Polymer Processing Instabilities Control and Understanding

Polymer_Processing_Instabilities_Control_and_Understanding/list.txtPolymer Processing Instabilities Control and Understanding\dk1205_fm.pdf Polymer Processing Instabilities Control and Understanding\DK1205_INTRO.pdf Polymer Processing Instabilities Control and Understanding\DK1205_PartA.pdf Polymer Processing Instabilities Control and Understanding\DK1205_PartB.pdf Polymer Processing Instabilities Control and Understanding\DK1205_PartC.pdf

Polymer_Processing_Instabilities_Control_and_Understanding/Polymer Processing Instabilities Control and Understanding/dk1205_fm.pdfPolymer ProcessingInstabilities

Control and Understanding

DK1205_half-series-title 10/18/04 10:46 AM Page A

Copyright 2005 by Marcel Dekker. All Rights Reserved.

CHEMICAL INDUSTRIES

A Series of Reference Books and Textbooks

Consulting Editor

HEINZ HEINEMANNBerkeley, California

1. Fluid Catalytic Cracking with Zeolite Catalysts,Paul B. Venuto and E. Thomas Habib, Jr.

2. Ethylene: Keystone to the Petrochemical Industry,Ludwig Kniel, Olaf Winter, and Karl Stork

3. The Chemistry and Technology of Petroleum,James G. Speight

4. The Desulfurization of Heavy Oils and Residua,James G. Speight

5. Catalysis of Organic Reactions, edited by William R. Moser

6. Acetylene-Based Chemicals from Coal and OtherNatural Resources, Robert J. Tedeschi

7. Chemically Resistant Masonry,Walter Lee Sheppard, Jr.

8. Compressors and Expanders: Selection andApplication for the Process Industry, Heinz P. Bloch,Joseph A. Cameron, Frank M. Danowski, Jr., Ralph James, Jr., Judson S. Swearingen, and Marilyn E. Weightman

9. Metering Pumps: Selection and Application,James P. Poynton

10. Hydrocarbons from Methanol, Clarence D. Chang

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11. Form Flotation: Theory and Applications,Ann N. Clarke and David J. Wilson

12. The Chemistry and Technology of Coal,James G. Speight

13. Pneumatic and Hydraulic Conveying of Solids,O. A. Williams

14. Catalyst Manufacture: Laboratory and CommercialPreparations, Alvin B. Stiles

15. Characterization of Heterogeneous Catalysts,edited by Francis Delannay

16. BASIC Programs for Chemical Engineering Design,James H. Weber

17. Catalyst Poisoning, L. Louis Hegedus and Robert W. McCabe

18. Catalysis of Organic Reactions, edited by John R. Kosak

19. Adsorption Technology: A Step-by-Step Approach to Process Evaluation and Application, edited by Frank L. Slejko

20. Deactivation and Poisoning of Catalysts, edited byJacques Oudar and Henry Wise

21. Catalysis and Surface Science: Developments in Chemicals from Methanol, Hydrotreating ofHydrocarbons, Catalyst Preparation, Monomers and Polymers, Photocatalysis and Photovoltaics,edited by Heinz Heinemann and Gabor A. Somorjai

22. Catalysis of Organic Reactions, edited by Robert L. Augustine

23. Modern Control Techniques for the ProcessingIndustries, T. H. Tsai, J. W. Lane, and C. S. Lin

24. Temperature-Programmed Reduction for SolidMaterials Characterization, Alan Jones and Brian McNichol

25. Catalytic Cracking: Catalysts, Chemistry, and Kinetics,Bohdan W. Wojciechowski and Avelino Corma

26. Chemical Reaction and Reactor Engineering,edited by J. J. Carberry and A. Varma

27. Filtration: Principles and Practices: Second Edition,edited by Michael J. Matteson and Clyde Orr

28. Corrosion Mechanisms, edited by Florian Mansfeld29. Catalysis and Surface Properties of Liquid Metals

and Alloys, Yoshisada Ogino

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30. Catalyst Deactivation, edited by Eugene E. Petersenand Alexis T. Bell

31. Hydrogen Effects in Catalysis: Fundamentals and Practical Applications, edited by Zoltn Pal and P. G. Menon

32. Flow Management for Engineers and Scientists,Nicholas P. Cheremisinoff and Paul N. Cheremisinoff

33. Catalysis of Organic Reactions, edited by Paul N. Rylander, Harold Greenfield, and Robert L. Augustine

34. Powder and Bulk Solids Handling Processes:Instrumentation and Control, Koichi Iinoya, Hiroaki Masuda, and Kinnosuke Watanabe

35. Reverse Osmosis Technology: Applications for High-Purity-Water Production, edited by Bipin S. Parekh

36. Shape Selective Catalysis in Industrial Applications,N. Y. Chen, William E. Garwood, and Frank G. Dwyer

37. Alpha Olefins Applications Handbook, edited byGeorge R. Lappin and Joseph L. Sauer

38. Process Modeling and Control in Chemical Industries,edited by Kaddour Najim

39. Clathrate Hydrates of Natural Gases,E. Dendy Sloan, Jr.

40. Catalysis of Organic Reactions, edited by Dale W. Blackburn

41. Fuel Science and Technology Handbook,edited by James G. Speight

42. Octane-Enhancing Zeolitic FCC Catalysts,Julius Scherzer

43. Oxygen in Catalysis, Adam Bielanski and Jerzy Haber44. The Chemistry and Technology of Petroleum:

Second Edition, Revised and Expanded,James G. Speight

45. Industrial Drying Equipment: Selection and Application, C. M. vant Land

46. Novel Production Methods for Ethylene, LightHydrocarbons, and Aromatics, edited by Lyle F. Albright, Billy L. Crynes, and Siegfried Nowak

47. Catalysis of Organic Reactions, edited by William E. Pascoe

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48. Synthetic Lubricants and High-Performance FunctionalFluids, edited by Ronald L. Shubkin

49. Acetic Acid and Its Derivatives, edited by Victor H. Agreda and Joseph R. Zoeller

50. Properties and Applications of Perovskite-Type Oxides,edited by L. G. Tejuca and J. L. G. Fierro

51. Computer-Aided Design of Catalysts, edited by E. Robert Becker and Carmo J. Pereira

52. Models for Thermodynamic and Phase EquilibriaCalculations, edited by Stanley I. Sandler

53. Catalysis of Organic Reactions, edited by John R. Kosak and Thomas A. Johnson

54. Composition and Analysis of Heavy PetroleumFractions, Klaus H. Altgelt and Mieczyslaw M. Boduszynski

55. NMR Techniques in Catalysis, edited by Alexis T. Belland Alexander Pines

56. Upgrading Petroleum Residues and Heavy Oils, Murray R. Gray

57. Methanol Production and Use, edited by Wu-Hsun Cheng and Harold H. Kung

58. Catalytic Hydroprocessing of Petroleum and Distillates, edited by Michael C. Oballah and Stuart S. Shih

59. The Chemistry and Technology of Coal: Second Edition, Revised and Expanded,James G. Speight

60. Lubricant Base Oil and Wax Processing, Avilino Sequeira, Jr.

61. Catalytic Naphtha Reforming: Science and Technology, edited by George J. Antos, Abdullah M. Aitani, and Jos M. Parera

62. Catalysis of Organic Reactions, edited by Mike G. Scaros and Michael L. Prunier

63. Catalyst Manufacture, Alvin B. Stiles and Theodore A. Koch

64. Handbook of Grignard Reagents, edited by Gary S. Silverman and Philip E. Rakita

65. Shape Selective Catalysis in Industrial Applications:Second Edition, Revised and Expanded, N. Y. Chen,William E. Garwood, and Francis G. Dwyer

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66. Hydrocracking Science and Technology, Julius Scherzer and A. J. Gruia

67. Hydrotreating Technology for Pollution Control:Catalysts, Catalysis, and Processes, edited by Mario L. Occelli and Russell Chianelli

68. Catalysis of Organic Reactions, edited by Russell E. Malz, Jr.

69. Synthesis of Porous Materials: Zeolites, Clays, and Nanostructures, edited by Mario L. Occelli and Henri Kessler

70. Methane and Its Derivatives, Sunggyu Lee71. Structured Catalysts and Reactors, edited by

Andrzej Cybulski and Jacob A. Moulijn72. Industrial Gases in Petrochemical Processing,

Harold Gunardson73. Clathrate Hydrates of Natural Gases: Second Edition,

Revised and Expanded, E. Dendy Sloan, Jr.74. Fluid Cracking Catalysts, edited by Mario L. Occelli

and Paul OConnor75. Catalysis of Organic Reactions, edited by

Frank E. Herkes76. The Chemistry and Technology of Petroleum:

Third Edition, Revised and Expanded, James G. Speight

77. Synthetic Lubricants and High-Performance FunctionalFluids: Second Edition, Revised and Expanded, Leslie R. Rudnick and Ronald L. Shubkin

78. The Desulfurization of Heavy Oils and Residua,Second Edition, Revised and Expanded, James G. Speight

79. Reaction Kinetics and Reactor Design: Second Edition, Revised and Expanded, John B. Butt

80. Regulatory Chemicals Handbook, Jennifer M. Spero,Bella Devito, and Louis Theodore

81. Applied Parameter Estimation for Chemical Engineers,Peter Englezos and Nicolas Kalogerakis

82. Catalysis of Organic Reactions, edited by Michael E. Ford

83. The Chemical Process Industries Infrastructure:Function and Economics, James R. Couper, O. Thomas Beasley, and W. Roy Penney

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84. Transport Phenomena Fundamentals, Joel L. Plawsky85. Petroleum Refining Processes, James G. Speight

and Baki zm86. Health, Safety, and Accident Management in the

Chemical Process Industries, Ann Marie Flynn and Louis Theodore

87. Plantwide Dynamic Simulators in Chemical Processingand Control, William L. Luyben

88. Chemicial Reactor Design, Peter Harriott89. Catalysis of Organic Reactions, edited by

Dennis G. Morrell90. Lubricant Additives: Chemistry and Applications,

edited by Leslie R. Rudnick91. Handbook of Fluidization and Fluid-Particle Systems,

edited by Wen-Ching Yang92. Conservation Equations and Modeling of Chemical and

Biochemical Processes, Said S. E. H. Elnashaie andParag Garhyan

93. Batch Fermentation: Modeling, Monitoring, and Control, Ali inar, Glnur Birol, Satish J. Parulekar,and Cenk ndey

94. Industrial Solvents Handbook, Second Edition,Nicholas P. Cheremisinoff

95. Petroleum and Gas Field Processing, H. K. Abdel-Aal,Mohamed Aggour, and M. Fahim

96. Chemical Process Engineering: Design and Economics,Harry Silla

97. Process Engineering Economics, James R. Couper98. Re-Engineering the Chemical Processing Plant: Process

Intensification, edited by Andrzej Stankiewicz and Jacob A. Moulijn

99. Thermodynamic Cycles: Computer-Aided Design and Optimization, Chih Wu

100. Catalytic Naptha Reforming: Second Edition, Revisedand Expanded, edited by George T. Antos and Abdullah M. Aitani

101. Handbook of MTBE and Other Gasoline Oxygenates, edited by S. Halim Hamid and Mohammad Ashraf Ali

102. Industrial Chemical Cresols and DownstreamDerivatives, Asim Kumar Mukhopadhyay

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103. Polymer Processing Instabilities: Control and Understanding, edited by Savvas Hatzikiriakos and Kalman B . Migler

104. Catalysis of Organic Reactions, John Sowa105. Gasification Technologies: A Primer for Engineers

and Scientists, edited by John Rezaiyan and Nicholas P. Cheremisinoff

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Polymer ProcessingInstabilities

Control and Understanding

edited by

Savvas G. HatzikiriakosThe University of British Columbia

Vancouver, British Columbia, Canada

Kalman B. MiglerNational Institute of Standards and Technology

Gaithersburg, Maryland, U.S.A.

Marcel Dekker New York

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5396-0_Hatzikiriakos_Prelims_R2_100604

Although great care has been taken to provide accurate and current information, neither the

author(s) nor the publisher, nor anyone else associated with this publication, shall be liable for

any loss, damage, or liability directly or indirectly caused or alleged to be caused by this book. The

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MD: HATZIKIRIAKOS, JOB: 04351, PAGE: ii


Preface

Polymer processing has grown in the past 50 years into a multi-billion dollarindustry with production facilities and development labs all over the world.The primary reason for this phenomenal growth compared to other materialsis the relative ease of manufacture and processing. Numerous methods havebeen developed to process polymeric materials at high volume and at rela-tively low temperatures.

But despite this success, manufacturing is limited by the occurrence ofpolymer processing instabilities. These limitations manifest themselves in twoways; rst as the rate limiting step in the optimization of existing operations,and second in the introduction of new materials to the marketplace. Forexample, it is natural to ask what is the rate limiting step for processingoperations; why not run a given operation 20% faster? Quite frequently, theanswer to this question concerns ow instabilities. As the polymeric materialis processed in the molten state; it retains characteristics of both liquids andsolids. The faster one processes it, the more solid like its response becomes;seemingly simple processing operations become intractably dicult and thepolymer ow becomes chaotic and uncontrollable.

A second problem concerns the development of new materials; this isparticularly pressing because new materials with enhanced properties oerrelief from the commodities nature of the polymer processing industry. But inorder to gain market acceptance, they must also enjoy ease of processability

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and there are numerous examples of new materials which processed poorlyand suered in the market.

The past decade has seen great progress in our eorts to understand andcontrol polymer processing instabilities; however much of the success isscattered throughout the scientic and technical literature. The intention ofthis book is to coherently collect these recent triumphs and present them to awide audience. This book is intended for polymer rheologists, scientists,engineers, technologists and graduate students who are engaged in the eld ofpolymer processing operations and need to understand the impact of owinstabilities. It is also intended for those who are already active in elds suchas instabilities in polymer rheology and processing and wish to widen theirknowledge and understanding further.

The chapters in this book seek to impart both fundamental and practicalunderstanding on various ows that occur during processing. Processes whereinstabilities pose serious limitations in the rate of production includeextrusion and co-extrusion, blow molding, lm blowing, lm casting, andinjection molding. Methods to cure and eliminate such instabilities is also ofconcern in this book. For example, conventional polymer processing addi-tives that eliminate ow instabilities such as sharkskinmelt fracture, and non-conventional polymer processing additives that eliminate ow instabilitiessuch as gross melt fracture are also integral parts of this book. Materials ofinterest that are covered in this book includemost of the commodity polymersthat are processed as melts at temperatures above their melting point (poly-ethylenes, polypropylenes, uoropolymers, and others) or as concentratedsolutions at lower temperatures.

Greater emphasis has been given to the ow instability of melt fracturesince such phenomena have drawn the attention of many researchers inrecent years. Moreover, these phenomena take place in a variety of processessuch as lm blowing, lm casting, blow molding, extrusion and variouscoating ows. Equally important, however, instabilities take place in otherprocessing operations such as draw resonance and the dog-bone eect inlm casting, tiger-skin instabilities in injection molding, interfacial instabil-ities in co-extrusion and several secondary ows in various contraction ows.An overall overview of these instabilities can be found in the introduction(Chapter 1).

It is hoped that this book lls the gap in the polymer processing lite-rature where polymer ow instabilities are not treated in-depth in any book.Research in this eld, in particular over the last ten years, has produced asignicant amount of data. An attempt is made to distil these data and todene the state-of-the art in the eld. It is hoped that this will be useful toresearchers active in the eld as a starting point as well as a guide to obtain


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helpful direction in their research. It would be almost impossible to include allthe knowledge generated over the past fty-to-sixty years in a single book. Assuch, we would like to apologize for not citing several important reports andcontributions to the eld.

Savvas G.HatzikiriakosKalman B.Migler


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Contents

PrefaceContributors

INTRODUCTION

1. Overview of Processing InstabilitiesSavvas G. Hatzikiriakos and Kalman B. Migler

PART A: VISCOELASTICITY AND BASIC FLOWS

2. Elements of RheologyJohn M. Dealy

3. Secondary Flow InstabilitiesEvan Mitsoulis

4. Wall Slip: Measurement and Modeling IssuesLynden A. Archer

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PART B: MELT FRACTURE AND RELATEDPHENOMENA

5. Sharkskin Instability in ExtrusionKalman B. Migler

6. StickSlip InstabilityGeorgios Georgiou

7. Gross Melt Fracture in ExtrusionJohn M. Dealy and Seungoh Kim

8. Conventional Polymer Processing AdditivesSemen B. Kharchenko, Kalman B. Migler,and Savvas G. Hatzikiriakos

9. Boron Nitride Based Polymer Processing AidsSavvas G. Hatzikiriakos

PART C: APPLICATIONS

10. Draw Resonance in Film CastingAlbert Co

11. Fiber Spinning and Film Blowing InstabilitiesHyun Wook Jung and Jae Chun Hyun

12. Coextrusion InstabilitiesJoseph Dooley

13. Tiger Stripes: Instabilities in Injection MoldingA.C.B. Bogaerds, G.W.M. Peters, and F.P.T. Baaijens

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Contributors

Lynden A. Archer Cornell University, Ithaca, New York, U.S.A.

F.P.T. Baaijens Eindhoven University of Technology, Eindhoven, TheNetherlands

A.C.B. Bogaerds Eindhoven University of Technology, Eindhoven, andDSM Research, Geleen, The Netherlands

Albert Co University of Maine, Orono, Maine, U.S.A.

John M. Dealy McGill University, Montreal, Quebec, Canada

Joseph Dooley The Dow Chemical Company, Midland, Michigan, U.S.A.

Georgios Georgiou University of Cyprus, Nicosia, Cyprus

Savvas G. Hatzikiriakos The University of British Columbia, Vancouver,British Columbia, Canada

Jae Chun Hyun Korea University, Seoul, South Korea

Hyun Wook Jung Korea University, Seoul, South Korea

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Semen B. Kharchenko National Institute of Standards and Technology,Gaithersburg, Maryland, U.S.A.

Seungoh Kim Verdun, Quebec, Canada

Kalman B. Migler National Institute of Standards and Technology,Gaithersburg, Maryland, U.S.A.

Evan Mitsoulis National Technical University of Athens, Athens, Greece

G.W.M. Peters Eindhoven University of Technology, Eindhoven, TheNetherlands

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Polymer Processing Instabilities Control And UnderstandingChemical IndustriesPrefaceContentsContributors

Polymer_Processing_Instabilities_Control_and_Understanding/Polymer Processing Instabilities Control and Understanding/DK1205_INTRO.pdf1Overview of Processing Instabilities

Savvas G. Hatzikiriakos

The University of British Columbia, Vancouver,British Columbia, Canada

Kalman B. Migler

National Institute of Standards and TechnologyGaithersburg, Maryland, U.S.A.

1.1 POLYMER FLOW INSTABILITIES

Hydrodynamic stability is one of the central problems of uid dynamics. It isconcerned with the breakdown of laminar ow and its subsequent develop-ment and transition to turbulent ow (1). The ow of polymeric liquids dierssignicantly from that of their low-viscosity counterparts in several ways and,consequently, the nature of ow instabilities is completely dierent. Mostnotably, whereas for low-viscosity uids it is the inertial forces on the uidthat cause turbulence (as measured by the Reynolds number), for high-viscosity polymers, it is the elasticity of the uid that causes a breakdown inlaminar uid ow (asmeasured by theWeissenberg number). Additionally, the

MD: HATZIKIRIAKOS, JOB: 04351, PAGE: 1

Ocial contribution of the National Institute of Standards and Technology; not subject to

copyright in the United States.

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high viscosity and the propensity for the molten uid to slip against solidsurfaces contribute to a rich and diverse set of phenomena, which this bookaims to review.

In this introductory chapter, an overview of most ow instabilities thatare discussed and examined in detail in this book is presented. Bothexperimental observations as well as modeling of ows with the purpose ofpredicting ow instabilities are of concern in subsequent chapters. In addi-tion, ways of overcoming these instabilities with the aim of increasing the rateof production of polymer processes are also of central importance [e.g., use ofprocessing aids to eliminate surface defects (melt fracture), adjustment ofmolecular parameters, and rational adjustment of operating procedures andgeometries to obtain better ow properties].

1.2 PART A: THE NATURE OF POLYMERIC FLOW

Polymeric liquids exhibit many idiosyncracies that their Newtonian counter-parts lack. Most notably are: 1) the normal stress and elasticity eects; 2)strong extensional viscosity eects; and 3) wall slip eects. These properties ofpolymeric liquids change dramatically the nature and structure of the owcompared to the corresponding ow of Newtonian liquids. Chapter 2 of Ref.2 presents a comprehensive overview of such striking dierences in New-tonian ow compared to their non-Newtonian counterparts. Part A of thisbook (Chapters 24) is thus devoted to the fundamental aspects of polymericow from the perspective of processing regime, where the above eects aremost manifest.

Chapter 2 by Dealy introduces the reader to the concepts of polymerrheology that are most important to polymer processing and denes a largenumber of the terms needed to read the rest of the book. It describes how torelate the stress on a uid to an imposed strain, rst for the simple linear caseand then for the cases more relevant to processing, such as nonlinear ows(such as shear thinning) and also extensional ows (which are prevalent inmost processing operations.) It carefully denes the Weissenberg number,which characterizes the degree of nonlinearity or anisotropy exhibited by theuid in a particular deformation, as well as the Deborah number, which is ameasure of the degree to which amaterial exhibits elastic behavior. It cautionsthe reader that our knowledge and measurement ability in this area are still intheir infancy.

Chapter 3 by Mitsoulis describes secondary ows in processing oper-ations, in particular those in which vortices or helices appear. This is a classiccase in which a ow phenomenon (vortices), which occurs for Newtonianows, also occurs for polymeric ows, but the nature of the ow is quitedistinct. For polymeric uids, the transition to vortices is governed largely by



the Deborah number. Depending on the response of the uid to extensionalows, the occurrence and magnitude of the vortices can be quite dierent.This chapter describes the occurrence of these ows in a number of processingows including extrusion, calendaring, roll/wire coating, and coextrusion.

Chapter 4 by Archer describes the phenomenon of wall slip, where theuid velocity does not go to zero at a solid wall. Again, although thisphenomenon has been reported for Newtonian uids, it is relatively weakand dicult to observe. However, for polymeric liquids, wall slip can be quitelarge; it signicantly impacts on the ow behavior (and necessary modeling)of processing operations. This chapter describes the signicant advancesmade recently on the measurement of wall slip, on the molecular theorydescribing it, and on the relationship at the interface between a polymer and asolid substrate. Although wall slip has been studied extensively in regard to itseects on extrusion instabilities, it plays a critical role in other processingoperations as well.

Unlike Newtonian uids, polymer melts slip over solid surfaces whenthe wall shear stress exceeds a critical value. For example, Fig. 1.1 illustratesthe well-known HaagenPoiseuille steady-state ow of a Newtonian uid(Fig. 1.1a) and two typical corresponding proles for the case of a moltenpolymer (Fig. 1.1b and c), indicating a small deviation from the no-slipboundary condition and plug ow. Although the classical no-slip boundarycondition applied in Newtonian uid mechanics (perhaps with the exceptionof rareed gas dynamics where the continuum hypothesis does not apply),wall slip is typical in the case of molten polymers.

For the case of a passive polymerwall interface where there is nointeraction between the polymer and the solid surface, de Gennes (4)proposed an interfacial rheological law suggesting that a melt would slip atall shear rates. This theory was extended by Brochard-Wyart and de Gennes(5) to distinguish the case of a passive interface (no polymer adsorption) fromthat of an adsorbing one. It has been predicted that there exists a critical wall



FIGURE 1.1 Typical velocity profiles of polymer melt flow in capillaries: (a) no-slip; (b) partial slip; and (c) plug flow.


shear stress value at which a transition from a weak to a strong slip takesplace. These predictions have been suggested to be true through experimen-tation (6,7), and Chapter 4 contains a further enhancement of the theory byArcher. The various mechanisms of wall slip in the case of ow of moltenpolymers are still under debate and many observations need better andcomplete explanation.

1.3 PART B: MELT FRACTURE IN EXTRUSION

Part B is devoted to describing extrusionone of the simpler processingoperations but is critically important because it is ubiquitous in polymermanufacturing and exhibits a full range of instabilities (Chapters 59). Herethe instabilities revolve around the phenomena ofmelt fracture, wall slip, andpolymer elasticity. The phenomena of melt fracture and wall slip have beenstudied for the past 50 years but have not yet been explained (3). Not only arethese phenomena of academic interest, but they are also industrially relevantas they may limit the rate of production in processing operations. Forexample, in the continuous extrusion of a typical linear polyethylene at somespecic output rate, the extrudate starts losing its glossy appearance; instead,a matte surface nish is evident and, at slightly higher output rates, small-amplitude periodic distortions appear on its surface (see Fig. 1.2). Thisphenomenon, known as sharkskin or surface melt fracture, is described inChapter 5 by Migler.

At higher values of the output rate, the ow ceases to be constant;instead, the pressure oscillates between two limiting values and the extrudate


FIGURE 1.2 Typical extrudates showing: (a) smooth surface; (b) shark skin meltfracture; (c) stickslip melt fracture; and (d) gross melt fracture.


surface alternates between relatively smooth and distorted portions(Fig. 1.2c). These phenomena are known as oscillating, stickstick, or cyclicmelt fracture, and are discussed in Chapter 6 by Georgiou. The chapterfocuses on the critical stresses for the instability as well as the eects ofoperating conditions and molecular structure. A one-dimensional phenom-enological model successfully describes much of the data.

At even higher values of the output rate, a new instability known asgross melt fracture (GMF) occurs, which is described in Chapter 7 by Dealyand Kim.Whereas sharkskin and stickslip instability are associated with thecapillary tube, GMF originates in the upstream region where the polymer isaccelerated from a wide-diameter barrel to a narrow-diameter capillary ororice. The extensional stress on the polymer associated with such a ow canrupture it, leading to a chaotic appearance when it nally emerges from thecapillary. This chapter presents a number of observations and contains acritical review of our understanding of GMF from the perspective of ourlimited understanding of what causes the rupture of molten polymers.

To increase the rate of production by eliminating or postponing themeltfracture phenomena to higher shear rates, processing additives/aids must beused. These aremainly uoropolymers and stearates, which are widely used inthe processing of polyolens such as high-density polyethylene (HDPE) andlinear low-density polyethylene (LLDPE). They are added to the basepolymer at low concentrations (approximately 0.1 %), and they eectivelyact as die lubricants, modifying the properties of the polymerwall interface(increasing slip of the molten polymers). Chapter 8 by Kharchenko,Hatzikiriakos, and Migler discusses processing additives with particularemphasis on uoropolymer additives. These uoropolymer additives havebeen known for a long time, but until recently, one could only speculate as toprecisely how they function. Recently, through visualization methods, theprecise nature of the coating and how it is created through the ow havebecome clear. Although these uoropolymer additives are eective for shark-skin and stickslip instability, they remain ineective for the case of GMF.

It has been recently discovered that compositions containing boronnitride can be successfully used as processing aids to not only eliminatesurface melt fracture, but also to postpone gross melt fracture and therebypermit the use of signicantly higher shear rates. These processing aids can beused for a variety of important extrusion processes, namely, tubing extrusion,lm blowing, blow molding, and wire coating. The mechanisms by whichboron nitride aects the processability of molten polymers and other impor-tant experimental observations related to the eects of boron nitride-basedprocessing aids on the rheological behavior and processability of polymerscan be found in Chapter 9 by Hatzikiriakos.




1.4 PART C: APPLICATIONS

1.4.1 Draw Resonance in Film Casting

In lm casting as well as in extrusion coating of polymeric sheet, a polymericmelt curtain is extruded through a narrow die slot, across an air gap or a liquidbath, and then onto a pair of (or just a single) take-up or chill rolls (seeFig. 1.3). Eorts to increase production speed and/or reduce lm thickness bygoing to higher draw ratio (take-up speed/extrudate speed) are hampered byedge neck-in and bead formation, but mainly by process instabilities (drawresonance and edge weave) (7), which give rise to spontaneous thickness andwidth oscillations (Fig. 1.4). Due to the high melt viscosity, the liquid curtainis usually pulled downstream by the drum, resulting in a long eective castingspan. As a result, the curtain necks-in at the edges giving rise to a nonuniformgauge prole with a characteristic dog bone shape (Fig. 1.4). On the otherhand, draw resonance is accompanied by spontaneous thickness and widthoscillationseects that are undesirable in lm production.

Chapter 10 by Co summarizes experimental observations of suchinstabilities that might occur in lm casting and extrusion coating ofpolymeric materials. In addition, modeling and stability analysis of drawresonance with the aim of predicting such instabilities are also thoroughlydiscussed. As will be seen in this chapter, computational uid mechanicsmodeling can help predict such instabilities and extend the parameter range ofstable and defect-free operation (so-called process operability window).



FIGURE 1.3 Schematic of a typical film casting process.


1.4.2 Fiber Spinning Instabilities

A simple schematic of the ber spinning process is depicted in Fig. 1.5. Thepolymer melt is pumped by an extruder and ows through a plate containingmany small holesthe spinneret. Fig. 1.5 shows the ow of the ber throughone such hole. The extruded lament is air-cooled by exposure to ambient air,and it is stretched by a rotating take-up roll at a point lower to its solidicationpoint.

As in all other polymer processing operations discussed before, the rateof production is limited by the onset of instabilities, and these are discussed inChapter 11 by Jung and Hyun. Three types of instability might occur in berspinning (8). The rst instability is called spinnability, which is dened as theability of the polymer melt to stretch without breaking (8). Necking mightoccur due to capillary waves or a brittle type of fracture due to crystallizationinduced by stretching. The second type of instability is referred to as drawresonance, which manifests itself as periodic uctuation of the cross-sectionalarea in the take-up area. This latter instability is similar to draw resonanceoccurring in the lm casting process of polymers (see Fig. 1.4). Finally, meltfracture phenomenon, as with other types of instabilities that may limit the



FIGURE 1.4 Instabilities associated with the film casting process of polymers.




rate of production such as ber spinning, involves extrusion through dies athigh shear rates.

Experimental observations in ber spinning application for variousresins are discussed in conjunction with modeling techniques of the processwith a focus on draw resonance. Attempts to predict such instabilities with theaim of predicting operability windows for increasing the rate of productionare also discussed. The rheological properties of the melts that play a role instabilizing/destabilizing the process are also relevant in such discussions.

1.4.3 Film Blowing Instabilities

A simple schematic of the lm blowing process is depicted in Fig. 1.6. Anextruder melts the resin and forces it to ow through an annular die. The meltextruded in the form of a tube is stretched in the machine direction by meansof nip roles above the die, as shown in Fig. 1.6. Air ows inside the lm bubbleto cool down the hotmelt. Thus, a frost line is established. The ratio of the lmvelocity to the average velocity of the melt in the die is referred to as the drawdown ratio (DDR).

The lm blowing instabilities are discussed in great detail in Chapter 11by Jung and Hyun. As in all polymer processes discussed so far, several

FIGURE 1.5 Schematic of a typical fiber spinning process.




instabilities might result as the extrusion rate (production rate) and DDR areincreased. First, as the extrusion rate is increased (shear rate in the die), meltfracture might result depending on the molecular characteristics of the resin.These are manifested as loss of gloss of the lm, small-amplitude periodicdistortions, or more severe form of gross distortions. These instabilitiesinuence seriously the optical and mechanical properties of the nal product.Melt fracture is discussed in detail in Chapters 57 (present volume). Ways ofpostponing these instabilities to higher shear rates by means of usingprocessing aids are discussed in Chapters 8 and 9 (present volume).

Draw resonance also occurs in this process as the DDR ratio increases(9). In the present process, draw resonance appears as periodic uctuations ofthe bubble diameter known also as bubble instability shapes (10). Anothertype of draw resonance that might occur in lm blowing is in the form of lmgauge (lm thickness) nonuniformity. This typically occurs at high produc-tion rates when the cooling rate requirements are increased (11). Molecularorientation during stretching of the lm and the extensional properties of theresin in the melt state are important factors inuencing the quality of the nalproducts (11).

1.4.4 Coextrusion and Interfacial Instabilities

Coextrusion instabilities are discussed in depth in Chapter 12 by Dooley.Coextrusion refers to the process when two or more polymer liquid layers areextruded from a die to produce either a multilayer lm or a ber. There arethree main problems associated with coextrusion. First, depending on the

FIGURE 1.6 Schematic of a typical film blowing process.


materials to be processed, melt fracture phenomenon might appear. This hasalready been discussed as these phenomena occur in most polymer processesthat produce extrudates at high production rates.

More importantly are the instabilities caused by dierences in theviscous and elastic properties of the components (see Fig. 1.7). If there is asignicant viscosity dierence between two liquids (i.e., two-layer ow), thenthe uid having the lower viscosity will tend to encapsulate the uid havingthe higher viscosity (9). For this to occur, the length-to-gap ratios should berelatively high, which gives enough time for rearrangement. Finally, even inthe absence of a viscosity mismatch, the interface can become wavy. Theseinstabilities are collectively known in coextrusion as convective interfacialinstabilities, or simply interfacial instabilities. These instabilities are demon-strated in Chapter 12 to be caused by viscoelastic mismatches between theuids coextruded, together with the geometry used for the process (1114).For example, the point of layermerging or the point on the interface at the exitmight cause interfacial waviness. The phenomena are complicated and notcompletely well understood as there are multiple factors involved.

1.4.5 Injection Molding and Tiger Stripes Instability

In injection molding, the polymer melt is softened rst in an extruder andpumped forward through a runner to ll in a mold that is in the shape of theproduct article. The challenge is to produce a product that is free of voids, hasa smooth and glossy surface, exhibits no warpage, and has sucient mechan-ical strength and stiness for its end use. The latter is signicantly inuencedby residual stresses due to the viscoelastic nature of ow, as well as due to



FIGURE 1.7 Instabilities in coextrusion: (a) gradual encapsulation of a viscousfluid (2) by a less viscous fluid (1) as both flow in a circular tube; and (b)interfacial instability in the form of a wavy interface.




shrinkage during cooling of the mold. As in most polymer processingoperations, several problems (instabilities) might appear.

First, if the cavity to be lled has an insert, then the melt ow splits andows around the obstacle. Consequently, these moving fronts meet again toresult a weld line or knit line. In other cases, multiple runners are used to ll inthe mold and, therefore, multiple weld lines result (see Fig. 1.8a). Lack ofsucient reentanglement of molecules across this line would lead to poormechanical strength of the nal product.

Jetting is another problem that might occur in injection molding. Whenthe size (gap or diameter) ismuch smaller than themold gap, themelt does notproperly wet the entrance to the mold in order to ll it gradually. Instead, itsnakes its way into the gap. This is shown in Fig. 1.8b.

Flow instabilities during injection molding (mold lling) also result insurface defects on polymer parts. For example, in lled polypropylenesystems, the regular dull part of nished parts is broken by periodic shinybands perpendicular to the ow direction (15). The appearance of thesestriped surface defects is known as tiger stripes. At the leading edge of the owduringmold lling, the polymer undergoes what is known as fountain ow.Great recent progress in this area is summarized in Chapter 13 by Bogaerds,Peters, and Baaijens, in which they demonstrate that there is a viscoelasticow instability at this airpolymerwall juncture that leads to the tiger stripephenomenon.

REFERENCES

1. Drazin, P.G.; Reid, W.H.Hydrodynamic Stability; Cambridge University Press:

New York, 1981.

FIGURE 1.8 Instabilities in injection molding: (a) formation of a weld line; and(b) the phenomenon of jetting.


2. Bird, R.B.; Armstrong, R.C.; Hassager, O. Dynamics of Polymeric Liquids: Vol.1. Fluid Mechanics; John Wiley and Sons: New York, 1987.

3. Pearson, J.R.A. Mechanics of Polymer Processing; Elsevier: New York, 1985.

4. de Gennes, P.G. Viscometric ows of tangled polymers. C. R. Acad. Sci. Paris,Ser. B 1979, 288, 219222.

5. Brochard-Wyart, F.; de Gennes, P.G. Shear-dependent slippage at a polymer/

solid interface. Langmuir 1992, 8, 30333037.6. Migler, K.B.; Hervet, H.; Leger, L. Slip transition of a polymer melt under

shear stress. Phys. Rev. Lett. 1993, 70, 287290.

7. Wang, S.-Q.; Drda, P.A. Stickslip transition in capillary ow of polyethylene.2. Molecular weight dependence and low-temperature anomaly. Macro-molecules 1996, 29, 41154119.

8. Petrie, C.J.S.; Denn, M.M. Instabilities in Polymer Processing. AIChE 1976,22, 209.

9. Baird, G.B.; Kolias, D.I. Polymer Processing; Butterworth-Heinemann:Toronto, 1995.

10. Kanai, T.; White, J.L. Kinematics, dynamics and stability of the tubular lmextrusion of various polyethylenes. Polym. Eng. Sci. 1984, 24 (15): 11851201.

11. Rincon, A.; Hrymak, A.N.; Vlachopoulos, J.; Dooley, J. Transient simulation

of coextrusion ows in coat-hanger dies. Society of Plastics Engineers. Proc.Annu. Tech. Conf. 1997, 55, 335350.

12. Tzoganakis, C.; Perdikoulias, J. Interfacial instabilities in coextrusion ows of

low-density polyethylenes: experimental dies. Polym. Eng. Sci. 2000, 40, 10561064.

13. Martyn, M.T.; Gough, T.; Spares, R.; Coates, P.D. Visualization of melt

interface in a co-extrusion geometry. Proceedings of Polymer Process Engineer-ing Conference; Bradford, 2001, 3745.

14. Dooley, J. Viscoelastic Flow Eects in Multilayer Polymer Coextrusion. Ph.D.Thesis, Technical University of Eindhoven, 2002.

15. Grillet, A.M.; Bogaerds, A.C.B.; Bulters, M.; Peters, G.W.M.; Baaijens, F.P.T.An investigation of ow mark surface defects in injection molding of polymermelts. Proceedings of the XIIIth International Congress on Rheology: Cam-

bridge, 2000; Vol. 3, 122.




ContentsIntroductionChapter 1 Overview of Processing Instabilities1.1 POLYMER FLOW INSTABILITIES1.2 PART A: THE NATURE OF POLYMERIC FLOW1.3 PART B: MELT FRACTURE IN EXTRUSION1.4 PART C: APPLICATIONS1.4.1 Draw Resonance in Film Casting1.4.2 Fiber Spinning Instabilities1.4.3 Film Blowing Instabilities1.4.4 Coextrusion and Interfacial Instabilities1.4.5 Injection Molding and Tiger Stripes Instability

REFERENCES

Polymer_Processing_Instabilities_Control_and_Understanding/Polymer Processing Instabilities Control and Understanding/DK1205_PartA.pdf2Elements of Rheology

John M. Dealy

McGill University, Montreal, Quebec, Canada

2.1 RHEOLOGICAL BEHAVIOR OF MOLTEN POLYMERS

Flow instabilities that occur in melt processing arise from a combination ofpolymer viscoelasticity and the large stresses that occur in large, rapiddeformations. This is in contrast to ows of Newtonian uids, where inertiaand surface tension are usually the driving forces for ow instabilities. Boththe elasticity and the high stresses that occur in the ow of molten polymersarise from their highmolecular weight (i.e., from the enormous length of theirmolecules). The high stresses are associated with the high viscosity of moltencommercial thermoplastics and elastomers. Typical values range from 103 to106 Pa s, whereas the viscosity of water is about 103 Pa sec. An easilyobserved manifestation of melt elasticity is the large swell in cross section thatoccurs when a melt exits a die.

The way melts behave in very small or very slow deformations isdescribed quite adequately by the theory of linear viscoelasticity (LVE), andone material function, the linear relaxation modulus G(t), is sucient todescribe the response of a viscoelastic material to any type of deformationhistory, as long as the deformation is very small or very slow. This informa-tion is useful in polymer characterization and determination of the relaxationspectrum. However, there is no general theoretical framework that describes




viscoelastic behavior in the large, rapid deformations that arise in polymerprocessing (i.e., there exists no general theory of nonlinear viscoelasticity).Nonlinear phenomena that arise in large, rapid ows include the strongdependence of viscosity on shear rate and the very large tensile stresses thatarise when long-chain branched polymers are subjected to extensionaldeformations.

Flow instabilities nearly always occur in response to large, rapiddeformations. This means that a ow simulation able to predict when a owwill become unstable must be based on a reliable model of nonlinearviscoelastic behavior. The lack of such a model at the present time greatlylimits our ability to model instabilities. Also lacking are generally accepted,quantitative criteria for the occurrence of instabilities, for example, in termsof a dimensionless group.

Although there remain major diculties in the modeling of melt owinstabilities, many experimental observations have been reported, and em-pirical correlations with various rheological properties have been proposed.In order to understand these, some knowledge of the rheological behavior ofmolten polymers is needed. It is the purpose of the present chapter to providea brief overview to prepare the uninitiated reader to understand the laterchapters of this book and the current stability literature.

2.2 VISCOELASTICITYBASIC CONCEPTS

Molten polymers are elastic liquids, and it is their elasticity that is the rootcause of most ow instabilities. Thus, these are usually hydroelastic ratherthan hydrodynamic instabilities. Melts of high polymers can store elasticenergy and, as a result, they retract when a stress that has been applied to themis suddenly released. However, they do not recover all the strains undergoneas a result of this stress, as they are liquids, and their deformation alwaysentails some viscous dissipation. These materials are thus said to have a fadingmemory.

The elasticity of polymers is intimately associated with the tendency oftheir molecules to become oriented when subjected to large, rapid deforma-tions. This molecular orientation gives rise to anisotropy in the bulk polymer,and manifestations of this anisotropy include ow birefringence and normalstress dierences.

A variable of central importance in the rheological behavior of poly-mers is time. Any system whose response to a change in its boundaryconditions involves both energy storage and energy dissipation must haveat least onematerial property that has units of time. Examples of such systemsare resistancecapacitance electrical circuits and the suspension systems ofautomobiles, which include springs (to store energy) and shock absorbers (to




dissipate energy). A system often used to demonstrate this fact consists of alinear (Hookean) spring and a linear dashpot, connected in series as shown inFig. 2.1. This system is called a Maxwell element, and we will see that itsbehavior has proven very useful in describing the viscoelastic behavior ofpolymers.

The force in the spring is proportional to the distance throughwhich it isstretched (Fs = KeDXs), and the force in the dashpot is proportional to thevelocity withwhich its ends are separated [Fd=Kv(dXd/dt)]. In the absence ofinertia, the forces in the two elements are equal, and the displacement of oneend relative to the other is DX = DXs + DXd. From this, it is easy todemonstrate that if the assembly is subjected to sudden stretching in theamount of DX0, the force will rise instantaneously to KeDX0 and then decayexponentially, as shown by Eq. (2.1):

Ft KeDX0exp tKe=Kv 2:1

We see that the ratio (Kv/Ke) is a parameter of the system and has units oftime. It is thus the relaxation time of the mechanical assembly.

2.3 LINEAR VISCOELASTICITY

Linear viscoelasticity is a type of rheological behavior exhibited by polymericmaterials in the limit of very small or very slow deformations. Although notdirectly applicable to the deformations that give rise to gross melt fracture,



FIGURE 2.1 The simplest mechanical analog of a viscoelastic liquid; the singleMaxwell element consisting of a linear spring in series with a linear dashpot.


LVE behavior is important for two reasons. First, information about linearviscoelastic behavior is very useful in the characterization of a polymer (i.e., indetermining its molecular structure). Second, it provides information aboutthe linear relaxation spectrum, which is an essential element of a model ofnonlinear viscoelastic behavior. A comprehensive discussion of the linearviscoelastic properties of polymers can be found in the book by Ferry (1).

The most basic experiment in viscoelasticity is the measurement of thetransient stress following a sudden step deformation. In the case of moltenpolymers, this is nearly always a shearing deformation, and the measuredstress is reported in terms of the relaxation modulus G(t). This function oftime is dened as the stress divided by the amount of shear strain c imposed onthe sample at t= 0:

Gturt=c 2:2A function such as this, which is a characteristic of a particular material, iscalled a material function.

The basic axiom of linear viscoelasticity, the Boltzmann superpositionprinciple, tells us that this function is independent of c and contains all theinformation needed to predict how a viscoelastic material will respond to anytype of deformation, as long as this deformation is very small or very slow.This principle can be stated in terms of an integral equation as follows forsimple shear deformation:

rt ttVl

Gt tVdctV 2:3

where r is the shear stress at time t, and dc(tV) is the shear strain that occursduring the time interval dtV. This can easily be generalized for any kinematicsby use of the extra stress tensor and the innitesimal strain tensor, whosecomponents are represented by rij and cij, respectively:

rijt ttVl

Gt tVdcijtV 2:4

and, in terms of the rate-of-strain tensor, this is:

rijt ttVl

Gt tVc: ijtVdtV 2:5

The extra stress is that portion of the total stress that is related todeformation. We recall that for in an incompressible uid, an isotropic stress(i.e., one that has no shear components andwhose normal components are thesame in all directions) does not generate any deformation. For example, the




Earths atmosphere at sea level, when completely still, is under a compressivestress of one atmosphere. Thus, there is an isotropic component of the totalstress tensor that is not related to deformation in an incompressible uid. Andthe innitesimal strain tensor is a very simple measure of deformation that isvalid only for very small deformations. The rate-of-deformation tensor isrelated to the velocity components by Eq. (2.6):

cijuBviBxj

BvjBxi

2:6

Eqs. (2.4) and (2.5) are alternative, concise statements of the Boltzmannsuperposition principle. A very simple model of linear viscoelastic behaviorcan be obtained by inserting into Eq. (2.5) the relaxation modulus that isanalogous to a single Maxwell element (i.e., a single exponential):

Gt Get=sr 2:7whereG is the instantaneous modulus and sr is a relaxation time. This is calledtheMaxwell model for the relaxationmodulus. Inserting this into Eq. (2.4), weobtain the simplest model for linear viscoelasticity:

rijt GttVl

ettV=sr dcijtV 2:8

The stress relaxation in an actual polymer can only very rarely beapproximated by a single exponential, and G(t) is usually represented eitherby a discrete or a continuous relaxation spectrum. In the case of a discretespectrum, the relaxation modulus is represented as a sum of weightedexponentials, as shown by Eq. (2.9):

Gt XNi1

Giet=si 2:9

ComparingEqs. (2.1) and (2.9), we see that this is analogous to the response ofamechanical assembly consisting of a series ofMaxwell elements connected inparallel, so that the displacements of all the elements are the same, and thetotal force is the sum of the forces in all the elements. Thus, Eq. (2.9) is calledthe generalized Maxwell model for the relaxation modulus.

The continuous spectrum F(s) is dened in terms of a continuous seriesof exponential decays, as shown by Eq. (2.10):

Gt l0

Fset=sds 2:10




It turns out to be more convenient to work with the time-weighted spectrumfunction H(s), which is dened as sF(s), as shown by Eq. (2.11):

Gt ll

Hset=sd ln s 2:11

It has been found that frequency (oscillatory shear) domain experimentsare much easier to perform than time domain (step strain) experiments, andthese provide an alternative means of establishing the linear behavior of apolymer. In oscillatory shear, the input strain is sinusoidal in time:

ct co sinxt 2:12As long as the response is linear (i.e., if co is suciently small), the shear stressis also sinusoidal, but with a phase shift d, as shown by Eq. (2.13):

rt ro sinxt d 2:13The response could be characterized by the amplitude ratio (ro/co) and thephase shift d as functions of frequency, but it is more informative to representthe response in terms of frequency-dependent in-phase (GV) and out-of-phase(G) components, as shown by Eq. (2.14):

rt roGV sin xt GVVcos xt 2:14The twomaterial functions of this relationship areGV(x), the storagemodulus,andGU(x), the loss modulus. The most common way of reporting linear visco-elastic behavior is to show plots of these two functions. Ferry (1) providesformulas for converting one LVE material function into another.

An alternative approach to viscoelastic characterization is to impose aprescribed stress on the sample andmonitor the strain. If a stress ro is imposedinstantaneously at time zero, and the resulting shear strain is divided by thisstress, the resulting ratio is the creep compliance J(t):

Jtuct=ro 2:15If the imposed stress is suciently small, the result will be governed by theBoltzmann superposition principle, and the creep compliance can, in princi-ple, be calculated if the relaxation modulus G(t) is known.

One can also impose a sinusoidal stress andmonitor the time-dependentdeformation as an alternative technique to determine the storage and lossmoduli.

2.4 VISCOSITY

Viscosity is dened as the steady-state shear stress divided by the shear rate ina steady simple shear experiment. For a Newtonian uid, the viscosity isindependent of shear rate, and this is also the prediction of the theory of linear




viscoelasticity. This means that a molten polymer will exhibit a shear rate-independent viscosity at a suciently small shear rate, although such a low-shear rate may be dicult to access experimentally, especially in the case ofpolymers with broad molecular weight distribution or long-chain branching.Outside this range, the viscosity is a strong function of shear rate, and thedependency of viscosity on shear rate g(c ) is an example of a nonlinearmaterial function.

2.4.1 Dependence of Viscosity on Shear Rate, Temperature,and Pressure

At suciently high-shear rates, the viscosity often approaches a powerlawrelationship with the shear rate. Figure. 2.2 is a plot of viscosity vs. shear ratefor a molten polymer, and it shows both a low-shear rate Newtonian regionand a high-shear rate powerlaw region. These data were reported byMeissner (2) some years ago and represent the ultimate in rheometricaltechnique. The variation of g with c implies the existence of at least onematerial property with units of time. For example, the reciprocal of the shearrate at which the extrapolation of the powerlaw line reaches the value of g0 isa characteristic time that is related to the departure of the viscosity from itszero-shear value. Let us call such a nonlinearity parameter sn.

The viscosity falls sharply as the temperature increases, as shown in Fig.2.2. Two models are widely used to describe this dependency: the Arrhenius



FIGURE 2.2 Viscosity vs. shear rate at several temperatures for a low-densitypolyethylene. (From Ref. 2.)


equation and the Williams-Landel-Ferry (WLF) equation. These are given instandard rheology references (1,3,4).

2.4.2 Dependence on Molecular Structure

The limiting low-shear-rate viscosity g0, the zero-shear viscosity, increaseslinearly with weight-average molecular weight when this is below an entan-glementmolecular weightMC, whereas above this value, the viscosity increasesin proportion to (Mw)

a, where a is usually around 3.5. This is one of severaldramatic rheological manifestations of the phenomenon called entanglementcoupling, which has a very strong eect on the ow of high-molecular-weightpolymers. Although it was once thought that these eects were caused byactual physical entanglements between molecules, it is now recognized thatthey are not the result of localized restraints at specic points along the chain.Instead, it is understood that they result from the severe impediment to lateralmotion that is imposed on a long molecule by all the neighboring molecules,although the term entanglement continues to be used to describe this eect. Athigher shear rates, the eect of molecular weight on viscosity decreases, so it isthe zero-shear value g0 that is most sensitive to molecular weight.

2.5 NORMAL STRESS DIFFERENCES

The shear rate-dependent viscosity is one manifestation of nonlinear visco-elasticity, and there are two additional steady-state material functionsassociated with steady simple shear when the shear rate is not close to zero.These are the rst and second normal stress dierences N1(c) and N2(c). Thethree material functions of steady simple shear g(c ), N1(c ), and N2(c ) areknown collectively as the viscometric functions.

These are dened using the standard frame of reference for simple shearshown in Fig. 2.3. The shear stress r is r21 (equal to r12), and the three normalstresses are: r11, in the direction of ow (x1); r22, in the direction of thegradient (x2); and r33, in the neutral (x3) direction. As this is, by denition, a



FIGURE 2.3 Standard coordinate definitions for simple shear.


two-dimensional ow, there is no velocity and no velocity gradient in the x3direction.

In an incompressible material, normal stresses are themselves of norheological signicance because as long as they are equal in all directions, theycause no deformation. However, dierences between normal stress compo-nents are signicant, because they do cause deformation. For steady simpleshear, the two rheologically signicant dierences are dened by Eq.(2.16a,b):

N1c:ur11 r22 2:16aN2c:ur22 r33 2:16b

In the limit of zero-shear rate (i.e., for linear viscoelastic behavior), these twomaterial functions approach zero, and as the shear rate increases, they are atrst proportional to the square of the shear rate. Thus, although the shearstress becomes linear with shear rate as the shear rate approaches zero,N1 andN2 are second order in c in this limit. This dependency inspired the denitionof the two alternative material functions dened by Eq. (2.17a,b):

W1c:uN1=c:2 2:17aW2c:uN2=c:2 2:17b

In other chapters of this book, it will be shown that these functions are relatedto several types of ow instability that occur in ows of viscoelastic melts.

2.6 TRANSIENT SHEAR FLOWS USED TO STUDYNONLINEAR VISCOELASTICITY

The response of a molten polymer to any transient shear ow that involves alarge or rapid deformation is a manifestation of nonlinear viscoelasticity.Some examples of ows used to characterize nonlinear melt behavior aredescribed in this section.

For large-step stress relaxation, the relaxation modulus is a function ofthe imposed strain as well as time: G(t,c). Except at very short times, thenonlinear relaxation modulus is often found to exhibit timestrain separabil-ity, which means that it can be represented as the product of the linearrelaxation modulus and a function of strain h(c) called the damping function,as shown by Eq. (2.18):

Gt; c Gthc 2:18The damping function is thus a material function that is wholly related tononlinearity in a step strain test. As the strain approaches zero, h(c) obviouslyapproaches one, and as the strain increases, it decreases.




The damping eect can be understood in terms of the tube model of thedynamics of polymeric molecules. In this model, the constraints imposed on agiven polymer chain by the surrounding molecules and that give rise toentanglement eects are modeled as a tube. In response to a suddendeformation, the tube is deformed, and the relaxation of the molecule ofinterest is constrained by its containment in its tube. When the imposeddeformation is very small, two mechanisms of relaxation occur: equilibrationand reptation. Equilibration involves the redistribution of stress along thechain within the tube. Further relaxation can only occur as a result of themolecule escaping the constraints of the tube, and this requires it to slitheralong or reptate out the tube. This is a much delayed mechanism, and this isthe cause of the plateau in the relaxation modulus for polymers with narrowmolecular weight distributions. If the molecular weight is not narrow, theshortermolecules making up the tube will be able to relax fast enough to causea blurring of the tube. This phenomenon is called constraint release and speedsup the relaxation of a long molecule in its tube.

The relaxation processes described above apply to linear viscoelasticbehavior. If the deformation is not small, there is an additional relaxationmechanismretraction within the tube. This is a fast relaxation, and once it iscompleted, the remainder of the relaxation process occurs as in the case of alinear response. It thus results in a relaxationmodulus curve that has an early,rapid decrease due to retraction, followed by a curve that has the same shapeas that for linear behavior. Thus, except for the very short-term relaxation, therelaxation modulus can be described as the linear modulus multiplied by afactor that accounts for the relaxation by retraction. This factor is thedamping function.

In start-up of steady simple shear, the measured stress is divided by theimposed constant shear rate to obtain the shear stress growth coecient,which is dened as follows:

gturt=c 2:19And the similarly dened shear stress decay coecient g(t) describes stressrelaxation following the cessation of steady simple shear.

2.7 EXTENSIONAL FLOWS

Most experimental studies of melt behavior involve shearing ows, but weknow that no matter how many material functions we determine in shear,outside the regime of linear viscoelasticity, they cannot be used to predict thebehavior of a melt in any other type of ow (i.e., for any other owkinematics). A type of ow that is of particular interest in commercialprocessing and the instabilities that arise therein is extensional ow. In this




type of ow, material points are stretched very rapidly along streamlines. Animportant example is the ow of a melt from a large tube into a much smallerone. In order to satisfy mass continuity, the velocity of a uid element mustincrease markedly as it ows into the smaller tube, and this implies rapidstretching along streamlines. This is shown schematically in Fig. 2.4.

Although entrance ow subjects some uid elements to large rates ofelongation, the rate of elongation is not uniform in space, so this ow eld isnot useful in determining a well-dened material function that describes theresponse of a material to extensional ow. It turns out to be quite dicult tosubject a melt to a uniform stretching deformation, and this is why reports ofsuch measurements are much rarer than those of shear ow studies.

The experiment usually carried out to study the response of a melt touniaxial extension is start-up of steady simple extension at a constantHenckystrain rate e. TheHencky strain rate is dened as follows, in terms of the lengthL of a sample:

e d ln L=dt 2:20Note that the length of a sample subjected to a constant Hencky strain rateincreases exponentially with time. This strain rate is a measure of the speedwith which material particles are separated from each other. The nonlinearmaterial function most often reported is the tensile stress growth coecient,which is dened as the ratio of the tensile stress to the strain rate, as shown byEq. (2.21):

gE t; e:urEt; e:=e: 2:21



FIGURE 2.4 Sketch of entrance flow showing stretching along streamlines: (a)without corner vortex; (b) with corner vortex.


2.8 DIMENSIONLESS GROUPS GOVERNING THE BEHAVIOROF VISCOELASTIC FLUIDS

When making a general statement about the ow behavior of polymers, it isuseful to represent the results of theoretical treatments and experiments interms of dimensionless variables. Two dimensionless groups are often used todescribe the rate or duration of an experiment. One of these, the Weissenbergnumber (Wi), is a measure of the degree of nonlinearity or anisotropyexhibited by the uid in a particular deformation, and the second, theDeborah number (De), is a measure of the degree to which a material exhibitselastic behavior. We will see in later chapters that these groups are useful fordescribing ow instabilities.

The Deborah number (De) is a measure of the degree to which the uidwill exhibit elasticity in a given type of deformation. More specically, itreects the degree to which stored elastic energy either increases or decreasesduring a ow. In steady simple shear, at steady state, when all stresses areconstant with time, the amount of stored elastic energy is constant with time,so the Deborah number is zero. Thus, it is only in transient ows that theDeborah number has a nonzero value. Here transientmeans that the stateof stress in a uid element changes with time. This can arise as a result of atime-varying boundary condition, in a rheometer, or of the ow from onechannel into a smaller one so that acceleration is involved. This dimensionlessgroup is the ratio of a time arising from the uids viscoelasticity (i.e., itsrelaxation time) to a time that is ameasure of the duration of the deformation.

For example, consider the response of a uid to the start-up of steadysimple shear in which a shear rate of c is applied to a uid initially in its reststate. In the initial stages of the deformation, the shear stress will increase withtime and will eventually reach a steady value. At times t after the shearing isbegun, the stress will continue to change with time until t is long compared tothe relaxation time of the uid. Thus, the Deborah number for this defor-mation is sr/t; when this ratio is large (short times), the amount of storedelastic energy is increasing, but when it is small (long times), it becomessteady, andDe approaches zero. In this example, the Deborah number varieswith time.

The Deborah number is only zero in deformations with constant stretchhistory (steady from the point of view of a material element). It is dicult towrite a concise denition of a time constant that governs the rate at whichstored elastic energy changes in a given deformation without reference to aspecic rheological model, and we therefore give a denition of the Deborahnumber in general terms:

De usr

characteristic time of transient deformation2:22




Turning to the Weissenberg number, its use can be demonstrated byreference to steady simple shear ow. We have seen that any uid that has ashear rate-dependent viscosity or is viscoelastic must have at least onematerial constant that has units of time and is characteristic of rheologicalnonlinearity, and we have called this time sn. Furthermore, the type ofbehavior exhibited by a particular material depends on how this time constantcompares with the reciprocal of the rate of the deformation. For example, letus say that a non-Newtonian uid has a nonlinearity time constant sn of 1 sec.This implies that if the shear rate c in steady simple shear is much smaller thanthe reciprocal of this time, its viscosity will be independent of shear rate. Thissuggests the denition of theWeissenberg number (Wi) as follows:

Wi u c:sn 2:23

(The symbolsWe andWs are also used for the Weissenberg number.) Thus,this dimensionless group is a measure of the degree to which the behavior of auid deviates from Newtonian behavior. For single-phase, low-molecular-weight uids, the time constant of the material is extremely short, so that theWeissenberg number is always very small for the ows that occur undernormal circumstances. But for molten, high-molecular-weight polymers, sncan be quite large. For polymeric liquids, the Weissenberg number alsoindicates the degree of anisotropy generated by the deformation. And inthe case of steady simple shear, the normal stress dierences are manifes-tations of anisotropy and thus of nonlinear viscoelasticity. Therefore, theWeissenberg number also governs the degree to which the normal stressdierences will dier from zero.

To summarize, when the Weissenberg number is very small in a simpleshear ow, the viscosity will be independent of shear rate, and the normalstress dierences will be negligible. These phenomena reect the fact that inthe limit of vanishing shear rate, the shear stress is rst order in the shear rate,whereas the normal stress dierences increase with the square of the shearrate. The Weissenberg number is easily dened for any ow with constantstretch history. For example, for steady uniaxial extension, we need onlyreplace the shear rate by the Hencky strain rate e.

In general, the degree to which behavior is nonlinear depends on theWeissenberg number. At very small deformation rates,Wi is much less thanone, and the stress is governed by the Boltzmann superposition principle. Athigher deformation rates,Wi increases and nonlinearity appears, as reected,for example, in the dependence of viscosity on shear rate and the appearanceof normal stress dierences. The two characteristic times dened above (snand sr) are closely related, and in recognition of this, the subscripts will bedropped in the rest of this discussion.






There are several pitfalls in the use of the two dimensionless groupsdened above to characterize the response of a material to a given type ofdeformation. First, there are hardly anymaterials whose viscoelastic behaviorcan be described using a single relaxation time. More typically, a spectrum oftimes is required, and this causes diculty in the choice of a time for use indening the Deborah number. The longest relaxation time is oftenidentied as the appropriate one for dening these groups, but for highlypolydispersed or branched polymers, it may be impossible to identify alongest time.

Another problem in the use of dimensionless groups to characterizedeformations is that for a melt consisting of a single polymer, in several owsof practical interest,Wi andDe are directly related to each other. This causesconfusion, as authors of books and research papers tend to use the two groupsinterchangeably in all circumstances. Some examples will help to illustrate thecorrect use of the Weissenberg and Deborah numbers.

A ow that has been of great interest to both experimental andtheoretical polymer scientists is the entrance ow from a circular reservoirinto a much smaller tube or capillary. A Weissenberg number can readily bedened for this ow as the product of the characteristic time of the uid andthe shear rate at the wall of the capillary. However, entrance ow is clearly nota ow with constant stretch history, and the Deborah number is thus nonzeroas well. Furthermore, as demonstrated by Rothstein and McKinley (5), insuch ows, the two groups are directly related. This can lead to confusionwhen experimental data and ow simulation results are compared (6). Notethat for fully developed capillary ow, De= 0.

On the other hand, a ow in which the two numbers can be variedindependently is oscillatory shear. As used for the determination of the linearviscoelastic behavior of polymers, this deformation is carried out at very smallstrain amplitudes, so that theWeissenberg number will be much less than one.As the frequency is increased from zero, the Deborah number, dened here asxs, is at rst very small, and the response is purely viscous. BecauseWi is alsovery small, the melt behaves like a Newtonian uid. If now the frequency isincreased, the Deborah number increases and the importance of elasticitygrows, and at very high De, the behavior becomes almost purely elastic.

If, however, the strain amplitude co is increased, then the strain rateamplitude c.ou xco, and the Weissenberg number, which is equal to c

.os, also

increases. By changing the frequency or the amplitude, Wi and De can bevaried independently. A convenient way of representing the parameter spaceof large-amplitude oscillatory shear is a Pipkin diagram, which is a graph ofWeissenberg number vs. Deborah number (4,7). In such a diagram, thebehavior in the lower left-hand corner (Wi b 1; De b 1) is that of aNewtonian uid. Near the vertical axis (Deb 1), the behavior is nonlinearbut inelastic and is governed by the viscometric functions. Near the hori-




zontal axis (Wib 1), linear viscoelastic behavior is expected, and far to theright (DeH 1), the behavior becomes indistinguishable from that of an elas-tic solid.

2.9 EXPERIMENTAL METHODS IN RHEOLOGYRHEOMETRY

The objective of rheometry is to make measurements whose data can beinterpreted in terms of well-dened material functions, without making anyassumptions about the rheological behavior of the material. Examples ofmaterial functions are the viscosity as a function of shear rate and therelaxation modulus as a function of time. These functions are physicalproperties of a material, and the detailed method by which they are deter-mined need not be reported in order for them to be properly interpreted. Incontrast to this type of measurement are empirical industry tests that yieldnumbers that can be used to compare materials but are not directly related toany one physical property. Such test data are widely used as specications forcommercial products and for quality control. An important example thatinvolves the ow of molten plastics is the melt ow rate, often called the meltindex (8).

In an experiment designed to determine a material function, it isnecessary to generate a deformation in which the streamlines are known apriori (i.e., that are independent of the rheological properties of the material).Such a deformation is called a controllable ow, and the number of suchdeformations is very limited. In fact, the only practically realizable control-lable ows are simple shear, simple (uniaxial) extension, biaxial extension,and planar extension. And the last two of these are suciently dicult togenerate that they are rarely used. Additional limitations on our ability todetermine rheological material functions in the laboratory are imposed byvarious instabilities that occur even in these very simple ows.

There are two ways of generating a shear deformation in a rheometer:drag ow and pressure ow. In drag ow, one surface in contact with thesample moves relative to another to generate shearing. In pressure ow,pressure is used to force the uid to ow through a straight channel, whichmay be a capillary or a slit. Drag ow can be used to determine a variety ofmaterial functions, including the storage and loss moduli as functions offrequency, the creep compliance as a function of time, the viscosity andnormal stress dierences as functions of shear rate, and various nonlineartransient material functions. Pressure-driven rheometers are useful primarilyfor the measurement of viscosity at high-shear rates.

Extensional rheometers are most often designed to generate uniaxial(tensile) extension in which either the tensile stress or the strain rate ismaintained constant.


Below are presented brief overviews of the way melt rheometers areused.More detailed information about experimental rheology can be found invarious rheology books (3,4).

2.9.1 Rotational Rheometers

Rotational rheometers can be classied according to the type of xture usedand by the variable that is controlled (i.e., the independent variable). Twotypes of xture are commonly used with molten polymers: cone-plate andplate-plate. The ow between a cone and a plate, one of which is rotating withrespect to the other at an angular velocity X, closely approximates uniformsimple shear, as the shear rate at a radius r is the local rotational speed Xr ofthe rotating xture at r, divided by the gap between the xtures at this value ofr, which we call h. In the cone and plate geometry, this distance is linear with r,with the result that rX/h (i.e., the local shear rate) is uniform. This feature ofcone-plate ow makes it useful for studies of nonlinear viscoelasticity. Cone-plate xtures are used to determine the viscosity and rst normal stressdierence as functions of shear rate at low-shear rates, as well as the responseof the shear and normal stresses to various transient shearing deformations.

The equations for calculating the strain rate and stresses of interest areas follows for the case of a small cone angle Ho:

c: X=Ho 2:24r 3M=2pR3 2:25aN1 2F=pR2 2:26

whereX is the angular velocity of the rotating xture; r is the shear stress;M isthe torque measured on either the rotating or the stationary shaft; F is thetotal normal thrust on the xtures; and R is the radius of the xtures.

Uniform simple shear is very well approximated by the ow between acone and a plate; if the cone angle is small, and the shear rate is not very high.Starting at moderate shear rates, however, various types of instability renderthe ow unsuitable for rheological measurements. Thus, cone-plate xturesare useful for determining moderate departures from linear viscoelasticity.The major concerns in using this technique are calibrating the sensors,avoiding degradation of the sample, and recognizing when an instabilityhas occurred.

Although cone-plate xtures can be used to determine the materialfunctions of linear viscoelasticity, it is more convenient to use plate-platextures for this application because the preparation and loading of samples,as well as the setting of the gap between the two xtures, are much simplied.Although the local shear strain between the parallel plates varies linearly with




radius, as long as the viscoelastic behavior is linear, the local stress isproportional to the strain, and the variation poses no problem in theinterpretation of the data. The equations for calculating the storage and lossmoduli are as follows:

GV 2Moh cos dpR4/o

2:25b

GW 2Moh sin dpR4/o

2:25c

where Mo is the amplitude of the torque signal; /o is the amplitude of theangular displacement of the oscillating shaft; and d is the phase angle betweenthe angular displacement and the measured torque.

The primary concerns in making reliable measurements are calibratingthe sensors, avoiding thermo-oxidative degradation of the sample, andensuring that the amplitude selected for the deformation or torque does nottake the sample out of its linear range of response.

Rotational rheometers can also be classied according towhich variableis controlled. Thus, there are controlled strain (actually controlled angularmotion) and controlled stress (actually controlled torque) instruments. Bothcan be used to determine linear viscoelastic properties, although controlledstress instruments give better results at low frequencies, whereas controlledstrain instruments are preferred at high frequencies. Thus, the two types ofrheometer provide information that is, to some degree, complementary.

2.9.2 Sliding Plate Rheometers

Sliding plate melt rheometers were developed to make measurements ofnonlinear viscoelastic behavior under conditions under which cone-plate owis unstable (i.e., in large, rapid deformations) (9). The sample is placedbetween two rectangular plates, one of which translates relative to the other,generating, in principle, an ideal rectilinear simple shear deformation. Thereare signicant edge eects at large strains, and to minimize the eect of theseon the measurement, the stress should not be inferred from the total force onone of the plates but measured directly in the center of the sample by a shearstress transducer. Such instruments have been used to determine the shearstress response to large, transient deformations (9,10). In addition, they havebeen used to determine the eect of pressure on the viscosity and nonlinearbehavior of melts (11,12). Their advantage over capillary instruments forhigh-pressure measurements is that the pressure and shear rate in the sampleare uniform.

Phenomena that limit their use are slip, cavitation, and rupture, whichinterrupt experiments at suciently high strains and strain rates. At the same




time, however, sliding plate rheometers have been found to be useful tools forthe study of melt slip (13).

2.9.3 Capillary and Slit Rheometers

Pressure-driven rheometers, particularly capillary instruments, are the workhorses of plastics rheology, as they are relatively simple and easy to use. Inmost capillary rheometers, the ow is generated by a pistonmoving in a roundreservoir to drive the melt through a small capillary, which often has adiameter of about 1 mm. After a short entrance length, the ow becomes fullydeveloped (i.e., the velocity prole and shear stress become independent ofdistance from the entrance). For a Newtonian uid, the fully developedvelocity distribution is parabolic, and the wall shear rate can be calculated byknowing only the volumetric ow rate Q and the radius of the capillary R.However, in the case of a non-Newtonian uid, the velocity distributiondepends on the viscosity function, which is initially unknown. It has provenuseful in polymer characterization, however, to use the formula for New-tonian uids to dene an apparent wall shear rate c.A. The wall shear rate (Bv/Br)r= R is actually negative because the velocity is zero at the wall and positiveat the center, but it is the magnitude of this quantity that is universally used indiscussing capillary and slip ow. Thus, the apparent wall shear rate is thepositive quantity dened as follows:

c:Au4Q

pR32:27a

Techniques for determining the true wall shear rate cw can be found in variousrheology books (3,4).

If the pressure gradient in the region of fully developed ow is known,the wall shear stress can be readily c

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