Rheology

1 Continuum Mechanics as a Foundation of Rheology

1.1 Stresses

1.1.1 General theory

1.1.2 Law of equality of conjugated stresses

1.1.3 Principal stresses

1.1.4 Invariants of a stress tensor

1.1.5 Hydrostatic pressure - spherical tensor and deviator

1.1.6 Equilibrium (balance) equations

1.2 Deformations

1.2.1 Deformations and displacements

1.2.1.1 Deformations

1.2.1.2 Displacements

1.2.2 Infinitesimal deformations: principal values and invariants

1.2.3 Large (finite) deformations

1.2.4 Special cases of deformations - uniaxial elongation and simple shear

1.2.4.1 Uniaxial elongation and Poisson's ratio

1.2.4.2 Simple shear and pure shear

1.3 Kinematics of deformations

1.3.1 Rates of deformation and vorticity

1.3.2 Deformation rates when deformations are large

1.4 Summary - continuum mechanics in rheology

1.4.1 General principles

1.4.2 Objects of continuum as tensors

References

Questions for Chapter 1

2 Viscoelasticity

2.1 Basic experiments

2.1.1 Creep (retarded deformation)

2.1.2 Relaxation

2.1.3 Fading memory

2.2 Relaxation and creep - spectral representation. Dynamic functions

2.2.1 Retardation and relaxation spectra - definitions

2.2.2 Dynamic functions

2.3 Model interpretations

2.3.1 Basic mechanical models

2.3.2 Complicated mechanical models - differential rheological equations

2.3.3 Non-mechanical models

2.4 Superposition - The Boltzmann-Volterra principle

2.4.1 Integral formulation of the superposition principle

2.4.2 Superposition principle expressed via spectra

2.4.3 Simple transient modes of deformation

2.4.3.1 Relaxation after sudden deformation

2.4.3.2 Developing stresses at constant shear rate

2.4.3.3 Relaxation after steady shear flow

2.4.3 Relationship between relaxation and creep functions

2.4.4 Relaxation function and large deformations

2.4.3.4 Relationship between relaxation and creep functions

2.4.3.5 Relaxation function and large deformations

2.5 Relationships among viscoelastic functions

2.5.1 Dynamic functions - relaxation, creep, and spectra

2.5.2 Constants and viscoelastic functions

2.5.3 Calculation of a relaxation spectrum

2.5.3.1 Introduction - general concept

2.5.3.2 Kernel approximation - finding a continuous spectrum

2.5.3.3 Computer-aided methods for a discrete spectrum

2.6 Viscoelasticity and molecular models

2.6.1 Molecular movements of an individual chain

2.6.1.1 A spring-and-bead model ("free draining chain")

2.6.1.2 Model of a non-draining coil

2.6.1.3 Model of a rotating coil

2.6.2 Relaxation properties of concentrated polymer solutions and melts

2.6.2.1 Concept of entanglements

2.6.2.2 Two-part distribution of friction coefficient

2.6.2.3 Non-equivalent friction along a chain

2.6.2.4 Viscoelastic entanglements

2.6.2.5 Rubber-like network

2.6.2.6 "Tube" (reptation) model

2.6.2.7 Some conclusions

2.6.3 Viscoelasticity of polydisperse polymers

2.7 Time-temperature superposition. Reduced ("master") viscoelastic curves

2.7.1 Superposition of experimental curves

2.7.2 Master curves and relaxation states

2.7.3 "Universal" relaxation spectra

2.8 Non-linear effects in viscoelasticity

2.8.1 Experimental evidences

2.8.1.1 Non-Newtonian viscosity

2.8.1.2 Non-Hookean behavior of solids

2.8.1.3 Non-linear creep

2.8.1.4 Non-linear relaxation

2.8.1.5 Non-linear periodic measurements

2.8.2 Linear - non-linear correlations

2.8.3 Rheological equations of state for non-linear viscoelastic behavior

2.8.3.1 The K-BKZ model

2.8.3.2 The Wagner models

2.8.3.2 The Leonov model

2.8.3.4 The Marrucci models

2.8.4 Comments - constructing non-linear constitutive equations and experiment

References

Questions for Chapter 2

3 Liquids

3.1 Newtonian and non-Newtonian liquids. Definitions

3.2 Non-Newtonian shear flow

3.2.1 Non-Newtonian behavior of viscoelastic polymeric materials

3.2.2 Non-Newtonian behavior of structured systems - plasticity of liquids

3.2.3 Viscosity of anisotropic liquids

3.3 Equations for viscosity and flow curves

3.3.1 Introduction - the meaning of viscosity measurement

3.3.2 Power-law equations

3.3.3 Equations with yield stress

3.3.4 Basic dependencies of viscosity

3.3.4.1 Viscosity of polymer melts

3.3.4.2 Viscosity of polymer solutions

3.3.4.3 Viscosity of suspensions

3.3.5 Effect of molecular weight distribution on non-Newtonian flow

3.4 Elasticity in shear flows

3.4.1 Rubbery shear deformations - elastic recoil

3.4.2 Normal stresses in shear flow

3.4.2.1 The Weissenberg effect

3.4.2.2 First normal stress difference - quantitative approach

3.4.2.3 Second normal stress difference and secondary flow

3.4.3 Normal stresses and elasticity

3.4.4 Die swell

3.5 Structure rearrangements induced by shear flow

3.5.1 Transient deformation regimes

3.5.2 Thixotropy and rheopexy

3.5.3 Shear induced phase transitions

3.6 Limits of shear flow - instabilities

3.6.1 Inertial turbulency

3.6.2 The Toms effect

3.6.3 Instabilities in flow of elastic liquids

3.6.3.1 Dynamic structure formation and secondary flows in elastic fluids

3.6.3.2 Secondary flows in the flow of elastic fluids

3.6.3.3 Shear banding

3.7 Extensional flow

3.7.1 Model experiments - uniaxial flow

3.7.2 Model experiments - rupture

3.7.3 Extension of industrial polymers

3.7.3.1 Multiaxial elongation

3.7.4 The tubeless siphon effect

3.7.5 Instabilities in extension

3.7.5.1 Phase transitions in extension

3.7.5.2 Rayleigh instability

3.7.5.3 Instabilities in extension of a viscoelastic thread

3.8 Conclusions - real liquid is a complex liquid

References

Questions for Chapter 3

4 Solids

4.1 Introduction and definitions

4.2 Linear elastic (Hookean) materials

4.3 Linear anisotropic solids

4.4 Large deformations in solids and non-linearity

4.4.1 A single-constant model

4.4.2 Multi-constant models

4.4.2.1 Two-constant potential function

4.4.2.2 Multi-member series

4.4.2.3 General presentation

4.4.2.4 Elastic potential of the power-law type

4.4.3 The Poynting effect

4.5 Limits of elasticity

4.5.1 Standard experiment - main definitions

4.5.2 Plasticity

4.5.3 Criteria of plasticity and failure

4.5.3.1 Maximum shear stress

4.5.3.2 The intensity of shear stresses ("energetic" criterion)

4.5.3.3 Maximum normal stress

4.5.3.4 Maximum deformation

4.5.3.5 Complex criteria

4.5.4 Structure effects

4.5.4.1 Strengthening

4.5.4.2 Thixotropy

References

Questions for Chapter 4

5 Rheometry. Experimental Methods

5.1 Introduction - Classification of experimental methods

5.2 Capillary viscometry

5.2.1 Basic theory

5.2.2 Corrections

5.2.2.1 Kinetic correction

5.2.2.2 Entrance correction

5.2.2.3 Pressure losses in a reservoir of viscometer

5.2.2.4 Temperature correction

5.2.2.5 Pressure correction

5.2.2.6 Correction for slip near a wall

5.2.2.7 Adsorption on a channel surface

5.2.3 Flow in incompletely filled capillary

5.2.3.1 Motion under action of gravitation forces

5.2.3.2 Motion caused by surface tension forces

5.2.4 Limits of capillary viscometry

5.2.5 Non-viscometric measurements using capillary viscometers

5.2.6 Capillary viscometers

5.2.6.1 Classification of the basic types of instruments

5.2.6.2 Viscometers with the assigned load

5.2.6.3 Cup viscometers

5.2.6.4 Glass viscometers

5.2.7 Viscometers with controlled flow rate

5.2.7.1 Instruments with a power drive

5.2.7.2 Instruments with hydraulic drive

5.2.7.3 Extrusion rheometers

5.2.7.4 Technological capillary tube viscometers

5.3 Rotational rheometry

5.3.1 Tasks and capabilities of the method

5.3.1.1 Viscometric and non-viscometric measurements

5.3.1.2 The method of a constant frequency of rotation

5.3.1.3 The method of a constant torque

5.3.2 Basic theory of rotational instruments

5.3.2.1 Instruments with coaxial cylinders

5.3.2.2 Instruments with conical surfaces

5.3.2.3 Bi-conical viscometers

5.3.2.4 Disk viscometers

5.3.2.5 Viscometers with spherical surfaces

5.3.2.6 End (bottom) corrections in instruments with coaxial cylinders

5.3.2.7 On a role of rigidity of dynamometer

5.3.2.8 Temperature effects

5.3.3 Limitations of rotational viscometry

5.3.4 Rotational instruments

5.3.4.1 Introduction - general considerations

5.3.4.2 Rheogoniometers and elastoviscometers

5.3.4.3 Viscometers with assigned rotational speed

5.3.4.4 Rotational viscometers for special purposes

5.3.4.5 Rotational instruments for technological purposes

5.3.5 Measuring normal stresses

5.3.5.1 Cone-and-plate technique

5.3.5.2 Plate-and-plate technique

5.3.5.3 Coaxial cylinders technique

5.3.5.4 Hole-pressure effect

5.4 Plastometers

5.4.1. Shear flow plastometers

5.4.2 Squeezing flow plastometers

5.4.3 Method of telescopic shear

5.4.3.1 Telescopic shear penetrometer

5.5 Method of falling sphere

5.5.1 Principles

5.5.1.1 Corrections

5.5.2 Method of rolling sphere

5.5.3 Viscometers with falling sphere

5.5.4 Viscometers with falling cylinder

5.6 Extension

5.6.1 General considerations

5.6.2. Experimental methods

5.6.2.1 The simplest measuring schemes

5.6.2.2 Tension in a controlled regime

5.6.2.3 Tubeless siphon instruments

5.6.2.4 Flow in convergent channels

5.6.2.5 High strain rate methods

5.6.3 Biaxial extension

5.7 Measurement of viscoelastic properties by dynamic (oscillation) methods

5.7.1 Principles of measurement - homogeneous deformation

5.7.2 Inhomogeneous deformations

5.7.3 Torsion oscillations

5.7.4 Measuring the impedance of a system

5.7.5 Resonance oscillations

5.7.6 Damping (free) oscillations

5.7.7 Wave propagation

5.7.7.1 Shear waves

5.7.7.2 Longitudinal waves

5.7.8 Vibration viscometry

5.7.8.1 Torsion oscillations

5.7.8.2 Oscillation of a disk in liquid

5.7.8.3 Oscillations of sphere

5.7.8.4 Damping oscillations

5.7.9 Measuring viscoelastic properties in non-symmetrical flows

5.7.9 About experimental techniques

5.7.9.1 Rotational instruments

5.7.9.2 Devices with electromagnetic excitation

5.7.9.3 Torsion pendulums

5.8 Physical methods

5.8.1 Rheo-optical methods

5.8.1.1 Basic remarks

5.8.1.2 Stress - optical rules for polymer melts

5.8.1.3 Stress-optical rule for polymer solutions

5.8.1.4 Viscometers for optical observations

5.8.1.5 Polarization methods for measuring stresses

5.8.1.6 Visualization of polymer flow in dies

5.8.2. Velocimetry

5.8.3 Viscometers-calorimeters

References

Questions for Chapter 5

6 Applications of Rheology

6.1 Introduction

6.2 Rheological properties of real materials and their characterization

6.2.1 Polymer materials

6.2.2 Mineral oils and oil-based products

6.2.3 Food products

6.2.4 Cosmetics and pharmaceuticals

6.2.5 Biological fluids

6.2.6 Concentrated suspensions

6.2.7 Electro- and magneto-rheological materials

6.2.8 Concluding remarks

6.3 Rheokinetics (chemorheology) and rheokinetic liquids

6.3.1. Formulation of problem

6.3.2. Linear polymerization

6.3.3 Oligomer curing

6.3.3.1 Viscosity change and a gel-point

6.3.3.2 Curing at high shear rates

6.3.3.3 Curing after gel-point

6.3.4 Intermolecular transformations

6.4 Solution of dynamic problems

6.4.1 General formulation

6.4.2 Flow through tubes

6.4.3 Flow in technological equipment

6.4.3.1 Pumping screw

6.4.3.2 Calendering and related processes

6.4.3.3 Extension-based technologies

6.4.3.4 Molding technologies

6.4.3.5 Compression molding

6.4.3.6 Injection molding

6.4.3.7 Injection-compression molding

References

Questions for Chapter 6