MS516KineticProcessesinMaterialsLectureNote

4.SurfacesandInterfaces—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)

CourseInformationMeltingtemperatureofnanoparticles

Buffat andBorel,PhysRevB13,2287(1976)

Meltingtemperatureofgoldparticlesasafunctionofsize

TM ofbulkgold~1064oC(1337K)

• Thermodynamicpropertiesofnanoparticlescanbeverydifferentfromthebulkduetotheincreasinginfluenceofsurfaces

• Godmadethebulk;thesurfacewasinventedbythedevil.--WolfgangPauli

CourseInformationThermodynamicsofinterfaces(surfaces)1st lawofthermodynamics:Foranywell-characterizedchangeofastate,

ΔU =Q ̶̶WChangeininternalenergyofthestate

Heatabsorbedbythesystem

Workdonebythesystemonthesurroundings

Forreversibleprocesses:δQ =TdS dU =T dS ̶̶ dWrev

SeparatedWrev intochemicalcontributionsandnonchemicalcontributions:

𝑑𝑊#$% ='𝜇)𝑑𝑁)

+

),-+'𝐹)𝑑𝑥)

1+

),-µ:chemicalpotentialc:numberofchemicalcomponentsnc:numberofnon-chemicalforces(pressure,gravity,electricandmagneticfields,etc.)Fi:aforcee.g.,(pressure,gravity,electricandmagneticfields,etc.)dxi:adisplacemente.g.,(volume,height,polarization,etc.)

AnotherpossibilityforFidxi isσ dA (σ:surfacefreeenergy,A:surfacearea)

CourseInformationThermodynamicsofinterfaces(surfaces)

𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 + 𝐹)𝑑𝑥) + 𝜇)𝑑𝑁)

T,µi,andFi areintensivevariables:independentofsizeofsystem(mass)

intensive extensive

Integratebybringingtogethermanysmallidenticalsystemswiththesameintensivevariables:

𝑈 −𝑈 0 = 𝑇𝑆 − 𝑃𝑉 + 𝐹)𝑥) + 𝜇)𝑁); 𝑈 0 = 0

Taketotaldifferential:

𝑑𝑈 = 𝑇𝑑𝑆 + 𝑆𝑑𝑇 − 𝑃𝑑𝑉 − 𝑉𝑑𝑃+ 𝐹)𝑑𝑥) + 𝑥)𝑑𝐹) + 𝜇)𝑑𝑁) + 𝑁)𝑑𝜇)

(*)

Comparisonwith(*)(1st law):

0 = 𝑆𝑑𝑇 − 𝑉𝑑𝑃+'𝑥)𝑑𝐹)

1+

),-+'𝑁)𝑑𝜇)

+

),-Gibbs-DuhemEquation

CourseInformationGibbs-Duhemequation

Gibbs-DuhemEquation

• Gibb-Duhemequationsaysintensive variablesarenotallindependentlyvariablewithinaphase.IfyouchangeT and{Fi}andallµi exceptone,thelastoneissetautomatically.

• Thisequationcanbeappliedtoeachphaseseparately,ortothewholesystemconsistingofmultiplephases(phasesandinterfacestogether).

Aside:fromthiscomesthesimplestderivationoftheGibbsphaserule• OneGibbs-Duhemequationforeachofp phases• VariablesareT,P,andonechemicalpotentialforeachofc species(ignoringother

non-chemicalcontributions)à p equations&(c+2)unknownsà #ofvariations(dF)inintensiveparameterswhichmaybemadearbitrarilywhileremaininginequilibriumisc +2– p

dF =c +2– pe.g.,eutectic“point”isalineifpressureisallowedtovary.IfP isconstant;dF =c +1– p

0 = 𝑆𝑑𝑇 − 𝑉𝑑𝑃+'𝑥)𝑑𝐹)

1+

),-+'𝑁)𝑑𝜇)

+

),-

CourseInformationThermodynamicsofinterfaces(surfaces)

phaseα

phaseβ

interface(surface)

z,distance

Extensivethermodynamic

quantity(suchasU,S,V,…)

• Gibbs-Duhem foreachofhomogeneousregions(phaseα,phaseβ,andinterface):

𝑆;𝑑𝑇 − 𝑉;𝑑𝑝 +'𝑁);𝑑𝜇)

+

),-

= 0

𝑆 𝑑𝑇 − 𝑉 𝑑𝑝 +' 𝑁) 𝑑𝜇)

+

),-

+ 𝑑𝜎 = 0

𝑆>𝑑𝑇 − 𝑉>𝑑𝑝 +'𝑁)>𝑑𝜇)

+

),-

= 0

(phaseα)

(phaseβ)

(interface)

Layerincludinginterface(arbitrarilychosenaslongasthickenoughtoincludeinhomogeneousregionsaffectedbythepresenceofinterface)

• 3(#ofphasesp+1)equations&c+3(T,P,σ)variablesàdegreesoffreedom,dF =c – p + 2

• [S]and[V]dependonthechoiceofthelayerthickness,butσshouldnot

(extensivevariableperunitareaoftheinterface)

0 = 𝑆𝑑𝑇 − 𝑉𝑑𝑝+'𝑥)𝑑𝐹)

1+

),-+'𝑁)𝑑𝜇)

+

),-

includesA∙dσ fortheinterfacelayer

CourseInformationThermodynamicsofinterfaces(surfaces)• Fortwo-components(i=1,2)system(c =2):

/

Q:Wouldthesebeindependentofthearbitrarychoiceofthelayerthickness?They’dbetterbe!

,=- ΔS =ΔV

excess duetotheinterface(differenceb/wsystemwithandwithout aninterface)

CourseInformationThermodynamicsofinterfaces(surfaces)• Iftheinterfaciallayerconsistsofnα units(#)ofα phaseandnβ unitsofβphases,

• Excessentropy,ΔS (S oftheinterfacialregionminus S ofthesameregionwithouttheinterface,i.e.entropyassociatedwiththeinterface)

= −𝜕𝜎𝜕𝑇 @

• Q:Whyisthefollowingexpressionphysicallywrong(althoughmathematically right)?

independentoflayerthickness

dependentoflayerthickness

totalentropyoftheinterfacialregion

entropyofasystemconsistingofthesame#ofnα ofα andnβ ofβwithoutanyinterface

CourseInformationGibbsabsorptionequation• Grainboundaryinasinglecomponentsystem(p =1,c =1,dF =2)

𝑆;𝑑𝑇 − 𝑉;𝑑𝑝 + 𝑁-;𝑑𝜇- = 0

𝑆 𝑑𝑇 − 𝑉 𝑑𝑝 +𝑁-;𝑑𝜇- + 𝑑𝜎 = 0

𝑑𝜎 = − 𝑆 −𝑁-𝑁-;

𝑆; 𝑑𝑇 + 𝑉 −𝑁-𝑁-;

𝑉; 𝑑𝑝 = − Δ𝑆 𝑑𝑇 + Δ𝑉 𝑑𝑝

excessentropyassociatedwithg.b

excessvolumeassociatedwithg.b

Comparewithfreeenergyofasinglebulkphase,dG =– SdT +Vdp

• Grainboundaryinabinarysinglebulkphasesystem(p =1,c =2,dF =3)

𝑑𝜎 = − 𝑆 −𝑁-𝑁-;

𝑆; 𝑑𝑇 + 𝑉 −𝑁-𝑁-;

𝑉; 𝑑𝑝 − 𝑁B −𝑁-𝑁-;

𝑁B; 𝑑𝜇B

𝜕𝜎𝜕𝜇B C,E

= − 𝑁B −𝑁-𝑁-;

𝑁B; = −∆𝑁B;interfaceexcess ofcomponent2(solute)

CourseInformationGibbsabsorptionequationSoluteconc.

(e.g.CinFe)

?

grainboundary

𝜕𝜎𝜕𝜇B C,E

= − 𝑁B −𝑁-𝑁-;

𝑁B; = −∆𝑁B;

positive∆𝑁B;

negative∆𝑁B;Asolutepreferentiallyabsorbs(‘segregates’)ataninterfacewhenthefreeenergyoftheinterfacedecreaseswithanincreaseinthesolutechemicalpotential

Ifcomponent2(solute)isdiluteinphaseα:

𝜇B; = 𝜇B;,G + 𝑅𝑇 ln𝑋B; 𝑑𝜇B; =

𝑅𝑇𝑋B;

𝑑𝑋B;è

𝑑𝜎𝑑𝑋B;

= −𝑅𝑇𝑋B;

(∆𝑁B;)

Examples:• Soapinwater:dσ /dX2 largeandnegative(verylargepositiveexcesssoaponthewatersurface)

• NaCl inH2O:dσ /dX2 smallandpositive(smallnegativeexcessNaCl,i.e.lesssalty,onthewatersurface)

CourseInformationGibbs’originalapproach

phaseα

phaseβ

Gibbsdividingsurface

z,distance

Extensivethermodynamic

quantity(suchasU,S,…)

Volume

• Phasesdividedbyadividingsurfacewhich,asfarasonecantell,runsparalleltothephysicallyperceivedsurface.

• Excessquantity:actualquantityofthetotalsystem(includingtheinterface)minusquantitiesthatthephaseswouldhaveiftheywereuniformrightuptothedividingsurface

• Gibbs’choicesinlocatingthedividingsurface:noexcessvolumeandnoexcesssolvent(component1)

•

• Itwasshownthatindividualexcessquantitiesaredependentonthelocationofthedividingsurface,butσ isnot.

σ = 𝑈OP − 𝑇𝑆OP −'𝜇)

+

),-

𝑁)QR

(extensivequantitiesperunitinterfacearea)

CourseInformationSurfaceenergyvs.surfacestress

I:2σ AII:(FB+2lfS)δx

𝐴T = 𝑙 𝑥 + 𝛿𝑥 = 𝐴 1 +𝛿𝑥𝑥𝜎T ≈ 𝜎 +

𝑑𝜎𝑑𝜖

𝛿𝑥𝑥 ,

I+II=III+IV

𝒇𝑺 =1𝑙𝛿𝑥 𝜎 +

𝑑𝜎𝑑𝜖𝛿𝑥𝑥 𝐴 1 +

𝛿𝑥𝑥 − 𝜎𝐴 ≈

1𝑙𝛿𝑥 𝜎 +

𝑑𝜎𝑑𝜖 𝐴

𝛿𝑥𝑥 = 𝝈 +

𝒅𝝈𝒅𝝐

III:FB δxIV:2σ’A’

forceperunitlengthexertedbythenewsurface:surfacestress

CourseInformationSurfaceenergyvs.surfacestress

• Youcancreatechargedsurfacesthatwanttospreadout(negative surfacestress,dσ/dε <−σ)

• Thesurfacewillstillbehappierifitdisappears(positive surfaceenergy)• Q:Isnegativesurfacestresscompressiveortensile?• Q:Issurfaceenergy=surfacestressinliquid?

• Surfacestress:Forcerequiredtoinfinitesimallydeformthesurfaceatconstantnumberofsurfaceatoms

• Surfaceenergy(surfacetension):WorkrequiredtocreatemoresurfaceatomsCleaveGaAsalong(111)

negativelychargedorbitalsatsurface

emptyorbitals

anneal:wafercurlsdown(becauseofnegativesurfacestress)!

CourseInformationSurface(free)energy:enthalpyσ = ∆𝐻P − 𝑇∆𝑆P,

∆𝐻P= 𝐹 `∆ℎ%bc𝐶e =𝐹 ` ∆ℎ%bc𝑉fB/h

enthalpyofvaporizationperatom

densityofatomsonthesurface(#/cm2)

(energy/unitarea)

fractionofbrokenbondsforanatomonthesurface

Meidema,Z.Metallkde69,287(1978)

• Estimationof<FBB>basedonthesimplebrokenbondscountingisreasonable~1/6.

avg.fractionofbrokenbondsforanatomonthesurface

CourseInformationSurface(free)energy:enthalpy

a

𝐶e 0 =𝑁Area =

1𝑎B

𝐶e 𝜃 =𝑁 + (Area ∗ tan 𝜃)/𝑎B

Area/cosθ

=1𝑎B cos𝜃 + sin 𝜃

∆𝐻P= 𝐹 `∆ℎ%bc𝐶e ≈16𝑎B cos𝜃 + sin𝜃 ∆ℎ%bc

Angulardependenceofsurfaceenthalpy

• Cuspsinsurfaceenthalpyatθ =0,whichhasthesmallestnumberofbrokenbonds

CourseInformationSurface(free)energy:entropyTwocontributionstotheentropypersurfaceatom:• Thesurfaceatomshavemorethermalfreedomduetothereductionofgeometricconstraints

• Extraconfigurationalentropyassociatedwithsurfacedefects

CourseInformationSurface(free)energy• Dostepsformonanoriginallyflatsurface?

Consider Estimateenergyforformationof(A,V)pairon(111)FCCsurface:

formvacancy:breakbondsformadsorbedatom:breakbonds

net:breakbonds

CohesiveenergyinFCC,UC =(1/2)z *bondenergy=6bondsperatomà Socostinenergyforeithervacancyoradsorbedatomseparately:UC/2~1eV

à Concentration:,oneAforeachV

à EstimatemagnitudeatRT:,lessthanoneperm2

𝑛x = 𝑛f = 𝑛e exp −𝑈{/2𝑘`𝑇

𝑛x𝑛e

≈ exp −1𝑒𝑉140 𝑒𝑉

= 10�-�

à Surfaceissmooth (nostepsspontaneouslyform)

CourseInformationSurface(free)energy• Dokinksformonagivenstraightstep?

upperplane

lowerplanestep

Motionofoneatomsà 4kinksEstimateenergyforformationofkinkson[110]ledgeon(111)plane

+:removalofatom:breakbonds- :additionofatom:formbonds

net:breakbondsorperkink:breakbonds=UC /12

Concentration:,one+foreach–

Estimate:at300K

Stepisrough

#ofsitesperunitlengthofledge

𝑛� = 𝑛� = 𝑛� exp −𝑈{/12𝑘`𝑇

𝑛�𝑛�≈ exp −

1/6𝑒𝑉1/40𝑒𝑉 = 10�h

(At1000oC:kBT~1/10eV=>n+/nl~0.2)

CourseInformationWulff plotWulff plot(Plotofσ(θ)inpolarcoordinates) Equilibriumshape

Innerenvelopeofperpendicularstoσ(θ)fromorigin

Equilibrium:∫∫σdA minimized;constraintoffixedtotalvolume

AthighT:Entropylowersσ ofnon-close-packedfaces

Notethatgrowthformwillingeneraldifferfromequilibriumform!

CourseInformationCapillarypressure(Laplacepressure)Considerisolated2-phasesysteminequilibrium:

α

β𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃;𝑑𝑉; − 𝑃>𝑑𝑉> + 𝜎𝑑𝐴 +'𝜇)𝑑𝑛)

+

),-dU oftotalsystem=0dS oftotalsystem=0(Sismaximized)dni oftotalsystem=0(closedsystem)

⇒ 0 = −𝑃;𝑑𝑉; − 𝑃>𝑑𝑉> + 𝜎𝑑𝐴 (𝑑𝑉; = −𝑑𝑉>)

⇒𝑃>−𝑃; = 𝜎𝑑𝐴𝑑𝑉>

generalexpression

𝑃>−𝑃; =2𝜎𝑟

Laplaceformulaforsphericalsurfaceofradiusr

fornon-spherical,curvedsurfaceswithprincipleradiiofcurvaturer1 andr2

𝑃>−𝑃; = 𝜎1𝑟-+1𝑟B

= 𝜎𝜅

Pβ:Pressureexperiencedbymoleculesinβ phase

CourseInformationSizeeffectonchemicalpotential

𝑑𝜇>

𝑑𝑃 �C

= 𝑉�> (molarvolume)

𝜇�> − 𝜇�,�> = � 𝑉�>𝑑𝑝 =e���� E���)+��E��ee���,E�,

>������E��ee���,E�,E�

(assumingVmβ isindependentofP)

Liquiddropletsurroundedbyvapor(1component)

Moleculeinβ feelshigherpressurethanPαbutdoesnotknowifitcomesfromsurfacetensionorfromaphaseorothersurroundingsà foramoleculeinβ,mustbe>𝜇�

>

Crystal particles

𝜇�> = 𝜇�> + 𝑉�>𝜎𝑑𝐴𝑑𝑉> = 𝜇�> + 𝑉�> 𝑑

𝑑𝑉>' 𝐴)𝜎)

�����+��e(facetedcrystal)

𝜇�> = 𝜇�> + 𝑉�> 𝜎 +𝑑B𝜎𝑑𝜃-B

𝜅- + 𝜎 +𝑑B𝜎𝑑𝜃BB

𝜅B(continuouslycurvedcrystalshape)

𝜇�,�>

CourseInformationGibbs-ThompsonEffectSmallparticlehaslowermeltingpointthanthebulk

liquid

bulksolid

T

µ

smallparticles

TM,∞TM,rΔT

∆𝜇 = Δ𝑆�Δ𝑇

∆𝑇 = −𝜎𝜅ΩΔ𝑆�

(peratom)𝑇�,� = 𝑇�,� 1−

𝜎𝜅ΩΔ𝐻�

= 𝑇�,� 1 −𝜎Δ𝐻�

2Ω𝒓

forsphericalparticles

∆𝜇 = 𝜎𝜅Ω

(Ω:atomicvolume)

CourseInformationSharpvs.diffuseinterface

G

XXα Xβ

(A-rich) (B-rich)

α βSystemwithmiscibilitygap:• AA,BBbondspreferredoverABbonds

• 𝜀 = 𝜀f` −12 𝜀ff + 𝜀`` > 0 X

z

sharp

X

z

diffuse,thenhowdiffuse?

vs.

WeseekforX(z)thatminimizesinterfacialfreeenergy,σ

X:molefractionofB

lessnegative

I IIα

β

α

β

α

β

IIIα β

α β

AAbonds(1– Xα)2 (1– Xβ)2BBbondsXα2 Xβ2ABbonds2Xα (1– Xα)2Xβ (1– Xβ)

fractionatα-α interface fractionatβ-β interface

𝐻�� − 𝐻� = −𝐴𝑎B { 1 − 𝑋;

B𝜀ff + 𝑋;B𝜀`` + 2𝑋; 1 − 𝑋; 𝜀f`

+ 1 − 𝑋>B𝜀ff + 𝑋>B𝜀`` + 2𝑋> 1− 𝑋> 𝜀f`}

𝜀ff:bondingenergyofA-A(negative)

𝐻��� − 𝐻��= +

2𝐴𝑎B { 1 − 𝑋; 1− 𝑋> 𝜀ff + 𝑋;𝑋>𝜀`` + (1 − 𝑋;)𝑋> + 𝑋; 1 − 𝑋> 𝜀f`}

𝑋;(𝑋>):molefractionofBinα-phase (β-phase)

Sharpvs.diffuseinterface

𝐻��� − 𝐻� = 2𝐴Δ𝐻R

Δ𝐻R =1𝑎B 𝑋> − 𝑋;

B𝜀f` −

12 𝜀ff + 𝜀`` = 𝜀

Δ𝑋𝑎

B

*CahnandHilliard,J.Chem.Phys.28,238(1958)

= 𝜀 𝛻𝑋 Bor𝑎�𝜀 𝛻𝑐 B gradientenergy*:excessfreeenergycomingfrom“wrong”bonds

X

z

Howtominimizegradientenergy?

interfaceconsistingofn atomiclayers

ΔXΔ𝐻R~𝑛𝜀

∆𝑋𝑛𝑎

B

=𝜀 ∆𝑋 B

𝑎B𝑛

• Theinterfacewantstospreadasmuchasallowed!(nà∞)

• Whatpreventsitfromspreadinginfinitely?

(ΔHS:perunitareaofinterface)

Sharpvs.diffuseinterface

Bulkfreeenergyfromnon-equilibriumcomposition

G

XXα Xβ

diffuseinterface,width~lα β

Xα

XβX’

• Consideraninfinitesimalvolumeelementwithinthediffuseinterfaceregionwhichhasauniformcomposition X’;freeenergyofthisvolumeelementisG(X’)

• ThisvolumeelementcouldloweritsfreeenergytoG0(X’)ifitphase-separateed intoα phase(withcompositionXα)andβ phase(withcompositionXβ).

• Penaltyinbulkfreeenergybyhavinga“wrong” uniformcompositionX’,G(X’)– G0(X’)

• Totalpenaltyofthediffuseinterface,

• Consideringbulkfreeenergycomingfromanon-equilibriumcomposition, anabruptinterfaceispreferred.

X’

� [𝐺 𝑋(𝑧) − 𝐺G 𝑋(𝑧) ]¥¦���

𝑑𝑧 = � ∆𝐺 𝑋(𝑧)¥¦���

𝑑𝑧

z

Sharpvs.diffuseinterface

G(X’)

G0(X’)

• Gradientenergywantstospreadtheinterfacewidth• Bulkfreeenergyoftheinterfacewantstoshrinktheinterfacewidth• Determinedbyminimizinginterfacialfreeenergy,

Interfacewidth,l

𝜎(𝑋 𝑧 ) = � [∆𝐺 𝑋 𝑧 + 𝐾 𝛻𝑋 B]�

��𝑑𝑧gradientenergycoefficient∝εexcessfreeenergy(peratom)ofa

solutionofuniformcompositionX

• Interfacewidthincreaseswithtemperatureandapproachesinfinityatacriticaltemperature,TC.Why?

Sharpvs.diffuseinterface