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KINETICS IN NANOSCALE MATERIALS KINETICS IN NANOSCALE MATERIALS KING-NING TU ANDRIY M. GUSAK CoverDesign:Wiley CoverImage:Courtesyoftheauthors Copyright©2014byJohnWiley&Sons,Inc.Allrightsreserved. PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey. PublishedsimultaneouslyinCanada. Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformor byanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,exceptas permittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfeeto theCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,MA01923,(978)750-8400,fax (978)750-4470,oronthewebatwww.copyright.com.RequeststothePublisherforpermissionshould beaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ 07030,(201)748-6011,fax(201)748-6008,oronlineathttp://www.wiley.com/go/permission. 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Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmay notbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitourwebsiteat www.wiley.com. LibraryofCongressCataloging-in-PublicationData: Tu,K.N.(King-Ning),1937- Kineticsinnanoscalematerials/byKing-NingTu,AndriyGusak. pagescm Summary:“Astheabilitytoproducenanomaterialsadvances,itbecomesmoreimportanttounderstand howtheenergyoftheatomsinthesematerialsisaffectedbytheirreduceddimensions.Writtenbyan acclaimedauthorteam,KineticsinNanoscaleMaterialsisthefirstbooktodiscusssimplebuteffective modelsofthesystemsandprocessesthathaverecentlybeendiscovered.Thetext,forresearchers andgraduatestudents,combinesthenoveltyofnanoscaleprocessesandsystemswiththetransparency ofmathematicalmodelsandgeneralityofbasicideasrelatingtonanoscienceand nanotechnology”–Providedbypublisher. “PublishedsimultaneouslyinCanada”–Titlepageverso. Includesbibliographicalreferencesandindex. ISBN978-0-470-88140-8(hardback) 1. Nanostructuredmaterials.2. Chemicalkinetics.3. Nanostructuredmaterials–Analysis. 4. Nanostructuredmaterials–Computersimulation.I.Gusak,AndriyM.II.Title. TA418.9.N35T82014 620.1′1599–dc23 2013042096 PrintedintheUnitedStatesofAmerica. 10987654321 CONTENTS PREFACE ix CHAPTER1 INTRODUCTIONTOKINETICSINNANOSCALEMATERIALS 1 1.1 Introduction 1 1.2 Nanosphere:SurfaceEnergyisEquivalenttoGibbs–ThomsonPotential 3 1.3 Nanosphere:LowerMeltingPoint 6 1.4 Nanosphere:FewerHomogeneousNucleationanditsEffectonPhaseDiagram 10 1.5 Nanosphere:KirkendallEffectandInstabilityofHollowNanospheres 13 1.6 Nanosphere:InverseKirkendallEffectinHollowNanoAlloySpheres 17 1.7 Nanosphere:CombiningKirkendallEffectandInverseKirkendallEffecton ConcentricBilayerHollowNanosphere 18 1.8 NanoHole:InstabilityofaDonut-TypeNanoHoleinaMembrane 19 1.9 Nanowire:PointContactReactionsBetweenMetalandSiliconNanowires 21 1.10 Nanowire:NanogapinSiliconNanowires 22 1.11 Nanowire:LithiationinSiliconNanowires 26 1.12 Nanowire:PointContactReactionsBetweenMetallicNanowires 27 1.13 NanoThinFilm:ExplosiveReactioninPeriodicMultilayeredNanoThinFilms 28 1.14 NanoMicrostructureinBulkSamples:Nanotwins 30 1.15 NanoMicrostructureontheSurfaceofaBulkSample:SurfaceMechanicalAttrition Treatment(SMAT)ofSteel 32 References 33 Problems 35 CHAPTER2 LINEARANDNONLINEARDIFFUSION 37 2.1 Introduction 37 2.2 LinearDiffusion 38 2.2.1 AtomicFlux 39 2.2.2 Fick’sFirstLawofDiffusion 40 2.2.3 ChemicalPotential 43 2.2.4 Fick’sSecondLawofDiffusion 45 2.2.5 FluxDivergence 47 2.2.6 TracerDiffusion 49 2.2.7 Diffusivity 51 2.2.8 ExperimentalMeasurementoftheParametersinDiffusivity 53 2.3 NonlinearDiffusion 57 2.3.1 NonlinearEffectduetoKineticConsideration 58 2.3.2 NonlinearEffectduetoThermodynamicConsideration 59 2.3.3 CombiningThermodynamicandKineticNonlinearEffects 62 References 63 Problems 64 v vi CONTENTS CHAPTER3 KIRKENDALLEFFECTANDINVERSEKIRKENDALLEFFECT 67 3.1 Introduction 67 3.2 KirkendallEffect 69 3.2.1 Darken’sAnalysisofKirkendallShiftandMarkerMotion 72 3.2.2 BoltzmannandMatanoAnalysisofInterdiffusionCoefficient 76 3.2.3 ActivityandIntrinsicDiffusivity 80 3.2.4 Kirkendall(Frenkel)VoidingWithoutLatticeShift 84 3.3 InverseKirkendallEffect 84 3.3.1 PhysicalMeaningofInverseKirkendallEffect 86 3.3.2 InverseKirkendallEffectontheInstabilityofanAlloyNanoshell 88 3.3.3 InverseKirkendallEffectonSegregationinaRegularSolution Nanoshell 90 3.4 InteractionBetweenKirkendallEffectandGibbs–ThomsonEffectintheFormation ofaSphericalCompoundNanoshell 93 References 97 Problems 97 CHAPTER4 RIPENINGAMONGNANOPRECIPITATES 99 4.1 Introduction 99 4.2 Ham’sModelofGrowthofaSphericalPrecipitate(C isConstant) 101 r 4.3 Mean-FieldConsideration 103 4.4 Gibbs–ThomsonPotential 105 4.5 GrowthandDissolutionofaSphericalNanoprecipitateinaMeanField 106 4.6 LSWTheoryofKineticsofParticleRipening 108 4.7 ContinuityEquationinSizeSpace 113 4.8 SizeDistributionFunctioninConservativeRipening 114 4.9 FurtherDevelopmentsofLSWTheory 115 References 115 Problems 116 CHAPTER5 SPINODALDECOMPOSITION 118 5.1 Introduction 118 5.2 ImplicationofDiffusionEquationinHomogenizationandDecomposition 121 5.3 SpinodalDecomposition 123 5.3.1 ConcentrationGradientinanInhomogeneousSolidSolution 123 5.3.2 EnergyofMixingtoFormaHomogeneousSolidSolution 124 5.3.3 EnergyofMixingtoFormanInhomogeneousSolidSolution 126 5.3.4 ChemicalPotentialinInhomogeneousSolution 129 5.3.5 CoherentStrainEnergy 131 5.3.6 SolutionoftheDiffusionEquation 134 References 136 Problems 136 CHAPTER6 NUCLEATIONEVENTSINBULKMATERIALS,THINFILMS, ANDNANOWIRES 138 6.1 Introduction 138 6.2 ThermodynamicsandKineticsofNucleation 140 CONTENTS vii 6.2.1 ThermodynamicsofNucleation 140 6.2.2 KineticsofNucleation 143 6.3 HeterogeneousNucleationinGrainBoundariesofBulkMaterials 148 6.3.1 MorphologyofGrainBoundaryPrecipitates 150 6.3.2 IntroducinganEpitaxialInterfacetoHeterogeneousNucleation 151 6.3.3 ReplaciveMechanismofaGrainBoundary 154 6.4 NoHomogeneousNucleationinEpitaxialGrowthofSiThinFilmonSiWafer 156 6.5 RepeatingHomogeneousNucleationofSilicideinNanowiresofSi 160 6.5.1 PointContactReactionsinNanowires 161 6.5.2 HomogeneousNucleationofEpitaxialSilicideinNanowiresofSi 164 References 168 Problems 168 CHAPTER7 CONTACTREACTIONSONSi;PLANE,LINE,ANDPOINT CONTACTREACTIONS 170 7.1 Introduction 170 7.2 BulkCases 175 7.2.1 Kidson’sAnalysisofDiffusion-ControlledPlanarGrowth 175 7.2.2 SteadyStateApproximationinLayeredGrowthofMultiplePhases 178 7.2.3 MarkerAnalysis 179 7.2.4 InterdiffusionCoefficientinIntermetallicCompound 182 7.2.5 WagnerDiffusivity 186 7.3 ThinFilmCases 187 7.3.1 Diffusion-ControlledandInterfacial-Reaction-ControlledGrowth 187 7.3.2 KineticsofInterfacial-Reaction-ControlledGrowth 188 7.3.3 KineticsofCompetitiveGrowthofTwo-LayeredPhases 193 7.3.4 FirstPhaseinSilicideFormation 194 7.4 NanowireCases 196 7.4.1 PointContactReactions 197 7.4.2 LineContactReactions 202 7.4.3 PlanarContactReactions 208 References 208 Problems 209 CHAPTER8 GRAINGROWTHINMICROANDNANOSCALE 211 8.1 Introduction 211 8.2 HowtoGenerateaPolycrystallineMicrostructure 213 8.3 ComputerSimulationofGrainGrowth 216 8.3.1 AtomisticSimulationBasedonMonteCarloMethod 216 8.3.2 PhenomenologicalSimulations 217 8.4 StatisticalDistributionFunctionsofGrainSize 219 8.5 Deterministic(Dynamic)ApproachtoGrainGrowth 221 8.6 CouplingBetweenGrainGrowthofaCentralGrainandtheRestofGrains 225 8.7 DecouplingtheGrainGrowthofaCentralGrainfromtheRestofGrainsinthe NormalizedSizeSpace 226 8.8 GrainGrowthin2DCaseintheNormalizedSizeSpace 229 8.9 GrainRotation 231 8.9.1 GrainRotationinAnisotropicThinFilmsUnderElectromigration 232 viii CONTENTS References 237 Problems 238 CHAPTER9 SELF-SUSTAINEDREACTIONSINNANOSCALEMULTILAYERED THINFILMS 240 9.1 Introduction 240 9.2 TheSelectionofaPairofMetallicThinFilmsforSHS 243 9.3 ASimpleModelofSingle-PhaseGrowthinSelf-SustainedReaction 245 9.4 ASimpleEstimateofFlameVelocityinSteadyStateHeatTransfer 250 9.5 ComparisoninPhaseFormationbyAnnealingandbyExplosiveReaction inAl/Ni 251 9.6 Self-ExplosiveSilicidationReactions 251 References 255 Problems 256 CHAPTER10 FORMATIONANDTRANSFORMATIONSOFNANOTWINS INCOPPER 258 10.1 Introduction 258 10.2 FormationofNanotwinsinCu 260 10.2.1 FirstPrincipleCalculationofEnergyofFormationofNanotwins 260 10.2.2 InSituMeasurementofStressEvolutionforNanotwinFormationDuring PulseElectrodepositionofCu 264 10.2.3 FormationofNanotwinCuinThrough-SiliconVias 266 10.3 FormationandTransformationofOrientedNanotwinsinCu 269 10.3.1 FormationofOrientedNanotwinsinCu 270 10.3.2 UnidirectionalGrowthofCu–SnIntermetallicCompoundonOriented andNanotwinnedCu 270 10.3.3 Transformationof⟨111⟩OrientedandNanotwinnedCuto⟨100⟩Oriented SingleCrystalofCu 274 10.4 PotentialApplicationsofNanotwinnedCu 276 10.4.1 ToReduceElectromigrationinInterconnectTechnology 276 10.4.2 ToEliminateKirkendallVoidsinMicrobumpPackagingTechnology 277 References 278 Problems 278 APPENDIXA LAPLACEPRESSUREINNONSPHERICALNANOPARTICLE 280 APPENDIXB INTERDIFFUSIONCOEFFICIENTD̃ =C MG′′ 282 B APPENDIXC NONEQUILIBRIUMVACANCIESANDCROSS-EFFECTSON INTERDIFFUSIONINAPSEUDO-TERNARYALLOY 285 APPENDIXD INTERACTIONBETWEENKIRKENDALLEFFECTAND GIBBS–THOMSONEFFECTINTHEFORMATIONOFA SPHERICALCOMPOUNDNANOSHELL 289 INDEX 293