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20 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY Thus to compute the stress an expression for the second moment of the Rouse mode amplitude is required. Note that cross correlations between the modes (p ≠ q) do not contribute to the stress. In fact an independent set of ODEs for 〈X p X p 〉 can be found and so the cross correlations can be ignored. The set of differential equations is obtained from equation 2.18 by using an Ito-Stratonovich relation (see appendix 2.I for a discussion of this result) 〈g p (t)X p (t)〉 = k BT 2ζ 0 N I ≈ . (2.22) Taking the second moment of equation 2.18 and substituting equation 2.22 gives ∂ 〈X p X p 〉 ∂t = κ ≈ . 〈X p X p 〉 + 〈X p X p 〉 .κ ≈ T − 2p2 τ R 〈X p X p 〉 + k BT Nζ 0 I ≈ . (2.23) By writing σ ≈ p = 2π2 Nb 2 p 2 〈X p X p 〉 the stress can expressed as a sum of mode contributions σ ≈ = 3k BT c N ∑ σ (2.24) ≈ p. where each of these components obeys an independent upper convected Maxwell (UCM) equation (see section 1.6.1 for an empirical argument which leads to a UCM model for polymer fluids) with relaxation time τ p = 1/2p −2 τ R . p ∇ σ = − (σ 2p2 − 1 ) ≈ p τ R ≈ p 3 I . (2.25) ≈ Thus each stress component makes an equal contribution to the stress but higher modes have much faster relaxation times (τ p ∼ p −2 ). This result justifies the assumption that small length-scales of the molecules are locally at equilibrium as used in the derivation of an expression for stress. It is worth noting that the above result can be derived without using the Ito- Stratonovich relation (equation 2.22). This is achieved by direct integration of equation 2.18 over time and then taking the square of the resulting expression before averaging. However, the Ito-Stratonovich relation is useful in that it allows the derivation of differential, as opposed to integral, constitutive equations. 2.2.8 Validity of the Rouse model As discussed above, some, apparently drastic, simplifications are made in deriving the Rouse model. However, under some circumstances these approximations appear to be valid. Although the neglect of hydrodynamic interactions causes the model to fail for dilute solutions, experimental evidence suggests that these interactions are screened in concentrated polymer fluids [Ferry (1980)]. Yet applying the model to the dynamics of concentrated fluids seems to bring the phantom chain assumption into question.
2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 21 This problem can be averted by considering certain specific regimes of melt dynamics. Where polymer chains are shorter than some critical mass or time-scales are shorter than some critical time then the Rouse model appears to describe these systems well. However in more general situations one must explicitly take into account the topological interactions of neighbouring chains. The standard model when taking this approach is outlined in section 2.3. Thus the surprising conclusion is that the simple model of Gaussian chains forms a sound basis for modelling concentrated polymer fluids. 2.3 Doi-Edwards model of entangled polymers In this section I will introduce the Doi and Edwards tube model [Doi and Edwards (1978)]. This is the current standard approach to dealing with polymeric fluids in which there is significant inter-chain overlap. I will then outline some recent refinements that aim to either improve the model’s quantitative predictions or generalise the model to different molecular architectures. Measurements of the zero shear viscosity of linear polymer melts show a transition in behaviour at some critical molecular weight, M c [Berry and Fox (1968), Colby et al. (1987)]. Below this weight the viscosity scales linearly with molecular weight as predicted by the Rouse model. This is part of the experimental verification of the Rouse model for low molecular weight chains. Above the threshold the scaling increases sharply to an exponent which is widely reported to be ≈ 3.4. The tube model, developed by Doi, Edwards and de Gennes [de Gennes (1971), Doi and Edwards (1978)], aims to describe polymer dynamics for chains with a mass in excess of M c . The observed change in scaling behaviour is attributed to interactions of the surrounding chains influencing the single chain dynamics. The critical chain mass is identified as the molecular weight at which significant inter-chain overlap begins to occur. Specifically, the dominant chain interaction is taken to be the topological constraint that one chain may not pass through another. Rather than attempting to solve the complicated many body problem of an entangled fluid directly, the tube model adopts a mean field approach in order to reduce the problem to that of a single chain. The topological interactions are modelled as a set of constraints that confine the chain to a tube-like region. This tube allows the chain to move freely along its own contour length, but lateral movement of the chain is severely restricted (see figure 2.5). The molecule is able to relax its configuration by diffusing back and forth along its own contour length. As the free ends of the chain emerge from the tube, they are able to explore any direction without passing through another chain. In this way the whole chain renews its configuration with a relaxation time controlled by the time taken for the chain to escape the whole of its original tube. The “primitive path” of the chain along the tube is defined to be the shortest path
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 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: 2.2. GAUSSIAN CHAINS 19 equilibrium
Page 37 and 38: 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 63 and 64: 3.1. INTRODUCTION 49 principle exte
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