Pair Correlation Function Characteristics of Nearl

arXiv:cond-mat/0408550v1[cond-mat.stat-mech]25Aug2004PairCorrelationFunctionCharacteristicsofNearlyJammedDisorderedandOrderedHard-SpherePackingsAleksandarDonev,1,2SalvatoreTorquato,1,2,3,∗andFrankH.Stillinger31PrograminAppliedandComputationalMathematics,PrincetonUniversity,PrincetonNJ085442MaterialsInstitute,PrincetonUniversity,PrincetonNJ085443DepartmentofChemistry,PrincetonUniversity,PrincetonNJ085441AbstractWestudytheapproachtojamminginhard-spherepackings,and,inparticular,thepaircorre-lationfunctiong2(r)aroundcontact,boththeoreticallyandcomputationally.Ourcomputationaldataunambiguouslyseparatesthenarrowingdelta-functioncontributiontog2duetoemerginginterparticlecontactsfromthebackgroundcontributionduetonearcontacts.Thedataalsoshowswithunprecedentedaccuracythatdisorderedhard-spherepackingsarestrictlyisostatic,i.e.,thenumberofexactcontactsinthejamminglimitisexactlyequaltothenumberofdegreesoffreedom,oncerattlersareremoved.Forsuchisostaticpackings,wederiveatheoreticalconnectionbetweentheprobabilitydistributionofinterparticleforcesPf(f),whichwemeasurecomputationally,andthecontactcontributiontog2.Weverifythisrelationforcomputationally-generatedisostaticpackingsthatarerepresentativeofthemaximallyjammedrandomstate.WeclearlyobserveamaximuminPfandanonzeroprobabilityofzeroforce,sheddinglightonlong-standingquestionsinthegranular-medialiterature.Wecomputationallyobserveanunusualpower-lawdivergenceinthenear-contactcontributiontog2,persistenteveninthejamminglimit,withexponent−0.4clearlydistinguishablefrompreviouslyproposedinversesquarerootdivergence.Additionally,wepresenthigh-qualitynumericaldataonthetwodiscontinuitiesinthesplit-secondpeakofg2,anduseashared-neighboranalysisofthegraphrepresentingthecontactnetworktostudythelocalparticleclustersresponsibleforthepeculiarfeatures.Finally,wepresentthefirstcomputationaldataonthecontact-contributiontog2forvacancy-dilutedFCCcrystalpackingsandalsoinvesti-gatepartiallycrystallizedpackingsalongthetransitionfrommaximallydisorderedtofullyorderedpackings.Unlikepreviousstudies,wefindthatorderinghasasignificantimpactontheshapeofPfforsmallforces.∗Electronicaddress:[e-mail]d-spherepackingshasbeenstudiedintenselyinthepastyears.Inthispaper,weinvestigatethepaircorrelationfunctiong2(r)oftheclassicalthree-dimensionalhard-spheresystemnearajammingpointforbothdisordered(amorphous,oftencalledrandom)aswellasordered(crystal)jammedpackings.ThebasicapproachfollowsthatofRef.[1],developedfurtherforcrystalpackingsofrods,disksandspheresinRef.[2].Wefocusonfinitespherepackingsthatarealmostcollectivelyjammed[3,4],inthesensethattheconfigurationpointistrappedinaverysmallregionofconfigurationspacearoundthepointrepresentingthejammedidealpacking[4].Difficultieswithextendingtheresultstoinfinitepackingswillbediscussedinwhatfollows.Intheidealjammedpackingparticlecontactsnecessarytoensurejammingareexact,andtheparticlescannotatalldisplace,evenviacollectivemotions.Suchidealjammed(orrigid)packingshavelongbeenthesubjectofmathematicalinquiry[5];however,theyarenotreallyattainableinnumericalsimulationswhereproducedpackingsinvariablyhavesomeinterparticlegaps(eventakingintoaccounttheunavoidableroundofferrors).Itisthereforeinstructivetobetterunderstandtheapproachtothisidealjammedstatecomputationallyandtheoretically,whichistheprimaryobjectiveofthispaper.Wechooseasourmaintoolofexplorationtheshapeofthevenerableorientationally-averagedpaircorrelationfunctiong2(r)aroundcontact.Thisisbecausethisfunctionisasimpleyetpowerfulencodingofthedistributionofinterparticlegaps.Inthejamminglimit,itconsistsofadeltafunctionduetoparticlecontactsandabackgroundpartduetoparticlesnotincontact.Asthejamminglimitisapproached,itisexpectedthatthedelta-functioncontributionwillbecomemorelocalizedaroundcontact.Wederivethefirstexacttheoreticalmodelforthisnarrowingforisostaticpackings(definedbelow),connectingg2totheprobabilitydistributionofinterparticleforcesPf,andverifytherelationnumerically.Inthiswork,wepresentcomputationaldatawithunprecedentedproximitytothejamminglimit,forthefirsttimeclearlyseparatingthenarrowingdelta-functioncontributionfromtheapparentlypersistentdivergingbackgroundcontribution.Thedatashowthatourdisorderedpackingshaveanexactlyisostaticcontactnetworkinthejamminglimit,butwithanunusualmultitudeofnearlyclosedcontacts.Westudythepropertiesofthecontactnetworkandfind,contrarytopreviousstudies,notracesofpolytetrahedralpacking,butratheracomplexlocal3geometry,indicatingthatthegeometricfrustrationduetotheconstraintsofglobaljammingonthelocalgeometryisnontrivial.II.THEORYApackingofNhardspheresofdiameterDind-dimensionalEuclidianspaceischarac-terizedbythe(Nd)-dimensionalconfigurationvectorofcentroidpositionsR=(r1,...,rN).Herewefixthecenterofmassofthepacking(withperiodicboundaryconditions),sothatinfacttheconfigurationspaceisofdimension(N−1)d.However,wewillusuallyneglectorderunitytermscomparedtoN.Theboundaryconditionsimposeddeterminethevolumeoftheenclosing“container”Vandthepacking(covering)fraction,ordensity,φ.Ajammedpackingi