LAOS is a convenient tool to measure the nonlinear viscoelastic behaviour of viscoe-lastic materials. Various viscoelastic models have been studied to interpret or predict nonlinear viscoelastic behaviours under LAOS. Nonlinear viscoelastic models con-sist of linear and nonlinear viscoelastic pa-rameters. Linear viscoelastic parameters are able to be determined by use of several al-gorithms which have been suggested by many researchers. However, there is no effi-cient way to determine the nonlinear param-eter for LAOS data. Previous researchers determined the nonlinear parameter relying on the other viscoelastic tests such as steady shear test. However, it was found that de-termined parameters cannot provide qualita-tive predictions for LAOS data1,2. Dynamic regression is suggested by Calin et al.3 to determine the parameter from the LAOS data, but it demands a lot of numerical computations and experimental data.

It is obvious that the analytical solutions of viscoelastic models help to determine nonlinear viscoelastic parameters for LAOS flow as well as understand the nonlinear behaviour of the materials. Analytical solu-tions for nonlinear viscoelastic models have been studied by use of power series expan-sion or asymptotic solutions4,5. However, there exists inevitable limitation comes from radius of convergence. Consequently, the solutions are valid for restricted region of oscillatory amplitudes.

Various nonlinear viscoelastic models have been studied to investigate the nonlin-ear behaviour of viscoelastic materials. The Giesekus and the PTT models are popularly

used to elucidate the nonlinear flow under LAOS. The constitutive equations of the Giesekus model can be written in terms of extra stress T as below:

DTTG

TT G21

=⋅λ

α+

λ+

∇

(1)

where ∇ means upper convected deriva-tives and α is the nonlinear parameter the model. G is modulus and D is deformation rate tensor. It is known that α of Giesekus model is a constant between 0 and 1. The extra stress of PTT model can be written as

( ) DTTT GG

2trexp1

=⎥⎦

⎤⎢⎣

⎡α

λ+

∇

(2)

It is known that α of PTT model is order of

210 − for solution system and 110 − for pol-ymer melt system6. Bae and Cho7 deter-mined α of Giesekus and PTT models as 0.54 and 0.14 for PEO aqueous solution based on the semianalytical equations of LAOS.

Cho8 adopted the perturbation method to calculate analytical solutions of the non-separable viscoelastic models. The per-turbation method is applied to calculate the analytical solutions of the Giesekus and the PTT models. It is remarkable that it opens a new way to calculate the analytical solution of not only shear stress but also normal stress for LAOS flow.

When Gα≡ε = 0, the solutions of the Giesekus and the PTT models 0T are equiv-

Analytical Solutions of the Giesekus and the Phan-Thien and Tanner Models for LAOS

Jung-Eun Bae and Kwang Soo Cho

Department of Polymer Science and Engineering, Kyungpook National University, Daegu,

Korea 41566

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. 24, 2016

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 flowsis here presented. The main goals of thestudy are to analyse the nonlinear dynam-ics of extrusion instabilities and to deter-mine the first normal stress difference inthe presence of an induced streamline cur-vature via the so-called ’hole effect’. Anoverview of the system, general analysisprinciples, preliminary results and overallframework are herein discussed.

INTRODUCTIONCapillary rheometry is the preferredrheological characterisation method forpressure-driven processing applications,e.g. extrusion, injection moulding. Themain reason is that capillary rheometry isthe only method of probing material rheo-logical properties in processing-like condi-tions, i.e. high shear rate, nonlinear vis-coelastic regime, albeit in a controlledenvironment and using a comparativelysmall amount of material.1 Thus, it isof paramount importance to develop newtechniques to enhance capillary rheome-ters for a more comprehensive probing ofmaterial properties. Extrusion alone ac-counts for the processing of approximately35% of the worldwide production of plas-tics, currently 280⇥ 106 tons (Plastics Eu-rope, 2014). This makes it the most im-portant single polymer processing opera-

tion for the industry and can be found ina variety of forms in many manufacturingoperations. Extrusion throughput is lim-ited by the onset of instabilities, i.e. prod-uct defects. Comprehensive reviews on thesubject of polymer melt extrusion insta-bilities can be found elsewhere.4,6 A re-cent method proposed for the detectionand analysis of these instabilities is that ofa high sensitivity in-situ mechanical pres-sure instability detection system for cap-illary rheometry.8,10 The system consistsof high sensitivity piezoelectric transducersplaced along the extrusion slit die. In thisway all instability types detectable, thusopening new means of scientific inquiry. Asa result, new insights into the nonlinear dy-namics of the flow have been provided.9,14

Moreover, the possibility of investigatingthe reconstructed nonlinear dynamics wasconsidered, whereby a reconstructed phasespace is an embedding of the original phasespace.2,14 It was shown that a positive Lya-punov exponent was detected for the pri-mary and secondary instabilities in lin-ear and linear low density polyethylenes,LDPE and LLDPE,.14 Furthermore, it wasdetermined that Lyapunov exponents aresensitive 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

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alent to that of the upper convected Maxwell (UCM) model. Because G is sufficiently large for polymer melts or concentrated polymer solutions, 0T can be remedied by the perturbation method with parameter ε .

!+ε+ε+= 22

10 TTTT . (3)

Substitution of lower order solutions pro-vides the first and the second solutions of the Giesekus model as

G1

2011 RTTT −≡−=+λ

∇ (4)

G2011022 RTTTTTT −≡⋅−⋅−=+λ

∇. (5)

The constitutive equations of the PTT model provide the analytical solutions as below:

( ) PTT10011 tr RTTTT −≡−=+λ

∇ (6)

( ) ( )PTT2

1000122 tr2tr

R

TTTTTTT

−≡

−+−=+λ∇

(7)

The perturbation approximation can be

easily calculated by use of integrating transform8. It is remarkable that this ap-proach facilitates systematic calculation for analytical solutions of normal stress differ-ences as well as shear stress under LAOS.

REFERENCES 1. Giacomin, A. J., Jeyaseelan, R. S.,

Samurkas, T. and Dealy, J. M. (1993), “Validity of separable BKZ model for large amplitude oscillatory shear”, J. Rheol., 37, 811.

2. Nam, J. G., Ahn, K. H., Lee, S. J., and

Hyun, K. (2010), “First normal stress difference of entangled polymer solu-tions in large amplitude oscillatory shear flow”, J. Rheol., 54, 1243.

3. Calin, A., Wilhelm, M., Balan, C.

(2010), “Determination of the nonlinear parameter (mobility factor) of the Giesekus constitutive model using LAOS procedure”, J. Non-Newtonian Fluid Mech., 165, 1564.

4. Gurnon, A. K. and Wagner, N. J.

(2012), “Large amplitude oscillatory shear (LAOS) measurements to obtain constitutive equation model parameters: Giesekus model of banding and non-banding wormlike micelles”, J. Rheol., 56, 3331.

5. Khair, A. S. (2016), “Large amplitude

oscillatory shear of the Giesekus mod-el”, J. Rheol., 60, 257.

6. Baik, E. S., “Model predictions of large

amplitude oscillatory shear behavior of complex fluids,” thesis, Seoul Na-tionalUniversity Library, Seoul, 2006.

7. Bae, J.-E., and Cho. K. S., (2015)

“Semianalytical methods for the deter-mination of the nonlinear parmeters of nolinear viscoelastic constituttive equa-tions from LAOS data,” J. Rheol., 59, 525.

8. Cho, K. S. (2015) “Viscoelasticity of

Polymers”, Springer

J.-E. Bae and K. S. Cho

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