Statistical Model - latech.edukmomeni/Courses/Multiscale Material Design/Lecture8... · PI: K....
-
Upload
nguyenhanh -
Category
Documents
-
view
213 -
download
0
Transcript of Statistical Model - latech.edukmomeni/Courses/Multiscale Material Design/Lecture8... · PI: K....
Statistical Model
Dr. Kasra Momeniwww.KNanoSys.com
PI: K. Momeni
Outline• Diffusion
• ContinuumDescription• DiffusionCoefficient• SurfaceDiffusionModel
• RandomNumberGenerators• RandomWalkModel
• Pseudocode• Statistics
• Homework
AdHiMad Lab 2
PI: K. Momeni
Diffusion• Itisoneofthekeymechanismsformasstransportwithinmaterials
• Applications:• Carburisation:Carbonisdiffusedintothesurfaceofsteel
• NuclearWaste:Preventingradioactivewasteleakageduetodiffusion
• Semiconductors:Dopingprocess• Phase-Transformation:Diffusionofanexternalelementinthehostlattice
• Corrosion:Diffusionofthereactingelementwithinthemetals
AdHiMad Lab 3
https://www.doitpoms.ac.uk/http://elementsmagazine.org/http://volga.eng.yale.edu/MaterialsScienceandEngineering:A510(2009):342-349.https://upload.wikimedia.org/
PI: K. Momeni
Diffusion• Experimentalcalculationofthediffusionratesisverycumbersomeandinsomecasesnotpossible
• Computersimulationsprovideanalternativetool,withhigheraccuracy,andenablingstudyoftheunderlyingmechanisms
• Differentdiffusionmechanismsbasedondimensionality• BulkDiffusion:Alloys,phase-transformation,dislocationmotion• SurfaceDiffusion:Growthoffilmsand2Dmaterials• Edgediffusion:Morphologyof2Dmaterials
AdHiMadLab 4
2DMaterials,3(4),041003.2DMaterials,4(2),025029.ChemistryofMaterials 26.22(2014):6371-6379.
PI: K. Momeni
Diffusion• Diffusionisafunctionofdifferentphysicalparametersincluding:temperature,initialconcentration(initialvalue),surfaceconcentration(boundaryvalues),time,anddiffusivity(diffusioncoefficient)
• Diffusion(ContinuumFormulation)– Fick’sLaw!" = −%"&'" 1*+,-./'"/+
= −& 0 !" 223,-.
!" isthefluxofelementi,%" isitsdiffusioncoefficient,and'" istheconcentrationofi’th specie.• CombiningFick’s1st and2nd laws:
/'"/+
= & 0 %"&'"
AdHiMad Lab 5
PI: K. Momeni
Probability and Average • Whatisthelikelihoodofheadvstailwhenyouflipacoin?
50%-50%• Whatisthelikelihoodof1whenyourolladice
1/6~17%• Approximatelyhowmany1sappearsifwetossthedice120000times?
• Generalcase:npossiblestates,N(>>n) #trials,mi #state i• Probabilityofstatei
4" =5"
6
AdHiMad Lab 6
https://img.clipartfest.com/https://images.clipartof.com/
PI: K. Momeni
Probability and Average • Thesumofallprobabilitiesoverdifferentstatesshouldbeunity
74"
8
"9:
=75"
6
8
"9:
=1675"
8
"9:
= 1
• Weightedaverage:AssumingA tohasanonunity valueinstatei
; =1675"
8
"9:
0 ;" =74"
8
"9:
0 ;"
• Equivalentcontinuumequationsare:; = <4(>) 0 ; > 3>
�
�• Calculatingtheprobabilityfunctionaliskeyforfindingavalueofaproperty.
AdHiMad Lab 7
PI: K. Momeni
Probability in a Diffusive Process• ThediffusionofaspecieisgovernedbyFick’slaw
/'"/+
= & 0 %"&'"• Inaspecialcaseofasinglespecie,1D,sphericallysymmetricsystemwithconstantD,anda' >, 0 = C > :
/'/+= %&D';&D =
1FD
/D
/FDFD
/D
/FD
• Thesolutionwillbe' G, + =
18 I%+ J/D L
MNO PQR⁄
• Whatisthetotalconcentrationastimepasses?
<' G, + 4IFD3F�
�
= 1
AdHiMad Lab 8
Shewmon,P.,&Janßen,M.(2016).DiffusioninSolids.Springer.
PI: K. Momeni
Probability in a Diffusive Process• Whatistheaveragedisplacementofatoms?(Similarto2stmomentofarea)
F = <F' G, + 4IFD3F�
�
=4I�
%+�
• Whatistheaveragesquaredisplacementofatoms?(similartomassmomentofinertia)
FD = <FD' G, + 4IFD3F�
�
= 6%+
Thus% =
16+
FD
• WhileD isamaterialproperty, FD canbemeasuredfromsimulations• WehadassumedD=cte,thus FD mustbeproportionaltot
AdHiMad Lab 9
PI: K. Momeni
Notes on Diffusion • Imperfectionsinthematerialaffectthediffusionrate
• Defects,dislocations,grainboundaries,…• Defectscommonlyresultintheincreaseofparentlatticevolumeà enhancethediffusionrate
• Diffusioncoefficientaffectedbydifferentdefects:%VWRR"XY < %["\V]XWR"]8 < %^_ < %\`NaWXY
• Diffusionisanactivatedprocess• DEnn <DEnnn
• Wecanonlyconsidernnjumps• Rateofjumpa exp(-DE/kBT)
AdHiMad Lab 10
DE
PI: K. Momeni
Random Walk• Amotioninspacewherethereisnorestrictionsonthedirectionofthenextjump,i.e.alljumpsareequallyprobable
• Arandomwalkproblemwillbedefinedknowing• Latticestructureofthematerial• Rateofjumps
vTrackingthepositionofatoms• Consideralattice– e.g.square• Assumejumpswithnormalizedratesr– up,down,left,right• Performnjumps• Calculate<R2>
AdHiMad Lab 11
PI: K. Momeni
Random Walk• Afternjumps
b 2 = F: + FD +⋯+ F8 =7F"
8
"9:• Forcalculatingthediffusioncoefficientweneedtocalculatethe b 2 0 b 2
b 2 0 b 2 =7F"D
8
"9:
+ 27 7 F⃗" 0 F⃗f
8
f9"g:
8M:
"9:
• Forasquarelattice• Lengthofalljumpsarea• Theonlydifferenceisinthedirectionofthejump
AdHiMad Lab 12
F⃗" 0 F⃗f= F⃗" F⃗f cos(k"f)
PI: K. Momeni
Random Walk• Thusforasquarelatticewehave
b 2 0 b 2 = 2-D + 2-D7 7 cos(k"f)8
f9"g:
8M:
"9:
• Theaverage b 2 0 b 2 = 2-D + 2-D ∑ ∑ cos(k"f)8f9"g:
8M:"9: = 2-D +
2-D ∑ ∑ cos(k"f)8f9"g:
8M:"9: = 2-D + 2-D ∑ ∑ cos(k"f)8
f9"g:8M:"9:
• Sincethejumpsareequallylikelyinalldirections
7 7 cos(k"f)8
f9"g:
8M:
"9:
= 0
• Thus b 2 0 b 2 = 2-D = Fmn5o+-D
AdHiMad Lab 13
PI: K. Momeni
Diffusion Coefficient – Random Walk• Wehadprovedthat% = :
pRbD ,andnowwefound bD = Fmn5o+-D
thus
% =16+
Fmn5o+-D =Fmn5o-D
6• CAUTION:Alljumpsareequallyviable• Inrealworldreversejumpsaremoreprobable
• WeaddacorrectionfactorK,thus% = q Nf`rsWO
p,whereK=1forrandomwalk
andK<1 forarealisticjump• K canbefoundusingatomisticcalculationsknowingtheinteratomicpotential
AdHiMad Lab 14
PI: K. Momeni
Implementation• Forimplementingtherandomwalkwefollowthesesteps
• Determinethelattice(possiblejumps:directionsanddistanceperjump)1. Generatethesequenceofrandomwalks2. Calculatethenewposition3. CalculatetheR(n)andbD 2• Performsteps1-3formanyrandomwalks• UsetheBinningprocesstocalculatetheaverage<R2>
AdHiMad Lab 15
PI: K. Momeni
Random Number Generators • Althoughcomputersaredevelopedtoperformaccuratecalculations,wemayusesomealgorithmstogeneratesaseriesofnumbersthatlook random
• Differentrandomnumbergeneratorshavedifferentqualities• Therearelibrariesoffunctionsthatyoumayuseforgeneratinghighqualityrandomnumbers
• drand48()• use<random>libraryinC++11• MathematicaRandomReal• Matlab:rand,haltonset
AdHiMad Lab 16
PI: K. Momeni
Random Number Generators - Park-Miller• Park-MillerGenerator
tfg: = -tfmod5• wherea andm areconstants,modisthereminderofLHSwrt RHS,andI1 istheinitialorseednumber
• Itresultsalongsequenceofsudo randomnumbersifwechoose- = 7x5 = 2J: − 1
• Maximumvalueofthisseriesism• CAUTION:Theserieswillreturnthesamesetofnumbersunlessyouchangetheseednumber
• Inrandomwalksyouneedtochangetheseed,otherwiseyougetthesamesetoverandover
AdHiMad Lab 17
Park,&Miller(1988)CommunicationsoftheACM,31(10),1192–1201
PI: K. Momeni
Random Walk – pseudo code• Definethesetofpossiblejumps
• e.g.[1,0][-1,0][0,1][0,-1]• Defineasetofpathsofrandomjumps
• e.g.P1:LLLUUDDRRDULDD;P2:LDDULDRRLDRRLDL;…• Findthenewposition,r(n)pi ,anditssquare,r2(n)pi,aftereachjump• Findtheaverageof< r(n)pi>and <r2(n)pi>betweenmanypathsPi• Plot<r2(n)pi>vsn• Thelargerthen,thecloserthe<r2(n)pi>toaline
AdHiMad Lab 18
PI: K. Momeni
Random Walk – Statistics • Considerjumpsonlyin1D• Astimepassestheatomattheoriginjumpsinalldirectionswithequalprobability
• Theaverageofthedisplacementswillbezero• Thereisahigherprobabilityoffindingatomsatthecenter
• ItisproventhattheprobabilityofarandomwalkisgivenbyaGaussiandistributionasfollows
t >8 =3
2I2-Dz.x
L>o −3>8D
22-D• ItshouldbenotedthattheGaussianfunctiongoestozeroasxgoesto∞,whichisnotrealistic
• Theexactprobabilisticfunctionshouldbezerobeyondna
AdHiMad Lab 19
PI: K. Momeni
Random Walk – Statistics • Theprobabilityinalldirectionsis
} b = t >8 t ~8 t �8 =3
2I2-DJ/D
L>o −3
22-D>8D + ~8D + �8D
• Theprobabilityforend-to-enddistance
} b = t >8 t ~8 t �8 =3
2I2-DJ/D
4IbDL>o −3bD
22-D
• Theprobabilityof} b canbefoundusingtheBinningmethodandplottingthehistogram
AdHiMad Lab 20
PI: K. Momeni
Homework• Writearandomwalkcodeandprovidethefollowingitemsforn=1000jumpsaveragedover10,100,and1000steps
• P(r) vsr• < r(n)pi>vs n• <r2(n)pi>vsn
AdHiMad Lab 21
PI: K. Momeni AdHiMad Lab 22
Questions