MonteCarloSimulationofDefectDiffusion

inFCCCrystals

Florida Society for Materials Simulations REU Program

Research Report

CheraRogers

WestVirginiaWesleyanCollege

Buckhannon,WestVirginia

Host:Dr.AnterEl‐Azab

DepartmentofScientificComputing

FloridaStateUniversity

Tallahassee,Florida

Abstract

Westudiedthediffusionofpointdefectsona2DsquarelatticebyMonteCarlo

method(Randomwalktheory).Inthisstudywehavestudiedtheeffectofensemble

sizeonacalculationofdiffusioncoefficient.Thesimulationsshowthatwhenwe

increasetheensemblesize,weconvergetostatisticallyreliableestimateofdiffusion

coefficient.Thisworkisafirststeptowardsunderstandingtheclusteringofdefects

insolids.

Introduction

Inaperfectcrystal,massandchargedensityhavetheperiodicityofthe

lattice.Theatomsinsolidsarrangethemselvesintoacrystallinestructure;perfect

crystalshavetheiratomsarrangedalongaperiodiclattice.Howeversolidsin

naturearenotperfect.Theyhavedefects.Thecreationofapointdefector

extendeddefectdisturbsthisperiodicity.Twoimportanttypesofpointdefectsare

vacancies,whereanatomismissingfromalatticesite,andinterstitials,wherean

atomisplacedbetweenlatticesites.Insolids,pointdefectsarescattered

throughoutthematerial,andtheirconcentrationisexponentiallydependentonthe

temperatureandtheenergyittakestoformthecrystal.Foraparticularmaterial

theformationenthalpy,Gfj,isconstant. Theconcentrationforpointdefectoftype

jisgivenbythefollowing.

C j expG j

f

kBT

Asyoucanseefromtheconcentrationequation,asweincreasethe

temperature,T,theconcentrationalsoincreases.Soinorderfortheatomsto

overcometheenergybarrierkeepingthemintheirlatticesites,weneedtoincrease

thetemperaturetocausetheatomstobecomemoreenergetic.

Inordertounderstanddefectsincrystalsweneedtounderstandhowthey

moveandspreadthroughthecrystal,soweneedtounderstanddiffusion.

Diffusion

Diffusionisthespreadofparticlesthroughrandommotionfromregionsof

highconcentrationtoregionsoflowconcentration.Diffusionisallaroundus,from

microscopicsystemstosituationsinoureverydaylives.Forillustration,imaginean

elevatoriscrowdedwithpeople(highconcentrationregion).Assoonasthe

elevatordoorsopen,thepeoplepushoutthroughthedoorsanddisperseinthe

emptyhall(lowconcentrationregion).Diffusionisalsopresentwhenmixingtwo

misciblefluids,asininkinwaterorcreaminyourcoffee.Intheseexamples,the

diffusingbodiesmoveinanydirectionandrandomly.Incontrast,inacrystalline

material,themovementofdiffusingdefectsisrestrictedbythesurroundinglattice.

AdolfFickfirststudiedtheprocessofdiffusion,andhedevelopedthe

mathematicalframeworktodescribethephenomenonofdiffusion.Heintroduced

theconceptofdiffusioncoefficient,D,andsuggestedalinearresponsebetweenthe

concentrationgradientandflux,J.ThisisknownasFick’sfirstlaw:

CDJ

Fick’sfirstlawissimilartothelawsgoverningothertypesofflowinnaturesuchas

Fourier’sLawandOhm’slaw.

Inthediffusionprocessthenumberofdiffusingparticlesisconserved,which

meansthatthedifferencebetweenthenumberofparticlesflowingintoaregionand

outofaregionresultsinanaccumulationofparticlesintheregion.Therateofnet

inflowtoaregiongivesthetimerateofchangeofconcentration;thisisthe

continuityequation.

t

CJ

BycombiningFick’sfirstlawwiththecontinuityequation,weobtainFick’ssecond

law,alsoknownasthediffusionequation.

C

t (DC)

DiffusionMechanismsinSolids

Diffusionincrystalshasdifferenttypesofmechanisms.Thesimplest

mechanismsareexchangeandringmechanisms.Exchangemechanismisthe

exchangeoflatticepositionsoftwoatomslocatedinadjacentsites.Ringmechanism

requiresthecoordinatedcirculatingmovementofthreetofiveatoms.However,

sincetheenergyrequiredfortheExchangeandRingmechanismsistoohighthey

arenotlikelytooccur.

OthertypesofdiffusionmechanismsareVacancy,Interstitial,and

Interstitialcymechanisms.Vacancymechanismoccurswhenanatomjumpsfrom

itslatticesitetoavacantsite.Interstitialmechanismiswhenanatombetween

latticesitesjumpstoanotherinterstitialsite.Interstitialcymechanismismuchlike

aninvasion;it’swhenaninterstitialatombumpsaneighboringlatticeatomoutof

itsplaceandthentakeitsplace.

Thediffusionofdefectsinthecrystalisgovernedbytheenergylandscape

seenbythedefects.Toillustratethis,thediagrambelowshowsthatinorderforan

atominsiteAtogettositeB,itmuchcrosstheenergybarrierofheightGM,called

themigrationoractivationenergy.Thehigherthebarrierthemoredifficultitisto

cross.ThelikelihoodofdiffusionisexpressedbythediffusivityD,andisdependent

onthemigrationenergyandthetemperatureT.

Tk

GDD

B

m

exp0

DisproportionaltotheBoltzmannprobabilityandD0isaproportionalityconstant.

RandomWalkTheory

Einsteinreasonedthatmoleculesarealwayssubjecttothermalmovements

ofstatisticalnatureduetotheirBoltzmanndistributionofenergy.Hederivedthat

themeansquaredisplacement,<R2>,ofarandomlymovingatomwasrelatedtothe

diffusioncoefficient,D.isthetimeelapsedfortherandommotion.Einstein’s

relationisgivenby

D R 2

4

Inordertocalculate<R2>,aseriesofrandomwalksofidenticalatomscanbe

analyzedaccordingtoRandomWalktheory.TheRandomWalktheoryreasonsthat

thediffusioninsolidsresultsfromparticlesjumpingfromsitetositerandomlyas

showninthisdiagram.

Inordertofindthetotaldisplacementbytheparticle,addalltheindividual

vectorstepstakenbytherandomlywalkingparticle.Thesquareddisplacement

wouldthentakethisform.

R r1l1

nstep

R2 rl2

l1

nstep1

rl rjJ l1

nstep

l1

nstep

Thisexperimentisrepeatedmanytimes,andanensembleaverageisfound.

R2 rl2 2

l1

nstep1

rl rJJ l1

nstep

l1

nstep

ThediffusioncoefficientDcanbere‐expressedinanotherusefulform.The

traveltimeisrelatedtotheaveragenumberofsteps<n>,thejumprate,andthe

coordinationnumberZ.

n

Z ,

BysubstitutingintoEinstein’srelation,Dcannowbewritinglikethis:

D R2

n

ThejumprateinthediffusionequationdependsontheactivationenergyG,the

attemptfrequency0,andthetemperatureT.

vo exp G

kBT

NumericalApproachtoCalculatingD

AMonteCarlomethodwasusedtosimulatetherandomwalkofdefectsona

2Dsquarelattice.Thejumpdistanceisalwaysthesame,butthedirectionofeach

jumpwasdecidedbythevalueofarandomnumbersampledfromauniform

distribution.Therandomwalkercangoalonganydirection[+x,‐x,+y,‐y]withequal

probability[1/4].ThisisshownintheflowdiagramfortheMatlabcode

implementationwrittenforthisproject.

MonteCarloSimulationResults

SamplesofParticleTrajectories

Thefollowingfiguresshowsnapshotsofrandomwalkon2Dlattice.Plotted

herearetheparticletrajectoriesofdifferentwalks.Thefirstisforonewalk,then

increasingto10,then50.Fromthesnapshots,itcanbeseenthatasthenumberof

walksisincreased,thepathsoftherandomwalkersseemtoconcentrateinamore

orlesscircularregionwiththecenteratthestartingpoint.

Nwalk=1:

Nwalk=10,Nwalk=50:

EffectofEnsembleSize

Whenweincreasetheensemblesize,statisticallytheresultshouldbemore

accurate.WecalculatedDfordifferentsizedensemblesofwalksasshowninthe

followingplot.

Whentheensemblesizeislargeenough,thevalueofDbecomesmoreorless

constant.

Asweincreasedthenumberofwalks,theensembleaveragedresultantX&Y

componentsapproachedzero,asexpected.Wefindthatbeyond15,000walksthe

fluctuationsconvergetozero.

HistogramofresultantX&Ycomponents

ToseethespreadoftheX(orY)componentsaboutthemean,theX(orY)

directionwasdividedintoseveralbins,andthenumberoftimesaresultantX(orY)

component“fell”inthebinwascounted.Fornwalk=15,000,wehaveplotteda

histogramoffrequencywithrespecttovariousbins:

WeseethatthefrequencydistributionisclosetoaGaussiandistribution.

Also,thehistogramforresultantXandYvaluesareplacedsymmetricallyaboutthe

meandisplacementcomponent(0).Ifweweretocontinuetoincreasethenumber

ofwalks,thehistogramwouldapproachaperfectGaussiandistribution.

HistogramofDisplacementdistance

Ifweplotthehistogramofdisplacementdistanceinthesamewayasbefore,

weexpecttogetaskeweddistributioninsteadofGaussiandistributionasshownin

belowfigure.Thisisbecauseallthevaluesofthedisplacementdistanceare

positive.Themostfrequentdistanceis15angstromsinallthreeensembleswith

thesamenumberofsteps/walkwhichisconsistentwiththeconclusionthat

fluctuationsinensembleaveragedvaluesapproacheszerowhenthenumberof

walksisgreaterthanorequalto15,000.

Conclusion

WeusedaMonteCarlomethodtocalculatediffusioncoefficientofFCCCuby

usingrandomwalktheory.

Fromtheresultswesawthatwhenweincreasetheensemblesizethediffusion

coefficientbecomesmoreorlessconstant.Anaturalextensionofthisworkwould

betostudyclusteringofpointdefectsinsolids.

Acknowledgements

SpecialthankstoDr.AnterEl‐Azabandhisgraduatestudentsforguidance,

referencematerials,andinformativediscussions.Thisworkwasdoneaspartofthe

FloridaSocietyforMaterialsSimulationsREUProgram.Iwouldalsoliketothank

thefacultyandstaffoftheDepartmentofScientificComputingatFloridaState

Universityfortheireffortstomakemysummerresearchexperiencemore

enjoyable.