A Model for Hybrid Simulations of Molecular Dynami

arXiv:0803.0099v3[cond-mat.soft]13Mar2008AModelforHybridSimulationsofMolecularDynamicsandCFDShugoYasuda∗andRyoichiYamamoto†DepartmentofChemicalEngineering,KyotoUniversity,Kyoto615-8510,JapanandCREST,JapanScienceandTechnologyAgency,Kawaguchi332-0012,Japan.(Dated:March13,2008)AbstractWeproposeamethodformulti-scalehybridsimulationsofmoleculardynamics(MD)andcom-putationalfluiddynamics(CFD).Inthemethod,usuallattice-meshbasedsimulationsareappliedforCFDlevel,buteachlatticeisassociatedwithasmallMDcellwhichgeneratesa“localstress”accordingtoa“localflowfield”givenfromCFDinsteadofusinganyconstitutivefunctionsatCFDlevel.WecarriedoutthehybridsimulationsforsomeelementalflowproblemsofsimpleLennard-JonesliquidsandcomparedtheresultswiththoseobtainedbyusualCFDswithaNew-tonianconstitutiverelationinordertoexaminethevalidityofourhybridsimulationmethod.Itisdemonstratedthatourhybridsimulationssuccessfullyreproducedthecorrectflowbehaviorob-tainedfromusualCFDsasfarasthemeshsizeΔxandthetime-stepΔtofCFDarenottoolargecomparingtothesystemsizelMDandthesamplingdurationtMDofMDsimulationsperformedateachtimestepofCFDs.Otherwise,simulationsareaffectedbylargefluctuationsduetopoorstatisticalaveragestakenintheMDpart.Propertiesofthefluctuationsareanalyzedindetail.PACSnumbers:31.15.xv46.15.-xKeywords:multi-scalesimulation,hybridsimulation,moleculardynamicssimulation,computationalfluiddynamics,fluctuatinghydrodynamics,complexfluids∗Electronicmail:[e-mail]†Electronicmail:[e-mail]sofcomplexfluidsareofparticularimportanceinvariousscienceanden-gineeringfields,suchasfluidmechanics,softmatterscience,mechanicalengineering,chem-icalengineering,andsoon.Becauseofthecomplicatedcouplingsbetweeninternaldegreeoffreedomsofcomplexfluidsandtheirflowbehavior,conventionaltreatments,basedonusualassumptionssuchasnon-slipboundaryconditionsandlinearNewtonianconstitutiverelations,areofteninvalid.Strikingexamplescanbeseeninsystemssuchascolloidaldisper-sions,polymericliquids,granularmatters,andliquidcrystals.Thosesystemsareknowntoexhibitpeculiarflowbehaviors,e.g.,shearthinningorthickening,viscoelasticity,jamming,flowinducedphasetransition,etc.Althoughthereexitshugeaccumulationofexperimentalandtheoreticalstudiesontherheologyofcomplexfluids,performingcomputationalfluiddynamics(CFD)simulationsarenotyetcommonforcomplexfluidssincereliableconstitutiveequationsareoftenunknownforthosesystems.Ontheotherhand,thereexistsadifferentproblemalsoformicroscopicapproachessuchasmoleculardynamics(MD)simulations,whileconstitutiveequationsarenomorenecessaryinthiscase.Thecharacteristictimeandlengthscalesofcomplexfluidseasilybecomeseveralorderslargerthanthoseofmicroscopicscales.Therefore,mosthydro-dynamicproblemsofcomplexfluidsareyetoutofreachofmicroscopicMDsimulations.Toovercomethoseseriouslimitationsmentionedabove,weaimtodevelopanewmulti-scalemethodwhichisforperforminghybridsimulationsofMDandCFDvalidforcomplexfluidswithoutanyconstitutiveequations.VariousmethodsforhybridsimulationsofMDandCFDhasalreadybeenproposedbyseveralresearchers.Mostofthosemethodsarebasedon“domaindecomposition”forwhichMDsimulationsareappliedonlyaroundthepointsofinterest,i.e.,inthevicinityofdefects,boundaries,interfaces,wheredetailsofmolecularmotionsareimportant,whiletheremainingregionsaretreatedonlybyCFD.1,2,3,4,5,6,7,8,9,10ExchangeofinformationbetweenMDandCFDisperformedinacouplingregionswhereeachsystemissubjectedtosomeconstrainstotaketheconsistencyofthetwosystems.Thiskindofhybridmethodisexpectedtobeusefulspeciallyforproblemsincludinginterfaces,suchasadhesion,friction,anchoringofcrystalliquids,stick-slipmotions,etc.InordertoapplyhybridmethodsofMDandCFDtohydrodynamicsofcomplexfluids,2adifferenttypeofapproachisprobablyneeded.Ourstrategyforthisisstraightforward.Wetrytodevelopamulti-scalehybridmethodbasedonthelocalequilibriumassumption.HereCFDisusedasafluidsolver,whileMDsimulationsareusedonlytogeneratelocalproperties,suchasconstitutiverelationsofthefluidunderconsideration,byperforminglocalstatisticalsamplinginaconsistentmatter.Thenumericalalgorithmisrathersimple.Weperformusuallattice-meshbasedCFDsimulationsatanupperlevel,buteachmesh-nodeisassociatedwithasmalllowerlevelMDcellwhichpassesa“localstress”toCFDaccordingtoa“localflowfield”givenfromCFDtoMDinsteadofusinganyconstitutivefunctionsatCFDlevel.MDsimulationsthushavetobeperformedatallnodepointsandateverytimestepsofCFD.OnemightthinkthatthesimulationswouldbebemuchfasterifweconstructtabulardatabaseoftheconstitutiverelationsbyperformingMDsimulationsinadvanceundermanydifferentsimulationparametersandreferthetablefromCFD.The“tabularapproach”worksmucheffectiveforsimplefluidsforwhichtheconstitutiverelationsdependonlyonafewparameters,suchasthedensity,temperature,andshearrate.Inthecaseofcomplexfluids,however,thenumberofparameterstobeconsideredcanbehugedependingonthelocalquantitiestobeconsidered.Inthecaseofchargedsystemsforexample,thelocalstressdependsalsoonthelocalcompositionsandchemicalpotentialsofionsandthelocalelectricfield,etc.AlthoughweusedonlysimpleLennard-J