Morphological evolution of immiscible polymer blends in simple shear and elongational flows

J. Non-Newtonian Fluid Mech. 86 (1999) 37±59

Morphological evolution of immiscible polymer blends in simple shear and elongational flows C. Lacroix, M. Grmela, P.J. Carreau* Center for Applied Research on Polymers, CRASP, Department of Chemical Engineering, Ecole Polytechnique, P.O. Box 6079, Stn Centre-Ville, Montreal, QC, Canada H3C 3A7 Received 4 May 1998; received in revised form 10 November 1998

Abstract The evolution of the morphology of the dispersed phase of immiscible polymer blends is studied in both shear and extensional flows. In the simple shear flow, predictions of the Lee and Park [H.M. Lee, O.O. Park, Rheology and dynamics of immiscible polymer blends, J. Rheol. 38 (1994) 1405±1425] and of the modified Lee and Park as well as the modified Grmela and Ait-Kadi [M. Grmela, A. Ait-Kadi, Comments on the Doi±Ohta theory of blends, J. Non-Newtonian Fluid Mech. 55 (1994) 191±195] models ([C. Lacroix, M. Grmela, P.J. Carreau, Relationships between rheology and morphology for immiscible molten blends of polypropylene and ethylene copolymers under shear flow, J. Rheol. 42 (1998) 41±62] are compared to stress growth data. The size of the dispersed phase is strongly affected by moderately finite strain imposed during the stress growth experiments. The morphological changes after stress relaxation are well predicted by the models for a polypropylene (PP), ethylene vinylacetate (EVA) and ethylene methylacrylate (EMA) blend [C. Lacroix, M. Grmela, P.J. Carreau, Relationships between rheology and morphology for immiscible molten blends of polypropylene and ethylene copolymers under shear flow, J. Rheol. 42 (1998) 41±62]. The prediction is less satisfactory for a polystyrene (PS)/ polyethylene (PE) blend. The morphological evolution for the elongation flows of the PP/EVA/EMA blends has also been investigated. The morphologies of samples extracted before and after extrusion through an hyperbolic shaped (nozzle) die show that the elongational flow induces fibrillar structures. The extensional viscosity data, needed to solve the interface governing equations of the Lee and Park model, have been obtained from entrance pressure drop measurements. These equations are shown to qualitatively describe the transition from a spherical to a fibrillar morphology. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Shear flow; Elongational; Extensional; Viscosity; Immiscible; Polymer

1. Introduction The design of new polymer blends of required properties is intimately related to the control of the morphology. In almost all cases, the components of the blend are immiscible and thus the resulting morphology is a two-phase morphology. More complex situations can arise when more than two ÐÐÐÐ * Corresponding author. 0377-0257/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 8 ) 0 0 2 0 1 - 8

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polymers are mixed together or when chemical reactions occur. In this article we will restrict our analysis to the study of binary blends. A simple way of classifying the flows is to divide them into two major classes: simple shear flow and elongational flow. Components of both flows are encountered for instance in extrusion, injection moulding, film blowing. Flow conditions can greatly affect the morphology and skin-core effects are frequently observed. To stabilize the morphology, copolymers are frequently added or generated in situ via chemical reactions. Therefore, predicting the evolution of the morphology under processing conditions is of major interest for optimizing the properties of new materials. The purpose of this article is to investigate the relationships between the flow conditions and the resulting morphology for different kinds of immiscible polymer blends. This study is based on theories recently developed to relate the morphological evolution of immiscible polymer blends in relation to their rheological properties [1±4]. This article is divided into two main parts. The first one concerns the rheology/morphology relationships induced by simple shear flow whereas the second one focuses on the morphologies obtained from dominant uniaxial elongational flows. 2. Relationships between rheology and morphology In processing conditions, under an applied flow, the complex microstructure of immiscible molten blends evolves and this in turn affects the rheology of the fluid. In contrast to small amplitude oscillatory flow, large deformations can generate breakup and/or coalescence phenomena for uncompatibilized blends. This dynamics of morphological changes induced by flows has been considered by Doi and Ohta [4]. They derived for a mixture of two Newtonian fluids new constitutive equations for describing evolution of the interface that takes into account breakup and coalescence. These interface governing equations are applicable for any type of flow. In simple shear flow, this theory is now well documented and has been investigated with different kinds of systems [5±12]. Lee and Park [1] have modified the stress equation of Doi and Ohta [4] to take into account the mismatch of viscosities of the different polymers. They have also introduced additional terms regarding the evolution of the interface. This model has been discussed and analyzed by Lacroix et al. [13] in the linear viscoelastic domain as well as in the nonlinear viscoelastic domain [3,11]. A modification of the mixing rule included in the Lee and Park model [1] has been proposed by Lacroix et al. [13] to better describe linear viscoelastic data for polypropylene (PP)/ethylene vinyl acetate (EVA) blends and satisfy limiting cases. In a very recent publication [3], we have used this modified model, called the LPL model, to predict the morphological evolution of a polypropylene (PP)/ethylene vinyl acetate (EVA)±ethylene methyl acrylate (EMA) blend under simple shear flow. In this model, the time evolution of the anisotropy tensor q and the time evolution of the interfacial area per unit volume Q are those of Lee and Park [1] whereas the extra stress tensor is expressed by   1 ‡ 3=2H

_ ij ÿ qij ; (1) ij ˆ M 1ÿH with H given by Hˆ

2…I ÿ M † : 2I ‡ 3M

(2)

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39

Here  is the interfacial tension, M and I the viscosities of the matrix and of the inclusions, respectively,  the volume fraction of the dispersed phase, _ ij ˆ ij ‡ ji are the components of the rate of deformation tensor and ij are the components of the velocity gradient tensor. In (1), the stress components are given by   6…I ÿ M † ij ˆ 1 ‡ (3)  M _ ij ÿ qij : 10…I ‡ M † The Lee and Park model will be called later in the article the LP model. We have also proposed a modification of the Grmela and Ait-Kadi model [2] to retrieve from thermodynamic considerations the governing equations developed by Lee and Park [1] from a dimensional analysis. In that case the major difference comes from the contribution of the interface which included nonlinear terms. With the mixing rule given in Eq. (1), we call this modified Grmela/Ait-Kadi model the LGC model. Finally, we have also used a simple linear mixing rule based on the volume fraction of each component called the LLP model. These different models will be compared with transient shear experiments. The development of the interface governing equations assumes affine deformation [1,4]. We introduce in this article governing equations including nonaffine deformations. We use the same notation as in our previous work [3] as well as in Grmela and Ait-Kadi [2]. Let us consider first the nondissipative part of the interface evolution equations (i.e. ˆ 0 in the notation used in [2,3]). We define a conformation tensor C that characterizes an element of surface. C is related to q and Q by 1 Qqij ˆ Cij ÿ Q2 ij : 3

(4)

The convection of this conformation tensor is governed by the lower convected derivative. Nonaffine deformation can be taken into account by introducing a slip parameter  in such a way that the convection of the tensor C becomes @Cij  ‡ Ckj ki ‡ Cki kj ÿ ‰Ckj …ki ‡ ik † ‡ Cik …kj ‡ jk †Š ˆ 0: 2 @t

(5)

We assume here, as in Doi and Ohta [4] and in Grmela and Ait-Kadi [2], that C is uniform, i.e. independent of the position vector. By introducing Eq. (4) into Eq. (5) the nondissipative part of the interface evolution equations is obtained, and we choose the dissipative function such that the relaxation mechanisms included in the Lee and Park model are retrieved [3]. The complete evolution equations are then expressed by     @qij     2 ‡ qki jk ÿ qkj ki 1 ÿ ‡ qkj ik ‡ …1 ÿ †ij lm qlm ˆ ÿqki kj 1 ÿ 2 2 2 2 3 @t     Q qlm lm   qlm qlm ÿ …1 ÿ † _ ij ‡ …1 ÿ † Qqij ÿ d1 d3 (6) qij ÿ d1 qij ; 3 M M Q Q @Q  2  Q ÿ d1 d3 qij qij ; ˆ ÿij qij …1 ÿ † ÿ d1 d2 @t M M

(7)

where d1, d2 and d3 (,  and , respectively, in [1]) are phenomenological parameters arising in , i.e. the dissipative part of the time evolution equation. They denote degrees of total relaxation, size

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C. Lacroix et al. / J. Non-Newtonian Fluid Mech. 86 (1999) 37±59

relaxation and breakup and shape relaxation. We keep the relaxation mechanisms proposed by Lee and Park [1]. The stress tensor components are, however, given by ij ˆ 2…1 ÿ †Cik

@ : @Ckj

(8)

We recall that one of the main advantages of the thermodynamic formulation is that a formula for the stress tensor arises as an integral part of the derivation of the governing equations. Setting  ˆ 0 in Eq. (6) allows to retrieve affine deformation. In that case, the equations are governed by the lower convected derivative whereas for  ˆ 2, the tensor evolution obeys an upper convected derivative. In the case when  ˆ 1, the flow is purely rotational and, as seen from Eq. (8), no contribution to the stress tensor comes from the interface. A similar observation has been made by Larson [14] when considering the Gordon±Schowalter derivative in another context. This set of equations (Eqs. (6)±(8)) can also be used in principle to obtain morphology changes in any flow situations. Few studies concern the rheological properties in extensional flows of immiscible blends in relation to their morphology. Gramespacher and Meissner [15] and Levitt et al. [16] have related the interfacial properties of different blends to their elongational properties but in very well defined extensional flow fields. Elongational flows are recognized to be very efficient for deforming and rupturing initially spherical droplets since the pioneering work of Taylor [17]. Nevertheless, most of the studies are restricted to Newtonian fluids and concern the mechanisms of deformation and rupture of a single droplet [18,19]. Real processing conditions imply in general the use of viscoelastic components, relatively concentrated systems with a micro-structure of a large number of droplets, and large deformations are involved. The viscoelastic nature of the components has a major influence on the morphology developments as shown by the works of Delaby et al. [20] and Mighri et al. [21] for elongational flows. Even if the Lee and Park [1] governing equations are an extension of the Doi±Ohta [4] theory derived for a mixture of Newtonian fluids, these and the modified equations presented here are the only semi-theoretical relations available for describing the morphological evolution under elongational flows. In this work we have used a converging die to determine the elongational properties of two different blends and assess the model capacity to describe the morphological evolution of the blends.

3. Experimental 3.1. Materials Two different systems have been investigated: blends of polystyrene and polyethylene (PS/PE) and polypropylene and ethylene vinyl acetate±ethylene methyl acrylate (PP/EVA±EMA). Since EVA and EMA are miscible, binary blends of PP and (EVA±EMA) are then obtained. The commercial polymers were a PP (PP3020GN3), an EVA (EVATANE 2805) and an EMA (LOTRYL 28MA07) supplied by ELF-ATOCHEM whereas a PS (STYRON D685) and a PE (LLDPE TUFLIN) were obtained from DOW CHEMICAL and UNION CARBIDE, respectively. Transient shear experiments have been carried out on a controlled strain rheometer (Bohlin VOR) under nitrogen atmosphere at a set temperature of 2008C for both PP/EVA±EMA and PS/PE blends. In that case, the blends have been

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41

prepared with an internal batch mixer. The blending conditions, the characteristics of the rheological measurements and of the image analysis used for this study are described elsewhere [3,13]. To determine the elongational flow properties, the blends as well as the `pure' components have been prepared with a twin screw extruder (Leistrizt 30±34) in a corotative mode. A small amount of antioxidant (Irganox B225 from Ciba-Geigy) was added to the blends and to the EVA±EMA. The extruded materials were then pelletized and fed to a single screw extruder (Killion). The characteristics of the different dies used in this study are described in the following section. 3.2. Technique Determining the elongational viscosity remains a very difficult task. Different techniques can be used as approximate methods for measuring the elongational flow properties, such as jet opposed nozzles, fibre spinning, converging flow. The more rigorous method developed by Meissner [22] using rotary clamps yields real uniaxial elongational measurements, but it is not easily accessible and trivial to use. It is also restricted to relatively low deformations. A technique which is frequently used is the pressure drop measurements for the flow through a contraction. This is particularly well adapted to the processing conditions. Several analyses have been proposed to correlate the flow rate and pressure drop measurements to the elongational viscosity. Cogswell [23] was the first in this context to study entrance pressure effects. Binding [24] and more recently Mackay and Astarita [25] revisited and extended the works of Cogswell [23,26,27]. Cogswell considered that the entrance pressure drop was due to two contributions, one to shear and another to elongation effects, whereas in Binding's analysis, minimization of the power consumption was used to obtain the velocity profile. In this study we will use the Cogswell [23], Binding [24] and Mackay and Astarita [25] analyses. It should be noted that alternative approaches have been proposed such as those of Gibson [28] or of Tremblay [29] based on the sink flow analysis. Most studies make use of capillary rheometers for determining the entrance pressure drops. But this is relatively far from the processing conditions for which the polymer is conveyed and pumped by an extruder and flows through a die. Therefore, a setup has been designed to measure shear as well as extensional properties using a single screw extruder (see Fig. 1). This setup allows us to obtain various flow rates by using a by-pass valve without modifying the high rotation screw speed (80 rpm). A static mixer was installed at the reservoir inlet to homogenize, as well as possible, the blend. Converging dies or capillary dies can be mounted on the extruder exit reservoir after the section where the pressure and

Fig. 1. Extrusion setup.

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C. Lacroix et al. / J. Non-Newtonian Fluid Mech. 86 (1999) 37±59

temperature are measured. The valve also allows us to extract samples before the converging section for morphological analysis. With this design, the morphology of the blends before die extrusion can be controlled by keeping the thermomechanical history unchanged. The morphology of the extruded samples was frozen by plunging the extrudate in cold water right at the die exit. The cooling rate was high enough to guarantee that no significant relaxation and morphological changes took place after extrusion. To measure a meaningful elongational viscosity, a constant and controlled elongational strain rate should be applied. This can be achieved by choosing correctly the shape of the converging section. Binding and Jones [30], and Kim et al. [31] have used planar hyperbolic shaped dies to measure the extensional flow properties of polymer melts or polymer solutions. James et al. [32,33] showed that a constant elongational strain rate could be obtained if the shape of a cylindrical converging die was given by R(z)2 ˆ constant, where R, the channel radius, depends on z. For an axi-symmetric geometry along the z-axis, the extensional strain rate is defined by _ ˆ …z†

dhVz i ; dz

(9)

where hVzi is the average velocity at a given z. From the expression of the volumetric flow rate Qf, it can be deduced that ! Qf d 1 _ ˆ …z† : (10)  dz R…z†2 Imposing a constant elongational strain rate results in the following shape   _ 1 ÿ1=2 z R…z† ˆ ‡ ; Qf R21

(11)

where R1 is the initial radius of the nozzle die. Two dies of different geometries have been micromachined. Both dies have the same entry radius (12.7 mm). The first one (D1) is 10 mm long with an exit radius of 1.995 mm whereas the second one (D2) is 20 mm long with an exit radius of 2.805 mm. A similar setup has been used by Crevecoeur and Groeninckx [34] to study the fibril formation of thermoplastic/TLCP blends. In our case, we were able to extract the samples just before and after the die so that the resulting morphologies could be analyzed in relation to the elongational effects generated by the die alone. Two different blends, a 70/30 PP/EVA±EMA and a 30/70 PP/EVA±EMA blend, are compared in extension to the properties of the unblended components. These blends are of particular interest because their zero shear viscosity ratios are either 0.1 or 10 depending on the matrix. Therefore, the effects of the viscosity ratio can be assessed in the dominant elongational flow. As it will be shown in the next section, shear viscosity data, and more precisely power-law parameters of the four different molten polymer or molten blends are needed to determine an equivalent extensional viscosity. Capillary rheometry has been carried out with the single screw extruder. The setup presented in Fig. 1 is flexible and allows the use of different capillaries instead of a converging section. Here, three different L/D ratios have been used (8, 16 and 24). The diameter of the dies was 1.6 mm. The whole system described in Fig. 1 was set at a temperature of 2008C. Before presenting the results in shear and

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43

extension, we will explicit the three different approaches (Cogswell, Binding, Mackay and Astarita) in the case of a constrained convergent geometry. 3.3. Cogswell analysis In Cogswell's [23] analysis, the entry pressure drop is divided into shear and elongational terms:
Pend ˆ
Ps ‡
Pe :

(12)

A power-law behavior for the shear properties of the material is assumed. For a converging die of radius R(z) with z the coordinate along the symmetry axis, the pressure drop due to shear in an element of length dz, assuming negligible inertial effects, is dPs ˆ

2w dz; R…z†

(13)

where w, the shear stress at the wall, is expressed for a power-law fluid by   3n ‡ 1 n n w ˆ K

_ a ; 4n

(14)

with

_ a ˆ

4Qf

R…z†3

:

(15)

In Eq. (14), K and n represent the consistency and index of plasticity, respectively, and g_ a is the apparent shear rate. By inserting the nozzle profile R(z) given by Eq. (11) and integrating along the die length,
Ps is expressed as   " 3…n‡1†  3…n‡1† # 4 KQf …3n ‡ 1†Qf n 1 1 ÿ ; (16)
Ps ˆ _ 3 …n ‡ 1† R0 R1 n where R0 and R1 are the outlet and inlet radii, respectively. The elongational contribution to the entrance pressure drop can be derived from energy considerations. If the flow is purely extensional, the pressure difference through a contraction is defined by Zld
Pe ˆ

…zz ÿ rr †_ zˆ0

R…z†2 dz; Qf

(17)

where ld represents the length of the die and zz ÿ rr is the primary normal stress difference. The elongational viscosity is defined by e ˆ

zz ÿ rr : _

(18)

Assuming that the elongational viscosity is independent of time or strain through the nozzle (notice that

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C. Lacroix et al. / J. Non-Newtonian Fluid Mech. 86 (1999) 37±59

for viscoelastic fluids, a constant elongational strain rate is a necessary but not a sufficient condition to have a constant e) and by substituting Eqs. (11) and (18) into Eq. (17) we obtain after integration   _d Qf =R21 ‡ l ; (19)
Pe ˆ e _ ln Qf =R21 where the elongational rate is based on the averaged velocity and is given by   dhVz i 2Vz ÿdR…z† _ ˆ  : R…z† dz dz Combining Eqs. (19) and (20), we get     Qf 1 1 R1 ÿ 2 ln :
Pe ˆ 2e 2 ld R1 R0 R0

(20)

(21)

The elongational viscosity can be calculated once we know the shear viscosity. 3.4. Binding's analysis Binding [24] extended the work of Cogswell [23] by using an energy balance and variational calculus. He assumed, moreover, a power-law form for the extensional viscosity which is then determined from minimization of the power consumption W. Assumption was also made that the converging angle is small enough so that (dR(z)/dz)2 or d2R(z)/dz2 can be neglected. Recirculating vortex or forced convergence profile are, therefore, involved in this analysis. The omission of (dR(z)/ dz)k with k > 1 is equivalent to make the lubrication approximation. The power consumption W is related to the excess (entrance and exit) pressure drop by
PendQf ˆ W and is due to shear, elongation and kinetic energy contributions. We will express more specifically these terms for an hyperbolic shaped die. The usual assumption in entry flows is that the velocity Vz for a power-law fluid is fully developed. In the converging section, Vz is given by !   3n ‡ 1 Qf r 1‡1=n 1ÿ : (22) Vz ˆ n ‡ 1 R…z†2 R…z† Following Binding's analysis [24] the rate of work per unit volume may be expressed by W ˆ Ws ‡ We ‡ Wkinetic energy :

(23)

The contributions due to shear and elongation are, respectively, defined by _ n‡1 ; Ws ˆ Kj j

_ t‡1 : We ˆ Ljj

(24)

In cylindrical coordinates, _ ˆ @Vz =@r and _ ˆ @Vz =@z and the elongational rate is obtained from the derivative of Eq. (22). This constitutes one of the differences with the `full technique' proposed by Mackay and Astarita [25] (see the next paragraph). By substituting the profile of the converging section

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(Eq. (11)) and integrating over the volume of the die we obtain   " 3…n‡1†  3…n‡1† # 4KQ2f …3n ‡ 1†Qf n 1 1 Wˆ ÿ _ R0 R1 3…n ‡ 1† n  t‡1  t   _ 3n ‡ 1 1 3 …3n ‡ 1†2 …1 ÿ 4 † ‡ Int Qf L ln 2 ‡ Q3f ; …2n ‡ 1†…5n ‡ 3† 2 R40 2 n‡1 2

45

(25)

where is the inverse of the contraction ratio ( ˆ R0/R1) and Int is defined by t‡1 Z1 3n ‡ 1 1‡1=n  d: Int ˆ 2 ÿ  n

(26)

0

From shear viscosity data and pressure drop data in the converging die L and t can be obtained and used to calculate the extensional viscosity ([35]). In the case of free convergence, it was assumed by Binding [24] that the fluid will create its own convergence profile so as to minimize the power consumption. When convergence is imposed, dR(z)/dz is a known function of z and lead to a power consumption W defined by Eq. (25). As pointed out by Binding and Jones [30], limiting conditions are implied by the use of Eq. (25). Actually, the rate of convergence imposed by the hyperbolic shaped die should be smaller than the one the fluid would adopt to minimize its energy. If not, Eq. (25) is no longer valid and the minimum power consumption will correspond to the following profile: !nÿt   ÿdR…z† t‡1 K…n ‡ 1†t‡1 …3n ‡ 1†Qf ˆ : (27) dz …3n ‡ 1†nn LtInt nR…z†3 The condition required to confine the fluid in the imposed geometry is  Qf 

2ld R20 1 ÿ 2

…1‡t†=…tÿn†

K…n ‡ 1†t‡1 …3n ‡ 1†nn LtInt

!1=…tÿn† 

 #…n‡1†=…tÿn† "  n 1 3=2 : 3n ‡ 1 ld R20

(28)

It was checked that for all the measurements this equation was satisfied. 3.5. Mackay and Astarita's analysis Following Binding's approach, Mackay and Astarita [25] derived recently an expression to estimate the extensional viscosity from the minimization of the dissipated power by using the Generalized Engineering Bernoulli Equations. They called this method a `full technique'. As already mentioned in the preceding paragraph, the major difference between the Binding and Mackay/Astarita analysis stems from the calculation of the extensional rate of strain. Mackay and Astarita have pointed out that the use of Eq. (22) implies that the velocity is zero at the edge of the vortex or of the converging profile. Evaluating @Vz/@z by taking the derivative of Eq. (22) leads to a negative nonzero value for the velocity gradient at the boundary. This is, therefore, in contradiction with the presupposed velocity profile. Instead of taking the derivative of Eq. (22), they used an average rate of elongation, as given by

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Eq. (20). The calculation of the dissipated power W for negligible kinetic energy is carried according to the following equation: tÿ1  2 nÿ1  2 ! Z dVz dVz dV dVz z dV: (29) Wˆ L ‡K dz dz dr dr Integrating over the volume of the whole system and taking into account the nozzle profile yields the following expression:     " 3…n‡1†  3…n‡1† #   4KQ2f …3n ‡ 1†Qf n 1 1 3n ‡ 1 t‡1 1 ‡ 2LQf Itn Wˆ ÿ ln 2 _t ; _ R0 R1 n‡1 3…n ‡ 1† n (30) with Itn ˆ

t…t ‡ 1†n…n ‡ 1† B…t; 2n=…1 ‡ n††; …t…n ‡ 1† ‡ 2n†……t ‡ 1†…n ‡ 1† ‡ 2n†

(31)

where B is the beta function defined by B…a; b† ˆ ÿ…a†ÿ…b†=ÿ…a ‡ b†;

(32)

with ÿ the gamma function. The power-law parameters L and t can be determined from an appropriate log±log plot once the shear viscosity power-law parameters are known. 4. Results and discussion 4.1. Morphology and simple shear flow properties For the two systems investigated (PS/PE and PP/EVA±EMA blends), shear stress growth data have been compared to the different model predictions. These start-up experiments are very useful to test the ability of the different models for predicting the morphological evolution of the blend under shear flow. Our first experimental results have already been published elsewhere ([3]). Fig. 2(a)±(c) report for a 80/ 20 PS/PE blend the stress growth data at a constant shear rate of 0.0507, 0.101 and 0.32 sÿ1, respectively. As for the PP/EVA±EMA blend [3] an overshoot is first observed before the transient viscosity reaches a constant value or before the sample is ejected from the geometry. No such overshoots could be observed for the unblended components at the different low shear rates investigated. For the PS/PE system, we have a situation analogous to that of the PP/EVA±EMA blend. The dispersed phase is less viscous than the matrix, which means that the droplets of PE can be deformed and eventually break. The flow was then stopped and the samples were allowed to relax. During this process, the deformed particles recovered their equilibrium shape under the effects of the interfacial tension. Fig. 2 also reports the predictions using the different models. The parameters used to fit the data with the different models for the PS/PE blend are reported in Table 1. The parameter determination is

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47

Fig. 2. Comparison between stress growth viscosity data and models predictions for a 80/20 PS/PE blend at T ˆ 2008C: (a)

_ ˆ 0:0507 sÿ1, (b) _ ˆ 0:101 sÿ1 (c) _ ˆ 0:3 sÿ1.

explained in detail in [3]. Note that d1 was determined from the linear properties and C2, appearing in the nonlinear term of the stress equation (Eq. (8)) and d2, parameter related to coalescence, were adjusted to describe the steady viscosity in stress growth experiments. The value of d2 was found to increase with shear rate and d3 was kept constant and equal to 1 ÿ , as done by Lee and Park [1]. The different abbreviations used to denote the models are explained in the first part of this article. As shown in Fig. 2(a) and (b), the LP, LLP, LPL and LGC models are able to describe the transient data for the 80/20 PS/PE blend. Even the simple linear mixing rule allows to get satisfactory results. The best description of the rheological data is obtained, however, with the LP model. This in contrast with the

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Table 1 Parameters used for predicting the transient rheological behavior of the 80/20 PS/PE blend with the different models and comparison between predicted radii values and experimentally determined values Parametersa

Calculated radiib

Shear rate (sÿ1)

d2 LP

d2 LLP

d2 LPL/LGC

C2 (N)

LP (mm)

LLP (mm)

LPL (mm)

LGC (mm)

Rv morphology (mm)

0.051 0.101 0.320

0.28 0.45 ±

0.16 0.24 ±

0.12 0.18 0.80

10ÿ11 10ÿ11 10ÿ11

4.3 4.1 ±

2.6 2.0 ±

2.0 1.6 4.6

2.0 1.6 4.6

2.8 1.6 1.2

a b

d1 ˆ 0.6, d3 ˆ 0.8,  ˆ ÿC1 ˆ 5 mN/m. Rv initial ˆ 2.4 mm.

results obtained for the PP/EVA±EMA system studied in [3] for which it was not possible to fit the data with the LP and LLP models. For the highest shear rate investigated (Fig. 2(c)), only the LGC model can describe these transient data. Even if these models are descriptive regarding the evolution with time of the transient viscosity, they can be used as predictive ones when considering the morphology. Nevertheless, depending on the chosen mixing rule, different parameters have to be used to fit the data. This choice will affect the predictions in droplets size changes during simple shear flow. Here we assess the ability of the models to predict qualitatively or quantitatively the morphological changes following stress growth and stress relaxation experiments. Indeed, the knowledge of the interfacial area Q allows us to determine an average diameter if the dispersed phase is present as spherical particles (which should be the case after relaxation). The results are reported in Table 1 for the PS/PE blend and are compared to the experimental determined values using scanning electron microscopy (SEM) of samples broken in liquid nitrogen. The corresponding results for the PP/EVA±EMA blend are given in [3]. For the PS/PE system, as well as for the PP/EVA±EMA, the predicted radius is very sensitive to the choice of the mixing rule. Table 1 shows that the LLP, LPL and LGC models give relatively satisfactory results for the lowest shear rates. Clearly the LP model overestimates the SEM determined values. But the best description of the rheological data is obtained with the LP model. For the highest shear rate investigated, it was not possible to obtain reasonable fits with the LP and LLP models. But in that case, the morphological predictions calculated with the LPL and LGC models are in total disagreement with the value determined from SEM. For this PS/PE blend, none of these models appear to be adequate for predicting the morphological evolution of the blend even if good descriptions of the viscosity data can be obtained. For all these models affine deformation has been assumed so far. For the highest shear rate (0.32 sÿ1), the LGC model predictions have been tested for nonaffine deformation. The results reported in Fig. 2(c) show that when increasing the value of the slip parameter , the peak of the overshoot is decreasing. For values of  higher than 0.2, oscillations start to appear. To describe the rheological data the parameters d1 and d3 have been kept constant but d2 had to be adjusted when changing the value of . Table 2 shows that when  is increased, d2 has to be decreased to describe the transient viscosity. The predicted radii range from 4.6 mm for  ˆ 0 to 2 mm for  ˆ 0.2. In this latter case, the predicted result is closer to the value determined from SEM. The results presented in Table 2 and in Fig. 2(c) show that  has a major influence on the calculated radii but it is of minor influence regarding the description of the rheological data.

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Table 2 Parameters used for predicting the transient rheological behavior of the 80/20 PS/PE blend and the 70/30 PP/EVA±EMA blend with the LGC nonaffine model PS/PEa

PP/EVA±EMAb



d2

Rv calculated (mm)

Shear rate (sÿ1)



d2

Rv calculated (mm)

Rv morphology (mm)

0 0.10 0.10 0.15 0.20

0.80 0.55 0.40 0.30 0.20

4.6 3.9 3.5 3.2 2.0

0.0126 0.126

0.125 0.050

0.135 0.45

6.6 3.0

6.0 3.0

a b

Shear rate ˆ 0.32 sÿ1. d1 ˆ 0.5, d3 ˆ 0.715,  ˆ ÿC1 ˆ 1.2 mN/m, C2 ˆ 10ÿ11 N.

The PP/EVA±EMA blend has been investigated at four different shear rates and the results for two different shear rates are presented in Fig. 3. The LPL and LGC models are able to describe qualitatively and in certain cases quantitatively the breakup and coalescence phenomena. The larger disagreement was obtained for the lowest shear rate (see Table 1 in [3]). Using the LGC model with nonaffine deformation can improve the predictions. As shown in Table 2, an appropriate choice of  and d2 allows to fit the morphological data. But as shown in Fig. 3, the rheological data are not better described by relaxing the affine deformation assumption. The nonaffine motion requires, moreover, the use of a supplementary parameter that has to be fitted, and the final morphology is required to determine this parameter. We did not succeed in relating the slip parameter  to quantities such as the viscosities of the phases and the volume fraction which should constitute factors influencing the deformation state of the interface. Table 1 shows that the mixing rule has a major influence on the calculated radius value. We have already pointed out that the discrepancies observed can arise from the lack of coupling effects between the orientation tensor and the bulk properties. The bulk and the interfacial properties are treated as separate entities. Even if the contribution of the interface is minor compared to the bulk properties, the effects of the interface deformation is not negligible. We believe that the simple mixing rules imbedded in the different equations for the stress tensor are valid for the linear viscoelastic domain, for which the deformations of the dispersed droplets remain small. But at moderately large deformations, the morphology (size, shape and orientation) of the dispersed phase will affect the rheology of the blend. The experiments show that large deformations have a great influence on the particles size (see Tables 1 and 2 and Table 1 in [3]). One has also to keep in mind that viscoelastic effects and mismatch of the viscosities of the two polymers will influence the behavior of the interface. More theoretical as well as experimental work are needed to clarify the role of the interface on bulk rheological properties on polymer blends under large deformation flow. Confrontation with experimental results forces us to let the phenomenological parameters d2, related to coalescence, and  depend on the velocity gradient. With this extension we still satisfy the requirement of the compatibility with thermodynamics provided that d2 remains positive and 0    1. The simplicity in solving the set of governing equations for complex flow situations is lost. Obviously, one has to include in the future a kinetic expression that will respect in a more faithful manner the physics involved, in particular related to the coalescence phenomena.

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Fig. 3. Comparison between stress growth viscosity data and models predictions for a 70/30 PP/EVA-EMA blend at T ˆ 2008C: (a) _ ˆ 0:0126 sÿ1, (b) _ ˆ 0:126 sÿ1.

In the model formalism presented here C characterizes an element of surface governed by a lower convected derivative, Cÿ1 is convected according to an upper derivative and represents the droplets. It is then possible to visualize the orientation of the droplets by plotting the conformation ellipses [36]. The conformation tensor is mapped onto a two-dimensional plane at different times. The principal axes of the ellipses are determined by the Eigenvectors of Cÿ1 whereas the Eigenvalues of Cÿ1 represent the major and minor axis, respectively. Fig. 4 illustrates the effects of an imposed shear rate of 0.126 sÿ1 on the evolution with the time of the conformation tensors for the 70/30 PP/EVA±EMA blend relatively to the evolution of the normalized interfacial area. Fig. 4(a) shows the evolution of the predicted normalized interfacial area in stress growth and relaxation experiment. An increase of Q indicates that breakup and deformation of droplets have occurred whereas a decrease of Q is associated with coalescence or with the recovery of the deformed droplets. Fig. 4(b) illustrates the different conformation states of the tensor Cÿ1 corresponding to the morphological changes shown in Fig. 4(a). At t ˆ 0 the isotropic state is represented by a circle. Upon the imposition of the shear rate, the droplets

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Fig. 4. (a) Evolution with the time of the normalized interfacial area predicted using the LGC model for the 70/30 PP/EVAEMA blend in stress growth and relaxation experiment at T ˆ 2008C with an imposed shear rate of _ ˆ 0:126 sÿ1; (b) Evolution of the conformation tensor Cÿ1 during stress growth and relaxation.

orient themselves along a preferred direction which is given by the major axis of the ellipse. The minor axis gives a qualitative idea of the spread in this preferred direction. Upon cessation of flow, the deformed droplets tend to recover their equilibrium shape under the effects of the interfacial tension. This representation remains, however, only indicative. 4.2. Morphology and elongational flow The first step for the estimation of the elongational viscosity according to Eqs. (16),(25) and (29) is to determine the power-law parameters of the shear viscosity. Results for the components and the blends (PP, EVA±EMA, 70/30 PP/EVA±EMA and 30/70 PP/EVA±EMA) are reported in Table 3. The range of shear rates covered by the capillary measurements was from 50 to 100 sÿ1 to about 500 sÿ1 depending on the polymer and the die used. The power-law parameters have been fitted for corrected (Bagley and Rabinovitsch corrections) data. For higher shear rates, outlet instabilities were very pronounced. Since the purpose of this work was not to study instabilities, the data were collected only when no extrudate distortion was visible and for stable extrusion pressures. The onset of the instabilities regions were

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Table 3 Power-law parameters in shear of the different systemsa K (Pa sn) n a

PP

EVA/EMA

70/30 PP/EVA±EMA

30/70 PP/EVA±EMA

6455 0.47

2884 0.44

5755 0.46

4730 0.43

_ n ÿ 1.  ˆ K| |

relatively different for the blends compared to their matrix. With the converging dies, no instabilities were observed whatever be the polymer and flow rate. As shown in Fig. 5, it was, however, not possible to verify adequately the Cox±Merz rule for the unblended components. The steady shear data are lower than the complex data and of slightly different slope, especially for the PP. This could be explained by the high viscous dissipation effects for all the extruded polymers investigated in this study. The melt temperature measured at the wall was much higher than the set point of 2008C. The temperature fluctuated between 2108C and 2258C depending on the flow rate. This variation of the bulk temperature is significative enough to affect the rheological measurements and the Bagley corrections, which are based on the assumption of isothermal flow. Another possible cause is polymer degradation during extrusion, especially in the case of PP which is sensitive to shear degradation. In the case of EVA± EMA, additional stabilizer has been added to avoid thermal degradation and cross-linking and the extrusion data are more in line with the cone-and-plate dynamic data. Nevertheless, only data obtained from the capillary and converging dies were used to assess the Lee and Park model for elongational flows. Fig. 6 reports the elongational viscosity as calculated using the Mackay±Astarita method, which we believe is the most appropriate. The vertical arrows in the figure indicate for which conditions the different samples have been extracted at the exit of the die for morphological analysis. The results obtained by using the Cogswell or Binding method were very similar to those reported here for the Mackay±Astarita method and only differed in the magnitudes of the elongational viscosities, as

Fig. 5. Complex and steady shear viscosities of the PP and the EVA±EMA at T ˆ 2008C.

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Fig. 6. Elongational viscosity of PP, EVA±EMA, 70/30 PP/EVA±EMA blend and 30/70 PP/EVA±EMA blend.

observed by Padmanabhan et al. [37] using the Cogswell and Binding methods. Notice that the contribution of the shear components is not negligible and represents about 40% of the total pressure drop. The data presented in Fig. 6 have been duplicated several times and even if temperature variations could be observed, the accuracy is believed to be good. Some discrepancies can be observed between the data obtained using the two different dies, D1 and D2, except for the 70/30 PP/EVA±EMA blend. These are, however, acceptable considering the difficulties of carrying meaningful measurements and estimating accurately the shear contribution to the pressure drop, mainly at low flow rates. For all the systems studied, the behavior at high extensional strain rate could be represented by a power-law expressions and the parameters L and t for the Mackay/Astarita analysis are reported in Table 4. The extensional behavior of both blends (Fig. 6) is in between those of the unblended components, dominated by that of the matrix as shown especially for the 30/70 PP/EVA±EMA blend. In this case, although the PP is considerably more viscous than EVA±EMA, its contribution to the viscosity of the Table 4 Power-law parameters in elongation for the Mackay/Astarita analysisa PP (D1) PP (D2) EVA/EMA (D1) EVA/EMA (D2) 70/30 PP/EVA±EMA 70/30 PP/EVA±EMA 30/70 PP/EVA±EMA 30/70 PP/EVA±EMA a

e ˆ L|eÇ|t ÿ 1.

(D1) (D2) (D1) (D2)

L (Pa st)

t

43 370 72 185 15 780 8735 30 960 30 474 17 815 10 410

0.50 0.35 0.69 0.97 0.56 0.56 0.68 0.92

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Fig. 7. Trouton ratio for the PP, EVA±EMA, 70/30 PP/EVA±EMA blend and 30/70 PP/EVA±EMA blend.

blend is negligible. These extensional viscosities are, obviously, apparent characteristics of the materials under elongational flows. They are nevertheless useful in comparing the processability of the blends with respect to their matrices. Fig. 7 reports the Trouton ratio for the four systems p investigated _ equal to 3_ for uniaxial as a function of the second invariant of the rate of deformation tensor, , elongational flow and to _ for simple shear flow. The shear data were obtained from capillary extrusion and the power-law parameters are given in Table 3. The discrepancies observed between the two die data in Fig. 6 are quite visible here for the unblended components. Nevertheless pronounced differences in elongational behavior compared to shear for the two different blends and their matrix are seen. In all cases, the Touton ratio far exceeds the limiting case of 3 expected for Newtonian fluids. The EVA/EMA is less strain-thinning than PP in elongation and very large values for the Trouton ratio are obtained at the largest elongational rates. This apparent strain-hardening is also visible for both blends, but the effect is less pronounced for the 70/30 PP/EVA±EMA blend, probably under the influence of the PP matrix which is more strain-thinning. As already mentioned, the setup presented in Fig. 1 allows us to extract the samples for morphological analysis before the converging section. For both blends, the dispersed phase (EVA± EMA or PP) was present as spherical droplets. But for the 30/70 PP/EVA±EMA blend, despite of the static mixer installed at the reservoir entry, the morphology of the extracted samples (arrows in Fig. 6 indicate the extrusion conditions) showed mainly droplets and also not very well defined domains (see Fig. 8). The volume average diameter of the droplets for both blends was about the same, i.e. independent of the zero shear viscosity ratio (see Table 5). The minus and plus signs appearing in Table 5 correspond for each die to the lowest and highest flow rate for which the morphological analysis has been carried out. As expected for the extrusion through a converging die, we observed a transition from an emulsion-type structure to a fibrillar structure as illustrated in Fig. 9. The fibril diameters were almost the same for both blends and the diameter did not appear to be affected by the extensional rate. For the 30/70 PP/EVA±EMA blend ellipsoids were also present. Several particular features can be extracted from these results. For both dies, D1 and D2, the Hencky strain is constant and

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Fig. 8. Micrographs of the blends before the converging section: (a) 70/30 PP/EVA±EMA blend, (b) 30/70 PP/EVA±EMA blend.

equal to 3.6 and 3.7, respectively. The resulting morphology is believed to be dependent on the Hencky strain and not on the extensional strain rate. Even if the initial particle diameter is relatively small, such dominant elongational flows are able to generate a fibrillar structure. But the flow is not the only variable to take into account. The large elongational viscosities and the elastic properties of both Table 5 Morphology obtained after extrusion through both hyperbolic shaped dies Fibrils diameter (mm) D1 70/30 PP/EVA±EMA, Rv 30/70 PP/EVA±EMA, Rv

ˆ 1.45 mm initial ˆ 1.45 mm ‡ domains initial

D2

(ÿ)

(‡)

(ÿ)

(‡)

0.80 1.0 ‡ Ellipsoids

0.60 1.1 ‡ Ellipsoids

0.80 0.90

0.70 0.90

56

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Fig. 9. Micrographs of the blends after the converging section with die D2 at the lowest flow rate: (a) 70/30 PP/EVA-EMA blend, (b) 30/70 PP/EVA-EMA blend.

components could play a dominant role, compared to the shear components, in the establishment of the final morphology [21,38]. The PP phase in the 30/70 PP/EVA±EMA blend was about ten times more viscous than the continuous phase and a fibrillar structure was still observed in a very short time (the residence time in the die is in the range of 0.1 to 1 s depending on the flow conditions). Eqs. (6) and (7) were used to predict the morphological evolution of the blends, assuming that the flow in the converging die was purely elongational and that the elongational viscosities of both components were constant through the die. Also the parameters determined for shear flow were initially used. The governing equations for the interface developed by Doi and Ohta [4] and extended by Lee and Park [1] describe the transition from a spherical morphology to a fibrillar structure. Nevertheless, the predictions remain only qualitative as the interfacial area is greatly overestimated. The calculated radii of fibers were of the order of nm whereas we determined experimentally radii of the order of 0.4 mm. Moreover, the time needed to reach a fibrillar structure was much longer than the residence time in the die and it increased when the slip parameter  was increased. Changing the value for d2, related to coalescence did not improve the predictions at least for the time required for the changes of

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structure. Therefore, we think that the discrepancies largely arise from the fact that the flow is not purely extensional. We recall that about 40% of the pressure drop is due to the shear effects. Moreover, Eqs. (6) and (7) may not be general enough to describe adequately the morphological evolution under elongational flows. More work is needed to characterize adequately these major transitions and test the relaxation mechanisms proposed by Lee and Park [1]. Vinckier et al. [11] have reported very recently that these mechanisms were not adapted for the breakup of highly extended droplets (fibrils) by endpinching or Rayleigh instabilities and only correctly predicted the relaxation of slightly deformed droplets. Neglecting the contribution of the interfacial term, the mixing rule included in Eq. (1) for the stress tensor slightly over-predicts the elongational viscosity data for the 70/30 PP/EVA±EMA For example, at an elongational rate of 10 sÿ1, the value predicted is 11 800 Pa s, compared to the experimental value of 11 200 Pa. However, both mixing rules (Eqs. (1) and (3)) under-predict the elongational viscosities for the 30/70 PP/EVA±EMA blend, but the best prediction is with the mixing rule of Eq. (3), with an elongational viscosity of 8200 Pa s at 10 sÿ1 compared to the experimental value of 8500 Pa s. More work will be carried on to fully investigate the contribution of the interface to the stress tensor and of coupling effects between the interface and the bulk properties of both components. 5. Conclusions The Lee and Park model [1] as well as modified versions [3] are shown to predict qualitatively the morphological changes occurring in both shear and elongational flows. The predictions of the different models for simple shear flow have been shown to be very sensitive to the choice of the mixing rule used to account for a mismatch of the both phase viscosities. We believe that coupling effects between the bulk properties and the interfacial tensor should be taken into account. This remains nevertheless an open subject. Adding an additional parameter for nonaffine deformations could lead to better results in certain cases but the predictions remain bounded to the chosen mixing rule. For elongational flows, the Lee and Park governing interface equations are able to qualitatively predict the transition from an initial spherical morphology to a fibrillar structure, observed using a nozzle shaped die, for which the flow is largely extensional. The extensional viscosity used in the Lee and Park equations have been calculated from entrance pressure drop measurements. The Cogswell [23], Binding [24] and Mackay/Astarita [25] analyses have been used for this purpose and all the three methods lead to values ranging in the same order of magnitude. The elongational behavior of both 70/30 PP/EVA±EMA and 30/70 PP/EVA±EMA blends are strain-hardening with respect to their shear properties as shown by large Trouton ratio values. The elongational properties of the EVA±EMA phase greatly influence those of the blends and appear to play a major role in the resulting morphology of the blends. Fibrillar morphology are indeed observed for blends having a zero shear viscosity ratio of 0.1 or 10. Acknowledgements The authors are thankful for the financial support received from ELF-ATOCHEM. Discussions with Prof. M. Moan are gratefully acknowledged. We are also thankful to Mr. L. Parent and Mr. C. Painchaud for their help during blending.

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