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

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Chapter 3 The pom-pom model in exponential shear. 3.1 Introduction In this chapter I analyse the viscometric flow, exponential shear. In particular, I examine exponential shear of branched polymer melts, using the pom-pom model of McLeish and Larson (1998) (see section 2.5.2 for an introduction to this model). I compare the flow with extensional flows and assess the practicality of using exponential shear as an alternative characterising flow for the multimode pom-pom model. Much of the method outlined in this chapter was published in the Journal of Rheology [Graham et al. (2001)], taking advantage of existing data from the literature. Subsequently, I retested the approach using some newly available experimental measurements by Suneel [Suneel et al. (Submitted)] and was able to confirm my findings on the degree of usefulness of exponential shear as a tool for characterising branched polymer melts. Exponential shear flows of polymer melts are interesting flows because they share properties in common with both planar extension and simple, constant rate, shear flows. In particular embedded points separate exponentially in time, like extensional flows, yet the flow direction and velocity gradient are perpendicular, as for all shear flows. Several experimental studies of the rheological behaviour of highly branched polymer melts in exponential shear, including comparisons to extensional and simple shear flows, have been conducted [Zülle et al. (1987), Venerus (2000)]. The geometry of an exponential shear flow is the same as simple shear but the shear rate grows exponentially with time. Typically the shear rate evolves as ˙γ(t) = α(e αt + e −αt ), (3.1) where α is some characteristic strain rate of the flow. Exponential shear has similar 48
3.1. INTRODUCTION 49 principle extension ratios to planar extension (see section 1.4.2). In both flows one direction extends exponentially in time, a second is unchanged and, to conserve volume, the final direction compresses exponentially. For example λ 1 (t) = e αt , λ 2 (t) = 1, λ 3 (t) = e −αt . (3.2) The principle extension ratios are found by taking the square root of the eigenvalues of the finger tensor, C −1 . Under some flow classification schemes this exponential ≈ growth of one of the principle stretch ratios classifies a flow as “strong” [Tanner and Huilgol (1975)]. A flow is classified as weak if it evolves in any other way. Hence under this and most other classification schemes extensional flow and exponential shear flows are regarded as strong. Simple shear is deemed to be weak since its largest principle stretch ratio grows linearly with time [Doshi and Dealy (1987)]. These schemes are based around the idea that a strong flow is likely to stretch the polymer chains of a melt. Venerus (2000) presents a more detailed summary of flow classification. A flow classification scheme can be given credibility if it is found to be consistent with rheological data. For example transient first normal stress growth coefficient data for branched polymer melts consistently rise above the predictions of linear viscoelastic theory at large strains. This is the so called strain hardening phenomenon. Conversely transient shear stress growth coefficient data typically fall below linear viscoelastic predictions at large strains and this is known as strain softening. In these cases the above flow classification scheme is consistent, in that the strong flows strain harden and weak flows soften. Dealy (1990) notes that there is some ambiguity in this use of linear viscoelastic theory as a reference curve for strain hardening since the phenomenon usually occurs at large strains where the theory is no longer valid. He suggests alternative reference curves, such as finite linear viscoelastic theory (FLV) 1 , but observes that all known flows thin relative to these curves. Continuing, he also points out further problems in choosing a suitable reference curve for exponential shear, particularly for first normal stress difference since linear viscoelastic theory predicts a zero value for this quantity. However, by careful choice of material function, Venerus (2000) has shown that exponential shear stress data fall below even linear viscoelastic theory and so has classified it as a weak flow. In this chapter I will solve the pom-pom equations as derived by McLeish and Larson (1998) for exponential shear and compare the solutions to those for simple shear and planar extension in section 3.2. The model will then be tested quantitatively 1 Otherwise known as first-order linear viscoelastic theory or the Lodge rubber-like model
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Molecular modelling of entangled po
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ii CONTENTS 2.3 Doi-Edwards model o
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List of Figures 1.1 Transient shear
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vi LIST OF FIGURES 3.12 Comparison
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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 and 54: 2.5. BRANCHED POLYMERS 39 By a forc
Page 55 and 56: 2.5. BRANCHED POLYMERS 41 density p
Page 57 and 58: 2.II. RESCALING A GAUSSIAN WALK 43
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Page 61: 2.III. OBSTRUCTED DIFFUSION 47 Comp
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
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Molecular modelling of entangled polymer fluids under flow The ...

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