Development of Stochastic Models for Dynamics of Self-Assembled Surfactant Systems

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Title:
Development of Stochastic Models for Dynamics of Self-Assembled Surfactant Systems
Physical Description:
1 online resource (159 p.)
Language:
english
Creator:
Ahn, Yong Nam
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Chemical Engineering
Committee Chair:
Kopelevich, Dmitry I
Committee Members:
Chauhan, Anuj
Butler, Jason E
Sinnott, Susan B

Subjects

Subjects / Keywords:
dynamics -- modeling -- stochastic -- surfactant -- transport
Chemical Engineering -- Dissertations, Academic -- UF
Genre:
Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
Understanding dynamic processes in surfactant self-assembled systems is crucial in various technological applications. The goal of this study is to develop stochastic models for dynamic processes in surfactant systems using coarse-grained molecular dynamics simulations.  Two types of processes are considered. The first of the investigated processes is molecular transport across oil-water interfaces covered by nonionic surfactants. Resistance of the surfactant monolayer to the solute transport is shown to be controlled by dense regions in the monolayer. Resistance to the transport of a hydrophobic (hydrophilic) solute increases as the length of the surfactant head (tail) group increases. Barriers for solute transport through surfactant monolayers are also influenced by the solute size. The non-adiabatic coupling of the monolayer with the solute position and configuration causes deviations of the system dynamics from the minimum energy path, which effectively leads to an increase of the barrier for the solute transport.  The second type of the considered processes is micellar self-assembly and disaggregation. We present a detailed model for two of the elementary steps involved in self-assembly of surfactants, namely addition/removal of a single surfactant molecule to/from a spherical micelle. Multi-dimensional free energy landscapes parametrized by monomer and micellar degrees of freedom are obtained using a series of constrained simulations. We observe that the system trajectory on the free energy landscape is multi-dimensional and cannot be reduced to motion along a one-dimensional path on this surface. In order to elucidate the collective dynamics of the multiple degrees of freedom, the most likely path is identified on the free energy landscapes.  Since the majority of natural and artificial surfactants are ionic, we also study the effects of electrostatic interactions on self-assembly of ionic surfactants. By solving the Poisson equation, it is shown that electrostatic potentials effectively increase (decrease) energy barriers of the addition (removal) of a surfactant molecule to (from) a micelle.  It is anticipated that the approaches discussed in this study can be extended to investigation of more complex dynamic processes in amphiphilic systems.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Yong Nam Ahn.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Kopelevich, Dmitry I.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-02-28

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Firstofall,Iwanttoexpressmydeepestgratitudetomyadvisor,Prof.DmitryIKopelevich.Heintroducedmetothefundamentalstudiesofsurfactantsystemsinmolecularlevelandtaughtmetorealizetheimportanceoffundamentalunderstandinginscienticresearches.HisconsistentguidanceandencouragementallowedmetocompletemyPhDstudy.DuringmyPhDcourse,IwasfortunatetoservetwosemestersasteachingassistantforthecoursesofElementaryTransportPhenomenaandComputermodelformulationsbyProf.AnujChauhanandProf.JasonEButler,respectively.Throughtheirguidance,Icouldenjoyteachingundergraduatestudentsanddevelopmyteachingskill.Theyalsogavemealotofimportantcommentsregardingmyresearch.IalsothankProf.SusanBSinnottforreadingmyresearchreports.IconveyspecialacknowledgementtoProf.SangHeupMoonatSeoulNationalUniversity.Heshowedmeastandardofanhonorablechemicalengineerthroughouthislife.IhopethatIwouldbecomeasgoodachemicalengineerasProf.Moon.ManythanksgotomyfriendsintheChemicalEngineeringDepartmentattheUniversityofFlorida,includingbutnotlimitedtoRobertColmayer,MarkPepple,WeiCheng,JunWu,HyunJungJung,GunjanMohan,AhishGupta,Chia-YiChen,andYoung-minBanfortheirfriendshipandsupport.Ithankmyparentsandsisterfortheirunaggingbeliefandloveforme.Myparentsandsisterdeservespecialmentionfortheirsupportandprayers.Iamalsogratefultomyparents-in-lawfortheirfaithfuladvice.Noneofmyachievementswouldhavebeenpossiblewithouttheloveandsupportofmywife,Jeeyoung.WheneverIwasmentallyandphysicallyexhaustedwithhardresearch,herencouragementgavemeanewdrivingforcetogoforward.Iexpressmyheartfulgratitudetoher.Mostimportantly,IwouldliketothankGodforthecompletionofmyPhDdissertation. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTIONS .................................. 14 1.1Background ................................... 14 1.2SpecicAims .................................. 15 1.3OverviewofDissertation ............................ 15 2METHODS ...................................... 18 2.1MolecularDynamicsSimulations ....................... 18 2.2Coarse-GrainedMolecularDynamicsSimulations .............. 21 2.3StochasticModelandConstrainedMDSimulations ............. 24 3MOLECULARTRANSPORTTHROUGHSURFACTANT-COVEREDOIL-WATERINTERFACE ..................................... 27 3.1Background ................................... 27 3.2ModelandSimulationsdetails ......................... 28 3.2.1CGMDMolecularModel ........................ 28 3.2.2DividingSurfaceofaMonolayer .................... 30 3.2.3PotentialofMeanForceandMinimalEnergyPath .......... 31 3.3MonolayerStructure .............................. 32 3.4BarrierstoSoluteTransport .......................... 34 3.4.1EffectsofSurfactantLengthonTransportBarriers .......... 34 3.4.2ComparisonofTransportBarriersforHydrophobic,HydrophilicandAmphiphilicSolute ......................... 36 3.4.3TransportBarriersforHydrophobicandHydrophilicOligomers ... 37 3.5DynamicBarriersforSoluteTransport .................... 45 3.5.1OrientationoftheAmphiphilicSolute ................. 45 3.5.2DynamicSolute-InterfaceCoupling .................. 50 3.5.3EffectsoftheSystemSize ....................... 61 3.6Discussion ................................... 65 3.6.1ValidationoftheSimulationTime-step ................ 65 3.6.2DemonstrationofIndependenceofResultsofConstrainedSimulationsoftheMethodofPreparationofInitialConditions .......... 66 3.7Summary .................................... 67 5

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................. 72 4.1Background ................................... 72 4.2ModelandSimulationsdetails ......................... 74 4.2.1EquilibriumMicelles .......................... 74 4.2.2CharacterizationofMicellarandMonomerCongurations ..... 75 4.2.2.1Principalradiianddirectionsofgyration .......... 75 4.2.2.2Micellarshells ........................ 76 4.2.3StochasticModelforMonomerAddition/Removal .......... 77 4.2.4ConstrainedSimulations ........................ 78 4.2.5UnconstrainedSimulations ...................... 80 4.2.6ChoiceofTranslationalDegreeofFreedom ............. 81 4.3Results ..................................... 82 4.3.1One-dimensionalFreeEnergyProle ................. 82 4.3.2CouplingbetweenMicellarShapeandMonomerMotion ...... 87 4.3.3AlternativeModelforMicellarStructure:HydrophobicPatches ... 101 4.3.4MonomerOrientation .......................... 107 4.4Discussion ................................... 112 5EFFECTSOFELECTROSTATICINTERACTIONSONFORMATIONANDDISINTEGRATIONOFSPHERICALMICELLES ................. 117 5.1Background ................................... 117 5.2ModelandSimulationsdetails ......................... 118 5.2.1CGMDMolecularModels ....................... 118 5.2.2PropertiesofEquilibriumMicelles. .................. 120 5.3FreeEnergyProles .............................. 122 5.4MonomerOrientations ............................. 124 5.5MicellarShape ................................. 125 5.6ElectrostaticInteractions ............................ 130 5.6.1AnalysisMethods ............................ 130 5.6.2ContributionofESPotentialtoForcesactingontheMonomer .. 133 5.6.2.1Averagechargedensity ................... 133 5.6.2.2Averageelectrostaticpotential ............... 133 5.6.2.3Timescaleofelectrostaticforceuctuations ........ 136 5.6.3EffectofChargeStrength ....................... 136 5.7Summary .................................... 138 6EFFECTSOFMULTI-DIMENSIONALTYOFFREEENERGYLANDSCAPEONRESULTSOFCONSTRAINEDSIMULATIONS ................ 142 7CONCLUSIONSANDDIRECTIONSOFFUTURERESEARCH ......... 146 REFERENCES ....................................... 152 BIOGRAPHICALSKETCH ................................ 159 6

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Table page 3-1BarriersG(inkBT)forsolutetransportacrosstheC3P3andC7P7monolayers. 35 3-2MeantransporttimefromzG,maxtozh,max(inpicoseconds). ............ 58 3-3Meantransporttimefromzh,maxtozG,max(inpicoseconds). ............ 58 7

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Figure page 3-1Schematicrepresentationofsolutesandsurfactantsconsideredinthecurrentwork. ......................................... 29 3-2Densityproleofahexadecane-waterinterfacecoveredbyC7P7surfactants. 32 3-3Comparisonofdensityprolesinvarioussurfactantmonolayers. ........ 33 3-4PotentialofmeanforceG(zs)actingonthehydrophobicmonomerC. ...... 36 3-5FreeenergyprolesforsolutesP,C,andCPintheC3P3andC7P7monolayers. 37 3-6FreeenergyprolesofhydrophobicandhydrophilicoligomersintheC3P3andC7P7monolayers. .................................. 38 3-7Contributions~G(zs,)oftheorientationofsoluteC2tothetotalfreeenergy. 40 3-8Roleofthesoluteandsurfactantsizesincreatingatransportbarrierforadimer. 41 3-9Contributions~Gi(zs,i)oforientationsofbondsC5Ci(i=1,...,4)tothetotalfreeenergyofthebranchedpentamerC5. ..................... 43 3-10Roleofthesoluteandsurfactantsizesincreatingatransportbarrierforabranchedsolute. ......................................... 44 3-11Contributions~G(zs,)oftheorientationoftheCPsolutetothefreeenergy. 46 3-12RatioR(z0)=(z0)=0(z0)ofthemeantimesofadsorptionofCPattheinterfacewithandwithouttheconstraintonthesoluteorientation. ............. 49 3-13ACFsoftherandomforce\(zs;t)actingontheC2soluteintheC3P3monolayer. 50 3-14Correlationtimesf(zs)oftherandomforcesactingonthesolutesinC3P3andC7P7monolayers. ................................ 52 3-15DependenceoftheaverageFouriermodes^h(0)kwithwavenumberk=0.65nm1onthesolutepositionzsforvarioussolutes. .................... 53 3-16DeformationoftheC3P3monolayercausedbytheC5solute. .......... 54 3-17Cross-sectionofthefreeenergyproleG(zs;f^hkg)oftheCsoluteintheC7P7monolayer. ...................................... 56 3-18CorrelationtimeskofthenormalmodesoftheC3P3andC7P7monolayers. 56 3-19PMFactingontheCsoluteintheC3P3monolayerobtainedfromsimulationsinthesmall(LLL)andlarge(2L2L2L)systems. ........... 62 3-20LargestinterfaceprotrusionsinducedbytheCsoluteintheC3P3monolayer. 62 8

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........... 63 3-22ComparisonofresultsofconstrainedsimulationsofthehydrophobicmonomerCintheC3P3monolayerperformedwithtime-stepsof40fsand10fs. ..... 66 3-23PotentialofmeanforceactingonP2soluteintheC3P3monolayerandC2soluteintheC7P7monolayer. ................................ 68 3-24CorrelationtimeoftheconstraintforceactingonP2soluteintheC3P3monolayerandC2soluteintheC7P7monolayer. ........................ 68 3-25DependenceoftheaverageFouriermodes^h(0)kwithwavenumberk=0.65nm1onthelocationofthesolutecenterofmass. .................... 69 4-1SchematicrepresentationofasingleC4P4surfactantandaninstantaneouscongurationoftheC4P4micelle. .......................... 75 4-2PotentialofmeanforceG0()andcorrelationtimeoftherandomforce)]TJ/F3 11.95 Tf 5.32 1.79 TD[(()actingonthereactioncoordinate. ......................... 83 4-3RepresentativeautocorrelationfunctionsC)]TJ/F3 11.95 Tf 5.32 1.79 TD[((;t)oftherandomforce\(t;)actingonthemonomerpartiallyinsertedintothemicelle. ............ 84 4-4Dependenceofthefrictioncoefcientonthereactioncoordinate. ...... 86 4-5Distributionofthemonomeradditiontimesobtainedfromtheunconstrainedsimulations. ..................................... 87 4-6AverageradiiofgyrationhRg,iiandaverageorientationscosioftheprincipaldirectionsdg,iofgyrationwithrespecttovectorRC. ................ 88 4-7Schematicrepresentationofeffectsofamonomeronthemicellarshapeduringthemonomeradsorption/desorption. ....................... 88 4-8AveragevaluesoftheLegendremodesforshellsS(C1),S(C4),S(P1),andS(P4). ......................................... 89 4-9ExtremaL(i)k,peakoftheaverageLegendremodes~L(i)k()inthecriticalregion. .. 92 4-10StandarddeviationsoftheLegendremodesforshellsS(C1),S(C4),S(P1),andS(P4). ...................................... 93 4-11StandarddeviationsoftheLegendremodesofanequilibriummicelles. ..... 94 4-12CorrelationcoefcientsC(i,j)k1,k2betweenLegendremodeswiththesamewavenumberkbutcorrespondingtoshellsS(i)andS(j)ofanequilibriummicelle. ...... 95 4-13-dependenceofcorrelationcoefcientsC(i,i)k1,k2betweenmodeswithdifferentwavenumberscorrespondingtothesameshellS(i). ............... 97 9

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...................................... 98 4-15AutocorrelationfunctionsoftherstfourLegendremodesoftheradiusofmicellarcore. ......................................... 99 4-16Correlationtimes(i)kfortheLegendremodesofshellsS(i)ofthemicelleatequilibrium. ...................................... 100 4-17Contributionofthemicellardeformationtothefreeenergy. ........... 101 4-18ProbabilitydistributionsP(A)oftheareaAofthecentralpatches. ....... 103 4-19Representativetimedependenceofthecentralpatchareacorrespondingtothemonomerconstrainedat=c. ........................ 105 4-20Systemcongurationscorrespondingtopointst=taandt=tbofFig. 4-19 106 4-21ACFsanddependenceofthecorrelationtimesofthecentralpatchareaonthemonomerposition. ............................... 107 4-22Contributions^G(,cosi)oftheorientationscosioftailandhead-groupofthemonomertothefreeenergy. .......................... 108 4-23CorrelationcoefcientbetweenorientationscosTandcosHofthemonomertail-andhead-groups. ................................ 109 4-24Projectionofatrajectoryobtainedfromtheunconstrainedsimulationsontothe-cosTplane. ................................. 110 4-25ContributionsG(dyn)(,cosi)oftheorientationscosioftailandhead-groupofthemonomertothedynamicfreeenergy. .................... 110 4-26CorrelationtimesandACFsofthemonomerDOFs. ............... 112 4-27CorrelationcoefcientsbetweenorientationcosTofthemonomertailandthemicellarLegendremodesL(i)2. ......................... 115 5-1Coarse-grainedmodelsofnonionicandionicsurfactants. ............ 119 5-2ExampleofaninstantaneouscongurationofC3P1andSDSmicelleA40. ... 121 5-3DensityprolesofC3P1andSDSmicellesofaggregationnumber40. ..... 122 5-4Paircorrelationfunctionofhead-groupsofC3P1andSDSmicellesofaggregationnumber40. ...................................... 122 5-5ComparisonofG0()inC3P1andSDSsystems. ................. 123 5-6Contributions^G(,cosT)ofthemonomerorientationstofreeenergyinC3P1andSDSmicellarsystem. .............................. 124 10

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.......................... 125 5-8AveragevaluesoftheLegendremodesforallshellsofC3P1andSDSmicelles. 127 5-9ThedifferenceL(i)k,peakofL(i)k,peakbetweenC3P1andSDSmicelles. ........ 128 5-10StandarddeviationshL(i)kiofLegendremodesL(i)kofequilibriumSDSandC3P1micelles. .................................... 129 5-11DependenceofBLmodes~lmforl=2onforhead-groupchargesandcounterioncharges. ........................................ 134 5-12ESpotentialscausedbytotalESeld,head-groupchargesandcounterioncharges. ........................................ 135 5-13FreeenergyproleofSDSmicellarsystemwithoutESpotential. ........ 135 5-14CorrelationtimesandstandarddeviationsofESforceandtotalforce. ..... 137 5-15PMFsofSDS-A40micellarsystemswith0.7eand1.0echargestrengths. ... 138 5-16Chargedensities(r,)forhead-groupsandcounterionswith0.7eand1.0echargestrengths. ................................... 139 5-17ESpotentials<(r,)>for0.7eand1.0echargestrengthsalongthepolaraxis. .......................................... 140 6-1PMFsactingonamonomerpartiallyinsertedintoanSDSmicellewithconstrainedheadbeadandCOMofthemonomer. ....................... 142 6-2Contributionsofthemonomerorientationtothefreeenergy^GC(C,)and^GH(H,). ....................................... 144 6-3ComparisonbetweenGC(C,)and~G(C,). ................... 145 7-1ResultsofconstrainedMDsimulationsofanSDSsurfactantinaDPPClipidbilayer. ......................................... 149 7-2AdsorptionofC4P4micelleintooil-waterinterface. ................ 151 11

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1 4 ].Therateofsolutetransportacrossadropletsurfaceiscontrolled,inpart,byasurfactantmonolayeradsorbedatthesurface.Theresistanceofthesurfactantmonolayertothesolutetransportisexpectedtobedeterminedbymultiplefactors,includingthesoluteandsurfactantsizeandstructure,densityofthemonolayer,andcurvatureoftheinterface.Inthisstudy,weinvestigatemechanismsofmoleculartransportacrossoil-waterinterfacecoveredbynonionicsurfactants.Theotherdynamicprocessstudiedinthisworkistheformationofsphericalmicelles.Inaqueoussolutions,surfactantsinamicelleareorientedwiththeirhydrophobictailstowardtheinteriorinordertominimizetheirexposuretowater.Micellarstructuredependsonseveralfactorssuchassurfactantstructure,surfactantconcentration,saltconcentration,temperature,pressureandexternalelds[ 5 ].Dynamicsofself-assembly 14

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4 6 8 ].Arecentstudy[ 9 ]demonstratedthatevensuchasimpleprocessasadditionofasinglesurfactantmonomertoamicelleinvolvesacomplexinterplaybetweenmicellarandmonomercongurations.Inthisstudy,weinvestigateeffectsofthesedegreesoffreedomonadditionofasurfactantmonomertoamicelle.Effectsoflong-rangeelectrostatic(ES)forceonmicellarstructureandself-assemblyprocessarealsostudied. 2 ,themethodsusedinsimulationsandanalysisoftheconsideredsystemsareintroduced.Inordertospeedupthesimulationsandfocusonthemainqualitativefeaturesofthesystems,acoarse-grainedmolecularmodel[ 10 ]isusedto 15

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3 ,mechanismsofsolutetransportthroughsurfactantmonolayersatoil-waterinterfacesareinvestigated.EnergybarriersalongMEPsarecalculatedforvarioussolutesandsurfactantmonolayers.Effectsofsolutecongurationsontheseenergybarriersarealsoexamined.Weshowthatattractiveinteractionbetweenthesoluteanditsfavorablephasecausessignicantchangesinthemonolayershapeinresponsetothesoluteposition.Thissolute-monolayercouplingprovidesanadditionalenergybarriertothesolutetransportcalledthedynamicenergybarrier.InChapter 4 ,weinvestigaterolesofvariousdegreesoffreedomonaddition/removalofasurfactantmonomerto/fromanonionicsphericalmicelle.First,thereactioncoordinateisdenedasthedistancebetweenthemonomerandthemicelleandafreeenergyproleisobtainedasafunctionofthereactioncoordinate.Thenweobtainmulti-dimensionalenergylandscapesparameterizedbythereactioncoordinateaswellasthemonomerorientationandmicellarshapeandassesseffectsoftheseadditionaldegreesoffreedomonthemonomeradsorptionintothemicelle.InChapter 5 ,effectsoflong-rangeESforceinanionicmicellarsystemisinvestigated.InordertoisolateESeffects,anionicmicelleiscomparedwithanonionicmicellewiththesameaggregationnumber.Withexceptionofthehead-groupcharges,structuresofthenonionicandionicsurfactantsareidentical.Duetothelong-rangeESforce,structureoftheionicmicellediffersfromthatofthenonionicmicelle.BysolvingPoissonequationwiththechargedensityobtainedfrommoleculardynamicssimulations,weexaminetheeffectsoftheESinteractionsontheionicmonomeradditionandremovalprocesses.InChapter 6 ,webrieydiscusseffectsofmulti-dimensionalityoffreeenergylandscapesonresultsofconstrainedsimulations.Itisshownthatconstrainingdifferent 16

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7 concludesthedissertationandpresentsabriefdiscussionoffutureresearchdirections. 17

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2 usingtheVerletleap-frogalgorithm[ 11 ], 2 andEq. 2 correspondtosolutionofNewton'sequationsofmotion,thetotalenergyofthesystemremainsconstant,whilethetemperatureofthesystemuctuatesduetochangesofkineticenergy.However,inrealphysicalsystems, 18

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12 13 ].Thisthermostatintroducesathermalreservoirandafrictiontermintheequationsofmotion(Eq. 2 ), 14 15 ]whichreplacesEq. 2 by Here,Misthefrictionparameterandmatrixbobeysthefollowingequationofmotion: 2NdfXi=1miv2i.(2) 19

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2NX1mivivi(2)and 2Xi
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2 intoashort-rangeandlong-rangepartsevaluatedintherealandreciprocal(Fourier)spaces,respectively, whereisaparameterthatdeterminestherelativeweightofthedirectandreciprocalsumsandm=(mx,my,mz)isawavevector.PerformanceofDiscreteFourierTransformation(DFT)usedinthereciprocalsummationforNchargesscalesasN2.TheperformanceisfurtherimprovedbytheParticleMeshEwald(PME)summationmethod[ 16 17 ]usedinthisstudy.InPME,thechargesareassignedtoagridusingcardinalB-splineinterpolation,thenFastFourierTransformation(FFT)isusedinthereciprocalsummation.TheperformanceofFFTscalesasNlnN,whereasthatofDFTscalesasN2whichsignicantlyreducesthecomputationalcostofthereciprocalsummationincomparisonwithEwaldsummation.AllsimulationsinthisstudyareperformedusingtheGROMACSMDpackage[ 18 ]. 10 ].Thismodelhasbeenshowntoaccuratelyreproduceanumberofphysicalpropertiesofwaterandalkanes,includingtheirdensitiesand 21

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19 22 ].However,wehavechosentousetheoldermodel[ 10 ]inthecurrentworkduetoitsrelativesimplicity,whichenablesustofocusongeneralqualitativefeaturesoftheconsidereddynamicprocessesinamphiphilicsystems.Inthecurrentwork,allmoleculesaremodeledusingtwotypesofcoarse-grainedbeads:hydrophobictailbead(denotedhereasC)andhydrophilicheadbead(denotedasP).OneCbeadrepresentsfourmethyleneormethylgroupsinanalkanechainandonePbeadrepresentsfourwatermolecules[ 10 ].Followingtheearlierworkofourgroup[ 9 23 24 ],wealsousethePbeadstorepresenthead-groupsofnonionicsurfactants.AsinglePbeadisexpectedtoapproximatetwoethoxygroupsand,hence,amodelCmPnsurfactantisexpectedtoapproximateanethoxylatedsurfactantC4mEO2n.Head-groupofanionicsurfactantandthecorrespondingcounterionconsideredinthisworkarealsomodeledbyasinglePbeadwithnegativeandpositivecharges,respectively.Thiscoarse-grainedmodelisexpectedtoapproximatethesodiumdodecylsulfate(SDS)ionicsurfactantwithsodiumcounterion.Interactionsbetweenchemicallyconnectedbeadsaremodeledbyharmonicpotentialsforthebondlengthandanglevibrations, 2Kbond(RRbond)2, 2Kangle(coscos0)2. ForceconstantsforthebondlengthandanglepotentialsareKbond=1250kJmol1nm2andKangle=25kJmol1rad2,respectively.TheequilibriumbondlengthRbondis0.47nmforallbondsandtheequilibriumbondangle0isdeterminedbythemoleculargeometry.Non-bondedinteractionsbetweenchemicallyconnectedbeadsareexcluded. 22

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23 ].Thediscrepancybetweenthemodelpredictionsandtheexperimentalvaluesislessthan25%.Aquantitativecomparisonbetweenthecomputedandexperimentallymeasuredsurfacetensionofsurfactantmonolayersisdifcultduetoanextremesensitivityofthesurfacetensiontosmallchangesofthesurfacecoveragewhenthelatterapproachesitsmaximumvalue[ 23 25 ].However,qualitativetrendspredictedbythecurrentmodelareinagreementwiththeexperimentalobservations[ 23 ].Specically,itisobservedthatincreasingthesurfacecoveragebysurfactantsleadstoloweringofthesurfacetension.Similarly,increasingthesurfactantlengthwhilekeepingthesurfacecoverageconstantleadstoloweringofthesurfacetension.Thismodelunderestimatesthecriticalmicelleconcentration(CMC)ofethoxylatedsurfactantsbyafactorofve[ 9 ].SandersandPanagiotopoulos[ 26 ]demonstratedthatabettermatchwiththeexperimentalCMCcanbeobtainedifoneincreasesthestrength 23

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27 ], 28 29 ], 2 24

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30 ].ThefreeenergygradientG0()correspondstotheaverageoftheforceFc(;t)necessarytoconstrain[ 31 ], 2 2 canbesolvedanalyticallytoobtainthetransitiontimesbetweendifferentmetastablestates.However,processesinamphiphilicsystemsdonotalwayssatisfythemainassumptionoftheone-dimensionalLangevinequation,namelyalargeseparationbetweenthetimescalesofthereactioncoordinateandthethermalbath.Inanearlierstudyofsolutetransportacrossaliquid-liquidinterface,ourgroupshowedthatthesolutetranslationalmotioniscoupledwithinterfaceuctuationsofcomparabletimescales[ 24 32 ].Thisrequiresonetoexplicitlyaccountfordynamicsoftheinterfaceand,hence,considerLangevinequationwithmultipledegreesoffreedom.Inthisdissertation,wedemonstratethatsimilarcouplingbetweendifferentdegreesoffreedomofcomparabletimescalesoccurforawiderangeofself-assemblyandtransport 25

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1.1 ,masstransferacrosssurfactantmonolayerscoveringmicroemulsiondropletshasnumerousapplication.However,understandingmolecularmechanismsofthistransportiscurrentlylimited.Mostmolecularmodelingstudies[ 33 36 ]ofsurfactant-coveredoil-waterinterfacesarefocusedonsuchpropertiesofsurfactantmonolayersasinterfacialtension,lateraldiffusionofsurfactantmoleculeswithinthemonolayer,andmonolayerstructure.Intheearlierstudies[ 23 24 ],ourgroupemployedCGMDsimulationstoexploreresistanceofsurfactantmonolayerstotransportofarelativelysimplesphericalhydrophobicsolute.Itwasshownthat,inadditiontohydrophobic/hydrophilicinteractions,theheightoftheenergybarriertothesolutetransportiscontrolledbystericeffectscausedbyvariationofthemonolayerdensityalongthepathofthesolute[ 23 ].Thedensityvariationincreaseswiththelengthofthesurfactantscomprisingthemonolayer,thusofferingoneofthemethodstocontrolthetransportrate.Inadditiontothisstaticbarrier,themonolayercreatesadynamicbarriertothesolutetransport[ 24 ].Thisbarrierarisesduetocouplingbetweenthesolutetranslationaldegreesoffreedomandthermaluctuationsoftheinterface.Thetimescaleoftheinterfaceundulationsiscomparablewiththetimescaleofthesolutemotion.ThisleadstoasubstantialdeviationofthesystemfromMEPasthesolutecrossestheinterface.Inordertocorrectlypredictthesolutetransportrate,itisnecessarytomodelthemotionofthesystemonthemulti-dimensionalfreeenergylandscapeparameterizedbyreactioncoordinatescorrespondingtothesolutetranslationaldegreesoffreedomandthemonolayershape.SuchdeviationsfromtheMEPmaybecomeevenmoresignicantfortransportofnon-sphericalsolutes,sinceinadditiontothesolutetranslationaldegreesoffreedom,thesolutecongurationmayalsocouplewiththedynamicsoftheinterface. 27

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10 ]toexploreeffectsofthesoluteandsurfactantsizeandstructureontheresistanceofthesurfactantmonolayerstothesolutetransport.Surfacesofthemicroemulsiondropletsaremodeledbyatoil-waterinterfacescoveredbynon-ionicsurfactants.Thelengthofthesurfactanthead-andtail-groupsisvaried,whilethemonolayerdensityiskeptconstant.Transportofhydrophobic,hydrophilic,andamphiphilicsolutesofseveralsizesandshapesisconsidered.ThegoalofthisChapteristosystematicallyexploreboththestaticanddynamicbarriersforvarioussolutesandmonolayers.Completereconstructionofthefreeenergylandscapeforthesecomplexsystemsisoutsidethescopeofthecurrentwork.MostoftheanalysisisperformedundertheassumptionthatthesystemdynamicsfollowsanMEP.Althoughthisassumptionisnotaccurateforsomesegmentsofthesolutetrajectory[ 24 ],thisanalysiswillenableustoidentifythesegmentsalongwhichtheassumptionfailsandobtaininformationnecessarytoamendthemodeltoaccountforthesystemdeviationfromtheMEP. 3.2.1CGMDMolecularModelWeconsidermoleculartransportacrossatwater-hexadecane(P-C4)interfacescoveredbysurfactantmonolayers.StructuresofsurfactantsandsolutesconsideredinthisworkareshowninFig. 3-1 .Inordertoassesseffectsofthesurfactantlengthonthesolutetransportrate,weperformadetailedanalysisoftransportacrossC3P3andC7P7monolayers.Inaddition,weevaluateeffectsofthesurfactantasymmetryonthesolutetransportbyperformingalimitedanalysisoftransportthroughC3P7andC7P3monolayers.Wecomparetransportpropertiesofhydrophilicandhydrophobicmonomersanddimers(P,C,P2,C2),anamphiphilicsoluteCP,andabranchedhydrophobicpentamerC5.Equilibriumbondanglesare90inC5and180inallotherconsideredmolecules.Inwhatfollows,wewillsometimesrefertosolutescomposed 28

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Schematicrepresentationofsolutesandsurfactantsconsideredinthecurrentwork.TheblackandwhitespheresrepresenttheCandPbeads,respectively. ofonetypeofbeads(i.e.,PorC)ashomogeneoussolutestocontrastthemwiththeamphiphilicsoluteCP.TheMDsimulationboxsizeis9.79.79.7nm3forsystemscontainingC3P3monolayersand9.79.726.2nm3forallotherconsideredsystems.Inordertoinvestigateeffectsofthesystemsizeontheobtainedresults,weperformanadditionalsetofconstrainedsimulationsoftheCsoluteintheC3P3monolayerusingthe19.419.419.4nm3simulationbox.Thepressureiskeptat1barandthetemperatureiskeptat300K,whichishigherthanthemeltingtemperature(290K)oftheconsideredcoarse-grainedwatermodel[ 10 ].Thesolventsareobservedtoremainliquidinallsimulations.Theintegrationisperformedwiththetime-stepof40fs.Thistime-stepiswithintherange(30to50fs)proposedintheoriginalworkofMarrinketal.[ 10 ].Ithasbeenrecentlyindicated[ 37 ]thatthistime-stepistoolargetoprovideareliableestimateofcertaindynamicproperties,suchasdiffusivityofsmallmolecules.However,ithasalsobeenobservedthatthisstep-sizeissufcienttoprovideaccurateinformationregardingthestructuralandenergeticproperties[ 37 ],i.e.propertiesofmaininterestinthecurrentwork.Inordertofurtherjustifyouruseofthe40fstime-step,weperformedanadditionalsetofsimulationswithasmaller(10fs)time-step.Resultsofthesesimulationsare 29

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3.6.1 .Theseresultsindicatethatthestructural,energetic,andevendynamicpropertiesofinterestinthecurrentworkcanbeaccuratelycomputedusingthelarger40fstime-step.Preparationofthesurfactant-coveredinterfacesisdiscussedindetailelsewhere[ 23 ].Allmonolayersconsideredinthecurrentworkhavethesamesurfacecoverage,2.0molecules/nm2,whichenablesustofocusoneffectsofthesoluteandsurfactantpropertiesonthetransportprocess.Atthiscoverage,theinterfacialtensionoftheC3P3,C7P3,C3P7,andC7P7monolayersis24,18,20,and10mN/m,respectively. 24 ],i.e.surfacespassingthroughmid-points(xj(t),yj(t),zj(t))ofbondsconnectingthesurfactanttail-andhead-groupsattimet.Inordertoinvestigatecouplingbetweenthemonolayerdeformationandthesolutetransport,wewillneedtoanalyzeFouriertransformsoftheinstantaneousdividingsurfaces.InstantaneousFourierharmonics^hk(t)areobtainedbyaleastsquarestoftheinstantaneousdividingsurfacetothetruncatedFourierseries, 3 ,Aistheareaoftheprojectionofthemonolayerontothexyplane,k=(kx,ky)isawavevector,and 30

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28 38 ]withthereactioncoordinatezsasdiscussedinSection 2.3 .Initialconditionsfortheconstrainedsimulationsaregeneratedbyintroducingthesoluteintotheoilphaseandthenpullingitacrossthemonolayerbyapplyinganarticialforce.Theconstrainedsimulationsforeachofthesolutelocationszsareperformedfor400nsandthestatisticalanalysisisperformedafter10nsofequilibration.Afterinitialtransient,theconstrainedsimulationsrelaxtowardstheMEP,i.e.towardsstatesminimizingthesystemfreeenergyforaspecicsolutelocationzs.Therefore,inadditiontothePMF,theconstrainedsimulationsallowustoobtaintheMEPandexploreitsneighborhoodbyanalysisofsteady-statethermaluctuationsofthesystem.Aswillbeshownbelow,therelaxationtimescalesoftheconsideredsystemsareontheorderof1ns,whichimpliesthatthe10nsequilibrationtimechoseninthecurrentworkissufcienttoensurethatthesystemhasrelaxedtowardstheMEPpriortotheproductionrun.Inordertofurtherconrmthat10nsisasufcientequilibrationtime,asetofadditionalsimulationsisperformed.Specically,fortwooftheconsideredsystems(P2inC3P3andC2inC7P7)weperformconstrainedsimulationswithinitialconditionspreparedbypullingthesolutefromwatertooil,i.e.inthedirectionoppositetothedirectionusedintheoriginalsimulations.ResultsofthesesimulationsarepresentedinSection 3.6.2 .Itisobservedthatthekeyenergeticandstructuralsystemcharacteristicsmeasuredinthecurrentwork,namelythepotentialofmeanforce,correlationtimeof 31

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23 ].Arepresentativedensityproleofasurfactant-coveredinterfaceisshowninFig. 3-2 anddensityprolesofallconsideredmonolayersarecomparedinFig. 3-3 Figure3-2. Densityproleofahexadecane-waterinterfacecoveredbyC7P7surfactants. Thesurfactantbondsclosesttothedividingsurfacetendtoorientthemselvesinthenormaldirectionwithrespecttothissurfaceinordertoreducetheinteractionbetweenthehydrophilicandhydrophobicbeads.Asthedistancebetweenabondandthedividingsurfaceincreases,theenthalpicconstraintsonthebondorientationbecomesignicantlyrelaxedandtheentropiccontributiontothefreeenergybecomesmoresignicant.Theentropicforcecausesthesurfactantstocoilupevenattheexpenseofalocaldensityincrease.Forlongersurfactants,theentropicforceislargerandis 32

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Comparisonofdensityprolesof(a)hexadecane,(b)water,(c)surfactanttails,and(d)surfactanthead-groupsintheC3P3,C7P3,C3P7,andC7P7surfactantmonolayers. capableofovercominglargerenthalpicrepulsionsbetweenthebeadscausinghigherlocalsurfactantdensity,seeFig. 3-3 candFig. 3-3 d.Thisstretchingofthesurfactantmoleculesnearthedividingsurfacefollowedbytheircompressionawayfromthesurfaceleadstoaminimumofthetotalmonolayerdensityatthedividingsurfaceandtwomaximasurroundingit[ 23 ](Fig. 3-2 ).Formonolayerscomposedofsymmetricsurfactants,thedensityprolesofthehead-andtail-groupsarenearlysymmetricwithrespecttoeachother.However,themaximumofthetotaldensityinthesemonolayersislargerintheP-regionthanintheC-region. 33

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3-3 indicatesthattheaverageconformationsofthesurfactanttailandhead-groupsareessentiallyindependentofeachother.Inotherwords,densityproleofamonolayercomposedofasymmetricsurfactantscanbepiecedtogetherfromdensityprolesofmonolayerscomposedofsymmetricsurfactants.Forexample,thedensityproleoftheC7P3monolayercanbeapproximatedbythedensityprolesoftheC3P3andC7P7monolayersintheP-andC-regions,respectively. 3-1 34

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BarriersG(inkBT)forsolutetransportacrosstheC3P3andC7P7monolayers. BarriersC3P3monolayerC7P7monolayer C0.812.71C21.004.20CP(headregion)0.892.84C51.157.76P0.000.77P20.001.23CP(tailregion)0.000.91 monomerCintheC3P3,C7P3,C3P7,andC7P7monolayers.ThereisaclearcorrelationbetweenthePMFandthelocalmonolayerdensity,ascanbeseenfromthefreeenergyanddensityprolesshowninFig. 3-4 .BarriersforthesolutetransportarecreatedbythehighdensityareasintheunfavorableP-regions,whiletheareasofhighdensityinthefavorableC-regionsoffernegligibleresistancetothetransport.SincethelocaldensitymaximumintheP-regionishigherformonolayerscomposedofsurfactantswithlongerhead-groups,thetransportbarrierscreatedbytheC7P7andC3P7monolayersarelargerthanthosecreatedbytheC3P3andC7P3monolayers.Recallthatdensityprolesofasymmetricmonolayerscanbepiecedtogetherfromsegmentsofthedensityprolesofsymmetricmonolayers(Fig. 3-3 andFig. 3-4 b).ThestrongcorrelationbetweenthedensityandthePMFactingonthesolutesuggeststhatthefreeenergyprolesofsolutesinasymmetricmonolayerscanalsobepiecedtogetherfromsegmentsofthecorrespondingfreeenergyprolesinsymmetricmonolayers.Indeed,theenergyprolesoftheCsoluteintheP-andC-regionsoftheC7P3monolayercanbeapproximatedbythefreeenergyprolesforthesamesoluteinthecorrespondingregionsoftheC3P3andC7P7monolayers,respectively.AnalogouspropertyisobservedfortheC3P7monolayer.Thisindicatesthatmechanismsofthesolutetransportinasymmetricmonolayerscanbeinferredfromthatinsymmetricmonolayers.Therefore,theremainderofthisstudyisfocusedonthesolutetransportinthesymmetric(C3P3andC7P7)monolayers. 35

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(a)PotentialofmeanforceG(zs)actingonthehydrophobicmonomerCintheC3P3,C7P7,C7P3,andC3P7monolayers;(b)Totaldensityprolesofthesemonolayers. 3-5 .ByanalogywiththehydrophobicsoluteC,itisexpectedthatthemainbarrierforthetransportofthehydrophilicsolutePacrossamonolayerwillbelocatedintheunfavorableC-region.However,sincethelocaldensityincreaseintheC-regionissmallerthanintheP-region(Fig. 3-4 b),thebarrierforahydrophilicsoluteissmallerthanthatforahydrophobicsoluteofcomparablesize,ascanbeseeninFig. 3-5 .Infact,sincethelocaldensitymaximumintheC-regionoftheC3P3monolayerisnegligible,thismonolayerimposesnobarrierintheC-regionforanyoftheconsideredhydrophilicandamphiphilicsolutes,seeTable 3-1 .TheC7P7monolayerdoesimposebarriersforthehydrophilicandamphiphilicsolutesintheC-regionbutthesebarriersaresignicantly 36

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Figure3-5. FreeenergyprolesforsolutesP,C,andCPintheC3P3andC7P7monolayers. Asexpected,theCPsolutehasastrongpreferencetobelocatednearthedividingsurfaceofthemonolayerandthecorrespondingfreeenergyG(zs)reachesitsminimumatzs0(Fig. 3-5 ).Uptoanadditiveconstant,thefreeenergyprolesfortheCPsoluteintheP-andC-regionsareingoodagreementwiththosefortheCandPmonomersintheirunfavorablephases,respectively.Thisisconsistentwiththenegligibleresistancetotransportofmonomersthroughtheirfavorablephases.Hence,totheleadingorder,thecontributionoftheP(C)beadtotheenergybarrierfortransportoftheCPdimerthroughtheP-(C-)regionofthemonolayerisnegligible. 3-6 andthevaluesofthetransportbarrierslistedinTable 3-1 37

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BFigure3-6. Freeenergyprolesof(a)hydrophobicand(b)hydrophilicoligomersintheC3P3andC7P7monolayers. However,thebarrierheightdoesnotincreaseasfastasthenumberofbeadsintheoligomer,especiallyfortransportthroughthethinnerC3P3monolayer.Forexample,thebarrierfortransportofthebranchedpentamerC5isonly1.4timeslargerthanthebarrierfortransportofthemonomerCthroughthismonolayer.TheincreaseofthetransportbarrierwiththesolutesizeismoresubstantialinthethickerC7P7monolayer.Forexample,thetransportbarrierforC5inthismonolayeris2.9timeslargerthanthatforthemonomerC.Inessence,thesoluteandsurfactantsizesactsynergisticallytoincreasethetransportbarrierforoligomers.Inordertounderstandtheroleofthesoluteandsurfactantsizesincreatingthetransportbarrier,letusconsidercongurationofanoligomerasitpassesthroughtheenergybarrier.Sincethemainsourceofthetransportbarrierisalocalmaximumofthemonolayerdensity,theoligomerisexpectedtoassumesuchacongurationthatitwouldminimizethenumberofbeadssimultaneouslypassingthroughthedensestmonolayerregion.Inparticular,theP2andC2dimersareexpectedtobeperpendiculartotheinterfaceastheypassthroughtheenergybarrier.Toconrmthis,considerthedimerorientation 38

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3-7 .Here~G(zs,)areplottedonlyfor0becausethebeadsofC2areindistinguishableandhence~G(zs,)areevenfunctionsof.Inwhatfollows,statementsregardingtheparallelandnormalorientationsofthesoluterefertoitsorientationwithrespecttothemonolayersurface.Thepreferredorientationofthesolutenearthedividingsurfacevariesbetweenparallelandnormal,dependingonthesolutelocation.Asthesoluteapproachesthedividingsurfacefromitsfavorablephase,italignsitselfparalleltotheinterfaceinordertoexposebothofitsbeadstothefavorablephase.Ifthesolutewerenormaltothemonolayeratthesezs,oneofitsbeadswouldbeexposedtotheunfavorablephase.Notethatthesoluteremainsparalleltotheinterfaceevenforsmallpositive

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24 ]. BFigure3-7. Contributions~G(zs,)oftheorientationofsoluteC2tothetotalfreeenergyin(a)C3P3and(b)C7P7monolayersareshownbythecontourplots.Energyvalues(inkBTunits)areindicatedbycolor.ThetopplotsshowthepotentialsofmeanforceG(zs)actingonthesolutewhosecenterofmassislocatedatzs.Preferredsoluteorientationsatvariouszsareshownschematicallyandbordersbetweendomainsofdifferentpreferredsoluteorientationsareindicatedbydashedlines.Thebordersaredenedasthepointsatwhichbothorientationsofthesoluteareequallyprobable. Asthesolutepassesthroughtheenergybarrier,itreorientsitselfinthedirectionnormaltotheinterface.Thisensuresthatonlyonebeadatatimemovesthroughtheenergymaximum.Fig. 3-7 alsoindicatesthatthereisamildpreferenceforthenormalorientationofC2inanarrowdomainintheC-region.Thispreferenceiscausedbythedenseregioninthefavorablephase.Sincethisregiondoesnotcreateasignicantenergybarrier(Fig. 3-4 ),wefocusourdiscussiononthesolutebehavioraroundthebarrierintheunfavorablephase. 40

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BFigure3-8. Roleofthesoluteandsurfactantsizesincreatingatransportbarrierforadimer.ThedashedandsolidlinesshowthefreeenergyprolesG(zs)forthemonomerCandthedimerC2in(a)C3P3and(b)C7P7monolayers.ThelledcirclesshowthemostprobablelocationsofbeadsofC2whenitscenterofmassislocatedattheenergymaximumzs=zG,max.TheopencirclesindicatevaluesofthePMFactingonthedimerbeadsattheselocations. PMFsactingonthedimerC2anditsindividualbeadsCaroundthetransportbarriersintheC3P3andC7P7monolayersareshowninFig. 3-8 .Sincethedimerisorientedinthenormaldirectionasitpassesthroughthetransportbarrier,thePMFactingonitatthebarriercanbeapproximatedasthesumofthePMFsactingonitsbeadslocatedatzG,maxRbond=2.HereRbondisthebondlengthandzG,maxislocationofthedimercenterofmasscorrespondingtothemaximumofthePMFactingonthedimer.ThewidthoftheenergybarrierforthemonomerintheC3P3monolayeriscomparablewiththedimerbondlengthRbond.Therefore,whileoneofthedimerbeadsislocatednearthetransportbarrierforthemonomer,theotherhasalreadypasseditandislocatedinalessdenseregion,seeFig. 3-8 a.ThePMFactingonthesecondbeadissignicantlysmallerthanthePMFactingontherstbeadandthusthetransportbarrierforthedimeronlyslightlyexceedsthetransportbarrierforamonomer.Ontheotherhand,theenergybarrierfortheCbeadintheC7P7monolayeriswideincomparisonwithRbond.Therefore,bothofthedimerbeadsarelocatedinthedense 41

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3-1 ).InordertoassessconformationsofC5asitpassesthroughtheenergybarrier,weconsiderorientationsiofitsbondsC5Ci,i=1,...,4,withrespecttothepositivedirectionofthez-axis.HereC5isthecentralbeadofthepentamerandthebonddirectionsaretakentobepointingfromC5toCi(i=1,...4).Sincethebondsareotherwiseindistinguishable,weenumeratethemintheorderofdescendingi.Accordingtothisordering,bonds1and4aretheclosesttothenormalorientation(=1)andbeadsC1andC4are,respectively,therstandlasttopassthroughthebarrierasthesolutemovesfromoiltowater.Contributions~Gi(zs,i)oforientationsofbondsC5Ci(i=1,...,4)tothetotalfreeenergyareshowninFig. 3-9 .PreferenceforaparticularbonddirectionisweakerinC5thaninadimer,sincethebondmovementismorerestrainedinthebranchedmolecule.Nevertheless,weobservesomesimilaritybetweenthebondorientationsinC5andC2.Specically,thepreferredorientationsofthepentamerbondsshifttowardsthemonolayernormalasthesolutepassesthroughtheinterface.Behaviorofbonds1and4exhibitsadditionalsimilaritieswiththedimer.Thesebondstendtotiltawayfromthemonolayernormalasthecorrespondingbeads(C1andC4)approachtheinterfacefromtheoil-richsideandtilttowardsthemonolayernormalinthedenseregionsonbothsidesofthedividingsurface. 42

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BFigure3-9. Contributions~Gi(zs,i)oforientationsofbondsC5Ci(i=1,...,4)tothetotalfreeenergyofthebranchedpentamerC5in(a)C3P3and(b)C7P7monolayers.Energyvalues(inkBTunits)areindicatedbycolor.Contourlinesforenergiesexceeding2kBTarenotshown. AlthoughthetendencyforthebondreorientationinC5isrelativelyweak,thetransportbarrierforC5throughthethinC3P3monolayerisalmostthesameasthatfortheC2dimer.Tounderstandthesituation,inFig. 3-10 weplotthemostprobablecongurationsofC5attheenergybarrierintheC3P3andC7P7monolayers.Inaddition, 43

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3-10 showsthePMFsactingonC5andC.ThedistancebetweenbeadsC1andC4inthez-directioniscomparablewiththewidthoftheenergybarrierforasingleCbeadintheC3P3monolayer.Therefore,whilebeadC4islocatedatthemaximumofitsPMFintheC3P3monolayer,beadC1islocatedoutsideofthedensemonolayerregion.Hence,thePMFactingontheentirepentameratitsenergymaximumissignicantlysmallerthanthePMFactingonvemonomersattheirenergymaximum.Ontheotherhand,thePMFactingoneachbeadofC5inthethickerC7P7monolayeriscomparablewiththePMFattheenergybarrierforasingleCbead.Thisleadstoasubstantialincreaseofthetransportbarrierforthepentamerincomparisonwiththebarrierforthemonomertransportthroughthismonolayer. BFigure3-10. Roleofthesoluteandsurfactantsizesincreatingatransportbarrierforabranchedsolute.ThedashedandsolidlinesshowthefreeenergyprolesG(zs)forthemonomerCandthebranchedpentamerC5in(a)C3P3and(b)C7P7monolayers.ThethicklinesandlledcirclesshowthecongurationofC5whenitscenterofmassislocatedattheenergymaximumzs=zG,max.Thez-coordinatesofthelledcirclescorrespondtothez-coordinatesofthemostprobablepositionsofthepentamerbeadswhenzs=zG,max.TheopencirclesindicatevaluesofthePMFactingonthebeadsattheselocations. Insummary,eventhoughbondsofabranchedoligomerdonotallaligninthenormaldirection,thebarrierforitstransportthroughathinmonolayermaystillbeonly 44

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3.4 ,weinvestigatedbarrierstosolutetransportassumingthatthesystemdynamicsisadiabatic,i.e.itfollowsanMEPparameterizedbythesolutetranslationaldegreeoffreedomzs.Thisassumptionholdsonlyifthetimescaleofzsismuchslowerthantimescalesofallotherdegreesoffreedom.InthisSection,weinvestigatevalidityoftheadiabaticassumptionanddiscussdynamiccontributionsofotherdegreesoffreedomtoresistancetosolutetransport. 3-11 .Thesoluteorientationisconsideredwithrespecttothepositivedirectionofthez-axisandthesolutedirectionistakentobethatofthevectorpointingfromitsCtoPbeads.Incontrastwiththeoligomers,thereisonlyonedomainofpreferredorientationfortheamphiphilicsolute.ThisdomainsurroundsthemonolayerdividingsurfaceandthepreferredCPorientationinthisdomainisnormaltotheinterfacewiththesolutehead-grouppointingintotheP-region.Thisorientationpreferenceoftheamphiphilicsoluteismuchstrongerthanthatofoligomers(Fig. 3-7 andFig. 3-9 ).IforientationoftheCPsoluteapproachingtheinterfaceisunfavorabletoadsorption,thesoluteencountersaveryhighenergybarrierandislikelytobereected.ThisbarrierisnotcapturedbythePMFG(zs)obtainedfromtheconstrainedsimulations,sincethelattermeasuretheforceexperiencedbythesolutewhenitsorientationuctuatesaroundthepreferredstate.Inotherwords,G(zs)doesnotaccountforadynamic 45

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BFigure3-11. Contributions~G(zs,)oftheorientationoftheCPsolutetothefreeenergyin(a)C3P3and(b)C7P7monolayersareshownbythecontourplots.Energyvalues(inkBTunits)areindicatedbycolor.ThetopplotsshowthepotentialsofmeanforceG(zs).Bordersofthedomainsofpreferredsoluteorientationareshownbydashedlines.Thesedomainsaredenedasdomainsinwhichtheenergydifferencebetweenthemostandleastpreferredsoluteorientationsexceedsthethermalenergy,i.e.j~G(zs,=1)~G(zs,=1)j>kBT. 46

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@(12)@ @P(zs,;t)(3)withboundaryconditions@P(zs,;t) 3 ,Eq. 3 andEq. 3 hasbeenanalyzedforparticleadsorptiononsphericalsurfaces[ 39 40 ].Inthelattercase,adsorptionrateiscomputedbysolvingasteady-statediffusionequationassumingauniformsoluteconcentrationatinnitedistancefromtheadsorbingsurface.However,theone-dimensionalgeometryofthetranslationaldegreeoffreedominEq. 3 precludesexistenceofanon-trivialsteady-statesolutionofthisequationwithnitesoluteconcentrationatinnity.Hence,wesolvethisequationinanitedomain.Weintroduceareectiveboundaryatsomepointzlintheoilphase, 47

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27 ],weobtainthefollowingequationforthemeantransporttime(z0,0) @z20+DR@ @0(120)@ @0=1.(3)Theboundaryconditionsfor(z0,0)inz0and0arethesameasforP(zs,;t)inzsand.TheapproachtosolutionofEq. 3 issimilartothatforasteady-statediffusionequationforaconguration-constrainedadsorptiononasphere[ 39 ].Thefunction(z0,0)isexpandedintermsofLegendrepolynomialsLk(0), 3-12 forarangeofinitialsolutepositionsz0andseverallocationszlofthereectiveboundary. 48

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RatioR(z0)=(z0)=0(z0)ofthemeantimesofadsorptionofCPattheinterfacewithandwithouttheconstraintonthesoluteorientation.Differentcurvescorrespondtodifferentlocationszlofthereectiveboundaryintheoilphase. AscanbeseenfromFig. 3-12 ,therelativecontributionoftheorientationconstrainttotheadsorptiontimeisfairlysmallanddecreasesastheinitialsolutepositionz0ismovedawayfromtheinterface.Forexample,thedelayofadsorptioncausedbytheorientationconstraintdoesnotexceed10%ifthesoluteisinitiallylocatedmorethan3nmawayfromtheinterface.ThesmalleffectoftheorientationconstraintonthetimescaleofCPadsorptioninthemonolayeriscausedby(i)weaknessoftheconstraint(=0.15)and(ii)relativelylargerotationaldiffusivityofthesolute(DR=DT19nm2).Itislikelythatlargersolutesexperienceamoresignicanteffectoftheorientationconstraintontheiradsorptiontimescale.TherotationalandtranslationaldiffusivitiesofrigidrodmoleculescomposedofNsphericalbeadsscaleaslnN=N3andlnN=N,respectively[ 41 ].Althoughthescalingmaybedifferentforexiblesolutes,thegeneraltrendisexpectedtoremainthesame:thetimescaleofthesoluterotationdecreasesfasterthanthatofitstranslationasthesolutesizeincreases.Theorientationconstraintmayalsobecomestrongerforlargersolutes.Nevertheless,thedynamiceffectoftheamphiphilicsoluteorientationisratherlimited.Evenifitsroleinthesoluteadsorptionisnon-negligible,ithasavanishingeffect 49

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3.5.2 playanimportantdynamicroleinallcases. 24 ]thatthetransportrateofthehydrophobicmonomerCacrosstheC3P3monolayerisaffectedbyanon-adiabaticsolute-interfacecoupling.Themainrequirementsforthiscouplingare(i)exibilityoftheinterfaceand(ii)differentafnitiesofthesolutefordifferentcomponentsoftheinterfaceand/oruidsseparatedbytheinterface.Therefore,thedynamicsolute-interfacecouplingisexpectedtobeageneralfeaturefortransportacrosssurfactant-coveredinterfaces. Figure3-13. ACFsoftherandomforce\(zs;t)actingontheC2soluteintheC3P3monolayer.Thesolutecenterofmassisconstrainedat(a)zs=0.0nm(dashedline)and(b)zs=1.0nm(solidline).TheinsetshowsC(zs;t)forsmallt.ACFsarenormalizedsothatC(zs;t=0)=1. 50

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3-13 .Afterfastinitialoscillations(t2ps),thedecayofC(zs;t)followseitherasingleoradoubleexponentiallaw,C(zs;t)=C1(zs)et=1(zs)or 3 holdsand2(zs)ifEq. 3 holds.AscanbeseenfromFig. 3-14 ,thevaluesoff(zs)areextremelysensitivetothesolutelocationinalloftheconsideredsolute-interfacesystems.Formostofthesolutelocations,f(zs)isontheorderofafewpicoseconds,whichcorrespondstothetimescaleofthermaloscillationsofindividualmolecules.However,inanarrowregionf(zs)exceedsthemoleculartimescalebyordersofmagnitude.Theslowuctuationsoftherandomforceactingonthesoluteinthisregionarecausedbycouplingbetweenthesolutemotionandtheinterfaceundulations.Thiscouplingisthestrongestintheneighborhoodofthefreeenergybarrierandthereforeitmaysignicantlyaffectthesolutetransportacrosstheinterface.Furthermore,theheightofthepeakoff(zs)increaseswiththesolutesize,whichsuggeststhatthecouplingisstrongerforlargersolutes.Thecrucialrequirementforthesolute-interfacecouplingisadeformationoftheinterfacecreatedbyasolutelocatedinsomeneighborhoodofthedividingsurface[ 24 ].WemonitorthedependenceofthemonolayershapeonthesolutelocationzsusingtheaverageFouriermodes^h(0)k(zs)h^hk(zs;t)iofthedividingsurface.Here^hk(zs;t)aretheFouriermodesoftheinstantaneousdividingsurfacesdenedbyEq. 3 andEq. 3 ;^hk(zs;t)areobtainedfromMDsimulationswiththesoluteconstrainedatzs. 51

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BFigure3-14. Correlationtimesf(zs)oftherandomforcesactingonthesolutesin(a)C3P3and(b)C7P7monolayers. DependenceoftheaverageFouriermodes^h(0)k(zs)withwavenumberk=0.65nm1onthesolutepositionzsisshowninFig. 3-15 .Thesituationisqualitativelysimilarformodeswithotherwavenumbers.Inallconsideredsystemsthemonolayerdeformsfromtheatshape(^h(0)k=0)inresponsetothesolutemotionthroughtheinterface.Tounderstandtheoriginofthemonolayerdeformationanditsroleinthedynamicsolute-interfacecoupling,considere.g.transportofahydrophobicsolutefromoiltowaterphase.SeveralsnapshotsofthisprocessareshowninFig. 3-16 .Asthesoluteapproachestheinterfacefromtheoilphase,theinterfacedeformsinordertomaximizewettingofthesolutebyhydrophobicgroups.TheinterfaceprotrusionintothewaterphasegrowsasthesolutemovesfurtherintotheP-region.Eventually,energypenaltyassociatedwithsuchdeformationsbecomestoolargetobecompensatedbytheattractiveinteractionbetweenthesoluteandthehydrophobicgroups.Asaresult,oncethesolutepassesthroughacriticalprotrusionpointzh,max,theprotrusionquicklydisappears.Smalldisplacementsofthesolutearoundthecriticalpointzh,maxcauselargechangesinthepreferredinterfaceshapeduetothelargesensitivityof^h(0)k(zs)to 52

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BFigure3-15. DependenceoftheaverageFouriermodes^h(0)kwithwavenumberk=0.65nm1onthesolutepositionzsforvarioussolutesin(a)C3P3and(b)C7P7monolayers.LocationsofmaximaofG(zs)areshownbycirclesandlocationsofboundariesofdomainsofthepreferredorientationoftheC2,P2,andCPsolutesareshownbytriangles.Inthesystemswithnotransportbarrier(i.e.,P,P2,andCPintheC-regionoftheC3P3monolayer),thecirclesshowpointsatwhichG(zs)reachesaconstantvalue. changesinzsaroundzh,max.However,thetimescaleoftheinterfacerelaxationtowardsthepreferredconguration^h(0)k(zs)ismuchslowerthanthetimescaleofthesmallsolutedisplacement.Therefore,theinstantaneousinterfaceshapewillbedifferentfromitspreferredshape.Theunfavorableinterfaceshapewillcausearestoringforcepullingthesolutetowardsthepreferredsolutepositioncorrespondingtothisshape.Itisthisrestoringforcethatcorrespondstotheslowcomponentoftherandomforce.Theargumentsaboveindicatethat,oncethesolutepassespointzh,maxandtheinterfacereturnstotheatstate,thesolutebecomessurroundedbyunfavorableenvironment.Therefore,zh,maxshouldbelocatedfairlyclosetopointzG,max.HerezG,maxdenoteslocationoftheenergybarrieror,forsystemswithnotransportbarrier,thepointatwhichG(zs)reachesaconstantvalue.ThisconjectureisconrmedbyFig. 3-15 whichshowsthatpointszG,maxarelocatedpreciselyatthepointscorrespondingtotheprotrusiondisappearance. 53

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B CFigure3-16. DeformationoftheC3P3monolayercausedbytheC5solutewiththecenterofmasslocatedat(a)zs=0.1nm,(b)zs=1.1nm,and(c)zs=1.4nm.Thethicklinesshowthedividingsurfaceandthespheresrepresentthebeadsofthesolute(green),oil(yellow),water(gray),andsurfactanttails(red)andheads(blue).ThesnapshotsofthemolecularsystemsareplottedusingtheVMDpackage[ 42 ]. Inordertoassesseffectsoftheinterfacedeformationonthesolutetransport,itisnecessarytoreconstructthefreeenergylandscapeparameterizedbydegreesoffreedomofboththesoluteandtheinterface.Inthecaseofasphericalsolute,theenergylandscapecanbeapproximatedas[ 24 ] 2Xk6=0k(zs)^hk^h(0)k(zs)2(3)Herekisthestrengthofcouplingbetweenthesoluteandtheinterfacialmode^hk;kcanbeobtaineddirectlyfromconstrainedMDsimulationsusingtheequipartitiontheorem, 54

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24 ].Inthecurrentwork,weappliedthesamevalidationproceduretoverifythevalidityofthismodelforboththehydrophobicandhydrophilicsphericalsolutesinC3P3andC7P7monolayers.Furthermore,weveriedthatthekeyfeatureofthismodel,namelyharmoniccouplingbetweenthesolutetranslationaldegreeoffreedomzsandtheinterfacialmodes^hk,holdsforallmulti-beadsolutesconsideredinthecurrentwork.Therefore,thefollowingdiscussionofthenon-adiabaticsolute-interfacecouplingmodeledbyEq. 3 isexpectedtoholdforallconsideredsolutesandinterfaces.Across-sectionofthefreeenergyproleisshowninFig. 3-17 .Eq. 3 impliesthattheMEPonthefreeenergysurfaceG(zs;f^hkg)isapathsuchthat^hk=^h(0)k(zs)forallkateverysolutepositionzsalongthepath.Thesystemfreeenergyonthispath,GMEP(zs;f^hkg)=G0(zs),doesnotdependonthedetailsofthecouplingbetweenthesoluteandtheinterfaceuctuations.However,thesharpchangesoftheinterfaceshapebetweenzh,maxandzG,maximplyexistenceofsharpturn(s)ontheMEP,whichmaycausethesystemdynamicstodeviatefromtheMEP.ThisdeviationissignicantifthetimescaleofthesolutemotionalongtheMEPiscomparablewithorisfasterthanthetimescaleofthesystemapproachtowardstheMEP.Thelatterischaracterizedbytherelaxationtimescaleofthosenormalmodes^hkofthemonolayerwhichexhibitstrongsensitivitytothesolutelocation.Ourcalculationsindicatethatthissensitivityisstrongforlongwavelengthmodes,suchasthemodeshowninFig. 3-15 .ThecorrelationtimeskofthenormalmodesoftheC3P3andC7P7monolayersareshowninFig. 3-18 andindicatethatthetimescaleofthelongmodesisontheorderofhundredsofpicoseconds.LetusnowestimatethetimescaleofmotionalongtheMEPbetweenthepointszh,maxandzG,max,i.e.intheregioncorrespondingtothestrongestsensitivityoftheinterfaceshapetothesolutelocation.WeassumethatthesystemdynamicsalongtheMEPisdescribedbytheone-dimensionalMarkovianLangevinequationEq. 2 with 55

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Cross-sectionofthefreeenergyproleG(zs;f^hkg)oftheCsoluteintheC7P7monolayer.Thecross-sectionisperformedbythezs^hkplanecorrespondingtotheinterfacialmodewithwavenumberk=0.65nm1.Energyvalues(inkBTunits)areindicatedbycolor.TheMEPisshownbythethickblackline.LocationofthesaddlepointofG(zs;f^hkg)isshownbythecircle.Hypotheticpathsforthesolutetransportfromwatertooilandfromoiltowaterareshownbyarrows. Figure3-18. CorrelationtimeskofthenormalmodesoftheC3P3andC7P7monolayers. thePMFG(zs)obtainedfromtheconstrainedsimulations.Therandomforceinthisequationisassumedtoreectonlycollisionsofthesolutewithindividualmolecules.In 56

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27 ] 2 usingtheACFoftherandomforcecontainingonlycontributionsofinteractionsbetweenindividualmolecules.ThisACFisobtainedbyremovingtheslowlydecayingexponentialtermfromtheACFC(zs;t)obtainedfromtheconstrainedMDsimulations.Themeantransporttimes[zG,max!zh,max]and[zh,max!zG,max]arelistedinTable 3-2 andTable 3-3 ,respectively.ThetransportfromzG,maxtozh,maxisveryfast,[zG,max!zh,max]<100psforallconsideredsystems,sinceitcorrespondstoadownhillmotiononthefreeenergylandscape.Moreover,theproximityofpointszh,maxandzG,maxensuresarelativelysmallfreeenergydifferencebetweenthesepoints.Hence,climbingthefreeenergygradientfromzh,maxtozG,maxtakesplacerelativelyfast:inalloftheconsideredsystems,[zh,max!zG,max]isontheorderofhundredsofpicosecondsor,atmost,afewnanoseconds.Therefore,thetimescaleofapproachtowardstheMEPiscomparablewithorisevenslowerthanthetimescaleofmotionalongtheMEPbetweenzh,maxandzG,max.Thus,thesystemisunlikelytofollowtheMEPintheneighborhoodoftheenergybarrier.ThisdeviationfromtheMEPleadstoanincreaseoftheeffectiveenergybarrierforsolutetransport.Forexample,ourearlieranalysis[ 24 43 ]oftransportofthehydrophobicmonomerCacrosstheC3P3monolayerindicatesthatthesenon-adiabaticeffectsleadtoafactoroffourdecreaseofthetransportrate.Itisimportanttonotethatthesamesolute-interfacesystemislikelytofollowdifferentpathsdependingonwhetherthesolutemovesfromoiltowaterorfromwatertooil.ThesituationisillustratedinFig. 3-17 forahydrophobicsolute.MotionofthissolutefromoiltowaterislikelytofollowtheMEPforzszh,max,sincethetimescaleforthesoluteclimbingthefreeenergyslopefromthefavorableoilphasetotheunfavorable 57

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MeantransporttimefromzG,maxtozh,max(inpicoseconds). MeantrasporttimeC3P3monolayerC7P7monolayer C2896C21661CP(headregion)2274C51364P3876P24251CP(tailregion)3962 Table3-3. Meantransporttimefromzh,maxtozG,max(inpicoseconds). MeantrasporttimeC3P3monolayerC7P7monolayer C162643C2116910CP(headregion)68382C51322905P163342P2272504CP(tailregion)160292 waterphaseisrelativelylarge.However,theforegoingdiscussionimpliesthattheinterfaceshapewillremainrelativelyunchangedduringthesolutemotionfromzh,maxtozG,max.Therefore,theinterfaceislikelytoremaindeformedevenafterthesolutehaspassedthroughzG,max.Ontheotherhand,theinterfaceisunlikelytoundergoanydeformationinresponsetothesolutetransportintheoppositedirection,sincethetimescaleoftransportfromtheunfavorablephasetothefavorablephaseisrelativelyfast.Itisexpectedthatthenon-adiabaticsolute-interfacecouplingwillhaveanevenmoresignigicanteffectonthetransportrateofoligomers.AscanbeseenfromFig. 3-15 ,boththemagnitudeoftheinterfacedeformationandthegradientof^h(0)k(zs)intheneighborhoodofzh,maxincreasewiththesizeofhomogeneoussolutes.Therefore,thedynamicsolute-interfacecouplingisstrongerforlargeroligomers,asconrmedbytheincreaseofthecorrelationtimef(zs)oftherandomforcewiththesolutesize(Fig. 3-14 ). 58

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3-15 ).Asaresult,intheseregionsthestrengthofthedynamicsolute-interfacecouplingfortheCPsoluteiscomparablewiththatforthePandCsolutes.Indeed,thecorrelationtimef(zs)oftherandomforceactingontheCPsolutehastwopeaksandtheirlocationandheightintheC-andP-regionsareconsistentwiththoseofpeaksoff(zs)forthePandCmonomers,respectively(Fig. 3-14 ).Effectsofthemonolayerpropertiesonthestrengthofthedynamicinterface-solutecouplingarelessobvious.Ononehand,protrusionsinducedbythesamesoluteintheC3P3andC7P7monolayersareverysimilar(Fig. 3-15 )whichwouldsuggestsimilarcouplingstrength.Ontheotherhand,thetimescaleofuctuationsoftheC7P7monolayerisslowerthanthatoftheC3P3monolayer(Fig. 3-18 ).Thisdifferenceisalsoreectedinthetimescalesofrandomforcesactingonthesolute(Fig. 3-14 ).ThesloweructuationsoftheC7P7monolayerimplytheslowerapproachtowardstheMEPduringthesolutetransportbetweenzh,maxandzG,maxthroughthismonolayer.However,thetimescaleoftransportalongtheMEPbetweenzh,maxandzG,maxisalsoslowerintheC7P7monolayerthanintheC3P3monolayer(Table 3-2 andTable 3-3 ).Therefore,amoredetailedanalysisofthesystemdynamicsisneededinordertodetermineeffectsofthemonolayerpropertiesonthesolute-interfacecoupling.Thisnon-adiabaticdynamics 59

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44 46 ].ItisexpectedthatthisapproachwillallowustodeterminetheprobabilitydistributionofthepathsonthefreeenergylandscapeandassessvariousfactorscontributingtodeviationsofthesystemdynamicsfromtheMEP.InconclusionofthisSection,wenotethatthesolute-interfacecouplingmaybefurtherinuencedbynon-adiabaticbehaviorofsoluteconguration.Forexample,itislikelythatthereorientationoftheC2andP2solutesaroundtheenergybarrierwillleadtoadditionaldeviationsofthesystemfromitsMEP.Aswediscussedearlier,thereorientationofthesesolutesleadstoreductionoftheenergybarrierstotheirtransportacrossthinmonolayers.Thisreorientationtakesplacewhenthemonolayerdeformationisthelargest,ascanbeseenfromlocationsoftheboundariesbetweendomainsofpreferredsoluteorientationshowninFig. 3-15 .Therefore,itisexpectedthatthesolutereorientationwillbedynamicallycoupledwithitstranslationandtheinterfacerelaxation,thuspushingthesystemfurtherawayfromtheMEP.Incontrast,dynamiccouplingbetweenorientationoftheamphiphilicsoluteCPandtheinterfaceundulationsisunlikely,eventhoughtheboundariesofthedomainofpreferredorientationofthissolutearelocatednearpointszG,maxandzh,max(Fig. 3-15 ).Thekeydifferencebetweentheamphiphilicandhomogeneoussolutesisthattheformerhasonlyonedomainofpreferredorientationanditsorientationdoesnotexperienceanyconstraintsoncethesoluteleavesthisdomain.Inparticular,theCPorientationcanremainthesameafterthesolutedesorptionfromthemonolayerasitwasbeforethedesorptionwithoutencounteringanyenergypenalty.Ontheotherhand,orientationofahomogeneousdimerpassingthroughtheenergybarrierneedstoundergoafastchangeinorderforthesystemtostayontheMEP.Asdiscussedabove,thisdimerreorientationisunlikelytohappensufcientlyfast,causingnon-adiabaticcouplingofthesoluterotationwithotherdegreesoffreedom. 60

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3-19 .Itisobservedthatthedifferencebetweenthesolutefreeenergiesinthebulkphasesisindependentofthesystemsize.However,theapparentbarrierimposedonthesolutetransportdecreasesasthesystemsizeincreases.Inordertounderstandtheoriginandimplicationsofthissystemsizedependence,itisnecessarytoconsiderthesolute-inducedinterfacedeformation.Interfaceprotrusionsinducedbythesoluteconstrainedatzs=zh,maxinthesesystemsareshowninFig. 3-20 .Itisobservedthatincreasingthesystemsizeleadstothefollowingchanges:(1)thesolutepositionzh,maxcorrespondingtothelargestinterfacedeformationisshiftedmoreintotheunfavorablephase(water,inthecurrentexample)and(2)themagnitudeoftheprotrusionisincreased.However,theshapeof 61

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PMFactingontheCsoluteintheC3P3monolayerobtainedfromsimulationsinthesmall(LLL)andlarge(2L2L2L)systems. theprotrusionremainsalmostunchangedasthesystemsizeincreases.Thissuggeststhatthecriticalcurvatureoftheinterface,i.e.thecurvatureatwhichtheenergypenaltyassociatedwiththeinterfacedeformationbecomestoolargetobecompensatedbytheattractiveinteractionbetweenthesoluteandthefavorablephase,isrelativelyindependentofthesystemsize. Figure3-20. LargestinterfaceprotrusionsinducedbytheCsoluteintheC3P3monolayer.Averagedividingsurfacesofthemonolayercorrespondingtothesoluteconstrainedatzs=zh,maxinthesmall(LLL)andlarge(2L2L2L)systemsareshownbythegrayandblacklines,respectively.Thecorrespondingsolutelocationsareindicatedbythegrayandblackcircles. 62

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3-21 .Ascanbeseen,thereisgoodagreementbetweenthemeanvaluesofthenormalmodesinthesystemsofdifferentsizes.Themostprominentdifferenceisaslightshiftofthelocationofthemaximumofthenormalmodewiththesystemsizeincrease,whichisconsistentwiththeshiftofzh,maxobservedinFig. 3-20 .Similarresultsareobservedforallmodeswithshorterwavelength.Furthermore,weobservethatothermodecharacteristics,suchasthestrengthkoftheircouplingwiththesoluteandtherelaxationtimeoftheirthermaluctuationsarealsonotsensitivetothesystemsize. Figure3-21. DependenceoftheaveragevaluesoftheFouriermodes^h(0)kwithwavelengthLonthepositionoftheCsoluteintheC3P3monolayerinthesmall(LLL)andlarge(2L2L2L)systems. Themaindifferencebetweenthelargeandsmallsystemsisthepresenceofadditionalinterfacialmodesinthelargesystemthatareabsentinthesmallsystem.Itisthisdifferencethatisresponsibleforthelargerinterfacedeformationinthelargesystem.WeobservethatbehaviorofthemodeswithwavelengthexceedingLisqualitatively 63

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3 .TheseobservationsallowustoconcludethatsimulationsofthesmallsystemareessentiallyequivalenttosimulationsofthelargesystemwiththenormalmodesofwavelengthexceedingLconstrainedatzero.Inparticular,Eq. 3 impliesthattheMEPinthesmallsystemis 3-19 .InordertoobtainanaccuratepredictionoftheenergyalongtheMEP,oneneedstoperformsimulationsinasystemofsize~Lchosensothatthenormalmodeswithwavelengthexceeding~LmakeanegligiblecontributiontoEq. 3 .Ourcalculationsindicatethatthisconditionisnotsatisedevenbythelarge2L2L2Lsystem.Therefore,itisnecessarytofurtherincreasethesystemsizeinordertocorrectlypredictthefreeenergyalongtheMEP.Thesystemsizessatisfyingthisconditionforlargersolutesareexpectedtobeevenlarger,thusmakingcalculationofthefreeenergyalongMEPsextremelytime-consuming.However,weexpectthatanaccuratepredictionofthesolutetransportratecanbemadewithoutthepreciseknowledgeoftheMEPandthefreeenergyalongit,sincetheactualpathtakenbythesystemislikelytodeviatefromtheMEP.AswediscussedinSection 3.5.2 ,thetimescaleoftheinterfacialmodesistooslowforthesystemtofollowtheMEPintheneighborhoodofzh,max,seealsoFig. 3-17 .Deviationoftheinterfacial 64

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3.6.1ValidationoftheSimulationTime-stepThesimulationsreportedinthecurrentpaperwereperformedwiththetime-stepof40fs.Thistime-stepiswithintherange(30to50fs)suggestedintheoriginalwork[ 10 ]ondevelopmentofthecoarse-grainedmodelemployedinthecurrentstudy.Recently,Wingeretal.[ 37 ]haveindicatedthatthistime-stepistoolargetoprovideareliableestimateofcertaindynamicproperties,suchasdiffusivityofsmallmolecules.However,ithasalsobeenobservedthatthisstep-sizeissufcienttoprovideaccurateinformationregardingstructuralandenergeticproperties[ 37 ],i.e.propertiesofmaininterestinthecurrentwork.Inordertofurtherjustifyouruseofthe40fstime-step,weperformedacontrolsetofsimulationswitha10fstime-steprecommendedbyWingeretal.[ 37 ].TheseadditionalsimulationswereperformedfortheCsoluteintheC3P3monolayer.Resultsofthesimulationswithtime-stepsof10fsand40fsarecomparedinFig. 3-22 .Specically,wecompare(1)meanvaluesoftheconstraintforce,(2)autocorrelationfunctionsoftheconstraintforce,(3)correlationtimesoftheconstraintforce,and(4)theaverageinterfacedeformation.Goodagreementbetweenthesimulationswiththesetime-stepsindicatesthatthelargertimes-stepusedinthecurrentworkissufcientlysmalltoaccuratelypredictthesystempropertiesinvestigatedinthecurrentwork. 65

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B C DFigure3-22. ComparisonofresultsofconstrainedsimulationsofthehydrophobicmonomerCintheC3P3monolayerperformedwithtime-stepsof40fs(solidgraylines)and10fs(dashedblacklines).(a)MeanconstraintforcehFc(zs)i;(b)Autocorrelationfunction(ACF)oftheconstraintforcecorrespondingtothesolutecenterofmassconstrainedatzs=0.84nm;TheinsetshowsC(zs;t)forsmallt.ACFisnormalizedsothatC(zs;t=0)=1;(c)Correlationtimeoftheconstraintforce;(d)AverageFouriermode^h(0)koftheinterface;thewavenumberoftheshownmodeisk=0.65nm1. 66

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3-23 ,Fig. 3-24 andFig. 3-25 .Itisobservedthatthekeyenergeticandstructuralsystemcharacteristicsmeasuredinthecurrentwork,namelythepotentialofmeanforce,correlationtimeoftherandomforce,andthemeaninterfacedeformation,areconsistentforbothmethodsofpreparationofinitialconditions.ThisindicatesthatthesystemsindeedrelaxtowardstheirMEPsonthetimescaleofthe10nsequilibration.Thisconclusionisconsistentwiththemeasuredrelaxationtimesoftheinterfacialmodes,whichareontheorderofhundredsofpicoseconds(Fig. 3-18 ).Thedynamicmembrane-solutecouplingleadstocomparablerelaxationtimesoftherandomforceactingonthesolute(Fig. 3-14 ).Sincetherandomforcecontainsacumulativeeffectoftheinterfacialuctuations,thecorrelationtimeoftherandomforceissomewhatlargerthanthatofindividualinterfacialmodes.Nevertheless,thecorrelationtimeoftherandomforceisstillseveraltimessmallerthantheequilibrationtimeusedinoursimulations. 67

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BFigure3-23. Potentialofmeanforceactingon(a)P2soluteintheC3P3monolayerand(b)C2soluteintheC7P7monolayer.Initialconditionsfortheconstrainedsimulationsarepreparedbypullingthesolutefromoiltowater(solidgrayline)andfromwatertooil(dashedblackline). BFigure3-24. Correlationtimeoftheconstraintforceactingon(a)P2soluteintheC3P3monolayerand(b)C2soluteintheC7P7monolayer.Initialconditionsfortheconstrainedsimulationsarepreparedbypullingthesolutefromoiltowater(solidgrayline)andfromwatertooil(dashedblackline). domainsofhighdensitycreatedbyhead-andtail-groupsofsurfactants.ThelocaldensityincreaseandthewidthofthehighdensitydomainintheP-(C-)regionarecontrolledbythelengthofsurfactanthead(tail)groupsanddeterminethemonolayerresistancetotransportofhydrophobic(hydrophilic)solutes.Thelocaldensityincrease 68

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BFigure3-25. DependenceoftheaverageFouriermodes^h(0)kwithwavenumberk=0.65nm1onthelocationofthesolutecenterofmass:(a)P2soluteintheC3P3monolayerand(b)C2soluteintheC7P7monolayer.Initialconditionsfortheconstrainedsimulationsarepreparedbypullingthesolutefromoiltowater(solidgrayline)andfromwatertooil(dashedblackline). createdbysurfactanthead-groupsislargerthanthatcreatedbysurfactanttail-groupsofthesamelength.Thisdifferencecausesanasymmetryintheheightsoftransportbarriersforhydrophobicandhydrophilicsolutes.Namely,thebarrierfortransportofahydrophobicsolutethroughmonolayerCmPnislargerthanthebarrierfortransportofahydrophilicsoluteofsimilarsizeandstructurethroughmonolayerCnPm.Theobservedcorrelationbetweenthesurfactantsizeandthemonolayerpermeabilityisinaqualitativeagreementwithexperimentsonuptakeofwaterinwater-in-oilmicroemulsion[ 47 ].Intheseexperiments,themonolayersarecomposedofAOTsurfactantsandalcoholsanditisfoundthatincreasingthetaillengthofalcoholssignicantlyreducestherateofwateruptake.Thecorrelationofthefreeenergyanddensityprolesisalsoobservedintheoreticalandcomputationalmodelingofotherprocesses,suchastransportacrossliquid-liquidinterfaces[ 48 ]andlipidbilayers[ 38 49 ]andsolubilizationinmicelles[ 50 ]. 69

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51 ]andMDsimulations[ 52 ]oftransportthroughlipidbilayers.Inadditiontothestaticbarrier,thesolutetransportrateisaffectedbyadynamiccouplingwiththemonolayeructuations.Thiscouplingiscausedbyasolute-inducedinterfacedeformationandrelaxationoftheinterfacetotheunperturbedstate.Thetimescaleoftheinterfacerelaxationiscomparablewithtimescalesofsolutetranslationalandrotationalmotion,whichleadstodeviationsofthesystemfromitsMEPandthuseffectivelyincreasesthebarrierforsolutetransport.Themechanismofthedynamicsolute-interfacecouplingisrathergeneralanditisexpectedtooccurinvariousinterfacialsystem.Forexample,thiscouplinghasbeen 70

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53 54 ]andunchargedmolecules[ 24 ]acrossaliquid-liquidinterface.Furthermore,thesolute-interfacecouplingislikelytotakeplaceduringtransportthroughlipidmembranes.SeveralMDstudiesofthistransport[ 38 49 55 ]reportasignicantlocaldecreaseofthesolutediffusivity,whichislikelytobecasedbyanon-adiabaticsolute-membranecoupling. 71

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25 56 57 ],theory[ 58 ],andsimulations[ 59 68 ].However,modelingdynamicsofself-assemblyandstructuraltransitionsinsurfactantstructuresismorechallenging.Sincetheseprocessesinvolverearrangementofindividualmolecules,itisnecessarytomodelthemusingMDsimulations.However,timescalesofmostself-assemblyprocessesisoutofreachofdirectMDsimulations.Althoughself-assemblyofmicellesfromaninitiallyrandomdispersionofsurfactantscanberoutinelyobservedinMDsimulations[ 68 69 ],dynamicsofself-assemblyobservedinthesesimulationsislikelytobedifferentfromthoseatcriticalmicelleconcentration(CMC).Thesimulationsareperformedinsmallsimulationboxes(withsidelengthontheorderoftensofnanometers),whichimpliesthatsurfactantconcentrationinthesesimulationssignicantlyexceedsCMC.Sinceself-assemblyismainlyadiffusion-controlledprocessandthesurfactantconcentrationinMDsimulationsisveryhigh,micellarformationobservedinMDsimulationstypicallytakesplaceonatimescaleO(100)ns,whichsignicantlyexceedsexperimentalresults[ 6 70 ](frommilisecondstominutes).Inreality,theself-assemblyprocesstakesamuchlongertimeandthecorrespondingindividualstepsofself-assemblywouldlikelybedifferentfromthoseobservedinMDsimulations[ 9 ].Moreover,micellardisintegrationistypicallynotobservedonthetimescaleofMDsimulations,sincethisisanactivatedprocesswithaveryhighactivationbarrier.Thegoalofthisstudyistodevelopastochasticmodelfortwoofthestepsinvolvedinmicellarformation,namelyadditionandremovalofindividualsurfactants(referredtoasmonomers)to/fromsurfactantclusters(micelles).Earlierworkofourgroup[ 9 ]demonstratedthat,despitetheirapparentsimplicity,theseprocessesmayinvolvea 72

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5 .Sincetheprocessesofsurfactantaddition/removalto/fromamicelleinvolvemultipleDOFs,weneedtoreconstructthemulti-dimensionalfreeenergylandscapeG(,,...)parameterizedbyallrelevantDOFs,,....AsarststepinthereconstructionofG(,,...)weperformconstrainedMDsimulationstoobtaindependenceofthefreeenergyonthedistancebetweenthemicelleandthemonomerwhenallotherdegreesoffreedomareequilibrated.TheobtainedfreeenergycanbeconnectedtothefreeenergyGMEPalongtheMEPonthefreeenergylandscape.Furthermore,analysisofprobability 73

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4.2.1EquilibriumMicellesThecurrentworkisfocusedontheprocessesofaddition/removalofasurfactantmonomerto/fromanequilibriummicelle.Inthecurrentwork,weconsiderC4P4surfactants(Fig. 4-1 a),whichcanbeconsideredtobeanapproximationtotheC16EO8ethoxylatedsurfactants.Ithasbeenshowninourearlierwork[ 9 ]thattheaggregationnumberofanequilibriumC4P4micelleis64.Therefore,weinvestigatethemonomerremovalfromamicelleofaggregationnumber64andadditionofamonomertoamicelleofaggregationnumber63.Themicellarstructureswerepreparedbyself-assemblyofaninitiallyrandomdispersionofthesurfactantmoleculesinwater.ThedetailscanbefoundinRef.[ 9 ].ExampleofaninstantaneouscongurationoftheobtainedequilibriumC4P4micelleisshowninFig. 4-1 b. 74

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BFigure4-1. (a)SchematicrepresentationofasingleC4P4surfactant.Thehead-andtail-groupdirectors,dHanddT,arealsoshown.(b)ExampleofaninstantaneouscongurationoftheC4P4micelleofaggregationnumber64.TailbeadsCandheadbeadsPareshownbytheblackandlightgreyspheres,respectively.Watermoleculesareomittedforclarity. 4.3.3 71 ] 75

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4-1 a). 76

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72 ]tocharacterizesurfaceofalipidbilayer.ItisshowninAppendixthatthisapproachyieldsthefollowingexpressionfortheinstantaneouscoefcientsL(i)k(t): 4 followsfromEq. 4 andthefactthatP1(x)=x.ThelastequalityholdsbecausetheorigincoincideswiththemicellarCOM(recallthatallbeadsofthemicellehavethesamemass).LegendreexpansionEq. 4 allowsustoreplacetheearlieradhocapproachtomodelingmicellarstructure[ 9 ].Thisapproachwasbasedonanalysisofhydrophobiccorepatchesexposedtothesolvent.Theinformationregardingthesepatchesisimplicitlycontainedinthelocaldensity(and,hence,Legendremodes)ofthemicellarcorona.Lowlocaldensityofallcoronashellsinsomeregionwouldindicatepresenceofahydrophobicpatchinthatregion. 77

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2 .However,thisDOFisnotsufcienttoaccuratelydescribethemonomeradditionprocessandoneneedstoaccountforthemonomerorientationandpossiblyotherDOFs,suchasthemicellarstructure.Nevertheless,thetranslationalDOFallowsustoparametrizetheminimumenergypath(MEP),i.e.apathsuchthatthefreeenergyisminimizedinalldirectionstransversaltothepathdirection.ThesystemwillfollowthispathifthetimescaleofmotionalongthepathismuchslowerthanrelaxationofallotherDOFstowardsthepath,i.e.ifallotherDOFsareslavedtothetranslationalDOF.Ifthisconditionisnotsatised,thesystemdynamicscorrespondstoamotiononamulti-dimensionalfreeenergylandscape,whichcanberepresentedasfollows: 4 ,itfollowsthat 78

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2 ,averageofFc(;t)correspondstothegradientofthepotentialofmeanforce(PMF)G0()actingon.Ontheotherhand,Eq. 4 impliesthatthemeanforceactingonis 30 ].Initialconditionsfortheconstrainedsimulationsareobtainedfromanequilibriummicellarstructure.Amonomerisrandomlychosenandispulledwithaconstantvelocityappliedtoitscenterofmass.Thecongurationsofthemicelleandthepulledmonomeraresavedfordifferentvaluesofthereactioncoordinate.Theseconformationsserveasstartingcongurationsfortheconstrainedsimulationsfordifferentvaluesofthereactioncoordinate.Thetotaltimeforeachconstrainedsimulationis800ns. 79

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80

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4-2 .Theinitialcongurationofthesystemisrandomizedbyassigningrandomvelocities(sampledfromtheMaxwelldistribution)toeachatom.Afterthis,aconstrainedsimulationisperformedfor10nswhichensuresrandomizationoftheinitialcongurationsofthemicelleandthemonomerwhilekeepingthemsufcientlyfarfromeachother.SincethetimescaleoftheDOFscharacterizingthemicellarandmonomercongurationsisO(1)nsorless(Section 4.3 ),thedurationofthisinitialconstrainedsimulationisexpectedtoproduceasufcientlyrandomizedsampleofinitialcongurations.Afterthis10nssimulation,theconstraintisremovedandunconstrainedsimulationsareperformeduntilthemonomerentersthemicelle.ThemonomerisconsideredtohaveenteredthemicellewhendistanceRCbetweentheCOMsofthemonomerandthemicelleis2nm.ThisvalueofRCcorrespondstotheequilibriumlocationofamonomerinsideamicelle.159numberofsuchunconstrainedsimulationswereperformed. 4.3.2 ).Ontheotherhand,inunconstrainedsimulationsthemicellarshapeismuchlesssensitivetothemonomerposition,sincethetimescaleofuctuationsofthemicellarshapeislargerthanthatofthemonomertranslation.Hence,amicelleislikelytomaintainanearlyequilibriumshapeduringamonomeradsorptionintoamicelle.ThisisindeedconrmedinSection 4.3.2 .Therefore,thesamevalueofRCcorrespondstodifferentdistancesbetweenthemonomerandthemicellarsurface,dependingonthesimulationprotocol(constrainedorunconstrained).Intheabsenceoflong-rangeinteractions,themonomer-micelle 81

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4.3.1One-dimensionalFreeEnergyProleDependenceofthepotentialofmeanforceG0()actingonamonomerpartiallyinsertedintothemicelleisshowninFig. 4-2 .Asexpected,thefreeenergyG0()decreasessignicantlyasthemonomerentersthemicelleandreachesitsminimumat=eq=0.21nmcorrespondingtotheequilibriumlocationofthemonomerinsidethemicelle.Thefreeenergyincreasesasthemonomerispulledoutofthemicelleandbecomesessentiallyindependentofthemonomerpositiononcethelattercompletelyleavesthemicelle.Moreover,G0()doesnotexhibitabarrierforthemonomeradditiontothemicelle.Instead,thePMFstartsmonotonouslydecreasingoncethemonomercomesincontactwiththemicelleat1.7nm.Wewillrefertothismonomerlocationasthecriticalpointanddenoteitasc.Themotivationforthisnotationisthatiftherewereanenergybarrier,itwouldbemostlikelylocatedaroundthepointofrstcontactbetweenthemonomerandthemicelle.Wewillalsorefertoasmallneighborhoodaroundcasthecriticalregion. 82

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PotentialofmeanforceG0()(solidline)andcorrelationtimeoftherandomforce,)]TJ/F3 11.95 Tf 5.31 1.8 TD[((),(dashedline)actingonthereactioncoordinate. Thisabsenceofabarrierforthemonomeradsorptionisdeceptive.AsdiscussedinSection 4.2.4 ,thePMFobtainedfromtheconstrainedsimulationscorrespondstothesystemmotionalongtheMEP.However,thesystemislikelytodeviatefromtheMEPduringthemonomeradditionduetolackofseparationoftimescalesbetweenthetranslationalDOFandthemonomerorientationandthemicellarshape.ThedeviationsfromtheMEParelikelytoleadtoanenergybarriernotdetectablebytheone-dimensionalenergyproleshowninFig. 4-2 .Inwhatfollowswesystematicallyinvestigatecontributionsofvariousdegreesoffreedomtotheenergylandscape.However,theimportanceofadditionaldegreesoffreedomcanbegaugedbymeasuringtimescalesoftheuctuationsoftherandomforce\(,t)actingon.Ifistheonlyrelevantreactioncoordinatedescribingtheprocessandallotherdegreesoffreedomcanbeneglectedinmodelingtranslationalmotionofthemonomer,thetimescaleof\(,t)shouldcorrespondtothetimescaleofcollisionsbetweenindividualatoms,i.e.picoseconds.TheACF,C)]TJ/F3 11.95 Tf 5.31 1.8 TD[((;t),of\(,t)decaysveryquicklyifthemonomerislocatedsufcientlyfarfromthemicelleorarounditsequilibriumpositioninsidethemicelle.ThiscanbeseenfromtheACFscorrespondingto=2.77nmand=0.33nm,respectively,showninFig. 4-3 .TheACFscanbe 83

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RepresentativeautocorrelationfunctionsC)]TJ/F3 11.95 Tf 5.32 1.8 TD[((;t)oftherandomforce\(t;)actingonthemonomerpartiallyinsertedintothemicelle. approximatedbyanexponentialdecaylaw,Eq. 3 ,wherethecorrelationtime)]TJ/F1 11.95 Tf -430.62 -22.11 TD[(denedas1isontheorderofafewpicoseconds.Thetimescaleoftheuctuationsof\(;t)whenthemonomerisinsidethemicelleissomewhathigherthanthatwhenthemonomerisinthebulksolution.Thisistobeexpected,sincetheBrownianforceactingonthemonomerinsidethemicelleiscausedbycollisionswithbulkier(andhence,slower)surfactantmolecules.TheACFchangesdramaticallywhenthemonomerislocatedaroundtheentrypointintothemicelle,seeACFat=1.41nminFig. 4-3 .Inthiscase,theACFfollowsadoubleexponentialdecaylaw,Eq. 3 ,wherethefastdecaytime1isstillsmall(10ps)andcorrespondstothethermalmolecularcollisions.Theslowdecaytime2isordersofmagnitudelargerthan1andcorrespondstocouplingofthereactioncoordinatewithslowcollectivemotionofmolecules,suchasuctuationsofthemicellarshape.WhentheACFfollowsthedoubleexponentialdecaylaw,wedenethecorrelationtime)]TJ/F3 11.95 Tf 5.32 1.79 TD[(()oftherandomforceas2(),i.e.thetimescaleoftheslowestdecayofC)]TJ/F3 11.95 Tf 5.32 1.79 TD[((;).Dependenceofthecorrelationtime)]TJ/F3 11.95 Tf 5.32 1.79 TD[(()onissuperimposedontheplotofG0()inFig. 4-2 .Thelargeincreasein)]TJ/F1 11.95 Tf 8.64 1.79 TD[(nearthepointofentryofthemonomer 84

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24 ],lipidbilayers[ 73 ]andsurfactantmonolayersatinterfacesinChapter 3 .Inthesesystemsthecorrelationtimeofthestochasticforceincreasesbyordersofmagnitudewhenthesoluteapproachesthemembraneortheinterfaceduetocouplingbetweenthesolutemotionandundulationsoftheinterfaceorthemembrane.Similarly,inSection 4.3.2 weshowthatthelargeincreaseof)]TJ/F1 11.95 Tf 8.64 1.79 TD[(duringthemonomeradditiontothemicellecorrespondstocouplingbetweenthemonomertranslationanductuationsofthemicellarshape.DependenceofthefrictioncoefcientonthemonomerpositioncomputedfromEq. 2 isshowninFig. 4-4 .Clearly,theslowdecayof\(;t)intheneighborhoodoftheentrypointintothemicelleleadstoatwoordersofmagnitudeincreaseofthefrictioncoefcient.Simplistically,onecansaythatthebarriertothemonomerentryintothemicelleisnotduetotheenergyincreasebutduetothefrictionincrease.However,amoreaccurateapproachwouldbetogobeyondtheone-dimensionalmodel,sincetheeffectivefrictionincreaseisduetotheslowuctuationsoftherandomforce,whichinturnviolatestheMarkovianassumption(Eq. 2 )oftheone-dimensionalmodel.Wenotethatthisisdifferentfromthatreportedintheearlierworkofourgroup[ 9 ].Thereasonisthatin[ 9 ],theintegrationoftheACFC)]TJ/F3 11.95 Tf 5.32 1.8 TD[((;)forthecalculationof()wastruncatedatasmall.However,amoredetailedanalysisofC)]TJ/F3 11.95 Tf 5.32 1.8 TD[(()indicatesthatC)]TJ/F3 11.95 Tf 5.32 1.79 TD[(()decaysratherslowlyinthecriticalregion,asseeninFig. 4-3 .And,althoughthevaluesofC)]TJ/F3 11.95 Tf 5.32 1.79 TD[(()aresmall,theslowdecayofC)]TJ/F1 11.95 Tf 8.64 1.79 TD[(leadstolargevaluesofinthecriticalregion.Aswillbeshownbelow,theslowcontributiontotherandomforcecanbeaccuratelyexplainedbycouplingbetweenmultipleslowDOFsofthesystem.Therefore,inordertodescribethetranslationalmotionofthemonomer,itisnecessarytoextract 85

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Dependenceofthefrictioncoefcientonthereactioncoordinate.Thinblacklineshowsobtainedbythedirectintegration.ThickgraylineshowsobtainedbyintegrationoftheACFwithremovedslowlydecayingcomponent. thecontributionofonlythefastcomponentof\(;t)tothefrictioncoefcient.ThiscontributionisshownbygraylineinFig. 4-4 .BeforeproceedingtoanalysisofotherDOFs,weshowthattheone-dimensionalmodelsignicantlyunderestimatesthemonomeradditionrate.Themeanadditiontimeaddcorrespondstothemeantimeofthemonomertransportfromsomeinitialpoint0totheequilibriumlocationeq=0.21nminsidethemicelle.Accordingtotheone-dimensionalmodel[ 27 ], 4-5 weshowdistributionofthemonomeradditiontimesobtainedfromtheunconstrainedsimulation.Thedistributioncanbemodeledbythe 86

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Distributionofthemonomeradditiontimesobtainedfromtheunconstrainedsimulations. exponentialdistributionwiththemeanadditiontimeadd=295.2124.09ns.Therefore,theone-dimensionalmodelunderestimatestheadditiontimebyafactorofsix.Thisresultprovidesfurthersupportforexistenceofanenergybarriernotcapturedbytheone-dimensionalenergyproleand,hence,importanceofotherDOFsduringthemonomeraddition. 4.3.1 ,thesignicantincreaseofthecorrelationtimeoftherandomforcewhenamonomerentersamicellesuggestscouplingbetweenthemonomermotionandthemicellarshape.ThisisconrmedbydependenceoftheaverageradiiofgyrationhRg,iiofthemicelleonthereactioncoordinateshowninFig. 4-6 a.Thelargestradiusofgyration,Rg,1,increasesinthecriticalregion,whilethesmallestradiusofgyration,Rg,3,decreases.Moreover,inthecriticalregiontheprincipaldirectiondg,1correspondingtothelargestradiusofgyrationbecomesalignedwiththevectorRCconnectingthemicellarandmonomerCOMs.ThiscanbeseeninFig. 4-6 ,whichshowstheaverageorientations,cosi,ofdg,iwithrespecttoRC.For>c,cosi0.5,i.e.theprincipaldirectionsofgyrationareorientedatrandom.However,as 87

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BFigure4-6. (a)AverageradiiofgyrationhRg,ii;(b)Averageorientationscosioftheprincipaldirectionsdg,iofgyrationwithrespecttovectorRC.TheradiiofgyrationareorderedsothatRg,1Rg,2Rg,3. themonomerapproachesthemicelle,theattractiveinteractionsbetweenthemonomertailandmicellarcorepullthemicelletowardsthemonomer,asillustratedinFig. 4-7 Figure4-7. Schematicrepresentationofeffectsofamonomeronthemicellarshapeduringthemonomeradsorption/desorption. Themonomer-induceddeformationofthemicelleisfurtherquantiedbyLegendremodesofthemicellarshells.Averagevaluesofthecoefcients~L(i)k()hL(i)ki()fortheleadingfourmodesofseveralshellsareshowninFig. 4-8 .Forreference,shapesrepresentedbythesemodesareshownasinsetsinFig. 4-8 .Magnitudes~L(i)0ofthezero-thmodescorrespondstotheaverageradiioftheshellsandothermodesdescribedeviationsofthemicellarshapefromthesphere.Asexpected,theaverageradiiofallmicellarshellsslightlyincreaseasthemonomerentersthemicelle(Fig. 4-8 a).However, 88

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B C DFigure4-8. AveragevaluesoftheLegendremodesforshellsS(C1),S(C4),S(P1),andS(P4):(a)~L(i)0~L(i)0()~L(i)0(+1),where~L(i)0(+1)istheaveragemagnitudeofthe0-thmodeofanequilibriummicelleintheabsenceofthemonomer;(b)~L(i)1,(c)~L(i)2,and(d)~L(i)3.Theinsetsshowshapesrepresentedbythemodes;inplots(b)-(d),theshapeofthe0-thmodeisshownbyadashedlineforcomparison. thisincreaseisnotmonotonousforthecoreshells.Thisislikelyduetothemicellarelongationwhichtakesplacewhenthemonomerentersthemicelle.Whenthemonomerissufcientlyfarfromthemicelle(2.5nm),theaveragemagnitudeofallmodesL(i)kwithk>0iszero,whichindicatesthat,onaverage,the 89

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4-8 c).Essentially,oncethemonomertailcomesintocontactwiththemicellarcore,itpullsthecoretowardsitself.Theelongationisthelargestnearthepointofrstcontactbetweenthemonomerandthemicelle.Asthemonomermovesfurtherintothemicelle,micellardeformationnecessarytomaintainattractiveinteractionsbetweenthemonomertailandthemicellarcoredecreases,whichisindicatedbythedecreaseof~L(i)2.Oncethemonomermovesbeyonditsequilibriumpositioneq,itpullsothersurfactantswithitsothatthemicellebecomescompressedinthedirectionofthepolaraxisRC.Qualitativesimilarityofthe-dependenceof~L(i)2forallmicellarshellsS(i)indicatesthatthemicelleasawholeundergoesanelongationduringthemonomeradsorption.However,themagnitudeofchangeof~L(i)2decreasesasthedistancebetweentheshellS(i)andthemicellarcenterincreases.Asaresult,elongationoftheoutermostcoronashell,S(P4),isverysmall.ThedifferencebetweenresponsesofthemicellarcoreandcoronatothemonomerismoreclearlydemonstratedbybehaviorofmodesL(i)kwithk6=0,2.Consider,e.g.,modesL(i)1andL(i)3describingasymmetricdeformationsalongthepolaraxis(Fig. 4-8 bandFig. 4-8 d).Forcoreshells,thesemodesexhibittrendssimilartothoseofL(i)2.Thisindicatesthat,inadditiontothesymmetricdeformationsharedbythecoreandthecorona,thereisalsoanasymmetricdeformationofthecorealonetowardsthemonomer.Incontrast,densityofthecoronabeadsintheneighborhoodofthemonomer 90

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4-8 indicates,themostsignicantdeviationsof~L(i)k()fromtheirequilibriumvaluestakeplaceinthecriticalregion.Extremaof~L(i)k()inthecriticalregionaresummarizedinFig. 4-9 .Wewillrefertotheseextremaasthepeakvaluesof~L(i)kanddenotethemasL(i)k,peak.ForallcoreshellsandcoronashellS(P1)closesttothecore,thepeakoflargestmagnitudeisobservedforthesecondmode.Inotherwords,deformationoftheseshellsisdominatedbytheoverallelongationofthemicelle.Forallothercoronashells,theelongationeffectiscomparablewithorislesssignicantthanthedepletionofthehydrophilicbeadsaroundthemonomer.SignicanceofthedepletionincreaseswiththedistanceoftheshellfromthemicellarCOM.Thisindicatesthatbeadsoftheoutershellsaremorereadilydeectedbythemonomer.Contributionofhigherordermodesisrelativelysmalland,forsufcientlylargek,jL(i)k,peakjdecreasesmonotonouslyaskincreases.Thisdecreaseisslowerforthecoronashellsandtherelativecontributionofthehigherordermodesismoresignicantfortheseshells.Thisindicatesthatsmall-scalefeaturesdescribedbythehigherordermodesaremoredominantforthecoronashells,whilethecoreshellsarerelativelysmooth.Thisistobeexpected,sincerepulsionofthehydrophobiccorefromthecoronaandwaterrequiresminimizationofthesurfaceareaofthecore.Thecoreshapeisdeterminedbyabalancebetweenthesurfacetensionandattractiontothemonomertail.Nosuchrestrictionsapplytothecoronashells,whichresultsintheirabilitytorespondtothemonomerpresensebyamorelocalizedshapechange.ProbabilitydistributionsofuctuationsofthemicellarmodesaroundtheirmeanvaluescanbeadequatelyapproximatedasGaussiandistribution.Therefore,inordertofullycharacterizethesedistributions,itissufcienttoconsiderstandarddeviations[L(i)k]

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ExtremaL(i)k,peakoftheaverageLegendremodes~L(i)k()inthecriticalregion.Forthe0-thmode,L(i)k,extremaof~L(i)0=~L(i)0()~L(i)0(+1)areshown.Dataforthecoreandcoronashellsareshownbydashedbluelineswithopensymbolsandsolidredlineswithlledsymbols,respectively. oftheLegendremodesandcorrelationsbetweenthem.Dependenceof[L(i)k]onissubstantiallyweakerthanthatof~L(i)k(),ascanbeseenfromFig. 4-10 .Magnitudeofuctuationsofsomemodesisessentiallyindependentofthemonomerposition.Othermodesexhibitamoderateincreaseoftheuctuationmagnitudeinthecriticalregion.Thisincreaseisassociatedwithuctuationsofthemicellebetweenthesphericalandelipsoidalshapes.Sincedeviationsof[L(i)k()]fromtheirequilibriumvaluesarerelativelysmall,thefollowingdiscussionofqualitativepropertiesof[L(i)k]isfocusedonanequilibriummicelle.Thevaluesof[L(i)k]atequilibriumareshowninFig. 4-11 .ThemagnitudeofuctuationsofmodesL(i)kwithwavenumbersk=0and1issimilarforthecoreandcoronashells.However,modeswithlargerwavenumbers(k3)undergomuchlargeructuationsinthecoronathaninthecore.Thisdifferenceiscausedbythesurfacetensionatthecore-coronainterfacewhichdampenslarge-wavenumbeructuationsofthemicellarcore.Moreover,thestandarddeviations[L(i)k]ofthecoronamodesincreaseasthewavenumberincreases,whichinidicateslargeructuationsofsmaller-scalefeaturesofthecoronastructure.Fluctuationsofsomecoreshellalso 92

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B C DFigure4-10. Standarddeviations[L(i)k]oftheLegendremodesforshellsS(C1),S(C4),S(P1),andS(P4):(a)[L(i)0],(b)[L(i)1],(c)[L(i)2],and(d)[L(i)3]. exhibitanincreasewiththewavenumberfork4.However,therateofthisincreaseisslowerthanthatforthecoronashells.FurtherinsightintomicellardynamicsisgainedbyconsideringcorrelationsbetweentheLegendremodes.Wedenotethecorrelationcoefcientbetweenmodesk1andk2ofshellsS(i)andS(j)byC(i,j)k1,k2.CorrelationcoefcientsC(i,j)k,kbetweenLegendremodesL(i)kwiththesamewavenumbercorrespondingtodifferentshellsofanequilibriummicelleareshowninFig. 4-12 .WiththeexceptionofmodesL(i)1,correlationsbetweenthecoreandcoronamodesarequiteweak.Ontheotherhand,modesL(i)1ofthecore 93

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StandarddeviationsoftheLegendremodesofanequilibriummicelles.Dataforthecoreandcoronashellsareshownbydashedbluelineswithopensymbolsandsolidredlineswithlledsymbols,respectively. andthecoronashellsexhibitstronganticorrelation.Thisimpliesthatanincreaseofthetailgroupdensity(formationofacoreprotrusion)inoneofthemicellarhemispherescausesadensitydecrease(formationofconcavity)inthecoronaofthishemisphere.Inotherwords,theprotrusioninacorepushesthecoronagroupsapartevenintheabsenceofanexternalperturbation.ThisbehaviorechoeseffectsofamonomerontheaveragevaluesofL(i)1(Fig. 4-8 b).Wenotethat,inpart,thisanticorrelationbetweenuctuationsofL(i)1isrequiredbyEq. 4 ,whichyields 4 followsfromageometricconstraintanddoesnotdependonphysicalpropertiesofamicelle.However,thisgeometricconstraintdoesnotspecifywhichofthecovariancesshouldbenegative.Hence,thefactthatthecorrelationcoefcientsarenegativeforcorona-coreshellpairsonlyandarequitelargeandpositivewithinthecoronaandcoreshellsreectsthephysicalpropertiesofthemicelle. 94

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CorrelationcoefcientsC(i,j)k1,k2betweenLegendremodeswiththesamewavenumberkbutcorrespondingtoshellsS(i)andS(j)ofanequilibriummicelle:(a)k=0,(b)k=1,(c)k=2,and(d)k=3 95

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4-13 .Foreachshell,thestrongestcorrelationsarebetweenmodesL(i)0andL(i)2,whichindicatescouplingbetweenuctuationsoftheaverageradiusofthemicelleanditselongation.Correlationsbetweenmodesofthecoronashellsareweakerthanthoseofthecoreshells.Moreover,forthecoronashells,correlationsbetweenmodesotherthanL(i)0andL(i)2arenegligibleforall.Forthecoreshells,correlationsbetweenotherpairsofmodesbecomesignicant.ParticularlystrongcorrelationsarebetweenmodeL(i)2andotherlow-wavenumbermodesL(i)k,k4.ThestrongestcorrelationsbetweendifferentmodesareobservedwithinshellS(C2).Theforegoingdiscussionallowsustodevelopamodelforthecontributionofthemicellarstructuretothefreeenergyofthesystem.Tothisend,weintroducevector 96

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4 .Wedenotethepreferredmicellarcongurationcorrespondingtothemonomerconstrainedatby~L().Analysisoftheconstrainedsimulationsshowsthat,foreachmonomerposition,theprobabilitydistributionoftheLegendremodescanbeapproximatedbyaGaussiandistribution, 2L()A()L(),(4)where 4-14 a.ThisplotshowsthesecondLegendremode(i.e.themodecorrespondingtoelongation)ofoneofthecoreshells.AccordingtomodelEq. 4 ,theminimumenergypath(MEP)correspondstothepathL()=~L(),i.e.theaveragemicellarshapesdiscussedabove 97

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BFigure4-14. Cross-sectionsof(a)equilibriumand(b)dynamicfreeenergylandscapesbytheL(C4)2plane(inkBTunits). correspondtothemicellarshapesalongtheMEP.Themonomer-inducedmicellardeformationleadstoasharpturnoftheminimumenergypath(MEP)inthecriticalregion.ThepreferredmicellarshapealongtheMEPquicklychangesfromspheretoellipsoidelongatedtowardthemonomeratthecriticalpointatwichthemonomertailstartsfeelingattractiontothemicellarcore.Iftheturnissufcientlysharpandthedifferencebetweenthetimescalesofmotioninthe-andL(i)k-directionsissufcientlylarge,itislikelythatthesystemwilldeviatefromtheMEP.InordertoestimatetimescaleoftheLegendremodes,weconsidertheirACFsC(t).TypicalexamplesofC(t)areshowninFig. 4-15 .ThedecayoftheACFscanbeapproximatedbyalinearcombinationofexponentialset=.Thisisconsistentwiththequadraticmodel,Eq. 4 fortheenergyoftheLegendremodes[ 27 ].Wedenethecorrelationtime(i)koftheLegendremodeL(i)kasthelargestfactorintheseexponentials,i.e.thecorrelationtimecorrespondstotheslowesttimescaleofuctuationsofL(i)k(t).CorrelationtimesoftheLegendremodesoftheequilibriummicelleareshowninFig. 4-16 .Correlationtimesofmostmodesexceed100psandtheyareevenlarger 98

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AutocorrelationfunctionsoftherstfourLegendremodesoftheradiusofmicellarcore. forthedominantmodes(L1,L2,andL3).Thesetimescalesarecomparablewiththoseoftheslowcomponentoftherandomforceactingonthemonomer(Fig. 4-2 ),whichprovidesanadditionalevidencethatthenon-Markoviannatureofthisforceisduetothecouplingbetweenthemonomertranslationanductuationsofthemicellarshape.SincethetimescaleoftheLegendremodesismuchslowerthanthatofthetranslationalmotionofthemonomer,themicellarmodesarenotlikelytorespondtosmallchangesofthemonomerpositioncorrespondingtolargechangesof~L()inthecriticalregion.Therefore,themostlikelypath(MLP)ontheenergylanscapecanbeapproximatedasL()=~L(+1).Thisisconrmedbyourunconstrainedsimulationsofthemonomeraddition.Across-sectionofthedynamicfreeenergyG(dyn)byaL(i)kplaneobtainedfromthesesimulationsisshowninFig. 4-14 b.Itisevidentthatthereisalmostnochangeinthemicellarshapeasthemonomerentersthemicelle.Therefore,freeenergyGMLP,in()alongthepathtakenbythemonomerasitentersthemicellecanbeapproximatedas 99

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Correlationtimes(i)kfortheLegendremodesofshellsS(i)ofthemicelleatequilibrium.Dataforthecoronashells(i=P1,...,P4)areshownbysolidredlinesandlledsymbolsandforthecoreshells(i=C1,...,C4)bydashedbluelinesandopensymbols AswediscussedinSection 4.2.4 ,thefreeenergyGMEPalongtheMEPdeviatesfromthePMFG0obtainedfromtheconstrainedsimulations.ItwasshowninRef.[ 73 ]thatharmoniccouplingEq. 4 requiresthefollowingcorrectiontoobtainGMEPfromG0: 4-17 ,thiscorrectionleadstoanincreaseofGMEPinthecriticalregion.Eq. 4 impliesthatthisincreaseisassociatedwiththeincreaseofthemagnitudeofthemicellaructuations(Fig. 4-10 ).DeviationsofthesystemfromtheMEPduringthemonomeradditionleadtoanadditionalincreaseofthefreeenergyinthecriticalregion.ItislikelythattheMLPforthemonomerdesorptionfromthemicellealsoexhibitsdeviationsfromtheMEP.Sincedesorptionisaslowactivatedprocess,themicellarshapewillhavesufcienttimetoadjustitselftothemonomerposition.Thelargestdeformationsofthemicelletakeplacerightbeforethecriticalpoint(Fig. 4-8 ).Oncethemonomerpassesthecriticalpoint,itsmotionbecomespurelydiffusive,i.e.fast.Asaresult,whenthemonomerpassesthecriticalpoint,themicellarshaperemainsconstant. 100

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4-17 .Wenotethattheaboveapproximationisinvalidforthemonomerinsidethemicelleandafteritmovedsufcientlyfarawayfromthemicelle,sinceatthesemonomerlocationsthemicellewillrelaxtowarditspreferredshape.Therefore,theonlytrustworthysegmentofGMLP,out()isforinthecriticalregion.Clearly,thenon-adiabaticcouplingbetweenthemonomerandthemicelleleadtoasubstantialincreaseofthebarrierforthemonomerremoval. Figure4-17. ContributionofthemicellardeformationtothefreeenergyGMEP()alongtheMEPaswellasthefreeenergiesGMLP,in()andGMLP,out()alongtheMLPsforthemonomeradsorptionanddesorptionto/fromthemicelle.PMFG0()obtainedfromtheconstrainedsimulationsisalsoshownforcomparison. 9 ].Here,weperformarigorousassessmentofthismodelandcompareitwiththemodeldevelopedinSection 4.3.2 .Withintheframeworkofthealternativemodel,eachpointonthemicellarsurfaceisconsideredtobeeitherhydrophilicorhydrophobic.AsurfacepointisdenedashydrophobicifthelinepassingthroughthispointandthemicellarCOMdoesnotintersectthemicellarcorona.Inotherwords,thehydrophobicpointscorrespondtothe 101

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4.2.2 ).Sincethemonomer-induceddeformationofthemicellarcoreisdominatedbythesecondLegendremode(Fig. 4-8 ),theseapproximationsareexpectedtoprovideanadequaterepresentationofthemicellarsurfacearea.Theprocessofmonomeradsorptionintoamicelleismostlikelytobeaffectedbythepatchclosesttothemonomer.Wewillrefertothispatchasthecentralpatch.Inordertolocateit,weconsiderthelineconnectingthemicellarCOMandtheC1beadof 102

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4.3.2 indicatesthatthemicellardegreesoffreedomexhibitthelargestdeviationfromtheequilibriumwhenthemonomerislocatedatthecriticalpoint,=c.Therefore,weinvestigatepropertiesofthecentralpatchcorrespondingtothemonomerconstrainedat=c.TheprobabilitydistributionP(A)ofthecentralpatchareaat=cisshowninFig. 4-18 .Forcomparison,thisgurealsoshowsthedistributionofareasofcentralpatchesinanequilibriummicelle,whicharedenedaspatchesclosesttoarandomlychosenaxis.Thisapproachtocomputingthepatchdistributioninanequilibriummicelleischosentomimicthatforanon-equilibriummicelle-monomersystem.Itdiffersfromasimplecountingofthenumberofpatchesofvariousareasforeachofthemicellarconguration:eventhoughthenumberofsmallpatchesonthemicellarsurfaceisrelativelylarge,theirsmallsizeimpliesthattheyarelesslikelytobetheclosestonestoapre-denedaxis.Indeed,Fig. 4-18 showsthat,uptoacertainsize,P(A)increasesasthepatchareaAincreases.P(A)reachesamaximumatA3nm2anddecreasesveryfastforAexceedingthisvalue. Figure4-18. ProbabilitydistributionsP(A)oftheareaAofthecentralpatchesinamicellewiththemonomerconstrainedatthecriticalpoint=c(solidline)andinanequilibriummicelle(dashedline). 103

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4-18 indicatesthatthecentralpatchdistributioninanequilibriummicelleisqualitativelysimilartothatinamicellewithapartiallyinsertedmonomer.However,thereisanoticeablequantitativedifferencebetweenthesedistributions.Specically,presenceofthemonomeratthecriticalpointincreasestheprobabilityofformationoflargerpatches.Thispartiallyconrmstheadhocassumptionmadeintheearlierwork[ 9 ]thatthemonomeradsorptiontakesplaceonlyiftheinitialpointofcontactbetweenthemonomerandthemicellebelongstoahydrophobicpatch.However,amoredetailedanalysisofthecentralpatchesindicatesthatthesituationismorecomplexthanassumedinRef.[ 9 ].Inparticular,inRef.[ 9 ]itwasassumedthatthecentralpatchremainsstableduringtheentireadsorptionprocess,i.e.asthemonomermovesfromthecriticalpointctoitsequilibriumlocationinsidethemicelle.Thetimescaleofthisprocessis200ps.Incontrast,weobservethatthepatchareauctuationsarequitelargeandfast,evenwhenthemonomerisconstrainedatthecriticalpoint(presumably,thelocationthatcorrespondstothemoststablepatchesincontactwiththemonomer).Thisisevidentfromtheevolutionofthecentralpatcharea,suchasthatshowninFig. 4-19 .Note,forexample,alargejumpinthepatchareaA(t)betweentimesta=225psandtb=237.5ps.ThesystemcongurationsatthesetimesareshowninFig. 4-20 .TherearetwoprocessesresponsibleforthelargejumpinA(t):(1)break-upofthelargecentralpatchduetodisappearanceofanarrowpassageconnectingtwoofitscomponents(withthesmallersegmentoftheoriginalpatchbecomingthenewcentralpatch)and(2)decreaseofthesizeoftheremainingcentralpatch.Bothoftheseeventsarecausedbyverysmallmovementofthemicellarmolecules:visually,themolecularcongurationscorrespondingtotimestaandtbarealmostidentical.Therefore,itwouldbedesirabletodevelopamethodforlteringoutthechangesofthepatchstructurecausedbysmallmolecularmovements. 104

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Representativetimedependenceofthecentralpatchareacorrespondingtothemonomerconstrainedat=c.Thesystemcongurationscorrespondingtot=taandt=tbareshowninFig. 4-20 aandFig. 4-20 b,respectively. InRef.[ 9 ],anadhocmethodforlteringwasproposed,basedonaveragingthepatchareasover200ps(themeantimeofthemonomertransportfromthecriticalpointtotheequilibriumpositioninsidethemicelle).Amorerigorousapproachshouldbebasedoninherenttimescalesofthepatchuctuations.ThesetimescalescanbeestimatedbyconsideringtheACFsofthecentralpatcharea.RepresentativeACFsofA(t)correspondingtodifferentmonomerlocationsareshowninFig. 4-21 a.TheACFscanbeapproximatedbydoubleexponentialsEq. 3 .Weconjecturethatthesmallercorrelationtime(1)correspondstothefastuctuationsoftheapparentpatchareacausedbybreak-upsandreconnectionsofpatches,similartothoseshowninFig. 4-20 .Thevaluesofthecorrelationtimes1and2forvariouslocationsoftheconstrainedmonomerareshowninFig. 4-21 b.NearthecriticalpointboththeslowandfastcomponentsoftheACFofA(t)increasebyatleastanorderofmagnitude.The-dependenceof1addsanadditionalcomplicationtodevelopmentofarobustmodelforthesurfacepatches,sinceitsuggeststhattheparametersforthelterofA(t)

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Systemcongurationscorrespondingtopointst=taandt=tbofFig. 4-19 .Leftpanel:molecularconguration;rightpanel:distributionofhydrophobicandhydrophilicpointsonthemicellarsurface.Thehydrophobicpatchesareshowninblackandthecentralpatchesarecircledbyredcurves. 106

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BFigure4-21. (a)ACFsofthecentralpatchareacorrespondingtothemonomerconstrainedatthreedifferentlocations,seelegend;(b)Dependenceofthecorrelationtimes1and2ofthecentralpatchareaonthemonomerposition. alsodependonthemonomerposition.Othercomplicationsincludenecessityto(1)developapopulationbalancemodelfortheformationanddisappearenceofthesurfacepatchesand(2)accountforthepatchshapeandlocation.ThismakesdevelopmentofamodelforthemicellarstructurebasedonthehydrophobicsurfacepatchesmuchmoreinvolvedthanmodelingofthemicellarstructureusingtheLegendreexpansionofthemicellarshells.Inparticular,thelattermodeldoesnotrequirelteringanddynamicsoftheLegendrecoefcientscanbedescribedbylinearLangevinequations.Moreover,themodelbasedontheLegendreexpansionaccountsforboththemicellarshapeanddynamicsofthehead-groupssurroundingthemonomer,whereasthemodelbasedonthesurfacepatchescanonlydescribethehead-groupdynamics.Therefore,intheremainderofthisChapter,wewillusetheLegendre-expansion-basedmodelforthemicellarstructure. 4-1 a).DirectionsofthesevectorsarechosensothatdTpointsfrom 107

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BFigure4-22. Contributions^G(,cosi)oftheorientationscosiof(a)tailand(b)head-groupofthemonomertothefreeenergy(inkBTunits). C1toC4anddHpointsfromP1toP4.Anglesbetweendirectorsdiandthepolaraxis(i.e.,vectorRCconnectingthemicellarandthemonomerCOMs)aredenotedbyi(i=TorH).Contributions^G(,cosi)ofthedirectororientationscositothefreeenergyareshowninFig. 4-22 andthecorrelationsbetweentheorientationsareshowninFig. 4-23 .Asexpected,awayfromthemicellethehead-andtail-groupdirectorsdonothaveapreferredorientation,asindicatedbyindependenceofthefreeenergyofcosTandcosH.Ontheotherhand,inthecriticalregionbothdTanddHexhibitstrongpreferencefortheorientationnormaltothemicellarsurface(cosT=cosH=1)andfreeenergyofunfavorableorientationsisprohibitivelyhigh.Thepreferenceforthenormalorientationstillholdswhenthemonomerisinsidethemicelle,albeittherangeoftheuctuationsaroundthepreferredorientationinsidethemicelleislargerthaninthecriticalregion.Moreover,uctuationsofdTanddHinsidethemicelleareessentiallyindependentofeachother,whereastheyarestronglycorrelatedinthecriticalregion(Fig. 4-23 ). 108

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CorrelationcoefcientbetweenorientationscosTandcosHofthemonomertail-andhead-groups. Thebottleneckinthefreeenergylandscapecausedbytheconstraintsonthemonomerorientationinthecriticalregionisexpectedtosignicantlyreducetherateofthemonomeradsorptionintothemicelle.Thisisconrmedbyourunconstrainedsimulations.ProjectionofatypicaltrajectoryontothecosTplaneisshowninFig. 4-24 .Themonomermakesmultipleattemptstoenterthemicellebutmostoftheseattemptsareunsuccessfulduetounfavorablemonomerorientation.Oncethemonomerapproachingthemicelleacquiresafavorableorientation,itentersthemicelleveryquickly.TheunconstrainedsimulationsaresummarizedbythedynamicfreeenergiesG(dyn)(,cosi)showninFig. 4-25 .Unlikethecaseofthemembranestructure,thedynamicandequilibriumfreeenergiesofthemonomerorientationsarefairlysimilar.Thisfurtherconrmsthatthedynamicsofmonomerorientationindeedplaysanimportantroleinthemonomeradditionprocess.Priortoestimationofeffectsoftheconstraintsonthemonomerorientationsonthemonomeradsorptionintothemicelle,weverifythateffectofothermonomerDOFsonthisprocessisnegligible.Thisisdonebycomparingthetimescalesoforientationsof 109

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Projectionofatrajectoryobtainedfromtheunconstrainedsimulationsontothe-cosTplane. BFigure4-25. ContributionsG(dyn)(,cosi)oftheorientationscosiof(a)tailand(b)head-groupofthemonomertothedynamicfreeenergy(inkBTunits). 110

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3 ,Eq. 3 ),andthecorrelationtimeofamonomerDOFisdenedasthelargestofthevaluesofkintheexponentialapproximationtotheACF.WedenotethecorrelationtimecorrespondingtoACFCiasi(i=T,H,TH,(H,int),and(T,int)).DependenceofthecorrelationtimesofthemonomerDOFsonisshowninFig. 4-26 a.CorrelationtimesTandHareverysimilar.Theyarefairlylarge(ontheorderof1ns)awayfromthemicelleandfurtherincreasebyanorderofmagnitudeinthecriticalregion.ThisprovidesanadditionalconrmationofimportanceofthedynamicsofdTanddHduringthemonomeraddition.Awayfromthemicelle,correlationtimesTH,T,int,andH,intareatleastanorderofmagnitudelowerthanTandH.ThisindicatesthatcosHTandtheinternalDOFsareslavedtocosTandcosHinthisregion.Inthecriticalregion,H,intremainssmall,whileT,intandTHbecomecomparablewiththecorrelationtimesofcosTandcosH.However,contributionsoftheslowcomponenttothedecayofCTHandCT,intaresignicantlysmallerthanthoseforCTandCH.ThisisevidentfromFig. 4-26 bwhich 111

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BFigure4-26. (a)Correlationtimesioforientationscosiofthemonomerdirectors(i=H,T,HT)andinternalDOFsofthemonomerhead-andtail-groups(i=(H,int)and(T,int));(b)ACFsCiofthemonomerDOFsinthecriticalregion. showstheACFsofthemonomerDOFsinthecriticalregion.Therefore,dynamicsofcosHandcosTplaysthedominantroleduringthemonomeradditionprocessandallothermonomerDOFscanbeassumedtobeslavedtocosHandcosTandmodeledasMarkovianthermalnoise. 9 ].ItisobservedthatmotionofthesystemalongtheMEPinvolvesmicellarshapedeformations.AnalysisoftimescalesofthemicellarstructureandthemonomertranslationindicatesthatthesystemtrajectoriesarelikelytodeviatefromtheMEPsduringadsorptionanddesorptionofthemonomer.ThistheoreticalpredictionisconrmedbyaseriesofdirectMDsimulationsofthemonomeradsorption.ItisshownthatthedeviationsfromtheMEPscauseanincreaseofthePMFactingonthe 112

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74 ], 4DTZ1cdexp(GMLP,in()=kBT) 75 ], 4-23 ),intherstapproximationweneglectthehead-groupdynamicsandfocusontheroleofthetail-groupintheadsorptionprocess.TheeffectofthemonomerorientationcosTonthemonomeradsorptionratecanbeestimatedbythefollowingsimplemodel.Weassumethatthemonomerundergoes 113

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@2@c(,) @DR12@c(,) 3.5.1 ,andc0isthebulkconcentration.Oncetheconcentrationc(,)isobtained,therateofmonomeradsorptioncanbecalculatedby[ 40 ] 114

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4-27 .Inthecriticalregion,thestrongestcorrelationsarebetweenthetail-grouporientationandtheelongationmodesL(i)2ofthemicelle.Thecorrelationisstrongerwiththecoreshells,sincethemicellarelongationiscausedbytheattractionbetweenthemonomertail-groupandthemicellarcore.Sincethesecorrelationsarenotnegligible,thesystemdynamicsshouldbeanalyzedbytakingthecouplingdynamicsbetweenthemicellarandmonomerDOFs.Inthefuturework,wewillusethepathintegrationapproach[ 44 46 ]formoredetailedanalysisofthesystemdynamics. Figure4-27. CorrelationcoefcientsbetweenorientationcosTofthemonomertailandthemicellarLegendremodesL(i)2(i=C1,C4,P1,P4). 4 forcoefcientsL(i)koftheLegendreexpansionofmicellarshellS(i).FollowingRef.[ 72 ],wedenedensityi(r;t)ofbeadsonthisshell, 115

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4 andEq. 4 inEq. 4 : 116

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7 ]anddrugdeliveryvehicles[ 76 ].Ionicmicellesareasubjectofactiveexperimental[ 25 56 57 ],theoretical,andcomputationalresearch[ 59 61 63 64 77 78 ].MDsimulationsproduceddetailedunderstandingofmolecular-scalefeaturesofionicmicellesatequilibrium.Forexample,MDsimulationsofYoshiietal.[ 77 78 ]investigatedsurfacestructure,criticalmicelleconcentration(CMC),andmicellarsizedistributionofequilibriumSDSmicelles.However,ourunderstandingofnon-equilibriumprocessesinvolvedinmicellarformationanddisintegrationislesscomplete.InChapter 4 ,wehaveshownthatevensuchsimpleprocessesasaddition/removalofasinglesurfactantmoleculeto/fromamicellemayinvolvecoupleddynamicsofmultipledegreesoffreedom,includingthemonomerorientationandthemicellarstructure.Ageneralapproachwasdevelopedwhichenablesonetodescribetheprocessesofthemonomeradditionandremovalbyasystemofstochasticequationsforrelevantdegreesoffreedom.InChapter 4 ,thisapproachwasappliedtoanonionicmicelle.ThegoalofthisChapteristoextendthismethodtoionicsurfactantsystems.Itisanticipatedthattheprocessofamonomeraddition/removalto/fromanionicmicellehasatleasttwonewfeatures.Firstly,thelong-rangeelectrostatic(ES)interactionsleadtotheenergybarrierforadditionoftheionicmonomertothemicelle[ 79 ].Thisbarrierdoesnotexistfornonionicmicellesanditislikelythatthepresenceofthebarrierwillshiftthebalancebetweenrelativeimportanceofvarious 117

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5.2.1CGMDMolecularModelsThemodelionicandnonionicsurfactantsareshowninFig. 5-1 .Tail-groupsofthesesurfactantsaremodeledaschainsofthreehydrophobicbeadsCandhead-groupsaremodeledbyasinglehydrophilicbead.ThepolargroupsoftheionicandnonionicsurfactantsaremodeledbybeadsoftypesQdaandP,respectively.TheonlydifferencebetweenthesebeadsisthenegativechargeoftheQdabead.SinceeachCbeadrepresentsfourmethylene/methylgroups,themodelionicsurfactantscanbeconsideredasamodelfortheSDSsurfactant,CH3(CH2)11OSO3Na.AsmentionedinSection 2.2 ,weassumethatthePbeadmodelstwoethoxygroups.Therefore,thenonionicC3P1surfactantcanbeconsideredasamodelfortheethoxylatedsurfactant,C12EO2.ThecounterionismodeledbythebeadofthesametypeastheSDShead-groupsbutwithapositivesign.Forbrevity,wewilldenotetheQda-typebeadsintheSDShead-groupsandcounterionsbyQandQ+,respectively.ThemagnitudeofchargeofQandQ+beadsisreducedfrom1.0eto0.7etoaccountforthehydrationshellsurroundingtheion[ 10 ].Thereductionofchargeofbothofthechargedbeadsisthesametoensureelectricalneutralityofthesystem.We 118

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Coarse-grainedmodelsofnonionicandionicsurfactantsconsideredinthiswork.Thecoarse-grainedbeadtypesaredenotedasfollows:C=apolar(4methylene/methylgroups),P=polar(2ethoxygroups),QandQ+=chargedbeadgroupwhichactsasahydrogenbonddonorandacceptor[ 10 ]. exploreeffectsofthechargereductioninSection 5.6.3 .Therelativedielectricconstantofthesolventmediumis20.Simulationsareperformedwithtime-stepof0.025ps.Theoriginalcoarse-grainedmodel[ 10 ]assumesstrongscreeningofchargeswitheffectiverangeofelectrostaticinteractions1.2nm.Thisassumptionsignicantlyspeedsupcomputationofelectrostaticforces.However,severalstudieshaveshownthatthisapproximationintroducesartifacts[ 80 81 ]andfailstocaptureessentialcharacteristicsofamphiphilicsystems.Forexample,Burovetal.[ 81 ]haveshownthatneglectingthelong-rangecontributionleadstounlimitedgrowthofmicellarstructureforionicsurfactants.LeeandLarson[ 80 ]investigatedeffectsoflong-rangeESpotentialoninteractionsbetweenadipalmitoylphosphatidylcholine(DPPC)bilayerandpolyamidoamine(PAMAM)dendrimer.Theyshowedthatwhilesomeofthesystemproperties(bilayerstructure,dynamicsoflipidsinsideabilayer)arethesameforlong-rangeandscreenedinteractions,neglectinglong-rangeinteractionsleadstoaqualitativediscrepancyforthemembranepermeabilitybydendrimers.Predictionsofthemodelwithlong-rangeESpredictthatsomePAMAMdendrimersinsertintotheDPPCbilayersandinduceporeformation,whichisconsistentwithexperiments. 119

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5-2 .SincetheSDSsystemincludeslong-rangeESinteractions,itisnecessarytoensurethatthesimulationboxissufcientlylargetopreventinteractionsbetweentheperidocimagesofthesystem.WeshowinSection 5.6 thattheelectrostaticeldgeneratedbythemicelleandcounterionsisnegligiblewhenthedistancefromthemicellarCOMexceeds5.5nmduetothemutualscreeningofthemicellarheadgroupsandthecounterions.Hence,oursimulationboxissufcientlylargetopreventelectrostaticinteractionsbetweenperiodicimagesofthesystem.AscanbeseenfromtheaveragedensityprolesshowninFig. 5-3 ,C3P1micellesaremoredenselypackedthanSDSmicelleswiththesameaggregationnumber.ThisisduetoelectrostaticrepulsionofthenegativelychargedSDShead-groups.Althoughthisrepulsionisexpectedtobepartiallyscreenedbythecounterions,itremainssignicant, 120

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BFigure5-2. Exampleofaninstantaneouscongurationof(a)C3P1and(b)SDSmicelleA40.TailbeadsC,chargedheadbeadsQ,andcounterionsQ+areshownbyblack,white,andgrayspheres,respectively.PolarheadbeadsParealsoshownbywhitespheres.Watermoleculesareomittedforclarity. asindicatedbytheheadgrouppaircorrelationfunctionsg(r)showninFig. 5-4 .Forthepurposesofcalculationofg(r),thedistancerbetweenthehead-groupsiscomputedalongthemicellarsurface.Therstpeakofg(r)forC3P1isthelargest,indicatingthatthehead-groupstendtoaggregate.Ontheotherhand,fortheSDSmicelle,thesecondpeakofg(r)islargerthantherstpeak,indicatingrepulsiveinteractionsbetweenthemicellarhead-groups.Thedistributionsofcounterionsandsurfactanthead-groupspeakatthesamedistancefromthemicellarCOM.However,therearesignicantdifferencesbetweentheshapesofthesedistributions.Thehead-groupshaveanarrowandsymmetricGaussiandistribution,whereasthecounteriondistributionishighlyasymmetricandextendswellintothebulksolvent.VisualinspectionoftheMDtrajectoriesshowsthatthecounterionsarehighlymobileandsomeofthemaredissociatedfrommicelles,seeFig. 5-2 .Toquantifyadegreeofdissociationofions,wesaythatanionisdissociatedfromamicelleifitsdistancefromthemostfarheadbeadfromthemicellarCOMexceedsLJdiameter. 121

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DensityprolesofC3P1(thickgraylines)andSDSmicelles(blacklines)ofaggregationnumber40. Figure5-4. Paircorrelationfunctionofhead-groupsofC3P1andSDSmicellesofaggregationnumber40. Weobservethat20to30%ofthecounterionsaredissociatedfromamicelleinaggreementwithothersimilarstudiesusingatomisticMD,CGMonteCarlosimulationsandexperiments[ 82 84 ]. 30 ].IncalculationofthecenterofmassweignoretheQ+counterions.Initialconditionsfortheconstrainedsimulationsareobtainedfrom 122

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ComparisonofG0()inC3P1andSDSsystems. anequilibriummicellarstructure.Amonomerisrandomlychosenandispulledbyaconstantvelocityappliedatitsheadgroup.Congurationsofthemicelleandthepulledmonomeraresavedfordifferentvaluesofthereactioncoordinate.Theseconformationsserveasstartingcongurationsfortheconstrainedsimulationsfordifferentvaluesofthereactioncoordinate.Allconstrainedsimulationsareperformedfor800nsandthestatisticalanalysisisperformedafter10nsequilibration.Fig. 5-5 showstheobtainedfreeenergyproles.Thereisadistinctequilibriumposition=01.7nmofamonomerinthemicellecorrespondingtothefreeenergyminimum,whichisaboutthesameforboththeC3P1andSDSmicelles.Thefreeenergyincreasesasthemonomerispulledoutofthemicelle.ThefreeenergyoftheC3P1systemisessentiallyindependentofthemonomerpositiononceitiscompletelypulledoutofthemicelle.Ontheotherhand,G0()intheSDSsystemremainsposition-dependentasthemonomerispulledoutofthemicelle.Theselong-rangeinteractionsareduetotheESinteractionsbetweenthemonomerandthemicelle,whichcreateabarrierforthemonomeraddition. 123

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BFigure5-6. Contributions^G(,cosT)ofthemonomerorientationstofreeenergyin(a)C3P1and(b)SDSmicellarsystem. 4.2.2 .ThemonomerorientationTistheanglebetweenthedirectordTandthevectorRCconnectingthemicellarandthemonomerCOMsasdenedinSection 4.3.4 .Thecontributions^G(,cosT)ofthemonomerorientationtothefreeenergyintheSDSandC3P1micellarsystemsarecomparedinFig. 5-6 .Whenthemonomerisfarfromthemicelle,ithasnopreferredorientationasindicatedbytheuniformdistributionofcosT.Nearthemicellarsurface,themonomershowsastrongpreferenceforthenormalorientationwithrespecttothemicellarsurfaceduetoattractionbetweenthemonomertail-groupandthemicellarcore.Asthemonomermovesfurtherintothemicelle,thispreferenceforthenormalorientationbecomesweakerbecausethemonomertail-groupcanbesurroundedbyhydrophobicbeadsevenifitsorientationdeviatesfromthenormalwithrespecttothemicellarsurface.Thisbehaviorofmonomerorientationsisgovernedbyhydrophobic/hydrophilicinteractions,notESinteractions.Therefore,^G(,cosT)inionicandnonionicsystemsarenearlyidentical. 124

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ComparisonofthecorrelationtimescosT()ofthemonomerorientationsinC3P1andSDSmicellarsystem. Fig. 5-7 comparesthecorrelationtimescosTofthemonomerorientationuctuationsinSDSandC3P1micellarsystems.ThecorrelationtimecosTisobtainedbyttingtheACFofcosTtosingleordoubleexponentials(Eq. 3 ,Eq. 3 ).AscanbeseenfromFig. 5-7 ,thecorrelationtimescosTintheSDSandC3P1systemsareingoodagreementfortheentirerangeof.ThisindicatesthateffectofESinteractionsondynamicsoftheuctuationsofmonomerorientationisalsonegligible. 4.3.2 showsthatamonomerpartiallyinsertedintoamicelleleadstodeviationsofthemicellefromasphericalshapeduetotheattractionbetweenthemonomertail-groupandthemicellarcore.Unlikethecaseofmonomerorientations,weexpecttoobserveatleastquantitativedifferencesofthemicellarshapedeformationbetweentheionicandnonionicmicellarsystems.InSection 5.6.2.2 ,wewillshowthatoneofthefactorsthatleadstothesmallerenergybarrierfortheionicmonomerremovalisthesmallerdensityoftheionicmicellecausedbyrepulsiveESinteractionsbetweenthehead-groups.Similarly,thedifferenceinmicellardensityisexpectedtocausesomequantitativedifferencesinmicellarshapedeformationsbetweentheionicandnonionicmicellesevenifthereisnodirecteffectofESinteractions. 125

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5-8 .Forreference,shapesrepresentedbythesemodesareshownasinsetsinFig. 5-8 .SomeofthemodesshowsomewhatdifferentdependencesonthereactioncoordinatewiththoseoftheC4P4micelle.Foreexample,thedependenceofL(Q)3exhibitsasimilarpatternwiththoseofL(i)3forthetailbeads(Fig. 5-8 d).Thisislikelyduetothesegregationofhead-groupsontheSDSmicellarsurfacebecauselocationofsegregatedhead-groupsismoreeasilyaffectedbytailconnectedtothem.However,generalqualitativefeaturesofmostofthemodesshowagoodagreementwithourearlierstudyoftheC4P4micelle.SincethesequalitativefeaturesarediscussedindetailinSection 4.3.2 ,herewefocusondifferencesbetweenSDSandC3P1micelles.MagnitudesL(i)0ofthezero-thmodescorrespondtoaverageradiioftheshells.AsshowninFig. 5-8 a,mostoftheshellsintheSDSmicellehavealargerradiusthanthoseintheC3P1micelle.ThedifferenceoftheshellradiusbetweentheSDSandC3P1micellesisthelargestfortheoutermostshell(i.e.S(P)andS(Q))anddecreasesfortheinnershells.Infact,thereisalmostnodifferencebetweenL(C1)0andL(C2)0.ThisobservationfurtherconrmsthefactthatSDSmicellehasasmallerdensitythanC3P1micelle.Asthemonomerapproachesthemicelle,allLegendremodesexhibitlargesensitivitytothemonomerpositionaroundtheentrypointintothemicelle.Wenotethatfork>1,deviationsofL(i)kfromtheirequilibriumvaluesarelargerfortheSDSmicelle(Fig. 5-8 candFig. 5-8 d).Thistrendismostevidentfortheoutershells(S(C3)andS(P)/S(Q)).ThisislikelyduetothesmallerdensityoftheSDSmicelle 126

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B C DFigure5-8. AveragevaluesoftheLegendremodesforallshellsofC3P1andSDSmicelles:(a)hL0i(b)hL1i,(c)hL2i,and(d)hL3i.Theinsetsshowshapesrepresentedbythemodes;inplots(b)-(d),theshapeofthe0-thmodeisshownbythedashedlineforcomparison. andsegregationofthehead-groupsoftheSDSsurfactants.ThelowerdensityoftheSDSmicelleimplieslargermobilityofsurfactantsinthedirectionnormaltothemicellarsurface,whichcausesalargerdeformationoftheoutershells.Inaddition,theeffectsoftailgroupsaremoresignicantonthesegregatedSDShead-groupsthantheaggregatedC3P1head-groups.Therefore,theSDShead-groupshellshowsaclosershapetothoseofthecoreshells. 127

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ThedifferenceL(i)k,peakofL(i)k,peakbetweenC3P1andSDSmicelles.ThedenitionsofL(i)k,peakandL(i)k,peakaregivenintext. Inconstrast,thevariationsof1stLegendremodeL(i)1aremoresignicantinthenonionicmicelle(Fig. 5-8 b).Thisislikelyduetoahigherdensityinthenonionicmicelle.Recallthatthe1stLegendremodeL(i)1representscollectivedeviationsofallsurfactantsfromasphereastheinsertshowsinFig. 5-8 b.SincetheC3P1surfactantsaremorecloselypackedwithinamicelle,thecoordinationbetweentheneighboringsurfactantsisstronger.Asaresult,thecollectivemotionofthesurfactantsismoresignicantfortheC3P1micelle.ThedifferencesbetweentheLegendremodesL(i)kintheC3P1andSDSmicellesaresummarizedinFig. 5-9 .Inthisgure,weplotthedifferenceL(i)k,peakofthepeakvaluesL(i)k,peakintheC3P1andSDSmicelles.Here,L(i)k,peakisdenedasthevaluescorrespondingtothelargestmagnitudeofhL(i)ki()inthecriticalregion.L(i)k,peakisdenedas 128

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StandarddeviationshL(i)kiofLegendremodesL(i)kofequilibriumSDSandC3P1micelles. fortheoutershells(i.e.S(C3)andS(P)/S(Q)).Infact,thedifferencesbetweentheinnershells(S(C1)andS(C2))arenegligiblefork3.ThestandarddeviationshL(i)kioftheLegendremodesfortheequilibriumC3P1andSDSmicellesareshowninFig. 5-10 .ThequalitativefeaturesofhL(i)kiinbothmicellesaresimilartothoseofC4P4micelles.Specically,themagnitudeoftheuctuationsislargerfortheshellscorrespondingtothesurfactanthead-groups.Fork>2,theuctuationsincreaseasthemodenumberincreases,whichsuggestslargeuctuationsofnedetails.WenotethattherearerelativelylargedifferencesbetweenhL(P)kiandhL(Q)ki.Fork<7,hL(Q)kiaresmallerthanhL(P)kibecausetheeffectsoftail-groupsarelargeronthesegregatedSDShead-groupsthantheaggregatedC3P1head-groups.Ontheotherhand,hL(Q)kiarelargerthanhL(P)kifornedetails(i.e.k>7).Thisislikelyduetothelargemobilityofcounterions.AdsorptionsanddesorptionsofindividualcounterionsonthemicellarsurfacewouldcauselargeuctuationsofnedetailsoftheSDSheadgroupstructure,hencelargehL(Q)ki. 129

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5.6.1AnalysisMethodsForagivenchargedensity,ESpotentialisobtainedbysolvingthePoissonequationinsphericalcoordinatesystemr=(r,,), 5-12 ).InordertosolveEq. 5 ,weexpand(r)and(r)intermsoftheeigenfunctionsfklm(r,,)oftheLaplacianoperatorinsphericalcoordinates, L,(5)wherelmisthem-throotofjl(r).Thesphericalharmonicsaredenedas @xh1x2@ @xi+1 1x2@2 130

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Lsatisfythefollowingequation: (lk)! and L,jllm0r LE=ZL0jllmr Ljllm0r Lr2dr respectively.Therefore,theeigenfunctionssatisfythefollowingnormalizationcondition: LPk0l0(x)jllm0r L (lk)!, wherexcos.Substitutingexpansionsof(r)and(r)intermsoftheeigenfunctions, L, L, 131

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5 andEq. 5 astheBessel-Legendre(BL)expansions.TheBLexpansionofthechargedensityisobtainedbyrepresentingthedensityofthechargesas (lk)!1XiqiPkl(cosi)eikijllmr L.(5)InordertoobtaintheESforceactingonthemonomer,weorientthepolaraxissothatthepolaraxiscorrespondstothereactioncoordinate,i.e.thevectorconnectingtheCOMofthemicelleandthehead-groupoftheconstrainedmonomer.Thepolaraxiscorrespondstocos=1.UsingthepropertyoftheassociateLegendrepolynomials, 21 2,k=00,k6=0,(5)theESpotentialalongthepolaraxiscanbeexpressedbyatruncatedBessel-Legendreexpansion, 21 2~0lmjllmr L,(5)andthecontributionofESforcetotheconstraintforceis 21 2~0lmlm 132

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5.6.2.1AveragechargedensityAsindicatedbyEq. 5 ,ESpotentialalongthereactioncoordinatedependsonlyonthemodes~0lmoftheBLseriesforchargedensity.Forsimplicity,wewillreferto~0lmand~0lmas~lmand~lm,respectively.ThestrongestdependenceonisexhibitedbytheBLmodesforl=2.AveragevaluesofBLmodesforl=2areshowninFig. 5-11 .BLmodesforbothhead-groupsandcounterionsexhibitsharpchangesnearthefreeenergybarrier.However,thevariationoftheBLmodesforthecounterionsisnotassharpasthatofthehead-groupmodes.AsshowninSection 5.5 ,theaveragemicellarshapeisalmostsphericalwhenthemonomerisfarawayfromthemicelleandmicelleundergoesasignicantdeformationwhenthemonomerapproachesthefreeenergybarrier.Sincethehead-groupchargesaredirectlyassociatedwiththemicellarshape,thesharpchangeofthehead-groupchargedensitycorrespondstothemicellarshapedeformationnearthefreeenergymaximum.Ontheotherhand,thecounterionsarerelativelyfreetodissociatefromthemicelle.Therefore,thereponseofthecounteriondensitytothemicellardeformationsisfairlyweak.Asaresult,thevariationofcounteriondensityisnotassignicantasthatofthehead-groupchargedensity. 5 isshowninFig. 5-12 .Toassesseffectsofthemicellarshape,wecomparethispotentialwith<0()>computedundertheassumptionthatthechargedensitiesaresphericallysymmetric.AsshowninFig. 5-3 ,boththehead-groupandthecounteriondensitiesgraduallyincreaseasoneapproachesthemicellarsurfacefromthebulkphase.However,thehead-groupdensityincreasesmorequicklythanthecounteriondensity.Asaresult,theimbalancebetweenthehead-groupandcounterionchargedensitiesbecomesmoresignicantasthemonomerapproachesthemicelleandthetotalESpotentialincreasesasthereactioncoordinatedecreasesasshowninFig. 5-12 a.Wenotethatthetotal 133

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BFigure5-11. DependenceofBLmodes~lmforl=2onfor(a)head-groupchargesand(b)counterioncharges. potentialisalmostnegligiblefor>5.5nmduetomutualscreeningofthehead-groupandcounterioncharges.Therefore,oursimulationboxsize13.413.413.4nm3isindeedsufcientlylargetoavoidESinteractionsbetweenperiodicimages.ThisalsoconrmsthatthesystemsizeL=25nmusedinEq. 5 doesnotintroduceanyboundaryeffects.<()>issmallerthan<0()>forentirerangeof.Thedifferencebetween<()>and<0()>isnoticeableforsmallanddecreasesasthemonomermovesfurtherawayfromthemicelle,seeFig. 5-12 a.Forsmallerandlarger,thedifferencesaremostlyduetothehead-groupsandcounterions,respectively(Fig. 5-12 bandFig. 5-12 c).Sincetheshapeofthecounterionclouddoesnotchangeasmuchasthatoftheheadgroups,thedifferencej<()><0()>jduetothecounterionsisnotaslargeasthatduetothehead-groups.InFig. 5-13 ,weestimatethePMFofSDSmicellewithoutESpotentialbysubtracting<()>fromG0()andcompareittotheoriginalPMFsintheSDSandC3P1micelles.Thenon-EScontributiontoPMFintheSDSmicelleisalmostindependentofwhenthemonomerisoutsideofthemicelle.ThisconrmsthatESinteractionsindeedcausetheenergybarrierfortheSDSmonomeraddition.Theenergy 134

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B CFigure5-12. ESpotentialscausedby(a)totalESeld,(b)head-groupchargesand(c)counterioncharges.Forcomparison,theESpotentialsobtainedfromthesphericallysymmetricchargedensitiesarealsoshown. Figure5-13. FreeenergyproleofSDSmicellarsystemwithoutESpotential.Forcomparison,pullPMFsofSDSandC3P1micellarsystemsarealsoshown. barriersforthemonomerremovalintheSDSandC3P1arealmostthesameiftheESpotentialisremovedfromthePMFoftheSDSmicelle.ThisindicatesthatthemainfactorwhichcausesthesmallerenergybarrierforthemonomerremovalfromtheSDSmicelleistherepulsiveESinteractionbetweenthemonomerhead-groupandtheothersurfactanthead-groups.WhenthemonomerisinsidetheSDSmicelle,theSDSforceseffectivelypushitoutofthemicelleduetointeractionsbetweenthemonomerhead-groupandhead-groupsofothersurfactants.ThesmallerdensityoftheSDSmicellefurtherreducesthebarrier.However,thiseffectisrelativelysmall. 135

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3 ,Eq. 3 ).Inwhatfollows,wewillreferESforcescausedbyhead-groupandcounterionasFE,headandFE,counterion,respectively.InFig. 5-14 a,weshowtheobtainedcorrelationtimesofFE,headandFE,counterion.Forcomparison,thecorrelationtimesofthetotalforce)]TJ/F1 11.95 Tf 10.09 0 TD[(isalsoshown.Thecorrelationtimesof)]TJ/F1 11.95 Tf 10.09 0 TD[(andFE,headincreasenearthefreeenergybarrier.Thisincreaseiscausedbythecouplingbetweenthemonomerandthemicellarshapeuctuations.Sincethecounterionsarenotdirectlyassociatedwiththemicelle,theeffectofmicellarshapeuctuationsonthecounterioncloudisrelativelyweak.Asaresult,thecorrelationtimeofFE,counteriondoesnotshowasignicantincreaseneartheenergybarrier.Wenotethatexceptfortheregionneartheenergybarrier,thecorrelationtimeoftheESforcesislargerthanthatofthetotalforce.ThisindicatesthattheeffectsoftheESforceuctuationsonthetotalforceuctuationsarenegligible.Infact,themagnitudeoftheESforceuctuationsistoosmalltoaffecttheuctuationsofthetotalforce(Fig. 5-14 b).ThisisfurtherconrmedbythefactthatthecorrelationtimesofthetotalforceintheSDSandC3P1micellarsystemsarealmostidenticalasshowninFig. 5-14 c. 10 ]employedinthisstudy,themagnitudeofchargeofchargedbeadsisreducedfrom1.0eto0.7etoaccountforthehydrationshellsurroundingthem.ThisreducedchargewasobtainedbyoptimizingpairdistributionfunctionsofaNaClsolutionpredictedbythecoarse-grainedmodeltoensureagreementwithanatomisticmodel[ 10 ].AsshowninSection 5.6 ,ESinteractionsleadtoseveraldifferencesbetweentheionicandnonionicmicellarsystems.Specically,theESinteractionscauseanincrease 136

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B CFigure5-14. (a)CorrelationtimesofESforceandtotalforceintheSDSmicellarsystem(b)StandarddeviationsoftheESforceandthetotalforceintheSDSmicellarsystem(c)CorrelationtimesoftotalforcesintheSDSandC3P1micellarsystems. andadecreaseoftheenergybarriersforthemonomeradsorptionandremoval,respectively.TherepulsiveESforcebetweentheionicsurfactanthead-groupsleadstosegregationofthehead-groupsonthemicellarsurfaceandsmallerdensityofsurfactanttailswithinthemicelle.InthisSection,weinvestigateeffectsofthechargestrengthontheseproperties.Tothisend,weconsidertheCGMDmodelwiththefullchargestrength1.0e.AllotherforceparametersoftheSDSsystemandallparametersofthesimulationsaresameasdescribedinSection 5.2.1 andSection 5.3 .ThePMFsoftheSDSmicellarsystemswith0.7eand1.0echargestrengthsarecomparedinFig. 5-15 .Surprisingly,weobservethattheenergybarrierforthemonomeradsorptiondecreasesandthatformonomerremovalincreasesasthechargestrengthisincreased.Inotherwords,thePMFoftheionicmicellewiththestrongerchargestrengthbecomesclosertothatofthenonionicmicelle.Inordertounderstandthisunexpectedphenomenon,weconsiderthehead-groupandcounterionchargedensities(r,cos=1)0(r)alongthepolaraxis.Bycalculating0(r)foreachvalueofthereactioncoordinate,weobtaintwodimensionalchargedensityproles(r,).Thedensityprolesfor0.7eand1.0echargestrengthsarecomparedinFig. 5-16 .Wedonotobservesignicantdifferencesbetweenthehead-groupdensities(Fig. 5-16 aandFig. 5-16 c).Sincethehead-groupchargesare 137

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PMFsofSDS-A40micellarsystemswith0.7eand1.0echargestrengths. directlyconnectedwiththesurfactantmoleculeswithinthemicelle,theyaredistributedaroundthemicellarsurfaceregardlessofthechargestrength.Ontheotherhand,thecounteriondensityaroundthemicelleincreasesduetothestrongerattractionbetweenpositivelyandnegativelychargedbeads(Fig. 5-16 bandFig. 5-16 d).Inadditiontotheoverallincreaseofthecounteriondensityaroundthemicellarsurface,weobservealocalincreaseofthecounteriondensityaroundthemonomerheadevenwhenthelatterisfarawayfromthemicellarsurface,seethecounteriondensityaroundtheblacklineinFig. 5-16 d.TheESpotentials<(r)>alongthepolaraxiscalculatedusingEq. 5 arecomparedinFig. 5-17 .Thesystemwith1.0echargestrengthexhibitsalocaldecreaseoftheESpotentialaroundthemonomerheadduetothelocallyincreasedcounteriondensity.Suchadecreaseisabsentinthesystemwiththeweakercharge.Thisexplainsthelargerenergybarrierforthemonomeradditioninthesystemwiththeweakercharge. 138

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B C DFigure5-16. Chargedensities(r,)for(a)head-groupsand(b)counterionswith0.7echargestrengthand(c)head-groupsand(d)counterionswith1.0echargestrength.Locationsofthemonomerhead-groupareshownbythesolidblacklines. employedinthisstudyensuresthatthedifferencesbetweentheconsideredionicandnonionicmicellarsystemsarecausedbyESinteractionsonly.Thelong-rangeESinteractionsinduceasmallerenergybarrierforthemonomerremovalandalargerenergybarrierofthemonomeraddition.RepulsiveESinteractionsbetweenhead-groupsoftheionicsurfactantsleadtosegregationofthehead-groupsonthemicellarsurface.Moreover,ionicmicelleshavesmallerdensitiesthannonionicmicellesduetotherepulsiveESinteractionsbetweenthehead-groups.Asaresult 139

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BFigure5-17. ESpotentials<(r,)>(inkBTunits)for(a)0.7eand(b)1.0echargestrengthsalongthepolaraxis.Locationsofthemonomerhead-groupareshownbythesolidblacklines. ofthisdensitydifferencecombinedwiththesegregatedhead-groups,ionicmicellesundergolargerdeformationsduringthemonomeradditionandremovalprocesses.OurstudyofC4P4micelleshowsthatthesemicellarshapedeformationsleadtocouplingbetweendynamicsofthemonomertranslationalmotionandmicellarshelluctuations.ThiscouplingresultsindeviationsofthemonomeradditionandremovalprocessesfromtheMEPs.Duetoabsentorsmallstaticenergybarrier,thedeviationfromtheMEPisexpectedtohaveamoresignicanteffectonthemonomeraddition.TheseobservationswouldsuggestthatdeviationsofthesystemtrajectoryfromtheMEPduringthemonomeradditionwouldbemoresignicantintheSDSmicellarsystem,sincelargermicellarshapedeformationsindicatelargercouplingstrengthwhenthereisnosignicantdifferenceinthemagnitudeoftheuctuations(Fig. 5-10 ).FormoreaccuratecomparisonbetweenthesystemtrajectoriesfortheSDSandC3P1monomeradditions,letusdeneL,maxandG0,maxaslocationsofthemaximumshapedeformationsandtheenergybarrier,respectively.ForC3P1system,G0,maxcorrespondstothepointatwhichG0()reachesaconstantvalue. 140

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4.3.2 .ThesecondorderLegendremode(k=2)hasthelargestcorrelationtime,140ps,forallshells.ThiscorrelationtimeissimilarinbothSDSandC3P1micelles.Moreover,boththesystemshaveconsistentmeanadditiontimefromG0,maxtoL,maxalongtheMEP.ThemeanadditiontimeestimatedfromEq. 3 isapproximately30ps,whichissignicantlysmallerthanthetimescalesofthemovementtowardtheMEP.TheseobservationssupportthatbothsystemsareunlikelytofollowtheMEPsandthedeviationfortheMEPislargerfortheSDSsystemduetothelargermicellarshapedeformations.Inconclusion,theinvestigationperformedinthisChaptergivesaqualitativeunderstandingofself-assemblyofionicsurfactants.Long-rangeESinteractionsplayanactiveroleinthedynamicsofionicsurfactantsbyshiftingactivationenergiesofaddition/removalofasurfactantmonomerto/fromamicelle.Itisexpectedthatlong-rangeESinteractionswillactivelyparticipateinsuchcomplexprocessesasthefusionbetweenionicmicelles. 141

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5 wediscussedresultsofconstrainedsimulationswiththereactioncoordinatedenedasthelengthofvectorRHconnectingthemicellarCOMwiththemonomerhead-group.ThischoiceisconvenientforanalysisoftheESeffects.Ontheotherhand,inChapter 4 ,wedenedthereactioncoordinateasthelengthofthevectorRCconnectingthemicellarCOMwiththemonomerCOM.InthisChapterwedemonstratethatthesereactioncoordinatesprovideaconsistentdescriptionofthemonomeraddition/removalprocessifoneaccountsforadditionaldegreesoffreedom.Asatestcase,weconsidertheSDSsystemdiscussedinChapter 5 .Theionchargestrengthis0.7eandallotherparametersarethesameasdescribedinChapter 5 .Weinvestigatetheprocessofadditionofamonomertoamicelleconsistingof39surfactants. Figure6-1. PMFsactingonamonomerpartiallyinsertedintoanSDSmicelle.ThePMFsareobtainedfromsimulationswithconstrainedheadbead(solidline)andcenterofmass(dashedline)ofthemonomer. Inwhatfollows,wewillusethefollowingnotationforthereactioncoordinates:HjRHjandCjRCj.PMFsobtainedfromMDsimulationswithconstrainedHandCaredenotedasGHandGC,respectively.ThesePMFsarecomparedinFig. 6-1 .It 142

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6 showsthatHisafunctionoftwoindependentDOFs,Candcos.Thissuggeststhatasinglereactioncoordinateisinsufcienttodescribethemonomeradsorptionanddesorptionandoneshouldconsidertwo-dimensionalfreeenergyproles,GC(C,)andGH(H,).Contributionsof^GCand^GHofthemonomerorientationtotheseenergyprolesarecomparedinFig. 6-2 .Freeenergy^GC(C,)exhibitsawiderbottleneckinthecriticalregion,whicheffectivelyoffsetsthehigherenergybarrierpredictedbytheone-dimensionalproleGC(C).Moreover,^GC(C,)predictsthatthemonomerismorelikelytoassumethe 143

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BFigure6-2. Contributionsofthemonomerorientationtothefreeenergy(kBTunits):(a)^GC(C,)and(b)^GH(H,). favorableorientation(=1)awayfromthemicelle.Toseethis,wenotethat^GC(C,)exhibitsaslightslopetowards=1forsufcientlylargeC(3nm
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BFigure6-3. (a)FreeenergyGC(C,)obtaineddirectlyfromsimulationswithconstrainedC;(b)Freeenergy~G(C,)obtainedfromsimulationswithconstrainedHbutparametrizedbythereactioncoordinateC(kBTunits). 145

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9 23 24 79 ]andprovidesnewinsightsintomolecularmechanismsoftheconsideredprocesses.Thecontributionsofthisstudyare: 23 24 ]investigatedtheeffectsofthesizeofsurfactantscomprisingthemonolayersontheenergybarriersforthetransportofasmallsphericalhydrophobicsolute.Asanextension,weinvestigatedeffectsofthesolutesize,structure,andhydrophobicityontheenergybarriersforthetransport.Inparticular,weshowedthattheoligomerschangetheircongurationinordertominimizetheenergypenaltyduringthetransport.Thedynamicsofsolutecongurationarecoupledwiththesolutetranslationandmonolayeructuations,whichdonotexistforthesphericalmonomericsolutes.Asaresult,deviationsofthesystemdynamicsfromtheMEPsaremoresignicantforthetransportoftheoligomersthanthatofthemonomericsolutes. 9 ].Thismodelidentieshydrophobicpatchesonthemicellarsurfaceandassumesthatthehydrophobicpatchesremainstableduringtheentiremonomeradsorptionprocess.Incontrast,weobservedthatthehydrophobicpatchesarenotstablebecauseevenverysmallmovementofthemicellarmoleculescaneasilycausethebreak-upofthepatches.WedevelopedanalternativemodelwhichusestheLegendrepolynomialexpansiontodescribeboththesurfacemicrostructureandshapeofthemicelle.Wereconstructedthemulti-dimensionalfreeenergylandscapesparameterizedbytheLegendrecoefcientsandotherreactioncoordinates(distancebetweenthemicelleandthemonomer,andthemonomerorientation).Wethenanalyzedthesystemtrajectoryontheseenergylandscapes,andshowedthattheenergybarriersforthemonomeradsorptionanddesorptionarehigherthanthosepredictedbytheone-dimensionalmodelwhichneglectsthecontributionsofthemicellarshape,microstructure,andthemonomerorientation.Wenotethatthelargestdeformationsofthemicelletakeplacerightbeforethecriticalpoint.Asaresult,evenmonomerdesorptionprocesscannotbedescribedbytheone-dimensionalmodel,andoneshouldtaketheeffectofthemicellarshapedeformationsintoaccount. 146

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79 ]hasdemonstratedthatthelong-rangeESinteractionsinionicmicellarsystemscreateanenergybarrierforthemonomeradsorption.Inthisstudy,weperformedamoresystematicanalysisofESinteraction.WeisolatedtheeffectsofESinteractionsbycomparingbehaviorofionicSDSsurfactantswiththatofnonionicC3P1surfactants.ThestructureandtheLennard-Jonesparametersoftheionicandnonionicsurfactantswereidenticalandtheonlydifferencesbetweenthemwerethechargesofsurfactanthead-groupsandpresenceofthecounterionsintheSDSsystem.Weshowedthatthenon-EScontributiontothePMFintheSDSsurfactantsystemisalmostidenticaltothatintheC3P1surfactantsystem.ThisconrmsthatESinteractionscreateanadditionalenergybarrierforthemonomeradsorption.AsurprisingdiscoveryofthisworkisthatthiseffectofESinteractionsontheenergybarrierbecomesweakerifthechargestrengthbecomesstronger(i.e.0.7e!1.0e).Thishappensbecausethecounterion-inducedscreeningofESinteractionsbetweenthesurfactanthead-groupsbecomesstrongerasthechargesincrease. 6 ,weshowedthatthePMFpredictedbytheconstrainedsimulationsforthemonomeradsorption/desorptionto/fromthemicelleissensitivetoaparticularchoiceofthereactioncoordinate.Specically,ifwedenethereactioncoordinateasthelengthofvectorCconnectingthemicellarandmonomerCOMs,theapparentenergybarrierfortheSDSmonomeradsorptionseemstobehigherthanthatwiththereactioncoordinatedenedasthelengthofvectorHconnectingthemicellarCOMandthemonomerhead-group.Wedemonstratedthatinordertoresolvethisinconsistency,oneneedstoaccountforanadditionaldegreeoffreedom,namelythemonomerorientation.Weshowedthatthetwo-dimensionalenergylandscapesintheC-andH-coordinatespredictconsistentenergybarriersforthemonomeradsorption/desorption.ThisfurtherconrmsthatcorrectpredictionoftransportratesrequiresacarefulanalysisofallrelevantDOF.Themethodsusedinthisstudyareexpectedtobeapplicabletootherself-assembledsystems.Forexample,themethodsofquanticationoftheshapesofsurfactantmonolayersandsphericalmicellescanbeappliedtootheratandsphericalmembranessuchaslipidbilayersandvesicles.Inwhatfollowswebrieydiscusspossibledirectionsoffutureresearchemployingthedevelopedtechniques.TranslocationofSurfactantsthroughaLipidBilayer.Itiswellknownthatadditionofsurfactants[ 85 87 ]tolipidmembranessignicantlyincreasesthemembranepermeability.Oneofthepossiblemechanismsofthispermeabilityincreaseisfrustrationofthetightlipidpacking[ 85 ],whichmakesiteasierforasolutetondapathwaythrough 147

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85 87 89 ].Recentexperiments[ 87 90 ]suggestthatinsomecaseslipidvesiclesdestabilizedbysurfactantsmaybeabletohealovertimeandnearlystoptheleakageoftheircontent.Understandingmechanismsofthemembranedestabilizationandhealingwillenablearationalchoiceanddesignofnon-toxicsurfactantsandwilladvancedevelopmentoftreatmentsoftoxicityduetolysisofcellmembranes.Oneofthepossiblemolecularmechanismsofthisdestabilizationandhealingofthelipidbilayeristhefollows.Initially,surfactantsareadsorbedintoouterleaetsofthebilayer.Thesurfactantsabsorbedintheouterleaetofthebilayerinduceastresswithinthebilayerduetodisparitiesin(i)densitiesandelasticpropertiesoftheinnerandouterleaetsand(ii)shapesofthesurfactantandlipidmolecules.Theformerdisparitycanbealleviatedbysurfactantip-oppingfromtheoutertotheinnerleaetofthebilayer.Thestraininducedbydifferencesintheshapesoftheamphiphilicmoleculesmayberelievedbyformationofsurfactantaggregates(rafts)withinthebilayer.Oncetheraftsareformed,thestrainwillbepresentmostlyalongtheirboundaries.Asarststepofthisstudy,wefocusontheadsorptionofsurfactantsintooneofthebilayerleaetsandip-oppingbetweendifferentleaets.Theremayexisttwopossibleroutesforthesurfactantstobeabsorbedintothebilayer.Therstpossiblerouteisthatthesurfactantsindividuallyabsorbedintothebilayerfromthebulksolventphase.WeperformedconstrainedsimulationsforanSDSsurfactantinaDPPCbilayer.ResultsofthesesimulationsareshowninFig. 7-1 .Here,thereactioncoordinatecorrespondstothedistancebetweentheCOMsofthebilayerandthehead-groupofthesurfactant.TheobtainedPMFsuggeststhatthereisessentiallynobarrierforthesurfactantadsorption.However,weobserveasubstantialdecreaseofthesurfactantdiffusivityD()asthesurfactantentersthebilayer.Ineffect,thebarrierforthesurfactantentryintothebilayerisnotduetoanenergyincreasebutduetothedecreaseddiffusivity. 148

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ResultsofconstrainedMDsimulationsofanSDSsurfactantinaDPPClipidbilayer.PotentialofmeanforceG()andlocaldiffusivityD()alongthisreactioncoordinateareshownbythesolidanddashedlines,respectively.Orientationsofthesurfactantmoleculeattheequilibriumpositionsinsidethemembraneareshownschematically. ThestudyinChapter 3 demonstratedthatsuchadecreaseoflocalmoleculardiffusivityiscausedbyadynamiccouplingbetweenthetranslationaldegreeoffreedomofthesolutemoleculeandslowmembraneundulations.Thisimpliesthatthemembraneshapeshouldbeconsideredasarelevantdegreeoffreedomtocorrectlydescribetheadsorptionprocess.Weanticipatethatatleasttwootherdegreesoffreedomsurfactantorientationandmicrostructureofthebilayersurfacewillplayasubstantialroleduringsurfactantadsorption.Forexample,thesurfactantismorelikelytobeadsorbedintothebilayerifitstailpointstowardsthebilayeranditapproachesanexposedareaofthehydrophobicbilayercore.SimilareffectswereobservedonadsorptionofsurfactantsintosphericalmicellesasshowninChapter 4 .Asimilarapproachcanbeappliedtoreconstructastochasticmodelforthesurfactantadsorptionintoabilayer.Thesecondpossiblerouteforsurfactantadsorptionintoabilayerisfusionofamicellewiththebilayer.OurpreliminaryMDsimulationssuggestthatthereisaverylargeenergybarrierforfusionbetweenamicelleandabilayerandthefusionprocessis 149

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4.2.1 .Thepreparedsystemisequilibratedfor100nsbyconstrainingthedistancebetweentheCOMsofthemicelleandtheoilphase.Duringtheequilibration,theboxsizereducestoequilibriumdimensionsof25.625.625.6nm3toachievethepressureof1bar.Aftertheequilibration,unconstrainedsimulationsareperformedfor500ns.Fig. 7-2 showsanexampleofmicellaradsorptionattheoil-waterinterface.Itisevidentthatthemicellerstexposesitshydrophobiccoretowardtheinterfacebyrearrangingthesurfactanthead-groups.Oncetheexposedhydrophobiccoreisabsorbedattheinterface,individualsurfactantsdiffusealongtheinterfacetoformauniformmonolayer.However,weobservethattheadsorptiondoesnottakeplaceinsomeMDsimulationsandtheinterfaceoftenrepelsthemicellewhenthelatterapproaches.Theseobservationssupportahypothesisthatthereisacriticalhydrophobicpatchsizeabovewhichtheadsorptiontakesplace.ItisexpectedthatthehydrophobicpatchanalysisdiscussedinSection 4.3.3 canbeappliedtoquantifythepatchsizeduringtheadsorption.Usingthepatchsizeasarelevantdegreeoffreedom,weexpectthatthekeyfeaturessuchasenergylandscapeanddiffusivityinthemulti-dimensionalparameterspacecanbeelucidatedbyaseriesofshort-timeMDsimulations[ 91 ]. 150

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AdsorptionofC4P4micelleintooil-waterinterface.Surfactanttail,headandoilbeadsareshownbyblack,whiteandgreycolors,respectively.Watermoleculesareomittedforclarity. 151

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YongNamAhnreceivedaB.S.degreewithhonorinChemicalandBiologicalEngineeringfromSeoulNationalUniversity,Seoul,SouthKoreain2007.HereceivedhisPh.D.inChemicalEngineeringfromUniversityofFloridainAugust2012.DuringhisPh.D.studies,heworkedunderthesupervisionofProf.DmitryIKopelevich.Hisresearchinterestsfocusonmodelingofdynamicsinself-assembledsurfactantsystems. 159