Injection Molding || Crystallization

Chapter 4Crystallization

4.1 Introduction

The majority of polymers used in industry are semicrystalline. Crystallizationoccurs during the processing, and consists of two stages:

1. Nucleation: Active nuclei are formed within the liquid phase, from whichcrystals can develop. If the nuclei appear spontaneously due to thermal fluc-tuations in the liquid, the nucleation is called homogeneous nucleation. If thenuclei are formed on the surface of foreign substances or crystals of the samematerial already present in the melt, the nucleation is called heterogeneousnucleation.

2. Growth: The nuclei develop into observable crystals.

In the formation of crystals, polymer chains fold back and forth to form thecrystalline lamellae. The crystalline lamellae and the amorphous phase arearranged in semicrystalline morphological entities, ranging from a micron toseveral millimeters in size. The most common morphologies that can be found ininjection-molded polymers are spherulites, which usually form under quiescentconditions, and shish-kebab structures, which may appear under shear flow [see,for example, Eder and Janeschitz-Kriegl (1997), Zuidema et al. (2001) andJaneschitz-Kriegl (2009)].

In the filling stage of injection molding, the polymer melt is subjected to highshear rates, which results in the so-called flow-induced crystallization (FIC). Themain features of FIC are the dramatic enhancement of the crystallization rate, andthe formation of oriented morphology. The effect of FIC continues after thecessation of flow.

Final properties of an injection-molded product, made of semi-crystallinepolymers, strongly depend on the flow-induced microstructure (crystallinity,morphology, orientation, etc.). Particularly, the enhancement in polymer crystal-lization rate crucially changes the solidification behavior, and the orientedmicrostructure leads to a local anisotropy in thermal and mechanical properties,

R. Zheng et al., Injection Molding, DOI: 10.1007/978-3-642-21263-5_4,� Springer-Verlag Berlin Heidelberg 2011

47

which can further cause anisotropic shrinkage and hence enhance warpage of theinjection-molded parts.

In this chapter, we shall focus on two relevant topics: the effect of flow oncrystallization kinetics, and the effect of crystallization on rheological and thermalproperties.

4.2 Crystallization Kinetics

4.2.1 The Kolmogoroff-Avrami Model

We model the effect of flow on crystallization kinetics by considering growth andnucleation. Following the work of Kolmogoroff (1937), we first consider anunrestricted growth of crystals and calculate a fictive volume fraction. LetG denote the rate of growth of the spherulite radius as a function of time. Thevolume of a phantom spherulite, nucleated at the instant s and grown up to the

instant t, is given by t1ðs; tÞ ¼ ð4p=3ÞR t

s GðuÞdu� �3

; the expression in the squarebrackets is the radius of the spherulite. The shish-kebab structure is represented bya cylindrical geometry. It is assumed by Eder and Janeschitz-Kriegl (1997) that thelateral growth rate is the same as the spherulite radial growth rate G, unaffected by

flow. Its volume is t2ðs; tÞ ¼ pls

R ts GðuÞdu

� �2; where ls is the length of the shish at

time t. Denoting the number density of the quiescent nuclei by Nq and the flow-induced nuclei number density by Nf, the unrestricted total volume fraction ofphantom crystals at time t is then given by

uðtÞ ¼Z t

0

_NqðsÞt1ðs; tÞdsþZ t

0

_Nf ðsÞ½xt1ðs; tÞ þ ð1� xÞt2ðs; tÞ�ds: ð4:1Þ

Here the first integral term on the right-hand side of the equation is the qui-escent contribution. The second integral term is the contribution of the flow-induced crystallization, where we introduce a two-value weight function x :

x ¼ 0 if _c[ 1=kR; andR t

0 g _c2dt [ wc

1 otherwise

(

; ð4:2Þ

where kR is the longest Rouse time, g is the shear viscosity, _c is the generalizedshear rate defined as _c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DijDij

p; with Dij ¼ ð1=2Þðoui=oxj þ ouj=oxiÞ being the

rate-of-deformation tensor, ts is the shearing time, the integration gives the specificwork done on the sheared polymer, and wc is the critical specific work. It has beennoticed by van Meerveld et al. (2004) that there is a minimum shear rate_cmin� 1=kR below which the orientated structure (shish-kebab) is unlikely to beformed. Janeschitz-Kriegl et al. (2003) suggests that it is the specific work done on

48 4 Crystallization

the sheared polymer melt that controls the resulting morphology. Experimentalobservations of Mykhaylyk et al. (2008) support this theory. They found that,above _c ¼ 1=kR ; a critical amount of work is necessary to cause a transition fromisotropic spherulites to oriented shish-kebabs.

In order to account for impingement, the fictive volume fraction should beconverted to the actual relative crystallinity, a, which is also called the degree ofspace filling, defined as the ratio of crystallized volume at a given time to the totalcrystallizable volume. According to Kolmogoroff (1937) and Avrami (1939), theactual crystal volume increase is equal to the liquid fraction of the fictitiousincrease, i.e., da ¼ ð1� aÞdu; which can be integrated to give

a ¼ 1� expð�uÞ: ð4:3Þ

This means that, to calculate the relative crystallinity, we have to calculate onlythe fictive volume fraction u. But to obtain u, we need to consider the growth rateand the nuclei number density.

4.2.2 Growth Rate

The increasing number of activated nuclei is the first noticeable effect induced byflow. The effect of flow on growth rate was also observed (Monasse 1995), but it isusually less important and can be neglected (Koscher and Fulchiron 2002), whilethe effect of flow on nucleation must be considered.

For a spherulite, the radial growth rate G can be calculated by applying theHoffman–Lauritzen theory (Lauritzen and Hoffman 1960)

GðTÞ ¼ G0 exp � U�

Rg T � T1ð Þ

� �

exp �Kg T þ T0

m

� �

2T2DT

� �

; ð4:4Þ

where U* is the activation energy of motion, often taken as a generic value6,270 J/mol (e.g., Koscher and Fulchiron 2002), Rg is the gas constant,T? = Tg – 30 with Tg being the glass transition temperature, DT ¼ T0

m � T is thedegree of supercooling, where T0

m is the equilibrium melting temperature and T isthe temperature at the crystallization. All the temperatures in this equation aregiven in K. The value of T0

m under atmospheric pressure can be determined bythe Hoffman–Weeks extrapolation method (Hoffman and Weeks 1962). In thismethod, the melting temperature Tm, measured using the Differential ScanningCalorimetry (DSC), is plotted as a function of the crystallization temperature, Tc,and the Tm = f(Tc) curve is extrapolated up to its intersection with the Tm = Tc

straight line. The intersection gives the value of T0m (Fig. 4.1). The Hoffman–

Weeks extrapolation method, however, is not a very accurate method. Improvedexperimental methods have also been discussed by Marand et al. (1998) andAl-Hussein and Strobl (2002).

4.2 Crystallization Kinetics 49

The equilibrium melting temperature may depend on pressure. Fulchiron et al.(2001) express the pressure dependence as a polynomial function

T0mðPÞ ¼ T0

mð0Þ þ a1Pþ a2P2; ð4:5Þ

where P is the pressure, and a1 and a2 are constants that can be determined fromthe Pressure–Volume–Temperature diagram.

To determine the parameters G0 and Kg, one needs to measure the growth rate G(T).For materials with slow crystallization kinetics, one can easily measure the spherulitegrowth rate as a function of temperature from micrographs (Fig. 4.2). Then G0 and Kg

are determined by plotting ln Gþ U�=RgðT � T1Þ against ðT þ T0mÞ=2T2DT:

For some industrial polymers, crystallization rates are too high so that theobservation of spherulite growth in the interesting temperature range is not exper-imentally possible. An approximate method is therefore useful. Van Krevelen(1976) proposed a semi-empirical equation for the growth rate of variety of commonpolymers as follows:

logG ¼ logG0 � 2:3T0

m

T

T0m

T0m � Tg

þ 50T0

m � T

; ð4:6Þ

with G0 = 7.5 9 108 lm/sFor shish-kebab structures, one assumes that the lateral growth rate is equal to

the corresponding radial growth rate for spherulites, but it is not yet possible todetermine the length growth rate. A useful approach to deal with the growth ofthread-like nuclei has been proposed by Liedauer et al. (1993). The methodinvolves the total length of threads per unit volume, which is given by

Ltot ¼Z t

0

_Nf ðsÞlsðt � sÞds: ð4:7Þ

Fig. 4.1 Determination of equilibrium melting temperature for a sample of iPP by the Hoffman–Weeks extrapolation method (From Zheng et al. (2010), with permission from SpringerScience ? Business Media B.V.)


The total length per unit volume has the following relationship with the averagedistance D between the threads:

Ltot ¼2ffiffiffi3p

D2; ð4:8Þ

The distance D is possible to be measured from electron micrographs, or pre-dicted from birefringence experimental data. The function of ls(t) is assumed to be

lsðtÞ ¼ gl kR _cð Þ2t; ð4:9Þ

where gl is a constant with the dimension m-2s-1. When experimental data for Ltot

are available, gl can be determined.

4.2.3 Nuclei Number Density

There are two important limiting cases of isothermal crystallization, namely theinstantaneous nucleation, where the nuclei are there from the beginning, and thesporadic nucleation, where the number of nuclei increases linearly with time.

Fig. 4.2 Growth of spherulites observed under microscopy for iPP at 135�C (From Zheng et al.(2010), with permission from Springer Science ? Business Media B.V.)


For quiescent crystallization under a constant temperature, in the case ofinstantaneous nucleation with a constant number density N0, one obtains NqðtÞ ¼N0HðtÞ for the activated quiescent nuclei number density, where H(t) is theHeaviside unit step function, zero for t \ 0 and unity for t C 0. Then the rate ofthe nuclei number density is _Nq ¼ N0dðtÞ; with dðtÞ being the Dirac delta functionconcentrated at t = 0. Equations 4.1 and 4.3 lead to the familiar Avrami equation:

aðtÞ ¼ 1� exp � 4p3

N0G3t3

: ð4:10Þ

For the case of sporadic nucleation, assuming _Nq ¼ constant; one has

aðtÞ ¼ 1� exp � p3

_NqG3t4� �

: ð4:11Þ

According to Koscher and Fulchiron (2002), for quiescent instantaneousnucleation, the value of N0 has an exponential dependence on the degree ofsupercooling DT:

lnN0 ¼ aNDT þ bN ; ð4:12Þ

where aN and bN are constants to be determined by fitting the equation to exper-imental data of N0(T). In case N0 cannot be counted by optical observations, it may

be evaluated by N0 ¼ 3 ln 2.ð4pG3t3

1=2Þ; where t1/2 is the half crystallization time,

corresponding to the time when the relative crystallinity a = 0.5. The data of halfcrystallization time under quiescent conditions can be measured from differentialscanning calorimetry (DSC) experiments.

The flow-induced nuclei number takes the following form as suggested by Ederand Janeschitz-Kriegl (1997):

_Nf þ1kN

Nf ¼ f ; ð4:13Þ

where kN is a nucleation relaxation time, which, according to Eder and Janeschitz-Kriegl, has a large value and may vary with temperature, and f is a function of flowvariables.

Several expressions for the function f have been proposed by different authorsbased on different assumptions about the driving force for the enhancement ofnucleation. The suggested driving forces include shear rate, recoverable strain, thefirst normal stress difference, the change in free energy induced by flow, the effectof the combination of shear rate and strain, etc.

Eder and Janeschitz-Kriegl (1997) consider a function of shear rate and assumethat

f ¼ gnðTÞ_c_cc

2

; ð4:14Þ


where _cc is a critical shear rate for the flow-induced nuclei generation, and gn is aparameter for the nucleation rate.

Eder and Janeschitz-Kriegl’s equation is based on the kinematics of the flowand not on the dynamics of molecules experiencing the kinematics. Zuidema(2000) and Zuidema et al. (2001) replaced the shear rate with the second invariantof the deviatoric part of the recoverable strain tensor. Koscher and Fulchiron(2002) considered the first normal stress difference as a driving force fornucleation.

Other authors, for example, Coppola et al. (2001) and Zheng and Kennedy(2004), considered the flow-induced change in free energy as a driving force. Thestarting point to construct a formulation is based on theories of Lauritzen andHoffman (1960) and Ziabicki (1996) for the quiescent nucleation rate, which isthen extended to include the flow-induced change of free energy. Coppola et al.(2001) propose the following rate function for the total nucleation rate:

_N ¼ C0kBTðDFq þ DFf Þexp � Ea

kBT

exp � Kn

TðDFq þ DFf Þn

; ð4:15Þ

where C0 includes energetic and geometrical constants, DFq is the Gibbs freeenergy difference between melt and crystalline phase under quiescent conditions,DFf is the flow-induced free energy change per unit volume (measured in J/m3), kB

is the Boltzmann constant, T is the absolute temperature, Ea is the activationenergy of the supercooled liquid-nucleus interface, and Kn is a constant containingenergetic and geometrical factors of crystalline nucleus. The power index n, alsoappearing as a subscript for K, accounts for the temperature region where thehomogeneous nucleation occurs.

Zheng and Kennedy (2004) proposed the following form for function f inEq. 4.13

f DFf ;T� �

¼C0kBTexp � U�

RgðT � T1Þ

� �

� DFq þ DFf

� �exp � Kg

T 1þ hDFf

� �T0

m � T� �

!(

�DFqexp � Kg

T T0m � T

� �

" #)

; ð4:16Þ

where h is given by h ¼ T0m=ðDH0TÞwith DH0 being the latent heat of crystalli-

zation (in J/m3). The FENE-P model has been used to calculate the flow-inducedchange in free energy and the subsequent nucleation enhancement. For low shearrates, the change in free energy due to flow is (Tanner and Qi 2005a)

DFf �12

n0kBTb

bþ 3

2

ðk _cÞ2; ð4:17Þ


and for high shear rates, one has

DFf �13

n0kBTðb� 2Þ lnðk _cÞ; ð4:18Þ

where n0 is the number density of the molecules (the n0kBT is a quantity normallythought as being folded into a shear modulus that can be estimated or measured),and b is the FENE-P model parameter.

Zheng and Kennedy (2004) have shown that the model is capable of describingthe Linkam shearing hot stage experiments for the dependence of the half crys-tallization time on shear rate and temperature reported by Koscher and Fulchiron(2002). Results are shown in Fig. 4.3.

Tanner et al. (Tanner 2003; Dai et al. 2006a; Tanner and Qi 2005, 2009)introduced a simple form for f in shear flows:

f ¼ A _cj jpc; ð4:19Þ

where A and p are constants, c ¼ _cts is the strain and ts is the shearing time.Wassner and Maier (2000) have measured the viscosity as a function of time underisothermal steady shearing at various shear rates for an isotactic polypropylene.Equation 4.14 fits the experimental data very well when p = 1.04. The equationalso shows a good agreement with small-amplitude oscillatory experiments. This

model may be formulated in an invariant manner by defining _c ¼ ð2D : DÞ1=2 and

c ¼ ðtrC� 3Þ1=2, respectively (see Eqs. 2.12 and 2.14) and thus can be used ingeneral flows in principle. Boutaous et al. (2010) have applied Eq. 4.19 in a non-isothermal crystallization simulation.

Fig. 4.3 The dependence of the half crystallization time on shear rate and temperature for asample of iPP. Curves are model predictions. Symbols are experimental results obtained fromLinkam shearing hot stage experiments (From Zheng and Kennedy (2004), with permission fromAmerican Institute of Physics for the Society of Rheology)


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Material characterization under controllable flow conditions are usually carriedout in rheometers and shearing hot stage. For example, in experimental studiesreported by Hadinata et al. (2005), a parallel plate rotational rheometer, theAdvanced Rheometric Expansion System (ARES), was used for viscosity mea-surements, and a Linkam CSS450 hot stage, as described by Mackley et al. (1999),was used in conjunction with a light intensity measuring device to observe theevolution of morphology.

4.2.4 Molecular Orientation

Mendoza et al. (2003) studied experimentally the influence of processing condi-tions on the spatial distribution of the molecular orientation in injection moldedisotactic polypropylene (iPP) plates. They found that the anisotropy of injectionmolded semi-crystalline polymers is governed by the orientation of the crystallinephase, and the distribution of the orientation strongly depends on the shear rate.Doufas et al. (2000), followed by Zheng and Kennedy (2001), have applied a rigiddumbbell model to simulate crystalline orientation in injection molded semi-crystalline polymers. The model reads, in the form used by Zheng and Kennedy(2001, 2004):

kcDhppi

Dtþ 2L:hppppi

þ hppi ¼ 13

I; ð4:20Þ

where p is the unit orientation vector, kc is the orientation relaxation time for thetumbling motion of p. The angular bracket denotes the ensemble average withrespect to the probability density function of the orientation state.

Mendoza et al. (2003) used the Hermans orientation factor describe the mea-sured results. The orientation factor is defined by

fc ¼3hcos2 hi � 1

2; ð4:21Þ

where h is the angle between the principal direction of the flow and the molecularchain axis of the crystal. In general, fc = -0.5 corresponds to perfect alignmentperpendicular to the flow direction, fc = 0 corresponds to random orientation, andfc = 1 corresponds to perfect alignment in flow direction. Note that hcos2 hi can beexpressed by hp1p1i: Thus we can write the orientation factor as

fc ¼3hp1p1i � 1

2: ð4:22Þ

Simulation results of Zheng and Kennedy (2001) compared with experiments ofMendoza et al. (2003) are shown in Fig. 4.4. The overall trend of the orientationdistribution is captured. The shearing region exhibits the highest level of orien-tation, which is solidified during filling. The orientation in the 1 mm-thick plate is


higher than in the 3 mm-thick plate. A gap-wise profile with a second hump ispredicted for the 1 mm-thick plate. The second hump corresponds to the orien-tation frozen in during the post-filling. In the post-filling stage, as the solid/meltinterface moves inward, a small flow rate in the narrower channel may still resultin a considerable deformation. In addition, due to the decreasing temperature withtime, even small shear rates could introduce high stresses and leads to a high levelof orientation.

4.3 Effect of Crystallization on Physical Properties

4.3.1 Effect of Crystallization on Rheology

The injection molding process includes both molten polymer and solidificationphase, where flow and crystallization may take place simultaneously. Therefore,for modeling injection molding, a rheological relation connecting the relativecrystallinity a to flow parameters is needed.

4.3.1.1 Two-Phase Model

Doufas et al. (1999, 2000) proposed a two-phase model based on a modifiedGiesekus model for the amorphous melt phase and a rigid dumbbell model for thesemi-crystalline phases. In the modified Giesekus model the relaxation time is afunction of relative crystallinity a:

kaða; TÞ ¼ kað0; TÞð1� aÞ2; ð4:23Þ

Fig. 4.4 Predicted crystalline orientation factor (Zheng and Kennedy 2001) compared withexperimental data (From Mendoza et al. (2003), with permission from Elsevier)


where the subscript a refers to the amorphous phase. The total extra stress tensor ofthe system is give by the additive rule:

s ¼ sa þ ssc; ð4:24Þ

where sa is the extra stress contributed by the amorphous phase calculated from thea modified Giesekus model, while ssc is the stress contributed by the semi-crys-talline phase calculated from the rigid dumbbell model.

This two-phase model is able to predict the locking-in of stresses and theorientation of crystallites.

Zheng and Kennedy (2004) followed the approach of a two-phase model, butin their model, the effect of crystallinity is not built-into the amorphousparameters. They view the system as a suspension of crystalline phase in amatrix of amorphous phase. The physical properties of the amorphous phase areindependent of a, while the viscosity of the whole system is a function of therelative crystallinity.

However, as pointed out by Tanner (2003), the stress law of (4.24) over sim-plifies the real picture of structure. The additive rule envisages parallel compo-nents of amorphous and crystalline phase at each point. Photographs ofcrystallizing polypropylene such as those shown by Koscher and Fulchiron (2002)and many others do not support this assumption.

4.3.1.2 Viscosity Model

Tanner and Qi (2005) treat viscosity of the whole system as a function of therelative crystallinity through a suspension-like expression:

gga¼ 1� a

A

� ��2; a\A; ð4:25Þ

where ga is the viscosity of the amorphous melt phase, A is a constant dependingon the crystal geometry, taking a value of 0.4–0.68.

Zheng and Kennedy (2004) proposed an alternative equation

gga¼ 1þ a=Að Þb1

ð1� a=AÞb; a\A; ð4:26Þ

where b and b1 are parameters to be determined by experiments, A is set to 0.44,corresponding to rough and compact inclusions.

Both of above models describe an upturn in viscosity, and show that g=ga !1when a! A.

Pantani et al. (2001b) suggest the following model

gga¼ 1þ b exp � b1

ab2

; ð4:27Þ

4.3 Effect of Crystallization on Physical Properties 57

The following expression

gga¼ exp bab1

� �ð4:28Þ

has been adopted by Zuidema et al. (2001) and Hieber (2002).Equations 4.27 and 4.28 also describe the upturn in viscosity. When a! 1;

both models show that the viscosity levels off and reaches a constant value.

4.3.2 Effect of Crystallization on Pressure–Volume–TemperatureRelations

In the governing equations for the injection molding simulation (as discussed inChap. 3), several important polymer material properties, such as density, com-pressibility, and thermal expansion coefficient, are related to the pressure–vol-ume–temperature (PVT) relation. Section 2.4 has shown the Tait equation, whichincludes several parameters to be determined from experimental data. For mostwidely used laboratory PVT apparatuses, data are measured under quasi-equi-librium state conditions, with few exceptions (e.g., van der Beek et al. 2006).Such conditions, however, never hold in real processing. In injection molding,polymers are subjected to flow and highly non-equilibrium cooling. For semi-crystalline polymers, the flow and thermal history strongly influence crystalli-zation which has a significant effect on the evolution of specific volume.Figure 4.5 shows a PVT diagram where the transition zone shifts to lowertemperatures with increasing cooling rate. The influence of flow on specificvolume will be opposite, shifting the transition zone up to higher temperatureswith increasing pressure. Experimental evidence has been reported by Luyé et al.(2001) and Forstner et al. (2009).

The specific volume v of a semi-crystalline polymer is a combination of theamorphous and crystalline structures. From the mass balance, one obtains

1v¼ 1� av1

vaþ av1

vc; ð4:29Þ

where va is the specific volume of the amorphous phase, vc is the specific volumeof the pure crystalline phase, and v? is the ultimate absolute crystallinity. Theproduct of the relative crystallinity and the ultimate absolute crystallinity av?denotes the volume fraction of crystallized material. If we assume that, in the solidzone of the PVT data, the crystallization has reached the ultimate state, i.e., therelative crystallinity is equal to one, we have the following expression for thespecific volume vs:

1vs¼ 1� v1

vaþ v1

vc: ð4:30Þ


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Eliminating vc from Eqs. 4.29 and 4.30, we obtain

1v¼ 1� a

vaþ a

vs: ð4:31Þ

Equation 4.31 is useful in simulations. Standard PVT data and the Tait equationcan be used for calculation of va at melt zone and vs at solid zone, while thespecific volume at transition zone can be calculated using crystallization kineticsand a linear interpolation between 1/va and 1/vs using Eq. 4.31.

It should be noted that some authors (e.g., Luyé et al. 2001) write the equationas v ¼ ð1� aÞva þ avs; where a is defined as a relative mass crystallinity (ratherthan the relative volumetric crystallinity as we have used); the expression isequivalent to Eq. 4.31.

4.3.3 Effect of Crystallization on Thermal Conductivity

Thermal conductivity is an important parameter used for material thermal calcu-lations in the injection molding simulation. A variation in the thermal conductivitywill alter the cooling rate and hence cause a variation in the temperature; this inturn will cause the viscosity, pressure, and frozen layer to vary. Classical Fouriertheory, which assumes thermal conductivity to be a constant scalar valuedepending only on temperature alone, is inadequate to describe the heat conduc-tion in deformed molten polymer. Van den Brule (1989, 1990) and van den Bruleand O’Brien (1990) suggested that heat transport mechanisms along the backboneof a polymer chain are more efficient than those between neighboring chains.Hence, the orientation of polymer chain segments induced by flow leads to ananisotropic thermal conductivity. To describe anisotropic thermal conduction, one

Fig. 4.5 Effect of cooling rate on PVT diagram (Reproduced from Luyé et al. (2001), withpermission from John Wiley and Sons)

4.3 Effect of Crystallization on Physical Properties 59

has to consider the non-Fourier law (see also Huilgol et al. 1992) with the thermalconductivity being written in a tensor form as

qi ¼ kijoT

oxj: ð4:32Þ

Van den Brule proposed a connection between the thermal conductivity tensorand the total stress tensor as follows:

kij �13

kkkdij ¼ Ctk0 rij �13rkkdij

; ð4:33Þ

where k0 is the equilibrium scalar thermal conductivity, dij is the unit tensor, and Ct

is the stress-thermal coefficient.Measuring the flow-dependent thermal conductivity is difficult. Nevertheless,

some experimental results were reported by Venerus et al. (1999, 2000, 2001 and2004) and Schieber et al. (2004) for amorphous polymers. The results indeedsupport the theory of van den Brule. In addition, they found that the product ofstress-thermal coefficient Ct and the melt plateau modulus GN is a nearly universalvalue, i.e., CtGN & 0.03.

What happens to semi-crystalline polymers is still to be explored. Dai andTanner (2006b) suggested that the Van den Brule theory might be used for shearedsemi-crystalline materials, with an enhanced value of Ct depending on theincreased crystallinity. The idea has been applied by Zheng and Kennedy (2006)and Zheng et al. (2010) in the injection molding simulation. They further assumek0 as a function of the relative crystallinity given by:

1k0ða; TÞ

¼ a

kðsÞ0 ðTÞþ 1� a

kðaÞ0 ðTÞ; ð4:34Þ

where a is the relative crystallinity, kðsÞ0 ðTÞ is the equilibrium scalar thermal

conductivity at solid state, and kðaÞ0 ðTÞ is the equilibrium scalar thermal con-ductivity at melt state. Measurement of k0 at multiple temperatures over therange including both the solid and melt states has been reported by Speight et al.(2010).

4.4 Influence of Colorants

Most studies on crystallization have considered virgin polymers without additives.However, in many injection-molded products, various additives such as colorantsare usually added and these are known to affect the crystallization behavior. Thetopic has been recently tackled by Hadinata et al. (2008), Zheng et al. (2008,2010), Lee Wo and Tanner (2010), and Zhu et al. (2009). These authors investi-gated the FIC behavior of an injection-molded isotactic polypropylene (iPP) mixed


with colorant additives. Two types of blue colorants were used in the study: one isthe Ultramarine Blue composed of Sodium Alumino Sulpho Silicate (UB) and theother is the PV Fast Blue composed of Cu-Phthalocyanine (CuPc).

Lee Wo and Tanner (2010) have carried out a steady shear experiment with aparallel plate rheometer to monitor the effect of crystallization kinetics on vis-cosity. Typical experimental results of the normalized viscosity gðtÞ=ga against thetime t at a temperature 140�C and two different shear rates, of 0.01 s-1 and0.1 s-1, are shown in Fig. 4.6, where three samples were used: (i) virgin iPP, (ii)iPP filled with 0.8% (by weight) CuPC colorant and (iii) iPP filled with 0.8% (byweight) UB-colorant. The CuPc colorant dramatically reduces the time of theupturn in the viscosity. Since the upturn in the viscosity indicates the onset ofcrystallization, the results suggest that the CuPC colorant behaves like an excellentnucleating agent. The UB colorant showed a small effect. Zhu et al. (2009)attribute the difference to the effects of the surface patterns, pigment sizes, degreeof aggregation and dispersion of the colorants. The CuPC colorant, which has theflat surface full of a nonuniform size distribution, activates effectively the heter-ogeneous nucleation. In contrast, the UB colorant, which has the curved surfacefull of a uniform size distribution, has failed to induce the heterogeneousnucleation.

To include the details of colorants in the numerical modeling is difficult, if notimpossible. Tactically, in a practical computation, the basic kinetic model formaterials with and without colorants can remain the same, while the modelparameters should be carefully determined for each colored polymer such that thenucleation and growth trends are correlated with the colorant effects.

Fig. 4.6 Experimental normalized viscosity against time for the pure iPP and the iPP mixed withCuPc and UB colorants at T ¼ 140�C and _c ¼ 0:01 s�1; 0:1 s�1: The concentration of eithercolorant is 0.8% by weight (provided by Dr. Duane Lee Wo)

4.4 Influence of Colorants 61

Results of numerical simulation and experiments of injection molding forisotactic polypropylene (iPP) with and without colorants have been reported byZheng et al. 2008, 2010). Cavity pressure profiles for injection-molded plates ofpure iPP, iPP with UB colorant and iPP with CuPC colorant are shown in Figs. 4.7,4.8 and 4.9, respectively. In spite of the fact that the concentration of the colorpigments is only 0.8% by weight, both experimental and predicted results showthat the CuPC colorant causes a distinctly earlier decay in pressure. In addition,shrinkage experiments and simulation show a high degree of anisotropic shrinkageexhibited by the CuPc-colored iPP, while the pure iPP and UB-colored iPP do notresult in highly anisotropic shrinkage. These results reveal the necessity ofincorporating the colorant effect in the simulation of flow-induced crystallizationin injection molding.

Fig. 4.7 Measured and predicted pressure profiles for the virgin iPP (From Zheng et al. (2010),with permission from Springer Science ? Business Media B.V.)

Fig. 4.8 Measured and predicted pressure profiles for the iPP with UB colorant (From Zhenget al. (2010), with permission from Springer Science ? Business Media B.V.)


4.5 Molecular Dynamics Simulation

In the previous sections we have dealt with some theories to solve problems forpolymer crystallization in processing. An alternative possibility is to use molecular-scale simulation to simulate directly the original system by first principles, ratherthan trying to obtain a simplified closed form expression to describe the behavior ofthe system. Molecular dynamics (MD) simulation is among the molecular-scalesimulation methods that integrates Newton’s equations of motion for a set ofmolecules (Allen and Tildesley 1989; Jabbarzadeh and Tanner 2006). The thermalenergy in an MD simulation is just the average kinetic energy of the atoms. In theMD simulation, the equations are deterministic, and the Brownian motion of themolecules is produced by a direct simulation of a huge number of intermolecularcollisions, very similar to the case in real fluids. It is this feature that distinguishesMD from other molecular simulation methods such the Brownian dynamics and theMonte Carlo methods. Jabbarzadeh and Tanner (2009, 2010) have used the MD tostudy the crystallization of polymers at rest, under flow and post-flow conditions.

In the work of Jabbarzadeh and Tanner, linear polyethylene CH3(CH2)nCH3

was modeled using a united atom model; for example, they have recently simu-lated a model polyethylene C162H326 (Jabbarzadeh and Tanner 2010). The groupsof CH2 and CH3 were treated as single interaction sites. Intra-molecular archi-tecture was used in the model, including bond stretching, angle bending anddihedral (torsional) potentials. Then the molecular positions, velocities and tra-jectories were calculated from a set of equations as the following:

_rðiÞ ¼ pðiÞ

mðiÞþ e1 _crðiÞ3 þ _erðiÞ; ð4:35Þ

_pðiÞ ¼ FðiÞ � e1pðiÞ3 _c� fpðiÞ � _epðiÞ; ð4:36Þ

Fig. 4.9 Measured and predicted pressure profiles for the iPP with CuPc colorant (From Zhenget al. (2010), with permission from Springer Science ? Business Media B.V.)

4.5 Molecular Dynamics Simulation 63

V ¼ 3_eV; ð4:37Þ

where r(i), p(i), and m(i) are position, momenta and mass of atom i; e is the unitvector in the flow direction; _c is the shear rate; F(i) is the total force applied by allother atoms in the system on atom i; _e is the dilation rate of the simulation boximplemented with a Nose–Hoover feedback scheme, and f is the Gaussianthermostating multiplier that keeps the temperature constant (Hoover 1985; Evansand Holian 1985); V is the volume of the simulation box. Periodic boundaryconditions in all three directions were applied for quiescent simulations. Lees–Edwards sliding brick periodic boundary conditions (Lees and Edwards 1972)were applied for planar shear simulations. Finally, the desired macroscopicproperties such as temperature, pressure, density, coefficients of thermal expan-sion, crystallinity and stresses were calculated from the microscopic information.The simulation is able to monitor the morphological transition from the amorphousstructure to the semi-crystalline structure under controlled temperature, pressure,cooling rate and flow conditions for desired polymer system (Fig. 4.10), and hashelped to unravel the effect of shearing or pre-shearing on the enhancement ofcrystallization speed.

Research and development of the MD simulation method is still progressing.Currently, MD simulations are computationally restricted to very tiny time scales(about tens of nanosecond) and very large shear rates (C107 s-1). The cost ofcomputer time increases dramatically with increasing number of atoms in themodel of molecular chains. The growth of computational power and the devel-opment of parallel algorithms will help researchers to simulate crystallization forlarge systems.

Fig. 4.10 Snapshot of the MD simulation of a model polyethylene, made of 686 molecules,C162H326: a before crystallization at T = 393 K and P = 1 atm, and b after crystallization,presheared at _c ¼ 109 s�1 for a total strain of c ¼ 5 then allowed to crystallize at 350 K for 58 ns(Reproduced from Jabbarzadeh and Tanner (2010), with permission from American ChemicalSociety)


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Injection Molding || Crystallization