Extrusion instability in an aramid fibre spinning process

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IN AN

ARAMID FIBRE SPINNING PROCESS

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus, prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties

in het openbaar te verdedigen op donderdag 9 juli om 15.00 uur

door

Sita DROST werktuigkundig ingenieur geboren op 18 oktober 1977

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Prof. dr. ir. J. Westerweel Prof. dr. S. J. Picken

Samenstelling promotiecommissie:

Rector Magnificus Voorzitter

Prof. dr. ir. J. Westerweel Technische Universiteit Delft, promotor Prof. dr. S. J. Picken Technische Universiteit Delft, promotor

Dr. ir. H. Boerstoel Teijin Aramid B. V.

Prof. dr. ir. M. T. Kreutzer Technische Universiteit Delft Onafhankelijke leden:

Dr. B. I. M. ten Bosch Shell Global Solutions

Prof. dr. J. J. M. Slot Technische Universiteit Eindhoven Prof. dr. ir. H. W. M. Hoeijmakers Universiteit Twente

The work in this thesis was financed by Teijin Aramid B. V. It was carried out partly at the Teijin Aramid research institute in Arnhem, and partly in the La-boratory for Aero- and Hydrodynamics of the faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology. Dr. ir. H. Boerstoel (Teijin Aramid B. V.) has, as a supervisor, significantly contributed to the pre-paration of this dissertation.

ISBN 978-94-6186-498-7

Printed by Ipskamp Drukkers B. V. Enschede

Cover: particle streak images of a viscoelastic Boger fluid flowing through an array of three outlets (see chapter 4).

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Samenvatting

De effici¨entie van polymeerextrusieprocessen kan behoorlijk worden beperkt door het optreden van viscoelastische extrusie-instabiliteiten. In het produc-tieproces van para-aramidevezels is bijvoorbeeld een extrusie-instabiliteit op micrometerschaal verantwoordelijk voor de verspilling van tonnen polymeer per jaar.

Er is tegenwoordig vrij veel onderzoeksliteratuur beschikbaar over dit soort extrusie-instabiliteiten. Deze literatuur is echter grotendeels van toepassing op isotrope vloeistoffen, terwijl de polymeeroplossing die wordt gebruikt voor de productie van para-aramidevezels vloeibaar kristallijn is. Vloeibaar kris-tallijne polymeren zijn anisotroop in rust, en het is bekend dat het stromings-gedrag ervan afwijkt van dat van andere viscoelastische vloeistoffen. Er was daarom aanvullend onderzoek nodig om de extrusie-instabiliteit in het para-aramide productieproces in kaart te brengen.

Het werk dat beschreven wordt in dit proefschrift houdt zich, ten eerste, bezig met de vraag in welke mate een contractiestroming van een nemati-sche aramideoplossing lijkt op contractiestromingen van isotrope polymeer-vloeistoffen. We hebben gekeken naar het gedrag van een nematische con-tractiestroming en de invloed hiervan op de ge¨extrudeerde polymeerstraal. Ten tweede is de invloed bestudeerd van de aanwezigheid van naburige uit-stroomopeningen op het gedrag van viscoelastische contractiestromingen, aan-gezien in een gemiddeld vezelspinproces zo’n 1 000 polymeerstraaltjes dicht op elkaar ge¨extrudeerd worden.

Voor het eerste gedeelte van het onderzoek is het aramide spinproces ge-modeleerd door een 100 µm diepe 100:1 vlakke contractiestroming met vrije

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uitstroom. De diepte van 100 µm is gekozen zodat de optisch anisotrope aramideoplossing transparant genoeg is om de stroming zichtbaar te maken, terwijl met deze diepte de drukval over de geometrie nog niet zo hoog is dat de glazen stromingscellen waarin de geometrie¨en ge¨etst zijn, zouden bescha-digen bij stromingssnelheden die realistisch zijn voor een vezelspinproces.

De contractiestroming van een nematische aramideoplossing is vergeleken met het gedrag van een PEG-PEO Bogervloeistof in dezelfde geometrie, met behulp van stromingsvisualisatie en Particle Image Velocimetry (PIV). Er is aangetoond dat er, onder vezelspinomstandigheden, viscoelastische wervels optreden in een nematische aramideoplossing. Net als in contractiestromin-gen van isotrope polymeervloeistoffen worden ook in aramideoplossing deze wervels groter als het debiet toeneemt, en worden ze kleiner als de ingang van de contractie geleidelijker gemaakt wordt (bijv. taps toelopend, of afgerond). Het snelheidsveld in de aramideoplossing bleek kenmerkend te zijn voor het shear-thinning gedrag van deze oplossing.

De invloed van de vloeibaar kristallijne defectstructuur in de aramideop-lossing was te zien in een golfpatroon in het stroomopwaartse kanaal, en in het optreden van gebieden met voorkeursstroming, in de eerste minuten na het aanzetten van de stroming.

Verder is aangetoond dat de heen-en-weer gaande beweging van de ge¨extru-deerde polymeerstraal gekoppeld is met asymmetrische snelheidsschomme-lingen in het stroomopwaartse kanaal. Alhoewel er in de experimenten geen extrusie-instabiliteit is waargenomen, laat toch het bestaan van een relatie tus-sen het gedrag van de polymeerstraal en het snelheidsveld stroomopwaarts, zien dat de stabiliteit van de stroming in het stroomopwaartse kanaal be-langrijk is voor de stabiliteit van de polymeerstraal.

Omdat aangetoond is dat contractiestromingen van nematische vloeistof-fen en isotrope polymeervloeistofvloeistof-fen voldoende op elkaar lijken, kon voor het bestuderen van de invloed van meerdere uitstroomopeningen gebruik ge-maakt worden van een modelvloeistof. Er zijn experimenten gedaan met een PEG-PEO Bogervloeistof, en numerieke simulaties met een FENE-CR model (Finitely Extensible Non-linear Elastic, Chilcott-Rallinson benadering), beide in contractiegeometrieën met één of drie uitstroomopeningen, en een grote of kleine gatafstand in het geval van drie uitstroomopeningen.

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drie uitstroomopeningen de kromming van de stroomlijnen in de richting van de uitstroomopeningen zorgt voor een horizontale drukgradi¨ent, waardoor de stroming ongelijk verdeeld wordt over de verschillende uitstroomopenin-gen. Doordat de kromming van de stroomlijnen verandert als de wervels op de randen van de uitstroomopeningen ontstaan en groter worden, is de ver-deling van de stroming over de verschillende uitstroomopeningen afhankelijk van het Weissenberg getal van de stroming.

Verder was te zien dat de grootte van de wervels vermindert door de aan-wezigheid van meerdere uitstroomopeningen, waarbij het verkleinen van de gatafstand zorgde voor een verdere vermindering van de wervelgrootte. Dit bleek te leiden tot een hogere maximale reksnelheid, een hogere drukval en verminderde stromingsstabiliteit. De schommelingen in de wervelgrootte in wat hier instabiele stroming genoemd wordt, lijken schommelingen in het de-biet en in de stromingsrichting in de uitstroomopeningen te veroorzaken.

De resultaten die in dit proefschrift worden gepresenteerd zijn relevant voor vezelspinprocessen, maar ook voor andere productieprocessen waarbij een viscoelastische vloeistof uit meerdere uitstroomopeningen ge¨extrudeerd wordt. Een logische vervolgstap zou zijn om experimenten te doen met een transparante modelvloeistof in een complexere, drie-dimensionale geometrie, om extrusieinstabiliteit te bestuderen in een geometrie met meerdere uitstroom-openingen en om de effici¨entie van dit soort extrusieprocessen te optimalise-ren.

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Summary

The efficiency of polymer extrusion processes can be severely limited by the occurrence of viscoelastic extrusion instabilities. In a para-aramid fibre spin-ning process, for example, a µm-scale extrusion instability is responsible for the waste of tons of polymer per year.

At present, a considerable amount of research literature is available on such viscoelastic extrusion instabilities. However, this literature largely ap-plies to isotropic polymers, whereas the polymer solution that is used for the production of para-aramid fibres is liquid crystalline. Liquid crystalline poly-mers (LCPs) are anisotropic at rest, and their flow behaviour is known to devi-ate from that of other viscoelastic fluids. Therefore, more research was needed to characterise the extrusion instability in the para-aramid fibre spinning pro-cess.

The work presented in this thesis deals, first, with the question to which extent the contraction flow of a nematic aramid solution is similar to contrac-tion flow of isotropic polymeric fluids. More specifically, we looked into the flow stability of nematic contraction flow, and its influence on the extruded aramid jet. Second, as a fibre spinning process typically involves the extrusion of around 1 000 closely spaced jets, the influence of the presence of neighbour-ing outlets on the behaviour of viscoelastic contraction flow is addressed.

For the first part of the research, the aramid fibre spinning process was modelled by a 100 µm deep 100:1 planar contraction flow with free outflow, which was designed to capture the essential features of the extrusion process. At a depth of 100 µm the optically anisotropic aramid solution is sufficiently transparent to allow for flow visualisation, while at the same time the pressure

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drop over the geometry permits flow velocities that are realistic for fibre spin-ning, without damaging the glass flow cells in which the planar contraction geometries were etched.

The contraction flow of a nematic aramid solution was compared with the behaviour of a PEG-PEO Boger fluid in the same geometry, using flow visu-alisation and Particle Image Velocimetry (PIV). It was shown that, under fi-bre spinning conditions, a nematic aramid solution shows viscoelastic vortex growth. Like in contraction flows of isotropic polymeric fluids, the vortex size in the aramid solution increases with increasing flow rate, and decreases when the contraction entrance is made more gradual (e.g. tapered, or rounded). The velocity field in the aramid solution was demonstrated to be characteristic of its shear-thinning behaviour.

The influence of the defect structure in the aramid solution was visible in a wavy instability in the upstream channel, and in the occurrence of regions with a higher velocity than the surrounding flow, in the first minutes after starting the flow.

The oscillation of the extruded jet was shown to be coupled to asymmetric velocity fluctuations in the upstream channel. Although no extrusion insta-bility was encountered in the experiments, the existence of a relation between the jet behaviour and the upstream velocity field implies that the stability of the upstream velocity field is important for the stability of the jet.

The similarity between the contraction flows of a nematic aramid solution and an isotropic viscoelastic fluid justifies the use of experiments with model fluids in the study of para-aramid fibre spinning. Therefore, the study of the influence of the presence of multiple outlets was carried out with a model fluid. Experiments using a PEG-PEO Boger fluid, and numerical simulations using a FENE-CR model (Finitely Extensible Non-linear Elastic, Chilcott-Ral-linson closure), were performed in contraction geometries with one or three outlets, and a large or small distance between the outlets in the three-outlet geometries.

The experiments and simulations show that in the three-outlet geometries, the curvature of the streamlines towards the outlets causes a horizontal pres-sure gradient in the upstream channel, resulting in the flow rate being dis-tributed unequally over the outlets. Because the streamline curvature changes with increasing lip vortex size, the distribution of the flow rate over the outlets

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depends on the Weissenberg number of the flow.

Furthermore, the vortex size was observed to decrease due to the pres-ence of multiple outlets, with a smaller distance between the outlets leading to smaller lip vortices. This was demonstrated to result in a higher maximum elongation rate, a higher pressure drop over the geometry, and a decreased stability of the flow. The fluctuations in vortex height in what is classified here as unstable flow seems to lead to fluctuations in flow rate and outflow direction in the outlets.

The results presented in this thesis are relevant for fibre spinning pro-cesses, but also for other production processes featuring multi-outlet extru-sion of viscoelastic fluids. A logical next step would be to do experiments with a transparent model fluid in a more complex, three-dimensional extru-sion geometry, to study extruextru-sion instability in a multi-outlet geometry and to optimise the efficiency of such extrusion processes.

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Nomenclature

Fibre spinning terms

Spinning The production process of yarn/fibres.

Spinneret Plate with a large number (O(1000)) of small holes (∅ ≈_{50 µm),}

from which polymer solution is extruded into jets.

Spin pack Assembly of spinneret, filters and complementary parts.

Filament Long, thin thread; an aramid yarn is made up of a bundle of fila-ments.

Fibre A thread or filament from which a vegetable tissue, mineral substance, or textile is formed (Oxford Dictionary). Often used as a synonym for yarn or filament.

Yarn Bundle of filaments, usually 500, 1000 or 2000.

Coagulation bath Bath containing non-solvent, in the case of aramid spin-ning usually water with a small amount of sulfuric acid, in which the yarn (up to then a bundle of liquid jets) solidifies.

Air gap Space between the spinneret and the coagulation bath surface, where the –still liquid– filaments are stretched, to increase their orientation.

Aramid spinning solution Nematic solution of 19.8 wt% PPTA in concen-trated, 99.8 wt% H2SO4, usually processed at temperatures between 80◦C and 90◦C.

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Nematic Form of orientational order in a liquid crystalline material, with one (average) direction of orientation and no long-range positional order.

Abbreviations

CFD Computational Fluid Dynamics

LCP Liquid Crystalline Polymer

LDV Laser Doppler Velocimetry

PIV Particle Image Velocimetry

PPTA Poly-paraPhenylene TerephtalAmide

Symbols

Symbol Description Units

Cxy (magnitude squared) coherence

-D diameter m

D rate of deformation tensor s−1

De Deborah number

-E elastic (or Young’s) modulus GPa

f frequency Hz

F body force N

gk k-th mode relaxation strength GPa

G shear modulus GPa

h half-depth m

I light intensity cd

K power law consistency index Pa·sn

L (characteristic) length m

L2 _{FENE model maximum extensibility}

-M magnification

-Mw (weight average) molecular weight g/mol

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-p pressure Pa

Pxy cross (power) spectral density power/Hz

Q (volume) flow rate m3/s

R radius m

Re Reynolds number

-S scalar order parameter

-t time s

T temperature ◦C

T characteristic time scale s

u, v, w velocity in x-, y-, and z-direction m/s

U characteristic velocity m/s Wi Weissenberg number -β contraction ratio -δ phase lag ◦ ϵ strain -˙ϵ elongation rate s−1 γ shear strain -˙γ shear rate s−1 η (apparent) viscosity Pa_·s ηe extensional viscosity Pa·s

ηp, ηs polymer and solvent viscosity Pa·s

θ half-angle of convergence ◦ λ relaxation time s λr retardation time s µ viscosity Pa·s ρ density kg/m3 σ surface tension N/m

σ (normal) stress tensor Pa

τ (shear) stress tensor Pa

ψ stream function m2_/s

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1. Introduction

Racing bikes, car tires, bullet-proof vests, cables, sailing boats, protective un-derwear... Something all these products have in common, is that they may contain fibres to improve their performance. Those fibres can, for example, be para-aramid fibres, such as Twaron and Kevlar. These high-performance fibres combine a high (tensile) strength, high stiffness and a good resistance against cutting, heat and chemicals. These properties make para-aramid fibres extremely suitable for the aforementioned applications and countless others.

With a global demand of around 50 000 tons per year, para-aramid fibres are the largest commercial application of so-called liquid crystalline polymers (see also section 2.4, for more explanation about liquid crystals).

Para-aramid fibres are produced through what is known as a dry jet wet spinning process. The poly-p-phenylene terephtalamide (PPTA) polymer is dissolved in sulfuric acid, and the resulting solution is extruded from a so-called spinneret, that is, a plate with up to two thousand small holes. The resulting jets of polymer solution are stretched and then led through a water bath – the coagulation bath. The water in the bath cools the jets, and part of the sulfuric acid in the jets diffuses into the water. As a result, the jets solidify. The remaining acid is removed in a subsequent number of washing steps, after which the bundle of filaments (called yarn) is dried and wound onto bobbins (figure 1.1).

The yarn on these bobbins is the base material for most applications of aramid fibres. It can, for example, be twisted into cords and cables, woven into fabrics, processed into composites, cut into staple fibre for protective clothing, or ground into pulp for applications like brake pads, paper or gaskets.

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Figure 1.1: Schematic representation of a typical dry jet wet spinning process. Via the spin pack (a.) the polymer solution is led to the spinneret (b.), through which it is extruded into jets. In the air gap (c.), the polymer jets are stretched, after which they are led through a water bath, the coagulation bath (d.). The resulting bundle of filaments – or yarn – is washed and dried (e.) and finally wound onto bobbins (f.). Note that the stretching in the air gap implies that the winding velocity is (much) higher than the extrusion velocity.

1.1 Incentive

A crucial step in the production of para-aramid fibres is the extrusion of the polymer solution into jets and the subsequent stretching of these jets, before they solidify in the coagulation bath. Both the extrusion and the stretching impose elongational flow in the polymer solution, aligning the rigid PPTA molecules along the axis of the jet, and hence giving the yarn its excellent mechanical properties.

During the start-up of the spinning process, the extrusion flow is often un-stable. Usually this instability diminishes over time, eventually allowing the remaining process steps to be engaged. However, sometimes the instability persists in some of the jets – or even in all of them – and the spinneret has to be replaced. The resulting waste of time and material is substantial. Clearly, this situation offers enormous potential for improvement.

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A closer look

Although considerable improvement can be achieved by more careful han-dling of the spinnerets and polymer solution throughout the spinning process, in part the extrusion instability appears to be inherent to the process. There-fore, prior to this research, some experiments were carried out to study the problem in more detail.

The behaviour of three free-falling jets of PPTA-solution was studied using a microscope lens. The diameter and velocity of these jets are similar to those in the production process. Figure 1.2 shows two sequences of photographs, in which a corkscrew-like instability appears in one of the jets (the other two jets are outside the field of view). In the top row, the instability disappears after a while, without causing trouble. In the bottom row, however, the corkscrew can be seen to hit the spinneret and stick to it, resulting in a lump of polymer solution. This lump keeps growing larger and eventually merges with adja-cent jets and disrupts the spinning process. Throughout the experiment, this type of instability appeared intermittently in all three jets, though not neces-sarily in all jets at the same time.

The helical distortions of the jet seem to originate from upstream, in or above the spinneret. Similar distortions can be found in literature (see also chapter 2), where they are commonly known as melt fracture. Alternative terms are solution fracture, wavy instability or volume instability. Melt frac-ture has been shown to be directly coupled with a viscoelastic instability in the flow near the entrance of the die, or spinneret. Exploratory experiments – described in chapter 2 – showed that the flow above a spinneret resembles this unstable entrance flow.

However, we cannot simply assume that the instability shown in figure 1.2 is a melt fracture-like phenomenon. The PPTA-H2SO4solution used for para-aramid fibre spinning is a so-called liquid crystalline polymer solution, with behaviour that deviates from that of most isotropic polymer melts and solutions in which melt fracture has been reported (see also chapter 2). Also, the extrusion instability shown in figure 1.2 differs from the melt fracture in-stability described in literature, in that it appears intermittently, whereas melt fracture is usually reported to set in above a certain critical flow rate, as a persistent distortion of the jet.

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ob-Figure 1.2: Onset of instability in a free-falling jet of PPTA solution, extruded from a∅_{70 µm orifice at a speed of around 0.5 m/s. Top row: Time sequence in which}

instability appears and disappears. Bottom row: time sequence in which instability appears and causes the jet to hit the spinneret surface and stick to it, forming a growing lump of polymer solution (images recorded at 15 fps, i.e. ∆t between images = 67 ms).

serve in the para-aramid fibre spinning process as melt fracture was deemed premature. Nonetheless, the idea of the extrusion instability originating from a flow instability upstream seemed sufficiently plausible to pursue further. Therefore, it was decided to investigate the flow above the spinneret, and its influence on the extruded jet, in more detail.

1.2 Objective and scope

The goal of the research described in this thesis is two-fold:

1. Investigate the stability of the flow of PPTA-H2SO4solution into a single spinneret hole, and study the influence of this flow on the stability of the jet downstream of this hole. In other words, determine whether or

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not the extrusion instability that is observed in the para-aramid fibre spinning process (figure 1.2) is a melt fracture-like phenomenon.

2. Study how the proximity of neighbouring holes in a spinneret influences the flow stability, and how possible interaction influences the overall pressure drop over the spinneret and the distribution of flow rate over the holes in the spinneret.

Our focus is on the problems caused by extrusion instability during start-up of the spinning process. Therefore, practically all experiments discussed in this thesis are with a freely falling jet, that is, without stretching the jet. Furthermore, the flow through the spinneret is considered incompressible and isothermal. As we are primarily interested in the influence of geometry on the stability of the flow, material properties, like polymer concentration and molecular weight distribution, are not considered as variables. Finally, the extrusion instability is regarded as a macroscopic flow instability. Therefore, we do not go into detail on any molecular phenomena.

Ultimately, we aim to use the results of this research to improve the effi-ciency of the production process, by reducing or even eliminating the extru-sion instability in the polymer jets, directly downstream of the spinneret. This ultimate goal is kept in mind during all phases of the research, and in the final chapter of this thesis (chapter 5), recommendations are given for further work in this direction.

1.3 Structure and approach

As a starting point for this research, we did some exploratory experiments and performed a literature study. Both are summarised in chapter 2, where also a brief explanation of viscoelastic flow, and a description of a number of viscoelastic constitutive equations are given. The subsequent chapters de-scribe how we worked towards the goals stated in the previous section: in chapter 3, the flow of PPTA-H2SO4solution through a single contraction is studied and compared with contraction flow of an isotropic model polymer solution. In chapter 4, interaction between multiple outlets is investigated and its effects on flow stability and flow rate distribution are discussed. Chapter 5 comprises the conclusions of this work, together with recommendations for

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future research and, as already mentioned in the previous section, directions for improving the efficiency of the aramid fibre spinning process.

The work in this thesis is mainly experimental, at some points supple-mented by relatively simple numerical simulations. Because it is not possible to study the flow in a realistic, three-dimensional spinning set-up, the experi-mental work was carried out in quasi-2D microfluidic flow geometries. By not only using PPTA-H2SO4solution, but also a transparent model fluid, which can be – and has been – studied in 3D, we hope to deduce at least a notion of what is going on in a real aramid fibre spinning process.

The numerical simulations in this thesis are primarily intended as support for the experimental work, to increase the understanding of experimental re-sults and to explore situations that are either too difficult, too time consuming, or too expensive to study experimentally. All simulations were carried out in simple, 2D geometries, using the FENE-CR constitutive equation (Finitely Ex-tensible Non-linear Elastic model, using an approximation by Chilcott and Rallinson, see also section 2.3). This way, the results are kept traceable and computation times acceptable.

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2. Background

The intermittent appearance of a corkscrew-shaped instability in the polymer jet, directly below the spinneret (figure 1.2), arouses curiosity about the flow further upstream. Where does this instability originate from? Is the upstream flow unstable as well? Are there bubbles or particles disturbing the flow? Un-fortunately, possibilities of viewing this flow are limited: the spinneret and spin pack are not transparent, and also the transparency of the polymer solu-tion is extremely low, due to its anisotropic, liquid crystalline character.

This chapter describes a series of exploratory experiments to study the in-stability of the jet in some more detail, and to get a first impression of the flow in the spin pack interior. After that, an introduction on viscoelastic flow is given, together with a description of a number of viscoelastic constitutive equations. The chapter ends with a literature review on extrusion instability and viscoelastic contraction flow, which was used to tentatively interpret the experimental results and to direct further research.

2.1 Exploratory experiments

Jet instability

The helical instability in the polymer jet (figure 1.2) seems to be a stochastic phenomenon; there are no conditions (at least, not known) under which it is guaranteed to occur. Even if it occurs, it is intermittently, and usually only a very limited number of jets in one spinneret are affected (typically less than 10, in a spinneret with 1 000 holes). The instability can persist for hours in one jet, but it can also disappear much sooner.

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This stochastic character makes it difficult to study the instability in a sys-tematic way. Nevertheless, some interesting observations could be done.

The first of these observations is that, in one recording, at one extrusion ve-locity, two distinct wavelengths occur in the helical instability. This is shown in figure 2.1: the wavelength in the top row (±6 jet diameters) is almost twice as large as the wavelength in the middle row (±3 jet diameters), while in the bottom row multiple wavelengths seem to be present.

The helix does not seem to move downstream with the jet. It starts directly downstream of the spinneret (figure 2.1, top and middle row, leftmost image), then briefly extends further downstream (figure 2.1, top and middle row, sec-ond and third image), and finally dies out (figure 2.1, top and middle row, rightmost image).

Furthermore, careful observation reveals that even in a seemingly stable jet, a slight oscillation is present; the outflow direction seems to rotate around the vertical axis (figure 2.2). This oscillation was analysed by tracking the po-sition of the jet in a video recording, and calculating its fast Fourier transform. The dominant frequency of the oscillation in this particular case – a single jet extruded from a∅_{80 µm hole – was found to lie around 1.5 Hz.}

Both the helical instability and the swaying motion of the jet seem to orig-inate from the flow further upstream; the influence of the coiling of the jet in the coagulation bath was found to be negligible.

Flow visualisation

Two advantages of PPTA-H2SO4solution are that it is solid at room temper-ature, and that it has a relatively long structure relaxation time (i.e. after ces-sation of flow, it takes relatively long – in the order of seconds to minutes – before the flow-induced orientation disappears, see Picken, [66]). This makes it possible to freeze the flow in a spin pack, and study cross-sections of it. To improve the visibility of the flow patterns, two colours of polymer solution were used, as explained in figure 2.3.

The most striking features of the flow are, firstly, that recirculation zones seem to be present near the entrance of the spinneret holes (figure 2.3, right), and secondly, that the flow appears to be periodic: the ribbon of fused poly-mer jets directly below the spinneret shows a fairly regular pattern of

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alter-Figure 2.1: Image sequences of the intermittent helical instability in a jet of aramid spinning solution (∅_{70 µm, extrusion velocity 0.5 m/s). The wavelength in the top}

row (±6 jet diameters) is larger than that in the middle row (±3 jet diameters). In the bottom row, a combination of wavelengths seems to be present.

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Figure 2.2: Schematic representation of low frequency oscillatory movement in an ex-truded jet of aramid spinning solution

nating yellow and black stripes, instead of being first yellow, then black, as expected from a steady, laminar flow (figure 2.4).

Further experiments showed that the pattern of recirculation zones – as we will call them for now – is influenced by the presence of neighbouring outlets and the distance between these. This is illustrated in figure 2.5, where the recirculation zones around a single outlet are clearly larger than those around outlets in an array. Also applying mild tension to the extruded jets increases the size of the recirculation zones, as shown in figure 2.6.

The rheological behaviour of the black polymer solution differs slightly from that of the original yellow polymer solution. As this might influence the flow pattern in the spin pack, also a number of experiments was carried out with only yellow polymer solution. The flow pattern is still visible, albeit less clearly, and it still shows the same recirculation zones along the edges of the array of spinning holes (figure 2.7). This gives us sufficient confidence that the flow patterns observed in the two-colour experiments are not an artifact of differences in rheological behaviour.

The flow visualisation experiments discussed here give a qualitative im-pression of the flow in a spin pack and the parameters influencing it. The re-sults show that the flow in a spin pack is more complex than initially expected, and that it is affected by the flow conditions further downstream, below the spinneret. The patterns that were observed suggest that the flow upstream of

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Figure 2.3: Visualisation of flow upstream of a spinneret. Two colours of polymer so-lution were pressed sequentially through a spinneret with a rectangular array of holes (left), after which the flow was stopped and rapidly cooled with liquid nitrogen, to freeze the flow pattern. Right: cross-section of solidified flow, showing a flow pattern that suggests the presence of recirculation zones, both at the entrance of a hole (detail) and along the edge of the array of holes.

the spinneret and the instability directly beneath it, are of a viscoelastic nature. The next sections go into some more detail on viscoelastic flow (2.2), and the numerical models that exist to describe viscoelastic behaviour (2.3).

To support the interpretation of the results of the exploratory experiments, and to direct further research, a literature study was performed, which is sum-marised in section 2.4. In this section, viscoelastic extrusion and contraction flow instabilities are discussed. Furthermore, a brief introduction is given on liquid crystalline fluids and their main differences with standard viscoelastic fluids.

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Figure 2.4: Ribbon of fused polymer jets, just below the spinneret. The regular pattern of black and yellow stripes suggests periodic flow in the spin pack.

2.2 Viscoelastic flow

Like many other polymeric liquids, aramid spinning solution shows viscoelas-tic flow behaviour under some conditions. As the term “viscoelasviscoelas-tic” already implies, viscoelastic behaviour comprises both viscous – fluid-like – and elas-tic – solid-like – elements. Very simpliselas-tically, ideal elaselas-tic behaviour can be described by a linear spring (figure 2.8, left; Hooke’s law), ideal viscous be-haviour by a dashpot (figure 2.8, right; Newton’s viscosity law), and viscoelas-tic behaviour by a combination of these.

The force needed to deform an ideal elastic solid is directly proportional to the extent of deformation. If the force is removed, the material instanta-neously returns to its original shape (e.g. stretching and releasing a rubber band). On the other hand, the force needed to deform an ideal viscous liq-uid is directly proportional to the rate of deformation. If the force is removed,

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Figure 2.5: The size of the recirculation zones is influenced by the presence of neigh-bouring outlets and the distance between them. Top: single outlet, middle: rectangular array of outlets, centre-to-centre distance = 1 mm, bottom: rectangular array of outlets, centre-to-centre distance = 0.5 mm. In all cases an extrusion velocity of 0.5 m/s was used.

the material stays in its deformed state (e.g. pouring water from a bottle into a glass). In mechanics these relations are commonly expressed in terms of (shear) stress, τ (force per unit area):

elastic: τ=Gγ, viscous: τ=η ˙γ.

Here γ is the deformation, or shear strain, ˙γ is the rate of deformation, or shear strain rate, and G and η are constants of proportionality: the shear modulus and the viscosity, respectively.

The stress in a viscoelastic material depends on both strain and strain rate. This can, for example, be illustrated by the response to a sinusoidal

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deforma-Figure 2.6: Applying tension to the extruded jets increases the size of the recirculation zones. Left: no tension was applied, right: flow pattern when mild drawing is applied to the extruded polymer jets. Rectangular array of∅_{65 µm spinning holes, extrusion}

velocity = 0.5 m/s.

Figure 2.7: Microscope image of a thin slice of solidified polymer solution. Flow through a rectangular, staggered array of outlets, ∅ _{59 µm, extrusion velocity =}

0.5 m/s. No coloured polymer solution was used in this case, but the recirculation zones are still visible. This demonstrates that the observed flow patterns in the two-colour experiments are not an artefact of small differences in rheological behaviour.

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Figure 2.8: Schematic representation of an ideal, Hookean spring (F= k·x), and an ideal, Newtonian dashpot (F =c· ˙x). A viscoelastic material can be represented as a combination of these two elements.

tion (figure 2.9). Applying a sinusoidal deformation to an ideal elastic material results in a sinusoidal stress of the same frequency, which is in phase with the deformation. In an ideal viscous liquid the stress depends linearly on the rate of deformation, so it will be 90◦out of phase with the deformation. For a (lin-ear) viscoelastic material, the response is the sum of an elastic and a viscous contribution:

γ(t) =γ0sin ωt ↔ σ(t) =Gγ0sin ωt+ηγ0ω cos ωt. With τ₀′ = Gγ0 (elastic stress amplitude), τ0′′ = ηωγ0 (viscous stress ampli-tude), and τ₀′′/τ₀′ = tan δ (i.e. τ₀′′ = sin δ, τ₀′ = cos δ), this can be re-written as

σ(t) =sin(ωt+δ).

That is, the response of a linear viscoelastic material will have a – frequency dependent – phase lag, δ, between 0◦ and 90◦(i.e. δ= 0◦in an elastic solid,

δ=90◦in a viscous fluid).

In describing the response of a viscoelastic fluid to oscillatory deformation, it is customary to denote the elastic shear modulus, or storage modulus, by G′. A viscous modulus, or loss modulus, is then defined by G′′ = τ₀′′/γ0. Likewise, denoting the viscosity by η′, the elastic contribution to the total, or

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viscoelastic viscous elastic deformation σ / σma x (-) ϵ (-) t (s) 0 π 2π 3π 4π 0 π 2π 3π 4π −1 0 1 −1 0 1

Figure 2.9: Response of an elastic solid, a viscous liquid, and a viscoelastic material (bottom plot), to a sinusoidal deformation (top plot).

complex, viscosity is η′′ = G′/ω. This way, complex notation can be used to describe the relation between deformation and stress, with G∗(ω) =G′(ω) +

iG′′(ω)and η∗(ω) =G∗(ω)/iω=η′(ω_{) −}iη′′(ω)the complex modulus and complex viscosity, respectively.

It should be noted that up to now, we discussed linear viscoelastic be-haviour. Most materials only show linear viscoelastic behaviour for suffi-ciently small deformation – for larger deformation the response becomes non-linear. In our sinusoidal deformation example, this means that the response will contain not only the frequency of the applied deformation, but also higher harmonics. Other manifestations of non-linear viscoelasticity are shear thin-ning or thickethin-ning, and shear induced normal stress.

In the limit of very slow deformation, a viscoelastic material will approach ideal viscous behaviour, whereas in the limit of very fast deformation, the material’s response will become perfectly elastic. An example of this limiting behaviour can be seen in Silly Putty: if left on a flat surface, it will form a puddle, but if a ball of Silly Putty is thrown onto the same surface, it will bounce.

Several dimensionless numbers exist to assess whether a material can be expected to behave predominantly viscous, predominantly elastic, or viscoelas-tic, under certain conditions. The most widely used dimensionless numbers

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are discussed in the next section.

Dimensionless numbers

To quantify the importance of elastic effects in a flow, two different dimension-less numbers are commonly used: the Deborah and the Weissenberg num-ber. The difference between these two numbers is subtle, and they are often (though not always correctly) used interchangeably. Instructive papers on this subject were written by Dealy, [26] and Poole [68]. The Deborah number is de-fined as the ratio of the relaxation time of the fluid, λ, and a characteristic time of the deformation process, or experiment,T:

De= λ

T . (2.1)

The Weissenberg number represents the ratio of elastic and viscous forces in a flow, and is usually defined as the product of the relaxation time of the fluid,

λ, and a characteristic deformation rate, ˙γ:

Wi=λ ˙γ. (2.2)

As can be seen in these definitions, the Weissenberg number is useful for steady flows, while the Deborah number should be used for unsteady flows. Here, steady is meant in a Lagrangian sense, that is, a flow with a constant stretch history. For example, the Weissenberg number would be used for steady pipe flow (De = 0 here), while for pulsating pipe flow the Deborah number would be used. If the diameter of the pipe varies along its length, the flow can be steady in a Eulerian sense (i.e. in the laboratory frame of refer-ence), but not in a Lagrangian sense (i.e. in a frame of reference moving with the flow). In such a case, again, the Deborah number can be used.

Sometimes complex flows can be characterised by a single length scale. In that case, it is possible to define a characteristic rate of deformation, ˙γ=U/L, while a characteristic time for the flow would beT =L/U. Consequently, De and Wi become identical. In many other complex flows, De and Wi differ only by a constant factor, which may be one of the reasons that these numbers are often used interchangeably.

For many practically relevant viscoelastic flows, it is difficult to unambigu-ously define a Deborah or Weissenberg number. To start with, most real

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vis-coelastic fluids have a spectrum of relaxation times, rather than one single relaxation time. Usually, a mathematical moment of this spectrum, or a value calculated from molecular theory is used. Furthermore, many practical flows are complex, and thus involve both shear and elongational deformation. Both can be used to estimate a characteristic deformation rate, often these deforma-tion rates are related by a constant, geometrical factor.

Although from the above the use of Wi and De in practical flows appears to be limited, these are still the most widely used dimensionless numbers for viscoelastic flow. Considering this, care should be taken when interpreting results from literature; one should always check how the Weissenberg or Deb-orah number is defined in a particular case.

Finally, as De and Wi are used to estimate the importance of elastic effects in a flow, we need to make sure that there are no other effects that dominate the flow behaviour. The most common effect to check for is inertia, for which the Reynolds number is used:

Re= ρUL

η . (2.3)

The Reynolds number compares inertial and viscous forces, with ρ the fluid density, U a characteristic velocity, L a characteristic length scale, and η the fluid viscosity.

2.3 Constitutive equations

As many flows of practical interest are viscoelastic, much effort is put into modelling viscoelastic flow behaviour. Like Newtonian flows, viscoelastic flows are governed by the conservation of mass and the Navier-Stokes equa-tions for conservation of momentum:

Dρ

Dt +ρ∇ ·u=0, (2.4a)

ρDu

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or, for incompressible flow:

∇ ·u=0, (2.5a)

ρDu

Dt = −∇p+ ∇ ·σ+F. (2.5b)

Here, ρ is the fluid density, t stands for time, u is the velocity vector, p is the pressure, σ is the deviatoric stress tensor, and F are the body forces (e.g. gravity). The operator D/Dt is the material derivative,

Dv Dt =

∂v

∂t +u· ∇v,

for any vector v (or scalar v).

The distinction between different types of flow behaviour lies in the ex-pression for the deviatoric stress: the constitutive equation. Newtonian fluids have a constant viscosity, µ. The deviatoric stress is then given by σ = 2µD, with D the rate of deformation tensor:

D= 1₂!∇u+ (∇u)T".

For a generalised Newtonian fluid, the deviatoric stress still depends only on the rate of deformation tensor, but now the viscosity is no longer constant. Rather, it is a function of a scalar measure of the rate of deformation, ˙γ =

2(D: D)12 (i.e. the second invariant of D): σ=2η(˙γ)D. This allows for shear thinning or shear thickening behaviour, for example.

In viscoelastic flow the deviatoric stress becomes time dependent – it not only depends on the current rate of deformation, but also on the deforma-tion history (which determines the current deformadeforma-tion of the material). This means that a differential or integral equation is needed to mathematically de-scribe σ.

Constitutive equations for viscoelastic flow can be based on a kinetic the-ory, that is, on a model consisting of springs and dashpots, or on a molecular theory. In the following, a short description is given of a number of widely used constitutive equations (more comprehensive descriptions can be found in e.g. [14,15], or [45]). All these equations are given in differential form, which means that they can be used in finite element or finite volume codes.

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In this thesis, we use viscoelasticFluidFoam, a viscoelastic flow solver in OpenFOAM, written by J. Favero, [34]. OpenFOAM is a free, open source finite volume software package. The viscoelastic flow solver contains the fol-lowing constitutive equations:

• Kinetic theory

– Maxwell linear

– Upper Convected Maxwell, Oldroyd-B

– White-Metzner (Larson, Cross, Carreau-Yasuda)

– Giesekus

– FENE-P

– FENE-CR

• Network theory of concentrated solutions and melts

– Phan-Thien Tanner (PTT) linear

– PTT exponential

– Feta PTT

• Reptation theory/tube models

– Pom-Pom model

– Double-eqation extended Pom-Pom (DXPP)

– Single-equation extended Pom-Pom (SXPP)

– Double convected Pom-Pom (DCPP)

Of these equations, a short description is given, together with the behaviour in simple shear and extension. In the descriptions a single relaxation time, λ, is used. This can be replaced by a discrete spectrum of relaxation times to obtain a multi-mode model. The stress is then given by the sum of the contributions from the different modes.

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Maxwell linear

The Maxwell linear equation models a spring and dashpot in series:

σ+λ∂σ

∂t =2GλD=2ηD, (2.6)

In steady shear and steady uniaxial elongation this gives

σ12=η ˙γ, σ11−σ22=3η ˙ϵ, (2.7)

respectively. That is, in steady state the linear Maxwell model gives New-tonian viscous flow behaviour. On the other hand, for very high deforma-tion rates the limit of a Hookean elastic solid is attained, with σ12 = Gγ and σ11−σ22=Eϵ (with E=3G).

The linear Maxwell equation is not frame invariant and can therefore only be used for small strains. Frame invariant versions can be derived by writ-ing the equation in a frame of reference that moves with a material element. The three most common frame invariant time derivatives are the upper- and lower-convected and the corotational time derivative.

Upper Convected Maxwell, Oldroyd-B

The Upper Convected Maxwell (UCM) model is derived by writing the linear Maxwell equation in a frame of reference that is convected with the material elements, with base vectors that are parallel to and deform with the material lines (i.e. contravariant base vectors):

σ+λ

▽

σ=2ηD (2.8)

whereσ▽is the upper convected derivative of σ:

▽

σ≡ Dσ

Dt − ∇u

T

·σ−σ· ∇u. (2.9)

Here, D/Dt is the substantial time derivative, that is, the time derivative in a frame that moves with the material element.

Dσ Dt ≡

∂σ

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If the stress is homogeneous, that is, spatially uniform, the convective trans-port term u· ∇σ =0.

Other, less popular frame invariant versions of the linear Maxwell model are the Lower Convected Maxwell model, with convected, covariant base vec-tors, and the Corotational Maxwell Model, with a reference frame that rotates with the angular velocity of a fluid element.

The behaviour of the UCM model in steady shear and steady uniaxial elon-gation is σ12=η ˙γ, σ11−σ22= # 2 1₋2λ ˙ϵ+ 1 1+λ ˙ϵ $ η ˙ϵ, (2.11)

respectively. This means that, like the linear Maxwell model, the UCM model has a constant viscosity in shear. However, the stress in uniaxial elongation becomes unbounded for elongation rates ˙ϵ≥1/2λ. This is because the spring in the model has a constant stiffness and can be extended indefinitely. To prevent this unphysical behaviour, a non-linear spring can be used, with a stiffness that increases to infinity with increasing extension. An example of a model that uses such a non-linear spring is the FENE-model, which is dis-cussed later in this section.

Oldroydderived frame invariant versions of the so-called Jeffreys equa-tion for polymer soluequa-tions:

σ+λ∂σ ∂t =2η # D+λr∂D ∂t $ , (2.12)

with λr the retardation time (will be explained further hereafter). This is the linear Maxwell equation with an extra term, arising from stresses in the sol-vent. An upper and a lower convected version of this equation can be de-rived: the Oldroyd-B and Oldroyd-A equation, respectively. Like the lower convected Maxwell model, the Oldroyd-A equation is hardly ever used. The Oldroyd-B equation is σ+λ ▽ σ =2η # D+λr ▽ D $ , (2.13) This is equivalent to σ =σ_p+σ_s, (2.14)

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where σpis the polymer stress, following from the UCM equation, and σs = 2ηsD is the Newtonian solvent stress. The retardation time λr and the total viscosity η then become

λr = ληs

ηs+ηp, η=ηp+ηs. (2.15)

In steady shear and steady uniaxial elongation, the Oldroyd-B equation gives similar results as the UCM equation:

σ12=η ˙γ, σ11−σ22=3ηs˙ϵ+ # 2 1₋2λ ˙ϵ+ 1 1+λ ˙ϵ $ ηp˙ϵ (2.16)

The Oldroyd-B equation is also obtained if a polymer solution is modelled as elastic dumbbells in a solvent. The integral version of the UCM equation, the Lodge equation, is obtained from both the Green-Tobolsky temporary net-work model and the Rouse elastic dumbbell model [45].

White-Metzner (Larson, Cross, Carreau-Yasuda)

The White-Metzner equation is a modified version of the UCM equation, with a relaxation time that depends on the rate of strain. In this way the equation allows for relaxation that is faster than any deformation, like in irreversible non-affine motion.

σ+λ(II_D)

▽

σ=GI, (2.17)

where λ is a function of IID, the second invariant of the rate of strain tensor D, IID=2D : D, and I is the unit tensor. The more common form of the White-Metzner equation is obtained by defining a new stress tensor, τ_≡σ−GI

τ+λ(II_D)

▽

τ=2η(II_D)D, (2.18)

with η(IID)≡Gλ(IID).

Non-affine motion, here modelled by the dependence of λ on IID, pro-duces shear thinning and strain softening. In OpenFOAM three different ex-pressions for λ(IID)are implemented. It should be noted that Favero uses a

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different definition of the second invariant of the rate of strain tensor, IID ≡

(2D : D)1/2, that is, the square-root of the usual definition.

Larson1 λ= λ0 1+λ0aIID , (2.19) Cross λ= λ0 1+ (LIID)1−n , (2.20) Carreau-Yasuda λ=λ0 % 1+ (LIID)b &n−1 b , (2.21)

with λ0, a, b and n constants.

In steady shear and steady uniaxial elongation, the White-Metzner-Larson equation gives σ12= η0˙γ 1+aλ0˙γ , σ11−σ22= G!1+a√3λ0˙ϵ " ˙ϵ 1+!a√3−2"λ0˙ϵ − G ! 1+a√3λ0˙ϵ " ˙ϵ 1+!a√3+1"λ0˙ϵ , (2.22) respectively, with η0 = Gλ0. If a > 2/ √

3, the UCM singularity in uniaxial elongation is eliminated. White-Metzner-Cross results in

σ12= η0˙γ 1+ (L ˙γ)1−n, (2.23) σ11−σ22= G # 1+!L√3 ˙ϵ"1−n $ 1+!L√3 ˙ϵ"1−n−2λ0˙ϵ − G # 1+!L√3 ˙ϵ"1−n $ 1+!L√3 ˙ϵ"1−n+λ0˙ϵ . (2.24)

And finally, White-Metzner-Carreau-Yasuda yields

σ12=η0˙γ % 1+ (L ˙γ)b& n−1 b , (2.25) σ11−σ22= G 1₋2 ˙ϵλ0 ' 1+!L√3 ˙ϵ"b (n−1 b − G 1+ ˙ϵλ0 ' 1+!L√3 ˙ϵ"b (n−1 b (2.26)

1_{In his book on constitutive equations, [45], Larson cites Ide and White, [40], as the source of} this expression

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a=10 a=1 η / η0 ˙γ 10−4 10−2 100 102 104 10−4 10−3 10−2 10−1 100 a=10 a=1 ηe /3 η0 ˙ϵ 10−4 10−2 100 102 104 10−4 10−2 100 102 104 L=1, n=0.8 L=10, n=0.1 L=1, n=0.1 η / η0 ˙γ 10−4 10−2 100 ₁₀2 ₁₀4 10−4 10−3 10−2 10−1 100 L=1, n=0.8 L=10, n=0.1 L=1, n=0.1 ηe /3 η0 ˙ϵ 10−4 10−2 100 ₁₀2 ₁₀4 10−4 10−2 100 102 L=1, n=0.8, b=2 L=1, n=0.1, b=10 L=10, n=0.1, b=2 L=1, n=0.1, b=2 η / η0 ˙γ 10−4 10−2 100 ₁₀2 ₁₀4 10−4 10−3 10−2 10−1 100 L=1, n=0.8, b=2 L=1, n=0.1, b=10 L=10, n=0.1, b=2 L=1, n=0.1, b=2 ηe /3 η0 ˙ϵ 10−4 10−2 100 ₁₀2 ₁₀4 10−4 10−2 100 102

Figure 2.10: Behaviour of the White-Metzner model with Larson’s (top), Cross’s (mid-dle) and the Carreau-Yasuda (bottom) expressions for λ, left column: steady shear flow, right column: steady uniaxial elongation. Larson’s expression yields shear thin-ning with a rate of -1, in shear the parameter a controls the value of ˙γ from which shear thinning sets in, while in uniaxial elongation it can be used to circumvent the singular-ity at high ˙ϵ. In the expressions of Cross and Carreau-Yasuda, L has the same function as a. Additionally, n determines the rate of shear thinning and b, in the Carreau-Yasuda expression, determines how sharp the transition to shear thinning is.

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Giesekus

In the Giesekus model, the polymer molecules are considered as dumbbells with anisotropic drag, due to the orientation of surrounding molecules. To represent this anisotropic drag, Giesekus replaced the reciprocal relaxation time in the UCM equation by an anisotropic mobility tensor that he assumed to be proportional to the state of stress in the material:

λσ▽+B· (σ−GI) =0, (2.27) with B−I=α!σ G−I " , (2.28) resulting in λτ▽+τ+ α G(τ·τ) =2GλD, (2.29) or, equivalently, λτ▽+τ+αλ η0 (τ·τ) =2η₀D, (2.30)

with again τ ≡σ−GI. For incompressible fluids, in which the stress is only

defined to within an isotropic constant, either σ or τ can be used and the Giesekus equation is equal to the UCM equation with an additional quadratic stress term.

The anisotropy is determined by the empirical constant α, with α = 0 in case of isotropic drag and α=1 in case of maximum anisotropy.

In steady shear the Giesekus model results in, [14],

η η0 = (1−f) 2 1+ (1−2α)f, (2.31) with f = 1−χ 1+ (1−2α)χ, χ2= ) 1−16α(1−α)(λ ˙γ)2₋₁ 8α(1−α)(λ ˙γ)2

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It can be seen that here η is singular for α=0 and α=1, that is, for minimum and maximum anisotropy, respectively. For α = 0.5, the viscosity simplifies into

η η0

= 2

1+)1+4 ˙γ2. (2.32)

To model a polymer solution instead of a melt, the solvent stress is in-cluded in the total stress,

τ=τ_p+τ_s, (2.33a) τ_s = −2η_sD, (2.33b) τ_p+λ₁ ▽ τ_p−αλ1 ηp * τ_p·τ_p+=2η_pD. (2.33c)

In steady shear flow, the viscosity now becomes (Bird, [14])

η η0 = λ2 λ1 + # 1₋λ2 λ1 $ (1−f)2 1+ (1−2α)f, (2.34)

with η0=ηp+ηs, λ1the polymer relaxation time and λ2=λ1ηηps the retarda-tion time.

In both the melt and the solution formulation, the model yields exponen-tial shear thinning behaviour, with a slope of -1. The addition of the retarda-tion time λ2in the solution formulation lets the viscosity level off to a constant value in the limit of high shear rates.

In steady uniaxial extension, the extensional viscosity is

ηe 3η0 = λ2 λ1 + 1 6α # 1−λ_λ2 1 $ # 3+ 1 λ1˙ϵ ', 1−4λ1˙ϵ(1−2α) +4λ2₁˙ϵ2 −,1+2λ1˙ϵ(1−2α) +λ21˙ϵ2 ($ . (2.35)

For low and high ˙ϵ, ηetends to a constant value.

FENE-P and FENE-CR

The unphysical behaviour of Maxwell-type models in elongational flow is caused by the infinite extensibility of the Hookean springs in these models.

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α=0.5, λ2/λ1=0 α=0.3, λ2/λ1=0 α=0.3, λ2/λ1=10−3 η / η0 ˙γ 10−2 ₁₀0 ₁₀2 ₁₀4 ₁₀6 10−5 10−4 10−3 10−2 10−1 100 ηe /3 η0 ˙ϵ 10−2 ₁₀0 ₁₀2 ₁₀4 ₁₀6 100.1 100.2 100.3

Figure 2.11: Behaviour of the Giesekus model, left: steady shear, right: steady uniaxial elongation (legend holds for both figures). The model shows shear thinning with a rate of -1. In shear, the value of α determines how sharp the transition to shear thinning is, while in uniaxial elongation it determines the plateau value of ηeat high ˙ϵ. If λ2/λ1is given a (small) non-zero value, also the shear viscosity levels off to a constant value at high ˙γ.

In reality, polymer chains can only be stretched up to a certain maximum length. The Finitely Extensible Non-linear Elastic (FENE) model incorporates this finite extensibility of polymer chains by using a non-linear equation for the spring force, which tends to infinity when the spring approaches its max-imum extensibility.

For a freely jointed chain with many bonds, this force-extension relation has been computed to be an inverse Langevin equation (see e.g. Larson [45]). However, usually the simpler approximation of a Warner spring is used,

F= FH

1− (R/R0)2

, (2.36)

where FH is the linear, Hookean spring force, R is the polymer end-to-end vector and R0is the end-to-end vector at full extension.

Use of the Warner spring results in an equation for the change of chain con-figuration distribution (Smoluchowski equation) that has no general analytic solution. Therefore, a closure approximation is needed. Well-known closure approximations are the one by Peterlin (FENE-P) and the one by Chilcott and Rallinson (FENE-CR). The latter is almost identical to the Peterlin closure, but adjusted so that the model gives no shear-thinning (useful when simulating

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e.g. Boger fluids).

The FENE-P and FENE-CR constitutive equations are, respectively, # 1+ 3 L2₋₃ + λ η0L2 tr(τ_p) $ τ_p+λ ▽ τ_p= 2η0D 1−3/L2, (2.37) -L2+ λ η0tr(τp) L2₋₃ . τ_p+λ ▽ τ_p=2 -L2+ λ η0tr(τp) L2₋₃ . η0D, (2.38)

where tr(τ_p)is the trace of the stress tensor, τ_ii, and L2 = HR2

0/kT is a di-mensionless maximum extensibility (H is the Hookean spring constant, k is Boltzmann’s constant and T is the temperature).

For steady flow, both sets of equations reduce to a set of coupled algebraic equations that can be solved analytically [15]. In steady shear flow, the FENE-P model gives for the only non-zero stress components, τ11and τ12,

τ11 = 2L4η0 3λ(L2₋₃₎ ' 1+cosh # 1 3arccosh # 1+27 ˙γ 2_λ2₍_L2₋₃₎2 L6 $$( , (2.39) τ12 = η₀2L4˙γ η0L4+λ(L2−3)τ11 . (2.40)

Using the definitions of cosh and arccosh,

cosh x= e

x₊_e_−x

2 ,

arccosh x=ln!x+)x2₋₁"_, it can be seen that

cosh(1 3arccosh x) = 12 '! x+)x2₋₁" 1 3 +!x+)x2₋₁"− 1 3( .

Now, from equation 2.40, it can be shown that at low shear rates the viscosity tends to a constant value, while at high shear rates it shows shear thinning behaviour, with an exponent of−2

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Considering the rather impressive load of mathematics required to obtain this result for the FENE-P model in steady shear, it is decided to use solu-tions from literature for the behaviour of the FENE-P model in steady uniaxial elongation and the FENE-CR model in shear and elongation. Herrchen and

¨

Ottinger, [38], compared the original FENE model with the FENE-P and the FENE-CR model for several standard flow types. Using a slightly different notation, including a solvent viscosity, they give series expansions of the solu-tions for steady shear and steady uniaxial elongation in the limits of low and high deformation rates. For the FENE-P model in steady uniaxial elongation they obtain for ˙ϵ→0 ηe−3ηs η0 = 3L L+3 (2.41) for ˙ϵ→∞ ηe−3ηs η0 =2L. (2.42)

The steady shear viscosity of the FENE-CR model is constant,

η−ηs η0

=1, (2.43)

while its behaviour in steady uniaxial elongation is

for ˙ϵ→0 ηe−3ηs η0 =3, (2.44) for ˙ϵ_→∞ ηe−3ηs η0 = 2L₋2. (2.45)

Phan-Thien Tanner

The Phan-Thien Tanner model is a network model in which a certain slippage of network junctions is allowed with respect to the surrounding continuum, by incorporating nonaffine motion of the Gordon-Schowalter type. The strand breakage probability in the network is non-constant and depends on chain ex-tension. Consequently, in the original formulation of this network model by Yamamoto, the stress tensor cannot be written as a function of kinematic vari-ables alone, that is, the model does not produce a closed constitutive equation.

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Figure 2.12: Behaviour of different FENE models, left: steady shear, right: steady uni-axial elongation (both dimensionless, b is the same as L2, from Herrchen and ¨Ottinger, [38]).

To still obtain a closed constitutive equation, Phan-Thien and Tanner ap-plied pre-averaging to the R2term in the chain configuration equation of Ya-mamoto’s model. This leads to the following constitutive equation

λ"τ+Y[trτ]τ=2η₀D, (2.46)

where Y[trτ]is a function of the trace of the stress tensor, τii, and "

τdenotes

the Gordon-Schowalter convected derivative, "

τ≡ Dτ

Dt − ∇u T

·τ−τ· ∇u+ξ(τ·D+D·τ), (2.47)

with ξ an empirical coefficient that determines the amount of nonaffine mo-tion (for ξ = 0, the motion is affine and the derivative is corotational, for

ξ _{= ±}1, the upper (+) and lower (-) convected derivatives are recovered, re-spectively). If Y[trτ] =1, the Johnson-Segalman model is obtained (for more details on this model, see for example Larson, [45]).

Phan-Thien and Tanner suggested two different possibilities for the func-tion Y[trτ]: linear Y[trτ] =1+ελ η0 trτ, (2.48a) exponential Y[trτ] =exp # ελ η0 trτ $ , (2.48b)

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with ε another empirical parameter. As long as ε is much smaller than ξ, the value of ξ controls the level of shear thinning, whereas the value of ε influ-ences the elongational viscosity. The two different possibilities for Y give dif-ferent behaviour in elongational viscosity. The linear version gives a mono-tonically increasing elongational viscosity that levels off to a constant value for high elongation rates. The exponential version results in an elongational viscosity with a maximum with increasing ˙ϵ (see for example Larson, [45], Bird, [15]).

A third possibility is the fixed eta, or feta-PTT model, with fixed shear viscosity, but allowing parameter selection based on elongational rheology. This model is essentially the same as the linear PTT model, but now with a λ and η0that depend on τ:

λ(τ) = λ 1+f, η0(τ) = η0 (1+g)b, with f = ελIτ η0 and g=A -I -Iτλ2 η₀2 .a .

Here, Iτand I Iτare the first and second invariant of the stress tensor, respec-tively.

Analytical solutions for steady shear and elongational flow will not be given here. Instead, the behaviour of the model will be discussed qualita-tively, based on several limiting cases.

If Y[trτ] = 1, the Johnson-Segalman equation is obtained. This equation predicts shear thinning and elongational thickening,

η= η0

1+ (1₋ξ2₎_λ2_˙γ2, ηe=

3η0

(1−2ξ ˙ϵ)(1+ξ ˙ϵ). (2.49)

Note that for ξ = 0 the results for the Upper Convected Maxwell model are recovered (no shear thinning and singularity in elongational flow) and that in shear the value of ξ determines the shear rate at which shear thinning sets in. In elongational flow, the value of ξ serves to shift the singularity in elonga-tional viscosity to higher extension rates.

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The behaviour of the PTT model is similar, but here the singularity in ex-tensional flow is blunted by the parameter ε. Furthermore, if ξ = 0, shear thinning is still predicted, because of the term involving ε (the model is then comparable with the Giesekus and FENE models). For ξ=0.2 and ε=0.01 or

ε=0.001, the behaviour in steady shear and elongation are shown in figures 2.13 and 2.14, respectively (from Phan-Thien and Tanner, [62, 63]).

Figure 2.13: Steady shear behaviour of the PTT model, for ξ = 0.2 and ε = 0.01 or

ε=0.001. From Phan-Thien and Tanner, [62, 63].

Pom-Pom

The Pom-Pom model is a model for branched polymers, intended to represent the combination of shear thinning and elongational thickening seen in such polymers. Like the Doi-Edwards model for entangled flexible linear poly-mers, it is a tube model. The polymers are modelled as idealised molecules, contained in tubes with a single backbone and multiple branches at each end. The viscoelastic stress is determined by the evolution of tube backbone stretch and tube backbone orientation:2

2_{Originally, the Pom-Pom model was formulated as a set of integral equations. Here the} (ap-proximate) differential form is given, which is also used in OpenFOAM, because of computational efficiency.

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Figure 2.14: Steady uniaxial elongation behaviour of the PTT model (qualitative). Elon-gational viscosity ηTas a function of elongation rate G. From Phan-Thien, [62].

evolution of orientation S ▽ A+ 1 λ0b ' A−1₃I ( =0, S= A IA , (2.50)

with A an auxiliary tensor and IA =tr(A)its first invariant. In terms of S the equation for the evolution of orientation becomes

▽ S+2[D: S]S+ 1 λ0bIA ' S−1₃I ( =0, (2.51)

evolution of backbone stretch DΛ Dt =Λ[D: S]− 1 λs [ Λ₋₁_]_, _(2.52) with λs =λ0se− 2 q (Λ−1) _∀_Λ_≤_q, viscoelastic stress τ= η0 λ0b ! 3Λ2S−I". (2.53)

Here, q is the number of branches on a tube. The factor 2_q in the exponent in the equation for λs can be seen as a measure of the influence of surrounding