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

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40 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY I now present a brief outline of the drag-strain coupling modification to the pompom equations suggested by Blackwell et al. (2000). The purpose of this modification is to incorporate the physical effect of the branch points withdrawing in to the backbone tube before the maximum stretch condition is reached. The effect of this modification is to smooth out the discontinuity in the gradient of the stress which is predicted by the original model when the maximum stretch is reached. This discontinuity is not seen in experimental data for mono-disperse H-polymers [McLeish et al. (1999)]. The effect of withdrawal of the branch-point into the backbone tube before the onset of maximum stretch is modelled by coupling it to the tension in the backbone through the backbone stretch, λ. This withdrawal shortens the effective length of the free arms since some of the arm material is now inside the tube. The average time taken for a free arm to retract up to the branch point is exponentially dependent on arm length, which in turn determines how frequently the branch point can take a diffusive step. Hence even a small degree of branch point withdrawal has a strong influence on the stretch relaxation time. By allowing the branch point to move in a harmonic potential the variation of stretch relaxation can be related to the value of the backbone stretch. This results in a modification of the stretch relaxation time as shown in table 2.2. The stretch relaxation time, which was constant in the original model, has a dependence on the backbone stretch. τ s → τ s e −ν∗ (λ−1) . (2.42) Where ν ∗ is a constant which is inversely proportional to the co-efficient of the harmonic potential in which the branch point moves. Hence as the stretch grows, the branch points become increasingly withdrawn producing exponentially faster stretch relaxation. Thus as the maximum stretch is approached the molecule is able to relax its stretch faster and so the growth rate of λ is reduced. The constant ν ∗ is expected to be inversely proportional to q since increased branching will increase the localisation of the branch points. For the multi-mode approach (see section 2.5.3) Blackwell et al. (2000) found that a material independent value of ν ∗ = 2/q was consistent with existing data. The pom-pom model has been tested against experimental data on model H polymers [McLeish et al. (1999)]. This study included both linear and non-linear rheology as well as SANS measurements under strong flow. The quantitative comparison is very reasonable. However, the availability of data on model H and pom-pom polymers is limited. 2.5.3 Randomly branched polymers The pom-pom equations can be generalised from mono-disperse melts of pom-pom molecules to model the rheology of industrial grade polymeric materials such as low
2.5. BRANCHED POLYMERS 41 density polyethylene (LDPE) using the multimode approach described by Inkson et al. (1999). Commercial LDPE is a polydisperse blend of molecules with multiple irregularly spaced long chain branches. Each section of this complex molecular architecture has its own time-scales for orientation and stretch relaxation. Under the multimode method these sections are modelled by a superposition of pom-pom molecules of differing relaxation times and arm numbers. The stress contribution of these pom-pom molecules is then summed to obtain the total stress, σ ≈ = n∑ n∑ σ = 3 g i λ 2 ≈ i S (2.43) i ≈ i. i=1 i=1 The approximation is the decoupling of the modes of a connected molecule. There ought to be interactions between the separate sections on the same molecule, however these are neglected. The moduli, g i , and backbone orientation times, τ bi for a particular melt can be determined from the linear viscoelastic behaviour of the material. However, the values of τ s and q must be determined from a non-linear flow experiment, usually uniaxial extension. In the multimode formulation of the pom-pom model the direct computation of timescales from molecular structure is replaced by fitting to experimental rheology. This set of variables {τ bi , g i , τ si , q i } can then by used to make successful predictions for other simple flows, such as shear and planar extension, and for complex geometries [Lee et al. (2001)]. The method is a generalisation of the practice of fitting a spectrum of Maxwell modes to linear oscillatory shear data. In this case the spectrum is extended to include non-linear parameters and the pom-pom model is used as the underlying theory. It should be noted that the range of phenomena captured by this approach is significantly larger than that of linear response. 2.5.4 Discussion of multimode pom-pom model On first inspection the decoupling of different sections of a connected molecule appears to be severe approximation. It is then, perhaps, surprising that the method works so well. Blackwell et al. (2001) have investigated this decoupling by deriving a more rigorous tube theory for symmetric molecules containing multiple layers of branching including the appropriate coupling through the branch points. They compared their predictions to those of the multimode pom-pom and managed to identify some discrepancies which also appear in the comparison of the pom-pom model with real data. Generally, the errors were not serious, though. It is likely that the errors associated with the decoupling occur a large strains, particularly in steady state. In this case the errors will be screened out by the viscous contribution from modes which are in linear response. This mechanism also explains why the multi-mode pom-pom can be used to model non-linear shear of branched polymer melts despite the omission of CCR (see
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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 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
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: 2.5. BRANCHED POLYMERS 39 By a forc
Page 57 and 58: 2.II. RESCALING A GAUSSIAN WALK 43
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Page 61 and 62: 2.III. OBSTRUCTED DIFFUSION 47 Comp
Page 63 and 64: 3.1. INTRODUCTION 49 principle exte
Page 65 and 66: 3.2. SINGLE MODE POM-POM MODEL 51
Page 67 and 68: 3.2. SINGLE MODE POM-POM MODEL 53 A
Page 69: 3.2. SINGLE MODE POM-POM MODEL 55 2
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Molecular modelling of entangled polymer fluids under flow The ...

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