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Toward understanding low surface friction on quasiperiodic surfaces

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Toward understanding low surface friction on quasiperiodic surfaces
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McLaughlin, Keith
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Quasicrystals
Atomistic simulation
Nano-tribology
Stochastic differential equations
Computational physics
Dissertations, Academic -- Physics -- Masters -- USF   ( lcsh )
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Abstract:
ABSTRACT: In a 2005 article in Science 45, Park et al. measured in vacuum the friction between a coated atomic-force-microscope tip and the clean two-fold surface of an AlNiCo quasicrystal. Because the two-fold surface is periodic in one direction and aperiodic (with a quasiperiodicity related to the Fibonacci sequence) in the perpendicular direction, frictional anisotropy is not unexpected; however, the magnitude of that anisotropy in the Park experiment, a factor of eight, is unprecedented. By eliminating chemistry as a variable, the experiment also demonstrated that the low friction of quasicrystals must be tied in some way to their quasiperiodicity. Through various models, we investigate generic geometric mechanisms that might give rise to this anisotropy.
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Thesis (M.S.)--University of South Florida, 2009.
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by Keith McLaughlin.
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Toward understanding low surface friction on quasiperiodic surfaces
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ABSTRACT: In a 2005 article in Science [45], Park et al. measured in vacuum the friction between a coated atomic-force-microscope tip and the clean two-fold surface of an AlNiCo quasicrystal. Because the two-fold surface is periodic in one direction and aperiodic (with a quasiperiodicity related to the Fibonacci sequence) in the perpendicular direction, frictional anisotropy is not unexpected; however, the magnitude of that anisotropy in the Park experiment, a factor of eight, is unprecedented. By eliminating chemistry as a variable, the experiment also demonstrated that the low friction of quasicrystals must be tied in some way to their quasiperiodicity. Through various models, we investigate generic geometric mechanisms that might give rise to this anisotropy.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
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Advisor: David Rabson, Ph.D.
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Quasicrystals
Atomistic simulation
Nano-tribology
Stochastic differential equations
Computational physics
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TowardUnderstandingLowSurfaceFrictiononQuasiperiodi cSurfaces by KeithMcLaughlin Athesissubmittedinpartialfulllment oftherequirementsforthedegreeof MasterofScience DepartmentofPhysics CollegeofArtsandSciences UniversityofSouthFlorida MajorProfessor:DavidRabson,Ph.D. BrianSpace,Ph.D. InnaPonomareva,Ph.D. MohamedElhamdadi,Ph.D. DateofApproval: November16,2009 Keywords:Quasicrystals,AtomisticSimulation,Nano-Trib ology,Stochastic DierentialEquations,ComputationalPhysics c r Copyright2009,KeithMcLaughlin

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ACKNOWLEDGMENTS ThisworkwassupportedbytheNationalDefenseScienceandE ngineeringGrant, NationalScienceFoundationS-STEMprogram,UniversityofS outhFloridaResearch ComputingandtheUniversityofSouthFloridaPhysicsDepar tment.AdditionallyI thankRomainPerriotandAaronLandervillefortheirhelpfu lsuggestions,myfriends andfamilyfortheircontinuedsupport,andmyadvisorDr.Da vidRabsonforhis guidanceinthiswork.

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TABLEOFCONTENTS LISTOFTABLES iii LISTOFFIGURES iv LISTOFABBREVIATIONS vi ABSTRACT vii CHAPTER1INTRODUCTION 1 1.1Park'sexperiment 3 1.2Proposedmodels 4 CHAPTER2WHATISAQUASICRYSTAL?7 2.1Anexample:theFibonaccisequence82.2Fourier-spacecrystallography 10 2.3Gauge-invariantquantities 12 2.4Calculationofspacegroups 13 CHAPTER3STOCHASTIC-ODEMETHOD21 3.1VanGunsterenandBerendsen'sAlgorithm243.2Testingthealgorithm 26 3.3QuasiperiodicS-ODE 30 CHAPTER4MDSIMULATIONSOFD-ALCONIAPPROXIMANTS36 4.1Harper'swork 37 4.2BeyondHarper 38 4.3Results 41 CHAPTER5ASIMPLERMODEL:FIBONACCIUM45 5.1ConstructingFibonaccium 45 5.2Fixedorder,variablemass 48 5.3Fixedmass,variableorder 48 CHAPTER6CONCLUSIONSANDFUTUREWORK51 6.1Futurework 53 REFERENCES 57 APPENDICES 62 AppendixALinearresponse 63 AppendixBS-ODEsolver 65 AppendixCLAMMPSinputles 76 i

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AppendixDGeneratingFibonaccium82 ii

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LISTOFTABLES Table2.1TheFibonaccisequence 9 Table4.1d-AlCoNimolecular-dynamicsparameters42Table5.1Fibonacciummolecular-dynamicsparameters49 iii

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LISTOFFIGURES Figure1.1Park'sschematic 4 Figure1.2Park'sfrictionmeasurements 5 Figure1.3Park'storsional-responsemeasurements6Figure2.1Icosahedraldiractionpattern 9 Figure2.2Glideplane 12 Figure2.3Five-foldwave-vectors 14 Figure2.4Five-folddensity-wavepatterns 15 Figure2.5Fourier-space p 51 m ( k ) 17 Figure3.1Tomlinson-PrandtlModel 22 Figure3.2Tshiprut'sresults 23 Figure3.3Brownianmotion 28 Figure3.4Lineartransport 29 Figure3.5ReproducingTshiprut'sresults 30 Figure3.6Quasiperiodic-approximantpotentials31Figure3.7S-ODEfrictionversustemperature32Figure3.8S-ODEtimeseriesdata 33 Figure3.9Frictionforincommensuratepotentials34Figure3.10Frictionforcommensuratepotentials35Figure3.11Positiontime-seriesforcommensuratepotentia ls35 Figure4.1Harper'sexperiment 37 Figure4.2Harper'sresults 38 Figure4.3d-AlCoNiT11sampleandtip 40 Figure4.4SurfaceterminationsfortheT11approximant41 iv

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Figure4.5Stick-sliponT11(a) 42 Figure4.6FrictionversusloadfortheT11approximant43Figure5.1BCCFibonaccium 46 Figure5.2SCpairpotentials 47 Figure5.3Fibonacciumwithvariablemassratio49Figure5.4Fibonacciumoforder55 50 Figure5.5Simple-cubicFibonacciumfrictionalresponse50Figure6.1ElectronicDoSforaFibonaccichain53Figure6.2FrictionversuslatticeconstantviaS-ODE55Figure6.3ElectronicDoSforaFibonaccichain56 v

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LISTOFABBREVIATIONS BCC Body-CenteredCubic D-ALCONI DecagonalAluminumCobaltNickel DoS DegreesofFreedom DMT Derjaguin-Muller-Toporov FCC Face-CenteredCubic GCC Group-CompatibilityCondition LAMMPSLarge-scaleAtomic/MolecularMassivelyParallelSi mulator MD MolecularDynamics PBC PeriodicBoundaryConditions QC Quasicrystal S-ODE StochasticOrdinaryDierentialEquation vi

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TOWARDUNDERSTANDINGLOWSURFACEFRICTIONON QUASIPERIODICSURFACES KeithMcLaughlin ABSTRACT Ina2005articleinScience[45],Parketal.measuredinvacu umthefrictionbetween acoatedatomic-force-microscopetipandthecleantwo-foldsu rfaceofanAlNiCoquasicrystal.Becausethetwo-foldsurfaceisperiodicinonedi rectionandaperiodic(with aquasiperiodicityrelatedtotheFibonaccisequence)inth eperpendiculardirection, frictionalanisotropyisnotunexpected;however,themagn itudeofthatanisotropyin theParkexperiment,afactorofeight,isunprecedented.By eliminatingchemistryasa variable,theexperimentalsodemonstratedthatthelowfri ctionofquasicrystalsmust betiedinsomewaytotheirquasiperiodicity.Throughvario usmodels,weinvestigate genericgeometricmechanismsthatmightgiverisetothisan isotropy. vii

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CHAPTER1 INTRODUCTION Discoveredin1982[59], 1 quasicrystals(QCs)exhibitspectacularpropertiesand havebeenthefocusofcountlesstheoreticalandexperiment alinvestigations.Nearly defect-freeQCsoncepiquedtheinterestofthermoelectrics researchersduetotheirremarkablylowthermalconductivities[18].Titanium-basedQ Cshavebeenfoundto adsorbhydrogenparticularlywell,makingtheseaninteres tingprospectinhydrogenstoragetechnology[18,32].Theirlow-frictionsurfaces,r esistancetocorrosionand abrasion,andnon-stickpropertiesmakeapplicationascook wareandsurgicalblades possible[18,48,63].Otherdevelopmentsincludethermalb arriers[10]andquasicrystallinereinforcedmetal-matrixcomposites[67].Moreover ,QCsareimmenselyexciting forthechallengesthey'vepresentedtotheory.Theclassi cationofsymmetrygroupsin QCshasinitiatedfurtherconsiderationofEwaldandBienen stock'sFourier-spacecrystallography[3,56],superspaceapproachesinvolvingproj ectionfromhigher-dimensional spaces[5,30],andmorerecently,applicationofgroupcoho mology[14,15,50,52],which providedasimpliedexplanationofsymmetry-inducedband-s ticking[34,35],aswellas anoveleectthatisnotyetunderstood[13,16,50].Modelsu singFibonaccichainshave beendevelopedtoinvestigatespectral,electronic,andph ononiceectsinQCs[4,12,47], whilethedevelopmentoflattice-gasmodels[41],applicati onoftilingtheory[51],and adaptationoflow-energyelectrondiractiontechniques[8 ]havebeenessentialtomakingprogresstowardsthedeterminationofQCatomicstructu re. AlthoughonlyafewofthepropertiesthatmakeQCsspecialar ewellunderstood,it hasbeenespeciallydiculttopinpointthephysicalmechan ismsthatarefundamentally responsibleforlowfrictiononQCsurfaces.Forsometime,l owfrictionwasbelieved toprimarilyresultfromhardnessorsurfacechemistry[18, 64].However,a2003experi1 Thoughquasicrystalswerediscoveredin1982,Shechtmandidnotpublis hhisndingsuntil1984. 1

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mentfoundatwo-folddierenceinfrictioncoecientsbetwe enanicosahedral-AlPdMn quasicrystalandanearbycrystallineapproximant[37],pe rhapssuggestingthatthe quasiperiodicityofasystemmayalsoplaysomeroleinfrict ion,whileservingtoeliminatesurfacechemistryasamajorcontributor.Then,in2005 ,Parketal.measuredan eight-foldfrictionalanisotropyalongthesurfaceofdecag onal-AlCoNiwhichpossessed coexistingperiodicandaperiodicaxes[44{46].Thisresul t,alongwiththenotionthat symmetrymayleadtosignicantconsequencesinthephononi candelectronicspectra ofsolids[4,12,16,26,34,35,47,50],seemstoimplicatequ asiperiodicityasbeinglargely responsibleforthelowfrictionobservedinthesesystems. BymodelingtheAlCoNisystemusedbyParketal.,ourgoalwas toinvestigate theimportanceoftheroleplayedbyquasiperiodicityasitr elatestothishugefrictional anisotropy,andlow-frictionQCsurfacesingeneral.Tothis end,weperformednumericalcalculationsonamodelstochasticdierentialequatio nandranmolecular-dynamics simulationsontwomodelsetsofquasicrystallinesystems: areplicaofAlCoNiapproximantsandasimple-cubicsystemwithquasiperiodicityimpos edbymodulatingatomic masses. Beforegoingintofutherdetail,wewillusetherestofchapt eronetoreviewthe experimentperformedbyParketal.andbrieryoverviewourp roposedmodels.To follow,chaptertwowillreviewsomeconceptsofquasicryst alsandquasicrystallography, includingFourier-spacecrystallography,classicationo fspacegroups,theFibonacci sequence,andanexamplespace-groupcalculation. Inchapterthree,wewillexamineastochasticordinarydie rentialequationusedby Tshiprutetal.tomodelthetemperaturedependenceofstick -slipfrictiononaperiodic surface[68].Wehavereproducedtheseresultsandwillelab orateonthealgorithmused beforeapplyingthismodeltoanaperiodicsurface.Chapter sfourandvewillpertain tomolecular-dynamicssimulationsontwoseperatesetsofmo dels.Finally,inchapter six,wewilldrawconclusionsfromthesumofourthreemodels anddiscussfuturework. 2

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1.1Park'sexperiment Althoughsymmetrywasknowntobeintimatelytiedtopropert iessuchasheatand electronictransport,priortothe2005publicationinScie ncebyParketal.,symmetry 2 wasnotstronglyimplicatedasbeingresponsibleforlowfri ctiononQCsurfaces.Rather, manyresearchersfocusedonincommensurabilitybetweenin terfacesandhardness[18, 37],inparticularthelatter'seectonsurface-to-surfacec ontactarea.Thoughthese propertiesmayplayarole,Park'sexperiment,whichcompar edthefrictioncoecient measuredalongaperiodicandanaperiodicaxisofdecagonal -AlCoNi,providedstrong evidencetosuggestthatsymmetrywasthetrueculprit[44{4 6]. Intheexperiment,asingle-grainAl 72 Ni 11 Co 17 3 QCwascuttoproduceasample withdimensions1cm 1cm 1 : 5mm,andsuchthatthelargestsurfacepossessed two-foldsymmetrywithaperiodicaxis(paralleltoaten-fold axis)andanaperiodic axis(paralleltothetwo-foldaxis);seegure1.1.Thetipwa scoatedwith50nm ofTiN,whichwasthenpassivatedwithC 16 alkanethiol;thenalproductmeasured 30-50nmpriortocontactandpossessedaspringconstantof2 : 5N/m.Thesampleand tipwerebothprepared,andtheexperimentwasperformed,in anultra-high-vacuum (UHV)chamberat1 : 0 10 10 Torr. Thecantileverwasscannedatsomeangle ,where =45 ( 45 )corresponded totheaperiodic(periodic)axis,seegure1.1, 4 andthetorsionalandderectionresponseweremeasured.Usingtheassumptionthatthefrictio nialforcewasafunction ofseparablevariables,scanningdirectionandappliedloa d,therelationshipbetween torsionalresponseandfrictionalforcewasderivedanduse dtodeterminethefriction forappliedloadsfrom 130to70nNandslidingvelocitiesfrom20to2000nm/s.Frictionalanisotropywithavalue = periodic = aperiodic =8 : 2 0 : 4wasfoundwithno signicantdependenceonloadorvelocity,where a isthecoecientoffrictionfor slidingalongdirection a ;seegures1.2and1.3.Themeasuredshearstressesof690(8 5 MPa)fortheperiodic(aperiodic)directionsappliedtothe Derjaguin-Muller-Toporov 2 Specically,thedierencebetweentranslationalsymmetryandqu asiperiodicity. 3 Thechemicalcompositionwasveredbyenergy-dispersivex-rayanal ysis. 4 Symmetricangleswereusedtoeliminatepossibledierencesinc antileverdeformationandbuckling. 3

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Figure1.1.(A)Illustrationofsinglegraind-AlCoNi.Frict ionmeasurementswereperformedonthetwo-foldsurface,alongthetwo-foldandten-fold axes.(B)Frictionis measuredbymovingthecantileveratsomescanningangle,an drecordingthetorsional andderectionresponse.OriginallyprintedinScience 309 .Reprintedwithpermission fromMiguelSalmeron.(DMT)model[7]yieldedthesolidlinesshowningure1.2ing oodagreementwiththe measuredfriction. Parketal.explainedthatbecausethecontactareawasfound tobeunaected byslidingangle,hardnesswaseliminatedasanexplanation forthishugefrictional anisotropy,atleastinthiscase.Slipplaneswerealsoelim inatedsincetherewasno obseveredplasticdeformationofthesample.BecausetheTi Ntipwasstructurallydifferentfromd-AlCoNi,andpossiblyamorphous,argumentsinv olvingcommensurability arenotwellfounded. 5 1.2Proposedmodels Thoughweinitiallysoughttoinvestigatead-AlCoNiapproxi mantusingMD,unexpectedresultsledustoformulatetwoadditionalmodels: aS-ODEapproachandour Fibonacciummodel. WewereoptimisticthatperformingMDsimulationsond-AlCoN iwouldyieldsome anisotropyinagreementwithParketal.Fromthere,wewould focusoninvestigating 5 Thoughsuchargumentsmaystillbeusefultounderstand other experiments. 4

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Figure1.2.Measuredfrictionasafunctionofslidingangle foraxedloadandsliding velocity.Thepeakcorrespondstoslidingparalleltothepe riodicaxis.Graphobtained fromreference[46].theexactmechanismoftheanisotropybymodelingphononpar ticipationratiosand dierencesinstick-slipalongeachaxis.Aftertweakingthe potentials,usingdierentapproximantsanddierentsurfaceterminations,andex ploringseveralpointsin theparameterspace,wewereunabletomeasureanisotropyin agreementwithPark's experiment. Asidefromtheexclusionofelectronicdegreesoffreedom,w efoundtwomajordecienciesinourd-AlCoNimodelwereitspoorstabilityevenat 0Kelvinandourinability touserealisticslidingvelocities.Thecorrectionofthes etwofailureswerethefocusof ourtwoalternativemodels.First,weusedaS-ODEmodel,deve lopedbyTshiprut et.al[68].Withthisapproachweloseatomicity,andthusan ynotionofphonon propagation.Thesilverliningisthatvelocitiesaslowasa picometer/secondcanbe explored.Second,weperformedMDsimulationsonaseriesof ctitioussolidswith cubicsymmetry,butwithmassesperturbedaccordingtotheF ibonaccisequence,such thatinthelimit,oursamplehasasurfacepossessingaperio dicaxisandaquasiperiodicaxis.Thestrengthhereisthatwehaveaccesstoaveryla rgenumberofstable approximantsofincreasingquasiperiodicity.Thestrengt hofthismodelisthatwecan 5

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Figure1.3.(A)Torsionalresponse,andthereforefriction ,doesnotvarysignicantly withintherangeofslidingvelocitiesexploredbyParketal .(B)Torsionalresponseand frictionalanisotropyasafunctionofappliedloadfor =+ = 45 .Graphsobtained fromreference[46].focusonthecoexistenceofquasiperiodicandperiodicaxes whileessentiallyignoring anyothervariables. Finally,weinvestigatedthepossibilityofusingalinear-r esponseapproach[1]but foundthatthiswouldnotadequatelydescribethestick-slip behaviorthatweobserve inourMDtrials.Thisisdiscussedinmoredetailinappendix A. 6

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CHAPTER2 WHATISAQUASICRYSTAL? We'vediscussedquasicrystalsforthelengthofthispaperw ithoutyetdiscussingin detailwhatexactlyismeantbyquasicrystal.TheInternati onalUnionofCrystallographystatesthataQCis\anysolidhavinganessentiallydiscr etediractionpattern"[43]. Thatmightnotbeveryhelpful,sowe'llcomebacktoitlater. Foremost,quasicrystalisshortforquasiperiodiccrystal .Clearly,thisimpliesthe lackoftruetranslationalsymmetry,afeaturethatisthede ningcharacteristicofany conventionalcrystal.QCs,however,exhibitanapproximat etranslationalsymmetry: takinganynitepatchofsize S fromaQC,onecanndanexactcopyatsomedistance L .Becauselim S !1 L = 1 ,thereisnotrueperiodicityinaQC;wecannottranslatean innitequasicrystalandbringitbackintoexactcoinciden cewithitself.Ontheother hand,thisapproximatesymmetryimpliesthatthereislong-r angeordering.Infact,if welifttherequirementforidentityundertranslationorot hersymmetryoperationand replaceitwith indistinguishability ,wemayrestorethenotionofsymmetryinQCs. Next,considerthatadiscreteperiodiclatticecanonlyexh ibit1,2,3,4,and6-fold symmetry.Proof 1 Considerthetwo-dimensionalcase.Supposewecangeneratea discretelattice with n -foldrotationalsymmetry,with n =5or n> 6, n 2 N .Takeonegenerating vectorofthelatticetobe a 1 =[ a x ; 0] 2 R 2 2 Applytherotationmatrix R ( )= 0B@ cos sin sin cos 1CA ; (2.1) 1 Althoughthisresultiswell-known,thisproofisoriginal. 2 Wearefreetochooseourcoordinatesysteminsuchawaytomakeonecomp onentof a 1 =0. 7

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where = 2 n foran n -foldrotation.Applyingtherotationgivestwolinearlyind ependentvectors, a 1 and a 2 = R ( 2 n ) a .Werequirethat a 2 beinthelattice,sinceitis symmetry-equivalentto a 1 .Since n 6 =1or2, a 1 and a 2 arenotco-linear;theyaresucienttogeneratethelattice.Applyingtheinverserotatio non a 1 yields a 3 = R ( 2 n ) a Sincethelatticeisdiscrete,andnotwoof a 1 ; a 2 and a 3 areco-linear, 3 werequirethat a 1 beanintegrallinearcombinationof a 2 and a 3 R ( ) a 1 = m 1 a 1 + m 2 R ( ) a 1 : (2.2) Takingthe x components, a x =( m 1 a x cos 2 n + m 2 a x cos 2 n )(2.3) cos( 2 n )= 1 m 1 + m 2 ; (2.4) 1 cos( 2 n ) 2 Z ; (2.5) butthisisnotthecasefor n =5or n> 6;thuswereachacontradiction. 4 Thisisaveryimportantresult,foritimpliesthatperiodic decagonal,octagonal, andpentagonalcrystalscannotexist.Howeverthisrestric tiondoesnotholdforquasicrystals.Infact,diractionpatternswitheight-foldan dten-foldsymmetryhavebeen observed.Unliketheirperiodicsiblings,thediractionp atternsofquasicrystalsare dense.However,ifwechoosesomeminimumintensity andonlyconsiderBraggpeaks withintensity > ,thenweareleftwithan\essentiallydiscrete"pattern;se eg.2.1. 2.1Anexample:theFibonaccisequence TheFibonaccisequenceisanorderedlistofcharacterstrin gswhoselimitdemonstratesquasiperiodicity.Itisconstructedbystartingwi ththestring S ,andthenapplyingthetransformationrule S L L LS ;seetable2.1.TheFibonaccisequence hasanintimaterelationshipwithboththeFibonaccinumber sandthegoldenmean 3 Thisistruesincewehaveexcluded n =1 ; 2 ; 4. 4 cos 2 n for n 6ismonotonicallydecreasingintheinterval cos 2 6 ; lim n !1 cos 2 n =(2 ; 1);therefore 1 = cos 2 6 ; lim n !1 1 = cos 2 n =( 1 2 ; 1). 8

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Figure2.1.Diractionpatternfromaquasicrystalwithico sahedralsymmetry.ImageobtainedfromRonLifshitz'swebpagehttp://www.lassp.cornell.edu/lifshitz/quasicrystals .html.Retrieved16Sep2009. = 1+ p 5 2 .Inparticular,thelengthofeachterm N i ( L )+ N i ( S )correspondstothe i thFibonaccinumber,andtheratio N i ( L )+ N i ( S ) N i ( L ) = N i 1 ( L ) N i 1 ( S ) yieldsthecontinued-fraction approximationstothegoldenmean. OrderLengthSequence N ( L )+ N ( S ) N ( L ) 11 S 1 21 L 1 32 LS 2 43 LSL 1 : 5 55 LSLLS 1 : 667 68 LSLLSLSL 1 : 6 713 LSLLSLSLLSLLS 1 : 625 821 LSLLSLSLLSLLSLSLLSLSL 1 : 615 934 LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS 1 : 619 11 :::LSLLSLSLLS::: Table2.1.TherstninetermsoftheFibonaccisequence.The lengthsoftheterms correspondtoFibonaccinumbers,andtheratio N ( L )+ N ( S ) N ( L ) convergestothegolden mean .Thelimitingtermisanexampleofaquasiperiodicsequence .Eachpreceding termistheunitcellforaquasiperiodicapproximant. Imposingperiodicboundaryconditions, 5 onendsthateachnitetermdisplays translationalsymmetry. 6 Thelimitingterm,however,isquasiperiodic:onecantake 5 So LSL becomes :::LSLLSLLSL::: 6 Theseareexamplesofquasiperiodicapproximants. 9

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anynitesubstringofsize l ,andtranslateitadistance d ,ndinganidenticalsubstring. Ofcourse,inanynitetermthispropertycanalsobeobserve dif l issucientlysmall. Forinstance,takingthetermoflength13,wecanchoosethe rstoccurenceof LSL as oursubstring.Translatingtotherightbythreecharacters wendanidenticalsubstring. However,ifwechoose LSLLSL instead,wemusttranslatebyvecharacters. Physically,wecaninvestigategenericpropertiesofquasi periodicstructuresand modeltheeectsofphasonrips[31]byconstructingasystem withthesymmetryofa termfromtheFibonaccisequence.Examplesincludetwoearl y90'spublicationswhere calculationsoftheelectronicandvibrationaldensitieso fstateandheatcapacitieswere performedfortheFibonaccichainandcomparedtoitsperiod iccounterpart[4,47].More recently,Engeletal.performedMDsimulationsofaFibonac cichainusingadouble-well potentialtoallowforphasonrips,ndingarelationshipbe tweenanharmonicityand theopeningofbandgapsinthevibrationaldensityofstates 2.2Fourier-spacecrystallography Thoughwemayobservethesymmetriesofquasicrystalssimpl ybyobservingtheir diractionpatterns,wemayuseRokhsar,WrightandMermin' sFourier-spacecrystallography[3,9,50]toclassifythesesymmetrygroupswithma thematicalrigor.Usually, wedenethepointgroupofadirectlattice 7 tobethesetofallisometriesaboutaxed pointthatleavethedirectlatticeinvariant.Instead,wer eplacethecriterionofcompletecoincidencewiththenotionof indistinguishability .Thatis,insteadofleavingthe directlatticeinvariant,werequirethatall n -bodycorrelationfunctionsofthedensity 8 beleftinvariant. 1 V n Z ( r r 1 ) ( r r n )d r = 1 V n Z 0 ( r r 1 ) 0 ( r r n )d r ; (2.6) 7 Inthissection,werefertothereal-spacelatticeasthe\directlat tice,"andtheFourier-spacelattice assimply\thelattice." 8 Thisdensitymaybemass,nuclear,electronic,etc. 10

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where ( r )isthedensityand 0 ( r )isthatsamedensityundersometransformation.By takingtheFouriertransformofeachside,wearriveatthefo llowingrelation: n X i =0 k i =0 ) ( k 1 ) ( k n )= 0 ( k 1 ) 0 ( k n ) ; 8 n 2 Z ; (2.7) fromwhichitfollowsthattwodensitiesareindistinguisha bileifthereexistsagauge function ( k ),linearonthelattice L ,suchthat 0 ( k )= ( k ) e 2 i ( k ) : (2.8) Itfollowsimmediatelythatforallpoint-groupoperationso nthelattice, g 2 G ,wehave ( g k )= ( k ) e 2 i g ( k ) ; (2.9) wherewecall g ( k )thephasefunction,whichisdenedmodulounityandisrequ ired tobelinearonthelattice L 9 Ifthevalueofthephasefunctioniszeroforall k 2L thenweknowthattheoperationisinthereal-spacepointgrou p.Ifitisnon-zerofor some k thenitispossiblethattheoperationmustbecombinedwiths ometranslationto recoverindistinguishability. 10 Wecanshowbytheassociativityofthegroupelements that gh ( k ) g ( h k )+ h ( k ) ; (2.10) where isequalitymodulounity.Wecallthisthegroup-compatibili tycondition (GCC).Finally,wehaveanadditionalfreedomtochooseagau ge,astwophasefunctions relatedby g ( k ) 0g ( k ) ( g k k )(2.11) describethesamesymmetry.Givenaparticulardensityfunc tion,wecanusegaugeinvariantlinearcombinationsoftheform P i g i ( k i ),where g i k i k i =0,touniquely determinethespace-group[50].Thespacegroupis symmorphic ifandonlyifallgaugeinvariantsareequaltozero.Wenormallyspecifyaspace-gro upoperationby f R ( g ) ; t g g 9 Foraperiodiccrystal,thephasefunctionissimply g = t g k 2 ,where t g correspondstothe symmetryoperation f g; t g g 10 Onecanshowthatatranslationinreal-spacechanges ( k )onlybyaphase. 11

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Figure2.2.Anexampleofanon-symmorphicsymmetry.Thereis amirrorsymmetry alongthedottedline,butthisrerectionalsorequirestran slationbyahalflatticevector upwardsordownwards.(Weassumethepatternextendstoinn ity.) where R ( g )isarotation,rerectionorinversion,and t g isacorrespondingtranslation.A symmorphicspacegroupisonewhere,withthecorrectchoice oforigin,allspace-group operationsareoftheform f R ( g ) ; 0 g .Ifaspacegroupisnotsymmorphic,wesaythat itisnonsymmorphic.Foranexampleofanoperationthatwoul dmakeaspacegroup nonsymmorphic,seeg2.2.2.3Gauge-invariantquantities Thesimplesttypeofgauge-invarianttakestheform g ( k ), g k = k g k = k implies that ( g k )= ( k ),buttherearecaseswhere g ( k ) 6 0.Insuchacase,symmetry requiresthat ( k )=0tosatisfyequation(2.9).Thisgauge-invariantofther stkindis calledasystematicextinctionandresultsinmissingBragg peaksindiractionpatterns. Ofthe230crystallographicspacegroups,onlytwo, I 2 1 2 1 2 1 and I 2 1 3,possessgaugeinvariantsofthesecondkind[50]. 11 HereweconsideraFourier-spacevector q ,which mayormaynotbeanelementof L ,andthegroupofsymmetryoptions g 2 G q G suchthat q g q = k g 2L .Ifwetaketwoelements g;h inthe littlegroup G q wend thatthequantity g ( k h ) h ( k g )isgauge-invariant.Itcanbeshownthatanon-zero 11 Althoughthereare157non-symmorphicspacegroups,onlythesetwocontain invariantsthatcannot bereducedtorst-order. 12

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valueforthisgauge-invariantrequiresallelectronicener gylevelstocrosstheBloch wavevector q inpairs[34,35]. Thethirdknowngauge-invariantquantityfallsoutofgroupc ohomology[16,52] andonlyappearsinquasicrystallographicspacegroups.In anunpublishedpaperby FisherandRabson[13],suchaspacegroupisconstructedfro mthepointgroup D 4 withthelatticegeneratedby b 1 = e i ;b 2 = e i and b 3 =^ z + 1+ i 2 ( b 1 b 2 ).Although therehasbeensomeheadwayinunderstandingthephysicalim plicationsofthethird invariant[26],thistopichasnotbeenexploredingreatdet ail. 2.4Calculationofspacegroups d-AlCoNiisanexampleofanaxialquasicrystal[24,40]consi stingofve-foldplanes suchthateachplaneisrotated36 withrespecttoitsneighbor[6].Clearlyinan analogouscrystallographicsystemthiswouldgiverisetoa screw-axissymmetry, 12 and wewillshowthatinthequasicrystallographiccasethisisn odierent. Westartwithasimplemodelforeachofthethreeve-foldplan egroups: p 5, p 5 m 1, p 51 m .Wewillchooseasmallnumberofwave-vectorsinFourier-spac eandassign ( k )onthesepoints 13 suchthatweachievethedesiredsymmetryandthereal-space densityisreal. 14 Thisisknownasadensity-wavepattern.Unlikeinatilingmod el,we truncatethenumberofwave-vectorswithnon-zeroweightandt hereforeloseatomicity inreal-space. Tostart,wetakethefollowingtwo-dimensionaldensities: 15 p 5 ( k s )= c ( k s )+ c 9 X l =0 i ( 1) l ( k s k l )+ ( k s k l k l +1 ) ;(2.12) p 5 m 1 ( k s )= c ( k s )+ c 9 X l =0 i ( 1) l ( k s k l )+ ( k s k l k l +1 ) ;(2.13) p 51 m ( k s )= c ( k s )+ c 9 X l =0 ( k s k l )+ i ( 1) l ( k s k l k l +1 ) ;(2.14) 12 Forinstance,acrystalwithhexagonalplaneswith30 rotations 13 ( k )=0elsewhere. 14 Asopposedtocomplex.Thisrequires ( k )= ( k ). 15 Bydeningourdensitiesinsuchawayweareimplicitlychoosing agauge,For p 5 m 1thischoiceis ( k )= r ( 1+ k 1 1 k 1 k ) 13

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Figure2.3.Wave-vectorsincludedin p 5 m 1 ( k ), p 51 m ( k )and p 5 ( k )densitywavemodels (lefttoright).Squaresrepresentwave-vectorswithrealw eights ( k )=Re( ( k )),while circlesandcrossesindicateaphaseof e i and e i ,respectively.Themirrorsymmetry in p 5 m 1isalongalinedrawnthroughtheoriginandconnectingtwoi nner-starpoints. In p 51 m weinsteadconnectalinethroughouter-starpoints.Imagesg eneratedusing Maple.where k l isaten-foldstarontheunitcircle,suchthat k 2 l and k 2 l +1 aredisjointve-fold starsand k l = k l +5 l 2 Z 10 ,and c isaconstantwithunitsofinversearea; 16 seegure 2.3. Aten-foldrotationonthesedensitieswouldrequire 10 ( k l ) 1 2 tosatisfyequation(2.9),butsuchaphasewouldviolatelinearityas0 10 (0) 10 ( P l k l ) 6 P l 10 ( k l ) 1 2 .Ontheotherhand, 5 ( k s ) 0worksneforve-foldrotations. TaketheFouriertransform ( s )= R ( k s ) e i s k s d 2 s foreachof(2.12)-(2.14)toobtain thefollowing2Dreal-spacedensities: p 5 ( s )= c 1+ 9 X l =0 i ( 1) l ( e i s k l + e i s ( k l + k l +1 ) ) ;(2.15) p 5 m 1 ( s )= c 1+ 9 X l =0 i ( 1) l e i s k l + e i s ( k l + k l +1 ) ;(2.16) p 51 m ( s )= c 1+ 9 X l =0 e i s k l + i ( 1) l e i s ( k l + k l +1 ) ;(2.17) seegure2.4. Next,weobtainthedesiredaxialquasicrystalbychoosingo neofthereal-space densitiesinequations(2.15){(2.17)andforminganABABst ackingwithinterlayer 16 Thismakes ( k s )unitlessandgives ( s )unitsofinversearea. 14

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Figure2.4.Densitywavesinrealspacefor p 5 m 1 ( s ), p 51 m ( s )and p 5 ( s )densitywave models(lefttoright).Mirrorsymmetrycanbefoundin p 5 m 1and p 51 m spacing a witheachadjacentlayerrotatedby180 : 17 Bythismethodwend ( r )= ( s ;z )= X m 2 Z ( s ) ( z 2 ma )+ ( s ) ( z (2 m 1) a ) ; (2.18) whereweuse s asavectorintheplanewith z alongtheperpendicularaxis.Now,to determinethespacegroupoftheseaxialstackings,wemusto btainthe3DFourier-space densitybymeansofFouriertransform: ( k )= ( k s ;k z )= 1 8 3 Z ( k ) e i k r d 3 r = 1 8 3 Z ( k ) e i ( k s s + k z z ) d 2 s d z: (2.19) Pluggingin(2.18)alongwithsomesimplication,wend ( k )= 1 8 3 Z ( ( s )+ e ik z z ( s )) e i k s s d 2 s X m 2 Z e 2 ik z ma : (2.20) Invokingtheidentity P m e imx = P n ( x 2 n )leadsto ( k )= 1 8 3 X n 2 Z ( k z n a ) Z ( ( s )+( 1) n ( s )) e i k s s d 2 s : (2.21) Wecanapplythisresulttoequation(2.15),whileusing k l = k l +5 .Uponrearrangementofthesumandsimplicationwend p 5 m 1 ( k )= 17 Thisissymmetryequivalenttoa36 rotation. 15

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c X n 2 Z n; even ( k s )+ 9 X l =0 h i ( 1) l n; odd ( k s k l )+ n; even ( k s k l k l +1 ) i : (2.22) Thus,wendthe p 5 m 1stackinghasinner-star( k s = k l )extinctionsonevenlayers, while k s =0andtheouter-star( k s = k l + k l +1 )arebothextinctontheodds.Similarly, wend p 51 m ( k )= c X n 2 Z n; even ( k s )+ 9 X l =0 h n; even ( k s k l )+ i ( 1) l n; odd ( k s k l k l +1 ) i ; (2.23) p 5 ( k )= c X n 2 Z n; even ( k s )+ i n; odd 9 X l =0 ( 1) l h ( k s k l )+ ( k s k l k l +1 ) i : (2.24) Wecanlookagaintogure2.3,butthistime,foreachsymmetr y,thesquares representextinctionsonoddlayers,whilethecrossesandc irclesareextinctoneven layers.Addtionally,gure2.5depictsafewlayersof p 51 m ( k ).Wecanuseequations (2.22){(2.24)todeterminethephasefunctionsandthevalu esofthegaugeinvariants forourmodelquasicrystals. Let h denoteamirrorinthexy-plane.Itiseasytoconvinceoneself that h 2 G ourthreedensities,butformally,wemustsolveforthephas efunction h ( k ).Wetake ( h k )= ( h ( k s + k z )).Since k s liesintheplaneofthemirror, h k s = k s ,while hk z = k z ,andwehave ( h ( k s + k z ))= ( k s k z )= ( k s + k z ) e 2 i h ( k s + k z ) .Substituting anyofequations(2.22){(2.24)givesthefollowingresult h ( k ) 0 8 k 2L : (2.25) Byconrming h 2 G weruleoutanyspace-groupsthatdonotcontain h .Still,wemust considerotherpossiblespace-groupoperationssuchasthos egeneratedby r and m where r k l 7! k l +1 ,correspondingtoa36 rotation,and m isaverticalmirrorleaving 16

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Figure2.5.AfewlayersoftheFourier-spacedensity p 51 m ( k ).Thesolidlayersareeven, checkeredareodd.Wave-vectorswithnon-zeroweightaremark edasbluespheres.Red starsindicateextinctwave-vectors.k l and k l +5 invariantforsome l 2 Z 10 .Startingwith p 5 m 1,wewillneedtodetermine thevaluesofthegauge-invariantquantity g ( a ^ z ). 18 Withthisinformation,wecan uniquelydetermineeachspacegroupusingtablesfromrefer ence[53]. Consider p 5 m 1 ( r k ). ( r ( k l + k l +1 ))= ( k l +1 + k l +2 )= ( k l + k l +1 ) e 0 ; (2.26) ) r ( k l + k l +1 ) 0;(2.27) ( r ( k l + ^ z=a ))= ( k l +1 + ^ z=a )= ( k l + ^ z=a ) e i ; (2.28) ) r ( k l + ^ z=a ) 1 = 2 : (2.29) Nowweapplylinearityandthefactthat l isarbitrary, 1 = 2 5 = 2 X l 2 Z odd10 r ( k l + ^ z=a ) 5 r ( ^ z=a ) ; (2.30) 18 Wechoosethisquantitybecauseitisinvariantforboth g = r and g = m 17

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where Z odd10 = Z 10 \f Odds g .Thuswend r ( a ^ z ) 1 2 19 Next,weconsider p 5 m 1 ( m k ). ( m ( k l + k l +1 ))= ( k l 0 + k l 0 1 )= ( k l + k l +1 ) e 0 ; (2.31) ) m ( k l + k l +1 ) 0;(2.32) ( m ( k l + ^ z=a ))= ( k l 0 + ^ z=a )= ( k l + ^ z=a )) e 0 ; (2.33) ) m ( k l + ^ z=a )) 0(2.34) where l and l 0 havethesameparity.Againemployinglinearity, 0 X l 2 Z odd10 r ( k l + ^ z=a ) 5( ^ z=a ) ; (2.35) andwehave m ( a ^ z ) 0,andnallyndthatthespacegroupfor p 5 m 1 ( k )is V ( r 1 = 2 ;h;m ) 20 orusingInternationalnotation P 10 5 m 2 m 2 c 21 Next,wetacklethep51m.Startingfrom p 51 m ( r k ), ( r ( k l + k l +1 + ^ z=a ))= ( k l +1 + k l +2 + ^ z=a )= ( k l + k l +1 + ^ z=a ) e i ; (2.36) ) r ( k l + k l +1 + ^ z=a ) 1 = 2;(2.37) ( r ( k l ))= ( k l +1 )= ( k l ) e 0 ; (2.38) ) r ( k l ) 0 : (2.39) Linearitygives r ( k l + k l +1 + ^ z=a ) r ( k l )+ r ( k l +1 )+ r ( ^ z=a ) (2.40) 0+0+ r ( ^ z=a ) r ( ^ z=a ) 1 = 2 : (2.41) 19 Since ^ z=a isinvariantunder r ,thisgauge-invariantexplainsthesystematicextinctionatwavevectorsoftheform k =(2 n 1) ^ z=a 20 Inthelanguageofreference[53]. 21 Itissimpletocheckthatthesephasefunctionssatisfythegroup-c ompatibilitycondition(2.10). 18

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Forwe p 51 m ( m k )wehave ( m ( k l + k l +1 + ^ z=a ))= ( k l 0 + k l 0 1 + ^ z=a )= ( k l + k l +1 + ^ z=a ) e i ; (2.42) ) m ( k l + k l +1 + ^ z=a ) 1 = 2;(2.43) ( m k l )= ( k l +1 )= ( k l ) e 0 ; (2.44) ) m ( k l ) 0;(2.45) m ( k l + k l +1 + ^ z=a ) 0+0+ m ( ^ z=a ) : (2.46) Weobtain r ( a ^ z ) 1 = 2and m ( a ^ z ) 1 = 2andthusthespacegroup V ( r 1 = 2 ;h;m 1 = 2 ) or P 10 5 m 2 c 2 m Finally,considerthep5-generateddensity, p 5 ( k ), ( r ( k l + ^ z=a ))= ( k l +1 + ^ z=a )= ( k l + ^ z=a ) e i ; (2.47) ) r ( k l + ^ z=a ) 1 = 2;(2.48) ( r ( k l + k l +1 + ^ z=a ))= ( k l +1 + k + l +2+ ^ z=a )= ( k l + k l +1 + ^ z=a ) e i ; (2.49) ) r ( k l + k l +1 + ^ z=a ) 1 = 2;(2.50) 0 r ( k l + k l +1 + ^ z=a ) r ( k l + ^ z=a ) r ( k l +1 ) : (2.51) Thus, r ( a ^ z ) 1 = 2.Meanwhile,weassumethat m isavalidgroupoperationandnd ( m ( k l + ^ z=a ))= ( k l 0 + ^ z=a )= ( k l + ^ z=a ) e 0 ; (2.52) ) m ( k l + ^ z=a ) 0;(2.53) ( m ( k l + k l +1 + ^ z=a ))= ( k l 0 + k + l 0 1+ ^ z=a )= ( k l + k l +1 + ^ z=a ) e i ; (2.54) ) m ( k l + k l +1 + ^ z=a ) 1 = 2 : (2.55) 19

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Combiningequations(2.52)and(2.54),wend m ( k l ) 1 = 2,butthisleadstoa contradictionas 0 m (0) m ( X l 2 Z odd10 k l ) 6 X l 2 Z odd10 m ( k l ) 1 = 2 : (2.56) Wehavededucedthat m ,infact,doesnotbelongtothepointgroupforourstacked p5model.Thespacegroupmustthenbe V ( r 1 = 2 ;h )or P 10 5 m Becauseeachstackingofve-foldplanesresultsinaten-fold screwaxis,weconclude thatthesysteminquestion,d-AlCoNi,shouldalsoexhibitth isten-foldsymmetry. 20

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CHAPTER3 STOCHASTIC-ODEMETHOD Molecular-dynamicssimulationsarecomputationallyexpen siveandhavemanylimitingfactors.Althoughsimulationscanbeperformedwiths everalmillionsofatoms, thismayonlycorrespondtomicrometerlengthscalesandpic osecondtimescales.AlthoughthisdoesnotmakeMDsimulationoffrictionimpossib le,manyshortcutsmust betaken.Forinstance,intheAlCoNisimulationswewilldis cussinthefollowingchapter,aslidingvelocityof5m/sisused.Thisisroughlysixor dersofmagnitudelarger thanthosevelocitiesusedinexperiment[44].Inaddition, largeapproximantunitcells mustbeavoided,elseweincurevenmorecomputationalcost. Forthisreason,appealingtoasimplermodelcanbeextremel ybenecial.One modelinparticularwasdevelopedbyPrandtlandTomlinson[ 2,17,38,66]inthe1920's andwiththeuseofmoderncomputationalpowerhasresultedi nseveralpublications [11,19,23,25,55,57,68].Themodelisdescribedbyanordin ary-dierentialequation 1 involvingaclassicalparticlewithmass m coupledtoasecondbodymovingwithvelocity v slide andsubjecttoacontactforceduetosurfacecorrugation, 2 m x + d d x U 0 cos( 2 x a )+ k ( x v slide t )=0 ; (3.1) where U 0 istheamplitudeofthesurfacecorrugation, a istheunitcellsize,and k isa springconstantforthecouplingbetweentheparticleandth eslidingbody;seegure 3.1. 3 Morerecently,studieshaveintroducedtwoadditionalterm sinordertoinclude theeectsofthermalructuations[11,57,68].AMarkovian-n oiseterm R ( t )anda dissipativeviscosity-liketerm mr x areincludedinthemodel,andourODEbecomesa 1 Partial-dierentialequationinthe2-Dcase. 2 PerhapsthetipandbaseofanAFMcantilever,respectively. 3 Wewilllaterreplacetheperiodicpotentialwithonethatisquasipe riodic. 21

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Figure3.1.Tomlinson'smodeldescribingatomic-scalefric tion.Theclassicalparticleis subjecttoacouplingforceandsurfacecorrugation.Figure obtainedfromreference[23]. Stochastic-ODEoftheform 4 m x + @ @x U 0 cos( 2 x a )+ k ( x v slide t )+ mr x + R ( t )=0 ; (3.2) where R ( t )obeystheructuation-dissipationrelation, 5 h R ( t ) R ( t 0 ) i =2 mrk B T ( t t 0 ) : (3.3) ThisiscalledthegeneralizedPrandtl-Tomlinsonmodelandh asbeenusedtomodelthe cantileverandtipusedinFFMmeasurements[11,55,57,68]. Ina2009PhysicalReviewLetter[68],Tshiprutetal.implem entedthisS-ODE approach,ndinganintriguingtemperaturedependenceofk ineticfriction,including non-monotonicityforcertainchoicesofparameters.Asecon dquantity,sliplength h L i |theaveragedisplacementduringasinglestick-slipevent| provedworthyofnotice. Foraparticularchoiceofparameters, 6 h L i isfoundtobelonginoneofseveralregimes. Forthelowesttemperatures,thedynamicswerecharacteriz edbytriple-slips, h L i 3 a Increasingto15 K ,amixtureoftriple-anddouble-slipswasfound.Asthetempe rature 4 S-ODE'softhisformarealsoknownasLangevinequations. 5 Thenotation h ::: i alwaysdenotesatimeaverageunlessotherwisenoted. 6 U 0 =0 : 26eV, a =0 : 3nm, k =1 : 5N/m, m =5 10 11 kg, r =1 10 5 s 1 ,and v slide =10nm/s. 22

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Figure3.2.Thetemperaturedependenceofnormalizedavera ge-sliplengthandfriction. Frictiontendstodecreaseastemperatureisincreased,but changesinslipregimemay leadtolocalmaximaandplateaus.Figureobtainedfromrefe rence[68]. continuedtobeincreased,transitionstodouble-slips,ami xofdouble-andsingle-slips, andnally,with T 300K,single-slipswereobserved, h L i a Theimportanceofthisstick-slipcharacterizingquantityi srevealedingure3.2. Witheachtransitiontodecreasingslip-lengths,onemaynd eitheranincreaseor plateauinthetemperaturedependenceinparticular,thepe akinthetemperaturedependenceofthefrictioncoincideswellwiththetransition fromthetriple-double-slipto thedouble-slipregime. Tshiprutetal.concludethattwocompetingeectscharacte rizethetemperature dependenceatconstantslidingvelocity.First,slip-lengt haside,thefrictiontendsto decreaseasthetemperatureincreasesasdescribedbySange tal.[57], F / const T 2 = 3 j ln v T j 2 = 3 .However,withrisingtemperature,slip-lengthalsodecrea ses,whichhas thereverseeectonfriction. InvestigationsofthePrandtl-Tomlinsonmodelandsimilarm odelstypicallyinclude aperiodicsurface-corrugationpotentialterm.Ourgoalint hischapteristoinvestigate theeectofreplacingthisperiodicpotentialwiththatofa quasiperiodicapproximant. 23

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Beforeprocedinginthisfashion,weshallrstdiscusstheS -ODEsolvingalgorithm used,andwe'llreviewsometestsweperformedonsimplermod els|Brownianmotion andlineartransport.WewillthenreproducesomeofTshipru t'sresults.Finally,we willperformcalculationsusingourquasiperiodicapproxi mantpotentialsanddiscuss ourresults.3.1VanGunsterenandBerendsen'sAlgorithm Tondasolutiontothedierentialequationinquestion,we mustperformintegrationofarandomvariable.Wemayusetraditionalmethods 7 onlyinthelimit t r 1 ; otherwiseweviolateructuation-dissipation,equation(3. 3)[61,70].InTshiprut'sproblem,theslidingvelocityforthecantilever v slide =10nm/s.Becausetheperiodofour surfaceis0 : 3nm,anditwouldn'tbeunreasonabletohaveresolutionover 1000points perperiod,weneed t 3 10 5 s r 1 .However,inthisproblem r 1 =10 5 s. Moreover,theslipeventswhichdominatethebehaviorofthe tipalsooccuronthe microsecondscale,reinforcingtheneedtomovetosmallert imesteps.Todoso,we mustemployanalternativetechnique. FollowingthestepsofvanGunsterenandBerendsenintheir1 982publication[70], webeginwiththeequation, m x = mr x + F ( x )+ R ( t ) ; (3.4) withmass m ,viscositycoecient r ,Markoviannoise R ( t ),andanexternalforcewith noexplicittimedependence F ( x ).Inaddition,weimposethefollowingconstraintson R ( t ): h R ( t ) R ( t 0 ) i =2 mrk B T ( t t 0 ) ; (3.5) W [ R ( t )]=[2 h R 2 i ] 1 = 2 exp f R 2 2 h R 2 i g ; (3.6) h x (0) R ( t ) i =0 ;t 0 ; (3.7) h F (0) R ( t ) i =0 ;t 0 ; (3.8) 7 SuchastheEulerorVerletalgorithms. 24

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whereangledbracketsdenoteequilibriumensembleaverage sand W [ R ( t )]istheGaussianprobabilitydistributionofthestochasticterm R ( t ). Bywriting v =_ x inequation(3.4),weobtain v ( t )= e r ( t t n ) v ( t n )+ 1 m Z t t n e r ( t 0 t n ) F ( t 0 )+ R ( t 0 ) d t 0 : (3.9) Atthispoint,manyalgorithmsrestrictthetimestep t tosatisfyructuation-dissipation, whichallowsR(t)tobetreatedasaconstantforeachindivid ualtimestep[61,65].Instead,wewilldirectlyintegratethestochasticforcewith respectto t .Weexpand F ( t ) intoitsTaylorseries F ( t )= F ( t n )+ F ( t t n )+ O [( t t n ) 2 ]thenintegratetond x ( t n + t )= x ( t n )+ Z t n + t t n v ( t )d t (3.10) = x ( t n )+ v ( t n ) r (1 e r t )+ F ( t n ) mr 2 r t 1+ e r t + F ( t n ) mr 3 1 2 ( r t ) 2 r t +1 e r t + 1 mr Z t n + t t n 1 e r ( t n t t ) R ( t )d t + O [( t ) 4 ] : (3.11) Usingthedenition, X n ( t ) ( mr ) 1 Z t n + t t n e t n t r t R ( t )d t; (3.12) andadding(3.11)toitselfbutwiththereplacement t t ,wend, x ( t n + t )= x ( t n )(1+ e r t ) x ( t n t ) e r t + F ( t n ) t mr (1 e r t ) + F ( t n ) t mr 2 r t 2 (1+ e r t ) (1 e r t )+ X n ( t )+ e r t X n ( t ) + O [( t ) 4 ] ; (3.13) 25

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whichreducestotheVerletalgorithminthesmall r limit.Bycalculatingthevalues h X 2 ( t ) i h X 2 ( t ) i and h X ( t ) X ( t ) i ,wendabivariatedistribution, 8 W X n 1 ; X n = 4 h X 2 n 1 i 2 h X 2 n i 2 (1 r 2 ) 1 = 2 exp h X 2 n i 2 X 2 n 1 2 h X 2 n 1 ih X 2 n i rX n +1 X n + h X 2 n 1 i 2 X 2 n 2 h X 2 n 1 i 2 h X 2 n i 2 (1 r 2 ) ; (3.14) where X ( t )= X ( t ). Nowthetaskissimple.Atsometimestep t n +1 ,weassume x ( t n ), x ( t n 1 ), X n 1 ( t ) and F ( t n 1 )areknown. F ( t n )canbeevaluatedfromthepotentialand F ( t n )canbe evaluedthroughdierencing.Next,wesample X n ( t )fromthebivariatedistribution andcalculate X n ( t )fromagaussiandistribution.Equation(3.13)thensupplie s x ( t n +1 ).Combining(3.11)withitself,inamannersimilartotheVe rletalgorithm, yieldsanexpressionforvelocity. Thecodewe'veimplementedcanbefoundinappendixB. 3.2Testingthealgorithm Aneccessarystepwhenimplementinganyalgorithmistosubj ectittovarioustests. Theobviousreasonforthisistoworkoutanybugsinthecodeo rimplementationand checkformistakesinthederivations.Moreover,wemustbes urethatthecodestill worksforverysmall r t ,andndwhenthecodebreaksdownforlarge r t .Inour case,wefoundthecodefailedforverysmall r t duetonumericalunderrow.Wewere requiredtoexpandseveraloperationsintotheirpowerseri es. 9 Wedon'twishtoboreouraudiencewithdetailsofeachminort est.Rather,wewill presentdetailsofthethreeprimarybenchmarks:agreement withknownresultsfrom Brownianmotion,lineartransportandTshiprutetal.'smod el. InBrownianmotion,weareinterestedinthediusionofapar ticleinasolvent.The particleisrepeatedlykickedbysolventparticlesinrando mdirections,whileundergoing 8 Thedistributionisbivariatebecause X ( t )and X ( t )areintegralsof R ( t )overthesametime intervals. 9 Theexpansionsusedcanbefoundintheappendixofreference[70]. 26

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anaresistiveforceduetoviscosity.In1D,wehavethedier entialequation, m x + mr x R ( t )=0 : (3.15) Multiplyingbothsidesby x andusingtheidentity d d t ( x x ) x 2 + x x leadsto m d d t ( x x )= m x 2 mrx x + xR ( t ) : (3.16) Now,takingtheexpectationvaluesofeachsideandinvoking equipartition, m d d t h x x i = k B T mr h x x i : (3.17) Substituting u = h x x i k B T mr wehave, u = ru; (3.18) whosesolutionis u = k B T mr e rt .Thisyields h x x i = k B T mr (1 e rt ).Substituting d d t h x 2 i =2 h xx i andintegratinggives h x ( t ) 2 i = 2 k B T mr ( t + 1 r e rt ) ; (3.19) whichisthedesiredresult. 10 Using r =1 10 7 ps 1 T =300Kand5 10 11 kg,weran1000trialswithour code,andfound h x 2 i ingoodagreementwiththeory,seegure3.3. Next,weaddedaconstantforcetoourdierentialequationt oobservelineartransport.StartingwiththeDE, m x + mr x F R ( t )=0 ; (3.20) wetaketheexpectationvalueofeachside,andset x =0,tondtheterminalvelocity, h v ( t ) i = F mr (3.21) 10 Thisleadstothefamilar h x 2 i =2 Dt inthelong-timelimit. 27

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Figure3.3.1000trialsofourS-ODEcodefoundthemean-square ddisplacementfora Brownianparticle(dotted)ingoodagreementwiththeory(s olid). Plottingtheaverageof1000trialswith F =0 : 001ev/ A,wealsondgoodagreement withthistheoreticaltrajectory;seegure3.4. Finally,wewishedtoreproducetheresultspresentedbyTsh iprutetal.Wereplaced theconstantpotentialwithonethatwasperiodic,andintro ducedaninteractionwith asecondparticle.Wecalltheoriginalparticlethe\tip"an dthesecondparticlethe \bob."Thebobhasthesameinitialpositionasthetip,butho ldsaconstantvelocity v slide = v bob .Thebobandtipinteractviaaspringwithforceconstant k .Theequation ofmotionforthebobis m x bob = k ( x tip v bob t )+ F app ; (3.22) where F app = F friction f istheforceappliedtokeep_ x bob = v bob constant.Thetip's motionisgovernedby m x tip = mrx tip @ @x tip U 0 cos 2 x tip a k ( x tip v bob t )+ R ( t ) ; (3.23) allvariableshavingtheirusualmeanings. 28

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Figure3.4.Comparisonof1000trialsofourS-ODEcode(dotte d)versusthetheoretical curve h x i = F mr t (solid)forBrownianmotionwithaconstantexternalforce. Sinceweareinterestedintheenergylosttofriction 11 wewishtocalculatethe spatial-averagedquantity h f i x bob .Usingthechainrule, h f i x bob = 1 L Z L 0 f ( x tip ;t )d x bob (3.24) = 1 L Z T f 0 f ( x tip ;t ) @x bob @t d t (3.25) = 1 L v bob Z T f 0 f ( x tip ;t )d t; (3.26) = 1 T f Z T f 0 f ( x tip ;t )d t = h f i t h f i ; (3.27) where T f isthetotaltime,and L = v bob T f Now,usingthefactthatthelefthandsideofequation(3.22) isidenticallyzero, h f i = k h x tip v bob t i ; = k ( h x tip i v bob T f = 2) : (3.28) 11 Energylossimplieswork. U loss = R f ( x )d x =( x f x 0 ) h f i x 29

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Figure3.5.Usingourcode,wereproducedresultsingoodagr eementwithTshiprutet al.Thepeakintheaveragefrictionforce(dashed)coincide swithasuddendecreasein sliplength(solid)at T 40K.Astraightforwardalgorithmforcalculatingslipleng th isgiveninappendixB.Thuswendthatforthismodel,wehaveaverysimpleandelega ntwayofcalculating theaveragefrictionalforce|simplybycalculating x ( t ). ApplingtheDE,equation(3.23),toourcodeandcalculating h f i asgivenbyequation (3.28),wefoundresultsinagreementwithTshiprutetal.:c omparegures3.5and3.2. 3.3QuasiperiodicS-ODE BecausethegeneralizedPrandtl-Tomlinsonmodelwaseecti veatpredictingsome interestingfrictionphenomena,webelievedthatitshould alsobeeectiveatmodeling frictioninthequasiperiodiccase.Thoughclearlythemode llackssomephysics|e.g., phonons|thereductionincomputationalcostrelativetoth ree-dimensionalmolecular dynamicsisenormous.Moreover,regardlessoftheoutcome, theresultsofthesesimulationsmayhelpdeterminewhethercertainpropertiesarer esponsibleforloweringthe friction,whilerulingoutothers. 30

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Figure3.6.Twoexamplesofthepotentialsusedinthesecalc ulations.For b=b 0 =3 = 2 (dotted)wehaveaperiodicunitcellof0 : 9nm,repeatingthreetimeswithinthebounds ofthegraph,while b=b 0 =144 = 89(solid)boastsacellofsize4 : 32nm. Togenerateourapproximantquasiperiodicsurfaceswerepl aceourperiodicpotential U = U 0 2 x tip a with U = U 1 cos 2 x tip b + U 2 cos 2 x tip b 0 : (3.29) Bysetting b=b 0 equaltothequotientoftwoconsecutiveFibonaccinumbersF ib n +1 = Fib n wendaseriesofquasiperiodicpotentialapproximants;se egure3.6.Ofcourse,inthe limit n !1 wehave b=b 0 = ,thegoldenmean,whichyieldsaperfectlyquasiperiodic potential. Wewillusetwodierentmethodstoinvestigatetheroleofqu asiperiodicityindeterminingfrictionforthismodel.First,wecanset U 1 = U 0 andrunsimulations overvariousvaluesof b=b 0 ,whilesearchingforanytrendsasourpotentialtendsto 31

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Figure3.7.Temperaturedependenceforselectquasiperiod icapproximantswithinin thegeneralizedPrandtl-Tomlinsonmodel.Weobserveveryli ttledierencebetweenall approximantsofthird-order( b=b 0 =5 = 3|notshown)andhigher. quasiperiodicity.Second,wecanlet b=b 0 = andvary U 1 and U 2 ,therebyintroducingquasiperiodicityperturbatively.Inallcaseswewi llimposeconstantpower P =lim x !1 1 x R x x U ( x 0 ) 2 d x 0 SettingourparameterstobethesameasthoseusedbyTshipru tetal.,with b = a = 0 : 3nmand P =0 : 676eV 2 ,weallow b=b 0 totakevalues1 = 1 ; 2 = 1 ; 3 = 2 ; 5 = 3 ; 8 = 5 ;:::; 144 = 89. Foreachchoiceofthisratio,weperformedcalculationsfor temperatures T 2f 0 ; 395 g K; seegure3.7.Thoughwendthatthereissomedeviationbetw eenthelowestorder approximantandtheothers,thisbehaviorsettlesandconve rgesby b=b 0 =3 = 2for hightemperatures,andforalltemperaturesby b=b 0 =8 = 5.Thetracesfor b=b 0 =8 = 5 andallhigher-orderapproximantsarevirtuallyindistingu ishable.Forexampleposition time-series,seegure3.8. Usingoursecondapproach,weseemuchmoreinterestingbeha vior.Here,weimpose contantpowerinourpotentialwhilevarying U 1 and U 2 with b=b 0 = ;seegure3.9. Wendthatforanunmixedhigh-frequencypotential, U 1 =0eV, U 2 =0 : 26andxed 32

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Figure3.8.Positiontime-seriesdatafor b=b 0 =3 = 2(left)and b=b 0 =8 = 5(right),at 395K.Bothcasesclearlydemonstratestick-slipbehavior,w iththeformerhavingfewer slipsaswellaslargeraveragemagnitude.T ,thereisanabsoluteminimuminaveragefrictionalforce.S trangely,wealsonda localminimumatsometemperature-dependentmixtureofthep otentials.At0K,the minimumoccurswhen U 2 =0 : 11eV,butthis\optimal"valuefor U 2 increasesasthe temperatureincreases,reaching U 2 =0 : 21eVfor T =200K,beforebeingcompletely smearedoutat300K. Ascompellingasthisresultmayappear,wendthatthiseec tisnotonlyduplicated,butalsomagnied,whenwechoosecommensuratepoten tials, b=b 0 =3 = 2;see gure3.10.Althoughitisdiculttodeduceintheincommens uratecase,herethepositiontime-seriesclearlyindicatedaphasetransitionoccur ingatthelocalminimum.At 0Kinthetime-seriesfor U 2 0 : 04eV,wendthetypicalstairpattern,corresponding to7 : 5 Aslips.However,lowering U 2 justbelowthiscriticalvalue,wendthateach slip-eventsplitsintoa3 : 5anda4 : 0 Aslip;seegure3.11.Theconsequenceofthis doublingofslipeventsisahugeincreaseinfriction.Offur therinterestisthatweseeno similarphasetransitionas U 2 becomeslarge(as U 1 becomessmall).Althoughweare cautiouslyexicitedaboutthiseect,itdoesnothingtohel pusunderstandlow-friction surfacesinquasicrystals. Perhapsanothercomparisonworthmakingisthatatlowtempe rature,wendthat thefrictionmeasuredwithcommensuratepotentialsisgrea terthanwhat'smeasuredin theincommensuratecase,exceptnearthephaseboundary.Un fortunately,thiseect becomescompletelywashedoutat T =100Kandabove. 33

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Figure3.9.Frictionmeasuredwhilevaryingthecoecients U 1 and U 2 oftwoincommensuratepotentials.Thelocalminimumforsmall(butnon-z ero) U 2 shiftstolarger U 2 asthetemperatureincreases,beforenallybecomingwashe doutcompletely. Despiteanystrongindicationthatquasiperiodicityleads tolowerfrictionwithinthis model,itisimportanttonotethatthisisnotapurelynegati veresult.Aswementioned previouslyinthischapter,thisS-ODEmodelleavesalotofph ysicsoutoftheequation. We'veyettoconsidereitherelectronicorphononiceects. Itisn'tparticularlyclearhow toimplementsuchphysicsinamodelwhichlacksatomicity.I nthefollowingchapter, wewilltrytomakeupforthesedecienciesbyimplementinga nMDmodel. 34

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Figure3.10.Frictionmeasuredwhilevaryingthecoecient s U 1 and U 2 oftwocommensuratepotentials.Thelocalminimumforsmall,non-zero U 2 ismuchmoreexaggerated thanintheincommensuratecase,andmucheasiertoundersta ndintermsoftheposition time-series. Figure3.11.Thepositiontime-seriesforcommensuratepote ntials b=b 0 =3 = 2,for U 2 =0 : 039(solid)and0 : 042(dotted)eV.Wendthatallvaluesof U 2 0 : 039look qualitativelylikethesolidtrace,whilethose U 2 0 : 042arequalitativelysimilartothe dottedtrace. 35

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CHAPTER4 MDSIMULATIONSOFD-ALCONIAPPROXIMANTS Movingtowardsamoreseriousmodeloffrictiononquasiperi odicsurfaces,wesought toqualitativelyreproducethefrictionalanisotropythro ughmolecular-dynamics(MD) simulations.Unfortunately,MDsimulationsarenotperfec treplicasofthephysical world.We'vealreadymentionedthatperformingsuchsimula tionssetsveryrestrictive limitsonthesizeofoursamplesandslidingvelocities.The formerfurtherrestricts ustorelativelysmallunit-cellapproximants.MDsimulatio nsalsocompletelyignore electronicdegreesoffreedom,whichmayplayanimportantr oleinthisphenonenon[64]. Moreover,thedynamicsproducedinMDsimulationsareonlya spreciseasthepotentials used.WhilewearegratefultoMarekMihalkovi^cforprovidi ngpairpotentials[41], bydenition,pairpotentialsdonotincludeanymany-bodyin teractions,whichare extremelyimportantindescribinginteractionsinmostsol ids. Ontheotherhand,MDsimulationshavebeenusedextensively throughoutthe literaturetomodeltribologyofperiodicsystems[21,22,2 9,33,60,62],duetotheirability toaccuratelyreproducesomecomplicateddynamicaleects ,particularlyincasesthat maybeotherwiseinaccessibletotheoryorothercomputatio nmethods,suchasMonte Carlo.Withthisinmind,webelieveextendingsuchsimulati onstothequasiperiodic caseprovidesawell-understood,atomisticapproachtoexam inethedynamicsofsuch systems. Inthischapter,wewillbebuildingontheworkofHeatherM.H arper[20],who gainedmuchexperienceperformingandtryingtoperfectMDs imulationsona25-atom unitcell(H1)d-AlCoNiapproximant[49].Wewillrstdiscus shersimulationdetails andresultsbeforediscussingourexperienceswiththe343-a tomT11approximant. 36

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Figure4.1.\Adamant"tipandd-AlCoNiapproximantH1arediv idedintofree,thermostatandrigidsections.GraphicsrenderedbyVMD[27].4.1Harper'swork Harperperformed34770-atomsimulationsofslidingfrictio nbetweenacrystalline tipandthetwo-foldfaceofad-AlCoNiapproximant.Thesample wasbuiltfrom25atomunitcellsoftheH1approximantphasesuppliedbyMikeW idom.Widom-Moriarty potentialswereemployed[42,72,73].Intheexperimentcon ductedbyParketal.,athiolpassivatedtitanium-nitridetipwasusedtoreduceadhesion withthesample.Rather thanincludethecomplicatedinteractionsoftheorganicmo lecules,Harperusedanearly rigid\adamant"FCCtip,whoseinteractionwiththesamplew aspurelyrepulsive. Boththetipandthesampleweredividedintothreesectionsb yplanesparallelto theinteractingsurfaces.Thesectionsofthetipandsample thatwereindirectcontact obeyedtheusualNVEdynamics.ALangevinthermostatwascou pledtothemiddle sectioninthetwodirectionsorthogonaltosliding.Finall y,thetwosectionsfarthest fromcontactwereheldcompletelyrigid;seegure4.1. Thesimulationswereperformedusingbetweenoneandtwomil liontimestepsof4 femtosecondsat0K.Thetipwasloweredbyimpartingaveloci tyontherigidlayer 37

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Figure4.2.Theaveragefrictionalforceasafunctionofloa dforceforaperiodic(dashed) andperiodic(solid)slidingdirections.Dataforthisgur eadaptedfromreference[20]. untilthedesiredloadwasachieved.Thesystemwascoupledt oathermostatuntilthe systemequilibratedat0K.Next,atransversevelocityof2m /swasassignedtothe rigidlayerofthetip.Theforcerequiredtomaintainthisve locityisequalandopposite tothefrictionalforce,andwasrecordedatxedintervals. Usingveslidingvelocities v slide 2 [0 : 04 ; 0 : 12] A/psandloads F load 2 (0 ; 100)nN, 1 Harperfoundtime-averagedfrictionalforcesontheorderof piconewtons,fourordersof magnitudelowerthanexperiment[46].Foreachslidingvelo city,thefrictioncoecient wasfoundtobelowerfortheaperiodiccase,thoughonlybyan averageof8%;see gure4.2.4.2BeyondHarper AlthoughHarper'sresultshadsomedegreeofqualitativeag reementwithexperiment,quantitativelytherewasmuchtobedesired.Inanatte mpttocorrectthis,we 1 Comparethesevaluesto v slide 2 [1 10 10 ; 2 10 8 ] A/psand F load 2 [ 150 ; 100]nNfrom experiment[46] 38

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carriedouttwosignicantchanges|thersttoincreasethe anisotropy,andthenext toincreasethefrictioncoecient. Experimenthasfoundthatthefrictionmeasuredonquasicry stalsislowerthanon nearbyapproximantphases[37].Extendingthisideatoours ituation,ifweswitchtoa largerunit-cellapproximant,wemayndgreaterfrictional anisotropybetweenthetwo slidingdirections.Tothisend,wereplacedHarper's25-ato mH1approximantunitcell witha343-atomT11unitcell. 2 Next,comparingHarper'ssimulationwiththegeneralizedT omlinson-Prandtlmodel fromthepreviouschapterrevealsasignicantdierence.I nthelatter,theAFMcantileverissimulatedbyconsideringaninteractingbodyatt achedbyaspringtoanoninteractingbodymovingataconstantvelocity. 3 Thespringconstantisafreeparameter thatcanbesettoavaluedeterminedbyexperiment.InHarper 'swork,onereplaces thistwo-bodysystemwithasingletipthatisrigidandincons tantmotionononeend, andfreeandinteractingontheother.Thedisadvantagehere isthatthespringconstant isnolongerdirectlyadjustable|itisafunctionofthecohe sivepotentialandgeometry ofthetip.Moreover,becausetheadamantpotentialisrathe rsteep,wecanexpectthis springconstanttoberatherlarge. Fromexperiment,itisknownthatasmallerspringconstanta llowsforamoresensitivemeasurementofthefrictionalforce[46].Infact,amol ecular-dynamicsinvestigation byShimizuetal.[60]foundthatthecomponentsofthespring constantinthesliding andloaddirectionsbothplayavitalroleindeterminingthe onsetofstick-slipandthe coecientoffriction:areductioninthesliding-direction springconstantbyafactorof fournearlydoublesthemeasuredfrictioncoecient. Withthisinmind,wechosetoreplaceHarper'sadamanttipwi thatip-and-\bob" combination.Thetipwasconstructedfromanaluminumunitc ellbutmadecompletely rigid.EachtipatomwouldinteractwiththesampleviaaLenn ard-Jonespotential. Meanwhile,thecenterofmassofthetipwouldbecoupledtoth ebobbyaspringwith constant k ;seegure4.3. 2 CourtesyofMikeWidom. 3 Thetipandbaseofthecantilever,respectively. 39

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Figure4.3.InoursimulationswiththeT11approximant,wer eplacedHarper'stipwith atipand\bob"combination.Thisallowsthespringconstant tobesettovaluestypical ofexperiment.Thesampleisstilldividedintorigid,therm ostat,andfreeregions. Becauseweswitchedtoalargerapproximant,wewerefacedwi ththeadditional complicationofdeterminingthecorrectsurfaceterminati on.Weselectedtwocandidate surfaceterminations,whichwewillcallT11(a)andT11(b), thatwereconsistentwith thedescriptiongiveninreference[44];seegure4.4.Some notablefeaturesinclude pentagonswithasinglevertexexposedtothesurface,disto rtedpentagonswithtwo verticiesonthesurface,onlyaluminumatomsexposed,andt heFibonaccisequence withlengths L =4 : 9 0 : 3 Aand S =2 : 8 0 : 2 Aparalleltothesurface,allofwhich wereabsentfromthesurfaceterminationusedinHarper'swo rk. Weperformedthebulkofoursimulationsonasmall2744-atomT 11samplewitha 340-atomBCCtipbutalsoperformedaseriesofrunsonalarger4 9392-atomsample witha1600-atomtip.Using0 : 002femtosecondtimesteps,with 10 7 10 8 totalsteps, thebobwasloweredadistance x 0 ,andthetipandsamplewereequilibratedata temperature T .Aconstantvelocity v slide intheslidingdirectionwasthenassignedto therigidlayerofthesample,whilethebobwasxedinspace. 4 Thespringforcebetween 4 Thisispreferredoverxingthesampleandslidingthe\bob"becau seofaLAMMPSbug. 40

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Figure4.4.(Top)Surfacestructuretakenfrom[44].Theblu eatomsarealuminum,and pinkaretransitionmetals.Darkeratomsareinthenearerof thetwolayers.Both, theT11(a)(middle)andT11(b)(bottom)surfaceterminatio nshaveseveralfeaturein commonwithPark'ssample.Heretheblueatomsarealuminum, whileredandgreen arethetransitionmetals,cobaltandnickel,respectively .Theasterisksdenoteatoms thatwereremovedduetoinstability.thetipandspringwererecordedevery10timesteps.Thecomp onentsofthisforcein thecompressionandslidingdirectionsweretheloadandfri ctionalforce,respectively. 4.3Results Simulationswererunforseveralchoicesintheparametersp ace,whicharetabulatedintable4.1.Typicalvaluesfortheloadforcesusedw ere F load 2 (0 ; 48)nN, correspondingtopressures P 2 [0 ; 10 : 8]GPa. 5 SeeappendixCforasampleinputle andsubmissionscript. Performingtimeaveragesoftheforcesonthebobwasnotasim plematter,especially indeterminingthefrictionalforce,sincetheructuations duetostick-slipwereofmuch largermagnitudethantheaveragefriction;seegure4.5.P erformingpeak-to-peak averageswasverydiculttoautomate,andpickingoutthese \peaks"isnotexceedingly 5 Comparethesepressurestothoseusedinexperiment P exp 2 [ 1 : 13 ; 0 : 87]. 41

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SeriesSampleAtoms T (K) k (eV/ A 2 ) v slide ( A/ps) (1)T11(a)3094010.05(2)T11(a)3094010.075(3)T11(a)309400.50.05(4)T11(a)3094010.033(5)T11(b)3094010.075(6)T11(b)309400.50.075(7)T11(b)3094010.05(8)T11(b)309430010.05(9)T11(b)5099330010.01(10)T11(b)50993010.01 Table4.1.Theparametersusedinmolecular-dynamicssimula tionsperformedonthe T11approximantwithtwosurfaceterminations(a)and(b).F rictionwasnotfoundto belowerinthequasiperiodicslidingdirectionforanychoi ceofparameterslisted. Figure4.5.Theforcetime-seriesfromseries(1)intable4.1 alongthequasiperiodic slidingdirection,with F load =3 : 12eV/ A. straightforwardsincetherearetwodistinctfrequenciesc ontributingtotheoscillationsin thefrictionforce:onecharacterizedbytheunit-celllengt handtheotherduetostick-slip andlargelydependentonthenormalforceandspringconstan t.Afterexperimenting withsomettingroutines,wedecidedtosimplyusethetime-a veragedvalueswith error-barscalculatedbytakingthestandarddeviationofth ecumulativeaverage. TherunsusingtheT11(a)surfaceterminationsturnedoutto beparticularlytroublesome,asseveralatomsdissociatedfromthesurface,andins omecases,werecompletely ejectedfromthesimulationbox.Thissurfacefeaturedanap proximatelytwo-foldfrictionalanisotropy,butwiththeperiodicslidingdirection havinglowerfriction,contrary 42

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Figure4.6.(Left)Frictionasafunctionofloadforceforse ries(1)intable4.1.Wend aclearanisotropy,butwiththeperiodic(dotted)slidingd irectionbeinglowerthanthe quasiperiodic(solid)directionbyafactorof 2 : 0.(Right)Frictionplottedagainstthe loadforceforseries(7)intable4.1.Herewefoundsomeprom ise,astheanisotropy betweentheperiodic(dotted)andquasiperiodic(solid)sl idingdirectionsincreasedwith largeload.totheexperimentalresults.Wedecidedtoremovetheproble maticatomsfromthe simulationentirely,butthishadlittleeectontheresult s;seegure4.6. TheT11(b)surfaceterminationprovedtobemorestable,but inthiscasewefound littletonofrictionalanisotropywhatsoever.Whenthe309 4-atomT11(b)runwith v slide =0 : 05 A/psnallygaveusaresultwithlowerfrictionontheaperio dicsliding directionforlargeloads,wedecidedtoprobeevenlargerlo adforces;seegure4.6.To doso,withoutplasticdeformation,wewererequiredtoincr easethesizeofthetipand sample,therebyincreasingthecontactareaanddecreasing thepressure.Also,because thiseectwasonlyseeninrunswithlowerslidingvelocitie s,wechosetolowerthis velocitybyanotherfactorofve,to0 : 1 A/ps.Atthistime,onlyresultsfromrunswith T =300Kareavailable,andunfortunatelythefrictionalanis otropywasnotreproduced here. Althoughtheseresultsdonotdomuchtoimproveontheworkdo nebyHarper,much ofparameterspacehasyettobeexplored.Specically,weha venotperformedanyruns withvelocitiesanywherenearthoseusedinexperiment.Tho ughitwouldbeimpossible tolowerourslidingvelocitytotheorderofmicrometersper second,velocitiesofseveral centimeterspersecondarewithinreach.Moreover,itisunc learwhetherthestick-slip 43

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mechanismplayedaroleintheexperimentalresults:thetip waspassivatedandthe contactareawaslargerbynearlyanorderofmagnitude.Itma ybeworthexploring morepointsintheparameterspaceusingamorerepulsivetip -surfaceinteractionor with k !1 Ontheotherhand,itispossiblethatthereissimplynothing tondhere.Between thisworkandHarper's,therehavebeenglimpsesofanisotro pyfavoringthequasiperiodicslidingdirection,butonlywithafactorof1 : 1 1 : 2,farfromtheexperimentally observedeight-foldanisotropy.Clearly,smallchangesint heparameterswillnotsuce. Rather,itismorelikelythatweneedtomovebeyondpair-pote ntials,perhapsevenincludingelectronicdegreesoffreedom.Anotheralternativ eistodevelopamodelwhich emphasizesthequasiperiodicityofthesysteminacontroll edmanner.Thisisexactly whatisdoneinthefollowingchapter. 44

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CHAPTER5 ASIMPLERMODEL:FIBONACCIUM Althoughwefailedtoreproducetheexperimentalresultsof Parketal.inourMD simulationsofd-AlCoNiapproximants,thetruefailurewaso urinabilitytodetermine why.Whywereweunabletoreproducetheeight-foldanisotrop y,orforthatmatter,any anisotropywhatsoever?Wecouldcontinueinthefashionofc hapterfour,continuing tomaketweaksandinvestigatealargersampleofparameters pace,butwithouta signicantbreakthrough,itisdoubtfulwhetheraclearund erstandingwillbeachieved. Ontheotherhand,wecoulddevelopanewmodelwiththisquest ioninmind.Rather thangenericallytryingtoreproducethefrictionalanisot ropy,wecanconstructamodel thatisolatesthefeatureofourprimaryinterest:quasiper iodicity. TheFibonaccisequencehasbeenusedinvariousworksasamod elofquasiperiodicity [4,12,36,47]andinsomecaseshasbeenexpandedtothetwo-di mensionalcase[28, 69].Here,wewilluseapproximantsfromtheFibonacciseque ncetogeneratethreedimensionalsolidswithvaryingordersofquasiperiodicit y.Withthis,wemaydirectly probetheroleofquasiperiodicitywhileminimizinginterf erencefromotherunrelated mechanisms.5.1ConstructingFibonaccium Asdiscussedinsection2.1,theFibonaccisequencecanbeus efulforreproducing somephysicsofquasiperiodicstructures.Aswithd-AlCoNi, wewantedtoconstructour modelwithtwoquasiperiodicandoneperiodicaxes.Westart edbytakinganexisting cubiclatticeandtriedtoperturbtheinteratomicdistance saccordingtotheFibonacci sequence.Aswebegantodeveloppotentialsforsuchasystem ,itbecameclearthat perturbingspringconstantsor,almostequivalently,mass eswouldnotonlysimplifythe 45

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Figure5.1.5th-orderFibonacciumbuiltfromiron(BCC)unit cells.Insimulationswith thisstructure,weemploytheembedded-atommethod(EAM)pot entialfromreference [39].Key: m blue = m LL m green = m LS = m SL m red = m SS modelagreatdeal,butalsoallowustoskippotentialdevelo pmentaltogethersincewe couldsimplypluginthepotentialsfromtheunperturbedsys tem. Ignoringtheperiodicaxisforamoment,wetookatwo-dimensi onalgridandlabeled eachcolumn|eitherLorS|accordingtosomeniteFibonacci sequenceapproximant,thenrepeatedforeachrow.Atthatpoint,eachgridce llhadatwo-character label:LL,LS,SL,orSS.Wethenconstructedatwo-dimensiona lsurfacewithtwo quasiperiodicdirectionsbyconvertingeachgridcellinto acubicunitcell(FCC,BCC orSC)containingatomswithatomicmassescorrespondingto thatcell'slabel.For instance,allcellslabeledSSwouldhaveatomswithmass m SS =50amu,whileLSand SLcellscontainedatomswithmass m LS = m SL =100amuandLLcellscontained thosewithmass m LL =200amu.Finally,wecompletedourconstructionof\Fibona ccium"byrepeatingthisstructureperiodicallyinthethird direction;seegure5.1.The codeusedtogeneratetheFibonacciumsampleissuppliedina ppendixD. Usingthisrecipe,weconstructedtwoseriesofFibonaccium approximants.Therst serieshadsamplesbuiltfrombody-centeredcubic(BCC)unit cells,withlatticeconstant 46

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Figure5.2.IsotropicpairpotentialsdevelopedbyRechtsm anetal.wereimplemented forsimulationsofSCFibonaccium[54].Thepotentialtakes theform V sc ( r )= 48572 r 12 0 : 25142exp( 10 : 761( r 3 p 2) 2 ).Theminimumisatthesecondnearest neighbordistancetoavoidtheformationofclosed-packedst ructures. a =2 : 86 A.Wechosethisstructure,asamany-bodyembedded-atom-metho d(EAM) potentialwasreadilyavailable[39].Thesecondserieswas constructedfromsimplecubic(SC)unitcells( a =3 : 0 A)tominimizethenumberofatomsrequiredforhigherorderapproximants.Forthisstructure,weusedisotropicp airpotentialsdeveloped byRechtsmanetat.[54];seegure5.2.Thispotentialistai loredtostabilizethe simple-cubicstructurebyplacingtheminimumatthesecond-n earest-neighbordistance andisscaledtogivethedesiredlatticeconstantandacohes iveenergycomparable tothatofourd-AlCoNiT11approximant:0 : 53eV/atom.Ineachcase,thetipwas constructedwithcubicsymmetryandlatticeconstantincom mensuratetothesample toavoidregistry. Wealsohadtwodierentwaysofcontrollingthedegreeofqua siperiodicityinour sample.First,wecanchooseasingleapproximantandset m LL = m SS ,therebyretainingacompletelyperiodicstructure. 1 Now,byincreasing m LL whilekeeping m SS xed,quasiperiodicitycouldbeintroducedtothesystemin acontrolledandcontin1 Wewillalwaysuse m LS = m SL = m LL m SS 2 47

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uousmanner.Second,wecanchoose m LL 6 = m SS andinsteadcontrolthedegreeof quasiperiodicitybytheorderoftheapproximantused.5.2Fixedorder,variablemass Therstsetofsimulationswereallperformedona1024-atoms ixth-orderBCC Fibonacciumsample,correspondingtotheFibonaccisequen ceoflength8.Thelight masswaskeptxedandequaltotheatomicmassofiron, m SS =55 : 845amu,while theheavymasswasvaried,takingoneofsixpossiblevalues: m LL 2 [55 : 85 ; 1406 : 25] amu.Themediummassesweresettotheaverageofthelightand heavymasses: m LS = m SL = m SS m LL 2 .Thesimulationswereperformedusingthesameprocedureas thatused onthed-AlCoNiapproximants,whichisoutlinedinchapter4, with26dierentload forces F load 2 [ 3 : 3 ; 5 : 8]eV/ Aforeachchoiceof m LL andslidingdirection. 2 Thespring constantwassetto1eV/ A 2 ,andthetipwasdraggedalongthequasiperiodic/periodic surfaceforatotalof3.6nanosecondsat0.05 A/psat T =0K.Asampleinputleis giveninappendixC. Theresultswerenotparticularlyinteresting.Forindivid ualchoicesofmass,the averagefrictionalforcewasabitnoisyasafunctionofload force,butthecoecientsoffrictionextractedvialinearregressionhaderro rsofonly4%.Weperformed thelineartsforbothslidingdirectionsandcalculatedth etotalfrictionalanisotropy = periodic = quasi periodic quasi q ( periodic periodic ) 2 +( quasi quasi ) 2 ,foreachchoiceofmass.Unfortunately,wefoundnoevidencethattheanisotropyhasanyde pendenceonthemass ratioforxed-orderapproximants;seegure5.3. Becausetheseresultsweren'tveryencouraging,wechoseto forgofurthercalculations withlargersamplesandinsteadfocusonoursecondapproach 5.3Fixedmass,variableorder Ratherthanxingapproximantorder,weperformedsimulati onsforxedmasseson SCandBCCFibonacciumapproximantsofuptotenth-order; 3 seegure5.4.Because 2 Theloadscorrespondtopressures P 2 [ 5 : 55 ; 9 : 76]GPa. 3 CorrespondingtotheFibonaccisequenceoflength55. 48

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Figure5.3.(Left)Frictionalanisotropyasafunctionofma ssratio.(Right)Theaverage frictionalforceasafunctionofloadfor m LL =m SS =25.Evenforsuchalargemass ratio,wendnoevidenceoffrictionalanisotropy. SeriesBCC/SCAtoms(approx.) T (K) k (eV/ A 2 ) v slide ( A/ps) (1)BCC120000100.05(2)BCC12000010.05(3)BCC1200015010.05(4)SC3100030010.05(5)SC3100030010.005(6)SC310003000.10.05(7)SC310003000.10.005(8)SC310001000.10.005(9)SC310003000.10.0005 Table5.1.Theseparameterswereusedinthemolecular-dynam icssimulationsonFibonaccium.Inallcases m SS =50and m LL =200wereused.Thenumberofatomsin eachcaseisapproximatebecauseitvariesdependingonthea pproximant. theapproximantunitcellsareofvaryingsizes,smallerapp roximantswererepeatedin thesimulationboxsoparticlenumber(andvolume)couldber oughlyconstantforeach sample.Thesimulationswereperformedintheusualway,wit hrunsatboth T =300 K andabsolutezero,andvaryingspringconstantsandsliding velocities.Choicesforthe experimentalparametersforeachseriesofrunsarearelist edintable5.1. Theresultsfromthesesetsofrunsdidnotshowmuchpromisee ither.Takeseries(7), forinstance.Werancalculationsforthirteenchoicesofco mpressions F load 2 [0 : 45 ; 17 : 5] eV/ A|correspondingtopressures P 2 [0 : 06 ; 2 : 43] GPa |forFibonacciumapproximantsoforderfour,seven,andten.Here,weobservednosign icantdierenceinthe 49

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Figure5.4.Simple-cubicFibonacciumoforder55.Thediere ntcolorsrepresentdifferentatomicmasses:darkblue=LL,lightblue=LS=SL,gree n=SS.Thetipis showninyellow.Theperiodicandquasiperiodicslidingdir ectionsaretotheupper-left andupper-right,respectively.Imagerenderedusingrasmol [58]. Figure5.5.Thefrictionalforce(versusload)forseries(7 )withsimple-cubicFibonaccium sampleoforder3and55,(left)and(right)respectively.De spiteaverysignicant dierenceintheunit-cellperiodicity,wedonotobserveany signicantdierencesin thefrictionalresponse.frictionalbehaviorbetweenthetwoslidingdirections,an d,ifanything,theanisotropy droppedfrom =1 : 05 0 : 12forfourth-orderto =0 : 90 0 : 08fortenth-order;see gure5.5.Wewillnotattempttoarguethatlarger-orderappr oximantsincreasingly favortheperiodicslidingdirectionsincetheanisotropie shaveoverlappingerror-bars, butcertainlythereisnoevidenceheretolinkquasiperiodi citytolowfriction. 50

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CHAPTER6 CONCLUSIONSANDFUTUREWORK Inthiswork,wehaveimplementedthreetechniquesinanatte mpttounderstand thelow-frictionphenomenononquasicrystalsandhowitrela testoquasiperiodicity.We modeledfrictionona1-Dquasiperiodicsystemusingastocha sticdierentialequation andperformedmolecular-dynamicssimulationsond-AlCoNiap proximantsandourown toymodel,Fibonaccium.Innocasesdidwendaloweringoffr ictionduetoquasiperiodicity.Thesendingsleadtotwopotentialconclusions.E itherthereisashortcoming (orshortcomings)inourmodels,orthereissimplynoeectt obeobserved.First,we shallassumetheformer. InourS-ODEapproach,thereareseveralaspectsthatdeserve scrutiny,butthemost obviousisthelackofatomicity.Withoutatomstherecannot beneitherphononsnor electronsbutratheraviscocity-likemechanismforenergyd iusion.Thoughitseems reasonabletomodelthetransferofenergyawayfromthesurf aceofoursamplevia thediusionofenergy, 1 thiseliminatesthepossibilityofinterferencebetweendi erent electronicandphononicmodesandcompletelyignorestheex istenceofextendedor algebraically-decayingstates.Asecond,possiblyrelated ,deciencybecomesevident whenoneconsidersthetimescalesinvolvedinthemodel.Bec auseoftheverylow slidingvelocity,thetimebetweeneachstick-slipeventist ypicallyontheorderoften milliseconds.Ontheotherhand,thetimescalefordampingi s 1 r = 1 10 7 ps =ten microseconds.Sincethemaindynamicalvariablerelatedto frictioninthisproblem, thevelocity_ x ,issoquicklydampedout,welackcouplingbetweenadjacent stick-slip events. 1 Intheformofheat. 51

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Onepossibleremedyforthesedecienciesistoaddamemoryt ermtothedierential equation,onethatcouplesstick-slipeventsemulatingthee ectofphonons.However, thecorrectwayofdoingthisisnotclear. Next,letusconsiderourmolecular-dynamicssimulations.T hemostobviousweaknesshere,asmentionedinpreviouschapters,isthelargesl idingvelocity.IntheexperimentconductedbyParketal.ond-AlCoNi,theslidingveloci tyneverexceeded10 6 A/ps.Evenwhenwepushourcomputationallimits,wecanonly reduceourvelocity downto5 10 4 A/ps|stilltwoordersofmagnitudetoofast.Itispossiblet hat thislargevelocitysimplywashesouttheeectsthatproduc ethefrictionalanisotropy. Ontheotherhand,wewouldexpecttheanisotropytogrowasth evelocityapproaches theexperimentalmagnitude,butwedonotobserveanysuchtr endsinthefrictional responsebetwen v slide =0 : 0005and v slide =0 : 5 A/ps.Moreover,itisunclearwhethera furtherreductionofvelocityshouldalterthephysicsinan yappreciableway,asweare alreadywellbelowthecharacteristicspeedsforoursystem :thelatticeconstanttime springfrequency a q k m bob =660m/s,andthespeedofsoundinFibonaccium,which weestimatetobeontheorderof10 5 m/s. Asecondmajordeciencyisourlackoflong-rangepotentials .TheRechtsman potentialdecaystozeroquicklyafterthelocalminimaatth esecondnearest-neighbor distance3 p 2 A[54],theWidom-Moriartypotentialsusedaretruncatedat7 Ato enhancestability[71],andtheEAMpotentialsarecutoat5 : 3 A.Clearlylongrangeinteractionsareresponsibleforthestabilityofrea l-lifequasicrystals,andsuch interactionsmayalsoenhancetheeectsof(approximate)q uasiperiodicityaseach atomwill\see"alargerportionoftheunitcell. Third,theMDcalculationslackelectronicandaccessiblep hasondegreesoffreedom. Theelectronicstatesmayplayaparticularlyimportantrol e,sinced-AlCoNiandother quasicrystalsaresemi-metals.Moreover,quasiperiodicit yaectselectronicstructurein apeculiarway,introducingafractal-likestructureriddle dwithVanHovesingularities [4];seegure6.1.Perhapsthisorsomeotherelectronicee ctisthetrueculpritbehind thelowfriction. 52

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Figure6.1.Theelectronicdensityofstatescalculatingus ingtight-bindingbyBottger andKasner.Imageobtainedfromreference[4]. Despitethesedeciencies,itisstillpossiblethatweared oingalltherightthings, andthereissimplynoanisotropyheretond.Thiswouldseem toexplainthefailureoftheFibonacciummodel,butwithoutmakingaccusation sofPark'sexperiment being\wrong,"itdoeslittletoexplainthefailureofoursi mulationsofthed-AlCoNi approximants.Whateverthecase,itisstillpossiblethatq uasiperiodicityisnotatall responsiblefortheselow-frictionsurfaces;howeverthere isnoothermechanismtowhich wecanreallyattributethisresponsibility.Asdiscussedi nchapter1,thereareinstances whereargumentssuchashardness,adhesion,orregistrymak esense,butnoneofthese mechanismscansingle-handedlyexplainalloccurrencesofl ow-frictionquasicrystalline surfaces.6.1Futurework Aswehavenotdrawnanystrongconclusionsfromthiswork,it isclearthatmore calculationsarerequired.Theobviousnextstepistoexplo remorepointsintheparameterspaceofourMDsimulations.Inparticular,wewould liketoperformmore 53

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simulationswithlowvelocities,largersamples,higher-or derapproximants,tipswithincreasedsurfaceareatoallowforlargeloadforces,andlarg espringconstantstodecrease thefrictionalcontributionfromstick-slip.Additionally ,wewouldliketoperformcalculationsofthephononspectrum,phononparticipationratio s,andthermalconductivity alongthetwoslidingdirectionsforboththed-AlCoNiapprox imantsandFibonaccium. Next,weareinterestedinperforming2-Dsimulationsoffric tionbetweenasingleatomtipandaFibonaccichain.Wecanimposeperiodicityont hisFibonaccichain inthesamemannerasusedforFibonaccium|byperturbingthe atomicmasses. Usingananharmonicpotential,andreferringtoreference[ 12],wecanchooseatomic massesthatopengapsinthephononspectrumatcertainfrequ encies;seegure6.3. Bymeasuringfrictionusingslidingvelocitiescorrespond ingtothesefrequenciesand comparingtocalculationswithunperturbedmasses,webeli evewemayuncoversome interestingbehavior.Furthermore,performingsimulatio nsonsuchacomputationally lightweightmodel,aswasthecasewiththeS-ODEmodel,alsoo pensupthepossibility ofexploringverysmallslidingvelocities. Onanothernote,unrelatedtoquasiperiodicity,weuncover edsomeinterestingfeatureswhileexperimentingwithourS-ODEmodel.First,aswed iscussedinchapter3 forthepotential, U = U 1 cos 2 x tip b + U 2 cos 2 x tip b 0 ; (6.1) taking b 0 =b 2 Q ,wefoundtwodistinctregimesoffrictionalresponse,sepa ratedby adeepminimumforlowtemperature;seegure3.10.Starting fromtheleftofthe minimum,thefrictionalresponseischaracterizedby7 : 5 Astick-slipevents.Upon crossing,theseslipeventssuddenlysplitintoatwosepara teeventsoflength3 : 5and4 : 0 A;seegure3.11.Aswemovedtowardsirrationalvaluesfor b 0 =b thiseectbeganto vanish;seegure3.9.Thenextstepinunderstandingthisph enomenonistoperform morecalculationsandtrytonarrowthetransitionboundary toassmallaregionas possible.Fromthere,wehopetouncoversomecluesbyperfor mingFourieranalysisof thepotentialagainstthetime-series. Finally,weperformedS-ODEcalculationswithaperiodicpot ential U 2 =0,but withvariableunit-celllength b .Plottingtheaveragefrictionalforceagainst b ,wefound 54

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Figure6.2.Frictionversusthelatticeconstantcalculate dusingtheS-ODEmethod fromchapter3.Onaverage,wendthatfrictiondecreasesat largelatticeconstant,we ndtwojumpdiscontinuitieswherethefrictionabruptlyin creasesbyseveralhundred piconewtons.twojumpdiscontinuities;seegure6.2.Atrstsight,onew ouldassumethatthese discontinuitieslikelycorrespondtosomeresonancebetwe entheslidingvelocityandthe springfrequency.Itturnsoutthatthisisnotthecase,as v slide =b isdierentfrom p k=m byseveralordersofmagnitude.Thenextstepistoperformca lculationsfor latticeconstantsnotcoveredbytheinterval[3 ; 6] A.Atthatpoint,wemaybeableto ndmorecluesinthepositiontime-seriesdata. 55

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Figure6.3.(Top)Thephononicdensityofstatesfortheharm onic(dotted)andanharmonic(solid)Fibonaccichains.(Bottom)Thecoherents tructurefactorforthe anharmoniccase.Imageobtainedfromreference[12]. 56

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APPENDICES 62

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AppendixALinearresponse GivenaLangevinequation,onemayuselinear-responsetheor ytoderiveanexpressionbasedonequilibriumcorrelationfunctions.From theequationforaBrownian particle m d v d t = rv + R ( t ),onendsthetime-averagedforce(friction)ontheparticl e isproportionaltothevelocity, h F i = r h v i .Withsomemanipulationwemaynd r = 1 2 k B T R 1 1 h R (0) R ( t ) i d t [1],where R ( t )isthestochasticforceinanequilibrium ensemble. Nowsuppose,ratherthanBrownianmotion,weareusingtheTo mlinson-Prandtl modeltocalculatethefrictionbetweenasurfaceandaslidi ngAFMtip;seeequation (3.1).Hereweshallndthatwhilethismethodmaystillbee ectiveforacalculation of r r isnolongerthedominantterminthefrictionalforce.Becau seofthis,weare nolongerabletocalculatefrictionbasedonequilibriumen sembleaverages,butrather, wemustperformdynamicsimulations. IntheTomlinson-Prandtlmodel,theAFMtipisconnectedtoa\ bob"viaaspring withconstant k .Thebobslidesataconstantvelocity,whilethetipissubje cttoa periodicsurfacepotentialandthermalructuations. Theequationofmotionforthebobis m x bob = k ( x tip v bob t )+ F app =0 ; (A.1) where k isthespringconstantand F app = F friction f istheforceappliedforceto keep_ x bob = v bob constant. 1 Thetip'smotionisgovernedby m x tip = rx tip @U ( x tip ) @x tip k ( x tip v bob t )+ R ( t ) ; (A.2) where U ( x )isaperiodicpotentialwhichdescribesthesurfacecorrug ationofthesample, and R ( t )isaMarkoviannoisetermobeying h R i =0and h R ( t ) R ( t 0 ) i =2 mrk B T ( t t 0 ). 1 Hereafter,whenwesay\friction"wemeanthemagnitudeoffriction. 63

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AppendixA(Continued) Sinceweareinterestedintheenergylosttofriction 2 weareinterestedincalcuating thespatial-averagedquantity, h f i x bob .Usingthechainrule, h f i x bob = 1 L Z L 0 f ( x tip ;t ) dx bob = 1 L Z T f 0 f ( x tip ;t ) dx bob dt dt (A.3) = 1 L v bob Z T f 0 f ( x tip ;t ) dt (A.4) = 1 T f Z T f 0 f ( x tip ;t ) dt = h f i t ; (A.5) where T f isthetotaltime,and L = v bob T f Usingthefactthattheleft-handsideofequation(A.1)iside nticallyzero, h f i t = k h x tip v bob t i : (A.6) Invokingequation(A.2), h f i t = m x tip + rx tip + @U ( x tip ) @x tip R ( t ) t = rx tip + @U ( x tip ) @x tip t ; = rv bob + @U ( x tip ) @x tip t (A.7) wherewehaveusedthefactthatthepotentialisperiodic,an dthatinthelong-time limitwehave h x tip i t 0and h x tip i t = h x bob i t Wendthatthersttermcontributeslinearlytothefrictio nasexpected;however thesecondtermcontributesincaseswhenthetipdoesnotmov eataconstantvelocity,in particular,whenstick-slipoccurs.Onemayexpectthatathi ghtemperatures,thestickslipbehaviormightvanish;howeverthisisnotthecaseeven atthehighesttemperatures exploredbyTshiprutetal.inreference[68].Thatis,using r =5x10 6 kg = s, U ( x )= U 0 sin(2 x=a ), U 0 =0 : 26eV, a =0 : 3nmand v bob =10nm = s,at400Kwendthat secondtermislargerbyafactorof4000. 2 Energylossimplieswork. U loss = R f ( x ) dx =( x f x 0 ) h f i x 64

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AppendixBS-ODEsolver WediscussedVanGunsterenandBerendsen'salgorithm[70]f orsolvingLangevintypetochasticordinarydierentialequationsinsection3 .1.Thisisourimplementation intheCprogramminglanguage. /*quasi-tsh.c *KeithMcLaughlin*2009JUNE23*/ #include#include#include#includestructconstants{ /*reciprocalmass*/doublerm;/*relaxationtime*/doubleeta;/*temperatureinenergyunits*/doublekT;/*slidingvelocity*/doubleVtip;/*averagepotential*/doubleUzero1;doubleUzero2;/*springconstant*/doubleKspring;/*latticeconstant*/doublea;/*goldenratioapproximation*/doubletau; };/*fromseperatesourcefile*/doublegaussd(doublemean,doublestd);voidusage(char*this){ fprintf(stderr,"usage:%sdtTmaxTempnum(phi)den(phi)U _0_1U_0_2\n",this); fprintf(stderr,"or\n");fprintf(stderr,"usage:%shelp\n",this);exit(-1); }voidprinthelp() 65

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AppendixB(Continued) { fprintf(stderr,"quasi-tsh.c\n");fprintf(stderr,"KeithMcLaughlin\n");fprintf(stderr,"2009JUNE23\n");fprintf(stderr,"\n");fprintf(stderr,"\nThisprogramisfreesoftware:youcanr edistributeitand/ormodify\n"); fprintf(stderr,"itunderthetermsoftheGNUGeneralPubli cLicenseaspublishedby\n"); fprintf(stderr,"theFreeSoftwareFoundation,eitherver sion3oftheLicense,or\n"); fprintf(stderr,"(atyouroption)anylaterversion.\n");fprintf(stderr,"\nThisprogramisdistributedinthehope thatitwillbeuseful,\n"); fprintf(stderr,"butWITHOUTANYWARRANTY;withouteventh eimpliedwarrantyof\n"); fprintf(stderr,"MERCHANTABILITYorFITNESSFORAPARTICU LARPURPOSE.Seethe\n"); fprintf(stderr,"GNUGeneralPublicLicenseformoredetai ls.\n"); fprintf(stderr,"\nI'llsolvetheLangevinequationfor1dfrictiononaquasiperiodic\n"); fprintf(stderr,"surface,usingamodelsimilartoTshipru tetal.inPRL102136102\n"); fprintf(stderr,"(2009).\n");fprintf(stderr,"\nm*dv/dt=-m*eta*v+R(t)-dU'(x)/dx-K *(x-V*t)\n"); fprintf(stderr,"whereR(t)isastochasticforceandKisas pringconstant\n"); fprintf(stderr,"RatherthanusingU~U_0*sin(2Pix/a),we willuse\n"); fprintf(stderr,"U'~U_0_1*sin(2Pix/a)+U_0_2*sin(2Pix /b),whereb=phi*a,wherephiis\n"); fprintf(stderr,"arationalapproximationtothegoldenra tio.\n"); fprintf(stderr,"\nSolutionisviaGunsterenandBerendse n,Mol.Phys.198245(637)\n"); return; }/*getmicrosecondprecisionseed-neededsincewewillbeus inggaussd*/ intinitrnd(){ intrval=0;structtimevalthetime;if((rval=gettimeofday(&thetime,(structtimezone*)0)) !=0) fprintf(stderr,"gettimeofdayreturnedanerror.\n"); srandom(thetime.tv_usec);returnrval; }/*calculatethesystematicforce*/doubleev_force(doublex,doublet,structconstants*c){ doubleTwoPiOverA=2*M_PI/c->a;doubleTwoPiOverB=2*M_PI/(c->tau*c->a);return-(c->Kspring)*(x-(c->Vtip)*t)+/*springforce*/ -(c->Uzero1)*cos(TwoPiOverA*x)*TwoPiOverA+/*atomicp ot1*/ -(c->Uzero2)*cos(TwoPiOverB*x)*TwoPiOverB;/*atomicp ot2*/ }voidsetconstants(double*A,double*EoverCpls,double*G overCpls,double*Cpls, 66

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AppendixB(Continued) double*fA4,double*fA5,double*fA8,structconstants*c, doubledt) { inti; /*usedtostorevariouspowersofc->eta*dt. emphasisonreadabilityofpowerseriesexpansions*/A[0]=1;for(i=1;i<=9;i++)A[i]=A[i-1]*c->eta*dt; /*Powerseriesexpansions*//*goodforc->eta*dt<<1*/ *EoverCpls=1./2.*A[3]+3./8.*A[4]+29./160.*A[5]+ 43./640.*A[6]+1831./89600.*A[7]+381./71680.*A[8]+235009./193536000.*A[9]; *GoverCpls=1./2.+3./8.*A[1]+21./160.*A[2]+19./640.* A[3]; *Cpls=2./3.*A[3]-1./2.*A[4]+7./30.*A[5]-1./12.*A[6] + 31./1260.*A[7]-1./160.*A[8]+127./90720.*A[9]; *fA4=A[1]-1./2.*A[2]+1./6.*A[3]-1./24.*A[4] +1./120.*A[5];/*eqA4*/ *fA5=1./12.*A[3]-1./24.*A[4]+1./80.*A[5] -1./360.*A[6];/*eqA5*/ *fA8=1./2.*A[2]-1./6.*A[3]+1./24.*A[4] -1./120.*A[5];/*eqA8*/ return; }voidprintdetails(doubledt,structconstants*c,doubleT ) { fprintf(stderr,"Running%dstepswith:\n",(int)(dt/T)) ; fprintf(stderr,"\tdt=%10lfm=%10lf\n",dt,1/c->rm);fprintf(stderr,"\tkT=%10lfeta=%10lf\n",c->kT,c->eta ); fprintf(stderr,"\ntx\n");return; }voidprintcomplete(doublet,structconstants*c,doublex sum, intcount,doubledt,doubleT) { fprintf(stderr,"=%lf(eV/A)\n", -c->Kspring*(xsum/count-(c->Vtip*t/2))); fprintf(stderr,"=%lf(pN)\n", -1602.17646*c->Kspring*(xsum/count-(c->Vtip*t/2))); fprintf(stderr,"dttmaxtempnphidphiU1U2\n");fprintf(stderr,"%lf%lf%lf%lf",dt,T,c->kT/8.6173423e -5, -1602.17646*c->Kspring*(xsum/count-(c->Vtip*t/2))); return; }intintloop(doubledt,doubleT,structconstants*c){ 67

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AppendixB(Continued) /*usedforcalculationof*/ intcount=0;doublexsum=0; /*dynamicvariables(andinit)*/ doublex[3],f[2],v,df,t;x[0]=x[1]=x[2]=f[0]=f[1]=v=df=t=0; /*randomvariables(andinit)*/ doubleXpls[2],Xneg[2];Xneg[0]=Xneg[1]=Xpls[0]=Xpls[1]=0; /*setsomeconstants*/ doubleA[9],EoverCpls,GoverCpls,Cpls,fA4, fA5,fA8,sig215,sig212; setconstants(A,&EoverCpls,&GoverCpls,&Cpls,&fA4, &fA5,&fA8,c,dt); /*usefulidentity*/ doubleeneg=exp(-A[1]); /*outputrundetails*/ printdetails(dt,c,T); /*calculatethefirstposition-weassumex(0)=0,v(0)=0*/ /*highestindexisthemostcurrent*/f[1]=ev_force(x[2],t,c);/*f(0,0,c)*/Xpls[1]=gaussd(0.,sig212);x[2]=c->rm*f[1]/(c->eta*c->eta)*(fA8)+Xpls[1];/*eq2 .26*/ /*iterate*/ for(t=0;trm*f[1]*d t/c->eta*(fA4) +c->rm*df/(c->eta*c->eta)*(fA5)+Xpls[1]+eneg*Xneg[1];/*useeq2.6*/ printf("%lf%lf\n",t,x[2]); /*usedtocalculate*/ xsum+=x[2]; 68

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AppendixB(Continued) count++; } /*dumpsomestufftofile*/ printcomplete(t,c,xsum,count,dt,T);fprintf(stderr,"=%lf(eV/A)\n", -c->Kspring*(xsum/count-(c->Vtip*t/2))); fprintf(stderr,"=%lf(pN)\n", -1602.17646*c->Kspring*(xsum/count-(c->Vtip*t/2))); fprintf(stderr,"dttmaxtempnphidphiU1U2\n");fprintf(stderr,"%lf%lf%lf%lf",dt,T,c->kT/8.6173423e -5, -1602.17646*c->Kspring*(xsum/count-(c->Vtip*t/2))); return0; }/*Receives:dt,T*/intmain(intargv,char**argc){ doubledt,T;/*ps*/structconstantsc;intnphi,dphi;/*initializeconstants*/c.rm=3.2e-13;/*(ev*ps^2/A^2)^-1*/c.eta=1e-7;/*ps^-1*/c.Kspring=0.0936226462;/*eV/A^2*/c.a=3.;/*latticeconstantinA*/c.Vtip=1e-10;/*A/ps*//*printhelpmessage*/if(((argv-1)>0)&&(argc[1][0]=='h')){ printhelp();usage(argc[0]); }/*someerrorchecks*/if((argv-1)!=7)usage(argc[0]);/*settimestep*/if((sscanf(argc[1],"%lf",&dt))<=0)usage(argc[0]);/*settotaltime*/if((sscanf(argc[2],"%lf",&T))<=0)usage(argc[0]);/*settemperature*/if((sscanf(argc[3],"%lf",&(c.kT))<=0))usage(argc[0] ); c.kT=c.kT*8.6173423e-5;/*convertfromkelvintoeV*/ 69

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AppendixB(Continued) /*goldenratioapproximation*/if((sscanf(argc[4],"%d",&(nphi))<=0))usage(argc[0]) ; if((sscanf(argc[5],"%d",&(dphi))<=0))usage(argc[0]) ; c.tau=(double)nphi/(double)dphi;/*Uzeroamplitudes*/if((sscanf(argc[6],"%lf",&(c.Uzero1))<=0))usage(arg c[0]); if((sscanf(argc[7],"%lf",&(c.Uzero2))<=0))usage(arg c[0]); /*intializetherandomnumbergenerator*/if(initrnd()!=0) fprintf(stderr,"initrandreturnedanerror.\n"); /*iterateandoutput*/if(intloop(dt,T,&c)!=0) fprintf(stderr,"intloopreturnedanerror.\n"); fprintf(stderr,"%d%d%lf%lf\n",nphi,dphi,c.Uzero1,c. Uzero2); return0; } quasi-tsh.c requiresrandomnumbersselectedfromanormaldistributio n. gaussd.c suppliesthis. /*gaussd.c *DavidA.Rabson*/ #include#include/*uniformdeviateontheinterval[0,1)--calledbygaussd( )below*/ staticdoubleuniform()return(double)random()/(double)((unsignedlong)0x800 00000); /*GettwoGaussiandeviatesofzeromeanandunitstd.,theng etdesired*/ doublegaussd(doublemean,doublestd){ staticintnotstored=1;/*0ifthere'savaluestorediny2*/doublew1,w2,uni=0;staticdoubley2;/*seeNum.Rec.p289*/if((notstored=!notstored)) returnstd*y2+mean;/*returnthestoredvalue*/ 70

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AppendixB(Continued) while(uni==0)uni=uniform();w1=sqrt(-2.*log(uni));/*seeNum.Rec.eqn.7.2.10(2ndCe d)*/ uni=0;while(uni==0)uni=uniform();w2=2.*M_PI*uni;y2=w1*cos(w2);returnstd*w1*sin(w2)+mean; } Onceeachcalculationiscompletedwecancountthenumberof stick-slipevents using countslips.c .Thisisperformedasseparatelyfromthecalculationitsel f,because itinvolvestweakingtwotolerances, xtol and ttol ,bothusedtopreventthedouble countingofevents. #include#include#include#defineBUFFSIZE300/*countslips.c *UniversityofSouthFlorida*Solid-StateTheory*KeithMcLaughlin*22Jun2009*/ /*Usedwithtshiprut.corsimilarprogramstocalculatethe average *sliplengthandthetotalnumberofslips.*/ intisnum(char*string){ doubledummy;if(sscanf(string,"%lf",&dummy)==0)return1;return0; }intisfile(char*filename){ FILE*filetest;if((filetest=fopen(filename,"r"))==NULL)return1;fclose(filetest);return0; } 71

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AppendixB(Continued) interrormsg(interrcode,char*string){ if(errcode==0){ fprintf(stderr,"Impropersyntax.\n");fprintf(stderr,"Usage:\n");fprintf(stderr,"%sinputfiletx\n",string);fprintf(stderr,"OR\n");fprintf(stderr,"%stx
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AppendixB(Continued) intdeleteme; };intreadfromstdin(structfilestruct*input){ intc;/*setthetmpfilename*/sprintf(input->name,"%s/tmp-XXXXXX",P_tmpdir);/*createatemporaryfile*/intfd=mkstemp(input->name);if(fd==-1)errormsg(2,NULL);/*openforreadingandwriting*/input->ptr=fdopen(fd,"r+");if(input->ptr==NULL)errormsg(3,NULL);rewind(input->ptr);/*readstdinintothetempfile*/while((c=getc(stdin))!=EOF) putc(c,input->ptr); /*incaseweforgettolater*/rewind(input->ptr);/*settheflagfordeletionwhenwe'redone*/input->deleteme=1;return0; }intcountslips(doublextol,doublettol,structfilestruc t*input) { /*inourcodesx(0)=0,andt(step=0)=0*/doublexlast=0,tlast=0;doublexread,tread;intcount=0;charbuffer[BUFFSIZE]; /*Readinthevaluesforxandtinthefile.IfDx>xtolthen weMIGHThaveajump.Wedon'twanttocountasinglejumpmultipletimesthough,sowemakesurethatacertainamountoftimehaspassedsincethelastjump.Ifthathasn'thappene d thenwedecidethatweweredoublecounting. */ while(NULL!=fgets(buffer,BUFFSIZE,input->ptr)) 73

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AppendixB(Continued) { sscanf(buffer,"%lf%lf",&tread,&xread);if(xread>xlast+xtol){ xlast=xread;if((tread>tlast+ttol)){ tlast=tread;count++; } } }/*needtoprintcountandavgsliplength*/fprintf(stderr,"totalslips\tavgslip\n");printf("%d\t%lf\n",count,xlast/count);return0; }intmain(intargc,char**argv){ structfilestructinput;input.deleteme=0;doublettol,xtol;/*tolerances*//*wewilleitherreadfromstdinorfromfile*/switch(argc-1){ case2: if(isnum(argv[1])==1)errormsg(0,argv[1]);if(isnum(argv[2])==1)errormsg(0,argv[1]);sscanf(argv[2],"%lf",&xtol);sscanf(argv[1],"%lf",&ttol);readfromstdin(&input);break; case3: if(isnum(argv[2])==1)errormsg(0,argv[1]);if(isnum(argv[3])==1)errormsg(0,argv[1]);sscanf(argv[3],"%lf",&xtol);sscanf(argv[2],"%lf",&ttol);if(isfile(argv[1])==1)errormsg(1,argv[1]); 74

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AppendixB(Continued) input.ptr=fopen(argv[1],"r");break; default: errormsg(0,argv[0]); }/*mainprogram*/countslips(xtol,ttol,&input);/*we'redonewiththis*/if(input.deleteme==1)remove(input.name);return0; } 75

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AppendixCLAMMPSinputles Thisisasampleinputleusedinourmolecular-dynamicssimu lations.Eachinput leismodiedby submit.sh toreplacetextenclosedbyangledbrackets\ hi "andsubmittedtotheSunGridEngine.Textenclosedbycurlybracket s\ fg "istobeeditted manuallybytheuser. ####template.in####FibonacciumFrictionSimulations##KeithMcLaughlin##SolidStateTheory##DepartmentofPhysics##UniversityofSouthFloridadimension3boundaryspp##twoperiodicboundaries##Metalunits:Angstroms,eV,picoseconds,protonmassunitsmetaltimestep0.002##They'reatoms!(norotationalDoF)atom_styleatomic##Atomiccoordinatesread_data{PATHTOSAMPLE}/##Typesofpotsusedpair_stylehybridtablelinear10000lj/cut6##1-3interactwith4viaLJpair_coeff1*34lj/cut0.052.672696154##1-3interactwitheachotherviatablepair_coeff1*31*3table{SC-POT-LOCATION}SC##somestuffdoesn'tinteractpair_coeff1*35nonepair_coeff4*54*5none##Defineregions##xloxhiyloyhizlozhi##UsingINFdoesn'tworkproperly,soIuse+/-300instead.regionQC_rigid_regionblock-300-24.00001-300300-3003 00unitsbox regionQC_tstat_regionblock-24-18.00001-300300-30030 0unitsbox regionQC_free_regionblock-180.00001-300300-300300un itsbox regionDUMP_regionblock-1810-300300-300300unitsbox##DefinegroupsgroupQCtype<=3groupTIPtype=4groupBOBtype=5##Wedon'tusenveonrigidbodies.groupnve_groupunionQCBOB 76

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AppendixC(Continued) ##Usedfordumpingatoms.weonlydumpregionsthatareinter esting groupDUMP_groupregionDUMP_region##Definebasedonatoms*INITIALLY*inspecifiedregions.##Itdoesn'tmatteriftheylatermoveout.groupQC_rigidregionQC_rigid_regiongroupQC_tstatregionQC_tstat_regiongroupQC_freeregionQC_free_region##NVEensemble.WewillcontrolTviathermostating.fixfix_nvenve_groupnve##MaketheTIPrigidfixTIP_rigidTIPrigidsingletorque*offoffoff##TethertheTIPtotheBOB;k=1fixTIP_tetherTIPspringcoupleBOB10000##SetforcesontheBOBtozerofixBOB_forceBOBsetforce000##SetforcesonthebaseoftheQCtozerofixQC_rigid_forceQC_rigidsetforce000##CalculatetemperatureofthedynamicregionoftheQCcomputetemp_QC_freeQC_freetemp##OutputOptions-forcompressionandrelaxationstepsthermo 20 thermo_stylecustomstepetotalc_temp_QC_free##COMPRESSION###################################### ############## ##thermostattheentireqc;relaxationtime=0.002timeste ps fixtstat_qc_allQClangevin000.002109232##Lowerthetip.10000timestepsper1A.velocity BOBset-0.0500unitsboxsumno run ##FULLRELAXATION###########################stoploweringthetip.velocity allset000unitsboxsumno ##relaxtheentireQCandTIPforawhile.unfixtstat_qc_allfixtstat_allalllangevin00198463##Run!run 10000 ##HEATTHESYSTEM###########################Removethethermostatfromthebobandrigidlayers. 77

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AppendixC(Continued) unfixtstat_all##HeatupthethermostatandfreelayersfixQC_0_T_tstatQC_tstatlangevin0{T_final}131222fixQC_0_T_freeQC_freelangevin0{T_final}131222##Run!run 10000 ##PREPAREFORSLIDING######################NowthatweareatT_finalunfixQC_0_T_tstatunfixQC_0_T_freefixQC_T_T_tstatQC_tstatlangevin300300200398128##Weonlywanttoconsiderperpendicularmotioninthetherm ostatapplication ##DIRECTIONcaneitherbe"01"or"10"computetemp_QC_tstat_partQC_tstattemp/partial1 fix_modifyQC_T_T_tstattemptemp_QC_tstat_part##slidingvelocityvelocity QC_rigidset0sumnounitsbox ##Run!run 20000 ##SLIDE###################################dumpatomcoordinatesdumpdump_xyzDUMP_groupcustom100001.dmpidxyztype##modifytheoutput.wewanttooutputtheforcesontheBOBthermo_stylecustomstepetotal\ c_temp_QC_freef_BOB_force[1]f_BOB_force[2]f_BOB_for ce[3] ##Movethetipthroughtheunitcellrun 2000000 Becausesuchalargenumberofrunsneedtobeperformedtodet erminethefriction coecients,weautomatethesubmissionprocessforseveral dierentcompressionsusing apairofBASHshellscripts, submit.sh and mpi sub.sh submit.sh preparestheinput lebymakingtheneccessarymodicationsto template.in andsetsuptheworking directory. #!/bin/bash##submit.sh##SUBMISSIONSCRIPTFORLAMMPSRUNSSERIES=RUNS 78

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AppendixC(Continued) ##Herewegivealistoftheapproximantsused##thefilenameforeachapproximantshould##be$APPROX.structAPPROXS="big-t11-Abig-t11-Bbig-t11-C"SLIDING_VELOCITY=0.01##Foreachapproximantforsamplein$APPROXS;do ##setthefilenameforthestructfilestruct=$sample.struct##createasubdirectoryforthisapproximantmkdir$samplecd$sample##wewillbeslidingintheyandzdirectionsfordirectioninyz;do if["$direction"="y"];then ##Settheslidingvelocity##andturnthethermostatoffalong##they-axisvel="$SLIDING_VELOCITY0"therm="01" fiif["$direction"="z"];then vel="00.01"therm="10" fi##Foreachcompression(intimesteps)forcompressionin20000400006000080000100000;do ##ifthedirectoryalreadyexists,skipitif[-d$direction\_$compression];then continueecho$direction\_$compressionskipped fi##ifitdoesnot,makeadirectoryforthe##slidingdirectionandcompressionmkdir$direction\_$compressioncd$direction\_$compression##preparetheinputfilebyreplacing##anglebracketedvariablessed-e"s||$therm|"\ -e"s||$vel|"\-e"s||$struct|"\-e"s||$compression|"\ <$CWD/template.in>run.in ##submittothegridengineviampi_sub.shecho-e"MD$direction\_$compression\n8\nlmp
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AppendixC(Continued) done donecd.. done Finally, submit.sh calls mpi sub.sh whichhandlestheactualsubmissionprocess. #!/bin/bash##mpi_sub.sh##generatesandsubmitssubmissionsciptsforGridEngineEMAIL={EMAILADDRESS}SHELL=/bin/bashLOG=sub.log##Isqstatinthepath?whichqstat2>/dev/null>/dev/nullif[!$?-eq0];then echo"error:qstatnotfound"1>&2exit1 fi##Checkcommandlineargumentsif[$#-eq4];then JOBNAME=$1NUMPROC=$2EXECUTE=$3HOWLONG=$4 else echo"JOBNAME?"readJOBNAMEecho"NumberofProcessors?"readNUMPROCecho"WhichBinary?"readEXECUTEecho"Howlong?(int)(hours)"readHOWLONG fi##createtheinputfiletmp=$(mktemp)cat>$tmp<
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AppendixC(Continued) ###Sendmemail#$-M$EMAIL#$-notify#$-mabe###Whichshell?#$-S$SHELL#$-oout.$JOBNAME.\$JOB_ID#$-peompi*$NUMPROC#$-lh_rt=$HOWLONG:00:00exportDEBUG_MPI=truesge_mpirun$EXECUTEEOF##Submittogridqsub<$tmp##Writethealoginpwdecho-ne`date`":$JOBNAME:">>$LOGtail-n1$tmp>>$LOG##Movethetempfiletopwdmv$tmp. 81

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AppendixDGeneratingFibonaccium AlthoughtherecipeforconstructingFibonacciumgivening iveninsection5.1is ratherstraight-forward,generatingthestructureleforL AMMPSisnotrivialtask. We'vesuppledthesourcecodeusedtocreatethesestructure lesforthesimple-cubic case.ThecodeexpectstheusertosupplytheFibonacciseque nceused,e.g.\LSLLS". /*create_fibs_sc.c **UniversityofSouthFlorida*Solid-StateTheory*KeithMcLaughlin*15JUL2009*/ #include#include#include#include#definebuffsize300#include"error.h"/*CreatesalammpsinputstructureforaBCCsampleswhosema ssesare* *modulatedaccordingtothefibonaccisequencealongtwoax es,andis* *periodicinthethird. */ /*Handlesallerrors*/voiderrormsg(interrcode,char*string){ if(errcode==0){ fprintf(stderr, "error:usage:%ssmall_massL/S_ratiolattice_constants equence\n" ,string); fprintf(stderr, "error:sequenceisgivenasastringofL'sandS's.Ex.\"LSL LS\"\n"); }if(errcode==1) fprintf(stderr,"error:L/S_ratiomustbegreaterthan1.\ n",string); if(errcode==2){ fprintf(stderr,"error:sequencemustconsistofonlyL'sa ndS's.\n"); fprintf(stderr,"error:yousupplied:%s.\n",string); }exit(1); } 82

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AppendixD(Continued) /*printheader*/intprintheader(char*sequence,double*doubles,intnato ms,doublecellsize) { printf("\"FibSquare\"BCCLattice%s,S=%lf,L/S=%lf, LatticeConst.=%lf\n",sequence,doubles[0],doubles[1] ,doubles[2]); printf("\n");printf("%datoms\n",natoms);printf("\n");printf("3atomtypes\n");printf("\n");printf("%lf0xloxhi\n",-cellsize);printf("%lf0yloyhi\n",-cellsize);printf("%lf0zlozhi\n",-doubles[2]);printf("\n");printf("Masses\n");printf("\n");printf("1%lf\n",doubles[0]*doubles[0]);/*S^2*/printf("2%lf\n",doubles[0]*doubles[0]*doubles[1]);/ *S*L*/ printf("3%lf\n",doubles[0]*doubles[1]*doubles[0]*do ubles[1]);/*L*L*/ printf("\n");printf("Atoms\n");printf("\n");return0; }/*generatestheheaderfortheoutputstructfile*/intgenheader(double*doubles,char*sequence){ char*sequence0=sequence;intnatoms=0;doublecellsize; /*Calc.thelengthoftheunitcellintheaperiodicdirectio n*/ while(*sequence0!='\0'){ natoms++;sequence0++; }cellsize=natoms*doubles[2];natoms=natoms*natoms;/*natomsinx&y,andtwolayersinz* / printheader(sequence,doubles,natoms,cellsize);return0; }/*generateandprinttheatomsfortheoutputstruct*/ 83

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AppendixD(Continued) intgenstruct(double*doubles,char*sequence){ intxflag,yflag;doublex=0;/*coordinatesxandy*/doubley=0;char*xseq=sequence;char*yseq=sequence;intnatom=1;intatomtype;doubles=doubles[0];/*smallmass*/doublel=doubles[0]*doubles[1];/*largemass*/doublea=doubles[2];/*latticeconst.*/while(*xseq!='\0') /*loopthroughthesequencewith2-variables->2dimension s*/ /*placeatomsinthelatticewithmassdependentontheseque nce*/ { switch(*xseq){ case'L':case'l':xflag=1;break;case'S':case's':xflag=0;break;default:errormsg(2,sequence); }/*nestedloop*/while(*yseq!='\0'){ switch(*yseq){ case'L':case'l': /*xflag+yflag+1=atomtype,ex.l+s+1=type2,l+l+1=type3 */ yflag=1;break;case'S':case's':yflag=0; 84

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AppendixD(Continued) break;default:errormsg(2,sequence); }/*addtwoatoms(bcc)*/atomtype=xflag+yflag+1;printf("%d%d%lf%lf%d\n",natom,atomtype,x,y,0);natom++; /*incrementybythelatticeconstant,forthenextatom*/ y+=a; /*movetothenextcharinthesequence*/ yseq++; }y=0;/*resety=0,forthenewvalueofx.*/x+=a;yseq=sequence;xseq++; }return0; }intmain(intargc,char*argv[]){ doublemass;doublemassratio;doublelength;doubledoubles[3];charsequence[buffsize];if((argc-1)!=4)errormsg(0,argv[0]);if(isnum(argv[1])==1)errormsg(0,argv[0]);if(isnum(argv[2])==1)errormsg(0,argv[0]);if(isnum(argv[3])==1)errormsg(0,argv[0]);mass=strtod(argv[1],NULL);massratio=strtod(argv[2],NULL);length=strtod(argv[3],NULL);/*tosimplifythepassingofvariables*/doubles[0]=sqrt(mass);doubles[1]=massratio;doubles[2]=length;/*MakesureLong>Short*/if(massratio<1)errormsg(1,NULL); 85

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AppendixD(Continued) sprintf(sequence,"%s",argv[4]);genheader(doubles,sequence);genstruct(doubles,sequence);return0; } 86