Rheol Acta (2010) 49:105–118DOI 10.1007/s00397-009-0395-3

ORIGINAL CONTRIBUTION

Measurements and model predictions of transientelongational rheology of polymeric nanocomposites

Mahmoud Rajabian · Ghassem Naderi ·Charles Dubois · Pierre G. Lafleur

Received: 4 December 2008 / Accepted: 13 October 2009 / Published online: 13 November 2009© Springer-Verlag 2009

Abstract Transient elongational rheology of two com-mercial-grade polypropylene (PP) and the organoclaythermoplastic nanocomposites is investigated. A specif-ically designed fixture consisting of two drums (SERUniversal Testing Platform) mounted on a TA Instru-ments ARES rotational rheometer was used to mea-sure the transient uniaxial extensional viscosity of bothpolypropylene and nanoclay/PP melts. The Henckystrain rate was varied from 0.001 to 2 s−1, and the tem-perature was fixed at 180◦C. The measurements showthat the steady-state elongational viscosity was reachedat the measured Hencky strains for the polymer and forthe nanocomposites. The addition of nanoclay particlesto the polymer melt was found to increase the elonga-tion viscosity principally at low strain rates. For exam-ple, at a deformation rate of 0.3 s−1, the steady-stateelongation viscosity for polypropylene was 1.4 × 104

Pa s which was raised to 2.8 × 104 and 4.5 × 104 Pa safter addition of 0.5 and 1.5 vol.% nanoclay, respec-tively. A mesoscopic rheological model originally de-veloped to predict the motion of ellipsoid particlesin viscoelastic media was modified based on the re-cent developments by Eslami and Grmela (Rheol Acta47:399–415, 2008) to take into account the polymer

M. Rajabian · C. Dubois (B) · P. G. LafleurCREPEC, Department of Chemical Engineering,Ecole Polytechnique, CP 6079, Succ. Centre-Ville,Montreal, H3C-3A7, Quebec, Canadae-mail: charles.dubois@polymtl.ca

M. Rajabian · G. NaderiIran Polymer and Petrochemical Research Institute,Tehran, Iran

chain reptation. We show that the orientation states ofthe particles and the rheological behavior of the layeredparticles/thermoplastic hybrids can be quantitativelyexplained by the proposed model.

Keywords Modelling · Elongational viscosity ·Nanocomposites

Introduction

Rheology of hybrid materials containing layered nano-sized particles in viscous and viscoelastic fluids hasbeen extensively studied in recent years (Eslami et al.2009; Rajabian et al. 2008a; Letwimolnum et al. 2007;Lin-Gibson et al. 2004; Ren et al. 2003; Galgali et al.2001; Solomon et al. 2001; Giannelis et al. 1999). Al-though most practical processes use shear for homog-enization, melt blending (for thermoplastics resins orsolution mixing for the thermosets), and other prepara-tion stages, shear-free flow properties are yet importantin certain applications where pure elongation or biaxialstretching is applied to the nanomaterials. Examplesinclude elongational mixers, fiber spinning, and filmstretching where strong shear-free fields are applied tothe material and one often encounters with the situa-tions that require knowledge of elongational viscositiesof the nanocomposite materials.

The experimental studies on shear-induced orienta-tion of the nanoplatelets suspended in a polymer showthat the alignment of the clay particles is sufficientlylow below a critical shear rate (Rajabian et al. 2008a;Lin-Gibson et al. 2004). At low shear rates, the polymerchains have time to diffuse into the clay layers withno significant changes in their alignment. Experimental

106 Rheol Acta (2010) 49:105–118

studies on the effect of shear on orientation in polymer–clay nanocomposites have been conducted by a numberof groups, and in most cases the full three-dimensional(3D) orientation of clay has been reported (Lin-Gibsonet al. 2004). The two-dimensional clay platelets canalign in three possible orientations referred as perpen-dicular, transverse, and parallel orientations. In a per-pendicular orientation, the vector normal to the platesurface aligns parallel to the vorticity vector which isthe direction perpendicular to the flow and velocitygradient directions. The particles lie within the planeof shear flow and velocity gradient. In the traverseorientation, the particles lie in the plane of vorticity andvelocity gradient with the surface normal parallel to theflow while in a parallel orientation the platelets lie inthe plane of flow and vorticity, and the surface normalaligns parallel to the velocity gradient axis. Althoughthe general response for a particle to the shear flow isexpected to be the parallel orientation, there is no clearevidence supporting this idea. Indeed, the orientationalstate for the lamellae has a distribution correspond-ing to the shear magnitude, interactions among theparticles and between the particles, and the polymerfluid and preorientation history. Instead, the particlesexhibit a mixed microstructural orientation within theshear flow.

Due to the incorporation of the solid nanoparticlephase within the macromolecular chains, the material isexpected to exhibit strong viscoelastic behavior. In thispaper, we present a model to investigate flow alignmentof small platelets in a viscoelastic fluid commonly usedin polymer nanocomposites. The quantities of interestare orientation of the particles with the applied flowfield, extension of polymer chains, and the measuredmacroscopic properties which, in case of uniaxial elon-gation flows, are the elongation viscosities. Here, forthe first time, we report the microstructural develop-ment exhibited by the thin-layered platelets dispersedin complex fluids under shear-free flows, and we sug-gest how the initial random alignment of the particleswithin the applied flow field are modified. The criticalrole of the particle alignment is assessed via a 3 × 3symmetric tensor which can be viewed as the mostlikely orientation of the group of disk-like nanoparti-cles, and the magnitude of its components suggests thearrangement within the suspended fluid which can beboth viscous (Newtonian case) and viscoelastic. Thequantity of fundamental interest for the polymer isagain a conformation tensor which depending on thetype and extent of flow conditions varies from zeroto a positive value corresponding to the maximumextension attained on the polymer chains. The particleorientation and polymer conformation tensors are used

to express the total free-energy function of the suspen-sion. In elongational flows, the original FENE-P modelhas limitations due to the absence of chain reptation.The macroscopic stress and hence elongation viscosityis a consequence of macromolecular chain orientationvector only, resulting in overestimations of the steady-state elongational viscosity at larger strain rates. Thepredictions can be modified by the addition of a new pa-rameter to the original FENE-P dumbbells which char-acterizes the backbone of the polymer chains (Eslamiand Grmela 2008). The combination of the new termsto the original FENE-P (rigid) dumbbells allows us tomodel chain reptation or stretching. Here, we presentan analysis of the motion of the highly coupled solidand symmetric ellipsoid particles in a viscoelastic fluidextended with the assumption of chain stretching andthen present a constitutive equation to describe time-dependent rheological response for the suspensions.Rheological predictions are then compared with the setof experimental observations measured with the SERfixture on the polypropylene and PP nanocomposites.

Following the theoretical section and description ofour model, we describe the experimental steps of thepresent study which include preparation and charac-terization of the test nanocomposite samples. We thenuse an SER test fixture mounted on a controlled strainrheometer to measure the extensional stress growthfor the PP and PP nanocomposite samples. We inves-tigate the startup elongational rheology over a range ofHencky strains measurable by this fixture.

Theory

The state variables used in this work are two symmetric3 × 3 tensors designated hereafter as C and A cor-responding to polymer conformation and particle ori-entation states, respectively. They simply describe theextension of the polymer chains and the orientation ofthin disks (layered organoclay particles), respectively.Eslami and Grmela (2008) have recently reformulatedthe well-known FENE-P model by introducing chainreptation into the original expression. The main ad-vantage of their work is that it is developed on themesoscopic description level and therefore can be usedin combination with other conformation tensors. Theso-called local FENE-P dumbbells model is derived byan additional dissipation term to model the chain ex-tension or reptation of the macromolecules. Thus, theintermolecular interactions among neighboring FENE-P dumbbells were expressed using an additional term inthe free energy of the macromolecule and, hence, in thegoverning equations. In the conformation tensor level,

Rheol Acta (2010) 49:105–118 107

the state variable for the polymer chains is described bya 3 × 3 tensor C(t, s) which is the second moment of thedistribution function

Cij =∫

ψ (R, s) Ri R j dR (1)

where R is the end-to-end vector of the polymer andΨ (R, s) denotes the configuration space distributionfunction of the macromolecule chains. t indicates timewhile s (−1 ≤ s ≤ 1) is an independent variable repre-senting the chain stretching. Assuming that clay layersare rigid and axisymmetric oblate particles uniformlydispersed in the viscoelastic matrix, a unit vector Palong the axis of the disk can be used to describeits orientation. A group of ellipsoid particles can berepresented by a probability distribution function ϕ(p),defined as the probability for a particle being alignedwithin an angular range dp of the direction p is equal to

ϕ(p)dp. The components of the orientation tensor canbe written as:

Aij =∫

pi p jϕ (p) dp (2)

Since the particles are assumed to be rigid and inex-tensible, trace of A is constant:

Aii = 1 (3)

Assuming the particles are evenly dispersed in thepolymer, the components of the orientation tensor arenot varied with the spatial position and only changewith time.

The overall free energy of the suspensions of el-lipsoidal particles in the viscoelastic fluids can be ex-pressed by tensors C and A, provided the terms influ-encing the rheology as follows (Rajabian et al. 2005,2008a, b):

�(u, A, C) =∫

u ju j

2ρdr − nm HR2

0 ln (1 − trC) + Ktr (Cs · Cs) − 1

2kBT

×[

nm ln (det C)

+np ln det A − 2Bppnp((tr A)2 − tr AA

) − Bpmnpnmtr (C · A)

](4)

where U is the momentum field, T temperature, andKB the Boltzmann constant. nm and np denote macro-molecules and particles number density, respectively.The first term in the above equation represents the totalkinetic energy of the suspension where p is the massdensity; the second term is associated with the FENE-P-type intramolecular interactions (Bird et al. 1980;Bird and DeAguiar 1983), and the terms inside thebracket are related to the entropy due to contributionsfrom the macromolecular chains, ellipsoidal particles,and the macromolecule–particles and particle–particleinteractions. In this expression, the third term in theright-hand side represents the energy associated withthe interactions of polymer chains, and K signifies theextent of such intermolecular interactions. Addition-ally, the interparticle interaction parameter denoted byBpp in Eq. 4 is a phenomenological parameter thatsignifies the extent of bonds and forces existing amongthe particles in the suspension (i.e., physical, chemical,etc.). The last term appearing in Eq. 4 models theparticle–macromolecule interactions with a parameterBpm representing the extent of such interactions. Bythe particle macromolecule interactions, we denoteboth chemical (i.e., covalent bonds) and physical bondsacross the interface of the particles and the polymericmatrix. It was shown in a recent investigation that theextent of interactions in polymer suspensions can havesignificant effects on viscoelastic behavior, and hence

the experimental data on rheology can be used to de-termine degree of interactions in polymer suspensions(Rajabian et al. 2008b). Before proceeding to derive thegoverning equations, we should mention here that theoverall free-energy function for the systems of suspen-sions of symmetric ellipsoidal particles homogeneouslydispersed in a polymer matrix was developed based on theGENERIC formulation on the conformation tensor space(Grmela and Ottinger 1997; Ottinger and Grmela 1997).

The equations governing the time evolution of theparticle orientation and polymer conformation tensorsare the standard equations derived in detail in a previ-ous paper as follows (Rajabian et al. 2005):

∂ Aij

∂t= − 1

2(�.A − A.�)ij

− 1

2λ

(−Ailklγ jk − A jlklγik + Ailkjγlk + A jlkiγlk)

− ∂�

∂�Aij

(5)

∂C∂t

=(

∂C∂t

)advective

+(

∂C∂t

)dissipative

= −1

2(� · C − C · �) + 1

2(γ · C − C · γ )

− ∂�

∂�C+

(∂�

∂ (�C)s

)s

(6)

108 Rheol Acta (2010) 49:105–118

where �ij and γij represent the vorticity and rate ofdeformation tensors, respectively, defined as:

�ij = ∂u j

∂xi− ∂ui

∂x j(7)

γij = ∂ui

∂x j+ ∂u j

∂xi(8)

It can be seen that Eq. 5 is a Jeffery-type equation(Jeffery 1922) initially developed to model the motionof ellipsoid particles in homogeneous flows that hasbeen extended by combining the dissipation terms andby allowing the fourth-order tensor Aijkl defined as:

Aijkl =∫

pi p j pk plψ (p) dp (9)

to be expressed by the second order orientation tensor.The procedure to estimate the fourth order as well ashigher-order orientation tensors from the second-orderorientation tensor known as closure approximation hasbeen the subject of various investigations in the pastwhich is out of the scope of this investigation (see forexample Advani and Tucker 1987, 1990; Folgar andTucker 1984). Among the closure approximation tech-niques found in the literature, we tested the quadratic,hybrid,and linear closures which all produced nonsense

oscillations and inaccurate results for the orientationtensor components at high shear rates. We have alsoinvestigated the natural and modified natural closure orinvariant-based optimal fitting which generated moreaccurate results but were very slow for the numericalcomputations. However, the orthotropic closure ap-proximation was found more attractive since it did nothave the drawbacks observed with the other closuretechniques and hence was selected for this work. Con-sequently, the modified form of orthotropic closure ap-proximation ORF originally developed by Cintra andTucker and enhanced by Werveyst was adopted for thisstudy (Cintra and Tucker 1995; Werveyst 1998). λ is amaterial constant that depends on the particle aspectratio:

λ = (l/d)2 − 1

(l/d)2 + 1

(10)

The last term in the right-hand side of Eq. 5 andthe last two terms in Eq. 6 are dissipation terms tobe defined when the dissipation potential is fixed.In this work, we use the following dissipation func-tion which combine the three driving forces, i.e.,�A, �C, and �Cs = ∂�C

∂s by means of the follow-ing relationship (Eslami and Grmela 2008; Rajabianet al. 2005):

� =(

�Aij −1

3tr�Aδij�cij

(�cij

)s

)⎛⎜⎝

p 0 00

m,11ijkl 0

0 0 m,22ijkl

⎞⎟⎠

⎛⎜⎝

�Akl − 1

3tr�Aδkl

�ckl(�ckl

)s

⎞⎟⎠ (11)

where p is the mobility parameter of the particles,and m,11 and m,22 are the macromolecular chains mo-bility tensors defined as functions of macromoleculesconformation tensor C and mobility parameters:

m,11 = m,110

[κ1C + κ1C · C

](12)

m,22 = m,220

[κ1C + κ1C · C

](13)

The mobility parameters m,110 and

m,220 are phe-

nomenological parameters which for a given polymerare to be determined by experimental observationsof the measurable quantities such as shear or normalstresses, viscosity, etc.

Note that in the expressions of the dissipation poten-tial, the free-energy derivatives �c and �A are given asfollows:

�cij = 1

2kBT

(nmb

1 − trCδij − nmC−1

ij + nmnp Bpm A)

−2KCssij (14)

�Aij = −1

2kBT

(np A−1

ij − 2np Bpp(δij − Aij)

−npnm BpmC)

(15)

Having defined all dissipation terms for the polymer,we can formulate them in the governing equation fortensor C:

(∂C∂t

)dissipative

= − ∂�

∂�C+

(∂�

∂ (�C)s

)s

= −4 m,110

[κ1 (C · �C + �C · C)

+2κ2C · �c · C]

+4 m,220

(1+

m,220

(trC−trCeq

)2)

× [κ1 (C · (�C)s + (�C)s · C)

+2κ2C · (�C)s · C]

s (16)

Rheol Acta (2010) 49:105–118 109

Therefore:

∂C∂t

= − 1

2(� · C − C · �) + 1

2(γ · C − C · γ )

− 4 m,110

[κ1 (C · �C + �C · C) + 2κ2C · �C · C

]

+ 4 m,220

(1 +

m,220

(trC − trCeq

)2)

× [κ1 (C · (�C)s + (�C)s · C)+2κ2C · (�C)s · C

]s

(17)

The governing equations for the particle orientationtensor after combining the terms arising from the dissi-pation function are:

dAij

dt= − 1

2

(ωik Akj − Aikωkj

)

+ 1

2λ

(γik Akj + Aikγkj − 2γkl Aijkl

)

− 1

3(γlmγml)

12 p

(AilφAlj + AljφAil

)

+ 2

9(γlmγml)

12 p AijφAkk (18)

To complete the model, we need to define the extrastress tensor which arises from the polymer chains de-noted as τm and from the particles τ p according to thefollowing expressions (Rajabian et al. 2008a, b; Eslamiand Grmela 2008; Grmela 1985):

τmij = −2

1∫

−1

Cik�Ckjds=−2

1∫

−1

(C · �C−C ·

(∂�

∂Cs

)s

)ds

= −nmkBT

1∫

−1

(b

1 − trCC − δ

)ds

+ 2K

1∫

−1

(C · Css + Css · C

)ds (19)

τp

ij = 1

2λ

(�Alk Alijk + �Alk Akijl − 2�Ami Amljl

+�Alk Aljik + �Ami Amljl − 2�Amj Amlil)

− γij

(γlmγml)12

p

(φAlm − 1

3φAkkδlm

)

× Amp

(φAlp − 1

3φAkkδlp

)(20)

The set of the partial differential equations for timeevolution Eqs. 17 and 18 and hence the extra stresstensors components (Eqs. 19 and 20) are solved hence-forth numerically for the selected flow conditions forboth independent variables 0 ≤ t and −1 ≤ s ≤ 1. Thenumerical code for computations of the conformationand orientation tensors were developed in Mathemat-ica 6, a special programming language operating inter-actively, which is capable of sophisticated numerical,symbolic (algebraic), and graphical computations. Assuggested by Eslami and Grmela (2008), due to theexpected symmetry of the solutions, the domain for sis reduced to 0 ≤ s ≤ 1. The initial conditions (t = 0)are the equilibrium solutions obtained after solving thefollowing equations:

∂�

∂Cij= 0 ; ∂�

∂ Aij= 0 (21)

Moreover, for the boundary conditions, the follow-ing equations are used in our calculations:

C (t, 1) = ∂2C∂s2

= 1

3 + bδ (22)

∂C∂s

(t, 0) = ∂3C∂s3

(t, 0) = 0 (23)

In the next section, we illustrate the model predic-tions computed for the simple shear and elongationalflows which are of potential interest in rheometry andprocessing of nanocomposite materials. However, themodel can be used for any types of flows by varyingthe flow conditions, although numerical solutions maybecome prohibitively expensive to reach for complexflows.

(a) The velocity gradient tensor for the simple shearflow is given by:

∂ui

∂x j= γ (t)

⎛⎝ 0 1 0

0 0 00 0 0

⎞⎠ (24)

and rate of deformation and vorticity tensors are:

γ = γ (t)

⎛⎝ 0 1 0

1 0 00 0 0

⎞⎠ ; ω = γ (t)

⎛⎝ 0 −1 0

1 0 00 0 0

⎞⎠ (25)

The material functions at the startup of shear floware written as follows:

η+ (t, γ ) = −τ12

γ; �+

1 (t, γ ) = − (τ11 − τ22)

γ 2(26)

110 Rheol Acta (2010) 49:105–118

Fig. 1 Maps of conformation tensor components for the polymer in a simple shear test, �v = 0.05; b = 10; Mw = 105; ρ = 1,000 kg/m3;K = 100; k1 = 1; k2 = 0; p,11 = 10−7;

m,110 = 2 × 10−8;

m,220 = 10−5; m,22 = 2 × 105; γ = 1 s−1

(b) In uniaxial elongation, the velocity gradient tensoris equal to:

∂ui

∂x j= ε(t)

⎛⎝ −1/2 0 0

0 − 1/2 00 0 1

⎞⎠ (27)

where ε(t) is its magnitude. The material func-tion which in this case is the startup elongationalviscosity (Bird et al. 1987a, b):

η+ (t, γ ) = −τ33 − τ11

ε(28)

Results and discussions

We illustrate the model predictions for tensors C and Ain the simple shear test in Figs. 1 and 2, respectively. Wenote that C11 and C12 increase to their stationary valueswhereas both C22 and C33 decrease until the steady-state plateau is attained. The variations of C33 is less

Fig. 2 Orientation tensor components for the nanoparticles in asimple shear test, �v = 0.05; b = 10; Mw = 105; ρ = 1,000 kg/m3;K = 100; k1 = 1; k2 = 0; p,11 = 10−7;

m,110 = 2 × 10−8;

m,220 = 10−5; m,22 = 2 × 105; γ = 1 s−1

Rheol Acta (2010) 49:105–118 111

significant than other components of the conformationtensor. It is interesting to note that chain reptation be-comes less important to the end point of the chain back-bone, i.e., s = 1, since the macromolecules moves morefreely at the end points and hence the tensor C is atthe equilibrium regardless of the time of flow (Eq. 22).

Figure 2 shows the components of the particleorientation tensor A for the oblate spheroids withL/d = 1/100 (d = 100 nm and L = 1 nm) in the simpleshear. The particles from a random 3D orientationwith (Aii = 1

3 ) are significantly aligned with the surfacenormal towards the velocity gradient axis X2. There isa minimum in A12 at t = 2 where the majority of theparticles are reoriented at 45◦ with respect to the shearplane. At t = 20, the steady state for the particles hasbeen reached.

Figures 3 and 4 plot diagonal components of tensorsC and A for the elongational flow, respectively. Here,in contrast to the shear flow, only three componentsexist, and the off-diagonal components are zero. Sincethe elongation is in the X3 direction, C33 quickly in-creases to its stationary value for the uniaxial stretchingflow while C11 and C22 due to the symmetry of flowfield decrease. Both A11 and A22 increase to theirstationary values of about 0.5, while A33 decreases tonearly zero. It shows that the particle axis from random3D orientation is aligned in the (X1, X2) plane whichis perpendicular to the elongation direction. It showsthat the applied elongation field can strongly createan orientation state for the thin disks; the axis of thedisks is randomly aligned in the (X1, X2) plane andperpendicular to the elongation direction.

Fig. 3 Maps of conformation tensor components for the polymer in a uniaxial elongation experiment, �v = 0.05; b = 10; Mw = 105;ρ = 1,000 kg/m3; K = 10,000; k1 = 1; k2 = 0; p,11 = 10−6;

m,110 = 10−8;

m,220 = 10−7; m,22 = 2 × 105; ε = 1 s−1

112 Rheol Acta (2010) 49:105–118

Fig. 4 Conformation tensor components for the nanoparticlesin a uniaxial elongation experiment, �v = 0.05; b = 10; Mw =105; ρ = 1,000 kg/m3; K = 10,000; k1 = 1; k2 = 0; p,11 = 10−6;

m,110 = 10−8;

m,220 = 10−7; m,22 = 2 × 105; ε = 1 s−1

Figure 5 shows the predicted rheological propertiesof the nanocomposite φv = 5% at the startup of shearflows. A pronounced overshoot in viscosity which coin-cides with the minimum in A12 is noticeable. The peakin viscosity of organoclay/PP nanocomposites has beenalready reported and was attributed to the alignment ofthe particles (Letwimolnum et al. 2007).

Figure 6a, b displays predictions of viscosities and thefirst normal stress difference coefficient at the startupof shear flows for different rates of deformation forthe polymer with the assumption of chain stretching.Figure 6a reveals a nearly Newtonian plateau at shearrates lower than 0.001 s−1. Albeit less pronounced inthis scale, the results shown in Fig. 6a exhibit a smallovershoot in viscosity. The time to attain steady stateincreases monotonically as the shear rate is decreased.

Fig. 5 Transient shear viscosity and first normal stress differencecoefficient results of the nanocomposite, �v = 0.05; b = 10;Mw = 105; ρ = 1,000 kg/m3; K = 100; k1 = 1; k2 = 0, Bpm =Bpp = 0; p,11 = 10−7;

m,110 = 2 × 10−8;

m,220 = 10−5; m,22 =

2 × 105; γ = 1 s−1

Fig. 6 Transient shear viscosity (a) and first normal stress dif-ference coefficient (b) results for the polymer, �v = 0.05; b = 10;Mw = 105; ρ = 1,000 kg/m3; K = 100; k1 = 1; k2 = 0; p,11 = 10−7;

m,110 = 2 × 10−8;

m,220 = 10−5; m,22 = 2 × 105; γ = 1 s−1

Figure 7a, b presents the transient viscoelastic proper-ties of the nanocomposites at the different shear rates.The addition of 5% nanoparticles to the polymer dra-matically modifies the properties especially at the lowershear rates. Both viscosity and first normal differencecoefficient are significantly increased over the entirerange of shear rates. The nanocomposite exhibits strongovershoots in the viscosity due to the presence of theparticles which align with the shear direction.

Figures 8 and 9 present the model results of elonga-tional viscosity for the polymer and the nanocomposite,respectively. The time-dependent increase of the elon-gational viscosity and the plateau values which are allpredicted by the model as will be seen in the last sectionare in qualitative agreement with the experimental data.

In Figs. 10 and 11, the effects of nanoparticle load-ings on transient shear and elongational flow propertiesof polymer are depicted. The calculations show that byincreasing the particle content in the polymer phase,the properties are considerably increased. Strong vis-cosity overshoots due to the rearrangement of the

Rheol Acta (2010) 49:105–118 113

Fig. 7 Transient shear viscosity (a) and first normal stress dif-ference coefficient (b) results for the nanocomposites, b = 10;Mw = 105; ρ = 1,000; K = 100; k1 = 1; k2 = 0, �v = 5%; p,11 = 10−7;

m,110 = 2 × 10−8;

m,220 = 10−5; m,22 = 2 × 105

particles with the applied shear field are observed forthe nanoclay polymer hybrids. The extent of the over-shoots is augmented with the particle content (Fig. 10a).It is seen that small concentrations of solid and non-deformable layered nanoparticles can significantly en-hance the shear-free viscosity of the pure polymer.

The effect of interactions parameter Bpp on shearviscosity and normal stress coefficient and elongationalviscosity is illustrated in Fig. 12a–c. This parameteris varied from 0 corresponding to the case of no in-teraction to −5 corresponding to more interactions.On the other hand, a positive value for Bpp acts bylowering the viscous dissipation. A physical interpre-tation of the interaction parameter can be expressedby bonds or forces existing between the componentsin the nanocomposites. A positive value signifies therepulsion between the particles which result in thereduction of viscosity and normal stresses. We note thatby decreasing Bpp both shear and elongational materialfunctions are increased significantly. This is due to thecouplings between the particles which influences the

Fig. 8 Elongation viscosity results for the polymer (FENE-Pdumbbells with the assumption of chain stretch); b = 50; Mw =105; ρ = 1,000; K = 500; k1 = 1; k2 = 0;

m,110 = 10−9;

m,220 =

10−7; m,22 = 2 × 105

viscous dissipations in the suspension. Similar effectsare observed on the predicted elongational viscosity re-sults. One can assume the interaction parameter as thephysical or chemical bonds among the particles as wellas that across the particle and polymer interfacial area.The existence of the bonds in suspensions yields highervalues for both viscosity and normal stresses (Bpp < 0).

In Fig. 13, we present the transient elongational vis-cosities of the nanocomposites as the particle–polymerinteraction parameter Bpm is varied. These interactionsare taken into account in the Helmholtz free-energyfunction by coupling the conformation and orientationtensors and are found useful to model the changes inthe interfacial bonds between the fibers and the poly-mer in fiber suspensions (Rajabian et al. 2005, 2008a, b).The parameter Bpm is varied from 0 to an appropriate

Fig. 9 Elongation viscosity results for the nanocomposites;b = 50; Mw = 105; ρ = 1,000; K = 500; k1 = 1; k2 = 0, �v = 5%; p,11 = 10−6;

m,110 = 10−9;

m,220 = 10−7; m,22 = 2 × 105

114 Rheol Acta (2010) 49:105–118

Fig. 10 Effects of the addition of nanoparticles on viscoelasticproperties of the polymer melt; b = 50; Mw = 105; ρ = 1,000;K = 500; k1 = 1; k2 = 0; p,11 = 10−7;

m,110 = 10−9;

m,220 =

10−7; m,22 = 2 × 105; γ = 0.01 s−1. a Transient viscosity; b firstnormal stress difference coefficient

positive value corresponding to less coupling. How-ever, the negative values significantly increase theelongational viscosity by coupling the particles andthe polymer matrix. The interparticle and particle–macromolecule interaction parameters can be viewedas the extent of bonds and forces among the compo-nents in the suspension which enhance both viscousdissipation and normal stresses.

Experimental

Materials

The Cloisite 15 A organoclay, a natural montmoril-lonite modified with a dimethyl dehydrogenated tallowquaternary ammonium was obtained from SouthernClay Products. Two commercial-grade polypropylenePro-fax 6523 and Pro-fax 6823 with the melt flow in-dexes (MFI) of 14 and 0.5 g/10 min, respectively, here-after called PP1 and PP2, were used for this investi-

Fig. 11 Effects of the addition of nanoparticles on transientelongation viscosity of the polymer melt, b = 50; Mw = 105;ρ = 1,000; K = 500; k1 = 1; k2 = 0, b = 50; Mw = 105; ρ =1,000; K = 500; k1 = 1; k2 = 0; p,11 = 10−6;

m,110 = 10−9;

m,220 = 10−7; m,22 = 2 × 105; ε = 0.01 s−1

gation. The maleic-anhydride-modified polypropyleneEpolene G3015 (acid number = 15, MA = 1%) wasobtained from Eastman Chemical. All materials weredried at 80◦C for 48 h prior to use in nanocompos-ite preparations. Two nanocomposites with nanoclayvolume factions of 0.5% and 1.5% were prepared at180◦C by a two-step melt mixing process in a HaakeRemix 600 internal mixer. In the first stage, the claypowder and PPMA pellets were carefully dry-mixed toensure an adequate dispersion of the particles at themacroscale level. The mixture was then melt-mixed at180◦C under nitrogen to obtain a master batch usingthe mixer at a rotor speed of 150 rpm for 10 min.For all master batches, the PPMA/clay weight ratiowas kept constant, i.e., 3:1. In the second step, themaster batch was then dry-blended with polypropyleneto give the desired composition and fed to the mixerby applying the aforementioned processing conditions.Figure 14 illustrates the TEM image of the nanocom-posite at φv = 1.5%. The picture proves a mixedexfoliated/intercalated dispersion state of the layeredsilicates in the polymer phase where most silicate layersare exfoliated with some remaining stacks of nanoclayplatelets.

Rheological measurements

Measurements of extensional viscosities were per-formed on an SER fixture model SER-HV-A01 (Xpan-sion Instruments LLC) specifically designed for use oncommercial torsional rheometers. The fixture consistsof two drums mounted on a system of gears and bear-ings connected to a drive shaft whose rotation results in

Rheol Acta (2010) 49:105–118 115

Fig. 12 Effects of the interaction parameter Bpp on transientshear and elongation flow properties of the nanocomposites, b =50; Mw = 105; ρ = 1,000; K = 500; k1 = 1; k2 = 0, b = 50; Mw =105; ρ = 1,000; K = 500; k1 = 1; k2 = 0; a shear viscosity; b firstnormal stress difference coefficient; c elongation viscosity

the desired Hencky strains on the two drums which ro-tate equally but in opposite directions. The fixture wasmounted on a controlled-strain ARES rheometer (Ad-vanced Rheometrics Enhanced System). The Henckystrain applied to the test sample can be expressed as:

εH = 2�RL0

(29)

Fig. 13 Effects of the particle–macromolecular chain interactionparameter Bpm on elongation viscosities of the nanocomposites,b = 50; Mw = 105; ρ = 1,000; K = 500; k1 = 1; k2 = 0; p,11 =10−6;

m,110 = 10−9;

m,220 = 10−7; m,22 = 2 × 105; ε = 0.01 s−1

where R is the radius of the windup drums; L0 is thefixed length of the specimen being stretched which isthe centerline distance between the drums, and � is aconstant drive shaft rotation rate. The torque recordedon the rheometer obviously displays the torque on thedrums due to the resistance of the specimen to elon-gation as a tangential force according to the followingrelationship:

τ (t) = 2RF (t) (30)

Therefore, the stress growth elongational viscosity ofthe stretched sample can be calculated by the followingexpression:

η+E(t) = F(t)

εH As(t)(31)

where As is the cross-sectional area of the film sample.Since the instantaneous cross area of a stretched moltenspecimen decreases exponentially with time and the

Fig. 14 TEM image of organoclay/PP nanocomposite, φv =0.5%

116 Rheol Acta (2010) 49:105–118

polymer samples exhibit a decrease in density uponmelting by volumetric expansion, the following expres-sion is used to calculate the cross section area at a giventime:

As (t) = A0

(ρs

ρm

) 23

Exp (−εHt) (32)

Where A0 is the initial cross-sectional area of thespecimen in the solid state, and ρs and ρm are thedensities in the solid and molten state, respectively.Further details on SER extensional rheometers can befound in references (Sentmanat et al. 2005).

Sample specimens of PP and PP nanocomposites forextensional rheological measurements were preparedby compression molding at 180◦C and a fixed time of10 min. The film sample after compression molding wascut to a rectangular section of approximately 17 by12.7 mm with a thickness of about 0.3 mm. All testswere performed at a constant temperature of 180◦C.The Hencky strains range selected for this work wasbetween 0.001 and 2 s−1.

Comparison with the model predictions

The startup shear viscosity results for the polypropy-lene melt and for the organoclays/polypropylene sus-pensions at three volume fractions are illustrated inFig. 15. The addition of a small fraction of rigid-layered silicate particles to the polypropylene signifi-cantly increases the shear viscosity. The experimentalresults reveal sensitivity of the rheological response to

Fig. 15 Measured startup shear viscosities for the PP/organoclaynanocomposites. The model parameters are: b = 30; Mw = 105;ρ = 760 kg/m3; K = 6,000; k1 = 1; k2 = 0; Bpp = −1; 1 = 10−9 m,d = 100 × 10−9 m; p,11 = 10−5; m,11

0 = 3 × 10−8; m,220 = 10−5;

m,22 = 106

Fig. 16 Experimental data versus model predictions for theelongation viscosities of the polypropylene melt PP1; MFI = 14.The model parameters are: b = 200; Mw = 105; ρ = 760 kg/m3;K = 5 × 104; k1 = 1; k2 = 0;

m,110 = 3 × 10−8;

m,220 = 10−5;

m,22 = 106

the presence of particles. The nanocomposites exhibitpronounced overshoots in transient viscosity due tostructure breakup and particle alignment. The peakin transient viscosity is related to the orientation oflayered particles and not to the viscoelasticity of thepolypropylene matrix. The model predictions underes-timate the overshoots even though the stationary valuesare well predicted by the model.

The uniaxial elongation flow behavior of the highand low MFI polypropylene melts are shown in Figs. 16and 17, respectively. The model predictions are alsopresented by solid lines for strain rates tested. The

Fig. 17 Experimental data versus model predictions for theelongation viscosities of the polypropylene melt PP2; MFI = 0.5.The model parameters are: b = 200; Mw = 105; ρ = 760 kg/m3;K = 5 × 104; k1 = 1; k2 = 0;

m,110 = 1.35 × 10−9;

m,220 = 10−7;

m,22 = 106

Rheol Acta (2010) 49:105–118 117

results of the elongational viscosities for the polypropy-lene show no strain-hardening behavior. The figuressuggest that good agreement is observed between theexperimental data and the model calculations particu-larly at the low rates of deformations. The stationaryvalues of elongational viscosities for the PP1 (MFI =14) vary from 8,000 to 16,000 Pa s for strain rates of 2and 0.03 s−1, respectively. η+ is observed to decreasewith increasing ε without ever going through a max-imum. Figure 16 displays the elongational viscositiesof the polypropylene with the low MFI (MFI = 0.5).The elongational viscosities are much higher than thoseof polypropylene grade with the higher MFI shownin Fig. 15. Since the measured torque was larger forthe PP2, the data are presented for a broader rangeof strain rates. The time to attain the steady state inthe elongational test is much longer than PP1, about1,000 s. The model predictions are well consistent withthe elongation viscosity data for all Hencky strains.

Figures 18 and 19 present the elongational viscositiesof the organoclay/PP1 nanocomposites at 0.5% and1.5% volume contents, respectively. The model calcu-lations are also shown by the solid lines after extensivecomputations to best fit the experimental data. It can beseen that the addition of 0.5% and 1.5% clay particlesto the polymer results in an increase of the elongationalviscosity of the pure polymer melt. The hybrid materi-als do not exhibit any strain-hardening behavior whichis related to the polymer matrix. As observed in Figs. 15and 16, the highly linear polypropylene does not illus-trate an abrupt upturn in elongational viscosity curves.In spite of excellent agreement observed between themeasured and predicted elongation viscosities at low

Fig. 18 Experimental data versus model predictions for thetransient elongation viscosities of the nanocomposites PPNC 1(PP1/organoclay φv = 0.5%). In addition to those for the polymermatrix, the model parameters for the nanoparticles are: 11 =10−5, 1 = 10−9 m, d = 100 × 10−9 m

Fig. 19 Experimental data versus model predictions for thetransient elongation viscosities of the nanocomposites PPNC 2(PP1/organoclay φv = 1.5%). In addition to those for the polymermatrix, the model parameters for the nanoparticles are: 11 =10−5, 1 = 10−9 m, d = 100 × 10−9 m

strain rates, both polymer and the hybrid materialsreveals disagreements at large strain rates. This can bein part related to the inaccuracies in our measurementespecially at high strain rates.

Figure 20 presents the measured elongational viscos-ity of the low-MFI polypropylene/organoclay hybridstogether with the model predictions. The figure revealsthat the predictions agree with the experimental data atlow strain rates. However, as the strain rate is increased,they deviate further. Similar trend was observed withthis polymer, suggesting the limitations of the modelparticularly at large strain rates where chain stretchingbecomes more significant.

Fig. 20 Experimental data versus model predictions for thetransient elongation viscosities of the nanocomposites PPNC 3(PP3/organoclay φv = 0.5%). In addition to those for the polymermatrix, the model parameters for the nanoparticles are: 11 =10−5, 1 = 10−9 m, d = 100 × 10−9 m

118 Rheol Acta (2010) 49:105–118

Conclusions

The elongational flow properties of two commercial-grade polypropylene melts and PP/nanoclay hybridsare measured experimentally using the SER geometry.The elongational viscosities of pure polymer melts canbe fairly explained by assuming chain stretching with anadditional reptation term to the FENE-P dumbbells asintroduced recently by Eslami and Grmela (2008). Thisidea then can be introduced in combination with thesolid ellipsoid particles to model the flow properties oflayered nanoparticles/polymer hybrids at homogenousflow fields. The bulk of experimental data as well as the-oretical analysis on hybrids of solid layered nanopar-ticles in polymer matrices found in the literature arein shear flows and thus it was found useful to under-take a study on shear-free flows of this unique classof materials currently in great expansion in materialsscience and processing. We explore the microscopicarrangements of the rigid nanoparticles in the polymerand the macroscopic viscosities of nanocomposites as afunction of strain rates, particle concentrations, and theapplied flow fields.

References

Advani SG, Tucker CL (1987) The use of tensors to describe andpredict fiber orientation in short fiber composites. J Rheol31:751

Advani SG, Tucker CL (1990) Closure approximations for three-dimensional structure tensors. J Rheol 34:367

Bird RB, DeAguiar JR (1983) An encapsulated dumbell modelfor concentrated polymer solutions and melts I. Theoreticaldevelopment and constitutive equation. J Non-Newton FluidMech 13:149

Bird RB, Dotson PJ, Johnson NL (1980) Polymer solution rhe-ology based on a freely extensible bead-spring chain model.J Non-Newton Fluid Mech 7:213

Bird RB, Armstrong RC, Hassager O (1987a) Dynamics of poly-meric liquids, vol 1, 2nd edn. John Wiley, New York

Bird RB, Hassager O, Armstrong RC, Curtis CL (1987b) Dynam-ics of polymeric liquids, vol 2, 2nd edn. Wiley, New York

Cintra JS, Tucker CL (1995) Orthotropic closure approximationsfor flow-induced fiber orientation. J Rheol 39:1095

Eslami H, Grmela M (2008) Mesoscopic formulation of repta-tion. Rheol Acta 47:399–415

Eslami H, Grmela M, Bousmina M (2009) A mesoscopic tubemodel of polymer/layered silicate nanocomposites. RheolActa 48:317–331

Folgar FP, Tucker CL (1984) Orientation behaviour of fibers inconcentrated suspensions. J Reinf Plast Compos 3:98

Galgali G, Ramesh C, Lele A (2001) A rheological study on thekinetics of hybrid formation in polypropylene nanocompos-ites. Macromolecules 34:852

Giannelis EP, Krishnamoorti R, Manias E (1999) Polymer–silicate nanocomposites: model systems for confined poly-mers and polymer brushes. Adv Polym Sci 138:10

Grmela M (1985) Stress tensor in generalized hydrodynamics.Phys Lett A 111:41

Grmela M, Ottinger HC (1997) Dynamics and thermodynamicsof complex fluids I. Development of a GENERIC formula-tion. Phys Rev E 55:6620

Jeffery GB (1922) The motion of ellipsoidal particles immersedin a viscous fluid. Proc R Soc Lond A, 102:161

Letwimolnum W, Vergnes B, Ausias G, Carreau PJ (2007) Stressovershoots of organoclay nanocomposites in transient shearflow. J Non-Newton Fluid Mech 141:167

Lin-Gibson S, Kim H, Schmidt G, Han CC, Hobbie EK (2004)Shear-induced structure in polymer–clay nanocomposite so-lutions. J Colloid Interface Sci 274:515

Ottinger HC, Grmela M (1997) Dynamics and thermodynamicsof complex fluids II. Illustrations of a general formalism.Phys Rev E 55:6633

Rajabian M, Dubois C, Grmela M (2005) Suspensions of semi-flexible fibers in polymeric fluids: rheology and thermody-namics. Rheol Acta 44:521

Rajabian M, Naderi G, Beheshty MH, Lafleur PG, Dubois C,Carreau PJ (2008a) Experimental study and modelingof flow behavior and orientation kinetics of layered sil-icate/polypropylene nanocomposites in start-up of shearflows. Int Polym Process 13:110

Rajabian M, Dubois C, Grmela M, Carreau PJ (2008b) Effects ofpolymer–fiber interactions on rheology and flow behavior ofsuspensions of semiflexible fibers in polymeric liquids. RheolActa 47:701–71

Ren J, Casanueva BF, Mitchell CA, Krishnamoorti R (2003)Disorientation kinetics of aligned polymer layered silicatenanocomposites. Macromolecules 36:4188

Sentmanat M, Wang BN, McKinley G (2005) Measuringthe transient extensional rheology of polyethylene meltsusing the SER universal testing platform. J Rheol 49:585

Solomon MJ, Almusallam AS, Seefeldt KF, SomwangthanarojA, Varadan P (2001) Rheology of polypropylene/clay hybridmaterials. Macromolecules 34:1864

Werveyst B (1998) Numerical predictions of flow-induced fiberorientation in three dimensional geometries. PhD The-sis, Department of Mechanical and Industrial Engineering,University of Illinois at Urbana-Champaign