Toward a viscoelastic modeling of anisotropic shrinkage in injection molding of amorphous polymers

Toward a Viscoelastic Modeling of Anisotropic Shrinkagein Injection Molding of Amorphous Polymers

Keehae Kwon, A. I. Isayev, K. H. Kim

Institute of Polymer Engineering, The University of Akron, Akron, Ohio 44325-0301

Received 12 August 2003; accepted 3 March 2005DOI 10.1002/app.22399Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: A novel approach to predict anisotropicshrinkage of amorphous polymers in injection moldings wasproposed using the PVT equation of state, frozen-in molec-ular orientation, and elastic recovery that was not frozenduring the process. The anisotropic thermal expansion andcompressibility affected by frozen-in molecular orientationwere introduced to determine the anisotropy of the lengthand width shrinkages. Molecular orientation calculationswere based on the frozen-in birefringence determined fromfrozen-in stresses by using the stress-optical rule. To modelfrozen-in stresses during the molding process, a nonlinearviscoelastic constitutive equation was used with the temper-ature- and pressure-dependent relaxation time and viscos-ity. Contribution of elastic recovery that was not frozen

during the molding process and calculated from the consti-tutive equation was used to determine anisotropic shrink-age. Anisotropic shrinkages in moldings were measured atvarious packing pressures, packing times, melt tempera-tures, and injection speeds. The experimental results of fro-zen-in birefringence and anisotropic shrinkage were com-pared with the simulated data. Experimental and calculatedresults indicate that shrinkage is highest in the thicknessdirection, lowest in the width direction, and intermediate inthe flow direction. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci98: 2300–2313, 2005

Key words: injection molding; orientation; polystyrene; rhe-ology; simulations

INTRODUCTION

Shrinkage prediction for injection molded parts is in-fluenced by the volumetric shrinkage, flow-inducedresidual stresses and orientation, flow-induced crys-tallization, and heat transfer. Since all these factors areinfluenced by the processing conditions, such as pack-ing pressure, packing time, melt temperature, moldtemperature, injection speed and material, andthermo-physical and rheo-optical properties, as wellas geometric constraints, the prediction of shrinkage,especially of anisotropic linear shrinkage, is quite acomplex issue. Clearly, the shrinkage anisotropy inmoldings cannot be predicted based on volumeshrinkage alone. Therefore, novel methodology foraccurate prediction of development of anisotropicshrinkage of molded products is required.

For amorphous polymers, it is well known that theflow and thermal stresses become residual due to thepassage through its glass transition temperature. Al-though the residual flow stresses are an order of mag-nitude smaller than the residual thermal stresses,1 onecannot neglect the flow contribution to the overallresidual stresses because the frozen-in orientation interms of residual birefringence during molding is

found to be closely related to the residual flow stressesvia the stress-optical rule.

Attempts to predict frozen-in orientation in mold-ings of amorphous polymers were carried out by sev-eral researchers.2–7 The empirical stress-optical rulewas used to connect the residual stresses to birefrin-gence. Isayev and Hieber2 calculated the three frozen-in birefringence components in injection moldingsduring the filling and cooling stage using the Leonovconstitutive equation. They approximated that theshear and normal stresses start to relax after filling ofthe mold due to cessation of shear flow. Flaman3

simulated the buildup and relaxation of molecularorientation in injection-molded products and investi-gated the influence of the processing conditions, moldelasticity, and pressure dependence of material func-tions on the pressure and birefringence profiles. Re-cently, Isayev et al. studied the residual birefringencein molded disks5,7 and thermal birefringence in freelyquenched plates6 of amorphous polymers. The mea-sured birefringence data were compared with the re-sults of numerical simulation based on the linear vis-coelastic and photoviscoelastic constitutive equations.

The main origin of the anisotropy of properties inmoldings is related to the orientation of molecularchains. Mechanical, optical, and other properties ofpolymeric products can be improved through the con-trol of molecular orientation. The thermal expansioncoefficient, compressibility, and elastic modulus of ori-

Correspondence to: A. I. Isayev ([email protected]).

Journal of Applied Polymer Science, Vol. 98, 2300–2313 (2005)© 2005 Wiley Periodicals, Inc.

ented polymers usually show the anisotropy closelyrelated to orientation. Hellwege et al.8 and Hennig9

studied the relation between thermal expansion anddraw ratio for amorphous polymers and found thelinear thermal expansion along the draw directiondecreases with draw ratio while that in the perpendic-ular direction increases. Hennig9 also showed that theorientation similarly affects anisotropy of the linearcompressibility and linear Young’s modulus. Further-more, Retting10 studied the effect of molecular orien-tation on the mechanical properties and calculated itsinfluence on the linear thermal expansion coefficientsin the parallel and perpendicular directions.

A rapid, nonhomogeneous cooling of a polymericmelt through the glass transition temperature, Tg, in-troduces nonequilibrium volumetric changes, frozen-in orientation, and residual stresses in final products.By following the PVT diagram from the glass transi-tion to ambient conditions, one can obtain the averagevalue of the final product volume.11–13 In particular,Hellmeyer and Menges11 were the first to approachthe modeling of shrinkage by the equation of state.However, their multilayer model for shrinkage duringthe holding pressure phase was found to be only inqualitative agreement with experimental data. Isayevand Hariharan12 incorporated the equation of state(PVT equation) with one-dimensional cavity fillingsimulation to model volumetric shrinkage. They alsoused the first-order rate theory for volumetric changesin conjunction with solving the transient one-dimen-sional heat-conduction equation with a convectiveheat-transfer boundary condition to simulate the den-sity variation. However, the approach to calculateshrinkage was limited because PVT diagrams are suit-able to describe isotropic shrinkage only.

Recently, several new approaches were developedto predict anisotropic shrinkage.14–21 Bushko andStokes14,15 used thermorheologically simple thermo-viscoelastic material model to predict the part shrink-age, warpage, and build-up of residual stresses in theinjection molding process. However, they neglectedflow effects and assumed that in-plane shrinkages areequivalent. Jansen and Titomanlio16–18 proposed asimple thermoelastic model taking into account ther-mal and pressure effect on in-plane shrinkage andPoisson’s ratio on thickness shrinkage. They assumedthat during solidification no relaxation and creep oc-curs and stresses start to build up as soon as thetemperature drops below the solidification tempera-ture. To consider shrinkage in the length and widthdirections, Jansen19 set up a model that predicts vari-ation in modulus, Poisson’s ratio, coefficient of ther-mal expansion, and compressibility in injectionmolded semicrystalline products. Hieber20 used thePVT behavior affected by crystallinity, dependentupon the thermal, pressure, and shear stress history.Based on the modified material constants, he calcu-

lated the time-dependent gapwise shrinkage of amolded specimen prior to ejection from the cavity andcompared it with measured data. Recently, Kennedyand Zheng21 used a residual strain and thermovis-coelastic stress model to calculate the shrinkage coef-ficient and to predict the in-plane shrinkage in theparallel and transverse to the flow direction. However,they applied a hybrid model that used measuredshrinkage to improve the theoretical prediction ofshrinkage and warpage.

In the present study, a new model to predict theanisotropic shrinkage of injection molded products ofamorphous polymer was proposed based on the ori-entation function and elastic recovery determinedfrom the nonlinear viscoelastic constitutive equation.Numerical simulation of the viscoelastic injectionmolding process including filling, packing, and cool-ing stages was carried out. As an example, theGiesekus–Leonov constitutive equation was used tocalculate the flow-induced residual stresses, frozen-inorientation, and elastic recovery. The predicted aniso-tropic shrinkages and birefringence were compared tothe experimental data measured on moldings ob-tained at different processing conditions.

THEORETICAL

Governing equations

The general behavior for an incompressible, noniso-thermal flow is described by transport equations:mass, momentum, and energy equation. The continu-ity and momentum equations are given by:

��

�t � �� 0 (1)

��

�t � � � �� P � � � � (2)

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

The Giesekus–Leonov multi-mode viscoelastic con-stitutive equation2,22 is used to describe the rheologi-cal behavior of polymer melts as:

C�

k �1

2�k�Ck

2 �13 (IIC

k � ICk )Ck � I� � 0 (3)

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

k

is the Jaumann derivative of the elastic strain tensor, Iis the identity tensor, and IC

k and IICk are the first and

the second invariant of the elastic strain tensor, Ck.In a nonisothermal flow under consideration, the

energy equation is

INJECTION MOLDING OF AMORPHOUS POLYMERS 2301

�Cp��T�t � � � �T� � k�2T � � (4)

where Cp is the specific heat, k is the thermal conduc-tivity, and � is the energy dissipation function forviscoelastic flow23 as

� � 2s0tr�e2�

� �k�1

Nk

4�k2 �1

3 trCk � {tr(Ck�1) � trCk} � tr(Ck

2) � 3�(5)

where s is the nondimensional rheological parameterbetween 0 to 1, e is the deformation-rate tensor, k and�k are the viscosity and relaxation time in the kth mode,and 0 is the zero-shear viscosity such as

0�T� �¥k�1

N k�T�

1 � s (6)

The temperature-dependence of viscosity and relax-ation time of the kth mode can be expressed by theWLF type eq. (2) as

k�T� � k�Tr� �aT

aTr

, �k�T� � �k�Tr� �aT

aTr

(7)

where logaT � � logaTg T Tg

��C1�T � Tr�

C2 � T � Tr� T � Tg � is the shift

factor, Tr is the reference temperature dependent onthe polymeric material, and C1 and C2 are constantsdependent on the reference temperature.

The pressure dependence of viscosity and relaxationtime can be expressed as24

k�T, P� � k�T� � exp��P�, �k�T, P� � �k�T� � exp��P�

(8)

where � is the pressure coefficient of viscosity.In this research, the following assumptions are

made for the simulation of the injection molding pro-cess:

a. The thin film approximation.b. No slip condition at the wall.c. No inertial and body force in the momentum

equation.d. Thermal conduction in the flow direction is

negligible with respect to conduction in thethickness direction.

e. No fountain flow effect at the melt front.

For one-dimensional incompressible flow in Carte-sian coordinates in the filling stage, eqs. (1) and (2)are expressed as:24

�

�x �S�P�x� � 0 (9)

where x is the flow direction and S is the fluidityexpressed as

S � �0

b z2

dz in Cartesian coordinate (10)

S �12 �

0

r r3

dr in cylindrical coordinate (11)

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

tensor has the form

Ck � � C11,k C12,k 0C12,k C22,k 0

0 0 1� (12)

Therefore, the stress tensor can be expressed by

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

�� 2 �

k�1

N

k�T�� C11,k C12,k 0C12,k C22,k 0

0 0 1� (13)

where �̇ � �(��x/�z) is the shear rate, k � (k/2�k) isthe modulus of the kth mode, and � (0/2�1).

The gapwise average velocity, ��x, is expressed as

�� x �1b �

0

b

�x dz (14)

The governing equation for Ck in eq. (3) can beexpressed as follows:

DC11,k

Dt � 2C12,k

��x

�z �1

2�k�C11,k

2 � C12,k2 � 1� � 0 (15)

DC12,k

Dt � C22,k

��x

�z �1

2�k�C11,k � C22,k�C12,k � 0 (16)

C11,kC22,k � C12,k2 � 1 (17)

where D/Dt is the material derivative operator.

2302 KWON, ISAYEV, AND KIM

The shear stress, �12, is expressed as

�12 � �xz � �̇, �x � ��P�x (18)

By integrating eq. (14) by part and eliminating �̇with eq. (18), the average velocity, ��x, is expressed as

�� x ��x

b �0

b z2

dz �

�x

b S (19)

From eqs. (13), (14), and (19), �x is given as

�x �

�2 s�� x � 2 ¥k�1N k �

0

b zC12,k

�1dz�

�0

b z2

�1dz

(20)

The elastic strain tensor components at a steadystate flow, Ck

st, are given by2:

C11,kst �

2 Xk

1 � Xk(21)

C12,kst �

2�̇Xk

1 � Xk(22)

C22,kst �

2

1 � Xk(23)

where Xk � 1 � 1 � 4(�̇�k)2.

The shear viscosity can be expressed as

� 0s � �k�1

N 2k

1 � 1 � 4��̇�k�2 (24)

During the packing stage, an extra material is forcedinto the cavity to compensate for volume shrinkagedue to solidification. The packing pressure is built upand the corresponding density is increased. To calcu-late pressure build-up during the packing stage andpressure decay during the cooling stage, the followingequations24 are used:

G�x, t��P�t �

1b

��S�x�

�x � �F�x, t� (25)

where

G�x, t� �1b � 1

P � P� � �0

b �1 ��

�� dz (26)

F�x, t� � �1b �

0

b �1 ��

�� ln T�t dz (27)

Equations (25)–(27) are the unified formulation forthe filling, packing, and cooling stages. During thefilling stage, F(x,t) and G(x,t) are negligible. Therefore,eq. (25) is reduced to eq. (9). During the packing stage,G(x,t) is important while F(x,t) is negligible. Duringthe cooling stage, both F(x,t) and G(x,t) become impor-tant.

Residual stress, orientation function, and elasticrecovery

From eq. (10), the normal stress differences and shearstress are calculated by:

N1�x, z, t� �xx � �zz � 2 �k�1

N

kC11,k�x, z, t�

� C22,k�x, z, t�� (28)

N2�x, z, t� �zz � �yy

� 2 �k�1

N

kC22,k�x, z, t� � 1� (29)

N3�x, z, t� �xx � �yy

� 2 �k�1

N

kC11,k�x, z, t� � 1� (30)

�12�x, z, t� �xz � 2 s�1�̇ � 2 �k�1

N

kC12,k (31)

In general, for amorphous polymers, the total residualbirefringence, �na, is the sum of the flow, �na

fl, andthermal birefringence, �na

th.

�na � �nafl � �na

th (32)

However, for simplicity here, it is assumed that thecontribution of residual thermal birefringence, �na

th, tothe total residual birefringence is negligible. Although,in general, this assumption is not necessarily true butfor some polymers, for example, polystyrene, such anassumption can be made. Therefore:


�na � �nafl (33)

The residual flow birefringence for amorphous poly-mers can be calculated by the stress-optical rule dur-ing the nonisothermal flow and the subsequent relax-ation. The flow birefringence in the x-z plane, �na

fl, isgiven by:

�nafl � C��T� � ��x, z, t�

� C��T�N12�x, z, t� � 4�12

2 �x, z, t� (34)

where C�(T) is the time-dependent stress-optical coef-ficient as a function of temperature. The value of fro-zen-in normal and shear stresses are calculated asdescribed earlier.2,4

Similarly, the birefringence in the z-y and x-y direc-tions is calculated as:

nzz � nyy � C��T� � N2

nxx � nyy � C��T� � N3 (35)

From the residual flow birefringence, one can cal-culate the amorphous orientation function by

for,1 ��na

�na0 , for,2 �

nzz � nyy

�na0 , for,3 �

nxx � nyy

�na0 (36)

where �na0 is the amorphous intrinsic birefringence.

The calculations of residual stresses and birefringenceare continued until the temperature at the particularposition drops below the glass transition tempera-ture, Tg.

The total strain during flow consists of the recover-able strain, �e, and the inelastic or plastic strain, �p, as:

�total � �e � �p (37)

The magnitude of the frozen-in recoverable strain,called the elastic recovery, is total shear strain recov-ered after unloading the shear stress. The transient, �t,and the ultimate, � , elastic recovery can be expressedas25:

�t�t� � �0

t

�̇e�� d� (38)

� � �0

�̇e�� d� (39)

The shear rate during recovery is calculated fromeq. (31) by letting shear stresses to zero as

�̇e�t� �¥k�1

N kC12,k

s�1(40)

Anisotropy of thermal expansion coefficient andcompressibility

In the case of uniaxial orientation, the linear thermalexpansion coefficients (LTEC) can be evaluated byknowing the orientation function. For uniaxially ori-ented polymers, the relation between the thermal ex-pansion in the direction parallel, �x, and perpendicu-lar, �y, to the flow and the orientation function is givenas9,26:

�x � �0�1 � for�, �y � �0�1 �for

2 � (41)

The linear compressibility, �x and �y, can also becalculated by relations9 similar to eq. (41):

�x � �0�1 � for�, �y � �0�1 �for

2 � (42)

In eqs. (41) and (42), �o and �o are the LTEC andlinear compressibility in the isotropic state.

In the case of injection moldings, the situation ismade complex since the orientation is three dimen-sional, being different in the length, width, and thick-ness directions. Under the assumption of biaxial planeorientation, from orientation functions, one can calcu-late the anisotropic linear thermal expansion coeffi-cient and compressibility in the length and width di-rections as:

�x � �0�1 � for,1�, �y � �0�1 � for,2� (43)

�x � �0�1 � for,1�, �y � �0�1 � for,2� (44)

Anisotropic shrinkage in injection moldings

The prediction of volumetric shrinkage has beenbased on the specific volume, V, history that a poly-meric melt passes through during the injection mold-ing process. The volumetric shrinkage can be calcu-lated as12:

SV �V� i � Vf

V� i

(45)

where V� i is the initial specific volume of melt and Vf isthe final specific volume at room temperature. Sincethe polymer melt undergoes severe pressure and tem-perature changes in a short time, the initial specificvolume cannot be assumed to be as that at constantpressure and temperature conditions. Two ap-


proaches were suggested to calculate this volume, in-cluding the average specific volume until the end ofthe packing stage12 and the specific volume at the endof the packing stage.27 In the present simulation, theinitial specific volume was calculated as12:

V� i �1

�tp � tf� �tf

tp

V� �t� dt (46)

where tf is the filling time, tp is the packing time, andV� (t) is the specific volume averaged in the gapwisedirection at any time for the particular cross sectionsuch that

V� �t� �1b �

0

b

V�z, t� dz (47)

The specific volume for amorphous polymers can becalculated from the Spencer–Gilmore equation ofstate28 as

V�T, P� �1

��T, P��

1��

�R�

P � P�T (48)

where

�� l, R� � R� l, P� � P� l if T � Tt

�� s, R� � R� s, P� � P� s if T � Tt

with Tt � b5 � b6P.where b5, b6 are the parameters of the pressure-depen-dent transition temperature, Tt, and �� l, R� l, P� l and ��s, R� s,P� s are the material parameters that are obtained by thefitting of eq. (48) to the experimental data on thespecific volume measured above and below the glasstransition temperature for amorphous polymers (asshown in Figure 7).

The shrinkage changes in the injection moldedproducts are determined by two effects: shrinkage dueto cooling effect and shrinkage due to pressure effect.The cooling effect causes all layers to experience thesame thermal contraction after ejection. In addition tothis, each cross section solidifies under the differentpressures and tends to expand proportional to thatsolidification pressure. Therefore, the total shrinkagecan be calculated as follows16–19:

Si � �i�Ts � T � � �iP� s, i � x, y (49)

where Ts is the solidification temperature (Tg for amor-phous polymers), T is the ambient final temperature,and P� s is the average solidification pressure at eachcross section, which is calculated as

P� s�x� �1

�tp0 � tf� �tf

tp0

P�x, t� dt (50)

or18

P� s�x� �1b �

0

b

Ps�x, z� dz (51)

where tP0 is the time at which the pressure goes to zeroand Ps(x, z) is the pressure when the local temperatureat each z reaches Tg.

However, it was found (as shown below) that thiscompressibility contribution to anisotropic shrinkagedue to the second term in eq. (49) with the solidifica-tion pressure calculated via eqs. (50) or (51) is negli-gible due to the small difference in the anisotropiccompressibility introduced by the low value of orien-tation function developed during molding. Therefore,anisotropic in-plane shrinkages are calculated basedon the thermal effect only as

Si � �i�Ts � T �, i � x, y (52)

At the end of the packing stage, at the positionwhere T � Tg, a recovery of elastic strain, introducedduring flow, occurs due to abrupt pressure decaycaused by release of the packing pressure. This elasticrecovery contributes to the total shrinkage, causing anabrupt planar dimension change at the time of the endof packing. It can be calculated as the total elasticrecovery accumulated until the end of the packingstage as

S� � �

0

tP

�̇e�t� dt (53)

Without affecting any volumetric shrinkage, thiselastic recovery can be applied to the length and widthshrinkages only, due to freezing of the extensive skinlayer that causes geometric constraints that preventthickness shrinkage during the packing stage. Due toheat transfer inside the mold cavity, the wall region isalready solidified, while the core region still remainsin the melt state due to the slow cooling rate in thisregion. This solidified wall region acts as a constraint,not allowing further shrinkage in the thickness direc-tion. Therefore, the contribution of elastic recoverythat is not frozen can be applied to cause shrinkage inthe length direction and simultaneously applied tocause expansion in the width direction. At this mo-ment, the thickness shrinkage remains constant. Thisway, the volumetric shrinkage is preserved. The ex-pansion caused by the pressure removal and its con-


tribution to the anisotropic shrinkage is very smalldue to the very small value of compressibility. There-fore, the contribution is significantly smaller than thatof elastic recovery change at the end of the packingstage.

Therefore, the final length and width shrinkages arecalculated by

Sx � �x�Ts � T � � S� (54)

Sy � �y�Ts � T � � S� (55)

Then, the shrinkage in the thickness direction can becalculated based on the volumetric shrinkage andanisotropic in-plane shrinkage as

Sz � S� � �Sx � Sy� (56)

EXPERIMENTAL

Materials

In this study, polystyrene, Styron 615-APR suppliedby Dow Chemical Company, is used as a representa-tive of the amorphous polymers. The rheologicalmodel parameters with three relaxation modes wereobtained by curve fitting of the viscosity data, mea-sured by Lee et al.,29 to eqs. (7) and (24). Experimentaldata by Lee et al. were obtained by using two instru-ments: Rheometrics Mechanical Spectrometer and In-stron Capillary Rheometer. This was done to cover awide range of shear rates required for injection mold-ing simulation. This fit and experimental data areshown in Figure 1. The specific volume data for poly-styrene was obtained from Oels and Rehage30 andfitted to the Spencer–Gilmore equation, eq. (48). This

fit and experimental data are shown in Figure 2. Thematerial parameters for PS 615 are shown in Table I.

Injection molding

The injection molding experiments were carried outon a Van Dorn 55F screw injection molding machine.An ASTM Charpy impact bar cavity, shown in Figure3, was used. The injection molding experiments werecarried out under different processing conditions with

Figure 1 Flow curves at various processing temperaturesas a function of shear rates for PS 615. Symbols represent theexperimental data,29 and lines indicate the nonlinear regres-sion fit to eqs. (6), (7), and (17).

Figure 2 Specific volume versus temperature for PS 615 atdifferent pressures. Symbols represent the experimental da-ta,30 and lines indicate fit by the Spencer–Gilmore equation.

TABLE IMaterial and Model Parameters of PS 615

PS Ref.

WLF Equation:C1 3.6C2 (K) 131.9 [5]Tr (K) 473.5

Pressure-Dependent viscosity� (Pa�1) 4.3e�8 [24]

Leonov model:s 0.001Tr (K) 473.151 (Pa � s), �1 (s) 2709.7, 0.18182 (Pa � s), �2 (s) 186.8, 0.01073 (Pa � s), �3 (s) 96.7, 3.1861e�4

Spencer–Gilmore equation:�� l, ��s (Kg/m3) 1.3005e�3, 1.1480e�3R� l, R� s (J/Kg K) 2.0445e�2, 0.4157e�2P� l, P� s (Pa) 3.6600e�8, 1.3962e�8b5 (K) 3.6236e�2b6 (K/Pa) 3.0333e�7

�na0 �0.195 [31]

C� (Pa�1) 4.8e�9 [32]�0 (K�1) 0.75e�4 [33]�0 (Pa�1) 0.0667e�9 [33]


varying packing pressure, packing time, melt temper-ature, and injection speed, as shown in Table II. Themold temperature of 25°C and cooling time of 30 s areused.

Anisotropic shrinkage and birefringencemeasurements

The shrinkage in the three different directions, that is,the length, width, and thickness directions, of injectionmolded samples is measured by a Digimatic caliper(Mitutoyo) for the length direction, and a Digimaticmicrometer (Starett) for the width and thickness direc-tions. The part dimensions are divided by the molddimensions to calculate the percent shrinkage. Thewidth and thickness dimensions are measured at threedifferent positions along the flow direction.

To measure the birefringence distribution of injec-tion molded PS samples, an Optical Microscope ofLeitz Laborlux 12 POL S was used equipped with a 4th

order compensator (Leitz Laborlux). The specimenswere prepared by cutting the injection molded bars.The sample of 10 mm thickness was cut at the midwayof a molded sample by using a diamond saw (BuehlerIsomet). Then, the cut sample was cut parallel to thex-z plane by a diamond saw. The obtained slice of thesample has a thickness of around 300 m. Birefrin-gence was determined by the phase difference (retar-dation) between two perpendicular plane-polarizedwave motions through a sample and calculated asfollows:

�n ��

d (57)

where � is the retardation and d is the sample thick-ness.

Meshes and numerical scheme

The numerical simulations of the injection moldingprocess of PS were carried out by the finite differencemethod using ANSI C programming language. Themesh was generated over the mold cavity and thedelivery system, which were divided into 12 segmentsand equally-spaced 182 nodes in the flow direction.The half thickness in the delivery system and thecavity was discretized by 65 nodes equally spaced.

In the filling stage, the temperature is assumed to beuniform and equal to inlet melt temperature, To. First,flow at the melt front was assumed to be the fullydeveloped Poiseuille-type as the front progresses fromthe sprue to the end of the cavity. At the melt frontnodes, x � xf, the elastic strain tensor, Cij,k(xf, zj, t), canbe calculated by using the steady state formulations,eqs. (21)–(23), and satisfying the shear rate and pres-sure gradient, eqs. (18) and (20). Then the shear rate,�̇(xf, zj, t), can be obtained by solving eqs. (18) and(20)–(23) by means of the Newton–Raphson iterativemethod. The melt front moves along the flow directionuntil it reaches the end of the cavity. The elastic straintensor, Cij,k(xi, zj, t), and shear rate, �̇(xi, zj, t), at thelocations other than the melt front, were determinedby solving eqs. (15)–(18) and (20) by means of theNewton–Raphson iterative method. The values at theprevious time are used as an initial guess for the shearrate and elastic strain tensor. After the mold cavity isfilled completely, the packing stage starts.

In the packing stage, an additional melt is injectedinto the mold cavity to compensate the specific vol-ume change due to cooling in this filling stage. Theflow rate during the packing stage was calculated by

Figure 3 Dimensions of the sprue, runner, gate, and moldcavity for the ASTM impact bar mold. Cross section of themelt delivery system is circular, and that of the gate andcavity is rectangular. Thickness of cavity and gate is 3.18mm. (All dimensions in mm).

TABLE IIProcessing Conditions for Injection Molding Process

Packingpressure

(MPa)

Packingtime(sec)

Flow rate(cm3/sec)

Melttemperature

(°C)

1 34.5 5 35.9 2202 20.67 5 35.9 2203 68.9 5 35.9 2204 34.5 2 35.9 2205 34.5 10 35.9 2206 34.5 15 35.9 2207 34.5 5 8.95 2208 34.5 5 107.4 2209 34.5 5 35.9 20010 34.5 5 35.9 240


using the volume change between the mold cavity andthe injected material at the pressure determined fromeqs. (25)–(27). From the flow rate during packing, theshear rate and elastic strain tensor developed duringthe packing stage were obtained.

In the cooling stage, the velocity and shear rates areset to zero and the flow stresses developed during thefilling and packing stages relax and are governed bythe viscoelastic constitutive equation of eqs. (15)–(17).It is evident that this simulation neglects a possiblebackflow arising from the sudden release of the pack-ing pressure.

RESULTS AND DISCUSSION

Simulated pressure

The calculated pressure traces obtained at two posi-tions (see Fig. 3) in the runner (position A) and mid-way of the cavity (position M) are shown in Figure 4.The filling, packing, and cooling stages of pressurechange can be distinguished in this Figure. The releaseof the packing pressure causes a fast decay of pressurein the cavity and runner.

Orientation function and anisotropic properties

Figure 5 shows the measured and calculated frozen-inbirefringence at the midway of moldings obtained atdifferent injection speeds. During the filling stage, thehigh shear rate near the wall lets polymer moleculesbe aligned along the flow direction. Therefore, thebirefringence is mainly developed near the wall regiondue to the high shear rate and fast cooling in thisregion. The birefringence developed during the fillingstage near the wall region gets frozen-in when melt

passes through the glass transition temperature due tothe fast cooling rate. In the packing stage, the birefrin-gence starts to develop in the core region due toadditional packing flow. The birefringence layer nearthe wall due to the flow effect is broad and low at lowinjection speed, while it is narrow and high at highinjection speed. As the injection speed increases, themaximum birefringence near the wall increases due toenhanced flow by the high shear flow.

From the calculated birefringence, the orientationfunctions can be calculated. In this study, both as-sumptions on uniaxial and biaxial orientation are usedto find the anisotropic thermal expansion and com-pressibility as a function of orientation. In the injectionmolding process, the orientation is mainly developedin the flow direction. Under the assumption of uniax-ial orientation, the orientation function in the x-z di-rection was used. Under the assumption of biaxialorientation, orientation functions in the x-z and y-zdirections were considered.

Figure 6(a) illustrates the variation of orientationfunction under the uniaxial orientation assumptioncalculated by eq. (36) at different packing times. Theorientation function increases due to the increasedfrozen-in birefringence as the packing time increases.However, the value of the orientation function devel-oped in amorphous polymers is very low (��1), asshown in Figure 6(a). Under the assumption of biaxialorientation, orientation functions in the length andwidth directions were calculated at different packingtimes, and results are shown in Figure 6(b). Both ori-entation functions increase with increasing packingtime due to an increase of birefringence. However, thevalue of the orientation function is lower in the widthdirection since birefringence in that direction is low.

Figure 7 shows the calculated linear thermal expan-sion coefficients (LTEC) in the length and width direc-

Figure 4 Calculated pressure traces in runner (position A)and midway of cavity (position M), as shown in Figure 2.Processing conditions: PP � 34.5 MPa, tP � 5 s, Tm � 200°C,and Q � 35.9 cm3/s.

Figure 5 Gapwise distribution of measured (symbols) andcalculated (lines) birefringence at midway of the cavity atvarious injection speeds. Processing conditions: PP � 34.5MPa, tP � 5 s, and Tm � 200°C.


tions based on the orientation functions obtained atdifferent packing times using the assumption of uni-axial and biaxial orientation, respectively. As shownin Figure 7(a), in the case of the uniaxial orientationassumption, the LTEC in the length and width direc-tions are calculated using one orientation function andtheir values have different dependence on orientation.The LTEC in the flow direction decreases, while theone in the width direction increases with the orienta-tion function, as also indicated by eq. (41). The LTECin the flow direction is lower than in the width direc-tion at the same amount of orientation. The linearcompressibility would vary with the orientation func-tion similarly as the LTEC, as given by eq. (42). How-ever, in the case of the biaxial assumption, the LTEC inboth the flow and width directions were calculatedusing two different orientation functions, but theyhave the same dependence on the orientation func-tion, as indicated in Figure 7(b). The LTEC in both theflow and width directions decreases with the orienta-

tion functions, as given by eq. (43). In fact, the LTEC inthe flow direction is lower than the LTEC in the widthdirection because of the higher orientation function inthe former direction. However, it is seen from Figure7 that the LTEC in the flow and transverse directionsis not significantly different.

Also, it should be mentioned that the value of thedeveloped orientation function in amorphous poly-mers in injection moldings is too small to distinguishthe differences between the uniaxial and biaxial ori-entation assumptions.

Volumetric and anisotropic shrinkages

Using the fitted parameters shown in Table I, thevariation of specific volume and resulting volumetricshrinkage were calculated during the molding pro-cess. Figure 8(a) shows the measured and calculatedvolumetric shrinkage in moldings as a function of

Figure 6 Predicted orientation functions as a function ofpacking time using the assumption of uniaxial (a) and bi-axial (b) orientation. Processing conditions: PP � 34.5 MPa,Tm � 200°C, and Q � 35.9 cm3/s.

Figure 7 Predicted linear thermal expansion coefficients asa function of orientation function using the assumption ofuniaxial (a) and biaxial (b) orientation obtained at differentpacking times. Processing conditions: PP � 34.5 MPa, Tm �200°C, and Q � 35.9 cm3/s.


packing times. Calculations were carried out with andwithout inclusion of the pressure effect on viscosity(see eq. (8)). As the packing time increases, the volu-metric shrinkage decreases due to the fact that morematerial is injected into the cavity at longer packingtimes acting during the molding process. With inclu-sion of the pressure-dependent viscosity, the volumet-ric shrinkage slightly increases since the flow duringthe packing stage is hindered by increased viscosity.Similar to the packing time effect, the volumetricshrinkage also decreases as the packing pressure in-creases, as shown in Figure 8(b).

Comparisons between the measured and predictedanisotropic shrinkages are shown in Figures 9 to 12. Inthese Figures, the effect of processing conditions onanisotropic shrinkage, developed in molded PS strips,

is shown. In particular, Figure 9 shows the length,width, and thickness shrinkages as a function of pack-ing time calculated based on inclusion of both thermaland compressibility effects, as indicated by eqs. (49)and (56). Again, calculations were carried out withand without inclusion of the pressure effect on viscos-ity. But this effect is found to be insignificant. Theanisotropic shrinkage as a function of packing timewas also calculated based on inclusion of the thermaleffect, but without inclusion of the compressibilityFigure 8 Measured (symbols) and predicted (lines) volu-

metric shrinkage with (solid line) and without (dashed line)inclusion of pressure-dependent viscosity as a function ofpacking time at processing conditions of PP � 34.5 MPa, Tm� 200°C, and Q � 35.9 cm3/s (a) and packing pressure atprocessing conditions of tP � 5 s, Tm � 200°C, and Q � 35.9cm3/s (b).

Figure 9 Measured (symbols) and predicted (lines) aniso-tropic shrinkages due to contributions of the thermal andcompressibility effect as a function of packing time with(dotted dashed line) and without (dotted line) inclusion ofpressure-dependent viscosity at processing conditions of PP� 34.5 MPa, Tm � 200°C, and Q � 35.9 cm3/s.

Figure 10 Measured (symbols) and predicted (lines) aniso-tropic shrinkages due to contribution of the thermal effectalone as a function of packing time with (dotted dashed line)and without (dotted line) inclusion of pressure-dependentviscosity at processing conditions of PP � 34.5 MPa, Tm� 200°C, and Q � 35.9 cm3/s.


effect, as indicated by eqs. (52) and (56). The results areshown in Figure 10. All three anisotropic shrinkagesdecrease with increasing packing time, with the thick-ness shrinkage most strongly affected by the packingtime among all the shrinkages. The length and widthshrinkages show a little difference with packing timesince the orientation function developed in an amor-phous polymer is too small, although it increases withpacking time, as evident from Figure 6a. The thicknessshrinkage shows the slightly better agreement withexperimental data in the case of without inclusion ofcompressibility, as indicated by Figure 10. However,the predicted length and width shrinkage do not showany difference. They do not predict well the experi-mental behavior because the developed orientation istoo small to affect anisotropic shrinkage.

To predict experimentally observed difference be-tween the length and width shrinkages, the contribu-tion of elastic recovery to the in-plane shrinkage isintroduced, as given by eqs. (53)–(55). At the end ofthe packing time, an abrupt release of packing pres-sure occurs, flow ceases, and the pressure in the cavitygradually decays. At that time, an unfrozen core layer,where its temperature is above the glass transitiontemperature, still remains, as schematically indicatedby Figure 11. As the pressure in the cavity decays, theelastic recovery in this unfrozen layer can recover veryquickly and introduce a dimension change in themoldings. Due to the heat transfer inside the moldcavity and the following geometric constraint, theelastic recovery that was not frozen during the pack-ing stage contributes to shrinkage in the length direc-tion and expansion in the width direction. Therefore,the final anisotropic shrinkages in the length andwidth directions were calculated by inclusion of ther-mal and elastic recovery effects given in eqs. (53)–(55).

Figures 12(a) and b show the anisotropic in-planeshrinkage developed in moldings calculated by usingthe uniaxial and biaxial orientation assumption, re-spectively, along with experimental data. Due to thevery small amount of orientation achieved in amor-phous polymers, the calculated shrinkages based onthe uniaxial and biaxial orientation do not show muchdifference. Therefore, further calculations in this study

were conducted based on the uniaxial assumption oforientation.

Figure 13 shows the measured and calculated aniso-tropic shrinkages at different processing conditions.Figures 13(a,b) show the shrinkages in the three dif-ferent directions of the molded parts as a function ofpacking time and packing pressure, respectively. Asthe packing time increases, shrinkage in the thicknessdirection decreases due to reduced volumetric shrink-age since more material is injected to compensate forthe shrinkage during the packing stage. Shrinkage inthe flow and width directions decreases slightly. Thisis due to the fact that the orientation function devel-oped during the molding process of amorphous poly-mers is not high, as evident from their values depictedin Figure 5(a). Similar to the packing time effect, thethickness shrinkage decreases and the length and

Figure 12 Measured (symbols) and predicted (lines) lengthand width shrinkages due to contributions of the thermaland elastic recovery effect as a function of packing timeusing the assumption of uniaxial orientation (a) and biaxialorientation (b). Processing conditions: PP � 34.5 MPa, Tm� 200°C, and Q � 35.9 cm3/s.

Figure 11 Schematic diagram for the frozen layer in themold cavity.


width shrinkage slightly decreases as the packingpressure increases. The thickness shrinkage is similarto that calculated based on inclusion of the thermaleffect alone, as shown in Figure 10.

On the other hand, the melt temperature and injec-tion speed have little effect on shrinkage in the length,width, and thickness directions, as noted from Figures13(c,d). The predicted shrinkage values in the length,width, and thickness directions show a fair agreementwith the experimental data. Therefore, the calculationsof anisotropic shrinkages in moldings by using eqs.(53)–(56) are a suitable way to their predictions.

In many cases, it is well known that the thicknessshrinkage is one order of magnitude larger than thein-plane shrinkage. In all experiments, the thicknessshrinkage is always much larger than the length andwidth shrinkage. Also, the length shrinkage is slightlylarger than the width shrinkage. In simulated results,the most important processing condition to controlshrinkage and orientation is packing time and packing

pressure. Especially, the thickness shrinkage isstrongly affected by the packing time and packingpressure. With the variation of melt temperature andinjection speed, the thickness shrinkage is onlyslightly affected. The length and width shrinkages arenot significantly affected by processing conditions, in-cluding packing time and packing pressure. Accord-ing to the comparison between experimental and sim-ulated results, the thickness shrinkages show goodagreement with experimental results. The length andwidth shrinkage calculated including elastic recoverycontribution is also in a fair agreement with the ex-perimental data.

CONCLUSIONS

A novel approach to predict anisotropic shrinkage ininjection-molded parts of amorphous polymers wasproposed based on the frozen-in orientation function

Figure 13 Measured (symbols) and predicted (lines) anisotropic shrinkage as a function of packing time (a), packingpressure (b), melt temperature (c), and injection speed (d) with (dotted dashed line) and without (dotted line) inclusion ofpressure-dependent viscosity. Processing conditions: PP � 34.5 MPa, tP � 5 s, Tm � 200°C, and Q � 35.9 cm3/s.


and elastic recovery. The predicted results were com-pared with experimental data obtained at differentprocessing conditions, such as packing time, packingpressure, melt temperature, and injection speed. Tointroduce anisotropy into the shrinkage calculation,the anisotropic linear thermal expansion coefficientwas calculated as a function of orientation function.The orientation function was calculated using the bi-refringence obtained by the stress-optical rule and theviscoelastic constitutive equation. The explicabilitiesof uniaxial and biaxial orientation assumptions tomoldings were tested, and their effect on anisotropicin-plane shrinkage was investigated. The predictedlength and width shrinkage show only a little differ-ence due to the small amount of orientation achievedin moldings of amorphous polymers. To predict theexperimentally observed difference between thelength and width shrinkages, the contribution to theanisotropic shrinkage of elastic recovery that was notfrozen-in during the process was considered. In agree-ment with experiments, the predicted thicknessshrinkage was much higher than the predicted lengthand width shrinkages, with the length shrinkage beinghigher than the width shrinkage. The packing timeand packing pressure were found to be the most im-portant parameters affecting thickness and volumetricshrinkage. All predicted anisotropic shrinkages wereshown to be in fair agreement with experimental re-sults. The effects of the pressure-dependent viscosityand compressibility on anisotropic shrinkage werefound to be insignificant.

This work was supported by grant DMI-0322920 from theNational Science Foundation, Division of Engineering.

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Toward a viscoelastic modeling of anisotropic shrinkage in injection molding of amorphous polymers