A model for the oxidation of ZrB2, HfB2 and TiB2

www.elsevier.com/locate/actamat

AmodelfortheoxidationofZrB2,HfB2andTiB2T.A.Parthasarathy

ba,*a,R.A.Rappb,M.Opekac,R.J.Kerans

ddUES,Inc.,Dayton,OH45432,USA

TheOhioStateUniversity,Columbus,OH,USA

cNavalSurfaceWarfareCenter,CarderockDivision,WestBethesda,MD,USA

AirForceResearchLaboratory,MaterialsandManufacturingDirectorate,AFRL/MLLN,Wright-PattersonAFB,OH45433-7817,USA

Received16March2007;accepted7July2007

Availableonline4September2007

Abstract

AmechanisticmodelthatinterpretstheoxidationbehaviorofthediboridesofZr,HfandTiinthetemperaturerangeof$1000–1800°Cwasformulated.Availablethermodynamicdataandliteraturedataforvaporpressuresanddiffusivitieswereusedtoevaluatethemodel.Goodcorrespondencewasobtainedbetweentheoryandexperimentsforweightgain,recessionandscalethicknessasfunc-tionsoftemperatureandoxygenpartialpressure.Attemperaturesbelowabout1400°C,therate-limitingstepisthediffusionofdissolvedoxygenthroughafilmofliquidboriaincapillariesatthebaseoftheoxidationproduct.Athighertemperatures,theboriaislostbyevaporation,andtheoxidationrateislimitedbyKnudsendiffusionofmolecularoxygenthroughthecapillariesbetweennearlycolumnarblocksoftheoxide,MO2.

Ó2007ActaMaterialiaInc.PublishedbyElsevierLtd.Allrightsreserved.

Keywords:Model;Oxidation;ZrB2;HfB2;TiB21.Introduction

Componentsinhigh-flowenvironments,suchastheleadingedgeofahypersonicvehicle,aresubjectedtoveryhightemperaturesandrequirethatthegeometricintegritybemaintainedduringservice[1,2].Thehightemperaturesandthelow-pressureenvironmentmakeoxidationandevaporativelosscriticalfactorsinmaterialselection[3].ThediboridesofZrandHf,whicharetermedultra-high-temperatureceramics(UHTCs),havehighmeltingpointsandareamongthemostpromisingmaterialsforlong-termserviceunderhypersonicconditions[3,4].Thediboridesarealsoattractiveduetotheirhighthermalconductivity,whichisusefultooffsetthestringentthermalgradientsimposedbyaerothermalheating.Thesediborideshavealsobeenconsideredascoatingsforprotectionofcarbon-basedcomposites[5]andtitaniumalloys[6].

Correspondingauthor.

E-mailaddress:Triplicane.parthasarathy@wpafb.af.mil(T.A.Partha-sarathy).

*AhistoryofstudiesontheoxidationofdiboridesisgivenbyOpekaetal.[3].Basedonthepromiseshownfromearlyworks,agoodoxidationscreeningstudyoftherefractorydiboridesofTi,Zr,Hf,NbandTawasconductedbyKauf-manetal.[7,8]andlatersummarizedbyFenter[9].TheyreportedthatthediboridesofZrandHfarethemostoxida-tionresistant.TheyfurtherreportedthatadditionsofSiCupto20%byvolumefurtherimprovedtheoxidationresis-tanceofthesematerials.Opekaetal.[3,4,10]pointedoutthatthehighmeltingpointsandlowvaporpressuresoftheoxidesandsub-oxidesofZrandHf,andthebeneficialvaporpressurevs.PO2relationshipsforBOxspecies,areresponsibleforthesuperiorhigh-temperatureresistanceofthesematerials.Usingvolatilitydiagrams,theyfurthershowedthatthediboridesofZrandHfappeartohavetheleastproblemwithrespecttodisruptionoftheproductscalebyvaporspecies.ZrandHfcarbidesarelimitedbyCOpar-tialpressuresthatexceed1atmabove1730°C,yieldingaporousandnon-protectiveoxidescale.Silicahasverylowdiffusivitiesforoxygenandisaveryprotectivescaleatmoremoderatetemperatures,butactiveoxidationtogaseousSiO

1359-6454/$30.00Ó2007ActaMaterialiaInc.PublishedbyElsevierLtd.Allrightsreserved.doi:10.1016/j.actamat.2007.07.027

6000T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–6010

disruptsthescaleaboveapproximately1800°C[3,4,10].Consequently,SiC,Si3N4andtransitionmetalsilicidesarenotveryusefulatultra-hightemperatures.Incontrast,boricoxideisglassy,flowstofillporesintheoxidescaleandexhibitsa1atmpartialpressure(forgaseousB2O3)onlyat1950°C,despiteitswell-knowntendencytoevaporateatmoderatetemperatures(above$1200°C).ThegaseousB2O3evaporatesatopensurfacesiteswithoutdisruptingthescale.WhenSiCisaddedtothesystem,furtherimprove-mentsarerealizedsincethesilica-basedglassoffersgreaterresistancetoevaporationthanB2O3alone[11].

Thusfar,theunderstandingofthesuperioroxidationbehaviorofthediborideshasbeenprimarilyqualitative.Forengineeringdesignandapplications,aquantitativemodelisneededthatpredictsallaspectsofoxidation.Itisdesirabletopredictfactorssuchasscalethickness,sub-straterecessionandweightchange,underasetofcomplexconditionsoftemperatureandenvironment.Manyexperi-mentalworkshavebeenconductedunderisothermalcon-ditionsandinalaboratoryenvironment,whilerealengineeringconditionsinvolvethermalgradients,environ-mentalgradientsandturbulentgasflow.

Thelong-termobjectiveofthisworkistodevelopamodelthatcanbeappliedtoaidengineeringdesignforhypersonicenvironments.However,thecomplexnatureofthematerialsmakesitnecessarytostartwithamodelforsimplersystemsandconditions,wherereliableandwell-definedexperimentaldataareavailable.Inthispaper,wepresentamodelthatisfoundtopredictthescalethick-ness,metalrecessionandweightchangesfortheoxidationofZr,HfandTidiboridesunderisothermalandslowgasflowconditions.Variousfactorssuchasresidualboriacon-tentanddependenciesontemperatureandoxygenpartialpressurearealsopredictedandfoundtobeinreasonableagreementwithexperiments.Thismodelprovidesareason-ablebasisforfuturework.

2.Conceptualframework

Themodelreportedinthisworkisbasedonconceptsoftheoxidationprocessderivedfrommicrostructuraldetailsavailableintheliterature.Themorphologyoftheoxidationproduct,aswellasthephasespresentandtheirdistribution,hasbeenreportedbyseveralresearchers[3,4,9,10,12].ForZrB2(orHfB2),oxidationinairresultsinadenseadherentoxidescaleconsistingoftwophases,ZrO2(orHfO2)andB2O3.Nointermediatephasesorboratesofthesesystemsareobservedattheinterfaces.Thedistributionofthephasesinthescalevarieswithtem-perature.Atlowertemperatures([1000°C),aglassyB2O3filmisobservedontopofthe(ZrO2+B2O3)scale,butthisexternalborialayerisabsentathighertempera-tures.Atalltemperaturesaporouszirconiascaleisonecomponentoftheoxidationproduct.Atlowandinterme-diatetemperatures,theporesinthezirconiaarefilledwithboria.Further,themicrostructureoftheZrO2appearstochangefromequiaxedgrainsattemperaturesbelow$1000°C,tocolumnargrainsathighertemperatures.Athighertemperatures,theoutersurfaceofthescaleisnodularZrO2,andthecross-sectionrevealsacolumnarstructureofZrO2withaglassyB2O3inbetweenthecolumnargrains.Asimilartransitioninoxidationrateshasbeennotedwithrespecttooxygenpartialpressure.Thedependenceoftheoxidationrateontheoxygenpar-tialpressurechangesfromlineartonodependenceasthetemperatureisincreasedabove1150°CforZrB2[13].Inadditiontothesetwoexperimentallyobservedtempera-tureregimes,itisexpectedthatahigh-temperatureregimeexistswherethereisverylittleornoboriainthescale.ThethreeregimesareshownschematicallyinFig.1.Inthiswork,wefocusontheintermediatetemperatureregime,whichistheregimeofengineeringinterest.Themodeldevelopedherepredictsthetemperaturerangeof

Intermediate Temp (~1000 to ~1800 °C)ZrB2B2O3(g)ZrO2B2O3(l)Low Temp < ~ 1000°CZrB2B2O3(g)Very High Temp > ~1800 °CZrB2B2O3(g)ZrO2B2O3(s,l)ZrO2Fig.1.TheoxidationproductsformedduringoxidationofZrB2inthreetemperatureregimes.T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–60106001

theintermediateregimeandtheoxidationbehaviorwithinthisintermediateregime.

TheconceptualframeworkfortheoxidationintheintermediatetemperatureregimeisshownschematicallyinFig.2a,andthestepsconsideredinthemodelareshowninFig.2b.Foroxidationatintermediatetemperatures,oxi-dationofthemetal(inthediboride)resultsinanevapora-tion-resistantrefractoryoxidethatformsaporousskeleton.Theoxidationoftheboronresultsinaglassyboricoxidethatflowstofillthebaseoftheporousskele-ton.Atthesurface,theboriaevaporates.Bothoxidationreactionsoccurattherefractorysubstrate–scaleinterfacebytheinwarddiffusionofoxidantspecies.Atsteadystate,oxidationtakesplacebybothgaseousdiffusionofoxygenthroughtheopenporesintheoxideskeleton,andbydis-

solvedoxygendiffusingthroughtheliquidboria(atthebaseoftheporousskeletonbeneaththesurface)toreachtheZrB2–scaleinterface.Thetworeactionsare:5

ZrB2þO2!ZrO2þB2O3

2

B2O3ðlÞ!B2O3ðgÞ

ð1Þð2Þ

TherateofdissolutionofoxygenintoB2O3isnottakentoberatelimitinginthemodeldevelopedbelow.Thetransportofoxygenthroughtherefractoryoxideisassumedtobenegligiblecomparedtothatthroughtheporesfilledwithglassyboria.Attemperaturesofinterest(above1000K),B2O3(v)hasbeenshowntobethedomi-nantvaporspeciesatoxygenpartialpressuresgreaterthanthatinequilibriumwithZrB2andZrO2[12].

aScale thickness, LB2O3(l) layer thickness, hB2O3(l)−>B2O3(g)O2(g)ZrB22ZrBO2in B2O3(l)O2(g)B2O3(l)0.2 O20.8 N2(1 atm)B2O3(g)B2O3(g)Recession, R5ZrB2+O2−>ZrO2+B2O3(l)2ZrO2bZrB2B2O3(l) formationB2O3(g) OUT O2(g) INB2O3(l)O2N2O2air(0.2 O2, 0.8 N2)B2O3perfect sinkfor B2O3(g)ZrO2(inert)O2permeation through B2O3(l) IN PB2O3(at l-ginterface)= Vapor Pressure in eqbmwith B2O3(l)Fig.2.(a)SchematicsketchofmechanismsinvolvedoxidationofZrB2inair,intheintermediatetemperatureregime(1000–1800°C).(b)Schematicofthemechanisticstepsconsideredinthemodel.6002T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–6010

3.Themodel

Theobjectiveofthemodelistopredictthekineticsofscalethickness,substraterecessionandweightchangeasafunctionoftimeattemperatureandoxygenpartialpres-sure.AschematicforthemodelisshowninFig.2b.DenseZrO2isassumedtobeabarriertooxygenpermeation,duetoitsnegligiblerateofambipolardiffusion.ThevalidityofthisassumptionispresentedintheAppendix.TheporositybetweenZrO2columnsisassumedtobecontinuous,andtortuosityisneglected.

TheambientairistakentoconsistofN2andO2only.Afraction,f,ofthescaleistakentobeporousandpro-videacontinuouspathwayforgaseousdiffusionandfortheevaporationofliquidB2O3atthesurface,whichistakentobeaperfectsinkforB2O3.Therelativeratesoffillingtheporebyoxidationatthesubstrate–scaleinterfaceandtheevaporationofB2O3determinestheextenttowhichtheporesarefilledwithB2O3.Atagiventime,t,thetotalscalehasathicknessofLanditsinnerportionisfilledwithliquidB2O3toathickness,h.ThedistanceL–histhediffusiondistanceintheporeoverwhichgaseousoxygenmustdiffusetoreachtheB2O3.TheoxygenisassumedtodissolveinB2O3andcontinuetodiffusetothesubstrate–scaleinterfacewheretheoxi-dationreactionoccurs.AsexplainedintheAppendix,thedensezirconiacolumnsareconsideredtobeimpervi-oustooxygen.

Atthesubstrate–scaleinterface,reaction(1)takesplace.Thus,atsteadystate,thediffusionalfluxofoxygenmustbebalancedbytheformationratesofZrO2andB2O3.jJ5O2j¼2n_5B2O3¼2

n_ZrO2:ð3Þ

ThefluxofgaseousoxygenintheporouschannelthatisL–

hlongisgivenby:

i

jJCaOO2ÀCO2

2j¼fDO2

LÀh

;ð4Þ

whereDisthediffusioncoefficient,Ctheconcentration,thesuperscript‘‘i’’referstotheinterfacebetweenB2O3(l)andthegasphase,whichisatadistancehfromthesubstrate–scaleinterface,andthesuperscript‘‘a’’referstothescale–ambientinterface.Similarly,thefluxofB2O3(v)fromthesurfaceofB2O3(l)totheambient,isgivenby:

JCiBOB2O3ÀCaB2O3

23¼fDB2O3LÀh:ð5Þ

TakingJB2O3¼nB2O3(quasi-steadystate),andcombiningEqs.(3)–(5)theconcentrationofmolecularoxygenattheB2O3liquid–vaporinterfaceisobtainedas:

Ci5DB2O3i

O2¼CaO2À2DðCB2O3ÀCaB2O3Þ:ð6ÞO2TheconcentrationCaB2O3attheambientsurfaceistakentobezero.TheconcentrationCiB2O3oftheB2O3(v)atthisinterface,whichdoesnotdependonthelocaloxygenpres-sure,isobtainedfromvaporpressuredata[14]:

Ci¼PB2O

3B2O3

RT

;

PB2ðatmÞ¼3Â108exp󰀂À45;686

󰀃

O3

T

:

ð7Þ

ThepermeabilityofoxygenthroughtheliquidB2O3dependsontheoxygenactivitygradientacrosstheliquidlayerofthicknesst.Fromtheidealgaslaw,theoxygenpar-tialpressureattheB2O3liquid–vaporinterfaceisgivenby,PiiO2¼RTCO2:

ð8Þ

Theactivityofoxygenatthescale–B2O3interfaceisgivenby:5=2

K¼aZrB2ðPsÞreaction1

O2a:

ð9Þ

B2O3aZrO2

UsingthethermodynamicdataofBarin[14]forreaction(1),andequatingtheactivitiesofthesolidphasestounity,Eq.(9)givesthefollowingfortheoxygenactivity,PsthesubstrateZrBO2,at2–B2O3interface(s):

Ps

10󰀂99;967󰀃

O2ðatmÞ¼5Â10expÀT:ð10Þ

ThevaluesgivenbyEq.(10)areconsistentwiththemapspresentedbyFahrenholtz[12].ThediffusivityofoxygenthroughboriahasbeenmeasuredbyTokudaetal.[15],whofoundalineardependenceoftheperme-ationfluxonoxygenpartialpressureanddeducedthatmolecularoxygendiffuses.Theoxygenfluxacrosstheli-quidboria,JO2ðB2O3Þ,isrelatedtothepartialpressuregradientthroughtheoxygenpermeabilitycoefficient,P,as[16,17]:

jJPOO2–B2O32ðB2O3Þj¼hfhPis

iO2ÀPO2:ð11ÞTheoxygenpermeabilitycoefficient,PO2–B2O3,isobtainedfromtheliterature[15,17]as:

P=msatmÞ¼0:15exp󰀂À16;000

󰀃O2–B2O3ðmolT:ð12Þ

SubstitutingEq.(12)intoEq.(11)andequatingthefluxofoxygenthroughthegaseouslayer,givenbyEq.(4),anequationrelatingtheborialayerthickness,h,tothescalethickness,L,isobtained:sh¼qL;

q¼

PO2–B2O3ðPiO2ÀPO2Þ

DOiis

2ðCaO2ÀCO2ÞþPO2–B2O3ðPO2ÀPO2Þ

:ð13Þ

Theequationsfortherateofchangeofscalethickness(L)andmass(W)ofthescalearegivenbyaccountingforthefluxbalanceusingEq.(3)

dL󰀂1󰀃󰀂1󰀃dWZrO2

dt¼¼

󰀂1ÀfqZrO2dt

1󰀃

21Àf5JMZrOO2

2q;ð14ÞZrO2

T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–60106003

whereqreferstodensityandMtomolarvolume.Combin-ingwithEq.(4),oneobtains:

dL2󰀂MZrO2󰀃󰀂f󰀃Cai

O2ÀCOdt¼5qDO2

2ð1ÀqÞ:ð15ÞZrO21ÀfLIntegrationofEq.(15)givesaparabolicequationforgrowthofthescale:

\"L2¼22󰀂MZrO2󰀃󰀂f󰀃CaOÀCi

#O2

5qfDO22ð1ÀqÞt:ð16Þ

ZrO21ÀThemagnitudeofrecessionofthesubstrateisgivenby:R¼Lð1ÀfÞ

MZrB2=qZrB2

M:

ð17Þ

ZrO2=qZrO2

Thetotalweightchangeperunitareaisgivenby:DW

A

¼LqZrO2ð1ÀfÞþhfqB2O3ÀRqZrB2:ð18Þ

Itremainstoevaluatethediffusioncoefficientofgaseousoxygenthroughtheporousregionfilledwithoxygen,B2O3(g)andnitrogen.Thediffusioncoefficientinamul-ti-componentgaseoussystemcanbeapproximatedby[18]:

D1;ð2;...;iÞ¼P1i¼1

;

xni

i¼i

ðxi=D1ÀiÞ

Pj¼1

j

n;ð19Þ

j

whereDreferstodiffusivity,ntomolefraction,subscript‘‘1,(2,...,i)’’referstothediffusivityofspecies1inamed-iumofispecies,subscript‘‘1Ài’’referstothediffusivityofspecies1inabinarymixtureofspecies1andi.Thediffu-sivityD1–2ofspecies1inabinarygasmixturewithspecies2isgivenby[19]:

D0:0018583T3=2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1=M1Þþð1=M2Þ

ffi

1–2¼r2Xð20Þ

12DPwhereD1–2isthegasdiffusivityincm2/s;

X1:060360:1930001:03587

D¼

TÃ0:15610þ

expð0:47635TÃÞþexpð1:52996TÃÞ

þ

1:76474

expð3:89411TÃÞ

;TükTpe;e12¼ffiffiffiffiffiffiffie1e2ffi;r12¼0:5ðr1þr2ÞinA˚;P¼pressureinatm;12

Mi¼molecularweightðg=molÞ.

TheparametersneededfortheaboveexpressionscanbeobtainedfromtheworkofSvehla[20].Whenthesizeoftheporeissmall,themeanfreepathislongerthanthecharac-teristicdimensionofthesystem,andthediffusionofgasesisgovernedbyKnudsendiffusion[21]:

D4󰀂8RT󰀃1=2r

k¼

3pM2;ð21ÞwhereDkistheKnudsendiffusivity,Mthemolecularmassofthediffusingspeciesandrtheradiusoftheporouspath-way.Theeffectivediffusivityisgivenby[21]:

Deff¼ðDÀ1À1

1

kþD1–2ÞÀ:

ð22Þ

UsingEqs.(19)–(22),Eqs.(16)–(18)givethescalethick-ness,recessionandweightgainasafunctionoftempera-ture,timeandpartialpressureofoxygen.4.Modelpredictionscomparedwithliteraturedata4.1.ZrB2ThemodelwasverifiedbycomparingitspredictionswithexperimentaldatareportedintheliteraturefortheoxidationofZrB2.AscanbeseenfromthedevelopmentinSection3,mostoftheparametersneededforthepredictioncanbeobtainedfromtheliterature.Twoparametersinthemodelthatarenotwellknownaretheporefractionandporeradius,andtheyarelikelytovarybetweenexperimentsdependingonstartingmicrostructureandoxidationcondi-tionssuchastemperatureandenvironment.Thevaluesof0.05and0.5lm,respectively,werechosentogetthebestfitwithalloftheexperimentaldataforZrB2,butwerekeptthesameforHfB2andTiB2.Fromwhatcanbeinferredfrommicrographsofoxidescalespresentedintheliterature,thesevaluesarequiteplausible.Thesensitivityofthepredictionstothechoiceofthesevariableswasexaminedandispre-sentedinalatersection.Theothersignificantunknownsarethevelocitiesandwatercontentsoftheatmospheresintheexperiments.Futureexperimentersareencouragedtoreporttheseparameters;theycanhavelargeeffects.

Extensivethermogravimetricanalysis(TGA)dataonweightchangevs.timein250TorrofpureoxygenwasreportedbyTrippandGraham[22].TheycomparedtheirresultswiththeparabolicrateconstantdatafromBerko-

-4.01400 °C1000 °C-4.5Kuriakose-Margrave-5.0Berkowitz-MattuckTripp-Graham)s--5.54m/2-6.0gk(-6.5 pk -7.0goL-7.5-8.0-8.5ZrB2250 Torr of Oxygen-9.00.000550.000750.000951 / Temperature (K-1)Fig.3.Theexperimentaldatafromthreedifferentworksintheliterature[12,22,23]ontheparabolicrateconstantofmeasuredweightgainofsamplesin250Torrofpureoxygenatdifferenttemperaturesareshowncomparedwiththemodelpredictions.Theporosityinthescale,f,wastakentobe0.05andtheporeradiustobe0.5lm.6004T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–6010

witz-Mattuck[13]andKuriakoseandMargrave[23].Fig.3showstheresultsfromallthreeexperimentalworks,com-paredwiththemodelpredictions,foranassumedporefractionof0.05andporeradiusof0.5lm.TheweightgaindataofTrippandGrahamarelowerthanthemodelpre-dictionsattemperaturesabove1200°C.Thelowervaluesofexperimentaldatacouldarisefromalowgasflowrateusedintheexperiment,whilethemodelassumesaperfectsinkforB2O3(v).However,whenthemodeliscomparedwithallofthereporteddata,thecorrespondenceistakentobegoodinthatitmatcheswellatlowtemperaturesandisagoodrepresentationofthewidelyscattereddataathightemperatures.

Fentersummarizedthemetalrecessionofseveraldibo-ridesinair,asafunctionoftemperature[9].TheresultsforZrB2inFig.4aarecomparedtothemodelprediction.Themodelcomparesverywellwithdataupto$1850°C,abovewhichthemodelpredictsthatthehigh-temperatureregimedominatesandthatalloftheboriawillevaporate

assoonasitforms.Theexperimentaldatashowasignifi-cantenhancementinrecessionatthistemperature.Themodelpredictsthetransitiontemperaturecorrectly.Thereasonfortheupwardtrendinrecessionisnotclear;onepossibilityissignificantspallationofMO2duetotheabsenceofboriaathighertemperature.

Finally,Opekaetal.[3,10]andTalmyetal.[24]havereportedthemasschangeandoxidelayerthicknessesofZrB2oxidizedinairorAr/O2mixtureswithanoxygenpar-tialpressureof0.2.TheirdataarecomparedwiththemodelpredictionsinFig.4bandc.Again,thecorrespon-denceistakentobequitereasonablegiventheassumedgeometricalfactors.4.2.HfB2Themodeliseasilyextendedtootherdiboridesbysim-plyusingtheappropriatethermodynamicdataandtheappropriatephysicalpropertiessuchasmolecularweight,

a100001800 °C1600 °C ZrB2, air, 1hrRecession, μm1000100Model predictionExperimental (Fenter)100.000450.000500.000551 / Temperature (K-1)0.00060bMass Change per unit area, kg/m21.E+001550 °C1150 °CcOxidized layer thickness, μm1400 °C1000 °C1.E-01ZrB2, O2/Ar, 2hr1001.E-02Model PredictionExperimental data (Talmy)Experimental data (Opeka)ZrB2, air, 5hrModel PredictionExperimental data (Opeka)1.E-030.00040.00050.00060.00070.000810.00040.00060.00080.00101 / Temperature (K-1)1 / Temperature (K-1) Fig.4.(a)ModelpredictionsoftherecessioninairasafunctionoftemperatureisshowncomparedwithdataofKaufmanandClougherty[7,8]andFenter[9].(b)ThemodelpredictionsformasschangeareshowncomparedtoexperimentaldataobtainedinAr/O2gasin2h[10].(c)Thescalethicknessobtainedafter5hinairbyOpekaetal.isshowncomparedtothepredictions.Theporosityinthescale,f,wastakentobe0.05andtheporeradiustobe0.5lm.T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–60106005

aRecession, microns1650 1400 ºC1000Model predictionData (Kaufman-Fenter)b1650 1400 1150 ºC-3.0Model prediction-4.0Log kp (kg2/m4-s)Data (Berkowitz-Mattuck)-5.0-6.0-7.0HfB2, 250 Torr of Oxygen 100HfB2, air, 1 hr100.00050.00060.00070.00081 / Temperature (K-1)-8.00.00050.00060.00070.00081 / Temperature (K-1)Fig.5.ComparisonofmodelpredictionswithexperimentaldataonoxidationofHfB2.(a)RecessioninflowingairfromKaufmanandClougherty[7,8]andFenter[9].(b)ParabolicrateconstantforweightgainasafunctionoftemperaturefromBerkowitz-Mattuckin250Torrofoxygen[13].Theporosityinthescale,f,wastakentobe0.05andporeradiustobe0.5lm.Weight Change, kg/m2density,etc.ForHfB2,oneobtainstheoxygenpartialpres-sure,PsO2,attheHfB2–B2O3interfaceas:

󰀂󰀃

100;0709s

PO2ðHfB2ÞðatmÞ¼6Â10expÀ:ð23Þ

TTworeportsintheliteraturehavecharacterizedtheoxi-dationbehaviorofHfB2.Fenter[9]summarizedthemea-suredrecessionratesinflowingair,whileBerkowitz-Mattuck[13]reportedontheparabolicrateconstantsin250Torrofpureoxygen.Thesedataarecomparedwiththemodelpredictionsusingaporefractionof0.05andporeradiusof0.5lminFig.5.Thepredictedrecessionishigherthanthemeasuredvalues,whilethepredictedpara-bolicrateconstantsarelower.4.3.TiB2ForTiB2,oneobtainstheoxygenactivity,PsO2,attheTiB2–B2O3interfaceas:

󰀂󰀃

95;12610s

PO2ðTiB2ÞðatmÞ¼5Â10expÀ:ð24Þ

TByuseoftheappropriatephysicalpropertiesforTiO2andTiB2theoxidationbehaviorwaspredicted,onceagainassumingaporefractionof0.05andaporeradiusof0.5lm.Thepredictionswereevaluatedbycomparingwithexperimentaldataavailableonweightchangeandreces-sionreportedbyKohetal.[25].ThiscomparisonisshowninFig.6.

5.Parametric/sensitivitystudies

Aparametricstudywasconductedtoassessmodelsen-sitivityandtodeterminetheeffectsofexperimentalvari-ables.Fig.7showsthetemperaturedependenceoftheparabolicrateconstantforoxidationofZrB2.Aclear

a0.15TiB2, air1200 °C0.100.05800 °C0.000 5Time, hrs10bScale thickness, microns140120100806040200TiB2, air1200 °C800 °C0 5Time, hrs10Fig.6.ComparisonofmodelpredictionswithexperimentaldataonTiB2.(a)Weightgainwithtimeinairand(b)scalethicknessafteroxidationinair.Theporosityinthescale,f,wastakentobe0.05,withaporeradiusof0.5lm.ThedatawereobtainedfromKohetal.[25].6006T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–6010

a1400 °C-3.0ZrB2, air-4.0b1ZrB2, air0.81000 °CRelative fraction of B2O3Log kp (kg2/m4-s) -5.00.6-6.00.41800 °C0.2-7.0Activation energy changes ....0.00061 / Temperature (K-1)0.0008... as B2O3isexhausted by evaporation0.00061 / Temperature (K-1)0.0008-8.00.000400.0004Fig.7.ResultsofaparametericstudyontheeffectoftemperatureonoxidationbehaviorofZrB2.(a)Theparabolicrateconstantand(b)therelativefractionofliquidboriainthescale.Themodelpredictsachangeintheapparentactivationenergyfortheparabolicrateconstantthatcorrelateswithlossofboriafromthescale.a10000 ZrB2, air, 1hrbRelative fraction of B2O31ZrB2, airRecession, μm10001001.50.50.150.050.80.51.51000.60.150.40.05100Pore radius μmPore radius μm0.200.0004100.00040.00050.00060.00071 / Temperature (K-1)0.00050.00060.00071 / Temperature (K-1)cOxidized layer thickness, μm10000 ZrB2, air, 1hr1001.50.50.150.05dLogkp (kg/m-s) -41000-51001.50.50.150.05ZrB2, air24100Pore radius mμ-6Pore radius μm100.00040.00050.00060.0007-11 / Temperature (K)-70.00040.00050.00060.00071 / Temperature (K-1)Fig.8.Theeffectofporeradius(Knudseneffect)on:(a)recession,(b)boriafractioninscale,(c)oxidethicknessand(d)parabolicrateconstant.T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–60106007

changeinapparentactivationenergyisseenataround1400°C.Thefigureincludesthepredictedfractionofboria(h/L)inthescale.Thechangeinactivationenergyclearlyarisesfromthelossofboriafromthescale.

Next,themodelwasusedtostudytheeffectofvaryingtheporefractionandporeradius,toassessitssensitivity.Theseparametersaredifficulttomeasureandhavenotbeenreportedexperimentally.Further,sincethemodelneglectsthetortuosityoftheporousscale,theporeradiusandporefractionusedinthemodelmustbeconsideredtobe‘‘effec-tive’’parametersratherthantheactualphysicalvalues.Itisquitepossiblethattheseparametersthemselvesvarywithtemperatureorotherexperimentalconditionssuchasenvi-ronment.Thusitisusefultoconductasensitivitystudy.Fig.8showstheeffectofporeradiusontherecession,scalethickness,parabolicrateconstant,andthethicknessfractionofboriainthescale.Thechoiceofporeradiusissignificantwhentheporeradiusissmallerthanabout1lm.However,theeffectisinsignificantattemperaturesbelowabout1400°C,whichcorrespondstothetempera-tureatwhichboriaevaporationbecomessignificant.Fig.9showstheeffectofporefractionontheoxidationbehaviorofZrB2inair.Therateconstant,scalethicknessandrecessionallincreasebyaboutanorderofmagnitudewhentheporefractionisincreasedfrom0.025to0.2.Thustheeffectisnearlylinear.

ComparingFigs.8and9toFigs.3–5itcanbeseenthatveryclosecorrespondencebetweenthemodelandexperi-mentaldatacouldbeobtainedifporefractionand/orporeradiusaretakentovaryratherthanremainconstantforallscales.Thescatterintheexperimentalresultsaloneimpliesthatthesegeometricalparametersmayvarydependingontheexposuretemperature,and/orexperimentalconditionssuchasflowratesofgases,andhumidity,andtheymayevenchangesomewhatduringthecourseofanexperiment.Theliteraturedataistypicallyincompletewithrespecttodetailsoftheenvironment,especiallyhumidity,andthescalemicrostructures.

Fig.10asummarizestheeffectofoxygenpartialpressureinO2/N2gasmixtureof1atmtotalpressure,onthepara-bolicrateconstantasafunctionoftemperature.Thedependenceislinearat1000°Candbecomesnegligibleat

aLog kp (kg2/m4-s)-3-4-5-6-7Pore fraction0.20.1250.0750.050.025ZrB2, air -80.00040.00050.0006-10.00071 / Temperature (K)bOxidized layer thickness, μm100000.20.1250.0750.050.025c10000 ZrB2, air, 1hr ZrB2, air, 1hr1000Recession, μm10000.20.1250.0750.050.025100Pore fraction100Pore fraction100.0004100.00050.0006-10.00070.00040.00050.00060.00071 / Temperature (K)1 / Temperature (K-1)Fig.9.Theeffectofporefractionon:(a)rateconstant,(b)oxidethicknessand(c)recession,forachangeinporefractionfrom0.025to0.2.6008T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–6010

aLog Kp (kg2/m4-s)-4.0-5.0-6.0-7.0-8.0-9.0-10.00.011600 °C1400 °C1200 °C1000 °CZrB20.1 1Oxygen partial pressure, atmbLog Kp (kg2/m4-s)-4Pore radius μm1600oCcLog Kp (kg2/m4-s)-4Pore fraction-51001.50.5-50.20.1250.0750.050.025-60.150.05-6ZrB2Ptot=1atm1600oCZrB2Ptot=1atm -70.0010.010.1 1-70.0010.010.1 1Oxygen partial pressure, atmOxygen partial pressure, atmFig.10.Parametricstudiesofthedependenceof:(a)parabolicrateconstantontheoxygenpartialpressureinO2/N2gasmixture(1atmtotalpressure).ThePO2dependenceitselfisdependenton(b)theporesizeand(c)porefraction.1600°C;thisisconsistentwiththeexperimentalobserva-tionsofBerkowiz-Mattuck[13].Fig.10bshowsthattheporeradiusispredictedtohaveasignificanteffectatlowoxygenpartialpressuresandtheporefractionhasanear-lineareffect(Fig.10c)asmightbeexpected.6.Summary

Aphysicalmodeltopredicttheoxidationbehaviorofrefractorydiboridesintheintermediatetemperatureregime(1000–1800°C)ispresented.Themodelassumesthattherefractoryoxideproductdoesnotsupportsignificantoxy-gendiffusionduetoalowoxygenvacancyconcentrationandtoalackofsufficientelectronicconductivity.Theporosityinthebaseoftheoxideisassumedtobefilledwithliquidboriawhich,however,evaporatesfromthesurface.Acomparisonofthemodelwithexperimentaldataintheliteratureshowsreasonableagreementinpredictingweightchange,metalrecession,oxidescalethicknessandthetem-peraturedependenceoftheparabolicrateconstantforZrB2,evenwithpostulatedvaluesforporegeometryandunknownexperimentalflowrates.Comparisonwithlim-itedliteraturedataforHfB2andTiB2showthatthemodelmayhaveageneralapplicabilitytorefractorydiborides,

althoughdataonoxygendiffusionandelectronicconduc-tivityoftheoxides,especiallyathightemperatures,willberequiredtorefinethemodel.Acknowledgements

ItisapleasuretoacknowledgeusefuldiscussionswithDr.I.TalmyofNavalSurfaceWarfareCenter,CarderockDivision,WestBethesda,MD,whosharedsomeofherunpublishedwork.WeacknowledgediscussionswithalltheparticipantsofaworkshoponUHTC,sponsoredbyDr.JoanFulleroftheAirForceOfficeofScientificRe-search(AFOSR)in2005.ThisworkwassupportedinpartbyUSAFContractNo.FA8650-04-D-5233.Appendix.Discountingambipolaroxygendiffusioninzirconia

Inthemodelforrefractorydiborideoxidation,thediffu-sionofoxygen,viaambipolardiffusion,isconsideredtobenegligiblebecauseofthelowmagnitudesofoxygenvacan-ciesandp-typeelectronicconductioninthezirconiaoxida-tionproduct.Intherelativelyhighoxygenactivitiesencounteredintheoxidation,onlyp-typeelectroniccon-

T.A.Parthasarathyetal./ActaMaterialia55(2007)5999–60106009

ductionneedstobeconsidered;n-typeconductionisnegli-gible.Infact,dataarenotavailableforthemagnitudeofthep-typeelectronicconductivityinrelativelypureZrO2,butsomemeasurementshavebeenmadeforcalcia-stabilized-zirconia,e.g.Zr0.85Ca0.15O1.85,asafunctionoftemperature.TheinterpretationofdefectequilibriabyLas-kerandRapp[26]permitsageneralevaluationofp-typeconductivityasafunctionofdopantconcentrationandoxygenactivitytoprovidemagnitudesfortheionicandp-typeelectronicconductivitiesforcompositionswithmuchlowerdopantconcentrations.

Considerthedefectequilibriumforzirconia:OxOþ2h_()12

O2ðgÞþVO€;ðA1ÞwhereOxOisanoxygeniononanoxygenlatticesite,V€isadoublyionizedoxygenvacancyandh

_isapositiveO

hole.Fromthelawofmassaction:1=2

K¼½VO€󰀆PA1

O2½h

_󰀆2:

ðA2Þ

Ifthezirconiaweredopedbysubstitutionofanaliovalentsolute,e.g.Ca2+ions,ontheZr4+latticesites,thenthesim-plifiedelectricalneutralityconditionwouldread:2½Ca00Zr󰀆¼2½VO

€󰀆(½h_󰀆:ðA3Þ

UponsubstitutionofEq.(A3)intoEq.(A2),twopredic-tionsarise:

½V€󰀆/½Ca00

OZr󰀆¼fðPO2ÞðA4aÞ½h

_󰀆/½Ca00Zr

󰀆1=2ðPO2

Þ1=4:ðA4bÞ

Fromtheusualjustifiedassumptionofconcentration-independentmobilitiesoftheoxideionsandpositiveholes,theionicconductivityandp-typeelectronicconductivityobeythesamedependenciesasEqs.(A4a)and(A4b).Patterson[27]haspresentedanevaluationoftheionictransferencenumberforZr0.85Ca0.15O1.85(CSZ)asafunc-tionoftemperaturetoestablishtheelectrolyticdomain.Fromaknowledgeoftheionicconductivity,providedbyPattersonetal.[28]andPatterson’stransferencenumbers[27],thep-typeelectronicconductivityforCSZcanbecal-culated.Then,usingEqs.(A4a)and(A4b),thefunctionscanbeappliedtonearlypureZrO2,e.g.containing150ppmdivalentimpurityconcentration.

FromPattersonetal.[28]for0.15CaO-stabilizedZrO2(CSZ),usingtheionicconductivityat1273Kandthereportedactivationenergy,theoxygenpartialpressure-independentionicconductivityisgivenas:

r1cmÀ1Þ¼2323exp󰀂À123;500

󰀃

ionðXÀ8:314T:ðA5Þ

Similarly,thep-typeelectronicconductivityinCSZcanbederived.Thereportedholeconductivityis2·10À4XÀ1cmÀ1at1273KandPO2¼106withtion=0.99.Further,theionicconductivityequalstheholeconductivityat(1/T,PO2)valuesof(0.0003,103)and(0.0012,1024).

Fromthesedata,andcombiningwithEq.(A5),forcondi-tionswhererion¼rh_,theholeconductivityisobtainedas:

r_ðX

À1

11=4󰀂235;200󰀃

hcmÀÞ¼23;280PO2expÀ8:314T:ðA6ÞFromLaskerandRapp[26],thedependenceon[Ca]isgi-venas:

r_/½Ca00󰀆

1=2

hZr;rion/½Ca00󰀆1

Zr:

ðA7Þ

CombiningEqs.A5,A6,A7oneobtains:

rÀ1cmÀ1Þ¼½C󰀆󰀂123;500󰀃

ionðXdopant

0:152323expÀ8;ðA8Þ

rðXÀ1

cmÀ1Þ¼󰀂½Cdopant󰀆󰀃:314T1=20:1523;280P1=4󰀂235;200󰀃h_O2expÀ8:314T

:ðA9ÞConsideraZrO2compositiondopedwith150ppmofadivalentimpurity.At1273K,theionicandholeconductivitiesare2·10À5XÀ1cmÀ1and1.64·10À7XÀ1cmÀ1,respectively,withatransferencenumbertion=0.9918.Thustheassumptionofnegligibleambipolardiffu-sionisvalidatedforthistemperature.

At1873K,theionicandholeconductivitiesare8.35·10À4XÀ1cmÀ1and2.03·10À4XÀ1cmÀ1,respec-tively,withatransferencenumbertion=0.804.Thus,atthesehighertemperatures,p-typeconductivitybecomesmoreimportantandambipolardiffusionofoxideionscan-notbeneglected.However,athightemperatures,theporesinthezirconiaareexpectedtobedevoidoftheboriaglassduetoevaporation.Thusitisunlikelythatambipolaroxy-gendiffusioncouldcontributeafluxthatiscompetitivewithrespecttogaseousdiffusionofoxygenmoleculesinthegasphase.Theassumptionofatleast150ppmofaliovalentimpurityinthezirconiaoxidationproductisnotunrealistic.TheremainingissuewiththisanalysisistheassumptionthatthelightlydopedzirconiaproductwouldexistinaCaF2-typelatticestructureforthescale;however,ifthezirconiascaleexistedonadifferentcrystallinestate(e.g.tetragonalormonoclinic),theoxygendiffusivitywouldcertainlybenegli-gibleregardlessoftheelectronicconductivity.References

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