Flow-Induced Crystallization in the Injection Molding ofPolymers: A Thermodynamic Approach
Kyuk Hyun Kim, A. I. Isayev, Keehae Kwon
Institute of Polymer Engineering, University of Akron, Akron, Ohio 44325-0301
Received 7 May 2004; accepted 6 July 2004DOI 10.1002/app.21228Published online in Wiley InterScience (www.interscience.wiley.com).
ABSTRACT: The prediction of the crystallinity and micro-structure that develop in injection molding is very importantfor satisfying the required specications of molded prod-ucts. A novel approach to the numerical simulation of theskin-layer thickness and crystallinity in moldings of semi-crystalline polymers is proposed. The approach is based onthe calculation of the entropy reduction in the oriented meltand the elevated equilibrium melting temperature by meansof a nonlinear viscoelastic constitutive equation. The eleva-tion of the equilibrium melting temperature that resultsfrom the entropy reduction between the oriented and unori-ented melts is used to determine the occurrence of ow-induced crystallization. The crystallization rate enhanced by
the ow effect is obtained by the inclusion of the elevatedequilibrium melting temperature in the modied HoffmanLauritzen equation. Injection-molding experiments at vari-ous processing conditions were carried out on polypro-pylenes of various molecular weights. The thickness of thehighly oriented skin layer and the crystallinity in the mold-ings were measured. The measured data for the microstruc-tures in the moldings agree well with the simulated results. 2004 Wiley Periodicals, Inc. J Appl Polym Sci 95: 502523, 2005
Key words: crystallization; injection molding; layer growth;poly(propylene) (PP); rheology
In polymer processing operations such as injectionmolding, ber spinning, lm blowing, and casting, themolten polymer is subjected to intense shear and elon-gational ow and crystallizes during the imposition ofow. Because of the ow-induced crystallization, themorphology of the semicrystalline polymer that de-velops in the nal product is typically very differentfrom what is observed as a result of quiescent crystal-lization of the same polymer. The reduction of entropybetween the oriented and unoriented melts, which isdue to the molecular chain orientation of the melts,increases the equilibrium melting temperature (Tm
0 ),and so the crystallization behavior becomes differentfrom that of the unoriented polymer melts. The extentof the Tm
0 increase depends on the degree of orienta-tion in the polymer melt. In other words, the crystal-lization kinetics are affected by a ow eld in the caseof an oriented polymer melt.
A number of mathematical models have been pro-posed for the simulation of quiescent crystalliza-tion.18 Most of the nonisothermal crystallization the-ories24 have been developed from the AvramiKol-mogoroff theory for isothermal crystallization.58
Schneider et al.4 proposed a system of differentialrst-order rate equations for describing the noniso-thermal crystallization kinetics by combining the workof Avrami and Kolmogoroff.58 They developed thekinetics by considering both the formation and growthof nuclei in terms of the system of rate equations. Inthe limiting case of very fast nucleation with respect tothe growth rate, the system of rate equations wouldreduce to a single rate equation, that is, the Avramiequation for isothermal crystallization.
Nakamura and coworkers2,3 extended the theoriesof isothermal crystallization of Avrami and Kolmog-oroff58 and Evans9 on the basis of isokinetic condi-tions. They suggested a kinetic model of nonisother-mal quiescent crystallization, which is customarilycited in the literature as the Nakamura model.
Most of the scientic studies on ow-induced crys-tallization have been mainly concerned with experi-mental elucidation and qualitative understanding. Al-though the basic features of ow-induced crystalliza-tion are well known, the quantitative modeling of theprocess has not been developed signicantly. Experi-mental investigations and theoretical work concerningthe development of the morphology and kinetics ofow-induced crystallization have been made bySchultz and coworkers,1013 McHugh and cowork-ers,1418 Ziabicki and coworkers,1922 and Janeschitz-Kriegl and coworkers.2326
The theoretical work devoted to studying the devel-opment of the microstructure associated with ow-
Correspondence to: A. I. Isayev (email@example.com)
Journal of Applied Polymer Science, Vol. 95, 502523 (2005) 2004 Wiley Periodicals, Inc.
induced crystallization can be broadly classied intothree categories: statistical mechanical models de-voted to the equilibrium properties of stretched poly-mer networks,27,28 statistical mechanical and classicalthermodynamic models devoted to the kinetic prop-erties of the crystallization process,19 and models con-necting the evolution of the morphology with the owand transport phenomena.13
The statistical mechanical model expressing themelting temperature (T0) elevation was rst quantiedby Flory27 in his classic derivation of the equilibriumtransition temperature for a stretched, crosslinked sys-tem. This model was later modied by Gaylord,28 whoapplied the concepts of irreversible thermodynamicsto determine the crystallization rates.
Ziabicki19 and later Janeschitz-Kriegl et al.25 devel-oped continuum models for ow-induced crystalliza-tion based on modications of the Avrami equation,which contains a phenomenological orientation factorfor the effect of ow. Although this approach repre-sents an important advance with respect to the T0elevation theory, the resulting models do not couplethe rheology of a semicrystalline system with the crys-talline kinetics in a predictive way.
Advances in connecting the morphology with themacroscopic properties were made by Schultz,13 whotreated the heat transfer away from a growing crystalfront as a rate-determining step.
Shimizu et al.,29 Katayama and Yoon,30 and Chenand Spruiell31 developed mathematical models forow-induced crystallization to simulate the high-speed ber-spinning process.
McHugh et al.14,15,18 developed a model for ow-induced crystallization with the Hamiltonian bracketformulation with the Avrami equation. The model hasa number of parameters that can be obtained fromrheological measurements and quiescent crystalliza-tion experiments.
Yeh and coworkers3235 carried out research onow-induced crystallization from a morphologicalpoint of view. They observed a thermally reversibletransformation from a brillar morphology to a lamel-lar crystalline morphology in strain-crystallized poly-mers. They discovered the existence of a nodularstructure within perpendicularly oriented lamellae ina glassy, amorphous state.
The ow-induced crystallization behavior is of greatindustrial importance because solidication phenom-ena usually take place from a strained melt. A com-prehensive study of polyethylene lms crystallizedfrom a strained melt has shown a shish-kebab struc-ture similar to that of samples obtained from a stirreddilute solution.36
According to the rubber elasticity theory,37 the mo-lecular origin of elastic force exhibited by a deformedelastomeric network can be expressed by the summa-tion of the internal energy and entropy change contri-
bution. For the formation of temporary entanglementsin an elastic liquid, the change in the conformationalentropy makes an overwhelming contribution to thevariation of the free energy.37,38
Haas and Maxwell39 and Ishizuka and Koyama40
approached ow-induced crystallization phenomenafrom a thermodynamic point of view. When polymerchains are under stress inuence, entropy reductiontakes place. In an oriented melt state, this results in theelevation of Tm
0 . By using this elevated melting tem-perature (Tm), they could determine the ow-en-hanced crystallization rate constants.40
A great deal of related research has been conductedby Isayev and coworkers.4146 They rst proposed amethod41,42 for predicting the skin-layer thickness inthe injection moldings of isotactic polypropylene (iPP)with the modication of the Janeschitz-Kriegl modelof ow-induced crystallization2325 and with the Na-kamura model of nonisothermal quiescent crystalliza-tion.2,3 They assumed that during the injection-mold-ing process at any material point, there existed a com-petition between the initiation of ow-induced andquiescent crystallization. In addition, they4750 sug-gested a unied approach to crystallization phenom-ena under processing conditions. In their model, purequiescent crystallization is a special case of ow-in-duced crystallization.
According to their model, the ow-induced crystal-lization process may take place under or after shear-ing, being dependent on the effect of supercooling andthe intensity of shearing or the evolution of the shear-ing prehistory during the induction period of crystal-lization. Flow-induced crystallization can produce ahighly oriented shish-kebab crystallite microstructure.However, in the Janeschitz-Kriegl model and its de-rivative attributable to Isayev et al.,41,42 ow-inducedcrystallization is specically used to refer to cases inwhich highly oriented lamellar crystallites are pro-duced. In addition, the determination of the modelparameters related to ow-induced crystallization re-quires a tremendous amount of experimental workwith a special kind of extrusion experiment.
Recently, Isayev and coworkers51,52 approached theow-induced crystallization phenomena from a ther-modynamic point of view. They calculated Tm due tothe reduction of entropy between the oriented andunoriented melts with the nonlinear viscoelastic con-stitutive equation.53 They showed some preliminaryresults for the prediction of the skin-layer thicknessand crystallinity development where the ow-inducedcrystallization takes place during the lling stage ininjection moldings.51,52
In this research, a more detailed account is providedon the further development of the model of ow-induced crystallization based on a thermodynamicpoint of view and a nonlinear viscoelastic constitutiveequation along with an increased crystallization rate.
FLOW-INDUCED CRYSTALLIZATION 503
In agreement with Flory,27 the elevation of Tm0 , which
results from the entropy reduction between the unori-ented and oriented melts, has been used to determinethe occurrence of ow-induced crystallization and theenhancement of the crystallization rate by ow. Thecrystallization rate enhanced by ow has been in-cluded by the incorporation of Tm into the HoffmanLauritzen equation.1 A nonlinear viscoelastic constitu-tive equation53 has been used to determine the en-tropy change and to calculate Tm.
The general behavior of the process is described bytransport equations: conservation of mass, momen-tum, and energy. The continuity equation for the owis
t v 0 (1)
where is the density and v is the velocity vector.The momentum equation in a uid with no body
vt v vP (2)
where P is the pressure and is the stress tensor.A Leonov multimode nonlinear constitutive equa-
tion has been used to describe the rheological behaviorof polymer melts:53
Ck2 13 IICk ICk Ck I 0 (3)where I is the identity tensor; IC
k and IICk are the rst
and second invariants of the elastic strain tensor (Ck),respectively; k is the relaxation time; and C
k is theJaumann derivative.
In the nonisothermal ow under consideration, theenergy equation is
CpTt v T k2T Q (4)where T is the temperature, Cp is the specic heat, k isthe thermal conductivity, is the energy dissipation,and Q is the rate of heat release due to crystallizationper unit of volume. This latter quantity is denedaccording to the crystallization kinetics as follows:
Q XHcddt (5)
where X is the ultimate degree of crystallinity, Hc isthe heat of fusion for the pure crystal, and is therelative degree of crystallinity.
For a viscoelastic ow, is54
2 ICk IICk ICk 3 trCk2 3(6)
where E is the deformation-rate tensor, s is the dimen-sionless rheological parameter lying between zero andunity, k is the shear viscosity of the kth mode, and 0is the zero-shear-rate viscosity.
E and 0 may be expressed as follows:
E 12 v vT (7)
N kT1 s (8)
k and k are assumed to have an Arrhenius-typetemperature dependence:
kT AkexpTbT (9)kT BkexpTbT (10)
where Tb is the temperature sensitivity of the param-eters and is related to the activation energy and Ak andBk are constants.
These values may increase dramatically as the poly-mer crystallizes.55 However, for simplicity, the crys-tallinity effect on the viscosity and relaxation time hasnot been considered in this study.
For simple shear, Ck has the following form:
Ck C11,k C12,k 0C12,k C22,k 00 0 1
(11)In this research, the following assumptions have beenmade for the simulation of the lling stage in aninjection-molding process:
1. The thin-lm approximation is employed.2. There is a no-slip condition at the wall.3. The inertial and body force in the momentum
equation are neglected.4. The pressure is independent of the thickness di-
504 KIM, ISAYEV, AND KWON
5. Thermal conduction in the ow direction is neg-ligible with respect to conduction in the thicknessdirection.
6. No fountain ow effect at the melt front is con-sidered.
For one-dimensional ow in Cartesian coordinatesin the lling stage, the conservation of mass and mo-mentum may be expressed as follows:56
x S Px 0 (12)where x is the ow direction and S, the uidity, isdened as
dy in Cartesian coordinates (13)
dr in cylindrical coordinates
The boundary and initial conditions for the tempera-ture and ow velocity are given by
Ty, t 0 T0, Tb, t 0 Tw,
Ty 0, t 0, vxb, t
vxy 0, t 0 (14)
where b is the half-gap of the cavity, Tw is the walltemperature, and vx is the velocity in the ow direc-tion.
The average velocity (U) can be expressed as fol-lows:
According to the N-mode Leonov model, is thengiven by
y, t 21Ts 0 1 01 0 00 0 0
kT C11,k C12,k 0C12,k C22,k 00 0 1
(16)where (vx/y) is the shear rate and k k/(2k) is the modulus of the kth mode [
0/(21)].The governing equations for Ck in eq. (3) can be
expressed as follows:
2 1 0 (17)
C11,k C22,kC12,k 0 (18)
C11,kC22,k C12,k2 1 (19)
where D/Dt is the material derivative operator.With the addition of a pressure gradient (x), the
shear stress (12) can be expressed as follows:
12 xy , x Px (20)
On the other hand, by integrating eq. (15) by part andeliminating with eq. (20), we nd that
xb S (21)
Accordingly, from eqs. (15), (16), and (20), x can beexpressed as follows:
2sbU 2 k1N k 0
The elastic strain tensor components at a steady-stateow (Ck
st) can be expressed as follows:
2Xk1 Xk (23)
21 Xk (25)
where Xk is equal to 1 1 3k2.The shear viscosity () can be expressed as follows:
N 2k1 1 4k2 (26)
For the cal...