Thermal Analysis.docx

Thermal Analysis.docx

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ThermalAnalysis

12

Thermalanalysis

12.1INTRODUCTION

Thermalanalysisembracesallmethodsinwhichmeasurementsaremadeofaphysicalpropertythatchangesasthetemperatureisvaried.Anumberofthetechniquescanbealsocomplementedbytheadditionoftimeoroscillatoryvariationtoenhancetheinformationthatcanbeobtainedfromthesemeasurements.Theexperimentscanusuallybedividedintoisothermalinwhichcontinuousmeasurementsasafunctionoftimeand/orfrequencyareperformedataconstanttemperatureandprogrammedtemperaturemeasurementswherethetemperatureisvariedinawelldefinedmanner.Techniquessuchasdifferentialscanningcalorimetry,DSC,ordifferentialthermalanalysis,DTA,primarilymeasurethethermalpropertiesofthematerialandallowcalculationoftheenthalpy（ΔH）orentropy（ΔS）changesthatoccurwhentransformationssuchascrystalmeltingoccur.Measurementsofthevariationinphysicalpropertiessuchasthemodulusindynamicmechanicalthermalanalysis,DMTA,andelectricalpermittivityindielectricthermalanalysis,DETA,arebothusefulmethodsforcharacterizingpolymersandcanalsoprovidevaluableinformationonthemoleculardynamics.Modulusandelectricaldataareoftenvaluabledesigndataandcanbeuseddirectly.Bymakingmeasurementsatdifferentfrequenciesthemethodscanbeusedfortheestimationofactivationenergiesoranalysedusingthetime-temperaturerelationshipsbasedonfreevolumeexemplifiedintheWilliams-Landel-Ferry（WLF）equation.

12.2DIFFERENTIALTHERMALANALYSIS（DTA）AND

DIFFERENTIALSCANNINGCALORIMETRY（DSC）

TheDTAandDSCtechniquesareverysimilarandmaybediscussedtogether.TheessentialfeaturesoftheDTAapparatusareshowninFig.12.1.ThesampleisplacedinacellS,locatedinablockwhichcanbeheated（orcooled）ataprogrammedrate.

AreferencesampleinanidenticalcellRislocatedclosetothesamplecellintheuniformtemperatureblock;itstemperatureisrepresentedbyTr.Thereferencesamplemayeitherhaveaprecisetransitionintheregionofobservation（e.g.naphthalenewithameltingpointof~80℃）orhaveafairlyconstantheatcapacity（e.g.analuminiumdiscorpowder）.Thepurposeofthereferenceistoprovideadirectcomparatorfortemperaturemeasurementforthesample,assistingminimizationofinaccuracies（notcorrect）duetothermallag（todevelopmoreslowlythanothers）intheequipment.Whenthesamplepassesthroughatransitionalstateitstemperature（Ts）departsfromthatofitssurroundings.Iftheprogrammeissetforheating,atanendothermictransitionsuchasacrystalmeltingtransition,TsfallsbelowtheprogrammetemperatureandthereferencetemperatureandΔT（=TsTr）isnegative.ThesizeofΔTdependsonthethermalpropertiesoftheequipment,particularlythethermalcapacityofthecell,aswellasthemassofsampleand,forfinitesamples,thethermalconductivity.ForthisreasonitisdifficulttoextractquantitativemeasurementsofthethermalpropertiesofthesampleusingDTA,thoughthetemperaturesatwhichtransitionsoccurcanbelocatedfairlyaccurately.

Fig.12.1SchematicofaDTAapparatus;risreference;sisspecimen.WrandWsareconstantinstrumentfactorsthatdependonthethermalcharacteristicsofequipment.

Carefulconsiderationmustbegiventothehandlingofthedata,foritiseasytouseerroneouscriteriaforlocatingaspecifictemperatureofinterest,e.g.theglasstransitiontemperature（seeSection12.2.2）.MoredirectmeasurementofthermalpropertiesispossibleusingDSCanditisgenerallypreferredforquantitativeanalysis.Inthismethodthesampleandreferenceareprovidedwithindependentheaters,Fig.12.2.Backgroundheatingoftheblockisusuallyprovidedseparatelysothatthemicroheatersaresensitivetotherequirementsofthesampleandreferencecellattheprogrammetemperature（Tp（t））.ThetemperatureofeachcellismeasuredcontinuouslyandcomparedwiththeinstantaneousvalueofTp（t）.Itisarrangedthatthepowerdeliveredtothesampleandreferencecellsviatheindividualheatersisafunctionofthedeparturefromtheprogrammetemperature,i.e.Ws（TsTp）andWr（TrTp）respectively.Thedifferentialpowerrequirement{Ws（TsTp）Wr（TrTp）}isthequantityplottedandcanbepresentedasafunctionofTp,TrorTs.WiththisarrangementTp,TrorTsshouldbeveryclosetogetherevennearatransitionandthereforemuchcloserthanTsandTr,intheDTAmethodwheneverthermalchangesaretakingplace.

12.2.1QUANTITATIVEANALYSISOFDTAANDDSC

Fig.12.2SchematicofaDSCapparatus:

risreference;sisspecimen.WrandWsareconstantinstrumentfactorsthatdependonthethermalcharacteristicsoftheequipment.

Fig.12.3HeattransferinaDSCcell:

Rrepresentsthethermalresistancetoheatflowbetweencellandtheblock.

ToanalysetheDTAorDSCexperimentsonemustfirstlyconsidertheheatflowbetweentheblockandthesample.AssumethattheblocktemperatureisTpandthatofthesampleTsandletthetotalresistancetoheatflowbetweencellandblockbe'R',Fig.12.3.Whenheatflowsintothesamplefromthesurroundings（ataratedQs/dt）theenergybalancegives:

（12.1）

whereHsistheenthalpyofthesampleandCsistheheatcapacityofthesampleplusthecell.Therateofheatnowcanalternativelybegivenas

（12.2）

Hence

（12.3）

Ifasuitableinternalreferencesamplehasbeenselected,theequivalentexpressionforthereferencecellcanbewrittenas

（12.4）

wheresubscriptrstandsforthereferencecellandRisassumedequalforthe（identical）sampleandreferencecells:

i.e.

（12.5）

whereΔT（=TsTr）andthesubscriptsisdroppedfromHsinceitisredundantwhenchangesinenthalpyoccuronlyinthesample.RememberingthattheDTAmethodproducesaplotofΔTversusT（ort）then,inprinciple,equation（12.5）canbeusedforquantitativeanalysis.Inpracticeitisneitherconvenientnoraccuratetodothis,foritrequiresaknowledgeofR,thethermalresistance,whichdependsonseveralthingsincludingtheconductivityofbothsampleandreference.Thesevaluesnotonlyalterwithtemperature,butshowmarkedchangesoneithersideofatransition.

ForDSCwecantakeequation（12.1）andtheequivalentequationforthereferencecellandfind

（12.6）

whereΔQisthedifferenceinheatsuppliedtothetwocells,i.e.

（12.7）

Nowfromequation（12.2）wehave

and

sothat

（12.8）

Substitutionof（Tr-Ts）fromequation（12.8）intoequation（12.7）gives

（12.9）

IfRismadesufficientlysmallthenthefinalterminequation（12.9）canbemadenegligible;thiscanbeachievedwithoutaffectingthesensitivityofthemethod,whereasinspectionofequation（12.6）showsthatwithDTAthesensitivitydependsonR（i.e.ΔT∝R）.UsingDSC,ifΔCisthedifferenceinheatcapacitybetweenthesampleandreferencecellthenthemeasuredheatflow（Q1）whenbothpansareemptywillbe

Q1=KΔC（12.10）

whereKisaconstantfortheapparatus.Ifthesamemeasurementprocedureisnowusedwiththesampleinpositionthedifferenceinheatcapacitybetweenthetwocellsbecomes（ΔC+msCp,s）,wheremsisthemassofthesampleandCp,sisthespecificheatcapacityofthesampleandthecorrespondingmeasurementis

Q2=K（ΔC+msCp,s）（12.11）

Ifnowthesampleisreplacedbyacalibrant（c）（e.g.alumina）andtheprocedureisrepeatedthemeasurementbecomes

Q3=K（ΔC+mcCp,c）（12.12）

Fromequations（12.10）,（12.11）and（12.12）itfollowsthat

（12.13）

ThusCp,sversustemperaturecurvescanbeobtained.Thesecanbeintegratedtogiveenthalpychanges（seeSection12.2.2（a））.

12.2.2MODULATEDDSC（MDSC）

Inequation（12.13）theheatcapacityofthesampleisdeterminedwiththeassumptionthattheheatingorcoolingisalinearramp.Forasimplefirstorderthermodynamictransitiontheformofthecurve,itsmagnitudeandlocationarethesameforbothcoolingandheatingcycles.Thesystemisthereforethermodynamicallyreversible.However,foranumberofprocessessuchastheglasstransitionandpolymerizationofathermosetresintheheatcapacitycanbethermodynamicallyirreversible.Theglasstransitionisnormallyconsideredtobeauniquelydefinedtemperaturehoweverinpracticetheprecisevaluedoesreflecttheeffectsofdisorderwhichmaybefrozenasaconsequenceofrapidcoolingofthesample.TheeffectofirreversibleprocessesonthemeasuredDSCtraceisthatthefirstmeasuredcurveisoftendifferentfromthesecondandsubsequenttracesobtainedbycoolingandre-heatingthesample.Thisfactcanbeusedtodeterminetheextentofresidualmonomerinafabricatedsampleandalsothefrozen-inentropyinasupercooledglassymaterial.Themodulatedtechniquehasbeendevisedtoallowseparationofreversibleandirreversiblecomponentstothetotalmeasuredheatcapacity.Superpositionofanoscillatoryheatingandcoolingcycleonthelinearrampallowsseparationofthesecomponents.ThesampleisheatedtoatemperatureT1'thatisabovethelinearrampvalueT1andthencooledtoT1"whichisbelowthelinearrampvalue.Thetemperatureseenbythesamplewillthereforehavetheformshownbelow（Fig.12.4）.

Fig.12.4SchematicrepresentationofamodulatedDSCexperiment.

Theperiodoftheoscillationcanbeadjustedtoincreasethesensitivityoftheexperiment.TypicallythefrequencyofoscillationisaboutIHzandtheamplitudeoftheoscillationadjustedtobeabout5º.Theprecisevaluescanbeadjustedtoaparticularexperimentalsituation.Themodulationppliedtothesamplesisthencomparedwiththesimilarvariationappliedtoareferenceandtherealandimaginarycomponentsofthedifferen