A new approach to modeling the cure kinetics of epoxy/amine thermosetting resins. 2. Application to a typical system based on bis[4-(diglycidylamino)phenyl]methane and bis(4-aminophenyl) sulfone

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Macromolecules, 24, 11, pp. 3098-3110, 1991-05-01

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A new approach to modeling the cure kinetics of epoxy/amine

thermosetting resins. 2. Application to a typical system based on

bis[4-(diglycidylamino)phenyl]methane and bis(4-aminophenyl) sulfone

Cole, K. C.; Hechler, J. J.; Noel, D.

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https://nrc-publications.canada.ca/eng/view/object/?id=7b54d765-52b1-4e33-bd84-e08028aec153 https://publications-cnrc.canada.ca/fra/voir/objet/?id=7b54d765-52b1-4e33-bd84-e08028aec153

1991, 24, 3098-3110

A New Approach

to

_{Modeling the}

Cure

Kinetics

of

Epoxy Amine

Thermosetting

Resins.

2.

_Application

to

a

_Typical

_System

Based

on

Bis

_[4-

_{(diglycidylamino)phenyl]}

methane and

Bis(4-aminophenyl)

Sulfone

K.

C. Cole,*J.-J. Hechler, and D. Noel

Industrial MaterialsResearch_{Institute, National}Research _{Council of}Canada,

75boulevarddeMortagne, Boucherville,Québec, CanadaJ4B6Y4 Received_July31,1990;RevisedManuscriptReceivedNovember9, 1990

ABSTRACT: Differentialscanningcalorimetry in dynamicandisothermalmodeswasused_{to study the}cure

kinetics ofthe_{commercial epoxy system Narmco}5208, whosemain componentsare

bis[4-(diglycidylamino)-phenyljmethaneand bis(4-aminophenyl) sulfone. Thedatawere analyzedintermsofa new mechanistic approach describedinthe preceding paper. The treatment explicitlytakesintoaccountboththe

epoxide-aminereactions andthe subsequentetherification reaction. The kineticscan be_completelydescribedin

terms ofthreerateconstants, whichobeytheArrhenius relationship. Excellentagreementwiththe experimental

dataisobtainedif theetherificationreactionisassumedtobefirstorderwithrespecttothe concentrations

ofepoxide groups,hydroxylgroups, and thetertiaryamine groups formedintheepoxide-aminereaction.

This_{model applies}over the whole_rangeofconversion uptothepointwherethe resinvitrifiesandthe reaction

becomesdiffusion-controlled. Theeffectofthediffusion controlis_{described very well by}an _approachbased on_{simple equations proposed}inthe_{literature. Altogether,}the model allows accurateprediction ofthe degree

ofconversionover the_{whole range}ofcure andover _{the temperature}range160-200 °C,whichcovers the usual curing conditions. Although the rate constants derived are _{specific to Narmco} 5208, the modelitself is

generally applicabletootherepoxy aminesystems.

Introduction

Background.

Mathematical modelingofthe_forming

process in composite materials can lead to significant

savingswithrespectto the amountofexperimental work

involved_{in determining the optimumprocessing} condi-tionsfora_{particularpart.1} For thermoset matricessuch

as _{epoxy resins, probably the most important aspect} of

the model is an accurate_description of_the polymeriza-tionreactionkinetics. Becauseof_{the complexity} ofthe chemistry involved, existingmodelsare _usually

semiem-piricalapproximations, which workwithvaryingdegrees ofsuccess. The work tobedescribed hereinwas under-takenwiththe aimofdevelopinganaccuratekineticmodel

forthe commercial_productNarmco_Rigidite 5208.

Be-cause the existingmodelsthatwere tried were found to worklesswell thandesired,a new mechanistic approach

was _developedas described in_{the precedingpaper.2} In

thispaper,wedescribe_{its application to}theNarmco5208

system and demonstrate that it provides an excellent

description ofthe curing kinetics.

Narmco 5208. Narmco _Rigidite 5208, produced by BASF Narmco Division, is reported to consist ofthree

components.3 The mainone isan _{epoxy resin based}on

tetraglycidyldiaminodiphenylmethane(TGDDM);a

well-known commercial product of this _type is _{Ciba-Geigy’s} AralditeMY 720. Thesecond_componentistheprimary

amine curingagentdiaminodiphenylsulfone (DDS),also

available_{from Ciba-Geigy} as Hardener HT 976. These two monomers also form the basis of several other commercial products. The third componentofNarmco

5208is an _epoxy resinbasedon a _{bisphenol A Novolac;}

sucha _productissoldbyCelaneseunder thename

Epi-Rez SU-8.

PreviousWork. Thegeneralnatureofthe_{epoxy amine}

reactionandsome well-known equationsusedtodescribe

the kineticshave been discussed_{in part}1 ofthisseries.2

*_{Author towhom correspondence shouldbeaddressed.}

Barton4hasreviewedmuchoftheliterature. Therehave beennumerous studiesofmixturesof TGDDMandDDS

andofcommercial systemscontainingtheseproducts.5*21

These have involved different_approaches to describing

the kinetics,with_varyingdegreesofsuccess. Insome cases, a_goodfit to experimental datawas obtainedinthe early

stagesofthereaction,butdeviationswere observedinthe later _stages. In other cases, reasonably good overall

agreementwas obtainedbutonlybytakinga

phenome-nological approach in which empirical parameters are introduced and varied to

fit

the data.

It

_appears to be

well establishedthattheetherificationreactionis signif-icantandcannotbeignored.4 Onequestionthathasnot

been_{satisfactorily}resolved,however,isthatoftherelative reactivitiesofthe_primaryaminegroups

initially

present

andthe_secondaryaminegroupsformedinthe reaction.

While it is _frequently stated that the _secondary amine

groupsare muchslowertoreact,theevidencefor thisis

not entirely convincing. Gupta and co-workers8 used

differentialscanningcalorimetry(DSC),Fourier transform infraredspectroscopy(FT-IR),and_{electron spin}resonance (ESR) to study mixtures involvingTGDDM,DDS, and

thesecondaryamineDMDDS

(bis[4-(methylamino)phe-nyl]sulfone). They concludedthatin the earlystagesof

cure themain reactionisbetweenprimaryamineand

ep-oxide(PA-E), thatthereactionbetween secondary amine

and epoxide (SA-E) is negligible, and that the cure is

completedbythehydroxyl-epoxide(OH-E)reaction.The rate constant forthelatterwas estimated tobean order

ofmagnitude smaller than thePA-E rate constant, and the enthalpy ofreactionabout eight timesless. _However,

Barton17 has _{provided conflicting data showing} that

TGDDM-DDSmixtureswith_{amine/epoxide ratios}

rang-ing from0.61to1.14havea_{constant enthalpy}ofreaction

ofabout110

_kJ/mol

ofepoxide. Thissuggeststhatboth

primaryand _{secondary amine groups} react completely

with_epoxideand that residual _epoxideis consumed by

the OH-Ereaction, all these _{reactions having}

approxi-mately equal exothermicity. The calorimetric data of

0024-9297_{/91/2224-3098$02.50/0} Published 1991 _{by the American Chemical Society} Downloaded via NATL RESEARCH COUNCIL CANADA on September 11, 2018 at 20:39:54 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

Macromolecules, Vol. 24,No. 11, 1991 Modeling the Cure Epoxy

Gupta et al. _mayalso be interpreted in this way.

Fur-thermore,theirresults concerning theratioofthesulfone

peaksat1150and1105 cm"1intheIRspectrum mayalso

be_{subject to other interpretations. Morgan}andothers9"11

have alsostudied themechanismofTGDDM-DDScuring

by means ofFT-IR _{spectroscopy.} Their dataindicated

thatthe epoxidehomopolymerization reactionissome 200

timesslowerthan thePA-Ereaction. TheSA-Ereaction

was assumedtobeaboutten times_slower,on thebasisof

data_{obtained by}Bell22 forthe reactionof DGEBA

(di-glycidyl etherofbisphenol A)with MDA (methylenedi-aniline). Similarly,theOH-Ereactionwas also assumed

tobetentimes slower,on thebasisoftheabove-mentioned

work of_{Gupta et}al.7-8 While these_{assumptions appear}

to be in agreement with their results, this does not

constituteproof that theyare _valid,sincethe data _may notbe_sufficientlysensitive. Infact,some of theirdata11

suggestthattheSA-Ereactionisrather significant, and

therecentwork_{by Zukas et}al.12indicatesthatthe_primary

and secondary amine groups have approximatelyequal

reactivity. Thisimportantquestionwillonlyberesolved

by further experimental work, preferably involving a techniquethatcan _{directly quantify}thedifferent_types

ofaminegroups.

Publisheddata for Narmco5208are ratherlimited.

Ciz-mecioglu and Gupta6studiedthe kinetics by dynamicDSC

and analyzed the results in terms ofa _single nth-order

reaction. This isan _{oversimplification,}since

it

hasbeen

clearly establishedthatthe epoxy-amine reaction is

au-tocatalyticandthat etherificationcan alsobe_important.

Recently,Chiao21has_analyzedtheTGDDM-DDSdata

of Mijovic et al.15 in terms of a mechanistic model incorporating an _{epoxide-hydroxyl} reaction in addition to the epoxide-amine reactions. The kineticequations were not solved_{analytically,}so anumericalsolutionwas

usedto

fit

thedata,andgoodagreementwas obtained.No assumptionwas madeconcerning the relative reactivity

ofthe secondary andprimaryamine_{groups. However,}in

orderto

limit

thenumberof_parameterstobefittedtoa

reasonable _{number, the ratios}ofthe rate constants for

catalysisbyhydroxylgroupsformed in the reactionand

forcatalysis byimpurities

initially

presentwere fixed at

values_{obtained by analyzing earlier}data23forthereaction

between _{phenyl glycidyl} ether and diethylamine or

n-butylamine. The apparent activationenergiesobtained

forthePA-E,SA-E,andOH-Ereactionswere _respectively

55.1, 71.6,and 97.2_kJ/mol.

Experimental

Section

Materials. In addition tothe composite Narmco5208,some

mixturesoftheindividualcomponents knowntobe_presentin this system were _{studied. Samples} of Araldite MY 720 and

HardenerHT976were obtainedfromCiba-Geigy andEpi-Rez SU-8 from Celanese. The epoxide content ofthe resins was

determinedby reacting the resinwithexcess hydrochloricacid

in acetone/water solutionandthen_titratingtheexcessacidwith

sodiumhydroxide. Theequivalentweights obtainedwere 130.2

(

= _2.1%)forMY _720and227.3

(

= _{0.4%)forSU-8. These}

are in_goodagreementwiththe_typicalvaluesreportedby the

manufacturersforthesematerials(117-134forMY720,215for

SU-8). FortheHardener HT976,whichisnear 100% pure, the

equivalentweightwas calculated tobe62.1. Thisis basedonthe assumptionthateachofthefourN-H bonds presentcan react

withan _{epoxidegroup.}

The_prepregstudiedwas _{Narmco Rigidite5208/WC3000-42}

fromNarmcoMaterialsInc. It consistsofwoven carbon-fiber reinforcement_impregnatedwithNarmco5208resintoanominal

resincontentof42 %by weight. Therecommendedcure schedule

involvesa 1-hdwellat135°C_{followed by}a2-hcure at177°C. Thevolatiles_{content, measured by weighing samples before and}

afterheatingat177°Cfor20min,wasfoundtobe 1.90±0.03 %, basedon total_{prepreg weight.}

The_prepregasreceivedispartlypolymerized,or _{“B-staged".} Aspointed outby Roberts,1foraccurate modelingofthecure it isessentialtocharacterize theinitialchemical stateoftheresin,

i.e.,itscomposition andinitialdegreeofadvancement (polym-erization). Thiswas donebyreverse-phaseliquid

chromatog-raphy.24 Different mixtures of MY720,SU-8, andHT976were

preparedinacetonesolutionandtheacetonewas removedunder

vacuum withoutheating. Sampleswere _{then polymerized to two}

differentdegreesofadvancement by heating at120°Cfor30and

80min. _AnalysisbyRPLC_{gave chromatograms}in whichthe

areas ofthefollowing fourpeaks couldbemeasured: TGDDM

(main component of MY 720), DDS (HT 976), DGEBA = diglycidyletherof_bisphenolA_{(major component}ofSU-8), and theinitialreactionproductbetweenTGDDMandDDS. These

quantitieswere then related to themixture composition and

degreeofadvancement. On analyzing the Narmco5208resin and comparing its chromatogram to that of the calibration mixtures, itscompositionwas estimatedtobe65.9% by weight

MY720,13.2% SU-8, and20.9% HT976,andtheinitial_degree

of polymerizationa0in theprepregasreceivedwas 0.038. This

composition correspondstoan initial_{amine-to-epoxide}ratioB

equal to0.60, basedon the epoxide_{equivalentweightsreported} above.

Somebinaryandternary mixtures of MY720,SU-8, andHT

976were also_{studied by}DSC. Thesewere preparedby dissolving precisely weighedquantities ofthematerialsinacetone,pouring

thesolutionintoashallowaluminumdish, and evaporatingthe

acetoneundervacuum without_heating.

Thermal Analysis. The_prepregcure was studied_byboth

dynamicand isothermalDSC4 on a Setaram Model CDP111

calorimeter. Dynamicruns were doneonbothpreparedmixtures

and prepreg. Approximately50mgofsamplewas placedinan

open platinum crucible that was suspended vertically in the

calorimeter. An empty cruciblewas usedasthereference. The

samplewas heatedina_{nitrogenatmosphere}atafixedrateof

5 _°C/minfromroom _{temperature to}around300 °C,atwhich pointsome _{decomposition}isobserved. Inthecase of_prepreg

samples, the exactweight ofresin presentafterthe DSCrun was

determined_bythermogravimetricanalysis(TGA)on aSetaram B70_{thermobalance equipped}witha_{graphite resistance furnace.}

The samplewas heatedinan _{oxygen atmosphere}at10_°C/min

fromroom _{temperature to}9006C.26

IsothermalDSCruns were doneon _prepregonly,at temper-atures rangingfrom160to210°C. Forthese experimentsthe

samplesizewasaround100 mgand unsealedaluminumcrucibles

were used. Inorderto minimizethe_{time required for}

equili-bration after _{the sample}is introducedinto thecalorimeter, a

warmingoven was addedto preheatboth_{sample and reference}

for about 10min at125°C before introductionintothemain

oven. RPLC_analysisindicatedthatlessthan0.5% ofthe

ep-oxide_{groupsreact during this treatment.} Thereference consisted

ofacruciblecontainingfullypolymerized prepreg, and sample andreferenceweightsaswellas_{the corresponding crucible weights} were matchedas_closelyas_possible. The calorimetersignal (heat

flux=

dQ/dt)was measuredforabout6-7h, exceptat210°C where1.5-2hwassufficient. Aftereachisothermal run,adynamic

run was _performedon thesame _{sample to complete the}

polym-erizationanddeterminetheresidual heatofcuring. Threeruns were doneateachtemperature.

Thedigitizeddatawere _acquiredbyaHewlett-Packard

HP-85_{computer and subsequently}transferred toaVAX_11/750or an _{Apple Macintosh for further treatment.}

In isothermal DSC, there is a brief_{period oftemperature}

equilibration afterthe sampleisintroducedintotheinstrument.

Inorder to estimate thetime requiredby thecalorimeter togive a_{meaningfulsignal,}a_{few samples}were run _{twice, the}firstrun

givingtheraw polymerizationcurve,thesecond_givingthe

cor-respondingexperimentalbaseline, whichisthenusedtocorrect the raw curve.4 It was foundthat the_shapes ofthe raw and

corrected curves are identical after about 2 minandare _only

slightly different before the point where _{the signal} becomes

exothermic. Since thetime requiredtoobtaintheexperimental

baselinecan bequitehigh (up to 6-7 h), theextrapolationmethod

Cole Macromolecules, Vol. 24, 11, 1991

Table I

DynamicDSCData for theMixtures InvolvingMY 720,HT 976,and SU-8

mixture MY720, % HT976, % SU-8,%

amine-to-epoxideratio 7W,eC _Mí,aJ/g

AH,kJ/mol ofepoxide 1 77.97 22.03 0 0.59 252 643.7(5.4) 107.5 2 77.02 22.98 0 0.63 250 637.9 (3.9) 107.8 3 73.17 26.83 0 0.77 234 605.2(5.9) 107.7 4 68.99 31.01 0 0.94 220 563.3(3.7) 106.3 5 67.58 32.42 0 1.01 218 556.3(4.5) 107.2 6 77.78 13.17 9.05 0.33 267 657.5 (4.8) 103.2 7 68.92 22.05 9.03 0.62 250 595.7(4.1) 104.7 8 59.49 31.54 8.97 1.02 217 533.3(4.6) 107.4 9 68.80 13.13 18.08 0.35 268 623.5 (3.0) 102.6 10 59.73 22.27 18.00 0.67 248 565.6(7.8) 105.2 • _{Valuesinparentheses}

correspond tostandard deviationsbasedon threedeterminations.

obtainedbyhorizontal extrapolationusing the last tenpoints of thecurve, and thepointwhereitintersectswiththe onsetofthe

reaction exothermwastaken tobe_{the starting time of}thereaction

(t = _0). Thetotal_heatsof polymerization_{obtained bythe two}

methodsofbaselinecorrectionwere identicalwithinlessthan

1%. Consequently, the extrapolation methodwas chosen for

correctionof alltheexperimentalcurves. These experiments showedthatthepoints obtainedwithinthefirst 2 minofthe

polymerizationrun are_subjecttosome error, andtheir ordinate values_(dQ/dt)are lowerthanthe real values.

Results and Discussion

Mixtures. Dynamic

DSC. Inorder to relatethe

ep-oxide conversion to the heat evolution in the DSC

experiment,

it

is necessarytoassume thattheheatreleased

on reactionofan _epoxidegroupisthesame regardlessof

the_{type ofepoxy}or thenatureofthereaction. To confirm

this_hypothesis,severalmixturesofMY720,

HT

976,and

SU-8 were _{analyzed by dynamic} DSC. The results are

given inTable I. The mixturescover a _rangeof compo-sitionsandamine-to-epoxide ratios. Attemptstoanalyze

some _binary mixtures ofSU-8 and

HT

976were

unsuc-cessful. First, it is

difficult

to remove all the acetone

without_{heating the mixture.} Second,it isbelievedthat

the_{higherfunctionality of}SU-8leads_{to rapid}vitrification,

whichcauses the reaction to_stopwellbeforecompletion. Figure 1 shows the DSC curves for three different mixturesofMY720andHT 976withvarying

amine-to-epoxideratios. The effecton the nature ofthe reaction

is evident. Mixture 5 _{(Figure la)} is a stoichiometric

mixture in which the number of _{epoxide rings} is _just sufficienttoreactwith allthe

N-H

bonds present: hence

the_{amine-epoxide}reactionisbelievedtobethedominant one. TheDSCcurve showsa_{strong exothermic}peakwith

amaximum at218°C. In mixture3 _(Figure

_lb),

thereis

a30%excess ofepoxidegroupswithrespect to amine, and

theseare believed_{to polymerize by etherification.} This

occurs atasomewhat higher_temperatureand_{substantially}

broadens the DSC exotherm peak; the maximum now

occurs at234°C. In mixture2_(Figurelc),thereisa60%

excess of _{epoxide groups.} The etherification reaction

becomeseven more _importantand shifts the _peak

max-imumto 250 °C.

The other mixturesgavesimilarcurves,andtheresults

are summarized in_{Table I. The temperature} at which the _peak maximum occurs is _{highly dependent} on the overall amine-to-epoxide ratio but is not _{significantly}

affected by the replacementof part of the MY 720 by

SU-8. A_plotofTm..versus the amine-epoxideratio_gives a_{goodlinear relationship(squaredcorrelationcoefficient}

0.982). The enthalpies of polymerizationforthedifferent

mixtureswere _{determined by computer integration}ofthe

exotherm_{peak using}alinearbaseline. _{The results,}based

on threetofour determinations foreach_mixture,are listed

Figure1. _DynamicDSCcurves formixturesof MY720andHT 976with_{amine-to-epoxide ratios}of (a) 1.01,(b) 0.77, and (c) 0.63;heating rate5_°C/min.

in thetable. Thevaluesvarywiththe mixture,but if they

are _{converted to kilojoules per equivalent} of _epoxide

present,aconstantvalueofabout107

_kJ/mol

isobtained.

This is in excellentagreementwith the results reported

byBarton.17 The factthatthesame resultisobtainedfor

mixture 5,wheremostofthe epoxide groupsreactwith

amine, and mixture 2,where at leastone-third react by

etherification, indicatesthatthe enthalpies forthesetwo reactions are _{very similar. Likewise, comparison} of mixtures1and10_suggeststhatthe_{enthalpy for the}SU-8

is not muchdifferentfrom that for the MY 720. Both

mixtureshavesimilar_{amine-to-epoxide ratios,}but mixture 1containsno _SU-8,whilein mixture10,14% ofthe epoxy

groupscome fromSU-8. The two mixtureswithverylow

amine-to-epoxide ratios(0.33and0.35)giveslightlylower

enthalpiesofreaction. Thiscouldbedue_{to incomplete}

Macromolecules, Vol. 24,No. 11, 1991 Modeling the Cure Epoxy

Prepreg.

Dynamic

DSC. FivespecimensofNarmco 5208_prepregwere _{analyzedbydynamic}DSCandTGAas

describedin_{the Experimental}Section. TheDSC curve

isverysimilarto Figure

lc,

withamaximum around 250

°C. Theaverageweightloss_{during the}DSCexperiment

was 2.1 ± _{0.5%, which} is close to the value of 1.9%

determinedfor the volatiles content. Thisindicatesthat

the amountof_degradationthat occurs is_negligibleand

thattheweightloss ismainlyduetononreactivevolatiles

suchaswaterand acetone,whichare known tobe_present

inthe _prepreg. Foreach specimen, the weightofresin

remainingaftertheDSC experimentwas determined_by

TGAandthiswas taken to bethe_{true weight}ofresinin

the prepreg, exclusive ofvolatiles. The _{energy released} on _{polymerization}was determined_{byintegrating}the DSC

peak andwas thendivided bythisweight togivethe

en-thalpy of polymerization AH. The average of the five valueswas 620.4_{J/g, with}astandarddeviationof7.5_J/g.

It

was _{pointed out} inthe Experimental Sectionthat this resinis_partlypolymerized,or _B-staged,to the extentof

0.038;inotherwords, 3.8% ofthe epoxide groups

initially

present in the _{system have} already reacted. Thus, for completely unreacted Narmco5208,AHwouldbe_620.4/ (1

-0.038)or 645.1_J/g. On thebasisoftheother results given inthe Experimental Section,the enthalpy per

ep-oxide group is calculated to be 114.3 _{kJ/mol. This} is slightlyhigher thanthe valueof107-108_{kJ /mol reported}

in Table

I

for prepared mixtures but within the range

100-118

_kJ/mol

typically observed for _{epoxide ring}

opening.26 The averagefibercontent ofthe prepregfor

thefivespecimensanalyzedwas 59.0 ± 0.9% by weight.

Prepreg.

Isothermal

DSC. Three experimentswere performed ateachofsixtemperatures ranging from 160

to210°Cin_10-degincrements. A_{typicalcurve,showing}

heatflow_(dQ/dt)as afunction oftime at160°Cis_given in_Figure2a. Thiscurve hasbeen_{corrected by}means of the extrapolation method describedinthe Experimental

Section. _{Integrationgives}theevolvedheatas afunction

oftime:

Q(t) =

J/dQ/dt

dt (1)

Thetotalheatevolvedduringtheisothermal experiment

is _{given by the}totalarea under the isothermalcurve:

Qi=

J0ifdQ/dt dt (2)

where

tf

is the time corresponding to the end of the

experiment, when the rate of heat evolution becomes

negligible. Since_{the polymerizationmay}notbe_complete aftertheisothermalrun,

it

is_{completed in thesubsequent}

dynamicrun, as shownin_Figure2b. _{Integration of}this curve _gives the residual heat of _{polymerization Qr.}

At

higher temperatures, the isothermal cure occurs much faster,a_higherdegreeofpolymerizationis_{achieved, and} Qrislower.

In order to analyze the kinetics, the curves must be

expressedin termsofa,thedegreeofpolymerization. To

do_{this, the}usual_assumptionwas made,namely,thatthe

heatevolvedis_{directly proportional}tothenumberof

ep-oxide_{groups reacted, regardless} ofthe typeofreaction.

As discussed above,the results for_{the various prepared}

mixtures supportthis _assumption.

Inmostanalysesofthekineticsofprepregsystems,it is assumedthatthe extent of_B-stagingis_negligible,that

is,thatthedegreeofpolymerization at the beginningof

the cure (ao) is equal to zero. In this work, the initial

degreeofcure, determined byliquidchromatographyto

Figure2. DSCcurves forNarmco5208_prepreg: (a)isothermal

run at160°C; (b)dynamiccurve recordedafter isothermalrun, heating rate 5_°C/min.

beequalto0.038,was takenintoaccountinthe_analysis.

Thisleadsto the following equationsforderivinga and

da/dt

asfunctions oftime:

“(í)

= ao+

_(¿+^)Q(í)

da _

(

1~ ao

\dQ

dt ~ \Q¡+

Q,/dt

(3) (4)

Withthese data

it

is _possible to constructa curve of

da/dt

versus a foreach_temperature. Theseare thenused

to test thekineticmodel. Oftheseveral_{thousand points}

ina _{given curve, 50-100}were selected at_equalintervals

for this analysis.

Analysis of

the

Data

by Means

of

Some

_Existing

Models. Thefirst _{test applied to the data involved the} simplifiedHorie equation:27

da/dt

=

{

₊

2 ){1- ) ß- )

₍₅₎

where the rateconstantsK\andK2correspondrespectively

to catalysis by groups

initially

presentin the resin and

catalysisbyhydroxylgroupsformedinthe reactionand

Bis the ratioof primary amine

N-H

bonds to epoxide

groupsinthe

initial

mixture. Figure3shows_plotsofthe reducedrate function¥\{oi) = ₍₁

)-1(

-a)-1(da/dt)

versus a forthree different_{temperatures.} This should

givea_straightlinewith intercept

_K\

andslopeK2. The

valueofBwas fixed at_0.6,asdetermined by_liquid chro-matography. Except for the

first

few points, linear behavioris_{observed up}toaabout_0.2,with_{slight deviation}

between0.2and0.3_{and significant deviation beyond}this

point. The deviationobserved athigher temperatures for thefirstfew_pointscan beexplainedby the factthatthese

occur withinthefirst2min oftheexperiment, before the

system has

_fully

_{equilibrated. The deviation} observed

when a > 0.3 isattributed to thelack_{of validity of}the modelin this_{region, where}theetherificationreactionis

believed to become _{significant. Linear regression} was performedby using the pointsshownbyfilledsymbolsin

the figure. The intercepts of these lines were used to determine the values of

_da/dt

corresponding to a = _0.

3102 Cole etal. Macromolecules, Vol. 24, 11, 1991

0.0 0.1 „ 0.2 0.3 0.4

Figure3. Plotsof Fi(a)- ₍₁

-a)~l(B

-a)-1(da/dt)as afunction

ofa for_threedifferent_{temperatures;}unitss"1x 1000.

Anextension of_{this approach,}used_{by Springer et}al. forthe commercial _productHercules3501-6,®separates

the reaction intotwo distinct_stages.

It

isassumedthat inthebeginning (a<0.3)onlytheepoxide-amine reaction

needbeconsidered, andeq 5 isused. Whenthe

epoxide-aminereactionis_complete,the epoxide-hydroxyl reaction

takes over, and, sincethe_hydroxylconcentrationisnow constant, the reaction is assumed to be

first

order with

respecttoepoxideconcentration. Thusitcan bedescribed

by thefollowing equation:

da/dt

-K3(1

-a) (6)

In thiscase thereducedfunction

F^a)

= ₍₁

-a)-1(da/d£) should be constant and equal to K¡. To test this

hypothesis, the quantity Fi(a) was _{plotted against}a as

shownin Figure4. _{Interesting behavior}isobserved. _Only

at the higher temperatureof200 °Ccould _the approxi-mationbeconsideredtobe_valid,and,even _{then, only up} toacertain point. Generally_speaking,the_quantityFata)

risesto a maximum valueat arounda = _0.3,butinstead

ofremaining constantitgraduallydeclines

until

areaches

about0.7 or 0.8.

At

_{this point}there is a rather _abrupt dropandthe reaction rate falls to zero beforecompletion.

This is believedto be due to vitrification ofthe_system

andtheonsetofdiffusion control. This willbediscussed

in more detail later.

It

is _{interesting to note} that in Springer’s work,although thisapproachgavea_reasonably

goodfitto the experimental data fora _{up to}about_0.6,in order to achieve it it was _necessary to assume _negative

valuesforthe rate constant Kzl The problemliesinthe

factthatHercules 3501-6isa_complexmixture_containing not onlyTGDDMandDBSbutalsotwootherepoxiesas wellas aborontrifluoridecatalyst. Asa_{result, the}curve

Figure4. PlotsofFi(a) = ₍₁

-a)-1(da/dt)as a function ofa

forthreedifferenttemperatures;unitss"1 x 1000.

of

_da/dt

versus a does _{not clearly} show the maximum

typical of an _{autocatalytic reaction, which eq} 5 was

designedto represent.

An attempt was also made to

fit

the data with the semiempirical equationproposed byKamal:28

da/dt

=

(Kr+ K2am)(1

-a)n (7)

The fourunknownswere determinedsothatthecalculated

and experimental data matched with respect to the

followingfour values: (1) the

initial

value of

_da/dt,

as determined_{from the intercepts}in_Figure3; (2) amax,the locationofthemaximum valueof

_da/dt;

(3) (da/dt)™.».

the maximumvalue;(4)thevalueof

_da/dt

ata = _0.5.The

resultsare shownin Figure5.

At

160°C,an excellent

fit

was obtained up to thepoint of vitrification.

At

180and

200°C,the

fit

was_gooduptoa= _0.5but

poorbeyondthat point. Furthermore, the values of m and n varied

significantly withtemperature,asindicatedinthe figure. Equation7 isthus lessthan ideal for this system.

Analysis

of the Data

_by

Means of the New Approach. The model_developedinthe_{precedingpaper} involvesa mote _{rigourous approach than}thosethathave

justbeen_applied. It isamechanistic approachinwhich the Horie modelisextendedtoincludetheetherification

reaction. The conversion of epoxide groups intoether groups isdescribed by thefollowingkineticexpression:

dE_ d[ether] _ „

“dt~

di

-fe^A3

(8)

where E, [ether], H, andA3are the concentrationsof

ep-oxide, ether,hydroxyl,andtertiaryamine groups,

respec-tively. The exponents m and n _{(which should not} be

Macromolecules, Vol. 24,No. 11, 1991

Figure5. Resultsobtainedon _{fittingexperimental data forda/} dtversus a with_eq7.

Table II

Different Possibilities fortheEtherification Reaction Mechanism

case featerm m n z natureof reaction 1 EH 1 0 _{epoxide-hydroxyl, uncatalyzed} 2 EH1 2 0 epoxide-hydroxyl,catalyzedby

hydroxyl

3 _EHAS 1 1 1.5 _{epoxide-hydroxyl,catalyzedby}

alltertiaryamine groups 4 _EHAS 1 1 0 _{epoxide-hydroxyl,catalyzed}

only bytertiaryamine groups

formed in reaction

5 E 0 0 _{homopolymerization,}firstorder inepoxide

6 _EA3 0 1 1.5 _{homopolymerization,catalyzed}

by alltertiaryamine groups 7 EAs 0 1 0 _{homopolymerization,catalyzed}

onlybytertiaryamine groups

formedin reaction

scenariosforthe reaction mechanismasshownin Table

II.

The

first

corresponds to the case of an

epoxide-hydroxylreactionthatoccurs eitherwithout_catalysisor

withcatalysisbygroups whoseconcentrationisconstant and therefore maybe_incorporatedintothe rate constant

*3· The secondis similarexceptthata second_hydroxyl

groupaccelerates_{the reaction, giving}risetoan EH?term rather than EH. The

third

and fourth scenarios

corre-spond to an _{epoxide-hydroxyl} reaction catalyzed by

tertiaryamine_groups. In thiscase,adistinctionismade betweenthe_{tertiaryamine groups}

_initially

_{present in the}

system (in this case, in the TGDDM resin) and those

produced by the epoxide-amine reaction. The coefficient

Zisdefined2as _ZA^/BEq,whereA30andEqare theinitial concentrations of tertiaryamine_{and epoxide.} Onthebasis

Modeling the Cure Epoxy

oftwoTAgroupsforeveryfourTGDDMepoxide groups

(butnone forSU-8) and B= _0.60,Zisestimatedtobe 1.5

forNarmco5208. Case3_(withZ= _1.5)includes_thesein

the calculation. Case 4_{(with Z}setto0) assumes thatonly the_{tertiaryamine groupsproduced}in_{the epoxide-amine}

reaction_catalyze the _{epoxide-hydroxyl}reaction. Cases

5-7 correspond to etherificationwithouttheparticipation of hydroxyl groups, i.e., homopolymerization. This is

assumedto occur either bya _simplefirst-orderreaction

with_respectto_epoxide(case5)or withthe_{participation}

of tetiaryamine_groupsas_justdiscussed(cases6 and7).

The kineticsare describedinterms ofthe_{following two} variables: 8 II (-* (9) 2Al+ A2

H

-H0 GO)

/3~1

2A10 " 2A10

where A\isthe concentrationof primary amine_groups,

Aa is the concentrationofsecondary amine_groups,and the subscript “0” _designates the

initial

value of the concentration to whichit isattached. The variable a is the usual overall extentofconversionof_{epoxide groups.}

The variable ßisequalto thefraction of primaryamine

N-H

bonds that have reacted with _{epoxide groups} to

produce linkagescontaininghydroxylgroups;thusitisa measure of the extent of the epoxide-amine reaction. Together these two variables cover the two reaction

pathways.

Thekineticequationscan besolved2togivethe following

two _equations: =

ß

+ K

*3bk [CJ

+ C2d2+ C3In(1+

RT'fi

+ C4In(1 -ß)] (11)

da/dt

=

[ {

+

_2ß)(1

-ß)+

3( +ß +ß2 (1-«)

(12)

where

is the rate constant for _{the epoxide-amine}

reaction_catalyzedbygroups

initially

present(including hydroxyl),K2 isthe rate constant forthesame reaction

catalyzedbyhydroxylgroupsformedinthe reaction,K3

isthe rate constant forthe etherificationreaction, Y =

Hq/BEo, and R =

K1/BK2. The coefficients Ci to C4

dependontheparticularvaluesofmandn andare _given

in Table

II

ofthe_{precedingpaper.}

With

these equations,

although

da/dt

cannotbeexpressedanalytically interms

ofa, the relationship betweenthe two can be expressed

intermsofa setof_{calculated points.}

Regardless ofthe particular scenario, there are three

rate constants that must be determined to best fit the experimental datafor

da/dt

as afunctionofa. Because the dependent variable

da/dt

cannotbe expressedas a simple function ofthe independent variable a,

it

is not

possibletouse_{standard regression techniques}todetermine

theconstants. Instead,an iterative_procedurewas used.

After

initial

estimatesfor

K\,

K2,and K3were _chosen,for

each_{experimental value}ofa the corresponding valueof

j3wascalculatedby numerical solution

ofeqll.

Thevalues

ofaand_ßwere thenusedineq12to calculate

_da/di.

The coefficient Y (relatedtothe

initial

hydroxylcontent)was

assumedtobe_{negligible. The calculated}valuesof

_da/dt

were _comparedwith_{the experimental}valuesandthe rate

constantswere _adjusted until the calculated _and exper-imentalcurves _{matched according to the following three} criteria: (1) They shouldhave thesame value of

_da/dt

3104 Coleetal. Macromolecules, 24,No. 11,1991

Figure6. (a,Top)Comparisonofexperimental valuesofda/dtwithcurves _{calculated according}tocases1,5,and6ofTableII,for one _experimentat180°C. (b,Bottom)Comparisonof experimentalvaluesof_da/dtwithcurves _{calculated according to}cases 2-4 and7 of TableII,forone _experimentat 180°C.

were _{obtained by extrapolation,}basedon the intercepts

ofthelines givenin Figure

3.(2)

The maximum calculated value of

_da/dt

should be the same as the maximum experimentalvalue(butnot_necessarilyat thesame value

ofa). (3)The twocurves shouldhavethesame valuesof

da/dt

ata = _{0.5. Once}the threerate constantsthat _meet thesecriteriawere _{found, the overallquality of}thefitwas judged by graphically comparing the calculated and

experimentalsets ofdata.

Figure 6 shows the results for the seven different

scenarios_appliedto one setofdata obtained at 180°C.

Threeofthecases,showninFigure6a, gaverather poor fits. These include a bimolecular_{epoxide-hydroxyl} re-action(case 1),uncatalyzedhomopolymerization(case5),

and homopolymerizationcatalyzedbyall tertiaryamine

groups(case6). Ofthe_remainingcases,depicted in Figure

6b,three_gavea_goodfit butone was

_virtually

perfect.(As

already mentioned, the deviation observed forthe first point or two is due _{to incomplete equilibration of}the

calorimeter, and the deviationobserved_{at high}valuesof

a isduetovitrification ofthe resin.) The perfectfit

cor-respondsto case 4,wherethe reactionisfirst-order with

respect to epoxide groups,hydroxylgroups, andthe tertiary amine groups formedin thereaction(butnotthose present

in theTGDDM). Whether the slight differencebetween

this case and theother three in Figure6b is significant

(i.e.,whethercase 4_correspondstothe true mechanism)

needs to be _{confirmed by} further experimentation. A

possibleexplanationis_giveninthe nextsection. Because

the case 4_{mechanism gives} the _{best agreement,} it was

used to analyze the data and determine the three rate

constantsfor all18experiments (three ateachofthe six temperatures studied).

Asmentionedabove,the deviationobserved_{at higher} valuesofa_{(seeFigure6b,}case 4) isdue tothevitrification oftheresin. Theeffectsofthisvitrificationare even more evident in Figure 4, where the function F¡(a) shows a

Macromolecules, Vol. 24,No. 11, 1991 Modeling the Cure Epoxy

a

Figure7. “Diffusion factor"obtained bydividing experimental

values_forda/dtbyvaluespredicted according to chemicalkinetic model. Symbols correspond to experimental points, lines to

regressionfit obtainedwitheq15.

andtheresincross-links, theglasstransitiontemperature

Tg ofthe system rises. When it approaches the curing

temperature, the resinpasses froma_{rubbery to}a _glassy

state,the_{mobility of}thereactinggroupsishindered,and

the reaction is _{controlled by} diffusion rather than by chemicalfactors. Various workershave_attempted

math-ematical treatments of thisphenomenon.29"35 Atypical approach31 isto express Tgin termsofa _usingDi

Bene-detto’s equation and then to express the

diffusion-controlled rate constantintermsof_T-Tgbya Williams-Landel-Ferry typeequation. Thus theoverallrelationship

is rather complex. Chern and Poehlein have recently

proposedasomewhat_{simpler semiempiricalrelationship}

basedon freevolumeconsiderations.30 When thedegree

ofcure reachesacertaincriticalvalueac,diffusioncontrol

takesover andthediffusion-controlled rate constant K¿

is _{given by}

Ká=

Kcexp[-C(a

-ac)] (13)

whereKc istherateconstant fornon-diffusion-controlled

(chemical) kinetics and C is a constant. This _equation

correspondstoarather_abruptonsetofdiffusion control ata = _ac. In reality,_{theonsetissomewhatmore}

gradual andthereisa _regionwhere both chemicalanddiffusion

controlare _{significant. According to Rabinowitch,36}the overalleffective rate constantKe can beexpressedin terms

ofK¿andKcas follows:

1—L+-L

Ke Kd kc

(U)

Combiningeqs13and14,we can definea“diffusionfactor”

/(a) as follows:

_1_

1+ _exp[C(a

-ac)] (15)

The effective reactionrateisequal to the chemical reaction

rate_multipliedbythisfactor, which followsan _S-shaped curve. Forvaluesofa _{significantly}lowerthan_ac,f(a) is approximatelyequal to1anddiffusion controlis_negligible.

Whena _{approachesac,f(a) beginstodecrease,reaching}

0.5whena = _ac.

Beyond thispoint

it

continuestodecrease,

eventually approachingzero,sothatthereactionbecomes very slow andeffectivelystops. In thiswork, data for f(a) were _{obtained bydividingthe experimental}values_{of da/}

di

bythosepredictedon thebasisofthechemicalkinetic

model. _Figure7showsthe resultsforone _{experiment at}

eachtemperature. Thedropoffin rate due to theonset

ofdiffusion controlisobvious. _{An interestingphenomenon}

isobserved athigher temperatures. At200°C, whena is

greater than0.6 the reaction rate increases _{slightly (by}

about 10%)abovethat_{predicted by the model,}beforeit

dropsoffbecauseof diffusion control.

At

210°C,asimilar effectisobservedbut itismuchmore _pronounced. This

isbelievedtobeduetoanother reaction, not included in

themodel,whichbecomesimportantonly at higher

tem-peratures. Thesame _phenomenonhasbeen_{reported by}

Mijovicet al.15foran

MY

_720/HT976mixturecuredat

210 °C. On the basisof FT-IR results, Morgan et al.10

have suggestedthatat temperatures greater than200°C

two hydroxylgroups combinewithlossofwater to form an ether_group,andthiscouldexplain theresults. Values

ofCand acineq 15 were _{determined by applying} non-linearregressionto the datafor_Ke/Kcfortemperatures

between160and200°C. Exceptforthe slight deviation

at 200 °C _{just mentioned, very} good agreement was obtained,as shownin Figure 7.

Figure8showsthe resultsforone _{experiment at}each

temperaturewhen_{the experimental}valuesof

_da/dt

are compared tothosecalculated by the model(case4),with

thediffusion_{factor included. Except}forthe deviations previously mentioned, the agreementis excellent. The

average constants obtained from the analysis are

sum-marizedinTable

_III,

_{together with}the standard deviations.

The corresponding Arrhenius plotsofInKversus 1_/Tare

shownin Figure9. Individualdata points forthethree

experiments at eachtemperature are shown in order to

givean indicationofthescatter. Goodlinear correlations are observed,whichlead to theparametersgiveninTable

IV. The activation energies for the X-catalyzed and

hydroxyl-catalyzed epoxide-amine reactionsare 61.4and

72.5 _{kJ/mol, respectively, while} that forthe

etherifica-tionreactionis_{significantlyhigher}at101.4_kJ/mol. These valuesare verysimilartothoseobtained byChiao.21 The

variation of_{the quantities}acand Cwithtemperatureis

shown _{in Figure} 10. For ac, the relationship can be

consideredlinearwithin experimentalerror, and regression

givesthe following equation (squaredcorrelation

coeffi-cient0.987):

ac= 0.005376T-0.1350 ₍₁₆₎ whereTis_expressedin_degreesCelsius. Thusthediffusion factor defined ineq 15isalsoafunctionof_temperature. For the coefficientC,thereis_considerablymore _scatter,

andno discernible trend with _temperatureis observed.

The_{average valuefor the experiments from}160to190°C

is 30.1 (standard deviation 5.1). Thevalues for200 °C

were not used in the _averaging because of the slight

deviationobservedatthistemperature,attributedto the

hydroxyl-hydroxylcondensationreaction.

Summary oftheModel andGeneralDiscussion.

It

hasbeenshownthatan excellent

fit

to_{the experimental}

DSC datafor_{the curing}ofNarmco5208in_the

temper-aturerange 160-200 °Cisobtained byusingamodel based on the following _{assumptions: (i)} a reaction between

primaryamine_{and epoxide groups}(PA-E) thatproduces secondary amine andhydroxylgroups andthatis_catalyzed both_bygroups

_initially

_{present in the}resin(rate constant

Ki)

and by the _{hydroxylgroups} formed inthe reaction

(rate

constant^);

(ii)asimilarreaction between secondary

amine and epoxide groups(SA-E),whoserateconstant is

one-halfofthat forthe previous reaction; (iii)areaction

between epoxide andhydroxylgroupsthatis_{catalyzed by}

the _tertiaryamine groups formed in the SA-Ereaction (rateconstantKz);(iv)adiffusion control correction factor

for which a _{semiempirical equation} is _given and which

becomes _important when the _degree of conversion

ap-proachesacertaincriticalvalueac,whichisdependenton the temperature.

Cole et Macromolecules, Vol. 24,No. 11, 1991

0.14

0.0 0.2 0.4 0.6 0.8 1.0 0.20

0.0 0.2 0.4 0.6 0.8 1.0

Figure8. Comparisonofexperimental valuesofda/dtwith control, forsix temperatures.

0.0 0.2 0.4 0.6 0.8 1.0

calculated according to the proposed model,including diffusion Table III

AverageConstants_(withStandardDeviationsin Parentheses)for theProposed Model*

temp, “C Ki,s'1 X1000 Ki,s"1X 1000 K3,s'1 X1000 «C C

160 0.134(0.011) 1.232(0.057) 0.062(0.008) 0.725 (0.012) 34.7(7.2) 170 0.238 (0.014) 1.682(0.056) 0.112(0.007) 0.779 (0.002) 26.7(1.2) 180 0.321(0.026) 2.750(0.053) 0.218 (0.023) 0.827(0.005) 27.0(2.0) 190 0.474 (0.025) 4.199(0.235) 0.444(0.070) 0.898 (0.016) 31.9(4.3) 200 0.602 (0.026) 6.288(0.231) 0.637(0.061) 0.934(0.020) 40.6 (9.0) 210 0.836 (0.004) 9.503(0.037) 1.130 (0.038) «_Valuesin

parentheses correspond to standard deviationsbasedon threedeterminations.

Theprogressofthe_{polymerization}isdescribedin terms

oftwo_parameters,a and _ß. The

first

_(a) is the overall

degreeofconversionas_expressedin terms ofthefraction

ofepoxidegroups reacted. Thesecond _{(ß) is}thefraction

of amine

N-H

bonds that have reacted. These two

quantitiesare_closelyinterrelatedandtheirevolutionwith

time is_{given by the followingequations:}

ß/dt

=

[(K,

+

₎₍

-ß)](1 -<x)f(a,T) (17)

da/dt

=

[B^

+

2ß)(

₁

-ß)

+

_Kg/fKl

-a)f(a,T) (18)

where thediffusioncontrol factor_f(a,T) is _givenby f{a,T) =

[1+ exp(30.1a + 4.06

-0.1617T)]"1 (19) with Texpressedindegrees Celsius. Equations17and18 correspondto case 4of_{the general model (m}= =

1)with

Y= Z= _{0. Equation19is}obtained by combining

eqs15

and 16. If the reaction is carried out at constant tem-perature, therelationshipbetweena and_ßisdescribed_by

eq 11. ThequantityBistheratio ofamineN-H bonds to epoxide rings, and fora _{typical lot of}Narmco 5208it is_equalto0.60. Also,theinitial _degreeofconversionao in Narmco5208isnotzero becauseofresinadvancement

duringthemanufacturingprocess(B-staging) and possibly

duringsubsequenthandling. Typicalvaluesfora_newly

received batch are ao = _0.038 and ß = _a0/B = _0.063.

Accuratevalues_maybedeterminedfora_givenbatch by physicochemicalanalysis.

The three rate constants obeythewell-known Arrhe-nius relationship

= _A¡

exp(-E¡ú/RT) and may be

calculatedfrom the parameters giveninTableIV.

The amountof_{heat produced by} 1_gofNarmco5208

resinon _goingfroma = 0toa = lis _645J. Thisis based

on the true resin_weight,exclusiveofvolatile components (solventandmoisture).

Somediscussionisin order_concerningthe_assumptions

ofthe model. Assumption ii referred to above implies

thatthe secondary amine groups showthesame reactivity

withrespecttoepoxideasthe_primaryamine groups.This importaint questionhasbeen_{extensively studied,}andthe results have been summarized by various authors.4-26’37 While many ofthe experimental values that have been

reportedfor this ratioare closeto0.5,others are

signif-icantlylower, especially inthecase ofaromatic amines.

However,as was _{recently pointed}out_{byCharlesworth,38}

Macromolecules, Vol. 24,No. 11,1991

Figure9. Arrhenius plots for therate constants determined

from theanalysis.

150 160 170 180 190 200 210 Temperature (°C)

Figure 10. _Dependenceof«„andthecoefficientC (eq 15)on

temperature, T.

Table IV

Arrhenius Parametersforthe_ProposedModel

rateconst,

a-1_InA_£,/£,

K

£„

kJ/mol

K, 8.243 7389 61.4

K2 13.36 8719 72.5

K3 18.53 12221 101.4

solutionsinalcohol, anditisknownthatalcohols havea strongcatalytic effecton _{epoxideringopening}reactions.

Hence the situation in alcohol solution may be _quite

differentfromthatinbulk_{mixtures involving}no solvent.

Infact, data obtained byCharlesworth37-38_{and by other} workers39·40on bulkmixturesofepoxyresin and aromatic

Modeling the Cure Epoxy

Figure 11. Structure of thecross-linked polymer formedby

reaction betweenTGDDMandDDS.

amines suggest that the assumption of_{equal reactivity}

can be_quite valid in such cases. In _{the specific}case of

TGDDMand DDS,asmentionedintheIntroduction,there

are _conflictingresultsintheliteratureand_{the question}

will_only beresolved _throughfurther work.

Assumptioniii impliesthattheetherificationreaction

proceeds through a _{ternarycomplex involving epoxide,}

hydroxyl, andtertiaryamine_groups. Thisisnot

unrea-sonable, since bothhydroxyland _tertiaryamine_groups

are knowntoacceleratetheetherificationreaction.41"44In

fact it is the same _type of _{complex proposed} for the hydroxyl-catalyzed epoxide-amine reaction, exceptthat

whentheaminegroupbearsahydrogen atom the reaction

followsadifferent_path.

It

is _interesting,however, that

thebest

fit

to the experimental data isobtained

if

it

is

assumedthatthe_tertiaryamine_{groupsproduced}inthe

reaction are more effective in catalyzing the etherifica-tionthanarethose_initiallypresentintheTGDDM. Figure

11 shows the structure of the polymer formed by the

TGDDM-DDS reaction. The _tertiary amine _groups formedin_{the reaction originate from the DDS and} are attachedtoa _{phenylring with}an SO¡group inthe para position. Thegroups

initially

present, however,are inthe

TGDDMand are attachedto a _{phenyl ring with} a GHz groupinthe para position. The_{strong effect}ofsuch_groups

on kinetics,even _{though they}maybefarfrom thereactive

site,is well-known. Forexample, Tanakaet al.42found

thatthe rateofreactionof_{the epoxy compound phenyl} glycidyl ether with different substituents in the para positionvariedbyafactor of4. Thus

it

isnotunreasonable

to postulatethatthe two typesof tertiaryamine_groups

shownin_Figure 11_might have_widelydifferent

reactiv-ities, given thedifferentelectronicnatureoftheSO2and

GHz groups. If thisis thecase, though, it is

difficult

to

arguethatthe_{tertiaryamine groups}from theDDSshould

bemore _{active, becausethe electron-withdrawing nature}

oftheSO2groupswouldbe _expectedto makethemless

nucleophilic. A more _{plausible explanation may exist}

based on steric arguments. The newly formed _tertiary

amine _groups are located in a _position wherethey can bring about intramolecular catalysis ofreactionsofthe remainingepoxidering,asshownat thebottomof_Figure

11. Dusek and_Matéjkahave usedthis_{argument to explain}

the apparentenhanced reactivity ofthe_{second epoxide} ringin N^V'-diglycidylamines.46·46 Their_{detailed study}

of NJV'-diglycidylaniline showed that intramolecular

etherificationis_significantandoccurs faster than the in-termolecularreaction. Similarresultshave beenobtained byothers47and

it

has_{been suggested}thatintramolecular reaction and _{ring formation}are _{quite important in} the

TGDDM-DDSsystem.11 Theseargumentsprovidesome support forassumptioniii ofour model.

Although vitrification was found to have a _profound effecton the kinetics,no effectswere observedthat_might

3108 Coleet al. Macromolecules, 24, 11, 1991

Figure12. Calculatedcurves for_da/dtas afunctionofaat180

°C, showing the separatecontributionsfromthePA-E, SA-E,

andOH-Ereactions.

bedue to gelation, whichoccurs at lower valuesofa.

It

may be that because _{the epoxide-amine reaction} is

predominantbefore_gelationandetherification afterward, the corresponding rate constants reflect thestateofthe resin at_{each stage.} However,

it

ismore _likelythat,ashas

been suggested by others, diffusion control becomes

important only on

vitrification

and not on _{gelation.48·49}

It

is _{interesting to} compare our results withthose of Chiao.21 Inbothcases,agoodfitwas obtained by varying threerate constants,oneof_{which corresponds to the}

ether-ification reaction. However, the assumptionsmade are

different. In our _case, we assume _{equal reactivity of}

primaryand secondary amine groups;thismakesitpossible

to_partlysolvethekinetic_{equationsand develop}certain analyticalexpressionsto model thecure. InChiao’scase, heassumes afixed value for theratiooftherateconstants

forhydroxyl-catalyzed and noncatalyzed amineaddition, on thebasisofresultsobtainedon adifferent_{epoxy amine}

system. Thisis_questionablebecause_{the “noncatalyzed”}

reaction, at leastin thecase of_{commercial materials,}is

generallyacknowledgedto involvecatalysisbyimpurities

present in theresin,andtheir_typeandconcentration_may vary substantially fromone _systemtoanother. Thus both treatmentsare _{subject to}some _question,andthe factthat

agoodfit isobtainedinbothcases _suggeststhattheDSC

dataare _{not greatly sensitive to either. Only}furtherwork

will

determine which approach is more valid for the

TGDDM-DDS system.

Twofactorshave beenneglectedinour _{treatment, which} mayhavesome influenceon theresults. Thefirst isthe

presence oftheminor epoxy resininNarmco 5208;

it

is

known tobesomewhatmore reactivethan_{the major}resin.3

Thesecondisthepresencein the kinetics equationsofthe

coefficient Y,whichisrelatedto the concentrationof

hy-droxylgroups

initially

present in theresin. In thiswork

itwas assumedtobe_negligible,as no _experimentalvalue

was available. It would be _interesting to further refine the_{model by doing}a_studyofthe binaryTGDDM-DDS

system in which the

initial

concentration of hydroxyl

groups isdetermined and includedin themodel.

Calculated

Results.

With

the_{proposed model,}

it

is

possible to perform calculations in order to better

un-derstand how the _{system behaves} in different

circum-stances. For_{example,Figure}12shows_{the predicted}curve

for

_da/dt

as afunction ofa forisothermal curingat 180

°C, including the effects Of _{vitrification,} which occurs aroundac = _0.83. The_{curve isdecomposed}to_showthe

contributions from the threedifferentreactionsinvolved,

namely,PA-E,SA-E,andOH-E. The_figureclearlyshows

why the approximation mentioned earlier,in which the

30 60 90 120 150 180 210 240 Time (min) -b 0.8 -LU

Í

0.6 -$

I

0.4 -J 0.2 -0.0 \ Epoxide

\z

Hydroxyl x / / Ether

/

>.-180°C 30 60 Time (min)

Figure13. Calculated concentrationsof_{epoxide,hydroxyl,}and ether groupsas afunctionoftimeforisothermalcuring at160, 180,and200°C.

reactionis_separatedintotwodistinctstages, isnot really

applicable. Thegeneralbehavioristhesame _regardless

ofthetemperature. Fora = 0-0.2theOH-E contribution

is_{negligible and theapproximation}isvalid. Between0.2

and0.8,however,allthreereactionsare _{significant. From}

0.2to0.6,the epoxide-aminereactionsare more _important than the OH-E reaction.

At

a = _0.6the

PA-E

reaction

becomesnegligibleandthe OH-Ereaction becomesthe

most _{important. Finally,} at a around _0.8, the SA-E reactionbecomes_{negligible. Only then couldeq 6}serve as a _{goodapproximation,} but

it

cannotbe usedbecause

in the real_systemvitrificationoccurs near this pointand

the reactionbecomes_{diffusion-controlled. Thus,}inspite

of_{the significant difference} inactivation_energyforthe amine and etherification reactions, the two cannot be

clearlyseparatedintime. Thisdescriptionofthe_curing

process isingeneral agreementwiththefindingsofMorgan

and MonesforTGDDM-DDS,asillustratedin Figure10

ofref 11.

It is _{also possible to predict}the evolutionofthe cure

with_{time, by applyingeqs}17and 18 on an incremental

basis. The concentrationsofthedifferentspeciescan be

calculatedfroma and_ßbymeans ofthe_{equations given} in the preceding paper.2 Figures 13 and 14 show the predicted concentrations of the different species as a

function oftimefor isothermal curingat 160,180,and200

°C. In all threecases,thereis a_{“turning point”}around a = _0.45. This _occurs after₇₁ min at _160°C,30min at

180_°C,or 14min at 200°C. _{At this point,} over 90% of

the _primary amine groups have disappeared but ether

formationis_{just starting to}becomeimportant. However,

theSA-E reaction, whichproducestertiaryamine groups,

Macromolecules, 24, 11, 1991

Time (min)

Figure14. Calculatedconcentrationsofprimaryamine (PA), secondary amine (SA), and tertiary amine (TA) groups as a

function of time for isothermal curing at160,180, and200°C.

effectof_temperatureisthat,beyond theturningpoint, at higher temperatures theconversion ofepoxideto ether

proceedsfaster and to a _{greater extent.}

At

160°C, the

criticalvaluefor diffusion control(ac = _{0.72) is}reached

after218min.

At

this point,58% ofthe epoxide groups havereactedwithamine,14% have reactedwith_hydroxyl,

and 28% remain unreacted.

At

180 °C, ac (=0.83) is

reached after 117 _{min, at which point} 59% of epoxide

groups have reactedwithamine, 24% have reactedwith

hydroxyl, and17% are unreacted. Finally,at200°C,ac (=0.94) is reached after 73 _min, where 59% of_epoxide

groups have reactedwithamine, 35% have reactedwith

hydroxyl,and6%are unreacted. Thus at the_pointwhere

vitrification slows down the reaction significantly, the epoxide-amine reactionis

_virtually

_{complete,regardless} oftemperature, but theepoxide-hydroxyl reactionis_only partiallycomplete. Athigher temperatures, more ether linkagesare formed,as_expected. After vitrification,the cure proceedsatan _{ever-decreasing}rate,sothat_complete cure _{may not}beachieved. Aftertwice the time required toreachac,thedegreeofcure iscalculatedtobe78% at

160°C, 89% at 180 °C,and 98% at 200 °C.

Finally,toillustratethe overallaccuracyofthemodel, Figure 15 shows the evolution ofa with time at three

differenttemperatures, comparingthe_{behavior predicted} by the general model with the results for the three

individual DSC experiments.

Conclusion

Amechanistic modelhasbeen proposedfor the kinetics

ofpolymerizationofNarmco5208,whichconsistsmainly

oftheepoxy resinbis[4-(diglycidylamino)phenyl]methane

Modeling the Cure Epoxy

Figure IS. Evolutionofa with time at threedifferent

tem-peratures, as _{predicted by} the general model (line) and as

measuredin three separate DSC experiments(O, X,

).

(TGDDM) and theaminehardener bis(4-aminophenyl)

sulfone (DDS). This model represents a _significant

improvementover muchofthe_{previouslypublished work}

inthat

it

explicitlytakesintoaccountboth_the

epoxide-amine reactions and the subsequentetherificationreaction.

The kineticsare _completelydescribed interms ofthree

rate constants,whichwere determined tofitthedata.The

best fit was obtained on the assumptionthatthe

ether-ificationreaction_{is catalyzedby thetertiary}amine groups

formed in the epoxide-amine reaction; further experi-mental workis_requiredtoconfirmthis. Therate constants obeythe Arrhenius relationship and values ofthe pre-exponential factor A and the activation_energyE& have

beendetermined. Theseare _{specific to the Narmco}5208

system,butthegeneralapproach couldbeapplied toany

amine-curedsystem.

Apartfromthe three rate constants,no other_parameters were varied to

fit

thedata. _{The amine-to-epoxide}ratio

Bwas _assignedavaluedetermined by independentmeans

(liquidchromatography) andwas not varied as in some previouswork.5 Furthermore, the kineticswere assumed

tobe

first

orderwith_respectto_{all participating}species, and

it

was not_necessarytointroduce nonintegerexponents

asin the widelyused_eq7. _{Although the}modelisthusless

empirical than previousones, it givesan excellentfit to the experimental data for all values of the degree of conversiona_{up to the}critical pointwherethe resinvitrifies

andthe reactionbecomesdiffusion-controlled. Todescribe

the cure in this _region, a diffusion factor has been

introducedon thebasisofacombinationoftwo equations

proposedby otherworkers.30-36 Withthe inclusionofthis factor, it is_possibleto calculatewithexcellent precision thedegreeofconversionover thewhole rangeofcure and