product of combustion and thus the formation of an acid (HF) would depend on the efficiency of the hydrolysis process between the ambient water from air and Teflon1 (TFE) vapors. The hydrolysis process appears to be inefficient. The CI value decreases with increase in the amount of non-halogenated additive (PVC-1 to -4). 53.9.2 Smoke Damage The fire property associated with smoke damage is the ratio of the yield of smoke-to-the yield of CO2 . The ratio increases with increase in the equivalence ratio or ventilation restriction. The yield of smoke is proportional to the smoke generation rate and the yield of CO2 is proportional to the chemical heat release rate. The higher the ratio of the yield of smoke to the yield of CO2 , higher the damage due to smoke relative to the damage due to heat. Smoke is a mixture of black carbon (soot) and aerosol [26,27]. It has been suggested that soot nucleation and growth occur near the highly ionized regions of the flames in combustion processes, and that some of the charges are transferred to smoke particles. Multimodal distributions show that the soot particle radii belong to three ‘‘modes’’ [26]: 1. ‘‘Nuclei mode’’ has a geometric mean radius between 0.0025 and 0:020 m and probably results from the condensation of gaseous carbon moieties. 2. ‘‘Accumulation mode’’ encompasses particles in the size range 0:075---0:25 m and apparently results from the coagulation and condensation of the ‘‘nuclei mode’’ particles. 3. ‘‘Coarse mode’’ at several microns that is attributed to the precipitation of fine particles on the walls of vehicle exhaust systems and a subsequent entrainment in the issuing gases. In fires, large variations in smoke particle size are due to coagulation and condensation. Data from various fires show that initially the smoke particles are in the coarse mode. As the smoke moves away from the fire origin, large particles settle down to the floor, leaving small particles having radii of 0:04---0:09 m (accumulation mode). It thus appears that smoke damage in the room of fire origin is expected to be due to particles of several microns in radius in the coarse mode, whereas smoke damage downstream of the fire is expected to be due to particles with radius less than 0:1 m in the lower end of the accumulation mode. Soot is an efficient absorber of HCl. In the combustion of 79.5%

PVC-20.5% PE, 19 mg of HCl/g of smoke is loosely bound and 27 mg of HCl/gm of smoke is tightly bound to soot [28]. Smoke damage in industrial and commercial occupancies is considered in terms of discoloration and odor of the property exposed to smoke, interference in the electric conduction path and corrosion of the parts exposed to smoke is a carrier of the corrosive products.

53.10 FIRE PROPERTIES ASSOCIATED WITH FIRE SUPPRESSION/EXTINGUISHMENT Several fire properties are associated with fire suppression/extinguishment by active and passive fire protection techniques. Changes in the values of the properties are used to assess the effectiveness of the techniques. Passive fire protection techniques enhance resistance to: (1) pyrolysis, ignition, combustion, and fire propagation processes, and (2) generation of heat and products. Active fire protection techniques provide hinderance to the growth of the fire by: (1) interacting with the burning polymer in the solid phase (mainly removal of heat) [29]; (2) reducing the availability of the oxygen to the fire (creation of nonflammable mixture); and (3) removal of heat from the flame and interference with the chemical reactions within the flame [30]. 53.10.1 Passive Fire Protection Passive fire protection is provided by various chemical and physical means. Increasing the Resistance to Ignition and Fire Propagation by Increasing the Values of CHF and TRP The CHF and TRP values can be increased by modifying the pertinent parameters such as the chemical bond dissociation energy and thermal diffusion (combination of the density, specific heat and thermal conductivity). Decreasing the Values of the HRP and the Flame Heat Flux The heat release rate is equal to the Heat Release Parameter (HRP) times the net heat flux [Eq. (53.22)]. Decrease in the HRP value would decrease the heat release rate. The HRP value can be decreased by decreasing the heat of combustion and/or increasing the heat of gasification by various chemical and physical means. An examination of the data in Table 53.9 for heat of combustion show that introduction of oxygen, nitrogen, sulfur, halogen, and other atoms into the chemical structures of the polymers reduces the heat of combustion. For example, the heat of combustion decreases when the hydrogen atoms attached to

FLAMMABILITY carbon atoms in polyethylene are replaced by the halogen atoms, such as by fluorine in TFE. The chemical heat of combustion decreases from 38.4 MJ/kg to 4.2 MJ/kg and the HRP value decreases from 17 to 2 (Table 53.9). The HRP values can also be reduced by increasing the heat of gasification and decreasing the heat of combustion by retaining the major fraction of the carbon atoms in the solid phase, a process defined as charring. Several passive fire protection agents are commercially available to enhance the charring characteristics of materials. The effect on flame heat flux by passive fire protection is determined by using the radiation scaling technique, where combustion experiments are performed in oxygen concentration higher than the ambient values. As discussed previously, liquids which vaporize primarily as monomers or as very low molecular weight oligomer, the flame heat flux values are in the range of 22---44 kW=m2 , irrespective of their chemical structures. For solid polymers, which vaporize as high molecular weight oligomer, the flame heat flux values increase substantially to the range of 49---71 kW=m2 , irrespective of their chemical structures. Passive fire protection agents which can reduce the molecular weight of the pyrolysis products of the polymers would be effective in reducing the flame heat flux and complement the active fire protection agents.

Changing the Melting Behavior of Materials The chemical heat release rate increases very rapidly as a polymer changes from a solid to a boiling liquid pool, creating dangerous conditions and presenting a serious challenge to the active fire protection agents. Inert passive fire protection agents added to the polymer which would eliminate the boiling liquid pool would be effective in complementing the active fire protection agents.

Decreasing the Value of the Product Generation Parameter (PGP) Nonhalogenated passive fire protection agents which reduce or eliminate the release of halogenated and highly aromatic products and enhance release of aliphatic products, rich in hydrogen and oxygen atoms but poor in carbon atoms, would be effective in reducing the nonthermal damage due to smoke and corrosion. Some of the passive fire protection agents, available commercially, interact with the polymers in the solid as well as in the gas phase during pyrolysis and combustion. The critical parameter that needs to be examined in the presence and absence of the passive fire protection agents is the ratio of PGP (smoke, CO, corrosive and toxic products) to HRP. The effectiveness of the passive fire protection agent would be reflected in the small values of the ratios at fire control, suppression, and/or extinguishment stage.

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53.10.2 Active Fire Protection Active fire protection is provided by applying agents as liquids, gases, solid powders, or foams to the flame and/or to the surface of the burning polymers.

Flame Suppression/Extinguishment by Liquid Vapors and Gaseous Agents A flame will extinguish when the time required for the chain reaction which sustains the combustion process exceeds the time it takes to replenish the necessary heat and reactants [30]. The most commonly used liquid and gaseous chemical inhibition agents at the present time are: Halon1 1211 (CBrClF2 ), 1301 (CBrF3 ), and 2402 (CBrF2 CBrF2 ). Because of the contribution of Halons to depletion of the stratospheric ozone layer, they will, however, not be used in the future [30]. There is thus an intense effort underway to develop alternative fire suppressants to replace ozone layer depleting Halons. The Halon alternatives belong to one of the following classes: (1) Hydrobromofluorocarbons (HBFC); (2) Chlorofluorocarbons (CFC); (3) Hydrochlorofluorocarbons (HCFC); (4) Perfluorocarbons (FC); (5) Hydrofluorocarbons (HFC); and (6) Inert gases and vapors. The most common test to screen the Halon alternates is the ‘‘cup burner’’ test, where concentration of Halons or alternates required for extinction of a small laminar diffusion flame are determined. Table 53.13 lists the concentrations of Halon 1301 and alternates required for heptane flame extinction in the ‘‘cup burner’’ test. Acceptable total flooding agents in normally occupied areas are indicated by bold letters and numbers. When the amount of an agent applied to a burning polymer is close to the amount required for flame extinction, first flame instability sets in, followed by flame liftoff from the surface and finally the flame is extinguished, as indicated in Fig. 53.7 for the flame extinction of PMMA by Halon 1301. Initially there is a rapid decrease in the chemical heat release rate as Halon is added to the flame. There is a gradual increase in the chemical heat release rate between 5.40% and 6.25% of Halon unto flame extinction. The increase in the chemical heat release rate appears to be due to increase in the flame luminosity (increase in the flame radiative heat flux transferred back to the fuel surface). Figure 53.8 shows that the generation efficiencies of CO, mixture of hydrocarbons, and smoke increase significantly with increase in the Halon concentration. The effect of Halon on the generation efficiencies is strong for CO and the mixture of hydrocarbons and weak for smoke. This type of combustion behavior of PMMA is similar to one found with the ventilation controlled combustion, i.e., increasing preference of fuel carbon atom to convert to CO and the mixture of hydrocarbons rather than to smoke. It thus appears that the chemical interruption processes for flame extinction by Halon and reduced oxygen are very similar.

912 / CHAPTER 53 TABLE 53.13. Concentrations of halon 1301 and alternates required for flame extinction in the heptane ‘‘cup burner’’ test.a Agent Name

Formula

Nitrogen Carbon dioxide Helium Argon Silicone Containing Agent Silicone tetrafluoride Sodium Containing Agent Sodium bicarbonate (10---20 mm) Halon Halon 1301 HFC HCFC-22 (Du Pont FE 232) HBFC-22B1 (Great Lakes FM 100) HFC-23 (Du Pont FE13) HFC-32 HCFC-124 HBFC-124B1 HFC-125 (Du Pont FE 25) HFC-134 HFC-134a HFC-142b HFC-152a HFC-218 HFC-227ea (Great Lakes FM 200) Trifluoromethyl Iodide 1311 FC-14 FC-116 C318 FC-5-1-14 (3M PFC 614)

Relative concentration

Inert Agents N2 CO2 He Ar

32 23 31 41

11.0 7.9 10.7 14.1

SiF4

36

12.4

NaHCO3

3:0b

CF3 Br

2.9

CHCIF2 CHBrF2 CHF3 CH2 F2 CHCIFCF3 CF3 CHFBr3 CF3 CHF2 CHF2 CHF2 CF3 CH2 F CCIF2 CH3 CHF2 CH3 CF3 CF2 CF3 CF3 CHFCF3 CF3 I CF4 CF3 CF3 C4 F8 C4 F10

— 1.0 4.00b 1.52 4.28 3.03 2.83 0.97 3.24 3.86 3.62 3.79 9.31 2.10 2.10b 1.03 4.76 2.69 2.52 1.90b

11.6 4.4 12.4 8.8 8.2 2.8 9.40 11.2 10.5 11.0 (calc) 27.0 (calc) 6.1 6.1 3.0 13.8 7.8 7.3 5.5

From Refs. 30 and 31. Acceptable total flooding agents in normally occupied areas [32].

0

200

400 600 800 Time (second)

1000

5.97% 5.40% 4.77%

0.30

Hydrocarbons CO

0.20

4.33%

Generation Efficiency

Flame Extinction

100

0

0.40

6.25% Flame Liftoff

200

Flame Instability

5.40%

300

5.97%

4.77%

400

4.33%

Chemical Heat Release Rate (Kw/m2)

OC CDG

6.25%

0.50 500

Flame Extinction

a b

Concentration (volume %)

0.10 Smoke

0.00

1200

FIGURE 53.7. Chemical heat release rate versus time for the combustion of 100 mm  100 mm  25 mm thick horizontal slab of polymethylmethacrylate exposed to 40 kw=m2 in coair flow with varying Halon 1301 concentration at a velocity of 90 mm/s in the ASTM E 2058 FPA. Numbers and their locations represent Halon 1301 concentrations in volume percents and application times. Times for flame instability, liftoff, and extinction are also indicated.

0

200

400 600 800 Time (second)

1000

1200

FIGURE 53.8. Generation efficiencies of CO, mixture of hydrocarbons, and smoke versus time for the combustion of 100 mm  100 mm  25 mm thick horizontal slab of polymethylmethacrylate exposed to 40 kw=m2 in co-air flow with varying Halon 1301 concentration at a velocity of 90 mm/s in the ASTM E 2058 FPA. Numbers and their locations represent Halon 1301 concentrations in volume percents and application times. Times for flame instability, liftoff, and extinction are also indicated.

FLAMMABILITY This experimental finding is consistent with the concept that a critical value of the Damkohler number for flame extinction [30]. The existence of the critical conditions at flame extinction has also been postulated by the ‘‘Fire Point Theory’’ [29] and supported by the experimental data for the critical mass pyrolysis and heat release rates [2]. The fire property associated with flame extinction by gaseous agents thus would be the critical value of the HRP.

Flame Suppression/Extinguishment by Liquids At the flame extinction condition, the critical mass pyrolysis and heat release rates can be expressed as: 00

00

m_ cr ¼

00

00

00

q_ e þ q_ f
q_ rr
q_ w , DHg

(53:27)

  DHch 00 00 00 00 00 _ (q_ e þ q_ f
q_ rr
q_ w ), Qcr ¼ DHg

(53:28)

00

where m_ cr is the critical mass pyrolysis rate for flame 00 extinction (kg=m2 -s), q_ e is the external heat flux 00 (kW=m2 ), q_ f is the flame heat flux transferred back to the 00 surface (kW=m2 ), q_ rr is the surface re-radiation loss 00 (kW=m2 ), DHg is the heat of gasification (kJ/kg), Q_ cr is the critical value of the chemical heat release rate for flame extinction (kW=m2 ), DHch is the chemical heat of combus00 tion (kJ/kg), DHch =DHg is the HRP, and q_ w is the heat flux removed from the surface of a burning polymer by a liquid such as water, as a result of vaporization expressed as: 00

00

q_ w ¼ «w m_ w DHw ,

(53:29) 00

where «w is the water application efficiency, m_ w is the water application rate per unit surface area of the polymer (kg=m2 -s), and DHw is the heat of gasification of water (2.58 MJ/kg). If only part of the water applied to a hot surface evaporates and the other part forms a puddle, such as on a horizontal surface, blockage of flame heat flux to the surface and escape of the fuel from the polymer surface are expected. Eq. 53.27 thus is modified as: 00

00

q_ w ¼ m_ w («w DHw þ dw ),

(53:30)

where dw is the energy associated with the blockage of flame heat flux to the surface and escape of the fuel vapors per unit mass of the fuel (MJ/kg). From Eqs. 53.27 and 53.30, with no external heat flux: 00

00

m_ w ¼

00

00

m_ cr DHg q_ f
q_ rr
: «w DHw þ dw «w DHw þ dw

(53:31)

The first term on the right-hand side takes into account the effects of the physical differences on flame extinction such as the fire size and polymer shape, size, and arrangement. The second term on the right-hand side takes into account

/

913

the effects of the chemical differences on flame extinction such as the chemical structures of the polymers and additives. In large-scale fires, the second term is negligibly small and water application rate required for flame extinction depends mainly on the flame heat flux, surface re-radiation loss, and mode of water application. The critical values of the mass pyrolysis rate, heat release rates, and water application rates for flame extinction for polymers, are listed in Table 53.14. For the polymers listed in the table, the critical values of the heat release rates do not depend on the generic natures of the polymers. The average critical values of the chemical, convective, and radiative heat release rates are 100+7, 53+9, and 47+10 kW=m2 , respectively. The critical water application rate required for flame extinction is: polyoxymethylene, polymethylmethacrylate and polyethylene with 25% chlorine (2:1---2:5 g=m2 -s)<polyethylene and polypropylene (3:5---4:1 g=m2 -s)<polystyrene (5:1 g=m2 -s). 53.11 STANDARDS AND TESTING OF POLYMER PRODUCTS AND MATERIALS Polymer products are used in a variety of applications in residential, private, government, industrial, transportation, and manufacturing occupancies. Consequently, for the assessment of the fire hazards of polymer products large numbers of fire scenarios need to be considered for testing. To avoid this problem of large number of fire scenarios to be considered for testing, two types of standard test methods have been developed: 1. Test methods to comply with specific regulations or voluntary agreements: these types of test methods are usually larger than laboratory-scale tests and are included in the prescriptive (specification)-based fire codes1. Generally, products in their end-use configurations are tested under a defined fire condition. 2. Small-scale standard test methods: these types of test methods have been developed based on qualitative experiences as well as on the understanding of fire stages and associated hazards. Relatively simple types of measurements are made for various fire properties of the polymeric materials for each fire stage. These types of standard test methods are useful for the performancebased fires codes which are being considered to augment or replace the prescriptive-based fire codes2 [33–35]. Both types of standard test methods for products in their end-use configurations and polymeric materials used for the construction of products are promulgated by various 1

The codes reflect expectations for the level of fire protection. An example of the prescriptive-based code for passive fire protection is the specified fire resistance rating for an interior wall, whereas for the performance-based code it would be a prediction for the desired passive fire protection based on the engineering standards, practices, tools, and methodologies. 2

914 / CHAPTER 53 TABLE 53.14. Critical mass pyrolysis, heat release, and water application rate.a Critical values for flame extinction m_ cr (kg=m2 -s)  103

Q_ ch (kW=m2 )

00 Q_ con (kW=m2 )

00 Q_ rad (kW=m2 )

00 W_ w (kg=m2 -s)  103

Polyoxymethylene Polymethylmethacrylate Polyethylene Polypropylene Polystyrene

4.5 3.2 2.5 2.7 4.0

(65) 77 96 104 108

50 53 55 61 44

(14) 24 42 43 64

2.3 2.5 3.8 3.0 5.1

Polyethylene foams 1 2 3 4 Average

2.6 2.6 2.5 2.6 2.6

— — — — 88

— — — — 51

— — — — 38

— — — — 3.8

Chlorinated polyethylenes 25% chlorine 36% chlorine 48% chlorine

6.6 7.5 7.6

95 — —

48 — —

47 — —

2.1 — —

Expanded polystyrene GM47 GM49 GM51 GM53 Average

6.3 4.9 6.3 5.7 5.8

— — — — 108

— — — — 44

— — — — 64

— — — — 5.1

Polyurethane foams (flexible) GM21 GM23 GM25 GM27 1=CaCO3 Average

5.6 5.3 5.7 6.5 7.2 6.1

— — — — — 101

— — — — — 48

— — — — — 53

— — — — — —

Polyurethane foams (rigid) GM29 GM31 GM35 Average

7.9 8.4 6.9 7.7

— — — 102

— — — 44

— — — 58

— — — —

Polyisocyanurate foams (rigid) GM41 GM43 Phenolic foam

6.8 5.5 5.5

— — —

— — —

— — —

— — —

Polymers

a

00

00

From the data measured in the ASTM E 2058 FPA at FMRC; -: no data or considered in the average data.

national and international standards organizations and government and private agencies, for example the following [36,37]. 1. Australia (Standards Australia, SA); 2. Canada (Canadian General Standards Board, CGSB); 3. China-People’s Republic (China Standards Information Center, CSIC); 4. China-Republic of China-Taiwan-(Bureau of Standards, Metrology and Inspection, BSMI);

5. Europe (International Electrotechnical Commission, IEC; European Committee for Electrotechnical Standardization, CENELEC; European Committee for Standardization, CEN, International Standards Organization, ISO); 6. Finland (Finnish Standards Association, SFS); 7. France (Association Europeene Des Constructeurs De Materiel Aerospatial, AECMA; Association Francaise De Normalisation, AFNOR);

FLAMMABILITY 8. Germany (Deutsches Institut Fur Normung, DIN); 9. India (Indian Standards Institution, ISI); 10. Israel (Standards Institution of Israel, SII); 11. Italy (Ente Nazionale Italiano Di Unifacazione, UNI); 12. Japan (Japanese Standards Association, JSA); 13. Korea (Korean Standards Association, KSA); 14. New Zealand (Standards New Zealand, SNZ); 15. Nordic Countries (Nordtest: Denmark, Finland, Greenland, Iceland, Norway, and Sweden); 16. Russia (Gosudarstvennye Standarty State Standard, GOST); 17. South Africa (South African Bureau of Standards, SABS); 18. United Kingdom (British Standards Institution, BSI; Civil Aviation Authority, CAA); 19. USA (examples of government agencies: department of transportation, DOT; military-MIL; National Aeronautical and Space Administration, NASA. Examples of private agencies: American National Standards Institute, ANSI, American Society for Testing and Materials, ASTM; Building Officials & Code Administrators International Inc., BOCA; Electronic Industries Alliance, EIA; FM Approvals; Institute of Electrical and Electronics Engineers, IEEE; National Fire Protection Association, NFPA; Underwriters Laboratories, UL). Each national and international standards organization, government, and private industries from each country, listed above and others, use their own standard test methods for the evaluation of the products and materials. Consequently, there are literally thousands of standard test methods used on a worldwide basis [37–41]. The national and international standards organizations list their test methods in catalogues for standards such as: the European Committee for Standardization, CEN [42], FM Approvals [43], Underwriter’s Laboratories (UL) [44], International Standards Organization (ISO) [45], American Society of Testing and Materials (ASTM) [46] and others. Because of the use of thousands of standard testing methods, products accepted in one country may be unacceptable in the other, creating confusion and serious problems for the manufacturers and fire safety regulator. Vigorous efforts are thus being made, especially in Europe, to harmonize the standard test methods3. Recently the European Commission’s, single burning item (SBI) and reaction to fire classification [42] is the best example of harmonizing hundreds of 3

ISO, IEC, Nordtest, CEN, US Federal Aviation Administration’s (FAA) standards criteria are internationally acceptable for regulations, and others.

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European standard testing methods for building products into a single standard test method. The single burning item test method (EN 13823) for testing the fire safety of construction products will be widely used by the manufacturers to allow for the affixing of ‘‘C’’ marking that will indicate compliance with the ‘‘Essential Requirements of the Union Directive 89/ 106/EEC’’. In addition, new regulations, Euroclasses4, and test methods designated EN ISO, are in a process of being introduced that will be used throughout Europe [47,48]. Further harmonization is expected as many regulatory agencies are considering augmenting or replacing the prescriptive-based fire codes (currently in use) by the performance-based fire codes. In the performance-based fire codes, engineering methods are used that need data for the fire properties [33–35]. The data for the fire properties can be obtained from many standard test methods currently in use worldwide by modifying the test procedures and data acquisition methodology. Since fire properties will be measured quantitatively, the standard test methods will be automatically harmonized worldwide and the assessment for the fire resistance of materials and products will become reliable, as it will be subject to quantitative verification. Following sections describe some commonly used standard test methods: 53.11.1 Standard Tests for the Ignition Behavior of Polymer Materials Standard test methods have been developed for examining the ignition behavior of polymeric materials. Some test methods provide qualitative data, while others provide partial or complete quantitative data for the ignition resistance of materials (Section 53.3, Tables 53.3 and 53.4). The following are examples of the common standard test methods used for examining the ignition resistance of materials: 1. ISO 871 (Tig , ignition temperature in the hot oven) [45]; 2. ASTM D 1929 (Tflash , flash ignition temperature) and Tig (spontaneous) [46]; 3. ASTM E 1352 (qualitative-cigarette ignition of upholstered furniture) [46]; 4. ASTM E 1353 (qualitative-cigarette ignition resistance of components of upholstered furniture) [46]; 5. ASTM F 1358 (qualitative-effects of flame impingement on materials used in protective clothing not designed primarily for flame resistance) [46]; 6. ASTM C 1485 (CHF value of exposed attic floor insulation using an electric radiant heat energy source) [46]; 4 There are seven main Euroclasses for building materials for walls, ceiling, and floors: A1, A2, B, C, D, E, and F [47,48]. A1 and A2 represent different degrees of limited combustibility. B to E represent products that may go to flashover in a room within certain times [47,48]. F means that no performance is determined [47,48]. Thus there are seven classes for linings and seven class for floor coverings [47,48]. There are additional classes of smoke and any occurrence of burning droplets [47,48].

916 / CHAPTER 53 7. ASTM E 648 (CHF value of floor covering systems using a radiant heat energy source) [46]; 8. ASTM E 1321 and ISO 5658 (CHF and TRP values) [45,46]; 9. ASTM E 1354 and ISO 5660 (CHF and TRP values) [45,46]; 10. ASTM D 1929 (Tig values) [46]; 11. ASTM E 2058 (CHF and TRP values) [46]. Tests performed in the apparatuses specified in the three standards listed above, i.e., ASTM E 1321/ISO 5658 (LIFT apparatus), ASTM E 1354/ISO 5660 (cone calorimeter), and ASTM E 2058 (fire propagation apparatus) provide complete set of fire properties for the assessment of ignition behavior of polymer products. These apparatuses also provide data in a format that is useful for the engineering methods in the performance-based fire codes. Examples of the data for CHF and TRP values are listed in Table 53.3. Polymer products with high CHF and TRP values have high resistance to ignition. 53.11.2

Standard Tests for the Combustion Behavior of Polymer Materials

The burning behaviors of polymeric materials are examined by measuring the release rates of material vapors, heat, and chemical compounds including smoke in the apparatuses specified in the standard test methods. From these measurements, the following fire properties are derived: 1. Heat of gasification and heat losses (Table 53.2);

following these standard test methods as well by the ASTM E 662 (smoke density Chamber) [46]. ASTM D 5865/ISO 1716: Test Method for Gross Heat of Complete Combustion [45,46] This standard test method incorporates the fundamental principles for the energy associated with the complete combustion of materials and thus is independent of fire scenarios [49]. Gross and net of complete combustion of materials are used in the performance-based fire codes for the assessment of fire hazards associated with the use of products and protection needs. The gross heat of complete combustion is measured in the oxygen bomb calorimeter. In Europe, gross heat of complete combustion (gross calorific potential, PCS), measured by following the ISO 1716 standard test method is used for the classification of reaction to fire performance for construction products (prEN 13501-1) [42]: 1. Construction products excluding floorings: – Class A1: PCS # 1.4–2.0 MJ/kg. – Class A2: PCS # 3.0–4.0 MJ/kg. 2. Floorings: – ClassA1fl : PCS # 1.4–2.0 MJ/kg. – ClassA2fl : PCS # 3.0–4.0 MJ/kg. The gross heat of complete combustion is used to determine the net heat of complete combustion5 defined as the quantity of energy released when a unit mass of specimen is burned at constant pressure, with all the combustion products, including water, being gaseous.

2. Chemical, convective, and radiative heats of combustion (ratio of the summation of the heat release rate to the summation of the release rate of material vapors) (Table 53.9);

ASTM E 136/ISO 1182: Standard Test Method for Behavior of Materials in a Vertical Tube Furnace at 750 8C [45,46]

3. Yields of various chemical compounds (ratio of the summation of the release rate of each compound to the summation of the release rate of material vapors, Table 53.9).

This standard test method specifies the use of a small-scale apparatus to assess the noncombustibility behavior of building construction materials under the test conditions. The standard test apparatus consists of two concentric, vertical refractory tubes, 76-mm and 102-mm (3 and 4-in.) in inside diameter and 210 to 250-mm (8.5–10-in.) in length. Electric heating coils outside the larger tube are used to apply heat. A controlled flow of air is admitted tangentially near the top of the annular space between the tubes and passes to the bottom of the inner tube. The top of the inner tube is covered. Temperatures are measured by thermocouples at the center: (1) between the two concentric tubes, (2) close to specimen location, and (3) sample surface.

4. Combustion efficiency (ratio of the heat of combustion to the net heat of complete combustion); 5. Generation efficiency of chemical compounds (ratio of the yield of a compound to the maximum possible stoichiometric yield of the compounds based on the elemental composition of the material). The heat of complete combustion is measured according to ASTM D 5865/ISO 1716 test methods [45,46]. The release rates of material vapors, heat, and various chemical compounds (including smoke) are measured according to ASTM E 906 (the Ohio State University Heat Release Rate, OSU-HRR, Apparatus), ASTM E 2058 (fire propagation apparatus) and ASTM E 1354/ISO 5660 (cone calorimeter) [45,46]. Smoke released in flaming and nonflaming fires of materials is also characterized

5 If the percentage of hydrogen atoms in the sample is known: net heat of complete combustion in kJ/g ¼ gross heat of complete combustion in kJ/g
0.2122  mass percent of hydrogen atoms, where heats of combustion are in kJ/g [46]. If the percentage of hydrogen atoms is not known: net heat complete of complete combustion in kJ/g ¼ 10. 025 þ (0.7195) gross heat of combustion in kJ/g [46].

FLAMMABILITY Test specimens are used in granular or powdered form contained in a 38-mm  38-mm  51-mm holder. The specimen in the holder is placed in the center of the inside vertical refractory tube after the temperature at the specimen location is maintained at 750  5:5 C for 15 min. The test is continued until all the temperatures have reached their maximum values. Visual observations are made throughout the test on the specimen behavior, combustion intensity, smoke formation, melting, charring, etc. The specimen is weighed before and after the test. The data measured in the test are used to assess the following specimen behaviors: 1. Weight loss, Dm # 50%; 2. Surface and interior temperature, DT # 30 8C;

/

917

to heat in a natural gas or propane fueled furnace with temperature increasing in the following fashion: 5 min 538 8C 10 min 704 8C 30 min 843 8C 1h 927 8C 2 h 1,010 8C 4 h 1,093 8C $ 8 h 1,260 8C The standard test method has been designed to test the following building materials and assemblies in the furnace6: 1. Bearing and nonbearing walls and partitions: the area exposed to fire is $ 9-m2 (100-ft2 ) with neither dimension less than 2.7-m (9-ft); 2. Columns: the length of the column exposed to fire is $ 2.7-m (9-ft);

3. There is either no flaming, i.e., flaming duration, tf ¼ 0, or there is no flaming after the first 20 seconds, tf #20 s.

3. Protection for structural steel columns: the length of the protected column is $ 2.4-m (8-ft) held in a vertical orientation. The column is exposed to heat on all sides;

In Europe, data from ISO 1182 are used for the classification of reaction to fire performance for construction products (prEN 13501-1) [42]:

4. Floors and roofs: the area exposed to fire is $ 16-m2 (180-ft2 ) with neither dimension $ 3.7-m (12-ft);

1. Construction products excluding floorings: – Class A1: DT#30  C, Dm # 50 %, and tf ¼ 0. – Class A2: DT#30  C, Dm # 50 %, and tf #20 s. 2. Floorings: – Class A1fl : DT#30  C, Dm # 50 %, and tf ¼ 0. – Class A2fl : DT#30  C, Dm # 50 %, and tf #20 s. ASTM E 906, ASTM E 2058, and ASTM E 1354/ISO 5660: Standard Test Methods for Release Rates of Material Vapors, Heat, and Chemical Compounds [45,46] These standard test methods specify the use of small-scale apparatus to quantify the fire properties of materials. The apparatuses specified are: 1. ASTM E 906 (the OSU-HRR Apparatus); 2. ASTM E 2058 (the fire propagation apparatus, FPA); 3. ASTM E 1354/ISO 5660 (cone calorimeter).

5. Loaded restrained and unstrained beams: the length of the beam exposed to fire $ 3.7-m (12-ft) and tested in a horizontal position; 6. Protection for solid structural steel beams and girders: the length of beam or girder exposed to the fire is $ 3.7-m (12-ft) tested in a horizontal position; 7. Protective members in walls, partition, floor, or roof assemblies: the sizes used are same as above for the respective specimens. Various criteria are used for the acceptance of the specimens: 1. Sustains itself or with the applied load without passage of flame or gases hot enough to ignite cotton waste or the hose assembly for a period equal to that for which classification is desired; 2. There is no opening that projects water from the stream beyond the unexposed surface during the time of water hose stream test;

One of the standard test apparatuses (the FPA is shown in Figs. 53.1).

3. Rise in the temperature on the unexposed surface remains # 139 8C above its initial temperature;

ASTM E 119: Standard Test Methods for Fire Tests of Building Construction and Materials-The Fire Endurance Test [46]

4. Transmission of heat through the protection during the period of fire exposure for which classification is desired maintains the average steel temperature # 538 8C (measured temperature # 649 8C);

This standard test method specifies use of large-scale furnace for testing of walls, columns, floors, and other building members, under high fire exposure conditions. Fire resistance is expressed in terms of time to reach critical point, i.e., ‘‘1/2-h (hour)’’, ‘‘2-h’’, ‘‘6-h’’, and other ratings of building materials and assemblies as they are exposed to heat. The building materials and assemblies are exposed

5. For steel structural members (beams, open-web steel joists, etc), spaced more than 1.2-m (4-ft), the average temperature of steel # 593 8C (measured temperature # 704 8C) during the classification period. 6 As needed, load is applied to the specimens throughout the test to simulate a maximum load condition in their end use application.

918 / CHAPTER 53 ASTM E 1529: Standard Test Methods for Determining Effects of Large Hydrocarbon Pool Fires7 on Structural Members and Assemblies [46] The standard test method specifies large-scale test, similar to ASTM E 119, except that exposure of specimens consist of rapidly increasing heat flux. In the test, specimen surface is exposed to an average heat flux exposure of 158 kW=m2  8 kW=m2 attained within the first 5-min and maintained for the duration of the test. The temperature of the environment reaches $ 815 8C after the first 3-min of the test and remains between 1010 8C and 1180 8C at all times after the first 5-min. This standard test method is used to determine the response of columns, girders, beams or structural members, and fire-containment walls, or either homogeneous or complete construction exposed to rapidly increasing heat flux. In this standard test method, combination of heat flux and temperature for the control is specified compared to ASTM E 119, where only temperature is specified. Performance is defined as the period during which structural members or assemblies will continue to perform their intended function when subjected to fire exposure. The results are reported in terms of time increments such as 1/2-h (hour), 3/4-h, 1-h, 1.5-h, and others. The tests are performed in a fashion similar to that in the ASTM E 119, except for the heat flux and temperature profiles. For example, in this standard test method, a heat flux exposure of 158 kW=m2 to the specimen surface is specified within first 5-min of the test. In the ASTM E 119, a heat flux exposure of 35 kW=m2 at 5-min and 118 kW=m2 at 60-min to the specimen surface is specified. In this standard test method, conditions are simulated to test the performance of structural members and assemblies exposed to fire conditions resulting from large, freeburning (outdoors), fluid-hydrocarbon-fueled pool fires. This information is needed for the design of facilities for the hydrocarbon processing industry (oil refineries, petrochemical plants, offshore oil production platforms, and others) and chemical plants. In the future, this information may also be used in the design of high rise buildings because of the extreme terrorist act that occurred in New York City on September 11, 2001. There was a complete collapse of the World Trade Center Towers due to exposure to very hot pool fires from the large spillage of aviation gasoline.

53.11.3 Standard Tests for the Flame Spread Behavior of Polymer Materials In the standard test methods, specifications are made for making visual observations for movement of flame and char during the test and measurements for the surface temperature and release rates of material vapors, heat, and chemical compounds, including smoke. Both small-scale and largescale flame spread and fire growth tests are performed using materials and products. Following are some of the popular standard test methods for characterizing flame spread and fire growth behaviors of materials and products. The following are some of the popular standard test methods for the flame spread behaviors of the polymeric materials.

prEN ISO/FDIS 11925-2: Reaction to Fire Tests for Building Products-Part 2: Ignitability When Subjected to Direct Impingement of Flame [42,45] The apparatus consists of a stainless steel 800-mm high, 700-mm long, and 400-mm wide chamber with an exhaust duct attached at the top of the chamber. In the test, 250-mm long and 180-mm wide specimen with thickness # 60-mm is used. The specimen is placed in a holder consisting of a double U-shaped frame made from 15-mm wide and 5-mm thick stainless steel sheets hanging vertically inside the stainless steel chamber. The holder is 370-mm long and 110-mm wide with a 80-mm wide open mouth. The specimen is placed between two halves of the holder that are held together by screws or clamps. The holder can move closer to or away from a 458-propane gas burner (similar to a Bunsen burner). A 100-mm  50-mm  10-mm deep aluminum foil tray containing filter paper is placed beneath the specimen holder and replaced between the tests. The flame from the burner is applied for 15 or 30 s and the burner is retracted smoothly. For 15 s flame application, the test duration is 20 s after flame application. For 30 s flame application time, the test duration is 60 s after flame application. The following observations are made in the test: 1. Ignition of the specimen; 2. Fs: Flame spread up to150-mm and time taken; 3. Presence of flaming droplets; 4. Ignition of the filter paper below the specimen.

7 A large pool fire is defined as that resulting from hundreds (or thousands) of gallons of liquid hydrocarbon fuel burning over a large area (several hundred to thousand square meters) with relatively unrestricted airflow and release of chemical compounds. A range of temperatures, velocities, heat fluxes, and chemical conditions exist and vary dramatically with time and spatial location.

In Europe, data from ISO 11925-2 are used for the classification of reaction to fire performance for construction products (prEN 13501-1) [42]: 1. Construction products excluding floorings: – Class B: Fs # 150-mm within 60 s for 30-s exposure. – Class C: Fs # 150-mm within 60 s for 30-s exposure.

FLAMMABILITY – Class D: Fs # 150-mm within 60 s for 30-second exposure. – Class E: Fs # 150-mm within 20 s for 15-s exposure 2. Floorings – Class Bfl: Fs # 150-mm within 20 s for 15-s exposure. – Class Cfl: Fs # 150-mm within 20 s for 15-s exposure. – Class Dfl: Fs # 150-mm within 20 s for 15-s exposure. – Class Efl: Fs #150-mm within 20 s for 15-s exposure.

UL 94: Standard Test Methodology for Flammability of Plastic Materials for Parts in Devices and Appliances [44] This standard test method is similar to prEN ISO/FDIS 11925-2 test. In the test, both horizontal burning (HB) and vertical burning (V) behaviors of 127-mm (5-in) long, 13-mm (0.5-in) wide, and up to 13-mm (0.5-in) thick material samples are examined. Horizontal burning test is performed for 94HB classification of materials. The sample used is placed on top of a wire gauge and ignited by a 30-s exposure to a Bunsen burner at one end. The material is classified as 94HB if over 76-mm (3.0-in) length of sample: (1) flame spread rate < 38.1 mm/min for 3–13-mm thick sample and < 76-mm/min for 3-mm thick sample or flame spread is < 102-mm (4.0-in.). Vertical burning test is performed for the 94V-0, 94V-1, or 94V-2 classification of materials. The bottom edge of the sample is ignited by a 5-s exposure to a Bunsen burner with a 5-s delay and repeated five times until the sample ignites. The 94V-0, 94V-1, and 94V-2 material classification criteria are listed in Table 53.10. The relative resistance of materials to flame spread and burning according to UL94 is HB < V-2 < V-1 < V-0. The ordinary polymeric materials, which generally have low fire resistance, are classified as HB. Most of the high temperature and halogenated polymeric materials, that generally have high fire resistance, are classified as V-0.

ASTM D 2863 (ISO 4589): Test Methodology for Limited Oxygen Index [46] The test is described in Section 53.6.1. The LOI values and UL 94 classification of materials are interrelated. The LOI values for V-0 materials are $ 35, whereas the LOI values are < 30 for materials classified as V-1, V-2, and HB. The standard test method has not been developed to predict the fire behavior of materials expected in actual fires, bur rather to screen materials for low and high resistance to fire propagation. For the majority of high temperature and highly halogenated materials, the LOI values are $ 40. These polymers have high resistance to ignition, combustion, as well as fire spread, independent of fire size and ignition source strength.

/

919

ASTM E 162 (D 3675): Standard Test Method for Surface Flammability Using a Radiant Energy Source [46] In this small-scale test method, 460-mm (18-in.)  150mm (6-in.) wide and up to 25-mm (1-in.) thick vertical sample is used. The sample is exposed to a temperature of 670 + 4 8C at the top from a 300-mm (18-in.)  300mm (12-in.) inclined radiant heater with top of the heater closest to and the bottom farthest away from the sample surface. The sample is ignited at the top and flame spreads in the downward direction. In the test, measurements are made for the arrival time of flame at each of the 75mm (3-in.) marks on the sample holder and the maximum temperature rise of the stack thermocouples. The test is completed when the flame reaches the full length of the sample or after an exposure time of 15-min, whichever occurs earlier, provided the maximum temperature of the stack thermocouples is reached. Flame spread index (Is) is calculated from the measured data, defined as the product of flame spread factor, Fs, and the heat evolution factor, Q. Many polymeric materials and products have been tested using this standard test method. The Is values vary from 0 to 2,220, suggesting large variations in the fire spread behavior of materials. Many regulations and codes specify the Is value as an acceptance criterion of materials and products. For example, for structural composites inside naval submarines and for passenger cars and locomotive cabs [50,51], the following Is values are specified for the acceptance of the materials: 1. Is < 20 for structural composites inside naval submarines; 2. Is # 25 for cushions, mattresses, and vehicle components made of flexible cellular foams for passenger cars and locomotive cabs and thermal and acoustic insulation for buses and vans; 3. Is # 35 for all vehicle components in passenger cars and locomotive cabs and for seating frame, seating shroud, panel walls, ceiling, partition, windscreen, HVAC ducting, light diffuser, and exterior shells in buses and vans; 4. Is # 100 for vehicle light transmitting polymers in passenger cars and locomotive cabs. The above listed criteria for the Is values (< 20) suggest that structural composites for inside naval submarines are expected to have high resistance to flame spread and heat release if exposed to heat flux values similar to those used in the ASTM E 162. Also, materials used in passenger cars, locomotive cabs, buses, and vans with Is values # 25 as well as # 35 are expected to have relatively higher resistance to fire spread and heat release rate compared to the ordinary materials with Is values # 100 under low heat exposure conditions.

920 / CHAPTER 53 ASTM E 1321 (ISO 5658): Standard Test Method for Determining Material Ignition and Flame Spread Properties (Lateral Ignition and Flame Spread Test, LIFT) [45,46] This standard test method is discussed in Section 53.6.2. The test has been developed to provide pertinent data needed by the prescriptive-based and performance-based fire codes for the fire hazard analyses and protection needs for residential, private, government and industrial occupancies, transport and manufacturing, and others. ASTM E 648 (ISO 9239-1): Standard Test Method for Critical Radiant Flux of Floor-Covering Systems Using a Radiant Heat Energy Source [45,46] This standard test method specifies the use of an intermediate-scale test method for testing, similar in principle to ASTM E 1321 (ISO 5658). A 1.0-m (39.4-in.) long and 0.20-m (7.9-in.) wide horizontal sample is exposed to radiant heat flux in the range of 1–11 kW/m2 from a 308-inclined radiant panel all contained inside a chamber. The heat flux is at 11 kW/m2 at the sample surface that is closer to the radiant heater. The radiant flux decreases as the distance between the sample surface and the radiant heater increases to the lowest value of 1 kW/m2. A pilot flame ignites the sample surface exposed to 11 kW/m2, and flame spread is observed until the flame is extinguished at some downstream distance due to decrease in the radiant flux. The radiant flux at this distance is defined as the critical radiant flux (CRF) of the sample: 00

00

CRF ¼ q_ cr
q_ f (x)

(53:32)

00

where q_ f (x) is the flame heat flux at distance x where flame is extinguished (kW/m2). Thus, materials and products for which radiant fraction of the flame heat flux is higher would have lower CRF values. Materials with higher radiant fraction of the flame heat flux have lower resistance to flame spread due to efficient heat transfer ahead of the flame front. This test method was developed as a result of need for flammability standard for carpets and rugs to protect the public against fire hazards [52]. Consequently, several carpet systems were tested by this standard [52–54]. This standard test method (ASTM E 648) is specified for the classification of the interior floor finish in buildings in the NFPA 101 Life Safety Code [55]: 1. Class I: interior floor finish: CRF > 4.5 kW/m2; 2. Class II: interior CRF280 kW over a period of 30 s; 2. Sampling duct temperature >400 8C at any instant or >300 8C over a period of 30 s; 3. Material falling onto the sandbox burner substantially disturbs the flame of the burner or extinguishes the burner by choking. The test data are used to obtain the following parameters to rank the fire performance of the specimens: 1. FIGRA index, 2. SMOGRA index9, 3. THR600s : total heat released within 600 s, 4. TSP600s : total smoke released within 600 s, 5. LFS: lateral flame spread, 6. Flaming/nonflaming droplets/particles and ignition of the paper10 (prEN ISO 11925–2). In Europe, data from prEN 13823 are used for the classification of reaction to fire performance for construction products (prEN 13501–1) [42]: 1. Construction products excluding floorings: – Class A2: FIGRA # 120 W/s; LFS < edge of specimen, THR600s # 7:5 MJ, smoke production and melting/burning drops. – Class B: FIGRA # 120 W/s; LFS < edge of specimen, THR600s # 7:5 MJ, smoke production and melting/ burning drops. – Class C: FIGRA # 250 W/s; LFS < edge of specimen, THR600s # 15 MJ, smoke production and melting/ burning drops – Class D: FIGRA # 750 W/s, smoke production and melting/burning drops. The use of this standard test method for regulatory purposes is very similar to that of the ASTM E 84 standard test method. The intent of this standard test method is to separate materials and products with higher flame spread resistance from those with lower resistance. It has not been designed to predict the flame spread behavior of materials and products in actual fires.

3. Horizontal flame spread observed visually, i.e., time taken to reach the extreme edge of the main 1.5-m  1.0-m sample panel; 4. Falling molten droplets and particles. The performance of the specimen is evaluated over a period of 20 min. However, the test is terminated earlier if any of the following conditions occur:

9

2

s1 ¼ SMOGRA # 30 m2 =s and TSP600s # 50 m2 ; s2 ¼ SMOGRA # 180 m2 =s2 and TSP600s # 200 m2 : s3: neither s1 nor s2. 10 d0 ¼ no flaming droplets/particles in prEN 13823 within 600s; d1 ¼ no flaming droplets/particles persisting longer than 10 s in prEN 13823 within 600 s; d2 ¼ neither d0 nor d1 (ignition of paper in prEN ISO 11925–2 results in a d2 classification).

FLAMMABILITY 53.12 APPENDIX 53.12.1 Nomenclature A CHF CI cp CDG FPI 00 G_ j GTR DHi

DHco DHg DHw  DHco  DHco2

DHo HRP Kthick Kthin m_ a 00 m_ f 00

m_ w OC PGP 00 q_ e 00 q_ f 00 q_ rr 00 Q_ i 0 Q_ i S DTig TRP u V_T

Wf Wj yj Yo

total exposed surface area of the material (m2 ) critical heat flux (kW=m2 ) ˚ /min)/(g=m3 ) corrosion index (A specific heat (MJ/kg-K) carbon dioxide generation calorimetry fire propagation index 0 [1000 (0:42Q_ ch )1=3 ={DTig (kpcp )1=2 }] mass generation rate of product j (kg=m2 -s) gas temperature rise calorimetry heat of combustion per unit mass of fuel pyrolyzed (MJ/kg) heat of complete combustion of CO (MJ/kg) heat of gasification of the polymer (MJ/kg) heat of gasification of water (2.58 MJ/kg) net heat of complete combustion per unit mass of CO generated (MJ/kg) net heat of complete combustion per unit mass of CO2 generated (MJ/kg) net heat of complete combustion per unit mass of oxygen consumed (MJ/kg) heat release parameter (DHch =DHg ) thermal response parameter for thermally thick polymers (kW-s1=2 =m2 ) thermal response parameter for thermally thin polymers (kJ=m2 ) mass flow rate of air (kg/s) gasification rate of the polymer or the mass loss rate (kg=m2 -s) water application rate per unit surface area of the material (kg=m2 -s) oxygen consumption calorimetry product generation parameter {yj =DHg } (kg/MJ) external heat flux (kW=m2 ) flame heat flux (kW=m2 ) surface re-radiation loss (kW=m2 ) heat release rate per unit sample surface area 00 (m_ DHch ) (kW=m2 ) heat release rate per unit sample width (kW/m) stoichiometric mass air-to-fuel ratio (
) ignition temperature above ambient (K) Thermal Response Parameter fire propagation rate (mm/s) total mass flow rate of the fire product-air mixture (m3 =s) volumetric total mass pyrolyzed in the pyrolysis or combustion of the polymer (kg) total mass of product j generated in the pyrolysis or combustion of the polymer (kg) yield of product j(Wj =Wf ) (kg/kg) mass fraction of oxygen (
)

/

923

Greek a b j F d dw

«w w r cj

ventilation correlation coefficient for nonflaming region (
) ventilation correlation coefficient for transition region (
) ventilation correlation coefficient for the equivalence ratio (
) 00 equivalence ratio (Sm_ p A=m_ air ) thickness or depth (m) energy associated with the blockage of flame heat flux to the surface and escape of the fuel vapors per unit mass of the fuel (MJ/kg) water application efficiency (
) 00 00 combustion efficiency [Q_ ch =m_ DHT ] 3 density (kg=m ) stoichiometric yield for the maximum conversion of fuel to product j (
)

Subscript a ch con corr cr e ex f fc fr g i ig j m n o rad stoich rr s v w 1

air or ambient chemical convective corrosion critical external flame extinction flame or fuel flame convective flame radiative gas or gasification chemical, convective, radiative ignition fire product melting net initial radiation stoichiometric for the maximum possible conversion of the fuel to the product surface re-radiation surface ventilation-controlled fire water well-ventilated

Superscripts . ’ ’’

per unit time (s
1 ) per unit width (m
1 ) per unit area (m
2 )

Abbreviations ABS CPVC

acrylonitrile-butadiene-styrene chlorinated polyvinylchloride

924 / CHAPTER 53 CR CSP, CSM, CLS-PE

neoprene or chloroprene rubber

chlorosulfonated polyethylene rubber (Hypalon) CTFE chlorotrifluoroethylene (Kel-F)1 E-CTFE ethylene-chlorotrifluoroethylene (Halar)1 EPR ethylene propylene rubber ETFE ethylenetetrafluoroethylene (Tefzel)1 EVA ethylvinyl acetate FG fiber glass reinforced FR fire retarded FEP fluorinated polyethylene-polypropylene (Teflon1) IPST isophthalic polyester PAN polyacrylonitrile PC polycarbonate PE polyethylene PEEK polyether ether ketone PES polyethersulphone PEST polyester PET polyethyleneterephthalate (Melinex1, Mylar1) PFA perfluoroalkoxy (Teflon1) PMMA polymethylmethacrylate PO polyolefin PP polypropylene PPS polyphenylene sulfide PS polystyrene PTFE polytetrafluoroethylene (Teflon1) PU polyurethane PVEST polyvinylester PVCl2 polyvinylidene chloride (Saran1) PVF polyvinyl fluoride (Tedlar1) PVF2 polyvinylidene fluoride (Kynar1, Dyflor1) PVC polyvinylchloride Si silicone SBR styrene-butadiene rubber TFE tetrafluoroethylene (Teflon1) XLPE crosslinked polyethylene XLPO crosslinked polyolefin Related information can be found in Chapter 43. REFERENCES 1. A. Tewarson, J. Fire Science 10, 188 (1992). 2. A. Tewarson, SFPE Handbook of Fire Protection Engineering. (The National Fire Protection Association Press, Quincy, MA, 1995) pp. 3-53–3-124. 3. A. Tewarson, J. Fire Science 12, 329 (1994). 4. A. Tewarson and R. F. Pion, Combustion and Flame, 26, 85 (1976). 5. A. Tewarson, ‘‘Fire Hardening Assessment (FHA) Technology for Composite Systems’’, Technical Report ARL-CR-178, Contract DAAL01-93-M-S403, prepared by the Factory Mutual Research Corporation, Norwood, MA for the U.S. Army Research Laboratory, Watertown, MA., November 1994. 6. A. Tewarson and S. D. Ogden, Combustion and Flame, 89, 237 (1992). 7. A. Tewarson and M. M. Khan, ‘‘Flame Propagation for Polymers in Cylindrical Configuration and Vertical Orientation,’’ Twenty-Second

8. 9. 10. 11. 12. 13.

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FLAMMABILITY

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Research Institute, (Center for Global Environmental Technologies, Albuquerque, NM, 1994). V. Beck, ‘‘Performance-Based Fire Engineering Design and Its Application in Australia’’, Fire Safety Science, Fifth International Symposium, pp. 23–40, International Association for Fire Safety Science, edited by Y. Hasemi, Japan, 1997. B.J. Meacham, ‘‘Concepts of a Performance-Based Building Regulatory System for the United States’’, Fifth International Symposium, pp. 701–712, International Association for Fire Safety Science, edited by Y. Hasemi, Japan, 1997. ASTM’s Role in Performance-Based Fire Codes and Standards, edited by J.R. Hall, ASTM STP 1377, The American Society for Testing and Materials, West Conshohocken, PA. 1999. Worldwide Standards Service for Windows, HIS, Englewood, CO, 2002. J. Troitzsch, ‘‘International Plastics Flammability Handbook-Principles, Regulations, Testing and Approval’’, (Macmillan Publishing Co., Inc., New York, NY 1983). A.H. Landrock, ‘‘Handbook of Plastics Flammability and Combustion Toxicology-Principles, Materials, Testing, Safety, and Smoke Inhalation Effects’’, (Noyes Publications, Park Ridge, NJ, 1983). ‘‘Flammability Testing of Building Materials-An International Survey’’, Document NO. TH 42126, British Standards Institution, London, UK, 2000. A.D. Makower, ‘‘Fire Tests-Buildings Products, and Materials’’, British Standards Institution, London, UK. C.J. Hilado, ‘‘Flammability Test Methods Handbook’’, Technomic Publication, Westport, Conn, 1973. European Committee for Standardization (CEN) (http://www. cenorm.be/). FM Approval Standards (http://www.fmglobal.com/research_standard _testing/product_certification /approval_standards.html). UL Standards (http://ulstandardsinfonet. ul.com/catalog/). ISO Standards (http://www.iso.ch/iso/en/isoonline.frontpage). ASTM Standards (http://www.astm.org). B. Sundstrom, ‘‘European Classification of Building Products’’, Interflam’99, 8th International, Fire Science & Engineering Conference, 2, pp. 769- Interscience Communications, London, UK, 1999. B. Sundstrom, and S.D. Christian, ‘‘What are the New Regulations, Euroclasses, and Test Methods Shortly to be Used Throughout Europe’’, Conference Papers, Fire and Materials, pp. 117–127, January 22–24, 2001, Interscience Communications, London, UK, 1999.

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925

49. F.B. Clarke, ‘‘Issues Associated with Combustibility Classification: Alternate Test Concepts’’, Fire Safety Science, Proceedings of the Fifth International Symposium, pp. 165–175, International Association for Fire Safety Science, edited by Y. Hasemi, Japan, 1997. 50. ‘‘Test Procedures and Performance Criteria for the Flammability and Smoke Emission Characteristics of Materials Used in Passenger Cars and Locomotive Cabs’’, Federal Register, Rules and Regulations, 64(91), Wednesday, May 12, 1999. 51. Department of Transportation, Federal Transit Administration, Docket 90-A ‘‘Recommended Fire Safety Practices for Transit Bus and Van Materials Selection’’, Federal Register, 58(201), Wednesday, October 20, 1993. 52. I.A. Benjamin, and C.H. Adams, ‘‘The Flooring Radiant Panel Test and Proposed Criteria’’, Fire Journal, 70 (2), 63–70, March 1976. 53. S. Davis, J.R. Lawson, and W.J. Parker, ‘‘Examination of the Variability of the ASTM E 648 Standard with Respect to Carpets’’, Technical Report NISTIR 89–4191, National Institute of Standards and Technology, Gaithersburg, MD, October 1989. 54. K. Tu, and S. Davis, ‘‘Flame Spread of Carpet Systems Involved in Room Fires’’, Technical Report NBSIR 76–1013, National B. 55. NFPA 101 Life Safety Code, Chapter 10 Annex, National Fire CodesA Compilation of NFPA Codes, Standards, Recommended Practices and Guides, 5, pp. 101–306 to 101–307, National Fire Protection Association, Quincy, MA 2000. 56. NFPA 318 ‘‘Standard for the Protection of Cleanrooms’’, National Fire Codes, 6, pp. 318–1 to 318–22, National Fire Protection Association, Quincy, MA, 2000; 57. J.S. Newman, and A. Tewarson, ‘‘Flame Spread Behavior of CharForming Wall/Ceiling Insulation’’, Fire Safety Science, Third International Symposium, pp. 679–688, International Association for Fire Safety Science, edited by G. Cox and B. Langford, (Elsevier Applied Science, New York, NY, 1991). 58. B. Sundstrom, P.V. Hees, and P. Thureson, ‘‘Results and Analysis from Fire Tests of Building Products in ISO 9705, the Room/Corner Test; The SBI Research Program’’, Technical report SP-RAPP, 1998:11, Swedish National Testing and Research Institute, Fire Technology, Boras, Sweden 1998. 59. J.S. Newman, and J. Steciak, ‘‘Characterization of Particulates from Diffusion Flames’’, Combustion and Flame, 67, 55–64, 1987. 60. S.E. Dillon, ‘‘Analysis of the ISO 9705 Room/Corner Test: Simulation, Correlations, and Heat Flux Measurements’’, NIST-GCR-98–756, National Institute of Standards and Technology, Gaithersburg, MD, August 1998.

CHAPTER 54

Thermal-Oxidative Stability and Degradation of Polymers Vladyslav Kholodovych and William J. Welsh Department of Pharmacology, University of Medicine & Dentistry of New Jersey (UMDNJ) – Robert Wood Johnson Medical School (RWJMS) and the UMDNJ Informatics Institute, Piscataway, NJ 08854

54.1 54.2 54.3 54.4 54.5 54.6 54.7 54.8

Basic Definitions and Modes of Degradation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure–Property Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degradation Reaction Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additives for Enhanced Thermal-Oxidative Stability. . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Methods of Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tabulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Science Tools on the World Wide Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

927 928 929 930 933 933 934 936 938

In chain depolymerization (also known as chain depropagation or ‘‘unzipping’’), a given main chain is reduced in length by the sequential removal of monomer units from chain termini or at ‘‘weak links’’. A ‘‘weak link’’ may be a chain defect, such as an initiator fragment, peroxide, or an ether linkage arising as impurities from polymerization in the presence of oxygen. The slightly higher activity of a tertiary H atom may also provide a site for the initiation of the degradation process. Chain depolymerization exhibits three characteristic features: (1) the major product (volatile or not) is monomer, (2) the decrease in bulk-polymer molecular weight is initially negligible, and (3) the rate of conversion gradually decreases. Chain depolymerization can be regarded as the opposite of addition (chain-growth) polymerization. A specific example is poly(methyl methacrylate) (PMMA). In random scission, chain breaking occurs at random points along the chain. Random scission exhibits the following characteristic features: (1) the major products are typically fragments of monomer, dimer, trimer, etc., up to molecular weights of several hundred; (2) the decrease in molecular weight is initially appreciable; and (3) the rate of degradation is initially rapid and approaches a maximum. Random scission, as exemplified by polyethylene (PE) and polypropylene (PP), can be viewed as the reverse of

54.1 BASIC DEFINITIONS AND MODES OF DEGRADATION Thermal stability refers to the ability of a material to maintain desirable mechanical properties such as strength, toughness, or elasticity at a given temperature. At the other extreme, thermal degradation can be defined functionally as the deterioration of those properties of polymers which make them useful commercially as rubbers, plastics, and fibers. Degradation reactions are most important in two phases of the life of a synthetic polymer: (1) during fabrication when both thermal and oxidative reactions can occur, and (2) during service life under prolonged exposure to light and oxidation. Symptoms of polymer degradation include hardening, brittleness, softening, cracking, discoloration, as well as alteration of specific polymer properties, e.g., mechanical and thermodynamic properties. In cases where molecular weight decreases, such molecular weight-sensitive properties as mechanical strength, elasticity, solution viscosity, and softening point will suffer most dramatically. Thermal degradation of organic polymers typically begins around 150– 200 8C, and the rate of degradation increases as the temperature increases. The types of polymer degradation can be divided into three general categories: chain depolymerization, random scission, and substituent reactions [1–11]. 927

928 / CHAPTER 54 condensation (step-growth) polymerization. In random scission, the polymer radical is both highly reactive and surrounded by an abundance of secondary hydrogens. This type of thermal degradation will therefore be favored if transfer is significant. Transfer reactions, in which a long-chain radical attacks another chain (intermolecular) or itself (intramolecular), produce fragments larger than monomer and promote random chain scission. In both chain depolymerization and random scission, thermal degradation is a free-radical chain reaction. Initiation, which is the splitting of the chain to form radicals, may occur at chain ends, at ‘‘weak links’’, or at random points along the chain structure. Radical degradation often leads to crosslinking which can be visualized as resulting from the combination of radical sites on adjacent chains. Chain cleavage can occur either by primary homolytic skeletal cleavage or by an intramolecular attack by a terminal radical unit on its own chain. It is possible to differentiate between chain depolymerization and random scission in some cases by following the molecular weight of the residue as a function of the extent of reaction. Specifically, the ultimate product of random scission is likely to be a disperse mixture of fragments of molecular weight up to several hundred, whereas chain depolymerization yields large quantities of monomer. In degradation by substituent reactions, the substituents attached to the polymer-chain backbone are modified or eliminated. Any volatile products evolved will therefore be chemically unlike monomer. The most prominent example of degradation via substituent reaction is poly(vinyl chloride) (PVC). Like all thermoplastics, PVC is processed at about 200 8C at which temperature it loses HC1 quite rapidly and is converted to a deeply colored polyene polymer, i.e., ---CH2 ---CHCl---CH2 ---CHC1 ! ---CH ¼ CH---CH ¼ CH---þ 2HCl:

The actual degradation mechanism is more complex than implied by this simple reaction. If substituents reactions occur, they generally ensue at temperatures (T < 150 8C) below that of degradation reactions in which the backbone bonds are broken. Consequently, the reactivity of the substituents relative to that of the polymer backbone will largely dictate whether a particular polymer undergoes thermal degradation by substituent reactions or by reactions involving the backbone (e.g., chain depolymerization and random scission) [1–11].

54.2 STRUCTURE–PROPERTY RELATIONSHIPS Polymers decompose at significantly lower temperatures than model compounds, perhaps by as much as 200 8C. The main reasons are twofold: (1) polymer molecules often incorporate reactive structural abnormalities (‘‘weak links’’) absent in the model compound; and (2) polymer degradation can lead to chain processes, not accessible to

model compounds, which accelerate the degradation reaction. The limited thermal stability of organic high polymers is due to several factors, including: (1) C–C bonds are relatively weak and oxidatively unstable; (2) fragmentation of the polymer during degradation is entropy favored; and (3) the presence of terminal catalytic sites, reactive atoms (e.g., tertiary H atoms), and ‘‘weak links’’ (e.g., branch points) along the chain which initiate decomposition [1–11]. The thermal stability and mode of decomposition of a polymer are determined by both physical and chemical factors [1–11]. In many cases, the maximum service temperature of polymers is limited not by the breaking of chemical bonds but rather by changes in physical characteristics at elevated temperatures. While retaining their chemical structures, they become weak, soft, and eventually fluid. The physical requirement of a thermally stable polymer is that it has high melting or softening temperature. The same factors that raise Tg and Tm , namely, chain rigidity and strong interchain forces, also raise thermal stability. Chain rigidity can be conferred by ring structures linked by collinear or para chain-extending bonds, while strong interchain attractions are attained by (intermolecular) dipolar and hydrogen-bonding interactions. The introduction of polar groups (e.g., CN, Cl, F) and hydrogen-bonding groups (e.g., –OH, –C(O)NH–) will often raise the melting and softening points appreciably. Stereoregularity in a vinyltype polymer can produce a dramatic positive effect on thermal stability. For example, atactic polystyrene is amorphous with a Tg of about 80 8C while isotactic polystyrene is crystalline with a Tm of about 230 8C. The regular structure of the latter fits more readily into a crystalline lattice, and intermolecular forces are more difficult to overcome. Short bulky sidegroups (e.g., ---CH3 in polypropylene) can actually increase the melting point by reducing chain mobility, but long bulky sidegroups tend to reduce the melting point by disrupting the efficiency of chain packing. Crystalline forms of polymers are more resistant to oxidation than amorphous forms due to oxygen-permeability differences. For amorphous polymers, polymers oxidize more rapidly above than below their Tg due to the faster rate of diffusion of oxygen. Surface regions are particularly susceptible to oxidative degradation. The chemical factors which influence thermal stability are more diverse than the physical factors. Of primary importance, heat-resistant polymers require bonds of high dissociation energy. For example, poly(tetrafluoroethylene) (PTFE) is superior to PE and many other polymers in terms of thermal stability. The stability conferred by fluorine substitution is clearly associated with the relatively high value for the dissociation energy of C–F bonds. In fact, PTFE [---CF2 CF2 ---] is the most stable and most widely applied of the fluorinated polymers. Since the strong C–F bond renders transfer unlikely, chain depolymerization of PTFE gives high yields of monomer. Van Krevelen [12] found a reasonably linear correlation between the half-decomposition temperature T1=2 and the

THERMAL-OXIDATIVE STABILITY AND DEGRADATION OF POLYMERS TABLE 54.1. Typical bond dissociation energies (kJ/mol). Bond C–C C¼¼C C

C C–H C–Cl C–F C–O C–N C¼¼N N–H ROO–H CH3 C(O) – H

Aromatic or heterocyclic

Aliphatic

Reference

410 — — 427–435 — — 448 460 — — — —

284–368 615 812 381–410 326 452 350–389 293–343 615 390 377 368

[1,22,23] [22] [22] [1,22,23] [22] [23] [22] [22] [22] [22] [1] [1]

bond dissociation energy Ediss of vinyl polymers, i.e., T1=2 ¼ 1:6Ediss (in kJ/mol)þ140. The bond dissociation energy (Table 54.1) of the bond in question depends on its bond order (i.e., single, double, triple), on resonance effects, on steric strain induced by bulky neighboring groups, and on the rigidity of their own or adjacent valence structures. Steric strain from crowded methyl groups, for example, makes polyisobutylene less stable to heat than PE. Most heat-resistant polymers, other than some inorganic and fluorinated polymers, have wholly aromatic chains like poly(p-phenylene). A rigid crosslinked network will also improve thermal stability. Crosslinked thermosets, such as phenolic, melamine, and epoxy plastics, are more resistant to heat than general purpose thermoplastics. Whereas thermoplastics are limited in use by the temperatures at which they soften, thermoset materials are limited by temperatures at which bonds begin to break [13]. Two additional chemical factors that are important in determining thermal stability are the reactivity of the depropagating radical and the availability of reactive hydrogen atoms for transfer. Reactive tertiary H atoms are important for the production of oligomers, whereas methylene or benzene H atoms are relatively inert. In 1,1-disubstituted vinyl polymers (e.g., poly (vinylidene cyanide): [---CH2 --- C(CN)2 ---]), the degrading radical is relatively unreactive by virtue of being trisubstituted. Since there are no reactive hydrogen atoms, transfer is suppressed and monomer production is dominant. The influence of radical stability is emphasized by a comparison of the behaviors of PE and PP with the polydienes [---CH2 ---CR ¼ CH---]. While PE and PP engage overwhelmingly in transfer (i.e., random scission) due to high radical reactivity, the polydienes engage in chain depolymerization due to the high relative stability the allylic radical. The relative reactivity of C–H bonds in polymers follows the order: allylic > tertiary > secondary > primary. Polystyrene and polyisobutylene are exceptions to this rule in that the benzylic and secondary H atoms, respectively, are shielded by relatively inert phenyl and methyl groups [1–13].

/

929

Degradation rates of polymers in air at temperatures below 150 8C depend on the reactivities of the peroxy radicals formed. In polymers most resistant to oxidation, H atoms are either totally absent or appear in unreactive methyl and phenyl groups. Polymers containing unsaturated linkages, such as polyisoprene or polybutadiene rubbers, can be attacked by atmospheric ozone as well as by oxygen. Polarity effects usually dominate in polymers containing heteroatoms, hence the rate of oxidation decreases along the series: CH2 > CHC1 > C(H)COOCH3 > C(CH3 ) COOCH3 > CH(CN) > CF2 ---CF2 . Heteroatoms affect the strength of neighboring C–H bonds mainly by modifying the polar properties of transition states. Since the peroxy radical is electrophilic, the oxidation of ethers, aldehydes, amines, and sulfides occurs through abstraction of H atoms on carbons adjacent to the unshared electron pair on the heteroatom. Conversely, electron-deficient groups tend to stabilize neighboring H atoms. Few polymers can withstand temperatures above 200 8C in air. Exceptions include aromatic, heterocyclic, and socalled ladder polymers (Fig. 54.1) [13,14–19]. Appropriately named, ladder polymers will degrade into fragments only if two parallel main-chain bonds (the ‘‘rungs’’ of the ladder) break [13]. Since this event is unlikely, ladder polymers like the two benzimidazobenzophenanthrolines designated BBB and BBL [17] (Fig. 54.1) possess exceptional thermal-oxidative stability. Moreover, recombination (‘‘healing’’) of the broken bond is facilitated by the remaining intact bond which holds the severed bond in close proximity for recombination. These rigid aromatic and ladder polymers can be ‘‘articulated’’ by linking the rigid units together by ether [–O–], ester [–C(O)O–], amide [–C(O)NH–], or sulfone [---SO2 ---] groups. The insertion of these flexible units between the rings imparts added flexibility but at the cost of reduced thermal stability [13]. In summary, the basic requirements for heat-resistant polymers are: (1) high bond-dissociation energies (i.e., strong primary bonds); (2) chain rigidity supplemented by resonance stabilization; (3) high melting or softening points (i.e., strong secondary bonds); (4) structures resistant to free-radical chain processes; (5) low permeability and chemical reactivity (especially to oxygen) by virtue of crystallinity, crosslinking, and efficient chain packing; and (6) elimination (during synthesis and processing) of ‘‘weak links’’ in the chain where free-radical degradation often initiates.

54.3 DEGRADATION REACTION MECHANISMS The oxidative degradation of polymers involves freeradical chain reactions. For example, degradation of polyolefins such as PE is commonly initiated by hydroperoxide impurities incorporated during synthesis and processing.

930 / CHAPTER 54 N

N

O

O

n

N

S

S

N

PBO O

O

N

n

PBT(PBZT) F

F

F

F

N n n

O Polyimide

O

Poly(phenylene)

n Poly(perfluorophenylene)

O

O

O

O

N

N

N

N

N

N

N

N

n

BBB

n

BBL

FIGURE 54.1. Examples of aromatic, heterocyclic, and ladder polymers.

Polymers may be attacked by molecular oxygen, ozone, or by indigenous free radicals in the polymer. Thermal-oxidative degradation of polyolefins in air is autocatalytic, i.e., the rate is slow at first but gradually accelerates to a constant value. According to the three-step mechanism outlined below, the RO2 peroxy radicals formed (Step 1) are sufficiently reactive to attack some primary CH bonds of the chain R’H (Step 2). The peroxy radical RO2 is thus reformed (Step 3) and can attack another CH bond. This chain reaction continues until termination occurs (Step 4) [1–11]. Step 1: polymer ! RO2 (Initiation)  0 (slow) Step 2: RO2 þ R H ! ROOH þ R0 (Propagation) (fast) Step 3: R þ O2 ! RO2 Step 4: 2RO2 ! unreactive products (Termination) Step 3 is accelerated by the decomposition of the hydroperoxide products ROOH to form additional free radicals, i.e., ROOH ! RO þ OH. Degradation is also accelerated by the presence of even a small number of reactive tertiary H atoms but sometimes secondary H atoms. Evidence indicates that a plethora of free-radicals including peroxy RO2 , hydroperoxy HO2 , oxyradicals RO, hydroxy HO, and alkyl R are capable of formation and thereby initiating thermaloxidative degradation of the polymer [1–11]. 54.4 SPECIFIC EXAMPLES An exhaustive survey of the thermal stabilities and degradation processes of the multitude of polymer families is beyond the scope of this work. Instead, the polymers selected for discussion below are both familiar and representative of the wide range of thermal-oxidative behavior exhibited by polymers [1–13].

54.4.1

Polyethylene (PE)

PE is thermally stable to about 290 8C, above which it undergoes a decrease in molecular weight with little volatilization. Above 360 8C, volatilization is rapid. The polymer also undergoes some crosslinking when heated at elevated temperatures. The rate of oxidation is related to the degree of chain branching since this gives rise to susceptible tertiary hydrogens. Small concentrations of C¼¼C and C¼¼O double bonds or peroxides along the chain will activate H atoms on neighboring bonds, thus complete saturation (no double bonds) improves oxidation resistance. The volatile products consist of a continuous spectrum of hydrocarbons ranging from C1 to C70 or higher. This suggests a random-scission degradation mechanism initiated at the weak links followed by inter- and intramolecular chain transfer. Low-density polyethylene (LDPE) contains more chain branching than high-density polyethylene (HDPE). Therefore, the order of increasing oxidation is HDPE < LDPE. Few additives to impart thermal stability are compatible with PE in amounts larger than 1 % or so. 54.4.2 Polypropylene (PP) Thermal degradation of PP starts at about 230 8C by a random scission process which yields virtually no monomer up to about 300 8C. Similar to PE, the degradation products of PP span a range of unsaturated hydrocarbons up to C70 and higher. PP is much more susceptible than PE to oxidation because PP has branch points on alternate carbon atoms. The greater availability of reactive tertiary H atoms explains why the temperature at which degradation initiates is lower for PP (230 8C) than for PE (290 8C).

THERMAL-OXIDATIVE STABILITY AND DEGRADATION OF POLYMERS 54.4.3 Polystyrene (PS) PS exhibits a maximum in the rate of degradation and a rapid decrease in molecular weight, both of which are characteristic of a random scission process. Evidence suggests that the decrease in molecular weight is the result of scission of a limited number of ‘‘weak links’’ in the polymer structure. The volatile products of thermal degradation of PS are monomer (42%) with progressively decreasing amounts of dimer, trimer, tetramer, and pentamer. Thermal degradation initiates along the chain at weak links, which might be unsaturated bonds or perhaps CH2 ---CHPh---CHPh---CH2 --(Ph ¼ C6 H5 ) sequences resulting from head-to-head addition of monomer units during polymerization. 54.4.4 Poly(vinylchloride) (PVC) PVC is relatively unstable to heat above 250 8C, even in the absence of oxygen. The substituent reaction is initiated by scission of the weakest C–Cl bonds, which are characteristically located at the chain ends since double bonds are formed as a result of disproportionation or transfer to monomer during polymerization. The chlorine radical Cl so formed abstracts an H atom to form HC1. The resulting chain radical then reacts to form a double bond with regeneration of a chlorine radical. The reaction is accompanied by embrittlement and dramatic discoloration of the material, arising from light absorption by the conjugated backbone (C–C¼¼C–). The polymer yellows when there are seven conjugated double bonds and discolors through brown to black with increasing extension of the conjugated doublebond system. Stabilizers which are invariably added to improve the heat and light stability include inorganic and organic derivatives of lead as well as organic derivatives of barium, cadmium, zinc, and tin. 54.4.5 Poly(acrylonitrile) (PAN) Like PVC, PAN discolors thermally at 175 8C due to the linking of nitrile groups to form conjugated carbon–nitrogen sequences. Consistent with degradation by substituent reaction, the color of degrading polymer progresses through the spectrum from yellow to red and the decrease in molecular weight is initially negligible. 54.4.6 Poly(tetrafluoroethylene) (PTFE) PTFE is a highly crystalline polymer that is devoid of crosslinks and branching. PTFE undergoes nearly 100% conversion to monomer at elevated temperatures. Thermal degradation by chain depolymerization at the chain ends probably starts at low temperatures (250–350 8C), while random-scission cleavage likely becomes more pronounced at higher temperatures. Although PTFE is the most stable of

/

931

the vinyl polymers, it cannot withstand prolonged exposure to temperatures above about 350–400 8C. The much greater strength of the C–F bond over the C–H bond explains why transfer processes, which largely control the thermal degradation of PE, are virtually absent in the thermal decomposition of PTFE. The degradation process is more complicated in the presence of air than in vacuum. 54.4.7 Polyamides (PAs) Degradation of PAs can occur at melt-spinning and molding temperatures. Residual water plays an important role, initiating hydrolysis of peptide linkages followed by decarboxylation of the resulting carboxyl groups. The principal volatile products of thermal degradation are carbon dioxide and water. 54.4.8 Heat-Resistant Polymers Many of the emerging technologies, particularly in the realm of electronics and aerospace science, require processable polymers endowed with superior mechanical properties and thermal-oxidative stability [13,16]. The structural feature common to such high-performance polymers is an aromatic backbone associated with high-bond dissociation energies, rigidity, and resonance stabilization. The mechanism of polymer degradation is principally oxidative in nature, hence incorporation of heterocyclic units further improves the thermal stability by increasing the char yield at very high temperature. The most successful of the new high-temperature polymers are those containing aromatic units in the chain backbone. For example, the polypyromellitimides (more commonly known as polyimides) (Fig. 54.1) show considerable promise as temperature-resistant plastics. The commercial polyimide Kapton is extremely heat stable, retaining more than 50% of its original tensile strength after 1,000 hours in air at 300 8C. The fluorination of aromatic structures provides additional thermal-oxidative stability. The parent structure, polytetrafluorophenylene, is stable to 500 8C in vacuum [1–11]. The aromatic heterocyclic rodlike polymers poly(pphenylenebenzobisoxazole) (PBO) and poly(p-phenylenebenzobisthiazole) (PBZT or PBT) [14–20] possess rigid rodlike structures which provide superior tensile properties and excellent thermal stability. Thermal analysis of PBO and PBT reveals minimal weight loss in air at 316 8C. Thermal decomposition of both polymers begins at 600 8C and reaches a maximum between 660 and 700 8C. The total weight loss for both PBO and PBT is about 28% at 1,000 8C [16]. Unfortunately, wholly aromatic and/or heterocyclic polymers are notoriously difficult to process because they: (1) exhibit low solubilities in common organic solvents and (2) typically start to decompose at a lower temperature than they melt. Attempts to improve the processing characteristics of

932 / CHAPTER 54 temperature-resistant polymers contain wholly inorganic backbones with high bond energies, such as the polyphosphazenes [–P(RR’)¼¼N–] and the polysiloxanes [SiRR’–O] [9]. Some polyorganosilanes [–SiRR’–] are thermally stable to temperatures above 250 8C (>350 8C under inert conditions). This thermal stability is consistent with the strengths of silicon–silicon (80 kcal/mol) and carbon–silicon (90 kcal/mol) bonds [20]. A clever strategy for imparting thermal-oxidative stability in a polymer is exemplified by the so-called ‘‘ladder polymer’’ (Fig. 54.1) [13,17]. As the name implies, the chain of a ladder polymer can be broken only if at least two bonds on the same ring are severed. The likelihood that this will happen is low. Moreover, the broken bond has a high probability of reconnecting since the other ‘‘rung’’ of the ladder will hold the atoms of the severed bond in close proximity for bond reformation. Owing to these design features, the thermal stability of ladder polymers is often superior to the usual single-stranded types. Based on the extensive experimental analysis of numerous heat-resistant polymers, Arnold [13] proposed a set of generalizations regarding correlations between polymer structure and thermal stability. Summarizing the more notable points, the highest stabilities were found for ladder-type polymers (e.g., BBB and BBL) and those containing heterocyclic or aromatic conjugated rings (e.g., polyimides, polyphenylenes, perfluoropolyphenylenes, PBO, PBT) (Fig. 54.1). The stability of polymers containing fused rings decreases as the number of fused chain segments increases. With few exceptions, most high-temperature polymers start to decompose at nearly the same temperatures

these polymers have focused on inserting flexible ‘‘spacer’’ groups (e.g., amides, esters, ethers, sulfones) into the otherwise rigid chain backbone. The incorporation of even a small number of such spacer groups will increase the polymer’s conformational flexibility and entropy and thus improve its tractability by allowing mutual rotation of adjacent chain elements about the flexible moieties. At the same time, these spacer groups will often alter the colinearity of the otherwise rigid chain thereby lowering the melt temperature. In general, the thermal-oxidative stability of these polymers diminishes as the ratio of flexible-to-rigid moieties increases [13]. The so-called ‘‘articulated’’ PBO and PBT, in which 3,3’biphenyl or 4,4’-(2,2’-bipyridyl) moieties have been incorporated into the otherwise rodlike backbone, are appreciably more stable than those containing diphenoxybenzene (Ph–O–Ph) segments (Fig. 54.2). While PBO and PBT articulated with diphenoxybenzene units experience significant weight losses at 316 8C, those articulated with biphenyl and bipyridyl units are largely unaffected at that temperature and display thermo-oxidative stability comparable to the parent PBO and PBT polymers. While the biphenyl unit appears to give better stability than the dipyridyl unit, the stability of the articulated PBO and PBT polymers decreases with increased content of the flexible unit in the backbone [14]. A number of techniques, including addition of stabilizers and crosslinking, are used to extend thermal stability. Some polymers, for example PEEK (polyaryletherether ketone) and poly(phenylene sulfide), gain their thermal stability by virtue of their high degree of crystallinity. Other

N

N N

N N

N

N

N

N

N

N

O

O

FIGURE 54.2. Examples of flexible spacer groups.

THERMAL-OXIDATIVE STABILITY AND DEGRADATION OF POLYMERS in both air and nitrogen. For polymers containing phenylene groups, the order of stability is para > meta > ortho. Crosslinking generally results in enhanced stability. Copolymerization can yield enhanced thermo-oxidative stability as, for example, imide copolymers of various heterocyclics are oxidatively more stable than the imide homopolymer. In terms of flexible spacer groups, the most stable are perfluoroaliphatics like ---CF2 --- followed by –O–, –S–, –CONH–, and –CO–. The least stable were alkylene linkages, ---SO2 ---, ---NH---, Cl-containing groups, and alkylene groups. However, any flexible spacer unit inserted into the backbone of aromatic or heterocyclic polymers can be expected to diminish both short-term and long-term stability.

R3 P þ ROOH ! R3 PO þ ROH (tertiaryphosphine): The inclusion of very small quantities of ethylene or propylene (1–3%) in poly(vinyl chloride) has resulted in copolymers of greatly improved heat stability relative to the parent PVC. Since degradation of PVC involves loss of HC1, compounds that react with the HC1 to form stable products, such as metal oxides, are used as stabilizers.

Oxidative degradation of polymers typically follows a free-radical mechanism involving crosslinking and/or chain scission initiated by free radicals from peroxides formed during the initial oxidation step [1–11]. Enhanced stability has been achieved by the use of additives which are frequently called antioxidants or heat stabilizers. One approach employed to reduce the oxidation of polyolefins like PE and PP is to terminate the chain reaction by introducing an antioxidant with a greater affinity than a polyolefin for the peroxy radical RO2 . Such antioxidants (AH) function by reacting with RO2 to form a relatively inactive radical A, i.e.,

54.6 EXPERIMENTAL METHODS OF ANALYSIS Polymer degradation can be monitored by measurement of molecular weight using viscometry, osmometry, light scattering, ultracentrifuge, and gel-permeation chromatography (GPC). GPC (more generally called size-exclusion chromatography) can be used in estimating the effect of degradation on molecular-weight distribution (MWD). Spectroscopic probes of thermal degradation include UV spectroscopy, IR spectroscopy, NMR spectroscopy, electron-spin resonance spectroscopy (ESR, EPR), and mass spectrometry (MS). Multiple internal reflectance infrared spectroscopy (MIRS) allows a very thin surface layer to be examined. Another method is flash pyrolysis in which the polymer’s temperature is raised very rapidly to 500 8C or more at which the molecules are broken down into small fragments. The fragment pattern can be analyzed by gas chromatography (pyrolysis-GC) and mass spectrometry (pyrolysis-MS), either separately or in combination (pyrolysis-GC/MS) [2–6,13]. Several thermal techniques are commonly employed to monitor the thermal stabilities of polymers [2–6,13]. In thermogravimetric analysis (TGA), a sensitive balance is used to follow the weight change of the sample in a specified environment (vacuum, air, or inert atmosphere)

RO2 þ AH ! ROOH þ A While amines and some annular hydrocarbons are suitable chain terminators, hindered phenols such as di-t-butyl-pcresol (alias butylated hydroxytoluene or BHT) are most popular because they avoid discolorization and they eliminate two free radicals per BHT molecule (Fig. 54.3). The resonance-stabilized aryloxy radical is protected by the bulky electron-releasing t-butyl groups in the 2 and 6 positions, so the hindered phenol can combine with a second peroxy radical but cannot combine readily with molecular oxygen or with another aryloxy radical nor abstract H atoms from the polymer to initiate a new free-radical chain reaction.

O· C(CH3)

(CH3)C

(CH3)C

OH C(CH3)

(CH3)C

C(CH3)

2R'O2

CH3

933

Oxidative free-radical degradation by hydroperoxides can be catalyzed by certain transition metal ions, especially those of copper, cobalt, and manganese. To reduce the rate of free radical formation, two classes of additives are used: (1) organic phosphines, amines, and sulfides which catalyze the decomposition of the hydroperoxides to nonradical products, and (2) metal-ion chelators (e.g., Ph– CH¼¼NNH–CO–CO–NHN¼¼CH–Ph). Tertiary phosphines are thus oxidized to phosphine oxides, tertiary amines to amine oxides, and sulfides to sulfoxides, e.g.,

54.5 ADDITIVES FOR ENHANCED THERMAL-OXIDATIVE STABILITY

OH

/

R'OOH CH3

H3C

OOR'

FIGURE 54.3. Illustration of the function of the hindered phenol di-tert-butyl-p - cresol (BHT).

934 / CHAPTER 54 as a function of time or temperature. Thermomechanical analysis (TMA) measures the mechanical responses of a polymer as a function of temperature. Typical measurements include: expansion properties, tension properties (elastic modulus), dilatometric properties (specific volume), single-fiber properties (single-fiber modulus), and compression properties. In isothermogravimetric analysis (IGA), weight loss as a function of time is recorded at a specified temperature. At lower temperatures, IGA is a valuable supplement to TGA in obtaining data on long-term stability. Thermal volatilization analysis (TVA) records the evolution of volatile products by measuring the pressure of volatile degradation products continuously in an evacuated system. According to Arnold [13], the preferred method of determining the relative short-term thermal or thermo-oxidative stability of high-temperature polymers is dynamic TGA. Longer-term stabilities are most conveniently defined by IGA if the temperature is properly chosen. Combination of these tests with TMA, which provides data on the Tg and softening behavior, gives a complete picture of the thermal limitations of most polymers. Accelerated aging tests, such as the familiar ‘‘air-oven test’’, have aided the investigation of thermal-oxidative degradation. The air-oven test involves subjecting a polymer sample to temperatures ranging from 70 8C to 150 8C with air flowing over the surface of the sample. The change in stress–strain behavior (e.g., tensile modulus, tensile strength, elongation at break) is measured on samples removed from the oven at intervals until the point of failure is reached. The rationale behind accelerated testing is basically that the results can be extrapolated in time to simulate actual service conditions. In reality, most accelerated aging tests are therefore a compromise between convenience and reliability [2]. In organic polymers, the progress of oxidation reactions can be followed using infrared (IR) spectroscopy. IR absorption bands of interest in PE and related polymers are C–H stretching (3:4 mm), C–H bending of CH2 groups (6:8 mm) and CH3 groups (shoulder at 7:25 mm on an amorphous band at 7:30 mm), and CH2 rocking in sequences of methylene groups (13:9 mm). Other key absorption bands include C¼¼C in natural rubber (6:1 mm), C¼¼O and ether in PMMA (5.8 and 8:9 mm, respectively), aromatic structures in PS (6.2, 6.7, 13.3, and 14:4 mm), C–Cl in PVC (14:5 mm), peptide groups in nylon (3.0, 6.1, and 6:5 mm), and CF2 in PTFE (8:2---8:3 mm) [1–11].

54.7 TABULATED DATA There seems to be no accepted standard way of quantifying the thermal-oxidative stability and/or degradation of polymers. Therefore, different sources of data will often provide different criteria for describing the absolute or relative stability of polymers. Tables 54.2–54.5 summarize thermal-stability data extracted from a variety of sources

TABLE 54.2. Half-decomposition temperature T1=2 a and monomer yield for selected polymers. Polymer Poly(tetrafluoroethylene) (PTFE) Poly(p-phenylene methylene) Polymethylene Polybutadiene Polyethylene (PE) (branched) Polypropylene Polystyrene (PS) Polyisobutylene Poly(ethylene oxide) Poly(methyl acrylate) Poly(methyl methacrylate) (PMMA) Poly(propylene oxide) (isotactic) Poly(propylene oxide) (atactic) Poly(vinyl acetate) Poly(vinyl alcohol) Poly(vinyl chloride) (PVC)

T1=2 ( C)b

Monomer yield (%)

509 430 414 407 404 387 364 348 345 328 327 313 295 269 268 260

> 95 0 < 0.1 95 1 1 0 0 0

a

Data taken from [12]. Temperature at which the polymer loses 50% of its weight, if heated in vacuum for 30 min.

b

TABLE 54.3. Typical values of the upper use temperature ( 8C) for several familiar and commercial polymers. Polymer Natural rubber SBR Acrylate Butyl Chlorosulfonated polyethylene EPDM Epichlorohydrin Fluorinated rubbers Neoprene Nitrile Polybutadiene (cis-1,4) Polyisoprene (cis-1,4) Polysulfide Silicone Poly(vinyl chloride) (PVC) Polystyrene (PS) Polymethacrylates Polyolefins Polyamides Epoxy resins Polycarbonate Poly(phenylene oxide) (PPO) Polysulfone Polyfluorocarbons Aromatic polyamides Polyimides Poly(tetrafluoroethylene) Polybenzimidazole (PTFE) Polyurethanes

Upper use temperature (8C) Reference 80 110 150 100 120 150 120 230 100 120 100 60–80 80 230 60 60 60–80 60–90 80–100 80–110 100–135 130–150 130–150 150–220 180–230 180–250 180–250 250–300 70–110

[11] [11] [11] [11] [11] [11] [11] [11] [11] [11] [11] [11,12] [11] [11] [12] [12] [12] [12] [12] [12] [12] [12] [12] [12] [12] [12] [12] [12] [11]

THERMAL-OXIDATIVE STABILITY AND DEGRADATION OF POLYMERS

/

935

TABLE 54.4. Thermal stability of selected heat-resistant aromatic, heterocyclic, and ladder-type polymers in an inert atmosphere. Polymer F

F

F

F

PDT (8C)a

Reference

720

[13]

690–710

[13]

690–710

[13]

660–700

b

700

b

685–700

[13]

685–700

[13]

660

[13]

650

[13]

650

[13]

650

[13]

650

[13]

n O

O

N

N

N

N

O

O

N

N

N

N

n

n

N

N O

O

N

S

S

N

n

n

N

N

S

S

n

N S

n

n N

N

N H

N H

N

N

N H

N H

n

n

N N H

n

N N H a

N O

N H

n

Polymer decomposition temperature. number of relevant articles on PBO and PBT and related rodlike polymers can be found in Macromolecules, 14, 891 (1981) and in the Dec. 1980 and March 1981 special issues of the Brit. Polym. J.

bA

936 / CHAPTER 54 TABLE 54.5. Initial temperature reported for thermal decomposition of selected common polymers. Polymer Poly(acetylene) Poly(butadiene) Poly(chloroprene) Natural rubber Poly(ethylene) Poly(propylene) Poly(acrylonrtrile) Poly(methacrylic acid) Poly(vinyl acetate) Poly(vinyl alcohol) Poly(vinyl chloride) Poly(styrene) Phenol-formaldehyde resin Cellulose Cellulose triacetate Ethyl cellulose

Initial decomposition temperaturea (8C) 650 325 170 287 264 120 235 200 213 240 200 300 250 250 250 306

a

Data taken from [21]. Value given for each polymer represents the lowest decomposition temperature for which decomposition products are given in [21].

in the literature on selected familiar and commercial polymers. Table 54.6 compares the relative stability of several flexible linking groups. For a comprehensive listing of polymers, including a description of the products of thermal degradation, the reader is directed to Grassie [21]. Related information can be found in Chapter 53.

TABLE 54.6. Thermal and thermal-oxidative stability of some simple flexible linking groupsa. Group –CO– –CONH– ---(CF2 )3 --–COO– –S– ---CH2 CH2 -----CH2 --–O–

Thermal stabilityb (8C)

Thermal-oxidative stabilityc (8C)

500 500 469 457 436 429 408 —d

389 431 —d 447 418 383 —d 368

a

Data taken from [13]. Temperature for 25% weight loss in 2 hours in inert environment. c Temperature for 25% weight loss in 2 hours in air (oxygen). d Data not available. b

54.8 MATERIAL SCIENCE TOOLS ON THE WORLD WIDE WEB Today more and more information is taken from on-line resources. The World Wide Web has become a popular and reliable tool for research in many disciplines including polymer science. Rather than an exhaustive overview of the available web-based resources for polymer scientists, this section is interned to point the scientist to a few notable Internet portals that were useful in preparing this chapter.

FIGURE 54.4. A screenshot of the main page of the MatWeb portal. Reprinted with permission ß (1996-2006) by Automation Creations, Inc.

THERMAL-OXIDATIVE STABILITY AND DEGRADATION OF POLYMERS

/

937

FIGURE 54.5. Example of the Excel spreadsheet generated after MatWeb search. Reprinted with permission ß (1996–2006) by Automation Creations, Inc.

MatWeb, http://www.matweb.com, is a searchable database of over 46,000 metals, plastics, ceramics, and composite materials. It allows search by material type, trade name, range of values, composition, UNS number (Unified Numbering System for Metals and Alloys) and even system of units (metric, common US units). An example of searchable materials includes thermoplastic and thermoset polymers such as ABS, nylon, polycarbonate, polyester, polyethylene, and polypropylene; metals such as aluminum, cobalt, cop-

per, lead, magnesium, nickel, steel, superalloys, titanium, and zinc alloys; ceramics; plus semiconductors, fibers, and other engineering materials. For registered users, all data retrieved from searches can be exported to an Excel spreadsheet for further analysis (Fig. 54.4, 54.5). Another web resource, Omnexus can be found at the following address: http://www.omnexus.com/index.aspx (Fig. 54.6).

FIGURE 54.6. A screenshot taken of the front page of the Omnexus website. Reprinted with permission from Omnexus.com.

938 / CHAPTER 54 This site contains very useful information, such as current news in polymer science, information about on-line seminars and scientific conferences, and numerous material databases. Databases are searchable by various criteria, such as physical and chemical properties, molecular weight or density. The search output also contains information about manufacturer and on-line vendors. This site requires registration for full access, but registration is free. REFERENCES 1. L. D. Loan and F. H. Winslow, in Macromolecules: An Introduction to Polymer Science, edited by F. A. Bovey and F. H. Winslow, (Elsevier Science & Technology, Amsterdam, 1982) p. 576. 2. N. Grassie and G. Scott, Polymer Degradation & Stabilisation (Cambridge University Press, Cambridge, 1988) p. 222. 3. L. Reich and S. S. Stivala, Elements of Polymer Degradation, (McGraw-Hill Book Company, New York, 1971) p. 361. 4. N. Grassie, in Encyclopedia of Polymer Science and Technology, vol.4, (Interscience Publishers, New York, 1966) p. 647. 5. L. I. Nass, in Encyclopedia of Polymer Science and Technology, vol.12, (Interscience Publishers, New York, 1966) p. 725. 6. J. E. Mulvaney, in Encyclopedia of Polymer Science and Technology, vol. 7, (Interscience Publishers, New York, 1966) p. 478. 7. R. B. Seymour and C. E. Carraher, Jr., Structure-Property Relationships in Polymers, (Plenum Press, New York, 1984) p. 246. 8. C. Hall, Polymer Materials, (Halsted Press: John Wiley & Sons, New York, 1989) p. 243. 9. H. R. Allcock and F. W. Lampe, Contemporary Polymer Chemistry, third edition, (Prentice-Hall, Inc., Upper Saddle River, NJ, 2003) p. 832.

10. H.-G. Elias, Macromolecules, second edition, (Plenum Press, New York, 1984), p. 564. 11. F. W. Billmeyer, Jr., Textbook of Polymer Science, third edition (Wiley-Interscience: John Wiley & Sons, New York, 1990) p. 578. 12. D. W. van Krevelen, Properties of Polymers: Their Correlation with Chemical Structure: Their Numerical Estimation and Prediction from Additive Group Contributions, third edition, (Elsevier Scientific, Amsterdam, 1997), p. 875. 13. C. Arnold, Jr., J. Polym. Sci.: Macromolecular Rev. 14, 265 (1979). 14. W. J. Welsh, D. Bhaumik, H. H. Jaffe, et al. Polym. Eng. Sci. 24, 218 (1984). 15. R. C. Evers and G. J. Moore, J. Polym. Sci.: Part A: Polym. Chem. 24, 1863 (1986). 16. W. J. Welsh, in Current Topics in Polymer Science, vol. I, edited by R.M. Ottenbrite, L. A. Utracki, and S. Inoue, (Hanser Publishers, Munich, 1987) p. 217. 17. F. E. Arnold and R. L. Van Deusen, Macromolecules, 2, 497 (1969); R.L. Van Deusen, O. K. Goins, and A. J. Sicree, J. Polym. Sci. A-l, 6, 1777 (1968). 18. W. J. Welsh, D. Bhaumik, and J. E. Mark, J. Macromol. Sci. Phys. 20, 59 (1981). 19. W. J. Welsh and J. E. Mark, in Computational Modeling of Polymers, edited by J. Bicerano, (Marcel Dekker, Inc., New York, 1992) p. 648. 20. R. D. Miller and J. Michl, Chem. Rev. 89, 1359 (1989). 21. N. Grassie, in the Polymer Handbook, fourth ed., edited by J. Brandrup (Editor), Edmund H. Immergut, Eric A. Grulke, Akihiro Abe, Daniel R. Bloch, (Wiley-Interscience: John Wiley & Sons, New York, 2003) p. 2336. 22. J. H. Noggle, Physical Chemistry, third ed., (Pearson Education, New York, 2002), p. 1108. 23. B. E. Douglas, D. H. McDaniel, and J. J. Alexander, Inorganic Chemistry, 2nd ed., (John Wiley & Sons, Inc., New York, 1993) p. 78.

CHAPTER 55

Synthetic Biodegradable Polymers for Medical Applications Laura J. Suggs*, Sheila A. Moorey, and Antonios G. Mikosy y

*Department Biomedical Engineering, University of Texas at Austin, Austin, TX 78712 Department of Bioengineering, Rice University, PO Box 1892, MS-142, Houston, TX 77251-1892

55.1 55.2 55.3 55.4 55.5 56.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biodegradable Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

939 939 947 947 947 948 949

55.1 INTRODUCTION

55.2 BIODEGRADABLE POLYMERS

Biodegradability has been the primary consideration in the development of biomedical materials due to problems associated with the biocompatibility of long-term, nondegradable polymer implants. Biodegradable polymers have been formulated for uses such as sutures, drug delivery devices, scaffolds for tissue regeneration, vascular grafts and stents, artificial skin, orthopedic implants, and others. The purpose of this overview is to elucidate the characteristics of several synthetic biodegradable polymers for medical applications, which include degradation modes and rates and their relationship to physicochemical, thermal, and mechanical properties. Polymers mentioned in the chapter are poly (a-hydroxy esters), poly(e-caprolactone), poly (ortho esters), polyanhydrides, poly(3-hydroxybutyrate) polyphosphazenes, polydioxanones, fumarate-based polymer, polyoxalates, poly(amino acids), and pseudopoly (amino acids). The synthesis, medical uses, and processing techniques of these polymers are not discussed in detail, but additional references are given for each polymer as well as several comprehensive review articles [1–8].

55.2.1 Poly(a-Hydroxy Esters) Poly(Glycolic Acid) Poly(glycolic acid) (PGA) is a highly crystalline, hydrophilic, linear aliphatic polyester (Structure 1). As such, it has a high melting point and a relatively low solubility in most common organic solvents. At room temperature, PGA is soluble in hexafluoroisopropanol, a highly toxic solvent. It degrades primarily by bulk erosion through random hydrolysis of its ester bonds. Reed and Gilding [9] report that the degradation kinetics is biphasic, with the first phase of degradation occurring by diffusion of water to the amorphous regions and subsequent hydrolysis. The second phase begins as water penetrates and hydrolyzes the more crystalline regions. The molecular weight distributions, which show two degradation phases, are given in Fig. 55.1 [9]. For PGA surgical sutures, mass loss occurs primarily during the second phase, completing the entire process between weeks 4 and 12. The rate of hydrolysis can be controlled

939

940 / CHAPTER 55 14 days 7 days 21 days

56 days

28 days

'0' weeks 0 days '2' weeks

1.5 High MW

FIGURE 55.1. Molecular weight distributions as a function of degradation time for PGA sutures at pH 7 and 37 8C. The formation of a bimodal distribution is evident at large times due to the biphasic degradation of PGA. (Reprinted with permission from [9].)

in vitro by varying the pH [10]. Any large deviation from neutral pH drives hydrolytic cleavage. In addition, the degradation rate can be affected by the degree of crystallinity or ‘‘curing time’’ of PGA, as shown in in vivo studies [11]. The crystallinity of PGA is typically between 46% and 52% [9], the maximum crystallinity during degradation occurring in the time between the two degradation phases. The values of crystallinity are not only influenced by the quenching or ‘‘curing’’ process but also the molecular weight of the polymer [12]. PGA (crystallinity 50%) loses most of its mechanical strength over the first 2–4 weeks of degradation [9]. This is asynchronous with the mass loss which begins at approximately week 4. This is due to the bimodal degradation distribution. The amorphous regions are hydrolyzed first which results in loss of mechanical strength, while the degradation and diffusion of low molecular weight chains later result in significant mass loss. The stress/strain curves showing the effect of degradation on mechanical strength are given in Fig. 55.2 [9]. Poly(Lactic Acid) Poly(lactic acid) (PLA) (Structure 2) is also a linear polyester, but the presence of an extra methyl group makes it more hydrophobic than PGA. Its water uptake in thin films is approximately 2% [13]. The methyl group contributes to a more amorphous character as well as increasing its solubility in organic solvents. In addition, this group creates a chiral center which results in two different enantiomeric forms of the polymer, P(D)LA and P(L)LA. The racemic mixture of the two is abbreviated as P(D,L)LA. The most commonly used form is P(L)LA which, like all poly(lactic acids), releases lactic acid upon degradation. PLA is frequently cast from common solvents. These include: chloroform, methylene chloride, methanol, ethanol, benzene, acetone, dioxane, dimethylformamide, and tetrahydrofuran [14–16]. PLA has also been shown to degrade by a homogeneous, hydrolytic erosion [17–19]. For example, P(D,L)LA degrades in a conventional two-stage process where the majority of molecular weight loss occurs in the first stage, and the subsequent loss in mass and tensile strength begins in the second stage at a number average

kgm−2(x 10−7)

Low MW

1.0

'4' weeks

0.5

'6' weeks 0

10

20 % elongation

30

FIGURE 55.2. Stress/strain curves showing the loss of mechanical strength with degradation time for PGA sutures at pH 7 and 37 8C. (Reprinted with permission from [9].)

molecular weight of 15,000 [9]. P(L)LA of molecular weight 95,000 degraded in vivo by 56% in 6 months based on peak molecular weight (Mp ) [20]. For P(D,L)LA between 58,000 and 87,000, 49% degraded in vivo in 1 month, also based on Mp . A half-life of 6.6 months by mass was reported [11] for P(L)LA of molecular weight 85,000. In vitro studies [9] showed a 50% loss in weight average molecular weight (Mw ) in 16 weeks with a concurrent loss of 10–15% by mass. The degradation rate of PLA also varies with varying pH [21,22]. The amount of lactic acid released during the course of PLA degradation is very small but increases rapidly as PLA is broken down to low molecular weight oligomers. A sudden rise in the lactic acid concentration in vivo can render the local environment acidic and induce an inflammatory reaction or even tissue necrosis. The use of polydispersed PLA can result in distribution of the lactic acid production over time [23]. Thermal and mechanical properties of both P(L)LA and P(D,L)LA of various molecular weights are given in Table 55.1. Additional thermal properties of PLA are found in Lu et al. [24]. Poly(Lactic-co-Glycolic Acid) Copolymers The advantage of copolymerizing poly(a-hydroxy esters) is the ability to control physical and mechanical properties;

SYNTHETIC BIODEGRADABLE POLYMERS FOR MEDICAL APPLICATIONS

/

941

TABLE 55.1. Thermal and mechanical properties of respective synthetic biodegradable polymers [4,24,48,74,75,95,96].

Polymer

Glass DecompoWeight sition Heat of Tensile Tensile Flexural Elongation Elongation average transition Melting temp. fusiona strength modulus modulus at yield molecular temp. temp.a at break (8C) (8C) weight (8C) (Jg
1 ) (MPa) (MPa) (MPa) (%) (%)

PLGA P(L)LA P(L)LA P(L)LA P(D,L)LA P(D,L)LA P(D,L)LA Poly(e-Caprolactone)

50,000 50,000 100,000 300,000 21,000 107,000 550,000 44,000

35 54 58 59 50 51 53
62

210 170 159 178 A A A 57

Poly(a-Hydroxy Ester) 254 71 — 242 41 28 235 20 50 255 39 48 255 A — 254 A 29 255 A 35 350 34 16

— 1,200 2,700 3,000 — 1,900 2,400 400

— 1,400 3,000 3,250 — 1,950 2,350 500

— 3.7 2.6 1.8 — 4.0 3.5 7.0

P(CDM-co-HD) 35:65 P(CDM-co-HD) 70:30 P(CDM-co-HD) 90:10

99,700 101,000 131,000

55 84 95

A A A

Poly(Ortho Esters) 358 A 20 362 A 19 338 A 27

820 800 1,150

950 1,000 1,250

4.1 4.1 3.4

— — — — — —

60 47 4 0 15 96

86 66 178 200 205 240

— — — — — —

— — — — — —

370,000 450,000 529,000 227,000

1
1 2
5

Poly(3-Hydroxybutyrates) Copolymers 171 252 51 36 2,500 160 243 32 24 1,400 145 235 12 20 1,100 137 251 7 16 620

2,850 1,600 1,300 750

PTMC

48,000


15

A

Polydioxanones 261 A

PBPA PDTH

105,000 101,000

69 55

A A

Pseudopoly(Amino Acids) 135 A 50 138 A 40

2,600 2,600 2,600 8,200 14,200 8,050 13,090

11.2 11.2 11.2
54.1
44.6
43.5
46.1

A A A 26.5 39.7 25.0 27.7

PSA P(CPP-co-SA) 22:78 P(CPP-co-SA) 41:59 P(CPP-co-SA) 60:40 P(CPP-co-SA) 80:20 PCPP

PHB P(HB-co-HV) 93:7 P(HB-co-HV) 89:11 P(HB-co-HV) 78:22

PPF:PPF-DA 1:2b PPF:PPF-DA 1:1 PPF:PPF-DA 2:1 P(PF-co-EG)c 33:66 P(PF-co-EG) 33:66 P(PF-co-EG) 66:33 P(PF-co-EG) 66:33

Polyanhydrides — 153 — 64 — 8 — 25 — 34 — 111

Poly(Fumarates) — A — A — A — 17.5 — 20.1 — 0.2 — 9.9

— — — — — —

0.5

61 70 64 0.23 0.32 1.06 0.91

3

—

2,150 1,630

2,400 —

857 3,124 923 2,644 806 2,206 2.16 0.87 1.9 3.87 11.02 1.69 5.05 2.39

— — — — — —

2.2 2.3 5.5 8.5

20

— 6 3.3 2 — 6 5 80

220 180 7

— — — — — —

2.5 2.8 17 36

160

3.5 3.5

4 7

5.6 4.3 4.3 — — — —

10.8 11.3 12.9 — — — —

a

The symbol A designates amorphous polymer. PPF:PPF-DA 1:2 refers to the ratio of double bonds present in each monomer. c P(PF-co-EG) was crosslinked with poly(N-vinyl pyrrolidinone). b

however, there is no linear relationship between the physical properties of the constituent homopolymers and their copolymers. Most of these copolymers are amorphous (between approximately 24 and 67 mol% glycolic acid) [13], and therefore, degradation rates are highly dependent on the

relative amount of each comonomer. Copolymers with high or low comonomer ratios are less sensitive to hydrolysis than copolymers with a more equimolar ratio, due to their greater crystallinity. Half-lives for various PLA and PGA ratios are depicted graphically in Fig. 55.3 [11].

942 / CHAPTER 55 [27], vascular grafts [28], drug carriers [29,30], and scaffolds for tissue engineering [31,32]. This is due in part to the FDA approval of these polymers for certain medical applications.

11/2 months

6

4

55.2.2 Poly(«-Caprolactone)

2

0 0

100

PLA PGA copolymer ratio

100

0

FIGURE 55.3. Variation of half-life of PLGA copolymers with the lactic acid and glycolic acid copolymer ratio in vivo. (Reprinted with permission from [11].)

Due to the dependence of the degradation rate of poly (lactic-co-glycolic acid) (PLGA) copolymers on pH, a phenomenon known as autocatalysis occurs where the carboxylic acid monomers released during degradation reduce the pH and further induce degradation [22–25]. For large-scale polymers, autocatalysis causes a heterogeneous degradation where the pH decreases in the center of the polymer, and a differential in the degradation rate is created [26]. Multiple uses of poly(lactic acid), poly(glycolic acid), and their copolymers have been described including sutures

Poly(e-caprolactone) (PCL) is a semicrystalline, aliphatic polyester (Structure 3). It is soluble in tetrahydrofuran, chloroform, methylene chloride, carbon tetrachloride, benzene, toluene, cyclohexanone dihydropyran, and 2-nitropropane; and only partially soluble in acetone, 2-butanone, ethyl acetate, acetonitrile, and dimethyl fumarate [33]. PCL is also capable of forming blends as well as useful copolymers with a wide range of polymers [34]. PCL has been shown to degrade by random hydrolytic scission of its ester groups, and under certain circumstances, by enzymatic degradation [33]. It is similar to P(D,L)LA, in that it degrades in a two-phase process with the molecular weight loss occurring primarily in the first phase, and the major mass and strength loss at the onset of the second at a number average molecular weight of 5,000 [35]. However, PCL degrades almost three times slower than P(D,L)LA [4]. A graph of molecular weight versus time showing the degradation of PCL capsules in vivo is given in Fig. 55.4 [35]. The crystallinity of PCL increases with decreasing molecular weight with polymers of molecular weight above 100,000 being about 40% crystalline. This value increases to about 80% for molecular weights of 5,000 [35]. As a result, PCL behaves like PGA in that the residual crystallinity increases

MOLECULAR WEIGHT (Mn)

105

104

103 50

100 TIME (WEEKS)

150

FIGURE 55.4. Decrease of molecular weight of PCL with the degradation time for PCL capsules loaded with various drugs in vivo. (Reprinted with permission from [35].)

SYNTHETIC BIODEGRADABLE POLYMERS FOR MEDICAL APPLICATIONS as the polymer degrades. The degradation rate of PCL can be increased by forming a copolymer with DL-lactide [36]. In addition, PCL is affected by acidic conditions consistent with an autocatalytic degradation mechanism, and it is also influenced by the addition of small molecules such as ethanol, pentanol, oleic acid, decylamine, and tributyiamine [37]. PCL has a low glass transition temperature of
62 8C, existing always in a rubbery state at room temperature, and a melting temperature of 57 8C. It has been postulated that these properties lead to a high permeability of PCL for controlled release agents. Other thermal and mechanical properties are listed in Table 55.1.

55.2.3 Poly(Ortho Esters) Poly(ortho esters) are amorphous, hydrophobic polymers containing hydrolytically labile, acid-sensitive, backbone linkages (Structures 4, 5, 6). Due to their hydrophobicity, they can easily dissolve in organic solvents including: chloroform, methylene chloride, and dioxane. However, it can be difficult to remove the solvent in a situation such as a solvent casting [38]. In addition, these polymers are not inherently susceptible to degradation in the presence of water, although they can be if anhydrides (acid excipients), glycolic acid, or lactic acid are incorporated. They are susceptible to thermal degradation and must be processed accordingly. Poly(ortho esters) are a class of polymers which can degrade heterogeneously by surface erosion [39]. These

/

943

polymers lose material from the surface only, while retaining their original geometry. As such, their primary use is in drug delivery [40]. The first class of poly(ortho esters), as shown in Structure 4, generates a carboxylic acid upon hydrolysis which then further catalyzes the acid-sensitive cleavage. A basic salt such as Na2 CO3 or Mg(OH)2 is usually incorporated to neutralize the acid product, however, this creates a diffusion-limited system which exhibits nonzero-order drug release characteristics. The second class, represented by Structures 5 and 6, does not produce acidic hydrolysis products, and its degradation can be controlled by the incorporation of either acidic or basic excipients. In the case of acid addition, water penetrates, ionizes the acid, and reduces the pH. This then catalyzes the hydrolysis, resulting in a hydration front and an erosion front. For a basic excipient, water must penetrate, elute, or neutralize the base, and then allow erosion to occur, decreasing the rate of hydrolysis [41]. According to the choice of additive, degradation rates can be varied from several days to years. Acid excipients can also be incorporated into the polymer itself as pendant chains which are solubilized upon cleavage [42]. For example, the degradation rates are enhanced for polymers containing trans-cyclohexanedimethanol (CDM) and 1,6hexanediol (HD) when acidic functionalities, 9,10-dihydroxy-stearic acid (DHSA) [43], are incorporated, as shown in Fig. 55.5. The polymer can also be crosslinked at temperatures as low as 40 8C with an excipient stabilized interior [44].

100

WEIGHT LOSS - PERCENT

80

60

40

20

0

0

10

20

30

40 TIME - DAYS

50

60

70

80

FIGURE 55.5. Variation of cumulative weight loss with time for poly(ortho esters) containing trans-cyclohexanedimethanol (CDM), 1,6-hexanediol (HD), and 9,10-dihydroxy-stearic acid (DHSA) (in the form of 6  0.5 mm disks at pH 7 and 37 8C). S ¼ 58% CDM, 38% HD, 4% DHSA; E ¼ 59% CDM, 39% HD, 2% DHSA; G ¼ 59.5% CDM, 39.5% HD, 1% DHSA; C ¼ 59.7% CDM, 39.75% HD, 0.5% DHSA. (Reprinted with permission from J. Heller, D. W. H. Penhale, and S. Y. Ng. in Long-Acting Contraceptive Delivery Systems, edited by G. I. Zatuchni, A. Goldsmith, J. D. Shelton, and J. J. Sciarra, (Harper and Row, New York, 1984), p. 127.)

944 / CHAPTER 55

55.2.4 Polyanhydrides

100 90

20:80

80

45:55

70 80:20 60 PCPP 50 40 30 20 10 0 0

2

4

6

8

10

12

14

Time, Weeks

FIGURE 55.7. Change of cumulative percent of polymer degraded with degradation time for P(CPP-co-SA) copolymers in the form of compression-molded disks of 1.4 cm diameter and 1 mm thickness at pH 7.4 and 37 8C. (Reprinted with permission from K. W. Leong, B. G. Brott, and R. Langer, J. Biomed. Mater. Res. 19, 941 (1985).) 14

120

pH 10.0 Cumulative Percent Degraded

GLASS TRANSITION TEMPERATURE - °C

Polyanhydrides are a class of hydrolytically unstable polymers that are usually either aliphatic, aromatic, or a combination of the two. Two general representations are given in Structures 7 and 8. These polymers dissolve in common organic solvents including chloroform and methylene chloride and are extremely sensitive to aqueous environments. In addition, they are very reactive and can react with amine or other nucleophilic groups that are present in drugs intended for controlled release. This is true especially at elevated temperatures, for example, as occurs during polymer processing [48].

The degradation of polyanhydrides can be varied from days to years depending on the choice or combination of choices of backbone structure [49,50]. The degradation rate of several different combinations of the aliphatic monomer, sebacic acid (SA), and the aromatic monomer, bis(p-carboxyphenoxy)propane (CPP), is given in Fig. 55.7. The polymer primarily degrades by surface erosion [51–53]. As such, it is a candidate for drug delivery, eliminating the need for additional excipients. Its degradation rate is also sensitive to changes in pH, typically increasing with increasing pH as shown in Fig. 55.8 [50].

Cumulative Percent Degraded1

Functionalizing the third class with lactic acid or glycolic acid produces an autocatalytic polymer [45,46]. Degradation is mediated by surface and bulk erosion, which is controlled by the concentration of the a-hydroxy acid segments [47]. There is a linear relationship between weight loss and lactic acid release suggesting surface erosion, also molecular weight decreases signifying bulk erosion. Unlike PLGA and PLA the bulk of the material does not become acidic; the acid products from hydrolysis are concentrated at the surfaces and are easily diffused away [47]. The mechanical and thermal properties of these polymers can also be varied over a wide range by the selection of starting materials with differing compositions and molecular weights. The tripolymerization of 3,9-bis(ethylidene 2,4,8,10-tetraoxaspiro[5,5]undecane) with mixtures of the rigid diol CDM and the flexible diol HD allows preparation of polymers with controlled glass transition temperature [40] (Fig. 55.6). Other thermal and mechanical properties for P(CDM-co-HD) copolymers are listed in Table 55.1.

100 80 60 40 20

12 10

pH 8.0

8

pH 7.4

6 4 2 0

0 0

20

40

60

80

100

pH 9.0

0

50

100

150 200 Time, Hours

250

300

350

PERCENT 1,6-HEXANEDIOL

FIGURE 55.6. Glass transition temperatures of poly(ortho esters) consisting of 3,9-bis(ethylidene 2,4,8,10 tetraoxaspiro [5,5]undecane) with trans-cyclohexanedimethanol and 1,6hexanediol as a function of the 1,6-hexanediol content. (Reprinted with permission from [40].)

FIGURE 55.8. Change of cumulative percent of polymer degraded with degradation time at varying pH levels for PCPP in the form of compression-molded disks of 1.4 cm diameter and 1 mm thickness at 37 8C. (Reprinted with permission from K. W. Leong, B. G. Brott, and R. Langer, J. Biomed. Mater. Res. 19, 941 (1985).)

SYNTHETIC BIODEGRADABLE POLYMERS FOR MEDICAL APPLICATIONS

+ x

100 % of initial weight

There are a wide variety of processing techniques available for forming polyanhydrides, however, care must be taken in incorporating controlled release agents at high temperatures because of the reactivity of the polymer with the drug and the instability of the polymer itself. The mechanical properties of polyanhydrides are generally poor, tending to be brittle with minimal fiber-forming abilities. Forming copolymers of polyanhydrides increases the mechanical properties, while maintaining their degradation characteristics [54,55]. Copolymers of methacrylated sebacic acid (MSA) and 1,6-bis(carboxyphenoxy) hexane (MCPH) have been shown to have similar mechanical properties of cortical and trabecular bone [56]. These copolymers degrade by surface erosion allowing the scaffold to maintain its structural integrity [56]. In addition, polyanhydrides have been shown to have excellent in vivo biocompatibility [57]. The thermal properties of representative P(CPP-co-SA) copolymers are given in Table 55.1. A detailed presentation of thermal properties is given in Domb et al. and Tamada and Langer [48,58].

+ x

/

945

+ x

+ x

90 80 70 60 50

0

10

20

30

40 Time (d)

50

60

70

FIGURE 55.9. Kinetics of percent of initial weight loss for P(HB-co-HV) copolymers of different copolymer ratios and molecular weights in the form of solvent-cast disks of 2 cm diameter and 0.15 mm thickness at 70 8C and pH 7.4. D ¼ 10% HV, Mw ¼ 750,000; I ¼ 20% HV, Mw ¼ 300,000; C ¼ 12% HV, Mw ¼ 170,000; G ¼ 12% HV, Mw ¼ 100,000; C ¼ 20% HV, Mw ¼ 36,000. (Reprinted with permission from [60].)

55.2.5 Poly(3-Hydroxybutyrate) Copolymer

55.2.6 Polyphosphazenes

Poly(3-hydroxybutyrate) (PHB) is crystalline, thermoplastic polyester made by micro-organisms as an energy storage molecule (Structure 9). As such, it can be enzymatically degraded by certain bacteria. It is often copolymerized with hydroxyvaleric acid (Structure 10) to create poly(3hydroxybutyrate-co-3-hydroxyvalerate), P(HB-co-HV). Solvent casting has been described from solution in chloroform, methylene chloride, and tetrahydrofuran [59,60]. The degradation of PHB produces D-3-hydroxy butyric acid, normally found in human blood, which may contribute to its low toxicity. There is evidence for both enzymatic and hydrolytic degradation in vivo [61]. In vitro studies [59, 60] suggest that PHB and P(HB-co-HV) copolymers degrade by hydrolysis in a multistage process where the majority of the molecular weight loss occurs before any significant mass loss. A graph of weight loss for various P(HB-co-HV) copolymers is given in Fig. 55.9 [60]. The copolymerization of hydroxybutyric acid with hydroxyvaleric acid increases the percentage of amorphous regions compared to PHB, which are readily attacked by hydrolytic degradation thereby increasing degradation rates. In addition, elevated temperatures and alkaline conditions have been shown to increase degradation rates. The crystallinity and mechanical properties of the P(HBco-HV) copolymer can be varied by modification of the percentages of the respective monomers. The higher the percentage of hydroxyvalerate, the less crystalline and the more elastic the polymer becomes. Some thermal and mechanical properties are presented in Table 55.1. A study of thermal characteristics in vivo is given in Gogolewski et al. [61], and a mechanical evaluation in vivo and in vitro is found in Miller and Williams [62].

Polyphosphazenes consist of a backbone of alternating nitrogen and phosphorus atoms (Structure 11). The R and R’ groups on either side of the phosphorus can be widely varied depending on the route of synthesis. The choice of functional groups determines the physical and chemical properties of the polymer [63,64]. Some important types of polyphosphazenes that have been synthesized are nonhydrolyzable, hydrophobic polymers; nonhydrolyzable, hydrophilic polymers; and hydrolyzable polymers. Those in the first class include polymers with side fluoroalkoxy, aryloxy, or organosilicon hybrid groups. These polymers are usually elastomers with water contact angles on the order of poly (tetrafluoroethylene) [65]. The second class consists of polymers with alkylamino, alkylether, alcohol, carboxylic acid, glyceryl, or glucosyi functionalities. These can be quite hydrophilic and are often crosslinked to form hydrogels. The third class of polymers includes those that can be hydrolyzed to form phosphate and ammonia derivatives. Some important side groups include: amino acid esters, steroidal groups, imidazolyl groups, and other bioactive molecules. In addition, the surface can also be activated for use in controlled release.

55.2.7 Fumarate-Based Polymers The following polyesters are based on fumaric acid, a naturally occurring substance found in the Krebs cycle [8]. Three types of fumarate-based polymers are discussed: poly(propylene fumarate) (PPF), poly(propylene fumarateco-ethylene glycol) (P(PF-co-EG)), and oligo(poly(ethylene glycol) fumarate) (OPF).

946 / CHAPTER 55 Poly(Propylene Fumarate) Poly(Propylene Fumarate) (PPF) is a linear, unsaturated, hydrophobic polyester (Structure 12) containing hydrolyzable ester bonds along its backbone. PPF is highly viscous at room temperature and is soluble in chloroform, methylene chloride, tetrahydrofuran, acetone, alcohol, and ethyl acetate [66]. The double bonds of PPF can form chemical crosslinks with various monomers, such as N-vinyl pyrrolidone, poly(ethylene glycol)-dimethacrylate, PPF-diacrylate (PPFDA), and diethyl fumarate [67,68]. The choice of monomer and radical initiator directly influence the degradative and mechanical properties of the crosslinked polymer. Once crosslinked, PPF forms a solid material with mechanical properties suitable for a range of bone engineering applications. PPF crosslinked with either thermal- or photo-initiators exhibits a biphasic degradation at 37 8C. During the initial phase of degradation, PPF’s mechanical strength increases, whereas the mechanical strength diminishes in the second phase [69,70]. This phenomenon can be explained by the fact that, at 37 8C, enough energy is provided for the entrapped initiators to sustain the crosslinking reaction [70,71]. To produce a crosslinked polymer of composition similar to that of the uncrosslinked polyester, diethyl fumarate or a derivative of PPF, PPF-diacrylate (PPF-DA) is used as a crosslinker [71,72]. Particulate ceramics such as b-tricalcium phosphate (b-TCP) can also be incorporated within the network to modify the crosslinked polymer’s mechanical properties [67]. Hybrid alumoxane nanoparticles can also be incorporated in PPF to provide mechanical reinforcement [73]. Poly(Propylene Fumarate-co-Ethylene Glycol) Poly(propylene fumarate-co-ethylene glycol) (P(PF-coEG)), (Structure 13), is an amphiphilic block copolymer of PPF and poly(ethylene glycol) (PEG). P(PF-co-EG) is soluble in toluene, N, N-dimethylformamide, tetrahydrofuran, and acetone [74]. Similar to PPF, P(PF-co-EG) degrades via hydrolysis of the ester bonds found along its backbone [74]. Unlike PPF, the crosslinked P(PF-co-EG) forms hydrogels. Increasing the amount of PEG within the copolymer increases its hydrophilicity, thus encouraging an influx of water within the network and inducing the material to swell [75]. Similarly, increasing the concentration and/or molecular weight of the PPF block reduces the degree of swelling [75]. The relative amount of the PPF block also affects the mechanical properties of the crosslinked P(PF-co-EG). PPF is the only portion of the copolymer that can form covalent bonds for crosslinking, so more PPF block result in more possible crosslinks, yielding a stronger material [75]. Additionally the hydrophobic PPF moieties can interact with each other, forming secondary interactions that further

strengthen the material. A compilation of thermal and mechanical properties for P(PF-co-EG) are listed in Table 55.1. Oligo (Poly(Ethylene Glycol) Fumarate) The final type of fumarate-based polymer discussed, oligo (poly(ethylene glycol) fumarate) (OPF) (Structure 14), is a highly hydrophilic, linear, unsaturated polymer, composed of alternating PEG and fumarate moieties [76]. OPF is soluble in aqueous and organic solvents [76]. Like all fumarate-based polymers, crosslinking occurs through the fumarate groups and degradation is mediated by hydrolysis of the ester bonds. Similar to P(PF-co-EG), the PEG block gives OPF its hydrophilicity. In addition, OPF’s properties are controlled by the ratio of fumarate to PEG and the molecular weight of the PEG. Increasing the molecular weight of the PEG produces a less crosslinked, and more swollen hydrogel [76,77]. Moreover, increasing the fumarate to PEG ratio increases the number of crosslinks within the network and decreases the swelling of the hydrogel [76]. Due to their high hydrophilicity, OPF hydrogels have been used to encapsulate mesenchymal stem cells for bone engineering applications [78,79]. 55.2.8 Polydioxanones and Polyoxalates Four important classes of polymers from dioxane-diones and oxalates are poly(l,4-dioxane-2,5-diones), polyoxalates, poly(l,3-dioxane-2-one) and poly(l,4-dioxane-2,3-dione), and poly(p-dioxanone). Representative diagrams are given in structures 15, 16, 17, and 18, respectively. The first class has been produced with an alternating glycolide/lactide sequence. Both PGA and PLA have been mentioned previously, and the physical properties of the alternating copolymer are a weighted average of the two homopolymers. Secondly, a polyoxalate has been reported [80] with an ester backbone, which can be hydrolytically cleaved to produce propylene glycol and oxalic acid. The predicted degradation rate is faster than PGA owing to its lower degree of crystallinity and less hydrophobic character. The third class primarily consists of polymers of 1,3dioxane-2-one otherwise known as trimethylene carbonate (TMC) and its copolymers with glycolide and lactide. PTMC degrades at a much slower rate than PGA. In addition, it softens between 40 8C and 60 8C, has low mechanical strength [5], and is reported to improve handling properties in copolymers with PGA [4]. Some thermal and mechanical properties of PTMC are shown in Table 55.1. Lastly, poly(p-dioxanone) is thought to degrade by a mechanism similar to PGA [81]. The backbone is hydrolytically cleaved in a bulk erosion process with the major weight loss occurring between weeks 12 and 18 [82]. It has superior strength characteristics compared to PGA as well as high crystallinity up to 37%.

SYNTHETIC BIODEGRADABLE POLYMERS FOR MEDICAL APPLICATIONS

/

947

55.2.9 Poly(Amino Acids)

55.3 SUMMARY

Poly(amino acids) as shown in Structure 19 are synthetically derived polymers which can be prepared from a variety of amino acids. The physical and chemical properties depend, in large part, on the functionalities of their respective side chains, however, poly(amino acids) have some common features. Most are highly insoluble in organic solvents, and they tend to swell in aqueous solution. They have poor mechanical properties and are difficult to process. In addition, the hydrolysis of the amide bond has an enzymatic contribution that is difficult to predict or control in vivo. The degradation products, amino acids, are natural components of proteins and should not cause a toxic response upon degradation, however, polymers containing three or more amino acids can elicit a strong immunologic response [83]. Additional properties for specific combinations of amino acids are given in Banera et al. [84]. Certain side chain modifications have been made in order to avoid some of these limitations. Poly(L-lysine) [85,86] and poly (L-glutamic acid) [87] have both been modified through their chemically reactive side chains to produce hybrids with bioactive molecules.

While the previous list summarizes most of the currently used biodegradable polymers as well as some new materials, and while it describes the state-of-the-art at this time, it is certainly not exhaustive. There are many new products being developed as well as novel modifications of the polymers described with in the chapter. Ideally, polymers can be chosen and tailored for a specific application based on their physical and chemical properties. We have shown properties that are crucial to the function of the polymer in question and also give sources where additional information can be found.

55.2.10 Pseudopoly(Amino Acids) Pseudopoly(amino acids) are polymers derived from amino acids with nonamide linkages; these are represented by the wavy line in Structures 20, 21, and 22. This is usually done by the polymerization of trifunctional amino acids by reaction with side chain functional groups. Three important categories include: serine derived polyesters [88] hydroxyproline derived polyesters, and tyrosine-derived polymers. The first has not been widely used as a biomaterial [89]. The second group consists of poly(N-acyl-hydroxyproline esters) from N-protected hydroxyproline. These polyesters are soluble in benzene, toluene, chloroform, dichloromethane, carbon tetrachloride, tetrahydrofuran, and dimethylformamide. They are thermally stable up to 300 8C, have glass transition temperatures ranging from 71 8C to 157 8C, and are easily processed [89]. The third group consisting primarily of modified polycarbonates [90] can be derived from diphenols such as hydroquinone or Bisphenol A [91] (BPA), or a tyrosine dipeptide such as desaminotyrosyl-tyrosine hexyl ester (DTH). PDTH was found to be relatively strong with good biocompatibility [89,92,91]. The degradation of the tyrosine-derived polycarbonates is controlled by the hydrolysis of the ester and carbonate bonds [93]. Carbonate bonds will hydrolyze at a faster rate than the ester bonds, which leads to an initial reduction of molecular weight without mass loss [93,94]. Its reported half-life is 26 weeks, but it can take up to 4 years before the polymer is completely resorbed [91,93]. Additional thermal and mechanical data is given in Table 55.1

55.4 ACKNOWLEDGMENTS We acknowledge support by the National Institutes of Health and the Alliances for Graduate Education and the Professoriate related to synthetic biodegradable polymers for medical applications. 55.5 APPENDIX Abbreviations b-TCP BPA CDM CPP DHSA DTH HD Mp Mw OPF PBPA P(CDM-co-HD) PCL PCPP P(CPP-co-SA) P(D)LA P(D,L)LA PDTH PEG PGA PHB P(HB-co-HV) PLA PLGA P(L)LA PPF PPF-DA

b-tricalcium phosphate Bisphenol A trans-cyclohexanedimethanol bis-(p-carboxyphenoxy)propane 9,10-dihydroxy-stearic acid desaminotyrosyl-tyrosine hexyl ester 1,6-hexanediol peak molecular weight weight average molecular weight oligo(poly(ethylene glycol) fumarate) poly(Bisphenol A) poly(trans-cyclohexanedimethanolco-1,6-hexanediol) poly(e-caprolactone) poly(bis-(p-carboxyphenoxy)propane) poly(bis-(p-carboxyphenoxy)propaneco-sebacic acid) D enantiomer of poly (lactic acid) racemic mixture of D and L enantiomers of poly(lactic acid) poly(desaminotyrosyl-tyrosine hexyl ester) poly(ethylene glycol) poly(glycolic acid) poly(3-hydroxybutyrate) poly(3-hydroxybutyrateco-3-hydroxyvalerate) poly(lactic acid) poly(lactic-co-glycolic acid) L enantiomer of poly(lactic acid) poly(propylene fumarate) PPF-diacrylate

948 / CHAPTER 55 P(PF-co-EG)

Structure 8: Polyanhydride

poly(propylene fumarate-co-ethylene glycol) poly(sebacic acid) poly(trimethylene carbonate) sebacic acid trimethylene carbonate

PSA PTMC SA TMC

O

O

O

R O

O n

Structure 9: Poly(3-hydroxybutyrate)

56.6 CHEMICAL STRUCTURES Structure 1: Poly(glycolic acid)

O O O

O n

n

Structure 2: Poly(lactic acid)

Structure 10: Poly(3-hydroxybutyrate)

O

O

O

O

n

n

Structure 3: Poly(ε-caprolactone) O

Structure 11: Polyphosphazenes

R'

O 5

P

n

N

Structure 4: Poly(ortho ester)

R' O

O

n

R

Structure 12: Poly(propylene fumarate)

O

O n

O

HO

Structure 5: Poly(ortho ester)

OH

O O

R'

n

O

Structure 13: Poly(propylene fumarate-co -ethylene glycol) O

R

O

O

n

Structure 6: Poly(ortho ester)

O

O n

R'

O

O O

O O

m

Structure 14: Oligo (poly(ethylene glycol) fumarate) O

O

O

O

R

O

n

H

O

O n

Structure 7: Polyanhydride

O O

O

H

n m

O

O

O

R n

n

O

R'

O

O

O

SYNTHETIC BIODEGRADABLE POLYMERS FOR MEDICAL APPLICATIONS

949

Structure 22: Pseudopoly(amino acid)

Structure 15: Poly(1,4-dioxane) R'

/

R

O

R

R" O O

N H

R'

N H n

O

R'"

O

O

n

Structure 16: Polyoxalate

REFERENCES

O O

O R O n

Structure 17: Poly(1,3-dioxane-2-one)

O R O

O n

Structure 18: Poly(p-dioxanone)

O

R' R' O

O R' R

R' n

Structure 19: Poly(amino acid)

R

N H O n

Structure 20: Pseudopoly(amino acid) O

O R

R

NH

NH n

Structure 21: Pseudopoly(amino acid)

COOH

N H

R

COOH

N H

R n

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CHAPTER 56

Biodegradability of Polymers Anthony L. Andrady Engineering & Technology Dirsian, RTI International, Research Triangle Park, NC 27709

56.1 56.2 56.3 56.4 56.5 56.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Environmental Biodegradability of Biopolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem of Assessing Biodegradability of Synthetic Polymers . . . . . . . . . . . . . . . . . Biodegradability of Synthetic Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Modifications to Enhance Biodegradability of Polymers . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

951 953 955 956 957 961 961

even whole leaves might be used as reference materials in biodegradation studies. In general, biodegradation of polymers occurs as an extracellular process (because macromolecular dimensions do not permit their transport across cell membranes), catalyzed by enzymes. A number of such enzymes are known and are classified on the basis of the degradation reaction step they catalyze. Thus, hydrolases, esterases, isomerases (or transferases), oxido-reductases, hydrogenases, and ligases can increase the rates of respective reactions by 6– 20 orders of magnitude even under ambient temperatures [6]. Enzymes that specifically catalyze the breakdown of naturally occurring polymers such as cellulose, lignin, chitin, and proteins are readily available in nature. For synthetic polymers, with a much shorter history of less than half a century of use, appropriate enzymes are more difficult to find in nature. Given the impressive diversity of microbiota, as yet unknown enzyme systems for synthetic polymers might very well exist. Exceptions to the rule of recalcitrance of synthetic polymers include aliphatic polyesters, polyethers, some polyamides, and poly(vinyl alcohol). As most of the readily biodegradable polymers are waterinsoluble, the degradation reaction must be heterogeneous, initially localized at the surface of the polymer. Close contact between the biota and polymer is generally a prerequisite to ensure the high concentration of enzymes to enable these reactions. With biopolymers, the general process involves both exo- and endo-enzymes. The former yields fragments from chain ends, while the latter causes random

56.1 INTRODUCTION Biodegradation might be conveniently defined as ‘‘a chemical change in polymer facilitated by living organisms, usually micro-organisms’’ [1]. This definition is somewhat restricted, however, in that it excludes chemical breakdown processes on polymer substrates mediated by manmade enzymes. Inspite of its emphasis on microbial processes, the definition covers both key areas of interest: environmental biodegradability of polymers, and in vivo biodegradability of polymers in mammalian systems. The present discussion is limited to the environmental biodegradation of polymers. Adopting a definition for practical purposes, particularly to delimit those polymers that are ‘‘environmentally biodegradable,’’ is more complicated and several definitions have been proposed [2]. There is renewed interest in the use of such polymers in disposable packaging materials to ensure their degradation in post-consumer litter and waste streams. All organic materials must invariably biodegrade (despite the extremely slow kinetics in the case of most synthetic polymers) in the environment; however, to be of practical benefit a readily biodegradable polymer must break down due to biotic causes in a reasonable timescale. With no agreed benchmark available to indicate what such a ‘‘reasonable’’ rate of biodegradation might be, the use of natural biopolymers as standard biodegradable materials [3] is a common trend reported in the literature. Because of their rapid breakdown, regenerated cellulose or filter paper [4], wood pulp [5], and 951

952 / CHAPTER 56 main-chain scission. The fragments, such as cellobiose in the case of cellulose, might then be further biodegraded by specific enzymes. A distinction needs to be made between true biodegradation and biologically mediated disintegration or volume reduction of polymers, which does not amount to biodegradation. The attack of polyethylene by insects [7,8] for instance, belongs to this latter category. In spite of the ‘damage’ suffered by the polymer, the predominant change is physical, and the indigestible polymer is merely reduced in particle size at the end of process. The blends of a biologically inert thermoplastic, such as polyethylene, and a readily biodegradable substance, such as starch, also belong to the same class of biodeteriorable materials. On biodegradation of surface starch, a thin film of the composite material disintegrates into small particulates without substantial chemical breakdown of the polymer. A second class of biodegradable polymers of interest are those used in the human (or animal) body. These polymers include those used in artificial organs, other implants, and controlled release devices for delivery of pharmaceuticals. Being placed in contact with the tissue environment, they can potentially biodegrade. In products such as biodegradable sutures and bioerodible drug-delivery matrices, such breakdown in the body may be undesirable. Experimental data reported on biodegradation of polymers are somewhat limited. However, from a consideration of the available data and the characteristics of the biodegradation process, several factors that affect the environmental biodegradability of polymers might be identified.

56.1.1 Molecular Weight Long chain-like molecular geometry and high molecular weights do not necessarily preclude biodegradation. Both biopolymers (cellulose, chitin), as well as some synthetic polymers (i.e., polycaprolactone) are readily biodegradable. However, a general relationship does exist between the average molecular weight of polymers and their amenability to biodegradation: the shorter the chains, the higher the likelihood of biodegradation [9–11]. Not only does a lower degree of polymerization yield a higher concentration of chain end groups, but it also discourages the formation of crystalline domains that are generally difficult to biodegrade. High chain-end concentrations [12] promote exotype reactions, and noncrystalline regions are known to be preferentially biodegraded in synthetic polymers [13–16] as well as in the biopolymer lignocellulose [17]. Even with polyethylenes generally regarded as being bioinert, the lower molecular weight fractions biodegrade and yield carbon dioxide product at a measurable rate. Gel permeation chromatography (GPC) was recently used to demonstrate the biodegradability of polyethylene wax exposed to bacteria and fungi. The beta oxidation rates for the low-molecular weight polyethylene exposed to bacterial

consortia were 36 times higher compared to that exposed to Aspergillus sp. isolated from soil [150]. Several researchers [151,152] reported the average molecular weight of polyethylenes exposed to biotic environments to slightly increase compared to those incubated in sterile media. The observation is likely a result of the lower molecular weight fraction of the sample with a higher concentration of chain ends being selectively biodegraded. (However, the alternative explanation of the formation of a surface biofilm that limits oxygen availability to the polymer has also been proposed [153].) Even with LDPE exposed to biotic environments (cultured compost microorganisms were used at 95 8C) it is the short chain branches on the chains that are preferentially biodegraded [154]. Observations on common plastics (LDPE, PVC, PS, and urea formaldehyde) subjected to long-term (32 years) soil-burial studies show only the LDPE films to be surface-degraded to any significant extent. The surface molecular weight of these samples decreased to almost 50% of that of the bulk [155,156]. Partially photodegraded polymers contain low-molecular weight fraction of degradation products and therefore biodegrade at a faster rate compared to the virgin material [157,158]. Similarly prethermal degradation of biodegradable polymers such as PHB, PHBV, and PCL copolymer also was reported to increase the rate of biodegradability under compost conditions [159]. The same was also reported for starch/LDPE blends [160]. Degradation of the polymer yields hydrophilic carboxylic acids, susceptible to easy biodegradation, as a major degradation product. These functional groups appear to be preferentially biodegraded under biotic exposure. Unlike with the thermo-oxidative degradation of LDPE where the concentration of carbonyl groups increase with exposure, biodegradation results in a decrease of these hydrophilic groups with the duration of exposure [161,162].

56.1.2 Structural Complexity In most cases, the biodegradability implies the existence of a set of micro-organisms able to utilize the polymer substrate as a carbon and energy source. Since this has to be accomplished with minimum expenditure of energy by the organism, complex polymers requiring numerous enzyme-mediated steps for their breakdown represent a poor substrate [18]. Often, the required ensemble of enzymes is not available from a single species of micro-organism, and the substrate requires several different organisms to act in concert to effect biodegradation. Increased structural complexity of a substrate generally leads to recalcitrance in the environment. Persistence of soil humic acids, naturally occurring biopolymers in soil, are attributed to their structural complexity [19,20]. This assumes that biologically mediated breakdown of organic compounds always involves the use of substrate as a source of energy. Co-metabolism is an important exception in which the biodegradation does not yield any energy

BIODEGRADABILITY OF POLYMERS for use by the organism contributing the enzyme. Nevertheless, co-metabolism is common in nature, and is a true biodegradation process to the extent that it depletes the substrate polymer. 56.1.3 Hydrophilicity Water-soluble synthetic polymers such as poly(vinyl alcohol) [21], poly(acrylic acid) [6], and polyethers tend to be more biodegradable than water-insoluble polymers of comparable molecular weight. Increasing the hydrophilicity of a polymer by chemical modification also increases its biodegradability [22]. The functional groups that impart watersolubility may also contribute to ready biodegradability of these systems. From a microbiological standpoint, the presence of a dissolved substrate may induce the production of necessary enzymes within the micro-organisms. 56.2 ENVIRONMENTAL BIODEGRADABILITY OF BIOPOLYMERS Unlike xenobiotic substrates, biopolymers such as cellulose have been in the eco-system for a very long time, allowing the evolution of efficient enzymatic pathways specific for the breakdown of these substrates. Common biopolymers therefore readily undergo biodegradation in a wide variety of environmental conditions ranging from aerobic compost heaps to anoxic deep-sea marine sediments. 56.2.1 Cellulose The average degree of polymerization (DP) of cellulose depends upon the source; values of 153 300 for California cotton [23] and 2000–6000 for Valonia sp. [24], have been reported. Pulp and viscose (cellophane) are processed celluloses with drastically reduced DP ranging in the low thousands at best. The lower-DP polymer is generally more readily biodegraded. The common micro-organisms involved in cellulose biodegradation are summarized in Table 56.1. These include both bacteria and fungi; the deleterious effect of white-rot and brown-rot fungi on lignocellulose is well known. Several enzymes act synergistically in the breakdown of cellulose in a series of hydrolysis reactions [25,26]. Endocellulases attacking the amorphous regions of the celluloses cause random chain scission. The exo-cellulases act at terminals of chains splitting off cellobiose units that, in turn, are hydrolyzed by b-glucosidase. Lignin component, often found associated with cellulose, also can biodegrade via oxidative pathways. Relevant enzymes (such as lignases, laccase, alcohol oxidase) have been reported [27,28]. Cellulose fillers [172,173] including some types of wood fibers [174] have been used as a filler with thermoplastics to obtain biodegradable materials with improved film quality. Crude cellulose in the form of surface-modified flax fibers

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reinforce biodegradable polyesters [175]. Particularly interesting are cellulose filled composites of biodegradable resins such as poly(propylene carbonate) filled with short lignocellulosic fibers. Cellulose being less hydrophilic compared to biodegradable additives such as starch, yields materials of good mechanical properties [176]. Cellulose is often found in close association with lignin fibers (hence is strictly lignocellulose) in mechanical pulps or flax fiber material. In these, the cellulose component is rapidly biodegradable in the environment while lignin biodegrades but at a much slower rate. However, lignin has also been used in plastic materials to impart some degree of biodegradability. Lignin grafted to PVAc, and PVA enhanced the biodegradability of these materials while that grafted to the readily biodegradable PLC decreased the materials overall its biodegradability [177]. The effect of compatibilizers on cellulose acetate – organoclay was recently reported [178].

56.2.2 Chitin and Chitosan Chitin that occurs in the exoskeleton of invertebrates (such as mollusks and arthropods) is composed of N-acetyl-D-glucosamine residues linked by 1,4 b-linkages. A partially deacetylated chitin also occurs naturally as chitosan. Microbial species responsible for the breakdown of chitin and chitosan have not been comprehensively studied. Micro-organisms found in a variety of environments (for instance, in fresh water [29], marine sediment [30], garden soil [31], and even anaerobic environments [32]) are known to produce chitinases and/or chitosanases. Table 56.1 shows a listing of some reported species of bacteria and fungi that yield these enzymes and are therefore, able to biodegrade these polysaccharides.

56.2.3 Starch A polysaccharide made of linear amylose chains and branched amylopectins, starch is well known to be readily biodegradable [33]. The a 1–4 linkages in both components are easily hydrolyzed by amylases while the a 1–6 links at branch points in amylopectin are attacked by glucosidases. Biodegradable multiphase systems that include starch as one of the phases has been reviewed [163]. (Interestingly, proteins such as crosslinked furfural-soy protein concentrates have also been used in place of carbohydrate polymers for biodegradable polymers [164] or as biodegradable fillers in polyesters [165]. It is important to note that it is only the starch content of the composite that is biodegradable. The biodegradable systems typically include a pro-oxidant additive to facilitate rapid thermooxidative breakdown of the synthetic polymer [166]). To that extent, these systems display not only biodegradation of the starch but concurrent thermooxidative degradation of the synthetic polymer.

954 / CHAPTER 56 Systems where the synthetic polymer matrix is also biodegradable have been reported in recent literature. The morphology and interface properties [167] as well as the biodegradation [168] of starch with poly(lactic acid) was recently reported. A silica filled crosslinked starch/polyacrylamide composite showed superabsorbancy as well as enhanced biodegradability by sewage sludge inoculum as well as specific microorganisms (Bacillus cereus and

E. coli), (note, however, that polyacrylamide is not an enhanced biodegradable polymer). Starch–poly (propylene carbonate) composites not only yielded a fully biodegradable composite material, but also a composite with improved the mechanical properties [169]. PVA–starch composites were processed into a foam with biodegradable as well as good mechanical properties [170]. PVA is biodegradable, but at a much slower rate compared to PCL or

TABLE 56.1. Microbial biodegradation of cellulose and chitin. Substrate Cellulose

Class Bacteria

Fungi

Ascomycetes and fungi imperfecti

Chitin

Bacteria and fungi

Chitosan

a

[W] and [B] refer to white-rot and brown-rot fungi, after [98].

Micro-organism

Reference

Cellvibro gilvus Clostridium thermocellum Bacteroides succinogenus Ruminococcus albus Psudomonas fluorescence var cellulosa Sporocytophaga myxococcides Coriolus vesicolor [W]a Phanerochaete chrysosprium [W]a Irpex lacteus [W]a Schizophyllum commune [W]a Fomes annosus [W]a Stereum sanguinolentum [W]a Peurotus ostreatus [W]a Polyporus schweinitzii [B]a Poria Placenta [B]a Poria Vailantii [B]a Coniophora cerebella [B]a Tyromyces palustris Serpula lacrymans [B]a Lentinus lepideus [B]a Chaetomium globosum Chaetomium thermophile Trichoderma viride Trichoderma reesei Trichoderma koningii Penicillium funiculosum Fusarium solani Aspergillus aculeatus Aspergillus niger Sporotrichum thermophile Myrothecium verrucaria Myxobacteria spp. Psudomonas spp. Serratia spp. Bacillus spp. Pseudmonas spp. Flavobacterium spp. Streptomyces antibioticus Streptomyces griseus Penicillium oxalicum Streptomyces erythaeus Trichoderma harzianum Streptomyces orientalis Aspergilus niger Myxobacter Streptomyces griseus Streptomyces spp.

[53] [54] [55] [56] [57] [58] [59,60] [61] [62] [63] [64] [65] [66] [67] [68,69] [70] [71] [71,72] [73] [74] [75] [76] [77,78] [77,78] [77,78] [79] [80] [81] [82] [83] [84] [85] [86,87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97]

BIODEGRADABILITY OF POLYMERS cellulose. Lactide fillers have also been successfully used in place of starch with a biodegradable polymer matrix (copolymers of 1,3,trimethylene carbonate) to obtain environmentally biodegradable materials [171]. 56.2.4 Polyhydroxyalkanoates (Bacterial polyesters) These polymers are produced as intracellular storage materials in a variety of bacteria grown under physiologically stressed conditions. Specific species, such as Alcagenes eutrophus (cultured under ammonium-limited growth conditions), and Rhodobacter sphaeroides (cultured under phosphate or sulfate-limited growth conditions) can yield as much as 80% dry weight of the polyester. The use of mixed organic carbon sources during bacterial fermentation allows the production of a variety of polymers and copolymers of this class. Poly-(hydroxybutyrate), PHB, and the random copolymer of poly(hydroxybutyrate-co-valerate), PHBV, have been the most studied of this class of biopolymers [34]. The structure of the 100% isotactic polymer is given below [6].

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56.3 PROBLEM OF ASSESSING BIODEGRADABILITY OF SYNTHETIC POLYMERS Synthetic polymers generally biodegrade very slowly, but several exceptions exist. These exceptions undergo biodegradation in the environment at a measurable rate and are commonly referred to as ‘‘biodegradable plastics.’’ Several experimental approaches to establishing the rates and extents of their biodegradation under specific exposure conditions are available. To quantify the rates of breakdown, it is important to define a criterion for assessment of polymer biodegradability. With a hypothetical polymer, it is convenient to represent the biodegradation process by the following generalized sequence [1]. High polymer

Low polymer

Organic intermediate

O2 CO2 + H2O + Energy

O Biomass Respiration OCHCH2C

R=-(CH2)x-CH3, x=0-8 or higher.

R

Thermal and mechanical properties of several of the copolymers have been reported [35,36]. Biodegradation of these polymers by bacterial esterases yield monomers, dimers and trimers split off from the hydroxyl-terminal of the polymer chain. 45.2.5 Other Filler Materials A variety of fillers are used with biodegradable polymer materials to obtain the mechanical properties needed for specific applications. As reported in the literature these additives do not generally alter the biodegradability of the polymer matrix. The effect of the new nanosized filler materials in this regard, however, is interesting and has not been extensively studied. Because of their superior reinforcing characteristics [179] nanofillers such as montmorillonite (nanoclays) and hectile silicates are likely to be a popular additive and its impact on biodegradability needs to be studied. With silica fillers in a range of biodegradable polymers (layered nanocomposites), the general trend was toward increased biodegradability [180]. With nanoclays as well, the same trend was observed for both photo- and biodegradation [181].

The sequence suggests that the criterion for assessment of biodegradation depends upon the particular definition of biodegradation adopted. For instance, the rate of depletion of the polymer substrate might be adopted as the approach; alternatively, the rate of carbon dioxide generation might be used in its place. Under identical conditions, the rates of biodegradation of the same substrate, obtained using these two approaches, will be quite different. The choices of tests available are listed in Table 56.2, along with references on their use to determine the environmental biodegradability of polymers (or organic substrates). Each approach focuses on a different stage of the biodegradation process [37]. Consequently, the results from different tests on the same substrate are not comparable. This is demonstrated in a comparison of the test data on aliphatic polyesters. Lenz [6] compared the data by Potts et al. [38] for surface colonization of polymers by micro-organisms (i.e., biomass yield) with data on weight loss in soil burial and on hydrolysis by fungal lipases, for the same polymers. As expected, the rankings of five polyesters in terms of their biodegradability, estimated using two different criteria, were quite different. Recent work by Yakabe et al. [39] showed that for PHBV, cellulosics and poly(caprolactone) substrates biodegradability as measured by the MITI standard test, and fresh sewage sludge exposure test, showed poor agreement. However, they noted the halflifetimes for the substrates in sewage sludge exposure, and soil burial exposure to show a moderate degree of correlation.

956 / CHAPTER 56 TABLE 56.2. Summary details for six biodegradability tests based on OECD guidelines. Test method (OECD)

[S]

Units

[I]

Die-away (301 A)

10–40

mg DOC/l