florida society for materials simulations reu … carlo simulation of defect diffusion in fcc...
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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.