198 Applied RheologyVolume 16 · Issue 4

Viscoelastic Effects in Multilayer Polymer Extrusion

Patrick D. Anderson1*

, Joseph Dooley2, Han E.H. Meijer

1

1 Materials Technology, Eindhoven University of Technology, P.O. Box 513, WH 4.142,5600 MB Eindhoven, The Netherlands

2 The Dow Chemical Company, 433 Building, Midland, Michigan, USA

* e-mail: p.d.anderson@tue.nlFax: x31.40.2447355

Received: 25.2.2006, Final version: 31.5.2006

Abstract:

The effect of viscoelasticity on multilayer polymer extrusion is discussed. In these coextrusion processes prede-termined patterns are created with a remarkable breadth of complexity even in geometrically simple dies viaelastic rearrangements caused by the second-normal stress differences. A computational method is offered,based on the mapping method, which quantitatively describes the flow-induced patterns. Besides that theresults are esthetically beautiful, they are also relevant for practice, since process and die design optimizationis now possible. Not only to minimize interface distortion, but potentially also to deliberately create new process-es and products based on this flow-induced patterning of polymers.

Zusammenfassung:

Der Einfluss der Viskoelastizität auf die Mehrschichtextrusion bei Polymeren wird diskutiert. Bei diesen Koex-trusionsprozessen werden vorher bestimmte Muster mit einer bemerkenswerten Komplexitätsvielfalt erzeugt,sogar in geometrisch einfachen Düsen mit Hilfe elastischer Umlagerungen, die durch die zweite Normalspan-nungsdifferenz erzeugt werden. Eine numerische Methode, die auf der Abbildungsmethode basiert, wird dar-gestellt, die die strömungsinduzierten Muster quantitativ beschreibt. Die Resultate sind nicht nur ästhetischschön, sondern auch für die Praxis relevant, da die Optimierung des Prozess- und Düsendesigns nun möglich ist– nicht nur um die Verzerrung der Grenzfläche zu minimieren, sondern um möglicherweise auch gezielt neueProzesse und Produkte zu schaffen, die auf der strömungsinduzierten Musterbildung bei Polymeren basieren.

Résumé:

L’effet de la viscoélasticité sur l’extrusion de polymères multicouches est discuté. Dans ces procédés de co-extru-sion, des empreintes prédéterminées sont créées qui possèdent une gamme remarquablement étendue de com-plexité et ceci même pour des filières à géométrie simple. Ces empreintes sont associées à des réarrangementsélastiques causés par la seconde différence de contraintes normales. Une méthode de calcul numérique est pro-posée, basée sur la méthode de cartographie qui décrit quantitativement les empreintes induites par l’écoule-ment. En plus de l’obtention de résultats esthétiquement magnifiques, ceux-ci sont pertinents pour ce qui concer-ne la pratique, puisque l’optimisation des procédés ainsi que des conceptions des filières est maintenant possible.Non seulement est-il possible de minimiser les distorsions d’interface, mais aussi de nouveaux procédés et pro-duits peuvent être potentiellement créés de manière délibérée grâce à cette technique d’empreintes de poly-mères induites par l’écoulement.

Key words: extrusion, rheology, secondary flow, normal stresses, pattern formation

1 INTRODUCTIONMultilayer coextrusion is a process in whichpolymers are extruded and joined together in afeedblock or a die with the purpose of forminga single structure with multiple layers. Theattraction of coextrusion is both economic andtechnical as it is a single-step process. Startingwith two or more polymer materials, that aresimultaneously extruded and shaped in a sin-gle die, a multilayer sheet or film can be formed.Coextrusion avoids the costs and complexitiesof conventional multistep lamination and coat-

ing processes, where individual plies must bemade separately, and then primed, coated, andlaminated. Moreover, coextrusion readilymakes it possible to manufacture productswith layers thinner than can be made and han-dled as an individual ply. Consequently, onlythe necessary thickness of a high performancepolymer is used to meet a particular specifica-tion of the product. In fact, coextrusion hasbeen used commercially to manufactureunique films consisting of hundreds of layerswith individual layer thicknesses less than 100

© Appl. Rheol. 16 (2006) 198–205

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nm. In an academic setup in our Eindhoven lab-oratory even individual layer thicknesses lessthan 40 nm have been reached using a multi-flux static mixer. It is difficult to imagine anoth-er practical method of manufacturing thesenanolayered structures.

Layers may be used to place colors, bury recy-cle, screen ultraviolet radiation, or to provide bar-rier properties, minimize die-face buildup, and tocontrol film-surface properties, for example.Additives, such as antiblock, antislip, and anti-static agents, can be placed at specific layer posi-tions. High melt strength layers can carry lowmelt strength materials during fabrication.Unfortunately, the best designed die or feed-block does not necessarily ensure a commercial-ly acceptable product. Layered melt streamsflowing through a coextrusion die can spreadnonuniformly or can become unstable leading tolayer nonuniformities and even intermixing oflayers under certain conditions. The causes ofthese instabilities are related to non-Newtonianflow properties of polymers and viscoelasticinteractions.

Coextruded layers normally should haveuniform thicknesses throughout the structurefor optimal performance. However, layer thick-ness non-uniformities have been observed inmany commercial coextruded structures. Previ-ous work has shown that layer thickness varia-tions can occur for many reasons.

Several of the primary causes of layer thick-ness variations are interlayer instabilities, vis-cous encapsulation, and elastic layer rearrange-ment. Interfacial instability is an unsteady-stateprocess in which the interface location betweenlayers varies locally in a transient manner. Vis-cous encapsulation is a phenomenon in which aless viscous polymer will tend to encapsulate amore viscous polymer as they flow through achannel. Elastic layer rearrangement occurswhen elastic polymers flow through non-radial-ly symmetric geometries producing secondaryflows which drive rearrangement of the layerthicknesses. These layer thickness variationshave been studied experimentally [1 - 4] andnumerically [5 - 9].

Interfacial instability in a number of coex-truded polymer systems has been experimental-ly correlated with viscosity ratios and elasticityratios [10], and a simplified rheology review hasbeen given [11]. Other studies have looked at vis-

cosity differences [12 - 14], surface tension [15],critical stress levels [16 - 18], viscosity model para-meters [19 - 21], and elasticity [22, 23]. The workof Dooley clearly separates the effects interlayerinstabilities, viscous encapsulation, and elasticlayer rearrangement on layer thickness varia-tions via a systematic approach [24].

In this paper we examine the layer unifor-mity of coextruded structures with similar vis-cosities. The finite element method is used tosimulate the flow of viscoelastic polymers in dif-ferent channel geometries and the location ofthe coextruded interface is determined using themapping method. Finally, a quantification of theinfluence of second-normal stress differences onthe resulting secondary flow is given.

2 MATHEMATICAL EQUATIONS ANDNUMERICAL METHODThe momentum and continuity equation for thesteady state flow of an incompressible vis-coelastic fluid are given by

(1)

(2)

where p is the pressure field, tt is the viscoelasticextra-stress tensor, and u is the velocity field.Inertial and volume forces are assumed to benegligible.

If a discrete spectrum of N relaxation timesis used then tt can be decomposed as follows:

where tti is the contribution of the ith relaxationtime to the viscoelastic stress tensor. For theextra stress contributions tti, a constitutive equa-tion must be chosen.

A realistic viscoelastic equation that at leastdescribes second-normal stress differences inflow is the Giesekus constitutive equation thattakes the form

(3)

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where li is the relaxation time, hi is the viscosityfactor of the ith mode. D is rate of deformationtensor,

and the upper triangle stands for the upper-con-vected time derivative operator defined as,

(4)

The symbol I denotes the unit tensor. In Eq. 3, aiare additional material parameters of the model,which control the ratio of the second to the first-normal stress difference. In particular, for lowshear rates, a1 = - 2N2/N1, where a1 is associatedwith the highest relaxation time li and N1, N2 rep-resent the first and second-normal stress differ-ence, respectively. The set of partial differentialEqs. 1 - 3 are completed with initial and boundaryconditions.

In the community of viscoelastic flow com-putations great effort has been made to developnumerical methods to accurately solve the fullset of equations for increasing levels of elastici-ty. In the absence of a solvent viscosity themomentum equation looses its ellipticity whichhas direct consequences for any finite elementmethod. The constitutive equation is a hyper-bolic partial differential equation which requiresspecial attention. Two suitable techniques todeal with the constitutive equations are discon-tinuous Galerkin (DG) and streamline upwinding(SUPG). In this work the latter method is used.

One of the standard approaches to retainellipticity of the momentum equation is theDEVSS formulation, see Problem 1 below. A greatadvantage of this method over other techniques(like EVSS) is that the objective derivative of therate of strain tensor is avoided and the methodis not restricted to a particular class of constitu-tive equations. In the discrete momentum Eq. 6,an elliptic operator 2 h–(D - D–) is introduced, whereD– is a discrete approximation of the rate-of-straintensor D obtained from Eq. 8. If the exact solu-tion is recovered, this elliptic operator vanishes.However, in a finite element calculation this isgenerally not the case.

Problem 1 (DEVSS-G/SUPG)Find G, tt, u, p such that for all admissible weight-ing functions E, S, v, and q:

(5)

(6)

(7)

(8)

If DEVSS-G/SUPG and a continuous interpolationof the extra stress tensor are used, the strain ratetensor (or the velocity gradient tensor) is inter-polated in the same way as the extra stress ten-sor. The most commonly applied element for theDEVSS-G/SUPG method has a bi-quadratic veloc-ity, bi-linear pressure, stress and strain rate (orvelocity gradient) interpolation. This combina-tion of interpolation spaces is also used in thiswork.

3 THE MAPPING METHODSeveral computational techniques can be used totrack interfaces in fluid flows [25, 26]. These tech-niques can be separated into two broad cate-gories: front capturing and front tracking. In thefront capturing technique, massless markers aredistributed within the fluid domain or a markerfunction is advected with the flow. The most dif-ficult task within these approaches is to deter-mine the location of the interface. Usually, it isrecovered or “captured” using the calculated val-ues of the marker function. The front trackingmethod on the other hand uses a separate mov-ing mesh to describe the interface. Its location isaccurately known at each time step. A disadvan-tage of both the front capturing and front track-ing technique is that all computational work hasto be repeated when the initial location of theinterface changes. For passive interfaces, anoth-er method can be used to determine the evolu-tion of the coextruded interface: the mappingmethod [27, 28].

The mapping method is based on fronttracking, but the main difference is that this tech-nique does not track each material volume overthe total length z of the flow separately but

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instead creates a discretized mapping from a ref-erence grid to a grid deformed during a relative-ly short representative length span Dz of the flow[29]. In the mapping method, the flow domain isdivided into non-overlapping subdomains withboundaries. This subdivision is fully decoupledfrom any velocity field discretization. The bound-aries of the subdomains are tracked, using fronttracking, in a flow field over a distance fromz = z0 to z = z0 + Dz. Then the mapping from theinitial grid to the deformed grid is constructed bydetermining the fraction of each cell of thedeformed grid back in to the initial cells. Theresult is stored in a matrix Fwith size n¥n, wheren is the number of cells in the grid. The accuracyof the method depends on the accuracy of thevelocity field, and on the grid size n and mappingstep Dz [27].

Once the matrix F is known, the advectionof any initial color distribution, stored in a vectorC0 of length n, is easily determined by the matrixvector multiplication providing the structureafter Dz, yielding C1. The advection after 2Dz, C2,is determined by multiplying F with C1, and so onfor NDz to arrive at the total flow length Dz. Usingthis technique, the interface location for coex-truded structures with identical materials in eachlayer can easily be determined. In this work themapping method is applied to determine the pro-gression of the coextruded interface for thesquare and rectangular channel geometries.

4 RESULTS

4.1 FLOWFigure 1 shows the predicted secondary flows ina square and in a rectangular channel for a vis-coelastic fluid. For the fluid flowing through a

square channel contain eight recirculation zonesor vortices of roughly the same size are found,two each per quadrant. For the flow in the rec-tangular channel four larger and two smaller vor-tices appear. The flow patterns appear to corre-spond well with the interface deformationshown for the polystyrene resin in Fig. 2 whereexperimental results are shown for the flow in asquare and rectangular channel [30, 31]. Here,two-layer coextruded structures were madeusing the same polymer in each layer with dif-ferent colored pigments added to each layer todetermine the location of the interface. A seriesof experiments were conducted that showedthat the addition of the pigments at the loadingsused in these experiments did not affect the flowproperties of the resins and that the startinginterface was indeed flat [32]. The interestingaspect of the layer rearrangement shown in Fig.2 is that everywhere in the duct the same mate-rial flows and that thus viscous encapsulation isnot the driving force. In case of viscous encapsu-lation the materials flow down the channel as theless viscous material encapsulates the more vis-cous material and an energetically preferredstate is reached.

4.2 APPLICATION OF THE MAPPING METHODThe polystyrene resin of Figs. 1 and 2 was simu-lated using the Giesekus constitutive Eq. 3. Basedon the dynamic rheological properties of thepolystyrene resin at a temperature of 204˚C, a

Figure 1 (left):Secondary flow in the axialcross section of the slit.

Figure 2:Experimental results show-ing the evolution of materi-al in a square and rectangu-lar channel.

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Based on the determined velocity field themapping technique was applied and Figs. 3 and 4illustrates they way the technique works by show-ing the deformation of a (in this example verycoarse) grid consisting of 20 ¥ 20 cells after 5 L/Dand 10 L/D, where D is the width of the channel andL its length. The deformation of the grid follows thesecondary flow field patterns shown in Fig. 1. Notethat a comparison of a single cell in the three gridsshows that the area of a cell is not preserved dur-ing deformation, however the flux through eachcell is preserved. The actual computations are doneon a much finer grid, i.e. 400 ¥ 400 cells, and themapping matrices created after 5 L/D and 10 L/D arecombined to determine the evolution.

4.3 MAPPING RESULTS FOR THE SQUARECHANNELResults of the mapping operation on different ini-tial layer configurations in the square channel are

Table 1:Material parameters for the Giesekus model with five relaxation times forpolystyrene.

Figure 3 (left above):The figure on the left shows a coarse initial mapping grid consisting of 20 ¥20 cells. The figures in the middle and the right show the deformation of thegrid in (a) after 5 L/D and 10 L/D respectively. The actual mapping grid con-tains 400 ¥ 400 cells.

Figure 4 (left below):Schematic figure showing the initial grid and the deformed grids after L/D = 5and L/D = 10 in the channel.

Figure 5 (right above):Interface deformation evolution of six different initial (column wise) struc-tures in a square channel. Each subsequent figure in a row shows the progres-sion after a multiple of 5 L/D. The last column contains the extruded struc-ture after 50 L/D. The flow rate was 0.72 cc/s.

Figure 6 (right below):Interface deformation evo-lution of six different initial(column wise) structures in asquare channel. Each subse-quent figure in a row showsthe progression after a mul-tiple of 5 L/D. The last col-umn contains the extrudedstructure after 50 L/D. Theflow rate was 2.48 cc/s.

discrete spectrum of five relaxation times li rang-ing from 10-2 to 102 seconds was chosen. The cor-responding partial viscosities hi were fitted onthe basis of the dynamic properties of the stor-age and loss moduli (G’ and G’’, respectively)while the ai parameters were selected based onthe viscosity [33]. Table 1 shows the values for thematerial properties used in the model. Two dif-ferent volumetric flow rates were considered: alow flow rate of 0.72 cc/s and a high flow rate of2.48 cc/s.

A finite element method was applied todetermine the velocity field in the channels andthe DEVSS/G-SUPG technique in combinationwith a theta-scheme was used to march in timeto the steady state in the channel. Details of thismethod can be found in [34]. The finite elementmesh consisted of 20 ¥ 20 ¥ 2 elements and thevolumetric flow rate was prescribed via aLagrange multiplier.

I li hi ai [s] [Poise]

1 102 23740 0.862 101 97670 0.483 100 163500 0.564 10-1 44640 0.535 10-2 8540 0.51

L/D=10

L/D=5

L/D=0

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shown in Fig. 5 (low flow rate) and Fig. 6 (high flowrate), respectively. These initial layer configurationsare based on experimental structures developedand evaluated previously [24]. These results showthe effect of secondary flows on viscoelastic poly-mers as they flow through a square channel. Allresults are computed quickly (typically less than 1CPU second on a standard PC) once the mappingmatrix is determined. As was concluded fromexperimental results in previous work, the depen-dence of layer deformation on the flow rate is small[24]. Furthermore, it can be concluded that by apply-ing the mapping method, the details of the defor-mation patterns are accurately resolved.

4.4 MAPPING RESULTS FOR THE RECTANGULARCHANNEL

Figure 7 shows the results of the layer interfacedeformation in the rectangular channel. Theseresults are similar to those in the square channelbut the location and magnitude of the secondaryflows have changed. As the aspect ratio of thechannel gets larger, the recirculation zone asso-ciated with the longer side becomes larger whilethe recirculation associated with the smaller sideis diminished. This shows how the secondaryflows are affected by the aspect ratio of the chan-nel. We have not investigated the influence ofthe aspect ratio of the channel in great detail. Thetopological structure of the secondary flowchanges upon increase of the aspect ratio; theeffect of the second-normal stresses can bereduced greatly. The effect of more practical dieshapes, such as circular, teardrop, and leakingrectangular shaped channels was investigatedexperimentally already by Dooley [24].

4.5 COMPARISON OF EXPERIMENTAL ANDCOMPUTATIONAL RESULTS

Figure 8 shows experimental and numericalinterface deformations for the progression of a

49-strand polystyrene structure as it flows downa square channel. The top row images are exper-imental samples showing the progression of a49-stand polystyrene structure as it flows downthe square channel. The bottom row are thenumerical predictions at the same channel crosssections. The first column are images taken nearthe entry of the channel while the images in thelast column are located near the exit of the chan-nel.

This figure shows excellent agreementbetween the experimental and numericalresults. This is the most complex structure of thethree examined since an interface location mustbe determined for each of the 49 strands as theyflow down the channel. However, even thoughthis is the most complex structure, it appears thatthe mapping method was able to predict thestrand deformation very accurately at each loca-tion down the channel.

4.6 INFLUENCE OF SECOND-NORMAL STRESSDIFFERENCEIn order to determine whether the present exper-imental technique could, ultimately, be used asa measure of the second-normal stress differ-ence at high flow rates, a number of simulationswere performed with different levels of vis-coelasticity. The square channel geometry waschosen for these simulations since it producesstrong secondary flow effects. For these simula-tions, the channel width and height were set to1, the dimensionless flow rate was set to 1 as wasthe relaxation time. The level of viscoelasticitywas changed by changing the non-linear a para-meter in the single mode Giesekus model from0.2 to 0.8 with steps of 0.2. The mapping matri-ces were computed for L/D = 10. Mapping wasapplied up to 20 times, yielding results up to L/D= 200. The initial (color) distribution was chosensuch that the most reliable deformation patterns(from the corners to the center) were visualized.

The results of these simulations are shown inFig. 9. The pointed parts of the lines on the diago-nals are numerical artifacts that could be solvedby grid refinement. It can be concluded fromnumerical experiments like these that the relative

Figure 7 (left):Interface deformation evo-lution of three different ini-tial (column wise) structuresin the rectangular channel.Each subsequent figureshows the progression aftera multiple of 5 L/D. The lastcolumn contains the extrud-ed structure after 50 L/D.The flow rate was 2.48 cc/s.

Figure 8:Comparison of experimen-tal (top row) and numericalresults (bottom row) for theprogression of a 49-strandpolystyrene structure as itflows down a square chan-nel. Excellent agreement isobserved.

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difference in the refractive indices of the differ-ent structures to control optical properties.

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level of viscoelasticity can indeed be determinedbased on the amount of layer deformation causedby second-normal stress difference driven flows.These results indicate that the technique can beused to determine the second-normal stress dif-ference under realistic flow rates by applying hetechnique in an inverse way. It is suggested bythese results that (for this flow rate) a channellength of 50 < L/D < 100 should be chosen to dif-ferentiate the level of viscoelasticity. Via an itera-tive numerical and experimental approach, theseflows could even help to design improved consti-tutive equations that can quantitatively predictsecond normal stresses over a broad range ofdeformation rates. Note that the secondary flow-induced deformations are (sensitive) integrals ofthe velocity field over time, rather than (non-sen-sitive) differentials, which are stresses.

5 CONCLUSIONSThe mapping method was used to determineinterface locations for coextruded polystyrenestructures as they flow through square and rec-tangular channels. Excellent agreement betweenthe experimental and numerical results wasobtained. These results show the power of themapping technique in determining interfacedeformation in monolithic coextruded structures.Moreover, the simulation results suggest that thetechnique can be used to determine the second-normal stress difference under realistic flow rates.

Clearly, the flow-induced patterningmethod which was demonstrated here would bemore useful if part of the polymer is providedwith additional functionality, such as electricalconductivity to be used for LED or capacitor appli-cations. Other possibilities include a controlled

Figure 9:Numerical simulations com-paring layer rearrangementas a function of distancetraveled down a squarechannel. The viscoelasticitywas changed by changingthe non-linear a parameterin the one mode Giesekusmodel from 0 to 0.8 withsteps of 0.2.

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