Dissipative Particle Dynamics with Energy Conserva

arXiv:cond-mat/0002237v1[cond-mat.stat-mech]16Feb2000DynamicandTransportPropertiesofDissipativeParticleDynamicswithEnergyConservation.J.BonetAvalosandA.D.MackieDepartamentd’EnginyeriaQu´ımica,ETSEQUniversitatRoviraiVirgili.CarreteradeSalous/n,43006Tarragona(Spain)SimulationresultsofthethermalconductivityLofDissipativeParticleDynamicsmodelwithEnergyConservation(DPDE)arereported.WealsopresentananalysisofthetransportequationsandthetransportcoefficientsforDPDEbasedonalocalequilibriumapproximation.Thisapproachisvalidwhentheparticle-particlethermalconductivityλandthefrictioncoefficientζarelarge.Aqualitativederivationofthescalingformofthekineticcontributionofthetransportofenergyisderived,yieldingtwodifferentformsforthekineticcontributiontotheheattransport,dependingonthevalueofλ.WefindagreementbetweenthetheoreticallypredictedvalueforLandthesimulationresults,forlargeλandmanyparticlesinteractingatonetime.Significantdifferencesarefoundforsmallnumberofinteractingparticles,evenwithlargeλ.Forsmallervaluesofλ,theobtainedmacroscopicthermalconductivityisdominatedbydiffusivetransport,inagreementwiththeproposedscalingform.I.INTRODUCTIONThemodelforthesimulationofthedynamicsofcomplexsystems,knownasDissipativeParticleDynamics(DPD),hasundergoneanimportantdevelopmentinrecentyearssinceitwasintroducedin1992byHoogerbruggeetal.[1].Thisoff-latticesimulationmethodologyisespeciallyaddressedatthemodelingofthebehavioroffluidsatascalewherefluctuationsareimportant.Thus,themethodissuitableforboththestudyofthedynamicsofsmallsystemsandinsituationsdominatedbyBrownianmotion,forinstance.Thepotentialofthismethodologywasrealizedfromthebeginningand,asaconsequence,DPDhasalreadybeenappliedtothestudyofthedynamicsofsystemsofpracticalinterestsuchaspolymericmaterials[4–6],polymeradsorption[7],colloidalsuspensions[8],multicomponentsystems[9],amongothers.Despitetheadvancescarriedouttowardsacompleteunderstandingofthefeaturesofthisnewmethodology,therearestillunclearaspects,mainlyconcernedwiththelarge-scaletransportpropertiesofthemodel.Thestudyoftheseproperties,relativetotheextendedversionoftheDPDalgorithmwhichincludesheattransport[2,3],isthesubjectofthispaper.IntheDPDmodel,anensembleofparticles,viewedasmesoscopicrepresentationsofflocksofphysicalmolecules,interactwitheachotherthroughconservativeaswellasdissipativeforces.Randomforcesarealsoexplicitlyconsideredtomaintaintheparticlesinconstantagitationevenintheabsenceofexter-nalforcefields.Inthismethod,theparticle-particleinteractionsaresuchthatthetotalmomentumisconserveddespitethepresenceofdissipativeandrandomforcesand,therefore,themacroscopichydro-dynamicbehavioratlong-wavelengthandlong-timescalesisguaranteed.Inaddition,sincethemodelisnotdefinedonalattice,itisGalileaninvariantandhasnoextraconservationlawsapartfromthosethatarephysicallyrelevant.Togetherwiththetheoreticaladvantages,theDPDsimulationmethodologycanbeimplementedusingonlylocalinteractions,whichmakesitfasttorunonacomputer,asinLatticeGascellularautomata[10],forinstance.TheoriginalDPDmethodwasonlyconcernedwiththeconservationofthetotalmomentumoftheparticlessothatthemodelcouldreproducecontinuityandNavier-Stokesequationsatthemacroscopiclevel.Severalrefinements,suchastodemandthattheequilibriummomentumdistributionoftheparticlesbeoftheMaxwell-Boltzmanntype[11],wereaddedaimingatacloserrelationshipofthemodelwithrealphysicalsystems.However,theenergytransportand,therefore,theheatflow,couldnotbemodeledsincetheDPDmodelwasessentiallyisothermal.Anextentionofthemodeltoalsoincorporatetheconservationofthetotalenergyintheparticle-particle1interaction,hasbeenproposedinref.[2],andlaterinref.[12].Morerecently,wehaverefinedtheoriginalalgorithmtoensureenergyconservationateverytime-step,insteadofinthemean[3].InourversionoftheDPDmodelwithenergyconservation(referredtoasDPDEfromnowon),togetherwiththepositionandmomentum,aninternalenergyofeachparticleisexplicitlyconsideredasarelevantvariable.Withthedefinitionofaparticle’sinternalenergy,anenergybalancecanbeestablishedsothattheworkdonebythedissipativeandrandomforcesisstoredorreleasedfromtheparticle’sinternalenergy,andthusthetotalenergyisheldconstantintheparticle-particleinteraction.Therefore,asaconsequence,theadditionoftheenergyconservationtothemomentumconservationintheparticle-particleinteractionallowstheDPDEmodeltodescribeheattransporttogetherwiththemomentumtransport.Hence,thefivehydrodynamicfields,density,momentaandenergy,issuedfromtheconservationlawsatamicroscopiclevel,canbeproperlyreproducedbytheDPDEmodel.InthispaperweaddressthestudyofthetransportpropertiesoftheDPDEmodelfromboth,theoreticalaswellassimulationpointsofview.InthecaseoftheisothermalDPD,muchworkhasbeendevotedtotheanalysisofthetransportpropertiesofthemodel,especiallywithregardtothecalculationoftheshearviscosity[13–15],towardsaqualitativeandquantitativeunderstandingofthelong-wavelengthlong-timedynamicbehaviorofthemodel.SignificantprogresshasbeendoneinthisdirectionbytheformulationofBoltzmann-typeequations,whichhavebeensolvedbymeansofdifferentapproximations.Inparticular,inref.[13]thetransportequa