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

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38 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY tube model [Doi and Edwards (1986)]. A pom-pom molecule comprises of two q-armed Figure 2.13: A three armed pom-pom molecule (q=3) stars connected by a backbone section. This backbone section has no free ends so that reptation and retraction as outlined above are prevented by the branch points. The molecule relaxes through a hierarchical series of processes. First the free arms begin to relax in the same manner as a star polymer. The subtle difference here is that part of the the entanglement network, the backbone sections, are fixed so are not removed by dynamic dilution. Each time the arm retracts fully the branch point takes a diffusive hop. The dynamics of the backbone sections are controlled by these diffusive hops. At time-scales longer than the arm retraction time the motion of the backbones becomes that of a linear polymer in a tube formed by self entanglements of the backbone segments. The friction is concentrated at the ends of the chain and the diffusion rate is controlled by the free arm relaxation rate. Thus the molecular time-scales are influenced by the number of arms, q, through the dilution calculation. Variations of q change the relative concentration of arm and backbone material which has a secondary effect on both the chain end friction and the number of mutual entanglements between backbone sections. McLeish and Larson (1998) derived a detailed linear version of the model which explicitly computes the stress relaxation contribution from all parts of the molecule. However, for the purposes of this thesis their non-linear model is of more interest. Under non-linear response the dominant contribution to the deviatoric stress can be shown to come from the backbone sections. Since the friction felt by the backbone is concentrated at the branch points the chain stretch is uniform along the backbone. Counterintuitively, this means that a simpler model can be used for stretch in branched molecules than linear molecules. The pom-pom model describes the effect of an imposed deformation on a melt of identical pom-pom molecules. The configuration of the melt is described by three dynamical variables S, λ and s c . The tensor S =< uu > ≈ ≈ describes the pre-averaged backbone tube orientation (u is a unit vector parallel to a tube segment) and λ is the pre-averaged stretch of the backbone section which is defined as the ratio of the current length of the backbone to their equilibrium length.
2.5. BRANCHED POLYMERS 39 By a force balance at the branch points, it can be shown that the maximum tension which the backbone can sustain is determined by the number of arms, q. This results in the condition that λ cannot exceed q. When λ reaches q the backbone stretch becomes fixed and the arms segments are drawn into the tube. The length of arm material drawn into the tube is denoted by s c (see figure 2.13). Blackwell et al. (2000) have modified this onset of maximum stretch and their adaption will be discussed below. McLeish and Larson (1998) provided two non-linear versions of their model. In the full model the orientation equation is the integral equation which was originally derived by Doi and Edwards (1986) in the context of reptation of linear polymers. To ease the computational effort an approximate closed form differential equation for the dynamics of S ≈ was suggested by Harlen. This differential approximation has the same asymptotic behaviour under many simple flows as the integral version. I will use the differential approximation throughout. This approach disregards the contribution to stress of oriented arm segments hence removing s c as a dynamical variable. The evolution of S ≈ and λ are governed by differential equations in table 2.2 and the resulting stress tensor, σ ≈ , is calculated from these quantities. An important feature of this model, which Table 2.2: The pom-pom constitutive equation- differential approximation Orientation S ≈ = A ≈ Tr A ≈ D Dt A ≈ = κ ≈ · A ≈ + A ≈ · κ ≈ T − 1 τ b (A ≈ − I ≈ ) τ b = backbone orientation relaxation time Stretch D Dt λ = κ : Sλ − eν∗ (λ−1) ≈ ≈ τ s (λ − 1) τ s = backbone stretch ν ∗ = relaxation time drag-strain coupling constant Stress σ ≈ = 3G 0 φ 2 b λ2 S ≈ G 0 = plateau modulus φ b = fraction of molecular weight in the backbone distinguishes it from other commonly used constitutive equations, is the introduction of two distinct time-scales for the backbone, its orientation relaxation time, τ b , and its stretch relaxation time, τ s . Tube dynamics requires that the orientation time is larger than the stretch time, a fact which is generic to all entangled polymers.
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 51: 2.5. BRANCHED POLYMERS 37 From this
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
Page 69: 3.2. SINGLE MODE POM-POM MODEL 55 2

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