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12 Applied RheologyVolume 15 · Issue 1

Capillary Break-up Rheometry of Low-Viscosity Elastic Fluids

Lucy E. Rodd1,3, Timothy P. Scott3, Justin J. Cooper-White2, Gareth H. McKinley3*

1 Department of Chemical and Biomolecular Engineering, The University of Melbourne,VIC 3010, Australia

2 Division of Chemical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia

3 Hatsopoulos Microfluids Laboratory, Department of Mechanical Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139, USA

*Email: [email protected]

Received: 9.8.2004, Final version: 11.11.2004

© Appl. Rheol. 15 (2005) 12-27

Abstract:We investigate the dynamics of the capillary thinning and break-up process for low viscosity elastic fluids suchas dilute polymer solutions. Standard measurements of the evolution of the midpoint diameter of the neckingfluid filament are augmented by high speed digital video images of the break up dynamics. We show that thesuccessful operation of a capillary thinning device is governed by three important time scales (which charac-terize the relative importance of inertial, viscous and elastic processes), and also by two important length scales(which specify the initial sample size and the total stretch imposed on the sample). By optimizing the ranges ofthese geometric parameters, we are able to measure characteristic time scales for tensile stress growth as smallas 1 millisecond for a number of model dilute and semi-dilute solutions of polyethylene oxide (PEO) in water andglycerol. If the final aspect ratio of the sample is too small, or the total axial stretch is too great, measurementsare limited, respectively, by inertial oscillations of the liquid bridge or by the development of the well-knownbeads-on-a-string morphology which disrupt the formation of a uniform necking filament. By considering themagnitudes of the natural time scales associated with viscous flow, elastic stress growth and inertial oscilla-tions it is possible to construct an “operability diagram” characterizing successful operation of a capillary break-up extensional rheometer. For Newtonian fluids, viscosities greater than approximately 70 mPas are required;however for dilute solutions of high molecular weight polymer, the minimum viscosity is substantially lowerdue to the additional elastic stresses arising from molecular extension. For PEO of molecular weight 2 · 106 g/mol,it is possible to measure relaxation times of order 1 ms in dilute polymer solutions with zero-shear-rate viscosi-ties on the order of 2 – 10 mPas.

Zusammenfassung:Wir untersuchen die Dynamik der Kapillarverdünnung und des Zerreissens niedrig-viskoser elastischer Fluidewie verdünnte Polymerlösungen. Die standardisierten Messungen der Veränderung des Durchmessers in derMitte des einschnürenden flüssigen Filaments werden durch Hochgeschwindigkeitsdigitalvideoaufnahmen derZerreissdynamik verbessert. Wir zeigen, dass eine erfolgreiche Bedienung eines Kapillarverdünnungsgeräts vondrei wichtigen Zeitskalen bestimmt wird (die die relative Bedeutung von trägen, viskosen und elastischen Pro-zessen charakterisieren), und darüber hinaus von zwei wichtigen Längenskalen (die die Ausgangsgrösse derProbe und das Verstreckverhältnis der Probe spezifizieren). Durch Optimierung dieser geometrischen Parame-ter sind wir in der Lage, die charakteristischen Zeitskalen des Zugspannungswachstums bis 1 ms für mehrereModelllösungen aus Polyethylenoxid (PEO) in Wasser und Glyzerin zu messen. Wenn das Aspektverhältnis derProbe zu klein ist oder die absolute axiale Verstreckung zu gross, werden die Messungen durch Trägheitsoszil-lationen der flüssigen Brücke oder der Entwicklung der bekannten Perlen-auf-der-Kette-Morphologie begrenzt,die die Bildung eines gleichmässig einschnürenden Filaments verhindern. Durch Betrachtung der Grösse dernatürlichen Zeitskalen, die mit dem viskosen Fliessen, dem Wachstum der elastischen Spannungen und denTrägheitsoszillationen verbunden sind, ist es möglich, ein "Bedienungsdiagramm" zu erstellen, das die erfolg-reiche Bedienung eines Kapillaraufbruchdehnrheometers darstellt. Für Newtonsche Fluide sind Viskositätengrösser als ca. 70 mPas erforderlich; für verdünnte Lösungen von hochmolekularen Polymeren ist die minimaleViskosität jedoch wesentlich kleiner aufgrund der zusätzlichen elastischen Spannungen, die aus der molekula-ren Verstreckung resultieren. Für PEO mit Molekulargewicht 2 x 10**6 g/mol ist es möglich, Relaxationszeitenin der Grössenordnung von 1 ms in verdünnten Polymerlösungen mit Schernullviskositäten der Grössenordnungvon 2 – 10 mPas.

Résumé:Nous avons étudié la dynamique des mécanismes d’amincissement et de rupture capillaire dans le cas de flu-ides élastiques de faible viscosité tels que des solutions diluées de polymère. Des mesures standard de l’évolu-tion du diamètre médian du filament de fluide sous striction sont augmentées par des images vidéo digitalisées

Applied Rheology Vol.15-1.qxd 18.03.2005 11:18 Uhr Seite 12

13Applied RheologyVolume 15 · Issue 1

à grande vitesse qui rendent compte de la dynamique de rupture. Nous montrons que le fonctionnement cor-rect d’un appareil d’amincissement capillaire est gouverné par 3 échelles de temps importantes (qui caractérisentl’importance relative des mécanismes inertiels, visqueux et élastiques), et aussi par deux longueurs caractéris-tiques importantes (qui spécifient la taille initiale de l’échantillon et l’étirement total imposé à l’échantillon). Enoptimisant les portées de ces paramètres géométriques, nous sommes capable de mesurer des montées en con-trainte de tension sur des échelles de temps aussi petites que la microseconde, pour un certain nombre de solu-tions standards diluées et semi diluées d’oxyde de polyéthylène (PEO) dans de l’eau et du glycérol. Si le rapportd’anisotropie de l’échantillon est trop petit, ou si l’étirement axial est trop grand, alors les mesures sont limitéesrespectivement par les oscillations inertielles du pont liquide ou par le développement de la célèbre morpholo-gie de perles-sur-une-corde qui empêche la formation d’un filament de constriction uniforme. En considérantles ordres de grandeur des échelles de temps naturel associés avec l’écoulement visqueux, la montée en con-trainte élastique et les oscillations inertielles, il est possible de construire un «diagramme d’opérabilité» qui car-actérise le bon fonctionnement d’un rhéomètre extensionnel à rupture de capillaire. Pour des fluides New-toniens, des viscosités supérieures à environ 70mPa sont requises; cependant pour les solutions diluées depolymères de haut poids moléculaire, la viscosité minimale autorisée est significativement plus petite, à causedes contraintes élastiques additionnelles qui ont pour origine l’extension moléculaire. Pour un PEO possédantun poids moléculaire de 2x106 g/mol, il est possible de mesurer des temps de relaxation de l’ordre de la mil-liseconde pour des solutions diluées possédant une viscosité statique de l’ordre de 2-10 mPa.s.

Key words: capillary thinning, extensional rheometry, viscoelastic filament

1 INTRODUCTION

Over the past 15 years capillary break-up elonga-tional rheometry has become an important tech-nique for measuring the transient extensionalviscosity of non-Newtonian fluids such as poly-mer solutions, gels, food dispersions, paints, inksand other complex fluid formulations. In thistechnique, a liquid bridge of the test fluid isformed between two cylindrical test fixtures asindicated schematically in Fig. 1a. An axial step-strain is then applied which results in the forma-tion of an elongated liquid thread. The profile ofthe thread subsequently evolves under theaction of capillary pressure (which serves as theeffective ‘force transducer’) and the necking ofthe liquid filament is resisted by the combinedaction of viscous and elastic stresses in thethread.

In the analogous step-strain experimentperformed in a conventional torsional rheometer,the fluid response following the imposition of astep shearing strain (of arbitrary magnitude g0)is entirely encoded within a material functionreferred to as the relaxation modulus G(t, g0). Byanalogy, the response of a complex fluid follow-ing an axial step strain is encoded in an apparenttransient elongational viscosity function hE(e· , t)

which is a function of the instantaneous strainrate, e· and the total Hencky strain (e = Úe· dt’) accu-mulated in the material. An important factorcomplicating the capillary break-up technique isthat the fluid dynamics of the necking processevolve with time and it is essential to understandthis process in order to extract quantitative val-ues of the true material properties of the testfluid. Although this complicates the analysis, andresults in a time-varying extension rate, this alsomakes the capillary thinning and breakup tech-nique an important and useful tool for measur-ing the properties of fluids that are used in free-

Figure 1:Schematic of the CapillaryBreakup ExtensionalRheometer (CaBER)geometry containing a fluidsample a) at rest and b)undergoing filamentthinning for t > 0.



surface processes such as spraying, roll-coating orink-jetting. Well-characterized model systems(based on aqueous solutions of polyethyleneoxide ) have been developed for studying suchprocesses in the past decade and we study thesame class of fluids in the present study [1, 2].

Significant progress in the field of capil-lary break-up rheometry has been made in recentyears since the pioneering work of Entov and co-workers [3, 4]. Capillary thinning and break-uphas been used to measure quantitatively the vis-cosity of viscous and elastic fluids [5, 6]; explorethe effects of salt on the extensional viscosity forimportant drag-reducing polymers and otherionic aqueous polymers [7, 8], monitor the degra-dation of polymer molecules in elongational flow[4] and the concentration dependence of therelaxation time of polymer solutions [9]. Theeffects of heat or mass transfer on the time-dependent increase of the extensional viscosityresulting from evaporation of a volatile solventin a liquid adhesive have also been considered[10]; and more recently the extensional rheologyof numerous inks and paint dispersions havebeen studied using capillary thinning rheometry[11]. The relative merits of the capillary break-upelongational rheometry technique (or CABER)and filament stretching elongational rheometry(or FISER) have been discussed by McKinley [12]and a detailed review of the dynamics of capil-lary thinning of viscoelastic fluids is providedelsewhere [13].

Measuring the extensional properties oflow-viscosity fluids (with zero-shear-rate viscosi-ties of h0£ 100 mPa s, say) is a particular challenge.Fuller and coworkers [14] developed the opposedjet rheometer for studying low viscosity non-New-tonian fluids, and this technique has been usedextensively to measure the properties of variousaqueous solutions (see for example Hermansky etal. [15] or Ng et al. [16]). Large deformation rates(typically greater than 1000 s-1) are required toinduce significant viscoelastic effects, and at suchrates inertial stresses in the fluid can completelymask the viscoelastic stresses resulting from mol-ecular deformation and lead to erroneous results[17]. Analysis of jet break-up [18, 19] and drop pinch-off [20, 21] have also been proposed as a means ofstudying the transient extensional viscosity ofdilute polymer solutions. After the formation of aneck in the jet or in the thin ligament connecting

a falling drop to the nozzle, the dynamics of thelocal necking processes in these geometries arevery similar to that in a capillary break-up rheome-ter. However, the location of the neck or ‘pinch-point’ is spatially-varying and high speed photog-raphy or video-imaging is required forquantitative analysis. One of the major advan-tages of the CABER technique is that the minimumradius is constrained by geometry (and by the ini-tial step-strain) to be close to the midplane of thefluid thread, unless very large axial strains areemployed and gravitational drainage becomesimportant [22].

For low viscosity non-Newtonian fluidssuch as dilute polymer solutions, the filamentthinning process in CABER is also complicated bythe effects of fluid inertia which can lead to thewell-known beads-on-a-string morphology [23,24]. Stelter et al. [7] note that such processes pre-vent the measurement of the extensional vis-cosity for some of their lowest viscosity solutions.With the increasingly widespread adoption ofthe CABER technique, it becomes important tounderstand what range of working fluids can bestudied in such instruments. If the fluid is not suf-ficiently viscous then the liquid thread under-goes a rapid capillary break-up process before theplates are completely separated. The subsequentthinning of the thread can thus not be moni-tored. The threshold for onset of this processdepends on the elongational viscosity of the testfluid and is frequently described qualitatively as‘spinnability’ or ‘stringiness’. The transient elon-gational stress growth in the test fluids dependson the concentration and molecular weight ofthe polymeric solute as well as the backgroundviscosity and thermodynamic quality of the sol-vent. In the present note we investigate the lowerlimits of the CABER technique using dilute solu-tions of polyethylene oxide (PEO) in water andwater-glycerol mixtures. In order to reveal thedynamics of the break-up process we combinehigh-speed digital video-imaging with conven-tional laser micrometer measurements of themidpoint radius Rmid(t). We explore the conse-quences of different experimental configura-tions and the roles of solvent viscosity and poly-mer concentration. The results can beinterpreted in terms of an ‘operability diagram’based on the viscous and elastic time scales gov-erning the filament thinning process.



2 EXPERIMENTAL METHODS ANDDIMENSIONLESS PARAMETERS

2.1 FLUIDS

In this study we have focused on aqueous solu-tions of a single nonionic polymer; polyethyleneoxide (PEO; Aldrich) with molecular weightMw = 2.0 · 106 g/mol. Solutions were prepared bymixing the polymer into deionized water at con-centrations of 0.10 wt%, and 0.30 wt% usingmagnetic stirrers at slow/moderate speed set-tings. In order to explore the effects of the back-ground solvent viscosity, an additional solutionwith 0.10 wt% PEO dissolved in a mixture of 55%glycerol in water was also prepared. Additionalexperiments exploring the role of PEO concen-tration in the Capillary Break-up Rheometer havebeen performed by Neal and Braithwaite [25].The results of progressive dilution of a highmolecular weight polystyrene dissolved inoligomeric styrene have also been investigatedrecently using capillary break-up rheometryby Clasen et al. [26]. In this latter study, thesolvent viscosity of the oligomer is hs ≥ 40 Pa s;these solutions are therefore significantly moreviscous than the aqueous solutions discussed inthe present work.

The important physico-chemical andrheological properties of the test fluids are sum-marized in Table 1. The steady-shear viscosity ofeach fluid was measured using a double gap Cou-ette fixture with an AR2000 rheometer. Thesteady-state values of the surface tension weredetermined using a Krüss K-10 tensiometer witha platinum du Nouy ring. It is known that aque-ous PEO solutions are weakly surface active andthat the dynamic surface tension decreases withtime after a fresh interface is created [21, 27].However, the variation in s is small (typicallyDs £ 10 · 10-3 N/m) and drop pinch-off/breakup

experiments show that this difference does notsignificantly affect the dominant balance drivingthe thinning process [27]. We have alsoperformed additional capillary breakup experi-ments to investigate the role of dynamic sur-face tension and these are discussed briefly inSection 3.3.

For PEO, the characteristic ratio is C∞ =4.1 [28], the repeat unit mass is m0 = 44g/mol andthe average bond length is l = 0.147 nm. The meansquare size of an unperturbed Gaussian coil is·R2Ò0 = C∞ (3Mw/m0)1l2 and we thus obtain c* =Mw/(NA·R2Ò03/2) ª 2.53 · 10-3 g/cm3 for this mol-ecular weight. However, water is known to be agood solvent for PEO, so that the polymer coilsare extended beyond the random coil configura-tion and the above expression is an overestimateof the coil overlap concentration. Tirtaatmadja etal. [27] summarize previous reported values ofthe intrinsic viscosity for numerous high molec-ular weight PEO/water solutions. The measure-ments can be well described by the followingMark-Houwink expression

(1)

with the intrinsic viscosity [h] in units of cm3/g.The solvent quality parameter can be extractedfrom the exponent in the Mark-Houwinkrelationship [h] = K Mw(3n-1) to yield 3n-1 =0.65 î n = 0.55.

Combining this expression with Graess-ley’s expression for coil overlap [29] we findthat for our PEO sample c* = 0.77/[h] ª8.6 · 10-4 g/cm3 (0.086 wt%). The two solutionsconsidered here are thus weakly semi-dilutesolutions.

The longest relaxation time of amonodisperse homopolymer in dilute solution isdescribed by the Rouse-Zimm theory [30] andscales with the following parameters:

h⎡⎣ ⎤⎦ = 0 072 0 65. .Mw

Fluid c/c* s h0 tRayleigh tvisc Oh l[mN/m] [mPas] [ms] [ms] [ms]

0.1wt% PEO 1.16 61.0±0.1 2.3±0.2 20.9 1.61 0.077 1.50.3wt% PEO 3.49 60.8±0.2 8.3±1.0 20.8 5.78 0.27 4.40.1wt% PEOin Gly/Water 1.16 58.0±0.1 18.2±0.5 23.0 13.3 0.58 23.1

Table 1:The physico-chemical andrheological properties of theaqueous polyethylene oxide(PEO) solutions utilized inthe present study. Themolecular weight of thesolute is Mw = 2 · 106 g/mol.



(2)

where the Mark-Houwink relationship has beenused in the second equality. The precise value ofthe prefactor in the Rouse-Zimm theory dependson the solvent quality and the hydrodynamicinteraction between different sections of thechain; however it can be approximately evaluat-ed by the following expression [27]:

(3)

where

represents the summation of the individualmodal contributions to the relaxation time. For n= 0.55 the prefactor is 1/z(3n) = 0.463. The longestrelaxation time for the PEO solutions utilized inthe present study is thus l = 0.34 ms. Christantiand Walker [31] use a different prefactor in Eq. 3but report very similar values of the Zimm timeconstant for PEO solutions of the same molecu-lar weight (but in a more viscous solvent).

This value of the relaxation time repre-sents the value obtained under dilute solutionconditions and characteristic of small amplitudedeformations so that the individual chains do notinteract with each other. However the solutionsstudied in the present experiment are in factweakly semidilute solutions and the extensionalflow in the neck results in large molecular defor-mations. Numerous recent studies with dilutesolutions of high molecular weight polymers [8,9, 19, 27] have shown that the characteristic vis-coelastic time scale measured in filament thin-ning or drop break-up experiments is typicallylarger than the Zimm estimate and is concentra-tion dependent for concentration values sub-stantially below c*. The Zimm time-constantshould thus be considered as a lower bound onthe polymer time scale that is measured duringa capillary thinning and break-up experiment.

z nn

( )3 13

1=

=

∞

∑ ii

lz n

h h= ( )

⎡⎣ ⎤⎦13

s wMRT

lh h h n⎡⎣ ⎤⎦ =∼ s w s wM

RTK M

RT

3 2.2 INSTRUMENTATION

In the present experiments we have used a Capil-lary Break-up Extensional Rheometer designedand constructed by Cambridge Polymer Group(www.campoly.com). The diameter of the endplates is D0 = 6 mm and the final axial separationof the plates can be adjusted from 8 mm to 15 mm.The midpoint diameter is measured using a near-infra-red laser diode assembly (Omron ZLA-4) witha beam thickness of 1mm at best focus and a lineresolution of approximately 20 mm. High resolu-tion digital video is recorded using a Phantom V5.0high speed camera (at 1000 frames/second) witha Nikon 28 - 70 mm f/2.8 lens. Exposure times are214 ms per frame. The video is stored digitally usingan IEEE1394 firewire link and individual frames arecropped to a size of 512 x 216 pixels. The resultingimage resolution is 26.8 mm/pixel and the overallimage magnification is 1.7 x.

2.3 LENGTH-SCALES, TIME-SCALES ANDDIMENSIONLESS PARAMETERS

The operation of a capillary-thinning rheometeris governed by a number of intrinsic or natural-ly-occurring length and time scales. It is essentialto understand the role of these lengthscales andtimescales in controlling the dynamics of thethinning and break-up process. We discuss eachof these scales individually below:

The Sample Aspect Ratio: L(t) = h(t)/2R0As indicated in Fig. 1, the initial sample is a cylin-der with aspect ratio L0 = h0/2R0. Exploratorynumerical simulations for filament stretchingrheometry [32, 33] show that optimal aspectratios are typically in the range 0.5 £ L0 £ 1 inorder to minimize the effects of either an initial‘reverse squeeze flow’ when the plates are firstseparated (at low aspect ratios L(t) << 1) or sag-ging and bulging of the cylindrical sample (athigh aspect ratios). The final aspect ratioLf = hf/2R0 attained when the plates are sepa-rated controls the total Hencky strain that isapplied and is discussed further below.

The Capillary Length: lcap = ÷(s/rg)Surface tension is employed in order to hold theinitial fluid sample between the two endplates.



It is thus essential that the initial plate separa-tion h0 is small enough to support a static liquidbridge. Although the classical Plateau-Rayleighstability criterion h0 £ 2πR0 is well-known [34,35], this result only applies in the absence ofbuoyancy forces. For the case of axial gravity, thesituation is more complex and the maximum sta-ble sample size depends on both the fluid volumeand the Bond number Bo = rgR02/s [36]. In orderto keep the initial configuration close to cylindri-cal (with little axial ‘sagging’) capillary breakuptests typically employ axial separations h0 £ lcapor equivalently h0/R0 £ 1/÷Bo. For water (r =1000 kg/m3; s = 0.072 N/m), the capillary lengthis lcap = 2.7mm.

The Viscous Break-up Timescale: tvFor a viscous Newtonian fluid undergoing capil-lary thinning, a simple force balance shows thatthe break-up process proceeds linearly with time[37]; and close to break-up the filament profile isfound to be self-similar [38 - 40]. These observa-tions can be combined to provide a means ofextracting quantitative values for the capillaryvelocity ncap = s/m, or correspondingly the fluidviscosity (if the surface tension is determinedindependently). Provided gravitational effectsare not important (so that Rmid << lcap), the evo-lution in the midpoint radius is given by McKin-ley and Tripathi [5]:

(4)

The characteristic viscous time scale for thebreak-up process is thus tv = 14.1 mR0/s.

The Rayleigh Time Scale: tRFor a “low viscosity fluid” (to be defined more pre-cisely below), the linear stability analysis of LordRayleigh [35] for break-up of an inviscid fluid jetis an appropriate starting point. Dimensionalanalysis shows that the characteristic time scalearising in inertio capillary processes is a time con-stant now defined as the Rayleigh time, tR =÷(rR03/s). For a filament or jet of water withradius 3mm, the Rayleigh time scale is extreme-ly short, tR = 0.02 s (20 ms), so that necking andrupture proceeds extremely rapidly. This timescale plays a key role in controlling the operabil-

R t R tmid ( ) = −0 14 1s

m.

ity of filament thinning devices as we showbelow in the Discussion section.

The question that naturally arises in acapillary-thinning test is what constitutes a “lowviscosity” or, conversely, a “high viscosity” fluid?This can be answered by considering the similar-ity solutions obtained in the past decade for thenonlinear evolution and rupture of viscous andinviscid liquid threads. Specifically we wish tocompare the time scale for viscocapillarybreakup, tV, to the critical time for necking andbreakup of an inviscid fluid thread which we shalldenote ti. The similarity solution for inertiocapil-lary pinching of a liquid thread is nonlinear intime, and of the form [40, 41]:

(5)

This expression is in good agreement with exper-imental measurements of drop pinchoff [27, 40].The critical time to breakup is found by settingthe radius Rmid(t = 0) = R0 to give

(6)

The resulting ratio of time to breakup for thevisco-capillary and inertio-capillary processes isrelated to a dimensionless number known as theOhnesorge number:

(7)

Note that here we have retained the numericalfactors obtained from the similarity solutions forvisco-capillary and inertio-capillary breakup inthe definition. Neglecting these factors leads toquantitative errors in the viscosity obtained fromobservations of filament thinning [7, 42] and aninaccurate estimate for the relative balances ofterms controlling capillary break-up processes. A“low-viscosity fluid” in terms of capillary breakupelongational rheometry thus implies tv < ti (i.e.Oh < (7.23)-1 ); for aqueous solutions (with s≈ 0.07

Viscous Time ScaleInertial Time Scale

tt

v

i

= =114 1

1 957 230

03 1 2

. /

. /./

m s

r s

R

ROh

( )=

tR

tRi = ( )⎛⎝⎜

⎞

⎠⎟=−0 64 1 953 2 0

3 1 2

. ./

/r

s

R t t tmid i( ) = ( ) −( )0 64 1 3 2 3. / / /s r


N/m; R0 ≈ 3 mm) this corresponds to m < 0.063Pa s.

The Opening Time dt0 and the Imposed AxialStrain efIn a torsional step strain experiment, the shearstrain is considered theoretically to be appliedinstantaneously. In reality, the step response ofa conventional torsional rheometer is on theorder of 25 - 50 ms and the torsional displacementis approximately linear with time. By analogy, theaxial step strain imposed during a capillarybreak-up test is typically considered in a theo-retical analysis to be imposed instantaneously. Inexperiments, however, the plate separationoccurs in a finite time, denoted dt0. If a servo-sys-tem is used to stretch the liquid filament, thenthis time can be varied and the displacement pro-file may be linearly or exponentially increasingwith time. However, as a result of inertia in theplate & drive subsystem it typically is constrainedto be dt0 ≥ 0.050 s. Because the filament mustnot break during the opening process we mustrequire tv ≥ dt0. This criterion sets a stringentlower bound on the Newtonian viscosity that canbe tested in a CABER device as we show below.

The initial rapid separation of the end-plates also results in the imposition of an initialHencky strain (a ‘pre-strain’) of magnitude ef =ln(hf/h0) = ln(Lf/L0). As a consequence of the no-slip boundary conditions, the deformation of thefluid column is not homogeneous (i.e. the sam-ple does not remain cylindrical); this axial mea-sure of the strain is thus not an accurate measureof the actual Hencky strain experienced by fluidelements near the midplane of the sample. If theinitial radius of the sample at time t0 is R0 andthe midpoint radius of the filament at time t1 =t0 + dt0 is denoted R1, then the true Hencky pre-strain imposed during the stretching process is e1= 2ln(R0/R1) . It is not possible to predict this finalradius R1 without choosing a constitutive modelfor the fluid; however, for many test samples(with tv >> dt0), the midpoint radius of the sam-ple at the cessation of the stretching is given bythe lubrication solution for a viscous Newtonianfluid [43]:

(8)R R L Lf1 0 0

3 4≈ ( )−

//


The Polymer Relaxation Time: lIf the test fluid in a capillary thinning test is apolymer solution, then non-Newtonian elasticstresses grow during the transient elongationalstretching process. Ultimately these extensionalstresses grow large enough to overwhelm theviscous stress in the neck. An elastocapillary forcebalance on a uniform cylindrical thread of radiusR1 then predicts that the filament radius decaysexponentially in time:

(9)

The additional factor of 2-1/3 in the prefactor ofEq. 9 is missing in the original theory [37] due toa simplifying approximation made in derivingthe governing equation [26]. This simplificationhowever does not change the exponential factorthat is used to measure the characteristic timeconstant of the polymeric liquid. This exponen-tial relationship between the neck radius andtime has been utilized to determine the relax-ation time for many different polymeric solu-tions over a range of concentrations and molec-ular weights [3, 4, 6, 7, 42]

Note that although this time constant isreferred to as a ‘relaxation time’ – because it isthe same time constant that is associated withstress relaxation following cessation of steadyshear – in a capillary-thinning experiment, thestress is not relaxing per se. In fact the tensilestress diverges as the radius decays to zero. Thetime constant obtained from a CABER test is thusmore correctly referred to as the ‘characteristictime scale for viscoelastic stress growth in a uni-axial elongational flow’. This is, of course, pre-cisely the time constant of interest in commer-cial operations concerned with drop break-up,spraying, mold-filling, etc.

For low viscosity systems, however, thisexponential decay becomes increasingly difficultto observe due to the formation of the well-known beads-on-a-string morphology [23, 24].The elastic stresses in the necking filament growon the characteristic scale l and must grow suf-ficiently large to resist the growth of free-surfaceperturbations, which evolve on the Rayleigh timescale, tR. In the same manner that comparison of

R tR

GRtmid ( )

=⎛⎝⎜

⎞⎠⎟

−⎡⎣ ⎤⎦1

11 3

23

sl

/

exp /


the viscous and Rayleigh time scales resulted ina dimensionless group (the Ohnesorge number)so too does comparison of the polymer time scaleand the Rayleigh time scale. This dimensionlessratio may truly be thought of as a Deborah num-ber [44] because it compares the magnitude ofthe polymeric time scale with the flow time scalefor the necking process in a low viscosity fluid:

(10)

Note however that because the necking filament isnot forced by an external deformation, it selfselectsthe characteristic time scale for the neckingprocess. This Deborah number is thus an ‘intrinsicquantity’ that cannot be affected by the rheologist;except in so far as changes in the concentration andmolecular weight of the test fluid change the char-acteristic time constant of the fluid.

This is not the only possible dimension-less measure of viscoelastic effects. The defor-mation rate in an exponentially-necking threadis given (using Eq. 9) by e· ∫ -(2/Rmid)dRmid/dt =2/(3l) . The product of the relaxation time and thedeformation rate is thus a constant that may bedefined as a Weissenberg number, Wi ∫ le· = 2/3.As noted by Entov and Hinch [37], this valueexceeds the critical value Wi = 1/2 for the coilstretch transition in uniaxial flow in order tomaintain the squeezing flow and ensure the elas-tic stress balances the ever-growing capillarypressure. However, once again this value can notbe externally varied and is determined by thepolymer relaxation time and the fluid surfacetension.

As we have shown above the Rayleightimescale is very small for aqueous polymer solu-tions. Inertio-capillary thinning thus results inrapid stretching in the fluid filament with localstrain rates on the order of e· ~ tR -1 ≥ 50s-1. Itshould thus be possible to test low viscosity flu-ids with small relaxation time constants. Thequestion is how small? In the experimentsdescribed below, we seek to determine for whatrange of Deborah numbers it is possible to usecapillary breakup extensional rheometry todetermine the relaxation time of low viscosityfluids.

Det RR

≡ =l l

r s03 /

3 RESULTS

3.1 BEADS ON A STRING ANDINERTIO-CAPILLARY OSCILLATIONS

In Fig. 2 we present a sequence of digital videoimages that demonstrate the time evolution inthe filament profile for the 0.10 wt% PEO solu-tion; corresponding to a very low Deborah num-ber, De = 0.072. In all of the experiments pre-sented in this paper we define the time origin tobe the instant at which axial stretching ceases,so that t = tlab - dt0 .The first image at timet = - 0.05 s thus corresponds to the initial config-uration, of the liquid bridge with L0 = 3 mm/6 mm = 0.5. We also report the total time for thebreak-up event to occur as determined fromanalysis of the digital video sequence; with thepresent optical and lighting configuration theuncertainty in determining the break-up time isapproximately ± 0.005 s. For consistency we thenshow a sequence of five images that are evenlyspaced throughout the break-up process. Thehorizontal shaded region indicate the approxi-mate width of the laser light sheet that is pro-jected by the laser micrometer.

From Fig. 2, it is clear that initially, dur-ing the first 25 ms of the axial stretching phase,the filament profile remains almost axially-sym-metric about the midplane and a neck starts toform near the middle of the fluid thread asexpected. However, this axial symmetry is notmaintained at the end of the stretching sequenceand a local defect or ‘ligament’ forms near thelower plate. Following the cessation of stretch-ing, the filament rapidly evolves into a charac-teristic beads-on-a-string structure with a pri-mary droplet and several smaller ‘satellitedroplets’. The hemispherical blobs attached toeach end plate oscillate with a characteristic time


Figure 2: Formation of abeads-on-string and dropletin the 0.1% PEO fluidfilament for L = 2.0 andho = 3 mm, in whichtevent = 50 ms.


scale that is proportional to the Rayleigh timeconstant, tR = 22 ms.

The strong top-bottom asymmetry inthe axial curvature that can be observed in thethin ligament which develops at t = 0 is a hall-mark of an inertially-dominated break-upprocess [39, 45]; the viscous time-scale is only tv= 1.6 ms for this low viscosity fluid, hence we findOh << 1 and the absence of an axially-uniform fil-ament near the midplane suggests additionallythat elastic effects are weak and De << 1. It is thusnot easy to use such experiments to extract theviscoelastic time constant of the fluid. Elasticstresses only become important on very smalllength scales and very short time scales when thestretching rate associated with Rayleigh break-up becomes sufficiently large. From Eq. 10 wemay estimate this length scale by equating thepolymeric and inertial (Rayleigh) time scales (i.e.by setting De ª 1) to find Rligament ~ (l2s/r)1/3 .Conversely, the effective relaxation time may beestimated from observing the size of the vis-coelastic ligament that initially forms when elas-tic effects first become important. From Fig. 2cwe estimate Rligament ≈ 0.3mm, thus indicatingthat l = (sRligament 3 /r)1/2 ≈ 1 ms. Such a mea-surement is clearly imprecise; but serves to pro-vide an a priori estimate that can be used to com-pare with better measurements we make below.

A second example of difficulties that canbe encountered with CABER measurements isshown in Fig. 3 for the 0.1 wt% PEO solution inwater/glycerol. The increased viscosity of thefluid delays the break-up event substantially andthe increased relaxation time of the polymerleads to the formation of an axially uniform fluidligament as desired. However, inertial oscilla-tions of the hemispherical droplets attached toeach endplate still occur. The low aspect ratio ofthe selected test configuration (hf = 8.46 mm; Lf= 1.41) results in these oscillations intruding into

the observation plane of the laser micrometer.The period of these fluctuations may be esti-mated from the Rayleigh theory for oscillationsof an inviscid liquid drop (see Chandrasekhar,1962 for additional details). The fundamentalmode has a period tosc ∫ 2p/wosc = (p/÷2)tR. Forthe 0.1 wt% PEO solution this gives tosc ≈ 46 msin good agreement with the experimental obser-vations.

The consequences can be seen in Fig. 4awhich shows the evolution in the midpoint diam-eter Dmid(t) = 2Rmid (t). The oscillations can beclearly seen in the data for the 0.1 wt% glyc-erol/water solution; however the exponentialdecay in the radius at long times can still be clear-ly discerned; and the data can be fitted to a decay-ing exponential of the form given by Eq. 9. Thevalue of the characteristic time constant for eachmeasurement is shown on the figure. The lowerviscosity 0.1 wt% and 0.3 wt% solutions breakvery rapidly; typically within the period of a sin-gle oscillation.

The effects of varying the imposed stretch,i.e. varying the final aspect ratio Lf = hf/2R0, on theevolution of the midplane diameter is shown inFig. 4b. At the highest aspect ratio (Lf = 2), cor-responding to the high-speed digital imagesshown in Fig. 2, the measurements do not show


Figure 3 (left below):Periodic growth and

thinning of the filamentdiameter due to the inertial

oscillation of the fluidend-drops seen in the

0.1% PEO/glycerol solutionat early times, for L = 1.41

and h0 = 3 mm.

Figure 4: Exponential decayof the fluid filament diameterfor a (right above) 0.1% PEO,

0.3% PEO and 0.1%PEO/glycerol solutions at an

aspect ratio of 1.41, andb (right middle) 0.1% PEO

solution for h0 = 3mm andL = 1.41, 1.61, 1.79 and 2.


exponential thinning behavior as a consequenceof the large liquid droplet passing through themeasuring plane. As the aspect ratio isdecreased, the data begins to approximate expo-nential behavior and regression of Eq. 9 to thedata results in reasonable estimates of the relax-ation time.

3.2 SAMPLE SIZE AND VOLUME

As we noted above in Section 2, the initial sam-ple configuration can play an important role inensuring that capillary break-up rheometryyields reliable and successful results. By analogy,in conventional torsional rheometry it is key toensure that the cone angle of the fixture is suffi-ciently small or that the gap separation for a par-allel plate fixture is in a specified range. In Figs. 5- 7 we show the consequences of varying the ini-tial sample gap height, as compared to the cap-illary length lcap = ÷(s/rg). In each test we usethe 0.30 wt% PEO solution and a fixed finalaspect ratio of L=1.61; corresponding to a finalstretching length hf = 1.61(2R0) = 9.7 mm.

If h0/lcap < 1 then the interfacial forcearising from surface tension is capable of sup-porting the liquid bridge against the sagginginduced by the gravitational body force. Conse-quently the initial sample is approximately cylin-drical and the initial deformation results in a top-bottom symmetric deformation and theformation of an axially-uniform ligament att = 0 when deformation ceases, as shown in Fig. 5.However, if the initial gap is larger, as shown inFig. 6 (corresponding here to h0 = 3mm) andexceeds the capillary length scale (h0/lcap = 1.19),then asymmetric effects arising from gravita-tional drainage become increasingly important.Even under rest conditions (as shown by the firstimage in Fig. 6), gravitational effects result in adetectable bulging in the lower half of the liquidbridge; as predicted numerically [36]. This asym-metry is amplified during the ‘strike’ or gap-opening process as indicated in the 2nd and 3rd

frames. However as viscoelastic stresses in theneck region grow and a thin elastic thread devel-ops, the process stabilizes and exponential fila-ment thinning occurs once again. In Figure 7 weshow an even more pronounced effect when theinitial gap is 4 mm (corresponding to h0/lcap =

1.58). The asymmetry of the initial condition andthe extra fluid volume (corresponding to a vol-ume of V ª pR02h0 = 113 ml; i.e. twice the fluid vol-ume in Fig. 5) is sufficient to initialize the forma-tion of a ‘bead’ or droplet near the middle of thefilament at t = 25 ms (0.2tevent), which subse-quently drains into the lower reservoir. A distinctuniform axial thread only develops for timesgreater than t ≥ 0.3tevent≈ 0.04 s. This severelylimits the useful range of measurements.

The measured midpoint diameters forthe conditions in Figs. 5 - 7 are shown in Fig. 8.The progressive drainage of the primary dropletthrough the measuring plane of the laser


Figure 5 (above): Filamentthinning of the 0.3% PEOsolution for an initial gapheight of h0 = 2 mm andL = 1.61, in which the totaltime of the event,tevent = 100 ms.

Figure 6 (middle): Filamentthinning of the 0.3% PEOsolution for an initial gapheight of h0 = 3 mm andL = 1.61, in which the totaltime of the event,tevent = 110 ms.

Figure 7 (below): Filamentthinning of the 0.3% PEOsolution for an initial gapheight of h0 = 4 mm andL = 1.61, in which the totaltime of the event,tevent = 125 ms.

Figure 8: Exponential decayof the fluid filament diame-ter for the 0.3% PEO solu-tion for L = 1.6 and initialsample heights of h0 = 2, 3and 4 mm.


micrometer can be clearly seen in the data forh0 = 4 mm. Although an exponential regime(corresponding to elasto-capillary thinning withapproximately constant slope of the form givenby Eq. 9) can be seen for the intermediate sepa-ration (h0 = 3 mm), the effects of the initial asym-metry in the sample – coupled with the perturb-ing effects of axial drainage during the plateseparation – result in fluctuations in the diame-ter profile and an under-prediction in the longestrelaxation time. The smallest initial gap setting(h0 = 2 mm), however, results in steady expo-nential decay over a time period of approxi-mately Dt = 40 ms; corresponding to Dt/3l ≈ 1.8and, consequently from Eq.9 a diameter decreaseof more than a factor of 6. This is of a sufficient-ly wide range to satisfactorily regress to theequation.

One important feature to note from acareful comparison of Figs. 7 and 8 is the differ-ence in spatial resolution offered by the digitalimaging system; the laser micrometer has a cal-ibrated spatial resolution of ca. 20 mm [6] whichis reached after a time interval of approximatelyDt ≈ 50 ms; hence Dt/tevent ª 50/125 = 0.4. By con-trast, a thin elastic ligament can still be visuallydiscerned for another 50 ms. The performance offuture Capillary Break-up Extensional Rheome-ters may thus be enhanced by employing lasermicrometers with higher spatial resolution orusing analog/digital converters with 16 bit or 20bit resolution. Such devices however typicallybecome increasingly bulky and expensive.

3.3 THE ROLE OF FLUID VISCOSITY ANDASPECT RATIO

As we noted in Section 2.1, the longest relaxationtime and also the zero-shear rate viscosity of adilute polymer solution both vary with the vis-cosity of the background Newtonian solvent and

also with the concentration of the polymer insolution. The characteristic viscous and elastictime scales associated with the break-up processalso increase and so do the dimensionless para-meters Oh and De. Inertial effects thus becomeprogressively less important and capillary break-up experiments become concomitantly easier.An example is shown in Fig. 9 for the 0.1 wt% PEOsolution in glycerol/water at a high aspect ratio(Lf = 2.0). The equivalent process in a purelyaqueous solvent has already been shown in Fig.2 and resulted in a beads-on-string structure thatcorrupted CABER experiments. However, byincreasing the background solvent viscosity thisbreak-up process is substantially retarded (thetotal time for break-up increases from 50 ms toover 400 ms) and a uniform fluid filament isformed between the upper and lower plates. Thecorresponding midpoint diameter measure-ments for each of the test fluids (in this case witha reduced aspect ratio of Lf = 1.6 and an initialgap of h0 = 3mm) are shown in Fig. 10a. For the0.1 wt% PEO solution in glycerol/water a statisti-cally significant deviation from a pure exponen-tial decay can be observed for t ≥ 0.18s. Theoret-ical considerations show that this deviationshould correspond to the onset of finite extensi-bility effects associated with the PEO moleculesin the stretched elastic ligament attaining full


Figure 9 (left below):Thinning of fluid filamentfor the 0.1% PEO/glycerol

solution for L = 2 and h0 = 3mm, in which the total

event time, tevent = 420 ms.

Figure 10: a (right above): Exponential

decay of the fluid filamentdiameter for L=1.61 and

h0 = 3mm andb (right middle): relaxation

time as a function of aspectratio, for the 0.1% PEO,

0.3% PEO and 0.1% PEO/glycerol solutions in which

h0 = 3mm.


extension [37]. In this final stage of break-up,numerical simulations with both the FENE-P andGiesekus models show that the filament radiusdecreases linearly with time [46].

Finally, our results for the measuredrelaxation times of the three test fluids are sum-marized in Fig. 10b. Each point represents theaverage of at least three tests under the speci-fied experimental conditions. No data could beobtained with the 0.1 wt% PEO/water solution ataspect ratios L ≥ 1.8 due to the inertio-capillarybreak-up and beads-on-a-string morphologyshown in Fig. 2. It can be noted that the measuredrelaxation times vary with aspect ratio only veryweakly. This is reassuring for a rheometric deviceand indicates that relaxation times as small as l≈ 1 ms can successfully be measured using capil-lary thinning and break-up experiments. Aver-age values of the measured relaxation times aretabulated in the final column of Table 1.

We have also investigated the role ofdynamical variations in the surface tension byrepeating the experiments in the 0.1 wt% glyc-erol/water solution after addition of 0.4%2-butanol. Dynamic surface tension measure-ments using a maximum bubble volume ten-siometer show that this small molecule migratesto the interface very rapidly (in less than 1 ms) andleads to a surface tension coefficient that is con-stant over the time of the breakup process [27].Capillary breakup measurements show that therelaxation time remains essentially unchangedwith mean values (n = 5) of l = 18 ± 1 ms (nobutanol added) to 17 ± 1 ms (with 0.4% 2-butanoladded).

4 DISCUSSION & CONCLUSION

In this paper we have performed capillary break-up extensional rheometry (CABER) experimentson a number of semi-dilute polymer solutions ofvarying viscosities using cylindrical samples ofvarying initial size and imposed stretches of dif-ferent axial extent leading to various imposedaxial strains. High speed digital imaging showsthat changes in these parameters may changethe dynamics of the filament thinning and break-up process for each fluid substantially.

By considering the natural length scalesand time scales that govern these dynamics, we

have been able to develop a number of dimen-sionless parameters that control the successfuloperability of such devices as extensional rheome-ters; the most important being the Ohnesorgenumber and a natural or ‘intrinsic’ Deborah num-ber. In addition, decreasing the Bond number, orequivalently the dimensionless aspect ratio h0/R0£ 1/÷Bo, can help extend the operating range byminimizing the effects of gravitational drainage.These constraints can perhaps be most naturallyrepresented in the form of ‘operability diagrams’such as the ones sketched in Figure 11. In Fig. 11awe select the dimensional parameters corre-sponding to the zero-shear-rate viscosity, h0, ofthe solution and the characteristic relaxationtime, l, as the abscissa and ordinate axes respec-tively. A more general version of the same nomo-gram is shown in Fig. 11b in terms of the Ohnesorgeand Deborah numbers.

For Newtonian fluids (corresponding tol = 0) we require, at a minimum, that tv ≥ ti (orequivalently from Eq. 7 that Oh ≥ 7.23-1 ª 0.14 ) inorder to observe the effects of fluid viscosity onthe local necking and break-up. As we discussedin Section 2.3 for the present configuration thisgives a lower bound on the measurable viscosityof 63 mPa s. However, the device also takes afinite time (which we denote dt0) to impart theinitial axial deformation to the sample. An addi-tional constraint is thus tv ≥ dt0 or


Figure 11: An ‘operabilitydiagram’ for capillarybreak-up elongationalrheometry.a (above): dimensional formshowing the minimumvalues of viscosity, m, andrelaxation time, l, requiredfor successful measurementof the capillary thinningwith presently availableinstrumentation;b (below): dimensionlessform of the nomogramshowing fundamentallimitations to the operationof such devices.


For a prototypical Newtonian fluid with s ≈0.060 N/m, a plate size of R0 = 3mm and an open-ing time of dt0 = 50ms we find m ≥ 0.071 Pa s. Thisdefines the intersection of the operabilityboundary with the abscissa. Increasing the dis-placement rate of the linear motor in order toreduce the opening time would enable some-what lower viscosity fluids to be tested; howev-er the critical time scale ti for inerto-capillarybreak-up of a Newtonian fluid thread will ulti-mately limit the range of viscosities that can besuccessfully tested.

The dilute polymer solutions tested inthe present study obviously have viscosities sig-nificantly less than this value, and viscoelasticityfurther stabilizes the filament against breakup.The simplest estimate for the range of relaxationtimes that can be measured is to require that theintrinsic Deborah number is of order unity, orequivalently l ≥ tR = ÷(rR13/s) ª 6 ms. Howeverthis estimate is based on an elasto-capillary bal-ance in the thread of radius R1 that is formed atthe instant that the imposed stretching ceases(see Eq. 8). In reality we are able to resolve thin-ning threads of substantially smaller spatialscale. Closer analysis of the digital video fromwhich the images in Fig. 2 are taken (specifically,the frames between times t = - 25 ms and 0 mswhich are not presented here) shows that a neckfirst forms at t = - 5 ms, when the thread diame-ter at the neck is approximately 200mm; the min-imum resolvable viscoelastic relaxation timeshould thus be l >> ÷((103)(2 · 10-4)3/(0.06)) ª0.4 ms.

However, just as in the above argumentsregarding the minimum measurable Newtonianviscosity, the capabilities of the instrumentationalso play a role and may serve to further constrainthe measurable range of material parameters.More specifically, the minimum measurable radius,the total imposed stretch and the sampling rate willall impact the extent to which a smoothly decayingexponential of the form required by Eq. 9 can beresolved. In the present experiments we have sam-pled the analog diameter signal from the lasermicrometer at a rate of 1000 Hz ( dts = 0.001s), andthe minimum radius that can be reliably detected

ms

d≥14 1 0

0. Rt by the laser micrometer is Rmin ª 20 mm. If we

require that, as an absolute minimum, we monitorthe elastocapillary thinning process long enough toobtain 5 points that can be fitted to an exponentialcurve, then the measured radius data must spanthe range Rmin £ Rmid(t) £ Rmine+5dts/3l. Howeverthe radius of the neck at the cessation of theimposed stretching (t = 0) is given (at least approx-imately) by Eq. 8. Combining these expressions wethus require that

Rearranging this expression gives:

(11)

For an axial stretch of Lf = 1.6, a sampling timeof 1 ms, and a minimum detectable radius of20 mm we obtain a revised estimate of the mini-mum measurable relaxation time l ≥ 0.36 ms,which is in agreement with our present observa-tions. In reality, Eq. 8 is an overestimate of theneck radius, R1, at the cessation of the stretchingphase, since the lubrication theory from which itis derived implicitly assumes viscous effects arefully developed throughout the axial stretchingprocess. The data in Fig. 4 show that, in general,for low viscosity fluids the exponential neckingphase starts at a somewhat lower value of themeasured radius. This will increase the lowerbound given by Eq. 11; however the weak loga-rithmic dependence of this expression on theprecise value of the radius makes such correc-tions small.

This estimate of the minimum vis-coelastic time scale denotes the limiting boundof successful operation for a very low viscosity(i.e. an almost inviscid) elastic fluid; correspond-ing to the ordinate axis (Oh ô 0) of Fig.11. Theshape and precise locus of the operability bound-ary within the two-dimensional interior of thisparameter space will depend on all three timescales (viscous, elastic and inertial) and also onthe initial sample size (h0/lcap) and the total axialstretch, Lf , imposed. It thus needs to be studied

ld

≥⎡⎣ ⎤⎦

−

5 3

03 4

tR R

s

f

/ln //

minL

R e R Rtf

smin

/ /5 31 0

3 4d l ≤ = ( )−L




in detail through numerical simulations. Howev-er, our experiments indicate that it is possible,through careful selection of both the initial gap,h0, and the final strike distance, hf, to success-fully measure relaxation times as small as 1 msfor low viscosity elastic fluids with zero-shear-rate viscosities as small as 3 mPa s. Near theboundary of the operating space it is importantto perform multiple experiments in order toobtain reliable measures of the mean value andstandard deviations of the measured relaxationtimes. By analogy, in shear rheometry of low vis-cosity fluids it is essential to perform multipleexperiments and average the measured stresssignal in order to obtain reliable values of the vis-cometric properties.

A final practical use of an ‘operabilitydiagram’ such as the one sketched in Fig. 11 is thatit enables the formulation chemist and rheolo-gist to understand the consequences of changesin the formulation of a given polymeric fluid. Thechanges in the zero-shear-rate viscosity andlongest relaxation time that are expected fromdilute solution theory and formulae such as Eq. 3are indicated by the arrows. Increases in the sol-vent quality and molecular weight of the solutelead to large changes in the relaxation time, butsmall changes in the overall solution viscosity (atleast under dilute solution conditions). By con-trast, increasing the concentration of dissolvedpolymer into the semi-dilute and concentratedregimes leads to large increases in both the zero-shear-rate viscosity and the longest relaxationtime. It should be noted that the dynamics of thebreak-up process can change again at very highconcentrations or very high molecular weightswhen the solutions enter the entangled regime(corresponding to cMw ≥ rMe, where Me is theentanglement molecular weight of the melt).Although capillary thinning and break-up exper-iments can still be successfully performed, thedimensionless filament lifetime tevent/l (asexpressed in multiples of the characteristicrelaxation time) may actually decrease from thevalues observed in the present experiments dueto chain disentanglement effects [47]; i.e. a con-centrated polymer solution may actually be lessextensible than the corresponding dilute solu-tion. Capillary thinning and break-up experi-ments of the type described in this article enablesuch effects to be systematically probed.

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