Helmut Münstedt Friedrich Rudolf Schwarzl Deformation and Flow of Polymeric Materials Deformation and Flow of Polymeric Materials Helmut Münstedt Friedrich Rudolf Schwarzl • Deformation and Flow of Polymeric Materials 123 Helmut Münstedt Friedrich Rudolf Schwarzl Lehrstuhl fürPolymerwerkstoffe Friedrich-Alexander-Universität Erlangen-Nürnberg Erlangen Germany ISBN 978-3-642-55408-7 ISBN 978-3-642-55409-4 (eBook) DOI 10.1007/978-3-642-55409-4 Springer Heidelberg NewYork Dordrecht London LibraryofCongressControlNumber:2014940158 (cid:2)Springer-VerlagBerlinHeidelberg2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Duringthelastfewdecades,theuseofpolymericmaterialshasgrownsothatthey form the most important class of materials if counted by volume and it can be regardedascertainthattheirgrowthrateswillremainhighinthefuture.Whileat the beginning of the era of polymers Chemistry was the predominant science, EngineeringandPhysicshavebecome moreandmorerequisitefor the successful development of polymeric materials. Engineering, as the efficient production of materialsofgoodqualityandtheirhighlysophisticatedprocessinghavedeveloped into key factors of economic success; and Physics, as material properties have become decisive for specific applications and the continuing innovation and improvement of products. These requirements cannot be fulfilled without a fundamental knowledge of polymeric materials and a profound understanding of the relations between the molecular structure of polymers and their physical and processing properties. These developments and the increasing number of joint uses of various mate- rials in highly sophisticated technical products led to the foundation of special institutes devoted to materials science at some universities about 40 years ago. Besides classical materials like metals, glass, and ceramics, polymers became a central point of academic teaching and research. This book has its origin in lec- tures for students in the field of polymeric materials within the Department of MaterialsScienceandProcessingattheFriedrich-Alexander-UniversityErlangen- Nürnberg. Its main intention is to teach the basics about polymers necessary for everybody working with these materials. One part is based on the textbook ‘‘Polymermechanik,’’ which appeared in 1990, the other stems from more recent lectures. Thebookfollows twomainguidelines.Oneisaquantitativedescriptionofthe molecular structure, though this has sometimes had to be simplified due to its complexnature.Theotherpresentsrelationshipsbetweenmolecularquantitiesand materialproperties,whichcoverthesolidandthemoltenstate.Thetemperatureis the key external parameter according to which the mechanical behavior is com- prehensively discussed. Nevertheless, this work has to be regarded neither as a complete text book on the mechanics of polymer materials nor on their rheology. Rather, it is meant to discuss these fields from a common viewpoint, which encompasses the transitions between the solid and molten states. v vi Preface ‘‘Tools’’aredescribedinsofarastheyarebelievedtobenecessaryforadeeper understanding of the results presented. They comprise measuring devices as well as mathematical formalisms required for dealing with large deformations. Muchspaceisdevotedtothelineartheoryofviscoelasticityasitiscoherentin itself and allows quantitative insights into relations between properties and molecular structure. Theories covering the nonlinear viscoelastic regime which dominates processing are not yet in a state of development enabling quantitative descriptionsofpropertiesandprocesseswithageneralitycomparabletothelinear behavior. This has to be said, although many theories have been published since 1990, the year of the appearance of ‘‘Polymermechanik.’’ Nevertheless, an over- viewofconstitutiveequationsbasedonvariousmodelsisgivenandsomeoftheir predictions are compared with experimental findings. Inmanypartsofthebook,resultsofinvestigationsperformedattheInstituteof Polymer Materials are presented. They are examples of how research and estab- lished knowledge can become complementary parts of teaching. Regarding the references,longlists,whichcouldeasilybeobtainedtodayfromvariouselectronic bibliographies, have deliberately been avoided. Instead, a selection of original literature is cited, which opens the door to deeper information for those who are interested in more details. The book is written from our experience of teaching the knowledge on poly- mericmaterials,which wethinktobeuseful forpeopleinterestedandengagedin this class of materials. We hope it will be helpful for students to consolidate and broaden their knowledge, to researchers in the field of polymers at various insti- tutions, and even to those working in industry, whenever they would like to get some fundamental questions answered, which arise from dealing with polymers. As it is obvious from the reference lists, the originality and actuality of the results presented in this book are based, to a high degree, on the research of doctoral students under the guidance of the authors of this book. Particularly appreciated are the contributions from the theses of: Dr. Dietmar Auhl Dr. Robert Greiner Dr. Hans-Jürgen Grieß Dr. Joachim Kaschta Dr. Daniela Hertel Dr. Günther Link Dr. Jens Hepperle Dr. Walter Pfandl Dr. Claus Gabriel Dr. Michael Wolf Dr. Ute Maria Kessner Dr. Franz Zahradnik Dr. Stefan Kurzbeck Dr. Julia Maria Resch Dr. Martin Schwetz Dr. Florian Stadler Dr. Jens Stange Dr. Thomas Steffl Dr. Erik Wassner Dr. Friedrich Wolff Preface vii H.MünstedtthanksDr.UteMariaKessner,Dr.MartinSchwetz,andDr.Friedrich Wolffforreadingpartsofthemanuscript,Dr.FlorianStadlerforfruitfuldiscussions andM.Sc.UteZeitlerforhersupportinpreparingcomputer-basedversionsofmany ofthefigures. F.R.SchwarzlwishestorememberthelateDr.ir.J.Heijboerwithwhommany fruitful discussions in the field of secondary relaxation mechanisms of polymers were held. He wishes also to thank Prof. Dr. H. Janeschitz-Kriegl for a life-long friendship and scientific cooperation in Graz, Delft, Linz, and Erlangen, Prof. Dr. ir.L.C.E.Struikfortheverystimulatingdiscussionsconcerningthesolidstateof polymersinDelftandProf.Dr.M.H.Wagner,TechnicalUniversityBerlin,forthe introduction to his theories on polymer rheology. Erlangen, March 2014 Helmut Münstedt Friedrich Rudolf Schwarzl Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 General Aspects of Polymeric Materials . . . . . . . . . . . . . . . . 1 1.2 Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 General Classification of Polymeric Materials . . . . . . . . . . . . 4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Physical Structure of Macromolecules. . . . . . . . . . . . . . . . . . . . . 7 2.1 Structure and Brownian Motion of Macromolecules. . . . . . . . 7 2.2 Molar Mass and Molar Mass Distribution . . . . . . . . . . . . . . . 10 2.3 The Random Walk Problem in Three Dimensions . . . . . . . . . 19 2.4 Macromolecules in Solution. . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Statistical Shape of Linear Macromolecules in H-Solution . . . 31 2.6 Statistical Shape of Macromolecules in Good Solvents. . . . . . 35 2.7 Analysis of Branched Macromolecules . . . . . . . . . . . . . . . . . 37 2.8 Size of Macromolecules in the Glassy and Molten State. . . . . 40 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Experimental Methods to Determine Molecular Quantities . . . . . 43 3.1 Osmometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Viscometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Gel Permeation Chromatography . . . . . . . . . . . . . . . . . . . . . 68 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Structure and States of Polymers . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1 Classification of Polymeric Materials . . . . . . . . . . . . . . . . . . 77 4.2 Molecular Structure of Amorphous Polymers. . . . . . . . . . . . . 87 4.3 States of Order of Uncross-Linked Amorphous Polymers . . . . 89 4.4 Influence of Molar Mass and Cross-Linking Density . . . . . . . 94 4.5 Semicrystalline Polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.5.1 Features of Crystallinity . . . . . . . . . . . . . . . . . . . . . 99 4.5.2 States of Order. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5.3 Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 ix x Contents 4.6 The Specific Volume of Polymers . . . . . . . . . . . . . . . . . . . . 111 4.6.1 The Specific Volume of Amorphous Polymers. . . . . . 111 4.6.2 The Free Volume Theory . . . . . . . . . . . . . . . . . . . . 115 4.6.3 Volume Relaxation and Physical Aging . . . . . . . . . . 117 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5 Linear Viscoelastic Deformation Behavior in Simple Shear . . . . . 121 5.1 Theoretical Description of the Deformation Behavior of Polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2 Creep, Creep Recovery, and Stress Relaxation. . . . . . . . . . . . 122 5.3 The Principle of Superposition. . . . . . . . . . . . . . . . . . . . . . . 127 5.4 Relaxation and Retardation Spectra . . . . . . . . . . . . . . . . . . . 130 5.5 The Creep Recovery Experiment . . . . . . . . . . . . . . . . . . . . . 137 5.6 The Creep Compliance of Amorphous Polymers . . . . . . . . . . 140 5.7 Relations Between Creep and Stress Relaxation. . . . . . . . . . . 149 5.8 Oscillatory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.9 Approximate Relations Between Measurable Viscoelastic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.10 The Viscoelastic Behavior of Amorphous Polymers in Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6 Time-Temperature Shift of Mechanical Properties. . . . . . . . . . . . 189 6.1 The Significance of the Time-Temperature Shift for the Description of the Deformation Behavior of Polymers . . . . . . 189 6.2 The Time-Temperature Shift Principle . . . . . . . . . . . . . . . . . 190 6.3 The Time-Temperature Shift of the Glass-Rubber Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.4 The Time-Temperature Shift in the Flow Region of Amorphous Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.5 The Time-Temperature Shift in the Flow Region of Semicrystalline Polymers. . . . . . . . . . . . . . . . . . . . . . . . . 210 6.6 The Time-Temperature Shift of Secondary Transitions in the Glassy State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7 Linear Viscoelastic Deformation Under Three-Dimensional Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.1 The Stress Tensor and the Equations for the Balance of Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.2 The Strain Tensor for Small Deformations . . . . . . . . . . . . . . 227 7.3 The Rheological Equation of State for Isotropic Linear Elastic Materials (Hookean Theory of Elasticity). . . . . 230 Contents xi 7.4 The Rheological Equation of State for Isotropic Linear Viscoelastic Materials at Small Deformations . . . . . . . . . . . . 231 7.5 Simple Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.6 Isotropic Compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.7 Uniaxial Tensile Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.8 The Viscoelastic Functions for Amorphous Uncross-Linked Polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8 Fundamentals of the Rheology of Large Deformations. . . . . . . . . 243 8.1 Kinematics of Large Deformations. . . . . . . . . . . . . . . . . . . . 243 8.2 Deformation Gradient and Finite Strain Tensors. . . . . . . . . . . 247 8.3 Relative Deformation Gradient and Relative Strain Tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 8.4 The Rate of Strain Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . 254 8.5 Dynamics of Deformable Bodies . . . . . . . . . . . . . . . . . . . . . 256 8.6 Eigenvalues and Invariants of the Stress Tensor. . . . . . . . . . . 259 8.7 Transformation of the Strain Tensors and the Rate of Strain Tensor to Principal Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 8.7.1 Time-Dependent Simple Shear. . . . . . . . . . . . . . . . . 268 8.7.2 Multidimensional Time-Dependent Incompressible (Isochoric) Extension . . . . . . . . . . . . 271 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9 Large Deformations of Polymers. . . . . . . . . . . . . . . . . . . . . . . . . 275 9.1 Stress-Strain Behavior of Polymeric Materials. . . . . . . . . . . . 275 9.2 Rheological Equation of State for Isotropic Elastic Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9.3 Rheological Equation of State for the Ideal Rubber . . . . . . . . 284 9.4 Statistical Theory of Rubber Elasticity . . . . . . . . . . . . . . . . . 289 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 10 Equations of State for Polymer Melts . . . . . . . . . . . . . . . . . . . . . 297 10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.2 Rheological Equation of State for the Elastic Liquid After Lodge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10.2.1 The Constitutive Equation. . . . . . . . . . . . . . . . . . . . 298 10.2.2 The Lodge Liquid in Time-Dependent Simple Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 10.2.3 The Lodge Liquid in a Stressing Experiment in Simple Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 10.2.4 The Lodge Liquid in Shear Creep and Creep Recovery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305