Molecular modelling of entangled polymer fluids under flow The ...

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2 CHAPTER 1. INTRODUCTION extensional properties of the material this effect can be avoided. Theoretical rheology is also increasingly being used to gain information about the molecular structure of a material from its rheological behaviour. The degree of branching in a polymer molecule is known to have strong influence on its rheology. For example highly branched polymer melts such as low density polyethylene (LDPE) exhibit strong strain hardening under extensional flows. Strain hardening is defined as an increase in viscosity with strain and is a desirable processing property. However, melts comprising of mostly linear molecules rarely show strain hardening and are often strain softening. Thus the arrangement branch points or topology of a molecule can dominate its rheological behaviour. 1.2 The stress tensor The goal of theoretical rheology is to relate the deformation history to macroscopic properties of the material. The mechanical stress exerted by the material in response to the deformation governs the flow of the material and so is one of the most frequently measured properties. Rheological constitutive equations make predictions of the stress tensor, σ. The Cartesian components, α, β, of the stress tensor are defined as the force ≈ per unit area in the α direction acting across a plane whose normal vector is in the β direction. In most material the stress tensor is symmetric. Asymmetry implies that microscopic points in the material experience a non-zero torque. 1.3 Viscoelasticity Traditionally, the distinction between a liquid and a solid is clear. The response of an ideal solid to deformation may be modelled by Hooke’s law. The stress response is proportional to the imposed strain and is independent of the strain rate. Thus the elastic energy supplied by the deformation is completely conserved. An ideal liquid has a Newtonian viscosity where the stress response is proportional to the imposed deformation rate and the total strain is irrelevant. In a Newtonian liquid the energy of deformation is completely dissipated. Polymer liquids, amongst others, show behaviour which is part-way between these two extremes. This is demonstrated in figure 1.1 which compares the transient shear rheology of a polymer melt at low shear rates with the two ideal limits, showing elastic behaviour at early times, giving way to viscous behaviour at longer times. Note that at low rates different deformation rates superimpose onto a single, rate independent curve; this is not the case for more rapid deformations.
1.4. DEFORMATION KINEMATICS 3 10 5 Newtonian Viscosity 10 4 η 0 (t) σ xy / γ . [Pa-sec] 10 3 Hooke’s Law 10 2 10 -2 10 -1 10 0 10 1 time [sec] Figure 1.1: Transient shear viscosity of a polymer melt at low shear rates compared to a perfectly viscous liquid and an ideal elastic solid. 1.4 Deformation kinematics Before attempting to derive a constitutive equation the deformation history imposed on the material must be defined. The response of a material depends on the geometry of the imposed deformation, therefore an adequate mathematical description of the deformation is essential. One way to achieve this is to specify the velocity field, v, imposed by the deformation on a material element at point r. However, only relative motion of material points are relevant, hence the velocity gradient tensor, κ ≈ , is usually a more relevant measure, κ (r, t) = (∇v(r, ≈ t))T . (1.1) If κ ≈ does not depends on spatial position r then the flow is deemed a simple flow. A range of useful flows such as shear and extension fall under this definition. Simple flows are also more easily analysed theoretically and so data on these flows is relatively abundant and reliable. In this thesis I will test constitutive models under simple flows against experimental data as a prerequisite to understanding more complicated deformations. constitutive equations. The tensor κ ≈ is widely used to define the deformation in differential An alternative description of the deformation is the deformation gradient tensor, E ≈ , which relates the vector connecting two embedded points before the deformation, X, and after the deformation, X ′ , X ′ = E.X. (1.2) ≈ The deformation gradient tensor is more commonly used in integral constitutive equations or to describe step deformations. Care must be exercised in the use of E since it contains information not only about the stretching caused by the deformation but also
Page 1: Molecular modelling of entangled po
Page 4 and 5: ii CONTENTS 2.3 Doi-Edwards model o
Page 6 and 7: List of Figures 1.1 Transient shear
Page 8 and 9: vi LIST OF FIGURES 3.12 Comparison
Page 10 and 11: List of Tables 1.1 The tensorial de
Page 12 and 13: Acknowledgements I am very grateful
Page 15: Chapter 1 Introduction 1.1 Overview
Page 19 and 20: 1.4. DEFORMATION KINEMATICS 5 flow
Page 21 and 22: 1.5. LINEAR RHEOLOGY 7 1.5.1 Linear
Page 23 and 24: 1.6. NON-LINEAR RHEOLOGY 9 1.5.2 Li
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Page 45 and 46: 2.4. CHAIN STRETCH AND CONSTRAINT R
Page 51 and 52: 2.5. BRANCHED POLYMERS 37 From this
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Page 63 and 64: 3.1. INTRODUCTION 49 principle exte
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3.2. SINGLE MODE POM-POM MODEL 53 A
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3.2. SINGLE MODE POM-POM MODEL 55 2
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