Continuum model for the simulation of ﬁber spinning, with ... · PDF filea set of rate equations, ... Schematic illustration of the melt spinning process. ... of diameter D0 at a

J. Non-Newtonian Fluid Mech. 150 (2008) 177–195

Continuum model for the simulation of fiber spinning, withquiescent and flow-induced crystallization

Jan van Meerveld a, Markus Hutter a,∗, Gerrit W.M. Peters b

a Institute of Polymers, Department of Materials, ETH Zurich, 8093 Zurich, Switzerlandb Materials Technology, University of Technology Eindhoven, Eindhoven, The Netherlands

Received 25 April 2006; received in revised form 25 June 2007; accepted 27 October 2007

Abstract

A macroscopic model for the description of low- and high-speed fiber spinning of crystallizing polymers is proposed. The model incorporates airdrag, air cooling, gravity, surface tension and the viscoelastic rheological behavior of the melt. The crystallization dynamics are formulated througha set of rate equations, which describe the quiescent nucleation and growth of spherulites, as well as the flow-induced nucleation and longitudinalgrowth of fibrils. The influence of the crystallization, as well as the glass transition, on the rheological response is accounted for through a viscousstress contribution. The proposed model realistically describes the velocity, temperature, stress, apparent viscosity and crystallinity profile alongthe spinline under different processing conditions. Moreover, using a single set of model parameters a fair description of the experimental velocityand temperature profiles is obtained for both low- and high-speed spinning conditions of nylon 66. For poly(ethylene terephthalate) the modeldescribes high-speed spinning conditions only.© 2007 Elsevier B.V. All rights reserved.

Keywords: Flow-induced crystallization of polymer melts; Fiber spinning; Separation of crystal nucleation and growth; Crystal morphology; Nylon 66; Poly(ethyleneterephthalate)

1. Introduction

The crystallization dynamics of polymers strongly dependson the temperature, cooling rate and the application of a flowfield, see, for example, [1–4]. All aspects are of relevance tohigh-speed fiber spinning, where the melt is quickly cooledfar below the melting temperature and the polymers becomestretched and oriented in the extensional flow field, see, forexample, [5]. The change in crystal morphology with increasingtake-up speed is generally believed to affect the (mechanical)properties of as-spun fibers. The crystallization rate increases byseveral orders of magnitude for relatively large take-up speeds(>4000 m/min for nylon and PET), which is generally assumedto be connected to flow (or stress)-induced crystallization[1,5–8]. A characteristic feature of high-speed fiber spinning ofcrystallizing polymers is the formation of a neck in the filament[5,7,9,10]. The systematic experimental analysis of Haberkornet al. [5] provides physical insight in the neck formation for

∗ Corresponding author. Tel.: +41 44 632 45 44; fax: +41 44 632 10 76.E-mail address: [email protected] (M. Hutter).

nylon 6 and nylon 66. Two observations are of particularimportance, which are illustrated with the help of Fig. 1. First,Haberkorn et al. [5] showed that only in the neck region theextension rate, ε, exceeds the reciprocal of the weight averagedreptation time of the melt. Hence, only in the neck region signifi-cant orientation and stretching of the chains can be expected. Forlow take-up speeds, in general, and in the draw-down regime forhigh take-up speeds the orientation and stretching of the chainsis limited. Second, the onset of the crystallization occurs at theend of the neck region, but the majority of the crystallizationoccurs after the neck is formed. Similar findings are reportedfor PE [9] and PET [7,11]. These two experimental findingssuggest that only at the end of the neck the chains are sufficientlystretched to enhance the crystallization rate by several ordersof magnitude. The formation of a semi-crystalline morphologyallows to bear the external load on the filament, which leads tothe stabilization of the deformation. The necking behavior isfurthermore complicated by the presence of radial variations ofthe temperature and stress, which is reflected by the presence ofthe so-called “sheet–core structure” of as-spun fibers [12,13].These studies of the cross-section of the fiber show that the outer

0377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jnnfm.2007.10.020

178 J. van Meerveld et al. / J. Non-Newtonian Fluid Mech. 150 (2008) 177–195

Fig. 1. Schematic illustration of the melt spinning process. Parameters in the figure are defined in Section 2.

region (the sheet) of the fiber consists of highly aligned crystals,whereas in the middle region (the core) a more spheruliticsemi-crystalline structure is observed. For further details onexperimental and modeling aspects of fiber spinning the readeris referred to Chapter 12 of [1] and [12,14], respectively.

Next, we turn to the model description of flow-inducedcrystallization (FIC) during fiber spinning. The simulation oflow-speed fiber spinning of slowly crystallizing polymers is notaffected by the crystallization dynamics, and for a review werefer to [15]. Based on the analysis of Ziabicki [14]1D radi-ally uniform models for the FIC during fiber spinning wereproposed by Spruiell and coworkers [16,17]. A drawback ofthese models is a purely Newtonian rheological description ofthe melt and consequently neck formation is not predicted. Aviscoelastic description of the melt is incorporated in the mod-els developed by Ziabicki et al. [18], Joo et al. [19], Kulkarniand Beris [20] and Doufas et al. [21]. The models of Ziabickiet al. [18] and Joo et al. [19] are ‘single’ phase models in thesense that the rheology of the semi-crystalline melt remainsviscoelastic, whereas the viscosity (relaxation time) changeswith the degree of crystallinity. However, neck formation isnot reported or predicted. Kulkarni and Beris [20] consider atwo-phase model where the contribution of the amorphous meltand a stiff crystalline phase to the total stress are weightedby means of the degree of crystallinity. A similar approachis more recently taken by Kannan and Rajagopal [22]. Alter-natively, Doufas et al. [21] consider that the total stress hasadditive contributions from the amorphous and crystalline phase,without an explicit weighting of the stress contributions by thecrystallinity. The stress contribution of the crystalline phase is

described by a model for liquid crystalline polymers. Here,it is noted that for large extension rates the stress contribu-tion for liquid crystalline polymers is dominated by a viscousstress [21,23]. In the model of Doufas et al. [21] the relax-ation time (and viscous stress) of the crystalline phase stronglyincreases with the degree of crystallinity. The models devel-oped by Kulkarni and Beris [20], Kannan and Rajagopal [22]and Doufas et al. [21] are able to predict neck formation. Thisobservation suggests that the contribution of the amorphous andcrystal phase to the mechanical behavior needs to be reflectedby two individual contributions to the stress in the momentumbalance.

In FIC models for fiber spinning the Nakamura equation[24,25] is generally used to describe the crystallization dynam-ics, see for example [16,17,19–21]. However, the use of theNakamura equation is inappropriate as, firstly, the ‘isokinetic’assumption is not satisfied during FIC, and, secondly, a changein number density of crystals cannot be described [2]. For thesereasons, FIC dynamics needs to be formulated through a set ofrate equations, which allow to describe the enhanced nucleationof spherulites [4,26,27], or the formation of the shish-kebab mor-phology [2,28–30]. The description of the degree of crystallinityby means of a set of rate equations is identical [2,31,32] tothe crystallization theory of Kolmogorov [33], Avrami [34–36],Johnson and Mehl [37] and Evans [38] and has two advan-tages compared to the Nakamura equation. First, nucleation andgrowth dynamics are separated. Second, the formation of differ-ently shaped crystals can be incorporated naturally [2,30,32]. Inprinciple, the use of the rate equations is an attractive approach todescribe FIC on a continuum level, as changes in the crystalliza-

J. van Meerveld et al. / J. Non-Newtonian Fluid Mech. 150 (2008) 177–195 179

tion dynamics under general flow conditions can be incorporatedin a more transparent and considerably more detailed fashioncompared to the Nakamura equation.

Based on a non-equilibrium thermodynamics analysis of theFIC of polymer melts [39] we propose a continuum model todescribe high-speed fiber spinning, which is in line with thephysical understanding gained by the experiments of Haberkornet al. [5]. The major difference with previous models lies in thedescription of the crystallization dynamics. In the present modelwe consider that the flow-induced nucleation and growth of fib-rils occurs due to the stretching of the amorphous chains, inaddition to the quiescent nucleation and growth of spherulites.To implement this approach we use a set of rate equations tomodel the evolution of the morphology during fiber spinning. Inprevious models the crystallization dynamics are described bymeans of a Nakamura equation and consequently one can, firstly,not predict a morphology, secondly, not distinguish betweennucleation and growth dynamics, and thirdly, not include theformation of differently shaped crystals.

2. Balance equations

The process of melt spinning a single filament is illustrated inFig. 1. The polymer melt exits the spinneret through a capillaryof diameter D0 at a mass throughput W and a temperature T0.The fiber is drawn continuously at a take-up velocity vL by thetake-up role, which is located a distance L from the spinneret.After the polymer leaves the spinneret it initially swells, but aftera few diameters downstream the fiber continuously reduces indiameter due to the rotation of the take-up role. For high take-up speeds a neck is formed, which is a localized reduction indiameter to a constant value. The fiber is cooled by a quenchflow of temperature Ta and velocity vc in the regime z1 < z < z2.Downstream of the extrudate swell, the downward velocity vz isapproximately radially uniform at all positions z, because of thesmall variation in radius with the distance from the spinneret andthe large radius of the longitudinal curvature compared to thefilament radius. Hence, after the extrudate swell the so-calledthin-filament approximation may be applied.

2.1. Consistency with non-equilibrium thermodynamicprinciples

The model proposed in the present paper is based on themacro- and microscopic evolution equations resulting from anon-equilibrium thermodynamic analysis of the FIC of poly-mer melts [39], within the GENERIC framework [40–42]. Thisnon-equilibrium thermodynamic analysis provides a consistentcoupling between the influence of the mass transfer betweenthe melt and crystalline phase and the chain configuration onthe thermodynamic driving force for crystallization and, viceversa, the influence of crystallization on the mass transfer andthe chain configuration of the amorphous part of the chains.The final thermodynamic driving force is similar to that orig-inally derived by Flory [43], but guarantees that the influenceof the chain configuration reduces to zero provided the chainconfiguration is identical to the equilibrium configuration. Com-

pared to previous thermodynamic analyses of FIC [43,44], it isconvenient to consider a configuration tensor of the amorphouspart of the chain rather than to consider a configuration tensorfor the entire polymer chain, i.e., amorphous plus crystallineparts [44]. For this reason a term describing the change in chainconfiguration due to crystallization is incorporated into the evo-lution equation of the chain configuration of the amorphous partof the chain, see Eq. (6). However, the relevance of the chainconfiguration on the thermodynamic driving force is negligibleduring high-speed fiber spinning. The reason for this observa-tion is that for high-speed fiber spinning of PET and nylons thecrystallization typically occurs, roughly, 100◦ below the meltingtemperature of the particular polymer [5,7,45], and consequentlythe additional contribution from the chain extension to the totalthermodynamic driving force for the crystallization dynamicsis minimal. This observation indicates that the main effect offlow on the crystallization dynamics is reflected by a changein the kinetic aspects for the crystallization dynamics with thechain extension. These kinetic aspects are reflected by a changein the characteristic timescale for nucleation and growth, whichis practically unconstrained by non-equilibrium thermodynamicformalisms. For this reason we omit a detailed discussion of thenon-equilibrium thermodynamic analysis in the present paperand refer the interested reader to Chapter 7 of [39].

In view of the relation between flow-induced crystallizationand thermodynamics we wish to point out three main discrep-ancies between our treatment and the model for fiber spinningproposed by Kannan and Rajagopal [22]. First, in their model therate of crystallization (see their Eq. (27)) does not automaticallyvanish for the conditions at which crystallization is initiated (seetheir Eq. (22)). Second, using their material parameters it seemsthat the equilibrium crystallization temperature as obtained fromtheir Eq. (22) is approximately 150 K below the experimentalvalue. And third, the Nakamura equation with exponent m = 1 isused in [22], which corresponds to the one-dimensional growthof heterogeneously nucleated crystals. In contrast, in our treat-ment presented in the following the rate of crystallization departsfrom zero at the initiation of crystallization, the experimentalvalue for the equilibrium melting temperature is correctly repro-duced, and a hierarchy of rate equations is used to model theeffect of flow both on crystal nucleation and growth.

2.2. Macroscopic balance equations

As mentioned above, below the extrudate swell regime thethin-filament approximation may be applied, where the flow fieldis locally approximated as an uniaxial extensional flow. Here,the velocity profile is assumed to be independent on the radialposition in the filament, whereas the circumferential variationsare absent due to the axisymmetric configuration of the fiberspinning process. As a result, the mass balance equation, themomentum balance and the temperature equation can be radiallyaveraged over the cross-section of the filament, see [15,46–48].

We consider that the total density of the crystallizing meltremains constant during the crystallization, which implies theassumptions that, firstly, the melt and crystalline phase areincompressible and, secondly, the density of the melt and crys-


talline phase are identical. Based on these assumptions the totalmass balance reduces to

W = ρvzA = ρvz

π

4D2, (1)

with A and D the cross-section and diameter of the filament atposition z, respectively.

The momentum balance equals

Wdvz

dz= d

dz[A(τzz − τrr)] − Cf ρav

2z + ρgA + 1

2πσ

dD

dz. (2)

The term on the left hand side corresponds to the inertia of afluid element. The terms on the right hand side correspond to thetensile force on the fiber at position z, the air drag exerted on thefiber, the gravitational force and the interface tension betweenthe filament and the surrounding air, respectively. Here, τzz, τrr

are the longitudinal and radial component of the stress tensorτ, Cf is the air drag coefficient, ρa the density of air, g is theacceleration due to gravity and σ the interfacial tension. Theobservation that the isotropic fluid pressure is absent in Eq. (2)results from a subtle radial averaging procedure of the z and rcomponent of the momentum balance, in conjunction with theboundary conditions at the surface of the filament [46–48].

The temperature (or energy) equation equals

ρCpvz

dT

dz=− 4

Dh(T − Ta)+(τzz−τrr)

dvz

dz+ρ�h(T )φ∞vz

dφ

dz.

(3)

The terms on the right hand side correspond to the heat exchangebetween the fiber and the quench flow of temperature Ta, the‘viscous heat dissipation’ and finally the release of latent heat isdue to the crystallization. Here, Cp is the heat capacity at con-stant pressure per unit mass, h is the heat transfer coefficient,�h(T ) the latent heat per unit mass at temperature T, and φ∞is the final mass fraction of the crystalline phase. The influenceof lateral and radial heat conduction is neglected in the tem-perature equation given in Eq. (3) for simplicity. The use of aradially averaged temperature is restricted to low values of theBiot number, i.e. the heat conductivity of the polymer is muchlarger compared to h D [49]. The requirement of a low Biot num-ber is not consistent with realistic fiber spinning conditions, andleading order radial variations in the temperature are present,see, for example [19,49] and references therein. As a result, thepresent model is a simplification of reality and, for example, theeffect of a ‘sheet–core’ structure [12,13] on the neck formationis not described.

In line with the thin-filament approximation the flow fieldis considered to be a homogeneous axisymmetric uniaxial flowfield, and assuming a constant total density, the velocity gradient,in cylindrical coordinates, equals

∇v =

⎡⎢⎢⎢⎢⎢⎢⎣

dvz

dz0 0

0 −1

2

dvz

dz0

0 0 −1

2

dvz

dz

⎤⎥⎥⎥⎥⎥⎥⎦

, (4)

with dvz/dz the local extension rate. For axisymmetric reasonsthe stress tensor τ adopts the form:

τ =

⎡⎢⎣

τzz 0 0

0 τrr 0

0 0 τθθ

⎤⎥⎦ , (5)

with τrr = τθθ .

2.3. Rheological description

The rheological behavior of the amorphous part of the melts isdescribed by a micro-rheological model based on the dimension-less configuration tensor of the amorphous part of the chain, A,as derived in Chapter 7 of [39]. Here, we briefly describe the indi-vidual contributions to the dynamics in A. The components of Aare made dimensionless with respect to the equilibrium configu-ration of a completely amorphous chain, A = 〈rara〉/(NKl2K/3),with 〈〉 denoting an average over the configuration space, ra isthe end-to-end vector of the amorphous part of the chain, NK isthe total number of Kuhn segments per chain and lK is the Kuhnlength. Considering upper convective Maxwell behavior for thechange of A in the flow field, the Giesekus mobility matrix forthe relaxation dynamics, finite extensibility and the reduction ofra due to crystallization we arrive at the dynamic equation forA of the form [39],

DADt

= (∇v)T · A + A · (∇v)

− 1

τrelax((1 − αG)I + αGA)(K(A, ms)A − I)

−1 + f (A)

1 − msAφ∞

Dφ

Dt, (6)

K(A, ms) = 1

1 − ms

1

1 + 1/((1 − ms)NK) − tr[A]/(3NK(1 − ms)2)

,

(7)

with D/Dt = ∂/∂t + v · ∇ the material derivative, τrelax therelaxation time of the melt, αG the Giesekus mobility param-eter, 0 ≤ αG ≤ 1, ms the mass fraction of the crystalline phase,ms = φ∞φ with φ the relative crystallinity and φ∞ the finalmass fraction of crystalline material in a melt crystallized sam-ple. The change of A due to the crystallization dynamics isaccounted for by the last term in Eq. (6), where f (A) is definedas f (A) = (3/2)(A/tr[A] − I/3) : (A/tr[A] − I/3). Provided Ais equal to the equilibrium configuration, i.e. A = Aeq = (1 −ms)I, f (A) = 0 and the change in A is identical to the changeof Aeq due to crystallization. Alternatively, for a large extensionthe reduction of A increases and approaches the limiting casewhere the chains are in the fully extended state. The influenceof the finite extensibility and the change in the equilibrium con-figuration of the amorphous part of the chains (of (1 − ms)NKKuhn segments) are accounted for in the first and second factorof the spring constant K(A, ms), respectively.

The stress tensor associated to the amorphous phase, τa, takesthe form

τa = G(K(A, ms)A − I), (8)


with G the shear modulus of the melt. Under non-crystallizingconditions the total stress is identical to that of the amorphousphase only, i.e. τ = τa for T > Tm0 with Tm0 the quiescent melt-ing temperature.

In addition to the stress from the amorphous melt, τa, weconsider a purely viscous stress contribution:

τgc = ηgc[∇v + (∇v)T], (9)

which accounts, firstly, for a viscous stress contribution observedfor the mechanical behavior of polymer materials in the vicin-ity of the glass transition and, secondly, in a phenomenologicalfashion for the influence of the crystallization on the rheologicalbehavior. These two contributions are reflected by the expressionof the ‘glassy-crystalline’ viscosity ηgc based on the followingarguments.

The total stress of polymers deformed at temperatures aroundthe glass transition temperature, Tg, has a contribution fromthe deformation of the chains, as for a polymer melt or rubber,and a viscous contribution, as for simple fluids [50]. Actually,the viscous contribution dominates the mechanical response atlarger extension rates for temperatures close to Tg, which isreflected by a (pronounced) failure of the stress-optical-rule[50–53]. The variation of the viscous stress with temperaturefollows the time–temperature-superposition principle, whichis well described by the William-Landel-Ferry (WLF) equa-tion [50,53]. These experimental observations suggest that, asa first approximation, ηgc ∼ ηg0(Tref)aWLF

T (Tref), with ηg0 the‘glassy viscosity’ at the reference temperature Tref and aWLF

T thetime–temperature shift factor according to the WLF equation,log aWLF

T = −c1(T − Tref)/(c2 + T − Tref).Now we turn to the influence of crystallization on the

rheology. Experiments indicate that the viscosity and linearviscoelastic behavior of a crystallizing melt changes, rather dras-tically, during different stages of the crystallization process, see,for example, [54–57]. However, to date a general quantitativedescription is not yet available [26,58]. Therefore, we take aphenomenological approach and consider that the viscous con-tribution increases according to the relationship ηgc ∼ exp(Fφ),with F a phenomenological parameter. Combining these twocontributions we arrive at the expression:

ηgc = ηg0(Tref)aWLFT (Tref) exp(Fφ), (10)

which has the following desirable features. First, for φ = 0 onerecovers the ‘glassy’ viscosity. Second, for temperatures wellabove Tg the viscosity is negligible compared to the viscosityof the melt, as it should, while it has approximately the sametemperature dependency as predicted by the time temperatureshift factor according to the Arrhenius law. Third, the influenceof the crystallization results in an instant, and large, increase inηgc. This last feature is essential to account for the influence ofthe crystallinity on ‘stabilizing’ the deformation of the filament,which lead to the formation of the ‘neck’ for high take-up speeds.

It may be noted that from a rheological point of view theintroduction of τgc is in practice similar to the representationof the crystalline phase by liquid crystalline polymers in themodel of Doufas et al. [21,59]. There, the viscous stress con-

tribution for liquid crystalline polymers is, firstly, of the order(10−3 − 10−2)η0(T ) exp(Fφ)(∇v + (∇v)T)/2, with η0(T ) thetemperature-dependent zero shear rate viscosity of the melt, and,secondly, dominates for large extension rates [23]. As a result,the influence of the crystallization on the rheology in the presentmodel and that proposed by Doufas et al. [21] are similar inpractice, although the physical motivation differs.

Under crystallization conditions, i.e. T < Tm0, the total stresshas two additive contributions and equals τ = τa + τgc. Hence,in the present model we consider a single momentum balance,Eq. (2), with a stress tensor which has two additive contributionsassociated to the stress in the amorphous melt, τa, and a viscousstress, τgc, reflecting the effect of the crystallization and theincrease of viscous forces when the melt approaches the glasstransition temperature.

2.4. Crystallization dynamics

In the present model the total crystallization dynamics isbased on the contributions from quiescent, and flow-inducedcrystallization. We formulate these two contributions in termsof a set of rate equations.

The quiescent crystallization is based on the nucleation andthree-dimensional growth of spherulites. This contribution isdescribed by the following set of rate equations [2,31,32]:

DΨq0

Dt= Grq(T )A′Ψq

1 , (11)

DΨq1

Dt= Grq(T )A′Ψq

2 , (12)

DΨq2

Dt= Grq(T )A′Ψq

3 , (13)

DΨq3

Dt= 8πζ, (14)

where Ψq0 is the total volume of spherulites per unit volume,

Ψq1 is the total (growth) surface of the spherulites per unit vol-

ume, Ψq2 equals 8π times the sum of the radii of spherulites per

unit volume, Ψq3 corresponds to 8π times the number density

of nuclei, Grq(T ) is the quiescent radial growth rate, ζ is thenucleation rate due to thermal and athermal nucleation. Hetero-geneous nuclei are reflected by a non-zero initial value of Ψ

q3 ,

Ψq3,0. All parameters Ψ

qi are ‘unrestricted’ quantities in the sense

that the influence of impingement of crystals is not accounted for.The incorporation of the term A′ in Eqs. (11)–(13) is of impor-tance to guarantee that the present model is compatible withthe non-isothermal FIC model developed within the GENERICframework in [39]. Based on this non-equilibrium thermody-namic analysis the radial growth rate Gr equals Gr = RA =R(Amacro + Amicro), withR a rate constant andA the total ther-modynamic driving force for crystallization. The terms Amacroand Amicro express the contributions to A associated to thechange in the macroscopic variables (e.g., temperature and pres-sure) and the (microscopic) chain configuration, respectively.Under quiescent conditions Amicro = 0 and consequently Gr =Grq = RAmacro. Hence, using a (phenomenological) expression


for the variation of Grq with temperature, we can express theradial growth rate under general conditions as Gr(T, A, ms) =Grq(T )(1 + Amicro/Amacro) = Grq(T )A′. In the present studywe consider heterogeneous nucleation only, i.e. ζ = 0 and Ψ

q3 =

Ψq3,0.

The expressions for Amacro and Amicro equal [39]

Amacro ≈ ρ(�h(T )/T − �h(Tm0)/Tm0), (15)

Amicro = ρNAkB

Me

[3

2(1 − ms)− 3NK

2ln[h(A, ms)]

+ 1

h(A, ms)

(3

2(1 − ms)− tr[A]

(1 − ms)2

)

−(1+f (A))

(3

2(1−ms)− 1

2h(A, ms)

tr[A]

(1−ms)2

)],

(16)

h(A, ms) = 1 + 1

NK(1 − ms)− tr[A]

3NK(1 − ms)2 , (17)

with Tm0 the (quiescent) melting temperature and Me the molec-ular weight per entanglement. For Amacro we have adopted the(usual) assumption that the difference in the specific Gibbsfree energy of the fluid and solid phase may be expressed asμf − μs ≈ �h(T ) − �h(Tm0)T/Tm0. The lengthy expressionfor Amicro is a direct result of considering the configurationentropy potential for a finite extensible chain instead of thatcorresponding to a Gaussian chain [39].

It is generally believed that a fibrillar morphology developsduring high-speed fiber spinning [60]. In this case the morphol-ogy consists of fibrils, which are oriented parallel to the flowdirection (fiber axis). In a small angle X-ray scattering study,Murthy et al. [61,62] report that for as-spun nylon fibers the fib-rils have a length of 100–300 nm and a diameter of 5–20 nm.Visual inspection of the TEM analysis of fast drawn PB-1 [63]and iPP [64,65] films from the melt reveal a fibril length ofabout 250 nm. Hence, based on the present experimental data,one finds that after (different) fast extensional flows the averagelength and diameter of the fibrils are approximately similar fordifferent polymers.

Based on these experimental observations we assume that thefibrillar morphology develops due to the flow-induced nucle-ation and longitudinal growth of fibrils. On the continuum levelthe development of the fibrils is represented by the flow-inducednucleation and longitudinal growth of cylinders of constantcross-section, Afibril = πr2

fibril. In this case the fibril formationcan be described by the following two rate equations:

DΨF0

Dt= Afibrilglg(A, T, ms)Ψ

F1 , (18)

DΨF1

Dt= gng(A, T, ms), (19)

where ΨF0 is the unrestricted volume of fibrils per unit volume

and ΨF1 equals the number density of fibrils per unit volume. The

constants gn and gl describe the sensitivity of the flow-inducednucleation and longitudinal growth of the fibrils, respectively,

on the chain configuration at temperature T, which is describedby the parameter g(A, T, ms). Hence, gng(A, T, ms) is the nucle-ation rate of fibrils and glg(A, T, ms) is the longitudinal growthrate (including both ends). The parameter g(A, T, ms) accountsfor both thermodynamic and kinetic contributions to the fib-ril formation. Unfortunately, the influence of temperature andthermodynamic driving force on the flow-induced nucleation isin general not resolved both experimentally and theoretically.For this reason, we adopt a phenomenological approach andassume that the change in thermodynamic driving force on theflow-induced nucleation and longitudinal growth of the fibrils issimilar to that observed for the radial growth rate, leading to

g(A, T, ms) ∼(

Grq(T )

Gmaxrq

)A′, (20)

with Gmaxrq the maximum quiescent growth rate. This Ansatz for

the thermodynamic contribution to g(A, T, ms), firstly, indicatesthat the temperature dependence on the nucleation and longitudi-nal growth of the fibrils is similar to the radial growth rate underquiescent conditions and, secondly, guarantees compatibility ofthe present model with non-equilibrium thermodynamics [39].However, the increase in A′ due to chain stretching proved tobe insufficient to describe the flow-induced fibril formation, asdiscussed before. Therefore, we consider that the chain config-uration also enhances the kinetic contribution to g(A, T, ms),which is proportional to the second invariant of configurationtensor of the amorphous part of the chain, J2,

J2(A) = 1

2

(A − 1

3tr[A]I

):

(A − 1

3tr[A]I

). (21)

The parameter J2 has been used before to describe the flow-induced nucleation and growth of ‘shish’ in the FIC modelproposed by Zuidema et al. [30]. It should be noted that theparameter J2 has a stronger influence of chain stretching, J2 ∼tr[A]2, compared to the Legendre polynomial P2, P2 ∼ tr[A],which describes the average orientation of the segments. Froma kinetic perspective one may envisage that the relative orienta-tion of the Kuhn segments is of relevance for the fibril formation.This feature may be better reflected by the stronger influence ofchain stretching in the parameter J2 compared to the P2. Thissuggestion is partly supported by experiments, as the crystal-lization dynamics of PET is found to scale with a power of P2,namely P4

2 [66], which may correspond to having both the nucle-ation rate and the longitudinal growth of fibrils proportional toJ2 ∼ tr[A]2. However, further investigations are needed to gaina better microscopic understanding of the influence of chainstretching on the flow-induced crystallization.

Combining the kinetic and thermodynamic contribution wearrive at the following expression for g(A, T, ms),

g(A, T, ms) = J2(A)

(Grq(T )

Gmaxr

)A′. (22)

Finally, we need to account for the impingement of the crys-tals, and its effect on φ. At present no rigorous theory existswhich describes the impingement of differently shaped and ori-ented crystals. Here, we follow Eder and Janeschitz-Kriegl [2]


and use the expression according to Avrami to relate the truevolume fraction of crystals, φ, to the total unrestricted volumefraction of spherulites and fibrils, φ = 1 − exp(−Ψ

q0 − ΨF

0 ).(This expression is in accordance with the Boolean grain modelfor randomly positioned and randomly oriented convex grains.)Using Eqs. (11) and (18) the dynamic equation for φ equals

Dφ

Dt= (1 − φ)(Grq(T )A′Ψq

1 + glg(A, T, ms)ΨF1 ). (23)

In summary, the total crystallization dynamics are describedby the time evolution of Ψ

q1 , Ψ

q2 , Ψ

q3 , ΨF

1 and φ, which areexpressed through Eqs. (12)–(14), (19) and (23). Here, the con-tribution from the growth of spherulites and the development offibrils to the total unrestricted volume of crystals is additive. Themagnitude of Ψ

q0 is primarily determined by the non-isothermal

conditions experienced till a specific time, as A′ is only weaklyinfluenced by chain stretching during fiber spinning conditions.While the effect of the temperature on the fibril formation issimilar to that of the spherulites, chain orientation and stretchingresult in a pronounced increase in the nucleation and longitudi-nal growth rate of the fibrils due to kinetic effects. Therefore,it depends on the transient non-isothermal conditions and chainconfiguration whether the crystallinity is predominantly deter-mined by the development of spherulites or fibrils.

In Section 5.1 the relative contribution from quiescent andFIC is analyzed, and the average length of the fibrils is com-pared to experimental data. In order to do so we calculate thecontribution from FIC from simultaneous, independent, simu-lations, where we consider that the fibril formation is the onlycrystallization process. In this case the restricted volume fractionof fibrils follows from the equation:

DφF

Dt= (1 − φF)Afibrilglg(A, T, ms)Ψ

F1 , (24)

used in conjunction with Eq. (19). Separately, to calculate thetrue (restricted) number density of fibrils, ΨF

1r (note difference toΨF

1 ), we need to account for the ‘swallowing’ of potential fibrilnuclei [2], which results in a modification of Eq. (19), namely,

DΨF1r

Dt= (1 − φF)gng(A, T, ms). (25)

Based on φF and ΨF1r the average length of a fibril, Lfibril, is

estimated according to the relationship Lfibril = φF/(AfibrilΨF1r).

In the present FIC model, the constants gn and gl need to befitted to describe experiments. In this respect it may be notedthat the crystallinity, φ, depends only on the magnitude of theproduct gngl. Alternatively, specification of gn and gl determinesthe morphology in terms of the number density of fibrils and theaverage length per fibril, Lfibril = φF/(AfibrilΨ

F1r).

3. Simulation procedure

For the simulations we first formulate the general dynamicequations for the rheological behavior and crystallizationdynamics for the stationary, axisymmetric conditions of fiberspinning. Under stationary conditions and the assumptions asso-ciated to the thin-filament approximation the material derivative

D • /Dt reduces to vzd • /dz in Eqs. (6), (12)–(14), (19) and(23). Due to the axisymmetric flow field one has Arr = Aθθ ,and consequently only Azz and Arr need to be determined. As aresult, the dynamic equations for Azz and Arr read,

dAzz

dz= 2

vz

Azz

dvz

dz− 1

vzτrelax(1−αG+αGAzz)(K(A, ms)Azz−1)

− (1+f (A))

1−msAzzφ∞

dφ

dz, (26)

dArr

dz= − 1

vz

Arr

dvz

dz− 1

vzτrelax(1 − αG + αGArr)(K(A, ms)Arr − 1)

− (1 + f (A))

1 − msArrφ∞

dφ

dz. (27)

Similarly, the dynamic equations for the crystallizationdynamics can be formulated for stationary conditions and thethin-filament approximation.

The contributions to the extensional stress from τa andτgc equal τa,zz − τa,rr = GK(A, ms)(Azz − Arr) and τgc,zz −τgc,rr = 3ηgcdvz/dz. Substituting these expressions into Eqs. (2)and (3) and using Eqs. (26) and (27), the momentum balance andtemperature equation can be cast into the form that dvz/dz anddT/dz are functions of vz, Azz, Arr and T for T > Tm, and asa function of vz, Azz, Arr, T, φ, Ψ

q1 , Ψ

q2 , Ψ

q3 , ΨF

1 , for T < Tm.The resulting set of coupled ordinary differential equations aresolved as an initial value problem combined with a shootingmethod.

As initial conditions for vz and T we use the cross-sectionaveraged velocity vz0 = W/(ρπD2

0/4) and the temperature, T0,at the spinneret. For the fiber spinning conditions under con-sideration here, the relaxation time τrelax of nylon 66 and PETis of the order of 1–10 ms when the melt leaves the spinneret[5,12,14]. As the extension rate in the extrudate swell regionand the beginning of the draw-down region are of the order of0.1 s−1, Newtonian behavior can be expected at small distancesfrom the spinneret. In the case of stationary Newtonian flowconditions the initial values for Azz and Arr are related to eachother according to the expression [21]:

Arr,0 − 1

Azz,0 − 1= −0.5, (28)

which follows from solving Eq. (6) for a stationary homoge-neous flow field in the Newtonian regime, i.e. A ≈ I. As a result,we can use Azz,0 as the shooting parameter in the present study,and Arr,0 follows from Eq. (28).

Following Doufas et al. [21], we consider a switch at T = Tm0as τzz − τrr = τa,zz − τa,rr for T > Tm0 and τzz − τrr = (τa,zz −τa,rr) + (τgc,zz − τgc,rr) for T < Tm0. At this ‘switch point’, theinitial value for dvz/dz is based on the simulation results for apure amorphous melt at T = Tm0, the presence of heterogeneousnuclei is reflected by a non-zero initial value for Ψ

q3 , Ψq

3 = Ψq3,0,

whereas for all other crystallization parameters the initial valueis equal to zero, φ = ΨF

1 = Ψq1 = Ψ

q2 = 0.


4. Model, material and physical parameters

4.1. General material and physical parameters

The heat capacity at constant pressure, Cp, is taken as themass averaged value over the fluid and solid phases,

Cp = Cpsφ∞φ + Cpf(1 − φ∞φ). (29)

The variation of the heat capacity per unit mass at constant pres-sure of the solid and fluid phases, Cpf and Cps, respectively, withtemperature is given by the expression [21]

Cpi = C1pi + C2

piT, (30)

with the parameters Cjpi given in Table 1 and T is in ◦C.

The latent heat �h(T ) is identical to the difference in theenthalpy per unit mass of the fluid and solid phases at tempera-ture T [21]:

�h(T ) = �h(0) + (C1pf − C1

ps)T + (C2pf − C2

ps)T2

2, (31)

with T in ◦C and �h(0) is the latent heat at the reference tem-perature of T = 0 ◦C.

The expression for the drag coefficient Cf equals

Cf = 0.27 Re−0.61D . (32)

Here, ReD is the diameter-based Reynolds number, defined asReD = vzD/νa with νa the dynamic viscosity of air. The pref-actor ‘0.27’ is taken from [67,68] and kept constant for allsimulations.

The heat transfer coefficient h in Eq. (3) is given by [69]:

NuD = hD

ka= Nu0

D Re1/3D

[1 +

(8vc

vz

)2]1/6

, (33)

with NuD the diameter-based Nusselt number, ka the thermalconductivity of air, and vc the quench air velocity. The magni-tude of the prefactor Nu0

D is taken equal to 0.42 [69], or statedotherwise.

The variation of ka and νa with temperature are given by therelationships [21]:

ka = 1.88 × 10−4 T 0.866f [J/m s K], (34)

νa = 4.12 × 10−9 T 2.5f

Tf + 113.5[kg/m3], (35)

where the fiber surface temperature Tf is in K and defined as thearithmetic mean of T and Ta, Tf = (T + Ta)/2.

4.2. Material parameters for nylon 66

For nylon 66 the zero shear rate viscosity, η0(T ), equals, [59]

η0(T ) = η0(T = 280 ◦C) aT (T ), (36)

aT (T ) = exp

[Ea

R

(1

T− 1

553.15

)], (37)

with aT (T ) the time–temperature shift factor with respect toa reference temperature of T = 280 ◦C and Ea the activationenergy of Ea = 56 kJ/mol [21]. Following Doufas et al. [21,59]the relaxation time of the melt equals τrelax = η0(T )/G with Gthe shear modulus of the melt given in Table 1.

For nylon 66 no data are available for the WLF parameters andthe magnitude of the ‘glassy’ viscosity, ηg0. For nylon 66 we con-sider that ηg0 = 0.001η0 at T = 280 ◦C and the WLF parametersin Table 1 represent that the glassy contribution becomes relevantfor T > Tg + 50 K. This choice is motivated by the quantitativefindings for PET, which are discussed in the next section. In thepresent paper, the WLF parameters for nylon 66 are determinedby fitting the velocity profile of run S12, as the influence of crys-tallization on the low take-up speeds data is minimal for run S12.

The motivation to describe the variation in τrelax and ηg0 withtemperature by the Arrhenius law and the WLF equation, respec-tively, is based on the following. It is known that the variation inrelaxation times with temperature is accurately described by theArrhenius law at high temperature, whereas the WLF equationis only appropriate for Tg < T < Tg + 100 K, see, for exam-ple, [70,71]. In order to realistically and accurately describethe influence of the temperature on the rheological behavior

Table 1Physical parameters for nylon 66 and PET

Parameter Symbol Unit Nylon 66 PET

Shear modulus G kPa 110 [21] 95.2 [97]Giesekus mobility parameter αG – 0.75 0.75Number of statistical links NK – 200 [21] 200 [45]Final degree of crystallinity φ∞ – 0.4 [59] 0.3 [45]Quiescent melt temperature Tm0

◦C 265 [14] 280 [14,98]Glass transition temperature Tg

◦C 45 [14] 85Heat capacity parameter Cpf C1

pf J/kg K 2100 [21] 1356 [45]Heat capacity parameter Cpf C2

pf J/kg K2 1.9 [21] 2.36 [45]Heat capacity parameter Cps C1

ps J/kg K 1240 [21] 1047 [45]Heat capacity parameter Cps C2

ps J/kg K2 0.83 [21] 2.93 [45]Latent heat parameter �h(0) kJ/kg 209.2 [21] 125 [45]Interfacial tension σ N/m 0.036 [14] 0.035 [14]WLF parameter c1 – 1.8 1.5 [78]WLF parameter c2 K 295 237.8 [78]


in the draw-down and neck regime we use the Arrhenius lawto account for the change in τrelax with temperature. In con-trast, for Tg < T < Tg + 100 K the variation in ηg0 with T is ofimportance, which we therefore describe by means of the WLFequation. An accurate description for the influence of ηg0 withtemperature at high temperature is not of relevance, as ηg0 ismuch smaller than the zero shear rate viscosity in this tempera-ture range. Another, practical, reason for this choice is that forcrystallizing polymers the activation energy Ea is well docu-mented, whereas WLF parameters are scarce, or not available[71].

To describe the quiescent radial growth rate, Grq(T ), weconsider the expression according to Lauritzen and Hoffmann[72–74]:

Grq(T ) = Gr0 exp

(U∗

R(T − (Tg − 30))

)

× exp

(K

T (Tm0 − T )

Tm0 + T

2T

). (38)

The parameters U∗ and K are obtained in the usual way, see,for example, [1], using experimental data for the growth rate ofnylon 66 reported in the literature [75–77]. The value of Gr0 isthen taken such that the maximum growth rate is identical to thatreported in Refs. [21,75]. As a result, Gr0 = 3.32 × 10−4 m/s,U∗ = 1.04 × 103 J/mol and K = 9.50 × 104 K2. Due to thelack of experimental data on the nuclei density of nylon 66we consider the development of spherulites from heteroge-neous nuclei, described by a constant value of Ψ

q3 = Ψ

q3,0 =

3.33 × 1016 m−3[21].

4.3. Material parameters for PET

For PET the zero shear rate viscosity, η0(T ), equals [45]

η0(T ) = η0(T = 280 ◦C) aT (T ), (39)

aT (T ) = exp

[7(280 − T )

T + 45

], (40)

with T the temperature in ◦C. As for nylon 66, the relaxationtimes equals τrelax(T ) = η0(T )/G.

For PET the ‘glassy’ viscosity ηg0 is estimated based onthe stress-optical measurements reported by Ryu et al. [52]. AtT = Tg = 85 ◦C the viscous stress contribution is about 1 MPafor a constant uniaxial extension rate of 0.25 s−1, giving a ‘glassyviscosity’ of ηg0 ≈ 1.3 × 106 Pa s. Using the WLF parametersfor PET reported in Ref. 12 of [78], see Table 1, ηg0 = 0.16 Pa sat T = 280 ◦C. As a result, ηgc is about a factor of 0.001 smallercompared to the viscosity of the melt for T = 280 ◦C, andbecomes important when the temperature approaches Tg alongthe spinline.

The quiescent radial growth rate for PET is described accord-ing to the phenomenological relationship by Ziabicki [14],

Grq(T ) = Gmaxrq exp

[−4 ln 2

(T − Tmax)2

D2av

], (41)

with Gmaxrq = 7.45 × 10−8 m/s, Tmax = 180 ◦C and Dav =

64 ◦C [14,79,80]. Under quiescent conditions the nuclei densityof PET is essentially constant [80], suggesting that growth ofspherulites primarily results from heterogeneous nuclei. Basedon these findings we consider that Ψ

q3 = Ψ

q3,0 = 1016 m−3[80].

4.4. Other model parameters

In the present study the value of the Giesekus mobilityparameter, αG, is taken equal to 3/4. In this case the stationarynon-linear extensional viscosity shows weak hardening behav-ior for τrelaxε ≈ 1 and thinning behavior for τrelaxε > 1 [81],representative for linear polymer melts [82]. The rheologicalbehavior for this value of the Giesekus mobility parameter isrepresentative for the polymers considered in this work, as thenylon 66 and PET are condensation polymers with a relativelynarrow molecular weight distribution which generally have amoderate degree of strain hardening and reveal thinning of theextensional viscosity.

For low-speed spinning conditions of nylon 66, quiescentcrystallization prevails and the velocity profile reaches a con-stant value at temperatures far above Tg. As the contributionfrom FIC is negligible, but quiescent crystallization is dominant,the velocity profile at low take-up speeds are fitted by adjustingthe magnitude of F only. The magnitude of F is kept constant for

Table 2Processing conditions of nylon 66 melt, nomenclature as in [59]

Test D0 (mm) vL (m/min) W (g/min) T0(◦C) L (m) z1 (m) z2 (m) vc (m/s) Ta(◦C)

S1 0.216 5700 2.22 290 1.60 0.04 1.16 0.3045 21S2 0.216 5700 2.22 290 1.60 0.04 1.16 0.5080 21S3 0.216 5700 2.22 290 1.60 0.04 1.16 0.7112 21S12 0.381 1006 1.42 277 3.00 0.0 1.43 0.3393 11.1S13 0.381 396 2.85 299 3.00 0.0 3.00 0.4785 11.1S14 0.330 396 1.43 298 3.00 0.0 3.00 0.4785 11.1S15 0.254 5300 1.902 287 1.35 0.045 3.16 0.3045 24S16 0.254 5300 2.038 287 1.35 0.045 3.16 0.3045 24S17 0.254 5300 2.265 287 1.35 0.045 3.16 0.3045 24S18 0.254 5300 2.492 287 1.35 0.045 3.16 0.3045 24S19 0.254 5300 2.627 287 1.35 0.045 3.16 0.3045 24S20 0.254 5300 2.944 287 1.35 0.045 3.16 0.3045 24

The melt used in runs S1–S3, S12–S14 and S15–S20 have different physical properties and model parameters, see Table 3.


Table 3Material properties [59] and the magnitude of gngl and Nu0

D for nylon 66 melts used in the individual runs

Test η0 (Pa s) Density, ρ (kg/m3) gngl (×1016 m−2 s−2) Nu0D

S1–S3 126 1106 0.65(fit:0.225) 0.42S12–S14 229 1106 0.65 0.42S15–S20 217 980 0.65 0.36 (fit:0.345)

The value of η0 is at T = 280 ◦C.

all other simulations of nylon 66. Subsequently, this value is alsoused for the simulations of PET. The parameters Afibril, gn andgl are determined as follows. Based on experimental data of as-spun fibers we consider for the cross-section of the fibrils a valueof Afibril = 10−16 m2, corresponding to a fibril diameter of about11.3 nm, which is of the proper order compared to availableexperimental data [61–65]. To determine gn and gl it is conve-nient that φ only depends on the magnitude of gngl, whereasthe number density and average length of the fibrils follow fromspecifying gn and gl. Given that the rheology is only influencedby φ (through Eq. (10)) the velocity profile for large take-upspeeds can be described by adjusting the quantity gngl. There-fore, the quantity gngl is the principle parameter to describethe influence of the flow on the total crystallization dynamicsin the present model. Subsequently, the values of gn and gl arechosen in order to obtain a realistic description of the morphol-ogy of as-spun fibers. To this end, we compare the predictedfibril length Lfibril = φF/(AfibrilΨ

Flr ) to those reported in the

literature [61–65]. Using gn/gl = 0.25 × 1030 m−4satisfactoryagreement with experimental data is obtained, see Fig. 10, andthis value is kept constant for all simulations.

5. Model predictions and comparison with experimentaldata for nylon 66

In the present section model predictions are presented fornylon 66. First, the influence of the take-up velocity on thevelocity profile is analyzed, qualitatively, and specific char-acteristic features of the model are illustrated. Subsequently,the influence of the quench flow and mass throughput for agiven take-up velocity is compared to experimental spinline data

reported by Doufas et al. [59]. We found that a fair descrip-tion of the experimental velocity profiles can be obtained forgngl = 0.65 × 1016 m−2 s−2and F = 30. For this reason we usethese values for the qualitative analysis in Section 5.1. Themodel, physical and processing conditions of the nylon meltsconsidered in this section are given in Tables 1–3 and Section 4.

5.1. Influence of the take-up velocity

The influence of the take-up velocity for constant massthroughput, W, is analyzed qualitatively. In order to do so,we consider the processing conditions and material parame-ters for run S18 and the model parameters gngl = 0.65 × 1016

m−2 s−2, F = 30 and Nu0D = 0.42 for take-up velocities in the

range vL = 2000–6000 m/min. The predicted velocity profilesdiverge with increasing distance from the spinneret; the neckmoves closer to the spinneret for increasing take-up velocities,see Fig. 2. For low take-up velocities, vL < 2000 m/min (notshown), the velocity increases in a smooth fashion with distancefrom the spinneret towards a plateau value (the take-up veloc-ity). In contrast, for large take-up velocities (vL > 4000 m/min)one observes a fast increase of vz during a relatively small dis-tance along the spinline till vz reaches, rather abruptly, a plateauvalue. As a result of the fast local increase in vz, also a pro-nounced reduction in the diameter of the filament is observed,till it reaches a constant value, see Fig. 2. This behavior is calledneck formation. In the remainder of this paper, we denote theposition where vz reaches the plateau value the ‘solidificationpoint’. In agreement with experiments [5,7,9], the model pre-dicts that the solidification point moves towards the spinneretfor increasing take-up velocity at constant mass throughput.

Fig. 2. Effect of take-up velocity, vL, on the velocity profile (left) and diameter profile (right) for constant mass throughput of W = 2.492 g/min. Processing conditionscorrespond to run S18 and Nu0

D = 0.42.


Fig. 3. Effect of take-up velocity, vL, on the temperature profile for constantmass throughput of W = 2.492 g/min. Processing conditions as in Fig. 2.

The predicted temperature profiles are shown in Fig. 3.For intermediate take-up velocities, vL = 2000–4000 m/min,a weak plateau in the temperature profile is observed. Thelocal temperature increase is more pronounced for large take-upvelocities, vL > 5000 m/min, and in agreement with experi-mental data [5] a local maximum is found. Similar experimentalobservations are reported for PE [9] and PET [83]. The localincrease in the temperature profile results from the release ofthe latent heat due to crystallization. The stronger increase in thetemperature profile with increasing take-up velocity is directlyconnected to faster crystallization dynamics and an increasein the degree of crystallinity with increasing take-up velocity,see Fig. 4. The intimate connection between the FIC and theformation of the neck is reflected by the observation that the posi-tion of the solidification point coincides with that for the localtemperature increase, as reported experimentally by [5,9,83].Additional systematic and accurate experimental work on singlefiber configurations is needed to provide an accurate tempera-ture, fiber diameter and crystallinity profile along the spinline fora clear resolution of their intimate relationship in flow-inducedcrystallization. These experimental results can then provide anexcellent benchmark for additional model validation.

Fig. 4. Effect of take-up velocity, vL, on the crystallinity profile for constantmass throughput of W = 2.492 g/min. Processing conditions as in Fig. 2.

Fig. 5. Effect of take-up velocity, vL, on the profile of the tensile stress, τzz − τrr ,and apparent viscosity, ηapp, for constant mass throughput of W = 2.492 g/min.Processing conditions as in Fig. 2.

The tensile stress, τzz − τrr, and the apparent viscosity, ηapp,defined as ηapp = (τzz − τrr)/(dvz/dz), are shown in Fig. 5. Thetensile stress and viscosity profiles diverge for increasing dis-tance from the spinneret, similar to the velocity profiles. ForvL = 2000 m/min one finds a gradual increase of the tensilestress and ηapp [16,84]. For vL ≥ 4000 m/min the tensile stressincreases rapidly in the neck region till the solidification point isreached, reflecting the rapid deformation of the polymer chains.For vL = 6000 m/min the apparent viscosity initially increases,goes through a maximum value, before it rises rather sharplyat the solidification point, see Fig. 5, in qualitative agreementwith experiment results for nylons [5] and PE melts [9]. Thestrong increase of ηapp around the solidification point for alltake-up velocities is due to fact that dvz/dz reduces essentiallyto zero after the solidification point (not shown). The observationof a local maximum in ηapp is connected to the transient non-isothermal rheological response characteristics for high-speedfiber spinning conditions. We address this issue in further detailin Section 7.

Next we turn to the crystallization dynamics. Nylon 66 is a rel-atively fast crystallizing polymer [85] and for vL < 2000 m/minthe influence of quiescent crystallization dynamics to the totalcrystallinity is of relevance, see Fig. 6. For vL = 4000–6000m/min the crystallinity is nearly zero before the solidificationpoint, increases strongly around the solidification point andfinally levels of before the take-up role is reached, see Fig. 4.All observations are consistent with experimental findings fornylon 66 [5], PE [9] and PET [11]. The change in predictedcrystallinity at the take-up role, z = L, for increasing vL isshown in Fig. 6. For vL < 1500 m/min the crystallization isdominated by the quiescent crystallization dynamics and thefinal crystallinity decreases with increasing vL. This trend isin agreement with experiments for low-speed fiber spinning[86,87]. For vL > 2000 m/min the total crystallinity increasesdue to the formation of fibrils and reaches a plateau value ofabout 0.95 for large take-up speeds. Based on the expressionLfibril = φF/(AfibrilΨ

F1r) one finds that the predicted fibril length

increases with increasing vL and reaches a plateau value of about185 nm for vL > 5000 m/min, see Fig. 6. This value is of the


Fig. 6. Predicted crystallinity (a) and average fibril length Lfibril (b) at z = L with increasing take-up velocity.

proper order compared to experimental findings for differentpolymers [61–65] and supports the Ansatz for the ‘driving force’g(A, T, ms) for the nucleation and longitudinal growth of fibrils.Finally, it is noted that the effect of flow on A′ is negligible, i.e.A′ ≈ 1, as suggested before [4,88,89].

5.2. High-speed spinning: influence of quench air velocity

The influence of the quench air velocity, vc, on the pre-dicted velocity profile is compared with experimental data forruns S1–S3, reported in [59]. Using the general parameter setgngl = 0.65 × 1016 m−2 s−2 the predicted velocity profile iswithin the range of experimental data, although better agreementis found by an independent fit for gngl = 0.25 × 1016 m−2 s−2,see Fig. 7. A different value for gngl is justified, as the poly-mer melt in runs S1–S3 differs from that considered in runsS15–S20, which are analyzed in the next section. Increasing vcresults in a faster cooling of the filament, the relaxation timeτrelax increases more rapidly, and consequently the non-linearrheological behavior becomes of relevance at smaller distancesfrom the spinneret. As a result, the position of the solidificationpoint shifts towards the spinneret. Reducing gngl results into two

Fig. 7. Velocity profiles for runs S1–S3. Full lines: model predictions for gngl =0.65 × 1016 m−2 s−2, dashed lines: model predictions for gngl = 0.225 ×1016 m−2 s−2, symbols: experimental data from [59].

changes in the temperature profile, see Fig. 8. First, the positionwhere the temperature increases moves towards the spinneretfor increasing gngl, just as for the solidification point. Second,the slower fibril formation and smaller total crystallinity withdecreasing gngl results in a weaker local increase in the tem-perature. In both cases, the temperature corresponding to thelocal minimum in the temperature profile is approximately con-stant as a function of vc, in agreement with experiments [5]. Thepredicted tensile stress at the solidification point is independentof vc and gngl (not shown). This observation is consistent withfindings of Haberkorn et al. [5], who report that the tensile stressat the solidification point is primarily governed by the take-upvelocity.

5.3. High-speed spinning: influence of mass throughput

The influence of the mass throughput, W, at constant take-upvelocity on the predicted velocity profile is compared toexperimental data for runs S15–S20, see Fig. 9. Comparedto runs S1–S3 the experimental data for runs S15–S20 aretaken from filaments located “a number of rows back from thequench screen” [59]. Due to the stronger interactions betweenthe filaments, the temperature and axial velocity profile differ

Fig. 8. Predicted temperature profiles for runs S1–S3. Top: T for gngl = 0.65 ×1016 m−2 s−2. Bottom: T − 25 ◦C for gngl = 0.25 × 1016 m−2 s−2.


Fig. 9. Velocity profiles for runs S15–S20. Model predictions for gngl = 0.65 ×1016 m−2 s−2. Full lines: model predictions for S15–S20 with Nu0

D = 0.36,dashed lines: model prediction for runs S19–S20 with Nu0

D = 0.345, symbols:experimental data [59].

considerably compared to a filament close to the quench screen[59]. As a result, the relationships for Cf and NuD may not beused, as they are applicable to single filaments only [90]. For thesimulation of runs S15–S20, we consider the average quenchconditions from the exit and keep all model parameter constantas specified before. Only the prefactor Nu0

D is reduced to avalue of 0.345–0.36, in order to obtain good overall agreementbetween the model predictions and the measured temperatureprofile for 0.15 m < z < 0.5 m, see Fig. 10.

For gngl = 0.65 × 1016 m−2 s−2 and Nu0D = 0.36 a good

overall agreement is found between the experimental and pre-dicted velocity profiles, see Fig. 9. The solidification point movesaway from the spinneret with increasing W, in agreement withexperiments [5,10,59]. It should be noted that for runs S19 andS20 good quantitative agreement between model predictions andexperimental data is obtained for Nu0

D = 0.345, as illustrated bythe dashed curves in Fig. 9. As expected from the findings in Sec-tion 5.2, a reduction in Nu0

D moves the solidification point away

Fig. 11. Model predictions for runs S15–S20 of the extensional stress pro-file, τzz − τrr , and apparent viscosity, ηapp, for gngl = 0.65 × 1016 m−2 s−2andNu0

D = 0.36.

from the spinneret. Given that runs S15–S20 are performed ona single polymer melt the magnitude of gngl is kept constant,because it is a material parameter, and a change in Nu0

D reflectsdifferences in the cooling conditions associated to the differ-ent positions of the filaments in the spinneret. In our view, thisapproach is more appropriate compared to that taken in Doufaset al. [59], who change the model (material) parameters associ-ated to the FIC and the relaxation time of the crystalline phasein their model to describe the velocity profiles for the individualruns, i.e. their parameters F and ξ vary to describe runs S15–S20(see Table 3 of [59]).

The predicted temperature profile for runs S15–S20 areshown in Fig. 10, and illustrates that for Nu0

D = 0.345–0.36 thepredictions are in good agreement with available experimentaldata for 0.15 m < z < 0.5 m. In agreement with experiments [5],the temperature at the solidification point is almost independentof the mass throughput and primarily determined by vL.

The predicted tensile stress and apparent viscosity are shownin Fig. 11. The tensile stress at the solidification point is nearlyindependent on the W and of the proper magnitude as foundexperimentally by Haberkorn et al. [5], τzz − τrr ≈ 107 Pa. In

Fig. 10. Temperature profiles for runs S15–S20. Left: model predictions for Nu0D = 0.36. Right: Comparison of model predictions with experimental data [59] for

z = 0.15–0.55 m: model predictions for S15–S20 with Nu0D = 0.36, full lines, for runs S19–S20 with Nu0

D = 0.345, dashed lines.


Fig. 12. Velocity profiles under the low-speed spinning conditions of runsS12–S14 for z = 0–2 m. Lines: model predictions for run S12 (—), S13 (– –)and S14 (– ·–), symbols: experimental data from [59].

all cases the apparent viscosity goes through a maximum, showsa weak thinning behavior before the solidification point, andincreases strongly after the solidification point.

5.4. Low-speed spinning

In this section the model predictions for low-speed spinningconditions of nylon 66 are compared with experimental data forruns S12–S14. Quiescent crystallization dynamics are importantfor take-up velocities for runs S12–S14, see Fig. 6. As discussedin Section 4.4, the parameter F can be obtained independentlyfrom the description of the velocity profile for low take-up veloc-ities, and is then used for high-speed spinning conditions, seeSections 5.2 and 5.3. In the present case, a parameter value ofF = 30 is obtained by fitting the velocity profile for run S13,which is then used for the model predictions of runs S12 andS14.

For runs S12–S14 the model predictions and experimen-tal data for the velocity profile are in good agreement witheach other, and show a gradual smooth increase towards vL,see Fig. 12, characteristic for low-speed spinning conditions

Fig. 13. Temperature profiles under the low-speed spinning conditions of runsS12–S14 for z = 0–2 m. Lines: model predictions for run S12 (—), S13 (– –)and S14 (– ·–), symbols: experimental data from [59].

Fig. 14. Model predictions for the crystallinity, φ, profiles under the low-speedspinning conditions of runs S12 (—), S13 (– –) and S14 (– ·–) for z = 0–2 m.

[16,84]. Also favorable agreement between the predicted andexperimental temperature profiles are obtained, see Fig. 13.Here, we observe the smooth decrease in temperature alongthe spinline, which is typically observed for low-speed spin-ning conditions [16,84]. The predicted degree of crystallinity,φ, increases smoothly from zero to a plateau value, see Fig. 14,where the crystallization dynamics varies according to the (bell-shaped) temperature-dependent radial growth rate between Tm0and Tg. At low-speed spinning conditions, the final crystallinityof the filament is related approximately to the residence time ofthe fluid element between Tm0 and Tg. For example, the smallerdegree of crystallinity for run S12 compared to run S14 primar-ily results from a shorter residence time between Tm0 and Tgfor run S12 compared to S14. (The reduction in the crystallinityalong the spin line with increasing vL is also reported experi-mentally by on-line X-ray scattering studies for PE, PVDF [86]and nylon 6 [87].) Alternatively, the larger final crystallinity forrun S13 compared to S14 is primarily connected to the differ-ence in the temperature profile of these two runs. Finally, thepredicted tensile stress profile increases in a smooth fashion tilla magnitude of about 0.1–0.5 MPa, and subsequently a weakgradual increase is observed as a result of air drag, see Fig. 15.

Fig. 15. Model predictions for the tensile stress profiles, τzz − τrr , under thelow-speed spinning conditions of runs S12 (—), S13 (– –) and S14 (– ·–) forz = 0–2 m.


Table 4Material and processing conditions of PET [7]

η0 (Pa s) ρ (kg/m3) D0 (mm) vL (m/min) W (g/min) T0(◦C) L (m) z1 (m) z2 (m) vc (m/s) Ta(◦C)

R1 331 1360 3.8 5947 4.7 310 1.40 0.0 1.40 0.1 24R2 331 1360 3.8 5490 2.8 310 1.40 0.0 1.60 0.1 24

The value of η0 is at T = 280 ◦C.

6. Comparison with experimental data for high-speedspinning of PET

Besides nylons also poly(ethylene terephthalate), PET, is apolymer used frequently for fiber spinning. PET is a slowlycrystallizing polymer and for low and intermediate spinning con-ditions the spun PET fiber is amorphous at the take-up role, see,for example, [1,6,7], which is a major difference compared tonylon 66 analyzed in the previous section.

Here, we consider the experimental data for the diameter andtemperature profile of the filament during high-speed spinningconditions of PET reported by Vassilatos et al. [7]. The pro-cessing conditions, material parameters and model parametersare given in Tables 1 and 4 and Section 4. As PET is amor-phous for low and intermediate take-up velocities, the procedureemployed above to determine the model parameter F cannot beapplied. For the simulation of PET we use F = 30, just as fornylon 66. For gngl = 10 × 1016 m−2 s−2we find rather goodagreement between model predictions and the experimental datafor the diameter and temperature profiles, see Figs. 16 and 17,under different processing conditions. The temperature at thesolidification point corresponds to T ≈ 180 ◦C for run R1 andT ≈ 160 ◦C for run R2, respectively, which is well above Tg.This illustrates that the formation of the neck is connected tothe influence of crystallization on the rheology and is not asso-ciated to the glass transition. Finally, it should be noted that thestabilization of the deformation behavior corresponds to a crys-tallinity of about 0.1, see Figs. 16 and 17. On the other hand, thetemporary rise in temperature at the solidification point is moresensitive to the overall crystallization dynamics. In summary, thepresent FIC model is able to describe high-speed spinning con-ditions of both nylon 66 and PET. Up to now, it was not possible

Fig. 16. Filament diameter, temperature and crystallinity profiles under high-speed spinning conditions of PET in run R1. Model predictions for the filamentdiameter (—) temperature (– –), and crystallinity (– ·–) and experimental datafor the diameter, (�), and temperature, (©) [7].

Fig. 17. Filament diameter, temperature and crystallinity profiles under high-speed spinning conditions of PET in R2. Model predictions for the filamentdiameter (—) temperature (– –), and crystallinity (– ·–) and experimental datafor the diameter, (�), and temperature, (©) [7].

to predict low- and high-speed spinning of PET with a singleset of model parameters. Finally, we note that the difference inchemical structure of nylon 66 and PET implies different crys-tallization paths under quiescent and flowing conditions [1,91].Consequently, it is to be expected that the nucleation and longitu-dinal growth dynamics of the fibrils will differ quantitatively. Asa consequence, the particular values of gn, gl and Afibril will dif-fer. However, to our knowledge, there is no experimental data toconfirm this. In this respect we recall that for continuum macro-scopic variables such as temperature, total crystallinity and fiberdiameter profile, only the variable gnglAfibril is of relevance,while the ratio gl/gn enters the average length of fibrils.

7. Discussion

As shown in the previous section the present model realisti-cally describes the velocity, temperature, stress and crystallinityprofile along the spinline for different process conditions anddifferent polymers. Model predictions of the velocity profiles ofnylon 66 are similar to those predicted by the model proposedby Doufas et al. [21,45,59], who consider a modified Nakamuraequation to describe the crystallization dynamics. Alternatively,in the present model the individual contributions from the qui-escent crystallization of spherulites and the FIC of fibrils duringfiber spinning are described through two hierarchies of rateequations. For nylon 66, quiescent crystallization dominates thecrystallinity at the take-up role for vL < 2000 m/min, whereasfor vL > 2000 m/min the formation of fibrils determinesthe crystallinity. Moreover, the present model describes themorphology of as-spun fibers, in agreement with experimentalobservations [61,62]. This is the primary advantage of the


present model compared to previous FIC models for fiberspinning, because a morphology can, by definition [2], not bepredicted by the Nakamura equation. The present description ofthe FIC is kept simple, to avoid the introduction of a large numberof (phenomenological) parameters. As the difference in gn andgl is based on independent experimental data for the fibril length,the velocity profile can be described by adjusting gngl only. Themagnitude of F is obtained from spinline data for low take-upvelocities and kept constant for all remaining simulations. As aresult, the model parameters associated to FIC can be obtainedin a transparent fashion. To our knowledge, detailed experi-mental data is missing that would allow to carefully validate orextend the present approach. Experimental information on thenumber density and shape of crystals can be obtained by wayof transmission electron microscopy techniques of the as-spunfiber [61,62] or by combined on-line small and wide angle X-rayscattering techniques [86]. The shape of discrete particles leavessignificant fingerprints in scattering experiments, namely in theform factor [92]. In general, the Porod limit and integral of theform factor in scattering experiments are related to the surfacearea and volume of the particles [92–94]. The experimentallydetermined form factor can also be compared to ones that havebeen pre-calculated for specific parameterized shapes, see [32]for the appropriate references. In any case a combined scatteringand transmission electron microscopy study is needed, as foran assembly of particles of unknown (but common) shape andpolydisperse size, the simultaneous determination of the shapeand size distribution is impossible [93].

Besides the ability to predict a morphology, we also foundthat a fair description of the velocity profiles for nylon 66 can beobtained for a single set of model parameters. Though this is anencouraging finding, ‘universal’ empirical values for gn and glcannot be given, as the position of the filaments in the spinneretof runs S1–S3 and S15–S20 differ. As a result, also the coolingand air drag on the filament differs, which result in differentpositions of the solidification point [59], see Sections 5.2 and 5.3.In addition, the present one-dimensional model does not accountfor radial variations in temperature, crystallization dynamics, themolecular orientation of the amorphous phase and the viscousstress. The absence of these radial variations is partly reflectedin the quantitative value of the phenomenological parametersfor the flow-induced crystallization, the effect of crystallinity onthe stress and the relationship for the glassy viscous stress.

In order to predict the neck formation, a viscoelastic rheologi-cal description of the melt is essential. The viscoelastic behaviorof the melt is, perhaps, illustrated best by the thinning behaviorof ηapp in the draw-down and neck region. However, the originof the thinning of ηapp is not well understood. Haberkorn et al.[5] considered that the thinning in ηapp could be connected tothe thinning of the stationary extensional viscosity or the onsetof a transient non-linear rheological response. First, we addressthe possibility that the thinning in ηapp is connected to the thin-ning in the stationary extensional viscosity. In order to do so,we calculated the stationary extensional viscosities based on thetemperature, T, and the extension rate, dvz/dz, at position z in theNewtonian regime, ηN

E (T ) = 3η0(T ), and the Giesekus model,ηG

E (T, dvz/dz). (To calculate ηGE (T, dvz/dz) we use the analyti-

Fig. 18. Model predictions of the apparent viscosity, ηapp, the stationary vis-cosities based on T and dvz/dz at position z according to the Giesekus model,ηG

E , and Trouton ratio, 3η0, and the Deborah number, De = τrelax dvz/dz, till thesolidification point of run S18.

cal expression for the Giesekus model reported in Table 7.3-5of [81], as it can be shown that for αG = 3/4 the influence offinite extensibility is negligible.) Fig. 18 illustrates that, firstly,for run S18 the difference between ηN

E (T ) and ηGE (T, dvz/dz)

is quite limited for αG = 3/4, and, secondly, that the increasein ηN

E (T ) and ηGE (T, dvz/dz) primarily results from the decreas-

ing temperature. This observation reveals that the decrease ofthe stationary extensional viscosity at high deformation ratescannot account for the difference between ηN

E and ηapp. (Sim-ilar observations apply to the other runs.) Hence, the thinningof the stationary extensional viscosity at high extension rates[82,95] cannot explain the thinning behavior in ηapp. Alterna-tively, the connection between a transient non-linear rheologicalresponse and the thinning in ηapp is supported by the observationthat, firstly, ηapp < ηG

E (T, dvz/dz) and, secondly, the position inthe local maximum in ηapp corresponds to the condition thatDe = τrelax(T )dvz/dz > 1. Hence, in the neck region the poly-mer chains become strongly stretched, which is also reflected bythe rapid increase in τzz − τrr, see Figs. 5 and 11. However, sta-tionary conditions are not reached due to the continuous coolingof the filament. Finally, it is interesting to note that also for thelow flow rates, De < 1, encountered in the draw-down regimeone finds that ηapp < ηN

E . This observation shows the impor-tance of a viscoelastic description of the melt, and the closeinteraction between the transient viscoelastic behavior and thenon-isothermal conditions experienced during fiber spinning ingeneral.

Finally, we turn to the predictions of low-speed fiber spinning.The crystallinity profiles presented in Fig. 14 are qualitativelysimilar to the predictions based on the Nakamura equation byDoufas et al. [59]. The quantitative differences result from thefact that in the present study the (quiescent) crystallization ofspherulites from heterogeneous nuclei is considered, corre-sponding to an Avrami index of 3, whereas Doufas et al. [59] usean Avrami index of 1 in their calculations. Although an Avramiindex of 1 is questionable for low-speed spinning conditions,we want to focus on the impact of this difference in the value ofthe Avrami index. In particular, the crystallinity influences the


rheological response, through the exponential of Fφ in Eq. (10).It is apparent that if the viscosity is to increase substantially ata lower degree of crystallinity, one is required to take a largervalue for F, and vice versa, to describe the velocity profile ofrun S13. In addition, a variation in F also results in a differentvalue for gngl to describe the velocity profiles of runs S1–S3and S15–S20. In this respect one should realize that for run S12the position where vz reaches the plateau value in the experi-mental velocity profile also coincides with the glass transition.Similar observations apply to low-speed spinning conditionsof PET fibers [84], which are completely amorphous forvL < 3000–4000 m/min [1]. This observation suggests that itmay be more appropriate to account for the influence of the glasstransition on the rheological behavior of the fiber, rather thananticipating a strong influence of a small degree of crystallinityon the rheology. As discussed before, experiments indicate thata viscous stress can dominate the rheological behavior of poly-mer materials at temperatures close to Tg [50–53], which led tothe introduction of a viscous stress in the present model. Here,we consider a ‘Newtonian’ viscosity for simplicity, as the mainfocus of the present paper is a description of the flow-inducedcrystallization process during fiber spinning in terms of a set ofrate equations for the morphology. A more realistic descriptionfor the influence of the transition to the glassy state on themechanical behavior may appear in future modeling approaches.

8. Conclusion

A continuum model for the simulation of fiber spinning,with a viscoelastic description of the melt, quiescent crystal-lization and flow-induced crystallization (FIC), is proposed andvalidated against experimental data for nylon 66 and PET. Therheological behavior is described by a modified Giesekus model,which in addition to finite extensibility accounts for the reductionin the length of the amorphous part of the chain due to crys-tallization. The crystallization dynamics are expressed on thecontinuum level through a set of rate equations, which accountfor both the quiescent crystallization of spherulites and the FICof a fibrillar morphology. The present model realistically pre-dicts the velocity, temperature, tensile stress, apparent viscosityand crystallinity profile along the spinline, in semi-quantitativeagreement with experiments for low- and high-speed spinningconditions of nylon 66 and PET. Also, for a single set of modelparameters the model predictions are in favorable agreementwith experimental data for the velocity and temperature profileof nylon 66 melts spun at low and high take-up velocities underdifferent processing conditions. For PET quantitative agreementis obtained with experimental data during high-speed spinningconditions only.

The application of the rate equations in the present modelresults in a realistic and transparent description of the crys-tallization dynamics, in terms of the quiescent crystallizationof spherulites and the FIC of fibrils. As an example, we pre-dict that for nylon 66 the quiescent crystallization dominates fortake-up velocities up to 1500 m/min, whereas the fibril forma-tion determines the crystallization for take-up velocities largerthan 2000 m/min. In addition, the model predicts the morphol-

ogy of as-spun fibers. These two features are an advantage ofthe present model over the Nakamura equation which, firstly,cannot predict a morphology, and, secondly, cannot describe thesimultaneous formation of spherulites and fibrils.

For low- and high-speed spinning conditions of nylon 66 thevelocity profile reaches the plateau value at temperatures wellabove the glass transition temperature, reflecting the influenceof the crystallization and the rheology of the melt. Alterna-tively, for intermediate take-up velocities the velocity profilereaches the plateau value at T ≈ Tg for nylon 66. Similar obser-vations apply to PET for low and intermediate take-up velocities.Given that the degree of crystallinity is rather low for nylon66 and virtually zero for PET under low-speed spinning con-ditions, it appears realistic to account for the change in therheology associated to the transition to the glassy state, insteadof anticipating a strong influence of the crystallization. A morerealistic description of the interrelation between the glass tran-sition and the rheological response may be incorporated into afuture model. Alternatively, from the perspective of model vali-dation polyethylene may be more suitable as it is a relatively fastcrystallizing polymer and the glass transition is not reached dur-ing fiber spinning conditions. Indeed, a detailed experimentalinvestigation of the crystallization of linear polyethylene meltwith a well defined molecular weight distribution [96] underisothermal, non-isothermal conditions in quiescent, shear andelongational flow field-induced conditions would provide a uni-fied bench mark for future model validation of flow-inducedcrystallization behavior.

Acknowledgment

The authors thank Prof. Hans Christian Ottinger for helpfulcomments and stimulating discussions.

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