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MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces and Interfaces—Part II Byungha Shin Dept. of MSE, KAIST 1 2016 Spring Semester

MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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Page 1: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

MS516KineticProcessesinMaterialsLectureNote

4.SurfacesandInterfaces—PartII

Byungha ShinDept.ofMSE,KAIST

1

2016SpringSemester

Page 2: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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)

Page 3: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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

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𝜕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

Page 5: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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

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RayleighInstability

• Athinjetofliquidisunstableandbreaksintospherestoreducesurfaceenergy(acylinderhasalargersurfaceareathanarowofspheres);similarphenomenaoccursinsolidstoo.

• Whynotevolvingintoonebigsphere?• Wavelengthselection:toosmallà sphereshavelargersurfaceareathanthecylindertoolargeà kineticallylimited

• Whydoatomsdiffusefromthetroughstocrests?Needtoconsiderbothprinciplecurvatures(onealongthelengthandtheotherperpendiculartothelength)

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RayleighInstability

CunanoparticlechainsgeneratedbyreductionofCuO nanowiresembeddedinAl2O3Qinetal.NanoLett.8,p.114(2008)

Page 8: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

RayleighInstability

Page 9: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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

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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

Page 11: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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

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OstwaldRipeningOstwaldRipeningorcoarsening:reductionofthetotalinterfacialenergy=>volumefractionofminorityphaseremainsapproximatelyconstantbuttheaveragesizeofthedomainsincreases(biggerdomainsgrowattheexpenseofsmallerdomains).

αβ

Supposeasystemconsistingofadispersionofsphericalparticlesofβ embeddedinamatrixα

Assumption:- Molarvolumesofα andβ arethesame- Noelasticstressespresentinthesystem- Isotropicinterfacialenergybetweenα andβ- Intrinsicdiffusivitiesofeachcomponent

equalandisotropic(needtoconsideronlyonecomponent;Kirdendall effectignored)

- Fastinterfacialkinetics(motionoftheinterfacelimitedbydiffusion)

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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

Page 14: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

Concentrationfieldinthematrix(α)betweentwoparticles(β)ofdifferentsizes

OstwaldRipening

α

β β

• Non-uniformcompositionfield(becauseofthecompositioninthematrixandparticlesarefunctionsoftheradiusofthecurvatureoftheinterface,i.e.,Gibbs-Thompsoneffect)àDiffusionofmassfromthesmallertothelargerparticleà growthofthelargerparticleandshrinkofthesmallerparticle

Page 15: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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

Page 16: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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.

Page 17: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

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

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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

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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$

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SmallvolumefractionlimitParticlesizedistributionscaledbytheaverageparticlesize=>timeindependent

OstwaldRipening

Page 21: MS516 Kinetic Processes in Materials Lecture Note 4. Surfaces …energymatlab.kaist.ac.kr/layouts/jit_basic_resources/... · 2018-07-19 · MS516 Kinetic Processes in Materials Lecture

SmallvolumefractionlimitParticlesizedistributionscaledbytheaverageparticlesize=>timeindependent

MicrostructuresofsolidSn-richparticlesinaPb-richliquid

OstwaldRipening