# Constitutive Equations for Polymer Melts and Solutions

### Butterworths Series in Chemical Engineering

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Constitutive Equations for Polymer Melts and Solutions presents a description of important constitutive equations for stress and birefringence in polymer melts, as well as in dilute and concentrated solutions of flexible and rigid polymers, and in liquid crystalline materials. The book serves as an introduction and guide to constitutive equations, and to molecular and phenomenological theories of polymer motion and flow. The chapters in the text discuss topics on the flow phenomena commonly associated with viscoelasticity; fundamental elementary models for understanding the rheology of melts, solutions of flexible polymers, and advanced constitutive equations; melts and concentrated solutions of flexible polymer; and the rheological properties of real liquid crystal polymers. Chemical engineers and physicists will find the text very useful.

## Table of Contents

Preface

Chapter 1. Introduction to Constitutive Equations for Viscoelastic Fluids

1.1 Introduction

1.2 Viscoelastic Flow Phenomena

Rod-Climbing

Extrudate Swell

Tubeless Siphon

Vortex Formation in Contraction Flows

Other Examples

1.3 Viscoelastic Measurements

Shear Thinning

Normal Stresses in Shear

Time-Dependent Viscosity

Stress Relaxation

Recoil

Sensitivity to Deformation Type

1.4 Deformation Gradient, Velocity Gradient, and Stress

The Deformation Gradient

The Velocity Gradient

The State-of-Stress Tensor

1.5 Relating Deformation and Stress

Viscoelastic Simple Fluids

The Newtonian Limit

The Elastic Limit

Frame Invariance

Examples of the Finger Tensor

Relationship Between the Finger tensor and the Velocity Gradient

1.6 A Simple Viscoelastic Constitutive Equation

Integral Version

Differential Version

Predictions

1.7 Summary

Chapter 2. Classical Molecular Models

2.1 Introduction

2.2 The Equilibrium State

Configuration Distribution Function

Polymer Chains as Hookean Springs

2.3 The Stress Tensor

Derivation from Spring Force

Derivation from Virtual Work

2.4 Rubber Elasticity Theory

2.5 The Temporary Network Model

Derivation of Constitutive Equation

Assumptions of the Green-Tobolsky Model

Successes and Limitations of the Green-Tobolsky Model

2.6 The Elastic Dumbbell Model

The Langevin Equation

The Smoluchowski Equation

The Constitutive Equation

2.7 The Rouse Model

The Langevin Equation

Normal Mode Transformation

The Stress Tensor and Constitutive Equation

Approximation for Slow Modes

Assumptions of the Rouse Model

2.8 Linear Viscoelasticity

Distribution of Relaxation Times

Time-Temperature Superposition

Nonlinear Superposition

2.9 Summary

Chapter 3. Continuum Theories

3.1 Introduction

3.2 The Constitutive Equation of Linear Viscoelasticity

Shear

Other Deformations

3.3 Frame Invariance

3.4 Oldroyd's Constitutive Equations

Convected Time Derivatives

Upper- and Lower-Convected Maxwell Equations

Oldroyd's Simple Equations

Corotational Maxwell Equation

3.5 The Kaye-BKZ Class of Equations

The Strain Energy Function

The History Integral

Shear

Time-Strain Separability

Lodge-Meissner Relationship

Other types of Deformation

3.6 Other Strain History Integrals

Wagner's First Equation

Superposition Integral Equation

Tanner-Simmons Equation

3.7 Summary

Chapter 4. Reptation Theories for Melts and Concentrated Solutions

4.1 Introduction

4.2 Simplifying Features of Melts

Chains in melts are ideal

No Hydrodynamic Interaction in Melts

Stress-Optic Law for Melts

4.3 Crossover to Entanglement Effects

Appearance of a Plateau Modulus

Meaning of the Plateau

4.4 The Doi-Edwards Constitutive Equation

Reptation

Nonlinear Modulus

The Probability Distribution Function

The Free Energy and the Stress Tensor

The Constitutive Equation

Premises of the Doi-Edwards Model

4.5 Approximations to the Doi-Edwards Equation

Currie's Potential

Larson's Potential

Approximation Based on the Seth Elastic Strain Measure

Differential Approximation

4.6 Predictions of Reptation Theories

Molecular-Weight Dependence

Relaxation Spectrum

Nonlinear Viscoelasticity

4.7 Curtiss-Bird Theory

4.8 Summary

Chapter 5. Constitutive Models with Nonaffine Motion

5.1 Introduction

5.2 Gordon-Schowalter Convected Derivative

The Stress Tensor

The Convected Derivative

5.3 Johnson-Segalman Model

Elastic Strain Measure

Predictions of the Johnson-Segalman Model

Forcing Corotation of Principal Stress and Strain Axes

Time-Strain Separability

5.4 Partially-Extending Convected Derivative

Shear Damping Function

Predictions in Steady Flows

Integral Equation

5.5 Irreversibility of Nonaffine Motion

Reversing Deformations

The Tube Picture

Differential Formulation of Irreversibility

5.6 White-Metzner Equation

Steady-State Flows

Sudden Deformations

5.7 Summary

Chapter 6. Nonseparable Constitutwe Models

6.1 Introduction

6.2 Giesekus and Leonov Models

Giesekus Model

Leonov Model

Predictions of the Leonov and Giesekus Models

6.3 Network Models

Yamamoto's Model

Phan-Thien/Tanner Model

Criticisms of the Phan-Thien/Tanner Model

Model of Acierno, La Mantis, Marrucci, and Titomanlio

Other Structural Models

General Network Model: Differential and Integral Form

The Equations of Bird and Carreau

6.4 Configuration Distribution Functions

Green-Tobolsky Network Model

Rouse-Zimm Dumbbell Model

Other Models

6.5 Summary

Chapter 7. Comparison of Constitutive Equations for Melts

7.1 Introduction

Considerations Affecting the Choice of Constitutive Equation

Approach Taken in this Chapter

7.2 The Relationship Between Integral and Differential Constitutive Equations

Network Integral Equations

Differential Analogs for Separable Kaye-BKZ Equations

Differential Analogs for Nonseparable Kaye-BKZ Equations

Comparison of Kaye-BKZ Equations with their Differential Analogs

Comparison of Separable and Nonseparable Differential Constitutive Equations

Alignment Strength versus Flow Strength

7.3 Comparing Constitutive Equations to Melt Data

Differential Constitutive Equations

Alignment Strength and the Damping Function

Constitutive Equations with a Dependence on Alignment Strength

7.4 Summary

Appendix

Chapter 8. Viscoelasticity of Dilute Polymer Solutions

8.1 Introduction

8.2 Linear Viscoelasticity

Rouse model

Hydrodynamic Interaction

High frequency Behavior

8.3 Non-Newtonian Viscosity

8.4 Expressions for the Stress Tensor

Kirkwood-Riseman Expression

Giesekus Expression

8.5 Dumbbells with Shear Thinning

Dumbbells with Hydrodynamic Interaction

Dumbbells with Excluded Volume

Dumbbells with Finite Extensibility

Dumbbells with Internal Viscosity

Summary of Dumbbell Models

8.6 Extensional Flow

The Effect of Finite Extensibility

Dumbbells with Variable Drag

8.7 Suspensions of Rigid Particles

Rigid Dumbbells

Rigid Ellipsoids

8.8 Summary

Chapter 9. Constitutive Equations for Special Flows

9.1 Introduction

9.2 Flows of Constant Stretch History

Viscometric Flows

Steady extensional Flows

General Flows of Constant Stretch History

Planar Flows of Constant Stretch History

9.3 Retarded Motion Expansion

Conditions for the Validity of the Retarded Motion Expansion

Numerical Calculations with the Retarded Motion Expansion

9.4 Foundations of Constitutive Theory

Oldroyd's General 'Elastico-Viscous Liquid'

Coleman and No11's Viscoelastic 'Simple Fluid'

Perturbation Expansions for Small Strain Amplitudes

9.5 Summary

Chapter 10. Theories for Nondilute Solutions of Rodlike Molecules

10.1 Introduction

10.2 Semidilute Regime

Diffusion Coefficient

Doi-Edwards Constitutive Equation for Semidilute Solutions of Rods

10.3 The Isotropic to Nematic Transition

Onsager theory

Flory Lattice Theory

Maier-Saupe Theory

10.4 Doi Constitutive Equation for Nematic Polymers

Dynamic Equation for the Order Parameter

The Stress Tensor

Shear Viscosity Predictions

10.5 Statics of Liquid Crystals

Theory of Spatially Varying Orientation in Nematics

Magnetic Fields

Textures of Nematics

10.6 Viscous Flow of Nematics

Ericksen's Transversely Isotropic Fluid

Leslie-Ericksen Theory

Boundary Effects

10.7 Rheology of Liquid Crystal Polymers

Temperature-Dependent Rheology

Birefringence

Relationship Between Linear and Nonlinear Rheology

First Normal Stress Difference

Complex Time-Dependent Stresses

Domain Theories

10.8 Summary

Author Index

Subject Index

## Product details

- No. of pages: 380
- Language: English
- Copyright: © Butterworth-Heinemann 1988
- Published: April 7, 1988
- Imprint: Butterworth-Heinemann
- eBook ISBN: 9781483162867

## About the Author

### Ronald G. Larson

## About the Editor

### Howard Brenner

#### Affiliations and Expertise

Massachusetts Institute of Technology

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