The University of Leeds

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1 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.

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3 Contents Abstract Acknowledgements vi vii 1 Introduction Overview The stress tensor Viscoelasticity Deformation kinematics Volume conserving flows Some simple flows Flow in complex geometries Linear rheology Linear oscillatory shear Linear continuous shear Non-linear rheology A simple empirical non-linear model Alternative experimental techniques Introduction to molecular rheology Overview Gaussian chains Random walk model Justification of the model Bead spring model Rouse dynamics Continuous limit Molecular expression for stress Non-linear constitutive equation Validity of the Rouse model i

4 ii CONTENTS 2.3 Doi-Edwards model of entangled polymers Linear rheology Contour length fluctuations Non-linear rheology Chain stretch and constraint release The Milner McLeish and Likhtman model Comments on the MMcL model Branched polymers Star Polymers H polymers and the pom-pom model Randomly branched polymers Discussion of multimode pom-pom model Appendix 2.I The Ito-Stratonovich relation Appendix 2.II Rescaling a Gaussian walk Appendix 2.III Obstructed diffusion The pom-pom model in exponential shear Introduction Single mode pom-pom model Solutions to the orientation equation Solutions to the stretch equation Behaviour of shear stress in exponential shear The multimode method applied to exponential shear Predicting exponential shear data using non-linear spectra from extensional rheology Measuring the non-linear parameters from exponential shear A verification of the method for a different melt Discussion Theory of CCR and chain stretch Introduction Tube model for linear polymers with CCR and stretch Rouse retraction term Tube diameter under deformation CCR stretch relaxation Suppression of reptation due to stretch Langevin equation Equation for the tangent correlation function Number of entanglements

5 CONTENTS iii Constraint release rate Contour length fluctuations Thermal constraint release from contour length fluctuations Rouse motion on sub-tube diameter length scales Summary of model equations Real space solution Finite difference solution of CLF term Results Steady state in shear Transient start-up of simple shear Single chain structure factor Comparison with experimental data Determination of model parameters from linear rheology Parameter free comparison with non-linear data Improvement of high rate predictions Conclusions Appendix 4.I Derivation of stretch-ccr renormalisation term Appendix 4.II Modified CLF term for a stretched chain Appendix 4.III Fourier space solution Bimodal blends of linear polymer melts Introduction Existing data and theory Self-dilute high molecular weight additive Generalisation of stretching CCR theory to self dilute case Uniaxial extension of bimodal blends Non-linear shear of bimodal blends Discussion and future directions Conclusions Randomly branched polymers Monodisperse linear polymers Future work

6 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 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)] 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)] Sketch of an N-step freely jointed random walk In a Gaussian random walk monomers which are well separated along the chain may come into close contact The derivation of a molecular expression for stress 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) 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) A chain in an entanglement network (narrow line) and its corresponding primitive path (broad line) Relaxation of oriented tube segments by reptation after a step strain 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)] Theoretical predictions of relaxation after a step strain. [Reproduced from Doi and Edwards (1986)] iv

7 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 Schematic representation of a constraint release event Three relaxation mechanism available to an unbranched, entangled polymer chain: reptation (a), constraint release (b) and retraction (c) A three armed pom-pom molecule (q=3) Sketch of a random walk rescaled from N steps of length b to Z steps of length a Schematic showing a Brownian particle, subject to a spring force and moving in an array of vanishing and re-appearing obstacles 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 Evolution of S xy for nearly exponential shear with varying τ b values. α = 1sec Evolution of S xx S yy for a planar extensional flow, ɛ = 1sec Evolution of S xx S yy for an exponential shear flow, α = 1sec Evolution of backbone stretch for a simple shear flow. τ b = 3sec, τ s = 1sec and q = Evolution of backbone stretch for an exponential shear flow, α = 1sec 1, τ b /τ s = Evolution of backbone stretch for a planar extensional flow. τ b = 3sec, τ s = 1sec and q= Evolution of shear stress in an exponential shear flow, α = 3sec 1, τ b = 3sec, τ s = 1sec and G 0 φ 2 b = 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) Multimode pom-pom free parameter fit to planar extension data from Hachmann (1997) 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

8 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 Free parameter fit of non-linear parameters of melt 1 using only nearly exponential shear data collected by Zülle (1987) Comparison of pom-pom predictions using spec II with uniaxial extension data for melt 1 from Meissner (1972) The 8 modes of melt 1 (see table 3.1) in nearly exponential shear for α = 0.01sec The 8 modes of melt 1 (see table 3.1) in uniaxial extension for ɛ = 0.01sec 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) 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 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 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 The effect of CCR on an unstretched segment (a) and a stretched segment (b) Mechanism by which CCR relaxes chain stretch Theory predictions of steady state shear stress as a function of shear rate (c ν = 0.1) 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 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

9 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 Comparison with Menezes and Graessley (1982) PBB shear viscosity (a) and first normal stress difference (b) data using parameters obtained only from linear rheology 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 = 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 = 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 = Steady state shear stress and first normal stress difference of PS/TCP [Hua et al. (1999)] compared with model prediction for R s = Derivation of CCR term for a stretched chain 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 Uniaxial extension of a set of bimodal blends [Hepperle (2001)] at o C compared with model predictions (R s = 2.0) 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 sec 1. Theory curves have shear rates in the range γτ long R =

10 List of Tables 1.1 The tensorial description of some simple flows Summary of the transformation from a discrete system to a continuous variable The pom-pom constitutive equation- differential approximation Non-linear spectrum of melt 1 fitted to extensional rheology, from Inkson et al. (1999) Non-linear spectrum of melt 1810H obtained by fitting to planar extension data from Hachmann (1997) 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) 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) 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 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 viii

11 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

12 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

13 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.

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15 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 = repeat units can be synthesised and polymer of length N = 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

16 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.

17 1.4. DEFORMATION KINEMATICS Newtonian Viscosity 10 4 η 0 (t) σ xy / γ. [Pa-sec] 10 3 Hooke s Law 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

18 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 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 = Some simple flows Flow κ E Shear Uniaxial extension Planar extension 0 γ ɛ ɛ/ ɛ/2 ɛ ɛ γ e ɛ e ɛ/ e ɛ/2 e ɛ e ɛ 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

19 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 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

20 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.

21 1.5. LINEAR RHEOLOGY 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)

22 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 G (ω)/g, G (ω)/g G G (a) 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.

23 1.6. NON-LINEAR RHEOLOGY 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.

24 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) η 0 (t) 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 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

25 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)].

26 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 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

27 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) = 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

28 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)) 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)

29 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 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 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