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 xvAcknowledgments xixIntroduction xxiAbout the Companion Website xxxiPart I Linear Viscoelasticity and Experimental Methods 11 Phenomenological Description of Linear Viscoelasticity 31.1 Basic Modes of Deformation 31.1.1 Startup shear 41.1.2 Step Strain and Shear Cessation from Steady State 51.1.3 Dynamic or Oscillatory Shear 51.2 Linear Responses 51.2.1 Elastic Hookean Solids 61.2.2 Viscous Newtonian Liquids 61.2.3 Viscoelastic Responses 71.2.3.1 Boltzmann Superposition Principle for Linear Response 71.2.3.2 General Material Functions in Oscillatory Shear 81.2.3.3 Stress Relaxation from Step Strain or Steady-State Shear 81.2.4 Maxwell Model for Viscoelastic Liquids 81.2.4.1 Stress Relaxation from Step Strain 91.2.4.2 Startup Deformation 101.2.4.3 Oscillatory (Dynamic) Shear 111.2.5 General Features of Viscoelastic Liquids 121.2.5.1 Generalized Maxwell Model 121.2.5.2 Lack of Linear Response in Small Step Strain: A Dilemma 131.2.6 Kelvin–Voigt Model for Viscoelastic Solids 141.2.6.1 Creep Experiment 151.2.6.2 Strain Recovery in Stress-Free State 151.2.7 Weissenberg Number and Yielding during Linear Response 161.3 Classical Rubber Elasticity Theory 171.3.1 Chain Conformational Entropy and Elastic Force 171.3.2 Network Elasticity and Stress–Strain Relation 181.3.3 Alternative Expression in terms of Retraction Force and Areal Strand Density 20References 212 Molecular Characterization in Linear Viscoelastic Regime 232.1 Dilute Limit 232.1.1 Viscosity of Einstein Suspensions 232.1.2 Kirkwood–Riseman Model 242.1.3 Zimm Model 242.1.4 Rouse Bead-Spring Model 252.1.4.1 Stokes Law of Frictional Force of a Solid Sphere (Bead) 262.1.4.2 Brownian Motion and Stokes–Einstein Formula for Solid Particles 262.1.4.3 Equations of Motion and Rouse Relaxation Time τR272.1.4.4 Rouse Dynamics for Unentangled Melts 282.1.5 Relationship between Diffusion and Relaxation Time 292.2 Entangled State 302.2.1 Phenomenological Evidence of chain Entanglement 302.2.1.1 Elastic Recovery Phenomenon 302.2.1.2 Rubbery Plateau in Creep Compliance 312.2.1.3 Stress Relaxation 322.2.1.4 Elastic Plateau in Storage Modulus G’ 322.2.2 Transient Network Models 342.2.3 Models Depicting Onset of Chain Entanglement 352.2.3.1 Packing Model 352.2.3.2 Percolation Model 382.3 Molecular-Level Descriptions of Entanglement Dynamics 392.3.1 Reptation Idea of de Gennes 392.3.2 Tube Model of Doi and Edwards 412.3.3 Polymer-Mode-Coupling Theory of Schweizer 432.3.4 Self-diffusion Constant versus Zero-shear Viscosity 442.3.5 Entangled Solutions 462.4 Temperature Dependence 472.4.1 Time–Temperature Equivalence 472.4.2 Thermo-rheological Complexity 482.4.3 Segmental Friction and Terminal Relaxation Dynamics 49References 503 Experimental Methods 553.1 Shear Rheometry 553.1.1 Shear by Linear Displacement 553.1.2 Shear in Rotational Device 563.1.2.1 Cone-Plate Assembly 563.1.2.2 Parallel Disks 573.1.2.3 Circular Couette Apparatus 583.1.3 Pressure-Driven Apparatus 593.1.3.1 Capillary Die 603.1.3.2 Channel Slit 613.2 Extensional Rheometry 633.2.1 Basic Definitions of Strain and Stress 633.2.2 Three Types of Devices 643.2.2.1 Instron Stretcher 643.2.2.2 Meissner-Like Sentmanat Extensional Rheometer 653.2.2.3 Filament Stretching Rheometer 653.3 In Situ Rheostructural Methods 663.3.1 Flow Birefringence 663.3.1.1 Stress Optical Rule 673.3.1.2 Breakdown of Stress-Optical Rule 683.3.2 Scattering (X-Ray, Light, Neutron) 693.3.3 Spectroscopy (NMR, Fluorescence, IR, Raman, Dielectric) 693.3.4 Microrheology and Microscopic Force Probes 693.4 Advanced Rheometric Methods 693.4.1 Superposition of Small-Amplitude Oscillatory Shear and Small Step Strain during Steady Continuous Shear 693.4.2 Rate or Stress Switching Multistep Platform 703.5 Conclusion 70References 714 Characterization of Deformation Field Using Different Methods 754.1 Basic Features in Simple Shear 754.1.1 Working Principle for Strain-Controlled Rheometry: Homogeneous Shear 754.1.2 Stress-Controlled Shear 764.2 Yield Stress in Bingham-Type (Yield-Stress) Fluids 774.3 Cases of Homogeneous Shear 794.4 Particle-Tracking Velocimetry (PTV) 794.4.1 Simple Shear 804.4.1.1 Velocities in XZ-Plane 804.4.1.2 Deformation Field in XY Plane 804.4.2 Channel Flow 824.4.3 Other Geometries 834.5 Single-Molecule Imaging Velocimetry 834.6 Other Visualization Methods 83References 845 Improved and Other Rheometric Apparatuses 875.1 Linearly Displaced Cocylinder Sliding for Simple Shear 885.2 Cone-Partitioned Plate (CPP) for Rotational Shear 885.3 Other Forms of Large Deformation 915.3.1 Deformation at Converging Die Entry 915.3.2 One-Dimensional Squeezing 925.3.3 Planar Extension 955.4 Conclusion 96References 97Part II Yielding – Primary Nonlinear Responses to Ongoing Deformation 996 Wall Slip – Interfacial Chain Disentanglement 1036.1 Basic Notions of Wall Slip in Steady Shear 1046.1.1 Slip Velocity Vs and Navier–de Gennes Extrapolation Length b 1046.1.2 Correction of Shear Field due to Wall Slip 1056.1.3 Complete Slip and Maximum Value for b 1066.2 Stick–Slip Transition in Controlled-Stress Mode 1086.2.1 Stick–Slip Transition in Capillary Extrusion 1086.2.1.1 Analytical Description 1086.2.1.2 Experimental Data 1096.2.2 Stick–Slip Transition in Simple Shear 1116.2.3 Limiting Slip Velocity V∗s for Different Polymer Melts 1136.2.4 Characteristics of Interfacial Slip Layer 1166.3 Wall Slip during Startup Shear – Interfacial Yielding 1166.3.1 Theoretical Discussions 1176.3.2 Experimental Data 1186.4 Relationship between Slip and Bulk Shear Deformation 1206.4.1 Transition from Wall Slip to Bulk Nonlinear Response: Theoretical Analysis 1206.4.2 Experimental Evidence of Stress Plateau Associated with Wall Slip 1226.4.2.1 A Case Based on Entangled DNA Solutions 1226.4.2.2 Entangled Polybutadiene Solutions in Small Gap Distance H∼50 μm 1236.4.2.3 Verification of Theoretical Relation by Experiment 1266.4.3 Influence of Shear Thinning on Slip 1276.4.4 Gap Dependence and Independence 1286.5 Molecular Evidence of Disentanglement during Wall Slip 1316.6 Uncertainties in Boundary Condition 1346.6.1 Oscillations between Entanglement and Disentanglement Under Constant Speed 1346.6.2 Oscillations between Stick and Slip under Constant Pressure 1346.7 Conclusion 134References 1357 Yielding during Startup Deformation: From Elastic Deformation to Flow 1397.1 Yielding at Wi<1 and Steady Shear Thinning at Wi>1 1407.1.1 Elastic Deformation and Yielding for Wi<1 1407.1.2 Steady Shear Rheology: Shear Thinning 1417.2 Stress Overshoot in Fast Startup Shear 1437.2.1 Scaling Characteristics of Shear Stress Overshoot 1447.2.1.1 Viscoelastic Regime (WiR <1) 1457.2.1.2 Elastic Deformation (Scaling) Regime (WiR >1) 1457.2.1.3 Contrast between Two Different Regimes 1487.2.2 Elastic Recoil from Startup Shear: Evidence of Yielding 1497.2.2.1 Elastic Recoil for WiR >1 1497.2.2.2 Irrecoverable Shear at WiR <1 1497.2.3 More Evidence of Yielding at Overshoot Based on Rate-Switching Tests 1537.3 Nature of Steady Shear 1547.3.1 Superposition of Small-Amplitude Oscillatory Shear onto Steady-State Shear 1557.3.2 Two Other Methods to Probe Steady Shear 1577.4 From Terminal Flow to Fast Flow under Creep: Entanglement–Disentanglement Transition 1597.5 Yielding in Startup Uniaxial Extension 1637.5.1 Myth with Considère Criterion 1637.5.2 Tensile Force (Engineering Stress) versus True Stress 1647.5.3 Tensile Force Maximum: A Signature of Yielding in Extension 1657.5.3.1 Terminal Flow (Wi<1) 1667.5.3.2 Yielding Evidenced by Decline in σengr 1677.5.3.3 Maxwell-Like Response and Scaling for WiR >1 1707.5.3.4 Elastic Recoil 1737.6 Conclusion 1757.A Experimental Estimates of Rouse Relaxation Time 1757.A.1 From Self-Diffusion 1757.A.2 From Zero-Shear Viscosity 1767.A.3 From Reptation (Terminal Relaxation) Time τd 1767.A.4 From Second Crossover Frequency∼1/τe 176References 1768 Strain Hardening in Extension 1818.1 Conceptual Pictures 1818.2 Origin of “Strain Hardening” 1848.2.1 Simple Illustration of Geometric Condensation Effect 1848.2.2 “Strain Hardening” of Polymer Melts with Long-Chain Branching and Solutions 1858.2.2.1 Melts with LCB 1858.2.2.2 Entangled Solutions of Linear Chains 1878.3 True Strain Hardening in Uniaxial Extension: Non-Gaussian Stretching from Finite Extensibility 1888.4 Different Responses of Entanglement to Startup Extension and Shear 1908.5 Conclusion 1908.A Conceptual and Mathematical Accounts of Geometric Condensation 191References 1929 Shear Banding in Startup and Oscillatory Shear: Particle-Tracking Velocimetry 1959.1 Shear Banding After Overshoot in Startup Shear 1979.1.1 Brief Historical Background 1979.1.2 Relevant Factors 1989.1.2.1 Sample Requirements: Well Entangled, with Long Reptation Time and Low Polydispersity 1989.1.2.2 Controlling Slip Velocity 1999.1.2.3 Edge Effects 1999.1.2.4 Absence of Shear Banding for b/H≪1 2019.1.2.5 Disappearance of Shear Banding at High Shear Rates 2029.1.2.6 Avoiding Shear Banding with Rate Ramp-Up 2029.1.3 Shear Banding in Conventional Rheometric Devices 2039.1.3.1 Shear Banding in Entangled DNA Solutions 2039.1.3.2 Transient and Steady Shear Banding of Entangled 1,4-Polybutadiene Solutions 2049.1.4 From Wall Slip to Shear Banding in Small Gap Distance 2089.2 Overcoming Wall Slip during Startup Shear 2099.2.1 Strategy Based on Choice of Solvent Viscosity 2099.2.2 Negligible Slip Correction at High Wiapp 2139.2.3 Summary on Shear Banding 2139.3 Nonlinearity and Shear Banding in Large-Amplitude Oscillatory Shear 2149.3.1 Strain Softening 2149.3.2 Wave Distortion 2159.3.3 Shear Banding 215References 21710 Strain Localization in Extrusion, Squeezing Planar Extension: PTV Observations 22110.1 Capillary Rheometry in Rate-Controlled Mode 22110.1.1 Steady-State Characteristics 22110.1.2 Transient Behavior 22310.1.2.1 Pressure Oscillation and Hysteresis 22310.1.2.2 Input vs. Throughput, Entry Pressure Loss and Yielding 22410.2 Instabilities at Die Entry 22610.2.1 Vortex Formation vs. Shear Banding 22610.2.2 Stagnation at Corners and Internal Slip 22710.3 Squeezing Deformation 23010.4 Planar Extension 233References 23311 Strain Localization and Failure during Startup Uniaxial Extension 23511.1 Tensile-Like Failure (Decohesion) at Low Rates 23711.2 Shear Yielding and Necking-Like Strain Localization at High Rates 23911.2.1 Shear Yielding 23911.2.2 Constant Normalized Engineering Stress at the Onset of Strain Localization 24311.3 Rupture-Like Breakup: Where Are Yielding and Disentanglement? 24511.4 Strain Localization Versus Steady Flow: Sentmanat Extensional Rheometry Versus Filament-Stretching Rheometry 24711.5 Role of Long-Chain Branching 25011.A Analogy between Capillary Rheometry and Filament-Stretching Rheometry 250References 251Part III Decohesion and Elastic Yielding After Large Deformation 25512 Nonquiescent Stress Relaxation and Elastic Yielding in Stepwise Shear 25712.1 Strain Softening After Large Step Strain 25812.1.1 Phenomenology 25812.1.2 Tube Model Interpretation 26112.1.2.1 Normal Doi–Edwards Behavior 26112.1.2.2 Type C Ultra-strain-softening 26212.2 Particle Tracking Velocimetry Revelation of Localized Elastic Yielding 26512.2.1 Nonquiescent Relaxation in Polymer Solutions 26612.2.1.1 Elastic Yielding in Polybutadiene Solutions 26612.2.1.2 Suppression of Breakup by Reduction in Extrapolation Length b 26912.2.1.3 Nonquiescent Relaxation in Polystyrene Solutions 26912.2.1.4 Strain Localization in the Absence of Edge Instability 27012.2.2 Nonquiescent Relaxation in Styrene–Butadiene Rubbers 27212.2.2.1 Induction Time and Molecular Weight Dependence 27312.2.2.2 Severe Shear Banding before Shear Cessation and Immediate Breakup 27512.2.2.3 Rate Dependence of Elastic Breakup 27512.2.2.4 Unconventional “Step Strain” Produced at WiR <1 27812.3 Quiescent and Uniform Elastic Yielding 27912.3.1 General Comments 27912.3.2 Condition for Uniform Yielding and Quiescent Stress Relaxation 28012.3.3 Homogeneous Elastic Yielding Probed by Sequential Shearing 28112.4 Arrested Wall Slip: Elastic Yielding at Interfaces 28312.4.1 Entangled Solutions 28312.4.2 Entangled Melts 28312.5 Conclusion 286References 28713 Elastic Breakup in Stepwise Uniaxial Extension 29113.1 Rupture-Like Failure during Relaxation at Small Magnitude or Low Extension Rate (WiR <1) 29213.1.1 Small Magnitude (ε ∼ 1) 29213.1.2 Low Rates Satisfying WiR <1 29213.2 Shear-Yielding-Induced Failure upon Fast Large Step Extension (WiR >1) 29313.3 Nature of Elastic Breakup Probed by Infrared Thermal-Imaging Measurements 29713.4 Primitive Phenomenological Explanations 29813.5 Step Squeeze and Planar Extension 299References 29914 Finite Cohesion and Role of Chain Architecture 30114.1 Cohesive Strength of an Entanglement Network 30214.2 Enhancing the Cohesion Barrier: Long-Chain Branching Hinders Structural Breakup 306References 308Part IV Emerging Conceptual Framework and Beyond 31115 Homogeneous Entanglement 31315.1 What Is Chain Entanglement? 31315.2 When, How, and Why Disentanglement Occurs? 31515.3 Criterion for Homogeneous Shear 31615.4 Constitutive Nonmonotonicity 31815.5 Metastable Nature of Shear Banding 319References 32216 Molecular Networks as the Conceptual Foundation 32516.1 Introduction: The Tube Model and its Predictions 32616.1.1 Basic Starting Points of the Tube Model 32716.1.2 Rouse Chain Retraction 32816.1.3 Nonmonotonicity due to Rouse Chain Retraction 32816.1.3.1 Absence of Linear Response to Step Strain 32816.1.3.2 Stress Overshoot upon Startup Shear 32916.1.3.3 Strain Softening: Damping Function for Stress Relaxation 33016.1.3.4 Excessive Shear Thinning: The Symptom of Shear Stress Maximum 33116.1.3.5 Anticipation of Necking Based on Considère Criterion 33216.1.4 How to Test the Tube Model 33216.2 Essential Ingredients for a New Molecular Model 33316.2.1 Intrachain Elastic Retraction Force 33416.2.2 Intermolecular Grip Force (IGF) 33516.2.3 Entanglement (Cohesion) Force Arising from Entropic Barrier: Finite Cohesion 33616.2.3.1 Scaling Analysis 33716.2.3.2 Threshold for decohesion 33816.3 Overcoming Finite Cohesion after Step Deformation: Quiescent or Not 33916.3.1 Nonquiescence from Severe Elastic Yielding 33916.3.1.1 With WiR >1 33916.3.1.2 With WiR≪1 34016.3.2 Homogeneous Elastic Yielding: Quiescent Relaxation 34016.4 Forced Microscopic Yielding during Startup Deformation: Stress Overshoot 34116.4.1 Chain Disentanglement for WiR <1 34116.4.2 Molecular Force Imbalance and Scaling for WiR >1 34216.4.3 Yielding is a Universal Response: Maximum Engineering Stress 34616.5 Interfacial Yielding via Disentanglement 34616.6 Effect of Long-Chain Branching 34716.7 Decohesion in Startup Creep: Entanglement–Disentanglement Transition 34916.8 Emerging Microscopic Theory of Sussman and Schweizer 35016.9 Further Tests to Reveal the Nature of Responses to Large Deformation 35116.9.1 Molecular Dynamics Simulations 35216.9.2 Small Angle Neutron Scattering Measurements 35316.9.2.1 Melt Extension at WiR≪1 35316.9.2.2 Step Melt Extension With WiR >1 35416.10 Conclusion 354References 35517 “Anomalous” Phenomena 36117.1 Essence of Rheometric Measurements: Isothermal Condition 36117.1.1 Heat Transfer in Simple Shear 36217.1.2 Heat Transfer in Uniaxial Extension 36417.2 Internal Energy Buildup with and without Non-Gaussian Extension 36617.3 Breakdown of Time–Temperature Superposition (TTS) during Transient Response 36817.3.1 Time–Temperature Superposition in Polystyrene Solutions and Styrene–Butadiene Rubbers: Linear Response 36817.3.2 Failure of Time–Temperature Superposition: Solutions and Melts 36917.3.2.1 Entangled Polymer Solutions Undergoing Startup Shear 36917.3.2.2 Entangled Polymer Melts during Startup Extension 37017.4 Strain Hardening in Simple Shear of Some Polymer Solutions 37217.5 Lack of Universal Nonlinear Responses: Solutions versus Melts 37417.6 Emergence of Transient Glassy Responses 378References 38018 Difficulties with Orthodox Paradigms 38518.1 Tube Model Does Not Predict Key Experimental Features 38518.1.1 Unexpected Failure at WiR≪1 38718.1.2 Elastic Yielding Can Lead to Nonquiescent Relaxation 38718.1.3 Meaning of Maximum in Tensile Force (Engineering Stress) 38818.1.4 Other Examples of Causality Reversal 38918.1.5 Entanglement–Disentanglement Transition 39018.1.6 Anomalies Are the Norm 39018.2 Confusion About Local and Global Deformations 39118.2.1 Lack of Steady Flow in Startup Melt Extension 39118.2.2 Peculiar Protocol to Observe Stress Relaxation from Step Extension 39218.3 Molecular Network Paradigm 39218.3.1 Startup Deformation 39218.3.2 Stepwise Deformation 393References 39419 Strain Localization and Fluid Mechanics of Entangled Polymers 39719.1 Relationship between Wall Slip and Banding: A Rheological-State Diagram 39819.2 Modeling of Entangled Polymeric Liquids by Continuum Fluid Mechanics 39919.3 Challenges in Polymer Processing 40019.3.1 Extrudate Distortions 40119.3.1.1 Sharkskin Melt Fracture (Due to Exit Boundary Discontinuity) 40119.3.1.2 Gross (Melt Fracture) Extrudate Distortions Due to Entry Instability 40319.3.1.3 Another Example Showing Pressure Oscillation and Stick–Slip Transition 40319.3.2 Optimal Extrusion Conditions 40419.3.3 Melt Strength 405References 40620 Conclusion 40920.1 Theoretical Challenges 41020.2 Experimental Difficulties 413References 415Symbols and Acronyms 417Subject Index 421