Palierne Model

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POLYMER ENGINEERING AND SCIENCE, NOVEMBER 2002, Vol. 42, No. 11 2107

INTRODUCTION

The relationship between rheology and morphologyof emulsions and polymer blends is of impor-tance, theoretically and experimentally, in order tounderstand the evolution of the two-phase morphol-ogy during mixing, since this morphology governs the

final properties of the end product. By choosing the ad-equate composing polymers for the blend, it is possibleto obtain a large range of properties. Blend propertiesin the liquid/liquid phase are generally influenced byvarious factors such as the rheological characteristicsof the components, composition, interfacial tension, do-main structure etc. (13). For the linear regime of theviscoelastic behavior, the Palierne model (4) is knownto give rather good results in predicting the blendproperties from the component properties and the

Linear Viscoelastic and Transient Behavior

of Polypropylene and Ethylene Vinyl Acetate Blends:

An Evaluation of the Linear Palierne

and a Nonlinear Viscoelastic Model

for Dispersive M ixtures

SANDEEP TYAGI andANUP K. GHOSH*

Centre for Polymer Science an d En gineering

India n In sti tute of Technology, Delhi

Hau z Khas, New Delh i -11001 6, India

and

P. MONTANARI, G. W. M. PETERS, andH. E. H. MEIJER

Ma terials Techn ology

Dutch Polymer Insti tut e

Eindh oven Universi ty of Technology

P.O. Box 513, 56 00 MB Eind hoven, The Netherland s

Blends of polypropylene/ethylene vinyl acetate (PP/EVA) have been investigatedfor linear and transient characteristics. The emulsion model developed by Paliernein 1990 is used to characterize the linear viscoelastic properties of the blends.PP/EVA blends with the viscosity ratio of 0.26 and different compositions, such as90/10, 80/20 and 70/30 wt% PP/EVA have been studied. It was found that thePalierne Model predicts well the linear behavior of all the compositions studied. Atlow frequencies, some deviation in dynamic moduli was found in case of the 70/30composition. Structural changes are studied during transient shear flow (step-up)experiments. A nonlinear rheological model for blends, developed by Peters, Hansen

and Meijer (PHM model), is used to describe these transient rheological data. Over-shoots and undershoots observed in the experimental data are compared to numer-ical results obtained with PHM model and explained on the basis of the deforma-tion of the dispersed phase. A modification of the model is proposed in order to geta better description of the behavior of the viscoelastic blend. Predictions of the mor-phological evolution of the blends under stepwise increase in shear rate experi-ments were calculated from the modified model and are found to describe thebreak-up phenomenon under moderately high shear flow.

*Corresponding author: Email: [email protected]


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fraction of the dispersed phase. A vast amount ofwork has been carried out on immiscible blends sub-jected to the nonlinear regime shear flow and variousmodels have been proposed to understand the behav-ior of the blends during shear flow (512). Consider-able efforts were made to understand the mechanismof droplet deformation, break-up and coalescence(1316). Doi and Ohta (7) proposed a constitutive

model to describe the rheological properties of two-phase systems with Newtonian constituents, havingequal viscosities, mixed in a nearly equal ratio. In thismodel, droplet deformation, coalescence and break-upwere found to be dependent upon the stresses arisingfrom the interfacial tension. The model was extended,allowing for a wider range of blend properties, by Leeand Park (17), and their model was found to describethe change in morphology of PS/PE blend during dy-namic oscillatory flow rather well. Vinckier et al. (9,10) and van Puyvelde (11) carried out extensive workon the micro-structural analysis and rheology of amodel blend (PIB/PDMS) system. They correlated theevolution of morphology during start-up of shear andoscillatory flow, with the first normal stress differenceand dynamic moduli of the blends.

Peters et al. (18) recently proposed an extended ver-

sion of the Lee-Park model, which incorporates themorphology-rheology relationship of the blends. Themodel was developed to study the shear and normalstresses in liquid-liquid systems using existing modelsfor the evolution of the deformation of the dispersedphase, which was incorporated in order to remove theadjustable parameters of the Lee-Park model. Themodel predicts the viscosity and the first normalstress difference for blends if the size and volumefraction of the dispersed phase and rheological char-acteristics of the blend components are known. As themodel was originally developed for Newtonian liquid-liq-uid systems, the present work includes modificationsto incorporate shear thinning behavior and normalstresses of the components. In the present study, lin-ear and nonlinear viscoelastic behavior, with the cor-responding evolution of morphology in the case of

step-up experiments, have been investigated. The ma-terial used was a PP/EVA blend with different compo-sitions. Comparisons have been made with the simu-lations of both the Palierne Model, for small strainoscillatory flow, and the PHM Model, for step-up inshear rate experiments.

RHEOLOGICAL MODELS USEDPaliernes Model

Measurements under small amplitude oscillatoryflow provide useful information for describing the mi-croscopic state of a blend. Paliernes Model (4) hasbeen found to be successful in predicting the linearviscoelastic behavior of blends, under these flow condi-tions. In this model, viscoelasticity of both phases, the

hydrodynamic interactions, the droplet size and size

distribution and the interfacial tension are included.Effects of gravity and inertia are neglected. For thecase of an emulsion of two viscoelastic phases Pal-iernes Model can be written as:

(1)

where,

,

iis the volume fraction of the dispersed phase with aradius R; Gm*, Gd*, Gb* stand for the complex shearmoduli of the matrix, the dispersed phase and theblend respectively, at angular frequency and is theinterfacial tension between the two polymer blendcomponents. Paliernes Model predicts Gb*, withoutusing any adjustable parameter, once the variables

Gm*, Gd*, , and Rare known. The morphology of theblend affects the rheological properties to a large ex-tent and for some concentrated systems Paliernes

Model fails to predict the viscoelastic properties (19).

Peters-Hansen-Meijer Model

A new constitutive model for liquid-liquid systemswas developed by Peters et al. (18). It is based on uni-fying present phenomenological models and applyinga separate description of the microstructure of thedispersed phase.

According to Mellema and Willemse (20), whoadopted Batchelors approach, the bulk stress in adispersion, T, can be written for Stokes flow as,

(2)

where the angular brackets denote a quantity aver-aged over the local volume V, having the limitationthat it contains many droplets and that its statisticalproperties vary negligibly, Tsis a structure dependentstress resulting from interfacial tension, cis the vis-cosity of the continuous phase and T is referred to asthe viscosity ratio term (17), but it too is structuredependent.

The viscosity ratio term can be written in terms of aseries of surface integrals as

(3)

in which d is the viscosity of the dispersed phase, A iis the interfacial area of an individual droplet in V, dsis a differential element of the droplet interface, nis

the unit normal to the interface directed into the con-tinuous phase, and uis the local fluid velocity, includ-ing disturbance terms due to the dispersed phase, ofthe flow. The summation indicates that the quantity istaken over all the droplets, indexed by i, within V.

T

1d c

21

Va

A i1un nu

2ds

T PI 2cD Ts T

Gm* 1 2 4 316Gm*1 2 19Gd*1 2 4 3Gm* 1 2 4 316Gm* 1 2 19Gd* 1 2 4

Hi1 2 41>R2 32Gm*1 2 5Gd*1 2 4 3Gd*1 2

401>R2 3Gm*1 2 Gd* 1 2 4 32Gd* 1 2

Gb*1 2 Gm*1 21 3iHi1 21 2iHi1 2

Sandeep Tyagi et al.

2108 POLYMER ENGINEERING AND SCIENCE, NOVEMBER 2002, Vol. 42, No. 11


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The structure dependent stress, Ts, reflecting theanisotropy of the disperse phase microstructure, canbe expressed as, (21),

(4)

in which the interfacial tension.

The viscosity ratio term contains two contributions.An elastic component due to the interfacial tensionand another, pure viscous term, containing the rate ofdeformation tensor:

(5)

for high volume fraction, of the dispersed phase.The elastic forces resulting from interfacial stress

and unequal viscosities are combined through TETs, with

(6)

T is determined by

(7)

where Qis the total interfacial area per volume V, T s(1)

is the covariant (or lower) convected derivative of Tsand 4L is a fourth order tensor given by

(8)

The relaxation time for deformed droplets is given by

(9)

in which Q0 is the equilibrium area, 0 is the relax-ation time for a non-dilute dispersion of sphericaldroplets, shown by Graebling (22), to be

(10)

and fstruct is a function of the stretch ratio whichvalue should approach one for slightly deformeddroplets and was chosen by Peters et al. (18) as

(11)

The final constitutive equation is written in terms ofT ETs, as

(12)

While carrying out the simulations in the final equa-tion (12) the values of dand cwere not taken as thezero shear viscosities of the dispersed and continuous

phase but were replaced by shear dependent viscosi-ties using the Carreau-Yasuda model.

EXPERIMENTAL

In the present study, binary blends of polypropylene(PP) and ethylene vinylacetate (EVA) were investigated.PP (Stamylan P13E10, Mw 500 kg/mol, Mw/Mn 6)was supplied by DSM, Netherlands and EVA (PileneEVA 1802, MFI 2.0 g/min, Vinyl acetate content 18%) was supplied by National Organic Chemical In-dustries Limited (NOCIL), Mumbai, India. The rheo-logical properties of the PP and EVA are given in Fig.

1aand 1b.

Sample Preparation

For each type of blend, as well as for the neat poly-mers, the same blending procedure has been appliedso that the thermo-mechanical history of the blendsand that of neat polymers remain similar. The pelletswere first dried in an air oven at 60C for 3 hrs. Nextthe pellets were dry blended and fed in a mini-mixer(with a conical corotating twin-screw geometry and arecirculating flow) in a batch of 5 gms. The mixing wascarried out at a temperature of 200C, and rotor speedof 150 rpm for 200 sec. The samples were taken outfrom the mixer in the form of strands and quenched

in air. The strands were chopped into small granules

and compression molded at 200C for 8 min, to obtaincircular discs for the rheological measurements. Thesamples were quenched in the cold press of the com-pression molding machine. Blends with the composi-tion 90/10, 80/20 and 70/30 wt% PP/EVA were pre-pared.

Rheological Studies

Rheological measurements were carried out on theneat polymers and blends, using a Rheometrics Me-chanical Spectrometer (RMS 800) with a cone and plategeometry having a cone angle of 0.1004 rad. and a di-ameter of 25 mm. Linear viscoelastic properties of thesamples were investigated in a frequency range of

0.1100 Hz. The maximum applied total deformationwas kept below 0.5%. Transient shear experimentswere carried out in a range of 0.01 to 0.5 s1 shear rate.This shear rate range was restricted to this small valuesince at high shear rates edge failure occurred (only inthe case of neat PP we were able to go up to a shearrate of 0.7 s1). All the experiments were carried out at200C under nitrogen atmosphere. Before the finaltests were carried out, each sample was preconditioned

c2 101d c2211 2d 13 2 2cdcD0 T

T PI

fstruct 1 1 21.5 1

F1p, 2 119d 16c2 3211 2d 13 2 2c41011 2d 110 4 2c

0 3

4QF1p, 2;

Q0

Q0 fstruct

4L 8QE

154I

2

3IT

2

EQTT

T s112

c1

Dln 1Q2Dt

dT 4L:D0

E a 5c211 2d 13 2 2cb

211 2 1d c2211 2d 13 2 2cTs

T 101d c2

211 2d 13 2 2cc D0

Ts

Va A i

a 13I nnbd s

Linear Viscoelast ic and Transient Behavior, PP/ EVA Blends



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Fig. 1a. Viscosity of Stamy lan PP () and EVA 18 02 ().

Fig. 1b. Frequency sw eep test for Stamy lan PP (closed sy mbols) and EVA 1 802 (open sym bols) at 200C. (G*, G, G in Pa; * in Pa.s)


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by shearing at a low shear rate until a stable mor-phology was obtained. The minimum time required forpreconditioning has been established during prelimi-nary tests. With a shear rate of 0.01 s1 this time wasfound to be 800 s. The stable morphology was con-firmed by scanning electron microscopy for each com-position. Preconditioning is followed by a step-upshear rate experiment (Fig. 2a), which allows probing

the morphology development (i.e. stretching, break-upand coalescence) during flow (Fig. 2b). After pre-shearing of all blends (at 0.01 s1 for 800 s) step-up of0.1, 0.3 and 0.5 s1was applied in three different ex-periments. The experiments were repeated severaltimes, also using different batches of samples.

Morphology

The morphology of the samples was studied using aPhilips Environmental Scanning Electron Microscope(ESEM-FEG, V-type) in order to investigate the evolu-tion of morphology generated during transient shearexperiments. For this the RMS 800 Rheometer was

stopped and the sample quenched with liquid nitro-gen in order to arrest the actual morphology. Thesamples were removed from the cone geometry and

were fractured in liquid nitrogen, in specific directions(Fig. 3) in order to probe the morphology changes thatoccur during flow in different directions.

RESULTS AND DISCUSSION

Dynamic Experiments

The dynamic moduli (G, G) and complex moduli

(G*) of the pure components and their viscosities (*)are shown in Fig. 1b. Paliernes emulsion model wasused to estimate the droplet size of the dispersedphase, by substituting the values of the complex mod-uli of the matrix (Gm* ), the complex moduli of thedispersed phase (Gd* ) and the interfacial tension (1.2mN/m) into Paliernes equation. Lacroix et al. (3, 23)reported the range of interfacial tension for PP/EVAblend to be 0.81.2 mN/m and for a similar blend ofpolypropylene and a mixture of ethylene vinyl acetateand ethylene methylacrylate (PP/(EVA-EMA) they re-ported the value to be 1.2 mN/m. Figures 4 6showthe G and G for the 90/10, 80/20 and 70/30 wt%PP/EVA blends along with the prediction of PaliernesModel. Average droplet sizes of 1.2, 2.5 and 4.5 mare predicted, which are in good agreement with theexperimental values obtained from microscopy, see

Table 1.The predictions of frequency dependent mod-uli for the blends using Paliernes model are rathergood, except for the 70/30 blend, where a small devi-ation can be seen at the lowest frequency (Fig. 6),which becomes more pronounced at 0.01 rad/sec (notshown in Fig. 6) (3). Steric interactions, partial misci-bility as well as the droplet size distribution can bereasons for this deviation. Interestingly, the dropletsize decreased after a pre-shear (at 0.1 s1 for 600 sec)(Table 1) leading to an improved agreement in the lowfrequency region.

Transient Experiments

Transient shear rate experiments were carried outin order to study the first normal stress difference (N1)and the viscosity () of the polypropylene as well asthe PP/EVA blends. The samples were presheared at0.01 s1 shear rate for 800 s and then a step-up inthe shear rates to 0.1 s1, 0.3 s1 and 0.5 s1 was

given.Figure 7aand bplots vs. time and N1vs. time re-

spectively, obtained from the step-up experimentswith pure PP. The final (averaged) values of first nor-mal stress difference and viscosity are given in Table 2

for different shear rates. To determine the first normalstress difference N1 for the blend, the contribution ofthe matrix is added by weighting with the volumefraction. The effect of the dispersed phase on the totalfirst normal stress difference of the blend is assumedto be negligible, because the presence of surroundingdrops of the dispersed phase has not been consideredduring simulation of the first normal stress difference.

Figure 8 ishows the experimental values along withsimulated values obtained using the PHM model, forN1 vs. time during the step-up experiments for the



Fig. 2a. Schematic representation of a stepw ise increase inshear r ate (step-up experiment).

Fig. 2b. Schematic of f i lament break-up dur ing shear f low .


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three different blends. The initial droplet radius wasobtained from micrographs of the blend samples. Themodel gives a good prediction of the data at the lowand intermediate step-up in shear rates (i.e., step-upfrom 0.01 to 0.1 s1 and 0.3 s1) except for the 70/30

blend (Fi g. 8 i c ) for which N1 is under predicted.The differences found can partly be attributed to the

assumption of an initial equilibrium state of (slightly)deformed droplets with some average size, and therelaxation behavior. Both the shape and size of theinterface in the blends can be different from thatpredicted by the theory using kinetic equations. As ev-ident from the SEM micrographs of 70/30 composi-tions (Fig. 9), droplets as well as the stretched fila-ments are present in the initial sample, leading to thedifferent relaxation behavior than that compared to

other compositions. In the other blends only dropletswith different dimensions were found in the initialsample morphology.

In Fig. 8 i i, the experimental and predicted (usingPHM model) values of the viscosity vs. time are shown

for the various compositions. The viscosities of theblends were calculated using two different approaches.In the first case the model was applied using the zeroshear viscosity (Simulation I). In that case too highviscosities are found for the blends, as one could ex-pect, since the PHM model is developed for viscous-viscous system only.

In the second approach (Simulation II), the sheardependent viscosity of the matrix material is incorpo-rated, i.e., the viscosity for the given macroscopicshear rate. With this modification, the predictions



Fig . 3 . Frac tu red sur face of thes a m p l e t h a t w a s s t u d i e d b y

ESEM.


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Fig. 4. Comparison for exper imental and Pal iernes Model predict ions for 90/ 10 PP/ EVA blend. (G and G in Pa)

Fig. 5. Comparison for exper imental and Pal iernes Model predict ions for 80/ 20 PP/ EVA blend. (G and G in Pa)


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were good for every composition and for all the threestep-up in shear rate experiments. Thus it can be con-cluded that it is possible to describe the rheologicalbehavior of blends of viscoelastic polymers reasonably

well by the modified PHM model.For the 70/30 composition, the morphology evolu-

tion with increase in step-up ratio has been analyzed

using microscopy. Increase in step-up ratio leads todecrease in the droplet size, as evident from Fig. 10.Figure 10ashows that application of step-up in shearrate from 0.01 to 0.1 s1 leads to the formation ofdroplets of average size of 3.1 m with size distribu-tion skewed towards the right. Increase in the step-upratio from 0.01 to 0.5 s1 leads to the decrease in theaverage size with the size distribution skewed more

towards left. A summing-up of variation in N1, viscos-ity and average size with the step-up from 0.01 to 0.1,0.3, 0.5 s1 is given in Fig. 11.

CONCLUSIONSLinear viscoelastic properties of PP/EVA blends

were studied. Paliernes model was found to describe

rather well the linear viscoelastic properties of theblends. The model of Peters, Hansen and Meijer wasable to predict the first normal stress difference of theblends with some over- and under-predictions. Sincethe model is developed for viscous-viscous systemsthe predicted viscosities, using the zero shear viscos-ity of the matrix material, are much higher than theexperimentally obtained values. This problem can beovercome by taking the shear rate dependent viscosityof the matrix as measured independently. Moreover,predictions of the morphological evolution of the

blends under stepwise increase in shear rate were ob-tained from the modified model and are found to de-scribe the break-up phenomenon under moderatelyhigh shear flow rather well.

ACKNOWLEDGMENT

One of the authors (S.T.) is thankful to the DutchPolymer Institute (DPI) for providing the facility andsupport to carry out this work during his stay in TheNetherlands.



Fig. 6. Comparis on for experimental an d Paliernes Model predictions before () and a ft er pr e-shea r () fo r 70 / 30 PP/ EVA blend .(G and G in Pa)

Table 1. Predicted and Experimentally ObservedDroplet Size in PP/EVA Blends.

Droplet Size (m)Blend

Composition Predicted from Experimentally(PP/EVA) Paliernes Model obtained

90/10 1.20 1.3080/20 2.50 2.2570/30 4.50 4.8070/30 (after preshear) 2.80 3.10


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Fig. 7. Viscosity and firs t normal stress difference (N1) duri ng st ep -up experimen ts for Stamy la n PP.


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Table 2. N1 and for Polypropylene at Different Shear Rates.

Shear rate (s1) 0 0.1 0.3 0.5

N1(Pa) 0 795 3030 5300(Pa.s) 20,000 14,900 12,200 10,800

Fig. 8i . First normal di f ference (N1) of th e PP/ EV A b lend sdur ing step-up exper iments (a) 90/ 10 (b) 80/ 20 (c) 70/ 30.

Fig. 8ii. Viscosity () of th e PP/ EVA b lend s du ri ng st ep-u pexper iments (a) 90/ 10 (b) 80/ 20 (c) 70/ 30.


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Fig. 9. Evolut ion of the structure for of dispersed phase in 70/ 30 PP/ EVA blend w ith increasing rat io of step change in shear rate(arrow indicates the direction of flow ).


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Fig. 10. Morphology and size distr ibut ion of the PP/ EVA 1802 70/ 30 w t% blend, after step-up in shear rate (a) from 0.01 s1 to0 .1 s 1(b) from 0.01 s1to 0.3 s1(c) from 0.01 s1to 0.5 s1.


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Fig. 11. Varia tion of first norma l stress difference, viscosity and a verage droplet size w ith shear rate.

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