Chapter
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Part I Linear Viscoelasticity and Experimental Methods
Chapter 1 Phenomenological Description of Linear Viscoelasticity
1.1 Basic Modes of Deformation
1.1.2 Step Strain and Shear Cessation from Steady State
1.1.3 Dynamic or Oscillatory Shear
1.2.1 Elastic Hookean Solids
1.2.2 Viscous Newtonian Liquids
1.2.3 Viscoelastic Responses
1.2.3.1 Boltzmann Superposition Principle for Linear Response
1.2.3.2 General Material Functions in Oscillatory Shear
1.2.3.3 Stress Relaxation from Step Strain or Steady‐State Shear
1.2.4 Maxwell Model for Viscoelastic Liquids
1.2.4.1 Stress Relaxation from Step Strain
1.2.4.2 Startup Deformation
1.2.4.3 Oscillatory (Dynamic) Shear
1.2.5 General Features of Viscoelastic Liquids
1.2.5.1 Generalized Maxwell Model
1.2.5.2 Lack of Linear Response in Small Step Strain: A Dilemma
1.2.6 Kelvin–Voigt Model for Viscoelastic Solids
1.2.6.2 Strain Recovery in Stress‐Free State
1.2.7 Weissenberg Number and Yielding during Linear Response
1.3 Classical Rubber Elasticity Theory
1.3.1 Chain Conformational Entropy and Elastic Force
1.3.2 Network Elasticity and Stress–Strain Relation
1.3.3 Alternative Expression in terms of Retraction Force and Areal Strand Density
Chapter 2 Molecular Characterization in Linear Viscoelastic Regime
2.1.1 Viscosity of Einstein Suspensions
2.1.2 Kirkwood–Riseman Model
2.1.4 Rouse Bead‐Spring Model
2.1.4.1 Stokes Law of Frictional Force of a Solid Sphere (Bead)
2.1.4.2 Brownian Motion and Stokes–Einstein Formula for Solid Particles
2.1.4.3 Equations of Motion and Rouse Relaxation Time 𝛕R
2.1.4.4 Rouse Dynamics for Unentangled Melts
2.1.5 Relationship between Diffusion and Relaxation Time
2.2.1 Phenomenological Evidence of chain Entanglement
2.2.1.1 Elastic Recovery Phenomenon
2.2.1.2 Rubbery Plateau in Creep Compliance
2.2.1.3 Stress Relaxation
2.2.1.4 Elastic Plateau in Storage Modulus G′
2.2.2 Transient Network Models
2.2.3 Models Depicting Onset of Chain Entanglement
2.2.3.2 Percolation Model
2.3 Molecular‐Level Descriptions of Entanglement Dynamics
2.3.1 Reptation Idea of de Gennes
2.3.2 Tube Model of Doi and Edwards
2.3.3 Polymer‐Mode‐Coupling Theory of Schweizer
2.3.4 Self‐diffusion Constant versus Zero‐shear Viscosity
2.3.5 Entangled Solutions
2.4 Temperature Dependence
2.4.1 Time–Temperature Equivalence
2.4.2 Thermo‐rheological Complexity
2.4.3 Segmental Friction and Terminal Relaxation Dynamics
Chapter 3 Experimental Methods
3.1.1 Shear by Linear Displacement
3.1.2 Shear in Rotational Device
3.1.2.1 Cone‐Plate Assembly
3.1.2.3 Circular Couette Apparatus
3.1.3 Pressure‐Driven Apparatus
3.2 Extensional Rheometry
3.2.1 Basic Definitions of Strain and Stress
3.2.2 Three Types of Devices
3.2.2.1 Instron Stretcher
3.2.2.2 Meissner‐Like Sentmanat Extensional Rheometer
3.2.2.3 Filament Stretching Rheometer
3.3 In Situ Rheostructural Methods
3.3.1.1 Stress Optical Rule
3.3.1.2 Breakdown of Stress‐Optical Rule
3.3.2 Scattering (X‐Ray, Light, Neutron)
3.3.3 Spectroscopy (NMR, Fluorescence, IR, Raman, Dielectric)
3.3.4 Microrheology and Microscopic Force Probes
3.4 Advanced Rheometric Methods
3.4.1 Superposition of Small‐Amplitude Oscillatory Shear and Small Step Strain during Steady Continuous Shear
3.4.2 Rate or Stress Switching Multistep Platform
Chapter 4 Characterization of Deformation Field Using Different Methods
4.1 Basic Features in Simple Shear
4.1.1 Working Principle for Strain‐Controlled Rheometry: Homogeneous Shear
4.1.2 Stress‐Controlled Shear
4.2 Yield Stress in Bingham‐Type (Yield‐Stress) Fluids
4.3 Cases of Homogeneous Shear
4.4 Particle‐Tracking Velocimetry (PTV)
4.4.1.1 Velocities in XZ‐Plane
4.4.1.2 Deformation Field in XY Plane
4.5 Single‐Molecule Imaging Velocimetry
4.6 Other Visualization Methods
Chapter 5 Improved and Other Rheometric Apparatuses
5.1 Linearly Displaced Cocylinder Sliding for Simple Shear
5.2 Cone‐Partitioned Plate (CPP) for Rotational Shear
5.3 Other Forms of Large Deformation
5.3.1 Deformation at Converging Die Entry
5.3.2 One‐Dimensional Squeezing
Part II Yielding – Primary Nonlinear Responses to Ongoing Deformation
Chapter 6 Wall Slip – Interfacial Chain Disentanglement
6.1 Basic Notions of Wall Slip in Steady Shear
6.1.1 Slip Velocity Vs and Navier–de Gennes Extrapolation Length b
6.1.2 Correction of Shear Field due to Wall Slip
6.1.3 Complete Slip and Maximum Value for b
6.2 Stick–Slip Transition in Controlled‐Stress Mode
6.2.1 Stick–Slip Transition in Capillary Extrusion
6.2.1.1 Analytical Description
6.2.1.2 Experimental Data
6.2.2 Stick–Slip Transition in Simple Shear
6.2.3 Limiting Slip Velocity Vs* for Different Polymer Melts
6.2.4 Characteristics of Interfacial Slip Layer
6.3 Wall Slip during Startup Shear – Interfacial Yielding
6.3.1 Theoretical Discussions
6.4 Relationship between Slip and Bulk Shear Deformation
6.4.1 Transition from Wall Slip to Bulk Nonlinear Response: Theoretical Analysis
6.4.2 Experimental Evidence of Stress Plateau Associated with Wall Slip
6.4.2.1 A Case Based on Entangled DNA Solutions
6.4.2.2 Entangled Polybutadiene Solutions in Small Gap Distance H ∼ 50 µm
6.4.2.3 Verification of Theoretical Relation by Experiment
6.4.3 Influence of Shear Thinning on Slip
6.4.4 Gap Dependence and Independence
6.5 Molecular Evidence of Disentanglement during Wall Slip
6.6 Uncertainties in Boundary Condition
6.6.1 Oscillations between Entanglement and Disentanglement Under Constant Speed
6.6.2 Oscillations between Stick and Slip under Constant Pressure
Chapter 7 Yielding during Startup Deformation: From Elastic Deformation to Flow
7.1 Yielding at Wi < 1 and Steady Shear Thinning at Wi > 1
7.1.1 Elastic Deformation and Yielding for Wi < 1
7.1.2 Steady Shear Rheology: Shear Thinning
7.2 Stress Overshoot in Fast Startup Shear
7.2.1 Scaling Characteristics of Shear Stress Overshoot
7.2.1.1 Viscoelastic Regime (WiR >1)
7.2.1.2 Elastic Deformation (Scaling) Regime (WiR <1)
7.2.1.3 Contrast between Two Different Regimes
7.2.2 Elastic Recoil from Startup Shear: Evidence of Yielding
7.2.2.1 Elastic Recoil for WiR <1
7.2.2.2 Irrecoverable Shear at WiR <1
7.2.3 More Evidence of Yielding at Overshoot Based on Rate‐Switching Tests
7.3 Nature of Steady Shear
7.3.1 Superposition of Small‐Amplitude Oscillatory Shear onto Steady‐State Shear
7.3.2 Two Other Methods to Probe Steady Shear
7.4 From Terminal Flow to Fast Flow under Creep: Entanglement–Disentanglement Transition
7.5 Yielding in Startup Uniaxial Extension
7.5.1 Myth with Considère Criterion
7.5.2 Tensile Force (Engineering Stress) versus True Stress
7.5.3 Tensile Force Maximum: A Signature of Yielding in Extension
7.5.3.1 Terminal Flow (Wi < 1)
7.5.3.2 Yielding Evidenced by Decline in 𝛔engr
7.5.3.3 Maxwell‐Like Response and Scaling for WiR >1
7.A.1 From Self‐Diffusion
7.A.2 From Zero‐Shear Viscosity
7.A.3 From Reptation (Terminal Relaxation) Time 𝛕d
7.A.4 From Second Crossover Frequency ∼1/𝛕e
Chapter 8 Strain Hardening in Extension
8.2 Origin of "Strain Hardening"
8.2.1 Simple Illustration of Geometric Condensation Effect
8.2.2 "Strain Hardening" of Polymer Melts with Long‐Chain Branching and Solutions
8.2.2.2 Entangled Solutions of Linear Chains
8.3 True Strain Hardening in Uniaxial Extension: Non‐Gaussian Stretching from Finite Extensibility
8.4 Different Responses of Entanglement to Startup Extension and Shear
Chapter 9 Shear Banding in Startup and Oscillatory Shear: Particle‐Tracking Velocimetry
9.1 Shear Banding After Overshoot in Startup Shear
9.1.1 Brief Historical Background
9.1.2.1 Sample Requirements: Well Entangled, with Long Reptation Time and Low Polydispersity
9.1.2.2 Controlling Slip Velocity
9.1.2.4 Absence of Shear Banding for b/H ≪ 1
9.1.2.5 Disappearance of Shear Banding at High Shear Rates
9.1.2.6 Avoiding Shear Banding with Rate Ramp‐Up
9.1.3 Shear Banding in Conventional Rheometric Devices
9.1.3.1 Shear Banding in Entangled DNA Solutions
9.1.3.2 Transient and Steady Shear Banding of Entangled 1,4‐Polybutadiene Solutions
9.1.4 From Wall Slip to Shear Banding in Small Gap Distance
9.2 Overcoming Wall Slip during Startup Shear
9.2.1 Strategy Based on Choice of Solvent Viscosity
9.2.2 Negligible Slip Correction at High Wiapp
9.2.3 Summary on Shear Banding
9.3 Nonlinearity and Shear Banding in Large‐Amplitude Oscillatory Shear
Chapter 10 Strain Localization in Extrusion, Squeezing Planar Extension: PTV Observations
10.1 Capillary Rheometry in Rate‐Controlled Mode
10.1.1 Steady‐State Characteristics
10.1.2 Transient Behavior
10.1.2.1 Pressure Oscillation and Hysteresis
10.1.2.2 Input vs. Throughput, Entry Pressure Loss and Yielding
10.2 Instabilities at Die Entry
10.2.1 Vortex Formation vs. Shear Banding
10.2.2 Stagnation at Corners and Internal Slip
10.3 Squeezing Deformation
Chapter 11 Strain Localization and Failure during Startup Uniaxial Extension
11.1 Tensile‐Like Failure (Decohesion) at Low Rates
11.2 Shear Yielding and Necking‐Like Strain Localization at High Rates
11.2.2 Constant Normalized Engineering Stress at the Onset of Strain Localization
11.3 Rupture‐Like Breakup: Where Are Yielding and Disentanglement?
11.4 Strain Localization Versus Steady Flow: Sentmanat Extensional Rheometry Versus Filament‐Stretching Rheometry
11.5 Role of Long‐Chain Branching
Part III Decohesion and Elastic Yielding After Large Deformation
Chapter 12 Nonquiescent Stress Relaxation and Elastic Yielding in Stepwise Shear
12.1 Strain Softening After Large Step Strain
12.1.2 Tube Model Interpretation
12.1.2.1 Normal Doi–Edwards Behavior
12.1.2.2 Type C Ultra‐strain‐softening
12.2 Particle Tracking Velocimetry Revelation of Localized Elastic Yielding
12.2.1 Nonquiescent Relaxation in Polymer Solutions
12.2.1.1 Elastic Yielding in Polybutadiene Solutions
12.2.1.2 Suppression of Breakup by Reduction in Extrapolation Length b
12.2.1.3 Nonquiescent Relaxation in Polystyrene Solutions
12.2.1.4 Strain Localization in the Absence of Edge Instability
12.2.2 Nonquiescent Relaxation in Styrene–Butadiene Rubbers
12.2.2.1 Induction Time and Molecular Weight Dependence
12.2.2.2 Severe Shear Banding before Shear Cessation and Immediate Breakup
12.2.2.3 Rate Dependence of Elastic Breakup
12.2.2.4 Unconventional "Step Strain" Produced at WiR <1
12.3 Quiescent and Uniform Elastic Yielding
12.3.2 Condition for Uniform Yielding and Quiescent Stress Relaxation
12.3.3 Homogeneous Elastic Yielding Probed by Sequential Shearing
12.4 Arrested Wall Slip: Elastic Yielding at Interfaces
12.4.1 Entangled Solutions
Chapter 13 Elastic Breakup in Stepwise Uniaxial Extension
13.1 Rupture‐Like Failure during Relaxation at Small Magnitude or Low Extension Rate (WiR <1)
13.1.1 Small Magnitude (𝛆∼1)
13.1.2 Low Rates Satisfying WiR <1
13.2 Shear‐Yielding‐Induced Failure upon Fast Large Step Extension (WiR >1)
13.3 Nature of Elastic Breakup Probed by Infrared Thermal‐Imaging Measurements
13.4 Primitive Phenomenological Explanations
13.5 Step Squeeze and Planar Extension
Chapter 14 Finite Cohesion and Role of Chain Architecture
14.1 Cohesive Strength of an Entanglement Network
14.2 Enhancing the Cohesion Barrier: Long‐Chain Branching Hinders Structural Breakup
Part IV Emerging Conceptual Framework and Beyond
Chapter 15 Homogeneous Entanglement
15.1 What Is Chain Entanglement?
15.2 When, How, and Why Disentanglement Occurs?
15.3 Criterion for Homogeneous Shear
15.4 Constitutive Nonmonotonicity
15.5 Metastable Nature of Shear Banding
Chapter 16 Molecular Networks as the Conceptual Foundation
16.1 Introduction: The Tube Model and its Predictions
16.1.1 Basic Starting Points of the Tube Model
16.1.2 Rouse Chain Retraction
16.1.3 Nonmonotonicity due to Rouse Chain Retraction
16.1.3.1 Absence of Linear Response to Step Strain
16.1.3.2 Stress Overshoot upon Startup Shear
16.1.3.3 Strain Softening: Damping Function for Stress Relaxation
16.1.3.4 Excessive Shear Thinning: The Symptom of Shear Stress Maximum
16.1.3.5 Anticipation of Necking Based on Considère Criterion
16.1.4 How to Test the Tube Model
16.2 Essential Ingredients for a New Molecular Model
16.2.1 Intrachain Elastic Retraction Force
16.2.2 Intermolecular Grip Force (IGF)
16.2.3 Entanglement (Cohesion) Force Arising from Entropic Barrier: Finite Cohesion
16.2.3.1 Scaling Analysis
16.2.3.2 Threshold for decohesion
16.3 Overcoming Finite Cohesion after Step Deformation: Quiescent or Not
16.3.1 Nonquiescence from Severe Elastic Yielding
16.3.2 Homogeneous Elastic Yielding: Quiescent Relaxation
16.4 Forced Microscopic Yielding during Startup Deformation: Stress Overshoot
16.4.1 Chain Disentanglement for WiR <1
16.4.2 Molecular Force Imbalance and Scaling for WiR >1
16.4.3 Yielding is a Universal Response: Maximum Engineering Stress
16.5 Interfacial Yielding via Disentanglement
16.6 Effect of Long‐Chain Branching
16.7 Decohesion in Startup Creep: Entanglement–Disentanglement Transition
16.8 Emerging Microscopic Theory of Sussman and Schweizer
16.9 Further Tests to Reveal the Nature of Responses to Large Deformation
16.9.1 Molecular Dynamics Simulations
16.9.2 Small Angle Neutron Scattering Measurements
16.9.2.1 Melt Extension at WiR≪1
16.9.2.2 Step Melt Extension With WiR >1
Chapter 17 "Anomalous" Phenomena
17.1 Essence of Rheometric Measurements: Isothermal Condition
17.1.1 Heat Transfer in Simple Shear
17.1.2 Heat Transfer in Uniaxial Extension
17.2 Internal Energy Buildup with and without Non‐Gaussian Extension
17.3 Breakdown of Time–Temperature Superposition (TTS) during Transient Response
17.3.1 Time–Temperature Superposition in Polystyrene Solutions and Styrene–Butadiene Rubbers: Linear Response
17.3.2 Failure of Time–Temperature Superposition: Solutions and Melts
17.3.2.1 Entangled Polymer Solutions Undergoing Startup Shear
17.3.2.2 Entangled Polymer Melts during Startup Extension
17.4 Strain Hardening in Simple Shear of Some Polymer Solutions
17.5 Lack of Universal Nonlinear Responses: Solutions versus Melts
17.6 Emergence of Transient Glassy Responses
Chapter 18 Difficulties with Orthodox Paradigms
18.1 Tube Model Does Not Predict Key Experimental Features
18.1.1 Unexpected Failure at WiR≪1
18.1.2 Elastic Yielding Can Lead to Nonquiescent Relaxation
18.1.3 Meaning of Maximum in Tensile Force (Engineering Stress)
18.1.4 Other Examples of Causality Reversal
18.1.5 Entanglement–Disentanglement Transition
18.1.6 Anomalies Are the Norm
18.2 Confusion About Local and Global Deformations
18.2.1 Lack of Steady Flow in Startup Melt Extension
18.2.2 Peculiar Protocol to Observe Stress Relaxation from Step Extension
18.3 Molecular Network Paradigm
18.3.1 Startup Deformation
18.3.2 Stepwise Deformation
Chapter 19 Strain Localization and Fluid Mechanics of Entangled Polymers
19.1 Relationship between Wall Slip and Banding: A Rheological‐State Diagram
19.2 Modeling of Entangled Polymeric Liquids by Continuum Fluid Mechanics
19.3 Challenges in Polymer Processing
19.3.1 Extrudate Distortions
19.3.1.1 Sharkskin Melt Fracture (Due to Exit Boundary Discontinuity)
19.3.1.2 Gross (Melt Fracture) Extrudate Distortions Due to Entry Instability
19.3.1.3 Another Example Showing Pressure Oscillation and Stick–Slip Transition
19.3.2 Optimal Extrusion Conditions
20.1 Theoretical Challenges
20.2 Experimental Difficulties
Nonlinear Polymer Rheology
：Macroscopic Phenomenology and Molecular Foundation
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