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Xavier COLIN, Fatima ESSATBI, Justine DELOZANNE, Gurvan MOREAU - Towards a general
kinetic model for the thermal oxidation of epoxy-diamine networks. Effect of the molecular mobility
around the glass transition temperature Polymer Degradation and Stability Vol. 181, p.117
-2020
Any correspondence concerning this service should be sent to the repository
Towards
a
general
kinetic
model
for
the
thermal
oxidation
of
epoxy-diamine
networks.
Effect
of
the
molecular
mobility
around
the
glass
transition
temperature
Xavier
Colin
a,∗,
Fatima
Essatbi
a,
Justine
Delozanne
b,
Gurvan
Moreau
ba PIMM, Arts et Métiers Institute of Technology, CNRS, CNAM, HESAM University, 151 boulevard de l’Hôpital, 75013 Paris, France b Safran Composites, A platform of Safran Tech, 33 avenue de la Gare, 91760 Ittevile, France
Keywords: Epoxy-diamine network Thermal oxidation Kinetic model Oxygen consumption Carbonyl build-up Molecular mobility
a
b
s
t
r
a
c
t
The kineticmodel previouslyestablished fordescribing the thermal oxidationof polymethylenic sub-strateshasbeensuccessfullygeneralizedtoaseriesofsixepoxy-diaminenetworks(EPO-DA) character-izedbyverydifferentglasstransition temperatures.Thismodelis derivedfromthe so-called “closed-loop” mechanisticschemewhichconsists inaradical chainreactioninitiatedbythe decompositionof hydroperoxidesand propagatingviatheC-Hbondslocatedinα ofheteroatoms(NandO).The numer-ousmodelparametersweredeterminedbyapplyinga“stepbystep” procedurecombiningexperiment andsimulation.Ontheone hand,oxygentransportproperties(i.e.coefficients ofoxygen diffusionand solubility)wereestimatedfromacompilation ofliteraturedata.Ontheotherhand,rate constantsand formation yieldsweredetermined byinversesolving methodfrom themeasurements ofoxygen con-sumption andcarbonylbuild-up performedonsixdifferent EPO-DAnetworksbetween 25 and200°C andbetween0.16and 20barsofoxygenpartial pressureinourlaboratory orinthe literature.It was foundthatthemolecularmobilitymainlyaffectstherateconstantsoftheelementaryreactions involv-ingthereactivespeciesinthelowestconcentration,i.e.peroxyradicals.Infact,therateconstantk6of theapparentterminationofperoxyradicalsisreducedbyaboutfiveordersofmagnitudewhenpassing fromrubberytoglassystateduetothefreezingoflargeamplitudecooperativemolecularmovements. Incontrast,therateconstantk3 ofthepropagationofoxidation,involvingperoxyradicalsbutalsothe polymersubstrate,isonlychangedbyoneorderofmagnitudearoundtheglasstransitiontemperature. TheintroductionoftheeffectofmolecularmobilityintotheArrheniuslawsofk6 andk3allowsbuilding mastercurvesandfinally,proposingasinglekineticmodelforthewholefamilyofEPO-DAnetworks.
1. Introduction
Thevast majorityofcompositematerial structuresusedinthe civil aeronautical sector are made of diamine cross-linked epoxy matrix(EPO-DA)reinforcedwithcarbonfibers.Thedevelopmentof a numericaltool forpredictingthedegradationstate ofthis poly-merfamilyunderthecombinedeffectsoftemperature,oxygenand mechanicalstressremainsanopenissueinthissector.Indeed,this toolcouldhelpaeronauticalmanufacturerstotakeintoaccountthe possiblealterationofthethermomechanicalpropertiesof compos-itestructuresfromthemomentoftheirdesignandsizing,butalso toconsidertheuseofthesematerials inmoreaggressive thermo-chemicalenvironments.
∗ Corresponding author.
E-mail address: xavier.colin@ensam.eu (X. Colin).
Thereisalargeamountofliteratureworksdevotedtothe ther-maldegradationmechanisms[1-9]andkinetics[10-17]ofEPO-DA networks,showingthatoxidationisclearlythepredominantaging process. However, in thesepublications, each network is consid-eredasacompletelydifferentmaterialfromothernetworksofthe EPO-DA family, because it is assumedto be characterized by its own chemistry, its own oxidation mechanisms, its own degrada-tionkineticsanditsspecifickineticmodel.Therefore,thevaluesof therateconstantsdeterminedbyusingthesekineticmodelsasan inversesolvingmethodarefoundtobequitedifferent,butwithout givinganyrealexplanationexcept,perhaps,the possibleeffectof molecularmobility[15].
However, contrary to popular belief, there are not so great chemicaldifferencesbetweenallthenetworksoftheEPO-DA fam-ily,especially fromachemicalkineticspointofview,asshownin Table1.Indeed, letusrecall that, accordingtoKorceketal.[18],
Table 1
Main oxidation sites in EPO-DA networks.
Starting epoxy monomer Oxidation sites
DGEBA, DGEBF, DGEBD, TGPAP, TGTPM, TGMDA, etc…
TGPAP and TGMDA only
Abbreviations : Diglycidyl ether of bisphenol A (DGEBA), Diglycidyl ether of bisphenol F (DGEBF), Diglycidyl ether of bisphenol D (DGEBD), Triglycidyl ether of para-amino phenol (TGPAP), Triglycidyl ether of triphenyl methane (TGTPM), Tetraglycidyl ether of methylene dianiline (TGMDA).
oxidation propagates via the most labile hydrogens of the con-stitutive repeating unit. Thus, in EPO-DAnetworks, the main ox-idation sites are oxy-methylene (-O-CH2 -) and amino-methylene (>N-CH2 -)groups,whicharealreadypresentinthestartingepoxy monomers, but also amino-methylene (-CH2 -N<) and methanol groups(>CH-OH)whichareformedduringthepolymerization re-action between the epoxy monomer and diamine hardener. All theseoxidation sitesare reportedin Table 1,as well asthe cor-respondingnameofthestartingepoxymonomer.
Basedontheseobservations,itisdifficulttoimaginelarge fun-damental differences between the oxidation mechanisms of the differentEPO-DAnetworks.However,astheirglasstransition tem-peraturecanvaryinawidetemperaturerange,typicallybetween 0°C and 221°C [19], it is expected that the molecular mobility plays indeed a key role in their oxidation kinetics, in particular onbothsidesoftheglasstransitiontemperature. Inother words, atasameagingtemperature, itwouldnotbe surprisingtofinda large difference betweenthe oxidation ratesof two EPO-DA net-worksthatare indifferent physicalstates(i.e.in glassy and rub-bery states). Unfortunately, until now, there are too few experi-mentaldataavailableintheliteratureonthethermaloxidationof EPO-DAnetworkstodoubtlesscheckthisassumption.Inaddition, almostallstudiesarefocusedonthermalaginginglassystatethat is,of course, the usual temperature range for industrial applica-tionsofthesematerials.
The impact of molecular mobility on the oxidation kinetics was clearly put in evidence for another thermodynamic transi-tion:themelting,inanelectron-beamirradiationcross-linked lin-earpolyethylene(XLPE)[20,21].Ajumpofabouthalfadecadewas observedfor the pre-exponential factoraround the melting tem-perature(130°C)intheArrheniusdiagramoftheoxidation induc-tiontime.However,thisdiscontinuityisrelativelysubtleandthus, canbehiddenbythewiderscatteringofexperimentaldatainPE compilations[22].
The effect of glass transition on the oxidation kinetics is ex-pected to be much more marked because all the diffusion pro-cessesarefavoredbylargeamplitudecooperativemolecular move-mentsinrubberystate, whereasthey are almosttotallyfrozen in glassystate.Infact,asforacommonpolymerizationreaction,two kineticregimesshouldbedistinguished:i)OneaboveTg ,forwhich thereactionkineticsisgovernedbytheintrinsicreactivityof
reac-tivespeciesandthus,canbesatisfyinglydescribedbytheclassical rulesofchemicalkinetics;ii)AnotherbelowTg ,forwhichthe re-action kinetics is governed by the diffusivity of reactive species. Thislatterregime wasspecificallyinvestigatedby Waite [23]and Russianresearchers[24,25].
That is the reason why some aromatic polymers which con-tainsalso few hydrocarbongroupsreputed to be veryeasily oxi-dizable(e.g.methynicandmethylenicgroups),suchaspolyimides of poly(bismaleimide) type [26] or polymerization ofmonomeric reactant (PMR) type [27], can be considered as thermostable in glassy state. This is also the case for many aromatic epoxies [10,28,29].
The present article aims at laying down the main lines of a general kinetic model for the thermal oxidation of networks of the EPO-DA family and checking its validity. After having re-called its theoretical bases and structure, its different param-eters will be determined by applying a “step by step” pro-cedure combining experience and simulation. First of all, oxy-gen transport properties(i.e. coefficients ofoxygen diffusionand solubility) will be estimated from a compilation of literature data. Then, rate constants and formation yields will be deter-minedby inversesolvingmethodfromthemeasurements of oxy-gen consumption andcarbonyl build-upperformedon six differ-ent EPO-DA networks between 25 and 200°C and between 0.16 and 20 bars of oxygen partial pressure in our laboratory or in literature.
2. Theory
2.1. Oxidationsites
Asmentionedinintroduction,themainoxidationsitesin EPO-DAnetworksareoxy-methylene(-O-CH2 -),amino-methylene(> N-CH2 -) andmethanolgroups(>CH-OH)(seeTable 1).Indeed,C−H bonds are characterized by a much lower dissociation energy (ED ≈376kJ.mol−1 )whenthey arelocatedin
α
ofaheteroatom(O orN)ratherthaninpolymethylenicsequences(ED ≈ 393kJ.mol−1 ) [30].Table2recalls theArrheniusparameters andorders of mag-nitudeat30°Coftherateconstantk3 forthepropagationof oxi-dationdetermined intheprevious studiesconductedinour labo-ratory[22,31-37].Table 2
Arrhenius parameters and orders of magnitude at 30 °C of the rate constant k 3 for the propagation of oxidation on
the main types of methylenic C −H bonds.
C-H bond k 30 (L.mol −1 .s −1 ) E 3 (kJ.mol −1 ) k 3 at 30 °C (L.mol −1 .s −1 ) Polymer References
−CH 2 −CH 2 − 1.5 × 10 10 73 4.0 × 10 −3 PE [22 , 31 , 32]
9.9 × 10 10 78 3.6 × 10 −3 EPDM [33]
−CH = CH −CH 2 − 4.0 × 10 9 68 7.7 × 10 −3 BR [34]
−O −CH 2 − 1.8 × 10 9 63 2.5 × 10 −2 PET, EPO-AN [35 , 36]
> N −CH 2 − 1.8 × 10 9 63 2.5 × 10 −2 PA 6–6 [37]
Abbreviations: Polyethylene (PE), Ethylene-Propylene terpolymer (EPDM), Polybutadiene (BR), Poly(ethylene tereph- thalate) (PET), Anhydride cross-linked epoxy (EPO-AN), Polyamide 6–6 (PA 6–6).
Thus, as linear polyesters [35], epoxy-anhydride networks [36] andaliphaticpolyamides[37],EPO-DAnetworkswillbe sub-jectedtoamuchmoresevereoxidationthanPE [22,31,32].
2.2. Mechanisticscheme
Theso-called“closed-loop” oxidationmechanism,developedfor polymethylenicsubstratesinpreviousstudies[22,31-37],was nat-urallychosenforEPO-DAnetworksinthepresentstudy.Its gener-alizationtoEPO-DAnetworksrequiredare-adjustmentinthe con-sumptionyieldsofPHsubstrateinthedifferentbalancereactions (i.e.involvingseveralelementarychemicalevents).Itsfinalversion, alreadydetailedandexplainedinreference[37],isgivenbelow:
Initiation:
1u) POOH → 2P•+(1−
γ
CO )P-OH+γ
CO P=O+γ
1 S {−(2+γ
1 )PH}(k1u ) 1b) 2POOH → P•+PO2 •+(1−γ
CO )P-OH+γ
CO P=O+γ
1 S {−(1+γ
1 )PH}(k1b ) Propagation: 2) P•+O2 →PO2 •(k2 ) 3) PO2•+PH→POOH+P•(k3)(Terminatingandnon-terminating)bimolecularcombinations: 4) P•+P•→
γ
4P-P+(1−γ
4)PH+(1−γ
4)F+γ
4B{+(1−γ
4)PH}(k4 )
5) P•+PO2 • →
γ
5 POOP+(1−γ
5 )POOH+(1−γ
5 )F+γ
5 B(k5 )6a)PO2 •+PO2 •→[PO••OP]cage +O2 (k6a ) 6b)[PO••OP]cage →POOP+B(k6b )
6c)[PO••OP]cage →P-OH+P=O(k6c )
6d) [PO••OP]cage → 2P•+2(1−
γ
CO )P-OH+2γ
CO P=O+2γ
1 S{−2(1+
γ
1 )PH}(k6d )where PH;POOH; P•, PO2 • andPO•;POOP, P-OH andP=O;F; S and B designate an oxidation site; an hydroperoxide; alkyl, per-oxyandalkoxylradicals;peroxide,hydroxyl andcarbonylgroups; adoublebond;achainscissionandacrosslinknode,respectively. kiare thereactionrateconstantsand
γ
iare theformationyieldsofthedifferentinactiveproductsinthecorrespondingbalance re-actions.
Verybriefly,letusrecallthemaincharacteristicsofthis oxida-tionmechanism:
i) Thepropagationofoxidation,byabstractingthemostlabile hydrogenatoms(3),leadstotheformationofaveryinstable species:hydroperoxidegroupPOOH.Indeed,thedissociation energyoftheO−Obondisextremelylow:ED ≈ 150kJ.mol−1 , againstmorethan280kJ.mol−1 forallotherchemicalbonds (i.e.C−O,C−N,C−CandC−H)composingtheEPO-DA net-works [30]. Therefore, this main propagation product will be, either from the starting of exposure, or very quickly, themainsourceofradicals.This“closedloop” character al-lowsaccountingforthesuddenandsharpauto-acceleration ofoxidation attheendofan incubationperiod,commonly
called the “induction period” [38]. It is noteworthy that, in the kinetic model, the initial hydroperoxide concentra-tion [POOH]0 is kinetically equivalent to all structural de-fects(e.g.oxidationproductsformedduringthe polymeriza-tionreaction ofthemacromolecular network)andintrinsic species(e.g.catalytic residues,PH-O2 complexes,other im-purities,etc.)responsiblefortheveryfirstoxidationevents. ii) Thethermaldecomposition ofPOOHoccursintwo modes: unimolecular (1u) and bimolecular (1b). When the poly-merisnot toopre-oxidized(i.e.when[POOH]0 < [POOH]C ), thePOOHdecompositionstartsinunimolecularmode.Then, when [POOH] exceeds a critical value [POOH]C , which is a decreasing function of temperature obeying an Arrhe-nius law, the bimolecular decomposition becomes in turn thepredominantmode.ForEPO-DAnetworks, itwasfound that: [POOH]C=6.8 × 103 Exp
(
−40,000/RT)
. On thecon-trary,when[POOH]0 >[POOH]C ,oxidationisinitiatedbythe bimolecularmodethroughouttheexposureduration.Thisis generallythecaseforallpolymersatlowtemperatures (typ-icallyforT<100°C).
iii) The writingofthe two initiations (1uand1b) can be sur-prisingduetotheapparentnon-compliancewith stoichiom-etry. In fact, these initiations are balance reactions involv-ing several elementary chemical events. Their writing will not be detailed here, but the reader can consult reference [37]for further details.Letusjustpointoutthatthiswriting isentirelyjustifiedbecausethedissociationofO−Obondis thelimitingstep.Inaddition,thesameformationyieldscan beused forthe differentinactiveproducts (P-OH,P=Oand S) becausetheyresultfromthesame reactivespecies(PO• andPH)involved inthesetwo initiations. However,due to theignoranceoftheexactnatureofcarbonylgroupP=Oand thus,thevalueofitsmolarextinctioncoefficient,itwas nec-essarytointroduceanapparent formationyield
γ
CO andto distinguishitfromtheactualyieldofchainscissionsγ
1 .The sameconsiderationsapply to thenon-terminating bimolec-ularcombination(6d).iv) Terminationsbetweentwo alkyl radicalsP• (4)orbetween onealkyl P• andone peroxy radicalPO2 • (5)are also bal-ance reactions.Indeed, thesetwo reactions involvetwo el-ementarychemicalevents:couplinganddisproportionation. Thus,
γ
4andγ
5aretheformationyieldsincrosslinknodes.v) Infact,thecombinationbetweentwo peroxyradicals PO2 • (6) involves four competitive elementary chemical events (6ato6d).Thiswritingallowedhighlightingtheexistenceof anon-terminatingcombination,butalsobetter distinguish-ing crosslinking B from chain scissions S. Indeed, in poly-methylenic substrates,the termination of PO2 • is not very efficient[39,40].Asan example,itwasfound thatabout70 and85% of alkoxyl radicals escape from the cage to initi-atenewradicaloxidation chainsat160°C inPE andPA6– 6,respectively[37].However, forasakeofsimplicity,these four chemical events can be grouped into a single
appar-ent terminating combination(PO2 •+PO2 •→inactive prod-ucts+O2 )whosethecorrespondingrateconstantk6 canbe expressed asa functionofall elementaryrateconstants, as follows[22]:
k6 = 2
(
k6b +k6c)
2 k6a
(
k6b +k6c +k6d)
(
2k6b +k6c +k6d)
(1)vi) Finally, it is noteworthythat inactive molecular (P-OH and P=O)andmacromolecularproducts(SandB)areexclusively formed in initiations (1u and1b) and in the (terminating and non-terminating) bimolecular combinations of peroxy radicals(6cand6d).
2.3.2.3Kineticmodel
Asystemoftennon-lineardifferentialequations(SED)was de-rived from the previous mechanistic scheme using the classical rulesofthechemicalkinetics.Thissystemgivesaccesstothelocal concentration changes of all the reactive speciesinvolved in the mechanistic scheme (i.e.[POOH], [P•], [PO2 •], [PO••OP]cage , [PH] and[O2 ]=f(z,t)),butalsotothelocalconcentrationchangesofthe mostimportantinactiveproductsfromapracticalpointofview,as theycan beeasily measuredexperimentally.Inthisstudy,we fo-cusedon oxygen consumption ([O2abs]), carbonyl groups([P=O])
and macromolecular modifications (i.e. B and S). In the case of one-dimensionaloxygen diffusion (e.g.through thesample thick-ness),theresultingSEDcanbewrittenasfollows:
d[POOH] dt =−k1u[POOH]− 2k1b [POOH] 2 +k 3[PO•2 ][PH] +
(
1−γ
5)
k5 [P•][PO•2 ] (2) d[P•] dt =2k1u [POOH]+k1b [POOH] 2 − k2 [P•][O2 ] +k3 [PO•2 ][PH]− 2k4 [P•]2 −k5 [P•][PO•2 ]+2k6d [PO•OP]cage (3) d[PO•2 ] dt =k1b [POOH] 2 +k2 [P•][O2 ]− k3 [PO2 •][PH]− k5 [P•][PO•2 ] −2k6a [PO•2 ]2 (4) d[PO••OP]cage dt =k6a [PO • 2 ]2 −(
k6b +k6c +k6d)
[PO••OP]cage (5) d[PH] dt =−(
2+γ
1)
k1u [POOH]−(
1+γ
1)
k1b [POOH] 2 −k 3 [PO•2 ][PH] −2(
1+γ
1)
k6d [PO••OP]cage (6) d[O2 ] dt =DO2∂
2 [O 2 ]∂
z2 − k2 [P •][O 2 ]+k6a [PO•2 ]2 (7) d[O2 abs ] dt =RO2 =k2 [P•][O2 ]− k6a [PO • 2 ]2 (8) d[P=O] dt =γ
CO k1u [POOH]+γ
CO k1b [POOH] 2 +(
k6c +2γ
CO k6d)
[PO••OP]cage (9) dS dt =γ
1 k1u [POOH]+γ
1 k1b [POOH] 2 +2γ
1 k6d [PO••OP]cage (10) dB dt =γ
4 k4 [P •]2 +γ
5 k5 [P•][PO•2 ]+k6b [PO••OP]cage (11)whereDO2 is thecoefficient ofoxygen diffusion intothe EPO-DA networkandzisthespatialco-ordinate(i.e.thedepthbeneaththe samplesurface).RO2 istherateofoxygen chemicalconsumption, commonlycalledthe“oxidationrate”.
This SED was solved simultaneously in space (z) and time (t) withnumerical algorithms specifically recommendedfor“stiff problems” of chemical kinetics [41], asalreadyreported inmany previouspapers,forinstanceinreference[42].
Initialandboundaryconditionswere:
i) Whent=0,atanydepthzinthesamplethickness:
[POOH]
(
z, 0)
=[POOH]0 =10−4 mol.L−1[P•]
(
z, 0)
=[PO•2 ](
z, 0)
=[PO••OP]cage(
z, 0)
=0 [PH](
z, 0)
=[PH]0and[O2 ]
(
z, 0)
=[O2 ]0ii) Whent≥ 0,onbothsamplesurfaces(z=± L):
[O2 ]
(
−L, t)
=[O2 ](
+L, 0)
=[O2 ]0OxygenconcentrationobeystheclassicalHenri’slaw:
[O2 ]0 =SO2 .PO2 (12)
whereSO2 is thecoefficient ofoxygen solubilityinto theEPO-DA network and PO2 is the oxygen partial pressure in the exposure environment.Thesamplethicknessis2L.
3. Materialsandexperimentalprocedure
3.1. EPO-DAnetworks
The studyfocused on sixperfectEPO-DA networks (i.e. with-out dangling chains) characterized by very differentglass transi-tion temperatures, typically ranged between 95 and 263°C (see Table3).
Fourofthemresultfromthereactionofcommonbi-functional or tri-functional epoxy monomers with an aromatic diamine hardener: 9,9-bis(3-chloro-4-aminophenyl)fluorene (CAF) or 4,4 -diaminodiphenylsulfone(DDS).Overthepasttwentyyears,these four networkswere all considered, inturn, aspotential matrices ofcarbonfiberreinforcedcompositematerialsforstructural appli-cations atlow tomoderatetemperatures (typicallybetween70°C and150°C)in thecivilaeronauticalsector. Themarketing of net-work no. 5was progressivelyabandoned in the early 2000s,but itsthermaldegradationbetween150°Cand210°Cwasthesubject ofseveralpapersafewyearsearlier,forinstance[10-14].Network no.6wasconsideredintheearly2010sforapplicationsat temper-aturesabove200°C,forwhichthereisclearlyalackofsolutionsin organicmaterials inEurope.Its thermaldegradationwasalsothe subject offew papers, forinstance [15,16]. In contrast,networks no.3and4arecompletely newmaterials whosethethermal ag-ingbehaviorshavenotstillbeenpublishedtodate.
FilmsofthesefourEPO-DAnetworks,withthicknessestypically ranging between25 and100μm,were produced by compression moldingthen post-curedunder primary vacuum(i.e.10−3 bar) in accordancewiththe recommendedindustrial curecycle, inorder toreachthemaximumcrosslinkingdensitywhileavoidingany un-desiredpre-oxidationbeforeexposuretothermalaging.Thatisthe reasonwhyaverylow concentrationinstructuraldefectswill be chosenformodelingthethermalagingkineticsofthesenetworks inthepresentstudy,typically:[POOH]0 =10−4 mol.l−1 .
After processing, thefilms were characterized by conventional laboratorytechniques.Inparticular,theirglasstransition tempera-ture(Tg )wasdeterminedbydifferentialscanningcalorimetry(DSC, witha20°C.min−1 heatingrate,undernitrogen)andconfirmedby mechanicalspectrometry(DMA, intensilemode,witha 1Hz fre-quencyand a2°C.min−1 heatingrate, undernitrogen). The mea-surementswerefoundveryclosetothetheoreticalvaluesofTg
Table 3
Molar mass of the repetitive monomer unit (m UCR ), density ( ρ) and concentration in oxidation sites (PH)
for perfect EPO-DA networks.
Number Network m UCR (g.mol −1 ) ρ [PH] (mol.L −1 ) T g ( °C)
1 Epon 828-Jeffamine D230 910 1.16 24.2 95 2 Epon 828-Ancamine 2049 918 1.2 32.7 150–155 3 DGEBF-CAF 1041 1.25 14.4 158 4 DGEBA-CAF 1097 1.25 13.7 182 5 (a)DGEBF/TGPAP-DDS 1362 1.31 15.4 211 6 (b)Tactix 123/Tactix 742-DDS 1674 1.11 10.6 263
Abbreviations : Diglycidyl ether of bisphenol A (DGEBA, Epon 828 or Tactix 123), Diglycidyl ether of bisphenol F (DGEBF), Triglycidyl ether of para-amino phenol (TGPAP), Triglycidyl ether of triphenyl methane (TGTPM or Tactix 742), poly(oxypropylene)diamine (Jeffamine D230), diamino dimethyl dicyclo- hexyl methane (Ancamine 2049), 9,9-bis(3-chloro-4-aminophenyl)fluorene (CAF), 4,4 -Diamino diphenyl sulfone (DDS).
(a) Matrix commercialized under the reference 977–2 by Cytec Fiberite. (b) Matrix commercialized under the name Tactix by Hunstman.
(i.e.without danglingchains).As anexample,themethodforthe Tg calculationofnetworkno.6isdetailedinreference[29].
In addition, EPO-DA networks no. 1 and 2, resulting from the reaction of diglycidyl ether of bisphenol A (Epon 828) with an aliphatic diamine hardener: poly(oxypropylene) diamine (Jef-famine D230) and diamino dimethyl dicyclohexyl methane (An-camine 2049) respectively, were considered in the present study fortwomainreasons:
i) Their Tg is lower than those of all previous networks (see Table3),thusallowingtoanalyzethepossibleeffectof molecu-larmobilityontheoxidationkineticsoverawidertemperature range;
ii) Theiroxygenconsumptionkineticswasmeticulouslymeasured inrealtimeatmoderatetoverylowtemperaturescloseto am-bient(typicallybetween140°Cand25°C)[44].
The concentration inoxidation sites [PH] was estimated from the knowledge ofthe repetitive monomer unit ofthe sixperfect EPO-DA networks. In Table 3, itcan be seen that [PH] is signifi-cantlyhigherforepoxiescross-linked withanaliphatic(typically, between 24and 33mol.L−1 ) than an aromaticdiamine (between 10and15mol.L−1 ),whichmightalreadyexplain asignificant dif-ference inoxidizability betweenthesetwo subfamiliesofEPO-DA networks.Itshouldbepointedoutthat,whenDDSisthe crosslink-ingagent,thesulfonylgroupprotectstheamino-methylenegroups (-CH2 -N<),formedduringthepolymerizationreaction,againstthe radicalattack fromoxidation.Indeed,thesulfonylgroupisa high electron-attracting group whose the inductive effect through the aromatic rings of DDS leads to an increase in the strength of the C−H bond. This stabilizing effect of sulfonyl group was first evidenced by comparing the polymerization kinetics of different epoxy-diaminemixtures [45].This isthereasonwhy[PH]isonly about10mol.L−1 fornetworkno.6.
3.2. Thermalagingandexperimentalprocedures
The oxidation kinetics of networks no. 3, 4 and 6 was stud-ied at 120, 150, 180 and 200°C under an oxygen partial pres-surerangedbetween0.21(ambientair)and20barsinautoclaves. Given the high impact of oxygen partial pressure on the oxida-tion kinetics, the aging experiments at the highest temperature (200°C) were limited to a maximum pressure of 1bar so asnot to observehighconversionratios ofoxidationfromthe veryfirst samplings. At the opposite, at the lowest temperatures (120 and 150°C), theagingexperiments werecarried outabove 0.21barin order to detect significant structural modifications after a mini-mum duration of 3 months. All the films were periodically re-moved fromtheagingchambersandcooledtoroom temperature
Fig. 1. Changes in the FTIR spectrum of network no. 3 during its thermal aging at 150 °C under 0.21 bar of oxygen (i.e. ambient air). The final spectrum (in sky blue) was recorded after 156 hours of thermal exposure.
inadesiccatorcontainingsilicagelforpreventinganymoisture re-covery prior to be characterized by FTIR spectrophotometry. The FTIRspectrawere recorded inatransmission mode between400 and4000cm−1 withaPerkinElmerFrontierapparatus,after hav-ing averaged the 16 scans obtained with a minimum resolution of4cm−1 .Asreportedmanytimesinliterature [1-10,46-48],the mainstructuralmodificationswereobservedinthecarbonylregion wheretwonewwideabsorptionbandsappearedandgrewrapidly with exposure time: one centered around 1680–1690 cm−1 , and the other around 1720–1730 cm−1 . As an example, Fig. 1 shows thechanges over time inthe FTIRspectrum of network no.3 at 150°Cunder0.21barofoxygen(i.e.ambientair).Thesetwobands wereassignedtoamidesandothertypesofcarbonylproducts, re-spectively.Unfortunately,thegreatvarietyoftheselatterproducts (aldehydes, carboxylic acids, phenyl formates, etc.), often result-ingfromoxidationinducedchainscissions inthehydroxylpropyl ethersegment[47],didnotallowidentifyingthemprecisely.Their averageconcentrationthroughoutthefilmthickness[P=O]was de-terminedbyapplyingtheclassicalBeer-Lambert’slaw:
[P=O]=epOD
ε
(13)whereODistheopticaldensityoftheIRabsorptionbandcentered at1720–1730 cm−1 (dimensionless),
ε
isthecorresponding molar extinction coefficient (expressed in L.mol−1 .cm−1 ), and ep is the filmthickness(incm).Forcarbonyl products, typicalvalues of
ε
areranged between 150L.mol−1 .cm−1 (for ketones) and 850L.mol−1 .cm−1 (for car-boxylic acids) [49-53].In a first approximation, an average value of500L.mol−1 .cm−1 waschosenforε
inthepresentstudy.Fig. 2. Arrhenius graph between 10 and 110 °C for the coefficient of oxygen solubility of EPO-DA networks.
The oxidation kineticsof networkno. 5wasmeasured in real time at180 and 200°C underan oxygen partial pressure ranged between0.05and 1barwith a homemade thermogravimeter.All detailsontheapparatusandexperimentalprocedureareavailable inreference[10].
The oxidation kineticsofnetworksno.1 and2wasmeasured inrealtimebetween25and140°Cunder0.166barofoxygenwith anultrasensitivemanometricmethod.Alldetailsontheapparatus andexperimentalprocedureareavailableinreference[44].
4. Kineticmodeling
Thefirststepinkineticstudyconsistedintheseparate determi-nationofthenumerousparametersofthekineticmodel.Theseare thesevenrateconstants(k1u ,k1b ,k2 ,k3 , k4 ,k5 andk6 ), thefour yieldsofinactiveproductsformation(
γ
1 ,γ
CO ,γ
4 andγ
5 )andthe twooxygentransportproperties(SO2 andDO2 ).4.1.Oxygentransportproperties
Figs. 2and3respectively showthe compilationsof the litera-turevaluesofSO2 andDO2 determinedbyoxygenpermeability be-tween10and100°C forEPO-DAnetworks[10,44,54-59].The val-uesmeasuredwithSystech8001andMoconOx-Tran2–21devices at room temperature for network no. 6 in our laboratory were addedtothiscompilation.
It clearlyappears that SO2 erratically varieswithtemperature. Given the relatively large scattering of the experimental data, it wasconsidered,infirstapproach,thatSO2 isindependentof tem-perature.Incontrast,DO2significantlyincreaseswithtemperature
andseemstoobeyanArrhenius’law.Itshouldbepointedoutthat, forbothoxygen transportproperties,no discontinuityisobserved whenapproachingtheglasstransitiontemperature.
Two sub-families of EPO-DA networks are clearly put in evi-dence:
a) For epoxies cross-linked with an aliphatic diamine hardener (JeffamineD230,JeffamineD400,Ancamine2049orIPDA):
SO2 ≈ 5.1 10−8 mol.L−1 .Pa−1 (14)
DO2 =3.010−5 Exp−44RT600 m2 .s−1 (15)
b) And for epoxiescross-linked withan aromatic diamine hard-ener(DDMorDDS,butalsoCAFbygeneralization):
SO2 ≈ 1.4510−7 mol.L−1 .Pa−1 (16)
DO2 =4.710−5 Exp−20RT000 m2 .s−1 (17)
Ofcourse,networkno.6followsthislatterbehavior.Therefore, it wouldseem that the presence of aromaticstructures, i.e. very rigidstructures,inthemacromolecularnetworkpromotesthe dis-solutionofoxygen,butalsoconsiderablyslowsdownitsdiffusion. Atthisstageoftheinvestigations,morein-depthstudiesona se-riesofmodelnetworksarenecessarytoelucidatetherelationships betweenthemolecularandmacromolecularstructures ofEPO-DA networksandtheiroxygentransportproperties.
4.2. Rateconstantsandformationyields
Rate constants and formation yields were determined by in-versesolvingmethod,inparticularbysimulatingascloselyas pos-siblethemeasurementsofcarbonylbuild-upandoxygen consump-tionofthesixEPO-DAnetworksunderstudy.Giventheverylarge numberofparameterstobedetermined,itwasdecidedtoapplya “stepbystep” procedure.
Inafirststep,asthemainoxidationsitesareC-Hbondslocated in
α
ofheteroatoms(NandO),thevaluesdeterminedforaliphatic polyamides in a previous study [37] were attributed to the rate constantsofEPO-DAnetworks.However,itshouldberecalledthat allthesevaluesweredeterminedinrubberystate,whilethe EPO-DAnetworksunderstudyaremainly inglassy state.Therefore,inFig. 3. Arrhenius graph between 10 and 110 °C for the coefficient of oxygen diffusion of EPO-DA networks.
asecondstep,itwasnecessarytograduallymodifysomerate con-stants, in particular those for which the corresponding reactions involvereactivespeciesinverylowconcentration, suchasperoxy radicals PO2 •. Indeed, it was expected that these rate constants wouldbedirectlyimpactedbythefreezingoflargeamplitude co-operativemolecularmovements,whichcharacterizethemain me-chanicalrelaxation
α
associatedtoglasstransition.Inorderof pri-ority,thesearetherateconstantsk6i ofthe(terminatingand non-terminating) bimolecularcombinations,then the rateconstantk3of the propagationof oxidation,because this latteronly involves onePO2 • radical.
Figs. 4and 5show two examples ofnumericalsimulations in glassy state attemperatures well belowthe glasstransition tem-perature (typically when T< Tg – 60°C). Carbonyl build-up was calculatedwithEq.(9)at150°Cbetween0.21and20barsof oxy-gen(Fig.4),andbetween120and200°Cunder0.21barofoxygen for network no.6 (Fig. 5). It can be seen that the kinetic model satisfyinglyaccountsforboththeeffectsofoxygenpartialpressure andtemperatureontheoxidationkineticsofthisEPO-DAnetwork. Therateconstantsandformationyieldsusedforthese simula-tionsarereportedinTable4.Inaddition,thevaluesfoundat150°C foraliphaticpolyamidesinapreviousstudy[37]aregivenfor in-formation.Indeed,theselatterwillbe veryhelpfulforthe follow-ing discussion,becausethey were obtainedinrubberystate, well abovetheglasstransitiontemperatureofthesematerials(typically whenT≥ Tg+100°C).
Allthesevaluescallforseveralcomments:
i) For network no. 6, the initiation rate constants (k1u and k1b) are about 3–4 times higher compared to aliphatic
polyamides throughoutthe temperaturerangeunder inves-tigation.Thisresultsuggestsahigherinstabilityof hydroper-oxidesinEPO-DAnetworkspresumablyduetothepresence of heteroatoms (N or O) in a
β
position (see Table 1). As it will be seen in the next Tables 5 and6, this is general behavior for EPO-DA networks. The Arrhenius laws found for k1u and k1b forthe whole family of EPO-DA networksthroughoutthe25–200°Cintervalare:
k1u =1.31013 Exp−130RT000s−1 (16)
and
k1b =1.9109 Exp−90000
RT L.mol
−1 .s−1 (17)
ii) For network no.6,the rateconstant k6 of apparent termi-nationofperoxyradicals isfiveorders ofmagnitudelower comparedto aliphaticpolyamidesthroughoutthe tempera-turerangeunderinvestigation.Thislargegapwasmainly at-tributedtotheeffectofmolecularmobilityonthe bimolec-ular combinations of peroxy radicals on both sides of the glasstransitiontemperature.Incontrastandasexpected,the impactof molecular mobilityon the propagationof oxida-tionismuchmoremoderate.Indeed,fornetworkno.6,the rateconstantk3 isonlytwoordersofmagnitudelower com-paredtoaliphaticpolyamides.
iii) The formation yield incarbonyls
γ
CO isquite differentbe-tween network no. 6 and aliphatic polyamides throughout the temperature range under investigation. This difference couldbeexplainedbytheformationofamuchgreater vari-etyofcarbonylproductsinEPO-DAnetworks.Inaddition,as therelativepredominancebetweenthesedifferentcarbonyl products depends on temperature,
γ
CO varies throughout the temperatureinterval under investigation. The fact thatγ
CO decreases withtemperaturejustconfirmsthataldehy-desare muchlessoxidizable atlowtemperature.As a con-sequence,theaveragemolarextinctioncoefficient
ε
,usedto estimatetheglobalconcentration ofcarbonylproductsfrom FTIRmeasurements(seeEq.13),shouldbeprogressively de-creasedwithtemperatureinordertotakeintoaccountthis changesin thechemical composition. As it willbe seen in thenext Table5, thisalso seems to be a general behavior forEPO-DAnetworks.iv) Allotherrateconstants(k2 ,k4 andk5 )andformationyields (
γ
1 ,γ
4 andγ
5 ) were kept identical, if not very close, toFig. 4. Carbonyl build-up in 100 μm thick films of network no. 6 (Tactix 123/Tactix 742-DDS) at 150 °C between 0.21 and 20 bars of oxygen. Comparison between simulations with Eq. (9) (solid lines) and experimental data (points).
Fig. 5. Carbonyl build-up in 100 μm thick films of network no. 6 (Tactix 123/Tactix 742-DDS) between 120 and 200 °C under 0.21 bar of oxygen (ambient air). Comparison between simulations with Eq. (9) (solid lines) and experimental data (points).
Table 4
Values of the rate constants and formation yields used for modeling the thermal oxidation kinetics of network no. 6 between 120 and 200 °C. Comparison with the values previously determined at 150 °C for aliphatic polyamides [37] .
Polymer Tactix 123/Tactix 742-DDS [T g = 263 °C] PA [T g ≈ 40–55 °C]
T ( °C) 120 150 180 200 150 k 1u (s −1 ) 8.0 × 10 −5 1.3 × 10 −3 1.5 × 10 −2 6.5 × 10 −2 4.4 × 10 −4 k 1b (L.mol −1 .s −1 ) 2.1 × 10 −3 1.5 × 10 −2 8.1 × 10 −2 2.2 × 10 −1 3.5 × 10 −3 k 2 (L.mol −1 .s −1 ) 10 8 10 8 10 8 10 8 10 8 k 3 (L.mol −1 .s −1 ) 7.0 × 10 −2 2.9 × 10 −1 9.9 × 10 −1 2.0 3.0 × 10 1 k 4 (L.mol −1 .s −1 ) 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 k 5 (L.mol −1 .s −1 ) 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 5.0 × 10 11 k 6 (L.mol −1 .s −1 ) 3.0 × 10 4 6.8 × 10 4 8.2 × 10 4 8.7 × 10 4 7.4 × 10 9 γ1 (%) 60 60 60 60 100 γCO (%) 8.5 11.2 21.5 33.5 100 γ4 (%) 50 50 50 50 55 γ5 (%) 50 50 50 50 55 Table 5
Values of the rate constants and formation yields used for modeling the thermal oxidation kinetics of networks no. 3 and 4 at 120 and 150 °C. Comparison with the values previously determined at 150 °C for aliphatic polyamides [37] .
Polymer DGEBF-CAF [T g = 158 °C] DGEBA-CAF [T g = 182 °C] PA [T g ≈ 40–55 °C]
T ( °C) 120 150 120 150 150 k 1u (s −1 ) 8.0 × 10 −5 1.3 × 10 −3 8.0 × 10 −5 1.3 × 10 −3 4.4 × 10 −4 k 1b (L.mol −1 .s −1 ) 2.1 × 10 −3 1.5 × 10 −2 2.1 × 10 −3 1.5 × 10 −2 3.5 × 10 −3 k 2 (L.mol −1 .s −1 ) 10 8 10 8 10 8 10 8 10 8 k 3 (L.mol −1 .s −1 ) 2.5 × 10 −1 3.0 3.1 × 10 −1 2.0 3.0 × 10 1 k 4 (L.mol −1 .s −1 ) 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 k 5 (L.mol −1 .s −1 ) 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 5.0 × 10 11 k 6 (L.mol −1 .s −1 ) 7.7 × 10 4 4.3 × 10 6 5.5 × 10 4 3.0 × 10 5 7.4 × 10 9 γ1 (%) 60 60 60 60 100 γCO (%) 25 30 30 35 100 γ4 (%) 50 50 50 50 55 γ5 (%) 50 50 50 50 55 Table 6
Values of the rate constants and formation yields used for modeling the thermal oxidation kinetics of networks no. 1 and 2 at 125 and 140 °C, and network no. 5 at 180 and 200 °C.
Network Epon 828-Jeffamine D230 [T g = 95 °C] Epon 828-Ancamine 2049 [T g = 150–155 °C] DGEBF/TGPAP- DDS [T g = 211 °C] T ( °C) 125 140 125 140 180 200 k 1u (s −1 ) 1.3 × 10 −4 5.4 × 10 −4 1.3 × 10 −4 5.4 × 10 −4 1.5 × 10 −2 6.4 × 10 −2 k 1b (L.mol −1 .s −1 ) 3.0 × 10 −3 8.0 × 10 −3 3.3 × 10 −3 8.0 × 10 −3 8.1 × 10 −2 2.2 × 10 −1 k 2 (L.mol −1 .s −1 ) 10 8 10 8 10 8 10 8 10 8 10 8 k 3 (L.mol −1 .s −1 ) 2.1 4.6 3.0 × 10 −1 7.0 × 10 −1 8.0 × 10 −1 3.0 k 4 (L.mol −1 .s −1 ) 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 k 5 (L.mol −1 .s −1 ) 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 k 6 (L.mol −1 .s −1 ) 2.9 × 10 9 6.9 × 10 9 6.5 × 10 5 9.9 × 10 6 1.7 × 10 6 2.7 × 10 7 γ1 (%) 60 60 60 60 100 100 γ4 (%) 50 50 50 50 55 55 γ5 (%) 50 50 50 50 55 55
the valuesof aliphaticpolyamides [37].As it can be noted inTables4,5and6,alltheseparametersareindependentof temperature.
Figs.6and7showtwoexamplesofnumericalsimulationsstill inglassystate,butnow,uptotemperaturesneartheglass transi-tion temperature(typicallyforTg – 60°C <T≤ Tg – 10°C). Car-bonylbuild-upswerecalculatedwithEq.(9)at120and150°C be-tween 0.21and10bars ofoxygen fornetworksno.3and4.Here also, it can be seen that the kinetic model satisfyingly accounts forboththeeffectsoftheoxygenpartialpressureandtemperature ontheoxidationkineticsofthesetwoEPO-DAnetworks,despitea slightlylargerscatteringofexperimentalresults.
Therateconstantsandformationyieldsusedforallthese sim-ulations are reported inTable 5.In addition, thevalues found at
150°Cforaliphaticpolyamidesinapreviousstudy[37]arerecalled forinformation.
Thesenewresults call forthe followingadditional comments. Asexpected,theimpactofthemolecularmobilityonthe bimolec-ularcombinations ofperoxy radicals andthe propagation of oxi-dationissignificantlyreducedwhenincreasingtheexposure tem-peratureandapproachingtheglasstransitiontemperatureofboth EPO-DAnetworks.Asanexample,atT=Tg – 30°Cfornetworkno. 3,k6 isonlythreeordersofmagnitudelowercomparedtoaliphatic polyamides(againstfiveordersofmagnitudewhenT<Tg – 60°C). Similarly,atT=Tg – 30°Cfornetworkno.3,k3 isonlyoneorderof magnitudelower comparedtoaliphaticpolyamides (againstthree whenT<Tg – 60°C).
Finally,Fig.8showsone exampleofnumericalsimulations on bothsidesoftheglasstransitiontemperature(typicallywhenTg –
Fig. 6. Carbonyl build-up in 25 μm thick films of network no. 4 (DGEBA-CAF) at 120 °C between 0.21 (ambient air) and 10 bars of oxygen. Comparison between simulations with Eq. (9) (solid lines) and experimental data (points).
Fig. 7. Carbonyl build-up in 25 μm thick films of network no. 3 (DGEBF-CAF) at 150 °C between 0.21 (ambient air) and 10 bars of oxygen. Comparison between some simulations with Eq. (9) (solid lines) and experimental data (points).
Fig. 8. Arrhenius graphs between 25 and 200 °C of the maximal oxidation rate under 0.166 bar of oxygen for 100 μm thick films of networks no. 1 and 2 (Epon 828-Jeffamine D230 and Epon 828-Ancamine 2049, respectively), and 50 μm thick films of network no. 5 (DGEBF/TGPAP-DDS). Comparison between the simulations with Eq. (8) (solid lines) and literature data (points) [10 , 44] .
130°C≤ T ≤ Tg +45°C). Maximaloxidation rateswerecalculated withEq.(8)between25and140°Cunder0,166bar ofoxygen for networks no. 1 and2, and at180 and 200°C under 0,166bar of oxygen fornetworkno.5. Hereagain, itcan be seenthat the ki-netic modelsatisfyinglypredicts theeffectoftemperatureonthe oxidationkineticsofthesethreeEPO-DAnetworksthroughoutthis widetemperaturerange.
For a sake ofbrevity and clarity, only the rate constants and formationyieldsusedforthesimulationsatthetwohighest tem-peratures, for each EPO-DAunder study, are reportedin Table6. Theparametervaluesobtainedatallothertemperaturesare avail-ableinAppendixAfornetworksno.1and2.
These newresultscall for thefollowing additionalcomments. As expected, the gap betweenthe oxidationkinetics of the EPO-DAnetworksandaliphaticpolyamidescontinues tonarrowwhen increasingthetemperatureofexposure,andendsupdisappearing almost totallywhen equalizingandexceedingtheglass transition temperatureofEPO-DAnetworks.Asanexample,atT=Tg +30°C fornetworkno.1,k6 andk3 takealmostthesamevaluesthanin aliphaticpolyamides(atT≈ Tg +100°C).
5. Discussion
Thevaluesoftherateconstantk6 determined forthesix EPO-DA networksunderstudyhave beenplottedasa function ofthe reciprocaltemperature1/TinFig.9.Thevaluesfoundforaliphatic polyamidesinapreviousstudy[37]havebeenaddedfor compari-son.Theverticalsolidstraight-linesindicatethepositionoftheTg ofthedifferentmaterials.
Atfirstglance, itwouldseemthat theeffectofmolecular mo-bilitycausesonlyashiftofthedifferentcurvesalongtheabscissa axis.Thus,ifassumingthatk6 obeysanArrheniuslaw,the molecu-larmobilitywouldonlyaffectthepre-exponentialfactorofk6 (i.e.
thefrequencyofchemicaleventsbetweenperoxyradicals)andnot itsactivationenergy.
Tocheck thisassumption, the valuesof k6 were plotted as a functionofthedifference(1/T– 1/Tg ),whichamountsto position-ing1/Tg ofallmaterials ontheordinate axispassingthrough the origin(i.e.at1/T=0).Asexpected,theresultingshiftalongthe ab-sciseaxisofthedifferentcurvesallows obtaining asinglemaster curve fork6 (see Fig. 10), whosethe generalshape can be
satis-fyingly described by the statistical model of Mahieux and Reifs-nider[60,61].Althoughthismodelwasfirstlydevelopedfor simu-latingthetemperaturedependenceofthestoragemodulusinlarge temperaturerange,duetothepresenceofseveralthermodynamic transitionssuchas:secondaryrelaxations,glasstransition,melting, etc.,itseemstobeperfectlysuitedforthisstudy.Aroundtheglass transitiontemperature,thismodelcanberewrittenasfollows:
k6 =k6r +
(
k6v +k6r)
× Exp −11/T− 1/Tv /Tr − 1/T m (18)whereTv andTr are thetemperaturesdelimiting theactionzone oftheglasstransitionfromthefullyglassyandrubberydomains, respectively; m is a statisticalparameter relatedto the width of theactionzone; andk6v andk6r are thevaluesofk6 inthefully glassyandrubberydomains,respectively.
Itsnumericalapplicationto themastercurve ofFig.10allows the determinationof the statisticalparameter (m≈ 1.3) and the temperaturedependenceofk6 onbothsidesoftheglasstransition temperature.Itwasfoundthatk6 tendstobealmostindependent oftemperatureintherubberydomain:
k6r =1.41010 L.mol−1 .s−1 (19)
Incontrast,k6 obeysanArrheniuslawintheglassydomain:
k6v =1.4105 Exp
−16000 R 1 T− 1 Tg L.mol−1 .s−1 (20)Fig. 9. Arrhenius graphs between 25 and 250 °C of the rate constant k 6 for the six EPO-DA networks under study and aliphatic polyamides. The vertical solid straight-lines
indicate the position of the T g of the different materials.
Fig. 10. Master curve around the glass transition temperature for the rate constant k 6 of EPO-DA networks and aliphatic polyamides. Modeling of the impact of the thermo-
Fig. 11. Arrhenius graphs between 25 and 250 °C of the rate constant k 3 for the six EPO-DA networks under study and aliphatic polyamides. The vertical solid straight-lines
indicate the position of the T g of the different materials.
It hasbeentried to apply thesame methodologyto rate con-stant k3 .However, in this case, the molecular mobilitydoes not causeasimpleshiftofthedifferentcurvesalongtheabscissaaxis. Indeed, in Fig. 11, it can be noted that some curves are already very close to each other, while the corresponding EPO-DA net-workshaveverydifferentvaluesofTg .Thisthecasefornetworks no.2,3and4ononehand,andnetworksno.5and6ontheother hand.
AccordingtoKorceketal.[18],thepre-exponentialfactork30 of therateconstantk3 dependsaboveall onthedissociationenergy of theC-H, butalsosecondarily onthe reactivityof theinvolved peroxyradicalPO2 •.However,intheprevioussection,ithasbeen shown that their reactivity (throughk6) is significantly impacted
by the molecular mobility. As a consequence, k3 is expected to show alittle morecomplicatedvariationwithTg thank6 inboth therubberyandglassydomains(seeEqs.19and21).
Atfirst,the valuesofk3 were plottedasafunction ofthe dif-ference(1/T– 1/Tg ),aspreviouslydonefork6 .Asexpected,the re-sultingshiftalongtheabsciseaxisofthedifferentcurvesdoesnot allowobtainingasinglemastercurvefork3 (seeFig.12).However, iflookingatthevariation ofk30 withthe Tg ofthedifferent ma-terials (mostofthe dataare inglassy state),one foundsthat k30 increasesexponentiallywithTg
k30 ∝ Exp
Tg 30 (21)Then,thecurveofaliphaticpolyamideswaschosenasthe refer-encecurve.Eq.21wasusedtoshiftalongtheordinateaxisallthe curvesof EPO-DAnetworks, thus obtaining asinglemaster curve fork3 (seeFig.13).Asfork6 ,thegeneralshape ofk3 canbe sat-isfyinglydescribedby thestatisticalmodelofMahieuxand
Reifs-nider[60,61]: k3 =k3r +
(
k3v +k3r)
× Exp −1/T− 1/Tv 1/Tr − 1/T m (22)wherek3v andk3r arethevaluesofk3 inthefullyglassyand rub-berydomains,respectively.
Itsnumericalapplicationto themastercurve ofFig.13allows thedetermination of thestatistical parameter(m ≈ 0.9)and the temperaturedependenceofk3 onbothsidesoftheglasstransition temperature.Itwasfoundthatk3 obeysanArrheniuslawwiththe sameactivationenergyon bothsidesof theglasstransition tem-perature.Intherubberydomain:
k3r =1.010−1 Exp
Tg− 50 30 Exp −64000 R 1 T− 1 Tg L.mol−1 .s−1 (23)Intheglassydomain:
k3v =1.910−2 Exp
Tg − 50 30 Exp −64R000 1 T− 1 Tg L.mol−1 .s−1 (24)Thescatteringofdatainthe glassydomainappears tobe suf-ficientlynarrow(overonlyonedecade) toallowtheuseofa sin-glekineticmodelforthewholefamilyofEPO-DAnetworks.If ne-glectinganyexperimentalerror,thisscatteringcanbeduetothree sources:
i) Errors in the calculation of the concentration of oxidation sites.
ii) Differencesinthe reactivityofoxidationsites (i.e.between oxy-methylene,amino-methyleneandmethanolgroups)and
Fig. 12. Result of the shift along the abscise axis of the curves of Fig. 11 . It can be noted an increase k 30 with the T g of the different materials.
Fig. 13. Master curve around the glass transition temperature for the rate constant k 3 of EPO-DA networks and aliphatic polyamides. Modeling of the impact of the thermo-
therefore also,differences inthe reactivity ofthe resulting peroxyradicals.
iii) A possible loss of aromaticity leading to the formation of highlyoxidizablealiphaticstructuresduringthermalaging. Theimpactoftwofirstassumptionsonthevaluesofk3 should be very limited(of a factor lower than 2) and thus, not allow a full explanation of the width of data cloud. In contrast, the last assumption should be carefully considered, in particular epoxies cross-linkedwithanaromatichardener,becauseitcould theoreti-callyimpactbyseveralordersofmagnitudethevaluesofk3 .Such a radical recombination has already been evidenced in a highly aromaticpolymer,forinstanceinpoly(etheretherketone)[62].
6. Conclusion
Thekineticmodeldevelopedforthethermaloxidationof poly-methylenic substrates in previous studies [22,31-37]was success-fullygeneralizedtoEPO-DAnetworksinthepresentstudy.Its nu-merous parametersweredeterminedby applyinga“stepbystep” procedurecombiningexperimentandsimulation.
It wasfoundthat theoxygen transport propertiesdepend(i.e. oxygendiffusionandsolubility)onthearomaticityoftheEPO-DA network, whereas the rateconstantsof theapparent termination of peroxyradicals (k6 ) andthe propagationof oxidation (k3 ) are highlyimpactedby themolecularmobilityaroundthe glass tran-sition temperature(Tg ). The introductionoftheeffect of molecu-larmobilityintotheArrheniuslawsofk6 andk3 allowed building mastercurvesandfinally,proposingasinglekinetic modelforthe wholefamilyofEPO-DAnetworks.Inaddition,thesharpvariation ofbothrateconstantswithtemperaturewithintheglasstransition zone(i.e.whenpassingfromglassytorubberystate)was success-fully describedusing the statisticalmodel ofMahieux and
Reifs-Table A.1
Values of the rate constants and formation yields used for modeling the thermal oxidation kinetics of networks no. 1 between 25 and 140 °C. Network Epon 828-Jeffamine D230 [T g = 95 °C]
T ( °C) 25 37 50 65 80 95 110 125 140 k 1u (s −1 ) 2.6 × 10 −10 2.0 × 10 −9 1.5 × 10 −8 1.3 × 10 −7 8.9 × 10 −7 5.4 × 10 −6 2.8 × 10 −5 1.3 × 10 −4 5.4 × 10 −4 k 1b (L.mol −1 .s −1 ) 3.3 × 10 −7 1.3 × 10 −6 5.5 × 10 −6 2.4 × 10 −5 9.4 × 10 −5 3.3 × 10 −4 1.0 × 10 −3 3.0 × 10 −3 8.0 × 10 −3 k 2 (L.mol −1 .s −1 ) 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 k 3 (L.mol −1 .s −1 ) 5.0 × 10 −4 1.1 × 10 −3 2.7 × 10 −3 7.0 × 10 −3 2.8 × 10 −2 2.6 × 10 −1 1.0 2.1 4.6 k 4 (L.mol −1 .s −1 ) 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 k 5 (L.mol −1 .s −1 ) 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 k 6 (L.mol −1 .s −1 ) 4.9 × 10 4 5.9 × 10 4 7.8 × 10 4 1.1 × 10 5 1.6 × 10 6 1.8 × 10 8 1.2 × 10 9 2.9 × 10 9 6.9 × 10 9 γ1 (%) 60 60 60 60 60 60 60 60 60 γ4 (%) 50 50 50 50 50 50 50 50 50 γ5 (%) 50 50 50 50 50 50 50 50 50 Table A.2
Values of the rate constants and formation yields used for modeling the thermal oxidation kinetics of networks no. 2 between 25 and 140 °C. Network Epon 828-Ancamine 2049 [T g = 150–155 °C]
T ( °C) 25 37 50 65 80 95 110 125 140 k 1u (s −1 ) 2.6 × 10 −10 2.0 × 10 −9 1.5 × 10 −8 1.3 × 10 −7 8.9 × 10 −7 5.4 × 10 −6 2.8 × 10 −5 1.3 × 10 −4 5.4 × 10 −4 k 1b (L.mol −1 .s −1 ) 3.3 × 10 −7 1.3 × 10 −6 5.5 × 10 −6 2.4 × 10 −5 9.4 × 10 −5 3.3 × 10 −4 1.0 × 10 −3 3.0 × 10 −3 8.0 × 10 −3 k 2 (L.mol −1 .s −1 ) 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 10 8 k 3 (L.mol −1 .s −1 ) 6.0 × 10 −4 1.6 × 10 −3 4.0 × 10 −3 1.2 × 10 −2 3.0 × 10 −2 6.8 × 10 −2 1.5 × 10 −1 3.0 × 10 −1 7.0 × 10 −1 k 4 (L.mol −1 .s −1 ) 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 8.0 × 10 11 k 5 (L.mol −1 .s −1 ) 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 3.0 × 10 11 k 6 (L.mol −1 .s −1 ) 2.0 × 10 4 2.4 × 10 4 3.3 × 10 4 4.0 × 10 4 4.8 × 10 4 6.2 × 10 4 9.8 × 10 4 6.5 × 10 5 9.9 × 10 6 γ1 (%) 60 60 60 60 60 60 60 60 60 γ4 (%) 50 50 50 50 50 50 50 50 50 γ5 (%) 50 50 50 50 50 50 50 50 50
nider [60,61],although initially developedfor the storage modu-lus. It is thus clear that the molecular mobilityplays a key role inboththemechanicalbehaviorandinthedurabilityaround the glasstransitiontemperatureofEPO-DAnetworks.
Despitethisapparentsuccess,atleasttwomainchallenges re-main to be met. The effect of aromaticity on oxygen transport propertiesisnotwell understoodandwoulddeservean in-depth andspecificstudyonaseriesofmodelEPO-DAnetworks,butalso on a series of linearpolymers having almost the same chemical composition than EPO-DA networks (e.g. polycarbonate, polysul-fone,etc.).
Similarly, the effect of aromaticity on the thermal oxidation kinetics should be carefully considered because it could explain smallbehavioraldifferencesobserved betweenthedifferent EPO-DA networks in the glassy state. In particular, a possible loss of aromaticity could certainly allow reducing the scattering of data observedonthemastercurveofk3 (Fig.13).
Initscurrentform,thekineticmodelcanbenowusedto eluci-datetheoxidationbehaviorofmanyindustrialmaterialsexhibiting Tg gradients such as: thick structures in composite material and jointsofglueinassembliesofmetalpartswithcompositematerial structures.
DeclarationofCompetingInterest
Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.
AppendixA
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