Composite cure kinetic analysis of unsaturated polyester free radical polymerization

Composite Cure Kinetic Analysis of Unsaturated PolyesterFree Radical Polymerization

YOUNG-MIN YUN,1,* SEUNG-JONG LEE,1 KI-JUN LEE,1 YOUNG-KWAN LEE,2 JAE-DO NAM3

1 Department of Chemical Engineering, Seoul National University, Seoul, Korea

2 Department of Chemical Engineering and 3 Polymer Science and Engineering, Sung Kyun Kwan University,300 Chunchun Dong, Jangan-ku, Suwon 400-746, Korea

Received 10 December 1996; revised 15 April 1997; accepted 22 May 1997

ABSTRACT: Curing kinetics of unsaturated polyester resin system exhibiting apparentinduction periods was investigated by modeling free radical initiation and propagationprocesses. The isothermal curing induction time as well as the maximum-rate timeprovided the same activation energy in the Arrhenius relation, and therefore, the iso-thermal curing master curve was constructed by using the reduced time method. Twomodel elementary rate equations for radical and monomer were proposed to describethe free radical polymerization of unsaturated polyester resin systems. The power lawwas adopted to express the conversion dependence function of the initiation efficiencyand the monomer reaction rate. Demonstrating the capability of the developed model,the agreement between experimental and predicted data was excellent in both isother-mal and dynamic-heating conditions, even with the same model parameters in differentthermal conditions. q 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 2447–2456,1997Keywords: unsaturated polyester; free radical polymerization; curing kinetics; induc-tion time; kinetics model

INTRODUCTION molding compound, pultrusion, spray deposition,injection molding, etc. Depending on the natureof processing, various unsaturated polyester resinModeling methodology of thermoset resin curesystems have been developed by the variation ofwas intensively investigated to describe the reac-the type and quantity of individual componentstion progress as a function of time and tempera-generally composed of initiator/catalysis, inhibi-ture in designing and manufacturing thermoset-tor, low profile agent, filler, and base resin.1,2 Mostbased polymers and composites. The accurate con-variations of unsaturated polyester may be aimedtrol of the cure-reaction kinetics, which may beat controlling the gel time and the cure tempera-achieved by appropriate kinetic models, often de-ture, which may simply be termed as cure kineticstermines the process efficiency and product qual-of the resin system.ity in manufacturing and utilization of thermoset-

As with most thermoset polymers, the cure re-ting systems. Unsaturated polyester resin isaction of unsaturated polyester is complicated bywidely used in various composite manufacturingthe nature of various mixtures, often resulting inprocesses such as resin transfer molding, sheetnumerous side reactions. It is further complicatedby the free radical initiation/inhibition reactions

* Currently at KOLON Group Central Research Institute, as well as the physical changes of molecular struc-Yong-In, Korea. ture leading to diffusion-controlled reaction pro-Correspondence to: J.-D. Nam

cesses.3 The curing behavior of unsaturated poly-Journal of Polymer Science: Part B: Polymer Physics, Vol. 35, 2447–2456 (1997)q 1997 John Wiley & Sons, Inc. CCC 0887-6266/97/152447-10 ester was modeled using the concept of free radi-

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2448 YUN ET AL.

cal polymerization.4–7 The approach by the free EXPERIMENTALradical polymerization requires several assump-

A commercially available unsaturated polyestertions and approximations to simplify the physicalresin system supplied by Aekyung Chemical Co.complexity of the formation of three-dimensionalwas used in this study. The base resin systemnetworks and copolymerization reactions of un-(PC-106) was a mixture of linear unsaturatedsaturated polyester and styrene.6 Incorporatingpolyester and vinylester dissolved in styrenethe effect of diffusion of polymeric segments, themonomer in a ratio of 2 parts alkyd to 1 partrate constants in the free radical polymerizationstyrene. As a low profile agent, 33 wt % of polyvi-reactions have been modified by introducing thenylacetate/styrene solution (LP-169) was mixedmolecular parameters of free volume and glasswith PC-106 in a weight ratio of 70/30 for PC-106transition temperature.8–13 Although several ad-to LP-169. The peroxide initiator was tert-butylvantages of these mechanistic kinetic models haveperoxybenzoate (TBPB) and 1 phr of TBPB wasbeen pointed out, the model equations often in-used in this kinetic study. Other proprietary ad-clude many parameters, which must be deter-ditives were cobalt octoate as accelerator andmined with the aid of numerical optimizationmethyl-tertiary butyl quinone and tertiary butylschemes.6,13 When the model parameters are toocatecol as inhibitor. Otherwise stated, the compo-numerous to be redundant in fitting the experi-sition of the resin system investigated in thismental results, however, the characteristic fea-study would be 80/20 for PC-106 to LP-169 withture of the model prediction may be dependent1 phr of TBPB.on the optimization schemes or initial values in

The mixture of PC-106, LP-169 and TBPB wassolving the systems of nonlinear differential equa-stirred for more than 30 min and degassed in ations.vacuum oven at room temperature. DifferentialA semiempirical model, proposed by Kamal etscanning calorimetry (DSC) measurements wereal., has also been used to describe the cure kinet-performed with a TA Instruments DSC 2910 cou-ics of unsaturated polyester systems with a muchpled to TA Instruments 2000 controller. Experi-smaller number of model parameters.14,15 Withments were conducted in a flowing nitrogen envi-such an approach, the simplicity may be very at-ronment (100 mL/min) using liquid samples of 12tractive in the general application of cure kinetics { 1.0 mg size. For isothermal experiments the

modeling. However, it should be mentioned that instrument was preheated to experimental tem-the induction period, which is often observed in perature, ranging from 75 to 1307C. The sampleunsaturated polyester systems resulting from the was then quickly placed in the DSC cell. For dy-reactions of inhibitor and initiator, may not ad- namic heating experiments, several heating ratesmittedly be described by the model. When the in- were investigated for the validity test of the modelduction period is relatively significant in the over- equation and parameters.all reaction processes, for instance, the model ap-plication may be limited only to a certain rangeof temperature or conversion. RESULTS AND DISCUSSION

The primary objective of this study is to developa free radical polymerization kinetics model for Figure 1 shows a typical DSC exotherm measuredcuring reactions of thermoset polymers exhibiting at an isothermal temperature 807C. The timean apparent induction period. The present article needed for the exothermic reactions to occur mayalso discusses the master curve construction of be defined as the induction period, which may bethermoset curing exotherms including the induc- quantitatively obtained by the intersection of thetion periods as well as the maximum reaction DSC base line and the extrapolated line from therates appearing in the middle of curing processes. deflection point of the cure exotherm. As can beThe developed model consists of two rate equa- seen, the exothermic reaction is retarded for abouttions for the lumped radical and monomer concen- 22 min of induction period until the autocatalytic-trations, the parameters of which were deter- type curing exotherm appears. It is noted thatmined by the characteristic feature of the master such a marked induction period should be in-curve and the characteristic reaction times. For cluded in kinetics modeling, even though the ex-this study, a general-purpose unsaturated polyes- otherm is not detected in DSC experiments.ter was used for the DSC kinetic analysis and Table I summarizes the measured induction

times (ti ) , and maximum reaction-rate times (tm )model development.


UNSATURATED POLYESTER FREE RADICAL POLYMERIZATION 2449

Figure 1. Typical DSC thermogram of 1 phr TBPB/Figure 2. Activation energy obtained as 90.6 KJ/molunsaturated polyester system cured isothermally atfrom both tm and ti compared with model prediction.807C.

value of 333.9 { 37 J/g without significant varia-as well as heat of reactions (DH ) measured attion with isothermal temperature. In practice, thevarious isothermal temperatures. The heat of re-given unsaturated polyester system is recom-action has been suggested as being dependent onmended to be cured in the above range of tempera-the isothermal curing temperature because thetures. Consequently, the kinetic analysis of thisresidual ethylenic double bonds remain after iso-study was collectively made for the isothermal ex-thermal curing.3,14,16 When the curing thermalperimental data obtained in the temperaturecondition is changed, the reaction mechanisms ofrange, 757C ° T ° 907C, for the model develop-thermoset polymers are also often altered, re-ment and its verification.sulting in different values of reaction heat as well

As characteristic times of the reaction kinetics,as activation energy.17 In the experimental rangethe induction time (ti ) and the maximum reac-of this study, the heat of reaction is also observedtion-rate time (tm ) were chosen for this study, andto increase with the isothermal temperature.plotted as a function of inverse temperature inHowever, as seen in Table I, when focusing on aFigure 2. It may be reasonable to consider thatspecific temperature range, 757C ° T ° 907C, thethe characteristic times obtained from the experi-heat of reaction exhibits a seemingly constantmental data follow the Arrhenius-type relationproviding the activation energy.17–19 The applica-

Table I. Summary of DSC Isothermal Curing ble Arrhenius-type equation to fit ti and tm mayResults for 1 phr TBPB/Unsaturated be written asPolyester Resin System

tm Å tmoexp(E /RT ) (1)T DH tl tm (da/dt)max

(7C) (J/g) (min) (min) (min01)where E is the activation energy, R is the gasconstant, and T is temperature. As seen in Figure75 333.5 33.96 53.69 0.03367

80 296.9 21.94 35.98 0.05148 2, linear relations are found for both times provid-84 333.0 14.60 25.61 0.07266 ing the same slope, which, consequently, gives the87 336.2 11.63 19.03 0.10267 apparent activation energy of the model unsatu-90 370.0 8.13 14.95 0.11984 rated polyester system. The obtained activation

100 404.6 3.10 6.54 0.23076 energy for both tm and ti is 90.6 KJ/mol, and tmo110 424.5 2.16 2.76 0.91610 for tm is 1.35 1 10012 min. The reasoning for eq.120 420.3 1.14 1.28 2.56908 (1) will be discussed later.


2450 YUN ET AL.

cals may be expressed by the generation of thejth initiated particle I*j from the initiator Ij , andthe subsequent reaction of I*j with the polyester(E ) and styrene (S ) resulting in their radicals E*and S*. The initiator radicals I*j are inactivatedby the reactions with inhibitors and, also, withother radicals. The styrene monomer usually hasgreater mobility than the unsaturated polyesterchain and thus can be homopolymerized. Overall,the resulting elementary reactions for the unsatu-rated polyester radical polymerization are so com-plex that several assumptions have been made tosimplify the kinetic expressions.6,13 The assump-tions used in this study are the lumped concentra-tions: (1) the total amount of monomer M may beexpressed as the sum of E , and S ; (2) the totalamount of radical M* may be expressed as thesum of E* and S*.

Using the lumped concentrations, various ki-Figure 3. Comparison of experimental results super- netic model equations have been proposed, for in-imposed by reduced time tr Å t /tm and computed curing

stance, by different variations in initiation effi-master curve denoted by eqs. (14) and (15) in text.ciency, termination reaction, and diffusion-con-trolled reaction constants.4,6,13 In modeling of theunsaturated polyester system, however, the ele-Using the determined activation energy from

tm and ti , the reaction exotherms at different tem- mentary reaction rate for the monomer does notseem to have critical disagreement among re-peratures may be superimposed, as shown in Fig-

ure 3. For the superposition, the experimental searchers because it is considered reasonable toassume that the monomer reaction rate is propor-time is divided by tm providing the da /dtr vs. tr

plot, where the reduced time is defined as tr Å t / tional to the concentration of monomers and radi-cals.6,13,20,21 Using the first-order kinetic expres-tm .17 The superposed experimental result demon-

strates that there is only one characteristic quan- sion for the monomer and radical concentrations,the monomer rate expression has been written astity that determines the temperature dependence

of unsaturated polyester curing reactions, whichmay subsequently be ascribed to the activation 0 dM

dtÅ kmMM* (2)energy. This experimental observation may also

be supported by the same value of activation en-ergy obtained from both tm and ti . The rate expressions for initiators, inhibitors,

and radicals, however, have been modified by var-ious approximations. The most significant modi-Elementary Reaction Expressions of Unsaturatedfication may be the diffusion-controlling effects.Polyester SystemFor example, the influence of diffusion-controlled

The unsaturated polyester resins are cured by reaction rate has been introduced empirically asadding and activating the free radical initiators a conversion-dependent factor into the reactionto start the chain reaction of polymerization. The rate constants.6,13 The proposed expression for thederived free radicals activate the reactive carbon propagation as well as termination reaction is ex-double bonds and styrene monomers for chain pressed as a function of conversion, viz.,propagation reactions. Typically, the decomposedinitiator radicals are inactivated by adding inhibi-

k Å koexp(0E /RT )S M 0 M`

M0 0 M`Dn

(3)tors in most utilization of unsaturated polyesterresin systems. The major hypothetical elementaryreactions of the curing processes have been sum-marized by more than 15 reactions including vari- where ko and n are constant, E is the activation

energy, and M , Mo , and M` are transient, initial,ous initiation, inhibition, propagation, and termi-nation reactions.6 The initiation of the free radi- and final concentrations of monomer, respec-



tively. The power-law expression of the concentra- The conversion-dependent lumped efficiencyfunction, f (a ) , leads the above relation to repre-tions in the above equation is often referred to as

conversion. sent an apparent reaction rate of radicals. In thisapproach, the rate equations for initiation, inhibi-Another modification in the free radical poly-

merization may be the initiation efficiency factor. tion, and termination are implicitly lumped intoan appropriate form of the efficiency function. TheThe initiation decomposition rate is often ex-

pressed by using the efficiency factor f , which may efficiency function, of course, should be validatedby the capability of fitting the experimental re-be defined as the fraction of initiator radicals

formed from initiator to those initiating polymer- sults, especially in the early stage of the curingprocess exhibited by the induction period.ization. It is often referred to account for decompo-

sition products that recombine to give initiatoragain and/or for the mutual collision and trans-

Composite Kinetic Methodologyport probability of activated initiators.4,20–22 How-ever, the efficiency factor is a very elusive quan- When several elementary reactions occur in atity and little discussed in the open literature; complex way, the conversion dependent functions,thus, it may be accompanied by exponents or em- elementary reaction rates, or elementary conver-pirical expressions as a function of initiator and sions have successfully been expressed by themonomer concentrations.4,13,21 Several expres- composite kinetic methodology.17–19 In this ap-sions for the efficiency functions, for instance, proach, the weighting factors are defined eitherhave admittedly been suggested, accounting for by the overall material balance or the importancethe entrapped initiator radicals.6,13 Consequently, of elementary reactions to describe the overall re-the efficiency factor may reasonably be expressed action behavior in a continuous fashion. Taking aby an appropriate function of conversion (a ) , viz., similar approach of the composite methodology for

modeling free radical polymerization, it may bef Å f (a ) (4) assumed that the lumped monomers exist in three

distinct forms, of which the sum is conserved byAs previously discussed, the rate expressionsthe material balance: the unreacted monomer M ,of the unsaturated polyester system often includethe radical-state monomer M*, and the monomerempirical modifications as the reaction constantin the polymerized state { (M{M )n{. Fromand efficiency factor. Such a model modificationthe quantities of these monomeric forms, the con-usually increases the number of kinetic parame-versions of monomer to the polymer (a ) and toters, which may undesirably become redundantthe radical (a*) may be defined as follows:in fitting the experimental data. It should be

noted that the redundant model parameters tendto lead the model prediction to be dependent on

a Å Mo 0 M 0 M*Mo

(6)the optimization scheme or initial guessing of sys-tems of nonlinear differential equations. There-fore, it seems most desirable to minimize the num- a* Å M*

Mo(7)

ber of model parameters in the model-develop-ment stage.

According to the material balance, the totalAccordingly, in this study, it is considered de-conversion, or the conversion of monomers to ei-sirable, for example, to lump such complex andther radicals or polymer is the sum of a and a*.elusive elementary reactions of inhibitor and ini-Using these definitions, the monomer and the rad-tiator to the rate expression of radicals in a semi-ical concentration may be represented by the cor-empirical way. Instead of using such redundantresponding conversions as M /Mo Å 1 0 a 0 a*model equations and parameters, an appropriateand M*/Mo Å a*.form of initiation efficiency function (4) may be

As discussed earlier, an appropriate form of theintroduced in a semiempirical fashion to the rateefficiency function in eq. (5) is needed to describeexpression of radicals. In this study, the rate ex-the apparent rate of radical reaction, which obvi-pression of radical is assumed to be proportionalously results from complex reacting characteris-to the concentration of monomer, and also to thetics of initiation, inhibition, termination, etc. Inconversion-dependent efficiency function, viz.,this study, the efficiency function is assumed tobe represented by the power-law to the conver-dM*

dtÅ ki f (a )M (5)

sion, f (a ) Å am . The power law expression is con-


2452 YUN ET AL.

sidered as one of the most general expressions in conversions to the radical (a*i ) corresponding todescribing the characteristic features of various M*i may be expressed asreaction mechanisms. For example, the conver-sion rate exhibits a convex or concave shape as afunction of time for nú 1 and nõ 1, respectively,

a ÅMo 0 M 0 (

N

iÅ1M*i

Mo(10)each representing decelerating and accelerating

rate characteristics. Using the conversions de-fined in eqs. (6) and (7), and the power-law ex-

a*1 ÅM*1Mo

, a*2 ÅM*2Mo

, . . . , a*N ÅM*NMo

(11)pression for the efficiency function, the rate equa-tion (5) may be rewritten as

Subsequently, the rate expressions for a anda*i may be assumed as similar forms of rate ex-da*

dtÅ kra

m(1 0 a 0 a*) (8)pressions shown in eqs. (8) and (9), respectively.Additional elementary reactions for a*i may be

where m is the power-law index and kr Å kro used to describe more complex reacting systems,exp(0E /RT ). for instance, when several activation energies or

Similarly, using the analogy to the elementary their modifications are needed to construct therate expression (2), the conversion rate of mono- master curve.17,19 So far as an optimum numbermer to polymer may be described. In eq. (2), the of model parameter is concerned, it should bemonomer reaction rate is admittedly expressed mentioned that the additional rate expressions forby the first power of the radical concentration. radicals would increase the number of model pa-However, as with eq. (8), it may also be a more rameters.general way of expression to use the power-lawfor the radical concentration, thus broadening the

Model Parameter Investigationmodel applicability for more complex reacting sys-tems. Then, it may be assumed that the conver- The reduced master curve, constructed by tr , maysion rate of monomer to polymer or the curing be expressed by the model eqs. (8) and (9) byrate of monomer takes a similar form of (2) with multiplying eq. (1) for tm on both sides of the equa-a power-law index (n ) to the radical concentrations. Comparing the Arrhenius-type form of tmtion, viz: in eq. (1) and also the Arrhenius reaction con-

stants in eqs. (8) and (9), it can be noticed thatthe temperature-dependent exponential termsda

dtÅ kpa*n (1 0 a 0 a*) (9)

are eliminated. Accordingly, the resulting equa-tions, which can be used to fit the experimentalmaster curve, are expressed as a function of thewhere kp Å kpoexp(0E /RT ) .reduced time, tr Å t /tm , viz:The resulting model eqs. (8) and (9) can be

solved numerically with the initial values of ao

and a*o , which may physically be interpreted as da*dtr

Å Cram(1 0 a 0 a*) (12)initial concentrations of monomer and radical, re-

spectively. It has been pointed out that the heatof reaction from the decomposition reaction of per- da

dtrÅ Cpa*n (1 0 a 0 a*) (13)

oxide initiator is negligible to the heat liberatedfrom the chain propagation reactions.23 Thus, theheat of reaction detected from DSC may be as- where Cr and Cp are temperature-independent

constants: Cr Å krotmo and Cp Å kpotmo . These twosumed to be ascribed only to the conversion a de-fined in eq. (6). model equations may be used to describe the tem-

perature-independent master curve using theFinally, the modeling methodology expressedin eqs. (6) through (9) may be generalized, for model parameters: m , n , Cr , Cp , and a*o . The gov-

erning equations include only five model parame-instance, by differentiating specific radicals fromthe lumped radical concentrations, M*. Assuming ters and therefore the trial-and-error method is

likely to be the most efficient way of fitting thethat N distinct radicals, M*1 , M*2 , M*3 , . . . ,M*N , are produced at distinct reaction rates, the cure master curve. In this study, model eqs. (8) –

(9) for various thermal conditions and eqs. (12) –monomer conversion to the polymer (a ) and the



(13) for the master curve were solved numerically that Cp /Cr can be used to adjust (tr )max to beplaced at 1.0. Finally, the initial concentration ofby the fourth-order Runge–Kutta method.

The master curve has several characteristic the radicals, a*o , exhibits a parallel shift of themaster curve with respect to tr without anyfeatures to be described by the model equations

and their parameters. For example, the maximum changes of master curve shape, as can be seen inFigure 5.reaction rate (da /dtr )max should appear at tr Å 1,

and the experimental values of the maximum re-action rate and the induction period of the model

Model Application to Unsaturatedunsaturated polyester system are 1.88 and 0.55,Polyester Systemrespectively, as can be seen in Figure 3. In addi-

tion, the model parameters should fit the unique By constructing the master curve with a singleactivation energy, it can be reasonably assumedshape of the superposed reaction exotherms. To

fit these characteristic features of the master that the activation energies in eqs. (8) and (9)are 90.6 KJ/mol for both equations with differentcurve, at least five model parameters may be con-

sidered reasonable, and thus have been thor- values of preexponential factor. The reasoning isbased on the hypothesis that the activation ener-oughly investigated.

In the parameter investigation, it has been gies used in the reaction constants of the modelequations would result in tm complying with thefound that the reaction orders of m and n seem to

determine the overall shape of the master curve, Arrhenius equation. However, it should bepointed out that the model equations have notspecifically the broadness and the height of the

exotherm master curve. When the reaction order been verified yet to exhibit the characteristictimes of tm and ti providing the same activationm and n are increased, the value of (da /dtr )max is

decreased and (tr )max is increased, resulting in a energies as used in the model equations. To vali-date this hypothesis, the model equation wasbroad shape of model prediction.24 Accordingly,

the reaction orders may be used to describe the solved with the activation energy, 90.6 KJ/mol,for the reaction constants kp and kr . From theoverall shape of the master curve and simultane-

ously to adjust the master curve maximum to be calculation, tm was obtained at different isother-mal temperatures. As can be seen in Figure 2, theplaced at (tr )max Å 1.

Subsequently, the Cp and Cr may be used to calculated activation energies compare very wellwith the experimental results, demonstrating theadjust the maximum reaction rate at 1.88, while

keeping the time for the master curve maximum validity of the hypothesis used in this study.Fitting the superposed master curve data, theto be placed at (tr )max Å 1.0. It is interesting to

note that the ratio (Cp /Cr ) and the multiplication model parameters were determined and summa-rized in Table III. The accuracy of the model is(CpCr ) of these two parameters have distinct char-

acteristics that may be utilized for this purpose. illustrated in Figure 3. As can be seen, the charac-teristic shape of the master curve and the induc-Table II summarizes the model parameters inves-

tigated in this study. As can be seen in Figure tion period are expressed by the developed modelequations with excellent accuracy. Only five4(a), the variation of CpCr changes the value of

(da /dtr )max with small changes of (tr )max . Along model parameters have been used in this model-ing to describe the specific quantities of the exper-with little variation of CpCr , Figure 4(b) shows

Table II. Model Parameters Investigated in Figures 4 and 5

Curvea Cp Cr CpCr Cp/Cr a*0

C1 2.0 1 105 2.10 1 1003 4.20 1 102 9.52 1 107 1.0 1 1007

C2 1.0 1 105 1.05 1 1003 1.05 1 102 9.52 1 107 1.0 1 1007

C3 5.0 1 104 5.25 1 1004 26.25 9.52 1 107 1.0 1 1007

C4 1.0 1 104 1.05 1 1002 1.05 1 102 9.52 1 105 1.0 1 1007

C5 1.0 1 106 1.05 1 1004 1.05 1 102 9.52 1 109 1.0 1 1007

C6 1.0 1 105 1.05 1 1003 1.05 1 102 9.52 1 107 1.0 1 1006

C7 1.0 1 105 1.05 1 1003 1.05 1 102 9.52 1 107 1.0 1 1008

a Reaction orders m Å 0.8 and n Å 1.1.


2454 YUN ET AL.

mol, as determined in the master curve construc-tion and validated by the model equations. In Fig-ure 6, the model equations are compared with theisothermal curing experiments at 75, 80, 83, 87,and 907C. Table III gives the summary of themodel parameters used for Figure 6. As can beseen, the maximum reaction rate, maximum reac-tion time, and induction period are very well pre-dicted by the model equations, demonstrating thevalidity of the modeling methodology. It shouldbe mentioned that all the model parameters usedin this study are independent of experimentaltemperatures, which is not often the cases in mostthermoset kinetics modeling.

The conversion of monomer to radical, definedas a*, is shown in Figure 7, calculated for theexperimental conditions in Figure 6 and Table III.Corresponding to the lumped radical concentra-tion of monomer, a* maintains a low value of itsinitial concentration, especially during the induc-tion period of the unsaturated polyester cure. Inthe induction period of isothermal holding, theradicals are inactivated primarily by the inhibi-tor; therefore, the radical concentration is likelyto remain low. Comparing Figures 6 and 7, it canbe seen that a* increases after a certain period ofinduction time and, subsequently, accelerates thepolymerization reaction, seemingly because mostof the included inhibitor is finally consumed bythe reactions with radicals. This result seems toagree with the predicted monomer concentrationby Han et al.,13 and also with the physical inter-

Figure 4. Effect of parameter (a) CpCr and (b) Cp /Cr

on prediction of da /dt , where plot is based on Table IIin text.

imental master curve, for example, (tr )max Å 1,(da /dtr )max Å 1.88, and (tr )i Å 0.55 as well asthe broadness and skew symmetry of the mastercurve.

After fitting the master curve, the isothermalexotherms at different temperatures can be pre-dicted as a function of time by solving model eqs.(8) and (9). The preexponential factors kpo andkro can be calculated by Cp , Cr , and tmo , whichhave already been determined by fitting the mas-ter curve exotherm. The activation energy used Figure 5. Effect of parameter a*o on prediction of da /

dt , where plot is based on Table II in text.for the reaction constants, kp and kr , is 90.6 KJ/



Table III. Model Parameters Used for Predicting 1 phr TBPB/UnsaturatedPolyester Resin System Curing under Isothermal and Dynamic-Heating Conditions

E (KJ/Mol) m n kp0 (min01) kr0 (min01) a*0

90.6 0.8 1.1 3.21 1 1016 3.37 1 108 1.0 1 1007

pretation of unsaturated polyester free radical po- CONCLUSIONSlymerization.

To validate the developed kinetic model in dif- Cure reaction kinetics of free radical polymeriza-ferent thermal conditions, the model equations tion was investigated for the general-purpose un-were tested for dynamic-heating DSC experi- saturated polyester system. Constructing thements.17–19 Figure 8 compares the dynamic exo- master curve of the reaction exotherms, a singletherm measured at a constant heating rate of 27C/ activation energy of the reacting system was de-min with model prediction. For comparison, the termined as 90.6 KJ/mol for both tm and ti . Usingsame model equations and parameters were used the power-law equation for the initiation effi-as with the isothermal prediction in Figure 6. It ciency function, two elementary model equationscan be seen that agreement between the two is were proposed for the radical and monomer reac-very good, and the predicted cure exotherm peak tion rates. The developed model equations pre-appears slightly earlier in dynamic heating, say dicted isothermal as well as dynamic-heating cur-by ca. 57C. The exothermic reaction of unsatu- ing reactions very well with only five model pa-rated polyester systems often exhibit an explosive rameters without any modification in differentreaction rate in DSC dynamic-heating experi- thermal conditions.ments, resulting in a very sharp peak of reactionexotherm. Considering the narrow range of reac-

This work was supported by the Ministry of Educationtion temperature and the high magnitude of theResearch Fund for Advanced Materials in 1995. Sup-exothermic peak in dynamic-heating data, theport for this work was also provided by Hanhwa Co.model prediction of the dynamic-heating condi-through project and equipment support to Seoul Na-tion is considered excellent and accurate enoughtional University. The authors wish to express theirfor most practical applications.appreciation to D. Kamm of T. A. Instruments and Dr.

Figure 6. Comparison of model prediction with ex-perimental results at various isothermal cure tempera- Figure 7. Predicted profiles of a* at various isother-

mal cure temperatures.tures.


2456 YUN ET AL.

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17. J.-D. Nam and J. C. Seferis, J. Appl. Polym. Sci.,Bong-Kyu Jo and Dr. Tae-Hoon Kwak of Hanhwa Co.50, 1555 (1993).for instrument support and for helpful discussions.

18. J.-D. Nam and J. C. Seferis, Polym. Sci. Polym.,Phys. Ed., 29, 601 (1991).

19. J.-D. Nam and J. C. Seferis, Polym. Sci. Polym.,Phys. Ed., 30, 455 (1992).

REFERENCES AND NOTES 20. R. M. Noyes, in Encyclopedia of Polymer Scienceand Technology, vol. 2, H. F. Mark, N. G. Gaylord,and N. M. Bikales, Eds., Wiley–Interscience, New

1. R. B. Prime, in Thermal Characterization of Poly- York, 1965, p. 796.meric Materials, E. A. Turi, Ed., Academic Press, 21. E. A. Grulke, Polymer Process Engineering, Pren-New York, 1981. tice Hall, Englewood Cliffs, NJ, 1994.

2. J. A. Brydson, Plastics Materials, 5th ed., Butter- 22. M. P. Stevens, Polymer Chemistry, 2nd ed., Oxfordworths, London, 1989. University Press, New York, 1990.

3. K. Horie, I. Mita, and H. Kambe, J. Polym. Sci. A- 23. H. Kubota, J. Appl. Polym. Sci., 19, 2279 (1975).1, 8, 2839 (1970). 24. Y. M. Yun, K. J. Lee, K. S. Oh, and J.-D. Nam,

Hwahak Konghak, to appear.4. J. A. Bisenberger and D. H. Sevastian, Principles


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