Abstract Elongational flow behavior of w/o emulsionshas been investigated using a capillary breakup elonga-tional rheometer (CaBER) equipped with an advancedimage processing system allowing for precise assess-ment of the full filament shape. The transient neck di-ameter D(t), time evolution of the neck curvature κ(t),the region of deformation ldef and the filament lifetimetc are extracted in order to characterize non-uniformfilament thinning. Effects of disperse volume fractionφ, droplet size dsv , and continuous phase viscosity ηc
on the flow properties have been investigated. At acritical volume fraction φc, strong shear thinning, andan apparent shear yield stress τy,s occur and shear flowcurves are well described by a Herschel–Bulkley model.In CaBER filaments exhibit sharp necking and tc aswell as κmax = κ (t = tc) increase, whereas ldef decreasesdrastically with increasing φ. For φ < φc, D(t) datacan be described by a power-law model based on acylindrical filament approximation using the exponentn and consistency index k from shear experiments. Forφ ≥ φc, D(t) data are fitted using a one-dimensionalHerschel–Bulkley approach, but k and τy,s progres-sively deviate from shear results as φ increases. Weattribute this to the failure of the cylindrical filamentassumption. Filament lifetime is proportional to ηc atall φ. Above φc,κmax as well as tc/ηc scale linearly with
K. Niedzwiedz · H. Buggisch · N. Willenbacher (B)Institute of Mechanical Process Engineering and Mechanics,Karlsruhe Institute of Technology,76131 Karlsruhe, Germanye-mail: [email protected]
τy,s. The Laplace pressure at the critical stretch ratioεc which is needed to induce capillary thinning canbe identified as the elongational yield stress τy,e, ifthe experimental parameters are chosen such that theaxial curvature of the filament profile can be neglected.This is a unique and robust method to determine thisquantity for soft matter with τy < 1,000 Pa. For theemulsion series investigated here a ratio τy,e/τy,s =2.8 ± 0.4 is found independent of φ. This result is cap-tured by a generalized Herschel–Bulkley model includ-ing the third invariant of the strain-rate tensor proposedhere for the first time, which implies that τy,e and τy,s
Emulsions are mixture of two immiscible liquids, wherethe droplets of one phase are suspended in a continu-ous phase of the second one. These apparently simplematerials have found applications in many areas of ourevery day life, e.g. food industry, cosmetics, and phar-macy, cleaning industry and environmental technology.Quality aspects which are crucial for their applicabilitylike: ease of use, physical stability or skin, taste, andeven aesthetic perception are directly related to therheological behavior (Wilson et al. 1998; Welin-Bergeret al. 2001; Brummer and Godersky 1999; Foerster et al.1999; Penna et al. 2001). Flow at low shear relatesto physical stability (Zografi 1982; Dickinson et al.1993; Miller et al. 1999). Medium and high shear rates
1104 Rheol Acta (2010) 49:1103–1116
correspond to product behavior during spreading orpouring processes and the extensional flow is identifiedwith such processes as coating, extrusion, and spraying.
Emulsions are thermodynamically unstable, theirproduction requires input of external energy. This isusually provided by mixing. The mixing conditions de-termine the interfacial area between two phases andaccordingly the droplet size distribution. A change inthe droplet size or emulsion concentration can lead tostrong changes in rheological properties (Prud’hommeand Khan 1996; Lee et al. 1997; Rieger 1991).
At low concentrations emulsions typically exhibitNewtonian flow behavior and as long as Ca << 1 thedroplets are undeformed. Deviations from the viscosityof hard spheres suspensions are due to changes ofthe external flow field around a drop due to internalcirculation and depend on the viscosity ratio of disperseto continuous phase ηd/ηc. This is captured by the so-called Taylor equation, but up to volume fractions ofabout φ = 0.25 the emulsion viscosity is close to thatof hard spheres suspensions (Macosko 1994). At higherconcentrations differences between emulsion and sus-pension rheology become more and more pronounced,shear thinning occurs and the dependence of the zero-shear viscosity on volume fraction can be describedphenomenological by a modified Krieger-Doughertyequation including the viscosity ratio ηd/ηc (Pal 2001).Suspensions cannot flow anymore at concentrationsabove φc, when particles are densely packed. For theconcentrated emulsions with φ > φc this is different dueto droplet deformability. Steady shear flow behavior ischaracterized by strong shear-thinning and an apparentyield stress. In small amplitude oscillatory shear suchemulsions behave predominantly elastic (G′ >> G′′).Princen and Kiss proposed a model describing shearflow of such highly concentrated, monodisperse emul-sions (Princen 1983; Princen and Kiss 1986, 1989). Sincemacroscopic deformation and flow are closely relatedto the deformation of individual droplets, they presumethat the yield stress τ y,s and static shear modulus G0
scale with the internal Laplace pressure 2σ i/dsv of thedroplets, where σ i is the interfacial tension between theoil and water phase and dsv is the droplet diameter. Fur-thermore, τ y,s and G0 also depend on geometrical pack-ing, i.e., volume fraction of the disperse phase. Princenand Kiss report a value φc = 0.71. Investigations byPonton et al. (2001) and Jager-Lézer et al. (1998) oncosmetic w/o emulsions confirmed the predictions ofPrincen and Kiss. They found an agreement for theshear modulus measured for emulsions with differentdroplet sizes and volume fractions, but φc ≈ 0.67 wasfound. Mason et al. (1996) also confirmed that the yieldstress of w/o emulsions scales with internal Laplace
pressure. However, they found a stronger dependenceof τ y,s on φ and a lower value φc = 0.635.
Mason also introduced an effective volume fractionφef f including a thin surfactant layer on a droplet sur-face (Mason et al. 1996; Calderon et al. 1994). TheMason equation for the yield stress which will be usedbelow to describe the behavior of the emulsions inves-tigated here reads:
τy,s = 0.512σi
dsv(φeff − φc)
2 (1)
Beyond that, the rheology of emulsions is modifiedby droplet size distribution and repulsive (stericor electrostatic) colloidal interactions (Tadros 1994;Larson 1999), e.g., for polydisperse emulsions thesurface-volume mean diameter has to be used for thecalculation of the internal Laplace pressure enteringin Eq. 1. Moreover, additional effects can modify theflow behavior of emulsions, e.g., flocculation with theformation of a sample spanning network can give riseto a yield stress and pronounced elasticity even at a lowdisperse phase concentration when attractive dropletinteractions dominate. It is also possible to vary emul-sion rheology by modifying the continuous phase usingpolymeric thickeners or additional surfactants, which iscommonly used in rheology of commercial emulsions(Wilson et al. 1998; Welin-Berger et al. 2001; Miller andLöffler 2006; Ghannam and Esmail 2005).
In spite of a quite good understanding of deforma-tion and breakup of single droplets as well as emul-sification phenomena in shear and elongational flows(Schubert 2005), there is not much known about elon-gational flow behavior of concentrated emulsions. Toour knowledge the extensional rheology has not beeninvestigated systematically so far. In contrast to the caseof Newtonian fluids, for the emulsion systems thereis no universal relationship between flow resistanceunder shear and elongational deformation. Therefore,it is important to perform elongational measurementsindependently.
Here, we use the capillary breakup elongationalrheometer (CaBER) technique in order to charac-terize the elongational flow behavior of concentratedemulsions. In CaBER experiments the sample is filledbetween two coaxial parallel plates creating a fluid con-nection between them. The plates are rapidly movedapart from each other corresponding to a step straindeformation. The resulting liquid bridge deforms grad-ually under the action of visco-elastic and capillaryforces, contracts, and finally breaks. The advantageof this method is that it is fast, straightforward andrequires only a small amount of probing sample (V <
0.1 ml). It can be applied to fluids with viscosities
Rheol Acta (2010) 49:1103–1116 1105
between 50 and 104 m Pa s. Investigations on exten-sional viscosity using the CaBER technique startedwith dilute and semi-dilute polymer solutions. Suchsolutions form cylindrical filaments and their diame-ter decays exponentially in time (Kheirandish et al.2008; Oliveira et al. 2006; Stelter and Brenn 2000),i.e., they exhibit uniform deformation at a constantstrain rate and a characteristic elongational relaxationtime λE can be extracted directly from the transientfilament diameter D(t) (Rodd et al. 2005; Bazilevskiiet al. 1997, 2001). Experimental results show that, inthe dilute region relaxation time probed by CaBER isequal to the longest shear relaxation time, λS = λE.For semi-dilute solutions of linear, flexible polymers,but also for polyelectrolyte complexes and commercialacrylic thickeners formation of exponentially decaying,cylindrical filaments is observed, but elongational re-laxation time is much lower than shear relaxation time,λE << λS and the interpretation of λE is still underdiscussion (Kheirandish et al. 2009; Clasen et al. 2006;Willenbacher et al. 2008). In general, fluid filamentsare not cylindrical and their diameter does not decayexponentially, i.e., neither strain nor strain rate are con-stant or uniform throughout the filament. This has beenconfirmed experimentally (Kheirandish et al. 2009;McKinley 2005; Tiwari et al. 2009; Niedzwiedz et al.2009) as well as by theoretical calculations and numeri-cal simulations (Renardy 2002; Renardy and Renardy2004; Yildirim and Basaran 2001; Doshi et al. 2003;Webster et al. 2008). Even in the case of Newtonianliquids thinning is inhomogeneous, filament shape isslightly concave but its diameter decreases linearly intime and numerical solutions for specified boundaryconditions allow for determination of ηE directly fromD(t) (McKinley 2005; McKinley and Tripathi 2000).Capillary thinning experiments performed for power-law and plastic fluids provide filaments with highly non-uniform shapes (McKinley 2005; Tiwari et al. 2009;Niedzwiedz et al. 2009). Thinning often takes place onlywithin a restricted volume of the filament around theneck, and is hardly visible beyond this region. In suchcases, it is not sufficient to measure the decay of themidpoint diameter D(t) and the filament lifetime tc,but time evolution of full filament profiles needs to beanalyzed and it has been shown, that the time evolutionof filament curvature at the neck κ(t), its maximumvalue κmax = κ(t = tc), and the region of deformationldef are appropriate parameters for characterization ofnon-uniform filament thinning (Niedzwiedz et al. 2009).
In this paper, we discuss the elongational flow behav-ior of w/o emulsions. Droplet volume fraction is variedin a broad range from semi-concentrated well below thecritical volume fraction φc up to highly concentrated
close to unity. Variation of φ in this range correspondsto a variation of the rheological signature from purelyshear thinning to strongly elastic behavior and the oc-currence of an apparent yield stress in shear flow. Theeffect of volume fraction, droplet size, and continuousphase viscosity on time evolution of filament shape inCaBER experiments has been investigated. The subse-quent variations in extracted characteristic parametersD(t), filament lifetime tc, the maximum curvature atthe neck κmax and deformation length ldef are dis-cussed. Furthermore, we point at correlations existingbetween shear and elongational flow behavior. Finally,we present a method to determine the elongationalyield stress of concentrated w/o emulsions, which canalso be applied to other soft matter with an apparentyield stress.
Materials
Raw materials and preparation methods
There were two different procedures to produce thesamples. In a first set of experiments we have workedwith a commercial cosmetic emulsion. The sample ischaracterized by dsv = 0.9 μm, φ = 0.75 and continuousphase viscosity ηc = 22 m Pa s (at T = 20◦C). This wasused as a basis to create samples with other concentra-tions. Lower concentrations were obtained by dilutingwith paraffin oil (of the same viscosity) and by addingdistillated water we could increase the disperse volumefraction (sample of series A1). These emulsion mixtureswere shaken for 24 h at 3,500 rpm. Thus, we haveproduced the samples with φ in a range from about 0.5up to 0.9. One should note that neither dilution nor
Fig. 1 Comparison of particle size distribution for the represen-tative samples from series A1 (a) and series B2 (b)
1106 Rheol Acta (2010) 49:1103–1116
concentration has a remarkable effect on the dropletsize distribution, as shown in Fig. 1. In addition, bydilution of the commercial emulsion with high viscosityparaffin oil (ηc = 230 m Pa s) we could make a secondgroup of samples—series A2—for which φ and ηc varysimultaneously. The disadvantage of this method is thatηc is related to the degree of dilution and it increasesnon-linearly with decreasing φ. Diluting sample from0.75 up to 0.38 viscosity changes from 22 to 152 m Pa s.A third group of samples, series A3, was created bydilution of the commercial sample down to φ = 0.45using mineral oils of different viscosities, resulting in aset of samples of constant concentration, but ηc varyingform 71 to 261 m Pa s.
Another set of samples (series B1) was emulsifiedby ourselves using a simplified recipe including onlydistilled water, oil, and emulsifier, without using anyother additives. The oil phase consisted of a paraffinoil with ηc = 230 m Pa s and the water droplet vol-ume fraction was chosen to be φ = 0.7. Polyglyceryl-3 Polyricinoleate (Imwitor�R 600, SASOL, Germany),also known as E476 (Wilson et al. 1998), was chosen asan emulsifier. Polyglyceryl-3 Polyricinoleate is a non-ionic, PEG-free emulsifier, made by esterification ofpolyglycerin, and plant-derived condensed castor oilfatty acids. Its hydrophilic lipophilic balance value isHLB = 4. Thus, it is a powerful w/o emulsifying agentand is especially suitable for low viscosity and highlyconcentrated emulsions.
The emulsification procedure was as follows. Theoil phase was mixed with emulsifier and heated toT = 70◦C. The volume of added emulsifier was varieddepending on the desired droplet size, and was kept be-tween 5% and 20%, referred to the oil phase. A coarsepremix emulsion was obtained by slowly adding waterunder blade stirring conditions (10 min at 500 rpm).The final droplet size was adjusted by further stirring ofthis coarse emulsion premix. Time and speed of stirringvaried in the range from 5 to 30 min and from 100 to13,500 rpm, respectively, resulting in average dropletsize between 2.4 and 10.3 μm.
Finally, we have created a set of home-made samples(series B2) with different continuous phase viscosity,using paraffin oils with ηc = 22 m Pa s, ηc = 230 m Pa s,and a 1:1 mixture of these oils (ηc = 75 m Pa s). For thisgroup, φ = 0.7 and dsv ≈ 5.5 μm.
One should note that due to the relatively largedroplet size most of these emulsions were not verystable. Changes in the droplet size distribution startedto occur already after 3 days. Therefore, all measure-ments were performed within 2 days after preparation.The basic parameters for all investigated samples aresummarized in Table 1.
Surface and interfacial tension
Surface and interfacial tensions were measured accord-ing to the Wilhelmy plate method using a DCAT 11EC
Table 1 Characteristicparameters for allinvestigated emulsionsamples
dsv /μm ηc/m Pa s φ/−Series A1 0.9 22 0.53, 0.56, 0.60, 0.64, 0.68,
tensiometer (Dataphysics, Germany; McClements1999). Measurements were performed at roomtemperature. The results obtained for the surfacetension of the samples of different droplet size andconcentrations are nearly identical (we have measuredall samples except these with φ > 0.75, here theapparent yield stress prevents usage of this technique).The average value measured for samples from seriesA and series B are nearly identical and is σ s =36 ± 2 mN/m. This also corresponds to the interfacialtension σ i between oil and aqueous phase. However,by adding emulsifier the interfacial tension is stronglyreduced. Already 5% of Imwitor®R 600 given to theoily phase is sufficient to lower σ i by one order ofmagnitude. The interfacial tension for samples fromseries B is σ i = 3.6 ± 0.4 mN/m and stays unchangedwith further increase of emulsifier concentration. Forthe commercial sample (series A) σ i is even lower,σ i = 2.6 ± 0.3 mN/m.
Droplet size distribution
The droplet size distribution was measured by staticlight scattering in the Fraunhofer diffraction limitusing a HELIOS device with wavelength of 632.8 nm(Helium–Neon Laser Optical System, Sympatec,Germany). The volume fraction of particles withdiameter di is obtained from the angular dependenceof scattered light intensity. From this we calculate thenumber fraction fi of particles with di (Leschonski1984), and the surface-volume mean diameter, dsv isgiven by:
dsv =∑
fid3i∑
fid2i
(2)
The sets of series A samples are characterized by smalldroplets (dsv = 0.9 μm) and a narrow distribution.On the other hand, the samples from series B havebigger size and their distribution is slightly broader (seeFig. 1).
Experimental
Shear rheology
Shear experiments were performed using a con-trolled stress rheometer (HAAKE Rheostress RS 150,Thermo HAAKE, Germany). All emulsions were mea-sured at T = 20◦C using cone-plate geometry (diameter
cone angle 50 mm/1◦). The gap distance between coneand lower rheometer plate was 0.053 mm, which isat least five times larger than the largest dsv . Therewas no indication of slip effects on the plate surfaceand the measurements on a given emulsion were wellreproducible when samples were newly loaded to therheometer. For the determination of the apparent yieldstress we have performed creep tests, where τ increasedlogarithmically from 0.1 to 50 Pa, for samples withlow internal phase concentration (φ < 0.75) and from0.1 to 500 Pa for samples with higher concentrations.The sampling time in each single measurement was900 s.
Extensional rheology
Elongational experiments were performed on aCaBER device (CaBER_1, Thermo HAAKE,Germany). The experiments were carried out atroom temperature. The default plate separationdistances were h0 = 3.00 mm and h f = 16.64 mm, i.e.,Hencky strain, ε = 1.71. The plates were separated inthe cushioned stretching profile within ts = 40 ms. Thesubsequent filament thinning process was capturedby a high-speed camera with a sampling speed of1,000 frames per second, the telecentric optics ensuredspatial resolution of 16.13 μm and image analysisprovided the transient filament diameter over the fullfilament length (Niedzwiedz et al. 2009). Differentsettings for h0, h f , and ts have been chosen in theexperiments used to determine the elongational yieldstress. Rheological results obtained from CaBERexperiments must be carefully considered with respectto the preconditioning of the samples including theappropriate choice of the stretch parameters in theinitial step strain. This is particularly true for complexfluids like wormlike micellar solutions, polymer blends(Miller et al. 2009) and also emulsions (Niedzwiedzet al. 2009). Miller et al. (2009) observe a change indroplet diameter of the investigated polymer blendswhen a critical capillary number is exceeded. For theemulsions investigated here the capillary number,Ca = ηcε̇dsv/2σi in the initial step strain of the CaBERexperiment varies between 10−4 and 10−2 dependingon the continuous phase viscosity and droplet size.Therefore, we do not expect droplet deformation orbreakup to be relevant during filament formation. Thishas been confirmed by droplet sizing experiments onselected samples before and after CaBER experiments.Within the accuracy of the HELIOS device we didnot detect any significant differences in the dropletsize.
1108 Rheol Acta (2010) 49:1103–1116
Results and discussion
Shear measurements
Disperse volume fraction
The measured flow curves for the samples of series A1are shown in Fig. 2. The emulsions exhibit an apparentyield stress τ y,s and shear thinning behavior. Obviously,τ y,s as well as the degree of shear thinning increaseswith increasing φ. At low concentrations only negligiblysmall yield stresses are measured, but at φ ≈ 0.66 anabrupt increase of τ y,s is observed. Quantitative analy-sis of the flow curves was done by fitting experimentalresults with the Herschel–Bulkley model:
τ = τy,s + kγ̇ n (3)
In the fit procedure we have allowed for a variation ofall three parameters, τ y,s, k, and n. The results obtainedfor τ y,s, k, and n are shown in Tables 2 and 3. Thesedata will be discussed in more detail along with theresults from the elongational experiments presentedbelow.
Droplet size and continuous phase viscosity
Emulsions of series B1 with varying dsv but constantφ = 0.7 exhibit similar rheological features as the sam-ples of series A with φ > φc. The apparent yield stressand the degree of shear thinning weakly decrease withincreasing dsv according to the respective change ininternal Laplace pressure. The absolute values withinthis series differ by less than one decade. The resulting
Fig. 2 Flow curves measured for the w/o emulsions from seriesA1. Solid lines represent fit results with Herschel–Bulkley model,Eq. 3
Table 2 Comparison of the fit parameters obtained for thepower-law model from shear and elongational experiments foremulsion series A1 with φ < φc
τ y,s obtained from fitting the data with the Herschel–Bulkley model (Eq. 3) agree reasonably well with thepredictions of Eq. 1.
We have also investigated two sets of emulsionswith varying continuous phase viscosity, namely seriesA3 (φ = 0.45, dsv = 0.9 μm) and B2 (φ = 0.70, dsv ≈5.5 μm). As expected, viscosity and degree of shearthinning are larger for the series with higher φ than thatfor the low φ series, but the shape of the η(τ ) curves isindependent of ηc, which is just a constant prefactor inthe η(τ ) curves.
Elongational measurements
First, qualitative analysis of elongational flow prop-erties can be done comparing the filament shapescaptured by the high-speed camera. Image sequencesshowing effects of disperse phase concentration onfilament thinning are presented in Fig. 3. Typicallyfor the yield stress and power-law fluids the thinningprocess starts slowly and rapid deformation occurs onlyshortly before breakup. Thus, the image sequences arecompared with respect to the filament lifetime, tc. Itis clearly visible that the filament profiles of the sam-ples with higher disperse phase concentration exhibitstronger necking and the thinning process takes placeonly in a narrow region around a well-defined neckingpoint for φ > φc. As the disperse volume fraction de-creases the thinning region becomes larger, the filamentcurvature decreases and for φ < φc essentially resem-bles the shape observed for the pure continuous phase.At that point, one should note that for the samples withφ ≥ 0.8 the existence of high apparent yield stressesprevents the self-thinning process under given experi-mental conditions. Instead, the stretched droplet adaptsits static equilibrium shape.
To get more accurate information on filament thin-ning we have analyzed full image sequences and ob-served the time evolution of filament diameter overtheir full height. The images were analyzed basedon light intensity scans, automatically recognizing thefilament edges and calculating the diameter line by line
Rheol Acta (2010) 49:1103–1116 1109
Table 3 Comparison of thefit parameters obtained forthe Herschel–Bulkley modelfrom shear and elongationalexperiments for emulsionseries A1 with φ > φc
φ n Shear experiment Elongational experimentτ y/Pa k/Pa sn τ y/Pa k/Pa sn
at any filament height as described in Niedzwiedz et al.(2009). The representative results for transient filamentdiameter evaluated at the neck point are shown inFig. 4. Obviously, the neck diameter immediately be-fore failure is always around 20 μm corresponding tothe optical resolution of the objective used in this seriesof experiments. For φ < φc the filament lifetime tc isshort and varies in a narrow range between 30 and70 ms, but for φ > φc increases strongly up to severalseconds. Furthermore, also the curvature of D(t) curveschanges at φc. Below this value, the neck diameterdecays smoothly, but for φ > φc a sharp downward bentis visible in the D(t) curves.
The low-concentrated samples (φ < φc) do not ex-hibit a yield stress or its value is very small and inCaBER experiments their filament curvature is negligi-bly small. Therefore, we use a simple analytical modelto describe the time evolution of the neck diameterMcKinley (2005):
D (t) = D0(n)σs
k(tc − t)n (4)
This equation has been derived for power-law fluidsbased on a one-dimensional analysis of the CaBERexperiment assuming uniform deformation of cylindri-
Fig. 3 Filament profiles formed during capillary thinning inCaBER experiments for samples of different disperse phaseconcentration
cal filaments and a negligible axial stress τ xx ≈ 0. Theinvolved parameters are similar as in the Herschel–Bulkley model, i.e., k is the consistency factor andn is the power-law coefficient, D0 is the initialfilament diameter, and (n) = 0.071 + 0.239 (1 − n) +0.548 (1 − n)2 is a numerical factor depending on thedegree of shear thinning.
In Fig. 5 we compare the experimentally determinedD(t) curves with model predictions based on Eq. 4and the corresponding model parameters are summa-rized in Table 2. We have calculated D(t) using theparameters k and n obtained from independent shearexperiments and to a first approximation these calcu-lations agree quite well with the experimental data,i.e., the same power-law parameters can be used todescribe shear as well as extensional flow. This is ofcourse not trivial and as can be seen from Fig. 5 theagreement between experiment and model calculationscan be further improved, especially for the samples withhigher droplet concentration, if k and n are treated asfree fit parameters.
Nevertheless, the power-law model starts to fail atφ ≈ 0.65, which is not surprising since τ y,s is no longernegligible. In order to describe the filament thinning ofthe higher concentrated emulsions we need to accountfor τ y,s. An according solution was recently suggested
Fig. 4 Transient diameter measured at the neck of the filamentfor the w/o emulsions from series A1
1110 Rheol Acta (2010) 49:1103–1116
Fig. 5 The transient diameter at the neck measured for lowconcentrated emulsions from series A1. The solid lines representthe prediction of a one-dimensional power-law model (Eq. 4)using parameters n and k independently obtained from shearexperiments. The dashed lines show a fit of Eq. 4 to the ex-perimental data for φ = 0.6 and 0.64 where k and n are variedindependently
by Tiwari et al. (2009) investigating capillary thinningof carbon nanotube suspensions. Based on the consti-tutive model of Herschel and Bulkley and again as-suming that the self-thinning of filaments in CaBERexperiments is quasi one-dimensional, i.e., filamentsare cylindrical and thin uniformly. Assumed that thereis no flow in the end regions of filament thread andthat τ xx ≈ 0 they obtained the following differentialequation for the transient filament diameter:
3(n+1)/2k(
− 2D
dDdt
)n
+ √3τy,s − 2σs
D= 0 (5)
This equation is derived on the basis of the follow-ing tensorial form of the Herschel–Bulkley model(Alexandrou et al. 2003; Basterfield et al. 2005; Coussotand Gaulard 2005; Castro et al. 2010):
T = 2[
τy,s√|II| + k(√|II|
)n−1]
D (6)
where T is the extra-stress tensor, D the strain-rate ofstrain tensor. In simple shear flow, the second invari-ant of D, II = 1
2
[(tr2D)2 − tr (2D)2] = −γ̇ 2 and Eq. 6
reduces to Eq. 3.In uniaxial elongation II = −3ε̇2 and for ε̇ > 0 the
first normal stress difference reduces to:
τxx − τyy = √3
(τy,s + k
(√3ε̇
)n)(7)
Where γ̇ and ε̇ are the shear and elongation rate,respectively.
This corresponds to a constant relationship betweenelongational and shear yield stress:
τy,e = √3τy,s (8)
This relationship tacitly assumes that the extra-stresstensor is independent of the third invariant of thestrain-rate tensor III = det 2D and according to thereferences mentioned above Eq. 6 seems to hold forplastic solids or pastes with very high yield stresses. Thevalidity of this assumption for the emulsions investi-gated here will be discussed in more detail below.
Figure 6 compares the experimental D(t) curves forthe higher concentrated emulsions φ > φc with modelpredictions based on Eq. 5. Obviously, this simplifiedmodel describes the experimental data very well. In thefitting procedure n was fixed to the value determinedfrom shear experiments and σ s = 36 mN/m was takenfrom surface tension measurements. The apparent yieldstress τ y,s and k were left as free parameters.
For the samples with φ ≈ φc we obtain similar valuesfor τ y,s as in shear experiments, but k values are clearlyhigher than in shear experiments. At higher concentra-tions φ ≥ 0.74, where the samples exhibit pronouncedyield stress in shear experiments and strong filamentcurvature in elongational experiments, the fit resultsshow severe discrepancies also for τ y,s. In this con-centration range the yield stress values extracted fromelongational experiments seem to be about a factor oftwo higher than in shear experiments (see Table 3).
We attribute these deviations between experimentand model prediction to the slender filament approxi-mation. Pronounced necking is observed above φc and
Fig. 6 The transient diameter at the neck measured for therepresentative samples from series A1. The solid line representsfit with the Herschel–Bulkley model, Eq. 5
Rheol Acta (2010) 49:1103–1116 1111
Fig. 7 Transient curvature measured at the neck of the filamentfor the w/o emulsions from series A1
the filaments exhibit a strongly non-uniform thinningrestricted to a small region around the neck (see Fig. 3).Data presented in Fig. 7 show that the curvature atthe neck increases strongly upon filament thinning.Thus, we suppose that a better description of CaBERexperiments for yield stress fluids requires numericalsimulation of the full, three-dimensional flow problemwith appropriate boundary conditions. This is outsidethe scope of this paper, but in the following we willsystematically discuss the effect of emulsion parametersφ, dsv , ηc, and finally τ y,s on filament thinning and inparticular on filament lifetime tc, the maximum cur-vature at the neck κmax, and deformation length ldef .Finally, we will propose a new approach for a directdetermination of the extensional yield stress based onCaBER experiments.
Disperse volume fraction
Figure 8 compares the results obtained for the charac-teristic parameters, tc, κmax, and ldef measured in elon-gational experiments upon variation of disperse volumefraction. Although there are only minor changes in theinitial filament diameter, the filament lifetime stronglychanges with disperse volume fraction. We can clearlydistinguish two different regions. For the samples withlow concentration, φ < 0.66, the filaments are ratherunstable, their lifetime is relatively short, on the orderof tenths of milliseconds, at volume fractions φ > φc
filaments become much more stable and a rapid in-crease of tc up to several seconds is observed. Con-cerning κmax, again a crossover is observed at φ = φc.For φ < φc, κmax is nearly constant at the level of≈ 0.04 mm−1, and above φc it strongly increases.
Fig. 8 Influence of dispersed volume fraction on filament lifetime (a), maximum filament curvature (b), and region of de-formation (c). The presented results are for the emulsions fromseries A1
Accordingly, the region of deformation also varies. Atlow φ, where the filament thins, at its full length (as it isobserved for the pure continuous phase) ldef stays at itsmaximal level. Increasing concentration above φc leadsto an abrupt decrease of deformed area.
Droplet size and continuous phase viscosity
Droplet size has a significant influence on the shearflow behavior especially on the apparent yield stress ofemulsions. According to Eq. 1 and as confirmed by ourshear experiments τ y,s is proportional to the Laplacepressure of the droplets because their deformabilitygoverns the flow behavior at such high internal phasevolume fractions. CaBER experiments on samples ofseries B1 reveal that filament lifetime tc and maxi-mum curvature κmax increase linearly with the internalLaplace pressure of the droplets 2σ i/dsv and henceτ y,s, whereas the deformation length scales as ldef ∝(2σi/dsv)
uniformly essentially over their full length, whereasabove φc strong necking occurs and filament thinningis restricted to a narrow region around the neck. In-vestigations on samples series A3 (φ = 0.45, dsv ≈0.9 μm) and B2 (φ = 0.7, dsv ≈ 5.5 μm) show thatthese characteristic features are not affected by con-tinuous phase viscosity. Filament shape including initialfilament diameter remains the same within each groupof samples upon changing ηc. The maximum curvatureand the deformed region are independent of ηc andtc is proportional to ηc for both sets of emulsions. Asexpected, the proportionality factor is much higher forφ > φc than for φ < φc.
1112 Rheol Acta (2010) 49:1103–1116
Fig. 9 Comparison of the experimental results obtained fordifferent groups of emulsions (open circles series A1, full squaresseries A2, open triangles series B1). Figures show, respectively,influence of the apparent yield stress on filament life time (a),maximum filament curvature (b), and region of deformation (c).The solid lines show empirical scaling of the discussed parameters
Apparent yield stress
The apparent yield stress is a characteristic featurefor densely packed emulsions in shear flow and inCaBER experiments the characteristic parameters arelifetime and shape of the filament. We have seen thatall these parameters are strongly determined by theinternal structure of the emulsion. They are especiallysensitive to the variation of φ for φ > φc. Further-more, we have observed that the changes reported inthese two different experiments are correlated to eachother. Filament lifetime in extensional experiments isproportional to the continuous phase viscosity. In Fig.9 the correlation between yield stress and characteristicCaBER parameters κmax, tc/ηc, and ldef is shown for thethree different emulsion series A1 (variation of φ), A2(variation of φ and ηc), and B1 (variation of dsv). Fila-ment lifetime normalized by continuous phase viscosityand κmax are directly proportional to the apparent shearyield stress, whereas ldef scales with τ
−1/2y,s .
In the following we will describe how the elonga-tional yield stress τ y,e can be directly derived fromCaBER-type experiments. The method is based on ear-lier investigations showing, that stretched yield stressfluids can form stable bridges with lengths bigger than2π times the radius of the cylindrical sample and thatit is the yield stress which determines the onset offilament deformation (Lowry and Steen 1995; Mahajanet al. 1999). A sample droplet is placed between theplates of the CaBER device and then subsequentlystretched from its initial height h0 to different height h f
within a defined stretching time ts. At each stretch ratio
ε = h f /h0 the filament shape is monitored for a pre-selected time texp. Here we have chosen texp = 120 s andif no measureable change in filament shape is detectedwithin that time the filament is considered to be in staticequilibrium. Then the Laplace pressure is determinedfrom the radius R(x) = D(x)/2 and curvature κ(x) ofthe filament:
�p=σs
(1
R(x)−κ(x)
)
with κ (x)=∣∣∣∣∣
(∂2 R/∂x2
)
[1 + (∂ R/∂x)2]3/2
∣∣∣∣∣
(9)
But when a critical stretch ratio εc is exceeded, thesample starts to flow, the filament necks, and eventuallybreaks. Then �p at the neck rapidly increases withtime, the neck diameter decreases, and the curvatureat the neck increases as shown in Figs. 4 and 7.
For a cylindrical filament in static equilibrium theforce balance reduces to −τ yy = �p. If τ xx ≈ 0 isassumed and gravitational effects are neglected, thisleads to
τxx − τyy = �p (10)
Then �p|ε=εc can be identified as the elongational yieldstress τ y,e.
The described variation in the Laplace pressure atthe neck below and above εc is shown in Fig. 10 for theemulsion from series A1 with φ = 0.75 and τ y,s = 13 Pa.As already mentioned in the experimental section aswell as in (Niedzwiedz et al. 2009) the choice of the
Fig. 10 Laplace pressure obtained from the shape of a liquidthread in static equilibrium at different ε. Above a critical stretchratio εc the Laplace pressure exceeds the value of the elonga-tional yield stress leading to a progressive increase of �p andthread necking. Data shown here are for the w/o-emulsion ofseries A1 with φ = 0.75. Experimental parameters are h0 = 3 mmand ts = 40 ms
Rheol Acta (2010) 49:1103–1116 1113
Fig. 11 a Filament profiles formed during stretching of theemulsion from series A1 with φ = 0.75 in static experiments fordifferent stretching times ts and constant h0 = 3 mm, correspond-ing εc values are also given. b Critical Laplace pressure �p forε = εc (closed squares) and corresponding filament curvature κc(open triangles) and inverse radius R−1
c (open circles) obtainedfor different stretching times ts
initial step strain parameters can have a significanteffect on the shape of the filaments and their subse-quent thinning for the emulsions investigated here. Thevariation of the filament profile and the correspondingcritical stretch ratio εc with stretching time ts is shownin Fig. 11a. Obviously, εc decreases significantly as thestretching time ts increases, but more importantly thefilament shape also varies in a wide range. The valuesfor �p|ε=εc and the corresponding values for R−1
c andκc are shown in Fig. 11b. Despite of the large variationin filament shape the Laplace pressure at the neck isconstant within experimental uncertainty irrespectiveof the choice of ts and seems to be a characteristicmaterial parameter. Moreover, at small ts the filamentcurvature is weak and the contribution of κ to �p isonly around 10%. Similar results were obtained for theother samples of series A1 with φ > φc. In these casesEq. 10 is valid and �p|ε=εc is a good approximation forτ y,e. At this point it should be noted that the cylin-drical filament approximation may be applicable forthe filaments in static equilibrium if the experimentalconditions are chosen appropriately, but it will alwaysfail for yield stress fluids in classical CaBER experi-
ments, since especially in the final stage of filamentthinning κ strongly increases (typically more than orderof magnitude, see Fig. 7) and its contribution to �pcannot be ignored. The effect of gravity on filamentstability and sagging has also to be considered and itsrelevance compared to surface tension is characterizedby the Bond number Bo = ρgR2/σ s, where ρ is thefluid density and g is the acceleration due to gravity.A reasonable criterion for the determination of τ y,e
according to Eq. 10 is Bo < 0.1 and the initial samplevolume or plate separation has to be selected accord-ingly. In the experiments presented here h0 = 2 mmwas chosen for 1 Pa < τ y,e < 30 Pa and h0 = 3 mm forsamples with higher yield stresses. A further reductionof accessible yield stresses could be achieved by usingsmaller sample volume (i.e., choosing smaller h0). Theinfluence of gravity may be further reduced and evenyield stresses as low as 0.1 Pa should be accessible byembedding the sample into an immiscible fluid withsimilar density (Mahajan et al. 1999).
The experimental approach outlined above providesa unique, robust, and reliable access to the elonga-tional yield stress of soft matter with yield stresses τ y <
103 Pa. Recently, Macosko and co-workers (Castroet al. 2010) have discussed the determination of theelongational yield stress of soap pastes with τ y ≈ 105 Pafrom orifíce extrusion experiments, but this method isnot sensitive enough for soft matter like the emulsionsconsidered here.
The results for τ y,e obtained from Eq. 10 as describedabove and those for τ y,s extracted from the shear stressvs. shear rate data shown in Fig. 2 according to Eq. 3
Fig. 12 Apparent elongational (open circles) and shear yieldstresses (closed squares) as a function of droplet volume fractionφ for the w/o-emulsions of series A1. The solid line the predictionfor τ y,s given in Eq. 1 with σ i = 2.6 mN/m, dsv = 0.9 μm andφc = 0.66. The insert shows the ratio τ y,e/τ y,s
1114 Rheol Acta (2010) 49:1103–1116
are summarized in Fig. 12. The τ y,e and τ y,s data areshown as a function of droplet concentration φ andthe insert shows the ratio τ y,e/τ y,s for the samples ofseries A1. Both τ y,s and τ y,e increase monotonicallywith increasing φ and τ y,e is always larger than τ y,s.The data for τ y,s agree well with those predicted byEq. 1 (Mason et al. 1996) if σ i = 2.6 mN/m and dsv =0.9 μm as obtained from independent measurementsare used and φc = 0.66 is chosen. As expected thecritical volume fraction is slightly higher than that forperfectly monodisperse systems, φc,m = 0.635 (Masonet al. 1996). The ratio of elongational to shear yieldstress is constant within experimental uncertainty andτ y,e/τ y,s = 2.8 ± 0.4 is found for the emulsions seriesinvestigated here. This result is in contradiction to the√
3 ratio provided by Eq. 8 based on the tensorial formof the Herschel–Bulkley model given in Eq. 6. To ourknowledge such a deviation has been observed herefor the first time and it seems that this constitutiveequation is not sufficient to describe the yielding ofsoft matter and the elongational yield stress has to betreated as an independent rheological parameter. Thisdeficiency may be due to neglecting the contribution ofthe third invariant of the strain-rate tensor. Therefore,we propose a generalized Herschel–Bulkley constitu-tive equation including III, which fulfills the generalrequirements for a Reiner-Rivlin fluid (Graebel 2007):
T = 2
⎡
⎣τy,s
√|II| + m13
√III2
+ k
(√|II| + m2
3
√III2
)n−1⎤
⎦ D
(11)
In simple shear flow, II = −γ̇ 2, III = 0 and Eq. 11 re-duces to Eq. 3. In uniaxial elongation, II = −3ε̇2, III =2ε̇3, and for ε̇ > 0 the first normal stress differencereads:
τxx − τyy = 3τy,s√3 + m1
+ 3k(√
3 + m2
)n−1ε̇n (12)
This generalized Herschel–Bulkley constitutive modelincludes the material parameters m1 and m2, which areirrelevant in shear experiments, but result in a nontriv-ial relationship between the yield stresses determinedin shear and uniaxial extensional flow:
τy,e = 3τy,s√3 + m1
(13)
If the contribution of III is neglected (i.e., m1 = m2 =0) Eqs. 11–13 reduce to Eqs. 6–8, respectively. It shouldbe noted that Eq. 11 is just one example for a gener-alized Herschel–Bulkley model leading to the result,that τ y,s and τ y,e are independent parameters. The ex-
perimentally determined ratio τ y,e/τ y,s corresponds tom1 = −0.73 for the w/o emulsions investigated here.Such deviations from the standard tensorial formu-lation of the Herschel–Bulkley model (Eq. 6) havenot been reported before. Macosko and co-workersfound good agreement between the yield stress val-ues obtained from orifice flow and shear experimentsbased on Eq. 6 for their soap pastes with τ y ≈ 105
(Castro et al. 2010), also e.g. studies dealing with flowinstabilities of Herschel–Bulkley fluids in complex flowkinematics analyzing experimental data on the basis ofEq. 6 did not resolve such discrepancies (Alexandrouet al. 2003; Coussot and Gaulard 2005). On the otherhand, it is hard to rationalize that a unique relation-ship between shear and elongational yield stress likeEq. 8 should hold for all kinds of soft matter with anapparent yield stress including such a large variety ofdifferent materials like polymeric gels and networks,jammed systems like pastes or colloidal glasses, highlyconcentrated emulsions and foams, but also suspen-sions/emulsions with low internal phase volume frac-tion, where attractive interactions among droplets orparticles lead to sample spanning network structures(Coussot 2004; Larson 1999). This topic clearly needsfurther investigations and the method proposed here,seems to be an appropriate starting point.
Conclusions
The flow behavior of w/o emulsions under extensionaldeformation has been investigated using a capillarybreakup elongational rheometer. The CaBER deviceis equipped with an advanced image processing systemincluding a high-speed camera and telecentric optics al-lowing for precise assessment of the full filament shape.The transient neck diameter D(t), time evolution of theneck curvature κ(t), the region of deformation ldef andthe filament lifetime tc have been extracted in orderto characterize thinning and breakup of non-cylindricalfilaments, typical for concentrated emulsions. Addi-tionally, standard shear experiments were done for ref-erence. Effects of disperse volume fraction φ, dropletsize dsv and continuous phase viscosity ηc on the flowproperties have been investigated in a broad parame-ter range (0.45 < φ < 0.8, 0.9 μm < dsv < 10 μm,22 m Pa s < ηc < 261 m Pa s) using well-characterizedmodel systems.
At a critical volume fraction φc, when the dropletsare densely packed and start to deform, shear as well aselongational flow properties change drastically. Shearflow curves exhibit strong shear thinning and an ap-parent yield stress τ y,s. The latter varies proportional
Rheol Acta (2010) 49:1103–1116 1115
to the internal droplet pressure, 2σ i/dsv and (φ − φc)2
as proposed earlier (Mason et al. 1996) and withσ i = 2.6 mN/m and dsv = 0.9 μm we find φc ≈ 0.66 forthe systems investigated here. In CaBER experimentsfilaments exhibit sharp necking and the maximum cur-vature at the neck, κmax, as well as tc increase, and ldef
decreases drastically for φ > φc. Shear flow curves arewell described by a Herschel–Bulkley model over theentire volume fraction range. For φ < φc, D(t) datafrom CaBER experiments are well described by a one-dimensional solution (cylindrical filament approxima-tion) of the Herschel–Bulkley or power-law model us-ing the exponent n and consistency index k from shearexperiments. Around and above φc, D(t) data can stillbe fitted using this 1−d Herschel–Bulkley approach,but first k and then τ y,s progressively deviate fromshear results as φ increases. We attribute this to thefailure of the cylindrical filament assumption as neckingis more and more pronounced. The curvature at theneck strongly increases with time for emulsions withφ ≥ φc and especially in the final stage of filament thin-ning the axial curvature cannot be ignored. The totalfilament lifetime tc is proportional to the continuousphase viscosity ηc at low as well as at high volume frac-tion. Above φc, combined data for three different setsof emulsions covering a yield stress range of about twoorders of magnitude show that κmax as well as tc/ηc scalelinearly with τ y,s and thus inversely with droplet diam-eter dsv, which enters via its contribution to the Laplacepressure of the individual droplets, whereas ldef scaleswith τ
−1/2y,s . The elongational yield stress τ y,e can be
determined directly from the Laplace pressure at thecritical stretch ratio at which a filament starts to thin.Experimental parameters can be set such, that the cur-vature of the filament in static equilibrium as well as thecontribution of gravity to the stress within the filamentcan be neglected so that �p|ε=εc = τ y,e if τ xx = 0 isassumed. This procedure provides a unique and reliableaccess to the elongational yield stress of soft matterwith yield stresses τ y < 103 Pa. For the emulsion seriesinvestigated here, covering two orders of magnitude inthe yield stress, a constant ratio τ y,e/τ y,s ≈ 3 is found.This is captured by a generalized Herschel–Bulkleymodel that includes the third invariant of the strain-ratetensor, and in conclusion the elongational yield stresshas to be treated as an independent fluid parameter.
Acknowledgements We would like to thank Julia Weberlingand Bianca Cornehl for their help in sample preparation and per-forming CaBER experiments. Further, we acknowledge financialsupport by Beiersdorf AG and Kompetenznetz Verfahrenstech-nik Pro3 e.V.
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![Abstract arXiv:1604.00326v1 [cs.CV] 1 Apr 2016Ziad Al-Halah Rainer Stiefelhagen Institute for Anthropomatics and Robotics Karlsruhe Institute of Technology fziad.al-halah, rainer.stiefelhageng@kit.edu](https://static.documents.pub/doc/80x56/5f5ddd801a5f7812d8456f69/abstract-arxiv160400326v1-cscv-1-apr-2016-ziad-al-halah-rainer-stiefelhagen.jpg)
