Download pdf - Birefringence in injection-compression molding of amorphous polymers: Simulation and experiment

Birefringence in Injection-Compression Molding ofAmorphous Polymers: Simulation and Experiment

Nam Hyung Kim, Avraam I. IsayevDepartment of Polymer Engineering, The University of Akron, Akron, Ohio 44325-0301

The influence of the processing variables on the resid-ual birefringence was analyzed for polystyrene and pol-ycarbonate disks obtained by injection-compressionmolding under various processing conditions. Theprocessing variables studied were melt and mold tem-peratures, compression stroke, and switchover time.The modeling of flow-induced residual stresses andbirefringence of amorphous polymers in injection-com-pression molded center-gated disks was carried outusing a numerical scheme based on a hybrid finiteelement/finite difference/control volume method. Anonlinear viscoelastic constitutive equation and stress-optical rule were used to model frozen-in flow stressesin moldings. The filling, compression, packing, andcooling stages were considered. Thermally-inducedresidual birefringence was calculated using the linearviscoelastic and photoviscoelastic constitutive equa-tions combined with the first-order rate equationfor volume relaxation and the master curves for theYoung’s relaxation modulus and strain-optical coeffi-cient functions. The residual birefringence in injection-compression moldings was measured. The effects ofvarious processing conditions on the measured andsimulated birefringence distribution Dn and averagetransverse birefringence <nrr2nhh> were elucidated.Comparison of the birefringence in disks manufacturedby the injection molding and injection-compressionmolding was made. The predicted and measured bire-fringence is found to be in fair agreement. POLYM. ENG.SCI., 53:1786–1808, 2013. ª 2013 Society of Plastics Engineers

INTRODUCTION

Conventional injection molding (CIM) process [1] is

one of the most widely employed polymer processing

operations, being characterized by high degree of automa-

tion, high productivity and good dimensional stability of

moldings. The process can make products with complex

geometries in one production step. Many macro- and

microdevices, such as watches and camera components,

automotive crash and acceleration distance sensors, read/

write heads of hard disks, CD drives, medical sensors,

pump, surgical instruments, and telecommunications com-

ponents, have been successfully molded by CIM. Molded

part defects, such as uneven shrinkage, warpage, sink

marks, residual stresses, and nonuniformity of mechanical

properties, are affected by the entire injection molding

cycle. In molded optical plastic parts, the residual

birefringence, caused by the flow and thermal stresses,

strongly affects their performance. In particular, the bire-

fringence can significantly reduce an optical performance

of electronic devices such as liquid crystal displays,

camera lenses, CD, DVD and other optical products.

A compression molding (CM) is the widely used

process to manufacture various products [1–4]. In this

process, polymer melt is squeezed by the moving platen

to fill the mold. The melt is then continued to be com-

pressed by the pressure exerted from the mold wall of the

core side. This process provides a more uniform pressure

along the cavity wall and requires a low molding pressure

for the postfilling process resulting in less part warpage

and residual stress. However, this operation cannot

achieve high productivity due to the labor-intensive

charge installation. It also has a limitation for molding

large parts of complex shapes.

In recent years, manufacturing of precision plastic opti-

cal products with a strong quest for quality, such as

lenses, disk substrates and other optical components, is

continuously gaining more importance. Though precision

parts usually have simple geometrical shapes such as

disks, plates, or cylinders, their dimensional accuracy and

stability must be strictly satisfied [5]. The optical

performance of these moldings depends on the frozen mo-

lecular orientation, residual stresses and birefringence.

Several special injection molding techniques have been

developed in order to fulfill the rising requirements on the

part quality. Among them, the injection-compression

molding (ICM) is widely used for producing parts with

improved dimensional stability and surface accuracy. By

adding a compression stage after partial melt filling of

the cavity, ICM provides advantages, such as a lower

molding pressure [6] and clamp force (typically 20 to

50% lower) [7], reducing cycle time and residual stress,

minimizing molecular orientation and birefringence, pack-

ing evenly, reducing uneven shrinkage, overcoming sink

Correspondence to: Avraam I. Isayev; e-mail: [email protected]

Contract grant sponsor: NSF Division of Engineering; contract grant

number: DMI-0322920.

DOI 10.1002/pen.23429

Published online in Wiley Online Library (wileyonlinelibrary.com).

VVC 2013 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—-2013

mark and warpage, reducing density variation and increas-

ing dimensional accuracy [8]. Thus, ICM is a suitable

technique to manufacture high precision optical parts with

improved optical performance [9].

Klepek [10] used the ICM to mold thick optical lenses.

Yang and Lien [11] studied ICM to manufacture high pre-

cision lenses. They compared the quality of polystyrene

(PS) convex lenses made by ICM with that of PS lenses

made by CIM. They found that ICM produced lenses of a

better quality than those produced by CIM at the same

cooling time. They concluded that ICM was a better solu-

tion for molding of high-end plastic optical lenses. Yang

and coworkers [8, 12–14] experimentally investigated the

quality of ICM disks and showed that ICM enhanced the

dimensional accuracy, especially in the direction perpen-

dicular to compression. Shin et al. [15] carried out experi-

mental study on birefringence in optical disks under vari-

ous processing conditions and concluded that the birefrin-

gence was mostly affected by the mold temperature and

cooling time. Michaeli and Wielpuetz [16] investigated

effects of processing parameters of ICM and CIM on the

optical part quality of windscreen for window panes using

a design of experiments. The main parameters influencing

the optical part quality were the injection and compression

velocity in ICM and the injection velocity and the packing

pressure in CIM. Chen and Young [17] studied the effect

of adding the compression stage to CIM for manufacturing

disks and found the significant improvement in their con-

formity to cavity profile and a reduced warpage.

At the present time, the science-based technology for

manufacturing of the ICM optical products has not been

fully established. This is due to the lack of understanding

of viscoelastic mechanical and optical behaviors that

polymers experience during the injection and compression

stages. Numerical analysis of the process including the

viscoelastic effect is quite difficult. Isayev and Hieber

[18] were the first who proposed a theoretical approach to

relate the nonlinear viscoelasticity of polymers to the de-

velopment of frozen-in molecular orientation (birefrin-

gence) in CIM. Also, Isayev and Azari [19] and Isayev

et al. [20] carried out both theoretical and experimental

investigations of squeezing flow of melt using a nonlinear

viscoelastic constitutive equation. They considered the

shear-free flow and the channel flow with moving bound-

ary squeezing the melt similar to that occurring in the

compression stage of ICM. Osswald and Tucker [21] pre-

sented compression molding simulations for nonplanar

parts by combining a finite element (FE) method based

on a control volume for tracking the moving flow front

based on inelastic flow models. Kwon et al. [22, 23] stud-

ied experimentally and numerically the birefringence dis-

tribution in ICM center-gated disk based on a nonlinear

viscoelastic fluid model. Wang [24] used a Hele-Shaw

flow to predict the melt front advancement and the distri-

bution of pressure, temperature, and flow velocity during

the injection, compression and packing stages of ICM.

Park et al. [25] presented a numerical simulation for ICM

of a center-gated disk using finite difference (FD) method

using Leonov compressible viscoelastic model and the

Tait equation of state. Kim et al. [26] and Lee et al. [27]

developed a numerical analysis using FD for the solution

of governing equations of the radial flow and studied the

distribution of birefringence in CIM and ICM center-gated

disk using the Leonov model. Chen et al. developed a nu-

merical algorithm to simulate the filling stage of the

stamping type of ICM using an inelastic [28–30] and

viscoelastic [31] flow. In particular, the effect of the proc-

essing conditions on simulated and measured cavity pres-

sures was studied and a fair agreement between them was

achieved [28]. The simulated results indicated that ICM

significantly reduces shrinkage and improves the uniform-

ity of moldings. Also, they studied effects of various

processing parameters on the average transverse birefrin-

gence without considering the gapwise birefringence dis-

tribution. The results showed that the transverse birefrin-

gence decreases with an increase of the melt temperature

and a decrease of the mold closing velocity, but it is

insignificantly affected by the flow rate and the mold tem-

perature.

Friedrichs et al. [32] used an incompressible viscous

flow model to simulate the birefringence in ICM of the

stamping type. Kim et al. [26, 33] carried out a numerical

analysis to investigate the effects of the compression stage

on the birefringence in ICM of the stamping type using a

viscoelastic model. They found that, in comparison with

CIM, ICM reduced the birefringence and the mold tem-

perature had a significant effect on the density distribution

in comparison with other processing conditions. Fan et al.

[34] developed a code for the simulation of ICM of a

CD-R using a compressible viscous flow model. Young

[35] investigated the residual stresses and shrinkage of

ICM pickup lens having large thickness variations using a

Cross-WLF type equation for description of the viscosity

and a modified Tait equation for the density. The mold

temperature and compression time were found to be the

most important factors affecting the lens shrinkage in the

thickness direction, resulting in surface profile deviation.

Rohde et al. [36] carried out an experimental investigation

on the effect of process parameters on the replication of

various microstructures on a flat disk using micro ICM.

The dimensions of the microstructures on the inserts card

moldings were measured using a confocal profiler and a

transcription ratio was determined to assess the quality of

the replication. Huang and Chung [37] experimentally

studied the feasibility of CIM and ICM for fabricating

3.5-inch light-guide plates (LGPs) with V-grooves. They

found that the height of the V-grooved microfeatures

replicated by ICM more accurately than those replicated

by CIM. Min and Yoon [38] compared the measured bire-

fringence and extinction angle in CIM and ICM disks and

found more uniform gapwise distributions of these quanti-

ties in the ICM disks. Also, a graphical representation of

the optical refractive indicatrix was suggested. Nagato

et al. [39] carried out experimental investigation of the

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2013 1787

effects of the mold temperature and pattern shape on the

degree of replication of high-aspect-ratio nanostructures

in optical disks and Ni stamper by ICM. A higher mold

temperature resulted in better replication. Ho et al. [40]

performed a three-dimensional (3D) numerical simulation

of flow in ICM and compared the simulated results with

existing experimental data for optical lenses. 3D flow

effects were found to be significant, especially during the

compression stage, and ICM achieved more uniform dis-

tributions of the shear rates and stresses in comparison

with CIM.

In the present study, extensive simulations of the vari-

ous components of the flow birefringence developed dur-

ing the filling, compression, packing, and cooling stages

of the ICM of a center gated disk of polystyrene (PS) and

polycarbonate (PC) is carried out by using hybrid CV/FE/

FD method. The Tait equation [41] is used to describe the

P-V-T relationship. The compressible nonlinear visco-

elastic model [42] is used to calculate the flow stresses.

The flow birefringence was calculated through a linear

stress-optical rule [43]. The linear viscoelasticity was used

to calculate the thermal stresses [44–46]. The thermal

birefringence is calculated from the thermal stresses using

the linear photoviscoelasticity and free volume theory tak-

ing into account the density relaxation [44–46]. Although

the coupling effect between the flow and thermal stresses

is neglected in calculations of the total residual stresses

and birefringence, the present ICM study shows that the

thermal birefringence contribution is significant in the

core of PS disks and throughout the thickness for PC

disks.

MATERIALS AND EXPERIMENTAL PROCEDURES

Two polymers used in the experiments are PS (Styron

615-APR) supplied by Dow Chemical Company and PC

(Lexan 123) supplied by General Electric (GE) Plastics.

The thermophysical properties of polymers are listed in

Table 1 [45–47]. The apparent viscosity, g, of PS melt as

a function of the shear rate, _c, were taken from the refer-

ences [46, 48] at temperatures of 161, 180.5, and

200.58C. The apparent viscosity of PC melt as a function

of the shear rate was measured at temperatures of 260,

273.3, 286.7, and 3008C using an Advanced Capillary

Rheometer (RH7). These viscosity data are depicted in

Fig. 1 for PS (a) and PC (b). These data were fitted using

the two relaxation modes for PS [46] and six relaxation

modes for PC by means of nonlinear regression [49]. The

lines in these figures represent the nonlinear regression fit

of the experimental data to the following equation [18]:

Z ¼ Z0sþXNk¼1

2Zk

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4 gykð Þ2

q (1)

where gk and hk are the temperature-dependent viscosity

and relaxation time of kth relaxation mode according the

WLF equation, s is the parameter lying between 0 and 1.

The parameters obtained from the best fit are listed in

Table 2. The density of polymer melts as a function of

temperature and pressure was presented by the P-V-T

equation of state due Tait [41, 47]. These parameters are

listed in Table 3.

ICM Experiments

Center-gated disks were made by the ICM using Hull

Hydraulic Transfer Molding Press. The schematic diagram

of the apparatus used in ICM experiments is shown in

Fig. 2. The machine is a vertical type ram injection unit

with a clamping system actuated by the hydraulic pres-

sure. The diameter and thickness of the center-gated disks

TABLE 1. Physical properties of polymers.

Properties PS PC

q0 3 103 (kg/m3)a 1.04 [46] 1.16 [47]

q 3 103 (kg/m3)a 0.984 [46] 1.06 [47]

Cp (J/kg K) 1420 [46] 2150 [47]

K (W/m K) 0.17 [46] 0.234 [47]

h (J/s m2 K) 490 [45] 490 [45]

a q0 at T ¼ 258C and q at T ¼ 1808C.

FIG. 1. Measured (symbols) [43] and fitted (curves) viscosity of PS

[47] (a) and PC (b) as a function of shear rate.

1788 POLYMER ENGINEERING AND SCIENCE—-2013 DOI 10.1002/pen

are 12.7 cm and 0.3175 cm, respectively, with the inlet

diameter of 1.36 cm. The schematic mold geometry with

coordinates s, h, z is shown in Fig. 3. The data acquisition

system is used to record the pressure measured by a pres-

sure transducer (Dynisco PT435A) at the center of the

mold and the ram position measured by means of linear

velocity displacement transducer (LVDT).

The ICM experiments were carried out in the follow-

ing manner. PS and PC pellets were dried at 808C and

1058C, respectively, for 4 h under vacuum. The exact

amount of PS and PC required to fill the delivery system

and cavity are used. At first, PS and PC disks of a diame-

ter of 4.445 cm were compression molded at a tempera-

ture of 2008C and 2308C, respectively. Then, the disks

were placed into the cylinder of the machine, and melted.

Finally, the ICM experiments were conducted using dif-

ferent processing conditions by varying the mold and melt

temperatures, compression stroke and switchover time

from injection to compression stage. These processing

conditions for molding of PS and PC disks are shown in

Table 4.

Birefringence Measurements

The measurement of the average transverse birefrin-

gence, < nss � nhh >, in ICM and CIM PS and PC disks

is done on blocks of 1.5 cm in width removed from the

disks, as shown in Fig. 4. To measure the gapwise bire-

fringence distribution, Dn, slices of a thickness of 0.5 mm

and 1 mm were cut parallel to the s� z plane of disks, as

also shown in Fig. 4. A low-speed diamond saw (Iso-

metTM, Buechler) at 120 RPM is used to cut the slices.

The thickness of the slices was measured using a digital

micrometer with a resolution of 0.001 mm. The optical

retardation of the slices was measured using a cross-polar-

ized optical microscope (Leitz laborlux 12 POL, Leitz

Wetzlar) with a four-order tilting compensator for PS and

a thirty-order compensator for PC (1592k, Leitz Wetzlar).

To determine the gapwise position at which the retarda-

tion of the slices is measured, a scale on the microscope

eyepiece and a 403 magnification objective lens are used.

The birefringence measurements are typically started at

about 0.125 mm away from the surface of the disk, since

compensation of the retardation was difficult to achieve at

the position closer to the surface. The average transverse

birefringence, < nss � nhh >, was measured using a Gaert-

ner optical polariscope (model L305) equipped with a

seven-order Babinet compensator (model L-133-A). The

birefringence is calculated from the measured retardation

as Dn ¼ C=d, where C is the optical retardation and d is

the slice thickness.

To determine the sign of the birefringence in PS and

PC disks, the direction of the slow axis in each sample is

compared to the direction of the slow axis shown by the

four-order and 30-order tilting compensators which have

the direction opposite to the slow axis. In particular, the

slow axis of PS sample is perpendicular to the flow

direction due to the presence of the benzene ring. For PS

sample, the direction of the slow axis of the tilting com-

pensators coincides with the direction of the slow axis.

Therefore, the sign of the birefringence of PS is negative.

The direction of the slow axis for the PC sample is per-

pendicular to the direction of the slow axis of the tilting

compensators. Therefore, the sign of the birefringence of

PC is positive.

NUMERICAL ANALYSIS

To reduce computation time in simulation of the mold-

ing process, a quarter of the disk was discretized in plane

by FE triangles. Along the gapwise direction FD meshes

were used. In FE modeling, a finer mesh typically results

in a more accurate solution, however, at the expense of a

significant increase of the computation time. In the cur-

rent simulations the disk mold was divided into 315 ele-

TABLE 3. Material constants in Tait equation.

Tait equation PS [47] PC [47]

b1,l (m3/kg) 970.8 862.8

b2,l (m3/kg K) 0.5788 0.549

b3,l (Pa) 1.555 3 108 1.565 3 108

b4,l (K21) 3.019 3 1023 2.738 3 1023

b1,s (m3/kg) 970.8 862.8

b2,s (m3/kg K) 0.2429 0.2229

b3,s (m3/kg) 2.008 3 108 2.65 3 108

b4,s (K21) 1.38 3 1023 2.78 3 1023

b5 (K) 360.2 422.2

b6 (K/Pa) 3.20 3 1027 5.00 3 1027

TABLE 2. Material constants of PS and PC used in the CIM and ICM

molding simulation.

Material constants PS PC

WLF equation

C1 8.285 7.23

C2 (K) 131.5 163.4

Tr (K) 474.15 533.15

Leonov model

s 0.0048 0.0001

g1 (Pa s) 2228 30.1

g2 (Pa s) 446.8 9.11

g3 (Pa s) — 392

g4 (Pa s) — 766

g5 (Pa s) — 50.8

g6 (Pa s) — 18.4

y1 (s) 0.1466 2.1

y2 (s) 0.00489 0.104

y3 (s) — 0.011

y4 (s) — 0.00115

y5 (s) — 0.00012

y6 (s) — 0.000016

Stress-optical coefficient and

volume relaxation time

Cflr (Pa21) 25.2 3 1029 [45] 5.6 3 1029 [45]

sr (s) 0.04 [45] 0.3 [45]


ments and 184 nodes in the plane based on the previous

work [46] where the initial trial showed that the computa-

tion time is significantly increased with an increase in

number of elements and nodes. With these meshes, the

CPU time to carry out calculations for one processing

condition was 4 to 6 h on Pentium IV 3 GHz PC.

For numerical simulation in the filling, compression,

packing and cooling stages, a nonlinear viscoelastic model

is used [42]. The resulting set of nonlinear algebraic equa-

tions was solved with a globally convergent Newton’s

method [50, 51].

Governing Equations of ICM Process

Mold filling during the ICM process is comprised of

two stages: injection mold filling of a melt into a partially

open mold and subsequent compression mold filling by

closing the mold. Inelastic [52–54] and viscoelastic [46,

50, 54–57] simulations of the CIM have been already

developed based on the control volume approach [54].

The governing equations for flow of the viscoelastic melt

under nonisothermal conditions during the compression

stage are presented earlier [57]. This article has also pro-

vided the solution for the melt front advancement during

injection and compression stages. In particular, the volu-

metric flow rate during the compression stage is equal to

the rate at which the mold is closed displacing a volume

of the melt injected. For control volumes located on the

flow front, the filling parameter is calculated from the

occupied volume fraction and the additional volume frac-

tion due to compression flow at a given time [57]. It

should be noted that the study [57] has not considered the

packing and cooling stages of the ICM in the develop-

ment of the flow birefringence and also the contribution

of the thermal birefringence into the total birefringence.

In the ICM simulation, the density q T;Pð Þ is assumed to

follow the Tait equation [41, 47].

The parameters gk and hk of kth mode are temperature-

dependent quantities based on the WLF-type temperature

dependence [57, 58].

FIG. 2. Schematic diagram of ICM apparatus used in experiments.[Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com.]


The governing equation for the viscoelastic flow during

injection and compression stages of ICM are given in

Refs. 50, 55–58, respectively.

Numerical Implementation

On the basis of the formulations described earlier [50,

55–58], a numerical scheme using CV/FE/FD for a two-

dimensional viscoelastic flow in a center-gated disk is

developed. To solve the equations, a set of appropriate

boundary conditions is given first. The CV/FE [54] with

FIG. 3. Geometry of a center-gated disk with coordinate system.

TABLE 4. Processing conditions for manufacturing of ICM and CIM PS and PC disks.

Run no. Melt temp. Mold temp. Volume flow rate (cm3/s) Compression stroke (cm) Compression speed (cm/s) Switch over time (s)

PS

1 230 40 13 0.15 0.4 1.1

2 230 60 13 0.15 0.4 1.1

3 230 80 14 0.15 0.4 1.1

4 190 60 14 0.15 0.4 1.1

5 210 60 14 0.15 0.4 1.1

6 230 60 15 0.1 0.4 1.0

7 230 60 16 0.2 0.4 1.08 210 60 15 0 0 0

PC

1 260 40 23 0.15 0.4 1.12

2 260 60 23 0.15 0.4 1.0

3 260 80 26 0.15 0.4 1.12

4 240 60 22 0.15 0.4 1.0

5 280 60 24 0.15 0.4 0.86

6 260 60 23 0.1 0.4 1.2

7 260 60 22 0.2 0.4 1.0

8 260 60 23 0 0 0

FIG. 4. Cutting procedure used for preparing a specimen from an injec-

tion-compression molded disk, to measure average transverse birefrin-

gence < nss � nhh > (a), and birefringence Dn (b).


triangular elements and linear shape functions is used.

The boundary conditions in the injection, compression

and packing stages are different. In the injection stage,

the volume flow rates at the entrance nodes were speci-

fied, and the pressures at the melt front nodes were

assumed to be zero with the pressure at the entrance

nodes and flow rates at the melt front nodes to be solved.

In the packing stage, the entrance pressure was speci-

fied as an imposed packing pressure, with flow rates at

the entrance nodes to be determined. Besides, in the

impermeable boundary region, the melt is in contact with

the boundary of the mold, and the normal velocity com-

ponents vanish. In addition, symmetric boundary condi-

tions at the centerline and no slip at the solid wall of the

cavity were assumed. The additional boundary conditions

for the compression stage in ICM were specified [57]. In

ICM, the temperature at the entrance nodes is assumed to

be uniform and equal to the inlet melt temperature, T0.To solve the pressure equations, a quarter of the disk

cavity is discretized by a series of three-node triangular ele-

ments in the x-y plane. Only one control volume is associ-

ated with each node. For each node of a triangular element,

by connecting the centroid of each element to the midpoints

of the three corresponding sides, the region enclosed by a

contour in the counter-clockwise direction around each ver-

tex node of a triangular element is specified as the subcon-

trol volume for this node. The polygonal control volume

that surrounds each vertex node is defined by the summa-

tion of sub-control volumes, which contains node N [54].

Pressure Development. After introducing linear inter-

polation functions as well as a FD representation for the

time derivative and applying the Galerkin weighted-resid-

ual procedure [59–61] and mass conservation using a con-

trol-volume approach [62], and the governing equation for

pressure [57] in the whole cavity domain is obtained.

Elastic Strain Tensor. To calculate the flow-induced

stresses during the filling and packing stages, the elastic

strain tensor, Cij;k, for all relaxation modes of the Leonov

model has to be calculated at each time step. Therefore,

the governing equations for the elastic strain tensor along

the streamwise coordinate are discretized using a FD

approach. The time derivatives are discretized using for-

ward difference and space derivatives were discretized

using backward difference [63].

To determine the components of Cij;k tensor, the Glob-

ally Convergent Method for Nonlinear Systems of Equa-

tions [51] that is closely related to the quasi-Newton method

of minimization was implemented for the discretized sys-

tem of governing equations for the elastic strain tensor. This

method is powerful but the numerical computation of the Ja-

cobian matrix represents a disadvantage [51]. However, for

this case the Jacobian matrix is available analytically. In this

work, the subroutine ‘‘fdjac’’ which calculates the Jacobian

matrix numerically, was replaced by the function that pro-

vides the Jacobian matrix calculated analytically.

Temperature Distribution. For the solution of the energy

equation, the implicit FD method is used for the conduction

and time derivative terms. To calculate the convective term

in the energy equation during the filling and compression

stages, the temperature distribution in the flow direction, is

linearly interpolated within each element. The heat convec-

tive terms at a node were evaluated by taking the area

average of the values calculated at the center of the upstream

elements of the node under consideration [55, 56, 64].

Numerical Algorithm and Convergence

At each new time step, the velocity field and pressure field

are first solved with the elastic strain tensor. During the solu-

tion process, the temperature obtained in the time step is used

to calculate the various physical quantities. After obtaining

the velocity and pressure fields, the energy equation, with the

fixed pressure and velocity fields, is solved to obtain the tem-

perature field, and then the next time step is ensured.

The under-relaxation iteration method [55] was used to

solve the pressure and temperature at each time step. Dur-

ing iteration, new values of the pressure and temperature

are obtained. When the solution converges, the old and

updated values are almost same. Iterations terminate when

the absolute change in the each property is less than a

specified value. In particular, the successive under-relaxa-

tion method is also used for calculation of the pressure

and temperature profile due to numerical stability.

Residual Stresses and Birefringence in ICM Parts

Flow Stresses and Birefringence. In the cooling stage,

the shear rate, velocity and pressure are taken to be zero.

Accordingly, the convection, dissipation, and pressure

terms in the energy equation are omitted. The shear

stresses and normal stress differences are calculated

according to the Leonov constitutive equation [42]. The

residual flow birefringence for amorphous polymers can

be calculated by the stress-optical rule during the noniso-

thermal flow and the subsequent relaxation. The flow

birefringence in s� z plane, Dnfl, is given as

Dnfl ¼ Cs Tð Þ � Ds s; z; tð Þ (2)

where,

Ds s; z; tð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN21 s; z; tð Þ þ 4t2sz s; z; tð Þ

q(3)

and Cr is the temperature dependent stress-optical coeffi-

cient of polymer in the melt state [43].

Similarly, the birefringence in the s� h plane is given by

nflss � nflyy ¼ CsðTÞ � ðsss � syyÞ (4)

where the superscript fl indicates flow-induced birefrin-

gence components, Dn and nss � nhh are the birefringence


component in the planes s� z and s� h, respectively.

rii(i ¼ s; h) are the normal stresses in the s and h direc-

tions, respectively, and ssz is the shear stress in the s� zplane.

Thermal Stresses and Birefringence. The residual ther-

mal stresses and birefringence in ICM disks are treated

as those for freely quenched polymer plates and are

calculated based on the linear viscoelasticity and

photoviscoelasticity with the volume relaxation effects

included [45, 65]. As a first approximation, we treated the

residual thermal stresses and birefringence in ICM parts

as those developed in freely quenched plates [45]. Based

on the varying temperature fields, the changes in visco-

elastic properties are determined and utilized to calculate

the thermal stresses until the polymer reaches thermal

equilibrium with the molding system. For this case, the

thermal stress tensor is

sth ¼sthss 0 0

0 sthyy 0

0 0 0

24

35 (5)

where rthss ¼ rthhh.The refraction index tensor due cooling alone is:

nth ¼nthss 0 0

0 nthyy 0

0 0 nthzz

24

35 (6)

Total Stresses and Birefringence. To the best of our

knowledge, no single nonlinear constitutive equation is

available to describe the rheological behavior of a poly-

mer in the glassy and melt states and in the transition

region between them. Moreover, in the injection, com-

pression and cooling stages, the dominant terms in the

dynamic equations are different. Therefore, a further sim-

plifying assumption was made that the coupling effects

between the flow and thermal stresses are negligible, such

that they can be evaluated independently. The constitutive

equation used to calculate the flow and thermal stresses

are different, as mentioned earlier. Thus, the total residual

stresses are:

sres

¼ sflres

þ sthres

(7)

Similarly, the total refraction index tensor is,

n ¼ n0dþ nth þ nflnth ¼n0 þ nrrthþnflrr

0 nflrz

0 n0 þ nthyy þ nflyy 0

nflrz 0 n0 þ nthzz þ nflzz

264

375 ð8Þ

where subscripts res and superscripts fl and th stand for

residual, flow, and thermal, respectively.

The birefringence measured in the rz plane is then

Dn ¼ Dnrz ¼ Dnflrz� �2þ Dnthrz

� �2þ2 nflrr � nflzz� �

Dnthrz� �h i1=2

(9)

with the flow birefringence, Dnflrz, and the thermal bire-

fringence, Dnthrz, being

Dnfl ¼ Dnflrz ¼ Cfls sflrr � sflzz� �2þ4 sflrz

� �2h i1=2(10)

Dnthrz ¼ nthrr � nthzz (11)

In free quenching nthrr ¼ nthhh, the birefringence in the rhplane becomes

Dnry ¼ nthrr � nthyy� �

þ nflrr � nflyy� �

¼ nflrr � nflyy (12)

RESULTS AND DISCUSSION

Pressure Profile

The experimental (symbols) and simulated (solid line)

pressure traces and a ram displacement for the ICM PS

run 1 in Table 4 are given in Fig. 5. The measured pres-

sure trace was obtained directly from the pressure trans-

ducer located under the sprue at the center of cavity, as

shown in Fig. 2. The simulated pressure trace during the

filling, compression and at the beginning of the packing

stage is in a good agreement with the experimental data.

It can be seen that the pressure increases monotonically

during the filling stage. Also, it is noted that the pressure

drops during the switchover time of 1.1 s from the filling

to compression stage. During this time the plunger does

not move and stress relaxation occurs. Accordingly, this

FIG. 5. Measured (symbols) and simulated (solid line) pressure at the

center of the disk and displacement of plunger (dashed line) as a func-

tion of time for run 1 of PS: melt temperature ¼ 2308C, mold tempera-

ture ¼ 408C, volume flow rate ¼ 13 cm3/s, compression stroke ¼ 0.15

cm, compression speed ¼ 0.4 cm/s and switchover time ¼ 1.1 s.


was the only mechanism governing the behavior of the

pressure history during the switchover time. When the

compression stage starts, the pressure sharply increases

followed by the overshoot at the beginning of the packing

stage and then the packing pressure is maintained. The

measured pressure trace is taken as the pressure for the

packing simulation. As seen from Fig. 5, the time at the

end of injection stage was 2.874 s and the switchover

time is 1.1 s. The compression stage ends at time of

4.254 s and then the packing stage starts. In the cooling

stage which starts right after end of the packing stage, the

pressure decreases due to the release of the packing pres-

sure and further cooling.

Velocity and Shear Rate Profiles

Figure 6 shows the predicted gapwise velocity (a, b)

and shear rate (c, d) profiles at the end of the injection (a,

c) and compression (b, d) stages of ICM PS disk in run 2.

As expected, during the injection stage the velocity

decreases with radial position and the melt flow dimin-

ishes in the wall layer around z=h ¼ 0.8 due to cooling

from the mold wall, as seen from Fig. 6a. In contrast to

the injection stage, the velocity at the end of the compres-

sion stage during ICM increases with the radius, as shown

in Fig. 6b. Similar velocity profiles are also obtained by

Kim et al. [26], Chen et al. [31], and Lee and Isayev

[57]. The velocity behavior can be explained by consider-

ing the mass balance. Since the volumetric flow rate of

the melt through the surface increases with the radius, the

velocity through the section at a position far from the

inlet is greater than that near the inlet. The 150% reduc-

tion of the centerline velocity between r¼ 2.948 cm and

r¼ 4.46 cm occurs at the end of the injection stage (Fig.

6a), whereas the only 45% increase occurs at the end of

the compression stage (Fig. 6b). This is an indication that

the squeezing flow during the compression stage causes

the pressure distribution in the mold cavity to become

more uniform, resulting in a reduction of the velocity var-

iation with radial position.

Figure 6c and d show the predicted gapwise shear rate

profiles at the end of the injection and compression stage,

respectively. Similar to the velocity behavior, the shear rate

at the end of the injection stage decreases with increasing

radial positions from the gate. However, the shear rate at

the end of compression stage increases with the radius.

Birefringence in PS

In this section the influence of the processing conditions

on the simulated and measured birefringence, Dn, and aver-

age transverse birefringence, < nss � nhh >, for ICM PS

disks is presented. The total birefringence is comprised of

both flow- and thermally-induced birefringence. The simu-

lated results are compared with measurements.

FIG. 6. Predicted gapwise distribution of velocity vs(a, b) and shear rate (c, d) at various radial positions for

run 2 of PS at the end of injection (a, c) (t ¼ 2.64 s) and compression stages (b, d) (t ¼ 4.16 s) during ICM.


Simulation of Transient Birefringence. The flow-

induced birefringence is determined by the thermome-

chanical history that the polymer experiences during the

polymer processing [18]. The build-up and relaxation of

the shear and normal stresses taking place during molding

process lead to a transient birefringence history. Figure 7

shows the evolution of the predicted gapwise flow-

induced birefringence, Dnfl, distribution at various radial

positions at the end of the injection stage corresponding

to t ¼ 2.64 s (a) and during the compression stage at t ¼3.8 s (b), and t ¼ 4.11 s (c) and at the end of the com-

pression stage corresponding to t ¼ 4.16 s (d) for the

ICM PS disk in run 2. The maximum birefringence at the

end of the injection stage occurred at about z=b ¼ 0.7–0.8

due to the dominant effect of the first normal stress differ-

ence during the injection stage. The maximum decreases

in the magnitude with the radial position. During the

switchover time, the stresses relaxation occurs due to the

absence of flow, but it is retarded due to cooling. After

the switchover time, the predicted transient gapwise bire-

fringence distribution during the compression stage

increases with time (Fig. 7b and c). The birefringence

maximum also increases. However, in contrast to the

results obtained during the injection stage, the birefrin-

gence behavior during the compression stage is different.

The simulated transient birefringence, Dn, at the end of

the injection stage and after the switchover time, changes

significantly. In particular, the magnitude of Dn signifi-

cantly decreases at the radius over r ¼ 4.46 cm. This is

due to the fact that after the switchover time the tempera-

ture is decreased along the gapwise direction at the loca-

tion close to the melt front. The latter causes an increase

in the shear and normal stresses during the compression

stage due to the increase of viscosity and relaxation time

of the polymer melt upon cooling. The minimum and

maximum of the birefringence appear near the wall

region. This is caused by the change of the shear rate due

to narrowing of the flow channel by the solidification

leading to freezing of the chain orientation in the filling

and compression stages and a slow relaxation in the cool-

ing stage. At the end of the compression stage (Fig. 7d),

a maximum and minimum of the birefringence can be

seen with their magnitude determined by the shear and

normal stresses present at the time of melt solidification.

Figure 8 shows the predicted gapwise distribution of

the flow birefringence at the various radial positions at

the end of the packing (a) and cooling (b) stages for ICM

PS disk in run 2. It should be noted that, due to the me-

chanical limitation of the ICM machine used in the pres-

ent study, it was difficult to precisely control the high

packing pressure required to compensate the melt shrink-

age during cooling. Therefore, the packing stage in this

FIG. 7. Predicted transient flow-induced birefringence (Dnfl) distributions in the gapwise direction at various

radial positions at the end of the injection stage at t ¼ 2.64 s (a), during the compression stage at t ¼ 3.8 s

(b) and t ¼ 4.11 s (c) and at the end of the compression stage t ¼ 4.16 s (d) for the ICM PS run 2.


study does not play a major role in development of the

flow-induced birefringence. It is seen from Fig. 8 that the

birefringence near the wall does not change and becomes

frozen during the packing stage since the temperature near

the mold wall is below the glass transition temperature,

Tg. However, outside this region the temperature is still

high and the stress and, therefore, birefringence relaxation

is significant. This figure shows that the predicted flow

birefringence in the core is very low due to the fast relax-

ation of chain orientation immediately after completion of

the compression stage. At the end of the cooling stage

(Fig. 8b), the temperature in an extensive wall region is

below Tg, therefore, the distribution of the birefringence

is similar to that at the end of the packing stage (Fig. 8a).

Since the shear and normal stresses cannot relax anymore

below Tg, the residual flow stresses are frozen during the

cooling stage. However, the residual flow birefringence in

the core is zero, due the complete relaxation of the chain

orientation upon completion of the packing stage, since

the temperature is still high in this region. A shoulder in

the birefringence profile that appears during the packing

stage is absent after cooling due to the stress relaxation in

this region.

The Effect of Mold Temperature. The predicted and

measured residual gapwise distribution of the flow bire-

fringence Dnfl at various radial positions for ICM PS

disks in runs 1 and 3 are shown in Fig. 9 at mold temper-

atures of 408C (a) and 808C (b) and a melt temperature of

2308C. As the mold temperature increases the overall

flow-induced birefringence and the thickness of solidified

layer decreases due to a slower rate of cooling leading to

faster stress relaxation at the higher mold temperature.

The predicted and measured birefringence is seen to

decrease with increasing the radial location due to a lower

shear rate during the injection stage, as shown in Fig. 6c.

The development of the gapwise distribution of the bire-

fringence is determined by the combination of effects of

the shear rate variation and the temperature history. The

former is caused by the flow during the filling and com-

pression stages followed by the packing stage. The tem-

perature history of the melt is affected by the switchover

time. During the injection stage, the velocity and shear

rate decrease near the mold wall because of cooling lead-

ing to the solidification. In addition, during the switchover

time, the further cooling takes place, but without the melt

flow, causing the stress relaxation. During the compres-

sion stage, the gapwise shear rate distribution shows a

peak just outside the solidified region. As a result, at the

FIG. 8. Predicted transient flow-induced birefringence (Dnfl) distribu-

tions in the gapwise direction at various radial positions at the end of the

packing (a) and cooling (b) stages for the ICM PS run 2.

FIG. 9. Measured (symbols) and predicted (lines) residual flow-induced

birefringence Dnfl distribution in the gapwise direction at various radial

positions at the end of the cooling stage for the ICM disks obtained in

runs 1 and 3 corresponding to mold temperature of 408C (a) and 808C(b). The contribution of the thermal birefringence was not included in

the predicted birefringence.


end of the compression stage, a maximum and a mini-

mum in the birefringence profile is developed due to the

shear and normal stresses. It seems that the packing stage

does not significantly affect the flow-induced birefrin-

gence because the packing pressure used is low and not

sufficient to compensate the shrinkage.

In contrast to the measured birefringence showing a

nonzero value in the core, the predicted residual birefrin-

gence in the core is zero (Fig. 9). As will be shown

later, the thermally-induced birefringence dominates in

the core region. The linear viscoelasticity and photovis-

coelasticity with the volume relaxation effects are used

to calculate the residual thermal stresses and birefrin-

gence [45]. In calculating the thermal birefringence, the

relaxation modulus and strain-optical coefficient func-

tions reported earlier [65] are used. As the first approxi-

mation, it is assumed that the residual thermal stresses

and birefringence in CIM and ICM disks developed sim-

ilarly as in freely quenched slabs. The predicted residual

thermal birefringence distributions along the gapwise

direction of PS slabs quenched from an initial tempera-

ture of 2308C to different quenching temperatures are

shown in Fig. 10. An increase in the quenching tempera-

ture leads to a decrease in the thermally-induced bire-

fringence. As seen from comparison of Figs. 9 and 10

the thermally-induced birefringence, Dnth, near the wall

is significantly lower than the flow-induced birefrin-

gence, Dnfl. Also, in this region, the flow-induced bire-

fringence in ICM PS disks is negative, while the ther-

mally-induced birefringence is positive. However, in the

core of ICM disks the magnitude of the flow-induced

birefringence is zero since the flow-induced birefringence

completely relaxed. The total residual birefringence was

obtained by summation of the calculated residual ther-

mally induced and flow-induced birefringence. Figure 11

shows the measured and simulated total birefringence,

Dn in the ICM PS disks in runs 1 and 3 at various ra-

dial positions. It is seen that when the thermally-induced

birefringence is added to the flow-induced birefringence,

the simulated total birefringence provided a better

description of the measured birefringence in ICM PS

disks. Simulations show that the total birefringence near

the wall is mainly caused by the flow, since the ther-

mally-induced birefringence at this location is low. How-

ever, the birefringence in the core is caused by thermal

stresses developed during rapid cooling. Both the pre-

dicted and measured birefringence shows slightly con-

cave shape in the core with a maximum near the mold

surface and a maximum at a location of about z=b =

0.70–0.75. The location of the birefringence maximum

at a mold temperature of 808C is shifted toward the sur-

face, because of the smaller thickness of the solidified

layer developed at the higher mold temperature.

Some differences in the simulated and measured bire-

fringence are evident in the region between the location

of a maximum of Dn and the core. Generally, the simu-

lated results underestimate the measured gapwise birefrin-

gence because in simulations of the present study, the

flow in the delivery system is neglected. It is known that

the contraction and expansion flow provides an additional

contribution to the birefringence [1].

FIG. 10. Predicted residual thermal birefringence Dnth of PS plates

freely quenched from a melt temperature of 2308C to different quench-

ing temperatures of Ti.

FIG. 11. Measured (symbols) and predicted (lines) total residual bire-

fringence Dn distribution in the gapwise direction at various radial posi-

tions at the end of the cooling stage for the ICM disks obtained in run 1

and 3 corresponding to a mold temperature 408C (a) and 808C (b). The

contribution of thermal birefringence is included in the predicted bire-

fringence.


The Effect of Melt Temperature. Figure 12 shows the

predicted and measured gapwise distribution of the flow

birefringence Dnfl at various radial positions at the end of

cooling stage for ICM PS disks in runs 4 and 5 corre-

sponding to melt temperatures of 1908C (a) and 2108C(b) at a mold temperature 608C. It is seen that the distri-

bution of birefringence is very sensitive to the melt tem-

perature. Higher peak values of the birefringence are

obtained at a lower melt temperature. The outer peak is

due to the flow during the injection and compression

stages leading to the frozen-in orientation near the surface

and the subsequent relaxation of the molecular orientation

in the intermediate region. This flow-induced birefrin-

gence peak near the wall reduces in value at the higher

melt temperature. This is because a faster stress relaxation

at the higher temperature lowers shear and normal stresses

developed during flow at the given volumetric flow rate

during the filling and compression stages. Accordingly,

the simulation shows that the maxima of the flow-induced

birefringence Dnfl at the wall and near the wall signifi-

cantly decrease when the melt temperature is increased.

The predicted flow-induced birefringence Dnfl in the core

is zero due to the fast relaxation of the chain orientation

right after filling, since the temperature is still high in this

region. However, the measured birefringence is nonzero

in the core region due to the presence of the thermally-

induced birefringence. Thus, to obtain a better prediction,

the total residual birefringence was calculated by summa-

tion of the calculated residual thermal and flow birefrin-

gence, as mentioned earlier. These results are given in

Fig. 13a and b. Compared with a melt temperature of

1908C, the birefringence in the core does not change sig-

nificantly as the melt temperature is increased to 2108C.This is due to the fact that these melt temperatures are

much higher than Tg of PS such that the effect of the melt

temperature on the thermal birefringence becomes insig-

nificant [45, 66]. It is clearly seen from Fig. 13 that the

calculated total birefringence is in a better agreement with

the measured one.

The average transverse birefringence < nss � nhh > is

directly obtained by measuring the retardation of the nor-

mal incident laser beam. The predicted average transverse

birefringence was calculated by the integration of the

in-plane birefringence over the thickness of the disk. The

average transverse birefringence as a function of radial

location for melt temperatures of 1908C, 2108C, and

2308C at a mold temperature of 608C corresponding to

FIG. 12. Measured (symbols) and predicted (lines) residual flow-

induced birefringence Dnfl distribution in the gapwise direction at various

radial positions at the end of the cooling stage for the ICM PS disks

obtained in runs 4 and 5 corresponding to a melt temperature of 1908C(a) and 2108C (b). The contribution of thermal birefringence is not

included in the predicted birefringence.



tions at the end of the cooling stage for the ICM PS disks obtained in

runs 4 and 5 corresponding to melt temperatures of 1908C (a) and 2108C(b). The contribution of thermal birefringence is included in the pre-

dicted birefringence.


ICM PS disks in runs 4, 2, and 5 is shown in Fig. 14. As

the mold temperature increases from 1908C to 2308C, avalue of < nss � nhh > decreases significantly due to a

faster stress relaxation at higher melt temperatures. In

agreement with experimental data reported earlier [15,

67], the birefringence < nss � nhh > decreases in the ra-

dial direction However, the theoretical and experimental

data on < nss � nhh > of the present study are in contrast

with experimental data reported by Chen et al. [31] indi-

cating the presence of the birefringence minimum at some

radial location. Finally, it should be noted that the simu-

lated and experimental results on < nss � nhh > at all ra-

dial locations at various melt temperatures are in a fair

agreement.

The Effect of Compression Stroke. In comparison with

CIM, the ICM introduces new processing parameters

including the compression stroke and speed and switch-

over time. Therefore, it is of great importance to deter-

mine the effect of the compression stage on the total bire-

fringence in disks. During the injection stage, the melt is

injected into a partially open cavity with a thickness being

greater than the final thickness from 1.0 mm to 2.0 mm.

Figure 15 shows the predicted total and measured gapwise

birefringence, Dn, distribution at various radial positions

at the end of cooling stage in ICM PS disks in runs 6 and

7 corresponding to compression strokes of 1 mm (a) and

2 mm (b). As the compression stroke increases from 1

mm to 2 mm, the birefringence decreases and shows a

lower maximum. At the higher compression stroke the

melt experiences a less resistance during the compression

flow leading to a lower pressure and therefore, lower

stresses, and birefringence.

The Effect of Switchover Time. The switchover from

the injection to compression stage is not instantaneous

leading to a time lag defined as the switchover time. To

identify the effect of the switchover time in ICM, calcula-

tions of the birefringence are carried out at zero and non-

zero switchover times. Figure 16 shows the simulated

gapwise residual flow-induced birefringence, Dn, distribu-tion at various radial positions at the end of cooling stage

in ICM PS disks in run 2, but at zero switchover time.

Results of the simulated birefringence at a switchover

FIG. 14. Measured (filled symbols) and predicted (open symbols) aver-

age transverse birefringence, –< nss � nhh > as a function of the radial

position for ICM PS disks in runs 2, 6, and 7.



tions at the end of the cooling stage for the ICM PS disks obtained in

runs 6 and 7 corresponding to a compression stroke of 1 mm (a) and 2

mm (b). The contribution of thermal birefringence is included in the pre-


FIG. 16. Predicted residual flow-induced birefringence, Dnfl, distribu-

tion in the gapwise direction at various radial positions at the end of the

cooling stage for the ICM PS disks in run 2 without switchover time.


time of 1.1 s in ICM PS disk in run 2 are shown earlier

in Fig. 8b. In the presence of the switchover time, there

are two competing effects. One is the stress relaxation

during switchover time when cooling takes place in the

absence of flow. The other is the development of higher

stresses due to flow of colder and therefore, more viscous

melt in the compression stage. Obviously, the dominating

effect is determined by the duration of the switchover

time. At a shorter switchover time a less time is available

for the stress relaxation and cooling. At a longer switch-

over time the cooling may be significant and therefore,

the second effect becomes more important in determining

the final birefringence distribution. The simulated birefrin-

gence with and without the switchover time suggests that

the switchover time is a relatively important parameter

affecting the birefringence. The simulated results in Fig.

16 show a minimum and a maximum of the birefringence

at the switchover time of 0 s with their magnitudes deter-

mined by the stress evolution when the melt solidifies at

a gapwise position. However, when the switchover time is

employed, the maximum of the birefringence increases. In

addition, the gapwise position corresponding to this maxi-

mum of Dn is shifted further away from the surface, as

seen from comparison of data depicted in Figs. 8b

and 16.

Comparison between CIM and ICM. It is of great im-

portance to find the effect of the compression stage on

the birefringence in the molded parts. For this purpose,

simulations are carried out without imposing the packing

pressure for CIM and for ICM with the zero switchover

time. Figure 17 shows the simulated gapwise flow-

induced birefringence, Dnfl, distribution along the radial

direction for ICM (a) and CIM (b) PS disks at the end of

the cooling stage. It is seen that the birefringence in the

ICM disk is dramatically decreased in comparison with

that of the CIM one, especially near the mold surface.

From these results, it is clear that the ICM certainly

reduces the flow-induced birefringence level most notably

near the wall region. It is expected that a more drastic

decrease of the birefringence can be obtained if the injec-

tion stage is applied at a wider cavity opening such that

the compression stage becomes more dominant over the

injection stage. Therefore, the simulation shows that the

ICM is indeed more suitable process for manufacturing of

precision products of good optical quality than the CIM.

It is also interesting to find the difference in the average

transverse birefringence in the ICM and CIM moldings.

The predicted and measured average transverse birefrin-

gence, < nss � nhh >, as a function of the radial direction

is compared in Fig. 18 for both CIM and ICM disks at the

end of the cooling. This birefringence decreases with the

radial direction as earlier reported [15, 67]. It can be seen

that the average transverse birefringence along the radial

direction decreases with distance from the gate. The aver-

age transverse birefringence in the CIM disk is approxi-

mately twice larger than that in the ICM one. Regardless,

the overall predicted values of < nss � nhh > are in a quali-

tative agreement with the experimental results.

Birefringence in PC Disks

The following sections present the influence of the

processing conditions on the simulated total residual bire-

FIG. 17. Predicted residual flow-induced birefringence, Dnfl, distribu-

tion in the gapwise direction at various radial positions at the end of the

cooling stage in ICM (a) and CIM (b) PS disks for run 2 without switch-

over time and packing stage.


age transverse birefringence as a function of the radial position of PS

disks for run 5 (ICM) and run 8 (CIM).


fringence, Dn, and the average transverse birefringence,

< nss � nhh >, in PC ICM disks. The total birefringence

is comprised of both flow- and thermally-induced birefrin-

gence. The simulated and measured results are compared

and the similarities and differences in the birefringence of

ICM PS and PC disks are discussed.

Numerical Simulation of Transient Birefringence. Fig-

ure 19 shows the predicted gapwise flow birefringence,

Dnfl, distribution at the end of the compression (a), packing

(b), and cooling (c) stages at various radial positions for

ICM PC disk in run 2. Similar to the PS disk, in the PC

disk at the end of the compression stage, a maximum and a

minimum of the birefringence near the wall is observed.

These are caused by the shift of the shear rate maximum

toward the core due to solidification. It is noted from Fig.

19a that at a radius of above r ¼ 4.46 cm, the magnitude of

Dn decreased tremendously. It is due to the fact that, after

the switchover time, the melt is cooled along the gapwise

direction away from the melt front leading to an increase

of the shear and normal stresses during the compression

stage. At the same time after the switchover time, since the

temperature along the gapwise direction in the region close

to the melt front is still high, a decrease of the stresses

takes place during the compression stage in that region.

The predicted gapwise distribution of the flow birefrin-

gence at the various radial positions at the end of the

packing stage is presented in Fig. 19b. As noted earlier,

in the present study the packing stage does not play a

major role in the development of the flow-induced bire-

fringence because the packing pressure is too low to com-

pensate the shrinkage of the melt caused by cooling. It

may be noted that the birefringence near the wall does

not change and frozen during the packing stage since the

temperature near the wall is below Tg. However, in the

core, the stress relaxation is significant since the tempera-

ture is still high and the birefringence is completely

FIG. 19. Predicted flow-induced birefringence (Dnfl) distributions in the

gapwise direction at various radial positions at the end of the compres-

sion (a), packing (b), and cooling (c) stages for the ICM PC disks for

run 2.



radial positions at the end of the cooling stage for the ICM PC disks in

runs 1 and 2 corresponding to mold temperatures of 408C (a) and 608C(b). The contribution of the thermal birefringence is not included in the

predicted birefringence.


relaxed. During the cooling stage, as shown in Fig. 19c,

the birefringence in the core relaxed completely before

the melt temperature reaches Tg in that region. Away

from the core the birefringence remains almost the same

as that at the end of the packing stage because the

stresses do not relax anymore and become residual since

the melt temperature is already below Tg.

The Effect of Mold Temperature. Figure 20 shows the

measured and predicted gapwise birefringence at the end of

cooling stage at various radial locations at mold tempera-

tures of 408C (a) and 608C (b) and a melt temperature of

2608C for ICM PC disks in runs 1 and 2. It is seen from this

figure that an increase in the mold temperature from 408C to

608C leads to a decrease in the magnitude of the flow bire-

fringence along the gapwise direction. The difference of the

predicted flow birefringence between the mold temperatures

of 408C and 608C becomes significant near the wall.

Comparison of the measured and predicted flow-

induced birefringence distribution shown in Fig. 20 indi-

cates that the measured birefringence significantly differ-

ent in the skin and core regions of ICM PC disk. These

differences are evidently due to neglect the thermally-

induced stress and birefringence. In particular, the tensile

stresses in the core and the compressive stresses in the

skin, evidently contribute significantly to the birefrin-

gence. In order to prove this the residual thermal stresses

and birefringence are calculated based on the linear vis-

coelasticity and photoviscoelasticity with volume relaxa-

tion and free quenching assumption [45, 66]. In calculat-

ing the thermal birefringence relaxation modulus and

strain-optical coefficient functions, reported in Ref. 65,

are used. The predicted thermal birefringence distributions

along the gapwise direction of PC slabs freely quenched

from an initial temperature of 2608C to different quench-

ing temperatures are shown in Fig. 21. It is seen that an

increase of the cooling temperature reduces the thermal

birefringence. In the case of ICM PC disks, the simulated

results in Fig. 21 show that the thermal birefringence,

Dnth, near the wall is lower than the flow birefringence,

Dnfl, shown in Fig. 20. In this region, the flow birefrin-

gence is positive, while the thermal birefringence is nega-

tive. However, in the core, the magnitude of the flow

birefringence is negligible (Fig. 20). In this case the ther-

mal birefringence is positive and much higher than that in

the quenched PS slabs. When the simulated thermal and

flow birefringence is added, the measured and simulated

total birefringence is in much better agreement, as indi-

cated in Fig. 22. It is clear that the contribution of the

thermal birefringence to the total birefringence in ICM

PC disks is higher than that in ICM PS disks. The pre-

dicted and measured birefringence is seen to have a con-

cave shape in the core. The predicted birefringence maxi-

mum appears near the surface at a location of about z=b¼ 0.75–0.90 with its value being slightly higher at a

lower mold temperature. The location of this maximum at

a higher mold temperature is shifted toward the surface.

Some differences between the measured and predicted

birefringence are evident in the region between the loca-

tion of maximum of Dn and the core as seen from Fig.

22. The latter is evidently due to neglect in the present

simulation of the contribution of contraction and expan-

sion flow in a delivery system [1].

FIG. 21. Predicted residual thermal birefringence Dnth of PC plates

quenched from 2608C to different quenching temperatures Ti.

FIG. 22. Measured (symbols) and predicted (lines) total residual

birefringence Dn distribution in the gapwise direction at various radial

positions at the end of the cooling stage for the ICM PC disks in runs 1

and 2 corresponding to mold temperature of 408C (a) and 608C (b). The

contribution of thermal birefringence is included in the predicted

birefringence.


The predicted average transverse birefringence,

< nss � nhh >, is calculated by the integration of the in-

plane birefringence over the thickness of the PC disk. The

average transverse birefringence as a function of radial

location at mold temperatures of 408C, 608C, and 808Cand a melt temperature of 2608C in ICM PC disks corre-

sponding to runs 1, 2, and 3 is shown in Fig. 23. Again,

as in the case of ICM PS disks, as the mold temperature

increases from 408C to 808C, the average transverse bire-

fringence decreases due to a faster stress relaxation at the

higher mold temperature. As in the case of PS disks, the

birefringence in PC disks also decreases in the radial

direction [15, 67]. The simulated results are in a qualita-

tive agreement with the experimental data. However, sim-

ilar to ICM PS disks, the simulated and experimental bire-

fringence < nss � nhh > in ICM PC disks of the present

study are in contrast with experimental data reported by

Chen et al. [31] indicating a minimum birefringence at

some radial location.

The Effect of Melt Temperature. The measured and

predicted gapwise birefringence distribution at various radial

positions at the end of cooling stage in PC ICM disks in runs

4, 2, and 5 are shown in Figs. 24a, 20b and 24b, correspond-

ing to a melt temperature of 2408C, 2608C, and 2808C,respectively. It is seen that the distribution of birefringence

is very sensitive to the melt temperature. Both the simulated

and experimental results indicate that the maximum of the

birefringence is significantly higher at a lower melt tempera-

ture due to a slower stress relaxation after the injection and

compression stages. The measured birefringence in the core

is little affected by the melt temperature since it is due to

the thermal birefringence, as shown below. The contribution

of the thermal birefringence in ICM PC disks is relatively

high in comparison with little contribution in ICM PS disks,

as discussed earlier.

By adding the simulated thermal and flow birefrin-

gence, the better prediction of the measured residual bire-

fringence profiles at various radial positions is obtained.

These results are shown in Figs. 25a, 22b, and 25b for

melt temperatures of 2408C, 2608C, and 2808C, respec-

tively. It can be seen that increasing the melt temperature

leads to a decrease of the birefringence. It should be

noted that compared with the effect of the mold tempera-

ture the change of the melt temperature has a larger effect

on the values of the residual birefringence.

In general, the simulation tends to underestimate the

gapwise distribution of measured residual birefringence.

In particular, it is found that some differences between

experimental and predicted birefringence are evident in

the region between the location of maximum of Dn and

the core. Regardless, the predicted overall total values of

Dn are in a qualitative agreement with experimental

results.

The average transverse birefringence, < nrr � nhh >, as

a function of radius at melt temperatures of 2408C,2608C, and 2808C and a mold temperature of 608C in

ICM PC disks in runs 4, 2, and 5 is shown in Fig. 26. As

the melt temperature increases from 2408C to 2808C, thebirefringence < nrr � nhh > decreases significantly due to


age transverse birefringence < nss � nhh > as a function of the radial

position for ICM PC disks in runs 1, 2, and 3 corresponding to mold

temperatures of 408C, 608C, and 808C.



radial positions at the end of the cooling stage for the ICM PC disks in

runs 4 and 5 corresponding to melt temperatures of 2408C (a) and 2808C(b). The contribution of thermal birefringence is not included in the pre-



a more stress relaxation at the higher melt temperature.

This birefringence also decreases in the radial direction

[15, 67]. The behavior of the predicted and measured

birefringence < nss � nhh > of the present study is in con-

trast with experimental data reported by Chen et al. [31]

indicating a minimum birefringence at some radial dis-

tance. Generally, the simulated and experimental results

on the average transverse birefringence at various melt

temperatures are in good agreement. However, some devi-

ation between them is seen at radial positions close to the

gate. It is attributed to the neglect of a contribution to the

birefringence due to flow in a delivery system consisting

of contraction and expansion region [1].

The Effect of Compression Stroke. Figure 27a and b

shows the predicted and measured gapwise distributions

of the total birefringence, Dn, at a mold temperature 608Cat various radial positions at the end of cooling stage for

ICM PC disks in runs 6 and 7 corresponding to at com-

pression strokes of 1 mm and 2 mm, respectively. As a

compression stroke increases the birefringence decreases

because the resistance of melt to flow under large com-

pression stroke is less resulting in a low pressure and

stresses. In particular, the compression stroke of 1 mm

showed a higher maximum of the birefringence than that

of 2 mm.



tions at the end of the cooling stage for the ICM PC disks obtained in

runs 4 and 5 corresponding to melt temperature of 2408C (a) and 2808C(b). The contribution of thermal birefringence was included in the pre-




position for ICM PC disk in runs 4, 2, and 5 corresponding to melt tem-

perature of 2408C, 2608C, and 2808C.



tions at the end of the cooling stage for the PC ICM disks in runs 6 and

7 corresponding to the compression stroke of 1 mm (a) and 2 mm (b).

The contribution of thermal birefringence is included in the predicted

birefringence.


The average transverse birefringence, < nss � nhh >, as

a function of radial location in ICM PC disks at compres-

sion strokes of 1 mm, 1.5 mm, and 2 mm, a melt temper-

ature of 2608C and a mold temperature of 608C corre-

sponding to runs 6, 2, and 7 is shown in Fig. 28. Again

as the compression stroke increases from 1 mm to 2 mm,

the average transverse birefringence, < nss � nhh >,

decreases significantly because during the injection and

compression stages less stresses are developed in the melt

at a higher compression stroke. This birefringence also

decreases in the radial direction [15, 67] in contrast to the

experimental data reported by Chen et al. [31], indicating

a minimum birefringence at some radial distances. The

simulated and experimental results of the average trans-

verse birefringence at all radial positions with compres-

sion stroke of 1 mm, 1.5 mm, and 2 mm are in a qualita-

tive agreement. Again, some discrepancies between the

experimental and simulated results may be attributed to

the neglect of a contribution to the birefringence due to

the flow in the delivery system consisting of contraction

and expansion region especially near the gate [1].

The Effect of Switchover Time. The switchover time

in ICM process also of greatly affects the flow-induced

birefringence. Due to the limitation of control system, the

switchover time in this work was varied from 0.9 to 1.1 s

only. Figure 19c shows the predicted gapwise residual

flow birefringence Dn distribution at various radial posi-

tions for ICM PC disks in run 2 corresponding to the

switchover time of 1 s. This data can be directly com-

pared with Fig. 29 showing the calculated results for a

hypothetical case of the ICM PC disk in run 2 corre-

sponding to zero switchover time with other processing

conditions being similar to run 2. As stated earlier, during

the switchover time, two competing effects occur cooling

and stress relaxation in the absence of flow. Dominating

effect during the switchover time depends on the duration

of the switchover time. At the short switchover time, the

relaxation effect is dominating factor over the cooling.

Also, at the short switchover time the pressure is lower

and stress relaxation is less. On the other hand, at a long

switchover time used in this work, the cooling is dominat-

ing factor. Due to the cooling effect, the flow-induced

birefringence near the mold surface becomes higher due

to higher stresses at the given compression speed and

slower stress relaxation during the compression stage.

The simulated results in this work suggest that the

switchover time is an important parameter affecting the

birefringence of ICM PC disks. The simulated results

shown in Fig. 29 indicate the presence of a minimum and

a maximum of the birefringence at the switchover time of

0. When the switchover time was employed, more cooling

and higher shear rate at the given compression speed

occur resulting in higher stresses with the maximum of

Dn increasing and shifting further away from the wall as

seen from comparison of Fig. 19c.

The Comparison between CIM and ICM. In contrast

to ICM, in CIM compression stage is absent. It is impor-

tant to find effects of the compression stroke on the bire-

fringence distribution, Dn, and average transverse birefrin-

gence, < nss � nhh >, in ICM PC disks. In this regard,

calculations are carried out at zero switchover time. Fig-

ure 30 shows the predicted gapwise residual flow birefrin-

gence, Dn, distribution at various radial positions at the

end of cooling stage of CIM (a) and ICM (b) without

inclusion of the packing stage. As seen from comparison

of Fig. 30a and b, the birefringence is dramatically

decreased in ICM PC disk, especially near the mold sur-

face in comparison with CIM PC disk. In particular, the

birefringence of ICM PC disk increases and then decrease

slightly with the radial position, whereas that of CIM PC

disk continues to decrease significantly with the radial

position. This trend in ICM and CIM disks is explained

in detail [26, 57]. A reduction of birefringence in ICM

PC disk at a radial position of 2.948 cm compared with

that at a radial position of 3.704 cm is due to the fact that



position for ICM PC disks in runs 6, 2, and 7 corresponding to the com-

pression stroke of 1 mm, 1.5 mm, and 2 mm.

FIG. 29. Predicted residual flow-induced birefringence Dnfl distribution

in the gapwise direction at various radial positions at the end of the cool-

ing stage for the ICM PC disks without switchover time for run 2.


the velocity and shear rate increase with the radial posi-

tion during the compression stage. In contrast in CIM PC

disk, the velocity and shear rate decrease with the radial

position. From these results, one can conclude that the

compression stage in the ICM certainly reduces the mag-

nitude of the flow birefringence, most notably near the

mold wall.

It is also interesting to find the difference in the average

transverse birefringence < nss � nhh > in ICM and CIM

disks. The predicted and measured average transverse bire-

fringence, < nss � nhh > as a function of the radial location

for CIM and ICM PC disks at the end of cooling is com-

pared in Fig. 31. In both cases this birefringence decreases

in the radial direction [15, 67]. It can be seen that the value

of < nss � nhh > along the radial direction decreases with

distance from the gate with its value in CIM disk being

approximately twice larger than that in ICM disk. Regard-

less, the overall predicted and measured values of

< nss � nhh > are in a qualitative agreement.

CONCLUSIONS

Simulations of a two-dimensional flow in the ICM PS

and PC center-gated disk are carried to predict the flow

and thermal gapwise birefringence distribution and the av-

erage transverse birefringence. The CV/FE/FD numerical

method is employed using a nonlinear viscoelastic model

for flow and the linear viscoelastic and photoviscoelastic

models for the thermal stresses and birefringence. The re-

sidual flow birefringence in the molded disk is calculated

by considering the filling, compression, packing, and

cooling stages of ICM by using a compressible nonlinear

viscoelastic constitutive equation [42]. To take into

account the compressibility of polymeric melts, the Tait

equation is used. The thermal birefringence in ICM disks

was calculated using the viscoelastic and photoviscoelas-

tic models [45, 66] based on the free quenching approxi-

mation. Then, the total birefringence is calculated as a

sum of the flow and thermal birefringence. Effects of

various processing conditions on the birefringence are still

elucidated.

From the extensive simulation of the flow birefrin-

gence for ICM PS and PC disks, processing conditions

affecting significantly the birefringence are identified. The

melt temperature, compression stroke and switchover time

are the processing parameters that have a strong influence

on the birefringence Dn and average transverse birefrin-

gence < nss � nhh >. With an increase of the melt tem-

perature and compression stroke the flow residual birefrin-

gence and average transverse birefringence significantly

decrease. At zero switchover time, the maximum in the

gapwise birefringence distribution in ICM disks is signifi-

cantly reduced compared to that in CIM disks. When the

switchover time is employed, the flow birefringence and

average transverse birefringence are higher compared with

those of ICM disks without a switchover time, but they

are still lower compared to the CIM disks.

The thermal birefringence in ICM disks is treated as

that developed in freely quenched slabs but with the tem-

perature evolution taken from simulation of ICM disks. It

is shown that the thermal birefringence in ICM disks is

mostly influenced by the mold temperature. The effect of

the melt temperature on the residual thermal birefringence

is comparatively less than that of the mold temperature

due to the fact that molding is typically carried out at the

FIG. 30. Predicted residual flow-induced birefringence Dnfl distribution

in the gapwise direction at various radial positions at the end of the cool-

ing stage for PC disks made by the CIM in run 8 (a) and ICM in run 2

without switchover time (b).


age transverse birefringence as a function of the radial position for PC

disks made CIM in runs 8 and by ICM in run 2.


melt temperature significantly above Tg. At the same melt

and mold temperatures, the compression stroke does not

show much effect on the thermal residual birefringence.

The presence of the compression stage in the ICM

decreases the overall birefringence, but not the thermal

birefringence. Therefore, one may conclude that ICM is

more suitable for manufacturing of high quality optical

parts than the CIM.

It was found that the measured residual birefringence

distribution showed a maximum near the mold wall and

significant birefringence in the core. The simulated results

capture the essential features of the gapwise flow birefrin-

gence, Dn, and average transverse birefringence,

< nss � nhh >, distributions.

The simulation showed that for ICM PS disks, the bire-

fringence caused by the thermal stresses was negligible

compared with the flow birefringence because the stress-

optical coefficient of PS in the glassy state is much lower

than that in the melt state. For ICM PC disks, the thermal

birefringence is of similar magnitude as the flow birefrin-

gence. The simulation showed that the gapwise residual

flow birefringence is zero at the core due to a fast relaxa-

tion of stresses at high temperatures. A peak near the

mold surface appears as a result of the combination of

frozen-in chain orientation in the injection and compres-

sion stages and slow relaxation in the cooling stage. The

nonzero concave shape of the residual birefringence in

the core predicted and measured in ICM disks is due to

the residual thermal birefringence. The magnitude of the

average transverse birefringence in CIM disks is twice

higher than that of ICM disks. To the best of our knowl-

edge, the present study is the first extensive experimental

and simulation investigations of the frozen-in thermal and

flow birefringence in ICM disks. Although the simulations

do not fully describe the experimental observations, the

present study is a significant step toward understanding

complicated phenomena.

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