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