Residual thermal birefringence in freely quenched plates of amorphous polymers: Simulation and experiment

Residual Thermal Birefringence in Freely Quenched Platesof Amorphous Polymers: Simulation and Experiment

G. D. SHYU, A. I. ISAYEV, C. T. LI

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

Received 17 July 2002; revised 30 October 2002; accepted 20 November 2002

ABSTRACT: Free quenching experiments were performed on thin plates of polystyrene(PS) and polycarbonate (PC). The thermal birefringence distribution along the thick-ness direction of the plates was measured. The birefringence data were compared withthe results of a numerical simulation based on the linear viscoelastic and photovis-coelastic constitutive equations for the mechanical and optical properties, respectively,and the first-order rate equation for volume relaxation. The effects of the initialtemperature, quenching temperature, and quenching media on the development ofresidual thermal stresses and birefringence were evaluated. At higher initial temper-atures (�105 °C), the thermal birefringence in quenched PS plates was negative at thecenter and positive at the surface, whereas at lower temperatures (close to the glass-transition temperature), the birefringence became positive at the core and negative atthe surface or positive through the entire cross section of the plate. The birefringencein freely quenching PC plates was positive at the center and negative at the surface atany initial temperature. These observations were in fair agreement with predicteddata. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1850–1867, 2003Keywords: residual thermal stress; residual thermal birefringence; free quenching;amorphous; polystyrene; polycarbonates; volume relaxation; linear viscoelasticity;photoviscoelasticity

INTRODUCTION

During processing, polymers undergo rapidnonisothermal cooling and transformation fromthe molten state to the solid state, and this re-sults in inhomogeneous densification of the poly-mer articles and changes in viscoelastic proper-ties. As the cooling progresses, thermal stressesand birefringence develop and become residualwhen the polymer vitrifies.1,2 Residual stressesand birefringence strongly affect the end-useproperties and warping of polymeric products,and so their accurate prediction will allow one to

determine optimum processing conditions. Infact, the factors governing the development ofthermal residual stresses and birefringence dur-ing polymer processing have received much atten-tion from many researchers over the years.1–13

Isayev1 and Isayev and Crouthamel2 gave goodreviews on the relevant theoretical works aboutthe thermal stresses and birefringence inquenched amorphous polymers. Roughly speak-ing, we can divide the theories into two catego-ries. One is based on the instant-freezing assump-tion.1,2,4,5,8–10,14–16 The other is based on the vis-coelastic relaxation assumption affected by rapidcooling.1,2,6,12,13,17–25 Therefore, the thermal his-tories and viscoelastic effects should be taken intoaccount when the thermal stresses and birefrin-gence of the final products are calculated. Also,thermal equilibrium is seldom attained in the

Correspondence to: A. I. Isayev (E-mail: [email protected])Journal of Polymer Science: Part B: Polymer Physics, Vol. 41, 1850–1867 (2003)© 2003 Wiley Periodicals, Inc.

1850

polymer processing because of the existing tem-perature gradient and the rapid temperaturechange. Consequently, the nonequilibrium freevolume should also be included in the Williams–Landel–Ferry (WLF) equation based on the free-volume approach.26 In addition, the time-depen-dent volume relaxation27,28 should be taken intoaccount.

In this work, a methodology is developed tocalculate the transient and frozen-in thermal bi-refringence from mechanical and optical materialfunctions. The residual birefringence and stressesin freely quenching plates are simulated and pre-dicted based on stress relaxation experimentsover a wide range of temperatures with simulta-neous measurements of Young’s modulus and thestrain-optical coefficient (C�) conducted on poly-styrene (PS) and polycarbonate (PC) and mastercurves of C�, the stress-optical coefficient (C�),and Young’s relaxation modulus obtained in ourprevious article.29 Both the instant-freezing andfree-volume relaxation assumptions for inorganicglasses and amorphous polymers are used. Basedon the instant-freezing theory, the two-constantinstant-freezing (TCIF) model is used. In thismodel, material properties (both mechanical andoptical properties) are assumed to be constant buthave different values in the rubbery (or melt)state and the glassy state, with an instantaneouschange in properties taking place at the glass-transition temperature (Tg). In addition, twomodels based on the viscoelasticity and photovis-coelasticity, accounting for the time and temper-ature dependence of mechanical and optical prop-erties with (VPWV model) and without inclusion(VPNV model) of the volume relaxation, are con-sidered. The volume relaxation in the VPWVmodel is based on the first-order rate equation. Toverify the modeling, we performed free quenchingexperiments on thin PS and PC plates, and thebirefringence distributions along the thickness di-rection (z) of the plates were measured. The ob-tained data are compared with the results of anumerical simulation based on the VPWV model,and the effects of quenching conditions on theformation of residual thermal stresses and bire-fringence are evaluated.

THEORETICAL

Problem Setup

We consider here an idealized problem of the freequenching of a plate with the geometry and coor-

dinate system shown in Figure 1. The plate isassumed to be infinite (the lateral dimensions ofthe plate are much larger than its thickness 2b)and is maintained at a uniform initial tempera-ture (Ti). Then, it is subjected to symmetric cool-ing from both surfaces at a coolant temperature(T�) below Tg. Therefore, heat transfer occursonly in z. Thermal stresses are generated by theinteractions between nonhomogeneous thermalcontraction and changes in the polymer viscoelas-tic properties during cooling. Also, the thermalbirefringence is generated through changes in C�

and C� during cooling.

Constitutive Equations

The viscoelastic constitutive equation for an iso-tropic linear material in nonisothermal processcan be described as follows:17

�ij � �0

� �2G��

�� ij �13 �ij�kk�

� �ijK��

��kk � 3�T��d�� (1)

with the reduced time (�) given by

� � �0

t d

aT�T��(2)

where �ij is the stress tensor, �ij is the straintensor, �ij is the unit tensor, �T is the thermalstrain, G is the shear relaxation modulus, K is thebulk relaxation modulus, and aT is the shiftfactor.

Figure 1. Coordinate system for the free quenchingof a plate.

RESIDUAL THERMAL BIREFRINGENCE 1851

The strain-optical and stress-optical behaviorof an isotropic linear photoviscoelastic materialcan be described as follows:29–31

nij � �ijnkk/3 � �0

�

C��

��

��ij � �ij�kk/3�d�� (3)

� �0

�

C��

��ij � �ij�kk/3�d�� (4)

where nij is the refraction tensor.

VPWV Model

The shift factor is determined by the free volume,which, in the equilibrium state, is dependent onthe temperature only. However, when a polymerplate is quenched from above Tg to below Tg, theequilibrium volume is not attained immediately,and a time-dependent volume relaxation is ob-served. The nonequilibrium state produced by thefast cooling introduces more free volume into thepolymers than that available in the equilibriumstate at the same temperature. This additionalfree volume accelerates all relaxation processes.Therefore, the pure viscoelastic model should bemodified to include the effect of the nonequilib-rium free volume. The resultant volume contrac-tion in the quenching process can be viewed asbeing composed of two steps, an instantaneouscontraction and a gradual slow contraction to thefinal equilibrium value. Therefore, the free vol-ume in the volume contraction process becomestime-dependent; as such, the nonequilibrium freevolume should be included in the WLF equation.

Because of the volume relaxation, �T in anonisothermal process can be described as fol-lows:29,30,32

d�T

dt � �g

dTdt �

�T � �Te

raT(5)

where �T and �Te are the actual and equilibriumthermal strain, respectively, at this temperatureT; r is the relaxation time at the reference tem-perature (Tr); and �g is the linear thermal expan-sion coefficient in the glassy state. The first termon the right-hand side of eq 5 describes the in-stantaneous contraction, and the second term

gives the gradual contraction according to thefirst-order rate theory. �Te is given as follows:

�Te � �1�T � Ti� if T � T2 (6)

�Te � �1�T2 � Ti� � �g�T � T2� if T T2 (7)

where Ti is the initial temperature at t � 0, T2 isthe glass-transition temperature observed in ex-periments of infinite timescale, and �1 is the ther-mal expansion coefficient in the rubbery state.

aT follows a modified form of the WLF equa-tion:

aT � exp�c1�Teff � Tr�

c2 � Teff � Tr� (8)

where Teff is the effective temperature, the tem-perature of an equilibrium state that has thesame amount of free volume as the nonequilib-rium state, and is, therefore, related to the actualtemperature through

Teff � T ��T � �Tr

�1 � �gif T � T2 (9)

Teff � T2 ��T � �Tr

�1 � �gif T T2 (10)

VPNV Model

The VPNV model can be obtained from the VPWVmodel by a modification of the �T and aT functions.In the VPNV model, �T is simply

�T � �1�T � Ti� if T � Tg (11)

�T � �1�T2 � Ti� � �g�T � T2� if T Tg (12)

and aT still follows the WLF equation as eq 8,except that it is calculated at the actual temper-ature rather than at Teff. The WLF equation ob-tained in ref. 29 by the fitting of the experimentalvalues of aT at high temperatures and extrapo-lated values at low temperatures was used in thesimulation. However, it should be noted that atlow temperatures, the experimental value of aTdeviates from the WLF equation, as indicated inref. 29.

1852 SHYU, ISAYEV, AND LI

Stress and Strain Analysis

For an infinite plate subjected to symmetricallyuniform cooling, all shear stress components arezero:17,19

�xy � �yz � �xz � 0 (13)

Because of the material continuity across z, allshear strain components are also zero:

�xy � �yz � �xz � 0 (14)

Because no surface traction is acting on the facesof the plate, �zz�z�b is equal to 0. From the zcomponent of the dynamic equation, it followsthat

�zz 0 b � z � b. (15)

Therefore, the problem is reduced to one of planarstresses. Because the lateral dimensions aremuch larger than the plate thickness and theedge effects are neglected, the two non-zero nor-mal stress components and non-zero normalstrain components in the xy plane are equal, re-spectively, and the strain components are inde-pendent of z. Consequently, it can be written that

�xx � �xx�z, t� � �yy � �yy�z, t� (16)

�xx � �xx�t� � �yy � �yy�t� (17)

Because the plate is free of constraints (no exter-nal forces), the resulting force is zero through thecross section:

�b

b

�xx�z, t�dz � 0 (18)

It is seen that with this idealized problem of freequenching, the calculation of residual thermalstresses and birefringence is greatly simplified.

Heat-Transfer Analysis

For the idealized problem stated previously, tem-perature variation is governed by a one-dimen-sional heat conduction equation:

�cp

�T�t � kth

�2T�z2 (19)

with the initial and boundary conditions given by

T�t�0 � Ti (20)

�T�z

z�0

� 0 (21)

kth

�T�z

z�b

� h�T�z�b � T�� (22)

where � is the density, c� is the heat capacity, kthis the heat conductivity, � � kth/(�cp) is the ther-mal diffusivity, and h is the heat-transfer coeffi-cient.

Numerical Simulation

Numerical simulation schemes were formulatedto calculate the residual thermal stresses andbirefringence in freely quenched plates of PS andPC. Because eq 19 is independent of stress andstrain, the temperature field can be calculatedfirst by the finite difference method. After thethermal history was obtained, the first-order rateequation, eq 5, together with eqs 6–10 was solvedby the fourth-order Runge–Kutta method to ob-tain �T, Teff, and aT. Then, � was determined fromeq 2. With the piecewise linear approximation[21], the integrals in eqs 1, 3, and 4 were ex-panded and then substituted into eqs 15 and 18 indiscretized formulations, and the strain history�xx(t) and �zz(z,t) and, therefore, the stress history�xx(z,t) were solved. After the strain history in theplate was obtained, the birefringence was calcu-lated as follows:

�n � nzz � nxx � �0

�

C��

��zz � �xx�d��

(23)

� �k�2

j�xx

k � �xxk1 � �zz

i,k � �zzi,k1

�i,k � �i,k1 �� i,k1

� i,k

C��i,k � ��d�� (24)

where i denotes discretization for z and k denotesdiscretization in time.


EXPERIMENTAL

Free quenching experiments were carried out un-der various quenching conditions, as listed in Ta-ble 1. Thin plates of PS (Styron 615-APR, DowChemical Co.) and PC (Lexan 141-111, GeneralElectric Co.) were used in the quenching experi-ments. The effects of Ti, the quenching tempera-ture, and the quenching media on the residualthermal birefringence were studied. Water andsilicone oil were used as the cooling media.

Plates (20 � 12 � 0.2 cm3) were made by com-pression molding at 180 °C for PS and at 230 °Cfor PC, followed by slow cooling, and birefrin-gence-free slabs were obtained. The plates werethen cut into smaller plates (4 � 4 � 0.2 cm3) forthe quenching experiments. For quenching in sil-icone oil, samples were first put into a large bea-ker filled with silicone oil at the specified Ti forthe time needed to attain thermodynamic equilib-rium; they were then taken out of the beaker andquenched in silicone oil at 25 °C. For quenching inwater, the samples were first put into glycerol atthe given Ti for the time needed to attain thermo-dynamic equilibrium, and they were thenquenched into water at the specified temperature.

Strips 0.22 cm wide for PS and 0.35 cm wide forPC were cut from the quenched plate with a low-

speed diamond saw (Isomet low-speed saw,Buechler). The birefringence distribution along zwas then measured with a polarizing microscope(Leitz Laborlux 12 POL, Leitz Wetzlar) with acompensator (1–4 orders, 1592k, Leitz Wetzlar)within 2–3 days after quenching. The absoluteerror in the birefringence measurements was5.0 � 106.

RESULTS AND DISCUSSION

The physical properties of PS and PC used in thenumerical simulation are listed in Table 2. Themaster curves and the shift factors for C� andYoung’s modulus were taken from ref. 29. Theexperimental data of thermal birefringence arecompared with simulated results from the VPWVmodel at 2 � 105 s after quenching. Aging occur-ring during the 2 days following quenching isalready included in the simulation by the VPWVmodel.

According to ref. 29, the C� values of PS and PCare dependent on the temperature and time. Inparticular, for PS, C� is positive at low tempera-tures and small times. This coefficient decreaseswith an increase in temperature and time, passesthrough zero, and then becomes negative, reach-ing a minimum at a certain temperature andtime. Finally, it approaches zero at high temper-atures and large times. C� of PS is positive belowTg and small times, decreases with an increase intemperature and time, passes through zero, andbecomes negative, approaching a constant nega-tive value at high temperatures and times. How-ever, C� of PC is always positive. It decreases withtemperature and time, approaching zero at hightemperatures and times. C� of PC is always pos-

Table 1. Quenching Conditions of the FreeQuenching Experiments

Polymer Initial TemperatureQuenching

Temperature

PS Glycerol Water135 °C (5 min) 25 and 0 °C120 °C (5 min) 25 and 0 °C105 °C (40 min) 25 and 0 °C97 °C (120 min) 25 and 0 °C91 °C (360 min) 25 °C

Silicone oil Silicone oil135 °C (5 min) 25 °C120 °C (5 min) 25 °C105 °C (40 min) 25 °C97 °C (120 min) 25 °C

PC Glycerol Water190 °C (5 min) 60 and 25 °C175 °C (5 min) 60 and 25 °C160 °C (20 min) 60 and 25 °C150 °C (120 min) 60 and 25 °C

Silicone oil Silicone oil175 °C (5 min) 25 °C160 °C (20 min) 25 °C150 °C (120 min) 25 °C

Table 2. Physical Properties of PS and PC Used inthe Simulation

Properties PS PC

�1 0.0002134 0.00024

�g 0.00006634 0.0000674

� (m2/s) 6.13 � 108 34 7.31 � 108 34

Poisson ratio 0.3333 0.4133

WLF equationC1 10.6 8.82C2 (K) 57.9 40.2Tr (K) 370.0 420.5

T2 (K) 313.0 380.0


itive and exhibits a constant value at low temper-atures and times. It increases with an increase intemperature and time, approaching a constantvalue at high temperatures and times.

The birefringence is a result of anisotropy inthe dielectric properties of the material. For aninitially isotropic polymer in the glassy state, theanisotropy upon deformation results from bonddistortion and tilting of anisotropic side groups,whereas in the melt or rubbery state, the anisot-ropy is related mainly to chain segmental andmolecular orientation. Therefore, the change inthe sign of C� for PS can be explained as follows.The birefringence of PS is mainly determined bythe orientation of the phenyl groups. Upon defor-mation in the glassy state, polymer chains cannotmove around, and the phenyl groups are tiltedtoward the stretching direction; this results in apositive birefringence. In the rubbery state, poly-mer chains, to some extent, are able to move andtend to align along the stretching direction, mak-ing the plane of the phenyl groups lie preferen-tially perpendicular to the stretching directionand, therefore, leading to a negative birefrin-gence.

PS

In the simulation based on the VPWV model, r� 0.04 s at Tr � 97 °C for the volume relaxation

was used for PS. This value fit well all the mea-sured residual birefringence values. Heat-trans-fer coefficients of 490 and 120 J/s m2 K were usedfor quenching in water and silicone oil, respec-tively.

Measured (symbols) and predicted (lines) re-sidual birefringence distributions along z of PSplates quenched into 25 °C water from different Tivalues are shown in Figure 2. The predicted re-sults are based on the VPWV model. For Ti valuesabove 105 °C, the birefringence is negative in thecore and positive at the surface. It is shown thatthe higher Ti is, the larger the absolute value is ofthe birefringence at the center and the closer theposition is of zero birefringence to the surface. AsTi drops and approaches Tg, the residual birefrin-gence becomes positive at the center and negativeat the surface; this exhibits a distribution totallyopposite to that at higher Ti values.

For symmetric quenching, the birefringenceprofile is not balanced, that is, b

b �ndz � 0, eventhough the stress profile is balanced as requiredby the force balance equation (eq 18) and asshown later in Figure 5. This effect has beenobserved earlier in quenched PS, PC, and PMMAplates1 and is due to the fact that C� is not aconstant but a function of both time and temper-ature.29 The numerical predictions of the residualbirefringence profile by the VPWV model are infair agreement with the experimental data.

Figure 2. Measured (symbols) and predicted (lines) residual birefringence distribu-tions along z of PS plates quenched in water from different Ti’s to 25 °C.


Figure 3. Measured (symbols) and predicted (lines) residual birefringence distribu-tions along z of PS plates quenched in water from different Ti’s to 0 °C.

Figure 4. Measured (symbols) and predicted (lines) residual birefringence distribu-tions along z of PS plates quenched in silicone oil from different Ti’s to 25 °C.


The measured (symbols) and predicted (lines)residual birefringence profiles in PS platesquenched from different Ti’s into 0 °C ice waterand 25 °C silicone oil are shown in Figures 3 and4, respectively. The predicted data are based onthe VPWV model. As we can find in both cases,the shapes of the residual birefringence profiles

are similar to those of quenching into 25 °C water,as shown in Figure 2, but with somewhat largeror smaller absolute values of the residual birefrin-gence dependent on quenching conditions. Be-cause the heat-transfer coefficient for quenchingin silicone oil is smaller than that in water,smaller values of birefringence are obtained for

Figure 5. Predicted residual stress distributions of PS plates quenched from differentTi’s to cooling temperatures of 25 and 0 °C into (a) water and (b) silicone oil.


quenching into silicone oil. These effects of thequenching temperature and quenching media(meaning different heat-transfer coefficients) onpredicted residual stresses and birefringence arefurther confirmed in Figures 5 and 6. The resid-

ual stresses are always positive (tensile) in thecore and negative (compressive) at the surfacelayer. The higher Ti’s result in larger absolutevalues of residual stresses both in the core andat the surface. Besides, the independence of

Figure 6. Predicted residual stress distributions of PS plates quenched into (a) waterand (b) silicone oil at a quenching temperature of 25 °C.

Figure 7. Predicted evolution of temperature (a) the actual profiles and (b) theeffective profiles during quenching from Ti � 120 °C to 25 °C water.


residual stresses on Ti at very high Ti’s can alsobe observed in Figures 5 and 6, and this wasearlier confirmed experimentally by Isayev andCrouthamel.2 It is obviously revealed that thelower quenching temperatures and the largerheat-transfer coefficients of cooling media giverise to higher levels of the residual stresses infreely quenched samples.

To obtain a better understanding of how theresidual stresses and birefringence are built up,

one has to investigate the transient responses.The evolution of predicted actual temperaturesand Teff’s for quenching in 25 °C water from Ti� 120 °C is shown in Figure 7. At higher temper-atures above 95 °C, volume relaxation happens sofast that Teff is virtually identical to the actualtemperature. Below 95 °C, the difference betweenthe actual temperatures and Teff’s becomes moreand more prominent as the actual temperaturedrops. At times later than 400 s after quenching,

Figure 8. Predicted transient stress and birefringence profiles in PS plates freelyquenched from Ti � (a) 120 and (b) 97 °C to 25 °C water.


the Teff’s are practically uniform through theplate and still decrease slowly with time, evi-dently giving rise to the aging phenomenon.

The transient stress and birefringence profilesin a PS plate quenched into 25 °C water from Ti� 120 °C, predicted by the VPWV model, aregiven in Figure 8(a). At the beginning, thestresses are positive at the surface and negativeat the core, and the birefringence is positive at thecenter and negative at the surface because C� isnegative at high temperatures. Later on, thestresses are positive at the core and negative atthe surface, and the birefringence becomes nega-tive at the center and positive at the surface. Inaddition, as noted from a comparison of the stressprofiles at t � 400 and 2 � 105 s, the residualstresses have relaxed after the temperature of thewhole plate has cooled down to the quenchingtemperature, whereas the absolute value of resid-ual birefringence increases further. However, be-tween the 3rd and 20th days after quenching, theresidual birefringence relaxes less than 0.6% inthe core and about 1% at the surface, whereasresidual stresses relax much more, about 5%.30

The predicted transient stress and birefrin-gence profiles of PS plates quenched in 25 °Cwater from Ti � 97 °C are shown in Figure 8(b).Birefringence pictures different than those in Fig-ure 8(a) are seen because of the different Ti effectson birefringence, as discussed previously.

The effects of the volume relaxation time onthe predicted residual stresses and birefringenceare shown in Figure 9. A decrease in the volumerelaxation time leads to an increase in the mag-nitude of residual stresses and to a decrease inthe magnitude of residual birefringence.

Comparisons of the residual stresses and bire-fringence in PS plates predicted by the VPNV andTCIF models with the VPWV model are shown inFigures 10 and 11, respectively. Although boththe TCIF and VPNV models can give reasonablepredictions of the residual stresses like the VPWVmodel, they cannot predict correctly the mea-sured residual birefringence profile indicated inFigure 2. No matter what Ti is, the TCIF modelalways gives negative birefringence at the surfaceand, at low temperatures, negative birefringencethroughout the whole cross section of the sample.Moreover, the TCIF model predicts no residualstresses and birefringence for Ti’s below Tg. Athigh temperatures, the predicted birefringence bythe VPNV model is negative throughout thewhole cross section of the sample, including thesurface. Also, the VPNV model at every Ti alwaysgives negative birefringence at the surface.

The presence of negative birefringence at thesurface in the TCIF model can be explained asfollows. In the TCIF model, it is assumed that C�

� 5.2 � 109 Pa1 at T � Tg and C� � 1011

Pa1 at T � Tg. At the start of quenching, the

Figure 9. Effects of the volume relaxation time on the predicted residual stress andbirefringence distribution of PS plates quenched from Ti � 120 °C to 25 °C water.


stresses at the surface are tensile, that is, posi-tive. At high temperatures, C� is negative. There-fore, at the start of quenching, the birefringenceinduced at the surface is negative. At the laterstage of quenching corresponding to low temper-atures (�Tg), the surface stresses are compres-sive (negative) and C� is positive, and so the re-sultant birefringence after quenching is also neg-ative.

The reason for the inability of the VPNV modelto predict a positive residual birefringence at thesurface can be explained as follows. At a time of2.9 s, the actual temperatures and Teff’s are 54.2and 84.9 °C (Fig. 7), respectively. The shift factorscorresponding to these temperatures are 1024.5

and 102.65, respectively, with the correspondingC� values29 at 0.5 s being 9.2 � 1012 and 4.1� 1012 Pa1. Because the incremental stresses

Figure 10. Predicted residual stress and birefringence distributions in PS platesquenched from different Ti’s to 25 °C water with the VPWV and VPNV models.


produced at a time of 2.9 s are compressive (neg-ative) at the surface, the incremental birefrin-gence will be positive if volume relaxation is con-sidered and negative if the volume relaxation isneglected. On the basis of these considerations,one can realize why the VPNV model predicts a

negative residual birefringence at the surface(Fig. 10). However, if the volume relaxation isconsidered, as for the VPWV model, C� can be-come negative even at the later stage of quench-ing because Teff is still high (see Fig. 7), evenwhen the actual temperature comes close to the

Figure 11. Predicted residual stress and birefringence distributions in PS platesquenched from different Ti’s to 25 °C water with the VPWV and TCIF models.


quenching temperature. Therefore, positive bire-fringence at the surface would be attained, asobserved experimentally (see Figs. 2–4).

PC

In the simulation based on the VPWV model, r� 0.3 s at Tr � 147.5 °C for the volume relaxation

was used for PC. This value was found to fit wellall measured residual birefringence values. Heat-transfer coefficients of 490 and 120 J/s m2 K wereused for quenching in water and silicone oil, re-spectively. It should be mentioned in advancethat no single volume relaxation time can provideoverall satisfactory predictions of the residual bi-refringence in freely quenched PC plates.

Figure 12. Measured (symbols) and predicted (lines) residual birefringence distribu-tions along z of PC plates quenched in water from different Ti’s to (a) 25 and (b) 60 °C.


The residual birefringence profiles in the PCplates quenched into 25 °C water and 60 °C waterand into 25 °C silicone oil are shown in Figures 12

and 13, respectively. The predictions are based onthe VPWV model. The birefringence is positive atthe center and negative at the surfaces. The

Figure 13. Measured (symbols) and predicted (lines) residual birefringence distribu-tions along z of PC plates quenched in silicone oil from different Ti’s to 25 °C.

Figure 14. Predicted residual stress distributions of PC plates quenched into waterfrom different Ti’s to cooling temperatures of 25 and 60 °C.


higher Ti is, the lower the quenching temperatureis that leads to the higher birefringence in thecore. Also, water-quenched plates show higherbirefringence than silicone oil-quenched plates. Inthree cases, the position of zero birefringencemoves toward the center as Ti decreases. As forPS plates, the birefringence profiles are not bal-anced.

Figure 14 gives the predicted residual stressdistribution of PC plates quenched into water atdifferent temperatures. Again, it proves that thelower quenching temperature leads to the higherlevel of the residual stresses both in the centerand at the surface. The prediction of the residualstresses and birefringence in PC plates quenchedfrom different temperatures to 60 °C water, with

Figure 15. Predicted residual stress and birefringence distributions in PC platesquenched from different Ti’s to 60 °C water with the VPWV and VPNV models.


the VPNV and TCIF models compared with theVPWV model, is shown in Figures 15 and 16,respectively. At high Ti’s, the right trend of resid-ual stresses observed in the experiments is pre-dicted in all three cases. However, for Ti’s close toTg (for the VPNV model) or a little bit higher thanTg (for the TCIF model), negative birefringence at

the center by the VPNV and TCIF models is ob-served, and this contradicts the experimental ob-servations (Figs. 12 and 13). As for PS, the TCIFmodel for PC predicts no residual stresses andbirefringence at Ti’s close and below Tg. At highTi’s, the birefringence value shows an abruptchange.

Figure 16. Predicted residual stress and birefringence distributions in PS platesquenched from different Ti’s to 60 °C water with the VPWV and TCIF models.


Although the photoviscoelastic behavior of PSis more complex than that of PC, as described inref. 29, a better prediction of experimental datafor the residual birefringence is obtained in freelyquenched PS plates than in PC plates. This ispossibly due to the absence of the volume relax-ation data for PC. Also, it may be speculated thatthe first-order rate equation with multiple relax-ation times is required to improve the predictionof the residual birefringence.

CONCLUSIONS

A methodology was developed to calculate thethermal birefringence from our earlier measuredmechanical and optical relaxation functions. Freequenching experiments were performed on PSand PC plates. At temperatures higher than 105°C, the residual birefringence in freely quenchedPS plates is negative at the center and positive atthe surfaces, whereas at temperatures close to Tg,the birefringence becomes positive at the centerand negative at the surface. The birefringenceprofile in freely quenched PC plates is alwayspositive at the center and negative at the surface.In both cases, the residual thermal stresses arealways positive at the center and negative at thesurface.

Ti has a larger effect on the residual birefrin-gence profile than the quenching temperature forboth freely quenched PS and PC plates. Higherheat transfer at the surface results in a largermagnitude of the residual birefringence. Unlikethe residual stress profile, the residual birefrin-gence profile is not balanced.

Both the VPNV and TCIF models cannot pre-dict correctly the residual birefringence profiles infreely quenched PS plates. Only the VPWVmodel, which contains the volume relaxation, candescribe the observed residual birefringence dis-tributions in PS plates.

The VPWV model can only qualitatively pre-dict the residual birefringence in freely quenchedPC plates. The volume relaxation behavior of PCrequires more study to obtain a better predictionof the residual birefringence distributions.

REFERENCES AND NOTES

1. Isayev, A. I. In Injection and Compression MoldingFundamentals; Isayev, A. I., Ed.; Marcel Dekker:New York, 1987; Chapter 3, pp 227–328.

2. Isayev, A. I.; Crouthamel, D. L. Polym Plast Tech-nol Eng 1984, 22, 177.

3. Titomanlio, G.; Drucato, V.; Kamal, M. R. IntPolym Process 1987, 1, 55.

4. Wimberger-Friedl, R.; Hendriks, R. D. H. M. Poly-mer 1989, 30, 1143.

5. Wimberger-Friedl, R.; De Bruin, J. G. J Polym SciPart B: Polym Phys 1993, 31, 1041.

6. Shyu, G. D.; Isayev, A. I. Soc Plast Eng Annu TechConf Pap 1993, 39, 1673.

7. Boitout, F.; Agassant, J. F.; Vincent, M. Int PolymProcess 1995, 10, 237.

8. Bushko, W. C.; Stokes, V. K. Polym Eng Sci 1995,35, 351.



11. Ghoneim, H.; Hieber, C. A. J Therm Stresses 1996,19, 795.

12. Guo, X.; Isayev, A. I. Soc Plast Eng Annu Tech ConfPap 1999, 45, 1813.

13. Guo, X.; Isayev, A. I. J Appl Polym Sci 2000, 75,1404.

14. Bartenev, G. M. J Technol Phys 1948, 18, 383.15. Bartenev, G. M. J Technol Phys 1949, 19, 1423.16. Indenbom, V. L. J Technol Phys 1954, 24, 925.17. Morland, L. W.; Lee, E. H. Trans Soc Rheol 1960, 4,

233.18. Muki, R.; Sternberg, E. J Appl Mech 1961, 28, 193.19. Lee, E. H.; Rogers, T. G. J Appl Mech 1963, 30, 127.20. Lee, E. H.; Rogers, T. G.; Woo, T. C. J Am Ceram

Soc 1965, 48, 480.21. Narayanaswamy, O. S.; Gardon, R. J Am Ceram

Soc 1969, 52, 554.22. Narayanaswamy, O. S. J Am Ceram Soc 1971, 54,

491.23. Narayanaswamy, O. S. J Am Ceram Soc 1978, 61,

146.24. Ohlberg, S. M.; Woo, T. C. Rheol Acta 1973, 12, 345.25. Ohlberg, S. M.; Woo, T. C. J Non-Cryst Solids 1974,

14, 280.26. Williams, M. L.; Landel, R. F.; Ferry, J. D. J Am

Chem Soc 1955, 77, 3701.27. Kovacs, A. J. Adv Polym Sci 1964, 3, 394.28. Goldbach, G.; Rehage, G. J Polym Sci Part C:

Polym Symp 1967, 16, 2289.29. Shyu, G. D.; Isayev, A. I.; Li, C. T. J Polym Sci Part

B: Polym Phys 2001, 39, 2252.30. Shyu, G. D. Ph.D. Dissertation, University of Ak-

ron, 1993.31. Williams, M. L.; Arenz, R. J. Exp Mech 1964, 4,

249.32. Rusch, K. C. J Macromol Sci Phys 1968, 2, 179.33. van Krevelen, D. W. Properties of Polymers;

Elsevier: Amsterdam, 1976.34. Encyclopedia of Polymer Science and Technology;

Interscience: New York, 1989.


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Residual thermal birefringence in freely quenched plates of amorphous polymers: Simulation and experiment