Theoretical and experimental studies of anisotropic shrinkage in injection moldings of semicrystalline polymers

Theoretical and Experimental Studies of AnisotropicShrinkage in Injection Moldings of SemicrystallinePolymers

Keehae Kwon, A.I. Isayev, K.H. Kim, and C. van SwedenInstitute of Polymer Engineering, The University of Akron, Akron, Ohio 44325-0301

A novel approach to predict anisotropic shrinkage ofsemicrystalline polymers in injection moldings was pro-posed using flow-induced crystallization, frozen-in mo-lecular orientation, elastic recovery, and PVT equation ofstate. The anisotropic thermal expansion and compress-ibility affected by the frozen-in orientation function andthe elastic recovery that was not frozen during moldingswere introduced to obtain the in-plane anisotropicshrinkages. The frozen-in orientation function was cal-culated from amorphous and crystalline contributions.The amorphous contribution was based on the frozen-inand intrinsic amorphous birefringence, whereas thecrystalline contribution was based on the crystalline ori-entation function, which was determined from the elas-tic recovery and intrinsic crystalline birefringence. Tomodel the elastic recovery and frozen-in stresses re-lated to birefringence during molding process, a nonlin-ear viscoelastic constitutive equation was used withtemperature- and crystallinity-dependent viscosity andrelaxation time. Occurrence of the flow-induced crystal-lization was introduced through the elevation of meltingtemperature affected by entropy production during flowof the viscoelastic melt. Kinetics of the crystallizationwas modeled using Nakamura and Hoffman-Lauritzenequations with the rate constant affected by the ele-vated melting temperature. Numerous injection moldingruns on polypropylene of various molecular weightswere carried out by varying the packing time, flow rate,melt temperature, and mold temperature. The anisotropicshrinkage of the moldings was measured. Comparison ofthe experimental and simulated results indicated a goodpredictive capability of the proposed approach. POLYM.ENG. SCI., 46:712–728, 2006. © 2006 Society of Plastics Engineers

INTRODUCTION

Shrinkage of injection-molded parts of semicrystallinethermoplastics is affected by the volumetric shrinkage,

flow-induced residual stresses and orientation, flow-inducedcrystallization, and heat transfer. Since all these factors areaffected by the processing conditions such as packing pres-sure, packing time, melt temperature, mold temperature,injection speed, and material properties as well as geometricconstraints, the prediction of shrinkage, especially of aniso-tropic linear shrinkage, becomes a quite complex issue.Clearly, the shrinkage anisotropy in moldings cannot bepredicted based on volume shrinkage alone. Therefore,novel methodology for accurate prediction of anisotropicshrinkage in moldings is required.

For semicrystalline polymers, because of the flow-in-duced crystallization, the morphology developed in the finalproduct is typically very different from that observed as aresult of quiescent crystallization. The reduction of entropybetween the oriented and unoriented melts, which is due tothe molecular chain orientation of melts, increases the equi-librium melting temperature and therefore results in a dif-ferent crystallization behavior from that of the unorientedpolymer melts.

A number of mathematical models have been proposedfor the simulation of quiescent crystallization [1–6]. Mostof the nonisothermal crystallization theories [1–4] havebeen developed based upon the Avrami-Kolmogoroff the-ory for isothermal crystallization [5, 6]. Nakamura et al. [2,3] extended the theories of isothermal crystallization tononisothermal quiescent crystallization, which is customar-ily cited in literature as “Nakamura model.”

The statistical mechanical model expressing the meltingpoint elevation was first quantified by Flory [7] in his classicderivation of the equilibrium transition temperature for astretched, cross-linked system. Similar concepts of meltingtemperature elevation of flow-induced crystallization weredeveloped by other researchers [8–11]. Flory’s Gaussianexpression was later modified by Gaylord [11] who appliedthe concepts of irreversible thermodynamics to determinethe crystallization rates using non-Gaussian concepts. Dun-ning [8] was the first one who introduced the concept ofmelting temperature elevation, due to crystallization, intothe rate of crystal growth and nucleation.

Ziabicki [12] and later Janeschitz-Kriegl and coworkers

Correspondence to: A.I. Isayev; e-mail: [email protected] grant sponsor: Divison of Engineering, National Science Foun-dation; contract grant number: DMI-0322920.DOI 10.1002/pen.20546Published online in Wiley InterScience (www.interscience.wiley.com).© 2006 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—2006

[13] developed a continuum model for flow-induced crys-tallization based on the modification of the Avrami equa-tion, which contains a phenomenological orientation factorfor the effect of flow. McHugh et al. [14, 15] combined theHamiltonian Bracket Formulation with the Avrami equationto predict the degree of crystallinity, induction time, andbirefringence during flow-induced crystallization.

Much research for flow-induced and quiescent crystalli-zation has been done by Isayev and his co-workers [16–21].They proposed a method to predict the skin layer thicknessin the injection moldings of isotactic polypropylene usingthe modification of Janeschitz-Kriegl model of flow-in-duced crystallization [16, 17]. In addition, they suggested aunified crystallization model to describe the phenomenaunder nonisothermal flow conditions [18, 19]. According totheir model, the flow-induced crystallization process maytake place during or after shearing, depending on the effectof supercooling and the intensity of shearing or the evolu-tion of shearing prehistory during the induction period ofcrystallization. They first attempted to incorporate crystal-lization and its influences on the viscoelastic behavior ofsemicrystalline polymers. In this unified crystallizationmodel, it is assumed that the flow stress influences only theinduction period, and once the crystallization process starts,crystallization kinetics follows according to quiescent crys-tallization. Recently, Isayev and coworkers [20–23] ap-proached the flow-induced crystallization phenomena basedon the thermodynamical point of view. They calculated theelevated melting temperature due to the reduction of en-tropy between the oriented and unoriented melts using thenonlinear viscoelastic constitutive equation and incorpo-rated this temperature into the Nakamura nonisothermalcrystallization equation.

A rapid nonhomogeneous cooling of a crystalline poly-mer melt through the melting temperature introduces non-equilibrium volumetric shrinkage. By following the PVTdiagram from the melt temperature to ambient conditions,one can obtain the average value of the final product volume[24–26]. However, using this approach to calculate shrink-age is limited because the PVT diagram can describe onlyisotropic shrinkage.

Recently, several new approaches were developed topredict anisotropic shrinkage [27–35]. Bushko and Stokes[27, 28] used a thermoviscoelastic material model to predictthe product shrinkage, warpage, and build-up of residualstresses in the injection molding process. However, theyneglected flow effects and assumed that in-plane shrinkagesare equivalent. Jansen and Titomanlio [29–31] proposed asimple thermoelastic model taking into account thermal andpressure effect on in-plane shrinkage and Poisson’s ratio onthickness shrinkage. They assumed that during solidifica-tion, no relaxation and creep occurs and stresses start tobuild up as soon as the temperature drops below the solid-ification temperature. To consider the difference in thelength and width shrinkage, Jansen [32] set up a model thatpredicts variation in modulus, Poisson’s ratio, coefficient ofthermal expansion, and compressibility of injection-molded

semicrystalline products. Hieber [34] used the PVT behav-ior affected by crystallinity, which depends upon the ther-mal, pressure, and shear stress history. Based on the mod-ified material constants, he calculated the time-dependentgapwise shrinkage of the molded specimen prior to ejectionfrom the cavity and compared it with measured data. Re-cently, Kennedy and Zheng [35] used residual strain and athermoviscoelastic stress model to calculate the shrinkagecoefficient and to predict the in-plane shrinkage parallel andtransverse to the flow direction. They used a hybrid modelthat improves the theoretical prediction of shrinkage andwarpage.

In the present study, a novel approach to predict theanisotropic shrinkage of injection-molded products of semi-crystalline polymers was proposed based on the frozen-inorientation function and elastic recovery determined from anonlinear constitutive equation. Numerical simulation of theviscoelastic injection molding process including filling,packing, and cooling stages was developed. The crystalli-zation rate enhanced by flow effect was included by theincorporation of the elevated equilibrium melting tempera-ture into the Nakamura and Hoffman-Lauritzen equation.To calculate frozen-in orientation, elastic recovery, andelevated melting temperature, a viscoelastic constitutiveequation was used. The applicability of different orientationassumptions was tested. The predicted anisotropic shrink-ages were compared with the experimental results measuredat various processing conditions.

THEORETICAL

Governing Equations

The general behavior for a compressible, nonisothermalflow is described by transport equations: mass, momentum,and energy equation. The continuity and momentum equa-tion are given by

��

�t� �� 0 (1)

��

�t� �� P � � � � (2)

where � is density, � is the velocity vector, P is the pressure,and � is the stress tensor.

A multi-mode viscoelastic constitutive equation [36, 37]is used to describe the rheological behavior of polymermelts as

C�

k �1

2�k�Ck

2 �1

3�IIC

k � ICk �Ck � I� � 0 (3)

where Ck is the elastic strain tensor in the kth mode, C�

k isthe Jaumann derivative of the elastic strain tensor, I is the

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—2006 713

identity tensor, and Ick and IIC

k are the first and the secondinvariant of the elastic strain tensor, Ck.

In a nonisothermal flow under consideration, the energyequation is

�Cp��T

�t� ��T� � k�2T � � Q̇ (4)

where Cp is the specific heat, k is the thermal conductivity,and Q̇ is the rate of heat generated by crystallization as

Q̇ � �X��Hc

d�

dt(5)

where X� is the ultimate degree of crystallinity, �Hc is theheat of fusion per unit mass for the ideal 100% crystallinestate, and � is the relative crystallinity.

The energy dissipation function for viscoelastic flow [38]is calculated as

� � 2s0tr�e2� � �k�1

Nk

4�k2

� �1

3trCk�tr�Ck

1� � trCk2 � tr�Ck

2� � 3� (6)

where s is the nondimensional rheological parameter be-tween 0 and 1, e is the deformation-rate tensor, k and �k arethe viscosity and relaxation time in the kth mode and 0 isthe zero-shear viscosity such as

0�T� �¥k�1

N k�T�

1 � s. (7)

The temperature-dependence of viscosity and relaxationtime in the kth mode is expressed by the Arrhenius-typeequation:

k�T� � Akexp�Tb

T �, �k�T� � Bkexp�Tb

T � (8)

where Tb is the temperature sensitivity related to the acti-vation energy and Ak and Bk are constants.

The viscosity and relaxation time may increase dramat-ically as the polymer melt crystallizes. The crystallinitydependence of viscosity and relaxation time is expressed as[34]

k�T,� � � k�T�exp��2�, �k�T,�� k�T�exp��2� (9)

where � is a constant.In this research, the following assumptions were made

for the simulation of the injection molding process:

1. The thin film approximation2. No slip condition at the wall3. No inertial and body force in the momentum equation4. Thermal conduction in the flow direction is negligible

as compared to conduction in the thickness direction5. No fountain flow effect at the melt front.For one-dimensional incompressible flow in the filling

stage, Eqs. 1 and 2 are expressed as [39]

�

�t�S�P

� x� � 0 (10)

where x is the flow direction and S is the fluidity expressedas

S �1

2 �0

b z2

dz (in Cartesian coordinates) (11)

S �1

2 �0

r r3

dr (in cylindrical coordinates) (12)

where z and r are the thickness and radial directions.In case of simple shear flow, the elastic strain tensor has

the following form:

Ck � �C11,k C12,k 0C12,k C22,k 00 0 1

� (13)

Therefore, the stress tensor, �, is expressed as

�� x,z,t� � 2 �1�T�s�̇�0 1 01 0 00 0 0

� � 2 �k1

N

k�T�

� �C11,k C12,k 0C12,k C22,k 0

0 0 1� (14)

where �̇ � ��x

� zis the shear rate, k �

k

2�kis the

modulus of kth mode and �0

2�k.

The gapwise average velocity (v� x) is expressed as

v� x �1

b �0

b

vxdz . (15)

The governing equation for Ck in Eq. 3 is expressed asfollows:

714 POLYMER ENGINEERING AND SCIENCE—2006 DOI 10.1002/pen

DC11,k

Dt� 2C12,k

�vx

� z�

1

2�k�C11,k

2 � C12,k2 � 1� � 0 (16)

DC12,k

Dt� C22,k

�vx

� z�

1

2�k�C11,k � C22,k�C12,k � 0 (17)

C11,kC22,k � C12,k2 � 1 (18)

whereD

Dtis the material derivative operator.

The shear stress (�12) is expressed as

�12 � �x Z � �̇, �x � ��P

� x. (19)

By intergrating Eq. 15 by part and eliminating �̇ with aid ofEq. 19, the average velocity (V� x) is expressed as

v� x ��x

b �0

b z2

dz �

�x

bS. (20)

From Eqs. 14, 15, and 20, �x is given as

�x �

�2 sv� x � 2¥k�1N k�

0

bzC12,k

�1dz�

�0

b z2

�1dz

(21)

The elastic strain tensor components at a steady state flow�Ck

st) are given by [36]

C11,kst �

�2Xk

�1 � Xk

(22)

C12,kst �

2�̇Xk

1 � Xk(23)

C22,kst �

�2

�1 � Xk

(24)

where Xk � 1 � �1�4��̇�k�2.

The shear viscosity is expressed as

� 0S � �k�1

N 2k

1 � �1 � 4��̇�k�2. (25)

During the packing stage, extra material is forced into thecavity to compensate for volume shrinkage due to solidifi-cation. The packing pressure builds up and the correspond-

ing density increases. To calculate pressure buildup duringthe packing stage and pressure decay during the coolingstage, the following equation [39] is used:

G� x,t��P

�t�

1

b

��S�x�

� x� � F� x,t� (26)

where G�x,t� �1

b � �1

P � P��0

b�1 ��

��dz (27)

F� x,t� � �1

b �0

b�1 ��

�� lnT

�tdz. (28)

Eqs. 26–28 are the unified formulation for filling, packing,and cooling stages. During the filling stage, F� x,t� andG� x,t� are negligible. Therefore, Eq. 26 is reduced to Eq.10. During the packing stage, G� x,t� is important whileF� x,t� is negligible. During the cooling stage, both F� x,t�and G� x,t� are important.

Flow-Induced Crystallization

For an oriented polymer melt, the reduction in entropy,due to the orientation of polymer molecules under flow,causes an elevation of the equilibrium melting point, Tm

0 .According to Flory [7], the elevated melting temperature,Tm, is calculated as

1

Tm�

1

Tm0 �

�S

�Hf(29)

where �S is the change of entropy and �Hf is the change ofenthalpy of crystallization, which is defined as the differ-ence between the heat released by crystallization (�Hf

m) andthe created interfacial energy, �Hf

A as

�Hf � �Hfm � �Hf

A � fcr�Hfm . (30)

The crystallization factor (fcr) is introduced as the crys-tallization enthalpy change divided by the heat of crystalli-zation. Evidently, since the crystallization factor is affectedby the created interfacial energy, this value is very impor-tant in the flow-induced crystallization. Accordingly, thetemperature elevation by the oriented polymer melt is ex-pressed as [20–23]

1

Tm�

1

Tm0 �

�S

fcr�Hfm . (31)

The entropy reduction in Eq. 31 is related to the firstinvariant of kth mode of the elastic strain tensor, IC

k , as [20]


�S � �k

k

T��ICk � 3� (32)

where ICk � C11,k � C22,k � C33,k.

The flow-induced crystallization rate is introduced byreplacing the equilibrium melting temperature (Tm

0 ) in Na-kamura Eq. 2 by means of the elevated melting temperature,Tm as [21–23]

d�

dt� nKs��̇,T��1 � � � � ln�1 � � ��n1�/n (33)

where n is Avrami exponent and Ks�T,�̇� is the modifiedcrystallization rate constant affected by the shear rate as

Ks��̇,T� � �ln 2�1/n� 1

t1/ 2�

0

exp� � U*/R

T � T��exp

� �Kk

T�Tm��̇� � T� f� (34)

with

f �2T

T � Tm��̇�, T� � Tg � 30

where f is the correction factor for the reduction in the latentheat of fusion as the temperature is decreased, R is theuniversal gas constant, U* is the activation energy forsegmental jump of polymer molecules assigned a universal

value of 6284 J/mol, � 1

t1/ 2�

0

is the preexponential factor that

includes all terms influencing the crystallization process butindependent of temperature, and Kk is the nucleation expo-nent. During the molding process, the cooling rate, espe-cially near the mold surface, can be very high (more than100°C/s). This effect was included in the calculation ofcrystallinity during molding. However, it should be notedthat crystallization kinetics at such high cooling rates wasnot measured in view of the absence of suitable techniquefor its determination by using DSC method, as described inthe experimental section.

Residual Stress, Orientation Function, and ElasticRecovery

From Eq. 14, the normal stress differences and shear stressare calculated by

N1� x,z,t� � �xx � �zz � 2 �k�1

N

k C11,k� x,z,t�

� C22,k� x,z,t�� (35)

N2� x,z,t� � �zz � �yy � 2 �k�1

N

k C22,k� x,z,t� � 1� (36)

N3� x,z,t� � �xx � �yy � 2 �k�1

N

k C11,k� x,z,t� � 1� (37)

�12� x,z,t� � �xz � 2 s�1�̇ � 2 �k�1

N

kC12,k� x,z,t�. (38)

In the case of semicrystalline polymers, the overall re-sidual birefringence (�n) is the sum of amorphous (�na)and crystalline (�nc) contributions and form birefringence��nf� as [40]

�n � �nc � �na � �nf � �nc � �nafl. (39)

In this research, for simplicity, it is assumed that thecontribution of form birefringence (�nf) and thermal bire-fringence in the amorphous region (�na

th) to the total resid-ual birefringence is neglected. Therefore, the total residualbirefringence for the semicrystalline polymers is calculatedas [41]

�n � �fc�nc0 � �1 � ��na

fl (40)

n22 � n33 � �fc�nc0 � �1 � ��n22 � n33�a

fl (41)

where � is the absolute crystallinity, fc is the crystallineorientation function, and �nc

0 is the intrinsic crystallinebirefringence.

The residual flow birefringence in the amorphous regioncan be calculated by the stress-optical rule during thenonisothermal flow and the subsequent relaxation. The flowbirefringence in the x–z plane, �na

fl, and in the z–y plane,�n22 n33�a

fl, are given by

�nafl � C��N1

2 � 4�122 (42)

�n22 � n33�afl � C� � N2 (43)

where C� is the stress-optical coefficient. The value offrozen-in normal and shear stresses are calculated as de-scribed earlier [36, 42].

The crystalline orientation function is obtained throughan elastic recovery. The magnitude of the frozen-in recov-erable strain called the elastic recovery is calculated as thetotal shear strain recovered after unloading the shear stress.The transient (�t) and the ultimate (��) elastic recovery areexpressed as [43]


�t�t� � �0

t

�̇e��d�, �� 0

�

�̇e��d�. (44)

By taking shear stresses to zero, the shear rate duringrecovery is calculated from Eq. 38 as

�̇e�t� �¥k�1

N kCxy,k

s�1. (45)

The crystalline orientation function as a function of theprincipal extension ratio is calculated as suggested by Gay-lord [11]

fc �1

2� 3�3

2 � �3 � 1� (46)

where the principal extension ratio (�) is expressed in termsof the ultimate elastic recovery in the simple shear flow as[44]

� ��

2 � 4

2. (47)

Under the uniaxial orientation assumption, only one ori-entation function, for,1, is considered. However, the orienta-tion developed during the injection molding process is morecomplex and cannot be considered as a simple uniaxialorientation. So, a biaxial orientation is considered and twodifferent kinds of assumptions are made. The first assump-tion about a biaxial orientation is that the biaxial behavior isonly introduced by amorphous contribution. In this case,one crystalline orientation (fc) and two different amorphousbirefringence, �na and �n22 n33�a, in the flow and trans-verse directions are used. The second one is the biaxialorientation assumption that considers different crystallineorientations, fc,1 and fc,2, and two different amorphous bire-fringence, �na and �n22 n33�a, in the flow and transversedirections. In the present study, the orientation functions arecalculated as

for,1 ��n

�n0 for,2 �n22 � n33

�n0 (48)

where the birefringence components, �n and �n22 n33�,are determined by the following orientation assumptions:

1. Uniaxial orientation

�n��fc�nc0��1��na

fl (49)

2. Biaxial orientation (amorphous contribution is different,but crystalline contribution is equal)

�n��fc�nc0��1��na

fl (50)

n22n33��fc�nc0��1��n22n33�a

fl (51)

3. Biaxial orientation (both amorphous and crystalline con-tributions are different)

�n � �fc,1�nc0 � �1 � ��na

fl (52)

n22 � n33 � �fc,2�nc0 � �1 � ��n22 � n33�a

fl (53)

fc,1 � fc, fc,2 � 1 � fc. (54)

The total intrinsic birefringence of semicrystalline poly-mers, �n0, is defined as

�n0 � ��nc0 � �1 � ��na

0. (55)

Anisotropy of Thermal Expansion Coefficient andCompressibility

In the case of uniaxially oriented polymers, the relationbetween the linear thermal expansion coefficient (LTEC) inthe direction parallel (�x) and perpendicular (�y) to the flowthe orientation function is given as [45, 46]

�x � �0�1 � for,1�, �y � �0�1 �for,1

2 � . (56)

The linear compressibility, �x and �y, are also calculatedby relations similar to Eq. 56 as [45]

�x � �0�1 � for,1� �y � �0�1 �for,1

2 � . (57)

In Eqs. 56 and 57, �0 and �0 are the linear LTEC andcompressibility in the isotropic state.

In the case of injection moldings, the situation is morecomplex, since the orientation is three-dimensional, beingdifferent in the length, width, and thickness directions.Under assumption of biaxial plane orientation, anisotropicLTEC, and compressibility in the length and width direc-tions are calculated as

�x � �0�1 � for,1�, �y � �0�1 � for,2� (58)

�x � �0�1 � for,1�, �y � �0�1 � for,2�. (59)


Anisotropic Shrinkage in Injection Moldings

The volumetric shrinkage is predicted based on the spe-cific volume, V, and the history that a polymeric melt passesthrough during the injection molding process. The volumet-ric shrinkage is calculated as [25]

Sv �V� i � Vf

V� i

(60)

where V� i is the initial specific volume of melt and Vf is thefinal specific volume at room temperature. Since the poly-mer melt undergoes severe pressure and temperaturechanges in a short time, the initial specific volume can notbe assumed to be the same as that at constant pressure andtemperature conditions [25, 33]. In the present simulation,the initial specific volume is calculated as [25]

V� i �1

�tP � tf� �tf

tp

V� �t�dt (61)

where tf is the filling time, tp is the packing time, and V� �t�is the time-dependent specific volume averaged through thegapwise direction for a particular cross section such that

V� �t� �1

b �0

b

V� z,t�dz. (62)

The specific volume for semicrystalline polymers is calcu-lated from the Spencer-Gilmore equation of state [47] andrepresented as [39]

V�T,P� �1

��T,P��

1

�� i�

R� i

P � P� i

T �1

�� t(63)

if T � Tt

1

�� t� b7exp�bgT� � b9P�, �� i � �� l, R� i � R� l, P� i � P� l (64)

if T � Tt

1

�� t� 0, �� i � �� s, R� i � R� s, P� i � P� s (65)

with the transition temperature (crystallization tempera-ture), Tt � b5 � b6P, and T� � T b5.where b5,b6,b7,b8,b9 and �� l,R� l,P� l, and �� s,R� s,P� s are the ma-terial parameters obtained by fitting Eq. 63 to the specificvolume data measured above and below the crystallizationtemperature for semicrystalline polymers where subscript lrepresents liquid state and s represents a solid state. In Eq.

63, term of1�� t

provides contribution of crystallinity into

specific volume at time of crystallization. For amorphouspolymers, since there is no crystallization, this term is equalto 0. Although the effect of the cooling rate on PVT behav-ior is important, the present study does not consider thiseffect. Obviously, this effect can easily be incorporated intothe present approach of shrinkage calculation, if such datawas available. However, the present literature does notprovide a nonequilibrium PVT equation that is suitable formolding simulation. Also, available literature and thepresent approach do not incorporate the effect of orientationon PVT equation.

The shrinkage changes in the injection-molded productsare determined by two effects: shrinkage due to coolingeffects and shrinkage due to pressure effects. The coolingeffect causes all layers to experience the same thermalcontraction after ejection. In addition to this, each crosssection solidifies under the packing pressure and therefore,at the end of the packing stage, tends to expand proportionalto its value. By these two effects, the total shrinkage can becalculated by [29–32]

Si � �i�Tm � T�� iP� , i � x,y (66)

where Tm is the elevated melting temperature, T� is theambient final temperature, and P� is the packing pressure.

When the temperature of the polymer melt reaches theelevated melting temperature, Tm, crystallization starts totake place. At that time, a recoverable elastic strain, intro-duced during flow in both crystalline and amorphous re-gions, should be frozen-in due to crystallization. However,the recoverable strain in the amorphous region can relaxuntil the glass transition temperature is reached. At the endof packing stage, a recovery of elastic strain in the amor-phous region occurs due to an abrupt pressure decay causedby the release of the packing pressure. This elastic recoverycontributes to the total shrinkage causing an abrupt planardimension change at the end of packing time. The elasticrecovery contribution to the total shrinkage for semicrystal-line polymers is calculated based on the amorphous contri-bution to the total elastic recovery accumulated until theelevated melting temperature:

S�� 1 � �p� �0

tTm

�̇e�t�dt (67)

where �p is the absolute crystallinity at the end of packingtime and tTm is the time when the elevated melting temper-ature is reached.

Without affecting volumetric shrinkage, this elastic re-covery can only be applied to the length and width shrink-ages, since freezing the extensive skin layer puts a geomet-ric constraint that prevents the thickness shrinkage duringthe packing stage. Because of heat transfer inside the moldcavity, the wall region is already solidified (crystallized),while the core region remained in the melt state due to a


slow cooling rate in this region. This crystallized skin layerwas measured on polypropylene moldings in earlier work[22] and found to have substantial thickness. Therefore, thissolidified wall layer acts as a constraint, not allowing furthershrinkage in the thickness direction. Therefore, the elasticrecovery that can be recovered at the end of the packingstage can be applied to cause shrinkage in the length direc-tion and simultaneously applied to cause expansion in thewidth direction. At this moment, the thickness shrinkageremains constant and the volumetric shrinkage is preserved.

Therefore, the final length and width shrinkages arecalculated by

Sx � �x�Tm � T�� xP� � S��. (68)

Sy � �y�Tm � T�� yP� � S��. (69)

From the calculated length and width shrinkage, the thick-ness shrinkage is calculated as

Sz � Sv � �Sx � Sy�. (70)

The contribution of each of these three terms in Eqs. 68 and69 will be elucidated through comparison with experiments.The contribution of thermal expansion and compressibilityto the shrinkage is included during the packing and coolingstages before ejection when the material is in the mold.After ejection, only the contribution of thermal expansion isincluded until room temperature is reached. The contribu-

tion of elastic recovery is considered at the time of releaseof the packing pressure.

EXPERIMENTAL PROCEDURES

Materials and Their Characterization

In this study, various isotactic polypropylenes with dif-ferent molecular weights, namely Pro-fax PP-6523, PP-6723, and PP-6823, that are simply homopolymers with ageneral purpose stabilization package, supplied by Basell,were used. The general material characteristics, includingmelt flow index, polydispersity index and weight-averagemolecular weight (Mw), and the model parameters related tothe crystallization kinetics and rheological model are givenin Table 1. The rheological model parameters with threerelaxation modes were obtained by curve fitting of theviscosity data, measured by Guo [51], to Eqs. 8 and 25. Thisfit and experimental data are shown in Fig. 1. The crystal-lization kinetics parameters [50] are also given in Table 1.The material parameters for PP [48, 49] are shown in Table2. In addition, Table 2 provides the parameters for the PVTbehavior [52], obtained by fitting the experimental data withthe Spencer-Gilmore equations (Eqs. 63–65). These parame-ters were used to calculate specific volume during moldings.

Injection Molding

The injection molding experiments were carried out on aVan Dorn 85T injection molding machine with two test

TABLE 1. Material specification and model parameters for PP 6523, 6723 and 6823.

PP-6523 PP-6723 PP-6823 Ref.

Material specification MFI (dg/min) 4.1 0.92 0.51PI 4.3 3.9 3.9MW 351,000 557,000 670,000

Thermal properties k (W/mK) 0.193Cp � 103 (J/kg K) 2.14 [48,49]�(kg/m3) 900

Crystallization model parameters(1/t1/2)0 � 108 (s1) 5.459 3.858 1.528 [50]Kk � 105 (K2) 3.99 3.93 3.81 [50]n 3 3 3 [50]�H0

0 (J/g) 110.56 96.10 91.02 [50]X� 0.5290 0.4598 0.4355 [50]fcr 0.11 0.11 0.11 [20]Tm

0 (°C) 172 172 172 [48]Tc (°C) 10 10 10 [48]� 124.4 124.4 124.4 [34]

Rheological model parameterss 0.00324 0.000974 0.00118Tb (K) 4,794 4,374 4,548A1 (Pas) 1.48e 1 1.80 2.18A2 (Pas) 1.82e 1 1.12 1.98 [21,22]A3 (Pas) 2.12e 2 8.64e 2 6.18e 2B1 (s) 3.42e 4 5.84e 4 7.28e 4B2 (s) 1.18e 5 2.34e 5 4.48e 5B3 (s) 2.80e 7 8.48e 7 5.64e 7


molds. The standard dumbbell geometry of the mold withthe melt delivery system shown in Fig. 2 was used tomeasure the pressure traces. Three flush-mounted piezo-electric pressure transducers (Kistler 6153B) were installedto trace the pressure buildup and decay during the injectionmolding process and their positions are shown in Fig. 2. Adata acquisition system, which consists of an A/D converterand a PC with Labview software, was used to record thepressure traces during the molding cycle. An ASTM Charpyimpact bar cavity shown in Fig. 3 was used to measure theanisotropic shrinkage. The Charpy impact bar cavity has awidth to thickness ratio of 4, which is not large. Strictlyspeaking, in this case, a two-dimensional cavity fillingshould be considered. However, for simplicity, a one-di-mensional cavity filling was used here. The injection mold-ing experiments were carried out under different processing

conditions by varying packing time (tp), flow rate (Q), melttemperature (T0), and mold temperature (TW) as shown inTable 3. A packing pressure of 44.5 MPa and a cooling timeof 20 s were used.

Anisotropic Shrinkage Measurement

The shrinkage in the three different directions, i.e. thelength, width, and thickness directions of the Charpy impactbar were measured using a digital micrometer (MitutoyoDigimatic) for the length and width directions, and a digitalmicrometer (Monsanto) for the thickness direction. The partdimensions were divided by the mold dimensions to calcu-late the percent shrinkage.

Meshes and Numerical Scheme

The numerical simulations of the injection molding pro-cess of PPs were carried out by the finite difference methodusing ANSI C programming language. Two different molds,dumbbell shape and Charpy impact bar, were used in the

FIG. 1. Flow curves at various melt temperatures as a function of shearrates for PP 6523. Symbols represent the experimental data [51] while linesindicate the nonlinear regression fit to Eqs. 8 and 25.

TABLE 2. Optical properties and material constants in the Spencer-Gilmore PVT equation of state for PP.

PP Ref.

Spencer-Gilmore equation

�� l, �� s (kg/m3) 1.084e � 3, 1.005e � 3

R� l, R� s (J/kg K) 2.203e � 2, 1.029e � 2

P� l, P� s (Pa) 2.650e � 8, 2.402e � 8b5 (K) 3.960e � 2 [52]b6 (K/Pa) 2.250e 7b7 (m3/kg) 8.700e 5b8 (K1) 5.370e 1b9 (kg/m3) 1.260e 7�na

0 0.06 [49]�nc

0 0.029 [49]C (Pa1) 9.0e 10 [48]�0 � 104 (K1) 1.2404 [48]�0 � 109 (Pa1) 0.0763 [48]

FIG. 2. Dimensions of mold, runner system, and sprue for dumbbellshape mold. P1, P2 and P3 are the positions of the pressure transducers.Cross section of the delivery system is circular. Cross section of the gateand cavity is rectangular. (unit: mm).Thickness of cavity and gate is 3.175mm.


present simulation and experiments. The mesh was gener-ated over the mold cavity and the delivery system, whichwas divided into 12 segments and equally-spaced into 182nodes in the flow direction in the both molds. The halfthickness in the delivery system and the cavity was dis-cretized into equally-spaced 65 nodes.

In the filling stage, the temperature at the sprue entrance

is assumed to be uniform and equal to the inlet melt tem-perature, To. First, the flow at the melt front is assumed tobe the fully developed Poiseuille-type. The melt front, xf,progresses regularly one space of �x at the time step of �t,starting from the sprue entrance to the end of cavity. Theinitial guess for the shear rate is given by [36]

�̇� xf,z,j,0� �3v� x

b2 zj (Cartesian coordinate) (71)

�̇� xf,zj,0� �4v� x

b2 zj (cylindrical coordinate). (72)

At the melt front nodes (x � xf), the elastic strain tensor(Cij,k� xf,zj,t�) is calculated by using the steady state formu-lations, Eqs. 22–24, and satisfying the shear rate and pres-sure gradient (Eqs. 19 and 21). Then, the shear rates��̇� xf,zj,t�� are obtained by solving Eqs. 19 and 21–24 bymeans of the Newton-Raphson iterative method [53]. Themelt front moves along the flow direction until it reaches theend of the cavity.

The elastic strain tensor (Cij,k� xi,zj,t�) and shear rate��̇� xi,zj,t�� at the locations other than the melt front (i�f)are determined by solving Eqs. 16–19 and 21 by means ofthe Newton-Raphson iteration method. The values at theprevious time are used as an initial guess for the shear rateand elastic strain tensor. After the mold cavity is filledcompletely, the packing stage starts.

In the packing stage, additional melt is injected into themold cavity to compensate the specific volume change dueto cooling. The flow rate during the packing stage is calcu-lated using the volume change between the mold cavity andthe injected material at the pressure determined from Eqs.26–28 as

Qpack � Vmold � V�T�t�,P�t��

�t (73)

where Vmold is the total volume of mold cavity and V is thetotal volume of injected material. From this flow rate, theshear rate and elastic strain tensor components developedduring the packing stage are obtained.

In the cooling stage, the flow is ceased and therefore thevelocity and shear rates go to zero. The flow stresses de-veloped during the filling and packing stages relax andgoverned by the viscoelastic constitutive equation, Eqs.16–18. It is evident that in the present simulation, a possiblebackflow arising from sudden release of the packing pres-sure is neglected.

RESULTS AND DISCUSSION

Pressure

Figure 4 shows pressure traces during the injection mold-ing of the dumbbell shape cavity mold for filling stage and

FIG. 3. Dimensions of mold, runner system, and sprue for Charpy impactbar mold. Position A and M are the locations where the pressure iscalculated. Cross section of the melt delivery system is circular. Crosssection of the gate and cavity is rectangular. (unit: mm). Thickness ofcavity and gate is 3.175 mm.

TABLE 3. Processing conditions for the injection molding of Charpyimpact bars of PP.

Packing timetp (s)

Flow rateQ (cm3/s)

Melttemperature

Tm (°C)

Moldtemperature

TW (°C)

1 15 49.00 215 252 15 49.00 230 253 15 12.25 230 254 15 122.50 230 255 5 49.00 230 256 30 49.00 230 257 15 49.00 230 608 15 49.00 250 25

Packing pressure Pp, 44.5 MPa and cooling time, 20 s.


beginning of packing stage (a) and complete molding cycle(b) for PP-6523. The pressures obtained at three differentpositions (shown in Fig. 1) are measured at several process-ing conditions. However, in the present study, the results forone condition only, showing the best prediction of shrink-age, are shown. However, it should be noted that the pres-sure traces in the Charpy impact bar cavity were not mea-sured, since the pressure transducers were not installed inthis mold. Both experimental and predicted pressure tracesin the dumbbell shape cavity are shown in these figures withthe symbols representing the recorded pressure at threetransducer locations and lines representing the correspond-ing predicted pressure traces. Generally, the simulated pres-sure traces during the filling stage and at the beginning ofthe packing stage are in fair agreement with the experimen-tal data. It can be seen that the experimental pressure tracesrise quickly to the maximum at the end of the filling stage,overshoot at the beginning of the packing stage, and thenreach constant values corresponding to the packing pressureand maintain their values during packing time. This pres-sure overshoot is due to a sudden imposition of the packingpressure. Namely, it takes some time for the pressure toachieve the steady packing pressure that applied to the melt

at the beginning of the packing stage. However, in thesimulation, the packing pressure is taken as the input pa-rameter and therefore this overshoot is not seen.

In the cooling stage, the experimental pressure decreasesvery fast in comparison with the predicted pressure. Inmolding experiments, the screw suddenly retracts slightlyafter removing the packing pressure due to “spring back”effect, leading to the fast decay of the measured pressure inthe cavity in comparison with the simulated pressure. It isevident that this simulation neglects a possible backflowarising from the sudden release of the packing pressure. It ispossible that this backflow may affect the shrinkage calcu-lation. However, since the shrinkage calculation below in-dicates a fair agreement with experiment, one can infer thatthe effect of backflow on shrinkage in the present experi-ment was insignificant in comparison with other effects thatwere included in simulation.

Volumetric and Anisotropic Shrinkages

Effect of Processing Conditions. Figure 5 illustrates themeasured and calculated volumetric shrinkage in moldingsof Charpy impact bar of PP 6523 as a function of packingtime (a), flow rate (b), melt temperature (c) and moldtemperature (d). To calculate the variation in specific vol-ume, and resulting volumetric shrinkage during the moldingprocess, the material parameters of Spencer-Gilmore equa-tion shown in Table 2, were used. Calculations were carriedout with and without inclusion of the crystallinity effect onviscosity and relaxation time by using Eq. 9. When thecrystallization starts to develop in moldings, the viscosityand relaxation time abruptly increase. This high viscosityachieved during the crystallization of polymer melt mayhinder the flow during the packing stage and reduces theshear rate immediately. Accordingly, because of the hin-dered flow during the packing stage, less material is injectedinto the cavity and the total volumetric shrinkage is in-creased. Fig. 5a shows the calculated and measured volu-metric shrinkage without and with consideration of thecrystallinity effect on viscosity at different packing times.As the packing time increases, the volumetric shrinkagedecreases due to the fact that more material is injected intothe cavity during molding process. Without considering thecrystallinity effect on viscosity, the calculated volumetricshrinkage is slightly smaller than the measured one. Withinclusion of the crystallinity effect on viscosity, the totalvolumetric shrinkage slightly increases, since the flow dur-ing the packing stage is hindered by increased viscosity.This result is in fair agreement with experimental data.Figure 5b–5d shows, respectively, comparison between themeasured and calculated volumetric shrinkages as a func-tion of flow rate, melt, and mold temperature. The flow ratehas very little effect on the volumetric shrinkage. Increasingthe melt temperature and mold temperature introduced anincrease in the measured and calculated volumetric shrink-age. According to Isayev and Hariharan [25], volumetricshrinkage increases as the melt temperature increases due to

FIG. 4. Experimental (symbols) and simulated (lines) pressure traces ofdumbbell shape mold at locations P1, P2, and P3 of Fig. 2 for PP 6523 atprocessing conditions of Pp � 44.8 MPa, tp � 15 s, Q � 49.0 cm3/s, T0 �230°C, and Tw � 25°C.


the higher volume contraction caused by a larger differencebetween the melt and mold temperature. Simultaneously,shrinkage is reduced at high melt temperature, since theviscosity is low and more additional material can be sup-plied into the cavity during the packing stage. This addi-tional material may reduce the volumetric shrinkage duringmoldings. Therefore, there is a competition between thesetwo phenomena. In results, higher melt temperature showedlower volumetric shrinkage for polystyrene. However, inthis study on PP, both measured and calculated volumetricshrinkages show an increase as melt temperature increases.In contrast to amorphous polymers, semicrystalline poly-mers, due to the crystallization, have a larger volume con-traction and show a larger volumetric shrinkage. Therefore,because of this larger volume contraction, higher melt tem-perature gave higher shrinkage. In the case of mold tem-perature effect, Han and Wang [33] found similar resultsthat lower mold temperature gave lower volumetric shrink-age for PET molded samples. At higher mold temperature,there will be an additional cooling effect after ejection,namely cooling down to the room temperature. This addi-tional cooling introduces higher shrinkage at the highermold temperature. In all cases, the results with inclusion ofthe crystallinity effect on viscosity show larger shrinkagesthan the ones without crystallinity effect. The flow rate andmelt temperature show little effect on the volumetric shrink-age. As the mold temperature increases, the volumetricshrinkage increases. The calculated volumetric shrinkages

with inclusion of the crystallinity effect on viscosity showsa better agreement with the experimental results than thosewithout the crystallinity effect on viscosity. Therefore, allfurther results are calculated with inclusion of the crystal-linity effect on viscosity by using Eq. 9.

The measured and predicted anisotropic in-planeshrinkages based on the different assumptions of orien-tation are shown in Fig. 6. Calculations were carried outwith inclusion of the elastic recovery contribution at theend of packing stage, as indicated by Eqs. 68 and 69. Byusing various orientation assumptions, discussed earlier,the length shrinkage is the same, while the width shrink-age is changed. In all cases, the calculated length shrink-age decreases with increasing packing time, while thecalculated width shrinkage shows the opposite trend.This is due to the fact that the heat transfer in the widthdirection is not considered in this simulation. The widthshrinkage determined by using assumption 2, biaxialorientation is affected by only amorphous contribution,Eqs. 48, 50 and 51, and shows the best agreement withexperimental data. Therefore, the calculated results be-low were obtained based on assumption 2.

In Fig. 7, the anisotropic shrinkage is calculated withoutinclusion of the elastic recovery based on the biaxial orien-tation of assumption 2 (Eqs. 48, 50, and 51). The results arecompared to the case with inclusion of the elastic recoveryat the end of packing stage, shown in Fig. 6b. Withoutinclusion of the elastic recovery contribution, the third term

FIG. 5. The measured (symbols) and predicted (lines) volumetric shrinkage for PP 6523 as a function ofpacking time (a), flow rate (b), melt temperature (c), and mold temperature (d) at Pp � 44.8 MPa with andwithout inclusion of crystallinity effect on viscosity


in Eqs. 66 and 67, the predicted length and width shrinkagesdid not show any differences, in contrast to the experimentaldata.

In the case of polypropylenes, among various contribu-tions (LTEC, compressibility and elastic recovery) toshrinkage, the main contribution to the in-plane anisotropy

of shrinkage comes from elastic recovery followed by con-tributions from LTEC and compressibility.

To investigate the compressibility effect on shrinkage,the anisotropic shrinkage as a function of packing time werecalculated with and without inclusion of the compressibilityeffect. Calculations are carried out based on the thermaleffect and elastic recovery at the end of packing. The resultsare shown in Fig. 8 for shrinkage in the length (a), width (b),and thickness (c) directions. With inclusion of the com-pressibility effect, the second term in the Eqs. 68 and 69, thepredicted results show a better agreement with experimentaldata for various shrinkages. Therefore, the compressibilityeffect on shrinkage was found to be significant in theshrinkage of PP.

Finally, all three different anisotropic shrinkages arecalculated by using Eqs. 68–70. Comparisons of the mea-sured and predicted results are shown in Fig. 9 as a functionof packing time (a), flow rate (b), melt temperature (c), andmold temperature (d) for PP 6523. The biaxial orientationfunctions were calculated based on the biaxial amorphousbirefringence and one crystalline orientation function asindicated by Eqs. 48, 50, and 51 (assumption 2). Calcula-tions are carried out based on inclusion of the elastic recov-ery contribution at the end of packing stage and the com-pressibility effect, as indicated by Eqs. 68–70. Thecrystallinity effect on the viscosity and relaxation time wasalso included. First, the effect of packing time on the aniso-tropic shrinkage is shown in Fig. 9a. Since the anisotropicshrinkage is predicted based on the volumetric shrinkage,orientation function and elastic recovery and these proper-ties are functions of spatial position, the local anisotropicshrinkage can also be predicted by using the present ap-proach. However, because of the fact that experimental datafor local shrinkage values was not available, the averageshrinkage values in the flow, transverse and thickness di-rections are compared. Among all the shrinkages, the thick-

FIG. 6. The measured (symbols) and predicted (lines) anisotropic in-plane shrinkages for PP 6523 as a function of packing time with inclusionof elastic recovery at the end of packing stage and based on the uniaxialorientation (assumption 1) (a), biaxial orientation (assumption 2) (b), andbiaxial orientation (assumption 3) (c). Processing conditions: Pp � 44.8MPa, Q � 49.0 cm3/s, T0 � 230°C, and TW � 25°C.

FIG. 7. The measured (symbols) and predicted (lines) anisotropic shrink-ages for PP 6523 as a function of packing time without inclusion of elasticrecovery and based on the biaxial orientation. Processing conditions: Pp �44.8 MPa, Q � 49.0 cm3/s, T0 � 230°C, and TW � 25°C.


ness shrinkage is most strongly affected by the packingtime. The calculated thickness shrinkage shows a fair agree-ment with experimental data. However, at the longest pack-ing time of 30 s, the predicted thickness shrinkage shows alower value than the measured one. This is possibly due to

the fact that the simulation does not consider the gatefreezing effect, which may occur during this experimentalrun. The calculated length shrinkage shows a better agree-ment with experimental data. Namely, it decreases withincreasing packing time. However, the predicted widthshrinkage shows the opposite trend. It increases with pack-ing time. Evidently, this is due to the fact that heat transferin the width direction is not included in this simulation.Effects of the flow rate, melt temperature, and mold tem-perature on shrinkages in the length, width, and thicknessdirections are noted in Fig. 9b–9d, respectively. The flowrate shows little effect on all the anisotropic shrinkages. Thecalculated shrinkages in the length, width, and thicknessdirection are in a fair agreement with the experimental data.As the melt temperature increases, the thickness shrinkageincreases while the length and width shrinkages show a littledifference. This occurs because the thermal contraction isincreased due to larger difference between the melt andmold temperatures when higher melt temperature is used.Similarly, as the mold temperature increases, the thicknessshrinkage increases and the length and width shrinkagesshow a little difference, too. The predicted shrinkage valuesin the length, width, and thickness directions show a fairagreement with the experimental data. Therefore, the cal-culation of anisotropic shrinkage values in moldings bymeans of Eqs. 68–70 is a suitable approach for their pre-diction.

In many cases, it is well known that the thickness shrinkageis significantly larger than the in-plane shrinkage. In thepresent experiments, the thickness shrinkage is also alwaysmuch larger than the length and width shrinkage. Moreover,the length shrinkage is slightly larger than the width shrinkage.According to the simulated results, the most important pro-cessing parameter to control shrinkage and orientation is thepacking time. Especially, the thickness shrinkage is stronglyaffected by the packing time. At the same time, the melttemperature and mold temperature can also affect the thicknessshrinkage. As the melt and mold temperature increase, thethickness shrinkage also increases. With the variation of flowrate, the thickness shrinkage is only slightly affected. Thelength and width shrinkages are not significantly affected byprocessing conditions including packing time. According tothe comparison between the experimental and simulated re-sults, the thickness shrinkage with inclusion of the crystallinityeffect on viscosity shows a good agreement with the experi-mental results. The length and width shrinkage calculatedincluding the elastic recovery contribution and based on thebiaxial orientation assumption is also in fair agreement with theexperimental data.

Effect of Molecular Weights. To investigate the effectof molecular weight on the shrinkage, the volumetric andanisotropic shrinkage were calculated for three differentmolecular weight polypropylenes. Calculations were car-ried out based on the biaxial orientation functions, asindicated by Eqs. 48, 50, and 51, with inclusion of theelastic recovery, as indicated by Eqs. 68 –70. Figure 10

FIG. 8. The measured (symbols) and predicted (lines) length (a), width(b), and thickness (c) shrinkages for PP 6523 as a function of packing timewith and without inclusion of compressibility and based on inclusion ofelastic recovery and the biaxial orientation. Processing conditions: Pp �44.8 MPa, Q � 49.0 cm3/s, T0 � 230°C, and TW � 25°C.


FIG. 10. The measured (symbols) and predicted (lines) volumetric shrinkage for PPs of different molecularweights as a function of packing time (a), flow rate (b), melt temperature (c), and mold temperature (d) at Pp �44.8 MPa with inclusion of crystallinity effect on viscosity.

FIG. 9. The measured (symbols) and predicted (lines) anisotropic shrinkages for PP 6523 as a function ofpacking time (a), flow rate (b), melt temperature (c), and mold temperature (d) at Pp � 44.8 MPa with inclusionof elastic recovery and based on the biaxial orientation.


shows the measured and predicted volumetric shrinkagefor various iPPs of different molecular weight as a func-tion of processing conditions. As shown in Fig 10a,among various PPs, the lowest molecular weight polypro-pylene, PP 6523, shows the highest decrease of the mea-sured and predicted volumetric shrinkage with the in-creasing packing time. Except for the case of shortestpacking time, the measured and predicted volumetricshrinkage increased with increasing molecular weight.With the variation of flow rate, melt temperature andmold temperature, depicted in Fig. 10b–10d, the mea-sured and predicted volumetric shrinkages slightly in-creased as the molecular weight increases.

Finally, the molecular weight effect on the measured andpredicted anisotropic shrinkages is indicated in Fig. 11. Asmolecular weight increases, length and thickness shrinkagesslightly increased, while the width shrinkage slightly de-creased. It is seen that the predicted results are in good agree-ment with the experimental data.

CONCLUSIONS

A novel approach to predict anisotropic shrinkage in theinjection-molded parts of semicrystalline polymers was pro-posed based on the frozen-in orientation function and elasticrecovery. The predicted results were compared with experi-mental data obtained at the different processing conditionssuch as packing time, flow rate, melt temperature, and moldtemperature. To introduce anisotropy into in-plane shrinkage,anisotropic LTEC and compressibility as a function of orien-tation function were calculated. The total frozen-in orientationfunction was calculated based on the amorphous contributiondetermined from flow birefringence through the stress-opticalrule and the crystalline contribution determined from the crys-talline orientation function obtained from the elastic recovery.

The applicabilities of different orientation assumptions weretested and their effects on the anisotropic in-plane shrinkagewere investigated. To predict the experimentally observed dif-ference between the length and width shrinkages, amorphouscontribution of elastic recovery to the anisotropic shrinkagethat was not frozen-in during the crystallization of melt wasconsidered. In agreement with experiments, the predictedthickness shrinkage was much higher than the predicted lengthand width shrinkages. The length shrinkage was slightly higherthan the width shrinkage. The effect of the crystallinity onviscosity was found to be significant in the calculation of thevolumetric and thickness shrinkage. An inclusion of the crys-tallinity-dependent viscosity in calculations caused an increaseof the calculated volumetric and thickness shrinkage, leadingto a better agreement with the measured ones. The packingtime was found to be one of the most important parametersaffecting the anisotropic shrinkage. The length and widthshrinkage were hardly affected by various processing condi-tions. All predicted anisotropic shrinkages were shown to be ina fair agreement with the experimental results.

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Theoretical and experimental studies of anisotropic shrinkage in injection moldings of semicrystalline polymers