Simultaneous In-situ Analysis of Instabilities and First Normal StressDifference during Polymer Melt Extrusion Flows

Roland Kádár1,2, Ingo F. C. Naue2 and Manfred Wilhelm2

1 Chalmers University of Technology, 41258 Gothenburg, Sweden2 Karlsruhe Institute of Technology - KIT, 76128 Karlsruhe, Germany

ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 22, 2014

ABSTRACTA high sensitivity system for capillaryrheometry capable of simultaneously de-tecting the onset and propagation of insta-bilities and the first normal stress differ-ence during polymer melt extrusion flows ishere presented. The main goals of the studyare to analyse the nonlinear dynamics ofextrusion instabilities and to determine thefirst normal stress difference in the pres-ence of an induced streamline curvaturevia the so-called hole effect. An overviewof the system, general analysis principles,preliminary results and overall frameworkare herein discussed.

INTRODUCTIONCapillary rheometry is the preferredrheological characterisation method forpressure-driven processing applications,e.g. extrusion, injection moulding. Cap-illary rheometry is the only method ofprobing material rheological properties inprocessing-like conditions, i.e. high shearrate, nonlinear viscoelastic regime, albeitin a controlled environment and using acomparatively small amount of material.1Thus, it is of paramount importance to de-velop new techniques to enhance the cap-illary rheometers for a more comprehen-sive probing of material properties. Extru-sion alone accounts for the processing ofapproximately 35% of the worldwide pro-duction of plastics, currently 280 × 106tons (Plastics Europe, 2014). This makesit the most important single polymer pro-cessing operation for the industry and can

be found in a variety of forms in many man-ufacturing operations. Extrusion through-put is limited by the onset of instabili-ties, i.e. product defects. Comprehensivereviews on the subject of polymer meltextrusion instabilities can be found else-where.4,6 A recent method proposed for thedetection and analysis of these instabilitieswas that of a high sensitivity in-situ me-chanical pressure instability detection sys-tem for capillary rheometry.8,10 The sys-tem consists of high sensitivity piezoelec-tric transducers placed along the extrusionslit die. In this way all instability types de-tectable thus opening new means of sci-entific inquiry. As a restult, new insightsinto the nonlinear dynamics of the flow hasbeen provided.9,14 Moreover, the possibil-ity of investigating the reconstructed non-linear dynamics was considered, wherebya reconstructed phase space is an em-bedding of the original phase space.2,14It was shown that a positive Lyapunovexponent was detected for the primaryand secondary instabilities in linear andlinear low density polyethylenes, LDPEand LLDPE,.14 Moreover, it was deter-mined that Lyapunov exponents are sen-sitive to the changes in flow regime andbehave qualitatively different for the iden-tified transition sequences.14 It was alsoshown that it is possible to transfer thehigh sensitivity instability detection sys-tem to lab-sized extruders for inline ad-vanced processing control and quality con-trol systems,.13

A very recent possibility considered

153

concerns the use of the high sensitivityinstability detection system for the mea-surement of the normal stress differencesvia the hole effect. Errors in pressure mea-surements for polymeric liquids, i.e. withnon-zero normal components of the extra-stress tensor, we known for decades to oc-cur for offset positioned transducer withrespect to the capillary walls.5 This effectwas used by Lodge for the determinationof the first normal stress difference, i.e. theLodge stressmeter.5 However, many chal-lenges still remain for the implementationand to achieve a desired accuracy for suchmethods.7,12

EXPERIMENTALThe experimental setup is based on a Göt-tert Rheotester 2000 capillary rheometerfeaturing a custom slit die. The die isequipped with four high sensitivity piezo-electric pressure transducers, Fig. 1.a, andhas an aspect ratio of W/H = 10, where Wand H are the channel width and heightrespectively (W = 3mm, H = 0.3mm).The transducers are capable of deliveringremarkable time and pressure resolutions,namely 10−3 s and 10−5 bar at a nominalpressure of 500 bar. Three of the transduc-ers, Tr1-Tr3, are equally spaced along thechannel length, whereas the fourth, Tr4, isfacing Tr3 with flush or offset positioningwith respect to the flow channel. The Tr4offset can be accordingly adjusted to in-duced streamline curvature and thus cre-ate the so called ’hole effect’ for the esti-mation of the normal stress difference(s),Fig. 1.b. Complementary, an online visual-isation system is positioned at the die exitfor advanced image analysis.

The experimental protocols for the test-ing consist of quasi ramped input signals,similarly to a typical protocol for capillaryrheometry measurements. In contrast, thestabilisation times for each shear rates wereimposed via scripting in order to bypassall take-over procedures. Two experimen-tal protocols were applied to cover from

Tr1Tr2Tr3 26

mm

310

10Tr4

Tr0

v/vmax = 0, 1W

H

x1x3

(a)

Streamlines

Die swelling

Tr3Tr4 T

HW

t11

t11

x1x2

x1x3

(b)

Figure 1. (a) Schematic representation ofthe high sensitivity extrusion die and the

corresponding transducer placement,Tr1,2,3. Tr0 represents a conventional

pressure transducer. (b) Detailedpositioning of the facing Tr3,4 transducers

used to induced streamline curvature.

low to high apparent shear rates. A con-stant quasi-linear ramp rate of dγ̇a/dt =0.76 × 10−3 s−2 was applied in the formercase, with γ̇a = 6Q/WH2 being the appar-ent shear rate, whereas in the latter casea nonlinear protocol was used. Other pa-rameters varied are the hole depth, see lhin Fig. 1, operating bulk temperature anddraw ratio. The latter was imposed usinga Rheotens device however here only zerodraw ratios are presented.

154

Commercial samples with differentmolecular characteristics, i.e. linear / shortchain branched and branched topologieswere considered. In this publication twosamples are emphasised, a low densitypolyethylene (LDPE) having Tm = 111 ◦C,Mw = 186 kg/kmol and Mw/Mn = 12 anda high density polyethylene (HDPE) hav-ing Tm = 131 ◦C (data provided by themanufacturers).

The oscillatory shear measurementspresented were performed on a TA Instru-ments ARES-G2 rheometer.

RESULTS AND DISCUSSIONAn example of typical mechanical pressureoutput from the in-situ transducers is pre-sented in Fig. 2. The diagram features theraw mechanical pressure signals as well.Piezoelectric transducers are mainly dedi-cated to fast transient processes, therefore,additional data treatment was necessaryfor the present experiments. Thus, a steadystate load can be expressed using the con-dition dpM/dt = s, where s is the drift.Using the conventional mechanical pres-sure data, Tr0 in Fig. 1.a, a drift correc-tion was considered, dpM/dt = 0, whereby

Tr1Tr2Tr3Tr4

lh = 0, 140 °C, v2 / v1 = 0

Raw data

Drift-corrected

p M /

bar

0

100

200

300

400

500

Time / s0 2000 4000 6000 8000 10000 12000 1400016000

.

3.9 20 39 59 79 98 118 137 0

Figure 2. Example of output signalshowing the raw and the drift correctedsignals for all transducers. Data for an

LDPE having Tm = 110 ◦C, Mw =256 kg/kmol and Mw/Mn = 15, tested at

140 ◦C with lh = 0.

a linear function of slope s is subtractedfrom the raw data, i.e. drift correcteddata. More details on data pre-processingare provided elsewhere.9,14 Overall, the sig-nals in Fig. 2 contain two contributions,namely a nominal value of the mechani-cal pressure, based on which the hole pres-sure is determined, and the fluctuations su-perimposed on the signal, based on whichthe onset of instabilities can be in-situ de-tected. It should be noted that the datadisplayed in the figure already exhibits aninherent pressure difference between Tr3and Tr4, although the Tr4 was in flush po-sitioning, lh = 0. This residual differencewas subtracted from subsequent hole pres-sure measurements for the determinationof the first normal stress difference.

The time dependent mechanical pres-sure data showing the influence of holedepth, lh = H, 6H, 20H, is presentedin Fig. 3.a for both low and high appar-ent shear rate experimental protocols. Theonset of instabilities is marked on the dia-grams whenever available from the analy-sis of the in-situ mechanical pressure fluc-tuations and spatio-temporal inline visu-alisations. Spatio-temporal visualisations,or space-time plots, are constructed by ex-tracting a line of pixels out of a colleciotnof frames and then successively add themto a new visualisation, a common methodin fluid dynamics.11,14 An example of thelatter is presented in Fig. 3.b. With respectto the nominal pressure, it can be seen andincrease in the pressure difference betweenTr3 and Tr4 is increased between lh = Hand lh = 6H. However, for lh = 20Hthe mechanical pressure decreases, whichcould indicate the occurrence of secondaryflows. By comparing the low and high shearrate tests there it is apparent that thereis no significant influence of the dynamicalhistory, i.e. the quasi-linear ramp rate, inthe steady-state mechanical pressure. Thestationarity of the flow for each apparentshear rate step can be analysed with re-spect to the existence of mechanical pres-

155

p M /

a.u

.

0

50

100

150

200

250

p M /

a.u

.

Time / s1030 2 4 6 8 10 12

0

50

100

150

200

250

p M /

a.u

.

Tr1Tr2Tr3Tr4

0

50

100

150

200

250

lh=H

lh=6H

lh=20H

Onset (1)

Onset (1)

Secondary flow

3.9 20 39 59 79 98 118 137 0 0314295275255236216196157

A

0 2000 4000

Time / s

Onset (1)

Onset (2)

(a)

Ax2 t

Onset (2)# = # =

(b)

Figure 3. (a) Influence of of hole depth, lh on the time dependent in-situ mechanicalpressure for the two inout protocols used in the study, i.e. for the low shear rate andhigh shear rate range. (b) Spatio-temporal visualisation, corresponding to region A in

(a), showing the transition from the primary to the secondary instability type. Data foran LDPE having Tm = 111 ◦C, Mw = 186 kg/kmol and Mw/Mn = 12 , tested at 180 ◦C.

sure fluctuations associated to the insta-bilities observed in LDPE using recurrenceplots. Such an example is presented in Fig.4 and corresponds to the γ̇ = 79 s−1 stepof the lh = H test in Fig. 3.a. A recur-rence plot comprises of mechanical pres-sure points pMi , pMj who have an interdis-tance smaller than ∆pm/103 in a embed-ded vector. The embedding is performed inthe framework of Takens’ embedding the-orem.2 The depicted recurrence plot con-siders embedding dimensions equal to 2and 3, with the delay time being deter-mined based on previous results.14 For anunsteady flow the points are correlated onthe diagonal whereas for steady-state flowsthey spread over the adjacent domain.

Examples of transient Fourier trans-form (FT) analysis of Tr1,2,3,4 are shownin Fig. 5, corresponding to the lh = H me-chanical pressure date in Fig. 3.a for thelow shear rate test. Overall, there are twomain contributions to the FT spectra forsupercritical extrusion flows: (i) the char-acteristic pattern frequencies, i.e. the sur-face and/or volume extrudate distortionsobserved and (ii) a low frequency contri-bution to the spectra.14 Other miscella-neous identified contributions include theelectrical network frequency (50Hz for theEU) and the machine engine, e.g. see Fig.6. It can thus be seen that while the lowfrequency contributions to the spectra areidentifiable in all transducers it is difficult

156

to identify peaks that could be associatedto the inline optical observations of extru-date patterns.

In order to identify the pattern contri-butions a different approach can be appliedapplied. Thus, it is assumed that the pat-tern frequency contributions are detectedonly using the transducers positioned closeto the die exit.14 Therefore, the spectra ofTr1 can be subtracted from the Tr3,4, withan appropriate scaling coefficient. Further-more, to compensate for the shortcomingof using a linear scaling, a circular colourmap can be used. An example of suchdifferential transient FT diagrams is pre-sented in Fig. 6 for the low and high shearrate test, lh = H. The onset of extru-date pattern characteristic frequencies andthe supercritical bifurcation behaviour canthus be observed. In the case of a branchedstructure, i.e. LDPE, the nonlinear dynam-ics feature the onset of a first instabilitytype having one characteristic (extrudatepattern) frequency, fch1.14 With increasingshear rate a lower frequency component is

dE = 3

dE = 2

1050 1 2 3 4 5 6

105

0

1

2

3

4

5

6

i

j

Figure 4. Recurrence plot of the LDPEdata in Fig. 3.a, lh = H, γ̇ = 79 s−1

showing the onset of stationarity, wherei, j represent the indexes of points having

a distance smaller than ∆pm/103 in a2-dimensional and 3-dimensional

reconstructed phase space.

0 2000 4000 6000 8000Time /s

3.9 20 39 59 79 98 118 137 0

Tr1

Tr2

Tr3

Tr4

LDPE‡ 180℃, v2/v1=0, lh=H

fengine

50Hz

50Hz

50Hz

50Hz

fengine

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

Figure 5. Transient Fourier transformanalysis of the in-situ mechanical pressurefluctuations. The data corresponds to the

diagrams in Fig. 3.

superimposed on the base one, and the re-sulting pattern is identified as having twocharacteristic frequencies,14 fch1 and fch2,e.g. see also Fig. 3.b. At the highest shearrates higher harmonics of the fundamen-tal are also readily distinguishable. Overall,the increasing hole depth has a stabilisingeffect on the flow, i.e. the onset of instabil-ities is postponed. Moreover, the onset ofthe second instability type was not identi-fiable in the spectra for lh > H. Finally,it is important to note that the signal cor-responding to the onset and propagationof instabilities is read in both Tr3 and Tr4transducers regardless of the hole depth.

For contrast, similar data is briefly pre-sented for the HDPE sample mentioned.For such short chained branched/linear

157

0.001

0.002

0.003

0.004

0.005

3.9 20 39 59 79 98 118 0137

0.001

0.002

0.003

0.004

0.005

0 1000 2000 3000 4000 5000 6000 7000 8000Time /s

Tr3-bTr1

fch1

50Hz

Onset (1)

fch2

fch1

Tr4-bTr1

50Hz

fch1

Onset (1)

Onset (2)

(a)

0.001

0.002

0.003

0.004

0.005

0 200 400 600 800 1000 1200 1400 1600 1800Time /s

0.001

0.002

0.003

0.004

0.005

157 196 216 236 255 275 295 0314

Tr3-bTr1

50HzOnset (1)

Tr4-bTr1

50Hz

Onset (2)fch1

fch2

fch1

fch2

Onset (1)Onset (2)

(b)

Figure 6. Differential transient Fourier transform analysis of showing the onset ofinstabilities and supercritical bifurcations for the data in Fig. 3.a, lh = H in the data of

Tr3,4, using (a) low and (b) high shear rate experimental protocols.

molecular architectures, smooth extrudateflow is replaced with increasing appar-ent share rate with the sharkskin instabil-ity, having one characteristic pattern fre-quency. The secondary instability type de-tected is that of stick-slip for which threecharacteristic frequencies are identifiable inthe spectra.14 An example of such a tran-sition sequence is presented in Fig. 7 asspatio-temporal visualisation. The visual-isation corresponds to region B in Fig. 9.The influence of hole depth on the nomi-nal value of the mechanical pressure for theHDPE sample is shown for the low, Fig. 8,and high, Fig. 7, apparent shear rate test(lh = H). It can be observed that the in-creasing hole depth has a destabilising ef-

fect on the flow, i.e. the stick-slip regimeis triggered at lower shear rates. The ef-fect on the onset of the sharkskin insta-bility was no conclusive from the tests. Inaddition, a strong influence on the historyof deformation is observed between the lowand high apparent shear rate tests in thecase of HDPE. The normal stress differencecan be computed from the curved stream-line tests based on the hole pressure ∆ph =a1N1+a2N2 where ∆ph = pTr3−pTr4 is thehole pressure, N1 = t11− t22, N2 = t22− t33the first and second normal stress differ-ence and a1, a2 the corresponding propor-tionality coefficients. For a circular open-ing with neglectable N2 the first normalstress difference cans be determined as5

158

x1

B

Smooth extrudate Sharkskinn Stick-slip

t

Onset (1) Onset (2)

T ≅ Tm

Figure 7. Spatio-temporal visualisation corresponding to the region B in Fig. 9 showingthe transition from smooth extrudate to the primary instability, i.e. sharkskin, and

further to the secondary instability type, i.e. stick-slip.

Onset(2)Onset(1)

3.9 20 39 59 79 98 118 137 0

Onset (2)

lh=0

p M /

a.u

.

Tr1Tr2Tr3Tr4

0

100

200

300

400

500

p M /

a.u

.

0

100

200

300

400

500

lh=H

p M /

a.u

.

Time / s100 2 4 6 8 10 12

0

100

200

300

400

500 Onset (2)

lh=20H

No secondary flow

103

Figure 8. Influence of hole depth, lh onthe time dependent in-situ mechanical

pressure for the low apparent shear rateregion HDPE having Tm = 131 ◦C, tested

at 180 ◦C.

N1 ≈ 0.25 ·∆pM (a1 = 4, a2 = 0). For vali-dation, such data is compared with oscilla-tory shear measurement via the the Launrule:3

NLaun1 (γ̇) = 2G′(ω)

⎛⎝1 +

(G′(ω)

G′′(ω)

)2⎞⎠

0.7

(1)

where G′ and G” are the dynamic moduli

p M /

a.u

.

Time / s

Tr1Tr2Tr3Tr4

0 2000 40000

100

200

300

400

500

03142952752552362161961571

B lh=H

Figure 9. High shear rate mechanicalpressure time dependence for an HDPEhaving Tm = 131 ◦C, tested at 180 ◦C.

determined from oscillatory tests. Such acomparison in presented in Fig. 10 for theLDPE sample. It can be seen that a goodagreement is found for the a1 = 4 case andlh = 6H. It could thus be asserted that thefor lh = H the hole pressure recorded is notsufficient for accurate readings whereas forlh = 20H secondary flows may be respon-sible for altering the measurements.

SUMMARY AND CONCLUSIONSA high sensitivity piezoelectric system forcapillary rheometry capable of simultane-ous in-situ detection and analysis of insta-bilities and determination of the first nor-mal stress difference during polymer meltextrusion flows was briefly described in thispublication. The transition sequences de-tected in the presence of curved stream-lines were presented. The analysis of thein-situ hole pressure shows that with care-ful data processing procedures it is possi-ble to determine (at least) the first normal

159

Laun rule2G'lh = H lh = 6H lh = 20H

.

N 1 /

Pa

103

104

105

106

γ / 1/s10−1 1 101 102 103

Figure 10. Comparison between the firstnormal stress determined through

capillary rheometry with results obtainedfrom oscillatory shear tests via the Laun

rule the the LDPE sample at 180 ◦C.

stress difference using the high sensitivitysystem.

ACKNOWLEDGEMENTSThe study was financed by the German Sci-ence Foundation, DFG grant WI 1911/14-1. The authors are grateful to Dr. I. Vitto-rias of LyondellBasell and Dr. S. Filipe ofBorealis for providing the test samples.

REFERENCES1. Walters, K. (1975), “Rheometry”,

Chapman and Hall, London.2. Takens, F. (1981), “Detecting strange

attractors in turbulence”, in “LectureNotes in Mathematics, Dynamical Systemsand Turbulence”, Rand, D. and Young, L.-S.3. Laun, H. M. (1986), “Prediction of

Elastic Strains of Polymer Melts in Shearand Elongation”, Journal of Rheology, 30,459-501.4. Larson, R. (1992), “Instabilities in vis-

coelastic flows”, Rheol. Acta, 31, 213-263.5. Barnes, H. A., Hutton, J. F. and Wal-

ters, K. (1998), “An introduction to rheol-ogy”, Elsevier, Amsterdam [u.a.].6. Denn, M. (2001), “Extrusion Insta-

bilities And Wall Slip”, Ann. Rev. FluidMech., 33, 265-287.7. Baird, D. (2008), “First normal stress

difference measurements for polymer meltsat high shear rates in a slit-die using holeand exit pressure data”, J. Non-NewtonianFluid Mech., 148, 13–23.8. Palza, H., Naue, I. F. C. and Wilhelm,

M. (2009), “In situ Pressure Fluctuationsof Polymer Melt Flow Instabilities: Exper-imental Evidence about their Origin andDynamics”, Macromol. Rapid Commun.,30, 1799-1804.9. Palza, H., Filipe, S., Naue, I. and

Wilhelm, M. (2010), “Correlation betweenpolyethylene topology and melt flow in-stabilities by determining in-situ pressurefluctuations and applying advanced dataanalysis”, Polymer, 51, 522–534.10. Palza, H., Naue, I., Filipe, S., Becker,A., Sunder, J., Göttfert, A. and Wilhelm,M. (2010), “On-line Detection of Poly-mer Melt Flow Instabilities in a Capil-lary Rheometer”, Kautsch. Gummi Kun-stst., 63, 456-461.11. Kádár, R. and Balan, C. (2012), “Tran-sient dynamics of the wavy regime inTaylor-Couette geometry”, Eur. J. Mech.B. Fluids, 31, 158 - 167.12. Teixeira, P. F., Hilliou, L., Covas,J. A. and Maia, J. M. (2013), “Assessingthe practical utility of the hole-pressuremethod for the in-line rheological charac-terization of polymer melts”, Rheol. Acta,52, 661-672.13. Ratzsch, K.-F., Kádár, R., Naue,I. F. C. and Wilhelm, M. (2013), “ACombined NMR Relaxometry and SurfaceInstability Detection System for PolymerMelt Extrusion”, Macromol. Mater. Eng.,298, 1124-1132.14. Kádár, R., Grosso, M., Naue, I. F.and Wilhelm, M. (2014), “Reconstructeddynamics of in situ mechanical pressurefluctuations during the extrusion flow ofpolyethylene melts”, (Manuscript in prepa-ration).

160