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MS516KineticProcessesinMaterialsLectureNote
4.SurfacesandInterfaces—PartII
Byungha ShinDept.ofMSE,KAIST
1
2016SpringSemester
CourseInformationSyllabus1.Atomisticmechanismsofdiffusion (3classes)2.Macroscopicdiffusion
2.1.Diffusionunderchemicaldrivingforce (2classes)2.2.Otherdrivingforcesfordiffusion (2classes)2.3.Solvingdiffusionequations (2classes)
3.Diffusion(flow)inglassystates (2classes)4.Kineticsofsurfacesandinterfaces
4.1.Thermodynamicsofsurfacesandinterfaces (4classes)4.2.Capillary-inducedmorphologyevolution (2classes)
4.2.1.Surfaceevolution4.2.2.Coarsening
5.Phasetransformation5.1.Phenomenological theory (1class)5.2.Continuousphasetransformation (3classes)
5.2.1.Spinodal decomposition5.2.2.Order-disordertransformation
5.3.Nucleationandgrowth(Solidification) (3classes)
Viasurfacediffusion
atomicvolume
𝜇 𝑥 = 𝜇$ + Ω𝜎𝜅 = 𝜇$+ Ω𝜎 −𝑑+ℎ𝑑𝑥+
1 + 𝑑ℎ𝑑𝑥
./+ ≈ 𝜇$ − Ω𝜎𝑑+ℎ𝑑𝑥+
𝐽2 = −𝑐2𝐷2𝑘6𝑇
𝛻𝜇 =𝑐2𝐷2Ω𝜎𝑘6𝑇
𝑑.ℎ𝑑𝑥.
𝜕ℎ𝜕𝑡 = −Ω
𝜕𝐽2𝜕𝑥 = −
𝑐2𝐷2Ω+𝜎𝑘6𝑇
𝑑;ℎ𝑑𝑥;
RecallthefluxequationfromLectureNotePart2
Atomsormoleculeswilldiffusefrompeaks(positivecurvature)totroughs(negativecurvature)toeliminatethechemicalpotentialgradient,resultingintheflatnessofthesurface.
Surfacemorphologyevolution
h(x)
x
surfaceflux(#/cm-sec)
cS:surfacedensity(#/cm2)
chemicalpotentialofflatsurface
𝜕ℎ𝜕𝑡 = −𝐵=
𝜕;ℎ𝜕𝑥;
Thedependenceofh onthelengthscalesofthesurfaceroughnesscanbeanalyzedwithindependentFouriercomponentshavingtheform,h (x,t)=A(t)sin(2πx/λ)
𝐴 𝑡 = 𝐴 0 exp −(2𝜋);𝐵=
𝜆; 𝑡
ℎ 𝑥, 𝑡 = ℎ(𝑥, 0) exp −(2𝜋);𝐵=
𝜆; 𝑡
• Exponentialdecayofamplitude• Decayrate~(1/λ)4 (Lifetime~λ4)• Arbitraryinitialprofile (Fouriersuperposition)à onlyfundamentalfrequencysurvives
SurfacemorphologyevolutionViasurfacediffusion
Viaevaporationorbulkdiffusion
𝐽J = 𝑣LMNO 𝜇 𝑥 − 𝜇P = −𝑣LMNO𝜎Ω𝜕+ℎ𝜕𝑥+
kineticcoefficientofevaporation
chemicalpotentialofanatominthevapor
𝜕ℎ𝜕𝑡 = −Ω𝐽J = 𝑣LMNO𝜎Ω
𝜕+ℎ𝜕𝑥+ = 𝐵Q
𝜕+ℎ𝜕𝑥+
• Exponentialdecayofamplitude• Decayrate~(1/λ)2
Selectiveevaporation(and/orre-condensation)toeliminatethechemicalpotentialgradient.
Surfacemorphologyevolution
RayleighInstability
• Athinjetofliquidisunstableandbreaksintospherestoreducesurfaceenergy(acylinderhasalargersurfaceareathanarowofspheres);similarphenomenaoccursinsolidstoo.
• Whynotevolvingintoonebigsphere?• Wavelengthselection:toosmallà sphereshavelargersurfaceareathanthecylindertoolargeà kineticallylimited
• Whydoatomsdiffusefromthetroughstocrests?Needtoconsiderbothprinciplecurvatures(onealongthelengthandtheotherperpendiculartothelength)
RayleighInstability
CunanoparticlechainsgeneratedbyreductionofCuO nanowiresembeddedinAl2O3Qinetal.NanoLett.8,p.114(2008)
RayleighInstability
Interfacereaction-limitedvs.Diffusion-limitedkinetics
Steady-stategrowthofasingle-componentsolid(“β”)fromadilutesupersaturatedsolution(“α”)
Forthegrowthofβ phasetooccur:(i) non-zerosoluteconcentrationgradient(diffusion)(ii) 𝐶TU atleastinfinitesimallygreaterthanthe
equilibriumconcentration𝐶TJ (interfacereaction)
𝐽T = −𝐷𝐶TVWXY − 𝐶TU
𝐿Diffusivefluxofsoluteatoms:
Fluxacrosstheinterface:(smalldrivingforce)
𝐽U = −𝑀𝐶TU 𝛻𝜇6 = −𝑀𝐶TU Δ𝜇6/𝑎
(<0)
(<0)
β α
Δ𝜇6 = 𝜇T6 − 𝜇^6
𝐽U = 𝐽T: 𝐶TU = 𝐷 𝐿⁄ 𝐶TVWXY + ℳ𝐶TJ
𝐷 𝐿⁄ +ℳ,whereℳ = 𝑀𝑘6𝑇/𝑎
𝐷 𝐿⁄ ≫ ℳ(1)Interfacerxn-limited()𝐶TU ~𝐶TVWXY,
𝑣 = 𝐽UΩ =(𝐷 𝐿⁄ )ℳ𝐷 𝐿⁄ +ℳ (𝐶TVWXY − 𝐶TJ)Ω
𝑣 =ℳ(𝐶TVWXY − 𝐶TJ)Ω
(allinformationaboutthebulkdiffusivitydisappears)
𝐷 𝐿⁄ ≪ℳ(2)Diffusion-limited()𝐶TU ~𝐶TJ, 𝑣 = (𝐷 𝐿⁄ )(𝐶TVWXY − 𝐶TJ)Ω
(allinformationabouttheinterfacemobilitydisappears)
Interfacereaction-limitedvs.Diffusion-limitedkinetics
= 𝑘6𝑇 ln𝐶TU
𝐶TJ− 𝑘6𝑇 ln
𝐶U
𝐶J
= 𝑘6𝑇 ln𝐶TU
𝐶TJ≈ −𝑘6𝑇 ln
𝐶TJ
𝐶TU
limg→P
ln(1− 𝑥) ≈ −𝑥
β α
≈ −𝑘6𝑇𝐶TJ
𝐶TU− 1
OxidationofaSiwafer
Initially,thereactionislimitedbytheinterfacekinetics.
𝑑𝐿𝑑𝑡 = ℳ
Ω2 𝐶igUjJVWXY − 𝐶igUjJJ , 𝐿 𝑡 = 𝐾l + 𝐾+𝒕
𝑑𝐿𝑑𝑡 =
𝐷𝐿Ω2 𝐶igUjJVWXY − 𝐶igUjJJ , 𝐿 𝑡 = 𝐾. +𝐾;𝒕𝟏/𝟐
Oncetheinitialoxideformsatthesurface,furthergrowthrequiresthediffusionofoxygendownwardthroughtheoxidefilmtoreactwiththeburiedsilicon.Forsufficientlythickfilms,thereactionislimitedbythediffusionofoxygenthroughtheSiO2 film.
Completelygeneralexpression:
𝐿 𝑡 =𝐷ℳ −1 + 1 +
2ℳ+
𝐷 𝐶igUjJVWXY − 𝐶igUjJJ Ω2 𝑡
l/+
Interfacereaction-limitedvs.Diffusion-limitedkinetics
OstwaldRipeningOstwaldRipeningorcoarsening:reductionofthetotalinterfacialenergy=>volumefractionofminorityphaseremainsapproximatelyconstantbuttheaveragesizeofthedomainsincreases(biggerdomainsgrowattheexpenseofsmallerdomains).
αβ
Supposeasystemconsistingofadispersionofsphericalparticlesofβ embeddedinamatrixα
Assumption:- Molarvolumesofα andβ arethesame- Noelasticstressespresentinthesystem- Isotropicinterfacialenergybetweenα andβ- Intrinsicdiffusivitiesofeachcomponent
equalandisotropic(needtoconsideronlyonecomponent;Kirdendall effectignored)
- Fastinterfacialkinetics(motionoftheinterfacelimitedbydiffusion)
Concentrationfieldinthematrix(α)betweentwoparticles(β)ofdifferentsizes
𝑋 𝑅 ≈ 𝑋TJ exp1𝑘6𝑇
𝟐𝜴𝝈𝑹 ≈ 𝑋TJ 1 +
𝟐𝜴𝝈𝑘6𝑇𝑹
= 𝑋TJ +2𝑙vT
𝑅
OstwaldRipening
• Compositionattheinterfaceinthematrix,
𝑋 𝑅 = 𝑋TJ +1
(𝑋TJ − 𝑋^J)𝐺T xx𝟐𝜴𝝈𝑹 = 𝑋TJ +
2𝑙vT
𝑅Precisely,
α
β β
Concentrationfieldinthematrix(α)betweentwoparticles(β)ofdifferentsizes
OstwaldRipening
α
β β
• Non-uniformcompositionfield(becauseofthecompositioninthematrixandparticlesarefunctionsoftheradiusofthecurvatureoftheinterface,i.e.,Gibbs-Thompsoneffect)àDiffusionofmassfromthesmallertothelargerparticleà growthofthelargerparticleandshrinkofthesmallerparticle
Smallvolumefractionlimit• Considerthelimitwherethereisnodiffusionalinteractionbetweenparticlesàdistancebetweenparticlesisinfinite,orthevolumefractionofparticlesiszero(oversimplifiedbutessentialphysicswillbethere)
• GrowthrateofaparticleofsizeR;equationgoverningtheevolutionofaparticlesizedistribution;massconservation
1𝑟+
𝑑𝑑𝑟 𝑟+
𝑑𝑋𝑑𝑟 = 0;
Quasi-stationary(ripeningtendstobeslowprocess),
• X(R)givenbyGibbsThompsonequationinthepreviousslide• ataninfinitedistancefromaparticle,X∞ (t)
𝑋 𝑟 = 𝑋$ + 𝑋TJ − 𝑋$ +2𝑙vTΩ𝛾𝑅
𝑅𝑟
𝑑𝑅𝑑𝑡 =
𝐷𝑅(𝑋^J − 𝑋TJ)
𝑋$ 𝑡 − 𝑋TJ −2Ω𝜎
(𝑋TJ − 𝑋^J)𝐺Txx𝑅
• Massbalanceequation, 𝑋^J − 𝑋TJ𝑑𝑅𝑑𝑡 = −𝐷
𝜕𝑋𝜕𝑟
supersaturation
OstwaldRipening
(1)
(sphericalcoordinates)𝑋 𝑟 = 𝐴 +𝐵/𝑟
CriticalparticlesizewheredR/dt =0,
Smallvolumefractionlimit
𝑑𝑅𝑑𝑡 =
𝐷2Ω𝜎𝑅(𝑋^J − 𝑋TJ)+𝐺Txx
1𝑅$
−1𝑅
𝑅$ =2Ω𝝈
(𝑋TJ − 𝑋^J)(𝑿$ 𝒕 −𝑿𝜶𝒆)𝐺Txx
OstwaldRipening
• 𝑅$ increaseswithsurfaceenergy,𝜎anddecreaseswithsupersaturation,𝑋$ 𝑡 − 𝑋TJ
• Eachparticleisinaraceagainstthecriticalparticlesize(particlessmallerthanR∞ shrinks,largerthanR∞ grows)
• R∞ increaseswithtimeasthesupersaturationdecreaseswithtime.
Smallvolumefractionlimit• GrowthrateofaparticleofsizeR;equationgoverningtheevolutionofaparticlesizedistribution;massconservation
f(R,t)dR:numberofparticlesperunitvolumeofsizeR toR +dR
�𝜕𝑓(𝑅, 𝑡)𝜕𝑡 𝑑𝑅
��
��= 𝑓 𝑅l, 𝑡
𝑑𝑅𝑑𝑡 ��l
− 𝑓 𝑅+, 𝑡𝑑𝑅𝑑𝑡 ��+
= −�𝜕(𝑓 𝑅, 𝑡 𝑑𝑅
𝑑𝑡 )𝜕𝑅 𝑑𝑅
��
��
ParticlesofsizeR1 thatgrow–ParticlesofsizeR1 thatshrink
ParticlesofsizeR2 thatgrow–ParticlesofsizeR2 thatshrink
𝜕𝑓(𝑅, 𝑡)𝜕𝑡 +
𝜕(𝑓 𝑅, 𝑡 𝑑𝑅𝑑𝑡 )
𝜕𝑅 = 0
Continuityequationwhichgovernsthetimerateofchangeoftheparticlesizedistributionf (R,t)
(2)
OstwaldRipening
Smallvolumefractionlimit
• GrowthrateofaparticleofsizeR;equationgoverningtheevolutionofaparticlesizedistribution;massconservation
𝑋P = 1 −𝜙 𝑋T + 𝜙𝑋^ = 𝑋T +𝜙(𝑋^ − 𝑋T) ≈ 𝑋$ +𝜙(𝑋^J − 𝑋TJ)
volumefractionofβ
= 𝑋$(𝑡) + (𝑋^J − 𝑋TJ) ∫;�.𝑅.𝑓 𝑅, 𝑡 𝑑𝑅$
P
total#ofB inα
total#ofB inβ
(3)
Eq’s (1),(2),(3)à solveforf (R,t),numericallyandanalyticalsolutionpossible(seetheappendixoftheVoorheeshandoutifyou’reinterestedinthefullmathematicaltreatment;LSWtheory)
OstwaldRipening
Smallvolumefractionlimit
𝑑𝑅𝑑𝑡 =
𝐷2Ω𝜎𝑅(𝑋^J − 𝑋TJ)+𝐺Txx
1𝑅$
−1𝑅 =
𝐴𝑅
1𝑅 −
1𝑅
• A:time-independentmaterialsconstant; L /t~1/L2 =>L∝t1/3• 1/3:characteristicexponentofaninterfacialenergydrivencoarseningof3Dparticles processwherethekineticsoftheprocessarecontrolledbydiffusion
𝑅 𝟑(𝑡) =8Ω𝛾𝐷
9(𝑋^J − 𝑋TJ)+𝐺Txx𝑡
𝑋$ 𝑡 = 𝑋TJ +9(Ω𝛾)+
𝐷(𝑋^J − 𝑋TJ)+𝐺Txx+
l/.
𝑡�𝟏/𝟑
UsinganalysisgivenintheappendixoftheVoorheeshandout:
OstwaldRipening
Scalinganalysis:
𝑅 ~𝑅$
SmallvolumefractionlimitParticlesizedistributionscaledbytheaverageparticlesize=>timeindependent
OstwaldRipening
SmallvolumefractionlimitParticlesizedistributionscaledbytheaverageparticlesize=>timeindependent
MicrostructuresofsolidSn-richparticlesinaPb-richliquid
OstwaldRipening