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

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22 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY Figure 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). from one chain end to the other that has the same topology as the actual chain, with respect to the entanglement network. Figure 2.6 illustrates this idea and demonstrates how the actual chain may contain a considerable amount of slack in comparison to the primitive path. In order to make these ideas quantitative some assumptions about the nature of the entanglement network surrounding a chain must be made. The simplest of these is to assume that the melt can be characterised by a single length-scale. This quantity is known as the tube diameter, a, and interactions with surrounding chains are assumed to act on length-scales greater than this length. The nature of the tube diameter and entanglements themselves is not well understood theoretically, but see Everaers (1999) for a recent summary. However, considerable progress can be made by treating a as an empirical quantity. Formally, a is defined as the correlation length of the primitive path. In equilibrium the primitive path is a Gaussian random walk of step length a and has the same end to end vector as the original chain. This leads to the following relationship which introduces the number of entanglement segments, Z, Nb 2 = Za 2 . (2.26)
2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 23 a) b) Figure 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). Figure 2.6: A chain in an entanglement network (narrow line) and its corresponding primitive path (broad line). By introducing the number of monomers per entanglement segment, N e = N/Z, the following relation can be found N e = a2 b 2 . (2.27) Equivalently the molecular mass between entanglement segments, M e , is often quoted, giving Z = N/N e = M/M e . The tube diameter is taken to be independent of molecular mass and chain topology, depending only on the chemistry of the molecule. If the tube model can be made to agree simultaneously with a wide range of different data sets this is still a sufficiently demanding test to suggest that the tube idea has captured the dominant physics of the problem. At the moment there is no consensus on numerical values for a or a unique method of determination. Attempts to produce a more unified quantitative theory are a current focus of much theoretical work (see, for example, Pattamaprom et al. (2000) or Likhtman and McLeish (2002)).
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
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Page 33 and 34: 2.2. GAUSSIAN CHAINS 19 equilibrium
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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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Page 63 and 64: 3.1. INTRODUCTION 49 principle exte
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