Multiscale Modeling of Materials - Concepts and Il

arXiv:cond-mat/0507558v1[cond-mat.mtrl-sci]24Jul2005MultiscaleModelingofMaterials-ConceptsandIllustrationAditiMallik,KeithRunge,JamesW.Dufty,andHai-PingChengDepartmentofPhysics,UniversityofFlorida,Gainesville,FL32611(Dated:February2,2008)AbstractTheapproximaterepresentationofaquantumsolidasanequivalentcompositesemi-classicalsolidisconsideredforinsulatingmaterials.Thecompositeiscomprisedofpointionsmovingonapotentialenergysurface.Intheclassicalbulkdomainthispotentialenergyisrepresentedbypairpotentialsconstructedtogivethesamestructureandelasticpropertiesastheunderlyingquantumsolid.Inasmalllocalquantumdomainthepotentialisdeterminedfromadetailedquantumcalculationoftheelectronicstructure.Theformulationofthisdescriptionasasequenceofphysicalapproximationsisconsideredinsomedetail.Theprimarynewingredientsare1)adeterminationofthepairpotentialfromquantumchemicaldataforequilibriumandstrainedstructures,2)developmentof’pseudo-atoms’forarealistictreatmentofchargedensitieswherebondshavebeenbrokentodefinethequantumdomain,and3)inclusionofpolarizationeffectsonthequantumdomainduetoitsenvironment.ThisformalstructureisillustratedindetailforanSiO2nanorod.Foreachconfigurationconsidered,thechargedensityoftheentiresolidiscalculatedquantummechanicallytoprovidethereferencebywhichtojudgetheaccuracyofthemodeling.Theconstructionofthepairpotential,therod,thepseudo-atoms,andthemultipolesisdiscussedandtestedindetail.Itisthenshownthatthequantumrod,therodconstructedfromtheclassicalpairpotentials,andthecompositeclassical/quantumrodallhavethesameequilibriumstructureandresponsetoelasticstrain.Inmoredetail,thechargedensityandforcesinthequantumsubdomainareaccuratelyreproducedbytheproposedmodelingoftheenvironmentaleffectsevenforstrainsbeyondthelineardomain.Theaccuracyofthemodelingisshowntoapplyfortwoquitedifferentquantumchemicalmethodsfortheunderlyingquantummechanics:transferHamiltoniananddensityfunctionalmethods.1I.INTRODUCTIONThefieldofmultiscalemodelinghasopenedaneweratocomputationalscienceforstudy-ingcomplexphenomenalikefracture,hydrolysis,enzymaticreactions,solute-solventstudies,hydrogenembrittlement,andmanyotherchemo-mechanicalprocessesinmacroscopicsam-ples.Oftenthesephenomenarequireaveryaccuratedescriptionatonescale,whileatanotherscaleamuchcoarsermethodcanbeappliedtogetsatisfactoryresults.Infact,thecoarserdescriptionforthebulksampleisrequiredbecausethemorefundamentalmethodsarecomputationallytoointensivetobeappliedtotheentiresystem.Thesescalesmaybelengthortime,oracombinationofboth.Thisschemeofcombiningdifferentmodelsatdifferentscalestoachieveabalanceofaccuracy,efficiency,andrealisticdescriptionisknownasmultiscalemodeling.Itisaccomplishedbyapplyingthehighaccuracymethodonlyinasmalldomainwhereitisneededthemostandmoreapproximatemethodsfortherestofthebulkwheretheyareappropriate.Forexample,incrackpropagationonehastoapplydetailedquantummechanicsatthetipofthecrack.Herethebondsarebreakingbetweentheatoms,causingamarkeddeformationoftheelectroncloudandchargetransferbetweenions.Butfarfromthecracktipwheretheatomsarelessdeformedtheycanbedescribedbyclassicalmechanicswithappropriatepotentials.Thus,theproblemarisesoflinkingquitedifferentdescriptionsforthedifferentlengthscales1.Thereisabroaddiversityofotherexamplesrequiringmultiscalemodeling:chemo-mechanicalpolishing2,tidalwaveprediction3,atmosphericsciences,embrittlementofnu-clearreactors4,andmanybiologicalsystems5.Twoclassescanbedistinguished6,“serialmultiscalemodeling”wherethevariousscalesareweaklycoupledbutthecomputationofparametersatsmallerscalesisrequiredforitsuseinmorephenomenologicalmodelsatalargerscale,and“concurrentmultiscalemodeling”wherethevariousdescriptionsappliedondifferentscalesshouldallbenestedwithproperboundaryconditions.Themultiscalemodelingdiscussedhereisofthe“concurrent”type,althoughtherearecomponentsthatcanbeconsideredasthe“serial”type(e.g.,parameterizationoftheunderlyingapproximatequantummethodused).Threedifferentlengthscalelevelsaredistinguished,thenanoscalewherethedetailsofelectronstructureandquantumchemistryareimportant,theatomicormicroscalewhereap-propriateclassicalpairpotentialscapturethestructureaccurately,andthemacroscalewhere2acontinuummechanicaldescriptionapplies.Thepresentworkaddressesonlyonepartic-ularsubclassofmultiscalemodeling,knownashybridquantummechanical(QM)/classicalmechanical(CM)problems,andtheirapplicationtosolidinsulators-silicainparticular.Thefurtherembeddingoftheatomicscaleinamacroscalemodelisnotconsideredhere.EarlyworkonthisQM/CMtopicbeganwithWarshelandLevitt7andhasacceleratedoverthepastdecade8.ApplicationsofQM/CMincludebiologicalsystemswherethisschemeissometimesreferredtoasQM/MM(molecularmechanics)modeling(enzymes,DNAsandproteins)9,10,11,12,13,vibrationalspectroscopy14,electronicexcitations8,15,16,hydrolysisofsilica17,18,andsolute-solventproblems19,20,21.ThechallengeoftheQM/CMsimulationistomovefromonelengthscaletoanother,assmoothlyaspossible.Asolidorlargeclusterbehavesasasingle”molecule”sothepartitioningbecomesmorecomplicatedduetocovalentbondsthatarecutbetweentheQMandCMregions.T