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Polymers in extensional flow

A comparison of microscopic and macroscopic

simulations with birefringence experiments

Polymers in extensional flow

A comparison of microscopic and macroscopic

simulations with birefringence experiments

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.F. Wakker,voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 9 oktober 2001 om 16.00 uur

door

Joannes Leonardus Jurrian ZIJLwerktuigkundig ingenieur,

geboren te Zaanstad

ii

Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. B.H.A.A. van den Brule

Samenstelling promotiecommissie:

Rector Magnificus, voorzitterProf. dr. ir. B.H.A.A. van den Brule, Technische Universiteit Delft, promotorProf. dr. ir. F.P.T. Baaijens, Technische Universiteit EindhovenProf. dr. ir. J.J.M. Braat, Technische Universiteit DelftProf. dr. ir. J.J. Elmendorp, Technische Universiteit DelftDr. ir. M.A. Hulsen, Technische Universiteit DelftProf. dr. J. Mellema, Universiteit TwenteProf. dr. ir. F.T.M. Nieuwstadt, Technische Universiteit Delft

Copyright c© 2001 by J.L.J. ZijlAll rights reserved.

ISBN 90-6464-337-7

Printed by Ponsen & Looijen, The Netherlands.

Aan mijn Ouders

Contents

Summary ix

Samenvatting xi

1 Introduction 11.1 Fluid dynamics in daily life . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Modeling polymer flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Experiments in polymer flow . . . . . . . . . . . . . . . . . . . . . . . 41.4 Objective and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Experimental methods and set-up 92.1 Flow Induced Birefringence . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Fundamentals of optics . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Relationship between optical properties and stress . . . . . . . 14

2.2 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Mechanics of the flow cell . . . . . . . . . . . . . . . . . . . . . 182.2.2 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Experimental aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Alignment procedure . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2 Tuning the electronics . . . . . . . . . . . . . . . . . . . . . . . 262.3.3 Adjusting the PEM . . . . . . . . . . . . . . . . . . . . . . . . 272.3.4 Correcting the influence of edge effects . . . . . . . . . . . . . . 28

3 Fluid characterization 333.1 Sample preparation of a PIB solution . . . . . . . . . . . . . . . . . . . 333.2 Experiments in shear with the PIB solution . . . . . . . . . . . . . . . 34

3.2.1 Dynamic measurements . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Steady shear measurements . . . . . . . . . . . . . . . . . . . . 353.2.3 Start-up in shear flow . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 Measurements of the normal stress difference N1 . . . . . . . . 36

3.3 Experiments with a PS solution . . . . . . . . . . . . . . . . . . . . . . 38

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3.4 Viscoelastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.1 Giesekus Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.2 FENE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Determination of the constitutive parameters . . . . . . . . . . . . . . 433.5.1 Parameters for the multi-mode Giesekus model . . . . . . . . . 433.5.2 Parameters for the FENE model . . . . . . . . . . . . . . . . . 45

3.6 Determination of the stress optical constant . . . . . . . . . . . . . . . 463.7 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Calculations in a four-roll mill geometry 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Geometry and dimensionless numbers . . . . . . . . . . . . . . 544.2 3D Inelastic steady-state flow calculations . . . . . . . . . . . . . . . . 56

4.2.1 Input parameters for 3D flow calculations . . . . . . . . . . . . 584.2.2 Results of 3D flow calculations . . . . . . . . . . . . . . . . . . 59

4.3 Parameter study of 2D viscoelastic calculations . . . . . . . . . . . . . 614.3.1 Input parameters using the 1-mode Giesekus model . . . . . . 624.3.2 Influence of inertia . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.3 Influence of relaxation time . . . . . . . . . . . . . . . . . . . . 634.3.4 Influence of the parameter α . . . . . . . . . . . . . . . . . . . 644.3.5 Oscillating viscoelastic flow . . . . . . . . . . . . . . . . . . . . 66

4.4 Comparison of the macroscopic and microscopic models . . . . . . . . 684.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Experimental Results 835.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Stationary measurements . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2.1 Measurements in the stagnation point . . . . . . . . . . . . . . 845.2.2 A comparison between PS and PIB in the central stagnation

point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2.3 Calculations and measurements on the axes of symmetry . . . 87

5.3 Time-dependent measurements . . . . . . . . . . . . . . . . . . . . . . 915.3.1 Start-stop experiments . . . . . . . . . . . . . . . . . . . . . . . 915.3.2 Oscillatory experiments . . . . . . . . . . . . . . . . . . . . . . 915.3.3 The orientation angle change of the polymers . . . . . . . . . . 95

5.4 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Conclusions and recommendations 1016.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1.1 Conclusions of the calculations . . . . . . . . . . . . . . . . . . 1016.1.2 Conclusions of the experiments . . . . . . . . . . . . . . . . . . 102

6.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . 103

List of symbols 105

vii

Nawoord 109

Curriculum Vitae 111

Summary

This thesis is dealing about the flow behavior of polymers. This area of the fluiddynamics is called rheology. Rheology is the study of the flow of special kinds ofmaterials that do not follow the simple constitutive law, known as a Newtonian fluid.Most of these particular materials, have both elastic and viscous properties. The flowbehavior of polymer solutions and melts play an important role in the manufacturingand application of many consumer products.

The simulation of polymer flow is possible in to different ways: on a macroscopic leveland on a microscopic level. In the traditional macroscopic way of simulating, thepolymer fluid is considered as a continuum in the sense that no attempt is made tomodel the microscopic character of the polymers as individual particles or to simulatethem at a microscopic level. In the macroscopic approach one is only interested inthe effective bulk properties of the liquid. The state of the art is here to postulate acertain solution in which adjustable constants are related to the effective bulk prop-erties. A disadvantage of the macroscopic calculation method is that the solution isdependent of the flow geometry. This means that an accurate prediction of the stressin advance in a new geometry is often difficult to make.

In contrast to the macroscopic approach, it is the aim of the micro-rheology to findpolymeric stresses from a model for the micro-structure of the fluid. In the micro-scopic simulations, the bulk flow behavior should be calculated as the net effect ofthe dynamics of a large number of individual polymer molecules which are modeledas two beads connected to a non-linear spring (dumbbell). On this dumbbell thereare acting different forces: Brownian forces, intramolecular forces, and hydrodynamicforces. All these forces contribute to the stretch and the orientation of a single dumb-bell. Calculating the stress in a polymeric liquid, one should take the average overmany dumbbells which are suspended in the flow. The advantage of this microscopiccalculation method is that the modeling occurs at a more fundamental level. Whenthe essential dynamics of the real polymers is captured, then the micro-rheologicalapproach should be able to predict the stress independently of the flow geometry.

In this thesis, the two ways of modeling are compared by birefringence experiments.Therefore we have built a four-roll mill geometry in which we can create a station-ary and time-dependent plane elongational flow. For the liquid we used a 2% (w/w)

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long chainlenght polyisobutylene(PIB)/decalin solution. This liquid has been char-acterized in shear flow by means of a rheometer. The parameters obtained in thecharacterization are used as input parameters for the calculations. By using the flowinduced birefringence (FIB) properties of polymeric solutions we were able to calcu-late the first normal stress difference N1 and the shear stress τxy. For this we usedthe stress optic rule (SOR) which gives the relationship between the birefringence ∆nand the stress in a polymeric liquid.

A direct comparison between experiments and calculations was difficult because thecalculations could only be performed in two dimensions. We have developed a correc-tion method for correcting 3D flow data into 2D flow by using an adjustable bottom inour four-roll mill geometry. A direct comparison between experiments and the macro-scopic Giesekus model results in a very good agreement. The microscopic calculationsusing the FENE model, were difficult to compare with the experiments because theDeborah number in the experiments becomes too high for the microscopic calcula-tions. Therefore we compared the macroscopic Giesekus model and the microscopicFENE model at a low Deborah number. The differences in the stresses between themodels were quite big. This was mainly caused by the total different behavior ofthe elongational viscosity of the both models while the shear viscosity of the bothmodels was the same. This means that for accurate calculations, the liquid should becharacterized in shear flow and elongational flow.

Next to this we did also a study of the influence of the chain length of the polymermolecule on the birefringence in a stationary and time-dependent extensional flow.Therefore we used a 10 % (w/w) short chainlength polystyrene (PS)/decalin solutionand a PIB/decalin solution. Because of the chain length of the PS was twenty timessmaller than the chain length of PIB we could observe a saturation of the birefrin-gence ∆n of the PS solution. In the time-dependent experiments we saw a differencein re-orientation process of the polymer segments when a long chain polymer is com-pared with a short chain polymer.

Jurrian Zijl

Samenvatting

Dit proefschrift gaat over de reologie van polymeer oplossingen. Reologie is de weten-schap die de zogenaamde niet-newtonse vloeistofen bestudeert. Deze vloeistoffenhebben doorgaans totaal andere eigenschappen dan normale vloeistoffen zoals wa-ter. Zo kunnen ze viscoelastisch gedrag en/of afschuifverdunnend gedrag vertonen.Karakteristiek voor viscoelastische materialen is dat hun stromingsgedrag bepaaldwordt door de vervormings geschiedenis die ze ondergaan hebben. Viscoelastischestoffen, waarvan polymeer oplossingen en -smelten typische voorbeelden zijn, spe-len een belangrijke rol in de produktie van een groot aantal plastic produkten. Deverwerking van deze viscoelastische stoffen gebeurt meestal in de vloeibare fase endaarom is het van belang het stromingsgedrag te kunnen voorspellen zodat de eind-produkten van goede kwaliteit zijn. Binnen dit kader speelt de computer simulatievan viscoelastische stromingen een cruciale rol.

Het simuleren van deze stromingen kan op een macroscopische manier en op een mi-croscopische manier. Bij de traditionele macroscopische manier worden de eigenschap-pen van de bulk-vloeistof beschreven met een enkele fenomenologische constitutievevergelijking. In dit geval worden de vloeistof eigenschappen verdisconteerd via eenset parameters. De constitutieve vergelijking vormt samen met de behoudswettenvoor massa en impuls een volledig stelsel vergelijkingen waaruit de druk, snelheiden spanning kan worden opgelost. Als macroscopische model hebben we in dit on-derzoek het Giesekus model gebruikt. Bij de microscopische manier van modellerenworden de polymeer moleculen afzonderlijk gemodelleerd als twee bollen verbondenmet een niet-lineaire veer (FENE-dumbbell model) waar allerlei krachten op werken.Het visco-elastische gedrag van de stroming wordt nu beschouwd als de som van dekrachten van zeer veel polymeer molekulen. Het voordeel van deze modelleringswijzeis dat er veel meer fundamenteel inzicht kan worden verkregen wanneer alle krachtenop de juiste wijze zijn gemodelleerd.

In dit proefschrift worden deze twee rekenmethoden met elkaar vergeleken in com-binatie met experimenten. Daarvoor hebben we een vier-rollen apparaat gebouwdwaarin we een stationaire en een tijdsafhankelijke vlakke rekstroming kunnen maken.Als vloeistof gebruiken we een 2 %(w/w) polyisobutyleen (PIB)/decaline oplossing.Deze vloeistof hebben we gekarakteriseerd in afschuifstroming in een rheometer om deparameters te verkrijgen voor de berekeningen. Door gebruik te maken van stromings-

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dubbelebrekingsmetingen konden we N1 en de schuifspanning τxy meten. Hiervoorgebruiken we de ’stress optical rule’, die het verband aangeeft tussen dubbele brekingen spanning in een vloeistof.

Het direkt vergelijken van het experiment met de berekeningen was tamelijk moeil-ijk omdat de viscoelastische berekeningen alleen in 2D konden worden uitgevoerd.Daarvoor hebben we een correctie methode uitgedacht waarmee we de experimentele3D data konden omzetten in 2D data. Het macroscopische Giesekus model kwamgoed overeen met de experimenten. Tussen de modellen onderling bleken behoorlijkeverschillen in de berekende spanningen te zitten. Dit kwam voornamelijk doordat deeigenschappen van de gekarakteriseerde vloeistof in rekstroming voor beide modellenniet het zelfde is, terwijl deze eigenschappen in afschuifstroming goed overeen kwa-men. Verder is naar voren gekomen dat de microscopische berekingsmethode al vrijsnel onstabiel wordt wanneer berekeningen bij hoge stroomsnelheden worden uitgevo-erd. Het macroscopische Giesekus model blijft veel langer stabiel en de berekeningenmet dit model komen goed overeen met de experimenten.

Daarnaast hebben we onderzocht wat de invloed van de ketenlengte van het poly-meer op de stromingsdubbelebrekingsmetingen was. Daarvoor gebruikten we een 10% (w/w) polystyreen (PS)/decaline oplossing en een PIB/decaline oplossing. Om-dat de ketenlengte van de polystyreen (PS) ruim twintig keer korter is dan die vanpolyisobutyleen (PIB) kunnen we verzadigingsverschijnselen van de dubbele breking∆n bij de polystyreen oplossing waarnemen. In de tijdsafhankelijke metingen zienwe een duidelijk verschil in het her-orientatie proces van keten segmenten wanneer derekrichting plotseling 90 graden roteert. Hoe langer de polymeer ketens zijn, des tetrager verloopt het her-orientatie proces.

Jurrian Zijl

Chapter 1

Introduction

1.1 Fluid dynamics in daily life

Fluid dynamics is the science that is engaged in the research of the motion of gasesand liquids. This is already a very old topic in physics. Leonardo da Vinci (1452-1519) was already fascinated by a falling water jet in a bucket. His well-knowndrawing of this falling jet illustrated the complexity of fluid flow in a very simpleapplication. Nowadays the understanding of flowing fluids is still unclear and it isa grateful topic area for the academic world to discover the mysteries of the motionof fluids. But it is also a very important subject in the present-day society. Manypractical applications are based on the physics of fluid dynamics. A number of topicareas of fluid dynamics can be mentioned when a technical environment is considered:aerodynamics, turbulence, multi-phase flow, rheology and so on. In this thesis weconcern ourselves with rheology.

Rheology is a relatively young division of fluid dynamics. The development of thechemical industry at the beginning of the 20th century, followed by the advent oflarge-scale production of synthetic polymers, resulted in a host of new materials withstrange flow behavior. This strange flow behavior is expressed in for example a non-constant viscosity of a polymer liquid in a certain flow condition. Rheology is the studyof the flow of these kinds of materials that do not follow the simple constitutive law,known as a Newtonian fluid. Most of these particular materials generally have bothelastic and viscous properties. Polymers, one of these materials, play an importantrole in the manufacturing and application of many consumer products: in the moldingof plastic bottles, spinning synthetic fibers for clothing industries, and the productionof paints. Applications are also present in food-products industry where polymersare used to improve the structure and properties of food products. Because underprocessing conditions, the polymers usually are in a liquid or dissolved state, theirflow behavior is a topic in the polymer processing industry.

From the examples mentioned above, it is clear that the study of non-standard fluids

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is not only an academic topic but also something of relevance for the industry. In thisthesis the motion and deformation of polymers in a solution will be investigated.

1.2 Modeling polymer flow

Characteristic for a polymer is its huge molecular weight. This weight is much larger(104 − 106 g/mol) than many other molecules. The reason is that most polymersconsist of long chains of carbon atoms which can be seen in figure 1.1. Sometimesthese chains are branched and sometimes the polymer chains are connected with eachother forming a network. In this thesis we concern ourselves with polymers whichare dissolved in a solvent. The solvent molecules are much smaller than the poly-mer molecules. When polymers are in the dissolved state, they show a viscoelasticflow behavior. This means that this fluid is an intermediate of an elastic solid anda viscous fluid. In the no-flow case the orientation of carbon chains are randomlydistributed in the medium. When they begin to flow, the chains will be stretchedand oriented in a preferential direction due to the shear and elongational forces andstrange flow behavior appears. Simulating this viscoelastic flow is a challenging topicand it can be done at different levels of complexity. In figure 1.1 we see the levels ofcomplexity. The most basic level is the atomic level. At this level we consider the in-dividual carbon atoms as basic elements. For representing only one polymer moleculewe need at least a few thousand carbon atoms. It is clear that at this fundamental

C

CC

CC

C CC

C C

C

molecule

bead−spring

polymeric liquid

Atomic level

rigid segments

Microscopic level

Macroscopic level

Figure 1.1: The several ways to model the polymers at different levels of complexity.

Introduction 3

level, simulations for complex flows can not be done due to lack of computer power.Therefore we can simplify this atomic level to a microscopic level. At this level, a fewnumber of carbon bonds are replaced by a single rigid segment. Even at this level flowcalculations in complex geometries are difficult to perform and we must make anothersimplification. The next step we make is the replacement of a number rigid segmentsby one (non-)linear spring. At this level it is possible to perform flow calculations.The most coarse grained level is the macroscopic level. At this level we only look atthe bulk properties of the polymer solution, such as the viscosity etc.

Macroscopic modelingThere are two different ways to get an expression for the stress in a viscoelastic fluid.Here the stress is defined as the force which is acting on a small plane in the vis-coelastic fluid divided by the area of that plane. The first possibility is to considerthe polymeric liquid as a continuum in the sense that no attempt is made to modelthe microscopic character of the polymers as individual particles or to simulate themat a microscopic level. This macroscopic approach is to relate polymeric stressesto large scale properties such as velocity gradients. The state of the art is here topostulate a certain solution which satisfies some main properties in which adjustableconstants are related to experiments. In this macroscopic approach one is only inter-ested in the effective bulk properties of the liquid. As a consequence the stress-strainexpression, alternatively called a constitutive equation, is of phenomenological char-acter: it takes the form of a postulated equation, which contains several adjustableparameters. These parameters are unknown a priori, and they have to be determinedexperimentally in rheometrical flows. Subsequently the constitutive equation can beused for the simulation of complex flows. This is one of the most attractive aspectsof the macroscopic approach based on one simple constitutive equation. It makes themethod computationally very economic. A weak point of the method, is the fact thatthe success of a constitutive equation in the simulation of a given liquid is often foundto depend on the particular flow geometry for which the constants were determined.This in spite of the fact that the constitutive equation aims to describe the flow behav-ior of the material for all deformations. Therefore it is difficult to develop constitutiveequations in this way that are accurate for a wide range of flow conditions.

Microscopic modelingIn contrast to the macroscopic or phenomenological approach described above, it isthe aim of micro-rheology to find polymeric stresses from a model for the micro-structure of the fluid. In the microscopic simulations, the bulk flow behavior shouldbe calculated as the net effect of the dynamics of a large number of individual poly-mer molecules. In the micro-rheological approach the polymer molecules are modeledas relatively simple mechanical objects (beads connected with springs) which are be-lieved to capture the basic physics of the real polymers. A typical example is thedumbbell model. In this model a real polymer molecule is modeled as a spring con-necting two beads. In the microscopic approach the solvent molecules are not modeledas individual particles. Their influence is modeled as random Brownian forces which

4 Chapter 1

are acting on the dumbbell simulating the collisions of the solvent molecules withpolymer molecules. These random Brownian forces acting on the polymer moleculesresult in a random walk and a repeatedly changing configuration of the carbon chainsof the polymer. Using the principles of Brownian dynamics, we can describe the equa-tions that govern the evolution of the configuration of each of the model molecules.The configurations of the polymer molecules at a later time are found by integratingan individual evolution equation for each of them. Similar to real fluids, importantmacroscopic quantities like the stress tensor are obtained as averages over the set ofconfigurations defined by the model polymers.

Obviously, from a computational point of view a microscopic approach to viscoelasticflow simulations is much less attractive than an approach based on macroscopic con-stitutive equations because for each model polymer an individual evolution equationhas to be integrated. On the other hand the micro-rheological approach, however, ismuch more satisfying from a physical point of view. Despite the fact that the con-stitutive equation is replaced by a microscopic model for the polymers, the modelingnow occurs at a more fundamental level. Furthermore, when the essential dynamicsof the real polymers is captured, then the micro-rheological approach should be ableto accurately predict the stress independently of the flow geometry.

1.3 Experiments in polymer flow

For improving the viscoelastic models, a huge number of experiments in polymer flowhave been done in the past. Until now, viscoelastic models still have deficiencies anda continuous stream of experimental work should be done to prove and validate newmodels. In fluid dynamics experiments, a number of experimental techniques are used.The most popular measurement methods are the optical techniques. The advantageof all optical techniques is that they do not disturb the flow field due to the absenceof a measurement probe. The most well-known optical measurement techniques inrheology are: light scattering techniques and polarimetry [Fuller (1995)]. This lasttechnique using the properties of polarized light, is used in this thesis. We look atthe changes of the polarization state of light when it travels through a birefringentmedium by comparing the difference of the in-coming and out-coming light beam ofa sample. The birefringence means that the index of refraction in two directions isdifferent due to anisotropy in a medium. In our case this medium is a polymer so-lution and the anisotropy appears due to orientation and stretching of the polymermolecules in the flow field. This is called Flow Induced Birefringence (FIB). It isassumed that the birefringence is linearly correlated to the stress in the polymers bya relationship that is called the stress optical rule.

Birefringence experiments have been done in many geometries. Usually, these ge-ometries are relatively simple, for example simple shear flow. Baaijens and Schoonengive an extensive literature study of birefringence experiments in different geometrieswhich are more complex [Baaijens (1994), Schoonen (1998)]. They mention three

Introduction 5

2h

R

Figure 1.2: The four-roll mill geometry used for generating a elongational flow.

important types of complex flow: contraction flow, flow around a cylinder and stag-nation flow. In this thesis we will investigate the flow behavior in stagnation flows.Flows having stagnation points are of particular interest because of the large elonga-tional rates combined with large residence times of the polymer molecules in the flow.To generate such a flow, we use the so-called four-roll mill geometry in figure 1.2.This geometry consists of four cylinders in a square. The upper left and lower rightcylinder rotate in an opposite direction with respect to the upper right cylinder andlower left cylinder. In this case we can create a stagnation flow. When we oscillatethe cylinders we can create a time-dependent elongational flow which is a new aspectin the field of this research.

The reason why a more complex geometry is used is that a simple geometry is insuffi-cient for testing viscoelastic models for all flow geometries. Therefore we have to usemore complex geometry for testing advanced viscoelastic models.

1.4 Objective and Outline

In this thesis we will make a comparison between experiments and calculations. Wealso compare macroscopic calculations with microscopic calculations. Furthermore,we will investigate the behavior of polymers in a time-dependent extensional flow field.This thesis can be divided into two parts: experimental part and numerical part. Inchapter 2 the experimental technique used in this study will be described. Then thedesign and construction of the experimental set-up will be given. In chapter 3 the

6 Chapter 1

characterization of the fluid, which is a semi-dilute polymer solution, is described.The parameters obtained in the experiments in the rheometer will be used in thecalculations done later. In chapter 4, we will show the calculations. The influence ofthe fixed walls in the experimental set-up on the flow and a parameter study will beone of the topics. In this chapter, we also make a comparison between the macroscopiccalculations and microscopic calculations. Finally, chapter 4 will end in a discussionabout the advantages and disadvantages of the calculation methods. In chapter 5 wewill show the results of the measurements and calculations with parameters found inthe fluid characterization. To show the influence of the chain length of the polymermolecule we show experiments using two different types of molecules. Finally, inchapter 6 the conclusions and recommendations of this research will be given.

References

Baaijens, J.P.W. “Evaluation of constitutive equations for polymer melts andsolutions in complex flows.” PhD. thesis Eindhoven University of Technology,The Netherlands, (1994).

Fuller, G.C. Optical rheometry of complex fluids. Oxford University Press, NewYork (1995).

Schoonen, J.F.M. “Determination of Rheological Constitutive Equations usingComplex Flow.” PhD. thesis Eindhoven University of Technology, TheNetherlands (1998).

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Chapter 2

Experimental methods andset-up

In this chapter an overview is given of the experimental Flow Induced Birefringencemethod used in this thesis. Flow Induced Birefringence is a rheo-optical measurementtechnique used as a tool in non-Newtonian fluid mechanics. This method is suitablefor measuring the stresses by considering the changes of the polarization of light trav-eling through a fluid in motion. An extensive description of the experimental set-upthat we have used will be given. The total set-up can be divided into three parts.The mechanical, optical and electronical part. Each of these parts will be discussed.Finally, some practical aspects for performing FIB measurements will be considered.

2.1 Flow Induced Birefringence

Optical experiments using various techniques, are being used in macro-molecular sci-ence and polymer rheology. The large advantage of an optical measurement methodis that it does not disturb the flow. One of these methods is the Flow Induced Bire-fringence (FIB) method. It is a method for measuring stress distributions by usingthe property of birefringence of a flowing polymer solution. Birefringence means thatthe index of refraction along two orthogonal axes is different. When the differenceof the refractive indices along the two orthogonal principal axes is known, stressescan be obtained. Birefringence appears in solids with a crystal structure like Calcite,but it can also be present in flowing fluids. Flow induced birefringence can be gen-erated in fluids which consist of long flexible molecules (polymers) or in solutions ofcrystal-like particles. When the fluid is at rest, these molecules or particles are ran-domly oriented and the fluid is optically isotropic. When the fluid starts to move, theshear forces and elongational forces cause an alignment of the molecules or particlesin a preferential direction, and the fluid becomes birefringent. It is proposed thatflow birefringence variations are correlated to stress variations in the polymers due to

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10 Chapter 2

orientation and elongation. This is known as the stress optical rule which gives therelationship between stress and birefringence. The birefringence can be measured bymeans of polarized light. A short introduction on polarized light and birefringencewill be given below.

2.1.1 Fundamentals of optics

Polarized lightBefore explaining the use of the FIB measurement technique for our application, itis necessary for a good understanding to give an overview of the fundamentals ofpolarized light and birefringence. A more detailed description can be found in stan-dard texts on optics [Hecht and Zajac (1974)]. Light is a traveling electro-magneticwave which can be described with the Maxwell equations. An X-plane polarized lightbeam traveling in the z-direction can be represented by the following mathematicalexpression:

Ex(z, t) = Ex0 cos k(z − ωt). (2.1)

Whereas a Y -plane polarized light beam can be represented by

Ey(z, t) = Ey0 cos(kz − kωt− δ), (2.2)

where ω is frequency and ω/k the light velocity and δ the relative phase angle betweenEx(z, t) and Ey(z, t) and Ex0 and Ey0 are the amplitudes of the waves. A schematicillustration of X-polarized and Y -polarized light is given in figure 2.1.

To illustrate the importance of the relative phase δ of the components, consider thesituation of a purely X-polarized wave and a purely Y -polarized wave, with ampli-tudes Ex0 and Ey0. Assume that δ = 0. Light waves obey the principle of superposition, and the net effect of the two waves can be computed simply by vectoriallyadding equations (2.1) and (2.2). The sum of the two components is a linearly po-larized beam with an amplitude of (E2

x0 + E2y0)

1/2. This is illustrated in figure 2.2.Another special case of particular interest is when both waves have equal amplitudes(Ex0 = Ey0) and in addition their relative phase difference δ = π

2 . In this case it ismore difficult to see what the resultant is. It is not confined to a line in the XY -plane, but it is a circle of radius 1

2

√2Ex0. This is called circularly polarized light. In

general, when δ has an arbitrary value and the amplitudes Ex0 and Ey0 are different,the resultant is an ellipse in the XY -plane.

A mathematical description of polarized lightThe modern representation of polarized light actually had its origins in 1852 in thework of Stokes. He introduced four quantities which are functions only of variables ofany electro-magnetic wave. These quantities are called the Stokes parameters. Thefour Stokes parameters are :

I1 = E2x0 + E2

y0 (2.3)

Experimental methods and set-up 11

X

Y

X

Z

X

Y

Y

Z

X

Y

Figure 2.1: Schematic illustration of Y -polarized and X-polarized light.

I2 = E2x0 − E2

y0 (2.4)I3 = 2Ex0Ey0 cos(δ) (2.5)I4 = 2Ex0Ey0 sin(δ). (2.6)

Here Ex0 and Ey0 are the amplitudes of the light waves and δ the phase shift betweenthese waves. It should be noted that I1 represents the light intensity of the beam. Itwas pointed out by Mueller in 1941 [Bickel et al. (1985)] that the Stokes parameterscould be regarded as components of a vector matrix. A beam of elliptically polarizedlight could be represented by a 4 component vector matrix and any change of polar-ization would be represented by a change in this vector matrix. Mathematically, a4 × 4 matrix is an operator which changes a 4 component vector into another one.Hence any optical device (quarter-wave plate, half-wave plate, birefringent medium,etc.) may be represented by a 4× 4 matrix. The big advantage of this representationis, that the total effect of a series of optical devices traversed in turn by the beam,is given by the product of the matrices for the separate optical devices. The difficultmathematical analysis of a series of optical devices is reduced to the routine calcula-tion of matrix products. Mueller devised this matrix method for dealing with Stokesvectors. He derived for a lot of optical devices the 4 × 4 matrix operators. Theseoperators can be found in Fuller [Fuller (1995)] and Baaijens [Baaijens (1994)].

12 Chapter 2

X

X

Y

Z

Y

RESULTANT

Figure 2.2: The resultant of two linearly polarized light beams without relative phasedifference

BirefringenceMany crystalline substances are optically anisotropic. In other words, their opticalproperties are not the same in all directions within a given sample. A consequence isthat in an optically anisotropic medium, the refractive index (n) has different values indifferent directions. In general a birefringent medium has three principal orthogonalaxes. We can define a refractive index tensor n where n1, n2 and n3 are the valuesof the refractive indices along the principal axes.

n =

n1 0 0

0 n2 00 0 n3

(2.7)

However most of the birefringent crystals have one neutral direction which meansthat n2 = n3. Also in our fluid flow a neutral direction exists. This direction isperpendicular to the XY -plane because we assume a negligible velocity in the Z-direction of our flow geometry.

Consider a piece of birefringent material with two axes of refractive indices (see fig.2.3). The definition of birefringence ∆n now means a difference in the components of


axis 2

X

Y

χ

d

axis 1

Figure 2.3: A piece of birefringent material with thickness d. The angle between axis1 and the X-axis is called the orientation angle χ.

n along two orthogonal axes in a material:

∆n = n1 − n2. (2.8)

Here axis 1 lies in the direction of the largest value of n (n1) and axis 2 lies in thedirection orthogonal to axis 1, n2 has the smallest value of n. The angle between axis1 and the X-axis of the laboratory frame is called the orientation angle χ (see figure2.3). Now consider a linearly polarized beam traveling in the direction perpendicularto the XY -plane in figure 2.3. The plane of polarization of this beam is χ + 45degrees with respect to the X-axis. When this beam hits the birefringent mediumit will be vectorially split-up along the principal axes 1 and 2 in two components ofequal amplitude. We remind that the refractive index is the ratio of the light velocityin vacuum respect to the light velocity in a medium. Because there is a differencebetween the refractive indices, the components of the beam propagate at differentspeeds. The phase difference δ between the two components after passing through athickness d of a birefringent medium is equal to :

δ =2πn1d

λ− 2πn2d

λ

δ =2π∆nd

λ. (2.9)

Where λ is the wavelength of the light in vacuum. The quantity δ is called theretardation. When the beam leaves the birefringent medium, the two components

14 Chapter 2

coincide again and the beam is transformed from a linearly polarized beam into anelliptically polarized beam or a circularly polarized beam when δ = π

2 . This is how aretardation plate works.

In optics there are a lot of devices available to transform the polarization states oflight. Retardation plates, or phase shifters are elements used for synthesis and analysisof light in various polarization states. We described the working of a retardationplate above. A retardation plate that will cause a phase difference of π

2 is called aquarter wave plate or λ/4 wave plate. Two quarter wave-plates will be used in ourexperiments. They will be used for making a circularly polarized beam.

2.1.2 Relationship between optical properties and stress

Many polymer solutions which are optically isotropic in equilibrium become anisotropicand show optical characteristics similar to crystals when the polymer chains are de-formed. However the birefringence of polymer solutions can be classified in two cate-gories: intrinsic birefringence and form birefringence. These two types of birefringenceof flowing polymeric materials can be explained from microscopic parameters:

• The intrinsic birefringence originates from the difference in polarizability of apolymer monomer in two directions: along its backbone with respect to per-pendicular to that. The more a polymer chain is stretched the larger the po-larizability along its backbone respect to the perpendicular direction. Intrinsicbirefringence is proportional to the stress in the polymer chain.

• The form birefringence is caused by a refractive index difference on a muchlarger length scale and is related to the difference between the refractive indexbetween the polymers and solvent and to the anisotropy of the shape of thedissolved polymer molecules in the solvent [Copic (1957), Doi and Edwards(1986)]. When a bad solvent is used, polymer molecules will configure in amicelle-like shape in the solution. When the concentration of the polymers ishigher than the coil overlap concentration c∗, form birefringence becomes lessimportant. Form birefringence is not proportional to the stress and it increasesfast with decreasing concentration.

In our study only intrinsic birefringence is important because we use a more concen-trated solution.

There is a direct relationship between the stress and the optical properties of a bire-fringent polymer solution. The more the polymer chains are elongated, the higherthe stresses in these chains and the higher the level of birefringence. By measuringchanges in polarization relative to the incident light, it is possible to calculate thebirefringence (∆n) which is to a good approximation linearly related to the stresstensor (τ ) as follows.

τ =1C

n, (2.10)


where n is the refractive index tensor. This relation is called the stress optical rule.This means, that the principal directions of the stress tensor τ coincide with theprincipal directions of the refractive index tensor n and the value of the stress in theprincipal directions is proportional to the birefringence. The constant C is called thestress optical constant. This constant can be measured in advanced rheometers inwhich both mechanical and optical measurements can be done simultaneously. Thestress optical constant C is mainly a function of the specific solvent which is usedand the architecture of the polymer molecule. It is independent of the molecularweight, at least for flexible polymer chains. If a sufficiently high concentration is used(c > 10%) then C is also independent of the concentration.

If we would like to calculate the stresses directly from the birefringence measurements,then it is important to mention that this calculation method is only valid when a twodimensional (2D) flow is considered. In a three dimensional flow the birefringence∆n and orientation angle χ continuously change along the light path and opticalmeasurements will measure cumulative optical properties which finally results in aneffective orientation angle χ and birefringence ∆n. In our flow geometry we will createan almost 2D flow to overcome this problem. For the two dimensional case the stressoptical rule simplifies to :(

τxx τxy

τxy τyy

)=

1C

(nxx nxy

nxy nyy

)(2.11)

Where τxx − τyy is equal to the first normal stress difference N1 and τxy is the shearstress. If the orthogonal refractive index axis nxx has an orientation angle χ withrespect to the reference X-axis in figure 2.3 then the following expressions determinedfrom Mohr’s circle [Gere and Timoshenko (1989)] are valid :

τxy =∆n

2Csin(2χ)

N1 = τxx − τyy =∆n

Ccos(2χ). (2.12)

The derivation of these relations can also be found in for example the book by Ma-cosko [Macosko (1994)]. These equations are called the Stress Optical Relations.

By measuring the retardation δ and the orientation angle χ of a light beam travelingalong the Z-axis in a 2D flow, we can calculate ∆n with eq.(2.9). With the stressoptical relations eq.(2.12), it is possible to calculate the shear stress τxy and the firstnormal stress difference N1.

It is important to emphasize that the stress optical relations are not the consequenceof fundamental physical principles; rather it is a plausible hypothesis, which has anextensive experimental support for polymer solutions and melts and is understoodtheoretically from kinetic theory. The validity of the stress optical rule has beenproven valid for various polymeric materials in shear flows for a wide range of shearrates. In shear flows it is fairly easy to compare optical measurements with mechanical

16 Chapter 2

experiments and check the linearity of the rule, in contrast to elongational measure-ments which are more difficult. In the review of Mackay and Boger, [Mackay andBoger (1988)] several studies are listed that did attempt to validate the stress opticalrule in elongational flows. However results of measurements of mechanical stressesin elongational flows of low-viscosity solutions are not reliable or impossible due toexperimental difficulties [Walters (1992)]. Therefore in such flows a reliable directvalidation of the stress optical rule by comparison with mechanical measurements ispresently not feasible. Nevertheless, it is generally agreed that the stress optical rulewill fail in the case of large extensional deformation [Doyle et. al (1998), Spiegelberget. al (1996)]. Then the polymer chains are fully stretched and oriented which willresult in a saturation of the birefringence.

2.2 Experimental set-up

The aim of this research is to compare viscoelastic flow calculations with experimentsflowing through complex geometries. The choice of a particular complex geometry forcombined numerical and experimental research is not easy. Some factors are of keyimportance.

• The accessibility for experimental tools and numerical convenience.

• The flow geometry should be relevant in terms of present-day rheological re-search topics.

One of these items is the behavior of polymers in an extensional flow field. One of thegeometries that satisfies these requirements is the cross-slot flow geometry in figure2.4. In this geometry an almost perfect elongational flow can be created due to thehyperbolic shape of the fixed walls. The flow in this geometry is pressure driven. Dueto a pressure gradient, the fluid start to flow. The advantage of this geometry is thehigh strain rate which can be reached in the central stagnation point. However, onlya stationary flow field can be investigated.

Another geometry that satisfies these requirements is the four-roll mill geometry infigure 2.4 invented by Taylor in 1936. We will use this geometry in our research. Theflow in this geometry is shear-driven. It implies that when a strong shear-thinningfluid is used in this geometry, less fluid will flow to the central stagnation pointand a high strain rate will not be reached. In this geometry both stationary andtime-dependent bi-axial extensional flow fields can be generated in the center regionbetween the four rollers. This geometry is also suitable for calculations due to theabsence of sharp corners. However a perfect elongational flow field can not be created.Generating an extensional flow field as good as possible, we fit a hyperbola through3 points of the cylinder wall. When a hyperbolic streamline is forced to go throughthe points A, B and C the ratio h

R becomes 0.207 (see figure 2.5). A more detaileddescription and literature survey of the design of this four-roll mill apparatus can befound in van der Reijden-Stolk [van der Reijden-Stolk (1989)].


outflow

ω

inflowinflow

outflow

outflow

inflowinflow

stagnationpoint line

stagnationpoint line

outflow

Figure 2.4: The lay-out of a cross-slot geometry (upper figure) and a the four-rollmill geometry (lower figure). In the cross-slot geometry there is a pressure-drivenflow which results in high strain rates in the stagnation point. In the four-roll millgeometry there is a shear force-driven flow. The advantage of a four-roll mill geometryis the possibility to create an oscillating extensional flow field.

18 Chapter 2

Y

A

B

C

R

R+ha

h

a

R+h

X

Figure 2.5: Fitting a circle through the points A, B and C of a hyperbola, the ratioh/R becomes 0.207

First we will describe the mechanics of this geometry. Then the other parts of the ex-perimental set-up such as the optics and electronics, will follow. A schematic overviewof these three parts is plotted in figure 2.6.

2.2.1 Mechanics of the flow cell

The four-roll mill of Taylor consists of four cylinders in a square box. By rotating andoscillating these cylinders, as shown in figure 2.4, a stationary and time-dependentextensional flow field can be generated in between the region of the rollers. In ourdesign, the four-roll mill apparatus is fixed in a squared box of 320× 320× 120 mm.So the total system volume is about 12 liters, which is quite large. The advantageof a large system volume is the absence of temperature rising due to viscous heating.The radius of the aluminum cylinders is R = 30.0 mm and the distance between themidpoints of the cylinders is 72.0 mm. The length L of the cylinders is 110 mm. Theminimum distance between the cylinder walls is 2h = 12 mm (see figure 2.5). In thebox, we made an adjustable bottom. In that way it is possible to change the aspectratio (2h/L) and the length of the light path. This can be seen in figure 2.7.

Another special point in this design are the windows. These windows must be freefrom any stresses. Otherwise they become also birefringent and may disturb the


lightbeam

electrical signals

Optic module

Electronics

Mechanics

Figure 2.6: Schematic illustration of the three parts of the set-up.

measurements. In our set-up we used BK 7 grade A optical glass windows with aparallelism of 5 arc seconds. The diameter of the circular windows at the upper andlower bottom are 25 mm and 40 mm, respectively.

The total system is driven by a 200 Watt servo motor (Maxon). This servo motorcan be controlled by a pilot signal which is generated by a programmable functiongenerator. The maximum oscillating frequency of the total mechanical system waslimited to 4 Hertz. Between the servo motor and the four cylinders there is a delay-action of about 1 : 10 by means of driving belts. These driving belts are chosen asstiff as possible because they can influence the rotation speed during high oscillatingdynamic experiments. The total driving system can be seen in figure 2.7.

2.2.2 Optics

In this section we give a description of the measurement method used in this study.First we will explain the measurement method. After that we describe the opticalelements and lay-out of the optical part of the set-up. The measurement methodwhich we use, has been developed by Frattini and Fuller in 1984 [Frattini and Fuller

20 Chapter 2

Driving belt

Windows

Adjustable bottom

Servo motor

Motorspeed

signal to electronics

Windows

Figure 2.7: An overview of the flow cell and the driving system of the cylinders


(1984)]. The optical train and the orientation angle of these polarizing elements ofthis measurement method can be found in figure 2.8. This optical train consists ofseveral polarizers and wave plates to convert the polarization state of the light. Themost important element in this train is the photo elastic modulator (PEM). This isa device for modulating the polarization of the laser beam. It consists of a crystalwhich is compressed and decompressed with a certain modulation frequency ω. Atthe output an oscillating retardation signal δpem = Apem sin(ωt) as a function of timecan be seen. When the light intensity signal (I), which is the first Stokes parameter,is calculated by means of Mueller matrix calculations, we find :

I =I04

(1− sin(Apem sinωt) sin 2χ sin δ + cos(Apem sinωt) cos 2χ sin δ) . (2.13)

Where Apem is the amplitude of the oscillating retardation and ω is the oscillatingfrequency of the photo elastic modulator. I0 is the light intensity of the laser whenit leaves the cavity. The δ and χ are the retardation and orientation angle in thesample respectively. Looking at expression (2.13) there are two terms which can beexpanded into Bessel functions: [Abramowitz et al. (1972), Watson (1966)]

cos(Apem sinωt) = J0(Apem) + 2∞∑

m=1

J2m(Apem) cos(2mωt).

sin(Apem sinωt) = 2∞∑

m=0

J2m+1(Apem) sin((2m + 1)ωt). (2.14)

Substituting these expressions (2.14) into (2.13) and expanding up to the secondharmonic frequency, we find :

I =I04[1 + J0(Apem) sin δ cos 2χ− 2J1(Apem) sin δ sin 2χ sin(ωt)

+2J2(Apem) sin δ cos 2χ cos(2ωt)]. (2.15)

Where J0(Apem) ,J1(Apem) and J2(Apem) are Bessel functions of order 0, 1 and 2.If the photo elastic modulator is tuned in such a way that J0(Apem) is equal to zeroby adjusting the peak retardation level of the PEM to Apem = 2.405 radians, thenexpression (2.15) is simplified to :

I =I04

1−2J1(Apem) sin δ sin 2χ︸︷︷︸

Aω

sin(ωt) + 2J2(Apem) sin δ cos 2χ︸︷︷︸A2ω

cos(2ωt)

.

(2.16)With the lock-in amplifiers it is possible to measure the amplitudes Aω and A2ω offrequencies ω and 2ω. Then these values are normalized by the DC value of thissignal by dividing by I0/4. Furthermore Aω must be divided by 2J1(2.405) and A2ω

by 2J2(2.405) to produce the following normalized amplitudes. Then two equations

22 Chapter 2

with two unknowns appear :

Aωn = − sin δ sin 2χA2ωn = sin δ cos 2χ. (2.17)

Where Aωn and A2ωn are the measured normalized amplitudes of the frequencies ωand 2ω, and δ and χ the retardation and orientation angle in the sample, respectively.In the experiments Aωn and A2ωn are always smaller than one. So we can solve theseset of equations (2.17) and we find for δ and χ in the fluid:

χ =12arctan

−Aωn

A2ωn

δ =A2ωn

| A2ωn |√

Aωn2 +A2ωn

2. (2.18)

Substitution of δ and χ in the stress optical relations (2.12), the shear stress τxy andthe normal stress difference N1 can be found.

Now the specifications and working of the devices are explained. The optical trainconsists of the elements shown in figure 2.8. The total optical train is mounted on aXY-traverse system. The displacement accuracy of this system is 0.1 mm.

LaserThe laser we used in the experiments is a Melles Griot 10 mW HeNe laser with awavelength of λ = 632.8 nm. The reason that a laser and not a laser diode is usedis the quality of the beams. A laser diode has an elliptically shaped beam and is atmost times relatively thick and the beam diverges quite quickly. This problem canbe overcome by beam shaping optics. But these optics may change the polarizationstate of the light. These undesirable effects do not appear when a laser with a smallbeam and little divergence is used. In our set-up a laser with a beam diameter (e−2)of 0.3 mm and a divergence of 1.2 mrad is used.

Polarization filtersThe polarization filters that are used are sheet polarizers mounted in adjustable ro-tating holders. These filters perform a single sheet transmission of 36 percent andhave an extinction ratio of 10−4. It is also possible to use Glan Thompson prisms.These prisms have a transmittance of more than 90 percent and an extinction ratio of10−5. This is better than sheet polarizers but they are much more expensive. In ourexperiments the sheet polarizers gave reliable results so the cheaper sheet polarizerswere used.

Wave platesThe first-order wave plates (OptoSigma) which are used in the experiments are madeof quartz. They have an aperture of 17 mm × 17 mm and have a parallelism of 2 arc-sec. The wave plates are mounted in holders in which they can rotate. The angle ofrotation is adjustable with an accuracy of 1 degree.


135

Laser

Polarizer

Quarter waveplate

Quarter waveplate

Polarizer

PEM

Flowcell

Photodiode

0

45

45

90

Controllerunit PEM

reference signalsto lock-in amp.

Tra

nsla

tion

stag

e

Figure 2.8: The optical train and the orientation angle of the elements

Photo Elastic Modulator PEMThe photo elastic modulator is an instrument used for modulating the polarizationof a beam of light. The principle of operation is based on the photo elastic effect,in which a mechanically stressed sample exhibits birefringence proportional to thestrain. In its simplest form the PEM consists of a rectangular bar of a transparentmaterial (quartz) attached to a piezoelectric transducer. The bar vibrates along itslong dimension at a frequency determined by the length of the bar and the speed ofthe longitudinal wave in the optical element material. The oscillating birefringenceeffects is at its maximum at the center of the bar. The PEM ( PEM-90TM Hindsinstruments) used in the experiments oscillate with a fixed manufacturer-determinedfrequency of 42 kHz. By means of an electronic controller unit the amplitude of theoscillating birefringence can be tuned. This controller unit also generates the referencesignals used for the lock-in amplifiers.

2.2.3 Electronics

In figure 2.9 a schematic drawing is given of the electronics and data-acquisitionboard. Three signals are coming from the other parts of the set-up and are drawnwith dotted lines in figure 2.9. One signal comes from the servo motor and indicatesthe rotation speed of the rollers, the other two signals come from the photo elasticmodulator and are reference signals used for the lock-in amplifiers. The elements of

24 Chapter 2

Lock-in

1 from PEM

Amp. F Amp. 2F

Lock-in Low-pass

Filter

Photodiode

Data-acquisitionBoard

Motorspeedsignal

Ref 2Reference signal

Figure 2.9: An overview of the electronics used in the set-up.

the electronic system will described below.

Photo diodeThe function of the photo diode is the transformation of light intensity signal into anelectronic signal. It consists of a piece of light sensitive material (active area) whichgenerates a current when light falls on it. A photo diode with a fast response timeshould be used in order to follow the fast fluctuations of the light intensity which maybe caused by the change of birefringence due to the deformation of the polymers. Weused a photo diode with a response time of 1 µs. The active area of the photo diodeshould not be too big. A large active area introduces a lot of noise and the responsetime of the photo diode becomes large due to the large electrical capacity of thisactive area. On the other hand the area should not be too small because in this casethe photo diode becomes extremely sensitive for misalignment. In the experiments aself-made photo diode with an active area of 3 mm2 is used.

Lock-in amplifierThe lock-in amplifier measures essentially the RMS value of an AC signal at a givenfrequency and gives an output in the form of a DC voltage proportional to the RMSvalue of the AC signal. These kind of amplifiers are extremely selective, they lock toand measure the particular frequency of interest ignoring all other signals at the input.In this set-up the RMS values of the frequencies 42 kHz and 84 kHz were measuredbecause the first harmonic frequency of our PEM was 42 kHz. In our experimentswe used two single phase lock-in amplifiers (LIA BV-150-H FEMTO MesstechnikGmbH).

Low pass filterIn the experiments we used a common first order low pass filter made by Krohn-Hite(model 3940). Using this filter, the DC-value of the signal for normalizing the RMSvalues, is obtained. The cut-off frequency was set on 3333 Hz. In this case the timeconstant of the filter was equal to the time constant of the lock-in amplifiers. For amore detailed explanation of these settings, one is referred to the next section.


Data acquisition systemThere are 4 channels that need to be measured and stored: Channels 1 and 2 measurethe RMS-values of the frequencies ω and 2ω, channel 3 measures the DC-value of thelight intensity signal and channel 4 measures the roller speed of the cylinders. Inour experiments a National Instruments data acquisition system is used (PC-516) incombination with Labview software. The resolution of this system is 16 bits and thereare 8 input channels. The maximum sample rate of this system was 5 kHz for fourchannels. The smallest time interval which can be measured then is 0.2 ms. This timescale is small enough to see what happens at the smallest time scales of the polymerflow.

2.3 Experimental aspects

When performing experiments, there will always be a number of additional aspectswhich should be taken into account. Here it is impossible to document all the aspectsof performing FIB experiments. Therefore, we restrict ourselves to the description ofonly the most important aspects. The aspects that will be discussed are the align-ment of the optics, the tuning of the electronics, the adjustment of the photo elasticmodulator and transferring 3D flow data to 2D flow data.

2.3.1 Alignment procedure

One of the most important aspects in our experiments, is the correct alignment ofthe optical elements, because a misalignment in the optics causes a large error in themeasurements. A problem related to this and really hard to overcome is misalignmentdue to the vibrations made by the servo motor in our set-up. These vibrations maycause a drift of the optics which result in a non-constant measured signals when astationary flow is studied. The problem of vibrations can be solved partly by fixatingthe optical train and the flow cell on two different frames, which are not mechanicallyconnected. This was done in our experiments.

During the alignment procedure, the laser and the photo elastic modulator are switchedon and the flow cell is removed from the optical train. The signal of the photo diodeis monitored on a oscilloscope. Looking at the shape of this signal on the oscilloscope,it is possible to see when a correct alignment has been reached with the inserting ofthe optical devices. When all optical devices are aligned well, one should only see ahorizontal line on the oscilloscope. Then the flow cell can be replaced back into theoptical train.

One final remark about the alignment should be mentioned. Every optical elementthat is hit by the laser beam, will cause a reflection. The alignment is perfect whenthis reflection is bounced back in the cavity of the laser. However in that case manyHeNe lasers become unstable and the light intensity fluctuations will be large. Soalways a little misalignment is present in the experiments.

26 Chapter 2

2.3.2 Tuning the electronics

In investigating the dynamics of the polymer solution, it is important to separate thetransient response of the fluid and the transients in the electronics. Interpreting thedata becomes very difficult otherwise. In this part it will be described how to dealwith this problem in order to get high quality data. A lot of electronic equipment isused for the measurement method presented in this thesis. Two elements of the totalequipment are important: the lock-in amplifier and the low pass filter. In eq.(2.16)the output signal of the light intensity as a function of time has been given:

I =I04(1−Aω sinωt+A2ω cos 2ωt), (2.19)

where Aω and A2ω are dependent on the retardation δ and the orientation angle χin the fluid flow. Looking at expression (2.19), the DC-value shows independence ofδ and χ. In practice, there are always some fluctuations in the laser intensity. TheDC-value of the signal is determined by a low pass filter. The RMS-value of the sinωtand cos 2ωt is measured by two lock-in amplifiers. When determining the retardationδ and orientation angle χ, the amplitudes Aω and A2ω must be normalized by dividingby the DC-value. Therefore it is convenient to have a constant DC-value. The firstorder low-pass filter always has a certain delay time before it reaches its final value.This delay time is expressed in the so called time constant. In the same way thelock-in amplifiers will respond with a certain delay time due to the built-in low-passfilter of their electronic circuit. If the dynamics of polymers should be measured, thetime constants of the electronics should be smaller than the smallest time scales ofthe flow (τflow).

Another important aspect of tuning the electronics is that the response time of am-plifiers and low-pass filter should be the same. If the two lock-in amplifiers do nothave the same time constant, then strange results are obtained when the retardationδ and orientation angle χ are calculated by means of eq. (2.17). Sometimes it is evenimpossible to calculate δ and χ because the normalized amplitudes are larger thanone and the solution of eq. (2.17) does not exists.

Finally the time constant of the photo diode(τph) should be smaller than the rest ofthe electronics. Then the following criterium should be satisfied:

τph < τlp = τlock1 = τlock2 < τflow. (2.20)

Where τlp, τlock1 and τlock2 are the time constants of the low-pass filter and the lock-inamplifiers respectively. In the experiments τph = 1 µs and the time constants of thelow-pass filter and the lock-in amplifiers are 0.3 ms. So the smallest time scales whichcan be observed in the flow without almost transient response of the electronics areabout 1.2 ms. Then a maximum error of 2% (e−4) is made.


LP:filter

t

laser P PPEMphotodiode

0-45 Oscilloscopeωpol

Figure 2.10: The set-up for zero setting of the Bessel function consists of two polarizersone fixed at orientation angle of −45 degrees and one rotating.

2.3.3 Adjusting the PEM

In the previous section, it was shown that it is important to minimize the fluctuationsof the DC value of the signal due to the finite response time of the low-pass filter. Away to minimize these fluctuations is to adjust the photo elastic modulator (PEM)in the right way. In eq. (2.15) we found for the light intensity signal:

I =I04[

DC−term︷︸︸︷1 + J0(Apem) sin δ cos 2χ−2J1(Apem) sin δ sin 2χ sin(ωt)

+2J2(Apem) sin δ cos 2χ cos(2ωt)]. (2.21)

In this equation the DC-term shows a dependence on the retardation δ and the orien-tation angle χ in the flow cell unless the 0th order Bessel function is adjusted to zero.Theoretically it means that Apem is set to 2.405, because J0(2.405) = 0. In practice,this can be done by a set-up as shown in figure 2.10. This set-up consists of onepolarizer oriented at −45 degrees, a photo elastic modulator oriented at zero degreesand a rotating polarizer with an angular velocity of ωpol. The modulating frequencyof the photo elastic modulator is ω. When the light intensity signal is calculated bymeans of Mueller matrix calculus, we find for the first Stokes parameter:

I =I04

(1− sin 2ωpolt cos(Apem sinωt)) , (2.22)

where Apem is the peak retardation level of the photo elastic modulator. Expandingthis expression into Bessel function series, we find:

I =I04

(1− J0(Apem) sin(2ωpolt)− 2J2(Apem) sin(2ωpolt) cos(2ωt)− .....) . (2.23)

If this signal is measured after passing a low-pass filter with a cut-off frequency atabout 10ωpol, the expression 2.23 will reduce to an ordinary sine function superim-posed on a DC-value:

I =I04

(1− J0(Apem) sin(2ωpolt)) . (2.24)

Checking this signal on an oscilloscope, the PEM can be tuned such that the amplitudeof the signal in expression (2.24) at the oscilloscope becomes zero. When we have set

28 Chapter 2

J0(Apem) to zero experimentally, the optical train in figure 2.10 is transformed into theoptical train in figure 2.8 without changing the PEM settings. It should be mentionedthat this method is only valid when the angular velocity of the rotating polarizer ωpol

is much smaller than the modulation frequency ω of the photo elastic modulator.In the experiments, the rotational speed of the polarizer was about 100 Hz and themodulation frequency was 42 kHz.

2.3.4 Correcting the influence of edge effects

Due to the presence of a bottom and top in the flow geometry, boundary layers nearthese fixed walls appear. These boundary layers cause a non-uniform birefringenceprofile along the entire light path. Therefore an adjustable bottom was made inorder to correct for this. One of the reasons for correction is, that the viscoelasticcalculations could only be performed in two dimensions.

The correction method goes as follows: When birefringence is measured, an integralalong the light path is measured. When strain rates are small and the stress opticallaw is still valid, the retardation δ is equal to :

δ(d) =∫ z=d

z=0

c1 · εx(z)dz. (2.25)

Where d is the length of the light path, εx is the strain rate and c1 is a constant. Infigure 2.11 we have plotted the retardation δ(z) for three different light path lengthsof L1, L2 and L3 respectively. The first part of the curves (OA) is the same for everylight path length because the boundary layers are the same in every case. If thebirefringence ∆n is calculated from this figure using a light path length of L1 we find:

∆n =λδ1

2π · L1. (2.26)

This is equal to the slope of line OB multiplied by a factor λ2π . In the same way

we will find for the birefringence ∆n at a cylinder length of L2, the slope of line OCmultiplied by a factor λ

2π . When the cylinder length goes to infinity then the slopeof that line approaches the slope of line AE. And the slope of line AE multiplied byλ2π is the birefringence ∆n in a perfect two dimensional flow.

In the experiments only the total retardation, corresponding to the points B ,C andD in figure 2.11 can be measured. Drawing a curve through these points will give astraight line which is parallel to the line AE. In that case the ∆n in two dimensionalflow can be determined. Essentially, we must do two measurements at light pathlengths L1 and L2 and the following calculation to obtain the birefringence for the2D flow ∆n2D:

∆n2D =λ(δ2 − δ1)

2π(L2 − L1). (2.27)

If the length of the cylinders is smaller than the thickness of the boundary layers thiscorrection method does not work. Furthermore the boundary layer thickness must


Distance z

Ret

arda

tion

δ

A

B

C

D

E

OL L L1 2 3

δ

δ

δ

1

2

3

Figure 2.11: Retardation δ as function of light path length at cylinder lengths of L1,L2 and L3.

be independent of the cylinder lengths. In our experiments, flow data were collectedat 55 and 80 mm light path lengths. After that, they were corrected by the methoddescribed above.

30 Chapter 2

References

Ambramowitz, M. and I.A. Stegun, (editors) Handbook of Mathematical Func-tions ninth edition, Dover Publications Inc., New York (1972).

Baaijens, J.P.W. “Evaluation of constitutive equations for polymer melts andsolutions in complex flows.” PhD. thesis. Eindhoven University of Technol-ogy, The Netherlands (1994).

Bickel, W. and W. Bailey, “Stokes Vectors, Mueller Matrices and PolarizedScattered Light.” Am. Journal. Phys, 53 468-483 (1985).

Copic, M. “Streaming birefringence of polymer solutions: anisotropy of internalfields.” J. Chem. Physics, 26 1382-1390 (1957).

Doi, M. and S.F. Edwards, The theory of polymer dynamics. Oxford UniversityPress Inc., Oxford, (1986).

Doyle, P., E.S.G. Shaqfeh, G.H. McKinley and S.H. Spiegelberg, “Relaxationof dilute polymer solutions following extensional flow.” J. Non-NewtonianFluid Mech. 76 79-110 (1998).

Frattini, P.L and G.G. Fuller, “A note on phase-modulated flow birefringence:a promising rheo-optical method.” J. Rheology, 28 61-81 (1984).

Fuller, G.G. Optical Rheometry of complex fluids Oxford University Press Inc.,Oxford (1995).

Gere, J.M. and S.P. Timoshenko, Mechanics of Materials second SI edition,VNR International Co. Ltd, Hong Kong (1989).

Hecht, E. and A. Zajac, Optics. Addison-Wesley, London (1974).

Mackay, M.E. and D.V. Boger, Flow visualisation in rheometry. In A.A.Collyer and D.W. Clegg, Rheological measurement, in chapter 14. ElsevierAmsterdam (1988).

31

32 Chapter 2

Macosko, C.W. Rheology principles,measurements and applications VCH Pub-lishers Inc. New York (1994)

Reijden-Stolk van der, C. “A study on deformation and break-up of dispersedparticles in elongational flow.” PhD thesis Delft University of Technology,The Netherlands (1989).

Spiegelberg, S.H., D.C. Ables and G.H. McKinley. “The role of end-effects onmeasurements of extensional viscosity in viscoelastic polymer solutions witha filament stretching rheometer.” J. Non-Newtonian Fluid Mech. 64 229-267(1996).

Walters, K. Recent developments in rheometry. In Proc. XIth Int. Congr. onRheology, Elsevier Science Publishers B.V., Brussels (1992)

Watson, G.N. A treatise of the theory of Bessel functions. second edition,Cambridge University Press, Cambridge (1966).

Chapter 3

Fluid characterization

In this chapter a description will be given of the characterization of the polyisobuty-lene (PIB)/decalin solution and a polystyrene (PS)/decalin solution. First the samplepreparation process of the PIB solution will be described. Next we will show the ex-periments in shear flow. After that, we will only mention the main properties ofthe Newtonian PS solution. Then an extensive description of two viscoelastic mod-els (Giesekus model and FENE model) will be given. After we have explained themodels, we will obtain the fit parameters for these models from the characterizationexperiments of the PIB solution. Two different models are fitted: the macroscopic4-mode Giesekus model and the microscopic FENE model. Finally the results of thedetermination of the stress optical constant of the PIB solution will be given. Thischapter will end with conclusions of the results.

3.1 Sample preparation of a PIB solution

Preparing a suitable fluid for flow induced birefringence experiments, a number ofproblems have to be overcome. First of all the solution should be transparent. Secondthe polymer molecules should generate a measurable amount of birefringence duringflow. The last criterium can only be resolved by doing some experiments, because theamount of birefringence depends on the type of solvent and the concentration. Whena possible combination of polymer and solvent has been found, further problems areencountered during mixing of the solvent and the polymers. By stirring the solutionwith a fast rotating propeller, polymer chains will be chopped into pieces and thedegradation process will be accelerated. The choice of a high viscosity solvent willlead to bubbles in the solution during mixing which will never leave the fluid. Thisis not acceptable for laser experiments because bubbles will scatter the laser beam.

In our experiments a low viscosity solvent, which evaporates hardly due to its lowvapor pressure, and a high-molecular polymer has been used to obtain a high enoughlevel of fluid elasticity. For the solvent we have used decalin, also called decahydron-

33

34 Chapter 3

aphthalene, (Merck, purity 95 %) which is a Newtonian solvent with a viscosity of 4mPa s. It is a lightly flammable and slightly corrosive fluid but not extremely toxicfor ones health. For the polymer we selected polyisobutylene (PIB) with a molec-ular weight of Mw = 4, 300, 000 g/mol (BASF Oppanol 100). In the experiments a2%(w/w) solution was used.

The PIB was cut into small pieces of size 1 cm3 and put into a large drum (15 liters)filled with decalin. The mixing occurred at room temperature by slowly rotating (20rev./min) the drum for at least 180 hours in order to obtain a homogeneous solution.

The concentration of the solution is too high to be considered as a dilute solution. So-lutions are considered concentrated when their concentration exceeds a critical valuec∗. This is called the polymer-coil overlap concentration. Above this concentrationmolecular chains start to interact with each other. For polyisobutylene (PIB) solu-tions with a molecular weight of order 1 · 106 this concentration is of order c∗ = 0.2%(w/w) [Quinzani et al. (1990)]. From a theoretical point of view, it is better to dealwith dilute solutions because it simplifies the assumptions and calculations. Froman experimental point of view however a higher concentration is preferred becausethen form birefringence effects (see subsection 2.1.2) can be neglected. Also, a higherconcentration will generate more viscoelastic behavior and show more birefringenceduring flow. In our experiments we have to come to a compromise and a higherconcentration was chosen.

3.2 Experiments in shear with the PIB solution

Before showing the results of the calculations, the constitutive parameters shouldbe determined. These parameters are the dominant relaxation times, the relaxationmoduli and parameters for shear thinning behavior. For the characterization of thesolution in shear flow, a Rheometrics ARES rheometer was used. The experimentswere performed in a Couette geometry with a bob diameter of 32 mm. For the mea-surements of the first normal stress difference N1 a cone-plate geometry with a coneangle of 0.0408 radians and a diameter of 50 mm was used. For the characterizationof the solution several experiments were done: steady-state shear, start-up flow inshear and dynamic experiments for determining the storage modulus G′(ω) and theloss modulus G′′(ω). All experiments were performed at a temperature of 21 C.

The calculations were performed with the DYNAFLUID program which was devel-oped in-house [Hulsen (2000)]. This program calculates the time evolution of thestresses by solving the (stochastic) differential equations or the constitutive equa-tions. In our calculations the macroscopic 4-mode Giesekus model and the microscopicFENE model were used. The motivation for using these models and a description ofthese models can be found in a later section of this chapter. In section 3.5 the valuesof constitutive parameters of the models will be shown and details about the fittingprocedure will be given. In this section only the experiments and data fits will beshown. In a later section we will give the parameters for these data fits. In the fol-

Fluid characterization 35

G′′ measurementsG′′ fitG′ measurementsG′ fit

Frequency ω (rad/s)

Mod

uliG

′an

dG

′′(P

a)

10001001010.10.010.001

100

10

1

0.1

0.01

0.001

Figure 3.1: Dynamic measurements of the PIB solution together with a 4-modeMaxwell fit at a temperature of 21 C

lowing sections 3.2.1 to 3.2.4, experiments are plotted with dots and the calculationswith the fitted parameters are plotted with lines. First we will show the linear vis-coelastic properties of the solution by plotting G′(ω) and G′′(ω). Second the steadyshear viscosity as a function of shear rate will be shown. Finally the experiments ofstart-up behavior and N1 will be given.

3.2.1 Dynamic measurements

Figure 3.1 shows the storage and loss moduli G′(ω) and G′′(ω) as a function of theoscillation frequency ω. Below a frequency of 0.1 rad/s the signal was out of therange of the transducer of the rheometer. The applied strain during the experimentswas 5%. For the value in this range the linear viscoelastic regime is valid. This waschecked by performing oscillation experiments for various strains.

3.2.2 Steady shear measurements

In figure 3.2 the viscosity as a function of shear rate is illustrated. The viscositydecreases more than a factor ten when the shear rate γ increases from 0.1 to 100 s−1.The zero-shear-rate viscosity of the 2% PIB solution is 1.21 Pa s. Both models givean almost perfect fit to the data.

36 Chapter 3

4-mode Giesekus model1-mode FENE modelExperiment PIB solution

Shearrate γ (s−1)

Visco

sity

η(P

as)

1001010.10.010.001

10

1

0.1

0.01

Figure 3.2: Steady shear viscosity of the PIB solution measured at a temperature of21 C. The lines are fits of the models

3.2.3 Start-up in shear flow

In the figures 3.3 and 3.4 we have plotted the start-up behavior of both our experi-ments and calculations of the FENE model and 4-mode Giesekus model respectively.Measurements of the transient viscosity η+(t, γ0) are performed for four shear ratesγ0 = 0.3, 1, 10 and 100 s−1. The 4-mode Giesekus model is in excellent agreementwith the measurements over more than two decades in shear rate while the micro-scopic FENE model shows a large overshoot. For times smaller than 0.05 seconds,deviations become large in both the 4-mode Giesekus model and the FENE model.In our opinion, this is caused by the inertia of the rotating cup of the rheometer andthe finite response time of the electronic system. Therefore the values of the first datapoints in these figures are not reliable.

3.2.4 Measurements of the normal stress difference N1

Finally the measurements of the first normal stress difference N1 are shown in figure3.5. These measurements were performed in a cone-plate geometry. Care must betaken during sample loading of the rheometer. When the solution is squeezed betweenthe cone-plate geometry, normal stresses will be introduced. Therefore after loadingwe waited for more than 10 minutes for the relaxation of all stresses in the samplebefore the measurements start.In this plot we plot two sets of experimental data (+ and ×). These data points are


1-mode FENE modelγ0=100γ0=10γ0=1γ0=0.3

Time (s)

Tra

nsient

viscos

ityη+(t,γ

0)(P

as)

1001010.10.01

10

1

0.1

0.01

Figure 3.3: Transient shear viscosity measured at a temperature of 21 C and calcu-lated with the FENE model.

4-mode Giesekus modelγ0 = 100γ0 = 10γ0 = 1γ0 = 0.3

Time (s)

Tra

nsient

viscos

ityη+(t,γ

0)(P

as)

1001010.10.01

10

1

0.1

0.01

Figure 3.4: Transient shear viscosity measured at a temperature of 21 C and calcu-lated with the 4-mode Giesekus model

38 Chapter 3

1 mode FENE model4 mode Giesekus model

Shearrate γ (s−1)

First

Nor

malstress

diffe

renc

eN

1(P

a)

1001010.10.01

100

10

1

0.1

0.01

0.001

0.0001

Figure 3.5: Steady state measurements of N1 with a cone plate geometry and calcula-tions with 4-mode Giesekus and FENE model. The parameters for these models aredetermined from the steady shear viscosity and start-up measurements. So the linesin the plot are not fits of the data.

two sets of similar measurements. They show a large spreading at higher shear rates.This illustrates the complexity of doing N1 measurements in experiments. It shouldbe noticed that the lines are not fits of the data. The parameters for these calculationsare determined from the steady shear viscosity and start-up measurements.

3.3 Experiments with a PS solution

The second fluid we have used in our research is a polystyrene (PS)/decalin solution.From a rheological point of view however, this fluid is not of any interest because it isan almost inelastic fluid. We only mention the main-features of this fluid and we willnot show any measurements and data fits of this solution. In chapter 5 we will makea qualitative comparison between a PIB/decalin solution and a PS/decalin solutionand therefore we will give some information of this solution. We made a 10%(w/w)solution by mixing little pieces of PS for 200 hours on a rollerbank. The molecularweight of the polystyrene was Mw = 380000 and thus the total chain length of themolecule relatively short for a macromolecule. The solution behaves like a Newtonianfluid and its zero-shear-rate-viscosity is 0.69 Pa s. The relaxation time λ < 0.04seconds. Due to this behavior we must conclude that this solution is not suitable for


making a comparison between experiments and viscoelastic calculations.

3.4 Viscoelastic models

We have shown the experiments in shear flow. We will fit the data with two viscoelas-tic models, after we have described these models in detail in this section. In this studywe will use the Giesekus model and the FENE model. Viscoelastic behavior of fluidsis already known for more than one hundred years. In 1867 Maxwell tried to describethis behavior by modeling a lump of fluid with a linear spring (spring constant G)connected in combination with a dash pot (viscosity η). When a certain displacementof this system is assumed, the force balance can be described by :

η

G

dF

dt+ F = η

dγ

dt. (3.1)

Where F is the force in the spring and γ the displacement.In terms of fluid dynamics, force F can be translated into the stress τ . The changesof displacement are translated into the strain and the quantity η/G is called therelaxation time λ. Rearranging and writing in a tensor form, we obtain :

λτ+ τ = 2ηD. (3.2)

Where τ is the stress tensor and D the rate-of-deformation tensor. The symbolis theupper convected time derivative. This model is called the upper convected Maxwellmodel and is very simple macroscopic model. A more extensive description of thismodel can be found in every textbook about rheology (see for example [Macosko(1994)]).

3.4.1 Giesekus Model

The Maxwell model is one of the basic viscoelastic models. A total overview of manymodels can be found in the book written by Larson [Larson (1988)]. One of the generalapplied models is the Giesekus model. Due to its age of more than thirty years, manyreferences to this model can be found in literature and therefore it will be describedbriefly in this thesis. This model will be also used in our calculations. Characteristicfor the Giesekus model is that it is based on anisotropic drag. The motivation for themodel comes from considering how the relaxation of a polymer molecule is modifiedwhen it is surrounded by other molecules that have other orientations. The moleculesare described by elastic dumbbells. Starting from this molecular level, Giesekus de-rived the following constitutive equation, modeling the bulk behavior of a viscoelasticfluid [Giesekus (1982)]:

λτ+ τ +

λα

ητ 2 = 2ηD, (3.3)

where λ is the relaxation time, τ the stress tensor, D the rate-of-deformation ten-sor and η the viscosity. In this equation anisotropic drag is taken into account by

40 Chapter 3

means of the parameter α. Giesekus speculated that the surrounding sea of orientedmolecules creates an anisotropic drag. When all dumbbells are oriented in one direc-tion the drag along this orientation direction is much less than perpendicular to it.This direction dependency is expressed by the parameter α which is also called themobility parameter. The maximum and minimum anisotropies correspond to α = 1and α = 0. When α does not lie in this range stresses can grow to infinity. Whenα is set to zero the upper convected Maxwell model (eq. 3.2) is recovered. In orderto obtain realistic calculations α is set between 0 and 0.5. Setting α > 0.5 the elon-gational viscosity decreases for increasing strain rate [Hulsen (1988)], which is veryunrealistic for solutions used in this study.

Despite of the age, the Giesekus model is still one of the best models for calculationsof steady shear viscosity and the first and second normal stress differences. Espe-cially when a multi-mode Giesekus model is used. Drawback of using this multi-modeGiesekus model is the physical interpretation of the different modes. Using a 4-modeGiesekus model in cross-slot flow geometry [Schoonen (1998)] results are in a agree-ment with experiments and calculations. In this study more or less a same geometryand solution are used and thus we have the possibility to check the experiments withcalculations.

3.4.2 FENE model

A different approach to model viscoelasticity is a microscopic description at molec-ular level. In the microscopic way of modeling, it is assumed that the polymersare dissolved in a Newtonian fluid. The polymer molecules can be represented by achain of beads connected with springs. The simplest way of modeling is to consider amolecule as two beads connected with a spring, the so-called dumbbell model. Duringflow several forces are working on a dumbbell :

• Hydrodynamic forces. These forces are caused by drag of the beads during flow.This force is given by the Stokes friction force.

• Brownian forces. These forces are caused by collisions of solvent molecules withthe beads.

• Intramolecular forces. Forces caused by stretching and bending the polymerchain. They are represented by the spring connecting the beads.

• Intermolecular forces. Forces caused by friction with other polymer molecules.These forces become important for concentrated solutions and melts.

• Inertia forces.

During flow, these forces are in equilibrium. Most of the times, the last two forces canbe neglected when dilute solutions is the object of study. In the simplest dumbbellmodel, the Hookean dumbbell model, a linear spring connects the two beads. This


model however has some disadvantages, the shear viscosity is constant and the exten-sibility of the dumbbell is infinite. Replacing the linear spring by a non-linear springa more sophisticated model called the FENE-model, is obtained. The FENE model(Finite Extensible Non-linear Elastic) was first used by Warner [Warner (1972)]. Inthis model the intermolecular forces are also not taken into account. In the FENEmodel the spring force F is given by :

F =HQ

1−Q2/Q02 . (3.4)

In this expression H is a measure for the spring stiffness, Q is the connector vectorbetween the beads, Q is the length of the spring and Q0 is the maximum possiblespring length. During flow, the dynamics of a suspension of dumbbells in a solventcan be described by a stochastic differential equation. This equation describes thetime evolution of the dumbbell connector vector Q(t). The time evolution of theinternal configuration of a single FENE dumbbell is governed by [Ottinger (1996)] :

dQ(t) =[κ(t) · Q(t)− 2

ζF(t)

]dt +

√4kTζ

dW(t). (3.5)

Looking at expression (3.5), we see two contributions on the right-hand side which canchange the dumbbell configuration dQ(t). In the first term, which has a deterministiccharacter, κ is the transpose of the velocity gradient, ζ is the friction coefficient of abead and F is the spring force calculated in equation (3.4). The second term describesthe Brownian forces and this term is of stochastic character. The vector dW(t) iscalled the Wiener increment, this is the Brownian step made in time step dt. Thecomponents of vector dW(t) are Gaussian distributed random numbers. The factorin front of dW(t) is a measure of the strength of the Brownian forces, where k isBoltzmann’s constant and T is the absolute temperature. For calculating the stressτ we use the Kramer’s expression for the stress averaging over N dumbbells [Bird(1987)]:

τ =1N

N∑i=1

QiFi. (3.6)

In literature, the quantities in equations (3.4) and (3.5) are often made dimensionless.Lengths will be measured in terms of

√kT/H. Time will be measured in terms of

ζ/4H which will be denoted as λH and can be considered as the relaxation time ofthe polymer molecule. Another parameter b found in literature is the so-called finiteextensibility parameter. The definition of b is :

b =HQ0

2

kT. (3.7)

In practice the parameter b influences the onset of shear thinning and strain harden-ing behavior. When the value of b is chosen large, the onset of shear thinning will

42 Chapter 3

start at high shear rate levels.

The description of the stochastic differential equation (3.5) above is valid for a singleFENE dumbbell. In the simulations, all macroscopic quantities of interest such as thepolymer stress, are determined from the configurations of a set of dumbbells. There-fore a large number of dumbbells dissolved in the flow must be simulated to obtaina statistically reliable value of the bulk properties. Ottinger and Laso introduced anew method in the literature, the so-called CONNFFESSIT method (Calculation ofNon-Newtonian Flow: Finite Elements and Stochastic Simulation Techniques), whichcombines microscopic Brownian dynamics simulations with the macroscopic finite el-ement method [Ottinger and Laso (1992)]. The idea is to disperse a large number ofmodel polymers in the flow domain at the start of the simulation. These polymersare convected and deformed by the flow field and for each of them the trajectory ofthe center of mass is calculated and simultaneously the evolution equation governingtheir configuration along these trajectories is computed. Thus, given the momentaryvelocity field, the new position and the new configuration of each of the model poly-mers is obtained. Next, the new polymer stress at each position is calculated from theupdated polymer configurations at that position. The resulting stress is subsequentlyused in the finite element part to solve for the momentum equation in order to obtainthe new update for the velocity and the pressure fields.

From a computational point of view, this method is far from optimal and the ma-jor drawback is that, when there are no dumbbells at a certain position in the flowfield, no stresses can be calculated. Recently, a study of van Heel [van Heel (2000)]appeared, in which more computationally efficient algorithms were described for themicroscopic simulation of viscoelastic flows.

Here we used one of these algorithms, the so-called Brownian configuration fieldsmethod [Hulsen et al. (1997)]. In this approach, instead of simulating individualdumbbells, the polymer configurations are stored in a vector field Q(x, t) ,the so-calledconfiguration field. This field can be thought of as a kind of continuous interpolationbetween the configurations of the dumbbells. Now in every point of the flow field thestress can be found. This simulation algorithm can be described as follows: Assumea certain velocity field like drawn in figure 3.6. This velocity field deforms the con-figurations of the dumbbells in each configuration field. Also the random Brownianmotion deforms the dumbbell configurations. All these deformations are stored in aconfiguration field. Using a constitutive equation for a single dumbbell, stresses in allconfiguration fields can be determined. Averaging over all these fields, a stress fieldcan be obtained. The more configuration fields are used, the less the noise due to theBrownian forces in the stress field can be seen. Using this stress field for the input ofbalance equations, a new velocity field is obtained and the cycle starts again. In oursimulations we used 2000 configuration fields to get a smooth stress field. Using thissimulation algorithm and updating the stress field for every time step a time evolutionof stresses can be found.


Figure 3.6: Simulation algorithm of the microscopic configuration fields method inflow around a sphere (Courtesy of Dr. E.A.J.F. Peters)

3.5 Determination of the constitutive parameters

The determination of the constitutive parameters of the PIB solution will be the inputfor the calculations in the next chapter. A description of the models used in this studyhas been given in the previous section. In this section only the parameter fit processwill be described.

3.5.1 Parameters for the multi-mode Giesekus model

To fit a multi-mode Giesekus model, we first fitted the the dominant relaxation timesλi and relaxation moduli Gi on the measurements of G′(ω) and G′′(ω) in figure 3.1.Fitting the storage (G′)and the loss (G′′) moduli, the following relations from thetheory of linear viscoelasticity [Bird et al. (1987)] are used :

G′(ω) =p∑

i=1

Giω2λi

2

1 + ω2λi2 , (3.8)

G′′(ω) =p∑

i=1

Giωλi

1 + ω2λi2 . (3.9)

Where p is the number of modes. For a given number of modes an optimal setof relaxation times λi and moduli Gi can be calculated using a non linear fitting

44 Chapter 3

algorithm. In this algorithm a least-square approximation is used fitting the twomoduli G′(ω) and G′′(ω) simultaneously to the data. In our algorithm the criterium

N∑(logG′

mes − logG′fit)

2 + (logG′′mes − logG′′

fit)2 (3.10)

is minimized. Here N is the number of data points and G′fit, G

′′fit and G′

mes , G′′mes are

respectively the fitted and measured moduli.When this fitting algorithm is used one should pay attention to the following issues.First, the number of modes should be as small as possible and second the fit on thedata should be smooth. Playing around with these criteria a four mode fit gave asmooth and accurate fit of the data as can be seen in figure 3.1. The parameter valuesηi, λi and αi are given in table 3.1

Other parameters in the Giesekus model should be determined next. The Giesekusmodel written in a general multi-mode form looks like :

λiτ i

+ τ i +λiαi

ηiτ i

2 = 2ηiD. (3.11)

Where λi is the relaxation time, τ the stress tensor, D the rate of deformation tensor,ηi the viscosity and αi is the mobility parameter and i is the number of the mode. Forevery mode there are three parameters, λi,ηi and αi. Two parameters for each modei are already known from the the data fit of the moduli: λi and ηi because ηi = Giλi.The fit of the third parameter αi is determined by trying to find the best set of αi’suntil the pretty good fit of the steady shear viscosity and transient shear viscositywas found. Most authors [Baaijens (1994), Schoonen (1998)] use the same αi for eachmode. But this does not give the best fit for the start-up behavior in shear. In thisthesis for each mode another αi is used which results in an excellent agreement ofthe experiments and calculations (figure 3.4). Writing a fitting algorithm for αi andimplement it into a computer program may give better results for the fit. Howeverour data fits are already very good, therefore we decided that this was not useful.The values of the fit parameters can be found in table 3.1 below.

Table 3.1: The constitutive parameters for the 4-mode Giesekus model.

Mode ηi [Pa s] λi [s] Gi [Pa] αi

1 0.663325 5.210315 0.12731 0.182 0.403795 0.818716 0.49320 0.503 0.117298 0.114166 1.02743 0.484 0.025726 0.011750 2.18935 0.15


3.5.2 Parameters for the FENE model

In the FENE model the polymers in the solution are represented by two beads con-nected by a non-linear spring. There are only three parameters to describe thisso-called dumbbell model. Those parameters are: the relaxation modules G = nkT ,the relaxation time λH and the parameter b for the non-linearity of the spring whichis called the finite extensibility parameter.

Fitting the microscopic FENE model by using a least square fit was impossible dueto the absence of a closed form constitutive equation. Therefore we used anothermethod. In DYNAFLUID the average stress of 25000 dumbbells after reaching thesteady state condition was calculated for one shear rate. Doing this for many shearrates, curves like figure 3.2 were obtained. So several shear viscosity functions werecalculated and then the best fit was selected. Again a fitting program might givebetter results. For this application the data fit in figure 3.2 is already good enough.For the first iteration the parameters of the 4-mode Giesekus model with the largestrelaxation time were used (nkT = 0.12731 and λH = 5.210315). However for thesevalues, the finite extensibility parameter b has to be chosen b = −11.15 in order toobtain the right value of 1.21 Pa.s for zero-shear-rate viscosity found in the experi-ments. The zero-shear-rate viscosity of a FENE model is given by [Bird et al. (1980)and Bird et al. (1987) ]:

η0 =b

b+ 5nkTλH . (3.12)

Choosing a negative value for b is not realistic so for the first iteration b was set tob = 100. Varying these constitutive parameters λH , nkT and b and taking care of theright value of the zero-shear-rate viscosity η0 using different sets of parameters, wefound:

Table 3.2: The constitutive parameters for the FENE model.

nkT [Pa] λH [s] b [-]0.135 10.0 50

We also tried to fit the transient shear viscosity by adjusting the parameters. Butthis was impossible to do without changing the steady shear viscosity function. Thelarge overshoots in the calculations (see figure (3.3) were still present using a widerange of the parameters nkT , λH and b.

Of course, it would be possible to achieve a better description of the transient shearviscosity data by fitting two or more FENE modes in a similar empirical way asused for the Giesekus model. However this is inconsistent with the spirit in whicha dumbbell model is being used. The FENE dumbbell model is considered to bea model for exploring the consequences of a non-linear relationship between force

46 Chapter 3

and overall molecular extension. These physical insights into the role of extensibilitywould become unclear when multiple modes are used.

3.6 Determination of the stress optical constant

Determining the stress optical constant (C) (see chapter 2) provides the possibilityto get the relation between birefringence and stress. In order to obtain this constant,both birefringence (∆n), orientation angle (χ) and stress (τ) have to be measured si-multaneously. Most of the time the stress optical constant (C) is determined in shearflow due to many experimental difficulties which will appear when an elongation flowis used [Macosko chap. 9 sec. 4.3,(1994)].

In our case a Couette cell was used. The measurements have been done on a Rheo-metrics ARES rheometer with an optical module also made by Rheometrics. In the1 mm gap between the bob and cup a shear flow is created. Considering the stressoptical relation

τxy =∆n

2Csin(2χ), (3.13)

two optical quantities (∆n and χ) and one mechanical quantity (shear stress τxy) haveto be measured. Measuring ∆n and χ is relatively simple using the optical moduleof Rheometrics. Determining the shear stress τxy from torque measurements is alsopossible. Plotting ∆n sin(2χ) against 2τxy results in a line having a slope of the stressoptical constant (C). However the experiments were difficult to execute. Due to thedesign of the optical Couette cell high shear rates could not be reached. At these highshear rates, the solution starts to flow over the upper glass window of the Couettecell and then the laser beam will be refracted. Using this low shear rate range, thelevel of birefringence of the solution was a few times above noise level of the opticalmodule. Measuring long time series (5 minutes) and time averaging resulted in usefulresults. The experiments were done at shear rates of γ is 3, 5.5, 10, 13 and 20 s−1

at 23 C. In figure 3.7 the results can be seen. A least square fit through the datapoints was made. From the slope of this line, we found C = 2.72 · 10−9 which is equalto the stress optical constant. The order of magnitude of this value is in agreementwith values found in literature for polymeric liquids. However care must be takento compare the stress optical constants in literature. Mackay [Mackay et al. (1988)]warned for a deviation up to 20 % in the stress optical constant of any polymericliquid between different batches. A saturation of the birefringence ∆n can not beseen due to the low shear rates used in the experiments. Extrapolating this line tothe origin, we see that it does not intersect the origin. This may be caused by thepresence of some residual birefringence in the glass windows of the Couette cell.


FitMeasurements

2τxy (Pa)

∆nsin(

2χ)(-)

4.543.532.521.5

1.3e-08

1.2e-08

1.1e-08

1e-08

9e-09

8e-09

7e-09

6e-09

5e-09

4e-09

Figure 3.7: Optical measurements done at a temperature of 23 C. The slope of theline is equal to the stress optical constant (C = 2.72 · 10−9 Pa−1)

3.7 Discussion and conclusions

At the end of this chapter, one important point has to be discussed. The character-ization of the fluid was done in shear flow. Shear flow is one of the basic flows inrheology for determining the fluid characteristics. However, recent insights show theimportance of a characterization in elongational flow as well. Measurements of theelongational viscosity of polymeric liquids are extremely difficult. In our laboratorythere is an an extensional rheometer, creating an uniaxial elongation. A descrip-tion of this apparatus can be found in Nieuwkoop [Nieuwkoop et al. (1996)]. But thezero-shear-rate viscosity of the PIB/decalin solution is too low to measure a detectableforce when a droplet is stretched to a filament. This was also the problem of Schoonenwhen he tried to characterize a tetradecane/PIB solution using the same rheometer[Schoonen (1998)]. Despite these difficulties of doing elongational experiments, anaccurate characterization of the polymeric solutions in both shear and elongationalflow will be crucial, for making progress in microscopic modeling in the future. Nowonly predictions of the elongational viscosity based on parameters found in shear flowcan be made. These predictions will be shown in chapter 4 when a comparison willbe made between the Giesekus model and FENE model.

After showing the results of the fluid characterization the main conclusions of thischapter will be given on the next page.

48 Chapter 3

• The 2%(w/w) solution of PIB and decalin is suitable for doing flow induced bire-fringence experiments. It is important to avoid bubbles during the preparationof the fluid.

• The 10%(w/w) PS/decalin solution does not show any viscoelasticity at inter-mediate timescales. This solution is not suitable for our research.

• The macroscopic 4-mode Giesekus model is in excellent agreement with boththe steady state and start-up measurements in shear flow.

• The microscopic FENE model only agrees with the steady state measurementsin shear flow.

• The moduli G′(ω) and G′′(ω) are fitted using 4 modes. This results in anaccurate and smooth fit

• The measurements of N1 deviate a little, but are acceptable for the calculationswith the 4-mode Giesekus model and FENE model.

• We have selected the non-linear parameters of the models by trial and errorbased on the graphs. The resulting parameters are considered good enough forthe numerical simulations

• For a slight improvement of the data fits, computer programs may give betterresults.

• Although the experiments were difficult to perform, the stress optical constantcould be determined. The value of C = 2.72 · 10−9 Pa−1 is in agreement withvalues found in literature.

• For making progress in modeling polymeric solutions, a characterization in bothshear and elongational flow must be made. However from an experimental pointof view elongational experiments are extremely difficult to do for the polymersolutions considered here.

References

Baaijens, J.P.W, “Evaluation of constitutive equations for polymer melts andsolutions in complex flows.” PhD thesis, Eindhoven University of Technology,The Netherlands (1994).

Bird, R.B., P.J.Dotson and N.L. Johnson, “Polymer solution rheology based ona finitely extensible bead-spring chain model.” Journal of Non NewtonianFluid Mech. 7 , 213-235, (1980).

Bird, R.B., F.C.Curtiss, R.C. Armstrong and O.Hassager. Dynamics of poly-meric liquids, Vol.2 John Wiley & Sons, New York, (1987).

Feigl, K., M. Laso and H.C. Ottinger, “CONNFFESSIT approach for solving atwo-dimensional viscoelastic problem” Macromolecules, 28, No. 9, (1995).

Frattini, P.L and G.G. Fuller, “A note on phase-modulated flow birefringence:a promising rheo-optical method.” J. Rheology, 28 61-81 (1984).

Fuller, G.G. Optical Rheometry of complex fluids Oxford University Press, Inc.Oxford (1995).

Giesekus, H. “A simple constitutive equation for polymer fluids based on theconcept of deformation-dependent tensorial mobility.” J. Non-NewtonianFluid Mech. 11 69-109, (1982).

Heel van, A.P.G, “Simulation of viscoelastic fluids. From microscopic modelsto macroscopic complex flows.” PhD thesis, Delft University of Technology,The Netherlands (2000).

Hulsen, M.A. “Analysis and numerical simulation of the flow of viscoelasticfluids.” PhD. thesis Delft University of Technology , DUP Delft (1988).

Hulsen, M.A. ,A.P.G van Heel and B.H.A.A van den Brule, “Simulation of vis-coelastic flows using Brownian configuration fields.” J. Non-Newtonian FluidMechanics, 70 79-101, (1997)

49

50 Chapter 3

Hulsen, M.A. “DYNAFLOW, a finite element program for viscous and viscoelas-tic fluid flow problems.” Reference manual, Delft University of Technology,The Netherlands (2000).

Horst, ter J.W., “Simulation of viscoelastic flow in a journal bearing geometryusing Brownian configuration fields” MSc thesis internal report MEAH-172 lab. for Areo and hydrodynamics, Delft University of Technology (1998).

Joseph, D.D. Fluid dynamics of viscoelastic liquids Applied MathematicalSciences, Vol. 84, Springer-Verlag, New York (1990).

Larson, R.G. Constitutive equations for polymer melts and solutions Butter-worths (1988).

Laso, M. and H.C. Ottinger, “Calculation of viscoelastic flow using molecularmodels: the CONNFFESSIT approach” J. Non-Newtonian Fluid Mechanics,47 1-20, (1993)

Mackay, M.E. and D.V. Boger, Flow visualization in rheometry. In A.A. Collyerand D.W. Clegg, Rheological measurement, Chapter 14. Elsevier, Amsterdam(1988).

Macosko, C.W. Rheology principles, measurements and applications. VCHPublishers Inc., New York (1994).

Nieuwkoop, J.E. van, and M.M.O Muller von Czernicki. “Elongation andsubsequent relaxation measurements on dilute polyisobutylene solutions.”Journal of Non Newtonian Fluid Mech. 67 , 105 (1996).

Ottinger, H.C. and M. Laso, ““Smart” polymers in finite-element calculations”.In Moldenaers, P. and Keunings, R., editors, Proc. XIth Int. Congr. onRheology, Elsevier Science Publishers B.V., Brussels, Belgium, 286–288(1992).

Ottinger, H.C. Stochastic Processes in polymeric fluids., Springer Verlag, Berlin,(1996).

Quinzani, L.M., G.H. McKinley, R.A. Brown and R.C Amstrong. “Modelingthe rheology of polyisobutylene solutions.” Journal of Rheology, 34, 705-748,(1990).

Schoonen, J.F.M. “Determination of Rheological Constitutive Equations usingComplex Flow.” PhD thesis. Eindhoven University of Technology, TheNetherlands (1998).


Walters, K. Recent developments in rheometry. In Proc. XIth Int. Congr. onRheology, Elsevier Science Publishers B.V., Brussels (1992).

Warner, H.R. “Theory and rheology of dilute suspensions of finitely extensibledumbbells.” Ind. Eng. Chem. Fundam., 11 379-387, (1972).

52 Chapter 3

Chapter 4

Calculations in a four-roll millgeometry

In this chapter we give an overview of the calculations performed in a four-roll millgeometry. First a three dimensional (3D) numerical investigation of the boundarylayers caused by the bottom and top of the flow cell will be presented. In this way wecan validate the correction method which was proposed in chapter 2. Second, a param-eter study, using a simple macroscopic viscoelastic model will be used in order to getmore physical insight in the interaction of shear thinning behavior and strain harden-ing behavior. Also the contribution of inertia effects in the flow will be investigated.Third, a comparison of the macroscopic calculations and microscopic calculations willbe presented. Finally this chapter will end with a discussion of the results.

4.1 Introduction

For the calculations on the four-roll mill, we use the same dimensions of the grid asused in the experimental set-up. The lay-out of this geometry can be found in figure4.1. First we will present 3D inelastic steady state calculations for the research ofthe boundary layers and the validation of the proposed correction method in chapter2. In section 4.3 we will focus on two dimensional (2D) steady state viscoelasticcalculations. We will investigate the influence of the Deborah number on the flow.A simple 1-mode Giesekus model is used for a parameter study to discover the mainfeatures of fluid flow in the four-roll mill geometry. In section 4.3.5 instationaryviscoelastic calculations will be shown. Calculations of a harmonic oscillation of thecylinders will be investigated. Important are the different times scales which appear.In section 4.4 a comparison between the microscopic calculations and the macroscopiccalculations will be made. For these calculations we will use the parameters foundin the fluid characterization which we described in chapter 3. Finally this chapterwill end with a discussion on the calculations. But first we will give an overview

53

54 Chapter 4

of the geometry and define some dimensionless numbers which are important in thecalculations.

4.1.1 Geometry and dimensionless numbers

A global overview of the geometry can be seen in figure 4.1. In this figure, the maindimensions are marked. For the length L and width B we have L = B = 150 mmand for the radius of the cylinders we have R = 30 mm. The smallest slit distancebetween the cylinders is D = 12 mm. For the calculations in three dimensions, wevaried the height. We did calculations for H = 32, 55, 80 and 110 mm.

There are a few dimensionless numbers which are important for the characterizationof this geometry and the flow in this geometry. The dimensionless number which isimportant for the 3D calculations, is the aspect ratio A given by :

A =H

D. (4.1)

Where H is the height of the flow container and D is the smallest slit distance be-tween the cylinders. The larger A, the more a 2D flow is approximated. In our 3Dcalculations we varied A between 2.67 and 9.17.

Another dimensionless number which we used in the calculations is the Reynoldsnumber. For this geometry we used the following definition:

Re =ρωRD

η0. (4.2)

Where ρ is the density, ωR is the tangential velocity at the wall of the cylinders, D isthe slit width between the cylinders and η0 is the zero-shear viscosity. The Reynolds

Central stagnationpoint

L

H

D/2 R

D/2ω

B

Figure 4.1: A quarter part of the total geometry, used in the calculations.

Calculations 55

number is a measure of the ratio of inertia forces and viscous forces. In our calcula-tions the Reynolds number was always smaller than 2.

Most of the measurements were performed in the central stagnation point and there-fore, we focused our calculations on this point too. We introduce the ratio of thestrain rate and the angular velocity:

S =εx

ω, (4.3)

where εx is the strain rate in the stagnation point and ω is the angular velocity of thecylinders. The value of S depends on the geometry. For a Newtonian fluid, neglectinginertia, S is a constant.

Finally we will introduce a set of dimensionless numbers, called the Deborah num-bers. The Deborah number expresses the ratio of the viscoelastic time scale and theresidence time of a polymer in the flow. A unique definition is difficult to give becauseit is dependent on the flow geometry and the position. Here we will use for the globalDeborah number:

Deg = λω. (4.4)

Where λ is the largest relaxation time of the polymer solution. For the Deborahnumber at a specific location we introduce the local Deborah number:

Del = λεx = SDeg. (4.5)

Here εx is the strain rate in the center of the four-roll mil geometry. In section 4.3.5we will show viscoelastic instationary calculations. Then the angular velocity of thecylinders will move with a periodic harmonic oscillation.

ω(t) = Ωmax sin(ωosct). (4.6)

Then two useful numbers can be defined. First we define a Deborah number basedon the oscillation frequency ωosc:

Deosc = λωosc. (4.7)

Second, we define the maximum possible angle of rotation of the cylinders of thefour-roll mill:

Φ =∫ t= π

ωosc

t=0

Ωmax sin(ωosct)dt. (4.8)

Φ =2Ωmax

ωosc. (4.9)

The meaning of those two numbers is important. They express the different regimes ofviscoelasticity. When Φ is very small, the solution shows linear viscoelastic behavior,while for large values of Φ non-linearity is important. The value of Deosc shows theratio of elasticity and viscosity in a flow similar to the oscillating frequency in dynamicmeasurements in a rheometer.

56 Chapter 4

4.2 3D Inelastic steady-state flow calculations

There are two fixed walls at the bottom and top of the experimental set-up whichinfluence the flow between the cylinders. The other fixed side walls do not influencethe flow near the rotating cylinders because they are far away. When the cylindersrotate, fluid will be pushed to the small slit between the cylinders with a velocitywhich is more or less equal to the tangential velocity of the cylinders. Near the bot-tom and top, this velocity is much lower due to the no-slip boundary condition of thefixed walls. These two walls will introduce boundary layers. When the length of thecylinders is large, the influence of the walls on the velocity of the flow halfway thecylinders is negligible. In that case a 2D flow is reached halfway the cylinders.

Nowadays, performing 3D viscoelastic calculations is possible, but calculations doneon a mesh containing more than a few thousand elements are still impossible dueto the lack of computer power. So at the moment, fully 3D viscoelastic calculationscan not be done for this geometry and we have to restrict ourselves to 2D viscoelasticcalculations. Because we would like to compare 3D measurements and 2D viscoelasticcalculations, a method for converting 3D flow data into 2D flow data was described inchapter 2. By means of numerical simulation of 3D inelastic flow, we can investigatethe order of magnitude of the thickness of the boundary layer in our flow container.When these dimensions are known, a minimal cylinder length can be calculated inorder to get a 2D flow halfway the cylinders. One of the aims of this section is theinvestigation of this boundary layer and the validation of the proposed correctionmethod.

Looking at studies of channel flow, an aspect ratio A = 10 gives a reasonable ap-proximation of a two dimensional flow [Wales (1976)]. In these studies, the instreamvelocity profile is plotted as a function of the channel height. When a flat velocityprofile is reached that is large compared to the boundary layers, the flow is called twodimensional. When the average velocity over the whole channel is compared to thevelocity on the centerline of the channel, a deviation of 10 percent was found whenan aspect ratio A = 10 was used [Wales (1976)]. Also Schoonen [Schoonen (1998)]did 3D calculations in a cross-slot flow geometry with an aspect ratio A = 8. Incontrast to Wales, he compared the first normal stress difference N1 in stead of thevelocity. He found a maximum error of 11 % with respect to the 2D flow case in thefirst normal stress difference N1.

In our case, the central stagnation point is the most important point in the flow wheremost of the measurements have been done. In this point all velocities are equal tozero. So we have to search for another method for the investigation of the boundarylayer. Looking at the velocity profile in the stagnation point is not an option. Moni-toring the stresses in this stagnation point is also impossible due to the inelastic flowcalculations.

Another way of investigating the boundary layer is looking at the strain rates in thestagnation point along the arrow line H in figure 4.1. The value of εx is non-zero and

Calculations 57

X

Y

Z X

Y

Z

XY

Z XY

Z

Figure 4.2: Grid used for 3D steady state calculations. The number of cells used inthe z-direction depends on the height of the cylinders. Small cells can be found inregions where velocities and velocity gradients become high.

58 Chapter 4

it will vary from zero at the walls to a maximum halfway the cylinders. Travelingalong arrow line H in figure 4.1, which is parallel to the rotation axis of the cylinders,a strain rate profile in the stagnation point can be obtained. We will use this methodin this study to investigate the boundary layers in the stagnation point. If the dimen-sions of the boundary layers are known, a minimum distance between bottom and topcan be obtained, in order to get a 2D flow field in the region halfway the cylinders.

4.2.1 Input parameters for 3D flow calculations

The 3D calculations are performed using the computer program CFX (version 4.2).CFX uses a finite volume method (FVM) for solving the equations of motion. It isa commercial available program made by AEA Technology Inc. This program canbe used for fully three dimensional non-Newtonian (inelastic) flow calculations usinga Carreau model for the viscosity. In all calculations, steady state conditions areassumed. We did calculations on three kinds of structured grids with 45000 , 102500and 222500 cells, respectively. We looked at the convergence and the results of the lasttwo grids were almost the same. So we used a grid containing 102500 cells, having thesame dimensions of the experimental set-up, which can be seen in fig. 4.2. It shouldbe mentioned that here half of the total geometry was used. From this point of viewwe did not use the computer memory very efficiently because only one-eighth of thegeometry is sufficient. However, using one-eighth part gave some problems with thecomputer program CFX. Due to a bug in the code the velocities in the stagnationpoint became not equal to zero. So it was decided to take half of the geometry. Inthat case the velocities in the stagnation point became zero. In CFX there is thepossibility to use the Carreau viscosity model which has the following expression forthe viscosity :

η(γ) = η0(1 + (Knγ)2)n−1

2 . (4.10)

Where η0 is the zero-shear-rate viscosity, Kn the consistency parameter, γ the shearrate and n the power law index. We used η0 = 1 Pa s, Kn = 1.1 and n = 0.85in the calculations. The shear thinning behavior of the Carreau model using theseparameters starts at γ = 0.1 s−1. The shear thinning behavior of our PIB/decalinsolution starts also at a shear rate of γ = 0.1 s−1 (see chapter 3). The density of thefluid was set to 950 kg/m3. This is roughly the density of many polymer solutionsbased on an organic solvent. In this study a few parameters were varied. First theangular velocity ω of the cylinders was varied. Then we can investigate whether theangular velocity changes the thickness of the strain rate boundary layers. The twocylinders in the grid (See figure 4.2) are counter rotating. In that case, if the gridgeometry is mirrored in the x-axis, an elongational flow is obtained.

The effect of the cylinder length is also investigated. The aim of these calculationsis to obtain the minimal distance between the top and bottom for maintaining a 2Dflow halfway the cylinders.

Calculations 59

4.2.2 Results of 3D flow calculations

In figure 4.3 the strain rate εx in the stagnation point is plotted as a function of z forthree different values of ω: 0.6, 1.2 and 2.4 rad/s. Then the Reynolds number willvary between 0.20 and 0.82. In this figure the length of the cylinders is 80 mm. Wedo not see a change of the thickness of the boundary layer. When the dimensionlessnumber S (eq. 4.3) is plot against z, all curves will coincide. This means there is alinear relationship between ω and the strain rate εx and the influence of inertia effectsand shear thinning on the flow are negligible. Defining the start of the boundary layerat 95 percent of the maximum value, we see that a boundary layer of approximately17 mm appears.

The change of strain rate εx along the cylinder in the stagnation point at differentlengths can be seen in figure 4.4. The angular velocity of the cylinders ω is 1.2 rad/s.We chose four different lengths H : 32 ,55, 80 and 110 mm. Then the aspect ratioA will vary between 2.67 and 9.17. When the cylinder length is chosen too small, aflat strain rate profile halfway the cylinders is never reached. Starting from a cylin-der length of 55 mm, a flat strain rate profile is reached. Now in the middle of thecylinder a region of approximately two dimensional flow is present. However, a littleovershoot in the strain rate at a cylinder length of 110 mm can be seen in figure 4.4.Fortunately this overshoot in εx is only a 3 %, so the flow can still be considered astwo dimensional. Certainly it is not caused by a bad grid cell distribution becauseextensive calculations of varying the number of grid cells in the z-direction, still showthis overshoot. An explanation for this overshoot may be the shear thinning effectsnear the walls.

A remarkable fact which appeared in the calculations was the presence of secondaryflows. In 2D flow only velocities in the x and y directions are expected. In our 3Dcalculations there is also a velocity component in the z direction. Near the bottomand top of the cylinder little eddies appear due to the no-slip boundary condition atthe top and bottom of the flow container. Calculations showed that the maximumupward or downward velocity has a value of three percent of the tangential velocity ofthe cylinders. In figure 4.5 these regions can be seen. The most black coloured spotsare regions with a downwards velocity in the z direction. Grey spots with a lightcoloured halo around are regions with an upward velocity. Although these velocitiesare small, they imply that a fully two dimensional flow can never reached in thisgeometry. In the central stagnation point, the deviation from 2D flow is very small.The larger the distance from this point, the larger the deviation from 2D flow.

Now we will summarize the main conclusions of the 3D calculations. After calculatingthe strain rate profiles, we found that the shape of this boundary layer of this profileis independent of the angular velocity ω and the height of the flow container. Theminimal distance to obtain a 2D flow halfway the cylinders is 55 mm. The proposedcorrection method, mentioned in chapter 2, has been validated numerically. Thereare little secondary flows in the flow container. These flows can influence the mea-surements. However, near the stagnation point the influence of these secondary flows

60 Chapter 4

ω = 0.6ω = 1.2ω = 2.4

Distance (mm)

Stra

inra

teε x

(s−

1)

80706050403020100

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Figure 4.3: Strain rate εx along the z direction at different rotational speeds (ω is 0.6,1.2 and 2.4 rad/s) at fixed cylinder height of 80 mm.

length 110 mmlength 80 mmlength 55 mmlength 32 mm

Distance (mm)

Stra

inra

teε x

(s−

1)

120100806040200

1

0.8

0.6

0.4

0.2

0

Figure 4.4: Strain rate εx along the z direction at different cylinder lengths at fixedω of 1.2 rad/s.

Calculations 61

Figure 4.5: An intersection of a horizontal and vertical plane in the four-roll millgeometry. Secondary flows can be seen near the top and bottom of the cylinders.Black coloured spots are regions with a downward velocity in the z-direction whilethe other spots represent the upward velocity

on the measurements will be very small.

4.3 Parameter study of 2D viscoelastic calculations

For the viscoelastic calculations done in this study, we used the finite element programDYNAFLOW, which is an extension of the SEPRAN package. DYNAFLOW hasbeen developed by Dr. M.A. Hulsen [Hulsen (2000)] in the laboratory for Aero-and Hydrodynamics at the faculty of Mechanical Engineering of Delft Universityof Technology. For the spatial discretisation a finite element method was used. TheDiscrete Elastic-Viscous Split Stress formulation (DEVSS) [Guenette et al. (1995)] forthe discretisation of the linear momentum balance and continuity equations was usedin order to obtain a better stability. The discontinuous Galerkin (DG) formulation[Fortin et al. (1989)] was used for the discretisation of the constitutive equation inthe macroscopic case and the equation for the configuration fields in the microscopiccase [Hulsen et al. (1997), van Heel (2000)]. For the time integration a second orderexplicit scheme was used [Hulsen (1996), Karniadakis et. al (1991)].Only 2D calculations will be performed. The computations were performed on aCompaq Alpha Server DS20 which contains two 64-bits RISC processors.

62 Chapter 4

4.3.1 Input parameters using the 1-mode Giesekus model

A parameter study is often done when physical insight is needed. For better values ofpressures, velocities and stresses, more complex calculations, using for example multi-mode models should be used. In this section the contribution of inertia to the flow, theinfluence of relaxation time and shear thinning behavior will be investigated for whichwe will use a single mode Giesekus model (See chapter 1). When we did calculationsusing this macroscopic model, we used about 40 Mb internal memory of the computer.The total CPU time for one run was about 2 hours. We focus on the central stagnationpoint of the four-roll mill geometry. Before starting the calculations, an almost gridindependent solution must be found first. The dimensions of the grid are equal to thedimensions of the experimental set-up. Due to symmetry, only one fourth of the totaldomain can be used. The gradients of the velocities and stresses near the cylinderwalls will be large, so the element size is chosen small. The calculations were doneon meshes containing 1032, 1932 and 2958 elements. Then stresses along the axes ofsymmetry were compared. The maximum difference in stresses between the meshescontaining 1932 and 2958 elements was 5 %. For the velocities a maximum differenceof 1 % was found. For the stationary calculations we used a time step of λ/500.Where λ is the relaxation time of the polymers. In this case the Courant number wassmaller than one. After approximately 5000 time steps the calculations convergedto a stationary solution. The mesh on which all calculations were performed can befound in figure 4.6.

4.3.2 Influence of inertia

Inertia effects are usually neglected when polymer flows are investigated because ofthe low Reynolds number. In this part we will investigate the contribution of inertiaon the stresses and velocity gradients. In equation 4.2 we gave the definition of theReynolds number. For the viscoelastic calculations we choose for the density ρ = 950kg/m3, D = 12 mm and η = 0.69 Pa s for the viscosity. The Reynolds number atan angular velocity of the cylinders ω = 3.5 rad/s is 1.7. For the relaxation time wechoose λ = 0.4 and for the shear thinning parameter α = 0.05. The maximum globalDeborah number reached in these calculations is Deg = 1.4.

For the stationary solution (cylinders rotating with a constant angular velocity), thefirst normal stress difference N1 as a function of the angular velocity in the stagnationpoint can be seen in figure 4.7. The difference due to inertia is small. Increasing theroller speed, the difference becomes larger. So the assumption of neglecting inertia inthe equations of motion is justified for the stationary solution.

In figure 4.8 we see the strain rate εx as a function of the angular velocity of thecylinders for a Newtonian fluid and a viscoelastic fluid. We observe a saturationin the strain rate for the viscoelastic fluid, while the Newtonian fluid represents astraight line. This saturation in the first normal stress difference N1 can also bee seenin figure 4.7. We expect a straight line when inertia and viscoelasticity are neglected.

Calculations 63

Due to strain hardening effects, the fluid tends to avoid the central stagnation pointand N1 will saturate. We will explain this effect in detail in the next section.

4.3.3 Influence of relaxation time

In figure 4.9 the influence of an increasing global Deborah number Deg on the dimen-sionless number S can be seen. The strain rate εx in the stagnation point at a fixedangular velocity ω = 3.32 rad/s was calculated. For this calculation the mesh in figure4.6 was used and also inertia was taken into account. For the shear thinning parame-ter the value α = 0.05 was used. The relaxation time varies from 0 (Newtonian fluid)to 1.5 seconds. Globally we see a decreasing S when the global Deborah number Deg

is increased. However, S has a maximum at Deg = 0.45. This strange behavior canbe explained by making a force balance between strain hardening effects and shearthinning effects. If the cylinders of the four-roll mill rotate at a fixed angular velocity,and the relaxation time is increased (and thus the Deg is increased), then the onsetof shear thinning behavior will start sooner and sooner. This can we see in the vis-cometric functions in figure 4.10, where the shear viscosity and elongational viscosityare plotted for three different relaxation times. Due to its local lower shear viscosityat the cylinder walls, more fluid will be pushed to the stagnation point which results

Figure 4.6: Mesh containing 1932 elements on which the calculations were performed

64 Chapter 4

in a higher strain rate in the stagnation point. If there is a higher strain rate then, theelongational viscosity will increase. Until shear thinning effects are larger than strainhardening effects an increase of S can be seen. When the relaxation time is increasedmuch more, the elongation viscosity increases faster at lower strain rates (see figure4.10) and then the strain rate and also S in the stagnation point will decrease. Infigure 4.10 the elongational viscosity and shear viscosity for the one-mode Giesekusmodel are plotted for different relaxation times at fixed α=0.05. Analytical solutionsfor these viscometric functions can be found in the book of Bird [Bird et al. (1987)].By an increasing relaxation time the shear viscosity decreases while the elongationalviscosity increases at a fixed shear rate resp. strain rate. For this geometry a max-imum in S is reached when Deg=0.45. The shape of the curve in figure 4.9 is alsodependent on the geometry. Putting the cylinders closer together higher strain rateswill be obtained and thus the elongational viscosity will soon play a more dominantrole and the maximum of S will shift to the left.

4.3.4 Influence of the parameter α

The onset of shear thinning behavior and strain hardening behavior in the Giesekusmodel is taken into account by the parameter α (See chapter 3). The smaller α thehigher the shear rates are where shear thinning appears and the higher the plateau-value of the elongational viscosity ηE for high strain rates. For α = 0 we have a fluidwith a constant shear viscosity. In this section we will investigate the influence of

with inertiawithout inertia

Angular velocity ω (rad/s)

N1in

stag

nation

point(P

a)

3.532.521.510.50

14

12

10

8

6

4

2

0

Figure 4.7: Stationary solution of N1 in the stagnation point with and without inertia.

Calculations 65

Newtonian fluid without inertiaViscoelastic fluid without inertia


Stra

inra

teε x

instag

nation

point(s

−1)

3.532.521.510.50

2.5

2

1.5

1

0.5

0

Figure 4.8: The strain rate ε in the stagnation point as a function of the angular ve-locity ω of the rollers. A large difference between a Newtonian fluid and a viscoelasticfluid can be seen.

parameter α on the local Deborah number Del in the stagnation point and the strainrate angular velocity ratio S. The results are plot in the figures 4.11 and 4.12. In figure4.11 we see S as a function of Deg for α = 0.05 and α = 0.11. The main differenceis the slower decay of the curve for α = 0.11. The maximum S is for α = 0.11 alittle bit lower than for α = 0.05. Larger differences can be seen in figure 4.12. Inthis figure the local Deborah number Del in the stagnation point is plot as a functionof the global Deborah number Deg. Striking is the presence of a maximum Del forα = 0.05, while for α = 0.11 a maximum Del is not reached. The explanation for thisbehavior was already given in the previous section. It is caused by the contributionof the force balance between shear thinning effects and strain hardening effects.

From these calculations we conclude that a higher angular velocity ω of the cylindersdoes not always lead to an increase of the strain rate in the central stagnation point.This can be considered as a major drawback of a four-roll mill geometry. Obtaininghigher Deborah numbers one can better use a cross-slot flow geometry.

66 Chapter 4

α = 0.05 and ω = 3.32 rad/s

Global Deborah number Deg

Dim

ension

less

numbe

rS

54.543.532.521.510.50

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 4.9: Stationary solution of the dimensionless number S in the stagnation pointat fixed ω =3.32 rad/s

4.3.5 Oscillating viscoelastic flow

In this section we will show calculations of oscillating flow. The cylinders of thefour-roll mill will make a harmonic oscillation.

ω(t) = Ωmax sin(ωosct). (4.11)

In our calculations we choose Ωmax = 9.96 rad/s, ωosc = 2.5 rad/s and ωosc = 0.75rad/s. For a harmonic oscillating flow, there are two dimensionless numbers whichplay an important role: Φ andDeosc (See equations 4.7 and 4.9). Φ is a measure for thenon-linearity of the viscoelasticity andDeosc is a measure for the instationary behaviorin the flow. We are interested in the non-linear viscoelastic flow behavior, so we chooseΦ = 7.97 and Φ = 26.6. For Deosc we choose Deosc = 0.3 and Deosc = 1.0. Therelaxation time was λ = 0.4 s. The maximum Reynolds number in these calculationswas 2.8. We plot the first normal stress difference N1 in the stagnation point as afunction of time and ω(t) in figure 4.13 for two Deosc numbers. Figure 4.13 showsthe non-linear behavior clearly. The response N1(t) does not approximate the shapeof ω(t). Furthermore we see large differences between calculations with and withoutinertia when Deosc is increased: the maximum value of N1 becomes larger and thephase shift of N1(t) with respect to ω(t) becomes larger too when inertia is taken intoaccount. Traveling waves phenomena can be observed during the start-up of the flow.Then N1 is almost equal to zero during the first time for a penetration time period

Calculations 67

λ = 10 sλ = 1.0 sλ = 0.1 s

Strain rate ε (s−1)

Elong

ationa

lvisco

sity

η E(P

as)

1010.10.010.001

100

10

1

λ = 10 sλ = 1.0 sλ = 0.1 s

Shear rate γ (s−1)

Shea

rviscos

ityη S

(Pas)

1010.10.01

1

0.1

0.01

Figure 4.10: The uniaxial elongational viscosity ηE (upper figure) and shear viscosityηS (lower figure) for the one mode Giesekus model with parameters η0 = 0.69 Pa s andα = 0.05. With increasing relaxation time λ, the shear viscosity decreases while theelongational viscosity increases at a constant shear rate and strain rate, respectively.

68 Chapter 4

α = 0.11α = 0.05


S

76543210

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 4.11: The influence of shear thinning parameter α for the one mode Giesekusmodel

after start-up.

4.4 Comparison of the macroscopic and microscopic

models

In this section we will made a comparison of the microscopic calculations and macro-scopic calculations. For the microscopic calculations we use a FENE model and a4-mode Giesekus model is used for the macroscopic calculations. The model param-eters for these two models were given in chapter 3 in table 3.1 and table 3.2. Onlythe stationary solution is studied here because instationary calculations failed whenthe model parameters of the tables 3.1 and 3.2 were used. For the microscopic com-putations we used the Brownian configuration fields method, described in chapter 3.For this microscopic calculation we used 800 Mb internal computer memory and thetotal CPU time for one run was about 100 hours. In contrast to the microscopic cal-culations, the macroscopic calculations were less time consuming. Only 4 hours CPUtime and 50 Mb internal memory were required. We chose for the angular velocity ofthe cylinders ω = 0.05 rad/s. The global Deborah number Deg becomes Deg = 0.50for the microscopic calculations and Deg = 0.26 for the macroscopic calculations.Higher values of Deg could not be reached for the microscopic calculations due toinstabilities.

Calculations 69

α = 0.11α = 0.05


Loc

alDeb

orah

numbe

rDe l

6543210

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 4.12: The local Deborah number Del in the stagnation point as a functionof the global Deborah number Deg. Striking is the presence of a maximum localDeborah number Del in the stagnation point.

In our experiences, the 4-mode Giesekus model showed a better stability than theone mode microscopic FENE model. The explanation for it is that stresses will besmeared out over 4 modes which results in a better stability. This experience is inagreement with Li [Li et al. (2000)]. They made a comparison using the 1-modeFENE-P model and a 2-mode Giesekus model.

A direct comparison of experiments and microscopic calculations is very difficult. Fora minimal detectable amount of birefringence (in order to calculate stresses), a mini-mum angular velocity of the cylinders ω = 0.5 radians per second was needed. Thenthe Deborah number Deg will become De = 5. This is even large for macroscopiccalculations and calculations will fail. So the only strategy left for a comparison, isto use the fit parameters found in the fluid characterization and doing calculations atlow Deborah numbers.

In the following a comparison between the 4-mode Giesekus model and the micro-scopic FENE model is made. The contour plots of the streamlines, pressure, velocitygradient εx and stresses will be shown. Note, that we use the same contour levels forboth models in all figures. As a result, some figures will have some empty regions. Ashort explanation of the plots will be given on the next page.

70 Chapter 4

without inertiawith inertiaoscilating function Ωmax = 9.96 rad/s

Period

Nor

mal

stress

diffe

renc

eN

1(P

a)

21.510.50

30

20

10

0

-10

-20

-30

without inertiawith inertiaoscilating function Ωmax = 9.96 rad/s

Period

Nor

mal

stress

diffe

renc

eN

1(P

a)

21.510.50

40

30

20

10

0

-10

-20

-30

Figure 4.13: Solution of N1 in oscillating elongational flow, with and without inertia.In the upper figure Deosc = 0.3 and in the lower figure Deosc = 1.0.

Calculations 71

StreamlinesIn figure 4.14 the streamlines can be seen. Remarkably is the presence of two counterrotating vortices one caused by the cylinder and the other in the upper right corner.Due to these counter rotating vortices two more stagnation points appear on the axesof symmetry. The streamlines for both models are in excellent agreement with eachother.

PressureThe contour plots of the pressure can be seen in figure 4.15. The two plots are more orless the same. The FENE calculations show some higher pressures near the cylinderwalls. Furthermore both contour plots show smooth curves which is a measure of thequality of the calculations.

Velocity gradientThe velocity gradient εx can be seen in figure 4.16. Both plots show a very goodsimilarity and this means the velocity field is also similar for both models. These cal-culations show the independence of the velocity field of the two models using differentparameters. Thus the velocity field is a bad distinguishing parameter for investigatingdifferent models.

StressesThe contour plots of the stresses can be found in figures 4.17, 4.18 and 4.19. In thesefigures quite large deviations in stresses can be seen. In figure 4.20, we plot an inter-section of τxx and τyy along the horizontal axis of symmetry (arrow line L in figure4.1). Then the differences between the two models become more clear. The maximumstresses in the FENE-model are larger than in the Giesekus model. Of course thisis caused by the large difference in the dominant relaxation time (λgiesekus = 5.21,λfene = 10) but it might also be caused by other effects in the FENE model. One ofthese unmodeled effects is the interaction of the FENE dumbbells to each other. An-other unmodeled aspect in the FENE model is the presence of branches in a polymermolecule. One of the challenging objects in future is to capture all these effects byusing for example multi bead-spring systems in different configurations [Denneman(1998)].

72 Chapter 4

11 1.100E-05

2

3

6

78

9

10

LEVELS

1 -8.100E-06 2 -6.190E-06 3 -4.280E-06 4 -2.370E-06 5 -4.600E-07 6 1.450E-06 7 3.360E-06 8 5.270E-06 9 7.180E-0610 9.090E-0611 1.100E-05

2

3

6

78

9

10

LEVELS

1 -8.100E-06 2 -6.190E-06 3 -4.280E-06 4 -2.370E-06 5 -4.600E-07 6 1.450E-06 7 3.360E-06 8 5.270E-06 9 7.180E-0610 9.090E-06

Figure 4.14: Streamlines using the 4-mode Giesekus model (upper) and FENE model(lower) at counter clockwise rotating ω = 0.05 rad/s.

Calculations 73

11 0.400

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4

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4

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5

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5

66

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7 7

7

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8

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9

910 LEVELS

1 -0.300 2 -0.230 3 -0.160 4 -0.090 5 -0.020 6 0.050 7 0.120 8 0.190 9 0.26010 0.33011 0.400

23

34

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5

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5

66

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6

77

777

7

88

8

8

9 9

9

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11

11

LEVELS

1 -0.300 2 -0.230 3 -0.160 4 -0.090 5 -0.020 6 0.050 7 0.120 8 0.190 9 0.26010 0.330

Figure 4.15: Contour lines of pressure using the 4-mode Giesekus model (upper) andFENE model (lower) at counter clockwise rotating ω = 0.05 rad/s.

74 Chapter 4

11 0.100

1

1

2

2

2

2

3

3

3

3

4

4

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LEVELS

1 -0.100 2 -0.080 3 -0.060 4 -0.040 5 -0.020 6 0.000 7 0.020 8 0.040 9 0.06010 0.08011 0.100

1

1

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LEVELS

1 -0.100 2 -0.080 3 -0.060 4 -0.040 5 -0.020 6 0.000 7 0.020 8 0.040 9 0.06010 0.080

Figure 4.16: Contour lines of velocity gradient εx using the 4-mode Giesekus model(upper) and FENE model (lower) at counter clockwise rotating ω = 0.05 rad/s.

Calculations 75

13

1

1

1

1

2

2

2

3 3

3

3

4

4

4

4

5

LEVELS

1 -0.050 2 0.012 3 0.073 4 0.135 5 0.196 6 0.258 7 0.319 8 0.381 9 0.44210 0.50411 0.56512 0.62713 0.68814 0.750

1

1

1

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89

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LEVELS

1 -0.050 2 0.012 3 0.073 4 0.135 5 0.196 6 0.258 7 0.319 8 0.381 9 0.44210 0.50411 0.56512 0.62713 0.68814 0.750

Figure 4.17: Contour lines of stress component τxx using the 4-mode Giesekus model(upper) and FENE model (lower) at counter clockwise rotating ω = 0.05 rad/s.

76 Chapter 4

14 0.200

4

5

5

6

6

7

7

7

88

8

9

91011

LEVELS

1 -0.200 2 -0.169 3 -0.138 4 -0.108 5 -0.077 6 -0.046 7 -0.015 8 0.015 9 0.04610 0.07711 0.10812 0.13813 0.16914 0.200

1

1

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1314

LEVELS

1 -0.200 2 -0.169 3 -0.138 4 -0.108 5 -0.077 6 -0.046 7 -0.015 8 0.015 9 0.04610 0.07711 0.10812 0.13813 0.169

Figure 4.18: Contour lines of stress component τxy using the 4-mode Giesekus model(upper) and FENE model (lower) at counter clockwise rotating ω = 0.05 rad/s.

Calculations 77

11

1

1

2

2

2

3

3

4

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4

LEVELS

1 -0.050 2 0.027 3 0.104 4 0.181 5 0.258 6 0.335 7 0.412 8 0.488 9 0.56510 0.64211 0.71912 0.79613 0.87314 0.950

1 1

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1 -0.050 2 0.027 3 0.104 4 0.181 5 0.258 6 0.335 7 0.412 8 0.488 9 0.56510 0.64211 0.71912 0.79613 0.87314 0.950

Figure 4.19: Contour lines of stress component τyy using the 4-mode Giesekus model(upper) and FENE model (lower) at counter clockwise rotating ω = 0.05 rad/s.

78 Chapter 4

FENE model4-mode Giesekus model

Distance from central stagnation point (mm)

τ xx(P

a)

160140120100806040200

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1


Distance from central stagnation point (mm)

τ yy(P

a)

160140120100806040200

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

-0.05

-0.1

Figure 4.20: An intersection of τxx and τyy along the horizontal axis of symmetry.There are large differences in the microscopic FENE calculations and the macroscopicGiesekus model calculations.

Calculations 79

4.5 Discussion

In this chapter many calculations have been presented and the main results will besummarized. For the three dimensional inelastic calculations, the dimensions of theboundary layer of the strain rate profile in the stagnation point was an importantobject of study. Roughly the thickness of a boundary layer is 17 mm for a fluid hav-ing a zero-shear-rate viscosity of 1 Pa s. This thickness is independent of the angularvelocity of the cylinders. Of course the question might raise if this thickness is of thesame dimension when a viscoelastic fluid was used. Schoonen [Schoonen (1998)] did3D flow calculations in a decoupled way. First using the Carreau viscosity model forcalculating the velocity field and then using this field for determining stresses. Basedon his results, we expect that the thickness of the edges may differ a little bit but notan order of magnitude. Roughly we expect the minimal distance of 55 mm betweentop and bottom of the cylinders to be sufficient to get two dimensional flow half waythe cylinders.

From the parameter study, information was obtained on the inertia effects in station-ary and instationary flow. For instationary flow the contribution of inertia to the flowbecomes more important. Also the importance of distinguishing the shear viscosityηS and the elongational viscosity ηE became clear and they lead to a remarkable ef-fect. Due to this effect the strain rate in the central stagnation point does not alwaysincrease when the angular velocity ω increases. There is an certain maximum strainrate which can obtained in the stagnation point.

The final comparison of the microscopic FENE model and the 4-mode macroscopicGiesekus model was made in section 4.3. It was a pity that experiments can not becompared directly with experiments due to large Deborah numbers, but even whenDeborah numbers are small, still large differences in stress fields can be seen. Howeverthe differences in the velocity gradient fields were small. This means that the velocityfield is a bad distinguishing parameter for investigating different models.The large stress differences of the microscopic model and macroscopic model is mainlycaused by totally different curves for the planar elongational viscosity ηPE , which canbe seen in figure 4.21. In this figure we calculated the planar elongational viscosityusing the parameters found in the fluid characterization. Even when strain rates aresmall (order 0.1) the planar elongational viscosity for the 4-mode Giesekus model andFENE model differs by a factor of 10. So for the characterization of a fluid, not onlydata fits in shear flow are enough. In chapter 3 we showed an almost perfect fit of thesteady shear data for the 4-mode Giesekus model and FENE model while calculationsof the elongational viscosity showed dramatic differences.

Solving this problem will become a challenging aim for the future. Actually there aretwo ways of solving this problem in order to get a better microscopic description ofthe model. The first way is using a multi-mode FENE model and fitting the shearviscosity data and elongational viscosity data and the dynamic data simultaneously.This will be a fit calculation and perhaps it will result in a good data fit but physical

80 Chapter 4



Plana

relon

gation

alviscos

ityη P

E(P

as)

1001010.10.010.001

1000

100

10

1

Figure 4.21: A large difference in stresses between the 4-mode Giesekus model andFENE model is caused by totally different curves of the elongational viscosity ηPE

insight in the total system will be lost. Probably this approach will work in engineer-ing applications but from a physical point of view this gives less satisfaction.The second and much more difficult way, is to capture all physical aspects into themicroscopic model. Molecular interaction, chain length distribution and the archi-tecture and configuration of the polymer molecule should be taken into account inorder to contain all physics. Then these physical aspects should be modeled by forexample multi bead-spring chains with or without branches dependent on the con-figuration of the polymer molecule. Although microscopic simulations are makingprogress, this study demonstrates that there are still a lot of problems to overcomeuntil this microscopic way of modeling is applicable to real polymer solutions. But infuture, computer power will increase and microscopic simulations are expected to bea powerful tool in simulation of polymer flow in complex geometries.

References

Bird, R.B., F.C. Curtiss, R.C. Armstrong and O. Hassager. Dynamics ofpolymeric liquids. Vol 1, John Wiley & Sons, (1987).

Denneman, A.I.M. “An extended bead-spring model for application in rheology.”PhD. thesis Twente University of Technology, The Netherlands (1998).

Fortin, M. and A. Fortin. “A new approach for the FEM simulation of viscoelas-tic flows.” J. Non-Newtonian Fluid Mech. 32 295-310 (1989).

Guenette, R. and M. Fortin. “A new mixed finite element method for Computingviscoelastic flows. ” J. Non-Newtonian Fluid Mech. 60 27-52, (1995).

Heel van, A.P.G, “Simulation of viscoelastic fluids. From microscopic modelsto macroscopic complex flows ” PhD. thesis Delft University of Technology,The Netherlands (2000).

Hulsen, M.A. “Stability of the implicit/explicit extension of the stiffly schemesof Gear.” Internal report MEAH-138, Laboratory for Aero- and Hydrody-namics, Delft University of Technology. (1996).

Hulsen, M.A. ,A.P.G van Heel and B.H.A.A van den Brule, “Simulation ofviscoelastic flows using Brownian configuration fields.” J. Non-NewtonianFluid Mechanics, 70 79-101, (1997).

Hulsen, M.A. “DYNAFLOW, a finite element program for viscous and viscoelas-tic fluid flow problems.” Reference manual, Delft University of Technology,The Netherlands (2000).

Karniadakis, G.E., M.Israeli and S.A. Orszag. “High-order splitting methodsfor incompressible Navier-Stokes equations.” J. Comput. Phys., 97 414-443,(1991).

Li,Jing-Ming, W.R. Burghardt, B.Yang and B. Khomami, “Birefringenceand computational studies of a polystyrene boger fluid in axisymmetric

81

82 Chapter 4

stagnation flow.” J. Non-Newtonian Fluid Mech. 91 189-220, (2000).

Schoonen, J.F.M. “Determination of Rheological Constitutive Equations us-ing Complex Flow.” PhD. thesis Eindhoven University of Technology, (1998).

Wales, J.L.S. “The application of flow birefringence to rheological studies ofpolymer melts.” PhD. thesis Delft University of Technology, DUP, Delft(1976).

Warner, H.R. “Theory and rheology of dilute suspensions of finitely extensibledumbbells.” Ind. Eng. Chem. Fundam., 11 379-387 (1972).

Chapter 5

Experimental Results

In this chapter results of the flow induced birefringence measurements will be pre-sented. We will investigate two different solutions: a polystyrene (PS)/decalin solu-tion and a polyisobutylene (PIB)/decalin solution. The chain length of these poly-mers are completely different and this results in a different behavior in a stationaryand time-dependent flow. We will also present a comparison between calculations andexperiments with the PIB solution.

5.1 Introduction

In this chapter we will show the results of the flow induced birefringence measure-ments done in this research project. We will make a distinction between stationaryand time-dependent measurements. The stationary measurements are performed onthe axis of symmetry of the four-roll mill while the time-dependent measurements areonly performed in the central stagnation point.

We also make a comparison between stationary calculations and measurements. Asmentioned in chapter 4, an extensive comparison between experiments and calcula-tions is difficult. For the experiments, a minimal angular velocity ω of the cylinders isneeded to create birefringence and for the computations we have to restrict ourselvesto low Deborah numbers. We are able to make a comparison between the Giesekusmodel and the experiments, but the comparison of the microscopic FENE model cal-culations unfortunately fails because the Deborah number Deg becomes too high atthe angular velocity ω where the experiments have been performed.

Finally we show the difference between two types of polymers in this four-roll mill ex-periment. We use the PIB solution and a PS solution described in chapter 3. The aimof this comparison is to see what the influence of chain length is on the birefringencemeasurements.

83

84 Chapter 5

80 mm PIB solution60 mm PIB solution

Angular velocity cylinders ω (rad/s)

Retar

dation

δ(d

egrees)

1086420

10

8

6

4

2

0

Figure 5.1: The retardation of the PIB solution in the stagnation point as functionof ω at two different path lengths of 60 mm and 80 mm. These kinds of experimentsare useful for determining the birefringence in fully 2D flow.

5.2 Stationary measurements

In this section we will present the steady-state measurements, i.e. the angular velocityis kept constant during the measuring time. We will first show the results of themeasurements done in the stagnation point and then the measurements done on theaxes of symmetry of the four-roll mill geometry.

5.2.1 Measurements in the stagnation point

For a comparison of the calculations with experiments, we have to measure the bire-fringence ∆n as a function of the angular velocity ω for two different cylinder lengths(See section 2.3.4). This has to be done in order to obtain the birefringence ∆n2D

for the two dimensional flow case. When this is known, we are able to compare theexperimental results with the 2D flow calculations. Figure 5.1 shows the retardationδ as a function of the angular velocity ω at two different cylinder lengths (60 and80 mm). The fluid used in this experiment was the 2% (w/w) PIB/decalin solution.The data follow a straight line even when the angular velocity is speed up to ω = 15rad/s. From these measurements we can calculate by means of equation (2.27) thebirefringence caused by the 2D flow. To reduce the influence of noise we first makea least-square line fit through the data points in figure 5.1. Subtracting these line

Results 85

functions from each others, will result in the retardation δ caused by the 2D flow.We used the stress optical relations and the value C = 2.72 · 10−9 Pa−1 for the stressoptical constant. For the orientation angle χ we have chosen χ = 0, after we havechecked the measured values. In the measurements χ was changing between -0.2 and0.2 degrees. Now we can calculate N1 in the stagnation point.

N1 =∆n2D

Ccos(2χ). (5.1)

The results of these calculations are illustrated in figure 5.2. In this figure, a compar-ison of N1 in experiments and calculations is made at different angular velocities ω.We see that the measured values of N1 are in very good agreement with the calculatedvalues of the 4-mode Giesekus model. However deviations become larger at larger ωwhen the Deg becomes larger. Calculations will fail when ω > 2.5 rad/s which isequivalent to Deg > 13.

It is amazing that the measurements of N1 are linear correlated to the angular velocityω. This is partly caused by the data fitting process, but at a first glance one shouldsay the PIB solution behaves like a Newtonian fluid. For investigating this behaviorwe did some calculations of the strain rate εx in the stagnation point as a function ofthe angular velocity ω. Figure 5.3 shows the results and we see an increasing strainrate εx when the angular velocity ω is increased. But we see also some saturation ofthe strain rate by an increasing value of the angular velocity. This means that we alsoshould see a saturation of N1 when the planar elongational viscosity ηPE is constant.However when we look at figure 5.4 where we have plotted the planar elongationalviscosity ηPE as a function of the strain rate ε, we see a transition regime betweenstrain rates 0.1 to 1 s−1 The stain rates we look at are all lying in this regime. So anincrease of the angular velocity ω will lead to a saturation of the strain rate ε but alsoan increase of the planar elongational viscosity ηPE . Therefore a linear relationshipbetween N1 and the angular velocity ω can be maintained.

5.2.2 A comparison between PS and PIB in the central stag-nation point

In this section we investigate the difference of using a polystyrene (PS)/decalin so-lution and a polyisobutylene (PIB)/decalin solution in the FIB experiments in thefour-roll mill geometry. When we look at figure 5.5, we see that the length of onemonomer is the same for polystyrene and polyisobutylene. However the PS moleculeconsists of about 3500 monomers while the PIB molecule consist of 76000 monomers.So the length of the PIB polymer is about twenty times longer than the PS polymer.This fact influences the birefringence measurements. In figure 5.6 we have plot theobservations of the birefringence ∆n as a function of the angular velocity ω for thePIB solution and PS solution. We see large differences between these two solutions.First we see a negative value of the birefringence ∆n for the PS solution when theangular velocity is positive, while the PIB solution shows a positive value of ∆n when

86 Chapter 5

Calculations using the 4-mode Giesekus modelMeasurements PIB solution


First

norm

alstress

diffe

renc

eN

1(P

a)

6543210

18

16

14

12

10

8

6

4

2

0

Figure 5.2: The measured and calculated values of N1 in the central stagnation pointusing the PIB solution. The calculations failed when the angular velocity ω becomehigher than 2.5 rad/s. The Deborah number Deg at this rotational speed is 13.

ω is positive. This fact is caused by the negative value of the stress optical constantC for polystyrene [Li et al. (1995)]. The PIB solution shows a more or less linearrelationship between the birefringence ∆n and angular velocity ω. This means thatthe PIB polymers are still not fully stretched even when the angular velocity ω = 15rad/s. In contrast to the PIB solution, the PS solution shows a saturation in thebirefringence ∆n. The explanation for the behavior of this Newtonian fluid, is thetotal orientation of the PS molecules in the stagnation point. The total number ofmonomers of the PS molecule is much smaller than the PIB molecule (see fig. 5.5).Due to the relatively short chain length of the PS molecule, chains are oriented andstretched faster than the PIB molecule. This leads to a saturation of the birefrin-gence when ω is increased. The same saturation effects in the birefringence were alsofound by Fuller [Fuller et al. (1981)] where a concentrated solution of water andpolyethylene oxide was studied in a four-roll mill geometry.

Results 87

5.2.3 Calculations and measurements on the axes of symmetry

In figure 5.7, we have plotted the first normal stress difference on the axes of symmetryof the four-roll mill geometry. We can only measure 9 mm from the stagnation pointbecause of the limited size of the window. The upper-right cylinder of the four-roll millwas counter clockwise rotating at an angular velocity ω = 1.5 rad/s. We measuredthe birefringence ∆n for two cylinder lengths of 53 and 80 mm along the vertical(Y-axis) and horizontal axis (X-axis) of symmetry which can be seen in figure 5.8.From these two sets of measurements we obtained the ∆n2D. The stagnation pointof the four-roll mill geometry is located at the origin of figure 5.7. When we look atthe calculations and measurements and follow a the path of a flowing polymer alongY-axis and after that along the X-axis. We see an increasing value of N1. Near thestagnation point, N1 suddenly increases fast. This is due to the very low velocity ofthe polymer and a very long residence time of the polymer molecule in this region ofstrong elongational flow. It has a lot of time to be stretched out and this results ina high value of N1. After the polymer molecule has passed the stagnation point, itwill flow along the X-axis. The calculations with the 4-mode Giesekus model followthe measurements quite well but the measurements show a steeper increase of N1

near the stagnation point. This is probably caused by the secondary flows whichappear in the 3D calculations. Striking is that the measured value of N1 on thenegative part of the X-axis deviates from the calculations. The asymptotic value

Calculations 4-mode Giesekus model


Stra

inra

tein

stag

nation

pointε(s

−1)

2.521.510.50

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Figure 5.3: Calculations of the strain rate εx as a function of the angular velocity ωusing the 4-mode Giesekus model.

88 Chapter 5

4-mode Giesekus model


Plana

relon

gation

alviscos

ityη P

E(P

as)

1001010.10.010.001

10

9.5

9

8.5

8

7.5

7

6.5

6

5.5

5

4.5

Figure 5.4: The planar elongational viscosity ηPE as a function of the strain rate εcalculated with the 4-mode Giesekus model.

of N1 should theoretically be equal to zero. The measurements tend to go to theasymptotic value N1 = 1 Pa. Probably this is caused due to larger timescales of thestress relaxation in the polymers which are also important but which are not found inthe fluid characterization in shear flow. These timescales are larger than the period inwhich the polymer makes one revolution around one of the cylinders. When we makean estimation of this timescale, we will use the rotational speed of the rollers, whichrotate at a speed of 1.5 radians per second. A polymer molecule which is attached atthe cylinder wall will make one revolution in 4.2 seconds. Polymer molecules whichare traveling just above the axis of symmetry will make one revolution in 4 to 6 timesthe rotation period of the cylinders. This means that the largest relaxation time ofthe polymers is at least of order 17 seconds.

The general conclusion of this comparison is that the 4-mode Giesekus model is ableto predict the first normal stress difference N1 in this geometry quite well. Thedeviations of the calculations respect to the experiments are small and are of order ofthe accuracy of the measurement method. However the Deborah number Deg is notvery high in these calculations.

Results 89

H

H H

Decalin

H

H

H

H

H H

H

H

H

H

H

H

H

H

H

Polystyrene monomer

C C

H

H

CH

CH

3

3

C C

H H

H

n

n

Polyisobutylene monomer

Figure 5.5: The structure of the solvent Decalin and the polystyrene monomer andpolyisobutylene monomer. In these experiments the PS molecule consists of 3500monomers and the PIB contains 76000 monomers.

PS solutionPIB solution

Angular velocity cylinders ω (rad/s)

Birefring

ence

∆n

(-)

151050-5-10-15

6e-07

5e-07

4e-07

3e-07

2e-07

1e-07

0

-1e-07

-2e-07

-3e-07

-4e-07

-5e-07

Figure 5.6: Birefringence as function of angular velocity measured in the stagnationpoint. Saturation effects can be seen for the PS solution due to the short chains ofthe polystyrene used in this experiment.

90 Chapter 5

Measurements PIB solutionN1 along X-axisN1 along Y-axis

Distance (mm)

First

norm

alstress

diffe

renc

eN

1(P

a)

86420-2-4-6-8

6

5

4

3

2

1

0

Figure 5.7: The measurements and calculations of the first normal stress differenceN1 with the 4-mode Giesekus model on the axes of symmetry at an angular velocityω = 1.5 rad/s. The negative values of the X-axis in this figure correspond to thenegative values of the Y-axis in figure 5.8.

Y−axis

X−axis

edge of the cylinder

ωflow direction

0

−8

8

Figure 5.8: The region around the central stagnation point. Measurements were donealong the X-axis and Y-axis.

Results 91

5.3 Time-dependent measurements

Now we will consider the time-dependent measurements. In this experiment we focusourselves on two kinds of motions of the cylinders of the four-roll mill: The start-stopexperiment and the oscillatory experiment. During the start-stop experiments thecylinders are rotating with a constant angular velocity ω and suddenly they stop. Wewill measure the birefringence ∆n in the stagnation point as a function of time. Inthe oscillatory experiments, the cylinders of the four-roll mill are moving in a pureharmonic oscillation. Again we will make a comparison between the PS solution andthe PIB solution.

5.3.1 Start-stop experiments

In figure 5.9 we have plotted the relaxation of the birefringence ∆n for the PIB andPS solution. The cylinders of the four-roll mill rotate at an angular velocity of 16.1rad/s and 16.6 rad/s for the PS solution and the PIB solution respectively. Wescale the birefringence ∆n in such a way that it is equal to 1 when the cylinders arerotating stationary. The bottom to top distance was 48 mm in both experiments.This figure shows some remarkable things. First, there appear two timescales inboth the PS solution and the PIB solution. A fast relaxation time can be seen justafter when the cylinders stopped and a much slower relaxation process after wards.In our opinion, the fast relaxation time is caused by a kind of coil-retraction andshrinking process of the polymers. The slower process is caused by the re-orientationof the chain segments due to reptation. This process describes the motion of a longpolymer chain in a concentrated solution where it is obstructed in its motion byother polymer chains. A short description of the reptation theory can be found in forexample the book of Larson [Larson (1999)]. Peters also gives a extended discussionof this model in his thesis [Peters (2000)]. In figure 5.9, we also plot some simpleexponential decaying functions. In the upper plot of figure 5.9 we have plotted adecaying function, having a time constant λ = 0.028 sec. This is more or less equal tothe largest timescale of the PS solution and is in agreement with estimations of thedominant relaxation time in chapter 3. In the lower plot of figure 5.9 we have plottedtwo simple decaying functions having time constants of λ = 5 and λ = 32 secondsrespectively. In chapter 3 we measured the dominant relaxation time for the PIBsolution and we found λ = 5.21 sec. Here another timescale of λ = 32 sec. appearswhich we did not found in rheometer experiments. This large time constant is notfound because the sensitivity of the rheometer was not enough for detecting this.

5.3.2 Oscillatory experiments

In this section we will present the birefringence measurements in a time-dependentelongational flow. In these experiments, the angular velocity ω of the cylinders isdescribed by the following function:

ω(t) = Ωmax sinωosct. (5.2)

92 Chapter 5

f(t) = c · e(−t/0.028)PS solution ω = 16.1 rad/s

Time (sec)

Scaled

Birefring

ence

∆n

(-)

0.10.090.080.070.060.050.040.030.020.010

1

0.1

0.01

g(t) = c · e(−t/32)f(t) = c · e(−t/5)

PIB solution ω = 16.6 rad/s

Time (sec)

Scaled

Birefring

ence

∆n

(-)

100806040200

1

0.1

0.01

Figure 5.9: The relaxation of the birefringence ∆n for the PS solution (upper figure)and the PIB solution (lower figure). There are two timescales which determine therelaxation process. The small one is caused by coil retraction and the large one iscaused by re-orientation due to molecular diffusion.

Results 93

angular velocity ωretardation linear response Φ = 1.7 radiansretardation non-linear response Φ = 8.5 radians

Time (sec)

Retar

dation

δ(d

egrees)

543210

10

5

0

-5

-10

Figure 5.10: The retardation δ as a function of the time plotted for two values of Φmeasured at a light path length of 55 mm for the PS solution (Deosc < 0.04). Whenthe total angle of rotation Φ is small, we see a linear response of retardation δ on theangular velocity ω.

The important parameters of this oscillation function are Deosc (see eq. 4.7) and themaximum angle of rotation Φ (see eq. 4.8). In figure 5.10 we see a time series of theretardation δ and the angular velocity ω of the cylinders as a function of time. Inthis figure ωosc = 3.77 rad/s and the values of Φ are 1.7 and 8.5 radians respectively.When the total angle of rotation Φ is small the response of the the retardation δ(t) onthe angular velocity ω(t) is linear because a cosine-shaped oscillation function resultsin a more or less cosine-shaped retardation function. We also see a phase shift ofabout π

2 radians between the angular velocity ω(t) and the retardation δ(t). Theretardation δ is linearly correlated to the stress σ and thus there is a phase shiftbetween the stress σ(t) and the angular velocity ω(t). We can explain this phase shiftby using a very simple Maxwell model which is already mentioned in chapter 3. Thestress σ in a Maxwell element (spring-dashpot element) is given by:

λdσ

dt+ σ = η

dγ

dt, (5.3)

were λ is the relaxation time and η the viscosity and γ the displacement. Whenthe cylinders of the four-roll mill start to oscillate with a harmonic function ω(t) =sin(ωosct), then the stretch of the polymer chains γ(t) in the region of the centralstagnation point will also oscillate with a function γ(t) = sin(ωosct). Now we take the

94 Chapter 5

first time derivative of γ(t) and substitute this expression in differential equation 5.3.

λdσ

dt+ σ = ηωosc cos(ωosct). (5.4)

Solving this differential equation using σ(0) = 0 as initial condition, we find:

σ(t) =ηωosc

1 + (λωosc)2(−e−t/λ + cos(ωosct) + λωosc sin(ωosct)

). (5.5)

Now consider only the stationary part of the solution and re-arrange the sine andcosine function:

σ(t) =ηωosc√

1 + (λωosc)2cos(ωosct− arctan(λωosc)). (5.6)

If we consider an almost non-elastic fluid like the PS solution then λωosc 1 andthus arctan(λωosc) ≈ 0. For σ(t) we find a cosine function and for ω(t) we used a sinefunction. This means that there is a phase shift of π

2 radians between σ(t) and ω(t).This is in agreement with the measurements of the PS solution in figure 5.10 where

angular velocity ωretardation almost linear response Φ = 26.6 radians

Time (sec)

Retar

dation

δ(d

egrees)

2520151050

15

10

5

0

-5

-10

-15

Figure 5.11: The retardation δ as a function of time plotted for the PIB solutionmeasured at a light path length of 80 mm (Deosc = 3.9). We see a little phaseshift between the two signals. This is caused by the viscoelastic behavior of the PIBsolution. For a pure viscous fluid like the PS solution, this phase shift is about π/2radians, which can be seen in figure 5.10.

Results 95

PS Φ = 52.90PS Φ = 31.74PS Φ = 10.57PS Φ = 2.65

angular velocity ω (rad/s)

Retar

dation

δ(d

egrees)

151050-5-10-15

10

5

0

-5

-10

Figure 5.12: The retardation δ as a function of the angular velocity ω for the PSsolution for four different values of Φ.The oscillation frequency ωosc = 3.77 rad/s inall cases. The smaller the value of Φ, the smaller the closed curve is.

we see a phase shift of about π2 radians. If we consider a viscoelastic fluid in a way

that λωosc 1 then σ(t) and ω(t) are in phase which we can see in figure 5.11 wherethe phase shift is very small.

Now if we present a parameter plot of the retardation δ against ω. We obtain anellipse when we are in the linear viscoelastic regime. When Φ is increased, the regimeof linear viscoelasticity will be left and higher harmonic frequencies become important.In this case, the retardation δ as a function of ω is no longer an ellipse. This is whathas been illustrated in figure 5.12. We have plotted the the retardation δ as a functionof ω for 4 different values of Φ for the PS solution and the PIB solution. The largerΦ, the larger the deviation from a ellipsoid. We can also see that the PIB solution hasa larger range of linear response than the PS solution. This is caused by the muchlarger chain length of the PIB molecules compared to the PS molecules.

5.3.3 The orientation angle change of the polymers

In figure 5.14 we plot the orientation angle χ as a function of time for the PS solutionand the PIB solution. In this experiment, we oscillate the cylinders very slowly witha harmonic oscillation and period of 10 seconds. Every time when the direction ofrotation changes sign, polymer chains will re-orient from parallel to the X-axis infigure 5.8 to parallel to the Y-axis. In the measurements shown in figure 5.14 this

96 Chapter 5

PIB Φ = 31.74PIB Φ = 10.57

angular velocity ω (rad/s)

Retar

dation

δ(d

egrees)

151050-5-10-15

10

5

0

-5

-10

Figure 5.13: The retardation δ as a function of the angular velocity ω for the PIBsolution for two values of Φ.

becomes visible in a kind of block function which changes from χ = 0 degrees toχ = 90 degrees. When we look at the two solutions, we see some differences. Firstwe see that the re-orientation of the PS chains goes much faster than the PIB chains.The PS polymers are all oriented within one second. After that time the orientationangle χ is almost constant until the sign of the angular velocity ω changes. Lookingat the re-orientation process of the PIB polymers, we see a much more irregular shapeof the curve. It is questionable if a steady-state value of χ is reached in a period of 5seconds when a PIB solution is considered. Also the reproduction of the signal of χ forthe PIB solution is not very well. There are large differences when the different cyclesare compared. The re-orientation process of the PS polymers goes much faster dueto the shorter polymer chains. The differences between the measurements of the twosolutions is caused by the difference in chain length of the PS and the PIB polymers.The larger the chain lengths the slower is the re-orientation. The reproduction of themeasurements of the orientation angle χ may be dependent on the concentration andthe solvent quality.

5.4 Conclusion and discussion

In this chapter we have shown the results of the measurements in the four-roll millgeometry. We were able to make a comparison between the macroscopic 4-mode

Results 97

PS solution

Time (s)

Orien

tation

angleχ

(deg

rees)

2520151050

140

120

100

80

60

40

20

0

-20

-40

PIB solution

Time (s)

Orien

tation

angleχ

(deg

rees)

2520151050

140

120

100

80

60

40

20

0

-20

-40

Figure 5.14: Time series of the orientation angle χ for the PS solution (upper figure)and the PIB solution (lower figure). Due to the much shorter chain length of the PSmolecule the re-orientation process is much faster than for the PIB molecule whenthe rotation direction of the four-roll mill goes in the opposite direction.

98 Chapter 5

Giesekus model and the experiments. The calculations of N1 compared by the mea-surements in the stagnation point and axes of symmetry are in agreement with eachother. However when large Deborah numbers (Deg) are used, deviations becomelarger. We validated the proposed correction method for 3D flow effects and it wasfound to work satisfactorily.

We compared a concentrated PS solution with a concentrated PIB solution. The chainlength of the PIB polymer is more than twenty times longer than the PS polymer.The short chain length of the PS polymer leads to a saturation of the birefringence ∆nin the stagnation point when the cylinders are rotating at a high angular velocity ω.The PIB polymers do not show any saturation effects, even when the angular velocityof the cylinders ω = 15 rad/s. This means that they are still not fully stretched whenthey are staying in a strong elongational flow for a long time.

For the time-dependent measurements we did a start-stop experiment and an oscilla-tory experiment. From the start-stop experiments, we see that the relaxation processof ∆n consists of two timescales: a small and a large one. The small timescale corre-sponds to the timescale in which the polymers suddenly shrink when the flow stops.The large relaxation time is caused by re-orientation of the polymer chains due tomolecular diffusion. The longer the chains are the larger the relaxation time. Themeasured values of the dominant relaxation times in the rheometer are in agreementwith measurements in the FIB experiments. In the oscillatory measurements we lookat the regime of linear viscoelasticity. This regime becomes smaller when polymerchains are shorter. Furthermore we show results of the orientation angle χ as a func-tion of time in an oscillating flow. The re-orientation of the polymer chains when thecylinders suddenly rotate into the opposite direction, goes slower when polymers arelonger. Also the reproduction of the different oscillation cycles becomes worse whenpolymers are longer.

Interpreting the time-dependent measurements is very difficult and it is obvious clearthat a microscopic description of one polymer molecule by a dumbbell model is in-sufficient for a time-dependent flow. It is also questionable if the macroscopic opticalproperties like the orientation angle of the refractive index axis n1 with respect tothe laboratory frame, is still equal to the orientation angle χ of the polymers when atime-dependent flow is considered.

References

Anna, S.L., G.H. McKinley, D.A. Nguyen, T. Sridhar, S.J. Muller, Jin Huangand D.F. James. “An interlaboratory comparison of measurements fromfilament-stretching rheometers using common test fluids.” J.Rheology 4583-114 (2001).

Fuller, G.G. and L.G. Leal “Flow birefringence of concentrated polymer solutionsin two dimensional flows.” J. Polymer Science 19 557-587 (1981).

Larson, R.G. The structure and rheology of complex fluids Oxford UniversityPress, New York (1999).

Li, Ji-Ming and W.R Burghardt. “Flow birefringence in axisymmetric geome-tries.” J.Rheology 39 743-765 (1995).

Peters, E.A.J.F, “Polymers in flow. Modelling and simulation.” PhD. thesisDelft University of Technology, Delft (2000).

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100 Chapter 5

Chapter 6

Conclusions andrecommendations

In the last chapter of this thesis we will summarize the main results and conclusionsbased on the work presented in this thesis. Occasionally, recommendations for furtherresearch will be given.

6.1 Conclusions

The aim of this research was to make a comparison between microscopic calcula-tions, macroscopic calculations and experiments for a flow of a viscoelastic fluid in acomplex geometry. We have built an experimental set-up in which a stationary anda time-dependent elongational flow can be created. For this we used the so-calledfour roll mill geometry. This geometry is complex enough for testing new modelsnumerically and experimentally. We used a FEM code for calculating the viscoelasticflow in the four-roll mill geometry using a macroscopic 4-mode Giesekus model anda microscopic FENE model. By means of birefringence experiments, we were ableto measure stresses. For the calculations, we first characterize the polyisobutylene(PIB)/Decalin solution in shear flow by means of a rheometer. Using the parametersfound in the characterization, we made a numerical comparison between microscopiccalculations and macroscopic calculations. Finally we made a comparison betweena concentrated polystyrene (PS)/decalin solution and the characterized PIB/decalinsolution. In the next sections we will mention the conclusions of the calculations andexperiments separately.

6.1.1 Conclusions of the calculations

• In the 3D inelastic calculations we found a minimum length H for the cylindersin order to get a 2D flow midway the cylinders. These calculations also vali-

101

102 Chapter 6

date the correction method for obtaining 2D flow data. Furthermore we foundsecondary flows near the bottom and top of the flow container.

• The parameter study shows that there is a certain maximum local Deborahnumber Del which can be reached in the central stagnation point of the four-roll mill geometry. This is a serious drawback of this geometry when polymerflow at high strain rates needs to be investigated. The higher shear thinningthe fluid, the lower the maximum Del in the stagnation point is.

• The viscoelastic calculations show the significance of making a distinction be-tween the shear-viscosity ηS and the elongational viscosity ηE . Using a Giesekusmodel and FENE model with the same shear viscosity ηS does not imply thatthe elongational viscosity ηE for both models is the same.

• A good data fit of the macroscopic 4-mode Giesekus model and the microscopicFENE model in shear flow is not sufficient for accurate calculations in complexgeometries. Also a characterization in planar and uniaxial elongation flow isneeded.

• The velocity field is a bad discriminating parameter for a comparison of themicroscopic calculations and macroscopic calculations.

6.1.2 Conclusions of the experiments

• The correction method for correcting 3D flow data in 2D flow data, proposed inthis thesis, has been validated experimentally. It will give useful results whenthe aspect ratio is 1:5.

• The characterization of the 2% (w/w) PIB/Decaline solution in shearflow hasbeen done and the stationary experiments can be very well fitted by the FENEmodel and the 4-mode Giesekus model. However the start-up measurementswhen compared to the FENE calculations show some deviations.

• The measured values of N1 in the four-roll mill geometry are in agreement withthe calculated values of the 4-mode Giesekus model. A comparison betweenexperiments and microscopic calculations with the FENE model could not bemade due to the high Deborah number which are reached in the computations.

• The architecture of a polymer molecule has large influence on the birefringence.Saturation of birefringence will be shown earlier in short chain length moleculeslike the PS solution.

• The longer the polymer chains are, the larger the regime of linear viscoelasticityis.

• The re-orientation process of the polymers in a time-dependent flow goes fasterwhen the length of the polymers become smaller.

Conclusions and recommendations 103

• It is questionable if the orientation of long polymer chains in a semi-dilutesolution in a time-dependent flow can be described by a dumbbell model.

6.2 Recommendations for future research

In this thesis we investigated the behavior of polymeric liquids in elongational flowby means of experiments and numerical calculations. Still there are a lot of questionsleft and they can be used for determining a direction for future research. Generally,the experiments of the dynamic behavior should be extended and for making progressin the simulations we have to wait for faster computers appears on which fully 3Dcalculations with a high resolution can be performed. We will give some recommen-dations for the experiments and calculations separately.

Experiments

• For a good comparison of different types of models, first a characterization inboth shear flow and elongational flow must be available. For measuring theelongational viscosity of solutions, it is crucial to have a reliable extensionalrheometer [Anna et al. (2001)]. The design and development of this instrumentwould be an object of study for the future.

• The dynamic behavior of polymer solutions at high strain rates should be in-vestigated more for better understanding the mechanism of orientation andstretching of molecules.

• A more detailed study of the relationship between stress and birefringence intime-dependent flows should be done.

Numerical simulations

• The development of computer codes for three dimensional viscoelastic calcula-tions.

• The development of stable numerical schemes at high Deborah numbers so thatindustrial problems can be calculated.

• The development of models where the shear viscosity as a function of shear ratecan be adjusted independently from the elongational viscosity as a function ofstrain rate.

• The microscopic FENE model used in this thesis should be extended for moreconcentrated solutions and the architecture of the polymer molecule should beincorporated in this model.

104 Chapter 6

List of symbols

symbol description unit

b finite extensibility parameter -c∗ coil-overlap concentration kg/kgd light path length mk Boltzmann’s constant J/Kn number of polymers per unit volume -n power law index -n1 index of refraction along axis 1 -∆n birefringence -∆n2D birefringence in 2D flow -n refractive index tensor -t time su velocity in x direction m/sv velocity in y direction m/sw velocity in z direction m/s

Apem peak retardation level PEM radAωn, A2ωn normalized amplitudes -C stress optical constant Pa−1

D slit distance between cylinder walls mD rate of deformation tensor -F force NF spring force in FENE spring NGi relaxation modulus of ith mode PaG′ Storage modulus PaG′′ Loss modulus PaH spring stiffness N/m

105

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symbol description unit

I light intensity of the laser WI0 maximum laser light intensity WI1 . . . I4 Stokes parameters -Jn(x) Bessel function of nth order -Kn consistency parameter Carreau model -L length of cylinders mN1 first normal stress difference PaQ connector vector length mQ0 maximum connector vector length mQ(t) configuration dumbbell as function of time -Q(x, t) configuration field -R radius of the cylinders mT absolute temperature K

α shear thinning parameter Giesekus model -αi shear thinning parameter of ith mode -γ displacement mγ shear rate s−1

δ retardation degrees,radiansε strain rate s−1

ζ friction coefficient -η, ηS shear viscosity Pa sηE elongational viscosity Pa sη0 zero-shear-rate- viscosity Pa sηi shear viscosity of ith mode Pa sκ transpose of velocity gradient tensor -λ wavelength of laser light mλi relaxation time of ith mode sλH relaxation time FENE model sν frequency of light Hzρ density of the fluid kg/m3

τ stress tensor Paτxy shear stress Paτlock time constant lock-in amplifier sτlp time constant low-pass filter sτph time constant photo diode sχ orientation angle degreesω angular velocity rad/sΦ maximum angle of rotation rad

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Abbreviations

FEM Finite Element MethodFENE Finite Extensible Non-linear ElasticFIB Flow Induced BirefringencePEM Photo Elastic ModulatorPIB PolyisobutylenePS PolystyreneSOR Stress Optical Rule

Nawoord

Een proefschrift schrijven is een enorme klus. Ontzettend veel gezwoeg en het in-caseren van vele (kleine) tegenslagen zijn eerder regel dan uitzondering bij het totstand komen van het proefschrift. Gelukkig waren er in mijn omgeving altijd mensendie mij in de minder leuke momenten konden steunen. Linda bedankt dat je almijn verhalen geduldig hebt aangehoord en mij altijd gesteund hebt op de kritischemomenten. ’t Was jammer dat je door de week vaak ver weg was maar daar gaatbinnenkort verandering in komen. Mijn familie, in het bijzonder mijn vader wil ikbedanken voor het geven van adviesen op het gebied van het organiseren van eenonderzoek en voor de pep-talks die zo nu en dan nodig waren. Mijn vrienden Martijnvan den Boogaart, Arjan Gooijer, Wilbert Kalis Hans van ’t Spijker, Alwin Kaashoeken Han Slootweg wil ik ook bedanken voor hun belangstelling tijdens mijn onder-zoek en voor de broodnodige afleiding daarnaast. Het was prettig om tijdens onzeeetafspraken onze vreugdevolle en minder vreugdevolle dingen met elkaar te delen endeze te verluchtigen met het geestrijke vocht.

Het werken op een laboratorium is een unieke ervaring die ik niet had willen missen.Ons laboratorium wordt bevolkt door een bonte verzameling mensen. De combinatievan humoristische collega’s gepaard met de grote vrijheid resulteerde in een uiter-mate goede sfeer binnen de groep. Ik heb vele goede herinneringen aan de BBQ’s,borrels en etentjes en de vele nuttige en onnuttige discusies die we samen gevoerdhebben. Al mijn collega’s wil ik bedanken voor de fijne tijd die ik de afgelopen jarenop het lab met jullie heb meegemaakt. Met Michiel van Bokhoven, Cas van Doorne,Geert Brethouwer en Alja Vrieling heb ik heel wat middagpauzes lopend langs de kadevan de Schie doorgebracht om even een frisse neus te halen alvorens we weer achterde computer doken. Mijn kamergenoten Theo Barenbrug en Jose Hernandez wil ikbedanken voor de discussies over polymeerfyscia. Daarnaast waren jullie gezelligekamergenoten waarmee ik heel wat afgelachen heb en waar ik veel van geleerd heb.

Ook op vakinhoudelijk gebied ben ik veel dank verschuldigd aan Ben van den Bruleen Martien Hulsen. Mijn promotor Ben van den Brule wil ik bedanken voor de zeergrote vrijheid tijdens het promotie onderzoek en de mogelijkeid die je me gebodenhebt om me bezig te houden met de experimentele reologie. Martien Hulsen wil ikbedanken voor de hulp bij het computer programma DYNAFLOW en de kennis dieje hebt overgedragen op het gebied van nummerieke simulaties van viscoelastische

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stromingen. Verder waren jouw kritische opmerkingen over de inhoud en opbouw vanmijn proefschrift van groot belang voor de leesbaarheid van dit boekje. Frans Nieuw-stadt wil ik bedanken voor de begeleiding tijdens de schrijffase van mijn proefschrift.

Het OBP personeel Roland van der Velden, Cor Gerritsen, Bert van der Velden enJoop Bodde wil ik bedanken voor de vele uren die zij in de werkplaats hebben doorge-bracht om mijn proefopstelling draaiend te krijgen. ’t Was zeker niet altijd makkelijk.Niettemin is er uiteindelijk wel iets moois uitgekomen. Ria van der Brugge-Peters enEveline van der Veer wil ik bedanken voor de noodzakelijke administratieve zakenzodat deze veel sneller geregeld werden dan wanneer ik alles alleen had moeten uit-zoeken. Tenslotte wil ik mijn Panta Rhei mede-bestuurders Ton van Heel en FrankPeters en later Theo Treurniet en Cas van Doorne bedanken voor de prettige samen-werking tijdens mijn bestuursjaren. Panta Rhei was voor mij bijzonder nuttig omnaast het gewone stromingleer onderzoek ook met totaal iets anders bezig te zijn.

Jurrian

Curriculum Vitae

Jurrian Zijl werd geboren op 5 april 1974 te Zaanstad. Toen hij 5 jaar oud wasverhuisden zijn ouders naar Bennekom, de plaats waar hij zijn verdere jeugdjarendoor bracht. Van 1986 tot en met 1992 volgde hij onderwijs aan het Ichthus Collegete Veenendaal. Daar behaalde hij in juni 1992 het VWO diploma. Van september1992 tot juni 1997 studeerde hij Werktuigbouwkunde aan de Technische UniversiteitDelft. Tijdens zijn studie heeft hij stage gelopen bij het Maritiem Research Insti-tuut (MARIN). Zijn afstuderen deed hij onder leiding van Prof.dr. R.V.A. Oliemansen ging over filmdiktemetingen langs een Taylorbel in slugstroming met behulp vaneen optische meettechniek (LIF). Dit werk werd gedaan in de vakgroep algemene stro-mingsleer van de faculteit Werktuigbouwkunde. Na zijn studie begon hij in september1997 als promovendus bij Prof.dr.ir. B.H.A.A. van den Brule. Daar deed hij reologischonderzoek naar rek verschijnselen in polymeer oplossingen waarvan dit proefschrift hetresultaat is. Vlak voor het afronden van dit proefschrift is hij in juni 2001 getrouwdmet Linda van den Bosch.

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