Chapter 1 Introduction This thesis is concerned with the behaviour of polymers in flow. Both polymers in solutions and polymer melts will be discussed. The field of research that studies the flow behaviour of these kinds of liquids is called rheology. In fact rheology comprises the study of all so-called non-newtonian liquids. This description emphasises the existence of Newtonian liquids. The flow properties of Newtonian liquids are fully characterised by a constant viscosity. Examples are water and glycerine. Non-Newtonian differ from Newtonian liquids in that they have a (micro)structure that can be influenced by flow. In the case of polymeric liquids the conformation of the polymers can change when the fluid is deformed. In equilibrium a polymer is coiled. When a fluid element is deformed, the polymer will uncoil. When subsequently deformation is stopped, the polymer will tend to recoil. This recoiling gives rise to elastic effects. In shear flow the polymer will align more and more with the flow direction when increasing the shear rate. Because of this alignment the shear force will decrease. The viscosity becomes dependent on the flow rate. The way polymers effect the fluid properties is called visco-elasticity. When describing the flow of a visco-elastic fluid one has to solve balance equations. At least the equation of conservation of mass and the conservation of momentum have to be solved. In situations where there is also heat transport conservation of energy as well as extra, thermodynamic, equations have to be obeyed. In this thesis this case will not be considered. To be able to solve the balance equation for momentum, one has to know all the forces that are present in the fluid, and how one fluid element acts on its neighbouring elements. The forces that are transmitted from one fluid element to the next are characterised by a quantity called the stress tensor. Rheology is mainly concerned with computing this stress tensor as function of the deformation history of a fluid element. The way the deformation history influences the stress is via the micro-structure of the fluid. When a fluid element is deformed the micro-structure changes. The way forces are transported through a fluid element depends on the structure of, in our case, the polymers. If these are more stretched in a certain direction larger forces will be transported in this direction. This phenomenon is comparable with a rubber band that is pulled. The classical approach of rheological modelling is based on continuum mechanics. 1
2 CHAPTER 1. INTRODUCTION Here some measure of deformation is constructed. The stress is related to this measure of deformation. Sometimes it is even possible to write down a (differential) equation using only the stress tensor and the rate-of-deformation tensor. These kind of equations are called (closed-form) constitutive equations. The fact that the material has a microstructure is only used in a phenomenological way. The equations are constructed such that they show elasticity, shear thinning, normal stress differences etc.. To describe a specific material parameters have to be fitted with experiment. This macroscopic approach has not led to constitutive equations that are fully satisfactory. One might be able to pick the parameters of a constitutive equation such that a certain flow type, e.g. shear flow, is described well. This is, however, no guarantee that other flow types, such as elongational flow, are predicted correctly. The ultimate goal is to use constitutive equations to describe visco-elastic flow in complex geometries. In such flows it is important that several flow types, and also complicated deformation histories, are described well. The macroscopic approach does not seem to be able to create a tool that performs well in all situations. The last three decades the research effort is shifting from macro to micro-rheology. In micro-rheology the micro-structure of the material is described in some detail. It is an experimental observation that the exact chemical details of a macromolecule do not influence the rheological behaviour. What is important is the overall architecture (e.g. whether a polymer is linear or branched), and properties like persistence length and solvent quality. An important textbook in this field of research is that of Bird et al. [1]. In this book the framework used by most people working in polymer micro-rheology is set out. This framework is that of bead-spring (and beadrod) systems. The macromolecules are modelled as beads connected by springs. The beads interact with the flow field. The springs model large parts of macromolecules, comprising many monomer units. The thermodynamic tendency of these parts to recoil is modelled by the springs. For micro-rheological systems the evolution of a polymer is a competition between deformation caused by flow and thermal fluctuations. In the bead-spring formalism this thermal motion is modelled by a Brownian force. The mathematical framework used in Bird et al. is that of Fokker-Planck or diffusion-convection equations. The same systems can also be described using stochastic differential equations. This formulation is better suited for doing computer simulations then the Fokker-Planck approach. The simulation method is called Brownian dynamics. The standard text on this methodology applied to polymer rheology is the book by Öttinger [2]. A large part of this thesis is concerned with Brownian dynamics simulations of bead-rod chains. The behaviour of polymeric fluids can not be described by one generic model. Depending on matters as molecular architecture and polymer concentration other phenomena are more relevant. In this thesis only linear polymer chains will be considered. When going from very dilute to concentrated solutions (and melts) the importance and the nature of polymer-polymer interaction changes. In very dilute systems the polymers do not really influence each other. When concentration increases, the polymers will influence each other through hydrodynamic interaction. This means that the changes in
1.1. OUTLINE 3 the flow field, caused by interaction with one polymer, are felt by neighbouring chains. When concentration increases still further hydrodynamic interaction is screened. Because of the high concentration of polymeric material the perturbation in the flow field are hindered to propagate. In highly concentrated solutions and melts the main polymer-polymer interaction is of a topological nature. Polymers can not move through each other. Instead of describing the interaction of all neighbouring polymers Doi and Edwards introduced the tube picture [3]. The Doi-Edwards theory describes the deformation and further evolution of an imaginary tube surrounding each polymer molecule. This tube is formed by topological constraints caused by neighbouring polymers. The conformation of the chain inside the tube determines the stress it exerts on its surroundings. An important goal in rheology is to perform non-newtonian flow simulations. These simulations are already highly complicated when using closed-form constitutive equations. Using kinetic models (i.e. bead-spring models) adds an extra complication because this description is much more detailed. It therefore creates much higher memory and CPU demands. A method to couple Brownian dynamics simulations of the microstructure with macroscopic flow simulations was introduced by Laso and Öttinger. This methodology was much improved by Hulsen et al. [4] with the introduction of the Brownian configuration field method (see also [5]). Brownian dynamics simulations are not the most efficient way to simulate the equations that arise from the kinetic modelling of concentrated melts. The Doi-Edwards equation, and also improvements on this equation (such as the one given in chapter 6), is best expressed as a so-called integral constitutive equation. In these equations important macroscopic quantities are described as integrals over the deformation history. In the final chapter of this thesis we will describe a methodology to incorporate this class of equations into macroscopic flow simulations, namely the deformation fields methodology (see also [6, 7, 8]). 1.1 Outline This thesis treats a number of subjects within the field of micro-rheology. This field of research tries to make a connection between the micro-structure of materials and their flow properties. All chapters deal with micro-rheological aspects of polymeric liquids. The second common factor is that all chapters present numerical studies. Apart from these still quite general classifications, the different subjects treated in the thesis are not much related. One can make a division into three main subjects: chapter 2 to 5 deal with Brownian dynamics simulation of bead-rod chains, chapter 6 treats reptation theory and chapter 7 treats a method for performing macroscopic flow simulations. The relation between chapter 6 and chapter 7 is that the developed reptation theory of chapter 6 is very well suited for implementation into the novel deformation fields method that is treated in chapter 7. Even the four chapters on the simulation of bead-rod chains are not as intimately connected as one might expect. In the first two chapters the bead-rod chain will be
4 CHAPTER 1. INTRODUCTION barely mentioned. Here the bead-rod chain is the motivation behind developing the theory and the numerical methods. Aside from this, the material presented in these chapters is much more general and can be used in many other applications. The first chapter is an introductory chapter on stochastic differential equations. Stochastic differential equations are used to model thermal fluctuations. These are important on polymer length scales. Some basic concepts and details of the numerical implementation of this type of equations, i.e. Brownian dynamics, are treated. The motivation for writing this chapter is that the details of stochastic differential equations are not too widely known. Furthermore, chapter 3 presents a thorough analysis of the stochastic motion that is subjected to rigid constraints. A solid background in the field of stochastic differential is needed to be able to comprehend the material presented there. Besides this, a view on the use of stochastic differential equations is developed, which is continued in chapter 3 and chapter 4. Bead-rod chains are chains formed by beads connected by rigid rods. These rigid rods form rigid constraints on the stochastic differential equations that describe the motion of the beads. This is the direct motivation for developing the theory treated in chapter 3. However, the reader is warned that the treatment goes much beyond the level of understanding that is strictly needed for the development of a Brownian dynamics code for freely-jointed bead-rod chains. The chapter can be viewed as an attempt to push modelling by using stochastic differential equations, to its limits. Stochastic differential equations are equivalent to Fokker-Planck equations, which are partial differential equations. Stochastic differential equations, or Langevin equations, are much more intuitive, in the way that they describe the motion of individual particles. They are also much easier and cheaper to implement numerically. However, for modelling purposes they are often judged unreliable. The common procedure is to first derive Fokker-Planck equations and then derive the valid stochastic differential equation from this. It is shown that, by a careful and precise treatment, stochastic differential equations can be used very well for modelling purposes, even in the difficult case where rigid constraints are present. The benefits are, firstly, that the derivation remains intuitive and physical. Secondly, the final equations are much easier to implement numerically, than the (fully equivalent) stochastic equations that are obtained when one starts from the Fokker-Planck equation. As a result of the detailed analyses this chapter is of a highly mathematical nature. Chapter 4 deals with development of a simulation algorithm for so-called freelydraining freely-jointed bead-rod chains. Using the general theory of chapter 3, also some aspects of the more general case of bead-rod chains with hydrodynamic interaction are discussed. In this chapter it is made clear that the theory based on the modelling approach using stochastic differential equations only, has large benefits for developing algorithms. This becomes especially clear by the treatment of the computation of polymer stresses. In an appendix some other algorithms developed in literature are discussed. The comparison is very favourable for the algorithm developed here. In chapter 5 simulation results are presented. The developed algorithm is used as a tool. The preceding chapters are not needed for reading this chapter. The chapter
1.1. OUTLINE 5 presents a detailed study of the conformational behaviour of an ensemble of bead-rod chains in an elongational flow. Three basic conformations, with their own particular dynamics, are studied. These conformations are the coiled chain, the stretched chain and the kinked chain. In strong uniaxial elongational flow the chain is squeezed into a one-dimensional structure. The dynamics of this structure is called kink dynamics. A semi-analytical theory is presented that describes this dynamics. The predictions of the theory compare well with the simulation results. Furthermore, it is shown that the fingerprints of the dynamics is found in experiments. How to introduce the kink dynamics mechanism into a coarse grained description will be discussed at the end of the chapter. Chapters 6 and 7 deal with melts. The basic concepts are very different from those used in the modelling of dilute polymeric liquids. Here integral constitutive equations are used, instead of stochastic differential equations. In chapter 6 we develop a constitutive equation for monodisperse linear melts. This theory is an extension of the Doi-Edwards reptation theory. The Doi-Edwards constitutive equation, for the evolution of the tube formed by surrounding polymers, contains many approximations. Especially the fact that connections between consecutive tube segments are neglected is problematic. We introduce a new approach that is aimed at repairing this shortcoming. Besides the treatment of connectivity also other, often discarded, phenomena such as so-called contour length fluctuations and chain stretch are included in the description. The most important feature of the approach presented is that it results in a constitutive equation that can still be incorporated into macroscopic flow simulations. During the development of the equation we found that the reptation theory has still some fundamental problems. Some of these are discussed in the appendices. Of all the material presented in this thesis, the content of the last chapter will likely have the most (short term) impact on the field of rheology. It presents a novel method to incorporate integral constitutive equations into macroscopic flow simulations. The basic quantities are Finger tensor fields which characterise deformation with respect to some time in the past. In many theories, such as reptation theories and network theories, these Finger tensor fields are the main quantities needed to perform stress calculations. The deformation fields method contains an efficient discretisation scheme for these Finger tensor fields. The method itself is described and many numerical aspects of the approach are treated. Benchmarking problems are solved numerically using the new method, and the results are checked against literature. Finally it is shown how the method can be easily generalised for the simulation of more advanced reptation theories.
6 CHAPTER 1. INTRODUCTION
Bibliography [1] R.B. Bird, C.F. Curtiss, R.C. Armstrong, and O. Hassager. Dynamics of Polymer Liquids. Vol. 2. Kinetic Theory, John Wiley, New York, 2 edition, 1987. [2] H.C. Öttinger. Stochastic Processes in Polymeric Fluids. Springer Verlag, Berlin, 1996. [3] M. Doi and S.F. Edwards. The theory of polymer dynamics. International series of monographs on physics, no. 73. Clarendon, Oxford, 1986. [4] M.A. Hulsen, A.P.G. van Heel, and B.H.A.A. van den Brule. Simulation of viscoelastic flows using brownian configuration fields. J. Non-Newtonian Fluid Mech., 70(1-2):79 101, 1997. [5] A.P.G. van Heel. Simulation of viscoelastic fluids. From microscopic models to macroscopic complex flows. PhD thesis, Technical University of Technology Delft, 2000. [6] E.A.J.F. Peters, M.A. Hulsen, and B.H.A.A van den Brule. Instationary eulerian viscoelastic flow simulations using time separable Rivlin-Sawyers constitutive equations. J. Non-Newtonian Fluid Mech., 89(1-2):209 228, 2000. [7] A.P.G. van Heel, M.A. Hulsen, and B.H.A.A. van den Brule. Simulation of the Doi-Edwards model in complex flow. J. Rheology, 43(5):1239 1260, 1999. [8] E.A.J.F. Peters, A.P.G. van Heel, M.A. Husen, and B.H.A.A. van den Brule. Generalisation of the deformation field method to simulate advanced reptation models in complex flow. J. Rheology, 44(4):811 829, 2000. 7
8 BIBLIOGRAPHY