Nonlinear polymer rheology : macroscopic phenomenology and molecular foundation

Content: Preface xv Acknowledgments xix Introduction xxi About the CompanionWebsite xxxi Part I Linear Viscoelasticity and ExperimentalMethods 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 ElasticityTheory 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 BrownianMotion and Stokes   Einstein Formula for Solid Particles 26 2.1.4.3 Equations of Motion and Rouse Relaxation Time   R 27 2.1.4.4 Rouse Dynamics for UnentangledMelts 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 DifferentMethods 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 ofWall 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 toWall 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 fromWall Slip to Bulk Nonlinear Response:Theoretical Analysis 120 6.4.2 Experimental Evidence of Stress Plateau Associated withWall 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   m 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 duringWall 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 ShearThinning at Wi>
1 140 7.1.1 Elastic Deformation and Yielding for Wi<
1 140 7.1.2 Steady Shear Rheology: ShearThinning 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 Considere 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   engr 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   d 176 7.A.4 From Second Crossover Frequency   1/  e 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  a1 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 FromWall Slip to Shear Banding in Small Gap Distance 208 9.2 OvercomingWall 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 MolecularWeight 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 ArrestedWall 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 (       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 InfraredThermal-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, andWhy 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 ShearThinning:The Symptom of Shear Stress Maximum 331 16.1.3.5 Anticipation of Necking Based on Considere 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  a1 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  a1 353 16.9.2.2 Step Melt ExtensionWith 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  a1 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