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

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Chapter 2 Introduction to molecular rheology 2.1 Overview In this chapter I will introduce some ideas used in the prediction of rheological properties from microscopic theories of the motion of polymer molecules. This section presents established knowledge, much of which is contained in the books of de Gennes (1979), Doi and Edwards (1986), Larson (1988) and Cates and Evans (2000). 2.2 Gaussian chains Many of the more advanced models for polymer dynamics make use of the statistics of Gaussian chains. Gaussian chains have a useful mathematical simplicity and, despite the seemingly crude assumptions necessary for their use, they form a valid model under many circumstances. 2.2.1 Random walk model The random walk model views a polymer chain as a series of N connected, straight bonds of fixed length, b. Each of these bonds points in a random direction chosen from an isotropic distribution. Thus each step is totally uncorrelated with the previous step. This type of model is known as a freely jointed random walk and is shown in figure 2.1. Using the fact that each step is independent of the previous step it is straightforward to show that the end to end vector r has the following average properties. 〈r〉 = 0 〈 r 2 〉 = Nb 2 . (2.1) 12
¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 2.2. GAUSSIAN CHAINS 13 ¡ ¡ ¡ ¡ ¡ ¡ ¡ b Figure 2.1: Sketch of an N-step freely jointed random walk R In addition, r is the sum of many random vectors of fixed length so, if N is sufficiently large, the probability density of r can be shown to tend to the following Gaussian distribution. Ψ(r) = 2.2.2 Justification of the model ( ) 3 3/2 ) 2πNb 2 exp (− 3r2 2Nb 2 . (2.2) Some limitations of using Gaussian statistics to model the static properties of a polymer chain are immediately apparent. A random walk of N steps of length b has a maximum possible end to end vector length of Nb corresponding to the case in which all of the bonds point in the same directions. Yet equation 2.2 assigns a non-zero probability to vector lengths in excess of this value. In practice, as long as N is reasonably large the probability of achieving these unphysically large end to end vectors is small enough to have a negligible effect on the chain properties. Under very large and rapid deformations a chain can be unravelled sufficiently to approach its maximum extensibility and in this case a more detailed counting of microstates must be used. Why take each monomer step to be freely jointed? There are examples of more detailed models of long chain molecules in which each bond direction is constrained to be related to the previous bond in the chain. However, so long as these correlation decay over some distance along the chain and the chain length is long compared with this distance, the value of 〈 r 2〉 still scales with the first power of N. The effect of these local bond correlations merely renormalises the bond length, b. This renormalised step length is known as the Kuhn statistical step length. The results presented so far still hold for weakly correlated random walks provided b is Kuhn step length rather than the raw bond length value. Another key assumption for the use of Gaussian chains is that interactions between two points which are well separated along the chain are neglected. Although plausible
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: 1.7. ALTERNATIVE EXPERIMENTAL TECHN
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
Page 59 and 60: ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
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
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