Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx

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Journal of Non-Newtonian Fluid Mechanics

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On the ‘‘viscosity overshoot’’ during the uniaxial extension of a lowdensity polyethylene

Teodor I. Burghelea a,⇑, Zdenek Stary b, Helmut Münstedt b

a Université de Nantes, Nantes Atlantique Universités, CNRS, Laboratoire de Thermocinétique de Nantes,UMR 6607, La Chantrerie, Rue Christian Pauc, B.P. 50609, F-44306 Nantes Cedex 3, Franceb Institute of Polymer Materials, Friedrich-Alexander-University, Martensstrasse 7, D-91058 Erlangen, Germany

a r t i c l e i n f o

Article history:Received 13 April 2011Received in revised form 1 July 2011Accepted 15 July 2011Available online xxxx

Keywords:Uniaxial extensionViscosity overshootHomogeneity of deformation

0377-0257/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jnnfm.2011.07.007

⇑ Corresponding author.E-mail address: Teodor.Burghelea@univ-nantes.fr

Please cite this article in press as: T.I. BurgheleaNewtonian Fluid Mech. (2011), doi:10.1016/j.jn

a b s t r a c t

An experimental investigation of the viscosity overshoot phenomenon observed during uniaxial exten-sion of a low density polyethylene is presented. For this purpose, traditional integral viscosity measure-ments on a Münstedt-type extensional rheometer are combined with local measurements based on thein-situ visualization of the sample under extension. For elongational experiments at constant strain rateswithin a wide range of Weissenberg numbers (Wi), three distinct deformation regimes are identified. Cor-responding to low values of Wi (regime I), the tensile stress displays a broad maximum, but such max-imum is observed with various polymeric materials deformed at low rates and it should not be confusedwith the ‘‘viscosity overshoot’’ phenomenon. Corresponding to intermediate values of Wi (regime II), alocal maximum of the integral extensional viscosity is systematically observed. Moreover, within thisregime, a strong discrepancy between integral measurements and the space average of the local elonga-tional viscosity is observed which indicates large deviations from an ideal uniaxial deformation process.Images of samples within this regime reinforce this finding by showing that, corresponding to the max-imum of the integral viscosity, secondary necks develop along the sample. The emergence of a maximumof the integral elongational viscosity is, thus, related to the distinct inhomogeneity of deformation statesand most probably not to the rheological properties of the material. In the fast stretching limit (high Wi,regime III), the overall geometric uniformity of the sample is well preserved, no secondary necks areobserved and both the integral and the space averaged transient elongational viscosity show nomaximum. A detailed but yet incomplete comparison of the experimental findings with results fromthe literature is presented and several open questions are stated.

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1. Introduction

Uniaxial extension is the dominant flow in many industrial pro-cesses and, therefore, accurate measurements of the extensionalrheological properties of polymer melts are very important. Froma more fundamental point of view, reliable measurements of therheological properties in extension are crucial for validating exist-ing theoretical models and suggesting new approaches. In spite ofthe universally recognized need for reliable elongational measure-ments of polymer melts, the development of extensional rheomet-ric equipment has progressed slowly during the past three decades.A reliable design of an extensional rheometer has met several prac-tical difficulties and one of the toughest is to generate a homoge-neous extensional flow. Several techniques to measure theelongational properties of polymer melts have been proposed: the

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et al., On the ‘‘viscosity oversnfm.2011.07.007

Rheometrics Melt Extensiometer (RME) by Meissner [9], the sup-porting oil-bath design by Münstedt [10], the Sentmanat exten-sional rheometer (SER) [21,20] and the Filament Stretching device(FISER), [23,1]. A comprehensive review of these different ap-proaches to extensional rheology of polymer melts is beyond thescope of this investigation and can be found in [19] and more re-cently in [25]. We note that for each of these approaches the homo-geneity of deformation states is crucial for reliably assessing theelongational properties of the material. Though previously recog-nized by most experimentalists, it is our belief that this issue didnot receive the proper attention. Only very recently the true dangerof sample non-uniformity during elongation has been made expli-cit, by measuring locally both the stresses and the strain and show-ing that in the case of strongly nonuniform samples the classicalextensional measurements become completely unreliable, [3]. In[3] it has been demonstrated by combined traditional integral vis-cosity measurements and local viscosity measurements based onin-situ local measurements of the sample diameter that geometricnon-uniformities of the sample under elongation typically result in

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

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unreliable viscosity data. Moreover, it has been shown that eveninitially homogeneous (perfectly cylindrical) samples loose theiruniformity at high enough Hencky strains and thus, the impact ofsample non-homogeneity on the viscosity measurements is alwaysan issue to worry about during extensional tests. Also very recently,a full numerical simulation of the extension process in a SER rheom-eter demonstrated that the loss of sample homogeneity duringdeformation leads to a strong strain localization along the samplewhich ultimately translates into unreliable measurements of thetransient elongational viscosity [31,6].

Here a more elaborated version of the method proposed in [3] isemployed to investigate a long standing problem in the exten-sional rheology of polymer melts, the viscosity (or tensile stress)overshoot observed during the uniaxial extension of some polymermelts at a constant rate of deformation.

Since the early days of extensional rheometry it has been ob-served that some strain hardening materials under uniaxial exten-sion display a clear maximum in the transient extensional viscosityright before (typically within less than a Hencky strain unit) thephysical rupture of the sample [5,15,9,12]. There is only one exper-imental paper we are aware of [16], which also presents an ex-tended (over several Hencky strain units prior to the physicalrupture of the sample) plateau after the stress maximum. Duringthe experiments presented in [16] the geometric uniformity ofthe sample has been locally monitored by means of a laser sheetwhich images a (presumably) small portion of the sample aroundits middle, measures in situ the mid-point point diameter of thesample and passes this information to a feedback loop which main-tains a constant rate of deformation at the mid-point of the sample(this procedure is explained in detail in [1]). However, the authorsof this study present no experimental evidence (based on videoimaging of the entire sample under deformation) on the homoge-neity of the sample during the elongation process but monitorthe sample homogeneity only in a finite region around its middleusing a laser technique described in detail in [1].

A local maximum in the tensile stress (transient elongationalviscosity), followed or not by a plateau, has been coined as ‘‘viscos-ity overshoot’’. The existence of a true viscosity overshoot is impor-tant from both a practical and a fundamental point of view. Inmany extension dominated processing and industrial operationssuch as extrusion, film blowing, melt spinning and thermoformingit is important to know whether a true steady state behavior can bereached under extension at a constant rate and if not, to under-stand how this fact influences the physical rupture of the material.From a theoretical point of view, in our opinion, this phenomenonis not yet fully understood. The POM-POM model for branchedpolymer melts [8] and the molecular stress function (MSF) model[28] predict a monotone increase of the transient extensional vis-cosity. Other theoretical works, however, are able to predict anovershoot in viscosity, [26,27]. Even more worrying, recent theo-retical models seem to be able to fit both a maximum and/or asteady state of the transient extensional viscosity [27]. This simplymeans that the phenomenology behind the stress maximum/over-shoot remains elusive.

The implications of the stress overshoot phenomenon duringuniaxial extension are, in our opinion, even more important. Re-cently, based on the observation of the stress overshoot phenome-non in both uniaxial extension of polymer melts [2,30] and startupshear of entangled polymer solutions [17], a universality claim‘‘Entangled liquids are solids’’ has been very recently formulated[29] for the case of shear flows. Though we do understand howimportant and appealing a universal behavior is and we do acceptthat some similarities between polymer melts under elongationand entangled solutions in startup shear may exist, we believe thatthe claim above should be still considered very cautiously, at leastbecause of the reasons below:

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1. The term ‘‘overshoot’’ implies, to our best understanding, a localmaximum followed by a plateau corresponding to lower values.Whereas such a plateau for the tensile stress has been observedfor entangled solutions [17], the data concerning polymer meltspresented in [2,30] display only a maximum but no plateau: thesample breaks just after the maximum. A true overshoot behav-ior (a maximum followed by a plateau) has been observed in[16] but at much higher Hencky strains.

2. The stress overshoot during the uniaxial extension of entangledpolymer melts reported in [2,30] refers to the so-called ‘‘engi-neering stress’’ which is not a real stress but just a tensile forcenormalized by a constant (the initial area of the sample underinvestigation). The (physical) true stress (which is calculatedby dividing the tensile force by the actual cross section of thesample) does not always exhibit an overshoot behavior and ifit does (e.g. for the strain hardening materials at high enoughrates of deformation) this occurs at significantly larger Henckystrains.

In the view of the remarks above, we believe that before ananalogy between the stress maximum (and only rarely a true stressovershoot, [16]) observed for polymer melts under extension andthe stress overshoot observed for entangled polymer solutions instartup shear is stated, a deeper understanding of each phenome-non is needed.

2. Description of the experiments

2.1. Experimental apparatus and techniques

The experiments have been conducted with a Münstedt typeextensional rheometer built in the house which is illustrated inFig. 1(a). A detailed description of this device can be found else-where, [11]. The specimen S under investigation is clamped be-tween the plates P1 and P2 of the rheometer and immersed in asilicone oil-bath C to minimize gravity and buoyancy effects,Fig. 1(a).

While the bottom plate P2 is stationary, the top plate P1 is movedvertically by an AC-servo motor M, controlled by an analogue to dig-ital converter installed on the computer PC1. The sample is illumi-nated from behind by two linear light sources LS1 and LS2

disposed as shown in the schematic top view presented inFig. 1(b). The idea behind the back-light illumination arrangementis to obtain a maximum of brightness only on the edges of the sam-ple and thus to allow accurate identification of the sample edges andreliably measure its diameter. A major difficulty in imaging a con-siderably elongated sample comes from the high aspect ratio(height to width) of the corresponding field of view, which duringextensional experiments at large Hencky strains may be as largeas 50:1. If a regular entocentric lens (with the entrance pupil locatedinside the lens) is used both the resolution and the level of geomet-rical distortion are unsatisfactory for high accuracy measurementsof the sample diameter. Additionally, corresponding to large Henckystrains, both the frame brightness and the degree of focusing be-come uneven through the field of view if the sample is imaged indivergent light. To circumvent these problems, we use in our studya high resolution telecentric lens with the entrance pupil located atinfinity, (VisionMes 225/11/0.1, Carl Zeiss) which images the sam-ple in parallel light and delivers frames with very uniform bright-ness and free of distortions (geometrical aberrations), perspectiveerrors and edge position uncertainties. A typical image of the sam-ple is presented in Fig. 1(c). Images of the sample under elongationare acquired in real time using a high resolution (3000 by 1400 pix-els full frame, which translates into roughly 60 mum spatial resolu-tion) low noise camera (Pixelink from Edmunds Optics) at a speed of

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

Fig. 1. (a) Schematic view of the experimental apparatus: C – oil-bath, P1 and P2 – top and bottom plates of the rheometer, S – the sample under investigation, M – AC servomotor, D – the control drive of the rheometer, PC1;2 – personal computers, TL – telecentric lens, CCD – video camera. (b) Sample illumination and imaging: LS1 and LS2 – linearlight sources, S – the sample under investigation. (c) Example of a telecentric sample image corresponding to �H ¼ 2:7. The field of view was actually larger but the image hasbeen cropped for clarity reasons. (d) Principle of the local measurements of the extensional viscosity. The vertical dotted lines represent the contour of an ideal uniformsample.

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3 frames per second. The video camera is installed on a second com-puter, PC2. The image acquisition is digitally synchronized with therheometer via a transistor–transistor logic (TTL) trigger signal sentby the rheometer drive D to the camera.

2.2. Materials and their rheological properties

The material used in this study is a low-density polyethylenefrom Lyondell Bassel with the trade name Lupolen 1840 D. Severalmolecular and rheological characteristics of the material are sum-marized in Table 1. LDPE 1840 D has a branched molecular struc-ture which has been systematically characterized by Nordmaierand co-workers [13,14]. It has a broad molar mass distributionwith a rather large molar mass, Mw, and a pronounced high molarmass tail.

The influence of the broadly distributed molar mass and thebranched molecular structure of the material on its rheologicalproperties in shear has been recently investigated experimentally[18]. Due to the broadly distributed molar mass and the degreeof chain branching, the maximum relaxation time of the material,

Table 1Molecular and rheological characteristics of Lupolen 1840 D at T = 140 �C. k ¼ J0

eg0 isthe largest polymer relaxation time.

Mwðkg=molÞ Mw=Mn g0ðPa sÞ J0e ð10�4Pa�1Þ k (s)

377 18 8,33,000 13.5 1100

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k, is quite high. It is calculated as k ¼ J0e � g0, where J0

e and g0 are thesteady-state recoverable compliance and the zero shear viscosity,respectively.

2.3. Preparation of the samples

For elongational measurements in the Münstedt rheometercylindrical specimens were used. The main batch of samples inves-tigated has been prepared as follows. At first a strand was extrudedthrough a capillary at 190 �C using a piston extrusion machine. Thediameter D of the die was 4.6 mm, the length L = 18.4 mm, and theapparent shear stress applied was 43.7 kPa. These extrusion condi-tions give rise to a strand with a diameter of about 8 mm afterannealing. The strand was extruded into a vessel containing an eth-anol–water mixture (90/10 vol. %) in order to ensure homogeneousstrand diameter along the axes of the extrusion. This procedureleads to specimens with a relative deviation of a diameter smallerthan 2%. After extrusion the strand was annealed in a silicone oil-bath at 150 �C for 20 min. This step ensures a complete relaxationof the thermal stress accumulated within the sample, which is nec-essary in order to suppress any stress history effects and obtainreliable and accurate rheological data. The initial length and diam-eter of each sample were D0 = 8 mm and L0 = 5 mm, respectively.The specimen’s surface was etched by air plasma at room temper-ature in order to increase its surface energy. The samples wereglued to aluminum clamps using a two-component epoxy resinadhesive, Technicoll 8266/67. These clamps serve to fix the

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specimen to the pulling rod and the force transducer of the rheom-eter. At last, the specimens were kept in an oven at 80 �C for 2 h fora complete curing of the glue.

2.4. Data analysis

The first step of our data analysis procedure was to interpolateboth the image sequence and the data acquired by the Münstedtrheometer on a common time axis, so a direct comparison betweenthe integral viscosity measurements and the shape of the sampleunder deformation can be made. Prior to analysis, each imagehas been compensated for non uniform brightness using a standardadaptive histogram equalization algorithm implemented underMatlab�. By identifying the edges of the sample from each image,the distribution of diameters along the actual length of the sampleis measured. This allows a statistical analysis of the stress distribu-tion along the sample under extension. The axial stress distributionis characterized by the first order of the distribution (the mean)and the second order moment of the distribution (the root meansquare deviation, rms).

Fig. 2. Transient elongational viscosities at various rates of deformation: (1) –_� ¼ 0:002 s�1, (2) – _� ¼ 0:015 s�1, (3) – _� ¼ 0:02 s�1, (4) – _� ¼ 0:025 s�1, (5) –_� ¼ 0:09 s�1. Each data set has been acquired until the physical rupture of thesample occurred. The data were measured at T = 140 �C.

3. Results

3.1. On the relation between a maximum in the tensile stress andsample uniformity

A first argument that the maximum of the elongational viscos-ity may not be a real material property but due to an experimentalartefact was published by Münstedt et al. [11]. In this paper it wasfound that for the same LDPE discussed here (Lupolen 1840 D) theviscosities obtained from the experimentally proven steady state increep experiments (Fig. 8 in [11]) agree very well with those foundfrom the maxima in stressing measurements (Fig. 9 in [11]). As thegeometric homogeneity of deformation states was well preservedduring creep experiments in a wide window of Hencky strains[11], this fact suggests that the viscosity decrease observed duringstressing experiments might be related to a severe necking of thesample at high elongation.

We address in the following a question which is decisive for afundamental understanding of the elongational behavior of poly-mer melts and its theoretical description:

Is a maximum of the transient tensile stress compatible with ahomogeneous deformation process?

To address this question, we focus in the following on a uniaxialdeformation process at constant rate, _� ¼ 1

LdLðtÞ

dt . Here L(t) and L0

stand for the actual and initial length of the sample, respectively.By uniform deformation we understand an uniaxial deforma-

tion process for which the diameter of the sample, D(t), is constantalong the actual length of the sample L(t) (see Fig. 1(d)) and, basedon the incompressibility condition, is given by DuðtÞ ¼ D0e� _�t=2. TheMünstedt rheometer measures an integral value of the tensilestress defined as ruðtÞ ¼ 4FðtÞ

pD2uðtÞ

and the transient elongational vis-cosity is further assessed as lþu ðtÞ ¼

ruðtÞ_� . Here F(t) stands for the

tensile force which is measured by the force transducer installedon the bottom plate P2 of the oil-bath rheometer. Such integralmeasurements are reliable only if the deformation of the sampleis homogeneous. If the sample deforms inhomogeneously, boththe rate of deformation and the tensile stresses become stronglylocalized along the sample and consequently, measurements oflþu ðtÞ become highly unreliable, [3].

Corresponding to a local maximum of the transient elongationalviscosity, the condition dlþu ðtÞ

dt ¼ 0 has to be fulfilled. If one assumesthat the deformation is homomgeneous, the diameter of the sampleand the rate of deformation do not depend on the axial coordinate zand, by an elementary algebraic manipulation, the viscosity maxi-

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mum condition can be rewritten as, dFðtÞdt þ FðtÞ _� ¼ 0. According to

this last equation, if the sample deforms homogeneously, in theneighborhood of a local maximum of the transient elongational vis-cosity, the tensile force should scale exponentially, FðtÞ / exp ð� _�tÞ.

The validity of the analytical condition for a maximum in stressis discussed detail in the next Section, 3.2.

3.2. The transient tensile force and tensile stress in different regimes ofextension

Integral measurements of the transient elongational viscosityl+(t) at a constant rate of deformation _� are presented in Fig. 2.The integral viscosity is obtained by measuring the transient ten-sile force, F(t), using the assumption that the diameter of the sam-ple is independent of the vertical coordinate, D(z, t) = Du(t). Eachdata set has been acquired until the physical rupture of the sampleoccurred. Except for the linear range of deformation, �H < 1, theshape of the transient elongational viscosity depends considerablyon the (constant) rate at which the material is deformed, _�. Thus,depending on the rate of deformation, the integral transient viscos-ity may display either a clear maximum (curves 1–4, Fig. 2) or amonotonic increase (curve (5), Fig. 2). As suggested by the simplediscussion presented in Section 3.1, in order to understand thephysical reasons underlying the viscosity maximum visible forthe curves (1–4), one has to focus not only on the tensile stressbut on the tensile force as well.

In Fig. 3 the transient tensile forces and stresses measured forthree different values of the Weissenberg number, Wi, are pre-sented. The Weissenberg number is defined as Wi ¼ _� � k. Corre-sponding to Wi = 1.1, the tensile stress displays a broad maximum,Fig. 3(a).

In order to connect the emergence of the stress maximum withthe discussion presented in 3.1, we need to discuss the homogene-ity of the sample around the stress maximum. The force maximumvisible in Fig. 3(a) (which corresponds to a shoulder in the tran-sient tensile stresses) represents the onset of a primary non-unifor-mity of the specimen, as predicted by the Considère criterion, [4,7].Such geometric non-uniformity of the sample (initially localizednear the plates P1;2 of the rheometer) occurs in most of the exten-sional experiments with materials of different kinds and it is re-lated to the rigid boundary conditions near the clamping pointsof the sample under investigation. Thus, the emergence of this ef-fect depends little on the molecular structure of the material: it can

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

Fig. 3. Transient tensile forces and stresses corresponding to different regimes of extension: (a) _� ¼ 0:001 s�1ðWi ¼ 1:1Þ, (b) _� ¼ 0:05 s�1ðWi ¼ 55Þ, (c)_� ¼ 0:3 s�1ðWi ¼ 330Þ. The dash-dotted lines are e� _�t . The insets in panels (a and b) display the images of the sample corresponding to the stress maximum and the insetin panel (c) displays the image of the sample prior to its physical rupture. The data were acquired at T = 140 �C.

T.I. Burghelea et al. / Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx 5

be observed for rubbers, for linear polymer melts and even duringcold drawing experiments (Ref. [22] and the references therein).With increasing time the tensile stress reaches a maximum around�H � 3.

It is interesting to note in Fig. 3(a) that in the neighborhood ofthe stress maximum the tensile force scales exponentially. This factdeserves a brief discussion.

Direct visualization of the sample under extension around thestress maximum shows that, corresponding to the Hencky strainwhere the broad stress maximum is observed, the deformation ofthe sample is not homogeneous, see the inset in Fig. 3(a). There-fore, the nearly exponential scaling of the tensile force around thisstress maximum cannot be explained by the simple derivationbased on the homogeneity assumption presented in Section 3.1.As we have observed such a maximum for a significant numberof various polymeric materials (with molecular structures signifi-cantly different from that of LDPE) during extensional tests con-ducted at low Wi,1 we believe that it should not be confused withthe viscosity overshoot phenomenon which was observed in a fasterregime of stretching where significant strain hardening effects werepresent, [26,16].

A stress maximum is observed at a larger rate of deformation,_� ¼ 0:05 s�1ðWi ¼ 55Þ, Fig. 3(b). We note that we do not observea true stress overshoot in the sense that the local stress maximum

1 Such a broad stress maximum has been observed at the Institute of PolymerMaterials for polystyrene melts and several polymer blends as well during elonga-tional experiments at low rates of deformation.

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is not followed by a plateau. Based on the derivation presented inSection 3.1, one can easily conclude that, corresponding to the localmaximum of the tensile stress, the deformation is inhomogeneous.Indeed, as shown in Section 3.1 if a homogeneous deformation isassumed then, corresponding to the stress maximum, the tensileforce should decay exponentially. This is clearly not the case forthe data presented in Fig. 3(b). Corresponding to _� ¼ 0:3 s�1

ðWi ¼ 330Þ, a local maximum of the tensile stress is no longer ob-served: the sample breaks before the tensile stress reaches either amaximum or a steady state.

In order to get a more complete picture of how the shape of thetransient tensile force/stress is influenced by the forcing conditionsand identify the deformation regime where a stress maximum isobserved, measurements similar to those presented in Fig. 3(a–c)were performed in a wide range of Weissenberg numbers, span-ning nearly three decades.

The results of these measurements are summarized in Fig. 4which presents theWi dependence of the Hencky strains corre-sponding to a maximum in the tensile stress (the squares) and tothe physical rupture of the sample (the circles). The Hencky strainspresented in Fig. 4 have been identified using integral measure-ments of the transient tensile force and stress, similar to those pre-sented in Fig. 3.

Based on a careful inspection of the dependencies F = F(t) andr = r(t), three different regimes of extension can be distinguished,Fig. 4. For Wi 6 10 the Hencky strains corresponding to a stressmaximum and to the physical rupture of the sample are practicallyindependent of Wi. In the deformation regime (I), the maximum of

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Fig. 4. Dependence of the Hencky strain corresponding to the stress maximum(squares) and physical rupture of the sample (circles) on the Weisenberg number,Wi. The vertical dotted lines delineate the extension regimes (I), (II) and (III). Thedata were acquired at T = 140 �C.

Fig. 5. Comparison between the integral (full lines) elongational viscosity l+(t)andthe mean (along the sample) of the point-wise elongational viscosity lþav ðtÞ in eachof the deformation regimes presented in Fig. 4: squares – Wi = 2.2 (regime I), circles– Wi = 16.5 (regime II), triangles – Wi = 300 (regime III). The error bars are definedby the root mean square deviation of the local elongational viscosity along thesample, lþrmsðtÞ. The vertical arrows indicate the physical rupture of the sample. Theinset presents a magnified view of the data acquired in regime II. The dash-dottedline represent shear measurements of 3 g0(t) in a linear range. The data wereacquired at T = 140 �C.

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the tensile stress and the physical rupture of the sample occur at anearly constant Hencky strain each and they are separated byroughly 0.8 Hencky strain units. We once more emphasize thatthe broad stress maximum observed within this regime can be ex-plained as a result of an inhomogeneous deformation process atsmall deformation rates and is not necessarily related to themolecular structure of the material.

As the rate of deformation is increased (Wi > 10) a second defor-mation regime is observed. The Hencky strains corresponding to astress maximum and to the physical rupture of the sample dependsignificantly on the Weissenberg number and they get progressivelycloser to each other as Wi is increased. This finding suggests that theemergence of a stress maximum and the physical rupture of thesample are interconnected phenomena. An argument supportingthis hypothesis is that, corresponding to the lower bound of the sec-ond deformation regime, (II), the time scale of the flow (estimatedhere as sf ¼ 1= _� � 100 s) is significantly smaller than the largestrelaxation time of the material, k, given in Table 1 as 1100 s. Thus,the deformation state corresponding to a maximum in the tensilestress is likely to be ‘‘remembered’’ until the physical rupture ofthe sample occurs. Within the deformation regime (II), the integralelongational viscosity displays a local maximum and its distributionaround the local maximum narrows as the rate of deformation (Wi)increases. Ultimately, if the Weissenberg number is increased evenfurther, Wi > 200, a third deformation regime, (III), is observed.Within this deformation a local maximum of the tensile stress isno longer observed and the Hencky strain corresponding to thephysical rupture of the sample becomes practically independentof Wi. A more detailed characterization of the deformation regimes(I–III) in connection with different failure mechanisms will be pre-sented elsewhere. This paper is mostly dedicated to the deformationregime (II) with a particular focus on the maximum of the transienttensile stress or extensional viscosity, respectively.

3.3. Comparison between the integral measurements and the statisticsof stress (viscosity) distribution along the sample

In subsequent experiments performed at the same temperatureand rate of deformation, we have observed that initially long andthin samples deform more homogeneously than initially shortand thick ones.

A recent 3D modeling of dual wind-up extensional rheometersindicates a geometrically inhomogeneous deformation if initially

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thick samples are used. This leads to significant deviations of thecross sectional area from the ideal uniaxial one which affect boththe strain and the measured viscosity, [31].

To get a clearer insight into the role of homogeneity for the reli-ability of the integral measurements, we compare in the followingintegral viscosity measurements with the mean of the viscositydistribution along the sample, defined as lþavðtÞ ¼ lþðz; tÞz. Herel+(z, t) stands for the local transient elongational viscosity definedas lþðz; tÞ ¼ 4FðtÞ

p _�D2ðz;tÞ. Certainly, if the deformation process is homo-geneous, the two measurements should coincide and thus, one canquantitatively assess the deviations from a homogeneous deforma-tion by monitoring the difference between the two types of mea-surements. The distribution of the locally measured stresses(elongational viscosities) is also characterized by its second ordermoment, defined as lþrmsðtÞ ¼ ðlþðz; tÞ � lþav ðtÞÞ

2z .

Results of such a comparison corresponding to each of thedeformation regimes previously discussed are presented in Fig. 5.The data presented in Fig. 5 were measured on samples with thesame initial aspect ratio, x0 = L0/D0 = 0.62, which is comparable tothe initial aspect ratio of the samples used in Ref. [16].

The consistency of the integral measurements and the mean ofthe viscosity distribution along the sample in a linear range ofdeformation is confirmed by the fulfillment of the Trouton rela-tionship: the local extensional data overlaps in a linear range withthe shear measurements of 3g0(t) (the dash-dotted line in Fig. 5). Inthe nonlinear range, the differences between integral and spaceaveraged measurements of the transient elongational viscosityseem to increase with the rate of deformation. Corresponding tothe second deformation regime (II) (Wi = 16.5) where the integraltransient elongational viscosity passes through a maximum, thespace averaged elongational viscosity has no maximum but seemsto reach a plateau, as visible in the inset of Fig. 5. This result hasbeen systematically reproduced over the entire second regime ofdeformation (data not shown here), suggesting once more thatmaximum of the transient elongational viscosity might not be atrue rheological feature of the material but merely an artifact re-lated to loss of homogeneity of the sample under deformation.

In the range of high Wi, (regime III) neither a maximum nor aplateau of the transient elongational viscosity is observed. Within

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

T.I. Burghelea et al. / Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx 7

this deformation regime, the integral and the space averaged vis-cosity measurements do agree qualitatively, though they are quan-titatively different.

3.4. Geometric non-uniformity of the sample and its relation with thestress maximum

The synchronization the image acquisition system with the oil-bath rheometer allows one to monitor the geometric homogeneityof samples under deformation simultaneously with the integralmeasurements of the transient elongational viscosity. Of particularinterest are the images of the sample acquired around the pointwhere a local maximum of the integral viscosity is observed, Figs.2 and 3.

In Fig. 6 we display images of the samples corresponding toeach of the deformation regimes presented in Fig. 4(b)at severalHencky strains. The images corresponding to the highest Henckystrains (the last column in Fig. 6) are the last images acquiredprior to the physical rupture of the sample. The images presentedin Fig. 6 have been rescaled in order to enhance the clarity of thepresentation. We note, however, that the rescaling does not alterthe main message concerning the geometric uniformity of thesample.

Within the first regime of deformation (I) (first row at the top ofFig. 6), the shape of the sample deviates strongly from a cylindricalone. The onset of these geometric non-uniformities (the primaryneck extended over the entire length of the sample) occurs atlow Hencky strains (�H 6 1) and, according to the Considère crite-rion, [4,7], is related to the local maximum in the tensile force ob-served in Fig. 3(a). In a range of high Hencky strains (�H � 3.5) priorto the physical rupture of the sample, secondary necks develop inthe proximity of the midpoint of the sample as visible in row (I)panels (e–g) of Fig. 6. The exact location of these secondary necks

Fig. 6. Sequence of specimen images under deformation at different Wi. The image rows(region III). The aspect ratio of each image has been modified in order to enhance the clindicated on the top of each image. The data were acquired at T = 140 �C.

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is not reproducible in subsequent experiments. In regime (II) ofdeformation (second row from the top in Fig. 6) the geometricinhomogeneity of the sample becomes even more pronouncedthan in regime (I): above the onset of the primary necking, thediameter of the sample is non constant over the entire length ofthe sample. Just after a local maximum in the viscosity is observedat �H � 3.3, a secondary neck emerges slightly below the centerpoint of the sample, second row, panel (a), Fig. 6.

A magnified view of these necks is presented in Fig. 7.As the Hencky strain increases, the secondary neck becomes

sharper (its local diameter decreases rapidly) and moves slowlyalong the sample. Another localized neck is formed at �H � 3.59and this ultimately leads to the physical rupture of the sample ina finite time. The monotonic increase of the error bars with strainduring integral viscosity measurements within the regimes I, IIcorresponding to large Hencky strains, Fig. 5, can now be easily ex-plained as a result of a systematic increase of the sample inhomo-geneity due to the emergence of secondary necks.

The emergence of secondary necks can also explain the discrep-ancy between the integral viscosity measurements and the spaceaverage of the viscosity distribution observed within the secondregime of deformation (the circles in Fig. 5 and the inset). Indeed,after the secondary necks are formed along the sample, the integralviscosity measurement which uses a position-independent value ofthe sample diameter, DuðtÞ ¼ D0 exp ��H=2ð Þ, systematically over-estimates the actual average sample diameter, D(t) = hD(z, t)iz. As aconsequence, above the onset of the secondary necking, the inte-gral transient elongational viscosity decreases and a viscosity max-imum is observed. On the other hand, if the emergence of thesecondary necks is accounted for by averaging the locally mea-sured elongational viscosity along the actual length of the sample,no decrease of viscosity is observed and the data seem to approacha steady state instead (the inset in Fig. 5).

(from top to bottom) correspond to: Wi = 2.2 (region I), Wi = 16.5 (region II), Wi = 99arity. The dotted squares indicate the location of the necks. The Hencky strains are

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

Fig. 7. Magnified views of the secondary necks highlighted in Fig. 6 corresponding to regime II (second row).

8 T.I. Burghelea et al. / Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx

These experimental findings suggest that the long debatedmaximum of the transient extensional viscosity does not reflecttrue rheological features of the material and is solely related toa severe inhomogeneity of deformation states due to the emer-gence of secondary necks along the sample. This conclusion isconsistent with the discussion presented in Section 3.1: if theviscosity maximum would emerge as a true rheological feature(that is in the absence of geometric inhomogeneities) than,corresponding to this maximum, the tensile force should scaleexponentially which, as already discussed above, is not the casewithin regime (II).

Finally, we turn our attention to the evolution of the sampleinhomogeneity during measurements of the transient elongationalviscosity in regime (III). Within this deformation regime, the over-all homogeneity of the sample is higher than within the regimes (I)and (II), though curvature effects are visible in the proximity of theplates of the rheometer, row (III), Fig. 6. In spite of a better samplehomogeneity (no secondary necks are observed in this deformationregime), however, the differences between the space averaged andthe integral measurements of the viscosity are significant (the tri-angles, Fig. 5), indicating that deviations from an homogeneousdeformation still exist and they increase with the Hencky strain.As at higher rates the deformation is more homogeneous (comparerows 2 and 3 from the top in Fig. 6), this fact is counterintuitive andit deserves a brief explanation.

It can be easily shown (see the derivation in Appendix A) thatthe relative difference between integral viscosity measurementslu(t) and the space averaged viscosity lþav :

lþavðtÞ � lþu ðtÞlþu ðtÞ

�������� � 4n / exp _�=2tð Þ ð1Þ

where n = D(z, t)/Du(t) < 1 with Dðz; tÞ ¼ Dðz;tÞ�DuðtÞ2 . The scaling rela-

tion above explains the increase of the relative viscosity error withthe rate of deformation at a fixed time instant observed in Fig. 5.

3.5. Comparison with results from the literature

In the following, a comparison of our experimental findingswith experimental work performed by others is presented.

The observation of a local maximum of the integral transientelongational viscosity is entirely similar to the several previousexperimental observations, [5,15,9]. Consistently with the previousobservations, after reaching a maximum, the integral extensionalviscosity decays monotonically until break of the sample occurredwithout reaching a plateau and, therefore, no true overshootbehavior is observed. The consistency with the previous observa-tions can be explained the similarity of the experimental ap-proaches: the rate of deformation is controlled at an integralscale and the transient elongational viscosity is assessed usingthe assumption that the sample deforms homogeneously.

The recent work by Rasmussen et al. [16] presents a detailedexperimental observation of a true viscosity overshoot (i.e. a max-imum in the extensional viscosity followed by an extended pla-teau). Whereas in our experiments a local maximum of theintegral elongational viscosity was found during each experiment

Please cite this article in press as: T.I. Burghelea et al., On the ‘‘viscosity oversNewtonian Fluid Mech. (2011), doi:10.1016/j.jnnfm.2011.07.007

conducted in the deformation regime II (Fig. 4), a plateau followingsuch maximum has never been observed.

In an attempt to understand this discrepancy, we compare ourdata analysis procedure (see the description in Section 2.4) withthe procedure described in [16]. There exists a fundamental differ-ence between the two approaches.

Whereas we have defined the Hencky strain using the actual

length of the sample, �HðtÞ ¼ ln LðtÞL0

h i, and measured it accordingly

by monitoring the position of the top plate of the rheometer,

Rasmussen et al. have defined it as ��HðtÞ ¼ �2ln DmidðtÞD0

h i, using the

middle plane diameter of the sample, Dmid(t). Unlike in the caseof our experiments where the rate of deformation is controlled atan integral scale, the filament stretching device used in [16] main-tains constant rate of deformation at the mid-point of the sample(with an accuracy better than 1%, [1]) via a feedback loop thatcontrols in real time the motion of the top plate of the device.

It is obvious that in the case of a uniaxial extension at a constantrate of deformation (in time and along the entire sample, i.e. thesample deforms homogeneously), the two ways of calculating theHencky strain are entirely equivalent. In the case of the experi-ments presented in this paper, however, one clearly deals with ageometrically non-uniform deformation process which ultimatelycan translate into a strong deviation from the idealized uniaxialcase. This experimental fact is illustrated in Fig. 6 where one canclearly see that, within the second deformation regime (the secondrow from the top), the sample is far from being cylindrical when aviscosity maximum is observed. The impact of the geometricnon-uniformity of the sample on the kinematics of the deformationprocess is illustrated in Fig. 8 for two values of the Weissenbergnumber within the second deformation regime.

Above the onset of the primary non-uniformity of the sample(the first maximum of the tensile force panels (b and d) Fig. 8),the strain becomes strongly localized along the sample. This canbe clearly noticed in Fig. 8(a and c) where the time dependenciesof the minimum sample diameter Dmin, the averaged (along the ac-tual length of the sample) diameter Dav and the maximum samplediameter Dmax are displayed together with the diameter corre-sponding to a uniform deformation at a constant rate (the full line).The deviations of Dav from the exponential dependence corre-sponding to the uniform deformation become even more pro-nounced above the onset of the secondary necks illustrated inFigs. 6 and 7 and indicated by the dashed arrows in Fig. 8.

A comparison between the Hencky strains �H and ��H is pre-sented in Fig. 8 (b and d). At late stages of the deformation process(after the viscosity overshoot is observed as indicated by thedashed arrows) the local slope dDminðtÞ

dt increases drastically suggest-ing that the highest rate of material deformation corresponds tothe necked region of the sample. As a consequence, the terminalvalues of ��H are considerably larger than those of �H.

During our experiments, the neck is roughly located around themiddle (but not exactly at, which in principle breaks the mid-planesymmetry) of the sample (though the exact location of the neck isnot reproducible in subsequent experiments), which is preciselythe point where Rasmussen et al. measure the diameter of thesample, [16]. In cases when several secondary necks emerge, their

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

Fig. 8. (a) Time dependence of the minimum diameter of the sample (circles), the average diameter (squares) and the maximum diameter (triangles) at_� ¼ 0:015 s�1ðWi ¼ 16:5). (b) Time dependence of the Hencky strain (calculated using the minimum diameter of the sample, DminÞ; ��H at _� ¼ 0:015 s�1 (Wi = 16.5). (c)Time dependence of the minimum diameter of the sample (circles), the average diameter (squares) and the maximum diameter (triangles) at _� ¼ 0:025 s�1 (Wi = 27.5). (d)Time dependence of the Hencky strain (calculated using the minimum diameter of the sample, Dmin), ��H at _� ¼ 0:025 s�1ðWi ¼ 27:5). The full line in panels (b and d) is theHencky strain measured using the actual length of the sample, �H. The insets in panels (b and d) present the tensile force. The full arrows indicate the onset of the primarygeometric inhomogeneity of the sample and the dashed arrows indicate the onset of the secondary necking instability.

T.I. Burghelea et al. / Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx 9

locations are not symmetric with respect to the mid-point of thesample (panels (d–g), second row in Fig. 6) and the mid-plane sym-metry of the sample is obviously broken.

Although in Ref. [16] a feedback mechanism has been em-ployed to ensure a constant rate of decay of the mid samplediameter, the assessment of the transient elongational viscosityremains in our opinion conceptually problematic, as long as noproof of the homogeneity of deformation states is provided. Fol-lowing the theoretical analysis of the force balance in a filamentstretching device provided in [23], keeping a constant rate at themid-point of the sample via a feedback mechanism that controlsthe motion of tho two plates of the rheometer does not suffice.The derivation presented in [23] uses the assumption that themid-plane symmetry of the sample is preserved during the defor-mation process. Therefore, within this theoretical framework, thehomogeneity of the sample deformation and its mid-plane sym-metry should also be demonstrated by an in-situ imaging of thesample during extension.

To better understand the physical origins of a true viscosityovershoot behavior we analyze in the following our data using aprocedure similar to some extent to that employed in Ref. [16].2

The result of such an analysis is presented in Fig. 9 (circles) togetherwith measurements of the space averaged extensional viscosity.

2 Unlike a filament stretching device, the oil-bath rheometer used in this studycannot maintain a constant rate of deformation at a given point along the sampleduring the entire duration of the experiment and therefore a one to one comparisonwith [16] was not feasible.

Please cite this article in press as: T.I. Burghelea et al., On the ‘‘viscosity oversNewtonian Fluid Mech. (2011), doi:10.1016/j.jnnfm.2011.07.007

Data extracted manually from Fig. 4 in [16] and correspondingto ��H ¼ 0:01 s�1ðWi ¼ 18:75) are also displayed (the triangles,Fig. 9) for comparison.

The extensional viscosity lþ1 is defined as:

lþ1 ðtÞ ¼4FðtÞ

p DmidðtÞð Þ2 d ��HðtÞð Þ

dt

: ð2Þ

Above the onset of the primary geometric non-uniformity (indi-cated by the full arrows in Fig. 9) the two data analysis proceduresapplied to the same raw data (namely the same force signal and thesame sequence of sample images) yield strikingly different results.Whereas the space averaged measurements reach a plateau (thesquares, Fig. 9), the point-wise measurements similar some extentto those reported in Ref. [16] lead to a clear viscosity overshootbehavior: a viscosity plateau following the viscosity maximum isnow visible up to ��H ¼ 5 for _� ¼ 0:015 s�1 and ��H ¼ 4:6 for_� ¼ 0:025 s�1. The magnitude of the overshoot behavior increasesas the Weissenberg number is increased most probably due to a lar-ger geometric inhomogeneity of the sample. We point out that theviscosity data extracted from Fig. 4 in [16] (the triangles in Fig. 9(aand b)) are qualitatively similar to the data obtained following aprocedure similar to that employed in [16].

Quantitative differences between the two data sets are mostprobably related to differences between the grades of LDPE usedand the magnitude of the Weissenberg number. However, in spiteof these differences, it is quite clear that using a data reductionprocedure similar to that employed in [16] near the point where

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

Fig. 9. Comparison between the integral viscosity measurements (full lines), space averaged viscosity (squares) and the elongational viscosity obtained following a proceduresimilar (see text) to that of Rasmussen et al. [16] (circles): (a) _� ¼ 0:015 s�1ðWi ¼ 16:5) (b) _� ¼ 0:025 s�1ðWi ¼ 27:5). The triangles are data extracted manually from Fig. 4from [16] and they refer to _��H ¼ 0:01 s�1ðWi ¼ 18:75). The full arrows indicate the onset of the primary geometric inhomogeneity of the sample and the dashed arrowsindicate the onset of the secondary necking instability.

10 T.I. Burghelea et al. / Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx

the secondary necks emerge (see Fig. 7) yields an entirely similarovershoot behavior of the transient elongational viscosity.

As a conclusion, the discrepancy between our transient viscos-ity measurements and the results presented by Rasmussen et al.originates from the differences between the two approaches:whereas we have used an integral definition for the Hencky strainand averaged the tensile stress along the sample, Ref. [16] used lo-cal values for both the Hencky strain and the stress.

4. Conclusions

A systematic experimental investigation of the long debated‘‘viscosity overshoot’’ during the uniaxial extension of a strainhardening polymer melt was presented.

The experiments reported in this paper have been conductedusing an oil-bath rheometer which controls the rate of deformation_� at an integral scale and assesses the transient elongational viscos-ity by real time measurements of the tensile force applied to thetop plate of the device.

As such measurements rely on the assumption that the sampledeforms homogeneously (its shape remains cylindrical during theentire deformation process and its diameter decays exponentiallyin time at a rate equal to the half of the integral rate), we havealternatively employed a recently implemented technique able tomeasure locally (at any point along the sample) the transient elon-gational viscosity by an in-situ analysis of the shape of the sampleunder extension, [3]. This method allows one to characterize thestatistics of the stress distribution along the sample, which pro-vides a quantitative measure of the deviations from an ideal uniax-ial deformation process. Unlike in [3] where the opacity of thesamples allowed one to track marks inscribed onto them and mea-sure the rate of extension locally, the integral rate of extension(calculated from the motion of the top plate of the rheometer)has been used.

An accurate synchronization between the oil-bath device andthe in-situ imaging system allowed the following measurements/investigations to be carried out simultaneously:

1. Traditional measurements of the integral extensional viscositylþu ðtÞ based on the measurements of the tensile force F(t) andusing the assumption that the sample deforms homogeneously,@Dðz;tÞ@z ¼ 0. It is this type of integral measurements which led to

the very first experimental observation of a viscosity maximum,[15,26].

Please cite this article in press as: T.I. Burghelea et al., On the ‘‘viscosity oversNewtonian Fluid Mech. (2011), doi:10.1016/j.jnnfm.2011.07.007

2. A qualitative assessment of the geometric homogeneity of thesample under elongation with a particular focus on the emer-gence of necks along the sample under extension.

3. A systematic characterization of the statistics of the spatial dis-tribution of the local elongational viscosity and its comparisonwith the integral measurements.

4. Point-wise measurements of both the kinematics (the rate ofextension) and the extensional stress (and viscosity) at themid-point of the sample.

The mathematical condition for the integral extensional viscos-ity to have a local maximum assuming a geometrically homoge-neous deformation process is presented in Section 3.1. Accordingto this simple derivation, a local viscosity maximum may be ob-served during a homogeneous deformation process only if the ten-sile force scales exponentially around this maximum. If thedeformation process is not homogeneous, a stress maximum andan exponential scaling of the tensile force may still be observedif the rates of deformation are small. These theoretical consider-ations are investigated experimentally corresponding to variousdeformation regimes. Depending on the magnitude of the Weiss-enberg number, we identify three distinct deformation regimes.At low Wi, (regime (I), Fig. 4) the integral tensile stress displays abroad maximum, Fig. 3(a). In the neighborhood of the stress max-imum, the tensile force decays nearly exponentially with a rate setby the integral deformation rate, _�. As within this regime the defor-mation is inhomogeneous (Fig. 6, row I), this nearly exponentialscaling cannot be explained by the simple derivation based onthe homogeneity assumption presented in Section 3.1. The stressmaximum observed in regime I can be observed for various otherpolymeric materials deforming at low Wi and it should not be con-fused with the viscosity overshoot phenomenon, which was ob-served at substantially larger Wi, [26,16].

We observe such a viscosity maximum for intermediate valuesof Wi, in regime II, Fig. 4. The existence of such a maximum is notconsistent with a homogeneous deformation process, because inthe neighborhood of this maximum the tensile force does not scaleexponentially, Fig. 3(b). As suggested by the convergence of thestress maximum and physical rupture lines (the Hencky strainscorresponding to the physical rupture of the sample) visible inFig. 4 within regime (II), the two phenomena are interconnected:the viscosity maximum is just a precursor of the physical ruptureof the sample. Indeed, real time imaging of the sample confirmsthat right above the stress maximum, secondary necks develop

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

3 We note, however, that in the neighborhood of the viscosity maximum, the localrate of extension is roughly constant, Fig. 8(b and d).

4 We emphasize once more that, according to the theoretical derivation by Szabo[23,24] regarding the Filament Stretching approach, the mid-plane symmetry of thesample is an essential assumption in relating the measured tensile forces and thelocal deformation of the sample to the extensional viscosity of the material.

T.I. Burghelea et al. / Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx 11

along the sample leading to sample’s rupture, Fig. 6, row II. Basedon the images of the sample, we measure alternatively the statis-tics of the stress distribution along the sample. Whereas in regime(II) the integral elongational viscosity displays a clear maximum,the space averaged viscosity reaches a plateau instead. Thereforewe conclude that the viscosity maximum observed in the tradi-tional measurements (which, once more, rely on the homogeneityof the sample) might be only an experimental artifact introducedby the strong geometric inhomogeneity of the sample. In the faststretching limit (regime III, Fig. 4), the homogeneity of the sampleis better preserved (Fig. 6, row III) and no viscosity maximum is ob-served, Fig. 3(c). The better homogeneity of the sample within thisregime and the lack of a viscosity maximum may be attributed tothe stabilizing role of the strain hardening effect which becomesincreasingly significant as Wi is increased.

Finally, our experimental findings are compared with a recentexperimental investigation of the viscosity overshoot phenomenonby Rasmussen et al. [16].

The discrepancy between the extensional viscosity measure-ments obtained by averaging the stresses along the sample pre-sented in this paper and the results presented in Ref. [16] isexplained by differences in the data analysis procedure which re-sults from differences between the rheometers used in the two stud-ies. As illustrated in Fig. 9, using a procedure similar to some extentto that employed in [16] (except for the local rate of deformationwhich was not constant during our experiments), one can qualita-tively reproduce a viscosity overshoot behavior as well, Fig. 9.

We summarize the main experimental facts regarding the exis-tence of a viscosity maximum and/or overshoot:

1. Within the second regime of deformation illustrated in Fig. 4, alocal maximum of the integrally measured transient elonga-tional viscosity is systematically observed. No true viscosityovershoot has been observed, in the sense that no plateau hasbeen reached after the maximum and these results reproducethe early observations of a viscosity maximum, [15,26].

2. The magnitude of the viscosity maximum observed in stressingexperiments (at a constant integral rate of deformation) per-formed in the second regime of deformation is consistent withsteady state viscosity measurements performed in creep exper-iments (at constant stress), [11]. As the homogeneity of thedeformation states is better preserved in creep experimentsthan in stressing experiments, the viscosity decrease which fol-lows the local maximum observed during stressing experimentsmay be connected to geometric non-uniformities of the sampleunder extension.

3. By synchronizing the traditional integral viscosity measure-ments with in-situ visual analysis of the (entire) sample underextension, it has been shown that the viscosity maximumemerges when a secondary necking instability develops onthe free surface of the sample, Fig. 7. As such strong geometricinhomogeneities are certainly not accounted for by the tradi-tional integral viscosity measurements, we consider this as anadditional argument that the viscosity maxima are highly ques-tionable as a material property.

4. An inconsistency between integral measurements of the tran-sient elongational viscosity and the spatial average of the localtransient elongational viscosity is found, indicating a significantdeparture of the extensional process from an ideal uniaxialextension.

5. A true overshoot behavior (a plateau followed by a maximum)can only be observed during point-wise measurements of thetransient elongational viscosity conducted at the mid-point ofthe sample. Although this procedure is not entirely similar tothat employed in [16] (because the oil-bath rheometer usedin our study does not allow to maintain a locally constant rate

Please cite this article in press as: T.I. Burghelea et al., On the ‘‘viscosity oversNewtonian Fluid Mech. (2011), doi:10.1016/j.jnnfm.2011.07.007

of deformation during the entire duration of the experiment,3)the similarity of the local viscosity data presented in Fig. 9 withthe overshoot behavior illustrated in Fig. 4 in [16] is quite surpris-ing. As in our case the secondary necks which accompany theovershoot emerge fairly close to the mid-point of the sample,we can once more associate the presence of the overshoot withthe necking instability.

Before closing this report, the experimental results presentedabove prompt us to address two more general questions regardingthe extensional rheology of polymer melts:

1. Is there a way to eliminate the geometric inhomogeneities duringextensional tests?

2. To what extent can one reliably assess rheological properties ofpolymer melts in the context of a geometrically inhomogeneousextension?

The answer to the first question is, in our opinion, negative. Atransition from a homogeneous deformation (cylindrical shape ofthe sample) to an inhomogeneous one (hyperboloidal shape ofthe sample) will always occur in a regime of low Hencky strains(typically �H 6 1) according to the Considère rule, [4]. Althoughthis primary inhomogeneity cannot be suppressed, its magnitudeand impact on the viscosity measurements can be, however,diminished by using long and thin samples. Depending on boththe properties of the polymeric material and the conditions ofextension (temperature, rate of deformation), secondary necksmay emerge in a range of high Hencky strains, prior to the physicalrupture of the sample. Unfortunately, we do not know neither howto suppress this instability nor how to control it. Additional studies(both experimental and numerical) are needed to clarify this point.

As the second question above is concerned, the answer is onlypartially positive and has to be presented in the context of theexperimental technique employed. Regardless the technique em-ployed, which can be either an integral one (such as the oil-bathapproach, the RME or the SER) or a point-wise one (such as the Fil-ament Stretching approach), the measurements performed at mod-erate Hencky strains seem to be reliable: the Trouton relationshipis well validated and consistent and qualitative assessments of thestrain hardening effect have been made with each of these devices.

Measurements in a regime of high Hencky strains are problem-atic due to an increased inhomogeneity of the sample which, insome cases may become critical (e.g. secondary necks superposedonto the primary neck). Whereas a Filament Stretching devicemight in principle circumvent the problem of the rate of strainlocalization by locally maintaining a constant rate of deformation,it still has to be verified experimentally (e.g. by an in-situ visuali-zation) that secondary necks do not emerge and the mid-planesymmetry of the sample is not broken.4 To our best knowledge, astudy that combines the Filament Stretching approach with an in-situ analysis of the homogeneity of the sample is still missing.

As a final conclusion, in the view of the points discussed above,neither a local maximum of the transient elongational viscosity nora true viscosity overshoot behavior seem to be, according to thisstudy, real rheological features but they emerge in connection withthe strong geometric non-uniformities of the sample at high Hen-cky strains. The main issue responsible for the emergence of a vis-cosity maximum and/or overshoot is the geometric inhomogeneity

hoot’’ during the uniaxial extension of a low density polyethylene, J. Non-

12 T.I. Burghelea et al. / Journal of Non-Newtonian Fluid Mechanics xxx (2011) xxx–xxx

of the sample which becomes critical when secondary necks areformed along the sample. Existing experimental work on exten-sional rheology of polymer melts in a non-linear range should bereconsidered particularly in relation with the inhomogeneity ofsample deformation. Future theoretical studies concerning with aviscosity overshoot should take these findings into account.

Acknowledgements

T. B. and Z.S. gratefully acknowledge the financial support fromthe German Research Foundation (Grants MU1336/6-4 andSTA1096/1-1, respectively). T. B. and Z.S. thank Mrs. MagdalenaPapp for her assistance during some of the experiments presentedin this study. One of us (T.B.) thanks Alfred Frey for valuable tech-nical advice, assistance with the Münstedt rheometer, and for theimplementation of the digital trigger for the camera. We are grate-ful to Mrs. Jennifer Reisser for her support in preparing part of thesamples used in this study and her assistance with part of the mea-surements we have reported.

Appendix A

Using an integral representation of the average of the distribu-tion of the viscosity along the sample and recalling the definitionlþu ðtÞ ¼

4FðtÞp _�D2

uðtÞfor the extensional viscosity corresponding to a cylin-

drical sample with the diameter Du ¼ D0e��H=2, one can write

lþavðtÞ � lþu ðtÞ ¼ �4FðtÞ

p _�D2uðtÞþ 4FðtÞ

p _�LðtÞ

Z LðtÞ

0

1D2ðz; tÞ

dz ð3Þ

Bearing in mind that the first term in the right hand side of theequality above does not depend on the vertical coordinate z, theequality above may be written:

lþavðtÞ � lþu ðtÞ ¼4FðtÞp _�LðtÞ

Z LðtÞ

0

1D2ðz; tÞ

� 1D2

uðtÞ

!dz ð4Þ

If one introduces Dðz; tÞ ¼ Dðz;tÞ�DuðtÞ2 to quantify the deviation of the

sample shape from the ideal cylindrical form, the last equality canbe rearranged:

lþavðtÞ � lþu ðtÞ ¼ �4FðtÞ

p _�D2uðtÞ

L�1ðtÞZ LðtÞ

0

Dðz; tÞ Dðz; tÞ þ DuðtÞ½ �DuðtÞ þ 2Dðz; tÞ½ �2

dz

ð5Þ

Finally, if one divides both sides of the equation above by lþu ðtÞ, oneobtains:.

lþavðtÞ � lþu ðtÞlþu ðtÞ

¼ �4L�1ðtÞZ LðtÞ

0

Dðz; tÞ½DuðtÞ þ Dðz; tÞ�½DuðtÞ þ 2Dðz; tÞ�2

dz ð6Þ

Assuming n = D(z, t)/Du(t) < 1, it can be easily shown that, to a lead-

ing order in n2;lþav ðtÞ�lþu ðtÞ

lþu ðtÞ

��� ��� � 4n / exp _�=2tð Þ.

Please cite this article in press as: T.I. Burghelea et al., On the ‘‘viscosity oversNewtonian Fluid Mech. (2011), doi:10.1016/j.jnnfm.2011.07.007

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