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

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List of Figures 1.1 Transient shear viscosity of a polymer melt at low shear rates compared to a perfectly viscous liquid and an ideal elastic solid. . . . . . . . . . . 3 1.2 a) The storage and loss modulus for a single Maxwell mode with relaxation time τ. b) Linear rheology of a real polymer melt fitted with a spectrum of Maxwell modes [Venerus (2000)]. . . . . . . . . . . . . . . . 8 1.3 Non-linear shear and uniaxial extension of an LDPE melt 1810H showing extension hardening (solid shapes) and shear thinning (open shapes) [Suneel et al. (Submitted)]. . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Sketch of an N-step freely jointed random walk . . . . . . . . . . . . . . 13 2.2 In a Gaussian random walk monomers which are well separated along the chain may come into close contact. . . . . . . . . . . . . . . . . . . . 14 2.3 The derivation of a molecular expression for stress . . . . . . . . . . . . 17 2.4 Scaling of linear viscosity of a range of linear polymer melts against X w which is proportional to molecular weight. The scaling switches from a slope of 1 to the 3.4 “law” for entangled polymer melts. From Berry and Fox (1968). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 The many body problem of an entangled melt (a) reduced to a single chain problem by replacing the individual entanglements with a confining tube (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 A chain in an entanglement network (narrow line) and its corresponding primitive path (broad line). . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Relaxation of oriented tube segments by reptation after a step strain . . 24 2.8 a) Dynamic modulus as calculated by the pure reptation model. b) Experimental storage modulus for a range of narrow distribution polystyrenes. Z ranges between ≈ 44 and ≈ 0.6 entanglements [Onogi et al. (1970)]. . 26 2.9 Theoretical predictions of relaxation after a step strain. [Reproduced from Doi and Edwards (1986)] . . . . . . . . . . . . . . . . . . . . . . . 28 iv
LIST OF FIGURES v 2.10 Comparison of predicted and measured damping functions after a large step strain for different linear polystyrene solutions [Osaki et al. (1982)]. The open and filled circles are data for different molecular weights and the tick directions indicate variations in concentration. . . . . . . . . . . 29 2.11 Schematic representation of a constraint release event. . . . . . . . . . . 31 2.12 Three relaxation mechanism available to an unbranched, entangled polymer chain: reptation (a), constraint release (b) and retraction (c). . . . 33 2.13 A three armed pom-pom molecule (q=3) . . . . . . . . . . . . . . . . . . 37 2.14 Sketch of a random walk rescaled from N steps of length b to Z steps of length a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.15 Schematic showing a Brownian particle, subject to a spring force and moving in an array of vanishing and re-appearing obstacles. . . . . . . . 45 3.1 Evolution of S xy for simple shear shear, ˙γ = 1sec −1 . Affine deformation corresponds to τ b = ∞ . The thin solid line corresponds to the value of τ b for which the steady state value of S xy is maximised. . . . . . . . . . 51 3.2 Evolution of S xy for nearly exponential shear with varying τ b values. α = 1sec −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Evolution of S xx − S yy for a planar extensional flow, ˙ɛ = 1sec −1 . . . . . 53 3.4 Evolution of S xx − S yy for an exponential shear flow, α = 1sec −1 . . . . . 54 3.5 Evolution of backbone stretch for a simple shear flow. τ b = 3sec, τ s = 1sec and q = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.6 Evolution of backbone stretch for an exponential shear flow, α = 1sec −1 , τ b /τ s = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.7 Evolution of backbone stretch for a planar extensional flow. τ b = 3sec, τ s = 1sec and q=4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.8 Evolution of shear stress in an exponential shear flow, α = 3sec −1 , τ b = 3sec, τ s = 1sec and G 0 φ 2 b = 1. . . . . . . . . . . . . . . . . . . . . . . . . 58 3.9 Pom-pom predictions compared to the experimental data for melt 1 of Zülle et al. (1987). Filled shapes are nearly exponential shear data points and open shapes are true exponential shear data points. Solid lines are nearly exponential shear predictions and dashed lines are true exponential shear predictions for both (a) and (b). . . . . . . . . . . . . . . . . . 59 3.10 Multimode pom-pom free parameter fit to planar extension data from Hachmann (1997). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.11 Comparison of multimode pom-pom predictions to experimental data for shear stress in true exponential shear from Venerus (2000). Solid curves are pom-pom predictions, shapes are data points and the dashed curve is the linear viscoelastic curve for simple shear. . . . . . . . . . . . . . . 62
Page 1: Molecular modelling of entangled po
Page 4 and 5: ii CONTENTS 2.3 Doi-Edwards model o
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 and 16: Chapter 1 Introduction 1.1 Overview
Page 17 and 18: 1.4. DEFORMATION KINEMATICS 3 10 5
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
Page 25 and 26: 1.7. ALTERNATIVE EXPERIMENTAL TECHN
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Page 29 and 30: 2.2. GAUSSIAN CHAINS 15 This leads
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Page 43 and 44: ¡ ¢¤£ ¥ 2.3. DOI-EDWARDS MODEL
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Page 51 and 52: 2.5. BRANCHED POLYMERS 37 From this
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Page 55 and 56: 2.5. BRANCHED POLYMERS 41 density p
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2.II. RESCALING A GAUSSIAN WALK 43
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2.III. OBSTRUCTED DIFFUSION 47 Comp
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3.1. INTRODUCTION 49 principle exte
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3.2. SINGLE MODE POM-POM MODEL 51
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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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