Quasi-one and two-dimensional transitions of gases

arXiv:cond-mat/0008206v1[cond-mat.stat-mech]14Aug2000Quasi-oneandtwo-dimensionaltransitionsofgasesadsorbedonnanotubebundlesS.M.Gatica1,M.J.Bojan2,G.Stan3,andM.W.Cole41DepartamentodeF´ısica,FacultaddeCienciasExactasyNaturales,UniversidaddeBuenosAires,1428,BuenosAires,Argentina.2DepartmentsofChemistryPennsylvaniaStateUniversityUniversityPark,PA16802,USA3InstituteforPhysicalScienceandTechnologyandDepartmentofChemicalEngineering,UniversityofMaryland,CollegePark,MD20742,USA.4DepartmentsofPhysics,PennsylvaniaStateUniversityUniversityPark,PA16802,USA(February1,2008)AbstractGrandcanonicalMonteCarlosimulationshavebeenperformedtodeter-minetheadsorptionbehaviorofArandKratomsontheexteriorsurfaceofarope(bundle)consistingofmanycarbonnanotubes.Thecomputedadsorp-tionisothermsrevealphasetransitionsassociatedwiththesuccessivecreationofquasi-onedimensionallinesofatomsnearandparalleltotheintersectionoftwoadjacentnanotubes.I.INTRODUCTIONMuchattentionhasfocusedrecentlyontheproblemofgasadsorptionwithinbundles(“ropes”)ofcarbonnanotubes[1–13].Forsmallatomsandmolecules,boththecylindricalspaceswithinindividualtubesandtheinterstitialchannelsbetweentubesareregionsofveryattractivepotentialenergyforadsorption.Largermoleculesexperienceasimilarattractionwithinthetubes(butnotwithintheinterstitialregion,unlessswellingoccurs).Ineithercase,thereoccurssignificantuptake,withintheserespectiveregions,atpressuresPbelowsaturatedvaporpressureP0(svp).ItwasnotedrecentlybyWilliamsandEklund[9]thatfinitebundlesofcarbonnanotubesoughttoexhibitadditional,significantadsorptionontheexternalsurfaceofthebundles.Inthethermodynamiclimitofaninfinitearray,suchexternaladsorptionrepresentsanegligiblefractionofthetotal.Typicalfinitebundles,however,haveradiusR∼100˚Aandareexpectedtomanifestsignificantexternaladsorption.Indeed,experimentsofTalapatraetal.[14]havebeeninterpretedasprovidingevidenceofsuchexternaladsorptioninthecaseofCH4molecules,NeandXeatoms.Inthispaperweformulateasimplemodelofthissystemwithwhichweexplorethephenomenonofexternaladsorption.WesupposethatthenanotubebundleofFig.1isoursubjectofinvestigation.Thisassumptionprovidesuswithanormalizationfactor(essentiallyasurface/volumeratio)requiredforaspecificpredictionoftheadsorbatemasspermassofsubstrate(andnothingmore).Anyalternativegeometryrequiresa“renormalization”1factor,discussedbelow.Ouractualcalculationsofexternaladsorptionareperformedwithasimplifiedabstractionoffigure1.Specifically,themodelemploysaperiodic,planararrayofparallelcylindersrepresentingthenanotubes.Thisplanarmodelshouldbeanadequateapproximationtotherealsituation(tubesattheperimeterofarope)inanexperimentalsituationforwhichRgreatlyexceedstheradius(∼7˚A)ofanindividualtube.Ourmethodofevaluatingtheadsorptioniscomputersimulation,usingthegrandcanonicalMonteCarlo(GCMC)method.Withthiswelldevelopedtechniqueweareabletodemonstratetheex-istenceofseveralphasetransitionsinthefilm.Theseincludebothtwo-dimensional(2D)transitions(analogoustolayeringtransitionsonflatsurfaces)andquasi-1Dtransitions(anal-ogoustoonesonsimilarquasi-1Dgeometries,suchasasurfacefacetcutclosetothatofalowindexsurface).[15]Theoutlineofthispaperisthefollowing.Thefollowingsectiondescribesourmodelpotentialandcomputationalmethod.SectionIIIdescribesourresults.InsectionIV,wedescribeimplicationsforexperimentsstudyingsuchsystems.II.METHODAsinmostadsorptionstudiesongraphiteorcarbonnanotubes,theadsorptionpotentialassumedhereisapairwisesumoftwo-bodyinteractionsU(x)betweenamoleculeandthenanotube’scarbonatoms[16,17]:V(r)=XiU(r−Ri)(1)whereristhepositionofthemoleculeandRiisthepositionofaCatom.Thepairpo-tentialisassumedtobeisotropicandofLennard-Jonesform:U(r)=4ǫ[(σ/r)12−(σ/r)6].Althoughweneglectheretheeffectofanisotropyforadsorptionongraphite[18],thepo-tentialconsideredissuitableforourqualitativestudy.TheLJparametersfortheAr-CandKr-CinteractionareobtainedwithsemiempiricalcombiningrulesfromthecorrespondingLJparameters:[19–22]:σgC=σgg+σCC2(2)ǫgC=√ǫggǫCCTheparametervaluesusedinthisstudyarelistedintheTableI.Anothersimplifyingassumptionemployedhereisthereplacementofdiscretecarbonatomsbyacontinuouscylindricalsheetofmatter;thisshouldnotdrasticallyaffecttheadsorbate’sbehaviorexceptwhentheadjacenttubesareinperfectregistry,anunlikelysituation.Theresultingpotentialatdistancerfromthenanotubeaxis,outsideofthetube,is[24]:V(r;R)=3πθǫσRh2132σr11M11(x)−σr5M5(x)i(3)whereθ=0.38˚A2isthesurfacedensityofCatomsandRistheradiusofthenanotube.Weusetheintegrals2Mn(x)=Zπ0dϕ1(1+x2−2xcosϕ)n/2.(4)Finally,theadsorptionpotentialontheexternalsurfaceofthenanotubebundleisobtainedbysumminguptheinteractionsofthemoleculewithallthenanotubesinthebundle.WeperformGCMCsimulationsofArandKrinteractingwiththeouterwallofabundleof(10,10)tubeslineduptocreatea“flat”surfacewithgrooves.IntheGCMCsimulations,thechemicalpotential,temperatureandvolumeareheldconstantwhilethenumberofparticlesvaries.Thistechniqueisstandardandhasbeendescribedintextbooks[25,26],soonlythedetailswillbegivenhere.Threekindsofmovesareperformed:displacementofamolecule,creationofaparticleanddeletionofapar