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14 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY Figure 2.2: In a Gaussian random walk monomers which are well separated along the chain may come into close contact. arguments for neglecting bond correlations along the chain can be made this is insufficient to allow the total neglect of all inter-monomer interactions. As a chain loops over on itself is possible for two monomers from separate parts of the chain to come into close contact. Figure 2.2 shows such a configuration. In fact Gaussian chains have no mechanism to prevent monomers from passing through each other. If the chain is, more realistically, to be considered to consist of monomers with a finite width as well as length, then configurations such as that in figure 2.2 should be discounted as allowable microstates of the polymer chain. This considerably changes the static properties of the chain. For example if a chain has a small value of r 2 then it is likely to contain many loops in which separate monomers are close to each other. Many of the microstates corresponding to this value of r 2 , which were allowed for phantom chains, must be neglected if the chain is self-avoiding. In contrast extended configurations lose fewer microstates this way since they contain fewer loops. Thus the chain end to end distribution function becomes more biased towards extended distributions and the chain becomes swollen. This is, indeed, the case for single chains, however, in melts and concentrated solutions the chain must also avoid interactions with neighbouring chains as well as with itself. If the monomer density is constant throughout the melt then configurations are lost equally for all end to end vectors and so Gaussian statistics are maintained. This unexpected result was first understood by Flory (1953) and has been verified experimentally by scattering experiments on linear polymer melts (see for example Cotton et al. (1974)). 2.2.3 Bead spring model Equation 2.2 can be used to derive a force extension law for a Gaussian chain. For a given end to end vector, r, the number of chain arrangements that achieve this vector, Ω, is given by Ω(r) = Ω total Ψ(r). (2.3)
2.2. GAUSSIAN CHAINS 15 This leads to an expression for the chain entropy, S = k B ln Ω, as a function of end to end vector. S = S 0 − 3k Br 2 2Nb 2 . (2.4) Since each Kuhn segment is freely jointed there is no internal energy cost associated with any distribution of the chain. As a consequence, the chain’s free energy, F, is purely entropic. F(r) = 3k BT r 2 2Nb 2 − F 0 . (2.5) This free energy is converted to a force by applying the grad operator, F(r) = − 3k BT r. (2.6) Nb2 If the chain is divided into N + 1 beads each connected by a spring then the forceextension law for each spring is F(r n ) = − 3k BT b 2 r n . (2.7) where r is the vector connecting beads n and n − 1. Thus the global properties of a Gaussian random walk are the same as a series of N + 1 beads connected by springs with spring constants given by equation 2.7. 2.2.4 Rouse dynamics With the bead spring model established as valid for the static properties of polymer chains it becomes natural to use the same ideas to model polymer dynamics. This model was originally proposed by Rouse (1953) and still underpins many modern theories. The polymer chain is modelled as a collection of N + 1 beads connected by springs with a spring constant of 3k B T/b 2 . The drag on a bead due to the surrounding solvent is proportional to the relative velocity of the bead and its surrounding solvent, with the friction constant ζ 0 per monomer. Interactions between the beads mediated by solvent, namely hydrodynamics interactions, are neglected. Topological interactions between the chains on different parts of the same chain are also neglected. Thus the beads are not prohibited from passing through each other. This seemingly unphysical model is still valid and useful under certain conditions (see section 2.2.8 for a discussion of this). The bead positions are labelled R 0 ....R N and the force due to the nth spring is calculated using r n = R n − R n−1 . Each bead experiences forces due to: the adjoining springs, drag from its surroundings and random Brownian collisions. Thus the force
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 31 and 32: 2.2. GAUSSIAN CHAINS 17 L α β Fig
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Page 45 and 46: 2.4. CHAIN STRETCH AND CONSTRAINT R
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
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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