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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
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ForEsther.
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
