Thermal birefringence in freely quenched multilayered slabs of amorphous polymers: Experiment and simulation

Thermal Birefringence in Freely Quenched MultilayeredSlabs of Amorphous Polymers: Experimentand Simulation

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

The simulation of the gapwise distribution of thethermally-induced residual birefringence and stressesin freely-quenched PS-PC-PS and PC-PS-PC multi-lay-ered slabs in water was carried out to calculate thegapwise distribution of the transient and residual bire-fringence. The modeling was based on the linearviscoelastic and photoviscoelastic constitutive equa-tions combined with the first-order rate equation forvolume relaxation. The master curves for the Young’srelaxation modulus and strain-optical coefficient func-tions obtained earlier for PS and PC were used in thesimulations. The obtained numerical results providedthe evolution of the thermally-induced stress and bire-fringence with time during and after quenching. Thepredicted gapwise residual birefringence distribution inthese slabs was found to be in a fair agreement withthe measured results. In addition, the gapwise distribu-tion of the thermally-induced residual birefringence inthe multi-layered PS-PMMA-PS, PMMA-PS-PMMA,PMMA-PC-PMMA, and PC-PMMA-PC slabs quenchedfrom different initial temperatures was measured.Explanations were provided for the observed gapwisedistribution of the thermal residual birefringence ineach layer of these slabs including the effect ofthe initial temperature. POLYM. ENG. SCI., 54:2097–2111,2014. VC 2013 Society of Plastics Engineers

INTRODUCTION

During polymer processing, the polymer undergoes

simultaneous mechanical and thermal influences in the

fluid, rubbery, and glassy states. These effects introduce

orientation, residual stresses, and shrinkage in the prod-

ucts which in turn affect the physical, optical and

mechanical properties, dimensional stability, and appear-

ance of the finished products. The frozen orientation man-

ifests itself in the frozen-in birefringence phenomenon.

The residual stresses and birefringence in polymer proc-

essing operations can be attributed to two main sources.

One is the flow-induced residual stresses including the

shear and normal stresses during the nonisothermal flow.

The other is the thermally induced stresses induced during

the cooling process and inhomogeneous densitification

even in the absence of flow [1–7]. The thermal stresses

occur in parts that experience inhomogeneous temperature

variations. In the case of elastic materials subjected to

free quenching, a temperature gradient induces the tran-

sient thermal stresses that vanish when the sample reaches

a homogeneous temperature. This occurs only if there is

no plastic deformation and the material properties

throughout the sample are homogeneous at the initial and

final states [6, 8]. Thermal stresses arise due to the com-

bination of an inhomogeneous temperature distribution

and a strong dependence of the mechanical properties of

the material on temperature. In the case of polymeric

materials, the combined effect of the non-equilibrium

density variation and the viscoelastic behavior of the

polymer taking place during cooling through the glass

transition temperature (Tg) results in the thermal stresses

and birefringence. Due to low thermal conductivity of

polymers, the cooling process generates a large tempera-

ture gradient along the sample thickness such that each

position reaches the final temperature at different times.

As a consequence, each point experiences different ther-

mal history. During cooling, due to the temperature gradi-

ent, the polymer contracts at different proportion causing

the thermal stresses. These stresses partially relax in

regions where the temperature is elevated. However,

when the polymer cools down from above to belowTg, its

Young’s modulus increases by several orders of magni-

tude [9]. Therefore, part of the thermal stresses and bire-

fringence generated during solidification will not relax

resulting in the residual stress and birefringence distribu-

tion in the final product [6]. However, the thermal bire-

fringence is not a linear function of the thermal stress,

since the stress-optical or strain-optical coefficient shows

a photoviscoelastic behavior and becomes a function of

both the time and temperature in the glass-to-rubber tran-

sition zone. Therefore, the memory effect in the optical

behavior of the polymer becomes significant and a photo-

viscoelastic relation should be utilized in correlating the

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.23748

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

VC 2013 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—2014

thermal birefringence with thermal stresses [10–14]. How-

ever, very little work is available in the existing literature

concerning this aspect.

Osaki and coworkers [15–18] performed the dynamic

and stress relaxation measurements for polystyrene (PS)

and polycarbonate (PC) over the temperature range of 90

to 115�C for PS and 142 to 156�C for PC. In the analysis

of the data, they utilized a so-called modified stress-

optical rule earlier developed by Read [19]. For PS, the

master curves of the Young’s relaxation modulus at the

reference temperature of 100�C, as well as the strain-

optical coefficient at the reference temperature of 115�Cwere obtained. Shyu and Isayev [12, 20] carried out the

stress relaxation measurement over the temperature range

of 22 to 123�C for PS and 22 to 153�C for PC to obtain

the master curve of the time-dependent relaxation modu-

lus and strain-optical coefficient functions along with

their respective shift factors from which the stress-optical

coefficient functions for both PS and PC were calculated.

Over the years the factors governing the development

of the thermal residual stresses and birefringence during

the polymer processing have received much attention

from many researchers [5, 10, 21–32]. In particular,

Isayev and Crouthamel [5] and Isayev [21] gave reviews

on the relevant theoretical works about the thermal

stresses and birefringence in quenched amorphous poly-

mers. The theories that have been attempted can be clas-

sified into two categories. One is based on the instant-

freezing assumption [5, 21–23, 27, 28, 30, 33–35], which

states that above and below Tgthe polymer can respec-

tively be treated as an idealized fluid and an idealized

elastic material [33]. The other is based on the visco-

elastic constitutive equation relating the stresses to the

strains with an inclusion of the free volume relaxation

affected by rapid cooling [5, 10, 21, 24, 32, 36–44].

Therefore, the thermal histories and viscoelastic effects

should be taken into account when the thermal stresses

and birefringence of the final products are calculated. In

the polymer processing because of the high temperature

gradient through the part thickness and the rapid tempera-

ture change, the nonequilibrium free volume should be

included in the Williams-Landel-Ferry (WLF) equation

based on the free-volume approach [45]. In addition, the

time-dependent volume relaxation [46, 47] should be

taken into account. Bartenev [34, 35] was evidently the

first to propose the “instant freezing” theory for descrip-

tion of the thermal stresses in inorganic glasses. It treats

the glass above Tg, as an ideal fluid bearing no stresses

due to low viscosity. On the other hand, below Tg, glass

is solidified and treated as an elastic material with no

flow and stress relaxation as a result of high viscosity.

Since the instant freezing assumption is too crude to

describe the time-temperature behavior of polymeric

materials in the transition zone, significant deviation of

the predictions from the experimental results was

observed. Santhanam [48] predicted the thermal stresses

in injection molded parts by introducing the linear visco-

elastic model. However, the predicted residual stresses

were much higher than experimental results. Bushko and

Stokes [22, 23] briefly reviewed the development in mod-

eling of the thermal stresses, studied the residual stresses

and dimensional changes caused by solidification using a

thermoviscoelastic model and thought this model could

be used to assess the packing pressure effect in injection

molded strip. But these studies did not take into account

the density relaxation phenomena of polymeric materials.

Schwarzl and Staverman [49] classified “thermorheo-

logically simple” materials as materials in which all

molecular changes are affected by temperature in the

same way and specified the condition for the material

relaxation function to satisfy the time-temperature equiva-

lence principle. Consequently, only amorphous thermo-

plastics are expected to exhibit such a thermorheological

simplicity since in the semicrystalline polymers both the

change of the crystalline and amorphous structure will

certainly depend on the temperature in a way different

from that of the viscosity in the amorphous polymers.

The most appealing equation to many researchers was

derived by Morland and Lee [36] who extended the linear

viscoelasticity theory to account for the time- and

temperature-dependent variations of properties by intro-

ducing the concept of the pseudotime or material time

which is in general a function of both time and space var-

iables. The model was applied to a cylinder of the incom-

pressible linear thermorheologically simple material

subjected to a steady-state temperature field. However,

the thermal stresses due to nonhomogeneous thermal

expansion were neglected in this model because of the

assumption of incompressibility. Based on the free vol-

ume theory, Shyu and Isayev [4, 10, 12, 50, 51] devel-

oped a physical model to predict the thermally-induced

birefringence. They also performed experimental meas-

urements of the birefringence in freely quenched samples.

Ghoneim and Hieber [52] attempted to predict the thermal

residual stresses including density relaxation phenomena,

indicating that the density relaxation has a significant

effect on the evolution of the residual stresses. However,

in their work, no attempt was made to compare numerical

results with experimental data. Guo and Isayev [24, 32]

compared the thermoelastic model and thermoviscoelastic

model to calculate the residual thermal stresses in freely

quenched slabs of semicrystalline polymers, based on the

modifications of the Indenbom theory [33] and Morland

and Lee viscoelastic constitutive equation [36] with crys-

tallization phenomena taken into account.

Recently, Min and Yoon [53] performed frequency

sweep tests to obtain the stress optical behavior in a wide

range of frequency and temperature including rubbery,

glassy, and glass transition regions for PS, PC, cyclic ole-

fin copolymer (COC). In particular, they found that for

COCs of different compositions exhibiting different val-

ues of Tg the stress-optical coefficients in rubbery and

glassy states was almost unaffected by the difference in

composition.

2098 POLYMER ENGINEERING AND SCIENCE—2014 DOI 10.1002/pen

The objective of this study was to elucidate the ther-

mally induced birefringence in freely quenched multilay-

ered slabs made of various layer combinations. To the best

of our knowledge, this study is the first attempt in available

literature where simulations and measurements of the gap-

wise distribution of the thermally-induced birefringence Dnin freely quenched multilayer slabs is considered. The pres-

ent modeling is based on the linear viscoelasticity and lin-

ear photoviscoelasticity accounting for the time and

temperature dependence of mechanical and optical proper-

ties with the volume relaxation based on the first-order rate

equation [25]. The numerical formulation is based on the

quasi-static analysis with the hypothesis of infinitesimal

deformations of linear viscoelastic multilayered slabs expe-

riencing one-dimensional heat conduction along their thick-

ness direction only. To verify the modeling, free quenching

experiments were performed on thin multilayered slabs at

different initial temperatures using various combinations of

polymer layers and their birefringence distributions along

the thickness direction were measured. This comparison is

useful for applications of the model to predict the residual

birefringence in co-injection molded and co-extruded parts.

MATERIALS AND EXPERIMENTAL PROCEDURES

Multilayered slabs were prepared from a variety of

material combinations under quenching conditions, as

listed in Table 1. Single component slabs of PS (Styron

615-APR, Dow Chemical Co.), PC (PC123, General Elec-

trical Co.), and polymethyl methacrylate (PMMA) (Per-

spex CP-51, INEOS Acrylics) with dimensions of 12 3

10 3 0.15 cm3 were first manufactured by compression

molding at 185�C for PS, 190�C for PMMA and 225�Cfor PC, followed by slow cooling for approximately 10 h

inside the compression molding machine to obtain the

stress-free slabs. The prepared slabs were stacked in the

rectangular mold to make multi-layered slabs with dimen-

sions of 12 3 10 3 0.45 cm3. The mold with the stacked

slabs was placed into vacuum oven for 45 min at a tem-

perature of 225�C for the PC being the outer layer and

210�C for the PS or PMMA being the outer layer fol-

lowed by slow cooling. These multilayered slabs were

then cut into dimensions of 5 3 5 3 0.45 cm3 for carry-

ing out free quenching experiments. Before quenching,

the birefringence measurements for one multilayered slab

were carried out showing negligible birefringence (of the

order of 1027). Initial temperatures of 150, 160, 170, and

180�C were employed for material combinations of PS-

PC-PS, PC-PS-PC, PMMA-PC-PMMA, and PC-PMMA-

PC, and 110, 130, 150, and 170�C for material combina-

tions of PS-PMMA-PS and PMMA-PS-PMMA. A

quenching temperature of 25�C was used for all material

combinations. For quenching, the slab was placed in a

beaker filled with silicone oil at the required initial tem-

perature. After reaching equilibrium the multilayered slab

was quickly taken out of the beaker and immediately

quenched in water at 25�C.

Based on the magnitude of the thermally induced bire-

fringence in the multilayer slabs, a strip of a thickness of

0.2 cm for PS-PC-PS and PC-PS-PC slabs and a strip of

a thickness of 0.2 cm or 0.4 cm for PMMA-PC-PMMA,

PC-PMMA-PC, PS-PMMA-PS, and PMMA-PS-PMMA

slabs were cut from the quenched slabs by a low speed

diamond saw (ISOMET/BUECHLER). The birefringence

distribution along the slab thickness direction was then

measured by a polarizing microscope (Leitz Laborlux 12

POL/LEITZ WETZLAR) with a compensator (4th or 30th

order, 1592K/LEITZ WETZLAR) within 2 to 3 days after

quenching. Figure 1 shows the schematic diagram of the

experimental procedure.

THEORETICAL

An idealized problem of the free quenching of a multi-

layered slab with the geometry and coordinate system

shown in Fig. 2 was considered. The slab was assumed to

be infinite in x-y plane with its thickness being 2b. Ini-

tially, the slab is at a uniform initial temperature of Ti

above Tg. Then, it is subjected to symmetric cooling from

both surfaces at a coolant temperature of T1 below Tg.

Therefore, heat transfer occurs only in z direction. The

thermal stresses and birefringence are generated due to

the interactions between nonhomogeneous thermal con-

traction and changes in the polymer viscoelastic and opti-

cal properties during cooling.

Temperature Distribution

The temperature distribution of the multilayered slab

was calculated by solving the one-dimensional heat trans-

fer equation:

@T

@t5aj

@2T

@z2

� �(1)

where aj5kj

qjCp;jis the thermal diffusivity with q, Cp, and

k being the density, heat capacity, and the thermal

TABLE 1. Multilayered slabs and quenching conditions.

Material combination Initial temperature Quenchingy temperature

PS-PC-PS Silicone oil Water

PC-PS-PC 150 �C (90 min) 25 �C160�C(15min) 25 �C

170 �C (8 min) 25 �C180�C (5 min) 25 �C

PS-PMMA-PS Silicone oil Water

PMMA-PS-PMMA 110�C (60 min) 25 �C130 �C (30 min) 25 �C150 �C (15 min)

170 �C (5 min)

25 �C

PMMA-PC -PMMA Silicone oil 25 �C Water

PC-PMMA-PC 150 �C (90 min) 25 �C160 �C (15 min) 25 �C170 �C (8 min) 25 �C180�C (5 min) 25 �C

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—2014 2099

conductivity with the indice j representing skin and core

polymer melts including the interface.

The initial and boundary conditions are

Tjt50 5 Ti (1a)

@T

@z

��z50

50 (1b)

For the quenching experiments, the cooling medium was

water which was in contact with the surfaces of the multi-

layer slab. Therefore, a convective boundary condition

between the polymer and water was imposed at the surfaces:

2kj@T

@z

��z56b

5h Tjz56b2T1ð Þ (1c)

where h is the heat transfer coefficient.

Constitutive Equations

To calculate the thermally-induced stresses and bire-

fringence in polymers, several approaches suggested [10,

23, 33, 36, 48, 49, 52, 54, 55]. The approach employed in

this work is the linear viscoelastic theory proposed by

Morland and Lee [36] and Lee et al. [38]. This theory is

based on the linear viscoelastic theory for thermorheolog-

ically simple materials and incorporates the shear and

bulk relaxation modulus functions. The constitutive equa-

tion for an isotropic linear viscoelastic material is given

[36, 37]:

rij5

ðt

0

2G t2sð Þ @@s

eij21

3Iedij

� �ds1dij

ðt

0

K t2sð Þ @@s

Ie½ �ds

(2)

where, rij is the stress tensor, eij is the strain tensor, dij is

the unit tensor, G and K is the time-dependent shear and

bulk relaxation modulus functions, and Ie is a bulk strain

defined as:

Ie5tr eð Þ5exx1eyy1ezz (3)

where, exx, eyy, and ezz are the strains in x, y, and z direc-

tions, respectively. The two terms on the right hand side

of Eq. 3 represent the deviatoric and dilatational response

of the viscoelastic material with respect to the elapsed

time t2sð Þ.In comparison with the mechanical viscoelastic behav-

ior, the photoviscoelastic behavior is less known. The

photoviscoelasticity means that the stress-optical coeffi-

cient obtained from creep experiments and the strain-

optical coefficient obtained from stress-relaxation experi-

ments, similar to the relaxation modulus, vary with

temperature and time. The photoviscoelastic phenomenon can

be clearly observed in the glass-to-rubber transition zone.

The first basic mathematical approach of the photovis-

coelasticity was given by Mindlin [56] who used a four-

element mechanical model for the incompressible material

and assumed that only the spring contributes to the bire-

fringence. Later, Read [57–59] extended Mooney’s

approach of the molecular theory of viscoelasticity for

bulk polymer under finite deformation to describe the

FIG. 2. Coordinate system for the free quenching of a multi-layered

slab.

FIG. 1. Free quenching experiments on multilayered slabs and their

cutting procedures.


optical behavior. Dill [60] considered the refraction index

tensor, nij, to be a functional of the strain (or stress) ten-

sor. After expanding the functional in power series, con-

sidering the case of small strain and then imposing the

time-temperature superposition, he obtained [11, 20, 50]

nij tð Þ5nodij1

ðt

0

Ce t2sð Þ @@s

eij21

3dijIe

� �ds

1dij

ðt

0

De t2sð Þ @@s

Ief gds

(4)

where no is the average refractive index, Ce and De is the

shear and bulk strain-optical coefficient, respectively.

Free volume of polymers in an equilibrium state is

dependent on temperature and pressure. When a polymer

is quenched from above to below Tg, the equilibrium vol-

ume is reached gradually leading to the volume relaxation

process. The nonequilibrium state produced by the fast

cooling introduces more free volume into the polymer

than that available in the equilibrium state at the same

temperature. This additional free volume accelerates all

relaxation processes. Therefore, the linear viscoelastic

models should be modified to include the effect of the

nonequilibrium free volume. The volume contraction

from the nonequilibrium state can be divided in two parts:

an instantaneous contraction and a gradual contraction to

the final equilibrium value. Therefore, the free volume

becomes time-dependent, and the nonequilibrium free vol-

ume should be included in the WLF equation.

The thermal strain due to the volume relaxation, eT in

the nonisothermal process can be described as [10, 25,

61]:

deT

dt5bg

dT

dt2

eT2eTe

sraT(5)

where eT and eTeare the actual and equilibrium thermal

strain, respectively, at temperature T. The value of sr is

the volume relaxation time at a reference temperature Tr,

and bg is the linear expansion coefficient in the glassy

state. Physically, the first term on the right-hand side

describes the instantaneous contraction and the second

term gives the gradual contraction according to the first-

order rate theory. The equilibrium thermal strain is given

as

eTe5b1 T2Tið Þ if T � T2 (6)

eTe5b1 T22Tið Þ1bg T2T2ð Þ if T < T2 (7)

where Ti is the initial temperature at t 5 0, T2 is the

glass-transition temperature observed in experiments of

infinite time scale at which the free volume becomes

zero, and bl and bg is the thermal expansion coefficients

in the rubbery and glassy states, respectively.

Leaderman [62] proposed that polymers exhibit the

time- and temperature-dependent mechanical properties

following the time-temperature superposition principle.

This principle states that a uniform shift in the relaxation

modulus, viscosity, and other characteristic functions of

the material is observed with a change of temperature.

This shift factor, aT , follows a modified form of the WLF

equation as:

log aT52B T2Trð Þ

af Tr2T2ð Þ Tr2T2ð Þ1T2Tr½ � (8)

where B is a constant [63], af is the thermal expansion

coefficient of the fractional free volume above T2 and Tr

is the reference temperature. This equation has a form

identical to the WLF equation if C1 and C2 are defined

as:

C15B

af Tr2T2ð Þ (9)

C25 Tr2T2ð Þ (10)

Eq. 9 is valid for polymer melts between Tg and

Tg 1 100�C in which the constants C1 and C2 are eval-

uated by fitting the experimental data of the shift factors

above Tg. However, at temperatures below Tg the none-

quilibrium free volume rather than the equilibrium vol-

ume should be considered because the WLF equation can

not describe the relaxation behavior in the glassy state.

To solve the problem, Rusch [61] suggested the concept

of the “effective temperature” which is the temperature

corresponding to the temperature of an equilibrium state

that has the same amount of the free volume as the none-

quilibrium state. This effective temperature is related to

the actual temperature through

Teff5T1eT2eTr

bl2bs

if T � T2 (11)

Teff5T21eT2eTr

bl2bs

if T < T2 (12)

Therefore, the shift factor function becomes:

log aT52C1 Teff2Trð Þ

C21 Teff2Trð Þ (13)

Eq. 13 is the same as Eq. 8 except that the effective

temperature, instead of the actual temperature, is used

due to the nonequilibrium free volume.

According to the time-temperature superposition princi-

ple, the shear relaxation modulus, G, after certain time t at a

temperature T corresponds to that at a pseudotime t=aT Tð Þat the reference temperature Tr. The term t=aT Tð Þ is called

the reduced time n and is defined as:

n5t

aT Tð Þ (14)

For the nonisothermal process, the time-temperature

superposition can be extended to a variable temperature

field by a new definition of the reduced time [38] which


means that the thermal history and the time dependence

of G are accounted by a single variable n defined as:

n5

ðt

0

dt0

aT Tðt0Þ½ � (15)

By introducing the reduced time into Eqs. 2 and 4, the

following equations to calculate the thermally-induced

stresses and the refractive index tensor in terms of the

reduced time were, respectively, obtained [38].

rij nm tð Þð Þ5ðnm

0

2G nm2n0m @

@n0meij n0m

21

3Ie n0m

dij

� �dn0m

1dij

ðnm

0

K nm2n0m @

@n0mIe n0m

23eT n0m

dij

� �dn0m

(16)

nij nc tð Þð Þ5nodij1

ðnc

0

Ce nc2n0c @

@n0c

eij n0c

21

3dijIe n0c

� �dn0c

1dij

ðnc

0

De nc2n0c @

@n0cIe n0c

23eT n0c

dij

�dn0c

(17)

where nm is the reduced time for the modulus and nc is

the reduced time for the strain-optical coefficient. It

should be noted that Ie23eT means the strains resulting

from a temperature change and it does not contribute to

the pressure and birefringence. Eqs. 16 and 17 use the

different reduced time for the shear relaxation modulus

and strain-optical coefficient because their shift factors

may be different.

NUMERICAL IMPLEMENTATION

Temperature History

Equation 1 with initial and boundary conditions were

numerically solved to determine the temperature history

using the Crank-Nicolson scheme [64] by means of a

finite difference method, as shown in Appendix A. After

obtaining the thermal history for the multilayered slab,

the first order rate equation, Eq. 5, together with Eq. 11through Eq. 13 for the skin and core layers was solved by

the fourth order Runge-Kutta method [64] to obtain the

thermal strain eT , the effective temperature Teff , and the

shift factor aT for the skin and core layers. It should be

noted that values of eT , Teff , and aT for the skin and core

layers were averaged to obtain the value at the interface.

Then the value of n was determined from Eq. 15.

Stress and Strain Analysis

For an infinite multilayered slab subjected to symmet-

ric cooling, only volume contraction occurs upon cooling.

Therefore, it was assumed that no shear stress and strain

components are imposed such that

rij5eij50 i 6¼ jð Þ (18)

Moreover, no surface traction is acting on the interface

and free surfaces of the slab in x-y plane.

rzz5 0 2b � z � b (19)

Therefore, the problem is reduced to a planar

stress problem. Since the lateral dimensions are much

larger than the thickness (the edge effect is neglected),

the two non-zero normal stress and strain components are

equal to each other, and the strain components are inde-

pendent on z. Consequently, it can be written that

rxx5rxx z; tð Þ5ryy5ryy z; tð Þ (20)

exx5exx tð Þ5eyy5eyy tð Þ (21)

Because the multuilayered slab is free of constraints

(no external forces), the resulting force is zero through

the thickness.

ðb

2b

rxx z; tð Þdz50 (22)

It is seen that with this idealized problem of free

quenching, the calculation of the thermal stresses and

birefringence is greatly simplified.

Thermal Stresses and Birefringence

Numerical simulation schemes were formulated to cal-

culate the residual thermal stresses and birefringence in

the freely quenched multilayered slab.

Expanding Eq. 16 yields the following equation:

rxx sxy sxz

syx ryy syz

szx szy rzz

��

��5

ðnm

0

2G nm2n0m @

@n0m

exx exy exz

eyx eyy eyz

ezx ezy ezz

��

��2

1

3

Ie 0 0

0 Ie 0

0 0 Ie

��

��

266664

377775dn0m

1

ðnm

0

K nm2n0m @

@n0m

Ie 0 0

0 Ie 0

0 0 Ie

��

��2

3eT 0 0

0 3eT 0

0 0 3eT

��

��

266664

377775dn0m

(23)


Substituting the volume bulk strain Ie given by Eq. 3into Eq. 23, the strain components resulting from only

temperature change become:

exx21

3Ie5

1

32exx2eyy2ezz

(24)

eyy21

3Ie5

1

32eyy2exx2ezz

(25)

ezz21

3Ie5

1

32ezz2exx2eyy

(26)

Ie23eT5exx1eyy1ezz23eT (27)

where the thermal strain eT is given by Eq. 5.

Substituting Eq. 24 to Eq. 27 into Eq. 24 and rearrang-

ing them using the stress analysis, Eq. 19 and Eq. 20, the

stress components were obtained as:

rxx5ryy5

ðnm

0

2G nm2n0m @

@n0m

1

3exx n0m

21

3ezz n0m � �

dn0m

(28)

rzz505

ðnm

0

2G nm2n0m @

@n0m

2

3ezz n0m

22

3exx n0m � �

dn0m

1

ðnm

0

K nm2n0m @

@n0m2exx n0m

1ezz n0m

23eT

� �dn0m

(29)

Numerical solutions for the stress and strain tensor

components are given in Appendix B.

After obtaining the strain history, exx tð Þ and ezz z; tð Þ, in

the multi-layered slab, the birefringence was calculated as

follows:

Dnth5nxx2nzz5

ðnc

0

Ce nc2n0ð Þ @@n0

exx n0ð Þ2ezz n0ð Þ½ �dn0 (30)

Similar to the stress calculation, using the piecewise

linear approximation, Eq. 30 was discretized to obtain the

explicit formulation for the birefringence calculation as

follows:

Dnth5nxx2nzz5Xj

k52

ekxx2ek21

xx 2ei;kzz 1ei;k21

zz

ni;kc 2ni;k21

c

3

ðni;kc

ni;k21c

Ce ni;kc 2n0

dn0

(31)

SIMULATED RESULTS AND COMPARISON WITHEXPERIMENTS

The physical properties of PS and PC used in the

numerical simulation are listed in Table 2. The Young’s

relaxation modulus, E, strain-optical coefficient, Ce,

stress-optical coefficient, Cr functions with the corre-

sponding shift factors, aT , are required in the linear visco-

elastic and photoviscoelastic constitutive equations. These

functions were obtained by performing tensile stress and

birefringence relaxation experiments and reported in ear-

lier studies [20, 50]. The residual birefringence distribu-

tion in freely quenched multilayered slabs was measured

within two to three days after quenching. The obtained

data were compared with the simulated results based on

the viscoelasticity and photoviscoelasticity with inclusion

of the volume relaxation till time of 2 3 105 s after

quenching. Since the volume relaxation was considered,

aging that occurred during two days following quenching

was included in the simulated results.

PS-PC-PS Slabs

Experiments have shown that no birefringence appears

when light is passed perpendicular to the x-y plane of the

multilayered slab, indicating that the birefringence in the

other two planes is equal. Thus, the birefringence in x-zor y-z planes was measured.

The measured and simulated thermally-induced resid-

ual birefringence distributions along the thickness direc-

tion of PS-PC-PS plates quenched in water at a

quenching temperature of 25�C from different initial tem-

peratures are shown in Fig. 3. The value of Cr for PC

used as the core layer is positive being, respectively 100

and 5600 Brewsters for the glass and melt state as deter-

mined from the tensile relaxation experiment [20, 50].

TABLE 2. Physical properties of PS and PC used in the simulation.

Properties PS Ref. PC Ref.

bl,(1/�K) 0.00066 [65] 0.00067 [65]

Bg,(1/�K) 0.00021 [65] 0.0002 [65]

a (m2/s) 6.13 1028 [65] 7.31 3 1028 [65]

h (J/s.m2,K) 490 [50] 490 [50]

sr (s) 0.04 [50] 0.3 [50]

Poisson ratio 0.33 [66] 0.41 [66]

WLF equation: C1 10.6 [50] 8.82 [50]

c2 (K) 57.0 [50] 40.2 [50]

Tr (K) 370.0 [50] 420.5 [50]

FIG. 3. Predicted and measured thermal birefringence distributions

along the thickness direction in PS-PC-PS multilayered slabs quenched

in 25�C water from different initial temperatures.


Therefore, the calculated and measured thermal birefrin-

gence in the PC core layer in quenched PS-PC-PS slabs

is always positive throughout the core thickness. It is

seen that the measured and simulated thermally-induced

birefringence of the PC core layer increases with an

increase of the initial temperature. In the PS skin layers,

at initial temperatures of 150 and 160�C the thermal bire-

fringence is negative throughout their thickness. It is due

to the fact that when the thermal stresses in the PS skin

layers of the quenched PS-PC-PS slabs are compressive

in the surface region and tensile away from the surface

(as shown in Fig. 4 for the thermal stress distribution),

the stress-optical coefficient, Cr, of PS changes sign dur-

ing cooling being negative in the melt state and positive

in the glassy [20, 50]. The compressive stress near the

surface and the positive stress-optical coefficient Cr of

PS layers in quenched PS-PC-PS slabs produces negative

birefringence in the case of initial temperatures of 150

and 160�C. Moreover, the compressive stress near the

surface and the negative stress-optical coefficient Cr of

PS after short quenching time produces negative birefrin-

gence for the case of initial temperatures of above 160�C.

On the other hand, for the core region of the PS skin

layers, the tensile thermal stresses and the negative stress-

optical coefficient Cr of PS after long quenching time

produces the negative birefringence at all the initial tem-

peratures, as shown in Fig. 3. Therefore, the sign for the

stress-optical coefficient Cr of PS results in the sign

reversal in the birefringence in the PS skin layers com-

pared with the PC core layer showing the positive bire-

fringence due to positive values of Cr of PC in the melt

and glassy states.

As the initial temperature increases from 160�C to

170�C and 180�C, the thermal birefringence becomes pos-

itive at the surface and negative in the extensive region

till the interface with position of zero birefringence being

near the surface of the PS layers. The difference between

birefringence distribution for the PC and PS layers can be

attributed in large part to the corresponding stress-optical

coefficient Cr of these polymers. It is known that the

value of Cr for PS is time- and temperature-dependent

and, negative and positive above and below Tg, respec-

tively. Its value for the melt is 25200 Brewsters [20, 50]

and 10 Brewsters at room temperature [20, 50, 67]. Such

a reversal in sign of the stress optical coefficient of PS

was caused by the orientation of the phenyl groups and

their polarizabilities [67–69].

It should be noted that the simulated results were

based on the volume relaxation times sr 5 0:04 s at 97�Cfor PS and sr 5 0:3s at 147.5�C for PC. These values

were found to fit all the measured thermal birefringence

in quenched single component PS and PC slabs [12, 25,

50]. However, it should be noted that no single volume

relaxation time can give the overall satisfactory predic-

tions of the residual birefringence in freely quenched PS

and PC slabs since the volume relaxation is governed by

a spectrum of relaxation times. The heat transfer coeffi-

cient, h, that best fit the experimental results was 490 J/s

m2 K for quenching of single component PS and PC slabs

in water [12, 25, 50]. This value of h was also used for

simulation of quenching of the multilayered slabs in this

study.

It should be noted that, similar to birefringence distri-

bution in the single polymer slabs subjected to the sym-

metric cooling, in the present case of the multilayered

slabs the thermal birefringence profile is not balanced,

that isÐ b2b Dndz 6¼ 0. The birefringence imbalance is due

to the fact that the stress-optical coefficient is not a con-

stant, but a function of both the time and temperature.

The simulated results of the thermal birefringence are in

qualitative agreement with the experimental data.

The predicted thermal stress distributions in the

PS-PC-PS slabs quenched in 25oC water from different

initial temperatures are plotted in Fig. 4. The thermal

stresses in the PC core layer are seen to be tensile (posi-

tive) and much lower than that in the PS skin layers. For

the PS skin layers, the thermal stresses are tensile (posi-

tive) near the interfaces, but they are compressive (nega-

tive) near the surface. Higher initial temperatures result in

slightly higher values of the thermal stresses in the PC

core layers and PS skin layers. This is in contrast to the

thermal birefringence in the PC core layer in Fig. 3 show-

ing a significant increase in the thermal birefringence

with the initial temperature. The latter is due to the

dependence of the stress optical coefficient on the time

and temperature. The fact that the effect of high initial

temperatures on the thermal stresses was insignificant is

in agreement with earlier observation on the PS and PC

slabs [5].

To get a better understanding of how the thermal

stresses and birefringence are built up, the evolution of

the actual and effective temperatures during quenching of

PS-PC-PS slab in 25�C water from the initial temperature

of 170�C was investigated (Fig. 5). As seen from this fig-

ure above an initial temperature of 145�C, the volume

FIG. 4. Predicted thermal stress distributions along the thickness direc-

tion in PS-PC-PS multilayered slabs quenched in 25�C water from dif-

ferent initial temperatures.


relaxation of PC core layer is very fast such that the

effective temperature in the PC core layer is virtually

identical with the actual temperature. The volume relaxa-

tion in the PS skin layer is also fast above 95�C with the

effective temperature in the PS skin layers being virtually

same as the actual temperature. Below 145�C a compari-

son of the actual and effective temperatures in the PC

core layer indicates that the difference between these tem-

peratures becomes more and more prominent with the

actual temperature being lower. For the PS skin layers, at

the actual temperatures below 95�C the difference

between the actual and effective temperatures becomes

large with the actual temperature also being lower than

the effective one. At 44 s after quenching, the gapwise

distribution of the effective temperature in the PC core

layer becomes uniform, but in the PS skin layers it still

decreases toward the surface. At 450 s after quenching,

the effective temperatures in both PC core and PS skin

layers become practically uniform, but still decreases

slowly with time due to the aging phenomenon.

PC-PS-PC Slabs

In this section, results obtained in quenched multilay-

ered slabs containing the PS core layer sandwiched

between the two PC skin layers. Figure 6 shows the

measured and simulated residual thermal birefringence

distributions along the thickness direction of the PC-PS-

PC slabs quenched in water at 25�C from different initial

temperatures. It is seen that, in contrast with the PS-PC-

PS slabs freely quenched from the high initial tempera-

ture, the thermal birefringence throughout the PS core

layer of freely quenched PC-PS-PC slabs is negative at

any initial temperature. This is because the stresses in the

quenched sample are tensile in the core layer, as was

shown in Fig. 4, and the stress-optical coefficient Cr for

the PS melt is negative in the melt state. Clearly, the

higher initial temperature leads to the larger negative

birefringence values throughout the whole PS core layer.

In contrast, in the PC skin layers, the sign of birefrin-

gence changes from positive near the interface to negative

at the surface. It is due to the fact that the stress-optical

coefficient Cr of PC in the melt and glassy states is posi-

tive. With the thermal stresses in quenched slabs being

compressive in the surface region and tensile in the core

region of the PC skin layers this leads to change of the

birefringence sign. The position of zero birefringence

changes with the initial temperature in a peculiar way. At

high initial temperature, the position of zero birefringence

is close to the surface and moves toward the interface as

the initial temperature decreases. The higher initial tem-

perature results in higher absolute values of the thermal

birefringence at the surface and near the interface. As

seen from Fig. 6, the simulated thermal birefringence,

similar to the measured one, shows the same trend with

change of the initial temperature. However, the predicted

thermal birefringence is only in a qualitative agreement

with the experimental data, especially for PC skin layers

at the low initial temperatures being close to its Tg.

In the following sections only experimental data for

the birefringence in the multilayered slabs containing

PMMA in the skin or core are presented. The simulations

were not carried out due to the absence of complete infor-

mation concerning the time- and temperature-dependence

of the stress- or strain-optical coefficients of PMMA in

the fluid, rubbery and glassy states.

PS-PMMA-PS Slabs

The measured thermal birefringence distributions along

the thickness direction of the PS-PMMA-PS slabs

quenched in water at 25�C from different initial tempera-

tures are shown in Fig. 7. The difference between bire-

fringence distribution for the PMMA and PS layers can

be attributed in large part to their corresponding stress-

optical coefficientCr. As mentioned earlier, the value of

Cr above Tg for PS is time- and temperature-dependent

being negative and positive below Tg. The stress-optical

FIG. 5. Evolution of the actual (a) and effective (b) temperature distri-

butions along the thickness direction during quenching of the PS-PC-PS

multilayered slab. Quenching conditions: Ti 5 170�C, T15 25�C.

FIG. 6. Predicted and measured thermal birefringence distributions

along the thickness direction in PC-PS-PC multilayered slabs quenched

in 25�C water from different initial temperatures.


coefficient Cr of PMMA in melt state changes sign from

negative to positive at high temperatures [71–75]. Similar

to the PS-PC-PS multilayered slabs freely quenched from

the high initial temperature, the thermal birefringence dis-

tribution in freely quenched PS-PMMA-PS slabs is posi-

tive throughout the PMMA core layer with its value

being very low in comparison with the value seen in the

PC core layer in Fig. 3. It is due to the fact that the ther-

mal stresses in quenched slabs are tensile in the core

region, therefore, with the positive sign of the stress-

optical coefficient of PMMA in the melt state at these ini-

tial temperatures the thermal birefringence in PMMA

core layer becomes positive. Moreover, since the stress-

optical coefficient of PC in the melt state is much larger

than that of PMMA, the thermal birefringence in the

PMMA core layer in PS-PMMA-PS slabs is more than

one order magnitude smaller than that in the PC core

layer of PS-PC-PS slabs. This is seen from comparison of

data depicted in Figs. 7 and 3. Also, it is seen from Fig.

7, the higher initial temperature leads to the larger value

of the thermal birefringence although the effect is not sig-

nificant in the PMMA core layer.

In the PS skin layers, the thermal birefringence is neg-

ative near the interface and in the extensive region away

from the interface and positive at the surface with the

position of zero birefringence moving toward the surface

with increasing initial temperature. As shown for the PS

skin layer in the PS-PC-PS slabs, change of the sign of

the stress-optical coefficient Cr of PS results in the sign

reversal in the thermal birefringence in the PS skin layer

compared with the PMMA core layer. The positive value

of Cr in the core layer of the PMMA melt does not cause

sign reversal in the thermal birefringence of the PMMA

core layer in the presence of the tensile thermal stresses.

Also, it is seen from Fig. 7 that areas of the positive and

negative birefringence of the PS-PMMA-PS slabs is

unequal with the inequality increasing with an increase of

the initial temperature. Moreover, significantly less mag-

nitude of the birefringence in the PMMA core layer in

comparison with that in the PS skin layers is due to the

fact that the stress-optical coefficient of the PMMA is

approximately one order magnitude lower than that of PS

[76].

PMMA-PS-PMMA Slabs

Figure 8 shows the measured thermal birefringence

distributions along the thickness direction of PMMA-PS-

PMMA slabs quenched in water at 25�C from different

initial temperatures. In contrast with the PS-PMMA-PS

slabs freely quenched from the high initial temperature,

the thermal birefringence in the freely quenched PMMA-

PS-PMMA slabs, similar to that in the PC-PS-PC slabs, is

negative throughout the PS core layer with value being

substantially higher compared with that in the PMMA

skin layer. This is because, even though the thermal

stresses in quenched slabs are tensile in the core layer,

the stress-optical coefficient Cr for the PS melt is nega-

tive. Therefore, the magnitude and the pattern of the bire-

fringence in the PS core layer in PC-PS-PC plates and

PMMA-PS-PMMA plates are similar to each other. The

higher initial temperature leads to the larger birefringence

value.

For the PMMA skin layers, the sign of birefringence

changes from positive near the interface to negative at the

surface with the position of zero birefringence slightly

moving toward the surface with increasing initial temper-

ature. It is due to the fact that the sign of the stress-

optical coefficient of the PMMA and PS in the melt and

glassy states are totally opposite. However, as the stress-

optical coefficient of PS is much larger than that of

PMMA, the thermal birefringence in the PMMA skin

layers in the PMMA-PS-PMMA slabs is less than that of

the PS skin layer in the PS-PC-PS slabs.

It is also noted in Fig. 8 that areas of the positive and

negative regions in the PMMA skin layers and the PS

core layer are not balanced. Moreover, the difference

between areas of positive and negative thermal

FIG. 7. Measured thermal birefringence distributions along the thick-

ness direction in PS-PMMA-PS multilayered slabs quenched in 25�Cwater from different initial temperatures.


ness direction in PMMA-PS-PMMA multilayered slabs quenched in

25�C water from different initial temperatures.


birefringence increases at the higher initial temperature.

In particular, the value of Cr for PS is negative and posi-

tive above and belowTg, while that of PMMA is positive

for the melt and negative at room temperature. The sign

change of Cr for PMMA was discussed by Tsvetkov and

Verkhotina [77] and Read [78]. According to Kock and

Rehage [74], the positive sign of the birefringence of

PMMA in the melt state is associated with a continuous

transition from hindered rotation to free rotation of the

ester side group. In other words, the optical anisotropy of

the PMMA monomer unit is mainly caused by the ester

side group. The contribution of the ester group to the

optical anisotropy of the statistical segment depends on

the orientation of the –COO plane with respect to the axis

of the segment. Calculations of the optical anisotropy by

Tsvetkov and Verkhotina [77] and Read [78] based upon

the additivity of bond polarizabilities show that the values

of the optical anisotropy differ for different positions of

the –COO plane. In the case of a fixed trans-position

where the –COO plane is perpendicular to the axis of the

segment and the optical anisotropy of the monomer unit

is negative. In the case of free rotation the calculations

show that the optical anisotropy is positive. The free rota-

tion means that there is no preferred position of the –

COO plane with respect to the axis of the segment. Thus

a temperature dependent change of the mean position of

the –COO plane can explain the temperature dependence

of the optical anisotropy as well as the change of the bire-

fringence sign in PMMA.

The stress-optical coefficient Cr of PMMA is approxi-

mately one order of magnitude lower than that of PS. It

can explain the lower values of the thermal birefringence

in the PMMA skin layers in comparison with PS core

layer, as shown in Fig. 8.

PMMA-PC-PMMA Slabs

The measured thermal birefringence distributions along

the thickness direction of the PMMA-PC-PMMA slabs

quenched in water at 25�C from different initial tempera-

tures are shown in Fig. 9. Similar to the PC core layer in

the PS-PC-PS slabs freely quenched from the high initial

temperature, the thermal birefringence distribution in the

PC core layer in freely quenched PMMA-PC-PMMA

slabs is positive throughout the layer with very high value

compared with that in the PMMA skin layers. The higher

initial temperature leads to the larger value of the thermal

birefringence in the PC core layer.

However, opposite to the case of the PS skin layers in

PS-PC-PS slabs, in the PMMA skin layers of the PMMA-

PC-PMMA slabs the sign of the thermal birefringence

changes from positive at the interface to negative at the

surface with the position of zero birefringence moving

slightly toward the surface with increasing initial temper-

ature. It is due to the fact that the sign of the stress-

optical coefficient of PMMA changes at high temperature

above Tg [69–73]. This trend is opposite to the sign of

the stress-optical coefficient for the melt and glassy states

of PS. Therefore, the pattern of the thermal birefringence

distribution in the PMMA skin layer of PMMA-PC-

PMMA slabs in Fig. 9 and in the PS skin layer of PS-PC-

PS slabs in Fig. 3 is totally opposite to each other.

The magnitude of the thermal birefringence in the PC

core layer is more than one order of magnitude larger

than that the PMMA skin layers. This is due to the fact

that the stress-optical coefficient of PC is significantly

larger than that of PMMA. The parabolic shape of the

thermal birefringence in the PMMA and PC layers and

unequal areas of the positive and negative regions in

PMMA-PC-PMMA slabs is observed. The difference in

areas of the negative and positive birefringence increases

with an increase of the initial temperature.

PC-PMMA-PC Slabs

Figure 10 shows the measured thermal birefringence

distributions along the thickness direction of PC-PMMA-

PC slabs quenched in water at 25�C from different initial

temperatures. The difference between the thermal bire-

fringence distribution in the PMMA and PC layers of PC-


ness direction in PMMA-PC-PMMA multilayered slabs quenched in

25�C water from different initial temperatures.


ness direction in PC-PMMA-PC plates quenched in 25�C water from

different initial temperatures.


PMMA-PC slabs can be attributed in large part to the dif-

ference in behavior of the stress-optical coefficient Cr of

these polymers as discussed earlier. The thermal birefrin-

gence in the PMMA core layer in PC-PMMA-PC slabs is

positive with its value being very low compared to that in

PC skin layers. It is due to the fact that the tensile ther-

mal stresses in the PMMA core layer of quenched slabs

with the positive sign of the stress-optical coefficient of

the PMMA melt would lead to the positive thermal bire-

fringence due to the positive sign of the stress-optical

coefficient of PMMA. However, as the stress-optical

coefficient of PMMA is much smaller than that of PC,

the thermal birefringence in the PMMA core layer in PC-

PMMA-PC slabs is more than one order of magnitude

smaller than that in the PC skin layers. Again the higher

initial temperature leads to larger value of the thermal

birefringence but the effect is not significant.

The sign of the thermal birefringence in the PC skin

layers changes from positive near the interface to negative

at the surface with the position of zero birefringence mov-

ing toward the surface with increasing initial temperature.

It is due to the fact that the stress-optical coefficient Cr of

PC in the melt and glassy states is positive with the thermal

stresses in quenched slabs being compressive in the surface

region and tensile in the core region in the PC skin layers.

The higher initial temperature results in the higher thermal

birefringence in the surface and near the interface.

CONCLUSIONS

Free quenching experiments from the different initial

temperatures into the water of 25�C were carried out on

PS-PC-PS, PC-PS-PC, PMMA-PC-PMMA, PC-PMMA-

PC, PS-PMMA-PS, and PMMA-PS-PMMA multilayered

slabs and the thermal birefringence distributions along the

thickness direction were measured.

In the PC core layer of the quenched PS-PC-PS and

PMMA-PC-PMMA slabs, the thermal birefringence is

positive with its value increasing with the initial tempera-

ture. In the PS skin layers, the thermal birefringence is

negative throughout the PS layers at an initial temperature

of 150�C, but at higher initial temperatures the positive

thermal birefringence at the surface and negative at the

interface and in the extensive skin layers was observed.

In the PS core layer of the quenched PC-PS-PC and

PMMA-PS-PMMA slabs, the thermal birefringence is

negative with its value increasing with the initial tempera-

ture. In the PC skin layers of the PC-PS-PC slabs and in

the PMMA skin layers of the PMMA-PS-PMMA slabs,

the thermal birefringence is negative near the surface and

positive near the interface. The position of zero thermal

birefringence is shifted toward the surface with an

increase of the initial temperature.

In the PMMA core layer of the quenched PS-PMMA-

PS and PC-PMMA-PC slabs, the thermal birefringence is

positive with its values increasing with the initial temper-

ature. In the PS skin layers of the PS-PMMA-PS slabs,

the thermal birefringence is positive at the surface and

negative near the interface. In the PC skin layers of the

PC-PMMA-PC slabs the thermal birefringence is negative

at the surface and positive near the interface. With an

increase of the initial temperature the position of zero

birefringence moves close to the surface.

Methodology was presented to calculate the thermal

birefringence in the multilayered PS-PC-PS and PC-PS-

PC slabs based on the mechanical and optical relaxation

functions of PS and PC measured earlier. The simulated

and measured birefringence distributions in the thickness

direction of the multilayered PS-PC-PS and PC-PS-PC

slabs freely quenched from different initial temperatures

into water at 25�C were found to be in a fair agreement.

However, for the multilayered slabs containing PMMA

the simulations were not carried out since complete infor-

mation on the time and temperature dependence of the

strain- or stress-optical coefficient of PMMA in the

glassy, rubbery and melt states is currently not available.

APPENDIX A: NUMERICAL SOLUTION FOR THETEMPERATURE HISTORY DURING QUENCHING

Discretizing the derivatives in Eq. 1 using forward differ-

ence for the first order derivative and the central differ-

ence for the second order derivative yields:

@T

@t5

1

DtTn11

i 2Tni

(A1)

aj@2T

@z25

1

2

aj

Dz2fðTn11

i21 22Tn11i 1Tn11

i11 Þ

1ðTni2122Tn

i 1Tni11Þg

(A2)

where subscript i and n stands for the finite difference

node along the thickness direction and time step,

respectively.

Substituting Eqs. A1 and A2 into Eq. 1 and grouping

terms yields the following equations for the interior nodes

along the thickness direction.

2~rj

2Tn11

i21 1 11~rj

Tn11

i 2~rj

2Tn11

i11 5~rj

2Tn

i21

1 12~rj

Tn

i 1~rj

2Tn

i11

(A3)

where

~rj5ajDt

Dz2(A4)

This equation is valid for i 5 2 to i max 21 with i 51

and i max corresponding to the boundaries.

For the skin layer,

~rj5~rs5asDt

Dz2(A5)

Where


as5ks

qsCp;s(A6)

For the interface between the skin and core layers,

~rj5~r int 5aint Dt

Dz2(A7)

Where

aint 5ac1as

2(A8)

For the core layer,

~rj5~rc5acDt

Dz2(A9)

Where

ac5kc

qcCp;c(A10)

To derive the first row of the band matrix, the boundary

condition at the center is given by the symmetry condi-

tion providing the absence of gradient at the center of the

slab. Discretization of the derivative in Eq. 1b using the

central differences of the first order yields:

@T

@z

��z50

51

2

1

2DzTn11

i11 2Tn11i21

1 Tn

i112Tni21

�50 (A11)

Tni215Tn

i111 Tn11i11 2Tn11

i21

(A12)

Substituting Eq. A11 into Eq. A3, at i50 corresponding to

the centerline, z50, the following equation is obtained for

the first row of the band matrix:

11~rcð ÞTn110 2~rcTn11

1 5 12~rcð ÞTn01~rcTn

1 (A13)

For the last row of the band matrix, the convective

boundary condition at the surface of the slab is given in

Eq. 1c. For the discretization, the equation is rewritten as:

kj@T

@z5hT12hTi at z56b for t > 0 (A14)

Discretization of the derivative and Ti in Eq. A13 yields

kj@T

@z5kj

1

4DzTn11

i11 2Tn11i21

1 Tn

i112Tni21

�� (A15)

2hTi52h

2Tn11

i 1Tni

(A16)

After substituting Eqs. A15 and A16 into Eq. A14 and

rearranging it for Tn11i11 the following equation is obtained:

Tn11i11 5Tn11

i21 2 Tni112Tn

i21

1CT12

C

2Tn11

i 2Tni

(A17)

Where

C54Dzh

ks(A18)

Substituting Eq. A17 into Eq. A3 yields the following

equation for the last row of the band matrix for the con-

vective boundary condition:

2~rsTn11i max 211 11~rs1

~rsC

4

� �Tn11

imax 5~rsTnimax 21

1 12~rs2~rsC

4

� �Tn

imax 1~rsC

2T1

(A19)

Equations A3, A13, and A19 are a set of algebraic equa-

tions that can be arrayed in a band matrix of the form

Ai Bi Ci½ � Ti

� �5 RHSi½ �, where Ti

� �is the solution of

the temperature at the present time. The discretized equa-

tions were solved by means of a tridiagonal solver using

the equilibrium method [64].

APPENDIX B: NUMERICAL SOLUTION FOR THESTRAIN AND STRESS HISTORIES

Using the piecewise linear approximation [64], the inte-

grals in Eqs. 28 and 29 were expanded as:

ðn

0

A n2n0ð Þ @B n0ð Þ@n0

dn05Xj

k51

B nk

2B nk21

nk2nk21

ðnk

nk21

A nj2n0

dn0(B1)

in which B nj

at n5nj(j denotes the current time step) is

unknown and B nk

at k < j (k denotes a previous time

step) was solved in the previous time step. Then the strain

ezz z; tð Þ is obtained by Eq. 29 and the strain exx tð Þ is

derived by substituting Eq. 28 into Eq. 22. After calcula-

tions of exx tð Þ and ezz z; tð Þ for the skin and core regions,

the exx tð Þ and ezz z; tð Þ at the interface is obtained by aver-

aging the value of exx tð Þ and ezz z; tð Þ for a node of the

skin and a node of the core near the interface of skin and

core regions. Then the stress history rxx z; tð Þ can be

solved by Eq. 28 using exx tð Þ and ezz z; tð Þ calculated ear-

lier. The final discretized formulation becomes:

Gi;k5

ðni;k

ni;k21

G ni;k2n0

dn0" #�

ni;k2ni;k21

(B2)

Ai;j52Gi;j ei;j21zz 2ei;j21

xx

1Xj21

k52

Gi;k ei;kzz 2ei;k21

zz 2ei;kxx 1ei;k21

xx

� �(B3)

Bi;j52 2Kl24Gi;j=3

= Kl14Gi;j=3

(B4)

ki;j523ei;j

T Kl if T > Tg

23ei;jT Kg2 Kg2Kl

ei

xx12eizz23ei

T

� �jT5Tg

if T > Tg

8<:

(B5)


Ci;j5 4Ai;j=32ki;j

= Kl14Gi;j=3

(B6)

ei;jzz52

ðb

2b

2 Ai;j2Gi;jCi;j

dz

� �� ðb

2b

2 Gi;j 11Bi;j � �

dz

� �(B7)

ei;jxx52Bi;jei;j

zz1Ci;j (B8)

ri;jxx52Gi;j 11Bi;j

ei;j

zz12 Ai;j2Gi;jCi;j

(B9)

where the superscript i denotes discretization in z direc-

tion and k or j denotes discretization in time. Values of

eixx12ei

zz23eiT

� �jT5Tg

were evaluated at the discretized

position i when the temperature at that position becomes

equal to Tg

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Thermal birefringence in freely quenched multilayered slabs of amorphous polymers: Experiment and simulation