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36 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY chains retract at the same rate and this retraction provides the dominant contribution to the total constraint release rate. The assumption is certainly not true for thermal constraint release where many processes occur simultaneously. For stretching flows in the distribution of retraction rates may also be broader. Blended systems where each species may have a different retraction rate and branched systems are also likely to violate this condition. A wide range of constraint release mobilities was used in the linear theory of Likhtman and McLeish (2002) using an algorithm which was first suggested by Rubinstein and Colby (1988). However, there is no clear generalisation of this approach to non-linear flows. 2.5 Branched polymers Branched polymers are significant from both an industrial and theoretical point of view. In many cases the processibility of branched polymers is superior to linear polymers. In terms of modelling, branched polymers provide a further test of the tube model, particularly since the molecular parameters ought to be independent of chain architecture. For these reasons understanding both the linear and non-linear rheology of branched polymers is a prominent task. 2.5.1 Star Polymers The tube concept highlights the importance of topology in the dynamics of entangled polymers and can be generalised to branched architectures. The simplest branching structure is the star, in which three or more linear arms meet at a single branch point. Tube theories for star polymers in linear response are well developed and have achieved quantitative agreement with experimental data [Pearson and Helfand (1984) Ball and McLeish (1989) Milner and McLeish (1997) Milner and McLeish (1998a)]. I will give a brief summary of the theoretical methods. Star polymers have a structure in which each arm has only one free end. Once entangled the branch point is pinned since the entropic penalty for dragging multiple arms into the same segment of tube is prohibitively large. As a consequence of the pinned branch point, reptation is impossible. Instead the arms relax by successively deeper incursions of the chain end towards the branch point. In effect, the whole arm must relax by contour length fluctuations of the primitive path. By this process, each arm relaxes independently and so the rheology and relaxation times of a symmetric star are independent of number of arms. To achieve a deep retraction the chain must occupy fewer tube segments than in equilibrium. This incurs an entropic penalty equivalent to that of a particle moving in a quadratic potential. Thus the arm relaxation time is found by solving a first passage problem for a Brownian particle in a harmonic potential.
2.5. BRANCHED POLYMERS 37 From this calculation an exponential dependence of the segment relaxation time on distance from the free end is obtained. This calculation is insufficient to reproduce experimental complex modulus data. The approach correctly predicts a widening of the relaxation spectrum relative to that of a linear polymer but it overestimates the terminal time by several orders of magnitude. This is resolved by the concept of dynamic dilution [Ball and McLeish (1989)]. Since the relaxation time is exponentially dependent on position along the arm, in the time taken for a particular segment to relax all segments that are closer to the chain end will have relaxed many times over. These faster relaxing segments do not constrain the motion of deeper segments and so when considering the relaxation of a particular segment all faster relaxing segments behave as a solvent. Consequently, as the relaxation proceeds the tube diameter progressively dilates so that the effective entanglement network consists only of unrelaxed material. This modification brings the theory into quantitative agreement with experimental data in the linear regime. Despite the theory producing reliable agreement with data there is still current debate over the validity of some of the assumptions that are inherent to this approach. The influence of the higher Rouse modes of the chain on the first passage problem may be non-negligible. Including these modes significantly complicates the mathematics of the problem. The relaxed material may also influence the drag experienced by the remaining sections of chain. Viovy et al. (1991) demonstrated this to be the case for bimodal blends of linear polymers. For an exploration of these recent developments see, for example, work by McLeish (Submitted) or a recent discussion of these issues by Likhtman (2002) at the ITP of the University of California, Santa Barbara. 2.5.2 H polymers and the pom-pom model In terms of non-linear rheology the presence of chain sections with no free ends is significant. This is because, under non-linear flow, the accumulation of chain stretch has a strong influence on the material’s rheological response. An arm in a star polymer can rapidly relax stretch by retraction of its free end. Such retraction cannot occur in a chain section that has no free ends since both ends of the section are pinned by the branch points. The simplest topology which contains such sections is the H polymer. Recognising that in stars the number of arms does not affect the relaxation time, one might expect to be able to model a slightly more general molecular shape which has more than two free arms at each branch point. These molecules are known as pom-poms (see figure 2.13). Here I present a brief review of the pom-pom model as derived by McLeish and Larson (1998). The model is based on the dynamics of pom-pom molecules, a rather generic long-chain branched architecture. It uses a generalisation of the Doi-Edwards
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
Page 27 and 28: ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡
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 47 and 48: 2.4. CHAIN STRETCH AND CONSTRAINT R
Page 49: 2.4. CHAIN STRETCH AND CONSTRAINT R
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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