Processing-Induced Properties in Glassy Polymers ...mate.tue.nl/mate/pdfs/6235.pdfProcessing-Induced Properties in Glassy Polymers: Application of Structural Relaxation to Yield Stress

Processing-Induced Properties in Glassy Polymers:Application of Structural Relaxation to Yield StressDevelopment

TOM A. P. ENGELS, LEON E. GOVAERT, GERRIT W. M. PETERS, HAN E. H. MEIJER

Dutch Polymer Institute (DPI), Section Materials Technology (MaTe), Eindhoven University of Technology,NL-5600 MB, Eindhoven, The Netherlands

Received 27 October 2005; revised 18 January 2006; accepted 19 January 2006DOI: 10.1002/polb.20773Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: A method is presented to predict the yield stress distribution throughoutan injection-molded product of an amorphous polymer as it results from processingconditions. The method employs the concept of structural relaxation combined with afictive temperature, following the Tool–Narayanaswamy–Moynihan formalism. Thethermal history, as it is experienced by the material during processing, is obtained bymeans of numerical simulation of the injection molding process. The resulting predic-tions of yield stress distributions are shown to be in excellent agreement with experi-mental findings, both for different mold temperatures and for different part thick-nesses. VVC 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1212–1225, 2006

Keywords: glassy polymers; physical aging; yield stress

INTRODUCTION

In the current engineering practice there is agreat variety of numerical tools available todesign polymer products. Examples include com-mercial codes that aid the optimization process intool design. In the field of injection molding, thesecodes (for example, Moldex3D, Moldflow) not onlyallow the analysis of mold filling, but in the caseof glassy polymers, also give access to numericalevaluation of residual stresses and dimensionalstability (shrinkage, warpage).1–3 However, infor-mation concerning mechanical properties, suchas the yield stress distribution in a product ortime-to-failure when placed under a static load, isnot provided by these codes. Nevertheless, theseare of high interest for the use of polymers instructural, load bearing applications, since they

determine the lifetime of the final product andthe corresponding loading range. With properknowledge of these properties, true product opti-mization becomes possible.

Long-term failure of glassy polymers understatic loading is dominated by plastic instabilitiesleading to plastic strain localization.4–9 Theseplastic instabilities are the same as those foundfor short-term failure, observed in tensile testingunder constant strain rate.10 In the last 15 years,a lot of effort has been invested by a number ofgroups at different universities into the develop-ment and validation of 3D constitutive modelsthat can describe this localization behavior ofglassy polymers, for example, in the group ofMary Boyce at MIT,11–13 the group of Paul Buckleyin Oxford14–16 and in our Eindhoven group.17–19

It was shown in subsequent quantitative stud-ies10,19–24 that it is the large strain intrinsicbehavior (yield, strain softening, and subsequentstrain hardening) of the polymer that determinesthe macroscopic localization behavior, and thusfailure. The model developed in our group proved

Correspondence to: L. E. Govaert (E-mail: [email protected])

Journal of Polymer Science: Part B: Polymer Physics, Vol. 44, 1212–1225 (2006)VVC 2006 Wiley Periodicals, Inc.

1212

to be capable of also predicting the long termstatic failure,25,26 including a static fatigue limitwhen aging kinetics were taken into account.

Here, we build on the constitutive model devel-oped in our group. In its latest form19 it canaccount for aging kinetics far below the glass-transition temperature, that is, the increase ofyield stress with time, and predict long-term fail-ure of glassy polymers. The rate at which agingtakes place depends on both the temperature andthe stress.19,27 Moreover, it has been shown thatupon plastic deformation all prior effects of agingcan be erased, and a rejuvenated material isobtained.18,28 Both thermal and mechanical his-tory are captured by the single state parameter S.This parameter thus describes aging and effec-tively determines the value of the yield stress as afunction of the time, temperature, and stress. Thestate parameter is defined with respect to thefully rejuvenated material as a reference state(S ¼ 0). The effects of aging and rejuvenation arecombined through a simple multiplicative relation,S ¼ SaRc, where Sa represents the aging kineticsandRc represents the rejuvenation kinetics.

In a previous study29 we showed that it is pos-sible to predict the yield stresses, Sa, of injection-molded products to a very good degree of agree-ment with experimental results. These predic-tions are based on a phenomenological approachin which we assume that the kinetics of aging,measured by annealing treatments well below theglass-transition temperature, Tg, and written inan analytical form, can be applied to the transienttemperature history that follows from the injec-tion molding process analysis.19 By integration ofthe analytical formulation of the aging kineticswith respect to temperature, the local values of Sa

can be obtained. Since the annealing treatmentswere performed well below Tg no equilibriumkinetics were taken into account (the increase inyield stress cannot be endless; it will eventuallyarrive at its equilibrium value). The developmentof the yield stress was initiated when passing Tg,which is a fixed temperature point in this case.This is, however, far from reality. The glass-tran-sition temperature is not a single temperaturepoint, but rather a temperature region in whichthe equilibrium melt falls from equilibrium.30

Strongly coupled to these equilibrium kinetics isthe phenomenon of physical aging.

Aging (and relaxation) of glassy polymers hasattracted a lot of attention in literature, and a num-ber of reviews are available.31–33 Although agingand relaxation are used to describe time-dependent

phenomena of a great variety of different proper-ties, for example, volume, enthalpy, yield stress,and viscoelastic properties, they are in nature thesame. All are a result of the kinetically governednonequilibrium state of the polymer below theglass-transition temperature, which leads to time-dependent behavior.30With respect to themodelingof aging phenomena, two main topics can be distin-guished. The first one is the development of me-chanical properties during use, that is, propertiesas they result from processing are not modeled, butrather taken as an initial state of the material, andthe evolution of properties over long periods of timeare followed. Illustrative examples are the develop-ment of linear viscoelastic properties34 and yieldstress27 as a function of elapsed time after quench-ing. The second topic is related to the modeling ofthe properties as they develop during different (of-ten complex) thermal histories, that is, the evolu-tion of properties through the Tg region are fol-lowed and modeled. Examples of this approach canbe found for volume and enthalpy relaxation,where a number of models are available that quiteadequately describe the nonequilibrium kine-tics.35–40 To our knowledge, such an approach hasnever been attempted with respect to the develop-ment of yield stress. Bauwens-Crowet and Bau-wens27 used a one-parameter relaxation modelapproach to describe the increase in yield stressupon annealing, which was somewhat similar tothe one-parameter modeling approach of Hutchin-son and Kovacs41 to describe the increase in en-thalpy overshoot, but this approach is limited in itsusability and can certainly not describe the devel-opment of yield stress as a result of processing con-ditions. Since it is the yield stress, and the relatedstrain softening, that determines failure, it is im-portant to be able to accurately describe the agingkinetics of polymers and to use these kinetics topredict the yield stress distribution from process-ing conditions.

In this study we present a method to predict theyield stress as a function of thermal history as aresult of processing conditions, which incorporatesthe equilibrium kinetics associated with the glasstransition. To do so we will adopt the TNM (Tool–Narayanaswamy–Moynihan)35,36,42 formalism todescribe the aging kinetics of the yield stress, ormore correctly, the yield stress retardation.

MODELING

Structural relaxation models have been used suc-cessfully in the past to describe volume and en-

YIELD STRESS DEVELOPMENT IN GLASSY POLYMERS 1213

Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb

thalpy relaxation of amorphous polymers. Theessence of these models is that they recognizethe nonequilibrium state of an amorphous glassbelow the glass-transition temperature and usethe distance from equilibrium as a driving forcefor the relaxation kinetics. The two main ap-proaches are the TNM model36 and the KAHR(Kovacs–Aklonis–Hutchinson–Ramos) model.38

Both are multiparameter models and are, in prin-ciple, equivalent. The TNM model uses the fictivetemperature, Tf, as defined by Tool,42 to fullydefine the thermodynamic state of the material,whereas the KAHR model uses the distance ofthe property under investigation from its equilib-rium value as a state parameter. Here, we willadopt the TNM model approach to describe theaging and equilibrium kinetics of amorphous poly-mers. Rather than directly describing the yieldstress retardation, we will apply the TNM modelto the zero viscosity, which is rate and loading ge-ometry independent, unlike the yield stress itself;for example, the yield stress measured in com-pression is not equal to the one measured in ten-sion. Adaptation was done such that the model isapplicable in a 3D constitutive model. Note thatthis so-called zero viscosity, the viscosity at zeroshear rate, is associated to segmental chain mo-bility (related to yield), and should not be con-fused with the zero shear viscosity used by rheol-ogists to describe the melt viscosity at zero shearrate.

TNM Model

The TNM model uses a fictive temperature, Tf, tofully define the state of the material, and the dif-ference between this fictive temperature and thecurrent temperature is the force driving the none-quilibrium glass toward equilibrium. For a givenproperty, P, (e.g., volume or enthalpy) at a givennonequilibrium temperature, T, the fictive tem-perature, Tf P, can be seen as the equilibrium tem-perature at which this property has the samevalue as that at the given nonequilibrium tem-perature (Fig. 1). The property P is given by:

PðT; nÞ ¼ PeðT0Þ þ aPlðT � T0Þ þ aPsðTf ðT; nÞ � TÞ¼ PeðTÞ þ aPsðTf ðT; nÞ � TÞ ð1Þ

where aPs ¼ aPl � aPg and n is the reduced time,defined later.

In eq 1, T0 is a temperature well above theglass-transition temperature, Tg, aPl is the slopeof property P in the liquid state, aPg is the slope of

property P in the glassy state, and Pe signifies thatP is in the equilibrium of the liquid state. Theslopes of the liquid and glassy state are hereassumed to be constant but can very well be a func-tion of temperature. In that case eq 1 is written as:

PðT; nÞ ¼ PðT;1ÞþZ Tf

T

aPsðT0ÞdT0: ð2Þ

In the case of constant slopes, eq 1, it is easilyconcluded that the relation basically describes twostraight lines, that is the liquid state and the glassstate, where the lines themselves depend on tem-perature, T. The fictive temperature, Tf, deter-mines on which of the two curves we are at a spe-cific temperature. The equilibrium state is herealso called liquid state, although in the case of poly-mers, the nomenclature ‘‘the melt state’’ wouldalso be appropriate. (The TNMmodel finds its ori-gins in the field of anorganic glasses, and there-fore, the term liquid is mostly used. We will usethe terms liquid and melt here interchangeably.)

The relaxation of property P now follows fromthe relaxation of the fictive temperature. Thisrelaxation is equivalent in nature to stress relaxa-tion, and therefore, can be described analogously.Assuming thermorheological simple behavior andusing the linearity of the reduced time, the relax-ation of the fictive temperature is given by:

Tf ðT; nÞ ¼ T �Z n

0

MPðn� n0ÞdTdn0

dn0 ð3Þ

where MP is the relaxation function and n thereduced time.

The relaxation function is defined by a stretchedexponential to account for a distribution of relax-ation times. This is not a true distribution in theform of a discrete spectrum as is used in the caseof the KAHR model,38 but rather a mathematicalconvenient way which is mostly attributed toKohlrausch43 and Williams and Watts.44 The re-laxation function and reduced time are given by,respectively:

MPðT; nÞ ¼ exp � nsPr

� �b !

; nðtÞ ¼ sPr

Z t

0

dt0

spðT;Tf Þð4Þ

where sPr is the relaxation time at a referencetemperature, b is a parameter that describes thenonexponentiality or width of the distribution of

1214 ENGELS ET AL.


relaxation times, and sP is the current relaxationtime.

The widely and successfully used definition ofthe relaxation time, sP, of the TNM model isapplied here. This definition, however, lacks theo-retical justification and reflects in essence a phe-nomenological approach. It should, however, bepossible to use other definitions of the relaxationtime which do have physical relevance, such asthe Adam–Gibbs relation.45,46

The relaxation time, sP, is defined by:

sPðT;Tf Þ ¼ A expx�H

RTþ ð1� xÞ�H

RTf

� �ð5Þ

where A is a pre-exponential factor, x defines thedegree of nonlinearity, DH is the activation en-thalpy, and R is the ideal gas constant.

Application to the Zero Viscosity of SegmentalChain Mobility

To describe the evolution of the yield stress, weapply the structural relaxation kinetics to thezero viscosity as illustrated in Figure 2 (note that

the logarithm of the viscosity is plotted versus theinverse of the temperature). For a uniaxial ten-sile test the relation between yield stress, ry, andzero viscosity, g0, is given by:

ry ¼ rs þ rr ð6Þrs ¼ 3g e

�0; rr ¼ Grðk2y � k�1

y Þ ð7Þ

where rs is the stress contribution arising fromsecondary interactions, rr is the stress contribu-tion of the entanglement network, g is the stress-and pressure-dependent viscosity, _e0 is the strainrate applied, Gr is the strain hardening modulus,and ky is the draw ratio at yield. A more elaboratederivation is given in the Appendix.

The viscosity, g, is defined by:

gðT;p;�s;SÞ ¼ Arejs0 exp�U

RT

� �exp

lPs0

� �expðSÞ

� �s=s0sinhð�s=s0Þ ¼ g0ðT;p;SÞ

�s=s0sinhð�s=s0Þ ð8Þ

where

Sðt;T;�cPÞ ¼ Saðt;TÞR�cð�cPÞ: ð9Þ

Figure 2. Nonequilibrium kinetics of the viscosity.

Figure 1. Nonequilibrium kinetics of property P anddefinition of fictive temperature, Tf, as a result of (tran-sient) temperature and annealing temperature, TA.



In eq 8, p is the hydrostatic pressure, �s is theequivalent stress, S is the state parameter, Arej isa pre-exponential factor related to the rejuven-ated state, s0 is the characteristic stress or modu-lus, DU is the activation energy, and l is the pres-sure dependence. Equation 9 is already explainedin the Introduction and basically describes theaging dependence of the viscosity through thefunction Sa(t,T) and the rejuvenation dependencethrough Rcð�cPÞ, where �cP is the equivalent plasticstrain. The definitions of the equivalent stressand equivalent plastic strain are given in the Ap-pendix (eqs A11 and A12).

The idea of using structural relaxation todescribe the effect of physical aging on the yieldstress is not new. Buckley and Jones14 alreadyproposed in their paper on the constitutive model-ing of amorphous polymers to use the fictive tem-perature to describe the structural state of thematerial. In a later publication,15 Buckley alsoadded the notion that there should be an evolu-tion contributed to the fictive temperature. Theuse of the fictive temperature by Buckley in thissense is the same as the use of the state parame-ter Sa

19 to describe the thermomechanical historyof the material as done here.

The use of structural relaxation models todescribe the zero viscosity of a material is some-what trivial, since the basis of these models is therelaxation kinetics as governed by the viscosity ofthe material. It is therefore surprising that notmore use has been made of these models todescribe aging phenomena of mechanical proper-ties, such as the increase of yield stress with time,or to use them to predict the mechanical proper-ties after a ‘‘complex’’ thermal history, such as thethermal history resulting from processing.

Our constitutive model in its current formuses the rejuvenated state as a reference state.This is the state in which all effects of prior ther-mal history have been erased.13,28 Although therejuvenated state is a well-defined state from amechanical viewpoint. From the viewpoint ofstructural relaxation and equilibrium kineticsthis state is poorly defined and is the nonequili-brium state, furthest form equilibrium, whichcan only be attained by mechanical deformation.In practice, the value of Arej (see eq 8), whichdefines the rejuvenated state, is effectively aninput parameter for the model and thereforeknown. To be able to fit the relaxation kineticsas derived in this study to the current constitu-tive framework, we will omit working from therejuvenated state, and therefore will use A0(Sa)

¼ Arej exp(Sa) for the pre-exponential parameter.To determine the value of the state parameterfrom the zero viscosity as it follows from thekinetics developed here will be a trivial exercise,which yields the following definition of the zeroviscosity:

g0ðT;p;SaÞ ¼ A0ðSaÞRTV� exp

�U þ lpV�

RT

� �ð10Þ

where s0 ¼ RTV� has been substituted to account

for the temperature dependence of the character-istic stress, and V* is the specific activation vol-ume. Note that S has been replaced by Sa, sincewe are only interested in the initial yield stressand thus Rcð�cPÞ is not yet developing, assumingthat plastic deformation only starts to developafter the point of yield.

To use the structural relaxation framework forthe zero viscosity, we need the temperature de-pendence of the zero viscosity both above andbelow the glass-transition temperature. The tem-perature dependence of the nonequilibrium vis-cosity below Tg is known to follow an Arrheniusform. Well above Tg (in the range of Tg þ 10 8C <Tg < Tg þ 100 8C) the temperature dependence ofthe equilibrium polymer melt in general followsVTF (Vogel–Tammann–Fulcher) or WLF (Wil-liams–Landel–Ferry) behavior.30,47 Since we areinterested in the mechanical properties as theydevelop below Tg as a result of nonequilibriumkinetics, the correct description of the tempera-ture dependence of the viscosity well above Tg isnot relevant here. In the transition region, how-ever, it was shown48,49 that the temperature de-pendence of the equilibrium glass shifts towardan Arrhenius dependence. This Arrhenius behav-ior of the equilibrium glass was verified only in atemperature window of a couple of tens of degreesbelow Tg, since below, relaxation times becomelarge and equilibrium cannot be obtained on an ex-perimentally acceptable timescale. In our previ-ous study on the development of the yield stressas a result of processing conditions29 we showedthat all structure development takes place in a tem-perature window of approximately Tg > T > Tg �20/30 8C. The use of an Arrhenius type of temper-ature dependence of the equilibrium viscositybelow Tg therefore appears to be acceptable.

Using the Arrhenius type of temperature de-pendence of the zero viscosity both in the liquidstate, DUl, as in the glassy state, DUg, and assum-ing that DU » lpV*, eq 1 in combination with eq10 can be rewritten to:

1216 ENGELS ET AL.


lnðg0ðT; nÞÞ ¼ lnA0ðSaÞR

V�

� �þ lnðTÞ þ �Ul

R

� �1

T

þ �Ul ��Ug

R

� �1

Tf ðnÞ �1

T

� �ð11Þ

where the first three terms on the right handside correspond to Pe(T) and the last term toaPs(Tf � T).

Together, eqs 3–5 and 11 form the basis of theframework that is needed to apply the concept ofstructural relaxation to the zero viscosity. Nowthe appropriate input parameters for the modelneed to be determined.

EXPERIMENTAL

Tensile bars were directly injection molded, oralternatively machined from injection-moldedplates. Both types of bars were made from thesame grade of polycarbonate, Lexan 141R, sup-plied by General Elastic Plastics (Bergen opZoom). Before injection molding, the polycarbon-ate was dried under vacuum at 80 8C for a periodof 24 h. All injection molding was performed onan Arburg 320S all-rounder 500-150.

The molded tensile bars were shaped accordingto the ISO 527 norm. The mold, manufactured byAxxicon Mold Technology (Helmond, the Nether-lands), was kept at a temperature of 90 8C. Thesespecimens were used for annealing experimentsclose to Tg. The samples were predried at 80 8Cfor 24 h, and subsequently annealed in a Heraeushot air circulation oven at 135, 140, and 145 8Cfor times up till 9 days. Before testing sampleswere allowed to cool to room temperature.

Next to the tensile bars, rectangular plateswith dimensions 70 � 70 � 1 mm3 and 70 � 70� 4 mm3 were injection molded (Fig. 3). The mold(from Axxicon Mold Technology) had a V-shapedrunner of 4 mm thickness and an entrance of 70

� 1 mm2 for the 1-mm-thick plates and an en-trance of 70 � 2 mm2 for the 4-mm-thick plates.The shape of the runner in combination with theentrance size caused the material to fill the cavityuniformly along the width of the cavity, whichwas proven by several short shot experiments.Melt temperature was set to 280 8C and injectionrates were 90 and 50 cc/s for the 1-mm and 4-mmplates, respectively. A packing pressure of 500bar was used to minimize shrinkage. Variation ofthe packing pressure from 25 to 900 bar proved tohave no effect on the measured properties of theplates. To investigate the influence of the temper-ature history, the mold cavity temperature wasvaried from 30 to 130 8C in steps of 10 8C. Coolingtimes were 60 and 150 s for the 1-mm- and 4-mm-thick plates, respectively. From the 1-mm-thickinjection-molded plates, rectangular bars of 70� 10 mm2 were cut parallel and perpendicular tothe flow direction. From the 4-mm-thick plates,bars were only taken parallel to the flow direc-tion. Gauge sections of 33 � 5 mm2 were thenmachined into these bars (Fig. 3).

Uniaxial yield stresses for both specimen geo-metries were determined on a Zwick Z010 univer-sal tensile tester. All measurements were per-formed at a linear strain rate of 10�3 s�1; themeasured engineering yield stress was convertedto true yield stress under the assumption ofincompressibility up to the point of yield.

NUMERICAL

Numerical Implementation

The evolution of the zero viscosity for the injec-tion-molded tensile bars was calculated assumingconstant cooling rates and homogeneous temper-ature conditions. This was done because the exactprocessing conditions for these bars were un-known. Furthermore, the annealing process can

Figure 3. Injection-molded plate and tensile specimen machined thereof.



be considered as an isothermal process in whichno temperature distributions exist in the sam-ples. Calculations were performed by evaluationof eqs 3–5 and 11. For a detailed description ofthe numerical implementation of the TNM modelthe reader is referred to ref. 39. Temperature his-tories for the rectangular plates were obtained bymeans of numerical simulation of the injectionmolding process, using the commercial injectionmolding simulation package Moldflow MPI(release 5.0). A 2.5D approach was used, since thewidth vs. thickness ratio of the plates is veryhigh, that is, pressures are calculated 2D and thetemperature and velocity field fully 3D. Processvariables were taken the same as mentioned inthe Experimental section. Material properties ofLexan 141R were taken from the standard Mold-flow library. The Moldflow simulations providedtemperature histories for a number of layers overthe thickness of the sample. The calculated ther-mal histories were exported and postprocessed(Matlab) to determine the evolution of the zeroviscosity using eqs 3–5 and 11. The temperaturehistories of the surface layers could not be used,since cooling was so fast that at the first incre-ment the temperatures were already below Tg

(Fig. 8).

Numerical Strategy

The numerical results are obtained using a set ofrelaxation time parameters that were obtainedby applying an optimization routine, nonlinear

least squares, to the yield stress results of theannealing treatments (Fig. 6) and the injectionmolding samples of 1 mm thickness (Fig. 9). Theparameter set is given in Table 2. All modeldescriptions and predictions are made with thissingle parameter set. The results for the 4-mm-thick injection molding samples are true predic-tions.

RESULTS AND DISCUSSION

Determination of the NonequilibriumTemperature Dependence

To derive the temperature dependency of the zeroviscosity in the glassy state, use is made of litera-ture results of yield stress vs. strain rate mea-sured at different temperatures for two differentmaterial states (that of Bauwens-Crowet andBauwens,27 (Fig. 4) and similar data measured byKlompen50). Furthermore, data by Bauwens-Cro-wet et al.51 of yield stress vs. temperature over alarge temperature range (from �120 to þ120 8C),measured at a single strain rate, are used.

Using the 3D constitutive framework and theboundary conditions that apply to uniaxial exten-sion under constant strain rate, the followingrelation can be derived to determine the zero vis-cosity from uniaxial yield stresses:

g0 ¼ RT

V�1ffiffiffi3

p 1

_e0sinh

ry � rrðkyÞffiffiffi3

p V�

RT

� �ð12Þ

where _e0 is the strain rate applied, ry is the yieldstress, and rr(ky) is the hardening contribution atthe point of yield. For a complete derivation ofthis experimental relation the reader is referredto the Appendix.

To be able to apply this relation, a number ofparameters are required. Most parameters aretaken to be the same as used in our previous pa-

Figure 4. Yield stress vs. strain rate at differenttemperatures (as indicated) and for two different ini-tial material states, as-received (!) and annealed(&). Lines are a guide to the eye; taken from ref. 27.

Table 1. Material Parameters Obtained forPolycarbonate19

K (MPa) 3750G (MPa) 308Arej at 295 K (1011) (s) 3.0l 0.08Sa –r0 0.965r1 50r2 �5Gr (MPa) 26

1218 ENGELS ET AL.


per29 (Table 1), but the determination of the acti-vation energy and activation volume is done hereagain, since the literature values gave unsatisfac-tory results.

By using the temperature dependence of thezero viscosity as follows from eq 10, an activationenergy, DU, of 326 kJ/mol was found. The activa-tion volume was determined from the yieldstresses, and was found to be 3.50 � 10�3 m3/mol.These values were in good agreement with litera-

ture values,27,50,52 although the activation energywas slightly higher.

Applying eq 12 to the yield stresses of Figure 4gives the zero viscosities as shown in Figure 5(a).Note that since the zero viscosity is strain-rate in-dependent, the four different strain rates pertemperature reduce to one zero viscosity. Tr is anarbitrary reference temperature and is taken tobe 150 8C; a literature value for the glass-transi-tion temperature of polycarbonate. The tempera-ture dependency for the different initial materialstates can be seen to be the same. In Figure 5(b),the literature yield data of Bauwens-Crowetet al.51 and Klompen50 are added. It can be seenthat all the yield data result in zero viscositieswhich are in good agreement with each other overa wide temperature range.

Determination of the Equilibrium TemperatureDependence

To determine the temperature dependence of thezero viscosity in the liquid state, use is beingmade of equilibrium yield data as measured byperforming annealing experiments close to Tg

(Fig. 6). Equilibrium data implies the plateau val-ues of the yield stress, which are obtained afterannealing up to equilibrium. Similar plateaus arefound in literature for the enthalpy overshoot forpolycarbonate53 and polystyrene54,55 when theyare annealed for long periods at temperatures nottoo far below Tg.

Figure 5. (a) Zero viscosities determined from theyield data from Figure 4; two different initial materialstates: quenched (!) and annealed (&). Lines areArrhenius fits with an activation energy of 326 kJ/mol. (b) Zero viscosities determined form yield data ofBauwens27,51 (&, !) and Klompen50 (*), showinggood correspondence of the temperature dependencybetween different studies for quenched material.

Figure 6. Yield stress vs. annealing time for threedifferent annealing temperatures; 135 8C (&), 140 8C(̂ ), and 145 8C (4); solid lines are model predictions.



Since the yield stresses are measured at roomtemperature, they must be shifted back to the ele-vated temperatures at which equilibrium wasobtained. This can be done by rewriting eq 10with respect to two temperatures, giving the ratiobetween the viscosities at those temperatures:

g0ðT1Þg0ðT2Þ ¼

T1

T2exp

�Ug þ lpV�

RT

� �1

T1� 1

T2

� �� ð13Þ

Since DUg » lpV* this reduces to:

g0ðT1Þg0ðT2Þ ¼

T1

T2exp

�Ug

RT

� �1

T1� 1

T2

� �� ð14Þ

where DUg is the activation energy of the glassystate.

In Figure 7 the values for the equilibrium vis-cosities as determined from our own annealingexperiments are shown (closed symbols), as wellas values for the equilibrium viscosities, whichwere determined from literature equilibriumyield stresses measured by Bauwens-Crowet andBauwens27 (open symbols). An activation energyof 1.50 MJ/mol is found for our equilibrium vis-cosities, whereas the activation energy of the lit-erature values seems somewhat higher. Since theexact experimental conditions and data of the lit-erature equilibrium values are unknown, we donot want to speculate as to where the differencein activation energies comes from. Possible molec-

ular weight effects, for instance, should be exam-ined in the future. For all model predictions theequilibrium activation energy of 1.50 MJ/mol wasused.

Now that the temperature dependencies ofboth the equilibrium glass and the nonequili-brium glass are established, we have a frame-work that allows us to apply structural relaxationto the zero viscosity. From the evolution of thezero viscosity, we should be able to predict the de-velopment of the yield stress as a result of pro-cessing conditions. The parameters that describethe relaxation times that dominate the zero vis-cosity are found by means of an optimization rou-tine applied to both annealing and processingresults and are given in Table 2.

Annealing Experiments up to Equilibrium

Tensile bars were annealed up to equilibrium(Fig. 6). To predict these annealing results, con-stant cooling and heating rates have beenassumed. It was shown, however, that the finalequilibrium curves (the total curves depicted inFig. 6) are not influenced by the processing condi-tions. Different cooling rates will result in differ-ent nonequilibrium yield stresses, but whenannealing is performed at temperatures close toTg up to equilibrium, where the relaxation timesare much shorter, the final response is hardlyinfluenced by the prior thermal history. As can beseen, the model descriptions fit the experimentalresults well.

Prediction from the Injection Molding Process

Figure 8 shows the predicted distribution of theyield stress over the thickness of the sample forvarious mold temperatures. The values at thesurface layers could not be calculated due to thelimited temperature information at these posi-tions. Already in the first time-increment of the

Figure 7. Equilibrium zero viscosities derived fromthe equilibrium yield stresses from Figure 6 (^) andequilibrium zero viscosities taken from ref. 27 (^);dashed lines are Arrhenius fits, and the solid line is amodel prediction.

Table 2. TNM Model Parameters

Zero viscosity parametersDUl (MJ/mol) 1.50DUg (kJ/mol) 326V* (10�3) (m3/mol) 3.5

TNM parametersln(A) (s) �247.6X 0.25DH (kJ/mol) 865.4b 0.54

1220 ENGELS ET AL.


simulations the surface layers adopt a tempera-ture below Tg, making an evolution through thetransition region impossible. It can be seen thatwith increasing mold temperature a more uni-form stress distribution is obtained. These resultsare in qualitative agreement with the results ofyield stress distribution in our previous study.29

From the yield stress distributions over thethickness of the samples, an area-weighted yieldstress can be calculated. The resulting averageyield stresses vs. mold temperatures, rangingfrom 30 to 130 8C, are given in Figure 9. For the1-mm-thick samples, yield stresses both paralleland perpendicular to the flow direction areshown, whereas for the 4-mm-thick samples onlyyield stresses parallel to the flow direction areshown. From the results of the 1-mm-thick sam-ples it can be seen that there is a minor influenceof orientation (<0.5 MPa on average), given thedifference between the results of the yieldstresses parallel and perpendicular to flow. How-ever, it has to be noted that the processing condi-tions for the 1-mm-thick samples are the mostextreme as to expected differences in yieldstresses due to molecular orientation. Thereforethe influence of orientation is omitted in thisstudy and isotropic conditions are assumed.

As noted earlier, the parameter set used in thisstudy was obtained by a fitting routine on boththe annealing results of Figure 6 and the resultsfor the 1-mm-thick samples of Figure 9. The dis-crepancies between model descriptions and ex-perimental results for the 1-mm-thick samples

are a direct result of using two data sets. Theannealing results pose somewhat differentdemands on the parameters than the processingresults. The model results for the 4-mm-thicksamples of Figure 9 are, however, true predic-tions.

It is interesting to see that slope of the pre-dicted yield stress vs. mold temperature shows adecrease at 130 8C for both sample thicknesses.This decrease in slope can also be seen in the ex-perimental results. The decrease can be attrib-uted to the fact that at high mold temperaturesthe yield stress is still evolving when the productis ejected from the mold. The rapid cooling on air,which then follows, prohibits further evolution ofthe yield stress.

Application to Annealing

In Figure 10 the literature zero viscosities of Fig-ures 6 and 7 are reproduced. The solid lines inthis figure are model predictions and the dottedline is the equilibrium state. The model predic-tions are, since the thermal histories of the sam-ples are unknown, calculated by using constantcooling rates. Effectively, the initial cooling rate,from point A to point B, was used as a fitting pa-rameter. To predict the right values for thequenched zero viscosities in the glassy state, acooling rate of 1.0 8C/s was used, which is a quiterealistic cooling rate.

In Figure 11 the evolution of the fictive temper-ature, Tf, vs. the applied temperature correspond-ing to the evolution of the zero viscosity of Figure

Figure 8. Distribution of the yield stress over thenormalized thickness, from center to surface, of theinjection-molded samples for various mold tempera-tures. Normalization was performed with respect tohalf the thickness (the total thickness being 1 mm).

Figure 9. Experimental yield stresses for samples4-mm thick (&) and 1-mm thick (*), versus modelpredictions (lines).



10 can be seen. The prescribed temperature his-tory starts at an initial temperature, T0, wellabove Tg(A

0). In the equilibrium state the fictivetemperature is equal to the prescribed tempera-ture (A ¼ A0). First, a constant cooling rate of1.0 8C/s is applied (B, B0). As soon as the relaxa-tion times become too long, it can be seen that thefictive temperature deviates from the prescribedtemperature and remains almost constant. Next,an annealing period of several years is applied,but since the relaxation times are extremely longat such low temperatures, there is no effect onthe fictive temperature. Subsequently, an anneal-ing step is applied, in which the temperature isincreased with a constant heating rate of 1.0 8C/sto an annealing temperature of 120 8C (B0–C0).The annealing period was 46 h, exactly the sameas the one Bauwens-Crowet and Bauwens usedexperimentally. During this annealing period, thefictive temperature can be seen to evolve towardequilibrium (C–D). Finally a last cooling step isapplied (D0–E0). The resulting predictions for theannealed zero viscosities can be seen to beslightly higher than the experimental values cor-responding to an overestimation of the yieldstress (Fig. 10).

RESULTS AND DISCUSSION

It is shown that the yield stresses as a result ofprocessing conditions, that is, mold temperature,can be predicted in good agreement with experi-mental results. The yield stress as a result of an

annealing treatment can also be described quitesatisfactory in a temperature window of 10–20 8Cbelow Tg. At temperatures further below Tg

model predictions showed to be in less agreementwith experimental findings (Fig. 10). This couldhowever be expected, since the TNM model isreported to be valid only in a small temperaturewindow of a couple of tens of degrees around Tg.

Further we want to note that the parameterset that was obtained in this study to describe theretardation of the zero viscosity (Table 2) is differ-ent from the parameter set found by Hodge56 todescribe the enthalpy relaxation of polycarbonateby means of DSC measurements. This indicatesthat the timescale of zero viscosity retardation isdifferent from the timescale of enthalpy retarda-tion. Differences in relaxation time scales for dif-ferent structural properties have been a topic ofdiscussion in many different studies and it is gen-erally accepted that time scales are different.31,40

We only wish to mention here that, although thetime scales are different, the nonlinearity param-eter, x, and the nonexponentiality parameter, b,show to be, within experimental uncertainty, thesame as the ones found by Hodge.56 This suggestsa possible relation between the relaxation timespectrum of both processes. It was already shownby Adam et al.57 and Bauwens-Crowet and Bau-wens27 that the increase in yield stress and theincrease in enthalpy overshoot are qualitativelyrelated. It should proof interesting to determineto what degree parameters from enthalpy retar-dation studies can be used for the prediction ofzero viscosity retardation.

Figure 10. Evolution of the viscosity as a functionof the prescribed temperature history from Figure 11.

Figure 11. Fictive temperature vs. prescribed tem-perature. Cooling and heating rates are constant.

1222 ENGELS ET AL.


CONCLUSIONS

The TNM structural relaxation model has beensuccessfully applied to the retardation of the zeroviscosity of polycarbonate to predict the evolutionof the yield stress. It is shown that by means ofsimulation of the injection molding process andthe thermal histories derived thereof, the yieldstress resulting from processing can be predictedfor a simple geometry. Furthermore, it is shownthat annealing of polycarbonate up to equilibriumin a temperature region close to Tg can bedescribed as well. Annealing at lower tempera-tures, however, is shown to be described less well.

The ability of predicting the yield stress of aproduct based only on the thermal history as aresult of processing, obtained through simulationof this process, opens new routes to true optimiza-tion of polymer products. The knowledge of theyield stress of a product showed to be enough tonumerically predict its long-term static fatiguelifetime.26 Currently, work is being performed toexplore the capabilities of numerical prediction ofdynamic loading conditions and impact condi-tions. If all aspects of polymer failure can beaccounted for by means of numerical simulationof the processing step and the subsequent numer-ical evaluation of the loading conditions, a prod-uct can be designed for performance without everdoing a single experiment.

APPENDIX

In the 3D constitutive model, the total stress isthe sum of two contributions; a driving stress, rs,and a hardening stress, rr:

r ¼ rs þ rr: ðA1Þ

The driving stress accounts for the intermolec-ular interactions, and is decomposed into a devia-toric part (d) and an hydrostatic part (h):

rs ¼ rds þ rhs : ðA2Þ

The deviatoric part is given by:

rds ¼ GB�

de ¼ 2gDp ðA3Þ

where G is the shear modulus, B�de is the devia-

toric part of the isochoric elastic left Cauchy-Green strain tensor, Dp is the plastic deformationrate tensor, and g is the viscosity.

The viscosity is described by the Eyring rela-tionship,58 but also incorporates pressure de-pendence18 and intrinsic strain softening andaging19:

gðT;p;�s;SÞ ¼ A0ðSÞs0 exp�U

RT

� �exp

lps0

� ��s=s0

sinhð�s=s0Þ ¼ g0ðT;p;SÞ

� �s=s0sinhð�s=s0Þ ðA4Þ

where p is the hydrostatic pressure, �s is theequivalent stress, A0(S) is a pre-exponential fac-tor related to the rejuvenated state in this paperdepending upon the state parameter S, s0 is thecharacteristic stress, DU is the activation energy,and l is the pressure dependence.

The state parameter, S, is decomposed to thefactor Sa(t,T), which describes aging kinetics as afunction of time and temperature, and Rcð�cpÞdescribes the softening kinetics19:

Sðt;T;�cpÞ ¼ Saðt;TÞRcð�cpÞ: ðA5Þ

The hydrostatic part is given by:

rhs ¼ KðJ � 1ÞI ðA6Þ

where K is the bulk modulus, J is the volumechange factor, and I is the unity tensor.

The hardening stress accounts for the molecu-lar network and is given by:

rr ¼ Gr B�

d ðA7Þwhere Gr is the strain hardening modulus and B

�d

is the deviatoric part of the isochoric left Cauchy-Green deformation tensor.

When incompressibility is assumed, the totalstress can be given by:

r ¼ �pIþ 2gDp þGrBd: ðA8Þ

For uniaxial deformation under a constant strainrate, the resulting Cauchy stress and hydrostaticpressure can be given by:

r ¼ 3g_ep þGrðk2 � k�1Þ ¼ rs þ rr ðA9Þ

p ¼ �g_ep � 1

3Grðk2 � k�1Þ ¼ � 1

3r ðA10Þ

where _ep is the plastic strain rate and k the drawratio.



Definition of equivalent measures:

�s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2trðrds � rds Þ

r¼ 1ffiffiffi

3p rs ðA11Þ

�c:

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2trðDp �DpÞ

q¼

ffiffiffi3

p_ep: ðA12Þ

In combination with eq A3 this gives:

�s ¼ g�c:

p ðA13Þ

rs ¼ 3g_ep ðA14Þ

By evaluation of eqs A9 and A10 at the point ofyield (ry,ky), assuming that at the point of yield itholds that:

! _ep ¼ _e0; _e0 being the applied strain rate! R�cð�cp ¼ 0Þ ¼ 1, thus S ¼ Sa(t,T)! �s >> s0 and therefore sinhð�s=s0Þ � 1

2 expð�s=s0Þ and therefore the viscosity reduces to:

g¼ s0ffiffiffi3

p_e0

ln 2ffiffiffi3

pA0ðSaÞ_e0

� �þ�U

RTþ lp

s0

� �: ðA15Þ

The uniaxial yield stress can be given by:

ry ¼ rsð_e0Þþ rrðkyÞ ðA16Þ

where

rsð_e0Þ ¼ 3s0ffiffiffi3

p þ llnð2

ffiffiffi3

pA0ðSaÞ_e0Þ ðA17Þ

rrðkyÞ ¼ffiffiffi3

pffiffiffi3

p þ lGrðk2y � k�1

y Þ: ðA18Þ

By combining eqs A14 and A16 and substitu-tion of s0 ¼ RT

V� , the zero viscosity can be directlydetermined from the yield stress by:

g0 ¼RT

V�1ffiffiffi3

p 1

_e0sinh

ry � rðkyÞffiffiffi3

p V�

RT

� �ðA19Þ

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