<%BANNER%>

Thermal Transport in Nanostructured Materials

Permanent Link: http://ufdc.ufl.edu/UFE0022640/00001

Material Information

Title: Thermal Transport in Nanostructured Materials
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Chen, Chia-Yi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: breather, nanoporous, nanotubes, sorbates, zeolites
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Thermal transport in nanostructured materials often exhibits significant deviations from predictions of the classical Fourier's law for thermal conductivity. The deviations occur because the length of the mean free path of heat-carrying phonons is comparable with characteristic length-scale of these materials. Therefore, it is necessary to develop a theory for thermal transport applicable to nanomaterials. In this study we investigate thermal conductivity in two classes of nanomaterials, namely quasi-one-dimensional materials and nanoporous materials with adsorbed guest molecules. For quasi-one-dimensional (Q1D) materials, we aim to understand nonlinear dynamics involved in heat transfer using a combination of molecular dynamics simulations and bifurcation theory. In non-equilibrium molecular dynamics simulations, we observe ballistic propagation of energy packets in model Q1D systems as well as in carbon nanotubes, which suggests the significance of ballistic heat transfer mechanism. To decipher structure of the waves propagating in the lattice, we obtain nonlinearlattice vibration modes by solving fundamental equations of motion numerically, without ignoring any structural details. We focus on localized nonlinear vibration modes, and investigate their properties and stability on the structured details of the lattice, and potential energy of interaction between lattice atoms. In the second part of this work, we investigate the effect of sorbate-lattice interaction in thermal transport for nanoporous materials. There is increasing evidence that thermal conductivity of nanoporous materials can be significantly affected by adsorption of guest molecules. These molecules serve as moving defects and provide additional scattering centers for heat-carrying phonons. In order to understand the sorbate-phonon interactions, we first perform molecular dynamics simulations of a realistic system, namely sodalite zeolite with small molecules (argon, xenon, and methane) encapsulated in its cages. We observe that the phonon lifetime often increases upon encapsulation of a sorbate into the zeolite which suggests that the sorbate-phonon interactions are qualitatively different from phonon scattering by point defects fixed in the lattice. We then proceed to develop a model for the sorbate-lattice interaction. For simplicity, we consider a one-dimensional lattice system. We investigate the role of the sorbate in the energy exchange between lattice modes and observe that even a weak interaction between the sorbate and lattice induces dispersion of energy over a wide spectrum of normal modes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chia-Yi Chen.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Kopelevich, Dmitry I.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022640:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022640/00001

Material Information

Title: Thermal Transport in Nanostructured Materials
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Chen, Chia-Yi
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: breather, nanoporous, nanotubes, sorbates, zeolites
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Thermal transport in nanostructured materials often exhibits significant deviations from predictions of the classical Fourier's law for thermal conductivity. The deviations occur because the length of the mean free path of heat-carrying phonons is comparable with characteristic length-scale of these materials. Therefore, it is necessary to develop a theory for thermal transport applicable to nanomaterials. In this study we investigate thermal conductivity in two classes of nanomaterials, namely quasi-one-dimensional materials and nanoporous materials with adsorbed guest molecules. For quasi-one-dimensional (Q1D) materials, we aim to understand nonlinear dynamics involved in heat transfer using a combination of molecular dynamics simulations and bifurcation theory. In non-equilibrium molecular dynamics simulations, we observe ballistic propagation of energy packets in model Q1D systems as well as in carbon nanotubes, which suggests the significance of ballistic heat transfer mechanism. To decipher structure of the waves propagating in the lattice, we obtain nonlinearlattice vibration modes by solving fundamental equations of motion numerically, without ignoring any structural details. We focus on localized nonlinear vibration modes, and investigate their properties and stability on the structured details of the lattice, and potential energy of interaction between lattice atoms. In the second part of this work, we investigate the effect of sorbate-lattice interaction in thermal transport for nanoporous materials. There is increasing evidence that thermal conductivity of nanoporous materials can be significantly affected by adsorption of guest molecules. These molecules serve as moving defects and provide additional scattering centers for heat-carrying phonons. In order to understand the sorbate-phonon interactions, we first perform molecular dynamics simulations of a realistic system, namely sodalite zeolite with small molecules (argon, xenon, and methane) encapsulated in its cages. We observe that the phonon lifetime often increases upon encapsulation of a sorbate into the zeolite which suggests that the sorbate-phonon interactions are qualitatively different from phonon scattering by point defects fixed in the lattice. We then proceed to develop a model for the sorbate-lattice interaction. For simplicity, we consider a one-dimensional lattice system. We investigate the role of the sorbate in the energy exchange between lattice modes and observe that even a weak interaction between the sorbate and lattice induces dispersion of energy over a wide spectrum of normal modes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chia-Yi Chen.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Kopelevich, Dmitry I.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022640:00001


This item has the following downloads:


Full Text

PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

Itakethisopportunitytostatemydeepsenseofgratitudetomyadvisor,Dr.DmitryI.Kopelevich,forintroducingmetoeldofmodelingandnumericalsimulations.Iamdeeplyindebtedtohispatienceteaching,guidanceandconstantencouragementoutofhisbestinterestforme.IamthankfulforhiseortinadvisingthreePhDstudentsatthesametime,workingwithuslateandbeingtirelesstoimprovemytechnicalwriting.IwouldalsoliketothankDr.AntonyLaddforhisencouragementwhenIwashisteachingassistant,suggestionsforresearchmethodandhiseorttosetupthecomputationalclusterinourdepartment.Ialsoappreciatethemembersofmycommittee,Dr.JasonWeaver,Dr.AntUralfortheiradviceandavailability.Inaddition,IexpressmysincerethankDr.ChauhanandDr.Tsengfortheiradviseformyfuturecareerpath;encouragementandinspirationinbothmyprofessionalandpersonallifeduringthelaststageofmyPhDstudy.IamgratefultothepeopleIhavechancetoworkwithduringthis5years,includingGunjanMohan,AshishGupta,BenjaminJames,Young-MinBan,ChrisCookandYoung-NamAhn.EspeciallyGunjanMohanforassistingmeonthenumericalmethodandprogrammingtechniques.Iwouldliketoexpressmygreatappreciationformyroommate,HanChang,forheremotionalsupportandfriendshipinthelastyearofmystudy.IthankJamieWang,PastorStevePettit,EllaPettit,andPattiBuckelewBryantfortheirlovingwordsandfaithfulprayersatalltime.ThisacknowledgementwouldnotbecompletewithoutDr.Keesling,Pei-HsunWu,ourfriendlysta,facultiesinChemicalEngineeringdepartmentandallmyfriendsinUSandTaiwanformakingmystudyabroadexperienceinGainesvilleanunforgeableexperience.AlsoIacknowledgethefundingfromUniversityofFloridaandcomputationalresourcesofUFHighPerformanceComputingCenter.FormyfamilyinTaiwan:myMother,sistersandbrother,Iamextremelygratefulfortheirlove,understanding,comfortandconstantsupportformethroughnumerousphone 4

PAGE 5

5

PAGE 6

page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 2NUMERICALMETHODS .............................. 18 2.1MolecularDynamicsSimulations ....................... 18 2.2TemperatureCouplinginMolecularDynamics ................ 18 2.2.1BerendsenThermostat ......................... 19 2.2.2Nose-HooverThermostat ........................ 19 2.2.3LangevinThermostat .......................... 20 2.3SteadyStateSolutions ............................. 21 2.4StabilityAnalysis ................................ 22 3NONLINEARLATTICEVIBRATIONMODESINMODELSYSTEMS .... 24 3.1One-DimensionalFPUsystem ......................... 24 3.1.1ThermalRelaxationSimulation .................... 24 3.1.2SteadyStateSolutions ......................... 26 3.1.2.1FPU-model ......................... 26 3.1.2.2FPU-model ......................... 27 3.2LatticeModelSystemsInHigherDimensions ................. 30 3.2.1SingleChainSysteminTwoDimensionalSystem .......... 31 3.2.2TwoCoupledFPUChains ....................... 32 3.2.3BodyCenteredCubicStructure .................... 34 3.3HexagonalTubeModel ............................. 36 3.3.1DispersionRelationship ......................... 38 3.3.2SteadyStateSolutions ......................... 40 3.3.3Conclusions ............................... 43 4NONLINEARLATTICEVIBRATIONALMODESINCARBONNANOTUBES 46 4.1SystemConguration .............................. 47 4.2Non-EquilibriumMDSimulation ....................... 51 4.3SteadyStatesSolutionsinCNTs ....................... 51 4.4Results ...................................... 54 4.4.1Conclusions ............................... 60 6

PAGE 7

...................................... 63 5.1ModelDetails .................................. 65 5.2Simulationdetails ................................ 68 5.3Normalmodesofsodalitecrystal ....................... 70 5.4Nonlinearphononandsorbatedynamics ................... 73 5.5Phononstatistics ................................ 78 5.5.1Conclusions ............................... 86 6MODELDEVELOPMENTFORSORBATEMOLECULESIN1DSYSTEM 88 6.1ThermalConductivityfromNEMDSimulations ............... 88 6.2SorbateinaHarmonicLattice ......................... 91 6.2.1ScatteringofaPhononWavepacket .................. 92 6.2.2ScatteringofaPlaneWave ....................... 96 6.2.3PossibleTheoreticalApproach:Multi-scaleExpansion ........ 98 6.2.4Conclusions ............................... 101 7CONCLUSIONSANDPOSSIBLEDIRECTIONSOFFUTURERESEARCH 102 REFERENCES ....................................... 103 BIOGRAPHICALSKETCH ................................ 111 7

PAGE 8

Table page 3-1Theneighborlistandequilibriumbondlengthfortheparticlesinhexagonalsystem. ........................................ 38 4-1ParametersforBrenner-Tersointeractionpotential. ............... 49 5-1ParametersforthelatticepotentialenergymodelEq. 5{1 ............ 68 5-2Lennard-Jonesparametersforsorbate-sorbateandsorbate-latticeinteractions. 68 5-3Normalizedaveragesofphononamplitudes,Qjk==jk. ........ 79 8

PAGE 9

Figure page 3-1LocalenergyevolutionanddetailofbreatherinteractioninaFPUchainusingtheprotocolofReigadaetal.[ 38 ]fork==1=2,=0. ............. 25 3-2Floquetmultipliesrandtheeectofunstableperturbationofbreathersolutions. 28 3-3Familiesofbreatherswithdierentcongurations. ................. 28 3-4AfamilyofbreathersolutionswithSTmodecongurationandtheirstabilityanalysisforaFPU-system. ............................ 29 3-5ResultsofthestabilityanalysisoftheSTmodebreathersforarangeof,!,andxed=1. .................................... 30 3-6NonlinearvibrationmodeanditsstabilityanalysisforasingleFPUchainintwo-dimensionalsystem. ............................... 32 3-7ThecongurationfortwocoupledFPUchains. .................. 33 3-8NonlinearvibrationmodeanditsstabilityanalysisfortwocoupledFPUchainsintwo-dimensionalsystemwithkFPU=kcoupling=1,==0. ......... 33 3-9NonlinearvibrationmodeanditsstabilityanalysisfortwocoupledFPUchainsintwo-dimensionalsystemwith(kFPU;kcoupling;FPU)=(1;1;1). ......... 34 3-10Thecongurationforbody-centeredcubicstructuresystem. ........... 34 3-11Continuationcurvesandcongurationofnonlinearsolutionscorrespondingtodierentamplitudesofthephononmodes. ..................... 36 3-12Nonlinearvibrationmodeoftwo-dimensionalbody-centeredcubicstructurefor(k;)coupling=(1:10;1:0) ............................... 37 3-13Structureofaunitcellofthemodelhexagonalsysteminthreedimensions. ... 38 3-14Thedispersioncurvesforthree-dimensionalhexagonaltubesystem. ...... 39 3-15Oneofthephononmodesofthehexagonalsystemanditsstabilityanalysiswith(k;)coupling=FPU=(1;0). ............................... 40 3-16InitialguessfortheNewton'smethodfor3Dhexagonalsystem. ......... 41 3-17Dependenceofamplitudeandfrequencyofasteady-statemodeonthestrengthofcouplingbetweenchainsinhexagonalsystem. .................. 41 3-18Congurationofthenonlinearsolutionswith(k;)coupling=(1;1). ........ 42 3-19ThecomparisonofsolutionswithandwithoutRWA. ............... 43 9

PAGE 10

........................................ 43 3-21Stabilityanalysisforthesolutionswith(without)RWAapproximation.Equivalenteigenvectorsareobserved. .............................. 44 3-22Stabilityanalysisforthesolutionswith(without)RWAapproximation.Dierenteigenvectorsareobserved. .............................. 44 4-1PreparationofananotubebyrollingthegraphitesheetinadirectionspeciedbythechiralvectorCh 48 4-2ComparisonofratioofthemagnitudeofthenonlinearandlinearforceofCNTandFPUsystem ................................... 50 4-3TemperatureproleandlocalenergyevolutionofNEMDsimulationsofasegmentofa100unit-cell(5,0)CNT. ............................. 52 4-4Phonondispersioncurvesofthe(5,0)carbonnanotube. .............. 53 4-5ThesummaryofthenumericalproceduretoobtainFourierseriesexpansionforacomplexpotential. ................................. 54 4-6Comparisonbetweenalinearphononmodeofthe(5,0)carbonnanotubeandthenonlinearmodeobtainedfromthismodebythecontinuationmethod. ... 56 4-7DependenceofthemodeenergyandfrequencyoftheTaylorsolutionsonforsolutionsshowninFigure 4-6 ............................ 57 4-8Thesolutionsof(5,0)CNTintwounitcellssystemstartingwiththeamplitudecorrespondingtothreedierenttemperature. ................... 58 4-9DependenceofthemodeenergyandfrequencyoftheTaylorsolutionsstartingwiththreeinitialthermalenergyonforsolutionsshowningure 4-8 ..... 59 4-10Thecomparisonthedisplacementforthe7thatomineachunitcellfor=1nonlinearmodeandstartingphononmodeina24unitcellssystemwiththesimpliedpotentialunderRWAapproximation. .................. 60 4-11Thenonlinearvibrationmodeswithsimpliedpotentialforthesystemsupto24unitcells. ...................................... 61 4-12Congurationofalinearlystablenonlinearsolutionbasedonsimpliedpotentialfunctionforafourunitcellssystem. ........................ 61 4-13Thenonlinearsolutionbasedonsimpliedpotentialfunctionforafourunitcellssystem. ........................................ 62 4-14TheeigenvectorcorrespondingtheFloquetmultipliershownastheopencircleinFigure 4-13 B. ................................... 62 10

PAGE 11

...... 66 5-2Dispersionrelationshipsforsodalite. ........................ 72 5-3ExamplesofautocorrelationfunctionsCjk()ofphononsinasorbate-freesodalitecrystal. ......................................... 76 5-4PowerspectraSjk(!)correspondingtothephononautocorrelationfunctionsshowninFigure 5-3 ................................. 77 5-5PowerspectraSV(!)ofsorbatevelocitiesinrigidzeolitecages. ......... 78 5-6Phononlifetimesjkinsorbate-freesodalitelattice. ............... 80 5-7Eectsofsorbatesonphononlifetimes. ....................... 83 5-8Relativedierences!jkbetweentheanharmonic(!ajk)andharmonic(!hjk)frequenciesinasorbate-freesodalitecrystal. .......................... 84 5-9Relativedierences!sjkbetweenphononfrequenciesinsodalitewithencapsulatedsorbates(!sjk)andinthesorbate-freesodalite(!ajk). ................ 84 5-10Magnitudesofcorrelationcoecients(jk;j0k0)betweenphononmodesjkandj0k0inasorbate-freesodalitecrystal. ........................ 85 5-11Eectofsorbatesonphonon-phononcorrelations. ................. 86 6-1EstablishedtemperatureproleandheatuxinNEMDsimulationsfor1Dsorbate-freelatticesystem. .................................. 89 6-2Dependenceofthermalconductivitiesonsorbate-latticeinteractionparameters. 90 6-3SteadystatetemperatureproleinNEMDsimulationwithimposedtemperaturegradientdT dx=0:002. ................................. 90 6-4Simulationsnapshotsofawavepacketscatteringsimulationforwavenumbercenteredaroundq0=0:1,(;)=(2;0:44)andMs=1. .................. 93 6-5Dependenceoftransmissionratiosonsorbatemassforthreedierentwavenumberswavepackets. ..................................... 93 6-6Scatteringofwavepacketswithwavenumbersq=0:475byasorbatewithxedsigma=0:44and(Ms,epsilon)=(0.5,5)and(1,1). .............. 95 6-7Evolutionofmodeenergyforincidentplanewavewithwavenumberq=0:245interactingwithdierentsorbatemolecules. .................... 96 6-8Fouriertransformofthesorbatedisplacementfortheplanarwavescatteringsimulations. ...................................... 97 11

PAGE 12

............................. 97 6-10ComparisonofFkandGkbetweenmodelpredictionandsimulationresults. ... 100 6-11Comparisonoffrequenciesobtainedfromdegenerateperturbationmethodandregulareigenvaluesolver. ............................... 101 12

PAGE 13

13

PAGE 14

14

PAGE 15

1 ].Toimprovefootprintofelementsofintegratedcircuits,non-volatilememorydevicesusing1Dstructuressuchasnanowiresandnanotubeshavedrawnsignicantattention.Inordertoensuretheperformanceandstabilityofincandesces,itisnecessarytoassessthermalpropertiesofnewnanoscalematerials[ 2 3 ].TheclassicalFourier'slawofthermalconductivity,statesthattheheatuxJislinearlyproportionaltothetemperaturegradientrT, 4 ]: 1{1 )and( 1{2 )arethat(i)thephononmeanfreepathlismuchsmallerthanthecharacteristicsizeofthematerialand(ii)thetemperaturegradientissucientlysmallsothatcollisionsbetweenphononsmaintainlocalequilibrium.Bothoftheseassumptionsarelikelytofailinnanosomiasystems.Forexample,themeanfreepathofheat-carryingphononsinsiliconat300Kis300nmwhereasthecurrentdimensionofathinsiliconlmisunder100nm[ 2 ]andthecharacteristicsizeofahotspotinatransistorcanbeassmallas10nm[ 5 ].Therefore,itisnecessarytoconsidertheeectofboundaryscatteringandphononconnement[ 6 { 8 ]explicitly.Theseissuesare 15

PAGE 16

9 { 11 ],makeastrongcontributiontothethermalenergytransferandleadtoqualitativelydierentthermalpropertiesinlowdimensionalmaterials.Infact,ithasbeenshownboththeoretically,usingtheself-consistentmodecouplingtheory[ 12 13 ],andbynumericalsimulations[ 13 { 15 ]thatthebulkheatconductivityexhibitsanomalousdependenceonthelatticesizeforone-andtwo-dimensionallattices.Inthecurrentwork,weinvestigatetheprocessofthermalconductivityinone-andquasi-one-dimensionalsystemssuchascarbonnanotubes(CNTs)[ 16 ].Duetotheiruniquemechanicalandelectronicproperties,CNTshavedrawnsignicantattentionforawidevarietyofpotentialapplications,[ 17 ]andarecurrentlybeingintegratedintoelectronicdevicessuchashigh-performanceballisticeld-eecttransistors,nanotuberandomaccessmemory(NRAM)andassemblingintegratedlogiccircuitonanindividualcarbonnanotube[ 18 { 20 ].CurrentlythereisonlyincompleteunderstandingregardingthermaltransportforCNTs,whichisextremelyimportantforheatmanagementissue.Recentexperiments[ 21 22 ]andmoleculardynamicssimulations[ 23 { 26 ]reportanomalousthermalconductivityofthenanotubes.However,therearesignicantdiscrepanciesbetweenresultsoftheexperimentalmeasurementsandthesimulations.Moreover,discrepanciesbetweenresultsobtainedusingdierentsimulationtechniquesconrmthattheconventionalthermalconductivitytheoryisinvalidforthissystemandthereisaneedforabetterdescriptionofthermaltransportphenomenaintheseQ1Dstructures.Inthecurrentwork,weusethefollowingtwoapproachestoinvestigatethethermaltransportin 16

PAGE 17

27 28 ]leadstoanorderofmagnitudedecreaseofitsthermalconductivity,andthatinturnincreasesitsthermoelectricgureofmerit.Aseriesofexperimentalandcomputationalstudiesindicatethatoscillations("rattling")ofguestmolecules(sorbatesorcations)withinzeolitesandskutteruditeshavesignicantimpactonthermalconductivityofthesecrystals.Murashovetal.[ 29 ]performedMDsimulationstoshowthatpresenceofcationscanleadtoeitheranincreaseoradecreaseofthermalconductivity,dependingontheirmass.Availabledatasuggestscomplexdependenceofthethermalconductivityonthenatureofaguestmoleculeandahostcrystal.Hencefundamentalunderstandingoftheeectofguestmoleculesonhostmatrixisnecessaryinordertoengineerheatconductancebytheuseofhost-guestinteractions. 17

PAGE 18

un=Fn(fung);(2{2)whereforcesFnaregivenby 2{2 usingthevelocityVerletalgorithm[ 30 ], 2F(t)t2v(t+t)=v(t)+1 2(F(t)+F(t+t))t;(2{4)wherer(t)andv(t)aretheatomicpositionandvelocity,respectively. 2miv2i=kBT 18

PAGE 19

31 ],Nose-Hoover[ 32 ],orLangevinthermostats. T(T0 dt=T0T T:(2{7) 19

PAGE 20

@qipi; dt=1 NotethatthevalueofQshouldbetheinverseofthecharacteristictimescaleofthesysteminordertoavoidinecientthermostatscheme(toolargevalueofQ)orhighfrequencytemperatureoscillations(toosmallvaluesofQ)inthesystem[ 33 ]. un=Fn_un+(t);(2{13)whereisthefrictioncoecientand(t)istherandomforceduetotheparticlecollisions.Therandomforcehasazeromeananditsautocorrelationfunctionsatisestheuctuation-dissipationtheorem, 2{13 implementedinthisworkfollowsthemethodofErmaket.al[ 34 ]. 20

PAGE 21

2{2 .Thecorrespondingsteadystatesolutionscanbethoughtofasperiodicorbitsinthe2Nd-dimensionalphasespace.ThecoordinatesofaphasepointYinthisphasespacearethenormalizedpositionsandvelocitiesoftheatoms,i.e.Y=fu1;:::;uN;_u1;:::;_uNg,whereunand_unared-dimensionalvectors.ThenumericalmethodusedtoobtainthesteadystatesolutionsofEq. 2{2 isbasedontheapproximationoftheperiodicsolutionsbyatruncatedFourierseriesexpansion, 2{15 intoEq. 2{2 ,weobtaintheequationsforxn;j 2{16 usingNewton'smethodtoobtainx.IterationsofNewton'smethodaregivenbyxn+1=xn+x,wherexisasolutionofthefollowinglinearsystemofequations: 2{16 .ThelatticesystemsaretranslationallyinvariantleadingtosingularityintheJacobianmatrix.Inordertoremovethesesingularities,weusethesingularvaluedecomposition(SVD)methodtosolvethesystemoflinearEq. 2{17 .InthismethodmatrixJisdecomposedas 21

PAGE 22

2{17 liesintherangeofJ,thesystemofEq. 2{17 hasmorethanonesolutions,sinceanyvectorinthenullspaceofJcanbeaddedtoasolutionx.Physically,thiscorrespondstoanarbitrarytranslationand/orrotationoftheentiremolecularsystem.SVDenablesustondthesolutionswiththesmallestnorm,i.e.preventstherotationandtranslationofthesystembyignoringthenullspacebasis.TheinverseofmatrixJis 35 ].Considerasteady-stateperiodicsolutionY(0)(t)=fu(0)1(t);:::;u(0)N(t);_u(0)1(t);:::;_u(0)N(t)gofEq. 2{2 ;theperiodofthissolutionisT=2=!.Forconvenience,letusrewritetheequationsofmotionEq. 2{2 asarst-ordersystemofequations, _Yn(t)=fn(t);(2{21) 22

PAGE 23

@Yn;n=1;:::;N: NowconsideraperturbationofY(0)(t)oftheformY(t)=Y(0)(t)+Y(1)(t),1.Theequationslinearizednearthesteady-statesolutionare _Y(1)n=XmLnm(Y(0)(t))Y(1)m;n=1;:::2N;(2{24)wherethetime-periodicddmatricesLnmaregivenby 2{24 ).Thej-thcolumnofthismatrixcorrespondstothesolutionwiththeinitialconditionoftheformY(1;j)(t=0)=(0;0;:::;0;1;0;:::;0)withtheonlynon-zeroelementofthevectorY(1;j)(t=0)locatedatthej-thposition.Thesesolutionsareobtainedbythenumericalintegrationofthesystemoflinearizedequations( 2{24 ).Oncethefundamentalsolutionsareobtained,wecomputetheFloquetmultipliers,i.e.theeigenvaluesj(j=1;:::;2Nd)ofthematrixF(t)attimet=T.Thesteady-statesolutionisstableonlyifalloftheFloquetmultipliershavemagnitudelessthanorequalto1.SincethesystemconsideredinthisworkisHamiltonian,foreacheigenvaluejthenumbersjand1=jarealsoeigenvalues[ 11 ](here,asteriskdenotescomplexconjugation).Thustheneutralstabilityofthesteady-statesolutionrequiresthatallthemultiplierslieontheunitcircleofthecomplexplane. 23

PAGE 24

36 ]system,i.e.alinearanharmonicchainofatomswiththefollowingHamiltonian: 9 11 37 ],DBsareintrinsiclocalizednonlinearvibrationmodesthatarequalitativelydierentfromthephononmodesandcanbeobservedinthermalrelaxationofthelattice.MDsimulationsofthermalrelaxationofFPUsystemhavebeenperformedbyReigadaetal.[ 38 ].Theyprepareda30-sitelatticeinitiallythermalizedatT=0:5byLangevinthermostatandafterthesystemreachedequilibrium,theydisconnectedthethermalizedlatticesystemfromtheheatbathandconnectedtheendsofthechainviaafrictiontermtodissipatetheenergyintothethermalreservoirofzerotemperature.HereweshowtheresultsofoursimulationsfollowingtheirprotocolforaN=50latticewithk==1=2,=0.Theevolutionofthelocalenergyofthissystemduringrelaxation 24

PAGE 25

B C D EFigure3-1. (A)ThebreathersinaFPUchainobtainedusingtheprotocolofReigadaetal.[ 38 ]fork==1=2,=0.Thegrayscaleinthisgurerepresentsthelocalenergymagnitude,withdarkershadingcorrespondingtomoreenergeticregions.Thehorizontalaxisindicatesthepositionalongthechainandtheverticalaxiscorrespondstotime.Theenergydensityisshownbyagrayscalefrom0(white)tothemaximumenergyrecordedduringthesimulation(black).Theenergylocalizesintonarrowbreatherswhichareseenastheblacklinesonthisplot.Thenoisyphononmodescorrespondtorandomlyshadedareas.B),Detailsoftheenergylocalizationandbreatherinteraction.C),D)andE)aresnapshotsfortheprocessofbreather-breathercorrespondingtothreedierenttimesinrelaxationsimulationindicatedbythedashedlinesinplotB. processisplottedinFigure 3-1 A),wherethedarkerregionrepresentshigherenergy.Weobservetheballisticenergytransferclearlyafterthedissipationsofphononmodes.Thebreathersareseentomovewithessentiallyconstantspeedandappeartobestablewithrespecttothelow-energynoisyphononmodes.Detailsofbreather-breatherinteractionsareshowninplotBandsnapshotsforthisprocessareshowninplotsC-E. 25

PAGE 26

39 ],inwhichcoecientswithj2inEq. 2{15 areneglected.Therefore,thenonlineartermsintheequationsofmotionareapproximatedasfollows: 2;cos3(!t)'3 4cos(!t)(3{2)WecompareeachoftheFouriercoecientsinEq. 2{2 tosetuptheequationsforNewton'smethod.SincequadraticforcetermisapproximatedbyaconstantunderRWAapproximation,itleadstonon-zeroaveragedisplacementsxn;0ofatoms. 3-3 A)(lowerinsetplot)isreferredtoasPmode(abbreviationforPage[ 40 ]).TheDBswithsymmetriccongurationinthelowerinsetplotofFigure 3-3 B)arereferredtoasSTmode(abbreviationforSieversandTakeno[ 41 ]).TheinitialguessforNewton'smethodaretwoofthehighfrequencyphononmodeswithwavevectork a,shownintheupperinsetsinFigure 3-3 A)andFigure 3-3 B).Weperformcontinuationofsolutionbygraduallyincreasingand!by=0:05and!=3103until=1.ThefrequencyofDBsshouldbeoutsideofthephononfrequencybandtoavoidresonancewithphononsolutions[ 9 ].ThesetwodegeneratenormalmodesleadtotwofamilyofPmodeandSTmodebreathers.ThedependenceofthemodeenergyonareshowninFigure 3-3 .Linear 26

PAGE 27

2.4 .RecallthatavibrationmodeisstableonlyifallofitsFloquetmultipliersjlieontheunitcircleofthecomplexplane.Theclosedandopencirclesinthesetwoguresrepresentthestableandunstablesolutions,respectively.ResultsofFloquetanalysisforastableSTmodeDBwith=1isshowninFigure 3-2 A),wherejarelocatedontheunitcircle.However,PmodesarelinearlyunstablewithunstablemultipliersbeingpurerealnumbersasshowninFigure 3-2 B).Figure 3-2 C)showstheeigenvectorcorrespondingtomultiplierindicatedbyanopencircleinFigure 3-2 B).Inordertounderstandtheeectofthisperturbationvector,weperformMDsimulationwithinitialconditionasPmodebreatherwithperturbationvector.TwosnapshotsareshowninplotDandE.Initially,thisperturbationresultsintheDBmovingtotherightandeventuallyleadingtoatransitionbetweenPmodeandSTmodecongurations. 2{2 ),weturntotheFPU-model.WeseeksolutionswiththeRWAapproximation,Eq. 3{2 .Now,thesolutionswillcontainastaticcomponent, 3-4 A).WeobservethatthebreatheramplitudeAincreaseswithincreaseofthemagnitudeofthecubicnonlinearity.ThedependenceofAonforaxedfrequency!=2:89andquarticterm=1isshowninFigure 3-4 B).Weobtainedbreatherfamiliesforarangeof!between2.2and2.93(whilekeeping=1),andobservedtheincreasingofAwhileincreasingvalueaswell.Interestingly,weobservethatintroductionofthecubicnonlinearitydestabilizesthebreathermodes.ThestableandunstablebreathersareshownrespectivelybysolidandopencirclesinFigure 3-4 B).Tosummarizetheresultsof 27

PAGE 28

B C D EFigure3-2. FloquetmultipliersDBs:A)STmodeandB)Pmodesolutions.(C)showstheDBcongurationaswellastheunstabledisplacement(ex)andvelocity(ev)perturbationcorrespondingtotheopencircleinplotBThesnapshotsofMDsimulationsforperturbedPmodebreathers.(D)Initialconguration(E)showsthetransitionfromPmodetoSTmode. BFigure3-3. Twofamilyof(A)STmodesand(B)PmodeDBs.Theclosed(open)circlesrepresentthelinearlystable(unstable)modes. 28

PAGE 29

B CFigure3-4. (A)AnSTmodefor=0:2,=1,!=2:87.Boththestatic()anddynamic()atomdisplacementsareshown(B)APmodebreatherfamilyfor!=2:89,=1,withvarying.(C)Resultsofthestabilityanalysisofthebreathersofthefamilyshowninplot(B).ThestabilityparameterisdenedbyEq. 3{5 .Intheseplots,theclosed(open)circlesdenotethelinearlystable(unstable)modes. theFloquetanalysisweintroducethequantity 3-5 B)showsthedependenceofonthevalueofcubicnonlinearityfor!=2:89and=1.Itisclearthatthebreathermodebecomesunstableasincreases.Weextendedthisanalysistobreatherscorrespondingtoothervaluesoffrequency!.TheresultsaresummarizedinFigure 3-5 C).Thebreathermodesareobservedtobeunstableforallvaluesofif!2:85,whereasforlargervaluesof!themodesbecomemorestableforatleastsomerangeof.Ingeneral,wedestabilizingeectsofthecubicnonlinearityonthebreathermodes.However,thedetailedstabilitypropertiesofthebreathersaremorecomplex.Forexample,thereare\islands"ofstabilityatlargervaluesoffor!=2:88and!=2:92.Thisagreeswithpublishedreportthatshowscubicanharmonicityreducesthethermalconductivity[ 42 ].TheinstabilityoflocalizedbreathersinFPU-indicatestheshortlifetimeofballisticenergypacketduringMDsimulation,whichimpliesthereductionofthermalconductivity[ 42 ]. 29

PAGE 30

ResultsofthestabilityanalysisoftheSTmodebreathersforarangeof,!,andxed=1.Theclosed(open)circlesdenotethestable(unstable)modes. 10 43 { 53 ].Thebreathershavealsobeenobservedinmodelsofrealphysicalsystem,suchasamolecularmodelforarowofatomsinasemiconductorcrystalGaN[ 54 ].Inadditiontoperiodicatomicchains,breathershavebeenobservedinsimulationsofdisorderedsystems[ 44 55 ],whichhasimportantimplicationsforthermalconductivityinpolymersandbiologicalsystems.Moreover,severalexperimentalstudiesindicateexistenceofbreathersinmolecularsystems.Forexample,spectroscopicstudiesoflaser-inducedvibrationsinaquasi-one-dimensionalchainofhalogen-bridgedmixedPtcomplex[ 56 57 ]reportaRamanspectrumcharacteristicoflocalizednonlinearvibrationmodes.Theintrinsiclocalizedmodeshavealsobeenexperimentallyobservedinmyoglobin[ 58 ].IntheprevioussectionwediscussedFPUmodelwithatomsallowedtomoveonlyalongthechain.Itismorerealistictoallowvibrationsinmorethanonedirectionandsobelowweinvestigatetheexistenceandstabilityanalysisfornonlinearlatticevibrationmodesintwoandthree-dimensionallatticesystemswithhighaspectratio.Weperformthenumericalcalculationforfewdierentmodelsystems:asingleFPUchainintwo-dimensionalsystem;twocoupledFPUchainsintwodimensionalsystem,body-centeredcubicstructureintwo-dimensionalsystem,andathree-dimensionalhexagonaltubeformed 30

PAGE 31

2"lXj=1knj un=lXj=1[k(rnjdj)+(rnjdj)3]rnj 3.1.2 WeconsideroneFPUchainwhichextendsalongtheaxialxdirectionwithparticlevibrationsinbothxandradialydirections.Figure 3-6 A)showsthevibrationmodewithbreathercongurationforxdirectiondisplacement,whichsatisestheequationsofmotion.However,thestabilityanalysisplottedinFigure 3-6 B)revealsthatthissolutionislinearlyunstable.TheeigenvectorscorrespondingtotheunstableFloquetmultipliersaretheperturbationsalongtheradialdirection.AnexampleispresentedinFigure 3-6 C).Tounderstandthecauseofinstability,asimpleforceanalysisisdoneforthreeatomsx1;x2;x3connectedbyFPUinteractionpotential.Weconsiderthecasewhereonlythecenterparticle(x2)hasdisplacement(dx;0)fromitsequilibriumposition,hencetheforceactingonx2alongydirectionisexpressedas 31

PAGE 32

B CFigure3-6. A)ThenonlinearvibrationmodeforasingleFPUchainintwo-dimensionalsystemB)Stabilityanalysisofthissolution;C)TheunstableeigenvectorcorrespondingtotheopencircleinB),whichindicatesthatthenonlinearsolutionisunstableagainsttheperturbationalongradialydirection. Thisdisplacementwillbestableagainstasmallperturbationalongydirectiony2ifthestablecriterion 3{8 intoEq. 3{9 andnotethatr12req=reqr23forthisconguration.Afteralgebraiccalculation,wehavethestabilitycriterionas 3{10 willnotbesatisedforsmallvaluesofy2,thisindicatesthatthedisplacementofthecentralparticleinydirectionalwaysincrease. 3-7 .TheforceconstantsalongeachFPUchainaredenotedwiththesubscriptFPUandtheforceconstantsbetweenthesetwoFPUchainsaredenotedwiththesubscript 32

PAGE 33

ThecongurationfortwocoupledFPUchains. BFigure3-8. A)SteadystatesolutionfortwocoupledFPUchainswithkFPU=kcoupling=1,==0.B)Stabilityanalysisofthissolution. 3-8 A)andB),respectively.ToexaminetheexistenceofnonlinearvibrationmodeswithcongurationsimilartoDBs,weobtainthenonlinearsolutionsbyusingbreatherswith(k;;)FPU=(1;0;1)astheinitialguessineachFPUchain,andthengraduallyincreasethevaluesofthecouplingforceconstants.AnexampleofsuchasolutionisshowninFigure 3-9 alongwithitsstabilityanalysis,whichindicatesthatthissolutionislinearlyunstable.WeplotthecongurationofunstableperturbationeigenvectorinFigure 3-9 C),whichcorrespondsto 33

PAGE 34

B CFigure3-9. A)SteadystatesolutionsfortwocoupledFPUchainswith(kFPU;kcoupling;FPU)=(1;1;1)withtheinitialguessshowninFigure 3-6 .B)StabilityanalysisforsolutionA)C)TheunstableeigenvectorcorrespondstotheopencircleinB),whichindicatesthatthenonlinearsolutionisunstableagainsttheperturbationalongradialydirection. Thecongurationforbody-centeredcubicstructuresystem. theopencirclemultiplierinFigure 3-9 B).Itshowsthattheinstabilityisstillcausedbyaperturbationinydirection. 3-10 .Werefertotheparticlesintheupperandlowerchainsastherstandsecondparticlesoftheunitcellandtothecenterparticleasthethirdparticle.Insection 3.2.2 ,theinitialguessesforNewton'smethodarelocalizedvibrationmodesforeachFPUchain.Wereachedthenalnonlinearsolutionsbygraduallyincreasingthecouplingstrength.However,thisapproachcannotbeusedforallthesystems,suchascarbonnanotubesduetothecomplexstructureoftheirunitcell.Ontheotherhand, 34

PAGE 35

3-8 astheinitialguessfortheNewton'smethod.Duetothedegeneracyofthephononmodesinthislatticesystem,weapplyadegenerateperturbationmethodtothisphonontoobtaintheinitialguessforNewton'smethod.Inaddition,dierentfromthepreviouscalculationwherefrequency!isagivenandxedparameterinNewton'siteration,hereweallow!tobeoneoftheunknownsinthenumericalprocess.Hence,weneedtoaddonemoreequationinordertoutilizeNewton'smethod.Thiscanbedonebyxingthecenterofmassinthesystem,i.e. 3-11 A),andthecongurationoffullnonlinearsolutions(=1)fromvariousinitialamplitudesareshowninFigure 3-11 B).Inthismodel,weobservedthatdegenerateperturbationwithappropriatemodeenergywillallowustoreachthelocalizedsolutions.Wecontinuetoincreasethecouplingstrengthto(k;)coupling=(1:1;1:0),andperformthestabilityanalysisoftheobtainedsolution.Inordertoexaminethestabilizationeectduetotheinteractionwithcenteredparticle,wecomparethesolutioncongurationandFloquetmultipliersfortwosetsofparameter.TheFloquetmultipliersforthesecondsetofparameters(k;)coupling=(0:55;0:55)ispresentedinFigure 3-12 E).ThereisasignicantdecreaseofthenumberandmagnitudeofunstableeigenvaluesincomparedtoFigure 3-12 D)duetostrongercouplingwiththecentralparticle.Moreover,weobservethedisappearanceofthehighlyunstableperturbation 35

PAGE 36

B C DFigure3-11. A)Familiesofnonlinearsolutionscorrespondingtodierentamplitudesofthephononmodes.B)nonlinearvibrationmode(=1)forA=1:2C)nonlinearvibrationmode(=1)forA=2:0D)nonlinearvibrationmode(=1)forA=2:9 withmultipliermagnitudeclosetozero(shownasanopencircle),whichcorrespondstoperturbationalongaxialdirection. 3-13 .TheparticleslocatedatthecornersofthehexagonarereferredtoastypeA,theparticleslocatedatthefacecentersarereferredtoastypeB,andtheparticlelocatedatthecenterofthehexagonisreferredtoastypeC.Weassigneachparticleanindex(i;Kj),whichindicatestheparticleislocatedintheithunitcellandbelongstojthparticleoftypeK(K=A,BorC;j=1;:::;6ifK=AorBandj=1ifK=C).Thehexagonsidesandthebond 36

PAGE 37

B C D EFigure3-12. Thenonlinearvibrationmodeoftwo-dimensionalbody-centeredcubicstructurefor(k;)coupling=(1:10;1:0).A),B)andC)showthedisplacementalongtheupper,lower,andcentralchains,respectively.D)Stabilityanalysisforthesolutionwith(k;)coupling=(1:10;1:0).E)Stabilityanalysisforthesolutionwith(k;)coupling=(0:55;0:45). 37

PAGE 38

Structureofaunitcellofthemodelhexagonalsysteminthreedimensions.6particleslocatedatcornersofthehexagon(shownbyclosedcircles)arereferredtoastypeA,6particleslocatedatthecentersoffaceplane(shownbyopencircles)arereferredtotypeB,andtheparticleatthecenterofthehexagonisreferredtoasC.seetextfordetail. Table3-1. Theneighborlistandequilibriumbondlengthfortheparticlesinhexagonalsystem. Particleindex Neighborindexes Equilibriumlength (i,C1), (i,Aj) (i,Bj),(i,Bj1),(i-1,Bj),(i-1,Bj1), (i,C1),(i+1,C1), (i,Bj) (i,Aj),(i,Aj+1),(i+1,Aj),(i+1,Aj+1), 2 (i,A16) (i,B16),(i-1,B16) lengthsintheaxialzdirectionhaveaunitlength.TheneighborlistofeachparticleandequilibriumlengthsbetweentheinteractingneighborsarelistedinTable 3-1 .SinceweareinterestedinthedierenceandtransitionofDBsin1DandQ1Dlaticesystem,eachAjparticlealongtheaxialdirectioncanbeconsideredasone1DFPUchain,withtypeBandCparticlesservingasconnectorstocoupletheindependentFPUsystems. 3-14 .Thereare4acousticbranchesinthissystem;2doublydegeneratetransverse 38

PAGE 39

Thedispersioncurvesforthree-dimensionalhexagonaltubesystem. acoustic(TA)modes,whichhavexandyvibrationsperpendiculartotheaxial(z)direction.Singlehighestenergyacousticmodeisthelongitudinalacoustic(LA)modeintheaxialdirection.Thefourthacousticmodeisrelatedtoarotationaroundtheaxis,whichiscalledatwistingmode(TW). 39

PAGE 40

BFigure3-15. A)Oneofthelinearphononmodesofthehexagonalsystemwith(k;)coupling=FPU=(1;0).B)Stabilityanalysisforthissolution. 3.2.3 ,werstobtainthesteadystatesolutionsforlinearsystem(==0),whichisobtainedviaNewton'smethodwithphononmodesolutionsasinitialguess.ThesolutionanditsFloquetmultipliersareplottedinFigure 3-15 .Comparedtothelinearsolutionsintwodimensionalsystem,thelinearsolutionsarelinearlystableduetothepresenceofhigherconstraintinbothaxialandradialdirections.Moreinteresting,thegapinthisunitcircleisonlyobservedinthissystemanditexistsforthreeofthelinearvibrationmodeswecomputed.Toobtainnonlinearlocalizedsolution,weassignthebreathersolutionswith(k;)FPU=(1;2)totheaxialdirectiondisplacementineachchain.Inthecalculation,wegraduallyincreasethevaluesofkcouplingandcouplingwithgivenincreasing!value.ThestructureforinitialguessisshowninFigure 3-16 .Thedependenceofamplitudeandfrequencyonvaryingkcoupling(coupling)whilekeepingcoupling(kcoupling)xedisshowninFigure 3-17 .Asitisshowninthisplot,increasingkcouplingwilldecreasethelocalizationofthesolutionsandwhileincreasingcouplingsupportslocalizedvibrationmodes.Thisrelationcanbeillustratedinthecomparisonofthecongurationsofnonlinearmodeswhere(k;)coupling=(0:1;0:1),(1,0.1)and(1,1)areplottedinFigure 3-20 A),B)andC)respectively.Itisclearthatincreasingkcouplingweakensthelocalizationofthesolutions;whilesolutionwithlargervaluesofcouplingsupportshighlylocalizedstructures. 40

PAGE 41

InitialguessfortheNewton'smethodforhexagonalsystem.Thisvibrationmodecorrespondstobreathersolutionin1DFPUsystemwithk=1and=2. BFigure3-17. Dependenceofamplitudeandfrequencyofasteady-statemodeonthestrengthofcouplingbetweenchainsinhexagonalsystem.A)couplingisvaried;kcoupling=1isxedB)kcouplingisvariedwithcoupling=1xed. Detailedstructureofoneoftheconvergednonlinearsolutioncorrespondingto(k;)coupling=(1;1)isshowninFigure 3-18 alongwithitsstabilityanalysis.ItshowsthatincreasingcouplingstrengthreducesthemagnitudeofunstableFloquetmultipliers.ThestabilityanalysisinplotDindicatesthatthismodeisstilllinearlyunstablesolution.Sinceweremovethecauseofinstabilityfromsystemconguration,hereweexaminetheinstabilityfromRWAapproximation.Therefore,weseekthesolutionwithoutusingRWAapproximationandcomparethesetwosolutions.Inotherwords,inthesystemwithoutapplyingRWAapproximations,wesolvethesystemofequationsforxn;1andxn;3using 41

PAGE 42

B C DFigure3-18. Oneofthenonlinearsolutionswith(k;)coupling=(1;1).PlotsA),B)andC)showthedisplacementinx,yandzdirectionrespectively.DStabilityanalysisforthissolution. theNewton'smethod.TheinitialguessarethesolutionsunderRWAapproximation.TheresultsfrombothapproachesareplottedinFigure 3-19 .Thecomparisonofxn;1vectorforbothsolutionsisshowninFigure 3-19 A)andthehigherfrequencyterm,xn;3isplottedinFigure 3-19 B).Firstly,themagnitudeofxn;3is2orderssmallerthanxn;1,andquantitativesimilarityforxn;1intwosolutionsindicatetheappropriateassumptionofignoringhigherfrequencytermsinthissystem.InordertoensurethevalidityofRWAapproximationinhigherdimensionalsystems,wecomparetheirFloquetmultipliersinFigure 3-21 A),wherethemultipliersforthesolutionusingandnotusingRWAapproximationareshownbydiamondsandpluses,respectively.ThecomparisonoftwopairsofeigenvectorscorrespondingtocloseFloquetmultipliersareshowninFigure 3-21 andFigure 3-22 .Itisclearthattherstpaircorrespondtothesameperturbationvector.Forthesecondpair, 42

PAGE 43

BFigure3-19. ThecomparisonofsolutionswithandwithoutRWA.A)showsthecos(!t)vectorB)thecos(3!t)vector B CFigure3-20. Comparisonofnonlinearmodesofhexagonalsystemforthreedierentstrengthsofcouplingbetweenchainsofatoms:A)(k;)coupling=(0:1;0:1);B)(k;)coupling=(1:0;0:1);C)(k;)coupling=(1:0;1:0);Thelinearandnonlinearforcesweakenandstrengthenlocalizationofthesolutionstructure,respectively. evenithasqualitativedierentvaluebutthesetwovectorshavethesameperiod,whichimpliestheRWAapproximationshouldstillbevalidinthissystem. 43

PAGE 44

BFigure3-21. A)Stabilityanalysisforthesolutionswith(without)RWAapproximationbydiamond(cross)markers.B)Comparisonofthez-directioneigenvectorsfortwocloseeigenvaluesshownasopentriangleandcircularmarkers.Equivalentvectorsareobserved. BFigure3-22. A)Stabilityanalysisforthesolutionswith(without)RWAapproximationbydiamond(cross)markers.B)Comparisonofthez-directioneigenvectorsfortwocloseeigenvaluesshownasopentriangleandcircularmarkers.Dierentperturbationvectorsareobserved. potentialfunction.Duetothehighfrequencyvalueforbreathersolutions,theyhaveextremelylonglifetimeinNEMDsimulations.Inhigherdimensionsmodelsystem,weshowedtheexistenceoflocalizednonlinearvibrationmodesinbothtwoandthree-dimensionalsystems.However,therearenolinearlystablenonlinearvibrationmodesundertheparticularinteractionparametersthatweexploredinthischapter.We 44

PAGE 45

45

PAGE 46

21 22 ]signicantlydierfromtheresultsofthemoleculardynamicssimulations[ 23 { 26 59 ].Moreover,dierentMDsimulationstechniquesleadtodierentresults.ThesestudieshaveusedtwocomplementaryMDtechniques:thenon-equilibriumandequilibriummoleculardynamicssimulations(referredtoasNEMDandEMDrespectively).TheNEMDsimulationsconsistofimposingthetemperaturegradientalongthenanotubeaxisbycouplingtheopposingendsofthenanotubetothethermalbathsatdierenttemperatures.Thiscouplingistypicallyimplementedbyusingoneofthethermostattechniques[ 60 ].ThethermaluxbetweenthenanotubeandthethermalbathiscomputedandthethermalconductivityisobtainedfromtheFourier'slaw( 1{1 ).TheEMDsimulationsarebasedonsimulationsinanequilibriumensembleandthethermalconductivityiscomputedfromtheuctuationsofthethermaluxinthesystemusingtheGreen-Kuboformulaofthelinearresponsetheory[ 4 ].Hence,boththeNEMDandEMDmethodsarederivedintheframeworkofthelinearresponsetheory,whichassumesthattherelationshipbetweentheperturbation(i.e.temperaturegradient)andtheresponse(i.e.theheatux)islinear.Thediscrepancybetweentheresultsofthesesimulationssuggeststhattheassumptionoflinearresponseisnotvalidinthecaseofthecarbonnanotubesandnonlineareectsplayasignicantrole.Recentanalyses[ 61 62 ]haveshownthatnonlinearlatticevibrationsincarbonnanotubesmayleadtoformationofstronglynonlinearlocalizedwaves(solitons).Thesestudieshaveconsideredlatticevibrationsinthecontinuumlimitandapproximatednanotubesbyaone-dimensionalchainofatoms,thusneglectingthedetailsofthelatticevibrationinthedirectionsnormaltothenanotubeaxis.Undertheseassumptions,it 46

PAGE 47

1{1 .Wenotethatthistreatmentissomewhatapproximatesinceinrealnanotubestheheattransportwillbeduetoamixoftheballisticanddiusiveeects.Inparticular,itisexpectedthatthenonlinearlocalizedstructureswilldierfromtheidealizedKdVsolitonsinthatthecollisionsbetweenthemwillbenon-elasticandthuswillleadtothetransferofenergybetweenthelocalizedvibrationmodes.Inthischapter,wepresenttheinvestigationofthenonlinearlocalizedvibrationmodesincarbonnanotubes,whileaccountingforthemoleculardetailsofthenanotubestructureintoaccount. 4-1 )intoacylinder[ 63 ].Thepositionrofeachcarbonatomontheunrolledgraphitesheetcanbedescribedbytwobasisvectorsa1anda2, 4-1 ,wherel1,l2areintegers.Thestructureofacarbonnanotubeisspeciedbythevector(!OA)whichcorrespondstoasectionofthenanotubeperpendiculartothenanotubeaxis(!OB).AcarbonnanotubeisconstructedbyrollingthegraphitesheetsothatpointsOandAcoincide.Thevectors!OAand!OBdenethechiralvector,ChandtranslationvectorTinaxialdirection.Thischiralvectorcanbeexpressedintermsofthebasisvectorsa1anda2, 47

PAGE 48

4-1 B). BFigure4-1. A)PreparationofacomputationalmodelforananotubebyrollingthegraphitesheetinadirectionspeciedbythechiralvectorCh(n;m)[ 63 ];B)nanotubewithchiralvectorCh(5;0). WehaveimplementedtheBrennerparametrization[ 64 ]oftheTersopotential,referredtoasVB,todescribetheinteractionbetweencarbonatoms[ 65 66 ].Thispotentialhasbeenusedinthepastforthermalconductivitycalculations,aswellasforinvestigationsofotherpropertiesofCNTs[ 67 { 69 ].Theexplicitpotentialfunctionis 48

PAGE 49

ParametersforBrenner-Tersointeractionpotentialbetweencarbonatoms. Parameter Value 6.0eV 1.22 R2 a0 presentedinEq. 4{3 andtheparametersarelistedinTable. 4-1 2[1+cos(rijR1 S1exp(p S1exp(p 70 ]toadjusttheequilibriumatomcoordinatessothatthetotalpotentialenergyofthenanotubeisminimized.Sinceourcalculationrequiregraduallyincreasethenonlinearityinthepotentialfunction,weapproximatetheBrennerpotentialbyaTaylorseriesuptothefourthpower, un=Xm@2V @un@umum"1 2Xm;l@3V @un@um@ulumul+1 6Xm;l;k@4V @un@um@ul@ukumuluk#;(4{4)sothatwecangraduallyincreasethenonlinearityforatomicinteraction.The2nd,3rd,and4thderivativesofthepotentialenergyfunctionVBthatarenecessaryintheanalysisarecomputednumericallybythecentralnitedierencescheme. 49

PAGE 50

@un@umum 2Xm;l@3V @un@um@ulumul+1 6Xm;l;k@4V @un@um@ul@ukumuluk: Here,theatomicdisplacementsfungaretakentobethoseofthenormalphononmodeswiththeamplitudeassigned,accordingtotheBoltzmanndistributionatthetemperatureT=300K,i.e, 1 2mXn_un2=1 2kBT:(4{7).TheforceratiosjFnj=jFljforthenanotubephononmodeswithinasingleunitcellsystemareshowninFigure 4-2 .Forcomparison,weshowsimilarratiosfortheFPUsystemdiscussedinchapter 3 .Itisclearthatthemagnitudesofnonlinearityofthesetwosystemsarecomparableandthereforeweexpectthatthecarbonnanotubespossessthenonlinearvibrationmodesqualitativelydierentfromthenormalphononmodes. Figure4-2. RatioofthemagnitudeofthenonlinearjFnjandlinearjFljforcesactingontheequilibriumlinearphononmodesofthe(5,0)nanotube(})andtheFPUlatticeat==1().Here,!maxisthemaximumphononfrequencyofthesystem. 50

PAGE 51

nm.ThedevelopedtemperatureproleisshowninFigure 4-3 A);theevolutionoflocalenergydistributionisshownintheFigure 4-3 B).Thelocalenergyisthesumofthekineticandpotentialenergyofanindividualunitcell.Thisrevealsthatapartofenergyistransferredbylocalizedpacketsandthatballistictransportmechanismdoesplayanimportantroleincarbonnanotubesystems.Todeterminethecorrespondingheatcarriersforthisphenomena,wesolvetheequationsofmotiontoobtainthesteadystatesolutionsincarbonnanotubes. 2 ,weobtainsteady-statesolutionsusingNewton'smethod.SinceVBpotentialiscloselyapproximatedbyVTpotentialat=1,initialguessesfortheBrennermodesaretakentobetheTaylormodeswith=1andthelatterareobtainedbyperturbingthelinearnormalmodes(=0)withanalyticalapproximationsfromperturbationtheory,andusingtheseapproximationsasinitialguessforpotentialVT,followedbyacontinuationofthesolutionbyincreasinguntilitreaches1.Thenormalmodesanddispersioncurveareobtainedfromthefollowingeigenvalueproblem[ 71 ]: @un@umumF(un)(4{8)where!isthevibrationfrequency.Wecomputedthephonondispersioncurvesforthe(5,5)carbonnanotubes,andobservedthattheseareingoodagreementwiththeresultsof 51

PAGE 52

BFigure4-3. NEMDsimulationsofasegmentofa(5,0)CNT.Thesystemcontains100unitcellsandcontainsaheatsourceatthecenter(inthe50-thunitcell)andaheatsinkintherstcell:A)Temperatureproleofthesystem;B)Thelocalenergytransportinasteady-statesystem moleculardynamicssimulationsreportedinRef.[ 26 ].Thedispersioncurveforthe(5,0)carbonnanotubesispresentedinFigure 4-4 52

PAGE 53

Phonondispersioncurvesofthe(5,0)carbonnanotubeunderBrenner-Tersopotential. Theunknownsforequationsofmotionaretheatomicdisplacement,whichareexpressedusingtheFourierseriesasinEq. 2{15 .Thefrequency!isaxedandgivenparameterduringtheiterationsoftheNewton'smethod.WeimposeperiodicboundaryconditionsintheaxialdirectionandalsoapplyRWAapproximation,i.e,thecoecientsxn;jwithj2areneglected.ThenumericalsolutionofEq. 2{2 requiresafunctionwhich,giventheFourierseriesexpansionsfortheatomdisplacementsul(t),computestheFourierseriesexpansionsfortheforceFn(u)actingontheatoms.ForsucientlysimplepotentialsFn(u)canbeobtainedanalyticallybydirectsubstitutionofuintoanexpansionforF.Formorecomplexpotentials,suchastheBrennerpotential,weusethefollowingproceduresummarizedinFigure 4-5 .GivenFouriercoecientsxforatomicdisplacementsu,weperforminverseFouriertransformtoobtainvaluesofu(t)atevenlyspacedtimestjwithinoneperiodofoscillations.Thenwecomputeforcesfk(tj)=fk(u(tj))actingoneachatomandperformFouriertransformontheforcevectorsattimesoft1;:::;tj.Practically,wesolvetheequationsofmotionsetupforzerothandrstFouriermodecoecients.Thenonlinearsolutionsx()areobtainedthroughcontinuationbyNewton'smethodfromnormalmodessolutions,whichrepresent=0solutions.TheJacobianmatrixinEq. 2{17 isobtainedfromcentralnitedierencescheme.WewillrefertononlinearmodesofCNTsobtainedwithpotentialsVTandVBasTaylorandBrennermodes(solutions),respectively. 53

PAGE 54

ThesummaryofthenumericalproceduretoobtainFourierseriesexpansionforacomplexpotential.Here,fftandifftdenotefastFouriertransformandinversefastFouriertransform. ApplicationofNewton'smethodtopotentialVBisextremelytimeconsuming,mainlyduetothenecessityofusinganite-dierenceschemetocomputeJacobianmatrixJ.Ontheotherhand,JforVTpotentialcanbeobtainedanalytically,whichallowsustoobtainthesolutionsrelativelyfasteveninlargesystems.Therefore,nonlinearsolutionsforperiodsexceeding2areobtainedforVTpotentialonly.Moreover,forsystemswithperiodexceeding2,weneglectthecubictermsandapproximateVTpotentialasasumofquadraticandquarticterms.Thesolutioncorrespondingtosuchapotentialdoesnotincludeastaticdisplacementterm(seeEq. 3{2 ),whichreducesthenumberofunknowns.Thissimplicationsallowsustoobtainthesteadystatesolutionswithperiodsupto24unitcells. 4-6 isobtainedusingthephononmodewithfrequency!0=3:1251014Hzasthestarting 54

PAGE 55

4{4 withgraduallyincreasingmagnitudeofnonlinearity.Thestepforischosentobe=0:02.Theinitialconditionforthesolutionofthenonlinearsystemat=0:02wasobtainedfromthelinearmodeusingtheregularperturbationanalysis.Weobtained!(=0:02)=1:251011Hzfromthisperturbationanalysis.Duringtheprocessofthecontinuationbyparameter,!waschangedateverystepbythesamevalue,!=1:251011Hzupto=0:68andtheniskeptasconstantto=1.Wenotethatalthoughfortherststepthevalueof!wasdictatedbytheperturbationmethod,thechangesin!intheconsecutivestepsweresomewhatarbitrarywiththegoaltoobtainasolutionwithfrequencysucientlydierentfromthatofphononmodes.Oncethecontinuationreachedthevalueof=1,weusedtheNewton'smethodagaintoobtainthevibrationmodecorrespondingtothecompleteBrenner-Tersopotential.ThenonlinearmodeobtainedbythismethodisshowninFigure 4-6 andiscomparedwiththecorrespondinglinearphononmodeinthesamegure.Thedisplacementsinthexandydirectionsareverysimilarforatomsofthetwodierentringsofthenanotubeunitcell(bothforlinearandnonlinearmodes),andthus,weshowthesedisplacementsforonlyoneoftherings.Theseresultsindicatethatthesystemdescribedbythecompletenon-linearcarbonnanotubepotentialpossessesvibrationmodesqualitativelydierentfromthoseofthelinearizedsystems.Inadditiontothelargedierenceinthedynamicdisplacementofthemodes,thenonlinearmodesexhibitlargestaticdisplacement(seeFigure 4-6 B,E)thatisabsentinthelinearmodes.ThecorrespondingmodeenergyalongthecontinuationbyparameterisalsopresentedinFigure 4-7 .Thelefthandsideandrighthandsideyaxisrepresentsmodeenergyandthefrequencyrespectively,xaxisisthevalueofnonlinearity.Even!valueinthiscalculationisgivenarbitrarily,fromtheinitialdropofmodeenergyforsolutionwith=0:02andtheconstantmodeenergywhile!iskeptconstantfor>0:68,whichimpliesthedependenceofmodeenergyon!. 55

PAGE 56

B C D E FFigure4-6. Comparisonbetweenalinearphononmodeofthe(5,0)carbonnanotubeandthenonlinearmodeobtainedfromthismodebythecontinuationmethod.Therstrowofplots[A),B),C)]showsthedisplacementofatomsinthezdirection.Forclarity,theatomsbelongingtothesameringareplottedonthesameline.Thesecondrowofplots[D),E),F)]showstheatomdisplacementinthexandydirectionsforoneofthenanotuberings.Thesecondringexhibitssimilardisplacementsandhenceisnotshown.Therstcolumnofplots[A),D)]showsthelinearsolution;thesecond[B),E)]andthethird[C),F)]columnsshowrespectivelythestaticandthedynamicdisplacementsofthenonlinearmode.Thedisplacementsareshownbyarrowswhichforclarityaremagniedbyafactorof20[allplotsexceptF)]orbyafactorof100[plotF)]. Inthetwounitcellssystem,weuse!obtainedfromdegenerateperturbationfortherststepinthecontinuationcurve.Duringtheprocessofthecontinuationbyparameter,!ischangedonlyattherststep.ThenonlinearmodecongurationshowninFigure 4-8 isobtainedusingthephononmodewithfrequency!0=1:2109Hzasthestartingpointwithmodeenergycorrespondingto350Kand!(=0:02)=3:1108Hz.Thevalueoffrequencyiskeptconstantuntilreaches1.Inaddition,toexploretheenergythresholdofinitialmodeenergyforexcitingthelocalizednonlinearvibration 56

PAGE 57

DependenceofthemodeenergyandfrequencyoftheTaylorsolutionsonforsolutionsshowninFigure 4-6 .Themodeenergyisshownbydiamonds.Thevalueoffrequency!isshownbypluses. modesasweobservedinsection 3.2.3 inQ1Dmodelsystem,westartwiththenormalmodeswiththermalenergycorrespondingto200K,250Kand350K.Belowweshowthedisplacementfortherstcarbonringandonlycomparethedynamicaldisplacementbetweennonlinearandlinearsolutionsforthethreetemperatures.TheresultsforbothTaylorandBrennersolutionsandthecorrespondingmodeenergycontinuationcurvesareshowninFigure 4-8 .PlotsAandDarethedisplacementforlinearsystem;BandEaretheTaylorsolutionsforthreedierentinitialthermalenergy;CandFareBrennersolutions.Thisnonlinearvibrationmodeshavecongurationsimilartothephononmodewherewestartedthecalculation.TheresultsstartingwithdierenttemperaturesleadtosolutionswithsimilarcongurationandBrennersolutionisverysimilartoTaylorsolutions,whichjustiesournumericalprocess.ThemodeenergyforthesethreesolutionbranchesalongthecontinuationbyparameterarepresentedinFigure 4-9 .Because!nliscloseto!0weobtainthevibrationmodeswithcongurationqualitativelysimilartothephononmodes.Forsystemwithsimpliedpotential,weobtainnonlinearvibrationmodesfor4,8,16and24unitcells.Inthispotentialfunction,weareabletoreachnonlinearvibration 57

PAGE 58

B C D E FFigure4-8. Thesolutionsof(5,0)CNTintwounitcellssystemstartingwiththeamplitudecorrespondingtothreedierenttemperature.Therstrowofplots[A),B),C)]showsthedisplacementofatomsxandydirectionsforoneofthenanotuberings.tothesameringareplottedonthesameline.Thesecondrowofplots[D),E),F)]showstheatomdisplacementinthezdirection.Forclarity,theatomsbelongingTherstcolumnofplots[A),D)]showsthelinearsolution;thesecond[B),E)]andthethird[C),F)]columnsshowrespectivelytheTaylorandBrennersolutions.Thedisplacementsareshownbysolidlineswhichforclarityaremagniedbyafactorshownintheleftcornerineachsubgures. modesfor=1and!=8:61011HzinonestepofNewton'smethodcalculation.Anexampleofthenonlinearvibrationmodeswithlongestlength,24unitcellsisshowninFigure 4-10 ,inwhichthesolutionisobtainedusingthephononmodewithfrequency!0=1:581014Hz.Sinceitisalongsystem,wepresentthisnonlinearvibrationmodebyplottingthedynamicaldisplacementforthe7thatomineachunitcell.PlotsA),B)andC)aretheatomicdisplacementforthenonlinearvibrationmodealongx,yandzdirections,respectively;andplotsD),E)andF)arethedynamicaldisplacementforlinearphononmodealongx,yandzdirections,respectively.Thenonlinearsolutionhas 58

PAGE 59

DependenceofthemodeenergyandfrequencyoftheTaylorsolutionsstartingwiththreeinitialthermalenergyonforsolutionsshowningure 4-8 .Themodeenergyisshownbydiamonds.Thevalueoffrequency!isshownbypluses. thequalitativelysimilarcongurationcomparedtotheinitiallinearphononmode.Weobservethatinthiscasethesolutionisqualitativelydierentfromtheinitialphononmode.Inordertosummarizethenonlinearsolutionsforthispotentialfunction,wemarktheconvergedsolutionsonthedispersioncurveinFigure 4-11 .PlotAshowsthesolutionfrequency,open(closed)markerrepresentthesolutionswithdierent(same)wavenumber(periodicity)fromtheinitialphononmodes.Inordertoquantifythedierencebetweennonlinearmodesandphononmodes,wenormalizethenonlinearsolutionsandcomputethenormofthedierencebetweenthesetwosolutions,theresultsareplottedinplotB.Itisnotedthatlargedierencesoccursforsolutionsforwhichthewavenumberequals 4-12 andFigure 4-13 ,respectively.Toinvestigatethecausefortheinstability,weplottheeigenvectorcorrespondingtotheopencircleinFigure 4-13 BinFigure 4-14 .Westillneedtoanalyzethiseigenvectortofullyunderstandtheeectofthisunstableperturbationonthenonlinearvibrationmode. 59

PAGE 60

B C D E FFigure4-10. Thecomparisonthedisplacementforthe7thatomineachunitcellfor=1nonlinearmodeandstartingphononmodeina24unitcellssystemwiththesimpliedpotentialunderRWAapproximation.Therstrowofplots[A),B),C)]showsthedisplacementofatomsforsolution.Thesecondrowofplots[D),E),F)]showstheatomdisplacementforthestartingphononmode.Therst,secondandthirdcolumnofplotsshowsthedisplacementinx,yandzdirection.displacementfortherst,fourthand7thatomineachunitcell. 60

PAGE 61

BFigure4-11. Thenonlinearvibrationmodeswithsimpliedpotentialforthesystemsof4,8,16and24unitcells,whicharerepresentedbycircles,triangles,diamondsandsquaresrespectively.Openandclosedmarkersrepresentthenonlinearvibrationmodeshavingwavenumberdierentfromthephononmodeswestartedthecalculation.A)locationof!value.B)normofthedierencebetweennormalized=1and=0solutions. BFigure4-12. Alinearlystablenonlinearsolutionbasedonsimpliedpotentialfunctionforafourunitcellssystem.A)thesolutioncongurationintherstcarbonringB)Floquetmultipliersofthissolution 61

PAGE 62

BFigure4-13. Thenonlinearsolutionbasedonsimpliedpotentialfunctionforafourunitcellssystem.FigureA)showstheatomicdisplacementanditsFloquetmultipliersareshowninplotB) BFigure4-14. TheeigenvectorcorrespondingtheFloquetmultipliershownastheopencircleinFigure 4-13 B. 62

PAGE 63

6 .Zeolitesaremicroporousalumino-silicatecrystalswithporesizescomparabletomoleculardimensions.Theyarewidelyusedasmolecularsieves,sorbents,catalysts,andionexchangers.Inaddition,severalpossibleapplicationsofzeolitescombiningadsorptionofguestmoleculeswithtemperaturecontrolareemerging.Examplesincludesolaradsorptionheatpump[ 72 ]andadsorptionchillersformicroelectronicdevices[ 73 ].Awidevarietyofavailablezeolitestructuresprovidesalargerangeofexibilityinne-tuningtheirthermalpropertiestoaspecictask.Zeolitethermalconductivitycanalsobealteredbyintroducingpointdefectsintothecrystallattice,e.g.,bysubstitutionofsomeofthesiliconatomsbyaluminum[ 74 ].Thepointdefectsprovideadditionalscatteringcentersforphononsthusreducingthecrystalthermalconductivity[ 75 ].Moreover,thenanoporousstructureofzeolitesprovidesanadditionalopportunitytocontrolthezeolitethermalpropertiesthroughintroductionofo-frameworkguestspecies(sorbatesorcations)intothecrystal.Infact,itiswellknownthatstronginteractionbetweenzeolitelatticevibrationsandguestmoleculessignicantlyaecttransportpropertiesoftheguestmolecules(seee.g.[ 76 { 78 ])aswellasthedynamicsofoscillationofsorbatemoleculesattheiradsorptionsites[ 79 ].Evidenceisaccumulatingthatpresenceofguestmoleculeswithinzeolitesalsoaectsthedynamicsofzeolitelatticevibrationsleadingtochangesinthermalpropertiesofzeolites[ 29 80 ].Similarhost-guestinteractionsplayasignicantroleinthermalconductivityofothernanoporousmaterialsoftechnologicalimportance.Forexample,skutteruditesarepromisingcandidatesfordevelopmentofecientthermoelectricmaterials,i.e.materials 63

PAGE 64

27 28 ]thatadditionofionstovoidsinskutteruditesleadstoanorderofmagnitudedecreaseoftheirthermalconductivitythusincreasingthethermoelectricgureofmerit.Availabledatasuggestcomplexdependenceofthethermalconductivityonthenatureofaguestmoleculeandahostcrystal.Asdiscussedabove,additionofionstoskutteruditesreducestheirthermalconductivity.Similarly,encapsulationofanatominaGeclathrateleadstoanorderofmagnitudereductioninthermalconductivity[ 81 ].Inaddition,moleculardynamics(MD)simulations[ 29 ]showadrasticreductionofthermalconductivityofzeoliteLTA-SiO2inthepresenceofheavycations.Theseresultsseemtosuggestthat\rattling"oftheguestspeciesinsideacrystalleadstoscatteringofphononsthusreducingtheirmeanfreepathandleadingtothedecreaseofthermalconductivity.However,otherobservationscontradictthispicture.Forexample,MDsimulationsofxenoninzeoliteLTA-SiO2indicateanincreaseofthezeolitethermalconductivityduetotheguest-hostinteractions[ 29 ].Inaddition,experimentsofGreensteinetal.[ 80 ]showthattheconductivityofzeoliteMFIissubstantiallyhigherwhenanorganictemplatecationtetrapropylammonium(TPA)ispresentinit,ascomparedtoasamplewithremovedtemplates.Therefore,thephonon-guestinteractionsmaybequalitativelydierentfromthephononscatteringbypointdefectsxedinthelatticeduetostronglynonlinearoscillationsofguestmoleculesinsidethecrystalpores.Interactionsofguestmoleculeswithhostlatticevibrationshavebeenextensivelymodeledinrecentyears[ 78 79 82 { 84 ].Typically,thegoalofthesestudiesistounderstandeectsofthelatticevibrationonthesorbatedynamicsinsidecrystalsandthelatticevibrationsarefrequentlymodeledasathermalbath.Inthecurrentwork,weareaimingatunderstandingthereverseprocess,i.e.eectsofthesorbate\rattling"onthecrystallatticevibrations.Inthischapterwepresentresultsofourinvestigationsofeectsofsorbatemoleculesondynamicsofindividualphonons.Itisexpectedthatunderstandingofsorbate-phononinteractions 64

PAGE 65

5-1 .Thiszeolitepossessesacubicsymmetryanditslatticeparameteris8.83A,see[ 85 ].Asodaliteunitcellconsistsofacageshapedlikeatruncatedcuboctahedronboundedbysix4-ringwindows(i.e.windowsformedbyfouroxygenandfoursiliconatoms)andeight6-ringwindows.Diameterofthe4-ringwindowsisverysmallandthesewindowsareimpermeablebythesorbatesconsideredinoursimulations.Inaddition,transportratesofargonandlargermolecules(methaneandxenon)throughthe6-ringwindowsareordersofmagnitudeslowerthanthephonon-phononandphonon-sorbateinteractions[ 77 ]andasorbateremainsinsideacageonthetime-scaleofinterest.Silicasodaliteisusuallysynthesizedbygrowingthecrystalaroundorganictemplatemoleculeswhichbecomeencapsulatedinthesodalitecagesafterthesynthesisiscomplete[ 85 86 ].Inthecurrentwork,weneglectthepresenceofencapsulatedtemplatesandperformMDsimulationsofeitherbaresilicasodalitecontainingonlySiandOatomsinitslatticestructureorsilicasodalitewithencapsulatedargon,methane,orxenonmolecules.TheequilibriumlatticecongurationandthepotentialmodelforzeolitelatticevibrationsusedinthisstudyarethesameasthoseusedbyKopelevichandChang[ 77 ] 65

PAGE 66

Ablockof222sodaliteunitcellscontainingninesodalitecages.Siliconandoxygenatomsareshownaslargerandsmallerspheres,respectively.InMDsimulationsofsorbate-latticesystems,thesorbatesarelocatedineightcornercagesofthisblock. inastudyofsorbatetransportthrough6-ringwindows.ZeolitelatticevibrationsaremodeledbyatruncatedversionofananharmonicpotentialenergyeldproposedbyNicholasetal.[ 87 ], 2Kr(rrOSi0)2(5{2)isthepotentialenergyofstretchingoftheO-Sibondr, 2K(0)2(5{3) 66

PAGE 67

2hK(1)(0)2K(2)(0)3+K(3)(0)4i(5{4)isthepotentialenergyofbendingoftheSi-O-Sibondangle,and 2KUB(rrSiSi0)2(5{5)istheUrey-BradleytermwhichrepresentslengtheningoftheSi-ObondastheSi-O-Sianglebecomessmaller.InEq. 5{5 ,rdenotesdistancebetweentwosiliconatomsofaSi-O-Siangle.ThepotentialmodelEq. 5{1 neglectssmallercontributionsincludedintheoriginalmodel[ 87 ]suchastorsionenergyofthedihedralSi-O-Si-Oangle,nonbondedLennard-Jonesinteraction,andelectrostaticinteractionbetweenzeoliteatomsduetothepartialchargesofSiandOatoms.Inprinciple,long-rangeelectrostaticinteractionsmayhaveasignicanteectonthelatticedynamics.However,ithasbeenshownin[ 87 ]thatelectrostaticinteractionshavelittleeectonthestructureordynamicsofthesilicasodalitelatticeduetohighsymmetryofthiscrystalandchargeneutralityofeachSiO2group.Moreover,sincethesorbatesconsideredinthecurrentworkareelectricallyneutral,electrostaticinteractionswiththepartialchargesofthelatticeatomsareexpectedtohavenegligibleeectsonthesorbatedynamics.Infact,thistruncatedmodelhasbeenshowntoyieldsgoodagreementbetweencomputedandexperimentalvaluesoftransportratesofinertgasesinsodalite[ 77 ].Sincethistransportprocessinvolveslargedeformationsof6-ringwindows,itisexpectedthat,despitetheintroducedapproximations,themodelEq. 5{1 providesanaccuratedescriptionofanharmoniclatticedynamics.ThevaluesoftheforceconstantsKaswellasthevaluesoftheequilibriumdistances(rOSi0andrSiSi0)andequilibriumbondangles0and0aresummarizedinTable 5-1 .TheforceconstantsandtheequilibriumbondanglesweretakenfromthepaperofNicholasetal.[ 87 ].TheequilibriumdistancesrSiSi0andrOSiwereobtainedfrom 67

PAGE 68

ParametersforthelatticepotentialenergymodelEq. 5{1 Si-O O-Si-O Table5-2. Lennard-Jonesparametersforsorbate-sorbateandsorbate-latticeinteractions. Interaction Ar-Ar[ 88 ] 1183.0 3.350 Ar-O[ 88 ] 1028.0 3.029 CH4-CH4[ 89 ] 1139.0 3.882 CH4-O[ 90 ] 1108.3 3.214 Xe-Xe[ 88 ] 3437.0 3.849 Xe-O[ 91 ] 1133.1 3.453 arequirementthattheequilibriumcrystalstructurepredictedbythepotentialeldcoincideswiththestructureobtainedexperimentallybyRichardsonetal.[ 85 ].ThevalueofrOSi0usedinthisworkissomewhatdierentfromthatproposedbyNicholasetal.[ 87 ]duetothedierencesinthepotentialasdiscussedindetailinRef.[ 77 ].Inordertoassesssorbatesizeeectsonthephonon-sorbateinteractions,weconsiderthreedierentsorbates,namelyargon,methane,andxenon.AllthesesorbatesaremodeledassphereswhichinteractwitheachotherandthezeolitelatticeatomsthroughtheLennard-Jonespotential.ThevaluesoftheLennard-JonesparametersandusedinourcalculationsarelistedinTable 5-2 .Thesorbate-latticeinteractionsaremodeledusingthecommonassumption[ 92 ]thattheinteractionbetweensorbatesandlatticesiliconatomscanbeneglectedandtheonlycontributiontothesorbate-latticepotentialenergyisduetointeractionbetweensorbatesandlatticeoxygen. 68

PAGE 69

31 ]withthetimeconstant1ps.Afterthisequilibration,oneormoresorbatemoleculeswereaddedtoeachofthecornercageofthe222sodaliteblock(seeFigure 5-1 )andthesorbate-latticesystemwasequilibratedusingtheNVTsimulationsforanadditional2ns.Theinitiallocationsforthesorbatesweretakentocorrespondtotheminimumofthesorbate-zeolitepotentialenergy.Duetosmallsizeofthesodalitecages,onlyonexenonmoleculeandnomorethantwoargonormethanemoleculescanbeplacedintoasinglecage.Therefore,weconsiderthefollowingvesorbate-latticesystems:onesorbate(Ar,CH4,orXe)perunitcellandtwosorbates(ArorCH4)perunitcell.Wewilldenotethesesystemsas1Ar/cage,1CH4/cage,1Xe/cage,2Ar/cage,and2CH4/cage.Theequilibrationwasfollowedbya5nsproductionrunofNVEsimulationsforeachofthesevesorbate-latticesystemsandthesorbate-freezeolite.Sinceoneofthemaingoalsofthisworkistoassesschaoticnonlineardynamicsofphonons,wechoseafairlysmallstepsize,t=0:1fs,forthemicrocanonicalsimulations.Thisstepsizeensuresthatthetotalenergyuctuationsareontheorderof0.001%.Inordertodemonstratethatthesorbate\rattling"insidethezeolitecageisqualitativelydierentfromharmonicornearlyharmonicoscillationsofpointdefectsinthelattice,weperformedadditionalsimulationsofsorbatedynamicsintheabsenceofthesorbate-phononinteractions.Inthesesimulations,thezeoliteatomswerexedattheirequilibriumpositions.Thesorbateswereinitiallyplacedatpositionscorrespondingtotheminimumofthesorbate-zeolitepotentialenergyandtheirvelocitywassampledfromtheMaxwelldistribution.Thesystemwasthenequilibratedfor2nsusingtheNVTsimulations,whichwerefollowedbya5nsNVEproductionrun.Theparametersofthese 69

PAGE 70

5.4 and 5.5 ,webrieyreviewbackgroundinformationonharmoniclatticedynamicsandcalculationofsodalitenormalmodes.ConsideracrystalmodeledbyaperiodicallyrepeatedblockofL1L2L3unitcells.EachunitcellcontainsNatomsandtwovectorsetsfa1;a2;a3gandfb1;b2;b3gformbasesoftheunitcellandthereciprocallattice,respectively;aibj=2ij.Letintegervectorl=(l1;l2;l3)specifytheunitcellwithcoordinates 71 ].Here,jisanumberoftheeigenmodeandkisawavevector, 5{7 )willbewrittenask=[h1h2h3].ElementsofthematrixD(k)aregivenby @r(l)@r(l00)r=reqeikr(ll0);;=1;2;3;;0=1;:::;N:(5{8) 70

PAGE 71

71 ] 2j_Qjkj2+1 2!hjk2jQjkj2:(5{11)Hereandintheremainderofthechapter,weusesuperscripthtodistinguishfrequencyofanormalmodeinaharmoniclatticefromthatinananharmoniclattice.ThesodaliteunitcellcontainsN=36atoms.Therefore,thereare108normalmodescorrespondingtoeachwavevectork.Sinceinthecurrentworkweconsidera222blockofunitcellsandsodalitepossessesacubicsymmetry,thereareonlyfourindependentwavevectors,k=[000];[100];[110],and[111],inourMDsystem.Forreference,thedispersionrelationships!hj(k)forsodaliteforwavevectorskpointingindirections[100],[110],and[111]areshowninFigure 5-2 .NormalmodesaccessiblebytheMDsimulationscorrespondtothesmallestandthelargestvaluesofjkjineachofthesethreeplots.Ingeneral,encapsulationofasorbateinsideazeolitecagemayaectthelinearizedlatticedynamicsandleadtochangesintheeigenvectorsofthedynamicalmatrix,whichwouldrequireonetousedierentnormalmodesintheanalysisofzeolitelatticevibrationinthepresenceofsorbates.However,thiseectissignicantonlyinthecaseofstronginteractionbetweenphononsandasorbatelocatedatanequilibriumadsorptionsite.Thissituationwouldoccurifthesorbatetstightlywithinazeolitecageorifthereare 71

PAGE 72

B CFigure5-2. Dispersionrelationshipsforsodaliteindirections(A)k=[100],(B)k=[110],and(C)k=[111]. long-rangeelectrostaticinteractionsbetweenthesorbateandthezeolitelattice.Ithasbeenshownin[ 84 ]thatcouplingbetweenthephononmodesandsmallelectricallyneutralsorbates,suchasthoseconsideredinthecurrentwork,isnegligiblewhenthesorbatesarelocatedatadsorptionsitesinsidesodalitecages.Forthesesystems,thesorbate-phononcouplingbecomessignicantonlywhenthesorbateapproachesthezeolitewall.Therefore,thepresenceofthesorbatesdoesnotaectthedynamicalmatrixofthecrystal.Thenonlineareectsofthesorbates,suchasthechangeinphononlifetimeandthenonlinearcorrectionstothephononfrequencyareanalyzedinthenexttwosections.Theinsensitivityoftheharmonicapproximationtothelatticedynamicstothesorbatesconsideredinthecurrentworkallowsustousethenormalmodesofthesorbate-free 72

PAGE 73

75 ] 93 ].Phonon-phononinteractionsduetolatticeanharmonicity,defects,andotherfactorsaremodeledbythecollisionintegralintheright-hand-sideofEq. 5{12 .Calculationofthecollisionintegralinexactformisverychallenginganditisusuallyapproximatedbyvariousmodels,suchasthesinglemoderelaxationtime(SMRT)approximation, 93 ]ortheharmonicenergy( 5{11 )ofthenormalmode[ 94 ].TheanalysisofthephonondynamicsbasedonBTEapproachhasseveraldrawbacks.Therelaxationtimeapproximationmaynotbeadequatetomodelphonondynamicsincomplexmaterials.AlthoughSMRTapproximation( 5{13 )canbeextendedtoaccountformultiplerelaxationtimesduetodierentphononscatteringmechanisms,theseextensionsrequirethephononautocorrelationfunctiontobeasumofmultipleexponentials.However,aswillbeshownbelow,someofthesodalitephononmodesdonot 73

PAGE 74

12 ], Qjk+jk_Qjk=(!ajk)2(QjkhQjki)+jk(t):(5{14)Here,hQjkiisthemeanvalueofthenormalmodecoordinateQjk,whichmaydierfromzeroduetoanharmonicityofthesystem,!ajkistheanharmonicfrequency,jkisthefrictioncoecient,andjk(t)istherandomforcewithzeromeanwhichisrelatedtothefrictioncoecientbytheuctuation-dissipationtheorem 5{14 )is[ 95 ] ~!jk=q 74

PAGE 75

5{16 demonstratesthat,similarly,toBTEwithSMRTapproximation,theLangevinmodel( 5{14 )predictsanexponentialdecayofthenormalmodeautocorrelationfunction.However,theLangevinequationprovidesmoreexibilityandallowsonetoperformrelativelysimpleadjustmentsofthemodeltotobservedphonondynamics.Thiscanbedonebyincludinganexplicitanharmonictermintotheequationormodifyingstatisticsoftherandomforce.Forexample,non-exponentialbehavioroftheautocorrelationfunctioncanbemodeledbyanon-Markovianrandomforce[ 96 ].ExamplesofphononautocorrelationfunctionsobtainedfromourMDsimulationsofsorbate-freesodaliteareshowninFigure 5-3 .Mostofthesefunctions,suchasthatshowninFigure 5-3 a,areingoodagreementwithEq. 5{16 .However,somemodesexhibitsignicantdeviationsfromthepredictionsofthisMarkovianLangevinequation.Autocorrelationfunctionsofallthesenon-MarkovianmodesarequalitativelysimilartooneoftheautocorrelationfunctionsshowninFigure 5-3 b-d.Thesemodespossesssecondary(slow)oscillationswhicharequalitativelysimilartooscillationsoftheautocorrelationfunctionoftheenergyofanentiresodalitecageobservedbyMcGaugheyandKaviany[ 97 ].Thesesecondaryoscillationswereinterpretedascorrespondingtolocalizationofenergyinsodalitecages.Ouranalysisofindividualphononmodesshowsthatonlyafractionofopticalphononsmodespossessesthissecondarytime-scale.InthecurrentworkweassumethattheMarkovianLangevinequation( 5{14 )providesanadequatemodelforthephonondynamicsandleavedevelopmentofitsextensionstoacountforsecondaryoscillationstofuturestudies.Therefore,thelifetimesandthefrequenciesofallmodesareobtainedbyapplyingEq. 5{16 toanalysisoftheirautocorrelationfunctionInparticular,Eq. 5{16 impliesthatthepowerspectrumSjk(!)ofthephononmodejkattainsitsmaximumat!=~!jk:Therefore,wedenetheapparentfrequencyofvibration,~!jk,asthelocationofthemaximumofSjk(!)forallmodes,includingthose 75

PAGE 76

ExamplesofautocorrelationfunctionsCjk()ofphononsinasorbate-freesodalitecrystal.Thenormalmodenumbersjareshowninthecorrespondingplots;thewavevectorisk=[000]inallfourexamples.InplotsA)andB),envelopesEjk()oftheautocorrelationfunctionsareshownbydashedlines. exhibitingsignicantdeviationsfromEq. 5{16 .TypicalexamplesofpowerspectraofthenormalmodesareshowninFigure 5-4 .Weobservethatformostmodes,eventhosenotsatisfyingtheLangevinmodel( 5{14 ),themaximumofSjkcorrespondstothehighestfrequencyofoscillationswhichweassociatewiththeapparentphononfrequency.Inafewcases,suchasthatshowninFigure 5-4 c,themaximumofSjkcorrespondstoslowersecondaryoscillations.Inthesecases,theapparentphononfrequencywasdenedasthefrequencyofalocalmaximumofSjk(!)withthelargestvalueof!.Inordertoestimatethelifetimeofaphononmode,wedeneanenvelopeEjk(t)ofitsautocorrelationfunctionCjk(t)asalineconnectinglocalmaximaofCjk(t),asshownbydashedlinesinFigure 5-3 a,b.ThefunctionEjkisthenttedtoanexponential.Inthecasesofnon-exponentialdecayofEjk,thephononlifetimeisobtainedbyttingitsinitial(quicklydecaying)segmenttoanexponential. 76

PAGE 77

PowerspectraSjk(!)correspondingtothephononautocorrelationfunctionsshowninFigure 5-3 .Thenormalmodenumbersjareshowninthecorrespondingplotsandthewavevectorisk=[000]inallfourexamples. Oncetheapparentphononfrequency~!jkandthelifetimejkareobtained,theanharmonicphononfrequency!ajkiscomputedfromEq. 5{19 .ThisequationiscorrectonlyforphononnormalmodesthatsatisfytheMarkovianLangevinequation( 5{14 ).However,thedierncebetween~!jkand!ajkisnegligibleif~!jk>>1=jk.Aswillbeshowninsection 5.5 ,phononlifetimesrangefrom0.4to30ps.Therefore,thecorrectionoftheapparentfrequency( 5{19 )issignicantonlyforphononswithverysmallfrequencies.Ourresultsindicatethatdynamicsofthelow-frequencymodesareingoodagreementwiththepredictionsofEq. 5{14 ,seee.g.Figure 5-3 a.Thecorrectionoftheapparentfrequencyisaccurateforthesemodes.Ontheotherhand,thiscorrectionisnegligibleforthehigh-frequencymodeswhichdonotsatisfytheMarkovianmodel( 5{14 ).AutocorrelationfunctionsCjkofaphononmodejkinasorbate-freesodaliteandasodalitecontainingsorbatesarequalitativelysimilarforthesamevaluesofjandk.Thisprovidesanadditionalconrmationthatsmallneutralsorbatesdonoalterthephononeigenmodes(seediscussionattheendofsection 5.3 ).Thesorbatesaectsuchphonon 77

PAGE 78

PowerspectraSV(!)ofsorbatevelocitiesinrigidzeolitecages. propertiesastheirlifetimeandfrequency.Itwillbeshowninthenextsectionthatsomeofthesechanges,namelyanincreaseoflifetimeofsomephononmodes,arequalitativelydierentfromthoseexpectedfromthesimplephononscatteringpicture.Thisimpliesthatthesorbatedynamicsisqualitativelydierentfromthatofpointdefectscoupledtothelatticebyanearlyharmonicpotential.Infact,thesorbatedynamicsischaoticevenintheabsenceofthethermalinteractionwiththelatticevibration.ThisisconrmedbythepowerspectraSV(!)ofthesorbatevelocitiesintherigidzeoliteshowninFigure 5-5 .Thesepowerspectraareratherwide,implyingchaoticsorbatedynamicsduetostronglynonlinearinteractionsbetweenthesorbatesandthezeolitewalls.Thenonlineareectsareevenstrongerforsystemswithtwosorbatespercage,asindicatedbythewidersorbatevelocityspectrainthesesystems. 78

PAGE 79

Normalizedaveragesofphononamplitudes,Qjk==jk.Onlymodeswithsucientlylargeaveragedeviationsfromzero,Qjk0:04,areshown.NormalmodesarelistedintheorderofdescendingQjk.Harmonicphononfrequencies!hjkcorrespondingtothelistedmodesarealsoshown. Qjk Sorbates/cage Barelattice 1Ar 1CH4 2Ar 2CH4 9.6 4.00 3.98 4.03 3.95 4.00 4.33 63[000] 98.4 -3.14 -3.21 -3.18 -3.18 -3.23 -3.69 48[000] 80.6 { { { { 0.26 0.83 60[000] 93.5 { { { { -0.048 -0.152 1[110] 4.3 { { { { { 0.048 78[000] 142.0 { { { { { -0.043 sucientlylargerelativedeviationoftheirmeanfromzero, Qjk=hQjki 5-3 .InEq. 5{20 ,jkdenotesthestandarddeviationofthenormalmodeuctuations.Inthesorbate-freelattice,modesQjkwithj=7andj=63andk=[000]deviatefromzeroby4and3standarddeviations,respectively.Allothermodeshavemuchsmallerdeviations,Qjk<0:02.AdditionofonesorbatepercageessentiallydoesnotchangethevaluesofQjk.However,additionoftwosorbatespercageleadstoasubstantialshiftofaveragevaluesofseveraladditionalmodes,implyingthattheequilibriumlatticecongurationslightlychangesduetothepresenceofthesorbates.Thischangeislargerinthecaseof2CH4/cage.Additionof2CH4/cageshiftsaveragesofseveralnormalmodesawayfromzeroaswellasfurtherincreasesQjkofthemodesj=7andj=63(k=[000])thatwerealreadyshiftedinthesorbate-freelattice.Nevertheless,eventhestrongestsorbateeectsonthemeannormalcoordinatesseeninthecaseof2CH4/cagearesignicantlyweakerthaneectsofintroductionofanharmonicitytoaharmonicsodalitelattice. 79

PAGE 80

Phononlifetimesjkinsorbate-freesodalitelattice. ItisinterestingtonotethatallsubstantialshiftsofhQjkitakeplaceforopticallatticemodeswithk=[000].Thelargestrelativedisplacementofanacousticmodeisrathersmall(Qjk=0:048)andisobservedformodej=1;k=[110]when2CH4/cageareadded.Phononlifetimesjkinsorbate-freeanharmoniclatticerangefrom0.4to30ps,asshowninFigure 5-6 .Themodeswithintermediatefrequencies,70ps1!hjk150ps1,possessshortlifetimes.Thephononswithhighfrequencies,!hjk>150ps1,correspondmostlytofastvibrationsofindividualbonds.Interactionsbetweenthesemodesandothermodesinthesystemareweakleadingtolargephononlifetimesforthehigh-frequencymodes.Somemodeswithlowfrequencies,!hjk<70ps1,alsopossesslonglifetimes.However,mostofthelow-frequencymodespossessrelativelyshortlifetimesindicatingthattheyarestronglycoupledwithothermodesinthesystem.Forcomparison,analysisofMcGaugheyandKaviany[ 97 ]basedonadecompositionoftheheatcurrentautocorrelationfunctionpredictsdecaytimeforheattransferassociatedwithlong-rangeacousticmodesinsodalitetobe1.67ps.Inaddition, 80

PAGE 81

80 ]haveestimatedphononrelaxationtimeinMFIzeolitetobe9.2psbyttingatheoreticalexpressiontoexperimentalthermalconductivitydata.Thisestimateisbasedonanassumptionthattherelaxationtimeisthesameforallphononmodes.Bothoftheseestimationsarewithintherangeofthephononlifetimesobservedinthecurrentwork.Relativechangesofphononlifetimes, 5-7 .InEq. 5{21 andelsewhereinthischaptersuperscriptsreferstoapropertyrelatedtoazeolitewithencapsulatedsorbates.ThedistributionsofjkshowninFigure 5-7 aarealmostidenticalforallthreesystemswith1sorbate/cage.Thesedistributionsaresymmetricwithrespectto=0,whichimpliesthatthephonon-sorbateinteractionsareequallylikelytodecreaseaswellasincreasethephononlifetime.Theincreaseofthephononlifetimecontradictsasimplepictureofphononscatteringbysorbatesandimpliesthatamorecomplexsorbate-phononinteractionisinplay.Whentwosorbatespercageareaddedtothesystem,thedistributionofjkbecomesskewedtowardsaveragedecreaseofthephononlifetime.Thistrendisespeciallypronouncedinthecaseofalargersorbate,namelymethane.Thiscanbeexplained,inpart,byatightertofthesorbateswithinthecagesleadingtoasmalleramplitudeofthesorbateoscillations,whichsuggestsmoresimilaritiesbetweensorbatesinthesesystemswithpointdefects.However,asFigure 5-5 indicates,thesorbatedynamicsinthe2sorbate/cagesystemsismorechaoticthaninthe1sorbate/cagesystems.Therefore,theanalogybetweenthe2sorbates/cagesystemsandcrystalswithpointdefectsisnotcomplete.Indeed,Figure 5-7 showsthatsomeofthephononmodesinthe2sorbates/cagesystemsundergoasignicant(ontheorderof100%)increaseoftheirlifetimeandhencethescatteringmodelisstillnotapplicabletothiscase. 81

PAGE 82

5-7 bweplotjkforeveryphononmode.Forclarity,onlytwoextremecasesareshown:1Ar/cageand2CH4/cage.For1Ar/cage,thechangesofphononlifetimesareevenlydistributedamongdierentfrequencies.Inthe2CH4/cagesystem,themodeslyinginthesmallandlargefrequencyregionsexperience,onaverage,largerchangeoftheirlifetimesthanthemodeswithintermediatefrequencies.Eectsofanharmonicityonthephononfrequencyinasorbate-freesodalitecrystalaresummarizedinFigure 5-8 whichshowsrelativedierences, 5-9 .Thefrequencychanges!sjkaresubstantiallylargerwhenmorethanonesorbatepercageisintroduced.Similarlyto!jk,!sjktendstoincreasewiththedecreaseofthephononfrequency.However,thechangesofphononfrequenciesduetoadditionofsorbatesaresmallerthanthechangesduetointroductionofanharmonicitytoasorbate-freeharmoniclattice.Uptothispoint,wehaveinvestigatedeectsofanharmonicityandsorbate-phononinteractionsonindividualphononmodes.Tocompletethepicture,wenowconsidercorrelationsbetweendierentphononmodes.ItisexpectedthatthisinformationwillbehelpfulindevelopingamorepreciseformoftheLangevinmodel( 5{14 )forphonon 82

PAGE 83

BFigure5-7. Eectsofsorbatesonphononlifetimes.A)DistributionsP(jk)oftherelativedierencesjkbetweenthephononlifetimessjkinsodalitewithencapsulatedsorbatesandthephononlifetimesajkinasorbate-freesodalite.B)Relativechangesjkoflifetimesofindividualmodesjkforthecasesof1Ar/cageand2CH4/cage. 83

PAGE 84

Relativedierences!jkbetweentheanharmonic(!ajk)andharmonic(!hjk)frequenciesinasorbate-freesodalitecrystal. Figure5-9. Relativedierences!sjkbetweenphononfrequenciesinsodalitewithencapsulatedsorbates(!sjk)andinthesorbate-freesodalite(!ajk).Forclarity,only!sjkwithmagnitudesgreaterthan102areshown. 84

PAGE 85

Magnitudesofcorrelationcoecients(jk;j0k0)betweenphononmodesjkandj0k0inasorbate-freesodalitecrystal.Darkersymbolscorrespondtostrongercorrelations.Forclarity,correlationcoecientsforjk=j0k0arenotshownandonlycorrelationsofmagnitudegreaterthan0.1areplotted. dynamicssinceitwillallowustoassesswhichphononsmakedominantcontributionstostochasticforcesactingoneachofthephononmodes.Themagnitudesofthecorrelationcoecients 5-10 .Forclarity,onlycorrelationsofmagnitudeexceeding0.1areplotted.Mostcorrelationcoecientsbetweenindividualphononsareratherweak.Correlationsbetweenlow-andmid-frequencymodesconstituteanotableexceptions:somecorrelationsoflow-frequencymodeswithmodesoffrequency!hjk100ps1havecorrelationcoecientaslargeas0.5.Theseobservationsareconsistentwiththemeasuredphononlifetimes,seeFigure 5-6 .Specically,phonon-phononcorrelationsinvolvingthehigh-frequencymodesareveryweak,whichismanifestedinlonglifetimesofthesemodes.Themodeswithintermediatefrequencies,70ps1!hjk150ps1,arestronglycorrelatedwithsomeofthelowfrequencymodes,whichleadstosmalllifetimeofthesemodes. 85

PAGE 86

Eectofsorbatesonphonon-phononcorrelations.Onlythosepairs(jk;j0k0)areshownforwhichadditionofsorbatescreatesacorrelationwithcoecientgreaterthan0.1orchangesthemagnitudeofanexistingcorrelationbymorethan0.1.Magnitudesofthechangedcorrelationcoecientsareindicatedbycolor. Eectofthesorbatepresenceonthephonon-phononcorrelationsisshowninFigure 5-11 .Additionofasinglesorbatepercageleadstorelativelysmallchangesinphonon-phononcorrelations,withnophononpairswithcorrelationcoecientgreaterthan0.1beingaected.Additionof2Ar/cagecreatesonenewphonon-phononcorrelationwithmagnitudeslightlylargerthan0.1.Incontrast,additionof2CH4/cagecreatesseveralstrongcorrelationsandmagniessomecorrelationswhichwerepresentinthesorbate-freelattice.Thelargestchangesofthecorrelationcoecientsuponadditionof2CH4/cagearebetweenthelow-frequencyacousticmodeswithwavevectork=[110]andotherphonons.Thechangeofthecorrelationcoecientinthesecasesis0.27. 86

PAGE 87

5{14 )providesanadequatemodelfordynamicsofmostofthephononmodes.However,dynamicsofsomephononmodesdoesnotagreewithpredictionsofEq. 5{14 .Thesemodesexhibitsecondaryoscillations(seeFigure 5-3 b-d)whicharemostlikelyrelatedtothetotalenergyuctuationsinsideasinglesodalitecage[ 97 ].Encapsulationofsorbatesinzeolitecagesdoesnotleadtoqualitativechangesofphonondynamics,ascanbeconcludedfromphononautocorrelationfunctions.However,dependingonthetypeandnumberofsorbatesinsidesodalitecage,someorallofthefollowingaspectsofphonondynamicsarechanged:(i)themeanvaluesofthenormalmodecoordinates,(ii)phononlifetimes,(iii)phononfrequencies,and(iv)phonon-phononcorrelations.Thelargestchangesofphonondynamicsareobserveduponencapsulationoftwomethanemoleculespersodalitecage.Thestrongesteectofallconsideredsorbatesonlatticedynamicsisasignicantchangeofthephononlifetimes.Fromthescatteringpictureofphonon-sorbateinteractionsitisexpectedthattheseinteractionswilldecreasephononlifetimes.However,weobservethatencapsulationofsorbatesleadstoanincreaseoflifetimesofalargefractionofphonons.Therefore,developmentofamoredetailedmodelisnecessarytounderstandthecomplexnonlinearsorbate-phononinteractions. 87

PAGE 88

98 ].Toinvestigatetheinuenceofsorbatesonlatticesystemandtodevelopaheattransfermodelfornanoporousmaterials,weconsiderasimple1Dsystem.First,weinvestigatedependenceofthermalconductivityofthissystemonsorbatepropertiesusingfullscalenon-equilibriumMDsimulations.Togetinsightintothesorbate-latticedynamics,weinvestigatephonon-sorbatedynamicsinaharmoniclattice,whichallowsustofocusontheroleofthesorbate-latticeinteractionsintheenergyexchangebetweenphononmodes.Inconclusion,wediscusspossibleapproachestodevelopmentofatheoreticalmodel. 3.1 )consistingofN=1024particleswithforceconstantsk=10;=50;and=180.Thesorbateislocatedbetweenlatticesites256and257andinteractswiththelatticeatomsthroughLennard-Jones(LJ)potential: r12 r6;(6{1)whereristhedistancebetweenthesorbateandalatticeatom.Periodicboundaryconditionsareimposedinthesystem.HotandcoldregionswithtemperaturesTh=1:2andTc=0:8eachcontaining100particlesaresetupattheedge(atoms1;:::;100)andatthecenter(atoms512;:::;612)ofthechainusingtheNose-Hooverthermostat(seesection 2.2.2 ).WeusevelocityVerletalgorithmwithtimestept=5104timeunit.Thesimulationiscarriedoutfor225000timeunits.Afterthesystemreachesasteadystate,wecomputetheheatux 2Xm6=nvnfnm;(6{2) 88

PAGE 89

BFigure6-1. A)Establishedtemperatureproleinnon-equilibriumMDsimulations.Thesorbateislocatedbetweenlatticesitesandsorbate-latticeinteractionsaremodeledbyLJpotential.B)Establishedheatuxinnon-equilibriumMDsimulations. wherevnisthevelocityofthen-thatomandfnmistheforcewithwhichthem-thatomisactingonthen-thatom.ThethermalconductivityisthencalculatedusingFourier'slaw,Eq. 1{1 .WeperformaseriesofNEMDsimulationswithsorbatesofdierentmassMsandLennard-Jonesparameterofsorbate-latticeinteraction.Inallthesesimulations,theeectiveLJdiameteris=0:23.ExamplesofestablishedtemperatureandheatuxprolesareshowninFigure 6-1 .Comparisonbetweenthermalconductivitiesofasorbate-freelatticeandlatticecontainingvarioussorbatemoleculesisshowninFigure 6-2 .Eveninthissimplesystem,weobserveacomplexpatternofdependenceofonthesorbate-latticeinteractionparameters.Thepresenceofthesorbatemayeitherincreaseordecreasethethermalconductivity.Inaddition,sucientlylargetemperaturegradientleadstodevelopmentofadiscontinuityinthetemperatureproleacrosstheunitcellcontainingthesorbate,asshowninFigure 6-3 .ThisphenomenonissimilartoKapitzathermalboundaryresistanceataninterfacebetweentwodissimilarmaterials[ 99 ].Inthelattercase,atemperaturediscontinuityisdevelopedtomaintaincontinuityoftheheatuxthroughtheinterface. 89

PAGE 90

Comparisonofthermalconductivities,,obtainedfromNEMDsimulationsforasystemcontaining1024atoms.TemperaturegradientinallsimulationsisdT dx=0:001.SolidcirclesrepresentthevaluesofinthepresenceofasorbatemoleculewithmassMsandLJinteractionparameters.Thesolidlinecorrespondstothethermalconductivityoflatticewithnosorbatemolecules. Figure6-3. SteadystatetemperatureproleinNEMDsimulationwithimposedtemperaturegradientdT dx=0:002Thesorbateislocatedbetweenlatticeparticles128and129.Adiscontinuityinthetemperatureproleisdevelopedacrosstheunitcellcontainingthesorbate. 90

PAGE 91

2!2qjQqj2+j_Qqj2;(6{4)istheHamiltonianofaphononmodeQqwithwavenumberq, 6{4 ),!qisthefrequencyofthephononwithwavenumberq,whichisgivenbythefollowingdispersionrelationship, msinqa 6{5 ),Ms,xs,andvsarethesorbatemass,position,andvelocities,respectively,andU0(xs)isthepotentialofinteractionbetweenthesorbateandthelatticeatomswhenthelatterarexedattheirequilibriumpositions.Intheabsenceofthesorbate-latticecoupling,i.e.when^U(fQjg;xs)0,thesystemisintegrable;thephononsandthesorbateundergoperiodicmotion.Introductionofanon-zeropotential^U(fQjg;xs)perturbstheirperiodictrajectories.Itisknownfromthetheoryofdynamicalsystems[ 100 ]thatasmallperturbationofanintegrablesystemsmayleadtoacompletedestructionoftheperiodictrajectories.Inthiscase,thesystemtrajectorybecomeschaotic,whichfacilitatesfastenergyexchangebetweenthesystemdegreesoffreedom.Suchchaoticbehaviorwilltakeplaceiftheunperturbedsystemsatisesthe 91

PAGE 92

101 ]forscatteringofawavepacketbyaninterfacebetweentwodierentcrystals.Weconsiderone-dimensionalharmoniclatticecontainingN=2001atoms.Thesorbateislocatedbetweenlatticesites1000and1001.Thelatticespringconstantisk=156andthesorbate-latticeinteractionsaremodeledbyLJpotential,Eq. 6{1 .Themassofsorbatemolecule,Ms,rangesfrom1to6andrangesfrom1to10,whileeectiveLJdiameterisheldxedat=0:44.Thesorbatedoesnotmoveuntilacollisionwithawavepacket.Thesimulationsareinitializedwiththesorbateplacedatitsequilibriumpositionwithinaunitcellanditsinitialvelocitysettozero.Initiallatticecongurationconsistsofaphononwavepacketcenteredatwavenumberq0.Thiswavepacketisalinearcombinationofthenormalmodeswithwavenumberssucientlyclosetoq0.Therefore,thedisplacementulofthel-thlatticeatomis 92

PAGE 93

B CFigure6-4. Scatteringofawavepacketcenteredaroundwavenumberq0=0:1byasorbatewithLJparameters(;)=(2;0:44)andmassMs=1.Locationoftheunitcellcontainingthesorbatemoleculeisshownbythedashedline. Figure6-5. Transmissionratiosforscatteringofdierentwavepacketsbysorbatemoleculeswithdierentmasses.LJpotentialparametersare(;)=(2;0:44)inallshownsimulations. latticeatomsareobtainedasfollows.Wedecomposeulintothenormalmodesandthen,usingthenormalmodefrequencies,obtainthetimederivativesoftheatomdisplacements.AtypicalexampleofthewavepacketscatteringisshowninFigure 6-4 .Ascanbeseen,apartofthewavepacketistransmittedthroughtheunitcellcontainingthesorbatemoleculeandpartofthewavepacketisreected.Thefrequencyofthenormalmodescontainedinthesewavepacketsremainessentiallyunchanged.Transmissionratios(i.e.,ratiosoftheamplitudeofthetransmittedandtheincidentwavepackets)foraseriesofthescatteringsimulationsaresummarizedinFigure 6-5 .In 93

PAGE 94

2kLJ(rr0)2; Thewavepacketiscenteredaroundnormalmodeswithwavenumberq0=0:475and!=16:9withxedeectiveLJparameter=0:44.WeusetwodierentvaluesofMsandforthesorbatemolecule:(0.5,5)and(1,1).Werefertothesetwointeractionparametersassystemand,respectively.FromEq. 6{9 ,thesorbatefrequencyis!s=16:8and5:36,respectively.Hencethesorbatefrequencyinsystemisclosertotheresonancefrequencywiththewavepacket.TheinitialcongurationofthewavepacketisshowninFigure 6-6 A).ThesystemcongurationsafterthewavepacketpassthesorbatemoleculeinsystemandareshowninFigure 6-6 B)andC),respectively.Thereisaqualitativedierencebetweenscatteringresultsforthesetwowavepackets.Underresonancecondition(system),wavepacketchangesitsshape,whichsuggeststhatadditionalmodesareexcitedduetointeractionwiththiswavepacket.Inordertoobtainabetterviewofthescatteringprocess,weperformwindowed-Fouriertransform(WFT)ofthelatticeconguration.Thistransformallowsustoobtainlocalwavenumbersofthelatticemodesfordierent 94

PAGE 95

B C D E FFigure6-6. Scatteringofwavepacketswithwavenumberswithq=0:475withxedsigma=0:44and(Ms,epsilon)=(0.5,5)and(1,1).A)Initialwavepacketcongurationforbothsimulations.;B)Systemcongurationafterscatteringwithsorbatefor(Ms,epsilon)=(0.5,5);C)Systemcongurationafterscatteringwithsorbatefor(Ms,epsilon)=(1,1);D)toF)areWindowedFouriertransformsofcongurationsshowninA)toC);xaxiscorrespondstolatticesitesandyaxiscorrespondstolocalwavenumbersoflatticeoscillations. locationswithinthesystem.TheresultsofWFTsforFigure 6-6 A)-C)areshowninFigure 6-6 D)-F),respectively.Inthesystem(plotE),thewavepacketstructureandwavenumberremainthesameasintheinitialwavepacket(plotD).Thisisexpectedasthisvalueoftheinitialwavevectorissucientlyfarfromtheresonantcondition.Ontheotherhand,inthesystemweobservethatthesymmetricstructureofthewavepacketisdestroyedandnearbywavemodesareexcitedduringthecollision.Thisquickredistributionofenergybetweendierentmodesischaracteristicofaresonantsystem. 95

PAGE 96

Evolutionofmodeenergyforincidentplanewavewithwavenumberq=0:245interactingwithdierentsorbatemolecules.Ms=1;=0:23andthreevaluesof0:5;1:5;2. 6-7 showsthattheenergychangeinbothsystemsAandCareintheformofperiodicoscillationwhereasrapiddecayingbehaviorisobservedinsystemB.Notethatinthisharmonicsystemtheenergyexchangebetweendierentnormal 96

PAGE 97

B CFigure6-8. FouriertransformofthesorbatedisplacementwithMs=1,=0:23and=0.5A),1.5B)and2.0C)respectively. Figure6-9. Dispersionofincidentmodeenergy(q=0:245)amongotherphononmodesunderdierentvaluesofsorbatemass.Ms=1;=0:23andthreevaluesof0:5;1:5;2. modesonlyoccursthroughsorbate-phononinteraction.Thus,rapidlydecayingbehaviorindicatesthesorbate-phononinteractionallowssignicantdispersionofincidentmodeenergyamongtheothernormalmodes.Theconditionforoccurrenceofenergychangebetweentwodegreesoffreedomistheresonancecondition.HencewepresentthesorbatefrequencyspectruminFigure 6-8 .Asexpected,weobserveawidefrequencyspectrumforsystemB.Moreover,inordertoidentifythedispersionofenergyamongtheotherphononmodes,weplotthegainedenergyoftheotherphononmodesexcepttheincidentmodeinFigure 6-9 .InsystemA,thereisalmostnoenergytransfertakingplace,implyingweakinteractionbetweensorbateandotherphononmodes.Thisisalsoveriedbythe 97

PAGE 98

6-8 A.Ontheotherhand,theexcitationofmostphononmodesisobservedforsystemB(closedcircles).NotethatthoughinFigure 6-8 Bthemagnitudeathighsorbatefrequencyisnotassignicantasthatforlowerfrequency,thehigherfrequencyphononmodesareexcitedduetothesatisfactionofresonancecondition 98

PAGE 99

6{13 intoEq. 6{12 ,weobtaintherelation 6{15 ,therelationbetweenthesetwosetsofcoecients,Eq. 6{13 ,becomes 6{17 99

PAGE 100

BFigure6-10. A)ComparisonoflinearmodelpredictionofFkfromEq. 6{18 andsimulationresultB)ComparisonoflinearmodelpredictionofGkfromEq. 6{18 andsimulationresult wherethemodejhastosatisfyEq. 6{15 .TheresultfromEq. 6{18 alongwiththeMDsimulationresultisshowninFigure 6-10 .TheevolutionofFkandGkcoecientsarecapturedwell.Inordertocompletetheabovederivation,weneedtheperturbedeigenvalueandeigenvectors,whichisachievedthroughdegenerateperturbationmethod.Thenewsetsofeigen-frequencyobtainedfromdegenerateperturbationmethodaswellasthenormaleigen-problemsolvingfunctionareshowninFigure 6-11 andrepresentedbysolidcirclesandopencirclesrespectively.Thetwocurvesmatchwellexceptnearzerofrequency,wheretheperturbationassumptionisnotvalid.Extendingthisanalysistononlinearsystem dt2=H0(f)+N(f);(6{19) 100

PAGE 101

Frequenciesobtainedfromdegenerateperturbationmethodandregulareigenvaluesolverareshownusingclosedandopencirclesrespectively. whereNisanonlinearoperator.TherstexpansionofEq. 6{12 stillholds.Inaddition,weperformmulti-timescales(tand)analysisfortimederivative.Inotherwords, dt!@ @t+@ @:(6{20)SubstitutingEq. 6{20 intoEq. 6{19 andcomparedierentorderterm,weexpecttodevelopamodelwithbetterdescriptionthanthecurrenttheoryforperturbationcausedbythestaticdefectatoms. 101

PAGE 102

5{12 .Innanoporousmaterials,wehavedemonstratedtheanharmoniceectonphononinteractioninducedbyabsorbedsorbatemolecules(gasmolecules)inzeolitemoleculesviamoleculardynamicssimulations.Weobservebothincreasinganddecreasingphononlifetimecausedbythepresenceofsorbatemolecule.Inordertodevelopamodeltodescribetheeectofsorbatemoleculeonphonondynamics,weinvestigateasimple1Dsystem.FromNEMDsimulationsweobservedcomplexdependenceofthermalconductivityonsorbate-latticeinteractionparameters.Basedonresonancecondition,theincidentmodeenergycanbedispersedamongtheothernormalmodesthroughsorbate-latticeinteractioninaharmoniclatticesystem.Tocontinuethiswork,weneedtocomputeArnolddiusionandusegeneralizedLangevinequationtodescribethedynamicsoflatticemodesduetothepresenceofsorbatemolecule. 102

PAGE 103

[1] J.A.Hutchby,R.Cavin,V.Zhirnov,J.E.Brewer,andG.Bouriano,\EmergingNanoscaleMemoryandLogicDevices:ACriticalAssessment,"Computer,vol.41,no.5,pp.28{32,2008. [2] P.Heino,\Thermalconductionsimulationsinthenanoscale,"JournalofComputationalandTheoreticalNanoscience,vol.4,pp.896{927,2007. [3] D.G.Cahill,W.K.Ford,K.E.Goodson,G.D.Mahan,A.Majumdar,H.J.Maris,R.Merlin,andS.R.Phillpot,\Nanoscalethermaltransport,"JournalofAppliedPhysics,vol.93,no.2,2003. [4] D.A.McQuarrie,StatisticalMechanics,Harper&RowPublishers,NewYork,1976. [5] P.G.Sverdrup,S.Sinha,M.Asheghi,S.Uma,andK.E.Goodson,\Measurementofballisticphononconductionnearhotspotsinsilicon,"Appl.Phys.Lett.,vol.78,pp.3331{3333,2001. [6] P.G.Sverdrup,Y.S.Ju,andK.E.Goodson,\Sub-continuumsimulationsofheatconductioninsilicon-on-insulatortransistors,"J.HeatTransfer{TransactionsoftheASME,vol.123,pp.130{137,2001. [7] S.K.KimandI.A.Daniel,\Gradientmethodforinverseheatconductionprobleminnanoscale,"Int.J.NumericalMethodsinEngineering,vol.60,pp.2165{2181,2004. [8] S.V.J.Narumanchi,J.Y.Murthy,andC.H.Amon,\Boltzmanntransportequation-basedthermalmodelingapproachesforhotspotsinmicroelectronics,"HeatandMassTransfer,vol.42,no.6,pp.478{491,2006. [9] S.FlachandC.R.Willis,\Discretebreathers,"Phys.Rep.,vol.295,pp.181{264,1998. [10] R.Reigada,A.Sarmiento,andK.Lindenberg,\AsymptoticdynamicsofbreathersinFermi-Pasta-Ulamchains,"Phys.Rev.E,vol.66,pp.046607,2002. [11] J.L.Marin,S.Aubry,andL.M.Floria,\Intrinsiclocalizedmodes:Discretebreathers.existenceandlinearstability,"PhysicaD,vol.113,pp.283{292,1998. [12] S.Lepri,R.Livi,andA.Politi,\Thermalconductioninclassicallow-dimensionallattices,"Phys.Rep.,vol.377,pp.1{80,2003. [13] A.LippiandR.Livi,\Heatconductionintwo-dimensionalnonlinearlattices,"J.Stat.Phys.,vol.100,pp.1147,2000. [14] T.Hatano,\HeatconductioninthediatomicTodalatticerevisited,"Phys.Rev.E,vol.59,pp.R1{R4,1999. 103

PAGE 104

B.Hu,B.Li,andH.Zhao,\Heatconductioninone-dimensionalchains,"Phys.Rev.E,vol.57,pp.2992{2995,1998. [16] S.IijimaandT.Ichihashi,\Single-shellcarbonnanotubesof1-nmdiameter,"Nature,vol.363,pp.603{605,1993. [17] R.H.Baughman,A.A.Zakhidov,andW.A.deHeer,\Carbonnanotubes{theroutetowardapplications,"Science,vol.297,pp.787{792,2002. [18] A.Javey,J.Guo,Q.Wang,andM.Lundstrom,\Ballisticcarbonnanotubeeld-eecttransistors,"Nature,vol.424,pp.654{657,2003. [19] Z.H.Chen,J.Appenzeller,Y.M.Lin,J.Sippel-Oakley,A.G.Rinzler,J.Y.Tang,S.J.Wind,P.M.Solomon,andP.Avouris,\Anintegratedlogiccircuitassembledonasinglecarbonnanotube,"Science,vol.311,no.5768,pp.1735{1735,2006. [20] V.Meunier,S.V.Kalinin,andB.G.Sumpter,\Nonvolatilememoryelementsbasedontheintercalationoforganicmoleculesinsidecarbonnanotubes,"PhysicalReviewLetters,vol.98,no.5,2007,Meunier,VincentKalinin,SergeiV.Sumpter,BobbyG. [21] P.Kim,L.Shi,A.Majumdar,andP.L.McEuen,\Thermaltransportmeasurementsofindividualmultiwallednanotubes,"Phys.Rev.Lett.,vol.87,pp.215502,2001. [22] J.Hone,M.Whitney,C.Piskoti,andA.Zettl,\Thermalconductivityofsingle-walledcarbonnanotubes,"Phys.Rev.B,vol.59,pp.R2514,1999. [23] M.A.OsmanandD.Srivastava,\Temperaturedependenceofthethermalconductivityofsingle-wallcarbonnanotubes,"Nanotechnology,vol.12,pp.21{24,2001. [24] S.Berber,Y.-K.Kwon,andDavidTomanek,\Unusuallyhighthermalconductivityofcarbonnanotubes,"Phys.Rev.Lett.,vol.84,pp.4613{4616,2000. [25] J.Che,T.Cagin,andW.A.Goddard,\Thermalconductivityofcarbonnanotubes,"Nanotechnology,vol.11,pp.65{69,2000. [26] S.Maruyama,\Amoleculardynamicssimulationofheatconductionofanitelengthsingle-walledcarbonnanotube,"MicroscaleThermophysicalEngineering,vol.7,pp.41{50,2003. [27] G.S.Nolas,G.A.Slack,D.T.Morelli,T.M.Tritt,andA.C.Ehrlich,\Theeectofrare-earthllingonthelatticethermalconductivityofskutterudites,"JournalofAppliedPhysics,vol.79,no.8,pp.4002{4008,1996,Part1. [28] G.P.Meisner,D.T.Morelli,S.Hu,J.Yang,andC.Uher,\Structureandlatticethermalconductivityoffractionallylledskutterudites:Solidsolutionsoffullylledandunlledendmembers,"PhysicalReviewLetters,vol.80,no.16,pp.3551{3554,1998. 104

PAGE 105

V.V.Murashov,\Thermalconductivityofmodelzeolites:Moleculardynamicssimulationstudy,"J.Phys.:Condens.Matter,vol.11,pp.1261{1271,1999. [30] W.C.Swope,H.C.Andersen,P.H.Berens,andK.R.Wilson,\Acomputersimulationmethodforthecalculationofequilibriumconstantsfortheformationofphysicalclustersofmolecules{applicationtosmallwaterclusters,"J.Chem.Phys.,vol.76,pp.637,1982. [31] H.J.C.Berendsen,J.P.M.Postma,W.F.vanGunsteren,A.DiNola,andJ.R.Haak,\Moleculardynamicswithcouplingtoanexternalbath,"J.Chem.Phys.,vol.81,pp.3684{3690,1984. [32] S.Nose,\Constanttemperaturemoleculardynamicsmethods,"Prog.Theor.Phys.Supp.,vol.103,pp.1,1991. [33] DMBylanderandL.Kleinman,\EnergyuctuationsinducedbytheNosethermostat,"PhysicalReviewB,vol.46,no.21,pp.13756{13761,1992. [34] D.L.ErmakandH.Buckholz,\NumericalintegrationoftheLangevinequation:MonteCarlosimulation,"J.Comput.Phys.,vol.35,pp.169,1980. [35] F.Verhulst,NonlinearDierentialEquationsandDynamicalSystems,Springer-Verlag,BerlinHeidelberg,1990. [36] E.Fermi,J.Pasta,andS.Ulam,\LosAlamosReportNo.LA-1940(1955),unpublished,"inCollectedPapersofEnricoFermi,E.Segre,Ed.,vol.2,p.978.UniversityofChicagoPress,Chicago,1965. [37] S.Aubry,\Breathersinnonlinearlattices:Existence,linearstabilityandquantization,"PhysicaD,vol.103,pp.201{250,1997. [38] R.Reigada,A.Sarmiento,andK.Lindenberg,\Energyrelaxationinnonlinearone-dimensionallattices,"Phys.Rev.E,vol.64,pp.066608,2001. [39] S.R.BickhamandJ.L.Feldman,\Calculationofvibrationallifetimesinamorphoussiliconusingmoleculardynamicssimulation,"Phys.Rev.B,vol.57,pp.12234,1998. [40] J.B.Page,\Asymptoticsolutionsforlocalizedvibrationalmodesinstronglyanharmonicperiodicsystems,"Phy.Rev.B,vol.41,no.11,pp.7835,1990. [41] A.J.SieversandS.Takeno,\Intrinsiclocalizedmodesinanharmoniccrystals,"Phys.Rev.Lett.,vol.61,pp.970{937,1998. [42] R.H.H.PoetzschandH.Bottger,\Interplayofdisorderandanharmonicityinheatconduction:Molecular-dynamicsstudy,"PhysicalReviewB,vol.50,no.21,pp.15757{15763,1994. 105

PAGE 106

K.Ullmann,A.J.Lichtenberg,andG.Corso,\Energyequipartitionstartingfromhigh-frequencymodesintheFermi-Pasta-Ulam-oscillatorchain,"Phys.Rev.E,vol.61,pp.2471,2000. [44] G.KopidakisandS.Aubry,\Intrabanddiscretebreathersindisorderednonlinearsystems.II.Localization.,"PhysicaD,vol.139,pp.247,2000. [45] P.G.Kevrekidis,K..Rasmussen,andA.R.Bishop,\Two-dimensionaldiscretebreathers:Construction,stability,andbifurcations,"Phys.Rev.E,vol.61,pp.2006,2000. [46] P.G.Kevrekidis,K..Rasmussen,andA.R.Bishop,\Comparisonofone-dimensionalandtwo-dimensionaldiscretebreathers,"MathematicsandComputersinSimulation,vol.55,pp.449,2001. [47] M.JohanssonandS.Aubry,\GrowthanddecayofdiscretenonlinearSchrodingerbreathersinteractingwithinternalmodesorstanding-wavephonons,"Phys.Rev.E,vol.61,pp.5864,2000. [48] F.Zhang,D.J.Isbister,andD.J.Evans,\Nonequilibriummoleculardynamicssimulationsofheatowinone-dimensionallattices,"Phys.Rev.E,vol.61,pp.3541,2000. [49] I.Bena,A.Saxena,G.P.Tsironis,M.Iba~nes,andJ.M.Sancho,\Connementofdiscretebreathersininhomogeneouslyprolednonlinearchains,"Phys.Rev.E,vol.67,pp.037601,2003. [50] F.Piazza,S.Lepri,andR.Livi,\Coolingnonlinearlatticestowardenergylocalization,"Chaos,vol.13,pp.637,2003. [51] G.P.TsironisandS.Aubry,\Slowrelaxationphenomenainducedbybreathersinnonlinearlattices,"Phys.Rev.Lett.,vol.77,pp.5225,1996. [52] M.Peyrard,\Thepathwaytoenergylocalizationinnonlinearlattices,"PhysicaD,vol.119,pp.184,1998. [53] T.Cretegny,T.Dauxois,S.Ruo,andA.Torcino,\Localizationandequipartitionofenergyinthe-FPUchain:Chaoticbreathers,"PhysicaD,vol.121,pp.109,1998. [54] A.Franchini,V.Bortolani,andR.F.Wallis,\Surfaceandgapintrinsiclocalizedmodesinone-dimensionalIII-IVsemiconductors,"J.Phys.:Condens.Matter,vol.12,pp.1,2000. [55] G.KopidakisandS.Aubry,\Intrabanddiscretebreathersindisorderednonlinearsystems.I.Delocalization.,"PhysicaD,vol.130,pp.155,1999. 106

PAGE 107

B.I.Swanson,J.A.Brozik,S.P.Love,G.F.Strouse,A.P.Shreve,A.R.Bishop,W.-Z.Wang,andM.I.Salkola,\Observationofintrinsicallylocalizedmodesinadiscretelow-dimensionalmaterial,"Phys.Rev.Lett.,vol.82,pp.3288,1999. [57] K.Kisoda,N.Kimura,H.Harima,K.Takenouchi,andM.Nakajima,\Intrinsiclocalizedvibrationalmodesinahighlynonlinearhalogen-bridgedmetal,"J.Luminescence,vol.94,pp.743,2001. [58] A.Xie,L.vanderMeer,W.Ho,andR.H.Austin,\Long-livedAmideIvibrationalmodesinmyoglobin,"Phys.Rev.Lett.,vol.84,pp.5435,2000. [59] J.F.Moreland,J.B.Freund,andG.Chen,\Thedisparatethermalconductivityofcarbonnanotubesanddimondnanowiresstudiedbyatomisticsimulation,"MicroscaleTherm.Eng.,vol.8,pp.61{69,2004. [60] M.P.AllenandD.J.Tildesley,ComputerSimulationofLiquids,OxfordUniversityPress,NewYork,1987. [61] A.V.SavinandO.I.Savina,\Nonlineardynamicsofcarbonmolecularlattices:Solitonplanewavesingraphitelayersandsupersonicacousticsolitonsinnanotubes,"Phys.SolidState,vol.46,pp.383{391,2004. [62] T.Yu.Astakhova,M.Menon,andG.A.Vinogradov,\Solitonsininharmoniclattices:Applicationtoachiralcarbonnanotubes,"Phys.Rev.B,vol.70,pp.125409,2004. [63] R.Saito,G.Dresselhaus,andM.S.Dresselhaus,PhysicalPropertiesofCarbonNanotubes,ImperialCollegePress,London,1998. [64] D.W.Brenner,\Empiricalpotentialforhydrocarbonsforuseinsimulatingthechemicalvapordepositionofdiamondlms,"Phys.Rev.B,vol.42,pp.9458{9471,1990. [65] J.Terso,\Empiricalinteratomicpotentialforcarbon,withapplicationstoamorphouscarbon,"Phys.Rev.Lett.,vol.61,pp.2879{2882,1988. [66] J.Terso,\Newempiricalapproachforthestructureandenergyofcovalentsystems,"Phys.Rev.B,vol.37,pp.6991{7000,1988. [67] KMLiew,CHWong,andMJTan,\Tensileandcompressivepropertiesofcarbonnanotubebundles,"ActaMaterialia,vol.54,no.1,pp.225{231,2006. [68] H.RAFII-TABAR,\Computationalmodellingofthermo-mechanicalandtransportpropertiesofcarbonnanotubes,"Physicsreports,vol.390,no.4-5,pp.235{452,2004. [69] A.Kalra,S.Garde,andG.Hummer,\FromTheCover:Osmoticwatertransportthroughcarbonnanotubemembranes,"ProceedingsoftheNationalAcademyofSciences,vol.100,no.18,pp.10175,2003. 107

PAGE 108

W.H.Press,S.A.Teukolsky,W.T.Vetterling,andB.P.Flannery,NumericalRecipesinFortran.TheArtofScienticComputing,CambridgeUniversityPress,2ndedition,1992. [71] A.A.Maradudin,E.W.Montroll,G.H.Weiss,andI.P.Ipatova,TheoryofLatticeDynamicsintheHarmonicApproximation,AcademicPress,NewYork,1971. [72] M.TatlierandA.Erdem-Senatalar,\Theperformanceanalysisofasolaradsorptionheatpumputilizingzeolitecoatingsonmetalsupports,"ChemicalEngineeringCommunications,vol.180,pp.169{185,2000. [73] J.M.Gordon,K.C.Ng,H.T.Chua,andA.Chakraborty,\Theelectro-adsorptionchiller:aminiaturizedcoolingcyclewithapplicationstomicro-electronics,"InternationalJournalofRefrigeration-RevueInternationaleDuFroid,vol.25,no.8,pp.1025{1033,2002. [74] Y.Hudiono,A.Greenstein,C.Saha-Kuete,B.Olson,S.Graham,andS.Nair,\Eectsofcompositionandphononscatteringmechanismsonthermaltransportinmzeolitelms,"JournalofAppliedPhysics,vol.102,2007. [75] G.P.Srivastava,ThePhysicsofPhonons,AdamHilger,Bristol,1990. [76] T.R.ForesterandW.Smith,\Bluemoonsimulationsofbenzeneinsilicalite-1.Predictionsoffreeenergiesanddiusioncoecients,"J.Chem.Soc.,FaradayTrans.,vol.93,pp.3249{3257,1997. [77] D.I.KopelevichandH.-C.Chang,\Diusionofinertgasesinsilicasodalite:Importanceoflatticeexibility,"J.Chem.Phis.,vol.115,pp.9519,2001. [78] S.C.TuragaandS.M.Auerbach,\Calculatingfreeenergiesfordiusionintight-ttingzeolite-guestsystems:Localnormal-modeMonteCarlo,"J.Chem.Phys.,vol.118,pp.6512{6517,2003. [79] F.Jousse,D.P.Vercauteren,andS.M.Auerbach,\HowdoesbenzeneinNaYzeolitecoupletotheframeworkvibrations?,"J.Phys.Chem.B,vol.104,pp.8768{8778,2000. [80] A.M.Greenstein,S.Graham,Y.C.Hudiono,andS.Nair,\ThermalpropertiesandlatticedynamicsofpolycrystallineMFIzeolitelms,"NanoscaleandMicroscaleThermophysicalEngineering,vol.10,pp.321{331,2006. [81] J.J.Dong,O.F.Sankey,andC.W.Myles,\Theoreticalstudyofthelatticethermalconductivityingeframeworksemiconductors,"PhysicalReviewLetters,vol.86,no.11,pp.2361{2364,2001. [82] R.TsekovandE.Ruckenstein,\Stochasticdynamicsofasubsysteminteractingwithasolidbodywithapplicationtodiusiveprocessesinsolids,"J.Chem.Phys.,vol.100,pp.1450{1455,1994. 108

PAGE 109

P.DemontisandG.B.Suritti,\Structureanddynamicsofzeolitesinvestigatedbymoleculardynamics,"Chem.Rev.,vol.97,pp.2845{2878,1997. [84] D.I.KopelevichandH.-C.Chang,\Doeslatticevibrationdrivediusioninzeolites?,"J.Chem.Phys.,vol.114,pp.3776{3789,2001. [85] J.W.Richardson,J.J.Pluth,J.V.Smith,W.J.Dytrych,andD.M.Bibby,\Conformationofethylene-glycolandphase-changeinsilicasodalite,"J.Phys.Chem.,vol.92,pp.243,1988. [86] K.Knorr,C.M.Braunbarth,G.vandeGoor,P.Behrens,C.Griewatsch,andW.Depmeier,\High-pressurestudyondioxolanesilicasodalite(C3H6O2)2[Si12O24]{neutronandX-raypowderdiractionexperiments,"SolidStateCommun.,vol.113,pp.503,2000. [87] J.B.Nicholas,A.J.Hopnger,F.R.Trouw,andL.E.Iton,\Molecularmodelingofzeolitestructure.2.Structureanddynamicsofsilicasodaliteandsilicateforce-eld,"J.Am.Chem.Soc.,vol.113,pp.4792,1991. [88] S.ElAmrani,F.Vigne,andB.Bigot,\Self-diusionofrare-gasesinsilicalitestudiedbymoleculardynamics,"J.Phys.Chem.,vol.96,pp.9417,1992. [89] D.Keer,A.V.McCormick,andH.T.Davis,\Unidirectionalandsingle-lediusioninAlPO4-5:Moleculardynamicsinvestigations,"Mol.Phys.,vol.87,pp.367,1996. [90] S.J.Goodbody,K.Watanabe,D.MacGowan,J.Walton,andN.Quirke,\Molecularsimulationofmethaneandbutaneinsilicalite,"J.Chem.Soc.FaradayTrans.,vol.87,pp.1951,1991. [91] R.L.June,A.T.Bell,andD.N.Theodorou,\Transition-statestudiesofxenonandSF6diusioninsilicalite,"J.Phys.Chem.,vol.95,pp.8866,1991. [92] A.V.Kiselev,A.A.Lopatkin,andA.A.Shulga,\Molecularstatisticalcalculationofgasadsorptionbysilicalite,"Zeolites,vol.5,pp.261,1985. [93] A.J.C.Ladd,B.Moran,andW.G.Hoover,\Latticethermalconductivity:Acomparisonofmoleculardynamicsandanharmoniclatticedynamics,"Phys.Rev.B,vol.34,pp.5058,1986. [94] A.J.H.McGaugheyandM.Kaviany,\Quantitativevalidationoftheboltzmanntransportequationphononthermalconductivitymodelunderthesingle-moderelaxationtimeapproximation,"PhysicalReviewB,vol.69,no.9,2004. [95] G.W.Gardiner,HandbookofStochasticMethodsforPhysics,Chemistry,andtheNaturalSciences,Springer-Verlag,Berlin,1983. [96] J.Luczka,\Non-markovianstochasticprocesses:Colorednoise,"Chaos,vol.15,no.2,2005. 109

PAGE 110

A.J.H.McGaugheyandM.Kaviany,\Thermalconductivitydecompositionandanalysisusingmoleculardynamicsimulations.PartII.Complexsilicastructures,"Int.J.HeatMassTransfer,vol.47,pp.1799{1816,2004. [98] P.G.Klemens,\Thescatteringoflow-frequencylatticewavesbystaticimperfections,"ProceedingsofthePhysicalSocietyofLondonSectionA,vol.68,no.12,pp.1113{1128,1955,16. [99] WALittle,\THETRANSPORTOFHEATBETWEENDISSIMILARSOLIDSATLOWTEMPERATURES,"CanadianJournalofPhysics,vol.37,no.3,pp.334{349,1959. [100] AJLichtenbergandLiebermanMA,RegularandChaoticDynamics,Springer,1992. [101] P.K.Schelling,S.R.Phillpot,andP.Keblinski,\Phononwave-packetdynamicsatsemiconductorinterfacesbymolecular-dynamicssimulation,"AppliedPhysicsLetters,vol.80,no.14,pp.2484{2486,2002,12. 110

PAGE 111

Chia-YiChenwasborninTaiwanonSeptember4th,1981.ShereceivedtheBachelorofSciencefromNationalTaiwanUniversity,Taipei,TaiwaninJune2003.Aftercompletingherbachelor'sdegreeshejoinedtheDepartmentofChemicalEngineering,UniversityofFlorida,inAugust2003.Herresearchinterestsaremodelingofthermaltransportinnanostucturedmaterials. 111