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

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ii CONTENTS 2.3 Doi-Edwards model of entangled polymers . . . . . . . . . . . . . . . . . 21 2.3.1 Linear rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.2 Contour length fluctuations . . . . . . . . . . . . . . . . . . . . . 26 2.3.3 Non-linear rheology . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Chain stretch and constraint release . . . . . . . . . . . . . . . . . . . . 30 2.4.1 The Milner McLeish and Likhtman model . . . . . . . . . . . . . 32 2.4.2 Comments on the MMcL model . . . . . . . . . . . . . . . . . . . 35 2.5 Branched polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5.1 Star Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5.2 H polymers and the pom-pom model . . . . . . . . . . . . . . . . 37 2.5.3 Randomly branched polymers . . . . . . . . . . . . . . . . . . . . 40 2.5.4 Discussion of multimode pom-pom model . . . . . . . . . . . . . 41 Appendix 2.I The Ito-Stratonovich relation . . . . . . . . . . . . . . . . . . 42 Appendix 2.II Rescaling a Gaussian walk . . . . . . . . . . . . . . . . . . . . 43 Appendix 2.III Obstructed diffusion . . . . . . . . . . . . . . . . . . . . . . . 44 3 The pom-pom model in exponential shear. 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Single mode pom-pom model . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.1 Solutions to the orientation equation . . . . . . . . . . . . . . . . 50 3.2.2 Solutions to the stretch equation . . . . . . . . . . . . . . . . . . 54 3.2.3 Behaviour of shear stress in exponential shear . . . . . . . . . . . 57 3.3 The multimode method applied to exponential shear . . . . . . . . . . . 57 3.3.1 Predicting exponential shear data using non-linear spectra from extensional rheology. . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.2 Measuring the non-linear parameters from exponential shear. . . 62 3.3.3 A verification of the method for a different melt. . . . . . . . . . 68 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 Theory of CCR and chain stretch 74 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Tube model for linear polymers with CCR and stretch . . . . . . . . . . 74 4.2.1 Rouse retraction term . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.2 Tube diameter under deformation . . . . . . . . . . . . . . . . . 75 4.2.3 CCR stretch relaxation . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.4 Suppression of reptation due to stretch . . . . . . . . . . . . . . . 78 4.2.5 Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.6 Equation for the tangent correlation function . . . . . . . . . . . 79 4.2.7 Number of entanglements . . . . . . . . . . . . . . . . . . . . . . 81
CONTENTS iii 4.2.8 Constraint release rate . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Contour length fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.1 Thermal constraint release from contour length fluctuations . . . 83 4.3.2 Rouse motion on sub-tube diameter length scales . . . . . . . . . 84 4.4 Summary of model equations . . . . . . . . . . . . . . . . . . . . . . . . 84 4.5 Real space solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5.1 Finite difference solution of CLF term . . . . . . . . . . . . . . . 87 4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6.1 Steady state in shear . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6.2 Transient start-up of simple shear . . . . . . . . . . . . . . . . . 88 4.6.3 Single chain structure factor . . . . . . . . . . . . . . . . . . . . . 90 4.7 Comparison with experimental data . . . . . . . . . . . . . . . . . . . . 92 4.7.1 Determination of model parameters from linear rheology . . . . . 92 4.7.2 Parameter free comparison with non-linear data . . . . . . . . . 93 4.7.3 Improvement of high rate predictions . . . . . . . . . . . . . . . . 93 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Appendix 4.I Derivation of stretch-CCR renormalisation term . . . . . . . . 100 Appendix 4.II Modified CLF term for a stretched chain . . . . . . . . . . . . 101 Appendix 4.III Fourier space solution . . . . . . . . . . . . . . . . . . . . . . 102 5 Bimodal blends of linear polymer melts 104 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Existing data and theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 Self-dilute high molecular weight additive . . . . . . . . . . . . . . . . . 106 5.3.1 Generalisation of stretching CCR theory to self dilute case . . . . 106 5.3.2 Uniaxial extension of bimodal blends . . . . . . . . . . . . . . . . 107 5.3.3 Non-linear shear of bimodal blends . . . . . . . . . . . . . . . . . 109 5.4 Discussion and future directions . . . . . . . . . . . . . . . . . . . . . . . 109 6 Conclusions 112 6.1 Randomly branched polymers . . . . . . . . . . . . . . . . . . . . . . . . 112 6.2 Monodisperse linear polymers . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Page 1: Molecular modelling of entangled po
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 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
Page 27 and 28: ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
Page 29 and 30: 2.2. GAUSSIAN CHAINS 15 This leads
Page 31 and 32: 2.2. GAUSSIAN CHAINS 17 L α β Fig
Page 33 and 34: 2.2. GAUSSIAN CHAINS 19 equilibrium
Page 35 and 36: 2.3. DOI-EDWARDS MODEL OF ENTANGLED
Page 43 and 44: ¡ ¢¤£ ¥ 2.3. DOI-EDWARDS MODEL
Page 45 and 46: 2.4. CHAIN STRETCH AND CONSTRAINT R
Page 51 and 52: 2.5. BRANCHED POLYMERS 37 From this
Page 53 and 54: 2.5. BRANCHED POLYMERS 39 By a forc
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2.5. BRANCHED POLYMERS 41 density p
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2.II. RESCALING A GAUSSIAN WALK 43
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¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
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2.III. OBSTRUCTED DIFFUSION 47 Comp
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3.1. INTRODUCTION 49 principle exte
Page 65 and 66:
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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