反应产物形核多步热力学补充材料.docx
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反应产物形核多步热力学补充材料.docx
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反应产物形核多步热力学补充材料
SUPPLEMENTARYINFORMATION
EvidenceofMulti-stepNucleationLeadingtoVariousCrystallizationPathwaysfromanFe-O-AlMelt
G.C.Wang1,2,3*,Q.Wang1,2*,S.L.Li1,2,X.G.Ai1,2andC.G.Fan3
1KeyLaboratoryofChemicalMetallurgyEngineering,LiaoningProvince,UniversityofScienceandTechnologyLiaoning,Anshan,Liaoning,114051,China
2SchoolofMaterialsandMetallurgy,UniversityofScienceandTechnologyLiaoning,Anshan,114051,China
3JiangxiUniversityofScienceandTechnology,Ganzhou,341000,China
*Correspondingauthor:
Tel.:
+86-412-5929573
E–mailaddress:
wang_guocheng@;wangqi8822@
Ⅰ.SimulationMethodstoCalculateThermodynamicPropertiesUsingDensityFunctionalTheory
Allthesimulationsforcalculatingthermodynamicpropertiesinthepresentworkareperformedusingdensityfunctionaltheory(DFT)withthemolecularorbitaltheorycomputationalprogramDmol3,amoduleofthecommercialsoftwareMaterialsStudio(MS)6.0.TheinitialstructuresareestablishedusingtheVisualizermoduleofMS6.0.ThegeometryoptimizationisperformedusingtheBFGS(theabbreviationcorrespondstothefirstlettersofthenamesofthefollowingresearchers:
Broyden,Fletcher,Goldfarb,andShanno)method41basedonaquasi-Newtonalgorithm.ThehybriddensityfunctionalBLYP(theabbreviationcorrespondstothefirstlettersofthenamesofthefollowingresearchers:
Becke,Lee,YangandParr)method42,43usingthegeneralizedgradientapproximation(GGA)isusedastheexchange-correlationpotentialfunction.Thethermodynamicpropertiesofvariousstructuresarecalculatedusingtheatomicharmonicvibrationalfrequency.Thevibrationalfrequenciesarecomputedbydiagonalizingthemass-weightedsecond-derivativematrix,F42,whichisgivenby
(S1)
whereqiandqjrepresenttwoCartesiancoordinatesofatomsiandj,respectively,andmiandmjarethemassesoftheatoms.ThesquarerootsoftheeigenvaluesofFijarethevibrationalfrequencies.Thesecondderivativesarecomputedasthefinitedifferencesofthefirstderivatives.Thegradientsarecomputedatdisplacedgeometries,andthesecondderivativesarecomputednumericallyusingtwo-point(orcentral)differencing.Displacementsaretakentoobtainthesecondderivatives
(S2)
wherethetwotermsrepresenttheanalyticderivativesattheequilibriumgeometryandatageometrywithcoordinateqjdisplacedbydistanceΔ.
Therefore,theheatcapacity(CP)iscomputedusingthevibrationalfrequencies43:
(S3)
where
and
aretheheatcapacitiesoftranslation,rotationandvibration,respectively;Ristheidealgasconstant(8.314J·mol-1K-1);kistheBoltzmannconstant;hisPlanck'sconstant;Tistheabsolutetemperature;andνiisthevibrationalfrequency.
Theenthalpy(H)isgivenby45
(S4)
where
and
aretheenthalpiesoftranslation,rotationandvibration,respectively.
Theentropy(S)45isgivenby
(S5)
where
and
aretheentropiesoftranslation,rotationandvibration,respectively;wisthemolecularmass;pisthepressure;σisthesymmetrynumber;cisthemolarconcentrationofthemolecules;andIA(B,C)isthemomentofinertia.
Thevibrationalfreeenergy(GV)iscalculatedusingtheexpression45
GV=H–T·S(S6)
Ⅱ.DetailsoftheDmol3SimulationoftheAluminaClusters
Theinitialstructuresofthealuminaclusters,(Al2O3)nwithn=1-10,15and30,areestablishedusingVisualizer.TheprecisionoftheBFGSalgorithmissetasfollows:
energy≤2.0×10-5Ha,tension≤0.004Ha/Å,andshift≤0.005Å.Theself-consistentfield(SCF)methodisusedwiththeprecisionofthetotalenergyandchargedensitysetat1×10-5Ha,andthethermalsmearingeffectisusedataprecisionof0.005Ha.Thecut-offradiusoftheDNPbasissetofthedorbitalis3.5Å.Electronsoutsidetheatomicnucleusarehandledusingtheeffectivecorepotentials(ECP)method46,47.
Ⅲ.DetailsoftheDmol3Simulationoftheα-Al2O3Crystal
Theinitialstructureoftheα-Al2O3crystalisadoptedfromtheMS6.0structuraldatabase.TheprecisionoftheBFGSalgorithmissetasfollows:
energy≤2.0×10-5Ha,tension≤0.004Ha/Å,andshift≤0.005Å.TheBrillouinzoneintegraliscalculatedusingtheMonkhorst-Pack48,49methodwithagridsizeof3×3×2k-points.Theprecisionofthetotalenergyandchargedensityissetas1×10-5Ha,andthethermalsmearingeffectisusedataprecisionof0.055Ha.Thecut-offradiusoftheDNPbasissetofthedorbitalis3.5Å.ElectronsoutsidetheatomicnucleusarehandledusingtheECPmethod46,47.
Ⅳ.Structuresof(Al2O3)n
Thestablestructuresof(Al2O3)nwithn=1-10,15,and30(FigureS1a)areingoodagreementwiththosereportedinpreviousstudies36,37.(Al2O3)1isalinearstructure.Thesmallclusters[(Al2O3)nwithn=2-5]arecagestructureswithhighsymmetries.Inthesmallclusters(n=1-5),theAlandOatomsaretriple-anddouble-coordinated,respectively,andarelinkedtoneighborsviasinglebonds.Thelarger(Al2O3)nclusterswithn=6-10and15arebothcageandcage-dimerstructures.Thelargestcluster,(Al2O3)30,isashellstructuresimilartofullerene.Tocharacterizethesizeof(Al2O3)n,wedefinethedistancebetweentwoatomsthatisthelargestdistancebetweenanytwoatomsinaclusterastheapparentsize(FigureS1b)ofthiscluster.Thebondlengths,apparentsizesandsymmetriesofthe(Al2O3)nclustersareshowninTableS1.
TableS1Thebondlengths,apparentsizesandsymmetriesof(Al2O3)n
(Al2O3)n
dAl-O/Å
Dmax/nm
Group
(Al2O3)1
1.602,1.692
0.658
Dh
(Al2O3)2
1.737
0.410
C3
(Al2O3)3
1.715
0.444
D3h
(Al2O3)4
1.712
0.588
Oh
(Al2O3)5
1.707
0.626
C5h
(Al2O3)6
1.706
0.830
C2
(Al2O3)7
1.705
0.766
C3h
(Al2O3)8
1.702
0.779
S8
(Al2O3)9
1.700
0.808
Cs
(Al2O3)10
1.700
0.952
C5
(Al2O3)15
1.698
1.212
C2v
(Al2O3)30
1.696
1.690
Ih
FigureS1Thestructuresandapparentsizesof(Al2O3)n.(a)Thestablestructures.(b)Theapparentsizes.
Ⅴ.ThermodynamicPropertiesof(Al2O3)n
Thethermodynamicpropertiesof(Al2O3)ninthetemperaturerangefrom0-1000KareshowninFigureS2.TheCP,HandSof(Al2O3)nincreaseasthetemperatureincreases(FiguresS2a,b,c),whereasGVandG(theGibbsfreeenergy)decreaseasthetemperatureincreases(FiguresS2d,e,g).Thus,thechangeinthethermodynamicpropertiesof(Al2O3)nwithtemperatureissimilartothatofthebulkmaterial.Additionally,theCP,HandSof(Al2O3)nincreasewithincreasingn(FiguresS2a,b,c).ThereisaregioninwhichtheGVcurvesintersectinthetemperaturerangefrom350-600K(FiguresS2d,e).Belowthisregion,GVincreaseswithincreasingn;abovethisregion,GVdecreaseswithincreasingn.Thisresultindicatesthatthestabilityof(Al2O3)ncanbechangedwithavariationintemperature.Thezero-pointenergy,E(0K),of(Al2O3)ndecreaseswithincreasingn(FigureS2f).TherelationshipbetweenE(0K)andnisapproximatelylinear,andtheE(0K)valueofaclusterismuchlessthantheGVvalueofthecluster.
Thethermodynamicregularsolutionof(Al2O3)ncanbeelucidatedasfollows.Using
equation(S3)canbewrittenas
.Atlowtemperatures,
>>1andCpcanbeexpressedas
.TheCpincreaseswithincreasingtemperature.Athightemperatures,
<<1andCpcanbeexpressedas
whereNisthenumberofparticles.TheCpincreaseswithincreasingtemperature.IfNisconstant,therateofchangeofCpwithtemperatureincreaseswithincreasingtemperature(FigureS2a).ThederivativesofHwithrespecttotemperatureare
and
.ReferringtoFigureS2a,
and
;thus,
and
.Thatis,
isanincreasingandconvexfunctionoftemperature.Therefore,HincreaseswithincreasingNatconstanttemperature(FigureS2b).Similarly,Scanbeexpressedas
inwhichCisconstant.Atloworhightemperatures,
(FigureS2c).ThederivativeofGVwithrespecttotemperatureis
.Similarly,
.Atlowtemperatures,
.Athightemperatures,
;thatis,therateofchangeofGVwithrespecttotemperatureisnegative(FiguresS2d,e).Inaddition,
;thus,ifT=0K,GVincreaseswithincreasingNattemperaturesbelow350K.Attemperaturesgreaterthan600K,
andGVdecreaseswithincreasingN.TherelationshipamongG,GVandE(0K)isgivenbyG=GV+E(0K)(FigureS2g).
Figure.S2Thethermodynamicpropertiesof(Al2O3)n.(a)TheCPvaluesareplottedasagroupofcoloredsolidcurves.(b)TheHvaluesareplottedasagroupofcoloredsolidcurves.(c)TheSvaluesareplottedasagroupofcoloredsolidcurves.(d)TheGVvaluesareplottedasagroupofcoloredsolidcurves.(e)Amagnifiedviewofthesectionof(d)intheregioninwhichtheGVcurvesintersect.Thetemperaturerangeoftheregionoftheintersectionisapproximately350Kto600K.(f)Thezero-pointenergyatatemperatureof0K,E(0K).(g)TheGibbsfreeenergy,G,withG=GV+E(0K).
Ⅵ.ThermodynamicPropertiesofα-Al2O3
Figure.S3Thethermodynamicpropertiesofα-Al2O3(CP,S,HandGV)thatweresimulatedbyDFTareplottedascoloredsolidcurves,andtheexperimentaldataofCPandS38areplottedasblueandgreentriangles,respectively.ThevaluesofCPandSofα-Al2O3foundusingDFTarenearlyconsistentwiththeexperimentaldatainthetemperaturerangefrom0Kto1000K.TheE(0K)ofα-Al2O3is-1867993.697KJ·mol-1.
Ⅶ.GibbsFreeEnergyChangesoftheFormationof(Al2O3)nandofitsTransformationintoα-Al2O3
Thefirststep,theformationofapopulationofaluminaclusterswithstablestructuresviathereactionbetweenAlandOinanFe-O-Almelt,isexpressedas
2Al+3O=(1/n)(Al2O3)n(S7)
wherenisthenumberofAl2O3unitsinacluster.
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