Constitutive Equations for Polymer Melts and Solutions

Constitutive Equations for Polymer Melts and Solutions

Butterworths Series in Chemical Engineering

1st Edition - April 7, 1988

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  • Author: Ronald G. Larson
  • eBook ISBN: 9781483162867

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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


    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


    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


    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


    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


    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


    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


    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


    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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