Principles of Solidification Martin Eden Glicksman Principles of Solidification An Introduction to Modern Casting and Crystal Growth Concepts 123 MartinEdenGlicksman MaterialsScienceandEngineeringDepartment UniversityofFlorida Gainesville,FL32611-6400,USA [email protected]fl.edu ISBN978-1-4419-7343-6 e-ISBN978-1-4419-7344-3 DOI10.1007/978-1-4419-7344-3 SpringerNewYorkDordrechtHeidelbergLondon LibraryofCongressControlNumber:2010936894 (cid:2)c SpringerScience+BusinessMedia,LLC2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein connection with any form of information storage and retrieval, electronic adaptation, computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not theyaresubjecttoproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Tomywife,Lucinda Preface HowmanyhavetrulybenefitedfrommyclassesIcannot judge.ButIhavetheconsolationthatonepersonhaslearned muchthereby,andthatisme. LudwigBoltzmann,Vienna,1905[1] Thisbookisadistillationbasedonmorethanthreedecadesofteachinguniversity coursesandprovidingprofessionalseminarsonsolidification,castingandwelding ofmetals,andcrystalgrowth.Thecoursesthemselveswereofferedformanyyears atRensselaerPolytechnicInstituteand,mostrecently,attheUniversityofFlorida. Their purpose, consistently, was to present to a variety of engineering and science studentsalogicalprogressionoftheessentialelementsofmaterialssciencerelevant to molten phases and processes leading to their crystallization. This text provides acomprehensivesurveyofscientificandengineeringfundamentalsforunderstand- ing crystallization processes, dwelling especially on applications to pure materi- als, metallic alloys, oxides, semiconductors, and polyphase systems. The didactic approach adhered to derives in large measure from the author’s personal lecture notesthatwerepreparedannuallyforone-semestercoursespopulatedbyadvanced undergraduateandgraduatestudentsinmaterialsscience,chemicalandmechanical engineering, geology and physics. This book is designed for teaching this group, as well as for professionals interested in specific topics or exposure to integrated aspectsofsolidificationandcrystalgrowth. Thedevelopmentofmostchaptersincludedhereinfavors,whereappropriate,a quantitative approach, augmented by scientific descriptions anchored in materials science and condensed matter physics. Familiarity with the calculus, ordinary and partialdifferentialequations,andatleastintroductorychemicalthermodynamicsis assumed,allofwhichwillprovehelpfultomostreaders.Duecarehasbeentaken to provide the reader comprehensive, logical developments supported by citations of pertinent research papers and helpful reviews. Although the literature on solid- ification and crystal growth still lacks a consistent, canonical terminology, I have endeavored, with some personal biases, to apply best practices throughout, as I observed their application by other researchers and thoughtful authors attentive to thisfield. vii viii Preface The text’s present author, in various ways, has also attempted to mirror the approachandlogicalstructuresosuccessfullyusedinProfessorBruceChalmers’s highly regarded monograph by the same title, published almost 50 years ago [2], while Dr. Chalmers was Gordon Mackay Professor of Metallurgy at Harvard Uni- versity. I used Professor Chalmers’s book when first teaching the subject of solid- ification and crystal growth; his book was reprinted for my classes in later years throughthegraciouspermissionofhisestate.Thatworkremainedourbackground text,asIexpandedsometopicsusingmypersonalnotestoup-datethemortointro- ducenewones.Thepresentbook,asalreadymentioned,resultedfromthoseserial teachingefforts,throughsteadyimprovementsbasedonfeedbackfromstudentsand colleagues, and progress in the field. This book is written with the express desire thatitrecapturesatleastsomeofthespiritandpurposeoriginallystatedbyBruce Chalmers,namely, ...toprovideacriticalreviewofthestateofknowledgeandunderstandingoftheprocess ofsolidification,definedforthispurposeasthediscontinuouschangeofstatefromliquid tocrystallinesolid. Professor Chalmers’s personal charm, his many research contributions, his unique technical and expository style, and, of course, his book itself, were mainstays in the early development of solidification science. It is the author’s further hope that thecurrentmonographwillcontinuetoassiststudentsandpracticingscientistsand engineersinobtainingafirmergraspofthisinterestingandimportantfield,which hascontinueditsdevelopmentatabriskpaceeversincethepublicationofProfessor Chalmers’s,PrinciplesofSolidification. IowethankstoformerstudentsfortheirfeedbackeachyearthatItaughtthissub- ject, which collectively helped shape the present form and content of this volume. My deepest appreciation extends to my dear friend Professor Markus Rettenmayr, FriedrichSchillerUniversity,Jena,Germany,whoprovidednumeroususefulcom- ments, alternative perspectives, and detailed suggestions for improving the clarity and consistency of the book, and who was willing to read critically an early draft ofthiswork;tomyfriendandcollaboratorProfessorPauloRios,UniversidadFed- eral de Volta Redonda, Brazil, for his encouragement, discussions, and unflagging interest in my writing this book; and to my friend and colleague Professor Diran Apelian,WorcesterPolytechnicInstitute,whosuccessfullylocatedout-of-printref- erencematerialsthatprovedsouseful. SupportprovidedtotheauthorthroughtheFloridatwenty-firstCenturyScholar Program,andthewarmencouragementandinterestalwaysreceivedfromProfessor KevinJones,Chairman,DepartmentofMaterialsScience&Engineering,andfrom theCollegeofEngineering,UniversityofFlorida,aregratefullyacknowledgedand weremuchappreciatedduringmanymonthsofwriting. Finally, my heartfelt appreciation and love extend to my wife, Lucinda, whose understanding,forbearanceandtoleranceofthelengthyandoftenintrusiveprocess Preface ix ofbookwriting,alongwithherprovisioningcountlessnourishingsnacksandmeals thatkeptmegoingthroughthewritingofthiswork. Gainesville,FL MartinEdenGlicksman References 1. E.Broda,LudwigBoltzmann,Man·Physicist·Philosopher,OxBowPress,Woodbridge,CT, 1983,p.102. 2. B.Chalmers,PrinciplesofSolidification,Wiley,NewYork,NY,1964. Contents PartI IntroductoryAspects 1 CrystalsandMelts............................................. 3 1.1 Crystals ............................................... 3 1.2 Melts.................................................. 4 1.2.1 ViscosityofMelts .............................. 5 1.3 ComparisonofCrystallineandMeltStructures............... 8 1.3.1 RDFsandLocalAtomicDensities ................ 9 1.3.2 RadialDistributionFunctions(RDFs).............. 10 1.4 StructureofMelts ....................................... 10 1.4.1 RDFsinMonatomicMelts....................... 10 1.4.2 RDFsinMolecularMelts ........................ 12 1.5 TheoreticalEstimatesofRDFs ............................ 14 1.5.1 AtomicCoordinationinMelts .................... 15 1.5.2 AtomicPacking ................................ 17 1.5.3 BondinginLiquids ............................. 18 1.5.4 InfluenceofTemperatureonRDFs ................ 19 1.6 TheoriesofMeltingandtheLiquidState.................... 20 1.6.1 LatticeVibrationalInstability..................... 20 1.6.2 HoleTheoriesofLiquids ........................ 21 1.7 Summary .............................................. 23 References .................................................... 24 2 ThermodynamicsofCrystal-MeltPhaseChange .................. 27 2.1 Enthalpy: Heat and Work Exchanges in Solid-Liquid Transformations ........................................ 27 2.2 ComparisonofTermsinΔH ............................. 31 f 2.3 Co-existenceofSolidsandMelts .......................... 32 2.3.1 Solid-MeltCriticalPoint......................... 34 2.4 ThermalSupercoolingandMetastability .................... 34 2.4.1 CharacteristicMeltSupercooling.................. 37 2.5 FreeEnergyChangesDuringFreezingandMelting........... 38 xi xii Contents 2.5.1 ReversibleSolidification......................... 38 2.5.2 IrreversibleSolidification ........................ 39 2.6 PrincipalTypesofBinaryAlloySolidification ............... 41 2.7 IllustrativeExamples .................................... 43 2.8 Summary .............................................. 50 References .................................................... 51 3 ThermalConceptsinSolidification .............................. 53 3.1 Near-EquilibriumFreezing ............................... 53 3.1.1 PreliminaryDefinitions.......................... 53 3.1.2 EnergyBalancesWithinaFreezingSystem......... 56 3.1.3 EnergyConservation............................ 57 3.2 EnergyBalancesatCrystal-MeltInterfaces.................. 60 3.2.1 Stefan’sInterfaceEnergyBalance................. 60 3.2.2 Fourier’sLawofHeatConduction................. 62 3.2.3 ThermalConduction ............................ 64 3.2.4 Newton’sLawofCooling:HeatTransferCoefficients 65 3.2.5 InterfacialGradients ............................ 66 3.3 Summary .............................................. 68 References .................................................... 68 4 SolidificationofPureMaterials.................................. 69 4.1 Quasi-staticTheory...................................... 69 4.1.1 Introduction ................................... 69 4.1.2 Analysisof1-DimensionalFreezing............... 71 4.1.3 Quasi-staticResults............................. 75 4.1.4 SolidificationofaCircularCylinder ............... 76 4.2 MovingBoundaryAnalysis............................... 80 4.2.1 Introduction ................................... 80 4.2.2 Conduction-Advection .......................... 81 4.2.3 Neumann’sSolution:Semi-infiniteFreezing ........ 85 4.3 Summary .............................................. 96 References .................................................... 97 PartII Macrosegregation 5 SoluteMassBalances:Macrosegregation .........................101 5.1 LocalandGlobalInterfacialEquilibrium....................101 5.1.1 Introduction ...................................101 5.1.2 TheDistributionCoefficient,k ...................103 0 5.2 SoluteRejectionattheSolid–LiquidInterface ...............106 5.2.1 InterfacialSoluteBalance........................106 5.3 Gulliver–ScheilMacrosegregationTheory...................109 5.3.1 DeterminationoftheDistributionCoefficient .......110 Contents xiii 5.4 MacrosegregationwithSolid–StateDiffusion ................113 5.4.1 Solid-StateDiffusion............................113 5.4.2 MassBalanceswithSolid–StateDiffusion..........114 5.4.3 TimeDependenceofFractionSolidified............116 5.4.4 SolutalFourierNumber .........................118 5.5 LimitsoftheBrody–FlemingsSoluteBalance ...............120 5.6 BinaryAlloySegregationCurves ..........................121 5.6.1 Gulliver–ScheilSegregation:SolubilityLimits ......121 5.6.2 InfluenceoftheFourierNumber ..................122 5.7 PurificationviaFreezing .................................123 5.7.1 FractionalCrystallization ........................123 5.7.2 ChemicalPurity................................124 5.7.3 WhyPureMaterials?............................124 5.7.4 CyclicUnidirectionalSolidification ...............125 5.8 ZoneRefining ..........................................129 5.8.1 SingleZonePass:Pfann’sEquation ...............129 5.8.2 MultipassZoneRefining.........................133 5.9 Summary ..............................................143 References ....................................................143 6 Plane-FrontSolidification.......................................145 6.1 Introduction............................................145 6.1.1 Steady-StateMacrosegregation ...................146 6.1.2 Steady-StatePlane-FrontFreezing ................146 6.1.3 SoluteBoundaryLayers .........................152 6.2 TransientMacrosegregation...............................154 6.2.1 InitialTransientsinInfiniteSystems...............155 6.2.2 TransientsinFiniteSystems......................158 6.2.3 FinalSegregationTransient ......................159 6.3 NumericalStudiesinFiniteSystems .......................160 6.4 Summary ..............................................161 References ....................................................161 7 CompositionControl...........................................163 7.1 ConvectioninFreezingMelts .............................163 7.1.1 MixingLimits .................................163 7.1.2 TheBPSModel ................................166 7.1.3 SolutiontotheBPSTransportEquations ...........167 7.2 EffectiveDistributionCoefficients .........................171 7.2.1 Definition and Role of Effective Distribution Coefficients ................................... 171 7.2.2 ResultsfromBPSTheory........................173 7.3 Summary ..............................................175 References ....................................................176