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Multiscale Modeling of Self-Assembly in Surfactant Systems

Permanent Link: http://ufdc.ufl.edu/UFE0022571/00001

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Title: Multiscale Modeling of Self-Assembly in Surfactant Systems
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Mohan, Gunjan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: assembly, colloids, constrained, disintegration, dynamics, energy, formation, free, ionic, kinetics, micelle, molecular, multiscale, nonionic, self, stochastic, surfactant
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Dynamics of self-assembly and structural transitions in amphiphilic systems play an important role in numerous processes, ranging from production of nanostructured materials to transport in biological cells. Theoretical and computational modeling of these processes is extremely challenging due to the large span of length- and time-scales involved. This study is focused on developing multiscale models for self-assembly in surfactant systems, with the main emphasis on the development of a multi-scale model for formation and disintegration of nonionic and ionic spherical micelles. The study is performed under the assumption that the dominant mechanism of micelle formation disintegration) is a stepwise addition (removal) of single monomers to (from) a surfactant aggregate. Different scales of these processes are investigated using a combination of coarse-grained molecular dynamics simulations, analytical and numerical solution of stochastic differential equations, and a numerical solution of kinetic equations. The removal of a surfactant from an aggregate is modeled by a Langevin equation for a single reaction coordinate, the distance between the centers of mass of the surfactant and the aggregate, with parameters obtained from a series of constrained molecular dynamics simulations. We demonstrate that the reverse process of addition of a surfactant molecule to an aggregate involves at least two additional degrees of freedom: orientation of the surfactant molecule and micellar microstructure. Formation of the ionic micelles involves one more degree of freedom which describes collective dynamics of the charges in the system. Time-scales of the additional degrees of freedom are comparable with the time-scale of the monomer addition to a micelle and hence these degrees of freedom play an active role in the monomer addition process. We demonstrate that neglecting their contribution leads to qualitative discrepancies in predicted surfactant addition rates and propose a stochastic model for the monomer addition which takes the additional degrees of freedom into account. The model parameters are extracted from molecular dynamics simulations and the surfactant addition rates are determined from Brownian dynamics simulations of this model. The obtained addition and removal rates are then incorporated into the kinetic model of micelle formation and disintegration. It is expected that insights gained in the course of development of the multi-scale model for this relatively simple self-assembly process will aid in the development of models for dynamics of more complex processes in amphiphilic systems such as collision of reverse micelles involved in formation of nanoparticles, rheology of worm-like micellar solutions, and fusion of lipid bilayers.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Gunjan Mohan.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Kopelevich, Dmitry I.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022571:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022571/00001

Material Information

Title: Multiscale Modeling of Self-Assembly in Surfactant Systems
Physical Description: 1 online resource (96 p.)
Language: english
Creator: Mohan, Gunjan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: assembly, colloids, constrained, disintegration, dynamics, energy, formation, free, ionic, kinetics, micelle, molecular, multiscale, nonionic, self, stochastic, surfactant
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Dynamics of self-assembly and structural transitions in amphiphilic systems play an important role in numerous processes, ranging from production of nanostructured materials to transport in biological cells. Theoretical and computational modeling of these processes is extremely challenging due to the large span of length- and time-scales involved. This study is focused on developing multiscale models for self-assembly in surfactant systems, with the main emphasis on the development of a multi-scale model for formation and disintegration of nonionic and ionic spherical micelles. The study is performed under the assumption that the dominant mechanism of micelle formation disintegration) is a stepwise addition (removal) of single monomers to (from) a surfactant aggregate. Different scales of these processes are investigated using a combination of coarse-grained molecular dynamics simulations, analytical and numerical solution of stochastic differential equations, and a numerical solution of kinetic equations. The removal of a surfactant from an aggregate is modeled by a Langevin equation for a single reaction coordinate, the distance between the centers of mass of the surfactant and the aggregate, with parameters obtained from a series of constrained molecular dynamics simulations. We demonstrate that the reverse process of addition of a surfactant molecule to an aggregate involves at least two additional degrees of freedom: orientation of the surfactant molecule and micellar microstructure. Formation of the ionic micelles involves one more degree of freedom which describes collective dynamics of the charges in the system. Time-scales of the additional degrees of freedom are comparable with the time-scale of the monomer addition to a micelle and hence these degrees of freedom play an active role in the monomer addition process. We demonstrate that neglecting their contribution leads to qualitative discrepancies in predicted surfactant addition rates and propose a stochastic model for the monomer addition which takes the additional degrees of freedom into account. The model parameters are extracted from molecular dynamics simulations and the surfactant addition rates are determined from Brownian dynamics simulations of this model. The obtained addition and removal rates are then incorporated into the kinetic model of micelle formation and disintegration. It is expected that insights gained in the course of development of the multi-scale model for this relatively simple self-assembly process will aid in the development of models for dynamics of more complex processes in amphiphilic systems such as collision of reverse micelles involved in formation of nanoparticles, rheology of worm-like micellar solutions, and fusion of lipid bilayers.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Gunjan Mohan.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Kopelevich, Dmitry I.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022571:00001


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Iowemydeepestgratitudetoallthosewhohavemadethisdissertationpossible.Iwouldliketoexpressmydeepestgratitudetomyadvisor,Dr.DmitryKopelevich.Iwasfortunatetojoinhisgroupandworkonresearchofmathematicalandcomputationalmodelingofsurfactantsystems.Hegavemefreedomtoexploretheeldofself-assemblyofsurfactantsystemandmolecularmodelingtechniques.Hetaughtmetorealizetheimportanceoffundamentalresearchanditsbroadimpact.Hispatienceinhandlingsituations,generoussupportandcondenceinmyabilitiesmotivatedmetogivemybestinstudies.Iamalsothankfultohimforcarefullyreadingandcommentingoncountlessrevisionsofresearchreports,articlesandannualprogressreports.DuringmyPhDstudiesIhadthegreatexperienceofworkingasateachingassistantforthecourseHeatTransferOperationsbyDr.JasonButler.Hehasalsoguidedmyresearchworkthroughdiscussionsandclassroomtraining.HeintroducedmetothevariousmodelingtechniquesapplicabletoComplexFluidsandbecauseofhisteachingIcouldeasilydevelopaBrowniandynamicsmodelforadditionrateofamonomertoamicelle.Dr.AnujChauhanhasbeenalwaystheretolistenandgiveadvice.Iamdeeplygratefultohimforthelongdiscussionsandaskingmefundamentalquestionsthathasguidedtheresearchwork.Dr.Shahisasourceofinspirationtoallnewfacultymembersandstudents.Ioweallmypracticalknowledgeaboutsurfactantsandself-assemblytoDr.ShahandDr.Moudgil.IamalsothankfultoDr.SussanSinnottforreadingmyreportsandcommentingonmyviews.IamalsogratefultoMs.Kelly,Ms.SandovalandMs.AldrichattheDepartmentofChemicalEngineeringfortheirsupportduringmygraduatestudy.Ideeplyappreciatethehelpofallmyfriends(RameshBabu,ShaktiGupta,SaurabhAgarwal,DeepakLachwani,AtulBakale,DileepPerchani,RajniSanghani,RahulSharma,VikasKothari,KaushalMudaliar,ValereChia-YiChen,JasonNoel,BerkOUsta,PhilipCobb,JoontaekPark,RyanTasse,YashKapoor,GauravMishra,WyongMinBan,Yong 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1SpecicAims .................................. 12 1.2Introduction ................................... 12 1.3TheoryandBackground ............................ 14 1.3.1MolecularDynamicsSimulations .................... 14 1.3.2StochasticModelofSelf-Assembly ................... 17 1.3.3ReactionCoordinateandConstrainedSimulations .......... 18 2SELF-ASSEMBLYOFNONIONICSURFACTANTS .............. 21 2.1Introduction ................................... 21 2.2ModelandSimulationsDetails ......................... 23 2.2.1Coarse-GrainedModel ......................... 23 2.2.2FormationofMicelles .......................... 25 2.3ConstrainedSimulations ............................ 27 2.4AnalysisBasedonOne-DimensionalLangevinEquation ........... 28 2.4.1EquilibriumProperties ......................... 28 2.4.2Removal/AdditionofMonomersfrom/toSurfactantClusters. .... 30 2.5Multi-DimensionalModel ........................... 31 2.5.1ImportanceofAdditionalDegreesofFreedom ............ 31 2.5.2Time-scalesofAdditionalDegreesofFreedom ............ 33 2.5.3ModelforMonomerAddition ..................... 36 2.5.4BrownianDynamicsSimulations .................... 40 2.6EquilibriumClusterSizeDistributionPredictedfromtheKineticModel .. 44 2.7Discussion .................................... 46 3SELF-ASSEMBLYOFIONICSURFACTANTS .................. 60 3.1Introduction ................................... 60 3.2ModelandSimulationsDetails ......................... 61 3.2.1Coarse-GrainedModels ......................... 61 3.2.2SimulationsofSelf-Assembly ...................... 63 3.3ConstrainedSimulation:DynamicsofMonomerRemovalfromMicelle ... 64 3.3.1ElectrostaticContributiontoFreeEnergy ............... 66 3.3.2ChargeDistribution ........................... 67 3.4Discussion .................................... 68 6

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.......... 75 4.1DynamicsofRemovalofSmallClustersfromaMicelle ........... 76 4.2InteractionandFusionofTwoSurfactantClusters .............. 77 APPENDIX ASIMULATIONDETAILS .............................. 80 A.1BerendsenThermostat ............................. 80 A.2BerendsenBarostat ............................... 80 BDETAILSOFTHEORETICALMODELS ..................... 82 B.1KineticApproachtoRareEvents ....................... 82 B.2TheoryofStochasticProcesses. ........................ 82 REFERENCES ....................................... 87 BIOGRAPHICALSKETCH ................................ 96 7

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Figure page 2-1Coarse-grainedrepresentationoflinearH4T4surfactantandwatermolecules. .. 48 2-2Snapshotofasimulationofself-assemblyofH4T4surfactantsintosphericalmicelles. ............................................. 49 2-3RadiiofgyrationRgN;i,i=1;2;3,ofclusterswithaggregationnumberN. .... 49 2-4Autocorrelationfunctionoftherandomforce(t;)foramonomerpartiallyremovedfrommicelleA64andconstrainedat=1:598nm. ........... 50 2-5DependenceofthefreeenergyG(solidline)andthefrictioncoecient(dashedline)onthedistancebetweencentersofmassofthemonomerandmicelleA64. 50 2-6Prolesofvolumefractionsofsurfactanttails,head-groups,andsolvent(T,H,S)formicelleA64asafunctionofdistancerfromthecenterofthemicelle. ............................................. 51 2-7FreeenergyprolesforremovalofamonomerfromclustersofvariousaggregationnumbersN. ..................................... 51 2-8EquilibriumdistancesbetweenthecentersofmassoftheclusterANandthesurfactantswithintheclusterobtainedfromconstrainedandunconstrainedsimulationsareshownbythesolidanddashedlines,respectively. .............. 52 2-9Dierencebetweenthestandardpartsofthechemicalpotentials0Nand01ofasurfactantmonomerinsideclusterANandafreemonomerinsolution. ..... 52 2-10ClustersizedistributionattotalsurfactantconcentrationofXT=2107M. 53 2-11Dependenceoftheconcentrationsofthefreesurfactantmonomers,X1(dashedline),andthesurfactantscontainedinclusters,Xclus(solidline),onthetotalsurfactantconcentrationXTpredictedbythethermodynamictheory. ...... 54 2-12AveragetimeN!N1ofmonomerremovalfromsurfactantclustersANpredictedbyEq.( 1{23 ). ..................................... 54 2-13AveragetimeN!N+1ofmonomeradditiontosurfactantclustersANpredictedbyEq.( 1{23 ). .................................... 55 2-14DistancebetweencentersofmassofmicelleA87andamonomerfortwodierentMDsimulations.Thesimulationsareperformedinacubic101010nm3cellwiththeperiodicboundaryconditions. ....................... 55 2-15SnapshotsofMDsimulationsofattemptsofmonomerentryintomicelleA87. .. 56 2-16Probabilitydistributionoforientation1cosofamonomerconstrainedatanentrytomicelleA64. ............................... 56 8

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57 2-18FractionofthesurfaceareaofclustersANoccupiedbyhydrophobicpatches. .. 57 2-19Fractionoftimethatclustersurfacesareoccupiedbyhydrophobicpatchesduringaperiodof200ps. .................................. 58 2-20TranslationaldiusioncoecientsofclustersANobtainedfromMDsimulations(solidline)andpredictedbytheStokes-Einsteinrelationship(dashedline). ... 58 2-21RateconstantskN;N+1ofadditionofmonomerstoclustersANobtainedfromtheBrowniandynamicssimulations(solidline)andfromtheSmoluchowskimodelEq.( 2{16 )(dashedline). ............................... 59 3-1Detailedatomicstructuresandthecorrespondingcoarse-grainedmodelforsodiumdodecylsulfate. .................................... 69 3-2Snapshotofasimulationofself-assemblyofSDSsurfactantsintomicelles. ... 70 3-3Autocorrelationfunctionoftherandomforce(t;)actingonamonomerpartiallyremovedfrommicelleA30andconstrainedat=1:19nm. ............ 70 3-4DependenceofthefreeenergyG(solidline)andthefrictioncoecient(dashedline)onthedistancebetweenthecentersofmassofthemonomerandthemicelleA30. .......................................... 71 3-5ThedensitydistributionofchargesinandaroundmicelleA30.ThedistributionofQ+beadsisshownbythesolidlineandofQbeadisshownbythedashedline. .......................................... 71 3-6FreeenergyGobtainedfromconstrainedsimulations(solidline)andtheelectrostaticcontributiontothefreeenergyfor>critobtainedfromsolvingthePoissonequation 3{1 formicelleA30(dashedline). ..................... 72 3-7Freeenergyprolesforremovalofamonomerfromclustersofdierentaggregationnumbers. ........................................ 72 3-8Typesofthebeadsfarthestfromtheorigininthedirection(;),aroundthemicelleA30. ...................................... 73 3-9TheminimumdistancebetweenaQ+counterionandalltheQchargesonthesurfaceofmicelleA30asafunctionoftime. .................... 73 3-10A)TheprobabilitydistributionoftheminimumdistancebetweenaQ+counterionandalltheQchargesonthesurfaceofmicelleA30.B)ThepotentialofmeanforceofinteractionbetweenaQ+counterionandthesurfaceofmicelleA30. .. 74 4-1Fusionoftwoclusterstoformalargesphericalmicelle. .............. 78 4-2Repulsionoftwoclusters. .............................. 79 9

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1 2 ].Inadditiontoindustrialapplications,dynamictransitionsofself-assembledstructuresofamphiphiliclipidmoleculesplaycrucialroleintransportinbiologicalcells[ 3 ].Thisstudyfocusesonaspecicself-assembledstructures,thesphericalmicelles.Despitetherelativesimplicityofthesphericalmicelles,theinformationgainedinthisstudyisexpectedtobeusefulindevelopmentofmodelsformorecomplexsystems.Thespecicaimsofthisstudyare: 12

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4 { 7 ]havemodeledenthalpicandentropiccontributiontofreeenergyandhaveobtainedCMCandmicellesizedistribution.Inadditiontomicelles,surfactantsmayassembleintoavarietyofaggregateswhosemorphologydependsonthesurfactantstructure,surfactantconcentration,saltconcentration,temperature,pressureaswellasexternalforces.ThesurfactantstructurecanbecharacterizedbythepackingparameterP,whichplaysanimportantroleindeterminingtheaggregatemorphology[ 7 ].ThepackingparameterisdenedasP=v=al,wherevisthevolumeofthehydrocarbonchain,listhelengthofthehydrocarbonchainandaistheeectivecross-sectionareaoftheheadgroup.SurfactantswithpackingparameterP1=3formsphericalmicelles,surfactantsformcylindricalorworm-likemicellesif1=3
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18 { 22 ],soastoeectivelycontrolandenhancequalityofmicellarsolutionsandaidindevelopmentofnewproducts.Forexample,reversesphericalmicellesarecurrentlyusedasnano-reactorsforthepreparationofnano-particles[ 23 ].Possiblefutureapplicationsofsphericalmicellesincludedesignofcontrolleddrugdeliveryvehicleswithdrugencapsulatedinthemicellecoreandreleasedbydiusionorbybreakingofmicelles.Thepurposeofthisstudyistodevelopsimulationmethodswhichcanprobetheprocessofself-assemblyatdierentlength-andtime-scales.Themainchallengeincomputationalstudiesofmicellarsystemisthattheself-assemblyandstructuraltransitionsarerareevents[ 24 ]correspondingtotransitionbetweenstatesseparatedbyahighfreeenergybarrier.Inthisstudyweusemoleculardynamicssimulationsandstochasticmodelingtechniquestomodeltheself-assemblydynamics. 1.3.1MolecularDynamicsSimulationsTheself-assemblyprocessofamphiphilicmoleculescanbemodeledbymoleculardynamicssimulations.ForasystemofNinteractingparticles,moleculardynamicssimulationssolveNewton'sequationsofmotionforeachparticle, @ri:(1{2) 14

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25 ],whichprovidesaweakcouplingtoanexternalbath.Thepressureofthesystemalsouctuatesduetochangeinmomentumofthemovingparticles.Thepressurecanbekeptconstantbyusingpressurecoupling,whichleadstovolumeuctuations.InourworkweuseBerendsenpressurecoupling.ThedetailsoftemperatureandpressurecouplingschemesaregiveninappendixA.Inthecurrentwork,weuseacoarse-grainedmolecularmodel[ 26 ].Inthismodel,groupsofatomsandsmallmoleculesaremodeledbyasinglebead.Thenon-bondedinteractionbetweenbeadsiandjisdescribedbytheLennard-Jones(LJ)potential, 15

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2Kbond(RRbond)2;(1{6)andaweakharmonicpotentialofcosinetypeforbondangle, 2Kanglecos()cos(0)2:(1{7)Here,Rbondand0aretheequilibriumbondlengthandangleandKbondandKanglearethecorrespondingforceconstants.ThevaluesofthemodelparametersforspecicsurfactantsystemswillbepresentedintheChapters 2 and 3 .ThetotalelectrostaticinteractionenergyofNparticlesandtheirperiodicimagesisgivenby 16

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1{11 ).ThismethodiscomputationallyslowandvariousvariantsofEwaldsummationhavebeendevelopedtoreducecomputationaltime.OnesuchmethodisParticleMeshEwald(PME)summationproposedbyDarden[ 27 28 ]toimprovetheperformanceofthereciprocalspacesummation.Incalculationofthereciprocalspacesum,thechargesareassignedtoagridusingcardinalB-splinesinterpolation.ThisenablesapplicationoftheFastFouriertransformationtocomputethecontributionofthereciprocalsumtocalculatetheforceandenergyinthereciprocalspace.ForasystemcontainingNchargedparticles,performanceofthePMEalgorithmscalesasNlog(N)andissignicantlyfasterthantheoriginalEwaldsummationwhichscalesasN2. 29 ]andneglectthecontributionsofclusterfusionandssion.Thustheself-assemblyprocessisdescribedbythefollowingsetofkineticequations 17

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Here,CNisthemolarconcentrationofaggregatesAN.InordertopredictthedynamicsofmicelleformationanddisintegrationitisnecessarytocomputetherateconstantskN;N1ofaddition/removalofmonomersto/fromsurfactantclusters.Weobtaintheseratesundertheassumptionthatthemonomer/clustersystemcanbedescribedbyastochasticmodelforasmallnumberofreactioncoordinates.Thisassumptionisvalidifthereisasucientseparationbetweenthetime-scalesofthereactioncoordinate(s)andallotherdegreesoffreedom,whichcanthenbemodeledbythermalnoise.BelowwediscussastochasticmodelbasedonaLangevinequationforasinglecoordinate.Weshowinsection 2.4.2 thatthisassumptionholdsformonomerremovalbutfailsformonomeraddition. 18

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30 { 32 ], 30 31 ]toobtainthegradientofthefreeenergyG0()andthefrictionkernel~.SincethereactioncoordinateisthedistancebetweenalinearcombinationofCartesiancoordinatesofparticles,theaverageoftheconstrainingforce,Fc,correspondstothefreeenergygradient[ 33 34 ], 1{17 )isthenemployed[ 30 ]tocomputethefrictionkernel~(t;)fromtheautocorrelationfunctionof(t;).Itwillbeshownbelowthatinthecurrentcase,thecorrelationtimeofthestochasticforceisnegligiblecomparedtothetime-scaleofthedynamicsofthereactioncoordinate.Therefore,thememoryeectscanbeneglectedand,introducingthetime-independentfrictioncoecientby ~(;)=2()();(1{19)onecansimplifyEqs.( 1{16 ),( 1{17 )to Thefrictioncoecient()canbeobtainedfromtherelationship 19

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1{17 )and( 1{19 ).IftheinertialtermofEq.( 1{20 )canbeneglectedincomparisonwiththefrictionandstochasticforces,theaveragetransporttimeofthereactioncoordinatefrom=ato=bis[ 35 36 ] 37 ] lc0 2.3 20

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38 ])andviceversa(e.g.,macroscopicshearinducesrearrangementofsurfactantmoleculesthatleadstotransitionfromvesiclestomicelles[ 39 ]).Detailedunderstandingoftheseprocessesthereforerequiresdevelopmentofamodelwhichlinkstherelevantscales.Inthisstudyweinvestigatemultipletime-scalesofthedynamicsofself-assemblyanddisintegrationofsphericalmicellesanddevelopaconnectionbetweenthesescales.Eveninthisrelativelysimplesystem,thereisasignicantseparationbetweentime-scalesofmoleculardynamics,individualstepsinvolvedinthemicelleformation/disintegration,andtheentireprocessofthemicellarformation/disintegration.Itisanticipatedthatinsightsintotheself-assemblydynamicsgainedfromthissystemwillberelevanttoother,morecomplexsystems.Multipletheoreticalandcomputationalstudies[ 40 { 44 ]haveobtainedthedependenceofthefreeenergyonthesizeofasurfactantaggregate.Thefreeenergyhastwominimacorrespondingtothefreesurfactantmonomersandtheequilibratedmicelles.Theseminimaareseparatedbyafreeenergybarrierwhichcorrespondstosmallsurfactantclusters(premicelles).Thefreeenergycanbeusedtoassesstheactivatedtransitionbetweenfreemonomersandmicelles.However,predictionofthetransitiontimealsorequiresknowledgeofthedynamicsofthesystemmotionalongthisfreeenergyprole.Recently,Kopelevichetal.[ 36 ]havedevelopedsuchamodelforthe\dynamics"ofamicellarsystemmodeledbylatticeMonteCarlo(MC)simulations.ThismodelisbasedonaLangevinequationfortheaggregationnumberofsurfactantclusters.Boththefree 21

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29 ].Thekineticmodelcorrespondingtotheseprocesseshasbeenextensivelystudied[ 29 45 { 49 ].However,usuallytheratesofsurfactantadditionandremovalusedinthesestudiesarenotbaseddirectlyonmolecular-scaledynamics.Theadditionandremovalratesforblock-copolymermicellarsystemshavebeenobtainedbyHalperinandAlexander[ 50 ].Thisestimationisbasedonscalingargumentsinthelimitoflongpolymerchainsandisnotreadilyapplicabletotheshort-chainsurfactants.Morerecently,vonGottbergetal.[ 22 ]obtainedtheseratesfromBrowniandynamicssimulationsofamodelsurfactantsystembyusingtwocomplementarymethods:(i)monitoringthetimerequiredforsurfactantstoredistributeamongtheaggregatesand(ii)computingthemonomerremovalrateusingKramers'ratetheory[ 51 ].Therstapproachdoesnotdirectlyprovideinsightintomolecularrearrangementsinvolvedintheaddition/removalofthesurfactantmolecules.Thesecondapproachimplicitlyassumesthattheremovalofsurfactantscanbedescribedbyaone-dimensionalLangevinequationinthehigh-frictionlimit.ThevalidityofthelatterassumptiondependsonthechoiceofthefrictioncoecientfortheBrowniandynamicsmodel. 22

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35 ]ofKramers'ratetheoryfornon-parabolicfreeenergybarriers.Ontheotherhand,monomeradditiontomicellesinvolvesatleasttwoadditionaldegreesoffreedom,themonomerorientationandacollectivedegreeoffreedomrepresentingmicellarmicrostructure.Wedevelopamulti-dimensionalstochasticmodelwiththeparametersextractedfrommoleculardynamicssimulationsanduseBrowniandynamicssimulationstoobtainthesurfactantadditionratespredictedbythismodel.Thecomputedratesareveriedbycomparisonofthemicellarsizedistributionandthecriticalmicelleconcentration(CMC)obtainedfromtheequilibriumtheoryandthekineticmodel.Wedemonstratethattakingthemonomerorientationandthemicellarmicrostructureintoaccountleadstoaqualitativeimprovementintheobtainedratesoveraone-dimensionalmodelformonomeraddition.Itisanticipatedthatthedynamicsofthemicellarmicrostructurealsoplaysanimportantroleinmorecomplexprocesses. 2.2.1Coarse-GrainedModelThesimulationsofmodelnonionicsurfactantsinwaterareperformedusingacoarse-grainedmoleculardynamics(CGMD)model.Coarse-grainedmodelsapproximategroupsofatoms(suchasseveralmethylorethoxygroups)asasingleunitedatom(bead)whichallowsforasignicantincreaseintheeciencyofmoleculardynamics(MD)simulations.SeveralCGMDmodelshavebeenintroducedandappliedtosimulationsofvariouscomplexmolecularsystems[ 10 26 52 { 55 ].Inthecurrentwork,weusethemodel 23

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26 ].Thismodelyieldsgoodagreementwithexperimentsandatomisticallydetailedsimulationsforawiderangeofamphiphilicsystems.Inthecurrentwork,allmoleculeswillbemodeledusingtwotypesofbeads:hydrophobictailbead(denotedasT)andhydrophilicheadbead(denotedasH).ThisnotationisdierentfromthatofRef.[ 26 ],wherefourbeadtypesareintroducedinordertodescribeawiderangeofamphiphilicmolecules.ThebeadsdenotedinthisworkasHandTcorrespondtobeadsC(apolar)andP(polar)inRef.[ 26 ].Weusethisdierentnotationasitismorecommonintheliteratureonsurfactantsystems.AsingletailbeadTrepresentsfourmethyleneormethylgroupsofthesurfactanttailandasingleheadbeadHrepresentsfourwatermolecules[ 26 ].TheoriginalworkofMarrinketal.[ 26 ]doesnotcontainamodelforheadgroupsofnonionicsurfactants.BasedonthemassandhydrophilicpropertiesofbeadH,weassumethatasingleheadbeadHapproximatestwoethoxygroups.Therefore,amodelTnHmsurfactantisexpectedtoapproximateanethoxylatedsurfactantC4nH8n+1[OCH2CH2]2m-OH.Wenotethatunlikethemodelsforwaterandmethylene/methylgroups,thecurrentmodeloftheethoxygrouphasnotbeenoptimizedtomatchtheexperimentaldata.Moreover,themodeldoesnotaccountfortheterminal-OHgroupofethoxylatedsurfactants.Nevertheless,themodelandethoxylatedsurfactantsaresucientlysimilartojustifyperformingacomparisonbetweenthemodelpredictionsandtheexperimentalresultsfortheethoxylatedsurfactants,seesection 2.4.1 .Inthecurrentwork,weperformsimulationswithamodelH4T4surfactant.Theschematicsofthecoarse-grainedH4T4surfactantandwatermoleculesisshowninFigure 2-1 .BothHandTbeadsaresphericalandhavethesamemass,m=72amu.Theinteractionbetweentwonon-bondedbeadsismodeledbytheLennard-Jones(LJ)potentialwiththesameeectivediameter,=0:47nm,forallbeads.Thecharacterofinteractionsismodeledthroughthevaluesoftheenergyparameter.TheinteractionbetweentwoHbeadsishighlyattractive(HH=5kJ/mol),betweentwoTbeadsis 24

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56 ].InstantaneousmolecularcongurationsofthesystemareplottedusingtheVMDpackage[ 57 ]. 25 ]withthetimeconstants1psandtheisothermalcompressibility1bar1.Equilibrationofpressureofthesystemleadstoaslightdecreaseinthesimulationcellsize,to8:88:88:8nm3forthesmallercellandto13:413:413:4nm3forthelargercell.Thesimulationsofself-assemblyarecarriedoutfor400nswithatime-stepof0.025ps. 25

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2-2 .Randomlydispersedsurfactantmoleculesassembleinmicellaraggregateswithintensofnanoseconds.Theself-assemblytakesplaceonthisshorttime-scalebecauseitismostlyadiusion-controlledprocess.Surfactantmonomersandsmallsurfactantclusters(premicelles)arehighlylikelytoaggregateuponcollisionswithaverysmallenergybarrier.Sincethesurfactantsareplacedinarelativelysmallsimulationcell(atconcentrationsmuchhigherthanthecriticalmicelleconcentration,CMC),thebarrierfortheiraggregationisverysmall[ 43 ]andtheirdiusiontimeisveryshort,leadingtothequickaggregationandmicelleformation.Ontheotherhand,thetime-scalesofmicellardisintegrationarenotaccessiblebythecoarse-grainedMDsimulationsbecausetheprocessofmicellardisintegrationisanactivatedprocesswithaveryhighactivationbarrier.Togaingreaterinsightintomicellekineticsitisnecessarytodevelopamodelwhichiscapableofaccessingtheselargetime-scaleswhileadequatelymodelingtherelevantfastprocessesoccurringonshorttime-scales.Wedevelopthismodelfromtheanalysisofelementarystepsofaddition/removalofsurfactantmonomersto/fromsurfactantclusters(i.e.micellesandpremicelles)ofvarioussizes.Inordertoperformthisanalysis,itisnecessarytoprepareadatabaseofsurfactantclusters.Aclusterisdenedasagroupofsurfactantmoleculessuchthatatleastonetailbeadofeachsurfactantmoleculebelongingtotheclusteriswithinacertaindistance,rcl,fromatailbeadofanotheramphiphilebelongingtothesamecluster.Inthecurrentworkweusercl=0:70nm,whichisslightlygreaterthantheequilibriumseparationdistancebetweenthebeads,req0:53nm.TheclusterdatabaseisgeneratedbydirectMDsimulations.Structuresofclusterswithrelativelysmallaggregationnumbers,N10,aregeneratedbyrandomlyplacingNsurfactantmoleculesinasimulationcelllledwithwaterandperformingMDsimulations 26

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2.3 .TheradiiofgyrationRgN;i;i=1;2;3;ofaclusterwithaggregationnumberNarealmostequaltoeachotherforawiderangeofN,asdemonstratedbyFigure 2-3 .Thesurfactantclustersarethereforecloselyapproximatedbyspheres. 58 ].Atypicalautocorrelationfunctionoftherandomforce(t;)=hFc()iFc(t;)isshowninFigure 2-4 anddemonstratesthatthecorrelationtimeof(t;)isverysmall(lessthan1ps).Similarcorrelationtimesareobtainedforthestochasticforcesforallconsideredsurfactantclustersandvaluesofthereactioncoordinate.Therefore,theassumptionEq.( 1{21 )thattherandomforcecanbemodeledaswhitenoiseisvalid.Furthermore,theconditionsofthehigh-frictionlimitEq.( 1{25 )aresatisedforallconsideredclustersandvaluesof,Ki()0:1,i=1;:::;4.Therefore,Eq.( 1{23 )canbereadilyappliedtoobtainthetransportratesalongthereactioncoordinate. 27

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1{18 )and( 1{22 )areshowninFigure 2-5 .Forreference,thecorrespondingprolesofthevolumefractionsofthesurfactantsandwaterinthemicelleanditsneighborhoodareshowninFigure 2-6 .SimilartrendsareobservedforbothG()and()forotherconsideredsurfactantclusters,withaggregationnumbersrangingfromN=2toN=88,asdemonstratedforG()inFigure 2-7 .TheobtainedfreeenergyprolesagreequalitativelywiththoseobtainedbyvonGottbergetal.[ 22 ]andPoolandBolhuis[ 59 ]forsimilarcoarse-grainedmodels.Thefreeenergyandthefrictioncoecientapproachaminimumandamaximum,respectively,atthesamevalueofthereactioncoordinate=0.Thisvaluecorrespondstotheequilibriumlocationofthemonomerinthecluster.Thisisfurtherconrmedbyunconstrainedsimulationsforsurfactantclusters,asdemonstratedinFigure 2-8 .Asthedistancebetweenthemonomerandtheclustercentersofmassincreasesabove0,theexposureofthemonomertailtotheunfavorablepolarenvironmentgrows,leadingtoanincreaseofthefreeenergyG().Thereisasignicantenergybarrierforthemonomerremoval,butthebarrierforthemonomeradditionisnegligible. 2.4.1EquilibriumPropertiesInordertoestablishaframeofreferencefortheinvestigationofthekineticsofthemicellarsystem,werstdetermineitsequilibriumproperties.Thesepropertieswillalsohelpustoassessvalidityofthecoarse-grainedsystem.Itiswell-known[ 7 ]thatforsolutionswithlowsurfactantconcentration,theconditionofthermodynamicequilibriumleadstothefollowingexpressionforthemolarfraction~XNofsurfactantmoleculescontainedinclustersofaggregationnumberN 28

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2-9 .Theknowledgeofthestandardchemicalpotentials0NforarangeofNallowsonetocomputevariousequilibriumpropertiesofthesurfactantsystem.Forexample,theclustersizedistributionforagiventotalsurfactantconcentrationXTisobtainedbysolvingtheequations( 2{1 )coupledwiththeconstraintonthetotalsurfactantconcentration 2{2 ),Misasucientlylargeclustersize.Inourcalculations,weuseM=88whichissignicantlylargerthanthemostprobablemicellaraggregationnumber.TheclustersizedistributionobtainedforthetotalsurfactantconcentrationofXT=2107MisplottedinFigure 2-10 andshowsthatthemostprobablemicelleaggregationnumberisaround64.Inaddition,itisclearthatcontributionofmicelleswithaggregationnumbersabove80tothetotalsurfactantconcentrationXTisnegligible,whichjustiestheuseofthecut-oaggregationnumberM=88inEq.( 2{2 ).Thecriticalmicellarconcentration(CMC)whichmarkstheonsetofmicellizationcanbedenedastheconcentrationX1ofthesurfactantmonomersatwhichX1coincideswiththeconcentrationXclusofsurfactantmoleculesinclusters[ 5 ], 29

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2-11 showsdependenceofX1andXclusonthetotalsurfactantconcentrationXTanddemonstratesthatCMCisX11:0107M.Forcomparison,theexperimentalvaluesofCMCforC16E7andC16E9at298KareX11:74106MandX12:09106M,respectively[ 60 ].Therefore,thetheoreticalpredictionunderestimatestheexperimentalvaluesforthemodeledethoxylatedsurfactantsbyalmostanorderofmagnitude.Inprinciple,thecoarse-grainedmodelcanbeadjustedtoimprovetheagreementwiththeexperimentsbutinthecurrentworkwehaverefrainedfromperformingtheseadjustments. 1{23 )forclusterswithaggregationnumbersNrangingfrom2to88areshowninFigure 2-12 .N!N1approachesitsmaximumof29.6msformicellescontaining64surfactants.Figure 2-7 showsthatthefreeenergybarriersforadditionofmonomerstosurfactantclustersareverysmall,whichseemstoimplythatthemonomerwillenteraclusterassoonasitapproachesone.Indeed,applyingequation( 1{23 )tocomputetheaveragetimeT(free!0)ofmonomerentryintoacluster(correspondingto0)aftertheirrstcontact(correspondingtofree),weobtaintheadditiontimesontheorderof200ps,seeFigure 2-13 .Therefore,theadditionofamonomertoasurfactantclusterappearstobecompletelycontrolledbydiusionsothatthefusionbetweenamonomerandaclustertakesplaceassoonastheyapproacheachother.Infact,thisisoftenassumedinthestudiesofmicellarkineticsandisabasisforapplicationoftheSmoluchowskitheoryofcolloidalaggregation[ 61 ]tomicellarformation[ 21 59 62 ]. 30

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2-14 ,whichshowsthedistancebetweenthecenterofmassofmicelleA87andthecenterofmassofamonomerfortwodierentMDsimulations(referredtoassimulationsIandII).Thesimulationsareperformedinacubiccellwithsidesof10nmandperiodicboundaryconditions,whichlimitsthedistancebetweenthecentersofmassto8.7nm.Theuctuationsofthedistancearound5-6nmcorrespondtoafreemonomeroatingessentiallyindependentlyofthemicelle.Themonomerentryintothemicellecorrespondstoasharpdecreaseinthemonomer-micelledistancetoapproximately2nm.ThisdecreaseisobservedforsimulationIattimet7ns,whichissignicantlylargerthantheestimate87!88220psbasedontheone-dimensionalLangevinequation(seeFigure 2-13 ).Moreover,forsimulationIIthemonomerdoesnotenterthemicellewithintheentiresimulatedtimeintervalof75ns. 2.5.1ImportanceofAdditionalDegreesofFreedomAdetailedexaminationofthetrajectoriesshowninFigure 2-14 revealsthatduringthesimulatedtimethemonomermakesmultipleattemptstomergewiththemicelle.However,thevastmajorityoftheseattemptsareunsuccessfulandthemonomerentersamicelleonlyifthefollowingtwoconditionsaresatised:(i)thehydrophobictailofthemonomerispointingtowardsthemicelleand(ii)themonomerapproachesthemicellenearanexposedareaofthehydrophobiccore.AnexampleofafavorablecongurationofthesystemisshowninFigure 2-15 A.ThiscongurationcorrespondstopointaofFigure 2-14 andleadstomonomerentryintothemicelle.Pointsb-dinFigure 2-14 provideexamplesofunsuccessfulattemptsofthemonomerentryintothemicelle.ThecorrespondingcongurationsofthesystemareshowninFigure 2-15 B-Dandrevealunfavorablecongurationsofthemonomerand/orthemicelle.Figure 2-15 Bshowsamonomerapproachinganexposedhydrophobicpatchonthemicellarsurface.However,thehydrophilichead-groupofthemonomerispointing 31

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2-15 Cshowsarepulsionofamonomerapproachingthemicellewithafavorablemicrostructurebutwithanunfavorableorientation(itsignicantlydeviatesfromthenormaltothemicellarsurface).Finally,ifthemonomerhasafavorableorientationbutitreachesthehydrophilicregionofthemicellarsurface,itisrepelled,asshowninFigure 2-15 D.Forillustrativepurposes,hereweconsideredarelativelylargemicelle,A87.Similardynamicsofmonomeradditionareobservedforsmallersurfactantclusters.TheserepulsivestericinteractionsarenotcapturedbythefreeenergyprolesshowninFigure 2-7 becausethelatterwerecomputedundertheassumptionofasinglereactioncoordinate.However,thediscussionaboveclearlydemonstratesthatitisnecessarytoincludeatleasttwomorereactioncoordinatesintoconsideration,themonomerorientation1andtheclustermicrostructure2.Theseadditionaldegreesoffreedomplayasignicantroleonlyduringtheadditionofthemonomertotheclusterandarenotrelevantduringtheremovalofthemonomerfromtheclusterbecause,aswillbedemonstratedinsection 2.5.2 ,thetime-scaleofthemonomerremovalismuchslowerthantherelaxationtimesof1and2.Therefore,duringmonomerremovalthemodes1and2areslavedtotheoriginalreactioncoordinateandthelattercoordinateissucienttodescribethemonomerdynamics.Ontheotherhand,thetime-scalesofthemonomerrotationandclustermicrostructurearecomparableorevenslowerthanthetime-scaleofthemonomermotionalongthecoordinateduringitsadditiontoacluster.Therefore,thesetwoadditionaldegreesoffreedomarenecessarytoaccuratelydescribethefreeenergybarrierforthemonomerentry.ThefreeenergycurvesofFigure 2-7 shouldbereplacedbythree-dimensionalhypersurfaceswhichprescribeahighfreeenergybarrierforthemonomerentranceintotheclusterifthemonomerapproachestheclusterwithanunfavorableorientationoranunfavorableinternalclustermicrostructure.Thisbarrierisnotvisibleontheone-dimensionalfreeenergycurvesofFigure 2-7 sincethesecurvesineectarecomputed 32

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2-3 ),weapproximatetheclustersurfacesbyspheres.Thesurfactantmoleculesarefairlyexibleandthemajorcontributiontotheadditionprocessismadebythesurfactanttail,whichmotivatesthedenitionoftheadditionaldegreeoffreedom1astheorientationofthesurfactanttailwithrespecttoanormaltotheclustersurface,1=cos.Here,istheanglebetweenvectors!T1T4and!OT4;T1andT4arerespectivelytherstandthelasttailbeadsofthemonomertailandOistheclustercenterofmass.Theprobabilitydistributionof1foramonomerconstrainednearanexitfrommicelleA64isshowninFigure 2-16 anddemonstratesthat,formostofthetimeduringtheconstrainedsimulations,themonomerisnormaloralmostnormaltothemicellarsurfacewithitstailpointingtowardsthemicellarcenterofmass.Therefore,onthetime-scalesoftheconstrainedsimulationsandthemonomerremovalfromthemicelle,theorientation1isslavedtothereactioncoordinateevenneartheexitfromthemicelle.Thereorientationduringtheconstrainedsimulationsleadstoaverysmallapparentbarrierfortheentryintothemicellepredictedbytheconstrainedsimulations.However,analysisoftheautocorrelationfunctionoftheorientationofthemonomerlocatedcompletely 33

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2-17 .ThefractionoftheclustersurfaceareaoccupiedbyhydrophobicpatchesdecreasesastheclusteraggregationnumberNincreases,asdemonstratedinFigure 2-18 .ThissuggeststhattheprobabilityofadditionofamonomertoclusterANdecreaseswithincreasingN.ThepatchdistributionsshowninFigure 2-17 areobtainedforoneparticularmomentoftimeand,ofcourse,highmobilityofsurfactantmoleculesleadstothepatchuctuations.Inordertoassessthetime-scaleandthemagnitudeoftheseuctuations,wemonitorthepatchlocationsoveraperiodoftime.Figure 2-19 showsthefractionoftimeeachsurfacepointisoccupiedbyahydrophobicpatchwithinaperiodof=200ps.Thischoiceoftimeismotivatedbythetime-scaleofadditionofamonomertoacluster 34

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2-17 issmallerthan200ps.Nevertheless,itisclearthatsomeofthepatchesremainstablewithinthe200psperiod,whichimpliesthattheywillremainstableduringmonomeraddition.Wenextassessthetime-scaleofthepatchuctuationsandcompareitwiththetime-scaleofthemonomerremoval.Oneofthepossiblemethodstoestimatethistime-scaleistomeasurepatchlifetimebytrackingapatchfromthemomentofitsformationtoitsdisintegration.Thetrackingisperformedbyconsideringsurfacepatchesattimeinstancesspacedbyt=25ps.Apatch(t+t)isconsideredtobeacontinuationofapatch(t)iftheintersection(t)\(t+t)isnotempty.Basedonthisdenition,weobservethatalifetimeofapatchforlargemicellesAN,60N70,isontheorderof1nswhereasforsmallclustersAN,10N20,itisontheorderof100ns.However,thepatchtime-scaleestimatebasedontrackingpatchesdoesnotfullyaddresstheirhighlydynamicnature.Infact,eventhoughthelifetimeofapatchonthesurfaceofasmallclusterisO(100ns),thelocationandtheshapeofthepatchchangemultipletimesduringthisperiod.Therefore,weadaptadierentstrategytoassessthetime-scaleofthemicrostructureuctuationsandrelateittothediusiontime-scaleofindividualsurfactantswithinacluster.Weconsidertwosurfactantmoleculeswhichareinitiallylocatednexttoeachotherwithintheclusterandcomputethecorrelationtimeoftheirrelativeorientation.Thiscorrelationtimeisontheorderof10nsforclusterswithaggregationnumberNrangingfrom9to64andonlyslightlyincreaseswiththeincreaseofN.Thistogetherwiththeresultsofthepatchtrackingimpliesthatevenamoderatechangeintherelativedisplacementofsurfactantmoleculeswithinalargemicelleleadstodestructionofapatch. 35

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1{20 )andincorporatetheadditionaldegreesoffreedom1and2toenableanaccuratepredictionofthefreeenergybarrierforthemonomerentry.Ingeneral,solutionofthismulti-dimensionalLangevinequationrepresentsanon-trivialtask.However,sincethesystemofnonionicsurfactantsconsideredinthecurrentworkdoesnotcontainlong-rangeinteractions,itisreasonabletoapproximatethesedegreesoffreedomasindependentuntilthemonomercomesintoacontactwiththecluster.Amonomeradditiontoanonionicsurfactantclustercanthereforebeadequatelydescribedbythefollowingmodel.Themonomerandtheclustertranslateandrotateindependentlyofeachother.Interactionbetweenthemoccursonlyifthemonomercomesintoacontactwiththeclustersurface.Ifthemonomerhasafavorableorientationandcontactstheclusteratahydrophobicpatch,thereactiontakesplaceinstantaneously.Otherwise,themonomerisrepelled.Thefavorableorientationofthemonomerisdeterminedbytheprobabilitydistributionofthemonomerorientationsobtainedfromtheconstrainedsimulations,suchasthatshowninFigure 2-16 formicelleA64.Basedonthesedistributions,inthecurrentworkweassumethatthemonomerorientationisfavorableiftheanglebetweenitstailandthenormaltotheclustersurfacedoes 36

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2.5.2 .Thepatchlocationisassumedtobeindependentfromthemonomerpositionuntilthemonomercomesintocontactwithit.Inordertocompletethemodel,weneedtoobtainthediusivitiesofthesurfactantclusterandthemonomer.ThetranslationaldiusivityDTNofclusterANisobtainedfrommoleculardynamicssimulations.Thesimulationsareperformedincubiccellswiththesideoflengthrangingbetweenh=9and10nmandwiththeperiodicboundaryconditions.SincetheaverageradiusRNofclusterANisofthesameorderofmagnitudeash(e.g.R642:15nm),itisnecessarytocorrecttheobtaineddiusivityvaluestoaccountforhydrodynamicinteractionsbetweenperiodicimagesoftheclusters.Forthepurposeofthiscorrection,weapproximatetheclustersbyrigidspheresandusetherelationshipbetweentheresistance(frictioncoecient)perofarigidsphereinaperiodicarrayandtheresistance1ofthissphereinaninnitedomain[ 63 ],1=Qper.ThefunctionQ=Q(R=h)isspeciedinRef.[ 63 ]anddependsontheratioofthesphereradiusRandtheunitcellsizeh.ThisallowsustoestimatethebulkdiusivityoftheclustersasD1=Dper=QusingthediusivityDperobtainedfromtheMDsimulationswiththeperiodicboundaryconditions.TheclusterradiusRNiscomputedfromitsprincipalradiusofgyrationRgN,RN=p 2-20 andarecomparedwiththepredictionoftheStokes-Einsteinrelationship,DTN=kBT=6RN,whereisthesolventviscosity.Recallthatinthecurrentcoarse-grainedmodelwaterisrepresentedbytheuidofhydrophilicbeads.Extrapolationofliteraturedata[ 64 ]forviscosityofLennard-Jonesuidstotheconditionsofoursimulationsyields7:1103cP,whichisingoodagreementwiththeviscosityofwater,H2O

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2-20 indicatesthattheStokes-Einsteinrelationshipprovidesgoodapproximationforthetranslationaldiusivityofthesurfactantclusters.ThisagreementallowsustoapplytheDebye-Stokes-Einsteinrelationship,DRN=kBT=6R3N,forrotationaldiusivityofrigidspherestothesurfactantclusters.Weobservethattheeectsofrotationofaclusterasawholearenegligibleincomparisonwiththediusionofindividualsurfactantswithinthecluster.TheDebye-Stokes-EinsteinrelationshipyieldsDRN=O(102ns1)forsmallerclustersandDRN=O(103ns1)forlargerclusters.Ontheotherhand,aswehaveseeninsection 2.5.2 ,thecorrelationtimeoftherelativeorientationoftwosurfactantswithinaclusterisontheorderof10ns,implyingthattherelativediusionofthesurfactantswithintheclusterisoneortwoordersofmagnitudefasterthantheclusterrotationaldiusion.Inasense,thenotionoftherotationaldiusionofasurfactantclusterismeaninglesssincetheclusterstructureisrearrangedmuchfasterthantheclusterwouldrotateasawholeifitwerearigidsphere.Therefore,theclusterrotationaldiusionisneglectedinourmodel.Thesurfactantmonomerismodeledbyarigidrodintheslenderbodyapproximation[ 65 ].Thetranslationaldiusionofsucharodisanisotropic,DT1=(DTk+2DT?)=3,whereDTkisthediusivityinthedirectionparalleltotherodandDT?=DTk=2isthediusivityinthedirectionnormaltotherod.ThetranslationandrotationdiusioncoecientsofthemonomerobtainedfromMDsimulationsareDT15:41010m2/sandDR16:37108s1,respectively.Therefore,DTk8:101010m2/sandDT?4:051010m2/s.Forcomparison,thetranslationandrotationdiusivitiesofthemonomerpredictedbytheslenderbodyapproximation[ 65 ]areDTk=kBTln(L=d)=2L7:121010m2s1,DT?=kBTln(L=d)=4L3:561010m2s1,andDR1=3kBT(ln(L=d))=L35:79108s1.Inthesecalculations,thelengthoftherodListakentobetwicethelargestradiusofgyrationofthemonomer,L1:6nm,theroddiameterdistakentobethatofthesurfactantbeads,d=0:47nm,and=0:8. 38

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Here,Iistheidentitymatrix,xNandx1arethecoordinatesofthecentersofmassoftheclusterANandthemonomer,uisthemonomerorientation,denedasaunitvectorpointingfromthetailtotheheadofthemonomer;N,T,andRaretheresistancematricesforthetranslationalmotionoftheclusterANandthetranslationalandtherotationalmotionsofthemonomer;N(t)andT(t)arethestochasticforcesdrivingthetranslationofthecentersofmassoftheclusterANandthemonomer,respectively,andR(t)istheweightedrandomforcedrivingthemonomerrotation[ 66 67 ].Allstochasticforcessatisfytheuctuation-dissipationtheorem, Theresistancematricesaregivenbythefollowingexpressions[ 65 66 ] DTNI; DTkuu+kBT DT?(Iuu); DR1(Iuu): Thereaction(i.e.,monomerentryintothecluster)maytakeplaceonlyiftherodisincontactwiththehydrophobiccore.Themonomerisconsideredtobeincontactwiththeclustercoreifoneoftherodendsiswithinthecoreradiusofgyration,Rcore,from 39

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2.5.4 .Stochasticequationsthat,similarlytoEqs.( 2{4 )-( 2{6 )and( 2{12 ),describediusion-controlledbimolecularreactionsinsomecasescanbesolvedanalytically.Forexample,therateoffusionoftwosphereswithnostericbarrierispredictedbytheSmoluchowskimodel[ 61 ].SeveralextensionsoftheSmoluchowskimodelhavebeenproposedtotakeintoaccountdierentreactionconditionsleadingtoassociationofspecies[ 68 { 71 ].Thesemodelsassumerelativelysimplegeometry:reactionstypicallytakeplacebetweensphericalparticleswithasinglesymmetricreactionpatch.Forcomplexboundaryconditionsinvolvingmultipledynamicpatches,ananalyticalsolutionoftheseequationsisunfeasible.Inaddition,weanticipateanextensionofthecurrentmodeltoamoredetailedmodelwithevenmorecomplexboundaryconditions(seesection 2.7 )andtoionicsurfactantsystemswithlong-rangeelectrostaticinteractionswhichwouldrequireustoconsiderdynamiccouplingbetweentheclustermicrostructureandthemonomer.Inthelattercase,thepurelydiusivemotionswillbereplacedbydynamicsdrivenbybothpotentialandrandomforces,therebymakingitevenmorediculttosolvethesystemanalytically.InthecurrentworkwethereforeuseBrowniandynamicssimulationstosolveEqs.( 2{4 ){( 2{6 )withreactionconditions( 2{12 )andobtainthemonomeradditionrates. 72 ]forabimoleculardiusion-controlledreaction.Thismethodallowsonetopredictthereaction 40

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72 ]: 1[1pq(b)][kD(b)=kD(q)]:(2{15)Theabovemethodwasdevelopedforreactionoftwosphericalparticleswithstationaryreactionsites.Itneedstobemodiedtoaccountfor(i)anon-sphericalshapeofthereactant(i.e.,themonomer)and(ii)dynamicsofreactivesites(i.e.,hydrophobicpatches)ontheclustersurface. 41

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2{14 )and( 2{15 )aretheSmoluchowskirateskD(b)andkD(q)whicharecomputedanalytically.ItiseasytoshowthattheSmoluchowskirelationship( 2{13 )stillholdsfortheprobabilityuxofarodrelativetoaspherewiththetranslationaldiusivityoftherodgivenbyD1=(2D?+Dk)=3.Thesecondmodicationissomewhatmorecomplicated.Recallthatdynamicsofthesurfacepatchisslowerthanthemonomeradditionunderfavorableconditions(seesection 2.5.2 ).Therefore,weassumethatthesurfacepatchdistributionremainsstableduringthemonomerapproachandanattemptedentry.Ifanattemptedmonomerentryisnotsuccessful,thereisalargeprobabilitythatitwillbefollowedbyanotherattemptatanearbypoint.Thetimebetweenthesetwoattemptsisshortanditisexpectedthatthecongurationoftheclustersurfacewillremainrelativelyunchanged.Hence,inthecurrentmodelwekeepthesurfacestructureconstantduringthesequentialcollisionswiththemonomeruntilthelattereitherleavesaneighborhoodoftheclusterorentersthecluster.Thesurfacestructureisreplacedeverytimethedistancebetweenthecentersofmassoftheparticlesbecomesr=q.ThesurfacepatchdistributionisrandomlyselectedfromadatabasewhichispreparedpriortoBDsimulationsfromtheresultsofMDsimulations.Thedatabaseispreparedasfollows.ForeachclusterAN,weperformMDsimulationsandsplittheobtainedtrajectoryintosegmentsoflengthseg=200ps.Thissegmentlengthapproximatelycorrespondstothetimeofadditionofthemonomertotheclusterwhenthealignmentofthemonomerorientationandthesurfacestructureisfavorable(seesection 2.4.2 andFigure 2-13 ).Foreachofthesesegments,weobtainthehydrophobicareasthatareexposedtowaterfortheentiresegperiod.Theseareascorrespondtohydrophobicpatchesthatareexpectedtoremainstableduringthemonomerentryinto 42

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2{4 ),( 2{5 )fortranslationofthecenterofmassofthemicelleandthemonomeraresimulatedusingthealgorithmproposedbyErmakandMcCammon[ 73 ].Theequation( 2{6 )forthemonomerrotationissimulatedusingthealgorithmofCobbandButler[ 67 ].Theinitialorientationsofboththemonomerandtheclusterarerandomandtheinitialseparationbetweenthecentersofmassofthemonomerandtheclusterisb=8nm.Thisvalueofbissucientlylargesothatthereactiveuxthroughthesurfacer=bhasnoangulardependenceandEq.( 2{13 )issatised.ThisisconrmedbyadditionalBDsimulationswiththeinitialparticleseparationsb=6nmandb=10nmwhichyieldthesamereactionratesassimulationswithb=8nm.Theobservedangle-independenceofthereactiveuxthroughthesurfacer=bevenforrelativelysmallradiibisduetoahomogenizingeectofthefrequentchangesinthereactivepatchesontheclustersurface.Theradiusoftheoutersphereqischosentobeq=30nm.ThisvalueissucientlylargecomparedtobsothatasignicantfractionofBDtrajectoriesissuccessful.Thereactionconditionsareimplementedasfollows.Ifthedistancebetweenthecentersofmassofthemonomerandtheclusterislessthan(Rcore+L=2),wecheckifthehydrophobicendofthemonomeriswithin0.2nmfromtheclustercoresurface.Ifso,themonomerandthecoreareconsideredtobeincontactand,ifthereactionconditions( 2{12 )arealsosatised,thereactionisconsideredtohavetakenplace.Otherwise,aBDstepisperformedtorepelthemonomerfromtheclusterinarandomdirection.BDsimulationwereperformedforclustersANwithaggregationnumberNrangingfrom9to88.TheobtainedadditionratesareshowninFigure 2-21 .Forcomparison,thisplotalsoshowspredictionsoftheSmoluchowskimodel, 43

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1{14 ),( 1{15 )andcompareitwiththeclustersizedistributionobtainedinsection 2.4.1 fromtheconditionofthermodynamicequilibrium.Forthepurposeofsolvingthemasterequations( 1{14 ),( 1{15 ),wetruncatethisinnitesystemofequationsatasucientlylargeaggregationnumber,M=88,andrearrangetheequationsasfollows: (2{17) Thissetcontains(M1)equationswithMunknownsC1;:::;CM.Theremainingequationisprovidedbytheconstraintonthetotalsurfactantconcentration, 2{17 ),( 2{18 ),weassignavalueC(0)1toC1,solvetheequations( 2{17 )and( 2{18 )forCN(C(0)1);N=2;:::;M,andobtainthetotalsurfactantconcentration,XT=XT(C(0)1),aposteriori. 44

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2{17 ),( 2{18 )isconvertedtoasingleequationforC2asfollows.ForgivenvaluesofC1=C(0)1andC2,wecomputeCN=CN(C(0)1;C2),N=3;:::;M,fromEq.( 2{18 ).SubstitutingthesevaluestogetherwithC2inEq.( 2{17 )andsolvingthelatterequationforC1,weclosethesystembyrequiringthattheobtainedrootofEq.( 2{17 ),C1C(i)1(C2;C3;:::;CM),coincideswiththegivenvalueC(0)1.Inotherwords,theproblemreducestosolvingasingleequation, 2{20 )issolvedusingNewton'smethod.OncethefunctionsCN(C1);N=2;:::;MandXT(C1)areobtained,wecomputetheCMCvalueusingtheapproachdiscussedinsection 2.4.1 .BDsimulationstoobtaintheadditionrateskN;N+1wereperformedforclusterswithaggregationnumberN9.ForsmallerN,thesphericalapproximationoftheclustersurfacebecomesinaccurate.Moreover,thesurfacebecomeslessdenedforthesesmallclustersandhencelocationofthehydrophobicpatchescannotbeadequatelydelineatedbymappingthemtotheclustersurface,aswasdoneforthelargerclusters.Therefore,inthecurrentwork,theratesofmonomeradditiontoclustersANwithN<9areapproximatedbyextrapolationoftheratesobtainedbyBDsimulationsforthelargerclusters.TheclustersizedistributionatXT=2107Mobtainedfromthesolutionofthemasterequations( 2{17 )and( 2{18 )iscomparedwiththepredictionofthethermodynamictheoryinFigure 2-10 .Theclustersizedistributionpredictedbythekineticequationsissomewhatwiderandtheaggregationnumberofthemostprobablemicelle,N=70,slightlyexceedsthepredictionofthethermodynamictheory,N=64.TheobtainedvalueforCMCis1:06108M,whichisanorderofmagnitudesmallerthanthepredictionofthethermodynamictheory,1:0107M.Reasonsforthisdiscrepancyandpossibleimprovementofthestochasticmodelarediscussedinthenextsection. 45

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2-10 .ThisqualitativedierencearisesbecausetheSmoluchowskiadditionratesgrowwiththeaggregationnumberNwhenN16,whereasthestericconstraintsdueto1and2leadtothedecreaseoftheadditionrateswithincreaseofN,seeFigure 2-21 46

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2{12 ).Inthecurrentmodelweassumethattherangeofthemonomerorientationsleadingtoasuccessfulentryintotheclusteristhesameforallpatches.However,ourobservationsofunconstrainedMDsimulationsofmonomerentryindicatethattherangeofthefavorablemonomerorientationsdependsonthepatchareaandonwhetherthemonomercontactsthepatchinitscentralorborderregion.Inaddition,amonomerwhoseorientationsignicantlydeviatesfromthenormaltotheclustersurfaceislesslikelytoentertheclusterthanamonomerorientedalongthenormal.Amoreaccuraterepresentationofthereactionconditionswouldassignprobabilitiesofasuccessfulmonomerentryintotheclusterbasedonthemonomerorientation,patchsize,andthelocationofthepatch-monomercontact.Moreover,inthecurrentmodelweassumethatthepatchshapeandlocationdonotchangeduringseveralconsecutivecontactswiththemonomer.Amoreaccuratereactionmodelwouldincludethechangesinthepatchshapeandlocationduetouctuationsofthesurfactantswithintheclusterandtheclusterinteractionwiththemonomerduringanentryattempt.Inthecurrentwork,thereactionconditionsweredeterminedfrom(i)constrainedsimulations(monomerorientation)and(ii)equilibriumMDsimulationsofindividualclusters(patchdynamics).Toimplementmostofthecorrectionsoutlinedaboveitisnecessarytodeciphershort-scaledynamicsofinteractionsbetweenthemonomerandthecluster.Themethodsusedinthecurrentworkarenotapplicabletoinvestigationsofthesenon-equilibriumprocesses.OnepossibleapproachistofollowthemethodofRef.[ 24 36 ]andobtainthereactionconditionsbasedonaseriesofshort-scalenon-equilibriumsimulationsinitializedatvariousinitialsurfactantandclustercongurations.Multiplerepetitionofsuchsimulationswiththesameinitialconditionforthereactioncoordinates(;1;2)butdierentinitialthermalbathcongurations(e.g.,locationsofthesolventmolecules)willyieldprobabilityP(;1;2)ofthemonomerentryintotheclusterforthegivenvaluesofthereactioncoordinates.Suchprobabilitiesobtainedforarangeofthe 47

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2{12 ).Similarly,onecanusetheseshort-scalesimulationstoassesseectsofthemonomer-clusterinteractionsonthepatchshapeandlocation.Anotherpossiblesourceofthediscrepancybetweenthemicellarsizedistributionsobtainedfromtheconditionofthermodynamicequilibriumandthekineticmodelistheassumptionthatthestep-wiseadditionandremovalofmonomersaretheonlymechanismsofthemicelleformationanddisintegration.Tocompletethismodel,itisnecessarytotakethefusionandssionofsurfactantclustersintoaccount.Asinthecaseofthemonomeradditiontoclusters,thedynamicsofthehydrophobicpatchesontheclustersurfacesplaysacrucialroleintheclusterfusion.OurMDsimulationsdemonstratethat,priortothefusionoftwoclusters,themoleculeswithinthemrearrangethemselvessothatthehydrophobicpatchesoftheclustersareexposedtoeachother.Therefore,understandingoftheclustersurfacedynamicsshouldenableadevelopmentofamodelforclusterfusion.Theclusterfusionissimilartoalargenumberofotherprocessesoccurringinself-assembledsystems,suchastransitionsfromsphericaltoworm-likemicellesandfusionoflipidvesicles.Understandingoftherelativelysimpleprocessofadditionofasinglemonomertoasurfactantclustersisthereforeexpectedtoaidinthedevelopmentofmodelsfordynamicsofthesemorecomplexsystems. Figure2-1. Coarse-grainedrepresentationoflinearH4T4surfactantandwatermolecules. 48

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Snapshotofasimulationofself-assemblyofH4T4surfactantsintosphericalmicelles.Thesurfactantvolumefractionis0.133.Theequilibratedsimulationcellisa8:88:88:8nm3cube.Hydrophilicheadbeadsandhydrophobictailbeadsareshownbylightgrayandblackspheres,respectively.Watermoleculesareomittedforclarity.Initiallythesimulationcellcontainsarandommixtureofsurfactantandwatermolecules. Figure2-3. RadiiofgyrationRgN;i,i=1;2;3,ofclusterswithaggregationnumberN. 49

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Autocorrelationfunctionoftherandomforce(t;)foramonomerpartiallyremovedfrommicelleA64andconstrainedat=1:598nm. Figure2-5. DependenceofthefreeenergyG(solidline)andthefrictioncoecient(dashedline)onthedistancebetweencentersofmassofthemonomerandmicelleA64. 50

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Prolesofvolumefractionsofsurfactanttails,head-groups,andsolvent(T,H,S)formicelleA64asafunctionofdistancerfromthecenterofthemicelle. Figure2-7. FreeenergyprolesforremovalofamonomerfromclustersofvariousaggregationnumbersN. 51

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EquilibriumdistancesbetweenthecentersofmassoftheclusterANandthesurfactantswithintheclusterobtainedfromconstrainedandunconstrainedsimulationsareshownbythesolidanddashedlines,respectively. Figure2-9. Dierencebetweenthestandardpartsofthechemicalpotentials0Nand01ofasurfactantmonomerinsideclusterANandafreemonomerinsolution. 52

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ClustersizedistributionattotalsurfactantconcentrationofXT=2107Mobtainedfrom(a)thermodynamictheory(solidline),seesection 2.4.1 .(b)kineticequations( 1{14 ),( 1{15 )withtheadditionratesdeterminedfromtheBrowniandynamicssimulationsofthemulti-dimensionalstochasticmodel(dashedline),seesection 2.6 ,and(c)kineticequations( 1{14 ),( 1{15 )withtheadditionratespredictedfromtheSmoluchowskitheoryofcolloidalaggregation(dottedline),seesection 2.6 53

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Dependenceoftheconcentrationsofthefreesurfactantmonomers,X1(dashedline),andthesurfactantscontainedinclusters,Xclus(solidline),onthetotalsurfactantconcentrationXTpredictedbythethermodynamictheory. Figure2-12. AveragetimeN!N1ofmonomerremovalfromsurfactantclustersANpredictedbyEq.( 1{23 ). 54

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AveragetimeN!N+1ofmonomeradditiontosurfactantclustersANpredictedbyEq.( 1{23 ). Figure2-14. DistancebetweencentersofmassofmicelleA87andamonomerfortwodierentMDsimulations.Thesimulationsareperformedinacubic101010nm3cellwiththeperiodicboundaryconditions. 55

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SnapshotsofMDsimulationsofattemptsofmonomerentryintomicelleA87.Hydrophilicheadbeadsandhydrophobictailbeadsareshownbylightgrayandblackspheres,respectively.PlotsA-Dcorrespondtopointsa-dofFigure 2-14 :A)Boththemonomerandthemicelleareinafavorablecongurationleadingtoasuccessfulentry;B),C)Monomerwithanunfavorableorientationapproachesanexposedhydrophobicpatchonthesurfaceofthemicelle;D)Monomerwithafavorableorientationapproachesthehydrophiliccoronaofthemicelle. Figure2-16. Probabilitydistributionoforientation1cosofamonomerconstrainedatanentrytomicelleA64.Thedistancebetweenthecentersofmassofthemonomerandthemicelleis=3:65nm. 56

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InstantaneousdistributionsofhydrophobicpatchesonsurfacesofclustersAN:A)N=16,B)N=32,C)N=64,andD)N=88.Thehydrophobicpatchesareshowninblackandthehydrophilicareasareshowninwhite.Thesphericalanglesandareinradians. Figure2-18. FractionofthesurfaceareaofclustersANoccupiedbyhydrophobicpatches. 57

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Fractionoftimethatclustersurfacesareoccupiedbyhydrophobicpatchesduringaperiodof200ps.TheinstantaneouspatchdistributionsshowninFigure 2-17 correspondtooneofthepointswithinthistimeinterval.ClusteraggregationnumbersareA)N=16,B)N=32,C)N=64,andD)N=88.Thedarkerareascorrespondtothehigherfractionofoccupationtime. Figure2-20. TranslationaldiusioncoecientsofclustersANobtainedfromMDsimulations(solidline)andpredictedbytheStokes-Einsteinrelationship(dashedline). 58

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RateconstantskN;N+1ofadditionofmonomerstoclustersANobtainedfromtheBrowniandynamicssimulations(solidline)andfromtheSmoluchowskimodelEq.( 2{16 )(dashedline). 59

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2 ],stabledispersedpharmaceuticalformulations[ 74 ]andnoveldrugdeliverymethods[ 75 ].Sincemostamphiphilicmoleculesfoundinbiologicalsystemshaveelectrostaticcharges,thestudyofself-assembledstructuresofionicsurfactantswillalsoaidinunderstandingofcomplexbiologicalprocesses[ 76 ].Theequilibriumpropertiesofself-assembledstructuresformedbyionicsurfactantshavebeenextensivelystudiedexperimentally[ 62 77 ]aswellasusingtheoreticalandcomputationalmodels[ 44 78 { 92 ].Forexample,Yoshiietal.[ 44 ]haveobtainedthefreeenergyofmicelleformationforSodiumdodecylsulfate(SDS)surfactantusingmoleculardynamicssimulationscombinedwiththermodynamicintegrationmethodandhavecalculatedcriticalmicelleconcentration(CMC)andmicellesizedistributioningoodagreementwiththeexperimentalresults.However,modelingdynamicsofself-assemblyandstructuraltransitionsinsurfactantstructuresismorechallengingsincetheseprocessesinvolveawidespanoftime-scales.Currentlyavailableexperimentalandtheoreticaltoolstostudytheseprocessesaddressdynamicsataparticularscale.Forexamples,atomisticmoleculardynamicsimulationswhicharelimitedtonanosecondsofsimulationtimeareoftenusedtoobtainpropertiessuchasdiusioncoecientandmorphologyofpreassembledstructures[ 87 { 91 ].AdvancementofcomputationalprocessingpowerrecentlyenabledatomisticMDsimulationsofself-assemblyofionicsurfactantsstartingfromrandominitialconguration[ 20 92 ].SincethesesimulationsareperformedinsmallsimulationboxeswithsizesofsidelengthO(10)nm,thesurfactantconcentrationinthese 60

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3.2.1Coarse-GrainedModelsCoarse-grainedMDsimulationscaneasilyachievemicrosecondsofsimulationtimeandhavesuccessfullyshownformationofself-assembledstructuresofsurfactantsstartingfromrandominitialdispersionofsurfactants.Coarse-grainedstudieswithimplicitandexplicitsolventhavebeencarriedouttostudytheself-assemblyofionicsurfactantsinsolvent[ 21 93 94 ].Coarse-grainedMDsimulationsisarealisticapproach[ 95 ]tostudytheself-assemblyofionicsurfactants.Thesesimulationspreservetherelevantmoleculardetailsandwhencombinedwithpotentialofmeanforcetechniquessuchasconstrainedsimulations,umbrellasamplingandthermodynamicintegrationmethodsyieldsignicantinsightintothedynamicsofself-assembly[ 96 ].Inthisworkweusethecoarse-grainedmodeldevelopedbyMarrinketal.[ 26 ].Themodelhasfourtypesofbeads,namely,apolar(C),non-polar(N),polar(P)andcharged(Q)beads.Agroupoffourmethylene/methylgroupsofahydrocarbonchainisrepresentedbyasingleapolarbeadC.Agroupoffourwatermoleculesisrepresentedby 61

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3-1 .Itshydrophobictailcontaining12methylene/methylgroupsismodeledbyachainofthreelinearlyconnectedbeadsoftypeC.Thesulfate(-SO4)headgroupismodeledbyasinglechargedbeadoftypeQa(chargedbeadsactingashydrogenbondacceptors).Sodiumcounterion(Na+)ismodeledasasinglechargedbeadoftypeQda(chargedbeadwhichactsasdonoraswellasacceptorofahydrogenbond).ThemagnitudeofchargeofQaandQdabeadsisreducedfrom1.0eto0.7etoaccountforthehydrationshellsurroundingtheion[ 26 ].Thereductioninchargeofbothofthechargedbeadsisthesametoensureelectricalneutralityofthesystem.Thedielectricconstantofthesolventmediumis20.Theinteractionsbetweennon-bondedbeadsaremodeledbytheLennard-Jones(LJ)potential.Thecharacterofinteractionismodeledthroughtheinteractionenergyparameter.TheP-P,P-Qa,P-Qda,Qda-QdaandQa-Qdaarehighlyattractive(=5kj/mol).Qa-QaandC-Cinteractionsareslightlyattractive(=3:4kj/mol),whereasC-P,C-QaandC-Qdainteractionsarealmostpurelyrepulsive(=1:8kj/mol).LJpotentialisshiftedsmoothlytozerousingthestandardGROMACSshiftfunction[ 56 ]startingfrom0.9nmtothecut-olengthof1.2nm2.5.Forsimplicity,wewillrefertoQda-typebeadmodelingsodiumcounterionasQ+andQa-typebeadmodelingsulfateheadgroupasQ.ThesimulationsareperformedintheNPTensembleattemperature300Kandpressure1bar.Berendsenthermostatandbarostatwithtimeconstants1psandisothermalcompressibility1bar1areusedtomaintainthetemperatureandpressureconstantinthesimulationboxduringMDsimulations.Simulationsareperformedwithtime-stepof0.025ps. 62

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26 ]assumesstrongscreeningofchargeswitheectiverangeofelectrostaticinteractions1:2nm.Thisassumptionsignicantlyspeedsupcomputationofelectrostaticforces.However,severalstudieshaveshownthatthisapproximationintroducesartifacts[ 97 { 101 ]andfailstocapturetheessentialcharacteristicsoftheevolutionanddynamicsoftheamphiphilicsystems.Burovetal.[ 101 ]haveshownthatneglectingthelong-rangecontributionleadstounlimitedgrowthofmicellarstructureforionicsurfactants.SchreiberandSteinhauser[ 99 ]haveshownthatthestabilityofan-helixstructureofapeptidechaindependsupontheeectiverangeofelectrostaticinteraction.Thereforeweassumethattheelectrostaticinteractionsarenotscreened,althoughthisassumptionprobablyleadstoanoverestimationofinteractionstrengthasthereissomedegreeofchargescreening.InthisstudyweuseParticleMeshEwald(PME)summationtoobtaintheelectrostaticinteractionpotentialandforcesactingonchargedparticles. 3.3.1 thattheelectrostaticinteractionbetweenamicelleandachargedmonomerbeadarescreenedbycounterionssothattheforcesbetweenamicelleandachargedbeadisnegligiblewhenthedistancebetweenthemexceeds4nm.Hence,eveninasmallsimulationboxtheinteractionsofbeadswiththeirperiodicimagesisexpectedtobenegligible.Theself-assemblysimulationsareperformedfor200SDSsurfactantsand16000waterbeadsinasimulationboxofdimension151515nm3.Toachievethepressureof1bartheboxsizereducestoequilibriumdimensions 63

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3-2 .Asinthecaseofnonionicmicellestheself-assemblytakesplaceonthisshorttime-scaleO(100)nsbecauseitismostlyadiusion-controlledprocessand,sincethesurfactantmoleculesarepresentinasmallsimulationbox,theydiuseandndothersurfactantmoleculesveryquickly.Inreality,theself-assemblyprocesswouldtakeamuchlongertimeandthecorrespondingindividualstepsofself-assemblywouldlikelybedierentfromthoseobservedinMDsimulations.Also,asinthecaseofnonionicsurfactants,wedonotobservedisintegrationofmicellesinoursimulationssincethetime-scalesofmicellardisintegrationarenotaccessiblebythecoarse-grainedMDsimulationssincetheprocessofmicellardisintegrationisanactivatedprocesswithaveryhighactivationbarrier. 64

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58 ].Themethodofconstrainedsimulationisdescribedindetailinsection 1.3.3 .Thetotaltimeforeachconstrainedsimulationis200ns.ConstrainedsimulationsareperformedformicellesANofaggregationnumberN=30,35and40.TheautocorrelationfunctionoftherandompartoftheconstrainingforceisshowninFigure 3-3 .Theautocorrelationfunctiondecaystozerowithin1ps,whichvalidatestheMarkovianLangevinmodelforthedynamicsoftheremoval/additionofmonomerfrom/tomicelle.ThefreeenergyproleobtainedbyintegrationofthemeanconstrainedforcefortheremovalofamonomerfromamicelleisshowninFigure 3-4 .Thereisadistinctequilibriumposition(min)ofamonomerinthemicellecorrespondingtothefreeenergyminimum.Thefreeenergyincreaseswithincreasingandreachesamaximumatacertaindistance=crit.Asthedistanceincreasesfurther,thefreeenergystartstodecrease.Thisdecreaseofthefreeenergyisabsentinnonionicmicelles[ 96 ]andinsection 3.3.1 wedemonstratethatitoccursduetolong-rangeelectrostaticinteractionbetweenthemonomerandthemicelle.DependenceofthefrictioncoecientforthemonomerinsidethemicelleonthereactioncoordinateisshowninFigure 3-4 .Thefrictioncoecientexhibitsamaximumat=mincorrespondingtothefreeenergyminimum.Thefrictioncoecientapproachesaminimumnearthefreeenergybarrierandquicklyincreasestoitsequilibriumvalue.Thisbehaviorofisqualitativelysimilartothatforthenonionicmicelles.Thefreeenergyandfrictioncoecientobtainedfromconstrainedsimulationsareusedtocalculatetheaveragetimeofremoval/additionofamonomerfrom/toamicelleusingEq. 1{23 .TheaveragetimeofremovalofamonomerfrommicelleA30,isabout13microseconds.The 65

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62 ]).Theestimatedaveragetimeforadditionofamonomerinitiallylocatedatdistance=4nmfromthecenterofmassofmicelleAN,(N=30,35and40)isO(1)ns,sincetheactivationbarrierformonomeradditionislow. 3-5 .DistributionoftheQchargeisaverynarrowGaussian,onthecontrarythedistributionofQ+counterionsisasymmetricaboutitspeakandextendswellintothebulksolvent.ThedistributionsofQchargeheadgroupsandQ+counterionspeakatthesameradialdistancefromcenterofmassofthemicelle.VisualinspectionoftheMDtrajectoriesshowsthatsomeofQ+counterionsaredissociatedfrommicellesandarerandomlydistributedatdierentdistancesfrommicellarcenters(Figure 3-2 ).Q+counterionsarehighlymobileandaredynamicallychangingthelocalenvironmentaroundthemicelle,inagreementwithobservationsofotherstudies[ 102 { 104 ].Weobservethat20to30%oftheQ+counterionsarenotboundtothemicelle.WemeasuredegreeofdissociationofQ+counterionsfromamicellebycalculatingthefractionofQ+counterionsinsphericalshellscenteredatthemicellecenter 66

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3-6 .ThedecayofthefreeenergyobtainedfromtheconstrainedsimulationscompareswellwiththesolutionofPoissonequationwhichconrmsourassumptionthatthisdecayiscausedbyelectrostaticinteractions.ThefreeenergyobtainedfromnumericalsolutionofPoissonequationisalmostconstantafter4nm.Thissuggeststhattheelectrostaticinteractionsbecomeveryweakafter4nm.However,thedecreaseinthefreeenergyobtainedfromtheconstrainedsimulationsislargerthanthedecreaseofthefreeenergypredictedbysolutionofPoissonequationforsphericallysymmetricchargedistribution.Thissuggeststhatassumptionofsphericalsymmetryofchargedistributiondoesnothold. 3-8 .Thisdistributionisobtainedforoneparticularmomentoftimeand,ofcourse,highmobilityofsurfactant 67

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3-9 .ThetaggedmonomerremainsassociatedwiththemicelleforO(10)ns.TheprobabilitydistributionpoftheminimumdistanceofQ+ionsfromtheQchargedbeadsonthesurfaceofthemicelleisshowninFigure 3-10 A.Thepotentialofmeanforce, 3-10 B. 102 { 104 ].Weconcludethattheprocessofremovalofasinglemonomerfromamicelleisanactivatedprocessandcanbedescribedbyasinglereactioncoordinatenamelythedistancebetweenthecentersofmassofthemonomerandthemicelle.Thepresenceofelectrostaticinteractionscreatesafreeenergybarrierfortheprocessofadditionofamonomertoamicelle.However,thisbarrierisrelativelysmallandotherdegreesoffreedom,suchasmicellarmicrostructureandcounteriondistribution,maybecomeimportant.An 68

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Figure3-1. Detailedatomicstructuresandthecorrespondingcoarse-grainedmodelforsodiumdodecylsulfate.Thecoarse-grainedbeadtypesaredenotedasfollows:C=apolar(4carbonatoms),Qa=chargedheadgroupwhichactsasahydrogenbondacceptor,Qda=chargedbeadgroupwhichactsasahydrogenbonddonorandacceptor[ 26 ]. 69

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Snapshotofasimulationofself-assemblyofSDSsurfactantsintomicelles.Simulationboxcontains200SDSmoleculesand16000Wbeads.Hydrophilicheadbeads(Q)andhydrophobictailbeads(C)areshownbyblackandwhitespheres,respectively.Sodiumcounterions(Q+)areshownbylightgrayspheres.Watermoleculesareomittedforclarity. Figure3-3. Autocorrelationfunctionoftherandomforce(t;)actingonamonomerpartiallyremovedfrommicelleA30andconstrainedat=1:19nm. 70

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DependenceofthefreeenergyG(solidline)andthefrictioncoecient(dashedline)onthedistancebetweenthecentersofmassofthemonomerandthemicelleA30. Figure3-5. ThedensitydistributionofchargesinandaroundmicelleA30.ThedistributionofQ+beadsisshownbythesolidlineandofQbeadisshownbythedashedline. 71

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FreeenergyGobtainedfromconstrainedsimulations(solidline)andtheelectrostaticcontributiontothefreeenergyfor>critobtainedfromsolvingthePoissonequation 3{1 formicelleA30(dashedline). Figure3-7. Freeenergyprolesforremovalofamonomerfromclustersofdierentaggregationnumbers. 72

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Typesofthebeadsfarthestfromtheorigininthedirection(;),aroundthemicelleA30.Theoriginofthecoordinatesystemisatthecenterofmassofthemicelle.Theregioncoveredbytail(C)beadsareshowninwhite,byheadbeads(Q)areshownbyblackandbycounterionbeads(Q+)areshownbygray. Figure3-9. TheminimumdistancebetweenaQ+counterionandalltheQchargesonthesurfaceofmicelleA30asafunctionoftime. 73

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A)TheprobabilitydistributionoftheminimumdistancebetweenaQ+counterionandalltheQchargesonthesurfaceofmicelleA30.B)ThepotentialofmeanforceofinteractionbetweenaQ+counterionandthesurfaceofmicelleA30. 74

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4.1 and 4.2 ,respectively.Inthesestudies,weconsideredsolutionsofH4T4surfactants. 75

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B.1 .Thisapproachinvolvesmultipleshorttimesimulationsinitializedatthesamevalueofreactioncoordinatebutdierentrandomcongurationsofdegreesoffreedomcorrespondingtoathermalbath.Thesystemevolveswithoutanyexternalconstrains.Andtheprocessisrepeatedusingsomeoftheobtainedcongurationsasinitialconditionsforanewsetofshort-scalesimulations.Thismethodisbenecialforunderstandingthekineticsof 76

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4-1 .Ataninitialstageofthefusionprocess,repulsionbetweenthepolarhead-groupsleadstolocaldeformationoftheclusters.Thehead-groupswithineachclustersrearrangetominimizetherepulsionandmoveapart,exposinghydrophobiccoresofthetwoclusters.Thehydrophobiccoresthenattracteachothercausingthefusionoftheclusters.Finally,themonomersrearrangewithinthefusedstructurewhichresultsinasphericalmicelle.Theentireprocesstakesontheorderof50ns.However,wedonotalwaysobservefusionofclustersandinsomeMDsimulationstwoclustersdonotundergostructuraltransitionsandrepeleachother,asillustratedinFig 4-2 .FromtheexampleshowninFigure 4-1 ,itisevidentthatthehydrophobicpatchdynamicsisactivelyinvolvedintheprocessofclusterfusion.Therefore,theresultsobtainedinthisdissertationresearchareexpectedtoprovideinsightsintothiscomplexprocess.Inaddition,weexpectthatunderstandingoftheclusterfusionwillrequireansweringthefollowingquestions: 1. Whatisthedynamicsofasinglemonomerinsideanaggregateandhowisitrelatedwiththepatchdynamics? 2. Whatisaprecursortoclusterfusion?Howdothemicelles\know"thattheyareclose? 3. Whathappenstowatermoleculesnearthemicellesduringfusion? 77

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Figure4-1. Fusionoftwoclusterstoformalargesphericalmicelle. 78

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Repulsionoftwoclusters. 79

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25 ]providesaweakcouplingtoanexternalthermalbath.Deviationofthecurrentsystemtemperature(T)fromaprescribedtemperature(T0)iscorrectedbyrescalingparticlevelocitiesbyafactor TT0 dt=T0T :(A{3) 25 ]theparticlecoordinatesandtheboxvectorsarerescaledaftereverystepbyamatrix.ThisresultsinanexponentialrelaxationofpressuretowardaprescribedpressureP0, dt=P0P :(A{4)Thescalingmatrixisgivenby 80

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81

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24 ],theeectiveFokker-Planckequationisconstructedbyaseriesofshort-timesimulationscapturingtheentiresystemdynamics.Thisisaveryusefultechniqueofobtainingsystemdynamicsfromsimulationdata.InordertounderstandtheFokker-Planckequationanditsconstituentcoecientletusrstbrieyreviewthemathematicaldescriptionofthestochasticprocess. 35 37 ] @t=1Xn=1@ @nDn(;t)W(;t)=LKMW;(B{1)whereDn(;t)arethedierentialmomentsofthetransitionprobabilityofthestochasticprocesswithafunctiondistributionatthestartingpointt;(t)=0.Thedierentialmomentsoftransitionalprobabilityaregivenby 82

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37 ].Foronedimensionalstochasticprocess,theFokker-Plankequationis @v()+@2 2@2(t;0) 83

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B{3 )yieldsequilibriumprobabilitydistributionWeq(),whichisrelatedtotheGibbsfreeenergybyWeq()/exp(G()=kBT).ThesteadystateFokker-Planckequation( B{3 )is B{5 )andsolvingforG(),weget _=1 35 ]: B{7 ).Therelationshipdependsonthemethodusedforcalculatingstochasticintegrals[ 105 ].Foronedimensionalprocess,ifoneusestheIt^omethodofintegration(whichusesthestartingpointvalueofthestochasticvariableinanintervalforcalculatingthestochastic 84

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22()kBT;(B{11) B{6 ).Theexpressionforfreeenergy( B{6 )and( B{14 )coincidesonlyforadditivewhitenoisei.e.positionindependentdiusioncoecient.Thedierenceissignicantformultiplicativewhitenoisei.e.positiondependentdiusioncoecient.TheuseoftheLangevinequationforthestochasticdescriptionofaprocessdependsontheinterpretationofthenoise(It^oorStratonovich)andrequiresthevaluesofthestochasticvariableatinnitelysmalltimeintervalsforanaccuratereconstructionofitsparameters.Itisimpossibletoobtainthevaluesatinnitelysmalltimeintervalsfrommoleculardynamicssimulations.WeusetheFokker-Planckdescriptionofastochasticprocesswhichdirectlyrelatesthedriftanddiusioncoecients 85

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86

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GunjanMohanwasborninIndia.HereceivedaBachelorofTechnologyfromIndiaInstituteofTechnology,Kanpur,Indiain2003.Aftercompletinghisbachelor'sdegree,hereceivedEngineeringAlumniFellowshiptopursueDoctoralstudiesattheDepartmentofChemicalEngineering,UniversityofFlorida,in2003.Hisresearchinterestfocusesoncomputationalmodelingofchemicalprocesses. 96