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

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16 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY balance on bead n is given by ζ 0 ( ∂Rn ∂t − κ ≈ .R n ) = − 3k BT b 2 (2R n − R n+1 − R n−1 ) + f n (t). (2.8) Here the viscous forces are assumed to dominate over inertial forces so the m ∂2 R term ∂t 2 is dropped. The random force experienced by each bead due to collisions with the surrounding is represented by the force f n . The beads at the chain ends are connected to just one other bead and so experience a spring force from just one neighbour. To ensure that equation 2.8 holds for the end bead, hypothetical beads at n = 0, N + 1 are included with zero extension. This leads to the following boundary conditions. R 0 = R 1 R N = R N+1 . (2.9) In principle equations 2.8 and 2.9 define a system that may be solved to give the trajectories of each beads within the Rouse chain. However, in practice it is often more convenient to take the continuous limit of this system (see section 2.2.5). In either case a further derivation is required to convert information about the chain configuration into a quantity which can be measured experimentally. I will demonstrate such a calculation in section 2.2.6 2.2.5 Continuous limit It is often more convenient to describe a chain contour in terms of a continuous variable rather than a set of discrete points. When describing a rouse chain by a set of N beads it is required that the system be independent of the number of beads used in the solution. One way of achieving this is to take the limit of the number of beads tending to infinity. This leads to a description of the chain contour in terms of a continuous variable rather than a set of discrete points. Thus R n becomes R(n), terms involving next neighbour differences become derivative with respect to the continuous variable n and sums can be replaced by integrals (see table 2.1) The derivatives are obtained Discrete Continuous R n → R(n) ∂R R n − R n−1 → ∂n R n+1 + R n−1 − 2R n → ∂ 2 R ∂n 2 δ nm → δ(n − m) ∑ m ′ n=m Table 2.1: Summary of the transformation from a discrete system to a continuous variable. → ∫ m ′ m dn
2.2. GAUSSIAN CHAINS 17 L α β Figure 2.3: The derivation of a molecular expression for stress through finite difference definitions of the derivative which become exact in the limit of large N. 2.2.6 Molecular expression for stress Equations 2.8 and 2.9 define a system that can be solved to find the molecular shape, R(s). From this information a variety of measurable quantities can be deduced, including the macroscopic stress tensor, σ. To compute σ for an ensemble of Gaussian ≈ ≈ chains the configuration of the chains must be known above some length-scale and chain segments on length-scales below this scale must be locally at equilibrium. In this calculation the cut off length-scale is taken to be the Kuhn step length, b, although often in practice, the molecule is only disturbed from equilibrium on length-scales much larger than this. This choice of length-scale corresponds to dividing the molecule into N subchains. An expression for the stress is obtained by considering a cube of length L, where L is large enough to be much greater than the end to end vector of a subchain and is
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: 2.2. GAUSSIAN CHAINS 15 This leads
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 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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