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

More documents

Recommendations

Info

42 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY section 2.4) since, again, the influence of CCR will be most prevalent in steady state of shear where the linear viscous modes dominate the multi-mode stress predictions. In this way it is possible that the broad distribution of relaxation times typical of polydisperse branched melts allows effective stress predictions to be made without a complete knowledge of the dynamics of the whole molecule at all times. However, it should be noted that this effect is dependent on the material having an appropriate molecular weight distribution and will not hold in general. In addition, the non-linear spectrum should not be regarded as denoting absolute values characterising the details of the molecule. This parameters are certainly related to the true molecular values and do provide some useful information about the general molecular structure. However, the approximations used in the multimode method sever the quantitative link between the parameters and the molecule. Read and McLeish (2001) have made some progress towards generating the non-linear parameters from a knowledge of the reaction kinematics rather than by fitting. A significant theoretical break through will be required to make this approach quantitative for non-linear flows with no parameter fitting. Rubio and Wagner (2000) have analysed the accuracy of the differential approximation for a range of simple flows. They identified a number of quantitative differences between the differential and integral versions of the model. Specifically the the differential approximation predicts much stronger alignment in shear than the full integral version. They claim that, if used for the multimode model, the integral version would overpredict the steady shear viscosity of LDPE relative to experimental data. This may be another example of a cruder mathematically solution of a model producing closer agreement with experimental data. Appendix 2.I The Ito-Stratonovich relation Consider a coupled system of variables X i that obey the following general form of stochastic equation dX p dt = F p (X i ) + g p (t), (2.44) where F p (X i ) is a set of functions of all of the X i which couples the dynamics of the X i . The stochastic noise term, g p (t) has the following average moments 〈g p (t)〉 = 0 〈 gp (t)g q (t ′ ) 〉 = Γδ(t − t ′ )δ pq . (2.45) where Γ is a constant with respect to time which may be different for each p. Thus each variable in the system is controlled by an independent stochastic force whose
2.II. RESCALING A GAUSSIAN WALK 43 correlations decay rapidly in time. F p (X i ) remains essentially fixed then For a time interval, t ′ = t − ∆t...t, over which ∫ t X p (t) = X p (t − ∆t) + ∆tF p (X i (t − ∆t)) + g p (t ′ )dt ′ . (2.46) t−∆t This allows the following useful correlation to be calculated ∫ t 〈X p (t)g q (t)〉 = X p (t − ∆t) 〈g q (t)〉 + ∆tF p (X i (t − ∆t)) 〈g q (t)〉 + = ∫ t t−∆t = 1 2 Γδ pq. Γδ(t − t ′ )δ pq dt ′ t−∆t 〈 gp (t ′ )g q (t) 〉 dt ′ The factor of 1/2 arises from the integration over half of the delta function. (2.47) This result hinges crucially on the existence of a time interval, ∆t, which is simultaneously small enough that the F p (X i ) do not change significantly and large enough for the correlations of the Brownian forces to decay. The separation of time-scales between these two processes is often wide enough that such a ∆t can be found. If the F p (X i ) are all linear functions then a more rigorous proof along the same lines as above can be made. Appendix 2.II Rescaling a Gaussian walk This problem is described in Doi and Edwards (1986) [section 2.1] and is recast in my notation in this section. Figure 2.14 demonstrates the division of the molecule from steps n = 0..N of length b to steps s = 0..Z of length a. Where each segment a contains b a Figure 2.14: Sketch of a random walk rescaled from N steps of length b to Z steps of length a.
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 and 52: 2.5. BRANCHED POLYMERS 37 From this
Page 53 and 54: 2.5. BRANCHED POLYMERS 39 By a forc
Page 55: 2.5. BRANCHED POLYMERS 41 density p
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?