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

Molecular modelling of entangled polymer fluids under flow
Richard Stuart Graham
Submitted in accordance with the requirements
for the degree of Doctor of Philosophy
The University of Leeds
Department of Physics and Astronomy
October 2002
The candidate confirms that the work submitted is his own and that appropriate
credit has been given where reference has been made to the work of others.

Contents
Abstract
Acknowledgements
vi
vii
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Deformation kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.1 Volume conserving flows . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.2 Some simple flows . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.3 Flow in complex geometries . . . . . . . . . . . . . . . . . . . . . 5
1.5 Linear rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5.1 Linear oscillatory shear . . . . . . . . . . . . . . . . . . . . . . . 6
1.5.2 Linear continuous shear . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Non-linear rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6.1 A simple empirical non-linear model . . . . . . . . . . . . . . . . 10
1.7 Alternative experimental techniques . . . . . . . . . . . . . . . . . . . . 11
2 Introduction to molecular rheology 12
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Gaussian chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Random walk model . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Justification of the model . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Bead spring model . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.4 Rouse dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.5 Continuous limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.6 Molecular expression for stress . . . . . . . . . . . . . . . . . . . 17
2.2.7 Non-linear constitutive equation . . . . . . . . . . . . . . . . . . 18
2.2.8 Validity of the Rouse model . . . . . . . . . . . . . . . . . . . . . 20
i

ii
CONTENTS
2.3 Doi-Edwards model of entangled polymers . . . . . . . . . . . . . . . . . 21
2.3.1 Linear rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Contour length fluctuations . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Non-linear rheology . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Chain stretch and constraint release . . . . . . . . . . . . . . . . . . . . 30
2.4.1 The Milner McLeish and Likhtman model . . . . . . . . . . . . . 32
2.4.2 Comments on the MMcL model . . . . . . . . . . . . . . . . . . . 35
2.5 Branched polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.1 Star Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.2 H polymers and the pom-pom model . . . . . . . . . . . . . . . . 37
2.5.3 Randomly branched polymers . . . . . . . . . . . . . . . . . . . . 40
2.5.4 Discussion of multimode pom-pom model . . . . . . . . . . . . . 41
Appendix 2.I The Ito-Stratonovich relation . . . . . . . . . . . . . . . . . . 42
Appendix 2.II Rescaling a Gaussian walk . . . . . . . . . . . . . . . . . . . . 43
Appendix 2.III Obstructed diffusion . . . . . . . . . . . . . . . . . . . . . . . 44
3 The pom-pom model in exponential shear. 48
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Single mode pom-pom model . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Solutions to the orientation equation . . . . . . . . . . . . . . . . 50
3.2.2 Solutions to the stretch equation . . . . . . . . . . . . . . . . . . 54
3.2.3 Behaviour of shear stress in exponential shear . . . . . . . . . . . 57
3.3 The multimode method applied to exponential shear . . . . . . . . . . . 57
3.3.1 Predicting exponential shear data using non-linear spectra from
extensional rheology. . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Measuring the non-linear parameters from exponential shear. . . 62
3.3.3 A verification of the method for a different melt. . . . . . . . . . 68
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Theory of CCR and chain stretch 74
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Tube model for linear polymers with CCR and stretch . . . . . . . . . . 74
4.2.1 Rouse retraction term . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.2 Tube diameter under deformation . . . . . . . . . . . . . . . . . 75
4.2.3 CCR stretch relaxation . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.4 Suppression of reptation due to stretch . . . . . . . . . . . . . . . 78
4.2.5 Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.6 Equation for the tangent correlation function . . . . . . . . . . . 79
4.2.7 Number of entanglements . . . . . . . . . . . . . . . . . . . . . . 81

CONTENTS
iii
4.2.8 Constraint release rate . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Contour length fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Thermal constraint release from contour length fluctuations . . . 83
4.3.2 Rouse motion on sub-tube diameter length scales . . . . . . . . . 84
4.4 Summary of model equations . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Real space solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.5.1 Finite difference solution of CLF term . . . . . . . . . . . . . . . 87
4.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.6.1 Steady state in shear . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.6.2 Transient start-up of simple shear . . . . . . . . . . . . . . . . . 88
4.6.3 Single chain structure factor . . . . . . . . . . . . . . . . . . . . . 90
4.7 Comparison with experimental data . . . . . . . . . . . . . . . . . . . . 92
4.7.1 Determination of model parameters from linear rheology . . . . . 92
4.7.2 Parameter free comparison with non-linear data . . . . . . . . . 93
4.7.3 Improvement of high rate predictions . . . . . . . . . . . . . . . . 93
4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Appendix 4.I Derivation of stretch-CCR renormalisation term . . . . . . . . 100
Appendix 4.II Modified CLF term for a stretched chain . . . . . . . . . . . . 101
Appendix 4.III Fourier space solution . . . . . . . . . . . . . . . . . . . . . . 102
5 Bimodal blends of linear polymer melts 104
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Existing data and theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 Self-dilute high molecular weight additive . . . . . . . . . . . . . . . . . 106
5.3.1 Generalisation of stretching CCR theory to self dilute case . . . . 106
5.3.2 Uniaxial extension of bimodal blends . . . . . . . . . . . . . . . . 107
5.3.3 Non-linear shear of bimodal blends . . . . . . . . . . . . . . . . . 109
5.4 Discussion and future directions . . . . . . . . . . . . . . . . . . . . . . . 109
6 Conclusions 112
6.1 Randomly branched polymers . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Monodisperse linear polymers . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

List of Figures
1.1 Transient shear viscosity of a polymer melt at low shear rates compared
to a perfectly viscous liquid and an ideal elastic solid. . . . . . . . . . . 3
1.2 a) The storage and loss modulus for a single Maxwell mode with relaxation
time τ. b) Linear rheology of a real polymer melt fitted with a
spectrum of Maxwell modes [Venerus (2000)]. . . . . . . . . . . . . . . . 8
1.3 Non-linear shear and uniaxial extension of an LDPE melt 1810H showing
extension hardening (solid shapes) and shear thinning (open shapes)
[Suneel et al. (Submitted)]. . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Sketch of an N-step freely jointed random walk . . . . . . . . . . . . . . 13
2.2 In a Gaussian random walk monomers which are well separated along
the chain may come into close contact. . . . . . . . . . . . . . . . . . . . 14
2.3 The derivation of a molecular expression for stress . . . . . . . . . . . . 17
2.4 Scaling of linear viscosity of a range of linear polymer melts against X w
which is proportional to molecular weight. The scaling switches from a
slope of 1 to the 3.4 “law” for entangled polymer melts. From Berry and
Fox (1968). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 The many body problem of an entangled melt (a) reduced to a single
chain problem by replacing the individual entanglements with a confining
tube (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 A chain in an entanglement network (narrow line) and its corresponding
primitive path (broad line). . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Relaxation of oriented tube segments by reptation after a step strain . . 24
2.8 a) Dynamic modulus as calculated by the pure reptation model. b) Experimental
storage modulus for a range of narrow distribution polystyrenes.
Z ranges between ≈ 44 and ≈ 0.6 entanglements [Onogi et al. (1970)]. . 26
2.9 Theoretical predictions of relaxation after a step strain. [Reproduced
from Doi and Edwards (1986)] . . . . . . . . . . . . . . . . . . . . . . . 28
iv

LIST OF FIGURES
v
2.10 Comparison of predicted and measured damping functions after a large
step strain for different linear polystyrene solutions [Osaki et al. (1982)].
The open and filled circles are data for different molecular weights and
the tick directions indicate variations in concentration. . . . . . . . . . . 29
2.11 Schematic representation of a constraint release event. . . . . . . . . . . 31
2.12 Three relaxation mechanism available to an unbranched, entangled polymer
chain: reptation (a), constraint release (b) and retraction (c). . . . 33
2.13 A three armed pom-pom molecule (q=3) . . . . . . . . . . . . . . . . . . 37
2.14 Sketch of a random walk rescaled from N steps of length b to Z steps of
length a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.15 Schematic showing a Brownian particle, subject to a spring force and
moving in an array of vanishing and re-appearing obstacles. . . . . . . . 45
3.1 Evolution of S xy for simple shear shear, ˙γ = 1sec −1 . Affine deformation
corresponds to τ b = ∞ . The thin solid line corresponds to the value of
τ b for which the steady state value of S xy is maximised. . . . . . . . . . 51
3.2 Evolution of S xy for nearly exponential shear with varying τ b values.
α = 1sec −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Evolution of S xx − S yy for a planar extensional flow, ˙ɛ = 1sec −1 . . . . . 53
3.4 Evolution of S xx − S yy for an exponential shear flow, α = 1sec −1 . . . . . 54
3.5 Evolution of backbone stretch for a simple shear flow. τ b = 3sec, τ s =
1sec and q = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 Evolution of backbone stretch for an exponential shear flow, α = 1sec −1 ,
τ b /τ s = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.7 Evolution of backbone stretch for a planar extensional flow. τ b = 3sec,
τ s = 1sec and q=4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8 Evolution of shear stress in an exponential shear flow, α = 3sec −1 , τ b =
3sec, τ s = 1sec and G 0 φ 2 b
= 1. . . . . . . . . . . . . . . . . . . . . . . . . 58
3.9 Pom-pom predictions compared to the experimental data for melt 1 of
Zülle et al. (1987). Filled shapes are nearly exponential shear data points
and open shapes are true exponential shear data points. Solid lines are
nearly exponential shear predictions and dashed lines are true exponential
shear predictions for both (a) and (b). . . . . . . . . . . . . . . . . . 59
3.10 Multimode pom-pom free parameter fit to planar extension data from
Hachmann (1997). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.11 Comparison of multimode pom-pom predictions to experimental data for
shear stress in true exponential shear from Venerus (2000). Solid curves
are pom-pom predictions, shapes are data points and the dashed curve
is the linear viscoelastic curve for simple shear. . . . . . . . . . . . . . . 62

vi
LIST OF FIGURES
3.12 Comparison of multimode pom-pom predictions to experimental data
for first normal stress difference in true exponential shear from Venerus
(2000). Solid curves are pom-pom predictions, shapes are data points
and the dashed curve is the FLV curve for first normal stress difference
in simple shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.13 Free parameter fit of non-linear parameters of melt 1 using only nearly
exponential shear data collected by Zülle (1987). . . . . . . . . . . . . . 66
3.14 Comparison of pom-pom predictions using spec II with uniaxial extension
data for melt 1 from Meissner (1972). . . . . . . . . . . . . . . . . . 67
3.15 The 8 modes of melt 1 (see table 3.1) in nearly exponential shear for
α = 0.01sec −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.16 The 8 modes of melt 1 (see table 3.1) in uniaxial extension for ˙ɛ =
0.01sec −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.17 Linear response of two batches of melt1810H: a)data collected by Venerus
(2000) (melt 1810H) b) data collected by Suneel et al. (Submitted) (melt
1810Hb). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.18 Experimental data and predictions for uniaxial extension of melt 1810Hb
(filled shapes) and simple shear (open shapes) made using spec Ib which
was obtained by fitting only to exponential shear data. . . . . . . . . . . 72
3.19 Experimental data and predictions for true exponential and (filled shapes)
and simple shear (open shapes) of melt 1810Hb made using spec IIb
which was obtained by fitting only to uniaxial extension data. . . . . . . 73
4.1 Two possibilities for the effect of a step deformation on the entanglement
network. (a) The number of entanglements points is fixed and so the tube
persistence length grows. (b) The tube persistence length remains fixed
so Z grows in proportion with the primitive path length. . . . . . . . . . 76
4.2 The effect of CCR on an unstretched segment (a) and a stretched segment
(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Mechanism by which CCR relaxes chain stretch . . . . . . . . . . . . . . 78
4.4 Theory predictions of steady state shear stress as a function of shear rate
(c ν = 0.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.5 Transient predictions for shear stress and normal stress against strain,
γ, for start-up of simple shear. Model parameters: Z = 20, c ν = 0.1
with shear rates from ˙γτ R = 21 to linear response. . . . . . . . . . . . . 90
4.6 S(q) in steady shear for a range of shear rates. The two higher rate Rouse
Weissenberg number are 0.42 and 6, respectively. Model parameters:
Z = 20 and c ν = 0.1. Contours lines map the same value on each plot. . 91

LIST OF FIGURES
vii
4.7 Comparison with the Menezes and Graessley (1982) linear oscillatory
shear data for three polybutadiene solutions: PBB, PBC and PBD, with
c ν = 1.0 (a) and c ν = 0.1 (b). The model parameters, listed in the figure
and in table 4.2, are the same for each molecular weight. . . . . . . . . . 93
4.8 Comparison with Menezes and Graessley (1982) PBB shear viscosity (a)
and first normal stress difference (b) data using parameters obtained
only from linear rheology. . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.9 Comparison with Menezes and Graessley (1982) PBB shear viscosity (a)
and first normal stress difference (b) data using parameters obtained
from linear rheology and R s = 2.0. . . . . . . . . . . . . . . . . . . . . . 96
4.10 Comparison with Menezes and Graessley (1982) PBD shear viscosity (a)
and first normal stress difference (b) data using parameters obtained
from linear rheology and R s = 2.0. . . . . . . . . . . . . . . . . . . . . . 97
4.11 Comparison with Hua et al. (1999) PS/TCP shear viscosity (a) and first
normal stress difference (b) data using parameters obtained from linear
rheology and R s = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.12 Steady state shear stress and first normal stress difference of PS/TCP
[Hua et al. (1999)] compared with model prediction for R s = 2.0. . . . . 99
4.13 Derivation of CCR term for a stretched chain . . . . . . . . . . . . . . . 101
5.1 A self dilute bimodal blend. The HMW chains are sufficiently rare that
they do not self entangle. All constraint release events are produced by
motion of the short chains. . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Uniaxial extension of a set of bimodal blends [Hepperle (2001)] at 173.5 o C
compared with model predictions (R s = 2.0). . . . . . . . . . . . . . . . 108
5.3 A qualitative comparison of data (1) and theory (2) for bimodal blends of
entangled polymer solutions under strong shear including shear stress (a)
and first normal stress difference (b). Experimental data by Osaki et al.
(2000b) (on blend f80/850) with shear rates ranging from 1.165 − 0.0086
sec −1 . Theory curves have shear rates in the range ˙γτ long
R
= 1 − 450. . 110

List of Tables
1.1 The tensorial description of some simple flows . . . . . . . . . . . . . . . 4
2.1 Summary of the transformation from a discrete system to a continuous
variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 The pom-pom constitutive equation- differential approximation . . . . . 39
3.1 Non-linear spectrum of melt 1 fitted to extensional rheology, from Inkson
et al. (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Non-linear spectrum of melt 1810H obtained by fitting to planar extension
data from Hachmann (1997). . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Criterion of equation 3.22 applied to the linear spectrum of melt 1 and
two non-linear spectra for melt 1. Spec I is a fit to uniaxial extension
[Inkson et al. (1999)] and Spec II is a fit to the transient nearly exponential
shear stress data collected by Zülle et al. (1987) . . . . . . . . . 65
3.4 Criterion of equation 3.22 applied to the linear spectrum of melt 1810Hb
and two non-linear spectra. Spec Ib is a fit to exponential shear data and
Spec IIb is a fit to the transient uniaxial extensional data from Suneel
et al. (Submitted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.1 Closed system of equation including describing the dynamics of an ensemble
of entangled linear polymers including: contour length fluctuations,
retraction, constraint release and variable number of entanglements. 85
4.2 Material parameters for two polybutadiene solutions [Menezes and Graessley
(1982)] and a polystyrene solution [Hua et al. (1999)]. Material parameters
are as quoted in the original papers, fitted parameters are obtained
from linear oscillatory shear and calculated parameters are computed
from the other parameters. . . . . . . . . . . . . . . . . . . . . . . 93
viii

Abstract
The aim of this thesis is to investigate the use of microscopic molecular models of
entangled polymer fluids to predict the bulk properties of these materials. This is
achieved by using and modifying refined versions of the tube model of Doi and Edwards
to study both linear and branched polymers.
I investigate long chain branched polymers using the “pom-pom” model of McLeish
and Larson. In particular, I use this model as a tool to characterise industrial long chain
branched materials using experimental data for exponential shear rheology. I highlight
the successes of this approach and investigate the limitations of shear rheology relative
to extensional measurements in this context.
Expanding on recent work concerning the non-linear rheology of model, linear polymeric
fluids I develop a detailed microscopic model for the dynamics of these materials.
The influence of chain stretch and contour length fluctuations are added to the Milner,
McLeish and Likhtman implementation of convective constraint release. These modifications
allow an effective comparison with experimental data for model entangled
polymer solutions to be made. From this comparison I am able to draw conclusions
concerning the ability of the tube model to predict non-linear rheology and to indicate
flow regimes in which new physical insight appears to be necessary. I discuss possible
future modifications to the model. Finally, I generalise the monodisperse model
to cover non-linear flows of self-dilute bimodal blends and compare predictions with
experimental data in both shear and extension.
ix

Acknowledgements
I am very grateful to my supervisors, Tom McLeish and Oliver Harlen, for their
help, support and guidance throughout my PhD. I have benefited greatly from their
supervision. I also owe a large debt of gratitude to Alexei Likhtman with whom I have
been collaborating for the last three years. I have learnt a considerable amount about,
not only polymer dynamics, but research methodology from Alexei and the standard
of my research has been considerably enhanced by Alexei’s influence.
I would also like to thank Richard Blackwell for answering many of my questions
about tube theory, particularly for branched polymers and for helping me to understand
the tube model in the early part of my PhD. I am grateful to Peter Olmsted for general
guidance during my PhD and for agreeing to act as my internal examiner. Thanks
to Daniel Read for helping me attain greater appreciation of my work through his
questioning and insightful suggestions.
I also appreciate useful discussions with numerous researchers during my time at
the KITP at the University of California, Santa Barbara. In particular I am grateful
to Scott Milner and Ron Larson for guidance during this time. I would like to thank
Professor Marrucci and Giovanni Ianniruberto for taking the time to look over my work.
I am also grateful to Professor Marrucci for agreeing to act as external examiner.
Thanks to Suneel for providing experimental rheology on melt1810Hb and collaborating
with me during the analysis of these data. I am also grateful to David Venerus
for kindly supplying his rheological data on both a model entangled linear solution and
an industrial branched melt, both sets of data were particularly helpful in the development
of this work. Thanks to Jens Hepperle for providing his blend data and for
spending time discussing his work during his visit to Leeds and to Akanari Minegishi
for kindly providing his blend data.
I would also like to acknowledge all of the people involved in the µPP project. My
involvement in this project has been useful in my development as a researcher. In
particular, thanks to Timothy Nicholson for running this project. I also acknowledge
Nat Inkson for help with the multimode pom-pom model.
Thanks to Maureen Thompson and Beverly Robinson for help and support during
my time at Leeds and to all other friends and colleagues at Leeds, particularly Anna
x

ACKNOWLEDGEMENTS
xi
Maidens, Stuart Hill, Carole Whiting, Mike Ries, Alessio de Francesco and Simon
Marlow.
I also acknowledge financial support from the EPSRC and BP Chemicals and I
would like to thank my contacts at BP, Choon Chai and Les Rose, for helping me to
understand the industrial relevance of my work and for providing a focus for my work.
Finally, thanks to my family for help and support during my studies.

Chapter 1
Introduction
1.1 Overview
Rheology is the study of the deformation of matter. In particular, it is the term
used to describe the study of complex fluids such as polymer melts, polymer solutions
and colloidal suspensions. The aim of theoretical rheology is to develop constitutive
equations that relate stress within the material to its deformation history. Constitutive
equations together with mass and momentum conservation can be used to predict
the flow of the material. Molecular rheology aims to derive and understand these
constitutive equations from the underlying microscopic physics of the material.
Polymer are large macromolecules. They consist of many chemical repeat units
covalently bonded into long chains. Chain with N = 10 2 − 10 4 repeat units can be
synthesised and polymer of length N = 10 9 −10 10 occur in nature. The topology of the
chain can vary from a simple linear chain to a complex branched structure. Chemically
identical materials with the same molecular weight but different topologies often have
radically different rheology. Conversely materials with different chemistries but with
molecules of globally the same shape often exhibit evidence of universal behaviour.
Molecular rheology of polymers has a long history [Bird et al. (1977), Larson (1988),
de Gennes (1979)]. However, work in this area has intensified in the last twenty years.
Understanding polymer rheology is important not only from the point of view of fundamental
science but because its applications often have very useful industrial consequences.
A good understanding of the link between the molecular constituents of a
polymer liquid and its rheological behaviour is a long term goal of molecular rheology.
This would allow the production of materials with a rheology that is tailored to their
application.
There are many processing problems which are thought to be avoidable if a material
has the correct rheology. For example, in a film blowing process small areas of thinning
in the film may grow in amplitude leading to rupture of the film. By changing the
1

2 CHAPTER 1. INTRODUCTION
extensional properties of the material this effect can be avoided. Theoretical rheology
is also increasingly being used to gain information about the molecular structure of a
material from its rheological behaviour.
The degree of branching in a polymer molecule is known to have strong influence on
its rheology. For example highly branched polymer melts such as low density polyethylene
(LDPE) exhibit strong strain hardening under extensional flows. Strain hardening
is defined as an increase in viscosity with strain and is a desirable processing property.
However, melts comprising of mostly linear molecules rarely show strain hardening
and are often strain softening. Thus the arrangement branch points or topology of a
molecule can dominate its rheological behaviour.
1.2 The stress tensor
The goal of theoretical rheology is to relate the deformation history to macroscopic
properties of the material. The mechanical stress exerted by the material in response
to the deformation governs the flow of the material and so is one of the most frequently
measured properties. Rheological constitutive equations make predictions of the stress
tensor, σ. The Cartesian components, α, β, of the stress tensor are defined as the force
≈
per unit area in the α direction acting across a plane whose normal vector is in the β
direction. In most material the stress tensor is symmetric. Asymmetry implies that
microscopic points in the material experience a non-zero torque.
1.3 Viscoelasticity
Traditionally, the distinction between a liquid and a solid is clear. The response of
an ideal solid to deformation may be modelled by Hooke’s law. The stress response
is proportional to the imposed strain and is independent of the strain rate. Thus the
elastic energy supplied by the deformation is completely conserved. An ideal liquid
has a Newtonian viscosity where the stress response is proportional to the imposed
deformation rate and the total strain is irrelevant. In a Newtonian liquid the energy of
deformation is completely dissipated. Polymer liquids, amongst others, show behaviour
which is part-way between these two extremes. This is demonstrated in figure 1.1 which
compares the transient shear rheology of a polymer melt at low shear rates with the two
ideal limits, showing elastic behaviour at early times, giving way to viscous behaviour
at longer times. Note that at low rates different deformation rates superimpose onto a
single, rate independent curve; this is not the case for more rapid deformations.

1.4. DEFORMATION KINEMATICS 3
10 5
Newtonian Viscosity
10 4
η 0
(t)
σ xy / γ . [Pa-sec]
10 3
Hooke’s Law
10 2
10 -2 10 -1 10 0 10 1
time [sec]
Figure 1.1: Transient shear viscosity of a polymer melt at low shear rates compared to
a perfectly viscous liquid and an ideal elastic solid.
1.4 Deformation kinematics
Before attempting to derive a constitutive equation the deformation history imposed
on the material must be defined. The response of a material depends on the geometry
of the imposed deformation, therefore an adequate mathematical description of the
deformation is essential. One way to achieve this is to specify the velocity field, v,
imposed by the deformation on a material element at point r. However, only relative
motion of material points are relevant, hence the velocity gradient tensor, κ
≈
, is usually
a more relevant measure,
κ (r, t) = (∇v(r,
≈ t))T . (1.1)
If κ
≈
does not depends on spatial position r then the flow is deemed a simple flow. A
range of useful flows such as shear and extension fall under this definition.
Simple
flows are also more easily analysed theoretically and so data on these flows is relatively
abundant and reliable. In this thesis I will test constitutive models under simple
flows against experimental data as a prerequisite to understanding more complicated
deformations.
constitutive equations.
The tensor κ
≈
is widely used to define the deformation in differential
An alternative description of the deformation is the deformation gradient tensor,
E
≈
, which relates the vector connecting two embedded points before the deformation,
X, and after the deformation, X ′ ,
X ′ = E.X. (1.2)
≈
The deformation gradient tensor is more commonly used in integral constitutive equations
or to describe step deformations. Care must be exercised in the use of E since it
contains information not only about the stretching caused by the deformation but also

4 CHAPTER 1. INTRODUCTION
the rotation. Thus if the stress is a functional of E it is not guaranteed that a purely
rotational deformation will not induce a stress in the material! A safer description is
the finger tensor, C
≈
−1
C
≈ −1 = E
≈ T .E
≈
. (1.3)
This remains invariant under a solid body rotation. The velocity gradient tensor and
the deformation gradient tensor are related by the following expression
1.4.1 Volume conserving flows
∂
∂t E ≈ (t′ , t) = κ
≈
.E
≈
. (1.4)
In polymer fluids the bulk modulus, which controls the response to a change in volume,
is typically many orders of magnitude large than the moduli for volume conserving
deformations. Thus considerable deformation can be achieved with imposed stresses
that are much smaller in magnitude than the bulk modulus. These deformations can
be taken to be volume conserving. In terms of the deformation gradient tensor the
condition det E
≈
= 1 implies a volume conserving deformation. The same constraint on
the velocity gradient tensor is Tr κ
≈
= 0.
1.4.2 Some simple flows
Flow κ
≈
E
≈
Shear
Uniaxial extension
Planar extension
⎛
⎝
⎛
⎝
0 ˙γ 0
0 0 0
0 0 0
⎞
⎠
˙ɛ 0 0
0 −˙ɛ/2 0
0 0 −˙ɛ/2
⎛
⎝
˙ɛ 0 0
0 −˙ɛ 0
0 0 0
⎞
⎠
⎞
⎠
⎛
⎝
⎛
⎝
1 γ 0
0 1 0
0 0 1
⎞
⎠
e ɛ ⎞
0 0
0 e −ɛ/2 0 ⎠
0 0 e −ɛ/2
⎛
⎝
e ɛ 0 0
0 e −ɛ 0
0 0 1
⎞
⎠
Table 1.1: The tensorial description of some simple flows
A number of different simple flows can be realised experimentally on polymer fluids,
with varying degrees of difficulty. The most straightforward is a shear flow in which
a uniform velocity gradient, ˙γ is imposed throughout the material. The shear strain,
γ, is the total accumulated deformation (γ = ∫ ˙γ(t ′ )dt ′ ). The usual convention is for

1.4. DEFORMATION KINEMATICS 5
flow to be in the x direction, the velocity gradient to act in the y direction, and the z
direction to be parallel to the vorticity. Experimental data typically measure a range
of relevant components of the stress tensor. The easiest component to measure is the
force which directly opposes the shear, namely the shear stress, σ xy . Polymer fluids
also typically exert a force normal to the shear plane. This stress is measured relative
to atmospheric pressure and is expressed as differences between diagonal components of
the stress tensor. Two independent stress differences can be defined: the first normal
stress difference, N 1 = σ xx − σ yy and second normal stress difference, N 2 = σ yy −
σ zz . Considerable experimental effort is required to produce reliable normal stress
measurements in shear [Meissner (1972)]. Since shear contains both extensional and
rotational characteristics, the principle stretching direction rotates as the flow proceeds.
As a result, despite being a comparatively simple experiment, it can pose theoretical
difficulties.
Extensional flows are rotation free. They may be achieved by increasing the length
of a sample exponentially in time, producing a linearly increasing velocity profile. This
constant velocity gradient is the extension rate, ˙ɛ, and from this the Hencky extensional
strain, ɛ can be defined as ɛ = ∫ ˙ɛ(t ′ )dt ′ . The actual extensional strain is exp( ∫ ˙ɛdt).
Conventionally, extension is taken to occur in the x direction. To maintain a fixed volume
the two remaining directions can be allowed to contract equally, which is known
as uniaxial extension. Alternatively, one direction can be held fixed, forcing the final
direction to contract sufficiently to maintain the volume, which is called planar extension.
For both flows the first normal stress difference can be measured and in planar
extension there is also a second normal stress difference. These extensional flows, particularly
planar extension, are difficult experiments and the necessary equipment is only
available in a limited number of laboratories. Other extensional flows are feasible, such
as bi-axial flows, however data for such deformations are rare. The velocity gradient
and deformation gradient tensors for these simple flows are shown in table 1.1.
1.4.3 Flow in complex geometries
The aim of many constitutive equations is to produce reliable quantitative predictions
for as many of the above simple flows as possible, using the same parameters for each
flow. This is, of course, conditional on the availability of suitable data for comparison.
However, almost all flows which are of industrial relevance are complex flows. Many
complex flows will be a combination of shear and extensional deformations and so a
model which captures these simple flows might be expected to perform well for flows
under a complex geometry if a suitable numerical implementation can be found. However,
this conjecture is by no means a guarantee. For example, many complex flows

6 CHAPTER 1. INTRODUCTION
have areas of reversing flow, which is not probed by the above flows. For an example
of such a computation using a finite element technique and a molecularly based
constitutive equation which explicitly takes into account reversing flow see Lee et al.
(2001).
1.5 Linear rheology
Linear rheology refers to experiments in which the applied strain is small (γ ≪ 1).
These measurements are useful for a variety of reasons. They are relatively easy to
realise experimentally and an isotropic material’s response is often insensitive to the
geometry of the deformation. The experiments can also probe the material over a very
wide range of timescales. When devising theories various linearised approximations are
valid, which simplify the mathematics and allows more detailed theoretical ideas to be
investigated.
For sufficiently small strains the relationship between the stress and strain in a
polymer liquid will be approximately linear. Also, the stress relaxation is characterised
by a scalar function G(t) which is independent of the imposed strain. For a small step
strain imposed at time t = 0 the stress at time t is given by
σ
≈
(t) = C
≈ −1 G(t). (1.5)
Thus the stress contribution at time t due to a small strain at time t ′ is
dσ(t) = d
≈ dt ′ C −1 (t ′ )G(t − t ′ )dt ′ . (1.6)
≈
In the limit of small strains the time derivative of the the finger tensor can be written
as
d
dt C −1 = κ + κ T . (1.7)
≈ ≈ ≈
Integrating equation 1.5 over the whole deformation history (t ′ = −∞...t) gives
σ
≈
(t) =
∫ t
∞
(
)
G(t − t ′ ) κ
≈ (t′ ) + κ(t ′ ) T dt ′ . (1.8)
≈
Coleman and Noll (1961) demonstrated that if G(t) has “fading memory” then this is
sufficient to produce the two extremes of elastic and viscous behaviour as outlined in
section 1.3. Fading memory is defined by the conditions that G(t) is integrable and
tends to zero sufficiently quickly as t → ∞ .

1.5. LINEAR RHEOLOGY 7
1.5.1 Linear oscillatory shear
Equation 1.5 can, in principle, be used to measure the relaxation modulus, G(t), directly.
However, the step strain is never completely instantaneous so measured data at
early times are unreliable and at long times the signal to noise ratio is weak, making the
terminal behaviour difficult to obtain. A more effective approach is to use a continuous
oscillating shear strain history. In complex notation this is expressed as
γ(t) = R(γ max exp(iωt)). (1.9)
Under this deformation history equation 1.5 gives
σ xy (t) =
∫ t
−∞
G(t − t ′ ) dγ
dt ′ (t′ )dt ′ . (1.10)
Substituting in the form of the oscillating shear history (equation 1.9) and changing
the variable of integration gives
(
σ xy (t) = R iωγ(t)
∫ ∞
0
)
exp(−iωs)G(s)ds . (1.11)
which can be rewritten as
σ xy (t) = R (γ(t)G ∗ (ω)) . (1.12)
where G ∗ (ω) is known as the complex modulus. Thus
G ∗ (ω) = iω
∫ ∞
0
e −iωt G(t)dt. (1.13)
When the complex modulus is written as G ∗ = G ′ + iG ′′ it can be seen that G ∗ consists
of a component which is in phase with the strain and one which is out of phase. The
in phase part, G ′ , is known as the storage or elastic modulus and the out of phase
part, G ′′ , is the loss or dissipative modulus. A perfectly elastic solid of modulus G 0
would have G ′ = G 0 and G ′′ = 0. In the case of a viscous liquid with viscosity η then
G ′ = 0 and G ′′ = ωη since σ xy is in phase with the shear rate.
For a viscoelastic
material both G ′ and G ′′ are functions of the applied frequency, ω. In general, the
loss modulus dominates at low frequencies, while the elastic modulus dominates at
high frequencies. The material crosses over from viscous behaviour elastic behaviour at
some intermediate frequency where G ′ = G ′′ . A simple form of the relaxation modulus,
the Maxwell model, where G(t) = G 0 exp(−t/τ) which is characterised by a relaxation
time, τ. The complex modulus for this model is given by,
G ′ (ω) = G 0
ω 2 τ 2
1+ω 2 τ 2 , G ′′ (ω) = G 0
ωτ
1+ω 2 τ 2 . (1.14)

8 CHAPTER 1. INTRODUCTION
In this case this cross-over frequency of G ′ and G ′′ is the exact reciprocal of the characteristic
time.
Although this simple model predicts the correct qualitative behaviour, to capture
the quantitative behaviour of a real polymer fluid it is typically necessary to use a
superposition of Maxwell modes.
G(t) = ∑ i
g i exp(−t/τ i ). (1.15)
The set of moduli and corresponding times scales {g i , τ i } is known as the relaxation
spectrum. A comparison of the linear rheology of a single exponential fluid to that of
a real polymer melt fitted with a relaxation spectrum is shown in figure 1.2. The melt
is a polydisperse branched material known as melt 1810H [Venerus (2000)]. Note that
the real melt has a considerably broader spectrum than the single Maxwell mode. In
this more general case G ′ and G ′′ do not cross over at the reciprocal terminal time.
Nevertheless, this cross over is often associated with a “characteristic relaxation time”
of the material. It should be noted that the decomposition of G(t) into discrete Maxwell
modes is not unique.
10 1
G’(ω)/G, G’’(ω)/G
10 0
10 -1
10 -2
G’
G’’
(a)
10 -3
10 -2 10 -1 10 0 10 1 10 2 10 3
ωτ
(b)
Figure 1.2: a) The storage and loss modulus for a single Maxwell mode with relaxation
time τ. b) Linear rheology of a real polymer melt fitted with a spectrum of Maxwell
modes [Venerus (2000)].
For a material with a wide range of relaxation times a correspondingly wide range
of oscillation frequencies is needed to characterise the material fully. In practice, this
is achieved, by appealing to the empirical principle of time-temperature superposition.
This assumes that changing the experimental temperature is equivalent to shifting
the frequency of the experiment.
Thus an instrument’s limited range of frequency
measurements can be extended by producing results at a range of temperatures and
then shifting the data to produce a single temperature master curve. Materials which
obey this principle are said to be thermo-rheologically simple.

1.6. NON-LINEAR RHEOLOGY 9
1.5.2 Linear continuous shear
Equation 1.10 can also be used to model a constant rate forward deformation provided
that the deformation rate is small in comparison to the longest relaxation time of the
material. For a continuous shear deformation commencing at time t = 0 the model
predicts,
σ xy (t) =
∫ t
0
G(t − t ′ ) ˙γdt ′ . (1.16)
In these shear experiments the transient shear viscosity, η + (t) = σ xy (t)/ ˙γ, is often
plotted as a function of time. In this plot a Newtonian fluid would show a constant
response at all deformation rates (see figure 1.1). For a linear viscoelastic fluid, whose
constitutive equation is given by equation 1.8, the transient shear viscosity, η 0 (t), will
be independent of the applied shear rate. This behaviour is seen in polymeric fluids
at low shear rates (less than the reciprocal of the longest relaxation time) and also at
early times when the applied strain, ˙γt, is small. Under these conditions the material
is described as being in linear response. The master curve can be obtained from the
linear relaxation spectrum. If G(t) is taken to be the sum of independent Maxwell
modes (equation 1.15) then equation 1.16 gives
η 0 (t) = ∑ i
g i τ i [1 − exp(−t/τ i )] . (1.17)
At higher deformation rates the transient curves of polymeric fluids often deviate from
linear response at strains of order one. The form of the deviation can be used to classify
the material’s non-linear response.
A similar transient viscosity, based on the first normal stress difference, can be
defined for extensional flows η + E (t) = N 1/˙ɛ. Using equation 1.8 it can be shown that,
for a linear viscoelastic fluid in uniaxial extension, η + E (t) = 3η 0(t).
1.6 Non-linear rheology
Non-linear rheology refers to flows in which the strain rates and accumulated strains
are large. More specifically, the strain rate must be faster than some characteristic
time of the material, usually the terminal time. Strains must be of order one or larger
to observe non-linear effects. These flows are useful for a number of reasons. Nonlinear
measurements on polymer liquids often show very striking behaviour and can,
consequently, be more sensitive to molecular details than weaker deformations. For
example, the extensional rheology of a polydisperse system can be strongly dependent
on the high molecular weight tail of the distribution. A material’s response to different
deformation geometries is often qualitatively different under non-linear deformations.

10 CHAPTER 1. INTRODUCTION
This provides a good testing ground for any non-linear theory. For example polymer
liquids are often strain hardening in extension and thinning in shear. In figure 1.3 at
10 6
η Shear
, η Extension
(Pa-s)
10 5
10 4
10 3
3η 0
(t)
0.003
0.01
0.03
0.1
0.3
1.0
3.0
10.0
10 -1 10 0 10 1 10 2 10 3 10 4
t (s)
η 0 (t)
Figure 1.3: Non-linear shear and uniaxial extension of an LDPE melt 1810H showing
extension hardening (solid shapes) and shear thinning (open shapes) [Suneel et al.
(Submitted)].
small strains the extension viscosity is three times the shear viscosity. However, at high
rates the extension data rise above the low rate extension curve whereas the high rate
shear data fall below the corresponding limiting curve. Many industrially relevant flows
involve very large strains and strain rates. To model these flows non-linear rheology
must be understood.
1.6.1 A simple empirical non-linear model
A simple non-linear equation can be derived from the approach used in section 1.5 by
avoiding the linear approximation of the time derivative of finger tensor. Equation 1.6
can be integrated by parts to give
∫
d
t
dt σ dG(t − t ′ )
= −
≈
∞ dt ′ C −1 (t ′ )dt ′ . (1.18)
≈
which is known as the Lodge equation. Furthermore this equation can be differentiated
with respect to t to obtain a differential equation. For simplicity I also take the form
of the relaxation modulus to be G(t) = G exp(−t/τ)
d
dt σ − κ.σ − σ.κ T = − 1 ) (σ − GI . (1.19)
≈ ≈ ≈ ≈ ≈ τ ≈ ≈

1.7. ALTERNATIVE EXPERIMENTAL TECHNIQUES 11
This can be written more compactly as
∇
σ = − 1 ) (σ − GI . (1.20)
≈ τ ≈ ≈
Where the operator ∇ is known as an upper convected Maxwell derivative and the
general form of constitutive equation is called an upper convected Maxwell equation. In
the non-linear regime the choice of equation 1.5 as the starting point of the derivation is
arbitrary and other combinations of the finger tensor are equally permissible. Under this
empirical approach these choices can only be vindicated by comparison with observed
phenomena.
If the relaxation modulus is taken to be a sum over exponential Maxwell modes
(equation 1.15) then equation 1.20 can be solved for an independent set of non-linear
Maxwell modes to produce stress predictions.
1.7 Alternative experimental techniques
Although the measurement of mechanical stresses is the most common and arguably
the most industrially relevant measurement there are a range of additional experimental
techniques which can be used to provide information about polymer fluids under flow.
From an empirical point of view these measurements provide additional phenomena
that must be explained. However, they are particularly useful in the field of molecular
rheology since many of the experiments offer a more direct probe of the molecular
dynamics. Examples of these experiment include small angle neutron scattering (SANS)
[Müller et al. (1993), McLeish et al. (1999)], NMR [Cormier and Callaghan (2002)],
neutron spin echo [Wischnewski et al. (2002)] and dielectric relaxation [Watanabe et al.
(2002)]. The use of these measurements in the verification and development of molecular
models is a relatively new approach but they appear to provide new insight into the
behaviour of polymers under flow [McLeish (2002)].

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

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

16 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
balance on bead n is given by
ζ 0
( ∂Rn
∂t
− κ
≈
.R n
)
= − 3k BT
b 2 (2R n − R n+1 − R n−1 ) + f n (t). (2.8)
Here the viscous forces are assumed to dominate over inertial forces so the m ∂2 R
term
∂t 2
is dropped. The random force experienced by each bead due to collisions with the
surrounding is represented by the force f n . The beads at the chain ends are connected
to just one other bead and so experience a spring force from just one neighbour. To
ensure that equation 2.8 holds for the end bead, hypothetical beads at n = 0, N + 1
are included with zero extension. This leads to the following boundary conditions.
R 0 = R 1 R N = R N+1 . (2.9)
In principle equations 2.8 and 2.9 define a system that may be solved to give the
trajectories of each beads within the Rouse chain. However, in practice it is often more
convenient to take the continuous limit of this system (see section 2.2.5). In either case a
further derivation is required to convert information about the chain configuration into
a quantity which can be measured experimentally. I will demonstrate such a calculation
in section 2.2.6
2.2.5 Continuous limit
It is often more convenient to describe a chain contour in terms of a continuous variable
rather than a set of discrete points. When describing a rouse chain by a set of N beads
it is required that the system be independent of the number of beads used in the
solution. One way of achieving this is to take the limit of the number of beads tending
to infinity. This leads to a description of the chain contour in terms of a continuous
variable rather than a set of discrete points. Thus R n becomes R(n), terms involving
next neighbour differences become derivative with respect to the continuous variable
n and sums can be replaced by integrals (see table 2.1) The derivatives are obtained
Discrete
Continuous
R n → R(n)
∂R
R n − R n−1 →
∂n
R n+1 + R n−1 − 2R n →
∂ 2 R
∂n 2
δ nm → δ(n − m)
∑ m ′
n=m
Table 2.1: Summary of the transformation from a discrete system to a continuous
variable.
→
∫ m ′
m
dn

2.2. GAUSSIAN CHAINS 17
L
α
β
Figure 2.3: The derivation of a molecular expression for stress
through finite difference definitions of the derivative which become exact in the limit
of large N.
2.2.6 Molecular expression for stress
Equations 2.8 and 2.9 define a system that can be solved to find the molecular shape,
R(s). From this information a variety of measurable quantities can be deduced, including
the macroscopic stress tensor, σ. To compute σ for an ensemble of Gaussian
≈ ≈
chains the configuration of the chains must be known above some length-scale and
chain segments on length-scales below this scale must be locally at equilibrium. In this
calculation the cut off length-scale is taken to be the Kuhn step length, b, although
often in practice, the molecule is only disturbed from equilibrium on length-scales much
larger than this. This choice of length-scale corresponds to dividing the molecule into
N subchains.
An expression for the stress is obtained by considering a cube of length L, where
L is large enough to be much greater than the end to end vector of a subchain and is

18 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
small enough that all quantities can be considered to be homogeneous inside the cube.
This situation is illustrated in figure 2.3. A subchain inside such a cube has its end to
end vector denoted by r n where n labels the bead number. This subchain will cross a
plane with normal in the α direction with probability r nα /L (α and β denote Cartesian
components). If the chain carries a tension F n then the force in the β direction caused
by this chain is F nβ . Note that both the end to end vector and the tension depend
upon bead number. The total polymer contribution to stress is obtained by summing
over all subchains inside the cube. This is achieved by summing over all chains and all
bead numbers,
σ αβ = 1 L 3
∑
chains,n
r nα F nβ . (2.10)
If the cube contains enough chains the sum over chains can be replaced by an ensemble
average 〈...〉. This average is over all subchains with the same bead position within the
same course-graining volume. An average over bead positions is not taken. The cube
will contain cL 3 /N chains (c is the monomer density) thus the polymer contribution
to stress inside this cube is
σ αβ = c ∑
〈r nα F nβ 〉 . (2.11)
N
n
The tension, F n , is a known function of the local chain configuration (equation 2.7).
Taking the continuous limit allows the sum over beads to be replaced by an integral
along the chain contour and r n by the first derivative of the chain space-curve. This
leads to the final expression for the stress
σ αβ = c N
3k B T
b 2
∫ N
0
〈 ∂Rα
∂n
〉
∂R β
dn. (2.12)
∂n
2.2.7 Non-linear constitutive equation
A non-linear constitutive equation can be constructed from the Rouse model in the
following way [Larson (1988)]. Beginning with the continuous representation of equation
2.8
( )
∂R
ζ 0
∂t − κ .R = 3k BT
≈ b 2
∂ 2 R
+ f(n, t). (2.13)
∂n2 The quantity f(n, t) represents the local Brownian forces acting on an individual beads,
which are assumed to be random and isotropically distributed. Thus the forces act
equally in all directions and are uncorrelated in time and position along the chain.
The size of the force is fixed by insisting that all points on the chain are in thermal

2.2. GAUSSIAN CHAINS 19
equilibrium with the surroundings. This gives
〈f(n, t)〉 = 0
〈
f(n, t)f(n ′ , t) 〉 = 2ζ 0 k B T δ(n − n ′ )I
≈
.
(2.14)
In the continuous limit the boundary conditions of equation 2.9 become
∂R
∂n
⏐ = 0. (2.15)
n=0,N
It is convenient to take a Fourier series of equation 2.13 in order to obtain a system of
ODEs. The equation is diagonalized by the following transformation.
X p (t) = 1 N
∫ N
0
R(n, t) cos
( pπn
)
dn. (2.16)
N
which has the inverse transform
R(n, t) = X 0 + 2
∞∑ ( pπn
)
X p cos . (2.17)
N
p=1
Thus the spacecurve describing the chain contour is decomposed into a Fourier series
with vector amplitudes, X p . Applying this transformation to equation 2.13 gives
∂X p
∂t
= κ
≈
.X p − p2
τ R
X p + g p (t). (2.18)
where g p (t) is the p th Fourier component of the random force f(n, t) and the Rouse
time, τ R , is given by
τ R = ζ 0b 2 N 2
3π 2 k B T . (2.19)
The second moment of the action of Brownian forces on the Rouse modes is obtained
by
〈
gp (t)g q (t ′ ) 〉 = 1
ζ 2 0 N 2 ∫ N
0
∫ N
= k BT
ζ 0 N δ(t − t′ )δ pq I
≈
.
0
〈
f(n, t)f(n ′ , t ′ ) 〉 ( pπn
)
cos cos
N
In terms of the Rouse modes, the stress (equation 2.12) becomes
( qπn
′
N
)
dndn ′
(2.20)
σ = 3k BT c 2π 2 ∑
p 2
≈ Nb 2 〈X p X p 〉 . (2.21)
N

20 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
Thus to compute the stress an expression for the second moment of the Rouse mode
amplitude is required.
Note that cross correlations between the modes (p ≠ q) do
not contribute to the stress. In fact an independent set of ODEs for 〈X p X p 〉 can be
found and so the cross correlations can be ignored. The set of differential equations is
obtained from equation 2.18 by using an Ito-Stratonovich relation (see appendix 2.I for
a discussion of this result)
〈g p (t)X p (t)〉 = k BT
2ζ 0 N I ≈ . (2.22)
Taking the second moment of equation 2.18 and substituting equation 2.22 gives
∂ 〈X p X p 〉
∂t
= κ
≈
. 〈X p X p 〉 + 〈X p X p 〉 .κ
≈ T − 2p2
τ R
〈X p X p 〉 + k BT
Nζ 0
I
≈
. (2.23)
By writing σ
≈ p
= 2π2
Nb 2 p 2 〈X p X p 〉 the stress can expressed as a sum of mode contributions
σ
≈
= 3k BT c
N
∑
σ (2.24)
≈ p.
where each of these components obeys an independent upper convected Maxwell (UCM)
equation (see section 1.6.1 for an empirical argument which leads to a UCM model for
polymer fluids)
with relaxation time τ p = 1/2p −2 τ R .
p
∇
σ = −
(σ 2p2 − 1 )
≈ p τ R ≈ p 3 I . (2.25)
≈
Thus each stress component makes an equal
contribution to the stress but higher modes have much faster relaxation times (τ p ∼
p −2 ). This result justifies the assumption that small length-scales of the molecules are
locally at equilibrium as used in the derivation of an expression for stress.
It is worth noting that the above result can be derived without using the Ito-
Stratonovich relation (equation 2.22). This is achieved by direct integration of equation
2.18 over time and then taking the square of the resulting expression before averaging.
However, the Ito-Stratonovich relation is useful in that it allows the derivation of
differential, as opposed to integral, constitutive equations.
2.2.8 Validity of the Rouse model
As discussed above, some, apparently drastic, simplifications are made in deriving the
Rouse model. However, under some circumstances these approximations appear to be
valid. Although the neglect of hydrodynamic interactions causes the model to fail for
dilute solutions, experimental evidence suggests that these interactions are screened in
concentrated polymer fluids [Ferry (1980)]. Yet applying the model to the dynamics
of concentrated fluids seems to bring the phantom chain assumption into question.

2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 21
This problem can be averted by considering certain specific regimes of melt dynamics.
Where polymer chains are shorter than some critical mass or time-scales are shorter
than some critical time then the Rouse model appears to describe these systems well.
However in more general situations one must explicitly take into account the topological
interactions of neighbouring chains. The standard model when taking this approach
is outlined in section 2.3. Thus the surprising conclusion is that the simple model of
Gaussian chains forms a sound basis for modelling concentrated polymer fluids.
2.3 Doi-Edwards model of entangled polymers
In this section I will introduce the Doi and Edwards tube model [Doi and Edwards
(1978)]. This is the current standard approach to dealing with polymeric fluids in which
there is significant inter-chain overlap. I will then outline some recent refinements that
aim to either improve the model’s quantitative predictions or generalise the model to
different molecular architectures.
Measurements of the zero shear viscosity of linear polymer melts show a transition
in behaviour at some critical molecular weight, M c [Berry and Fox (1968), Colby
et al. (1987)]. Below this weight the viscosity scales linearly with molecular weight
as predicted by the Rouse model. This is part of the experimental verification of the
Rouse model for low molecular weight chains. Above the threshold the scaling increases
sharply to an exponent which is widely reported to be ≈ 3.4. The tube model, developed
by Doi, Edwards and de Gennes [de Gennes (1971), Doi and Edwards (1978)], aims
to describe polymer dynamics for chains with a mass in excess of M c . The observed
change in scaling behaviour is attributed to interactions of the surrounding chains influencing
the single chain dynamics. The critical chain mass is identified as the molecular
weight at which significant inter-chain overlap begins to occur. Specifically, the dominant
chain interaction is taken to be the topological constraint that one chain may not
pass through another.
Rather than attempting to solve the complicated many body problem of an entangled
fluid directly, the tube model adopts a mean field approach in order to reduce the
problem to that of a single chain. The topological interactions are modelled as a set of
constraints that confine the chain to a tube-like region. This tube allows the chain to
move freely along its own contour length, but lateral movement of the chain is severely
restricted (see figure 2.5). The molecule is able to relax its configuration by diffusing
back and forth along its own contour length. As the free ends of the chain emerge from
the tube, they are able to explore any direction without passing through another chain.
In this way the whole chain renews its configuration with a relaxation time controlled
by the time taken for the chain to escape the whole of its original tube.
The “primitive path” of the chain along the tube is defined to be the shortest path

22 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
Figure 2.4: Scaling of linear viscosity of a range of linear polymer melts against X w
which is proportional to molecular weight. The scaling switches from a slope of 1 to
the 3.4 “law” for entangled polymer melts. From Berry and Fox (1968).
from one chain end to the other that has the same topology as the actual chain, with
respect to the entanglement network. Figure 2.6 illustrates this idea and demonstrates
how the actual chain may contain a considerable amount of slack in comparison to the
primitive path.
In order to make these ideas quantitative some assumptions about the nature of
the entanglement network surrounding a chain must be made. The simplest of these is
to assume that the melt can be characterised by a single length-scale. This quantity is
known as the tube diameter, a, and interactions with surrounding chains are assumed
to act on length-scales greater than this length. The nature of the tube diameter and
entanglements themselves is not well understood theoretically, but see Everaers (1999)
for a recent summary. However, considerable progress can be made by treating a as
an empirical quantity. Formally, a is defined as the correlation length of the primitive
path. In equilibrium the primitive path is a Gaussian random walk of step length a
and has the same end to end vector as the original chain. This leads to the following
relationship which introduces the number of entanglement segments, Z,
Nb 2 = Za 2 . (2.26)

2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 23
a)
b)
Figure 2.5: The many body problem of an entangled melt (a) reduced to a single chain
problem by replacing the individual entanglements with a confining tube (b).
Figure 2.6: A chain in an entanglement network (narrow line) and its corresponding
primitive path (broad line).
By introducing the number of monomers per entanglement segment, N e = N/Z, the
following relation can be found
N e = a2
b 2 . (2.27)
Equivalently the molecular mass between entanglement segments, M e , is often quoted,
giving Z = N/N e = M/M e .
The tube diameter is taken to be independent of molecular mass and chain topology,
depending only on the chemistry of the molecule. If the tube model can be made to
agree simultaneously with a wide range of different data sets this is still a sufficiently
demanding test to suggest that the tube idea has captured the dominant physics of the
problem. At the moment there is no consensus on numerical values for a or a unique
method of determination. Attempts to produce a more unified quantitative theory are
a current focus of much theoretical work (see, for example, Pattamaprom et al. (2000)
or Likhtman and McLeish (2002)).

24 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
2.3.1 Linear rheology
The tube model can be used to provide predictions for the linear relaxation modulus
after a small step strain, G(t). This is achieved by assuming that shortly after a
step strain the polymer stress is due to chains trapped in oriented tube segments. A
tube segment’s orientation is completely renewed when a free end of the chain passes
through it. This relaxation process is illustrated in figure 2.7. Immediately after the
a)
b) c)
Figure 2.7: Relaxation of oriented tube segments by reptation after a step strain
deformation all of the chain is trapped in deformed tube (a). As the chain begins to
move one chain end passes through some tube segments and these sections of oriented
tube are lost (b). Since the new configurations taken by the emerging chain end are
chosen without any constraint on their direction, they carry no stress. As the chain
moves back and forth under diffusion tube segments are destroyed from both ends (c).
The rate of relaxation can be obtained by solving a first passage problem of a chain in
a tube. Doi and Edwards (1986) performed this calculation under the assumption that
the primitive path length is fixed at its equilibrium value. These assumptions reduce
the system to a one-dimensional first passage problem for the diffusion of the centre of
mass of the chain. The centre of mass diffusion constant, D c , is take from the Rouse
model
This gives the relaxation function as
G(t) = G 0
∑
D c = k BT
ζ 0 N . (2.28)
p odd
( )
8
p 2 π 2 exp − p2 t
. (2.29)
τ d
The time τ d is known as the reptation time and is related to the molecular parameters
by
τ d = ζ 0N 3 b 2
π 2 k B T
( a
2
b 2 )
= 3Zτ R . (2.30)

2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 25
The constant of proportionality, G 0 , relating the stress to the fraction of unrelaxed
tube is called the plateau modulus. The concept is generalised from calculation of the
modulus in a cross-linked network. The plateau modulus is related to the molecular
parameters by
G 0 = 4 ρRT
. (2.31)
5 M e
The experimental value of the plateau modulus is often used as a method of determining
M e , or equivalently the tube diameter. This can be problematic since the experimental
plateau in G ′ (ω) is only independent of chain length in the limit of very highly entangled
materials. For example see figure 2.8(b). The plateau can be seen by eye since data
are available on an exceptionally well entangled material (Z ≈ 44). However, it would
be less clear if the data for the higher molecular weight melts were not available. The
factor of 4/5 arises because the entanglements allows longitudinal motion of the chain
along the tube to relax stress. The proportion of stress relaxed in this way is 1/5 of
the total stress and in this sense entanglements are distinct from crosslinks since this
longitudinal motion is not possible in crosslinked systems. This factor has resulted in
wide spread confusion and contradiction. For an explanation and a proposed solution
to this see Likhtman and McLeish (2002). I will adopt their conventions in this thesis.
In particular the definition of the entanglement modulus G e = σ xy (τ e )/γ, the relaxation
modulus a t = τ e after a small step shear is safer. This is directly equivalent to the
cross linked version since no relaxation on length-scales longer than the tube diameter
will have occurred. In terms of the molecular parameters G e is given by
G e = ρRT
M e
. (2.32)
Equation 2.30 demonstrates the strong influence of entanglements on the chain
dynamics, in that it they increase the characteristic relaxation time by a factor of
3Z. The above result can also be derived directly by solving a microscopic Langevin
equation as in the derivation of the Rouse model (see section 2.4.1).
The relaxation spectrum derived above can be used with equation 1.14 to produce
predictions for the complex modulus of a linear polymer fluid under linear oscillatory
shear.
These model predictions, together with some experimental data on a range
of nearly monodisperse polystyrene melts are shown in figure 2.8.
The qualitative
similarities between the theory and data are very encouraging. Both sets show a plateau
region in G ′ , that increases in width with molecular weight. In both cases the plateau
height is independent of molecular weight. The experimental data show an upturn at
high frequencies that is not present in the theory. This can be rectified by taking into
account the free Rouse motion of the chain at length-scales which are shorter than the
tube diameter. Similar qualitative agreement can be seen for G ′′ (ω).

26 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
10 1
10 0
G’/G 0 , G’’/G 0
10 -1
10 -2
10 -3
G’(ω)
G’’(ω)
10 -4
(a)
10 -5
10 -2 10 -1 10 0 10 1 10 2
0 ωτ d
(b)
¢¡¤£¦¥¨§©
Figure 2.8: a) Dynamic modulus as calculated by the pure reptation model. b) Experimental
storage modulus for a range of narrow distribution polystyrenes. Z ranges
between ≈ 44 and ≈ 0.6 entanglements [Onogi et al. (1970)].
The model does, however, have some quantitative failings. The general shape of
the predicted G ′ and G ′′ curves do not stand up to a direct quantitative comparison,
particularly for G ′′ . In addition the model predicts the steady state zero shear viscosity
to be
η 0 = π2
12 G 0τ d ∼ M 3 . (2.33)
whereas experimentally the scaling exponent is closer to 3.4 for moderately entangled
melts. The solution to this discrepancy is discussed in the next section.
2.3.2 Contour length fluctuations
While the basic DE model discussed above provides a qualitative explanation for wide
range of observed phenomena, the prediction of an incorrect scaling exponent for viscosity
with molecular mass is an importunate problem.
The most widely favoured
explanation for this discrepancy is that the approximation that the length of primitive
path is constant is too crude. The length of primitive path of a chain comprising
of beads and springs ought to be continually fluctuating about its equilibrium length
under the influence of thermal fluctuations. These fluctuations of the chain ends will
accelerate the rate at which tube segments are visited by these ends; accelerating the
relaxation relative to the pure reptation model. The relative importance of contour
length fluctuations (CLF) depends on the chain length, with CLF being more significant
for shorter chains. This explains the change in the scaling of the viscosity since
the chain terminal time is affected by this process. Mathematically, this requires the
inclusion of the higher Rouse modes in the first passage problem instead of merely the
lowest mode as in the above calculation. This deceptively simple extension has been
stubbornly resistant to a rigorous solution even in the linear regime. Various authors
have presented implementations of this physical argument [Doi (1981), Doi (1983), des

2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 27
Cloizeaux (1990), Milner and McLeish (1998b)], however each of these approaches is
either an incomplete solution or is based around uncontrolled assumptions. Recently
Likhtman and McLeish (2002) obtained a solution via a combined stochastic and analytic
method. In this approach analytic arguments are used to derive expressions for
the relaxation but with unknown coefficients. Numerical values for these coefficients
are found by fitting the results to stochastic simulations of the first passage problem of
a full Rouse chain in a tube. If thermal constraint release (see section 2.4) is included as
well, this method produces good agreement with the viscosity measurements of Colby
et al. (1987), which cover a particularly wide range of molecular weights spanning either
side of M c . This theory has also been shown to produce excellent predictions for the
shape of G ′ (ω) and G ′′ (ω) of monodisperse linear polymer melts.
2.3.3 Non-linear rheology
The DE model can be generalised to non-linear flows. This has been achieved, with some
success, for a range of flow situations. For example continuous forward deformations
can be modelled by assuming the deformation rate to be small with respect to the chain
Rouse time. This still allows a window of non-linear rates since the reptation time and
the Rouse time are fairly well separated in an entangled melt (τ d /τ R = 3Z). In this
case the chain retraction can be assumed to be instantaneous. The model thus consists
of three processes: affine deformation of the tube due to convection, instantaneous
retraction along the tube and reptation, as described above. This model has been
solved using a variety of different pre-averaging approximations. The most widely used
of these is the independent alignment approximation (IAA) which disregards the fact
that, as the chain retractions, orientation information is passed from one tube segment
to the next. This considerably simplifies the solution of the model. The model predicts
a number of qualitative features that are in agreement with experimental data. These
were considerable improvements over preceding models. In extension the model predicts
extension thinning in steady state, in agreement with data on monodisperse samples.
In non-linear shear the model predicts a transient overshoot in shear stress, but not
in normal stress, and strong shear thinning at high shear rates. Qualitatively, all of
these features are observed experimentally. The issue of shear thinning is particularly
significant. The Rouse model, outlined in section 2.2.4, has a constant shear viscosity,
and hence the shear thinning phenomenon can be attributed to the influence of chain
entanglements. However, the DE model predicts too much shear thinning which itself
can be problematic (see section 2.4).
An even greater degree of success was achieved in the prediction of stress relaxation
after a large step shear deformation. The model assumes a perfectly affine deformation
followed by two relaxation processes. The chain relaxes its primitive path length back

28 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
to equilibrium through Rouse-like motion without any change in chain orientation. The
chain then returns to an isotropic state via reptation. The usual measured quantity in
this case is the non linear relaxation modulus, G(t, γ), defined as
G(t, γ) = 1 γ σ xy(t, γ). (2.34)
At large strains these processes produce the following shape of G(t, γ). For t τ R
there is a rapid relaxation to a finite plateau. This plateau region extends until t τ d
when reptation relaxes the remaining stress, see figure 2.9) These qualitative features
Figure 2.9: Theoretical predictions of relaxation after a step strain. [Reproduced from
Doi and Edwards (1986)]
are seen in experimental data (see for example Osaki et al. (1982)). Furthermore
the model produces accurate quantitative predictions for a very specific measurement,
the damping function, h(γ). Beyond some critical time experimental data for G(t, γ)
displays time strain separability so that the relaxation modulus may be factorized as
G(t, γ) = h(γ)G(t). (2.35)
where G(t) is the relaxation modulus in the limit of small strains. This factorisation is
predicted by the DE model and thus experimental and theoretical damping functions
can be compared. This comparison is spectacularly successful (see figure 2.10). The
theory and data are in near perfect agreement up to large strains even with variations
in the molecular weight and polymer concentration. The agreement is achieved despite
the complete absence of any fitting parameters. While this result is very encouraging
it should be remembered that the damping function is sensitive only to the height of
the plateau relative to its linear value. Thus this measurement side steps many of the
difficulties with the tube model. For example, the damping function is insensitive to the
shape of the terminal relaxation. Time-strain separability demands that theory curves

¡ ¢¤£
¥
2.3. DOI-EDWARDS MODEL OF ENTANGLED POLYMERS 29
¦¨§©
¦¨§©©
Figure 2.10: Comparison of predicted and measured damping functions after a large
step strain for different linear polystyrene solutions [Osaki et al. (1982)]. The open and
filled circles are data for different molecular weights and the tick directions indicate
variations in concentration.

30 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
must superimpose with each other at long times but not with the data. This explains
why the DE model is successful despite the omission of contour length fluctuations.
Also, since the damping function is based on a ratio of values it is insensitive to the
plateau modulus. Thus the need to pick a molecular weight independent value for G 0
or to understand its variation with concentration is bypassed. Finally, one should note
that the calculation using the independent alignment approximation is closer to the
data than the more accurate solution of the model, although the variation between
the two methods of solution is small. Nevertheless this data comparison is a strong
confirmation of concept of the chain retraction along the tube contour.
2.4 Chain stretch and constraint release
The tube theory of Doi and Edwards (1986) is remarkably successful in describing a
wide range of qualitative features of the rheology of entangled polymer melts, yet despite
ongoing progress a definitive theory for non-linear flows remains elusive. In particular
it has proven difficult to find consistency in the model parameters used to fit linear and
non-linear data when the assumptions of the Doi-Edwards approach suggest that this
ought to be possible. These problems are notoriously manifest in a qualitative failing
of the Doi-Edwards (DE) theory in steady state of shear. When steady state shear
stress, σ SS
xy, is plotted as a function of shear rate, ˙γ, the model predicts a shear stress
maximum occurring when the shear rate exceeds the inverse reptation time ( ˙γ 1/τ d ).
However, experimental data in this regime indicate that shear stress is a monotonically
increasing function of shear rate (see for example Bercea et al. (1993)). Worse still,
a consequence of the shear stress maximum is that the DE model predicts a striking
shear banding instability occurring in moderately non-linear flows. Such a feature is
not observed in experiments.
Two possible mechanisms to rectify this problem are chain stretch and constraint
release. Both processes were discussed by Doi and Edwards but omitted from their constitutive
model. Chain stretch refers to configurations in which the length of occupied
tube exceeds its equilibrium value. In the DE model chain orientation relaxes on the
time-scale of the reptation time, while chain retraction, which is unhindered by tube
constraints, occurs at a rate determined by the Rouse time. In entangled systems these
two times-scales are reasonably well separated, τ d /τ R = 3Z where Z is the number of
chain entanglements. Consequently, it is, in principle, possible to model moderately
non-linear flows, in which ˙γτ d 1 and ˙γτ R ≪ 1, by assuming retraction occurs instantaneously.
However, the effect of chain stretch becomes significant when ˙γτ R 1, a
regime which is accessible in well controlled experiments on entangled polymers, particularly
in shear. A refinement to the DE model known as the DEMG theory [Marrucci
and Grizzuti (1988), Pearson et al. (1991), Mead and Leal (1995)] adds stretch to the

2.4. CHAIN STRETCH AND CONSTRAINT RELEASE 31
basic DE model. The inclusion of stretch improves transient predictions in start-up of
shear in several ways. The DEMG model predicts transient overshoots in shear stress
and normal stress which grow in size with shear rate. In addition, the strain at peak
stress of these overshoots increases with shear rate. All of these features are observed
experimentally. The DEMG theory is less successful in steady shear. In many circumstances
the shear stress maximum remains. In fact, any approach which merely
adds chain stretch, relaxing via Rouse retraction, to the DE theory is doomed to suffer
a similar fate. The reasons for this are two-fold. In rapid shear flows the DE orientation
tensor predicts strong steady state chain alignment along the shear direction.
As a consequence, the highly aligned chains present a very slim profile to the velocity
gradient and so predicted steady state stretch values are modest even at high stretch
Weissenberg numbers (W s = ˙γτ R ). A more fundamental problem is that the separation
of orientation time, τ d , and stretch time, τ R , is fixed at 3Z. Hence if Z is set to a large
enough value the onset of chain stretch is delayed to shear rates well in excess of 1/τ d
and the bare DE behaviour, including the stress maximum, is recovered at shear rates
around 1/τ d . The degree of entanglement necessary to see this effect is within the range
of existing experiments.
A second possible solution is constraint release. This is an additional relaxation
mechanism which recognises that whenever a chain end passes through a tube segment
the constraint that was imposed by this chain on a neighbouring chain is lost. Hence
the neighbouring chain is free to explore a wider region via lateral motion (see figure
2.11). Constraint release is a self consistent closure of the mean field approximation of
Figure 2.11: Schematic representation of a constraint release event.
the tube model. In the linear regime constraint release events are caused by reptation
of the surrounding chains. This is known as reptative or thermal constraint release.
Since reptative constraint release occurs on the time-scale of the reptation time of the
whole chain and one event only relaxes a small part of the chain, Doi and Edwards

32 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
argued that it has a negligible effect on relaxation. However, Likhtman and McLeish
(2002) demonstrated that constraint release has significant effects near the terminal
time. Additionally, constraint release becomes increasingly important in the non-linear
regime. In a crucial insight, Marrucci (1996) demonstrated that in non-linear flows
chain retraction also contributes to the constraint release rate. The effect of chain
retraction can greatly increase the rate of constraint release. The release rate grows
with the convection rate, becoming of order of the shear rate at high rates. This process
is known as convective constraint release (CCR) since tube constraints are swept away
by the convection of the flow. The mechanism becomes significant as ˙γ approaches
1/τ d , precisely the rate as which the Doi Edwards model begins to fail.
There have been various attempts to incorporate CCR into a constitutive model
both without chain stretch[Ianniruberto and Marrucci (1996), Ianniruberto and Marrucci
(2000)], and more recently with chain stretch [Ianniruberto and Marrucci (2001),
Mead et al. (1998)]. However, all of these theories model the effect of CCR by directly
modifying the overall chain relaxation time. Effectively the relaxation time becomes
dependent on the molecular response to the deformation. While the arguments for this
modification are molecularly motivated, the connection between constraint release and
the global relaxation time is, in the end, heuristic. Additionally, the assumption that
CCR acts in a global manner destroys any molecular detail on length-scales of less than
the overall chain length. Thus only rudimentary predictions for quantities that are sensitive
to finer molecular structure such as the single chain structure factor can be made.
New experimental techniques, in addition to mechanical stress measurements, are proving
to be useful tests of molecular based theories [McLeish et al. (1999), McLeish (2002),
Wischnewski et al. (2002), Müller et al. (1993), Watanabe et al. (2002)]. A consideration
of the local influence of constraint release is essential in this context. The idea
of the tube itself experiencing constraint release being modelled as a Rouse object was
postulated by de Gennes (1975). It utilises the fact that the Rouse model generically
describes the global behaviour of the local jump model for a connected object. The
approach allows a description of constraint release down to the length-scale of the tube
diameter. Viovy et al. (1991) formulated these ideas to model the linear rheology of
bimodal blends. With a careful choice of blend composition Rouse tube motion can
be directly observed in experimental data in the linear regime [Rubinstein and Colby
(1988)]. Recently Milner et al. (2001) and Likhtman et al. (2000) derived a non-linear
constitutive model by treating CCR as local Rouse-like tube motion. Their model is
intended to cover only non-stretching flows in order to consider the CCR mechanism
in isolation. The approach successfully eliminates the shear stress maximum without
relying on chain stretch. In addition, since the model is able to make detailed predictions
for the single chain structure factor, it offers an explanation for the relatively

2.4. CHAIN STRETCH AND CONSTRAINT RELEASE 33
low anisotropy observed in sheared melts [Müller et al. (1993)] when compared to the
predictions of the DE theory.
2.4.1 The Milner McLeish and Likhtman model
In this section I present a brief review of the model of Milner McLeish and Likhtman
(MMcL) for convective constraint release. The model describes the dynamics of a
monodisperse melt of entangled, linear polymer molecules under a strong deformation.
The chain is confined to a tube of diameter a due to constraints formed by surrounding
chains and the aim of the model is to derive dynamic equations for the entire configuration
of a single chain down to the length-scale of the tube diameter. The configuration
is described by the space curve R(s, t) which denotes the position vector of tube segment
s at time t. The tube comprises of Z = M/M e segments and s spans the chain
length, running from 0..Z. From a knowledge of the chain shape various macroscopic
quantities can be deduced.
The model accounts for a range of sources of motion and each process has a corresponding
term in the Langevin equation which models the dynamics of the entire
chain. The simplest of these is convection due to the applied deformation. The deformation
is described by the velocity gradient tensor, κ
≈
, and all points on the chain
move affinely with the flow. Relaxation is then relative to this affine motion. Three
relaxation mechanisms act on each tube segment. The first of these is reptation which
is curvilinear diffusion of the entire chain along its own contour. This processes relaxes
stress since the chain ends escaping the tube are free to choose any new orientation.
The expression for reptation is taken directly from the original DE model. The second
process is CCR which is assumed to act at an equal rate at all points along the chain.
It is modelled by Rouse tube hops of length a and frequency ν. Finally, retraction acts
along the tube contour, holding the total length fixed at its equilibrium value. The
retraction rate is proportional to the distance of the segment from the chain centre and
the constant of proportionality, λ, is chosen at each instant to maintain the chain at
its equilibrium length. Figure 2.12 shows these processes schematically.
Collecting all terms together into a Langevin equation for a single chain gives
(
R(s, t + ∆t) = R(s + ∆ξ(t), t) + ∆t κ .R + 3ν
≈ 2
∂ 2 R
∂s 2
( ) )
Z ∂R
+ g(s, t) + λ 2 − s .
∂s
(2.36)
The terms in equation 2.36 represent reptation, convection, CCR and retraction, respectively.
Appendix 2.III contains a derivation of the term corresponding to Rouse
like motion due to constraint release. Equation 2.36 contains two noise terms: ∆ξ(t)
describes Brownian forces leading to motion along the tube (reptation) and g(s, t) describes
motion due to random constraint release events.
Both terms are mean zero

¡ ¡
¡ ¡
34 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
(c)
¡ ¡
(b)
(a)
Figure 2.12: Three relaxation mechanism available to an unbranched, entangled polymer
chain: reptation (a), constraint release (b) and retraction (c).
Gaussian random variables and have the following second moments.
〈
∆ξ(t)∆ξ(t ′ ) 〉 2
=
3π 2 δ(t − t ′ )
Zτ e
〈
gα (s, t)g β (s ′ , t ′ ) 〉 = νa 2 δ(s − s ′ )δ(t − t ′ )δ αβ .
(2.37)
The angular brackets denote averages over an ensemble of chains and the indices α and
β denote Cartesian components. The local time-scale for the model is set by τ e which is
the Rouse time of a single entanglement segment. The second moment of the reptative
noise is obtained by rewriting equation 2.28 in terms of τ e . It should be noted that this
noise term sets the diffusion rate in terms of tube segments and so is a factor of 1/a 2
lower than the rate in real space given equation 2.28. This leaves the constraint release
rate, ν, as the only remaining unknown quantity. It can be determined self-consistently
from the retraction rate, λ, via the equation
ν = c ν
(λ +
)
4
π 2 Z 3 . (2.38)
τ e
Equation 2.38 counts the contributions to constraint release from retraction and from
reptation respectively with the parameter c ν determining the number of retraction
events necessary to result in one tube hop. Milner et al. (2001) argue that entanglements
result from “the mutual, delocalized topological interaction of many structures” and so
several retraction events are required to produce a tube hop of length a. Consequently,
a value of c ν ≤ 1 is expected.
The stress and single chain structure factor both follow from a knowledge of the
chain configuration, R(s, t). The stress tensor, σ, is given by
σ αβ = c N
3k B T
a 2
∫ Z
0
〈 ∂Rα (s)
∂s
〉
∂R β (s)
ds. (2.39)
∂s

2.4. CHAIN STRETCH AND CONSTRAINT RELEASE 35
Here c/N is the polymer chain concentration.
obtained from
S(q) =
∫ Z ∫ Z
0
0
⎛
exp ⎝− ∑ α,β
q α q β
2
∫ s ′ ∫ s ′
s
s
〈 ∂Rα (s 1 )
∂s
The single chain structure factor is
⎞
〉
∂R β (s 2 )
∂s ′ ds 1 ds 2
⎠ dsds ′ . (2.40)
〈 〉
Equations 2.39 and 2.40 show that a knowledge of the function f αβ(s,s ′ ) = ∂Rα(s) ∂R β (s ′ )
∂s ∂s ′
is sufficient to evaluate both the stress and the single chain structure factor. By taking
suitable averages of equation 2.36 a deterministic PDE for f
≈
(s, s ′ ) can be obtained. In
deriving an expression for f αβ(s,s ′ ) the following closure approximation is necessary.
〈R α (s, t)R β (s ′ , t)λ(t)〉 ≈ λ(t)〈R α (s, t)R β (s ′ , t)〉. (2.41)
Milner et al. (2001) verified this approximation by direct stochastic simulation of equation
2.36 and found it to be valid whenever strong CCR is present. The resulting PDE
for f (s, s ′ ) is then solved by converting to a Fourier sine series.
≈
2.4.2 Comments on the MMcL model
The MMcL model is significant in that it demonstrated that a full Rouse-like treatment
of constraint release is sufficient to produce a monotonically increasing steady
state shear stress as a function of shear rate. This was achieved without needing to
include chain stretch, something which had not been managed previously with simpler
implementations of CCR. If c ν > 0.07 this theory predicts that all linear polymers
have no shear banding instability regardless of the number of entanglements. However,
a quantitative comparison of this theory with non-linear shear experiments is difficult.
In experimental studies, the large build up of normal stresses under shear limits
the range of shear rates. The usual resolution is to perform experiments on entangled
solutions which have lower absolute stresses in comparison with equivalent melts.
To achieve entangled solutions very high molecular weight polymers are used. The
maximum molecular weight which can be used is limited by the ability to synthesise
monodisperse samples in sufficient quantities. Thus the number of entanglements in
the measured systems is not huge. For examples of these studies see Menezes (1980),
Menezes and Graessley (1982) or Bercea et al. (1993). In this case the window of shear
rates which are simultaneously non-linear and non-stretching is very small. The effect
of CLF and thermal constraint release make the range smaller still. This indicates the
need to add chain stretch to the theory.
Another significant assumption of the model is that the system can be described
by a single constraint release time. This is valid for the window of flow rates to which
the MMcL theory pertains since the simulations of Milner et al. (2001) confirm that all

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

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.

40 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
I now present a brief outline of the drag-strain coupling modification to the pompom
equations suggested by Blackwell et al. (2000). The purpose of this modification is
to incorporate the physical effect of the branch points withdrawing in to the backbone
tube before the maximum stretch condition is reached. The effect of this modification
is to smooth out the discontinuity in the gradient of the stress which is predicted by
the original model when the maximum stretch is reached. This discontinuity is not
seen in experimental data for mono-disperse H-polymers [McLeish et al. (1999)]. The
effect of withdrawal of the branch-point into the backbone tube before the onset of
maximum stretch is modelled by coupling it to the tension in the backbone through
the backbone stretch, λ. This withdrawal shortens the effective length of the free arms
since some of the arm material is now inside the tube. The average time taken for a
free arm to retract up to the branch point is exponentially dependent on arm length,
which in turn determines how frequently the branch point can take a diffusive step.
Hence even a small degree of branch point withdrawal has a strong influence on the
stretch relaxation time. By allowing the branch point to move in a harmonic potential
the variation of stretch relaxation can be related to the value of the backbone stretch.
This results in a modification of the stretch relaxation time as shown in table 2.2. The
stretch relaxation time, which was constant in the original model, has a dependence on
the backbone stretch.
τ s → τ s e −ν∗ (λ−1) . (2.42)
Where ν ∗ is a constant which is inversely proportional to the co-efficient of the harmonic
potential in which the branch point moves. Hence as the stretch grows, the
branch points become increasingly withdrawn producing exponentially faster stretch
relaxation. Thus as the maximum stretch is approached the molecule is able to relax
its stretch faster and so the growth rate of λ is reduced. The constant ν ∗ is expected
to be inversely proportional to q since increased branching will increase the localisation
of the branch points. For the multi-mode approach (see section 2.5.3) Blackwell et al.
(2000) found that a material independent value of ν ∗ = 2/q was consistent with existing
data.
The pom-pom model has been tested against experimental data on model H polymers
[McLeish et al. (1999)]. This study included both linear and non-linear rheology
as well as SANS measurements under strong flow. The quantitative comparison is very
reasonable. However, the availability of data on model H and pom-pom polymers is
limited.
2.5.3 Randomly branched polymers
The pom-pom equations can be generalised from mono-disperse melts of pom-pom
molecules to model the rheology of industrial grade polymeric materials such as low

2.5. BRANCHED POLYMERS 41
density polyethylene (LDPE) using the multimode approach described by Inkson et al.
(1999). Commercial LDPE is a polydisperse blend of molecules with multiple irregularly
spaced long chain branches. Each section of this complex molecular architecture
has its own time-scales for orientation and stretch relaxation. Under the multimode
method these sections are modelled by a superposition of pom-pom molecules of differing
relaxation times and arm numbers. The stress contribution of these pom-pom
molecules is then summed to obtain the total stress,
σ
≈
=
n∑ n∑
σ = 3 g i λ 2
≈
i S (2.43)
i ≈ i.
i=1 i=1
The approximation is the decoupling of the modes of a connected molecule. There
ought to be interactions between the separate sections on the same molecule, however
these are neglected. The moduli, g i , and backbone orientation times, τ bi for a particular
melt can be determined from the linear viscoelastic behaviour of the material.
However, the values of τ s and q must be determined from a non-linear flow experiment,
usually uniaxial extension. In the multimode formulation of the pom-pom model
the direct computation of timescales from molecular structure is replaced by fitting to
experimental rheology. This set of variables {τ bi , g i , τ si , q i } can then by used to make
successful predictions for other simple flows, such as shear and planar extension, and for
complex geometries [Lee et al. (2001)]. The method is a generalisation of the practice
of fitting a spectrum of Maxwell modes to linear oscillatory shear data. In this case the
spectrum is extended to include non-linear parameters and the pom-pom model is used
as the underlying theory. It should be noted that the range of phenomena captured by
this approach is significantly larger than that of linear response.
2.5.4 Discussion of multimode pom-pom model
On first inspection the decoupling of different sections of a connected molecule appears
to be severe approximation. It is then, perhaps, surprising that the method works
so well. Blackwell et al. (2001) have investigated this decoupling by deriving a more
rigorous tube theory for symmetric molecules containing multiple layers of branching
including the appropriate coupling through the branch points. They compared their
predictions to those of the multimode pom-pom and managed to identify some discrepancies
which also appear in the comparison of the pom-pom model with real data.
Generally, the errors were not serious, though. It is likely that the errors associated
with the decoupling occur a large strains, particularly in steady state. In this case the
errors will be screened out by the viscous contribution from modes which are in linear
response. This mechanism also explains why the multi-mode pom-pom can be used to
model non-linear shear of branched polymer melts despite the omission of CCR (see

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.

44 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
N e segments of length b, leading to N e b 2 = a 2 . As long as N e is reasonably large the
fluctuations in the length a will be small. Under this rescaling, the probability density
of the end to end vector R can be shown to be (see Doi and Edwards (1986) equation
2.36)
Ψ(R) =
This is has the same form as equation 2.2 with
( ) 3 3/2 )
2πZa 2 exp
(− 3R2
2Za 2 . (2.48)
b → a
N → Z.
(2.49)
Hence all of the results presented in this chapter can be re-derived for a Gaussian chain
which is coarse grained on the tube diameter, a, by using the above result.
For example, the Rouse spring force term can be rewritten for a chain which is
course grained to step length a.
This is needed later on see appendix 2.III and in
chapter 4 where I derive a generalised version of the MMcL model. Each bead now
contains N e monomers and so the friction per bead becomes ζ 0 N e . Using the above
result equation 2.13 rescales to
∂R
ζ 0 N e
∂t = 3k BT
a 2
∂ 2 R
+ ... (2.50)
∂s2 where R continues to ( measure a real ) distance. Introducing the Rouse time of an
entanglement segment τ e = ζ 0b 2 Ne
2
3π 2 k B
and using the relation a 2 = N
T
e b 2 gives
∂R
∂t = 1 ∂ 2 R
+ ... (2.51)
πτ e ∂s2 Alternatively the same result can be obtained by the following method.
Equation
2.13 can be rewritten as
ζ 0
∂R
∂t = 3k BT
b 2 N 2 e
which is consistent with equation 2.51.
∂ 2 R
∂(n/N e ) 2 + ...
= 1
πτ e
∂ 2 R
∂s 2 + ... (2.52)
Appendix 2.III
Obstructed diffusion
In this appendix I will derive the Rouse-like Langevin equation for a Rouse chain in
a tube experiencing constraint release at a single rate.
The problem was solved by

¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
2.III. OBSTRUCTED DIFFUSION 45
Milner et al. (2001) but the derivation was not included in their paper. The rescaled
Rouse term (equation 2.51) is needed. Each point on the chain can be considered as
Brownian particle in a series of obstacles which disappear and re-appear with frequency
ν. The boxes are a distance a apart. It is assumed that after the removal of an obstacle
the particle has time to fully equilibrate before the obstacle reappears. The particle
experiences a force of F in the x direction at all points in space. Thus after a constraint
ν
F
x
a
Figure 2.15: Schematic showing a Brownian particle, subject to a spring force and
moving in an array of vanishing and re-appearing obstacles.
is removed the probability of accepting the hop can be found by solving the zero flux
Smoluchowski equation,
k B T ∂Ψ
∂x
− FΨ = 0 (2.53)
the solutions of which have a Boltzmann distribution
( ) Fx
Ψ(x) = A exp . (2.54)
k B T
When a constraint to the right of the particle is removed the particle is localised to the
region 0...2a, hence the normalised solution for the probability density is
where α =
right is
Ψ + (x) = e αx α
e 2αa − 1 . (2.55)
F
k B T
. Consequently, the probability that the particle accepts the hop to the
P + (a) =
=
∫ 2a
Ψ + (x)dx
a
(2.56)
1
1 + e −αa .

46 CHAPTER 2. INTRODUCTION TO MOLECULAR RHEOLOGY
Note that in the limit of a large potential F ≫ 1 the particle will always accept the
opportunity to hop to the right. Similarly, a hop to the left is evaluated by normalising
the Boltzmann distribution between −a...a and finding the probability of the particle
being between −a...0. This gives
P − (a) =
1
. (2.57)
1 + eαa From these probabilities the first and second moments of the particle displacement
distribution can be found.
〈x(t)〉 = νt [P + (a) − P − (a)]
[ e αa ]
− 1
= νta
e αa + 1
Taking αa to be small and expanding the exponentials to first order.
≈ νta
[ αa
]
2
= νta2
2k B T F
Using F = 3k BT
a 2
∂ 2 R x(s)
∂s 2
from equation 2.50 gives
〈x(t)〉 = 3ν 2
∂ 2 R x (s)
∂s 2 t (2.58)
The second moment follows similarly
〈
x 2 (t) 〉 = νta 2 [P + (a) + P − (a)]
= νta 2 .
(2.59)
These moments can be mapped to an effective Langevin equation
where
dx
dt = 1
ζ eff
F eff + g(t). (2.60)
〈
g(t)g(t ′ ) 〉 = 2k BT
ζ eff
δ(t − t ′ ). (2.61)
The unknown quantities ζ eff and F eff are chosen so that equation 2.60 has the same
distribution of x(t) as the obstructed system solved above. Taking a direct average of
equation 2.60 gives
〈x(t)〉 = F eff
ζ eff
t. (2.62)

2.III. OBSTRUCTED DIFFUSION 47
Comparing this with equation 2.58 gives
F eff
ζ eff
= 3ν 2
∂ 2 R x (s)
∂s 2 . (2.63)
An expression for 〈 x 2 (t) 〉 is obtained by squaring and averaging the time integral of
equation 2.60 in the absence of an external force F = 0
〈
x 2 (t) 〉 =
∫ t
∫ t
〈
g(t ′ )g(t ′′ ) 〉 dt ′ dt ′′
0
0
= 2k BT
ζ eff
t.
(2.64)
Comparing the above equation with equation 2.59 gives
ζ eff = 2k BT
νa 2 . (2.65)
and hence
〈
g(t)g(t ′ ) 〉 = νa 2 δ(t − t ′ ). (2.66)
Repeating this approach in the two remaining Cartesian direction, and assuming
that constraint release is not correlated between the perpendicular directions or with
any other point on the chain gives the final expression as
∂R(s)
∂t
= 3ν 2
∂ 2 R(s)
∂s 2 + g(s, t). (2.67)
with
〈
g(s, t)g(s ′ , t ′ ) 〉 = νa 2 I
≈
δ(t − t ′ )δ(s − s ′ ). (2.68)
which is the result used by Milner et al. (2001).

Chapter 3
The pom-pom model in
exponential shear.
3.1 Introduction
In this chapter I analyse the viscometric flow, exponential shear. In particular, I examine
exponential shear of branched polymer melts, using the pom-pom model of McLeish
and Larson (1998) (see section 2.5.2 for an introduction to this model). I compare the
flow with extensional flows and assess the practicality of using exponential shear as
an alternative characterising flow for the multimode pom-pom model. Much of the
method outlined in this chapter was published in the Journal of Rheology [Graham
et al. (2001)], taking advantage of existing data from the literature. Subsequently,
I retested the approach using some newly available experimental measurements by
Suneel [Suneel et al. (Submitted)] and was able to confirm my findings on the degree
of usefulness of exponential shear as a tool for characterising branched polymer melts.
Exponential shear flows of polymer melts are interesting flows because they share
properties in common with both planar extension and simple, constant rate, shear flows.
In particular embedded points separate exponentially in time, like extensional flows, yet
the flow direction and velocity gradient are perpendicular, as for all shear flows. Several
experimental studies of the rheological behaviour of highly branched polymer melts in
exponential shear, including comparisons to extensional and simple shear flows, have
been conducted [Zülle et al. (1987), Venerus (2000)]. The geometry of an exponential
shear flow is the same as simple shear but the shear rate grows exponentially with time.
Typically the shear rate evolves as
˙γ(t) = α(e αt + e −αt ), (3.1)
where α is some characteristic strain rate of the flow. Exponential shear has similar
48

3.1. INTRODUCTION 49
principle extension ratios to planar extension (see section 1.4.2). In both flows one
direction extends exponentially in time, a second is unchanged and, to conserve volume,
the final direction compresses exponentially. For example
λ 1 (t) = e αt ,
λ 2 (t) = 1,
λ 3 (t) = e −αt . (3.2)
The principle extension ratios are found by taking the square root of the eigenvalues
of the finger tensor, C −1 . Under some flow classification schemes this exponential
≈
growth of one of the principle stretch ratios classifies a flow as “strong” [Tanner and
Huilgol (1975)]. A flow is classified as weak if it evolves in any other way. Hence under
this and most other classification schemes extensional flow and exponential shear flows
are regarded as strong. Simple shear is deemed to be weak since its largest principle
stretch ratio grows linearly with time [Doshi and Dealy (1987)]. These schemes are
based around the idea that a strong flow is likely to stretch the polymer chains of a
melt. Venerus (2000) presents a more detailed summary of flow classification.
A flow classification scheme can be given credibility if it is found to be consistent
with rheological data. For example transient first normal stress growth coefficient data
for branched polymer melts consistently rise above the predictions of linear viscoelastic
theory at large strains. This is the so called strain hardening phenomenon. Conversely
transient shear stress growth coefficient data typically fall below linear viscoelastic predictions
at large strains and this is known as strain softening. In these cases the above
flow classification scheme is consistent, in that the strong flows strain harden and weak
flows soften. Dealy (1990) notes that there is some ambiguity in this use of linear viscoelastic
theory as a reference curve for strain hardening since the phenomenon usually
occurs at large strains where the theory is no longer valid. He suggests alternative
reference curves, such as finite linear viscoelastic theory (FLV) 1 , but observes that all
known flows thin relative to these curves. Continuing, he also points out further problems
in choosing a suitable reference curve for exponential shear, particularly for first
normal stress difference since linear viscoelastic theory predicts a zero value for this
quantity. However, by careful choice of material function, Venerus (2000) has shown
that exponential shear stress data fall below even linear viscoelastic theory and so has
classified it as a weak flow.
In this chapter I will solve the pom-pom equations as derived by McLeish and
Larson (1998) for exponential shear and compare the solutions to those for simple
shear and planar extension in section 3.2. The model will then be tested quantitatively
1 Otherwise known as first-order linear viscoelastic theory or the Lodge rubber-like model

50 CHAPTER 3. THE POM-POM MODEL IN EXPONENTIAL SHEAR.
against experimental data from Zülle et al. (1987), Venerus (2000) and Suneel et al.
(Submitted) using the multimode generalisation [Inkson et al. (1999)] in section 3.3.
3.2 Single mode pom-pom model
Zülle et al. (1987) make a distinction between two types of exponential shear: true
exponential shear as defined in equation 3.1 and nearly exponential shear. Nearly
exponential shear has a shear rate that grows as ˙γ(t) = αe αt . It differs from true
exponential shear by a decaying term of αe −αt and hence the two flows converge when
αt ≫ 1. Note that nearly exponential shear does not have a purely exponential growth
of the extension ratio, except in the limit of long times where the behaviour approaches
true exponential shear. Consequentially the two flows are qualitatively similar and the
data converge for large strains [Zülle et al. (1987)]. In this section the solutions to the
pom-pom equations for nearly exponential shear will be analysed. Since most of this
analysis will be in the t ≫ 1/α limit the results will also hold for true exponential shear.
The solutions to the pom-pom equations for simple shear and planar extension have
been previously studied [see McLeish and Larson (1998), Bishko et al. (1999), Inkson
et al. (1999), Blackwell et al. (2000)]; they will serve as comparisons to the solutions in
exponential shear.
3.2.1 Solutions to the orientation equation
The orientation equation in table 2.2 can be solved analytically for simple shear, nearly
and true exponential shear and planar extension. Throughout the chapter I will be
assume that, for shear flows, the flow is in the x-direction and that the velocity gradient
is in the y-direction. For planar extension the principle stretch will always occur in the
x direction with the z-direction neutral. Simple shear flow is usually described by the
transient stress growth co-efficient, which is defined as η + (t, ˙γ) = σ xy / ˙γ. There is some
ambiguity in the definition of an equivalent material function for exponential shear. In
previous studies a variety of material functions have been used, including dividing the
shear stress by the instantaneous shear rate, ˙γ(t). In this chapter I will simply divide
by the initial shear rate, ˙γ 0 . Note that for true exponential shear ˙γ 0 = 2α and for
nearly exponential shear ˙γ 0 = α
In shear flows the rate at which the flow stretches the backbone segments, κ : S,
≈ ≈
is equal to ˙γS xy . In addition, S xy is also the component of the orientation tensor that
appears in the shear stress. Figure 3.1 shows the solution to the orientation equation for
simple shear with a variety of backbone orientation times. For shear rates greater than
1/τ b , S xy rises to a maximum before approaching its steady state value. By maximising
lim t→∞ S xy (t) with respect to τ b the value of τ b which produces the largest steady state

3.2. SINGLE MODE POM-POM MODEL 51
γ . [sec -1 ]
0.25
0.2
1.225
0.7
3.0
5.0
0.25
affine
S xy
0.15
0.1
0.05
0
0 5 10 15 20
time [sec]
Figure 3.1: Evolution of S xy for simple shear shear, ˙γ = 1sec −1 . Affine deformation
corresponds to τ b = ∞ . The thin solid line corresponds to the value of τ b for which
the steady state value of S xy is maximised.
value of S xy can be shown to be
τ b(max) =
√
3/2
. (3.3)
˙γ
This is shown as the thin solid line in figure 3.1. The dashed curves in figure 3.1, which
correspond to values of τ b other than τ b(max) , all fall below the τ b(max) curve in steady
state. When τ b < τ b(max) the rapid reptation means that the flow is not fast enough,
on average, to orientate the backbones significantly before they relax. For τ b > τ b(max)
the backbones are orientated through the optimum angle but the flow then continues
to orientate them towards the x-axis before the slow orientation relaxation produces
a dynamic equilibrium.
The contribution of a single backbone to S xy is maximised
when it lies in the x-y plane, making an angle of 45 o with the x-axis. At this angle
the contribution to S xy is 1/2. Note that even in the affine case S xy never reaches this
value since, due to the isotropic initial distribution of backbones, all of the backbones
will never simultaneously point in the same direction.
For nearly exponential shear the full analytic solution for S xy is
(1 + 2 α τ b ) ( e αt − 1 )
S xy =
),
(6 ατ b + 2ατ b e 2αt + 2ατ b e − t
τ b − 4ατ b e (α− 1 )t − τ b + 2e t
τ b − 2e (α− 1 )t τ b + 9 +
3
ατ b
(3.4)
with α as defined in equation 3.1. Figure 3.2 shows plots of S xy for varying backbone
orientation times. The solutions for S xy separate into two distinct regimes. For 1/τ b < α

52 CHAPTER 3. THE POM-POM MODEL IN EXPONENTIAL SHEAR.
S xy
0.25
0.2
0.15
τ b
[sec]
3
1
0.3
0.1
0.03
0.01
0.1
0.05
0
0 2 4 6 8
time [sec]
Figure 3.2: Evolution of S xy for nearly exponential shear with varying τ b values. α =
1sec −1 .
the orientation behaves essentially as for an affine deformation since the exponentially
growing shear rate rapidly becomes very large compared to τ b . In this limit S xy is very
well approximated by the affine orientation solution
S xy (t) =
e α t − 1
4 + e 2 α t − 2 e α t . (3.5)
This expression is obtained by taking the τ b → ∞ limit of equation 3.4 or by solving
the orientation equation with the relaxation terms removed.
In the opposite limit where 1/τ b ≫ α the expression for S xy can be simplified to
S xy (t) =
ατ b e αt
2α 2 τ 2 b e2αt + 3 . (3.6)
In this limit the initial shear rate is too small to produce any appreciable orientation.
The backbone section must “wait” until the instantaneous shear rate is comparable to
the reciprocal of its orientation time. Eventually the shear rate will become sufficiently
large to orientate the backbone section and so S xy grows, peaks and then tends to zero
at long times. In this limit the solutions for different values of τ b and α are related by
S xy
( t
α 1
, τ b1 , α 1
)
= S xy
( t + ∆t
α 2
, τ b2 , α 2
)
where ∆t is a time shift factor given by
∀t. (3.7)
( )
τb1 α 1
∆t = ln . (3.8)
τ b2 α 2

3.2. SINGLE MODE POM-POM MODEL 53
At constant α, decreasing τ b shifts the S xy curve to the right but does not alter its
shape since at lower orientation times it takes longer for the shear rate to reach a
sufficient size to orientate the backbone. Decreasing α broadens the curve because the
time interval between beginning orientation to completely orientating the backbone is
dilated. For intermediate relaxation times, where α ∼ 1/τ b , the solution is complicated
by the discontinuity in shear rate at t=0. This is observed in the rapid rise in degree
of orientation at early times followed by a shallowing of the gradient as the effect of
reptation becomes significant seen most clearly in the τ b = 0.1sec curve. The main
difference in behaviour of S xy between simple and exponential shear is that in the
exponential case the rising shear rate will always become large enough to orientate the
backbone sections and to cause S xy to tend to zero at long times regardless of how
small the backbone orientation time is.
In planar extension with a velocity field v x = ˙ɛx, v y = −˙ɛy and v z = 0, The rate
at which the flow stretches the backbone segments is ˙ɛ(S xx − S yy ). Figure 3.3 shows
the solutions to the pom-pom equations for orientation difference, S xx − S yy , in planar
extension. For ˙ɛ > 1/2τ b the backbone sections become fully aligned along the direction
1
0.8
0.6
S xx - S yy
0.4
0.2
τ b
[sec]
0.03
0.1
0.3
0.4
1.0
3.0
0
0 2 4 6 8
time [sec]
Figure 3.3: Evolution of S xx − S yy for a planar extensional flow, ˙ɛ = 1sec −1 .
of extension so that S xx − S yy approaches 1 at large times. For ˙ɛ < 1/2τ b the strain
rate is not sufficiently fast to align the backbones fully before they relax. The plateau
value of S xx − S yy falls with decreasing orientation time. Rapidly reptating backbones
hardly orientate at all in slow flows.
Figure 3.4 shows the predicted behaviour of S xx − S yy in exponential shear. When
α > 1/τ b the deformation is essentially affine. The resulting curves have some similarities
with those of planar extension. As is the case with extensional flows the molecular
segments align parallel to the x-axis. For α < 1/τ b the initial shear rate is too small to

54 CHAPTER 3. THE POM-POM MODEL IN EXPONENTIAL SHEAR.
orientate the molecules before they relax. Eventually, due to the exponentially growing
shear rate, the shear rate does become large enough to align the molecules and so
S xx − S yy will always tend to 1 however small the orientation relaxation time. This is
in contrast to planar extension where the faster reptating modes never fully orientate
for small values of ˙ɛτ b .
1
0.8
0.6
S xx - S yy
0.4
0.2
τ b [sec]
3.0
1.0
0.3
0.1
0.03
0.01
0
0 2 4 6 8
time [sec]
Figure 3.4: Evolution of S xx − S yy for an exponential shear flow, α = 1sec −1 .
3.2.2 Solutions to the stretch equation
For all three flows the stretch equation in table 2.2 needs to be solved numerically. This
was done using a standard adaptive Runge-Kutta approach and the results are shown
below. In this section the drag-strain coupling correction was not used (ν ∗ = 0) since
it only acts to smooth out the onset of maximum stretch and so does not qualitatively
affect this analysis. Figure 3.5 shows the solution to the stretch equation in simple
shear. Note that when the shear rate is comparable or larger than the reciprocal of
the stretch relaxation time ( ˙γ > 1/τ s ) simple shear can cause a significant amount
of stretching but this always decays to a smaller steady state value. When ˙γ < 1/τ s
the stretch relaxes before any significant transient stretch can accumulate. Figure 3.6
shows the evolution of backbone stretch for an exponential shear flow. Unlike simple
shear this flow is able to produce sustained backbone stretching that can lead to some
modes reaching maximum stretch. Analysis of the asymptotes of the stretch equation
in simple and exponential shear leads to the condition for sustained stretch to occur.
The behaviour of the stretch equation is governed by the κ
≈
: S
≈
term which corresponds
to the effect on the molecular stretch of the branch points deforming affinely with the
flow. In shear flows κ
≈
: S
≈
= ˙γS xy . Whether the stretch grows or decays is determined

3.2. SINGLE MODE POM-POM MODEL 55
2
1.8
γ . [sec -1 ]
backbone stretch, λ
1.6
1.4
3
1
0.3
0.1
1.2
1
1 10 100
time [sec]
Figure 3.5: Evolution of backbone stretch for a simple shear flow. τ b = 3sec, τ s = 1sec
and q = 5.
by the size of this term relative to the stretch relaxation of the molecule. In simple
shear S xy tends a constant value of
S xy →
τ b ˙γ
3 + 2 ˙γ 2 τ 2 b
as t → ∞. (3.9)
Thus, in the limit τ b ˙γ ≫ 1, S xy
becomes,
→ 1/2 ˙γτ b for large t and so the stretch equation
d
dt λ = [ 1
2τ b
− 1 τ s
]
λ + 1 τ s
. (3.10)
Since the pom-pom model requires that τ b > τ s , the stretch will always tend to an
equilibrium value of
1
λ = , (3.11)
1 − τs
2τ b
[ ] 1
− 2τ
[Inkson et al. (1999)]. Any additional transient stretch decays as e
1 t b τs towards this
equilibrium value. In the simple shear case the equilibrium value of λ is independent of
˙γ in the ˙γ ≫ 1/τ b limit, hence sustained stretching will never be produced regardless
of the size of the shear rate.
In exponential shear S xy decays as
S xy → 2ατ b + 1
2ατ b
e −αt t → ∞. (3.12)
Note that this behaviour is only weakly dependent on the orientation relaxation time
since at long times the shear rate is large enough for the deformation to be essentially