Size dependent interface energy and its applications

Size dependent interface energy and its applications

SurfaceScienceReports63(2008)

427–464

ContentslistsavailableatScienceDirect

SurfaceScienceReports

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Sizedependentinterfaceenergyanditsapplications

Q.Jianga,b, ,H.M.Lua,b,c

abc

KeyLaboratoryofAutomobileMaterials(JilinUniversity),MinistryofEducation,China

DepartmentofMaterialsScienceandEngineering,JilinUniversity,RenminStreet5988,Changchun130022,China

DepartmentofMaterialsScienceandEngineering,NationalLaboratoryofSolidStateMicrostructures,NanjingUniversity,Nanjing210093,China

articleinfoabstract

Reducingthesizesoflowdimensionalmaterialsleadstodramaticincreaseintheportionofsurface/interfaceatoms.Thepropertiesofasolidareessentiallycontrolledbyrelatedsurface/interfaceenergies.Althoughsuchchangesarebelievedtodominatebehaviorsofnanoscalestructures,littleexperienceorintuitionfortheexpectedphenomena,especiallyforthesize-dependenceoftheenergiesandtheirpracticalimplications,aremodeledanalytically.Inthiscontribution,theclassicthermodynamicsasapowerfultraditionaltheoreticaltoolisusedtomodeldifferentbulkinterfaceenergiesandthecorrespondingsizedependences.Duringthemodeling,anemphasisonsizedependencesoftheinterfaceenergiesisgiven,whichisinducedbysizedependenceofcoherentenergyofatomswithinnanocrystals.Itisfoundthatsolid–vaporinterfaceenergy,liquid–vaporinterfaceenergy,solid–liquidinterfaceenergy,andsolid–solidinterfaceenergyofnanoparticlesandthinfilmsfallastheirdiametersorthicknessdecreasetoseveralnanometerswhilethesolid–vaporinterfaceenergyratiobetweendifferentfacetsissize-independentandequalstothecorrespondingbulkvalue.Thepredictionsoftheestablishedanalyticmodelswithoutanyfreeparameters,suchassizeandtemperature,dependencesofthesefourkindsofinterfaceenergiesandrelatedsurfacestress,correspondtoexperimentalorothertheoreticalresults.Theaboveestablishedmodelsaresuitableforlow-dimensionalmaterialswithdifferentdimensionsanddifferentchemicalbondnatures.Moreover,severalrelatedapplicationsinthefieldsofnanophasetransitions,nanocrystalgrowth,andself-diffusionofliquidsareprovided.

©2008ElsevierB.V.Allrightsreserved.

Articlehistory:

Accepted14July2008editor:W.H.Weinberg

Keywords:

Solid–liquidinterfaceenergySolid–solidinterfaceenergySolid–vaporinterfaceenergyLiquid–vaporinterfaceenergy

SizeandtemperaturedependencesThermodynamics

Contents1.2.

Introduction........................................................................................................................................................................................................................428Solid–liquidinterfaceenergy............................................................................................................................................................................................4302.1.Thebulksolid–liquidinterfaceenergyγsl0..........................................................................................................................................................430

2.1.1.γsl0(Tm)of62metalliccrystals...............................................................................................................................................................4312.1.2.γsl0(Tm)of38organiccrystals................................................................................................................................................................4332.1.3.γsl0(Tm)inmetals:fccversusbcc...........................................................................................................................................................434

2.2.Thesizedependenceofsolid–liquidinterfaceenergyγsl(D)..............................................................................................................................4352.3.Nucleus–liquidinterfaceenergyofelements......................................................................................................................................................436Solid–solidinterfaceenergy..............................................................................................................................................................................................4373.1.Thebulksolid–solidinterfaceenergyγss0............................................................................................................................................................4373.2.Thesizedependenceofsolid–solidinterfaceenergyγss(D)...............................................................................................................................437Solid–vaporinterfaceenergy............................................................................................................................................................................................4384.1.Thebulksolid–vaporinterfaceenergyγsv0..........................................................................................................................................................438

4.1.1.Solid–vaporinterfaceenergyofelementarycrystals...........................................................................................................................4394.1.2.Solid–vaporinterfaceenergyofseveralceramicswithNaClstructure..............................................................................................442

4.2.Thesize-dependentsolid–vaporinterfaceenergyγsv(D)...................................................................................................................................443

3.

4.

Correspondingauthorat:DepartmentofMaterialsScienceandEngineering,JilinUniversity,RenminStreet5988,Changchun130022,China.Tel.:+8643185095371;

fax:+8643185095876.

E-mailaddress:jiangq@(Q.Jiang).

0167-5729/$–seefrontmatter©2008ElsevierB.V.Allrightsreserved.doi:10.1016/j.surfrep.2008.07.001

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428Q.Jiang,H.M.Lu/SurfaceScienceReports63(2008)427–464

5.

6.

7.

Liquid–vaporinterfaceenergy..........................................................................................................................................................................................4445.1.Thebulkliquid–vaporinterfaceenergyγlv0........................................................................................................................................................444

5.1.1.Determinationofγlv0(Tm)values..........................................................................................................................................................4465.1.2.Determinationofγl v0(Tm)values..........................................................................................................................................................4475.1.3.Estimationofγlv0(T)andγl v0(T)functions..........................................................................................................................................448

5.2.Thesizedependenceofliquid–vaporinterfaceenergyγlv(D)...........................................................................................................................449Applications........................................................................................................................................................................................................................4516.1.Thermodynamicphasestabilitiesofnanocarbon................................................................................................................................................4516.2.StatichysteresisofsolidtransitionofCdSenanocrystals...................................................................................................................................4536.3.Thecriticallayernumberofepitaxiallygrownfilms..........................................................................................................................................4546.4.Criticalmisfitofepitaxialgrowthmetallicthinfilms.........................................................................................................................................4556.5.Reconstructionpossibilityoffccmetallicsurfacesatroomtemperature..........................................................................................................4566.6.Temperature-dependentself-diffusioncoefficientinliquidelements..............................................................................................................457Summaryandfurtherprospects.......................................................................................................................................................................................458Acknowledgements............................................................................................................................................................................................................459References...........................................................................................................................................................................................................................459

1.Introduction

Wheninterfacesizeatleastinonedimensionisatnanometersizerange,thecorrespondingmaterialspropertiescouldnotbereadilyinterpretedbasedon‘‘classical’’atomicorsolidsolutiontheories,andtheregionsofspaceinvolvedwerebeyondthescopeofexistentexperimentaltechniques[1].Thesciencetointerpretthesephenomenahastakenafirmtheoreticalandexperimentalholdonthenatureofmatteratitstwoextremes:atthemolecular,atomic,andsubatomiclevels,andintheareaofbulkmaterials,includingtheirphysical,chemicalandelectricalproperties.Betweenthetwoextremesliesthenanometersizerange,orsocalledmesoscale.Evenwiththelatestadvancedtechniquesforstudyingthephasesandtheregionbetweenphases,agreatmanymysteriesremaintobesolved.That‘‘region’’ofthephysicalworldnotonlyrepresentsabridgebetweenchemicalandphysicalphases,butalsoplaysavitalbutoftenunrecognizedroleinotherareasofphysics,chemistry,materialsscience,biology,medicine,engineering,andotherdisciplines[1].

‘‘Interfacial’’phenomenamaybedefinedasthoserelatedtotheinteractionbetweenonephase(solidorliquid)andanother(solid,liquid,gasorvacuum)inanarrowregionwherethetransitionfromonephasetoanotheroccurs.Aswillquicklybecomeapparent,thetwoclassesofphenomenaareintimatelyrelatedandoftencannotbedistinguished[1].

Ourunderstandingofthenatureoftheinterfacialregionandthechangesortransitionsthatoccuringoingfromonechemi-cal(orphysical)phasetoanotherhashistoricallylaggedbehindmanyotherscientificareasbecauseofthedevelopmentandim-plementationofboththeoreticalandpracticalconcepts.Inthelatenineteenthandearlytwentiethcenturies,greatachievementsinthermodynamicshavebeenmadeinthetheoreticalunderstand-ingofinteractionsatinterfaces[1].Moderncomputationalandan-alyticaltechniquesavailableinthelasttwentyyearshaveledtosignificantadvancestowardamorecompleteunderstandingoftheinimitablenatureofinterfacesandtheinterfaceinteractionsduetotherapidincreaseofcomputationability/priceratioofcomput-ers[2].Theso-calledcomputationmaterialsscienceconsiderstheinterfacepropertiesfromthreedifferentsizescales[2]:

(1)Fromatomicscalewiththeabinitiocalculationbasedon

thefirstprinciples,whichconsidersmany-bodyinteractionbehaviorofseveraltentohundredsmoleculestogether;

(2)FromnanosizescalewithmoleculardynamicsandMonteCarlo

methods,whichstudiesmany-bodyinteractionbehaviorofseveralthousandstoseveralmillionmolecules;

(3)Engineeringbehavioroflarge-scaleconstructionproblems,

orbulkmaterialsusingfiniteelementmethodswhereaveragingconstitutivelawsareusedtoincorporateinterfacemicrostructures.However,becauseoftheunusualandsometimescomplexcharacterofinterfacesandassociatedphenomena,thedevelop-mentoffullysatisfyingtheoreticalmodelshasbeenslow.Thereexistsagreatdealofcontroversyinmanyareasrelatedtointer-faces[1].Also,theclassicthermodynamicshasmoreorlessne-glectedtointerprettheinterfacephenomenainmodernscienceduetotheappearanceofcomputersimulationtechnique.Notethatmanypresentcontroversiesintheoriesandexperimentsininterfacephenomenaarenotbad,sincetheyrepresentthefuelforcontinuedfundamentalandpracticalresearches.However,forthepractitionerwhoneedstoapplythefruitsoffundamentalre-search,suchuncertaintycansometimescomplicateattemptstosolvepracticalinterfacialproblems[1].

Nanoscienceandnanotechnologyarerapidlydevelopedre-searchfieldsinmaterialsscienceandengineeringinrecentyears.Assizeofmaterialsdropstonanometersizerange,inter-face/volumeratioincreasesandthusinterfaceeffectonmaterialspropertiesbecomesevident.Whensample(grain,domain)sizesbecomecomparablewiththespecificphysicallengthscalesuchasthemeanfreepath,thedomainsizeinferromagnetsorfer-roelectrics,thecoherencelengthofphonons,orthecorrelationlengthofacollectivegroundstateasinsuperconductivity,thenthecorrespondingphysicalphenomenonwillbestronglyaffected[3],andeventhequantumeffectispresent.

Theinterfacephenomena,whichaffectmaterialsbehaviors,areinnatureproducedbydifferentenergeticstatesofmoleculesontheinterfaceincomparisonwiththosewithinthematerials.Sincethemolecularinteractionsontheinterfacesdifferfromthoseontheinteriorofphases,theexcessspecificGibbsfreeenergyorinterfaceenergyformoleculesofunitinterfaceareaexists,whichisequaltothedifferencebetweenthetotalGibbsfreeenergyofinterfacemoleculesandthatwithinthephasesperunitarea.Theinterfacesandcorrespondinginterfaceenergiessometimeshavedifferentnames.Thefollowingareseveralexamplesofthem.Whentheinterfaceiscomposedofthesamesolids,uallysolid–vaporinterfaceenergyandliquid–vaporinterfaceenergyareconsideredtohavethesamesizeofsolid-vacuuminterfaceenergyandthatofliquid-vacuuminterfaceenergysincetheirdifferencesareoftenatleasttwoordersmallerwhentheexternalpressureisambientonealthoughitisknownthatthisdifferencejustleadstotheabsorptionofvaporoninterfaces.Thesolid–vaporinterfaceenergyisalsonamedassurfaceenergywhiletheliquid–vaporinterfaceenergyismadeassurfacetension[4].

Althoughliquid–vaporinterfaceenergycanmeanbothliquid–vaporinterfaceenergyandliquid–vaporinterfacestresswiththesamesizeandmeaning,thisidentityfailsforsolids[5].Theformerdescribesthereversibleworkperunitareatoformanewsolid

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Q.Jiang,H.M.Lu/SurfaceScienceReports63(2008)427–464429

NomenclatureaLatticeconstantbBurgersvectorASurfacearea

A0Surfaceatomdensity

ALAreaoftwo-dimensionalunitcellofliquidASAreaoftwo-dimensionalunitcellofsolidb Cut-offdistance

bccBodycenteredcubicstructureB14Svib(1 θ)/[3R(7 6θ)]BBuckyB (diamond2Svib1 θ)/(3Rcc=c H

/Hθ)

c Stephan’sv

constantvCNCoordinationnumber

CNTClassicalnucleationtheorydDimensionofcrystaldcDiamond-typestructureDDiameterDDiamond

DnCriticaldiameterofthenucleation

D0CriticaldiameterbetweensolidandliquidDcCriticaldiameterofphasetransition

ee= 4uS/(3λN1/32/3

aVs)EbBulkcohesiveenergyEfYoungmodulusoffilm

EVanderWaalsinterlayerattractionfvf=(ff+fr)/2

fSurfaceorinterfacestress

ffInterfacestressofforwardtransitionfrInterfacestressofreversetransitionfccFacecenteredcubicstructureFFullerenes

gl(r)Liquidradialdistributionfunction

gmGibbsfreeenergydifferencebetweenbulkliquidandcrystalG

Graphite

G

GibbsfreeenergychangeG ShearmodulusGsSurfacefreeenergyGeElasticfreeenergy

GVolumeGibbsfreeenergy

GviNon-coherentinterfacefreeenergyGdMisfitdislocationenergy

hcpHexagonalclosepackedstructurehfAtomicdiameteroffilm

hsAtomicdiameterofsubstrateHmg-atommeltingenthalpyH MolarmeltingenthalpyHmsSolidtransitionenthalpy

HHeatofevaporationatTmorTbHv HeatofevaporationatT=0KkvBoltzmann’sconstantk1k1=ZSk k /ZB

l11=ZS/ZLengthofB

adislocation

mm=(2 k1/2

1 km Layernumber

1)/2m CriticalLayernumber

McAtomicormolecularweightnNumberofatomsinamoleculeNaAvogadroconstantNdDislocationnumberO

Carbononions

pp=1/[mHv/(TmS) 1]PP=(Pf+Pr)/2

Pe

Necessarypressureforthesolidtransitioninthermodynamicequilibrium

PfForwardtransitionpressurePinInternalpressurePrReversetransitionpressurePwStaticpressurehysteresiswidthqq=(dρL/dT)[Tm/ρL(Tm)]rRadiusRIdealgasconstantReEquimolarradiusRsRadiusofsurfacetensionscSimplecubicstructureSSurfaceentropySbBulksolid–vaportransitionentropySmMeltingentropySsSolidtransitionentropySvibVibrationalentropytTimetfFilmthicknessTbBulksolid–vaportransitiontemperatureTcCriticaltemperatureTgGlasstransitiontemperatureTkKauzmanntemperatureTmMeltingtemperatureTnCriticaltemperatureofthenucleationTtTriplepointtemperatureuu=(dρ(L/dT)/ρL(Tm)ur)PotentialenergyfunctionudMisfitdislocationenergyofasingledislocationuveElasticv=freeenergyofunitvolumemHMolarvolumev TmSV VlG-atomvolumeofliquidV sG-atomvolumeofcrystal

V(P)Pressure-dependenttransitionvolumedifference

perg-atom

Vh(P V) V(P)ofhigh-pressurephaseVl(P) VwV(P)oflow-pressurephase0=0=Vw

γh(P=0) Vl(P=0)sv0/γlv0

WAsystem-dependentexponentZBNearestCNofinterioratom

Z Next-nearestCNofinterioratomZBhklBrokenbondnumber

ZSNearestCNofsurfaceatom

Z Next-nearestCNofsurfaceatomαS

α αβα ≈≈[27η/(32π)]1/36α/5χβδStability=p 2q/3

δTolman’sparameter

length,δ=Re Rsδ∞Tolman’slengthwhenD→∞εminMinimumvalueofδφStrain

γGeometricfactor

γexpExperimentalvaluesofinterfaceenergyγsl0Bulksolid–liquidinterfaceenergyγminMinimumvalueofγγmaxγsl0Maximumvalueofsl0

γss0Bulksolid–solidinterfaceenergyγsv0Bulksolid–vaporinterfaceenergyγlv 0

Bulklv0(T)

γ liquid–vaporinterfaceenergylv0(T)=dγlv0(T)/dT

Size dependent interface energy and its applications

430Q.Jiang,H.M.Lu/SurfaceScienceReports63(2008)427–464

ηη Packingdensity

Viscositycoefficientoffluid

Totalbondstrengthratiobetweennext-nearestκneighborandnearestoneκCompressibility

κhκλl κvalueofhigh-pressurephaseλλ value=(oflow-pressureη/π)2/3phase81/2/3)(6µcCriticalνµ～misfit=/[4π(1 νf)]θfPoissonratiooffilmρUndercollingdegreeσLDensityofliquid

τStressυTurnbullinduced Acoefficient

bystrainωBondconstantξωξ==|energy

relatedtoCNγlv0(Tm) γee

lv0(Tm)|/γlv0(Tm)ψ2 1/6h

ζEffectiveΦThedislocationstressfieldradius∞

Self-diffusionangle

coefficientBulksize

surface,whereasthelatterdenotesthereversibleworkperunitareaγduetoelasticdeformation,whichisequaltothederivativeofwithrespecttothestraintangentialtothesurface.

Thevaluesofallaboveinterfaceenergiesmaychangeasmate-rialsdimensionandbrokensymmetryvary.Thedimensionalityisdictatedbythesizeinacertaindirectionoftheobjectcomparedtoanappropriatelengthscaleforthesystemsdiscussed.Ifonlyonedimensionofthesampleisevidentlysmallerthanthislengthparameter,theobjectistwo-dimensional.Similardefinitionscanbegivenforone-dimensionalandzero-dimensional.Notealsothatthedimensioncouldevenbefractalwhenaninterfacehasanirreg-ularshape.

Sinceinthenanosizerangetheabinitiocalculationcannotberealizedduetothecomputingabilityofpresentcomputers,thecomputersimulationsremainmanyuncertainties.Analternativeistheclassicthermodynamics,whichstillholdsimportancetomodeltheabovephenomena.Thisisbecausenotonlytheclassicthermodynamicshasbeenwidelyappliedforbulkmaterials,butalsopeopleareveryfamiliarwiththisuniquetechnique.

Thelawsofthermodynamicswereformulatedasphenomeno-logicallawsaboutobservableandoperationallydefinedquantitieswherethestatisticsmechanicsisthebasisofthetheory.Hence,thethermodynamicapproachisapplicabletoabigsystemcon-sistingofgreatnumberofatoms/molecules[6].Anensembleofmanysmallobjectswillalsobeamacroscopicsystem,whichmayberatherspeculativeorcorrespondtoarealdispersedsystem(aerosol,micro-emulsion,composedmaterialetc).However,ex-tensionofthermodynamicmethodstoverysmallobjects,suchasnanoparticles,stillfacesmanyprincipaldifficulties[7].

Itisnoteworthythat,inthecontextofsomemodernexperi-mentaltechniquessuchastheatomicforcemicroscopy,thether-modynamicmethodisequallypromisingasatheoreticalandempiricaldescriptionoftheexperimentallyinvestigatednanosys-tems.Indeedinmanycases,experimentaldataonnanoparticlepropertiesareratherscantyandcontradictorysinceexperimen-talarrangementscanaffectmeasuringresultswhentheparticleisnotentirelyrigid[8].Thus,thermodynamicconsiderationonnanoscalebecomesanurgenttaskduetotherequirementofthedevelopmentofnanoscienceandnanotechnology.

Inthiscontribution,theinterfaceamountsmentionedabovearethermodynamicallymodeledbasedonessentiallytheauthor’s

ownrecentworksonthesizedependencesofinterfaceenergies.Thismodelsrelatethenanothermodynamicsdirectlytothecorre-spondingbulkamounts.FromSection2toSection5,themodelingsofsolid–liquidinterfaceenergyγsl,solid–solidinterfaceenergyγss,solid–vaporγinterfaceenergyγsv,andliquid–vaporinterfaceenergylvarepresented.SomecorrespondingapplicationsarelistedinSection6.Section7summarizesandprovidesafutureprospect.2.Solid–liquidinterfaceenergy

2.1.Thebulksolid–liquidinterfaceenergyγsl0

Thebulksolid–liquidinterfacefreeenergyγsl0,whichisdefinedasthereversibleworkrequiredtoformortoextendaunitareaofinterfacebetweenacrystalanditscoexistingfluidplastically,isoneofthefundamentalmaterialsproperties[9–16].Itplaysakeyroleinmanypracticallyimportantphysicalprocessesandphenomenalikehomogeneousnucleation,crystalgrowthfromthemelt,surfacemeltingandsurfaceroughening,etc.Thus,aquantitativeknowledgeofγsl0valuesisnecessary.However,directmeasurementsforγsl0arenotatalleasyevenforelementsincontrastγtothecaseofinterfaceenergyofbulkliquid–vaporγlv0[10,11,13].Varioustechniqueshavebeenappliedtoestimatesl0fromexperimentaldatalikemaximumsupercooling[9,10],dihedralangles(DA)[17,18],contactangles(CA)[18],thedepressionofmeltingpointsofsmallcrystals(DMP)[11,18,19],theshapeofthegrain-boundary-grooves(GBG)[16],andmoleculardynamicssimulations(MD)[20–24],etc.Despiteoftheefforts,theobtainedvaluesshowaconsiderablescatteringwithconflicts.Therefore,atheoreticaldeterminationofγsl0isofvitalimportance.Sometheoreticalattemptshavebeenmadetoobtainγsl0[9–11,15,17,25–29].Basedonthenucleationexperimentsandtheclassicalnucleationtheory(CNT),Turnbullproposedanempiricalrelationshipthatγsl0isproportionaltoitsmeltingenthalpyHm[10],

γCNT=τHm/(V21/3

sNa)(2.1)

whereTurnbullcoefficientτisconsideredtobe0.45formetals(especiallyclosed-packedmetals)and0.34fornonmetallicelementsatabout20%ofundercoolingbelowthemeltingpointTm,Vsistheg-atomvolumeofthecrystal,andNaistheAvogadroconstant.TheγCNTvaluesmeasuredbyTurnbullarerecognizednowtobelowerthanrealγslonesformetals[27–29].AccordingtothereviewpapersofEustathopoulosandKelton,τ=0.55±0.08[11]andτ=0.49±0.08[14]formetalswhileτvalueincreasesnoticeablyformoleculeshavingmoreasymmetry[15].Sinceτisafittedamount,thetheoreticalmeaningofthisempiricalruleisweakened.Moreover,Eq.(2.1)overlookssomeimportantpiecesofphysicsandwillconsideredlater.

AlthoughEq.(2.1)underestimatedγslvalues,theformofEq.(2.1)hasbeenrationalizedintermsofinterfacialbondingmodels[17,27–29].TheformulationdevelopedbyEwingis[27],

γ

bsl=hHm/(4Vs)+(Nabk/Vl)

gl(r)lng(r)dr

(2.2)

wherehistheatomicdiameterincrystals,gl(r)istheliquidradial(distributionfunction,b isthecut-offdistancebeyondwhichglr)showsnosignificantdeviationfromunity,kisBoltzmann’sconstant,andVlistheg-atomvolumeoftheliquid.TheapproachofMiedemaanddenBroederresultsin[28],

γsl=0.211Hm/(V2sNa)1/3+0.52×10 7T/V2/3

s(2.3)

whilethatofGránásyandTegzeyields[17,29],

γsl=φ(Hm+TSm)/[2(V21/3s/Na)].

(2.4)

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InEq.(2.4),Smismeltingentropyandφisageometricfactorrangingfrom0.29and0.63forcubicorhcpstructure,dependingontheinterfaceorientation.

Itisknownthatthemostpowerfulmethodavailablefortheoreticallyestimatingγslistheso-calledGibbs–Thomsonequation(knownalsoastheKelvinequation)[11],whichdescribestheequilibriumbetweenasmallsolidnucleusinasphericalshapeandtheinfiniteamountofitsliquid,

γsl=DnHm(Tn)(1 Tn/Tm)/(4Vs)

(2.5)

whereDnandTnarethecriticaldiameterofthesolidnucleusandthenucleationtemperature.Eq.(2.5)isvalidatT<Tmwheretheequilibriumsizeofthesolidphasehasafinitepositivevalue.ThephysicalmeaningofEq.(2.5)isthatthegrowthofthesolidphasewillbethermodynamicallymorefavorablethanitsdissolutionforsolidnucleuswithsizeDlargerthanDnatTn.Thus,theγslvaluecanbedeterminedintermsofEq.(2.5)aslongastheDnvalueatTnisknown.

ThroughcombiningEq.(2.5)andamodelforthesize-dependentmeltingtemperature,γsl0isalsoreadas[15],

γsl0=2hSvibHm/(3RVs)

(2.6)

whereRistheidealgasconstantandSvibisthevibrationalpartoftheoverallmeltingentropySm.

Recently,theCNTwasre-examinedthroughintroducingsize-dependentγslwheretheGibbsfreeenergychange G(D,T)ofthemetallicassemblyconsistingofthenucleusandtherestfluidcanbeexpressedasfollowing

[30],

G(D,T)

1 3h/D

7TSvib

m T

h3H=

m3VsD/(2h)Tm+6TR

T2Tm

×

7TT(D/h)3.

(2.7)

m+6T

Obviously,Dncanbedeterminedthroughletting G(D,T)/ D=0byEq.(2.7).

θ=Letting T =Tm TndenotetheamountofundercoolingandT/TmthenormalizedamountofundercoolingwithSvib/R≈0.95formetalsintermsofTable1ofRef.[30],itisfoundintermsofEq.(2.7)that,Dnθ/h≈2.1.

(2.8)

Foranypure,isotropic,spherical,condensedmaterial,hinEq.(2.8)canbeestimatedfromtheknownVsvalueandthepackingdensityηofthesolidcrystalsas,h=[6ηVs/(πNa)]1/3.

(2.9)

TakingV1/3

sdescribedbyEq.(2.9)andsubstitutingitandEq.(2.8)intoEq.(2.5),thereis,

γsl(T)=αHm(T)/(V2s/Na)1/3

(2.10)

withα≈[27η/(32π)]1/3.

/IntermsofHelmholtzfunction,HmdTwhereg(T)=gm(T) Tdgm(T)m(T)isthetemperature-dependentGibbsfreeenergydifferencebetweentheliquidandthecorrespondingcrystalinbulk,whichhasthefollowingformformetalsandsemiconductors[31,32],

gm(T)=7HmT(Tm T)/[Tm(Tm+6T)],

(2.11a)gm(T)=HmT(Tm T)/T2

m.

(2.11b)

Thus,Hm(T)functionformetalsandsemiconductorscanbereadas,

Hm(T)=49HmT2/(Tm+6T)2,(2.12a)Hm(T)=Hm(T/Tm)2.

(2.12b)

Notethatgm(T)reachesitsmaximumwhenTreducestotheidealglasstransitiontemperatureorKauzmanntemperatureTk≈0.27TmformetalsbutTk≈0.5T)/mforsemiconductors[31],whichisdeterminedbylettingdgm(TdT=0.BelowTk,theliquidmusttransformtoglasswherethespecificheatdifferencebetweenthecrystalandglassapproaches0<T<T(zero.SinceHmT)atT<T(T)isinducedbythespecificheatdifference,Hmkwillremainconstant,namelyHm(k)≈Hm(Tk).

CombiningEqs.(2.10)and(2.12),γsl(T)formetalscanalsobereadas,

γsl(T)=α Hm[7T/(Tm+6T)]2/(V2/3s/Na)1.

(2.13)

2.1.1.γsl0(Tm)of62metalliccrystals

γIntermsofEq.(2.13),Table1givesthemodelpredictionsforsl0valuesof62metalsatTmwheretherelatedparametersarealsolisted[33–35].Becausemanyoftheelements,i.e.thosewithasteriskincolumnofTable1,cantransformfromonecrystallineformtoanotheroneonheatingfrom0KtoT=m[34]whileγsl0iscalculatedatTTm,theηvaluesofthehightemperatureallotropicformsareemployed.Incaseofhexagonalclose-packed(hcp)elements,anidealhcpstructure(alltheatomicdistancesareuniformandη=0.74)hasbeensupposedforsimplification.Asacomparison,othertheoreticalresultsofγ1byEq.(2.4)[16],γ2byEq.(2.3)[28],γ3byEq.(2.2)[36],γ4byEq.(2.6),andavailableexperimentalorMDdata[11,16–19,22–24]arealsoshowninTable1.Itwasthoughtthatthefreeenergyofthesystemdependsnotonlyonγsl0butalsoontheenergyofthegrainboundary(whichvariesγwiththerelativeorientationofthegrains),ascatteringofsl0couldthusbeexpectedduetotherandomorientationoftheneighboringgrains[17].Asresults,twoquantities—theminimumandthemaximumofγsl0,takenasγminandγmax,characterizingfacetdependenceofγsl0,arepresentedinthecolumns3and4ofTable1intermsofEq.(2.4).Notethattoidentifydifferentγsl0results,numericseriesofsubscriptsareusedasillustratedinthecaptionofTable1.

InTables1and17metalshaveexperimentalorMDresults,whicharetogetherdenotedasγexp.AsshowninTable1,theagreementbetweenthepredictedγsl0valuesandγexpvaluesisgenerallygoodexceptforTlwhereγγsl0valueofTlisobviouslysmallerthanthecorrespondingexpone.TheγexpvalueofTlwasdeterminedbyasimilarmethodinestimatingγexp(Pb)=76mJ/m2[17],whichisconsiderablyhigherthantheresultsfromotherexperiments[17,18].Moreover,othertheoreticalresultsintermsofEqs.(2.2)–(2.4)arealsosmallerthanthisγexpvalue.Notethatγsl0valuesofAuandFe(177and241mJ/m2)correspondtothelatestMDresults(190and221mJ/m2)[18,22]butareevidentlysmallerthantheearlyexperimentalresults(270and336mJ/m2)[11,17].

γWhenγsl0valuesarecomparedwithothertheoreticalresultsof1,γ2,γ3,andγ4,severalobservationsareobtained:

(1)AsshowninTable1andFig.1a,γγsl0correspondswelltoγ1whereallsl0valueslieinthesizerangebetweenγminandγmax.Moreover,themostγsl0valuesareveryclosetothoseofγmaxandevidentlyhigherthanthoseofγmin.ItisinterestingthattheγmindataareinagoodagreementwiththosefromthesupercoolingexperimentsofTurnbull[10].

(2)AsshowninTable1andFig.1b,generallyγ2>γsl0exceptforAl,Er,Ge,Sbwhereγsl0ofthesefourelementsareonly2.7%,0.5%,4.8%and1.5%largerthanthecorrespondingγ2values,respectively.However,onlyγ2valuesofthirty-threeelementsarewithin15%ofthepredictedγsl0values.

(3)Oncontrarytotheγ2values,generallyγ3<γsl0exceptforTi,Pt,Fe,Zr,andMnasshowninTable1andFig.1c.Fortheformertwoelements,thedifferencebetweenγ3andγsl0issmallerthan5%;whileforthelatterthreeelements,thedifferences

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Table1

Comparisonsofγsl0(Tm)of62metalsamongthemodelpredictionsγsl0byEq.(2.13),γ1byEq.(2.4)[17],γ2byEq.(2.3)[28],γ3byEq.(2.2)[36],andavailableexperimentalorMDresultsγexp[11,16–19,22–24]

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Table1(continued)

433

parisonsofγγsl0(Tm)for62metals:(a)betweenthemodelpredictions

0byEq.(2.13)andγ1ebyEq.(2.4)wherethesymbols and respectivelydenotethemaximumandminimumvaluesofγ1[17];(b)betweenγ0andγ2byEq.(2.3) [28];(c)betweenγ0andγ3byEq.(2.2)( )[36].

are9.5%,17.1%and21.8%,respectively.Notethatγ3valuesaredirectlycitedfromRef.[36]duetothecomplexnumericalcalculationsoriginatedfromthederivativeofgl(r)functioninEq.(2.2).Moreover,amongthetwenty-onemetalspredictedbyEq.(2.4),eighteenofγ3valuesarewithin15%ofthepredictedγsl0values(theexceptionsareGe,MnandZr).InsertingEq.(2.9)intoEq.(2.6),thereis,

γsl(T)=α Hm(T)/(V2s/Na)1/3

(2.14)

whereα ≈6α/5withtheapproximationofSvib/R≈0.95asmentionedabove.Thus,γα 4isgenerally20%higherthanγsl0.Interestingly,valuesoffacecenteredcubic(fcc),bodycenteredcubic(bcc),simplecubic(sc)anddiamond-type(dc)structuresareabout0.67,0.66,0.60and0.52,whichcorrespondtotherespectivemaximumvaluesofδ(0.59,0.63,0.58and0.50)obtainedbyGránásyγetal.[17,29].Namely,γ4correspondstothemaximumof1asshowninFig.1ortheupperlimitofγsl0,whichhasalsobeenproposedbyDigilov[37].

Accordingtotheaboveanalyses,itcanbeconcludedthatγsl0correspondswelltotheavailableexperimentalandMDresultsaswellasthetheoreticalresultsofγ1,andgenerallyliesinthesizerangebetweenγ3andγ2orγ4.

Moreover,asshowninTable1,twentyelementstakefccorhcpstructureswithη=0.74;thirty-sixelementstakebccstructureswithη=0.68;oneelement,i.e.Hg,takesaslightlydistortedscstructurewithη≈0.52;andoneelementGetakesdcstructurewithη=0.34.SubstitutingthesefourηvaluesintoEq.(2.9)leadstothattheαvaluesoffcc,bcc,scanddcstructuresareabout0.561,0.545,0.499and0.433,whichareinexcellentagreementswiththerespectiveaveragesofφ(0.561,0.546,0.500and0.433)overallorientationsproposedbyGránásyetal.[17,29].Because0.34<α<0.74,themeanvalueofαforallmetalsisthus0.50±0.07,whichcorrespondswellto0.49±0.08ofKelton[14].Inaddition,theobtainedvalueofα≈0.56forfccmetalsisalso

inagreementwith0.55obtainedbyEustathopoulos[12]whileisslightlysmallerthan0.60derivedbyGránásyetal.[38]andlargerthan0.51estimatedbyLaird[16]fromthehard-spheremodel[39].2.1.2.γsl0(Tm)of38organiccrystals

Organiccrystalsasmolecularcrystalsdifferfrommetallicones,whosechemicalbondsarecovalentwithinmoleculeswhilevanderWaalsforcesorhydrogenbondsareamongmolecules.Theformerisresponsibleforstabilityofindividualmolecules,whilethelatterisprimarilyresponsibleforbulkpropertiesofmatter,suchasγsl0althoughtheformerismuchstrongerthanthelatter.SincebondstrengthsofvanderWaalsforcesorhydrogenbondsareweakerthanthoseofmetallicbonds,γsl0valuesoforganiccrystalsshouldbesmallerthanthoseofmetalliccrystalsduetothebonddeficitnatureofγsl0,suchasγsl0≈40±7mJ/m2ofPb[11],whichisoneofthesmallestγsl0valuesamongcrystalsexceptforalkalimetalswhereabnormallysmallγsl0values(≤30mJ/m2)resultfromsmallerHmandunnaturallylargerVsvaluesasshowninTable1.Moreover,γsl0valuesofcompoundscomposedoffullhydrogenbonds,suchasH2Owithγsl0≈44±10mJ/m2obtainedfromGBGexperiments[11],shouldalsobelargerthanthoseoforganiccrystalssincemoleculesoforganiccrystalsatmostonlypartlyconsistofhydrogenbondswhiletheothersarevanderWaalsforces.Thus,γsl0<40mJ/m2forusualorganiccrystals.Althoughsomemeasuredγsl0valuesoforganiccrystalscomposedofchainmoleculesaremuchlargerthantheabovelimitsasshowninTable2[40,41],theirrealvaluesshouldbesimilartothosecomposedofsphericalmoleculessinceγsl0denotesexcessenergyofunitareawheremolecularweighthasnegligibleeffectonit.Evenifchainmoleculesmaycontainoneormorehydrogenbonds,γsl0valuesstillvarylittlesincethemostbondsofthemoleculesarestillvanderWaalsforces.Thisanalysisalsoimpliesthatanisotropyofγsl0oforganiccrystalsissmall,althoughthisissueisstilldebateduptonow[11,13,15].

AlthoughEq.(2.13)isdeducedintermsoftheGibbs–Thomsonequationandrelateddiscussionforhomogeneousnucleationofmetallicelements,thisequationshouldbealsoapplicabletoothertypesofmaterials,aniccrystals.Althoughθ≈0.21andSvibθ//Rh≈≈02..951isarestillnotreasonable.suitableforBecauseorganicθvaluescrystals,oforganictheequationcrystalsofDnareunavailableforuswhilewaterconsistsoffullhydrogenbonds,wateristhustakenforexample.Sinceθ≈0.143[10],S..vib/R≈089andh≈0096[15],Eq.(2.8)canalsobeobtainedandisstillvalid.Notethatη=1inthiscasesincethecorrelationbetweenmoleculardiameterhandVgisunrelatedwithηforsphericalorganiccrystals.

Table2showsacomparisonofγsl0(Tm)for38differentorganicmolecularcrystalsbetweenthemodelpredictionsγsl0andthecorrespondingexperimentaldataγexp[11,13,40–45]wheretherelatedparametersarealsolisted[65–68].Itisfoundthatγsl0correspondtothetheoreticalresultsγ4[64]andthelatestexperimentaldata[13,44,45]whilearesmallerthantheearlyexperimentalones[11,40,41].Moreover,othertheoreticalresultsofγ1calculatedbyRaietal.[46–64],γ2toγ4arealsolisted

forcomparisonpurpose.NotethatinthiscasetheunitsofH

mandV aretakenasJ/molandcm3/molratherthanJ/g-atomand

cm3/g-atomlistedinTable2[69],andH

m=nHmandV =nVg

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Table2

Comparisonsofγsl0(Tm)of38organiccrystalsamongthemodelpredictionsγsl0byEq.(2.13),experimentaldataγexp[11,13,40–45],andothertheoreticalresultsγ1[46–63],

andγaredeterminedintermsofEq.(2.1)withτ=0.30and0.35,aswellasγintermsofEq.(2.6)[64]

withnbeingthenumberofatomsinamolecule.Thus,althoughαislargerthanτ,γsl0maybelargerorsmallerthanγ2andγ3becauseoftheexistenceofn.Generally,γsl0isingoodagreementwithγ2andγ4whileitissmallerthanγ1.Althoughγ1wasclaimedtobecalculatedintermsofEq.(2.1)withτlyingbetween0.30and0.35(τvaluewasnotdefinitelymentioned)[46–48,51–57,59–63],γ1valuesoffifteenorganiccrystalsarelargerthanγ3,whichisdeterminedbyEq.(2.1)withτ=0.35.Sinceγsl0correspondstoγ2withτ=0.30,thecasethatγsl0issmallerthanγ1isobviousandmayhardlyillustratetheincorrectnessofEq.(2.13).

Moreover,foratypicalfcccrystal,thebondnumberdecreaseofmoleculesonasolid–liquidinterfaceisusually1–2,whilethatonaliquid–vaporinterfaceis3–4.Ifthebondnumberisproportionaltothecorrespondinginterfaceenergyvalue,withthenotethatthebondstrengthdifferenceofmoleculesbetweensolidandliquidstatesisonlyseveralpercentofthebondstrength,γsl0<γlv0whileγlv0iseasytomeasurewithbetteraccuracy.Althoughtheaboveanalysisismadeinlightoftherelationbetweeninterfaceenergyandbondnumberformetals,thisrelationshouldbealsoapplicabletoothertypesofmaterials.Thus,theγlv0valueofthesamesubstanceisagoodreferenceasanupperlimitonγsl0.Thepredictionsγsl0intermsofEq.(2.13)areingoodagreementswiththisprinciple.

Sincetheearlymeasuredγsl0values,mainlythedataforLauric,MyristicandStearicacids[40,41],areevidentlylargerthanthecorrespondingγlv0values,thedisagreementsbetweenγsl0andearlyγexponlyforthesethreekindsofacidsmayscarcelyillustratetheinvalidityofEq.(2.13).Intermsoftheabovediscussion,the

predictionsofγsl0areatleastqualitativelycorrectinorderfororganiccrystals.

2.1.3.γsl0(Tm)inmetals:fccversusbcc

Thedependenceofγsl0oncrystalstructureiscrucialtounder-standtheroleofmetastablestructuresinnucleationpathways.In1897,Ostwaldformulatedhis‘‘steprule’’[70],whichstatesthatnucleationfromthemeltoccurstothephasewiththelow-estactivationbarrier,butnotnecessarilytothethermodynami-callymoststablebulkphase.Inthecaseofthenucleationoffcccrystals,thereisevidencethatcrystallizationoftenproceedsfirstthroughtheformationofbccnuclei,whichtransformtofcccrystalslaterinthegrowthprocess.Thisphenomenonhasbeenobservedinexperimentsonmetalalloys[71],incomputersimulationsofLennard-Jonesparticlesandweaklychargedcollides[72,73],andinclassicaldensity-functionaltheorystudiesofnucleationinLennard-Jones[74].Theseresultscouldbeinterpretedasthatifγsl0forbcccrystalsweresignificantlylowerthanthoseforfcccrystalsinthesesystemsasthatwouldleadtosubstantiallyloweractivationbar-riers[70].Usingasimplemodelofinterfacialstructure,SpaepenandMeyerpredictedthatγsl0forbcc-meltinterfaces(γbcc)shouldbeabout20%lowerthanthatforfcc-meltinterfaces(γfcc),basedonpackingconsiderations[75].Inarecentpaper,Sunetal.deter-minedγbccandγfccvaluesofironbyMDsimulations,obtainingthatγbcc/γfcc=65%–70%[23,76].

However,itisalsoeasytounderstandthedifferenceofγsl0betweenbcc-andfcc-interfacesintermsofEq.(2.13).Sincethe

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differenceofVbetweenbccandfccironisonly1.3%[23],theeffect(ofthechangeofVcanbeneglected.Thus,γbccHbcc

m/Hm

)(ηbcc/ηfcc)1/3/γfcc≈

fccintermsofEq.(2.13).WiththeknownvaluesofHbcc

/Hfccm≈0.72[23]andηbcc/ηfcc≈0.92,1 γbcc/γfcc≈m

0.30,whichisconsistentwiththereportedvalues30%–35%[23,76].Moreover,fromtheviewpointoftheinternalenergychange,thechangefromthemelttobccstructureshouldbesmallthanthattofccstructure,whichleadstoalsosmallactivationbarrier.

2.2.Thesizedependenceofsolid–liquidinterfaceenergyγsl(D)AsmentionedinSection2.1,γsl0valuesobtainedbyEq.(2.1)arerecognizednowtobelowerthanrealonesformetals[26–28],whichmayresultfromtwoapproximationsoftheCNT:(i)Thespecificheatdifferencebetweensolidandliquid ly,theinfluenceof Cpisneglected,whichresultsinthetemperatureindependenceofHmD(T);(ii)Thenucleus–liquidinterfaceenergyγsl(n,Tn)istreatedasthevalueforatemperature-independentplanarinterfaceγsl0,i.e.thecapillarityapproximation[10].SincethevaluesofγslinEq.(2.1)areinitiallyobtainedfornuclei-liquidinterfacewhileanynucleusduringsolidificationisinnanometersizerange,γsl(D),notγsl0,hastobeconsideredfortheoreticalpredictionofactualnucleationandgrowthprocesses.

Todeterminetheγsl(D)function,weconsideracompressiblesphericalparticle,oracubewithcubesidetakenasD,immersedinthecorrespondingbulkliquid.AccordingtoLaplace–Youngequation[73],theseis,

P=2fA/(3Vs)=4f/D

(2.15)

whereAisthesurfaceareaoftheparticle, Pisthedifferenceinpressureinsideandoutsideoftheparticleandfistheinterfacestress.ε=Usingthedefinitionofcompressibilityκ= V /(VP),D/D= A/(2A)= V/(3V)undersmallstrainandA/V=6/Dwhere denotesthedifference,

ε= 4κf/(3D).

(2.16)

Intermsofascalardefinitionoff[77,78],f= Gs/ A= (γslA)/ A

=γsl+A γsl/ A≈γsl+A γsl/ A

(2.17)

whereGγs=γslAstatesthetotalexcessGibbssurfacefreeenergy,or sl=( A/A)(f γsl).

TofindmathematicalsolutionsoffandγslorγslD)∞areneeded.Anunderstandable(D),twoboundaryconditionsofγsl(asymptoticlimitisthatasD→,γsl(D)→γsl0.WhenD→∞,let

γsl=γsl(D) γsl0.

(2.18)

SubstitutingEq.(2.18)intoEq.(2.17)andtakinginmindthatV/A=D/6and A/A=2ε= 8κf/(3D)intermsofEq.(2.16),itreads[79],

γsl(D)/γsl0=[1 8κf2/(3γsl0D)]/[1 8κf/(3D)].

(2.19)

Eq.(2.19)isconsistentwithgeneralcalculationsofthermody-namicsandquantumchemistryforparticles([20,80–82].

AnotherboundaryconditionofγslD)isγslvalueatthesmallestsizeD0wherealmostallatomsofalow-dimensionalcrystalimmersedinfluidarelocatedonitssurfaceandthecrystalisassumedtobeindistinguishablefromthesurroundingfluid.Thus,thesolid–liquidinterfaceisatalldiffusewherethecrystalissimilartoaclusterproducedbyanenergeticfluctuationofthefluid.Asresults,γsl(D0)→0wherehA/V=1 Vi/V=1 [(D0 2h)/D0]3 d=1[79]withVibeingtheinteriorvolumeofthe

crystalanddbeingthedimension.D0dependsontheexistenceofcurvature[83].Whenacrystalhasplanesurface,suchasfilms,D0=2h.Forcrystalswithcurvedsurfaces,suchasparticlesorwires,D0>2h.Asafirstorderapproximation,D0≈3hwherehA/V=26/27forasphericalparticle.Insummary,D0≈3h,(2.20a)D0=2h

(2.20b)

withEq.(2.20a)forparticlesorwiresandEq.(2.20b)forfilms.Eq.(2.19)canthusberewrittenas,

γsl(D)/γsl0=[1 D0/D]/[1 γsl0D0/(fD)]

(2.21)

with8κf2/(3γsl0)=D0orf=±[(3γsl0D0)/(8κ)]1/2.Thedifferentsignsoffcorrespondtothetensile(+)andcompressive( )stressonthesurface.Thepossiblephysicalbackgroundofthepositiveornegativefcouldbeillustratedbasedonthefollowingmechanism:Atomsattheinterfacesufferacoordinationnumber(CN)reduction,bondcontractsspontaneously,whichleadstotheenhancementoftheatomicbindingenergyandhencethetensilestress,orf>0[84].Thisisthecaseoffreenanoparticles.Whentheinterfaceatomsofdifferentelementsareintermixing,CNorthebondstrengthmayincrease,suchasalloyingorcompoundformation.Thealloyingandchemicalreactionmayalternatetheatomicvalences,whichmayintroducerepulsivestressamongtheionsorelectronicchanges.

Theabovededucedfisinfactthesolid–liquidinterfacestressfsl,whichdiffersfromthesolid-vacuumsurfacestressf(sv.Itisassumedthatf≈fslTm)≈fsv(Tm)becausetheinterfacestressinducedbytheelasticstrainofasolidremainsalmostconstantwhenthevacuumissubstitutedbytheliquidonthesolidsurfacewheretheliquidaffectslittletheelasticstrainofthesolid.Forthemostelements,fisoneorderlargerthanγsl0[85–88].AlthoughsomeMDworksbasedonahard-spheremodelshowthatfhasthesamemagnitudeofγsl0[89],thehard-spheremodelitselfmayleadtothisresultwherestrainisabsent,whichco-existswithf[84].Thus,intermsoftheconsiderationthatf>10γsl0,Eq.(2.21)maybesimplifiedasafirstorderapproximation,

γsl(D)/γsl0≈1 D0/D.

(2.22)

Eq.(2.22)denotesthatthedecreaseofγsl(D)isproportionaltosurface/volumeratiowitha1/Drelationship.Fig.2givesanexampleforcomparisonbetweenEq.(2.22)andthecomputersimulationresultofγsl(D)/γsl0[20]whereγsl(D)valuereducesasDdecreases.Themodelpredictionisconsistentwiththecomputersimulationresult.InFig.2,sincethecitedfcccrystalhasnotgivenanidiographicsubstance[20],thesimplifiedformofEq.(2.21),namelyEq.(2.22)isemployed.

γsl(D)valuesoffiveorganicnanocrystalshavealsobeencalculatedintermsofEq.(2.21)andshowninTable3.Thepredictionsareconsistentwiththeexperimentalobservationwiththenotethatthereexistsasizedistributionofnanocrystals[12].AsDdecreases,γsl(D)drops.AtD=4nm,thedecreaseinγsl(D)fordifferentsubstancesisdistinct,whichismainlyinducedbydifferentD0orD0/Dvalues.Therelativedropofγsl(D)valuesincomparisonwiththecorrespondingbulkvaluesreaches20%–40%whereD0maybenear4nm.Thus,theenergeticresistanceforthenucleationprocessinliquidmaybesmallerthanwhattheCNThasestimated[79].Inaddition,thesuccessofpredictionforγsl(D)valuesinreturnconfirmsagainthatHmdeterminesthesizesofγsl0valuesasshowninEq.(2.13).

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427–464

Fig.2.γsl(D)/γsl0asafunctionofD0/DaccordingtoEq.(2.22).Thesymboldenotesthecomputersimulationresultofγsl(8h)andγsl(8h)/γsl0=0.58[20].D0=3hintermsofEq.(2.20a).

Table3

Thecomparisonofγsl(D)/γsl0valuesbetweenthemodelpredictionsintermsofEq.(2.21)withD0=3handthecorrespondingexperimentalresults[13]withκin

110 8

2.3.Nucleus–liquidinterfaceenergyofelements

CombiningEqs.(2.6),(2.7),(2.12)and(2.22),theintegratedsize-andtemperature-dependentinterfaceenergycanbedeter-minedas[90],

γsl(D,T)/γ

3h 7T 2

sl0(Tm)=1 DT6T(2.23a)

γ

3h

m+T 2

sl(D,T)/γsl0(Tm)=1

D

T.

(2.23b)

m

SubstitutingEq.(2.23)intoEq.(2.7),Dncanbedeterminedbyletting G(D,T)/ D=0[90],Dnθ/h=2(B+

2 3Bθ/2)(2.24a)Dnθ/h=2(B +

BB 2 3B /2)

(2.24b)

withθ=Tm T,B=14S

vib1 θ2S

1 θ3R76,andB =vibTm3R.SubstitutingEq.(2.24)intoEq.(2.23)withexperimentallydeterminedθvalues,γsl(rn,Tn)atthenucleus–liquidinterfacecanbedeterminedquantitatively.

Fig.3showsacomparisonbetweenthepredictedγsl(Dn,Tn)valuesintermsofEq.(2.23)andexperimentallydeterminedγCNTvalues[9]withgoodagreement,whichimpliesthattheγCNTvaluebyEq.(2.1)isnotγsl0,butγCNT.

Combiningourpreviouswork[91],themodelpredictionsforbothTnandγsl(Dn,Tn)correspondwelltotheundercoolingexperimental(resultsalthoughboththeexpressionsofγslT)functionsdiffer

fromthoseintheCNT.Thepossible(D,T)andgmreason)maybethemutualcompensationbetweenγslfunctions.However,thereisabout40%difference(D,Tin)andgm(Tthe

valueofDnbetweentheCNT(Dn≈8h)andpresentmodel

Fig.3.Thecomparisonbetweenγsl(Dn,Tn)intermsofEq.(2.23)foravarietyofelementalsystemsandγCNTcalculatedfromtheCNT[10].

Fig.4.γsl(Dn,Tn)asafunctionofHm/VsforavarietyofelementalsystemsintermsofEq.(2.25a)(thesolidline)wherethesymbols and denotetheγsl(Dn,Tn)valuesofmetalsandsemiconductors,respectively.

(Dn≈11h),whichmayresultfromtheneglectofthederivativeofγsl(D,T)toD.Althoughwecannotconfirmtheabovedifference

fromexperimentsduetotheexperimentaldifficulties,webelievethatalittlelargerDnismorereasonable.Moreover,accordingtoEq.(2.24),thesize,ofDT)function,nisdecidedbySanditincreasesvibandθduetotheintroductionofγsl(DwithanincreaseinSviboradecreaseinθ.SinceSvibandθvaluesforelementswithdifferentbondnaturesaresimilar,Dnisinfactindependentontheelementaltypes.Thus,Dn≈11hforallconcernedelementshereisreasonable.

γItisobviousfromEq.(2.1)thatthesizeofhalsoaffectsCNTvalue.Toaccuratelyestimatetheinfluenceofh,hofallelementsshouldbeunifiedtoh wheretheelementswithdifferentstructureshavethesamecoordinationnumber(CN)of12.According()totheGoldschmidtpremiseforlatticecontraction[92],1 h/h willbe3%,4%,and12%ifCNoftheatomreducesfrom12to8,6,and4,respectively.Theadditionalcorrelationbetweenhandh fortheelements,whoseCNisnot4,6or8,canbefoundinRef.[35].Replacinghwithh ,Eq.(2.1)canbesimplifiedas,

γCNT=τ2Hm/Vs(2.25a)withτ2≈0.11±10%nmexceptsemi-metalsPb,SnandGaas

showninFig.4.

Inasimilarway,Eq.(2.6)canalsobesimplifiedas,

γsl0(Tm)=τ3Hm/Vs(2.25b)withυ3=2Svibh /(3R)≈0.18±15%nmexceptFe,AlandGa.

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437

Fig.5.γsl(D T

n,n)asafunctionofHm/Vswithθ=0.1foravarietyofelementalsystemsintermsofEq.(2.26)(thesolidline)wherethesymbols and denotetheγsl(D,T)valuesoffccandnon-fccelements,respectively.

Eq.(2.25)canrepresentγCNTorγsl0(Tm)withauniqueτ2orτ3ratherthantwodifferentτvaluesbyEq.(2.1)duetothedifferentbond naturesoftheelements.Moreover,thedisappearanceofhorhinEq.(2.25)impliesthattheuniqueparameterdecidingγslisHm.SubstitutingEq.(2.24)intoEq.(2.23)andplottingthefunction,thecurvecanbelinearlyregressedasafunctionofθ,whichleadsto,

γ

10sl(D n,Tn)≈(1.78 3.83θ)×10×Hm/Vs

(2.26)

whereT

nisanynucleationtemperatureandDnisthecorrespond-ingdiameterofanucleus.Thestandarddeviationsfor1.78and3.83areonly0.01and0.16,respectively.Eq.(2.26)indicatesthat:Eqs.(2.25a)and(2.25b)denotetwoextremecaseswhereθ=θn(themaximumdegreeofundercooling,whichisnearlyaconstantof0.18±0.02forthemostelements[93])andθ=0,respec-tively.Infact,atanyθvalue,γslV(D

n,Tn)isalwaysproportionaltoHm/s.Asanexample,Fig.5showssucharelationshipatθ=0.1wheretheslopelinearlyregressedisequalto1.40asindicatedbyEq.(2.26).

Fig.5impliesthatHmisrelatedtobondenergyofcrystallineatomswhileγsldenotesthebondenergydifferencebetweensurfaceatomsandinterioronesofacrystal.Sinceγsl∝γsv∝Eb[90],γsl∝Eb.Thesize-dependentEbhasbeendeterminedas[94],

E(D)/E1

2Sb1

b=1

2D/h 1

exp

3R2D/h 1

(2.27)

whereSb=Eb/Tbisthebulksolid–vaportransitionentropyofcrystalswithTbbeingthebulksolid–vaportransitiontemperature.Fig.6showsacomparisonbetweenγ(sl(D,T)/γsl0(T)functionintermsofEq.(2.22)andED)/EbfunctionbyEq.(2.27)(Sb≈12RisassumedexceptforSbandBi[33])withgoodagreement.

Anotherlinearrelationshipbetweenγsl(Dn,Tn)orγCNTandTmwithlargescatteramongdifferentgroupsofelements[10]hasbeenrejectedasthebasisforanempiricalruleinfavorofthecorrelationwithHm[39].However,directcalculationsforγCNTvaluesoftransitionfccmetalswithhard-spheresystemshaveshownastrongcorrelationwithTm[39,95],whichcanalsobeextendedtoγsl0(Tm)forallfccmetalssinceHm=TmSmandυ3Sm=1υ.59±7%nmJ/molKintermsofEq.(2.25b).Forotherelements,3Smvaluesshowlargescatterinarangeof1.04to5.44nmJ/molKduetothescatterofSmvalues.Thus,itisnotTm,butHm,thatgenerallycharacterizesγsl0(Tm)valuebetter.

Notethatalloftheaboveconsiderationsforthemonotonesize-dependentγsl(D,T)functionisbasedonanessential

assumption

parisonbetweenγsl(D)/γsl0shownassolidlineintermsofEq.(2.22)andE(D)/EbshownasdotintermsofEq.(2.27).

onE(D)/Ebfunction,whichsupposesthatasDdecreases,the

increase(ordecreaseinabsolutevalues)ofthecoherentenergyofmaterialsisinducedbytwoeffects:Oneistheincreaseofthesurface/volumeratio,andanotheristheincreaseoftheinterioratomsofthematerials,theincreaseofthelaterisstrongerthantheformer[94].Ifthelaterisneglected,thisisdoneespeciallybycomputersimulationtechnique,wecouldobserveaslightincreaseofγsl(D,T)asDdecreases[96].3.Solid–solidinterfaceenergy

3.1.Thebulksolid–solidinterfaceenergyγss0

Forcoherentorsemi-coherentsolid–solidinterface,itsin-terfaceenergycanbedeterminedbysomeclassicdislocationmodels[97].Whileforsemi-coherentinterface,atomicdiame-termisfitontheinterfacemustbesmallerthan0.15–0.25,γss0isverystronglydependentonthemis-orientationofthetwocrys-talhalvesandremainschallenge.However,forhighanglegrainboundary,γss0isalmostaconstant,whichwillbeconsideredhere.Sinceliquidmayberegardedasasolidwithsuchhighconcentrationofdislocationcoresthattheyareincontacteverywhere[105],solid–solidinterfaceenergyγss0atTmisconsidered)tobetwiceofγsl0approximately[106],γγbiningwithEq.(2.6),

(Tm)≈2sl0(Tmγss0(Tm)≈4hSvibHm/(3VsR).

(3.1)

AsshowninTable4,theγss0(Tm)valuesbasedonEq.(3.1)forelevenelementalcrystals(Cu,Ag,Au,Al,Ni,Co,Nb,Ta,Sn,PbandBi)andtwoorganiccrystals(PivalicacidandSuccinonitrile)

correspondtoavailabletheoreticalvaluesγ

Inaddition,thelisteddataofγ

ss0[44,45,98–104].

thess0andγequilibratedsl0forPivalicacidandSuccinonitriledeterminedbygrainboundarygrooveshapesalsoconfirmthevalidityofγss0(Tm)≈2γsl0(Tm)withtheabsolutedeviationbeingsmallerthan5.5%.Thus,Eq.(3.1)canbeusedtoquantitativelycalculatetheγss0(Tm)values,atleastformetalsandorganiccrystals.

3.2.Thesizedependenceofsolid–solidinterfaceenergyγss(D)Inthedeductionofγγss(D),thewayissimilartothedeductionofsl(D)whileEqs.(2.15)and(2.16)mustbemodifiedduetodifferentinterfaceconditions[79].Weassumethatfasafirstorderapproximationkeepsconstantforbothsolid–liquidandsolid–solidinterfacesatTm,whichleadstothesamestrainonbothsides

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Table4

Comparisonofγss0(Tm)betweenγss0byEq.(3.1)andavailabletheoreticalorexperimentalresultsγ ss0formetals[98–104]andorganiccrystals[44,45]withγinmJ

m2Fig.7.γss(D)/γss0asafunctionofD0/DaccordingtoEq.(3.4).ThesolidlineandthesegmentlineareobtainedbyuseofnegativeandpositivefdeterminedbyEq.(2.17),respectively.ThesymboldenotesthecomputersimulationresultsofCu[106].D0=3hintermsofEq.(2.20a).

oftheinterfacewhenthegrainsareisotropic.However,elasticmodulusofgrainboundariesshouldbelargerthanthatofthesolid–liquidinterfacewithlessstrainunderthesamestress.ThisisintroducedbythefactthatA=3V/Dforsolid–solidinterfacebecausetwosolid–liquidinterfacesofparticlescombinetoformonegrainboundarywith,P=2fA/(3V)=2f/D,(3.2)

and

ε= 2κf/(3D).

(3.3)

Bothequationsindicatethatthestrainonthegrainboundaryisonlyahalfofthatonthesolid–liquidinterfaceandthus A/A=2ε= 4κf/(3D).Now γss=γss(D) γss0=( A/A)(f γ ss)=4κf/(3D)(f γss),or

γss(D)/γss0=[1 D0/(4D)]/[1 γss0D0/(4fD)].

(3.4)

Similartothesimplificationofγsl(D),Eq.(3.4)canalsobesimplifiedas

γss(D)/γss0=1 D0/(4D).

(3.5)

Fig.7showsanagreementbetweenthepredictionsofEq.(3.4)andthecomputersimulationresultsforCu[107].Itisinterestingthattheuseofanegativeinterfacestressforγss(D)inEq.(3.4)leadstoafullagreementwiththecomputersimulationresults,whichimpliesthatf<0atthegrainboundary.

IfwecompareEqs.(2.22)and(3.5),γsl(D)/γsl0<γss(D)/γss0.Thisresultimpliesthatthestiffersurroundingofgrainsbrings

outlessdropoftheinterfaceenergyasDdecreases.Forthegrainboundaries,evenifwhenD→D0,γss(D)/γss0≈75%whileγsl(D)/γsl0=0.SincewhenD≈2D0,thegrainsarenomorestableandwilltransformtoamorphoussolidsintermsofthecomputersimulationresults[107],thesmallestvalueofγss(D0)/γss0couldbeabout85%.

4.Solid–vaporinterfaceenergy

4.1.Thebulksolid–vaporinterfaceenergyγsv0

Thesolid–vaporinterfaceenergyγsv0,usuallydefinedasthedifferenceoffreeenergybetweensurfaceatomsandinteriorones,isoneofthebasicquantitiestounderstandsurfacestructuresandphenomena[108].Despiteofitsimportance,γsv0valueisdifficulttodetermineexperimentally.Themostexperimentsareperformedatγhightemperatureswhereliquid–vaporinterfaceenergyofliquidlv0ismeasured,whichareextrapolatedtozeroKelvin.Thiskindofexperimentscontainsuncertaintiesofunknownmagnitudeandcorrespondstoonlyγsv0valueofanisotropiccrystal[109–111].Moreover,manypublisheddatadeterminedbythecontactangleofmetaldropletsorbypeeltestsdisagreewitheachother,whichcanbeinducedbythepresenceofimpuritiesorbymechanicalcontributions,suchasdislocationsliporthetransferofmaterialacrosstheboundary.Inaddition,therearehardlytheexperimentaldataonthemoreopensurfacesexceptfortheclassicmeasurementsonAu,PbandIntoourknowledge[112,113].Therefore,atheoreticaldeterminationofγsv0valuesespeciallyforopensurfaceisofvitalimportance.

Duringthepastfewyearstherehavebeenseveralattemptstocalculateγsv0valuesofmetalsusingeitherabinitiotech-niques[114–116]withtight-binding(TB)parameterizations[117]orsemi-empiricalmethods[118].γsv0values,workfunctionsandrelaxationforthewholeseriesofbccandfcc4dtransitionmet-alshavefirstlybeenstudied[114]usingthefull-potential(FP)linearmuffin-tinorbital(LMTO)methodinconjunctionwiththelocal-spindensityapproximationtotheexchange–correlationpo-tential[119–121].Inthesamespirit,γsv0valuesandtheworkfunctionsofthemostelementalmetalsincludingthelightac-tinideshavebeencarriedoutbytheGreen’sfunctionwithLMTOmethod[115,116,122,123].Recently,thefull-chargedensity(FCD)Green’sfunctionLMTOtechniqueintheatomic-sphereapproxi-mation(ASA)withthegeneralizedgradientapproximation(GGA)hasbeenutilizedtoconstructalargedatabasethatcontainsγsv0valuesoflow-indexsurfacesof60elementsintheperiodicta-ble[121,124–127].Theresultsdenoteameandeviationof10%forthe4dtransitionmetalsfromFPmethods[128].Thisdatabaseinconjunctionwiththepair-potentialmodel[129]hasbeenfurtherextendedtoestimatetheformationenergyofmono-atomicstepsonlow-indexsurfacesforanensembleofthefccandbccmet-als[130].

Ontheotherhand,thetraditionalbroken-bondmodelisagainsuggestedtoestimateγsv0valuesofthetransitionmetalsandthenoblemetalswithdifferentfacets[112,131,132].Thesimplestapproachtogetaroughestimationofγsv0valuesatT=0KistodeterminethebrokenbondnumberZhklforcreatingasurfaceareabycuttingacrystalalongcertaincrystallographicplanewithaMillerindex(hkl).Zhkl=ZB ZSwhereZSandZBarecoordinationnumbers(CN)ofsurfaceatomsandinteriorones,respectively.MultiplyingthisnumberwiththecohesionenergyperbondEb/ZBforthenon-spin-polarizedatomat0K,γsvisdeterminedby[133],

γsv0=(1 k1)Eb/(NaAS)

(4.1)

wherek1=ZS/ZBandASdenotestheareaofthetwo-dimensionalunitcellofsolid.

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InEq.(4.1),EbisindependentoncrystallinestructuresasafirstorderapproximationsinceenergydifferencesamongcrystalstructuresareseveralordersofmagnitudesmallerthanEbwhenthebondtyperemainsunaltered.

Althoughthebroken-bondruleseemstocontradictthebasicknowledgeabouttheelectronicstructuresinceEb,ingeneral,doesnotscalelinearlywithZS,theaboveestimationprovidestheorderofmagnitudeofγsv0andshowsapossiblerelationshipbetweenγsv0andatomicbindingstrength.Despitetheabsenceofverificationfromexperiments,sucharulehasbeenusedtogiveareasonabledescriptionofγsv0valueofAl[113].

SincethebondstrengthincreasesforanatomwithasmallerCN,thisCN-bond-strengthrelationcanbequantifiedusingtight-bindingapproximation.Inthesecond-momenttight-bindingapproximation,thewidthofthelocaldensityofstatesonanatomscaleswithZS,leadstoanenergygaintobeproportionaltothesquarerootofZSduetothelowingoftheoccupiedstates[131].

Neglectingtherepulsiveterms,thereisE1/2

b/ZB∝ZS.ByassumingthatEbofanatomisasumofcontributionsofallbonds,γsv0issuggestedas[131],

γsv0=(1 k1/2

1)Eb/(NaAS).

(4.2)

Eq.(4.1)doesnotconsiderthevariationofbondstrengthwithCNwhileEq.(4.2)hasneglectedtherepulsivetermsofpotentials[131],ly,theformerneglectswhilethelatteroverestimatestheeffectofrelaxationonγsv0[134],whichbringsoutthatEq.(4.1)canbeanacceptableconceptforstronglycovalentcrystalswhileEq.(4.2)isespeciallysuitablefornoblemetals.

AlthoughadirectutilizationofEq.(4.1)orEq.(4.2)isreason-able,oneofthemcouldnotalonegivesatisfiedpredictionsforγsv0valuesincomparisonwiththeexperimentalandtheoreticalre-sults[112].Toobtainamoregeneralformula,wearbitrarilyas-sumethatbothofEqs.(4.1)and(4.2)couldmakeupthedeficiencyeachotherwiththesameweighttobothformulae.Thus,γsv0val-uesmaybedeterminedbyanaveragedeffectofthemwithoutelab-orateestimationontherelaxationenergy[134],

γsv0=(2 k1

k1/21

)Eb/(2NaAS).

(4.3)

Eq.(4.3)impliesthatγsv0valuesstilldependonthebond-broken

rulealthoughtheyarescaledbetweenZ1/2

SandZS.

InEq.(4.3),ZSisdeterminedbythecrystallinestructurewiththecorrespondingZhklintermsofageometricconsideration[135,136].Forfccorhcpstructure,ZB=12;Forabcclattice,althoughZB=8istakenaccordingtothenearest-neighbordefinitionbysomeauthors(probablythemajority),othersprefertoregardZB=14sincethedifferencebetweenthenearestneighborbondlengthandthenext-nearestneighborbondlengthissmall[137].Here,thelatterisacceptedalthoughthenext-nearestneighborbondlengthisonlytakenasafractionofthenearestneighborbondlength.Byassumingthatthetotalenergyofasurfaceatomisthesumofcontributionsfrombothofthenearestneighborandthenext-nearestneighboratoms,Eq.(4.3)shouldberewrittenforbccmetalsafternormalization[134],

γ1/2 1/2

sv0=[(2 k1 k1)+ (2 k 1 k1)]Eb/[(2+2 )NaAS]

(4.4)

wherethesuperscriptcommadenotesthenext-nearestCNona

surfaceand showsthetotalbondstrengthratiobetweenthenext-nearestneighborandthenearestneighbor[138].

4.1.1.Solid–vaporinterfaceenergyofelementarycrystals

Toroughlyestimatethesizeof ,theLennard-Jones(LJ)potentialisutilized.Thepotentialisexpressedasu(r)= 4 [(ξ/r)6 (ξ/r)12]withεbeingthebondenergyandξinsuringdu(r)/dr(r=h)=0,i.e.ξ=2 1/6hwhereristheatomicdistance.Forfcccrystal,h=21/2a/2andh =a,respectively.Letr=a, ≈ /4,and ≈[(1/4)×6]/12=1/8.Thus,theeffectofthenext-nearestCNcanbeneglectedasafirstorderapproximation,ly,Eq.(4.4)canbesimplifiedasEq.(4.3)forfccandhcpcrystals;Forbcccrystal,h=31/2a/2andh =a,respectively.Letr=a, ≈2 /3.Thus, =[(2/3)×6]/8=1/2.AddingthisvalueintoEq.(4.4),

γsv0=[3 k1/2

1 k1 k1/2 (k 1/4)1/2]Eb/(3NaAS).

(4.5a)

NotethatthebondingoftheLJpotential,whichisutilizedtojustify valueinEq.(4.4),differsfromthemetallicbondinitsnature.Forinstance,inaLJbondedsystem,thesurfacerelaxationisoutwardswhilstinthetransitionmetalsitisinwards.However,thisdifferenceleadstoonlyasecondordererrorinourcaseandhasbeenneglected.

Theeffectofnext-nearestCNalsooccursforscanddcstructurecrystalsbecausetherearetwiceandthriceasmanythesecondneighborsasthefirstneighbors,respectively.Similartotheaboveanalysis,forsccrystals: ≈ /4and =[(1/4)×12]/6=1/2,whichisthesameforbccandthusEq. (4.5a)canalsoholdforsccrystals.Fordcstructurecrystals, ≈ /10and ≈[(1/10)×12]/4=3/10Withthis value,Eq.(4.4)isrewrittenas,

γsv0=[26 10k1 10k1/2

/21 3k1 (9k 1)1]Eb/(26NaAS).(4.5b)

Zhklcanbedeterminedbysomeknowngeometricalrules.Foranysurfaceofafccstructurewithh≥k≥l[134,135],Zhkl=2h+kforh,k,lbeingodd(4.6a)Zhkl=4h+2kfortherest.

(4.6b)

Inasimilarway,Zhklforanysurfaceofabccstructureisdeterminedwiththeconsiderationofthenext-nearestCN[134],Zhkl=2h+(h+k+l)forh+k+lbeingeven,(4.7a)

Zhkl=4h+2(h+k+l)forh+k+lbeingoddandh k l≥0,

(4.7b)Zhkl=2(h+k+l)+2(h+k+l)forh+k+lbeingoddand h+k+l>0

(4.7c)

wherethe2nditemoftheright-handsideofEq.(4.7)denotesthebrokenbondnumberofthenext-nearestneighbors.

Forsccrystals,Zhkilvaluesofthenearestandthenext-nearestatomsare1and4for(100)surfaceaswellas2and5for(110)surface,respectively.Fordcstructurecrystals,Zhklvaluesofthenearestandthenext-nearestatomsare1and6for(110)surface.Forseveralsurfacesofahcpstructure,Zhkilisobtainedby[136],Zhkil=4(h+k)+3lfor(0001)

(4.8a)Zhkil=4(h+k)+(8h+4k)/3for(1010

¯)(4.8b)wherethe1stitemoftheright-handsideofEq.(4.8)denotestheaveragenumberofbasalbrokenbondswhilethe2nditemisthatofnon-basalbrokenbonds.

Table5showssomenecessaryparametersinEqs.(4.3)and(4.5),andTables6–8givethepredictedγsv0valuesforfcc,bcc,hcp,dcandscstructurecrystalsintermsofthesetwoequationswheretherelatedparametersaretakenfromreferences[34,139].Asacomparison,thefirstprinciplecalculationsγ γ shown[109–111sv0andtwosetsofexperimentalresultssv0arealso].Notethattheexperimentalresultsarenotorientation-specificbutareaveraged

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Table5

AS,ZSandZBvaluesfordifferentsurfacesandstructureswhereaislatticeconstant

valuesofisotropiccrystals.Thus,theyshouldbeclosetothoseof

themostclose-packedsurface.

Forbothnobleandtransitionmetals,thepredictionsγsv0agreenicelywiththeexperimentalresultsandFCDcalculationsasshowninTables6–8althoughγsv0fortransitionmetalshaveslightlylargerdeviationsthanthoseforthenoblemetalsduetothefactthattheird-bandsarenotfullyfilledandtheypresentpeaksattheFermilevel,whichcanslightlychangefromonesurfaceorientationtotheotherandconsequentlytheenergyneededtobreakabondchangesalsoalittle.

Asshowninthesetables,γsv0valuesoftransitionmetalsincreasealonganisoelectronicrowwhereaheavierelementhasalargerγsv0value.Thisisbecausethed-levelofaheavierelementishigherinenergyandthecorrespondingd-wavefunctionwithastrongerbondingismoreextended.Thisisalsotrueforelementsinthesamerowintheperiodictablewhereaheavierelementhasγmored-electrons[110].AnexceptionisinVAserieswheresv0valueofNbissmallerthanthatofVpossiblyduetotherehybridizationofNbwhereNb,whosedshellislessthanthehalf-full,rehybridizesintheoppositedirection,i.e.,depletestheird2orbitalsbasedonachargedensitydifferenceanalysis[140–142].zγsv0valuesforspmetalsexceptforBearesmallerthanthoseford-metalsduetotheirbondnatureofs-andp-electrons,whicharemoremobilethanthelocalizedd-electronsandthereforelessenergyisneededtobreakthesebonds.

σForfccmetalsexceptCa,SrandAl,themean-squarerooterrorbetweenthepredictedandtheexperimentalresultsforthemostclose-packed(111)isabout7.5%.ForAl,thedegreeofcovalentAl–AlbondingincreasesorthenatureofthebondingchangeswithreducedCN[143],whichleadstomodelpredictiondeviatingfromtheexperimentalresultssinceourformulaneglectsthevariationofbondingtype.However,thereasonsofdeviationsforCaandSrareunclear.

Forhcpmetals,σ≈10%exceptMg,Zn,CdandTl.ForCdandTl,bothofthepredictionsandFCDcalculationsdeviateevidentlyfromtheexperimentalresults.InthecaseofZnandCd,c/aratios(1.86and1.89)arelargerthantheidealvalueof(8/3)1/2.Thus,thenearestCNvaluediffersfromtheidealcondition,whichshouldcontributethedeviations.

Forscmetals,σ=2.6%whereSbandBiwiththerhombohedralstructureareassumedtohaveslightlyadistortedscstruc-ture[110].

Forbccmetals,σ=10%.Thesmallestvalueofσinallconsideredσ=.structuresappearsfordiamondstructurecrystalswith14%,whichimpliesthatthepurecoherentbonddoesnotchangeafteraCNdeduction.

Notethatthetemperaturedependenceofγsv0isignoredherealthoughtheexperimentalresultslistedinTables6–8arecalculatedat0Kwhilethemostlatticeconstantscitedaremeasuredatroomtemperature.Thistemperatureeffect

Table6

Comparisonofsurfaceenergiesoffccmetalsamongthepredictedvaluesγsv0ofEq.(4.3),FCDcalculationsγ sv0[110],andexperimentalresultsγ sainnm[34],andγinJ/m2,whicharethesamev0[109,110].EbinkJ/g-atom[134],asthosein

deterioratesthepredictionaccuracyandcanbepartlyresponsibleforthedisagreementwithotherexperimentalandtheoreticalresults.

IntheFCDcalculations,thereareoftenexceptionsthatthemostclose-packedsurfacedoesnothavethelowestγsv0valueorthereexistsaweakorientation-dependence[111].Thesephysicallyunacceptableresultsarefullyavoidedhere.Moreover,theanisotropyofγsv0isperfectlyconsidered.Theratios1.16and1.27of(100)and(110)to(111)facetsforfccmetalsagreewiththeoreticalvaluesof1.15and1.22[112,132]whilethatof(100)to(110)facetsformonovalentspmetalsmetalsis1.16,whichiscomparablewith1.14basedonthejelliummodel[144].

γIftheexperimentalresultsaretakenasreferences,60%ofsv0valuesofthemostclose-packedsurfacesof52elementsshowninTables6–8havebetteragreementswithexperimentalonesthanthoseoftheFCDcalculationsdowhile20%oftheFCDcalculationsareinreverse.Notethatlocaldensityapproximation(LDA)isimpliedherewhileGGAisusedinFCD.Recently,ithasbeenshownthatbothmethodsneedtobecorrectedduetotheneglectofsurfaceelectronself-interactionswhereGGAisworsethanLDA[140–142,145].ThisissurprisingbecauseGGA

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441

Table7

Comparisonofsurfaceenergiesofcrystalsinbcc,scanddiamondstructuresamongthepredictedvaluesγsv0basedonEq.(4.5),FCDcalculationsγ s[109,110]

v0[110],andexperimentalresultsγ

isgenerallyconsideredtobethesuperiormethodforenergeticcalculations[110].

Theformulaforthetransitionmetalsandnoblemetalsworksbetterthanforothersasthegreatestcontributiontobondingisfromthes–dinteractionandtheorbitalsofthelatterlocalize,whichismorelikeapairinteraction.AccordingtoTables6–8,thepredictedγsv0valuesofdivalentspmetalshavebadcorrespondencewiththeexperimentalresultssincethemanybody(e.g.trimer)termsareherecriticaltounderstandthecohesiveenergy.Thus,theusedpairpotentialsphysicallymaybenotfullycorrect.Possiblythebackgroundoftheformula,i.e.thebroken-bondmodel,isnotuniversallyapplicablealthoughthelattice

Table8

Comparisonofsurfaceenergiesofhcpmetalsamongthepredictedvaluesγsv0ofEq.(4.3),theFCDcalculationsγ

[110],andtheexperimentalresultsγ [109,110]

constantsusedinEqs.(4.3)and(4.5)havemeasuringerrorofabout2%[109,125].

Accordingtothefirstprinciplescalculations,theeffectofrelaxationonthecalculatedγsv0valueofaparticularcrystallinefacetmayvaryfrom2%to5%dependingontheroughness[141,146].Thesemi-empiricalresultsindicatefurtherthatthesurfacerelaxationtypicallyaffectstheanisotropybylessthan2%[118].Surfacerelaxationsforvicinalsurfaceshavebeenstudiedmainlyusingsemi-empiricalmethodsduetothecomplexityarisenbythesimultaneousrelaxationofalargenumberoflayers[110].Here,therelaxationeffectissimplyconsideredbyaddingEq.(4.2)intoEq.(4.1).AccordingtoTables6–8,thismeasureleadstosatisfactoryresults.

Asasimplemodelwithoutfreeparameters,theaboveformulasuppliesanewinsightandanotherwayforageneralestimationofγsv0ofelements,whichisdifficulttorealizebypresentfirstprinciplescalculation.Moreover,thismodeldevelopsabasisofcomparisonandsupplementforfurthertheoreticalandexperimentalconsiderationsonγsv0ofelements.

Recently,Lodzianaetal.haveproposedthatγsv0ofθ-aluminaisnegative[147].Theiruseoftheterm‘‘negativesolid–vaporinterfaceenergy’’canandhascausedconfusioninthescientificcommunity[148].Mathuretal.nicelysummarizedandclarifieddefinitionsofγsv0forsingle-andmulti-componentsystems[148]:γsv0forsingle-componentsystemsisalwayspositive,whereasformulti-componentsystemsitcanbecomenegativeduetochemicaleffects,whichhasbeenconfirmedexperimentally[149].

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Table9

Comparisonofγ100valuesineV/atomofTMCamongγ1ofEq.(4.3),γ2ofEq.(4.2),γ3byusingFP-LMTOcalculationsbasedonLDA,simulationresultsγ4usingGGA-PW91andγ

byLDA[159],theoreticalresultsγ[156],andotheravailableresultsγusingpseudopotentialplane-wave-basedDFT[154,155]

4.1.2.Solid–vaporinterfaceenergyofseveralceramicswithNaClstructure

Thealkalinemetaloxides(AMO)asoneofthethreesupportingindustriesinmaterialdomainholdthebalanceindailylifeandindustrialmanufacture[150].Thetransitionmetalcarbides(TMC)andthetransitionmetalnitrides(TMN)havebeenwidelyappliedasthelayersofcuttingtools,electricallyconductingdiffusionbarriersinelectronicdevices,incoatingsforsolarapplicationsandforcorrosionprotection[151,152].Alloftheseapplicationsduetotheiruniqueproperties(e.g.highhardnessandhighTm)arecloselyrelatedwithitssurfacestate.γsv0asanimportantconceptchangesgrowthrate,catalyticbehavior,adsorption,surfacesegregationandformationofgrainboundaries.However,theγsv0valuesoftheoxidesaredifficultexperimentallytomeasureduetotheirioniccharacterandthehardness,whichresultinlessreliableexperimentaldata[153]althoughcomputersimulationsforithavebeencarriedout[154–157].

AlthoughEqs.(4.3)and(4.5)arededucedforelementalcrystals,theyhavebeendevelopedforinsulatorormetalliccompoundsinliteratures[156,158].Themainsourceofthebondingintheseceramicsistheionicinteractionbetweenmetalandnon-metalatomsintheNaClstructure[154]whereZB=6andZSare5,4,and3respectivelyfor(100),(110)and(111)surfaces[159].Notethatbecausecomparedγsv0valuesofthesecompoundsareusuallyreportedinunitofev/atomobtainedbycomputersimulations,γsv0valueinthissectionisalsodenotedinthisunit.

Table9givesγ100valuesofTMCsintermsofEqs.(4.2)and(4.3).OthercomputersimulationandtheoreticalresultsforTMCsarealsoshownforcomparison[154–156,159].ItcanbefoundthattheagreementsamongdifferentmethodsforstableTMCsarebetterthanthatformetastableTMCs,suchasCoC,NiCandOsC.1TheFermilevelsofbothCoCandNiClieintheupperpartofthedregion[160],whichmayleadtothatthesimulationresultsaremuchlowerthanthoseintermsofEq.(4.3).WhileforOsC,the

Fermilevelcontainsthemostdstatesthatdonothybridizewithcarbonstates[160].Thus,thesimulationresultsrepresentdifferenttrendscomparedwiththetheoreticalones.ThetheoreticalresultsofEq.(4.2)areingoodagreementwithLDAsimulationoneswhilethoseofEq.(4.3)havegoodcorrespondenceswithothertheoreticalresultsbasedonmodificationofEboftheclassicbroken-bondrule[156].Itisfoundthatallγsv0valuescalculatedfromGGAaresmallerthanthosefromLDA,whichconfirmsothersimilarsimulationresults[157].

FromTable9,eventhelargestpseudopotentialresult(0.68eV/atom)fromLDA[161]isabout18.1%lowerthantheresult(0.83eV/atom)fromLMTO[156]forTiCduetodifferentapproachesduringsolvingtheDFTequations.ThemethodusedinRef.[156]treatstheeffectiveone-electronpotentialwithoutanyshapeapproximationandallowsonetosolvetheelectronic-structureproblemforbulkaswellasforthesurfacefromthefirstprincipleswithinasinglescheme[162].LMTOoftenemploystheatomicsphereapproximation(ASA)thatisclearlyinadequateforstructureswithlargeinterstitialregionsorwithlowcoordinationsymmetry,asinthepresentsurfacecalculations.Thus,theLMTOapproachisgenerallyregardedasefficientbutrelativelyinaccuratemethod[163].

Table10showsγ100valuesoftwelveexistedstableTMNswiththesamemethodsasstatedabove[154,155].2γ100valuesofthestableTMNsdeterminedbyGGA-PW91withthecorrespondingpotentialarenearlyhalfoftheresultsintermsofEq.(4.3)whilearelargerthanothersimulationresults[154,155].LDAresultsarerelativelylargerandinagreementwiththemodifiedbrokenbondmodelbetter.Sincetheexistingresultsarelittle,itisdifficulttoevaluatetheaccuracyofdifferentsimulationmethods.

ForTMCsandTMNs,theexistenceofthed-electronoftransitionmetalsleadstoalmostthesamelatticeconstantsforthesamegroupwhentheperiodnumberoftransitionmetalsincreases.Atthesametime,theirEbincreasesgradually.Thus,γsv0valuesincreasealittleastheperiodnumberincreases.Intheotherside,

13dtransitionmetalcompoundsfromScCtoVCarestable[156];4dtransitionmetalcompoundsfromYCtoTcCarestable[156];5dtransitionmetalcompoundsfromLaCtoWCcanberegardedasstable[156].TMCsconsistingofthegroupsVIIIBaremetastablewhentheyhaveaNaClstructure.

100valuesofmetastablephaseshaveworseaccuracy,

partlyduetotheirmetastablenatureswithpossiblyunsuitablepotentials.Thus,thecorrespondingcalculationsarenotshown.

2Thecalculatedresultsforγ

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443

Table10

Comparisonγ100valuesineV/atomofTMNsamongγ1ofEq.(4.3),γ2ofEq.(4.2),simulationresultsγ3usingGGA-PW91andγ4usingLDA[159],andothersimulationresults

[154,155]

Fig.8.γsv(D)asafunctionof1/DintermsofEq.(4.12)(solidlines)and(4.14)(segmentlines)fornanocrystalsBe,Mg,NaandAlwithdifferentfacets.Thesymbols , and denotetheexperimentalresultsforBeandMg(0001)[175],thetheoreticalvaluesforNa(110)[176]andAl(110)[177].

asthegroupnumberinthesameperiodvaries,amaximumofEbvaluesisreachedatVBorIVBgroup.Thus,thecorrespondingγsv0valuesalsoapproachtheirmaxima.Itisknownthatthevariationofγthebandwidthofcompoundsshouldbeproportionaltothatofsv0values[156,157],whichhasbeenconfirmedfor4d-TMCbyanalyzingthedensityofstates(DOS)andtheFermienergyusingLMTOmethod[156].ItisfoundthatthebandwidthdevelopedanearlymaximumforZrCandNbCaround11eV,afterwhichitremainsfairlyconstant,andthenshiftsdownwardsforAgConlyaround5eV.Thesametrendisfoundinthe3d,4dand5dcarbides[156].ConsideringthechemicalsimilaritiesbetweenTMNsandTMCs,thevariationsofγsv0valuesinTMNseriesshouldfollowthesamerule.

Table11givesthecalculatedγsv0valuesofAMOsintermsofEq.(4.2),(4.3),LDAandGGA-PW91[159],andothertheoreticalresults[157].Thepackingdensitiesof(100),(110)and(111)surfacesofNaClstructureareroughlyintheratioof1:0.71:0.58,whichsuggestsasequenceofγ100<γ110<γ111[164]wherethe(100)surfaceisthemoststablesurfaceandthe(111)oneisthemostreactive[158].Asexpected,thecalculatedresultsagreewiththeabovesequence.

Inourknowledge,γ111valueshavenotbeendetermineduptonowexceptγ111valueofMgOwithγ111=2.54eV/atom[165],whichwasobtainedbytheDFTbased,full-potentiallinearizedaugmentedplane-wavemethod.Whencomparedwithγ100=0.26eV/atom,γ111isoneorderlargerthanγ100.Thiscanbeinducedbyitspolarnaturewithinstability.FromTable11,comparisonbetweenEq.(4.3)andthesimulationresults[159]showthatthedeviationbetweenthemincreasesalongtheseries

fromMgOtoBaO.Thismaybeduetotheincreaseofthecrystallinelatticeconstantwithdropofthebindingenergyinducedbytheincreaseofcationrepulsionanddecreaseofanion–anionoverlap.Foracertainsurface,γsv0valuesdecreasealongthesequencegoingfromMgOtoBaOwheretheenergylossofMgOismostobvious.MgOhasmuchbroaderbandwidthandhigherdispersionthantherestoftheseries.TheDOSanalysisindicatesthattheO(2p)bandbecomesnarroweralongtheseriesgoingfromMgOtoBaO[159].OnlythebandwidthofMgOisprominentbroaderwhilethatofCaO,SrO,andBaOareroughlythesame[157].Thesecouldbethechangecauseofγsv0alongtheperiodnumbersofthecorrespondingmetals.TheabovefactcanalsobeenconsideredbyEq.(4.3).Thereverseratiobetweentheperiodnumberofthemetalsandγsv0valuesofAMOsisinducedbyincreaseofaduetotherapidincreaseofs-electronorbitalastheperiodnumberincreaseswhileEbremainsalmostconstantforthesamegroupofelements.

4.2.Thesize-dependentsolid–vaporinterfaceenergyγsv(D)Thethermodynamicbehaviorofnanocrystalsdiffersfromthatofthecorrespondingbulkmaterialsmainlyduetotheadditionalenergetictermofγsv(D)A.Thistermsignificantlychangesthethermalstabilityofthenanocrystalsduetothelargesurface/volumeratioofnanocrystalsorA/V∝1/D[166–169].Whenthesurfacesofpolymorphsofthesamematerialpossessdifferentinterfacialfreeenergies,achangeinphasestabilitycanoccurwithdecreasingD[170].Despiteoftheimportanceofγsv(D),fewreliableexperimentalortheoreticalvaluesareavailable[64,111γ]whiletheeffectsofsizeandsurroundingofnanocrystalsonsv(D)arestudiedonlyalittle[79,96,170–172].

However,inmesoscopicsizerange,γlv(D)wasthermodynam-icallyconsideredfiftyyearsagobyTolmanandBuff,respec-tively[108,173].Thefinalformoftheanalyticalequationisasfollows[108],

γlv(D)/γlv0=1 4δ/D+···

(4.9)

whereδdenotesaverticaldistancefromthesurfaceoftensiontothedividingsurfacewherethesuperficialdensityoffluidvanishes.Asafirstorderapproximation,althoughthereisnodirectexperimentalevidencetosupportEq.(4.9),Eq.(4.9)shouldalsobeapplicabletopredictingγsv(D)sincethestructuraldifferencebetweensolidandliquidisverysmallincomparisonwiththatbetweensolidandvapororbetweenliquidandvapor.Inaddition,itisunknownwhetherDinEq.(4.9)canbeextendedfrommicronsizeγtonanometersize.Hence,atheoreticaldeterminationofsv(D)ismeaningful.

AlthoughboththeexpressionsofEq.(4.1)to(4.5)andthecorrespondingresultsaredifferent,allofthemindicatethat,

γsv=υEb/(NaAS)

(4.10)

whereυ<1isafunctionofCN.

Ifthenanocrystalshavethesamestructureofthecorrespond-ingbulk,υissize-independent.Thus,Eq.(4.10)maybeextendedtonanometersizeas[174],

γsv(D)=υEb(D)/(NaAS).

(4.11)

CombiningEq.(4.11)withEq.(2.27),thereis[174],

γ

1

2Sb1

sv(D)/γsv=1

2D/h 1

exp

3R2D/h 1

.

(4.12)

IntermsofEq.(4.12),comparisonsofγsv(D)ofBe,Mg,Na,AlthinfilmsandAuparticleswithdifferentfacetsbetweenmodelpredictionsandexperimentalorothertheoreticalresults[175–178]areshowninFigs.8and9wheretherelatedparametersarelistedinTable12.Itisevidentthatourpredictionsarein

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444Q.Jiang,H.M.Lu/SurfaceScienceReports63(2008)427–464

Table11

Comparisonofγsv0valuesineV/atomofAMOsamongγ1ofEq.(4.3),γ2ofEq.(4.2),simulationresultsγ3usingGGA-PW91andγ4usingLDA[159],andothertwoseriesofavailableLDAresultsγ

andGGAresultsγ[157]

Table

12

NecessaryparametersinequationswithD[175–178]andh[33]innm,E[33]inkJ/mol,T[33]inK,SinJ/mol-K,and[110,112]in

Jm2Sbh/(12R)≈hwhenSb≈ly,thetransitionzoneseparatingasolidphaseandavaporphaseisonlyoneatomiclayer,whichisanunderstandableresult.Thisdeterminedδvalueisexpectedsincewhentheatomicdistanceislargerthanh,thebondenergydecreasesdramatically.Thus,Eq.(4.9)canberewrittenas,

γlv(D)/γlv0≈γsv(D)/γsv0≈1 4h/D.

(4.14)

Itisknownthatγsv(D)ratiobetweendifferentfacetsisamoreimportantparameterindeterminingthecrystallineshapes.Eq.(4.12)indicatesthat,

γs1γs1v(D)v0=2

2svDγsv0

Fig.9.γsv(D)asafunctionof1/DfornanocrystalsAuwithdifferentfacetsinterms

ofEq.(4.12)(solidlines)andEq.(4.14)(segmentlines).Thesymbols , and denotethecalculatedresultsfor(111),(100)and(110)facetsintermsofamodifiedembedded-atom-methodpotential[178].

(4.15)

agreementwiththeexperimentalvaluesofBeandMg(0001),othertheoreticalresultsforNa(110),andthoseforthreelow-indexsurfacesofAu.Thedeviationsinallcomparisonsaresmallerthan5%exceptthatforAl(110)withadeviationofabout10%.

AsshownintheFigs.8and9,γsv(D)decreaseswithadecreaseinsize.ThistrendisexpectedsinceE(D)ofthenanocrystalsincreasesasDdecreases.Inotherwords,γsv(D)asanenergeticdifferencebetweensurfaceatomsandinterioratomsdecreasesasenergeticstateofinterioratomsincreases.

Consideringthemathematicalrelationofexp( x)≈1 xwhenxissmallenough,Eq.(4.12)canberewrittenas,

wherethesuperscripts1and2denotedifferentfacets.Eq.(4.15)impliesthatalthoughγsv(D)issize-dependent,thesolid–vaporinterfaceenergyratiobetweendifferentfacetsissize-independentandisequaltothecorrespondingbulkratio.Eq.(4.15)canalsobecomparedwiththetheoreticalresultsforAu.Forexamples,γs(v100)(3.8nm)/γs(v111)(3.8nm)≈1.24andγs(v110)(3.8nm)/γs(v111)(3.8nm)≈1.28[178],whichcorrespondwelltothecorrespondingbulkratiosof1.27and1.32[110].

SincethemetalsshowninFigs.8and9havedifferentstructures,themodelshouldbeapplicableforallcrystallinestructureswithdifferentfacets.Recently,themodelhasbeenextendedtosuccessfullydeterminethesize-dependentsurfaceenergyofnanocavities[179].Thus,Eq.(4.12)suppliesasimplewaytodetermineγsv(D)valuesofdifferentfacetswithoutanyfreeparameter.

5.Liquid–vaporinterfaceenergy

5.1.Thebulkliquid–vaporinterfaceenergyγlv0

Temperature-dependentγlv0(T)functionisknowntobeoneofthefundamentalandimportantquantitiesinthetheoryandpracticeofmaterialsprocessing(e.g.,crystalgrowth,welding,and

γsv(D)/γsv≈1 Sbh/(3RD).

(4.13)

Eq.(4.13)correspondstothegeneralconsiderationthatthedecreaseoftheanysize-dependentthermodynamicquantityisproportionalto1/D[30].Ifγsv(D)functionofEq.(4.13)andγlv(D)functionofEq.(4.9)havethesamesizedependence,δ=

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Q.Jiang,H.M.Lu/SurfaceScienceReports63(2008)427–464445

sintering),anditstemperaturecoefficientγ governsthewell-knownMarangoniconvectionlv0(T)=dγlv0(T)/dTonthesurfaceofmelt.Thereareseveralcharacteristicsoftheliquidsurface.First,theliquidsurfaceusuallytakesanequilibriumconfigurationwiththeminimumenergyduetothehighmobilityofliquidmolecules.Second,becausetheliquidfailswithrespecttoelasticdeforma-tionresistance,γlv0(T)equalssurfacestresswhensurfacead-sorptionisnottakenintoaccount[5].Althoughearlymethodsofmeasurementofγlv0(T)aresufficientlyprecise,thereisstilluncertainty )regardingitsabsolutevaluesandparticularlyregard-ingγlv0(Tfunctionmainlyduetotheeffectofimpurities,whichstronglychangesthemeasuredresults.Therefore,considerableef-fortshaverecentlybeen)directedtowardstotheexperimentalde-terminationsofγlv0(Tandγ lv0(T)ofmetals,andprogresshasbeenachievedwiththeadventoflevitationprocessingandoscillatingdroptechniques[180,181].However,suchanexperimentoftensuffersfromambiguitiesintheinterpretationoftheresultingfre-quencyspectra.Itisalsounlikelythatexperimentalmeasurementswilleverencompassallpossibletemperaturerangesofinterestandforallmetals.

Incontrasttothedeterminationofγlv0(Tm)values,γ knownexperimentallyevenforelementallv0(Tm)valuesarenotwellmetallicliquids.Arecentanalysisofexistingdatashowsthatthisquantityisknownwithaccuracybetterthan50%foronly19metalswhiletheaccuracyisworseforother28metals.Fortherest18metals(mainlyrefractorymetals),thereareevennoexperimentalresults[182].

ComputerstimulationswithMonteCarloormoleculardynam-icsmethodsareconsideredtobeoneofthereliablemethods[183],withwhichγlv0canbecalculatedeitherusingthemechanicalex-pressionforthesurfacestressorfromtheviewpointofγsv0.Unfor-tunately,theformerapproachsuffersfromratherhighfluctuationandstatisticaluncertainty,whilethelatterintroducesadditionalcomplexityintoperformance.Thus,thedemandofdevelopingre-liablepredictionmethodshasneverdeclined.

Semiempiricalpredictionsbasedonthecorrelationbetweenthesurfaceandbulkthermodynamicpropertiesarealwaysactive[184–187].Stephanfirstlylinksγlv0totheheatof

evaporationHv

atT=0K[184],

γ lv0(Tm)=c Hv/V2/3

l

(5.1)

withc beinganunknownconstant.Eq.(5.1)seemstoapplyonlytotransitionmetals.AlthoughEq.(5.1)hasexistedformorethan100years,attemptstotheoreticallydeterminec valuearerare.

Ontheotherhand,γlv0(T)ofpuresubstancesmaybeevaluatedfromvaluesofcriticaltemperatureTcbytheEötvosorGuggenheimempiricalequations[188],

γ2/3

lv0(T)Vl

=Q(1 T/Tc)

(5.2a)

or

γlv0(T)/γlv0(Tm)=(1 T/Tc)W

(5.2b)

wherethecoefficientQandexponentWaresystem-dependent,e.g.,W=4/5forstronglyhydrogenbondedsubstances,W=11/9forH2,N2andCO,etc.However,Wvalueforliquidmetalshasnotbeendeterminedtothebestofourknowledge.Moreover,unlikethoseoforganicliquids,TcvaluesofliquidmetalsareonlyavailableforalkalimetalsandHg[33],whichseverelyrestrictstheuseofEq.(5.2a).

Whenγlv0(Tm)andγ nearlylv0(Tm)avaluesconstantarebeingknown,equalundertoγ the

assumptionthatγ

lv0(T)islv0(Tm)whenTisnearTm,theγlv0(T)functionisexpressedsimplyinadifferentialform[189],

γlv0(T)=γ lv0(Tm)+γlv0(Tm)(T Tm).

(5.3)

γHowever,Eq.(5.3)hasnotbeenstrictlyexamined.Thus,both

lv0(T)andγ lv0(T)functionsneedtobefurtherconsidered.

AsstatedinSection4.1,Eq.(4.4)canbeusedtocalculateγsv0ofelements[133].DuetothestructuralsimilarityoftheliquidandsolidγatleastnearTm,Eq.(4.4)forγsv0shouldgivesuggestionsforlv0modelingandanalyticaldeterminationofc valueinEq.(5.1).About60yearsago,notingthatfusionhasonlysmalleffectsonvolume,cohesiveforces,andspecificheatofsubstance,Frenkelreachedtheconclusionthat‘‘thecharacteroftheheatmotioninliquidbodies,atleastnearthecrystallizationpoint,remainsfundamentallythesameasinsolidbodies,reducingmainlytosmallvibrationsaboutcertainequilibriumposition’’[190].Theveryslightchangeinvolumeonmeltingisalsothoughttoimplythattheatomsinaliquidaretightlyboundtooneanotherlikethoseinacrystallinesolid[191].Thus,thestructuralandenergeticdifferencesbetweenasolidandaliquidareverysmallincomparisonwiththosebetweenasolidandavapororbetweenaliquidandavapor.Consequently,Eq.(4.4)canbeextendedtodetermineγlv0withseveralmodifications[192]:

(i)SinceT≥Tm 0Kforliquids,Ebat0Kshouldbereplaced

byHv(T)andASshouldbesubstitutedwithAL(T),wherethesubscriptLdenotesliquid;

(ii)TheinfluenceofsurfaceentropyS(T)shouldcontributeγlv0

duetothehightemperature;

(iii)CNofaliquidisusuallydeterminedbyintegratingtheradial

distributionfunction(RDF)uptothefirstminimumwhilethedistanceofthesecondminimumofRDFisapproximatelytwicethatofthefirstone,theeffectofthenextnearest-neighborsthusmaybeneglectedintermsoftheLJpotentialfunction,namely, ≈0.Thus,Eq.(4.4)canberewrittenfordeterminingγlv0(T)[192],

γlv0(T)=[mHv(T) TS(T)]/[NaAL(T)]

(5.4)

withm=(2 k1/2

1 k1)/2.

Sincemetallicliquidiscloselypacked,thepackingdensityofarandomclosepacking(ηL=0.637)canbeemployedfortheliquid[35].AsdiscussedinRef.[35],thevolumechangeonmelting V/Vsisnotsolelydeterminedbythedifferenceinηbetweentwophases,andtheGoldschmidtpremiseforlatticecontractionshouldalsobeconsidered.Otherwise,whenηL=0.637iscomparedwithηfcc=0.74, V/Vs=ηfcc/ηL 1≈16%,whichislargerthanηtheexperimentallyobservedvalueof2%–6%[35].NotethatL=0.637isthemaximalvaluethatthesingle-componentliquidcantake,whichleadstothefactthatthespecificvolumedifferencebetweenasolidwithbccstructureandthecorrespondingliquidisonly0.2%intermsofEq.(2)ofRef.[35].Whetherametalcanbeundercooleddependsontheenergeticnucleationbarrier.Whenthevolumechangeoncrystallizationissmall,thecorrespondingnucleationbarriershouldbealsosmall.Thisleadstoasmalldegreeofundercooling[193].Ifthisrulecanalsobeappliedtoelements,thelocalorderinthemetallicliquidisverysimilartothebcc-typeshort-rangeorder(SRO)[194],suchasliquidZr[195].Correspondingly,thisconsiderationcanalsobeappliedonthesurfacestructureofliquidmetals.Notethatthesurfaceofbccstructureis(110)facettoensuretheminimumoftheproductofγsv0ΣAbcc[182].Thus,theexpressionofAbccisassumedtobeapplicablealsototheliquidasafirst-orderapproximation.Asaresult,k1=3/4andAL=81/2h2/3.Foranypure,isotropic,condensedmaterial,h=(6ηV/π)1/3.VcanbecalculatedfromtheatomicweightMandρ(T)byV=M/[Naρ(T)].Thus,AL(T)canbedeterminedas,AL(T)=λ {M/[NaρL(T)]}2/3

(5.5)

withλ =(81/2/3)(6η/π)2/3.SincedρL/dT≈dρL(Tm)/dTforliquidmetalsinthetemperaturerangeofTm 2Tm,ρL(T)isequal

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toρL(Tm)+(dρL/dT)(T Tm)withdρL/dTbeingthetemperaturecoefficientofliquiddensity.Thisrangecouldbeupto3TmforRbandCsand4TmforLiandK[67].

ItisknownthatHv(Tc)=0formostsubstancesandreachesthemaximumatthetriplepointTtwhereTtisveryclosetoTmfor(metals[196].Recently,anempiricalequationHit+jhasbeenproposedforliquidshavingatriplev(T)/Hpointv(Tm1 t))=wheret=(T Tm)/(Tc Tm),i=0.44andj= 0.137[197].IntermsoftheknownTm,TbandTcvaluesforalkali)metals(Hgisnotinvolvedinthiswork),itisfoundthatHv(TfunctionbetweenTmandTbisveryweakfunctionof)T(<2%)andthusnegligible.Moreover,itisreportedthatHv(TmvaluesofCd,Cr,MnandZnare101,344,234and115kJ/molwhilethecorrespondingHand119kJ/mol[33,196].Theirdifferencesv(Tb)valuesare100,339,226aresmallerthan3.5%.Thus,therelationHdoesnotleadtobigdeviationandv(Tm≤T≤Tb)≈Hcanbeacceptedasv(Tm)≈Hafirst-ordervapproximation.

InSkapski’smodel[185],themaincontributiontoSresultsfromthechangeinoscillationfrequencyofatomsinthesurface.NotethataccordingtoLindemann’scriterion[198],themeanvalueofoscillationfrequencyreachesacertainvalueatTm,whichleadstoaconstantSvalueatTm.ItisknownthatHliquidatoms,which)furtherdeterminesvdeterminesthebondstrengthofthesizeofoscillationfrequency.SinceHv(T(varieslittlebetweenTmandTb,thetemperaturedependenceofST)isthusnegligibleandS(T)≈S(Tm)≈Scanalsobeassumed.

WiththeseconsiderationsandinsertionofEq.(5.5)intoEq.(5.4),γlv0(T)betweenTmandTbcanbedeterminedas[192],

γ

)

2/3

lv0(T)=

mHv TS

ρL(TλN13

aM

(5.6a)

or

γlv0(T)

T

2/3

=1+p p

q

Tlv0TmT1 q+m

T(5.6b)

m

wherep=1/[mHv/(TmS) 1]andq=(dρL/dT)[Tm/ρL(Tm)]are

constantsforcertainmetals.

DerivativeofEq.(5.6a)withrespecttoTbringsout,

γ (T)

1T

dρL

lv0(T)=

γlv0T

mHvTS1

23LTdT

.

(5.7)

5.1.1.Determinationofγlv0(Tm)values

Table13givesthecomparisonbetweenthepredictedγlv0(Tm)valuesfor48liquidmetalsintermsofEq.(5.6a)andavailablemeanvaluesofexperimentalγelv0(Tm)results[182,189,199].TheseexperimentaldataaremainlyobtainedbythemaximumbubblepressuretechniqueforoxidizablemetalswithlowTmlikeNa,thesessiledroptechniqueformetalswithmoderateTmlikeCu,andthedropweighttechniqueemployedattheextremityofapendantwirewithelectronbombardmentheatingforrefractorymetalslikeW.

Itisfoundthatω=|γlv0(Tm) γee

lv0(Tm)|/γlv0(Tm)<10%forγ40elements(fromCutoBa;seeTable13).Notethatalthough

evmJ/m2

l0(Tm)=867wasproposedforAl[182],severalmeasurementssuggestedthatthemostdataforγetooxygen-saturatedmaterialandthatforpurelv0(Tm)ofAlpertainAlcouldabout～1070mJ/m2[197,200].Ifthisresultisused,ωforAlwillonlybe3.6%.FordivalentmetalsMg,ZnandCd,thepredictionsareevidentlysmallerthanγeBoom[199],thesethreemetalslv0(Tm).AccordingtoMiedemaandhaveanexceptionallystablefreeatomicconfiguration,whichisclosetothatofraregas.Thus,smallerγlv0(Tm)valuesintermsofEq.(5.6a)maybereasonable.AlthoughωvaluesofTa,Nb,Li,Be,andLarangefrom13%to22%,thecausesareunknown.

ThedatagivenaboveimplythatEq.(5.6a)issuitableforallmetalsalthoughtheerrorsfromtransitionmetalsareslightlylargerγthanthoseforothermetals.

lv0(Tm)valuesoftransitionmetalsincreasealong(anTisoelec-tronicrow,whereaheavierelementhasalargerγlv0m)value.Thisisbecausethedlevelofaheavierelementishigherinenergyandthecorrespondingdwavefunctionswithstrongerbondingaremoreextended.TwoexceptionsarePdandZr.ForPd,thefull-filleddorbitaldropsthesystemenergyintermsofHunt’srule,whichmakesthatitsHofPdarevvalueonlyapproachesthatofNi.SinceVmandTmvaluesobviouslylargerthanthoseofNi,γlv0(Tm)ofPdisthussmallerthanthatofNiintermsofEq.(5.6a);ForZr,itsHandVofHfwhileitsTvmvaluesapproachthoseγmvalueisobviouslysmallerthanthatofHf,lv0(Tm)ofZristhereforelargerthanthatofHfintermsofEq.(5.6a).ThereasonoflargerHvvalueofZrisunclear.

γlv0(Tm)valuesofspmetalsexceptthatofBearesmallerthanthoseofdmetalsduetothebondnatureofsandpelectrons,whicharemoremobilethanthelocalizeddelectrons.Moreover,incontrasttothetransitionmetals,γlv0(Tm)valuesofspmetalsdecrease alonganisoelectronicrow.Thisarisesbecausetheoutmostnselectrons(thenumberofperiodn =2–6)areprogressivelyboundmorelooselyastheyarescreenedfromthenucleusbytheincreasingnumberoffilledinnershellsintheioniccore.

NotethatassumingthegeometricstructureinliquidSiandGetobeclosetothatof(100)faceofscstructure,theirγlv0valuescanbecalculatedusingtheabovemethodandreasonablecorrespondenceswithexperimentalorothertheoreticalresultshavebeenfound[201].

TofindthesimilaritybetweenEqs.(5.6a)and(5.1),theprefactorc

inEq.(5.1)canbedeterminedbyrearrangingEq.(5.6a)atT=Tm,

γlv0(Tm)=cHv/V2/3

s(5.8)

with

c=(m TmS/Hv)/(λN1/3

a).

(5.9)

Eq.(5.9)hasdeterminedc valueinEq.(5.1)althoughEq.(5.8)

differsfromEq.(5.1)alittleduetothedifferenceofH

c=c HvandHv,v

/or

Hv.

Fig.10givesaplotofγlv0(T2/3

m)againstHtermsofEq.(5.8)withalinearlyv/Vsforconsideredliquidmetalsinregressedslopeofc=0.174×10 8mol1/3wherethecorrelationcoefficientofthefitis0.998.Allmetallicelements/canthusbeestimatedbythesamecvalue,whichimpliesthatTmHvisalmostaconstant(sinceSvaluehasbeentakenasaconstantaccordingtoLindemann’scriterion,m≈0.19andλ≈1.08)intermsofEq.(5.9),whichalsoconfirmsthecorrectnessofEq.(5.8).ItisknownthatH=1/Sv∝Hm[187]andTm/Hmm.SinceSmisalmostaconstantformetallic

elements,cvalueasaconstantisreasonable.Intermsofthe

datalistedinTable1ofRef.[199],itisfoundthatH

Hv

1.09)fortransitionmetals.Thus,c=v/Hcv isaconstant(≈Hv

/nearly

Hv=

0.174×10 8mol1/3wherec =0.16×10 8

mol1/3asdetermined.IncontrarytoEq.(5.1),whichisconsideredtobeonlysuitablefortransitionmetals[182,187,199],allmetalsareinvolvedinFig.10.

ThisimprovementisonlyachievedbyreplacingHatombondingofstableliquidv

byHHandcanv(Tmbe

).exactlyv(Tm)describesthemeasuredwhileHv

canbeobtainedonlybyextensionofexperimentalresults.Inaddition,sincethedifferencebetween0KandTmfortransitionmetalsarelargerthanthatfornon-transition

metals,whichleadstoalargerdifferencebetweenHv(Tm)andH

fortransitionmetalsthanfornon-transitionmetals.ThisresultsinvsmallersuitabilityrangeofEq.(5.1)thanthatofEq.(5.8).

γIntermsofEq.(5.6a),theintroductionofSdropsthevalueoflv0(T).AtTm,thedecreasingextentsrangefrom8%(forLaandCe)to20%(forMgandSr).

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Q.Jiang,H.M.Lu/SurfaceScienceReports63(2008)427–464447

Table13

2

Comparisonsofγlv0(Tm)inmJ/m2forliquidmetalsbetweenγlv0ofEq.(5.6a)andexperimentalresultsγlev0[182,189,199],aswellascomparisonsofγlv0(Tm)inmJ/mK

e

betweenγofEq.(5.7)andexperimentalorestimatedresultsγ[182,189,192,199]

5.1.2.Determinationofγl v0(Tm)values

Table13alsoshowsthecomparisonbetweenγl v0(Tm)valuesintermsofEq.(5.7)andtheavailableexperimentalorestimated

e

resultsforγl v0(Tm)[182,189,192,199].Goodagreementsarealsofound,whichindicatesthatEq.(5.7)providesasatisfactorydescriptionforγl v0(Tm).

Eq.(5.7)atT=Tmcanbewrittenas,

γl v0(Tm)=(p 2q/3)γlv0(Tm)/Tm.

1/3

(5.10)

Intermsoftheexpressionsforcandp,p=m/(cλNa)≈0.19.Ifq=(dρL/dT)[Tm/ρL(Tm)]= 0.17istakenasameanvalue,slopeβ=p 2q/3≈0.30.Therelationbetween γl v0(Tm)andγlv0(Tm)/Tmforthefourth,fifth,andsixthperiodsareplottedinFig.11intermsofEq.(5.10)withβ=0.30,where γl v0(Tm)functionsincreasealmostlinearlywithanincreaseinγlv0(Tm)/TmfortheAfamilymetalsinthesameperiod,andthesequenceisnearlythesameasthatinthePeriodicTableoftheElements,althoughsomedeviationsappear.Thisis

understandablesincetheiroutmostelectronicconfigurationsofs+delectronsundergonearlythesamesituationfromtheleftmost(IAmetals)ofonetotherightmost(VIIIAmetals)oftenintheseperiods.Theexceptionsareasfollows:(i)Inthefourthperiod(fromKtoNi),theanomaliesofMnandCrarepresentwheretheir3dorbitalishalf-filled;(ii)Similarly,theappearanceofthefull4dorbitalalsoresultsintheanomalyofPdoftheFifthPeriod(fromRbtoPd).Onthecontrary,theoccurrencesofthehalffull4dorbitalinMoandthehalffull5dorbitalofReintheSixthPeriod(fromCstoPt)donotchangethesequence.Thesemaybeexplainedasthefollowing:IntermsofHund’srule,thehalfandthefullfillingsofanorbitalusuallyresultindropofthesystemenergywhilethefullfillingismoreeffective.Forexample,theHvvaluesofCrandMnareevidentlysmallerthantheneighborelementsVandFeasshowninTable13.ItisalsoapplicabletoPdincomparisonwithRh(AgisnotinvolvedbecauseitisaBfamilymetal).Whiletheincreaseofelectronicshelldecreasestheeffectofelectricconfiguration,theHvvalueofMo(Re)isthusinbetweenthoseofNb(W)andTc(Os).

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