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Kinetics of Aluminum Lithium Alloys

Permanent Link: http://ufdc.ufl.edu/UFE0024666/00001

Material Information

Title: Kinetics of Aluminum Lithium Alloys
Physical Description: 1 online resource (160 p.)
Language: english
Creator: Pletcher, Ben
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aluminum, coarsening, diffusion, experimental, lithium, microstructural, multiparticle, ostwald, ripening, screening, simulation, tem
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Aluminum lithium alloys are increasingly used in aerospace for their high strength-to-weight ratio. Additions of lithium, up to 4.2 wt% decrease the alloy density while increasing the modulus and yield strength. The metastable, second phase Al3Li or 0 is intriguing, as it remains spherical and coherent with the matrix phase, , well into the overaged condition. Small interfacial strain energy allows these precipitates to remain spherical for volume fractions (VV ) of 0 less than 0.3, making this alloy system ideal for investigation of late-stage coarsening phenomena. Experimental characterization of three binary Al-Li alloys are presented as a critical test of di usion screening theory and multi-particle di usion simulations. Quantitative transmission electron microscopy is used to image the precipitates directly using the centered dark- eld technique. Images are analyzed autonomously within a novel Matlab function that determines the center and size of each precipitate. Particle size distribution, particle growth kinetics, and maximum particle size are used to track the precipitate growth and correlate with the predictions of screening theory and multi-particle di usion simulations. This project is the rst extensive study of Al-Li alloys, in over 25 years, applying modern transmission electron microscopy and image analysis techniques. Previous studies sampled but a single alloy composition, and measured far fewer precipitates. This study investigates 3 alloys with volume fractions of the 0 precipitates, VV =0.1-0.27, aged at 225C for 1 to 10 days. More than 1000 precipitates were sampled per aging time, creating more statistically signi cant data. Experimental results are used to test the predictions based on di usion screening theory and multi-particle aging simulations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ben Pletcher.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Glicksman, Martin E.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024666:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024666/00001

Material Information

Title: Kinetics of Aluminum Lithium Alloys
Physical Description: 1 online resource (160 p.)
Language: english
Creator: Pletcher, Ben
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aluminum, coarsening, diffusion, experimental, lithium, microstructural, multiparticle, ostwald, ripening, screening, simulation, tem
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Aluminum lithium alloys are increasingly used in aerospace for their high strength-to-weight ratio. Additions of lithium, up to 4.2 wt% decrease the alloy density while increasing the modulus and yield strength. The metastable, second phase Al3Li or 0 is intriguing, as it remains spherical and coherent with the matrix phase, , well into the overaged condition. Small interfacial strain energy allows these precipitates to remain spherical for volume fractions (VV ) of 0 less than 0.3, making this alloy system ideal for investigation of late-stage coarsening phenomena. Experimental characterization of three binary Al-Li alloys are presented as a critical test of di usion screening theory and multi-particle di usion simulations. Quantitative transmission electron microscopy is used to image the precipitates directly using the centered dark- eld technique. Images are analyzed autonomously within a novel Matlab function that determines the center and size of each precipitate. Particle size distribution, particle growth kinetics, and maximum particle size are used to track the precipitate growth and correlate with the predictions of screening theory and multi-particle di usion simulations. This project is the rst extensive study of Al-Li alloys, in over 25 years, applying modern transmission electron microscopy and image analysis techniques. Previous studies sampled but a single alloy composition, and measured far fewer precipitates. This study investigates 3 alloys with volume fractions of the 0 precipitates, VV =0.1-0.27, aged at 225C for 1 to 10 days. More than 1000 precipitates were sampled per aging time, creating more statistically signi cant data. Experimental results are used to test the predictions based on di usion screening theory and multi-particle aging simulations.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Ben Pletcher.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Glicksman, Martin E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024666:00001


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KINETICSOFALUMINUMLITHIUMALLOYS By BENA.PLETCHER ADOCTORALDISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c r 2009BenA.Pletcher 2

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ToAlejandraCarina,mydaughter 3

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ACKNOWLEDGMENTS Mydeepestappreciationtomylovingwife,Judith,forherhelpandsacrifceduring myyearsingraduateschool.Yoursupportandpatiencehasmadethispossible.Maywe continueourlearningandprovidethebestopportunitiesforourgrowingfamily.Ithank ProfessorGlicksmanforhisfnancialsupport,guidance,andindependencegrantedinmy thesiswork.Yourinruenceonthedevelopmentofmyresearchandcommunicationskills havecompletelychangedmylife.Theexperimentalworkwouldnothavereachedsucha levelofsuccesswithoutthediscussionsandprogrammingknowledgeofJamesLeBeau. MyappreciationtoDr.KegangWang,whohadthepatiencetoanswermyquestions andguidemethroughhismulti-particlediusionsimulationcode.Iwouldliketothank KerrySiebeinandGeraldBournefortheirhelpanddiscussionsconcerningmicroscopy. Thankyoutomycolleagueswhohaveprovidedlaboratoryassistanceandtechnicaladvice. Finally,Iwouldliketoextendaspecialthankstomyuncle,Dr.JohnPletcher,andaunt, BonnieFox,fortheirencouragementandfnancialsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 ABSTRACT........................................11 CHAPTER 1INTRODUCTION..................................13 2BACKGROUND...................................17 2.1Thermodynamics................................18 2.2ClassicalCoarseningTheory ..........................22 2.2.1ParticleSizeDistribution .......................25 2.2.2TemporalGrowth ............................27 2.2.3ExperimentalComparisonwithLSWTheory .............28 2.2.4ModifcationstoLSWtheory ......................29 2.3DiusionScreeningTheory ...........................31 2.4MultiparticleDiusionSimulation .......................38 2.5Aluminum-LithiumAlloys ...........................40 3EXPERIMENTALPROCEDURE .........................46 3.1HeatTreatment .................................47 3.2SamplePreparation ...............................49 3.3Microscopy ...................................50 3.4ImageProcessing ................................51 3.5DataAnalysis ..................................54 3.6SmallAngleX-rayScattering .........................55 3.7MultiparticleDiusionSimulation .......................55 4EXPERIMENTALRESULTS ............................56 4.1PhaseDiagram .................................57 4.2VickersMicrohardness .............................62 4.3MicrostructuralObservations .........................63 4.3.1Precipitate-FreeZones .........................63 4.3.2ParticleInteractions ...........................64 4.4QualitativeGrowthComparison ........................69 4.5ImageAnalysis .................................80 4.6ParticleSizeDistribution ............................84 4.7PrecipitateStatistics ..............................94 5

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4.8SelfSimilarityoftheParticleSizeDistribution ................95 4.9MaximumPrecipitateDiameter ........................95 4.10ComparisonoftheParticleSizeDistribution .................97 4.11Kinetics .....................................99 4.12Smallanglex-rayscattering ..........................101 5DISCUSSION .....................................105 5.1SamplePreparation ...............................105 5.2EectivenessoftheImageProcessingFunctions ...............106 5.2.1OverlappinginProjection .......................108 5.2.2ImageQuality ..............................109 5.3AnalysisandComparison ...........................112 6CONCLUSIONSANDFUTUREWORK ......................114 APPENDIX ADIFFRACTIONPATTERNINDEXING ......................117 A.1IntensityCalculations ..............................117 A.2IndexingtheDiractionPattern ........................119 BIMAGEANALYSISMATLABSOURCE ......................121 B.1fullanalyze.m ..................................121 B.1.1thresholder.m ..............................124 B.1.2particleStats.m .............................126 B.1.2.1CircularHoughGrd.m .....................128 B.1.2.2diaMeasure.m .........................145 B.2postProcess.m ..................................148 CCALIBRATIONOFMICROSCOPE ........................152 REFERENCES .......................................155 BIOGRAPHICALSKETCH ................................160 6

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LISTOFTABLES Table page 3-1Chemicalcompositionofthealuminumlithiumalloysusedforexperimentation. 47 4-1Particlesevaluatedforeachalloyperagingtime ..................60 4-2Particlesevaluatedforeachalloyperagingtime. .................80 4-3Meanprecipitatediameterasafunctionofalloyandagingtime.Alloycompositions inwt.% ........................................94 4-4Maximumnormalizedprecipitateradius, max ,aspredictedbyscreeningtheory andfoundbyexperiment. ..............................97 4-5Maximumprecipitatediameteraspredictedbyscreeningtheoryandfoundby experiment.Alloycompositionsinwt.%. ......................100 4-6ComparisonofprecipitatesizebetweenTEMandSAXSanalysis .........104 A-1Comparisonofinterplanarspacings,measuredversuscalculated,fortheAl-3.16 Lialloy ........................................120 7

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LISTOFFIGURES Figure page 2-1Diusionpotential, #(r ),forsphericalparticles ..................25 2-2NormalizeddistributionfunctionG()derivedbyLSW,Eq.2{21. ........28 2-3PSDpredictedbyLSWtheorycomparedwithexperimentalwork ........29 2-4PSDpredictedbyLSWtheorycomparedwitharangeofPSD'sforvarying V V fromdiusionscreeningtheory ...........................37 2-5Aluminum-LithiumBinaryPhaseDiagram .....................42 2-6Aluminum-Lithiumphasediagramshowingthemetastable 0 / solvus .....43 3-1Schematicofexperimentalfurnace .........................48 4-1Diractionpatternforatypicalbinaryaluminum-lithiumalloyagedat225C ..57 4-2DiractionpatternfromtheAl-3.16wt%Li ....................58 4-3Al-1.7Liagedat180Cshowingpreferrednucleationondislocationsgivinganon homogeneouspopulationof 0 precipitates .....................59 4-4Al-Liphasediagramshowingthemetastablesolvusfrompriorexperimentand thealloysusedinthecurrentstudy .........................60 4-5Coarseningofaluminumlithiumalloysagedat180Cshowsnon-steadystate coarseningbehaviorforatleast200hours .....................61 4-6TEMsamplesremovedfromrandomlocations,precipitateanalysisindicated uniformlithiumconcentration ............................62 4-7VickersmicrohardnessforAl-3.16%Liagedat225C,from0-240hours. .....63 4-8 0 precipitatesgrowinguptoandoneithersideofatiltgrainboundary,which doesnotexhibitaprecipitate-freezone.12houraging,225C ...........64 4-9PrecipitatecoalescenceinAl-3.66Lishowingnon-sphericalforms .........65 4-10Electronmicrographsshowingcoalescedandoverlappedparticlesbyprojection viewing(a)Overlappingparticles48Hr,Al-3.16wt.%Li,225C(b)Coalesced andoverlappedparticles,48hour,Al-3.16wt.%Li,225C(c)Closeupofcoalesced particles,48hour,Al-3.16wt.%Li,225C ......................67 4-11Precipitatesthathavegrowntogether,buthavenotcoalescedduetoalackof latticeregistry,Al-3.16Li,144hour,225C .....................68 4-12ElectronmicrographsforAl-2.1wt.%Li,225C(a)24Hour(b)36Hour(c)48 Hour ..........................................70 8

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4-13ElectronmicrographsforAl-2.1wt.%Li,225C(a)72Hour(b)96Hour(c)120 Hour..........................................71 4-14ElectronmicrographsforAl-2.1wt.%Li,225C(a)144Hour(b)192Hour(c) 240Hour.......................................72 4-15ElectronmicrographsforAl-2.45wt.%Li,225C(a)24Hour(b)36Hour(c)48 Hour..........................................73 4-16ElectronmicrographsforAl-2.45wt.%Li,225C(a)72Hour(b)96Hour(c)120 Hour..........................................74 4-17ElectronmicrographsforAl-2.45wt.%Li,225C(a)144Hour(b)192Hour(c) 240Hour.......................................75 4-18ElectronmicrographsforAl-3.16wt.%Liat200kXmagnifcation,225C(a)24 Hour(b)36Hour(c)48Hour ...........................76 4-19ElectronmicrographsforAl-3.16wt.%Liat200kXmagnifcation,225C(a)72 Hour(b)96Hour(c)108Hour ...........................77 4-20ElectronmicrographsforAl-3.16wt.%Liat200kXmagnifcation,225C(a)144 Hour(b)196Hour(c)240Hour ..........................78 4-21ResultsfromtheMatlabimageprocessingfunctions(a)Imported,36hour,225C (b)Blurred,grayscaleimage(c)Binaryimage(d)Centroidsandoutlineofthe binaryimagesuperimposedovertheoriginalimage ................82 4-22Particlediametersusingthecentroidssuperimposedovertheoriginalimage ..83 4-23ParticlesizedistributionsforAl-2.1wt.%Liagedat225C(a)24hour(b)36 hour(c)48hour ...................................85 4-24ParticlesizedistributionsforAl-2.1wt.%Liagedat225C(a)72hour(b)96 hour(c)120hour ..................................86 4-25ParticlesizedistributionsforAl-2.1wt.%Liagedat225C(a)144hour(b)196 hour(c)240hour ...................................87 4-26ParticlesizedistributionsforAl-2.45wt.%Liagedat225C(a)24hour(b)36 hour(c)48hour ...................................88 4-27ParticlesizedistributionsforAl-2.45wt.%Liagedat225C(a)72hour(b)96 hour(c)120hour ..................................89 4-28ParticlesizedistributionsforAl-2.45wt.%Liagedat225C(a)144hour(b)196 hour(c)240hour ...................................90 4-29ParticlesizedistributionsforAl-3.16wt.%Liagedat225C(a)24hour(b)36 hour(c)48hour ...................................91 9

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4-30ParticlesizedistributionsforAl-3.16wt.%Liagedat225C(a)72hour(b)96 hour(c)108hour ..................................92 4-31ParticlesizedistributionsforAl-3.16wt.%Liagedat225C(a)144hour(b)196 hour(c)240hour ...................................93 4-32Cumulativehistogramofallagingtimesfrom24to240Hours(a)Al-2.1wt.%Li, 225C(b)Al-2.45wt.%Li,225C(c)Al-3.16wt.%Li,225C ..............96 4-33Comparisonoftheparticle-sizedistributionfromexperiment,multi-particlesimulation andscreeningtheory(a)Al-2.1wt.%Li,225C(b)Al-2.45wt.%Li,225C(c)Al-3.16wt.%Li, 225C ..........................................98 4-34 R 3 versustime,exhibitingthepredictedlinearity .................99 4-35Scatteringcross-sectionsforAl-2.1Li,agedat225C ................102 4-36Scatteringcross-sectionsforAl-2.1wt.%Li,agedat225C .............103 5-1Exampleoftheflloperationon 0 particleswhoseoverlapisdark(a)AsScanned andCropped(b)Blurred,grayscaleimage(c)BinaryImage(d)Centroidssuperimposed ontheoriginalimage(e)Measureddiametersandcentroidssuperimposedon theoriginalimage ...................................109 5-2Exampleofhighdensityimagesandtheirdeleteriouseectsontheimageanalysis function ........................................111 A-1IndexeddiractionpatternfromtheAl-3.16wt%Li ................119 C-1CalibrationoftheCM-12microscopeusingstandard,90nmlatexspheres ....152 C-2CalibrationoftheJEOL200CXmicroscopeusingstandard,90nmlatexspheres 153 C-3CalibrationoftheJEOL2010Fmicroscopeusingstandard,90nmlatexspheres 154 10

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AbstractofDoctoralDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulfllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy KINETICSOFALUMINUMLITHIUMALLOYS By BenA.Pletcher August2009 Chair:MartinGlicksman Major:MaterialsScienceandEngineering Aluminumlithiumalloysareincreasinglyusedinaerospacefortheirhighstrength-to-weight ratio.Additionsoflithium,upto4.2wt%decreasethealloydensitywhileincreasingthe modulusandyieldstrength.Themetastable,secondphaseAl 3 Lior 0 isintriguing,asit remainssphericalandcoherentwiththematrixphase, ,wellintotheoveragedcondition. Smallinterfacialstrainenergyallowstheseprecipitatestoremainsphericalforvolume fractions(V V )of 0 lessthan0.3,makingthisalloysystemidealforinvestigationof late-stagecoarseningphenomena.ExperimentalcharacterizationofthreebinaryAl-Li alloysarepresentedasacriticaltestofdiusionscreeningtheoryandmulti-particle diusionsimulations.Quantitativetransmissionelectronmicroscopyisusedtoimage theprecipitatesdirectlyusingthecentereddark-feldtechnique.Imagesareanalyzed autonomouslywithinanovelMatlabfunctionthatdeterminesthecenterandsizeofeach precipitate.Particlesizedistribution,particlegrowthkinetics,andmaximumparticlesize areusedtotracktheprecipitategrowthandcorrelatewiththepredictionsofscreening theoryandmulti-particlediusionsimulations. ThisprojectisthefrstextensivestudyofAl-Lialloys,inover25years,applying moderntransmissionelectronmicroscopyandimageanalysistechniques.Previousstudies sampledbutasinglealloycomposition,andmeasuredfarfewerprecipitates.Thisstudy investigates3alloyswithvolumefractionsofthe 0 precipitates,V V =0.1-0.27,agedat 225Cfor1to10days.Morethan1000precipitatesweresampledperagingtime,creating 11

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morestatisticallysignifcantdata.Experimentalresultsareusedtotestthepredictions basedondiusionscreeningtheoryandmulti-particleagingsimulations. 12

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CHAPTER1 INTRODUCTION Multi-phasealloysarecommonlyusedforhighstrengthapplicationsemployingthe compositelikenatureofthedualmicrostructure.Aluminumandnickel-basealloysare twoofthemainalloysystemsthatrelypredominantlyontheprecipitationofasecond phasefortheirstrength.Typicallyinthefabricationofthesealloys,strictmeasuresare takentomonitorthethermalcyclesinheatingandquenching.Thisensuresthatthe secondphasewillnucleateandgrowuniformlytoaspecifedsizeanddistribution.For everyprecipitationhardenedalloy,signifcantmechanicaltestingisperformedtofndthe optimalagingtime,andbalancestrengthagainsttoughness.Adefnitecorrelationexists betweenthetimeatelevatedtemperaturesandthemechanicalproperties,whichcanbe exploitedinthedesignforhigh-temperatureapplications.Thesechangesinmechanical propertiesarecorrelatedtothegrowthofthesecondphase,alsoknownasmicrostructural evolution.Althoughthesizeanddistributionofthesecondphaseisrecognizedasthe causeofmechanicalbehavior,itremainsunderstoodinlimiteddetail.Eachalloysystem isstudiedindependentlyforitsresponsetotimeatelevatedtemperatureswithouta universaldescriptionofthesecondphasedistributionandgrowthkinetics.Understanding thecorrelationsamongtheatomicmodeling,thesecond-phasedistribution,thegrowth andprecipitateevolution,andthemechanicalproperties,wouldprovideanimportantand universallinkagefortwo-phasealloys.Thiscorrelationcouldeventuallyleadtoimproved alloydesign,moreaccuratepredictionsofstrengthandhardness,andmuchmorereliable estimatesofservicelife. Microstructuralevolutionaectsthematerialpropertiesandperformanceofany alloy.Thegoalofmanyresearchersinmaterialsscienceistounderstandandaccurately modelthefundamentalgrowthkineticsapplicabletoalltwo-phasealloys.Nucleation, growth,andcoarsening,comprisethethreemainstagesofmicrostructuralevolution. Nucleationandgrowtharethefrststageswherethesecondphaseinitiatesandbeginsits 13

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growth.Coarseningoccursinthelatestagesasthesystemdrivestowardaminimumof theresidualfreeenergy.Itisinthis`latestage'thatthespatialandtemporalcorrelation inthemicrostructureremainpoorlyunderstood.Recentadvancementsinboththeoryand modelingencouragefurthermodernexperimentationinthephasecoarseningoftwo-phase alloysystems.Screeningtheorypredictsthepopulationorparticle-sizedistributionofthe secondphaseasafunctionofvolumefraction.Ascreeninglengthoverwhichthediusion potentialinteractswithitsneighborsincorporatesvolumefractionasafundamental variable.MultiparticlediusionsimulationsdevelopedbyProfessorKeGangWangare capableofmonitoring2000particlesasanin-situcomputer`experiment'.Observationof thestochasticeectsincoarseningcandevelopabetterunderstandingoftheprecipitate neighborhood,orlocale.Experimentalworkisneededtoprovidethecriticaltestof bothscreeningtheoryandcomputersimulationinworkingtowardabetterfundamental understandingofmicrostructuralevolution. Thecurrentresearchprojectwasbasedonthisneedofcorrelatingmodernsimulation andtheorywithexperiment.Thegoalistocreateaquantitativelinkbetweenexperimental coarseningkinetics,coarseningsimulationandtheory,whilecorrelatingthembothtothe material'smechanicalproperties.Themethodologyofexperimentationisbasedonprior studies,withthedesiretoimproveexperimentalanalysisbyuseofmoderntechniques, thuscreatingamoreaccurateevaluationstandard.Useofautonomouscomputer-driven imageanalysiswasfoundtobenecessaryforuseasanobjectiveflterandmeasurement criterion.Inaddition,alargeparticlesamplesizewasdesiredtorepresentmorecloselythe sizeanddistributionofthesecondphase.Theseimprovementsaswellastheconsideration ofalargersamplingofagingtimeswouldsignifcantlyimprovethepresentdataavailable. AluminumisthesecondmostabundantmetallicelementontheEarth,theusesof whichhavemorethanquadrupledoverthepast6decades[1].Increasedrefningability andresearchinstrengtheningmechanismshaveresultedinaluminumalloyswithstrength exceedingmanystructuralsteels.Itshighstrength-to-weightratiomakesaluminum 14

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abetterselectionforweight-sensitivedesigns,asitisone-thirdthedensityofsteel. Inadditiontoitsmechanicalproperties,aluminumalsoexhibitssubstantialcorrosion resistanceinabroadrangeofenvironmentsandcanbefabricatedeasily.Weightand materialreductionindesignisimportantforallengineeredproducts,butbecomesmost criticalinthetransportationindustry.Withincreasingenergycostsandtheneedfor alternativeenergysources,itismostimportanttocreatelightweightstructuralmaterials thatmaybeusedinsafetyapplications.Forthesehigh-strengthapplications,precipitation hardenedalloysarecommonlyused.Naturally,aluminumalloysareanoptimalchoicefor investigationbothfortheireconomicinterestandtheiroft-usedtwo-phaseprecipitation hardenedmicrostructure. Historically,agreatvarietyofaluminum-basedalloysweredevelopedempirically, therefore,addinglithium|thelightestmetallicelement|waslogical,ifnotcompelling. Lithiumwasfrstusedasastrengtheningmechanisminaluminumin1945[2].Analloy ofaluminum,copper,lithium,manganeseandcadmiumwasalsousedformanyyears throughoutthe1950stothe1960swithoutmajorproblemsinaircraftcomponents[2]. Itwasdiscoveredthatforeachweightpercentlithiumaddedtherewasasubsequent3% decreaseinalloydensityand5%increaseinmodulus,uptothesolubilitylimitof4.2% Li[3].Thisuniqueabilitytoincreasethemoduluswhiledecreasingthedensityledto signifcantexperimentationwithaluminum-lithiumalloysinthe1960saslightweight replacementsinaerospaceframes.Thesealloys,however,werenotusedinthebinary state.Additionsofmanganese,orzirconium,createdamixedmicrostructureofcoherent andincoherentprecipitates,resultinginanoptimalbalancebetweenstrengthand toughness.In1959,themetastable,precipitate,Al 3 Li,termed 0 phase,wasdiscovered bySilcock[4].Itisthebinaryaluminumlithiumalloysthataremostinterestingtophase coarseningstudies,duetotheiruniquemicrostructuralevolution.Unlikeothermetastable aluminumprecipitates,whichuponcoarseninglosetheircoherency,themetastableAl 3 Li phaseremainscoherentwiththealuminummatrixafterextensiveaging.Inaddition, 0 15

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phaseishomogeneouslyprecipitatedandmaintainsasphericalshapeaftersignifcantly longagingtimes,evenatelevatedtemperatures. 16

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CHAPTER2 BACKGROUND Microstructuralevolutionexperimentationhaschangedfromaqualitativeobservation toaquantitativedescriptionofthemicrostructurealongwithitstemporaltransformations. Phasecoarseningtheory,oncerudimentary,ismoresolidlybasedonmaterialsproperties andincorporatesmoreaccurategrowthstatisticsandparticleinteractions.Advancements incomputationalpowerhaveallowedforthecreationoflarge-scalethree-dimensional multiparticlediusionsimulation.Inthefuture,multiparticlediusionsimulationmayuse materialspropertiestopredictthesizeanddistributionofthesecondphaseinanalloy andprovideaccurateestimatesofitsmechanicalproperties.Itisadvantageoustoknow themechanicalpropertiesofanalloybeforedesigntodelimititsapplicationsandexpected servicelife.Althoughsignifcantprogresshasbeenmadeintheareasofmicrostructural evolutionoverthepasthalfcentury,considerableworkremainsinunderstandingand capturingthefundamentalprinciples. Phasecoarsening,thelatestageofmicrostructuralevolution,wasfrstdescribed qualitatively,byphysicalchemistW.Ostwald,asthedecreaseintotalenergybyan increaseintheaverageparticlesize[5].Heobservedthatthesolubilityofchemical substancesincreasedastheparticlesizedecreased.Thesituationinamicrostructurethat correspondstotheenhancementofsolubilityisthetendencyforlargeparticlestogrow byconsumingthemoresolublesmallerparticles.Constantoverallchemicalcomposition (i.e.,massconservation)imposesthatthevolumeofthesecondphaseremainsconstant. Astheprecipitateevolvesataconstantvolumefraction,phasecoarsening,perse,results inadecreaseinthetotalinterfacialarea.Interfacialareaisproportionaltotheinterfacial energy`stored'betweenthephasesasthesystemattemptstoreachequilibrium.Most ofthesystemfreeenergyisalreadyreducedbynucleationandgrowth,leavingmuchless thanonepercentoftheinitialfreeenergytodrivethecoarseningprocess.Inessence, phasecoarseningisthefnalreversionofatwo-phasemixture(microstructure)toward 17

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thermodynamicequilibrium.Ostwaldripeningisonlyobservedinalloysatlongtimes,at elevatedtemperatures,andintheoveragedcondition.Theend-stateofOstwaldripening, orphasecoarsening,whichisneverreachedinpractice,isgrossseparationofthephases. [6]Abriefdiscussionofthethermodynamicsdrivingthecoarseningprocessispresented inthenextsection,followedbydevelopmentincoarseningtheoryandtheselectionof aluminumlithiumalloys. 2.1Thermodynamics Thecoarseningprocessiscontrolledbythethermodynamicsofthesystem.Interfacial curvatureisthecontrollingfactorinlocalizedchangestothethermodynamicproperties. Thisinterfacialregionbetweenthetwophasesisontheorderofafewatomiclayers. Withinthislocalizedregion,thermodynamicpropertiessuchassolubilitydierfrom thebulk.Themostsignifcantthermodynamicpropertyataninterfaceisthesurface freeenergy, r ,whichhasunitsofenergyperunitarea.Thus,asystemwithafnely dispersedsecondphase, f ,withinthematrix, ,hasalargeamountofinterfacialarea and,thereby,alargeexcesssurfacefreeenergy.Thesenumerousinterfacescreatesubtle spatialructuationsinthelocalfreeenergydensityofthesystem[7]. Thisinterfacialorcapillaryeectisdescribedasashiftinthelocalchemical potential, ,attheinterface.Foraspherical,isotropicsystem,thischemicalpotential shiftiswritten, )Tj /T1_2 11.955 Tf 11.955 0 Td ( 0 = 2r R n(2{1) where isthechemicalpotentialofthematrixphaseatthecurvedinterface, 0 isthe chemicalpotentialataplanarinterfaceofthematrixphase,n,isthemolarvolumeofthe f precipitatephase, r isthesurfacefreeenergy,and R istheradiusofthespherical particle.Theratioofchemicalpotentialandcompositionshiftareapproximately equalforsmallchangesinconcentrationcausedbyphasecurvature.Inamultiphase systemconsistingofparticleswithinamatrix,theselocalshiftsinchemicalpotential createlocalshiftsintheequilibriumconcentration.Thesolubilitychangeresultingfrom 18

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curvatureisknownastheThompson-Freundlicheect[8].RewritingEq. 2{1 torerect theconcentrationshiftaroundaspherical f -particleofradius, R ,givesthewell-known Thompson-Freundlichequation, C B (R )= C B exp 2r n R g T 1 R C B 1+ 2r n R g T 1 R : (2{2) Here C B (R )isthesolubilityofcomponent B inthe matrixatthesurfaceofa f particle withradius, R C B isthebulksolubilityinthe matrixata planar )Tj /T1_1 11.955 Tf 12.368 0 Td (f interface, R g istheuniversalgasconstant,and T istheabsolutetemperature[8].AlinearizationofEq. 2{2isvalidifthecurvature H =1 =R (2r n)=(R g T ).Thesmallfractionalincreasein concentrationaroundasphericalprecipitateof f isfoundfromEq. 2{2 tobe: C B (R ) )Tj /T1_1 11.955 Tf 11.956 0 Td (C B C B = 2nr R g T 1 R : (2{3) Soluteconcentrationatthesurfaceofasphericalinterfaceis,thus,afunctionof theinverseradius.Particlesofthesecondphasewithlargeradiiwillhavelittlerelative increasefromaplanarsurface,whereasasmallparticlewillhaveasignifcantincreasein soluteconcentration.Apolydispersedistributionofasecondphaseprecipitatewillhave arangeofsizes,therebygivinglocalsolubiltyinhomogeneitiesthroughoutthesystem. `Normal'diusionoccurswithsoluterowingdownthegradient,thismeansthatsolute willtendtorowfromregionsofhighinterfacialcurvaturetoregionsoflowcurvature. Thiscreatesadiusiveruxbetweenthesmallandlargeparticlesthatprovidesthe fundamentalbasisforthegrowthprocess.Smallparticleswillshrinkanddisappear,while largeparticlestendtogrow.Foraunitvolume,thetotalnumberofparticlesdecreases overtime,whereasthetotalvolumeofthesecondphaseremainsfxed.Theneteect resultsinanoverallreductionofaverageinterfacialcurvatureandinterfacialsurfacearea. Itismoreconvenienttobringtheabovethermodynamicderivationsintoadimensionless form.ThedimensionlesssolubilitychangeinEq. 2{3 becomesthecapillarydiusion 19

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potential #: #(R )= C B (R ) )Tj /T1_1 11.955 Tf 11.955 0 Td (C B C B ; (2{4) where #(R )istheequilibriuminterfacialvalueofthedimensionlesspotentialby concentrationinthematrixaboutthespherical f precipitateofradius R .Thelength scaleisnon-dimensionalizedbythecapillarylength, l c ,thatwillbeusedtoscaleall lengths,whichisdefnedas l c = 2r n R g T : (2{5) Adimensionlesstimemayalsobeintroducedas = l 2 c DC B n ; (2{6) where D isthediusioncoecient,and C B istheequilibriumsoluteconcentrationof component B ataplanarinterface[8]. Astheconcentrationchangesaboutthecurvedparticlesaresmall,aquasi-static approximationisvalid,simplifyingtheapproach.Thelargenumberofparticlesinthe microstructure,however,createsacomplexdiusionfeld.Eachparticle,dependingonits size,yieldsalocalboundaryvalueforthedimensionlessconcentrationfeld,thatmaybe rewrittenusingEq.2{3andfurthersimplifedbythecapillarylengthas, #(R )= 2nr R g T 1 R = l c R : (2{7) Thermodynamicprinciplesofthecoarseningprocessshowhowcurvedinterfaces changethelocalconcentrationequilibriumandcauselocalsolutegradients.Describing thesediusionalinteractionsinasystemofrandomlypositionedparticlesisdicult. Astatisticalsolutionisrequiredtodescribetherandompositionsandsizesofthe second-phaseparticles.RegardlessofthemethodusedtoexplainOstwaldripeningor coarsening,threefundamentalequationsmustbeestablishedinprinciple:thekinetic growthequationforaparticle,thecontinuityofparticlesthroughsizespace,andthe `global'massbalanceequationsthatcombinetoformtheso-called`master'equation[8]. 20

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KineticEquation: Thisequationdescribesthegrowthorshrinkagerateofan individualparticleofagivensize, R .Theconcentrationfeldequationdescribing massrow,whichissolvedforbothphasesisthequasi-staticdiusionequation,or Laplaceequation, r 2 C =0 : (2{8) Neglectingthetimedependentconcentrationfeldisjustifedbythesmallinterfacial velocitiesandsupersaturationofthesurroundingsolutefeld.Thevisualizationof thecoarseningsolutefeldisextremelydicultasitisafreeboundaryproblem, involvingmanyseparatedinterfaces.BoundaryconditionsaregivenbyEquation 2{2,indescribingtheconcentrationsofboththe and f phasesatthecurved interfaces.Dierencesincurvaturefromsmallandlargeparticlescreatethediusion potentialthatdrivestheinterfacialvelocity. ContinuityEquation: Astheindividualparticlesaregrowingandshrinkingbasedondiusionkinetics, thereisasubsequent`row'ofparticleswithrespecttotimethroughsizespace.The continuityequationfollowsthekineticsofeachindividualparticletodescribethe systemasawhole.Byfollowingthenumberofparticlesperunitvolumeintime, F (R;t),acontinuityequationintimeisgivenas, @F (R;t) @t + @ @R F (R;t) @R @t =0 : (2{9) Here,thegrowthrate, dR=dt,isgivenbytheabovekineticequation,where t is time.Thecontinuityequationisbasedontheassumptionthatthegrowthprocess issmoothandcontinuous,andoccurswithoutanyadditionalnucleationorparticle coalescence.Thelatterassumptionisconsistentwiththefactthatthefreeenergy availableforphasecoarseningismuchtoosmalltocausenucleation.Aparticle sizedistribution(PSD)isgivenbythecontinuityequationasitdescribesthesize 21

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distributionofthesecondphasewithinaunitvolumeforagiventime.ThePSDis usedtodescribethemicrostructureandlengthscalesamongtheparticles. MassConservationEquation: Conservationoftotalsoluteinthesystemmust besatisfed.Themassconservationequationincorporatesboththekineticand continuityequations, Q = G Z 1 0 R 3 f (R;t)dR: (2{10) Equation2{10relatesthevolumefraction, Q ,ofthesecondphasetoageometrical factor, G,basedontheformofthesecondphaseandthePSD, F (R;t)[6]. 2.2ClassicalCoarseningTheory Thefrstmajortheoreticalcontributiontophasecoarseningwasdevelopedby Lifshitz,Slyozov,andlaterbyWagnerin1961[9, 10].Thistheory,referredtohereinafter asLSW,givesastatisticalmechanicaldescriptionofcoarseningbasedoncontinuum mean-feldtheory.LSWdevelopedthefrstquantitativetheory,whichcouldgivephase coarseningpredictions[9, 10],byproposingasolutionforthetimeindependenceofthe particlesizedistributions.VoorheeshasreviewedtheLSWtheoryindepth,andconcisely enumeratesthefollowingassumptionsmadebyLSWintheformulationoftheirtheory [11]. Asingle,spherical,secondphaseparticleexistsandsensesa`meanfeld'formedby aninfnitelydilutedpopulationofparticleswithanaverageradius, hRi. Particlesarefxedinspace. Boththeparticlesandthematrixarerepresentedasruids,i.e.,arestrainfree. Diusionisquasi-staticandoccursbyadiusionlimitedprocess. Coalescenceornucleationeventsarenotconsidered. LSWtheoryprovidesthefundamentallimitformoderncoarseningtheories;therefore,it isimportanttounderstanditshistoricalbackground.LSWtheorypredictsthetemporal growthrate,thePSDandthemaximumnormalizedparticlediameter.Themorphology ofaprecipitated,spherical,secondphasewillbecharacterizedintermsofaparticle 22

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sizedistribution, f (R;t),where f isdefnedasthenumberofparticlesperunitvolume attime, t,inasizeclassof R to R + dR .Representingaparticlesizedistributionas acontinuousfunctionmeansthattheremustbesucientnumbersofparticlesinthe systemtojustifyusingacontinuumapproach.Thisassumptionisvalidsincetheparticle densitiesinmostcoarseningsystemsisontheorderof10 8 to10 14 particles/cm 3 [12].The ruxofparticlespassingthroughasizeclass R to R + dR is f R where R = dR=dt. Thus,thetimerateofchangeof f isgivenbythegeneralcontinuityequation,Eq. 2{9. InLSWtheory,therateofchangeofradius, R (R ),wasdeterminedbyevaluatingthe growthordissolutionofanisolatedsphericaldomainintoasupersaturatedmedium, knownasamean-feldapproach.Usingthequasi-staticLaplacefeldapproximationfor thedimensionlessdiusionpotential, #,(Eq. 2{4)thediusionfeldisgovernedby r 2 #(R )=0 ; (2{11) alongwiththeboundaryconditionsattheparticle-matrixinterfaceasgivenbythe Thompson-Freundlicheect.Thediusionfeldswithinthemultiparticlecoarsening systemarespatiallycomplexwithsomanyinteractinginterfaces.Eachinterface, basedonitsmeancurvature,1/R ,providesalocalboundaryvalueforthedimensionless concentrationfeld.Theseboundaryconditionsaregivenas, #(R )= l c R ; (2{12) and #(R ) lim r !1 = 1 hR i : (2{13) Eq.2{12givesthesoluteconcentrationatthesurfaceofasphericalparticle.Thesecond boundarycondition,Eq.2{13,usesthetypicalmeanfeldassumptionthataparticle interactswiththe average diusionpotentialestablishedbytotalpopulationthroughout thematrix.Heretheangularbracketsindicateanaverageoverthepopulation.Using theseboundaryconditionswithEq. 2{11 givesthesolutiontothequasi-staticdiusion 23

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equation,namely, #(r )= 1 r 1 )Tj /T1_1 11.955 Tf 17.8 8.088 Td (R hR i + 1 hR i : (2{14) ThecapillarydiusionpotentialisdimensionlessandbasedontheThompson-Freundlich eectthatsphericalparticleswillchangetheirlocalsolubilityinthematrixaccording totheircurvature[8].Anormalizedgraphofthediusionpotentialforvariousradii isshownplottedinFig. 2-1.Observethatthemedianparticlesizewithanormalized radius, = R=hR i,ofunityhasasurroundinggradientofzero.Thisisconsistentwith themean-feldapproach,andtheunderstandingthattheinstantaneousaverageorcritical particleneithergrowsnorshrinks,duetothemomentarylocalequilibriumwiththe matrixdiusionpotential.Particleslargerthanthecriticalradiushavesmalldecreases totheoveralldiusionpotential.However,particlessmallerthanthecriticalsize,havea diusionpotentialthatrapidlyincreasesastheyshrinkinradius. Thegrowthorshrinkagerateoftheparticleisrelatedtothediusionpotential gradientattheinterfaceandsolvedas, d# dr j r =R = 1 R 2 R hR i )Tj /T1_0 11.955 Tf 11.955 0 Td (1 : (2{15) Making R thedimensionlessparticleradius,andusingthedimensionlesstimevariable, thegrowthratecanbewrittenas, dR d = d# dr r =R = 1 R 2 R hR i )Tj /T1_0 11.955 Tf 11.955 0 Td (1 : (2{16) AsshownusingthediusionpotentialdefnedinEq. 2{14,particlesgroworshrink withrespecttotheaveragesize.Dependenceontheaverageparticleisclearlyshown byEq. 2{16 wherethescalingby hR i willdrivetheresultpositive(growth)ornegative (dissolution).Theseare`expected'asthismethodispurelystatisticalanddeterministic, andcannotrepresentthedynamicsofrealprecipitatesinstochasticsystems[12].The LSWmean-feldapproachisconsistentforasysteminthelimitofzerovolumefraction, 24

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Figure2-1.Diusionpotential, #(r ),forsphericalparticleswithvariousradiiversus normalizedradialdistance.Radiianddistancesareexpressedinunitsofthe averageradius hR i.[8] andprovideslimitingpredictionsfortheparticlesizedistributionandcoarseningratefor systemsunderextremedilution. 2.2.1ParticleSizeDistribution LSWfndsthatthePSDisindependentoftheinitialconditionsandisselfsimilar (ane)whennormalizedwiththeaverageparticlesize.Particleradiiinthesteadystate arenormalizedbythegrowingaverageradius hR i.Anormalizedradiusisdenotedby 25

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anddefnedasanyparticleradiusdividedbythetime-dependentaverageradius, hR i(t), = R hR (t)i : (2{17) PerthediscussioninSection 2.1,acontinuityequation(Eq. 2{9)willbeusedtorelate thepopulation'sdistributionfunction, F (R;t),asparticlesgroworshrink.Usingthe continuityequationinconjunctionwithourdimensionlessvariablesforparticlesize, (Eq. 2{17),andtime, (Eq. 2{6)resultsinananeformofthecontinuityequation, @f @ + @f @ + f (; ) hR (t)i @ R @ =0 : (2{18) Theleadingterm,_ ,inEq. 2{18 isthetimerateofchangeofthenormalizedparticleradii andisfoundtobe[8], = 1 hR i 1 )Tj /T1_2 11.955 Tf 11.955 0 Td ( 2 )Tj /T1_2 11.955 Tf 13.151 8.088 Td ( 3 dhR (t)i 3 d : (2{19) LSWtheoryshowsthatinsolvingEq 2{19 thereareseveraldierentbehaviorsofthe populationbasedontherateofchangeoftheaverageradius.Thecubicrelationof hR i anditstimederivativecreatethreedierenteigenvaluesorpopulationbehaviors[8]. dhR i 3 =d> 4 9 : Allparticleshavenegativegrowthandthepopulationeventually disappears.Thisisinconsistentwithmassconservationconstraints,thus,kinetic constantsgreaterthan4/9arenotacceptable. dhR i 3 =d< 4 9 : Particleswithintherangeof1 << 4:5grow,whileallothersshrink. Thisisinconsistentwithexperimentalandthermodynamicdataforachievinga stablerealisticpolydispersepopulationfromapolydispersepopulation. dhR i 3 =d = 4 9 : At =3 =2thereisazerorateofshrinkagecausingthistobethe maximumparticlesize.Thisgivesastableconfgurationwiththerangeof0 < < 1:5.Theparticleswhichdissolveprovidethemassforgrowingparticles. LSWconcludethattheaverageparticlegrowsvolumetricallyatadimensionless steady-staterateofexactly4/9. 26

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Sincetheaverageparticledeterminesthepopulationbehavior,thedistributionfunctionis givenwithrespecttoaverageparticlesize.Particlesgrowingandshrinkingwithrespect totheaverageobeyaself-similaroranedistribution,whichremainssteadyexceptfora growingscalefactorgivenbytheaverageradius.LSWtheoryexpressedthedistribution functionasaproduct-function f (; )= g () h( ); (2{20) where g ()representsthenormalizeddistributionfunctionand h( )representsthetime dependentportion.Solvingforthenormalizeddistributionfunctionandevaluatingthe resultanalyticallyyieldsthefollowinganedistributionfunction[8], g ()= 4 9 2 3 3+ 7=3 3 3 )Tj /T1_0 11.955 Tf 11.955 0 Td (2 11 =3 e )Tj /T1_3 7.97 Tf (2=(3 )Tj /T1_3 7.97 Tf (2) 0 3 2 : (2{21) ThetimeindependentPSDisshowninFigure2-2.OnenotesthatLSWtheory predictsamaximumnormalizedparticleradiusof max = 3 2 ,andisheavilyskewedtowards thelargersizes. 2.2.2TemporalGrowth LSWtheorywasfrsttopredictakineticrelationshipbetweenaverageparticlesize andtheagingtime.Alinearrelationshipispredictedbetweenthecubedrootoftimeand theaverageradius,givenas, 3 (t))Tj /T1_1 11.955 Tf 25.952 0 Td [( 3 (0)= 4 9 t: (2{22) Thecoecient 4 9 isthedimensionlesscoarseningrateaspredictedthroughLSWtheory fromitsassumptionofzerovolumefraction.Thekineticcoecient K LSW = 4 9 is experimentallyunattainableastheconditionsofzerovolumefractionarenotphysically possible.Thereis,however,confrmedlinearitybetweenthecubedradiusandtimeby experimentation[13{15].Thekineticcoecient, K ,hasbeenshowntobedependent onboththevolumefractionandtheagingtemperatureforanyalloythathasbeen experimentallytested[6]. 27

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Figure2-2.NormalizeddistributionfunctionG()derivedbyLSW,Eq. 2{21. 2.2.3ExperimentalComparisonwithLSWTheory Experimentationonrealalloys,withnon-zerovolumefractionofprecipitates, hasshownthatthePSDandkineticcoarseningratepredictedfromLSWareinvalid. ExperimentalPSD'sexhibitabroader,lessskeweddistribution,andawiderrangeof particlesizesthanthatpredictedbyLSWtheory.Figure 2-3 comparesthePSDpredicted byLSWwithexperimentalresultsfromtheAl-LiandFe-Ni-Al-Mosystems.Thisfailure topredictthecorrectPSDdevolvesfromthecriticalassumptionofaninfnitesimally smallvolumefractionandinfnitelylargeinter-particledistances.Byneitherconsidering thevolumefraction,particleinteractions,norvariationsinlocalenvironments,agreat disparitybetweenthetheoreticalassumptionsandtheexperimentalproceduredevelops. TheprincipleerroristhattheassumptionsmadebyLSWdonotrepresentareal, physicallyattainablemicrostructure.Attemptstocorrectforanon-zerovolumefraction andinteractionsamongsecond-phaseparticlesresultedinmanymodifcationsdeveloped 28

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Figure2-3.PSDpredictedbyLSWtheorycomparedwithexperimentalworkfromAl-Li [14](N)andFe-Ni-Al-Mo[6]( )systems,showingthatexperimentsyielda broaderandmoresymmetricdistribution. tobroadenthePSD.ModifcationsusedthebasicassumptionssetforthbyLSW,namely, thataparticleinteractswithanaveragediusionfeldormeanfeld.Theattempts toaccountforanon-zerovolumefractionemployedvariousmethodsofmodifyingthe interactionfeld,mainlyusingad-hocassumptionsthatbroadenthePSD.Itisimportant tonotethatallmodifcationstotheoryhaveshownthattobevalid,they must reduceto thepredictionsofLSWinthelimitofzerovolumefraction. 2.2.4ModifcationstoLSWtheory ModifcationsusedthebasicassumptionssetforthbyLSW,namely,thataparticle interactswithanaveragediusionfeldormeanfeld.Theattemptstoaccountfor anon-zerovolumefractionusedad-hocassumptionsthatbroadenthePSDthrough changestotheboundaryconditionsofthediusionpotential.Therearemanyproposed modifcationsofLSWtheory,themostwidelyusedandcitedmodifcationsarethe ModifedLSW(MLSW)andtheTsumurayaandMiyata(T-M)models. 29

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ProfessorArdellstudiedtheeectsofcoarseningonnickel-basealloysandproposeda modifedLSWgrowthrateequationfornon-zerovolumefractions(MLSW)[16].MLSW theoryusestheboundaryconditionthattheaveragesoluteconcentrationofthematrix occursatadistanceofhalfthemeanparticlespacingaboutthesphere.Thiscausesthe mean-feldinteractiontooccuratdecreasingdistanceswithincreasingvolumefraction. Thus,withincreasingvolumefraction,thecoarseningrateincreasesandthePSDbroadens [16]. TsumurayaandMiyatadevelopedsixdierentheuristicmodelstoexplainshapes ofthePSDobtainedexperimentally[17].SimilartoArdell,eachT-Mmodelmodifes theboundaryconditionofwheretheequilibriumconcentrationmatrixoccursaboutthe sphericalparticle.Eachmodelusesadierentdistance,looselytiedtoexperimentalPSD data.ThesemodelsdoincreasethecoarseningrateandbroadenthePSD,however,they arenotbasedonfundamentalphysicsofthesystem[17].Essentially,onewouldcompare theirexperimentaldatawitheachofthesixmodelstoobservewhichmodelbestft theirdata.Foramoredetailedsummaryofmean-feldmodifcationstoLSWfromthe twentiethcentury,refertotheworkofBaldan[6]. MarquseeandRossdevelopedafundamentallydierentapproachtoaccountfor thediusionfeld|thefrstmajorvariantfromtheLSWtheory[18].Ratherthanusing amean-feldapproachwithLaplace'sequationprovidingaquasi-static`cut-o',those authorsemployed`screening'overadispersedsystemofdiusionsourcesandsinks.Using Poisson'sequationtodescribediusioninanactivemedium,akineticexpressionwas developedtofndthegrowthratesinan`eectivemedium'.Theyfoundthemaximum particlesize,coarseningrateandanePSDasfunctionsofthevolumefraction.Their approachwasthefrstsignifcantdeparturefromLSWtheory,andusedasystematic statisticalmechanicmethodology,ratherthanapuremean-feldapproachtothediusion feld.TheworkfromMarquseeandRosswas,unfortunately,notwellexplained,and remainedunuseduntilthelate1990s[19]. 30

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NeighborinteractionswereincludedinanLSWmodifcationbyMarshand Glicksman,usinga`feldcell'approachbasedonthesizeclassofeachparticle.Each sizeclassofparticlewassurroundedbyauniquediusionfeldbasedontheLaplacian ormean-felddiusionpotential.Theirmodelapproximatescrystalliteswithvolume fractionsbetween0.3and0.6,andgivesresultsclosertorealitythanpriormodels[20, 21]. Althoughthisapproachremainslimitedbyusingthemean-feldassumption,itprovided animportantsteptowardsdevelopmentofthediusionscreeningtheory. 2.3DiusionScreeningTheory RecentdevelopmentsbyGlicksmanetal.[22, 23]havetakentheideaofastatistical feldcell[20]andapplieditusingtheDebye-Huckelequation,creatinganapproachtermed diusionscreeningtheory.Diusionscreeningtheoryisthefrstmajorfundamental changefromthemeanfeldLSWtheory.ByapplicationoftheDebye-Huckelequation, typicallyknownforusewithionicsolutions,aradiusof`inruence'iscorrectlyintroduced overwhichaparticlesenses,orignores,itsneighbors.Inananalogousmannertoionic solutions,thediusionfeldisrepresentedbythesecond-phaseparticlesactingaseither sourcesorsinks.Theabilitytodeterminestatisticallythelengthoverwhichinteractions takeplaceforagivenvolumefractionisattainable,andthislengthiscalledtheDebye diusionscreeninglength.RatherthanresortingtojustaLaplacianapproximation,used bymean-feldmodels,aPoissonapproximationisusedtolimitinteractiontoalength scalesetbytheDebyescreeningdistance|adistancefoundthroughexactstatistical physics. Sphericalprecipitatesaredescribedbytheirdimensionlessradii, R ,andtheassociated dimensionlessvolumepersteradianofthree-dimensionalspace, V .Alllengthscales arenon-dimensionalizedasgivenaboveinSection 2.1,usingthecapillarylengthand characteristictime.Typicaltoallcoarseningtheoriesthereoccursacriticalparticlesize suchthatatanyinstantintimetheaveragegrowthrateofthesecriticalparticlesiszero. Theassociatedcriticalradiusofprecipitatesisdenotedbythetime-dependentfunction 31

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R ? (t).Therenormalizedradiusofanyparticleofradius, R ,is ,defnedastheratioof R=R ? (t). Themainchallengeofanyphasecoarseningtheoryiscreatinganaccurateexpression fortherenormalizedgrowthrate,_ ,ineachsizeclassofthedispersedprecipitate.There areinteractionsanddiusiveexchangesbetweentheparticlesandthesedependontheir relativesizesandspatialpositions.Creatingadetaileddescriptionthatincorporatesall particle-particleinteractionswouldbeprohibitivelycomplexwithinarealmicrostructure. Onemethodofincorporatingparticle-particleinteractionsduringphasecoarsening, atleasttofrstorder,isbyrepresentingtheindividualinteractionbya`diusion screeninglength', R 0 .Thediusionscreeninglengthisacollectivepropertyofthe particlepopulationanditsetstherangeoverwhichinteractionsoccur;beyondthislength interactionseectivelycease.Thenormalizeddiusionscreeninglengthisdefnedas 0 = R 0 =R ? [22]. Analysisofascreenedcoarseningsystemofpolydispersesphericalparticlessuspended inathree-dimensionalmatrixusesthepreviouslydiscusseddistribution F (R;t),which isdefnedasthetotalnumberofparticlesperunitvolumeattime, t,withradiibetween R and R + dR .Emissionofsolutefromdissolvingparticles,orabsorptionbygrowing particlesismodeledmathematicallyasadistributionofsourcesandsinkswithinthe two-phasemicrostructure.Asthesystemisan`activemedium',thereappearsalocal diusionfeldthatsurroundstheparticleswhich,onaverage,isrestrictedbythediusion screeningprocess.Screeningtheorydoesnotincludeanystochasticaspectsoftheparticle interactions.TheonsetofscreeningrequiresthatthePoissonequationreplacetheusual Laplacianapproximationforquasi-staticdiusion[22].Thisprocedureyieldstheclassical Debye-Huckelequation: r 2 '( ~r ) )Tj /T1_0 11.955 Tf 11.955 0 Td ( 2 ('( ~r ) )Tj /T1_0 11.955 Tf 11.955 0 Td (' 1 )=0 ; (2{23) where '( ~r )=( C ( ~r ) )Tj /T1_0 11.955 Tf 12.818 0 Td (C 0 )=C 0 defnesadimensionlesspotential. C ( ~r )isthephysical concentrationfeldinthecontinuousmatrixsurroundingtheparticles, C 0 isthe 32

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equilibriumsolubility,and = R )Tj /T1_3 7.97 Tf (1 0 isintroducedasthediusionanalogofthereciprocal screeninglength.Theterm 1 isthecontinuous`background'diusionpotentialinthe matrix.ThegeneralsolutiontotheDebye-Huckelequationcanbeexpressedintheformof aYukawapotential,orexponentiallydampedCoulombpotential,as[19], '(r )= A )Tj /T1_1 11.955 Tf 13.151 8.087 Td (B r exp()Tj /T1_1 11.955 Tf (r ): (2{24) Here A and B areconstants,and r istherunningvariablegivingthedistancefromthe centerofthe ithparticletoanyfeldpoint.Animportantaspectisthattheterm B=r inEq.2{24correspondswiththewell-knownCoulombicorLaplacianpotentialusedin mean-feldtheory.ThisLaplacianpotentialiseectivelyscreenedor,truncated,bythe theexponentialtermoverthescreeninglength.Constants A and B canbedetermined fromtheThompson-Freundlichlocalequilibriumrelationusingthefollowingboundary conditions:1)thedimensionlesssurfacepotential '(R i )issetbytheparticle'smean curvature, '(R i )=1 =R i and2)theouterboundaryconditionissetbythediusion screeningdistanceoftheaverageparticle, '(R 0 )=1 =R ? [19].Insertingtheseboundary conditionsintoEq. 2{24 givetheradiallyaveragedPoissondiusionfeldas '(r )= 2 R ? + B R 0 exp()Tj /T1_1 11.955 Tf (R 0 ) )Tj /T1_1 11.955 Tf 13.15 8.088 Td (B r exp()Tj /T1_1 11.955 Tf (r ); (2{25) withtheconstant B givenby B = 2(R )Tj /T1_1 11.955 Tf 11.955 0 Td (R ? ) R ? [exp()Tj /T1_1 11.955 Tf (R ) )Tj /T1_0 11.955 Tf 11.955 0 Td ((R=R 0 )exp( )Tj /T1_1 11.955 Tf (R 0 ) : (2{26) Thephysicalmeaningof B isdimensionlessvolumeruxpersteradian[22].This makesthetime-rateofchangeofthedimensionlessvolume,persteradian,ofaspherical crystallite, V ,relatedtoBsuchthat V = B (2{27) 33

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Ifthevolumefractionissmall,bothterms R 1and R 0 1.Thisallowsforthe exponentialtermstobeexpandedinaTaylorseriesthatarewellapproximatedtofrst orderas, B = 2(R )Tj /T1_1 11.955 Tf 11.955 0 Td (R ? ) R ? (1 )Tj /T1_1 11.955 Tf 11.955 0 Td [(R=R 0 ) : (2{28) Animportantnoteisthatinthedilutelimit,thePoissondiusionfeldreducesto theLaplacianresult.Thisisvalidsinceasvolumefractionapproacheszero,screening mustvanish,astheDebyelength )Tj /T1_3 7.97 Tf (1 !1.Aconsistentdefnitionoftheouterfeld-cell radius, R 0 ,requiresthatitalsoapproachesinfnity.Inthiscase,Eq. 2{28 reducestothe resultfromLSW, B B LSW 2(R=R ? )Tj /T1_0 11.955 Tf 12.295 0 Td [(1).Thisisthedimensionlessvolumeruxper steradianinaninfnitesimallydilutesystem[19].Itisimportanttonotethatthescreening modelyieldstheLSWgrowthrateintheasymptoticlimitofzerovolumefraction,adding confdencetothegeneralityofthistheory. Wenowtakeacloserlookatthephysicalmeaningofthediusionscreeningdistance, R 0 .Ifthevolumefractionofthesecondaryphaseissucientlysmall, R=R 0 1,thena TaylorseriesexpansionofEq. 2{28 maybeusedtofrstorderas B = B TLS (1+ R=R 0 ): (2{29) TheGibbs-Thompsonlocalequilibriumrelationprovidestheboundaryconditions, givenearlier,forthebackgrounddiusionpotentialthatyieldsthefollowing, B = B TLS exp(R ): (2{30) Thiscanbesimplifedusingtheassumptionthatthevolumefractionisreasonablysmall, so R 1,resultinginthefrst-orderapproximation, B = B TLS (1+ R ): (2{31) ComparingEq. 2{29 whichresultsfromcombiningPoissondiusionwiththe statisticalmodelandEq. 2{31,whichderivesfromthegeneralkineticrelationshipof 34

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interactionswithinanactivemedium,showsthatatsmallvolumefractionstheyare equivalent.Theouter`cuto'radiusmaybeinterpretedastheDebyescreeninglengthof theactivemicrostructure, )Tj /T1_3 7.97 Tf (1 [22].Knowingthatthevolumefractionofthesecondphase is V V =4 =3hR 3 iN V ,showsthattheDebyescreeningparameter, ,iseasilyrelatedtothe measuredglobalparametersofthemicrostructure[ 19], = 3V V hR i hR 3 i 1=2 : (2{32) The`cuto'radius, R 0 = )Tj /T1_3 7.97 Tf (1 ,isdirectlyrelatedtotheratioofmomentsofthesecondary phasePSD,andthesquarerootofthevolumefractionas, R 0 = hR 3 i 3hR i 1=2 V )Tj /T1_3 7.97 Tf (1=2 V : (2{33) ThisindicatesthattheDebyescreeninglengthcanbedirectlymeasuredbyexperiment. Experimentationshowsthatthetheoreticalratio hR 3 i=hR i varieslittlewithvolume fractionandremainsofunitorder.Thisfndingshowsthatscreeninglengthisproportional to V )Tj /T1_3 7.97 Tf (1=2 V atsmallvolumefractions.ThegrowthrateofaprecipitateisderivedfromEq. 2{28 intermsofitsnormalizedradiusas: d d = 1 K ? ( )Tj /T1_0 11.955 Tf 11.955 0 Td (1) 2 (1 )Tj /T1_1 11.955 Tf 11.955 0 Td (p=p 0 ) )Tj /T1_1 11.955 Tf 11.955 0 Td (; (2{34) where, K ? = 1 3 d(R ? ) 3 dt ; (2{35) andthedimensionlessdiusiontimeisdefnedas = 1 3 ln(R ? ) 3 : (2{36) FollowingtheclassicalLSWapproach,thedistributionfunction F (; ),satisfesthe continuityequation @F (; ) @ + @ @ d d F (; ) =0 : (2{37) 35

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Inthelatestagesofphasecoarsening,thedistributionfunctionbecomesaneandthe functioncanberewrittenintheself-similarform, F (; )= G()H ( ): (2{38) Herethefunction H ( )isthetime-dependentportionthatspecifesitstemporalbehavior. G()isthetimeindependent,normalizedPSDandsatisfes d d d d + 1 G() d d dG() d = : (2{39) where istheseparationconstant,oreigenvalue.Thegeneralsolutiontotheabove equationisgivenasthespatialdistribution, G()= A d=d exp Z 0 1 d=d d ; (2{40) where hasbeenincludedinthenormalizationconstant A.Comparingthenormalized, time-independentPSDspredictedbyLSWandscreeningtheory,Figure 2-4,showsthat increasingvolumefractionwidensthePSD.Screeningtheory,incorporatingeectsof volumefraction,yieldsaPSDthatisnotasstronglyskewedtowardslargerparticlesas predictedbyLSW. LifshitzandSlyozofshowedthatsteady-statesolutionsareonlypossibleif K ? remains constant.Itsvaluecanbedeterminedbyapplyingthefollowingsimultaneousstability conditionsbasedonmassconservationinnormalizedradii: d d j = max =0(2{41) d d d d = max =0 : (2{42) 36

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Figure2-4.PSDpredictedbyLSWtheorycomparedwitharangeofPSD'sforvarying V V fromdiusionscreeningtheory Applicationofthesestabilityconditionsleadtothecoarseningrateconstant[19], K ? = 2 )Tj /T1_0 11.955 Tf 11.956 0 Td ((1 )Tj 11.955 9.225 Td (p 3V V )(1 )Tj /T1_4 7.97 Tf 22.349 4.707 Td (1 p 3V V + q 1 3V V + 1 p 3V V +1 (1 )Tj /T1_4 7.97 Tf 22.35 4.707 Td (1 p 3V V + q 1 3V V + 1 p 3V V +1) 3 : (2{43) Inaddition,thisalsoyieldsthemaximumallowedprecipitateradius, max ,as, max =1 )Tj /T1_0 11.955 Tf 25.2 8.088 Td (1 p 3V V + s 1 3V V + 1 p 3V V +1 : (2{44) Itiseasiertomeasuretheaverageradius hR i andthenusethekineticcoarsening equationintermsofaverageradiusratherthancriticalradius,namely, hR (t)i 3 )-222(hR (0)i 3 =3 K (V V )t; (2{45) where K (V V )=3K ? hi 3 : (2{46) hR (0)i istheaverageradiusat t =0,and K (V V )isthecoarseningrateconstantata volumefraction V V .Theratioof K (V V )to K (0),whichistheLSWresultof K (0)=8 =9 37

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isexpressedas K (V V ) K (0) = 27 8 K ? hi 3 : (2{47) UsingEq.2{43withEq. 2{47 givestherelativecoarseningrateintermsofthekey microstructurallengthscales, K (V V ) K (0) = 27 8 2 4 2 )Tj /T1_0 11.955 Tf 11.955 0 Td ((1 )Tj 11.955 9.225 Td (p 3V V )(1 )Tj /T1_4 7.97 Tf 22.35 4.707 Td (1 p 3V V + q 1 3V V + 1 p 3V V +1) (1 )Tj /T1_4 7.97 Tf 22.349 4.707 Td (1 p 3V V + q 1 3V V + 1 p 3V V +1) 3 3 5 hi 3 : (2{48) Eq.2{48showsthattheeectsofscreeninglengtharebroughtthroughitsinruenceon thevolumefraction, V V ,andthecubeoftheaveragenormalizedradius hi.Thisresult allowsforadirectcomparisonbetweentheoryandexperimentallymeasureddata. 2.4MultiparticleDiusionSimulation Multiparticlesimulationsrelyuponmonitoringeachindividualparticleinthesystem. Improvementsincomputertechnologyhaveincreasedthenumberofparticlesobserved fromonlyseveralin1973byWeinsandCahn[24]tohundredsbyVoorheesandGlicksman in1984[25, 26]tothousandsofparticlesbyrecentstudiesbyWangetal.[19].Themost advancedandmodernmultiparticlediusionsimulationisthatofProfessorKeGangWang [27].Aspecifednumberofparticlesarerandomlydistributedthroughathree-dimensional cell,withtheremainingspacebetweenparticlesflledbythematrixphase,throughwhich diusionoccurs.TheparticlesarelabeledbytheirlocationinCartesiancoordinates andtheirradiiareoriginallychosenfromarelativelynarrowGaussiandistribution. Twoassumptionsaremadeinthesimulation:1)thekineticsofcoarseningisbasedonly onvolumediusionthroughthematrix,and2)thevelocityoftransportisslowand consideredquasi-static[19].Thediusionfeldutilizesthequasi-staticapproximation describedinSection 2.1 andgivenas r 2 #( ~r )=0 : (2{49) 38

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Simulationsuseadiscretefnitesystemof n sphericalparticles.Theboundaryconditions ofthesphericalinterfaceremainasspecifedbytheThompson-Freundlicheect,written as #(R i )= 1 R i : (2{50) Thesolutiontotheconcentrationfeldfor n particlesisgivenbythesuperpositionof n concentrationfeldssummedoverthesystemofparticles[19]: #( ~r )= n X i=1 B i j ~r )Tj /T1_1 11.955 Tf 12.438 0 Td [(~r i j + # inf : (2{51) Eachparticle'sdiusionpotentialisdependentonitsneighbor'sthroughsummation oftheruxfeldsthroughoutthematrix.Thismethodprovidesanaccurate,indeed detailed,descriptionofeachlocale,orneighborhood,surroundingeveryparticle,more accuratelyrepresentingthestochasticnatureofarealsystem[28, 29].Eachtime-step up-datestheradiiforallparticlesandtheircoordinates.Trackingtheparticlesizesand changesintheirenvironmentaddsimportantmicrostructuralinformationtothediusion solution.Thisisnotpossibleinmean-feldtheoreticalapproaches.ProfessorWang's simulationseectivelytestsimulatedmicrostructureswithvolumefractions, V V 6 0:3, containinginitialpopulationsoftwothousandprecipitates.Ithasbeenshownthrough comparisonwithexperimentthatthemultiparticlesimulationscansimulateandpredict thePSD,maximumparticleradius,andcoarseningrateconstants[19, 23].Foramore detaileddescriptionofthesemulti-particlesimulationsanddevelopmenttheinterested readerisdirectedtoreferences[23, 27, 30]. Mean-feldtheorieshavebeenrepetitivelytestedwithavarietyofalloysystems, andshowedlittlecomparativesuccessbetweentheoryandmicrostructures.Diusion screeningtheorymorecloselycapturesthefundamentalinteractionbehavioramong particlesand,thus,exhibitsagreaterchanceofbeinguniversallyapplicablewithreal alloys.Multiparticlediusionsimulationshavealsoshowntheabilitytosimulate three-dimensionalsystemsbycapturingthekineticsandPSDsofrealsystems.Initial 39

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comparisonsamongthediusionscreeningtheory,multiparticlediusionsimulation,and priorexperimentalstudiesshowacceptableagreement[19, 23].Furthercomparisonis desirableusingexperimentaldatafromamoreidealcoarseningmicrostructure.Modern characterizationofthemicrostructureisrequiredtoobtainamoreincisivecomparison amongtheory,simulationsandexperimentation.Advancedcharacterizationtechniques, digitalimageacquisition,andenhancedcurrentcomputationalpowerallallowthe experimentaliststheabilitytocaptureeverlargersamplesizes,thereby,trackingmore usefulvariables,suchasmaximumparticlesizes,andthelocationoftheparticles,leading towardabettercharacterizationofthesystemandcriticaltestoftheory.Thebestknown solid-statesystemfortestingcoarseningtheory,intheauthor'sopinion,isthebinary aluminum-lithiumalloys. 2.5Aluminum-LithiumAlloys Eventhoughpreviouscoarseningstudieshaveusedaluminum-lithiumalloys,itis importanttonotewhytheyarenearlyidealandwhyfurtherexperimentationisneeded. Signifcantexperimentationandphase-coarseningstudieshavebeencompletedonother alloysystems.Baldan[15]reviewedtheworkoncorrelatingthemean-feldtheoriestothe growthof r 0 innickel-basesuperalloys.Thedicultywiththe r 0 phase,foundinthese alloys,isitsstrongelasticinteractionwiththematrix. Asystemthatavoidsthedicultywithelasticinteractionsisbinaryaluminum lithium.EarlyworkinunderstandingthenatureoftheAl 3 Liprecipitatewascompletedby NobleandThompsonandWilliamsandEdington[31{33].Theyusedelectronmicroscopy andplasmaenergylossestoquantifythemicrostructureandprecipitatedispersion. Theseinitialstudiesintoboththekineticgrowthfactorsandtheeectoflithiumon themechanicalpropertiesledtomorein-depthstudiesinthemidtolate1970 0 s.A combinationoftheoilcrisisandthegrowingneedforincreasedrocket-bornepayloadsled tofurtherfundamentalresearchinaluminum-lithiumalloysintheareasofmechanical propertiesandfracturebehavior[34]. 40

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Thealuminum-lithiumsystemdoesnotexhibitstrongelasticstrainsbetween precipitatesandmatrixbecauseofthematchinlatticeparameterforthe 0 precipitate as4.02 Aandaluminumas4.04 A[35].Thisprecipitate-to-matrixmisftresultsin aninterfacialstrainofonly )Tj /T1_0 11.955 Tf (0:08% :02%bywayofdark-feldTEMimageanalysis [33,36].Suchsmallmisftandlowstrainallowforthe 0 phasetoremaincoherentwith thealuminummatrixthroughsignifcantover-aging,and,moreover,continuetogrow inasphericalmanner.Theenergylosttostrainisminimal,unliketypicalcommercial nickelbasealloys[37],makingtheAl-Libinarysystemnearlyidealforexperimental characterizationofphasecoarsening.Thebinaryphasediagram,Figure 2-5,shows thatthemaximumsolidsolubilityoflithiuminaluminumis4.2wt%at600C[38, 39]. Inspectionofthephasediagramshowshowalloyscontaininglessthat4.2wt%Licanbe solutionheattreatedtoasingle -phaseabove580C.Itisthedecreaseinsolubilitywith decreasingtemperaturebetweenthe andthe +AlLiphasesthatallowsforarapid quenchandsubsequentsupersaturationoflithium. Byreheatingtoalowertemperature,thenon-equilibriumphaseAl 3 Linucleates andbeginstogrowinanattempttoreducetheenergyofthesystem.Themetastable 0 phaseisdenotedbythedottedline.Closerexaminationofthisregionusingpublished experimentaldatagivesagoodunderstandingofthemetastablesolvusbetween 0 and Figure 2-6. Thefrststudydefningthe 0 phasewascompletedbySilcockin1959usingaLaue camerawithstandardXRDmethods[4].Withadvancementsintransmissionelectron microscopy,thispowerfulcharacterizationtoolwassoonusedtoimagethe 0 precipitates inthemid1970's[31{33].Theseinitialstudiesresolvedthefne-scaleprecipitates,but weremainlydirectedtowarddefningthemetastablesolvus.Complementingthework ofTEM,small-angleX-rayscattering(SAXS)methodswereappliedtostudythe 0 phase[41].Bythelate1970stherewassuchalargeinterestinAl-Lialloysthat`The Minerals,MaterialsandMetalsSociety'(TMS)organizedasymposiaonthisalloysystem. 41

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Figure2-5.Aluminum-LithiumBinaryPhaseDiagram[38, 39] Thisbroadsymposiaincorporatesfundamentalresearch,mechanicalpropertiesand alloydevelopment,furtherinformationcanbefoundintheTMSproceedings[43].This wouldbethefrstofsixsymposiaonthistopic,thelastofwhichoccurredinthefall of1991.Moreinformationcanbefoundat[42{47].Variousstudiesonawiderangeof binaryandternaryAl-Libasedalloyswerecompletedduringthe1980susingtheTEM [13, 14, 40, 48{50].Coarseningstudies,specifcally,werefrstcompletedonalargescale inthemid1980sbyGuandLiedl[13].AnAl-2.8wt%Li-0.3wt%Mnalloywasstudied usingquantitativeTEMtofndtheparticle-sizedistributionsandkineticfactors.This studyconcludedthatalinearrelationshipexistsbetweenthecubedradiusandtime,but notwith K LSW .ThePSDsaremoresymmetricalandbroaderthanpredictedbyLSW theory,showingamuchlargervaluefor max =1.75.Afurtherstudy,byMahalingam, wascompletedwitharangeofbinaryAl-Lialloys,containing2.4to4.5wt%Li,using quantitativeTEM[14].Experimentalresultsfromthisstudywerethemostcomprehensive 42

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Figure2-6.Aluminum-Lithiumphasediagramshowingthemetastable 0 / solvus; Williams[32],Jensrud[40],Cocco[41],Noble[31],Williams/Edington[33], / 0 [38],Al-LiIII[42] forthebinaryAl-Lisystem,bothinitsalloyrangesandagingtemperatures.Theidentical failingswithrespecttotheLSWtheorywerefound,andlaterworkwasattemptedto modelthePSDsusingaWeibulldistribution[49].Theexperimentalstudiesreported sueredasaresultoftheinconsistenciesbetweenthetheoreticalassumptionofmean-feld theoryandactualexperimentalprocedure.PriorTEMinvestigationswerelimitedby microscoperesolution,asprecipitatessmallerthan40nmwerenoteasilymeasured. Laboriousimageanalysistechniqueslimitedthenumberofparticlesthatcouldbefeasibly counted,thereby,allowingsignifcantstatisticalerrorintheresultinganalysis.Asaresult, Mahalinghamconsideredonly4agingtimesandanaverageof500particles.Hislimited 43

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analysisadmitslargeerrorinestimatingcoarseningratesandparticlesizedistributions (PSDs). Small-anglescatteringmethodsprovideattractivecharacterizationtechniquesfor analysisoftwophasemicrostructures.Samplepreparationiseasy,thesamplevolume examinedisanorderofmagnitudegreaterthanthatinTEM,andthesamplingstatistics thereforearemuchfrmer.Thedisadvantageofsmall-anglemethodsisinthedata interpretation.WithTEMthereisadistinctadvantageofdirectimagingthatcaptures theprojectedparticlesize.SAXSmeasuresX-rayscatteringcausedbythevariationsin theelectrondensitydistribution.Asaresult,aproperanalysismodelmustbedeveloped tointerpretthescatteringprofle.Guinierdevelopedoneoftheearlymethodstoextract phaseinformationformetallicsystemsfromtheSAXSprofle[51].Triolo,etal.,frst appliedSAXSmethodologytoanAl-Lialloy,usingCu K radiation[52].Small-angle methodscanbeenhancedbyusingeitheraneutronsourceorsynchrotronsourcewhich allowsforgreaterpenetrationandcollimationofthebeam.Inaddition,thedetectorcan beplacedfurtherfromthesampleallowingforgreaterresolutionasthescatteringangle approacheszero, k ( A )Tj /T1_3 7.97 Tf (1 ) 0.Thefrstknownstudyusingsmall-angleneutronscattering wascompletedbyAbis,et.al[35],furtherstudiescanbefound[35, 53{61].Small-angle scattering(SAS)analysismethodsmaycomplementTEM,butshouldnotbeusedas thesolecharacterizationmethod.Therehavebeenvariousstudiesofaluminum-lithium alloysusingSASmethods,however,eachstudyislimitedtooneortwoalloysystemswith littlecomparisonwithdirectcharacterizationmethods.ComparisonsbetweenearlierTEM andSAXSstudiesarenotpossible,asexperimentalproceduresandalloychemistryvary betweenthem. Althoughthereisacurrentbodyofexperimentalresearchintocoarseningphenomena ofaluminum-lithiumalloys,therearenoextantcomprehensiveresults.Consistentresults amongthestudiesdogiveanideaofthemetastable / 0 solvus,showninFigure 2-6. Inaddition,theyprovideaguideforfuturestudieswithregardtoagingtemperature 44

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andexperimentalprocedure.Lithiumadditionsrequiregreatercareinprovidinganinert environmentduringheattreating.QuantitativeTEMstillremainsthebestmethodof characterization,however,SAXSprovidesanindependenttechniquewhichimproves confdence. 45

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CHAPTER3 EXPERIMENTALPROCEDURE Asevenkilogramwaeingotofaluminum-5wt.%lithiumwasdonatedforthe purposesofthisresearchbyKBAlloys.Thelithiumcontentwasgreaterthanthe maximumsolidsolubilityandwouldleadtoboththeAlLiequilibriumstructure, ,and 0 Concentrationsoflithiumhigherthanthemaximumsolidsolubilityarenotcommercially viableanddonotprovidegoodcandidatesforcoarseningstudies.AsdiscussedinSection 2.5,priorresearchhasshownthatalloyswithintherangeofAl-2.0wt%LitoAl-3.8wt%Li wouldbeacceptableforexperimentation[14, 33].Inordertodilutethelithiumcontent, itwasdeterminedthataportionofthealloycouldberemeltedwithpurealuminum tocreatealloysof0.7,1.0,1.4,1.7,2.1,2.45,3.1,and3.7wt%lithium.Alloyswere madewithlowerlithiumcontentthanshownbypriorresearchtotestthelimitsofthe metastable 0 solvus.Researchintotheworkabilityoflithiumshowedthatduetoits highreactance,specialconsiderationneededtobetakeninthemeltingandforgingof thealuminum-lithiumalloys.SophisticatedAlloys,Inc.specializesincustomalloysand wasabletoremeltproperly,dilutewithpurealuminum,andthencastthenewalloys. Theyemployedasmallvacuumfurnacethatisfrstevacuatedandreflledwithhigh purityargontoeliminateanylossofthelithiumcausedbyreactionwithotherelements, suchasoxygen,duringthemelting.Thewaeingotwascutintopiecesweighing0.75kg usingaportablebandsawandsenttoSophisticatedAlloys,Inc.Afterremeltingand dilution,thealloyswerecastascylindersof2.5cmdiameterby25cmlong.Thechemical analysisshowninTable 3 demonstratesthatthealloysusedwererelativelyfreeof impurities.ChemicalanalysiswassuppliedbySophisticatedAlloys,Inc.,usingthemethod ofinductively-coupledplasmamassspectrometry,withastandarddeviationof 0.02wt%. Fromthesecylinders,2mmthickdiskswerecutusingahorizontalbandsawwith mineraloilusedasthelubricatingmedia.Thesecirculardiskswerethencutintohalf circles,2mmthickforthethermaltreatments. 46

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Table3-1.Chemicalcompositionofthealuminumlithiumalloysusedforexperimentation. Lithium(wt%)Silicon(wt%)Iron(wt%)Copper(wt%)Aluminum 1.7 <0.001 <0.003 <0.002Balance 2.1 <0.001 <0.003 <0.002Balance 2.45 <0.001 <0.003 <0.002Balance 3.16 <0.001 <0.003 <0.002Balance 3.66 <0.001 <0.003 <0.002Balance 3.1HeatTreatment Athree-zonehingedtubefurnace,modelHTF55000,manufacturedbyLindberg/Blue wasusedforthesolutionheattreatmentandartifcialagingthermalcycles.A6.3cm diameterquartztubewasavailableforsamplecontainmentwithinthefurnace.The totalinnerlengthofthetubecontainedwithintheMoldathermIIinsulationwas61 cm.Theinsulationismoldedaroundthetubeandextendsfor13cmoneachsideof thefurnace.Thecontrolregionofuniformtemperatureextended15.3cminthecenter ofthefurnaceandwascontinuouslymonitoredbyanindependentthermocouplearray. Fivethermocoupleswereusedwithoneinthecenter,twoatdistances 6.3cmandtwo at 22.8cmfromthecenter.Theywereplacedatthebottomofthequartztubeand weremonitoredbyapersonalcomputerusingNationalInstrumentsVirtualBench-Logger program,Figure3-1.Thetemperaturewasstableto 2Cinthecenterregion. Previousstudieshaveshownaconcernwithreactionofthelithiumatelevated temperatures.Aneortwasmadetoprovideaninertatmosphereduringallheat treatments,thuseliminatinganylossoflithiumfromthebulkalloys.High-purityargon wasusedtomaintainaninertpositivepressurewithinthequartztubeduringheating.To ensureahigh-qualityargonstream,aDRIERITEgaspuriferwasusedtocreateadryness of0.005mg/l,andtoremoveimpuritiestoaneectivemoleculardiameterof5angstroms. Furthermore,anAlltechOxy-Trap,whichremovesoxygenfrominertgases,wasinstalled downstreamfromthegaspurifer.Oxygenisreducedtobetterthanonepartperbillion. Thus,theOxy-Trapeectivelyeliminatedbulklithiumlossbyeliminatingexposureto oxygen.Theargondeliverylineswereconnectedusing1/4inch316stainlesssteeltubing 47

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Figure3-1.Schematicofexperimentalfurnaceset-upshowingsampleplacementand independentthermocouplesforin-situmonitoring[62] withSwagelokfttingstoreducesignifcantlyanydiusionofoxygenintothesupplyline. Inadditiontoprovidingahighlyinertenvironmentduringfurnacetreatment,thesamples werecuttothicknessof2mmallowingremovalof0.75mmoneachsidepriortoanalysis. Thisensuresthatanylithium-depletedareaswereremoved,whichwereobservedtonot extendpast500micronsinthemostextremeconditions[42]. Analuminaboatwasusedfordeliveryofthesamplesintothefurnace.Toensure precise,repeatabledeliverytothecenteroftheheatingelementsamarkedglasspushrod wasused.Thefurnacewasboughtuptotemperaturewiththeemptyaluminaboatin placeandallowedtostablizefor1hour.Thiseliminatedanysignifcanttemperaturedrops whenloadingthesamplesintothefurnace.Within5minutesafterloading,thesample reachedthedesiredheattreatingtemperature. Allsampleswerefrstsolutionheattreatedat580Cfor1hourinthepreheated aluminaboat.Uponremovaltheywereimmediatelyquenchedinwaterat15C,followed byanacetonewash.Theresultingrapidcoolingproducedsamplescooltothetouch 48

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withinonesecond.Thesampleswerestoredatroomtemperaturewhilethefurnacewas allowedtoequilibratetotheagingtemperature.Agingtemperaturesof180Cand225C wereused.Thesampleswerethenloadedasbeforeandagedtotheirrespectivetimes between6and240hours.Thesampleswereremovedasbefore,quenched,washedin acetoneandstoredinmineraloilatroomtemperature. 3.2SamplePreparation Afterheattreatment,thehalf-circulardisksweresectionedintoquarters,with onequarterprocessedfortransmissionelectronmicroscopy(TEM)andtheothertobe processedforopticalmicroscopyandhardnessmeasurement.TEMisnecessaryasthe averageprecipitatesizeislessthan100nm.Adiamondsaw,usingmineraloilforcooling andlubrication,wasusedtocutthesamples.Adiamondsawwaschosenbecauseofits slowabrasivecutting,whichcreatesminimaldeformationandlimitstheintroductionof dislocationsintothesample.ThequartersampleforTEMwasthenmountedona25 mmdiameterplatentobeuniformlythinnedbyamanualhandpolisher.Siliconcarbide polishingpapersof180,240,400and600gritswereusedforlarge-scalematerialremoval. Thequartersweremechanicallythinnedwithparallelsidestoathicknessof250microns. Mountingwax,MWSO52,obtainedfromSouthBayTechnologywasselectedbecause ofitslowmeltingpointtemperatureof52Ctoadherethesampletotheplaten.Low temperaturewaxwasnecessarytoavoidadditionalcoarseningduringsamplemounting. Thethinnedquarterswerethentransferredandmountedtoaglassslidefortheremoval of3mmdisksbyanabrasiveslurrydisccutter.Theabrasiveslurrycutterusedadiamond coatedtipwithaninnerdiameterof3mm.Mineraloilwasusedasthelubricatingmedium duringcutting.Uponremoval,the3mmdisksweremountedandpolishedusingaGatan DiscGrinder,Model623,toathicknessof50-75microns.Thiswasachievedusingfne gritsiliconcarbidepolishingpaper,withafnalpolishata5micronfnish.Thisprovided sucientsurfaceroughnessforfurtherelectrolyticpolish. 49

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Automatictwinjetelectropolisherswereusedforthefnalpolish.Thisresearchbegan withaFischioneInstrumentsjetpolisherandlatersubstitutedaTenupol2,byStruers, forTEMsamplepreparation.Bothusedanelectolyticbathof20%nitricacidwiththe balanceofmethanol.Theelectrolyticbathwascontainedinadewarcooledtoarangeof -30Cto-20Cwithliquidnitrogen.Voltagewassetto12.3Vandadjustedasnecessary tomaintainacurrentrangeof10-15mAacrossthesample.Thesamplewaspolishedto perforationobtainingaregionofsucientthinnesstoviewasinglelayerof 0 precipitates. ThesampleswereexaminedintheTEMpreferablywithin24hoursofpolish. Theremainingquarterfromeachsamplewasmountedinepoxyforstandard metallographytechniques.Eachsamplewaspolishedusingsiliconcarbidepaperwith gritsof180,240,400,and600withparallelfacesmaintainedonthesample.Afterthe coarsepolishing,3MWetordry TM polishingpaperwasusedwithaluminumoxidegritsof 20,10,and3micronstopolishthesamplefurther.Lastly,a0.3micronaluminumoxide pasteonafeltwheelwasusedtocreateamirrorfnish.Vickersdiamondmicrohardness testsweredoneonthesamples.A200grammasswasusedforallthetests.Three indentationsweredonepersampletoaveragetheresultsandobtainageneraltrendofthe samplehardnesswithagetime. 3.3Microscopy Qualitativetransmissionelectronmicroscopy(TEM)wasusedasthemethodof distinguishingandcapturingthetwo-phasemicrostructureofthesamples.TEMisa usefultoolforthematerialscharacterizationoftwo-phasemicrostructures.APhilips CM-12,120kV,JEOL200CX,200kVandaJEOL2010F,200kVwereusedforimaging. Calibrationstandardswereusedaswellascomparativetestswithexactsamplesbetween themicroscopestoensurethattheimagingisprecise.Usingthestructuredierence betweenthealuminumand 0 phases,centereddark-feldimagingmaybeused.The 0 phaseistheAl 3 Liintermetallic,whichhasthecrystalstructureL1 2 andapm 3mspace group.Itissimilartotheface-centeredcubicstructureofaluminum,however,lithiumnow 50

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occupiesthecornersoftheunitcellandaluminumthefacecenters.Diractionfroma face-centeredcubiccrystal(FCC)structure,suchasaluminum,resultsinextinctionsfor anylatticeplanewithindicesthatareamixtureofevenandoddnumbers.The 0 phase issimilartotheFCCbutduetotheatomicdierencebetweenthealuminumandlithium atomsdiractionpatternsfromplanesofmixedindicesareresolved,albeitwithless intensity.Appendix A discussesindepththestructurefactoranditsuseindetermining bothintensityandindexing. Thecentereddark-feldimagingtechniquereliesuponcenteringtheobjectiveaperture aroundthedirectbeamandbringingaselecteddiractionspottothecenter.Inthiscase adiractedbeamfromthe 0 phasewasusedtocreateahighcontrastimagethatshows theAl 3 Liprecipitates.Byusingadoubletiltholder,themaximumintensitywasachieved bytiltingtoatwo-beamcondition,creatingahigh-contrastcentereddark-feldimage fromthelowerintensity 0 phase.Usingthistechnique,imagestakenwiththeprecipitates areshownwithasharpcontrastdierencefromthesurroundingaluminummatrix.This methodallowsforeasyanalysisofthe 0 precipitates. ThePhillipsCM-12imageswerecapturedusingKodakElectronMicroscopeFilm 4489,8.3x10.2cmwithautomaticexposurethroughtheCM-12feedbackcontrolled interface.AlloyAl-3.16wt.%LiwastheonlyalloyimagedwiththeCM-12.TheJEOL 200CXwasusedforthebalanceoftheresearch,whichisequippedwithaGatan Multiscan,Model791digitalaquisitioncamera.Exposuretimesweredeterminedby experiencedependingonthesampleandrangedbetween1-3seconds.Digitalimageswere savedingray-scaleformatinTaggedImageFileFormat(TIFF),whichcanbeanalyzedby anydigitalimageprocessingsystem. 3.4ImageProcessing Theexposedflmsweredevelopedbycommonmanualprocessingtechniques.Thedry flmswereimmediatelyplacedinNegaFilepolyviewpagestoprotectthem.Thesepages werethenplacedinanarchivalbinderforlong-termstorage.Negativeswerescannedinto 51

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digitalformatbyuseofaMicrotekScanMaker8700withtheaccompanyingScanWizard Pro7.20software.Thenegativescanningbedwasusedwitha300dpiresolutionandthe imagesweresavedinasRGBimagesintheTaggedImageFileFormat(TIFF).Images acquiredwithadigitalcameradidnotrequireanyprocessingpriortoanalysis.Allfles wereplacedintoanarchivalsystembasedontheiralloyandagingtime. Matlabwaschosenasthecomputermathpackagetoprocessandanalyzetheimages. TheprogramminglanguagewithinMatlabissimilartoClanguage,andcanbeusedto callmanybuilt-inmath,imageprocessingandotherfunctionsincludedinthesoftware package.SinceMatlabcanberunoneitherWindows,Macintosh,orLinuxplatforms,itis universalandcanberunfromanyworkingcopyofMatlab.Themaingoalindeveloping anautomatedimageanalysisfunctionistoeliminateanyhumaninterfaceinmeasuring theparticlesizes.Thebeneftishavingareproducibleandobjectivecomparatorforeach particle,whilemaintainingrapidrepetetiveanalysiscapableofanalyzinglargequantities ofimages. Ahierarchyofcallfunctionshasbeendevelopedwhereeachfunctioncompletesa specifctask.Themainfunctionthattheusercallsandprovidesthelistoffoldersfor evaluationiscalledfullAnalysis.Itisdesignedtoaccesstheimageflesandprovidesthe mainframeworkbywhichtheseparatefunctionsarecalled.Aswell,itsavestheenhanced gray-scaleandbinaryimagesandthecentroidsandparticlediameterforeachimage analyzed.ByinitiatingfullAnalysis,theprogramlocatesthedesiredfolderandopens theimagesandsendsthemthroughfourseparatefunctions:thresholder,particleStats, CircularHoughGrd,anddiaMeasure.Allfvefunctionsareexplainedingreaterdetailand shownentirelyinAppendix B. thresholder: Thresholderisdesignedtoenhancetheimageforoptimalanalysis bycreatingauniformintensityandeliminationofnoise.Imageenhancement wascompletedbyadaptedMatlabfunctionscalledinanorderfoundtoprepare theimagesbestforanalysis.Thereareawidevarietyofproceduresforcreating 52

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high-contrastimageswithclearlydefnedboundaries.Imageenhancementis necessarytoeliminateintensityructuations,scannedartifacts,cameradefects, andnoise.Itisimportanttoimplementamethodthatwillnotalterthesizeof theprecipitates,butwillonlycreateauniformcontrastdierencebetweenthe precipitatesandthesurroundingmatrix[63, 64].Thresholderhasforitsoutputa grayscaleandbinaryimage.Thegray-scaleimagehasanenhancedgradientwhich makesiteasierforfunctionCircularHoughGrdtofndthecentroids.Thebinary imageisusedbydiaMeasuretofndthediameterofeachparticle. particleStats: ParticleStatswilltaketheenhancedimageandperformstatistical analysistodeterminethevolumefraction.Itisalsocapableofanalyzingthe imagebasedonthepointandlinealmethodsofstereology.Methodsofstereology eliminatetheneedofanalyzingeachindividualprecipitateasitusesastatistical representationtodeterminebothaparticlecountandtheirrespectivesizes.These methodsweredevelopedtomakeanalysispossiblewhencomputationalspeedsand accuracywerelimited[65 ].Howevertheywerenotfollowedinthisanalysisasthe programdevelopedcandeterminethelocationandareaofeachprecipitatewith bothspeedandaccuracy.ParticleStatscallstwofunctions:CircularHoughGrd anddiaMeasure.Itpassesthegrayscaleandbinaryimagetoeachfunction respectively.Italsoreturnsthevolumefractionofthesecondphaseintheimage andthestatisticsfromthepointandlinealstereologicalmethods. CircularHoughGrd: CircularHoughGrdusesaHoughTransformobtainedfrom theMathWorkswebsite[66].TheHoughTransformusestheblurred,gray-scale imagebyfollowingthegradientbetweenthetwophases.Bytrackingwherethese gradientsintersect,thecentroidsofthesecond-phaseparticlesaredetermined.The functionisquitecapableofnotonlydeterminingindividualparticlesbutalsothose whichareoverlappingasitdesignedtofndanycircularobject.Theabilityto determinethecentroidsallowsforcalculationofthe 0 particleareas.Thisfunction 53

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receivesthegray-scaleimagefromthefunctionthresholderandreturnsthecentroids oftheparticlesinCartesiancoordinates. diaMeasure: Thisfunctioncreatesacircularmaskateachcentroidlocation.The centroidsfromfunctionCircularHoughGrdarereceivedandanalyzedindividually. Thecircularmaskiscomparedwiththebinaryimagetomatchtheproperdiameter accurately.Themaskradiusisincreasedbyonepixelandcompareduntiltwo percentoftheblackpixels(aluminummatrix)isincludedinsidethecircularmask. Oncethisoccurs,thepixeldiameterisrecordedandsavedwithitscorresponding centroid.Atthispointtheareaofthecircleisrecordedandthediameterisfound. Thediameteristhenrecordedwithitsrespectivecentroidandsavedtoadatafle foreachimage. 3.5DataAnalysis DatareturnedfromtheMatlabscriptfullAnalyzeisinthreeforms:binaryimage, gray-scaleimage,andparticlelocationanddiameter.Thebinaryandgray-scaleimages areobservedforsucientcontrastintheoriginalimageandthatindividualparticles weredierentiated.Iftheseimagesshowagglomeration,theyarediscardedandnot includedinthefnalanalysis.Combinationofallparticlecentroidsanddiametersfor agivenalloywerethencompiledintoasingletabularformatforeachrespectiveaging time.TheseflesareimportedintoMatlabandanotherfunctioncalledpostProcess determinestheparticlestatistics.Calibrationofthemicroscopeisnecessarytoobtain preciseknowledgeofthelengthscale.Latexspheres,90nmindiameter,wereusedto determinethemagnifcationcalibrationspecifcforthemicroscopesused.Fordiscussion onthecalibrationandspecifcationsseeAppendix C.PostProcessproducesthenormalized PSD,themeanparticlediameter, max ,themedianparticlediameter,thestandard deviation,andthecoecientofvariation.Allresultsarederivedfromknowingthesize andspatialarrangementoftheprecipitates. 54

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3.6SmallAngleX-rayScattering SamplesweremountedandevaluatedusingaBrukerAXSNanostarsystemwitha 2-dimensionaldetector.ThesampleswerepreparedasaboveforTEMcharacterization, stoppingbeforetheelectropolishingprocedure.Sampleswerethus3mmindiameterand mechanicallythinnedto75microns.Thesample-to-detectordistanceis1meterandyields ascatteringvector, k ,for k> 0:012 A )Tj /T1_4 7.97 Tf (1 .Thissystemiscapableofmeasuringspherical particlesupto50nm.Thetwo-dimensionalspectraareprocessedandintegratedby usingtheBrukersoftwarepackagetoyieldacross-sectionofthespectra.Analysisofthe resultingintensitycross-sectionemployedtheGuinierMethod[51].Small-anglescattering methodsprovideanindependentcharacterizationmethodtoTEMandsamplemorethan 50timesthenumberofparticlesobservedwiththeTEM. 3.7MultiparticleDiusionSimulation SimulationsuseasourcecodewrittenbyProfessorKegangWang,lastmodifedon November5 th 2003.SourcecodeiswritteninFortran,andcanbemodifedtosimulate dierentvolumefractionsandparticlepopulations.Thesimulationsrelyonarandom initialdistributionofparticlesbasedonanarrowgaussiandistribution.Simulations completedforthisexperimentwereconstructedbasedontheexperimentalalloys,using volumefractions, V V =0.10,0.15,and0.26.Simulationswerecompletedforinitial populationsof500and2000precipitates,completingupto100,000computertimesteps. 55

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CHAPTER4 EXPERIMENTALRESULTS Theexperimentsreportedinthisthesisproducedahomogeneous 0 precipitate structureforalloysAl-2.1wt.%Li,Al-2.45wt.%Li,andAl-3.16wt.%Li.Useofcentered dark-feldimagingonthetransmissionelectronmicroscope(TEM)withthe 0 superlattice rerectionscreatedhigh-contrastimagesforanalysis.Aqualitativecomparisonofeach sampleat330kXshowsasignifcantincreaseinparticlesizeoverthetotalagingtime rangeof240hours.ImageswereanalyzedwithinMatlabwithnovelfunctionstofnd thecentroidanddiameterofeachparticlepreciselywithinthedigitizedTEMimages. MeasurementsfromtheMatlabfunctionshowthattheexperimentachievedsteady-state coarseningandexhibitedstablePSDsforagingtimeswithintherange24-240hoursat 225C.Furthermore,Vickersmicrohardnesstestsshowthatpeakhardnessoccurredbefore the6houragingtime,furtherindicatingtheonsetofcoarseningby24hours.Agingtimes includedinthiskineticstudyrangedfrom24-240hours,whichensuredthatsteady-state conditionswerereached.QuantitativeTEMmeasurementsareinagreementwithsmall angleX-rayscatteringmeasurementsusingGuinieranalysistogivethemeanprecipitate radius. Crystalstructuredierencebetweenthefacecenteredcubic(FCC) -matrixand theL1 2 ,Al 3 Li,resultsindiractionrerectionsuniquetothe 0 phase.AtypicalX-ray diractionpatternshowsindependentsuperlatticererectionsmadeentirelyofthe 0 Figure4-1.Peakscommontobothphasesoverlapbecauseofclosecorrelationsintheir latticeparameters, a =4 :03 Aand a 0 =4 :01 A[67,68]. Thecentereddark-feldimagingtechniquewasusedforalloftheimagestaken forprecipitateanalysis.Exploitingthecrystalstructuredierencesbetweenthe 0 and aluminumphases,thesuperlatticererectionswereusedtocreatethecentereddark-feld image.Beforethesespotswereused,theywereindexedtoensurethatwhatwasobserved wasindeedthe 0 phase.TheseplaneswereindexedasshowninFigure4-2.Thespotsof 56

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Figure4-1.Diractionpatternforatypicalbinaryaluminum-lithiumalloyagedat225C lesserintensityareattributedtothemetastableAl 3 Li,andwouldnotbeseeninapure FCCmaterial.Furtherexplanationoftheindexingandintensitycalculationsisgivenin AppendixA.Usingthistechnique,imagestakenwiththeprecipitatesareshownwitha sharpcontrastdierencefromthesurroundingaluminummatrix. Bright-feldimagingwasalsousedforobservationofthesamples,asitgivesa uniqueviewofthesample.Eachsamplewasthoroughlyinspectedforuniformityin distributionandmicrostructurebeforeimagesweretakenforanalysisofthe 0 precipitates. Thesamplewasfrstsearchedtofndalargeregionofsucientthinnesswherea singleplaneofprecipitatescouldbeimaged.Thesethinareaswerethencheckedto ensurehomogeneousprecipitationdevoidofeitherprecipitate-freezonesalongthegrain boundaries,orexcessivedislocationdensityfrommechanicaldeformation. 4.1PhaseDiagram Earlyreportsshowednon-homogeneousnucleationandgrowth,whichmayoccurin alloyschosentooclosetothemetastablesolvusline.Thisstudyselectedarangeofalloys from0.7wt%Li-3.7wt%Litotestthephasediagrambasedonpriorstudies.Previous 57

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Figure4-2.DiractionpatternfromtheAl-3.16wt%Li studiesonthisalloysystemshowedthatthemaximumfunctionalagingtemperaturethat canbeusedwithoutdevelopingtheequilibrium f phaseis225C[14].Twotemperatures wereusedininitialtesting,180Cand225C.At225C,the 0 phasedevelopshomogenously foralloysAl-2.1wt.%Li,Al-2.45wt.%Li,Al-3.16wt.%Li,andAl-3.66wt.%Li.Agingatthe lowertemperatureof180C,alloyAl-1.7wt.%Lialsoexhibitedprecipitationof 0 ,however, itwasinhomogenous.Therewaspreferrednucleationaboutthedislocationsandgrain boundaries,whichdevelopedanon-uniformmicrostructure.Figure4-3showsregionsof largerprecipitatesonandarounddislocationlineswithsurroundingneighborhoodsof smallerprecipitates.Non-homogenouspopulationdistributionscannotbeaccurately characterizedwithregardstoprecipitatesizeandwerenotconsideredforthisstudy. ThepreferrednucleationondislocationsindicatesthatAl-1.7wt.%Liliesextremely closetothemetastablesolvus.Observationsfromthisstudyareconsistentwiththe metastable / 0 solvusfrompreviousexperiments,Figure 4-4.Atanagingtemperature 58

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Figure4-3.Al-1.7Liagedat180Cshowingpreferrednucleationondislocationsgivinga nonhomogeneouspopulationof 0 precipitates of225C,Al-1.7wt.%Lidoesnotdevelopthe 0 ,whereas,at180Citdoesform,albeit inhomogeneously,indicatingitisnearthesolvusline. Alloyswithlesslithiumcontentdonotexhibitanyprecipitationofthe 0 phaseper TEMandXRDanalysis.Alloysforconsiderationarethus:Al-2.1wt.%Li,Al-2.45wt.%Li, Al-3.16wt.%LiandAl-3.66wt.%Li.Thesealloyswereagedatboth180Cand225Cfor timesrangingfrom3to240hours. Analysisofthe 0 employsthemorecommonterm`volumefraction'whenreferringto thesecondphase.Inthismanner,defningthealloysintermsoftheir 0 volumefraction ismoreuseful.Thephasediagramisusedtogiveagoodestimateofthephaseorvolume fractionusingtheequilibriumleverruleandthemetastablesolvusline.Thismetastable solvusiswelldefnedfrompriorexperimentation,anditsaccuracyisreinforcedbythe currentstudy.ObservationofFigure 4-4 showsthealloysandagingtemperaturesusedfor thisstudydenotedontheexperimentalphasediagram.Usingthismetastablesolvusyields 59

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Figure4-4.Al-Liphasediagramshowingthemetastablesolvusfrompriorexperimentand thealloysusedinthecurrentstudy aresultforthecorrelationbetweenthealloysinweightpercentandthecorresponding volumefractionofthe 0 phaseinTable4-1. Table4-1.Particlesevaluatedforeachalloyperagingtime Alloy(wt%)VolumeFraction V V Al-1.7LiNA Al-2.1Li0.10 Al-2.45Li0.15 Al-3.16Li0.27 Al-3.66Li0.40 Agingtemperaturemustbecarefullycontrolledaslowtemperaturesresultin nucleationoccurringpreferentiallyalongdislocationsorgrainboundaries.Aging 60

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temperatureshigherthan240Ccauseequilibrium f ,AlLi,toformatthegrainboundaries andcauseprecipitate-freezonesintheadjacentregions[33].Loweragingtemperatures slowthekineticsandgiveaccesstoawiderrangeofvolumefractions.Thisexperiment attemptedtocharacterizetheAl-Lialloysat180C.Unfortunately,agingatthelower temperatureof180Cslowsthekineticssuchthatsteadystateisnotreacheduntilafter 240hours.Figure4-5showsthemeancubedradiusversusagingtimewhichshouldexhibit linearityinthelatestagesofcoarseningwhensteadystateisreached.Linearityisnot observedandthecubicrelationshipbetweenradiusandtimeindicatesthatagingtimes ontheorderof1000hourswouldbeneededtoproperlyobservethelatestagecoarsening phenomena. Figure4-5.Coarseningofaluminumlithiumalloysagedat180Cshowsnon-steadystate coarseningbehaviorforatleast200hours Thisfurtherrefnedourexperimentalparametersbyonlyconsideringanaging temperatureof225CforalloysAl-2.1wt.%Li,Al-2.45wt.%Li,Al-3.1wt.%Li,and Al-3.66wt.%Li,givinganexperimentalvolumefractionrangeof V V =0.10-0.40.Withthese refnedexperimentalparametersnowdelineated,itisimportantthatthemicrostructure 61

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isuniformandhomogeneousthroughouttheentiresample.Thiswasaccomplishedby samplingthroughouttheas-castsample,bothwithintheequiaxedandcolumnargrains. TEMdisksremovedfromtheas-castcross-sections,Figure4-6,wereimagedandfoundto haveuniformprecipitatesizethroughoutthesample.Precipitatesizeisafunctionofboth volumefractionandagingtemperature,therefore,auniformdistributionofprecipitates throughoutthemicrostructureindicatesahomogenouschemistry. Figure4-6.TEMsamplesremovedfromrandomlocations,precipitateanalysisindicated uniformlithiumconcentration 4.2VickersMicrohardness Vickersmicrohardnesstestsshowthatthealloysalreadypassedtheirpeakhardness in6hours.ArepresentativeplotforAl-3.16wt%Li,Figure 4-7,showsahardnessincrease fromsolutionheattreatmenttothefrstagingtimeof3hours.Thelargeincreaseis attributedtotheevolutionofthesecond-phase, 0 .Increasingtheagingtimebeyond threehoursresultsinahardnessdecrease.Thelongestagetimesof196and240hours yieldahardnessthatapproachesthatofthesolutionizedheattreatment.Hardness steadilydecreaseswithintherangeof24-240hoursindicatingthesampleisoveragedand coarsening.Steady-statecoarseningistheonlymodeofevolutionoccurringduringthese agetimes. 62

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Figure4-7.VickersmicrohardnessforAl-3.16%Liagedat225C,from0-240hours. 4.3MicrostructuralObservations BeforeimageacquisitionintheTEM,thesampleswereobservedforhomogeneity inthe 0 phase,precipitatesmustbewelldistributedandsphericalinshape.Ifany in-homogeneitiesareobserved,additionalsampleswouldbeimagedtodetermine thecause.Allalloysconsideredinthefnalanalysiswerefreefrom`detrimental' microstructuresandexhibitedahomogeneousdistributionofthe 0 phase. 4.3.1Precipitate-FreeZones Precipitate-freezones(PFZs)mostoftenformwithanagingtemperaturethatistoo high,causingformationofanequilibriumphasealongthegrainboundarythatismore energeticallyfavorable.DuetothelowerenergyoftheequilibriumAlLi,thisphasewill growpreferentially,causingdepletionoflithiumintheadjacentarea.ThesePFZscanbe detrimentaltothestudyofphasecoarsening,astheylocallyalterthevolumefractionof 63

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the 0 phase.ObservationofthesamplesdidnotrevealanyPFZsinasinglesample.On thecontrary,itwasobservedthattheprecipitatesnucleateandgrowrightuptothegrain boundarywithoutanydepletionoccurringoneitherside.Atypicalcentereddark-feld TEMmicrographofthe12houragedAl-3.16wt.%Lisample,takenatagrainboundary, canbeobservedinFigure 4-8.Precipitatesareshowninthisimagetoexistrightupto thegrainboundarywithnoPFZ.Thetiltofthislow-anglegrainboundarywassuchthat byrotationofthesample,rerectionsfrombothgrainswerecaptured.Thismicrographis indicativeofwhatwasobservedatalloftheagingtimes.ItwasdeterminedthatPFZs werenotpresent,andthattheagingtemperatureof225Cwasindeedacceptableforthe experimentation. Figure4-8. 0 precipitatesgrowinguptoandoneithersideofatiltgrainboundary,which doesnotexhibitaprecipitate-freezone.12houraging,225C 4.3.2ParticleInteractions As 0 beginstocoarsen,precipitatesinthesameneighborhoodmaybeginto growandimpingeoneachother.Itwasobservedthatfewerthan3%oftheparticles encounteredotherparticlesforalloysAl-2.1wt.%Li,Al-2.45wt.%LiandAl-3.16wt.%Li. 64

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AlloyAl-3.66wt.%Liexhibitedexcessiveparticlecoalescence,Figure 4-9,indicates precipitategrowththatisnotstrictlydiusionlimited.Thatis,particlesenlargeby joiningwiththeirneighbors.Inthismanner,theydonotexhibitclassical,diusion-limited coarseningandwerenotconsideredfurtherinthisproject.Asaresult,itwasdetermined Figure4-9.PrecipitatecoalescenceinAl-3.66Lishowingnon-sphericalforms thatthemaximumvolumefractionthatmaybeconsideredforcoarseningexperimentsin Al-Lialloysis V V =0.30.Therefore,alloysAl-2.1wt.%Li,Al-2.45wt.%LiandAl-3.16wt.%Li remainforthecriticalagingexperimentsconductedat225C. Itisimportanttodistinguishbetweenparticlesthatareincontactandparticles thatappearoverlappedasaresultoftheimageprojectionbytheTEM.Particleswhich appearoverlappedoftenresultfromathickTEMdiskcontainingmorethanasingle planeofprecipitates.ObservedintheTEMistheprojectionofoneparticleontopof another,whichliesbelowitwithinthesample.Precipitatesoverlappedbyprojection andcoalescedparticlescanbedistinguishedbythepresence,ornon-presence,ofaunion betweentheparticles.Insuchaprojectionoverlap,adistinctvariationintheintensity fromoneparticletothenextisobserved.Oftentheoutlineofeachprecipitateremains visiblewithintheunion.ObservationofFigure 4-10A showsseveraloverlappedparticles 65

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wherethecompletionofeachparticlecanbeseenwithintheunion,indicatingthatthey are,infact,nottouching.Itisalsoobservedthatsomeoftheunionsshowadarkregion causedbythesubsequentscatteringofthebeamdiractedbythefrstprecipitate.These darkregionscancausedicultyinanalysisandwillbediscussedinSection 5.2. Iftwoparticlesdoencountereachotherandcoalescenceoccurs,asmoothtransition isobservedsimilartosinteredparticles.Figure 4-10B isanimagetakenfromthe48hour agingtreatmentshowingacoalescenceinthecenterofthemicrograph.Observethat thereisnowasmoothtransitionbetweentheprecipitates,andsharpcontrastchange doesnotappearintheunionbetweenthem.Closerexaminationshowsthatattheupper rightofthecoalescedprecipitatesanapparentoverlapappearsbetweentheseparticles. Adistinctcontrastdierenceisseenwithintheunionshowingacompletionofeach respectiveparticle.Thisseriesofcoalescedandoverlappedparticlesisseenatgreater magnifcationinFigure 4-10C.Distinctionbetweenthesetwocasesisnecessaryinorder nottooverestimatetheencountersandcoalescencesofprecipitates. 66

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AOverlapbyProjection BParticleCoalescence CParticleCoalescence Figure4-10.Electronmicrographsshowingcoalescedandoverlappedparticlesby projectionviewing(a)Overlappingparticles48Hr,Al-3.16wt.%Li,225C(b) Coalescedandoverlappedparticles,48hour,Al-3.16wt.%Li,225C(c)Close upofcoalescedparticles,48hour,Al-3.16wt.%Li,225C 67

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Coalescencedoesnotalwaysoccurwhendierentprecipitatesarebroughtinto intimatecontact.Theremustbelatticeregistrybetweentheindividualparticlesforthem tocoalesce.Priorstudiesofthecoalescenceofparticleshaveshownthatanti-phase boundariesdonotform,signifyingthatonlyprecipitateswithnear-perfectlattice registrycanachievecoalescence[13].Latticeregistrywillonlyoccurtwentyfvepercent ofthetimefortheL1 2 crystalstructure,concludingthatcoalescenceisnotamajor contributiontophasecoarseninginthesealloys.Whenthereisnotcloselatticeregistry, aratteningoftheinterfaceisobservedbetweentheprecipitates.Thisisobservedunder themicroscopeassmallthinregionscontainingthealuminummatrixexistingbetweenthe particles,Figure4-11.Withcontinuedgrowth,arattenedinterfacedevelopsbetweenthe precipitatescausingsignifcantchangetotheshape.Thisdeviationfromsphericalshapeis duetothelackofgrowthalongtheinterfacebetweentheprecipitates.Itisobservedthat thelocaldiusionfeldismodifedasconcentrationchangesarenotexhibitedwiththis regionbetweenprecipitates. Figure4-11.Precipitatesthathavegrowntogether,buthavenotcoalescedduetoalackof latticeregistry,Al-3.16Li,144hour,225C 68

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Initialtestingrefnedthestudyparametersforagingtemperatureandalloysfor consideration.Threealloyswereshowntobeviable,namely,Al-2.1wt.%Li,Al-2.45wt.%Li andAl-3.16wt.%Li.Agingtemperaturewasselectedtobe225C,whichgaveanobservable coarseningratefortimesrangingfrom24-240hours. 4.4QualitativeGrowthComparison Onegoodvisualmethodtoobservetheparticlegrowthasafunctionoftimeisto vieweachagingtimeatthesamemagnifcation.Inthismanneronecanobservethe increaseinthelengthscaleofthe 0 precipitatephaseasafunctionoftime.Aqualitative growthcomparisonusingtransmissionelectronmicrographsat330kXmagnifcationis showninFigures 4-12 to 4-20.AlloysusedinthefnalanalysiswereAl-2.1wt.%Li,Figures (4-12 to 4-14),Al-2.45wt.%Li,Figures(4-15 to 4-17)andAl-3.16wt.%Li,Figures(4-18 to 4-20).Theimagesareshownasaqualitativecomparisonofparticlesizeonlyandwerenot specifcallyusedforparticleanalysisateachagingtime.Ourobservationsshowthatthere isanoptimalprecipitatedensityforanalysisoftheparticles. AlloyAl-2.1wt.%Lihasthesmallestcontentoflithiumand,thus,thesmallest volumefractionof V V =0.10.Imagingiseasilyaccomplishedasthepopulationiswell dispersedanditisrelativelyeasytocreateathinTEMsamplecontainingasingleplane ofprecipitates.ObservingFigure 4-23A itisclearthatthefnedistributionofprecipitates exhibitarangeofsizes,whichyieldsanaverageparticlediameterof41nanometers(nm). Astheagingprocesscontinues,theprecipitatesgrow,whilestillmaintainingadistribution ofsizes,Figure 4-13B.Byholdingthemagnifcationconstant,itisobservedthatthereare fewerandfewerparticleswithinthefeldofviewforlongeragingtimes.Ultimately,at240 hoursaging,thereareonly4particleswithintheimage,Figure 4-14C,withanaverage precipitatediameterof83nm. 69

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A24Hour B36Hour C48Hour Figure4-12.ElectronmicrographsforAl-2.1wt.%Li,225C(a)24Hour(b)36Hour(c)48 Hour 70

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A72Hour B96Hour C120Hour Figure4-13.ElectronmicrographsforAl-2.1wt.%Li,225C(a)72Hour(b)96Hour(c)120 Hour 71

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A144Hour B192Hour C240Hour Figure4-14.ElectronmicrographsforAl-2.1wt.%Li,225C(a)144Hour(b)192Hour(c) 240Hour 72

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IncreasingtheamountoflithiumtoalloyAl-2.45wt.%Li,increasesthevolume fraction, V V =0.15.The 0 phaseincreasecanbeobservedbothbyanincreaseinthe populationforagivenunitvolumeandtheincreaseinsize.ComparingFigure 4-12A for Al-2.1wt.%LiwithFigure 4-26A showsaqualitativeincreaseinprecipitatesize,andthese numbersareverifedthroughquantitiveanalysistobeanincreasefrom41nmto45nm, respectively.Aspredicted,withincreasingagingtimetheprecipitatescontinuetogrow reaching90nmatanagingtimeof240hoursat225C. A24Hour B36Hour C48Hour Figure4-15.ElectronmicrographsforAl-2.45wt.%Li,225C(a)24Hour(b)36Hour(c)48 Hour 73

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A72Hour B96Hour C120Hour Figure4-16.ElectronmicrographsforAl-2.45wt.%Li,225C(a)72Hour(b)96Hour(c) 120Hour 74

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A144Hour B192Hour C240Hour Figure4-17.ElectronmicrographsforAl-2.45wt.%Li,225C(a)144Hour(b)192Hour(c) 240Hour 75

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AlloyAl-3.16wt.%Lifurtherincreasesthevolumefractionof 0 phase, V V =0.27and makesimagingasingleplaneofprecipitatesmoredicult.Theeectofincreasinglithium contentisobviousfromcomparisonofthe24hragingtimebetweenAl-2.1wt.%Liand Al-3.16wt.%LiinFigures4-12Aand4-18A.Increasesinthelithiumcontentrequires TEMsamplepreparationtobeexecutedexceedinglycarefully,inordertocreatearegion asthinastheprecipitatesthemselves.Samplepreparationiscriticaltoachieveclear, high-contrastimagesthathaveminimaloverlapoftheprecipitates.Thisrequiresthatthe sampleshaveathicknessofapproximately70nm. A24Hour B36Hour C48Hour Figure4-18.ElectronmicrographsforAl-3.16wt.%Liat200kXmagnifcation,225C(a)24 Hour(b)36Hour(c)48Hour 76

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A72Hour B96Hour C108Hour Figure4-19.ElectronmicrographsforAl-3.16wt.%Liat200kXmagnifcation,225C(a)72 Hour(b)96Hour(c)108Hour 77

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A144Hour B196Hour C240Hour Figure4-20.ElectronmicrographsforAl-3.16wt.%Liat200kXmagnifcation,225C(a) 144Hour(b)196Hour(c)240Hour Reachingthelongestagingtimeofthisstudy,240hours,the 0 phaseremains coherent,asevidencedbyitssphericalshape.Itisobservedthatinordertomaintainthis coherency,particlesmayexhibitstackingfaultsasameansofloweringthestrainenergy withthesurroundingmatrix.Inadditiontothestackingfaults,dislocationsappearwith morefrequencyatlongeragetimes.Dislocationsmayformasexcessvacanciesarecreated inthematrixphasewhenthelithiumisremoved.Figure 4-20B showsaparticleonthe 78

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rightsideoftheimagewithasharpcusp.Thisisbelievedtooccurfromtheprecipitate encounteringadislocationandencompassingitwhilecontinuingtogrow. Therearesomegeneralobservationsnotedfromthemicrographsshownforthe qualitativecomparison.Theseobservationsledtoourmodifyingthebasisofdata collectionforoptimumanalysis.Observationofphasecoarseningatdivisionsintime showsthecontinuityofgrowthwhereprecipitatesarecontinuallygrowingorshrinking. Thefollowingobservationsarenoted: 1.Arelativedistributionbetweengrowingandshrinkingprecipitatesisobservedatall timescales. 2.Thesphericalnatureofthe 0 phaseremainsconstantthroughalltimescales. 3.Encountersbetweenparticles,eithercoalescenceorrepulsion,arenotcommon. 4.Imagesmustcontainasinglecross-sectionofprecipitatestoavoidparticleoverlapby projection. 5.Magnifcationgivingaprecipitatedensityof30-50perimageclearlyshowsthe distributionofthethe 0 phase. 79

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4.5ImageAnalysis Imageanalysisitselfissensitivelydependentontheprecisionandqualityofimages takenfromtheTEM.Usingofthedouble-tiltsamplestageallowsformaximumsample rotation,andthus,canmaximizetheintensityandcontrastdierenceatatwo-beam condition.Inaddition,thecorrectmagnifcationshouldbeselectedforidealprecipitate density.Theoperatormustalsoselectthethinnestareasrepresentativeoftheentire microstructure,whichminimizesoverlappingparticles.Whenthisisachieved,images areacquiredwiththegoalofcapturing1000individualparticlesthatcanbemeasured forboththeirpositionintheimageandfortheirrespectivesize.Byensuringthebest possibleimagequalityuponacquisitionintheTEM,thebestresultswillbeobtainedfrom thecomputeranalysissoftware.Alistofthealloysandtheirresultantparticlecountper agingtimeareshowninTable 4-2. Table4-2.Particlesevaluatedforeachalloyperagingtime. Agingtime(Hr)Al-2.1wt.%LiAl-2.45wt.%LiAl-3.16wt.%Li 2416621418996 36161712381012 48102013971288 72137915801208 96141211001882 120123416241013 144180110681039 19223831394965 24013001624631 TheimageprocessingfunctionscontainedandcalledbyfullAnalyzewithinMatlab werepresentedinSection 3.4.Theformulationofthesefunctionsbecameaniterative processleadingtoanoptimalresultbycontinualcomparisonbetweentheinputflesand theoutputresults.Assuch,itisnecessarytoshowtheaccuracyofeachofthefunctions toprovetheireectiveness.Beingdevelopedspecifcallyfortheanalysisofthesespherical precipitatestakenfromTEMmicrographsitwillbeshownthatitisauniversallyapplied andrepeatableseriesoffunctions. 80

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Thefrstresulttobeobservedisthatgeneratedbythefunction`threshold'.It receivestheoriginalscannedimageasinput,enhancestheimageandcreatesablurred, grayscaleandbinaryimage.Theissuesofconcernarethatthedilationanderosion processesusedforedgedefnitiondonotaddto,ordeletefrom,anysignifcantinformation toprecipitatestructure.Thedilationprocessgrowstheparticlesuniformly,whereasthe erosionprocesstakesawayfromtheparticles.Theeectsofthesholderarebestobserved byviewingboththeinputandoutputfromthefunction.AnimagefromAl-3.16wt.%Li after36hoursagingwillbeobservedthroughthisprocess;theas-scannedandcropped image,ascalledbyfullAnalyze,isshowninFigure 4-21A.PassingFigure 4-21A through thesholdereliminatesnoiseandcreatesauniformimagecontrastproducingagray-scale andbinaryimage.Thegray-scaleimage,Figure 4-21B,iscreatedwithsignifcantblur aroundtheparticles.Thisblur,orintensitygradient,aroundtheparticlesiswhat thecontinuingfunctionCircularHoughGrdwillusetodeterminethecentroidsofeach precipitate.Thebinaryimage,Figure 4-21C,istobeablackandwhitecopyofthe originalimage.Itisimportanttomaintainthecorrectsizeoftheparticlesbetweenthe binaryandoriginalimagetoensurethatthemeasureddiametersarerepresentativeofthe truemicrostructure.Theaccuracyofthebinaryimagetocapturetheboundariesofthe originalimageisshowninFigure 4-21D.Thisshowstheoriginalimagewherethewhite outlineistheborderbetweenthealuminumand 0 phaseinthebinaryimage.Thenearly identicalcorrelationbetweenthebinaryandoriginalimagedemonstratesthefdelityofthe enhancementprocedurecompletedbythefunctionthresholder. Thesecondresultistheaccuracyoffndingthecentroidsoftheprecipitatesusingthe gray-scaleimage.ThefunctionCircularHoughGrdtrackstheintensitychangearoundeach oftheparticlesandfndsthecentroids.Theaccuracyofthisfunctionisshownbyplotting thecentroidsontheoriginalimage,showninFigure 4-21D.ObservethattheHough transformiscapableofdefningthecentroidsofbothindividualandoverlappingparticles. Itisclearthatnotalltheparticlesarecaptured,butingeneralthose,whicharenot 81

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found,arenotdefnedclearly.Theylackasharpboundary,whichisprobablyindicativeof precipitateremnantsthathavebeenpartiallyremovedfromthesurfaceduringpolishing. AAsScannedandCropped BGrayscaleimage CBinaryimage DCentroidsontheoriginalimage Figure4-21.ResultsfromtheMatlabimageprocessingfunctions(a)Imported,36hour, 225C(b)Blurred,grayscaleimage(c)Binaryimage(d)Centroidsand outlineofthebinaryimagesuperimposedovertheoriginalimage 82

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Thethirdresultisthemeasurementofparticlediameterbycomparingagrowing circularmaskonthebinaryimage.Thiscomparatoracceptsadiameterwhentwopercent ofthesurroundingblack -matrixiscontainedwithinthecircularmask.Particlesthat intersecttheedgesoftheimagearenotconsidered.Theaccuracyofthemeasurements isshownbysuperimposingthecentroidsandtheirrespectivediametersontheoriginal image,Figure4-22.Thereisexcellentcorrelationbetweenthemeasureddiametersandthe originalimage.ItappearsthattheimageprocessingfunctionscreatedinMatlabclearly captureboththeprecipitatelocationsandtheirdiameters.Theabilitytoflterthedata forclaritybydeterminingwhichparticlesareclearlydefned,andfndingtheircentroids anddiameters,havebeenclearlyachievedbyanautonomouscomputerfunctionwrittenin Matlab.Themethodologyusedinthisthesisisconsideredtobemoreprecisethandirect humanmeasurement,duetothecomputerobjectivityandfxedfltercriteria. Figure4-22.Particlediametersusingthecentroidssuperimposedovertheoriginalimage 83

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4.6ParticleSizeDistribution PSDhistogramsarethebestwaytovisualizethesizeclassofprecipitateswithinthe sample.ThePSDisnormalizedonthehorizontalaxiswithrespecttothemeandiameter. Thisnormalizationallowsforadimensionlesscomparisonacrossalltemporalandlength scales.ThePSDsforeachalloyandagingtimeareshowninFigures 4-23 to 4-32.Itis notedthattoobtainsmoothdistributions,alargesamplesizemustbeconsidered.For thisstudy,itwasdeterminedthat1000particleswouldconstitutethe minimum sample sizeneededtogenerateacceptablestatistics.Thewidthsofthebinsinthehistogram aredeterminedbythesamplesizeandhasthefollowingrelationshipwithsamplesize, N: W = N 1=3 [69].Asaresult,abinwidthofW=0.1wasused.ThePSDsarewell behaved,exhibitinganearlysymmetricalandbroaddistribution.Withincreasinglithium contentorvolumefractionthePSDbecomesslightlymoresymmetricalresultinginan evenbroaderdistribution.ThePSDsareremarkablysimilarintheirappearanceacross alltheagingtimesfortheirrespectivealloy.Themaximumparticlesizevariesslightly betweentheagingtimesandalloys,butitisobviousthattheyarelargerthan1.5,as predictedfromLSWtheory.Al-2.1wt.%LishowsaslightnegativeskewinthePSD.The similarityamongthePSDsseemstoindicatethatthedistributionofparticleschanges linearlywiththelengthscale.Anincreaseinparticlesizeisaccompaniedbyanincrease intheinter-particledistances,thusresultinginanidenticalnormalizedPSDregardlessof agingtime. 84

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A24Hour B36Hour C48Hour Figure4-23.ParticlesizedistributionsforAl-2.1wt.%Liagedat225C(a)24hour(b)36 hour(c)48hour 85

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A72Hour B96Hour C120Hour Figure4-24.ParticlesizedistributionsforAl-2.1wt.%Liagedat225C(a)72hour(b)96 hour(c)120hour 86

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A144Hour B196Hour C240Hour Figure4-25.ParticlesizedistributionsforAl-2.1wt.%Liagedat225C(a)144hour(b) 196hour(c)240hour 87

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A24Hour B36Hour C48Hour Figure4-26.ParticlesizedistributionsforAl-2.45wt.%Liagedat225C(a)24hour(b)36 hour(c)48hour 88

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A72Hour B96Hour C120Hour Figure4-27.ParticlesizedistributionsforAl-2.45wt.%Liagedat225C(a)72hour(b)96 hour(c)120hour 89

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A144Hour B196Hour C240Hour Figure4-28.ParticlesizedistributionsforAl-2.45wt.%Liagedat225C(a)144hour(b) 196hour(c)240hour 90

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A24Hour B36Hour C48Hour Figure4-29.ParticlesizedistributionsforAl-3.16wt.%Liagedat225C(a)24hour(b)36 hour(c)48hour 91

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A72Hour B96Hour C108Hour Figure4-30.ParticlesizedistributionsforAl-3.16wt.%Liagedat225C(a)72hour(b)96 hour(c)108hour 92

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A144Hour B196Hour C240Hour Figure4-31.ParticlesizedistributionsforAl-3.16wt.%Liagedat225C(a)144hour(b) 196hour(c)240hour 93

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4.7PrecipitateStatistics ThecalibrationfortheTEMperAppendix C isappliedtothemeasuredcentroids anddiameterstofndtherealdimensionsoftheprecipitates.Themetricusedmostoften forcomparisonisthemeanparticlesize,asitistheeasiesttofndandtheleastproneto error.MeanprecipitatediametersareshowninTable 4-3 forallsamples. Table4-3.Meanprecipitatediameterasafunctionofalloyandagingtime.Alloy compositionsinwt.% Agingtime(Hr)Al-2.1wt.%Li, ,nm Al-2.45wt.%Li, ,nm Al-3.16wt.%Li, ,nm 2441.544.8 54.5 36 48.8 55.4 57.3 48 51.3 57.6 67.1 72 62.3 63.1 69.7 96 65.3 70.7 78.3 120 68.9 74.3 84.3 144 71.9 78.1 87.5 192 77.7 81.8 94.9 240 83.1 90.1 95.2 Observationsofphasecoarseningshowedtheprecipitatediameterdoubleformostofthe alloysbetweenthefrsttolastagingtime.Themeandiametershowsuniformgrowth exhibitedbysizeincreaseduetoagingtimeandthereisanaccompanyingincreasein volumefractionwithincreasedlithiumcontent.Thereisaclosecorrelationbetweenthe meanandmedianprecipitatesize,indicatinganearlysymmetricaldistribution. Thecoecientofvariation,defnedasthestandarddeviationoverthemean,isa dimensionlessmetricindicatingtheuniformityofthedistribution.Thevaluesforthe coecientofvariancearesmallandareconstantperalloyoveragingsequence.This indicatesthattheparticledistributionscanbeconsideredtobeatsteadystate.The standarddeviationdoesnotremainaconstantbutratherexhibitsalinearincreasewith themeanradius.Aconstantvalueforthecoecientofvariationisastrongindication thatthePSDisane.Thisrelationshipwiththestandarddeviationhasbeenshown beforeinaluminumlithiumalloysbyMahalingametal.[14]. 94

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4.8SelfSimilarityoftheParticleSizeDistribution ThePSDsofallofthealloysinFigures 4-23 to 4-32 qualitativelyshowssimilarity betweenthem,respectively.Aquantitativecomparisonisshownbythewidthofthe distribution,wherethemeanmaximumparticlesize, max ,issimilaramongtheaging times,Section 4.6 .Thecoecientofvariationisstableacrossalloftheagingtimes exhibitingminimalchangeandsteady-statecoarsening.Similarobservationsweremadeby GuandLiedl[13]throughcoarseningstudiesonanAl-2.8Li-0.3Mnalloy.PSDsthatare identical,independentofhistory,arecalledself-similar,orane,distributions. Allthenormalizeddiameterswerecombinedfromthevariousagingtimesresulting inatotalparticlecountlargerthan10,000foreachalloy.ThecumulativePSDs,Figures 4-32A to 4-32C,haveasmootherdistributionthanthosefortheindividualagingtimes dueprimarilytothelargersamplesize.Asaresultofthelargersamplesize,thebin widthischangedto0.5,whichcorrelatestotherelationshipsetforthbyScott[69]in Section 4.6.ThesymmetricalnatureofthePSDclearlydiersfromtheskewedPSD predictedbyLSWtheory.Theself-similarityindicatesthatregardlessofagingtime,asthe particlesincreaseinsize,thereisacorrespondingincreaseintheinter-particledistance. Particleneighborhoodsremainstatisticallythesame,theonlydierenceisthelength scale.ThePSDalsobroadenswithincreasingvolumefraction.Observingthemaximum frequencyfromthePSDsitisobservedthatAl-2.1wt.%Liexhibitsamaximaover1.6, Al-2.45wt.%Lioccursat1.45,andAl-3.16wt.%Liisjustunder1.4.Qualitatively,itis observedthatFigure 4-32C ismoresymmetricalthantheotherPSDs.Onenotesthe increaseinmaximumparticlesizewithanincreaseinthevolumefractionanditsrolein broadeningthePSD. 4.9MaximumPrecipitateDiameter Themaximumnormalizedparticlediameter, max ,isdiculttoobserve,because thereexistrelativelyfew`maximum'sizedparticles,andthusthemeasurementrequires anextremelylargesamplesize.Duetotheself-similarityofthemicrostructureand 95

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AAl-2.1wt%225C BAl-2.45wt%225C CAl-3.16wt%225C Figure4-32.Cumulativehistogramofallagingtimesfrom24to240Hours(a) Al-2.1wt.%Li,225C(b)Al-2.45wt.%Li,225C(c)Al-3.16wt.%Li,225C 96

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thenormalizeddiameters,theeectivesamplesizeperalloyisgreaterthan10,000 precipitates.Thisprovidesconfdenceinclaimingthatthelargestprecipitatemeasuredas max .Themaximumnormalizedparticlediameterfromexperimentisthelargestfound fromalloftheagingtimes,Table 4-4 .Thesevaluescanalsobecorrelatedwithdiusion screeningtheorywhichpredictsthemaximumnormalizedparticleradiusfromthevolume fraction.Diusionscreeningtheorypredicts max as, max =1 )Tj /T1_0 11.955 Tf 25.2 8.088 Td (1 p 3V V + s 1 3V V + 1 p 3V V +1 : (4{1) UsingthevolumefractionsforthealloysperTable 4-1 inEq. 4{1 yieldsthescreening theorypredictionsfor max asfoundinTable 4-4.Thispredictionisinagreementwith ourexperiments.Comparisonbetweenscreeningtheoryandexperimentshowsstrong correlation,withlessthan2%dierenceinvalue. Table4-4.Maximumnormalizedprecipitateradius, max ,aspredictedbyscreeningtheory andfoundbyexperiment. Al-2.1wt.%LiAl-2.45wt.%LiAl-3.16wt.%Li ScreeningTheory max 1.661.681.72 Experiment max 1.681.721.75 4.10ComparisonoftheParticleSizeDistribution ExperimentalPSDscorrelatewellwiththosepredictedfromscreeningtheoryand frommulti-particlesimulations.ThewidthsofthePSDscomparewellasdiscussedin Section4.9,however,observationoftheFigure4-33showsthatmoreofthelargersized precipitatesareobservedinexperimentthanpredicted.Thisdisparitybecomesmore pronouncedwithincreasedvolumefraction.Thisdiscrepancyisnotwhollyunexpected asincreasingvolumefractioncausesgrowthbycoalescence,whichisaprocessthatis notstrictlydiusionlimited.Althoughthereisagreaterpopulationoflargersizesthan predictedbyscreeningtheory,thebreadthandsymmetricalformofthePSDsarewell predictedbyscreeningtheory. 97

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AAl-2.1wt.%225C BAl-2.45wt.%225C CAl-3.16wt.%225C Figure4-33.Comparisonoftheparticle-sizedistributionfromexperiment,multi-particle simulationandscreeningtheory(a)Al-2.1wt.%Li,225C(b)Al-2.45wt.%Li, 225C(c)Al-3.16wt.%Li,225C 98

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4.11Kinetics Oneofthecommonwaystocharacterizephasecoarseningkineticsisbyusingthe linearrelationshipfrstpredictedwithLSWtheory,Equation(4{2).Althoughtheoriginal kineticfactor, K LSW ,isnotapplicable,thescalingrelationshipthatlinearityexists between R 3 andtimeremainsvalid,namely, R (t) 3 )Tj /T1_1 11.955 Tf 14.509 3.022 Td ( R (0) 3 = Kt: (4{2) UsingthemeandiametervaluesfromTable 4-3 foreachrespectiveagingtime,Figure 4-34 wasplottedas R 3 versustime.Linearityisindeedobserved,asexpectedfromEq.(4{2). Linearregressionwasusedtoftthisdata,wheretheslopescorrespondtothekinetic factors K (V V ).ConversiontostandardvaluesgivesthekineticcoecientsfoundinTable 4-5. Figure4-34. R 3 versustime,exhibitingthepredictedlinearity 99

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TheseresultsarecomparedwiththosemeasuredbyMahalingam,whostudied similaralloysofAl-2.4wt%LiandAl-3.10wt.%Liagedat225C.Mahalingamfound akineticcoecientforthesetobe K =134:3 10 )Tj /T1_4 7.97 Tf (24 cm 3 /sand K =163:1 10 )Tj /T1_4 7.97 Tf (24 cm 3 /s,respectively.ThekineticfactorsfromMahalingamarelargerthanthosefound withthisstudy.Itispuzzlingthatsuchalargedisparityhasarisenastheexperimental procedureandprecipitatesizesfortheagingtimesaresimilarinbothstudies.However, Mahalinghamstudiedtheinruencesforagingtimesof12,24,48,and72hoursforthese alloysandagingtemperature.Thepresentstudyconsidered9agingtimesbetween 24and240hours,withatargetofobtaining1000particlesforeachtime.Thegoalof Mahalingamwastoachievetheanalysisof500precipitateswithsemi-automatedanalysis. Thefrmerparticlestatisticsconsideredhere,basedonawiderrangeofexperimentaldata, lendcredibilitytothevaluesdeterminedinthepresentstudy. Table4-5.Maximumprecipitatediameteraspredictedbyscreeningtheoryandfoundby experiment.Alloycompositionsinwt.%. KineticFactorAl-2.1Li(V V =0.1)Al-2.45Li(V V =0.15)Al-3.16Li(V V =0.27) (K)*10 )Tj /T1_4 7.97 Tf (24 cm 3 =sec 80.8 97.1 124 K(V V )/K LSW Theory 1.49 1.79 2.29 K(V V )/K LSW Experiment1.52 1.87 2.39 Diusionscreeningtheorycannotonlypredict, max ,asshowninSection 4.7,butcan alsodeterminethekineticcoecientasafunctionofvolumefraction[19]asfollows, K (V V ) K (0) 6:41 2 6 4 2 )Tj /T1_0 11.955 Tf 11.955 0 Td ((1 )Tj 11.956 9.225 Td (p 3V V ) 1 )Tj /T1_4 7.97 Tf 22.349 4.707 Td (1 p 3V V + q 1 3V V + 1 p 3V V +1 1 )Tj /T1_4 7.97 Tf 22.35 4.707 Td (1 p 3V V + q 1 3V V + 1 p 3V V +1 3 3 7 5 : (4{3) Theratiocreatesanon-dimensionalmetricthatcanbecompareddirectlywith experimentalstudies.Figure4-34resolvestheexperimentalkineticfactorsbylinear regression,however, K (0)mustbecalculatedfrommaterialsconstants.Equation(4{4), wasusedforthecalculationalongwithmaterialsconstantsobtainedfrom[70], 100

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K (0)= 8 9 C e V 2 m rD RT : (4{4) Inthisequation, V m isthemolarvolumefractionof 0 (m 3 /mol), C e istheequilibrium solutecontentofthematrix(mol/m 3 ), r istheinterfacialsurfaceenergy(J/m 2 )and D is thediusioncoecient(m 2 /s).Thefollowingvalueswereused[14]: V m =3 :88 10 )Tj /T1_3 7.97 Tf (5 m 3 /mol C e =7:0 10 3 mol/m 3 r =0:130J/m 2 D=1 :84 10 )Tj /T1_3 7.97 Tf (19 m 2 /s TheLSWkineticcoecientisfoundfor225Ctobe K (0)=5:4 10 )Tj /T1_3 7.97 Tf (23 cm 3 /sec,yielding K (V V )=K (0) exp valuesshowninTable 4-5.Goodcorrelationexistsbetweenscreening theoryandexperimentalresultswithrespecttothekineticfactor. 4.12Smallanglex-rayscattering Smallanglex-rayscatteringyieldsatwodimensionalradialplot.Thetwodimensional plotshouldbeanalyzedtoensurethatthescatteringisisotropic,indicatingarandom microstructurewithoutpreferredorientation.Thisassumptionneedstobesatisfedasthe 0 phaseissphericalandrandomlydistributed.Allplotsconsideredinthisanalysiswere isotropicanduniformaboutthecenter. Thetwodimensionalplotisintegratedradiallyaboutthecentertoobtainascattering cross-section.Scatteringcross-sectionsforalloyAl-2.1wt.%Liagedat225Cfor12and36 hoursareshowninFigure 4-35.Typicalscatteringcrosssectionsareshownasscattering intensityversusthemagnitudeofthescatteringvector, j ~ k j,defnedas j ~ k j =4 sin =. Itisobservedthatwithincreasingagingtimetheinterferencepeakincreasesandshifts totheleft.Thisisexplainedbyreciprocalspaceasincreasingparticlesizewillleadto decreasingscatteringvectors.Analysisofthescatteringdistributionisnon-trivialand requiresmodelingoftheelectrondistributionoftheparticles.Thescatteringfunctionof 101

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Figure4-35.Scatteringcross-sectionsforAl-2.1Li,agedat225C aparticle, I (j ~ k j),isrelatedtoadistancepairdistributionfunction, p(r ),bytheFourier integraltransform, I (k )=4 Z inf 0 p (r ) sin(qr ) qr dr: (4{5) Guinierdrewonthecorrelationofelectrondensitytothemassmomentofinertiain mechanicstorelatethedistancedistributionfunction p(r )tothescatteringcurveas, R 2 = R 1 0 p(r ) r 2 dr 2 R 1 0 p(r )dr : (4{6) Solvingforthreedimensions,yieldsasolution,calledtheGuinierapproximation, I (h)= I (0)exp )Tj /T1_6 5.978 Tf 7.782 5.268 Td (k 2 R 2 G 3 : (4{7) 102

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Therelationshipbetweentheradiusofgyrationandtheparticledimensioncan besolvedforvarioushomogeneousbodies[51].Forsphericalscatteringparticlesthe relationshipisgivenby, R G = r 3 5 r: (4{8) UsingtheGuinierapproximationandthescatteringintensitytotherightofthe interactionpeak,theradiusofgyrationisfound.Creatingaplotofln( I (j ~ k j))versus j ~ k j 2 withinthelinearregiontotherightoftheinteractionpeakfndstheslopewhichis R 2 G =3. Figure 4-36 showsanexampleoftheGuinierplotforAl-2.1wt.%Liagedat225Cfor12 and36hours,linearregressionisusedtofndtheslope. Figure4-36.Scatteringcross-sectionsforAl-2.1wt.%Li,agedat225C Equations4{7and4{8 areusedtodeterminethesphericalprecipitatesize.The resultsarecomparedwiththoseobtainedfromTEMdirectanalysisinTable 4-6.Thereis goodcorrelationbetweenthequantitativeTEMresultsandGuinieranalysisfromSAXS. 103

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ThisisnotableasthesamplesizeforSAXSisanorderofmagnitudegreaterthanthatof TEM. Table4-6.ComparisonofprecipitatesizebetweenTEMandSAXSanalysis AlloyAgingtime(hr)TEM (nm)SAXS (nm) Al-2.1Li121616 Al-2.1Li3624.4 21.4 Al-2.45Li24 20.5 22.8 104

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CHAPTER5 DISCUSSION Itisobservedfromtheexperimentsreportedinthisthesisthattheexperimental procedureandanalysisproducedtheproperrangeofmicrostructures,eectivelycaptured thekeystatisticsofthe 0 precipitatepopulation,andyieldeddatathatcomparedwell withindependentpredictionsbasedscreeningtheoryandmulti-particlesimulations.In addition,experimentalresultsfromthisthesiscorrespondwellwithpriorexperimental studies.Automatedimageanalysisisdiculttoimplementwiththeaccuracyand perceptionattributabletothehumaneye.Assumptionsmustbemadeinitsformulation, withphysicallimitsimposedoninputimagequalityandontherangeoftheallowed particledensity.Theresultswerecomparedbothinternallywithstandardmetricmethods forselfconsistency,andwithdiractionresultsfromindependentSAXSresultsalso carriedout. 5.1SamplePreparation Signifcantcarewasexpendedincreatinguniformsamplesizebysectioning.This controlensuredthateachsampleundergoesanexactandreproducibleheatingand quenchcycle.Fortherelativelylongagingtimesthatwerestudiedintheseexperiments, heatingandquenchrateshaveminimalaects.Waterwasthequenchmediausedin theseexperimentsandwasshowntohavesucientthermalcapacityinprovidingarapid quenchrate.Fasterquenchratestendtosupersaturatethealuminummatrixwithlithium, andrepressthenucleationofthe 0 phase,atleastuntilheatingtothesubsequentaging temperature.Itisimportanttoeliminateanyadditionalnucleationandgrowthduringthe briefquench,andtoisolatethenucleationstructure.Properstatisticsrequirethateach samplehaveastatisticallyidenticalmicrostructureatthebeginningofeachagingcycle. Finally,controlofthethermalcyclesbecameofincreasinglyimportancefortheshorter agingtimes. 105

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TEMsamplepreparationwaseectiveinthemechanicalsectioningandpolish techniques.Samplesweredesignedtobe2mm,therebyallowingremovalofanysurface layerofdepletedlithiumcontent.Experimentshaveshownthatwhenexposedtothe atmosphere,theworstdepletionextendslessthanca.0.5mm[42 ].Thisstudyrelied onapureargonenvironment,andstillremovedasmuchas0.75mmfromthesurfaces studiedtoensureuniformityofthechemistry.Mechanicalprocessingandtwin-jet polishingmethodsweredevelopedempiricallywithlimitedadvicefromtheliterature [71].MechanicalpolishingtechniquesrepeatablycreatedTEMdiskswithathickness between25-50micronsinpreparationforthefnaljetpolishing.Thejetpolishing procedureitselfbecamearepeatablemethodologywithexperience,yetitstillrequired thepreparationofmultipleTEMdiskstoobtainonethatwasproperlypolished,and exhibitedacceptablethinnessfortheTEM.Perforationofthediskwouldnotconsistently occurinthecenterofthesamplewiththeFischioneequipment.UsingtheStruersjet polishingequipmentresultedinmorepreciselycenteredperforations.Asaresult,the TEMdiskwasmoreuniformlythinned,creatingwiderviewableareas,whereasingleplane ofprecipitatescouldbeimaged.ThesuperiorityoftheStruerspolishingequipmentis duetothecontinuouselectrodemakingcontactwiththesample.TheFischioneunitused awireelectrodethatisnotcontinuousaboutthesample,thusprovidinganon-uniform currentdistribution.Inaddition,itwasnotedthattherowvelocityofthepolishingjet wascontrolledbetterwiththeStruersunit. 5.2EectivenessoftheImageProcessingFunctions Theimageprocessingfunctionshavebeenshownwithrespecttotheirfunctionand theabilitytocapturetheprecipitatesizesinSection 4.5.Severalassumptionsweremade inthecreationoftheimageprocessingfunctions. PrecipitateDiameterisaMaximum: Theimagecaptureseachsphereatits maximumcross-section.Transmissionelectronmicroscopyisaprojectionfromthe samplebulk;therefore,itwasassumedthattheimagescapturetheentirethree 106

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dimensionalprecipitateatitsmaximumsizeintwodimensions[65,72].Assuch, eachradiusofthetwodimensionalprecipitateprojectionistakentobetheparticle's representativemeaninthreedimensions. SphericalPrecipitates: FullAnalyze,theimageanalysisfunction,assumesthatall 0 precipitatesarespherical.Themajorityoftheprecipitateswereindeedspherical, anditwasdeterminedthattherewasnotasignifcantdierencetowarrant measuringtheaspectratios.Thisexperimentshowedthatonlyalloyswithvolume fractionsgreaterthan0.3hadprecipitatesthatsignifcantlydeviatedfromsphericity. Atthesehighvolumefractions,diusionlimitedcoarseningwasnottheonlygrowth mechanism.Particlesbegintocoalesce,andthemicrostructureevolutioncould nolongerbedescribedsolelyasacoarseningphenomenon.Observationofthe 0 precipitatesfromAl-3.16wt.%Lishowdeviationsfromsphericalwerenotcommon. The 0 phasewasthereforetreatedasconsistingofsphericalprecipitates. Contributionofcoalescence: Coalescedprecipitatesrepresentlessthanone percentofthetotalfor V V < 0.3andhavebeendeemedtohavelittle,ifany, eectonthegrowthkinetics.Consistentwiththeassumptionthatallparticlesare spherical,theanalysisfunctionswerenotusedtoevaluateanycoalescedparticles. Imagequality: Imagesforevaluationarecorrectlyfocused,exhibitcontrast dierencebetweenphases,andhavelowprecipitatedensity.ThefunctionfullAnalyze wasdevelopedforTEMimagesofhighqualitythathavesharpboundariesbetween thetwophases.Theimagesmusthaveauniformintensityacrossthephasesanda signifcantcontrastdierencebetweenthem.Precipitatesmustbewell-distributed throughtheimagewithminimaloverlap.Asetmagnifcationcannotberequired, asthelengthscalesoftheprecipitatechangesteadilywithagingtime.Requirements werethereforesetwithregardtothenumberofprecipitatesobservedperimage. Observationofaround30precipitatesperimagehavebeenshowntobeanoptimal samplingforprocessingin`fullAnalyze'. 107

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5.2.1OverlappinginProjection Particlesthatoverlapintheimageduetoprojection,asdiscussedinSection4.3.2, aredicultforthecomputertoresolve.Humanobservationcanoftendistinguishthat theyarenotcoalescedbyseeingthestrongerintensityofoneparticleintheimagedunion. Computershavedicultyinrecognizingthethedierencebetweentheparticleswhere theyappearasaunionontheimage.Evenworseisthesituationwheretheunionbetween theoverlappedparticlesisdark,dueperhapstothesubsequentscatteringoftheinitial rerectionasitpassesthroughthesecondprecipitate.Anoverlapofthisnatureisshownin thelowerrighthandsideofFigure 4-21A andisenlargedinFigure 5-1A.Therearealways afewoverlappedparticlesperimageevenwiththethinnestsamples.ThefullAnalyze programiseectiveinlocatingcentroidsoftheoverlappedparticles,providedthatthe centroidsareseparatedbyatleastoneparticleradius.Measurementofthediameters ismoredicult.Overlapswhichresultinauniformcontrastbetweentheparticleare measuredwithoutdiculty,however,overlapsresultinginadarkimageunioncause thediameterstoappearsmaller.Iterativelycomparingacirclewiththebinaryimage causedearlyterminationduetotheinclusionofthedarkregionofoverlap,andthereby registeredabiastowardsmallerparticles.Theunderestimationwassignifcantandcaused askewinthePSD.Achangeintheenhancementtechniquewasemployedtorepairthese gapsbetweentheparticles,creatingamoreaccurateimage.Aftertheimageisdigitally dilated,areassurroundedbyallwhitespaceareflled,andtheimageisdigitallyeroded toitsoriginalsizedistribution.Theparticlesizesarethereforenotmodifed,onlythe darkunionsareflled,allowingthecomputertoseparateandmeasurethemautomatically withouthumaninteraction.Theblurred,gray-scaleandbinaryimagesthatresultare showninFigures 5-1B and 5-1C.Observethattheenhancementoperationhasjoined thetwoparticlesandeliminatedthedarkregionattheimageunion.Thecentroidsfor eachofthe 0 precipitatesareshowninFigure 5-1D.Accuracyofmeasurementcanbe 108

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qualitativelyobservedbysuperimposingthemeasureddiametersontheoriginalimage, Figure5-1.Themeasureddiametersappearslightlylargerthantheactualimage. AOriginal BGrayscaleimage CBinaryimage DCentroids EMeasuredDiameters Figure5-1.Exampleoftheflloperationon 0 particleswhoseoverlapisdark(a)As ScannedandCropped(b)Blurred,grayscaleimage(c)BinaryImage(d) Centroidssuperimposedontheoriginalimage(e)Measureddiametersand centroidssuperimposedontheoriginalimage 5.2.2ImageQuality Thedigitaldilation,fll,anderosionoperationshaveproventoworkwell,eliminating thedarkareasbetweenparticlesandmoreaccuratelyseparatingandmeasuringthe diameters.AsfullAnalyzeiscreatedforhigh-qualityimages,therearetwoimagingerrors whichcanadverselyaecttheexperimentalresults:1)poorimagefocus,and2)high particledensityimages. 109

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Poorimagefocus: LackofaproperfocusintheTEMcausesanapparentshadow aroundtheprecipitates.Theshadowcreatesbothablurringoftheparticlesizeand adecreaseintheuniformintensityacrosstheprecipitate.Blurringoftheparticle borderisnotdesired,asitcancauseaparticletoappearlarger.Measurements becomerawedasparticlesfromtheout-of-focusimagesappearlarger,thereby distortingthePSD.Sharpcontrastdierenceisneededbetweenthephaseswith uniformintensityacrosseachrespectivephase.Improperfocuscreatesanintensity gradientacrossthephasesmakingenhancementdicult. Properfocusisafunctionoftheoperatorerror.Ifsucienttimeisnottakento ensureoptimumsamplerotation,focallength,andintensity,anunacceptableimage results.Sincethegoaloftheautomatedimageanalysisistocreateaninvariable criterion,theprocedureisnotalteredtoaccommodatesub-parimages.Theoperator hasthesoleresponsibilityinmaintainingallcapturedimagesattheirhighestquality. Unfortunately,theTEMusedalsoexhibitedproblemsatmagnifcationshigherthan 300kX.Slightvibrationsintheroomduringimagecaptureledtomovementofthe TEMcolumnandsubsequentpoorimaging.Amovingimageresultsinanegative thatappearstobedoubleexposed,andcannotbeanalyzed.Signifcantcarewas employmenttoimageduringhoursthatwerenothigh-tractimesandthereby limitedvibrationsintheTEMlaboratory. Highparticledensityimages: HighprecipitatedensityimagesoccurwhentheTEM sampleistoothickorthemagnifcationistoolow.Ifthesampleistoothick, projectionthroughthesamplewillcapturemorethanoneprecipitateplanegiving overlapandanapparenthigherdensity.Likewise,usingamagnifcationthatis toolowwillalsoresultinahighapparentdensity.Figure5-2A showsasignifcant amountofoverlapamongtheparticles,especiallyonthelowerright.Ahighdensity duetooverlapgivesanincorrectcross-sectionofthevolumeandmisrepresentsthe neighborhoodofparticles.Furthermore,theautomatedanalysisislimitedbyits 110

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abilitytoseparatethem.Thedigitalflloperation,discussedinSection 5.2.1,can notmanageaclosearrayofparticles.Upondigitaldilation,iftheneighborhood istooclose,itwillfllinlargeregionsastheparticlesarealltouching.Dueto suchalargeflledarea,thesubsequenterosioncannolongerdistinguishindividual particles.Largeareasoftheimagearelostinboththegrayscaleandbinaryimages, Figures 5-2B and 5-2C.Unfortunately,thisisabyproductofdigitallyfllingin thedarkregionswithinprecipitateswhichoverlapbyprojection.Agglomeration ofparticleswerenotconsideredintheanalysisand,therefore,didnotaectthe experiment.Thehighparticledensityimagesarearesultofthesamplethicknessin theseexperiments.BycreatingTEMsamplesthatarethinner,thereislessoverlap andtherewillbenolossofinformation,dueforexampletoagglomerationcausedby toomuchfllingduringthedilationandflling. AAsScannedandCropped BBlurred,Grayscaleimage CBinaryimage Figure5-2.Exampleofhighdensityimagesandtheirdeleteriouseectsontheimage analysisfunction BoththefocusandparticledensitycanbeaddressedbysamplepreparationandTEM operation.Asaresult,theimageenhancementfunctionwasnotmodifedtoaccommodate poorlyfocusedorhigh-densityimages.Theimportanceofacontinuouscriterionin theimageanalysisprocessdidnotwarrantmodifcationforerrorsthatcanberesolved throughpreparationandmicroscopytechniques. 111

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Ifhighqualityimagesareprovidedtothe`fullAnalyze'functionitwillgiveaccurate andrepeatableresults.Thegoalwasmetofcreatingafullyautonomousprogramcapable ofreadinginimages,enhancingthem,fndingthecentroidsofcircularprecipitates,and determiningtheirdiameter.Carewastakeninensuringthattheoriginalprecipitate sizewasnotalteredduringenhancement.Thefunction`thresholder',canflteroutboth backgroundnoiseandextraneousprecipitatesnotfullycontainedintheTEMsample.By makingtheimageanalysisprocessautomatic,notonlywasauniversalcriterioncreated butrapidimageanalysisresulted.Anentireagingtime,forexamplethetwentyfve imagesfromthe12hourage,Al-3.16wt%Licouldbeanalyzedwith1031particlesinless thanonehour.Thistypeofrapidprocessingwouldbeimpossibleinasemi-automatic typeprocessrequiringhumanintervention. 5.3AnalysisandComparison Experimentationislimitedtovolumefractionslessthan0.30foraluminumlithium alloys,ashighervaluesleadtoexcessiveprecipitatecoalescence.Itisalsoobservedthat volumefractionslowerthan0.10arediculttoachieve.Approachingthemetastable solvusresultsinpreferrednucleationof 0 alongdislocationscausinganon-homogenous distribution.Loweragingtemperaturesof180Cresultinahomogenousdistributionof 0 butexcessivelyslowthekineticssuchthatimpracticallylongagingtimesarerequired toattainthesteady-statecoarseningprocess.Resultsweredevelopedforthreebinary aluminum-lithiumalloysranginginvolumefractionof 0 from0.10to0.27. Experimentalresultsareselfconsistentamongagingtimesandalloys.Thereis uniformgrowthoftheprecipitatewithastablecoecientofvariation.Theparticlesize distributionshavebeennormalizedbymeandiameterandcanbecomparedacrossall lengthscales.Ithasbeenshownthattheyareself-similaracrossallagingtimesandhave thusbeencombinedasacumulativePSDforeachalloy.CumulativePSDsprovidea betterstatisticalrepresentationofthepopulationstatisticsastheyincludeaminimum of10,000precipitates.Comparisonwithscreeningtheoryandmultiparticlesimulations 112

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demonstrategoodcorrelationwiththedistributions,Figure 4-33.Largerpopulationsof thelargeparticlesareobservedbyexperimentandsimulationsthanispredictedfrom theory,although,theorycorrelateswellwiththeobservedmaximumnormalizedparticle, max ,Table 4-4.Thisdierencebetweentheoryandexperimentation,however,increases withvolumefraction.Itisbelievedthatthisdierencemaybecausedbylargervolume fractions,whichcausedeviationtothediusion-limitedgrowthphenomenainthealloy samples.Inaddition,screeningtheoryisbasedonmany-bodyphysicsthattakesonlythe leading-orderinteractiontermintoaccount.Oneshouldnotexpectthetheorytoremain accurateatlargervolumefractions,e.g. V V > 0:2.Overall,thereisreasonablecorrelation amongthenormalizedPSDspredictedfromscreeningtheory,simulationsandexperiment. Aspredicted,thebroaderdistributionsandhighersymmetryareclearlyevidentasa functionofthevolumefraction. Comparisonoftheparticlegrowthrateisaccomplishedbynon-dimensionalizingthe kineticfactorwiththatofLSW,inthelimitasthevolumefractionapproacheszero.Table 4-5 showsthat,again,thereisagreementbetweenscreeningtheoryandexperimental results.Therearenotmajordiscrepanciesobservedinthecomparisonbetweenthe experiment,theoryandsimulation. 113

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CHAPTER6 CONCLUSIONSANDFUTUREWORK Thephasecoarseningexperimentsreportedinthisthesisshowthatarepeatable methodologywasused.Characterizationofthemicrostructurewasaccuratelycompleted usingobjectiveandadaptivecomputer-basedimage-analysisfunctions.Precipitate distributionsandgrowthkineticsarerepresentativeofpriorstudiesofaluminum-lithium binaryalloysandcomparewellwithmoderndiusionscreeningtheoryandmultiparticle simulations.Severalspecifcconclusionsmaybedrawnfromthisexperiment: ExperimentalProcedure: Ahomogeneousdistributionofsphericalprecipitates wasfoundforalloysAl-2.1wt%Li,Al-2.45wt.%Li,andAl-3.16wt.%Li.The precipitateswereconfrmedtobethe 0 ,orAl 3 Liphase,byindexingofthe diractionpatternsinbothTEMandXRD.Centereddark-feldimagingintheTEM byuseoftheAl 3 Lisuperlatticeprovidedhigh-contrastimagesoftheprecipitates. Theseexperimentsprovidethefrstmodern,quantitativeTEManalysisofAl-Li alloys. ImageProcessing: Matlabwassuccessfullyusedincreatingfunctionsforentirely automatedimageanalysis.ImagestakenfromtheTEMwereenhancedwithout modifyingthesizesoftheprecipitates.Thecentroidsofeachprecipitatewerefound, evenforprecipitatesthatwereoverlappedinprojection.Usingthesecentroids, thediameterofeachprecipitateparticlewascalculated.Boththediametersand centroidsaresuperimposedontheoriginalimageshowingnearlyperfectcomparison. AllresultsfromtheMatlabfunctionaresavedwiththeoriginalimagefle.No humaninterventionisrequiredwhenoperatingthisanalysisfunction,whichis capableofanalyzingmorethan25imagesperhour. Characterization: Mean 0 precipitateparticlesizedoubledbyaging24to240 hoursat225Cforallalloys.Thedistributionofparticlesizesisbroadforallof thesamples.Aminimumsamplesizeof1000precipitatesisneededtocreatean 114

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acceptablysmoothdistribution.ThePSDsfromalloftheagingtimesappearsimilar insymmetryandmaximumparticlesizeforeachalloystudied. ComparisonwithTheoryandModeling: Experimentalresultscompare extremelywellwithpredictionsfromdiusionscreeningtheoryand,independently fromlarge-scalemultiparticlecomputersimulations.Themaximumnormalized particlesize, max ,takenfromeachalloy,correlateswellwiththepredictionsfrom diusionscreeningtheory.Theparticlesizedistribution(PSD)isselfsimilar,or ane,acrossallagingtimes,andexhibitsenhancedbroadeningwithincreasing volumefraction.Coarseningratesagreeapproximatelywith,butarelowerthan,that thosereportedinpriorstudies,butcloselyagreewiththefundamentalpredictions basedondiusionscreeningtheory.Weconcurthatdiusionscreeningtheory providesafundamentalphysicsdescriptionoflate-stagephasecoarseningat low-to-moderatevolumefraction. Specifcally,thepredictionsbasedondiusionscreeningtheoryandmultiparticle diusioncomputersimulationsweretestedovertherangeof V V =0.10-0.27,using carefullyselectedhigh-puritybinaryAl-Lialloys.Resultsshowthatthereisagreement throughthecomparisonofthePSD,maximumparticlesize,andthecoarsening rate.Atlow-to-moderatevolumefractionsoftheprecipitatephase(0
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andSAXSanalysis.Thesamplesnearlycoverthecompleteagingrange,allowingfor comparisonofthecoarseningrate.SANStechniquesarerequired,duetotherelatively largeprecipitatesize,whichinturnrequirescatteringwavenumbersof j ~ k j! 0.006 A )Tj /T1_4 7.97 Tf (1 whichareotherwiseunattainablewithconventionalbench-topSAXSequipment. Althoughthisstudyfoundagreementbetweentheory,simulation,andexperiment forbinaryaluminum-lithiumalloys,furthercharacterizationofthespecifcalloysshould employhot-stagetechniqueswithinaTEM.In-situgrowthdatafrommicrostructures wouldprovideadeeperunderstandingoftheinruencesofstochasticphenomenon andprovidebettertestsimulations.Stochasticinformation,suchaslocale`noise'in microstructuresmayprovideenhanceinputtonon-deterministictheoriesneededat moderate-to-largevolumefractions.UseoftheLocalElectrodeAtomProbe(LEAP) wouldaddinterestingandimportantatomic-levelinformationabouttheprecipitateand itsinruenceonlocalshiftsinmatrixconcentration.Reconstructionofthechemicalmap wouldindependentlyyieldthethree-dimensionalprecipitatepopulation,andalsoelucidate thesoluteruxfeld.Initialtestinghasshownencouragingresultswithalloyssimilarto thoseusedinthisstudy[73]. 116

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APPENDIXA DIFFRACTIONPATTERNINDEXING Toverifythattheprecipitatesimagedwerethecorrectphaseofinterest, 0 ,Al 3 Li, adiractionpatternalongthe[101]zonewasindexed.Itisfurthershownhowitcan beclearlydeterminedthatthisisthe 0 precipitatesbytherelativeintensitydierence betweentheindexedspots. A.1IntensityCalculations Theintensityofthediractedpatternisdeterminedfromthestructurefactor.which isthederivedfromthesumofthescatteringfactors.TheAl 3 Liphasecrystallizesinthe Pm 3mspacegroupandstructureofL1 2 .Itissimilarinarrangementtothefacecentered cubiccrystalstructure.Theprimitivevectorsare ~a 1 = a ^ x (A{1) ~a 2 = a ^ y (A{2) ~a 3 = a ^ z (A{3) Thebasisvectorsare ~ b 1 =0(Li) (A{4) ~ b 2 = 1 2 b ^ x + 1 2 b ^ y (Al )(A{5) ~ b 3 = 1 2 b ^ x + 1 2 b ^ z (Al )(A{6) ~ b 4 = 1 2 b ^ y + 1 2 b ^ z (Al )(A{7) Theprimitiveandbasisvectorsfullydescribetheunitcellandcanbeusedtocalculate thestructurefactor.Thestructurefactorisasumofthecontributionfromeachatom withintheunitcellandisshowninitsgeneralforminequation A{8.Hereh,k,andlare thelatticeplanes,ordiractionspotsinreciprocalspace,andf n istheatomicscattering amplitude.Theatomicscatteringamplitudesforaluminumandlithiumare8.862and 117

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1.693respectively[74].Thesevaluesareusedbythestructurefactortodescribethe rerectionsfromaunitcell, F hkl = X f n exp2 i(hx i + ky i + lz i ): (A{8) Byapplyingthebasisvectorsabovetothestructurefactorequation,ageneralformis obtainedfortheAl 3 Liprecipitate.Thisgeneralstructurefactorhasarelationshipwith intensityasI(hkl) / F(hkl) 2 .Applicationofthebasisvectorswithinthestructurefactor calculation,yields, F hkl = f li + f al exp( i(h + k ))+ f al exp( i(k + l ))f al exp( i(h + l )): (A{9) Therelativeintensitycanbeusedtousedtodeterminenotonlywhatispresentin thediractionpattern,butalsowhatismissingthroughdestructiveinterference[75]. Extinctionsallowadeterminationofstructurebywhatisnotseen.Forthisexample,an observationofthefrstfourfamiliesofplaneswillbeused:100,110,111,200.Insertionof thesehklvaluesintoEq. A{9 yieldsthefollowing: F 100 = f Li )Tj /T1_1 11.955 Tf 11.955 0 Td (f Al (A{10) F 110 = f Li )Tj /T1_1 11.955 Tf 11.955 0 Td (f Al (A{11) F 111 = f Li +3 f Al (A{12) F 200 = f Li +3 f Al (A{13) Itisnotedthatinapurecrystalstructureofaluminumthelatticeplanes100and110 wouldresultinanextinction.Inthe 0 structuretheywillexistduetothedierence betweenthealuminumandlithiumelectronicstructure.Thisisevidencedbyanintensity dierencewherethe111and200planesare15.5timesgreatertheintensityofthe100and 110planes.Fromthisanalysisonecanvisuallyobservethequalitativedierencebetween rerectionsfromthealuminummatrixandthe 0 phase,Figure A-1.Theexistenceof 0 canbefurthershownquantitativelybyindexingofthediractionpattern. 118

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FigureA-1.IndexeddiractionpatternfromtheAl-3.16wt%Li A.2IndexingtheDiractionPattern Adiractionpatternofthe 0 phasewastakenalongthe[101]zoneaxisasshownin FigureA-1.ThecameralengthoftheCM-12transmissionelectronmicroscopewasfrst calibratedwithasinglecrystalgoldstandard.Byusingthecalibratedcameralength,the wavelengthforanacceleratingvoltageof120kV,0 :03349 A,andthemeasureddiraction spotdistancefromthetransmittedbeamtheinterplanarspacingscanbecalculatedby Bragg'sLaw, 2d hkl sin = n: (A{14) Theinterplanarspacingscanbecalculatedexactlybyusingthelatticeparameterand theplaneofinterestby, 1 d 2 = h 2 + k 2 + l 2 a 2 : (A{15) AcomparisonbetweenthecalculatedandmeasuredinterplanarspacingsinTable A.2, showsthatindeedtheprecipitatesarethe 0 phaseconsistingofAl 3 Li.Correlationofthe 119

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intensitydierenceandtheindexofthediractionpattern,showninFigure 4-2,shows thattheAl 3 Liprecipitatesareclearlydistinguishedbythelowerintensity. TableA-1.Comparisonofinterplanarspacings,measuredversuscalculated,fortheAl-3.16 Lialloy hklActual( A)Measured( A) 1004.013.95 1102.832.81 2002.011.98 1112.312.29 120

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APPENDIXB IMAGEANALYSISMATLABSOURCE Thefollowingcontainstheentiresourcecodeusedintheimageanalysis.Thecode isgivenandexplainedinorderofbeingcalledbythemainalgorithm.AnycoreMatlab functionsarenotpresentedordescribed. B.1fullanalyze.m Thisisthemainfunctionwhichtheusercallswithaflecontainingalistofallthe folderscontainingimagestobeanalyzed.Thefunctionwilllookupeachflewithinthe foldertoanalyzeandcreateseparatedirectorieswheretheresultswillbesaved.Once afleisopened,separatefunctionswillbecalledtofurtherprocesstheimages.The functionthresholder.mprocessestheimage,particleStats.mcallsseparatefunctionsto boththecentroidsandparticlediameters.Eachofthesesubfunctionswillbediscussed individuallyingreaterdetail.Allofthestatisticsarereturnedtotheirrespectiveflesto beaccumulatedandprocessedforthecumulativeanalysis. function[imageSpread]=fullAnalyze(folderNames) %Determinethenumberoffoldersthatneedtobeanalyzed. [numFolders]=size(folderNames); %Begintheanalysisroutineforeachfolder forj=1:numFolders %Createthenameoftheimagesspreadsheetwithin %eachfolderbasedonsecondcolumnwithinfolderNames imageSpread=strcat(folderNames(j,1),'/',folderNames(j,2)); %ReadinthedatafromtheExcel(mustbeexcel)spreadsheet 121

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[trashimageNames]=xlsread(imageSpread{1}); %Createfoldersfordatastorageaftergrabbing %thecurrentfoldername folderName=char(folderNames(j,1)); mkdir(folderName,'/Diameters'); mkdir(folderName,'/Images'); mkdir(folderName,'/Images/Hough'); mkdir(folderName,'/Images/BW'); %Determinenumberofimagestoanalyze [numImagesjunk]=size(imageNames); %Beginimageanalysissub-routine fori=1:numImages %Createthenameoftheimagetobyanalyzed charName=strcat(folderName,'/', char(imageNames(i)),'.tif'); charFile=char(imageNames(i)) %Runthethresholderroutinetoadjusttheinput %imagetobringoutparticles %outputistwovariables,oneforhoughtransform %andtheotherisforbwanalysis [imageHoughimageBW]=thresholder(imread(charName)); 122

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%Imagehoughisamatrixofnumbersbetween0and1, %needtoconverttoagrayscaleimage imageHough=255*imageHough; %Passthetwoimagestotheparticlestats results=particleStats(imageHough,imageBW); A=results.D; %Writeouttheresultsforthediametersandimages dlmwrite([charFile'_dia.txt'],A,';'); movefile([charFile'_dia.txt'], [folderName,'/Diameters/'charFile'_dia.txt']) imwrite(imageHough,[folderName,'/Images/Hough/' charFile'_hough.png']); imwrite(imageBW,[folderName,'/Images/BW/' charFile'_BW.png']); %Ifuncommented,thissectionwillproducean %overlayoftheboundariesovertheoriginalimage %[B,L]=bwboundaries(imageBW,'noholes'); %figure;imshow(imread(charName));holdon; %plot(results.D(:,1),results.D(:,2),'r+'); 123

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%fork=1:length(B) %boundary=B{k}; %plot(boundary(:,2),boundary(:,1),'w','LineWidth',2) %end end end B.1.1thresholder.m Thefunctionthresholderreceivestheimagespassedfromfullanalyze.m.Itconverts theimagetograyscalethenenhancesthecontrastbyequalizationofthehistogramand eliminatesanynoisebydeletinganyareaswithlessthan10pixels.Theparticlesarethen dilatedbyfvestepsfollowedbyafllingoperationandasubsequenterosion.Anynoise internaltotheparticlesiseliminatedaswellasanyoverlappedparticleswhoseunionis dark.Theaforementionedstepsarenecessarytofacilitatemoreaccuratecentroidfndings viatheHoughtransformation.Thegrayscaleimageisthensmoothedtoeliminateany blockyedgesremainingfromtheerosion-dilationprocess.Twoimagesarereturnedto fullAnalyze.m:agrayscaleimageforfndingthecentroidsandabinaryimagetobeused forthemaskcomparison. function[threshedImage,filledBW]=thresholder(RGBimageData) %ConverttheRGBimagetograyscale,adjustthehistogram,and %thresholdto %blackandwhite grayScale=rgb2gray(RGBimageData); histAdjust=adapthisteq(grayScale); threshold=graythresh(histAdjust)*.5; 124

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bwthresh=im2bw(histAdjust,threshold); %Getridofnoise bwopened=bwareaopen(bwthresh,10); imageOpen=bwopened; %Createastrelfordilation se1=strel('disk',5); %Runthroughdilation,bwmorphtoobtainthedesiredeffects, %andthenfillinanyholes imageOpen=imdilate(imageOpen,se1); imageOpen=bwmorph(imageOpen,'thicken',10); imageOpen=bwmorph(imageOpen,'bridge'); imageOpen=imfill(imageOpen,8,'holes'); %Newstrelforerosion,valueof7accountsforthickenoperation se2=strel('disk',7); filledBW=imerode(imageOpen,se2); %Forhoughimage,needtoconvertbacktoadouble %thenblurtheedgestoaidthecentroidfinder grayThresh=im2double(filledBW) PSF=fspecial('disk',10); threshedImage=imfilter(grayThresh,PSF,'circular','conv'); 125

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B.1.2particleStats.m ThisfunctionreceivestwoimagesfromfullAnalyze.m:abinaryandagrayscale imageprocessedfromthefunctionthresholder.m.Thegrayscaleimageispassedto CircularHoughGrd.mwhichfndsthecentroidsofcircularparticles.Thesecentroidsand thebinaryimagearepassedtotheseparatefunctiondiaMeasure.m,whichdetermines diametersoftheparticles. function[Statresults]=particleStats(particleImage,imagebw) %Definedivisionsforstereologicalmeasurements pointMethodDivisions=20; %Findthecentroids,thenreturnthem.Constantsdetermined %fromtrialanderror. %Thenpasstomeasurementfunctiontodetermineparticlediameters. [junkcentroids]= CircularHough_Grd(particleImage,[10,120],15,35); centerDia=diaMeasure(centroids,imagebw); %Usetheoriginalimagetocalculatetheexactvolumefractionof %particlesforthegivenarea. [x,y]=size(imagebw); 126

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counter=0; totalPoints=0; fori=1:x forj=1:y ifimagebw(i,j)>0 counter=counter+1; end totalPoints=totalPoints+1; end end volumeFrac=counter/(x*y); %Stereology,runthepointmethodandlinealmethods pointResult=pointMethod(imagebw,pointMethodDivisions); linealResult=linealMethod(imagebw,pointMethodDivisions); %Savetheresultsintoanobjectandsenditbacktothemainprogram Statresults.Point=pointResult; Statresults.Lineal=linealResult; Statresults.D=centerDia; Statresults.volFrac=volumeFrac; 127

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B.1.2.1CircularHoughGrd.m ThisfunctionwasfoundontheMatlabCentralFileExchange,donatedbyauthorTao Peng[66].Theprogramreliesonthegradientfeldsthroughoutthegrayscaleimagetofnd thecentroidsofacircularparticle.Theuserhascontrolovertheminimumandmaximum sizeoftheparticlestobefoundaswellastheaspectofratiosensitivity.Thefunctionwas usedasreceivedandonlyoptimizedforthisanalysiswithrespecttoparticleminimum andmaximum,aspectratioandthegradientfeldtobeconsidered.Thesevariablesare givenasuserdependent.TheoutputreturnedtoparticleStats.marethecentroidsgivenin cartesiancoordinateswithrespecttotheimage. function[accum,varargout]=CircularHough_Grd(img,radiusrange,varargin) %Detectcircleswithvariousradiiinagrayscaleimage % %[accum,circen,dbg_LMmask]=CircularHough_Grd( %img,radiusrange,grdthres,fltr4LM_R,fltr4accum) %CircularHoughtransformbasedonthegradientfieldofanimage. %NOTE:Operatesongrayscaleimages,NOTB/Wbitmaps. %NOloopsintheimplementationofCircularHoughtransform, %whichmeansfasteroperationbutatthesametimelarger %memoryconsumption. % %INPUT:(img,radiusrange,grdthres,fltr4LM_R,fltr4accum) %img:A2-Dgrayscaleimage(NOB/Wbitmap) %radiusrange:Thepossibleminimumandmaximumradiusofthecircles %tobesearched,intheformatof %[minimum_radius,maximum_radius](unit:pixels) %grdthres:(Optional,defaultis10,mustbenon-negative) %Thealgorithmisbasedonthegradientfieldofthe 128

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%inputimage.Athresholdingonthegradientmagnitude %isperformedbeforethevotingprocessoftheCircular %Houghtransformtoremovethe'uniformintensity' %(sort-of)imagebackgroundfromthevotingprocess. %inotherwords,pixelswithgradientmagnitudessmaller %than'grdthres'areconsiderednotbelongtoanycircle. %fltr4LM_R:(Optional,defaultis8,minimumis3) %Theradiusofthefilterusedinthesearchoflocal %maximaintheaccumulationarray.Todetectcircleswhose %shapesarelessperfect,theradiusofthefilterneeds %tobesetlarger. %fltr4accum:(Optional.Adefaultfilterwillbeusedifnotgiven) %Filterusedtosmooththeaccumulationarray.Depending %ontheimageandtheparametersettings,theaccumulation %arraybuilthasdifferentnoiselevelandnoisepattern %(e.g.noisefrequencies).Thefiltershouldbesettoan %appropriatelysizesuchthatit'sabletosuppressthe %dominantnoisefrequency. % %OUTPUT:[accum,circen,dbg_LMmask] %accum:TheresultaccumulationarrayfromtheCircularHough %transform.Theaccumulationarrayhasthesamedimension %astheinputimage. %circen:(Optional) %Centerpositionsofthecirclesdetected.IsaN-by-2 %matrixwitheachrowcontainsthe(x,y)positions %ofacircle. 129

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%dbg_LMmask:(Optional,fordebugpurpose) %Maskfromthesearchoflocalmaximaintheaccumulation %array % %EXAMPLE#0: %rawimg=imread('TestImg_CHT_a2.bmp'); %tic;[accum,circen]=CircularHough_Grd(rawimg,[1560]);toc; %figure(1);imagesc(accum);axisimage; %title('AccumulationArrayfromCircularHoughTransform'); %figure(2);imagesc(rawimg);colormap('gray');axisimage; %holdon;plot(circen(:,1),circen(:,2),'r+');holdoff; %title('RawImagewithCirclesDetected(centerpositionsmarked)'); %figure(3);surf(accum,'EdgeColor','none');axisij; %title('3-DViewoftheAccumulationArray'); % %COMMENTSONEXAMPLE#0: %Kindofaneasycasetohandle.Todetectcirclesintheimagewhose %radiirangefrom15to60.Defaultvaluesforarguments'fltr4LM_R' %and'fltr4accum'areused. % %EXAMPLE#1: %rawimg=imread('TestImg_CHT_a3.bmp'); %tic;[accum,circen]=CircularHough_Grd(rawimg,[1560],8,20);toc; %figure(1);imagesc(accum);axisimage; %title('AccumulationArrayfromCircularHoughTransform'); %figure(2);imagesc(rawimg);colormap('gray');axisimage; %holdon;plot(circen(:,1),circen(:,2),'r+');holdoff; 130

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%title('RawImagewithCirclesDetected(centerpositionsmarked)'); %figure(3);surf(accum,'EdgeColor','none');axisij; %title('3-DViewoftheAccumulationArray'); % %COMMENTSONEXAMPLE#1: %Theshapesintherawimagearenotverygoodcircles.Asaresult, %theprofileofthepeaksintheaccumulationarrayarekindof %'stumpy',whichcanbeseenclearlyfromthe3-Dviewofthe %accumulationarray.(Asacomparison,pleaseseethesharppeaksin %theaccumulationarrayinexample#0)Toextractthepeakpositions %nicely,avalueof20(defaultis8)isusedforargument'fltr4LM_R', %whichistheradiusofthefilterusedinthesearchofpeaks. %Thedetectedcircleat(55.5,103.3)isduetothearcshapeatthe %junctionoftwowhitediscsabove. % %EXAMPLE#2: %rawimg=imread('TestImg_CHT_b3.bmp'); %fltr4img=[11111;12221;12421;12221;11111]; %fltr4img=fltr4img/sum(fltr4img(:)); %imgfltrd=filter2(fltr4img,rawimg); %tic;[accum,circen]=CircularHough_Grd(imgfltrd,[1580],8,10);toc; %figure(1);imagesc(accum);axisimage; %title('AccumulationArrayfromCircularHoughTransform'); %figure(2);imagesc(rawimg);colormap('gray');axisimage; %holdon;plot(circen(:,1),circen(:,2),'r+');holdoff; %title('RawImagewithCirclesDetected(centerpositionsmarked)'); % 131

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%COMMENTSONEXAMPLE#2: %Thecirclesintherawimagehavesmallscaleirregularitiesalong %theedges,whichcouldleadtoanaccumulationarraythatisbadfor %localmaximadetection.A5-by-5filterisusedtosmoothoutthe %smallscaleirregularities.Ablurredimageisactuallygoodforthe %algorithmimplementedherewhichisbasedontheimage'sgradient %field. % %EXAMPLE#3: %rawimg=imread('TestImg_CHT_c3.bmp'); %fltr4img=[11111;12221;12421;12221;11111]; %fltr4img=fltr4img/sum(fltr4img(:)); %imgfltrd=filter2(fltr4img,rawimg); %tic;[accum,circen]=CircularHough_Grd(imgfltrd,[15105],8,10); %toc; %figure(1);imagesc(accum);axisimage; %figure(2);imagesc(rawimg);colormap('gray');axisimage; %holdon;plot(circen(:,1),circen(:,2),'r+');holdoff; %title('RawImagewithCirclesDetected(centerpositionsmarked)'); % %COMMENTSONEXAMPLE#3: %Similartoexample#2,afilteringbeforecircledetectionworksfor %noisyimagetoo. % %BUGREPORT: %Thisisanalphaversion.Pleasesendyourbugreports,commentsand %suggestionstopengtao@glue.umd.edu.Thanks. 132

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%Estimationoftheradiiofdetectedcirclesisplannedforthenext %version. % % %INTERNALPARAMETERS: %TheINPUTargumentsareonlypartoftheparametersthatareusedby %thecircledetectionalgorithmimplementedhere.Variablesinthecode %withaprefix'prm_'inthenamearetheparametersthatcontrolthe %judgingcriteriaandthebehaviorofthealgorithm.Defaultvaluesfor %theseparameterscanhardlyworkforallcircumstances.Therefore,at %occasions,thevaluesoftheseINTERNALPARAMETERS(parametersthat %areNOTexposedasinputarguments)needtobefinetunedtomake %thecircledetectionworkasexpected. %Thefollowingexampleshowshowchanginganinternalparametercould %influencethedetectionresult. %1.Changethevalueoftheinternalparameter'prm_LM_LoBndRa'to0.4 %(defaultis0.2) %2.Runthefollowingmatlabcode: %fltr4accum=[121;262;121]; %fltr4accum=fltr4accum/sum(fltr4accum(:)); %rawimg=imread('Frame_0_0022_portion.jpg'); %tic; %[accum,circen]=CircularHough_Grd(rawimg,... %[414],10,4,fltr4accum); %toc; %figure(1);imagesc(accum);axisimage; %title('AccumulationArrayfromCircularHoughTransform'); 133

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%figure(2);imagesc(rawimg);colormap('gray');axisimage; %holdon;plot(circen(:,1),circen(:,2),'r+');holdoff; %title('RawImagewithCirclesDetected(centerpositionsmarked)'); %3.Seehowdifferentvaluesoftheparameter'prm_LM_LoBndRa'could %influencetheresult. %Author:TaoPeng %DepartmentofMechanicalEngineering %UniversityofMaryland,CollegePark,Maryland20742,USA %pengtao@glue.umd.edu %Version:alphaRevision:Dec.04,2005 %%%%%%%%Argumentsandparameters%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Validationofarguments ifndims(img)~=2||~isnumeric(img), error('CircularHough_Grd:''img''hastobe2dimensional'); end if~all(size(img)>=32), error('CircularHough_Grd:''img''hastobelargerthan32-by-32'); end ifnumel(radiusrange)~=2||~isnumeric(radiusrange), error(['CircularHough_Grd:''radiusrange''hastobe',... 'atwo-elementvector']); end 134

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prm_r_range=sort(max([0,0;radiusrange(1),radiusrange(2)])); %Parameters(defaultvalues) prm_grdthres=10; prm_fltrLM_R=8; func_compu_cen=true; %Validationofarguments ifnargin>2, ifisnumeric(varargin{1})&&varargin{1}(1)>=0, prm_grdthres=varargin{1}(1); else error(['CircularHough_Grd:''grdthres''hastobe',... 'anon-negativenumber']); end end vap_fltr4accum=3;%filterforsmoothingtheaccumulationarray ifnargin>(1+vap_fltr4accum), ifisnumeric(varargin{vap_fltr4accum})&&... ndims(varargin{vap_fltr4accum})==2&&... all(size(varargin{vap_fltr4accum})>=3), fltr4accum=varargin{vap_fltr4accum}; else error(['CircularHough_Grd:''fltr4accum''hastobe',... 'a2-Dmatrixwithaminimumsizeof3-by-3']); end 135

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else %Defaultfilter(5-by-5) fltr4accum=ones(5,5); fltr4accum(2:4,2:4)=2; fltr4accum(3,3)=6; end vap_fltr4LM=2;%filterforthesearchoflocalmaxima ifnargin>(1+vap_fltr4LM), ifisnumeric(varargin{vap_fltr4LM})&& varargin{vap_fltr4LM}(1)>=3, prm_fltrLM_R=varargin{vap_fltr4LM}(1); else error(['CircularHough_Grd:''fltr4LM_R''hastobe',... 'largerthanorequalto3']); end end func_compu_cen=(nargout>1); %Reservedparameters dbg_on=false;%debuginformation dbg_bfigno=4; ifnargout>2,dbg_on=true;end %%%%%%%%Buildingaccumulationarray%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 136

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%Converttheimagetosingleifitisnotof %classfloat(singleordouble) img_is_double=isa(img,'double'); if~(img_is_double||isa(img,'single')), imgf=single(img); end %Computethegradientandthemagnitudeofgradient ifimg_is_double, [grdx,grdy]=gradient(img); else [grdx,grdy]=gradient(imgf); end grdmag=sqrt(grdx.^2+grdy.^2); %Getthelinearindices,aswellasthesubscripts,ofthepixels %whosegradientmagnitudesarelargerthanthegiventhreshold grdmasklin=find(grdmag>prm_grdthres); [grdmask_IdxI,grdmask_IdxJ]=ind2sub(size(grdmag),grdmasklin); %Computethelinearindices(aswellasthesubscripts)of %allthevotingstotheaccumulationarray. %TheMatlabfunction'accumarray'acceptsonlydoublevariable, %soallindicesareforcedintodoubleatthispoint. %Arowinmatrix'lin2accum_aJ'containstheJindices(intothe %accumulationarray)ofallthevotingsthatareintroducedbya 137

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%samepixelintheimage.Similarlywithmatrix'lin2accum_aI'. rr_4linaccum=double(prm_r_range); linaccum_dr=[(-rr_4linaccum(2)+0.5):-rr_4linaccum(1),... (rr_4linaccum(1)+0.5):rr_4linaccum(2)]; lin2accum_aJ=floor(... double(grdx(grdmasklin)./grdmag(grdmasklin))*linaccum_dr+... repmat(double(grdmask_IdxJ)+0.5,[1,length(linaccum_dr)])... ); lin2accum_aI=floor(... double(grdy(grdmasklin)./grdmag(grdmasklin))*linaccum_dr+... repmat(double(grdmask_IdxI)+0.5,[1,length(linaccum_dr)])... ); %Clipthevotingsthatareoutoftheaccumulationarray mask_valid_aJaI=... lin2accum_aJ>0&lin2accum_aJ<(size(grdmag,2)+1)&... lin2accum_aI>0&lin2accum_aI<(size(grdmag,1)+1); mask_valid_aJaI_reverse=~mask_valid_aJaI; lin2accum_aJ=lin2accum_aJ.*mask_valid_aJaI+ mask_valid_aJaI_reverse; lin2accum_aI=lin2accum_aI.*mask_valid_aJaI+ mask_valid_aJaI_reverse; clearmask_valid_aJaI_reverse; %Linearindices(ofthevotings)intotheaccumulationarray 138

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lin2accum=sub2ind(size(grdmag),lin2accum_aI,lin2accum_aJ); lin2accum_size=size(lin2accum); lin2accum=reshape(lin2accum,[numel(lin2accum),1]); clearlin2accum_aIlin2accum_aJ; %Weightsofthevotings,currentlyusingthegradientmaginitudes %butinfactanyschemecanbeused(applicationdependent) weight4accum=... repmat(double(grdmag(grdmasklin)),[lin2accum_size(2),1]).*... mask_valid_aJaI(:); clearmask_valid_aJaI; %BuildtheaccumulationarrayusingMatlabfunction'accumarray' accum=accumarray(lin2accum,weight4accum); accum=[accum;zeros(numel(grdmag)-numel(accum),1)]; accum=reshape(accum,size(grdmag)); %%%%%%%%Locatinglocalmaximaintheaccumulationarray%%%%%%%%%%%% %Stopifnoneedtolocatethecenterpositionsofcircles if~func_compu_cen, return; end clearlin2accumweight4accum; 139

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%Parameterstolocatethelocalmaximaintheaccumulationarray %--Segmentationof'accum'beforelocatingLM prm_useaoi=true; prm_aoithres_s=2; prm_aoiminsize=floor(min([min(size(accum))*0.25,... radiusrange(2)*1.5])); %--Filterforsearchingforlocalmaxima prm_fltrLM_s=1.35; prm_fltrLM_r=ceil(prm_fltrLM_R*0.6); prm_fltrLM_npix=max([6,ceil((prm_fltrLM_R/2)^1.8)]); %--Lowerboundoftheintensityoflocalmaxima prm_LM_LoBndRa=0.2;%minimumratioofLMtothemaxof'accum' %Smooththeaccumulationarray fltr4accum=fltr4accum/sum(fltr4accum(:)); accum=filter2(fltr4accum,accum); %SelectanumberofAreas-Of-Interestfromtheaccumulationarray ifprm_useaoi, %Thresholdvaluefor'accum' prm_llm_thres1=prm_grdthres*prm_aoithres_s; %Thresholdingovertheaccumulationarray accummask=(accum>prm_llm_thres1); 140

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%Segmentationoverthemask [accumlabel,accum_nRgn]=bwlabel(accummask,8); %SelectAOIsfromsegmentedregions accumAOI=ones(0,4); fork=1:accum_nRgn, accumrgn_lin=find(accumlabel==k); [accumrgn_IdxI,accumrgn_IdxJ]=... ind2sub(size(accumlabel),accumrgn_lin); rgn_top=min(accumrgn_IdxI); rgn_bottom=max(accumrgn_IdxI); rgn_left=min(accumrgn_IdxJ); rgn_right=max(accumrgn_IdxJ); %TheAOIsselectedmustsatisfyaminimumsize if((rgn_right-rgn_left+1)>=prm_aoiminsize&&... (rgn_bottom-rgn_top+1)>=prm_aoiminsize), accumAOI=[accumAOI;... rgn_top,rgn_bottom,rgn_left,rgn_right]; end end else %WholeaccumulationarrayastheoneAOI accumAOI=[1,size(accum,1),1,size(accum,2)]; end %Thresholdingof'accum'byalowerbound prm_LM_LoBnd=max(accum(:))*prm_LM_LoBndRa; 141

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%Buildthefilterforsearchingforlocalmaxima fltr4LM=zeros(2*prm_fltrLM_R+1); [mesh4fLM_x,mesh4fLM_y]=meshgrid(-prm_fltrLM_R:prm_fltrLM_R); mesh4fLM_r=sqrt(mesh4fLM_x.^2+mesh4fLM_y.^2); fltr4LM_mask=... (mesh4fLM_r>prm_fltrLM_r&mesh4fLM_r<=prm_fltrLM_R); fltr4LM=fltr4LM-... fltr4LM_mask*(prm_fltrLM_s/sum(fltr4LM_mask(:))); ifprm_fltrLM_R>=4, fltr4LM_mask=(mesh4fLM_r<(prm_fltrLM_r-1)); else fltr4LM_mask=(mesh4fLM_r
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aoi=accumAOI(k,:);%justforreferencingconvenience %Thresholdingof'accum'byalowerbound accumaoi_LBMask=... (accum(aoi(1):aoi(2),aoi(3):aoi(4))>prm_LM_LoBnd); %Applythelocalmaximafilter candLM=conv2(accum(aoi(1):aoi(2),aoi(3):aoi(4)),... fltr4LM,'same'); candLM_mask=(candLM>0); %Clearthemarginsof'candLM_mask' candLM_mask([1:prm_fltrLM_R,(end-prm_fltrLM_R+1):end],:)=0; candLM_mask(:,[1:prm_fltrLM_R,(end-prm_fltrLM_R+1):end])=0; %****Debugcode(begin) ifdbg_on, dbg_LMmask(aoi(1):aoi(2),aoi(3):aoi(4))=... dbg_LMmask(aoi(1):aoi(2),aoi(3):aoi(4))+... accumaoi_LBMask+2*candLM_mask; end %****Debugcode(end) %Groupthelocalmaximacandidatesbyadjacency,computethe %centroidpositionforeachgroupandtakethatasthecenter %ofonecircledetected [candLM_label,candLM_nRgn]=bwlabel(candLM_mask,8); 143

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forilabel=1:candLM_nRgn, %Indices(tocurrentAOI)ofthepixelsinthegroup candgrp_masklin=find(candLM_label==ilabel); [candgrp_IdxI,candgrp_IdxJ]=... ind2sub(size(candLM_label),candgrp_masklin); %Indices(to'accum')ofthepixelsinthegroup candgrp_IdxI=candgrp_IdxI+(aoi(1)-1); candgrp_IdxJ=candgrp_IdxJ+(aoi(3)-1); candgrp_idx2acm=... sub2ind(size(accum),candgrp_IdxI,candgrp_IdxJ); %Minimumnumberofqulifiedpixelsinthegroup ifsum(accumaoi_LBMask(candgrp_masklin))
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%****Debugcode(begin) ifdbg_on, figure(dbg_bfigno);imagesc(dbg_LMmask);axisimage; ifsize(accumAOI,1)==1, figure(dbg_bfigno+1); surf(candLM,'EdgeColor','none');axisij; end end %****Debugcode(end) %Output varargout{1}=circen; ifnargout>2, varargout{2}=dbg_LMmask; end B.1.2.2diaMeasure.m ThefunctionparticleStats.mpassesthematrixcontainingthecentroidsfoundby thehoughtransformandthebinaryimagecreatedbythresholder.DiaMeasurethensets upamaskfromastructureelementandcreatesacomparisonbetweenthebinaryimage andmask.Byincrementingtheradiusbyonepixel,aparticlediameterisdetermineda matchwhen2percentofthebackgroundisincludedwithinthemask.Thediameteris determinedbyusingthearearatherthanthediscretepixelmeasurement.Thediameters arereturnedtothefunctionparticleStats.m function[centroids]=diaMeasure(centroids,imagebw) 145

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%Determinenumberofparticlesintheimages [numParticlesjunkY]=size(centroids); %Settheboundssothattherearenotoutofboundserrors [boundiboundj]=size(imagebw); %Begintheroutinetomeasuretheparticlesizes fora=1:numParticles %Looptoexpandthetestcircle ford=10:1:1000 %Formthetestcircleandthensetthecentroidoftheparticle testCircle=imcircle(d); centeri=centroids(a,2); centerj=centroids(a,1); %Ifparticlecentroidiszero,ignoreit.Theseareproduced %fromparticlecoalescence %andotherradiithatdonotcorrespondtoactualparticles ifimagebw(round(centeri),round(centerj))==0 break; end %Setthestartingpositionforcomparisontothetestcircle. startj=round(centerj-d/2-1); 146

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starti=round(centeri-d/2-1); %Settheblackcounter black=0; %Routinetotraversetheareaofparticleimage correspondingtothetestcircle. %Movesintheydirection. fork=1:d curi=starti+k; %Movesinthexdirection forl=1:d curj=startj+l; %Ifstatementstodetermineifthecurrentpixel %intheparticleimageiswhiteandtomakesure %thatitisnotoutofbounds.Thisoccurs %forcentroidsneartheedge.Ifthepixel %isblack,accumulateontheblackcounter. if((curj<1|curi<1)) continue; elseif((curi>boundi)|(curj>boundj)) continue; elseif(testCircle(k,l)<1) continue; 147

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elseif(imagebw(curi,curj).02*bwarea(testCircle) centroids(a,3)=sqrt(bwarea(testCircle)/3.14159)*2; break; end end end B.2postProcess.m Uponanalysisofeachoftheindividualimages,allofthedataisstoredindependently foreachrespectivefle.Thematricesarecombinedaccordingtoagetime.Thecombined textfleisthenconvertedtoexcelformatthatisimportedinMatlab.ThepostProcess.m functionappliesacalibrationforthemicroscopeandconvertsthedatatakenaspixelsto theactuallengthscale.Thecumulativedataisthenprocessedandthefunctionreturns 148

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thenormalizedhistogramforeachagingtime,themeanradius,theinterfacialareadivided bythevolumefractionandotherstatisticaldatatobeusedforcomparison. functiondia=postProcess(data,mag,AgeTime) %ValueofmagnificationiskX [ij]=size(data); %Calibrationdataobtainedusing90nmlatexparticles. %TheseconstantsneedtobedeterminedforeachTEM. calibrate=(.853025*mag+18.4703)/90 diaRow=0; %Thisroutinegetsridofanydatapoints %wherethedataiszerofortheparticlediameter. %Thisisjustifiedbecauseanyzerostartsout %wherethecenteroidpointstoablackpixel. fork=1:i ifdata(k,3)>0 diaRow=diaRow+1; dia(diaRow,1)=data(k,3); end 149

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end %Normalizethediameterstothemean, %thencalculuatethehistogram normDia=dia/(mean(dia)); [n,xout]=hist(normDia,20); %Normalizethefrequencytothemaximum. C=n/max(n); %Calculatethephysicalradiibasedoff %magnificationcalibration R=(dia)/(2*calibrate); %Calculatetheaverager^3value R3=mean(R)^3 %CalculateVariousparameters inR=3./R; Sv=mean(inR) Dbar=mean(R.*2) Dmed=median(R.*2) s=std(R.*2) cv=s/Dbar A(:,1)=C; A(:,2)=xout; 150

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B(1,1)=R3; B(1,2)=Sv; B(1,3)=Dbar; B(1,4)=Dmed; B(1,5)=s; B(1,6)=cv; %Savethevariouscalculationstothespecifiedfiles. dlmwrite([AgeTime'Hour_normdia.txt'],normDia,';'); dlmwrite([AgeTime'Hour_hist.txt'],A,';'); dlmwrite([AgeTime'Hour_stats.txt'],B,';'); 151

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APPENDIXC CALIBRATIONOFMICROSCOPE ThePhillipsCM-12,JEOL200CX,andtheJEOL2010Ftransmissionelectron microscopeswerecalibratedusing90nmlatexspheresonacarbongrid.Imageswere takenatarangeofmagnifcations.Theimageprocessingprogramusedforanalysisofthe aluminum-lithiumstructurewasused.Thisallowedmultiplelatexspherestobemeasured ateachmagnifcationandmeandiameterusedforthecalculation. ForthePhillipsCM-12,theflmswerescannedinat300dpi,usingthesameprocess describedin3.4.Acorrelationwasthendevelopedbetweentheknowndiameterof90nm andthenumberofpixels.Thisisalinearcorrelationforthemicroscopeandthuscan beextendedforalllevelsofmagnifcation.Thepixeldiameterswereplottedversusthe magnifcationandlinearregressionwasusedtofndauniversalequationforcalibration, FigureC-1. FigureC-1.CalibrationoftheCM-12microscopeusingstandard,90nmlatexspheres Theresultingequationisgivenby, 152

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f = 0:853 mag +18:47 90nm (C{1) ThiscalibrationequationwasthenusedbythefunctionpostProcess.mtochangethe measuredpixelsintostandardunits. EquationC{1representstheweightedpixeldimensionsforallmagnifcationlevels. Moreover,thiscalibrationwasusedtodetermineallexperimentallengthscales. FortheJEOL200CX,imagesareacquireddigitallyinthetaggedimageformat (.tif),usingaGatandigitalcamera.Asdiscussed,thereisalinearrelationshipbetween magnifcationandthestandardsamplesizeuniquetothemicroscope,Figure C-2.The FigureC-2.CalibrationoftheJEOL200CXmicroscopeusingstandard,90nmlatex spheres resultingequationisgivenas: f = 0:675 mag +10:819 90nm (C{2) 153

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ThiscalibrationequationfortheJEOL200CXwasusedinthefunctionpostProcess.m forallimagestakenwiththismicroscope,therebyyieldingtherealdimensionsofthe precipitates. FortheJEOL2010F,imagesareacquireddigitallyinthetaggedimageformat (.tif),usingaGatandigitalcamera.Asdiscussed,thereisalinearrelationshipbetween magnifcationandthestandardsamplesizeuniquetothemicroscope,Figure C-3.The FigureC-3.CalibrationoftheJEOL2010Fmicroscopeusingstandard,90nmlatex spheres resultingequationisgivenas: f = 3:88 mag +31:6 90nm (C{3) ThiscalibrationequationfortheJEOL2010FwasusedinthefunctionpostProcess.m forallimagestakenwiththismicroscope,therebyyieldingtherealdimensionsofthe precipitates. 154

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BIOGRAPHICALSKETCH BenPletcherreceivedabachelorsdegreeinWeldingEngineeringfromLeTourneau Universityin2002.HeworkedfortwoyearswithChicagoBridgeandIron(CBI)asa weldingengineer.AssignedtotheregionofCentralandSouthAmerica,heworkedin theislandnationofTrinidadandTobago.Whilethere,hemetJudithRodriguez,anurse fromCBI'sVenezueladivision.WorkrelocationbroughtBentoJamaicawheretheywere subsequentlymarried.Pursuinghisdesireforagreaterunderstandingofmetallurgy,he enrolledatRensselaerPolytechnicInstitute.GraduatingwithaMasterofSciencedegree in2006,hefollowedhisadvisortotheUniversityofFlorida.UponcompletionofhisPh.D inthesummerof2009hewillrejoinCBIasthestametallurgist. 160