MS516 Kinetic Processes in Materials Lecture Note 4...
Transcript of MS516 Kinetic Processes in Materials Lecture Note 4...
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