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1、PHYSICALREVIEWB95,045423(2017)ElasticpropertiesandmechanicaltensionofgrapheneR.Ram´ırezandC.P.HerreroInstitutodeCienciadeMaterialesdeMadrid(ICMM),ConsejoSuperiordeInvestigacionesCient´ıficas(CSIC),CampusdeCantoblanco,28049Madrid,Spain(Received25November2016;published25January2017)Room-te
2、mperaturesimulationsofgraphenehavebeenperformedasafunctionofthemechanicaltensionofthelayer.Finite-sizeeffectsareaccuratelyreproducedbyanacousticdispersionlawfortheout-of-planevibrationsthat,inthelong-wavelimit,behavesasρω2=σk2+κk4.Thefluctuationtensionσisfinite(∼0.1N/m)evenwhentheexternal
3、mechanicaltensionvanishes.Transversevibrationsimplyaduplicityinthedefinitionoftheelasticconstantsofthelayer,asobservablesrelatedtotherealareaofthesurfacemaydifferfromthoserelatedtothein-planeprojectedarea.ThisduplicityexplainsthevariabilityofexperimentaldataontheYoungmodulusofgraphenebas
4、edonelectronspectroscopy,interferometricprofilometry,andindentationexperiments.DOI:10.1103/PhysRevB.95.045423I.INTRODUCTIONmembranemodelwithanomalousexponentsdeviatesfromthisrelation[14].Grapheneisasolidsurfaceinthree-dimensional(3D)Inthispaper,theanharmonicityofafree-standinggraphenespa
5、ce[1].Theareaperatom,A,isathermodynamicpropertylayerisstudiedbymolecular-dynamics(MD)simulationsinthatisdifficulttomeasure.Infact,theaccessibleobservableistheNτTensemble(Nbeingthenumberofatomsintheitsprojection,Ap,ontothemeanplaneofthemembrane,withsimulationcellandTthetemperature).Thefluc
6、tuationtension,ApA.Theequalityisachievedinastrictlyplanelayer.σ,andthebendingrigidity,κ,ofthelayerarestudiedat300KTheexistenceoftwodifferentareas,AandAp,suggestsaasafunctionofbothtensile(τ<0)andcompressive(τ>0)duplicityofphysicalproperties.Forexample,thenegative224stresses.Theanalyticl
7、ong-wavelimit,ρω=σk+κk,ofthermalexpansioncoefficientofgraphenerefersonlytotheZAphononsallowsustheformulationofafinite-sizeAp,whilethethermalexpansionofAispositive[2,3].Ancorrectiontothesimulations.TheamplitudeoftransverseinternaltensionconjugatedtotheactualmembraneareaA2fluctuatio
