Ostwald Ripening on Nanoscale

OstwaldRipeningonNanoscaleV.M.BurlakovDepartmentofmaterials,UniversityofOxford,ParksRoad,Oxford,OX13PH,UKandInstituteforSpectroscopyRussianAcademyofSciences,Troitsk,MoscowRegion,142190,RussiaAbstractApplicabilityofclassicalLifshitz-SlyozovtheoryofOstwaldripeningisanalyzedandfoundlimitedbyrelativelylargeclustersizesduetorestrictionsimposedbytheoreticalassumptions.Anassumptionaboutthesteadystateripeningregimeposesanupperlimit,whileanother,implicitassumptionofcontinuousdescriptionposesaclustersize-dependentlowerlimitonthesupersaturationlevel.Thesetwolimitsmismatchfortheclustersundercertainsizeinthenanometerscalemakingthetheoryinapplicable.Wepresentamoregeneric,moleculartheoryofOstwaldripening,whichreproducesclassicalLifshitz-SlyozovandWagnertheoriesinappropriateextremecases.Thistheoryhasawiderapplicabilitythanclassicaltheories,especiallyatlowersupersaturationlevels,andismoresuitablefornanoscalesystems.PACSNumbers:05.70.-a,64.60.My,81.30.-t1.IntroductionThecompetitivegrowthofnew-phaseclusters,namedOstwaldripening(OR)[1],isaphenomenonoftenobservedatlatestagesofmanyfirstorderphasetransformations,andplayingimportantroleinprecipitationhardeningofalloys[2-4],stabilityofemulsions[5-6],formationandstabilityofsurfacestructures[7-10],andsynthesisofnanoparticles[11].DuringORsmallclustersofatoms/moleculesdissolveandtransfertheirmasstobiggerclusters.Classicalmean-filedtheoryofORbyLifshitz-Slyozov-Wagner(SLW)wasV.M.Burlakov,OstwaldRipening.onNanoscale2developedforadilutesystemofclusterssuggestingthattheirvolumefractionisnegligiblylow[12-17].Themainpredictionsofthistheoryareconfirmedinnumerical[11,18-22]andexperimental[8-10,23-27]investigationsofthesystemssatisfyingthetheoreticalassumptions.Thereare,however,manyexperimentalstudiesreportingthecoarseningrateandtheclustersizedistributionsignificantlydifferentfromthosepredictedbyLifshitz-Slyozov[12,13]orWagner[14]theories.Thisisexplainedbythedeviationoftheexperimentalconditionsfromthoseassumedtheoretically,inparticular,byfinitevolumefractionofthenew-phaseclusters(seeRef26andreferencestherein).FurthertheoreticaldevelopmentsofORwerethereforefocusedondescribingthemorerealisticsystemstakingintoaccountfinitevolumefractionoftheclusters[18,28-29].BoththeclassicalLSWtheoryanditsfurtherimprovementsarebasedonsolvingdiffusionequationinthevicinityofclustersimplicitlyassumingthatmolecularconcentrationishighenoughtobeasmoothfunctionofspatialcoordinatesatleastonthescaleoftheclusterradius.Forsmallclustersizes(~10nm)thisassumptioncanbeviolated,astherequiredmolecularconcentrationappearstobeunrealisticallyhigh.Thispaperpresentsanalternative,molecularmeanfieldapproachtodescribingstructurecoarseningonnanoscale.ToillustratepeculiaritiesofthisnewapproachwefirstbrieflyoutlinetheclassicaldescriptionsofOR.2.ClassicaltheoriesofOstwaldRipeningClassicalmean-fieldtheoryofOstwaldripeningisderivedforasystemofsphericalclustersofcondensedphase,whichareinfinitelyfarfromeachother,andimmersedinavapor/solutionwithlowenoughsupersaturationsuchthatnofurtherclusterformation(nucleation)takesplace.Herewepresentaconciseandsimplifiedderivationofclassicalresults,payingparticularattentiontotheconditionsunderwhichtheyareapplicable.Forconveniencewerefertothecondensedphaseclustersas‘molecularclusters’(orjustV.M.Burlakov,OstwaldRipening.onNanoscale3‘clusters’),andtheinter-clusterspaceisreferredtoasvaporthoughitcanberepresentedbyatomic/molecularspeciesdissolvedinacondensedphasetoo.ConsiderthetotaldiffusionfluxC(R+r)ofmoleculestoward(oroutwardfrom)anisolatedmolecularclusterofradiusR,whichaccordingtotheGauss’theorem,doesnotdependontheradialdistancerfromtheclustersurface()()2()40dddnrCRrRrDdrdrdrπ+=⋅+⋅=(1)whereDismoleculardiffusioncoefficientandn(r)isthevaporconcentration.ThegenericsolutiontoEq.(1)hastheform()()()4CRnRrnDRrπ∞+=−+⋅+(2)withn∞beingthevaporconcentrationinfinitelyfarfromthecluster.TheboundaryconditionsforEqs(1)and(2)aredefinedbymolecularfluxdensityacrossthecluster-vaporinterface,20()()4inoutrdnRrCRDIIdrRπ=+==−(3)andbythevaporconcentrationattheclustersurface()()4CRnRnDRπ∞=−+⋅.(4)TheincomingfluxIinistakenaspositive,therebydefiningthesigninfrontofthederivativeinEq.(3),whereasthenetmolecularfluxdensityinoutII−isdescribedbytheWagneranzatz()()()inoutGTIIknRnR−=⋅−(5)intermsoftheconstantinterfacereactionratekandequilibriumvaporconcentration()GTnRattheclustersurface.Eqs(3)-(5)definetwosimultaneousequationsforC(R)andn(R),forwhichthesolutionC(R)isV.M.Burlakov,OstwaldRipening.onNanoscale4()()2()4GTknnRCRRDDkRπ∞⋅−=⋅⋅+⋅.(6)Ifweassociatethevolumevmwitheachmoleculeinacluster,thenthetimeevolutionoftheclusterradiusisgivenby()()2()4GTmmknnRdRvCRvDdtRDkRπ∞⋅−⋅==⋅⋅+⋅,(7)whichisthemainequationoftheclassicaltheories.Itcontainstwounknowns,()andGTnnR∞.Theparametern∞representsthemeanfield,whichmediatesmolecularexchangebetweendifferentclusters.Anequilibriummolecularconcentration()GTnRattheclustersurfaceisobtainedfromtheGibbs-Thomsonrelation,asdescribedinthenextsubsection.2.1Gibbs-ThomsonrelationTheGibbs-Thomsonrelationformsthebasisforallcl