How to Characterize Yield Stress Fluids with Capillary Breakup · First experiments on yield stress fluids have shown, that such fluids can form stable bridges with lengths bigger

41969-1Applied RheologyVolume 19 · Issue 4

How to Characterize Yield Stress Fluids with Capillary Breakup

Extensional Rheometry (CaBER)?

K. Niedzwiedz*1, O. Arnolds

1, N. Willenbacher

1, R. Brummer

2

1 Institute of Mechanical Process Engineering and Mechanics, Karlsruhe University,76131 Karlsruhe, Germany

2 Department of Rheology/Thermal Analysis, Beiersdorf AG, 20253 Hamburg, Germany

* Email: [email protected]: x49.721.6083758

Received: 23.1.2009, Final version: 7.5.2009

© Appl. Rheol. 19 (2009) 41969

Abstract:

Filament breakup of high viscosity fluids with apparent yield stress has been investigated and strategies for anappropriate characterization of their behavior in CaBER experiments are discussed. Filament profiles of such flu-ids exhibit significant concave curvature. Accurate determination of filament shape is mandatory for under-standing deformation behavior. Therefore, we have set up an optical train including high-speed camera, tele-centric objective and telecentric back-light illumination with a blue light emitting diode (LED) providing highcontrast filament shape imaging. Image analysis allows for diameter determination with an accuracy of3.55 mm/pixel. In addition to the transient filament diameter at the neck we have extracted the curvature at thispoint as a function of time and the region of deformation, in order to characterize the extensional flow behav-ior. We have investigated the time evolution of filament shape as a function of various experimental parame-ters like stretching time, velocity profile during stretching, stretching ratio and initial sample volume at con-stant stretching ratio. Filament thinning is independent of stretching time, ts and stretching velocity profile. Butwhen the same stretching ratio is applied at different initial volume fraction, filament curvature increases strong-ly with decreasing sample volume leading to an increase of filament life time according to the negative contri-bution of its curvature to the Laplace pressure inside the fluid.

Zusammenfassung:

Es wurde der Fadenabriss von hochviskosen Flüssigkeiten mit scheinbarer Fließgrenze untersucht und Strate-gien für die Charakterisierung ihres Verhaltens in CaBER Experimenten diskutiert. Das Fadenprofil solcher Flüs-sigkeiten zeigt eine signifikante konkave Krümmung. Daher ist die präzise Charakterisierung der Fadenkonturenfür das Verstehen des Deformationsverhaltens entscheidend. Aus diesem Grund wurde der optische Aufbau desCaBER Gerätes mit einer Hochgeschwindigkeitskamera, telezentrischen Objektiven und blauer, telezentrischerHintergrundbeleuchtung, die eine kontrastreiche Fadenkonturabbildung ergibt, ausgestattet. Die Bildanalyselässt eine hochauflösende Durchmesserbestimmung mit der Genauigkeit von ± 3.55 mm/pixel zu. Zusätzlich zumzeitlichen Verlauf des Durchmessers an der Fadeneinschnürung wurden auch der zeitliche Verlauf der Krüm-mung an dieser Stelle und der Formänderungsbereich berechnet, um das Dehnfließverhalten zu charakterisieren.Die zeitliche Entwicklung der Fadenprofile in Abhängigkeit von unterschiedlichen experimentellen Parameternwie z. B. Dehnzeit ts, Dehngeschwindigkeitsprofil, Dehnrate and Anfangsprobenvolumen bei konstanter Dehn-rate wurde untersucht. Die Fadenverdünnung ist unabhängig von ts und dem Dehngeschwindigkeitsprofil. Wirddie gleiche Dehnrate bei unterschiedlichem Anfangsprobenvolumen verwendet, steigt die Fadenkrümmung mitabnehmendem Probenvolumen stark an. Dies führt zur Erhöhung der Fadenlebensdauer entsprechend dem neg-ativen Beitrag der Krümmung zum Laplace Druck innerhalb der Flüssigkeit.

Résumé:

La rhéologie extensionnelle de fluides de viscosité élevée ayant une contrainte seuil a été étudiée dans le butdóbtenir des connaissances déterminantes sur leur comportement lors de ce type d´écoulement. Les expéri-ences ont été effectuées à láide du rhéomètre élongationel CaBER. Le profil des filaments obtenu pour de telsfluides montrent une courbure concave significative et une détermination précise de leurs formes est néces-saire pour comprendre leur comportement lors de la déformation. Pour cela nous avons équipé notre montageoptique dúne caméra à haute vitesse, dún objectif télécentrique et dún éclairage dárrière-plan télécentri-que de couleur bleue qui permet dóbtenir un contraste élevé de límage du filament. Lánalyse des images aupoint de courbure du filament permet une détermination du diamètre au cours du temps avec une précision delórdre de 3.55 mm/pixel. En plus de la détermination du diamètre, nous avons également étudié l´évolution dela forme de la courbure du filament en fonction du temps en variant divers paramètres expérimentaux tels que: la durée de l´élongation, le profile de vitesse durant l´élongation, le taux d´élongation et le volume initiale de

Applied_Rheology_Vol_19_4.qxd 13.08.2009 11:33 Uhr Seite 237

1 INTRODUCTION

Understanding of elongational deformationprocesses is of great importance in many indus-trial applications. Deformation and breakup ofliquid threads is crucial in such processes asspraying and atomization. But even in simpleevery-day actions e.g., filling in or out bottles,blending, or coating, extensional deformationplays an important role.

Capillary breakup was the subject of scien-tific discussion already in 19th century [1, 2]. Sincethan many studies emerged aiming at the under-standing of this phenomenon and later on givebeginning to extensional rheology. There are var-ious configurations of measurement instru-ments available for testing extensional defor-mation. Nowadays FiSER (filament stretchingextensional rheometer) [3, 4] and CaBER (capil-lary breakup extensional rheometer) are amongthe most common devices for measuring exten-sional flow properties of low viscosity fluids [5].In FiSER configurations exponential plate sepa-ration is imposed in order to produce elonga-tional deformation with constant deformationrate. The time evolution of tensile force acting onfluid filament and the midpoint filament diam-eter are measured. In CaBER experiments thesample is filled between two coaxial parallelplates creating a fluid connection between them.The plates are rapidly moved apart from eachother creating a step strain deformation. Theresulting liquid bridge deforms gradually underthe action of visco-elastic and capillary forces,contracts and finally breaks. Here, the only mea-sured parameter is the time evolution of mid-point diameter. The advantage of the CaBERtechnique is that it can be applied to liquids cov-ering a wide range of extensional viscosities,from about 5·10-3 Pa.s up to nearly 104 Pa·s. Thetechnique is straightforward, fast and requiresonly a small amount of probing sample (V < 0.1ml). In addition, for many fluids large Henckystrains up to e = 7 are attainable.

Investigations on extensional viscosity withFiSER and CaBER technique started with dilutepolymer solutions. Such solutions form uniformcylindrical filaments in CaBER experiments andtheir diameter decays exponentially [6 – 8]. Sur-prisingly, one has to dilute well below c* to findthat the corresponding relaxation time lE agreeswell with the shear relaxation time lS determinedfrom small amplitude oscillatory shear experi-ment [9]. FiSER experiments revealed significantstrain hardening for solutions containing evenlow concentration of high molecular weight poly-mer [4, 10]. Visco-elastic wormlike micelles solu-tions exhibit flow behavior similar to polymersolutions [11]. However, unlike polymers, wormlikemicelles are self-assembled, this means that theycan break apart under large elastic stresses result-ing in large differences between extensional vis-cosities deduced from FiSER and CaBER experi-ments [12]. Recently capillary breakup rheometryis more and more applied to investigate exten-sional properties of complex fluids [13 – 15]. Theseexperimental studies were supported by theoret-ical analysis and numerical simulations. Renardyperformed one dimensional analysis for the evo-lution of the fluid jet based on self-similarity solu-tions for power law fluids [16, 17], Giesekus [18] andPhan-Thien-Tanner (PTT) [19] models. Further in-vestigations developed more detailed analysis ofstretching deformation and breakup. Basaran etal. proved validation of one dimensional ap-proaches carrying out the comparison betweenone and two dimensional models concerningNewtonian and Carreau fluids. They performedcalculations for the time evolution of filament pro-files including two dimensional analysis of viscos-ity and velocity field through out the full length ofthe liquid bridge [20, 21]. Lately, Webster et al.modeled step-strain deformation of CaBER typeconfigurations [22]. Two dimensional analysis offilament profiles under uniaxial deformation wasperformed with special regard to differencesbetween Oldroyd-B, Giesekus and PTT models.

41969-2 Applied RheologyVolume 19 · Issue 4

l´échantillon à taux d´élongation constant. La réduction du filament est indépendante de la durée de l´élonga-tion ts et du profile de vitesse. Par contre pour un même taux d´élongation appliqué à un volume initial del´échantillon différent, la courbure du filament augmente fortement lorsque le volume diminue conduisant àune augmentation de la durée de vie du fil selon la contribution négative de sa courbure à la pression de Lapla-ce à líntérieur du fluide.

Key words: yield stress fluids, extensional rheology, CaBER, image analysis


Previous studies on the elongation breakupof complex fluids point out that, the shape of thefluid filament is not implicitly cylindrical, butseems to be material specific [13, 15]. Only weak-ly visco-elastic fluids like dilute and semi-dilutepolymer solutions undergo uniform cylindricalfilament thinning [14, 23, 24]. Even in the case ofthe Newtonian liquids, although filaments thinover their full length, thinning is inhomoge-neous [13, 25]. The filament shape is slightly con-cave, adjacent to the plates the diameter is thick-er than at the axial break point location. Forpower law and plastic fluids this effect is evenmore pronounced. As an illustration a compari-son of filament profiles commonly observed forvarious cosmetic products is shown in Figure 1.Thinning often takes place only within a restrict-ed volume of the filament, and is hardly visiblebeyond this region. For such filament profiles,the filament diameter and the filament life timestrongly depend on the height where they aremeasured. Neither e· (t) nor e(t) is constantthrough the filament length. In such cases, it isnot enough to measure the decay of the mid-point diameter in order to analyze the thinningprocess.

First experiments on yield stress fluids haveshown, that such fluids can form stable bridgeswith lengths bigger than 2p times the radius ofthe cylinder [26, 27]. Goldin et al. pointed at theexistence of a critical sample radius, Rc at the neckof filament [28]. As long as R > Rc the capillarypressure does not exceed the value of yield stress,i.e. it is not sufficient to generate flow. The fila-ment stays in static equilibrium and does notevolve in time. However, the processes of thin-ning and breakup occurring for that kind of liq-uids are not yet well understood.

We have chosen a highly concentrated w/oemulsion and a semi-dilute acrylic thickenersolution as model systems for yield stress fluids.The challenge of this work was to prove the appli-cation range of the CaBER technique, to investi-gate the influence of the experimental parame-ters on filament thinning of complex fluids withyield stress and to provide suggestions for ameaningful characterization of these fluids withCaBER. Similar studies were performed previous-ly by Rothstein et al. for surfactant and polyelec-trolyte solutions as well as polymer blendsrevealing a strong effect of step-strain parame-ters on CaBER measurements [29]. They observed

strong dependence of the imposed extensionrates e· = ln (hf/h0)/ts on capillary thinning and fil-ament breakup. In contrast, our experimentsshow that filament thinning of yield stress fluidsis almost independent of imposed stretchingtime ts or of stretching velocity profile, but itstrongly depends on the initial sample volume,even if the same stretching ratio (Hencky strain)is applied and this is related to the changes in fil-ament curvature. In Section 2 the high resolutionoptical set-up used to characterize the time evo-lution of the full filament shape is described. Sec-tion 3 concerns the influence of CaBER set-upparameters on filament thinning. Finally, in Sec-tion 4 concluding remarks and further perspec-tives are summarized.

2 EXPERIMENTAL

2.1 SAMPLES

Two distinct products have been used as modelsystems, a water in oil (w/o) emulsion and a highviscosity thickener solution. Droplet size distrib-ution of the emulsion was determined by staticlight scattering in the Frauenhofer diffractionlimit. Particle size distribution is narrow, themean particle diameter is d(0.5) = 0.911 mm, thevalues at 10 and 90 % are d(0.1) = 0.592 mm andd(0.9) = 1.378 mm, respectively. The phase volume,water:oil is 75:25. This means the emulsion iscomposed of a quasi-network of densely packedsmall water droplets in continues oil phase. Thelatter essentially consists of a paraffin oil withviscosity of hs = 22 mPa·s (at room temperatureT = 20˚C). The surface tension of the emulsionwas measured to be s = 52 mN/m.


Figure 1:Comparison of filamentprofiles for different cos-metic products: a) w/oemulsion, b) w/o emulsion,c) w/o creme, d) o/w creme,e) shower gel and f) sham-poo.


The thickener solution is based on Stero-coll®D (BASF SE, Ludwigshafen, Germany), whichis an alkali swellable polymeric thickener forpaper coating. Sterocoll®D is composed of acopolymer of ethylacrylate and (meth)-acrylicacid with molar ratio of acrylate to acid of about1:1. It is made by emulsion polymerization and issupplied as milky, aqueous dispersion with asolids content of 25 % and a pH of 2.2 - 2.6. Theaqueous solution of concentration 2 wt. % wasprepared by dilution with deionized water. ThepH value was adjusted to pH = 8.4 by slowlyadding 1 N NaOH. The solution was stirred atroom temperature for 24 h. Due to the covalentcrosslinks present in this polymer the dilute neu-tralized solution resembles the features of a dis-persion of highly swollen microgel particles [15].

Both samples exhibit plastic flow behaviorwith power law dependence in steady shear, t =ty +Kg· n with an apparent yield point ty ≈ 10 Pa.However, the low shear viscosity exhibited by theemulsion is three orders of magnitude higherthan the shear viscosity of thickener solution (seeFigure 2).

2.2 CABER SET-UP

The commercial CaBER 1 device (Thermo HaakeGmbH, Germany), where only the midpoint diame-ter is measured with a laser micrometer, has beencombined with an optical set-up allowing for assess-

ment of full filament shape. The CaBER device withthe optical train is shown in Figure 3. The high speedcamera, Fastcam-X 1024 PCI (Photron USA, Inc.) isplaced in front. It allows to take up to 1000 picturesper second with 1024 x 1024 pixel resolution. Thecamera is equipped with telecentric objectives.There are two objectives available, one with 1.06magnification and one with 5.00 magnification.This corresponds to pixel size of 16.13 mm and 3.55mm, respectively. Further improvement in resolutionis possible using objectives with higher magnifica-tion [30] at the expense of the length of the filamentthat is captured. To improve contrast, the fluid fila-ment is illuminated with a telecentric monochro-matic blue light. The illumination source is mount-ed on the back side of the CaBER device. Illuminationtakes place via a long hole through the clamp main-taining the plates between which the fluid is placed.To ensure parallel alignment required for telecentricconfiguration, the camera is mounted on a holder,which can be moved in three dimensions indepen-dently changing camera position in x, y and z direc-tions (see Figure 3).

It is worthwhile to note the importance ofthe telecentric setup for image processing [31]. Inconventional imaging systems with entocentricobjectives and diffuse illumination imagedefects appear, which can not be neglected for adetailed quantitative analysis. The common onesare monochromatic aberrations, curvature of thefield of image or image distortion and chromat-


Figure 2 (left above):Steady shear viscosity versusshear stress. Open symbolsrepresent the w/o emulsionand filled symbols correspondto the thickener solution.

Figure 3 (right above):CaBER set-up for imageanalysis.

Figure 4 (left below):Time evolution of filamentdiameter obtained from lasermeasurements (open symbols)and image analysis (filled sym-bols). Linear stretching profilewith stretching time, ts = 40 mswas used. The initial plate dis-tance was h0 = 3 mm, and endplate distance hf = 16.64 mmfor the emulsion (a) and hf =10.31 mm for the poly(ethyleneoxide) solution (b).

Figure 5 (right below):Comparison of measurementposition provided by laser andimage analysis. The line repre-sents spatial resolution gainedfrom image analysis (16.13 mmfor 1.06 magnification objec-tive). Height of filament can befreely adjusted to the filamentbreak point. The shaded areacorresponds to the 1 mm thicklaser beam always probing theaverage filament diameteraround the midpoint.


ic aberration. In telecentric systems only the lightthat is parallel to the optical axis passes throughand is analyzed. Size and shape of an imageformed by such a lens do not depend on theobjects distance from the lens and position in theview field. In this way most of the image defectsare eliminated. As long as the object stays with-in the telecentric field always the same size, thereal dimension of the object, is pictured. Our newCaBER set-up provides the possibility of simulta-neous measurement with laser micrometer andvideo imaging.

2.3 FILAMENT SHAPE ANALYSIS

Comparison of the results obtained for w/oemulsion from laser micrometer and imageanalysis is shown in Figure 4a. It can be seen that,though the reproducibility in both cases is com-parable, the filament diameters show differentthinning behavior. Filament diameter measuredby laser is thicker and the filament life time islonger compared with the results obtained fromimage analysis. This is different from the case ofcylindrical filaments where both methods are inperfect agreement (see Figure 4b). A 2 wt. % solu-tion of poly(ethylene oxide) (Mw = 2·106 g/mol) inwater was used as a weakly elastic reference flu-id forming uniform cylindrical filaments. In caseof non-cylindrical filaments the thinning place isnot fixed to the filament midpoint, so that theneck is not necessarily at the height of the laserbeam. Furthermore, the laser beam averagesover a height of 1 mm along the filament axis,compared to 16.13 mm camera resolution (for the1.06 magnification objective), as schematicallyshown in Figure 5. For that reason, the laser mea-surements lead to systematic deviations and areinappropriate to investigate non-cylindrical pro-files formed e.g. by emulsions with yield stress-es or highly visco-elastic polymer solutions [15].

Furthermore, it should be pointed out, that theanalysis of the filament diameter at the neck is not

sufficient to extract rheological parameters evenwith reasonable constitutive equations. Figure 6compares experimental results for the transientdiameter at the neck point measured for the w/oemulsion with theoretical predictions for power-law and Bingham fluids based on a simplified analy-sis assuming uniform deformation of cylindrical fil-aments [13]. The dashed line shows the predictionfor the power law model:

(1)

where f(n) = 0.0709 + 0.2388(1 - n) - 0.5479(1 - n)2

+ 0.2848(1 - n)3, D0 = 0.9 mm, filament life time, tc= 0.37 s and the power law parameters extractedfrom the shear experiment, K = 3.23 and n = 0.62.The solid line represents the Bingham model:

(2)

The calculation was done using the followingparameters extracted from shear experiments ty= 10 Pa and for m the limiting plateau viscosityreached at high shear rates was taken, m = 1 Pa·s.Obviously, the models investigated here can notdescribe the time evolution of the neck diameterfor this fluid. This failure is supposed to be main-ly due to the significant curvature of the fila-ments corresponding to a non-uniform strainand strain rate throughout the filament. Thedetermination of an elongational viscositywould require a full three dimensional simula-tion of the whole experiment using appropriateconstitutive equations and an additional mea-surement of the force acting on the end plates.But this is beyond the scope of this paper. There-fore, we discuss the determination of appropri-ate parameters describing extensional flow


Figure 6 (left):The transient diameter atthe neck measured for thew/o emulsion using linearvelocity stretching profilev = 0.34 mm/s and plateseparation distancesh0 = 3 mm and hf = 16.64mm. The dashed and solidlines show the predictionsfor the power law and theBingham models, respec-tively.

Figure 7:Typical light intensity pro-file perpendicular to the fil-ament axis.


nowill1

Notiz

the solvent viscosity µ is outside the square root in the exponential term

behavior of fluids with yield stress in CaBERexperiment. For quantitative comparison of non-cylindrical filaments, description of full filamentshape is needed.

For this issue an image analysis software hasbeen developed. The filament edges are detect-ed based on a scan of light intensity perpendicu-lar to the filament axis (see Figure 7). Pixels ofhigh and low light intensities represent thebright background field and dark filament pro-file, respectively. The edges of the filament areidentified, when the light intensity drop belowgiven threshold limit, which is chosen withrespect to background intensity and noise level.Due to telecentric optics and illumination fila-ment edges are well defined and can be deter-mined with an accuracy of ± 1 pixel. The filamentdiameter is calculated as a difference betweenthe two edges.

A typical result for the time evolution of thefilament shape is shown in Figure 8. It is notice-able that in the lower and in the upper parts ofthe filament, the diameter hardly changes. Thethinning process takes place in a very restrictedvolume. Therefore, it seems to be reasonable tocharacterize the thinning process by the regionwithin thinning takes place and the time evolu-tion of the filament curvature at the neck, in addi-tion to the transient neck diameter. This regionof deformation is characterized here by the dis-tance ldef between the upper and lower positionalong the filament axis where no change in diam-eter is detected. In cylindrical coordinates, thecurvature is given by:

(3)

where h is the filament height and R1 is the fila-ment radius. There is no analytical function to

describe the diameter as a function of verticalposition, thus it is more convenient to calculatethe derivatives numerically, directly from theexperimental data. In order to get reliable resultsit is recommended to smooth the data before dif-ferentiating. Here, we have used a conventionalcubic smoothing spline method [32].

3 RESULTS AND DISCUSSION

In the following the influence of the initial set-ting parameters of CaBER experiments is dis-cussed. We have varied:■ stretching time ts■ stretching velocity profile■ final plate separation hf at constant h0 (and

vice versa). This corresponds to a variation ofHencky strain

■ initial and final plate separation h0 and hf,respectively in such a way that e is kept con-stant, but the sample volume varies propor-tional to h0The capillary number, Ca = hse

· d/2s in the initialstep strain of the CaBER experiment is in order of10-5 for the emulsion investigated here. There-fore, we do not expect droplet deformation orbreakup to be relevant in this part of the CaBERexperiment.

3.1 INFLUENCE OF STRETCHING TIME

The influence of stretching time on the elonga-tion properties of the yield stress fluids was test-ed for stretching times between 20 ms (lowerdevice limit) and 60 ms, in steps of Dt = 5 ms. Theexperiments were performed using linearstretching profile. The plate distances were setto: initial gap h0 = 3.00 mm, final gap hf = 16.64mm. For each stretching time we have performedat least five measurements. The resulting repre-sentative curves for the time evolution of the fil-ament diameter at the neck corresponding toeach ts are shown in Figure 9. Though the dis-


Figure 8 (left):Typical results for time evo-lution of filament profile.

Figure 9:Influence of stretching timeon the time evolution of thediameter at the neck. Theexperiments were per-formed for w/o emulsionusing linear velocity stretch-ing profile and plate separa-tion distances h0 = 3 mmand hf = 16.64 mm.


nowill1

Notiz

the exponent 3/2 refers to the full term in square brackets in the denominator

crepancies between these results are remark-able, there is no systematic variation of filamentlife time, tc and region of deformation withstretching time. The mean values and corre-sponding standard deviation calculated from theresults of the repeated measurements are tc =0.321 s ± 12 % and ldef = 2.118 mm ± 6 %. To selectthe most appropriate stretching time for theexperiments one has to compromise. If the plateseparation is too rapid the fluid filament is unsta-ble and flatters. On the other hand, step strainexperiments require ideally infinitesimal shortdeformation time. Thus, most appropriate valuesare between 35 ms and to 50 ms.

3.2 INFLUENCE OF STRETCHING PROFILE

There are three stretching profiles available inthe commercial firmware for the CaBER 1 provid-ed by Thermo Haake. Linear, where the plates areseparated with constant velocity. Exponential,here the velocity of the plate increase exponen-tially. Lastly cushioned profile, here at the begin-ning the plates are separated with constantvelocity corresponding to the fastest stretchingtime, but in order to reduce problems caused bythe rapid deceleration, at the end plate motionslows down exponentially.

Measurements for each stretching profilewere performed with the stretching time 40 ms.The plates distances were again set h0 = 3.00 mmand hf = 16.64 mm. For the cushioned profile theneck is slightly higher compared to the two oth-er profiles. Nevertheless, the time evolution ofthe filament diameter at the neck is nearly iden-tical in all cases. Mean values and standard devi-ations of filament life time and deformation vol-ume as obtained from at least 5 repeatedmeasurements under the same experimentalconditions are shown in Table 1. Obviously, thecushioned profile yields the highest accuracy forthese parameters.

3.3 INFLUENCE OF PLATE SEPARATION

In this section influence of the geometricalstretching parameters on filament thinning ofthe yield stress fluids is discussed. A first set ofexperiments was performed with different ini-tial heights h01 = 1.50 mm, h02 = 2.00 mm, h03 =2.50 mm, h04 = 3.00 mm. The plates were alwaysmoved to the same end distance, hf = 11.17 mm.Accordingly, increase of h0 leads to a decrease ofthe Hencky strain during filament formation.Stretching time was constant for all measure-ments and equal to 40 ms. For good statistics, themeasurements were repeated several times. Therepresentative results for the time evolution offilament diameter at the neck is shown in Figure10a. As expected with increasing initial gap, i.e.increasing sample volume the initial filamentdiameter is larger and filament life time is longer.Moreover, it seems that the filament shape is notinfluenced by the Hencky strain. The curvaturemeasured at the neck shows similar time evolu-tion starting always with the same initial value(Figure 10b).

Similar effects are observed while varyingthe end distance, hf and keeping the initial gapconstant. Here, the constant sample volume isstretched more when hf is increased. Thus, the fil-ament diameter is smaller and filament life timeis shorter for larger hf.

Finally, the initial and the end distanceswere varied in order to keep the Hencky strainconstant. The measurements were performedfor the initial gap in the range between h0 = 1.50mm and h0 = 3.00 mm in steps of Dh0 = 0.25 mm(in the case of the thickener solution the small-est initial gap was h0 = 1.75 mm). The end distancewas adjusted in the way that the Hencky strain


linear exponential cushioned

tc 0.36 s ± 6 % 0.34 s ± 3 % 0.35 s ± 3 %ldef 1.97 mm ± 5% 2.23 mm ± 7% 2.16 mm ± 4%

Figure 10:Influence of imposedHencky strain, e on timeevolution of diameter (a)and curvature (b) at theneck. Experiments were per-formed for w/o emulsionusing linear velocity stretch-ing profile and stretchingtime ts = 40 ms, e waschanged by changing h0 ,while the end distancehf = 11.17 mm was kept con-stant. The inset in lowerright corner of figure (b)shows results rescaled withrespect to the filament lifetime, tc .

Table 1:Experimental results for fila-ment life time and deforma-tion volume measured withdifferent velocity stretchingprofiles.


was always equal to e = 1.71 for the w/o emulsionand e = 1.21 for the thickener solution. The mea-surements were performed using linear stretch-ing profile and stretching time ts = 40 ms. Asbefore the measurements were repeated sever-al times. Representative curves for the time evo-lution of diameter and curvature at the neck areshown in Figure 11. It can be seen that, althoughthe plate distance is varied, the initial value of thediameter at the neck D0 does not change whenHencky strain is constant. But the filament lifetime is systematically shorter for larger separa-tions. At the same time the curvature at the thin-ning point decreases with increasing plate sepa-ration.

The filament thinning process is determinedby the balance between visco-elastic and capil-lary forces. The visco-elastic forces are materialspecific, i.e. they are equal in each experiment.The capillary forces are determined by theLaplace pressure, which for curved filamentshape is given by:

(4)

here R1 is the filament radius and R2 = 1/k is theradius of curvature. For concave shape of the fil-ament, R2 is always negative. If plate distance ischanged in a way that keeps Hencky strain con-stant the initial filament diameter does notchange, but the initial curvature increases with

decreasing h0 (Figure 11). This corresponds to areduced Laplace pressure and hence filament lifetime is longer. The contribution of the filamentcurvature, 1/R2 to the Laplace pressure is shownin Figure 12.

On the other hand, when Hancky strain ischanged by varying either h0 or hf the initial diam-eter increases with decreasing e while the initialcurvature is essentially constant (Figure 10). HereLaplace pressure changes are due to the first termin Eq. 4 and Dp increases with increasing e andaccordingly, filament life time decreases.

4 CONCLUSION

Application range of the capillary breakup exten-sional rheometer for investigations of exten-sional flow properties of high viscosity yieldstress fluids was tested. A w/o emulsion and apolymeric thickener solution were used as mod-el systems. Both samples are characterized bysimilar apparent yield stress (ty ≈ 10 Pa) but theyhave distinctly different low shear viscosities.Both yield stress fluids exhibit inhomogeneousfilament thinning with a strongly concave fila-ment shape. The shape of the D(t) - curves is alsovery similar. However, the thickener solutionwith the lower steady shear viscosity has accord-ingly shorter filament life time. Analysis of fullfilament shapes requires an advanced set-up forimage acquisition and processing. This includesa high speed camera, telecentric objective andillumination, as well as an image analysis soft-


Figure 11:Influence of plate separa-tion distances, h0 and hf atconstant Hancky straine = 1.71 for the w/o emulsion(a) and (b) and e = 1.21 forthe thickener solution (c)and (d). Figures (a) and (c)show time evolution of thediameter at the neck andfigures (b) and (d) showaccordingly time evolutionof the curvature.


ware. The time evolution of the full filament con-tour is detected with resolution of ± 1 pixel.

The influence of the stretching parametersof the CaBER device on capillary thinning of flu-ids with an apparent yield stress was discussed.Neither stretching time nor stretching velocityprofile has significant influence on the capillarythinning of those fluids. Suggested initial set-upparameters are cushioned stretching profile,which provides the best reproducible results andoptimal stretching time is in the range between35 ms and 50 ms. Variation of Hencky strain,results in a decrease of initial diameter and fila-ment life time with increasing e, but it has noinfluence on the initial curvature. Moreover,diameter and curvature at the neck, measuredfor different e show similar time evolution. Last-ly, if plate separation is varied in a way that keepsHencky strain unchanged, the initial value ofdiameter remains constant but the curvatureincreases with decreasing initial gap separationand results in a deceleration of the thinningprocess according to the corresponding drop ofthe Laplace pressure inside the filament.

Although further efforts are required for adetermination of physical quantities like elonga-tional viscosity, the CaBER technique can be suc-cessfully applied to investigate elongational flowproperties of yield stress fluids. The time- andspace- evolution of the fluid filament can bequantified. The measurement technique giveswell reproducible results, variation in the char-acteristic parameters extracted from the fila-ment profiles do not exceed 10 %. Thus, futurework will address influence of fluid properties,e.g. volume fraction of disperse phase anddroplet size for the emulsions or polymer con-centration and pH value for the thickener solu-tions, on capillary thinning. The determination ofan elongational yield stress and the identifica-tion of rheological parameters characterizing thetransient extensional properties of such fluidswhich can be correlated to processing and appli-cation properties will be of particular interest.

ACKNOWLEDGMENT

We would like to thank Prof. Dr. Stephan Neser(Hochschule Darmstadt, Germany) for interest-ing discussions and advice regarding the opticalset-up and image analysis. We greatly acknowl-edge donation of Sterocoll®D by BASF SE and we

thank Anne Kowalczyk for her help in prepara-tion of Sterocoll®D solution and performingCaBER experiments on it. Further we thank Kom-petenznetz Verfahrenstechnik Pro3 e.V. andBeiersdorf AG for the financial support.

REFERENCES

[1] Plateau J: Experimental and theoretical research-es on the figures of equilibrium of a liquid masswithdrawn from the action of gravity, Annu. Rep.Smithsonian Inst. (1863) 207–285.

[2] Lord Rayleigh: On the stability of jets, Proc. Lon-don Math. Soc. 10 (1879) 4–13.

[3] Sridhar T, Tirtaatmadja V, et al.: Measurement ofextensional viscosity of polymer solutions, J.Non-Newtonian Fluid Mech. 40 (1991) 271–280.

[4] Tirtaatmadja V, Sridhar T: A filament stretchingdevice for measurement of extensional viscosi-ty, J. Rheol. 37 (1993) 1081–1102.

[5] Entov VM, Hinch EJ: Effect of a spectrum of relax-ation times on the capillary thinning of a filamentof elastic liquid, J. Non-Newtonian Fluid Mech. 72(1997) 1081–1102.

[6] Rodd LE, Scott TP, et at.: Capillary break-uprheometry of low-viscosity elastic fluids, Appl.Rheol. 15 (2004) 12–27.

[7] Basilevskii AV, Entov VM, et al.: Failure of PolymerSolution Filaments, Polym. Sci. A 39 (1997)316–324.

[8] Basilevskii AV, Entov VM, et al.: Failure of anOldroyd Liquid Bridge as a Method for Testing theRheological Properties of Polymer Solutions,Polym. Sci. A 43 (2001) 1161–1172.

[9] Clasen C, Plog JP, et al.: How dilute are dilute solu-tions in extensional flows?, J. Rheol. 50 (2006)849–881.

[10] Solomon MJ, Muller SJ: The transient extension-al behavior of polystyrene-based Boger fluids ofvarying solvent quality and molecular weight, J.Rheol. 40 (1996) 837–856.

[11] Rothstein JP: Strong flows of viscoelastic worm-like micelle solutions, Rheol. Rev. (2008) 1–42.

[12] Bhardwaj A, Miller E, et al.: Filament stretchingand capillary breakup extensional rheometrymeasurements of viscoelastic wormlike micellesolutions, J. Rheol. 51 (2007) 693–719.

[13] McKinley GH: Visco-elasto-capillary thinning andbreak-up of complex fluids, Rheol. Rev. (2005) 1–48.

[14] Kheirandish S, Guybaidullin I, et at.: Shear andelongational flow behavior of acrylic thickener


Figure 12:Influence of filament curva-ture on Laplace pressure.The open symbols show theLaplace pressure calculatedfrom Eq. 4 using the datafrom Figure 11 (a) and (b).The initial plate distancewas h0 = 1.5 mm, and endplate distance, hf = 16.64mm. Solid line was calculat-ed neglecting the filamentcurvature, i.e. R2 = 0.


solutions, Part I: Effect of intermolecular aggre-gation, Rheol. Acta 47 (2008) 999–1013.

[15] Kheirandish S, Guybaidullin I, et al.: Shear andelongational flow behavior of acrylic thickenersolutions, Part II: effect of gel content, Rheol. Acta48 (2009) 397–407.

[16] Renardy M, Renardy Y: Similarity solution forbreakup of jets of power law fluids, J. Non-New-tonian Fluid Mech. 122 (2004) 303–312.

[17] Renardy M: Similarity solution for jet breakup forvarious models of viscoelastic fluids, J. Non-New-tonian Fluid Mech. 104 (2002) 65–74.

[18] Renardy M: Self-similar breakup of Giesekus jet,J. Non-Newtonian Fluid Mech. 97 (2001) 261–293.

[19] Renardy M: Self-similar jet breakup for general-ized PPT model, J. Non-Newtonian Fluid Mech.103 (2002) 261–269.

[20] Yildirim OE, Basaran OA: Deformation andbreakup of stretching bridges of Newtonian andshear-thinning liquids: comparison of one- andtwo-dimensional models, Chem. Eng. Sci. 56(2001) 211–233.

[21] Ambravaneswaran B, Basaran OA: Effect of insol-uble surfactants on the nonlinear deformationand breakup of stretching liquid bridges,Phys.Fluids 11 (1999) 997–1015.

[22] Webster MF, Matallah H, et al.: Numerical mod-eling of step-strain for stretched filaments, J.Non-Newtonian Fluid Mech. 151 (2008) 38–58.

[23] Oliveira MSN, Yeh R, et al.: Extensional Rheologyand Formation of Beads-on-a-String Structures inPolymer Solutions, J. Non-Newtonian Fluid Mech.137 (2006) 137–148.

[24] Stelter M, Brenn G, et al.: Validation and applica-tion of a novel elongational device for polymersolutions J. Rheol. 44 (2000) 595–616.

[25] McKinley GH, Tripathi A: How to extract the New-tonian viscosity from capillary breakup measure-ments in a filament rheometer, J. Rheol. 44 (2000)653–670.

[26] Lowry BJ, Steen PH: Capillary Surfaces: Stabilityfrom Families of Equilibria with Application tothe Liquid Bridge, Proc. R. Soc. London, A 449(1995) 411–439.

[27] Mahajan MP, Tsige M, et al.: Stability of liquidcrystalline bridges, Phys. Fluids 11 (1999) 491–493.

[28] Goldin M, Pfeffer R, et al.: Break-up of a capillaryjet of a non-Newtonian fluid having a yield stress,Chem. Eng. J. 4 (1972) 8–20.

[29] Rothstein JP, Miller E, et al.: The Effect of Step-Stretch Parameters on Capillary Breakup Exten-sional Rheology (CaBER) Measurements, Proc. Int.Cong. Rheol. (2008).

[30] Sattler R, Wagner C, et al.: Blistering Pattern andFormation of Nanofibers in Capillary Thinning ofPolymer Solutions, Phys. Rev. Lett. 100 (2008)164502.

[31] Smith WJ: Modern optical engineering, 3rd Edi-tion, McGraw-Hill, New York (2000).

[32] Bowman AW, Azzalini A: Applied SmoothingTechniques for Data Analysis: the Kernel Ap-proach with S-Plus Illustrations, Oxford Universi-ty Press (1997).



Documents

How to Characterize Yield Stress Fluids with Capillary Breakup · First experiments on yield stress fluids have shown, that such fluids can form stable bridges with lengths bigger