基于液态合金短程序的混合构型熵新模型

物理化学学报(WuliHuaxueXuebao)

ActaPhys.-Chim.Sin.2012,28(7),1586-1592

doi:10.3866/PKU.WHXB201205042

July

www.whxb.pku.edu.cn

基于液态合金短程序的混合构型熵新模型

秦敬玉*

(山东大学材料液固态结构演变与加工教育部重点实验室,济南250061)

摘要:

基于液态合金化学短程序与拓扑短程序发展了一个新的混合构型熵计算模型,从这个模型可以导出用

来描述等原子直径随机混合物的理想混合熵.通过将该模型应用于一些理想的和真实的液态二元合金,可以看到化学短程序减小了混合构型熵,而原子尺寸差异的影响则较为复杂.当大原子进入小原子基体时,混合构型熵增大;而当小原子进入大原子基体时,混合构型熵减小.在这些合金中,共晶成分处并没有出现混合构型熵极大值.关键词:

液态合金;短程序;混合构型熵

O642;O645

中图分类号:

ANewModelfortheConfigurationalEntropyofMixinginLiquidAlloys

BasedonShort-RangeOrder

QINJing-Yu*

(KeyLaboratoryforLiquid-SolidStructuralEvolutionandProcessingofMaterials(MinistryofEducation),ShandongUniversity,

Jinan250061,P.R.China)

Abstract:Anewmodelhasbeendevelopedtocalculatetheconfigurationalentropyofmixinginliquidalloysinvolvingconsiderationofchemicalandtopologicalshort-rangeorder.Theentropyofmixingforanequiatomicrandommixturewasnaturallyreachedbythismodel.Theapplicationofthismodeltobothhypotheticandrealbinaryliquidalloysdemonstratedthatthechemicalshort-rangeorderalwaysdecreasedtheconfigurationalentropyofmixing,whereascomplicatedbehaviorwasfoundwiththeatomicsizeeffect.Theconfigurationalentropyofmixingincreasedwhenthelargeratomsenteredintothematrixofthesmalleratoms,whereasitdecreasedwhenthesmalleratomsweremixedintothematrixofthelargeratoms.Themaximumoftheconfigurationalentropyofmixingwasnotlocatedattheeutecticcompositioninthesealloys.KeyWords:

Liquidalloy;Short-rangeorder;Configurationalentropyofmixing

theconfigurationalentropyofmixingwiththeexcessentropyofmixingtobeparameterizedtofittheexperimentalinforma-tion.

Sincethesimpleliquidalloyscanbeviewedasclosepack-ingofhardspheres,someresearchers7-9developedamodelfortheentropyofmixinginliquidbinaryalloysfromthethermo-dynamicaspect.Insuchamodel,theentropyofmixingin-cludesfourpartswhichareresultedfromtheidealgas,thecon-

1Introduction

Theentropyandenthalpyofmixingareimportanttothema-terialresearch,suchasthecalculationofthephasediagram,1-3thepredictionoftheglassformingabilityofanalloy,4andthedesignationofthehigh-entropyalloys,5etc.Theenthalpyofmixingiswellmodeledbythequacichemicaltheory;however,theentropyofmixingisdifficulttodealwith.6Sofartheequi-atomicrandommixtureisstillthebestmodeltoaccountfor

Received:March31,2012;Revised:May4,2012;PublishedonWeb:May4,2012.∗

Correspondingauthor.Email:qinjy@sdu.edu.cn;Tel:+86-531-88392810.

TheprojectwassupportedbytheNationalNaturalScienceFoundationofChina(50971082).国家自然科学基金(50971082)资助项目

ⒸEditorialofficeofActaPhysico-ChimicaSinica

No.7QINJing-Yu:ANewModelfortheConfigurationalEntropyofMixinginLiquidAlloysBasedonShort-RangeOrder1587

figurationalentropyofmixingfromtheequiatomicrandommixture(hereafternamedidealmixingentropy),thesizeeffect,andthepackingfraction.Apartfromtheseterms,Hafner10sug-gestedthatforsomerealisticliquidalloysthecontributionfromelectrontotheentropyofmixingmustbetakenintocon-sideration.Infact,thestructureofmostrealisticliquidalloysismoresophisticatedthantheclosepackingofhardspheres.Boththetopologicalshort-rangeorder(TSRO)andchemicalshort-rangeorder(CSRO)areoftenthekeyfeaturesinmanyliquidalloys.Inordertotakethesizeeffectintotheentropyofmixing,Flory11modifiedtheformulaofidealmixingentropy.InordertoconsidertheCSRO,BhatiaandHergrove12devel-opedthemodelofassociatesinliquidalloyswhichdependsonseveraladjustableparameters.However,thesocalledassoci-atesofcompoundsaredifficulttobedescribedintheirsize,distributionoreventheirstoichiometriccompositionbyeitherradiationexperiments13ormoleculardynamicssimulations.14Infact,theCSROwhichisthestatisticalinformationoftheliquidalloyisdifferentfromtheassociateswhichhavethespecificcomposition.WitusiewiczandSommer15triedtocorrelatetheentropyofmixingtothemixingenthalpyinviewoftheCSROinliquidalloys.

Insolidalloys,theconfigurationalentropyofmixinghasbeenwellformulatedbythesublatticemodels.Recentlymanyresearchesonthermodynamicsofmaterialsemployedtheclus-tervariationmethod(CVM)16-18orcluster/siteapproximation1toevaluatetheconfigurationalentropyofmixing.Theexactconfigurationalentropyofmixingforsolidsolutionsorstoi-chiometriccompoundsisgenerallyreachedthroughverycom-plicatedprocedures.TheinfluenceofCSROontheconfigura-tionalentropyofmixingisoftendeterminedbythevariationofthefreeenergyontheinteractionparameterstoreachthemini-mum.Asfarasweknow,thereisalsoamodelontheconfigu-rationalentropyofmixingforcommonsolutions,19whichisinanalogywiththesublatticemodels.

Itistechnologicallyimportanttodevelopmodelsonthecon-figurationalentropyofmixingwhicharenotonlybeyondtheidealmixingentropybutalsomoresimplethanCVM.Asim-pleexpressionfortheconfigurationalentropyofmixingofmulticomponentinterstitialsolidsolutionsisrecentlydeducedbasedonaprobabilisticapproach.20However,duetothelackofsuchapplicablemodels,inthedesignofhighentropyalloys5theidealmixingentropyisstillemployed,whichrestrictsourunderstandingontheroleofchemicalandtopologicalshort-rangeorderonthepropertiesofthehighentropyalloys.

Researchersarefacingevenworseprobleminbulkmetallicglasses(BMGs)thaninhighentropyalloys.Duringthedevel-opmentofBMGs,someempiricalrulessuchastheBMGsfa-voringthemulticomponentalloysandlargeratomicsizediffer-enceamongthecomponentshavebeenveryhelpful.10,21,22Inor-dertoquantitativelyevaluatetheglassformingability,somere-searchers4triedtoincorporatetheseempiricalrulesintotheconfigurationalentropyofmixingbyEq.(1)11,16

Sm

m

mix=-kB∑xjln(xjVj/∑xiVi)

(1)

j=1

i=1

whereViandxiaretheatomicvolumeandatomiccompositionoftheithelementinthealloy,respectively,kBistheBoltzmannconstant,misthenumberoftheconstituents.Ifinabinaryal-loyV1=V2,thenEq.(1)becomestheidealmixingentropy.OnceV1≠V2theSmixfromEq.(1)isalwayslargerthantheidealmix-ingentropyatanycomposition.Thisunfortunatelymakestheideawidelyacceptedthatfromthethermodynamicpointofviewboththemultiplecomponentsandthelargersizediffer-enceamongtheconstituteelementsenhancetheconfiguration-alentropyofmixing,andhencereducetheGibbsfreeenergy.Thismisunderstandingfurtherblocksthediscoveryofthephys-icalnatureontheglassformingmechanismoutoftheempiri-calrules.Regardlessofthesizeeffect,itistruethatmulti-componentalloyingincreasesSmixwhosemaximumisreachedwheneachxjequalsto1/m.Inregardtotheatomicsizediffer-ence,however,wefinditisphysicallyincorrectinthatifoneoftheViʹstendstozeroSmixapproachestoinfinity.Forinstance,supposedthatweaddintooneliquidalloyanewcomponentwhosexjissmallenoughsoasnottomodifytheoriginalcom-positions,anditsatomicsizetendstozerosothatnottomodi-fytheoriginalvolumeeither,bydoingsotheoriginalliquidal-loydoesnotchangeinfact,butaccordingtoEq.(1)Smixbe-comesarbitrarilylarge,sothattheliquidalloybecomesinfi-nitelystable.Thisisanantinomy.Inordertoavoidtheperplex-itywemustabandontheformulaofEq.(1)whentakingtheatomicsizedifferenceintoconsideration,andneednewmod-elsbeyondtheidealmixingentropy.

TheCVMneglectedthesizeeffectofdifferentlatticesites,whilethemodel20ontheconfigurationalentropyofmixingofinterstitialsolidsolutionswasrestrictedtotheextremeoflargesizedifferencewiththeshort-rangeorderasanempiricallyad-justableparameter.Inaliquidalloyorasolidsolution,thechemicalinteractionandthesizeeffectareconvoluted,andbotharereflectedontothecoordinationnumbersandthepar-tialcorrelationfunctions.Sincealiquidalloyatequilibriummustbethestatewiththeminimumfreeenergy,theshort-rangeordersfromaliquidatequilibriumshouldgovernthecombinationalbehavioroftheatomswithoutthehelpofboththespecificknowledgeonthemixingenthalpyandtheregularnestedsublattices.

WithoutoptimizationontheSROjustlikeinCVMandoth-ermodels,therealisticCSROandTSROcanbeobtainedbyra-diationmeans13ormoleculardynamicssimulations.14BoththeCSROandtheTSROarenecessitiesindeterminingtheconfig-urationalentropyofmixing.Forabinaryalloy,theidealmix-ingentropyshouldhavethelowerlimitof0andtheupperlim-itofRln2(Rbeingthegasconstant)withonemoleofmateri-als.IftheCSROexists,thestructuralorderofthealloyshouldincrease,andthentheconfigurationalentropyofmixingmustdecrease,anyhow,themixingentropycannotbenegative.ThecoordinationnumberwhichisanimportantparameterinTSRO

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relatestotheatomicsizewhosedifferencebetweentheconsti-tutionalelementsoftenresultsinpositiveexcessentropyupontheidealmixingentropyinthealreadyexistedmodels.

Inthiswork,wefirstlywriteoutthecombinationalfactorsofaliquidalloybyconsideringbothCSROandTSRO,andthendeducetheconfigurationalentropyofmixingaccordingtotheBoltzmannrelationship.SeveralartificialliquidalloyswithassumedCSROandTSROareemployedtoshowtheuniquefeaturesofthismodel,andafterthat,themodelisappliedtofourrealisticliquidalloys.

2Modelandmethod

WestartfromalatticeoccupiedbyN1andN2atomsofabi-naryalloy,thesubscriptsmeantheatomtype.WehaveN=N1+N2(2)N1NN+2

N=x1+x2=1(3)whereNisthetotalnumberofatoms,xj(j=1,2)isthealloycomposition.

Atomsoftype1andtype2mayhavedifferentsizes,sotheyhavedifferentcoordinationnumbersnamedasn1andn2,re-spectively.Aroundanatomoftype1,therearen1f11atomsoftype1andn1f12atomsoftype2.Inanalogy,aroundanatomoftype2,therearen2f22atomsoftype2,andn2f21atomsoftype1,respectively.Sowehavethefollowingrelationship:f11+f12=f21+f22=1(4)wherefij(i,j=1,2)representsthechemicalenvironmentaroundatomoftypei.ThiskindofparametershasbeendeterminedbythevariationmethodontheGibbsfreeenergyorbysomespe-cificassumptionsinotherresearchworks.23Inourwork,theyarethecharactersofthepartialstructureinformationofliquidalloysfrommoleculardynamicssimulationsorradiationexper-iments.Itshouldbepointedoutthatfij≠xj(orfij=xj)meansthatthereis(ornot)CSROintheliquidalloys.

IftheCSROexists,toreachallthepossibleconfigurations,wemusttrytoarrange(N1n1f11+N2n2f21)atomsoftype1and(N1n1f12+N2n2f22)atomsoftype2toformabinarymixtureotherthantoarrangeN1andN2atoms.Sinceourstartpointisthesta-tisticalinformationfijsowethinkthattheinterchangeofalltheatomsoftypeiaroundtheatomsofthesametypeidoesnotcontributetotheconfiguration,sodotheatomsofthesametypeiaroundatomsoftypej.Howevertheinterchangeofiat-omsbetweenniandnjwillresultindifferentconfigurations.ThenthecombinationfactorisZ=Z1Z2(5)whereZ1andZ2arethecombinationfactorsofatomsoftype1andtype2,respectively,and

Z(N1n1f11+N2n2f21)!1=(N(6)

1n1f11)!(N2n2f21)!ZN1n1f12+N2n2f22)!2=

((N1n1f(7)

12)!(N2n2f22)!SowhensubstitutingEqs.(6,7)intoEq.(5),weget

Z=

(N1n1f11+N2n2f21)!(N1n1f12+N2n2f22)!

(N)!(Nf(8)

1n1f112n2f21)!(N1n112)!(N2n2f22)!Then,accordingtotheBoltzmannrelationshiptheconfigura-tionalentropyofmixingshouldbeSmix=kBlnZ(9)

Accordingtothesterlingformula,24,25

lnN!=NlnN-N,substitut-ingEq.(8)intoEq.(9),wegetatlast

SéêëNæN2n2f21öæ

N2n2f22ömix=kB1n1f11lnçè1+N÷+N1n1f11ø1n1f12lnçè1+N÷+

1n1f12øNfæN1n1f12öæ

N1n1f11öù2n222lnçè1+N÷+N2n2f22ø2n2f21lnçè

1+N÷2n2f21øúû(10)

Further,wecansimplifyEq.(10)intoEq.(11)byusingEqs.(2,3)

S=kéêæn2x2f21ön2x2f22öënlnçè1+n÷+næ

mixB1x1f111x1f12lnçè1+÷+

1x1f11øn1x1f12ønæn1x1f12öæ

n1x1f11öù2x2f22lnçè1+n÷+n2x2f22ø2x2f21lnçè

1+n÷2x2f21øúû(11)

Eq.(11)isnowthemoleconfigurationalentropyofmixingofthesystemcomposedofn1N1+n2N2atoms.Accordingtothead-ditivityofentropy,theoriginalvaluewithN1+N2atomsshouldbereducedbyafactorofx1n1+x2n2.Thentheconfigurationalen-tropyofmixingoftheoriginalsystemis

Skén2x2mix=Bxênæ

çf21ö1x1f11lnè

1+

÷+nxfln(1+1n1+x2n2ën1x1f11ø1112n2x2f22önx÷+næ

n1x1f12ö2x2f22lnçè

1+÷+

11f12øn2x2f22ønæn1x1f11öù2x2f21lnçè

1+n÷2x2f21øúû(12)Inthecompactform,itbecomes

éSkêæmix=Bm

êçç∑mnöù

lxlflk÷úl=1÷∑m∑nênjxjfjklnçççn÷ú(13)jxjfjk÷ixij,k=1ê÷úë

çè÷i=1øúûwheretheupperboundofeachsummationmis2forabinaryalloy.IfweanalyzeaternaryalloythroughtheprocedurefromEq.(2)to(13)again,wefindthatthemixingentropyhasthesameformulaasEq.(13),withmbeingthenumberofconstitu-entsofamulticomponentalloy.WethensuggestthatformulaofEq.(13)isapplicableformulticomponentalloys.Wewillil-lustratesomepropertiesofEq.(13)inthefollowing.Forsim-plicity,theupperboundofsummationmisomitted,andtherepresentationofthelowerboundofsummationisalsomodi-fied.

3Resultsanddiscussion

Inanequiatomicrandommixtureofamulticomponental-loy,thereisnoCSRO,sofij=xj,ni=nj=n,thenEq.(13)becomes

éS∑kêæç∑nxlflköùmix=B

nx∑êênxjfjklnççl÷÷ú÷ú

ij,kë

çnxèjfjk÷iøúûNo.7QINJing-Yu:ANewModelfortheConfigurationalEntropyofMixinginLiquidAlloysBasedonShort-RangeOrder1589

=∑kéB

xi∑êæêxç∑xlxköùjxklnçj,kçl÷úx÷jxk÷úiêë

çè÷øúû=kB∑xk∑xjln1x=-kB∑xjlnx(14)

kjj

j

j

DuringthedeductionofEq.(14),Eq.(3)isappliedforamulti-componentalloy.Eq.(14)isjusttheexpectedformulafoundin

manychemicalphysicstextbooks24,25fortheconfigurationalentropyofmixingofamulticomponentrandomsolution.Thismeansthattheequiatomicrandommixturemodelisonespe-cialcaseofourmodel.

InordertounderstandtheeffectofCSROontheconfigura-tionalentropyofmixing,weconstructanartificialbinaryalloyinwhichtheCSROisthestrongestattheequiatomiccomposi-tion,ormaybetheshort-rangeorderhasevolvedintothelong-rangeorder.TheB2typestructureunitsareassumedasthelo-calstructuretoaccountfortheCSROinsuchalloywhoseconstitutionalelementshavethesamesize.Then,theyhavethesamecoordinationnumbers,i.e.,n1=n2=n.Inparticular,theCSROinthisalloyissostrongthatonetypeofatomsaremost-lysurroundedonlybytheotherkindofatomsattheequiatom-iccomposition.Wethinkthatthefollowingfijisconvenientforthismodelalloy,f11=1-2x2,f12=2x2,f21=1,andf22=0,forx2<0.5;whileforx2>0.5,f11=0,f12=1,f21=2-2x2,andf22=2x2-1.Atx2=0.5,thealloyhasaB2typestructure,andtheconfigurationalentropyofmixingisexpectedtobezero.Inthefollowing,SmixisscaledbykBforconvenienceandhencethetermSmix/kBisdi-mensionless.

SubstitutingtheaboveparametersintoEq.(13),wehave

S/kx1-2x1(1-2x2)ln1xmixB=2

x-2x)+

1(12x1-2x2ln1x2

x,(x2<0.5)(15)

2

Smix/kB=0,(x2=0.5)(16)Sx-x2x2mix/kB=x1ln

12+2

x+

1

(2x2

x2-x2)ln1-x2+2x222x2,(x2>0.5)2-x(17)

2

Fig.1showstheconfigurationalentropyofmixingofthis

Fig.1

MixingentropyforthealloywithB2typelocalstructure

andtheequiatomicrandommixtureFig.2

Parametersoffijfortheassumedunlike-atom

coordinationtendency

artificialalloyaccordingtoEqs.(15-17).ItisshownthattheCSROreducestheentropyinthecompositionrangeabout0.08IftheCSROisnottoostrongtoformstoichiometriccom-pounds,butisstrongenoughtohaveunlike-atompreferentialcoordination,insuchcaseweemploytheparametersasf12=x2+0.8[0.25-(x2-0.5)2]≥x2,f11=1-f12≤x1,f21=x1+0.8[0.25-(x1-0.5)2]≥x1,andf22=1-f21≤x2.TheseparametersoffijareshowninFig.2.Theconfigurationalentropyofmixingforsuchcaseisplot-tedinFig.3.Itisfoundthattheconfigurationalentropyofmix-ingofsuchmodelalloyislowerthantheidealmixingentropy.Ifthealloyshavetheweaklike-atomcoordinationtendency,theparametersoffijcouldbeassumedasf22=x2+0.8[0.25-(x2-0.5)2]≥x2,f21=1-f22≤x1,f11=x1+0.8[0.25-(x1-0.5)2]≥x1,andf12=1-f11≤x2.Forsuchkindofalloys,theconfigurationalentro-pyofmixingisalsolowerthantheidealmixingentropyas

Fig.3Configurationalentropyofmixingofthemodelalloys

whichhaveunlikeorlikeatomcoordinationtendency

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showninFig.3.ThissurveysuggeststhatiftheCSROexists,thentheconfigurationalentropyofmixingmustbelowerthantheidealmixingentropyinthecompositionrangeof0.10.9,theequiatomicrandommixtureisalwaysagoodapproximationasdemonstratedbytheinsertofFig.3.Now,weturntotheeffectoftheatomicsize.Inthiscase,theCSROisneglected,firstly.Forsuchanalloy,thereisarela-tionshipoffij=xj,butthecoordinationnumbersfordifferentcomponentsdonotequaltoeachotheringeneral.SoEq.(13)changesinto

Smix/kB=ln(∑xini)-∑[xiniln(xini)]/∑xini

(18)

i

i

i

Eq.(18)isapplicabletomulticomponentalloys.Forabinaryal-loy,wesetn1=n,andn2=yn,inwhichyisarealnumberwithy=1meaningnosizeeffect.ThenEq.(18)isreformedas

Smin/kB=ln(x1+yx2)-[x1lnx1+yx2ln(yx2)]/(x1+yx2)(19)WecangetthefirstandsecondorderderivativesofEq.(19)ony,andfindthataty=x1/x2orx2=1/(1+y),Smix/kBhasthemaxi-mumofln2.Fig.4showsthatwhenychangesfrom0.1to10,theprofileofSmix/kBagainstcompositionchangessignifcantly.Ify=1,theconfigurationalentropyofmixingofthemodelal-loybecomestheidealmixingentropywhosemaximumisatx2=0.5,otherwise,theconfigurationalentropyofmixingdeviatesfromtheidealmixingentropyanditsmaximumshifttox2=1/(1+y).Inthemodelsdevelopedbyotherresearchers,theexcessentropyofmixingduetothesizeeffectisalwayspositiveandaddedontotheidealmixingentropy.However,ourmodelpre-dictsthatwhensmalleratomsenterintothematrixofthelageratoms,thesizeeffectmakestheentropyofmixinglowerthantheidealentropyofmixing,andviceversa.

Fig.4alsoshowsthatEq.(19)hastheuplimitofln2whatev-erthex1andyare.Thisisanotherfeaturethatcannotbefoundintheothermodels,whichisresultedfromthefactthatinourmodeltheCSROandtheTSROarecoherentduringmixing.Inthefollowingparts,wewillapplyourmodelofconfigura-tionalentropyofmixingonseveralgroupsofrealisticalloysagainstcomposition.Intheseliquidalloysalltheparametersof

Fig.4

Effectofatomicsizedifferenceontheconfigurational

entropyofmixing

x1:theatomicfractionofcomponent1inabinaryliquidalloy

Fig.5ConfigurationalentropyofmixinginliquidCu-Nialloys

xCu:theatomicfractionofCuinliquidCu-Nialloy

fijandniarecalculatedbyabinitiomoleculardynamicssimula-tion,anddetailsofoursimulationcanbereferredtoreferenc-es.26,27Accordingtothedifferentcharactersoftheirphasedia-grams,liquidCu-Ni,Pb-Sn,Ni-Zr,andFe-Sialloysareselected.ThephasediagramofliquidCu-Nialloysisatypicalmisci-blealloysystem.Accordingtothepaircorrelationfunctions,thediametersofNiandCuatomsintheliquidstateare0.242and0.245nm,respectively,sothesizeeffectontheentropyofmixingcouldbeneglected.Fig.5showsthattheprofileofcon-figurationalentropyofmixingofCu-Nialloysystemresem-blestheidealmixingentropyverywell.ThismeansthattheconfigurationalentropyofmixingoftheliquidCu-Nialloysiswelldescribedbytheidealentropyofmixing.

Fig.6showstheshapeoftheconfigurationalentropyofmix-inginliquidPb-Snalloyswhichbelongstoasimpleeutectical-loysystem.WhenxPb>0.8,theconfigurationalentropyofmix-ingissmallerthantheidealentropyofmixingduetothesizeeffect.Sincefromtheirliquidstructure,thesizesofPbandSnatomsarefoundtobe0.327and0.308nm,respectively.Theotherpartsofthecurveoftheconfigurationalentropyofmix-inginliquidPb-Snalloysarewellfittedbytheidealmixingen-tropy.

TheconfigurationalentropyofmixinginliquidNi-ZralloysisplottedinFig.7.ItisfoundthatintheliquidNi-Zralloys,

Fig.6ConfigurationalentropyofmixinginliquidPb-Snalloys

xPb:theatomicfractionofPbinliquidPb-Snalloy

No.7QINJing-Yu:ANewModelfortheConfigurationalEntropyofMixinginLiquidAlloysBasedonShort-RangeOrder1591

Fig.7ConfigurationalentropyofmixinginliquidNi-Zralloys

xNi:theatomicfractionofNiinliquidNi-Zralloy

thesizesofNiandZratomsare0.251and0.337nm,respec-tively.TheZratomislargerthantheNiatomby34.3%.Sup-posedthatthereisnoCSROinliquidNi-Zralloysi.e.,fij=xj.Withthecoordinationnumbersfromabinitiomoleculardy-namicssimulationbeingadaptedtoEq.(13),theprofileoftheconfigurationalentropyofmixingisalsoplottedinFig.7.BycomparisonofthecurveswithandwithoutCSRO,wefindthattheconfigurationalentropyofmixinginliquidNi-ZralloysispredominantlydeterminedbythelargesizedifferencebetweenNiandZratoms.ThecontributionofCSROonlylowersthecurvewithoutCSROslightlyaroundabout65%Ni.

InFe-Sialloysystem,therearestoichiometriccompoundsformedinthesolidstatebythetransitionmetalandthesemi-conductoratoms.Fromthepaircorrelationfunctions,thediam-etersofFeandSiatomsarebotharound0.258nm.Thereisnosizedifferenceseemingly;however,itmustberemindedthatthecoordinationnumbersmaybedifferentintheSi-richalloysandtheFe-richalloysduetothedifferentnatureoftheliquidstructure.27

TheCSRObetweenFeandSiatomsisverystrongwhenFecontentisbeyond0.5,27sothecurveofconfigurationalentropyofmixingislowerthantheidealmixingentropyasshowninFig.8.

Byapplyingourmodelontheseliquidalloys,itisfoundthat

Fig.8ConfigurationalentropyofmixinginliquidFe-Sialloys

xFe:theatomicfractionofFeinliquidFe-Sialloy

theCSROandthedifferenceoftheatomicsizebothcontributetotheconfigurationalentropyofmixing.WhenthedifferenceoftheatomicsizeorthecoordinationnumberinEq.(13)isnottoolargeandtheCSROisnottoostrong,theconfigurationalentropyofmixinginliquidalloyscanbewellmodeledbythemixingidealentropy.Whenthedifferenceoftheatomicsizeisverylarge,suchasintheliquidNi-Zralloys,itisthedominantfactorindeterminingtheprofileoftheconfigurationalentropyofmixingintheliquidalloys.

Attheeutecticpoints,theliquidalloysaremorestablethanthealloysatothercompositions.WehaveexpectedthatwhentheCSROandTSROaretakenintoconsiderationthereshouldbesomekindofindicationofeutecticalloyontheconfigura-tionalentropyofmixinginliquidalloyagainstcomposition.However,inthecurvesofconfigurationalentropyofmixinginliquidPb-Snalloys,Ni-Zralloys,andFe-Sialloys,noextraor-dinarybehaviorisperceivedattheeutecticcompositionsofthesealloys.Iftheconfigurationalentropyofmixingisthemainpartintheentropyofmixing,wesuggestthatthemixingenthalpymaytakethedominantpositionintheGibbsfreeener-gyofmixingintheliquideutecticalloys.Thisisstillopenandwillbepursuedfurtherinthefuture.

ThoughtemperatureisnotanexplicitparameterinEq.(13),itcanindirectlyinfluencetheconfigurationalentropyofmix-ingbychangingthefijandni.Forinstance,whentemperatureincreases,fijapproachesxj,andEq.(13)willapproachtoEq.(14).

ThismodeldependsontherealisticSROparameters.Byus-ingthewordrealisticwemeanthattheseparameterscande-scribetheliquidalloysatequilibriumwithoutadditionalas-sumptionsonthemixingenthalpy,andthenthereisnoneedofvariationonfreeenergylikeinBragg-Williamsmodel.Itcanalsobeappliedonthemetastablealloyssuchastheamorphousandsupercooledliquidalloys,oncetheSROparametersarereached.ComparedwithCVM,thelimitationofourmodelliesinthatitdoesnotincludethelong-rangeorderparameters,soonemustbecautioustoapplyittoorder-disordertransitioninsolidsolution.

4Conclusions

BytakingtheCSROandTSROintoconsideration,amodelfortheconfigurationalentropyofmixinginliquidalloysisde-veloped.Wedemonstratethattheequiatomicrandommixturemodelisnaturallyincludedinthismodel,andthedifferenceofatomicsizeandtheCSROcontributetotheconfigurationalen-tropyofmixingcoherently.TheexistenceofCSROalwayslowerstheconfigurationalentropyofmixing;whilethediffer-enceofatomicsizeinfluencestheconfigurationalentropyofmixinginacomplicatedmanner.Itlowerstheconfigurationalentropyofmixingwhenthealloyisrichinthelargeratoms,butenhancestheconfigurationalentropyofmixingwhenthealloyisrichinthesmalleratoms.Themaximumisnotlocated

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attheeutecticcompositioninthesealloys.Wealsosuggestthatthismodelmaybeappliedtonon-crystallinesolidsandsol-idsolutions.References

(1)Oates,W.A.;Zhang,F.;Chen,S.L.;Chang,Y.A.Phys.Rev.B1999,59,11221.doi:10.1103/PhysRevB.59.11221

(2)Johari,G.P.J.Chem.Phys.2010,132,124509.doi:10.1063/1.3364999

(3)Liu,Z.K.J.PhaseEquilib.Diffus.2009,30,517.doi:10.1007/s11669-009-9570-6

(4)

Jiang,Q.;Chi,B.Q.;Li,J.C.Appl.Phys.Lett.2003,82,2984.doi:10.1063/1.1571984

(5)Yeh,J.W.;Chen,S.K.;Lin,S.J.;Gan,J.Y.;Chin,T.S.;Shun,

T.T.;Tsau,C.H.;Chang,S.Y.Adv.Eng.Mater.2004,6,299.doi:10.1002/adem.200300567

(6)Wu,X.W.;Shi,J.ActaPhys.-Chim.Sin.1993,9,740.

[吴雄

武,时均.物理化学学报,1993,9,740.]doi:10.3866/PKU.

WHXB19930605

(7)Umar,H.;Meyer,A.;Watabe,M.;Young,W.H.J.Phys.F:Met.Phys.1974,4,1691.doi:10.1088/0305-4608/4/10/016(8)Hoshino,K.J.Phys.F:Met.Phys.1980,10,2157.doi:10.1088/0305-4608/10/10/013

(9)Singh,P.;Khanna,K.N.Pramana1984,23,511.doi:10.1007/BF02846627

(10)Hafner,J.Phys.Rev.A1977,16,351.doi:10.1103/PhysRevA.16.351

(11)Flory,P.J.J.Chem.Phys.1942,10,51.doi:10.1063/1.1723621(12)

Bhatia,B.;Hergrove,W.H.Phys.Rev.B1974,10,316.(13)Teng,X.Y.;Ye,Y.F.;Shi,Z.Q.;Wang,H.R.;Qin,J.Y.Acta

Phys.-Chim.Sin.2002,18,336.[滕新营,叶以富,石志强,

王焕荣,秦敬玉.物理化学学报,2002,18,336.]doi:10.3866/

PKU.WHXB20020410

(14)

Liu,R.S.;Liu,F.X.;Dong,K.J.;Zheng,C.X.;Liu,H.R.;Peng,P.;Li,J.Y.ActaPhys.-Chim.Sin.2004,20,1093.[刘让

苏,刘凤翔,董科军,郑采星,刘海蓉,彭

平,李基永.物理化

学学报,2004,20,1093.]doi:10.3866/PKU.WHXB20040907(15)Witusiewicz,V.T.;Sommer,F.J.Alloy.Compd.2000,312,228.

doi:10.1016/S0925-8388(00)01158-0(16)Oates,W.A.J.PhaseEquilib.Diffus.2007,28,79.doi:10.1007/s11669-006-9008-3

(17)Kikuchi,R.Phys.Rev.1951,81,988.doi:10.1103/PhysRev.81.988

(18)

Hao,S.M.J.Mater.Metall.2003,2,286.[郝士明.材料与冶

金学报,2003,2,286.]

(19)Abrams,D.S.;Prausnitz,J.M.AIChEJ.1975,21,116.doi:

10.1002/aic.690210115(20)Garcés,J.Appl.Phys.Lett.2010,96,161904.doi:10.1063/1.3400221

(21)

Inoue,A.;Zhang,T.;Masumoto,T.J.Non-Cryst.Solids1993,156/158,473.

(22)Waseda,Y.;Chen,H.S.;Jacob,K.T.;Shibata,H.Sci.Technol.

Adv.Mater.2008,9,023003.doi:10.1088/1468-6996/9/2/023003(23)Jiang,M.;Li,H.X.;Hao,S.M.J.Mater.Metall.2010,9,193.[蒋敏,李洪晓,郝士明.材料与冶金学报,2010,9,193.](24)Porter,D.A.;Easterling,K.E.PhaseTransformationsinMetalsandAlloys,2nded.;ChapmanandHall:London,1992.(25)

Xu,Z.Y.ThermodynamicsofMaterials,4thed.;HigherEducationPress:Beijing,2009.[徐祖耀.材料热力学.

第四版.北京:高等教育出版社,2009.]

(26)

Qin,J.Y.;Liu,H.;Bian,X.F.;Gu,T.K.J.Phys.:Condens.Matter2009,21,155106.doi:10.1088/0953-8984/21/15/155106(27)

Gu,T.K.;Qin,J.Y.;Xu,C.Y.;Bian,X.F.Phys.Rev.B2004,70,144204.doi:10.1103/PhysRevB.70.144204