Introduction

Blends of immiscible polymers display a complex andvariable rheological behaviour. It reflects the variousmorphological changes that can occur in such systemsduring and after flow (Tucker and Moldenaers 2002).The present study focuses on stress relaxation. In diluteblends, a typical droplet/matrix morphology is encoun-tered. During a stress relaxation experiment the shape ofthe droplets can evolve in different ways, depending ontheir initial geometry. For relatively small values of theaspect ratio, i.e. the ratio of the longest axis over theshortest axis of the droplet, the deformed droplet re-tracts to a spherical shape without breaking up. Morestrongly deformed droplets still retract without break-

ing, but they adopt a variety of intermediate shapes suchas cylinders or dumbbells (Yamane et al. 1998). Above acritical initial aspect ratio, approximately 10, break-upwill occur during relaxation: the ends of the dropletbecome bulbous and pinch off (Stone et al. 1986). Forvery long, fibrillar droplets, interfacial instabilities maygrow along the surface of the fibril, leading to simulta-neous break-up of the droplet into a string of fragments(Elmendorp 1986). During these relaxation processes,the orientation of the deformed droplets remains con-stant (Stone et al. 1986; Yamane et al. 1998). In con-centrated blends fibrillar morphologies develop morereadily (Jeon and Hobbie 2001; Huitric et al. 1998;Martin et al. 2000; Ziegler and Wolf 1999). The relaxa-tion of these more complex structures is not well

Thomas Jansseune

Paula Moldenaers

Jan Mewis

Stress relaxation after steady shear flowin immiscible model polymer blends

Received: 3 July 2003Accepted: 3 December 2003Published online: 8 April 2004� Springer-Verlag 2004

Abstract Stress relaxation in immis-cible blends is studied for a well de-fined shear history, i.e. afterprolonged steady state shearing.Model systems are used that consistof quasi-Newtonian liquid polymers.Hence the relaxation is dominatedby changes in the morphology of theinterface. Both shear stress and thefirst normal stress are considered.The measurements cover the entireconcentration range. For diluteblends the interfacial contribution tothe stress relaxation compares wellwith model predictions. Deviationsoccur when the matrix phase isslightly elastic. In that case the sim-ilarity between the relaxation ofshear and normal stresses is alsolost. The latter is attributed to a

wider drop size distribution.Increasing the concentration of thedisperse phase results in a complexevolution of the characteristicrelaxation times. The normal stressesrelax systematically slower than theshear stresses and the concentrationcurve includes two maxima. Evenfor equiviscous components theconcentration curves are not sym-metrical. It is concluded that even aslight degree of elasticity in the ma-trix phase drastically affects themorphology and the interfacialrelaxation of such blends.

Keywords Immiscible blends ÆMorphological models ÆRheology Æ Concentration ÆRelaxation

Rheol Acta (2004) 43: 592–601DOI 10.1007/s00397-003-0351-6 ORIGINAL CONTRIBUTION

T. Jansseune Æ P. MoldenaersJ. Mewis (&)Department of Chemical Engineering,K.U. Leuven, de Croylaan 46,3001 Leuven, BelgiumE-mail: jan.mewis@cit.kuleuven.ac.be

understood; various phenomena can contribute: e.g.retraction, coalescence and/or break-up (Elemans et al.1997).

The interface between the phases contributes directlyto the stresses, which explains the link between rheologyand microstructure in two-phase blends. This has beendemonstrated in particular for dilute blends of quasi-Newtonian fluids. Results have been reported for vari-ous types of flow histories, including steady state shearflow (Jansseune et al. 2000; Takahashi et al. 1994;Vinckier et al. 1997b), start-up of flow (Almusallamet al. 2000; Jansseune et al. 2001; Peters et al. 2001;Takahashi et al. 1994; Vinckier et al. 1997a), relaxation(Almusallam et al. 2000; Peters et al. 2001; Vinckier et al.1997b) and flow reversal (Minale et al. 1999). Veryrecently, Jackson and Tucker (2003) proposed a modelthat describes well transient droplet shapes over a widerange of kinematic conditions. Nearly all systematicstudies deal with dilute systems, often with up to 10% ofdispersed phase. The objective of the present work is toelucidate the effect of composition on relaxation in non-dilute systems. In such systems it is difficult to generate awell defined and reproducible morphology. Here, steadystate shearing has been used as a reference. Reaching thesteady state morphology turns out to be non-trivial andrequires extensive shearing.

Quasi-Newtonian components have been selected,so the stresses should be dominated by interfacialphenomena. Both shear stress and first normal stressdifference will be considered.

Theoretical

Models for droplet retraction during relaxation

Different models have been proposed to describe theshape evolution of moderately deformed dropletsduring retraction. In addition to the theory for smalldeformations (Rallison 1980), three other models will beconsidered: the Maffettone-Minale model (MM) (Maf-fettone and Minale 1998), the constrained volume model(CV) (Yamane et al. 1998) and semi-empirical modifica-tions of the Palierne model (Palierne 1990; Vinckier et al.1997b). The MM and CV models describe single dropletsas does the linear small deformation theory. Thesemodelsonly apply to dilute systemswithNewtonian components.The Paliernemodel is a rheological model. It incorporatesviscoelastic components as well as a limited degree ofdroplet-droplet interactions. From the morphologicalmodels the resulting interfacial stress contributions can becalculated (Doi and Ohta 1991).

Small deformation theory and the Palierne model Smalldroplet deformation theory, reviewed and completed byRallison (1980), provides solutions for the flow field and

the droplet deformation of a single, near-sphericaldroplet. The theory can be applied to steady state andtransient flows. The evolution of the droplet shape aftercessation of flow is described by analytical expressionsfor the major axes of the ellipsoidal droplet. The resultsare in good agreement with experimental observationswhen the capillary number of the preceding flow is below0.3 (Guido and Greco 2001). The dimensionlesscapillary number, Ca=gmR _c/a, expresses the balancebetween the shearing forces and the restoring interfacialforces, with gm the matrix viscosity, R the radius of theundeformed droplet, _c the shear rate and a the interfacialtension. The value of Ca controls the droplet deforma-tion during flow. When the capillary number reaches acritical value, Cacrit, the droplets become unstable andbreak up (de Bruijn 1989).

Palierne (1990) applied the small deformation theoryto oscillatory flow, including viscoelasticity of the com-ponents and some degree of droplet interaction. Fromhis analysis a relaxation time sd for droplet deformationcan be derived:

sd ¼Rgm

4a:19p þ 16ð Þ 2p þ 3� 2/ p � 1ð Þð Þ

10 p þ 1ð Þ � 2/ 5p þ 2ð Þ ð1Þ

where F is the volume fraction of the dispersed phaseand p the viscosity ratio, i.e. the ratio of the dropletviscosity over the matrix viscosity. The Palierne modeldescribes the linear viscoelastic behaviour.

The Maffettone-Minale model Maffettone and Minale(1998) proposed an evolution equation for the shapeand orientation of an isolated ellipsoidal Newtoniandroplet in a Newtonian matrix subjected to a generalflow field. Nonlinearities were introduced through aphenomenological term containing Ca. The modeldescribes the shape of the droplet without providingdetails of the flow in and around the inclusion. Forshear flow, the model can be applied as long asCa<Cacrit. The model predictions have been verifiedwith microscopic observations of the shape of singledroplets under, e.g. steady state shear flow (Guidoet al. 1999; Maffettone and Minale 1998), start-upflows (Almusallam et al. 2000) and flow reversals(Guido et al. 2000) with viscosity ratios between 0.1and 4. The model should not be used when the dropletevolution involves other shapes than ellipsoids, e.g.dumbbells. This was demonstrated by Almusallamet al. (2000) for the relaxation of considerably de-formed droplets, which can occur in step-strain exper-iments. The MM model has been successfully appliedto deduce the interfacial tension from microscopicobservations of retracting droplets in a variety ofpolymer blend systems (Mo et al. 2000). Because themodel adequately describes the shape of the interface,it can also be used to predict the interfacial contribu-

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tions to the stresses in dilute blends when the capillarynumber is not too high. This has been demonstratedfor steady state and start-up flows (Jansseune et al.2000, 2001).

The MM model describes the interface by a sym-metrical, positive-definite, second rank tensor S. Itseigenvalues are the squares of the three semi-axes of theellipsoid, scaled by the droplet radius. The evolution ofS under flow is given by

dSdt� X:Sþ S:X ¼ � f1

smS� g Sð ÞI½ � þ f2 D:Sþ S:Dð Þ

ð2Þ

with the appropriate expressions for the vorticity tensorW and the deformation rate tensor D of the governingflow field. The parameter sm is a time constant equal togmR/a. The function g(S) imposes droplet volume con-servation; f1 and f2 are derived from the linear theory byincorporating a phenomenological term containing Ca.

For steady state shear flow, an analytical solution forthe three different axes and for the orientation angle ofthe ellipsoidal droplet can be obtained from Eq. (2). Thedroplet deformation can be expressed by the aspect ratiorp as

rp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

f 21 þ Ca2 þ f2Ca

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

f 21 þ Ca2

q

f 21 þ Ca2 þ f2Ca

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

f 21 þ Ca2

q

v

u

u

u

t ð3Þ

During relaxation, after cessation of flow, Eq. (2)reduces to (Mo et al. 2000)

dSdt¼ � f1

smS� g Sð ÞI½ � ð4Þ

Equation (4) predicts a non-exponential relaxation withsr=sm/f1 as a characteristic time. In this equation theeffect of initial droplet deformation is taken intoaccount: the relaxation becomes slower for droplets thatare initially more deformed. The MM model reduces tosmall droplet deformation theory in the limit ofvanishing Ca, providing a relaxation time sd equal tothat of Eq. (1), evaluated for F=0.

Other models have been proven to be useful indescribing the morphology of blends in particular con-ditions. This is illustrated by the constrained releasemodel (Almusallam et al. 2000). Whereas in the MMmodel the deformation term is modified by a nonlinearterm in Ca, the CV model is based on affine deformationwith an empirical modification for the relaxation ofdroplets after step strain experiments. Almusallam et al.(2000) demonstrated a good agreement between themodel and experimental results for start-up and step-strain experiments for p=1 and Ca between 1.3 and 70.The proposed deformation term only applies in case ofaffine deformation and equiviscous components (Wetzel

and Tucker 2001). Different conditions will be consid-ered in the present work and therefore the model has notbeen used.

Stress response in dilute emulsions of Newtoniancomponents

As the morphological changes serve as a basis for thediscussion, only rheological equations are consideredthat are based on morphological models for dropletretraction. Other models for blends, e.g. those derivedfrom the Doi-Ohta formalism such as that by Bousminaet al. (2001), although rheologically useful, are thereforenot included.

The general expression for the total stress tensor rt indilute emulsions of Newtonian components (Kennedyet al. 1994; Onuki 1987) reduces, during relaxation, to

rt ¼ �PI

� gm � gdð ÞV

Z

unþ nuð ÞdS� aV

Z

nn� 1

3I

� �

dS

ð5Þ

where P is an isotropic pressure, I the unit tensor, V thetotal volume of the system, u the local velocity at thedroplet interface and gd the dispersed phase viscosity.The last term on the right hand side of Eq. (5) can becalculated directly when the shape, size and orientationof the interface are known, e.g. from one of the modelsdiscussed above. This term is therefore referred to as theinterfacial contribution to the stress. The second term iscalled here, by convention, the component contribution.It depends on the flow of the components in and aroundthe inclusions and is therefore determined by the vis-cosity of the components. During relaxation, the flow isdriven by the interfacial tension. Under these conditionsthe component contribution also scales with the inter-facial tension, exactly as the interfacial contribution.The component contribution to the stress relaxation isnormally ignored. This is correct when the viscosity ratiois equal to one, for other values of p positive or negativedeviations are possible. As the viscosity ratio remainsclose to unity in the present work, interference from thecomponents with the relaxation times are not consid-ered. This is also justified by numerical simulations(Cristini et al. 2002). Here, an exception is made for thepossible effect on the orientation angle of the inclusion,which is discussed in the next paragraph.

From the ratio of the interfacial contributions tonormal and shear stresses, the orientation angle of theinclusions can be deduced (Almusallam et al. 2000;Jansseune et al. 2000). It can be verified whether com-ponent contributions alter this relationship. The sym-metry of the flow field during relaxation can be exploitedto derive a relationship between the component contri-

594

butions and the droplet orientation. We consider adroplet in a Cartesian coordinate system, formed by axesin the 1 (velocity), 2 (velocity gradient) and 3 (vorticity)directions. The longest axis of the droplet lies in the 12plane and makes an angle h with the 1 axis. A secondrectangular coordinate system (1¢,2¢,3¢) can then bedefined by rotating the coordinate system around the 3axis over an angle h in such a manner that the 1¢ axiscoincides with the longest axis of the droplet and the 1¢3¢plane corresponds to a plane of symmetry of the droplet.The component contributions to the relaxation of theshear stress and first normal stress difference are calcu-lated by inserting the relations for u and n into Eq. (5).This can be done in both coordinate systems, yielding

r12;comp ¼ gd�gmð ÞV

R

u1n2 þ n1u2ð ÞdS

¼ gd�gmð ÞV

R

u01n01 þ u02n

02

� �

sin2hdSð6Þ

Making the same exercise for the first normal stressdifference reveals that the flow of Newtonian compo-nents during relaxation can also give rise to a first nor-mal stress difference. Together with Eq. (6), this resultsin a simple expression for the stress ratio of the com-ponent contributions during relaxation:

N1;comp

r12;comp¼ 2 cot 2hð Þ ð7Þ

This result is identical with the one that has been derivedfor the stress ratio of the interfacial contributions (Al-musallam et al. 2000; Jansseune et al. 2000). Hence, itcan be concluded that the total stresses during relaxationare related to the orientation angle in a similar fashion.Equation (7) is valid during relaxation in blends withNewtonian components whenever the drop shape ismirror-symmetric about the 1¢3¢ plane. In many droplet-matrix morphologies this can be assumed to be the case.Even for more complex microstructures, e.g. for fila-ments breaking up by interfacial instabilities, it mayprove useful. There are, however, other possible dropshapes, e.g. sigmoidal ones, that do not satisfy thesymmetry requirements.

Experimental

A model blend, containing polyisobutylene (PIB) andpolydimethylsiloxane (PDMS), has been selected for thiswork. The components are quasi-Newtonian and areimmiscible. As the relaxation time of a blend is pro-portional to the matrix viscosity, high viscosity grades ofthese polymers, i.e. PIB Parapol 1300 from Exxon andPDMS Rhodorsil 47V200.000 from Rhodia, were cho-sen. The components possess only a limited elasticity:the PDMS sample has a dominant relaxation time of theorder of 0.01 s in the terminal zone, whereas the PIBrelaxes even faster. The interfacial tension for this model

blend is 3 mN/m at room temperature (Guido et al.1999; Sigillo et al. 1997). Viscosities of PIB and PDMSat 293 K are 129 Pa.s and 208 Pa.s respectively. At287 K both polymers have a viscosity of 235 Pa.s.Experiments were performed at these two temperatures,resulting in viscosity ratios of 0.60 (PIB in PDMS293 K), 1 (all experiments at 287 K) and 1.61 (PDMS inPIB, 293 K). According to the empirical relation of deBruijn (1989), the critical capillary numbers for diluteblends of this system are 0.58 (p=1.61) or 0.46 (p=0.62)at 293 K, depending on the polymer that constitutes thematrix. At 287 K the critical capillary number becomes0.48. Blends were prepared with different amounts ofdispersed phase, covering the entire concentration range.

The measurements were performed on a RheometricsRMS 800, equipped with a cone and plate geometry(diameter 25 mm, cone angle 0.1 rad) to ensure a uni-form shear history throughout the sample. The tem-perature was controlled by means of a fluids bath. Toachieve reproducible initial conditions, the morphologyat the onset of the relaxation experiments has to becontrolled carefully. This can be done rather easily fordilute, non-compatibilized blends by shearing at a givenshear rate for a sufficiently long time. In such cases 3000to 4000 strain units often seem to be adequate (Minaleet al. 1997). In concentrated blends, however, effects ofthe flow history persist much longer, which could cause asubstantial scattering in the data (Vinckier et al. 1999).In the present study, shearing for up to 80.000 strainunits was found to be necessary for the most concen-trated systems. Similar long preshearing periods had tobe used by Astruc and Navard (2000) in blends ofPDMS and HPC. In Fig. 1 the reproducibility that canbe achieved in this manner for relaxation experiments isshown for a blend with a 50/50 composition.

The stress relaxation curves provide two kinds ofinformation. The instantaneous drop in stress at themoment the flow is arrested corresponds to the com-ponent contribution to the stress during flow, at least

Fig. 1 Reproducibility of relaxation measurements in concentratedblends (50% PIB/50% PDMS) after steady shearing at 5 s–1

(T=293 K)

595

with Newtonian components. In this manner the stressesduring shear can be separated in component and inter-facial contributions. The contributions to the steadystate stresses have been discussed elsewhere (Jansseuneet al. 2003). The part of the stress that relaxes graduallyreflects the shape relaxation of the dispersed phase(Vinckier et al. 1997b) and is the subject of the presentdiscussion.

Results and discussion

The stress relaxation after cessation of steady state flowhas been measured systematically over the whole con-centration range of the PIB/PDMS system describedabove. In the following section the resulting curves for adilute blend, with a dispersed phase volume fraction of0.1 will be compared with the model predictions forshape relaxation of single droplets. The effect of con-centration will be discussed on the basis of characteristicrelaxation times in the section after that.

Dilute blends

As a reference case for the more concentrated blends,dilute systems with 10% of disperse phase will be used.In principle, the models for single droplet morphologyshould describe reasonably well the resulting behav-iour(e.g. Friedrich et al. 1995; Jansseune et al. 2000). It isoften difficult to obtain the real steady state morphologyin dilute systems when the shear rate is below a criticalvalue. The pseudo steady state morphology, which isreached after shearing for a long time at a given shearrate, could still depend on the prior shear history of thesample. When starting initially with smaller droplets,coalescence will practically stop at a limiting size whichis below the smallest size that is reached by breaking upinitially larger droplets at that same shear rate. Thisresults in a hysteresis region for the morphology at lowconcentrations and shear rates (Minale et al. 1997). Fora blend with 10% PDMS in PIB this critical shear rate isapproximately 5 s–1 (Jansseune et al. 2000). An advan-tage of working in the hysteresis region is that a nar-rower droplet size distribution can be achieved (Grizzutiand Bifulco 1997).

First, an experiment with moderately deformeddroplets will be considered. In Fig. 2 the relaxationcurves of the shear stress and the first normal stressdifference are shown for the blend with 10% PDMSdispersed in PIB at 293 K. An initial average droplet sizeof 3 lm has been generated by pre-shearing at 5 s–1,corresponding to Ca=Cacrit (Jansseune et al. 2000).Subsequently, the flow is stopped and the droplets areallowed to retract. Next, the droplets are deformed byapplying a shear rate of 2.5 s–1, which corresponds to

Ca=0.29, i.e. half the critical value. As soon as thestresses level off, implying that the droplets have reachedtheir steady state deformation, the flow is stopped andthe stress relaxation is recorded. The shearing period iskept as short as possible, at most 6 s (c=15), to preventinterference from coalescence.

The aspect ratio of the droplets at Ca=0.5Cacrit iscalculated with the MMmodel to be 1.9. Upon cessationof flow a part of the shear stress, i.e. the componentcontribution to the stress during flow, relaxes instanta-neously (Fig. 2). Only after this sharp decrease theinterfacial stresses start to relax gradually. As the blendis composed of quasi-Newtonian liquids, there is nocomponent contribution to the first normal stress dif-ference during flow and therefore no sudden drop at theonset of the relaxation either (Fig. 2). As shown byVinckier et al. (1997b), the shear stress and the firstnormal stress difference show an exponential decreasewith identical relaxation times. From Eq. (7) it can beconcluded that the constant ratio between the first nor-mal stress difference and the shear stress means that theorientation angle h of the droplet does not change dur-ing relaxation. This is in agreement with the experimentsby Yamane et al. (1998).

Fig. 2a,b Comparison of the predictions of the MM model and thesmall deformation theory for: a the relaxation curve of the shearstress; b the first normal stress difference after shearing for 6 s at2.5 s–1 (10% PDMS in PIB at T=293 K)

596

The relaxation of the interfacial contribution to thestresses, calculated on the basis of the MM model, is inexcellent agreement with the experimental data, consid-ering that the model does not contain any fittingparameter. Although the small droplet deformationtheory could be expected to underestimate the dropletdeformation somewhat for Ca=0.29 (Guido and Greco2001), the model still fits the data during flow andrelaxation very well. Both models predict identicalrelaxation times for the shear stress and the first normalstress difference, i.e. 0.38 s. This value is also consistentwith the Palierne model, Eq. (1).

In Fig. 3 results are compiled of relaxation experi-ments after shearing at different shear rates and startingfrom different initial droplet deformations. Shear ratesinside as well as outside the hysteresis region were used.The droplet sizes and the extent of the hysteresis regionfor these experimental conditions are known from pre-vious work (Jansseune et al. 2000). The relaxation timesin Fig. 3 have been scaled with the time constant sd ofthe linear Palierne theory as derived from Eq. (1). Thetime required for the interfacial relaxation to decay to 1/e of its initial value was taken as the relaxation time. Forall experiments, except for the one at the highest shearrate (6 s–1), the relaxation curves were exponential.Values have been determined for both the shear stressand the first normal stress difference. For the samplesthat were generated in the hysteresis region, the tworelaxation times are identical. They increase withincreasing droplet deformation, consistent with the dataof Vinckier et al. (1997b). At small deformations, therelaxation times coincide with the Palierne values. Theaspect ratio rp has only a secondary effect, at least in thisrange, which the MM model seems to capture well. Anempirical modification of the Palierne time constant hasbeen proposed to describe the effect (Vinckier et al.

1997b). It was based on using a correction factorr\S(2/3;p), where rp was calculated from the Choi andSchowalter analysis (Choi and Schowalter 1975). Usingthe same correction factor with rp values derived fromthe more recent MM model does not produce satisfac-tory results.

Some of the data in Fig. 3 deviate from the modelpredictions. More specifically the relaxation times for N1

have a tendency to become larger than those for r12

when the droplets are initially more deformed. Accord-ing to Eq. (7), this should mean that the orientationangle of the droplets decreases in time. Such a conclu-sion is not supported by either theoretical or experi-mental evidence (Stone et al. 1986; Yamane et al. 1998).A possible explanation is provided by the broaderdroplet size distribution that is expected for blendsgenerated outside the hysteresis region. Smaller dropletsmake a larger angle with the flow direction, deform lessand retract faster. As a result the average orientationangle gradually decreases in time, even when the orien-tation angle of the individual droplets remains constant.The shear stresses are proportional to sin(2h); hencetheir value will be more affected by the smaller dropletsin the droplet distribution than by the larger ones, assmall droplets possess a larger orientation angle. Theopposite holds for the first normal stress difference.Because smaller droplets retract faster than larger ones,the shear stress will relax faster than the first normalstress difference.

The given profile of the relaxation curves is expectedwhenever blends are generated by a dynamic equilibriumbetween break-up and coalescence. Such results will bediscussed in the next section. As an example, the relax-ation after extensive shearing at 5 s–1 is shown forthe inverse system, i.e. 10% PIB in PDMS (Fig. 4), forwhich the critical shear rate is below 1 s–1. The dampedoscillation in the normal stresses is caused by thetransient in the transducer response. The faster initial

Fig. 3 Relaxation times for the shear stress (circles) and firstnormal stress difference (triangles) at different initial dropletdeformations for samples generated in the hysteresis region (opensymbols) and outside this region (filled symbols) (10% PDMS inPIB at T=293 K). The solid line is the prediction of the MM model

Fig. 4 Relaxation curves after shearing at 5 s–1 of a blendgenerated outside the hysteresis region (10% PIB/90% PDMS atT=287 K)

597

relaxation of the shear stress is clearly visible. It can alsobe seen that the curves of shear stress and normal stresstend towards the same slope near the end of the relax-ation. At this stage only the larger droplets are stillrelaxing and the evolution of the two stresses shouldbecome similar.

Concentrated blends

Increasing the concentration of the dispersed phase doesnot only accelerate coalescence (Minale et al. 1998); italso affects break-up (Jansen et al. 2001). The net resultis that the hysteresis region shifts to lower shear rates(Minale et al. 1998), the droplet size distributionbroadens (Friedrich et al. 1995; Sundaraj and Macosko1995) and more complex morphologies develop. Inparticular a fibrillar microstructure emerges in concen-trated blends (Martin et al. 2000; Ziegler and Wolf1999). Morphology hysteresis could only be detected inthe present case for 10 and 20% PDMS in PIB at 293 Kwith critical shear rates of approximately 5 and 2 s–1 forrespectively the 10 and 20% blends (Jansseune et al.2000). These results are consistent with the findings ofGrizzuti and Bifulco (1997) on a similar blend.

Experimental results for the characteristic relaxationtimes, covering the entire concentration range, areshown in Figs. 5 and 6 for respectively T=293 K andT=287 K. In all these experiments the blends weresheared until the morphology remained constant beforethe flow was stopped. Characteristic relaxation times fordilute blends as derived from the MM model are addedas horizontal bars. The droplet sizes were calculated onthe basis of the critical capillary number, using de Bru-ijn’s equation (de Bruijn 1989). For the samples thatdisplay a morphology hysteresis (Fig. 5a), the theoreti-cal relaxation times are given for the break-up as well asthe coalescence lines. As expected from the previoussection, predictions with the MMmodel are good for theblend of 10% PDMS dispersed in PIB with p=1.61,where a morphology hysteresis is indeed present. For theopposite case, i.e. 10% PIB in PDMS, deviations occur.In this case the relaxation times for N1 can be muchlarger than predicted, as already mentioned in the pre-vious section.

The concentration effects are qualitatively similar forthe shear stress and the first normal stress difference. Atrather low concentrations of the dispersed phase, there isgenerally an increase of the relaxation time with con-centration. A further increase in concentration of thedispersed phase results in a central concentration rangewith nearly constant relaxation time or with even twolocal maxima in relaxation time. The system with PDMSdispersed in PIB at 293 K at _c=1 s–1 (Fig. 5a) is anexception, because of the pronounced effect of mor-phology hysteresis. Plotting rheological parameters vs

composition can result in a single maximum, which isthen often associated with the point of phase inversion(Jeon and Hobbie 2001). When two maxima can be seen,it seems logical to assume a central concentration rangewith a co-continuous structure. There is, however, atpresent no direct evidence of a stable co-continuousmorphology during steady state shear flow (Veenstraet al. 1999; Jansseune et al. 2003). Also, such a mor-phology is expected to give rise to a strong increase inrelaxation times (Roths et al. 2002).

The interfacial contributions to the steady state shearstress and first normal stress difference do not display asingle maximum in the concentration curve either(Jansseune et al. 2003). Similar patterns have also beenreported earlier for the viscosity and the first normalstress difference (Huitric et al. 1998) and for the extent ofshear thinning (Ziegler and Wolf 1999). These have beenrelated to the presence of a fibrillar morphology in themiddle of the concentration range, in agreement withvarious direct observations (Jeon and Hobbie 2001;

Fig. 5a,b Concentration dependence of the relaxation times forshear stress (open circles) and first normal stress difference (opentriangles) after cessation of flow (T= 293 K): a shear rate=1 s–1; bshear rate=5 s–1. The small horizontal lines represent the relaxationtimes from the MM model. For the system with morphologyhysteresis both the relaxation times at the coalescence (lower one)and break-up line (upper one) are given

598

Huitric et al. 1998; Luciani and Jarrin 1996; Ziegler andWolf 1999). In concentrated systems fibrils could beoccasionally interconnected and hence their presencedoes not exclude a certain degree of co-continuity.

Freezing the structure immediately after shearing andobserving it by SEM suggests that, also in the presentcase, there is a fibrillar structure near the 50/50 com-position (Jansseune et al. 2003). Relaxation in suchsystems can proceed according to different mechanisms.Interfacial instabilities, resulting in a series of dropletsafter relaxation, can replace droplet retraction for fibrilsat all concentrations. This could be delayed in moreconcentrated blends, but such systems can also relaxby droplet coalescence (Elemans et al. 1997). This hasbeen observed during the annealing of fibrillar andco-continuous structures in a variety of blends withconcentrations of the less concentrated phase in excessof 30% (Bouilloux et al. 1997; Quintens et al. 1990;Willemse 1999). An indication for coalescence underquiescent conditions in the present systems is providedby the gradual change, during relaxation, of the dynamicmoduli (not shown). Coalescence under quiescent con-ditions is favoured by higher contents of the dispersed

phase, which could explain the minimum around 50/50as in Figs. 5 and 6.

At 287 K (Fig. 6) the viscosity ratio equals one,irrespective of which of the components is the continu-ous phase. For Newtonian components one wouldtherefore expect fully symmetric relaxation time/con-centration curves. This seems not to be the case here, inparticular for the first normal stresses at 5 s–1 (Fig. 6b).Under these conditions substantial differences are alsoobserved between the relaxation times for r12 and N1.The only possible reason for asymmetry seems to be theslight viscoelasticity of the PDMS used here. Two effectsmight play a role (de Bruijn 1989; Mighri et al. 1997;Tretheway and Leal 2001). First, component viscoelas-ticity could affect the droplet deformation during flowand during the subsequent retraction. A recent pertur-bation analysis suggested only a marginal shape effectbut a rather strong orientation effect for second orderfluids (Guido et al. 2002). Second, the droplet size dis-tribution might be altered by the viscoelasticity of thecomponents. Droplet elasticity is known to increase thecritical capillary number for break-up, whereas matrixelasticity tends to decrease it. The effect of elasticity oncoalescence is not well established.

Conclusions

The interfacial relaxation in a two-phasic blend is acomplex function of the morphology. This was investi-gated with relaxation experiments after steady stateshear flow. For dilute blends generated in the hysteresisregion, the rheological response could be adequatelymodelled on the basis of the Maffettone-Minale modelor, for Ca<0.3, with the linear theory for small dropletdeformations. The results for rather dilute blends, thatwere generated outside the hysteresis region, deviatedfrom the predictions. This was attributed to a broaderdistribution of droplet sizes. At the same time thestresses no longer decayed exponentially and the simi-larity between shear stress and first normal stress dif-ference was lost. This is consistent with a droplet sizedistribution. Smaller, less oriented droplets retract fasterthan larger, more oriented ones. Hence, the averageangle with respect to the flow direction decreases in time,which explains the relative changes of the stress com-ponents during relaxation.

Increasing the concentration of the dispersed phaseinitially slows down the relaxation because of hydrody-namic interactions between the droplets. A further in-crease in concentration can result again in shorterrelaxation times, producing relaxation time/compositioncurves with two maxima. The region between the max-ima seems to be associated with a morphology of closelypacked droplets or fibrils that relax, at least partially, bycoalescence. Experiments with equiviscous components

Fig. 6a,b Concentration dependence of the relaxation times(T=287 K): a shear rate=1 s–1; b shear rate=5 s–1); symbols seeFig. 5

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produce asymmetric curves of relaxation time vs com-position. This suggests that even a slight degree ofelasticity of the matrix could affect the relaxation pro-cess.

Acknowledgements This study was partially funded by a G.O.A.project (98/06) from the Research Fund of the Katholieke Uni-versiteit Leuven and by DSM Research (Geleen, NL). Stimulatingdiscussions with Prof. C Tucker are gratefully acknowledged.

References

Almusallam AS, Larson RG, Solomon MJ(2000) A constitutive model for theprediction of ellipsoidal droplet shapesand stresses in immiscible blends.J Rheol 44:1055–1083

Astruc M, Navard P (2000) A flow-inducedphase inversion in immiscible polymerblends containing a liquid-crystallinepolymer studied by in situ opticalmicroscopy. J Rheol 44:693–712

Bouilloux A, Ernst B, Lobbrecht A, MullerR (1997) Rheological and morphologi-cal study of the phase inversion inreactive polymer bends. Polymer38:4775–4783

Bousmina M, Aouina M, Chaudhry B,Guenette R, Bretas RES (2001) Rheol-ogy of polymer blends: non-linearmodel for viscoelastic emulsions under-going high deformation flows. Rheol.Acta 40:538–551

Choi SJ, Schowalter WK (1975) Rheologi-cal properties of nondilute suspensionsof deformable particles. Phys Fluids18:420–427

Cristini V, Macosko CW, Jansseune T(2002) A note on transient stress calcu-lation via numerical analysis. J Non-Newtonian Fluid Mech 105:177–187

de Bruijn RA (1989) PhD thesis, T.U.Eindhoven, Eindhoven

Doi M, Ohta T (1991) Dynamics and rhe-ology of complex interfaces I. J ChemPhys 95:1242–1248

Elemans PHM, van Wunnik JM, van DamRA (1997) Development of morphologyin blends of immiscible polymers. AI-ChE J 43:1649–1651

Elmendorp JJ (1986) A study on polymerblending microrheology. Polym Eng Sci26:418–426

Friedrich C, Gleinser W, Korst E, Maier D,Weese J (1995) Comparison ofsphere-size distributions obtained fromrheology and transmission electronmicroscopy in PMMA/PS blends.J Rheol 39:1411–1425

Grizzuti N, Bifulco O (1997) Effects of coa-lescence and break-up on the steady statemorphology of an immiscible blend inshear flow. Rheol Acta 36:406–415

Guido S, Greco F (2001) Drop shape underslow steady shear flow and duringrelaxation. Experimental results andcomparison with theory. Rheol Acta40:176–184

Guido S, Simeone M, Villone M (1999)Diffusion effects on the interfacial ten-sion of immiscible polymer blends.Rheol Acta 38:287–296

Guido S, Minale M, Maffettone PL (2000)Drop shape dynamics under shear-flowreversal. J Rheol 44:1385–1400

Guido S, Simeone M, Grecco F (2002) In:Martinez Boza FJ et al. (eds) Progress inrheology theory and applications. Gru-pa Esp. Reologia, Sevilla, pp 143–146

Huitric J, Mederic P, Moan M, Jarrin J(1998) Influence of composition andmorphology on rheological propertiesof polyethylene/polyamide blends.Polymer 39:4849–4856

Jackson NE, TuckerI CL III (2003) Amodel for large deformation of anellipsoidal droplet with interfacial ten-sion. J Rheol 47:659–682

Jansen KMB, Agterof WGM, Mellema J(2001) Droplet breakup in concentratedemulsions. J Rheol 45:227–236

Jansseune T, Mewis J, Moldenaers P, Mi-nale M, Maffettone PL (2000) Rheologyand rheological morphology determi-nation in immiscible two-phase polymermodel blends. J Non-Newtonian FluidMech 93:153–165

Jansseune T, Vinckier I, Moldenaers P,Mewis J (2001) Transient stresses inimmiscible model polymer blends dur-ing start-up flows. J Non-NewtonianFluid Mech 99:167–181

Jansseune T, Moldenaers P, Mewis J (2003)Morphology and rheology of concen-trated biphasic blends in steady shearflow. J Rheol 47:829–845

Jeon HS, Hobbie EK (2001) Shear viscosityof phase-separating polymer blendswith viscous asymmetry. Phys Rev E 63

Kennedy MR, Pozrikidis C, Skalak R(1994) Motion and deformation of li-quid drops, and the rheology of diluteemulsions in simple shear flow. ComputFluids 23:251–278

Luciani A, Jarrin J (1996) Morphologydevelopment in immiscible polymerblends. Polym Eng Sci 36:1619–1626

Maffettone PL, Minale M (1998) Equationof change for ellipsoidal drops in vis-cous flow. J Non-Newtonian FluidMech 78:227–241

Martin P, Carreau PJ, Favis BD (2000)Investigating the morphology/rheologyinterrelationships in immiscible polymerblends. J Rheol 44:569–583

Mighri F, Ajji A, Carreau PJ (1997) Influ-ence of elastic properties on dropdeformation in elongational flow.J Rheol 41:1183–1201

Minale M, Moldenaers P, Mewis J (1997)Effect of shear history on the morphol-ogy of immiscible polymer blends.Macromolecules 30:5470–5475

Minale M, Mewis J, Moldenaers P (1998)Study of the morphological hysteresis inimmiscible polymer blends. AIChEJ 44:943–950

Minale M, Moldenaers P, Mewis J (1999)Transient flow experiments in a modelimmiscible polymer blend. J Rheol43:815–828

Mo H, Zhou C, Yu W (2000) A newmethod to determine interfacial tensionfrom the retraction of ellipsoidal drops.J Non-Newtonian Fluid Mech91:221–232

Onuki A (1987) Viscosity enhancement bydomains in phase-separating fluids nearthe critical point: proposal of criticalrheology. Phys Rev A 35:5149–5155

Palierne JF (1990) Linear rheology ofviscoelastic emulsions with interfacialtension. Rheol Acta 29:204–214

Peters GWM, Hansen S, Meijer HEH(2001) Constitutive modeling of disper-sive mixtures. J Rheol 45:659–690

Quintens D, Groeninckx G, Guest M, AertsL (1990) Mechanical behavior related tothe phase morphology of PC/SANpolymer blends. Polym Eng Sci30:1474–1483

Rallison JM (1980) Note on thetime-dependent deformation of aviscous drop which is almost spherical.J Fluid Mech 98:625–633

Roths T, Friedrich C. Marth H, Honerk-amp J (2002) Dynamics and rheology ofthe morphology of immiscible polymerblends—on modeling and simulation.Rheol Acta 41:211–222

Sigillo I, di Santo L, Guido S, Grizutti N(1997) Comparative measurements ofinterfacial tension in a model immiscibleblend. Polym Eng Sci 37:1540–1549

Stone HA, Bentley BJ, Leal, LG (1986) Anexperimental study of transient effects inthe breakup of viscous drops. J FluidMech 173:131–158

600

Sundararaj U, Macosko CW (1995) Dropbreakup and coalescence in polymerblends: the effects of concentration andcompatibilization. Macromolecules28:2647–2657

Takahashi Y, Kitade S, Kurashima N,Noda I (1994) Viscoelastic properties ofimmiscible polymer blends under steadyand transient shear flows. Polym J26:1206–1212

Tretheway DC, Leal LG (2001) Deforma-tion and relaxation of Newtonian dropsin planar extensional flows of a Bogerfluid. J Non-Newtonian Fluid Mech99:81–108

Tucker CL, Moldenaers P (2002) Micro-structural evolution in polymer blends.Annu Rev Fluid Mech 34:177–210

Veenstra H, Norder B, Van Dam J,Posthuma de Boer A (1999) Stability ofco-continuous polystyrene/poly(ether-ester) blends in shear flow. Polymer40:5223–5226

Vinckier I, Moldenaers P, Mewis J (1997a)Transient rheological response andmorphology evolution of immisciblepolymer blends. J Rheol 41:705–718

Vinckier I, Mewis J, Moldenaers P (1997b)Stress relaxation as a microstructuralprobe for immiscible polymer blends.Rheol Acta 36:513–523

Vinckier I, Minale M, Mewis J MoldenaersP (1999) Rheology of semi-dilute emul-sions: viscoelastic effects caused by theinterfacial tension. Colloids Surf150:217–228

Wetzel ED, Tucker CL (2001) Dropletdeformation in dispersions with unequalviscosities and zero interfacial tension.J Fluid Mech 426:199–228

Willemse RC (1999) Co-continuous mor-phologies in polymer blends: stability.Polymer 40:2175–2178

Yamane H, Takahashi M, Hayashi R,Okamoto K, Kashihira H, Masuda T(1998) Observation of deformation andrecovery of poly(isobutylene) droplet ina poly(isobutylene)/poly(dimethylsiloxane) blend after application of stepshear strain. J Rheol 42:567–580

Ziegler V, Wolf BA (1999) Viscosity andmorphology of the two-phase systemPDMS/P(DMS-ran-MPS). J Rheol43:1033–1045

601