A new approach to modeling the cure kinetics of epoxy/amine thermosetting resins. 1. Mathematical development

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Macromolecules, 24, 11, pp. 3093-3097, 1991-05-01

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

thermosetting resins. 1. Mathematical development

Cole, K. C.

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1991,24, 3093-3097

A New Approach to Modeling the

Cure

Kinetics of

_{Epoxy Amine}

Thermosetting

Resins.

1.

Mathematical Development

K.

C.Cole

IndustrialMaterialsResearchInstitute, NationalResearchCouncil ofCanada, 75boulevarddeMortagne, Boucherville, Quebec,CanadaJ4B6Y4

ReceivedJuly31, 1990;RevisedManuscriptReceivedNovember9,1990

ABSTRACT: The Horiemodel for describing thecure kineticsofepoxy aminesystems isextendedtoexplicitly includetheetherificationreaction, whichbecomessignificantwhen thereisan excess ofepoxywithrespect

to amine and whenthecure is_{performed at higher temperatures. A}solution to thekineticequations is derivedthatmakesitpossibletodeterminetherelationshipbetweenthedegreeofconversionaandthe rate

ofconversion_da/dt. Differentpossible mechanismsforthe etherification reactionare considered. The

modelcan beusedtocalculatethe amountsof differentgroups formedinthe reaction andhence provide

information_{concerning the}network structure.

Introduction

The widespreaduse of_{epoxy-basedcomposites} in the

aerospaceindustry,coupledwiththe growthof computer-aideddesign andmanufacturing,hasresultedinincreased interestin _{modeling theprocessing} ofthese materials.1 The object is _{to optimize the processing parameters} in order _{to consistently obtain high-quality} parts and to minimize the experimental work required todesigna cure

cycleforanynew _partsthat_maybeintroduced.

With

a

goodmodel,

it

is _{possibleto predict}howthesystem

will

behave_duringcure and whatitsfinal condition

will

be.

Oneofthe mostimportantcomponentsofsucha model

isan accurate_descriptionofthecure _{kinetics. However,}

the chemistry involvedin_{the epoxycuringprocess is}rather complex, andinspiteoftheextensive researchthathas

been done over the years

it

is

still

not completely understood. As a _result,

it

is

_virtually

_{impossible to}

describe

it

rigorously, andexistingmodels alwaysinvolve certain assumptions and approximations.

Manycommercial_{composite systems consist}of amine-curedepoxy resins. The_epoxyamine reaction_produces hydroxylgroups,whichhavetwoeffects: (1)theycatalyze the reactionthatproducesthem and(2)theythemselves

can react with_{epoxy rings to form ether linkages.} The

progressofthecuring reactionisdescribed_{quantitatively} in termsofthefractionaldegreeofconversionofepoxide groups,usuallydesignateda. Tomodelthe kinetics

it

is

necessarytoderivean equationexpressing

da/dt,

therate ofchangeofa with_time,as afunction ofa andthe

tem-perature T. In _1970, Horieetal.,2 makinguse of mech-anisms proposedby earlierworkers, developedan_equation

todescribethe kineticsofthereaction_{between epoxy and} primaryamine. If

it

isassumedthatthe secondaryamine groups formed in the reaction show thesame _degree of

reactivity toward epoxy groups as the _primary amine groups

initially

present, theirequation simplifies to

da/dt

=

(Kx + K2a)(l

-a)(B

-a) (1)

where

_K\

isarate constantforthe reaction_catalyzedby

groups

initially

present in theresin,Kzisarateconstant for the reaction _{catalyzed by newly formed hydroxyl} groups,andBistheinitial ratio ofamine N-H bondsto epoxide rings. Thusthisequationtakesintoaccountthe autocatalytic nature ofthe _{epoxyamine reaction,} but it

doesnot allowforthe_{possibility of}other reactionsthat produce ethergroups (“etherification”). It was foundto fit experimental data well at lowlevelsofconversion, up toa = _{0.5or so.2,3} The deviations_{observed beyond}this

pointwere attributedtotheonsetof diffusioncontrolas a resultof_gelation ofthesystem.

Many aerospace materials contain aromatic amines, which require higher curing temperatures thanaliphatic

ones. _{Furthermore, there}isoftena_significantexcess of

epoxywithrespecttoamine. Underthese circumstances, theetherificationreactionbecomesmore _importantand the_{Horie approach}is lessvalid. _Attemptstoincludethe etherification_{reaction complicate the mathematics}

con-siderably,so an exact solutionofthe_{kinetic equations}has

not been obtained. In order to model cases where the Horie equation is _{inadequate, Kamal and} co-workers developedthe following semiempirical modification:4

da/dt

=

(Kx + K2am)(1

-a)n (2)

Theintroductionofthe_{variable exponents}mandn_usually

makes

it

possibleto obtaina_good

fit

to experimental_data,

andthisequationhasfound widesuccessful _application forboth_epoxyand polyestersystems. However, it does

not_explicitlytakeintoaccounttheetherification reaction,

so

it

does not provide a clear _description of_{the curing} process andits chemistry, whichis_{important for} under-standing the networkformationprocess. Theexponents

m andnare oftenfound tobetemperature-dependent,so

the_dependencymustbedeterminedover thewhole range of_temperaturesofinterest.

Recently therehave beenafewattempts to include the etherificationreactionintheanalysis.6"10 Whilethesehave

notmade_{the assumption}thattheprimaryandsecondary aminegroups showthesame _{reactivity,they}have made certain otherassumptionsand approximationsinorder to perform theanalysis. Forinstance, Zukasetal.6assumed that, like the epoxide-amine reaction, the epoxide-hy-droxylreaction involves two rate constants (correspond-ing to“uncatalyzed”andhydroxy-catalyzedreactions)and that their ratio is the same as for _{the epoxide-amine}

reaction. Inorderto obtainan _acceptable

fit

totheirdata, theyhad to_{introduce semiempirical modifications}such

as _letting some ofthe apparent reaction orders deviate

from1 or even _varywiththe_degreeofcure a. Riccardi

and Williams7·8 usedasimilarmechanismbutadifferent

mathematical treatment tostudyadifferent_{system. They}

obtained a _good fit to their data but some ofthe rate

constants,includingthosefor etherification,were

difficult

to determinewith_goodaccuracy becausethe datawere

not _sufficientlysensitive to them. Other workers have assumeda_{simpleepoxide-hydroxyl reaction}asthe ether-ification mechanism.9·10 Chern and Poehlein, in their 0024-9297/91/2224-3093$02.50/0 Published 1991_{by the American Chemical Society}

3094 Cole Macromolecules, Vol. 24,No.11, 1991 R’ I R I R‘ R 1 I 1 nh2 + 1 CH—CH —> 1 1 NH—CH-—CH— _OH Primary

Amine Epoxide Senary Hydroxyl

R R‘ I I R 1 R R· 1 1 R 1 1 1 HO-CH-CH-NH + CH-—CH —> H0-CH-CH-N-CH-CH-0H

V

2 2 Secondary _Epoxide

Amine TertiaryAmine Hydroxyl

R* R I I R R‘ I I R R | 1 1 NH-CH-CH-OH + 1 1 CH—CH —>

NH-V

CH—CH-O-CH 1 — _CH—OH

Hydroxyl Epoxide Ether Hydroxyl

R 1 R I n CH—CH -> — 1 -CH—CH —0-Epoxide Ether n

Figure1. Mainreactionsinvolvedinthecure of_{epoxy resins}

with primaryamine_{curingagents.}

analysis,9 determined only the rate constant for the reaction_{between epoxide and primaryamine;}allothers

were _expressed as a fixed multiple of thisone based on

data taken fromtheliteratureforsimilarsystems. Chiao10

also useddatafromothersystemsin order tofixtheratio ofcertainpairsofrateconstants. _Althougha_good

fit

and

reasonableresultswere obtained,thetransferofsuchdata fromone _system to anotheris_{subject to question,given}

that some of the rate constants are _dependent on the

amount_{of catalytic impurities in} theparticularsystem. In this paper, an alternative _{approach to solving the}

kinetic_equationsis_developed,in which the Horie treat-ment is extended to include theetherification reaction. Althougha_{simpleequation relating}

_da/dt

toacannotbe

obtained,

it will

beshownthat itis_possibletoindirectly determinethe exactrelationshipbetweenthe two. Dif-ferentpossible mechanisms fortheetherification reaction

are _considered,either involvingor not involving hydroxyl

and_tertiary aminegroups. Model Development

Epoxy Reactions. The cure of_epoxyaminesystems

has beenreviewed _by Barton11 and_Rozenberg.12 For a

primaryaminecuringagent,themain reactionsthatoccur are illustratedin Figure1. Their_{relative importance}has

been_{the subject ofmuch research,}butthesituationisfar from clearly understood. Thefirstreaction, whichoccurs

quite readily,isbetweenan_{epoxidering(E) and}a_primary

amine_group(PA)to producealink_containinga_secondary

amine group (SA) and a _{hydroxyl group (OH). The}

secondaryamine group formedin thisreactioncan react

further to give a _{tertiary amine group (TA)} and a new

hydroxyl group. As mentioned above, both of these reactions have beenshown to be catalyzedby hydroxyl groups. Anotherreactionthatcan occur isthatbetween

an _{epoxide ring}and ahydroxylgroup to form an ether

link

anda new _{hydroxylgroup,}whichisthenavailablefor further_{reaction. The epoxide-hydroxyl reactionis} gen-erallyslower_{than the epoxide-amine reaction and}becomes

importantonlywhenthecure is _{performed at high}

tem-peraturesor whenthereisan excess of_epoxywith_respect

to amine.11 Thesituation regarding the reactionbetween

Table I

Symbols Usedin theKinetic Analysis

group representation concn at time t concn at t-0 epoxide E E So hydroxyl H H Ho

primaryamine Ai A, A10

secondary amine A2 0

tertiaryamine As A3 Aao

ether ether [ether] 0

“impurity”catalyst X X X

epoxide and secondary amine is less clear. There is evidence that the _secondary aminegroups react at the

same rateasthe_primaryones,butthereis alsoevidence that they reactsignificantlymore slowly.12-13

Evenif no reactive amineis_{present,epoxy resins}

will

polymerize on theirown if heated toa _sufficientlyhigh temperature. This isattributedto a

“homopolymeriza-tion"reaction,which maybeinitiatedbyimpuritygroups present in the resin or _{by (nonreactive) tertiary} amine groups. Itis_generallyconsideredtobe

difficult

toachieve

unless specific catalystssuchasborontrifluoridecomplexes are _present.

Kinetic

_Equations. The resin system may be

con-sideredas a collectionof_{epoxide groups,} amine_groups, hydroxylgroups,ethergroups, andcatalysts. Thevarious

speciesand theirconcentrations maybe _{representedby}

thesymbols showninTable I. Weassume thatthe

un-reacted resin_{contains onlyepoxide,primary}and_tertiary amine,hydroxyl,and catalyst(otherthanhydroxyl)groups. The_primaryaminegroups come from the curing_agent.

Tertiaryamine groupsmaybe_presentas_partofthe_epoxy

molecule,so an

initial

concentrationterm Agoisincluded

toallowfortheirpresence; suchisthecase,forexample, in the well-known commercial productTGDDM (bis[4-(diglycidylamino)phenyl]methane). The “unknown" cat-alyst

X

representsimpurities thatmaybe_presentinthe resin; the concentration is taken to be constant. The proposedreactionscheme is as follows:

*1 E+ _Ax (+

)

— A2+ (+H) k\ E+

_Aj

_(+X)-*· A2+ (+X) *2 E+ _A2(+

_H)^A3

+

_H(+

_H) *'2 E+ _A2(+X)— A3+ (+ X) *3 E+ mH+ _nA3-»· ether+ mH+ _n A3

The first equation describesthe reaction between an

epoxide group and a _{primary amine group to form} a

secondaryamine group anda new _{hydroxylgroup.}

It

is

catalyzed by hydroxylgroups, H. These _{participate in} thereactionbutremain_unchanged,so_theyare shownin parentheses. The second equation describes the same

reactionbut_catalyzedbygroupsX

initially

present in the system. Thethirdandfourth_equationsare _{similar,except}

that _{they refer to the reactions between epoxide} and secondaryaminegroups toformtertiaryamine _and hy-droxylgroups. Thefirstfour equationsare _{equivalent to}

thoseusedbyHorie.2 Thefifth isnew _{and represents}the

etherification reaction. In _{the following treatment,} we use the term etherificationtorefer to_{both the} epoxide-hydroxylreaction and_{the homopolymerization reaction.} Inbothcases,the netresultisthesame: an _epoxidering istransformedintoanether_{linkage and}thereis no _change

Macromolecules, Vol. 24, No. 11, 1991 Modeling the CureKineticsofEpoxyResins.

Table II

Coefficients in Equation30Correspondingto Different PossibleMechanisms for theEtherification Reaction (Equation8)

term m n Ci c2 c3 c4

E 0 0 0 0 1 -1

EH 1 0 0 0 Y-R -(Y+1)

Em 2 0 -( +1) 0 _(Y-fi)2

-(Y+l)2

EA$ 0 1 -(«+ 1) 0 Z+R2 ~(Z+1)

3

1 1

(R+l)(R-

Y-l) _-V=(fl+1) _(Z+ R*)(Y-R) -(Z+1MY+1) inthenumberof_hydroxylgroups. However,inthefirst

case one_{hydroxylgroup}is_{destroyed and another}isformed. Inthesecond case,hydroxylgroups donot participate. To

cover thedifferentpossibilities, the equationiswrittenin

a _general form with coefficient “m” _{representing the}

number of hydroxylgroupsinvolved(aseither reactantor

catalyst) and“n"the number_{of tertiaryamine groups}(as

catalyst). Bothmandncouldbezero. _{(These exponents} m andn shouldnotbeconfusedwiththose usedineq 2.)

Thekinetic_equationsfortheabovereactionschemeare as follows:

dE/dt

= -k1HEA1 -k’1XEA1 -k2HEA2 -k'2XEA2-k3EHmA3n (3)

dAJdt

= -k1HEAl

-k\XEA1

(4) dA2/dt=

+klHEAl

+ _k'1XEAl -k2HEA2-k'2XEA2 (5) dAg/df = +k2HEA2 + k'2XEA2 (6)

dH/dt

= +k1HEAl +

k\XEAl

+ k2HEA2 + k'2XEA2 (7) d

_[ether]/di

= +k3EHmA3n (8)

The term ineqs3and8arisesfrom the etherification reaction.

It

is assumedto be

first

orderwith_respectto epoxide concentration, butthe exponents m and n _may be_0,1,or 2,dependingon theother_{speciesparticipating}

inthe reaction. _{By manipulation ofeqs}3-8,

it

shouldbe

possibletosolvefortheconcentrationofall_{six components} as a function of time. The concentration of catalytic impurities,

X,

is unknown but is assumed to remain constant_throughoutthecure. _{Thus the quantities}

k\X

and k'zX could be replaced by constants

k'\

and

k"i,

respectively. This is the same formulation that would

result

if

the reactionwere assumedtobe_uncatalyzed,so asfarasthe mathematical developmentis concerned,it

doesnot matter whether the reactionisconsideredtobe

uncatalyzed, X-catalyzed,or both.

Solution of

theEquations. Thereare three material

balance constraints inherent in _{these equations.} First, thelossofone

N-H

bond_alwaysresults inthe formation ofone

O-H

_bond,sothetotalnumberof

N-H

bondsplus O-H bonds is constant. Thus

2_{Aj + A2}+ = _constant =

2A10+ H0 so that

A2=

2A10+

H0-2A1-H

(9)

Second,thetotal numberofnitrogenatomsisconstant,

so that

Aj

+ _A2+ _A3= constant= A10+ A30 and A3— Ago+ A10

-Aj

-A2

Substituting forA2 fromeq 9 gives

A3=

Ago+ A1

-A10+

H

-H0 (10)

Third, thetotalnumberof_{oxygen atoms}isconstant, so

E+H+ _[ether] = _constant= E0+ H0 and [ether] = E0+

H0-E

- H (11)

On_{substitutingeqs}9and10intoeqs3,4, and7we are

left with_{the following three equations}

dE/dt

+

_dH/dt

= -k3EHm(,A30+A1 + -A10-H0)n (12)

dAJdt

+

_dH/dt

= _(k'2X+

k^I){2Aw

+

H0-2A1-H)E

(13)

dAJdt

=

-(k\X

_{+ k1H)EA1 (14)} Equations12-14involve only three unknowns(£,A\,and H) and may besolvedforthese as a functionoft. The

remaining three unknowns(A2, A3, and [ether]) maybe

obtainedfromthese_bymeans of_eqs9-11. Tosolveeqs

12-14,

it

is_helpfultomake_{the following transformation} todimensionless variables: = ₁

-Jr

or E = E0(

1-a)

(15) 2A, +A2

H-H0

0= 1--= _or H = _Ho+ 2Alo0 ₍₁₆₎ ¿ ^ 10 7= 1- T^" _or

Aj

= A10(l -7) (17)

The_quantityaisthe well-knownextentof_conversion,as

defined in termsof thefraction ofepoxidegroups reacted. The variable0 isthefractionofthe

N-H

bondsthathave reactedwithepoxide, and sincethelossofeach

N-H

bond resultsinformationofone O-H_bond,

it

isalsoameasure

ofthenumberof hydroxylgroupsproducedby the reaction. Finally,7 isa measure ofthenumber _{of primary}amine groups reacted. Whent = _{0(unreactedresin),a} =

ß- y

= _0,and whenthereactionis_completea = _ß= _y= _1.The

evolutionofthesethree variables with time _completely describes_{the curingprocess, sincetheycorrespond}to the reactionofepoxidewith primaryamine only (7),primary plus secondary amine (0), and primary amine plus secondaryamine plus hydroxyl (a). We alsodefine the followingconstants: B= _2Aio/£q= _{amine-to-epoxide ratio}

in unreactedresin; Y=

Hq/2Aiq=

Ho/BEo- measure of hydroxylcontent in unreactedresin; Z=

A30/A10= _2A30/

BEo - measure of tertiary amine content in unreacted resin. Ontransformation, eqs12-14 become

=

k3(BE0)m+n(.Y +

d)m((l/2)Z-

(1/2)

+

3096 Cole Macromolecules, Vol.24,No. 11, 1991

^-||*

=

WzXE,+

WoE,

+

_{^02 ß\}

X

(7-0)(l-«)

(19)

d7/dt

=

\k\XE0

+ _kxH0E0+ _kxE2B0\ X

(l-7)(l-«)

(20)

It

is_{impossible to}solvethese_equationswithoutmaking

some _{simplifications.}

Horie etal.madethe assumptionthatthe_{reactivity of} the_secondaryamine groups as _comparedto that ofthe

primaryamine groupsisthesame forboth the X-catalyzed

and_{the hydroxyl-catalyzed}reactions. In otherwords k2/k1 -

k'2/k\

= r (21)

and we obtain

da

_ 3( +ß (

+ ß2)"

ß

₍

+

_2ß)(1-ß)

Thisequation maybesolvedexactly,usingthemethodof integration bypartialfractions, togivea as a function of ß. For integralvalues ofm and n withm + 2n < 3, the

term onthe_rightmaybe_expandedinthe following form:

[Tl

+ + Kx +

2ß

+

W?]

wherethecoefficientsT¡dependon the_particularvalues

ofm andn. Suchan _expressionis _{easilyintegrated and}

thesolution to eq 29maybe written in the form

With

thisassumption,dividingeq 20intoeq 19leadsto

This_maybesolved(withthe

initial

condition7= ₀when

ß =

0) togive

^

=

+1^ ^(1-7)-(1-7)

(23)

2(1

~r)

Thisestablishestherelationbetweenßand7, and makes it_possibleto eliminateßfromeqs18-20,reducing them to two equations in two unknowns. Unfortunately, an

exactsolution is

still

impossible,andtoproceed further

it

isnecessaryto assume thatr =

x/z· This impliesthat the_secondaryamine

N-H

bonds reactwiththeepoxide ringsas_readilyasdotheprimaryamine

N-H

bonds.Since

a_{secondaryamine group}has_onlyone

N-H

_{bond, whereas} a_primaryaminegrouphastwo, theprobabilityofreaction

is halved. As already mentioned, there is conflicting experimentalevidenceon this_point,withcases wherethe assumptionhasbeenfound tobevalid andothers where it hasnot.12,13 Whenit is _valid,eq 23 _{simplifies to}

d=

l-(l-7)1/2

_{or 7} =

0(2-0)

₍₂₄₎

On_combiningeqs 19and 20_{and using}this _relationship to eliminate7, we obtain

d0/dt

= (Kx +

2ß)(1

-0)(1 -a) (25) where Kx=

_(l/2)E0(k\X

+ kxH0) and K2=

(1/2)E0\

₍₂₆₎

It

shouldbenoted

thatKi

combinestheeffectsofcatalysis byallgroups

initially

present,both unknown(X)and hy-droxyl(Ho). Although not specificallyconsidered here,if anyuncatalyzedepoxide-amine reactionwere _presentit

wouldalsobeincludedin

_K\

and_{eq 25}would

still

apply. Thesecond rate constant Kzcorresponds to catalysisby onlythose hydroxyl groupsformed inthe reaction.

On_eliminating7 fromeq 18, we obtain

^- &

=

3( +ß) (

₊

ß2 - )

₍₂₇₎

where

K3 =

k32~n(BE0)m+n (28)

Wenow haveasetoftwo _{equations (eqs 25}and27) in

two _{unknowns (a}andß). It is still impossible to obtain exact solutionsfora and/3as a function oft. However, if we divide_{eq 27by eq 25,}the ₍₁

-a)term cancels out

a =

ß

+

+ß [^ß

+

C/

+ _C3In(1 + RTlfi) + C4In

(1-0)]

(30)

whereR = _Kx/BKz. Theinitial condition _a = _0when

ß

= _{0hasbeenapplied. Expressions}for the coefficients_C\,

Cz, C3, and C4 are _given in Table II for five _possible

combinations ofm and n.

Combiningeqs 25and 27gives

da/dt

=

[B(KX +

2ß)(

1

-0)+ K3(Y+ 0)m(Z + 02)n] X

(l-«)

(31) It is_{impossible to}inverteq 30 toobtain0 intermsofa.

Consequently,eq31cannotbeconvertedintoan analytical expression relating

da/dt

to a. _However, the exact

relationshipbetween

_da/dt

anda can bedeterminedon a _{point-by-point} basis. Thus, for a _{given set} of rate constants

( ,

Kz, K3) anda _{given value}of_{0, eq 30may} beusedtocalculatethe correspondingvalueofa,followed

byeq31tocalculate

_da/dt.

Repeating thisprocedurefor

asetofvaluesof0rangingfrom0to1 _givesan exact set

ofpoints(a,

da/dt),

whichcan beusedto_plotacurve and comparewithexperimentaldata. The rate constantscan

thenbevaried

until

thebestmatchisobtained. Different reactionmechanisms fortheetherification reactioncan be investigated by using the appropriate values of the exponents m _{and n,} as _given in Table

II. It

should be

noted that this procedure allows no control over the

particularvaluesofaobtained. If itisdesiredtocalculate thevaluesof

_da/dt

_{corresponding toparticular}valuesof

a,thenan alternative procedurecan beusedinwhicheq 30is solved _numericallyfor0 andeq 31 is then usedto calculate

_da/dt.

Theuse ofthetwovariablesa and₀tomodelthecure

makesit_{possible to}followquantitativelythetworeaction paths, epoxy amine and etherification. The relative importanceofthese

will

vary dependingon the

temper-ature. Once thethree rate constantsandtheir temper-ature dependence have beendetermined, thecure can be

numerically modeled for any temperature program of interest byusingeqs 25and 31. If theratio ofhardener to epoxy in theresin ischanged,in principle thiscan be

takenintoaccount_{by calculating the}changesinBandEo

andmaking the appropriate adjustments to

K\,

Kz,and K3(eqs 26and28). Itshouldbenotedthateq 30,relating

a to_{ß,applies only}forthecase ofisothermal_{cure, where}

therate constants do not change. Itsmainusefulness lies in analyzing isothermal cure datato determine the rate

constants.

The concentrations ofthe various speciesinvolved in

the reactionsmay also berelated to a and_ß,bymaking

Macromolecules, Vol. 24,No. 11, 1991

initial

_epoxide concentrationEq,these are as follows Epoxide E/E0 = ₁ -a (32) Hydroxyl H/E0-

B{Y

+ ß) (33) Primaryamine A1/E0= (1/2)B(1 -ß)2 (34) Secondaryamine A2/Eo=

B0(l-0)

₍₃₅₎ Tertiaryamine (formedinreaction)

AJE0

=

(1/2)B02 (36)

Ether

[ether]/£o =

a-B0

₍₃₇₎

Equations25and30-37thus allowa_complete

descrip-tionofthecuringprocess. Furthermore,as_eq31is_simply adifferentrepresentationofeq3,itmaybebrokendown

as follows into contributions from the three different reactions involved:

PA-E

reaction

(da/dt)!

= BIKX +

)(

1-

a)(l

-ß)2 (38) SA-Ereaction

(da/dt)2 =

_BiKi

+

2ß)(

₁-

a)(l

-ß)ß (39)

Etherificationreaction

(da/dt)3

= 3( +ß) (

+

ß2 - )

(40)

It

is sometimes assumed that the reaction can be

approximately dividedintotwo_stages.

At

_{the beginning} ofthe cure, the amine-epoxide reaction dominates and etherificationis_{insignificant. Thus,}K$can be_{set equal}

tozero. _Equation 27thenreducesto =

ß,

and_{eq 25} reducestoeq1,the Horie equationwithr =

1/2. Toward

theendofthecure,the approximationcan bemadethat all the amine groups have reacted and the _hydroxyl concentrationis constant. In thiscase

0=1

andeq 31

reducesto

da/dt

= K3(Y+ 1)m(Z + 1)"(1 -a) = K'3{1 -a) (41)

Inotherwords,attheendofthecure the reaction tends

to become _simply first order with respect to epoxide

Modeling the Cure Kinetics ofEpoxy

concentration. This approximate approach has been appliedwith some success to one commercial_system.14

Conclusion

Anew _{approach to modeling the} cure of_epoxyresins

with primaryamineshas_{been developed.}

It

is basedon

the model_{proposedby Horie} but isextended to include the etherification reaction. The basic assumptions in-herentinthe modelare (i) the epoxide-amine reactions

are _hydroxylcatalyzed; (ii) thesecondaryaminegroups have thesame _{reactivity withrespect}to epoxide as the

primaryamine groups; and(iii)the etherification reaction isfirst orderwith respectto epoxideconcentration and mayalsoinvolvehydroxylgroups,tertiaryamine groups,

or both. Themodel makesuse ofno_{empirical parameters}

andno _{noninteger reaction}orders. Theevolutionofthe reaction isdescribed intermsoftwo variables, a and _0.

The

first

is the usual overall _{epoxide conversion.} The second isa measure oftheextentofthe epoxide-amine

reaction. Thus the two differentreaction pathwayscan be followed quantitatively. The kineticequationshave been solvedtoobtainan _{analytical relationship}between

a and0. Althoughan _explicitexpression cannotbederived torelate

_da/dt

toa, bothcan _{be expressed}intermsof_0.

Thus the exact _relationship between the two can be

calculatedon a_{point-by-point}basisfora_givensetofthe three rate constants involved. The best values for the rate constantscan be _{determined by calculating}a curve

of

_da/dt

versus a,comparingitto experimental data,and varying the rate constants until a suitable match is

achieved. Oncethe rate constantsare _{known, the curing}

processcan becompletely described, includingthe

evo-lution ofthe differentchemical_species as a function of

thedegreeofcure. The effecton thekineticsofa_change

inamine/epoxyratiocan alsobe_{predicted. Compared to}

previous models, thisnew _approach has the _important advantageofallowinganaccurate_{description of}the_curing

processover thewhole rangeofcure,without introducing

empirical parametersor _{makingapproximations}suchas

separating the reactioninto distinct regimes. Although developed forepoxy curedwith primaryamine,themodel couldbemodified tocover thecaseof_secondaryor mixed

amines. Inthesucceedingpaper, thesuccessfulapplication ofthemodeltoa_typicalcommercial productisdescribed.

References andNotes

(1) Roberts, R. W.SAMPE J. 1987,23(5) _(Sept/Oct.) 28. (2) Horie,K.; Hiura, H.;Sawada,M.;Mita,I.; Kamoe, H.J.Polym.

Sci.,PartA-l 1970, 8, 1357.

(3) Sourour,S.;Kamal, M.R.Thermochim.Acta 1976,14,41. (4) Kamal, M.R.Polym.Eng. Sci. 1974,14, 231.

(5) Zukas, W.X.;Schneider, N.S.;MacKnight,W.J.Polym.Mater. Sci. Eng. 1983, 49, 588.

(6) Zukas, W.X.; Dunn,D. A.;Gilbert, M.D.Polym. Mater.Sci. Eng. 1987,56, 346.

(7) Riccardi,C. C.;Williams,R. J. J.J._Appl.Polym.Sci. 1986,32, 3445.

(8) Riccardi,C. C.;Williams,R. J.J. _{Crosslinked Epoxies,}Proc. Discuss. Conf.,9th1986, 291.

(9) Chern, C. S.;Poehlein,G.W.Polym.Eng. Sci.1987,27, 788. (10) Chiao, L. Macromolecules1990,23, 1286.

(11) Barton,J.M.Ado.Polym.Sci. 1985, 72,111. (12) Rozenberg, B. A. Ado.Polym. Sci. 1986, 75, 113.

(13) Charlesworth,J._{J. Polym.} Sci.,Polym. Chem. Ed. 1980, 18, 621.

(14) Lee, W.L;Loos, A. C.;Springer,G.S.J.Compos.Mater.1982, 16, 510.