MS516 Kinetic Processes in Materials Lecture Note 5. Phase...

MS516KineticProcessesinMaterialsLectureNote

5.PhaseTransformation—PartI

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)

Consider:

IsothermalTransformationbyRandomNucleationandGrowth

Assumptions:a) Nucleiofβ appearrandomlyinspacewith

frequencyI perunitvolume(#ofnucleation/sec/cm3).

b) Growthrateisconstant,u (cm/sec),intime.c) Growthisisotropic

Fractionofα transformedtoβ:χ(t)≡Vβ /V,(whereVisthetotalvolume)

Transformationofundercooledα phaseto β phase; Novolumechange;Nocompositionchange;T uniformandconstant

PhenomenologicalTheory

Analysis(Johnson-Mehl-Avrami):1) Calculatevolumetransformedassumingconstantnucleationrate,evenin

already-transformedregions,andignoringgrainimpingement(“extendedvolume”)

2) Correctfor“phantomgrains”and overlap fromgrowth

Particlenucleatedatt =τ,itsextendedvolumeis 𝑑𝑉#$% ≡

43𝜋𝑢

+(𝑡 − 𝜏)+ fort>τ

Totalextendedvolumeis 𝑉#$% 𝑡 = 𝑉2 𝐼

43𝜋𝑢

+(𝑡 − 𝜏)+𝑑𝜏4

5=𝜋3 𝑉𝐼𝑢

+𝑡6

Extendedfractionofα transformedtoβ: 𝜒#$% 𝑡 ≡

𝑉#$% 𝑡𝑉 =

𝜋3 𝐼𝑢

+𝑡6


0fort<τ

𝑉#$% 𝑡 islargerthantheactualtransformedvolumesinceitincludes:• Nucleiwhichforminalreadytransformedregion(“phantomnuclei”)• Growthoccurringinpreviouslytransformedregions,asgrowingparticlesimpingeoneachother

(assumingconstantI&u)

Nowrelate𝑉#$% to𝑉% (actual).

Inatimeintervalwhentheextendedvolumeincreasesby𝑑𝑉#$% ,if

weassumerandomlylocatednucleationsites:

𝑑𝑉% = 1 −𝑉%

𝑉 𝑑𝑉#$%


Afraction(Vβ /V)oftheincreaseinextendedvolumewilloccurinpreviouslytransformedmaterial(alreadyβ phase)andtherefore,

Integrating, 𝑉#$%(𝑡) = −𝑉 ln 1 −

𝑉%(𝑡)𝑉

è 𝜒 𝑡 =𝑉% 𝑡𝑉 = 1 − exp −

𝑉#$%

𝑉 = 1 − exp −𝜋𝐼𝑢+𝑡𝟒

3

(Assumptionsforthelastequality:I isconstantandinterfacelimitedgrowth)

PhenomenologicalTheoryCasewherethereareafixednumberofrandomlydistributedheterogeneousnucleationsites

𝑑𝑁@ = −𝑁@𝜈@𝑑𝑡

Nucleationrateofagivensite#ofnucleationsitesperunitvol.

𝑁@ = 𝑁@5 exp( − 𝜈@𝑡)

𝐼 = −𝑑𝑁@𝑑𝑡 = 𝑁@5𝜈@ exp( − 𝜈@𝑡)

𝑉#$% 𝑡 = 𝑉2 𝐼

43𝜋𝑢

+(𝑡 − 𝜏)+𝑑𝜏4

5

(noassumptiononrandomnucleationandinterface-limitedgrowthneeded;however,3Disotropicgrowthisassumed)

= 𝑉8𝜋𝑢+𝑁@5𝜈@+

𝑒DEF4 − 1 + 𝜈@𝑡 −𝜈@H𝑡H

2 +𝜈@+𝑡+

6(3Disotropicgrowth+interface-limitedgrowth,u~constant)

FromSlide#4,

(<0)

PhenomenologicalTheory(1)Slownucleationrate(𝜈@𝑡 ≪ 1)

𝑒DEF4 ≈ 1 − 𝜈@𝑡 +𝜈@H𝑡H

2 −𝜈@+𝑡+

6 +𝜈@6𝑡6

24

SameresultsasSlide#5!

𝐼 = 𝑁@5𝜈@ exp( − 𝜈@𝑡) ≈ 𝑁@5𝜈@ = constant

𝑉#$% 𝑡 = 𝑉

8𝜋𝑢+𝑁@5𝜈@+


2 +𝜈@+𝑡+

6 ≈ 𝑉𝜋3 𝐼𝑢

+ 𝑡𝟒,

𝜒 𝑡 = 1 − exp −𝜋𝐼𝑢+𝑡𝟒

3

(2)Rapidnucleationrate(𝜈@𝑡 ≫ 1)

𝑉#$% 𝑡 = 𝑉

8𝜋𝑢+𝑁@5𝜈@+


2 +𝜈@+𝑡+

6 ≈ 𝑉𝑁@54𝜋3 𝑢+ 𝑡𝟑,

𝜒 𝑡 = 1 − exp −4𝜋𝑁@5𝑢+𝑡𝟑

3KeepingthetermthatishighestorderinνNt

PhenomenologicalTheoryAvrami expression: 𝜒 𝑡 = 1− exp − 𝑘𝑡 𝒏

• n =4correspondingtoaconstantnucleationrate• n =3correspondingtoanucleationratewhichdecreaseswithtime• n >4correspondingtoanucleationratewhichincreaseswithtime

,where3≤n ≤4.

• For2Dgrowth,à 2≤n ≤3𝑑𝑉#$% ≡ 𝜋𝑢H(𝑡 − 𝜏)H

• Forparticleswhichgrowonlyinonedirectionà 1≤n ≤2

Fractiontransformedaspredictedbyisothermaltransformation

ContinuousTransformation

• Initiallyunstable,aninfinitesimalvariationwillinitiatethetransformationandthedecreaseofbulkfreeenergy

• Beginningoftransformationinvolvesachangethatissmallindegree butlargeinextent

• Second-orderphasetransformation (secondderivativeoffreeenergydiscontinuous)

• Nolatentheat• Spinodal decomposition• Certainorder-disordertransition

(CuZn-type)

DiscontinuousTransformation

• Free-energybarriertoinfinitesimalvariation,systeminitiallymetastable.

• Beginningoftransformationinvolvesachangethatislargeindegree butsmallinextent(nucleationrequired)

• First-orderphasetransformation(firstderivativeoffreeenergydiscontinuous)

• Latentheat• Nucleation• Certainorder-disorder

transitions(suchasCu3Au-type)

ClassificationofPhaseTransformation

(a) First-orderphasetransition(b) Second-orderphasetransition

ClassificationofPhaseTransformation

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)

Spinodal decompositionAu– PtPhaseDiagram Freeenergydiagram

Outsidespinodal,

Insidespinodal,

𝑑H𝐺𝑑𝑋UH

> 0

𝑑H𝐺𝑑𝑋UH

< 0

Smallfluctuationsarelikelytodieout.Onlylargefluctuationslikelytogrow(nucleation&growth).

Smallfluctuationsarelikelytogrow.

Miscibilitygap𝜀Z[D\] −

12 𝜀Z[DZ[ + 𝜀\]D\] > 0

𝐷 = 𝐷∗ 1+𝑑 ln 𝛾U𝑑 ln𝑋U

RecallDarken’s equation:

insidespinodal1 +𝑑 ln 𝛾U𝑑 ln𝑋U

=𝑋U(1 − 𝑋U)

𝑘U𝑇𝑑H𝐺𝑑𝑥H ⇒ 𝐷 < 0

• IfD <0,anywavelengthfluctuationshouldbeamplified.Shortestwavelengthsshouldgrowfaster.

• However,observation:decompositionoccursonspatialscaleof~10nm• Gradientenergykillsfluctuationswithtooshortawavelength.(Review“sharpvsdiffuseinterface”ofLectureNotePart4-1)

K >0whenAandBdon’tlikeeachother(K∝ε = εeU −fH [εee + εUU]).

Spinodal decomposition

Forcompositionfluctuationsatconstantstructure:(1) WhichFouriercomponentshavethepossibilityofbeingamplified?(2) Ofthose,whichwillgrowfastest?

Smallfluctuations: LinearizethediffusionequationConsidereachFouriercomponentindividually

f0:Freeenergyofacompositionally uniformsystemwithcompositionCC0:averagecompositionΔC:amplitudeofsinusoidal fluctuation

AveragefreeenergyoftwocompositionallyuniformsystemsofcompositionC0+ΔCandC0-ΔC

Uniformsystem:

𝐹 = 2 𝑓5 𝑐5 𝑑Volnop

Non-uniformsystem:

𝐹 = 2 [𝑓5 𝑐 + 𝐾(𝛻𝑐)H+𝜂H𝑌(𝑐 − 𝑐5)H]𝑑Volnop

∆𝐹 = 2 [∆𝑓5 𝑐 + 𝐾(𝛻𝑐)H+𝜂H𝑌(𝑐 − 𝑐5)H]𝑑Volnop

“coherencystrainenergy”:elasticenergyterm,occursincrystalsonly

𝜂 =1𝑎5

𝑑𝑎5𝑑𝑐 , 𝑌 =

𝐸1 − 𝜐


++−

∆𝐹 = 2 [∆𝑓5 𝑐 + 𝐾(𝛻𝑐)H+𝜂H𝑌(𝑐 − 𝑐5)H]𝑑Volnop

Taylorexpandf0(c): 𝑓5 𝑐 = 𝑓5(𝑐5)+ (𝑐 − 𝑐5)𝑓5y 𝑐5 + f

H(𝑐 − 𝑐5)H𝑓5

yy 𝑐5

Consider 𝑐 𝑧 = 𝑐5 + ∆𝑐 sin2𝜋𝑧𝜆

Then: 𝑓5 𝑐 = 𝑓5(𝑐5)+ ∆𝑐 sin H~��

𝑓5y 𝑐5 + (∆�)�

HsinH H~�

�𝑓5yy 𝑐5

∆𝐹Vol =

12𝐿2 𝑑𝑧 𝑓5

yy (∆𝑐)H

2 sinH2𝜋𝑧𝜆 + 𝐾

2𝜋𝜆

H(∆𝑐)HcosH

2𝜋𝑧𝜆 + 𝜂H𝑌(∆𝑐)HsinH

2𝜋𝑧𝜆

�

D�

= 𝑓5yy (∆𝑐)H

4 + 𝐾2𝜋𝜆

H (∆𝑐)H

2 + 𝜂H𝑌(∆𝑐)H

2 =(∆𝑐)H

4 𝑓5yy + 2𝐾

2𝜋𝜆

H+ 2𝜂H𝑌

ΔF negativewhen 𝜆 > 𝜆� =8𝜋H𝑲

−𝒇𝟎yy − 𝟐𝜼𝟐𝒀


meaningonlycompositionalfluctuationwithwavelengthlargerthanλC amplifies!

(>0)

Stress-strainrelationsandelasticenergydensityforbinarystrainincubicsystem

𝐸 ≡𝜎$𝜀$

Young’smodulus

Whatisεy forbiaxialstress?

foruniaxialstressstate 𝜎�� =𝜎$ 0 00 0 00 0 0Poisson’sratio 𝜐 ≡

−𝜀�𝜀$ 𝜎$ 0 0

0 𝜎� 00 0 0

Linearelasticity=>addthecontributionstoεy fromeachstresscomponent

𝜀� =𝜎�𝐸 + −𝜐𝜀$ =

𝜎�𝐸 + −𝜐

𝜎$𝐸 =

1𝐸 (𝜎� − 𝜐𝜎$)

Forspecialcaseσx =σy,defineσbiax ≡σx =σy

𝜀� =𝜎��$𝑌

whereis“thebiaxialmodulus”𝑌 ≡𝐸

1 − 𝜈

bysymmetry,εx =εy;defineεbiax ≡εx =εy => 𝜀��$ =𝜎��$𝑌


Coherentstrainenergyduetocompositionchangeinslab

σx

CCC+ΔC

σx

σz=0

σy

σydefineη ≡d ln a0 /dc,wherec ≡molefractionof“solute”anda0 ≡stress-freelatticeconstant

𝜂∆𝑐 = ∆ ln 𝑎5 =∆𝑎5𝑎5

= 𝜀��$

dEnergyVolume =

1Area Length Force 𝑑(Displacement)

𝑑ℇ = 𝜎$𝑑𝜀$ + 𝜎�𝑑𝜀� = 2𝜎��$𝑑𝜀��$ = 2𝑌𝜀��$𝑑𝜀��$

ℇ = 𝑌𝜀��$H

ℇ = 𝑌(𝜂∆𝑐)H

Stressesimposedonslabbybulkmaterial


• Furtherinsidespinodal,f0’’morenegativeà paysforshorterλ (i.e.smallerλC)• Asyouapproachthespindal,λcà∞• Outsidespinodal,nucleationandgrowthistheonlyoptionàmustwaitforlargefluctuation


𝑓5yy + 2𝜂H𝑌 < 0

𝑓5yy < 0

Whichλ growsthefastest?


• Toosmallawavelengthà decaysbecausesmallerthanλC• Toolargeawavelengthà stillamplifiesbutveryslowduetoslowdiffusionkinetics

Interdiffusion fluxneededtoamplifycompositionalfluctuation

𝐽 = −𝑀𝑐5(𝛻𝜇U − 𝛻𝜇e)

(forsmallΔc,ℳ≈constant)


𝑐U

𝛻𝜇U < 0𝛻𝜇e > 0

= −𝑀𝑐5𝛻 𝜇U − 𝜇e

= −ℳ𝛻𝑑𝑓5𝑑𝑐 ¥

�¦�§

Cahnintroducesanewterminthediffusionequation,duetothegradientenergy(recallchemicalpotential~dG/dc;G hasadditionaltermsotherthanfo)

𝐽 = −ℳ𝑑𝑑𝑧

𝑑𝑓5𝑑𝑐 − 2𝐾

𝑑H𝑐𝑑𝑧H = −ℳ

𝑑𝑐𝑑𝑧𝑑H𝑓5𝑑𝑐H ¨

�¦�§

− 2𝐾𝑑+𝑐𝑑𝑧+

Classicaldiffusioneq.

Δc

𝜕𝑐𝜕𝑡 = −𝛻𝐽 =ℳ

𝑑H𝑓5𝑑𝑐H ¨

�¦�§

𝑑H𝑐𝑑𝑧H − 2𝐾

𝑑6𝑐𝑑𝑧6

Amplificationrate,R =Aq2 – 2Kq4 (q =2π /λ)


Forinitiallysinusoidalcompositionalfluctuation,𝑐 𝑧,0 = 𝑐5 + ∆𝑐 cos(2𝜋𝑧/𝜆)

Solutiontotheabovedifferentialeq:

𝑐 𝑧, 0 = 𝑐5 + ∆𝑐 cos(2𝜋𝑧/𝜆)exp(𝑅𝑡)

R =A(2π/λ)2

= ℳ𝑑H𝑓5𝑑𝑐H ¨

�¦�§

+ 2𝜂H𝑌𝑑H𝑐𝑑𝑧H − 2𝐾

𝑑6𝑐𝑑𝑧6 = 𝐴𝑐yy − 2𝐾𝑐′′′′

ifcoherencystrainexists

𝜆®�$ = 2𝜆�

Order-disordertransformationEarth-abundantsolarabsorberCu2ZnSnS4 (CZTS)

Showntobesubjecttotheformationofanti-sitedefects,CuZn- andZnCu+

Scragg etal.“Alow-temperatureorder-disordertransitioninCu2ZnSnS4 thinfilms”Appl.Phys.Lett.104,041911(2014)

Solidsolution:mixtureoftwo(ormore)atomsinonecrystallinephase(eithersubstitutionalorinterstitial)• aprimarysolidsolutionisstableinacompositionrangethatincludesthepurematerial(e.g.,dilutesolution,α-brass:Cu-Zn,0– 40at.%Zn,fcc)

• anintermediatesolidsolutionisstableinacompositionrangethatdoesnotincludethepurephase(e.g.,β-brass:Cu-Zn,45-55at.%Zn,bcc/CsCl)

Order-disordertransformation

Solidsolutionsareusuallydisorderedathightemperature(G =H – TS;entropydominatesathighT)


Belowacertaintemperature,thesolidsolutionseither:

• Order, iftheattractionbetweenunlikeatomsisgreaterthantheaverageattractionbetweenlikeatoms,εAB <½(εAA +εBB);Example:β-brass(Cu-Zn)

• Phaseseparate,ifεAB >½(εAA +εBB);Example:Cu-Ni;Au-Ni;Cr-W

Ifadisorderedsolidsolution isobservedtopersisttolowtemperature,itisduetokinetic factor(atommotionbecomestooslow:disorderedstructurefrozenin).

Describethestructureontwosublattices:α,β;bothsimplecubic;β pointsbodycenterofα lattice

• Orderedstate:allA atomsonα lattice;allB atomsonβlattice

• Disorderedstate:bothα andβ latticehaveequalnumberofA andB atoms

β-CuZn Ordering


Total#ofsitesinsystem:NTotal#ofsitesonα (orβ)lattice:N/2Total#ofA (orB)atoms:N/2

WeneedanorderparameterW,suchthatforfullorder:W =1forfulldisorder:W =0

So,#ofAatomsonα sites=(1+W)N /4check:orderW =1à #A atomsonα =N/2disorderW =0à #A atomsonα =N/4

α

β

Bysymmetry:#ofB atomsonβ sites=(1+W)N /4

Takingthedifferenceonα-lattice:#ofB atomsonα sites=N/2– (1+W)N /4=(1–W)N /4=#ofA atomsonβ sites

CalculatetheHelmholtzfreeenergyasafunctionofW:F =U – TSU =NAA εAA +NBB εBB +NAB εAB

nAA

NAA

=#ofAA bondsofanA atomonanα site=(#neighborsofA,i.e.allneighboringβ sites)⨯(fractionofβ siteswithA atoms)=8⨯[(1–W)N /4]/(N/2)

=4(1–W) #ofA atomsonβ sites

total#ofsitesonβ

=nAA ⨯#ofA atomsonα site=4(1–W)⨯(1+W)N /4=N (1–W2)

Order-disordertransformationβ-CuZn Ordering

Bysymmetry:NBB =N (1–W2)

Total#ofbondsinthesystem:(1/2)*(8N)=4NTakingthedifference:NAB =4N– NAA – NBB =2N(1+W2)

Ordering:ε <0;asWà 1U decrease!

Entropy:mixingofAandBontwosublattices (fraction:(1+W)/2,(1–W)/2)

𝑈 = 𝑁 1 − 𝑊H 𝜀ee + 𝜀UU + 2(1 +𝑊H)𝜀eU= 𝑁 𝜀ee + 𝜀UU + 2𝜀eU + 𝑊H𝑁[(2𝜀eU − 𝜀ee + 𝜀UU ]

𝑆 = −𝑁𝑘1 + 𝑊2 ln

1 + 𝑊2 +

1 − 𝑊2 ln

1 − 𝑊2

𝑑𝑆𝑑𝑊 = −

𝑁𝑘2 ln

1 +𝑊1 −𝑊

𝑑𝑈𝑑𝑊 = 2𝑁𝑊𝜀


𝜀

Entropyà disorderEnergyà order EquilibriumvalueofW

𝑑𝐹𝑑𝑊 =

𝑑𝑈𝑑𝑊 −𝑇

𝑑𝑆𝑑𝑊 = 2𝑁𝑊𝜀 +

𝑁𝑘𝑇2 ln

1 +𝑊1 −𝑊 = 0Atequilibrium,

𝑊 =𝑒D

6²³´µ − 1

𝑒D6²³´µ + 1

= tanh −2𝑊𝜀𝑘𝑇

−2𝑊𝜀𝑘𝑇 =

2𝑊 𝜀𝑘𝑇 ≡ 𝑥

𝑘𝑇2 𝜀 𝑥 = tanh𝑥

W1

TC

W

T1 T2


lim²→5

2𝑁𝑊𝜀 +𝑁𝑘𝑇2 ln

1 +𝑊1 −𝑊 ≈ 2𝑁𝑊𝜀 +

𝑁𝑘𝑇·2 2𝑊 = 0

TC (Wà 0)?

lim$→5

ln 1 + 𝑥 ≈ 𝑥

𝑇· = −2𝜀𝑘

W 1

TCT1 T2

• Entropychange(fullorder,W =1à fulldisorder,W =0)ΔS =ΔSmix (mean-field)=R ln 2=1.38cal /mol∙K

ΔS (experiment)=1.01cal /mol∙K

• Energychange(fullorderà fulldisorder)dU =2N W ε dW

Difference:duetoshort-rangeorder (A onα attractsB onβ,notrandomdistribution)

∆𝑈(meanfield) = 2 2𝑁𝑊𝜀𝑑𝑊²¦5

²¦f= −𝑁𝜀 =

12𝑁𝑘𝑇·

740KforCu-Zn

=740cal/mol

ΔU (experiment)=630cal/mol (difference:duetoshort-rangeorder)


Specificheat,CV

𝐶º =𝑑𝑈𝑑𝑇 =

𝑑𝑈𝑑𝑊

𝑑𝑊𝑑𝑇 = 2𝑁𝑊𝜀

𝑑𝑊𝑑𝑇 = −𝑁𝑘𝑊

𝑑𝑊𝑑(𝑇 𝑇·⁄ )

W1

TC

CV (calc)

CV (exp),short-rangeorderevenatT>TC

β-CuZn order-disordertransition:• F continuous

W (1st derivative)continuousCv (2nd derivative)discontinuous

• 2nd orderphasetransformation• Continuousphasetransformation• Nounder-coolingpossible


Order-disorder(discontinuousphasetransformation)

W

1

TC

Notallorder-disordertransitionis2nd order

MostsystemssuchasCuAu,Cu3Au,CuAu3:• F continuous;W (1st derivative)discontinuous• 1st orderphasetransformation• Discontinuousphasetransformation• Nucleationinvolved

W


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