纳米尺寸效应新应用单键比热强度及应变极限Word文件下载.docx

纳米尺寸效应新应用单键比热强度及应变极限Word文件下载.docx

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Ananalyticalsolutionshowsthatthemaximalstrainofanimpurity-freemetallicmonatomicchain（MC,oradefect-freenanowire（NW,variesin-apparentlywithmechanicalstressbutapparentlywiththeseparationbetweenthemeltingpoint（Tm（Kandthetemperatureofoperationintermsofexp{[Tm（K–T]-1},whereKisthedimensionoftheNW（foranMC,K=1.5.ReconciliationofthemeasureddataofAu-MCbreakinglimitsuggeststhatthediscrepancyinmeasurementarisesfromthermalandmechanicalfluctuationsneartheTmoftheMCthatis1/4.2foldofthebulkvalue.Findingsalsofavourthemechanismforthehigh-extensibilityofananograinedNWandfurtherindicatethatbondunfoldingofthelower-coordinatedatomsdominatethegrainboundaryactivitiesparticularlyattemperaturesapproachingtosurfacemelting.

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PACS:

61.46.+w;

82.45.Yz;

81.07.Bc

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Introduction

Metallicmonatomicchains（MCsandnanowires（NWshaveattractedtremendousinterestbecauseofthefundamentalsignificanceandthefascinatingpropertiesinquantumconductance,chemicalreactivity,thermalstability,mechanicalstrengthandductility.Thesearekeyissuesofconcerninupcomingtechnologiessuchasnano-device.AmetallicMCisanidealprototypeforextensibilitystudy,astheMCinvolvesmerelybondstretchingwithoutbondunfoldingoratomicglidingdislocations,asdoatomsinametallicNWconsistingofnanograinsuponbeingstretched.1Measuredusingtransmissionelectronmicroscopy（TEMatroomtemperatureundertensiontheAu-Aubondbreaksatalengththatvariesfrom0.29nm,20.36nm（±

30%,30.35~0.40nm4toevenasingleeventof0.48nm,5whileat4.2Kthebreakinglimitisreducedto0.23±

0.04nmasmeasuredusingscanningtunnellingmicroscopyandmechanicallycontrolledbreakjunction.6SophisticatedcalculationssuggestthattheAu-Auequilibriumdistance（withoutexternalstimuluscontractstoarangebetween0.232and0.262nm7fromthebulkvalue,0.2878nm,whereasthemaximalAu-Audistanceundertensionexceedsnolongerthan0.31nm.8TheAu-AubondintheMCistwicestrongerthanthatinthebulk.9DiscrepancycouldnotbetheoreticallysolvedunlessinsertingimpurityatomssuchasH,B,C,N,O,andSintotheAu-Auchainincalculations.10MetallicNWssuchasCuandAlshowextensibilitythatis101~3higherthanthebulkvaluesthoughtheMCsofCuandAlarehardtoformatambient.Thehighextensibilitywasattributedtoatomicdislocationsoratomicdiffusionatgrainboundariesthataresuggestedtobeeasier.11,12

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Combiningtheeffectsofthermalexpansion,mechanicalstretching,andtheatomiccoordination-number（CNimperfectioncausedbondcontractionwiththefactthatamoltenphaseisextremelycompressible,wehavederivedanumericalsolutiontosolvethediscrepancyinatomicseparationofanimpurity-freeMCwithandwithoutthermalandmechanicalstimuli.Extensionofthesolutiontoadefect-freemetallicNWsuggeststhatbondunfoldingoratomicslidingofthelower-coordinatedatomsatgrainboundariesdominatesthehighextensibilityofananograinedNWattemperaturesclosetothatforsurfacemelting.

Theory

Thebond-order-length-strength（BOLScorrelationpremise,whichhasbeendetailedinref13,indicatesthattheCNimperfectionofatomsatsitessurroundingdefectsornearthesurfaceedgecausestheremainingbondsofthelower-coordinatedatomstocontractspontaneouslyassociatedwithstrengtheningoftheshortenedbond.ThebondstrengtheningcontributestotheHamiltonianthatdictatestheentireband

gapexpansion.14Ontheotherhand,thestructuresuchasthecore-levelshift,13band

atomicCNimperfectionlowerstheatomiccohesiveenergythatdominatesthethermalstabilitysuchasmelting20andphasetransition15,16anddeterminestheactivationenergyforatomicdiffusion,atomicdislocation,andchemicalreaction.Thecompetitionbetweenenergydensityincreaseandtheatomiccohesiveenergysuppressionintherelaxedregiondominatesthemechanicalstrengthandcompressibilityofananosolid.17TheBOLScorrelationhasalsoenabledustodeterminetheidentitiesofaC-Cbondincarbonnanotubes,17theenergylevelsofan-4-

isolatedatom,13andthevibrationfrequencyofaSi-Sidimmerbond.18MatchingtheBOLSpredictionstothemeasurementsrevealsthatatthelowerendofthesizelimit（oneunitcellwithatomicCNof2,theAu4f-levelbindingenergyincreasesby~43%（=ci-1-1withrespecttothebulkvalueof-2.87eVandthemeltingpointofthesmallestAunanosolid,ortheAu-MC,decreasesfrom1337.33Kto320K,whichis1/4.2foldthebulkvalue.19

Figure1illustratesschematicallytheBOLScorrelationusingthepair-wiseinter-atomicpotential.WhentheCNofanatomisreduced,theequilibriumatomicdistancewillcontractfromone（unitind,beingtheequilibriumbondlengthinthebulktociandthebondenergywillincreaseinmagnitudefromone（unitinEb,beingthecohesiveenergyperbondinthebulktoci-m.Theciisthebondcontractioncoefficient.Theindexmisanadjustableparameterdependingonthenatureofthebond.Formetals,m=1;

forSiandC,mhasbeenoptimisedtobe4.8813and2.67,17respectively.Thesolidandthedottedu（rcurvescorrespondtothepair-wiseinter-atomicpotentialwithandwithoutCNimperfection.TheBOLScorrelationdiscussedherewithisconsistentwiththetrendreportedinRef.7albeittheextentofbondcontractionandenergyenhancementthatvariesfromcasetocaseinRef.7.TheBOLScorrelationformulatesthebondlength,di,thebondenergy,Ei（T=0,andthecohesiveenergy,EBi,peratomintheMCinthefollowingforms:

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⎧di（zid=ci（zi=21+exp（（12−zi/8zi]=0.6973⎪⎪−1⎨Ei（T=0=ciEb（T=0≅1.5Eb⎪⎪⎩Tm,i∝EBi=ziEi（zi=2

（1

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whereziistheeffectiveatomicCN.Tm,ibeingthemeltingpointoftheMCis

iandbdenotesproportionaltotheatomiccohesiveenergy,EBi=ziEi.20,21Subscript

aspecificithatomintheMCandanatominthebulk.TheBOLSpremisepredictsthatanAu-MC（zi=2bondcontractsby1-0.6973≅30%from0.2878to0.2007nmandthebondstrength（Ei/di=ci-2~2becomestwofoldsthebulkvalue.ThepredictedAu-MCequilibriumlengthisslightlyshorterthanthatmeasuredundertensionat4.2K,0.23±

0.04nm,6andthepredictedbondstrengthagreeswithreportedvalues.9SuchconsistenciesfurtherevidencethevalidityoftheBOLSconsiderationthatattributesthesizedependencyofananosolidtotheatomicCNimperfectionandtheincreasedportionofthelow-CNatomsofthenanosolid.

ThecharacteristicenergiesasindicatedinFig.1represent:

（iatE=0,thebondiscompletelybrokenwithzerointeratomicinteraction.（iiSeparationbetweenE=0andEi（TisthecohesiveenergyperCNatT,whichistheenergyrequiredforbondbreaking.（iiiThespacingbetweenEi（TandEi（0istheenergyofthermalvibration.IfonewantstomeltanatomwithzicoordinatesbyheatingthesystemfromTtoTm,i,oneneedstoprovidezi[Ei（Tm,i–Ei（T]=ziη1i（Tm,i−Tenergy.WhenTapproachestoTm,i,themechanicalstrengthapproachestozerowithinfinitecompressibility.

OnemayapplyatensilestressPtostretchabondintheMCfromitsoriginalequilibriumlengthatT,di（zi,T,0,tothebreakinglimit,diM（zi,T,P.MechanicallyrupturingofthebondattemperatureTneedsenergyη1i（Tm,i−T+η2ithatequalsthe1/zifoldthermalenergyforevaporatinganatominsolidstateatT:

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∫diM（zi,T,P

di（zi,T,0P（xdx=PdiM（zi,T,P−di（zi,T,0]=P∆di

=η1i（Tm,i−T+η2i

（2

Ideally,theslopeη1icorrespondstothespecificheatpercoordinate.Theconstantη2irepresents1/zifoldenergyrequiredforevaporatingamoltenatomoftheMC.η1iandη2icanbedeterminedwiththeknownci−mandthecorrespondingbulkvaluesofη1bandη2bthathavebeenobtainedasshowninRef.22indetail.

ConsideringtheeffectsofatomicCN-imperfection-inducedbondcontraction（cid,thermalexpansion（1+αT,withαbeingthelinearcoefficient,andmechanicalstretching（1+βi（zi,TP,withcoefficientβi,thedistancebetweentwonearestatomsintheinteriorofanMCcanbeexpressedas:

di（zi,T,P=d×

c（zi（1+αT[1+βi（zi,TP],

Themaximalstrainisthenexpressedas,

∆diM（zi,T,P=βi（zi,TPdizi,T,0

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wheredi（zi,T,0=d×

c（zi（1+αTisthebondlengthatTwithoutbeingstretched.OnecanapproximatethemeanPtotheP（xineqs（2and（3,asthediM（zi,T,Prepresentsthebreakinglimitandtheintegrationisaconstant.Combiningeqs.（2and（3,onehas,

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⎧η1i（Tm,i−T+η2i⎫P=±

⎨⎬βz,T×

dz,T,0iii⎩⎭2

（4Fortensilestress,P>

0,forcompressivestress,P<

0.Theextensibilityorcompressibility,β,ofasystemisexpressedas:

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⎡∂V∂u（r,T⎤=⎢−V∂V2⎥V∂PT⎣⎦

diτ（zi,T,0∝NiEiTm,i−EiT2−1βi（zi,T=−T

diτ（zi,T,0∝η1iTm,i−T（5

βistheinverseofYoung’smodulusorhardnessindimensionthatequalsthesumofbondenergyperunitvolume.24Thepowerindexτ=1,2,and3correspondtothedimensionalityofaMC,arod,andasphericaldot.Niisthetotalnumberofbondsindiτvolume.Oneneedstonotethatthebondnumberdensityintherelaxedregiondoesnotchangeuponrelaxation.Forinstance,bondrelaxationneverchangesthebondnumberbetweentheneighboringatomsinaMCwhethertheMCissuspendedorembeddedintheb