MS516KineticProcessesinMaterialsLectureNote

5.PhaseTransformation—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)

G

XXα1 Xα2

ThermodynamicsofnucleusformationG

XXL Xα

T2

at T2 GL

GS

X0

at T1

X0

spinodal,G”=0

ΔGV

ΔGV

• Drivingforceforphasetransformation(nucleation)?negativeΔGV• Nucleationinvolvestheformationofinterfacewhichcostsextrafreeenergy(positiveAσ)àsmallparticles,ΔG= VΔGV + Aσ >0;nucleationbarrierà particleslargerthanacriticalsizegrow

• Whydoparticlessmallerthanthecriticalsize(calledembryos)formatall?(Hint:vacancyformation)

αα

α

α1

α1

α1α2

L

ThermodynamicsofnucleusformationHomogeneousnuclei

α + α⟺β2β2 + α⟺β3

βi-1 + α⟺βi

··

iα⟺βi (clusterofβ structurewithi atoms)

Seriesofbimolecularreactions(VolmerandFrenkel)

Thermodynamicbalance:

Mixinginni clustersofsizei increasetheentropy ofthesystemà someclusterpopulationwillalwaysbepresent.

Dilutesolutiontheoryformixtureofn1,n2,………..,ni clustersChemicalpotentialofclusterofi atoms:

𝜇" = 𝜇"$ + 𝑅𝑇 ln𝑛"∑ 𝑛""

Inequilibrium:µi =i µ1

• µi ≡µ ofβi cluster(changeinGibbsfreeenergywhenanotherclusterofsizei isadded)≠µ ofanatominaβi cluster• µ1 ≡µ of“monomer”ofα𝜇" = 𝜇"$ + 𝑅𝑇 ln

𝑛",∑ 𝑛","

= 𝑖𝜇.

(ni:#ofclusterswithi atomspervolume)

𝜇" = 𝜇"$ + 𝑘0𝑇 ln𝑛",∑ 𝑛","

= 𝑖𝜇.𝑛",∑ 𝑛","

= exp −𝜇"$ − 𝑖𝜇"𝑅𝑇

Approximation:(#ofmonomersdominatesthesum)5𝑛" ≈ 𝑛.

𝑛", ≈ 𝑛.exp −𝜇"$ − 𝑖𝜇"𝑅𝑇 = 𝑛. exp −

∆𝐺"𝑘0𝑇

(“local”freeenergytotakei atomsandformoneclusterβi)

Assumptionof“classical”nucleationtheory(Gibbs):

∆𝐺" = 𝑎𝑖:/< + 𝑏𝑖= 𝜎𝐴 + Δ𝐺B𝑉D

assumedindependentofi (positive)

area

negative

volumeofβ phase

Thermodynamicsofnucleusformation

,where ∆𝐺" =𝜇"$ − 𝑖𝜇.𝑁FGHIJKLH

Sphericalclusters:

∆𝐺" = 𝑎𝑖:/< + 𝑏𝑖 =4𝜋𝑟<

3 Δ𝐺B + 4𝜋𝑟:𝜎

Criticalnucleus:𝑑∆𝐺"𝑑𝑟 = 0 ⇒ 𝑟∗ = −

2𝜎Δ𝐺B

,Δ𝐺∗ =16𝜋𝜎<

3Δ𝐺B:

Equilibriumconcentrationofcriticalnuclei: 𝑛∗ = 𝑛.exp −∆𝐺∗

𝑘𝑇

Thermodynamicsofnucleusformation

Clusterslargerthanther*grows.

Q:Whyisr*notdefinedbythesizeabovewhichΔG <0?

– +

Heterogeneousnuclei

Contactangleθ isdeterminedbythesurfaceenergybalance:σLM =σSM +σSL cosθ

Ifθ <180o (somewetting):• bettertohavesolid-moldinterfacethansolid-liquidinterface• ΔGiwouldbelowerinthepresenceofmold(byreplacing“expensive”S-Linterfacewith“cheap”S-Minterface)

• ΔGi (hetero.)=ΔGi (homo.)*S(θ)

• 𝑟∗ = −:YZ[\]̂

, Δ𝐺∗ = ._`YZ[a

<\]̂ b 𝑆(𝜃)

• 𝑆(𝜃) = (2 + cos𝜃) 1 − cos𝜃 :/4

Thermodynamicsofnucleusformation

σSL

σLM σSM

(forthederivationofS(θ),seeAppendixI)

Clustersizedistribution• Volmer-Webermodel:Anequilibriumdistributionofclusters(uptoandincludingi*;clusterslargerthani*growandareremovedfromthedistribution)isassumedtoexist

• Becker-Döring model:Steady-statedistribution intermsof(known)equilibriumdistribution

𝑛" = 𝑛.exp −Δ𝐺"𝑘0𝑇

ifi ≤i*,otherwisezero; 𝑛Bj∗ = 𝑛. exp −Δ𝐺∗

𝑘0𝑇

𝑛0k∗ =1𝑖∗

Δ𝐺∗

3𝜋𝑘0𝑇

./:

exp −Δ𝐺∗

𝑘0𝑇=1𝑖∗

Δ𝐺∗

3𝜋𝑘0𝑇

./:

𝑛Bj∗

Zeldovich factor

i /i*

n i

(ForfulltreatmentsofBecker-Döringmodel,seeAppendixII)

Growth:Attachmentrate

𝑘lmn = 𝑘$ exp −Δ𝐺op

𝑘0𝑇+−Δ𝑔r2𝑘0𝑇

− 𝑘$ exp −Δ𝐺op

𝑘0𝑇+Δ𝑔r2𝑘0𝑇

= 𝑘$ exp −Δ𝐺op

𝑘0𝑇exp

−Δ𝑔r2𝑘0𝑇

− expΔ𝑔r2𝑘0𝑇

≈ 𝑘$ exp −Δ𝐺op

𝑘0𝑇−Δ𝑔r𝑘0𝑇

negativeExamplesofΔgN:• Undercoolingbelowtheequilibriumtemperature,• Compressingbeyondtheequilibriumpressure,• Supersaturatingasolutionbeyondtheequilibriumconcentration,

Δ𝑔r 𝑇 = 𝑇 −𝑇m Δ𝑠tD =𝑇 −𝑇m

𝑇mΔ𝐻tD

Δ𝑔r(𝑃) = (𝑃− 𝑃m)ΔΩtDΔ𝑔r 𝐶 = −𝑘0𝑇 ln

𝐶 − 𝐶m

𝐶m

𝑣 =𝑏𝜏=

𝑏1 (𝑓𝑘lmn)⁄ = 𝑏𝑓𝑘lmn = 𝑏𝒇𝑘$exp −

Δ𝐺op

𝑘0𝑇−𝜟𝒈𝑵𝒌𝑩𝑻

τ:timeneededtoaddaMLf:thefractionofsitesatwhichthegrowthreactioncanoccurknet:netreactionrateconstantfortheatomistic attachmentatthegrowthsites

GrowthrateUniformgrowth(growthataroughsurface:f=1)

𝑣~𝐷"

𝑎 −∆𝑔r𝑘0𝑇

𝟏

= −𝐷"∆𝑠tD𝑎𝑘0𝑇

∆𝑇

𝑣 = 𝑏𝑘$ exp −Δ𝐺op

𝑘0𝑇−𝛥𝑔r𝑘0𝑇

∝ (−𝛥𝑔r)

Forsmallundercoolingofaliquid,

Stepnucleation-limitedgrowth(2Dnucleation-limitedgrowth)“Singular”interfaceofexactlytheclose-packedorientationàifnoextrinsicsupplyofsteps,monolayernucleationisrequiredforfurthergrowth

𝛥𝐺":k = 𝜋𝑟:𝑎∆𝑔rΩ + 2𝜋𝑟𝑎𝜎

𝑑∆𝐺":k

𝑑𝑟 = 0 ⇒𝑟∗ = −𝜎ΩΔ𝑔r

, Δ𝐺∗,:k = −𝜋𝑎𝜎:ΩΔ𝑔r

𝐼 = 𝑛∗,:k2𝜋𝑟∗𝑎𝑎: 𝑘$ exp −

Δ𝐺op

𝑘0𝑇=

Δ𝐺∗,:k

𝑖∗,:k𝑘0𝑇𝑒�

\]∗,b����

2𝜋𝑟∗

𝑎 𝑘$ exp −Δ𝐺op

𝑘0𝑇

Nucleationrate(steady-statedistributionofclusters,Becker-Döring):

total#of2Dcriticalnuclei total#ofsitesalongthevertical

surfaceofcriticalnucleus

Growthrate:stepnucleation-limited

attachmentrate

𝐼 ≈𝜋𝑘$𝑎 exp

𝜋𝑎Ω𝜎:

Δ𝑔r𝑘0𝑇, Δ𝑔r ∝ − ∆𝑇 𝐼 ∝ exp −

𝜋𝑎Ω𝜎:

∆𝑇 𝑘0𝑇

∆𝑇 ∆𝑇

Morenucleationàmorestepsonwhichtogrow

B-Ddistributionfor2Dnuclei

Growthrate:dislocation-assistedInsteady-stategrowth,stepwindstoformagrowthspiral.Theradiusofcurvatureofthemosthighlycurvedpartofthespiral≥R*

∆𝑅 = 𝐾ImH 𝑅∗ = 𝐾ImH𝜎Ω−∆𝑔r

𝑓m�� =𝑎∆𝑅 =

−∆𝑔r𝑎𝐾ImH𝜎Ω

geometricfactor~10

Inreality,theexponentn is1<n <2.

ΔR≈−∆𝑔r𝑎:

𝐾ImH𝜎Ω𝑘$ exp −

Δ𝐺op

𝑘0𝑇−𝛥𝑔r𝑘0𝑇

≈𝑎:

𝐾ImH𝜎Ω𝑘0𝑇𝑘$ exp −

Δ𝐺op

𝑘0𝑇−∆𝑔r 𝟐

𝑣 = 𝑏𝑓𝑘$ exp −Δ𝐺op

𝑘0𝑇−𝛥𝑔r𝑘0𝑇

Summaryofgrowthrate

Zonemelting

WilliamPfann (BellLab)

Alloysolidification

• Consider1Dsolidificationofliquidwithcomposition,x0

• Assumingpartitioncoefficient(k =xL /xS )isindependentofT

Forinfinitelyslowequilibriumsolidification:• AtT1,solidificationbeginswiththenucleationofsolidphasewithkx0 (soluteatomshavetoberejectedfromthesolidintotheliquid)

• AsT drops,molefractionofsoluteinsolid(inliquid)remainsuniformfollowingthesolidus(liquidus)

• SolidificationcompletesatT3.

AtT2

solid liquid

AlloysolidificationNodiffusioninsolidandperfectmixinginliquid

• AtT1,solidificationbeginswiththenucleationofsolidphasewithkx0

• AsT drops,liquidbecomesricherinsolutefollowingliquidus (perfectmixinginliquid)à thenextsolidslightlyricherinsolute

T

@T1

@TE

• Soluteconcentrationgradientinsolid(nodiffusioninsolid)

TE

𝒌

𝒌

𝒌localequil.atS-Linterface

AlloysolidificationNodiffusioninsolidandperfectmixinginliquid

T

@T1

@TE

𝒌

𝒌

𝒌localequil.atS-Linterface

• xS(z)? 𝑥� − 𝑥� 𝑑𝑓� = 1 − 𝑓� 𝑑𝑥�

soluterejectedfromthesolid

fractionofliquid

𝑥� = 𝑥�/𝑘 = 𝑥$(1 − 𝑓�)��.= 𝑥$𝑓���.

𝑥�(𝑧) = 𝑘𝑥$(1− 𝑓�)��.= 𝑘𝑥$ 1 −𝑧(𝑡)𝐿

��.

𝑥� =1𝑧� 𝑥��

$𝑧 𝑑𝑧 =

𝑥$𝑓�

1 − (1 − 𝑓�)�

L

increaseinliquidsoluteconc.

,wherez(t)istheposition ofS-L interface

(independentofz becauseofperfectmixing)

• Withk <1, xS andxL divergeasfS approachesto1(i.e.completionofsolidification)?• Practicallimits:xL =xE andxS =kxE.Why?

AlloysolidificationNodiffusioninsolidanddiffusionalmixinginliquid

• Initialtransient:soluterejectedfromthesolidwillonlybetransportedbydiffusionà rapidbuildupofsoluteinliquidneartheinterface(notperfectmixingintheliquid)

à rapidincreaseofsoluteconc.inthenextsolidformed

• Steady-statewhentheinterfacetemperaturereachesT3:liquidadjacenttosolidx0/k,solidformswithcompositionx0;soluterejectedfromthesolidtoliquid=solutetransportedawayfromdiffusioninliquid @TE

𝒌

initialtransient

steady-state

finaltransient

@T3

Eutecticsolidification

Nature405,434(2000)

Eutecticsolidification

Δ𝐺 𝜆 = 𝐺� − 𝐺� = Δ𝐺 ∞ +2𝜎tD𝜆

Là α +β

• Considerunidirectionaleutecticsolidificationresultinginlamellastructure.λ?

T1

xα xβxE

@T1

xα xβxE

G

(pervolume)– +

Δ𝐺 ∞ = −Δ𝐻� Δ𝑇

𝑇�, whereΔHF is(positive)heatoffusion(latentheat).

• Foragivenundercooling, Δ𝑇. = 𝑇� − 𝑇. ,whatisminimumλ thatmakesΔG(λ)≤0?

𝜆�"l =2𝜎tD𝑇�Δ𝐻� Δ𝑇

TE

Eutecticsolidification

• Δ𝐺t 𝜆�"l ,Δ𝐺D 𝜆�"l : freeenergyofα andβ includingthecontributionfromα−β interface

• Q:wherewouldΔ𝐺t 𝜆 > 𝜆�"l andΔ𝐺t 𝜆 < 𝜆�"l beintheG-x plot?

@T1

=2𝜎tD𝜆�"l

• Lamellastructurewithafiniteλàincreaseinfreeenergyofα andβàeffectivelylowering“TE”byΔT1

Eutecticsolidification• Withundercooling Δ𝑇 (= Δ𝑇. + Δ𝑇: ),lamellawithanyλ >λmin shouldbethermodynamicallystableandgrow.

• Drivingforceforthegrowth?• Withwhichλ,thelamellagrowsthefastest(largerλ:largerdrivingforcebutlongerdiffusionlength)?

freeenergy“consumed”toforminterface

“remaining”freeenergy:drivingforcefordiffusionalfluxtogrowthelamellastructure

xEΔ𝐺 ∞ −

2𝜎tD𝜆

=2𝜎tD𝜆�"l

−2𝜎tD𝜆

=Δ𝐻� Δ𝑇:

𝑇�

Eutecticsolidification

cLβ (xLβ/Ω):soluteB conc. intheliquidnearβ-L interface

cLα (xLα/Ω):soluteB conc.intheliquidnearα-L interface

slopem

Δ𝑇: =𝑚2(𝑥�t − 𝑥�D) =

𝑚2Ω∆𝑐

Δ𝑐 = 𝑐�t − 𝑐�D > 0à 𝐽¦ à growthwithv

y

𝐽¦ = −𝐷�𝑑𝑐𝑑𝑦

=2𝐷�∆𝑐𝜆

z

𝑣 =𝑑𝑧𝑑𝑡

=2𝐽¦

(𝑐� − 𝑐t�)=

8𝐷�(𝑐� − 𝑐t�)

Δ𝑇𝑚Ω

1𝜆1 −

𝜆�"l𝜆

#ofsolutethatshouldberejectedfromα togrowbydz

(𝑐� − 𝑐t�)𝑑𝑧 = 2𝐽¦𝑑𝑡#ofsolutediffusedawayinliquidnearα-Linterface

Mostlikely lamellaspacing,λ*(correspondingtothefastestgrowth,maxv),𝝀∗ = 𝟐𝝀𝒎𝒊𝒏

Heterogeneousnucleus

AppendixI:CalculationofS(θ)

S(θ)

AppendixI:CalculationofS(θ)

[4S(θ)]

S(θ)

S(θ)

Becker-Döring (Steady-statetreatmentofnucleationkinetics)Net rateoftransitionfromclustersofsize(i-1)toclustersofsizei:𝐼" = 𝑛"�.𝑎"�.𝑘­"�. − 𝑛"𝑎"𝑘�"

Togetk- intermsofk+ invokeequilibrium(net rate=0):Ii =0

𝑛®"�.𝑎"�.𝑘­"�. = 𝑛®"𝑎"𝑘�" 𝐼" = 𝑛®"�.𝑎"�.𝑘­"�.𝑛"�.𝑛®"�.

−𝑛"𝑛® "

Or,indifferentialform,

𝐼" = −𝑛®"𝑎"𝑘­"𝑑𝑑𝑖

𝑛"𝑛®"

givessteady-stateintermsofequilibriumdistribution

Insteady-state:Ii=constantwithrespecttoi =I (nucleationrate)

Integrate: 𝐼 �𝑑𝑖

𝑛® "𝑎"𝑘­"

0

"= � 𝑑

𝑛"𝑛® "

0

"=𝑛"𝑛®"−𝑛0𝑛®0

=𝑛"𝑛® "

(B:arbitrarylargenumberofi,whereni =0)

𝑛" = 𝑛®"𝐼�𝑑𝑖

𝑛® "𝑎"𝑘"

0

" 𝑘" ≡ 𝑘­" ;𝑛®" = 𝑛. exp −

∆𝐺"𝑘0𝑇

AppendixII:Becker-DöringModel

𝑛" = 𝐼 exp −∆𝐺"𝑘0𝑇

�exp ∆𝐺"

𝑘0𝑇𝑎"𝑘"

𝑑𝑖0

"

eΔGi/kBT hassharpmaximumaroundΔGi =ΔG*,sotakeaiki ~ai*ki*(constant)andexpandΔGi

ExpandΔGi inaTaylorseriesaboutΔG*

∆𝐺" = ∆𝐺∗ −Γ2 𝑖 − 𝑖∗ :;Γ = −

𝜕:∆𝐺"𝜕𝑖: "³"∗

𝑛" =𝐼 exp − ∆𝐺"

𝑘0𝑇𝑎"∗𝑘"

∗ exp −∆𝐺∗

𝑘0𝑇� exp −

Γ2(Δ𝑖):

𝑘0𝑇𝑑(Δ𝑖)

0�"∗

"�"∗

𝑛" =𝐼 exp − ∆𝐺"

𝑘0𝑇𝑎"∗𝑘"

∗ exp −∆𝐺∗

𝑘0𝑇2𝜋𝑘0𝑇Γ

:

Ifi smallenough,integrationlimitcanbetaken-∞to+∞

AppendixII:Becker-DöringModel

𝑛"𝑛® "=

𝐼𝑛.𝑎"∗𝑘"

∗ exp −∆𝐺∗

𝑘0𝑇2𝜋𝑘0𝑇Γ

:

Fori farawayfromi*,ori <i*– (kBT/Γ)1/2à 𝑛" = 𝑛®"

𝐼 = 𝑛.𝑎"∗𝑘"∗ 2𝜋𝑘0𝑇

Γ

:

exp −∆𝐺∗

𝑘0𝑇

𝐼0�k = 𝐼B�j2𝜋𝑘0𝑇Γ

:

Zeldovich factor

AppendixII:Becker-DöringModel