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Numerical Modeling of the Plasma-Particle Interactions of Aerosol Vaporization in a Laser-Induced Plasma

Permanent Link: http://ufdc.ufl.edu/UFE0043727/00001

Material Information

Title: Numerical Modeling of the Plasma-Particle Interactions of Aerosol Vaporization in a Laser-Induced Plasma
Physical Description: 1 online resource (167 p.)
Language: english
Creator: Jackson, Philip B
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: laser -- model -- plasma
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Laser-Induced Breakdown Spectroscopy (LIBS) is a powerful and well-established atomic emission diagnostic for the identification and analysis of unknown samples. Recent research efforts have shown that LIBS is useful for both qualitative identification and for the quantitative measurement of relative as well as absolute analyte concentration regardless of analyte state. More recently, much interest has been directed toward the use of LIBS in the analysis of aerosol systems, including those generated by laser ablation (LA-LIBS). While LIBS offers many advantages as a diagnostic tool, there are several difficulties that limit its capability and robustness. Chief among these are matrix effects and incomplete or inhomogeneous sample vaporization. In an effort to fully understand, and eventually mitigate, these difficulties, the current work seeks to design and implement a numerical model that describes the complex plasma-particle interactions that govern the LIBS of aerosol systems. The model incorporates the processes of heat transfer, hydrodynamics, mass diffusion, vaporization, and electromagnetism. The model considers the fundamental physics of three distinct regimes: the global plasma environment, the local particle behavior, and the initial nature of plasma inception.
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 Philip B Jackson.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Hahn, David W.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043727:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043727/00001

Material Information

Title: Numerical Modeling of the Plasma-Particle Interactions of Aerosol Vaporization in a Laser-Induced Plasma
Physical Description: 1 online resource (167 p.)
Language: english
Creator: Jackson, Philip B
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: laser -- model -- plasma
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Laser-Induced Breakdown Spectroscopy (LIBS) is a powerful and well-established atomic emission diagnostic for the identification and analysis of unknown samples. Recent research efforts have shown that LIBS is useful for both qualitative identification and for the quantitative measurement of relative as well as absolute analyte concentration regardless of analyte state. More recently, much interest has been directed toward the use of LIBS in the analysis of aerosol systems, including those generated by laser ablation (LA-LIBS). While LIBS offers many advantages as a diagnostic tool, there are several difficulties that limit its capability and robustness. Chief among these are matrix effects and incomplete or inhomogeneous sample vaporization. In an effort to fully understand, and eventually mitigate, these difficulties, the current work seeks to design and implement a numerical model that describes the complex plasma-particle interactions that govern the LIBS of aerosol systems. The model incorporates the processes of heat transfer, hydrodynamics, mass diffusion, vaporization, and electromagnetism. The model considers the fundamental physics of three distinct regimes: the global plasma environment, the local particle behavior, and the initial nature of plasma inception.
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 Philip B Jackson.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Hahn, David W.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043727:00001


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NUMERICALMODELINGOFTHEPLASMA-PARTICLEINTERACTIONSOFAEROSOLVAPORIZATIONINALASER-INDUCEDPLASMAByPHILIPB.JACKSONADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011PhilipB.Jackson 2

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ThisworkisdedicatedtoKnicole,whoseloveandsupportmadeitscompletionpossible. 3

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ACKNOWLEDGMENTS Iwouldrstliketothankallofmypastandpresentlabmatesfortheirfriendship,encouragement,andmostofall,fortheirhelp.IthankBretWindomandPrasoonDiwakar,whoarenotonlygreatresearchers,butwhowouldalsoprovidealaughandkindwordswhenIneededitmost.IthankKibumKimforbeingakindandhelpfulcolleague,roommate,andgolfpartner.IthankSoupyDalyanderandPatrickGarrityforthestudysessionsinpreparationforthequalifyingexam.IalsothankMichaelAsgill,MichaelBobek,andRichardStehlefortheirsupportduringthelastyearofmyresearch.IwouldespeciallyliketothankLeiaShanyfeltforbeingawonderfulfriendandcolleague,andforintroducingmetotwoofmynowfavoritepast-times,LostandWorldofWarcraft.IalsowouldliketothankmyparentsfortheirconstantencouragementandsupportduringmytimeattheUniversityofFlorida.Ithankmymotherforherunconditionalloveandpride,andforalwaysremindingmetousemycommonsense.Ithankmyfatherforhisseeminglyendlesswisdom.NomatterhowmuchIlearn,healwaysseemstocomeupwithnewinsightsIneverwouldhaveconsidered.IoweaspecialdebtofgratitudetoKnicoleColon.SomuchofwhatI'veaccomplishedoverthelasttwoyearsisduetoherinuenceinmylife.HerworkethicistomeastandardtowhichIwillalwaysseektoachieve.IthankDr.JillPetersonforherguidanceandsupportduringmymaster'sresearch.Ifshehadnotbelievedinme,IwouldnotbewhereIamtoday.Lastly,IwouldliketothankDr.DavidHahnforprovidingasmuchguidanceanddirectionasonlythemostdedicatedofmentors.Ithankhimforhisendlesswillingnesstoinspireandtohelpandmostlyforhispatienceoverthelastseveralyears. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 12 1.1LaserInducedBreakdownSpectroscopyofAerosolSystems ....... 12 1.2ThePhilosophyandDesignofaNumericalModel ............. 13 1.3ScopeoftheCurrentWork .......................... 16 2REVIEWOFLITERATURE ............................. 18 2.1Laser-InducedBreakdownSpectroscopy .................. 18 2.1.1Laser-InducedPlasmaDiagnostics .................. 18 2.1.2LocalThermodynamicEquilibrium .................. 20 2.2TheCurrentStateofAerosolLIBS ...................... 22 2.3Laser-InducedPlasmaModeling ....................... 24 2.4Inductively-CoupledPlasmaModeling .................... 28 2.5EarlyLaser-InducedPlasmaBehavior .................... 31 3COMPUTATIONALFUNDAMENTALS ....................... 34 3.1NumericalConsiderationsinAtomicEmissionSpectroscopy ........ 34 3.1.1TheBoltzmannDistributionandPartitionFunctions ......... 34 3.1.2TheSahaEquation ........................... 35 3.1.3DeterminingElectronDensityandIonizationStateDistributions .. 36 3.1.4SpectralLineBroadeningandtheCalculationofVoigtFunctions 44 3.2NumericalTechniquesfortheSolutionofPartialDifferentialEquations .. 48 3.2.1FiniteDifferenceMethodsversusFiniteElementMethods ..... 48 3.2.2TheExplicitFiniteDifferenceMethod ................. 49 3.2.3DerivingtheDiscretizationEquationsforOne-DimensionalConductionthroughaSphericallySymmetricMedium .............. 50 3.2.4TheImplicitFiniteDifferenceMethod ................. 55 3.2.5TheTridiagonalMatrixAlgorithm ................... 56 3.2.6TheSIMPLEAlgorithm ......................... 58 3.2.7TheSIMPLERAlgorithm ........................ 59 3.2.8SolvingforRootsofNon-LinearEquations .............. 61 3.2.8.1Thebisectionmethod .................... 61 3.2.8.2Fixed-pointiteration ..................... 62 5

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3.2.9CalculationofHigher-OrderLegendrePolynomials ......... 62 3.3AutomatedPeakDetectionAlgorithms .................... 65 3.3.1Smoothing ................................ 67 3.3.2BaselineCorrection ........................... 68 3.3.3PeakFinding .............................. 69 3.3.4PeakIdealization ............................ 70 4THESTATIC,CONDUCTIVEPLASMAMODEL .................. 76 4.1Overview .................................... 76 4.2TheProblemStatementandSimplifyingAssumptions ........... 76 4.3NumericalFormulationandImplementation ................. 78 4.3.1HeatTransfer .............................. 78 4.3.1.1Theexplicitnitedifferenceformulation .......... 79 4.3.1.2Theimplicitnitedifferenceformulation .......... 80 4.3.2MassDiffusion ............................. 81 4.3.2.1Theexplicitnitedifferenceformulation .......... 82 4.3.2.2Theimplicitnitedifferenceformulation .......... 83 4.3.3TemperatureDependentMaterialProperties ............. 84 4.3.3.1Density ............................ 85 4.3.3.2Specicheatcapacity .................... 86 4.3.3.3Thermalconductivity ..................... 86 4.3.3.4Massdiffusioncoefcient .................. 86 4.3.4DeterminingIonizationStateDistributions .............. 88 4.3.5SimulationofPlasmaRadiativeEmission .............. 90 4.4ResultsandDiscussion ............................ 90 4.4.1TheTemperatureField ......................... 90 4.4.2TheConcentrationField ........................ 91 4.4.3ElectronDensity ............................ 92 5MODELINGAEROSOLVAPORIZATIONWITHINTHELASER-INDUCEDPLASMA ....................................... 107 5.1OverviewoftheAerosolVaporizationProcess ................ 107 5.2InstantaneousAerosolVaporization ..................... 108 5.3LinearAerosolVaporization .......................... 109 5.4Heat-andMass-TransferModelingofAerosolVaporization ........ 110 5.4.1TemperatureIncreasetotheMeltingPoint .............. 111 5.4.2TheMeltingProcess .......................... 112 5.4.3TemperatureIncreasetotheBoilingPoint .............. 113 5.4.4TheVaporizationProcess ....................... 113 5.4.4.1Heattransferlimitedvaporization .............. 114 5.4.4.2Masstransferlimitedvaporization ............. 116 5.5ResultsandDiscussion ............................ 118 6

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6INVESTIGATIONOFPLASMAINCEPTION .................... 128 6.1IntroductionandMotivationforEarlyPlasmaStudies ............ 128 6.2ExperimentalApparatusandMethods .................... 129 6.3DataProcessingandAnalysis ......................... 131 6.3.1AutomatedPeakDetection ....................... 132 6.3.2PlasmaInceptionCharacteristics ................... 135 6.4ExperimentalResultsandDiscussion .................... 136 6.5TheoreticalConsiderationsandConclusions ................ 138 6.6ANoteonSphericalAberration ........................ 141 7CONCLUSIONS ................................... 157 7.1Summary .................................... 157 7.2SuggestionsforFutureResearch ....................... 158 REFERENCES ....................................... 160 BIOGRAPHICALSKETCH ................................ 167 7

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LISTOFTABLES Table page 4-1SummaryofparametersusedintheevaluationofdiffusioncoefcientbyChapman-Enskogtheory ............................. 93 8

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LISTOFFIGURES Figure page 2-1SchematicofatypicalLIBSexperimentalsetup. ................. 33 3-1ComparisonofDoppler,Lorentzian,andVoigtprolefunctions. ......... 72 3-2TheVoigtprolefunctionforvariousvaluesofthedampingparameter,a. ... 73 3-3Controlvolumeforageneralinteriornode. ..................... 74 3-4TherstsixLegengrepolynomialsoftherstkind. ................ 75 4-1Argongasdensity,,asafunctionoftemperature.SeeFujisaki(2002). .... 94 4-2Specicheatcapacity,Cp,ofargonasafunctionoftemperature. ........ 95 4-3Thermalconductivity,k,ofargonasafunctionoftemperature. ......... 96 4-4Massdiffusioncoefcientasafunctionoftemperature. ............. 97 4-5Plasmatemperaturedistributionevolutionwithtimeforaatinitialprole. ... 98 4-6Plasmatemperaturedistributionevolutionwithtimeforaparabolicinitialprole. ................................ 99 4-7Changeintemperaturewithtimeatthreelocationsintheplasma. ....... 100 4-8Concentrationdistributionofcadmiumatearlytimes. ............... 101 4-9Concentrationdistributionofcadmiumatlatertimes. ............... 102 4-10Temporalevolutionofcadmiumconcentrationatthreelocationswithintheplasma. .................................. 103 4-11Evolutionofelectrondensitywithtimeonalogarithmicscale. .......... 104 4-12Evolutionofelectrondensitywithtimeonauniformscale. ............ 105 4-13Temporalevolutionofelectronnumberdensityatthreelocationsintheplasma. ............................... 106 5-1Totalaerosolmassintheplasmavolume. ..................... 121 5-2Simulatedcadmiumconcentrationthroughouttheplasmaafter1s. ...... 122 5-3Simulatedcadmiumconcentrationthroughouttheplasmaafter5s. ...... 123 5-4Simulatedcadmiumconcentrationthroughouttheplasmaafter10s. ...... 124 5-5Simulatedcadmiumconcentrationthroughouttheplasmaafter15s. ...... 125 9

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5-6Simulatedcadmiumconcentrationthroughouttheplasmaafter20s. ...... 126 5-7Simulatedcadmiumconcentrationthroughouttheplasmaafter30s. ...... 127 6-1SchematicofexperimentalLIBSapparatusforplasmainceptionstudy. ..... 142 6-2Evolutionoflaser-inducedplasmainnitrogenoveritslifetime. .......... 143 6-3Laser-inducedplasmaformationinnitrogenatearlytimes ............ 144 6-4LineproleacrosstheCCDshowingearlyplasmainceptionfeaturesinnitrogen. ................................. 145 6-5LineproleacrosstheCCDshowingearlyplasmainceptionfeaturesinargon. ................................... 146 6-6LineproleacrosstheCCDshowingearlyplasmainceptionfeaturesinhelium. .................................. 147 6-7Collectionof30plasmainceptionimagesinnitrogeninrelationtothelaserbeamprole. ..................................... 148 6-8Collectionof30plasmainceptionimagesinargoninrelationtothelaserbeamprole. ......................................... 149 6-9Collectionof30plasmainceptionimagesinheliuminrelationtothelaserbeamprole. ......................................... 150 6-10Inrelationtothebeamprole,plasmainceptioneventsoccurpastthefocalpoint,wheretheplasmaformsatthefocalpoint. ................. 151 6-11Summaryofplasmainceptionstatisticsfornitrogen,argon,andheliuminrelationtothelaserbeamprole. .............................. 152 6-12SimulatedimageofthedistributionofphotondensityacrossseveralpixelsoftheCCD. ....................................... 153 6-13Simulateddistributionoftheprobabilityofamulti-photonionizationeventinnitrogen. ........................................ 154 6-14Simulateddistributionoftheprobabilityofamulti-photonionizationeventinargon. ......................................... 155 6-15Simulateddistributionoftheprobabilityofamulti-photonionizationeventinhelium. ........................................ 156 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyNUMERICALMODELINGOFTHEPLASMA-PARTICLEINTERACTIONSOFAEROSOLVAPORIZATIONINALASER-INDUCEDPLASMAByPhilipB.JacksonDecember2011Chair:DavidW.HahnMajor:MechanicalEngineering Laser-InducedBreakdownSpectroscopy(LIBS)isapowerfulandwell-establishedatomicemissiondiagnosticfortheidenticationandanalysisofunknownsamples.RecentresearcheffortshaveshownthatLIBSisusefulforbothqualitativeidenticationandforthequantitativemeasurementofrelativeaswellasabsoluteanalyteconcentrationregardlessofanalytestate.Morerecently,muchinteresthasbeendirectedtowardtheuseofLIBSintheanalysisofaerosolsystems,includingthosegeneratedbylaserablation(LA-LIBS).WhileLIBSoffersmanyadvantagesasadiagnostictool,thereareseveraldifcultiesthatlimititscapabilityandrobustness.Chiefamongthesearematrixeffectsandincompleteorinhomogeneoussamplevaporization.Inanefforttofullyunderstand,andeventuallymitigate,thesedifculties,thecurrentworkseekstodesignandimplementanumericalmodelthatdescribesthecomplexplasma-particleinteractionsthatgoverntheLIBSofaerosolsystems.Themodelincorporatestheprocessesofheattransfer,hydrodynamics,massdiffusion,vaporization,andelectromagnetism.Themodelconsidersthefundamentalphysicsofthreedistinctregimes:theglobalplasmaenvironment,thelocalparticlebehavior,andtheinitialnatureofplasmainception. 11

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CHAPTER1INTRODUCTION 1.1LaserInducedBreakdownSpectroscopyofAerosolSystems Laser-InducedBreakdownSpectroscopy(LIBS)isadiagnostictoolusedfortheidenticationandanalysisofunknownsamples.Sinceitsdiscoveryasananalyticalmethodintheearly1960s,LIBShasfoundever-increasingexposureinthelaboratoryandintheeld.Amongitsmanyadvantages,LIBSisareal-timetechniquethatcanbeappliedinsituwithlittleornosamplepreparation.Assuch,ithasthecapabilityofanalyzingsamplesinanystate,beitsolid,liquid,orgas.Recently,LIBShasbeenappliedtotheanalysisofaerosolsystemsaswell,includingaerosolsgeneratedbylaserablation,inatechniquecalledLA-LIBS. TheprimarychallengestotheaccuracyandrobustnessoftheLIBStechniquearedifcultiessuchasmatrixeffectsandfractionation.Matrixeffectsdescribeabroadclassofphenomenawherebythesignalbehavioroftheanalyteisaffectedbythepresenceofadditionalmatrixconstituents.Fractionationisessentiallytheincompleteorinhomogeneousvaporizationofasamplewithintheplasmaandresultsinananalyteresponsethatisnotreectiveofthetruesamplestoichiometry.Theanalytesignalthenprovidesamisleadingviewofsamplemakeup.Unfortunately,bothoftheseeffectsestablishalimittotheeffectivenessoftheLIBStechniqueinanalyzinggeneralsystems. Traditionally,researchershavereliedoncertainsimplifyingassumptionsinLIBSthatformafundamentalbasisonwhichthediagnosticisbuilt.WiththeconsiderationofseveraloftheaforementioneddifcultiesontheLIBSofaerosolsystems,itisbecomingincreasinglyapparentthattheseassumptionsmaywarrantreevaluationastotheirvalidity.Itmaybefoundthatnotonlydotheseassumptionsyieldaninexactpictureofthephysics,butitispossiblethattherelaxationoftheseassumptions,oreventheadoptionofnewones,mayleadtotheimprovementofthediagnostic. 12

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Chiefinthoseassumptionsarethattheprocessesofheattransferfromtheplasmaintothediscreteanalyteparticle,andmasstransferfromtheparticleintothelaser-inducedplasma,occurinstantaneously.Infact,however,theheattransferfromthelaserplasmatotheaerosolparticleoccursoveranitetime(Hohreiter,2006).Eventhoughthattimemaybesmallwhencomparedtotheplasmalifetime,itmaynotbesmallenoughtobeconsideredinstantaneous.Also,asmassisliberatedfromthesurfaceoftheparticleitdiffusesthroughouttheplasmavolumeoveranitetime.Althoughthediffusionofparticlemassisrapid,itmaynotbesorapidwhencomparedtothespeedofplasmaexpansionastobeassumedinstantaneous.Inlightoftheproblemsofmatrixeffectsandinhomogeneousvaporization,thetruetimescalesofheatandmasstransfermaynotonlyneedtobeaconsideration,butmayalsoleadtoanexplanationoftheirexistence. ReevaluationofthekeyassumptionsinLIBSmayprovideresearcherswithamorecompletepictureoftherapidandcomplexprocessesthatgovernthemethod.Inaddition,suchinsight,whileprovidingfundamentalknowledge,mayalsobeusedtocombatsomeofthedifcultiesofLIBS,andespeciallythoseofaerosolLIBS.Animprovedunderstandingofthefundamentalphysicsmayleadtomethodstolowerdetectionlimits,methodstoreduceuncertaintyinquantitativemeasurements,andtechniquestobuildmorerobusteld-deployablesystems. Theobjectiveofthecurrentresearchistodeveloparigorous,fundamentalmodeltodescribetheplasma-particleinteractionsofparticlevaporizationinLIBSinordertoprovidethecommunitywithmorecompleteknowledgeandultimatelyimprovetheeffectivenessofthediagnostic. 1.2ThePhilosophyandDesignofaNumericalModel Towardthisend,thecurrentstudyseekstodevelopandimplementacompletemathematicalmodelforthesynthesisofthevarietyofprocessesthattakeplaceduringtheLIBSofaerosolsystems.Theprocessesofheattransfer,hydrodynamics,mass 13

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diffusion,andevenelectromagnetics,eachdescribethemanydifferentphysicalphenomenaobservedinaerosolLIBS.First,thevariousmodesofheattransfermustbeexamined.Alaser-inducedplasmaisashort-lived,high-temperaturegasinwhichconduction,convection,andradiationmodesmayallplayappreciableroles.Furthermore,heattransferfromtheplasmatotheaerosolparticleisoneofthechiefmechanismsbywhichvaporizationoccurs.Itisnotedherethatbasedonthelargemismatchintheplasmavolume(thelarger)andthelaserfocalvolume(thesmaller)thatdirectlaser-particleinteractionsaremuchlesslikelythanplasma-particleinteractions.Highlycoupledtothetemperatureproblemarethehydrodynamicsofthesystem.Laser-inducedbreakdowninducesarapidplasmaexpansion,somuchsothatshockwavesareproduced.Thelargevelocitygradient,therefore,willhavesignicanteffectsonthetemperatureeldandthedistributionofmasswithintheplasma.Alsoimportanttothetransportofmaterialthroughouttheplasmavolumeismassdiffusionwhichgreatlyinuencesvaporizationintheimmediatevicinityofanaerosolparticle.Lastly,electromagneticforcesmaygreatlyaffecttheplasma'sbehavior,especiallywithregardtotheearlydynamics.Thelargeelectromagneticeldgeneratedfromtheincidentlaserpulseitselfinuencesthebreakdowneventandthereforetheinitialplasmacharacteristics. Withthisinmind,thecurrentmodelingeffortscategorizetheproblemintothreesub-modelsthatareimplementedindependently:aglobalmodel,alocalmodel,andaninitialmodel.Theglobalmodeldescribesthephysicalenvironmentthroughoutthelaser-inducedplasmaasdistributionsoftemperature,electrondensity,andmassthathasbeenliberatedfromanaerosolparticle.Oncetheglobalenvironmentisestablished,thelocalmodeldescribesthevaporizationkineticsofasingleaerosolparticlesubjectedtothelocalconditionsofcurrentplasmalocation.Whilethelocalmodeldependsupontheglobalmodel,theconverseisnottrue.Lastly,inordertodeterminethetemporalprogressionofbothglobalandlocalvariabledistributions,theinitialconditionsmust 14

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rstbeprescribed.Duetothecomplexityofmodelingconsiderationsduringtheearlytimesofplasmalife,whicharecharacterizedbynon-equilibriumdynamics,theinitialconditionsareprescribedbasedonempiricalobservations.Anexperimentalstudyintothegrowthandbehaviorofthelaser-inducedplasmainitsearlylifetimesisperformedtoprovideinsightintohowacompletemodelofaerosolLIBSmayincorporateadescriptionofplasmainception. Likeanynumericalmodeltherearetwochallengesthatmustbeaddressedwhenonediscussesthecorrectnessofthemodel:physicalcorrectnessandnumericalcorrectness.First,themodelmustbyphysicallycorrect.Thatis,thegoverningequationsandfundamentalprocessesconsideredmustindeedrepresentthecorrectphysicalprinciplesatwork.Muchcarehasbeentakentojustifytheuseofeachfundamentalprincipleandequationemployedinthecurrentmodelingtreatment,andeachisdiscussedastheyarise.Secondly,themodelmustexhibitnumericalcorrectness.Thatis,thesolutionproceduremustprovidenumericalvaluesthataccuratelysatisfytheequationsuponwhichtheyarebasedwithinacceptablenumericaluncertainty.Eachnumericaltechniquethatisusedhereiswidelyacceptedasacorrecttechniqueandisindependentlyveriedthroughtheuseofbenchmarkingexamples. Lastly,itisimportantthatanygoodnumericalmodelachievetwoobjectives:(1)itmustagreewithandsupport(orincertaincases,challenge)currentacceptedresearch,and(2)itmustbeabletomaketestablepredictions.MuchresearchiscurrentlybeingundertakentomorefullyunderstandthephysicsofaerosolLIBS.Assuch,muchdataexistsbywhichthecurrentmodelmaybeveried.Manymodeloutputquantitiesmaybecomparedwithvariouscurrentstudiestovalidatethemodel,suchas:temperaturemeasurements,diffusioncharacteristics,andevenspectralsignatures.Finally,oncethemodelhasbeenvalidated,itmaybeusedtoinvestigatenewsituationsthatinspirenewexperimentsinafurtherattempttoprovideinsightintothecomplicatedphysicsofthe 15

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phenomena.This,aboveall,isthemostimportantgoalbothofnumericalmodelingingeneralandofthecurrentefforts. 1.3ScopeoftheCurrentWork Thepresentstudyseekstoprovidethereaderwiththedescription,design,andimplementationofarigorousnumericalmodelfortheanalysisofaerosolLIBS.Thisstudyisorganizedinabottom-upfashionwitheachnewchapterbuildingupontheworkofeachpreviouschapter. Chapter2beginswithareviewofseveralimportant,fundamentaltopicsincludedbothforcompletenessandforreference.First,thebasicsofLIBSarecoveredalongwithadiscussionofafewimportantconceptsinthequanticationofatomicemissionspectroscopyingeneral.Second,severalbasicnumericaltechniquesareexaminedthatareimplementedthroughoutthepresentstudy.Thesetechniquesareprovidedhereintheirgeneralformssotheirimplementationinspecicfacetsofthemodelmaybebetterunderstood.Lastly,thechapterisconcludedwithadiscussionofautomatedpeakdetectionalgorithmsthatnduseintheanalysisofdatatakeninthecurrentexperimentalstudyofplasmainception. Afterthereviewofseveralfundamentalconcepts,Chapter3describesthecurrentstateofresearchintowhichthepresentstudyisplaced.First,currenttrendsinLIBSresearcharediscussed,asisthepresentroleofaerosolLIBS.Next,severalrecentmodelingeffortsinLIBSandrelatedtechniques,suchasInductively-CoupledPlasma,AtomicEmissionSpectroscopy(ICP-AES),arediscussed.Lastly,thechapterisconcludedwithadiscussionofthepresentunderstandingofearlyplasmabehaviorandnon-equilibriumconsiderations. Withthepreliminarybasisandmotivationforthecurrentstudyestablished,Chapter4beginsthedescriptionofmodelingeffortsbydetailingthemethodforsimulatingtheglobalplasmaenvironment.Thephysicsoftheglobalplasmamodelaredescribedindetail.Includedarediscussionsoftherelativeimportanceofconduction,convection, 16

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andradiationheattransfermodes,theeffectsoftemperaturedependentproperties,andthenecessaryconsiderationsfortheeffectiveimplementationofthesemodels.Theimplicationsofvariousphysicalphenomenaandmodelingmethodologiesarediscussedincludingtherolesofcompressibilityeffects,therolesofelectromagneticforces,andsingle-uidrepresentationsversusion-neutralrepresentations. Withtheglobalenvironmentestablished,Chapter5examinesthelocalenvironmentintheimmediatevicinityofasingleaerosolparticle.Thekineticsofaerosolvaporizationareinvestigatedalongwiththeireffect,ifany,ontheglobalenvironmentwithreferencetoacceptedmodelsofaerosolvaporization.Individualprocessesofmelting,evaporation,anddiffusionarediscussed.Thecompetingrolesofheattransfer-limitedvaporizationandmasstransfer-limitedvaporizationarealsodiscussed. Chapter6turnsattentiontotheinvestigationofthebehaviorofearlyplasmalifetimesandthestudyoftheplasmainceptioneventitself.Anexperimentalstudyispresentedtoinvestigatetheearliestbreakdowneventsandthesubsequentgrowthoftheplasmainseveraldifferentgases.Inthisstudynumerousimagesofinitialplasmabreakdownareautomaticallyprocessedtocompilestatisticsonthevariationsofearlybehaviorinthevariousgases.Theimplicationthisbehaviormayhaveonthecurrentunderstandingofplasmainceptionisintroduced. Lastly,Chapter7summarizesthemostimportantpointsandconclusionsofthepresentwork,suggestsrenementsthatmayimprovethesophisticationofthepresentmodelanddiscussesvariousavenuesofinterestthatmayinspirefuturework. 17

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CHAPTER2REVIEWOFLITERATURE 2.1Laser-InducedBreakdownSpectroscopy 2.1.1Laser-InducedPlasmaDiagnostics Inlaser-inducedbreakdownspectroscopy(LIBS),ahigh-energylaserpulseisfocusedtoapoint.Atthatpointthepowerdensitybecomessufcientlyhightoinducethebreakdownofwhatevermediumispresent,andahigh-temperatureplasmaresults.Theatomicemissionfromtheplasmaiscollectedandusedforvariousqualitativeandquantitativediagnostics.Figure 2-1 showsatypicalLIBSlaboratorycongurationwhereplasmaemissioniscollectedinbackscatterthroughtheuseofapiercedmirror. Therstlaser-inducedplasmatobeusedinthelaboratorywasproducedintheearly1960s(Miziolek,2006).Sincethen,theLIBStechniquehasfoundwidespreaduseintheanalyticallaboratoryasanattractivemethodforanalyzingmaterials.Likemanyothermethodsofatomicemissionspectroscopy,theprimarygoalofLIBSistheidenticationandanalysisofanunknownsample. OverthepastseveraldecadestheLIBSmethodhasproventobeusefulasarobustqualitativediagnosticforthedetectionofthepresenceofunknownsampleconstituents.Spectraofcollectedemissioncanbeobservedforthepresenceofpeaksatthecharacteristicwavelengthofagivenelement.LIBSuseslibrariesofelementalsignatures,andcombinationsofsuchsignatures,toidentifysamplesrangingincomplexityfromsingle-speciessamplestocomplexbiologicalsamples.Inmorerecentyears,LIBShasbeenshowntoprovidevaluablequantitativeanalysisaswell.Basedonrelativepeakintensitiesandspectrallinebroadening,researchershavebeenabletouseLIBStodeterminerelativeandevenabsoluteconcentrationsoftheconstituentsinasample(Miziolek,2006). Asananalyticaltool,LIBShasmanyadvantagesoverothermethodsofelementalanalysis.Firstofall,nosamplepreparationisrequiredforLIBSasitisatechniquethat 18

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canbeperformedonvirtuallyanysample,inanystate.LIBShasbeendemonstratedonsolidsurfaces,inliquids,ingases,and,mostrecently,onaerosolsystems(Hohreiter,2004).Moreover,LIBSproponentsstatethatitiscapableofinsituanalysis,inthatthelaserplasma,astheexcitationsource,isfocusedontothesample,ratherthanbringingthesampletotheexcitationsourceasinmanyotheratomicemissionspectroscopymethods.Theonlypreparationthatisrequiredisopticalaccesstothesample.Thisisespeciallybenecialinsituationsthatmaybehazardoustohumanlife.Lastly,aLIBSanalysisisfast.Duetotheaforementionedlackofsamplepreparationanddeliverytime,andthefactthatthelaser-plasmaitselfisshort-lived,asingleLIBSmeasurementcanbemadevirtuallyinstantly.ManyLIBSanalysesrequireanensembleofshotsandthenbatchprocessingoftheresultingdata.MostautomatedidenticationandchemometricroutinesarefastenoughthatLIBSisdescribedasareal-timetechnique. LIBSisnotaperfectdiagnostictool,however.Manychallengesstillexisttoimprovetherobustnessofthetechnique,especiallyinquantitativeanalysis.TherstchallengetoLIBSanalysisistheissueofsamplenon-homogeneity.TheLIBSplasmaissmall,and,assuch,probesasmallpointinspacethatmaynotcontainelementalconstituentsthatareperfectlyrepresentativeoftheoverallsample.Relatedtothisistheconceptoffractionation.Fractionationisessentiallythenon-uniformanalyteresponseofconstituentsintheplasma.Forexample,varyingvaporizationratesofplasmaconstituentsalterstheelementalexcitationoftheconstituents,andthereforenon-uniformvaporizationanddiffusioncanyieldmisleadingresultsfortherelativeconcentrationsthatarecalculated. MatrixeffectsalsolimittheLIBSdiagnosticasisthecaseinmanyotheranalyticalmethods.Matrixeffectsoccurwhenthepresenceofthevarioussampleconstituentsaffectsthesignalofthespecicelementofinterest.TwosamplesthatcontainthesameconcentrationofagivenelementmayeasilyyielddifferentabsolutesignalstrengthsinthesameLIBSsetupdependingonthestateofthesample.Matrixeffectsarenot 19

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completelylimiting,however.Oftenmatrix-dependantcalibrationisperformedtohelpmitigatetheseeffectsusingmatrix-matchedstandards. 2.1.2LocalThermodynamicEquilibrium Globalthermodynamicequilibriumexistsinamediumthatisinthermalequilibrium(constanttemperature),mechanicalequilibrium(constantpressure),andchemicalequilibrium(constantconcentration).Suchahomogeneousandconstantsystemallowsforseveralequilibriumrelationstobeemployedtodescribethesystem.Onamolecularlevel,thermodynamicequilibriumimpliesthatallcollisionalandradiativeprocessesbalanceoneanotherout.Inequilibrium,ionizationeventsareequallyfrequentasrecombinationevents,andradiationemittedisequaltoradiationabsorbed(Lochte-Holtgreven,1995). Globalthermodynamicequilibriumthereforeimpliesasystemisstaticandunchanging.Whilethisstatemayseemuninteresting,facetsofsuchaconceptmaybeemployedintrulydynamicsystems,allowingonetoaccuratelydescribeallthecomplexitiesofavaryingsystemwhilestilltakingadvantageofthesimpleequilibriumrelations.Suchisthecaseintheconceptoflocalthermodynamicequilibrium.Inlocalthermodynamicequilibrium(LTE),asinglepointinthesystemisassumedtobeinthermodynamicequilibriumwithsomesmallregionaboutthatpointintimeandspace.Inthissense,thermodynamicequilibriumholdsateachsinglepoint,whilestillallowingforthethermodynamicstatetovaryfromonepointtothenext. Fromamolecularviewpointinaplasma,localthermodynamicequilibriumnolongerrequirescollisionalandradiativeprocessestobalanceoneanother.Rather,collisionalprocessesareassumedtodominatetheplasmakinetics(Lochte-Holtgreven,1995). Thequestionremains,whenisthelocalthermodynamicequilibriumassumptionavalidone,andwhenisitnot?Iflocalthermodynamicequilibriumresultswhencollisionalprocessesdominateradiativeprocessesintheplasmakinetics,thenitisreasonabletoassumethatonemayrequiretheelectronnumberdensitytobesufcientlyhighto 20

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ensureahighcollisionrate.ThislineofthinkingleadstothepopularMcWhirtercriterion(Miziolek,2006)forestablishinglocalthermodynamicequilibrium: ne1.61012T1=2(E)3,(2) whereEistheenergytransitionofalineineV,andTisthetemperatureinK.ItisimportanttonotethattheMcWhirtercriterionisanecessary,butinsufcient,criterionforassuminglocalthermodynamicequilibrium(Tognoni,2006).Therehasbeenmuchrecentdiscussionondevelopingsufcientconditionsforwhichlocalthermodynamicequilibriumcancondentlybeassumedtohold.DespitethedifcultyinestablishingprecisemetricsfortheLTEassumption,researchersarecurrentlycondentthatlocalthermodynamicequilibriumholdsforallbuttheearliestofplasmalifetimes. Assuminglocalthermodynamicequilibriumholdsultimatelyallowsthestatisticsofmicroscopicstatestofollowcertainstandardrelations.Oncelocalthermodynamicequilibriumisestablished,thepopulationdistributionofexcitedstatesofaspeciesmaybedescribedbytheBoltzmannformula,andthepopulationdistributionofthedifferentionizationstatesofaspeciesmaybedescribedbytheSahaequation.Bothoftheserelationsarediscussedindetailinsubsequentsections.Indeeditisonlywhentheseandotherequilibriumconditionsholdthattemperaturemaybedenedasasingle,uniquequantityatapoint(Lochte-Holtgreven,1995). Variationsfromlocalthermodynamicequilibriumassumethatpopulationandvelocitydistributionsarenotgivenbytherelationsmentionedabove.Whenlocalthermodynamicequilibriumdoesnothold,theveryconceptoftemperatureiscalledintoquestion.Commonnon-equilibriummodelssimplifythisdifcultybyallowingfortwodistincttemperaturestoexistateachpoint,anelectrontemperature,Te,andaheavyparticletemperature,Tp,whicharedeterminedfromuniquedistributionrelationsforeachspecies(Povarnitsyn,2007). 21

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2.2TheCurrentStateofAerosolLIBS Thestudyoftheresponse,characteristics,andlatestimprovementofthelaser-inducedbreakdownofaerosolbasedsamplesisjustonesmallcorneroftheoverallLIBScommunity.Itis,however,aeldwithvastexposureintheliterature.Therstreportedcaseoftheuseofalaser-inducedplasmadiagnosticforthestudyofanaerosolsamplecanbetracedbacktoRadziemskietal.(1983) In1983Radziemskietal.developedtime-resolvedmeasurementsofthepresenceofseveralelementsinaerosols.Localthermodynamicequilibriumwasassumedthroughouttheirexperiment,withincreasedcondenceinthisassumptionaftertherst1s.Thecollectedspectrawereusedtocalculatetheplasmatemperatureandelectrondensity.Asimplehydrodynamicmodelwasalsoimplementedtopredictplasmatemperatureandsize.ThestudyalsorepresentstherstuseofLIBSforinsitumeasurementsofaerosols. Sincethen,theuseofLIBSonaerosolsystemshascontinuedtogrow.In1998,HahnstudiedtheuseofLIBSforthesizingofsingleaerosolparticles.OfparticularinterestwastheuseofLIBS,notjusttoqualitativelydeterminetheelementalcompositionofasingleaerosolparticle,buttoprovideaquantitativeanalysisofthemassconcentrationoftheparticle.Calibrationwasperformedasatwo-stepprocesswhereLIBSspectrawerecompared,rst,tothatofknownmassconcentration,andsecond,tothatofknownparticlesizeandcomposition. Later,in2001Carranza,etal.usedaerosolLIBStostudythedetectionoftraceconcentrationsoftheconstituentelements,suchasmagnesiumandaluminum,characteristicofreworks,inambientairfortheFourthofJulyholidayperiod.Increasesinsignalresponsefortheseelementswereobservedoverthreeordersofmagnitude.Themeasurementsalsoemployedareal-timeconditionaldataanalysisschemetoincreasetheeffectiveanalytesignal'sresponsebasedonwhetherornotanindividualLIBSmeasurement(i.e.thatfromasinglelaser-inducedplasma)couldbeclassiedas 22

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aparticlehit.Thisgreatlyreducedthenumberoftotalspectraintheensembleaverageandlimitedtheensembletospectrathatcouldyieldusefulinformation.Thereal-timenatureoftheexperimentandit'suseofconditionalanalysishasshownthatLIBSofaerosolshasbecomeamorecompetitivediagnosticovertheyears. In2002CarranzaandHahninvestigatedanupper-particlesizelimitforcompleteaerosolvaporization.Thesizelimitwasdeterminedbydeviationfromlinearmassresponseintheatomicemissionofsilicon.Inaddition,thefundamentalmechanismbywhichvaporizationoccursisassumedtobecontrolledbyplasma-particleinteractionratherthanbylaser-particleinteractionsbasedonthecomparisonofaerosolsamplingmeasurementswithPoissonstatistics.Assuch,thespatialandtemporalevolutionoftheplasmabecomesmoreimportanttotheoverallprocessandisdicussedindetail.ThermophoreticforcesandvaporexpulsiondynamicsarementionedtohaveimportantimplicationstoLIBS. ThefundamentalprocessesthatgoverntheLIBSofaerosolswasinvestigatedfurtherbyHohreiterandHahnin2004withtheultimategoaltounderstandandthusimprovethefactorsaffectingthequantitativeprecisionofthediagnostic.Spectralandtemporaleffectsofparticlepresenceorabsencewerestudied.Lasercavityseedingproducednosignicantimprovementoverthepossibleanalyteprecision,howevermarkedimprovementwasnoticedwhenconcomitantaerosolsfromthesamplestreamwereremoved.Theplasma-particleinteractionsinsimilarexperimentswerefurtherinvestigatedbytheauthorsin2006.Theinteractionbetweentheplasmaandindividualparticlemasscontrolstherateofparticlevaporizationanddiffusionthroughouttheplasmavolumetherebyinuencingthespectroscopicsignalmeasured.Finitetimescalesoftheseprocessesarediscussedalongwiththeissueofspatialnon-homogeneityandtheinuenceoflocalizedeffects. In2009Hahnsummarizesthecommunity'seffortsoverthepastdecadeunderstandandimprovetheuseofLIBSasadiagnosticforaerosolsystems.Theimportanceof 23

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understandingthemanyfundamentalprocessesthatgovernthecomplexplasma-particleinteractionsareemphasized.Also,Hahnchallengesseveralofthekeyassumptionsemployedduringtheearlierdaysofthediagnosticandsuggeststhatcriticalevaluationoftheassumptionsaretypicalatthisstageofascienticmethod'slifetime.Growthandimprovementofthediagnosticintothefuture,then,isassuredasmuchworkmuststillbedonetounderstandhowthefundamentalphysicsofaerosolLIBSultimatelyleadstoanalyteresponse. 2.3Laser-InducedPlasmaModeling Severalmodelshavebeendevelopedinrecentyearsinanattempttobetterunderstandandpredictvariousaspectsoflaser-inducedbreakdownspectroscopy.Whilemanyoftheseinvestigationsallinherentlyshareconsiderationofthesamephysics,thespecicsofeachmodelandtheirassumptionshavevariedsignicantly.ThisisexpectedsincethefundamentalprocessesthatgoverntheentireLIBSevolutionarenumerousandcomputationallycostly.Afullmodelthatseekstocontaineachfundamentalprocessforavarietyofspeciesovertheentireplasmalifetimewithdependenceonspaceandwavelengthisambitiousalmosttotheextentofbeingunwieldy.Despitethesemodelingdifculties,manysuccessfulLIBSmodelscanbefoundintheliterature. In1996,Hoetal.,publishedastudyonthenumericalmodelingoftheenergy-matterinteractionsofalaser-inducedplasmawithasolidsurface.WhileLIBSanalysisofsolidsurfaceshasbeencoveredintheliteratureingreatdetail,fewLIBSmodelsthatcouplemass,momentumandenergyconservationinmultiplephasesarefound.IntheHomodel,heatistransferredtothesolidsurfaceandphasetransitionsareallowedasthesolidconvertstoliquidandultimatelytothevaporphase.Severallayersareconsideredandthereforethetransportequationsaresolvedaspiecewisefunctionsthroughtheselayers.Radiationandabsorptionmechanismsareconsideredthroughouttheplasma,whilemaintainingtheassumptionoflocalthermodynamicequilibrium.Compressibility 24

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effectsarealsoconsideredandassuchthemodelproducesaneffectiveapproximationtothebehaviorofthesphericalshockwavepropagatingabouttheplasma. ThespecicproblemoftheexpandingplasmaandshockwaveinteractingwiththesurroundinggaswasstudiedbyItina,etal.in2003.Thegasdynamicsofthelaserplumeexpansionintobothvacuumanddensebackgroundgasareconsidered.Inaddition,twodifferentnumericalmethodsareusedtodevelopahybridmodelthatdescribesbothcontinuumandmolecularregimes.First,theauthorssolvethegasdynamicequationsofmass,momentum,andenergyconservation.Thisgivesaviewoftheproblemfromacontinuumormacroscopicviewpoint.Second,theauthorsusetheDirectSimulationMonteCarloapproachtoobtainamicroscopicviewofthephysics.Ofparticularinteresttotheauthorswasthemixingoflaserplumeandambientspeciestodescribeexperimentallyobservedphenomena. ThegasdynamicsofplasmaexpansionisagainconsideredbyMazhukinetal.(2003).Inthismodeltheplasmaisassumedtobenon-stationary,radiative,andrepresentedwithatwo-dimensionalaxiallysymmetricgrid.Theplasmaismodeledtoimpingeuponasolidsamplesurfacecomprisedprimarilyofaluminum.Theauthorsndthattheradiativecharacteristicsoftheplasmadominateoverconvectivemechanismsandthusdrivetheevolutionoftheplasmaexpansion.Non-equilibriumeffectsareconsideredonthespectraldependenceoftheradiationbothemittedandabsorbedbytheplasma.Theplasmaisassumedopticallythick. ArigorousplasmamodelwasdevelopedbyGornushkin,etal.rstin2001thatformsmuchoftheinspirationofthecurrentwork.Alsoassumedasopticallythick,therstplasmamodelenvisionedbyGornushkin,etal.considersbothconvectiveandradiativemodesofheattransport.Asinthepreviousmodelsconsideredsofarinthisreview,localthermodynamicequilibriumisconsideredthroughmuchofthework.Here,however,plasmaexpansionisnotfoundfromthesolutionofgoverninggasdynamicequationbutratherprescribedthroughsetfunctionswithempiricallychosen 25

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parameters.Theplasmaexpansionradius,aswellasthetemperatureprolethroughouttheplasmavolumeatanytimeisprescribedbasedonempiricalmeasurements.Basedonthislargelyempiricalmodel,thedistributionsofconstituentspeciesofsiliconandnitrogen,andtheirneutralandionizedstatesarecalculated.Ofprimaryinterestisthecalculationofthespectraldependenceoftheemittedradiation.Asaresult,theatomiclineprolesarecalculatedwiththeinclusionoflinebroadeningmechanismssuchasStarkbroadeningandDopplerbroadening.Theresultisaseriesofsyntheticspectrabasedonthemodel'sinputquantities. Sinceitsrstinception,themodelbyGornushkin,etal,hasundergoneseveralrevisionsinrecentyears.Ofparticularinterestisastudypublishedin2004wheremuchofthesemi-empiricalnatureofthemodelwasremovedinfavorofastrictsolutionofthegasdynamicequations.Again,radiativetransferandconvectiveheattransfermodesareconsideredtodominate.Thegasdynamicequationsaresolvedasalaser-inducedplasmaiscreatedonthesurface,andcompletelyvaporizesasphericalparticle.Theplasmaisassumedtobeinlocalthermodynamicequilibriumthroughout.Plasmaradiationiscalculatedasafunctionofspectraldependencetogeneratesyntheticspectra.Whilemuchoftheempiricalnatureofthemodelhasbeenremoved,someisstillretainedbywayoftheprescriptionofplasmainitialconditions.Themodelisdenedtobeginatsomesmalltimeafterbreakdownhasoccurred.Assuchtheinitialplasmatemperatureproleisprescribedalongwiththeinitialplasmaradiusandvelocity.Experimentalvericationofthemodelwasexhaustivelyperformedin2005. Morecasesofparticlesensitiveplasmamodelshavebeenfoundintheliteratureinmorerecentyears.Bleineretal.developedamathematicalmodeloflaser-assistedparticlesamplingin2004.Particlesofvarioussizedistributionsaremodeledinanexpandinglaserplumetoexaminetheirinuenceonmicro-particleformationandtheablationofsolidmaterial.Itwasfoundthatlocalplasmaconditionsdrivethekineticsofthemicro-processesratherthanbulklaser-plumecharacteristics.Theauthorspecically 26

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addressestheuseoflaserbasedtechniquesforthesamplingofdiscretepointsandthebenetsofmathematicallymodelingthebehavior. AfurtherrenementtotheworkofGornushkinetal.waspublishedbyKazakov,etal.in2006.Againthedynamicsofaconvective,radiativeplasmagasareconsidered,butinthiscase,theplasmaenvironmentexpandsnotintovacuum,butintoambientgas.Asaresult,themodelincludescompressibilityeffectsandisabletopredicttheformationofthesphericalshockwavethatpropagatesalongwithplasmaexpansion.Theinitialplasmadynamicsarestilldenedbasedonsemi-empiricalobservationandthemodelisonlyapplicableafterthelaserpulsehasvanished.Theevolutionsofatomicandioniclineprolesarealsocomputed. In2007,astudywasperformedbyPovarnitsym,etal.demonstratedseveralnon-equilibriumcharacteristicsoflaserplasmas,thoughthestudywasspecictothosecreatedfrompulsedlasersinthefemto-secondrange.Themodelassumedtheexistenceoftwoseparatetemperatures,theelectrontemperatureandtheheavyparticletemperature.Themodeldescribesthehydrodynamicmotionoftheplasmaandaccountsforlaserenergyabsorptionandconductionthroughasolidsampletarget.Phasetransitionsthroughoutthesampleareconsideredandtrackedusingahigh-ordermulti-materialGudunovmethod.Themodelisusedprimarytodescribetheablationandfragmentationofthetargetwithrespecttomeasuredstressesandobservedablationdepth. Morerecently,astudywasperformedbyDalyander,etal.thatalsoservedassignicantinspirationtothecurrentwork.Theauthorsdevelopanitedifferencesolutiontotheconductionequationtodescribethetemperaturedifferenceinastationarylaser-inducedplasmathatdoesnotexpandwithtime.Themodelwasdevelopedforthespecicpurposetounderstandtherolethatnitevaporizationanddiffusionratesplayinthenatureofaerosol-basedLIBSmeasurements.Aparticleconsistingofcadmiumandmagnesiumisintroducedintothecenteroftheplasmameshandisallowedto 27

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vaporizelinearlywithtime.Theresultingmassdiffusionthroughouttheplasmavolumeiscalculated.Basedonequilibriumconsiderationsthedistributionofneutralatomsandionsiscalculated.Fromthisatomicemissionisestimatedandusedtoassessthedistinctionbetweenglobaltemperatureevolutionandlocaltemperaturecharacteristics. 2.4Inductively-CoupledPlasmaModeling WhileLIBSistheatomicemissiondiagnosticthatisprimarilyunderconsiderationinthecurrentwork,thereareseveralothertechniqueswithinthewideeldofatomicemissionspectroscopywhosestudiesarealsorelevant.Inaddition,manyotherplasma-basedtechniquesexistintheanalyticalcommunity.Whileoperatingtemperatures,lifetimes,andothercharacteristicsofplasmascreatedfromthevarioussourcesmaydiffer,thefundamentalprocessesgoverningplasmasallsharecertaincommonphysics.AssuchthepresentresearchintheeldsofotherplasmatechniquesandvariousmodelfeaturesmayyieldusefulinsightintocurrenteffortsinLIBS. Aplasma-basedtechniquethatseeslargeexposureintheliteratureisInductively-CoupledPlasmaAtomicEmissionSpectroscopy(ICP-AES).Thecreationofaninductivelycoupledplasmaisdrasticallydifferentthantheformationofthelaserplasma.AnICPisasustainedplasmacreatedfromastrongelectromagneticeldthatinducesandmaintainsarelativelylarge(incomparisontoalaser-inducedplasma)plasmacore.CurrenteffortsinaerosolanalysisandalsomodelingintheeldofICP-AESlendmuchtothecurrentstudy. PerhapsthelargestcontributiontothepresentstudyfromtheICPcommunitycomesintheformoftheoreticalmodelsforthevaporizationkineticsofsoluteparticles.In1987,Hieftje,etal.developedtwocontrastingmodelsforthevaporizationofsingleparticlesentrainedinanalyticalamesorplasmas.Theformulationconsideredthatwhileheattransferandmasstransferwerebothimportantmechanismsinthevaporizationandliberationofmassfromasingleparticle,onlyonemechanismwouldberatelimitingandthereforesolelygoverntherateofparticleradiusdecrease.Their 28

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argumentsalsoconsideredtherolethattheparticlesizeplaysinthedeterminationoftheserateconstants.Infact,whetherheattransfer-limitedormasstransfer-limited,bothlargeparticleandsmallparticleregimesandexpressionweredenedforeachmechanism.Themodelfounddifcultiesindeterminingexactlyinwhatregimeagivenparticlemayfall,butitcreatedafoundationforaseriesoffollow-uptheoreticalformulationsthatsolvedtheproblemmoresuccinctly. In1998,HornerandHieftjedevelopedanumericalsimulationoftheICPenvironmentanditsinteractionwiththeirpreviouslyderivedaerosolvaporizationkinetics.Twotypesofsimulationswereperformed,onewheresingleaerosolswereentrainedintheICP,andonewheremany-particledistributionswereentrained.Theydeterminedthatchangestoplasmaoperatingconditions,andthusplasmapropertiesaffectedthevaporizationcharacteristicsappreciably.Itwasalsofoundthatfromthepreviousstudiesofvariousparticleregimesandmechanismsthatsmall-particleheattransferlimitedvaporizationseemedtodrivetheobservedbehavior.Themany-particlesimulationswereusedforcomparisondirectlywithexperimentalresults.Thechiefgoaloftheinvestigationissimilartothepresentstudy,namelytodeterminethemechanismbywhichmatrixinterferenceaffectsspectroscopicmeasurements. In2008,anadditionalrenementtotheaerosolvaporizationmodelwasmadebytheintroductionofamorerigorousdescriptionofthevaporizationkineticsofearlierphasetransitionsthantheevaporationphase.Particle-vaporizationkineticsaremodeledasaseriesofsequentialstepsthatdescribeeachtransitionfromsolidtoliquidandfromliquidtovaporindetail.Modelinputvaluesconsistofplasmaoperatingconditionsandlocationwithintheplasma,aswellascharacteristicsoftheparticlesthemselves,suchasdiameterandcomposition.Inaddition,theirearlierassessmentofwhatparticleregimeandwhatmechanismdominatesinanICPanalysisisrevisedshowingthateithermaybeimportantandcontrolling.Sinceeitherprocessmightlimittherate 29

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ofvaporization,bothareconsideredbasedonautomatedcriteriaduringsimulationexecution. TheeffectofaerosoldropletsandvaporizationmechanicsonanICPwerealsoinvestigatedbyHobbsandOlesikin1992.LargesignaluctuationsinanalyteresponsewereobservedduringICPmassspectrometry.Thesesignaluctuationswereinvestigatedexhaustivelyanditwasfoundthatthepresenceofincompletelydissolveddropletsandpartiallyvaporizedsolidparticlesaffectedtheanalyteresponseagreatdeal.Theauthorsalsofoundthattheseeffectsweredependentoncompositionandthatinsomecasesnoadverseorenhancingeffectswerefound.Insomecases,oppositeeffectsofsignalenhancementwereobserved.Theeffectswereattributedtoageneralclassofbehaviorsknownasmatrixeffects.Matrixeffectsarediscussedindetailpreviously.Studiessuchastheseprovidedusefulobservationsforthetheoreticalinvestigationofmatrixeffectsinatomicemissionspectroscopyinlateryears. In1997,Olesikdiscussedthemotivationsbehindtheoreticalinvestigationofindividualparticlehistories.OlesikstatedthattheanalyticalsignalsobservedduringICP-AESwereproductsofaseriesofkineticprocessesthatcontrolledthevaporizationofdropletsandparticlesfromwhichtheanalytescome.Particlesurfacetemperatureisrstraisedtothemeltingpoint,whenphasetransitiontoliquidoccurs.Theliquidparticlethenincreasesintemperatureuntiltheboilingpointisreachedwherebyparticleevaporationkineticstakeover.Particlevaporization,hereasoned,waslimitedeitherbyheattransfertothesurfaceoftheparticleorbymasstransfer.Thesevaporizationkineticsdependonlocalplasmaconditionsratherthanbulkproperties.Olesikalsodiscussedtheeffectsthatnon-idealvaporizationkineticshaveonanalytesignal. InthenextfewyearsseveralimagingstudieswereperformedtoobtainabetterpictureofthesekineticsdescribedbyOlesikandHeiftje.Houketal,in1997,performedaseriesofhighspeedphotographicstudiesofthehistoryofsolidparticlesandliquiddropletsinanICP.Theyfoundthatnotonlywereindividualparticlehistoriesimportant 30

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tothecontinuedstudyofICP-AES,butthatindividualparticlecalibrationwasdesiredfortheisolationofidealbehavior. SeveralotherICPstudieshaveproducedresultsthathaveinspiredinvestigationsinLIBSanalysis.In2006Hergenroderproposedthathydrodynamicsputteringisresponsibleforfractionationinvariousplasmastudies.Hismodelisbasedonthesolutionofathreedimensionalheatconductionequationwithmovinginterfaceboundaries.Particlevaporizationkineticsareconsideredwithspecicinterestonforcedinhomogeneousvaporizationwhereafractionofanalytematerialisevaporatedwhileafractionremainssolid.Themodelwasusedtoidentifyoptimaloperatingconditionstoavoidsuchbehavior.Inasimilarinvestigation,Bleineretal.studied,bynumericalsimulation,theeffectofsurfacemeltingandvaporizationduringlaserablation,alsowiththepurposeofexaminingfractionation.Itwasfoundthatathighirradiance,phaseexplosionanddropletexpulsiongreatlyenhanceablationrateandaffectidealsamplingconditions. 2.5EarlyLaser-InducedPlasmaBehavior Thestudyofearlylaserplasmabehaviorisanotherareawithlittleexposureintheliteratureinrecentyears.Non-equilibriumconsiderations,coupledwiththerapidtransientnatureoftheseregimesofplasmalife,makestudiesofearlyplasmadynamicsdifcult. Arelatedstudythathasceasedtobecommonintheliteratureinrecentyearsconcernthecharacterizationofplasmashape.OnesuchstudyperformedbyBeduneauandIkedain2003,whilenotspecicallyfocusedonearlyplasmalifetimes,lendsinformationtowardtheunderstandingtheplasmaformation.Inthestudyimagesandemissionspectrawerecollectedforavarietyoflaserenergiesandopticalcongurations.Itwasfoundthatnotonlywasgoodreproducibilityfoundforearlystagesofbreakdownbutthatthecharacteristicsofsizeandlocationdependgreatlyontheoperatingconditions.Highionizationlevelsintheearlyplasmawasfoundtobe 31

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conrmationoftheelectroncascademechanismforplasmaformation.Ionizationwasalsousedtoexplaintheasymmetryofplasmashape. In1988,astudybyCarlsandBrockwasperformedthatusedacomputermodeltoinvestigatelaser-inducedplasmaformationandtheexplosionofaerosoldropletswithinit.Themodeldescribedtheformationandevolutionoftheplasmaandtheuidowthatresults.Still,theonecomponentlackingistheinitialbreakdownevent,whichisinsteadrepresentedbyanempiricalinitialcondition. Lastly,in2008,astudywasperformedbyDiwakarandHahninwhichearlylaser-inducedplasmadynamicswereconsidered.Themotivationforthestudywasthatonlybyunderstandingthemechanismsofplasmacreationandevolutioncanthefundamentalprocessesoflaser-inducedbreakdownspectroscopybeunderstood.Therst100nsofplasmalifetimewereconsideredtodescribetheearlyplasma.Duringtherst50ns,signicantThomsonscatteringwasobservedandtheelectronnumberdensitywascalculated.Thehighlytransientnatureofelectrondensitywasusedtosuggestthatplasmadynamicsatearlytimeswereinfactnon-equilibriumdominated.AdditionalmeasurementsbyStarkbroadeningweremadeandseemedtocorroboratethisconclusion.Deviationfromlocalthermodynamicequilibriumwithintherst10nsofplasmalifetimewasthendiscussedasitpertainstotheplasma-particleinteractionspresentinLIBSmeasurements. 32

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Figure2-1. SchematicofatypicalLIBSexperimentalsetupwherecollectionistakeninback-scatter. 33

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CHAPTER3COMPUTATIONALFUNDAMENTALS 3.1NumericalConsiderationsinAtomicEmissionSpectroscopy 3.1.1TheBoltzmannDistributionandPartitionFunctions OneofthemostusefulrelationsthatmaybeemployedoncelocalthermodynamicequilibriumhasbeenestablishedistheBoltzmannequation.ForaspeciesinLTE,theBoltzmannequationrepresentsthedistributionofthepopulationateachexcitedstateforeachenergylevel.TheBoltzmannequationiscommonlywrittenas: ni n=gi U(T)exp)]TJ /F3 11.955 Tf 13.22 8.08 Td[(Ei kT,(3) wherenisthetotalnumberdensityfortheentirespeciesandniisthenumberdensityofthespeciesthatisexcitedtothei-thenergylevel.Thetermgiisthedegeneracyofthei-thlevel,U(T)isthespeciesinternalpartitionfunction,Eiistheenergyofthei-thlevel,kisBoltzmann'sconstant,andTisthetemperature(Lochte-Holtgreven,1995). TheinternalpartitionfunctionitselfisofinterestasitisthemostdifculttermoftheBoltzmanndistributiontocalculate.Thepartitionfunctionisthesumoverallpossiblemicrostatesandisgivenby: U(T)=Xigiexp)]TJ /F3 11.955 Tf 13.22 8.09 Td[(Ei kT.(3) Tocalculatethepartitionfunctionforaspeciesrequires,intheory,thesumoveraninnitenumberofenergylevels.Attemptstoperformsuchacalculationoftenproduceexorbitantlyhighvaluesforthepartitionfunctionandthesumdiverges.Thisandothercommondifcultiesincalculatingpartitionfunctionsarealleviatedwiththeuseofpolynomialapproximations.Irwinrepresentstheinternalpartitionfunctionsofseveralspecieswithpolynomialts(Irwin,1980)oftheform: 34

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lnU=5Xi=0ai(lnT)i,(3) wherethecoefcientsaiaretabulatedinthearticle(Irwin,1980).Thepolynomialapproximationsareconsideredaccurateandprovideacomputationallyinexpensivemethodforcalculatingpartitionfunctions. 3.1.2TheSahaEquation AusefulrelationfortherelativemagnitudeofconsecutiveionizationstagesofanyelementinaplasmaisgivenbytheSahaequation.Derivedin1920bytheastronomerMeghNadSaha,theSahaequationwasrstusedinthestudyofstellaratmospheres.TheSahaequationisderivedfromequilibriumconsiderations,andsoforittoholdtrue,theplasmaunderconsiderationmustbeassumedtobeinlocalthermodynamicequilibrium.Here,theplasma'skineticsareassumedtobedominatedbycollisionalinteractionsratherthanbyradiativeprocesses(Lochte-Holtgreven,1995). AcommonrepresentationoftheSahaequationis: nenz nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2Uz(T) Uz)]TJ /F9 7.97 Tf 6.59 0 Td[(1(T)2mekT h23=2exp)]TJ /F4 11.955 Tf 10.5 8.08 Td[(z)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(z)]TJ /F9 7.97 Tf 6.59 0 Td[(1 kT,(3) whereneistheelectronnumberdensityoftheplasma,andnzandnz)]TJ /F9 7.97 Tf 6.59 0 Td[(1arethenumberdensitiesofthez-thandz)]TJ /F6 11.955 Tf 12.83 0 Td[(1-thionizationstage,respectively.Here,Uzisthepartitionfunctionforthez-thionizationstage,meistherestmassoftheelectron,kisBoltzmann'sconstant,andhisPlanck'sconstant.Thetermz)]TJ /F9 7.97 Tf 6.59 0 Td[(1istheionizationenergyofthez)]TJ /F6 11.955 Tf 12.35 0 Td[(1-thstageandz)]TJ /F9 7.97 Tf 6.59 0 Td[(1isthereductionoftheionizationenergyduetothepresenceoftheplasmamicroeld.Notethataswrittenabove,therightsideoftheSahaequationisentirely(exceptforthez)]TJ /F9 7.97 Tf 6.59 0 Td[(1term)afunctionoftemperature,andcanbewritteninamoresuccinctform: nenz nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=Sz)]TJ /F9 7.97 Tf 6.59 0 Td[(1(T).(3) 35

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Alsonotethatintheaboveequationz=1correspondstotheneutralatom,z=2totherstionizationstate,andsoon.Hence,theexpressionz)]TJ /F6 11.955 Tf 12.32 0 Td[(1representsthechargeonthespecies. Forthecurrentpurposes,theSahaequationwillbeusedtosolvefortheionizationstatedistributionsofamulti-componentplasmawheremultipleionizationstatesareallowedtoexistinequilibrium.OneSahaequationmaythenbewrittenforeachelementalplasmaconstituentforeachpairofconsecutiveionizationstates.Forexample,inatwo-componentplasmawherethersttwoionizationstates(z=2,3)areconsidered,fourdistinctSahaequationscanbewrittenthatmustbesolvedsimultaneously,alongwithotherconservationequations,touniquelydeterminetheionizationstatedistributions.Thistopicwillbediscussedmorethoroughlyinthenextsection. Lastly,notethat,forthecurrentpurposes,thereductioninionizationenergy,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1,willbeneglected.Formostpracticalapplicationsoflaserplasmasthereductioninionizationenergyisonlyontheorderofabout0.1eV(Miziolek,2006).Neglectingthistermamountstoachangeinthetrueionizationenergyofonlyabout1%intheworstcase.Thissimplicationisjustiedwhenoneconsidersthatz)]TJ /F9 7.97 Tf 6.59 0 Td[(1isafunctionoftheelectronnumberdensity,ne.Whilemanyrelationsexisttodescribethisdependence(Lochte-Holtgreven,1995),thecomputationalcostofperformingthiscalculationwhilesolvingfortheionizationstatesisunwarrantedwhenoneconsidersitsnegligiblenumericaleffect. 3.1.3DeterminingElectronDensityandIonizationStateDistributions Ifboththetemperatureeldandtheconcentrationeldofspeciesareknown,thedistributionofneutralatomsandionscanbefound.Assumingtheplasmadynamicsarecollision-dominated(localthermodynamicequilibrium),therelationshipbetweenthenumberdensitiesoftwoconsecutiveionizationstatesisgivenbytheSahaequation(Radziemski,1989): 36

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nenz nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2Qz(T) Qz)]TJ /F9 7.97 Tf 6.58 0 Td[(1(T)2mekT h23=2exp)]TJ /F6 11.955 Tf 10.5 8.08 Td[(Ez)]TJ /F9 7.97 Tf 6.59 0 Td[(1 kT=Sz)]TJ /F9 7.97 Tf 6.59 0 Td[(1(T).(3) Here,z=1correspondstotheneutralatom,z=2correspondstotherstionizationstateandsoon,suchthattheexpressionz)]TJ /F6 11.955 Tf 12.39 0 Td[(1representsthechargeonthespecies.Also,neistheelectronnumberdensity,nzisthenumberdensityofspeciesz,Qz(T)istheinternalpartitionfunctionofspeciesz,meistherestmassoftheelectron,kisBoltzmann'sconstant,hisPlank'sconstant,andEzistheionizationenergyofspeciesz.SincetherighthandsideoftheequationiscompletelydenedbytemperatureonemayrepresenttheSahaequationby: nenz nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=Sz)]TJ /F9 7.97 Tf 6.59 0 Td[(1(T), where Sz)]TJ /F9 7.97 Tf 6.59 0 Td[(1(T)=2Qz(T) Qz)]TJ /F9 7.97 Tf 6.59 0 Td[(1(T)2mekT h23=2exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(Ez)]TJ /F9 7.97 Tf 6.59 0 Td[(1 kT. Asanexample,consideraplasmaenvironmentthatconsistsoftwoelements,argonandmagnesium,thatmayexistaseitherneutralorsinglyionizedatoms.InthiscaseonemaywritetwoSahaequations,oneforeachelement: neArII ArI=SAr,I(T)andneMgII MgI=SMg,I(T).(3) WhileSAr,IandSMg,Iarecompletelydeterminedbytemperature,thenumberdensitiesne,ArI,ArII,MgI,andMgIIareallunknown.Sincethetotalnumberdensitiesofeachspecies,irrespectiveofionizationstate,areknownfromtheconcentrationdistribution,onemayclosethesystemandsolveforalltheunknownsbyalsoconsideringtheconservationofspeciesandtheconservationofcharge.Conservationofspeciesforargonandmagnesiumaregivenbythefollowingtworelations: 37

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ArT=ArI+ArIIandMgT=MgI+MgII.(3) Conservationofchargeisthensimply: ne=ArII+MgII.(3) WiththetwoSahaequations,twoequationsfortheconservationofspecies,andasingleequationfortheconservationofcharge,allunknownscanbedetermined.Firstsolveequations 3 fortheneutralspeciestoget: ArI=neArII SAr,IandMgI=neMgII SMg,I.(3) Substitutingequations 3 into 3 gives: ArT=ArII1+ne SAr,IandMgT=MgII1+ne SMg,I.(3) Solvingeachofthesefortherstionizationstatesandsubstitutinginto 3 gives: ne=ArT 1+ne SAr,I+MgT 1+ne SMg,I.(3) Theonlyunknownintherelationaboveistheelectronnumberdensityne.Thisequationcanbesolvednumericallybyanumericalroot-ndingmethod.Moreover,auniquesolutionisguaranteedtobefoundfromthesetofpositiverealnumbersaswillbediscussedlaterinthissection.Oncenehasbeendetermined,alltheotherunknownnumberdensitiescanbefoundsequentiallyfromequations 3 and 3 Inasimilarfashion,thissystemmaybesolvedforanarbitrarynumberofparticipantspecieswithanarbitrarynumberofionizationstates.IngeneraltheSahaequationisgivenby: 38

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nenj,z nj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1=Sj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1,(3) wherenj,zisthenumberdensityofspeciesjinstatez.Notethatj=1,2,3,...,J,whereJisthetotalnumberofspeciespresent,andz=1,2,3,...,Z+1,whereZisthehighestionizationstateconsidered.WithJspeciesandZionizationstates,therearethenJZSahaequationsinoursystem(Gornushkin,2004). Conservationofspeciesisgivenby: Z+1Xz=1nj,z=Nj,(3) whereNjisthetotalnumberdensityofspeciesj.SincethereareJspeciesinthesystem,thereareJspeciesconservationequations. Conservationofchargeisthengivenby: JXj=1Z+1Xz=1(z)]TJ /F6 11.955 Tf 11.95 0 Td[(1)nj,z=ne.(3) ThesystemisnowclosedwithJZ+J+1equationsandJZ+J+1unknowns.Thesolutionofthesystembeginsbymultiplyingequation 3 byne, JXj=1Z+1Xz=2(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)nenj,z=n2e.(3) Next,multiplyequation 3 byz)]TJ /F6 11.955 Tf 11.95 0 Td[(1(nj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1)andsumoverallz'sandallj's,toyield: JXj=1Z+1Xz=2(z)]TJ /F6 11.955 Tf 11.95 0 Td[(1)nenj,z=JXj=1Z+1Xz=2(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)Sj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1.(3) Substitutingequation 3 intoequation 3 gives: n2e=JXj=1Z+1Xz=2(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)Sj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1.(3) Multiplyingequation 3 bynegives: 39

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Z+1Xz=1nenj,z=nenj,1+Z+1Xz=2nenj,z=neNj.(3) Substitutingequation 3 into 3 gives: nenj,1+Z+1Xz=2Sj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1=neNj.(3) Continuingtoexpandthissum,yields: nenj,1+Sj,1nj,1+Z+1Xz=3Sj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1=neNj,(3) nenj,1+Sj,1nj,1+Sj,2nj,2+Z+1Xz=4Sj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1=neNj.(3) Which,by 3 ,becomes: nenj,1+Sj,1nj,1+Sj,2Sj,1nj,1 ne+Z+1Xz=4Sj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1=neNj.(3) Continuing,thesystembecomes: nenj,1+Sj,1nj,1+Sj,2Sj,1nj,1 ne+Sj,3nj,3+Z+1Xz=5Sj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1=neNj,(3) nenj,1+Sj,1nj,1+Sj,2Sj,1nj,1 ne+Sj,3Sj,2nj,2 ne+Z+1Xz=5Sj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1=neNj,(3) nenj,1+Sj,1nj,1+Sj,2Sj,1nj,1 ne+Sj,3Sj,2 neSj,1nj,1 ne+Z+1Xz=5Sj,z)]TJ /F9 7.97 Tf 6.59 0 Td[(1nj,z)]TJ /F9 7.97 Tf 6.58 0 Td[(1=neNj.(3) Whichmoreconciselybecomes: 40

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nj,10BBBB@ne+Z+1Xz=2z)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yi=1Sj,i nz)]TJ /F9 7.97 Tf 6.59 0 Td[(2e1CCCCA=neNj.(3) Rearranginggives: nj,1=Nj 0BBBB@1+Z+1Xz=2z)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yi=1Sj,i nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1e1CCCCA.(3) Now,consideragainequation 3 .Expandingthesumwithrespecttozyields: ne=JXj=1nj,2+JXj=1Z+1Xz=3(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)nj,z.(3) Substituting 3 yields: ne=JXj=1Sj,1nj,1 ne+JXj=1Z+1Xz=3(z)]TJ /F6 11.955 Tf 11.95 0 Td[(1)nj,z.(3) Continuingtoexpandthesumyields: ne=JXj=1Sj,1nj,1 ne+JXj=12nj,3+JXj=1Z+1Xz=4(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)nj,z,(3) ne=JXj=1Sj,1nj,1 ne+JXj=12Sj,2nj,2 ne+JXj=1Z+1Xz=4(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)nj,z,(3) ne=JXj=1Sj,1nj,1 ne+JXj=12Sj,2 neSj,1nj,1 ne+JXj=1Z+1Xz=4(z)]TJ /F6 11.955 Tf 11.95 0 Td[(1)nj,z,(3) ne=nj,1Z+1Xz=2JXj=1(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)z)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yi=1Sj,i nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1e.(3) 41

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Substituting 3 into 3 nallyyields: ne=Z+1Xz=2JXj=1Nj(z)]TJ /F6 11.955 Tf 11.95 0 Td[(1)z)]TJ /F9 7.97 Tf 6.58 0 Td[(1Yi=1Sj,i nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1e0BBBB@1+Z+1Xw=2w)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yk=1Sj,k nw)]TJ /F9 7.97 Tf 6.59 0 Td[(1e1CCCCA.(3) Thisisanonlinearalgebraicequationfornewhosecoefcentsgrowincomplexityasoneincreasesthenumberoftheparticipatingspeciesandwhoseordergrowsasnewionizationstatesareadded.Theformoftheequation,however,suggeststhatundercertainconditionsonewillalwaysndaviablesolutionviaxed-pointiteration(Atkinson,1978).Sincethedesireistodeterminethedistributionofionizationstatesbasedoncalculatedtemperatureandconcentrationelds,theequationabovewillbeexecutedforavarietyofdifferentconditions.Thoseconditionsmayormaynotresultinanequationthataxed-pointiterationmethodisguaranteedtondasolutionforforagivenchoiceoftheinitialguess. Recallthatxed-pointiterationisaprocedureforsolvinganonlinearalgebraicequationintheform:xn+1=g(xn), ofwhichNewton'smethodisacommonexample.Atkinson(1978)describesconditionsforg(x)thatguaranteesxed-pointiterationwillconvergeuponauniquesolution. First,assumethatg(x)iscontinuouslydifferentiableon[a,b],thatg([a,b])[a,b],andthatMaxa
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Ifonetakes(0,1)asthedomain,thenitfollowsthatg([0,1])[0,1].Therefore,toshowthattherelationabove,ne=g(ne),hasauniquesolutionthatisguaranteedtobefoundbyxed-pointiteration(sinceg(ne)iscontinuouslydifferentiablein(0,1)),itmustbeshownthatMax0
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g0(ne)=Z+1Xz=2JXj=1)]TJ /F3 11.955 Tf 9.29 0 Td[(Nj(z)]TJ /F6 11.955 Tf 11.96 0 Td[(1)z)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yi=1Sj,i (z)]TJ /F6 11.955 Tf 11.95 0 Td[(1)nz+2Z)]TJ /F9 7.97 Tf 6.59 0 Td[(2e+Z+1Xw=2(z)]TJ /F3 11.955 Tf 11.96 0 Td[(w)nz+2Z)]TJ /F5 7.97 Tf 6.58 0 Td[(w)]TJ /F9 7.97 Tf 6.58 0 Td[(1ew)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yk=1Sj,k! nz+Z)]TJ /F9 7.97 Tf 6.59 0 Td[(1e+Z+1Xw=2nz+Z)]TJ /F5 7.97 Tf 6.59 0 Td[(wew)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yk=1Sj,k!2,(3) whichisarationalfractionwhosepolynomialorderinthedenominatorexceedsthepolynomialorderofthenumerator.Thefractionthentendsto0asnetendsto1.Thereforeitappearsthat,whilenotrigorouslyproven,Maxa
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ofthelifetimeeffectscomefromthedeactivationoftheexcitedstateduetocollisionsandistermedcollisionalbroadening. Collisionalbroadeningorpressurebroadeningwasrstdescribedin1905byH.A.Lorentzwhoshowedthatthewidthofspectralprolesisrelatedtothefrequencyofatomiccollisions(Lochte-Holtgreven,1995).Thetermcollisionalbroadeningisusedtodescribeeffectsfromcollisionsthatoccurbothbetweendifferentatomsaswellasbetweenlikeatoms.Mathematically,thespectralprolethatresultsfromcollisionalbroadeningtakestheformofaLorentzianfunctionthatcanbewritteninthefollowinggeneralform, SL()=2=(L) 1+[2(m)]TJ /F4 11.955 Tf 11.95 0 Td[()=L]2,(3) wheremisthecentralfrequencyandListhehalf-width.Theotherlifetimeeffects,suchasfromspontaneousorstimulatedemissioncanalsoberepresentedbyLorentzianprolesandoftenthemostdominantoftheseeffectscanbeassumedtobeindependent.TheresultisthatasingleLorentzianfunction,withanappropriatecompositehalf-width,canbeusedtomodelalloftheeffectstogether.Forexample,naturalbroadening,whichresultsfromthenaturaldecayoftheexcited-statepopulationduetospontaneousemission,isoftenanegligibleeffectincomparisontocollisionalbroadening. AnotherdominantsourceofspectrallinebroadeningcomesfromtheDopplereffect.Theatomsandionsthatarepresentinspectroscopicobservationsarealwaysinmotionwithsomedistributionofvelocities.BecauseoftheDopplereffect,thedistributionofvelocitiesresultsinthestatisticalvariationofobservedfrequencies.AccordingtoMaxwell'slawthedistributionofvelocitiesisGaussianinnature.IfitcanbeassumedthatthevelocityofasingleatomdoesnotchangewhileitradiatesthentheresultingdistributionoffrequenciesisalsoGaussian.AgeneralformforaspectrallineunderDopplerbroadeningis 45

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SD()=2p ln2 Dp exp)]TJ /F9 7.97 Tf 6.59 0 Td[(4(ln2)()]TJ /F10 7.97 Tf 6.58 0 Td[(m)2=(D)2,(3) whereDisthehalf-width.Whiletherearemanyotherphenomenathatleadtospectrallinebroadening,suchasStarkbroadeningthatresultsfromsystemswithpermanentdipolemoments,thecurrentstudywillchieyconsideronlycollisionandDopplerbroadening. Inreality,truespectralprolesareusuallyneitherpurelyGaussianinshapenorLorentzian.RatheracombinationofthetwoprolesisneededtoproduceabetterttingshapeandmodeltheeffectsofcollisionalandDopplerbroadeningsimultaneously.SuchacombinationisdescribedbytheVoigtfunctionwhichisnamedafterWoldemarVoigt'sworkofthelate19thcentury. TheVoigtproleisthereforeaconvolutionofLorentzianandGaussianproles,assumingthetwoeffectsareindependent,andisgivenbythefollowingformulation SV()=2p ln2 Dp K(a,r).(3) ThequantityK(a,r)isknownastheVoigtintegralandisdenedas K(x,y)=y Z1exp()]TJ /F3 11.955 Tf 9.3 0 Td[(t2) (x)]TJ /F3 11.955 Tf 11.96 0 Td[(t)2+y2dt,(3) wheretisadummyvariableofintegrationoverallfrequenciesandaisthedampingconstant.Thedampingconstant,a,isrelatedtotheLorentzianandDopplerhalf-widthsby a=p ln2L D.(3) TheVoigtfunctionrepresentsacombinationoftheLorentzianproleandtheDopplerprole.Thethreeprolesareshowntogetherandnormalizedin 3-1 .Qualitatively,thenormalizedLorentzianproleshapetendstofavoritstailsforadecreasein 46

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amplitudeaboutthemeanwhencomparedwiththeGaussianproleshape.TheVoigtfunction,asacombinationoftheLorentzianandGaussianproleshapes,allowsforanextradegreeofvariablitybyalteringthedominanceofeachcomponent.Thedampingparameter,a,determinestherelativeeffectsofeachproleasshownin 3-2 Fromapracticalperspective,theVoigtproleformulationaboveposesanadditionalchallengeinthatthenumericalcalculationoftheVoigtintegralcanbecostly.AnefcientmethodforcalculatingtheVoigtintegralisdesirableandnecessaryinanypracticalsituationwheremultiplehigh-resolutionVoigtfunctionsmustbecalculated. TherehavebeenmanystudiesthatdescribeefcientcalculationsoftheVoigtprolefunction,theoneadoptedhereisanimplementationoftheHumlicekalgorithmthathasbeenacceleratedbyKuntztottheneedsofopticalspectroscopy(Kuntz,1997).ThemethoddevelopedbyKuntzdividesthex-yplaneintofourregionsandapproximatestheVoigtintegralineachregionbyarationalpolynomialexpression.Forexample,Region1isdenedbytheexpressionjxj+y>15andthefollowingparameterswithinthisregionaredeveloped: a1=0.2820948y+0.5641896y3(3) b1=0.5641896y(3) a2=0.25+y2+y4(3) b2=)]TJ /F6 11.955 Tf 9.3 0 Td[(1+2y2(3) UsingtheseparameterstheVoigtfunctionisthenapproximatedbythefollowingrationalexpresssion: 47

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K(x,y)=a1+b1x2 a2+b2x2+x4.(3) ThisimplementationofHumlicek'salgorithmisconsiderablymoreefcientnumericallythanevaluationbyelementaryintegrationtechniques.Thisisespeciallybenecialinthecurrentstudy,whereroutinesaredevelopedtonumericallytexperimentallyobservedspectralwindowstosetsoftheoreticalprolesdescribedbyVoigtfunctions.AlgorithmsareimplementedtondthebesttofaVoigtproletoapeakofinterest.Thesemethodsareimplementedonahostofindividualpeaksovermanysetsofspectralwindowssuchthatthesavingsincomputationtimeareconsiderable. Overalltheinvestigationofspectrallinebroadeningisimportanttotheeldofquantitativespectroscopy.Theoreticalinvestigationsofaspectralpeak'swidthprovidesadditionalinformationthelocationandamplitudeofthepeakalonecannotprovide.Spectralprolewidthscanbeusedtoprovideestimatesofquantitativedatasuchasparticledensity,relativeconcentration,andtemperature. 3.2NumericalTechniquesfortheSolutionofPartialDifferentialEquations 3.2.1FiniteDifferenceMethodsversusFiniteElementMethods Incomputationaluiddynamicsandheattransfer,thetwomainchoicesforthemethodofformulationarenitedifferencemethods(FDM)andniteelementmethods(FEM).Bothmethodsdiscretizethepertinentpartialdifferentialequationsintoasystemofalgebraicequations,buttheunderlyingprinciplebywhichthisoccursisquitedifferent.Innitedifferencemethods,derivativeapproximationsareusedatnodalgridpointstoreducethepartialdifferentialequationtoanalgebraicone.Finiteelementmethodsmodelthefunctionitselfbetweengridpointsusingsometypeofproleassumption.Whilenitedifferencemethodstendtohavemorepopularityinthestudyofuidowandheattransfer,bothmethodshavemerit.Itisinterestingtonotethatmostresearchersinthesecomputationalarenasrarelycross-implementmethods(White,1974),andindeedthepracticaldifferencesbetweenthetwodonotwarrantadvantages 48

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ofoneovertheotherinconsideringaspecicproblem.Thus,toendeavortosolveaseriesofpartialdifferentialequationsmeanstomakeachoiceastothemethodofformulationonewillemploy. Whilethecomparativeadvantageofonemethodovertheotherdoesnotprohibitonemethodfromndinguseinanygiveneld,eachmethoddoeshaveitsspecicbenets.Foruidowandheattransferthenitedifferencemethodformulationstendstofollowamorelogicalderivation.Ontheotherhand,niteelementmethodformulations,whichusesomevariationalmethodintheirderivation,donotlendaseasilytoaphysicalinterpretation(Patankar,1980). Forthesereasons,onlythenitedifferencemethodhasbeenusedinthepresentstudytoformulatethesolutionofthepartialdifferentialequationsgoverningheattransfer,masstransfer,anduiddynamics. 3.2.2TheExplicitFiniteDifferenceMethod Finitedifferencemethodsfortimedependentpartialdifferentialequationsfallintoaspectrumofexplicit-nessintheirformulation.Anitedifferenceapproximationmaybefullyexplicit,fullyimplicit,ormayfallsomewherebetweenthetwoextremes,aformulationknownasaCrank-Nicholsmethod.Moreover,inthesolutionofmorecomplexsystemsofpartialdifferentialequations,solutionmethodsmayseetheuseofacombinationofexplicit,implicitorCrank-Nicholsmethodsforeachequationorfordifferenttermsinanygivenequation.Asaconsequence,bothexplicitandimplicitmethodsaredescribedhere. Explicitnitedifferencemethodscalculatethevaluesofthenodalunknownsforagiventimestepbasedpurelyontheirvaluesattheprevioustime.Implicitnitedifferencemethods,ontheotherhand,calculatethevaluesofthenodalunknownssimultaneouslyforanysingletimestep,andareonlyminimallydependentonthevaluesoftheprevioustimestep. 49

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Thediscretizationequationforanexplicitnitedifferenceschemewillhavethefollowingform(Patankar,1980): aiTp+1i=ai)]TJ /F9 7.97 Tf 6.58 0 Td[(1Tpi)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ai+1Tpi+1+bTpi+c. Asaconsequencethesolutionofsuchasetofdiscretizationequationsisquitesimpleandstraightforward.Beginningwithsomeinitialcondition,T0i,thevaluesofthenodalunknownsatthenexttimestep,T1i,aresimplycalculatedbyevaluatingeachdiscretizationequationexplicitly.Thereisnoneedforaniterativeprocedure.Eveninthecasewherethecoefcientsareafunctionofthedependentvariablethemselves,thecoefcientsareevaluatedasfunctionsofthevaluesattheprevioustimestep.SolvingtheequationsinthismannerrequirescomputationaltimeofcomplexityO(N). Itshouldbenotedthatonemajordisadvantageplaguesexplicitnitedifferencemethods.Thatis,ingeneral,explicitmethodsareonlyconditionallystable.Thetimestepsmustbesufcientlysmalltoguaranteeaphysicallymeaningfulsolutionisachievedfromsolvingtheequationsexplicitly.Forthediscretizationequationabove,thiscanbeachievedbyrequiringthateachcoefcientaiandbbepositive.Thismakessense,asoneexpectsanincreaseinanygivennodaltemperaturetoproduceadeniteincreaseinthenewnodalvalue.Inthecaseofone-dimensionalconductionincartesiancoordinates,forexample(Incropera,2002), Fo1 2 isasufcientcriterionforphysicallymeaningfulstability,withFobeingthenon-dimensionaltimedenedbyFo=t=L2c. 3.2.3DerivingtheDiscretizationEquationsforOne-DimensionalConductionthroughaSphericallySymmetricMedium Asarstapproach,thenitedifferenceequationsforaone-dimensionalsimplicationarederivedforthecurrentproblem.Theplasmawillbeassumedtobespherically 50

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symmetric,thatis,temperaturewillbedependentpurelyontheradialcoordinate.Conductionwillbetheonlymodeofheattransferconsidered,exceptfortheoutermostboundarywhichwillloseheatradiativelytotheenvironment.Furthermore,themediumisassumedhomogeneous,isotropic,stationary,freeofheatgeneration,andatlocalthermodynamicequilibrium.Nodesarenumbered0,...,mforatotalofm+1nodes,wherenode0isatthesymmetryboundary,orthecenteroftheplasmavolume,andnodemrepresentstheouteredgeoftheplasma. Threediscretizationequationswillbesolved:oneforthesymmetryboundary,onefortheouter,radiativeboundary,andonethatisvalidforallremaining,internalnodes.Theschemeisstartedbysolvingthediscretizationequationfortheinternalnodes1,...,n)]TJ /F6 11.955 Tf 12.19 0 Td[(1,n,n+1,...,m)]TJ /F6 11.955 Tf 12.19 0 Td[(1.Thecontrolvolumerepresentativeofeachoftheinternalnodesisgivenin 3-3 Writinganenergybalanceforacontrolvolumearoundnoden,yields: qjn)]TJ /F14 5.978 Tf 7.79 3.26 Td[(1 2+qjn+1 2=VCpTp+1n)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn dt,(3) whereqjn)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2isthetotalenergyenteringthecontrolsurfaceatn)]TJ /F9 7.97 Tf 13.72 4.71 Td[(1 2.FinitedifferencesimplicationsofFourier'slawthentakethefollowingform: qjn)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2=kn)]TJ /F14 5.978 Tf 7.79 3.26 Td[(1 2Tpn)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn dr4(rn)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2)2,(3) qjn+1 2=kn+1 2Tpn+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn dr4(rn+1 2)2,(3) Substituting 3 and 3 into 3 ,andintroducinganexpressionforthevolumeofthenitedifferenceelementgives: kn)]TJ /F14 5.978 Tf 7.78 3.25 Td[(1 2Tpn)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn dr4r2n)]TJ /F14 5.978 Tf 7.79 3.26 Td[(1 2+kn+1 2Tpn+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn dr4r2n+1 2=4 3(r3n+1 2)]TJ /F3 11.955 Tf 10.94 0 Td[(r3n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2)CpTp+1n)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn dt.(3) 51

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Notethat: r2n+1 2=rn+dr 22,(3) r2n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2=rn)]TJ /F3 11.955 Tf 13.15 8.08 Td[(dr 22,(3) r3n+1 2)]TJ /F3 11.955 Tf 11.95 0 Td[(r3n)]TJ /F14 5.978 Tf 7.78 3.25 Td[(1 2=rn+dr 23)]TJ /F11 11.955 Tf 11.95 16.86 Td[(rn)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dr 23.(3) Itfollowssincern=ndr: r2n+1 2=dr2n+1 22,(3) r2n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2=dr2n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22,(3) r3n+1 2)]TJ /F3 11.955 Tf 11.96 0 Td[(r3n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2=dr3"n+1 23)]TJ /F11 11.955 Tf 11.96 16.86 Td[(n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 23#.(3) Substituting 3 into 3 anddividingby4dr2yields: kn)]TJ /F14 5.978 Tf 7.78 3.25 Td[(1 2Tpn)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn drn)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22+kn+1 2Tpn+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn drn+1 22=1 3Cpdr"n+1 23)]TJ /F11 11.955 Tf 11.96 16.86 Td[(n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 23#Tp+1n)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn dt. (3) Rearranginggives: kn)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2dt Cpdr2(Tpn)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22+kn+1 2dt Cpdr2(Tpn+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)n+1 22=1 3"n+1 23)]TJ /F11 11.955 Tf 11.95 16.86 Td[(n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 23#(Tp+1n)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn). (3) Recallthenon-dimensionalFouriernumberiswrittenas: Fo=kdt Cpdr2.(3) 52

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Therefore Fon)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tpn)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22+Fon+1 2(Tpn+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)n+1 22=1 3"n+1 23)]TJ /F11 11.955 Tf 11.95 16.85 Td[(n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 23#(Tp+1n)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn). (3) Alsonotethatonecanreducethecubictermasfollows: "n+1 23)]TJ /F11 11.955 Tf 11.96 16.86 Td[(n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 23#=n3+3 2n2+3 4n+1 8)]TJ /F11 11.955 Tf 11.96 16.86 Td[(n3)]TJ /F6 11.955 Tf 13.15 8.09 Td[(3 2n2+3 4n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 8(3) "n+1 23)]TJ /F11 11.955 Tf 11.96 16.86 Td[(n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 23#=3n2+1 4.(3) Therefore,asdrbecomessmall: "n+1 23)]TJ /F11 11.955 Tf 11.95 16.86 Td[(n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 23#=3n2.(3) Onecanalsoreducethesquaredtermsinasimilarfashion: n+1 22=n2+n+1 4=n(n+1),(3) n)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22=n2)]TJ /F3 11.955 Tf 11.95 0 Td[(n+1 4=n(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1).(3) Substitutingtheseresultsintoequation 3 gives: Fon)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tpn)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)n(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1)+Fon+1 2(Tpn+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)n(n+1)=n2(Tp+1n)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn).(3) Rearrangingonearrivesatthenalresultforthediscretizationequationforeachoftheinternalnodes: Tp+1n=Tpn+1 nhFon)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tpn)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn)(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)+Fon+1 2(Tpn+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpn)(n+1)i.(3) 53

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Asimilarprocedurewillbefollowedforderivingthediscretizationequationforthesymmetricalboundarynode.Theenergybalanceforthesymmetrynodeis: qj1 2=VCpTp+10)]TJ /F3 11.955 Tf 11.96 0 Td[(Tp0 dt.(3) ByFourier'slaw: k1 2Tp1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp0 dr4dr 22=Cp4 3dr 23Tp+10)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp0 dt.(3) RearrangingandwritingintermsoftheFouriernumber,yields: Tp+10=Tp0+6Fo(Tp1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp0).(3) Lastlythediscretizationequationfortheouterboundarynodewillbederived.Theplasmaexchangesheatbyradiationtotheenvironment.Writinganenergybalanceforthiselementgives: qjm)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2+q"Rrr2m=VCpTp+1m)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpm dt.(3) Heretheheatuxduetoradiationisgivenby: q"R=(T41)]TJ /F3 11.955 Tf 11.96 0 Td[(T4m), whereistheemissivity(assumedheretobe1foraperfectblackbodyemitter),andistheStephan-Boltzmannconstant.Substitutingthisexpressionintotheaboveequation,alongwithFourier'slaw,resultsin: km)]TJ /F14 5.978 Tf 7.79 3.25 Td[(1 2Tpm)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpm dr4r2m)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2+(T41)]TJ /F3 11.955 Tf 11.96 0 Td[(T4m)4r2m=4 3(r3m)]TJ /F3 11.955 Tf 11.95 0 Td[(r3m)]TJ /F14 5.978 Tf 7.79 3.26 Td[(1 2)CpTp+1m)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpm dt.(3) Rearrangingandsimplifyinggives: 54

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km)]TJ /F14 5.978 Tf 7.79 3.26 Td[(1 2 CpTpm)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpm drr2m)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2+ Cp(T41)]TJ /F3 11.955 Tf 11.96 0 Td[(T4m)r2m=1 3(r3m)]TJ /F3 11.955 Tf 11.96 0 Td[(r3m)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2)Tp+1m)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpm dt,(3) km)]TJ /F14 5.978 Tf 7.79 3.26 Td[(1 2 CpTpm)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpm drdr2m)]TJ /F6 11.955 Tf 13.15 8.08 Td[(1 22+ Cp(T41)]TJ /F3 11.955 Tf 9.29 0 Td[(T4m)dr2m2=1 3dr3"m3)]TJ /F11 11.955 Tf 11.96 16.85 Td[(m)]TJ /F6 11.955 Tf 13.15 8.08 Td[(1 23#Tp+1m)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpm dt,(3) Fom)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tpm)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 10.74 0 Td[(Tpm)m)]TJ /F6 11.955 Tf 13.15 8.08 Td[(1 22+dt Cpdr(T41)]TJ /F3 11.955 Tf 10.74 0 Td[(T4m)m2=1 3"m3)]TJ /F11 11.955 Tf 11.95 16.85 Td[(m)]TJ /F6 11.955 Tf 13.15 8.08 Td[(1 23#(Tp+1m)]TJ /F3 11.955 Tf 10.74 0 Td[(Tpm).(3) Rearrangingonelasttimeonearrivesatthenalresultforthediscretizationequationfortheradiationboundarynode: Tp+1m=Tpm+3 hm3)]TJ /F11 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F9 7.97 Tf 13.15 4.7 Td[(1 23i"Fom)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tpm)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpm)m)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22+dt Cpdr(T41)]TJ /F3 11.955 Tf 11.95 0 Td[(T4m)m2#.(3) 3.2.4TheImplicitFiniteDifferenceMethod Itshouldbenotedthatthediscretizationequationsderivedintheprevioussectionwereanexampleofanexplicitnitedifferenceformulation.Intheimplicitnitedifferenceformulation,thenewnodalunknownsarewrittenintermsofeachotherandmustbecalculatedsimultaneously.Thediscretizationequationfortheimplicitformulationwillhavethefollowingform(Patankar,1980): aiTp+1i=ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1Tp+1i)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ai+1Tp+1i+1+bTpi+c. Hence,thenodalunknownsmustbesolvedsimultaneouslyforeachnewtimestep.Therearemanystrongdifferencesbetweentheimplicitandexplicitnitedifferenceformulations.First,sincetheimplicitmethodrequiresasimultaneoussolutionofthediscretizationequationsforeachtimestep,thecomputationexpensewill,ingeneral, 55

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begreaterthantheexplicitmethod.However,thebenetisthatimplicitschemesareunconditionallystable(Incropera,2002).Thatis,nomatterhowgreatthetimestep,physicallyrealisticsolutionsareguaranteedtobefound. Onemaywritetheresultsoftheprevioussection'sderivation,inthemanneroftheimplicitmethodasfollows.Thediscretizationequationfortheinternalnodesbecomes: Tp+1n=Tpn+1 nhFon)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tp+1n)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+1n)(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)+Fon+1 2(Tp+1n+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+1n)(n+1)i.(3) Thediscretizationequationforthesymmetryboundarynodeis: Tp+10=Tp0+6Fo)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Tp+11)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+10.(3) Andthediscretizationequationfortheradiationboundarynodeis: Tp+1m=Tpm+3 hm3)]TJ /F11 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.47 -9.68 Td[(m)]TJ /F9 7.97 Tf 13.15 4.71 Td[(1 23i"Fom)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tp+1m)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tp+1m)m)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22+dt Cpdr(T41)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+1m4)m2#.(3) While,ingeneral,schemestosolvematrixequationsresultingfromimplicitnitedifferencemethodshavecomputationalcomplexityO(N2)orO(N3),certainsimplealgorithmscanbefoundforlimitingcases.Suchanalgorithmisdescribedforpureconductioninthenextsection. 3.2.5TheTridiagonalMatrixAlgorithm Onceapartialdifferentialequationisapproximatedbyaseriesofnitedifferenceequations,whetherinanexplicit,implicit,orCrank-Nicholsmethod,thatsystemofequationsmustbesolvedsimultaneouslyfortheunknownvaluesofthedependentvariableateachnode.Therearenumerousgeneralmethodsthatmaybeusedtosolvesuchsystems,severalofwhichwillbediscussedinthepresentwork.Methodsforsolvingsystemsofalgebraicequationssimultaneouslycanbegroupedintotwo 56

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categories:directmethodsanditerativemethods.Directmethodsaresimplythoseemployinganite,deterministic,proceduralsolutiontothesystemthatrequiresnoiterativeconvergence.Iterativemethods,ontheotherhand,requireaproceduralcalculationtobeperformediterativelyuntilanacceptedconvergencehasbeenachieved. Therstmethodtobediscussedisadirectmethodthatmaybeusedtosolveasystemofequationsthat,whenwritteninmatrixform,produceatridiagonalmatrix.Suchsystemsarecommonlyencounteredinsolvingheatconductionequations. First,thediscretizationequationsarewritteninthefollowingform(Patankar,1980): aiTi=biTi+1+ciTi)]TJ /F9 7.97 Tf 6.58 0 Td[(1+di. Eachdiscretizationequationis,ingeneral,onlydependentonthreeconsecutivenodalunknowns,therebyproducingatridiagonalmatrixwhenwrittenasamatrixequation.Here,thenodesivaryas1,2,3,...,N.Notethatforthespecialcaseoftheboundaryequations,thecoefcientsc1andbNaresetas: c1=0andbN=0. Considerthediscretizationequationfortheboundaryatnode1.ThatequationhasasitsunknownsT1andT2.Thatrelationmaybesubstitutedintothediscretizationequationfornode2,resultinginanequationoftwounknowns,T2andT3.Ingeneraleachnodalequationcanthenberewrittenintheform: Ti=PiTi+1+Qi wherethecoefcientsPiandQiaregivenbythefollowingrelations: Pi=bi ai)]TJ /F3 11.955 Tf 11.96 0 Td[(ciPi)]TJ /F9 7.97 Tf 6.58 0 Td[(1 57

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Qi=di+ciQi)]TJ /F9 7.97 Tf 6.59 0 Td[(1 ai)]TJ /F3 11.955 Tf 11.95 0 Td[(ciPi)]TJ /F9 7.97 Tf 6.58 0 Td[(1 Thesolutionofthesystemofequationsisthenfoundasfollows(Patankar,1980). 1.CalculateallPi'sandQi'sfromi=1toi=Nfromtheequationsabove. 2.NotethatsincebN=0,thenPN=0andthereforeTN=QN. 3.SolvebackwardsforTifromi=N)]TJ /F6 11.955 Tf 11.96 0 Td[(1toi=1. Thissimplealgorithmisstraightforwardandrelativelyinexpensivecomputationally. 3.2.6TheSIMPLEAlgorithm TheTridiagonalMatrixAlgorithmisusedtosolveasystemofdiscretizationequationsforthespecialcaseinwhichtheyproduceatridiagonalmatrixequation.Suchasystemisoftenencounteredwhensolvingheatconductionormassdiffusionproblems.Ingeneral,thesolutionofapartialdifferentialequationthatcontainsconvectivetermsandnon-linearsourcetermsproducesasetofdiscretizationequationsthatarenotsoeasilysolved.Inaddition,manypracticalproblemsrequirethesolutionofmultiplepartialdifferentialequationssimultaneously.OneprocedureforsolvingsuchproblemsistheSemi-ImplicitMethodforsolvingPressure-LinkedEquations,orSIMPLE. TheSIMPLEalgorithmwasspecicallydesignedtosolvetheNavier-Stokesequationsfortheunknownvelocitydistributionwhenthepressureeldisalsounknown(Patankar,1980).Inatwo-dimensionalowsituation,forexample,thesystemofequationsconsistsofthecontinuityequationandtwomomentumequations(oneforeachcoordinatedirection).Thesethreeequationsarenecessarytosolveforthetwounknownvelocitycomponentsandthepressureeld.Theproblem'schiefdifcultyappearswhenoneattemptstosolvethediscretizationequationswithoutregardtothephysicsofthesituation.Caremustbetakenifoneistoobtainphysicallymeaningful,convergedsolutions. IntheSIMPLEalgorithm,arstguesstothepressureeldisusedtosolvethemomentumequations.Thecontinuityequationisthensolvedproducingacorrection 58

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tothepressureeld.Thecorrectedpressureisthenusedtocalculateacorrectedvelocitywhichbecomesthenalvalueforthevelocityattheendofagiveniteration.Theprocedureisrepeatediterativelyuntilconvergenceisfound.ThestepsintheSIMPLEalgorithmforaone-dimensionalcaseareoutlinedbelow(Patankar,1980). 1.Guessthepressureeldp. 2.Solvethemomentumequationtoobtainthevelocityeldu.aiui=ai+1ui+1+ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1ui)]TJ /F9 7.97 Tf 6.59 0 Td[(1+b+(pi)]TJ /F3 11.955 Tf 11.95 0 Td[(pi+1)Ai 3.Solvethecontinuityequationforthepressurecorrection,p0.aip0i=ai+1p0i+1+ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1p0i)]TJ /F9 7.97 Tf 6.59 0 Td[(1+b 4.Calculatethecorrectedpressurepi=p0i+pi. 5.Calculatethecorrectedvelocityui=ui+Ai ai(p0i)]TJ /F3 11.955 Tf 11.95 0 Td[(p0i+1). 6.Oncethevelocityeldisknown,solveforotherunknowns,suchasTintheenergyequation,etc. 7.Repeatfromstep2,untilaconvergedsolutionisachieved. TheSIMPLEalgorithmisapowerfultoolforsolvingnumerouspartialdifferentialequationssimultaneously.Inthepresentwork,theSIMPLEalgorithm,ormorepreciselythemodiedSIMPLERalgorithm,isusedtondtheunknownvelocity,pressure,andtemperaturebysolvingthemomentum,continuity,andenergyequationssimultaneously. 3.2.7TheSIMPLERAlgorithm TheSIMPLERalgorithmisausefulrevisiontotheSIMPLEalgorithmandstandsforSemi-ImplicitMethodforthesolutionofPressure-LinkedEquations,Revised.ThechiefadvantageofSIMPLERoverSIMPLEareitsimprovedconvergence.AlthoughthecomputationaleffortrequiredforoneiterationofSIMPLERislargerthanthatofSIMPLE,thefasterrateofconvergenceofSIMPLERresultsinfastertotalcomputationaltimesoverSIMPLE. 59

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ThemajordifferenceintheSIMPLERprocedureisthemannerinwhichthevelocityeldiscorrected.InSIMPLER,onestartswithaguessforthevelocityeldandusesthisvelocitytoapproximatethepressureeld.Withthepressureeldathand,themomentumequationsaresolvedforvelocity.Thevelocityeldisthencorrected,butitisnolongernecessarytocorrectthepressure.Theprocedureisthenrepeatedlyiterativelyuntilconvergence.ThestepsintheSIMPLERalgorithmareoutlinedinmoredetailbelow(Patankar,1980). 1.Guessthevelocityeld. 2.Calculatethepseudovelocityeld,^ui=ai+1ui+1+ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1ui)]TJ /F9 7.97 Tf 6.59 0 Td[(1+b ai 3.Solvethecontinuityequationtoobtainthepressureeld,p,wherethecoefcientsarecalculatedfromthepseudovelocities,aipi=ai+1pi+1+ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1pi)]TJ /F9 7.97 Tf 6.59 0 Td[(1+b 4.Withpknown,solvethemomentumequationforu,aiui=ai+1ui+1+ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1ui)]TJ /F9 7.97 Tf 6.59 0 Td[(1+b+(pi)]TJ /F3 11.955 Tf 11.95 0 Td[(pi+1)Ai 5.Solvethecontinuityequationtoobtainthepressurecorrection,p0,aip0i=ai+1p0i+1+ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1p0i)]TJ /F9 7.97 Tf 6.59 0 Td[(1+b 6.Correctthevelocityeld,butdonotcorrectthepressureeld.p=p,ui=ui+Ai ai(p0i)]TJ /F3 11.955 Tf 11.96 0 Td[(p0i+1) 7.Oncethevelocityeldisknown,solveforotherunknowns,suchasTintheenergyequation,etc. 8.Repeatfromstep2,untilaconvergedsolutionisachieved. 60

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NotethatSIMPLERdoesnotrelyonaguessedpressureeld,butratheraguessedvelocityeldasitsrststepasintheSIMPLEalgorithm. 3.2.8SolvingforRootsofNon-LinearEquations Manynumericalendeavorsrequiretheroutinesolutionfortherootsofnon-linearalgebraicequations.Assuch,itisnecessarytoincludeashortdescriptionoftworoot-ndingmethodsusedcommonlyinthepresentwork.Thebisectionmethodandxed-pointiterationwillbediscussed. 3.2.8.1Thebisectionmethod Thebisectionmethodisaprocedurethatguaranteesonetondarootgiventhatafunction,f(x),iscontinuousonaninterval[a,b],suchthatf(a)f(b)<0. Iftheinterval[a,b]canbechosensuchthatonlyonerootispresent,thenthebisectionmethodcanbeguaranteedtondit.Thestepsinthebisectionmethodareoutlinedbelow(Atkinson,1978). 1.Letc=(a+b)=2. 2.If(b)]TJ /F3 11.955 Tf 11.96 0 Td[(c)=ctolerance,thenroot=candexit. 3.Iff(b)f(c)0,thena=c,otherwiseb=c. 4.Returntostep1. Essentiallythebisectionmethodhalvestheintervalofinterestforeveryiterationthroughthealgorithm.Theintervalishalvedcontinuouslyuntilthedesiredtoleranceisachieved,calculatedasthepercentchangefromoneguessctothenext.Whenthetoleranceisreached,theguessedrootisthemidpointoftheintervalofinterestinwhichtherootisknowntolie. Thebisectionmethodisnotthefastestmethodofconvergence,butitisthemostdependableinthatitwillalwaysndarootinthegiveninterval[a,b]ifoneexists. 61

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3.2.8.2Fixed-pointiteration Thesecondroot-ndingmethodtobediscussedisthegeneralxed-pointiteration,ofwhichNewton'smethodisanexample.Inxed-pointiterationonesolvesanequationx=g(x)byperformingthefollowingiteration: xn+1=g(xn), wherex0isaninitialguess.Iterationoftheequationaboveisperformeduntiltheerror,jxn+1)]TJ /F3 11.955 Tf 12.1 0 Td[(xnj=xn+1,issufcientlysmall.Thebenetofxed-pointiterationisthatforcertainfunctionsitsconvergenceisquiterapid.Unfortunately,incertainsituations,themethodmayfailtondarootandsoadiscussionoftheuniquenessofsolutioniswarranted. Itcanbeshown(Gerald,1997)thatifg(x)andg0(x)arecontinuousonaninterval[a,b]andifjg0(x)j<1forallxin[a,b],thenthemethodofxed-pointiterationwillconvergetoarootinthatinterval.Thiscondition,whilesufcient,isnotalwaysnecessaryinthatarootmaystillbefoundevenifjg0(x)j>1.Forpracticalimplementationswherethisconditionmaynotapply,itisusefultoexamineifconsecutivexnvaluesconverge,thatis:jx3)]TJ /F3 11.955 Tf 11.96 0 Td[(x2j
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Legendrepolynomialsasbasisfunctions,itisnecessarytoevaluateaseriesofthepolynomials,Pn(x),toasufcientlylargentoguaranteeconvergence. AgeneralexpressionforthedenitionofeachLegendrepolynomialforanynisgivenbyRodrigues'formula Pn(x)=1 2nn!dn dxn(x2)]TJ /F6 11.955 Tf 11.96 0 Td[(1)n.(3) wherethedomainisusuallyjxj<1.TheLegendrepolynomialsarethensimpletodetermineouttoanynecessarynbyRodrigues'formula.TherstsixLegendrepolynomialsareshownbelowandplottedin 3-4 P0(x)=1(3) P1(x)=x(3) P2(x)=1 2(3x2)]TJ /F6 11.955 Tf 11.95 0 Td[(1)(3) P3(x)=1 2(5x3)]TJ /F6 11.955 Tf 11.95 0 Td[(3x)(3) P4(x)=1 8(35x4)]TJ /F6 11.955 Tf 11.95 0 Td[(30x2+3)(3) P5(x)=1 8(63x5)]TJ /F6 11.955 Tf 11.96 0 Td[(70x3+15x)(3) Theevaluationofthesepolynomialsistrivialandtheimplementationofthesecalculationswithinanumericalschemeisstraightforward.Adifcultyarises,however,whencalculatingtheLegendrepolynomialsforincreasinglyhighvaluesofn.InmanypracticalengineeringapplicationsthecalculationofonlythersttenLegendrepolynomialsin 63

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theseriesmaybeneededtoachievegoodconvergence.However,therearealsomanypracticalcases,suchasnearboundaryconditionsordiscontinuities,wheretheseriesmustbecarriedouttoanexcessof100termsormoreinordertoconverge. Consider,forexample,theLegendrepolynomialforn=17whosehighestordertermis P17(x)=1 229,3764,083,810,885x17+(3) ThemoststraightforwardprocedureforthenumericalcalculationoftheseriesofLegendrepolynomialswouldbetostoretheappropriatecoefcientsforeachtermandevaluatethestandardformofeachpolynomialdirectlyateachxrequired.Butalready,atn=17,theintegerpolynomialcoefcientsrequireatleasttendigitsofprecision.Moreover,theevaluationofthe17thpowerofanxwithinjxj<1mayeasilyfallclosetomachineprecision.Ultimately,calculatingtheLegendrepolynomialsinthisway,arguablythemoststraightforwardevaluationprocedure,isprohibitivepastn=17orn=18duetothelimitationsofmachineprecision,whichonmanycomputersisaround17to18digits. Instead,onecantakeadvantageofanalternateexpressionfordeterminingtheLegendrepolynomials.TheLegendrepolynomialsarealsoobtainablefromarecurrencerelationshipgivenby: Pn+1(x)=2n+1 n+1xPn(x))]TJ /F3 11.955 Tf 23.59 8.09 Td[(n n+1Pn)]TJ /F9 7.97 Tf 6.58 0 Td[(1(x).(3) UsingtherecurrencerelationwecancalculatetheLegendrepolynomialsatanyxforanysufcientlylargenwithouttheneedtoexplicitlydeneeachpolynomial.ItiseasytoseethatwithP0(x)=1andP1(x)=x,therepeatedapplicationof 3 n)]TJ /F6 11.955 Tf 12.94 0 Td[(1timesresultsinthedirectcalculationofPn(x)foranysinglex.Inaddition,sinceeachjPn(x)j<1,thereisnodangerofreachingmachineprecision. 64

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Itmay,atrst,appearthatcalculatingtheLegendrepolynomialsthroughtherecurrencerelationislessefcient,sinceinthedirectcalculationthepolynomialcoefcientsarealreadydened,storedinmemory,andmaybeaccessedthroughlook-up.Butthisisnotthecase.Themostefcientalgorithmsforthedirectcalculationofapolynomial,suchasHorner'smethod,haveatimecomplexityof(n)(Horowitz,1998).Itiseasytoseeuponinspectionthattheevaluationof 3 foranynisalsodonein(n)time.Inadditionthedirectcalculationofaseriesofpolynomialswhosecoefcientsarestoredinmemoryrequires(nlogn)spaceatbest,whereaswiththerecurrencerelationonly(1),orconstantspaceisneeded. Theuseoftherecurrencerelationship, 3 isthereforeamoreefcientmethod,bothinspaceandtime,forcalculatingLegendrepolynomialsforanyarbitrarynthatdoesnotencounterthemachine'slimitsofprecision. 3.3AutomatedPeakDetectionAlgorithms Itisoftendesirabletoautomatetheprocessofdetectingpeaksandrecordingtheircharacteristicsinanyspectroscopicapplication.Thisisespeciallydesirablewhenprecisepeakinformationmustbetakenfromensemblesthatcontainnumerousspectratotheextentthatpeakdetectionbyhandisnotpracticalpurelyfortimepurposes.However,automatedpeakdetectionmethodsarenotwithouttheirdifculties. Automatingpeakdetectionroutineshasmanyadvantagesanddisadvantages.Theadvantagesarethatpeakdetectionisautomatedandcanbecompletedinafractionofthetimeittakesthesameprocesstobedonebyhand,thehumanbiasisremoved(byagreatdeal,butnotperfectlyso)fromthepeakdetectionprocess,eachpeakisknownwiththesamecertainty,andthatcertaintycanbequotedcondentlytowithinfractionsofpixels.Thedisadvantageofpeakdetectionalgorithmsisthecomplexityofsaidalgorithmsthatisnecessarytoachievecondencethatonehaddetectedeveryimportantpeakandnotdetectedanyfalsepeaks.Removingthehumanbiasfrompeakdetectionisatwo-waystreet.Inordertoensurethatallimportantpeakshavebeen 65

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correctlyidentied,itisoftennecessarytone-tunecertainparametersofthealgorithm,whoseoptimumvaluesmightdifferfromonecasetoanother.Theseoptimumvaluesarecertainlynotknownbeforehand.Inaddition,theseparametersmustnotbedenedtooconservatively.Ifnotstrictenough,thealgorithm'sparametersmaydetecttoomanypeaksfromwhatitshouldrecognizeaslowerfrequencynoise. Allclassesofautomatedpeakdetectionalgorithmsmustconsistessentiallyofthreemaincomponents:smoothing,baselinecorrection,andpeaknding(Yang,2009).Allrawspectrathatonemightprocesswithanautomatedpeakdetectionroutinearepresumedtocomefromrealsources,suchasatomicemissionspectra,massspectra,orothers.Assuch,allrealrawspectraareknowntocontainsomelevelofnoiseatvaryingfrequencies.Thesmoothingprocessisessentiallydesignedtoremoveallnoiseaboveacertainfrequency.Byapplyingsometypeoflow-passsmoothingltertothedata,muchofthesmall,peak-likenoisecanberemoved. Baselinecorrectionisessentiallyameanstonormalizethespectra.Onewouldexpect,ordesire,thatnoisysectionsofdata,orsectionsthatcontainnopeakinformation,shouldbeclosetozero.Allrealdatacontainsomebaselineoffsetorcontinuumspectrathatmustberemovedtomaketheprocessofactuallyidentifyingpeakseasier.Oftenbaselinecorrectioncanbeachievedwithasimplesubtractionifbaselinedataisclosetouniform.Therearecases,however,whenbaselinesexhibitmonotonicallyincreasingordecreasingbehaviorandthealgorithmforbaselinereductiongrowsincomplexity. Finally,oncethebaselinehasbeenremovedandthedatasmoothedtoeliminateobviousnoisyuctuations,onlythencantheactualpeakndingroutinebeemployedwithrelativeease.Inallbutthemostidealorwellbehavedofcases,smoothingandbaselinecorrectionwillstillleavesomelocalmaximainthedatathatarenottruepeaksonewouldwanttodetect.Thepeakndingprocessusuallythenconsistsoftwosteps:identifyingalllocalmaximaandthendeterminingwhichofthelocalmaximaareimportantandwhicharenotimportant.Thestepofdeterminingwhichlocalmaximaare 66

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worthyofbeingfoundasapeakisaccomplishedbytheuseofathresholdvaluedenedinanynumberofways.Alllocalmaximaabovethisthresholdvaluearecountedasapeak,whilealltheremaininglocalmaximabelowthisvaluearenot. 3.3.1Smoothing OneofthesimplestltersusedforsmoothingdataistheMovingAveragelter.Eachdatapointisrecalculatedtobeamovingaverageofitssurroundingkdatapointsgivenbythefollowingformula, x0[n]=1 2k+1kXi=)]TJ /F5 7.97 Tf 6.59 0 Td[(kx[n)]TJ /F3 11.955 Tf 11.95 0 Td[(i], wherex[n]representsthedatabeforesmoothingandx0[n]representsthedataaftersmoothing.Theparameterkdeterminesthesizeofthelterwidthandthereforetheintensityofthesmoothingeffect.Thelterwidthisgivenbytheexpression2k+1,whichisthenumberofpointsincludedinthemovingaverage.Thegreaterthevalueofk,thegreaterthelterwidth,andthemoreintensethesmoothingeffect.Thechoiceoftheparameterkisthenparamounttotheeffectivenessofthesmoothingoperation.Toohighavalueofkmayreducefeaturesthatshouldbedetectedaspeaks,whiletoolowavalueincreasesthestrainonthepeakndingalgorithmperformedlaterandcouldpotentiallyresultinthedetectionofafalsepeak. Smoothingltersarealsowrittenasaconvolutionoftheoriginaldatavectortothelterwindowasseenbelow, x0[n]=x[n]w[n], wherew[n]isthelterwindow,whichforthemovingaveragelterisgivenby: w[n]=1 2k+1for)]TJ /F3 11.955 Tf 11.96 0 Td[(knk. 67

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Othersmoothingltersoffermuchinthewayofmorerobustsmoothing,wherevaluesoflterparameterscanbechosenthatproducegoodbehaviorforagivenclassofrawdata.Additionalsmoothinglterswillbediscussedinthefuture. 3.3.2BaselineCorrection Oncerawdataissmoothed,thenextstepistocorrectforthebaseline.Baselinecorrectionessentiallyconsistsoftwosteps:determiningthebaselineofthedataandthentheactualremovalofthebaseline.Thesecondstepisusuallyjustasimplesubtraction.TherstmethodwewilldiscussforthedetectionandremovalofthebaselineistheMonotoneMinimummethod(Yang,2009).TheMonotoneMinimummethodismostusefulforabaselinewhosebehaviorismonotonicallydecreasingfromthestarttotheendofthedata.ForoptimaleffectivenessoftheMonotoneMinimummethodonemaywishtoreordertherawdatadependingonthebaseline'sapparentbehavior.Asastartingpointwe'llsuggestthatthedatapointsbereversedifthenaldatapointisgreaterthantheinitialdatavalue(thiscorrection,ofcourse,assumesthatpeakinformationisnotcontainedintherstorlastdatapointinthespectra).Inotherwords,ifx[N]>x[0],thenlet x0[n]=x[N)]TJ /F3 11.955 Tf 11.96 0 Td[(n]. Thisreversalguaranteesthatifthebaselineshowseitheramonotonicallyincreasingordecreasingbehavior,thatthebaseline-correcteddatawillbeorderedappropriately. Todeterminethebaseline,thedifferencebetweeneachconsecutivedatapointisrstcalculatedtodeterminetheslopes[n]ateachpointngivenby: s[n]=x[n+1])]TJ /F3 11.955 Tf 11.96 0 Td[(x[n]. Nexttheslopevectorisscanned.Iftheslopeofapointisnegative,thevalueofthosepointswillbetakenasbaseline.Iftheslopeofapointisnotnegative,thevalueofthatpointwillbethebaselineforallsubsequentpointsuntiladatapointisfoundsuchthat 68

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itsvalueislowerthanthebaseline.Oncethebaselineisdetermined,thebaselinecorrecteddataisthengivenby: x0[n]=x[n])]TJ /F3 11.955 Tf 11.95 0 Td[(b[n], whereb[n]isthebaselinevector.Ifthedatavectorwasreversedtoensureamonotonicallydecreasingbaselineabove,thenthebaselinecorrectedvectormustbere-reversed,asanalstep,topreservetheoriginalpixelspace. 3.3.3PeakFinding Oncetherawdatavectorhasbeensmoothedandthebaselinecorrected,itisthenpossibletoidentifytheavailablepeaks.Smoothingandbaselinecorrectionmaybeappliedsuccessivelytoproduceasatisfactorilyconditioneddatavector.Hereonewillassumethatallsmoothingandbaselinecorrectionshavebeencompleted.Atthisstage,thepeakndingalgorithmconsistsoftwomainparts:determiningalllocalmaximaanddeterminingwhichlocalmaximaarepeaksandwhicharenot. Thedeterminationofalllocalmaximaisatthispointarelativelytrivialstep.Alocalmaximumpointisthepointwheretheslopechangesfrompositivetonegative.Oncealllocalmaximahavebeenidentiedonemustthenusesomecriteriatochooseifalocalmaximumisindeedworthyofbeingdesignatedasapeakorifthelocalmaximumisaremnantofsomelowerfrequencynoise.Usuallythisdecisionprocessisbasedonasimplethresholdvaluethatisacharacteristicoftherelativestrengthofapeak.Abovethisthresholdvaluethepeakisstrongenoughtobedetected.Belowthisthresholdvalue,thepeakisnotstrongenoughtobedetected. Thedecisioncriteriadiscussedhereistheshaperatio.Theareaunderthecurveforeachlocalmaximumwillbedetermined.Thecriteriawillbedeterminedbytheratioofeachareatothemaximumareafound.Putanotherway,if: An Amax>kT, 69

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thenAnrepresentsapeak.HerekTissomethresholdvaluethatmustbechosen.TypicallykTwillbethemostimportantfactorthatdeterminesthesensitivityoftheoverallpeakdetectionalgorithmandshouldbechosenwithcaution. Thereareseveralothercriteriabywhichonemaychooseifagivenlocalmaximumqualiesasapeak.Employingseveraldifferentcriteriasimultaneouslymayhelptoinstillcondencethatnotruepeaksareneglectedandnofalsepeaksaredetected.Absolutepeakintensityisoneadditionalcriteriathatmayused.Similartotheshaperatio,iftheabsoluteintensityratioofanypeaktothemaximumpeakissufcientlylarge,thatpeakmaybeatruepeak.Onemaychoosepeakwidth,ortheleft-handandright-handpeakslopesasthecriteria.Inthiscase,apeakmustbesufcientlywideincomparisontothewidestpeaktobeidentiedasatruepeak. 3.3.4PeakIdealization Onceapeakhasbeenidentieditisoftennecessarytodoadditionalprocessingonthatfeaturedependingontheapplication.Onemaywishtoworkwiththeoriginalorconditioneddataanditisnecessarytokeeptrackofseveralcharacteristicsofthepeakinadditiontosimplyitslocation,suchasitsFWHM,itspeakintensity,thelocationofitsendpointsandothers.Ausefulalternativeistotacharacteristicproletothepeakonceitisidentied.Itisadvantageousinmanyapplicationstohaveasimpleclosedexpressionforeachpeakratherthanadatavectordependingontheanalysistobedone.Thechoiceofthemathematicalformapeakshouldtakeissubjectiveandshouldbedeterminedbasedontheunderlyingphysicalbasisforthefeature.TheanalysisofphysicalimagesinthecurrentstudyproducepeaksthattendtotwellwithGaussianproles. EachpeakdetectedbythecurrentalgorithmthatcorrespondstoaphysicalfeatureisttoaGaussianfunctionoftheform: g(x)=Ae)]TJ /F5 7.97 Tf 6.58 0 Td[(b(x)]TJ /F5 7.97 Tf 6.59 0 Td[(x0)2 70

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whereAisthemaximumvalueofthepeak,bisavaluerelatedtothewidthofthepeak,andx0isthepeak'scenterlocation. TheanalysisofspectratendstoproducepeaksthatmaydeviatefrompureGaussianbehavior.InsteadVoigtproles,whicharecombinationsofbothGaussianandLorentzianproles,areusedtomodelpeaksdetectedfromspectroscopicdataandarediscussedin 3.1.4 71

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Figure3-1. ComparisonofDoppler,Lorentzian,andVoigtprolefunctions. 72

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Figure3-2. TheVoigtprolefunctionforvariousvaluesofthedampingparameter,a. 73

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Figure3-3. Controlvolumeforageneralinteriornode. 74

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Figure3-4. TherstsixLegengrepolynomialsoftherstkind. 75

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CHAPTER4THESTATIC,CONDUCTIVEPLASMAMODEL 4.1Overview Therststeptowardthedevelopmentofarigorousmodeloftheplasma-particleinteractioninaerosolLIBSisthedesignofamodeltodescribetheplasmaenvironment.Theplasmamodeldescribestheglobalenvironmentinwhichthevaporizationmodel,describedinthenextchapter,willbecontained.Thecompletemodelwillbeasynthesisofthesetworegimes:theglobalmodelandthelocalmodel. Theglobalplasmamodelbeginsasasimplecasetowhichadditionalcomplexitiesandsophisticationswillbeappliedgradually.Buildingasimple,andthereforesimplytestable,modelandincreasingsophisticationgraduallyisnecessarytoensurethemodelbehavesappropriately. Hereasimpleplasmamodelisimplemented,wheretheplasmaismodeledasastatic,conductivegas.Thetemperaturedistributioninspaceandtimeisfoundbysolvingtheequationofheattransfer.Thedistributionofspeciesconcentrationisfoundbysolvingtheequationsofmassdiffusion.TheionizationstatedistributionsandexcitedenergyleveldistributionsarefoundfromtheSahaandBoltzmannrelations.Finally,theemittedintensityiscalculatedandusedtosimulatetheexperimentalmeasurementoftemperature. ThenumericalformulationthatfollowsisimplementedintheC/C++programminglanguageandexecutedonamachineusinga2.6GHzIntelCore2Quadprocessor.Allpost-processingisdoneinMatlab. 4.2TheProblemStatementandSimplifyingAssumptions Theplasmaenvironmentismodeledasaone-dimensional,time-dependent,sphericallysymmetricsystem.Assuch,modelinputparametersandoutputquantitieswill,ingeneral,varywithbothradiusfromtheplasmacenterandtime.Thesystemisassumedtobestatic,thatis,thevelocityeldiszeroeverywhereandnoconvective 76

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termsappearineithertheenergytransportequationorthemasstransferequation.Localthermodynamicequilibriumisassumedtoholdatallnodesforalltimesthroughoutthemodelingprocess. Thetemperatureeldisfoundbysolvingtheequationofenergytransport,writtenforaone-dimensional,time-dependantspherically,symmetricsystemasshownbelow: 1 r2@ @rkr2@T @r+_q=Cp@T @t,(4) Hereitisassumedthatconductionistheonlymodeofenergytransport.Theconvectiveterms,whileplayingapotentialroleinthephysics,willnotbemodeledhereforthesakeofsimplicityandcomputationalcost.Manylaser-inducedplasmasaremodeledasopticallythin,andassuch,theradiativetermsintheenergyequationcanbeshowntobenegligibleforallbuttheearliestofplasmalifetimes(Gornushkin,2001).Theradiativetermsintheenergyequationwillbeaddressedagainduringthestudyofplasmainception. Thespeciesconcentrationdistributionisfoundbysolvingtheequationofmasstransfer,writtenforaone-dimensional,time-dependant,sphericallysymmetricsystemasshownbelow: 1 r2@ @rDABr2@CA @r+_NA=@CA @t.(4) Sincenobulkvelocityeldisassumedinthiscase,theonlymodeofmasstransportisthroughmassdiffusion. Thematerialcomprisingtheplasmaisassumedtobepureargongas.Thesolutionoftheenergyequationisthenastatementofargontemperatureateachpointintheplasma.Analytespeciesintheplasmamaybeeitheroftwocomponents:cadmiumormagnesium.Theseelementsareusedinthepresentstudyprimarilybecauseexperimentaldataexistsforpartialvalidation(Diwakar,2007).Thespecies 77

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concentrationofeachthroughouttheplasmavolumeisfoundindependentlyasarstapproximation. Lastly,itisimportanttonotethatallmaterialproperties,andhencethecoefcientsofthepartialdifferentialequations,areallowedtobefunctionsoftemperature.Thisamountstoanenergyequationwithnon-constantcoefcientsandamassdiffusionequationwithnon-constantcoefcientsthatiscoupledtotheenergyequation. Basedontheseconsiderations,theequationsofheattransferandmassdiffusionarethensolvedforthetemperatureandspeciesconcentrationdistributionsusingbothexplicitandimplicitnitedifferenceformulations. 4.3NumericalFormulationandImplementation 4.3.1HeatTransfer Theenergyequationtobesolvedisgivenby 4 inone-dimensional,sphericalcoordinates.Thepartialdifferentialequationissolvedusingnitedifferenceapproximations.Theproblemdomainisdenedtobeaspherewitharadiusof1.5mm,withtemperatureevaluatedat101nodes.Startingatthissizeneglectsapproximatelytherst100nsofrapidplasmaexpansion.Thegridspacingistherefore: r=1.5mm 101)]TJ /F6 11.955 Tf 11.96 0 Td[(1=15m(4) Thesimulatedtimeisallowedtoencompassatotalof30s,evaluatedat30,001temporalnodes.Thetimeresolutionistherefore: t=30s 30001)]TJ /F6 11.955 Tf 11.95 0 Td[(1=1ns(4) Sincetheenergytransportequationissecondorderinspaceandrst-orderintime,twoboundaryequationsandasingleinitialconditionarerequiredtouniquelysolveforthetemperaturedistribution.Theboundarynodeatr=0istakenasasphericalsymmetrycondition(i.e.,@T @rr=0=0),whichismathematicallyimplemented 78

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asaninsulatedboundary.Theboundarynodeatr=Rlosesheatbyradiationtotheenvironmentattemperature,T1, )]TJ /F3 11.955 Tf 11.95 0 Td[(k@T @rr=R=(T(R))]TJ /F3 11.955 Tf 11.95 0 Td[(T1)4(4) 4.3.1.1Theexplicitnitedifferenceformulation Theproblemisrstsolvedusinganexplicitnitedifferenceformulationforsimplicity.ThenitedifferenceequationsarederivedusingacontrolvolumemethoddescribedindetailinSection2.2.3.Thenitedifferenceequationfortheinternaltemperaturenodesaregivenbythefollowing: Tp+1n=Tpn+1 nhFon)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tpn)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1)+Fon+1 2(Tpn+1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpn)(n+1)i,(4) wheretheFouriernumberis: Fo=kt Cpr2.(4) Thediscretizationequationforthesymmetryboundarynodeis: Tp+10=Tp0+6Fo(Tp1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp0),(4) Thediscretizationequationfortheradiationboundarynodeis: Tp+1m=Tpm+3 hm3)]TJ /F11 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(m)]TJ /F9 7.97 Tf 13.15 4.7 Td[(1 23i"Fom)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tpm)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tpm)m)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22+dt Cpdr(T41)]TJ /F3 11.955 Tf 11.95 0 Td[(T4m)m2#.(4) Explicitnitedifferenceformulationsareattractiveastheirsolutionprocedureissimple.Foreachnewtimestep,thediscretizationequationscanbesolvedsequentiallyforeachnodewithouttheneedforiteration.Convergenceissuesaretherefore 79

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avoided.Stability,ontheotherhand,isnot.Explicitnitedifferenceschemesareconditionallystable,meaningthatthenumericalparametersmustbechosenwithspecicconsiderationstoavoidphysicallyunrealisticsolutions. Inthepresentcase,itisrequiredthateachcoefcientofthediscretizationequationsabovebepositive.ThisissatisedbyapplyingtheconditionthatFo1=2.Sincematerialpropertiescannotbeprescribedarbitrarily,thisisessentiallyalimitationonthetemporalandspatialgridspacing.Foraspatialgridspacingofr=15m,stabilitymaybeguaranteedfortimestepslessthan26ns. 4.3.1.2Theimplicitnitedifferenceformulation Thenitedifferenceapproximationwasalsoformulatedimplicitlyandcomparedtotheexplicitformulation.Ingeneral,implicitnitedifferenceformulationsaremorenumericallyaccuratetotruesolutionsandhavethebenetofbeingunconditionallystable.Unconditionalstabilityimpliesthatanychoiceofgridspacinginspaceortimewillyieldaphysicallyrealisticsolution.Thedrawbackofimplicitmethodsarethat,ingeneral,theymustbesolvedusingiterativemethodsandthereforemayrequiremorecomputationaltimethanexplicitformulations. Fortunately,manyimplicitnitedifferenceformulationsthatinvolveconductionordiffusiontermsonlyproducesystemsthatmaybesolvedbytheTridiagonalMarixAlgorithm(TDMA).Sincethecurrentcasefallsintothiscategoryofproblems,littleincreaseinexecutiontimewasfoundfromtheexplicittotheimplicitformulationsforanygiventimestep.Inaddition,sincetheimplicitformulationmaybecomputedoverfewertimestepsinthesamedomain,thetotalexecutiontimemaybereduced. TheimplicitnitedifferenceformulationisdescribedinSection2.2.4.Theresultingnitedifferenceequationforeachoftheinternalnodaltemperaturesisgivenby: Tp+1n=Tpn+1 nhFon)]TJ /F14 5.978 Tf 7.78 3.25 Td[(1 2(Tp+1n)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+1n)(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)+Fon+1 2(Tp+1n+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+1n)(n+1)i.(4) 80

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Thediscretizationequationforthesymmetryboundarynodeis: Tp+10=Tp0+6Fo)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Tp+11)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+10,(4) andthediscretizationequationfortheradiationboundarynodeis: Tp+1m=Tpm+3 hm3)]TJ /F11 11.955 Tf 11.96 9.69 Td[()]TJ /F3 11.955 Tf 5.47 -9.69 Td[(m)]TJ /F9 7.97 Tf 13.15 4.71 Td[(1 23i"Fom)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Tp+1m)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Tp+1m)m)]TJ /F6 11.955 Tf 13.15 8.09 Td[(1 22+dt Cpdr(T41)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp+1m4)m2#.(4) 4.3.2MassDiffusion Theproblemofmassdiffusionisdirectlycomparabletoheattransferastheirgoverningequationstakethesameform.Assuch,thesamemethodsusedforthesolutionofheattransferproblemsmaybeusedforthesolutionofmasstransferproblems.Here,themassdiffusionequationissolvedfortheconcentrationofseveralspecieswithintheplasmadomain.Themassdiffusionequationtobesolvedisgivenby 4 .Theproblemdomainisdenedinthesamemannerasthediscretizationusedforthesolutionoftheenergyequation,namely: r=15mandt=1ns(4) Theboundaryconditionatthecenter,r=0,is,again,denedtobyasymmetryboundaryconditiontopreserverthesphericalsymmetryofthesystem.Theboundaryconditionattheouternode,r=R,isdenedtobediffusionoutintoanenvironmentof0concentration. Thecurrentproblemconsidersthreeconstituentspeciestobepresent.Theplasmacarriergasispureargon.Aparticleofvaryingcompositionisplacedatthecenteroftheplasmaenvironmentandconsistsofsomemixtureofcadmiumandmagnesiumasnotedbefore.Oncematter,beitcadmiumormagnesium,isvaporizedandliberated 81

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fromtheparticle(aprocesstobediscussedindetailinChapter5)itdiffusesthroughouttheplasmaenvironment. Thecurrentmodelisthenthatofthediffusionoftwospeciesintoathird:(1)thediffusionofcadmiumintoargonand(2)thediffusionofmagnesiumintoargon.Eachprocesswillbesolvedindependentlyanditisassumedthatthepresenceofeitherspeciesdoesnoteffectthediffusionbehavioroftheother. 4.3.2.1Theexplicitnitedifferenceformulation Themassdiffusionequationisrstsolvedbywayofanexplicitnitedifferenceformulationinmuchthesamewayastheenergyequation.Thenitedifferenceequationfortheinternalnodesofthecadmiumconcentrationcanbeshowntobe: Cp+1Cd,n=CpCd,n+t nr2hDCd!Ar,n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(CpCd,n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(CpCd,n)(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)+DCd!Ar,n+1 2(CpCd,n+1)]TJ /F3 11.955 Tf 11.95 0 Td[(CpCd,n)(n+1)i.(4) Thediscretizationequationforthesymmetryboundarynodeis: Cp+1Cd,0=CpCd,0+6DCd!Art r2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(CpCd,1)]TJ /F3 11.955 Tf 11.96 0 Td[(CpCd,0.(4) Thediscretizationequationfortheouterboundarynode,forthediffusionofmassintozerocadmiumconcentration,is: Cp+1Cd,m=CpCd,m+t mr2hDCd!Ar,m)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(CpCd,m)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(CpCd,m)(m)]TJ /F6 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(DCd!Ar,mCpCd,m(m+1)i.(4) Thenitedifferenceequationsforthediffusionofmagnesiumintoargoncanbewrittensimilarly. Theissueofstabilitymustagainbeconsidered.Thechoiceofdiscretizationstepstoensurethestabilityoftheenergyequationtonotnecessarilyguaranteethestabilityofthemassdiffusionequation.Usingasimilarargumentasbefore,onendsthatto 82

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ensurethestabilityoftheexplicitschemeformassdiffusioneachcoefcientintheequationsabovemustbepositive.Fortypicalvaluesofthediffusioncoefcient,itmaybeshownthatstabilityisguaranteedfortimestepslessthan9ns.Whilethisisamuchmorestrictrequirementonthetimestepthanwasfoundforthestabilityanalysisfortheenergyequation,itisstillsatisedbythetimeresolutionof1nschosenabove. 4.3.2.2Theimplicitnitedifferenceformulation Theimplicitnitedifferenceformulationwasagainappliedtothemassdiffusionequationtoremovetherequirementofstabilityandreducethenumberoftimestepsnecessarytoarriveatanaccuratesolution.Sincethediffusionequationsofeachspeciesresultinmatrixsystemsthataretridiagonal,theTDMAmethodmaybeusedfortheirsolutionjustaswasdoneforthesolutionoftheimplicitdiscretizationoftheenergyequation. Theimplicitnitedifferenceequationforeachoftheinternalnodesforthemassdiffusionofcadmiumintoargonisgivenby: Cp+1Cd,n=CpCd,n+t nr2hDCd!Ar,n)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Cp+1Cd,n)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Cp+1Cd,n)(n)]TJ /F6 11.955 Tf 11.96 0 Td[(1)+DCd!Ar,n+1 2(Cp+1Cd,n+1)]TJ /F3 11.955 Tf 11.95 0 Td[(Cp+1Cd,n)(n+1)i.(4) Thediscretizationequationforthesymmetryboundarynodeis: Cp+1Cd,0=CpCd,0+6DCd!Art r2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Cp+1Cd,1)]TJ /F3 11.955 Tf 11.96 0 Td[(Cp+1Cd,0.(4) Thediscretizationequationfortheouterboundarynode,forthediffusionofmassintozerocadmiumconcentration,is: Cp+1Cd,m=CpCd,m+t mr2hDCd!Ar,m)]TJ /F14 5.978 Tf 7.78 3.26 Td[(1 2(Cp+1Cd,m)]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Cp+1Cd,m)(m)]TJ /F6 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(DCd!Ar,mCp+1Cd,m(m+1)i.(4) 83

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Thediscretizationequationsforthediffusionofmagnesiumintoargonmaybewrittensimilarly. 4.3.3TemperatureDependentMaterialProperties Finitedifferenceformulationsforthesolutionofpartialdifferentialequationsreachanextralevelofcomplexitywhenthecoefcientsoftheequationsarethemselvesfunctionsoftheunknownnodalquantities.Eachofthediscretizationequationswritteninthischaptermaybewritteninthefollowingform: aiTp+1i=ai)]TJ /F9 7.97 Tf 6.58 0 Td[(1Tpi)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ai+1Tpi+1+bTpi+c,(4) inthecaseofanexplicitformulation,andas, aiTp+1i=ai)]TJ /F9 7.97 Tf 6.59 0 Td[(1Tp+1i)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ai+1Tp+1i+1+bTpi+c,(4) inthecaseofanimplicitformulation. Ifthecoefcients,ai,intheseequationsareconstant,thenthesolutionproceduresthathavebeendescribedmaybeimplementedtoprovidephysicallyrealisticsolutions.Ifthecoefcientsarenotconstant,butfunctionsofthetemperature,ai=ai(Ti),thenadditionalconsiderationsmustbemade. Typically,theprocedureforthesolutionofnitedifferenceequationswithnon-constantcoefcientsfollowsthatforconstantcoefcients,exceptforoneadditionaliterativeprocedure.Ateachnewtimestep,thecoefcientsareevaluatedbasedonthetemperatureattheprevioustimeasaninitialguess.Thatis,api=api(Tp)]TJ /F9 7.97 Tf 6.59 0 Td[(1i).Thecoefcientsarecalculatedinthismannerandthenodaltemperaturesaresolved.Thenewnodaltemperaturewill,ingeneral,notbethesametemperatureasintheprevioustimestep.Thisnewtemperatureisusedtore-evaluatethecoefcientsandtheprocessissolvediterativelyinthismanneruntilthetemperaturenolongerchanges.Allowingforthecoefcientstobedependentupontemperatureisanintroductionofaniterativeprocedureateachtimestepregardlessofsolutionprocedure. 84

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Thisiterativesolutionmaybeavoided,however,ifthetimestepistakenassufcientlysmall.Ifthetimestepsaresmallenoughthatthecoefcientsdonotchangeappreciablyfromonesteptothenext,thenthecoefcientsmaybeapproximatedfromthetemperaturevaluesoftheprevioustimestep.Inthiscase,itissaidthatthecoefcientslagbehindthetemperaturesolutionbyonetimestep. Lastly,itisimportanttonotethatsincethestabilityofexplicitnitedifferenceformulationsdependonthevalueofthecoefcientsofthediscretizationequation,itisdesirabletoemployimplicitformulationswhenthecoefcientsarestrongfunctionsoftemperature.Iftemperaturevariesoverabroadrangeofvalues,asisthecaseinalaserplasma,thestabilitycriterionmaybedifculttoachieve,requiringprohibitivelysmalltimesteps.Usingimplicitnitedifferenceformulationsavoidstoproblemoftemperature-dependentcoefcientsfrombreakingthesolutionprocedure. Forthecurrentpurposes,temperaturedependantpropertiesyieldnon-constantcoefcientsinthediscretizationequations. 4.3.3.1Density Thetemperaturevaluesinalaser-inducedplasmavarygreatlyinashortdistanceandoverashortperiodoftime.Thepertinentproblemdomainwillseetemperaturesrangingfromroomtemperaturetotensofthousandsofdegrees.Becauseofthis,thematerialpropertiescannotbetakenasconstant.Insteadtheywillbeallowedtobefunctionsoftemperature. Therstpropertyexaminedistheargongasdensity.Fujisake(2002)implementsasimulationofanargonplasmausedinweldingthatusesatemperaturedependentmodelforargondensitygivenby: =1.783(273=T)]TJ /F6 11.955 Tf 11.95 0 Td[(2.0610)]TJ /F9 7.97 Tf 6.58 0 Td[(7T+6.7210)]TJ /F9 7.97 Tf 6.59 0 Td[(11T2)]TJ /F6 11.955 Tf 11.96 0 Td[(5.2110)]TJ /F9 7.97 Tf 6.58 0 Td[(15T3)(4) 85

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Thisdensitymodelforargonisonlyvalidbelow15000K.Itisdesiredtodeveloppropertymodelsforthecurrentpurposesallowingforamaximumtemperatureofabout30000K.Sincedensitydecreasesmonotonicallywithincreasingtemperature,densityismodeledtodecreasedowntoacriticalvalue,belowwhichthedensitycannotfall.Thisvalueistakenasthedensityat15000Kasgiveninthemodelabove.DensityisconstantforincreasingtemperaturebeyondthispointasshowninFigure 4-1 4.3.3.2Specicheatcapacity ThespecicheatcapacityforanargonplasmaisgivenbyMaouhoub(1999)basedonmeasurementstakeninplasmaarcsatatmosphericpressure.Localthermodynamicequilibriumisassumed.ThespecicheatvaluesusedforcalculationsinthecurrentstudyaretakenaspiecewiselineartstoMaouhoub(1999)andareshowninFigure 4-2 4.3.3.3Thermalconductivity ThethermalconductivityvaluesforargonusedinthepresentstudyaregivenbyAtsuchi,etal.(2005).There,theauthorsmodelaninductionthermalplasmainaninvestigationofnon-equilibriumbehaviorfortemperaturesrangingupto15000K.ThermalconductivityasafunctionoftemperatureforpureargonplasmasisshowninFigure 4-3 ThethermalconductivityiscalculatedasanapproximationtotheChapman-Enskogmethod.Thevaluesforthermalconductivityaremodeledasconstantabove15000K. 4.3.3.4Massdiffusioncoefcient Themassdiffusioncoefcientisingeneralafunctionoftemperatureandafunctionofthetwoconstituentsinthediffusionprocess.OftenthemassdiffusioncoefcientmaybemodeledasasimplepowerlawfunctionbasedonasinglereferencevalueasdescribedinIncroperaandDewitt(2002).Thisrelationshipiswrittenas: D(T)=DrefT Tref3=2.(4) 86

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Asarstapproximationthediffusioncoefcientiscalculatedbasedonthisrelationandisallowedtoholdforboththediffusionofcadmiumintoargonandthediffusionofmagnesiumintoargon.ReferencevaluesaretakenasTref=15000KandDref=0.04m2=sbasedonorderofmagnitudeestimates. WhilethistemperaturedependancesufcesfortemperaturesbelowTref,thevalueofthediffusioncoefcientquicklygrowsfortemperaturevaluesmuchlargerthanthis.Thesevaluesgrowrapidlyenoughtoinduceunstablebehaviorintheexplicitnitedifferencesolutionandleadtoimpracticallylowchoicesfortimeresolution.Inaddition,nousefulphysicsaremodeledbythisrelation. Chapman-Enskogtheoryisusedtomodeltheoreticalvaluesforthediffusioncoefcients.Thediffusioncoefcientiscalculatedby: DAB=3 16(4kBT=MWAB)1=2 (p=RuT)ABDfD,(4) whereMWABistheharmonicmeanofthemolecularweightsofspeciesAandB,ABisthearithmeticmeanofthehardspherecollisiondiametersofspeciesAandB,andDisadimensionlessempiricalttotemperature.TheparameterDisgivenby: D=1.06036 (T)0.15610+0.19300 exp(0.47635T)+1.03587 exp(1.52996T)+1.76474 exp(3.89411T).(4) Thenon-dimensionaltemperature,TiscalculatedfromtheLennard-Jonesenergyforeachspeciesby: T=kBT (AB)1=2.(4) ThepertinentpropertiesfortheevaluationofChapman-EnskogderiveddiffusioncoefcientsarepresentedinTable 4-1 ThediffusioncoefcientscalculatedforeachofthesemethodsareshowninFigure 4-4 overtheestimatedtemperaturerangeexpectedinalaser-inducedplasma. 87

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Itisalsoimportanttonotethatthemassdiffusionequationiscoupledtotheenergyequationduetothetemperaturedependenceofthemassdiffusioncoefcient.Sincetheconverseisnottrue,theenergyequationmaysimplybesolvedrst,ateachtimestep,andtheresultingtemperaturemaybeusedtoevaluatethediffusioncoefcientforthesolutionofthemasstransferequation. 4.3.4DeterminingIonizationStateDistributions Oncethetemperatureandspeciesconcentrationdistributionsareknown,onemaythencalculatetheionizationstatedistributionofeachspecies.Section2.1.5describesthisprocessindetail.Here,aspeciccaseisconsideredusingtheresultsofsection2.1.5asthesolution.Considerathree-componentplasma,whererst-andsecond-ionizationstatesareallowed(z=1,2,3).Inthiscase,onemaywritethreespeciesconservationequations: ArT=ArI+ArII+ArIII,(4) MgT=MgI+MgII+MgIII,(4) CdT=CdI+CdII+CdIII.(4) OnemaywritesixversionsoftheSahaequation,twoforeachspecies: neArII ArI=SAr,I(T),(4) neArIII ArII=SAr,II(T),(4) neMgII MgI=SMg,I(T),(4) 88

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neMgIII MgII=SMg,II(T),(4) neCdII CdI=SCd,I(T),(4) neCdIII CdII=SCd,II(T).(4) Lastly,thesystemofequationsisclosedbyconsideringtheconservationofcharge,whichissimply: ne=ArII+MgII+CdII+2ArIII+2MgIII+2CdIII Recallthegeneralsolutiontothesystemofequationsisgivenby: ne=Z+1Xz=2JXj=1Nj(z)]TJ /F6 11.955 Tf 11.95 0 Td[(1)z)]TJ /F9 7.97 Tf 6.58 0 Td[(1Yi=1Sj,i nz)]TJ /F9 7.97 Tf 6.59 0 Td[(1e0BBBB@1+Z+1Xw=2w)]TJ /F9 7.97 Tf 6.59 0 Td[(1Yk=1Sj,k nw)]TJ /F9 7.97 Tf 6.59 0 Td[(1e1CCCCA.(4) Theforpresentcaseunderconsideration,thisequationbecomes: ne=ArT 1+ne SAr,I+MgT 1+ne SMg,ICdT 1+ne SCd,I. Thisequationisnowafunctionofnealoneandmaybesolvedbyanumericalproceduresuchasthebisectionmethodorxed-pointiteration.Oncetheelectronnumberdensity,neisfound,theionizationstatedistributionscanbereadilycalculated. 89

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4.3.5SimulationofPlasmaRadiativeEmission Oncethetemperaturedistributionandconcentrationdistributionsofneutralatomsandionsareallknown,theplasmacompositionisthenfullydetermined.Theschemeisnowinapositiontosimulatetheactofspectroscopybycalculatingtheradiativeemissiononewouldmeasurewithaspectrometer.Thesimulatedemissioncanbeusedwithcommonlaboratorymetricstocalculatetemperatureandelectrondensityasonewoulddoinanexperiment.Withquantitiessuchastemperatureandelectrondensityknownfromtheory,onemaythenassessthevalidityofsuchmetrics. Theemittedintensityofaspeciesfromsomeexcitedstate,i,tothegroundstatemaybecalculatedfrom: Iij=Aijni(T,ne),(4) whereAijisthetransitionprobabilityandniisthenumberdensityofexcitedstatei.Withthetotalnumberdensityofeachneutralatomandionknown,thenumberdensityofeachspeciesineachexcitedstatemaybegivenbytheBoltzmannrelationintheassumptionoflocalthermodynamicequilibrium: ni n=gi U(T)exp)]TJ /F3 11.955 Tf 13.22 8.09 Td[(Ei kT.(4) Oncetheintensitydistributionisknown,thetotalintensityforeachtransitionofeachspeciesmaybecalculatedasavolume-weightedaverageoftheintensitydistribution(Dalyander2008). 4.4ResultsandDiscussion 4.4.1TheTemperatureField ThetemperaturedistributionasitchangeswithtimeisshowninFigure 4-5 ,wheretheinitialtemperatureproleisassumedtobeaconstant15000Kthroughouttheplasmavolume.Astheplasmalosesheatbyradiationtotheenvironmentattheouter 90

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boundaryasteeptemperaturegradientisobserved.Theinnermostboundaryexhibitsaatgradientconsistentwiththesymmetryconditionimposedatthatpoint. ThetemporalevolutionofthetemperaturedistributionforthecasewheretheinitialconditionisprescribedasaparabolicproleisshowninFigure 4-6 Figure 4-7 showsthetemporalevolutionoftheplasmatemperatureatthreepointswithintheplasma:theplasmacenter,halfwaybetweenthecenterandtheedge,andtheplasmaedge.Thebulktemperature,estimatedasanaveragevalueweightedbythevolumeofeachdiscretizedcontrolvolume,isalsoshown.Thetemperaturesmonotonicallydecaywithtimewiththevolume-weightedtemperaturemorecloselyfollowingthetemperatureoftheplasmacore. 4.4.2TheConcentrationField ThedistributionofcadmiumatomsasitchangeswithtimeisshowninFigure 4-8 forearlyplasmalifetimescorrespondingtothevaporizationphaseoftheparticle.Massenterstheplasmavolumefromthecenternode,diffusesthroughouttheplasmavolumeuntilnallydiffusingoutoftheplasmafromtheouterboundary.Athoroughdiscussionofparticlevaporizationisincludedinthenextchapter. Atlongertimes,aftertheparticlehasbeenfullyvaporized,theconcentrationeldbeginstosettle.Withnomorecadmiumatomsbeingaddedtothesystem,theconcentrationgraduallydecreasesthroughdiffusionfromtheouterboundaryandisshowninFigure 4-9 Figure 4-10 showsthechangeincadmiumconcentrationwithtimeatthreelocationsintheplasmavolume:attheplasmacenter,halfwaybetweenthecenterandtheedge,andattheplasmaedge.Thecadmiumconcentrationattheplasmacentergraduallyincreasesduetothenetincreaseincadmiumatomsgeneratedatthatlocationfromthevaporizationprocessanddiffusionofthoseatomstothesurroundingplasma.Amarkedchangeinbehaviorfortheplasmacenterconcentrationisseendue 91

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totheconclusionofthevaporizationprocess.Atthatpointthecadmiumconcentrationdecreasesmonotonically. Theconcentrationofcadmiumatomsattheplasmacenterandattheouteredgebothincreaserapidlyatearlytimesduetotheinuxofmassfromthecenternodeduetovaporization.Sometimeafterthevaporizationprocesscompletes,theconcentrationattheselocationsbeginstograduallydecrease.Therateofdiffusionofmassoutofthetotalplasmavolumeisobservedtobesignicantlylessthantherateofmassinuxthroughvaporization. Theconcentrationofmagnesiumatoms,whileatdifferentabsolutevales,followsthesamebehavior. 4.4.3ElectronDensity ThedistributionofelectronnumberdensityisshowninFigure 4-11 .Notethattheelectronnumberdensityishighlydependantontemperature. Figure 4-12 conveysthesameinformationasFigure 4-11 exceptthatthey-axisisgivenonauniformscaleratherthanlogarithmic.Theelectronnumberdensitydecaysrapidlywithtimeinasimilarfashionasthetemperatureprole.Electronnumberdensitydropsclosetozeroattheouterboundaryoftheplasmaandretainsazerogradientatthecentercorrespondingtothesymmetrycondition. TheelectronnumberdensityatthreelocationsintheplasmaareshowninFigure 4-13 .Thegureshowsthetemporalevolutionofelectronnumberdensityattheplasmacenter,athalfwaybetweenthecenterandplasmaedge,andattheplasmaedge.Electrondensitydecaysrapidlywithtime,withthecenterlinevaluesgreatlyexceedingthatoftheouteredge. 92

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Table4-1. SummaryofparametersusedintheevaluationofdiffusioncoefcientbyChapman-Enskogtheory iMWiii[g/mol][ang][K] Ar39.9483.408119.9Cd112.4112.6061227Mg24.30502.9261614 93

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Figure4-1. Argongasdensity,,asafunctionoftemperature.SeeFujisaki(2002). 94

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Figure4-2. Specicheatcapacity,Cp,ofargonasafunctionoftemperature.SeeMaouhoub(1999). 95

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Figure4-3. Thermalconductivity,k,ofargonasafunctionoftemperature.SeeAtsuchi(2005). 96

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Figure4-4. Massdiffusioncoefcientasafunctionoftemperature. 97

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Figure4-5. Plasmatemperaturedistributionevolutionwithtimeforaatinitialprole. 98

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Figure4-6. Plasmatemperaturedistributionevolutionwithtimeforaparabolicinitialprole. 99

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Figure4-7. Changeintemperaturewithtimeatthreelocationsintheplasma.Alsoshownisthevolumeweightedtemperature'sevolutionwithtime. 100

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Figure4-8. Concentrationdistributionofcadmiumatearlytimes. 101

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Figure4-9. Concentrationdistributionofcadmiumatlatertimes. 102

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Figure4-10. Temporalevolutionofcadmiumconcentrationatthreelocationswithintheplasma. 103

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Figure4-11. Evolutionofelectrondensitywithtimeonalogarithmicscale. 104

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Figure4-12. Evolutionofelectrondensitywithtimeonauniformscale. 105

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Figure4-13. Temporalevolutionofelectronnumberdensityatthreelocationsintheplasma. 106

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CHAPTER5MODELINGAEROSOLVAPORIZATIONWITHINTHELASER-INDUCEDPLASMA 5.1OverviewoftheAerosolVaporizationProcess TheplasmamodeldescribedinChapter4representsasimulationoftheplasmapropertiesastheyvarythroughoutitsvolumeandastimepasses.Energytransportandmasstransportbydiffusionareallowedtogovernthebehaviorandstateofthespecieswithin.Thecurrentmodelhasconsideredaplasmagascomprisedofargoninwhichcadmiumandmagnesiumatomsarediffused.Theplasmamodeldescribestheglobaldistributionoftemperatureandconcentrationwithoutregardtohowtheanalytespeciesofcadmiumandmagnesiumcometobepresent. Thischapterdiscussesamodelofaerosolvaporizationwithinthelaser-inducedplasmathatconsidersnottheglobalplasmaenvironment,butonlythoseconditionsatalocalpointthatwillgoverntheliberationofparticlemass.Theaerosolvaporizationmodelconsidersasingle,stationaryparticletobepresentattheplasmacenter.Theparticleiscomprisedofequalamountsofcadmiumandmagnesiumbymass.Theparticlewillvaporizegraduallyallowingmoreandmoremasstobeliberatedfromthesurfaceoftheparticle.OncethatmassisliberateditbecomespartoftheglobalplasmamodelandisallowedtodiffusethroughouttheplasmavolumebasedonthetheorydiscussedinChapter4. Theinterfacebetweenthelocalmodelofaerosolvaporizationandtheglobalmodeloftheplasmaenvironmentexistsinthegenerationtermsofthecentralnodediscretizationequations.RecallthecentralnodediscretizationequationsofChapter4forthedistributionofenergyderivedbasedonthesymmetryboundarycondition: Cp+1Cd,0=CpCd,0+6DCd!Art r2)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(CpCd,1)]TJ /F3 11.955 Tf 11.96 0 Td[(CpCd,0(5) Ifmassgenerationisallowedinthecentralnodeonly,thediscretizationequationbecomes: 107

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Cp+1Cd,0=CpCd,0+6DCd!Art r2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(CpCd,1)]TJ /F3 11.955 Tf 11.96 0 Td[(CpCd,0+_NCd(5) Inthiscase,thegenerationtermprovidesanincrease(ordecrease)intheconcentrationofthecentralnode.Theresultsofthischapterwillprovideadescriptionofthemassgeneratedasthatwhichisliberatedfromtheaerosolparticleduringthevaporizationprocess. Thevaporizationprocesswillbediscussedinthreecontexts:Instantaneousvaporization,linearvaporization,andakineticmodelofvaporization.Instantaneousvaporizationimpliesthatmassisliberatedfromtheparticle,nottrulyinstantaneously,butratherinstantaneouslyincomparisontotheanalyticaltimescalesofLaser-InducedBreakdownSpectroscopy.Next,linearvaporizationoftheaerosolparticlewillbediscussedwheremassisliberatedfromtheparticleataprescribedlinearratewithtimeforasimplecomparisonwiththeinstantaneousrate.Finally,arigorouskineticmodelofaerosolvaporizationwillbeconsideredwhereeachtransitionofaerosolphaseisconsidered. 5.2InstantaneousAerosolVaporization Whenoneconsidersinstantaneousvaporization,oranyinstantaneousprocess,itisunderstoodthattheprocessdoesnottrulytakeplaceinstantly,butratherveryquicklyincomparisontotheperiodoftimeunderconsideration.Nophysicalprocessthatinvolvesthetransportofmasscantrulyoccurinstantaneouslysincegeneralrelativitydictatesthatmassandinformationcantravelnofasterthanthespeedoflight. Inthecaseofaerosolvaporizationinalaser-inducedplasma,aninstantaneousvaporizationrateimpliesthattheprocesstakesplaceoveratimescalethatismuchsmallerthantheanalyticaltimescaleofspectroscopy.Ideally,thisishowtheanalyticalcommunityviewsthevaporizationofmassfromaerosolparticlesinLIBS,asaprocessthatcompletesrapidlyandfullybeforetheanalyticalsignaliscollected. 108

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Thisisassumedtobetheidealbehaviorforthesakeofdiagnosticfeasibility.Particlemassisliberatedfromtheaerosolrapidlyandthatmassisdistributedthroughouttheplasmaenvironmentsoquicklythattheanalytesignalreadfromspectroscopicmeasurementsisassumedtodescribeauniformconditionwithintheplasma.Theanalytesignal,then,mayberepresentedbyalinearfunctionofitsconcentrationwithintheplasma,andthereforedirectlyrelatedtotheparticlemassandsize. Thequestionofwhetheraerosolvaporizationoccurssorapidlyisthemajorissuedealtwithinthischapter.Itisassumed,thatwhileindeedrapid,theaerosolvaporizationprocessoccursatniteratesandthatthisdeviationfromidealbehaviordoesaffecttheanalytesignal. 5.3LinearAerosolVaporization Therststepinmodelingadescriptionofaerosolvaporizationmoredetailedthantheidealassumptionthatitoccursinstantaneouslyislogicallytoconsiderthatvaporizationoccurslinearlywithtime.Infact,manyrealkineticvaporizationprocesseswillshowstronglinearbehaviorincertainconditions.Here,asaninitialattemptatcomplexity,itwillbeconsideredthatasingleaerosolparticleinalaser-inducedplasmawilllosemasslinearlywithtimeataratethatwillbeprescribedbasedonempiricalobservations. Theaerosolparticlewillbeassumedtovaporizecompletelyoveraperiodoftime,tv,andthereforethechangeinparticlemassasafunctionoftime,t,isgivenbythefollowingexpression: dm dt=4 3r3ppt tv(5) wherem=m(t)istheparticlesmassasafunctionoftime,rpistheparticleradius,andpistheparticledensity. 109

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Theparticleunderconsiderationhereiscomposedofbothcadmiumandmagnesiumdenedtobeinequalamountsbymass.Theparticleradiusistakentoberp=100nmandthetimefortotalvaporizationissettobetv=15s.Withthisassumedlinearvaporizationmodel,theamountoftotalmassintheplasmavolumeisshowninFigure 5-1 Recallthatinstantaneousvaporization,consideredastheidealbehavior,assumesthatvaporizationandthediffusionofmassthroughouttheplasmavolumeisveryrapid.Assuchtheentireaerosolparticle'smasswouldbedistributedevenlythroughouttheplasmavolume.ThisisrepresentedbytheatlineinFigure 5-1 .Thecaseoflinearvaporizationwithanitediffusioncoefcient,asdescribedinChapter4,isgivenbythedottedline.Notethatthetotalmassintheplasmavolumeincreasesduringthevaporizationtime,tv,andthendecreasesafterwardasnomorematterisaddedtotheplasma,yetmatterisallowedtodiffuseouttotheenvironment.Thenon-linearnatureofthecurveduringthevaporizationperiodshowsthebalancebetweenthemassinuxfromvaporizationandthediffusionofmassoutoftheplasma.Toshowthatthevaporizationprocessisindeedlinear,anothercaseisconsideredwheremassisallowedtodiffusethroughouttheplasmavolume,butnotoutoftheplasmavolume.ThisisrepresentedinFigure 5-1 bythedashedline.Thetotalmassincreaseslinearlywithtime,untilthevaporizationtimeisexceededatwhichtimethetotalmassremainsataconstantvalueconsistentwiththevaluefromtheidealcase. 5.4Heat-andMass-TransferModelingofAerosolVaporization Whilemanyprocessesofaerosolvaporizationmayindeedyieldlinearbehaviorwithtime,themodelingoflinearvaporizationataprescribedratelackstherigorofthekinetictheoriesofheatandmasstransfer.Considerednextisthecompleteaerosolvaporizationprocessmodeledasaseriesof4steps.Eachtransitionisconsideredsequentiallyandisassumedtobeindependentofthenext. 110

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Atthestartofthesimulation,t=0theparticleisassumedtobeatauniformtemperatureequaltoroomtemperature.It'ssuddenintroductionintotheplasmaenvironment,whosetemperaturegreatlyexceedstheboilingpointoftheparticle,willinducephasechangeinthefollowing4steps: 1.Particletemperatureincreasestothemeltingpoint,Tm 2.Phasechangefromsolidparticletoliquidparticle 3.Particletemperatureincreasestotheboilingpoint,Tb 4.Phasechangefromliquidparticletovapor Particlediametersunderconsiderationhere,rp=100nm,aremuchsmallerthanthediscretizedspacialstepsassumedintheglobalplasmamodel,dr=15m.Assuchthegaseousparticlemassthatisliberatedfromtheparticlesurfaceinthelaststepofthevaporizationprocessyieldsthevalueofmassgenerationincludedintheglobalmodel. 5.4.1TemperatureIncreasetotheMeltingPoint Attime,t=0,theaerosolparticle,assumedtobeperfectlyspherical,isatauniformtemperatureequaltothatoftheambientenvironment,T1.Uponitsexposuretothelaser-inducedplasma,therststepthatitwillundergoistoincreaseitstemperaturetothemeltingpoint.Here,itisassumedthattheparticleremainsatauniformtemperaturethroughoutitsvolumeasthattemperatureincreases,sinceBi<<1,whereBi=(hD=3k)istheBiotnumber.Foragiventimestep,thetotalchangeinparticletemperatureduringthisprocessisgivenbythefollowingexpression: TP=(Tg)]TJ /F3 11.955 Tf 11.95 0 Td[(TP) e3hcMZ srCp,lVcc(5) whereTgisthelocalplasmatemperature,TPistheparticlesurfacetemperature,Misthemolecularweightoftheparticlespecies,Zisthegridspacing,sisthedensityofthesolidparticle,andVccisthevelocityoftheparticleduringthetimestepunderconsideration.Thequantity,hcisaheattransfercoefcientbasedonthemotionoftheparticlethroughtheplasmaenvironmentandisgivenbythefollowingexpression: 111

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hc=Kg 2r(2+0.515p Re)(5) whereKgisthethermalconductivityoftheplasmaevaluatedatTg(Horner2007). Thiskineticprocessassumesthatheattransferisthelimiting,andtherefore,governingmechanismfortemperatureincrease.Theprocessisassumedtobedrivenbythedifferenceintemperatures,Tg)]TJ /F3 11.955 Tf 12.17 0 Td[(TP,eventhoughinrealityasmalllayerofvaporatsometemperaturebetweenthetwosurroundstheparticle. Lastly,itisimportanttonotethatthisphasemayormaynotbesignicanttotheoverallvaporizationprocess.BasedonplasmatemperatureunderconsiderationinChapter4,thisstepintheprocessmaytakeaslittleasafewnanosecondsoruptoasmuchasseveralhundrednanosecondstocomplete. 5.4.2TheMeltingProcess Oncetheaerosolparticlehasreachedthemeltingpoint,thephasetransitionofsolidtoliquidoccurs.Thisprocessismodeledasasimplechangeofphasewithallotherthermodynamicandmechanicaltraitsremainingconstant.Theparticle'sshaperemainssphericalandtheparticledoesnotlosemass,itmerelychangesfromsolidtoliquid.Thistransitionisthereforesignicantlysimplertocalculatethanthetransitionfromliquidtogas,wheremassliberationdoesindeedoccur. Itisassumedhere,andthroughouttherestofthechapter,thatsublimation,thatisthetransitionfromsoliddirectlytogaseousspecies,doesnotoccur.Sublimationtypicallyoccursatpressuresmuchhigherthanthatexperiencedbythelaser-inducedplasmaatatmosphericpressure. Thetotaltimerequiredfortheparticletomeltfromsolidtoliquidisgivenbythefollowingrelation: tmelt=2srHfus (Tg)]TJ /F3 11.955 Tf 11.96 0 Td[(Tm)Mhc(5) 112

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whereTmisthemeltingpointoftheparticle,Hfusisthelatentheatoffusionfortheparticle,andhcisthesameheattransfercoefcientdescribedpreviously.Thetimerequiredforthistransitiontooccuristypicallygreaterthantheindividualtimestepsdescribedfortheglobalplasmamodel.Onemaythereforecalculatethetotalamountofmassthathasmeltedinagiventimestepbythefollowingequation: Mmelt=4s 3Z(Tg)]TJ /F3 11.955 Tf 11.96 0 Td[(Tm)Mhc 2VccsHfus3(5) Again,thismodelassumesthattheparticletemperatureisuniformthroughoutandequaltoTm. 5.4.3TemperatureIncreasetotheBoilingPoint Onlyaftertheparticlehascompletelychangedphasetoliquid,isthetemperatureincreasetotheboilingpointconsidered.Thisprocesscanbecalculatedinmuchthesamewayastheincreaseintemperaturetothemeltingpoint.Therelationdescribingthetemperatureincreaseinthisphaseisgivenbythefollowingrelation: TP=(Tg)]TJ /F3 11.955 Tf 11.95 0 Td[(TP) eh3hcMZ lrCp,gVcci(5) ThisrelationisalmostidenticaltoEquation 5 exceptthattheparticledensityisnowthatofaliquid,andthespecicheatisthatofagas. Thisprocess,muchlikethetransitiontothemeltingpointisusuallyrapidasitoccursbeforetheplasmatemperaturehasdecreasedsignicantlyeitherthroughthelossofenergytothepreviousvaporizationstepsortotheexpansionandcoolingoftheplasmatotheenvironment. 5.4.4TheVaporizationProcess Thelaststepintheoverallvaporizationprocessistheactualphasechangeofliquidparticlemasstogaseousparticlemassanditssubsequentliberationfromthesphereofinuenceoftheparticle.Theevaporationphaseisbyfarthemostcomplex,andthereforemostcomputationallytaxingportionoftheoverallvaporizationmodel. 113

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Thevaporizationprocessoccurswhenheatistransferredtomoleculesatthesurfaceoftheliquidparticle.Thoseliquidparticlesreachtheboilingpointandmoveawayfromthesurfaceatavelocitydeterminedbytheboilingpointandataratedeterminedbythelocalvaporpressureofthematerial.Theheatingofsurfacemoleculesandthesubsequentliberationofthosemoleculesisaprocessthatcombinesmasstransferandheattransfer.Sincetheseprocessesarecoupledtogether,thetotalprocessislimitedbytheslowerofthetwo.Theremainderofthischapterwillbedevotedtothedeterminationofwhichmechanismlimitsthevaporizationprocessandthereforedeterminestherateofmassevaporated. 5.4.4.1Heattransferlimitedvaporization First,oneconsidersthecasewherevaporizationislimitedbytheeffectsofheattransfer.Heatconductsfromthebulkplasmagastothesurfaceofthesphericalparticlewhichcausesitsradiustochangewithtime,oftenwrittenasaquadraticexpressionsimilartothefollowing: r2=r20)]TJ /F3 11.955 Tf 11.95 0 Td[(kHT,lt(5) wherekHT,listheheat-transferlimitedrateofvaporizationforlarge-particles.Thelarge-particlesqualierwillbediscussedbelow.Thedeterminationofthisrateconstantisfoundfromkineticargumentsthatmodelthetransferofheatfromtheplasmagasthroughavaporlayerandintothemoltenparticlemass.Thisrelationisgivenbelow: kHT,l=2MKg(Tg)]TJ /F3 11.955 Tf 11.96 0 Td[(TP) Hvapl(5) whereKgistheconductivityofeithertheplasmaorparticle,whicheverislowest.Thequantity,isthemasscounterowcoefcientandisgivenbythefollowingexpression: =ln1+Hov Hvap Hov Hvap(5) 114

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whereHovistheoverallheatofvaporizationgivenbythefollowingexpression: Hov=Cp,g(Tg)]TJ /F3 11.955 Tf 11.95 0 Td[(TP)+Hat+Hion(5) whereisthefractionofspeciesatomized,denedheretobeunityandisthefractionofsinglyionizedparticlescalculatedfromtheSaha-Boltzmannequation.Withtheheattransfer-limitedvaporizationrateconstantnowknownbasedontheseequations,themasslostbytheparticleperunittimeisgivenbythefollowingrelationship: dm dt=)]TJ /F6 11.955 Tf 9.29 0 Td[(2lkHT,lr=)]TJ /F6 11.955 Tf 10.49 8.08 Td[(4MKg(Tg)]TJ /F3 11.955 Tf 11.96 0 Td[(TP) Hvapr(5) Animportantdistinctionneedstobemadeinregardtothevalidityoftheserelations.Thisargumentfortheheattransferlimitedrateconstantforvaporizationrequiresthattheparticlebelargeincomparisontothemeanfreepathoftheplasmaenvironment,suchthatthesituationfallswithinacontinuumdescription.Ifaparticleissmallincomparisontothemeanfreepath,thenheattransferbehavesslightlydifferently,infactitisslowedincomparisontothecontinuumheattransfer.ThisphenomenonisknownastheKnudseneffect. TheKnudsennumberisanon-dimensionalratiorepresentingtherelationshipbetweentheparticlemeanfreepathoftheplasmaandthelengthscalecharacteristicofparticlediameter,andiswrittenas: Kn= 2r(5) TypicallyiftheKnudsennumberissmallerthanabout0.001itisstatedthattheparticlefallswithinthelarge-particleregime.IftheKnudsennumberislargerthanthisquantity,thensmall-particle,orKnudseneffect,considerationsneedtobemade.TheKnudseneffectisquantiedasacorrectiontotheheattransfercalculatedinacontinuumregime.Sincetheheattransfer-limitedrateofvaporizationisdirectlyrelatedtotheamountof 115

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heattransferred,onemaywritetheKnudseneffectintermsofthevaporizationrateconstantsas: kHT,l kHT,s=1 1+Z r(5) whereZisaquantityknownasthetemperaturejumpdistancethatdescribesthedistanceoverwhichthetemperaturechangesfromthatattheparticle'ssurfacetotheplasmagas.Thetemperaturejumpdistanceisgivenby: Z=2)]TJ /F3 11.955 Tf 11.96 0 Td[(a a 1+4Kg gvgCp,g(5) whereaisthethermalaccommodationcoefcient,takentobe0.8,andgammaisthespecicheatratio,whichforargongasis5/3. Together,thetemperaturejumpdistanceandKnudseneffectcompriseacorrectiontothepreviouslycalculatedvaporizationrateconstant. 5.4.4.2Masstransferlimitedvaporization Theevaporationprocessismostlikelyheattransferlimitedifthereisasteeptemperaturegradientaroundtheparticle,whichusuallyoccursiftheboilingpointiswellbelowtheplasmagastemperature.Iftheboilingpointiscloseto,orexceedsthelocalgastemperature,thenevaporationtransitionislikelymasstransferlimited.Liketheheattransfer-limitedvaporizationmechanism,themasstransferprocessoccursbydifferentkineticsinthelarge-particleinKnudsenregimes.Therefore,bothalarge-particlevaporizationrateconstantandasmall-particlevaporizationrateconstantwillbedeveloped. Ingeneral,thechangeinradiuswithtimeofaparticleundermasstransfer-controlledvaporizationisgivenbythefollowingexpression: dr dt=)]TJ /F3 11.955 Tf 9.3 0 Td[(MPs (2MRTg)1=2l(1)]TJ /F4 11.955 Tf 11.95 0 Td[(=2)1+vgr (1)]TJ /F10 7.97 Tf 6.59 0 Td[(=2)D12(5) 116

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whereistheevaporationcoefcient,Psisthesaturatedvaporpressure,andvgistheaerosolparticlevelocity.Thisexpressionmaybeevaluatedasgiven,orcanbesimpliedintolarge-particleandsmall-particleexpressions. Inthelarge-particleregime,forKnudsennumberssmallerthanabout0.001,itcanbeshownthat: vgr>>D12(1)]TJ /F4 11.955 Tf 11.96 0 Td[(=2)(5) Therefore,theparticleradiusasafunctionoftimecanbewrittensimilarlyasthelarge-particlecaseforheattransfer-limitedvaporizationas: r2=r20)]TJ /F3 11.955 Tf 11.95 0 Td[(kMT,lt(5) wherethevaporizationrateconstant,kMT,lisgivenby: kMT,l=2MsD12 lRTg(5) Inthecaseofsmallparticles,theoppositecondition,of 5 istrue,namely: vgr<
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TransitionregimesbetweenKnudsennumbersabout0.001and0.1aredifculttoplaceineitherthelarge-particleorsmall-particleapproximations.Therefore,forquestionableparticleKnudsennumbers,thegeneralequationforthechangeofparticleradiusfortime,givenby 5 ,mustbesolvedexplicitly. 5.5ResultsandDiscussion Thischapterhasoutlinedindetailamethodfordeningtheindividualtransitionsthattakeplaceduringtheaerosolvaporizationprocess.Aparticle,onceintroducedintotheplasmaenvironment,increasesintemperaturetoitsmeltingpoint,undergoesphasechangefromsolidtoliquid,thenincreasesintemperaturetoitsboilingpoint,andnallyundergoesphasechangefromliquidtovapor. Emphasishasbeenplacedonthefactthattheparticlevaporizationkineticsarealocalprocessandarethereforegovernednotbythebulkplasmaconditions,butonlythelocalconditionsinthevicinityofaparticle.Caremustbetaken,then,whenimplementingthecurrentvaporizationmodelwithinthecontextoftheglobalplasmamodelintroducedinChapter4.TheglobalplasmatemperatureproleissolvedforrstasdescribedinChapter4.Oncethetemperatureisknownnearthelocationoftheaerosolparticle,thechangeinstateoftheparticle,basedonthetransitionsdescribedinthischapter,isthencalculated. Aseachtimesteppasses,theparticle'shistoryprogressessequentiallyalongthefourtransitionstepsofthepresentkineticmodel.First,attheinitialmodeltimestep,theparticleisassumedtohavejustbeeninstantaneouslyintroducedintotheplasmaenvironment.Itstemperaturecorrespondstothatoftheambientenvironment,T1.Duringthersttimestep,itschangeintemperatureiscalculatedbasedontheequationsforthattransition.Eachnewtimestepincreasestheparticletemperatureuntiltheboilingpointhasbeenreachedandatthatpointtheprogramowdirectsthelocalmodelintothenexttransition.Eachtimestepmeltsmoreandmoreoftheparticle,untilitis 118

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completelymelted.Oncetheparticlehasbecomefullyliquid,programowdirectsthenexttimesteptothenalstepofvaporization. Oncetheevaporationstepisreached,onlythenisparticlemassaddedintothebulkplasmadiscretizationequationsbywayoftheirgenerationterms.First,theKnudsennumberiscalculatedfortheparticlebasedonitscurrentradiusatthattimestep.Basedonthisvalue,theproperregime,whetherlarge-particleorsmall-particleisassumed.Then,theheattransfer-limitedvaporizationrateconstantisdetermined,kHT,lorkHT,s.Next,themasstransfer-limitedvaporizationrateconstantisdetermined,kMT,lorkMT,s.Sincethevaporizationprocessisgovernedbywhichevermechanism,heattransferormasstransfer,isslowest,thelowerofthetworateconstantsischosenastheappropriaterateconstant. Basedonthischoice,theamountofparticlemassthatisliberatediscalculatedateachtimestepandusedasthevalueforthegenerationtermsintheglobalmodelofmassdiffusionasdiscussedpreviously.Theparticle'snewradiusiscalculatedandusedforthenexttimestepuntiltheparticleiscompletelyvaporized. Sincethediffusionofparticlemassthroughouttheplasmavolumeisdependentontheavailablemassthatiscloseto,butliberatedfrom,thevaporizingparticle,thediffusionisalsodependentonthemeansbywhichparticlevaporizationoccurs.TheatomicemissionandLIBSresponseofanaerosolsystemisthereforedependentuponthevaporizationprocessaswell.Andindeedthedifferentmethodsfornumericallymodelingvaporization,whetherinstantaneously,linearly,orfromarigorousheat-andmass-transferscheme,affecttheLIBSresponse. Figure 5-2 showstheresultingmassdiffusionthroughoutthesimulatedlaser-inducedplasmavolumeintherstmicrosecondfortheheat-andmass-transfervaporizationmodel.Thegurefollowstheradialsymmetryofthemodelandshowstheconcentrationofcadmiumliberatedfromasingleaerosolparticlelocatedinthecenteroftheplasmavolumeonalogarithmicscale. 119

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Astimepasses,particlevaporizationcontinuessimultaneouslywiththediffusionofmassintotheplasmaenvironmentasshowninFigure 5-3 after5s,inFigure 5-4 after10s,andinFigure 5-5 after15s. Byabout20saftertheinitiationofthelaser-inducedplasma,theparticlehasfullyvaporizedreleasingnonewcadmiumatomsintotheplasmavolumeasshowninFigure 5-6 .Thediffusionprocesscontinues,however,asthecadmiumconcentrationseeksequilibriumwiththesurroundings.After30s,asshowninFigure 5-7 thecadmiumconcentrationisapproachinguniformity. 120

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Figure5-1. Totalaerosolmassintheplasmavolume. 121

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Figure5-2. Simulatedcadmiumconcentrationthroughouttheplasmaafter1s. 122

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Figure5-3. Simulatedcadmiumconcentrationthroughouttheplasmaafter5s. 123

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Figure5-4. Simulatedcadmiumconcentrationthroughouttheplasmaafter10s. 124

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Figure5-5. Simulatedcadmiumconcentrationthroughouttheplasmaafter15s. 125

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Figure5-6. Simulatedcadmiumconcentrationthroughouttheplasmaafter20s. 126

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Figure5-7. Simulatedcadmiumconcentrationthroughouttheplasmaafter30s. 127

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CHAPTER6INVESTIGATIONOFPLASMAINCEPTION 6.1IntroductionandMotivationforEarlyPlasmaStudies Thepresentstudyhas,asmanybeforeit,soughttomodelandunderstandthevariousplasma-particleinteractionspresentinLIBSandotherplasmabasedtechniques.ThevariousmodelingeffortsdiscussedinChapter 2 haveallconsideredseveralofthemostimportantmechanismsintheprocessesoflaserplasmaexpansion,plasmacooling,aerosolvaporizationandradiativeemission.Eachofthesestudieshas,understandably,requiredtheuseofseveralsimplifyingassumptionstovalidatetheuseofmanyfundamentaltheoriesofspectroscopicapplications,suchastheassumptionoflocalthermodynamicequilibrium. Oneaspectofthelaser-inducedplasmabehavioranditsaffectonplasma-particleinteractionsthathaslargelybeenleftunconsidered,withrespecttothedevelopmentofarigorousanalyticalmodel,istheareaofplasmainceptionandofearlyplasmalifetime.Thereareseveralreasonswhythisisso.First,theinceptionandearlylifetimesofthelaser-inducedplasmaoccuronatimescaleontheorderoflessthan100ns.Thistimescaleisalmostalwayssignicantlyshorterthantheanalyticaltimescalesinvolvedinplasmadiagnostics.Thephysicalconsiderationsthatdominateduringthistimeperiodarethuslythoughttobegenerallylessimportantthanthoseinlatertimes. Secondly,whenonedoesconsiderthephysicsofplasmainceptionandearlyplasmabehavior,onenoticesthatnon-ideal,oratleast,non-equilibriumeffectsarelikelytodominateinthisregime.Assuchthemodelingofthephysicsaremuchmorecomplicatedandbasedonlessdirectordeterministicmethodsthanthemodelingofeffortsafterthistimeperiod.Itmaybeargued,then,thatmodelingeffortsinthisregimeofferlittletotheaccuracyofexistingplasmamodelsandwouldincurrelativelysignicantcomputationallyexpense. 128

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Thecontraryargumentismadehere,however.Theplasmamodelsthatignorethephysicsofplasmainceptionandthedynamicsofearlyplasmaexpansionmustbedependentuponsomeempiricalinformationonwhichtobasetheinitialconditionsofthemodel.Suchaprocedureiscertainlyvalid,however,whenthegoalistounderstandtheplasma-matterinteractionsonafundamentallevel,anydependenceonexperimentaldatatothemodelinputisalimitationtoitsscope.Thisisespeciallytruewhenoneconsiderstheextenttowhichthelongtermbehaviorofmanysuchnumericalsystemsaredependentupontheinitialconditions.Anyplasmamodelthatdoesnotconsiderthemechanismsofplasmainceptionandearlyplasmalifetimeisthereforeincompleteandopentorenementbasedontheseconsiderations. Ultimately,themannerinwhichalaser-inducedplasmaformsislikelytohavesomeeffectontheresultingdynamicsofplasma-materialinteractionandwillthereforeinuencetheLIBSresponse.Considerationofthenon-equilibriumbehaviorinearlyplasmaformationoffersinsightintothemorecomplexfeaturesoftheplasmaenvironmentthatareoftenassumedaway. Towardthisend,aseriesofinvestigationsisimplementedtoprobeandmodelthedynamicsofplasmaformation.Animagingexperimentisperformedtostudythebehaviorofearlyplasmainceptioneventsandthesubsequentplasmaformationatearlytimesinthreedifferentgases.Thebehaviorofinitialplasmainceptionisshowntovaryamongthethreegases:nitrogen,argon,andhelium.Analysisofthedifferencesinplasmaformationcharacteristicsforthethreegasessuggeststhatthechemicalpropertiesofthegasinuenceplasmainception.Atheoreticalinvestigationastowhythisissoiscarriedout. 6.2ExperimentalApparatusandMethods Animagingstudywasperformedtoprobethebehavioroflaser-inducedplasmaformationatitsearliestobservablelifetimes.TheexperimentalsystemforthisstudyisshowninFigure 6-1 .ForallexperimentsaQ-switchedNd:YAGlaser(Continuum) 129

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operatingatafundamentalfrequencyof1064nmwasusedasthelasersource.Furthermore,thelaserpowerwasabout400mJperpulse,witha10nspulsewidthand1Hzrepetitionrate.A75-mmfocalpointlenswasusedtofocusthelaserbeamtoapointtosufcientlycausebreakdownwithinthesix-facedsamplechamber.Imageswerecollectedwithtwoseparatecameras,andAndorICCDcameraandaPI-MaxICCDcamera,bothorientedperpendiculartothedirectionoflaserpropagation.Eachcamerawasconnectedtoalaboratorycomputerandimageswererecordedwithaccompanyingsoftwareasdatasetsconsistingoftwo-dimensionalarraysofnumberscorrespondingtopixelcountsacrosstheCCD.TheCCDchipsonbothcamerashavearesolutionof1024by1024pixels. TheICCDcameraandlaserQ-switchwerebothtriggeredfromafour-channeldigitaldelay/pulsegenerator(StanfordResearchSystems,ModelDG535).Thecameraandlaserwereeachtriggeredfromindividualchannelsofthedelaygeneratorwithasetdelaybetweenthetwotriggerssoastocaptureaspecictimeintheplasmalife.Delaygapswereadjustedsoastocaptureplasmaimagesbetween1nsand108.4nsafterthelaserpulse. Usingthedescribedexperimentalsetup,aseriesofplasmaformationimagesweretakenoverthreedifferentdaysandforthreedifferentambientgases:nitrogen,argon,andhelium.Between100and500imagesweretakenforeachgasoneachdaycreatinganensembleofplasmaformationimagesofabout1000imagesforeachgasorabout3000imagestotal.Inadditiontothesetofearlylifetimedata,imagesweretakenatvariousstagesinthetotallifetimeuptoextinctionforcomparison. Lastly,thelaserbeamprolewasmeasuredusinganink-ablationmethodinordertopositiontheplasmaformationimagesrelativetothebeamfocalpoint.Inkwasplacedonaseriesofcolorlessglassslidesandthenplacedinthepathofthelaserbeam.Theablatedareaoftheinkbythebeamwasusedtoprovideanestimateofthebeamprolediameter.Thepositionofeachslidewasvariedalongthedirectionofbeampropagation 130

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andrecordedusingtheCCD.Thisprovidedaplotofthebeamprolediameter,notonlyinrealspace,butalsointermsofthepixelcoordinatesoftheCCD. 6.3DataProcessingandAnalysis Theexperimentalprocedurediscussedintheprevioussectionwasusedtoinvestigatetheplasmainceptioneventforthreeambientgases:nitrogen,argon,andhelium,allatatmosphericpressure.Alargeensembleofimages,around1000,foreachgaswererecordedduringtheearlytimesofplasmaformation.Aseriesofimagesovertheentireevolutionofplasmalifetimeofeachgaswerealsotaken.Figure 6-2 showsacollectionofrawplasmaimagesthatweretakenatvariousstagesinthetotalevolutionoftheplasma'slifetimefornitrogen.Atearlytimes,lessthan100ns,theplasmaisformingfromsmall,discretebreakdownkernels.Theindividualkernelsgrowandcoalesce,formingthefulladultplasmaaround100ns.Atmuchlatertimes,ontheorderofafewmicroseconds,theplasmadeactivationbeginsastheexcitedstatesbegintorelaxandemissionfades. However,itistheearliestplasmalifetimesthatareofmostinterestinthepresentstudy.Figure 6-3 showsanotherseriesofplasmalifeevolution,butoveramuchshorterscalethanFigure 6-2 ,betterresolvingtheearlyprogressionofformation.InFigure 6-3 (a)theearliestbreakdowneventsareseenclearlyasindividualandseparatekernels.Afteronly10nsthekernelsbegintogrowtogetherandtheadultplasmabeginstoformtotheleft(whichistowardtheexcitationsource). Theensembleofdataimagesforearlyplasmaformationarealltakenpriorto10nsaftertheinitiationofthelaserpulseandthereforeresembleFigure 6-3 (a).Eachofthe1000imagesforeachgasshowacollectionofsmall,discretebreakdowneventsthatvaryinpositionandnumberalongthedirectionoflaserpropagation.Thecharacteristicsofeachimagevarysomewhatpredictablybygas.Eachensembleisprocessedasabatchtocalculatepertinentcharacteristicsofplasmaformationfeatures. 131

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Thelargeamountsofimagescollected,however,makesthemanualcalculationofthestatisticalcharacteristicsofeachimageunwieldy,andthereforeanautomatedprocedurewasdesignedtocarryouttheprocess.ThefundamentalsofthedesignandimplementationofthetechniquesofautomatedpeakdetectionarediscussedindetailinChapter2. 6.3.1AutomatedPeakDetection BasedonthetechniquesdevelopedinChapter2,anautomateddetectionschemewasdevelopedforthepurposeofcalculatingthestatisticalnatureoftheobservedplasmainceptionkernels.Thealgorithmsperformedseveralidenticalstepsforeachimagebasedonparameterschosenfromtheexaminationofseveraltestcases. Tostart,eachimagewasrecordedasatwo-dimensionalarrayofdatarepresentingthephotoncountateachpixelonthecharge-coupleddevice(CCD)camera.ItisthenassumedthateachimageresemblesFigure 6-3 (a)inthatitcontainsaseriesofbrightcollinearspotswhosenumberandgeometricalcharacteristicsaretobeextracted.First,thecenterlineoflaserbeampropagationisdeterminedbybinningeachrowofdataandndingtherowofmaximumcountintensity.Thisoperationidentiesthepixelrownumberofthecenterlineofthesetofcollinearspots. Thetwo-dimensionalarrayofimageinformationcannowbecondensedintoaone-dimensionalprolebasedonthecenterlinealongthedirectionofplasmapropagation.Theone-dimensionalstripusedforanalysiswastakenasthesumofthreerowprolessurroundingthecenterline.Theone-dimensionalprolecorrespondingtotheimageshowninFigure 6-3 (a)isshowninFigure 6-4 Aftertheone-dimensionalproleisextracted,itisthenanalyzedbasedontheautomatedpeakdetectionalgorithmsdescribedinChapter2andconsistofvemainsteps:pre-processing,smoothing,baselinecorrection,peak-nding,andoptimization. Therststepimplementedintheanalysisofrawimageprolesconsistsofseveralsimplepreliminaryroutinestoconditionthedatatoensuresuccessfulandrobust 132

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completionoftheanalysis.Preliminarytestsareconductedonthedatatoensurethatitcontainsvalidnumericaldatavalues,theappropriatelength(1024elements),andwellconditionedbounds(oftenspectraldatatakenfromaCCDchipmaycontainafewelementsoferroneousdataneartheedge).Othersimpleconditioningprocessesarealsoimplemented,suchas'cosmicray'removal.OftenCCDpixeloutliersmaybeobservedinrandompixelsattributedtorandomcosmicrayeventsfallingontotheCCD,fromimproperreadoutevents,orfrombleed-overfromadjacentsaturatedpixels.Thisoftenappearsasasinglebrightpixelamidsurroundingpixelswithsignicantlylessrecordedphotoncount.Suchphenomenon,whilerare,mayindeedaffectthecalculatedresults.Asimplealgorithmtoremoveanycosmicrayeventisimplementedthroughalterthatremovesallpeaksthathaveawidthofasinglepixel. Thedataisthensentthroughaseriesofsmoothingltersinanefforttoremovepixel-to-pixelvariationasasourceofnoise.Smoothingisperformedbywayofbothsecond-orderandthird-ordermoving-averagelters.Thesmoothingoperationisgenerallythatwhichrequiresthemostscrutinyandattentionfromtheuserasitisaroutinewithatendencytoalterworkingdatainanegativeway.Whileinsufcientsmoothingproducesadatasettoonoisytoextractmeaningfulfeaturesduringthelaststageofanalysis,toomuchsmoothingcandampenpeakvaluesand,inextremecases,eveneraseentirefeatures.Smoothingparametersarethereforere-evaluatedduringthelaststageofoptimizationandinvestigatedmanuallyforavarietyoftestcases. BaselinecorrectionwasperformedusingthemonotoneminimumtechniqueofSection2.3.2whichremovesamonotonicallyincreasingtrendbaselinefromthedata.Theentiredataensemblewaswell-behavedacrosstheCCDinthatbaselineswerelargelyuniformforeachimageandthereforeimplementationoftheremovalroutinewasrobust. Oncethedatawasproperlysmoothed,andthebaselineremoved,theidenticationofimportantpeakswascarriedoutbasedonseveralcriteria.Apreliminarylistofpeak 133

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featureswasdeterminedbasedontheareaundereachfeature.Anypeakwithanareagreaterthan1%ofthetotalareaisextractedasimportant.Thatlistisfurtherrenedoverseveralsteps.First,peakswithaninsufcientwidthareremovedasfalse-positives.Second,theabsolutemagnitudeofeachpeakiscomparedwiththestrongestfeatures.Peakswhosemagnitudeisacertainmultipliersmallerthanthestrongestaredisregardedasinsignicant. Thelaststepofdataprocessingconsistsofseveralsimpletechniquestoensuretheretrievalofmeaningfuldata.First,foreachimage,thenumberofpeaksdetectedisexamined.Athresholdvalueischosensuchthatifthenumberofpeaksdetectedisabovethisvalue,theimagemustundergoprocessingagainwithmorestringentlterparameters.Thisstephasshowntobemostnecessaryintheanalysisofheliumimageswherethelowmagnitudeoffeaturesinrelationtothenoiselevelgreatlyincreasesthedifcultyofndingusefulfeatures.Secondly,onceasetofpeakshavebeenidentied,theimageisprocessedagain,usingaslightlydifferentsetoflterparameterstodeterminehowthesetofdetectedfeatureswillchange.Generally,ifslightvariationofthelterparameterproducesthesame(orsimilar)setofdetectedfeaturesthenthecondencethatthosefeaturesaretrulyimportantisgreater.Caseswhereslightvariationinthelterparametersresultsinasignicantlydifferentsetofdetectedfeaturesareaggedformanualinvestigation.Lastly,eachpeakdetectedisttoaGaussianfunctioninordertodetermineitsfull-widthathalf-maximum(FWHM).ThevalueofFWHMisusedtodetermineifthefeature'swidthisproperforitsmagnitude.AnyfeatureswithaFWHMthatexceedsacertainthresholdareaggedasunlikelycandidates. TherawdataprolealongwiththeresultingprocessedresultsforasinglecaseinnitrogengasisshowninFigure 6-4 .Thesmoothingroutinehasremovedmuchofthehighfrequencynoise,whilestillretainingtheoverallphysicalcharacteristicsoftheinceptionkernels.Thebaselinehasbeenremovedproperlyandthealgorithmhas 134

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detectedthepresenceofeightpeaksinthistestcasefornitrogen.Manualinspectionoftheresultingconditionedproleillustratesthatahumanuserwouldidentifythesameeightpeaksasthealgorithmasimportantfeaturessuggestingcondenceinthealgorithm'sautomatedresults. AsinglerepresentativecasewasexaminedinargongasandtheresultsareshowninFigure 6-5 .Notethatthealgorithmisobservedtobesuccessfulinthedeterminationofthelargestcharacteristicpeaks,butfailstodetectafewofthesmallerfeatures.Thisdoesnotconstituteafailureonthepartofthealgorithmasthedisregardedfeaturesmayormaynotbeimportant.Notethatthespreadofpixelrangeoverthesepeaksisgreaterthanthatofnitrogengas. AnaltestcasewasexaminedinheliumgasandtheresultsareshowninFigure 6-6 .Thealgorithmdetectsthemajorfeatures,althoughalsodetectsasmallpeakthatmayormaynotbeatruefeature.Suchbehaviorispossibleduetothenatureoftheshaperatiocriterionforpeaknding.Alsonotethatasmall,single-pixel-widefeatureiscompletelyremovedfromtheconditioneddataasaresultofthecosmicraylter. 6.3.2PlasmaInceptionCharacteristics Withthealgorithmdevelopedandfunctioningproperlywithcondencefortheaforementionedtestcases.Theprocedureisimplementedtotheentireensembleof3000imagescollectedinthestudy.Severalmetricsforeachsetofdetectedfeaturesarechosentoberecordedforeachimageandthecollectivestatisticsforeachareexamined.Thecharacteristicsrecordedforeachimageareasfollows: 1.numberofpeaks 2.areaofeachpeak 3.full-widthathalf-maximumofeachpeak 3.pixelrangeoverallpeaks 4.minimumandmaximumseparationbetweenconsecutivepeaks 5.numberofresolvedpeaksversusnumberofcombinedpeaks 135

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Thevariationofthesecharacteristicsoverthethreechosengases:nitrogen,argon,andheliumyieldinsightintothehowchangesinchemistryaffectplasmaformation. 6.4ExperimentalResultsandDiscussion TheexperimentalprocedureoutlinedinSection6.2wascarriedoutanddataanalysiswasperformedontheresultingensembleasdiscussedinSection6.3.Asaresult,asetofplasmainceptionstatisticswascollectedforeachgas.Thebeamprolewasmeasuredandallpixeldatawasconvertedtorealspaceforcomparison. Typicalresultsfromtheensembleofabout1000imagestakeninnitrogenareshowninFigure 6-7 .Thegureshows30well-conditionedresultstakenatrandom,overallthreedays,fromtheensemble.InFigure 6-7 eachcollinearsetofpointsrepresentsaninceptionimage.Whileeachsetofinceptionpointswereinrealitylocatedalongthebeam'scenterline,theyareshownintheguredisplacedaboveandbelowforclarity.Thebeamprole,asmeasured,isshownbythedashedline,whileapolynomialtofthebeamproleisshownasasolidline. Nitrogenkernels,asshowninFigure 6-7 ,aretypicallyuniformlydistributedabouta3mmregiondownstreamofthebeam'sfocalpoint,awayfromtheexcitationsource.Onaveragebetween7and8plasmainceptioneventswererecordedforeachimageinnitrogen. Figure 6-8 showsasimilarplotfor30well-conditionedresultstakenatrandom,overallthreedays,fromtheensembleofimagesinargon.Eachcollinearsetofinceptionpoints,againrepresentsasingleimage,andtheyaredisplacedaboveandbelowthecenterlineforclarity.Here,itcanbeseethatthebehaviorofthedistributionofinceptioneventsinargondiffersmarkedlyfromthatofnitrogen.Figure 6-8 showsthatpeaksarespreadoverabout4mmstartingatthebeamfocalpointandcontinuingdownstream,awayfromtheexcitationsource.Insteadofauniformdistributionofinceptionpoints,however,theeventsaredistributedbi-modally.Onaveragebetween5and6inceptioneventswererecordedforeachimageinargon. 136

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Lastly,Figure 6-9 showsasimilarplotfor30well-conditionedresultstakenatrandom,overallthreedays,fromtheensembleofimagesinhelium.Eachcollinearsetofinceptionpoints,againrepresentsasingleimage,andtheyaredisplacedaboveandbelowthecenterlineforclarity.Whilethespatialspreadofplasmainceptioneventsinargonandnitrogenwerebothrelativelywide,thespreadofeventsinheliumissignicantlysmaller,coveringanareaofonlyabout2mmdownstreamfromthefocalpoint.Thedistributionofeventsinhelium,likenitrogen,washighlyuniformaboutthisregion.Onaveragebetween4and5plasmainceptioneventswererecordedforeachimage.Theensembleofimagesinheliumwereparticularlydifculttoproduceconsistentlyacceptableprocessedresultsbasedonthepreviouslydiscussedautomatedroutine.Manualinspectionoftheprocessedresults,therefore,suggeststhattheaveragenumberofplasmainceptioneventsforeachimageinheliumshouldactuallybebetween3and4. Whencomparingtheresultsoftheplasmainceptioncharacteristicsovereachambientgas,itisinterestingtonotethatmosteventsbegindownstream(awayfrom)ofthelaserbeamfocalpoint.Itrstglance,thisseemscounter-intuitive.Aplasmaisknowntoforminambientgaswhenthephotondensityinthelaserbeambecomessufcientlyhighenoughforbreakdowntooccur.Thelargestphotondensityinafocusedbeamoccursatthefocalpoint,anditisthereforeintuitivethattheplasmashouldformatthefocalpoint,notdownstreamfromthefocalpoint.However,considerFigure 6-10 thatshowsseveralplasmacontourswithinthebeamproleatvarioustimes.Attheearliesttimes,showninFigure 6-10 (a),individualplasmainceptioneventsformdownstreamofthebeamfocalpoint.Atlatertimes,however,inFigure 6-10 (b)and(c),theadultplasmagrowsfromtheinitialinceptioneventstowardstheexcitationsourceandultimatelyformsatthebeamfocalpointinaccordancewithintuition. Figure 6-11 showsthenalresultsoftheplasmainceptionstudybycomparingthedistributionofindividualbreakdowneventsforeachofthethreegases.Theminimum, 137

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average,andmaximumlocationsforbreakdownforeachgasareshownalongwiththebeamprole.Nitrogenandargonexhibitasimilarrangeofeventsspatially,thoughnitrogendoessouniformlyandargonbi-modally.Therangeofeventsinheliumarepackedsignicantlytighterinheliumandalsouniformlydistributed.Notethatforeachgas,thedistributionofbreakdowneventsishighlyrepeatable,butvaryfromgastogas.Thissuggeststhatthedifferenceinbehavioreachgasexhibitsisduethechemistryofthatgasratherthaninuencesfromthelasersourceoroptics. 6.5TheoreticalConsiderationsandConclusions Therearetwoprimarymechanismsforthegrowthoffreeelectronsintheformationofalaser-inducedplasma:cascadeionizationandmulti-photonionization(MPI).Incascadeionization,afreeelectronimpactsanatom,causingionization,producinganadditionalfreeelectron.Thisleadstoarapidgrowthoffreeelectrondensityinagasandplasmaformation.Inmulti-photonionization,multiplephotonsareallincidentonasingleatomatoncesuchthatthesumofthephotonenergiesexceedstheionizationenergyoftheatomandafreeelectronisproduced. Itisgenerallythoughtthatbothprocessesplayaseparateroleinlaser-inducedplasmaformation.Therapidgrowthoftheplasmaaftertheinitialbreakdowniscommonlyattributedtocascadeionization.Butforcascadeionizationtotakeplacetheremustalreadybefreeelectronspresent,oratleastarstfreeelectronpresenttoimpactanatom.Thatrstelectronisthoughttobeproducedbymulti-photonionization.Individualplasmainceptionevents,suchasthatshowninFigure 6-2 (a),maythencorrespondtoindividualinstancesofmulti-photonionizationthatcreatetheseedelectronsneededforcascadegrowth. Considerthelikelihoodofmulti-photonionizationingasessuchasnitrogen,argon,orhelium.Theionizationenergiesfornitrogen,argon,andheliumare1503kJ/mol,1520kJ/mol,and2372kJ/molrespectively.Relatingthesevaluestotheenergyinasinglephotonfromalasersourceoperatingat1064nm,itrequires14simultaneousphotons 138

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tocauseionizationinnitrogenandargon,and21simultaneousphotonstocauseionizationinhelium.Thelikelihoodforthistooccurcanbeevaluatedbyconsideringthedistributionofphotondensityinthelaserbeamalongwiththeprobabilityofphoton-atominteraction. Figure 6-12 showsasimulationofthedistributionofphotondensityinalaserbeamcorrespondingtotheprolemeasuredinprevioussections,withaGaussiandistributionofenergyacrossitsdiameterandapulseenergyof400mJ.Thisdistributionofphotonsexistswithinanexposuretimeof2.910)]TJ /F9 7.97 Tf 6.59 0 Td[(5ns,theamountoftimeittakesforaphotonoflighttotraverseonepixel.Thebottommostrowofpixelsinthegurecorrespondstothebeamcenterline.Themaximumphotondensitythereforeoccursalongthecenterlineatthepointofminimumbeamdiameterandisabout71015photons=mm3. Anaveragenumberofpixelsperatom(ormolecule)canbecalculatedbymultiplyingthephotondensitybythevolumeofasingleatom(ormolecule).Thevolumeofanatom(ormolecule),however,isnotastraightforwardpropertywhenconsideringthesphereofinuenceanucleusanditselectroncloudexhibitsonsurroundingphotons.TheVanderWaalsradiusisusefultomodelanatom(ormolecule)asahardsphere,butitisunlikelyagoodestimatorofthevolumeoverwhichaphotonmustbewithininordertobeinuencedbytheparticle.Forthepurposesoftheargument,aradiusofinuenceoftwicetheVanderWaalsradiuswillbeconsideredtodenetheappropriatevolumeinwhichaphotonmustbetobeinuenced,orabsorbed,byanatomormolecule.Theaveragenumberofphotonspernitrogenmoleculeatthepeakofphotondensitywouldbeabout1.610)]TJ /F9 7.97 Tf 6.58 0 Td[(3photons=moleculeforasingleexposure. Asingleexposure,however,representsonlyasmallfractionofthetimethataphoton,orgroupofphotonsmayinteractwithamolecule.Forthepurposesofthisdiscussion,anexposurerepresentstheamountoftimeittakesforaphotontotraversethelengthofonepixelinspace,about2.910)]TJ /F9 7.97 Tf 6.59 0 Td[(5ns.Theamountoftimerequiredformulti-photonionizationtoliberateanelectroncanbeestimatedtobeontheorderof 139

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about110)]TJ /F9 7.97 Tf 6.59 0 Td[(10s(Kulander,1987).Therefore,fromtheperspectiveoftheatomsormoleculespresentinasinglepixel,aphotonmaylingerwithinthesphereofinuenceoveraperiodofabout1000exposures.Thiseffectivelyincreasesthedensityofphotonsthatmayimpingeaparticlesimultaneouslybyafactorofabout1000.Thepeakvalue,therefore,fortheaveragenumberofphotonspernitrogenmoleculeatthepeakofphotondensityisabout1.6. Considerfurtherthatthearrivalofasingleormultiplephotonstoatarget,suchasaCCDpixelorparticle,isdescribedwellbyPoissonstatistics.Theprobabilitythatnphotonsarrivewithinatargetvolumesimultaneouslyisgivenby: Pn=ne)]TJ /F10 7.97 Tf 6.58 0 Td[( n!,(6) whereistheaveragenumberofphotonspertargetvolume.Theprobabilityofamulti-photonionizationeventinnitrogenoverasinglepixelcanthereforebeestimatedbysubstituting=1.6andn=14intheaboveequationandmultiplyingbythenumberofparticlesinasinglepixel.Thisgivestheprobabilityofamulti-photoneventatthepeakofphotondensitytobeontheorderofabout1. Asimulationofthedistributionoftheprobabilityofamulti-photonionizationeventinnitrogenisshowninFigure 6-13 .Herethebottommostrowcorrespondstothelaserbeamcenterline.SimilardistributionsfortheprobabilityofMPIeventsinargonandheliumareshowninFigures 6-14 and 6-15 ,respectively. ThedistributionofMPIprobabilitiesinheliumshowsadistinctlysmallervariationspatially,thaneitherargonornitrogen.Thisisprimarilyduetothedifferenceinionizationenergy.Notonlyisitmoredifculttofreeanelectronfromheliumbymulti-photonionization,buttheregioninspaceoverwhichthisispossibleissmalleraswell.Thisagreeswellwiththepreviousresultsforthedistributionofplasmainceptionevents. 140

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6.6ANoteonSphericalAberration Thedifferencesintheobservedspatialdistributionoftheindividualplasmainceptioneventsforeachgashasbeendiscussed.Whilethisdiscussionhasfocusedonwhythespatialdistributionvarieswiththegas,itisstilllefttobeconsideredastowhyitexistsatall.Apossible,andlikely,explanationoftheexistenceofthespatialdistributionofplasmakernelsisduetothepresenceofsphericalabberationonthefocusinglens.Asthelasersourcepassesthroughthefocusinglens,lightraysarerefractedtowardsthefocalpointasdescribedbySnell'sLaw.Sphericalaberrationisessentiallyduetothepresenceofthenon-idealcurvatureofthelensresultinginanimperfectfocalpoint.Infact,lensessufferingfromsphericalaberration,produce,notasinglefocalpoint,butaniteregionoverwhichthebeamdiameterisaminimum.Thisiscausedbythenon-uniformrefractionoflightraysimpingingthelensfartherfromitscenter.Lensesthatdonotsufferfromsphericalabberationareknownasasphericlensesandarecharacterizedbysurfaceprolesthatarenotsimplyportionsofspheresorcylinders.Suchlensesaremoredifculttomanufactureandarethusmorecostly. Thefactthatthelensusedforthecurrentstudywasspherical,andthereforesuffersfromsphericalabberationisonepossibleexplanationbehindtheexistenceofthespatialdistributionofplasmainceptionevents.Asthereisnotonesinglefocalpoint,butasmallrangeofminimumdiameter,thenthereisanentireregionofspacewheremorethanonemulti-photonionizationeventsmaytakeplace.However,mostLIBSexperimentsintheliteratureusesphericallensesandsuchapracticeisnotdetrimental. Thefactremainsthatthedistributionofindividualplasmainceptionkernelsdoeschangedependingonwhichambientgasisbeingobserved.Sowhiletheexistenceofthedistributionofkernelsmaybedueofsphericalabberationitisstillthegaschemistrythatisresponsibleforitscharacteristics. 141

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Figure6-1. SchematicofexperimentalLIBSapparatusforplasmainceptionstudy. 142

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Figure6-2. Evolutionoflaser-inducedplasmaoveritslifetime.(a)Earlyplasmaformation20nsafterpulse,(b)Earlyplasmaformation30nsafterpulse,(c)Earlyplasmaformation40nsafterpulse,(d)Fullyformedlaser-inducedplasma100nsafterpulse,(e)Plasmabeginstorelax1safterpulse,(f)Plasmadeactivatesanddecays2safterpulse. 143

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Figure6-3. Laser-inducedplasmaformationinnitrogenatearlytimes.(a)Earlyplasmainceptioneventsataveryshorttime(1ns)afterpulse,(b)Earlyplasmaformation10nsafterpulse,(c)Earlyplasmaformation20nsafterpulse,(d)Earlyplasmaformation30nsafterpulse,(e)Earlyplasmaformation40nsafterpulse. 144

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Figure6-4. LineproleacrosstheCCDshowingearlyplasmainceptionfeaturesinnitrogen.Theupperproleshowstheraw,unprocessedsignal,whilethelowerproleshowstheprocessedsignalwithpeaksidentied. 145

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Figure6-5. LineproleacrosstheCCDshowingearlyplasmainceptionfeaturesinargon.Theupperproleshowstheraw,unprocessedsignal,whilethelowerproleshowstheprocessedsignalwithpeaksidentied. 146

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Figure6-6. LineproleacrosstheCCDshowingearlyplasmainceptionfeaturesinhelium.Theupperproleshowstheraw,unprocessedsignal,whilethelowerproleshowstheprocessedsignalwithpeaksidentied. 147

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Figure6-7. Collectionof30plasmainceptionimagesinnitrogeninrelationtothelaserbeamprole.Eachsetofinceptionpointsoccuralongthecenterlineinreality,butareshowndisplacedforclarity. 148

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Figure6-8. Collectionof30plasmainceptionimagesinargoninrelationtothelaserbeamprole.Eachsetofinceptionpointsoccuralongthecenterlineinreality,butareshowndisplacedforclarity. 149

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Figure6-9. Collectionof30plasmainceptionimagesinheliuminrelationtothelaserbeamprole.Eachsetofinceptionpointsoccuralongthecenterlineinreality,butareshowndisplacedforclarity. 150

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Figure6-10. Inrelationtothebeamprole,plasmainceptioneventsoccurpastthefocalpoint,wheretheplasmaformsatthefocalpoint.(a)Earlyplasmainceptioneventsshortlyafterthepulse(1ns),(b)Earlyplasmaformation20nsafterpulse,(c)Earlyplasmaformation40nsafterpulse. 151

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Figure6-11. Summaryofplasmainceptionstatisticsfornitrogen,argon,andheliuminrelationtothelaserbeamprole. 152

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Figure6-12. SimulatedimageofthedistributionofphotondensityacrossseveralpixelsoftheCCD. 153

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Figure6-13. Simulateddistributionoftheprobabilityofamulti-photonionizationeventinnitrogen. 154

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Figure6-14. Simulateddistributionoftheprobabilityofamulti-photonionizationeventinargon. 155

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Figure6-15. Simulateddistributionoftheprobabilityofamulti-photonionizationeventinhelium. 156

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CHAPTER7CONCLUSIONS 7.1Summary Thecurrentstudyendeavorstounderstandandquantifythecomplexplasma-particleinteractionsthattakeplaceduringthelaser-inducedbreakdownspectroscopyofaerosolsystems.Importantly,applicationsextendtootheranalyticalmethodssuchasInductively-CoupledPlasmaAtomicEmissionSpectroscopy(ICP-AES)andLaser-AblationInductively-CoupledPlasmaMassSpectrometry(LA-ICP-MS),whereplasma-particleinteractionsintheICPareanalagoustothecurrentstudy.Thestudyoftheplasma-materialinteractionsisbeingaccomplishedthroughthedesignandimplementationofanumericalmodelthattakesintoaccounttheindividualprocessesofheattransfer,masstransfer,andvaporizationkinetics.Severaladvancementshavebeenmadetowardthisgoal. First,theglobalplasmaenvironmenthasbeenmodeledbysimulatingtheprocessesofheattransferandmasstransferthroughdiffusion.Basedonaprescribedinitialconditionandappropriateboundaryconditions,theenergyequationforconductionissolvednumericallyusinganimplicitnitedifferenceschemetoobtainthetemperatureeldasafunctionofplasmaradiusandtime.Massdiffusionisallowedthroughouttheplasmaenvironmentandthemasstransferequationissolvedthroughasimilarprocedureastheenergyequationtoobtaintheconcentrationeld,alsoasafunctionofplasmaradiusandtime,forthevariousplasmaconstituents.Oncethetemperatureandconcentrationeldsareknown,severalplasmapropertiesarecalculated,suchaselectrondensity,ionizationstatedistributions,andemissionintensity. Second,thelocalplasma-particleinteractionsaremodeledthroughvariousmethodstosimulatetheprocessesofaerosolparticlevaporizationanddissociation.Vaporizationisrstsimulatedtooccurataconstantprescribedrateasapreliminarymethodtoinvestigatetheeffectsofanitevaporizationrateversusaninstantaneous 157

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rate.Next,vaporizationismodeledasaseriesofdistinctstepsofmelting,evaporation,andspeciesliberation.Atomsareremovedfromtheaerosolparticleataratethatiseithercontrolledbyheattransferormasstransferdependingonthecurrentstateoftheenvironment. Modelingeffortsshowthattheparticlevaporization,massdiffusion,andheattransferprocessesthattakeplace,dosoovernitetimescales.Theseresultsshowthatwhileitisoftencommonplaceforresearcherstoassumethattheseprocessestakeplacewithsufcientratestobeassumedinstantaneous,thismaynotbethecase,especiallyforearlytimes.Furthermore,nitevaporizationanddiffusionratesaffecttheLIBSresponseandknowledgeoftheseprocessesmayleadtoanincreasedunderstandingofhowmatrixeffectsinuencethediagnostic.Resultssuggestthatsincethegoverningprocessesoccurovernite,butrapid,timescalesthatLIBSobservationshouldtakeplaceatlatertimestojustifythesimplifyingassumptionsandallowtimefortheanalytespeciestodiffusethroughandequilibratewiththeentireplasma. Lastly,anexperimentalstudyhasbeenperformedtoinvestigatetheearliesttimesofplasmaexistenceinordertofurthertheunderstandingofthephysicsofplasmainception.Plasmaswerecreatedinseveraldifferentgasesandtheirbehaviorattheearliestobservablelifetimeswasstudied.Atearlytimes,plasmasformnotfromasinglebreakdownevent,butfromseveralinitialbreakdownkernelslocateddownstreamfromthelaserfocalpoint.Thenumberandspatialdistributionofinitialbreakdowneventsvariesbymedium.Astimepassestheindividualbreakdownkernelsgrowandcoalescetowardthelasersource,culminatinginafullyformedplasmalocatedinthecenterofthelaserfocalpoint.Sphericalabberationofthefocallensandthevaluesoftheionizationenergyforthedifferentgasesareusedtoprovideanexplanationforthisbehavior. 7.2SuggestionsforFutureResearch Whiletherehasbeenmuchworkdonetowardsthefundamentalunderstandingofthecomplexplasma-materialinteractionsthatgoverntheLIBSofaerosolsystems, 158

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therearemanywaysinwhichthepresentresearchmaybeextendedtoprovidefurtherinsight.Basedonthepreviouslydiscussedresults,thefollowingeffortsareproposedforfutureresearch: Implementationofthesolutionofthevelocityeldbasedeitheronpoint-blasttheory,orafullsolutionoftheNavier-Stokesequations.Thevelocityeldmaythenbeusedtodeterminetheimportanceofconvectivetermsofheatandmasstransport.Itisdesiredtoalsoaccountforcompressibilityeffects,andthereforethepresenceoftheplasma'ssphericalshockwaveatearlytimes. Evaluationofradiativemodesofheattransferintheglobalplasmaenvironmentmodel. Investigationoftheeffectsofspectrallydependentquantitiesthroughtheimplementationoflinebroadeningmechanismsandthecalculationoflineprolefunctionstogeneratemodeloutputthatsimulatesspectra. Investigatetheeffectsofelectromagneticforcesduringthedurationofthelaserpulsetotheformationofthelaser-inducedplasma.Theelectricandmagneticforcetermsactassourcefunctionstodrivethehydrodynamicmotionduringtheperiodoftimewhenthelaserpulseisactivecreatingafullymagnetohydrodynamicmodeloflaser-inducedplasmabehavior. Introductionofatheoreticalmodelofplasmainception,therebyremovingthesemi-empiricalnaturefromthecurrentplasmamodel.TheplasmainceptioncharacteristicsmayexploredtheoreticallythroughtheintroductionoftheeffectoftheelectromagneticforcespresentintheexcitinglaserpulsetotheinitialconditionsofthesystemorbytheevaluationofaMonteCarlosimulationtotheassumptionoflocalthermodynamicequilibriumatearlyplasmatimes. Togetherwiththepreviouslyestablishedmodel,theseadditionsandrenementswillcompriseasophisticatedandinclusivedescriptionoftheprocessesimportanttoLIBSofaerosolsfromwhichmuchfundamentalknowledgemaybegleanedandusedforthebenetoftheLaser-InducedBreakdownSpectroscopyresearchcommunity. 159

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BIOGRAPHICALSKETCH PhilipJacksonwasawardedbachelor'sdegreesinbothAerospaceEngineeringandMechanicalEngineeringattheUniversityofFloridain2003.Hereceivedamaster'sdegreeinMechanicalEngineeringunderDr.JillPetersonattheUniversityofFloridain2005.HeiscurrentlyaresearchassistantintheLaser-BasedDiagnosticsLaboratoryattheUniversityofFloridawhilepursuingadoctoraldegreeunderDr.DavidHahn. 167