Structure and Rheology of Molten Polymers

Preface

Contents

1 Introduction

1.1 Melt Structure and its Effect on Rheology

1.2 Overview of this Book

1.3 Applications of the Information Presented

1.4 Supplementary Sources of Information

2 Structure of Polymers

2.1 Molecular Size

2.1.1 The Freely-Jointed Chain

2.1.2 The Gaussian Size Distribution

2.1.3 The Dilute Solution and the Theta State

2.1.4 Polymer Molecules in the Melt

2.2 Molecular Weight Distribution

2.2.1 Monodisperse Polymers

2.2.2 Average Molecular Weights – Moments of the Distribution

2.2.3 Continuous Molecular Weight Distribution

2.2.4 Distribution Functions

2.2.5 Narrow Distribution Samples

2.2.6 Bimodality

2.3 Tacticity

2.4 Branching

2.5 Intrinsic Viscosity

2.5.1 Introduction

2.5.2 Rigid Sphere Models

2.5.3 The Free-Draining Molecule

2.5.4 Non-Theta Conditions and the Mark-Houwink-Sakurada Equation

2.5.5 Effect of Polydispersity

2.5.6 Effect of Long-chain Branching

2.5.7 Effects of Short-Chain Branching

2.5.8 Determination of the Intrinsic Viscosity – Extrapolation Methods

2.5.9 Effect of Shear Rate

2.6 Other Structure Characterization Methods

2.6.1 Membrane Osmometry

2.6.2 Light Scattering

2.6.3 Gel Permeation Chromatography

2.6.4 Mass Spectrometry (MALDI-TOF)

2.6.5 Nuclear Magnetic Resonance

2.6.6 TREF and CRYSTAF

2.6.7 Molecular Structure from Rheology

2.7 Summary

3 Polymerization Reactions and Processes

3.1 Introduction

3.2 Classifications of Polymers and Polymerization Reactions

3.3 Structural Characteristics of Polymers

3.3.1 Introduction

3.3.2 Chemical Composition –Role of Backbone Bonds in Chain Flexibility

3.3.3 Chemical Composition – Copolymers

3.3.4 Tacticity

3.3.5 Branching

3.4 Living Polymers Having Prescribed Structures

3.4.1 Anionic Polymerization

3.4.2 Living Free Radical Polymerization

3.4.3 Analogs of Polyethylene for Research

3.5 Industrial Polymerization Processes

3.6 Free-Radical Polymerization of Low-Density Polyethylene (LDPE)

3.7 High-Density Polyethylene

3.7.1 Catalyst Systems

3.7.2 Branching in HDPE

3.7.3 Ultrahigh Molecular Weight Polyethylene

3.8 Linear Low-Density Polyethylene

3.9 Single-Site (Metallocene) Catalysts

3.9.1 Catalyst System

3.9.2 Long-Chain Branching in Metallocene Polyethylenes

3.10 Polypropylene

3.11 Reactors for Polyolefins

3.12 Polystyrene

3.13 Summary

4 Linear Viscoelasticity – Fundamentals

4.1 Stress Relaxation and the Relaxation Modulus

4.1.1 The Boltzmann Superposition Principle

4.1.2 The Maxwell Model for the Relaxation Modulus

4.1.3 The Generalized Maxwell Model and the Discrete Relaxation Spectrum

4.1.4 The Continuous Relaxation Spectrum

4.2 The Creep Compliance and the Retardation Spectrum

4.3 Experimental Characterization of Linear Viscoelastic Behavior

4.3.1 Oscillatory Shear

4.3.2 Experimental Determination of the Storage and Loss Moduli

4.3.3 Creep Measurements

4.3.4 Other Methods for Monitoring Relaxation Processes

4.4 Calculation of a Spectrum from Experimental Data

4.5 Moments of the Relaxation Spectrum as Indicators of Molecular Structure

4.6 Time-Temperature Superposition

4.7 Time-Pressure Superposition

4.8 Summary

5 Linear Viscoelasticity – Behavior of Molten Polymers

5.1 Introduction

5.2 The Zero-Shear Viscosity

5.2.1 Effect of Molecular Weight

5.2.2 Effect of Polydispersity

5.3 Relaxation Modulus

5.3.1 General Features

5.3.2 How a Melt Can Act Like a Rubber

5.4 The Storage and Loss Moduli

5.5 The Creep and Recoverable Compliances

5.6 The Steady-State Compliance

5.7 The Storage and Loss Compliances

5.8 Determination of the Plateau Modulus

5.9 The Molecular Weight Between Entanglements, Me

5.9.1 Definitions of Me

5.9.2 Effects of Molecular Structure on GN0 and Me

5.9.3 Molecular Weight Between Entanglements (Me) Based on Molecular

170

5.10 Rheological Behavior of Copolymers

5.11 Effect of Long-Chain Branching on Linear Viscoelastic

175

5.11.1 Introduction

5.11.2 Ideal Branched Polymers

5.11.3 Storage and Loss Moduli of Model Branched Systems

5.11.4 Randomly Branched Polymers

5.11.5 Low-Density Polyethylene

5.12 Use of Linear Viscoelastic Data to Determine Branching Level

5.12.1 Introduction

5.12.2 Correlations Based on the Zero-Shear Viscosity

5.13 The Cole-Cole Function and Cole-Cole Plots

5.13.1 The Complex Dielectric Constant and the Cole-Cole Function

5.13.2 Cole-Cole Plots for Characterizing Linear Viscoelastic Behavior

5.13.3 Van Gurp-Palmen Plot of Loss Angle Versus Complex Modulus

5.14 Summary

6 Tube Models for Linear Polymers – Fundamentals

6.1 Introduction

6.2 The Rouse-Bueche Model for Unentangled Polymers

6.2.1 Introduction

6.2.2 The Rouse Model for the Viscoelasticity of a Dilute Polymer Solution

6.2.3 Bueche’s Modification for an Unentangled Melt

6.3 Entanglements and the Tube Model

6.3.1 The Critical Molecular Weight for Entanglement MC

6.3.2 The Plateau Modulus GN0

6.3.3 The Molecular Weight Between Entanglements Me

6.3.4 The Tube Diameter a

6.3.5 The Equilibration Time te

6.4 Modes of Relaxation

6.4.1 Reptation

6.4.2 Primitive Path Fluctuations

6.4.3 Reptation Combined with Primitive Path Fluctuations

6.4.4 Constraint Release – Double Reptation

6.4.5 Rouse Relaxation Within the Tube

6.5 Summary

7 Tube Models for Linear Polymers – Advanced Topics

7.1 Introduction

7.2 Limitations of Double Reptation Theory

7.3 Constraint-Release Rouse Relaxation

7.3.1 Non-Self-Entangled Long Chains in a Short-Chain Matrix

7.3.2 Self-Entangled Long Chains in a Short-Chain Matrix

7.3.3 Polydisperse Chains

7.4 Tube Dilation or “Dynamic Dilution”

7.5 Input Parameters for Tube Models

8 Determination of Molecular Weight Distribution Using Rheology

8.1 Introduction

8.2 Viscosity Methods

8.3 Empirical Correlations Based on the Elastic Modulus

8.4 Methods Based on Double Reptation

8.5 Generalization of Double-Reptation

8.6 Dealing with the Rouse Modes

8.7 Models that Account for Additional Relaxation Processes

8.8 Prediction of Polydispersity Indexes

8.9 Summary

9 Tube Models for Branched Polymers

9.1 Introduction

9.2 General Effect of LCB on Rheology

9.3 Star Polymers

9.3.1 Deep Primitive Path Fluctuations

9.3.2 Dynamic Dilution

9.3.3 Comparison of Milner-McLeish Theory to Linear Viscoelastic Data

9.4 Multiply Branched Polymers

9.4.1 Branch-Point Motion

9.4.2 Backbone Relaxation

9.4.3 Dynamic Dilution for Polymers with Backbones

9.4.4 Predictions for Molecules with Moving Branch Points:

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9.5 Theories and Algorithms for Polydisperse Branched Polymers

9.5.1 Hierarchical Dynamic Dilution Model

9.5.2 Slip Link Simulations

9.6 Dilution and Combinatorial Rheology

9.7 Summary

10 Nonlinear Viscoelasticity

10.1 Introduction

10.2 Nonlinear Phenomena – A Tube Model Interpretation

10.2.1 Large Scale Orientation – The Need for a Finite Strain Tensor

10.2.2 Chain Retraction and the Damping Function

10.2.3 Convective Constraint Release and Shear Thinning

10.3 Constitutive Equations

10.3.1 Boltzmann Revisited

10.3.2 The Rubberlike Liquid

10.3.3 Wagner’s Equation

10.3.4 Other Integral Constitutive Equations

10.3.5 Differential Constitutive Equations

10.4 Nonlinear Stress Relaxation

10.4.1 Doi and Edwards Predictions of the Damping Function

10.4.2 Estimating the Rouse Time of an Entangled Chain

10.4.3 Damping Functions of Typical Polymers

10.4.4 Normal Stress Relaxation

10.4.5 Double-Step Strain

10.5 Dimensionless Groups Used to Plot Rheological Data

10.5.1 The Deborah Number

10.5.2 The Weissenberg Number

10.6 Transient Shear Tests at Finite Rates

10.6.1 Stress Growth and Relaxation in Steady Shear

10.6.2 Nonlinear Creep

10.6.3 Large-Amplitude Oscillatory Shear

10.6.4 Exponential Shear

10.7 The Viscometric Functions

10.7.1 Dependence of Viscosity on Shear Rate

10.7.2 Normal Stress Differences in Steady Simple Shear

10.8 Experimental Methods for Shear Measurements

10.8.1 Optical Methods

10.8.2 Generating Step Strain

10.8.3 Rotational Rheometers

10.8.4 Measurement of the Second Normal Stress Difference

10.8.5 Capillary and Slit Rheometers

10.8.6 The Cox-Merz Rule

10.8.7 Sliding Plate Rheometers

10.9 Extensional Flow Behavior – Introduction

10.10 Extensional Flow Behavior of Melts and Concentrated Solutions

10.10.1 Linear, Monodisperse Polymers

10.10.2 Effect of Polydispersity

10.10.3 Linear Low-Density Polyethylene

10.10.4 Model Branched Systems

10.10.5 Long-Chain Branched Metallocene Polyethylenes

10.10.6 Randomly Branched Polymers and LDPE

10.11 Experimental Methods for Extensional Flows

10.11.1 Introduction

10.11.2 Rheometers for Uniaxial Extension

10.11.3 Uniaxial Extension – Approximate Methods

10.11.4 Rheometers for Biaxial and Planar Extension

10.11.5 Extensional Rheometers – Summary

10.12 Shear Modification

10.13 Summary

11 Tube Models for Nonlinear Viscoelasticity of Linear and Branched Polymers

11.1 Introduction

11.2 Relaxation Processes Unique to the Nonlinear Regime

11.2.1 Retraction

11.2.2 Convective Constraint Release

11.3 Monodisperse Linear Polymers

11.3.1 No Chain Stretch: the Doi-Edwards Equation

11.3.2 Chain Stretch: the Doi-Edwards-Marrucci-Grizzuti (DEMG) Theory

11.3.3 Convective Constraint Release (CCR).

11.3.4 Differential Constitutive Equations Containing CCR

11.4 Polydisperse Linear Polymers

11.5 Comparison of Theory with Data for Linear Polymers

11.5.1 Shearing Flows

11.5.2 Extensional Flows

11.5.3 Processing Flows

11.5.4 Constitutive Instabilities and Slip

11.6 Polymers with Long-Chain Branching

11.6.1 The Pom-Pom Model

11.6.2 Revisions to the Pom-Pom Model

11.6.3 Empirical Multi-Mode Pom-Pom Equations for Commercial Melts

11.7 Summary

12 State of the Art and Challenges for the Future

12.1 State of the Art

12.2 Progress and Remaining Challenges

Appendix

A Structural and Rheological Parameters for Several Polymers

B Some Tensors Useful in Rheology

Nomenclature

Author Index

Subject Index