Shape transition of unstrained flattest single-walled carbon nanotubes - PDF document

Shape transition of unstrained flattest single-walled carbon nanotubes under pressure Citation: 115, 044512 (2014); doi: 10.1063/1.4863455 View online: View Table of Contents: Published by the Shapetransitionofunstrainedflattestsingle-walledcarbonnanotubesunderWeihuaMu,JianshuCao,andZhong-canOu-YangDepartmentofChemistry,MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USAStateKeyLaboratoryofTheoreticalPhysics,InstituteofTheoreticalPhysics,TheChineseAcademyofSciences,P.O.Box2735Beijing100190,ChinaKavliInstituteforTheoreticalPhysicsChina,TheChineseAcademyofSciences,P.O.Box2735Beijing100190,ChinaSingapore-MITAllianceforResearchandTechnology(SMART),Singapore138602CenterforAdvancedStudy,TsinghuaUniversity,Beijing100084,China(Received2December2013;accepted15January2014;publishedonline30January2014)Singlewalledcarbonnanotube’s(SWCNT’s)crosssectioncanbeattenedunderhydrostaticpressure.Oneexampleisthecrosssectionofasinglewalledcarbonnanotubesuccessivelydeformsfromtheoriginalroundshapetoovalshape,thentopeanut-likeshape.Atthetransitionpointofreversibledeformationbetweenconvexshapeandconcaveshape,thesidewallofnanotubeisattest.Thisattesttubehasmanyattractiveproperties.Inthepresentwork,anapproximateapproachisdevelopedtodeterminetheequilibriumshapeofthisunstrainedattesttubeandthecurvaturedistributionofthistube.Ourresultsareingoodagreementwithrecentnumericalresults,andcanbeappliedtothestudyofpressurecontrolledelectricpropertiesofsinglewalledcarbonnanotubes.Thepresentmethodcanalsobeusedtostudyotherdeformedinorganicandorganictube-likestructures.2014AIPPublishingLLCLLChttp://dx.doi.org/10.1063/1.4863455I.INTRODUCTIONCarbonnanotubes(CNTs)havesuperiormechanicalpropertiesduetostrongcarbon-carbonatomicinteractionsintheirhoneycomblattices.CNTs’highelasticmodulus,exceptionalaxialstiffness,andlowdensity,makethemidealfornanotechnologyapplications.CNTs’mechanicalproper-tiesarehighlyanisotropic,andhavebeenstudiedexten-Incontrasttothehightensilestrength,CNTsaresusceptibletomechanicaldistortionintheirradialdirectionsunderappliedhydrostaticpressureontheorderofGPa.Theradialdeformationcontrolledbytheappliedpressureprovidesanapproachtomodifytheelectronicpropertiesofsinglewalledcarbonnanotube(SWCNTs),consequently,radialdeformationofSWCNTscanbeobservedbyopticalspectroscopysincetheelectronicbandstructureofaSWCNTissensitivetoitsmorphologicaltransition.addition,aSWCNT’schemicalreactivitydependsonitsme-chanicaldeformations,whichplaysthekeyroleinthedesignofCNT-basedgassensors.AmongtheradiallydeformedSWCNTs,thefullycol-lapsedstructurewithtwostrainededgesbridgedbyaatmid-dlesectionattractsintenseinterestbecauseofitsphysicalandchemicalpropertiesassociatedwithitsatribbon-likemiddlepart.However,thefullycollapsedSWCNTsarestabilizedbyvanderWalls(vdW)interactionbetweentwoopposingatwalls(withtypicalinterlayerdistance3.4A)leadstoirreversiblecollapsing.ThevdWinterac-tionmayalsoinducetwistandbendingoftheatsectioninafullycollapsedSWCNT.Moreover,theedgesectionofacol-lapsedSWCNTishighlystrained.Asitiswidelyknown,withtheincreaseofhydrostaticpressure,acylindricalSWCNTatrstbecomesoval-shaped,andthenbecomespeanut-shaped.Betweenthesetwoshapes,thereexistsacriticalshapeundercertainpressurewhichistheunstrainedattestcongurationofSWCNTs.TheradiallydeformedSWCNTwithaatsec-tionthatissimilartofullycollapsedSWCNTshaveadvan-tagesoverfullycollapsedones:(1)itcanbeshiftedbacktothestatewithacircularcrosssectionreversibly;(2)thereisnotwistintheatsectionsincevdWinteractioncanbeneglected;(3)thereisnostraininthetwoedges,thereforeavoidingthestrain-inducedchangeoftheelectronicbandstructure.II.MODELInthepresentmanuscript,wewilltheoreticallydeter-minetheshapesoftheunstrainedattestSWCNTs.Althoughtheproblemhasbeenstudiedbymoleculardynamicalsimula-tionsandnumericalcalculations,therestilllackstheanalyt-icalexplicitexpressionswhichcanprovideadesignprincipleforCNT-baseddevices.Wewillaccomplishthistaskbycare-fullystudyingthetransitionoftheSWCNTdeformingfromaconvexshapetoaconcaveshape,andgivingananalyticalexpressionofthecriticalshapeforanunstrainedattesttube.Basedonthisanalyticalexpression,wecancalculatethecriti-calpressurefortheunstrainedattestSWCNTs.WewillalsodiscussthecurvatureeffectfortheorbitalhybridizationofcarbonatomsonthesidewalloftheSWCNT.Thelatterisim-portantforstudyingtheabsorptionofmoleculesonSWCNTs.AlthoughthedeformationoftheSWCNTinvesti-gatedinthepresentworkisinidealconditions,theresultscanbeusedasthetheoreticallimitsfortheactualCNT-devices.TheequilibriumshapeofdeformedSWCNTsisdeter-minedbyminimizingthefreeenergyundercertainconstraints.Inthepresentproblem,thefreeenergycontainstheelastic Electronicaddresses:whmu@mit.eduandmuwh@itp.ac.cnElectronicmail:jianshu@mit.edu0021-8979/2014/115(4)/044512/6/$30.002014AIPPublishingLLC,044512-1JOURNALOFAPPLIEDPHYSICS,044512(2014) energyandthepressureterm,.AlthoughbothbondbendingandbondstretchingcontributetotheelasticenergyofSWCNTs,attheenergyscale1eVisthebendingstiffnessofSWCNTs),thebondbendingeffectpredominates.Thebond-bendingenergyofaSWCNTcanbedescribedbyitscurvatures KdAHere,andarethemeancurvatureandGaussiancurvatureofthesurfacesofcarbonatoms,and1.17eV(Ref.)isconsistentwiththeresultofTersoffetal.Theexpressionoffreeenergycanbemappedto2D.ConsiderastraightSWCNTwithradius,withouttheinclusionofitstwoend-caps,thesurfaceofthetubecanbedescribedincylindri-calcoordinatesas,Þ¼fcossinHere,0,and0,withthelengthofstraighttubeaxis.Theoriginofisarbitrary.Thesurface’smeanandGaussiancurvatureare20.Comparingwiththerelativecurvatureofaplanecurve,thebendingenergycanberewrittenas,wherethearcparameterofboundarycurveandlineelementdTheequilibriumshapeofadeformedSWCNTisthesolu-tionofthe2Dvariationproblem0,with istheLagrangemultiplier,whichisintroducedtokeepthetubecircumferenceatconstant.Distortingthebyasmallperturbation)alongthenormaldirec-tionofthecurve,thevariationsinwhichcanbeobtainedinasimpleway,asshownin.Then,theequationfortheequilibriumshapeofSWCNTsisobtained Obviously,thereisaspecialsolutioncorrespondingtothetubeofcircularcross-section,,whichimpliesthenecessaryconditionformaintainingSWCNT’scircularcrosssection.TheinitialequationofEq. kc8k4rþ kc2kr0þ leadstotheequationof(6)Underhydrostaticpressure,thetubeofcircularcross-sectiontubecanbedeformedtoaconvexstructurewithlowersymmetry.Withtheincreaseofpressure,thecur-vaturechangescontinuallyfrom0(convexshape)to0(concaveshape).Acriticalshapeforthetransitionfromconvextoconcavesatises0,withtheminimalcurvaturebeingexactlyzero,0.ThecriticalshapeistheattestshapeforunstrainedSWCNTs.Inparticular,forthedeformedSWCNTwithsymmetry,theconvexshapeisthe“oval”shape,andconcaveshapeis“peanut-like”shape,asshowninFig..Equationcanbesolvednumericallybyiteration.Duetothesymmetryofthedeformedshape,onlyaquarterofthecurveneedtobecon-sidered,asshowninFig..Forconvenience,thelengthofisre-scaledto2.The0andtheinex-tensibleconditionleadtoaconstraint p2¼ð0kr; (7)Thedirectionoftangentiallinechangesfrom/2to0,asthearcparameterchangesfrom0to/2,whichprovidesanotherindependentequation p2¼ð0kr; (8) FIG.1.IllustrationofshapetransitionsforacrosssectionofSWCNTwithsymmetry.Thecriticalshapeforthetransitionfromtheovalshapetothepeanut-likeshapeistheunstrainedattestshapeofSWCNT. FIG.2.IllustrationofaquarterofthedeformedcrosssectionofaSWCNTsymmetry.Theistheanglebetween-axisandthedirectionofthecurve’stangentialline.Theisthepolarangle.044512-2Mu,Cao,andOu-YangJ.Appl.Phys.,044512(2014) canbecalculatedself-consistently.TherelationdisusedinthederivationofEq..Inprinciple,theattestshapewithhighordersymmetries3)canbeobtainedsimilarly.IntheapplicationofdeformedSWCNTrelateddevices,suchasagassensor,itisnecessarytoderiveanexplicitexpressionofthecriticalpressurefortheattestshapeasthedesignguidelineforworkingconditions.Inthisstudy,wewillgiveananalyticalsolutionforthat.III.CALCULATIONBasedonthenumericalresultof,weknowthatverysmall.Thus0isagoodinitialguessfortheself-consistentofshapeequations..Wegetanapproximatesolutionwith0,andthencompareitwiththenumericalsolutiontochecktheconsistency.For0,Eq.0,itcanbereducedto  withtheadditionalnecessaryconditionofequilibriumcriticalshape:reachesitsmaximumat/2,anditsminimumat.Thus,wehave,(seeAppendixforthedetails).Theconstantlengthof l02n¼ pq0naÞð1=3Þðp=2þp=np=2k1r;nðhÞdh¼ (10)Theconstants)canbederivedas  76 23 432F1 1; 56; 32;cos2 3p4n Here,isthehypergeometricfunction.)foraunstrainedattesttubewithsymmetriesarelistedinTable.Thecurvature2hastheformof ðnÞpq02=3 34hþ 5p6 Inparticular,4.3244,predictsthecriticalshapeforthetransitionfromoval-shapedtubetopeanut-likeshape.TheequilibriumshapeoftheunstrainedattestSWCNTswithsymmetrycanbedescribedbyparametricequations(seeAppendixforthedetails) pq0c2sin h4þ p41=3 3h4þ p4y2ðh pq0c2 sin2=3  h4 p41=3 3h4þ ThecomparisonoftheexplicitresultandtheexactresultisshowninFig.Thecross-sectionalareaofthetubeis 12 pq0c2ðpp=2 Comparingwiththeundistortedtube,,thegeometricconstantis78,whichisingoodaccordancewiththeexactvalue8195.ThecurvaturedistributioncharacterizestheatnessofthedeformedSWCNTs.Thecurvaturedistributionprovidesuse-fulinformationaboutthebondhybridizationofSWCNTs,whichgovernsthegas-absorbingabilityofSWCNTs.Wecomparethecurvaturedistributionsofourapproximateresultandtheexactsolution,asshowninFig..Theexactcurvaturedistributioncanbecalculatedself-consistentlyaccordingtoEqs.(7)and(8).ThehybridorbitalofcarbonatomsintheSWCNTissensitivetothelocalcurvatureofthetube.ForaSWCNT,inthelocalcoordinateofagivencarbonatom,thethreeneighboursofthecenteratomhavecoordinates ða0=RÞ23sin4hjcoshj;y0j¼sinhjþ ða0=RÞ26sin3hjcosð2hjÞ;j¼1;2;3;z0j¼ 3,and3beingtheanglesbetweenbondcurvesandthedirectionofthetubeaxis,atthepositionofatom1.42Aistheequilibriumbondlengthofcarbon-carbonbond,isthe FIG.3.ComparisonoftheexplicitsolutionandexactsolutionofdeformedSWCNTs’atthecriticalattestshapewithsymmetry.Thesolidlinecurveisthepresentexplicitapproximatesolutionoftheshapeequation,thedashed-dottedlinecurveistheexactnumericalsolution,andthedashedlineisunperturbedcircularshapeofaSWCNT.TheistheradiusoftheSWCNTwithoutthedeformation. TABLEI.ThevaluesfortheapproximatesolutionofaunstrainedattestSWCNTwhosecrosssectionhas23456789104.3243.7283.3723.1242.9362.7882.6652.5622.473044512-3Mu,Cao,andOu-YangJ.Appl.Phys.,044512(2014) radiusoftheSWCNT,characterizesthecurvaturedependentpropertiesofSWCNTs.Thedistancebetweenandtheirthreeneighbourscanbedescribedas .Thedirections(thedirectionofsym-metryaxis)forthethreeorbitscanbewrittenasInthelocalcoordinatesystemassociatedwiththegivecarbonatom,theorbitandthreeorbitscanbedecom-posedas,,and.with, 6a20323a0q  329a203ð323a20Þr 1p1 ,and.Therefore,Þi¼ p4a0j2s ,whichplaysakeyroleintheelectronicandchemicalpropertiesofSWCNTs.Þi¼j.Byadjustingappliedhydro-staticpressure,theatthemainpartofaSWCNTcanbeswitchedbetweenIV.CONCLUSIONInsummary,wehavetheoreticallyinvestigatedtheunstrainedattestshapeofSWCNTunderhydrostaticpres-sure,whichcanrecovertoitsoriginalcircularcross-sectionafterwithdrawingthepressure.WendagoodapproximatesolutionfortheshapeofthisattestSWCNT,andtheoreti-callydeterminethecurvaturedistribution.Wealsodiscussthecurvature-dependenthybridorbitaloftheSWCNT.Thepres-entanalyticalsolutionisingoodagreementwiththeexactnu-mericalsolution,anditcanbeusedasthedesignguidelineinCNT-basednano-electronicdevices.Ourapproachcanbegeneralizedtoinvestigateotherinorganicandorganicelasticmembranesystems,includingtheself-assembledpolymermaterialsandcolloidalaggregations.Intheactualdevices,thedeformedSWCNTmaybesupportedbythesubstrate.ThepresentresultsprovidethetheoreticalboundaryforthisCNT-baseddevices.ItisworthnotingthatthevariationsinEq.(3)canalsobeenobtainedinotherapproaches.ACKNOWLEDGMENTSW.MuandJ.CaoacknowledgethenancialassistanceofSingapore-MITAllianceforResearchandTechnology(SMART),NationalScienceFoundation(NSFCHE-112825).W.MuandZ-c.Ou-YangacknowledgethesupportofNationalScienceFoundationofChina(NSFC)(GrantsNos.11074259and11374310),andtheMajorResearchPlanoftheNationalNaturalScienceFoundationofChina(GrantNo.91027045).J.CaohasbeenpartlysupportedbytheCenterforExcitonics,anEnergyFrontierResearchCenterfundedbytheU.S.DepartmentofEnergy,OfceofScience,OfceofBasicEnergySciencesunderAwardNo.DE-SC0001088.W.MuthanksDr.LinyingCuiformodifyingthemanuscript.APPENDIXA:THEDETAILSOFVARIATIONALCALCULATIONSWewillshowthedetailsofderivingthevariationsin.ThecurveisshowninFig.Withthedistorsion)alongthenormaldirectionoftheclosedboundarycurve,theplaneboundarycurvebecomesthenormaldirectionofthecurve,andarcparameter.Thelineelement d~rðsÞdsþ Thelengthofthe FIG.5.Ageneralclosedcurve. FIG.4.Comparingtheapproximateandexactresultsofthecurvaturedistribu-tionofunstrainedSWCNTsatthetransitionfromovaltopeanutshape.Here,isthepolarangleasshowninFig..Thecurvatureisintheunitof1/,withbeingtheradiusofunperturbedcircularcrosssectionofaSWCNT.044512-4Mu,Cao,andOu-YangJ.Appl.Phys.,044512(2014) Here,wehaveusedtherelationbetweenthetangentandnormalvectorofaplanecurve denotedd/d.Thus,therstordervariationofthelengthofcurvehastheformWiththedistorsion,theisnotthearcparameterofthedeformedcurve,thecurvaturechangesto ~r0ðsr00ðsÞjj~r0ðs¼ Here,wehaveusedtherelationsÞ!ðwehaveTherstordervariationofisthereforeAsshowninFig.2,withthedistorsion,thedchangesto ^ez2ds12krðsÞwðsÞrðstðs0ðsÞ~rðsÞ ^nðsðsÞ^ezw2ðsÞThus,ðdA! ^ez2þds12krðsÞwðsÞrðstðsÞþw0ðsÞ~rðsnðsðsÞ^ezw2ðsÞ APPENDIXB:THEDERIVINGOF  canberewrittenas whichcanbereducedto Integratingthetermsonbothsides,wehavetheapproxi-mateexpressionof)asÞ¼ð0,atforthecriticalshape,we.ThencanbedeterminedbasedonEq..Theexplicitexpressionof)isshowninAPPENDIXC:THEDERIVINGOFEQ.Thecurvature)istheapproximateexplicitsolutionofEqs.(7),whichplaysthecenterroleindetermin-ingtheparametricequationsforthecurveforThecurve2)canbeexpressedassinHere,isthearcparameter.Theoriginissetatthecenterofthecrosssection,and0.Accordingtothedenitionofthecurvatureofthecurve,,thelineelementcanbedescribedby044512-5Mu,Cao,andOu-YangJ.Appl.Phys.,044512(2014) d~s¼ d~sdhdh¼ theequationcanbefurtherreducedto cos~hkr;2ð~hÞ;y2ðhhp=2d~h Substitutetheexplicitexpressionof)totheaboveequationandintegrate,wecanobtainEq.R.Saito,M.S.Dresselhaus,andG.Dresselhaus,PhysicalPropertiesofCarbonNanotubes(ImperialCollegePress,London,1998).A.Jorio,M.S.Dresselhaus,andG.Dresselhaus,CarbonNanotubes(Springer,Berlin,2008).J.Tang,L.-C.Qin,T.Sasaki,M.Yudasaka,A.Matsushita,andS.Iijima,Phys.Rev.Lett.,1887(2000).B.Anis,K.Haubner,F.Borrnert,L.Dunsch,M.H.Rommeli,andC.A.Phys.Rev.B,155454(2012).M.M.J.Treacy,T.W.Ebbesen,andJ.M.Gibson,Nature(London)678(1996).M.Yao,Z.Wang,B.Liu,Y.Zou,S.Yu,W.Lin,Y.Hou,S.Pan,M.Jin,B.Zou,T.Cui,G.Zou,andB.Sundqvist,Phys.Rev.B,205411C.Caillier,D.Machon,A.San-Miguel,R.Arenal,G.Montagnac,H.Cardon,M.Kalbac,M.Zukalova,andL.Kavan,Phys.Rev.B,125418G.Gao,T.Cagin,andW.A.Goddard,Nanotechnology,184(1998).A.L.Aguiar,E.B.Barros,R.B.Capaz,A.G.S.Filho,P.T.C.Freire,J.M.Filho,D.Machon,C.Caillier,Y.A.Kim,H.Muramatsu,M.Endo,andA.San-Miguel,J.Phys.Chem.C,5378(2011).J.A.Elliott,J.K.W.Sandler,A.H.Windle,R.J.Young,andM.S.P.Phys.Rev.Lett.,095501(2004).M.H.F.SluiterandY.Kawazoe,Phys.Rev.B,224111(2004).P.Tangney,R.B.Capaz,C.D.Spataru,M.L.Cohen,andS.G.Louie,Nano.Lett.,2268(2005).A.N.ImtaniandV.K.Jindal,Phys.Rev.B,195447(2007).W.Yang,R.-Z.Wang,Y.-F.Wang,X.-M.Song,B.Wang,andH.Yan,Phys.Rev.B,033402(2007).J.Zang,A.Treibergs,Y.Han,andZ.F.Liu,Phys.Rev.Lett.,105501D.Y.Sun,D.J.Shu,M.Ji,F.Liu,M.Wang,andX.G.Gong,Phys.Rev.,165417(2004).J.Z.Cai,L.Lu,W.J.Kong,H.W.Zhu,C.Zhang,B.Q.Wei,D.H.Wu,andF.Liu,Phys.Rev.Lett.,026402(2006).J.Tang,L.-C.Qin,T.Sasaki,M.Yudasaka,A.Matsushita,andS.Iijima,J.Phys.:Condens.Matter,110575(2002).K.Thirunavukkuarasu,F.Hennrich,K.Kamaras,andC.A.Kuntscher,Phys.Rev.B,045424(2010).S.Park,D.Srivastava,andK.Cho,NanoLett.,1273(2003).K.MylvaganamandL.C.Zhang,Nanotechnology,410(2006).Y.-F.ZhangandZ.-F.Liu,,928(2006).J.Kong,N.R.Franklin,C.Zhou,M.G.Chapline,S.Peng,K.Cho,andH.,622(2000).B.I.Yakobson,C.J.Brabec,andJ.Bernholc,J.Comput.-AidedMater.,173(1996).B.I.Yakobson,C.J.Brabec,andJ.Bernholc,Phys.Rev.Lett.,2511T.C.ChangandZ.R.Guo,NanoLett.,3490(2010).O.E.Shklyaev,E.Mockensturm,andV.H.Crespi,Phys.Rev.Lett.155501(2011).J.Liu,Arch.Appl.Mech.,767(2012).R.Martel,T.Schmidt,H.R.Shea,T.Hertel,andP.Avouris,Appl.Phys.,2447(1998).P.E.Lammert,P.H.Zhang,andV.H.Crespi,Phys.Rev.Lett.,2453J.TersoffandR.S.Ruoff,Phys.Rev.Lett.,676(1994).W.Mu,M.Li,W.Wang,andZ-c.Ou-Yang,NewJ.Phys.,113049O.-Y.Zhong-can,Z.B.Su,andC.L.Wang,Phys.Rev.Lett.,4055M.AbramowitzandI.Stegun,HandbookofMathematicalFunctionsNinthPrinting(DoverPublications,Inc.,NewYork,1972).W.Mu,G.Zhang,andZ.-C.Ou-Yang,Jpn.J.Appl.Phys.,Part1065101(2012).W.MuandZ.-C.Ou-Yang,“Chiralitydependentelasticityofsinglewalledcarbonnanotubes,”inNanowires:Properties,Synthesis,andApplications,editedbyV.Lefevre(NovaSciencePublishers,Inc.,Hauppauge,2012),pp.160–170.W.Mu,G.Zhang,andZ.-C.Ou-Yang,Appl.Phys.Lett.,053112J.WuandJ.Cao,J.Phys.Chem.B,21342(2005).J.WuandJ.Cao,PhysicaA,249(2006).Z.-C.Ou-Yang,J.-X.Liu,andY.-Z.Xie,GeometricMethodsintheElasticTheoryofMembranesinLiquidCrystalPhases,1sted.,AdvancedSeriesonTheoreticalPhysicalScienceVol.2(WorldScienticPublishingCompany,Singapore,1999).X.-H.Zhou,Int.J.Mod.Phys.B,587(2010).044512-6Mu,Cao,andOu-YangJ.Appl.Phys.,044512(2014)