Rheol Acta (2008) 47:643–665DOI 10.1007/s00397-008-0273-4

ORIGINAL CONTRIBUTION

Suspension-based rheological modelingof crystallizing polymer melts

Rudi J. A. Steenbakkers · Gerrit W. M. Peters

Received: 10 July 2007 / Accepted: 20 February 2008 / Published online: 29 April 2008© The Author(s) 2008

Abstract The applicability of suspension models topolymer crystallization is discussed. Although directnumerical simulations of flowing particle-filled meltsare useful for gaining understanding about the rheo-logical phenomena involved, they are computationallyexpensive. A more coarse-grained suspension model,which can relate the parameters in a constitutive equa-tion for the two-phase material to morphological fea-tures, such as the volume fractions of differently shapedcrystallites and the rheological properties of bothphases, will be more practical in numerical polymerprocessing simulations. General issues, concerning themodeling of linear and nonlinear viscoelastic phenom-ena induced by rigid and deformable particles, arediscussed. A phenomenological extension of linearviscoelastic suspension models into the nonlinearregime is proposed. A number of linear viscoelasticmodels for deformable particles are discussed, focusingon their possibilities in the context of polymer crys-tallization. The predictions of the most suitable modelare compared to direct numerical simulation results andexperimental data.

Keywords Polymer crystallization ·Structure development · Rheology ·Suspension models

Paper presented at the 4th Annual European RheologyConference (AERC), April 12–14, 2007, Naples, Italy.

R. J. A. Steenbakkers · G. W. M. Peters (B)Department of Mechanical Engineering,Eindhoven University of Technology (TU/e),5600MB Eindhoven, The Netherlandse-mail: g.w.m.peters@tue.nl

Introduction

The significant effects of flow on the crystallizationkinetics of polymers, specifically the increase of thenucleation density and the transition from sphericalto anisotropic growth, have incited a great deal ofscientific effort, both experimental and theoretical. Ex-perimental studies usually involve subjecting an under-cooled melt to a short, well-defined flow in the earlystage of crystallization, where nearly all of the materialis still in the amorphous phase, and monitoring thesubsequent structure development by any one of a va-riety of measurement techniques, including rheometry,microscopy, and scattering and diffraction methods, ora combination of these methods. Because our under-standing of the phenomena occurring in the early stage,which determine to a great extent the final semicrystal-line morphology, is still incomplete, it is not surprisingthat far less attention has been devoted to the influ-ence of structure development on the rheology of acrystallizing melt. However, once the mechanisms offlow-induced crystallization are known, this will be thefirst step in going from short-term flow to continuousflow experiments, where the local process of phasetransformation is affected by the development of semi-crystalline structures on an orders of magnitude largerlength scale and vice versa. These experiments will beuseful as validation for polymer processing simulations.

During the last decade, a number of concepts havebeen proposed that deal with the rheology of crystal-lizing polymer melts. Winter and coworkers (Horst andWinter 2000a, b; Pogodina and Winter 1998; Pogodinaet al. 1999a, b, 2001; Winter and Mours 1997) observedan apparent similarity to the rheology of chemicalgels, in which polymer molecules are connected by

644 Rheol Acta (2008) 47:643–665

permanent crosslinks into a sample spanning network.They considered crystallizing melts as physical gels, inwhich crystallites were connected by amorphous “tiechains.”

Janeschitz-Kriegl et al. (2003) estimated the fractionof chains involved in nuclei in their experiments andfound it to be so small that, during the major part of thecrystallization process, no interaction among the nucleior the resulting spherulites was to be expected. To ex-plain the observed nonlinear increase of the nucleationdensity as a function of the mechanical work supplied tothe melt, they introduced the concept of flow-inducedactivation of dormant nuclei (Janeschitz-Kriegl 2003;Janeschitz-Kriegl and Ratajski 2005).

Another explanation of the strong self-enhancingeffect of nucleation was proposed by Zuidema (2000)and Zuidema et al. (2001). They assumed that nucleilocally act as physical crosslinks, increasing the proba-bility that chain segments remain in an ordered statelong enough to serve as new nuclei. In other words,gel-like behavior is not caused by the formation ofa percolating network of semicrystalline domains butby effective branching of the amorphous phase. Low-frequency rheological measurements, supporting thisidea, were recently published by Coppola et al. (2006).

A few attempts have been made to capture thekinetics of flow-induced crystallization in a continuumdescription, embedded in a formal theoretical frame-work of nonequilibrium thermodynamics. For example,the Poisson bracket formalism (Beris and Edwards1994) was used by Doufas et al. (1999). In their model,which was applied to flow-induced crystallization dur-ing fiber spinning (Doufas et al. 2000a, b; Doufas andMcHugh 2001a, b) and film blowing (Doufas andMcHugh 2001c), details of the microstructure, e.g., sizeand shape of the crystallites, are not taken into account.The crystalline phase is simply modeled as a collectionof bead–rod chains. A Giesekus model is used for theamorphous phase, with the relaxation time dependingon the degree of crystallinity χ as

λam = λam,0 (1 − χ)2 (1)

to account for the loss of chain segments due to crystal-lization.

Hütter (2001) developed a flow-induced spheruliticcrystallization model based on the “general equationfor the nonequilibrium reversible–irreversible cou-pling” or generic (Grmela and Öttinger 1997; Öttingerand Grmela 1997). The microstructure enters his modelthrough the evolution of the interfacial area, obtainedfrom the Schneider rate equations (Schneider et al.1988). This gives rise to a pressure term in the mo-

mentum balance, related to the surface tension, as wellas to an interfacial heat flux in the energy balance.However, the extra stress tensor is written as the sumof the viscous stress contributions from the matrix andthe spherulites,

τ = τ am + τ sc , (2)as if the material were a homogenous mixture. Here,“am” stands for the amorphous matrix and “sc” forthe partially crystalline, partially amorphous materialinside the spherulites, which we call the semicrystallinephase. When both phases are incompressible, the par-tial stresses are given by

τ am = 2 (1 − φ) ηam D (3)and

τ sc = 2φηsc D , (4)where φ is the volume fraction of spherulites, or degreeof space filling, and D is the deformation rate tensor.Equations 2, 3, and 4 yield the effective viscosity

η = (1 − φ) ηam + φηsc . (5)Thus, no connection is made between rheologicalproperties and microstructural features. Van Meerveld(2005) and Van Meerveld et al. (2008) extendedHütter’s model with a description of the viscoelasticbehavior of the melt and used it to simulate fiber spin-ning. In contrast to Hütter et al. (2005), who developeda single set of rate equations, allowing for changesin crystallite shapes and growth directions, they usedtwo sets of rate equations to describe the evolutionof spherulites and oriented crystallites. Although mor-phology development is incorporated in these models,at the continuum level, the stress is determined bythe additive “rule of mixtures,” Eq. 2. The questionremains whether this is a realistic choice for describingthe rheology of crystallizing polymer melts.

The morphology that develops as nuclei grow intocrystallites with distinct shapes agrees with the basicconcept of a suspension: isolated particles (the crys-tallites) are scattered throughout a continuous matrix(the amorphous phase). It is well known that the ruleof mixtures fails to describe the volume fraction depen-dence of the rheological properties of suspensions. Thesame may hence be expected for crystallizing melts.Boutahar et al. (1996, 1998), Tanner (2002, 2003), andVan Ruth et al. (2006) therefore used ideas from sus-pension rheology to describe the evolution of linear vis-coelastic properties during crystallization, as a functionof the degree of space filling and the properties of theindividual phases.

Rheol Acta (2008) 47:643–665 645

Crystallizing polymer melts differ from ordinary sus-pensions in a number of ways. The crystallites grow,they can have different shapes depending on the flowhistory, and their properties evolve in time. The lattercan be shown by combined optical microscopy andrheological measurements during crystallization. Thedynamic modulus continues to increase after the com-pletion of space filling (Van Ruth et al. 2006). This isthe result of perfection of the semicrystalline phase,also referred to as secondary crystallization. In thispaper, crystallites are therefore treated as particleswhose properties depend on their internal degree ofcrystallinity,

χ1 = χφ

, (6)

thus providing the possibility to incorporate perfectionin the model. The surrounding amorphous phase actsas a matrix, whose properties change as well. As yetunpublished results (J.F. Vega, personal communica-tion) show strongly increased storage and loss moduli,measured at a constant frequency, directly after shortsteady shear flows. At the same time scale, no signifi-cant degree of space filling was observed by means ofoptical microscopy (D.G. Hristova, personal commu-nication). Therefore, these results cannot be explainedby particle-like effects of the crystallites on the overallrheology. Coppola et al. (2006) drew the same con-clusion from a comparison of dynamic measurementson partially crystallized melts and on an amorphousmelt filled with solid spheres. However, for the partiallycrystallized samples, the degrees of space filling wereprobably underestimated (see the “Frequency sweepsat different volume fractions” section).

To explain these observations, the amorphous matrixwill be described as a crosslinking melt, with flow-induced nucleation precursors acting as physical cross-links (Zuidema 2000; Zuidema et al. 2001). In the laterstages of crystallization, the flow is severely disturbedby the presence of crystallites. Both phenomena have anonlinear effect on the kinetics of flow-induced crystal-lization; furthermore, they are mutually coupled. Theinfluence of flow on the early-stage kinetics, related tostructure development within the amorphous matrix,will be discussed elsewhere. Two-dimensional (2D)simulations of flow-induced crystallization in a particle-filled polymer melt have already been performed with-out taking the physical crosslinking effect into account(Hwang et al. 2006). Here, we focus on the laterstages of crystallization, which are dominated by spacefilling and perfection of the internal structure of thecrystallites.

A suspension model for crystallization under realprocessing conditions has to meet at least the followingrequirements:

1. The model has to be applicable in the entire rangeof volume fractions, i.e., from the purely amor-phous state (φ = 0) to complete filling of the mate-rial by the crystallites (φ = 1). This rules out dilutesuspension theories, although an interpolation be-tween analytical results for φ → 0 and φ → 1 hasbeen applied with some success (Tanner 2003).

2. The possibility to incorporate differently shapedparticles is essential for describing different semi-crystalline morphologies. Here, spherulites and ori-ented crystallites are represented by spheres andcylinders, respectively, and we need a suspensionmodel that can deal with both.

3. To describe the evolution of linear viscoelasticproperties, as measured during crystallization, themodel must provide a relationship between theseproperties and morphological features.

4. Quantitative description of most manufacturingprocesses requires that the effect of crystallizationon the nonlinear viscoelastic behavior is capturedas well.

In the “Linear viscoelastic suspension rheology” sec-tion, we briefly review how the effective dynamic me-chanical properties of a linear viscoelastic suspensioncan be obtained from an elastic suspension model bymeans of the correspondence principle (Christensen1969; Hashin 1965, 1970a, b). The consequences ofmodeling crystallites as either rigid or deformable parti-cles are discussed. No specific suspension model is used;the discussion is of a general nature. A complemen-tary phenomenological modeling approach to nonlin-ear viscoelastic suspension rheology is introduced in the“Nonlinear viscoelastic suspension rheology” section.Its ability to qualitatively reproduce results from ex-periments (Mall-Gleissle et al. 2002; Ohl and Gleissle1993) and numerical simulations (Hwang et al. 2004a)is investigated. The properties of a specific linear visco-elastic suspension model are discussed in the “Historyand relation to other models” and “Influence of phaseproperties” sections. In the “Comparison to numericaland experimental data” section, its predictions arecompared to numerical (Hwang et al. 2004a) and ex-perimental (Mall-Gleissle et al. 2002) results for rigidparticle suspensions. In the “Application to crystalliza-tion experiments” section, they are compared to experi-mental data on quiescent and short-term shear-inducedcrystallization of different polymer melts (J.F. Vegaand D.G. Hristova, private communications; Boutahar

646 Rheol Acta (2008) 47:643–665

et al. 1996, 1998; Coppola et al. 2006). The conclusionsof this paper are summarized in the “Conclusions”section.

Modeling

Various constitutive models are available to describethe nonlinear viscoelastic behavior of the matrix ofthe suspension, i.e., the amorphous phase of the crys-tallizing melt. Differential models are most suited fornumerical simulations of complex flows. Some of themost advanced are the Rolie-Poly model (Likhtmanand Graham 2003) for linear melts and the Pom-Pom(McLeish and Larson 1998) and eXtended Pom-Pom(XPP: Verbeeten et al. 2004) models for branchedmelts. These and other differential models can bewritten in a general form, involving a slip tensor,which represents the nonaffine motion of polymerchains with respect to the macroscopic flow (Peters andBaaijens 1997).

The linear viscoelastic behavior of the matrix is char-acterized by the complex dynamic modulus, which is afunction of the frequency ω,

G∗0 (ω) = G′0 (ω) + jG′′0 (ω) (7)and which is fitted by an M-mode discrete relaxationspectrum, giving the storage modulus

G′0 (ω) =M∑i

G0,iλ20,iω

2

1 + λ20,iω2(8)

and the loss modulus

G′′0 (ω) =M∑i

G0,iλ0,iω

1 + λ20,iω2(9)

in terms of the moduli G0,i and relaxation times λ0,i.The influence of particles on the linear viscoelasticproperties of a suspension is discussed next.

Linear viscoelastic suspension rheology

Our point of departure is the general expression forthe effective shear modulus G of a suspension ofelastic particles dispersed throughout an elastic matrix(Torquato 2002),

(10)

where φ is the volume fraction of the dispersed phase;s∼ is an array of shape factors that define the parti-cle geometry; ν0 and ν1 are the Poisson ratios of the

continuous phase and the dispersed phase, respectively;and μ is the ratio of the shear moduli of the phases,

μ = G1G0

. (11)

In general, G0 and G1 only occur in suspension modelsvia this ratio. The dimensionless quantity fG = G/G0 isknown as the relative shear modulus. Expressions anal-ogous to Eq. 10 can be written down for the effectivebulk modulus K, Young’s modulus E, and Poisson ratioν (Torquato 2002). Any two of these properties deter-mine the mechanical behavior of an elastic material. Inviscous systems, the relative viscosity fη = η/η0 is used.

To describe suspensions where both the matrix andthe particles are linear viscoelastic, the effective dy-namic shear modulus is written in the same form as inthe elastic case,

(12)

with μ∗ = G∗1/G∗0. This implies that G∗0 and G∗1 areknown in the same range of frequencies. The relativedynamic shear modulus will later on be denoted byf ∗G(ω, φ) or, simply, by f

∗G. However, one should keep

in mind that, besides the frequency and the volumefraction, it also depends on the geometry of the par-ticles and the material properties of the phases. Thedynamic modulus ratio μ∗ governs the frequency de-pendence of f ∗G, which makes it a complex quantity,

f ∗G (ω, φ) = f ′G (ω, φ) + jf ′′G (ω, φ) . (13)The Poisson ratios may, in principle, also be com-plex. However, experiments on different thermoplasticpolymers have shown that the imaginary part of thecomplex Poisson ratio ν∗ = ν ′ − jν ′′ has a maximum atthe glass transition temperature Tg, where it is about anorder of magnitude smaller than the real part, i.e., ν ′′ ∼10−2, and that it decreases strongly upon departurefrom Tg (Agbossou et al. 1993; Waterman 1977). Wetherefore assume that, in the present case, all Poissonratios are real.

For a constant volume fraction, the correspondenceprinciple (Christensen 1969; Hashin 1965, 1970a, b)relates the relative dynamic shear modulus f ∗G to therelative shear modulus of an elastic suspension withthe same microstructure. In the case of a steady-stateoscillatory deformation with frequency ω, f ∗G is simplyobtained by replacing the moduli G0 and G1 in theelastic model by their dynamic counterparts G∗0 andG∗1. Of course, the volume fraction of crystallites ina crystallizing polymer melt is not constant. However,according to Tanner (2003), if φ changes slowly com-pared to the characteristic time scale of stress relax-

Rheol Acta (2008) 47:643–665 647

ation, the correspondence principle will still be a goodapproximation.

At this point, it should be noted that the densitydifference between the amorphous phase and the semi-crystalline phase of a polymer has an influence on thevolume fraction, which is given by

φ = φ̃ρamφ̃ρam +

(1 − φ̃

)ρsc

. (14)

Here, ρam and ρsc are the densities of the amorphousand the semicrystalline phases, respectively. The vol-ume fraction φ̃, uncorrected for the density difference,is calculated as the volume of transformed amorphousphase per initial unit volume of material. Equation 14can easily be included in the rate equations for thegrowth of the semicrystalline phase (Liedauer et al.1993; Schneider et al. 1988). In the “Application to crys-tallization experiments” section, where the actual vol-ume fraction is determined directly from microscopicimages, no correction is necessary.

Crystallites as rigid particles

Because, in general, the dynamic modulus of a polymerincreases by several orders of magnitude during crys-tallization, one may argue that the crystallites can beconsidered rigid. Any suspension model should makesure that, with this assumption, all occurrences of μ∗cancel each other out. This is trivial; if the infinitemodulus ratio remained, the effective modulus of thesuspension would already go to infinity when addingan infinitesimal amount of particles to the pure matrix,which is unrealistic. If the Poisson ratios are real, as weassume here, for rigid particles, the relative dynamicmodulus thus becomes real as well,

fG (φ) ≡ lim|μ∗|→∞ f∗G

(φ, μ∗ (ω)

). (15)

The effective storage modulus is then given by

G′ (ω, φ) = fG (φ)M∑i

G0,iλ20,iω

2

1 + λ20,iω2(16)

and the effective loss modulus by

G′′ (ω, φ) = fG (φ)M∑i

G0,iλ0,iω

1 + λ20,iω2. (17)

Hence, upon adding particles, all moduli increase by thesame amount, which, moreover, is independent of thefrequency, whereas the relaxation times remain equalto those of the matrix.

For suspensions in which the particles are essentiallyrigid, the validity of Eqs. 15, 16, and 17 has beenconfirmed by experiments, as well as numerical sim-ulations. Schaink et al. (2000) investigated the indi-vidual effects of Brownian motion and hydrodynamicinteractions on the viscosity of suspensions of rigidspheres by means of Stokesian dynamics simulations.They used a viscous fluid and a linear viscoelasticfluid as the matrix and found that the hydrodynamiccontributions in both cases were similar. Expressionsfor the components η′ = G′′/ω and η′′ = G′/ω of thedynamic viscosity, equivalent to Eqs. 16 and 17, wereobtained. Using the relative viscosity from the viscoussimulation results, Schaink et al. were able to reproducesome of the oscillatory shear data of Aral and Kalyon(1997) for suspensions of glass spheres in a viscoelasticfluid, namely, those with φ = 0.1 and φ = 0.2. See et al.(2000) subjected suspensions of spherical polyethyleneparticles in two different viscoelastic matrix fluids tosmall-amplitude oscillatory squeezing flow. They foundthat, indeed, independent of the frequency, the relativequantities η′(φ)/η′0 of one system and G

′(φ)/G′0 andG′′(φ)/G′′0 of the other system were all described bya single master curve in the examined volume fractionrange, 0 ≤ φ ≤ 0.4.

Because we want to be able, in a later stage, toextend our work with a model for perfection of thesemicrystalline phase and study its effect on mechanicalproperties, we prefer to treat crystallites as deformableparticles. In this way, the possibility to model relativelyweak (low χ1), as well as stiff (high χ1), semicrystallinestructures also remains. Moreover, in numerical poly-mer processing simulations, it is preferable to work witha dynamic modulus that remains finite. This is not thecase if crystallites are modeled as rigid particles up tolarge volume fractions.

Tanner (2003) proposed to use two separate models.The first gives fG for small volume fractions, assumingthe crystallites to be rigid, according to Eq. 15. From thesecond model, which describes the crystallizing melt atlarge volume fractions, the additional relative dynamicmodulus

h∗G =G∗

G∗1(18)

is obtained. Depending on the microstructure of thesystem, we could, for example, use a model for denselypacked particles, i.e., the crystallites, with the amor-phous phase filling the interstices, or a suspensionmodel with the amorphous phase as the particles andthe semicrystalline phase as the matrix. In any case,the relevant dynamic modulus ratio is now μ∗−1. It is

648 Rheol Acta (2008) 47:643–665

assumed that the amorphous phase essentially consistsof voids, so that

hG (1 − φ) ≡ lim|μ∗|−1→0 h∗G

(1 − φ, μ∗−1 (ω)) . (19)

An interpolation between the solutions of the small-and large-volume fraction models is necessary to insurea continuous transition at intermediate volume frac-tions. A linear interpolation has the general form

G∗ (ω, φ) = F (φ) G∗0 (ω) + H (φ) G∗1 (ω) (20)with

F (φ) = [1 − w (φ)] fG (φ) (21)and

H (φ) = w (φ) hG (φ) , (22)where w ∈ [0, 1] is an empirical weighting function.Tanner (2003) determined F and H directly, by fit-ting them to the oscillatory shear data of Boutaharet al. (1998) for a polypropylene melt containing dif-ferent volume fractions of spherulites. A qualitativeagreement with the shear-induced crystallization exper-iments of Wassner and Maier (2000) was found usingthese empirically determined interpolation functions. Itshould be noted that the experiments were limited tovery low shear rates (0.003 ≤ γ̇ ≤ 0.16).

If G∗1(ω) is known and is fitted by a discrete relax-ation spectrum of N modes, Eqs. 16 and 17 are nowextended to

G′ (ω, φ) = F (φ)M∑

i=1G0,i

λ20,iω2

1 + λ20,iω2

+H (φ)N∑

k=1G1,k

λ21,kω2

1 + λ21,kω2(23)

and

G′′ (ω, φ) = F (φ)M∑

i=1G0,i

λ0,iω

1 + λ20,iω2

+H (φ)N∑

k=1G1,k

λ1,kω

1 + λ21,kω2. (24)

In both the small-volume fraction model and the large-volume fraction model, all moduli change by the sameamount while the relaxation times do not change. Dueto the interpolation, however, the overall relaxation be-havior of the material varies with the volume fraction,unless M = N and λ0,i = λ1,i.

Although it is possible to capture, in this rather sim-ple way, the evolution of linear viscoelastic propertiesduring crystallization, we take a different approach.The linear viscoelastic modeling presented here will be

extended to the nonlinear viscoelastic regime for ap-plication in polymer processing simulations. The inter-polation method is not suited to this purpose becausethe optimal fitting parameters, defining the weightingfunction w(φ), probably change with the processingconditions.

Crystallites as deformable particles

In general, if G∗1 is finite, f∗G is complex and Eq. 12

yields for the effective storage modulus

G′ = ( f ′G − f ′′G tan δ0) G′0 (25)and for the effective loss modulus

G′′ =(

f ′G +f ′′G

tan δ0

)G′′0 (26)

with tan δ0 = G′′0/G′0 tangent of the loss angle of thematrix. The fact that the expressions between paren-theses in Eqs. 25 and 26 are different has an importantconsequence. Equation 25 can be written as

G′ =M∑

i=1

(f ′G −

f ′′Gλ0,iω

)G0,i

λ20,iω2

1 + λ20,iω2(27)

and Eq. 26 as

G′′ =M∑

i=1

(f ′G + f ′′Gλ0,iω

)G0,i

λ0,iω

1 + λ20,iω2. (28)

It is clear that, if the effective relaxation times λi arechosen equal to the relaxation times λ0,i of the ma-trix, G′ and G′′ can only be described by the samespectrum if f ′′G = 0. All moduli then increase by thesame amount f ′G relative to those of the matrix, so thatG′ and G′′ are shifted independent of the frequency,corresponding qualitatively to the behavior of a rigidparticle suspension. However, f ′′G = 0 only if μ∗ is real,i.e., if G∗1 is proportional to G

∗0 so that both have the

same frequency dependence, which is not the case insuspensions encountered in practice or in crystallizingpolymer melts.

If f ′′G = 0, f ∗G must be determined in the whole rangeof frequencies of interest, given the dynamic moduliG∗0(ω) and G

∗1(ω) of the individual phases. In numerical

simulations of crystallization during flow, G∗ can, at anytime step, be fitted by a new set of effective moduliand effective relaxation times, using the set from theprevious time step as a first estimate. If the number ofmodes is the same for each phase, they can be expressedin terms of the moduli and relaxation times of thematrix as

Gi (φ) = kG,i (φ) G0,i (29)

Rheol Acta (2008) 47:643–665 649

and

λi (φ) = kλ,i (φ) λ0,i (30)with 1 ≤ kG,i ≤ G1,i/G0,i and 1 ≤ kλ,i ≤ λ1,i/λ0,i. In thisway, a smooth transition from the matrix spectrum tothe particle spectrum is obtained. If the latter consists ofN < M modes, while going from φ=0 to φ=1, M−Nof the initial M modes should vanish. If N > M, N − Mnew modes should appear. To ensure consistency, asingle criterion must be used to choose the numberof modes in the phase spectra and in the effectivespectrum.

Thus, we use a single suspension model, in contrastto the interpolation method, where different models areused at small and large volume fractions. Therefore,we need a suspension model that is valid in the entirerange of volume fractions, as stated in the Introduction.This severely limits the number of suitable models. Wewill come back to this in the “Evaluation of a linearviscoelastic model” section.

Nonlinear viscoelastic suspension rheology

The correspondence principle is only valid in the linearviscoelastic regime because it relies on the fact thatthe stress evolution is given by a Boltzmann integral(Christensen 1969; Hashin 1970a, b). In the context ofmodeling flow-induced crystallization during process-ing, nonlinear effects will be important at least in theamorphous phase, where the largest deformations takeplace. In general, nonlinear viscoelastic constitutivemodels contain the moduli Gi and the relaxation timesλi of the linear relaxation spectrum plus a number ofadditional parameters. We assume that the correspon-dence principle still applies to the linear viscoelasticpart of the rheology. The effective moduli and relax-ation times are then related to those of the matrix byEqs. 29 and 30, respectively.

Experiments on suspensions of rigid particles in aviscoelastic matrix have shown that the maximum strainamplitude, below which linear viscoelastic behavior isobserved, decreases strongly with increasing particlevolume fraction (Aral and Kalyon 1997). Thus, eventhough the matrix is linear viscoelastic, at a certainvolume fraction, the behavior of the suspension willbecome nonlinear viscoelastic. This phenomenon mayalso be expected to occur if the particles are not rigid,although to our knowledge, no supporting data areavailable.

The experimental results of Ohl and Gleissle (1993)and Mall-Gleissle et al. (2002), in which suspen-sions of essentially rigid spheres in viscoelastic matrixfluids were subjected to simple shear flow, show a

pronounced influence of the volume fraction on thenormal stress differences. It was found that, for con-stant φ, the steady-state first normal stress differenceN1 = τ11 − τ22 correlated with the shear stress as N1 ∼τ n12, where 1.63 ≤ n ≤ 1.68. When the volume fractionof particles was increased at a constant value of theshear stress, they saw that the first normal stress dif-ference decreased. This means that the dependence ofN1 on φ is weaker than that of τ n12 on φ.

Hwang et al. (2004a), who simulated 2D suspensionsof rigid discs in an Oldroyd-B fluid, found a similarscaling of the time-averaged steady-state stress func-tions N1 and τ12 with an exponent n = 2. Furthermore,they showed that both the macroscopic shear viscosityη = τ12/γ̇0, where γ̇0 is the externally applied shearrate, and the macroscopic first normal stress coefficient�1 = N1/γ̇ 20 increase with γ̇0, as well as with φ. Mall-Gleissle et al. (2002) also observed that the magnitudeof the second normal stress difference |N2| = |τ22 − τ33|increased by the same amount as N1 upon increasingthe volume fraction at constant shear stress. This wasnot the case in the simulations of Hwang et al. (2004a)because the Oldroyd-B model does not predict a secondnormal stress difference in planar shear.

The dependence of N1 and τ12 on the volume frac-tion of particles can be reproduced, at least qual-itatively, by assuming that the “effective” velocitygradient tensor can be written as

L (φ, γ̇0) = kL (φ, γ̇0) L0 (31)to take into account that the macroscopic velocity fieldL0 is locally disturbed by the presence of particles. Theundisturbed shear rate, defined as

γ̇0 =√

2D0 : D0 (32)with D0 = 12

[L0 + LT0

]the undisturbed deformation

rate tensor, together with the volume fraction of parti-cles determines the strength of the disturbances kL ac-cording to Eq. 31. To illustrate this phenomenologicalmodel of particle-induced nonlinear effects, we choosea single-mode upper-convected Maxwell model. UsingEqs. 29, 30, and 31, the constitutive relation for theextra stress tensor becomes

+1

kλ λ0τ = 2 kG kL G0 D0 .

��

(33)

In a steady-state simple shear flow, Eq. 33 yields theshear stress

τ12 = kGkλkLG0λ0γ̇0 (34)and the first normal stress difference

N1 = 2kGk2λk2LG0λ20γ̇ 20 =2τ 212

kGG0. (35)

650 Rheol Acta (2008) 47:643–665

In accordance with the numerical simulations of Hwanget al. (2004a), the first normal stress difference, at agiven volume fraction, is proportional to the square ofthe shear stress. They used an Oldroyd-B model forthe matrix, which leads to equivalent results if usedin combination with the phenomenological nonlinearviscoelastic model discussed here. Furthermore, also inaccordance with these simulations and with the experi-ments of Mall-Gleissle et al. (2002), the ratio of the firstnormal stress difference and the nth power (here n = 2)of the shear stress, both normalized by their values atφ=0, is independent of the shear stress and shear rate:

β (φ) = N1 (φ, γ̇0) /N1 (φ = 0, γ̇0)[τ12 (φ, γ̇0) /τ12 (φ = 0, γ̇0)

]2 = 1kG (φ) . (36)

Figure 1 shows how τ12 and N1 change if the volumefraction is increased from φ1 to φ2 while the macro-scopic shear rate is kept constant. For rigid particles(kλ = 1) not disturbing the macroscopic velocity field(kL = 1), N1 increases linearly with τ12. However, theresults of Hwang et al. indicate that the dependenceof N1 on τ12, upon increasing the volume fraction ata constant shear rate, becomes stronger than linear.Here, this deviation is taken into account by the pa-rameter kL, which is a function of the volume fractionas well as the shear rate. Thus, with this parameter, ashear thickening is introduced, which is also in accor-dance with the simulations of Hwang et al. Moreover,it qualitatively agrees with the experiments of Ohl andGleissle (1993) involving rigid particle suspensions withshear thinning matrix fluids, where the shear thinningeffect was observed to decrease with increasing volumefraction at high shear rates.

log (τ12)

log

(N1)

φ 1 φ 2

log kG(φ 2)kG(φ1)

log kG(φ2)kG(φ 1)

log k (φ 2) kL(φ 2 , 0)k (φ 1)kL(φ 1 , 0)

2 log k (φ 2)kL(φ 2 ,γ 0)k (φ 1)kL(φ 1 , 0)

λ

λ

λ

λγ

γγ

Fig. 1 The volume fraction dependence of the first normal stressdifference and the shear stress at a macroscopic shear rate γ̇0

To describe the dependence of N2 on φ, a consti-tutive model should be chosen that predicts a nega-tive second normal stress difference in simple shearflow. One option is to use a model with a Gordon–Schowalter derivative (Larson 1988) and a nonzeroslip parameter ζ , like the Phan-Thien–Tanner (PTT)model. However, it can be shown that β then de-pends on the macroscopic Weissenberg number, Wi0 =λ0γ̇0, which contradicts the experimental results. Via adifferent approach, Tanner and Qi (2005) developeda phenomenological nonlinear viscoelastic suspensionmodel, showing reasonable agreement with experimen-tal data for N1 and N2. The stress tensor in theirmodel consists of two modes. One is described by aPTT model, with ζ = 0 and including a volume fractiondependence of the relaxation time, and the other by aReiner–Rivlin model with a volume fraction-dependentviscosity. The latter causes the second normal stress dif-ference. A definitive validation of the method proposedhere may be possible by starting with more advancedconstitutive models, like, for example, the Rolie-Poly(Likhtman and Graham 2003), Pom-Pom (McLeish andLarson 1998), or XPP (Verbeeten et al. 2004) models.

As shown in Fig. 1, shifting the shear stress andthe first normal stress difference by kG(φ), we shouldend up on the line with slope n corresponding to thevolume fraction φ. Hence, experimental data such asthose of Mall-Gleissle et al. can be used to validate anycombination of a constitutive model for the matrix anda suspension model for the influence of the particles onthe effective linear viscoelastic properties. Moreover,numerical results like those of Hwang et al. allow forthe independent validation of linear viscoelastic modelsfor suspensions of rigid particles because the same con-stitutive model for the matrix can be chosen as in thesimulations. Unfortunately, we are not aware of sim-ilar experimental or numerical results for deformableparticles.

The parameter kL, which describes the nonlinearviscoelasticity induced by the presence of particles, canbe found by fitting it to experimental or numerical data.This procedure is not independent of the constitutivemodel used for the matrix because different constitu-tive models may yield different values for the exponentn. For the upper-convected Maxwell and Oldroyd-Bmodels, where n = 2, we find

kL (φ, γ̇0) = N1 (φ, γ̇0) /N1 (φ = 0, γ̇0)τ12 (φ, γ̇0) /τ12 (φ = 0, γ̇0) . (37)

Figure 1 shows that this parameter determines the de-viation from the curve N1 ∼ τ12, when the volume frac-tion is increased at a constant macroscopic shear rateγ̇0. Hwang et al. showed the dependence of kL on the

Rheol Acta (2008) 47:643–665 651

volume fraction and the macroscopic shear rate in their2D simulations (Fig. 8 in Hwang et al. 2004a).

In the following, an elastic suspension model, takenfrom the literature, is transformed to a linear viscoelas-tic model by means of the correspondence principle.Its predictions are compared qualitatively to the nu-merical simulations of Hwang et al. and the experi-ments of Mall-Gleissle et al. in the “Comparison tonumerical and experimental data” section and quantita-tively to crystallization experiments in the “Applicationto crystallization experiments” section. A quantitativeevaluation of the phenomenological model of nonlinearviscoelastic suspension rheology, discussed above, isbeyond the scope of this paper.

Evaluation of a linear viscoelastic model

Analytical descriptions of the effects of particles onthe rheology of a suspension are generally restrictedto isolated particles or to interactions between pairsof particles and are, therefore, valid only in dilute orsemidilute conditions, respectively. In the case of a crys-tallizing polymer melt, however, we need a suspensionmodel that is applicable in the entire range of volumefractions. An appropriate choice might be one of the so-called self-consistent estimates, which have been usedfor quite some time in the mechanical modeling ofelastic composites. Essentially, the effective propertiesare found as follows: A stress or strain is prescribed atthe boundary of a unit cell, which gives a simplified pic-ture of the microstructure. The mechanical responseof the unit cell is calculated, and when this responsebecomes homogeneous, the effective mechanical prop-erties are found.

The generalized self-consistent method ofChristensen and Lo (1979) was claimed by theseauthors to be valid in the entire range of volumefractions. Furthermore, it gives solutions for suspen-sions of spherical particles and suspensions of longparallel cylindrical fibers, corresponding to the dif-ferent microstructures found locally in a crystallizingpolymer melt. The generalized self-consistent methodthus meets the first two requirements stated in theIntroduction. The third and fourth have already beendealt with in the “Modeling” section. We thereforediscuss this model in detail here.

History and relation to other models

The generalized self-consistent method was originallycalled three-phase model, but it was renamed byChristensen (1990) in reference to the self-consistent

method (Hill 1965a, b; Budiansky 1965). This model,which has an analogy in the theory of heterogenousconducting materials (Bruggeman 1935), considers asingle particle embedded in a homogeneous matrixwith the effective properties sought. The generalizedself-consistent method, on the other hand, uses a unitcell made up of a particle surrounded by a concen-tric shell of the matrix material. This coated particleis embedded in the effective homogeneous medium.The difference between the two models can be inter-preted as follows: “While the self-consistent methodseeks to predict the interaction of an inclusion and itsneighboring microstructure (the combined effect of thematrix and other inclusions), this model includes (ina certain approximate sense) the interaction betweenthe inclusion and the surrounding matrix, as well as theneighboring microstructure” (Nemat-Nasser and Hori1993). A coated particle unit cell was used earlier byFröhlich and Sack (1946) for elastic spheres in a viscousmatrix, by Oldroyd (1953) for viscous drops or elasticspheres in a viscous matrix, by Kerner (1956) for elasticspheres in an elastic matrix, and by Hermans (1967)for unidirectional elastic fibers in an elastic matrix. Twoversions of the generalized self-consistent method exist:a three-dimensional (3D) one in which the particle andmatrix domains of the unit cell are concentric spheresand a 2D one in which they are concentric circles.These give the solutions for spherical particles and longparallel cylindrical fibers, respectively.

Table 1 summarizes the main properties of the gen-eralized self-consistent method and some other suspen-sion models. Palierne (1990) developed a model forincompressible linear viscoelastic emulsions in whichthe drops are at least approximately spherical. For di-lute emulsions, he considered a single drop suspendedin the effective medium. Not surprisingly, neglectingthe effect of surface tension, the result is the same asthe analytical solution of the self-consistent method inthe dilute limit, taking the matrix as incompressible(Hill 1965b). In contrast to both the self-consistent andthe generalized self-consistent methods, the derivationof the nondilute Palierne model is based on a unit cellin which one particle is at the center of a sphere filledwith the matrix and other particles, which in turn issurrounded by the effective medium. If the effect ofsurface tension is again neglected, it turns out that theresult is exactly the viscoelastic analogue of the modelof Kerner (1956) for all volume fractions (Graeblinget al. 1993; Palierne 1990). Through a similar deriva-tion for an elastic suspension of spheres, Uemura andTakayanagi (1966) also arrived at the same effectiveshear modulus as Kerner, although a different expres-sion for the effective Poisson ratio was obtained.

652 Rheol Acta (2008) 47:643–665

Table 1 Attributes of some suspension models

Model Phases Particles Volume fraction Ref.

SCM

E

SpheresSmall or large

Hill (1965b)Cylinders Hill (1965a)

GSCMSpheres Arbitrary, size distribution

Christensen and Lo (1979, 1986), Christensen (1990)Cylinders should admit φ → 1

Torquato, exactArbitrary

Arbitrary Torquato (1997)Torquato, TOA Small to moderate Torquato (1998)

PalierneLVE Spheres Small to moderate

Palierne (1990)Bousmina Bousmina (1999)

SCM self-consistent method, GSCM generalized self-consistent method, TOA third-order approximation, E elastic, LVE linearviscoelastic

Christensen and Lo (1979) demonstrated that theelastic shear modulus predicted by the 3D general-ized self-consistent method lies between the classicalupper (Hashin 1962) and lower bounds (Hashin andShtrikman 1963; Walpole 1966) for all volume frac-tions, whereas Kerner’s result coincides with the lowerbound. Contrary to Palierne’s model, the 3D gen-eralized self-consistent method is only equivalent toKerner’s model for vanishing volume fraction ofspheres. Whereas Palierne did not consider drops closeto contact with each other, Christensen (1990) derived,by physical reasoning similar to that of Frankel andAcrivos (1967), the functional form of fG(φ) and ofthe relative transverse shear modulus fG23(φ) for rigidspheres and unidirectional rigid cylinders, respectively,when φ → 1. They were found to agree with the cor-responding asymptotic forms of the generalized self-consistent method, for a compressible matrix and foran incompressible matrix. Of course, the volume frac-tion can only go to one if the distribution of particlediameters is sufficiently broad, so that small particlescan fill the spaces between larger particles.

Bousmina (1999) proposed an emulsion model basedon the 3D generalized self-consistent method, extend-ing the particle modulus with a term due to surfacetension. In the coefficients of the quadratic function,one of whose roots is f ∗G (see Eq. 41 later on), onlyterms of order φ were retained. It is therefore notsurprising that only small differences with Palierne’smodel were observed. The expressions for the coeffi-cients given in Bousmina’s paper contain a few errors,apparently mostly because he was unaware of an erra-tum (Christensen and Lo 1986) to the original paperon the generalized self-consistent method; the correctexpressions are included here in the Appendix.

Christensen (1990) validated the 3D generalizedself-consistent method with respect to experimentaldata on suspensions of rigid spheres. The results proved

superior to those of two homogenization schemeswidely used at that time, i.e., the Mori–Tanaka method(Benveniste 1987) and the differential scheme (Phan-Thien and Pham 1997), especially for volume frac-tions φ > 0.4. However, Nemat-Nasser and Yu (1993)pointed out uncertainties in some of the experimentaldata used for comparison, which were compiled byThomas (1965). Segurado and Llorca (2002) performed3D numerical simulations of suspensions of spheres inan elastic matrix, where 0 ≤ φ ≤ 0.5. They comparedtheir results to the predictions of the Mori–Tanakamethod, the generalized self-consistent method, andthe third-order approximation (Torquato 1998) of anexact series expansion for the effective stiffness tensorof elastic two-phase media (Torquato 1997). The gener-alized self-consistent method performed just as well asthis third-order approximation when the particles weredeformable, except that the effective bulk modulus waspredicted slightly more accurately by the third-orderapproximation. For rigid spheres, the latter also yieldedsomewhat better results.

Because the generalized self-consistent method ismuch easier to implement than Torquato’s third-orderapproximation, we will use it to determine the linearviscoelastic properties of a suspension with the aid ofthe correspondence principle. As explained above, thesuspension is represented by a unit cell consisting of aparticle and a surrounding matrix shell. Their radii area and b , respectively. The volume fraction of particlesis given by

φ =( a

b

)3(38)

for the 3D coated sphere unit cell and

φ =( a

b

)2(39)

Rheol Acta (2008) 47:643–665 653

for the 2D coated cylinder unit cell. The unit cell issuspended in an infinitely extending effective medium,which has the effective properties of the suspension.These properties are found when the response of theunit cell to a given load equals the response of thehomogeneous effective medium. In the 3D generalizedself-consistent method, the relative shear modulus (seeEq. 10) is obtained from the quadratic equation

Af 2G + BfG + C = 0 , (40)

where the coefficients A, B, and C depend on φ, μ, ν0,and ν1. These coefficients are given in the Appendix.The relative bulk modulus was found to be the same asin the composite spheres model of Hashin (1962).

For elastic suspensions of long parallel cylindricalfibers, Hashin and Rosen (1964) derived the compo-nents of the fourth-order stiffness tensor, except theshear modulus in the transverse plane. The 2D general-ized self-consistent method gives the relative transverseshear modulus as the solution of a quadratic expressionsimilar to Eq. 40, but with different coefficients. Theseare also included in the Appendix.

In accordance with the correspondence principle, therelative dynamic modulus is obtained from

A∗ f ∗G2 + B∗ f ∗G + C∗ = 0 , (41)

where the complex coefficients A∗, B∗, and C∗ followfrom A, B, and C when μ is replaced by μ∗. Fora crystallizing polymer melt, we propose to calculatethe effect of the presence of spherulites by meansof the 3D generalized self-consistent method and to usethe resulting effective medium as the matrix in the 2D

0 0.2 0.4 0.6 0.8 110

0

101

102

103

ν0 = 0.500

ν0 = 0.499

ν0 = 0.497

ν0 = 0.490

ν0 = 0.470

ν0 = 0.400

ν0 = 0.100

φ [-]

f G[-

]

Fig. 2 Influence of the Poisson ratio of the matrix on the relativemodulus of an elastic suspension of spheres (μ = 103, ν1 = 0.5)

0.7 0.75 0.8 0.85 0.9 0.95 1

102

103

ν1 = 0.500

ν0 = 0.400

ν0 = 0.300

ν0 = 0.200

ν0 = 0.100

φ [-]

f G[-

]

Fig. 3 Influence of the Poisson ratio of the particles on therelative modulus of an elastic suspension of spheres (μ = 103,ν0 = 0.5)

generalized self-consistent method, which accounts forthe influence of oriented crystallites.

Influence of phase properties

First of all, let us take a look at the original 3D gener-alized self-consistent method for elastic suspensions ofspheres, according to which the relative shear modulusfG is obtained from Eq. 40 with real coefficients A,B, and C. The curve of log( fG) vs φ for μ = 103 andν0 = ν1 = 0.5, plotted in Figs. 2 and 3, has two inflectionpoints: one at φ ≈ 0.70 and the other at φ ≈ 0.95. Inbetween these points, the second derivative is negative,

∂2 log( fG)∂φ2

< 0 , (42)

and consequently, a “shoulder” appears in the curve.Beyond φ ≈ 0.95, fG swiftly approaches its final valuefG(φ = 1) = μ; note that fG at φ = 0.95 is still smallerthan μ/2.

The shape of the relative modulus curve dependsmost strongly on the Poisson ratio of the matrix andon the modulus ratio. Figure 2 shows that, upon low-ering ν0 while keeping ν1 = 0.5, fG decreases and theshoulder vanishes quickly: at ν0 = 0.49, it is not recog-nizable anymore. Decreasing ν1 while ν0 = 0.5 has amuch weaker influence on fG, as seen in Fig. 3, and theshoulder remains. Thus, even at large volume fractions,compressibility of the matrix has a more profoundinfluence on the results of the 3D generalized self-consistent method than compressibility of the particles.Furthermore, the shoulder diminishes at lower valuesof the modulus ratio, as shown in Fig. 4. It should benoted that the logarithmic scale used on the vertical

654 Rheol Acta (2008) 47:643–665

0 0.2 0.4 0.6 0.8 110

0

101

102

103

μ = 103

μ = 102

μ = 101

φ [-]

f G[-

]

Fig. 4 Influence of the modulus ratio on the relative modulus ofan elastic suspension of spheres (ν0 = ν1 = 0.5)

axes of these figures exaggerates the effects mentioned.For the second derivative of fG with respect to φ, onefinds

∂2 fG∂φ2

<1

fG

(∂ fG∂φ

)2(43)

if Eq. 42 is satisfied. Because the right-hand side ofEq. 43 is always positive, when fG is plotted on alinear scale, a smaller decrease of ν0 or μ suffices tomake the shoulder vanish: it is already gone at a matrixPoisson ratio ν0 = 0.498 for ν1 = 0.5 and μ = 103, andat a modulus ratio μ = 50 for ν0 = ν1 = 0.5.

To our knowledge, this peculiar feature of the gener-alized self-consistent method has never been reportedbefore. Looking at the material parameters in previouspublications, it is easy to see why. Christensen andLo (1979) simulated suspensions with modulus ratiosμ = 23.46 and μ = 135.14. According to Fig. 4, thelatter is high enough to cause an observable shoulderin a suspension with an incompressible matrix, butChristensen and Lo used a matrix with ν0 = 0.35, whichis too low even when μ = 103 (Fig. 2). In the later workof Christensen (1990), the compressibility effect wasobscured because only rigid particles were considered(μ → ∞). The high end of the curve is then stretchedto infinity, so that the shoulder is smoothed out.Segurado and Llorca (2002) used the 3D generalizedself-consistent method to simulate both suspensions ofrigid spheres and suspensions of deformable spheres. Inthe latter case, the matrix was, again, too compressible(ν0 = 0.38) and the modulus ratio too low (μ = 26.83).Apart from that, the maximum volume fraction used intheir calculations was φ = 0.5, which is below the rangewhere the shoulder develops.

When the 2D generalized self-consistent method isused to calculate the relative transverse shear modulusof an elastic fiber-reinforced material, the results showthe same dependence on the Poisson ratios and themodulus ratio. Applying the generalized self-consistentmethod to a suspension of linear viscoelastic materials,similar effects are observed in the storage modulus andthe loss modulus.

Comparison to numerical and experimental data

As explained in the “Nonlinear viscoelastic suspensionrheology” section, models for the effects of particleson the linear viscoelastic properties of a suspension,governed by the multipliers kG,i for the modulus andkλ,i for the relaxation time, can be validated by nu-merical simulations. Hwang et al. (2004a) presentedresults for a sheared 2D system, consisting of rigiddiscs suspended in an Oldroyd-B fluid. Their simu-lation method, based on sliding rectangular domainswith periodic boundary conditions, was extended tothree dimensions to describe suspensions of sphericalparticles (Hwang et al. 2004b). It was also modified for2D extensional flow, based on stretching rectangulardomains with periodic boundary conditions (Hwangand Hulsen 2006). An alternative method for 2D exten-sional flow, using a fixed grid, was developed recentlyby D’Avino et al. (2007a, b). The 2D shear resultsare considered here. Figure 5 shows the time-averaged

10–1

100

101

10–2

10–1

100

101

102

0 0.2 0.4 0.610

0

101

σ12 [Pa]

N1

[Pa]

φ [-]

k G[-

]

Fig. 5 The steady-state first normal stress difference and shearstress from simulations (lines, Hwang et al. (2004a)) and fromthe 2D generalized self-consistent method at an arbitrary shearrate (circles). Inset: kG(φ) from simulations (dots) and from thegeneralized self-consistent method (circles). The angle bracketsindicate that the steady-state properties were obtained from thesimulations by averaging over time

Rheol Acta (2008) 47:643–665 655

steady-state first normal stress difference 〈N1〉 vs thetime-averaged steady-state shear stress 〈σ12〉 (σ12 is thesum of the viscous mode and the viscoelastic mode inthe Oldroyd-B model). Each line corresponds to thesimulation results at a constant area fraction of disks,which is equivalent to a volume fraction of infinitelylong parallel cylinders, and different shear rates.

Because of the rigidity of the particles, kλ,i = 1 andkG,i = fG is a real number, which is obtained fromthe 2D generalized self-consistent method. Irrespec-tive of the shear rate, shifting τ12(φ = 0) to the rightand N1(φ = 0) upwards by the same factor fG(φ), weshould end up on the line corresponding to the areafraction φ (see Fig. 1). The circles in Fig. 5 indicatethe results of the 2D generalized self-consistent methodat an arbitrary constant shear rate. It turns out thatthese agree with the simulations up to φ ≈ 0.10. Atlarger area fractions, the 2D generalized self-consistentmethod predicts a much stronger increase of fG thanthe simulations. This is not entirely surprising becauseHwang et al. (2004a) determined the steady-state sus-pension properties from simulations with a single par-ticle in a periodic domain. The authors already notedthat this method does not give realistic results for highlyconcentrated systems. Nevertheless, an area fraction of10% is quite small.

We also compared the predictions of a single-modeupper convected Maxwell model, combined with the3D generalized self-consistent method, to the experi-mental data of Mall-Gleissle et al. (2002). As seen inFig. 6, the data are underpredicted already for φ = 0.05.Better results would probably have been obtained witha more advanced constitutive model. However, evenfor relatively simple ones, like the Giesekus model or

102

103

104

101

102

103

104

φ = 0φ = 0.05φ = 0.15φ = 0.25

τ12 [Pa]

N1

[Pa]

Fig. 6 The steady-state first normal stress difference and shearstress from experiments (symbols, Mall-Gleissle et al. (2002)) andfrom the 3D generalized self-consistent method (lines)

the PTT model, no analytical solutions can be derivedfor τ12 and N1. The relaxation behavior of the matrixhas to be known to calculate them numerically. Unfor-tunately, we do not have this information.

Application to crystallization experiments

We looked at two types of rheological measurementson crystallizing polymer melts to investigate whethersuspension models can indeed capture the phenom-ena observed in these experiments. In the first typeof experiment, after different short periods of shear,the evolution of the linear viscoelastic properties wasfollowed in time at a constant frequency (J.F. Vega,personal communication). In the second type, the linearviscoelastic properties were measured over a range offrequencies for different constant volume fractions ofcrystallites (Boutahar et al. 1996, 1998; Coppola et al.2006).

The results from these experiments were used tovalidate the generalized self-consistent method, as wellas the interpolation method, which was described inthe “Crystallites as rigid particles” section. While thelatter usually treats the highly filled polymer melt asa suspension of amorphous particles in a semicrystal-line matrix, the former always takes the crystallites asparticles. It has been mentioned in the “History andrelation to other models” section that, to allow φ → 1,the generalized self-consistent method assumes a broaddistribution of particle diameters. This is generally notthe case in a crystallizing polymer melt, but there,complete space filling is achieved in a different way.After impingement of the crystallites, further growthwill be restricted to the directions in which amorphousmaterial is still present, until all of it has been incor-porated in the semicrystalline phase. Formally, becausethe crystallites become irregularly shaped, the gener-alized self-consistent method does not apply anymore.However, we do not expect the rheological propertiesof a highly concentrated suspension to be very sensitiveto variations in particle shape.

Evolution of linear viscoelastic propertiesafter short-term shear

Flow-induced crystallization experiments, carried outin our own group (J.F. Vega, personal communication)are considered first. An isotactic polypropylene melt(HD 120 MO, supplied by Borealis) was subjectedto different short periods of shear. Subsequently, itslinear viscoelastic properties were monitored in time bymeans of oscillatory shear measurements. The resultsare shown in Fig. 7. It is clear that, immediately after

656 Rheol Acta (2008) 47:643–665

100

101

102

103

104

103

104

105

106

107

108

0

10

20

30

40

50

60

70

80

90

t [s]

G[P

a]

δ[

]°

Fig. 7 Evolution of the storage modulus (open symbols) andloss angle (filled symbols) during crystallization, measured underquiescent conditions (open and filled circles) and after shearing atγ̇ = 60s−1 for ts = 3s (open and filled triangles) and ts = 6s (openand filled squares). Part of the data points were omitted for thesake of clarity

the flow, the dynamic modulus of the material, whichwas then still largely amorphous, was already increasedsignificantly. The first values of G′ and G′′ measuredafter the flow were used as G′0 and G

′′0 in the model

calculations. The plateaus of G′ and δ, reached in thelate stage of crystallization, were extrapolated to theearlier stages by the functions

G′1 (t) ={

G′1 (t1) for t ≤ t1G′1 (t1)

[tt1

]mfor t > t1

(44)

and

δ1 (t) ={

δ1 (t1) for t ≤ t1δ1 (t1) + cδ ln

(tt1

)for t > t1

(45)

and these extrapolations were used as the linear vis-coelastic properties of the semicrystalline phase. Thevalues of the parameters in Eqs. 44 and 45 are givenin Table 2. The characteristic time t1, indicating thetransition from the space-filling stage to the perfectionstage, was defined as the intersection of the extrap-olated linear fits of the log(G′)–log(t) data in thesestages, as shown in Fig. 8.

102

103

104

107

108

t [s]

G' [

Pa]

Fig. 8 Close-up of the storage moduli from Fig. 7. Part of the datapoints were omitted and the curves corresponding to ts = 3s andts = 6s were shifted vertically by factors 1.2 and 1.5, respectively,for the sake of clarity. The solid lines are fits of the data in theplateau region and in the region of strong increase of G′. Thedashed lines indicate t1 and G′1(t1)

In the experiments with 3 and 6 s of shear, orientedcrystallites were observed. There, the effect of thespherulites was calculated by the 3D generalized self-consistent method, and the resulting effective rheolog-ical properties were used as matrix properties in the2D generalized self-consistent method. The evolutionof the volume fraction of spherulites was determinedfrom optical micrographs taken during the flow andthe subsequent crystallization (D.G. Hristova, personalcommunication). The result for the quiescent melt isshown in Fig. 9. The volume fraction of oriented crys-tallites could not be determined accurately in this way.Therefore, the degree of crystallinity χ , derived fromin situ wide-angle X-ray diffraction (WAXD) mea-surements (D.G. Hristova, personal communication),was used to estimate the total semicrystalline volumefraction φ. By definition, φ and χ are related as

φ (t) = χ (t)χ1 (t)

. (46)

The integrated intensities XW AX D of the diffractionpeaks, which are normalized by the total integratedintensities, are included in Fig. 9 for the quiescent

Table 2 Parameters for calculating the rheological properties and degree of crystallinity of the semicrystalline phase by means ofEqs. 44, 45, and 47

ts [s] t1 [s] G′1 (t1) [Pa] δ1 (t1) [◦] χ1 (t1) [-] m [-] cδ [◦] cχ [s−1]

0 2.89 × 103 4.83 × 107 3.48 – 9.00 × 10−2 −1.29 × 10−1 –3 4.25 × 102 4.80 × 107 4.83 4.98 × 10−1 7.02 × 10−2 −3.40 × 10−1 6.67 × 10−56 3.22 × 102 4.68 × 107 4.97 5.40 × 10−1 7.11 × 10−2 −3.40 × 10−1 0

Rheol Acta (2008) 47:643–665 657

100

101

102

103

104

0

0.2

0.4

0.6

0.8

1

t [s]

[-],

XW

AX

D[-

]

Fig. 9 Space filling during quiescent crystallization, obtainedfrom optical microscopy (crosses), along with the integratedWAXD intensity during quiescent crystallization (circles) andafter shearing at γ̇ = 60s−1 for ts = 3s (triangles) and ts = 6s(squares). Solid lines are fits of the WAXD data in the plateauregion. Dashed lines indicate t1 and χ1(t1)

and flow-induced crystallization experiments. Compar-ing the WAXD and optical microscopy results for thequiescent experiment, it is seen that XW AX D is close tobut slightly above φ up to φ = 0.3. Because the degreeof crystallinity is, by definition, always smaller than thedegree of space filling, we conclude that XW AX D > χ .On the other hand, the values of t1 derived from thestorage modulus measured during flow-induced crystal-lization (Table 2) correlate well with the onset times ofthe plateaus in the WAXD data. So at least the timescale is correct, provided that t1 corresponds to φ = 1as observed in the quiescent crystallization experiment.We assumed XW AX D to be an adequate measure of theinternal degree of crystallinity χ = χ1 in the plateauregion, where the highest signal-to-noise ratio was ob-tained. Unfortunately, however, the development of χand χ1 during the space-filling process could not bereconstructed from these data.

The integrated WAXD intensity in the plateau re-gion was fitted by a linear function of time and ex-trapolated to obtain χ1(t > t1). We assumed that nosecondary crystallization took place up to t = t1 in anyof the experiments considered here, so,

χ1 (t) ={

χ1 (t1) for t ≤ t1χ1 (t1) + cχ [t − t1] for t > t1 . (47)

Furthermore, for lack of better experimental data,XW AX D was taken as χ and the experimental data werescaled according to Eqs. 46 and 47 to obtain the totalvolume fraction. The results are shown in Fig. 10.

The volume fraction of spherulites φsph was deter-mined from optical microscopy in the same way as for

the quiescent experiment. The volume fraction ofspherulites in the amorphous phase,

xsph = φsph1 − φ + φsph , (48)

was used in the 3D generalized self-consistent methodto calculate the linear viscoelastic properties of theeffective matrix in the 2D generalized self-consistentmethod. There, the volume fraction of orientedcrystallites,

φori = φ − φsph , (49)was used to calculate the linear viscoelastic propertiesof the crystallizing melt.

The results of the generalized self-consistent methodare compared to those of the interpolation method inFig. 11. The main difference is that G′(t), accordingto the generalized self-consistent method, goes throughtwo inflection points before reaching the plateau G′ =G′1. This is due to the shoulder in f

∗G(φ), as discussed in

the “Influence of phase properties” section. The shoul-der is followed by a decreasing G′ for the experimentwith 6 s of shear. This is caused by our estimate ofthe volume fraction of oriented crystallites, which goesthrough a maximum and then drops back to zero. Themore pronounced local extrema in δ at large volumefractions (Fig. 12) are caused by local extrema in G′′,which, depending on the phase properties, may arise inthe linear viscoelastic generalized self-consistent meth-od. For the interpolation method and the generalizedself-consistent method, the data for the highest sheartime are not captured. This can be explained by the lackof information about the shape and orientation of par-ticles in the interpolation method and by uncertainties

100

101

102

103

104

0

0.2

0.4

0.6

0.8

1

t [s]

Fig. 10 Space filling, estimated from WAXD for the flow-induced crystallization experiments, as explained in the text

658 Rheol Acta (2008) 47:643–665

100

101

102

103

104

104

105

106

107

108

t [s]

G[P

a]

Fig. 11 Evolution of the storage modulus under quiescent con-ditions (circles) and after shearing at γ̇ = 60s−1 for ts = 3s (tri-angles) and ts = 6s (squares). Dashed lines were obtained by theinterpolation method, Eq. 23, using a hyperbolic tangent functionfor w(φ). Solid lines correspond to the generalized self-consistentmethod, using Eq. 41 with the coefficients given in Table 4 in theAppendix and Eq. 66 from the Appendix

in the estimate of the oriented volume fraction used inthe generalized self-consistent method.

The degree of space filling is often estimated by alinear scaling of the storage modulus (Gauthier et al.1992; Khanna 1993). This can be rewritten as

G′ (t) = G′0 (t) +[G′1 (t) − G′0 (t)

]φ (t) . (50)

Figure 13 shows that the rheological data are not re-produced from the microscopic images by Eq. 50. Alsoplotted is the logarithmic scaling law,

G′ (t) = G′0 (t)[

G′1 (t)G′0 (t)

]φ(t), (51)

which was used by Pogodina et al. (1999b). It performsmuch better than the linear scaling law, though not aswell as the generalized self-consistent method and theinterpolation method.

Frequency sweeps at different volume fractions

We first consider the experiments of Coppola et al.(2006). They cooled down a poly(1-butene) melt to atemperature Tc below the nominal melting tempera-ture Tm and let it crystallize at Tc for a controlledamount of time. Then, they applied an inverse quench(Acierno and Grizzuti 2003) to a temperature Tiq closeto, but still below, Tm and measured the rheologicalproperties by means of a multiwave technique, whichextended the experimentally accessible range of fre-quencies down to the order of 10−3 rad/s. This revealedthat the plateau in the storage modulus, observed at

100

101

102

103

104

0

10

20

30

40

50

60

t [s]

[°]

Fig. 12 Evolution of the loss angle, measured under differentconditions and calculated by the interpolation method and thegeneralized self-consistent method, indicated by the symbols andlines as in Fig. 11

low frequencies, did not continue all the way downto zero, as in a chemical or physical gel, but fell offat a certain frequency. This supports the idea thatcrystallization precursors and/or crystalline nuclei actas physical crosslinks but are too far apart to forma percolating network (Zuidema 2000; Zuidema et al.2001).

The results of Coppola et al. are reproduced inFig. 14. Different volume fractions of spherulites wereobtained by varying the duration of crystallization atTc. The rheological properties of the partially crystal-lized melts were compared to those of an amorphousmelt containing, according to the authors, a much largervolume fraction of solid spheres. The effect observed

100

101

102

103

104

104

105

106

107

108

t [s]

G[Pa]

Fig. 13 Evolution of the storage modulus, measured under dif-ferent conditions, indicated by the symbols and as in Fig. 11.Dashed lines correspond to the linear scaling law, Eq. 50, solidlines to the logarithmic scaling law, Eq. 51

Rheol Acta (2008) 47:643–665 659

10–4

10–3

10–2

10–1

100

101

102

103

100

102

104

106

108

ω [rad/s]

G[P

a]

Fig. 14 Storage modulus measured by Coppola et al. (2006)for crystallizing melts (open symbols) and an amorphous meltcontaining glass beads (crosses). Lines correspond to the 3Dgeneralized self-consistent method

in the low-frequency rheology was much smaller thanin the partially crystallized melts, from which the au-thors concluded that, indeed, the behavior of the lattercannot be explained by gelation. However, they de-termined the volume fraction by means of the linearscaling law, Eq. 50. Based on the results discussed inthe previous section, we believe that the actual volumefraction was much larger and we expect it to be moreaccurately estimated by the generalized self-consistentmethod. The results of this model are included inFig. 14. The low-frequency behavior was obviously not

Table 3 Volume fraction of spherulites, as obtained from thelinear scaling law, Eq. 50 (φG′(t)), from differential scanningcalorimetry (φDSC), and from the 3D generalized self-consistentmethod (φGSCM) for the two sets of experiments

φGSCM φG′(t) φDSC

Coppola et al. (2006)0.25 0.00580.30 0.01290.40 0.02801 1

φbeads

0.12 0.12Boutahar et al. (1996, 1998)

0.06 0.10.11 0.20.16 0.30.23 0.40.32 0.50.41 0.60.55 0.70.73 0.8

captured for the partially crystallized melts, nor forthe particle-filled melt. Therefore, the model was fittedto the data at high frequencies. The volume fractionsobtained through both approaches are listed in Table 3.According to the generalized self-consistent method,the samples were highly filled with spherulites. Unfor-tunately, no optical data are available to validate thisresult.

Experiments on two fundamentally different materi-als were published by Boutahar et al. (1996, 1998). Onematerial was a heavily nucleated polyethylene melt,whose morphology looked like that of a colloid. Thecrystalline nuclei were very small, close together, andhighly imperfect. The other was an isotactic polypro-pylene melt, containing large, well separated spheru-lites. We will consider the data for this suspension-likematerial here.

The evolution of the storage modulus and the lossmodulus is depicted in Figs. 15 and 16, respectively.From Boutahar et al. (1996), we estimated G′1 = 107 Paand G′′1 = 6 × 105 Pa. The results are not very sensitiveto the values used. Although, at the largest volume frac-tion, G′′ exceeds G′′1 around ω = 1rad/s, the absolutevalue |G∗| is always smaller than |G∗1|, as it should be.The generalized self-consistent method was again fittedto the data at high frequencies. Also for these experi-ments, the model fails to capture the observed strongincrease of the storage and loss moduli with increasingvolume fraction at low frequencies. However, the fittedvolume fractions correspond very well to the valuesobtained by Boutahar et al. from differential scanningcalorimetry (Table 3).

10–1

100

101

102

103

104

105

106

107

ω [rad/s]

G[P

a]

Fig. 15 Storage modulus measured by Boutahar et al. (1996,1998) (dots) and obtained from the 3D generalized self-consistentmethod (lines)

660 Rheol Acta (2008) 47:643–665

10–1

100

101

102

103

104

105

106

G[P

a]

ω [rad/s]Fig. 16 Loss modulus measured by Boutahar et al. (1996, 1998)(dots) and obtained from the 3D generalized self-consistentmethod (lines)

Modeling of low-frequency suspension rheology

A few models describe the development of a low-frequency plateau in G′ and G′′. In the Palierne model,it is governed by the surface tension. However, be-cause spherulites behave like solid particles rather thanliquid drops, the surface tension is probably negligi-ble in a crystallizing melt, at least for the relativelylarge volume fractions (corresponding to large radiiof spherulites) where the plateau is first observed.Van Ruth et al. (2006) carried out quiescent crystal-lization experiments, similar to those of Boutahar et al.(1996, 1998). They used the Palierne model with anequilibrium modulus,

Geq = c (φ − φ0)m , (52)added to G′, which could be fitted to the experimentaldata with realistic values of the parameters φ0 and m.However, the results of the Palierne model for G′′ werenot shown. A Cross model was used to fit the dynamicviscosity |η∗| for each volume fraction separately.

The self-consistent method predicts a percolationthreshold, where the modulus rises or drops steeply, ap-proaching the threshold from below or above, respec-tively. It diverges to infinity, respectively zero, whenboth phases are incompressible and the particles aremuch stiffer than the matrix. This happens at φ = 0.4for spheres (Hill 1965b) and at φ = 0.5 for cylindricalfibers (Hill 1965a). When applied to a linear viscoelasticsuspension by means of the correspondence principle,the effect is mainly observed at low frequencies asa plateau in G′. This was shown by Wilbrink et al.

(2005). However, the self-consistent method did notagree quantitatively with their suspensions, for whichthe percolation threshold was around φ = 0.10.

Albérola and Mélé (1996) incorporated the con-cept of percolation in the generalized self-consistentmethod. In their modified unit cell, the part of thematrix trapped inside particle clusters is representedby a sphere, which is surrounded by two concentricshells, representing the particles and the nontrappedmatrix. This double-coated sphere is surrounded bythe effective medium. Hervé and Zaoui (1993, 1995)extended the generalized self-consistent method to par-ticles coated by an arbitrary number of layers. Theirmodel was used by Albérola et al. to calculate the rhe-ological properties of a suspension with clustering de-formable particles. According to the simulation resultsof Hwang et al. (2004a, b, 2006), extensional stresses be-tween particles in a sheared viscoelastic matrix enhancetheir tendency to form clusters. Therefore, we ap-plied the modified generalized self-consistent methodto the experiments described in the previous section.However, it turned out that the percolation thresholdshifts f ∗G similarly for all frequencies. Hence, the shoul-der, observed experimentally at low frequencies, is notreproduced by this model.

Conclusions

The approach to suspension-based rheological mod-eling of crystallizing polymer melts, presented in thispaper, has several advantages. Linear viscoelastic sus-pension models are available to obtain the modulusand relaxation time multipliers kG,i and kλ,i. These ac-count for the influence of crystallites on the relaxationspectrum used in the constitutive model for the melt.The fact that crystallites can be described as nonrigidparticles prevents problems at high degrees of spacefilling. The phenomenological nonlinear viscoelasticsuspension model qualitatively reproduces results ofnumerical simulations but is much less expensive com-putationally and can therefore easily be used in indus-trial processing simulations. Only standard rheologicalexperiments are needed to quantify the phenomenolog-ical parameter kL.

After considering several linear viscoelastic suspen-sion models, we used the generalized self-consistentmethod (Christensen 1990; Christensen and Lo 1979,1986) because of its simplicity and because, in combina-tion with the nonlinear viscoelastic model, it satisfies allthe requirements stated in the Introduction. Combined

Rheol Acta (2008) 47:643–665 661

rheometry and optical microscopy showed that a widelyused linear scaling of the storage modulus (Gauthieret al. 1992; Khanna 1993) severely underpredicted thedegree of space filling. The generalized self-consistentmethod described the evolution of linear viscoelasticproperties during crystallization rather well, at least atmoderate to high frequencies; the low-frequency be-havior (Boutahar et al. 1996, 1998; Coppola et al. 2006)was not captured. An extension of the generalized self-consistent method, including percolation (Albérola andMélé 1996), did not solve this problem; the relative dy-namic modulus was merely shifted in a similar mannerfor all frequencies.

Acknowledgements We are grateful to Dr. J.F. Vega (ConsejoSuperior de Investigaciones Cientìficas Madrid, Spain) and Dr.D.G. Hristova (TU/e, The Netherlands) for providing some oftheir experimental results.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

Appendix

3D generalized self-consistent method and Bousmina’slinear viscoelastic model For the 3D generalized self-consistent method, the coefficients are summarized inTable 4. Some of the expressions given by Bousmina(1999) as first-order approximations of A, B, and Ccontain errors, most of which are due to misprints

in Christensen and Lo (1979). Bousmina’s Eqs. 54–59should read

A = K1 [24K2 − 150K3φ] , (53)

B = 12

K1 [9K2 + 375K3φ] , (54)

C = 14

K1 [−114K2 − 675K3φ] , (55)

K1 =[

5

2

(G∗1 + α/R

G∗0− 8

)(56)

+ 7(

G∗1 + α/RG∗0

)+ 4

], (57)

K2 =(

G∗1 + α/RG∗0

+ 32

), (58)

K3 =(

G∗1 + α/RG∗0

− 1)

. (59)

2D generalized self-consistent method Under the as-sumption of transverse isotropy, Hooke’s law for anelastic fiber suspension can be written in matrix formaccording to

⎡⎢⎢⎢⎢⎢⎢⎣

τ11τ22τ33τ12τ13τ23

⎤⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎣

C11 C12 C13 0 0 0C12 C22 C23 0 0 0C13 C23 C22 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C22 − C23

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

�11�22�33

2�122�132�23

⎤⎥⎥⎥⎥⎥⎥⎦

(60)

Table 4 Parameters used in the 3D generalized self-consistent method (Christensen and Lo 1979, 1986; Christensen 1990)

Shear modulus G, Eq. 40

ABC

⎫⎬⎭ = c1

(G1G0

− 1)

η1φ10/3 + c2

[63

(G1G0

− 1)

η2 + 2η1η3]φ7/3 + c3

(G1G0

− 1)

η2φ5/3 + c4

(G1G0

− 1)

η2φ + c5η2η3

with... ... for A: ... for B: ... for C:

c1 8 (4 − 5ν0) −4 (1 − 5ν0) −4 (7 − 5ν0)c2 −2 4 −2c3 252 −504 252c4 −50

(7 − 12ν0 + 8ν20

)150 (3 − ν0) ν0 −25

(7 − ν20

)c5 4 (7 − 10ν0) −3 (7 − 15ν0) − (7 + 5ν0)η1

(G1G0

− 1)

(7 − 10ν0) (7 + 5ν1) + 105 (ν1 − ν0)η2

(G1G0

− 1)

(7 + 5ν1) + 35 (1 − ν1)η3

(G1G0

− 1)

(8 − 10ν0) + 15 (1 − ν0)

662 Rheol Acta (2008) 47:643–665

Table 5 Parameters used in the composite cylinders model (Hashin and Rosen 1964) and the 2D generalized self-consistent method(Christensen and Lo 1979, 1986; Christensen 1990)

Longitudinal Young’s modulus E11, Eq. 68D1 1 − ν1 D4 2ν20 φ1−φD2

1+φ1−φ + ν0 F1

ν0φE1E0

+ν1(1−φ)ν1φ

E1E0

+1−φD3 2ν21 F2

ν1ν0

F1

Poisson ratio ν12, Eq. 70

L1 2ν1(1 − ν20

)φ + ν0 (1 + ν0) (1 − φ)

L2(1 − ν1 − 2ν21

)φ

L3 2(1 − ν20

)φ + (1 + ν0) (1 − φ)

Transverse shear modulus G23, Eq. 40

A 3c1c2φ (1 − φ)2 +(c2η0 − c3φ3

)(c1η0φ − c4)

B −6c1c2φ (1−φ)2+[c2 (η0−1)−2c3φ3

](c1φ+c4)

C 3c1c2φ (1 − φ)2 +(c2 + c3φ3

)(c1φ + c4)

c1 c2 c3 c4 η0 η1G1G0

−1 G1G0 +η1G1G0

η0−η1 G1G0 η0+1 3−4ν0 3−4ν1

with the index 1 corresponding to the direction of thefiber axes and the indices 2 and 3 corresponding to per-pendicular directions in the transverse plane. Hashinand Rosen (1964) conveniently selected the followingmoduli to describe the mechanical behavior of the sus-pension: the plane-strain bulk modulus

K23 = C22 + C232

, (61)

the transverse shear modulus

G23 = C22 − C232

, (62)

the longitudinal shear modulus

G12 = G13 = C44 , (63)

the longitudinal Young’s modulus

E11 = C11 − 2C212

C22 + C23 , (64)

and C11. In the case of a random arrangement of thefibers across the 23-plane, they found

K23K0

= (1 + 2ν0φ)K1K0

+ 2ν0 (1 − φ)(1 − φ) K1K0 + 2ν0 + φ

, (65)

G12G0

= (1 + φ)G1G0

+ 1 − φ(1 − φ) G1G0 + 1 + φ

, (66)

E11E0

=[(

E1E0

− 1)

φ + 1]

(67)

×[

D1 − D3 F1 + (D2 − D4 F2) E1E0D1 − D3 + (D2 − D4) E1E0

], (68)

and

C11 = E11 + 4ν212 K23 , (69)

where

ν12 = ν13 =L1φ E1E0 + L2ν0 (1 − φ)L3φ E1E0 + L2 (1 − φ)

(70)

is the Poisson ratio for uniaxial stress in the directionof the fiber axes. The parameters D1, D2, D3, D4, F1,F2, L1, L2, and L3 depend on the properties of theindividual phases and on φ and are given in Table 5. Thesame effective properties follow from the 2D gener-alized self-consistent method, which additionally givesthe relative transverse shear modulus G23/G0 fromEq. 40 with the parameters A, B, and C from Table 5.

Rheol Acta (2008) 47:643–665 663

References

Acierno S, Grizzuti N (2003) Measurements of the rheologicalbehavior of a crystallizing polymer by an “inverse quench-ing” technique. J Rheol 47:563–576

Agbossou A, Bergeret A, Benzarti K, Albérola N (1993) Mod-elling of the viscoelastic behaviour of amorphous thermo-plastic/glass beads composites based on the evaluation of thecomplex Poisson’s ratio of the polymer matrix. J Mater Sci28:1963–1972

Albérola ND, Mélé P (1996) Viscoelasticity of polymers filledby rigid or soft particles: theory and experiment. PolymCompos 17:751–759

Aral BK, Kalyon DM (1997) Viscoelastic material functions ofnoncolloidal suspensions with spherical particles. J Rheol41:599–620

Benveniste Y (1987) A new approach to the application of Mori–Tanaka’s theory in composite materials. Mech Mater 6:147–157

Beris AN, Edwards BJ (1994) Thermodynamics of flowing sys-tems with internal microstructure. Oxford University Press,Oxford

Bousmina M (1999) Rheology of polymer blends: linear modelfor viscoelastic emulsions. Rheol Acta 38:73–83

Boutahar K, Carrot C, Guillet J (1996) Polypropylene duringcrystallization from the melt as a model for the rheology ofmolten-filled polymers. J Appl Polym Sci 60:103–114

Boutahar K, Carrot C, Guillet J (1998) Crystallization of poly-olefins from rheological measurements—relation betweenthe transformed fraction and the dynamic moduli. Macro-molecules 31:1921–1929

Bruggeman DAG (1935) Berechnung verschiedener phy-sikalischer Konstanten von heterogenen Substanzen I.Dielektrizitätskonstanten und Leitfähigkeiten der Mis-chkörper aus isotropen Substanzen. Ann Phys Folge 5 24:636–679

Budiansky B (1965) On the elastic moduli of some heterogeneousmaterials. J Mech Phys Solids 13:223–227

Christensen RM (1969) Viscoelastic properties of heterogeneousmedia. J Mech Phys Solids 17:23–41

Christensen RM (1990) A critical evaluation for a class of micro-mechanics models. J Mech Phys Solids 38:379–404

Christensen RM, Lo KH (1979) Solutions for effective shearproperties in three phase sphere and cylinder models. JMech Phys Solids 27:315–330

Christensen RM, Lo KH (1986) Erratum: solutions for effectiveshear properties in three phase sphere and cylinder models.J Mech Phys Solids 34:639

Coppola S, Acierno S, Grizzuti N, Vlassopoulos D (2006) Vis-coelastic behavior of semicrystalline thermoplastic polymersduring the early stages of crystallization. Macromolecules39:1507–1514

D’Avino G, Maffettone PL, Hulsen MA, Peters GWM (2007a)A numerical method for simulating concentrated rigid par-ticle suspensions in an elongational flow using a fixed grid.J Comp Phys 226:688–711

D’Avino G, Maffettone PL, Hulsen MA, Peters GWM (2007b)Numerical simulation of planar elongational flow of con-centrated rigid particle suspensions in a viscoelastic fluid. JNon-Newton Fluid Mech 150:65–79

Doufas AK, McHugh AJ (2001a) Simulation of melt spin-ning including flow-induced crystallization. Part III: quan-titative comparisons with PET spinline data. J Rheol 45:403–420

Doufas AK, McHugh AJ (2001b) Two-dimensional simulation ofmelt spinning with a microstructural model for flow-inducedcrystallization. J Rheol 45:855–879

Doufas AK, McHugh AJ (2001c) Simulation of film blowing in-cluding flow-induced crystallization. J Rheol 45:1085–1104

Doufas AK, Dairanieh IS, McHugh AJ (1999) A continuummodel for flow-induced crystallization of polymer melts.J Rheol 43:85–109

Doufas AK, McHugh AJ, Miller C (2000a) Simulation of meltspinning including flow-induced crystallization. Part I: modeldevelopment and predictions. J Non-Newton Fluid Mech92:27–66

Doufas AK, McHugh AJ, Miller C, Immaneni A (2000b) Simula-tion of melt spinning including flow-induced crystallization.Part II: quantitative comparisons with industrial spinlinedata. J Non-Newton Fluid Mech 92:81–103

Frankel NA, Acrivos A (1967) On the viscosity of a concentratedsuspension of solid spheres. Chem Eng Sci 22:847–853

Fröhlich H, Sack R (1946) Theory of the rheological propertiesof dispersions. Proc R Soc Lond A 185:415–430

Gauthier C, Chailan JF, Chauchard J (1992) Utilisation del’analyse viscoélastique dynamique à l’étude de la crystalli-sation isotherme du poly (téréphtalate d’ethylène) amorphe.Application à des composites unidirectionnels avec fibres deverre. Makromol Chem 193:1001–1009

Graebling D, Muller R, Palierne JF (1993) Linear viscoelasticbehavior of some incompatible polymer blends in the melt.Interpretation of data with a model of emulsion of viscoelas-tic liquids. Macromolecules 26:320–329

Grmela M, Öttinger HC (1997) Dynamics and thermodynamicsof complex fluids. I. Development of a general formalism.Phys Rev E 56:6620–6632

Hashin Z (1962) The elastic moduli of h