Limitations of model-fitting methods for kinetic analysis: Polystyrene thermal degradation

LIMITATIONSOFMODEL‐FITTINGMETHODSFORKINETICANALYSIS:POLYSTYRENETHERMAL DEGRADATION.  PedroE.Sánchez‐Jiménez,LuisA.Pérez‐Maqueda,AntonioPerejón,JoséM.Criado InstitutodeCienciadeMaterialesdeSevilla,C.S.I.C.‐UniversidaddeSevilla,C.AméricoVespucio nº49,41092Sevilla,Spain Abstract Inthispaper,someclarificationsregardingtheuseofmodel‐fittingmethodsofkineticanalysisare providedinresponsetothelackofplotlinearityanddispersionintheactivationenergyvaluesfor thethermaldegradationofpolystyrenefoundintheliteratureandsomeresultsproposingannth ordermodelasthemostsuitableone.Inthepresentwork,twomodel‐fittingmethodsbasedon thedifferentialandintegralformsofthegeneralkineticequationareevaluatedusingboth simulatedandexperimentaldata,showingthatthedifferentialmethodisrecommendeddueto itshigherdiscriminationpower.Moreover,theintrinsiclimitationsofmodel‐fittingmethodsare highlighted:theuseofalimitedsetofkineticmodelstofitexperimentaldataandtheidealnature ofsuchmodels.Finally,itisconcludedthatachainscissionmodelismoreappropriatethanfirst order. Keywords:KineticAnalysis;IntegralMethods;ChainScission;KineticModel;Polystyrene  1. Introduction Animportantvolumeofworkhasbeendevotedtothedeterminationoftheactivationenergyand kineticmodeldescribingthethermaldegradationofpolystyrene(Bremsetal.,2011,Chenetal., 2007,ChenandVyazovkin,2006,Faravellietal.,2001,Petersonetal.,2001,Sanchez‐Jimenezet al.,2012,Snegirevetal.,2012,Sterlingetal.,2001,Westerhoutetal.,1997).However,results showawidedivergenceintheactivationenergyvaluesandthekineticmodelsproposed.Also, bothcomplexandone‐stepmechanismsaresuggested.Arecentpaperquestionstheapplication ofgeneralkineticmodelssuchasdiffusionornucleationforthekineticanalysisofpolymer pyrolysis(Bremsetal.,2011).Inthatpaper,thepyrolysisofpolystyreneisstudiedbyfittingthe experimentaldata,recordedunderdynamicheatingconditions,tosomeofthemostwidelyused   Corresponding author. Tel +34954489548 Fax +34954460665 e-mail address: [email protected] kineticmodelsbymeansoftheCoats‐Redfernintegralmodel‐fittingmethod(CoatsandRedfern, 1964).Alackofplotlinearityandinconsistentactivationenergyvaluesareobservedandthe authorsconcludethatsimplefirstorsecondordermodelsaregoodenoughfordeterminingthe activationenergyand,therefore,constitute“themostsuitabledesignapproach”.Otherworks employingsimilarsingle‐curvemodel‐fittingkineticproceduresreachedanalogousconclusions (Ahmadetal.,2010,AhmedandGupta,2009,KrishnaandPugazhenthi,2011).However,some otherworks,employingisoconversionalandnon‐linearmodelfittingmethods,haverevealedthe unsuitabilityofnthorderreactionmodelsandsuggestanacceleratorymodelinstead(Chenetal., 2007,Sanchez‐Jimenezetal.,2010,Snegirevetal.,2012,Vyazovkinetal.,2004).Suchwas confirmedbySanchez‐Jimenezetal,proposingarandomchainscissionmodel(Sanchez‐Jimenez etal.,2010,Sanchez‐Jimenezetal.,2012).Sinceseveralauthorshavefoundthattheactivation energydoesnotchangewiththereactedfraction(Petersonetal.,2001,Sanchez‐Jimenezetal., 2012,Snegirevetal.,2012),itcanreasonablybeassumedthattheprocessissinglestepand otherresultscanbeaconsequenceofbothinappropriatekineticproceduresornon‐adequately controlledexperimentalconditionssinceithasbeenprovedthatheatandmasstransfer phenomenacanaffectgreatlytothecalculatedkineticparameters. Here,weaimtoprovidesomeclarificationsregardingtheuseofmodel‐fittingmethodsthat mightexplainsuchdiscrepancy.Model‐fittingmethodsarefrequentlyusedforperformingthe kineticmodelingofthepyrolysisofpolymersandplasticwastebecauseoftheirsimplicity.A numberofmethodshavebeenproposed,butallbasicallyconsistonfittingtheexperimentaldata toaconversionfunction,alsoknownaskineticmodel,whichrelatestheextentofconversionwith thereactionrate(Perez‐Maquedaetal.,2005b).Kineticmodelsaremathematicalfunctions developedfromcertainphysico‐geometricalassumptionsregardingthereactantsshapeandthe reactiondrivingforce.Therefore,theycanbeidentifiedwithareactionmechanism(Khawamand Flanagan,2006).However,twoimportantlimitationsmustbeconsideredwhensuchmodelsare appliedtorealsystems.Firstly,thekineticmodelsarebuiltonstrictassumptionsthatmightnot befulfilledinarealprocess.Certainly,mostofthemodelswereproposedforsolidstatereactions ingeneralandsomeofthem,suchasthenucleationandgrowthorthegeometricalshrinkage ones,mightnotseementirelyappropriateforpolymerdegradationprocesses.Othermodelssuch asthosebasedonapowerlawweredevelopedbyassumingamechanismofnucleationand growthofnucleiwithoutoverlapofthegrowthnuclei(JacobsandTompkins,1955,Khawamand Flanagan,2006).Thus,thepowerkineticlawscannotbeappliedtothewholereactionrangebut onlytotheacceleratoryperiodatwhichtheoverlappingofnucleihasnotyetstarted.The potentiallawsbecomethewell‐knownAavrami‐Erofeevkineticmodelsiftheoverlappingof growingnucleiistakenintoaccount.Thus,Avrami‐Erofeevkineticequationsinsteadofpower lawsshouldbeusedforperformingthekineticanalysisonthewholereactedfraction.Secondly, anylistofmodelsisinevitablyincompleteanditisperfectlypossiblethatthestudiedprocessis notdescribedbyanyofthem.Forexample,akineticmodelofrandomscission,basedonthe cleavageandulteriorvolatilizationofpolymericchains,wasrecentlyproposed(Sanchez‐Jimenez etal.,2010).Thatmodelisespeciallysuitabletopolymericdegradationstudies,andhasaclear physicalmeaning.Actually,sinceitspublication,wehavefoundanumberofpolymerssuchas cellulose,polystyreneandpolybutylenterephtalate,thatdecomposeaccordingtoarandom scissionmodel(Sanchez‐Jimenezetal.,2010,Sanchez‐Jimenezetal.,2011,Sanchez‐Jimenezet al.,2012).Here,weattempttoprovidesomeinsightabouttheuseofmodel‐fittingmethodsand highlightsomeofitslimitationsinordertoclarifywhyinsomecasestheexperimentaldata cannotbeproperlyfittedbyanykineticmodel(Bremsetal.,2011).  2. Experimental ThepolystyrenestudiedwassuppliedbyGoodfellow(powder,productnumber261595). Thermogravimetric(TGA)measurementswerecarriedoutinaTAInstrumentsQ5000IR thermobalance(TAInstruments,Crawley,UK)connectedtoagasflowsystemtoworkininert atmosphere(150mLmin‐1).Smallmasssamples(∼10mg)wereusedinordertominimizemass andheattransferphenomena.Experimentswererecordedunderlinearheatingrateconditions, at0.5,1,2and5Kmin‐1.Theα‐Tplotsobtainedfromthesetwomethodsweredifferentiatedby meansoftheOriginsoftware(OriginLab)togetthedifferentialcurvesemployedinthekinetic analysis.  3. ResultsandDiscussion Figure1includesasetofcurvessimulatedaccordingtothefollowingkineticparameters: Ea=200kJ/mol,A=3*1015s‐1,andL2randomscissionkineticmodelf(α)=2(α1/2‐α).Suchwerethe parametersobtainedforthedegradationofpolystyreneinapreviousstudyandwillserveto compareexperimentalandsimulatedresults(Sanchez‐Jimenezetal.,2012).Thesimulations wereperformedbymeansofanumericalintegrationmethod(Runge‐Kutta)usingtheMathcad softwareandassumingheatingratesof0.5,1,2and5K/min.Theselectedconversionfunction, f(α),correspondstoaL2chainscissionkineticmodel,asdescribedelsewhere(Sanchez‐Jimenezet al.,2010).Allfoursimulatedcurveshavebeenfittedtosomeofthemostcommonkineticmodels usingthesameCoats‐RedfernintegralmethodemployedinBrems’work(Bremsetal.,2011).The resultingplotsareshowninFigure2.Whendealingwithrealexperimentaldataitiscommonto restrictthefittodatawithinalimitedrangeofconversionbecausethelowandhighendsofthe conversionrangeareoftendistortedduetoexperimentalerrors.Suchrestrictionwasalso employedinBrems’paper(Bremsetal.,2011).Thus,wehavecarriedoutthefittingprocedure usingdatawithintherange0.1≤α≤0.9.Figure2showstheplotsobtainedfromfittingthe simulatedcurvestothedifferentkineticmodelsemployed,alongwiththeactivationenergies deducedfromtheslopeofeachplot.Inprinciple,whenthedataisfittedtothecorrectkinetic model,astraightlineisexpected.However,allF1,A2andL2modelsyieldsalinearplot,butonly whenthedataarefittedtoL2,theactivationenergyyieldedbythemethodisthecorrectone,i.e. 200kJ/mol.Thus,unlesstheactivationenergyispreviouslyknown,itwouldbedifficultto discriminatethecorrectkineticmodelobeyedbytheprocessusingthismethodbecausemore thanonekineticmodelcanprovideagoodfittothedata.Additionally,itisclearthatthefitsof thedatasimulatedassumingdifferentheatingratestoacertainmodelareparallelineverycase. Consequently,sincetheactivationenergyisdeductedfromtheslopes,thevaluedeliveredbythe methodmustbethesamewhatevertheheatingrate. Analternativetothepreviousmethodconsistsonfittingtheexperimentaldatatothedifferent kineticmodelsusingdirectlythedifferentialformofthegeneralkineticequation.After reorganizingtermsandtakinglogarithms,inpresentsthefollowingform(Sanchez‐Jimenezetal., 2009): 󰇡∝  ⁄ 󰇛∝󰇜 󰇢 (1) Theplotofthelefthandsideoftheequationaboveversustheinverseofthetemperatureyieldsa straightlinewhentherightf(α)isselected.Moreover,whendatafromcurvesrecordedunder differentheatingratesareplottedtogether,allofthemwillliealongastraightlineiftherightf(α) isselectedsincetheinterceptnowdependsexclusivelyonthepre‐exponentialfactor.The reducedchanceofselectinganincorrectmodelconstitutesanimportantadvantageovermodelfittingmethodsbasedontheintegralformofthekineticequation.ThisisclearlyshowninFigure 3,whichincludestheplotsobtainedwhenthecurvesinFigure1arefittedtoseveralmodels accordingtoEq.(1).ItisevidentthatL2istheonlymodelthatcorrectlyfitsallthesimulateddata, yieldinganactivationenergyof200kJ/molandapreexponentialfactorof4.8*1015s‐1.Thus,both theactivationenergyandthekineticmodelcanbeunambiguouslydeterminedbythismethod.It isalsoworthnoticingthatwhenfittingtoinappropriatemodels,theplotscorrespondingto differentheatingratesdonotyieldstraightlinesbutleadtoparallelplots.Thisbehaviorallowsto concludethatthesameactivationenergywouldbeobtainedfromtheCoats‐Redfernmethod whenappliedtoindividualα‐Tcurvesrecordedatdifferentheatingrates Finally,asetofexperimentalcurvescorrespondingtothethermaldegradationofpolystyrene, oneofthetwopolymersstudiedinBrems’work,isusedtoconfirmtheobservationsmadewith thesimulatedcurves.Theexperimentalcurves,showninFigure4,wererecordedatheatingrates of0.5,1,2and5K/min(Sanchez‐Jimenezetal.,2012).WhentheCoats‐Redfernintegralmethod isusedtoperformthekineticanalysis,theplotsinFigure5areobtained.Ashappenedwiththe simulateddata,itisdifficulttodiscriminatebetweenthedifferentmodels.ThefitstoF1,R2,L2 andA2allgivegoodlinearityandreasonablevaluesoftheactivationenergy.Figure6showsthe plotsresultingfromfittingtheexperimentaldatatodifferentkineticmodelsusingthedifferential method(Eq.(1)).OnlywhentheexperimentaldataisfittedtoaL2randomscissionkineticmodel, anapproximatelystraightlineisobtained.Theslopeleadstoanactivationenergyof221kJ/mol, reasonablyclosetothevalueobtainedbyapreviousstudyusingadifferentmethodology (Sanchez‐Jimenezetal.,2012).Notethat,iftheL2modelisnotconsidered,noneoftheotherfive kineticmodelshereemployedreallymeetthelinearityrequirements.Thismightexplainwhyno kineticmodelswasfoundtoproperlyfitthedecompositionreactionsofpolystyreneand polyethyleneterephtalateatBrems’work(Bremsetal.,2011),sincetheoptionofachainscission modelwasnottested.Itisinterestingtonotethattheresultsprovidedbybothmethodsyields slightlydifferentactivationenergyvaluesfromthesameexperimentaldata.Thediscrepancyis duetothenatureofeachmethod.Whilethedifferentialmethodusesthedataasis,theintegral methodusesanapproximationtothetemperatureintegral,thatdoesnothaveananalytical solution(Perez‐Maquedaetal.,2005a,b).Itshouldalsobementionedthattheactivationenergy obtainedbythesefittingproceduresisalsoslightlydifferenttothatobtainedbyisoconversional andcombinedanalysismethodsinpreviousworks,i.e204kJ/mol,usingthesameexperimental data(Sanchez‐Jimenezetal.,2012).BothCoats‐Redfernanddifferentialmethodsconsiston fittingtheexperimentaldatatoatheoreticalkineticmodel,basedoncertainphysic‐geometrical assumptionsdifficulttofulfillinrealsystems.Therefore,deviationsfromidealityareexpected.On theotherhand,theisoconversionalmethodismodel‐freeandthecombinedmethoddeductsthe kineticmodeldirectlyfromtheexperimentaldatawithoutmakinganypreviousassumptionsoit canbeassumedthelattermethodsprovidemoreaccuratevaluesoftheactivationenergy. Finally,theactivationenergiesincludedinFigures5and6correspondtoanaveragevalue. Anyway,lessthan5%variationisfoundbecausetheplotscorrespondingtodifferentheating ratesarealmostperfectlyparallel,asitwasobservedwhenthemethodwasappliedtosimulated data.Thus,theactivationenergyyieldedbythismethodisconfirmedtobeindependentofthe heatingrateusedtoobtaintheexperimentaldata.Suchindependencyisnotobservedinthe valuesreportedinBrems’work,whereasignificantdispersionintheactivationenergiesobtained isfound.Thiscanbeattributedtodeficienciesintheexperimentalcurves.Theinfluenceof experimentalfactorssuchastheheatormasstransferisimportantandcannotbedisregarded, especiallywhenexperimentaldataarerecordedatveryhighheatingrates.Atthoserates,they canmasktherealprocessandcomplicatetheextractionofexperimentaldatathatarereally representativeofthereaction,leadingtotheinaccurateestimationofkineticparameters.Itis evenpossiblethatthereactionmechanismchanges.Thus,curvesrecordedatheatingratesas differentas15and120K/minareprobablynotdirectlycomparable.Theimportanceofobtaining accuratekineticparameterscannotbeoverstatedsincetheassumptionofincorrectmodelsor activationenergieswouldrenderuselessanyattempttopredictthebehaviorofthesystemto conditionsdifferentofthoseunderwhichtheexperimentaldatawasrecorded.  4. Conclusions Here,ithasbeenshownthat,duetotheintrinsiclimitationsofidealmodels,modelfitting methodsshouldbeemployedcautiouslybecauseitmightbepossibletheycannotexactly representarealprocess.Moreover,thepossibilitythatnomodelinasetcouldbeusedto describeaprocessshouldbetakenintoconsiderationwhenareasonablefitisnotfound. Additionally,recentpublicationsseemtosupportthattherandomchainscissionmodelmightbe veryadequatetodescribepolymerdegradationreactionsand,therefore,itshouldbeincludedin thelistoftestedmodelswhenstudyingsuchkindofprocesses.  ACKNOWLEDGEMENT FinancialsupportfromprojectsTEP‐03002fromJuntadeAndalucía,CTQ2011‐27626fromthe SpanishMinisteriodeEconomíayCompetitividadandFEDERfundsisacknowledged.Additionally, oneoftheauthors(PESJ)issupportedbyaJAE‐Docgrant(CSIC‐FSE).    References 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PolymerDegradationandStability.2001;73:75‐82.  VyazovkinS,etal.Kineticsofthethermalandthermo‐oxidativedegradationofapolystyrene‐clay nanocomposite.MacromolRapidCommun.2004;25:498‐503.  WesterhoutRWJ,etal.Kineticsofthelow‐temperaturepyrolysisofpolyethene,polypropene,and polystyrenemodeling,experimentaldetermination,andcomparisonwithliteraturemodelsand data.IndEngChemRes.1997;36:1955‐64.                   FigureCaptions. Figure1.Kineticcurvessimulatedaccordingthefollowingkineticparameters:Ea=200kJ/mol,A=3 *1015s‐1,andf(α)=2(α1/2‐α). Figure2.PlotsobtainedfromfittingthesimulatedcurvesinFigure1tosomeofthemostusual kineticmodelsbymeansoftheintegralmodel‐fittingprocedureemployedbyBrems’.(Bremsetal. ,2011)Theactivationenergyisobtainedfromtheslopeofthecurves. Figure3.PlotsobtainedfromfittingthesimulatedcurvesinFigure1tosomeofthemostusual kineticmodelsbymeansofadifferentialmodel‐fittingmethod(Eq.1).Theactivationenergyis obtainedfromtheslopeofthecurves. Figure4.Experimentalcurvescorrespondingtothethermaldegradationofpolystyrene.(Sanchez‐ Jimenezetal.,2012) Figure5.PlotsobtainedfromfittingthesimulatedcurvesinFigure4tosomeofthemostusual kineticmodelsbymeansoftheintegralmodel‐fittingprocedureemployedbyBrems’.(Bremsetal. ,2011)Theactivationenergyisobtainedfromtheslopeofthecurves. Figure6.PlotsobtainedfromfittingthethermaldegradationcurvesinFigure4tosomeofthe mostusualkineticmodelsbymeansofadifferentialmodel‐fittingmethod(Eq.1).Theactivation energyisdeducedfromtheslopeofthecurves.        