Melt Rheology and Its Role in Plastics Processing: Theory and Applications

In: Springer eBook Collection

Abstract

1. Introduction to Rheology -- 1.1 What is Rheology? -- 1.2 Why Rheological Properties are Important -- 1.3 Stress as a Measure of Force -- 1.4 Strain as a Measure of Deformation -- 1.4.1 Strain Measures for Simple Extension -- 1.4.2 Shear Strain -- 1.5 Rheological Phenomena -- 1.5.1 Elasticity; Hooke's Law -- 1.5.2 Viscosity -- 1.5.3 Viscoelasticity -- 1.5.4 Structural Time Dependency -- 1.5.5 Plasticity and Yield Stress -- 1.6 Why Polymeric Liquids are Non-Newtonian -- 1.6.1 Polymer Solutions -- 1.6.2 Molten Plastics -- 1.7 A Word About Tensors -- 1.7.1 Vectors -- 1.7.2 What is a Tensor? -- 1.8 The Stress Tensor -- 1.9 A Strain Tensor for Infinitesimal Deformations -- 1.10 The Newtonian Fluid -- 1.11 The Basic Equations of Fluid Mechanics -- 1.11.1 The Continuity Equation -- 1.11.2 Cauchy's Equation -- 1.11.3 The Navier-Stokes Equation 40 References -- 2. Linear Viscoelasticity -- 2.1 Introduction -- 2.2 The Relaxation Modulus -- 2.3 The Boltzmann Superposition Principle -- 2.4 Relaxation Modulus of Molten Polymers -- 2.5 Empirical Equations for the Relaxation Modulus -- 2.5.1 The Generalized Maxwell Model -- 2.5.2 Power Laws and an Exponential Function -- 2.6 The Relaxation Spectrum -- 2.7 Creep and Creep Recovery; The Compliance -- 2.8 Small Amplitude Oscillatory Shear -- 2.8.1 The Complex Modulus and the Complex Viscosity -- 2.8.2 Complex Modulus of Typical Molten Polymers -- 2.8.3 Quantitative Relationships between G*(?) and MWD -- 2.8.4 The Storage and Loss Compliances -- 2.9 Determination of Maxwell Model Parameters -- 2.10 Start-Up and Cessation of Steady Simple Shear and Extension -- 2.11 Molecular Theories: Prediction of Linear Behavior -- 2.11.1 The Modified Rouse Model for Unentangled Melts -- 2.11.1.1 The Rouse Model for Dilute Solutions -- 2.11.1.2 The Bueche Modification of the Rouse Theory -- 2.11.1.3 The Bueche-Ferry Law -- 2.11.2 Molecular Theories for Entangled Melts -- 2.11.2.1 Evidence for the Existence of Entanglements -- 2.11.2.2 The Nature of Entanglement Coupling -- 2.11.2.3 Reptation -- 2.11.2.4 The Doi-Edwards Theory -- 2.11.2.5 The Curtiss-Bird Model -- 2.11.2.6 Limitations of Reptation Models -- 2.12 Time-Temperature Superposition -- 2.13 Linear Behavior of Several Polymers 94 References -- 3. Introduction to Nonlinear Viscoelasticity -- 3.1 Introduction -- 3.2 Nonlinear Phenomena -- 3.3 Theories of Nonlinear Behavior -- 3.4 Finite Measures of Strain -- 3.4.1 The Cauchy Tensor and the Finger Tensor -- 3.4.2 Strain Tensors -- 3.4.3 Reference Configurations -- 3.4.4 Scalar Invariants of the Finger Tensor -- 3.5 The Rubberlike Liquid -- 3.5.1 A Theory of Finite Linear Viscoelasticity -- 3.5.2 Lodge's Network Theory and the Convected Maxwell Model -- 3.5.3 Behavior of the Rubberlike Liquid in Simple Shear Flows -- 3.5.3.1 Rubberlike Liquid in Step Shear Strain -- 3.5.3.2 Rubberlike Liquid in Steady Simple Shear -- 3.5.3.3 Rubberlike Liquid in Oscillatory Shear -- 3.5.3.4 Constrained Recoil of Rubberlike Liquid -- 3.5.3.5 The Stress Ratio (N1/?) and the Recoverable Shear -- 3.5.4 The Rubberlike Liquid in Simple Extension -- 3.5.5 Comments on the Rubberlike Liquid Model -- 3.6 The BKZ Equation -- 3.7 Wagner's Equation and the Damping Function -- 3.7.1 Strain Dependent Memory Function -- 3.7.2 Determination of the Damping Function -- 3.7.3 Separable Stress Relaxation Behavior -- 3.7.4 Damping Function Equations for Polymeric Liquids -- 3.7.4.1 Damping Function for Shear Flows -- 3.7.4.2 Damping Function for Simple Extension -- 3.7.4.3 Universal Damping Functions -- 3.7.5 Interpretation of the Damping Function in Terms of Entanglements -- 3.7.5.1 The Irreversibility Assumption -- 3.7.6 Comments on the Use of the Damping Function -- 3.8 Molecular Models for Nonlinear Viscoelasticity -- 3.8.1 The Doi-Edwards Constitutive Equation -- 3.9 Strong Flows; The Tendency to Stretch and Align Molecules -- References -- 4. Steady Simple Shear Flow and the Viscometric Functions -- 4.1 Introduction -- 4.2 Steady Simple Shear Flow -- 4.3 Viscometric Flow -- 4.4 Wall Slip and Edge Effects -- 4.5 The Viscosity of Molten Polymers -- 4.5.1 Dependence of Viscosity on Shear Rate -- 4.5.2 Dependence of Viscosity on Temperature -- 4.6 The First Normal Stress Difference -- 4.7 Empirical Relationships Involving Viscometric Functions -- 4.7.1 The Cox-Merz Rules -- 4.7.2 The Gleissle Mirror Relations -- 4.7.3 Other Relationships 176 References -- 5. Transient Shear Flows Used to Study Nonlinear Viscoelasticity -- 5.1 Introduction -- 5.2 Step Shear Strain -- 5.2.1 Finite Rise Time -- 5.2.2 The Nonlinear Shear Stress Relaxation Modulus -- 5.2.3 Time-Temperature Superposition -- 5.2.4 Strain-Dependent Spectrum and Maxwell Parameters -- 5.2.5 Normal Stress Differences for Single-Step Shear Strain -- 5.2.6 Multistep Strain Tests -- 5.3 Flows Involving Steady Simple Shear -- 5.3.1 Start-Up Flow -- 5.3.2 Cessation of Steady Simple Shear -- 5.3.3 Interrupted Shear -- 5.3.4 Reduction in Shear Rate -- 5.4 Nonlinear Creep -- 5.4.1 Time-Temperature Superposition of Creep Data -- 5.5 Recoil and Recoverable Shear -- 5.5.1 Creep Recovery -- 5.5.1.1 Time-Temperature Superposition; Creep Recovery -- 5.5.2 Recoil During Start-Up Flow -- 5.5.3 Recoverable Shear Following Steady Simple Shear -- 5.6 Superposed Deformations -- 5.6.1 Superposed Steady and Oscillatory Shear -- 5.6.2 Step Strain with Superposed Deformations -- 5.7 Large Amplitude Oscillatory Shear -- 5.8 Exponential Shear; A Strong Flow -- 5.9 Usefulness of Transient Shear Tests -- References -- 6. Extensional Flow Properties and Their Measurement -- 6.1 Introduction -- 6.2 Extensional Flows -- 6.3 Simple Extension -- 6.3.1 Material Functions for Simple Extension -- 6.3.2 Experimental Methods -- 6.3.3 Experimental Observations for LDPE -- 6.3.4 Experimental Observations for Linear Polymers -- 6.4 Biaxial Extension -- 6.5 Planar Extension -- 6.6 Other Extensional Flows -- References -- 7. Rotational and Sliding Surface Rheometers -- 7.1 Introduction -- 7.2 Sources of Error for Drag Flow Rheometers -- 7.2.1 Instrument Compliance -- 7.2.2 Viscous Heating -- 7.2.3 End and Edge Effects -- 7.2.4 Shear Wave Propagation -- 7.3 Cone-Plate Flow Rheometers -- 7.3.1 Basic Equations for Cone-Plate Rheometers -- 7.3.2 Sources of Error for Cone-Plate Rheometers -- 7.3.3 Measurement of the First Normal Stress Difference -- 7.4 Parallel Disk Rheometers -- 7.5 Eccentric Rotating Disks -- 7.6 Concentric Cylinder Rheometers -- 7.7 Controlled Stress Rotational Rheometers -- 7.8 Torque Rheometers -- 7.9 Sliding Plate Rheometers -- 7.9.1 Basic Equations for Sliding Plate Rheometers -- 7.9.2 End and Edge Effects for Sliding Plate Rheometers -- 7.9.3 Sliding Plate Melt Rheometers -- 7.9.4 The Shear Stress Transducer -- 7.10 Sliding Cylinder Rheometers -- References -- 8. Flow in Capillaries, Slits and Dies -- 8.1 Introduction -- 8.2 Flow in a Round Tube -- 8.2.1 Shear Stress Distribution -- 8.2.2 Shear Rate for a Newtonian Fluid -- 8.2.3 Shear Rate for a Power Law Fluid -- 8.2.4 The Rabinowitch Correction -- 8.2.5 The Schummer Approximation -- 8.2.6 Wall Slip in Capillary Flow -- 8.3 Flow in a Slit -- 8.3.1 Basic Equations for Shear Stress and Shear Rate -- 8.3.2 Use of a Slit Rheometer to Determine N1 -- 8.3.2.1 Determination of N1 from the Hole Pressure -- 8.3.2.2 Determination of N1 from the Exit Pressure -- 8.4 Pressure Drop in Irregular Cross Sections -- 8.5 Entrance Effects -- 8.5.1 Experimental Observations -- 8.5.2 Entrance Pressure Drop—the Bagley End Correction -- 8.5.3 Rheological Significance of the Entrance Pressure Drop -- 8.6 Capillary Rheometers -- 8.7 Flow in Converging Channels -- 8.7.1 The Lubrication Approximation -- 8.7.2 Industrial Die Design -- 8.8 Extrudate Swell -- 8.9 Extrudate Distortion -- 8.9.1 Surface Melt Fracture—Sharkskin -- 8.9.2 Oscillatory Flow in Linear Polymers -- 8.9.3 Gross Melt Fracture -- 8.9.4 Role of Slip in Melt Fracture -- 8.9.5 Gross Melt Fracture Without Oscillations -- References -- 9. Rheo-Optics and Molecular Orientation -- 9.1 Basic Concepts—Interaction of Light and Matter -- 9.1.1 Refractive Index and Polarization -- 9.1.2 Absorption and Scattering -- 9.1.3 Anisotropic Media; Birefringence and Dichroism -- 9.2 Measurement of Birefringence -- 9.3 Birefringence and Stress -- 9.3.1 Stress-Optical Relation -- 9.3.2 Application of Birefringence Measurements -- References -- 10. Effects of Molecular Structure -- 10.1 Introduction and Qualitative Overview of Molecular Theory -- 10.2 Molecular Weight Dependence of Zero Shear Viscosity -- 10.3 Compliance and First Normal Stress Difference -- 10.4 Shear Rate Dependence of Viscosity -- 10.5 Temperature and Pressure Dependence -- 10.5.1 Temperature Dependence of Viscosity -- 10.5.2 Pressure Dependence of Viscosity -- 10.6 Effects of Long Chain Branching -- References -- 11. Rheology of Multiphase Systems -- 11.1 Introduction -- 11.2 Effect of Rigid Fillers -- 11.2.1 Viscosity -- 11.2.2 Elasticity -- 11.3 Deformable Multiphase Systems (Blends, Block Polymers) -- 11.3.1 Deformation of Disperse Phases and Relation to Morphology -- 11.3.2 Rheology of Immiscible Polymer Blends -- 11.3.3 Phase-Separated Block and Graft Copolymers -- References -- 12. Chemorheology of Reacting Systems -- 12.1 Introduction -- 12.2 Nature of the Curing Reaction -- 12.3 Experimental Methods for Monitoring Curing Reactions -- 12.3.1 Dielectric Analysis -- 12.4 Viscosity of the Pre-gel Liquid -- 12.5 The Gel Point and Beyond -- References -- 13. Rheology of Thermotropic Liquid Crystal Polymers -- 13.1 Introduction -- 13.2 Rheology of Low Molecular Weight Liquid Crystals -- 13.3 Rheology of Aromatic Thermotropic Polyesters -- 13.4 Relation of Rheology to Processing of Liquid Crystal Polymers -- References -- 14. Role of Rheology In Extrusion -- 14.1 Introduction -- 14.1.1 Functions of Extruders -- 14.1.2 Types of Extruders -- 14.1.3 Screw Extruder Zones -- 14.2 Analysis of Single Screw Extruder Operation -- 14.2.