Nonlinear Polymer Rheology

SHI-QING WANG, PhD, is Kumho Professor of Polymer Science at the University of Akron. He has been teaching at the university level for more than 28 years and has over 150 peer reviewed publications. Dr. Wang is a reviewer for many journals and a Fellow of both the American Physical Society (APS) and American Association for the Advancement of Science (AAAS).

Preface xv

Acknowledgments xix

Introduction xxi

About the Companion Website xxxi

Part I Linear Viscoelasticity and Experimental Methods 1

1 Phenomenological Description of Linear Viscoelasticity 3

1.1 Basic Modes of Deformation 3

1.1.1 Startup shear 4

1.1.2 Step Strain and Shear Cessation from Steady State 5

1.1.3 Dynamic or Oscillatory Shear 5

1.2 Linear Responses 5

1.2.1 Elastic Hookean Solids 6

1.2.2 Viscous Newtonian Liquids 6

1.2.3 Viscoelastic Responses 7

1.2.3.1 Boltzmann Superposition Principle for Linear Response 7

1.2.3.2 General Material Functions in Oscillatory Shear 8

1.2.3.3 Stress Relaxation from Step Strain or Steady-State Shear 8

1.2.4 Maxwell Model for Viscoelastic Liquids 8

1.2.4.1 Stress Relaxation from Step Strain 9

1.2.4.2 Startup Deformation 10

1.2.4.3 Oscillatory (Dynamic) Shear 11

1.2.5 General Features of Viscoelastic Liquids 12

1.2.5.1 Generalized Maxwell Model 12

1.2.5.2 Lack of Linear Response in Small Step Strain: A Dilemma 13

1.2.6 Kelvin–Voigt Model for Viscoelastic Solids 14

1.2.6.1 Creep Experiment 15

1.2.6.2 Strain Recovery in Stress-Free State 15

1.2.7 Weissenberg Number and Yielding during Linear Response 16

1.3 Classical Rubber Elasticity Theory 17

1.3.1 Chain Conformational Entropy and Elastic Force 17

1.3.2 Network Elasticity and Stress–Strain Relation 18

1.3.3 Alternative Expression in terms of Retraction Force and Areal Strand Density 20

References 21

2 Molecular Characterization in Linear Viscoelastic Regime 23

2.1 Dilute Limit 23

2.1.1 Viscosity of Einstein Suspensions 23

2.1.2 Kirkwood–Riseman Model 24

2.1.3 Zimm Model 24

2.1.4 Rouse Bead-Spring Model 25

2.1.4.1 Stokes Law of Frictional Force of a Solid Sphere (Bead) 26

2.1.4.2 Brownian Motion and Stokes–Einstein Formula for Solid Particles 26

2.1.4.3 Equations of Motion and Rouse Relaxation Time τR27

2.1.4.4 Rouse Dynamics for Unentangled Melts 28

2.1.5 Relationship between Diffusion and Relaxation Time 29

2.2 Entangled State 30

2.2.1 Phenomenological Evidence of chain Entanglement 30

2.2.1.1 Elastic Recovery Phenomenon 30

2.2.1.2 Rubbery Plateau in Creep Compliance 31

2.2.1.3 Stress Relaxation 32

2.2.1.4 Elastic Plateau in Storage Modulus G’ 32

2.2.2 Transient Network Models 34

2.2.3 Models Depicting Onset of Chain Entanglement 35

2.2.3.1 Packing Model 35

2.2.3.2 Percolation Model 38

2.3 Molecular-Level Descriptions of Entanglement Dynamics 39

2.3.1 Reptation Idea of de Gennes 39

2.3.2 Tube Model of Doi and Edwards 41

2.3.3 Polymer-Mode-Coupling Theory of Schweizer 43

2.3.4 Self-diffusion Constant versus Zero-shear Viscosity 44

2.3.5 Entangled Solutions 46

2.4 Temperature Dependence 47

2.4.1 Time–Temperature Equivalence 47

2.4.2 Thermo-rheological Complexity 48

2.4.3 Segmental Friction and Terminal Relaxation Dynamics 49

References 50

3 Experimental Methods 55

3.1 Shear Rheometry 55

3.1.1 Shear by Linear Displacement 55

3.1.2 Shear in Rotational Device 56

3.1.2.1 Cone-Plate Assembly 56

3.1.2.2 Parallel Disks 57

3.1.2.3 Circular Couette Apparatus 58

3.1.3 Pressure-Driven Apparatus 59

3.1.3.1 Capillary Die 60

3.1.3.2 Channel Slit 61

3.2 Extensional Rheometry 63

3.2.1 Basic Definitions of Strain and Stress 63

3.2.2 Three Types of Devices 64

3.2.2.1 Instron Stretcher 64

3.2.2.2 Meissner-Like Sentmanat Extensional Rheometer 65

3.2.2.3 Filament Stretching Rheometer 65

3.3 In Situ Rheostructural Methods 66

3.3.1 Flow Birefringence 66

3.3.1.1 Stress Optical Rule 67

3.3.1.2 Breakdown of Stress-Optical Rule 68

3.3.2 Scattering (X-Ray, Light, Neutron) 69

3.3.3 Spectroscopy (NMR, Fluorescence, IR, Raman, Dielectric) 69

3.3.4 Microrheology and Microscopic Force Probes 69

3.4 Advanced Rheometric Methods 69

3.4.1 Superposition of Small-Amplitude Oscillatory Shear and Small Step Strain during Steady Continuous Shear 69

3.4.2 Rate or Stress Switching Multistep Platform 70

3.5 Conclusion 70

References 71

4 Characterization of Deformation Field Using Different Methods 75

4.1 Basic Features in Simple Shear 75

4.1.1 Working Principle for Strain-Controlled Rheometry: Homogeneous Shear 75

4.1.2 Stress-Controlled Shear 76

4.2 Yield Stress in Bingham-Type (Yield-Stress) Fluids 77

4.3 Cases of Homogeneous Shear 79

4.4 Particle-Tracking Velocimetry (PTV) 79

4.4.1 Simple Shear 80

4.4.1.1 Velocities in XZ-Plane 80

4.4.1.2 Deformation Field in XY Plane 80

4.4.2 Channel Flow 82

4.4.3 Other Geometries 83

4.5 Single-Molecule Imaging Velocimetry 83

4.6 Other Visualization Methods 83

References 84

5 Improved and Other Rheometric Apparatuses 87

5.1 Linearly Displaced Cocylinder Sliding for Simple Shear 88

5.2 Cone-Partitioned Plate (CPP) for Rotational Shear 88

5.3 Other Forms of Large Deformation 91

5.3.1 Deformation at Converging Die Entry 91

5.3.2 One-Dimensional Squeezing 92

5.3.3 Planar Extension 95

5.4 Conclusion 96

References 97

Part II Yielding – Primary Nonlinear Responses to Ongoing Deformation 99

6 Wall Slip – Interfacial Chain Disentanglement 103

6.1 Basic Notions of Wall Slip in Steady Shear 104

6.1.1 Slip Velocity Vs and Navier–de Gennes Extrapolation Length b 104

6.1.2 Correction of Shear Field due to Wall Slip 105

6.1.3 Complete Slip and Maximum Value for b 106

6.2 Stick–Slip Transition in Controlled-Stress Mode 108

6.2.1 Stick–Slip Transition in Capillary Extrusion 108

6.2.1.1 Analytical Description 108

6.2.1.2 Experimental Data 109

6.2.2 Stick–Slip Transition in Simple Shear 111

6.2.3 Limiting Slip Velocity V∗s for Different Polymer Melts 113

6.2.4 Characteristics of Interfacial Slip Layer 116

6.3 Wall Slip during Startup Shear – Interfacial Yielding 116

6.3.1 Theoretical Discussions 117

6.3.2 Experimental Data 118

6.4 Relationship between Slip and Bulk Shear Deformation 120

6.4.1 Transition from Wall Slip to Bulk Nonlinear Response: Theoretical Analysis 120

6.4.2 Experimental Evidence of Stress Plateau Associated with Wall Slip 122

6.4.2.1 A Case Based on Entangled DNA Solutions 122

6.4.2.2 Entangled Polybutadiene Solutions in Small Gap Distance H∼50 μm 123

6.4.2.3 Verification of Theoretical Relation by Experiment 126

6.4.3 Influence of Shear Thinning on Slip 127

6.4.4 Gap Dependence and Independence 128

6.5 Molecular Evidence of Disentanglement during Wall Slip 131

6.6 Uncertainties in Boundary Condition 134

6.6.1 Oscillations between Entanglement and Disentanglement Under Constant Speed 134

6.6.2 Oscillations between Stick and Slip under Constant Pressure 134

6.7 Conclusion 134

References 135

7 Yielding during Startup Deformation: From Elastic Deformation to Flow 139

7.1 Yielding at Wi<1 and Steady Shear Thinning at Wi>1 140

7.1.1 Elastic Deformation and Yielding for Wi<1 140

7.1.2 Steady Shear Rheology: Shear Thinning 141

7.2 Stress Overshoot in Fast Startup Shear 143

7.2.1 Scaling Characteristics of Shear Stress Overshoot 144

7.2.1.1 Viscoelastic Regime (WiR <1) 145

7.2.1.2 Elastic Deformation (Scaling) Regime (WiR >1) 145

7.2.1.3 Contrast between Two Different Regimes 148

7.2.2 Elastic Recoil from Startup Shear: Evidence of Yielding 149

7.2.2.1 Elastic Recoil for WiR >1 149

7.2.2.2 Irrecoverable Shear at WiR <1 149

7.2.3 More Evidence of Yielding at Overshoot Based on Rate-Switching Tests 153

7.3 Nature of Steady Shear 154

7.3.1 Superposition of Small-Amplitude Oscillatory Shear onto Steady-State Shear 155

7.3.2 Two Other Methods to Probe Steady Shear 157

7.4 From Terminal Flow to Fast Flow under Creep: Entanglement–Disentanglement Transition 159

7.5 Yielding in Startup Uniaxial Extension 163

7.5.1 Myth with Considère Criterion 163

7.5.2 Tensile Force (Engineering Stress) versus True Stress 164

7.5.3 Tensile Force Maximum: A Signature of Yielding in Extension 165

7.5.3.1 Terminal Flow (Wi<1) 166

7.5.3.2 Yielding Evidenced by Decline in σengr 167

7.5.3.3 Maxwell-Like Response and Scaling for WiR >1 170

7.5.3.4 Elastic Recoil 173

7.6 Conclusion 175

7.A Experimental Estimates of Rouse Relaxation Time 175

7.A.1 From Self-Diffusion 175

7.A.2 From Zero-Shear Viscosity 176

7.A.3 From Reptation (Terminal Relaxation) Time τd 176

7.A.4 From Second Crossover Frequency∼1/τe 176

References 176

8 Strain Hardening in Extension 181

8.1 Conceptual Pictures 181

8.2 Origin of “Strain Hardening” 184

8.2.1 Simple Illustration of Geometric Condensation Effect 184

8.2.2 “Strain Hardening” of Polymer Melts with Long-Chain Branching and Solutions 185

8.2.2.1 Melts with LCB 185

8.2.2.2 Entangled Solutions of Linear Chains 187

8.3 True Strain Hardening in Uniaxial Extension: Non-Gaussian Stretching from Finite Extensibility 188

8.4 Different Responses of Entanglement to Startup Extension and Shear 190

8.5 Conclusion 190

8.A Conceptual and Mathematical Accounts of Geometric Condensation 191

References 192

9 Shear Banding in Startup and Oscillatory Shear: Particle-Tracking Velocimetry 195

9.1 Shear Banding After Overshoot in Startup Shear 197

9.1.1 Brief Historical Background 197

9.1.2 Relevant Factors 198

9.1.2.1 Sample Requirements: Well Entangled, with Long Reptation Time and Low Polydispersity 198

9.1.2.2 Controlling Slip Velocity 199

9.1.2.3 Edge Effects 199

9.1.2.4 Absence of Shear Banding for b/H≪1 201

9.1.2.5 Disappearance of Shear Banding at High Shear Rates 202

9.1.2.6 Avoiding Shear Banding with Rate Ramp-Up 202

9.1.3 Shear Banding in Conventional Rheometric Devices 203

9.1.3.1 Shear Banding in Entangled DNA Solutions 203

9.1.3.2 Transient and Steady Shear Banding of Entangled 1,4-Polybutadiene Solutions 204

9.1.4 From Wall Slip to Shear Banding in Small Gap Distance 208

9.2 Overcoming Wall Slip during Startup Shear 209

9.2.1 Strategy Based on Choice of Solvent Viscosity 209

9.2.2 Negligible Slip Correction at High Wiapp 213

9.2.3 Summary on Shear Banding 213

9.3 Nonlinearity and Shear Banding in Large-Amplitude Oscillatory Shear 214

9.3.1 Strain Softening 214

9.3.2 Wave Distortion 215

9.3.3 Shear Banding 215

References 217

10 Strain Localization in Extrusion, Squeezing Planar Extension: PTV Observations 221

10.1 Capillary Rheometry in Rate-Controlled Mode 221

10.1.1 Steady-State Characteristics 221

10.1.2 Transient Behavior 223

10.1.2.1 Pressure Oscillation and Hysteresis 223

10.1.2.2 Input vs. Throughput, Entry Pressure Loss and Yielding 224

10.2 Instabilities at Die Entry 226

10.2.1 Vortex Formation vs. Shear Banding 226

10.2.2 Stagnation at Corners and Internal Slip 227

10.3 Squeezing Deformation 230

10.4 Planar Extension 233

References 233

11 Strain Localization and Failure during Startup Uniaxial Extension 235

11.1 Tensile-Like Failure (Decohesion) at Low Rates 237

11.2 Shear Yielding and Necking-Like Strain Localization at High Rates 239

11.2.1 Shear Yielding 239

11.2.2 Constant Normalized Engineering Stress at the Onset of Strain Localization 243

11.3 Rupture-Like Breakup: Where Are Yielding and Disentanglement? 245

11.4 Strain Localization Versus Steady Flow: Sentmanat Extensional Rheometry Versus Filament-Stretching Rheometry 247

11.5 Role of Long-Chain Branching 250

11.A Analogy between Capillary Rheometry and Filament-Stretching Rheometry 250

References 251

Part III Decohesion and Elastic Yielding After Large Deformation 255

12 Nonquiescent Stress Relaxation and Elastic Yielding in Stepwise Shear 257

12.1 Strain Softening After Large Step Strain 258

12.1.1 Phenomenology 258

12.1.2 Tube Model Interpretation 261

12.1.2.1 Normal Doi–Edwards Behavior 261

12.1.2.2 Type C Ultra-strain-softening 262

12.2 Particle Tracking Velocimetry Revelation of Localized Elastic Yielding 265

12.2.1 Nonquiescent Relaxation in Polymer Solutions 266

12.2.1.1 Elastic Yielding in Polybutadiene Solutions 266

12.2.1.2 Suppression of Breakup by Reduction in Extrapolation Length b 269

12.2.1.3 Nonquiescent Relaxation in Polystyrene Solutions 269

12.2.1.4 Strain Localization in the Absence of Edge Instability 270

12.2.2 Nonquiescent Relaxation in Styrene–Butadiene Rubbers 272

12.2.2.1 Induction Time and Molecular Weight Dependence 273

12.2.2.2 Severe Shear Banding before Shear Cessation and Immediate Breakup 275

12.2.2.3 Rate Dependence of Elastic Breakup 275

12.2.2.4 Unconventional “Step Strain” Produced at WiR <1 278

12.3 Quiescent and Uniform Elastic Yielding 279

12.3.1 General Comments 279

12.3.2 Condition for Uniform Yielding and Quiescent Stress Relaxation 280

12.3.3 Homogeneous Elastic Yielding Probed by Sequential Shearing 281

12.4 Arrested Wall Slip: Elastic Yielding at Interfaces 283

12.4.1 Entangled Solutions 283

12.4.2 Entangled Melts 283

12.5 Conclusion 286

References 287

13 Elastic Breakup in Stepwise Uniaxial Extension 291

13.1 Rupture-Like Failure during Relaxation at Small Magnitude or Low Extension Rate (WiR <1) 292

13.1.1 Small Magnitude (ε ∼ 1) 292

13.1.2 Low Rates Satisfying WiR <1 292

13.2 Shear-Yielding-Induced Failure upon Fast Large Step Extension (WiR >1) 293

13.3 Nature of Elastic Breakup Probed by Infrared Thermal-Imaging Measurements 297

13.4 Primitive Phenomenological Explanations 298

13.5 Step Squeeze and Planar Extension 299

References 299

14 Finite Cohesion and Role of Chain Architecture 301

14.1 Cohesive Strength of an Entanglement Network 302

14.2 Enhancing the Cohesion Barrier: Long-Chain Branching Hinders Structural Breakup 306

References 308

Part IV Emerging Conceptual Framework and Beyond 311

15 Homogeneous Entanglement 313

15.1 What Is Chain Entanglement? 313

15.2 When, How, and Why Disentanglement Occurs? 315

15.3 Criterion for Homogeneous Shear 316

15.4 Constitutive Nonmonotonicity 318

15.5 Metastable Nature of Shear Banding 319

References 322

16 Molecular Networks as the Conceptual Foundation 325

16.1 Introduction: The Tube Model and its Predictions 326

16.1.1 Basic Starting Points of the Tube Model 327

16.1.2 Rouse Chain Retraction 328

16.1.3 Nonmonotonicity due to Rouse Chain Retraction 328

16.1.3.1 Absence of Linear Response to Step Strain 328

16.1.3.2 Stress Overshoot upon Startup Shear 329

16.1.3.3 Strain Softening: Damping Function for Stress Relaxation 330

16.1.3.4 Excessive Shear Thinning: The Symptom of Shear Stress Maximum 331

16.1.3.5 Anticipation of Necking Based on Considère Criterion 332

16.1.4 How to Test the Tube Model 332

16.2 Essential Ingredients for a New Molecular Model 333

16.2.1 Intrachain Elastic Retraction Force 334

16.2.2 Intermolecular Grip Force (IGF) 335

16.2.3 Entanglement (Cohesion) Force Arising from Entropic Barrier: Finite Cohesion 336

16.2.3.1 Scaling Analysis 337

16.2.3.2 Threshold for decohesion 338

16.3 Overcoming Finite Cohesion after Step Deformation: Quiescent or Not 339

16.3.1 Nonquiescence from Severe Elastic Yielding 339

16.3.1.1 With WiR >1 339

16.3.1.2 With WiR≪1 340

16.3.2 Homogeneous Elastic Yielding: Quiescent Relaxation 340

16.4 Forced Microscopic Yielding during Startup Deformation: Stress Overshoot 341

16.4.1 Chain Disentanglement for WiR <1 341

16.4.2 Molecular Force Imbalance and Scaling for WiR >1 342

16.4.3 Yielding is a Universal Response: Maximum Engineering Stress 346

16.5 Interfacial Yielding via Disentanglement 346

16.6 Effect of Long-Chain Branching 347

16.7 Decohesion in Startup Creep: Entanglement–Disentanglement Transition 349

16.8 Emerging Microscopic Theory of Sussman and Schweizer 350

16.9 Further Tests to Reveal the Nature of Responses to Large Deformation 351

16.9.1 Molecular Dynamics Simulations 352

16.9.2 Small Angle Neutron Scattering Measurements 353

16.9.2.1 Melt Extension at WiR≪1 353

16.9.2.2 Step Melt Extension With WiR >1 354

16.10 Conclusion 354

References 355

17 “Anomalous” Phenomena 361

17.1 Essence of Rheometric Measurements: Isothermal Condition 361

17.1.1 Heat Transfer in Simple Shear 362

17.1.2 Heat Transfer in Uniaxial Extension 364

17.2 Internal Energy Buildup with and without Non-Gaussian Extension 366

17.3 Breakdown of Time–Temperature Superposition (TTS) during Transient Response 368

17.3.1 Time–Temperature Superposition in Polystyrene Solutions and Styrene–Butadiene Rubbers: Linear Response 368

17.3.2 Failure of Time–Temperature Superposition: Solutions and Melts 369

17.3.2.1 Entangled Polymer Solutions Undergoing Startup Shear 369

17.3.2.2 Entangled Polymer Melts during Startup Extension 370

17.4 Strain Hardening in Simple Shear of Some Polymer Solutions 372

17.5 Lack of Universal Nonlinear Responses: Solutions versus Melts 374

17.6 Emergence of Transient Glassy Responses 378

References 380

18 Difficulties with Orthodox Paradigms 385

18.1 Tube Model Does Not Predict Key Experimental Features 385

18.1.1 Unexpected Failure at WiR≪1 387

18.1.2 Elastic Yielding Can Lead to Nonquiescent Relaxation 387

18.1.3 Meaning of Maximum in Tensile Force (Engineering Stress) 388

18.1.4 Other Examples of Causality Reversal 389

18.1.5 Entanglement–Disentanglement Transition 390

18.1.6 Anomalies Are the Norm 390

18.2 Confusion About Local and Global Deformations 391

18.2.1 Lack of Steady Flow in Startup Melt Extension 391

18.2.2 Peculiar Protocol to Observe Stress Relaxation from Step Extension 392

18.3 Molecular Network Paradigm 392

18.3.1 Startup Deformation 392

18.3.2 Stepwise Deformation 393

References 394

19 Strain Localization and Fluid Mechanics of Entangled Polymers 397

19.1 Relationship between Wall Slip and Banding: A Rheological-State Diagram 398

19.2 Modeling of Entangled Polymeric Liquids by Continuum Fluid Mechanics 399

19.3 Challenges in Polymer Processing 400

19.3.1 Extrudate Distortions 401

19.3.1.1 Sharkskin Melt Fracture (Due to Exit Boundary Discontinuity) 401

19.3.1.2 Gross (Melt Fracture) Extrudate Distortions Due to Entry Instability 403

19.3.1.3 Another Example Showing Pressure Oscillation and Stick–Slip Transition 403

19.3.2 Optimal Extrusion Conditions 404

19.3.3 Melt Strength 405

References 406

20 Conclusion 409

20.1 Theoretical Challenges 410

20.2 Experimental Difficulties 413

References 415

Symbols and Acronyms 417

Subject Index 421