An Introduction to Kinetic Monte Carlo Simulations of Surface Reactions PDF
Preview An Introduction to Kinetic Monte Carlo Simulations of Surface Reactions
Lecture Notes in Physics Volume 856 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard B.-G.Englert,Singapore,Singapore U.Frisch,Nice,France F.Guinea,Madrid,Spain P.Hänggi,Augsburg,Germany W.Hillebrandt,Garching,Germany R.A.L.Jones,Sheffield,UK M.S.Longair,Cambridge,UK M.L.Mangano,Geneva,Switzerland J.-F.Pinton,Lyon,France J.-M.Raimond,Paris,France A.Rubio,Donostia,SanSebastian,Spain M.Salmhofer,Heidelberg,Germany D.Sornette,Zurich,Switzerland S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany H.vonLöhneysen,Karlsruhe,Germany W.Weise,Garching,Germany Forfurthervolumes: www.springer.com/series/5304 The Lecture Notes in Physics TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelop- ments in physics research and teaching—quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. 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Proposals should be sent to a member of the Editorial Board, or directly to the managingeditoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] A.P.J. Jansen An Introduction to Kinetic Monte Carlo Simulations of Surface Reactions A.P.J.Jansen ST/SKA EindhovenUniversityofTechnology Eindhoven,Netherlands ISSN0075-8450 ISSN1616-6361(electronic) LectureNotesinPhysics ISBN978-3-642-29487-7 ISBN978-3-642-29488-4(eBook) DOI10.1007/978-3-642-29488-4 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012940387 ©Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Preface Kinetic Monte Carlo (kMC) simulations form still a quite new area of research. Figure1showsthenumberofpublications(articlesorreviews)with“kineticMonte Carlo”inthetitleorabstractaccordingtotheabstractandcitationdatabaseScopus. There are two things to note. On the one hand it is not a very extensive area of researchyet.Averydiligentresearchercanstillkeeptrackofallpublicationsthat appear.Ontheotherhand,thenumberofpublicationsisrapidlygrowing. Figure 1 shows that there were no publications before 1993 that used the term kMC.ThisdoesnotmeanthattherehavebeennokMCsimulationsbeforethatyear. Therehavebeensomebutthetermwasnotusedyet.Infact,therearestillpeople, whodowhatwewillcallkMCsimulationshere,butwhodonotusetheterm.One mundanereasonforthatisprobablythattheyuseanalgorithmthattheyregardas one of many possible algorithms for doing Monte Carlo (MC) simulations. Why give it a special name? Another reason may be historical. Instead of kMC, people have used and still use the term dynamic MC. This is a term introduced by D.T. GillespieforhisalgorithmsthatuseMCtosolvemacroscopicrateequations.These algorithms are often almost identical to the ones we will describe in Chap. 3, and it seems reasonable to use the same term even when the algorithms are used for differentproblems.Therehasbeenatendencytobemorestrictintheterminology however.Forexample,thetermStochasticSimulationAlgorithmisnowoftenused whenusingMCforrateequations.ThereareevenpeoplethatrestrictthetermkMC tooneparticularalgorithm,theVariableStepSizeMethodinourterminology(see Sect.3.2),eventhoughallotheralgorithmsinChap.3giveexactlythesameresults. ButthetermkMChasalsobeenusedforrateequations.Sothesituationconcerning terminologyisstillfluent. SowhatdowemeanwhenweusethetermkMC?Therearealwaystwoaspects tokMCaswewilldiscussithere.Wewillregardasystemasasetofminimaofa potential-energysurface(PES).Theevolutionofasysteminrealtimewillthenbe regardedashopsfromoneminimumtoaneighboringone.Thesearetheelementary eventsofkMC.Thesecondaspectconcernsthealgorithms.ThehopsinkMCwillbe seentobestochasticprocessesandthealgorithmsuserandomnumberstodetermine atwhichtimesthehopsoccurandtowhichneighboringminimumtheygo.Thisis vii viii Preface Fig. 1 Number of publications (articles or reviews) with “kinetic Monte Carlo” in the title or abstractasafunctionoftheyearofpublication ourgeneraldefinitionofkMC.WewilluseithoweveronlyinChap.2andSect.8.4. In the rest of this book we will make an additional assumption. This is where the surfacereactionsinthetitleofthisbookcomein.Thesurfaceonwhichthereactions take place is often periodic and has translational symmetry in two directions. The minimaofthePESarerelatedtotheadsorptionsitesofthesurface.Thelatterform alatticeandthereactionscanbemodeledwithalattice-gasmodel.Wewillseethat thisisevenpossibleiftheperiodicityofthesurfaceisnotperfect.SokMCinthis bookstandsforalattice-gasmodelthatdescribestheevolutionofthesysteminreal timeandwithelementaryeventsthatarestochasticandthatcorrespondtoreactions andotherprocesses. This book has two objectives. First, it is about the kMC method. A derivation ofthemethodwillbegivenfromfirstprinciples,andwewilldiscussvariousalgo- rithmsthatcanbeusedtodoactualsimulations.Thismeansthatmuchofthebook is also supposed to be useful to people who use kMC for other systems than sur- facereactions.Forexample,thederivationofthemasterequationinChap.2,which forms the basis of our theory of kMC, does not use any information particular to surfacereactions.Itonlyassumesthatyouhaveasystemthatcanbedescribedbya single-valuedPES.Thisincludesaverylargemajorityofallsystemsoneencounters inchemicalphysics.Chapter8alsohasasectionthatdiscusseskMCforwhenthis isalloneknowsaboutasystem. Most of the book does however assume that a lattice-gas model is used, be- causethissimplifiestheapplicabilityofkMCenormously.However,thisstilldoes not restrict the usefulness only to surface reactions. In fact, most publications us- ing lattice-gas kMC are not about surface reactions. There are many applications Preface ix ofkMCincrystallinesolids,polymers,crystalgrowth,chemicalvapordeposition, molecular-beamepitaxy,ionimplantation,etching,nanoparticles,andnon-reactive processes.ThediscussionsofalgorithmsinChap.3andthewayprocessescanbe modeledinChaps.5,6,7,and8arejustasusefulforthoseapplicationsasforsur- face reactions. However, the second objective of this book is to show what kMC simulations can teach us about the kinetics of surface reactions that one finds in catalysisandsurfacescience.Thebookwasmainlywrittenwiththisinmind.This means that there are aspects that are relevant for the application of kMC to other areasthatwillnotbefoundhere,whereassomeaspectsthatarediscussedheremay notberelevantfortheseareas. The book is called an introduction because it is meant to give all information on kMC simulations of surface reactions that you need if you want to start from scratch. A lot of space is devoted to the basics, which are discussed in detail. The term“introduction”isnotmeanttoimplythateverythinginthisbookislowlevel oreasy.Somethingsarebutothersdefinitelyarenot.Itisforexamplequiteeasyto implementthealgorithmsofChap.3forasimplesystemofsurfacereactions,and theresultingcodewillprobablyyieldveryusefulandinterestinginformationonthe kineticsofthesystem.Writingageneral-purposecodehoweverismuchharder.Also thetheoreticalderivationofthemasterequationonwhichwebasekMC,advanced aspectsofthealgorithms,andcertainnewdevelopmentsinChap.8areanythingbut easy. Thestructureofthisbookisasfollows.Chapter1discusseswhyonewouldwant todokMCsimulations.Thekineticsofsurfacereactionsisnormallydescribedwith macroscopicrateequations.Therearedifferentwaysinwhichtheseequationscan beused,butitisshownthattheyallhavesubstantialdrawbacks. Chapter 2 deals with the basic theory. It introduces the lattice gas as the model forthesystemsinthisbook,anditgivesthederivationofthemasterequation.This isthecentralequationforkMC.ItformsthebasisofallkMCalgorithms,itrelates quantumchemicalcalculationsofrateconstantstokMC,anditrelateskMCtoother kinetictheorieslikemicrokinetics. Chapter 3 discusses kMC algorithms. kMC generates a sequence of configura- tionsandtimeswhenthetransitionsbetweentheseconfigurationsoccur.Thissolves themasterequation.Therearemanyalgorithmsthatyieldsuchasequenceofcon- figurationsandwhicharestatisticallyequivalent.Wediscussafewindetailbecause theyaretheonesthatareefficientformodelsofsurfacereactions.Time-dependent rateconstantsarediscussesseparatelyasthedeterminationofwhenprocessestake placeposespecialproblems.Parallelizationisdiscussedaswellassomeolderalgo- rithms.Someguidelinesaregivenofhowtochooseanalgorithmforasimulation. Chapter 4 shows how the rate constants that are needed for kMC simulations canbeobtained.Itshowshowrateconstantscaneitherbecalculatedorbederived fromexperimentalresults.Calculatingrateconstantsinvolvesdeterminingtheinitial andthetransitionstateofaprocess,theenergiesofthesestates,andtheirpartition functions.Thephenomenologicalormacroscopicequationistheessentialequation togetrateconstantsfromexperiments.Lateralinteractionscanaffectrateconstants substantially,butbecausetheyarerelativelyweakandspecialattentionneedstobe giventothereliabilityofcalculationsoftheseinteractions. x Preface Chapters5and6discusswaystomodelsurfaceprocesses.Thesechaptersdeal withthesametopic,butapproachitfromdifferentangles.Chapter5showsthetools thatwecanuseinmodeling.Forsimplesystemsthereisalatticecorrespondingto the adsorption sites and the labels of the lattice points describe the occupation of thesites.Thelabelscanhoweveralsobeusedtomodelstepsandotherdefectsand sites on bimetallic substrates. The lattice points don’t need to correspond to sites however,butcanalsobeusedtostoreotherinformationlikethepresenceofcertain structures in the adlayer. Processes need not always to correspond to reactions or otheractualprocesses,butwhentheyhaveaninfiniterateconstanttheycanbeused in a general-purpose code to handle exceptional situations that are normally hard- codedinspecial-purposecodes. Chapter6discussestypicalsurfaceprocessesandhoweachofthemcanbemod- eled in different ways using the tools from Chap. 5. The way to model many pro- cesses for kMC simulations is straightforward. There are however also processes that one encounters regularly and for which there are more modeling options and forwhichitisnotalwaysclearwhichthebest.Wediscussseveralofthem. Chapter 7 shows how the modeling of various surface processes can be inte- grated. We discuss a number of complete surface reaction systems and show the benefitsofkMCsimulationsforthem.Chapter8finallydiscussessomeaspectsof kMC that one might want to improve and some likely new developments. kMC is averyversatileandpowerfulmethodtostudythekineticsofsurfacereactions,but thereareneverthelesssomesystemsandphenomenaforwhichonewouldlikeitto beevenmoreefficientoronewouldliketoextendit. Geldrop,Netherlands TonekJansen Acknowledgements This book would never have been written without the support of many people. I would like to thank Rutger van Santen, Risto Nieminen, Juha-Pekka Hovi, Hans Niemantsverdriet, Vladimir Kuzovkov, Mark Koper, and Rasmita Raval for many discussionsonthekineticsofsurfacereactions.PeterHilbersandJohnSegerscon- tributed a lot to my understanding of the algorithms of kinetic Monte Carlo. The Ph.D.studentsandpostdocsRonaldGelten,RafaelSalazar,SilviaNedea,Cristina Popa,SandervanBavel,JorisHagelaar,MaartenJansen,ZhangXueqing,andMin- hajGhourihavetaughtmealotabouthowtomodelsurfacereactions.Inparticular theyhavetaughtmetotrustourcomputercodesalsoforverycomplicatedsystems. IamgratefultoIvoFilotforhiscriticalcommentsonChap.4.Specialthanksgoto ChrétienHermseandJohanLukkienforendlessdiscussions,unfailingsupportand enthusiasm even after they stopped working on kinetic Monte Carlo themselves. ChrétienhascomeupwithmanyofthemodelingtrickswithoutwhichChaps.5,6, and7wouldhavelookedverydifferent.Johanhasbeencrucialinthedevelopment ofthealgorithmsthatcanbefoundinChap.3.HehasalsowrittentheCarlospro- gramthathasbeenessentialforalmostalloftheworkinkineticMonteCarlothat I have done. Finally I would like to thank friends and family for bearing with me when I was preoccupied with matters kinetic Monte Carlo when I shouldn’t have been. Ihavetogivecreditformanyofthegoodideasinthisbooktothepeoplemen- tionedabove,butI doclaimthatthebadideas, errors, andothershortcomingsare allmine. xi
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