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BasicsofmoleculardynamicsEquationsofmotionforMDsimulationsTheclassicalMDsimulationsboildowntonumericallyintegratingNewton’sequationsofmotionfortheparticlesNixVxdtxdmijiiiji022Lennard-JonespotentialOneofthemostfamouspairpotentialsforvanderWaalssystemsistheLennard-Jonespotential6124)(ijijijrrrLennard-JonesPotentialDimensionlessUnitsAdvantagesofusingdimensionlessunits:thepossibilitytoworkwithnumericalvaluesoftheorderofunity,insteadofthetypicallyverysmallvaluesassociatedwiththeatomicscalethesimplificationoftheequationsofmotion,duetotheabsorptionoftheparametersdefiningthemodelintotheunitsthepossibilityofscalingtheresultsforawholeclassofsystemsdescribedbythesamemodel.DimensionlessunitsWhenusingLennard-Jonespotentialsinsimulations,themostappropriatesystemofunitsadoptsσ,mandεasunitsoflength,massandenergy,respectively,andimpliesmakingthereplacements:IntegrationoftheNewtonianEquationClassesofMDintegrators:low-ordermethods–leapfrog,Verlet,velocityVerlet–easyimplementation,stabilitypredictor-correctormethods–highaccuracyforlargetime-stepsBoundaryConditionPeriodicboundaryconditionSpecularboundaryconditionInitialstateAtomsareplacedinaBCC,FCCorDiamondlatticestructureVelocitiesarerandomlyassignedtotheatoms.Toachievefasterequilibrationatomscanbeassignedvelocitieswiththeexpectedequilibriumvelocity,i.e.Maxwelldistribution.Temperatureadjustment:Bringingthesystemtorequiredaveragetemperaturerequiresvelocityrescaling.Gradualenergydriftdependsondifferentfactors-integrationmethod,potentialfunction,valueoftimestepandambienttemperature.ConservationLawsMomentumandenergyaretobeconservedthroughoutthesimulationperiodMomentumconservationisintrinsictothealgorithmandboundaryconditionEnergyconservationissensitivetothechoiceofintegrationmethodandsizeofthetimestepAngularmomentumconservationisnottakenintoaccountEquilibrationForsmallsystemswhosepropertyfluctuateconsiderably,characterizingequilibriumbecomesdifficultAveragingoveraseriesoftimestepsreducesthefluctuation,butdifferentquantitiesrelaxtotheirequilibriumaveragesatdifferentratesAsimplemeasureofequilibrationistherateatwhichthevelocitydistributionconvergestotheexpectedMaxwelldistribution)2/exp()(21TvvvfdVelocitydistributionasafunctionoftimeInteractioncomputationsAllpairmethodCellsubdivisionNeighborlistsThermodynamicpropertiesatequilibriumThermophysicalandDynamicPropertiesatequilibriumExample:CalculationofthermalconductivityateqilibriumFollowingaretheequationsrequiredtocalculatethermalconductivity:ijjijjiiiiEtrvFvQ)(NjBpttjVTkk12)()0(1QQiNjtjitjiNti1))(()(1)()0(QQQQWhereQistheheatfluxWherekisthethermalconductivityNonequilibriumdynamicsHomogeneoussystem:nopresenceofphysicalwall,allatomsperceiveasimilarenvironmentNonhomogeneoussystem:presenceofwall,perturbationstothestructureanddynamicsinevitableNonequlibriummoreclosetothereallifeexperimentswheretomeasuredynamicpropertiessystemsareinnonequilibriumstatesliketemperature,pressureorconcentrationgradientExample:Calculationofthermalequilibriumatnonequilibrium(directmeasurement)TomeasurethermalconductivityofSiliconrod(1-D)heatenergyisaddedatL/4andheatenergyistakenawayat3L/4Afterasteadystateofheatcurrentwasreached,theheatcurrentisgivenbyUsingtheFourier’slawwecancalculatethethermalconductivityasfollowsStillinger-WeberpotentialforSihasbeenusedwhichtakescareoftwobodyandthreebodypotentialtAJz2zTJkz/Example:continued..200220240260280300320020406080100120140160180200Series1y=0.2686x+233.34R2=0.9384220230240250260270280405060708090100110120130MolecularDynamicsSimulationofThermalTransportatNanometerSizePointContactonaPlanarSiSubstrateCalculatedtemperatureprofileintheSisubstratefora0.5nmdiametercontactradius.Schematicdiagramofthesimulationbox.Initialtemperatureis300K.Energyisaddedatthecenterofthetopwallandremovedfromthebottomandsidewalls.ThermalconductivityofnanofluidsatEquilibriumSchematicdiagramNanoparticlesResultsResultscontinuedReferencesRapaportD.C.,“TheArtofMolecularDynamicsSimulation”,2ndEdition,CambridgeUniversityPress,2004W.J.MinkowyczandE.M.Sparrow(Eds),“AdvancesinNumericalHeatTransfer”,vol.2,Chap.6,pp.189-226,Taylor&Francis,NewYork,2000.Koplik,J.,Banavar,J.R.&Willemsen,J.F.,“MoleculardynamicsofPoiseuilleflowandmovingcontactlines”,Phys.Rev.Lett.60,1282–1285(1988);“Moleculardynamicsoffluidflowatsolidsurfaces”,Phys.FluidsA1,781–794(1989).