Rheol Acta (2014) 53:725–739DOI 10.1007/s00397-014-0789-8

ORIGINAL CONTRIBUTION

Experimental study on the capillary thinning of entangledpolymer solutions

Dirk Sachsenheimer · Bernhard Hochstein ·Norbert Willenbacher

Received: 14 February 2014 / Revised: 27 June 2014 / Accepted: 10 July 2014 / Published online: 1 August 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract The transient elongation behavior of entangledpolymer and wormlike micelles (WLM) solutions has beeninvestigated using capillary breakup extensional rheometry(CaBER). The transient force ratio X = 0.713 reveals theexistence of an intermediate Newtonian thinning region forpolystyrene and WLM solutions prior to the viscoelasticthinning. The exponential decay of X(t) in the first periodof thinning defines an elongational relaxation time λx whichis equal to elongational relaxation time λe obtained fromexponential diameter decay D(t) indicating that the initialstress decay is controlled by the same molecular relaxationprocess as the strain hardening observed in the terminalregime of filament thinning. Deviations in true and appar-ent elongational viscosity are discussed in terms of X(t). Aminimum Trouton ratio is observed which decreases expo-nentially with increasing polymer concentration leveling offat Trmin = 3 for the solutions exhibiting intermediate New-tonian thinning and Trmin ≈ 10 otherwise. The relaxationtime ratio λe/λs, where λs is the terminal shear relax-ation time, decreases exponentially with increasing polymerconcentration and the data for all investigated solutions col-lapse onto a master curve irrespective of polymer molecularweight or solvent viscosity when plotted versus the reducedconcentration c[η], with [η] being the intrinsic viscosity.This confirms the strong effect of the nonlinear deformationin CaBER experiments on entangled polymer solutions assuggested earlier. On the other hand, λe ≈ λs is found forall WLM solutions clearly indicating that these nonlinear

D. Sachsenheimer (�) · B. Hochstein · N. WillenbacherInstitute for Mechanical Process Engineering and Mechanics,Group Applied Mechanics (AME), Karlsruhe Instituteof Technology (KIT), Gotthard-Franz-Straße 3, 76131Karlsruhe, Germanye-mail: sachsenheimer@kit.edu

deformations do not affect the capillary thinning process ofthese living polymer systems.

Keywords CaBER · Filament stretching · Polymersolution · Wormlike micelles · Elongational viscosity ·Relaxation time

Introduction

Capillary breakup extensional rheometry (CaBER) is a com-mon technique for the determination of the elongationalbehavior of low-viscosity fluids. This simple and versa-tile method has been suggested more than 20 years ago(Bazilevsky et al. 1990; Entov and Hinch 1997; Bazilevskyet al. 2001); and is commercially available (Haake CaBER1, Thermo Scientifics). In contrast to other techniques,CaBER allows for large Hencky strains up to εmax ≈ 8...13which are of great significance to industrial practice. TheCaBER experiment is based on the creation of an instableliquid bridge. After applying a step strain within a shortstrike time in a range of typically ts ≈ 20...100 ms, thediameter of the liquid filament decreases due to the act-ing surface tension. Monitoring of the filament diameter asa function of time allows for the determination of charac-teristic elongational flow properties such as the apparentelongational viscosity (Anna and McKinley 2001a), theelongational relaxation time (Entov and Hinch 1997; Clasen2010), or the elongational yield stress (Niedzwiedz et al.2009; Niedzwiedz et al. 2010; Martinie et al. 2013).

In a commercial CaBER setup, the only measured quan-tity during filament thinning is the diameter decay D(t) ata single position in the filament. Different extensions ofthe experimental setup are reported in the literature suchas optical shape recognition using a (high-speed) camera

726 Rheol Acta (2014) 53:725–739

(Christanti and Walker 2001; Niedzwiedz et al. 2009;Nelson et al. 2011; Gier and Wagner 2012; Sattler et al.2012), force measurement during initial step strain (Klein etal. 2009) as well as during the whole capillary thinning pro-cess (Sachsenheimer et al. 2012). Unfortunately, controllingthe temperature of the thinning liquid thread is not possi-ble due to the lack of an appropriate temperature controlunit for CaBER and, therefore, capillary thinning is onlyperformed at room temperature. In a study on polyacryloni-trile solutions, samples have been preheated and an elevatedconstant temperature was assumed due to the short time ofexperimentation although these measurements were doneat room temperature (Tan et al. 2012). In contrast, exten-sional rheometry of polymer melts has been performed atelevated temperatures. The “universal extensional rheome-ter” of (Munstedt 1975, Munstedt 1979; Munstedt et al.1998) stretches polymer melt filaments in a tempered sil-icon oil bath in order to maintain an accurate temperaturecontrol spatially as well as temporally. Furthermore, a fil-ament stretching extensional rheometer (FiSER) has beenequipped with an oven (Bach et al. 2003) in order to deter-mine the elongational properties of polymer melts up toa temperature of T = 200◦C. The length of the filamentis continuously increased; therefore, FiSER experimentsrequire very large devices and an accurate temperature con-trol is complicated due to convection in the oven. To reducea vertical temperature gradient, (Bach et al. 2003) used sixtemperature transducers combined with eight heat elementsas well as a nitrogen flow through the oven smearing outtemperature gradients. To the best of our knowledge, nocapillary thinning experiments under controlled temperatureconditions other than room temperature have been reportedso far.

The thinning behavior of a liquid thread is controlled bythe capillary force and viscous or viscoelastic forces. Theinterplay of these forces results in characteristic diametervs. time curves observed experimentally and predicted the-oretically for different types of fluids, e.g., Bingham plastic,power law, Newtonian fluid, or viscoelastic fluid. However,the classical way of calculating the elongational viscosityfrom CaBER experiments is based on the assumption ofvanishing axial normal stress (axial force is only causedby the surface tension). Recently, we introduced the so-called tilted CaBER method (Sachsenheimer et al. 2012)to determine the true axial force during the capillary thin-ning process. In this type of experiment, the filament isstretched horizontally and the axial force can be determinedfrom gravity-driven bending of the filament in a range ofF = 1,000–0.1 µN.

Using the CaBER technique, we want to improveour knowledge of the self-controlled thinning behaviorof concentrated and mostly entangled polymer solutions.Therefore, we give first a brief introduction into the

theory of filament thinning and CaBER experiments ofnon-Newtonian fluids focusing on the elongational relax-ation time and the true elongational viscosity. Then, wepresent the experimental setup and give a short overviewof the samples used including preparation and characteri-zation. Following with the experimental part, we analyzethe effect of different geometrical setups. The transientforce ratio including its effect on the elongational viscos-ity is discussed. After this, we discuss the elongationalrelaxation time and the relaxation time ratio including a uni-versal scaling for polymer solutions. Finally, we compareCaBER results for some surfactant solutions with filamentstretching (FiSER) experiments and give a short conclusion.

Filament thinning and CaBER experiment

Diameter vs. time evolution during capillary thinning

Newtonian fluids form nearly cylindrical filaments. Themidpoint diameter Dmid of such a thread decreases linearlywith time t in a CaBER experiment according to the fol-lowing (Papageorgiou 1995; McKinley and Tripathi 2000):

Dmid(t) = D1 −��

ηst (1)

where D1 is the initial diameter of the linear filamentthinning region (this corresponds to time t = 0 in themodel equation), � is the surface tension, ηs is the shearrate-independent shear viscosity of the Newtonian liquid,and � is a constant numerical factor. Neglecting inertia,Papageorgiou (1995) predicted � = 0.1418 which wasconfirmed experimentally (McKinley and Tripathi 2000;Sachsenheimer et al. 2012) for the capillary thinning of aNewtonian fluid.

Weakly viscoelastic fluids like dilute and semi-dilutesolutions of linear, flexible polymers form perfect cylin-drical filaments, and their diameter decreases exponentiallywith time in CaBER experiments (Bazilevsky et al. 1990;Renardy 1994, 1995; Brenner et al. 1996; Bazilevsky et al.1997; Eggers 1997; Liang and Mackley 1994; Entov andHinch 1997; Kolte et al. 1999; Anna and McKinley 2001a;Rodd et al. 2005; McKinley 2005; Ma et al. 2008; Tuladharand Mackley 2008; Miller and Cooper-White 2009; Clasen2010; Arnolds et al. 2010; Vadillo et al. 2010, 2012; Campo-Deano and Clasen 2010; Sachsenheimer et al. 2012).Assuming, that the force in the filament is only caused bythe surface tension, i.e., no axial normal stresses are con-sidered (σzz = 0), the diameter vs. time curve is given by

D(t) = D1

(GD1

2�

)1/3

exp

(− t

3λe

)(2)

Rheol Acta (2014) 53:725–739 727

where D1 is the filament diameter at the time when theexponential filament thinning sets in (this corresponds totime t = 0 in the model equation), G is the elastic modulus,and λe is the so-called elongational relaxation time whichdepends on polymer concentration. The prefactor in Eq. 2can differ if the σzz = 0 assumption is not fulfilled (Clasenet al. 2006a). Evaluating the elongational relaxation time λeseems to be easy, but technically it is not always unambigu-ous. Generally, the exponential decay in the diameter vs.time curves occurs at an intermediate stage of filament thin-ning. Purely viscous stresses dominate initially, while finiteextensibility leads to an accelerated filament decay priorthe filament breakup (Entov and Hinch 1997). For entan-gled polymer solutions, the exponential thinning region isbarely observable (Clasen 2010) or even completely absent(Arnolds et al. 2010; Sachsenheimer et al. 2012) even if thefilament shape still remains cylindrical.

The elongational relaxation time λe has been comparedto a mean shear relaxation time defined as follows (Liangand Mackley 1994):

λs =∑

3giλi∑gi

(3)

which is calculated from the relaxation time spectrum whichcan be obtained from small amplitude oscillatory shear(SAOS) experiments without assuming a specific constitu-tive equation. However, the comparison with λe was notsatisfactory: for a series of polyisobutylene (PIB) solutions;they found λe ≈ 3λs but for the fluid S1, λe ≈ 15λs .

From a theoretical point of view, Entov and Hinch (1997)analyzed the effect of a relaxation time spectrum on thecapillary thinning of viscoelastic filaments using a multi-mode FENE model. For the so-called middle elastic timesresulting in a diameter decay given according to

D(t) = D1

[D1

2�

N∑i=1

gi exp

(− t

λi

)]1/3

(4)

where gi and λi are the strength and time parameter ofthe i-th component of the relaxation time spectrum andN is the number of elements. However, it is obvious thata simple exponential filament thinning as observed for,e.g., weakly viscoelastic polymer solutions, can only beobserved if all modes i > 1 are relaxed and (4) reducesto Eq. 2 with λe ≡ λ1 (longest relaxation time of thespectrum). The difference between the diameter calculatedfrom Eqs. 4 and 1 is indeed decreasing with time andless than 5 % for times t > 2.5λ1. Therefore, the highermodes should only affect the initial nonexponential thin-ning process. Using polystyrene-based Boger fluids, Annaand McKinley (2001a) also showed the equivalence of theZimm relaxation time λZ as, e.g., defined in Rubinstein and

Colby (2003) and the elongational relaxation time deter-mined from the exponential diameter decay. However, moredetailed experimental studies have found relaxation timeratios λe/λZ = 0.1 ... 30 for polyethylenoxied (PEO) andpolystyrene (PS) solutions with concentrations c ≤ c∗,where c∗ is the critical overlap concentration (Christanti andWalker 2001; 2002; Tirtaatmadja et al. 2006; Clasen et al.2006a; Campo-Deano and Clasen 2010; Vadillo et al. 2012;Haward et al. 2012a). For concentrations c ≤ c∗, relaxationtime ratios λe/λZ = 1...100 have been reported (Rodd etal. 2005; Clasen 2010; Haward et al. 2012a) for differentpolymer solutions.

Finally, λe has been related to the terminal shear relax-ation time λs estimated from SAOS in the terminal flowregion (ω → 0) according to

λs = limω→0

G′

ωG′′ . (5)

λe/λs ≤ 1 values have been documented for PEO and PSsolutions with concentrations c∗ < c < ce (Oliveira etal. 2006; Arnolds et al. 2010; Clasen 2010) where ce isthe entanglement concentration. Arnolds et al. (2010) havequantified the effect of nonlinear deformation in CaBERexperiments using a single factorizable integral modelincluding a damping function which has been determinedfrom steady shear data. The decrease of the λe/λs ratiowith increasing polymer concentration could be predictedquantitatively.

In short, the elongational relaxation time has been com-pared to different molecular and shear relaxation times, butno simple and universal correlation between these valuescould be found. However, different shear relaxation timesdiffer from each other. Using the well-known expressionsfor G′ and G′′ of the multimode Maxwell model with dis-creet relaxation time spectrum (gi , λi), the terminal shearrelaxation time defined by Eq. 5 reads

λs,Max =∑

giλ2i∑

giλi=

∑giλ

2i

η0, (6)

where η0 = ∑giλi is the zero shear viscosity. Accord-

ingly, λs can be interpreted as the centroid of the relaxationtime distribution (Bohme 2000). Obviously, λs,Max (givenby Eq. 6) differs from the mean shear relaxation time λssuggested by Liang and Mackley (1994) and also from thelongest relaxation time λ1 of the spectrum. Using the com-plex moduli G′ and G′′ of the Rouse and Zimm model(Rubinstein and Colby 2003), the characteristic model timeconstants can be related to the terminal shear relaxation time(5) according to λR = 2λs and λZ = [1 − 1/ (3ν)]−1 λs ,respectively. Here, ν is the scaling exponent associated withsolvent quality.

728 Rheol Acta (2014) 53:725–739

Determination of the elongational viscosity from capillarythinning experiments

The uniaxial elongational viscosity ηe is generally definedas follows (Schuemmer and Tebel 1983):

ηe = σzz − σrr

ε, (7)

where σzz is the axial normal stress in the fluid filament, σrris the radial normal stress, and ε is the elongational rate.

Taking into account mass conversation as well as axialand radial force balances on a cylindrical filament undergo-ing elongation but neglecting gravity and inertia effects, (7)yields (Sachsenheimer et al. 2012)

ηe = �

dD/dt− 2F

πD dD/dt, (8)

where � is the surface tension, and F the axial force inthe filament with diameter D. Unfortunately, F is not mea-sured in standard CaBER experiments. Instead, σzz = 0is assumed, and the axial force is calculated from thesurface tension alone (F = πD�). This results in asimplified expression for the elongational viscosity, the so-called apparent elongational viscosity (Anna and McKinley2001a)

ηe,app = − �

dD/dt(9)

which is widely used in the literature (Rothstein 2003;Yesilata et al. 2006; Oliveira et al. 2006; Bhardwaj et al.2007a; Sattler et al. 2007; Chellamuthu and Rothstein 2008;Kheirandish et al. 2008; Tuladhar and Mackley 2008; Chenet al. 2008; Yang and Xu 2008; Kheirandish et al. 2009;Arratia et al. 2009; Miller and Cooper-White 2009; Davidet al. 2009; Regev et al. 2010; Bischoff White et al. 2010;Clasen 2010; Becerra and Carvalho 2011; Erni et al. 2011;Rathfon et al. 2011; Nelson et al. 2011; Haward et al.2012a; Haward and McKinley 2012b; Gier and Wagner2012; Tembelya et al. 2012; Vadillo et al. 2012; Sankaranand Rothstein 2012).

Deviations from the σzz = 0 assumption obviously occurfor Newtonian liquids (Liang and Mackley 1994; Kolte andSzabo 1999; McKinley and Tripathi 2000; Sachsenheimeret al. 2012) and polymer solutions (Clasen et al. 2006a;Sachsenheimer et al. 2012), and this may be quantifiedusing the dimensionless force ratio (McKinely and Tripathi2000; Sachsenheimer et al. 2012)

X = F

πD�, (10)

where π�D is the force due to the surface tension of theliquid and the σzz = 0 assumption is fulfilled as long as theforce ratio is X = 1. However, the tilted CaBER method can

be used by measuring the true elongational viscosity ηe fromCaBER experiments using Eq. 8 without any assumptions.

Measurement techniques

Capillary thinning (CaBER) measurements

We have used the commercially available Haake CaBER 1(Thermo Scientifics, Karlsruhe, Germany) for our elonga-tional measurements. The fluid under test is placed betweentwo plates of diameter D0 = 6 mm. At time t = t0, a liq-uid bridge is created by separating the upper plate from ainitial displacement hi to a final displacement hf within aconstant strike time of ts = 40 ms using a linear stretch-ing profile (constant velocity). Plate separation parametershave been varied between 0.51 mm < hi < 3 mm and7.4 mm < hf < 12 mm. At time t1 = t0 + ts , the liq-uid thread with an initial diameter of D1 = D (t1) beginsto thin due to the acting surface tension until the filamentbreaks. Filament shapes were captured using an opticalsetup including a high-speed camera and telecentric illumi-nation (Niedzwiedz et al. 2009; Sachsenheimer et al. 2012).In order to determine the axial force in the filament, thewhole CaBER setup has been rotated by 90◦. Horizontal fil-ament stretching experiments have always been performedwith hi = 0.75 mm and hf ≈ 11.5 mm. The axial forceF was calculated from the bending line w of the liquidfilament (Sachsenheimer et al. 2012) according to

F = πρg

4w

x∫0

x∫0

D(x)2dxdx, (11)

where ρ is the density and g is the gravitational constant.For analyzing wormlike micelle solutions, the experi-

mental setup with a resolution of 16 µm has been changedusing a mirrorless interchangeable-lens camera (Nikon 1V2) with a 14-megapixel image sensor yielding a resolutionof 2.5 µm. Gentle stretching conditions with hi = 1 mm andhf = 8 mm have to be selected in order to create a filament.A higher initial stretching would not yield a homogenousliquid thread.

Temperature control unit

In order to investigate polymer solutions at different tem-peratures, a special temperature control unit has been con-structed. It consists of a temperature chamber mounted onthe CaBER device as shown in Fig. 1 and a temperaturesensor installed in the CaBER bottom plate next to thesample. Special glass windows allow for capturing the fila-ment shape using our optical setup as described above andrecording the diameter as a function of time using the laser

Rheol Acta (2014) 53:725–739 729

Fig. 1 Picture of the temperature cell mounted on a CaBER device

micrometer of the device. The cell can also act as a sol-vent trap. Therefore, solvent is filled into the cell creatinga saturated atmosphere to prevent evaporation of solventfrom the fluid filament even at elevated temperatures or longfilament lifetimes. In contrast to the oven of Bach et al.(2003) where several electric heating elements have beenused, our temperature control unit is connected to a waterbath thermostat. This allows for a very precise temperatureadjustment between 5 and 90◦C with an accuracy of 0.2◦C.

Supplementary measurements

Zero shear viscosity η0 and the longest relaxation time(according to Eq. 5) were determined from SAOS experi-ments using a Physica MCR501 (Anton Paar, Graz, Austria)equipped with a cone-plate geometry (50 mm diameterand 1◦ cone angle). Surface tension � has been deter-mined at T = 20◦C using a DCAT1 tensiometer (Data-Physics, Filderstadt, Germany) equipped with a platinum-iridium Wilhelmy plate within an experimental error of�� = ±1 mN/m. Density measurements have been per-formed using a pycnometer with a total volume of V =10.706 cm3at T = 20◦C.

Sample preparation and characterization

PS, PEO, and cetylpyridinium chlorid/sodium salicy-late/sodium chloride (CPyCl/NaSal/NaCl) solutions havebeen used as model systems to study the elongational behav-ior of viscoelastic, semi-dilute, and concentrated polymer

solutions (c > c∗). Polystyrene (Polymer Standards Ser-vice, Mainz, Germany) with a weight average molecularweight of Mw = 3 · 106 g/mol and a polydispersity indexof PDI = 1.17 has been dissolved in diethyl phthalate(Merck, Darmstadt, Germany) with a zero shear viscosityof η0 = 12.6 ± 0.3 mPas. Polyethylene oxide (Sigma-Aldrich, MO, USA) with weight average molecular weightof Mw = 1 · 106 g/mol, Mw = 2 · 106 g/mol, andMw = 4 · 106 g/mol has been dissolved in distilled water.Additionally, aqueous solutions of a low molecular weightPEO (Mw = 3.5 · 104 g/mol, in the following labeled asPEG) with concentrations c = 8 % (η0 = 17.3± 0.1 mPas)and c = 16.7 % (η0 = 97.0 ± 0.5 mPas) have been usedas a solvent for the PEO sample with Mw = 1 · 106 g/mol.Aqueous PEO solutions are in the θ -state at room temper-ature. Therefore, adding PEG will not change the solventquality. All polymer solutions were prepared by adding thepolymer powder to the solvent. Samples were homogenizedby means of shaking at room temperature until the solu-tions become totally clear. Long shaking times of approx-imately 1 month were needed for the highly concentratedPS solutions. The surfactant solution CPyCl/NaSal/NaClis common model systems for linear wormlike micelles(WLM) in the entangled state and corresponding linearviscoelastic data are, e.g., given by Berret et al. (1993).In addition to the reptation time scale λrep of covalentlybounded polymers, WLM break and reform with a charac-teristic time λbr . For λbr << λrep, the rheological behavioris described by a single Maxwell model with relaxationtime λs = √

λbrλrep (Rehage and Hoffmann 1988; Cates1987; 1988). The CPyCl/NaSal/NaCl solutions have beenprepared following Berret et al. (1993) at molar surfactantconcentrations between 40 and 120 mM and constant saltsurfactant ratio of R = 0.5. A 0.5-M NaCl solution has beenused as solvent to ensure a constant electrostatic screeninglength while varying the CPyCl and NaSal concentrations.NaSal is strongly binding to the surfactant, and both CPyCland NaSal are responsible for the creation of the wormlikestructures. Therefore, we have calculated the living polymerconcentration c as the sum of the mass concentrations ofCPyCl and NaSal.

Viscoelastic behavior can be observed in a CaBERexperiment if the time scale of the elastically controlledthinning (assumed to be the terminal shear relaxationtime λs) exceeds the viscous time scale (assumed to betv = η0D/2�) quantified using the dimensionless elasto-capillary number (Anna and McKinley 2001a; Clasen2010):

Ec = 2λs�

η0D. (12)

Since the diameter D is a function of time, hence theelasto-capillary number is a function of time, too. Figure 2

730 Rheol Acta (2014) 53:725–739

10-1 100 10110-1

100

101

102

PSPEO1MioPEO2MioPEO4MioCPC/NaSal/NaCl

elas

to-c

apilla

ry n

umbe

r Ec 0 [

- ]

concentration c [%]

Fig. 2 Initial elasto-capillary number as a function of polymer con-centration for solutions (squares), aqueous PEO solutions with Mw =1 · 106 g/mol (circles), aqueous PEO solutions with Mw = 2 ·106 g/mol (diamonds), aqueous PEO solutions with Mw = 4 ·106 g/mol (triangles), and CPyCl/NaSal/NaCl (hexagons)

shows the minimum elasto-capillary number Ec0 calculatedusing the initial diameter of D0 = 6mm for PS, aque-ous PEO, and CPyCl/NaSal/NaCl solutions investigated inthis study. The elasto-capillary number Ec decreases withincreasing polymer concentration in good agreement withthat in the literature (Clasen 2010; Sachsenheimer et al2012) since the zero shear viscosity varies stronger withpolymer concentration than the shear relaxation time. Forlow polymer concentrations, Ec is nearly constant. The crit-ical concentration where Ec begins to decrease is equal tothe entanglement concentration ce determined from shearviscosity data. These values are summarized in Table 1.

CPyCl/NaSal/NaCl solutions investigated here do notshow a constant Ec value for low concentrations, indicatingthat all surfactant solutions are in the entangled state.

However, Ec0 < 1 values may indicate that the begin-ning of the thinning process is controlled by viscous forces

Table 1 Entanglement concentrations ce and intrinsic viscosities [η]

Material Entanglement Intrinsic

concentration ce /% viscosity [η]/cm3 g−1

PS 2.5 295

PEO 1Mio 1.7 598

PEO 2Mio 1.3 1,027

PEO 4Mio 0.66 1,764

instead of viscoelastic as suggested by Clasen (2010) for PSsolutions.

Results and discussion

Effect of stretching ratio on the diameter vs. time curve

Low initial heights hi and high final heights hf are neededfor horizontal filament stretching experiments in order tocreate a measurable deflection of the liquid thread and pre-vent liquid dropping out of the initial gap of the plates.Dramatically changing of the stretching parameter mightinfluence the thinning behavior of a liquid thread. In addi-tion to Miller et al. (2009) and Kim et al. (2010), we studiedthese effects using PEG/PEO solutions. Figure 3 showsexemplarily the diameter vs. time curve for 1 % PEO ofMw = 1 · 106 g/mol dissolved in a 16.7 % PEG solu-tion using different initial (hi = 0.51 mm, hi = 0.75 mm,and hi = 3.01 mm) and final heights (hf ≈ 8 mm,hf ≈ 10 mm, and hf ≈ 12 mm). Filament lifetimetf il increases with increasing initial height hi, hi but sig-nificant differences between hi = 0.51 mm and hi =0.75 mm cannot be observed. The highest filament lifetimesare observed for the lowest stretching ratio (hi = 3.01 mm

and hf = 7.4 mm). Differences in tf il might be due todifferences in the capillary stress σrr (t1) = −2�/D1 rightafter the upper plate has reached the end position. Higherfilament lifetimes correspond to higher D1 values (see,for example, Fig. 3c) and therefore to lower initial radialstresses.

An exponential decrease of the filament diameter isclearly and easily visible allowing for a determination of theelongational relaxation time λe according to Eq. 2. How-ever, for higher polymer concentrations, the onset of theexponential decay regime is shifted to lower diameters andλe can only be determined in the final stage of thinningclose to filament breakup. Finally, at concentrations c > 4 %(Arnolds et al. 2010), the exponential diameter decay isno longer observable. In these solutions, a high force ratioX > 1 is present (Sachsenheimer et al. 2012) during thewhole thinning process. But for solutions with increasedsolvent viscosity (added PEG), λe can be determined fromexponential filament thinning up to 5 % (8 % PEG). Nev-ertheless, all experiments yield an identical value for theelongational relaxation time independent of hi and hf inagreement with those in the literature (Kim et al. 2010;Miller et al. 2009). The diameter Dexp,0 where the fila-ment begins to thin exponentially decreases with increasingstretching ratio. Therefore, the elasto-capillary number Ecat which the exponential decrease sets in is not a materialparameter as suggested by Clasen (2010) but also dependson the choice of hi and hf .

Rheol Acta (2014) 53:725–739 731

Fig. 3 Filament diameter as afunction of time for a 1 % PEOof Mw = 1 · 106 g/mol

dissolved in a 16.7 % PEGsolution with given initial andfinal heights

0.0 0.5 1.0 1.5

10-1

100

101

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5

CB

diam

eter

[mm

]

time [s]

A hi = 0.51 mm hf / mm

8.0 10.1 11.6

time [s]

hi = 0.75 mm hf / mm

8.0 10.2 11.9

hi = 3.01 mm hf / mm

7.4 10.1 11.6

time [s]

Transient force ratio and its relation to the elongationalviscosity

Figure 4 shows typical results for a 2 % PEG/PEO anda 4 % PS solution for the force ratio X as a function of

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

PEG/PEO 2%

PS 4%

force

ra

tio

X [

- ]

normalized time t/tfil [ - ]

exponential decay

Fig. 4 Transient force ratio X as a function of normalized time for a2 % PEO of Mw = 1 · 106 g/mol dissolved in a 16.7 % PEG solutionand a 4 % PS solution. The gray line illustrated the exponential decayin the force ratio fitting (15) with X = X∞ to the experimental data

the normalized time t/tf il , where tf il is the filament life-time. The force ratio decreases exponentially at the earlystate of the thinning process, indicating a characteristic ini-tial relaxation process. The sudden separation of the lowerand upper plates provokes a huge and rapid deformation ofthe liquid thread. The diameter of the filament decreasesfrom the initial diameter D0 = 6 mm to a diameter D1 ≈1 mm observed right after the upper plate has reachedits final position corresponding to an initial Hencky strainof ε0 ≈ 3.6. For a typical strike time of ts = 40 ms,the initial deformation occurs at an average elongation rate< ε0 > = ε0/ts ≈ 90 s−1. This initial step strain resultsin an (initial) axial normal stress σzz which relaxes after thestep strain. For PEO solutions with low polymer concentra-tions c ≤ 2.5 ce, the transient force ratio X(t) levels off toa final value of X∞ = 1.00 ± 0.04 independent of concen-tration. For higher PEO concentrations, the force ratio X∞increases monotonically with polymer concentration andthe exponentially decreasing diameter region is no longerobserved (Arnolds et al. 2010; Sachsenheimer et al. 2012).A force ratio X = 1 corresponds to a vanishing axial nor-mal stress σzz and, therefore, to an axial force F only givenby the surface tension � and the diameter D of the liquidthread.

The PS and CPyCl/NaSal/NaCl solutions show a morecomplex behavior. After the initial decrease, the transientforce ratio levels off to a much lower value of Xmin =0.75 ± 0.05 for PS and CPyCl/NaSal/NaCl independentof the concentration (except of the 1 % PS solution forwhich the force signal is too low and no Xmin could be

732 Rheol Acta (2014) 53:725–739

detected). These values agree well with the theoreticalvalue of XNew = 0.713 for a Newtonian liquid (Papa-georgiou 1995). A force ratio X < 1 corresponds to acompressive axial normal stress σzz in the fluid filament(Sachsenheimer et al. 2012). After passing the Xmin region,the force ratio starts to increase to a final force ratio ofX∞ = 0.98 ± 0.08 for all PS and CPyCl/NaSal/NaCl solu-tions investigated here. These force measurements confirmthe occurrence of an early visco-capillary-controlled regimeand a transition to an elasto-capillary-controlled regimefor PS and CPyCl/NaSal/NaCl solutions as suggested byEntov and Hinch (1997) as well as by Clasen (2010). Thevisco-capillary thinning regime occurs due to the low elasto-capillary numbers of these solutions. However, changingthe solvent viscosity of PEO solutions by adding PEG doesnot cause an early Newtonian thinning regime. The viscos-ity of 16.7 % PEG solution is about seven times higherthan the viscosity of diethyl phthalate but elasto-capillarynumbers Ec0 > 1 are found independent of solventviscosity ηsol .

The elongational viscosity ηe is often calculated asapparent value ηe,app using the σzz = 0 assumption(corresponding to X = 1) according to Eq. 9. Gener-ally, the elongation rate ε changes during capillary thin-ning in CaBER experiments, but it remains constant fora liquid thread with exponentially decreasing diameter.Then, the elongation rate is related to the elongationalrelaxation time according to ε = 2/ (3λe). The nor-mal stresses σzz and σrr (if present) also depend ontime. In consequence, the elongational viscosity ηe orηe,app are transient values even if the elongation rate ε isconstant.

Figure 5 shows exemplarily the true and apparent elon-gational viscosity as a function of time t for the 3 %PEG/PEO and the 4 % PS solution. In the first thinningperiod, ηe decreases with time according to the decreaseof the force ratio X(t), whereas ηe,app increases mono-tonically. As the capillary thinning proceeds further, thetransient elongational viscosity ηe goes through a distinctminimum with ηe,min and finally increases exponentiallyas expected for a Maxwell fluid with a single relaxationtime. This strain hardening is a consequence of the well-known entropy elasticity of polymer chains. During this laststage of thinning, ηe,app and ηe are equal as expected forX = 1. Similar results are found for all solutions inves-tigated in this study. For the PS and CPyCl/NaSal/NaClsolutions, the intermediate regime is controlled by visco-capillary thinning (X = 0.713) as discussed above yieldingan apparent elongational viscosity substantially higher thanthe true value.

The differences between true and apparent elongationalviscosities are caused by the axial normal stress σzz in thefilament which can be expressed in terms of the force ratio

0 1 2 3 4 5 6

102

103

104

PEG/PEO 3%

elongational viscosity

apparent elongational viscosity

elo

ng

atio

na

l vis

co

sity [

Pa

s]

time t [s]

t1 = 40 ms

0.0 0.2 0. 4 0.6 0.8 1.0 1.2

101

102

103

PS 4%

elongational viscosity

apparent elongational viscosity

elo

ng

atio

na

l vis

co

sity [

Pa

s]

time t [s]

t1 = 100 ms

Fig. 5 Elongational viscosity (squares) and apparent elongational vis-cosity (circle) as a function of time for a 3 % PEO of Mw = 1 ·106 g/mol dissolved in a 16.7 % PEG solution (top) and a 4 % PS(bottom) solution

X and the radial normal stress σrr according to the following(Sachsenheimer et al. 2012):

σzz = 2σrr (1 −X) . (13)

Rheol Acta (2014) 53:725–739 733

The ratio of the true elongational viscosity and the apparentelongational viscosity then reads as follows:

ηe

ηe,app= σzz − σrr

−σrr= 2X − 1. (14)

From Eq. 14, it becomes obvious that a force ratio X < 1,as observed for a Newtonian controlled thinning, results inan elongational viscosity smaller than the apparent valueand X > 1, as, e.g., observed during the initial period ofthinning or for highly concentrated PEO solutions (Sachsen-heimer et al. 2012), causes ηe > ηe,app. Both elongationalviscosities (true and apparent) are only equal if the forceratio is X = 1.

In addition, the uniaxial elongational viscosity of poly-mer solutions can be obtained from filament stretchingextensional rheometer (FiSER) measurements. In such anexperiment, the liquid under test is also placed between twoparallel plates, but the upper plate is separated continuouslyuntil the filament breaks (Tropea et al. 2007). Generally, twotypes of velocity profiles are possible (Kolte et al. 1997):The upper plate is separated with an exponentially increas-ing velocity or the plate velocity is controlled, such that thefilament diameter decreases exponentially corresponding toa constant elongational rate during the whole experiment(Kolte et al. 1997; Tropea et al. 2007). The transient elonga-tional viscosity is computed from the tensile force exertedby the fluid column on the endplate and the filament diam-eter at the axial midpoint (McKinley et al. 2001). Moredetails about FiSER are given by Tirtaatmadja and Sridhar(1993); Spiegelberg et al. (1996); Anna et al. (1999, 2001b);Orr and Sridhar (1999); McKinley et al. (2001); McKinleyand Sridhar (2002); Rothstein and McKinley (2002a;2002b); Rothstein (2003), and Tropea et al. (2007).

However, FiSER measurements yield higher values forthe maximum elongational viscosity ηmax

e than CaBER(Bhardwaj et al. 2007a). Similar results are obtained forCPyCl/NaSal/NaCl solutions investigated in this study (datanot shown). The observed differences between both tech-niques may be explained by the simplified CaBER analysisusing the σzz = 0 assumption. But as shown above, X = 1was found the final period of thinning for WLM solutions.Therefore, the σzz = 0 assumption is fulfilled and can-not be the reason for the differences in ηmax

e . It seemsto be more likely that differences in strain history andfailure mechanism affect the material behavior in elon-gational flow. Indeed, FiSER experiments of the 2.94 %CPyCl/NaSal/NaCl solution show a very high maximumforce ratio of XFiSER∞ = Dminσmax/4� ≈ 60 and a diame-ter Dmin ≈ 1 mm at filament rupture, clearly indicating thedifference in the thinning mechanism.

Figure 6 shows the minimum viscosity ratio T rmin =ηe,min/η0 for the investigated polymer solutions as a func-tion of polymer concentration c. The viscosity ratio T rmin

0 1 2 3 4 5 6100

101

102

PS PEG/PEO PEO 1Mio CPyCl/NaSal/NaCl

visc

osity

ratio

Tr m

in [

- ]concentration c [%]

Newtonian limit Tr = 3

Fig. 6 Minimum viscosity ratio T rmin = ηe,min/ηs as a function ofconcentration for PS (squares), PEO of Mw = 1 · 106 g/mol dis-solved in a 16.7 % PEG solution (circles), aqueous PEO of Mw =1·106 g/mol (triangles), and CPyCl/NaSal/NaCl (diamonds) solutions

decreases exponentially with increasing polymer concen-tration c according to T rmin ∝ exp (−0.74 ± 0.02 c/%)

and levels off at a constant value for polymer concentra-tions c > ce for all solutions including CPyCl/NaSal/NaCl.For PEO solutions with Mw = 1 · 106 g/mol, T rmin isbetween 60 and 70 for low concentrations and levels off toT rmin = 9.5 ± 0.8. Obviously, the initial step strain resultsin a nonlinear change of polymer configuration which doesnot relax and capillary thinning does not start from the equi-librium where Tr = 3 would be expected. Increasing thesolvent viscosity results in a slight increase of T rmin forhigh PEO concentrations (T rmin = 13.4 ± 0.8 for PEOwith Mw = 1 · 106 g/mol dissolved in an aqueous solutionof 16.7 % PEG) and in a similar shift of the concentra-tion at which the limiting value of T rmin is reached. Incontrast, for PS and CPyCl/NaSal/NaCl solutions, Tr levelsoff at T rmin = 3, indicating a relaxation of initial stressesand a pure Newtonian response for high polymer concentra-tions before the beginning of the elasto-capillary thinningprocess. Nevertheless, at low concentrations, T rmin >> 3is found even though an intermediate Newtonian thinningregime is observed for these solutions.

However, analyzing the minimum Trouton ratio T rmin

seems to be useful due to two reasons. The determinationof the minimum elongational viscosity ηe,min is very con-venient, whereas a determination of the maximum elonga-tional viscosity or maximum Trouton ratio often calculated

734 Rheol Acta (2014) 53:725–739

for surfactant solutions (Bhardwaj et al. 2007a, b) includeslarge experimental uncertainties. All solutions (includingthe CPyCl/NaSal/NaCl surfactant solutions) investigated inthis study have thin to very tiny diameters which are hard tobe determined accurately. The minimum elongational vis-cosity also indicates a transition in the thinning behavior. Inthe initial stage of the experiment, the thinning behavior iscontrolled by an initial relaxation of the axial normal stress(decrease of ηe) followed by a pronounced increase of theradial normal stress (increasing of ηe) at the final stage ofthinning.

Elongational relaxation time

In addition to the determination of elongational viscosities,CaBER experiments also allow for measuring the elon-gational relaxation time λe. Commonly, λe is determinedfrom the exponential diameter decay of the liquid filamentundergoing capillary thinning according to Eq. 2. Figure 7shows three independently determined elongational relax-ation times for the PEO (Mw = 1 · 106 g/mol) and PSsolutions investigated in this study: the elongational relax-ation time λe determined from the diameter vs. time curveof a vertically stretched filament, the elongational relax-ation time λe determined from the diameter vs. time curveof a horizontally stretched filament, and the force relaxationtime λX calculated from the initial exponential decay of thetransient force ratio in the initial thinning period. Althoughthe filament diameter does not decrease exponentially in thisregime, the force ratio follows an exponential decrease intime according to

X (t) = C exp

(− t

λX

)+ X, (15)

where C is a constant and X is the value of the force ratioafter the initial decay (X ≡ X∞ ≈ 1 for PEO solutionsand X ≡ Xmin ≈ 0.75 for PS and CPyCl/NaSal/NaClsolutions). A representative result as fit of Eq. 15 to exper-imental data for 2 % PEO dissolved in the 16.7 % PEGsolution is shown as gray line in Fig. 4. We also analyzedthe influence of the strike time 20 ms < ts < 480 ms on theforce relaxation time as shown in the inset of Fig. 7 (top)for the aqueous 4 % PEO (Mw = 1 · 106 g/mol) solu-tion. In the investigated parameter range λX is found to beindependent of the strike time. This indicates that the initialrelaxation of the force ratio X(t) is not influenced by iner-tia effects as expected since Reynolds numbers defined asRe = ρvD0/η0, where v is the velocity of the upper plateduring step strain v = (

hf − hi)/ts , are less than one. For

the 4 % PEO solution (inset in Fig. 7) with a zero shearviscosity of η0 = 8.6 Pas, the corresponding Reynoldsnumbers are between 0.016 and 0.38 for strike times vary-ing from 480 and 20 ms. However, considering our standard

1 2 3 4

10

100

10 100

100

200

x [m

s]

strike time ts [ms]

X

e tilted CaBER

e Arnolds et. all 2010

rela

xatio

n tim

e [m

s]

concentration c/ce [ - ]

PEO

1

10

100

PSX

e tilted CaBER

e upright CaBER

rela

xatio

n tim

e [m

s]

concentration c/ce [ - ]

Fig. 7 Elongational relaxation times from CaBER experiments foraqueous PEO of Mw = 1 · 106 g/mol (top) and PS solutions (bot-tom). Relaxation time values for PEO solutions in the upright CaBERexperiment are taken from Arnolds et al. (2010). The inset diagramrepresents the force ratio relaxation time as a function of the strike timefor a 4 % PEO solution. Lines are guide to the eye

setup with ts = 40 ms, the maximum Reynolds numbersof Re = 4 are found for solutions where a force relaxationtime could be determined. But even in these cases, inertiaeffects may be neglected since the filaments are stabilizedelastically (Sachsenheimer et al. 2012).

Rheol Acta (2014) 53:725–739 735

Nevertheless, too high strike times should not be chosenbecause a superposition of stretching and capillary thinninghas to be avoided. Therefore, we propose an appropriatestrike time of ts < 0.1 tf il .

All three elongational relaxation times agree well withinexperimental error irrespective of the polymer concentrationas long as X∞ = 1. The impressive agreement between λeand λX suggests that the first stage of capillary thinning,where the diameter does not thin exponentially, is controlledby the same relaxation process as the terminal thinningregime with its exponential filament decay. The excellentagreement between all three relaxation times also shows thatgravity has no influence on the elasto-capillary thinning ofviscoelastic fluids demonstrating the equivalence of uprightand tilted CaBER experiments.

Furthermore, the elongational relaxation times increasemonotonically with increasing polymer concentration c.The force relaxation times λ of PEO solutions with concen-trations c ≥ 4 % (corresponding to c > 2 ce) show a strongerincrease with concentration c than elongational relaxationtimes for concentrations c < 2 ce. However, a determinationof λ was not possible for these solutions due to the absenceof an exponentially decreasing diameter region (Arnolds elal. 2010). These solutions also exhibit a final force ratioX > 1 (Sachsenheimer et al. 2012).

Correlation between shear and elongational relaxation time

In Fig. 8, the relaxation time ratio λe/λs is shown as afunction of polymer concentration c. Data for PEO solu-tions with different molar masses and solvent viscosities aswell as PS solutions at different temperatures are displayed.In all cases, the relaxation time ratio decreases exponen-tially with increasing polymer concentration c covering upto 2 orders of magnitude. PEO solutions with higher PEOmolar mass show lower λe/λs values (at constant polymerconcentration c) and a stronger dependence on polymer con-centration. Investigations on PEO solutions shows that thesolvent viscosity does not affect the relaxation time ratioλe/λs . Varying the temperature of the PS solution between10 and 40◦C corresponds to a viscosity as well as a shearand elongational relaxation time change but does not affectthe relaxation time ratio. Slight changes in surface ten-sion or solvent quality upon temperature variation are notconsidered to be relevant for the λe/λs ratio.

Obviously, changing the solvent viscosity or the temper-ature affects the relaxation of dissolved polymer moleculesin the same way for elongational and shear flows. The elon-gational relaxation time λe, therefore, is a characteristicparameter of the polymer solution but not equal to the ter-minal shear relaxation time λs . Comparing λe to a meanshear relaxation time λs as suggested by Liang and Mack-ley (1994) according to Eq. 3 requires the knowledge of

Fig. 8 Relaxation time ratio as a function of concentration for differ-ent polymer solutions with different solvent viscosities and tempera-tures as shown in the legend of the diagram

the relaxation time spectrum which cannot be determinedunambiguously. However, the PS solutions investigated hereare thermo-rheologically simple so that the viscosity func-tion η (ω) of all investigated PS solutions (independent ofconcentration or temperature) can be plotted as dimension-less master curve by scaling the viscosity with the zeroshear viscosity η0 and the angular frequency with the ter-minal shear relaxation time λs . Accordingly, all relaxationtimes scales with λs and any average relaxation time like,e.g., λs , exhibit the same dependence on concentration ortemperature. As a consequence, the ratio of λe and anyaverage relaxation time should be constant independent ofconcentration or temperature. The strong decrease of λe/λsmust therefore be related to a nonlinear material property asalready suggested by Arnolds et al. (2010).

Finally, it is interesting to note that the exponentialdecrease of λe/λs with increasing concentration seems to bea universal feature of entangled polymer solutions. Figure 9displays the λe/λs values as a function of normalized con-centration c [η], where [η] is the intrinsic viscosity of thepolymer solution, for solutions of PEO and PS with differentmolecular weights, solvent viscosities, and temperatures.Obviously, all data collapse onto a single master curve whenplotted as a function of c [η] and follow a unique exponentialdecay.

For comparison, we have investigated a series ofCPyCl/NaSal/NaCl solutions which represent simpleMaxwell fluids characterized by one relaxation time (Cates1996; Berret et al. 1993) as shown in the inset of Fig. 10

736 Rheol Acta (2014) 53:725–739

0 10 20 30 40 5010-2

10-1

100

101

PS PEO 1Mio 10°C 0% PEG 20°C 8% PEG 30°C 16.7% PEG 40°C

PEO 2Mio 0% PEG 8% PEG

PEO 4Mio

rela

xatio

n tim

e ra

tio

e/s [

- ]

normalized concentration c[ ] [ - ]

Fig. 9 Relaxation time ratio as a function of normalized concentrationfor polymer solutions shown in Fig. 8

where G′ and G′′ data for two CPyCl/NaSal/NaCl solu-tions are shown. Solid lines represent the best fit of the

ω

±

1 2 3 4 5 610-1

100

101

e/ s = 1.17 0.05

10-1 100 10110-2

10-1

100

101

102

c = 1.64%

[1/s]

G', G'' [Pa]

c = 4.62%

rela

xatio

n tim

e ra

tio

e/s [

- ]

concentration c [%]

CPyCl/NaSal/NaCl

Fig. 10 Relaxation time ratio as a function of concentration for dif-ferent CPyCl/NaSal/NaCl solutions. The line represents the relaxationtime ratio mean value of λe/λs = 1.17 ± 0.05. The inset showsthe storage (squares) and loss modulus (circles) as a function ofangular frequency for the 4.42 % (filled symbols) and the 1.64 %CPyCl/NaSal/NaCl solution (open symbols). Best fits of experimen-tal data, using a single Maxwell model, are represented in the inset assolid lines

Maxwell model to the experimental data. The relaxationtime ratio for these simple Maxwell fluids is shown inFig. 10. Surprisingly, all solutions show a constant value ofλe/λs = 1.17± 0.05 independent of concentration, indicat-ing that the elongational behavior of WLM solutions is onlygiven by linear fluid properties determined in small ampli-tude oscillatory shear. This result is remarkable because therelaxation time of a wormlike structure strongly depends onthe mean length of the micelles. Finding λe ≈ λs suggeststhat the mean length does not change during the elonga-tional deformation as discussed theoretically (Vasquez et al.2007; Cromer et al. 2009; Germann et al. 2013). A changein micellar length due to a deformation-induced increase ofthe breakage rate would dramatically shorten the relaxationtime and λe/λs << 1 would be expected.

On the other hand, λe ≈ λs implies that nonlinear effectsare not relevant in CaBER experiments of these surfac-tant solutions. Identifying the shear relaxation time λs ascharacteristic relaxation time of the material and consider-ing ε = 2/ (3λe), the relaxation time ratio λe/λs mightbe interpreted as an inverse Weissenberg number Wi−1 =(ελs)

−1 = 3λe/ (2λs). For CPyCl/NaSal/NaCl solutions,Wi = 0.6 is independent of surfactant concentration indicat-ing a linear viscoelastic response. Unfortunately, the modelof Arnolds et al. (2010) taking into account the strong defor-mation during capillary thinning cannot be applied here.The damping function cannot be determined from inde-pendent steady shear experiments since shear banding isprominent in these flows (Bhardwaj et al. 2007a, b).

Conclusion

We have investigated capillary thinning of entangled poly-mer solutions focusing on weakly elastic systems showingexponential diameter decay.

The tilted CaBER method has been used to determinethe force ratio X and the true elongational viscosity ηe. Theforce ratio decays exponentially and levels off at X∞ = 1for the PEO solutions with high elasto-capillary numbersEc0. However, for the PS and CPyCl/NaSal/NaCl solutionswith Ec0 ≤ 1, an intermediate Newtonian thinning regimewas confirmed based on the measured force ratio X = 0.713as already suggested by Clasen (2010). The initial stressdecay after step strain filament formation is characterizedby a relaxation time λX equal to the relaxation time λedetermined from the terminal exponential thinning region.Obviously, the corresponding initial decrease of the elon-gational viscosity ηe is controlled by the same molecularprocesses like the increase of ηe (strain hardening) in thefinal regime of thinning. Since ηe goes through a distinctminimum during filament thinning, a characteristic min-imum Trouton ratio T rmin can be determined accurately.

Rheol Acta (2014) 53:725–739 737

This Trouton ratio decreases exponentially with increas-ing polymer concentration for all investigated solutions andlevels off at T rmin = 10 for all PEO solutions irrespec-tive of solvent viscosity. For the PS and CPyCl/NaSal/NaClsolutions characterized by low elasto-capillary numbers,T rmin = 3 is found.

The relaxation time ratios λe/λs decreases exponentiallywith increasing polymer concentration and the data for allinvestigates PEO and PS solutions collapse onto a singlemaster curve irrespective of polymer molecular weight, sol-vent viscosity, or temperature when plotted versus reducedconcentration c [η]. This decrease is due to the strong non-linear deformation in CaBER experiments as suggested ear-lier (Arnolds et al. 2010) but not due to a different weightingof relaxation times in the different flow kinematics.

On the other hand, λe ≈ λs is found for all investigatedCPyCl/NaSal/NaCl solutions, clearly indicating that nonlin-ear effects are not relevant in capillary thinning of these“living” polymer systems.

Acknowledgments The authors would like to thank Sonja Muller,Sebastian Bindgen, and Frank Bossler for their help in sample prepara-tion and performing the experiments. We would like to thank JonathanRothstein and Sunil Khandavalli (University of Massachusetts) for thepossibility to use the FiSER setup and for all help given. Financialsupport by German Research Foundation DFG grant WI 3138/13-1 isgratefully acknowledged.

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