Using Cluster Dynamics to Model Electrical Resisti

arXiv:cond-mat/0611524v1[cond-mat.mtrl-sci]20Nov2006UsingclusterdynamicstomodelelectricalresistivitymeasurementsinprecipitatingAl-ScalloysEmmanuelClouet∗andAlainBarbuServicedeRecherchesdeM´etallurgiePhysique,CEA/Saclay,91191Gif-sur-Yvette,FranceAbstractElectricalresistivityevolutionduringprecipitationinAl-Scalloysismodeledusingclusterdynamics.Thismesoscopicmodelinghasalreadybeenshowntocorrectlypredictthetimeevolutionoftheprecipitatesizedistribution.Inthiswork,weshowthatitleadstootoresistivitypredictionsinquantitativeagreementwithexperimentaldata.Weonlyassumethatallclusterscontributetotheresistivityandthateachclustercontributionisproportionaltoitsarea.Oneinterestingresultisthattheresistivityexcessobservedduringcoarseningmainlyarisesfromlargeclustersandnotreallyfromthesolidsolution.Asaconsequence,onecannotassumethatresistivityasymptoticbehaviorobeysasimplepowerlawaspredictedbyLSWtheoryforthesolidsolutionsupersaturation.Thisforbidsanyderivationoftheprecipitateinterfacefreeenergyorofthesolutediffusioncoefficientfromresistivityexperimentaldatainaphase-separatingsystemlikeAl-Scsupersaturatedalloys.Keywords:Precipitation,Resistivity,Kinetics,Aluminumalloys,ClusterdynamicsPACS:64.60.Cn,64.60.-i,64.70.Kb,64.75.+g1IntroductionResistivitymeasurementsareusuallyaconvenientwaytofollowprecipitationkineticsinphase-separatingsystems.Indeed,assumingthatresistivityispro-portionaltothesolidsolutionsolutecontentasitisusuallydone[1],onegetsdirectaccesstothesolutesupersaturation.Theso-calledLSWtheory(Lif-shitzandSlyozov[2]andWagner[3])asextendedbyArdell[4]allowsthen∗[e-mail]evier6February2008todeducefromthesemeasurementskeyparameterslikethesolubility,theprecipitateinterfacefreeenergyandthesolutediffusioncoefficient.NumeroussuchmeasurementsexistinAl-Scalloys[5–10].TheyhavebeenintendedtocharacterizetheprecipitationoftheAl3ScL12structureinalu-minumalloys.Inapreviousstudy[11],weusedclusterdynamicstomodelpre-cipitationinAl-ZrandAl-Scalloys.Thismesoscopicmodelingwasshowntoprovidepredictionsoftheprecipitatesizedistributionsinquantitativeagree-mentwithavailableexperimentaldata[12–14].Itisthereforeworthseeingifsuchanagreementcanbeobtainedalsowithresistivitymeasurements.Thepurposeismainlytoseehowpredictiveclusterdynamicscanbe,i.e.ifonecanobtainreliableinformationconcerningtheprecipitatesaswellasthesolidsolutioninthewholetimerangeofprecipitationkineticsandnotonlyintheasymptoticlimitofthecoarseningstagelikeLSWtheorydoes.ConfrontingclusterdynamicspredictionswithresistivitymeasurementswillvalidatetoothemultiscaleapproachdevelopedforprecipitationinAl-Zr-Scalloys[11,15–18].Theuseofresistivityexperimentstovalidateamultiscalekineticmodelinghasalreadyproveditsefficiencyforirradiatediron[19].Thefirstpartofthisarticleexplainshowresistivitycanbemodeledwithintheframeworkofclusterdynamics.Athoroughcomparisonbetweentheobtainedmodelpredictionsandavailableexperimentaldataisthenperformed.Thisallowsustoconcludeonthevalidityoftheinterfacefreeenergieswhichareobtainedthroughsuchmeasurements.2ClustergasmodelofelectricalresistivityClusterdynamicsmodelsthephase-separatingalloyasagasofsoluteclusterswhichexchangesoluteatomsbysingleatomdiffusion.Clustersareassumedtobesphericalandaredescribedbyasingleparameter,thenumbernScofsoluteatomstheycontain.Thetimeevolutionoftheclustersizedistributionisgovernedbyamasterequationwhichinputparametersarethesolutediffusioncoefficientandtheprecipitateinterfacefreeenergy.ForAl-Scalloys,theseneededparametersweredirectlydeducedfromanatomicdiffusionmodel[15,16,18]previouslybuiltforAl-Zr-Scalloys.Itshouldbestressedthat,althoughnoneoftheseinputparameterswereintentionallyfitted,clusterdynamicsmanagedtoreproducereasonablyexperimentaldataonthetimeevolutionoftheprecipitatesizedistribution.ThereaderisreferredtoRef.11forafulldescriptionofclusterdynamicsmodelinganditsapplicationtoAl-Scalloys.Wewillonlydescribeinthefollowinghowthistechniquecanbeadaptedtomodelelectricalresistivityofaphase-separatingAl-Scalloy.22.1ClustercontributiontoelectricalresistivitySoastouseclusterdynamicstosimulatetheevolutionoftheelectricalre-sistivityduringtheannealingofasupersaturatedsolidsolution,weassumethatthetotalresistivityoftheclustergasisgivenbythesumofeachclusterresistivity.Doingso,weneglectanyinterferencebetweenclusters.Therefore,theinputparametersofthemodelingarethecontributionsofeachcluster,i.e.ρnScforaclustercontainingnScsoluteatoms.Somecalculationsofthiscontributionexistintheliterature(forareview,seeRef.1).MostofthemhavebeenaimedtoexplaintheincreaseofresistivityatthebeginningoftheprecipitationkineticsinalloyslikeAl(Zn),Al(Ag),Al(Cu)orAl(Cr),showingthatthisresistivityincreaseisassociatedwiththeapparitionofsmallclusters(∼10−20˚A).Allthesecalculationsconsidertheelasticscatteringofvalenceelectronsbytheperturbingpotentialduetothepresenceofthesoluteatomscomposingthecluster.TheyshowthatsoluteatomswhichareagglomeratedinasmallclustercanhaveahigherresistivitythanthesameisolatedsoluteatomsbecauseofBraggscattering.Indeed,thesmallestthecluster,themostr