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Molecular Dynamics as a Foundation for Flux Prediction through Nanoporous Membranes

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Title:
Molecular Dynamics as a Foundation for Flux Prediction through Nanoporous Membranes A Vectorized, Constraint-Free Approach to Conservative Simulations
Creator:
Inman, Matthew C
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
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english
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1 online resource (165 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
LEAR,WILLIAM E,JR
Committee Co-Chair:
HAHN,DAVID WORTHINGTON
Committee Members:
ROY,SUBRATA
BRENNAN,ANTHONY B
CRISALLE,OSCAR DARDO
Graduation Date:
8/9/2014

Subjects

Subjects / Keywords:
Carbon ( jstor )
Carbon nanotubes ( jstor )
Conceptual lattices ( jstor )
Diameters ( jstor )
Diffusion coefficient ( jstor )
Modeling ( jstor )
Molecules ( jstor )
Simulations ( jstor )
Trajectories ( jstor )
Velocity ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
diffusion -- dynamics -- md -- nanotubes -- transport
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

Notes

Abstract:
A novel, open-cathode direct methanol fuel cell (DMFC) has been designed and built by researchers at the University of North Florida and University of Florida. Foremost among the advances of this system over previous DMFC architectures is a passive water recovery system which allows product water to replenish that consumed at the anode. This is enabled by a specially-designed water pathway combined with a liquid barrier layer (LBL). The LBL membrane is positioned between the cathode catalyst layer and the cathode gas diffusion layer, and must exhibit high permeability and low diffusive resistance to both oxygen and water vapor, bulk hydrophobicity to hold back the product liquid water, and must remain electrically conductive. Maintaining water balance at optimum operating temperatures is problematic with the current LBL design, forcing the system to run at lower temperatures decreasing the overall system efficiency. This research presents a novel approach to nanoporous membrane design whereby flux of a given species is determined based upon the molecular properties of said species and those of the diffusing medium, the pore geometry, and the membrane thickness. A molecular dynamics (MD) model is developed for tracking Knudsen regime flows of a Lennard-Jones (LJ) fluid through an atomistic pore structure, hundreds of thousands of wall collision simulations are performed on the University of Florida HiPerGator supercomputer, and the generated trajectory information is used to develop number density and axial velocity profiles for use in a rigorous approach to total flux calculation absent in previously attempted MD models. Results are compared to other published approaches and diffusion data available in the literature. The impact of this study on various applications of membrane design is discussed and additional simulations and model improvements are outlined for future consideration. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: LEAR,WILLIAM E,JR.
Local:
Co-adviser: HAHN,DAVID WORTHINGTON.
Statement of Responsibility:
by Matthew C Inman.

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Source Institution:
UFRGP
Rights Management:
Copyright Inman, Matthew C. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Classification:
LD1780 2014 ( lcc )

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MOLECULARDYNAMICSASAFOUNDATIONFORFLUXPREDICTIONTHROUGHNANOPOROUSMEMBRANES:AVECTORIZED,CONSTRAINT-FREEAPPROACHTOCONSERVATIVESIMULATIONSByMATTHEWCLAYINMANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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2014MatthewClayInman 2

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Tomywifeandfamily. 3

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ACKNOWLEDGMENTS FirstandforemostIwouldliketothankmyadvisorDr.WilliamLearfortakingachanceonastudentwithanacademicallycheckeredpastandhelpinghimdevelopasastudent,aresearcherandaperson.Ithasbeenmygreatpleasuretoworkwithyouoverthepast3yearsandIwouldnotbestandingwhereIamtodayhaditnotbeenforyourguidance.IwouldalsoliketogivemythankstotherestofcommitteefortheirpatienceandguidanceasIwadedmywaythroughtheliteratureinsearchofaresearchprojectthatIwasinspiredby.ToDr.AnthonyBrennanforhisinsightsonmaterialsprocessingandenthusiasmforourproject.Isincerelyhopethatwemaycollaborateonfutureresearch,asourdiscussionswerealwaysengaging,insightful,andaweeklyhighlight.ToDr.OscarCrisalleforhisadviceanddirection,especiallyearlyon,aswemadetripstoUNFtoworkwiththeDP4.Ilearnedmoreaboutcontroltheoryandfuelcellelectrochemistryduringthosecarridesthaninweeksofself-study,betterenablingmetobemoremultidisciplinaryandworkwithmycolleaguesonothersubsystems.ToDr.DavidHahnforhisincrediblelevelofsupportdespitehisunenviableworkloadasDepartmentChair.WhetheritwasofferingtowritemelettersofrecommendationforfellowshipsbeforeIcouldask,ororganizinggraduateteachingprogramsthatIhadthepleasureofparticipatingin,orjustteachingthebestdarnConductionHeatTransferclassontheplanet,havingDr.HahnasapartofthisworkissomethingIamparticularlyhappyaboutandproudof.AndtoDr.SubrataRoy,whoseknowledgeintheeldofmicro-andnanoscaleowsissomethingIamstrivingtoachieve.Ithasbeenapleasureworkingwithallofyou.MyworkhasbeennanciallysupportedthroughseveralsourcesandI'dliketoacknowledgetheUnitedStatesArmyCommunications-ElectronicsResearch,DevelopmentandEngineeringCenter(CERDEC)andtheUFMechanicalandAerospaceEngineeringDepartmentfortheircontributions.Iwouldalsoliketothank 4

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ProfessorRaphaelHaftkaforhisdonationstothedepartmentandfoundingtheteachingprogramforgraduatestudentsofwhichIhadtheprivilegeofparticipatingin.IhavealsobeenprivilegedtoworkwithanumberofindividualswhoIhavelearnedagreatdealfromovertheyearsonthisproject.AspecialthankstomyfellowDMFCresearchteammatesFennerColsonandSydniCredle.IwouldalsoliketoacknowledgeRafeBiswas,ShyamMudiraj,LukeNealandJohnCrittendenfromtheUFteam,andDr.JamesFletcher,Dr.PhilipCox,JasonHarrington,BenjaminSwanson,TaylorMaxwellandMarkButlerfromtheUNFresearchgroup.Frompumps,tocontrols,toexperimentationandmodeling,Ihavebecomeafarmorewell-roundedengineerinlargeparttoyourskills,yourintelligenceandyourwillingnesstoworkwithme.Iwon'tforgetit.Thankyou.IwouldalsoliketothankseveralamongthecurrentandformerfacultyandstaffatUFforhelpingmakemyexperienceattheUniversityofFloridasofantastic.JeffStudstillwasenormouslyhelpfulinallmannersofmaintenanceandinventoryandShellyBurlesonalwayshelpedkeepmeontrackandworkedpatientlywithmedespitehavinganearperpetuallineofstudentsoutherdoorstretchinghalfwaytoNebraska.Dr.LouCattafestagavememyrstshotatUndergraduateresearchanditchangedmyentireperspectiveonwhatIwantedtogetoutofmytimeatUF.LikewiseDr.DavidMikolaitisgavememyrstopportunitytoTAasanundergraduateandIknewfromthatmomentthatonedayIwantedtobeaprofessor.InadditionIwouldliketothanktheprofessorswhoseteachingstylesandsubjectmattertrulyinspiredmetobecomeabetterstudent:Dr.MarkSheplak,Dr.JamesKlausner,andDr.NormanFitz–Coy.AndlastlyI'dliketothankDr.JohnAbbittandDr.LarryUkeiley,whileactingasTAfortheircoursesItookeveryopportunityIcouldtolearnhowtoteachfromthem.Ihopetoapplythoselessonsinclassesofmyowndowntheroad.OnamorepersonalnoteI'dliketothankthefriendsI'vehadthroughoutandthoseIhavemadealongthewaywhohavehelpedkeepmesaneattendingschooltheselast9 5

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years(anactmanyofthesepeoplewouldconsiderinsane).HanifAli,DylanChaseandMichaelBobekdeservemuchmorethanksthanIcanpossibletintotwodimensionshere.Andthewholemathcrowd,Kevin,Erica,Lee,JillandKarly.Onthemostpersonalnoteofall,Isendmythanksandlovetomyfamily.Tomygrandparents,auntsandunclesfortheirsupport.Ihavecometorealizeonlyinthelast10yearshowgettingtoseeallofyoursooftenistheexceptionandnottherule.Iamtrulyluckytohavesuchatight-knitandlovingextendedfamily.Tomyfather,ClayInman,whocametoeverysoccergameandinstilledacuriosityinthesciencesinmefromaveryyoungage.Needlesstosay,I'mgladhedid.Tomymother,MaryInman,whohasfacedtrialsandsicknessesthatwouldhavebeatenmostbutremainstothisdaythestrongest,mostcaringandlovingpersonIknow.Tomyyoungerbrother,TomInman,whoiswithoutquestionthekindestandmostforgivingpersonIknow.Andtomywife,JessicaInman.Shehasbeenmypillarofsupportandmakesmylifebettereveryday.Ineverthoughtanybodywouldbeabletolovemesomuch.Iloveyouall.Thankyou. 6

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 10 LISTOFFIGURES ..................................... 11 NOMENCLATURE ..................................... 13 ABSTRACT ......................................... 22 CHAPTER 1INTRODUCTION ................................... 24 1.1Motivation .................................... 24 1.1.1ApplicationsofSelectiveMembranes ................. 24 1.1.2IntroductiontoDirectMethanolFuelCells .............. 24 1.1.3ExistingIssueswiththeOpenCathodeDMFCDesign ....... 25 1.1.4BenetsofanImprovedLiquidBarrierLayer ............. 28 1.2ResearchObjective .............................. 28 1.3DissertationOutline .............................. 29 2LITERATUREREVIEW ............................... 30 2.1FoundationalTheoryofMassTransportintheKnudsenandTransitionalRegimes ..................................... 30 2.1.1SeminalAnalysisofSub-ContinuumDiffusion ............ 30 2.1.2TheStefan-MaxwellApproachandtheDustyGasModel ...... 33 2.1.2.1DevelopmentoftheDustyGasModel ........... 35 2.1.2.2PrevalentresearchapplicationsoftheDustyGasModel . 37 2.1.2.3CriticalreviewoftheDustyGasModel ........... 39 2.1.2.4OthernotableStefan-Maxwellformulations ........ 40 2.2TransportModelsinthePresenceofanAdsorptionField .......... 42 2.2.1TheOscillatorModel .......................... 42 2.2.1.1Seminalmodeldevelopmentforlow–pressuresystemsandsubsequentmodelrevisions .............. 42 2.2.1.2Criticalreviewoftheuxhypothesisandmathematicalformulation .......................... 47 2.2.2TheDistributedFrictionModel ..................... 47 2.3MolecularDynamics .............................. 49 2.3.1IntroductiontoMolecularDynamicsandAtomisticSystems .... 49 2.3.1.1Argumentforsingle–walledcarbonnanotubesasatransportmedium ............................ 49 2.3.1.2Transportthroughsingle–walledcarbonnanotubesviaequilibriummoleculardynamics .............. 51 7

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2.3.1.3Transportthroughsingle–walledcarbonnanotubesvianonequilibriumandstatisticalmethods ........... 55 2.3.2SimulationofChainMoleculesandPolymers ............ 58 2.4ConcludingRemarks .............................. 61 3VECTORIZEDMOLECULARDYNAMICDIFFUSIONMODEL .......... 62 3.1OutlineforaCoupledEMD/KineticTheoryApproachtoFluxPrediction .. 62 3.2RationaleforaVectorizedApproach ..................... 63 3.3DevelopmentofModelSystems ........................ 64 3.3.1IntegrationProcedureforItinerantPointMasses ........... 64 3.3.2IntegrationProcedureforItinerantMolecularBodies ........ 70 3.3.2.1Watermodels ........................ 70 3.3.2.2Constraintsvs.quaternionsinrigidbodydynamics .... 73 3.3.2.3Fourth-orderquaternionintegration ............ 75 3.3.3ConvergenceCriteria .......................... 84 3.3.4ComparativeStudyofIntegratorOrdersofAccuracy ........ 84 4RESULTSANDDISCUSSION ........................... 94 4.1SimulationsintheInniteKnudsenNumberRegime ............ 94 4.2RariedMolecularTransportthroughSingle–WalledCarbonNanotubes . 99 4.2.1OxygenTransport ............................ 102 4.2.2ModelValidation ............................ 106 4.2.3ConvergenceCriteria .......................... 118 4.2.4WaterTransport ............................. 124 4.3SampleSelectivityAnalysis .......................... 126 5CONCLUSIONSANDFUTUREWORK ...................... 130 APPENDIX AFORCEFIELDDERIVATIONS ........................... 134 BRUNGE–KUTTACOEFFICIENTSFORTHEPOINTMASSCASE ....... 142 CRUNGE–KUTTACOEFFICIENTSFORRIGIDBODYCASE ........... 143 DDISTRIBUTIONFORMULATIONSFROMRAWTRAJECTORYDATA ..... 145 D.1RadialOccupancyDistribution ........................ 145 D.2Intra–CollisionalAxialDisplacementDistribution .............. 148 D.3SpecularityDistribution ............................ 148 ESAMPLEMATLABCODEFORFINITEKNUDSENMODEL ........... 151 E.1VectorizedCodeSamples ........................... 151 E.2DiscussionofAdaptiveTimeSteps ...................... 152 8

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REFERENCES ....................................... 155 BIOGRAPHICALSKETCH ................................ 165 9

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LISTOFTABLES Table page 3-1Approximateorderoferrorofvariousnumericalintegrationschemes ...... 68 3-2Watermodelparameters .............................. 71 4-1O2diffusioncoefcientandstreamingvelocitydependenceonSWCNTradius. 104 4-2Propertiesofgasesusedinthemodelvericationanalysis. ........... 107 4-3Experimentalconditionsofthedouble–walledCNTmembranediffusionexperimentofHoltetal.[ 60 ]. ................................... 107 4-4Comparesowcharacteristicspredictedforvariousdiffusingspeciesgivenstatisticsgeneratedfrom50kand25kwallcollisionsimulations. ......... 119 D-1OutputvectorsfromtheFKMsimulations. ..................... 145 10

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LISTOFFIGURES Figure page 1-1TraditionalandnovelDMFCwaterrecoveryarchitectures ............ 26 1-2DMFCsinglecellschematic ............................. 27 2-1CrosssectionalviewofpermeatetrajectoriesintheOscillatorModel ...... 44 2-2Commoncarbonnanotubelatticedefects ..................... 51 2-3DualControlVolumeGrandCanonicalMonte–Carlosimulationdiagram .... 57 2-4Polymerpackingmodeldiagram .......................... 60 3-1IllustrationoftheLorenz–BerthelotmixingrulesforLJpotentialinteractionsbetweendissimilarbodies .............................. 65 3-2Prevalentwatermodelarchitectures. ........................ 72 3-3EvolutionoftheradialnumberdensityproleofO2inan(8,8)SWCNT ..... 85 3-4LinearregressionoftheradialnumberdensitydistributionofO2inan(8,8)SWCNT ........................................ 86 3-5EvolutionoftheaxialcollisiondistancedistributionofO2inan(8,8)SWCNT .. 87 3-6LinearregressionoftheaxialcollisiondistancedistributionofO2inan(8,8)SWCNT ........................................ 88 3-7Comparisonofintegrationaccuracyfort=0.1fs ................ 90 3-8Comparisonofintegrationaccuracyfort=0.5fs ................ 91 3-9Comparisonofintegrationaccuracyfort=1.0fs ................ 92 3-10Comparisonofintegrationaccuracyfort=2.0fs ................ 93 4-1ConvergedradialoscillatorybehaviorofO2inastaticSWCNT ......... 96 4-2ColdWallEquilibriumModelpermeatetrajectoryevolution ............ 97 4-3ConvergedaxialoscillatorybehaviorofO2inastaticSWCNT .......... 98 4-4x)]TJ /F6 11.955 Tf 11.96 0 Td[(yprojectionofanO2moleculeightpathinafullydynamicSWCNT .... 100 4-5RelationshipbetweentheColsondiffusioncoefcient/averagestremingvelocityratioandporeraius. ................................. 104 4-6RadialandaxialmotionofanO2permeantina(10,10)SWCNTviatheFiniteKnudsenmodel. ................................... 105 11

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4-7O2trajectoryillustrationsthroughadouble–walledCNT. ............. 108 4-8Comparisonofmeasuredandpredictedvolumeowratesofvariousgasesthroughcarbonnanotubemembranes. ....................... 109 4-9DegreeofspecularityforO2throughadouble–walledCNT. ........... 112 4-10PolarplotoftheradialoccupancydistributionforO2ina(10,10)SWCNT. ... 114 4-11Maxwell–Boltzmanenergydistributionandadiscussiononitsvalidityinconnednanospaces. ..................................... 115 4-12Comparisonofsimulatedandmeasuredselectivityindouble–walledCNTs. .. 116 4-13EvolutionoftheradialnumberdensityprolesofO2inSWCNTsofvariouschiralities. ....................................... 120 4-14LinearregressionoftheradialnumberdensitydistributionofO2inSWCNTsofvariouschiralities. ................................. 121 4-15Evolutionoftheintra–collisionalaxialdisplacementdistributionsofO2inSWCNTsofvariouschiralities. ................................. 122 4-16Linearregressionoftheintra–collisionalaxialdisplacementdistributionsofO2inSWCNTsofvariouschiralities. .......................... 123 4-17Radialnumberdensityprolesofvarious3–bodiedwatermodelsat50Cina(10,10)SWCNT. ................................... 124 4-18jzjprolesofvarious3–bodiedwatermodelsat50Cina(10,10)SWCNT. .. 125 4-19Oxygenowrateandoxygen/watervaporselectivityasafunctionofporeradiusinsidesingle–walledcarbonnanotubes. ...................... 128 A-1SchematicofthepairwiseLennard–Jonesinteractionatlongdistance ..... 134 A-2SchematicofthesinglepairandhexagonalMorsebond–stretchinginteractions 136 A-3Schematicoftheharmonicbond–bendinginteractionsbetweenlatticetriplets . 137 A-4Illustrationoftherotatedframeofreferenceusedincalculatinginducedquadrupoleinterations. ...................................... 140 D-1Exampleradialnumberdensitydistribution. .................... 148 D-2Exampleintra–collisionalaxialdisplacementdistribution. ............. 149 D-3Illustrationofthewall–collisionalframeofreferenceusedindetermininghowspecularthereectionsareinsideacarbonnanotube. .............. 149 E-1Carbon–CarbonLennard–Jonespotentialenergyfunction ............ 153 12

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LISTOFSYMBOLS,NOMENCLATURE,ANDABBREVIATIONS Alphabetical SymbolDenitionpg.# AHeavisidefunction 48 cMolarconcentration 34 dParticlediameter 101 DDiffusioncoefcient 31 DStefan–Maxwelldiffusioncoefcient 34 DPorediameter 101 EEnergy 70 FComponentforce 43 FCartesianforcevector 67 HHamiltonian 43 IMomentofinertiatensor 77 JMolarux 32 JRunge–Kuttacoefcientvectorforpositionsecondderivatives 70 KRunge–Kuttacoefcientvectorforpositionrstderivatives 70 kBBoltzmann'sconstant,1.380610)]TJ /F2 7.97 Tf 6.58 0 Td[(23kgm2 s2K 31 kTThermaldiffusionratio 36 KnKnudsennumber 30 LPorelengthormembranethickness 31 Continuedonnextpage... 13

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Alphabetical–Continuedfrompreviouspage SymbolDenitionpg.# LRunge–Kuttacoefcientvectorforquaternionrstderivatives 80 mMass 43 MTotalmolecularmass 77 MRunge–Kuttacoefcientvectorforquaternionsecondderivatives 80 MMolecularmass 31 nNumberdensity 35 NAAvogadro'snumber,6.02211023mol)]TJ /F2 7.97 Tf 6.59 0 Td[(1 31 NCNumberofcarbonatoms 142 OOrderofmagnitude 55 pMomentum 43 PPressure 31 qChargeorpartialcharge 71 qQuaternioncomponent 75 [q]Quaternion 75 [_q]Firsttimederivativeofquaternion[q] 77 [q]Secondtimederivativeofquaternion[q] 78 QVolumetricowrate 31 RRotationmatrix 75 Continuedonnextpage... 14

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Alphabetical–Continuedfrompreviouspage SymbolDenitionpg.# rCartesianpositionvector 66 _rCartesianvelocityvector 41 rCartesianaccelerationvector 67 RPoreradius 31 RIdealgasconstant,8.314J molK 32 tTime 67 TTemperature 31 UPotentialenergy 43 vMassaveragedvelocity 34 Greek SymbolDenitionpg.# Morseexponentialcoefcient 66 Specicweight 31 Dustygasdiffusioncoefcientfactor 35 Error 68 "Lennard–Jonespotentialwelldepthofan–pairdeterminedfromtheLorenz–Berthelotmixingrules 66 Tortuosityfactor 37 Viscosity 41 Continuedonnextpage... 15

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Greek–Continuedfrompreviouspage SymbolDenitionpg.# Angle 66 #Specularreectionfactor 31 Quadrupolemomenttensor 140 Porosity 37 Meanfreepath 30 zIntra–collisionalaxialdisplacement 104 Chemicalpotential 34 Kinematicviscosity 37 Distributedfrictionmodelwallfrictioncoefcient 48 $Molaruxratio 33 Collisionfraction 114 Density 43 %Resistancefactor 41 Lennard–Jonesdiameterofan–pairdeterminedfromtheLorenz–Berthelotmixingrules 66 &Poredensity 106 Oscillatormodelhoppingtime 44 Torquevector 77 'Volumefraction 41 Continuedonnextpage... 16

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Greek–Continuedfrompreviouspage SymbolDenitionpg.# Molefraction 32 Shearstress 48 !Angularvelocityvector 77 _!Angularaccelerationvector 78 Thermaldiffusionfactor 36 Subscripts SymbolDenitionpg.# ABBinarydiffusionbetweenconstituentsAandB 33 dReferringto“dust”particlesinthedustygasmodel 35 DIndicatesadiagonalizedmatrix 79 eEquilibriumvalue 66 gCenterofmassproperty 77 KnAccordingtoclassicalKnudsenanalysis 31 LJReferringtotheLennard–Jonespotential 66 mMembrane 41 mpMostprobablevalue 115 MorseReferringtotheMorsebondstretchingpotential 66 QReferringtotheinducedquadrupolepotential 140 rRadialcomponentinpolarcoordinates 43 Continuedonnextpage... 17

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Subscripts–Continuedfrompreviouspage SymbolDenitionpg.# rmsRoot–mean–squaredvalue 115 SMAccordingtoclassicalStefan–Maxwellanalysis 33 TTotal 83 trTransitionregime 33 Azimuthalcomponentinpolarcoordinates 43 Referringtotheharmonicbondbendingpotential 66 Superscripts SymbolDenitionpg.# [i]ModiednotationfortheithapproximationtotheintegralasdenedinAppendix A andAppendix B 70 LJReferringtotheLennard–Jonespotentialmodel 67 MReferringtotheMorsebondstretchingmodel 67 >Matrixtranspose 78 Referringtotheharmonicbondbendingmodel 67 Conjugate 75 0Evaluatedinthebodyxedframeofreference 77 18

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Accents,Delimiters,andSpecialConventions SymbolDenitionpg.# hiAveragedvalue 35 kkL2norm 66 tTimestep 67 Grassmannproduct 77 Abbreviations AbbreviationDenitionpg.# ACLAnodecatalystlayer 27 ADLAnodediffusionlayer 27 BFBodyxed 77 BFMBinaryfrictionmodel 40 CCLCathodecatalystlayer 26 CDLCathodediffusionlayer 27 CFFCentralforceeld 60 CNTCarbonnanotube 49 COMCenterofmass 73 CVDChemicalvapordeposition 50 DCV-GCMCDualcontrolvolumegrandcanonicalMonte–Carlo 49 DGMDustygasmodel 35 Continuedonnextpage... 19

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Abbreviations–Continuedfrompreviouspage AbbreviationDenitionpg.# DMFCDirectmethanolfuelcell 22 EMDEquilibriummoleculardynamics 28 EOMEquationofmotion 28 FCFuelcell 24 FKMFiniteKnudsenmodel 99 FORFrameofreference 77 GCMCGrandcanonicalMonte–Carlo 55 GKGreen–Kubo 54 GLSGas–liquidseparator 26 HEMHighenergymode 95 LBLLiquidbarrierlayer 22 LJLennard–Jones 22 MDMoleculardynamics 22 MeOHMethanol 26 MTPMMeantransportporemodel 40 MVRK4Multivariate,4th–orderRunge–Kutta 67 NEMDNonequilibriummoleculardynamics 42 NGPC33rd–orderNordsieck/Gearpredictor-corrector 68 NGPCppth–orderNordsieck/Gearpredictor-corrector 68 Continuedonnextpage... 20

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Abbreviations–Continuedfrompreviouspage AbbreviationDenitionpg.# OMOscillatormodel 42 PEMProtonexchangemembrane 26 PEMFCProtonexchangemembranefuelcell 24 REBOReactiveempiricalbond-order 52 RKRunge–Kutta 68 SMStefan–Maxwell 33 SOFCSolidoxidefuelcell 38 SPCSimplepointcharge 71 SPC/EExtendedsimplepointcharge 70 SWCNTSingle–walledcarbonnanotubes 49 TIP3P3–point–transferableintermolecularpotential 71 TIP4P4–point–transferableintermolecularpotential 70 TIP5P5–point–transferableintermolecularpotential 71 UFUniversityofFlorida 25 UNFUniversityofNorthFlorida 25 VVVelocityVerlet 84 21

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMOLECULARDYNAMICSASAFOUNDATIONFORFLUXPREDICTIONTHROUGHNANOPOROUSMEMBRANES:AVECTORIZED,CONSTRAINT-FREEAPPROACHTOCONSERVATIVESIMULATIONSByMatthewClayInmanAugust2014Chair:WilliamE.LearMajor:MechanicalEngineeringAnovel,open-cathodedirectmethanolfuelcell(DMFC)hasbeendesignedandbuiltbyresearchersattheUniversityofNorthFloridaandUniversityofFlorida.ForemostamongtheadvancesofthissystemoverpreviousDMFCarchitecturesisapassivewaterrecoverysystemwhichallowsproductwatertoreplenishthatconsumedattheanode.Thisisenabledbyaspecially-designedwaterpathwaycombinedwithaliquidbarrierlayer(LBL).TheLBLmembraneispositionedbetweenthecathodecatalystlayerandthecathodegasdiffusionlayer,andmustexhibithighpermeabilityandlowdiffusiveresistancetobothoxygenandwatervapor,bulkhydrophobicitytoholdbacktheproductliquidwater,andmustremainelectricallyconductive.MaintainingwaterbalanceatoptimumoperatingtemperaturesisproblematicwiththecurrentLBLdesign,forcingthesystemtorunatlowertemperaturesdecreasingtheoverallsystemefciency.Thisresearchpresentsanovelapproachtonanoporousmembranedesignwherebyuxofagivenspeciesisdeterminedbaseduponthemolecularpropertiesofsaidspeciesandthoseofthediffusingmedium,theporegeometry,andthemembranethickness.Amoleculardynamics(MD)modelisdevelopedfortrackingKnudsenregimeowsofaLennard-Jones(LJ)uidthroughanatomisticporestructure,hundredsofthousandsofwallcollisionsimulationsareperformedontheUniversityofFloridaHiPerGatorsupercomputer,andthegeneratedtrajectoryinformationisusedtodevelopnumber 22

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densityandaxialvelocityprolesforuseinarigorousapproachtototaluxcalculationabsentinpreviouslyattemptedMDmodels.Resultsarecomparedtootherpublishedapproachesanddiffusiondataavailableintheliterature.Theimpactofthisstudyonvariousapplicationsofmembranedesignisdiscussedandadditionalsimulationsandmodelimprovementsareoutlinedforfutureconsideration. 23

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CHAPTER1INTRODUCTION 1.1Motivation 1.1.1ApplicationsofSelectiveMembranesTheneedforhighlyefcient,lowcostgasseparationprocessesisprevalentinawiderangeofapplicationsspanningfromtheenergyindustrytofood,andpermeablemembranespresentanattractivesolutiontomanyoftheseapplicationsduetotheirsimpliedowstructureandlowmaintenanceoperation.CurrentapplicationsofmembraneseparationinindustryincludetheseparationofairforN2production,theproductionofoxygen-enrichedairforhigh-temperaturefurnacesorcementkilns[ 7 ],O2separation,orthedehydrationofairforbreathingLithium-ionbatteries.TheseparationofinertgassessuchasH2andN2[ 100 ],orArgon[ 83 ]fromAmmoniaplantpurgegases,CO2andH2O()removalfromnaturalgasstreams[ 7 ],theseparationofchlorouorocarbonsfromairandlarge-scalerefrigerationfacilities[ 30 ],andpreferentialethylenepermeabilityinfoodpreservation[ 48 ]areafewmoreexamplestodemonstratethescopeofapplicability.Anotheruseisintheadvancementoffuelcelltechnology,wherepassivewaterreclamationtechniqueshavethepotentialtorevolutionizetheProtonExchangeMembraneFuelCell(PEMFC)industryandmakefuelcellsaviableoptionforpoweringportableelectronics. 1.1.2IntroductiontoDirectMethanolFuelCellsFuelcells(FC)arechemicalenergyconversiondevicescapableofgeneratingelectricitythroughthecatalysisofafuelsourceandchannelingtheliberatedelectronsthroughanexternalloadbeforerecombiningthemolecularconstituentstocreatelessenergeticbyproductsandclosethereactionloop.Unlikebatteries,fuelcellsrequireacontinuousstreamofreactanttooperatebutarebettersuitedforlongdurationoperationastheenergydensityofanyfuelcellsystemwillconvergetotheenergydensityofthefuelbeingfedthroughtheanodeloopgivenenoughtimeandhighenoughefciency. 24

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AmongthemostpromisingfuelcellarchitecturesforuseinportableelectronicsareDMFCsbecauseoftheirrelativelylowoperatingtemperature(70CforDMFCs,150CforreformingFCs,and600CforsolidoxideandmoltencarbonateFCs[ 137 ]),easeoffuelstorage(liquidmethanolatroomtemperatureasopposedtophosphoricacidorcompressedhydrogen),andthehighenergydensityofmethanolwhencomparedtostate-of-the-artLi-ionbatteries.ComprehensivecomparisonsofDMFCsagainstconventionalbatteriesforportableapplicationshavebeencompleted[ 62 ],forecastingalargemarketforminiaturizedhighenergypowersourcesthatstandardbatterieswillnotbeabletosatisfy,butthatDMFCshavethepotentialtoasthetechnologycontinuestomature. 1.1.3ExistingIssueswiththeOpenCathodeDMFCDesignWhiletheopencathode,DMFCarchitectureshowspromiseasaportablepowersource,manyofthecomponentsfoundinmostDMFCs,suchaspumps,heatexchangersandcondensers,donotscaledowninsizeandmaintainthenecessarylevelsofeffectivenessforuseinmanyportableelectronicsapplications.ThisobstaclehasbeenovercomebyresearchersattheUniversityofNorthFlorida(UNF)andUniversityofFlorida(UF)byremovingtheexternalwaterrecirculationloopseeninFigure 1-1A andinsteadreturningtheappropriateamountofproductwaterintotheanodestreaminternallyviaaLBLinsidetheFCstack(asshowninFigures 1-1B and 1-2 ).Thispassiveapproachtowaterrecoveryalsogreatlyreducesthesystemweightandparasiticload.However,despitethisadvancementseveralobstaclesstillremain,thegreatestofwhichismaintainingwaterbalancewhileoperatingatanearoptimumtemperature.ThecurrentphaseofdevelopmentforthisprojectfocusesontheidenticationofpotentialmembranematerialsandmorphologiesthatwillexhibitthedesiredO2/H2OselectivityatthedesiredtemperaturewithoutappreciablyloweringtheO2activityatthecathodecatalystsites. 25

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ATraditionalDMFC BNovelDMFCwithLBLFigure1-1. SystemarchitecturesfortraditionalDMFCswithanexternalwaterrecoveryloop( 1-1A )andforanovelDMFCdesignwhereinwaterisrecoveredinternallyandpassivelyviaaLBL( 1-1B ).TheseguresaretakenfromtheUNF/UFpostersessiondisplayatthe2012FloridaEnergySummit[ 44 ]. Directmethanolfuelcellsfeeddilutemethanol(MeOH)throughtheanodeowchannels,wherethereactantisallowedtodiffusenormaltotheanodeowstreamandtowardtheanodecatalystsites.Herethewaterandmethanolreacttoformatomichydrogenionswhichowthroughtheprotonexchangemembrane(PEM),freeelectronswhichowthroughthelow-resistanceexternalloadtogeneratetheelectricalcurrent,andcarbondioxidewhichisreturneddirectlyintotheanodestreamandlteredoutthroughagas-liquidseparator(GLS).OncethehydrogenionshavetraversedthePEMandreachedthecathodecatalystlayer(CCL)theyarerecombinedwiththeliberatedelectronsinanO2richenvironmenttoformatwophasemixtureofwaterandwatervapor.LocatedbetweentheCCLandthegasdiffusionlayerleadingouttotheambientowstreamistheLBL,whichholdsbacktheliquidwater,forcingitbackintotheanodeowstream,whileremainingelectricallyconductiveandpermeabletooxygen.Itisimportanttomaintainaconsistentoperatingtemperatureandconcentrationofmethanolintheanodestreaminordertooperateatthesystemmaximumefciencypoint,andtokeepthecatalystsandmembraneshydratedwithoutoodingthem.IntraditionalDMFCarchitecturesthiswasdonebyreturningafractionoftheproductwaterfromthecathodesideintotheanodestreamdosingsaidstreamwithneatMeOHsuchthatthemolarityofthemixtureisunchanged. 26

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Figure1-2. SchematicofasinglecellofaDMFC.Fromright(anodeowchannel)toleft(cathodeowchannel)thecellcomponentsaretheanodediffusionlayer,theanodecatalystlayer,theprotonexchangemembrane,thecathodecatalystlayer,theliquidbarrierlayerandthecathodediffusionlayer.Theionicandmoleculartransportthrougheachlayerareshownasistheexternalloadthroughwhichtheliberatedelectronsow.Notethatthegureabovedoesnotgivethestoichiometryoftheanodeandcathodein-owsandout-ows,butonlytheconstituentspresentineach. Theuseofacathoderecirculationloop(asshowninFigure 1-1A )givesthesystemnecontroloverhowmuchproductwateriscondensedandreturnedintotheanodeowstreamversushowmuchisevaporated,allowingforawiderangeofefcientoperatingtemperatures.Withouttheadditionalpumpingandcondensingcomponentsthetemperaturewindowforefcientoperation(centeredaround50Cforthenewarchitecture)isnarrowed,andthecurrenttemperaturerangeissolowthatbothtransportandelectrochemistryarehindered.Generally,DMFCsoperateinthe70C 27

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range,andasidefromtheaforementionedtransportandelectrochemicalbenets,thiswouldalsoallowforoperationathigherambienttemperatures. 1.1.4BenetsofanImprovedLiquidBarrierLayerThepositiveeffectsofanewLBLallowingforhighertemperatureoperationwhilemaintainingwaterbalancearenumerous.Higheroperatingtemperaturesresultinincreasedcatalystactivityandtransportpropertiesofthereactants[ 99 ],whichwillextendthekneeofthepolarizationcurveallowingforgreaterpowergenerationwhileavoidingtheconcentrationoverpotentialregime[ 8 ].Thisnecessitatesfewercellstoprovidethesamepoweroutput,decreasingthesystemsize,costandweight.Furthermore,increasingtheoperatingtemperaturemeansthatthetemperaturedifferencebetweentheFCstackandambientisgreater,yieldingmoreefcientheattransferandevaporationoftheunreturnedproductwater.Alloftheseeffectsresultinanincreaseinenergydensityandefciency. 1.2ResearchObjectiveTheobjectiveofthisresearchistodevelopanoveldesigntoolforuseinpredictingpermeabilityandselectivitypropertiesofnanoporousmembranes.Specically,thisresearchfocusesonthedevelopmentofahighlyefcientequilibriummoleculardynamics(EMD)modelwhichcalculatesthetrajectoriesofanisolatedpermeateparticleormoleculeasittravelsthroughanadsorptiveeldgeneratedbytheporestructure.Themodelistobeconservativeandunconstrained,meaningnoarticialforcesorenergycorrectorssuchasNose–Hooverthermostatsorbarostatsareused.MATLABandanin-housedevelopedvectorizedfourth–order,multivariateRunge–Kuttaalgorithm(notode45)areusedtosolvetheequationsofmotion(EOM),resultinginamuchhigherorderofaccuracyandcomparablecalculationtimestotraditionalVerletandGeartechniquessolvedinC.Thegeneratedtrajectorydataispost–processedtoyieldMonte–Carlo–likestatisticalensemblesfortheradialdistributionsofnumberdensity,streamingvelocity,intra–collisionalaxialdisplacementdistributions,andwallcollision 28

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frequencies.Classically,diffusioncoefcientsaredeterminedfromsuchsimulationsusingtheEinsteinorGreen–Kuborelations[ 3 ]whichrelatetheuxandconcentrationgradientatlongtimesthroughequilibriumcorrelationfunctions.Departingfromthisstandardapproach,anovelandrigorousanalysistocalculateKnudsenregimediffusioncoefcientsgivendistributionsofpermeatecollisiondistancesandaxialvelocities,twoparametersthatthesimulationsperformedaspartofthisdissertationgenerate,isused(ColsonF,privatecommunication,May6,2014). 1.3DissertationOutlineThisdissertationdetailsresearcheffortstowardstheadvancementofmoleculardynamicstechniquesandKnudsenscalediffusionanalysis.Chapter 2 providesthereaderwithacomprehensivesurveyofresearchconductedtodateonthetopicsofKnudsenandtransitionregimediffusion,Stefan–MaxwellmasstransportmodelingwithastrongfocusupontheDustyGasModel,proposedmodelswithabinitioinclusionofintermolecularandpermeate–porepotentials,andvariousmoleculardynamicstechniquesusedinthesimulationofgas/liquidmasstransportandpolymermatrices.Chapter 3 detailsthedevelopmentoftheEMDmodel,beginningbyexplainingtherationalebehindtheproposedsolutionprocedureandcodestructure,thendiscussingthecriteriaforstatisticalconvergence,andnallysummarizingtheevolutionofthemodelfromsingle–body,LJpotentialstorigidanddynamicmoleculesunderLJandinducedquadrupoleforces.Chapter 4 presentstheresults,discussingthepotentialandthelimitationsofeach,andprovidesavalidationanalysisperformedusingpreviouslypublisheddata.Chapter 5 presentstheconclusionsdrawnfromthevarioussimulatedresultsandafutureworksectionoutliningthemostadvantageousimprovementstotheproposedroutineforfutureconsideration. 29

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CHAPTER2LITERATUREREVIEW 2.1FoundationalTheoryofMassTransportintheKnudsenandTransitionalRegimesThissectionisdevotedtoreviewingthepioneeringanalysesofrariedgasdynamicsinhighlyconnedchannelsandpores.TheprogressionfromKnudsen's[ 78 ]seminalhard-spheremomentum-balanceandthedevelopmentofaconcentration-independenttransfercoefcientinthelimitofKn!0,totheuseoftheStefan-MaxwellmasstransportrelationstobridgetheKnudsenandnormaldiffusionregimes[ 40 ]isdetailed.Inallcasesadsorptionprocessesareconsiderednegligible,leadingtosimplemassratiorelationshipsforthesquareduxratio.DiffusioninthepresenceofanadsorptiveeldisdiscussedinSection 2.2 . 2.1.1SeminalAnalysisofSub-ContinuumDiffusionMolecularowphenomenawasrststudiedandcharacterizedbyKnudsenin1909.Inhisseminalwork[ 78 ]KnudsennotedtheexperimentallyobserveddepartureinuidowmeasurementsfromthatpredictedusingPoisseuille'sLawinincreasinglynarrowtubesforrariedgases,andattemptedtodenethistransitionfromfrictionaltomolecularowanddevelopapredictiveuxanalysisinthisnewregime.Knudsenthusassumedasysteminwhichthecharacteristicdimensionoftheowchannel,i.e.thediameterofthetube,wouldbemuchlessthanthemean-freepath()ofthediffusingparticle,therebyhypothesizingthattheparticlemotionwouldbedominatedbyinteractionswiththechannelwallasopposedtointermolecularinteractionsasdescribedinFickiandiffusion.Itmustbenotedherethatwhile!isthefundamentalassumptioninidealizedmolecularowanalysis,itprovestobeadangerousassumptionformoleculardynamicists,aslowparticleloadingcanleadtospurioustrappingofthesorbatewithintheprominentpotentialwellsofthesimulationspace.ThiswillbediscussedindetailinChapter 4 .Furtherassumingdiffusereectionsatthewall,rectilinearparticletrajectories,andacircularchannelcross-sectionofradiusR,Knudsen 30

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([ 78 ],[ 79 ])developedEquation( 2 )todescribethevolumetricowrateofaspecies(QKn)throughanarrowcylindricalporeinthepresenceofanaxialpressuregradientP.QKn=4 3s 2 R3P L (2)whereisthespecicweight,andListheporelength.ThisanalysiswasfurtherdevelopedbySmoluchowski[ 116 ]andKnudsen[ 80 ]throughamorerigorous,kinetictheoryapproachtodeterminethemolecularuxacrosssomearbitraryboundary,yieldingthearchetypalexpressionforthediffusioncoefcient,DKn(cm2/s)giveninEquation( 2 ).DKn=2R 3r 8kBNAT M (2)whereTistheabsolutetemperature,Misthediffusingspecies'molecularweight(kg/mol),NAisAvogadro'sNumber,andkBistheBoltzmannconstant.Smoluchowskialsointroducedaspecularreectionfactor,wheresomepercentage#ofthepermeateparticlesareassumedtoreectdiffuselyfromthewallwhiletheremainder,(1)]TJ /F9 11.955 Tf 12.73 0 Td[(#),reectspecularlythroughthetube.Thisfractionwasproposedasameansofsimulatingmomentumlossatthewalls.Meansforpredicting#arenotprovidedbySmoluchowski,however,thisfactorisusedintheDustyGasModelbyMasonandMalinauskas[ 93 ]totunethepredicteddiffusioncoefcientstomorecloselyresembleexperimentallymeasuredvalues.PollardandPresent[ 103 ]laterproposedamethodwithwhichtobridgeRandRdiffusionlimits,i.e.theresultsofKnudsenandSmoluchowskiandthoseofself-diffusionfromkinetictheoryrespectively.Resultslookedpromisingintherangefrom1R 5whencomparedwiththeBosanquetinterpolationrandom-walkmodel;alackofexperimentaldatainthisrangemademoreconventionalmodelvalidation 31

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impossible.PollardandPresentalsoconsideredtheeffectsofnitetubelengths,notingthatthemajorityoftheuxacrossagivenboundaryisgivenbytheparticletrajectoriesreboundingoffthewallwithinofthatboundary,andconcludingthatthediffusioncoefcientdescribinguxthroughnanoporesofnitelengthislessthantheinnitecasebyasmuchasafactorof1)]TJ /F2 7.97 Tf 13.15 4.71 Td[(3 4R L.FollowingtheworkofPollardandPresent,severalothergroupsexploredtransitionregiondiffusion.ScottandDullien[ 111 ]consideredanisothermal,isobaricopenbinarysystemandusedakinetictheorymomentumbalanceoneachspeciesofthemixturetodescribethenettransportofeachthroughacapillarytube.Thisapproachassumesidealgasbehaviorwithdiffusewallreections,negligibleaxialmomentumtransfertothewallfollowingacollision,thatDalton'slawofpartialpressuresholdsinthemoleculardiffusionregime,andthatthemomentumcontributionsduetobulkmoleculardiffusionandKnudsendiffusionareadditive.Thisapproachisamongtherstattemptsatafullycomprehensivediffusionmodel.RemickandGeankoplois[ 108 ]latermodiedandsoughttoexperimentallyvalidatetheScottandDullienapproachoverawiderangeofpressuresusingaWicke–Kallenbachdiffusioncell[ 107 ],inwhichseparatestreamsofH2andN2owoneithersideofabundleofnarrowglasscapillaries(I.D.=0.002in51m)underequalpressureconditions.Thecrossdiffusionisthendeterminedviaconcentrationmeasurementstakenbyadownstreammassspectrometer.ThenettransportequationsforKnudsenow(JKn),transitionowasmodiedbyRemickandGeankoplois(Jtr),andStefan-Maxwellmoleculardiffusion(JSM)ofspeciesA(subscripted)aregivenbyEquations( 2 )through( 2 )respectively.JKn,A=DKn,AP RTL(A,0)]TJ /F9 11.955 Tf 11.96 0 Td[(A,L) (2) 32

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Jtr,A=Dtr,AP RTdA dz (2)JSM,A=DABP $ARTL1)]TJ /F9 11.955 Tf 11.95 0 Td[($AA,L 1)]TJ /F9 11.955 Tf 11.96 0 Td[($AA,0 (2)wherePisthetotalpressure,isthemolefraction,DABisthebinarydiffusioncoefcientinabulkuid,and$istheratioofmolaruxesofAandB($A=JB=JA).Thetransitiondiffusioncoefcient(Dtr)wasformulatedinsuchawaythatintheKnudsenregimeDtrwouldconvergetoDKn,andlikewisewouldconvergetoDABintheFickiandiffusionregime,andtakesthefollowingform:Jtr,A=DABP ln1)]TJ /F23 7.97 Tf 6.59 0 Td[($AA,L 1)]TJ /F23 7.97 Tf 6.58 0 Td[($AA,0ln1)]TJ /F9 11.955 Tf 11.95 0 Td[($AA,L+DAB=DKn,A 1)]TJ /F9 11.955 Tf 11.95 0 Td[($AA,0+DAB=DKn,A (2)Bothstudiesshowmoderate-to-strongagreementinthetransitionregimeoverawiderangeofpressures,howeverbothsufferfromanexcessofoversimplifyingassumptions,specicallythesuperpositionofthediffusioncontributionsfrombulkandKnudsenscalediffusion.ThisassumptionwasalsousedintheDustyGasModelandhasbeenthesubjectofmuchcriticisminthelastdecade[ 19 ].Furthermore,recentstudiesbySoukupetal.haveshownthattheWicke–Kallenbachdiffusioncellstronglydeviatesfrompredicteduxbehaviorforporeslargerthan6minradiusdueto“intrudingpermeationtransportcausedbytheverysmallpressuredifferencebetweencompartments”ofthecell[ 119 ]. 2.1.2TheStefan-MaxwellApproachandtheDustyGasModelTheStefan-Maxwell(SM)equationsofbinary(andthegeneralizedn-componentmixture)diffusionbeginwithaforcebalanceoversomearbitrarydifferentialcontrolvolume.Inordertodescribetherelativemotionofthevariousmixtureconstituents 33

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somedrivingforcemustbeappliedtoeverymolecule.Consideringtheone-dimensional(z-axis)motionofspeciesi,assumingthesystemisisothermalandsomepartialpressuregradientexistsacrossthedifferentialcontrolvolume,thentheforcepervolumeisgivenas)]TJ /F10 7.97 Tf 10.49 5 Td[(dPi dz.Assumingtheuidactsasanidealgasmixturethentheforcepermoleofspeciesicanbewrittenas)]TJ /F8 7.97 Tf 10.49 4.71 Td[(RT PidPi dz=RTdln(Pi) dzwhichisthepartialmolalGibbsfunctionorthechemicalpotentialgradient)]TJ /F7 11.955 Tf 5.48 -9.68 Td[()]TJ /F10 7.97 Tf 10.49 5.25 Td[(di dz.Thisdrivingforcewillthenbebalancedbythefrictionalforcesactingbetweenthevariousspeciespresentinthemixturesuchthat:)]TJ /F5 11.955 Tf 10.49 8.09 Td[(di dz=RT Dijj(vi)]TJ /F12 11.955 Tf 11.95 0 Td[(vj) (2)where DijistheSMdiffusioncoefcient,cjisthemolarconcentrationofspeciesjandvi)]TJ /F12 11.955 Tf 12.03 0 Td[(vjisthemassaveragedvelocitydifferencebetweenspecies.NotingthatJi=ctivi,thenEquation( 2 )canberewrittenas:)]TJ /F9 11.955 Tf 14.91 8.09 Td[(i RTdi dz=jJi)]TJ /F9 11.955 Tf 11.96 0 Td[(iJj ct Dij (2)Inabinary,ideal,isothermalsystemliketheonedescribedabovetheSMdiffusioncoefcientisidenticaltotheFickiandiffusioncoefcientthatarisesfromtheassumptionthattheuxofagivenspeciesislinearlyproportionaltoitscompositiongradientandaveragevelocity.TheSMformulationcanbegeneralizedtoann-componentsystembyfollowingthesameprocedureandisgivenbyEquation( 2 ).)]TJ /F9 11.955 Tf 14.92 8.09 Td[(i RTdi dz=nXj=1,j6=ijJi)]TJ /F9 11.955 Tf 11.95 0 Td[(iJj ci Dij (2)AcomprehensivereviewonthesubjectoftheSMapproachtomasstransferanalysisisgivenbyKrishnaandWesselingh[ 82 ]. 34

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2.1.2.1DevelopmentoftheDustyGasModelTheDustyGasModel(DGM)wasrstproposedbyEvansetal.[ 40 ]in1961asameanstoanalyticallypredictgaseousmasstransportthroughporousmediaunderuniformpressureconditionsinalldiffusionregimesthroughasimpliedkinetictheoryanalysis.Assumingthatdiffusionthroughsaidmediumisonedimensionalandlimitedtotheporevolumeandhencenegligiblealongthesurfaceorthroughthebulk,theporestructureisviewedasjustonemoreconstituentofamulti-componentgasmixture.Theparticlesofthisnewconstituent,or“dust”particles,areheldstatic,areevenlydistributedthroughoutthematrixandareconsideredverylargewithrespecttotheotherdiffusingspecies.Theseassumptionsallowforthekeysimplifyingconditionshvdi=0and@nd=@z=0wherehvdiistheaveragedustvelocityandndisthedustnumberdensity(molecules/cm3).Furtherassumingfullydiffuseand“thermalized”wallcollisions,meaningthatboththereectedanglesandvelocitycomponentsarerandomizedwithrespecttoaMaxwelliandistribution,Evanswasabletodevelopamodelwhichmaintainedtheclassicaluxratio(J1=J2=)]TJ /F4 11.955 Tf 11.29 -.17 Td[((M2=M1)1=2)resultthroughalldiffusionregimes,independentlyre-derivedtheBosanquetinterpolationformulaofdiffusioncoefcientsandconvergedtotheclassicalKnudsenandFickiandiffusionresultsatthepressureextremes.Evansetal.[ 41 ]laterexpandedthisworktoincludeapressuregradientalongtheaxisofdiffusion.Byassumingnotemperaturegradients(andhencenothermaldiffusion)andbyarguingthattheforcedowcomponentoftheoveralluxcouldbeanalogizedtoanempiricallyalteredformulationoftheoriginalDGMuxequation,thetotaluxofamulti-componentmixtureinthepresenceofanaxialpressuregradientwasreducedtothefollowingrelationships:J1=)]TJ /F4 11.955 Tf 11.29 -.17 Td[((D1)e@n1 @z+1J1 (2) 35

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J)]TJ /F21 11.955 Tf 11.96 20.45 Td[("1)]TJ /F21 11.955 Tf 11.96 16.86 Td[(M1 M2 1 = 2 #J1=)]TJ /F21 11.955 Tf 11.29 16.86 Td[(C2 kBTdP dz (2)where(D1)eistheeffectivediffusioncoefcientaccountingforthemediumporosityandtortuosity,1isadiffusioncoefcientfactorwhichvariesfrom0intheKnudsenregimetounityinthebulkdiffusionregime,andC2isanexperimentallydeterminedempiricalconstantintendedto“trytohide[the]exclusivelydiffusiveowmechanism”intheDGMformulation.Thisapproachshowedreasonableagreementwiththelimitedavailableexperimentaldata,butclearlythedeviationfromanythingresemblingthetruephysicsofthesystemistroublinganddifculttojustify.TheDGMwasagainextendedbyMasonetal.[ 92 ]toincludetemperaturegradientinduceddiffusivetransportinbothporousmediaandincapillarysystems,inthepresenceofapressuregradient.Byassumingthatthetotaluxwasequivalenttothelinearsuperpositionoftheviscousowanddiffusiveowtermsthethermaltranspirationcanbecalculatedatanypressuresolongasthethermaldiffusionratio(kT)canbeapproximated.IntheKnudsenregime,assumingndnandMdM,thenkinetictheorytellsusthatthemixtureactsasaLorenziangas[ 27 ],andkT=1 2n n+nd (2)Likewise,inthelimitofnormaldiffusion,wherendnandMdM,themixturebehavesasaquasi-Lorenziangas[ 91 ]andthethermaldiffusionratiocanbeapproximatedas:kT=nnd (n+nd)2Q (2)whereQisthethermaldiffusionfactorgivenbyEquation( 2 )fora“dusty”gas. 36

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Q=9 16 8kBT1 2= ndK0 M1 2 (2)whereistheporosity,isthetortuosityfactor,isthegaskinematicviscosityandK0isaconstantdenedbythegeometricpropertiesoftheporeandthepercentageof“thermalized”wallcollisions[ 40 ].Usingtheselimitingconditionsasaguide,Mason[ 92 ]arguesthatkTcanbereasonablyapproximatedinanyregimebyEquations( 2 )and( 2 ).kT=nnd (n+nd)2 (2)1 =2d+ Q (2)ImplicitinthisformulationofthermallydrivendiffusionisthatintheKnudsenregimekTdoesnotvarywithtemperature.ItisnotedbyChapmanandCowling[ 27 ]thatthisassumptiondoesnotholdiftheparticlesaremovingthroughapotentialeld,andthereisadenitedependenceonintermolecularandmolecule-wallinteraction.Theabilitytoadequatelyaccountfordiffusionthroughporesinwhichthepermeateandthediffusingmediumhaveafnitiesforoneanotherisofgreatsignicancetotheapplicationofselectivemembranedesign,andmakestheDGMill-equippedtothetask. 2.1.2.2PrevalentresearchapplicationsoftheDustyGasModelTheDGMremainsoneofthemostprevalentanalyticaltoolsforpredictingtransportcharacteristicsthroughmedia,theseminalpapersfromEvansetal.([ 40 ],[ 41 ]),Masonetal.[ 92 ]andMasonandMalinauskas[ 93 ]garneringcitationsinover1400scienticpublicationsasofDecember2013,perGoogleScholar.WhiletheoriginalpurposeoftheDGMwastohelpexplainonedimensional,binarymasstransportphenomena, 37

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thescopeandapplicationofsaidtheoryhasbranchedoutfarbeyondthatofrariedgasowsthroughmicroscalecapillaries.WhencharacterizingthedehydrogenationofethylbenzenetostyreneinxedbedcatalyticreactorsElnashaieetal.[ 38 ]usestheDGMtodescribethesimultaneousdiffusivemasstransferandchemicalreactionsinthemesoscalecatalyticchannels.Later,Veldsinketal.[ 129 ]comparedthecatalysteffectivenessfactorcalculatedusingtheDGMandasimplerFickmodelformultiplecatalysissitegeometries,specicallyplanarand“point-of-symmetry”congurationsandcomposite-planarstacksresemblingcatalyticmembranes,andfoundtheDGMtobeasuperiorpredictor.Inamorerecentstudy,Suwanwarangkuletal.[ 121 ]comparedtheconcentrationoverpotentialinsolidoxidefuelcells(SOFC)predictedusingtheGeneralizedSM,DGMandFick'smodelsagainconcluding,despiteitsincreasedcomputationalcost,theDGMtobethemostaccurateapproachduetoitsinclusionofKnudsenoweffects.Extendingthisstudy,Tseronisetal.[ 127 ]developeda2dimensional,heterogeneousversionoftheDGMforuseincalculatingdetailedspeciescompositionprolesandconcentrationoverpotentialsinSOFCs,showingmarkedlyimprovedpredictivecapabilities.TheDGMhasalsocomparedfavorablywiththeMeanTransportPoreModel[ 37 ]inbinaryandternarycounter-owsimulationsofinertgases[ 55 ].TheDGMhasalsofoundseveralnichesoutsideofcatalysisresearch.Moonandcoworkers[ 95 ]turnedtotheDGMwhenlookingtodesigncarbonmonoxideltersforreforminghydrogenfuelcells,notingthatCOandotherreformerby-productstendtodegradecatalyticperformance.UsedinconcertwiththeGeneralizedSMandLangmuirisothermmodels,theDGMwasusedtodetermineH2andCOselectivityinmethyltriethoxysilanemembranes,foundinnaturalgasreformers,forbinaryandquaternarygasmixtures.TheDGMhasalsobeenusedinwatervaportransportanalysisinconstructionmaterials.Gudmundsson([ 50 ],[ 51 ])notedthatmoistureisthesinglemostcommoncauseofdamagetobuildingsanddevelopedanefcientsetof 38

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experimentsinvolvingtracergasmeasurementstocharacterizebuildingmaterials.ThesemeasurementswouldthenbeusedtocalculatetheDGMcoefcientstogivethepredictedwatervaporuxthroughsaidmaterial. 2.1.2.3CriticalreviewoftheDustyGasModelNumerouscriticalreviewsoftheDGMhavesurfacedintheliterature([ 77 ],[ 15 ],[ 16 ],[ 19 ])debatingthevalidityoftheassumptionsmadeandtheunderlyinganalysis.Ina1983monograph,MasonandMalinauskas[ 93 ]addressedseveraloftheseconcerns,specicallyhowtheDGMcouldbeusedtoanalyzediffusionregimesoutsideoftheKnudsenregimewhenthebasisofthemodelisChapman-Enskogtheory,andhowthesuperpositionofthediffusiveandviscousuxesarejustied.Totherstpoint,MasonarguesthatingeneraltheBoltzmannequationisappliedtoonlybinarycollisions,butthetheoryisjustasapplicabletoa“dusty”gassolongasthedustparticlesareofsignicantlygreatermassthenthatoftheotherdiffusingspecies.Inthiscase,ifmultiplegasmoleculescollidewithasingledustparticlesimultaneously,thecollisionsarestill“uncorrelated”,hencethetheorystillapplies.Astotheadditiveuxcontributions,therationaleislesssatisfactoryandadmittedlyincomplete.KerkhofandGeboers[ 77 ]describethissuperpositionas“arbitrary”,andultimatelycondemnthemodelforitsempiricalfoundationasopposedtobeingbuiltupfromrstprinciples.Alsonotedaretheunjustiedleapsbetween“point”uxesandaveragecross-sectionaluxeswhenspeciesvelocitiesarenotimplicitlyknowntobeconstantwithinsaidcross-section.Later,BhatiaandNicholson([ 15 ],[ 16 ],[ 19 ])pointoutthefailureoftheDGMtoaccuratelypredictuxesordiffusioncoefcientswhenuid-wallinteractionsaredominant,citingthattheDGMtendstooverpredictdiffusivitiesbyasmuchasandorderofmagnitude[ 70 ]undertheseconditions.ThesendingsleadtothedevelopmentoftheOscillatorModelinSection 2.2.1 andthesubsequentDistributedFrictionModelinSection 2.2.2 . 39

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2.1.2.4OthernotableStefan-MaxwellformulationsTheMeanTransportPoreModel(MTPM)wasproposedbySchneider[ 110 ]andwasmotivatedbytheunder-predictionofuxbytheSMapproachinporouscatalysts.Thiswasattributedtotheadditionalpressuregradientscreatedbythechemicalreactionsoccurringatthecatalystcite,andhencetheSMforcebalancewasmodiedtoincludea“forcedow”termaswellastheconcentrationgradientinducedow.Itisworthnotingherethatthisformulationwouldseemtodoublecountthepressuregradientcreatedbythechemicalreactions,asthechemicalpotentialgradientisafunctionofthepartialpressuredifferenceacrosstheentirecontrolvolume(i.e.thelengthofthepore).Theunder-predictionbytheSMequationisduetothefactthatthemolefractionsofeachconstituentarechangingduetothechemicalreactionswheretheoriginalformulationhasnomechanismtoaccommodatetheadditionandsubtractionofmoleculesofaspecicspecies.Therefore,theMTPMcanatbestbeconsideredacorrelationbetweentheSMandareactingsystemasthetruephysicsoftheproblemarenotadequatelydescribesintheformulation.Assumingisothermalconditions,Schneidersoughttocharacterizebinaryandternaryowsthroughcapillarytubesinthetransitionregimeinthepresenceofanexternalforce.ThestandardSMforcebalancegivenbyEquation( 2 )wasthenappropriatelymodiedtoequatethefrictionalforceswiththesuperposedcontributionsofthechemicalpotentialandexternalforce,whileaccountingfortheKnudsenregimeowviathemethoddescribedbyRemickandGeankoplois[ 108 ].AfterassumingthatthemomentumuxinthetransitionregimecanbeapproximatedbyascaledsummationofthepredictedmomentumuxesintheKnudsenandcontinuumregimes,anintegralexpressionforthetransitionregimetransportofagasmixtureinthepresenceofanexternalforcewasderived.LateramodiedSMapproachcalledtheBinaryFrictionModel(BFM),developedbyKerkhof[ 75 ],wasproposedinorderdescribethetransportofamixturethroughan 40

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inertmembranewhileaccountingforinter-speciesfrictionaleffectsandthosebetweenthediffusingmixtureandthewall.Assumingisothermal,idealgasbehaviorforabinary,onedimensionaluidowandaheterogeneousmixtureofmembraneparticlesactingasapseudo-uidinthespiritoftheDGM,theLightfootformulationoftheSMequationcanbewrittenas:)]TJ /F9 11.955 Tf 14.91 8.09 Td[(i RTdi dz+'i ctRTdP dz=nXj=1,j6=iiJj)]TJ /F9 11.955 Tf 11.95 0 Td[(jJi ct Dij)]TJ /F16 11.955 Tf 11.96 0 Td[(%imJi (2)where%imisthemembranewallresistancefactor,ctisthetotaluidconcentrationnotincludingthemembranepseudo-uidparticles,and'iisthevolumefractionofspeciesi.IfoneconsiderstransportinthelimitofKnudsenregimeowitcanbeseenthatthersttermontherighthandsideofEquation( 2 )goestozeroand%imcanbesolvedfordirectly.Asimilaranalysisisalsocarriedoutforconnedliquidows,albeitwithouttheslipowphenomenaseeningases,andwascomparedwiththeexperimentaldataofTruittetal.[ 126 ]fortheowofnoblegasesthroughmicroporousgraphiticmembranes.ThesameproblemwasfurtherevaluatedbyKerkhof,GeboersandPtasinski[ 76 ]consideringcomponentvelocities(_ri)intheviscousfrictiontermsinsteadofmassaveragedvelocities(v)asseeninEquation( 2 ),andtheradialformofthemodiedmomentumbalanceisgivenby:)]TJ /F16 11.955 Tf 10.49 8.09 Td[(@Pi @z)]TJ /F16 11.955 Tf 11.95 0 Td[(i1 r@ @rr@_ri @r=)]TJ /F6 11.955 Tf 9.3 0 Td[(PtnXj=1,j6=iij Dij(_ri)]TJ /F12 11.955 Tf 12.45 0 Td[(_rj) (2)inwhichiisthepartialviscosity.UtilizingaslipboundaryconditionatthewallandasymmetryconditionattheaxisananalyticalsolutiontoEquation( 2 )isderivedforthevelocityprolesofabinarymixture.ThesevelocityprolescanthenbeintegratedandusedtodetermineSMdiffusioncoefcientsviatheSMrelation.ItisnotedbyBhatiaandNicholson[ 18 ]thatthisapproachprovidesnomeansbywhichtodetermine 41

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thepartialviscositiesandthatthevelocityprolescanonlyberesolvedifsolvedsimultaneously,acomputationallyintensiveprocedure. 2.2TransportModelsinthePresenceofanAdsorptionFieldThissectionfocusesontheworkdonebytheBhatiagroupattheUniversityofQueenslandinBrisbane,inwhichnovelandefcientapproachestolowdensitygaseousows(OscillatorModel)andviscousows(DistributedFrictionModel)inthepresenceofaLJpotentialeldareproposed.Theseformulationsnaturallyaccountforadsorption-likeeffectsbytheirinclusionofVanderWaalsinteractionsbetweentheporewallandthediffusingspeciesintheequationsofmotion,insteadofsuperposingseparatelyconsidereddiffusiveandsurface-boundmotions.TherelativestrengthsandweaknessesofeachmodelarediscussedaswellasadetailedderivationandcritiqueoftheOscillatorModel. 2.2.1TheOscillatorModel 2.2.1.1Seminalmodeldevelopmentforlow–pressuresystemsandsubsequentmodelrevisionsTheOscillatorModel(OM)wasrstproposedbyJeppsandcoworkers[ 69 ]in2003.Notingthestronginuenceofthepore-permeateinteractionsinlowdensitygasesandtheinaccuraciesfoundinnonequilibriumthermodynamic[ 43 ]andsuperposedSM/Random-Walkmodels[ 93 ]insuchsystems,Jeppsetal.soughttodevelopanewanalyticalprocedurefordeterminingtransportcoefcientsfromclassicalHamiltonianMechanics,adoptingseveralnonequilibriummoleculardynamics(NEMD)conventionsformathematicalclosure.Itisassumedthatasinglemoleculeisreleasedinsideaninnitecylindricalporeandisacteduponbyapotentialeldsymmetricabouttheporeaxis.Thismolecule“oscillates”withinthepore(Figure 2-1 )experiencingaCosineLawdiffuse,thermalizedreection[ 42 ]witheachcollision.Thepotentialeldconsidered(U)isthatofadistributed6-12LJpotentialasgivenbyTjatjopoulosandFeke[ 125 ]inwhichthecombinedcontributionofindividualLJsitesareintegratedovertheentire 42

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poresurfacethendividedbysomeLJcitedensity,yieldinganexact,hypergeometricsolutiontothepermeatepotentialenergyasafunctionofposition.Itisworthnotingthatwhiletheexactsolutionisknown,theintegralsareinsteadevaluatednumericallyforcomputationalefciency.Itiseasilyshownthatsuchasymmetric,innite,integratedpotentialeldwillresultinazeronetforceonthefreemoleculeintheaxialdirection,andthisallowsforapurelyradialdescriptionofthepotentialenergy.However,thisalsoimpliesthattheaxialvelocity(vz)isinvariantbetweenwallcollisions,andthattheaverageaxialmomentumimpartedtothepermeatebythewallwillbe0.Thisleadstothepredictableresultofzeroux,asnooneparticlecausesux,butratheraconcentrationgradientofmanyparticles.Tocircumventthis,Jeppsproposesanaxialpseudo-force(Fz)beappliedtothefreemolecule,analogoustothatseeninNEMD,preferentiallydrivingthemoleculedownstream.Thispseudo-forceisanalogizedtothechemicalpotentialgradientwhichisnotedasbeingthedriverofdiffusiveprocesses[ 43 ],andthusissubstitutedintothephenomenologicalequationfordiffusivetransportinEquation( 2 ).D0=JMkBT 1 O=hvzikBT Fz (2)whereJMisthetotalmassux,istheuiddensity,Oisthechemicalpotentialgradient,andhvziistheaverageaxialvelocity.Notethathvzi=JM=anditistheonlyunknownontherighthandsideoftheequation.TheHamiltonian(H)ofthesystem,giveninEquation( 2 ),isthenseparatedinto(r,)andzcomponentsandintegratedtoobtainEquations( 2 )–( 2 ).H=U(r)+pr2 2m+p2 2mr2+pz2 2m)]TJ /F5 11.955 Tf 11.95 0 Td[(Fzz=Er,+Ez (2) 43

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Figure2-1. Cross-sectionalviewofacylindricalporeconsistingofLennard-Jonessites(graycircles)andthetrajectories(bluedottedlines)ofapermeatemolecule“oscillating”within.Theradiallocationsr+andr)]TJ /F1 11.955 Tf 10.41 1.79 Td[(aretheboundsofintegrationusedwhencalculatingthe“hoppingtime”,whichisthenusedtoevaluatethediffusioncoefcientD0. D0=kBT 2mt(r,pr,p)e)]TJ /F10 7.97 Tf 6.59 0 Td[(Er,=kBTdrdprdp te)]TJ /F10 7.97 Tf 6.59 0 Td[(Er,=kBTdrdprdp (2)(r,pr,p)=p 2mZr+r)]TJ /F5 11.955 Tf 13.24 11.87 Td[(Er,)]TJ /F6 11.955 Tf 11.96 0 Td[(U(r))]TJ /F5 11.955 Tf 20.97 8.09 Td[(p2 2mr2dr (2)wherer+andr)]TJ /F1 11.955 Tf 10.41 1.8 Td[(denotetheradialdistanceofthepermeatefromtheaxisatthestartandmid-pointofaoscillationrespectively(showninFigure 2-1 ),m=M=NA,andisthe“hoppingtime”.Similarly,aninniteslit-porecongurationismodeledwherethepotentialeldisconsideredtobeafunctiononlyofthefreeparticlesdisplacementnormaltotheporeplanes.Jeppsetal.[ 70 ]laterextendedtheslit-poremodeltouidnumberdensitiesgreaterthan1nm)]TJ /F2 7.97 Tf 6.59 0 Td[(3byincorporatingahydrodynamicslip-owmodelinwhichtheuiddensity 44

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(^)andviscosity(^)areallowedtovaryalongtheporecrosssectionasdescribedbyBitsanisetal.[ 21 ]andChungetal.[ 29 ]respectively.FollowingintegrationoftheHamiltonianEOMsthediffusioncoefcientcanbeexpressedasaconstantboundarytermandaspatiallyvariant“viscous”termwhichvanishesinthelimitoftheaverageuiddensityapproaching0(Knudsenregime).ReasoningthatthisanalysisshouldyieldidenticalresultstotheOMinthisextreme,theconstantboundarytermisequatedwiththeOMdiffusioncoefcientfromEquation( 2 )andthetotaldiffusioncoefcientisgivenbyEquation( 2 ).Notethatthenestedintegralsontheright–handsidegivethetotalproductofthespatiallyvariantdensityandstreamingvelocityterms,wherehvzi/ .D0=DOM0+2kBT h^iHZ0)]TJ /F10 7.97 Tf 6.59 0 Td[(x01 ^()Z0^()d2d (2)Similarly,theOMwasextendedtohigherdensityowsthroughcylindricalporesbyBhatiaetal.[ 11 ],butthistimealsoincorporatingsomestatisticalmechanicaltheorytoimprovethemodelcomputationalefciency.Understeadystateconditions,Bhatiaarguesthattheproductofthechemicalpotentialgradient(O),againequatedwiththeNEMDpseudo-force,andthemean“hoppingtime”(hi)inthephenomenologicalequationisrelatedtothemeanaxialmomentumgainofthefreemoleculeinasingle“oscillation.”Equatingthesetwoexpressionsthediffusioncoefcientcanbewrittenasafunctionofsaidmean“hoppingtime”throughEquation( 2 ):D0=kBT mhi (2)Themean“hoppingtime”isdeterminedbyintegratingtheHamiltonianEOMsassumingconstantcircumferentialmomentumbetweenwallcollisionsandusingtheintegrated6-12LJpotentialfunction[ 125 ]previouslydiscussed.Adetaileddescription 45

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ofthederivation,aswellasacritiqueontheOMformulationisgiveninsection 2.2.1.2 .UndertheseconditionsBhatiaisabletowriteasanintegralfunctionofr,pr,andpofthefreemolecule.Thenbyconsideringalargeensembleofmomentumvaluestakenfromacanonicaldistribution[ 120 ]representingalargesamplingofpermeate-wallcollisions,alargesampleof“hoppingtimes”canbecalculatedandaveragedforuseinEquation( 2 ).SimilartothemethodofJeppsetal.[ 70 ],athigheruidnumberdensitiesa“viscous”termisderivedusingthedensityfunctionaltheoryofBitsanisetal.[ 21 ]andsuperposedwiththeOMdiffusioncoefcientcalculatedfromEquation( 2 ).BhatiaandNicholson[ 13 ]furtherdevelopedtheOMtoaccountforporestructureswith“smoothenergylandscapes”likecarbonsorsilicainwhichasignicantfractionofthereectionsarespecularinnature.Simulationsofhydrogen,methaneandcarbontetraouridediffusingdownacylindricalsilicaporeoverawiderangeofnumberdensitiesarecomparedwithEMDandNEMDmodelsandshowstrongagreement.AnextensiontotheOMhasalsobeendevelopedformolecularly-narrowporestructures,meaningthecharacteristicdiameteroftheporeislessthan2moleculardiametersofthediffusingspecies,creatingnearsingle-leow[ 14 ].WhenconsideringKnudsenowphenomena,increasingthenumberdensityoftheuidincreasestheux,howeverinthecaseofexceedinglynarrowpores,increasingthenumberdensitydoesjusttheopposite.Thisisattributedtotheincreasednumberofwallcollisionsforeachparticleduetothehighlyrepulsivecentralregionofthepore.InpreviousOMformulations,increasingthenumberdensitymeantthattheviscous-likeeffectsoftheowneededtobeaccountedforviatheLocalAverageDensityModel,howeverinthisspecialcaseonlymonolayersurfaceowsarepossibleduetothehighlyconnedgeometry,makingviscousowmodelsinvalid.Therefore,fortransitionregimeowsinmolecularly-narrowporesa“nearestneighbor”modicationtotheOMisproposedinwhichtheuid-uidinteractionofthe2nearestuidmoleculesareincludedinthecalculationofthetotalpotentialenergyofthefreemolecule.Theaxiallocationsofthe 46

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“nearestneighbors”aredeterminedbasedupontheaverageuiddensityandtheporeradius,thenassignedrandompolarangleandradialcoordinatesbeforethestandardintegrationoftheHamiltonianEOMsasdiscussedearlier.ThisapproachcomparedfavorablywithEMDandNEMDsimulationsofCH4andCF4inamorphoussilicapores. 2.2.1.2CriticalreviewoftheuxhypothesisandmathematicalformulationThephenomenologicalequationisashorthandmeansofswitchingfromoneframeofreferencetoanother.InsolvingtheHamiltonianEOMs,thetrajectoriesofthefreemoleculearegiveninaLagrangianmethodofdescription.However,uxisauniquelyEulerianconcept,andonemustinvokeGauss'Lawinordertoperformsuchatransformation.InsteadEquation( 2 )isaproportionalityrelationship,drawingparallelsbetweentheEulerianuxterm(JM)andtheLagrangianaverageaxialvelocity(hvzi),whichcanbethoughtofastheuxcarrier,andrelatingthemthroughaconstantdiffusioncoefcient,therebylackingsomemathematicalrigor.Furthermore,thechemicalpotentialgradientusedinEulerianuxcalculationshasnophysicalanalogintheLagrangianequationsofmotion,makingthestraightsubstitutionofOforthenon-physicalNEMDpseudo-forceFzdifculttojustifyandtheratioO mhi hvzi=1alikelyoversimplication.Additionally,analogizingFztoOwouldnecessitateameansofchoosingFzbaseduponthesystemboundaryconditions,i.e.reservoirpressures,temperaturesandnumberdensitiesdeningthepartialmolalGibbsfunctions.Thatsaid,noindicationisgivenastohowFzisdetermined,statingsimplythat Fz = m wassettovaluesintherange0.002.02 nm = ps2 ,asthis“affordedlinearresponsebehavior”[ 69 ]withoutfurtherjustication. 2.2.2TheDistributedFrictionModelAllmodelsoftransitionregimemixturetransportdiscusseduptothispointhavereliedonsuperposedviscousanddiffusivetransporttermsstemmingfromtheDGMconvention([ 11 ],[ 40 ],[ 41 ],[ 69 ],[ 92 ],[ 93 ]),orhavefollowedfromtheSMformulationandattemptedtorelatethethermalandconcentrationgradientdrivenuxestoasingle 47

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mixtureshearstressterm([ 75 ],[ 76 ],[ 110 ],[ 134 ]).Thelattersetofmodelshavehithertoassumednoadsorptiveinteractionbetweenthewallandfreemolecules.Therstsuccessfulintegrationofsolid-uidpotentialsintoaSMtypemodelofgaseousmixturetransportwasgivenbyBhatiaandNicholson([ 15 ],[ 16 ])viaacouplednear-wall/far-eldapproachthatallowedformomentumexchangeofeachdiffusingspecieswiththewallinsidethehighlyrepulsiveregionoftheLJpotentialeld.Thederivationassumesaone-dimensionalowofann-componentgasmixturewithinacylindricalporeofradiusRexhibitingaradiallydistributedpotentialeldlikethatusedintheOM(andproposedbyTjatjopoulos[ 125 ]).TheequationofmotionfortheithcomponentofthemixtureisgivenbyEquation( 2 ).1 rd drridhvii dr=ni(r)di dz+ntkBTnXj=1ij(hvii)-222(hvji) Dij+inihviiA(r)]TJ /F5 11.955 Tf 11.96 0 Td[(r) (2)wheredi dzisthespeciesspecicchemicalpotentialgradient,iisthemolefraction,ni(r)andnt(r)arethenumberdensityoftheithcomponentandthatofthetotalmixturerespectively,DijisFick'sbinarydiffusivity,viandiarethespeciesspecicaverageaxialvelocityandpartialviscosityrespectively,iisthe“wallfrictioncoefcient”calculatedbaseduponthewallpotentialeldU(r),andAistheHeavisidefunctionequalto1ifr>randzerootherwise.ThisformulationmirrorsthatseenintheVelocityProleModel[ 76 ]withtwokeymodications.Therstbeingtheinclusionofthewallpotentialviatheiterm,andthesecondbeingthetypicalmixtureshearstressterm()beingreplacedbycomponentviscositiesaccordingtoEquation( 2 ).nXi=1idhvii dr=dhvi dr= (2)Thisconditionstipulatesthat“forahomogeneousuidthetotalshearstressonallthecomponentsisthatonthemixtureasawhole”andisnolongerafunctionofthe 48

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mass-averagedmixturevelocity[ 77 ].Furthermore,themixturecenterofmassisnolongerthechosenframeofreference,butrathertheconstituentcentersofmass,whichhasyieldedimprovedaccuracyinthepredictionoftheOnsagercoefcients.LikewiththeOM,thistheoryhasbeenvalidatedagainstEMDandNEMDsimulationswithstrongagreement,butusesthesamenon–physical,NEMDargumentsasdiscussedin 2.2.1.2 whendeterminingi. 2.3MolecularDynamicsThissectionfocusesonthedevelopmentofMDsimulationtechniquesusedintheanalysisofmasstransportthroughnanoporesandnanoporousmedia.Becausethetopicisnanoscalediffusion,thisreviewonlyconsidersatomisticpotentialeldsasopposedtosmearedpotentials([ 125 ],[ 11 ]),andtheLennard-Jonespotentialfunctionisthemodelofchoice.Asanintroductiontoatomisticmodelingsingle-walledcarbonnanotubes(SWCNT)arediscussedduetotheirhighly-orderedandwell-studiedlatticestructure.ResearchprojectsexaminingdiffusionthroughSWCNTsviaequilibriumandnonequilibriumMDmodelingarediscussed,asistheuseofdualcontrolvolumegrandcanonicalMonte-Carlo(DCV-GCMC)modelsinsuchsystems,andtheadvantagesanddisadvantagesofeachapproacharehighlighted.Thissectionisconcludedwithareviewofthesimulationofamorphousstructuresthroughconformationalenergyminimizationtechniques. 2.3.1IntroductiontoMolecularDynamicsandAtomisticSystems 2.3.1.1Argumentforsingle–walledcarbonnanotubesasatransportmediumCarbonnanotubes(CNT)wererstdiscoveredbyIijima[ 64 ]in1991,grownonthenegativeterminalofacarbonelectrodeduringanarc-dischargeevaporationprocessintendedfortheproductionofC60fullerenemolecules.ThestructuresobservedbyIijimaweremulti-walled,rangedfrom4to30nmindiameter,wereupto1microninlength,andweregrowninarandom,uncontrolledfashion.ItwasnotuntiltwoyearslaterthatIijimaandIchihashi[ 65 ]wereabletosynthesizeSWCNTstructures,thistimegrown 49

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inthegasphaseofanarcdischargeexperimentasopposedtogrowingdirectlyuponthecatalyst.OthernotablemethodsofsynthesizingCNTsarelaserablation[ 124 ]andcarbonmonoxidecatalysis[ 98 ],developedatRiceUniversitybytheresearchgroupofRichardSmalley,andchemicalvapordeposition(CVD),originallydevelopedbyEndo[ 39 ].TheprocessofCVDistheonlymethoddevelopedatpresentcapableofproducingCNTsofuniformcharacteristicsenmasse[ 31 ]andhasrecentlybeenusedtodevelophighlyordered,verticallyalignedCNTmembranes[ 135 ].Initialtestsperformedonthesenanotubebundlesdemonstratedhighgaspermeabilityandthepotentialforuseinselectivitydrivenapplications.Theseaspects,combinedwiththehighelectricconductivityofmanyCNTchiralitiesandthechemicalinertnessofthestructureinthepresenceofmethanolmakeCNTsanintriguingpossibilityasafutureLBLmaterial.Fromthestandpointofmolecularmodeling,thehighlyordered,repeatingstructureofSWCNTsmakethemastraightforwardchoiceformodelingatomisticpores.UsingGroupTheoryandthescrew-axissymmetryinherentinCNTs,Damnjanovicetal.[ 34 ]developedasetofequationstogivecylindricalcoordinatesofeveryatominthelatticeofadefect-freeSWCNTofagivenlengthandchirality.ThedefectfrequencyseeninmodernCNTsynthesistechniquesislow,themostcommonsinglevacancy,interstitialorStone–Wales5defects(seeFigure 2-2 )occurringnomorethanoncepermicron(lengthwise)[ 32 ].Thissuggeststhatthedefect-freeassumption,atleastinthecaseofdiffusingparticlescatteringangles,isfundamentallysound.Furthermore,thesignicantstructuralsimilaritiesbetweenSWCNTsandheavilystudiedgraphenesheetsallowsfortheuseofpreviouslyderivedpotentialmodelswhendetermininglatticedynamics. 50

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ADefect-free BSinglevacancy CStone–Wales5Figure2-2. Illustrationsofthemostcommongrapheneandcarbonnanotubelatticedefects.Figure 2-2A showsadefect-freelattice.Figure 2-2B isasinglevacancyor“pointvacancy”defectinwhichthreeC=Cbondsarebroken,freeingoneatomiccarbonfromthelattice(usuallyoccurspost-synthesis[ 32 ]).Figure 2-2C isaStone–Wales5defectinwhichabondrotationoccursandresultsinnegligiblechangesintheCNTphysicalproperties[ 32 ].Interstitialdefects,inwhichanadditionalatomiccarbonisbondedtoanotherwisedefect-freelattice,arealsocommonbutonlyinlayeredgraphenesheetsormulti-walledCNTs,formingasabondedcarbonconnectingthetwoadjacentlayers. 2.3.1.2Transportthroughsingle–walledcarbonnanotubesviaequilibriummoleculardynamicsEquilibriumMDmodelingoftransportthroughSWCNTswasrstpublishedbyTuzunetal.[ 128 ]in1996.Armchairtype,10and20nanotubes(13.6Aand27.1Aindiameterrespectively),oflength130Aand172Aweresimulatedwithbothstaticanddynamicwallmolecules.Forthedynamiccaseonlythebondstretching,bondbendingandstericvanderWaals(excludingthe1and1pairsperconvention[ 10 ])potentialswithinagivencut-offradius(20A)wereconsidered.Theprimaryjusticationforneglectingthetorsionalpotentialsisthecomputationalexpense,thoughitisalsostatedthattheeffectshouldbeminimalduetothelackof“kinking”seeninthenanotubelatticeduringsimulation,suggestingthatthereisalackofexcessivetorsionalmotions.Waltheretal.[ 131 ]latershowsthatthisassertionisnotentirelycorrect,demonstratingthattheinclusionoftorsionaleffectslendsasignicantamountofstiffnesstothelattice. 51

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Itisalsoworthnotingthattheharmonicforceconstantusedintorsionalpotential(andtorsionalspring)modelshadnotyetbeenapproximated,indeeditwasnotuntil5yearlaterinthepaperofWaltherandcoworkers[ 131 ]thatanysuchconstantswerederivedforCNTs.SaidapproximationwasmadebytakingadvantageofthesimilaritiescarbonhexagonalringstructurefoundinzigzagtypeSWCNTsandtetracene.Freemoleculesofheliumandargonwerebothinitiatedinsidethetubeandthetrajectoriesoftheparticlesweretracked.ThesesimulationswerecarriedoutusingtheCalvoandSanz-Sernaalternating,fourth-ordersymplecticintegrator[ 49 ]tosolvetheHamiltonianequationsofmotion.Forthesakeofcomputationaltractability,thenumberofparticlespresentinthetubewaslarge,largeenoughthatboththeheliumandargonbulkdensitiesweregreaterthantheirrespectiveliquidregimelimits.Andyet,eveninthisextremethesimulatedtimewasontheorderoftensofpicoseconds.Giventheshorttimescaleandtheunrealisticinitialconditions,thepublishedresultsprovidelittleinsighttothetruephysicsofthephysicalsystem.However,thisworkservedasalaunchingpointforotherresearcherslookingtoexploreMDsimulationsofCNTs,andhighlightedtheneedformoreefcientsimulationtechniques.MaoandSinnott[ 89 ]consideredasimilarprobleminwhichmethane,ethaneandethylenediffusethroughfullydynamicachiralnanotubesatroomtemperature.However,insteadofusingtheMorsebond-stretchingandaharmonicbond-bendingpotentialforthenanotube([ 128 ],[ 131 ])thebindingpotentialwascalculatedusingareactive,empiricalbond-order(REBO)technique,originallydevelopedbyBrenner[ 23 ]toinvestigatethegrowthofdiamondlmsinCVDexperiments.ThetechniquewaslaterrenedbyBrenneretal.[ 24 ]withanexpandedlibraryofexperimentalmeasurementsandmodiedanalyticfunctionstobetterrepresentbothhydrocarbonandsolidcarbonstructures.ThisREBOapproachisadditiveandpairwise,muchliketheLJpotential,andhencethenumberofinter-latticeinteractionsconsideredisgreatlyreduced.Furthermore,athird-orderNordsieckpredictor-correctorintegratorwasusedtosolve 52

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theNewtonianequationsofmotion,againsimplifyingtheprocessusedbyTuzunetal.Inter-nanotubeinteractionsinnanotubebundleswerealsoexamined.Evenwiththesesimplications,thesimulationtimesreportedwere15psorless,againmakingconclusionsdrawnfromtheresultslikelyextrapolations.InasubsequentpublicationMaoandSinnott[ 90 ]considermixturesofhydrocarbonsdiffusingthroughSWCNTslookingatseparationofspecies.Justasinthepreviouslydiscussedpublication[ 89 ]thediffusioncoefcientsweredeterminedusingoneofthreeaveragedtrajectoryproportionalityrelationshipsbaseduponwhethertheowwasdeemedtobeinthenormal,single-le,ortransitionregime,andthesimulationswherelengthenedto100ps.FluxwasthencalculatedusingsaiddiffusioncoefcientsandtheOnsagerrelationshipderivedfromirreversiblethermodynamics.However,thistheoryonlyholdstruewheninteractionsbetweenspeciescanbeneglected,whichbydenitionisnotthecaseinthetransitionandnormaldiffusionregimes.Also,preliminaryresultsfromthisproposalsuggestthat100psofsimulationtimedoesnotallowforstatisticalconvergenceoftheradialnumberdensityoraxialvelocityprolesintheKnudsenregime.Hummeretal.[ 63 ]studiedmolecularwatertravelingthroughasubmerged,nitelengthofnon-polarSWCNTusingEMDinanefforttodetermineifthestructureinteriorswouldwet.IndeeditwasfoundthattheCNT,6armchairtype13.4Ainlengthand8.1Aindiameter,transmittedanaverageof17moleculespernanosecondovera66nssimulationperiod.Inthisstudythelatticestretching,bendinganddihedralpotentialswereconsideredandthecorrespondingEOMsweresolvedusingtheAMBER6.0softwarepackage.Thewater–carboninteractionconsideredherewassimplytheLJpotentialbetweencarbonandatomicoxygen.Waltheretal.[ 131 ]alsolookedatwatertransportthroughfullydynamicSWCNT(dihedralpotentialsincluded)butusedamoresophisticatedwatermodelwhichincludedaquadrupoleinteractionbetweenthecarbonandthepartialchargesoftheH2Oconstituents(SPCwatermodelofTeleman 53

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andcoworkers[ 122 ]).ThissamemodelhasbeenusedbyWerderetal.[ 133 ]inthepredictionofwatercontactanglesonhydrophobicsurfaces.Skoulidasetal.[ 113 ]usedEMDtocomparetransportratesofmethaneandhydrogenthroughSWCNTsandzeolitesofcomparableporediameter,concludingthatthesmoothenergylandscapefoundinnanotubeporesresultinnearspecularpermeatereectionsanduxesfarexceedingthosefoundindisorderedsilicatemembranes.Theuxratiosweredeterminedviatheratiosoftherespectiveself-diffusivityandtransportdiffusivitycoefcients,evaluatedusingtheEinsteinrelationandthemethodofTheodorou[ 86 ]respectively.Einstein'sself-diffusivityequationrelatesthemean-squareddisplacementofaparticletoann–dimensionaltransportcoefcient(Ds)overlongtimethroughthefollowingexpression:hr2(t)i=2nDst (2)Thesimulateddisplacementsusedtocharacterizetheself-diffusivityofadsorbatesintheSWCNTbundlesandzeolitemembraneswereaveragedover2.3nsusingaconstantkineticenergyVerletleap-frog(third-order)algorithmwith5fstimestepsandaNose–Hooverthermostat.Anothermethodofdeterminingself-diffusivityisbyinvokingtheGreen–Kubo(GK)relationandevaluatingthetimeintegralofthevelocityautocorrelationfunctionperthemethodofMaginnandTheodorou[ 86 ].Thetransportdiffusivitycoefcient,equivalentinthiscasetotheFickiancoefcient,isdeterminedbyinvokingamodiedGKrelationshipwhereinsteadtheinnitetimeintegralofthespeciesvelocitycross-correlationisevaluatedtogivea“correcteddiffusivity”.ThentheDarkenequation[ 35 ]isused,relatingsaidcorrecteddiffusivitytothetransportdiffusivitythroughtheconstanttemperaturelogarithmicfugacitygradientwithrespecttothelogarithmicconcentrationgradientoftheadsorbate.Noconvergenceanalysisisprovidedtoensure 54

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thatthesimulationtimes(O(10)ns)aresufcientapproximationstotheGKinniteintegrals.Skoulidasetal.[ 115 ]laterexpandedthisstudytocarbondioxideandnitrogentransportthroughSWCNTsrangingfrom8.14A(6)to54.25A(40)indiameter,populatingthenanotubeinteriorwithenoughmoleculessoastobecomparabletothenitrogendiffusionmeasurementsofHindsetal.[ 56 ].Thepredicteduxusingthemethoddescribedaboveisafactorof30greaterthanthatmeasuredexperimentally. 2.3.1.3Transportthroughsingle–walledcarbonnanotubesvianonequilibriumandstatisticalmethodsNonequilibriumMDandgrandcanonicalMonte–Carlo(GCMC)simulationtechniquesarealsoprevalentintheeldofMDanddiffusionalprocesses.AcommonlycitedissuewithEMDisthattemperatureisnotaccountedforintheNewtonianequationsofmotion,makingthesimulationofnonequilibriumsystemscontainingheatowsorshockwavesimpossible.NEMDintroducesameansofincorporating“microscopicmechanicaldenitionsofmacroscopicthermodynamicandhydrodynamicvariables”[ 61 ].Forexample,thesuperpositionofanunidirectionalforceinparallelwiththeexpecteddirectionofdiffusionwilldrivetheparticlesdownstream,eveninthepresenceofdiffusereections,therebysimulatingachemicalpotential(orconcentration)gradient.Ifthesimulatedsystemistobeisothermal,thentheworkdoneonthesystembythisexternalpseudo-forceiscounterbalancedbyscalingtheparticlevelocitiessuchthatthedesiredthermalvelocityismaintained(Nose–Hooverthermostat)ormodifyingtherelativeparticlecenter-to-centerdisplacementssoastomaintainaconstantsystempressure(Berendsenbarostat).Nicholson[ 97 ]comparedNEMDandEMDcalculateddiffusioncoefcientsofamixtureofCO2andCH4throughgraphiticnanoporesofvaryingradii,concludingthattheywereequivalentwithintheerrorsofthecalculation.FollowingtheEMDapproach,usingafourth-orderGearpredictor–correctoralgorithmtoresolvethe 55

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trajectoriesthroughadiffuselyreecting,cylindricalpore,theaxial(“streaming”)velocitycross-correlationwasintegratedover4.2ns(1.05106timesteps,4fseach)andthermallyscaledfollowingtheGKmethodology.TheNEMDsimulationinsteadimposeda0.1 nm = ps2 accelerationuponeachpermeatemoleculeandmodiedtheparticlevelocitieswithaGaussianthermostat.Themethodinwhichthevalueoftheexternalaccelerationwaschosenisunclear,thoughtheauthordoesmentionthatotherdrivingaccelerationsweretestedandyielded“generallypooragreement”withtheEMDresults.Ifaphysics-basedapproachtodeterminingthisaccelerationisnotavailable,thenthistechniqueislikelyunsuitedformembranedesignpurposes.Kotsalis,WaltherandKoumoutsakos[ 81 ]usedNEMDtostudytwo-phaseowsofH2OandH2O/N2mixturesinSWCNTsofvaryingchirality(20,30,40)andvaryingtemperature(300K).Asbefore,aforceisappliedtothemoleculespreferentiallydrivingthemdownstreamcreatinganon-zeronetuxandatwodimensionalBerendsenthermostatisusedtomitigatetheresultantviscousheating.Whatisnovelaboutthisapproachistheexternalaccelerationisadaptive,changingsuchthatthetotalcenter-of-massvelocityofthemixtureisheldconstantatsomedesiredvalue.However,thisapproachstillsuffersfromthefactthattheimposedaccelerationisnotaprescribedvalueindeterminingtheuxofaspeciesthroughthepore.Analternativetoresolvingindividualparticletrajectories,GCMCsimulationsprovideameanstosamplealargeensembleofparticlesastheymovedownapotentialenergyslope(orupitwithinsomeprobability)inaconstanttemperature,pressure,andchemicalpotentialenvironment.Theformulationandevaluationofthisapproacharediscussedindepthelsewhere[ 3 ].AnintriguingapplicationofthistheoryisinthedevelopmentoftheDCV-GCMC.Cracknell,NicholsonandQuirke[ 33 ]developedamethodofdetermininguxthroughanyporegeometrydirectlybycountingparticlestravelingfromonereservoir 56

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toanother.EachreservoirispopulatedwithLJparticlesandthenumberofparticles(NAandNB,seeFigure 2-3 )uctuatesutilizingthegeneralGCMCschemesoastobeheldataspeciedchemicalpotential(AandB).IntheGCMCreservoirsthewallsnormaltothexandy–axesaresubjecttominimumimageboundaryconditions[ 3 ]andifaparticlecrossestheleftmostorrightmostboundariesthatparticleisdeletedfromthesimulation.BecausetheparticlesareallowedtoleavethesimulationbycrossingsaidboundariesorthroughstochasticdeletionsahighratioofstochastictodynamicoperationswererequiredtomaintainthedesiredO(between20-to-1and110-to-1).Ifaparticleentersthecentralsimulationspace,or“owregion”,thentheparticletrajectoriesweretrackedsubjecttointermolecularandpermeate–wallpotentialsandpurelydiffusereectionsafterwallcollisions.Thecaseofspecularreectionswasalsoexplored,thoughtheresultinguxeswereinpooragreementwithotherMDndings.Thisapproachsimulated4nsworthofmolecularmotionandrequired170hoursofCPUtimetocomplete. Figure2-3. IllustrationofthebasicarchitectureoftheDCV-GCMCsimulationspaceproposedbyCracknell[ 33 ].Theleftmostandrightmostcells(AandBrespectively)arereservoirsheldatconstantchemicalpotential()viaGCMCoperationsandthecentralcellistheregioninwhichmoleculartrajectoriesarecalculated.Fluxismeasuredbycountingthenumberofparticleswhichtravelfrom1reservoirtotheotherandiscompareddirectlytotheimposedchemicalpotentialgradient. Zhengetal.[ 136 ]modiedthismethodbyincludingbufferzonesbetweenthereservoirsandtheowregionandbyperformingMDtrajectorycalculationsinevery 57

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cellwhileincludingGCMCoperationswithinthereservoirsatsetintervalstomaintainthedesiredO.ThediffusateexaminedisadilutemixtureofmethanolandwaterandtheH2O/MeOHmolarpercentcompositionsmaintainedinthe2reservoirsare64/36and0(vacuum).Thecentralowregionsconsideredarearmchair-typeSWCNTwithchiralindicesrangingfrom6to10(diametersrangingfrom6.79to13.57A)modeledasstatic,hard-sphereLJlattices.Bothhydrophobicandhydrophilicowregionsweresimulated.NomodicationsweremadetotheCNTstructureforthehydrophobiccaseasSWCNTsarehydrophobicstructures([ 96 ],[ 106 ]).Inthehydrophiliccasecarboxylacidgroups(–COOH)arebondedtoonesideofagraphenelatticeinaprescribedsitedensityandthentheentireconstructionisrolledtoformthenanotube.Oncethe–COOHextrusionlocationshavebeenoptimized(tominimizethetotalpotentialenergy)stronglyhydrophilicinteractionsbetweenthewaterandnanotubecanbesimulated.ThebufferzoneslayoutsidetheGCMCregionsandarepresentinordertostudytheendeffectsofthediffusionprocess.Again,computationalexpenseheldthesesimulationstolengthsof3ns,andallbutthelast0.5nswasnecessaryforequilibrationoftheGCMCcells. 2.3.2SimulationofChainMoleculesandPolymersSingleWalledCNTsprovidemoleculardynamicistswithanidealizedcrystallinestructurewithareadilydenedequilibriumorientationandframeworkwithinwhichtosimulatebondedandnon-bondedinteractionsbetweenparticles.Themaindrawbackisthatsuchhighly-ordered,uniform,andcrystallinecylindricalporestructuresareuncommonandcostlytomanufacture.Thevastmajorityofmembranematerialspopulatedwithdisorderedmicro-andnanoscalediffusionpathsareamorphous,withsomeinherenttortuosity,PTFe(polymeric)andsilicatezeolites(graphitic)beingprimeexamples.Theincrediblearrayofporesizes,irregulargeometriesandcomplexinteractionsofgroupingsoflongpolymerchainsmakesMDasapredictivetoolformembraneuximpractical.However,whilediffusionthroughapolymermatrixmaybea 58

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computationallyexpensiveproblem,ifanorderedarrayofcylindricalporesweredrilledthroughthebulkmaterialthenanEMDapproachsimilartothatalreadydiscussedmaybeemployedtoresolvemoleculartrajectoriesthroughsaidstructures.Suchmembranesarealreadybeingdevelopedthroughblockcopolymerization[ 66 ].InMDsimulationspolymerchainsaregrowninsideacontrolvolumesubjecttomirroredboundaryconditions(seeFigure 2-4 )untilthedesireddensityofthematrixisreached.Successivepolymerunitsarebondedsuchthattheconformationenergyisminimizedduringthebuildingphase.Afterthevolumeelementhasbeenfullypopulatedthestructureneedstogothroughanequilibration,orrelaxationphase.Asthechainisbuiltthelongrangeinteractionsareignored,theequilibrationphaseadjuststhebackboneconformationanglesofthechaininordertominimizethetotalenergyofthematrix.ThisprocedurewasdevelopedbyTheodorouandSuter[ 123 ]andisthefoundationofpolymermolecularmodelingresearch.Concerningthestudyofmoleculartransportthroughpolymers,Fritzetal.[ 45 ]usedthismethodtomodelapolydimethylsiloxanemembraneandexaminedthediffusiveseparationofconstituentsofawater/ethanolmixture.Hofmannetal.[ 57 ]performedacomparativestudyofpenetrantdiffusionthroughstatic(glassy)anddynamic(rubbery)polyimideandpolysiloxanematrices,andcharacterizedtheresidencetimeoffreevolumevoids,drawingconclusionsastothemechanismgoverningpermeabilityinsuchsolids.LaterHofmannandBohningetal.[ 58 ]extendedthisstudytopoly(isobutylene),polyethylene,andpolypropylenestructuresanddevisedamodelsimilartotheDCV-GCMCapproachwherethepolymermatrixactsasthe“owregion”locatedbetweenaMCpopulatedsourceandsink.ThemaindifferenceisthatthemotionofthepenetrantmoleculesandthesurroundinglatticeisresolvedusingtheGusev–SutertransitionstateMonte–CarloprocedureasopposedtoVerletorGeartypenumericalintegratorsappliedovertheentiresystem.Thesimulationofacompositepolymer/zeolitemembranewasalsoexamined.MorerecentlyHofmann, 59

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Figure2-4. 2Dillustrationofthepolymer“growth”algorithmofTheodorouandSuter[ 123 ].Theaboveexampleisapolystyrene(C8H8)chain.Acharacteristicunitmoleculeofpolystyreneisplacedinthevolume(labeled1andcircled)andthenthechainisgrownoneunitatatime,eachnewlybondedmoleculeorientedsuchthattheconformationenergyofthenewbondisataminimum.Ifanyofthestructurescrossthecontrolvolumeboundary(asisthehexagonalhydrocarbonringof4andallof6)theyarereectedsothatmanyoftheseunitvolumesmaybestackedtogethertoformalarge,repeatingpolymermatrix. KuleshovaandD'Aguanno[ 59 ]studiedwatertransportthroughNaonmembranes,inparticularexaminingthelatticearchitectureatequilibrium,thetransportofprotonsasseeninDMFCs,theformingofhighlyreactivehydroniumions,andtheexpecteddiffusioncoefcients.Amongtheircontributionstothescienceisarevisedwaterforceeld,improvinguponthecentralforceeld(CFF)[ 85 ]byparameterizingtheCFFwith 60

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thepublishedneutrondiffractiondatasetsofSoper[ 118 ]ofwaterandicecompositionoverlargetemperatureandpressureranges. 2.4ConcludingRemarksTheabovereviewprovidesadetailedoverviewofnanoscalemasstransportanalysis,beginningwiththeseminalnon-interacting,momentumbalanceapproachofKnudsen,throughtheapplicationoftheSMequationsandthedevelopmentoftheDGM,andtheuseofMDtoshedlightontheinteractionbetweenpermeatemoleculesandthesurroundingporestructure.Ashasbeennoted,eachapproachhasitsdrawbacks,whetheritisthenon-inclusionorarbitrarysuperpositionofadsorptiveinteractionsseeninKnudsenandtheDGM,theuseofnon-physicalNEMDconventionsinthephenomenologicaluxrelationasseenintheOM,ortheprohibitivecomputationalexpenseofDCV-GCMCtechniques.Theworkpresentedherewillprovideastrongerrst-principlesgroundingfornanoscalediffusionprocessesthroughtheuseoflong-durationEMDsimulationsandpresentanefcientandnovelmeansforresolvingnanosecondsworthoftrajectorydata(O(107)timesteps)withouttheuseofathermostatoranyotherarticialmeansofenergyconservation. 61

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CHAPTER3VECTORIZEDMOLECULARDYNAMICDIFFUSIONMODELChapter 3 discussestheevolutionofthisresearchproject,beginningwiththedevelopmentofanovelandrigorousKnudsenregimeuxmodel(ColsonF,privatecommunication,May6,2014),thendetailingtherationalebehindthechoiceinprogramminglanguageandnumericalsolutionprocedureforevaluatingtheEOMs,thenreviewingthemodelsystemsconsideredandthesimulationresults.AmoredetaileddiscussionoftheresultsandconclusionsareprovidedinChapter 4 andChapter 5 respectively. 3.1OutlineforaCoupledEMD/KineticTheoryApproachtoFluxPredictionIngeneral,uxhasbeendeterminedfromMDdatainoneofthreeways:(1)thephenomenologicalequationderivedfromtheStefan–Maxwellrelations,(2)theGKvelocitycross–correlationtechnique,or(3)theEinsteinrelationofmean-squaredparticledisplacement.EachoftheseapproachesinvokecertainproportionalityargumentsinordertodeneadiffusioncoefcientwhichisthenrelatedtouxthroughclassicalDarkenandOnsagerrelationships.AmongtheprimarygoalsofthisworkistocontributetothedevelopmentofanimprovedKnudsenscaleuxequationmirroringclassicalkinetictheorydynamicsofcontinuumgases.Incontinuummechanicsdiffusionisacollisionalprocessandthemeanfreepath,denedas“theaveragedistanceamoleculetravelsbetweensuccessivecollisions”[ 130 ],isafundamentalquantityinevaluatingthetransportofaspecies.Similarly,collisiondistancesshouldplayastrongroleinKnudsenregimediffusion,butinthiscasethecollisiondistancewillbemeasuredbetweensuccessivewallcollisionsasopposedtointermolecularcollisions.ThissuppositionisadeningcharacteristicoftheOMdevelopedbytheBhatiaresearchgroup,discussedintheChapter 2.2.1 .InthelimitasKn!1intermolecularinteractionisnegligibleand,onaverage,astatisticallysignicantsamplingoftrajectoriesfromoneparticlewillcharacterizethebehaviorofallofthe 62

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freeparticleswithinthepore.Thisassumptionallowsforanexactsolutiontothetotaluxgivenadistributionofaxialdistancestraveledbetweensuccessivewallcollisionsandanumberdensitygradientofthemoleculesofinterestalongtheporeaxis.ThisanalysiswasdevelopedbyFennerColsonandthederivationandpreliminaryresultsaretobethefocusoffuturepublications.Whatislefttodetermineisthedistributionofaxialcollisionaldistances,andthisiswhereEMDbecomesapowerfultool.ThegoalistodevelopahighlyaccurateEMDmodelthatwillresolvethetrajectoriesofasinglepermeatemoleculeasittravelsthroughanatomisticporeofinnitelengthoverlongsimulationperiods(O(10)]TJ /F2 7.97 Tf 6.58 0 Td[(8)s).Collisionaftercollisionaresimulateduntiltheradialdistributionandaxialdistancestatisticsconverge(O(105)oscillations,O(107))timesteps),seeSection 3.3.3 foramorecompletediscussion.Topreservetheaccuracyofthegeneratedpositionandvelocitydatanoarticialenergymodiersareused. 3.2RationaleforaVectorizedApproachThemostfundamentaldenitionofMDisthesolutiontotheEOMsofann-bodyproblem.Withthesteadyincreaseincomputationalmusclemoleculardynamicistshavebecomemoreambitious,specicallybyincreasingntothepointthatliquid/vaporinterfaces,bulkliquidsandlarge,dynamicpolymermatricesarefullyresolvable.Theclassicapproachtosolvingthen-bodyproblem,allforcesbeingconsideredpairwiseandadditive,istostepthroughtheelementsofthesimulationone-by-oneviaafororwhileloopcodearchitectureinoneofthegeneralcodinglanguagessuchasC,C++,orFortran.Despitethefactthatthen-bodyproblemisastapleoflinearalgebra,thiselement-wiseapproachhasbeenthenormfordecades.Iftheindividualforcesactingbetweenparticlepairscanbesolvedforsimultaneouslythroughmatrixoperationsthentheentireprocesscanbemademoreefcient,andthisiswheretheuniquecolumnarmemoryallocationofMATLABroffersadistinctadvantageoverotherprogramminglanguages. 63

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MATLABbeganasaninterfacewiththeLINPACKFortranlibraryforperformingmatrixoperations.ThiscollectionofsubroutineswaslaterupdatedtobetterutilizetheCPUmemorysearchpatternsinordertomaximizethenumberofoatingpointoperationsperformedinagiventimeinterval.Thisupdate(LAPACK–LinearAlgebraPackage)alsooptimizedfunctionsforshared-memoryandparallelprocessingunitsandhasnowlargelyreplacedtheLINPACKlibrariesasthepreferredpackageforcallingBLAS(BasicLinearAlgebraSubroutines)andsolvingsimultaneousequationsinlinearsystems.Therefore,whileMATLABisstillperfectlycapableofsolvinglargesystemsofequationsthroughloop-basedmethodologies,itisnolongertheoptimalmeansbecauseoftheinherentvectorprocessing,memoryread/writehierarchy.MathWorksrhasdevelopedanumberof“toolboxes”forvariousapplicationsincludingimageprocessing,optimization,curvettingandpartialdifferentialequationsbuthasyettodevelopanypackagesforMD,andtotheauthor'sknowledgenomajorscholarlypublicationsregardingMDsimulationsofKnudsenscalediffusionprocessesmakeuseofMATLABdespiteitsfavorableprogrammingarchitectureforthisproblem.ThisprojecttakesfulladvantageofthevectorizedarchitectureofMATLABanddemonstratestheprogram'sviabilityasanext–generationsimulationtoolformoleculardynamicists. 3.3DevelopmentofModelSystems 3.3.1IntegrationProcedureforItinerantPointMassesThepermeatetrajectoriesareinuencedbythepotentialeldthroughwhichthebodyistraveling.Inthecaseofthestaticandphonon-activemodelsforO2,thisincludesonlythesuperposed6-12LJpotentialsofthesimulatedpermeate–porepairs.ThisresultsinatotalofNtotalforcespertimestep.AssumingthatthepermeatespeciesisO2andtheporewalliscomposedofpurecarbon,theLJpotential(ULJ)isgivenbyEquation( 3 ).BoththeLJdiameter()andpotentialwelldepth(")parametersaretabulatedasspeciesspecicvalues,andtheLorenz–Berthelotmixingrulesareused 64

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toapproximatethepotentialinteractionbetweendisparatemolecularpairs.Figure 3-1 demonstratestherelationshipbetweenamixedpotential(green)andits2element-wisecomponents.WhenconsideringthefullydynamiccaseMorsebond-stretchingandharmonicbond-bendingpotentials,givenbyEquations( 3 )and( 3 )respectively,arealsoincludedbetweenlatticemoleculeswithin1and2sequentialbonds.ThevanderWaalsinteractionsbetweenthelatticemolecules(excludingthe1-2and1-3pairs)foratotalofN2total+4Ntotalforcesarealsoincluded.IftorsionalpotentialswerealsoconsideredthenthetotalwouldincreasetoN2total+52Ntotal.InthecaseofarigidH2OmoleculewithinarigidCNT,boththerotationalandtranslationmotionofthebodymustbecaptured,aswellastheinuenceoftheinducedquadrupolesbetweenthemoleculeandthecarbonlattice,necessitating5Ntotalforcesandtorquestobecomputedpertimestep.Rigidbodydynamicsarecoveredindetailinsection 3.3.2.3 . Figure3-1. Lennard–Jonespotentialenergygivenasafunctionofcenter-to-centerdisplacementofcarbon–carbonandmolecularoxygenpairs,andthecarbon–oxygeninteractionascalculatedfromtheLorenz–Berthelotmixingrules. 65

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ULJ(ri,rj)=4"CO2"CO2 krijk12)]TJ /F21 11.955 Tf 11.96 16.86 Td[(CO2 krijk6# (3)UMorse(ri,rj)=KMorse)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(e)]TJ /F17 7.97 Tf 6.59 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re)2 (3)U(ri,rj,rk)=1 2K(cosijk)]TJ /F4 11.955 Tf 11.95 0 Td[(cose)2 (3)wherekrijk=kri)]TJ /F12 11.955 Tf 11.96 0 Td[(rjk=(xi)]TJ /F5 11.955 Tf 11.96 0 Td[(xj)2+(yi)]TJ /F5 11.955 Tf 11.96 0 Td[(yj)2+(zi)]TJ /F5 11.955 Tf 11.95 0 Td[(zj)21 2andcosijk=rijrkj krijkkrkjkIneachcase,ri=[xi,yi,zi]representingtheCartesiancoordinatesoftheithelement,rij=ri)]TJ /F12 11.955 Tf 12.08 0 Td[(rjrepresentingthevectorpointingfromthejthelementtowardstheithelement,KMorseandKaretherespectiveforceconstants,reandearetheequilibriumbondlengthsandbondangles,andistheMorseexponentialcoefcientwithunitsofA)]TJ /F2 7.97 Tf 6.58 0 Td[(1.Solvingforaspeciccomponentoftheoverallforcerequiresthepotentialfunctions,givenbyEquations( 3 )through( 3 ),tobepartiallydifferentiatedwithrespecttotheappropriatecoordinateofthecenter-to-centerdistancebetweentheparticlesofinterestandthensummed.Anexampleofthex-componentoftheLJforce(FLJx)actingbetweenanO2(componenti)andcarbonatom(componentj)isprovidedinEquation( 3 )below. 66

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FLJxi(ri,rj)=)]TJ /F16 11.955 Tf 13.02 8.09 Td[(@ULJ @krijk@krijk @xi=24"CO2xi)]TJ /F5 11.955 Tf 11.96 0 Td[(xj krijk2"2CO2 krijk12)]TJ /F21 11.955 Tf 11.96 16.86 Td[(CO2 krijk6# (3)Asexpected,FLJxi=)]TJ /F5 11.955 Tf 9.3 0 Td[(FLJxj.TheevaluationoftheresultantforcesfromtheMorseandharmonicpotentialfunctions(FMandFrespectively)canbefoundinAppendix A .Withalloftheforcesnowdened,theEOMofboththelatticeandpermeateparticlesisgivenbyEquation( 3 ).Noticethattheforcesactingontheithparticleareafunctionofthepositionsofeveryotherparticleinthesystem,andthatFMi=Fi=0forthestaticwallcases.miri=3Xj=1FMi(ri,rj)+3Xj=13Xk=1Fi(ri,rj,rk)+NLJXj=1FLJi(ri,rj) (3)Deviatingfromconvention,amultivariate,fourth-orderRunge-Kutta(MVRK4)solverfor2ndorderODEsisusedtoresolvetheparticlemotioninthepresenceofaspatiallyvariantforceeld.Runge–KuttasolvershavebeendiscussedasstrongcandidatesforMDintegration[ 67 ],buttheyaregenerallyavoidedinfavoroffasterVerletorGearapproachesthatrequirefewerevaluationsoftheforceeldandareadaptabletoleap-frognumericalschemes.However,thenumericalerroraccruedinthesetechniquesmakelongdurationsimulationsimpossiblewithoutemployingarticialenergyconstraints,suchasaNose–Hovervelocityscalingthermostat.Byimprovingtheefciencyoftheforcecalculationsviavectorizedprogramming,theMVRK4solvernowbecomesaviableoption.Table 3-1 givesalistofseveralcommonnumericalschemesforMDandtheapproximateorderoferrorforeach.AllenandTidesley[ 3 ]noteacurioustrendinthehigherorderGearapproachesregardingtheapparentdependencyoftheenergyuctuationsont.TheyshowthereexistsathresholdtimestepatwhichthehigherorderGearintegratorsbecomelessaccuratethanboththelowerordermethodsandtheVerletapproach.Acommon 67

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Table3-1. BreakdownofnumericalintegratorscommontoMDsimulations.Abalanceistypicallystruckbetweenthenumberofcalculationsperformedpertimestep(theforceeldFbeingthemostcomputationallyexpensive),thenumberofvectorsstoredineachtimestep,andtheapproximateorderofaccuracyofthetechnique(O()).Notethatthe3rd-order(mostcommon)andpth-order(generalized)Nordsieck/Gearpredictor-correctormethodsareabbreviatedNGPC3andNGPCprespectively. NumericalintegratorNo.Fcalcs.pertNo.storagevectorsO()Ref. Verlet14O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(t2[ 3 ],[ 10 ]Leap-FrogVerlet15O)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(t2[ 3 ],[ 10 ]VelocityVerlet26O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(t2[ 3 ],[ 10 ]NGPC318O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(t3[ 46 ],[ 47 ]NGPCp12(p+1)O(tp)[ 3 ]MVRK446O)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(t5[ 6 ] goalamongmoleculardynamicistsistousethelargesttimesteppossibleinone'ssimulations,andthisfactmakes4th-orderandhigherGearapproachesrareintheliterature.BelowtheprocedureforsolvingtheEOMsoftheFully-DynamicModel(consistingofNtotalcarbonatomsand1permeatemolecule)isoutlined.ThesimpliedsolutionsoftheColdWallEquilibrium,Phonon-ActiveandRandomizedPolymermodelsareequivalenttothosefoundinAppendix B ,howevernowthereisonly1particleofinterest(thepermeate)andhencetheMorseandHarmonicpotentialsareomitted.Attheirmostbasiclevel,Runge–Kutta(RK)solversgiveweightedaveragesolutionapproximationssuchthathigherordererrortermscancelout.A4th-ordersolverliketheoneproposedhereisaweightedaverageof4suchapproximations.Inordertoresolvethemotion(positionandvelocitycomponents)ofNtotalparticlesweneedtodene2newvectorfunctionsforeachparticle,presentedinEquations( 3 )–( 3 ).Letirangefrom1toNtotalandindicateforwhichparticletheEOMsarebeingsolved.Furthermore,let[r]Ntotal=1[x,y,z]Ntotal=1and[_r]Ntotal=1[u,v,w]Ntotal=1,givingtheCartesiancoordinatesandvelocitycomponentsofparticles1throughNtotal. 68

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_ri=fit0,[r]Ntotal=1,[_r]Ntotal=1 (3)ri=git0,[r]Ntotal=1,[_r]Ntotal=1 (3)Becausethisisa4thordersolver,thereare4RKcoefcientstodeneforeachcomponentoftheequationsabove.Thiswillresultisatotalof24NtotalRKcoefcients,buttheformofeachisidenticalfromparticletoparticle.Theresulting24uniquecoefcientsarederivedindetailinAppendix B .Thesecoefcients(Ki,1:4,Ji,1:4)eachhaveanx,yandzcomponentandareusedtoestimatethenewCartesianpositionandvelocitycomponentsafteratimestepoft.ri(t+t)=ri(t)+t 6(Ki,1+2Ki,2+2Ki,3+Ki,4) (3)_ri(t+t)=_ri(t)+t 6(Ji,1+2Ji,2+2Ji,3+Ji,4) (3)Equations( 3 )and( 3 )canberewrittenasshowninEquations( 3 )and( 3 )byusingthesimpliedformsoftheRKcoefcientsderivedinAppendix B .RecallthatRKalgorithmsmakenumerousapproximationsandaveragethemsuchthatthehigherordererrorsintheTaylorseriesexpansionscancelout.Inthecaseofspatiallyvariant,forceelddrivenmotioninaconservativesystem,thosevariousapproximationsaremadebyevaluatingtheforcesactingonparticleiattimest,t+t=2(twice,onceundershootingandonceovershootingthetruevalue),andt+t. 69

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ri(t+t)=ri(t)+t 6(_r[1]i+2_r[2]i+2_r[3]i+_r[4]i) (3)_ri(t+t)=_ri(t)+t 6(r[1]i+2r[2]i+2r[3]i+r[4]i) (3)Thisprocedureisrepeatedasthesimulationmarchesforwardintime.ToensurethatthetrajectoriesoftheparticlesarebeingcalculatedcorrectlythetotalenergyofthesystemisevaluatedandsummedateachtimestepasshowninEquation( 3 ).ET=NtotalXi=1 mi 2_ri_ri+3Xj=1UMorse(ri,rj)+3Xj=13Xk=1U(ri,rj,rk)+NtotalXj=1,j6=iULJ(ri,rj)! (3) 3.3.2IntegrationProcedureforItinerantMolecularBodiesThesolutionproceduretotheEOMsdiscusseduptothispointhasbeenthatofapointmasssystem.Inthecaseofwater,thepotentialfunctionsandtheintegrationprocedurewillneedtobeexpandedtoaccountforpolarityandthemolecularstructurerespectively.Section 3.3.2.1 willdetailthemostprevalentwatermodelstobefoundintheliterature,Section 3.3.2.2 coversthedescriptionofrigidbodymotionthroughconstrainandquaterniondynamics,andSection 3.3.2.3 providesanoutlineoftheRKintegrationschemeofthetranslationalandrotationalEOMs.TheremainderofthederivationisprovidedinAppendix C . 3.3.2.1WatermodelsAsinsection 3.3.1 ,modelsaredistinguishedbywhatpotentialfunctionsareincludedwhendeningthesimulationspace,butinthecaseofwaterthemoleculargeometryalsoplaysavitalrole.Thetwomostprominentbasemodelsforwaterfoundintheliteraturearetheextendedsimplepointcharge(SPC/E)modelbyBerendsenetal.[ 9 ]andthe4–point–transferableintermolecularpotential(TIP4P)modelbyJorgensen 70

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etal.[ 71 ].TheSPC/Eisa3-bodymodelinwhichtheequilibriumhydrogenbondlengthsandangleare1Aand109.47,andpartialchargesof+0.4238eand)]TJ /F4 11.955 Tf 9.3 0 Td[(0.8476eareassignedtothehydrogenatomsandoxygenatomrespectively.TheTIP4Phasasimilarbasearchitecture,howeveranadditionalmasslessfourthbodyisincludedalongthebisectionofthebondangle(seeFigure 3-2B )andamodiedpartialchargeoftheoxygenisappliedthere,alteringthedistributedchargeofthemoleculeandreportedlyresultinginamoreaccuratedensityproleinliquidwatersimulations[ 71 ].TheparametersfortheTIP4PandseveralothernotablemodelsareprovidedinTable 3-2 below. Table3-2. Alistofthepertinentparametersin5prominentwatermodelsandtheircorrespondingreferences.ThistableistobeusedinconcertwithFigures 3-2A – 3-2C toprovideafulldescriptionofeachmonomerarchitecture.Thesubscripted“O”,“H”and“M”denotetheoxygen,hydrogen,andthecharged,masslessdummyconstituentsrespectively. SPCSPC/ETIP3PTIP4PTIP5P OO(A)3.1663.1663.150613.153653.12"OO(kJ=mol)0.6500.6500.63640.648520.66944krOHk(A)1.01.00.95720.95720.9572HOH(deg)109.47109.47104.52104.52104.52qH(e)+0.41+0.4238+0.417+0.52+0.241qO(e)-0.82-0.8476-0.834––krOMk(A)–––0.150.70qM(e)–––-1.04-0.241MOM(deg)––––109.47Ref.[ 2 ][ 9 ][ 71 ][ 71 ][ 87 ] Clearlyeachnewarchitecture(eachhavingitsownuniquesetofLJandpartialchargeparameters)willaffectthemassandforcedistributionsonthebodyandthuswillaltertherotationalandtranslationalmotion.Additionally,theinclusionofintramolecularforceswillleadtodynamicmodelssuchastheexibleTIP4PofLawrenceandSkinner 71

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A B CFigure3-2. Agraphicallookatthebase3–,4–,and5–bodywatermodelsusedinMD.ThepartialchargesandLJparametersforeachconstituentareprovidedinTable 3-2 .Figures 3-2A , 3-2B and 3-2C showtheSPC/E,TIP4PandTIP5Pmodelsrespectively.Figure 3-2C isareplicationofthemolecularrepresentationgivenbyMahoney&Jorgensen[ 87 ]. [ 84 ]ortheSPCexbyAmiraetal.[ 4 ],amongnumerousothers.Foracomprehensivesurveyofwatermodelsdevelopedoverthelast30years,thereviewbyGuillot[ 53 ]isrecommended. 72

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Ina2008publicationAlexiadisandKassinos[ 2 ]performedacomparativestudyofwaterdensityprolesinbothrigidandfullydynamicCNTsusingbothrigidandexibleSPC,SPC/EandTIP3Pwatermodels,concludingthatexibilityofthecarbonlatticeandthatoftheH–O–Hbondangledonotsignicantlyalterthecalculateddensities,saveinthecaseofverynarrowCNTs(8achiralcasedemonstratedgreaterthanaverageexingoftheH–O–Hbondangle).Beingthatoneoftheobjectivesofthisdissertationistodevelopconvergedradialnumberdensityprolesforthediffusingspecies,andconsideringthenotedincreaseincomputationtimeandminimalcarbonatomdisplacementfrommyfullydynamicCNTsimulationperformedaspartofthisdissertation(discussedingreaterdetailinChapter 4 ),theremainderofthisworkwillfocusonthedynamicsofrigidwaterandCNTmodels. 3.3.2.2Constraintsvs.quaternionsinrigidbodydynamicsWhenconsideringrigidbodydynamics,moleculardynamicistshavetwochoiceswhenattemptingtosolvetheEOMs:(1)themethodofconstraints,or(2)concurrentlyresolvingthetranslationofandtherotationaboutthebodycenterofmass(COM).Inconstraintdynamicsonesolvesfortheinternalbondstressesnecessarytoensurethatthebondlengthbetween2constituentsremainsconstant,andthisisapowerfultechniqueifone'sgoalwastoensureonlypartialrigidity,aprominentexamplebeingifthegoalweretodevelopawatermodelinwhichthehydrogenbondlengthswereheldxed,butthebondanglewasallowedtochange.ThesecondoptionrequirestheCOMpositionandvelocityvectorsaswellastheorientation(Euleranglesorquaternions)andtherotationratetobecalculatedateachtimestep.Asaruleofthumb,AllenandTildesley[ 3 ]suggeststhat,withtheexceptionofdiatomicmolecules,low–orderleap–frogintegrationofthequaternionequationsofmotionprovidesthemostadvantageouscombinationofaccuracyandcomputationalefciencyforrigidbodies.However,leap–frogschemesrequiretheadditionaluseofacoupledthermalbathorgeneralizedthermostattoomaintainaconstantenergylevelandhenceareill-suited 73

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forthisanalysis.Acomparativestudyofvariousintegrationschemes,includingtheleap–frogVerletapproach,canbefoundinSection 3.3.4 .Forthisanalysisthegoalistochoosetheapproachwhichisbestsuitedtoahigher–orderintegrator,specicallythe4thorderMVRKschemeproposedintheprevioussection.WhendevelopingtheEOMsforarigidwatermolecule3bondlengthconstraintsand3relativevelocityconstraintsmustbeenforcedaccordingtoEquations( 3 )and( 3 ),resultingin6unknownLagrangemultipliersthatmustbesolvedforiteratively.Classically,theseequationsaresolvedusingtheSHAKE[ 109 ](Verlet)orRATTLE[ 5 ](VelocityVerlet)algorithmsdevelopedbyRyckaertetal.andAndersenrespectively.However,inthecaseofhigher–orderintegration,thesealgorithmswouldneedtoberunmultipletimes,4timesinthecaseofthe4thorderMVRKschemeproposedhere,pertimestep.Furthermore,becauseoftheenforcedlineardistancesbetweenbondedatomscalculatingsquarerootsisunavoidable.Allofthesefactorsservetodriveupthecomputationalexpenseoftheconstraintapproach.Contrastingly,inthecaseof4thorderintegrationoftherotationalEOMs,thepositionsandvelocitiesoftheindividualparticlesneednotbetracked,butonlythepositionandvelocityofthemoleculeCOM,theorientationalquaternions,andthebody–centeredangularvelocity.Whilethisapproachdoesrequiretheuseofmatrixarithmetic,thedecreaseinthenumberofstoredvariables,theeliminationoftheneedforanyiterativesolutionprocedure,andthefacttheMATLABisspecicallydesignedtohandlesuchproblemsefcientlymakesthistheclearchoice.Section 3.3.2.3 detailstheintegrationprocedure.Readersarereferredto[ 3 ]and[ 10 ]foramorecomprehensivereviewofconstraints.krijk2=d2ijfori,j=f1:3g&i6=j (3) 74

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k_rijkkrijk=0fori,j=f1:3g&i6=j (3) 3.3.2.3Fourth-orderquaternionintegrationPerhapsthemostfamousandintuitivemeanswithwhichtodescribethe3dimensionalorientationofanobjectisthroughtheuseofEulerangles,whichdene3successiverotationsthatalignboththespace–xedandbody–xedframesofreference.However,thereisopportunityforinversionoftherotationmatricestodivergeduetothelossofadegreeoffreedominaphenomenoncalledgimballock.InatypicalMDsimulationconsistingofhundredsofthousandsifnotmillionsoftimesteps,andsubsequentlyjustasmanymatrixinversions,thepossibilityofgimballockoccurringandcrashingthesimulationistoogreat.Therefore,quaternionshavebecomeafarmoreattractiveandwidelyusedconvention.Quaternions,representedhereas[q],arehypercomplexvariablesthatinessenceaddaredundancytotheEulerangleapproach,over-specifyingthesystemtoensuredivergence–freerotationsatthepriceofaddingoneextraparameter.Quaternionsarerelatedtoastandardrotationmatrix( R)throughsuccessiveGrassmannproducts()ofthequaternionanditsconjugateasshowninEquation( 3 ).[q]x[q]= R([q])x=2666664q20+q21)]TJ /F6 11.955 Tf 11.95 0 Td[(q22)]TJ /F6 11.955 Tf 11.95 0 Td[(q232(q1q2+q0q3)2(q1q3)]TJ /F6 11.955 Tf 11.96 0 Td[(q0q2)2(q1q2)]TJ /F6 11.955 Tf 11.95 0 Td[(q0q3)q20)]TJ /F6 11.955 Tf 11.96 0 Td[(q21+q22)]TJ /F6 11.955 Tf 11.95 0 Td[(q232(q2q3+q0q1)2(q3q1+q0q2)2(q3q2)]TJ /F6 11.955 Tf 11.96 0 Td[(q0q1)q20)]TJ /F6 11.955 Tf 11.96 0 Td[(q21)]TJ /F6 11.955 Tf 11.96 0 Td[(q22+q233777775x (3)where 75

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[q]=q0+q1i+q2j+q3k=26666666664q0q1q2q337777777775[q]=q0)]TJ /F6 11.955 Tf 11.95 0 Td[(q1i)]TJ /F6 11.955 Tf 11.95 0 Td[(q2j)]TJ /F6 11.955 Tf 11.95 0 Td[(q3k=26666666664q0)]TJ /F6 11.955 Tf 9.3 0 Td[(q1)]TJ /F6 11.955 Tf 9.3 0 Td[(q2)]TJ /F6 11.955 Tf 9.3 0 Td[(q337777777775andxisanarbitraryvectorsubjecttotherotation.TheGrassmannproductwillgureprominentlyinthefollowinganalysisandisprovidedbelowforeasyreference.Equation( 3 )givestheGrassmannproductforaquaternionpair,andEquation( 3 )isthatofaquaternion–vectorpair.Additionally,theidentitiesgiveninEquations( 3 )and( 3 )willbeinvoked.[a][b]=264a0A375264b0B375=264a0b0)]TJ /F12 11.955 Tf 11.96 0 Td[(ABa0B+b0A+AB375 (3)A[b]=2640A375264b0B375=264)]TJ /F12 11.955 Tf 9.3 0 Td[(ABb0A+AB375 (3)q20+q21+q22+q23=1 (3) 76

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d dt)]TJ /F6 11.955 Tf 5.48 -9.69 Td[(q20+q21+q22+q23=2(q0_q0+q1_q1+q2_q2+q3_q3)=0 (3)Toreiterate,inordertofullydenethemotionofarigidbodywemustresolvethetranslationofthebody'sCOMinaglobalframeofreference(FOR),andtherotationofthebodyaboutitsownCOMinabody-xed(BF)FOR,andthegoalistosolvethefollowingequationsusinga4thorderRKintegrator:Mrg=F (3)I0_!0+!0(I0!0)=0 (3)whereMisthetotalmassofthemolecule,rgisthelinearaccelerationofthemoleculeCOM,Fistototalexternalforceactingonthebody,Iisthemomentofinertiatensor,isthetorqueaboutthebodyCOM,and!istheangularvelocity,withthevaluesdenedintheBFFORgiveninprimednotation.BeforetheRKcoefcientscanbederived!0and_!0mustberelatedtothequaternionsandtheirderivativesthroughEquations( 3 )and( 3 ).[_q]=1 2[q]!0=1 226666666664q0)]TJ /F6 11.955 Tf 9.3 0 Td[(q1)]TJ /F6 11.955 Tf 9.3 0 Td[(q2)]TJ /F6 11.955 Tf 9.3 0 Td[(q3q1q0)]TJ /F6 11.955 Tf 9.3 0 Td[(q3q2q2q3q0)]TJ /F6 11.955 Tf 9.3 0 Td[(q1q3)]TJ /F6 11.955 Tf 9.3 0 Td[(q2q1q0377777777758>>>>>>>>><>>>>>>>>>:0!0x!0y!0z9>>>>>>>>>=>>>>>>>>>;=1 2 Q([q])2640!0375 (3) 77

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[q]=1 2([_q]!0+[q]_!0)=1 20B@ Q([_q])2640!0375+ Q([q])2640_!03751CA (3)FromEquation( 3 )thefollowingexpressionfor!0isdeveloped:2640!0375=2[q][_q]=226666666664q0q1q2q3)]TJ /F6 11.955 Tf 9.29 0 Td[(q1q0q3)]TJ /F6 11.955 Tf 9.3 0 Td[(q2)]TJ /F6 11.955 Tf 9.29 0 Td[(q2)]TJ /F6 11.955 Tf 9.29 0 Td[(q3q0q1)]TJ /F6 11.955 Tf 9.29 0 Td[(q3q2)]TJ /F6 11.955 Tf 9.3 0 Td[(q1q0377777777758>>>>>>>>><>>>>>>>>>:_q0_q1_q2_q39>>>>>>>>>=>>>>>>>>>;=2 Q>([q])26666666664_q0_q1_q2_q337777777775 (3)andEquation( 3 )canberearrangedasshowninEquation( 3 )soastoisolatetheBFangularaccelerationonthelefthandside.Thenbyinvokingthe!0relationshipgiveninEquation( 3 )therotationalequationsofmotioncanbefullyspeciedbythequaterniondynamics._!0=I0)]TJ /F2 7.97 Tf 10.08 0 Td[(1[0)]TJ /F18 11.955 Tf 11.96 0 Td[(!0(I0!0)] (3)TheentireexpressioncanbefurthersimpliedbyrecognizingthatintheBFFORIisconstantandisgivenbyEquation( 3 ),wherer0i=r0i)]TJ /F12 11.955 Tf 13.22 0 Td[(r0g= R>(ri)]TJ /F12 11.955 Tf 13.21 0 Td[(rg)denesthevectorbetweenatomiandtheCOMintheBFframe,and[x0i,y0i,z0i]givesthecorrespondingcoordinatesintherotatedFOR.IfthisexpressionisdiagonalizedasshowninEquation( 3 )thenexpressionfromEquation( 3 )collapsestogiveEquation( 3 ). 78

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I0=3Xi=1mi2666664(y0i)2+(z0i)2)]TJ /F5 11.955 Tf 9.3 0 Td[(x0iy0i)]TJ /F5 11.955 Tf 9.3 0 Td[(x0iz0i)]TJ /F5 11.955 Tf 9.3 0 Td[(y0ix0i(x0i)2+(z0i)2)]TJ /F5 11.955 Tf 9.3 0 Td[(y0iz0i)]TJ /F5 11.955 Tf 9.29 0 Td[(z0ix0i)]TJ /F5 11.955 Tf 9.3 0 Td[(z0iy0i(x0i)2+(y0i)23777775 (3)I0D=2666664I0D,x000I0D,y000I0D,z3777775 (3)_!0=26666640x I0D,x+I0D,y)]TJ /F10 7.97 Tf 6.59 0 Td[(I0D,z I0D,x!0y!0z0y I0D,y+I0D,z)]TJ /F10 7.97 Tf 6.58 0 Td[(I0D,x I0D,y!0z!0x0z I0D,z+I0D,x)]TJ /F10 7.97 Tf 6.59 0 Td[(I0D,y I0D,z!0x!0y3777775 (3)FollowingthesameprocedurediscussedinSection 3.3.1 ,fourgeneric,timedependentfunctionsaredeveloped,oneforeachoftherstderivativesoftheindependentvariables(rg,_rg,[q],[_q]),andthecorrespondingRKcoefcientsaredened.AcompletederivationofthecoefcientsisfoundinAppendix C ._rg(t)=f1(t,rg,_rg,[q],[_q])=f1(t,_rg) (3)rg(t)=f2(t,rg,_rg,[q],[_q])=f2(t,rg,[q]) (3) 79

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[_q](t)=f3(t,rg,_rg,[q],[_q])=f3(t,[_q]) (3)[q](t)=f4(t,rg,_rg,[q],[_q])=f4(t,rg,[q],[_q]) (3)rg(t+t)=rg(t)+t 6(K1+2K2+2K3+K4) (3)_rg(t+t)=_rg(t)+t 6(J1+2J2+2J3+J4) (3)[q](t+t)=[q](t)+t 6(L1+2L2+2L3+L4) (3)[_q](t+t)=[_q](t)+t 6(M1+2M2+2M3+M4) (3)Itisassumedri(t),_ri(t),rg(t),_rg(t),[q](t),and[_q](t)areallknownquantitiesandtheseinitialvalueswillbereferencedwithoutthe(t)designationfortheremainderofthisderivation.Todeterminetherst–ordercoefcientsthetotalexternalforceonthemoleculeintheglobalFORandthetotaltorqueintheBFFORmustbecalculated.Asbefore,withinthespatiallyvariantpotentialeldoftheCNTinterior,thetotalforceactingonafreebodyisafunctionoftherelativedistancesbetweensaidfreebodyandallofthecarbonatomsinthelattice.Thatmeansthepositionofeachconstituentof 80

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theitinerantmolecularbodymustbegivenintheglobalFOR(ri),otherwisedenedbyrgand[q].Thecarbon–waterinteractionincludesbothaLJpotentialbetweentheoxygenandthecarbon,aswellasaninducedquadrupolepotentialbetweencarbonatomsandthepartialchargesofthewaterconstituents.ThisisthesameapproachtakenbyWaltheretal.[ 131 ].AslightlysimpliedpotentialwasusedbyHummer,Rasaiah,andNoworyta[ 63 ]inwhichthequadrupolepotentials,notedforhavingonlyaminorinuenceontheadsorptionofoxygenongraphite[ 22 ],areneglected.Inthisanalysisbothhavebeensimulated.Thereaderisreferredto[ 131 ]foramoredetaileddiscussionofthequadrupolepotentialfunction.Thereforetheexternalforcesactingontheindividualatomsinthemolecule,thecorrespondingtotaltorque,andtheangularvelocityintheBFframeofthemoleculearegivenbyEquations( 3 )through( 3 ).TheangularaccelerationintheBFframecanbefoundbyapplyingthe01and!01valuesderivedinEquations( 3 )and( 3 )toEquation( 3 ).Fi,1=Fi(rg,[q])fori=1,2,3 (3)01=3Xi=1r0i8: R1Fi,19; (3)2640!01375=2 Q>([q])26666666664_q0_q1_q2_q337777777775 (3)where 81

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R1= R([q])and Q1= Q([q])andthesubscripted”correspondstothevariablespresentinthecalculationoftherstsetofRKcoefcients,whicharegivenbelow.K1=f1(t,_rg)=_rg (3)J1=f2(t,rg,[q])=1 M3Xi=1Fi,1=FT,1 M (3)L1=f3(t,[_q])=[_q] (3)M1=f4(t,rg,[q],[_q])=1 20B@ _Q12640!01375+ Q12640_!013751CA=[q]1 (3)where _Q1= Q([_q])Thesubsequentcoefcientsareevaluatedfromtheformer.The2ndsetofRKcoefcientsareprovidedbelowbywayofexample.ThefullderivationisavailableinAppendix C . 82

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K2=f1t+t 2,_rg+t 2J1=_rg,2 (3)J2=f2t+t 2,rg+t 2K1,[q]+t 2L1=FT,2 M (3)L2=f3t+t 2,[_q]+t 2M1=[_q]2 (3)M2=f4t+t 2,rg+t 2K1,[q]+t 2L1,[_q]+t 2M1=[q]2 (3)FollowingthisprocedurethroughthenumericalapproximationtotheintegrationoftherotationalEOMsyieldEquations( 3 )–( 3 ).Thisintegrationschemeisimplementedinawhileloopcodestructure,marchingforwardintimeuntiltheradialnumberdensityandaxialcollisiondistancedistributionshaveconvergedviatheconditionsdiscussedinthenextsection.rg(t+t)=rg(t)+t 6(_rg+2_rg,2+2_rg,3+_rg,4) (3)_rg(t+t)=_rg(t)+t 6M(FT,1+2FT,2+2FT,3+FT,4) (3)[q](t+t)=[q](t)+t 6([_q]+2[_q]2+2[_q]3+[_q]4) (3) 83

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[_q](t+t)=[_q](t)+t 6([q]1+2[q]2+2[q]3+[q]4) (3) 3.3.3ConvergenceCriteriaTheparticleresidencetimeinannuliofvaryinginnerandouterdiametersarealsorecorded,asistheaverageaxialvelocityatsaidradialintervals.Todeterminetheappropriatenumberofcollisionsneededtoaccuratelycharacterizemolecularmotionwithintheporethesedistributionsareconstantlyupdateduntilchangefromonetothenextiswithinadesiredtolerance.Furtherchecksarealsoconsidered,onebeingthattheradialnumberdensitydistributionshouldhaveitspeakcenteredonthezero-forceradiusoftheadsorptioneldandtheaveragez-velocity,hvzi,shouldbe0.Itisworthnotingherethatinthecaseofmoleculartransporttheorientationoftheitinerantbodyhasanon-negligibleeffectontheCNTpotentialeldleadingtothepresenceofazero–forceradialenvelopeasopposedtoasingleradiallocation.Figures 3-3 and 3-4 demonstratetheevolutionandconvergenceoftheradialnumberdensitydistributionof50CO2(Kn=77)diffusingdownan(8,8)SWCNTovera20ns,50kwallcollisionsimulation.Similarly,Figures 3-5 and 3-6 demonstratetheevolutionandconvergenceoftheaxialcollisiondistancedistributionforthesamesystem.FurtherresultsfromthissimulationandothersarelocatedinChapter 4 . 3.3.4ComparativeStudyofIntegratorOrdersofAccuracyTheorderofaccuracyofseveralcommonnumericalintegratorswerebrieydiscussedinSection 3.3.1 andsummarizedinTable 3-1 .Thissubsectionpresentsanextensionofthatsameinformationingraphicalformtofurtherestablishahigher–orderRKintegratorasnotjustaviableoptioninMD,butasasuperiorapproach.InthefollowinganalysisaLJO2particleinaninnite(10,10)SWCNTissimulatedover10ps.TheEOMsareevaluatedusingthevelocityVerlet(VV),3rdorderGearpredictor–corrector(GPC),and4thorder,multivariateRKmethods.Nothermostats 84

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Figure3-3. Evolutionoftheradialnumberdensityprolefrom5kwallcollisions(A)through50kwallcollisions(F)foranO2moleculeinan(8,8)SWCNTat50C.Foreachprole(bluediamonds)thex-axisgivestheradialpositionwhilethey-axisshowsthetimefractionaparticularradialbinispopulated.Thedottedredlineshowstheradialpositionofthezero–forceannulus. 85

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Figure3-4. ThissetofplotsshowsalinearregressionanalysisofplotsA–EinFigure 3-3 withthe50koscillationcase.Thesquaredresidualforeachcaseisprovidedand99%convergenceisachievedwithin25kwallcollisions. 86

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Figure3-5. Evolutionoftheaxialcollisiondistancedistributionfrom5kwallcollisions(A)through50kwallcollisions(F)foranO2moleculeinan(8,8)SWCNTat50C.Foreachhistogramthex-axisgivestheaxialdistancetraveledbetweenwallcollisionswhilethey-axisshowsthefractionofcollisionsthattraveledaparticularaxialdistance. 87

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Figure3-6. ThissetofplotsshowsalinearregressionanalysisofplotsA–EinFigure 3-5 withthe50koscillationcase.Thesquaredresidualforeachcaseisprovidedand95%convergenceisachievedwithin31kwallcollisions. 88

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orotherarticialenergyconservationtechniquesareused.Thelengthofthetimestepissetatagivenvalue(0.1,0.5,1,and2fs)andthetotalenergyiscalculatedaftereachtimestep.TheenergycurvesarethengraphedandtheRMSerroriscalculatedagainsttheinitialvalue.Theaverageper–timestepcomputationtimesarealsoreported.IneachcasetheGearmethodisthefastest,roughlytwicethespeedoftheRKmethod.However,theRKapproachisconsistently4to5ordersofmagnitudemoreaccuratefromanRMSerrorperspective,andthatnumberwillonlygetlargerasthesimulationtimesgoup.Furthermore,theaverageper–timestepcomputationtimesreportedheredonotincludeathermostatingcalculationstepwhichwillincreasethevaluesforboththeVVandGPCmethods.Thestep–likeproleoftheVVenergycurveisattributedtotheinabilityofa2ndordernumericalintegratortoaccuratelytraceahighlystifffunction,asthestepsinenergyoccurwhiletheO2trajectoryiswithinthehighlyrepulsive(krk12)regiononthepotentialeld.SimilarbehaviorisobservedwiththeGPCmethod,thoughtheenergytendstospikeatthecollisionpointofthetrajectorythenreturntoavaluemuchclosertoinitialenergystatethanisachievedthroughtheVVapproach.IndeedthesamecanbeseenwiththeRKapproach,thoughitisordersofmagnitudemoreaccuratethaneventheGPCmanagesforanequivalenttimestep.Itisworthnotingherethattimestepsof2fsarethemostcommon,anditisclearfromthisanalysisthatsuchtvalueswouldbeimpossibletoimplementinaconservativesimulationwithoutsomemeansofregulatingtheenergyifoneofthemoretraditionalintegrationschemesweretobeused.TheprogressionofFigures 3-7 through 3-10 ,steppingupfrom0.1fsto2fstimestepsrespectively,demonstratesthenumericalerrordependenceonboththetimestepandthetruncationorderoftheintegratorhintedatinTable 3-1 .TheargumentcanbemadethaterrorstemmingfromthenumericalintegrationoftheEOMsisoflittleimportwhenweighedagainsttheeffectsofanimperfectandincompletedescriptionofthepotentialeld.Byexample,the6LJpotentialisnotapopularMDconvention 89

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Figure3-7. Abovearethree10pssimulationsof50CO2diffusingthroughastaticSWCNT,eachwithatimestepof0.1fs,andeachusingadifferentintegrationschemetoresolvetheEOMs.Theexactsolutionwouldresultisperfectenergyconservationat4028.30J/mol,howeverstep-likedriftinthetotalenergycanbeseeninboththeVV(bluedotted)andtheGPC(greendashed)schemes.TheaccruedRMSerrorfortheRKschemeiseightordersofmagnitudelessthanthatoftheGPCmethod,withtwicethecostincomputationaltime.TheinlaidgraphshowsazoomedinviewoftheRKandGPCcurves,they-axisspanningfrom4027.95to4028.40J/mol. becauseofitsaccuracy(particularlyintheneareld),butbecauseitcanbeevaluatedwithoutcalculatingthreedimensionallineardistances,meaningcomputationally 90

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Figure3-8. Abovearethree10pssimulationsof50CO2diffusingthroughastaticSWCNT,eachwithatimestepof0.5fs,andeachusingadifferentintegrationschemetoresolvetheEOMs.TheaccruedRMSerrorfortheRKschemeissevenordersofmagnitudelessthanthatoftheGPCmethod,withtwicethecostincomputationaltime.TheinlaidgraphshowsazoomedinviewoftheRKandGPCcurves,they-axisspanningfrom4020to4029J/mol.Theinitialenergyis4028.30J/mol. expensivesquarerootscanbeavoided.Thatsaid,thisanalysisshowsthattheerrorcontributionfromthenumericsisfarfromnegligible.Furthermore,inthisanalysisthemoleculartrajectoriesareofparamountimportance,butthermostatsandbarostatsbothenforceenergyconservationbyalteringthecalculatedightpaths,makingthem 91

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Figure3-9. Abovearethree10pssimulationsof50CO2diffusingthroughastaticSWCNT,eachwithatimestepof1fsandadifferentnumericalintegrator.TheaccruedRMSerrorfortheRKschemeisveordersofmagnitudelessthanthatoftheGPCmethod,withtwicethecostincomputationtime.TheinlaidgraphshowsazoomedinviewoftheRKandGPCcurves,they-axisspanningfrom3970to4030J/mol.Theinitialenergyis4028.30J/mol. inadequateoptions.Forthiswork,theconstraint-freeRKapproachisthenaturalchoiceandisreadilyadaptabletomoreaccuratepotentialfunctions(suchastheBuckingham[ 10 ]or10carbon[ 22 ]potentials). 92

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Figure3-10. Abovearethree10pssimulationsof50CO2diffusingthroughastaticSWCNT,eachwithatimestepof2fsandadifferentnumericalintegrator.TheaccruedRMSerrorfortheRKschemeisfourordersofmagnitudelessthanthatoftheGPCmethod,withtwicethecostincomputationaltime.TheinlaidgraphshowsazoomedinviewoftheRKandGPCcurves,they-axisspanningfrom2800to4200J/mol.Theinitialenergyis4028.30J/mol. 93

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CHAPTER4RESULTSANDDISCUSSIONChapter 4 detailsthedevelopmentofthisproject,beginningwiththefullyindependent,inniteKnudsennumbersimulationattemptsandconcludingwiththeinducedquadrupoleH2O–CNTmodel,anddiscussestheresultsgleanedfromeach.Section 4.1 corroboratestheassertionthatfree–bodyloadingofthesimulationspaceisofcriticalimportancetotheequilibriumbehaviorofthesystem,asthepresenceofsteeppotentialwellstendtocapturemoleculesinaberrant,periodicorbits.Sections 4.2.1 and 4.2.2 discusstheFiniteKnudsenmodelsimulationresultsandcomparesthepredicteduxwithmeasurementsfoundintheliterature.Section 4.2.3 detailstheconvergenceanalysisoftherelevanttransportcharacteristicsandChapter 4 concludeswithanexaminationofprominentwatermodelsconsideredforsimulatedwatervaportransportinSection 4.2.4 . 4.1SimulationsintheInniteKnudsenNumberRegimeTherstsimulationsattemptedwerethoseofasingleO2LJparticlecareeningthroughastatic,defect–freeSWCNT.ThehypothesiswassimilartothatoftheOM[ 11 ],thatinthelimitofverylargeKnudsennumbersalargesamplingoftrajectoriesofasingleparticlewillberepresentativeoftheentireensemble.ThekeydifferencebetweentheapproachtakenhereandintheOMisthattheOMassumedadistributedLJpotentialforthewallasopposedtothatofanatomisticsummedpotential,andthisallowedforadecouplingoftheaxialandradialmotionofthepermeate.Assuch,2Dtrajectoriescouldbeusedtoevaluatethe“hoppingtimes”,buttheeffectsofthenear–eldonthereectionanglesarelost.InsteadtheOMdrawsinitialconditionsfromacanonicaldistributionbetweenwallcollisions.TheinniteKnudsennumbersimulationsreportedhereaimedtoresolvethefull3Dmotionthrough50ksuccessivewallcollisionswithoutassumingareectionconditiontoseeifarecognizablepatternintheradialnumberdensityandaxialtrajectorywouldemerge.However,asnotedbyKargeretal.[ 74 ],thelackofadditionalitinerantparticlestoperturbthestaticlattice 94

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potentialeldresultedinarticial“trapping”ofthefreebody.ThistrappingphenomenoncanbeseeninFigure 4-1 ,whereat15nstheradialoscillationprolecollapsesandtheparticleappearstofallintoaperiodic,oscillatorypattern.Figure 4-2 providesadown–axisviewofthetrajectoriestobetterobservethetendencyofthebodytofallintotheorbitalsinkcreatedbythesymmetryinherentinthesimulationspace,andFigure 4-3 showstheaxialmotionofthebodyoverthesametimeframe.Lookingtoovercomethechallengesposedbythecapturedpermeateparticlesthreeinitialattemptsweremadetointroducebothtemporalandspatialvarianceintheporepotentialeld:(1)modelingtheprominentphononmodesofthelatticeindependentofthepermeate,(2)dynamicallymodelingallofthecarbonbondsandstericinteractionsinthelatticesubjecttothepermeate–porepotential,and(3)representingtheporeasarigidpolymerwithmultipleconstituentsandaroughwall(meaningtheporeradiusisnon–constant).Fortherstcase,referredtohereasthePhonon–Activemodel,thewalldynamicsaremodeledusingamixedchiralityandtemperaturedependentradialbreathingmodeandachiralitydependenthighenergymode(HEM)asdescribedbyMohretal.[ 94 ].Thephononamplitudesandfrequenciesaretakenfrom[ 117 ]and[ 94 ].ItisassumedthatthecharacteristicdynamicsoftheSWCNTlatticewillnotbegreatlyaffectedbythepresenceofasinglefreeparticle,andsothemotionofthewallatomsisassumedindependentofthepermeatetrajectory.However,thisresultsinthewalldoingworkonthediffusatewhileignoringthereciprocalworkthatthediffusatewoulddoonthewall,yieldingnon–conservativeresults.Casetwowasdevelopedmoreasacheckastotheeffectofwalldynamicsondiffusionratherthanalegitimatedesigntool,asthenumberofcalculationsrequiredpertimestepmakesittoocomputationallyexpensivetogeneratethekindofsimulationlibraryintendedaspartofthisresearch.TheresultsgiveninFigure 4-4 arefroma25pssimulationofanO2moleculeinsidea(10,10)SWCNTat50Cwhosebondstretching, 95

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Figure4-1. ThreeviewsoftheradialoscillationsofaO2moleculeinitializedat50Cthermalvelocityinsideastatic(15,15)SWCNT,eachoveranarrowerwindowoftime.Thetopgureshowsall50koscillations,themiddlegureshowsthenal280collisionsandthebottomgureshowsthenal47.ThegreendashedlineshowstheradialpositionatwhichFLJr=0. 96

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A B CFigure4-2. Inalloftheplotsabovethesolidbluelinetracestheparticletrajectory,thedashedblackcurveistheFr=0annulus,andtheredcirclesarethecarbonLJsitesoftheSWCNT.Figure 4-2A showsthetrajectoriesfromto5000thtothe5050thwallcollision.Likewise,Figures 4-2B and 4-2C showwallcollisions10000-10050and20000-20050respectively. bondbendingandstericLJinteractionsareallcalculatedasdiscussedinsection 3.3.1 andAppendix A .Whatresultsisanimmediatedampingoftheradialoscillationsofthepermeate,aneffectnotedbybothTuzunetal.[ 128 ]andWaltheretal.[ 131 ],as 97

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Figure4-3. ThreeviewsoftheaxialmotionofaO2moleculeinitializedat50Cthermalvelocityinsideastatic(15,15)SWCNT,eachoveranarrowerwindowoftime.Thetopgureshowsall50koscillations,themiddlegureshowsthenal1200collisionsandthebottomgureshowsthenal280.ThegreendashedlineshowstheaxialpositionatwhichFLJz=0. 98

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thekineticenergyofthefreemoleculeistransferredintothelattice.Asthesimulationprogressesthelatticeisobservedtodeformslightly,creatingapockettoaccommodatethepermeateandtrapit.Asincaseone,withoutanoutsideforcefromanotherdiffusingmoleculetoupsettheforcebalancetheoriginalpermeatemoleculewillremaintrappedandaxiallystationaryfortheremainderofthesimulation.Casethreeexploredtheeffectofunevendistributionsofwallatoms,thickwalls(asopposedtotheatomicallythinwallsofSWCNTs),multiplewallconstituents,andnon-constantporediametersonsinglemoleculetransport.Asinthepreviouscases,thegoalwastocreateaporestructurewithinherentirregularityintheatomicconguration,therebybreakingtheaxialsymmetryoftheforceeldresponsibleforcapturingthepermeate.However,duetotherigidityofthepolymerasteep,stationarypotentialwellstillexistswithinthesystem,andlikebeforethefreemoleculeeventuallyfallsintoit.TheresultsobtainedfromthethreeSWCNTandtherigidpolymerporemodelsobviatethefactthatEMDsimulationsofasinglemoleculeinsideasemi-innite,repeatingstructureareinsufcienttodetermineKnudsenscalediffusionrates,asthesystemovertimegravitatestowardsanarticialaxiallystationarysteadystate.Thisleavestwokeyavenuestoexplore:(1)seedthetubewithalargenumberoffreemoleculeswhilemaintainingalargeKnudsennumberor(2)periodicallysimulateintermolecularcollisionsbaseduponthesystemKnudsennumber.ThelatterofthesetwooptionsledtothedevelopmentoftheFiniteKnudsenModel. 4.2RariedMolecularTransportthroughSingle–WalledCarbonNanotubesTheFiniteKnudsenModel(FKM)calculatestheightpathofabodyasittravelsthroughsomeprescribedpotentialeld.Simulationsofasingleitinerantbodydonotlendthemselveswelltotheconceptoftemperature,astemperatureisastatisticalpropertyarisingfromalargeensembleofparticles.Therefore,theFKMcanbeconsideredpartofthemicrocanonicalNVEensemble,meaningthenumberofparticles,thesystemvolume,andthetotalenergyofthesystemareconserved. 99

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Figure4-4. IllustrationofanO2moleculeinitializedat50Cthermalvelocityinsideafullydynamic(10,10)SWCNT.ThebluecircleistheO2moleculeposition12.5psintothesimulation,thebluedottedlineistheightpathfromt=0tot=12.5ps,andtheredcirclesarethepositionsofthecarbonatomsatt=12.5ps.Herethenormalcircularcross-sectionoftheSWCNTisskewedtobalanceouttheadditionalforceonthelatticefromtheO2molecule. Whensamplingalargenumberofindividualparticlesfromasingle–speciescontinuumataxedtemperature,thethermalenergydistributionwillfollowtheclassicalMaxwell–Boltzmanndistribution.Itissimilarlyexpectedthat,foraconned,molecularow,anon–uniformdistributionwillgoverntheenergy.Onlybyintegratingoverthe 100

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entireenergyspectrumwillthetotaluxbecalculable,asthediffusioncoefcientisproportionaltothesquarerootoftemperatureintraditionalKnudsenanalysis.ForthisdissertationtheMaxwell–Boltzmanndistributionisused,withtheaveragetemperaturedenedasthesystemoperatingtemperature,todescribetheenergeticsofthepermeant.Totheauthor'sknowledgethereisnotheoreticallyderived,broadlyacceptedenergydistributionforsub–continuumgasdynamics,however,theprocessofintegrationofthediffusioncoefcientrelationisfullygeneralizabletobewithrespecttoanyenergydistribution.ThesimulationscarriedoutbytheFKMareconservative,andthestatisticsgeneratedfromanindividualrunwilldeneaspeciesandtemperaturespecicdiffusioncoefcient.Inadditionalsimulations,withallotherparametersheldxed,changesintemperaturewilldenethediffusioncoefcientcurve.TheKnudsenderivedE/p Tisusedtodeveloppowerlawcurvesofbesttforourinitialanalysis.FurtherdiscussiononimprovedenergydistributionsandenergyproportionalitiesarefoundinChapter 5 .BesidestheuseoftheMaxwell–Boltzmanndistribution,theidealgas,continuumdenitionofthemeanfreepathisused.Foranidealgasofhard,sphericalbodiesthemeanfreepathcanbegivenas:=1 p 2d2n=KnD (4)wheredistheparticlediameter,Distheporediameter,andnisthepermeatenumberdensity[ 130 ].ForFKMsimulations,isttoanormaldistributionandaninitialvalueisdrawnattheoutset.Thenthecurvilineardistancetheparticletravelsisevaluatedeverytimestepuntilsaiddistanceisgreaterthanorequalto,atwhichpointaperfectlyelasticintermolecularcollisionoccurs,thepost–collisionvelocityvectorisrandomized,andanewvalueisdrawnfromthedistribution.Inthiswayexternalinuencescanbe 101

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includedataminimalcomputationalexpensethathelppreventthecaptureofpermeateparticles,andalsoprovidesameansforcapturedbodiestoescapetheorbitalsinks.OnemorefundamentaldifferenceexistsbetweentheinniteandniteKnudsenmodelsdiscussedthusfar.Intheinnitecasethesimulatedtrajectoryisthatofasingleparticlefromstarttonish.Inthenitecasethesimulatedintermolecularcollisions,inessence,marktheinitializationofanewparticleenteringthesameenergyclass.RecallthattheoriginalsimulationsconsideredaninnitelyrariedmolecularowthroughanequilibratedstaticCNT,meaningbothsorbate-sorbateinteractionsandsorbate-latticeworkexchangearenonexistent.Inthiscasetheonlybodywithpotentialandkineticenergyisthatofthesorbate,andinaconservativesystemthismeansthatthetotalenergyofthesinglesorbatemoleculewillremainconstant.Inatruesystem,asinglesorbatemoleculewillnotmaintainaconstantenergyduetoworkinteractionswiththediffusionmediumandtherarecollisionwithothersorbatemoleculesofhigher/lowerenergyclassesormasses.However,foraconservativesysteminequilibrium,foreveryparticleleavingaparticularenergyclass,anotherwillbeintroduced.Therefore,insteadofdevelopingstatisticsforaspecicparticleataspecictemperature,wearedevelopingstatisticsforaspecies-particularenergyclass.Thenthetotaluxisfoundthroughintegrationoftheindividual,energydependentuxpredictionswithrespecttosomeknownenergydistributionwithinthesimulationspace.Thisisasmalldistinction,butonethatneedstobeconsideredduetothediffusenatureofthesimulatedcollisions. 4.2.1OxygenTransportAsstatedinSection 4.2 ,theobjectiveoftheFKMistocontributetothedevelopmentofaKnudsenregimediffusionmodelthataccuratelypredictsmasstransportratesbyprovidingconstraint–freetrajectorydataofasinglepenetrantmoleculeoverastatisticallysignicanttimespan.EarlierattemptsoutlinedinSection 4.1 allfellvictimtothesameshortcoming,specicallythatthelackofadditionalitinerantmassesinthesimulationspaceresultedinwell–dened,stationarypotentialwellsthatthefree–body 102

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inquestionwouldgravitatetowardsandbecometrapped(seeFigure 4-3 ).EachoftheseearlierattemptseffectivelymodeledthemolecularbehaviorofagasofinniteKnudsennumber.WhatsetstheFiniteKnudsenmodelapartismanifestfromitsname;intermolecularinteractionsarenowsimulatedasre-randomizationsofthevelocityvectorafterthebodyhastraveledacurvilineardistanceequaltothemeanfreepathofthespeciesinacontinuumattheappropriatetemperatureandpressure.Thesesimulatedcollisionsareperfectlyelasticand,foranO2moleculeina(10,10)SWCNTatSTP,occuraboutonceevery130wallcollisions.Forallofthefollowingguresinthissubsection,thepermeantisanoxygenmolecule,whosetotalenergyisequaltothatofanaverageO2moleculeina50C,1atmensembleinfreespace,boundwithintheLJpotentialeldofatomisticSWCNTsofvariouschiralities.Table 4-1 providestheCNTdimensionsandprominentstatisticalvaluesfortheconsideredcases.Figure 4-5 showsthelinearrelationshipbetweenthediffusioncoefcient/streamingvelocityratioandtheporeradiusexpectedfromEquation( 2 ).Notethatthetheoreticalmeanmolecularvelocityincylindricalporesusesthe 3 = 8 factorintroducedbySmoluchowski[ 116 ]andthestandard 3 = 2 kBTenergyrelation,resultinginEquation( 4 )below.hv2zi=8kBNAT M (4)Figures 4-6A and 4-6B showtheradialandaxialpositionsofthepermeantforthe(10,10),50kwall–collisioncase.Notrappingphenomenaisobservedandthetotalenergydriftovertheentiretyofthesimulation(1.4107timesteps)is0.079 J = mol ,resultingina 1 = 500 %changefromthetotalinitialenergy.Similarly,noneoftheremainingcaseslistedinTable 4-1 resultedincapturedtrajectories.Sections 4.2.2 through 4.2.4 focusuponthemodelvalidationagainstpublisheddatasets,todemonstratethemodelaccuracyversusthetraditionalKnudsenapproach;themodelconvergencecriteria, 103

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Table4-1. Providesthesimulatedvaluesofporeradius,Knudsennumber,averageintra–collisionalaxialdisplacement,averagestreamingvelocity,andtheColsondiffusioncoefcientfor8SWCNTchiralities.Thesimulationtemperatureandpressureconditionsare50Cand1atmrespectively. Chirality(8,8)(10,10)(12,12)(15,15)(20,20)(25,25)(30,30)(40,40) RA5.436.788.1410.1813.5716.9620.3527.14Kn77.462.051.641.331.024.820.615.5hjzjiA3.514.194.785.265.045.785.586.45hjvzjiA ps9.197.747.657.555.524.984.694.32DC103A2 ps3.903.463.653.662.803.012.712.73 Figure4-5. TheratiooftheColsondiffusioncoefcientandtheaverageaxialstreamingvelocityareplottedwithrespecttotheCNTradius,showingthelinearrelationshippredictedfromtheory.They-axisisgiveninA. determiningthenecessarynumberofsimulatedwall–collisionsnecessarytodevelopaccuratedistributionsinradialnumberdensityandintra–collisionalaxialdisplacement;andtheapplicationofthisapproachtorigid,three–bodieswatermodels. 104

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A BFigure4-6. Plotsoftheradial( 4-6A )andaxial( 4-6B )movementintimeofanO2moleculetravelingthroughastatic(10,10)SWCNT,initializedat50Cad1atm,yeildingaKnudsennumberof62.Thereforeanintermolecularcollisionissimulatedafterthepermeanthastraveled,onaverage,42nm.Thissimulationiscarriedoutover14milliontimesteps,resultinginanaveragetof2fs,withoutanyenergyconstraints.Thetotalenergydriftis0.079 J = mol . 105

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4.2.2ModelValidationA2006studybyHoltetal.[ 60 ]providesbothadesignprocedureforCNTmembraneswithacharacteristicporesizeoflessthan20A,anduxmeasurementsforvariousgasesandliquidwaterthroughsaidmembranes.Theresultssupportthehypothesisthattheuxofaspeciesisproportionaltotheinversesquareofthemolecularweight[ 78 ],howevertheyalsoshowthatthetraditionalKnudsenmodel,givenbyEquation( 4 ),under–predictsthemeasuredowratesbymorethananorderofmagnitude.Equation( 4 )isacombinationofEquations( 2 )and( 2 )reportedinChapter 2 .Themembranesareadensearrayofverticallyaligned,CVDproduceddouble–walledCNTsgrownonanetchedsiliconwafer,suspendedwithinanimpermeablesiliconnitridematrix.Transmissionelectronmicroscopywasusedtodeterminetheuniformityofthenanotubeinnerandouterdiametersaswellastheaverageporedensity(&2.51011cm)]TJ /F2 7.97 Tf 6.59 0 Td[(2).Thenanotubediameterdistributionsarereportedandtheaveragesaregivenas1.6and2.3nmfortheinnerandoutercasesrespectively.ThegasesconsideredarereportedinTable 4-2 andthesimulationconditionsusedtomimicthoseoftheexperimentareprovidedinTable 4-3 .QKn=4 3p 2r kBNAT MR3 PP LAm&m (4)Double–walledCNT,25koscillationsimulationswereperformedontheUFHiPerGatorsupercomputerforeachspeciesandthesimulateduxandselectivityratioswerecomparedwiththosereportedin[ 60 ].ThediffusioncoefcientsanduxvaluesweredeterminedusingtheColsonmethod(ColsonF,privatecommunication,May6,2014)whichutilizestheintra–collisionalaxialdisplacementdistributiondeterminedfromtherawtrajectorydata.Figures 4-7A and 4-7B providealookatsaiddistributionanda25oscillationsampleoftheO2simulationvieweddownthesharedCNTaxis.Thedetailsbehindtheuxanalysisaretheprincipaltopicinaforthcomingdissertation 106

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Table4-2. Therstcolumnlistsallofthenon–hydrocarbongasesmeasuredin[ 60 ]andtheremainingcolumnsgivethesimulationparametersusedforeach.Theparametersincolumns2through4weretakenfrom[ 20 ]. GasMolecularweightLJdiameterLJpotentialwelldepthM(g/mol)(A)"(J/mol) H22.022.915315.95He4.002.57684.81Ne20.182.789296.83N228.013.667829.78O232.003.433939.53Ar39.953.4321017.69CO244.013.9961579.75Xe131.304.0091951.40 Table4-3. AlistofparametersusedtomimictheexperimentsconductedbyHoltetal.[ 60 ].Onlytheinnerandouterdiametersaregivenin[ 60 ],notthechiralindices.ForthesimulationscarriedouthereitisassumedtheCNTsarearmchairtypewithdiametersof16.28and23.06Arespectively.Notethat3thicknessvaluesweretestedwithallotherexperimentalparametersheldxed. ChiralityTemperaturePressureThicknessMembranePoreareadensityInnerOuterT(C)P(atm)L(m)Am(cm2)&m(cm)]TJ /F2 7.97 Tf 6.59 0 Td[(2) 2.0(12,12)(17,17)5012.81.7510)]TJ /F2 7.97 Tf 6.58 0 Td[(32.510113.0 107

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(UF,December2014)andreadersinterestedintheinnerworkingsandbackgroundmathematicsofthisnovelapproachtouxcalculationarereferencedthere. A BFigure4-7. Illustrationoftherawsimulationdataandthezdistributionobtainedviapost–processing. 4-7A givesadown–axisviewoftheO2trajectorybetweenits10,000thand10,025thwallcollision.TheredcirclesrepresenttheindividualcarbonLJsites,thebluelinedenestheO2ightpath,andthegreendashedlinedenesthetransitionfromanattractive(inner)torepulsive(outer)forceeld. 4-7B givestheintra–collisionalaxialdisplacementdistributionfortheentire25kwallcollisionsimulation. Figure 4-8 plotsthespeciesmolecularweightversusthecalculatedvolumetricowrateforboththetraditionalKnudsenapproachgivenbyEquation( 4 )andthosefromthesimulationsfora3.0mthickCNTmembrane.Themeasureduxvaluesarenotexpresslygivenin[ 60 ],onlytherangebywhichtheKnudsenapproachunder–predictedthemeasurements.Inthecaseofthe3.0mmembranethe“enhancement”factorrangesfrom20to80.ThereforetheresultsofHoltetal.arepresentedasanenvelopeofvaluesboundedbybluedashedlineslocatedat20and80timesthegraphedKnudsendatapointvalues.TheFiniteKnudsenModelsimulationsshowslightlyimprovedagreementwiththeexperimentalmeasurements,withan“enhancementfactor”rangingfrom7.2to51.5. 108

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Figure4-8. Comparisonoftheuxpredictedthrougha3.0mthickCNTmembraneusingEquation( 4 )(greentriangles)andtheFiniteKnudsenModelsimulations(redcircles)tothevaluesmeasuredbyHoltetal.[ 60 ](bluedashedlines),shownaboveasabandofvaluesspanning16to60timesthecalculatedKnudsenowrates. Theseresultsprovidesomepromisingstepsforwardformoleculardynamicists,andalsoservetohighlightseveralresearchtopicsthatrequiremorein-depthexamination.Ononehand,theresultsshowthatsingle–particleEMDwithsimulatedinter–particlecollisionscanreasonablymimictraditionaltheory,providinganattractivealternativetomorecomputationallycostlyhigh–loadingEMD,NEMD,andMCtechniquesfordevelopingnumberdensityandstreamingvelocityprolesinsystemswhereKnudsen 109

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diffusionanalysishasshownstrongagreement.AprominentexamplebeingtheworkofGruenerandHuber[ 52 ],whoperformedpermeancetestingofcrystallinesiliconmembranescharacterizedbylinear,irregularlyshapedporeswithanaveragediameterofroughly12nm.Conversely,thismodelhasdifcultymatchingthemeasurementsreportedbybothHoltetal.[ 60 ]andMajumderetal.[ 88 ]forCNTmembranes,whichbothpredictenhancementfactorsof16andgreater.IndeedEMDingeneralhasbeenunabletopredictthisprocesswell.TheclosestsimulatedresultsforthisenhancedowphenomenonaregivenbyAckermanetal.[ 1 ]andBhatiaetal.[ 12 ],whoeachmanagetofallwithinthecorrectorderofmagnitude.BothpapersperformedEMDsimulationsoflightLJgasesdiffusingthroughrigid(10,10)SWCNTsandreportedtheself-andtransportdiffusivitiesvs.thosetroughzeolitesofcomparabledimension.Theself-diffusioncoefcientsofArat1atmdeterminedusingtheFKMandthosereportedareofthesameorderofmagnitude,bothapproaching2.010)]TJ /F2 7.97 Tf 6.58 0 Td[(2 cm2 = s .However,thetransportdiffusivitiesareordersofmagnitudeapart.Toreiterate,inthisdissertationtheColsonmethodisusedtodeterminetransportpropertieswhich,intheKnudsenlimit,directlyrelatestheequilibriummotionofamoleculetouxcontributionthrougharigorousmathematicalformulation(ColsonF,privatecommunication,May6,2014).TheAckermanetal.approachusestheclassicalDarkenequation[ 35 ]torelatetheselfdiffusivitytoa“correcteddiffusivity”throughathermodynamiccorrectionfactorrelatedtothelogarithmicfugacitygradient.Therelationisbasedontheassumptionthatachemicalpotentialgradientdrivesdiffusionandconvergesto1asthesorbateconcentrationapproaches0.Thoughthismethodhasattractedwidespreaduseamongstequilibriummoleculardynamicists,itwasnevertrulymeantfornanoscaletransportproblemsasthefugacityisamacroscopicpropertyofanidealgas,andassuchwouldseemtofalloutsidethepurviewofsub–continuumdiffusionthroughsmallchiralitySWCNTs.TheapproachtakenbyBhatia,Chen,andSholl[ 12 ]todetermine 110

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thetotaltransportcoefcientmirrorthatoftheOMdeveloptedbyBhatiaetal.[ 11 ]andisdiscussedatlengthinChapter 2.2.1 .Ifindeedthethermodynamiccorrectionfactoristhemostaccuratemeansofderivingtransportpropertiesfromequilibriummotioninconnedsub–continuumspacesthenthesimulationscarriedouthereshouldallowforsimilarresultstothosepublished.However,theColsonmethodallowsfordirectdeterminationofthetransportcoefcientwithoutresolvingnumerousisothermsthroughcomputationallyexpensiveGCMCsimulations,anditavoidstheuseofanymacroscopicuidpropertiesthataredifculttojustifyformolecularows.ThemarkedunderestimationofuxseenintheKnudsenapproachhasbeenattributed,atleastinpart,tothediffusereectionassumptionmadeintheinitialformulation.GreaterthanpredictedmasstransportthroughCNTshasbeendiscussedpreviouslyinboththeliterature([ 13 ],[ 72 ],[ 113 ])andChapter 2 ;theprimarycausestemmingfromtheuniformityinthecarbonlatticeandlackoftortuositycreatinganearidealenvironmentforspecularbehavior.Figures 4-9A through 4-9C showthedegreeofspecularitydistributions,0indicatingafullyspecularreectionand1indicatinga90offsetfromthespecularreectionangle,fortherotatedcollisionaxesfortheO2simulation.Inallthreedimensions,particularlyaxially,thereectionanglesarehighlyspecular,enhancingthemasstransportrate.WhiletheFiniteKnudsensimulationsshowasmuchasafactorof8improvementinuxpredictionoverthatofEquation( 4 )thereisstillroomforimprovement.Theuseofthecontinuumconventionsforthemean–freepathtoestimatetheaveragedistanceanitinerantmoleculewouldtravelbeforeencounteringanotherisagrossviolationofseveraloftheassumptionsmadeinthederivationofEquation( 4 ),specicallytheuseofhardspheresinstraight–linemotion.WithasoftspheremodellikethatofLJsystemsthesphereofinuencebetweenbodiesbecomesinnite,asonebodyonlylosestrackofanotherinthelimitofthecenter–to–centerdistanceapproachinginnity.Generallyacutoffdistanceisdenedasthepotentialbetweenbodiesbecomesvanishinglysmall 111

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A B CFigure4-9. Visualizationofthefractionofwallcollisionsresultinginspecularreections.Thex0,y0,andz0axesoriginateatthecollisioncitewiththey0andz0axesorientednormaltothewallandparallelwiththeCNTaxisrespectively.Adegreeofspecularityof0representsaperfectlyspecularreection,whilevaluesof1describeareection90offsetfromspecular. atlargedistances,decayingatarateproportionaltotheinversedistancetothesixth,howeverthisstillmuddlesthederivationduetothecurvilinearmotioninducedby 112

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spatiallyvariantforcesactingonthebodiesasopposedtoindividualimpulsivecollisionsbetweenbodiesthatotherwiseareunawareofoneanother.Theradialnumberdensitygraphs(seeFigures 4-10A and 4-13 )suggestastrongtendencyofthepermeanttooccupyarelativelynarrowannulusencompassingthezeroforceorbital.Furthermore,thoughtheinnerdiameterofthe(12,12)–(17,17)double–walledCNTis16.3A,anO2LJmoleculeenergizedat25Cnevergetscloserthan3Afromthewalloverthecourseofa50kwallcollisionsimulation.Infact,theaverageradialpositionatthemomenttheradialvelocityvectorswitchessignfrompositivetonegativeis4.9A,creatinganeffectiveporediameterof9.8A.Thecombinedeffectsofthepermeantclusteringnearthenullorbitandthedecreasedeffectiveowareashouldservetodecreasetheexpecteddistancetraveledbetweencollisions.Asignicant,thoughlittlediscussed,contributortothediscrepancyinpredictedandmeasureduxistheuseoftheMaxwell–Boltzmanndistributionasthepredictedenergyprole.Formembranesofthistype,wheretheporediameterislessthanthemeanfreepathofthegas,theseedingoftheporesisakintoeffusion.Effusionratesareknowntoscalewiththeaverageparticlevelocity,andassuchitislikelythatadisproportionatenumberofparticlesfromthehigherenergystatesofthecontinuumdistributionenterthetubewithrespecttothemoreabundantlowerenergystates.Clearlythiswouldservetoaltertheenergydistributionwithintheentryandexitregionsofthepore,pushingtheaverageenergytotheright(seeFigure 4-11 ).Additionally,inthecentralregionoftheporethefreemoleculehasequilibratedwithboundlatticeparticlesofadifferentspeciesasopposedtootherlikefreemoleculesasassumedinMaxwell'soriginalderivation,againalteringthedistribution.AnotherslightoversimplicationinboththeevaluationoftheFiniteKnudsenresultsstemfromtheuseofonlytheaverageporeradiusintheuxcalculation.Inrealitythereexistsadistributionofnanotubechiralitiesandradiithroughwhichthediffusingspeciestravel.ByrevisitingEquation( 4 )weseethattheKnudsenowrate 113

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A B CFigure4-10. VisualizationsofseveralkeystatisticalquantitiesgleanedfromtherawtrajectorydataofapenetrantO2moleculewithina(10,10)SWCNT.Figure 4-10A isapolarcontourplotofthetimefractionanO2moleculeoccupiesacertainradialannulus,thevaluesofwhicharegivenbytheaccompanyingcolormap.TheCNTLJsitesaredepictedasredcircles.Figure 4-10B showstheaverageaxialspeedin A = s asafunctionofradialposition,giveninA.Thedashedgreenlineshowsthezero–forceradius.Figure 4-10C isahistogramofthefractionofcollisions()thattravelacertainintra–collisionalaxialdistance(z)giveninA.Eachoftheseplotsweregeneratedfromasimulationof50kwallcollisionsat50Cand1atm. scaleswithporediametercubed,andboththesimulationsconductedaspartofthisdissertationandthosefoundintheliterature[ 97 ]showanon-linearuxdependencyonporesize.Assuchthetotaluxcalculationshouldincludeintegratingovertheporediameterdistribution.ItisunclearwhetherornottheenhancementfactorreportedbyHoltetal.includesthisintegratedeffectoftheuxcontributionduetoporesize,butcalculatingtheowrateusingthepublishedporediameterdataradiusresultsina7%increasevs.usingonlythepublishedaverageporediameter.Furthermore, 114

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Figure4-11. AgraphicallookattheclassicalMaxwell–Boltzmanenergydistributionforasingle,hardsphereidealgas.Thex-axisisgivenintermsofeffectivetemperature(Te=E 3 = 2 kB)forclarity,andthemostprobable(Tmp),average(hTi),androot-mean-squaredtemperatures(Trms)areprovidedinred. thepublisheddiameterdistributionisbasedona391CNTsample,arelativelysmallsumconsideringthenumberofporesestimatedtobeinthehundredsofmillions.Itispossiblethatthisdistributionisnotwhollyrepresentativeofthetrueporeaverage.Again,uxwillscaleroughlywiththeaveragediametercubed,soevensmallchangesintheaveragediametercouldresultinsignicantchangestothepredictedux.Onenalnoteislargeuncertaintyassociatedwiththeporedensity,specicallyinaccuratelydeterminingthetotalnumberofopen-endedCNTsthatspantheentirelengthofthemembraneandremainunclogged.GreatcarewastakenbyHoltandcoworkers[ 60 ]tocharacterizetheporesizesandpotentialforblockages,andTEMimageswereextensivelystudiedtoestimatethe“activeporedensity”,butthisisdiscussedasthe“singlelargestuncertainty”associatedwithcomparingthemeasurementstothetheoreticalux.Figure 4-12 showstherelationshipbetweenthespeciesmolecularweightandtheselectivityofthemembranewithrespecttoheliumforboththeexperimentaland 115

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Figure4-12. Thepublishedselectivityvalues[ 60 ]ofadouble–walledCNTmembranearepresented(bluetriangles)forH2,Ne,N2,O2,Ar,CO2andXewithrespecttoHe.TheyshowremarkableagreementwiththeinversesquarerootproportionalitywithmolecularweightpredictedbyKnudsen.Thepowercurveofbestthasanexponentof-0.49.TheselectivitypredictedthroughtheuseoftheFKMandtheColsonmethodaregiveninredwithabest–tpowercurveexponentof-0.395andsquaredresidualof0.93.ThedeviationisconsistentwiththemeasurementsofMajumderetal.[ 88 ]whociteadependenceof-0.42.Theseresultsarefrom50koscillationsimulationscarriedoutatSTP. simulatedcases,andservestohighlightthedeviationfromconventionasthemolecularweightofthepermeateincreases.Theselectivityisgivenhereastheratioofdiffusionowrates,withQHeassignedtothedenominator,andtheexperimentaldatawere 116

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acquiredfromthepublishedmaterial[ 60 ]usingPlotDigitizerrv.2.0software.Bothdataarettedtopower–lawcurves(dottedlines),andtheexperimentaldatatstheinversesquarerootofmolecularweightrelationshippredictedinEquation( 4 )verywell.Holtetal.reportanexponentof-0.490.01forthepower–lawcurveofbestt[ 60 ],meanwhilethesameexponentforthesimulationcurveofbesttis-0.37withasquaredresidualvalueof0.93.ItisworthnotingthatsimilarmeasurementsmadebyMajumderetal.[ 88 ],forH,N2,O2andArdiffusionthrough7nminnerdiameterMWCNTmembranes,reportamolecularweightdependenceofM)]TJ /F2 7.97 Tf 6.58 0 Td[(0.42.Thenoteddeviationisstrongestforthespeciesofgreatermolecularweightandpotentialwelldepth.AsseeninEquation( 3 ),theforceactingbetweenthepermeantandadsorbatescaleslinearlywith",andasshowninTable 4-2 ,thevaluesforCO2andXeare1.5and2timesthatofO2respectively.Ifapower–lawcurveofbesttweretthroughthesamedataexcludingCO2andXethentheexponentclimbsto-0.45withasquaredresidualof0.95.Conceivably,thissuggeststhatatthesehighmolecularweightsthestaticwallassumptionbeginstofail,asthemagnitudeoftheLJforcebecomesrelevantwithrespecttothoseoftheinter–latticecarbonbonds.IftheseparticleswereallowedtodoworkontheCNTwall,thensomeoftheirkineticenergywouldbesiphonedoffintoexcitingvibrationalmodesinthelattice,decreasingtheirrespectivediffusioncoefcientsandbringingthesimulatedselectivitycurveexponentclosertotheKnudsenpredicted-0.5.Insummary,whilethelevelofagreementbetweenthesimulatedandexperimentalresultsislow,thelevelofuncertaintyinboththemodelandtheexperimentmustbeconsideredbeforethrowingouttheproverbialbabywiththebathwater.The6LJpotential,whileconsideredveryaccurateatevaluatingthelongrangedispersionrelationshipsbetweenparticles,poorlyaccountsfortheneareldeffects.MoreaccuratemodelssuchastheBuckinghampotentialandthe10-4BojanandSteelepotentialsexist,buteachhavesignicantcomputationaldrawbacks.Furtherdiscussion 117

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canbefoundinChapter 2 .Additionally,theuseoftheMaxwell–Boltzmanntheoryislikelyinerrorasitwasnotintendedtoaccuratelypredictenergydistributionsinconnedspaceswithstrongpotentialelds.Whenconsideringtheexperimentation,evensmallchangesintheaverageporediameter,theporediameterdistributionortheactiveporedensitycanchangetheuxpredictiondramatically.AsnotedbyChenetal.“thegreatestopportunity(orneed)in[CNTMD]isfordetailedexperimentalmeasurements,”andtothispointonlythepioneeringworkoftheLivermoreNationalLabandMechanicalEngineeringteamfromBerkley[ 60 ]andthemorerecenteffortsofMajumder,Chopra,andHinds[ 88 ]haveexploredthistopicexperimentally. 4.2.3ConvergenceCriteriaThemotionofanoxygenmoleculeboundwithintheLJpotentialeldsofatomisticSWCNTsofvariouschiralities,eachsimulatedat50Cand1atm,arereportedbelow.Figure 4-13 showstheevolutionoftheradialnumberdensityprolewithrespecttothenumberofsimulatedwallcollisionsfor(10,10),(20,20)and(30,30)SWCNTs.Here,thenumberdensityisevaluatedbyassigninganitenumberofradialbinswithspeciedinnerandouterradii,thendeterminingtheratioofthetotalighttimethepermeatemoleculespentineachannularbin.Ineachcase,asexpected,theannuluswiththemaximumprobabilityofoccupancyalsocontainsthenanotubenullforceorbit.Additionally,inallcasestheresidualerrorintheprolesreachesthedesired5%thresholdwithin25kwallcollisions,roughly2106timesteps.ThedevelopmentoftheresidualsaregiveninFigure 4-14 .Inasimilarfashion,Figures 4-15 and 4-16 showtheevolutionoftheintra-collisionalaxialdisplacementdistributionwiththenumberofsimulatedwallcollisions,andtherespectivelinearregressionanalyses.Againtheseplotsaregeneratedfor(10,10),(20,20)and(30,30)SWCNTsandagainweseeconvergence(5%residual)within25kwallcollisions.Aswouldbeexpectedfromthefollowinganalysis,consideringonlytherst25koscillationsinthedouble–walledCNTmodelvalidationcasesfromSection 4.2.2 118

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madenoappreciabledifferenceintheuxortheselectivitywithrespecttothemeasuredvalues.TheuxvaluesanddiffusioncoefcientsaregiveninTable 4-4 . Table4-4. Comparisonofowcharacteristicspredictedforvariousdiffusingspeciesgivenstatisticsgeneratedfrom50kand25kwallcollisionsimulations.Bothsimulationsarefromthesamedataset,the25kcasebeingthersthalfofthe50kcase.ThesimulationwascarriedoutatSTPwithina(12,12)–(17,17)double–walledCNT. Gas50kOscillations25kOscillationsDCSelectivityDCSelectivitym2 s10)]TJ /F2 7.97 Tf 6.59 0 Td[(7m2 s10)]TJ /F2 7.97 Tf 6.58 0 Td[(7 H29.621.7910.251.71He5.37–5.98–Ne2.840.532.920.49N23.480.653.840.64O23.170.593.400.57Ar2.850.533.090.52CO23.340.623.610.60Xe2.000.372.200.37 119

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A B CFigure4-13. ThissetofplotsshowstheevolutionoftheradialnumberdensityproleforO2travelingthrougha(10,10) 4-13A ,(20,20) 4-13B ,and(30,30) 4-13C SWCNTat50C.Foreachprole(bluediamonds)thex-axisgivestheradialpositionwhilethey-axisshowsthetimefractionaparticularradialbinispopulated.Thedottedredlineshowstheradialpositionofthezero–forceannulus.Foreachtwo–columnedsubplot,thedistributionisderivedfrom5k(A-topleft)to50k(F-bottomright)oscillationsimulations. 120

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A B CFigure4-14. ThissetofplotsshowsalinearregressionanalysisoftheevolutionoftheradialnumberdensityproleforO2travelingthrougha(10,10) 4-14A ,(20,20) 4-14B ,and(30,30) 4-14C SWCNT.Foreachsubplot,thelinearregressionprogressesfrom5k(A-top)to40k(E-bottom)oscillations,comparedagainstthe50kcase.TheR2valueforeachtisprovidedintheupperlefthandcornerofeachplot. 121

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A B CFigure4-15. Thissetofplotsshowstheevolutionoftheintra–collisionalaxialdisplacementdistributionforO2travelingthrougha(10,10) 4-15A ,(20,20) 4-15B ,and(30,30) 4-15C SWCNTat50C.Foreachhistogramthex-axisgivestheaxialdistancetraveledbetweenwallcollisionswhilethey-axisshowsthefractionofcollisionsthattraveledaparticularaxialdistance.Foreachtwo–columnedsubplot,thedistributionisderivedfrom5k(A-topleft)to50k(F-bottomright)oscillationsimulations. 122

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A B CFigure4-16. Thissetofplotsshowsalinearregressionanalysisoftheevolutionofthetheintra–collisionalaxialdisplacementdistributionforO2travelingthrougha(10,10) 4-16A ,(20,20) 4-16B ,and(30,30) 4-16C SWCNT.Foreachsubplot,thelinearregressionprogressesfrom5k(A-top)to40k(E-bottom)oscillations,comparedagainstthe50kcase.TheR2valueforeachtisprovidedintheupperlefthandcornerofeachplot. 123

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4.2.4WaterTransportThemotionofthree–bodiedwatermoleculesboundwithintheLJandinducedquadrupolepotentialeldsofatomisticSWCNTsofvariouschiralities,undervariousKnudsennumberconditionshavebeensimulatedandarereportedbelow.Figure 4-17 showstheradialnumberdensityprolesoffourseparatewatermodelswithina(10,10)SWCNT;theSPC,SPC/EandTIP3Pmodelswithinducequadrupoleandoxygen–carbonLJeffects,andaSPCmodelwithonlyLJeffects(labeledLJinthegurelegend).ThemodelparametersaregiveninTable 3-2 andFigure 3-2 . Figure4-17. Radialnumberdensityproleof4differentwatermodelswithina(10,10)SWCNT.TheSPC,SPC/E,andTIP3Pmodelsallincludeinducedquadrupoleinteractions,whiletheLJmodelhasthesamearchitectureastheSPCmodel,butonlyincludestheoxygen–carbonLJinteractions. Figure 4-18 showstheintra-collisionalaxialdisplacementdistributionforeachmodel,yieldingColsontransportcoefcientsof2.67,2.28,2.48,and2.5210)]TJ /F2 7.97 Tf 6.59 0 Td[(7 m2 = s fortheLJ,SPC,SPC/EandTIP3Pmodelsrespectively.FromacomputationalstandpointtheLJmodelisthemostattractive,butwithouttheadditionalinducedquadrupoleinteractionstheorientationdependenceoftheparticlewithrespecttothewallislost. 124

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A B C DFigure4-18. Intra-collisionalaxialdisplacementdistributionof4differentthree–bodiedwatermodelswithina(10,10)SWCNT.TheSPC,SPC/E,andTIP3Pmodelsallincludeinducedquadrupoleinteractions,whiletheLJmodelhasthesamearchitectureastheSPCmodel,butonlyincludestheoxygen–carbonLJinteractions. 125

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Holtetal.[ 60 ]andMajumderetal.[ 88 ]bothperformedmeasurementsofliquidwaterpermeationthroughCNTmembranes,buttotheauthor'sknowledgenowatervaportransportEMDsimulationsorexperimentationhavebeenperformed.ThereforeweturntotheworkofAlexiadisandKassinos[ 2 ],whosuggestthattheSPC/EmodelsimulationsofliquidwaterwithinarigidCNTframeworkprovidesasoundapproximationtothedensityproleobservedinsimulationswithfullydynamicmolecularandlatticestructures,andwillthereforebeusehere.ExtendedSPCwatersimulationswerecarriedoutunderthesameconditionsdescribedinSection 4.2.2 togetameasureofthewaterselectivityvs.thatpredictedforoxygen.TraditionalKnudsentheorywouldpredictthewatertransporttobegreaterbecauseitisoflessermolecularweight,andtheselectivitywouldbegivenas DK,H2O = DK,O2 =p MO2 = MH2O =1.33.However,ourresultsshowthatthewater–CNTafnityworkstocounterthistrend,resultinginaselectivityequalto DC,H2O = DC,O2 =0.78.ThisprovidesagreatdealofoptimismthatnanoporousmaterialshaveanopportunitytoprovidehighdiffusivitiesforO2withmoderateselectivitywithrespecttowatervapor,akeyfactorinthedevelopmentoffuturegenerationDMFCs. 4.3SampleSelectivityAnalysisWiththeadventofhighlyorderedphasesegregated,columnarmorphologiesinblockcopolymerlms[ 101 ]andhigh-volumeCVDproductionofverticallyalignedCNTbundles([ 60 ],[ 88 ])non–tortuous,nanoporousmembranesarebecomingeasiertoproduceandareattractingnoticefromtheresearchcommunityasapossibleavenuefornext–generationpassiveseparationprocesses([ 66 ],[ 135 ]).StandardtheorysuggeststhatKnudsenregimeselectivityisdrivenbythemolecularweightsofthediffusingspecies,thelighterspeciesdisplayinggreaterpermeability.Yuetal.[ 135 ]notedthereversetrendintheirtriisopropylorthoformate/hexanepervaporationexperimentsthroughCNTmembranesandsuggestthisresultmaybedueto“preferentialadsorption.”Amongtheprimarygoalsofthisprojectistocontributetothedevelopmentofatoolwith 126

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whichmaterialscientistsandmembranemanufacturerscouldmakepredictionsofseparationandpermeabilitycharacteristicsofproposednanoporousmembranedesignspriortofabrication.Specically,thisprojecthasbeenmotivatedbytheneedforanimprovedLBLforaportableDMFCwhichexhibitslowresistancetooxygentransport,moderateoxygenselectivityoverwatervapor,andhighelectricalconductivitywhileremainingchemicallyinertinthepresenceofmethanol.ThissectionismeanttoillustratehowtheniteKnudsenmodelmaybeemployedtoaidinthedevelopmentofsuchamembraneandshowthedeparturefromclassicalKnudsentheorycausedbythemembraneadsorptiveeld.Single–walledCNTsarechosenasthetransportmediumforthisexamplebothfortheirchemical/electricalpropertiesandtheirfavorableatomicstructureforefcientatomisticsimulations.ThediffusingspeciesconsideredareO2,modeledasaLJsphericalbody(seeTable 4-2 fortheparametervalues),andrigidSPC/EH2OmoleculesasdiscussedinSection 4.2.4 .Themembranethicknessissetto2m,consistentwiththeworkofHoltetal.[ 60 ].Theoxygenconcentrationgradientisderivedfromtheexpectedstoichiometryofa20WDMFCoperatingat50CandEquation( 4 )isusedforthecomparisontotheKnudsenprediction,whereAmand&mareneglectedbecauseonlyoneporeisbeingconsidered.Sevensimulationsarereportedforeachdiffusingspecies,withtheonlyvariablebeingthenanotubediameter.TheresultsarepresentedinFigure 4-19 .ThetotaldesiredowrateofO2willbedeterminedbythepoweroutputoftheFCplustheparasiticloadandthemembraneareawillbedeterminedbythesizeofthePEM.BysettingthenumberofcellsintheFCstackFigure 4-19 canbeusedtodeterminethenecessarynumberofporesperLBLasafunctionofporeradiusforsufcientoxygentransport.Furthermore,theselectivitycurvecanbeusedtodetermineacutoffradiusforpreferentialO2transport.ThisanalysissuggeststhatSWCNTswithdiametersgreaterthan50Awillnolongerexhibitoxygenselectivityoverwatervapor.Italsosuggeststhatastheporediameterincreases(andtheKnudsennumberdecreases) 127

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Figure4-19. SimulatedvaluesofoxygenvolumetricowrateandH2O/O2diffusioncoefcientratiosaregivenassolidreddiamondandbluecirclemarkersrespectively,andthecorrespondingKnudsenderivedpredictionsaregivenasdashedlinesofthesamecolor.EachisplottedwithrespecttotheCNTradius.Theselectivitycurveisofinterest,asitappearstobetrendingtowardstheKnudsenpredictedvalueastheKnudsennumberdecreases.TheapparentcutoffdiameterforapreferentialoxygentransportforaSWCNTmembraneis50A.TheO2dataabovearetakenfrom25kwall–collisionsimulations,thewatersimulationsalltimedoutafter60hours,completingbetweenonly21kand1.5kcollisions.ThisisduetotheincreasedcomputationalexpenseoftheinducedquadrupoleFORtransformations. themolecularweighteffectspredictedbytraditionalKnudsenanalysisofnon-interactinghardspheresbecomemoredominantwithrespecttothepreferentialadsorptioneffectsdiscussedbyYuetal.[ 135 ].Thisisillustratedbythelogarithmiccurve–t(blackline)graduallyapproachingtheKnudsenselectivitylimit(bluedashedline)inbothvalue 128

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andslopeastheporeradiusincreases.Thismakesintuitivesense,asthestrengthoftheinducedquadrupoleeffectsbetweentheCNTlatticeandthewatermoleculeisdeterminedbytheproximityofcarbonatomswithinthelatticetothesorbatemoleculeconstituents.IfawatermoleculeisapproachingthewallwithitshydrogenatomspointedtowardsthenanotubeaxisthenthequadrupoleforcesinducedontheoppositewallwillbefargreaterforthesmallerCNTs,asthecarbonsitesarecloserthaninthecaseofthelargerdiametertubes.Inthecaseoflargerpores,wherethewatermoleculesaremorefrequentlyorientedtowardsthewallofapproachduetotherelativelyweakattractionsfromthecross–diametercarbonsites,thebehaviortendstowardasimpleLJmolecularowwheretheaveragelinearvelocitymagnitudesapproachq 3kBT M.Thesesimulationscanprovideasolidinitialdesignpoint,andcanbeusediterativelywithexperimentationtodevelopamembranewithallofthedesiredcharacteristicsinanefcientmanner. 129

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CHAPTER5CONCLUSIONSANDFUTUREWORKAvectorized,constraint–freeKnudsenregimeMDalgorithmwasdevelopedexhibitingexceptionalenergyconservationandefcientcomputationthroughtensofmillionsoftimesteps.TheaccuracyofvarioustraditionalsymplecticintegratorsusedtoresolvetheEOMsareshowntobehighlydependentuponthetimesteporderofmagnitude,andtheproposedRKalgorithmusedinthisdissertationconsistentlydisplaysaccuracyaminimumoffourordersofmagnitudegreaterthanothertestedmethods,atlessthandoublethecomputationalexpense.IfoneaccountsforthenecessaryinclusionofthermostattingandbarostattingroutinesintheVerletandGearapproachesthatcanbeomittedhere,thenthedifferenceincomputationexpensebecomesevenmorecomparable.Thisapproachhasbeenadaptedforbothitinerantpointmassandrigidmolecularbodies,andisreadilyextendabletoexiblesystemswherethebonddynamicsareimportant.MATLABwasthecomputingenvironmentusedforthisproject,asthevectorizedstructureofthelanguageandefcientmatrixalgebraitaffordslendthemselveswelltoMD.Furthermore,MATLABisanuntappedresourceinMD,discardedinfavoroftraditional,andtypicallyfasterC++andFortranroutines.SimulatingtheproposedRKalgorithminbothC++andafullyvectorizedversioninMATLABshowsthat,whileC++stilloffersaslight(roughly7%)advantageinprocessingspeed,MATLABisaviableoptionformoleculardynamicists,particularlyinpost–processingandsortingdatasetswhenhighlyefcientlogicalindexingoflargearrayscanbeutilized.PartofthefutureworkproposedherewillbethedevelopmentofaMDtoolboxforMATLAB,asdensepackingoffreeparticlesintraditionalhard–spheresimulationsdemonstratesthecomputationaladvantagesofvectorizedprogramminglikefewotherproblems.Thenalresultdemonstratesconstraint–free,conservativesimulationstensofnanosecondsinlength,representingacombinedshowingofcomputationalefciency 130

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andaccuracyunmatchedintheliterature,andshowsthenotionofathermostatbeingrequiredforconservative,nanoscaleMDsimulationtobeobsolete.Thecomputationalapproachpresentedherehasmanypotentialapplications,oneofwhichistoprovidestatisticalinformationnecessaryasinputtoanewlyproposed,fundamentally–rigorousmethodologyforcalculationofKnudsenregimeuxanddiffusivity.Nanouidicsisafastgrowingeldinengineering,andtheabilitytoefcientlycharacterizethetransportpropertiesthroughafullyphysics–basedandrigorousapproachwillbekeytoitscontinuedgrowth.Thepossibilityofdeterminingsaidcharacteristicsfromsingle–moleculesimulationsunmodulatedbytraditionalenergyconstraintsprovidesapromisingstepinthatdirection.Manyavenuesremainopenforcontinuedexplorationofthistopic,whetheritbeinimprovingtheinteractionpotentials,expandingthesystemapplicabilitytopolarpolymericsystems,orimprovingtheprogramaccessibilitythroughthedevelopmentofintuitiveuserinterfaces.Asarststep,thedevelopmentofimprovedneareldpotentialfunctionsforCNTsandadditionalexperimentationofuxthroughCNTmembranes,particularlyaslargescalenanotubeproductionbecomesmoreuniformandcontrolled,areneeded.Oneinterestingoptionthatistobepartoffutureresearchisthemodicationofthe10-4GraphitepotentialofBojanandSteele[ 22 ].Thisisawidelyusedpotentialfunctionincarbonaceousslitporesandthepotentialwellandcarbondiameterparametersaregenerallyusedin6-12LJsimulations,buttotheauthorsknowledgeithasnotbeenusedinCNTsimulations.Doubtlessthisisduetotheincreasedcomputationalexpense,asutilizingthismodelwouldrequireasimilarproceduretothatperformedinthisstudyofthewater–carboninducedquadrupoleinteractions,whereinonewouldneedtodeneanewFORforeachcarbonsiteandrotatebetweenthoseandsomeglobalFORtodeterminethetotallattice–sorbateinteraction.However,simulationsofthiskindwhichwouldhavebeenconsidered 131

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intractableforsystemscomprisedofmanyparticles,becomecomputationallyviableforthesingleitinerantbodysimulationsproposedhere.Withincreasedcondenceinthedelityofthecarbon–sorbatepotentialthenextstepwillbetocontinuetobuildthelibraryoftrajectorydatatorenetheestimatesontherelationshipbetweentheColsondiffusioncoefcientandtheeffectivesimulationtemperature.TraditionalKnudsenanalysissuggestsasquarerootoftemperaturedependencebutthishasthepotentialtodistortasthemolecularmotiondeviatesfromtheassumedrectilineartrajectories,andbeginstofollowanear–wallhoppingpatternlooselycomparabletothatofsurfacediffusion.Concurrently,futureeffortswillneedtobefocuseduponthedenitionofanappropriateenergydistributionforconnedmolecularows.RecenteffortshavebeenmadebyWaltheretal.[ 132 ]tofullymodeltheexperimentsofHoltetal.forwaterowthrough1.6nmdouble–walledCNTs2minlength,anundertakingofstaggeringproportions,tobetterunderstandthemechanismsresponsiblefortheirreportedfasttransportcharacteristics.Whilenoinformationwaspresentedonthecomputationalrequirementsnecessaryforsuchanendeavor,thefactremainsthatmacroscale,atomisticsimulationsarenowarealityandrepresentavaluableresourceforresearcherlookingforinsightsintotransportproperties,suchassub–continuumendeffectsorenergydistributions,forwhichuniversallyacceptedtheoryhasprovedelusive.ContinueddevelopmentofFKMprogramisneededtomakethecodemoreaccessible,andpartofthefutureworkwillneedtoberelatedtobuildingaGUIenvironmentwithinwhichausercandenetheirsimulationspace,thediffusingspecies,thedesiredpotentialelds,andsoon.SuchanenvironmentwillbeanimportantsellingpointinpushingthisapproachasaviableoptionformoleculardynamicistsandafutureadditiontotheMATLABtoolbox.Twoadditionalundertakingsofgreatimportancetothenextphaseofthisprojectaretheextensionofthisapproachtopolymericsystemsandtheestablishmentofa 132

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noveltransportcoefcientcalculationforKnudsenregimeuxes.AsnotedbySkoulidasandSholl[ 114 ],thefrequentlyusedDarkenapproximationshouldnotbeexpectedtobe“quantitativelyaccurateforanysystemofadsorbatesandadsorbent,”asthecorrecteddiffusivityvaluesvaryunpredictablywiththesimulationloadingfromspeciestospecies.ConcurrentworkatUFisfocusedondeningatransportcoefcientinwhichthesingle–loaded,equilibriumsimulationsofthisworkgeneratetherequisitetrajectorydistributionsusedtoclassifyparticlesasuxcontributors,providingafundamentalapproachtoKnudsenregimeuxanalysisthatavoidstheuseofanymacroscopicthermodynamicproperties. 133

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APPENDIXAFORCEFIELDDERIVATIONSDenedbelow( A )istheLJpotentialfunctionactingbetweenconstituentsAandB,locatedatcoordinatesriandrjrespectively.Figurerefg:InducedDipoledepictsthelongrangeinduceddipoleinteraction. FigureA-1. IllustrationoftheinduceddipolevanderWaalspotentialbetween2non-bondedatomsusedtoapproximatethepotentialenergychangewhenthedistancebetweentheatomsisincreased/decreased.ThisisequivalenttotheLennard-Jonesmodelofa2-bodyproblem. ULJ(ri,rj)=4"AB"AB krijk12)]TJ /F21 11.955 Tf 11.95 16.86 Td[(AB krijk6# (A)wherekrijk=kri)]TJ /F12 11.955 Tf 11.96 0 Td[(rjk=(xi)]TJ /F5 11.955 Tf 11.96 0 Td[(xj)2+(yi)]TJ /F5 11.955 Tf 11.96 0 Td[(yj)2+(zi)]TJ /F5 11.955 Tf 11.95 0 Td[(zj)21 2TheresultingCartesiancomponentforcesoneachparticlearegivenbelow(equations( A )–( A )).FLJxi(ri,rj)=)]TJ /F16 11.955 Tf 14.85 8.09 Td[(@Ur @krijk@krijk @xi=24"CO2xi)]TJ /F5 11.955 Tf 11.95 0 Td[(xj krijk2"2CO krijk12)]TJ /F21 11.955 Tf 11.95 16.86 Td[(CO krijk6# (A)FLJxj(ri,rj)=)]TJ /F5 11.955 Tf 9.3 0 Td[(FLJxi(ri,rj) (A) 134

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FLJyi(ri,rj)=)]TJ /F16 11.955 Tf 14.86 8.09 Td[(@Ur @krijk@krijk @yi=24"CO2yi)]TJ /F5 11.955 Tf 11.95 0 Td[(yj krijk2"2CO krijk12)]TJ /F21 11.955 Tf 11.96 16.86 Td[(CO krijk6# (A)FLJyj(ri,rj)=)]TJ /F5 11.955 Tf 9.3 0 Td[(FLJyi(ri,rj) (A)FLJzi(ri,rj)=)]TJ /F16 11.955 Tf 14.86 8.09 Td[(@Ur @krijk@krijk @zi=24"CO2zi)]TJ /F5 11.955 Tf 11.95 0 Td[(zj krijk2"2CO krijk12)]TJ /F21 11.955 Tf 11.96 16.86 Td[(CO krijk6# (A)FLJzj(ri,rj)=)]TJ /F5 11.955 Tf 9.3 0 Td[(FLJzi(ri,rj) (A)Denedbelow( A )istheMorsebond–stretchingpotentialfunctionactingbetweenbondedparticleslocatedatcoordinatesriandrj.Figuresrefg:MorseA01andrefg:MorseA02depictthebondstretchingasharmoniclinearspringsandshowtheinteractionofasinglelatticeatomwithitsneighbors.UMorse(ri,rj)=KMorse)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F5 11.955 Tf 11.96 0 Td[(e)]TJ /F17 7.97 Tf 6.59 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re)2 (A)TheresultingCartesiancomponentforcesoneachparticlearegivenbelow(eqations( A )–( A )).FMxi(ri,rj)=)]TJ /F16 11.955 Tf 10.49 8.09 Td[(@UMorse @krijk@krijk @xi=2KMorsee)]TJ /F17 7.97 Tf 6.59 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re)xi)]TJ /F5 11.955 Tf 11.96 0 Td[(xj krijk)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(e)]TJ /F17 7.97 Tf 6.58 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re))]TJ /F4 11.955 Tf 11.96 0 Td[(1 (A)FMxj(ri,rj)=)]TJ /F5 11.955 Tf 9.3 0 Td[(FMorsexi(ri,rj) (A) 135

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A BFigureA-2. Figure A-2A isanillustrationofaMorsebond–stretchingpotentialbetween2adjacent,bondedatomstoapproximatethepotentialenergychangewhenthedistancebetweentheatomsisincreased/decreased.Figure A-2B depictsthe3Morsepotentialsactingonanarbitrarylatticeatom0(orange)fromitsneighboringatoms1,2,and3(blue). FMyi(ri,rj)=)]TJ /F16 11.955 Tf 10.49 8.09 Td[(@UMorse @krijk@krijk @yi=2KMorsee)]TJ /F17 7.97 Tf 6.58 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re)yi)]TJ /F5 11.955 Tf 11.95 0 Td[(yj krijk)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F17 7.97 Tf 6.59 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re))]TJ /F4 11.955 Tf 11.95 0 Td[(1 (A)FMyj(ri,rj)=)]TJ /F5 11.955 Tf 9.3 0 Td[(FMorseyi(ri,rj) (A)FMzi(ri,rj)=)]TJ /F16 11.955 Tf 10.49 8.08 Td[(@UMorse @krijk@krijk @zi=2KMorsee)]TJ /F17 7.97 Tf 6.58 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re)zi)]TJ /F5 11.955 Tf 11.95 0 Td[(zj krijk)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(e)]TJ /F17 7.97 Tf 6.59 0 Td[((krijk)]TJ /F10 7.97 Tf 10.82 0 Td[(re))]TJ /F4 11.955 Tf 11.95 0 Td[(1 (A)FMzj(ri,rj)=)]TJ /F5 11.955 Tf 9.3 0 Td[(FMorsezi(ri,rj) (A) 136

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Equation( A )istheHarmonicbond–bendingpotentialfunctionactingbetweenbondedparticleslocatedatcoordinatesri,rjandrk.Figuresrefg:BondBendA01throughrefg:BondBendA03depicttheharmonicbondbendingasharmonicleafspringsandshowsthe9suchinterationsanysinglelatticeatomhaswithitsneighbors. A B CFigureA-3. Figure A-3A illustratestheharmonicbondstretchingpotentialbetween3bondedatomstoapproximatethepotentialenergychangewhentheanglebetweentheatomsisincreased/decreased.Figure A-3B showsthe3bond–stretchingpotentialsactingonanarbitrarylatticeatom0,whereatom0isthecentralbodyinthebondedtriplet.Figure A-3C showstheremaining6interactionswhereatom0isaperiferalatominthetriplet. U(cosijk)=1 2K(cosijk)]TJ /F4 11.955 Tf 11.96 0 Td[(cose)2 (A) 137

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wherecosijk=~rij~rkj krijkkrkjkTheresultingCartesiancomponentforcesoneachparticlearegivenbelow(eqations( A )–( A )).Fxi(ri,rj,rk)=)]TJ /F16 11.955 Tf 25.55 8.09 Td[(@U @(cosijk)@(cosijk) @xi (A)=)]TJ /F5 11.955 Tf 9.3 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.95 0 Td[(cosekrijkkrkjk(xk)]TJ /F5 11.955 Tf 11.95 0 Td[(xj))]TJ /F4 11.955 Tf 11.95 0 Td[((rijrkj)krkjk krijk(xi)]TJ /F5 11.955 Tf 11.95 0 Td[(xj) krijk2krkjk2Fxj(ri,rj,rk)=)]TJ /F16 11.955 Tf 28.2 8.08 Td[(@U @(cosijk)@(cosijk) @xj (A)=)]TJ /F5 11.955 Tf 11.96 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.95 0 Td[(cosekrijkkrkjk(2xj)]TJ /F5 11.955 Tf 11.95 0 Td[(xi)]TJ /F5 11.955 Tf 11.95 0 Td[(xk))]TJ /F4 11.955 Tf 11.95 0 Td[((rijjrkj)krijk krkjk(xk)]TJ /F5 11.955 Tf 11.96 0 Td[(xj)+krkjk krijk(xi)]TJ /F5 11.955 Tf 11.95 0 Td[(xj) krijk2krkjk2Fxk(ri,rj,rk)=)]TJ /F16 11.955 Tf 25.54 8.08 Td[(@U @(cosijk)@(cosijk) @xk (A)=)]TJ /F5 11.955 Tf 9.29 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.96 0 Td[(cosekrijkkrkjk(xi)]TJ /F5 11.955 Tf 11.96 0 Td[(xj))]TJ /F4 11.955 Tf 11.96 0 Td[((rijrkj)krijk krkjk(xk)]TJ /F5 11.955 Tf 11.95 0 Td[(xj) krijk2krkjk2Fyi(ri,rj,rk)=)]TJ /F16 11.955 Tf 25.55 8.09 Td[(@U @(cosijk)@(cosijk) @yi (A)=)]TJ /F5 11.955 Tf 9.3 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.95 0 Td[(cosekrijkkrkjk(yk)]TJ /F5 11.955 Tf 11.95 0 Td[(yj))]TJ /F4 11.955 Tf 11.95 0 Td[((rijrkj)krkjk krijk(yi)]TJ /F5 11.955 Tf 11.95 0 Td[(yj) krijk2krkjk2 138

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Fyj(ri,rj,rk)=)]TJ /F16 11.955 Tf 28.2 8.09 Td[(@U @(cosijk)@(cosijk) @yj (A)=)]TJ /F5 11.955 Tf 11.96 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.95 0 Td[(cosekrijkkrkjk(2yj)]TJ /F5 11.955 Tf 11.95 0 Td[(yi)]TJ /F5 11.955 Tf 11.95 0 Td[(yk))]TJ /F4 11.955 Tf 11.95 0 Td[((rijjrkj)krijk krkjk(yk)]TJ /F5 11.955 Tf 11.96 0 Td[(yj)+krkjk krijk(yi)]TJ /F5 11.955 Tf 11.95 0 Td[(yj) krijk2krkjk2Fyk(ri,rj,rk)=)]TJ /F16 11.955 Tf 25.54 8.09 Td[(@U @(cosijk)@(cosijk) @yk (A)=)]TJ /F5 11.955 Tf 9.29 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.96 0 Td[(cosekrijkkrkjk(yi)]TJ /F5 11.955 Tf 11.96 0 Td[(yj))]TJ /F4 11.955 Tf 11.96 0 Td[((rijrkj)krijk krkjk(yk)]TJ /F5 11.955 Tf 11.95 0 Td[(yj) krijk2krkjk2Fzi(ri,rj,rk)=)]TJ /F16 11.955 Tf 25.55 8.08 Td[(@U @(cosijk)@(cosijk) @zi (A)=)]TJ /F5 11.955 Tf 9.3 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.96 0 Td[(cosekrijkkrkjk(zk)]TJ /F5 11.955 Tf 11.96 0 Td[(zj))]TJ /F4 11.955 Tf 11.95 0 Td[((rijrkj)krkjk krijk(zi)]TJ /F5 11.955 Tf 11.95 0 Td[(zj) krijk2krkjk2Fzj(ri,rj,rk)=)]TJ /F16 11.955 Tf 28.2 8.09 Td[(@U @(cosijk)@(cosijk) @zj (A)=)]TJ /F5 11.955 Tf 11.96 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.95 0 Td[(cosekrijkkrkjk(2zj)]TJ /F5 11.955 Tf 11.95 0 Td[(zi)]TJ /F5 11.955 Tf 11.95 0 Td[(zk))]TJ /F4 11.955 Tf 11.95 0 Td[((rijjrkj)krijk krkjk(zk)]TJ /F5 11.955 Tf 11.96 0 Td[(zj)+krkjk krijk(zi)]TJ /F5 11.955 Tf 11.96 0 Td[(zj) krijk2krkjk2Fzk(ri,rj,rk)=)]TJ /F16 11.955 Tf 25.54 8.09 Td[(@U @(cosijk)@(cosijk) @zk (A)=)]TJ /F5 11.955 Tf 9.29 0 Td[(Krijrkj krijkkrkjk)]TJ /F4 11.955 Tf 11.96 0 Td[(cosekrijkkrkjk(zi)]TJ /F5 11.955 Tf 11.96 0 Td[(zj))]TJ /F4 11.955 Tf 11.96 0 Td[((rijrkj)krijk krkjk(zk)]TJ /F5 11.955 Tf 11.96 0 Td[(zj) krijk2krkjk2 139

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TheinducedquadrupolepotentialforaSWCNTisanextensionofthequadrupolepotentialdevelopedbyHansenandBruch[ 54 ]forgraphene–adsorbateinteractions,inwhichanewFORisdenedforeachcarbonatomandeachFORisnormaltothegraphenesheet.Similarly,inthecaseofaSWCNTwithNcatomsinthelattice,NcFORsaredened,onecenteredoneachofthebondedcarbons,andeachnormaltothecurvatureofthenanotube,pointinginwardtowardthecentralaxis.Figure A-4 providesa2DviewofthetranslationandrotationofthequadrupoleFORandequation( A )givesthepotentialfunction.Thereaderisreferredto[ 131 ]forcontinueddiscussionofthemodel. FigureA-4. IllustrationoftheFORtranslationandrotationusedintheCNTinduceddipolepotentialenergycalculations. UQ(r0)=1 3q 40"x0x03x02)-221(kr0k2 kr0k5+y0y03y02)-222(kr0k2 kr0k5+z0z03z02)-221(kr0k2 kr0k5# (A) 140

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where0isvacuumpermittivity,isthequadrupolemomenttensor,andqisthepartialchargeoftheparticularconstituentofthemoleculeforwhichthepotentialisbeingcalculated.Thequadrupolemomenttensorvalues,takenfrom[ 54 ],areprovidedbelow.x0x0=)]TJ /F4 11.955 Tf 9.3 0 Td[(2y0y0=)]TJ /F4 11.955 Tf 9.3 0 Td[(2z0z0=)]TJ /F4 11.955 Tf 9.3 0 Td[(3.0310)]TJ /F2 7.97 Tf 6.59 0 Td[(40C2 (A) 141

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APPENDIXBRUNGE–KUTTACOEFFICIENTSFORTHEPOINTMASSCASEKi,1=fit0,[r]NC=1,[_r]NC=1=_r[1]i (B)Ji,1=git0,[r]NC=1,[_r]NC=1=r[1]i (B)Ki,2=fit0+t 2,[r]NC=1+t 2Ki,1,[_r]NC=1+t 2Ji,1=_ri+t 2Ji,1=_r[2]i (B)Ji,2=git0+t 2,[r]NC=1+t 2Ki,1,[_r]NC=1+t 2Ji,1=r[2]i (B)Ki,3=fit0+t 2,[r]NC=1+t 2Ki,2,[_r]NC=1+t 2Ji,2=_ri+t 2Ji,2=_r[3]i (B)Ji,3=git0+t 2,[r]NC=1+t 2Ki,2,[_r]NC=1+t 2Ji,2=r[3]i (B)Ki,4=fit0+t 2,[r]NC=1+tKi,3,[_r]NC=1+tJi,3=_ri+tJi,2=_r[4]i (B)Ji,4=git0+t 2,[r]NC=1+tKi,3,[_r]NC=1+tJi,3=r[4]i (B) 142

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APPENDIXCRUNGE–KUTTACOEFFICIENTSFORRIGIDBODYCASEK1=f1(t,_rg)=_rg,1 (C)J1=f2(t,rg,[q])=FT,1 M (C)L1=f3(t,[_q])=[_q]1 (C)M1=f4(t,rg,[q],[_q])=[q]1 (C)K2=f1t+t 2,_rg+t 2J1=_rg,2 (C)J2=f2t+t 2,rg+t 2K1,[q]+t 2L1=FT,2 M (C)L2=f3t+t 2,[_q]+t 2M1=[_q]2 (C)M2=f4t+t 2,rg+t 2K1,[q]+t 2L1,[_q]+t 2M1=[q]2 (C) 143

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K3=f1t+t 2,_rg+t 2J2=_rg,3 (C)J3=f2t+t 2,rg+t 2K2,[q]+t 2L2=FT,3 M (C)L3=f3t+t 2,[_q]+t 2M2=[_q]3 (C)M3=f4t+t 2,rg+t 2K2,[q]+t 2L2,[_q]+t 2M2=[q]3 (C)K4=f1(t+t,_rg+tJ3)=_rg,4 (C)J4=f2(t+t,rg+tK3,[q]+tL3)=FT,4 M (C)L4=f3(t+t,[_q]+tM3)=[_q]4 (C)M4=f4(t+t,rg+tK3,[q]+tL3,[_q]+tM3)=[q]4 (C) 144

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APPENDIXDDISTRIBUTIONFORMULATIONSFROMRAWTRAJECTORYDATAThisappendixprovidesadetailedbreakdownofhowthegeneratedtrajectorydataisefcientlysortedandbinnedsoastogeneratestatisticaldistributionreportedinthisdissertation.Section D.1 coverstheradialoccupancydistributionanditsproposedrelationtotheexpectednumberdensityprole,Section D.2 detailsthedevelopmentoftheintra–collisionalaxialdisplacementdistribution,andtheSection D.3 concludesthisappendixwithalookatthespecularityprolesreportedonpage 112 .ForasimulationofNttimesteps4primarymatricesaresavedandreturned.ThesearedetailedinTable D-1 andwillbeusedthroughoutthisappendix. TableD-1. OutputvectorsfromtheFKMsimulations. NameSizeDescription posNt3Givesthe[x,y,z]componentsofthepositionvectorforthepermeateparticleateverytimestep.velNt3Givesthe[x,y,z]componentsofthevelocityvectorforthepermeateparticleateverytimestep.Ham(Nt+1)2The“Hamiltonian”matrixreportsthetotalenergy(ET)andtime(t)ateachstepofthesimulation.Osc counterNosc1Thiscountermatrixreportsthetimeindexatwhichtheradialvelocityofthepermeatereversesfrompositivetonegative,signifyingawallcollision. D.1RadialOccupancyDistributionThemainassertionoftheFKMisthatthesimulationofasingleparticleoverastatisticallysignicantperiodoftimewillbeindicativeofthebehaviorofaconnedgassolongasthegasissufcientlyrariedortheconnesaresufcientlysmall,i.e.thesimulationisrmlyintheKnudsenregime.Ifthishypothesisholdstruethenifoneweretodeterminethetimefractionthesinglesimulatedparticlespentwithineachdifferentialannulusofthecapillary,thesamedistributionwouldholdtruefortheaggregate.Therefore,duetotheintermolecularindependenceofthemoleculargas,we 145

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havedevelopedaprobabilitydistributionfortheradiallocationsofthesorbateparticles.Ifthenumberdensityatagivenaxiallocationisknown,thenthatvaluetimestheradialprobabilitydistributionjustdiscussedwillgivetheradialnumberdensityprole.TherststepistodividetheporeinteriorintoNrconcentricannularregions,eachwitharadialthicknessofdR= R = Nr .Thentheradialcomponentsofdisplacement(r),initialvelocity(vr)andacceleration(ar)aredeterminedforeachtimestep(dt). dt=Ham(2:end,2)-Ham(1:end-1,2);r=(pos(:,1).+pos(:,2).)..5;vr=(pos(:,1).*vel(:,1)+pos(:,2).*vel(:,2))./r;ar=(vr(2:end)-vr(1:end-1))./dt;Thenextstepistoassigneveryrvaluetotheappropriateannularbin.Thenallofthebinassignmentsandcorrespondingtimesteps,velocitiesandaccelerationsarecollectedinasinglematrixALPHAforsorting.Notethatifbin1-bin0equalszeroforanygiventimestepthenthatmeansthattheitinerantbodyisstillwithinthesameannulusbetweensuccessivetimesteps. bin0=floor(r(1:end-1)/dR);bin1=floor(r(2:end)/dR);ALPHA=[bin0,bin1-bin0,dt,r(1:end-1),vr(1:end-1),ar,r(2:end)];ALPHAisthendividedusinglogicalsortingintomatricesinwhichthepermeatestaysinthesameannulusforagiventimestep(BETA)andoneinwhichthepermeatehasmigratedtooneofthoseadjacent(GAMMA).TheALPHAmatrixcannowbedeletedtoconservememory. BETA=ALPHA(ALPHA(:,2)==0,:);GAMMA=ALPHA(ALPHA(:,2)=0,:);TheGAMMAmatrixisfurthersortedtoisolatethecasesinwhichthepermeatetravelstoaneighboringannulusortraversesseveralwithindt.Notethatthe1psufx 146

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denotestheparticletraveling1binwithinthecorrespondingtimestep.AtthispointtheALPHAandoriginalGAMMAmatricescanbedeletedtoconservememory. GAMMA1p=GAMMA(GAMMA(:,2)==1,:);GAMMAnp=GAMMA(GAMMA(:,2)>1,:);GAMMA1n=GAMMA(GAMMA(:,2)==-1,:);GAMMAnn=GAMMA(GAMMA(:,2)<-1,:);UsinglogicaloperatorstosorttheselargematricesisoneofthegreatestadvantagesoftheMATLABenvironment,astheseoperationscanbeperformedvectoriallyinhundredthsofasecondformatriceswithtensofmillionsofdatum.Nowthatthedataisorganizedanewmatrixisdened(RND)thatwillrecordthetotalamountoftimetheparticlespentineachradialbin.Attheendofthisprocessthebinnedtimeswillbedividedbythetotalsimulationtimetogivethetimefractioneachannuluswasoccupied.TheRDNmatrixiseasiesttollfortheBETAcases.ForeachannulusBETAissearchedtoidentifywhichelementsbelongtothatradialbin,thenthecorrespondingdt'saresummed. RND=zeros(N_ann,1);forii=1:N_annbeta=BETA(BETA(:,1)==ii-1,:);RND(ii)=sum(beta(:,3));endInthecaseswheretheparticletravelstooneoftheneighboringannuliorbeyondtheconstantaccelerationequationsofmotionmustbesolvedtodeterminethetimefractionofdtthattheparticleoccupieseachbin.t=p jvr0+2ar(r1)]TJ /F5 11.955 Tf 11.96 0 Td[(r0)j)]TJ /F5 11.955 Tf 19.12 8.09 Td[(vr0 ar 147

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OncetheRNDmatrixhasbeengeneratedtheprolecanbeplottedvs.radialpositionasshowninFigure D-1 . FigureD-1. ThetimefractioninwhichaLJO2particleoccupiesaspecicradiallocationinastatic(10,10)SWCNTwithaneffectivetemperatureandpressureof50Cand1atmrespectivelyovera50kwallcollisionsimulation. D.2Intra–CollisionalAxialDisplacementDistributionTheintra–collisionalaxialdisplacementdistribution(lambda z)isafareasierdatasettodevelop,astheindicesatwhichtheitinerantbodyexperiencesa“wallcollision”arestoredintheOsc countermatrix,andallofthez-componentsofpositionarestoredinthe3rdcolumnoftheposmatrix.Assuch,thedistributioncanbegeneratedfromthefollowing: lambda_z=pos(Osc_counter(2:end),3)-pos(Osc_counter(1:end-1),3);andcanbereportedinahistogramasshowninFigure D-2 . D.3SpecularityDistributionToassesshowspecularthepost–wallcollisionreectionsareaspecularityfactor()isdenedinwhichavalueof0representsaperfectlyspecularreectionand1signiesareection90offsetfromspecular.ThenatranslatedarotatedFORis 148

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FigureD-2. Theintra–collisionalaxialdisplacementdistributionofaLJO2particleinastatic(10,10)SWCNTwithaneffectivetemperatureandpressureof50Cand1atmrespectivelyovera50kwallcollisionsimulation. denedateachcollisionsite,eachFORz0-axisbeingparalleltotheCNTaxisandeachFORy0-axisbeingnormaltotheCNTwallasshowninFigure D-3 . FigureD-3. Illustrationofthewall–collisionalFORusedindetermininghowspecularthereectionsareinsideaCNT. 149

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ThetwoincomingandtwooutgoingdatapointsarereferencedbyusingtheOsc countermatrixandusedtoapproximatetheincidenceandreectionangles.ThesearefurtherbrokendownintotheirCartesianconstituentsandthencanbeplottedforeachasshowninFigure 4-9 . 150

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APPENDIXESAMPLEMATLABCODEFORFINITEKNUDSENMODEL E.1VectorizedCodeSamples %Lennard-JonesPotentialFunctioninducedbythelattice[J/mol]ULJ=@(x,y,z,CPMx,CPMy,CPMz)4*eps*sum((sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).-...(sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).);%AccelerationComponentsinducedbyLennard-JonesForcesofthelattice[Angstrom/ps]%Note:The10-4istheconversionfactorfromJ/kg/AngstromtoAngstroms/psax=@(x,y,z,CPMx,CPMy,CPMz)24*eps*sum(((x-CPMx)./((x-CPMx).+(y-CPMy).+(z-CPMz).)).*...(2*(sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).-...(sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).))/MW*10-4;ay=@(x,y,z,CPMx,CPMy,CPMz)24*eps*sum(((y-CPMy)./((x-CPMx).+(y-CPMy).+(z-CPMz).)).*...(2*(sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).-...(sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).))/MW*10-4;az=@(x,y,z,CPMx,CPMy,CPMz)24*eps*sum(((z-CPMz)./((x-CPMx).+(y-CPMy).+(z-CPMz).)).*...(2*(sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).-...(sigma./((x-CPMx).+(y-CPMy).+(z-CPMz).)).))/MW*10-4;whilecounter(ii,1)
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CPM(:,1),CPM(:,2),CPM(:,3));ax3=ax(pos(i,1)+dt*(vel(i,1)+.5*dt*ax1),...pos(i,2)+dt*(vel(i,2)+.5*dt*ay1),...pos(i,3)+dt*(vel(i,3)+.5*dt*az1),...CPM(:,1),CPM(:,2),CPM(:,3));ay3=ay(pos(i,1)+dt*(vel(i,1)+.5*dt*ax1),...pos(i,2)+dt*(vel(i,2)+.5*dt*ay1),...pos(i,3)+dt*(vel(i,3)+.5*dt*az1),...CPM(:,1),CPM(:,2),CPM(:,3));az3=az(pos(i,1)+dt*(vel(i,1)+.5*dt*ax1),...pos(i,2)+dt*(vel(i,2)+.5*dt*ay1),...pos(i,3)+dt*(vel(i,3)+.5*dt*az1),...CPM(:,1),CPM(:,2),CPM(:,3));pos(i+1,:)=pos(i,:)+dt*(6*vel(i,:)+dt*[ax0+ax1+ax2,ay0+ay1+ay2,az0+az1+az2])/6;vel(i+1,:)=vel(i,:)+dt*[ax0+2*ax1+2*ax2+ax3,ay0+2*ay1+2*ay2+ay3,az0+2*az1+2*az2+az3]/6;E2=.5*MW*sum(vel(i+1,:).)*1e4+ULJ(pos(i+1,1),pos(i+1,2),pos(i+1,3),CPM(:,1),CPM(:,2),CPM(:,3));Ham(i+1,:)=[E2,t+dt];t=t+dt;i=i+1;end E.2DiscussionofAdaptiveTimeStepsAprevalentMDpracticeistochoosethelargesttimestepwhichwillallowforaccurateresolutionofthemolecularmotion,whichleadstotheconventionofchoosingthetimesteptobeanorderofmagnitudelessthanthesmallestoscillationfrequencyinthesystem.Therefore,foradynamicSWCNTtimestepsontheorderof0.5fsarecommon,asthisallowsformoderatelyaccurateresolutionoftheprominentradialbreathingmodesofthelattice.However,ifonewishedtoproperlyresolvethehydrogenbondoscillationsofagroupofwatermolecules,timestepsontheorderofattosecondswouldbenecessary(whichoftenleadstotheuseofaconstrainedbondmodels).AsshowninTable 3-1 ,theaccuracyofthenumericalintegrationschemescanberelatedtothetimestep,butanotherfactoristhestiffnessofthefunctionbeingintegrated.Figure E-1 showsthe6-12LJmolarpotentialenergyfunctionactingbetween2carbonatoms 152

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asafunctionofthecenter–to–centerdistancebetweennuclei(CC=4.02A,"CC=251.04 J = mol ). FigureE-1. ULJvs.krijkforapairofatomiccarbonsusingthe6LJmodel.Thetransitionfrompositivetonegativepotentialoccursatkrijk=CCandthepotentialenergyminimum(orwelldepth)isequalto"CC. Notethetransitionfromnegativetopositivepotentialatkrijk=CC,markingthetransitionfromattractivevanderWaalsforcedominantbehaviortostronglyrepulsiveBornforcesdominating.Asshowningure E-1 thistransitionoccursverysuddenlyandcanleadtolargenumericalerrorsiftoolargeatimestepistakeincloseproximitytothiszero–forceradius.Forthesakeofargumentallowt0beanappropriatelysmalltimestepforthenearCCregion,resultingin<0.001%changeinthetotalenergycalculation.Inthekrijk>CCregiont0nolongergivesanoptimaltimesteptonumericalerrorratio,andcantypicallybeincreasedbyanorderofmagnitude. 153

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Therefore,anadaptivetimesteproutinewasdevelopedinordertochoosethetimestepthatwouldresultinaspeciederrortolerancetomaximizethesimulationefciency.Thecalculationsaredonebeforetheactualsimulationbegins.Themoleculeofinterestispositionedradiallyoff–centeredwithintheporeandinitializedwitharandomizedvelocityvectorwhosemagnitudeisequivalenttothatofthethermalvelocity.Therstapproximationofthemoleculeaccelerationiscalculatedandanoverlylargetimestepischosenandusedinthefulltrajectorycalculations.Ifthetotalenergychangeisgreaterthansomeuserspeciedtolerancethenthetimestepisdecrementedby0.01%andtheprocessisrepeateduntilsaidtoleranceissatised,atwhichpointthemoleculeradialpositionisincreased,thevelocityisreinitializedandtheentireprocessisrepeateduntiltheentirerangeofpotentialradialpositionshavebeencovered.Thisgeneratesadiscretizedlookuptablethatcomparestheinitialaccelerationapproximation(ameasureofthestiffness)totheoptimalt.Asthesimulationisrunning,foreverynewpositionthistableislinearlyinterpolatedtoprovidetheappropriatetimestep. 154

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REFERENCES [1] AckermanDM,SkoulidasAI,ShollDS,JohnsonJK.DiffusivitiesofArandNeinCarbonNanotubes.MolSimulat.2003;29:677. [2] AlexiadisA,KassinosS.Inuenceofwatermodelandnanotuberigidityonthedensityofwaterincarbonnanotubes.ChemEngSci.2008;63:2793. [3] AllenMP,TildesleyDJ.ComputerSimulationofLiquids.Oxford:ClarendonPress;1987. [4] AmiraS,SpangbergD,HermanssonK.DerivationandevaluationofaexibleSPCmodelforliquidwater.ChemPhys.2004;303:327. [5] AndersenHC.Rattle:A“velocity”versionoftheshakealgorithmformoleculardynamicscalculations.JComputPhys.1983;52:24. [6] AtkinsonKE.AnIntroductiontoNumericalAnalysis(2ndedition)NewYork:JohnWiley&Sons;1989 [7] BakerRW.FutureDirectionsofMembraneGasSeparationTechnology.IndEngChemRes.2002;41:1393. [8] BarbirF.PEMFuelCells:TheoryandPractice.SanDiego:ElsevierAcademicPress;2005. [9] BerendsenHJC,GrigeraJR,StraatsmaTP.TheMissingTerminEffectivePairPotentials.JPhysChem.1987;91:6269. [10] BerendsenHJC.SimulatingthePhysicalWorld:HierarchicalModelingfromQuantumMechanicstoFluidDynamics.NewYork:CambridgeUniversityPress;2007. [11] BhatiaSK,JeppsO,NicholsonD.TractablemoleculartheoryoftransportofLennard-Jonesuidsinnanopores.JChemPhys.2004;120:4472. [12] BhatiaSK,ChenH,ShollDS.Comparisonsofdiffusiveandviscouscontributionstotransportcoefcientsoflightgasesinsingle-walledcarbonnanotubes.MolSimulat.2005;31:643. [13] BhatiaSK,NicholsonD.Transportofsimpleuidsinnanopores:Theoryandsimulation.AIChEJ.2006;52:29. [14] BhatiaSK,NicholsonD.Anomaloustransportinmolecularlyconnedspaces.JChemPhys.2007;127:124701. [15] BhatiaSK,NicholsonD.Frictionbasedmodelingofmulticomponenttransportatthenanoscale.JChemPhys.2008;129:164709. 155

PAGE 156

[16] BhatiaSK,NicholsonD.ModelingMixtureTransportattheNanoscale:DeparturefromExistingParadigms.PhysRevLett.2008;100:236103. [17] BhatiaSK.Modelingpuregaspermeationinnanoporousmaterialsandmembranes.Langmuir.2010;26:8373. [18] BhatiaSK,NicholsonD.Moleculartransportinnanopores:atheoreticalperspective.PhysChemChemPhys.2011;13:15350. [19] BhatiaSK,NicholsonD.SomepitfallsintheuseoftheKnudsenequationinmodellingdiffusioninnanoporousmaterials.ChemEngSci.2011;66:284. [20] BirdRB,StewartWE,LightfootEN.TransportPhenomena(2ndedition).NewYork:JohnWiley&Sons;2002. [21] BitsanisI,VanderlickTK,TirrellM,DavisHT.Atractablemoleculartheoryofowinstronglyinhomogeneousuids.JChemPhys.1988;89:3152. [22] BojanMJ,SteeleWA.InteractionsofDiatomicMoleculeswithGraphite.Langmuir.1987;3:1123. [23] BrennerDW.Empiricalpotentialforhydrocarbonsforuseinsimulatingthechemicalvapordepositionofdiamondlms.PhysRevB.1990;42:9458. [24] BrennerDW,ShenderovaO,HarrisonJ,StuartSJ,NiB,SinnottSB.Asecond-generationreactiveempiricalbondorder(REBO)potentialenergyexpressionforhydrocarbons.JPhys:CondensMatter.2002;14:783. [25] CelestiniF,MortessagneF.Cosinelawattheatomicscale:TowardrealisticsimulationsofKnudsendiffusion.PhysRevE.2008;77:021202. [26] CelestiniF,MortessagneF.Cosinelawattheatomicscale:TowardrealisticsimulationsofKnudsendiffusion.PhysRevE.2008;77:021202. [27] ChapmanS,CowlingTG.TheMathematicalTheoryofNon-UniformGases.NewYork:CambridgeUniversityPress;1952. [28] ChenH,JohnsonJK,ShollDS.TransportDiffusionofGasesIsRapidinFlexibleCarbonNanotubes.JPhysChemB.2006;110:1971. [29] ChungTH,AjlanM,LeeLL,StarlingKE.Generalizedmultiparametercorrelationfornonpolarandpolaruidtransportproperties.IndEngChemRes.1988;27:671. [30] ChungIJ,LeeK-R,HwangS-T.SeparationofCFC-12fromairbypolyimidehollow-bermembranemodule.JMembrSci.1995;105:177. [31] CollinsPG,AvourisP.Nanotubesforelectronics.SciAm.2000;283:62. 156

PAGE 157

[32] CollinsPG.DefectsandDisorderinCarbonNanotubes.In:NarlikarAV,FuYY.OxfordHandbookofNanoscienceandTechnology:FrontiersandAdvancesVol.2.Oxford:OxfordUniversityPress;2010. [33] CracknellRF,NicholsonD,QuirkeN.DirectMolecularDynamicsSimulationofFlowDownaChemicalPotentialGradientinaSlit-ShapedMicropore.PhysRevLett.1995;74:2463. [34] DamnjanovicM,MilosevicI,VukovicT,SredanovicR.Fullsymmetry,opticalactivity,andpotentialsofsingle-wallandmultiwallnanotubes.PhysRevB.1999;60:2728. [35] DarkenLS.Diffusion,mobilityandtheirinterrelationthroughfreeenergyinbinarymetallicsystems.TransAIME.1948;184:41. [36] DresselhausMS,EklundPC.Phononsincarbonnanotubes.AdvPhys.2000;49:705. [37] EhrhardtK,KlusacekK,SchneiderP.Finite-differenceschemeforsolvingdynamicmulticomponentdiffusionproblems.ComputChemEng.2000;12:1151. [38] ElnashaieSSEH,AbdallaBK,HughesR.Simulationoftheindustrialxedbedcatalyticreactorforthedehydrogenationofethylbenzenetostyrene:heterogeneousdustygasmodel.IndEngChemRes.1993;32:2537. [39] EndoM.TheProductionandStructureofPyrolyticCarbonNanotubes(PCNTs).Nature.1993;54:1841. [40] EvansRBIII,WatsonGM,MasonEA.GaseousDiffusioninPorousMediaatUniformPressure.JChemPhys.1961;35:2076. [41] EvansRBIII,WatsonGM,MasonEA.GaseousDiffusioninPorousMedia.II.EffectofPressureGradients.JChemPhys.1962;36:1984. [42] FeresR,YablonskyG.Knudsen'scosinelawandrandombilliards.ChemEngSci.2004;59:1541. [43] FittsDD.NonequilibriumThermodynamics:APhenomenologicalTheoryofIrreversibleProcessesinFluidSystems.NewYork:McGraw-Hill;1962. [44] FletcherJ,LearWE.AdvancedDirectMethanolFuelCellforMobileComputing.PosterPresentedat:FloridaEnergySummit;August,2012;Orlando,FL. [45] FritzL,HofmannD.Moleculardynamicssimulationsofthetransportofwater-ethanolmixturesthroughpolydimethylsiloxanemembranes.Polymer.1997;59:1035. 157

PAGE 158

[46] GearCW.Numericalinitialvalueproblemsinordinarydifferentialequations.EnglewoodCliffs:PrenticeHallPTR;1971. [47] GearCW.TheAutomaticIntegrationofOrdinaryDifferentialEquations.CommunACM.1971;14:176. [48] GrandisonAS.Membraneltrationtechniquesinfoodpreservation.In:ZeuthenP,Bgh-SrensenL.FoodPreservationTechniques.BocaRaton:CRCPressLLC;2003.263. [49] GraySK,NoidDW,SumpterBG.Symplecticintegratorsforlargescalemoleculardynamicssimulations:Acomparisonofseveralexplicitmethods.JChemPhys.1994;101:4062. [50] GudmundssonK.Anapproachtodeterminingthewatervapourtransportpropertiesofbuildingmaterials.NordicJBuildPhys.2003;3:1. [51] GudmundssonK.Alternativemethodsforanalysingmoisturetransportinbuildings:Utilizationoftracergasandnaturalstableisotopes[dissertation].Stockholm,Sweden:KungligaTekniskaHogskolan;2003. [52] GruenerS,HuberP.KnudsenDiffusioninSiliconNanochannels.PhysRevLett.2008;100:064502. [53] GuillotB.Areappraisalofwhatwehavelearntduringthreedecadesofcomputersimulationsonwater.JMolLiquids.2002;101:219. [54] HansenFY,BruchLW.Molecular-dynamicsstudyofthedynamicalexcitationsincommensuratemonolayerlmsofnitrogenmoleculesongraphite:Atestofthecorrugationinthenitrogen-graphitepotential.PhysRevB.1995;51:2515. [55] HejtmanekV,CapekP,SolcovaO,SchneiderP.Dynamicsofpressurebuild-upaccompanyingmulticomponentgastransportinporoussolids:inertgases.ChemEngJ.1998;70:189. [56] HindsBJ,ChopraN,RantellT,AndrewsR,GavalasV,BachasLG.Alignedmultiwalledcarbonnanotubemembranes.Science.2004;303:62. [57] HofmannDWM,FritzL,UlbrichJ,PaulD.Molecularsimulationofsmallmoleculediffusionandsolutionindenseamorphouspolysiloxanesandpolyimides.ComputTheorPolymSci.2000;10:419. [58] HofmannDWM,FritzL,UlbrichJ,SchepersC,BohningM.Detailed-atomisticmolecularmodelingofsmallmoleculediffusionandsolutionprocessesinpolymericmembranematerials.MacromolTheorySimul.2000;9:293. [59] HofmannDWM,KuleshovaL,D'AguannoB.MoleculardynamicssimulationofhydratedNaonwithareactiveforceeldforwater.JMolModel.2008;14:225. 158

PAGE 159

[60] HoltJK,ParkHG,WangY,StadermannM,ArtyukhinAB,GrigoropoulosCP,NoyA,BakajinO.FastMassTransportThroughSub-NanometerCarbonNanotubes.Science.2006;312:1034. [61] HooverWG.Nonequilibriummoleculardynamics:therst25years.PhysicaA.1993;194:450. [62] HoruzI,FletcherJ,KuoCC,CredleS,LearWE.ComparisonofDirectMethanolFuelCellsagainstConventionalBatteries.In:6thInternationalEnergyConversionEngineeringConference.Cleveland:AmericanInstituteofAeronauticsandAstronautics;2008.1. [63] HummerG,RasaiahJC,NoworytaJP.Waterconductionthroughthehydrophobicchannelofacarbonnanotube.Nature.2001;414:188. [64] IijimaS.Helicalmicrotubulesofgraphiticcarbon.Nature.1991;354:56. [65] IijimaS,IchihashiT.Single-shellcarbonnanotubesof1-nmdiameter.Nature.1993;363:603. [66] JacksonE,HillmyerM.Nanoporousmembranesderivedfromblockcopolymers:fromdrugdeliverytowaterltration.ACSNANO.2010;4:3548. [67] JanezicD,OrelB.ImplicitRunge–KuttaMethodforMolecularDynamicsIntegration.JChemInfCompuSci.1993;33:252. [68] JasperAW,MillerJA.LennardJonesparametersforcombustionandchemicalkineticsmodelingfromfull-dimensionalintermolecularpotentials.CombustFlame.2014;161:101. [69] JeppsOG,BhatiaSK,SearlesDJ.WallMediatedTransportinConnedSpaces:ExactTheoryforLowDensity.PhysRevLett.2003;91:126102. [70] JeppsOG,BhatiaSK,SearlesDJ.Modelingmoleculartransportinslitpores.JChemPhys.2004;120:5396. [71] JorgensenWL,ChandrasekharJ,MaduraJD,ImpeyRW,KleinML.Comparisonofsimplepotentialfunctionsforsimulatingliquidwater.JChemPhys.1983;79:926. [72] JosephS,AluruNR.WhyAreCarbonNanotubesFastTransportersofWater?NanoLett.2008;8:452. [73] KahnD,LuJP.Vibrationalmodesofcarbonnanotubesandnanoropes.PhysRevB.1999;60:6535. [74] KargerJ,RuthvenDM,TheodorouDN.Chapter8:MolecularDynamicsSimulations.In:DiffusioninNanoporousMaterials.Weinheim:WILEY-VCHVerlagGmbH&Co.KGaA;2012. 159

PAGE 160

[75] KerkhofPJaM.AmodiedMaxwell-Stefanmodelfortransportthroughinertmembranes:thebinaryfrictionmodel.ChemEngJ.1996;64:319. [76] KerkhofPJaM,GeboersMAM,PtasinskiKJ.Ontheisothermalbinarymasstransportinasinglepore.ChemEngJ.2001;83:107. [77] KerkhofPJaM,GeboersMAM.TowardaUniedTheoryofIsotropicMolecularTransportPhenomena.AIChEJ.2005;51:79. [78] KnudsenM.DieGesetzederMolekularstromungundderinnerenReibungsstromungderGasedurchRoheren.AnnPhys.1909;28:75. [79] KnudsenM,FisherWJ.TheMolecularandFrictionalFlowofGasesinTubes.PhysRevLett.1910;31:586. [80] KnudsenM.Thecosinelawinthekinetictheoryofgases.AnnPhys.1915;48:1113. [81] KotsalisEM,WaltherJH,KoumoutsakosP.Multiphasewaterowinsidecarbonnanotubes.IntJMultiphaseFlow.2004;30:995. [82] KishnaR,WesselinghJA.TheMaxwell-Stefanapproachtomasstransfer.ChemEngSci.1997;52:861. [83] KrishnamurthyR,LernerSL,MacLeanDL,inventors;TheBocGroup,Inc.assignee.Argonrecoveryfromammoniaplantpurgegasutilizingacombinationofcryogenicandnon-cryogenicseparatingmeans.USPatent4,752,311.June21,1988. [84] LawrenceCP,SkinnerJL.FlexibleTIP4Pmodelformoleculardynamicssimulationofliquidwater.ChemPhysLett.2003;372:842. [85] LembergHL,StillingerFH.Central-forcemodelforliquidwater.JChemPhys.1975;62:1677. [86] MaginnEJ,BellAT,TheodorouDN.Transportdiffusivityofmethaneinsilicalitefromequilibriumandnonequilibriumsimulations.JPhysChem.1993;97:4173. [87] MahoneyMW,JorgensenWL.Ave-sitemodelforliquidwaterandthereproductionofthedensityanomalybyrigid,nonpolarizablepotentialfunctions.JChemPhys.2000;112:8910. [88] MajumderM,ChopraN,HindsBJ.MassTransportthroughCarbonNanotubeMembranesinThreeDifferentRegimes:IonicDiffusionandGasandLiquidFlow.ACSNano.2011;5:3867. [89] MaoZ,SinnottSB.AComputationalStudyofMolecularDiffusionandDynamicFlowthroughCarbonNanotubes.JPhysChemB.2000;104:4618. 160

PAGE 161

[90] MaoZ,SinnottSB.SeparationofOrganicMolecularMixturesinCarbonNanotubesandBundles:MolecularDynamicsSimulations.JPhysChemB.2001;105:6916. [91] MasonEA.HigherApproximationsfortheTransportPropertiesofBinaryGasMixtures.II.Applications.JChemPhys.1957;27:782. [92] MasonEA,EvansRBIII,WatsonGM.GaseousDiffusioninPorousMedia.III.ThermalTranspiration.JChemPhys.1963;38:1808. [93] MasonEA,MalinauskasAP.Gastransportinporousmedia:Thedustygasmodel.Amsterdam:Elsevier;1983. [94] MohrM,MachonM,ThomsenC,MilosevicI,DamnjanovicM.Mixingofthefullysymmetricvibrationalmodesincarbonnanotubes.PhysRevB.2007;75:195401. [95] MoonJ-H,BaeJ-H,BaeY-S,ChungJ-T,LeeC-H.Hydrogenseparationfromreforminggasusingorganictemplatingsilica/aluminacompositemembrane.JMembrSci.2008;318:45. [96] MullerEA,RullLF,VegaLF,GubbinsKE.AdsorptionofWateronActivatedCarbons:AMolecularSimulationStudy.JPhysChem.1996;100:1189. [97] NicholsonD.Asimulationstudyoftheporesizedependenceoftransportselectivityincylindricalpores.MolPhys.2002;100:2151. [98] NikolaevP,BronikowskiMJ,BradleyRK,RohmundF,ColbertDT,SmithKA,SmalleyRE.Gas-phasecatalyticgrowthofsingle-walledcarbonnanotubesfromcarbonmonoxide.ChemPhysLett.1999;313:91. [99] NordlundJ,LindberghG.Temperature-DependentKineticsoftheAnodeintheDMFC.JElectrochemSoc.2004;151:A1357–A1362. [100] OstuniR,FilippiE,SkinnerGF,inventors;AmmoniaCasaleSA,assignee.HydrogenandNitrogenRecoveryfromAmmoniaPurgeGas.USPatentApp.13/639,860.February14,2013. [101] PhillipWA,RzayevJ,HillmyerMA,CusslerEL.Gasandwaterliquidtransportthroughnanoporousblockcopolymermembranes.JMembrSci.2006;286:144. [102] PoirierDR,GeigerGH.TransportPhenomenainMaterialsProcessing.Warrendale:TheMinerals,Metals&MaterialsSociety;1994. [103] PollardWG,PresentRD.OnGaseousSelf-DiffusioninLongCapillaryTubes.PhysRev.1948;73:762. 161

PAGE 162

[104] RaoAM,RichterE,BandowS,ChaseB,EklundPC,WilliamsKA,FangS,SubbaswamyKR,MenonM,ThessA,SmalleyRE,DresselhausG,DresselhausMS.Diameter-SelectiveRamanScatteringfromVibrationalModesinCarbonNanotubes.Science.1997;275:187. [105] RaoAM,BandowS,RichterE,EklundPC.Ramanspectroscopyofpristineanddopedsinglewallcarbonnanotubes.ThinSolidFilms.1998;331:141. [106] ReichS,ThomsenC,MaultzschJ.CarbonNanotubes:BasicConceptsandPhysicalProperties.Weinheim:WILEY-VCHVerlagGmbH&Co.;2004. [107] RemickRR.BinaryandTernaryGasDiffusionthroughFinePoresintheTransitionRegionbetweentheKnudsenandMolecularRegions[dissertation].Columbus,OH:TheOhioStateUniversity;1972. [108] RemickRR,GeankoploisCJ.BinaryDiffusionofGasesinCapillariesintheTransitionRegionbetweenKnudsenandMolecularDiffusion.IndEngChemFundam.1973;12:214. [109] RyckaertJ-P,CiccottiG,BerendsenHJC.Numericalintegrationofthecartesianequationsofmotionofasystemwithconstraints:moleculardynamicsofn–alkanes.JComputPhys.1977;23:327. [110] SchneiderP.MulticomponentIsothermalDiffusionandForcedFlowofGasesinCapillaries.ChemEngSci.1978;33:1311. [111] ScottDS,DullienFAL.Diffusionofidealgasesincapillariesandporoussolids.AIChEJ.1962;8:113. [112] ScottK,TaamaW,CruickshankJ.Performanceofadirectmethanolfuelcell.JApplElectrochem.1998;28:289. [113] SkoulidasAI,AckermanDM,JohnsonJK,ShollDS.RapidTransportofGasesinCarbonNanotubes.PhysRevLett.2002;89:185901. [114] SkoulidasAI,ShollDS.TransportDiffusivitiesofCH4,CF4,He,Ne,Ar,Xe,andSF6inSilicalitefromAtomisticSimulations.JPhysChemB.2002;106:5058. [115] SkoulidasAI,ShollDS,JohnsonJK.Adsorptionanddiffusionofcarbondioxideandnitrogenthroughsingle-walledcarbonnanotubemembranes.JChemPhys.2006;124:054708. [116] vonSmoluchowskiM.ZurKinetischenTheoriederTranspirationundDiffusionverdunnterGase.AnnPhys.1910;33:1559. [117] SokhanVP,NicholsonD,QuirkeN.Phononspectrainmodelcarbonnanotubes.JChemPhys.2000;113:2007. 162

PAGE 163

[118] SoperAK.Theradialdistributionfunctionsofwaterandicefrom220to673Kandatpressuresupto400MPa.ChemPhys.2000;258:121. [119] SoukupK,SchneiderP,SolcovaO.ComparisonofWicke–KallenbachandGraham'sdiffusioncellsforobtainingtransportcharacteristicsofporoussolids.ChemEngSci.2008;63:1003. [120] StoweK.IntroductiontoStatisticalMechanicsandThermodynamics.NewYork:JohnWiley&Sons;1984. [121] SuwanwarangkulR,CroisetE,FowlerMW,DouglasPL,EntchevE,DouglasMA.PerformancecomparisonofFicks,dusty-gasandStefanMaxwellmodelstopredicttheconcentrationoverpotentialofaSOFCanode.JPowerSources.2003;122:9. [122] TelemanO,JonssonB,EngstromS.Amoleculardynamicssimulationofawatermodelwithintramoleculardegreesoffreedom.MolPhys.1987;60:193. [123] TheodorouDN,SuterUW.DetailedMolecularStructureofaVinylPolymerGlass.Macromolecules.1985;18:1467. [124] ThessA,LeeR,NikolaevP,DaiH,PetitP,RobertJ,XuC,LeeYH,KimSG,RinzlerAG,ColbertDT,ScuseriaGE,TomanekD,FischerJE,SmalleyRE.CrystalineRopesofMetallicCarbonNanotubes.Science.1996;273:483. [125] TjatjopoulosGJ,FekeDL,AdinMannJJr.Molecule-MicroporeInteractionPotentials.JPhysChem.1988;92:4006. [126] TruittJ,SmithNV,WatsonGM,EvansRBIII,MasonEA.TransportofNobleGasesinGraphites(No.ORNL-TM-135).ProgressReportforthePeriodJanuary31,1961toJanuary31,1962.OakRigde,TN:OakRidgeNationalLab;1962. [127] TseronisK,KookosIK,TheodoropouloC.Modellingmasstransportinsolidoxidefuelcellanodes:acaseforamultidimensionaldustygas-basedmodel.ChemEngSci.2008;63:5626. [128] TuzunRE,NoidDW,SumpterBG,MerkleRC.Dynamicsofuidowinsidecarbonnanotubes.Nanotech.1996;7:241. [129] VeldsinkJW,vanDammeRMJ,VersteegGF,vanSwaaijWPM.Theuseofthedusty-gasforthedescriptionofmasstransportwithchemicalreactioninporousmedia.ChemEngJ.1995;57:115. [130] VincentiWG,KrugerCHJr.IntroductiontoPhysicalGasDynamics.Huntington:RobertE.KriegerPublishingCompany;1965. [131] WaltherJH,JaffeR,HaliciogluT,KoumoutsakosP.CarbonNanotubesinWater:StructuralCharacteristicsandEnergetics.JPhysChemB.2001;105:9980. 163

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[132] WaltherJH,RitosK,Cruz-ChuER,MegaridisCM,KoumoutsakosP.BarrierstoSuperfastWaterTransportinCarbonNanotubeMembranes.NanoLetters.2013;13:1910. [133] WerderT,WaltherJH,JaffeRL,HaliciogluT,NocaF,KoumoutsakosP.MolecularDynamicsSimulationofContactAnglesofWaterDropletsinCarbonNanotubes.NanoLetters.2001;1:697. [134] YoungJB,ToddB.Modellingofmulti-componentgasowsincapillariesandporoussolids.IntJHeatMassTransfer.2005;48:5338. [135] YuM,FunkeHH,FalconerJL,NobleRD.Highdensity,vertically-alignedcarbonnanotubemembranes.NanoLetters.2009;9:225. [136] ZhengJ,LennonEM,TsaoH-K,ShengY-J,JiangS.Transportofaliquidwaterandmethanolmixturethroughcarbonnanotubesunderachemicalpotentialgradient.JChemPhys.2005;122:214702. [137] EG>echnicalServiced,Inc.FuelCellHandbook(7thedition).ContractNo.DE-AM26-99FT40575.Morgantown,WV:U.S.DepartmentofEnergy;2004. 164

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BIOGRAPHICALSKETCH MatthewInmanwasborninOrlando,Floridain1987andisacurrentMechanicalEngineeringgraduatestudentattheUniversityofFlorida(UF).Asanundergraduatehesampledmanydegreepathsincludingarchitecture,mathematics,andcivilengineeringbeforelandingintheMechanicalandAerospaceEngineering(MAE)Departmentforgood.WhileearninghisBachelorofScienceinAerospaceEngineeringhevolunteeredandworkedasbothanundergraduateTAandRA,actedasTreasurerandEngineeringCouncilrepresentativefortheSmallSatelliteDesignClubandtookgraduatelevelcoursework.In2010hewasadmittedintotheUFMAEgraduateprogramandearnedhisMastersofScienceinAerospaceEngineeringwhileworkingwithboththeInterdisciplinaryMicrosystemsGroupandtheGasDynamicsLabonowcontrolandterrestrialcombustionprojectsrespectively.AsaMechanicalEngineeringPh.D.studentworkingintheFuelCellLabundertheguidanceofDr.WilliamE.Lear,hiscourseworkfocusedonmassandheattransport,hisresearchhasfocusedonmoleculardynamicssimulation,andhewasawardedagraduatestudentteachingpositiontogiveseveralthermalmanagementlecturesfortheseniorlevelundergraduateMechanicalDesigncourse.Recreationallyheenjoysindoorrockclimbing,readingJimButchernovels,brewingbeerwithhisbrotherandplayingUltimateFrisbeewithhiswifeandfriends. 165