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Role of Hydrodynamic Interactions in Dynamics of Semi-Flexible Polyelectrolytes

Permanent Link: http://ufdc.ufl.edu/UFE0042735/00001

Material Information

Title: Role of Hydrodynamic Interactions in Dynamics of Semi-Flexible Polyelectrolytes
Physical Description: 1 online resource (2 p.)
Language: english
Creator: KEKRE,RAHUL
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: DNA -- ELECTROHYDRODYNAMICS -- HYDRODYNAMICS -- MICROFLUIDICS -- MIGRATION -- POLYELECTROLYTE -- POLYMER -- SIMULATIONS
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: Experiments have shown that DNA molecules in capillary electrophoresis migrate across field lines if a pressure gradient is applied simultaneously. We suggest that this migration results from an electrically driven flow field around the polyelectrolyte, which generates additional contributions to the center-of-mass velocity if the overall polymer conformation is asymmetric. Numerical simulations and experiments have demonstrated that confined polymers migrate towards the center of the channel in response to both external forces and uniaxial flows. Yet, migration towards the walls has been observed with combinations of external force and flow. In this work, the kinetic theory for an elastic dumbbell developed by Ma and Graham Phys. Fluids 17, 083103 (2005) has been extended to account for the effects of an external body force. Further modifications account for counterion screening within a Debye-Huckel approximation for the specific case of applied electric field. The theory qualitatively reproduces results of both experiments for the migration of neutral polymers and polyelectrolytes. The favorable comparison supports the contention Long et al., Phys. Rev. Lett. 76, 3858 (1996) that the hydrodynamic interactions in polyelectrolytes decay algebraically, as 1/r^3, rather than exponentially. A coarse-grained polymer model, without explicit charges, is developed and integrated using Brownian-dynamics simulations in analogy with the kinetic theory. The novel feature of the simulations is the inclusion of hydrodynamic interactions induced by the electric field. This model quantitatively captures experimental observations Zheng and Yeung, Anal. Chem. 75, 3675 (2003) of DNA migration under combined electric and pressure-driven flow fields in absence of any adjusted parameters. In addition the model predicts dependence of electrophoretic velocity on the instantaneous length of the polyelectrolyte which has been verified by experiments of Lee et. al. Electrophoresis 31, 2813 (2010). The model also predicts phenomenons that are yet to be verified experimentally. These include decrease in diffusivity and increase in radius of gyration of the polyelectrolyte in high electric fields due to internal dispersion. The resulting change in orientation distribution at high electric fields decreases the extent of migration. Preliminary results from microfluidic experiments are presented in this dissertation demonstrating the saturation of migration. This dissertation also includes comparison of results from lattice-Boltzmann and Brownian dynamics simulations of a linear bead-spring model of DNA for two cases; infinite dilution and confinement. We have systematically varied the parameters that may affect the accuracy of the lattice-Boltzmann simulations, including grid resolution, temperature, polymer mass, periodic boundary size and fluid viscosity. For the case of a single chain Lattice-Boltzmann results for the diffusion coefficient and Rouse mode relaxation times were within 1?2% from those obtained from Brownian-dynamics. Results from both methods are also compared for polymer migration in confined flows driven by a uniform shear or pressure gradient. Center-of-mass distribution obtained from Lattice-Boltzmann simulations agrees quantitatively with Brownian-dynamics results, contradicting previously published results. The mobility matrix for a confined polymer was derived by applying Faxen?s correction to the flow-field generated by a point force bounded by two parallel plates. This formulation of the mobility matrix is symmetric and positive-definite for all physically accessible configurations of the polymer.
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 RAHUL KEKRE.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Butler, Jason E.
Local: Co-adviser: Ladd, Anthony J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042735:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042735/00001

Material Information

Title: Role of Hydrodynamic Interactions in Dynamics of Semi-Flexible Polyelectrolytes
Physical Description: 1 online resource (2 p.)
Language: english
Creator: KEKRE,RAHUL
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: DNA -- ELECTROHYDRODYNAMICS -- HYDRODYNAMICS -- MICROFLUIDICS -- MIGRATION -- POLYELECTROLYTE -- POLYMER -- SIMULATIONS
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: Experiments have shown that DNA molecules in capillary electrophoresis migrate across field lines if a pressure gradient is applied simultaneously. We suggest that this migration results from an electrically driven flow field around the polyelectrolyte, which generates additional contributions to the center-of-mass velocity if the overall polymer conformation is asymmetric. Numerical simulations and experiments have demonstrated that confined polymers migrate towards the center of the channel in response to both external forces and uniaxial flows. Yet, migration towards the walls has been observed with combinations of external force and flow. In this work, the kinetic theory for an elastic dumbbell developed by Ma and Graham Phys. Fluids 17, 083103 (2005) has been extended to account for the effects of an external body force. Further modifications account for counterion screening within a Debye-Huckel approximation for the specific case of applied electric field. The theory qualitatively reproduces results of both experiments for the migration of neutral polymers and polyelectrolytes. The favorable comparison supports the contention Long et al., Phys. Rev. Lett. 76, 3858 (1996) that the hydrodynamic interactions in polyelectrolytes decay algebraically, as 1/r^3, rather than exponentially. A coarse-grained polymer model, without explicit charges, is developed and integrated using Brownian-dynamics simulations in analogy with the kinetic theory. The novel feature of the simulations is the inclusion of hydrodynamic interactions induced by the electric field. This model quantitatively captures experimental observations Zheng and Yeung, Anal. Chem. 75, 3675 (2003) of DNA migration under combined electric and pressure-driven flow fields in absence of any adjusted parameters. In addition the model predicts dependence of electrophoretic velocity on the instantaneous length of the polyelectrolyte which has been verified by experiments of Lee et. al. Electrophoresis 31, 2813 (2010). The model also predicts phenomenons that are yet to be verified experimentally. These include decrease in diffusivity and increase in radius of gyration of the polyelectrolyte in high electric fields due to internal dispersion. The resulting change in orientation distribution at high electric fields decreases the extent of migration. Preliminary results from microfluidic experiments are presented in this dissertation demonstrating the saturation of migration. This dissertation also includes comparison of results from lattice-Boltzmann and Brownian dynamics simulations of a linear bead-spring model of DNA for two cases; infinite dilution and confinement. We have systematically varied the parameters that may affect the accuracy of the lattice-Boltzmann simulations, including grid resolution, temperature, polymer mass, periodic boundary size and fluid viscosity. For the case of a single chain Lattice-Boltzmann results for the diffusion coefficient and Rouse mode relaxation times were within 1?2% from those obtained from Brownian-dynamics. Results from both methods are also compared for polymer migration in confined flows driven by a uniform shear or pressure gradient. Center-of-mass distribution obtained from Lattice-Boltzmann simulations agrees quantitatively with Brownian-dynamics results, contradicting previously published results. The mobility matrix for a confined polymer was derived by applying Faxen?s correction to the flow-field generated by a point force bounded by two parallel plates. This formulation of the mobility matrix is symmetric and positive-definite for all physically accessible configurations of the polymer.
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 RAHUL KEKRE.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Butler, Jason E.
Local: Co-adviser: Ladd, Anthony J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042735:00001


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IwouldliketoacknowledgetheconstantsupportandencouragementprovidedbymyadvisorsProfessorTonyLaddandProfessorJasonButler.Withouttheirguidance,inspiration,andrefusaltoacceptmediocreworkthisthesisaswellasmyeducationwouldnothavereacheditsconclusionatUniversityofFlorida.IconsideritanhonortohaveaccomplishedmyDocotoraldegreeundertheirtutelage.Whiletheysharesimilargoalsinresearch,myworkexperiencewithbothofthemwasdistinctyetinbothcaseseducationalandenjoyable.IwouldespeciallyliketothankJasonforbeingafriendandmotivatorinthetoughestoftimeswhotookituponhimselftoensurethatIdidwellpersonallyandprofessionally.Tony,ontheotherhand,providedmewithmuchneededperspectiveandadviceatopportunemoments.IhopetoincorporateTony'sdevotiontoscienceandscienticmethodandJason'sexemplaryworkethicinallmyfutureendeavors.Iwouldliketoextendmygratitudetowardsmyothercommitteemembers,ProfessorSergeiObukhov,ProfessorDmitriKopelevichandProfessorAnujChauhanfortheiracademicinputsandpatiencewithmyresearch.IamthankfultoProfessorsAnujChauhan,DmitriKopelevich,LewisJohns,andSergeyVasenkovforputtinginextraefforttomakemyclassroomexperienceatUFLintellectuallystimulating.IacknowledgeProf.ArvindAsthagiriforgivingmetheopportunitytoworkasaTeachingAssistantinhiscourse.ThemicrouidicexperimentspresentedinthisthesiswouldnothavebeenpossiblewithoutthegenerouslendingofaNikonDiapohotmicroscopeandaQImagingcamerabyProfesssorRichardDickinson.IalsoextendmygratitudetowardsProf.ZHughFanforhishelpinprovidingthecastingmoldforthemicrouidicchanneland,toProfessorTanmayLeleforthepermissiontousethemicroscopicset-upinhislab.IwouldespeciallyliketothankfellowgraduatestudentCarlos-SilveraBatistaforintroducingmetotheworldofscienticexperimentationandteachingmethebasics 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 LISTOFSYMBOLS .................................... 12 ABSTRACT ......................................... 17 CHAPTER 1DYNAMICSOFPOLYMERSOLUTIONS ..................... 19 1.1Introduction ................................... 19 1.2HydrodynamicInteractions .......................... 21 1.2.1Effectsduetobodyforces ....................... 21 1.2.2Effectofelectriceld .......................... 23 1.3NumericalMethods .............................. 25 1.4MicrouidicsExperiments ........................... 27 2KINETICTHEORY .................................. 29 2.1LiteratureReview ................................ 29 2.2Model ...................................... 30 2.2.1VelocityExpressions .......................... 32 2.2.2ProbabilityDistribution ......................... 36 2.2.3Center-of-massDistribution ...................... 40 2.3ResultsandDiscussion ............................ 42 2.3.1Pressure-drivenFlow .......................... 43 2.3.2MigrationduetoanExternalForce .................. 45 2.3.3FlowandForceinConjunction .................... 47 2.3.4FlowandForceinOpposition ..................... 48 2.4Electrophoresis ................................. 49 2.4.1VelocityExpressions .......................... 50 2.4.2ProbabilityDistribution ......................... 52 2.4.3Center-of-massDistribution ...................... 53 2.4.4Results ................................. 55 2.5DiscussionofResults ............................. 58 3BROWNIANDYNAMICSSIMULATIONSOFELECTROPHORESISOFDNA 60 3.1IntroductoryRemarks ............................. 60 3.2SimulationTechniqueandDNAModel .................... 61 3.3Results ..................................... 63 6

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....................... 67 4MICROFLUIDICEXPERIMENTS .......................... 72 4.1Objective .................................... 72 4.2DetailsofExperiments ............................. 74 4.2.1PreparationofMicrochannel ...................... 74 4.2.2PreparationofExperimentalSamples ................ 76 4.2.3ExperimentalSetup .......................... 77 4.2.4ImageProcessing ........................... 78 4.3CalibrationofExperimentalSetup ...................... 80 4.3.1Theory .................................. 80 4.3.2ResultsfromCalibrationExperiments ................. 81 4.4ResultsandDiscussion ............................ 82 4.5Summary .................................... 85 5COMPARISONOFLATTICEBOLTZMANNANDBROWNIANDYNAMICSSIMULATIONS:INFINITEDILUTION ....................... 87 5.1MotivationforComparisons .......................... 87 5.2PolymerModelandSimulationMethods ................... 88 5.2.1BrownianDynamics .......................... 89 5.2.2LatticeBoltzmann ............................ 90 5.2.3Polymer-uidCoupling ......................... 93 5.3Results ..................................... 97 5.3.1StaticProperties ............................ 97 5.3.2DiffusionCoefcient .......................... 101 5.3.3RouseRelaxationTimes ........................ 105 5.4SingleMoleculeComparison ......................... 109 6COMPARISONOFLATTICEBOLTZMANNANDBROWNIANDYNAMICSSIMULATIONS:CONFINEDSYSTEM ....................... 111 6.1SimulationsinConnedSystem ....................... 111 6.2SimulationTechniques ............................. 112 6.2.1Polymermodel ............................. 112 6.2.2Lattice-Boltzmann ........................... 112 6.2.3Brownian-dynamics ........................... 113 6.3Results ..................................... 116 6.3.1Uniformshearow ........................... 118 6.3.2Pressure-drivenFlow .......................... 121 6.4RemarksonComparisonStudies ....................... 122 7CONCLUSIONS ................................... 123 APPENDIX 7

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................................ 128 BPRETABULATINGTHEGREEN'SFUNCTION .................. 131 REFERENCES ....................................... 135 BIOGRAPHICALSKETCH ................................ 140 8

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Table page 5-1Staticanddynamicpropertiesofapolymerchain.FluctuatingLBsimulationsfora10-segmentchainarecomparedwithBrowniandynamics. ......... 100 5-2Effectofsystemsizeonthediffusioncoefcientofa10-segmentchain. .... 103 9

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Figure page 1-1Stokesowduetoapointforce ........................... 22 1-2HydrodynamicInteractionsinducedbytheelectriceldactingonamacroion. 24 1-3-DNAmoleculesowinginamicrochannel. ................... 27 2-1Thegeometryusedincalculatingthedisturbancevelocityatrcausedbyapointforceatr0. ................................... 35 2-2Distributionofthecenter-of-massforpressure-drivenowonly. ......... 43 2-3Mechanismformigrationawayfromthewallduetoshearow. ......... 44 2-4Distributionofthecenter-of-massforpressure-drivenowonly(resultsfromtheory). ........................................ 45 2-5Distributionofcenter-of-massforanexternalforce ................ 46 2-6MechanismformigrationawayfromthewallduetoanexternalforceFE. .... 46 2-7Distributionn(y)fortheexternalforceandoweldactinginconjunction. ... 48 2-8Mechanismformigrationduetocombinationofowandanexternalforce. ... 48 2-9Distributionn(y)foranappliedexternalforceandimposedoweldactinginopposition. ...................................... 49 2-10Distributionn(y)fortheelectrophoreticandpressure-drivenowactinginconjunction. ...................................... 55 2-11Distributionn(y)fortheelectrophoreticandpressure-drivenowactinginopposition. ...................................... 58 3-1Illustrationofabeadinthebead-springmodelofDNA. .............. 62 3-2MigrationproleofDNAforappliedelectric-eldandapressuregradient. ... 64 3-3Comparisonofmigrationprolesobtainedfromsimulationswithexperimentalresults. ........................................ 65 3-4Effectofelectriceldonmigrationprole. ..................... 66 3-5Electrophoreticvelocityofdifferentlengthchainsinauniformshearow. ... 68 3-6Effectofelectriceldonpolyelectrolyteradiusofgyration ............ 69 3-7Effectofelectriceldonpolyelectrolytediffusivity ................ 70 4-1Mechanismillustratingdecreaseinlateralmigrationathigherelds ....... 73 10

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........................... 74 4-3Illustrationofsoftlithographyprocess ....................... 76 4-4Experimentalapparatusformicrouidicexperiments ............... 78 4-5Schematicdiagramoftheopticalsystem ..................... 79 4-6Imageof-DNAmoleculeowinginmicrochannelwithandwithoutelectriceld 80 4-7Illustrationofelectroosmoticowinmicrochannel ................. 81 4-8Resultsfromcalibrationexperimentsofuorescentmicrospheres ........ 83 4-9Saturationofcross-streammigrationof-DNA .................. 84 4-10Fullydevelopedmigrationproleof-DNA .................... 85 4-11Centerofmassdistributionformicrospheresand-DNA ............ 86 5-1ConformationalpropertiesofpolymerfromLBandBDsimulations. ....... 98 5-2DiffusioncoefcientsfromLBsimulationswithdifferentunitcellsizes. ..... 102 5-3DiffusioncoefcientsfromLBsimulationswithdifferentgridresolutions .... 104 5-4NormalizedautocorrelationfunctionsoftheRouse-modeamplitudes. ..... 106 5-5RelaxationtimesoftheRousemodesfromLBandBDforN=10. ....... 107 5-6RelaxationtimesoftheRousemodesfromLBandBDforN=20. ....... 108 6-1Illustrationofthecongurationofbeadsiandjbetweentwoparallelplates. 114 6-2Decayofthenormalizedcorrelationfunctionstodeterminetherelaxationtime. 117 6-3Centerofmassdistributionofaconnedpolymerdrivenbyuniformshearow. 119 6-4Distributionoftheend-to-endvectoracrossthechannel. ............. 120 6-5ResultsforLBandBDsimulationsofaconnedpolymerinapressure-drivenow. .......................................... 121 B-1Illustrationofsymmetrybasedonthecenterofthechannel. ........... 132 B-2Illustrationofcylindricalsymmetryinthesystem. ................. 133 11

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1 4 ]hasshowncross-streammigrationofDNAdrivenbyuniformshearinstronglyconnedsystems,withchanneldimensionsapproximatelytentimestheradiusofgyration(Rg)ofthemolecule.TheDNAmolecule,owingtoitselasticity,stretchesandorientsinthepresenceofthelocalshear.Disturbancesintheow,alsoknownashydrodynamicinteractions(HI),arereectedfromthewallsandliftthestretchedandorientedmoleculetowardsthecenter.InSection 1.2 ,Iintroducetheconceptoflong-rangehydrodynamicinteractionsforparticlessuspendedinuidandextendtheideatomodelingDNAasachainofbeadsconnectedbysprings.ExperimentsbyZhengandYeung[ 5 8 ]forlargechannelseparations,100Rg,demonstratethatifanexternalelectriceldisappliedparalleltothedirectionofow,theDNAmoleculesconcentrateatthecenterofchannel.Moreover,moleculeswithlongercontourlengthshadahigherconcentrationnearthecenter.Ifthedirectionoftheappliedeldisreversedrelativetotheow,theDNAmoleculescongregateatthe 19

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1.3 andSection 1.4 ,respectively.InChapter 2 kinetictheoryofaconneddumbbellispresented,exploringthelateralmigrationofexiblemoleculesdrivenbyacombinationofowandexternaleld.Thedumbbellmodelistheextendedtoamulti-beadmodelthatrepresentsa-DNAmolecule.ResultsfromBrownian-dynamicssimulationsofthismodelcomparefavorablywithexperimentandarediscussedindetailinChapter 3 .Furthermorethesimulationsalsopredictpolyelectrolytedynamicswhichhavenotyetbeenveriedexperimentally.Oneofthepredictionsisthatthemigrationlayerstartstodecreaseforverylargeelectricelds.Thisclaimisinvestigatedandpartiallyveriedthroughmicrouidicexperiments,detailsofwhicharepresentedinChapter 4 .Inaddition,twonumericalmethodsforsimulatingpolymersimulationslattice-Boltzmann(LB)andBrowniandynamics(BD),arecomparedthroughsimulationsofdiluteDNAsolutionsinfreespaceandinconnement.Thesetwosimulationmethodsemploycontrastingapproachestosolvetheuidowequations.Lattice-Boltzmannisanexplicitsolventmethodandthefar-eldhydrodynamicdisturbancesdevelopovertimethroughsuccessivecollisionsbetweenthesolventparticlesandthepolymermolecule.Allexplicitsolventmethodsareinherentlyrestrictedbyacontrolvolumewhichdictatesthesizeofthesimulation.ThesolventdegreesoffreedomareaccountedforimplicitlyintheBDsimulationsandthereforeperiodicboundaryconditionsarenotnecessary;far-eldHIin 20

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5 andthemigrationprolealongwithene-to-endstretchofthepolymerispresentedinChapter 6 .Apreviouscomparison[ 9 ]investigatingpolymermigrationinconnedchannelsshowedsimilartrends,butdifferentchoicesofpolymermodelandparametersprecludedaquantitativecomparison.Finally,theresultsofallthechaptersaresummarizedinChapter 7 .Futureworktofurtherenhanceourunderstandingofpolyelectrolytesisalsosuggested. 1.2.1EffectsduetobodyforcesHydrodynamicinteractions(HI)arecentraltothephenomenadescribedinthisdissertation.Themotionofmesoscaleparticlesinauidgeneratesfar-elddisturbances,knownashydrodynamicinteractions.ThismanuscriptfocusesontheeffectofHIinlowReynoldsnumberow,orStokesow.ConsidertheillustrationinFigure 1-1 whichdemonstratesapointforce,f,actingontheuid.DisturbancesinuidmotionduetothisforcearegovernedbytheStokesequationprovidedinertialeffectsareneglected.ThemathematicalformulationofStokesequationis, 21

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Stokesowduetoapointforce,f,actingontheuid. determiningthehydrodynamiceldsisknownastheOseentensorandisgivenby, 8r 1-1 .LinearityoftheStokesequationsallowsthesuperpositionofvelocityfrommultipledisturbances.Computationofthedisturbancevelocityincreasesincomplexitywiththeinclusionofno-slipboundariesandnitesizeeffects.Incertaincases,suchasniteparticlesize[ 10 11 ]orpresenceofano-slipboundary[ 12 13 ],theGreen'sfunctiondescribingtheowproleisdenedanalyticallyandisacomputationallyinexpensivecomponentofthesimulations.Inmorecomplexcasesthough,suchasaparallelplategeometry[ 14 ],numericalintegrationisrequiredtoobtainaccurateowelds. 22

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15 ].Ontheotherhand,hydrodynamicdisturbancesareneglectedintheRousemodelofpolymers,whichisapplicabletosimulationsofconcentratedpolymersolutionsandmeltswherehydrodynamicinteractionsarescreened. 2 4 ].Thesestudiesdemonstratethatthepolymerchainstretchesinthelocalshearrateandthedisturbanceintheuidowreectedfromthewallsliftsthecenter-of-massofthemoleculeawayfromthewalls.Recently,experimentsbyZhengandYeung[ 5 8 ]demonstratesthatifanelectriceldisappliedinadditiontotheow,thenweobservecross-streammigrationeveninabsenceofstrongconnements.Moreover,basedonthedirectionoftheelectriceldthepolymermotioncouldbedriventowardsorawayfromthecenter.TheseresultssuggestpresenceofhydrodynamicinteractionsinducedbyelectriceldandchallengethecommonlyheldbeliefthatsuggestsscreeningofHI.Usingtheresultsfromtheseexperimentsasastartingpoint,thisthesisprovidesevidencethathydrodynamicinteractionsdoexistinthepresenceofanexternalelectriceld. 23

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Illustrationofhydrodynamiceldgeneratedbyelectriceldactingonamacroion.Theguredepictsthedisturbanceintheuidduetotheeld. Itisfrequentlyassertedthatanelectriceldactingonachargedmacroiondoesnotgeneratealong-rangedisturbanceintheuidow[ 16 17 ].Thisassertionisusedtoexplaintheweakcontour-lengthdependenceoftheelectrophoreticmobilityofDNAandotherpolyelectrolytes.However,theactualsituationisnotsosimple[ 18 19 ];infactthereisanelectricallyinducedHIbetweenchargedmacroions[ 20 ],whichforlargeseparationsisofdipolarform.AlthoughtheOseeneldgeneratedbytheelectricforceonamacroioniscounterbalancedbyanopposingforceonthesurroundingcounterions,thereisaresidualowfromthequadrupolemomentofthechargedensityasillustratedinFigure 1-2 .Atdilutesaltconcentrations,theinducedHIchangesthestaticanddynamicpropertiesofDNAasdiscussedindetailinChapters 2 4 24

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21 ]thatamoleculeinastretchedstatehasdifferentresponsetotheappliedelectriceldwhencomparedwithmoleculesinacoiledstate.AmodelforDNAthatincludesHIgeneratedbyelectriceldandtheresultsdiscussedinthissubsectionarepresentedindetailinChapter 3 22 ],BD[ 23 ],DissipativeParticleDynamics(DPD)[ 24 ],andLB[ 2 ]simulations. 25

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3 .ThecomputationalcostofBDsimulationsofapolymerconsistingofNsegmentsscalesasO(N3).IntheuctuatingLBmethod,allinteractionsarelocal,leadingtoamuchmorefavorableO(N)scalingwiththenumberofbeads.Howeverthelargenumberofadditionaldegreesoffreedomandtheadditionalinertialtimescale,generatedbytheexplicitsolventmodel,increasethecomputationalcostofthesimulations.GenerallythecomputationaltimefordilutesystemsinunconnedgeometriesfavorsBD,whilethetimeformoreconcentratedsolutionsinconnedgeometriesfavorstheLBmethod.Inthisworkweareconcernedwithvalidation,ratherthancomputationalcost.AbriefintroductionoftheLBsimulationmethodanditscomparisonwiththeBDsimulationsofapolymerchaininfreeaswellasconnedspaceispresentedinChapters 5 and 6 26

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Pictureof-DNAmoleculesowinginamicrochannelofcross-section300x24m2duetoanappliedpressuregradientandanexternalelectriceld.Thelowerendofpicturecorrespondstolowerwallwhilethetopendrepresentsthecenter(atadistanceof150m)ofthechannel. 25 ],immunoassays[ 26 ],owcytometry[ 27 ],PCRamplication[ 28 ],andDNAanalysis[ 29 ].Manyoftheseapplicationshaveutilityforclinicaldiagnostics[ 30 ].Onecatalystfortheriseofmicrouidicsisthedemandinmolecularandcellbiologyforanalyticalmethodswithhigherthroughput,highersensitivity,andhigherresolutionthanconventionalmethods.Typically,thevolumeofuidsinvolvedisoftheorderofafewmicrolitersandthedeviceitselfisafewmillimetersinlength.Althoughtheuidpropertiesremainthesameatthemicroscale,surfacetension,viscosity,andelectricalchargescanbecomedominantforcesonauidbecausethesurface-to-volumeratioismuchgreaterthanformacro-scalesystems.Recentadvancesinphotolithographytechniqueshaveimprovedtheabilitytomanufacturemicrouidicdevices.Photolithographysharessomefundamentalprincipleswithphotographyinthatthepatternintheetchingresistiscreatedbyexposingittolight,eitherdirectly(withoutusingamask)orwithaprojectedimageusinganopticalmask.Usingcommerciallyavailablephotoresists,permanentmoldsarecreatedwhicharethenreplicatedusingsoft-lithographytechniques.Themostcommonly 27

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31 32 ]visualizedandmanipulated-DNAandhasrevolutionizedthestudyofpolymerdynamicsindilutesolutions.Theseexperimentsprovidedastrongthrusttodeveloplab-on-a-chiptechnologiesfocusedonDNAanalysis,separationandsequencing.VisualizationoftheDNAmoleculeisdoneusingepiuorescentorconfocalmicroscopywithuorescentdyeslikeYOYO-1andTOTO-1.ThesedyeshaveastrongbindingafnityfortheDNAbackbonethroughelectrostaticandstericconstraints.Furthermore,theelectrophoreticdynamicsofDNAisnotsignicantlyalteredbythepresenceofthedyes.Thesecharacteristicsmake-DNAanidealmoleculetostudy.Figure 1-3 depictsapictureofDNAmoleculesowinginamicrochannel.InChapter 4 ,Ipresentresultsfrommicrouidicexperimentsofdilutesolutionsof-DNAdrivenbyacombinationofexternalelectriceldandpressuregradient.TheexperimentsaremotivatedbythepredictionsmadebytheBDsimulationspresentedinChapter 3 .Forexample,themodelsuggestsadecreaseinlateralmigrationofDNAwithincreasingelectriceldprovidedthatthepressuregradientisheldconstant.Indeed,experimentswithweakelectricelds(Chapter 4 )suggestsaturationinmigrationproleof-DNA. 28

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33 ].Thenetmigrationofexiblepolymerstowardsthecenterofachannelinresponsetopressure-drivenowisawell-studiedcase[ 1 3 34 35 ].Inpressure-drivenow,hydrodynamicliftonapolymerinthevicinityofaboundingwallcausesthemigration[ 2 36 ].Thelocalshearowextendsthepolymer,generatingtensioninthechainandanadditionaloweldaroundthepolymer.Thisoweldbecomesasymmetricnearano-slipboundaryandresultsinanetdrifttowardsthecenterofthechannel.Recentsimulationsofaconnedpolymerdrivenbyanexternalforcealsofoundmigrationtowardsthecenterline[ 37 ],thoughthemechanismclearlydiffersfromshear-inducedmigration.Furthermore,simultaneousapplicationofanexternalforceandapressure-drivenowalteredthemigrationinanon-additivefashion.Whenforceandowwereappliedinconjunction,themigrationtowardsthecenterlinewasenhanced.However,applyingtheexternalforceintheoppositedirectionofthepressure-drivenowcausedamigrationeithertowardsorawayfromthewalls,dependinguponthemagnitudeoftheappliedforce.Inthischapter,migrationofapolymerchainundercombinationsofexternalforceandpressure-drivenowisexaminedusingthekinetictheoryofanelasticdumbbell.ThetheoryofMaandGraham[ 36 ]isextendedinSec. 2.2 toincludetheactionoftheexternalforceonthepolymer.ResultsfromthetheoryfordifferentcombinationsofforceandowarepresentedanddiscussedinSec. 5.3 .Thetheoryqualitativelyreproducesallthemigrationphenomenaobservedinthesimulations.Multiplemigrationmechanismshavingsignicanteffectsonthepolymerdistributionareidentied. 29

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6 ].Here,anelectriceldactinguponthepolyelectrolyteprovidestheexternalforceleadingtomigration[ 38 ].However,neitherthesimulations[ 37 ]northekinetictheorydevelopedinSec. 2.2 considertheeffectsofcounterionscreeningonthehydrodynamicinteractionsinducedbytheelectriceld.Modelsofelectrophoresistypicallyassumecompletescreening,[ 16 ]sothathydrodynamicinteractionsbetweenpolymersegmentscanbeignored.Thisfreedrainingassumptioniscritical,sincetherelevantmechanismsidentiedinSec. 5.3 resultfromhydrodynamicinteractions.However,withintheDebye-Huckelapproximation,thereremainsadipolarcontributiontothehydrodynamicoweldaroundascreenedpointcharge,[ 19 39 ]whichwasoverlookedintheoriginalwork[ 16 ].Moreover,experiments[ 5 ]onDNAincombinedelectricandpressure-drivenoweldsmeasurednon-uniformdistributionsofDNAtransversetotheowdirection.Thequalitativesimilaritiesbetweentheexperimentalresults,thenumericalsimulations[ 37 ],andtheorydevelopedinSec. 2.2 suggestthatthescreeningofthevelocitydisturbancecreatedbyelectrophoresisisnotcomplete.Consequently,modicationstothetheoryaremadeinSec. 2.4 toaccountforthecounterionscreeningofhydrodynamicinteractionswithintheDebye-Huckelapproximation.TheresultspresentedinSec. 2.4.4 demonstratethatmigrationcanoccurforcombinationsofelectrophoresisandpressure-drivenow,andthathydrodynamicinteractionsinpolyelectrolytesdecayalgebraically. 36 ]forapolymersubjectedtoalocalshearowisextendedtoincludebothbodyforcesandelectricelds,usingthesameapproximations:thepolymerdistributionfunctionisfactorizedintoaproductofcenter-of-massandorientationdistributions,theeffectsoftheno-slipboundaryconditionattheconning 30

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36 ],thedistributionisexpandedinapowerseriesintheforceandowelds.Thisadditionalapproximationenablesananalyticsolutionforthetimeindependentdistribution.Theevolutionofapolymerdumbbellinsolutioncanbedescribedbyacontinuityequationforthedistributionfunction,(rc,q,t),ofthecenter-of-massrcandend-to-endvectorq[ 40 ]: @rc(_rc)@ @q(_q).(2)Theprobabilitydistributionfunctionisseparatedintocenter-of-mass(n)andend-to-end()distributionfunctions, 2 )overqgivesthesteady-statedistributionofthecenter-of-mass, @rc(nh_rci)=0,(2)wheretheanglebracketshiindicateanensembleaverageoverq, @q(_q)=@ @rc(n_rc).(2)Approximating_rcbyh_rciinEq.( 2 )andmakinguseofEq.( 2 )gives @q(_q)=0.(2) 31

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2 ),Eq.( 2 )issolvedforthecenter-of-massdistribution.Inprinciple,computingsuccessiveapproximationstotherighthandsideofEq.( 2 )wouldimprovethesolutionsofandn.However,theorientationdistributionfunctionequilibratesmuchfasterthanthepolymermigratesanddiffusesacrossstreamlines,sotheapproximationmadeinEq.( 2 )isnotsevere[ 36 ].Additionally,comparingtheresultsofthetheorywithsimulations(Sec. 5.3 )demonstratesthaterrorsassociatedwiththisassumptionarequantitative,ratherthatqualitative. 2.4 .Inadditiontotheexternalforce,anintramolecularforce,FS=FS1=FS2,actsintheoppositedirectiononeachbead.Intheabsenceofinertia,thevelocityofeachbead(i)isdeterminedfromtheforcebalance @riln.(2)ImplicitintheexpressionfortheBrownianforceisaninstantaneousaverageoverthesolventdegreesoffreedom. 32

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8jrjI+rr 14 ]isusedtocalculatetheadditionaloweldarisingfromcancellationofthesourcevelocity[Eq.( 2 )]alongtheboundarysurfaces.Thecombinedeffectoftwoboundariesisapproximatedbysuperposingthecancelingoweldsfromeachsurface[ 36 ].Numericalevidence[ 34 41 ]suggeststhatthisapproximationisreasonablewhentheseparationbetweenthewallsisanorderofmagnitudelargerthantheradiusofgyrationofthepolymer.Thedisturbancevelocityatapositionrduetoapointforcelocatedatr0is[ 12 ] 33

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ThegeometryisillustratedinFig. 2-1 ;y0isthedistanceofthesourcepointfromtheboundaryaty=0.ThetwocontributionsoftheOseentensorwithinGareduetoimagesoftheStokesletabovethetopwallandbelowthebottomwall.Thepotentialdipoles,PD,andStokesletdoublets,SD,areincludedtoenforcethetangentialno-slipconditionsonthetopandbottomwalls,with 8jrj5jrj23r2x3rxry3rxrz3ryrxjrj2+3r2y3ryrz3rzrx3rzryjrj23r2z, 8jrj30rx0rx0rz0rz0. ThevelocitiesofthebeadsaredeterminedfromEqs.( 2 ),( 2 ),and( 6 ), 34

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Thegeometryusedincalculatingthedisturbancevelocityatrcausedbyapointforceatr0.Thecalculationincludestherstimagesourcesatr0+2(Hy0)eyandr02y0eywhichcancelthenormalvelocityofthesourceatthetopandbottomwalls.Additionalforcedipolesandstressdoubletsareneededtocancelthetangentialvelocitiesatthewalls. Thevelocityofthecenter-of-massofthedumbbell,_rc=(_r1+_r2)=2,andrateofchangeoftheconnectorvector,_q=_r2_r1,areevaluatedfromEq.( 2 ), 8qq:rru0+1 2FS+2 2 )and( 2 )arelinearcombinationsofij, 35

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321 (ycH)2,(2) 2 )intoEq.( 2 )andmultiplyingbyngivestheuxofthecenter-of-mass, kBTDKFE+nkBT @qhDKi@n @rc, whichmatchestheexpressioninEq.28ofMaandGraham[ 36 ]withanadditionalcontributionassociatedwiththeexternalforce,FE.Forsteadyow,theuxnormalto 36

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@ychSyyikBT Theexpansionsfortheinteractiontensors[Eqs.( 2 )-( 2 )]havebeensubstitutedintoEq.( 2 )togetherwithalinearspringforce,FS=q,togiveEq.( 2 ).Calculatingthedistributionacrossthechannel,n(yc),requirestheensembleaveragesoverqthatappearinEq.( 2 ).TheapproximationofEq.( 2 )isusedtodetermineandthencalculatetheensembleaverages.UsingEq.( 2 )inEq.( 2 )gives @qqru02 @qFE+kBT Inrecentwork[ 29 36 ],authorshaveignoredtheinuenceofhydrodynamicinteractionsontheorientationdistribution,,usedaFENE-Pspringforce,andthensolvednumericallyforthedeformationtensor,hqqi,asafunctionofthetransverseposition(yc)withinthechannel.Oncedetermined,thecomponentsofthedeformationtensorwereusedtoevaluatethequantitiesnecessaryforcalculatingn(yc).Inthepresentcircumstances,hydrodynamicinteractionsmustberetainedwithinEq.( 2 )topredictmigrationduetotheexternalforce.Specically,thetermFEwhichresultsfromtheinteractionwiththeboundingwallsmustbeincluded,thoughtheothercontributionsproportionaltoand^arestillneglected.AlinearspringforceisusedinsteadoftheFENE-PspringforceandsimplifyingEq.( 2 ),usingEq.( 2 )for,gives @qx2 3+q@ @q2kBT @q2+FEqy@ @qx+qx@ @qy.(2) 37

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_(2) kBTFE(2) 2 )is 2c4c1(qx)2+c2qxqy+c3qy2+c4(qz)2,(2)withcoefcients 8(F)_ 2_ 8(_)2+(F)_ 16(_)2, andnormalization 2 ), 38

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40 ].Fortheconditionsofinterest,termsproportionalto_andFareoforderoneorsmallerovermostofthechannelwidthwhichisaccessibletothepolymer.Thevaluesareinfactzeroatthecenterlineandthepolymercannotaccesspositionsnearthewallduetoexcludedvolume.Consequently,Eq.( 2 )isexpandedabout_=0andF=0,andtosecondorder, 21 2_+Fqxqy+1 32(_)21qy2+(qx)2+qxqy2+1 16_(F)2qy2+(qx)2+2qxqy2+1 8(F)21+qxqy2, wheretheequilibriumdistributionforthedumbbellis 2(qx)2+qy2+(qz)2.(2)Notethatthissameresultcanbederivedbyconstructingaperturbationsolutionforaround_=0andF=0,inthesamemannerasdoneforasimpleshearow[ 40 42 ],andthensolvingEq.( 2 )forthe0th,1st,and2ndordersolutions;thisprocedurecouldbefollowedforavarietyofnon-lineardumbbellmodels.Eq.( 2 )cannowbeusedtocalculatehSyxiandhSyyi, 60(2)3=2[_+2(F)] 840(2)3=256011(_)23(F)237_(F). Inaddition,q2x2q2y+q2zisrecalculatedusingEq.( 2 ), 8(_)2+(F)_2(F)2;(2)thisapproximationcanbecomparedtotheexactexpressioninEq.( 2 ). 39

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2 )-( 2 )withinEq.( 2 )andsettingtheuxinthetransverse(y)directiontozerogivesadifferentialequationfortheprobabilitydistributionofthecenter-of-massacrossthechannel, 30(2)3=2F[_+2(F)], 280p @y11(_)2+3(F)2+37_(F). Thepositionyc,heightH,andradiusahavebeenscaledbythelengthp 37 41 ],twodifferentPecletnumbershavebeendened.Onecomesfromtheratioofthemeanshearratetothediffusivityofthefreely-drainingdumbbell, 40

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RgH=H Rg. RewritingEq.( 2 )usingthedenitionsinEqs.( 2 )-( 2 )gives 315H2Pe2f2yH227a2H2 yyH, 315H2Pe2f2yH+27a2H2 Fortermshavingasignof,theminussigncorrespondstothecasewheretheexternalforceactsinthesamedirectionoftheoweld(FE>0)andtheplussigntoanexternalforceopposingtheoweld(FE<0).Notethatthefunctionaldependenceon 41

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2 )-( 2 ),incontrasttoEqs.( 2 )-( 2 )whichpresenttheresultsintermsofthelocalvaluesoftheshearrateandexternalforce. 2 )forthecasesofowonly,forceonly,andcombinationsofforceandow.Theheightofthechannelincomparisontotheradiusofgyration,H,wassettoeightandthehydrodynamicradius,a,to0.125forallresultsshown.Thesesameparameterswereusedinthenumericalsimulationsofmulti-beadchains[ 37 41 ],whichareherecomparedwiththeresultsofthetheory.TermswithinSandGproportionaltoa3wereignored.Thesetermsnegligiblyimpactthedistributioninthebulkofthechannel,thoughverynearthewallthesetermsdominate.However,aforcetomaintaintheexcludedvolumeofthepolymerinthevicinityofthewallwasnotincludedintheforcebalanceofEq.( 2 ).Consequently,theconcentrationnearthewallwassettozerobyperformingtheintegrationofEq.( 2 )betweeny=0.25Rgandy=H0.25Rg,ratherthan0andH.Thedistributionn(y)wasnormalizedbynt,where 42

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Distributionofthecenter-of-massforpressure-drivenowonly.a)Resultsfromthetheoryneglectinggradientsinthediffusivity.b)ResultsofnumericalsimulationsreportedbyUstaetal.[ 41 ]Resultsareplottedbetweentheboundingwallaty=0andcenterline(y=4)ofthechannel. 2 )reducesto 4y+lny 4Hy+lnHy 2-2 a.Thedepletionlayerresultsfromthehydrodynamicinteractionofthedumbbellwiththewall,aspreviouslyidentied[ 1 36 ].AsillustratedinFig. 2-3 ,astrongshearowtendstoorienttheconnectorvectorqsuchthatthepolymerisintension.ThetensioninthespringcreatesaforceFSactingoneachbeadandaconsequentvelocitydisturbanceontheotherbead.Duetothebreakinsymmetrycausedbythepresenceofthewall,thevelocitydisturbanceresultsinanetliftofthecenter-of-massawayfromtheboundary.TheresultsinFig. 2-2 adifferfromtheresultsofnumericalsimulations[ 1 37 ]intwoqualitativeaspects.Firstly,simulationresultsexhibitedasignicantdepletionnearthewallsatequilibriumwhichwasnotobservedintheanalyticalresultsdueto 43

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Mechanismformigrationawayfromthewallduetoshearow.Theoweldalignsandstretchesthedumbbell.Thespringforce,whichisintension,generatesadisturbanceowandanetvelocityawayfromtheno-slipboundary. theabsenceofanexcludedvolumeforcewiththewall.Secondly,inthesimulationsthemaximuminn(y)occursoff-center(seeFig. 2-2 b),implyingtheexistenceofasecondarymechanismpromotingmigrationtowardstheboundariesincompetitionwiththeshear-inducedmigration(Fig. 2-3 ).Thisadditionalmechanismarisesfromthevariationindiffusivityofthepolymercausedbythenon-uniformshearow[ 43 ].Thetransversediffusivityofapolymernearthecenterofthechannel,wheretheshearrateapproacheszero,ishigherthanapolymernearthewallthatishighlyextendedbytheshear;thegradientintransversediffusivityresultsinadriftofthepolymerawayfromthecenterline.Thiseffectiscapturedbythetheorywithinthegradient(G)terms.IncludingthesetermswiththoseinEq.( 2 )andintegratinggives 4y+lny 4Hy+lnHy 288 315H2yyH!#.(2)PlotsofthissolutioninFig. 2-4 demonstratethatthegradientindiffusivitymovesthemaximuminn(y)awayfromthecenterlineandtowardstheboundary.QuantitativedifferencesbetweenthenumericalandanalyticalsolutionsareapparentinFig. 2-2 .PredictingasimilardepletionlayernearthewallrequiresavalueofPefwhichisapproximatelyonethirdofthevalueusedinthesimulations.NotethatthehighestvalueofPef=30correspondsto_=20whenevaluatedaty=2.Thisvalue 44

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Distributionofthecenter-of-massforpressure-drivenowonly.Theoreticalresultsincludingthegradientinthediffusivity. maybetoolargefortheexpansioninEq.( 2 ),whichassumes_isoforderoneorless.However,thetheorycontainsnoadjustableparametersandisyetabletocapturetheessenceofthesimulationresults. 2 )reducesto yHy#.(2)Theexternalforce,ifsufcientlylargecomparedtothethermaluctuations,resultsinastrongmigrationofthepolymerawayfromthewallasseeninFig. 2-5 a. 45

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Distributionofcenter-of-massforanexternalforcewhenthereisnoimposedoweld(Pef=0).a)Resultsfromthetheoryandb)resultsfromnumericalsimulationsbyUstaetal.[ 37 ]areshown. MechanismformigrationawayfromthewallduetoanexternalforceFE.Theexternalforcerotatesadumbbelltoanorientationwhichresultsinadriftawayfromthewall,asshownin(c).Forexample,adumbbellperpendiculartothewall(a)rotatestowardsalignmentwiththewall,whereasadumbbellparallelwiththewall(b)rotatesawayfromthewall. Themigrationarisesfromtherotationofthedumbbellnearthewalltoanaverageorientationcenteredaboutthe45oline,whichresultsinadriftawayfromthewallundertheactionofanexternalforceasillustratedinFig. 2-6 .Therststepoftheprocess,therotation,occursdirectlyasaresultofthehydrodynamicinteractionwiththewall;thetermFEappearinginEq.( 2 )isresponsiblefortherotation,butiszeroifthewallisnotpresent.Whetheralignedperpendicularorparalleltothewall,thepolymerrotatestowardsthepreferredorientationshowninFig. 2-6 .Theorienteddumbbellthencoupleswiththeexternalforcethroughthehydrodynamicinteractionbetweenthebeadstomovethepolymerawayfromthewall. 46

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2 )aresmall.TheresultsfromthetheoryarequalitativelyconsistentwiththoseofthesimulationsuptoPe=400.BeyondPe=400,thedistributionpredictedfromthetheorycontinuestoshifttowardsthecenterlineofthechannelwithoutbound,whereasthesimulationsindicatedalimittotheextentofmigrationasPeincreased.Infact,thesimulationresultsforPe=1100andPe=1nearlymatched[ 37 ].Thelimitofmigrationinthesimulationsresultsfromhydrodynamicdispersionofachainofpointparticles,similartoasedimentingsuspension[ 44 ].Themodeloftheelasticdumbbelldoesnotincludethiseffect. 2-7 .Forexample,comparingthetheoreticalresultforPe=200andPef=5withtheresultforPe=200andPef=0(Fig. 2-5 a)showsthatthedepletionlayerhasextendedfromapproximatelyy=1toy=1.5andthecenterlinevaluehasincreasedfromn=nt=0.32ton=nt=0.45.Similarincreasesareobservedinthecorrespondingsimulationresults.Theenhancedmigrationtowardsthecenterresultsonlypartiallyfromtheadditiveeffectofthemigrationmechanismsforowandforce.Anothermechanism,proportionaltothecrossproductofthePecletnumbersisincludedalongwiththetermsassociatedwiththeexternalforceandshear-inducedmigrationmechanisms, 2-8 ,theshear 47

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Distributionn(y)fortheexternalforceandoweldactinginconjunction.a)ResultsfromthetheoryareplottedatdifferentPeforthecaseofPef=5andb)resultsofnumericalsimulations[ 37 ]withPef=12.5areshownforcomparison. Theshearowresultsinapreferredorientationofthepolymersimilartothatshown.Whenactinginthedirectionofow,theexternalforce(solidarrow)liftsthecenter-of-massupwards(awayfromthewall)withvelocity_ry.Theforce,actingcountertotheow(dashedarrow),resultsinadriftdownwards(towardsthewall).Notethatthismechanismdoesnotdependuponhydrodynamicinteractionwiththeboundingwalls. oworientsthebeads,onaverage,suchthatthepolymerisintensioneveninthebulk;theexternalforcethencausesamigrationofthepolymerinthedirectionoftheaverageorientation,towardsthecenterinthiscase.Thisforce-inducedmigrationofanorientedpolymerisadirectresultofhydrodynamicinteractionsbetweenthebeads. 2-8 ).Equation( 2 )describingthedistributionn(y)is 48

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Distributionn(y)foranappliedexternalforceandimposedoweldactinginopposition.(a)ResultsofthetheoryforfourdifferentPenumbersatPef=10and(b)numericalresults[ 37 ]withPef=12.5areshown. modiedbychangingthesignofthethirdterm, 2-9 ashowsresultsfromthetheoryataxedowrateofPef=10.ForrelativelysmallvaluesofPe,themaximuminthepolymerdistributionisclosertotheboundingwallthantothecenterline.Astheforce(Pe)increases,themaximumshiftstowardsthecenterline.ForsufcientlyhighPe,themigrationtowardsthecenterlineduetotheappliedforce(thePe2term)dominatesthemigrationtowardsthewallresultingfromthecouplingofthelocalshearowanddriftduetotheexternalforce.Simulationresults[ 37 ](Fig. 2-9 b)showaqualitativelysimilarbehavior.ThepredictedtransitionfrommigrationtowardsthewalltomigrationawayfromthewalltakesplaceatdifferentvaluesofPeandPefthaninthesimulations. 38 ],anelectriceldactinguponapolyelectrolyteprovidestheexternalforce.Undertheseconditions,screeningduetotheactionofthe 49

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16 ].However,recentvisualizationexperiments[ 5 6 ]onDNAinthesimultaneouspresenceofanelectriceldandapressure-drivenowproduceddistributionssimilartothoseappearinginFigs. 2-7 and 2-9 .Thisobservationsuggeststhathydrodynamicinteractionscreatedbytheelectriceldarenotfullyscreened.Indeed,calculations[ 18 19 39 ]basedonaDebye-HuckelapproximationindicatethattheGreen'sfunctioncontainsalgebraicallydecayingtermsinadditiontoexponentiallydecayingones.ThisGreen'sfunctionisusedheretodemonstratethatcomplexmigrationbehaviorduetohydrodynamicinteractionscanstilloccurunderelectrophoresis.Toenableananalyticalcalculation,thecounterion-distributionsurroundingachargedbeadisassumedtohaveaDebye-Huckelstructuredespitethepresenceofow.Expressionsforthevelocityofthecenter-of-massandend-to-endvectorsaredevelopedinSec. 2.4.1 andthenappliedinSec. 2.4.2 toderivethesteadystatedistributions,n(rc)and(rc,q). 45 ].BalancingforcesandneglectinginertiaasinSec. 2.2.1 ,thebeadvelocityisgivenby 6 )]andv0EisthevelocitydisturbanceinducedbytheelectriceldEactingontheotherbead.Theelectrostaticforcesactingbetweenthebeadsareassumedtobefully 50

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45 ].Likewise,theforceonabeadduetothepresenceofachargedwallisalsoneglected.Thevelocitydisturbanceinducedbytheelectriceld, 16 ].However,calculations[ 18 19 39 ]ofthehydrodynamicGreen'sfunctionrelatingthedisturbancevelocityatapointrtoanelectriceldactinguponascreenedpointchargeatr0givesanalgebraicallydecayingdipolareldaswellasexponentialterms.Retainingonlythelong-rangedipolartermgivesahydrodynamicinteractiongovernedby 2 )withinEq.( 2 )givesthevelocityofabead, 2 )and( 2 ),thevelocityofthecenter-of-massandconnectorvectorare 8qq:rru0+1 2FS+SEQE+kBT 51

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2 )and( 2 )intoEqs.( 2 )and( 2 )givesequationsforthecenter-of-massdistributionandend-to-enddistributionfunctions,n(yc)and.AsinEq.( 2 ),theuxinthey-direction, @ychSyyikBT vanishesatsteadystate.CalculatingtheensembleaveragesappearinginEq.( 2 )requiresasolutionoftheequationfortheorientationdistribution, @qx2 3+q@ @q2kBT @q2.(2)AsinEq.( 2 ),thecontributionsof^andhavebeenignoredandaHookeanspringforcehasbeenused.DuetothefactthatSEdoesnotincludetheasymmetryarisingfrominteractionswiththewalls,theelectricelddoesnotaffecttheend-to-enddistribution.Asaresult,thistheorywillnotpredictmigrationarisingfromtheelectriceldalone.ThesolutionofEq.( 2 )is 2d3(qx)2+d1qxqy+d2qy2+d3(qz)2,(2)withcoefcients 2_ 8(_)2 16(_)2, 52

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2 )about_=0tosecondordergives 4_qxqy+1 32(_)21qy2+(qx)2+qxqy2, fromwhichtheensembleaveragesinEq.( 2 )arecalculated.Theresultsare 8(_)2, 840(2)3=256011(_)2, 20(2)3=2(D)2_, where 2 ). 2 )andusingtheensembleaveragesascalculatedinEqs.( 2 )-( 2 )givesanequationgoverningthedistributionforthecenter-of-mass, 53

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10(2)3=2Fe_ 280p @y(_)2 andFehasbeendenedas 2 ),( 2 ),and( 2 )),butwithFsettozero.TheprimarydifferencewiththepreviousresultofEq.( 2 )appearsinEq.( 2 )forFe.Thistermincludestheeffectofthescreeningofthehydrodynamicinteractions,asapparentfromthequadraticdependenceontheDebyelength,1=D.TocomparewithresultspresentedinSec. 5.3 ,Eq.( 2 )isrewrittenintermsofthePecletnumbers.Thenon-dimensionalparametersa,H,y,andPefhavebeendenedasinSec. 2.2.3 .TheinverseDebyelengthisnon-dimensionalizedbytheradiusofgyration, 2 )becomes 54

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Distributionn(y)fortheelectrophoreticandpressure-drivenowactinginconjunction.Resultsareshownfor(a)Pef=15and(b)DPee=250. where 315H2Pe2f2yH9aH yyH, InEq.( 2 ),theminuscorrespondstothecaseofelectrophoresisinthesamedirectionastheoweldandtheplusforelectrophoresisopposingtheowdirection.ThequadraticdependenceofFeontheDebyelengthisnolongerexplicit,sincethedenitionofPeecontainsonefactorofD. 5.3 wererepeatedforidenticalconditions,substitutingelectrophoresisforthebodyforce.ThechannelwidthisH=8andthebeadradiusisa=0.125.Fortheresultsshown,thevalueofn(y)issettozeronearthewallbyperformingtheintegrationbetweeny=0.25andH0.25;thediffusivity,De,issettooneinEq.( 2 ).Whentheelectrophoreticforceactsinthesamedirectionasthepressure-drivenow,thedumbbellmigratestowardthecenterofthechannelasseeninFig. 2-10 .The 55

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2-10 awouldcollapseontoasinglecurvematchingtheresultsinFig. 2-4 forPef=15andPe=0.IgnoringthevelocitydisturbancesinducedbytheelectriceldsetsSEyxtozeroandeliminatescontributionsoftheelectriceldtothetransverseuxinEq. 2 .Consequently,FewithinEq. 2 wouldbecomezeroandreducethesolutionforthedistributionn(y)tothatgivenbyEq. 2 ,whichincludesonlytheeffectsofthepressure-drivenow.Physically,theenhancedmigrationtowardsthecenterofthechanneliscausedbyamechanismqualitativelysimilartotheonediscussedinSec. 2.3.3 andillustratedinFig. 2-8 .Theshearowalignsthepolymer,whichthendriftsintheappliedelectriceldduetothedipolarhydrodynamicinteractionsbetweenthebeads.ThiscouplingisapproximatedbythetermnQEinEq. 2 .Migrationtowardsthecenter,underthecombinedactionofowandelectrophoresis,hasbeenobservedexperimentally[ 5 ],butneitherownorelectrophoresisproducedsignicantmigrationinisolation.Inthelaboratoryexperiments[ 5 ],thechannelwasmuchwiderincomparisontothepolymersize,H100Rg,andtheowratewassmall,Pef<10.Undertheseconditions,migrationdrivenbyhydrodynamicinteractionswiththewallsissmallandthedominantcontributioncomesfromthecouplingbetweentherotatedandstretchedpolymerandthedipolardisturbanceeld.ThephysicsinthelaboratoryexperimentsarecontainedwithinasimpliedversionofEq. 2 56

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5 ]observedmigrationofthepolymertowardsthewall,whichisconsistentwiththephysicalpicturepresentedbyFig. 2-8 andasarguedinSec. 2.3.4 forthecaseofanexternalforceintheabsenceofanyscreening.Forelectrophoreticmotionopposingthedirectionofow,thecurrenttheorycanpredictmigrationtowardstheboundingwallonlyifthehydrodynamicinteractionsinducedbytheelectriceldareretained.Fig. 2-11 ashowstheresultofthecalculation.IncreasingD1Peeincreasesthemaximumvalueofn(y)andmovesthemaximumtowardstheboundingwallsaty=0.ZhengandYeung[ 5 ]suggestedthatthemechanismshowninFig. 2-8 wasresponsibleforthemigrationbehaviorobservedintheirexperiments,specicallydiscussingmigrationofanellipsoidalparticle.However,theydidnotaddressscreeningofthehydrodynamicinteractions,whichcanpotentiallyeliminatethecouplingbetweentheappliedelectriceldandthemigrationvelocity.Forexample,rigiddumbbells[ 46 ]andslenderrods[ 47 ]withauniformchargedistributiononthesurfaceoftheparticleandaninnitelythindoublelayerexperiencenotransversemotioninresponsetoanappliedelectriceld,regardlessoftheorientationoftheparticle.Fig. 2-11 ashowsthatthemaximumvalueofn(y)continuestoincreaseandshifttowardsthewallasD1Peeincreases.Thiscontrastswiththepreviouscase(Fig. 2-9 )ofabodyforce,wherethemaximumshiftedtowardsthecenterforasufcientlyhighforcing.Inthecurrentcase,theinteractionsoftheappliedforceswiththeboundingwallsarenotincludedintheanalysis,thereforeareversalisnotpredicted.However,migrationtowardsthecenterisobservedforsufcientlylargeowrates.AsseeninFig. 2-11 b,themaximumincreasesandshiftstowardsthecenterwithincreasingPefwhileholdingD1Peeconstant. 57

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Distributionn(y)fortheelectrophoreticandpressure-drivenowactinginopposition.Resultsareshownfor(a)Pef=15and(b)D1Pee=250. 2 37 ]ofpolymersinpressure-drivenandcentrifugalelds.Lackofquantitativeagreementarisesfromthesimplicationswithinthetheoryandthereducedconformationaldegreesoffreedomofthedumbbell.Thesimplicationsincludedsuperposingthefar-eldapproximationforthehydrodynamicinteractionswiththeboundingwalls,assumingthattheorientationdistributiondependsonlyparametricallyonthepositionofthecenter-of-mass,andapproximatingtheorientationdistributionbyapowerseriesintheforceandowelds.Nevertheless,thetheoryclearlyidentiesthemechanismscausingmigration.Forthecaseofapressure-drivenow,theinhomogeneousdistributionresultsfromthepreviouslyidentiedeffectsofgradientsindiffusivity[ 43 ]andshear-inducedmigration[ 36 ].Withanexternalforce,butnoow,apreferentialorientationinducedbythepresenceofthewallsandasubsequentdriftawayfromthewallscreatesanetmigrationtowardsthecenter.Couplingbetweenshear-inducedorientationofthepolymerandtheexternalforcecancauseeithermigrationtowardsthecenterortowardsthewalls,dependinguponthedirectionoftheforce. 58

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5 6 ],indicatingthatthehydrodynamicinteractionsinducedbytheelectriceldinpolyelectrolytesarenotfullyscreened. 59

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16 17 ],therebyexplainingtheweakcontour-lengthdependenceoftheelectrophoreticmobilityofDNAandotherpolyelectrolytes.However,theactualsituationisnotsosimple[ 18 39 ];infactthereisanelectrically-inducedhydrodynamicinteractionbetweenchargedmacroions[ 20 ],whichforlargeseparations,rij,isofdipolarform, 4r3ij3rijrij 3 ).Inthethindoublelayerlimit(aD)thehydrodynamicinteractionvanishes,f!0,butfordiffusedoublelayers(aD),f!1[ 20 ].ThemonomersizeofDNA,a=0.34nm,issignicantlylessthanthescreeninglengthinmillimolarsaltsolutions,D10nm,sothatthediffuselimitisappropriateinmoderatetolowsaltconditions.InviewofEq.( 3 ),electrophoreticallydrivenowsmayleadtonewphenomenainsituationswherethemeanpolymerconformationisaspherical.Infact,anumberofexperimentshaveshownthatDNAmoleculesincapillaryelectrophoresismigrateacrosstheeldlineswhenapressuregradientisappliedincombinationwiththeelectriceld[ 5 8 ].Interestingly,thepolymerconcentratesnearthewallwhenthepressure-gradientisappliedcountertotheelectriceld,butnearthecenterofthecapillarywhentheowandeldarealigned.Wehavepreviouslysuggestedthat 60

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37 48 ].Hereweshowthatacoarse-grainedmodelofapolyelectrolyte,withoutexplicitchargesbutwithamobilitytensorbasedonEq.( 3 ),canquantitativelyaccountfortheexperimentallyobservedmigrationwithoutanyttingparameters.Ourresultsshowthattheinsensitivityoftheelectrophoreticmobilityofpolyelectrolytestovariationsincontourlengthisduetosphericalaveragingofthepolymerconformations,ratherthanacompletescreeningofthehydrodynamicinteractions.Oursimulationspredictthattheelectrophoreticmobilityofashearedpolyelectrolytewillshowameasurabledependenceonchainlength,evenforlargemolecularweights. 23 49 ]witheachbeadrepresentinga1.06msegmentorabout6200basepairs.ThepotentialparametersaretakenfromRef.[ 50 ],exceptthatheretherepulsiveforcefromthewallhasbeendoubled,toeliminatecrossingswhenthecombinationofowandelectriceldsdrivethepolymertowardstheboundary.Themobilitytensordescribingtheelectrically-drivenhydrodynamicinteractionsbetweenthebeadsisconstructedbyassumingthateachblob(orbead)ofthecoarse-grainedmodelcontainsanelectricallyneutralbutpolarizablerandomcoilofQbbackbonechargesandassociatedcounterions;wefurtherassumethatthebackbonechargesareuniformlydistributedthroughouttheblob.Whenanelectriceldisapplied,eachbackbonechargeanditssurroundingionsgenerateadipolaroweldaccordingtoEq.( 3 )(withQ=eandf=1).Theelectrically-drivenoweldaroundablobofpolyelectrolyteisthenobtainedbysummingthecontributionsfromchargeswithin 61

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Illustrationofabeadinthebead-springmodelofDNA.Inadditiontothesemi-exibleDNAmolecule,eachbeadconsistsofcounterionsthatarepartofit. theblob.Asarstapproximationweneglectthedistancebetweeneachmonomerandthecenterofmassoftheblobsothatallthesourcetermsappeartobeatthecenter;thisgivesthecorrectlong-range(1=r3ij)interactionbetweendistantsegmentsofthepolyelectrolyte.EachblobthengeneratesaoweldoftheformofEq.( 3 )withQ=Qbandf=1.Itisimportanttonotethatapictureoftheblobasachargedparticlesurroundedbycounterionsisincorrect;thechargesaredistributedthroughoutthevolumeoftheblobasillustratedinFig. 3-1 .Thechargeassociatedwithasinglebead(orblob)istakenas, 51 ].Theelectrophoreticmobilityofthechainsegmentrepresentedbyasinglebeadis[ 16 ] 62

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52 ], 3 ),u0(r)istheambientoweld,FBisaBrownianforcewithvarianceFBiFBj=21ijt,andFCisthesumoftheconservativeforces,includinginteractionswiththeconningwalls.Thehydrodynamicmobilitytensor,ij,includestheself-interaction,theRotne-Pragerinteractionbetweenthebeads[ 10 ],andaregularizedinteractionbetweenabeadandaplanarboundary[ 50 ].InthisworkweuseasuperpositionapproximationtotheGreen'sfunctionforaconnedpairofparticles,whichisquiteacceptablewhentheshearratesaresmall[ 50 ].Theelectrophoreticinteraction,Eij,alsoincludesreectionsattheboundarieswithinthesamesuperpositionapproximation.Theself-interactionterms,iiandEii,includeboththesingle-particlemobilities,0andE0,andtheinteractionsbetweenthebeadandanearbyboundary. 1 35 ].Inapressure-drivenowthelocalshearratestretchesthepolymer,whichgeneratesanetliftforcefromtheboundary[ 36 48 53 ].However,thesemechanismsonlyactwithinthevicinityoftheboundary,ontheorderoftheradiusofgyration,Rg,andalwaysproducemigrationtowardsthecenterofthechannel.ExperimentsonDNAincombinedelectricandpressure-driven-owelds[ 5 6 ]showsignicantcross-channelmigrationoverlengthscalesoftheorderofthechanneldimension(100Rg),similartothesimulationresultsshowninFig. 3-2 .Thewidechannelandrelativelyweakeldsintheseexperimentsminimizetheliftfromtheboundaries,andnolateralmigrationwasobservedwheneithertheelectriceldorthepressuregradientwereappliedindividually. 63

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Centerofmassdistributionforconcurrent(opencircles)andcounter-current(solidcircles)applicationofanexternalelectric-eldandapressuregradient.Theinsetshowsresultswithandwithoutthecontributionsofthewallstothehydrodynamicinteractions(lowerhalfofthechannelonly). InFig. 3-2 weshowsimulationresultsforthesteady-stateconcentrationproleofa21mDNAstrand(20beads)connedwithinaatchannelofwidth75m(100Rg).Theuidisdrivenbyapressuregradient,@xp,alongtheaxial(x)directionandthepolymerisdrivenbyanadditionalbodyforceoneachbeadQbEx.Thescreeninglength,meanshearrateandelectriceldwerechosentomatchtheconditionslistedinFig.6ofRef.[ 5 ]:D=140nm,=4.09s1,andEx=62.5Vcm1.TheshearrateisthencharacterizedbyadimensionlessWeissenbergnumber,Wi==0.9,whereistheviscousrelaxationtimeofthepolymer[ 50 ].WeintroduceadifferentWeissenbergnumbertocharacterizetheelectrophoreticvelocity,WiE=E0QbEx=Rg=150.Simulationsincludingelectricallydrivenhydrodynamicinteractions,Eq.( 3 ),exhibitthesamequalitativebehaviorasthelaboratoryexperiments;thepolymermigratestowardsthecenterofthechannelwhentheowandelectriceldarealigned(concurrent),but 64

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Centerofmassdistributionforconcurrent(circles)andcounter-current(triangles)applicationofanexternalelectric-eldandpressuregradient.Simulations(solidsymbols)arecomparedwithexperimentalresults(opensymbols).ThemaingurecomparessimulationsinacylindricalcellwithFig.6ofRef.[ 5 ];D=140nm,Wi=0.90,WiE=150.TheinsetgurecomparessimulationsinasquarecellwithFig.3ofRef.[ 7 ];D=30nm,Wi=12.5,WiE=320.Thesimulationparameterswerematchedtothevaluesreportedfortheexperiments. towardsthewallswhentheyactinopposition(countercurrent).Ontheotherhand,iftheelectrically-drivenHIareneglectedthenthepolymerisuniformlydistributedacrossthechannel.Migrationofpolyelectrolytesincombinedshearandelectriceldscanbeunderstoodbyconsideringtheeffectofanelectriceldonapolyelectrolytechainthathasbeendistortedbyashearow.ThedistributionofcongurationsisapproximatelyGaussian,P(r)=(2jGj)3=2exp(rG1r=2),withagyrationtensorG(Wi)=N1PNi=1hririi;atequilibriumG(Wi=0)=(R2g=3)1.Inaweakshearow,thepolymerisstretchedandorientedwithrespecttothepressure-gradient,P(r)Peq(r)Gxyrxry=R4g, 65

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Thicknessofthepolymerlayer,lc,foraconcurrentapplicationofelectric-eldandpressure-drivenow(Wi=0.68)inaplanarchannel(D=140nm).Theconcentrationlayeristakenasthehalf-widthoftheregionthatconnes95%ofthepolymerconcentration(measuredfromthecenterline).TheinsettothegureshowsthecomponentsGxyandtr(G)oftheradiusofgyrationtensor,averagedoverthewidthofthechannel.Alllengthsarescaledbytheequilibriumradiusofgyration,Rg. whereGxy/WiR2g.Theelectrically-drivenhydrodynamicinteractionsthencauseacross-streammigrationoftheshearedpolymer,ratherlikethelateraldriftofasettlingpairofparticle,UEyQExWi2D=R2g,whereQ=NQbisthetotalchargeonthepolyelectrolyte.Thedirectionofmigrationdependsonthesignsofthepressuregradient(viaGxy),andthepotentialgradient(viaEx).TheinsettoFig. 3-2 comparesconcentrationprolesforsimulationsincludingandignoringtheeffectsoftheconningwallsonthemobilitiesijandEij.Thesimilarityoftheconcentrationprolesconrmsthatmigrationisdominatedbythecouplingbetweentheshearrateandtheelectriceld,anddiffersfromwhathasbeenobservedforneutralpolymers[ 1 41 ].Infact,thesmallowrate(Wi<1)makesthehydrodynamicliftfrom 66

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3-3 ,withparametersthatwerechosentoreproducetheconditionsoflaboratoryexperiments(Fig.6ofRef.[ 5 ]);D=140nm,Wi=0.90,WiE=150.Thesimulatedconcentrationprolesmatchtheexperimentaldatatowithinthestatisticalerrorsoftheexperiments(thestatisticalerrorsinthesimulationsarenegligible),forboththeconcurrentandcounter-currentows.Itissignicantthattherearenofreeparametersinthesecomparisons;alltheconstantsinEqs.( 3 )and( 3 )aredeterminedindependently.However,theelectriceldquotedinRef.[ 5 ]isapparentlymeasuredacrosstheelectrodesandtheeldinthecellitselfisnotknownprecisely;thusthelevelofagreementmaybefortuitous.Areasonableagreementbetweensimulationandexperimentpersistsdowntosmallerscreeninglengths,asshownintheinsettoFig. 3-3 ,whichcorrespondsto0.1mMsalt[ 7 ].InthiscasetheglasscellwasnotcoatedwithPVA(polyvinylalcohol)andtherewasasignicantelectroosmoticow,whichisnottakenintoaccountinthesimulation.Neverthelessthetrendsaresimilar,withastronglocalizationofpolymerconcentrationnearthecenterofthecell. 67

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Electrophoreticvelocityofdifferentlengthchainsinauniformshearow,u0x=y(D=140nm).Theelectrophoreticvelocity,UEoffourdifferentlengthchainsarecompared:L=5.3m(N=5),L=10.6m(N=10),L=21.2m(N=20),andL=42.4m(N=40).TheelectrophoreticvelocityoftheunshearedchainisUE0andtheshearrateisnon-dimensionalizedbytheviscousrelaxationtimeofthe20beadchain20. 3-4 .Theobservedvariationintheconcentration-layerthicknessislargelycontrolledbytheresponseoftheradiusofgyrationtensor,G,totheincreasingelectriceld.Inaweakelectriceld(WiE<1)Gxydecreaseswithincreasingelectriceld(insettoFig. 3-4 ),suggestingthatelectrophoresisrotatesthechainbacktowardstheaxis;howevertheproductGxyEcontinuestoincrease,drivingfurthermigrationtowardsthechannelcenter(decreasinglc).Nevertheless,atsufcientlyhighelds(WiE>10)thepolymersuddenlystartstoelongatetomuchlargerdimensions,asshownbytherapidincreaseintr(G)(insettoFig. 3-4 ).Thisadditionalstretchingfurtheralignsthepolymerwiththeow 68

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Asymmetricincreaseinradiusofgyrationofthepolyelectrolyteplottedversusthedimensionlesselectriceld. direction,despitethesimultaneousincreaseinGxy,andleadstoamaximummigrationaroundWiE=10,becausethehydrodynamicinteractions(decayingasR3)arethenspreadoveralargerdistance.Moreover,thestretchinthepolymerishigheralongthedirectionoftheappliedelectriceld(x-dir)asshowninFigure 3-6 .Theeffectoflong-rangehydrodynamiceldsisnotrestrictedtotheradiusofgyrationtensor;insteadastheelectriceldincreaseswealsoobserveachangeinthepolyelectrolytediffusivity,D.Thedipolarhydrodynamiceldactingbetweenthesubunitsofthepolyelectrolyteincreasesitseffectivetemperature,resultinginincreaseddiffusivity.Figure 3-7 Ademonstratesthisincreaseindiffusivityforadumbbellinperpendicularaswellasparalleldirectiontotheappliedeld.Asobserved,thediffusivityincrementislargeralongthedirectionoftheappliedeld.ThephenomenaofdispersionalsoexplainstheswellingofthepolyelectrolyteortheincreaseinGwithincreasingelectriceld(insetofFig. 3-4 ).Theincreasedhydrodynamicsizeisinversely 69

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DiffusivityofapolyelectrolytemoleculeplottedversusthedimensionlesselectriceldWiE.Inplot(A)resultsforadumbbellmodelofpolymerarepresentedinparallelandperpendiculardirectionstotheappliedelectriceldwhileinplot(B)resultsforachainwithNb=20forthreedifferentsaltconcentrationsarepresented.ThediffusivityismadedimensionlessusingDNb,0whichisthediffusivityofthechainatWiE=0andwhereNbcorrespondstothenumberofbeadsinthechain. proportionaltopolyelectrolytediffusivityandisthedominantfactorindeterminingpolyelectrolytediffusivityoflargerchains.Figure 3-7 BshowsthatDfor-DNA(Nb=20)decreaseswithincreasingelectriceldandmoreover,ifthehydrodynamiceldsinducedbytheappliedelectriceldarecurtailed,byincreasingtheelectrolyteconcentrationinthesolution,nochangeindiffusivityisobserved.Thechangeindiffusivityofapolyelectrolyteinpresenceofelectriceldreportedinthisworkdoesnotcontradictpreviousexperimentalwork[ 54 ].Theirexperiments,conductedatWiE10and[c]=20mMNaClconcentration,demonstratethatfree-solutiondiffusivityofDNAisinsensitivetoappliedelectricelds.Simulations 70

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16 17 ],However,ourmodelforacoarse-grainedpolyelectrolytesuggeststhatlength-dependentelectrophoresiswilloccurifthemeanpolymercongurationisnon-spherical.Thechain-lengthdependenceoftheelectrophoreticvelocityinauniformshearow,u0x=y,isshowninFig. 3-5 fordifferentlengthDNA.Intheabsenceofowthemobilityisindependentofchainlength,sinceinthiscasethemeanpolymerconformationremainsspherical.However,addingaweakshearowstretchesthepolymer,2GxxGyyGzz>0,andthentheelectriceldgeneratesanadditionalcontributiontoelectrophoresisfromhydrodynamicinteractions.Theincreaseintheelectrophoreticvelocityisinitiallyproportionalto2,buteventuallysaturatesataround1.1E0whentheWeissenbergnumberforthechain>10.Atsmallershearratestherearemeasurablevariationsinthemobilityofdifferentlengthchains,possiblyallowingforseparationbylengthinasinglecapillary. 71

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55 ].Onesuchmolecule,thedouble-stranded-DNA,isofparticularinteresttopolymerphysicistsandbiologistsalikeowingtoitsabundanceinnature,itsstabilitytolargeshearratesaswellaselectricelds,andtheabilitytoeasilyimagethemolecule.ThesefeaturesofthemoleculewereexploitedbytheexperimentsofPerkinset.al.[ 31 32 ]whichimagedandmeasuredtherelaxationofasingle-moleculeof-DNAfromaninitiallystretchedstateforthersttime.Theseexperimentsofferedadirectinsightintothedynamicbehaviorofmesoscalepolymersunlikeanypreviousexperimentsandopenedthedoorstonumeroustheoretical[ 56 57 ],computational[ 58 ],andexperimentalstudies[ 59 ]of-DNA.Recentexperiments[ 5 8 ]haveshownthat-DNAmigrateslaterallywhendrivenbycombinationsofapressuregradientandanelectriceld.Thiscross-streammigrationofDNAhasbeeninvestigatedandexplainedusingkinetictheory[ 48 ]andBrowniandynamicssimulations[ 60 ].TheexibleDNAmoleculestretchesandorientsduetothelocalshearrate.Thecenterofmassofthisorientedmoleculethendriftslaterallydependingonthedirectionoftheappliedelectriceld.Thelateralmotionofthemoleculeisaresultoflong-rangehydrodynamiceldsinducedbytheappliedelectriceld.ThemechanismisexplainedindetailinChapters 2 and 3 .Thischapterfocusesoninvestigatingaspecic,non-intuitive,behaviorincross-streammigrationof-DNAthroughmicrouidicexperiments.Simulationspredictanincreaseinradiusofgyration,G,ofthemoleculeinthepresenceofanelectriceldduetointernaldispersionofitssubunits.Furthermore,thedipolarformofthehydrodynamiceldresultsinhigherextensionofthepolymerinthedirectionoftheappliedeld.Thisasymmetricincreaseinradiusofgyration(Figure 3-6 )reducesthenetangleof 72

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Thisillustrationshowsthatasymmetricincreaseinradiusofgyrationofthepolyelectrolyteinhighelectriceldcandecreasethelateralvelocityofthecenterofmass. orientationofthepolyelectrolyteanddecreasesitsobservedlateralmigration.ThisphenomenaisillustratedinFigure 4-1 .Thesimulationspredictthatforaxedowratetheextentofcross-streammigrationattainsalocalmaximaatanoptimumelectriceldandthenstartsdecreasingagain(Figure 3-4 ).ApartfromvalidatingtheBrowniandynamicssimulations,thisworkalsolaysthefoundationforamorequantitativecomparisonbetweenexperimentsandsimulations.TheBrowniandynamicssimulationscomparefavorablywithexperimentsbyZhengandYeung[ 5 8 ]andmoreimportantly,forcertaincasesshowquantitativeagreementinabsenceofanyadjustableparameters.Theseresultsestablishsomecondenceinthesimulationseventhoughtheexperiments[ 5 8 ]focusedontechnologicalapplications,suchasDNAseparationbasedonlengthandlateralfocussingofmolecules,insteadofmethodicallyinvestigatingtheeffectofexperimentalparameterssuchastheelectriceldandelectrolyteconcentration.Ourexperimentalworkaimstosystematicallystudytheeffectofappliedelectriceldandelectrolyteconcentrationoncross-streammigration 73

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Illustrationofthemicrochannelusedformicrouidicexperiments.ThedimensionsofthechannelLHWarelabeled.Theelectrodes,markedinthegure,arealsotheuidinletandoutlet.(ThisgureisreproducedwithpermissionfromRef.[ 61 ].) andenhancetheunderstandingoftheowdisturbancesinducedinpresenceoftheelectriceld.Inaddition,calibrationoftheexperimentalsetupisperformedtoensurethatthestudyisquantitative.Thischapterisorganizedasfollows.Section 4.2 detailstheexperimentalmethodology,whichincludespreparationofthemicrochannelandthesolution,theexperimentalsetup,andtheprotocolforimageanalysis.Section 4.3 outlinesthetheorybehindthecalibrationexperimentswithuorescentmicrospheresanddiscussestheresultsofcalibration.Resultsfromcross-streammigrationof-DNAarepresentedinSection 4.4 .Finally,Section 4.5 concludesthischapterandrecommendsfuturework. 4.2.1PreparationofMicrochannelWeusemicrochannelsmanufacturedfromPDMS(Poly-dimethylsiloxane)andaglasscover-slideofthicknessof170microns.AschematicdiagramofthemicrochannelispresentedinFigure 4-2 .Thestainlesssteeluidoutletsarealsousedaselectrodesintheseexperiments.Thechannelshavealength,L=3cm,whiletheirwidth(W)and 74

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4-3 .ThepolymerbaseforPDMSanditscuringagent(commerciallyavailableasSylgard184SiliconeElastomerkit)aremixed,withaweightratioof10:1,usingastirringrod.Theairbubblestrappedinpre-curedPDMSaredegassedandtheelastomerispouredonthemastermold.ThecuringprocessofPDMSrequiresatleast2hours,afterwhichthePDMSstampcanbepeeledoffthemold.Thestampisthenbondedwiththeglassslidebyactivatingboththesurfacesusingaplasmatreater.Imagingoftheowsinsidethechannelisdonethroughtheglassboundary.Twodifferentcastingmoldsareusedintheseexperimentswithrectangularcrosssections,onemadeofnegativephotoresistSU-8withdimensions30024m2andtheothermadeofSiliconwithdimensions8080m2.TheSU-8moldwascreatedbyspin-coatinga24mthickphotoresistlayeronasiliconwafer.ThewaferwasthenexposedtoUVrayswithaphotomaskofthemicrochannel(width300m)protectingthephotoresist.SU-8isanegativephotoresist,andthereforetheregionexposedtoUVlightsolidies.Apermanentmoldwasobtainedbybakingthewaferfortheappropriatetime.Theparametersrequiredforthespincoatingprocessareprovidedindetailbythemanufacturerofthephotoresist(MicroChem).ThethickermoldwascreatedusingtheDeepReactiveIonEtch(DRIE)technique.Inthismethodthesiliconwaferiscoatedwitha20mlayerofSU-8andexposedtoUVlightinpresenceofan80mwidephotomask.DRIEusesanSF6basedplasmatoetchthesiliconwafer.Toobtainahighaspectratiostructure,theetchingprocessisinterruptedtocoatthesidewallswithapassivationlayer(consistingofC4F8).Afteranappropriatenumberofetchingandprotectioncycles,thedesired80metchisobtainedinthesiliconwafer.Theappliedphotoresistisnotaffectedbytheplasmaetchcycleand 75

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IllustrationofsoftlithographyprocesstocreatePDMSstampsofthemicrochannelfromthepermanentmold. mustberemovedusingacetonetoobtainthepermanentstructureetchedonthewafer.ThepermanentmoldcreatedinbothcasesthenneedstobetreatedwithasilanelayerusingeitherHMDS(Hexa-methyldisilazane)orTCMS(Tri-chloro-methylsilane)toinhibittheadhesionofPDMStosilicon.Theaveragethicknessofthethinnerandthethickermoldsis24.10.3mand78.74.3mrespectively.BothmoldswerecreatedintheNanoscaleFabricationfacilityatUniversityofFlorida(UF).The24mmoldwasmanufacturedbyCarlosSilvera-BatistaworkinginProf.KirkZeigler'slab,whilethe80mmoldwasmadebyPanGuworkinginProf.Z.HughFan'slabattheUniversityofFlorida. 76

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4-4 .Themicrochannelusedfortheseexperimentswassetuponaninvertedmicroscope(NikonDiaphot200)andtheowofuorescentmicrospheresand-DNAwasobservedusingepiuorescentmicroscopy.Theimagesarecapturedonadigitalcamera(MonochromeRetiga-SRV,byQ-Imaging)designedforhighspeedandhighsensitivityapplicationsandarestoredonacomputerforpostprocessing(Section 4.2.4 ).Peripheralequipmentrequiredfortheexperiment,thatisnotshowninthegure, 77

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Theexperimentalsetuputilizesaninvertedmicroscopeconnectedtoalightsource(Hglamp)andadigitalcameratocaptureimages.Themicrochannelispositionedasindicatedandtheexperimentsarerunbyconnectingthechanneltoasyringepumpandapowersupply(notshowninthegure).(ThisimageisreproducedwithpermissionfromRef.[ 61 ].) includesapositivedisplacementsyringepump(HarvardapparatusPump11PicoPlussyringepump)andanelectriceldgenerator(Agilent33210AFunction/ArbitraryWaveformGenerator).TheschematicdiagramoftheopticalsystemispresentedinFigure 4-5 .Weuseamercury-vaporlampasthelightsource.Theorescentimagingofmicrospheresand-DNArequiresanexcitationlightofwavelength491nmandemitsalightofwavelength509nm.Thisisenabledbymountingaband-passlterandadichroicmirrorintheopticalpathasillustrated.Weusea63XmagnicationLeitzWetzlarobjectivewithnumericalapertureof0.85forallresultspresentedinthischapter.Theresolutionoftheimagescapturedwiththissetupis0.60mperpixel. 78

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Schematicdiagramoftheopticalsystem.Theband-passlterfocusestheexcitationbeamwhilethedichroicmirrorallowstheemittedlightwavestobecapturedbythecamera.FlorescentparticlesusedinexperimentsincludethecarboxylatemodiedmicrospheresandYOYO-1dyestained-DNA.(ThisimageisreproducedwithpermissionfromRef.[ 61 ].) ImageJsoftwarewhichisautomatedtoidentifyindividualmoleculesanddistributethemintobinsbasedontheiry-coordinate.Eachexperimentaldatapointforcross-streammigrationofDNAisconstructedfromatleast40,000DNAmoleculesidentiedfrom2000imagesofthemicrochannel.Theseimagesarecapturedfromthez=H 4-6 .Theseimagesaretakenfromthechannelwithcross-section30024m2.Theguredemonstratescross-streammigrationof-DNAmoleculeswhenelectriceldisappliedparalleltotheow.Moreover,themigrationresultsinlargernumbersofmoleculesbeingcapturedperimage. 79

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Imageof-DNAmoleculesowingin30024m2cross-sectionchannel.ThegureonleftistheowonlycasewithWi=0.7,whiletheoneonrightappliesanelectriceld,E=4V=cm,inadditiontotheow. FordeterminingthevelocityproleofuorescentmicrospheresinthechannelweusetheManualTrackerpluginavailableforImageJ.Werecord250imagesofthemicrospheresdrivenbyacombinationoftheappliedvoltagedifferenceandapressuregradient.Theimagesarecapturedafterthemicrospherevelocitieshaveobtainedasteadystate.Thepressuregradientisappliedduringthecalibrationexperimentstoovercomeaggregationofthemicrospheresattheelectrode.Theobtainedimagesaresub-dividedinto8binsofequalsize.Approximately25microspheretrajectoriesareidentiedineachbintodeterminetheowprole. 4.3.1TheoryCalibrationexperimentswereperformedtoestimatetherelationshipbetweenthevoltagedifferenceappliedtotheelectrodesandtheactualelectriceldinsidethemicrochannel.Theseexperimentsinvestigatethemotionofuorescentspheresinthe8080m2microchannelinthepresenceofanelectriceldandapressuregradient.Theappliedelectriceldnotonlygeneratesadirectelectrophoreticvelocity,vE=EMSE,fromthemicrospheres,butalsoresultsinanelectroosmoticow,vEO,inthemicrochannel.Previousliterature[ 62 63 ]suggeststhatelectroosmoticvelocity 80

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Illustrationofthevariousowsobservedinsidethemicrochannel.Fornegativelychargedmicrospheres,theelectroosmoticow,vEOaddstotheelectrophoreticresponse,vE,toappliedelectriceld.Theoweldduetopressuregradient,vF,isparabolic. inaPDMS/Glasschannel,withdimensionsandelectrolyteconcentrationsimilartoourexperiment,isdirectlyproportionaltoappliedelectriceldandtheelectroosmoticmobilityisgivenbyEO=1.68104m2V1s1.NotethatvEOactsinoppositiontotheappliedelectriceldandincontextofthepresentexperiment,itresultsinincreasingthespeedoftheuorescentmicrospheres(illustrationinFig. 4-7 ).Theoverallvelocityofamicrosphereinthechannel,vMSisafunctionofy,itsdistancefromthechannelboundary,andisgivenby, 4-8 plotsthevelocityproleinthechannelforavolumetricowrateVF=0.01Lmin1inconjunctionwithfourdifferentvaluesoftheelectriceld.Thedirectionoftheappliedelectricelddrivestheuorescentmicrospheresinthedirectionoftheow.Theexperimentallyobservedvelocityvismadedimensionlessusingtheaveragemicrospherevelocity,vavg,andiscomparedwiththetheoreticallycomputedmicrosphere 81

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4-8 .Themotionofamicrosphere,onaverage,asobservedinthemicrochannelis 4-9 Ashowsthatthesteadystatedistributionproleof-DNAacrossthechannelsaturatesforWiE40.Theprobabilitydistributionfunctionisplottedinthegraphforsixdifferentvaluesoftheelectriceldactingparalleltoaxedowrateof0.03Lmin1whichcorrespondstoWi=0.7.Themigrationlayerthickness,lc,obtainedfromexperimentsandBrownian-dynamicssimulationsisplottedinFigure 4-9 B.Thesimulationresults[ 60 ]areforanidenticalowWeissenbergnumber,butforaparallelplategeometrywithH=75m.ThedecreaseinmigrationlayerindicatesthattheDNAmoleculesareobservedinathinnerbandclosetothecenterofthechannel,thereforeimplyinghighercross-streammigration.Bothsimulationsandexperimentsshowaninitialincreaseinmigration 82

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Velocityofuorescentmicrospheresversustheirpositioninthechannelfor4differentvalueofelectriceld.Thevelocitiesaremadedimensionlesswiththetheaveragevelocity,vavg,(Eq. 4 )insidethechannel. withincreasingeld,thoughmigrationpredictedbythesimulationsishigherthantheobservationmadeforexperiments.Moreover,thesimulationspredict(Fig. 4-9 B)anincreaseinlcathigherelds.Thiseffectisaresultoftheincreaseinradiusofgyrationofthepolyelectrolyteduetohydrodynamiceldsinducedbytheelectriceld.Thisincreaseinsizereducestheangleatwhichthemoleculeisorientedrelativetotheappliedelectriceldandhenceresultsinadecreaseinmigration.ThisphenomenaisexplainedindetailinSection 3.4 andRef.[ 60 ].Experimentsathigherelectriceldsareneededtoinvestigateapossibledecreaseinmigrationof-DNA.Migrationlayersobtainedfromtheexperimentsarefullydeveloped.Figure 4-10 plotsthesteadystatevalueofmigrationlayeratdifferentpositionsalongthelengthof 83

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Thegraphcapturesthesaturationofcross-streammigrationof-DNAwithincreasingelectriceldandaconstantowrate,Wi=0.7.Thetopgureplotsprobabilitydistributionfunctionfor6differentvaluesofelectriceld,rangingfromWiE=4to115inachannelwithcross-section8080m2.Theplotinthebottomcomparestheexperimentalresultsobtainedforthemigrationlayer,lc=Rg,withtheresultsobtainedfromBDsimulationsforaparallelplategeometrywithH=75m. thechannel.ResultsareplottedforaoweldwithWi=0.7inconjunctionwithanexternalelectriceldwithWiE=12andWiE=75.Thevalueoflc=Rgatthequarterlengthofthechannel,x=L=0.25,issimilartothevalueatthecenterofthechannel,x=L=0.50forthecaseofWiE=75.Forweakerelectriceld,WiE=12,themigrationproledevelopsmoreslowly;stilltheproleisfully-developedwhichcanbeinferredfromthevalueofmigrationlayerthicknessatx=L=0.375and0.5.Figure 4-10 alsoshowsauniformdistributionof-DNAattheentranceofthechannel,x=L=0forbothvaluesofWiE.SmallmigrationobservedforWiE=75attheentranceisaresultanalreadydevelopingmigrationprole.Theseobservationsconrmtheabsenceofanyendeffectsinthechannelthatmayexplaincross-streammigration. 84

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Plotofmigrationlayer,lc=Rg,versusthelocationalongthelengthofthechannel.ResultsarepresentedforaowratewithWi=0.7inadditionwithanelectriceldofWiE=12(circles)andWiE=75(squares). Figure 4-11 showthedistributionofcenterofmassofuorescentmicrospheresaswellas-DNAforavolumetricowrateof0.01Lmin1andanappliedelectriceldE=33.4VcmorWiE=75.Theplotshowssignicantlyhigherlateralmigrationfor-DNAwhencomparedwiththemicrospheres.Thisobservationstrengthensourhypothesis[ 48 60 ]thattheexibilityofamoleculeisimportantinobservingcross-streammigration.Notethatadepletionlayer,ld0.08Horroughly10timestheparticlediameter,isobservedforthemicrospheres.Wesuspectthatthisdepletionisacombinationofstericeffectsofthewallandopticalerrorsintheexperiments.Furtherinvestigationsareplannedtoaddressthisissue. 85

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Centerofmassdistributionformicrospheres(squares)and-DNA(circles)forconcurrentapplicationofelectriceldwithWiE=75andoweldwithWi=0.7.Theerrorsbars,whennotplotted,aresmallerthanthesizeoflabel. qualitativelywiththepredictionsmadebyBrownian-dynamicssimulationspresentedinChapter 3 .Thesimulationssuggestedthatathighelectriceldsthemigrationlayerstartsdecreasingbutthisphenomenahasnotbeencapturedinourstudyduetotherestrictionsimposedbytheequipment.Yet,theeffectofsaturationofthemigrationlayerfurthersolidiesthehypothesisthatlongrangeuiddisturbancesinducedbyelectriceldsexist.ThemountingevidenceofHIinducedbyelectriceldshouldencouragesingle-moleculestudiesof-DNAinvestigatingitsstaticanddynamicpropertiesduringelectrophoresis. 86

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64 ]hasbeenproposedasabasisfornumericalsimulationsofpolymersolutions[ 65 ],usingafrictionalcouplingbetweenthepolymerandthesurroundinguid.Singlemoleculedynamicsobtainedwiththisalgorithmcomparesfavorablywithcoarse-grainedmoleculardynamicssimulationsusingexplicitsolvent[ 66 ].Morerecently,thesameideahasbeeninvestigatedforconnedpolymers[ 2 ],asasimplerandpossiblymoreefcientalternativetoBrowniandynamics(BD)[ 23 ].Bothmethodshavebeenappliedtoproblemsofpolymermigrationinshearandpressure-drivenows[ 1 23 41 ]andshowedsimilartrendswithincreasingshearrate,butdifferentchoicesofpolymermodelprecludedaquantitativecomparison.HereweinitiateasystematiccomparisonofLBandBDmethods,beginningwiththepropertiesofanisolatedchain;subsequentlywewillextendtheinvestigationtoconnedpolymersinshearandpressure-drivenows(seeChapter 6 ).ThisworkcomplementsarecentstudybyPhametal.[ 67 ].InertialeffectsareneglectedinBrowniandynamics,resultingininstantaneouspropagationofmomentum.Althoughthesolventdegreesoffreedomaretherebyeliminated,theinteractionsbetweenthebeadsarelong-range,whichleadstoanO(N3)scalingofthecomputationalcostforapolymerconsistingofNsegments.IntheLBmethod,allinteractionsarelocalandthecomputationalcostscaleslinearlywiththevolume.However,thelattice-Boltzmannmethodintroducesanextra,inertialtimescale,duringwhichthehydrodynamicinteractionspropagatethroughouttheuidbyviscousmomentumdiffusion.Surprisinglythishaslittleeffectonthetimestep;LBsimulationsusecomparabletimestepstoBD,aswillbeseeninSec. 5.3.1 .Hydrodynamicretardation,sometimesthoughttobeapotentialsourceoferrorinlattice-BoltzmannsimulationsofStokesow[ 9 67 ],isinfacteasilymanaged.Nevertheless,the 87

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67 ].Quantitativecomparisonsrequireanidenticalmicro-mechanicalmodelofthepolymer.However,theBDsimulationsareforanisolatedchain,whiletheLBsimulationsuseperiodicboundaryconditions.Itisthereforenecessarytocorrectforthedifferencesinthelong-rangeoweldsinperiodicandunboundedsystems;inparticular,thediffusioncoefcientinaperiodicsystemhasacorrectionproportionaltoL1,whereListhelengthoftheperiodicunitcell.However,priorresearch[ 2 68 ]hasshownthatthesecorrectionscanbecalculatedquantitatively,basedonthehydrodynamictheoryforaperiodicunitcell[ 69 ].CorrectionstothecongurationalpropertiesandRouserelaxationtimesaremuchsmaller,O(L3);whentheboxlengthis510timesthepolymersize,deviationsfromtheinnitesystemarenegligible.ThediffusioncoefcientsandRouserelaxationtimeswerefoundtodependonthedegreeofdiscretizationofthelattice-Boltzmannuid.Ourresultsshownumericalconvergencewithincreasinggridresolution,andtheconvergedresultsagreewithBrowniandynamicswithin12%. 2r20ln1r2 5 ); 88

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23 49 ].Theidenticalpolymermodel,describedbyEqs.( 5 )and( 5 ),wasusedforbothBDandLBsimulations.ThebeadsarecoupledtotheuidwithaStokesfrictioncoefcient=6a,whereistheuidviscosityandthehydrodynamicradiusofthebeadsa=0.362b. 5 )and( 5 ).WeusetheRotne-Pragerregularizationofthemobilitymatrixofpointparticles[ 10 11 ],whichapproximatestheStokes-owresultinthelimitrija,butensuresthatthemobilitymatrixremainspositivedeniteforallrij: 4a rij+1 2a3 4a rij3 2a3 32rij 32rij 89

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52 ]forstochasticintegrationreducestoanexplicitEulerintegrationscheme, 70 ],whichscalesmorefavorablyasO(N2.25). 71 72 ]hasbeenusedtosimulatethedynamicsofdilutepolymersolutionsinperiodic[ 65 73 ]andconnedgeometries[ 2 37 41 ].IntheoriginalformulationoftheLBmodel[ 71 72 ],theviscousstresstensorwasassumedtouctuatearoundthelocalNavier-Stokesstress,butthismodelfailstosatisfytheuctuation-dissipationtheorematsmallscales[ 74 ],unlessthermaluctuationsinthenon-hydrodynamicmodesareincludedaswell.Herewesummarizetheimproved 90

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75 ].Includingthermaluctuationsinthenon-hydrodynamicmodesleadstosmall,butnoticeable,improvementsintheequipartitionofenergybetweentheuidandpolymerdegreesoffreedom(Sec. 5.3.1 ).Inthelattice-Boltzmannmodel,theuiddegreesoffreedomarerepresentedbyadiscretizedone-particlevelocitydistributionfunctionni(r,t),whichdescribesthemassdensityofparticleswithvelocityciatthepositionrandtimet.Thehydrodynamicelds,massdensity,andmomentumdensityj=u,aremomentsofthisvelocitydistribution, Thetimeevolutionofni(r,t)isdescribedbyadiscreteanalogueoftheBoltzmannequation[ 76 77 ], 78 ]wasused,whichincludesrestparticlesand18velocitiescorrespondingtothe[100]and[110]directionsofasimplecubiclattice.Thepopulationdensityassociatedwitheachvelocityhasaweightacithatdescribesthefractionofparticleswithvelocityciinauidatrest: 3,a1=1 18,ap 36.(5)Thedeterministiccollisionoperatorislinearizedaboutthelow-velocityequilibriumdistribution,neqi, 2c4s,(5) 91

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79 80 ], 79 80 ],suchthatthebacktransformationincludestheweightsaci[ 81 ], 82 ].Theshearviscosityisrelatedtoe, 81 82 ],althoughthatpropertyisnotessentialinthepresentcontext.ThekeydifferencebetweentheuctuatingLBwiththelattice-Boltzmannmodelisinthecollisionoperator.TheuctuatingLBcollisionoperatorcontainsrandomexcitationsofthenon-conservedmoments,c.f.Eq.( 5 )[ 83 ], x3k,(5) 92

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83 ],orlargechangesinmkwilloccasionallyoccur,leadingtonegativevaluesofni.Theamplitudeoftherandomforcingisdeterminedfromtheuctuation-dissipationrelation[ 72 ]andiscontrolledbythemassofanLBparticle,mp[ 75 ].Thermodynamicconsistencyrequiresthatmpisrelatedtotheeffectivetemperatureoftheuctuatinguid[ 75 83 ], 74 ],notjustthestress[ 64 ].Ithasbeenshowntheoretically[ 75 ]thattheexcitationofthenon-hydrodynamicmodesisessentialtosatisfytheuctuation-dissipationrelationatallscales,althoughatlongwavelengthsonlytheexcitationsinstressareimportant. 65 ].Sincetheuidsatisesitsownuctuation-dissipationrelation,Rihasalocalcovariancematrix 93

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2 ]).Sincethemonomersmovecontinuouslyoverthegrid,theuidvelocities,un,areinterpolatedfromneighboringgridpointstothebeadlocationri, 84 ] 83 ].Wewillpresentsomeresultswiththree-pointinterpolation, 31+p 21 653jujp 2juj3 203 2juj,(5)butmostoftheresultsuse2-point(linear)interpolation.Toconservemomentum,theaccumulatedforceexertedbythebeadontheuidisdistributedtothesurroundingnodeswiththesameweightfunction[ 65 83 ].Theinputfriction0=6a0isnotthesameastheeffectivefriction=6a,asmeasuredbythedragforceonthebeadorbyitsdiffusioncoefcient.Thisisbecause 94

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6xg,(5)wheregisanumericalfactor[ 83 ]thatisindependentoftheuidviscositybutdependsontheinterpolationfunction.Fromthediffusionofindividualmonomerswehavedeterminedvaluesofg=0.77forlinear(two-point)interpolationandg=1.0forthethree-pointinterpolation.TheLBresultsinthischapterarematchedtoBDsimulationswiththesameeffectiveradius,a.Thecoupledequationsofmotionfortheparticlesanduidaresolvedbyoperatorsplitting[ 83 ];typically,thethermodynamicforcesareintegratedwithasmallertimestepthanthehydrodynamicforcestomaintainstability.TheLBtimeintervaltisdecomposedintoMstepsoflengthh=t=M,whereMischosentobesufcientlylargethattheconservativeforcesareintegratedaccurately;typicallyM10inoursimulations.Thealgorithmusedinthisworkisasfollows: 1. AtthebeginningoftheLBstep,determinetheuidvelocitiesatthegridpoints. 2. UpdatethepolymerpositionsandvelocitiesoverMsub-cycles.ForeachsubcycleamodiedVerletalgorithmisusedtoupdatethepositionsandvelocitiesofthebeads: (a) Firststreamtheparticlepositionsandvelocitiesforhalfatimestep, wheretheconservativeforceF1i=rr(1)iisevaluatedfromthecoordinatesatthehalftimestep,r(1)i. (b) Usetheupdatedpositions,r(1)i,tointerpolatetheuidvelocitytothebeadlocations,Eq.( 5 ). 95

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Updatethebeadvelocitiesforafullsteph,usingamidpointapproximationtothefrictionaldragforceEq.( 5 ), (d) Redistributethemomentumtransferredbyparticle-uidcoupling (e) Streamtheparticlepositionsandvelocitiesforthesecondhalfstep, Theexactsequenceofupdatesisimportanttopreservethesecond-orderaccuracyoftheoperator-splittingmethod.ThealgorithmreducestotheVerletschemewhen0!0. 3. UpdateLBpopulationstoaccountformomentumtransferfromparticle-uidcoupling. 4. UpdateLBalgorithmforonetimestept.Thereareanumberofnearlyequivalentwaystobreakdownthecoupleddynamicsoftheparticle-uidsystem;thealgorithmdescribedaboveisthemostaccurateofthevariationswehaveinvestigated,althoughthedifferencesinlong-timeproperties(conformation,diffusion,Rouserelaxationtimes)aregenerallysmall.Themidpointmethodispreferabletoarst-orderupdate,eitherexplicitorimplicit,sinceneitheroftheseleadtoexactthermalizationofthekineticenergyoftheparticles.Forforce-freeparticlesitisstraightforwardtoshowthat[ 2 ] 96

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5 ),affectstheinterpolateduidvelocityaftereachsubstep(h)oronlyaftereachLBstep(t).Numericalresultsshowthatthepolymertemperatureisclosertotheuidtemperature(within0.3%)iftheinterpolatedvelocityisupdatedeverysubstep(h).WhenthevelocityisonlyupdatedattheLBsteps(everyt),thepolymertemperaturediffersfromtheuidtemperaturebyabout3%.TheresultspresentedinSec. 5.3 havetheinterpolateduidvelocityupdatedeveryh. 2(N+1)2Xijr2ij,(5)andend-to-endvector,R2e, 97

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Conformationalproperties,R2g=b2(closedsymbols)andR2e=b2(opensymbols),versusthedimensionlesstimestept=t0.TheLBdatausedagridresolutionx=1.29b;conformationalpropertieswithothergridresolutionsarestatisticallyindistinguishable(seeTable 5-1 ). ofthepolymerarecomparedinFig. 5-1 .Theconformationalpropertiesfromlattice-BoltzmannareindistinguishablefromBrowniandynamicswithinthestatisticalerrors(0.5%).Despitetheextrainertialtimescale,theLBsimulationsusecomparabletimestepstoBrowniandynamics;neithermethodshowsstatisticallysignicantdeviationsinRgandRewhenthetimesteptislessthan0.01t0.TheaccuracyofaBDsimulationdependsonlyonthetimestep,butresultsfromLBsimulationsmaydependonanumberofparameters:uctuationlevelortemperature(T),lengthoftheperiodicunitcell(L),gridresolution(x),uidviscosity(),andparticlemass(m).ResultsforarangeofvaluesoftheseparametersaresummarizedinTable 5-1 usingaFENEchain(Sec. 5.2 )of10segments.InLBsimulations,thetimescalet0===b2=Tiscontrolledbythetemperature,whichsetsthelevelofuctuationsintheuid,Eqs.( 5 )( 5 ),andparticles,Eqs.( 5 )( 5 ).The 98

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75 83 ];onedimensionlessmeasureistheparameter=hu21i=c2s(seeTable 5-1 ),whichrelatestheuidvelocityuctuationsinasingleLBcelltothesoundspeed.FortheLBmodeltoadequatelyrepresentanincompressibleuid(juj=cs<0.1),shouldbelessthan0.01.AtthehighesttemperatureshowninTable 5-1 ,=0.024,thepolymerisindeedslightlyswollen.Theuctuation-dissipationtheorem(FDT)shouldensurethatthepolymerthermalizestothesametemperatureastheuid;inotherwordsmhv2ii=Mhu21i,whereM=x3isthemassofuidinasingleLBgridcell.ThecolumnT=TinTable 5-1 measurestherelativedeviationsinthepolymertemperaturefromthermalequilibrium;theseareusuallysmall,oftheorderof0.2%.Largerdeviationsoccurwhenthetemperatureistoohigh(=0.024),orwhentheLBmodelisnotexactlythermalized(footnotes5and6).Ifthekinetic(orghost)modesarenotsubjecttorandomforcing,theuctuation-dissipationrelationisbrokenatshortlengthscales[ 74 75 ].Thiscausesasmallerrorinthesizeofthepolymer,12%,whencomparedtotheproperlythermalizedsimulations,withsimilardeviationsinthediffusioncoefcientandRouserelaxationtimes(footnote5).Largererrorsinbothstaticanddynamicpropertiesoccuriftheuiddynamicsispurelydissipative(footnote6),becausetheuctuation-dissipationrelationisthenbrokenatalllengthscales;resultsfromsimulationswithoutuiductuations,forexampleRef.[ 9 ],areinvalid.TheestablishmentofgoodthermalequilibriumbetweenthepolymeranduidrequiresexactthermalizationoftheLBuidandthecouplingalgorithmdescribedinSec. 5.2.3 .Theratios=Tand=Tmustbekeptconstantifthepolymerconformationsaretobeindependentofthedegreeofcoarsegrainingoftheuiddegreesoffreedom.Thedimensionlesstimestept=t0oftheLBsimulationinFig. 5-1 isthencontrolledbythetemperatureoftheuctuatinguid(Tor).SincetheviscosityisindependentoftemperatureintheLBmodel,theSchmidtnumberSc==D,variesinversely 99

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Staticanddynamicpropertiesofapolymerchain.FluctuatingLBsimulationsfora10-segmentchainarecomparedwithBrowniandynamics.TheresolutionoftheLBgrid,x,canbecomparedwiththeRMSdistancebetweenthebeadshr2i1=2=1.60b.Theparameter=hu21i=c2sisameasureofthetemperatureoftheuid,T=Mhu21i,whereM=x3isthemassofuidinasinglegridcell.ThedimensionlesstimestepintheLBsimulationsisrelatedtothetemperaturethroughthescalingwitht0===b2=T.Tisthedifferenceintemperaturesoftheparticlesanduid.Themassofabead,kinematicviscosityoftheuidandthelengthoftheperiodicunitcellarem=0.1M,=0.1x2=tandL=9.4Rg,unlessotherwiseindicated.Thestatisticalerrorsindiffusivity,D=D0,andRouse-moderelaxationtime,1=t0,arelessthan0.5%;D0=T=isthemonomerdiffusivity,andSc0andScaretheSchmidtnumbersbasedonthemonomerandpolymerdiffusivities.ErrorsinR2e=b2arementionedinbracketswhileinR2g=b2theyarelessthan0.1. 100

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5-1 showthatthestaticanddynamicpropertiesarebothinsensitivetoSchmidtnumber,buttherearesmalldeviationswhenSc<30,whichisconsistentwithearlierndings[ 2 ].OtherauthorshavesuggestedthattheSchmidtnumberbasedonthemonomerdiffusion,Sc0==D0,shouldbeinexcessof30[ 65 ],butourresultssuggestthatthismaybeoverlyrestrictive;wedonotndsystematicdeviationsineitherstaticordynamicpropertiesuntilSc0<10.Bothstaticanddynamicpropertiesarestatisticallyindependentofuidviscosity(footnote2)andparticlemass(footnote3)overtherangesstudied.Finally,wenotethattheconformationalpropertiesandRouse-moderelaxationtimesareindependentofthesizeoftheunitcell(footnote1).TheseresultsareconsistentwithrecentLBsimulations[ 67 ],whichfoundaweaksystemsizedependencewhenL<5Rg;inoursimulationsL10Rg.ThesystematicdependenceofthediffusioncoefcientonLhasbeenanalyticallycorrectedinTable 5-1 ,asdiscussedindetailinSec. 5.3.2 6d dthrc(t)rc(t)i,(5)ratherthantheslope,sincethederivativeasymptotesatmuchearliertimes.Theshort-timediffusivitydeterminedbyBrowniandynamicsisfoundfromEq.( 5 )inthelimitt!0.ItisequaltotheKirkwooddiffusivityandonlyslightlydifferent,by12%,fromthelong-timediffusivity[ 85 ].TheLBsimulationsareinertial,andherelimt!0D(t)=0.Nevertheless,inbothmethodsthediffusivityreachesitsasymptoticvalue,D,overatimeoftheorderoftheZimmtime,tZ=6R3g=T.OurinvestigationsshowthatthediffusioncoefcientinanLBsimulationdependsononlythreeparameters:thesizeoftheperiodicunitcell,L,thegridresolution,x=b, 101

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DiffusioncoefcientsfromLBsimulationswithdifferentunitcellsizes,L.Thecoarsestgridresolution,x=b=2.58,wasusedforcomputationalefciency. andthemethodofinterpolation.SincetheBDresultsareforanisolatedpolymer,theLBdatamustbecorrectedfornite-sizeeffects;hereweuseawell-establishedcorrectionforthediffusioncoefcient[ 68 ],whicheffectivelyeliminatesthedependenceofthediffusioncoefcientontheboxsize.Theself-diffusioncoefcientofasphericalparticleofradiusRinaperiodicsystemwithrepeatlengthLcanbewrittenas[ 68 ], L+4R3 102

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Effectofsystemsizeonthediffusioncoefcientofa10-segmentchain.TheresolutionoftheLBgrid,x=b=2.58.Theparameter=hu21i=c2sisameasureofthetemperatureoftheuid,T=Mhu21i,whereM=x3isthemassofuidinasinglegridcell.DListhediffusioncoefcientfromLBsimulationsandD1isthecorrectedvaluefromEq( 5 );resultsincludingthe(R=L)3correctionareindicatedbyD(3)1. ordercorrectionisindependentofR, 5-2 showtheexpectedlineardependenceonL1,withthesameasymptoticvalueofthepolymerdiffusivity(D=D0=0.1875)fromeitherextrapolation,ttingto4differentsystemsizes(9.420Rg)showsthatthelimiting,lowT

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DiffusioncoefcientsfromLBsimulationswithdifferentgridresolutions,x=b,arecomparedwithBrowniandynamics(dashedline);theverticalbarindicatesthestatisticaluncertaintyintheBDdata. diffusioncoefcient,asplottedinFig. 5-2 ,requiresatemperature48timessmaller,whichtranslatesto48timesmoreprocessingforthesamestatistics.Itispossibletofurtherrenethecorrectionfornite-sizeeffectsbyincludingthenextterminEq.( 5 ),butthisrequiresthepolymersize.Deningthediffusivityoftheisolatedchain,D1=T=6R1,intermsoftheeffectiveradiusR1,andrearrangingEq.( 5 )resultsinacubicequationforx=R1=L, 3x3(2.837+xL)x+1=0,(5)wherexL=RL=L,withDL=T=6RL.Sincetheadditionalcorrectionissmall,Eq.( 5 )canbesolvedforxinafewiterations.Thisleadstoslightlymoreconsistentdiffusioncoefcientsfromthesmallerboxsizes,asshowninnalcolumnofTable 5-2 .ThediffusivitiesinTable 5-1 includetheextracorrectionterm. 104

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2 65 ]suggeststhataratioofx=b12isadequateformostpurposes.Herethediffusioncoefcientof10segmentchainsareshowninFig. 5-3 forarangeofdifferentgridresolutions,0.650), hXp(0)Xp(0)i,(5) 105

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NormalizedautocorrelationfunctionsoftheRouse-modeamplitudes,Cp(t),Eq.( 5 ).Thetwolongestwavelengthmodes(p=1,2),anintermediatewavelength(p=4)andashortwavelength(p=8)areshownforthreedifferentgridresolutions;x=b=2.58(circles),x=b=1.29(squares),andx=b=0.86(triangles);thesolidlinesaretheBrowniandynamicsresults. decaysalmostexponentially 5-4 ;pdenestheRouserelaxationtimeofthepthmode.Thecorrelationfunctionsforthetwonestgridresolution,x=0.65b(notshown)andx=0.86b,agreealmostperfectlywithBrowniandynamics,whileforthecoarsestresolutionx=2.58btherearesignicantdeviationsinthelong-wavelengthmodes.Somewhatsurprisingly,theerrorsinthelessresolvedLBsimulationsdiminishwithincreasingmodenumber,sothatforp=4theresultsforallgridresolutionsareessentiallyindistinguishable.Howeverforstillshorterwavelengthsthediscrepanciesincreaseagainforthecoarsergrids,thistimeintheoppositedirection. 106

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RelaxationtimesoftheRousemodes,p,fromLB(solidcircles)andBD(solidtriangles).ResultsforN=10arecomparedforthreedifferentgridresolutions,x=2.58b(left),x=1.29b(center),andx=0.86b(right);resultsforx=0.65bareindistinguishablefromx=0.86b. Wehaveselectedoneparticulartime,t=12.6t0,wherethep=1modehasdecayedto45%ofitsinitialvalue,foramoredetailedcomparison.ForthethreeresolutionsshowninFig. 5-4 ,thedeviationfromBrowniandynamicsare11%(x=2.58b),5%(x=1.29b),and<0.5%(x=0.86bandx=0.65b),respectively.DatainFig.7ofRef.[ 67 ],takenatasimilartime,showdeviationsbetweenLBandBDoftheorderof2%.Thegridresolutioninthesesimulationscorrespondstohr2i1=21.1x,similartotheintermediateresolutioninourwork,x=1.29borhr2i1=21.2x.Ourresultsshowslightlylargerdeviations,possiblyduetotheshorterchainorthesofterexcludedvolumeforces,bothofwhichemphasizetheshort-rangehydrodynamicinteractions.Forthesystemsizesweused,theO(L3)correctionstotheRouse-moderelaxationtimes[ 67 ]aresmall;simulationswithx=1.29bandL=18.8Rg(insteadofL=9.4Rg)showasimilar(4%)deviationfromBrowniandynamicsatt=12.6t0. 107

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RelaxationtimesoftheRousemodes,p,fromLB(solidcircles)andBD(solidtriangles).ResultsforN=20arecomparedforresolutionsx=2.58b(left)andx=1.29b(right). TheRouse-moderelaxationtimeswerecalculatedfromtheintegraloftheautocorrelationfunctionsCp(t),Eq.( 5 ).Therstportionoftheintegralwascalculatedbynumericalquadrature,uptoatimewherethecorrelationfunctionhaddecayedtolessthan5%ofitsinitialvalue.Tominimizetheeffectofstatisticalerrorsontheintegral,wettedthelastportionofthecorrelationfunctiontoasingleexponentialandcalculatedthelong-timecontributionanalytically.Theoverallintegralisinsensitivetotheexactvalueoftherelaxationtimeofthettedexponential,whichwasdeterminedself-consistentlyfromthevalueofp.SincethedecayofCp(t)followsasingleexponentialalmostexactly,thisprocedureisquiteprecise;weusedthesameprotocolforbothLBandBDcorrelationfunctions.ThedatainFig. 5-5 showthatthelattice-BoltzmannmethodcanreproducethewholeRousespectrumwhensufcientlyresolved.Forthetwonestresolutions,x=

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5-6 wecomparetheRousespectrumforchainswith20segments.Theerrorsintherelaxationtimesaresimilartothoseobtainedfortheshorterchainswiththesamegridresolution. 67 ],throughasystematicinvestigationofvariationsintheLBmodelparameters.Ourresultsshowthathydrodynamicretardation,whichissometimessuggestedtobeareasonfordiscrepanciesbetweenLBandBDresults[ 9 ],isinfacteasilycontrolled;thediffusioncoefcientandRousespectrumareindependentofSchmidtnumberwhenSc0>10.Otherparameterssuchasuidviscosityandbeadmasshavelittleeffectontheresults.Somewhatdisappointingly,ahigher-orderinterpolationoftheuidvelocityelddoesnotleadtoimprovedagreementwithBrowniandynamics.Althoughresultswiththree-pointinterpolationconvergetothesamevaluesaswithlinearinterpolation,theconvergenceisslower,ratherthanfasterasonewouldhavehoped.Despitethesmootherinterpolationoftheoweld,theforceisdelocalizedoveralargervolumeandthisseemstoreducetheaccuracy,whilesimultaneouslyincreasingthecomputationalcost.Thecrucialparameteraffectingtheaccuracyofalattice-BoltzmannsimulationistheresolutionofthepolymerontheLBgrid.WhenthemeandistancebetweenneighboringbeadsismorethantwicetheLBgridspacing,theagreementbetweenLBandBDsimulationsisessentiallyexact.Howeverthecomputationalcostofanegridishigh, 109

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65 ].Theerrorsinthedynamicpropertiesarethenaround5%,whichissufcientformostpurposes.AtypicalLBsimulation,x=1.29b,L=9.4Rg,=0.003,runfor8105t0,requiresabout70hoursofcomputation.ComparableBrowniandynamicssimulationstakeapproximatelyonehour.However,simulationsofconcentratedsolutionsinconnedgeometriesaremorefavorableforlattice-Boltzmannmethods[ 67 ]. 110

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1 2 22 87 ]ofconnedpolymersdrivenbyrectilinearowshaveestablishedtheessentialroleofHIinpolymermigration.Thelocalshearrateextendsthepolymerandgeneratestensioninthechainwhichdisturbsthesurroundingow-eld.Inthevicinityofaboundary,thisdisturbanceowbecomesasymmetricandproducesahydrodynamicliftofthepolymerawayfromthewall.Migrationinuniformshear[ 3 ]andpressure-drivenows[ 4 ]hasbeensimulatedbylattice-Boltzmann[ 2 ]andBrownian-dynamics[ 1 ].Adirectcomparisonoflattice-BoltzmannandBrowniandynamicssimulationsofpolymermigrationshowedsignicantdifferencesintheconcentrationproles[ 9 ].However,inthatworktheLBmethodwasimplementedincorrectly,causingaviolationoftheuctuation-dissipationtheorem(seeSec. 5.3.1 ).Ontheotherhand,staticanddynamicpropertiesofanisolatedpolymercalculatedbyLBandBDdoagree,towithin12%[ 67 88 ].Therelaxationtimesoftheinternaldegreesoffreedomalsomatchovertheentirespectrum,showingthatthehydrodynamicinteractionsareidenticalatbothlargeandsmalllengthscales.ThischapterreportsquantitativecomparisonsbetweenLBandBDsimulationsofaconnedpolymerchaindrivenbyanexternalow.Steadystatedistributionsofthecenterofmassofthepolymeranditsend-to-endvectorarefoundtobeinquantitativeagreementforbothuniformshearandpressure-drivenows.Theseresultsareatvariancewithapreviousstudy[ 9 ]inwhichthetwomethodspredictedqualitativelydifferentmigrationprolesinapressure-drivenow.Inthatstudy,LBsimulationsofthepolymerconcentrationdidnotcapturethecharacteristicdoublepeakobservedinpressure-drivenowandshowedsignicantlylessmigrationthanBDsimulationsatasimilarWeissenbergnumber(Ref.[ 9 ],specicallyFigure4). 111

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13 ]oftheGreen'sfunctionforapointforceconnedbetweentwoparallelplates[ 14 ].Theregularizationaccountsfortheadditionaldisturbanceintheow-eldduetothenite-sizeofthesourceandreceiver;itgeneratesasymmetricandpositive-denitemobilitymatrixforallphysicallyrelevantcongurationsofthepolymer.Thedisturbanceowisdecomposedintoasuperposedsolutionfortheindividualwallsandacorrectionduetomultiplereections.Thesuperposedsolutioniscalculatedexactly,whereasthecomputationallyexpensivecorrectionisinterpolatedfromaprecalculatedlook-uptable. 6.2.1PolymermodelThebead-springmodelofDNAwithFENEspringsandexcludedvolumeinteractionsisdescribedinSec. 5.2 .WeuseN=10tomodelaDNAmoleculewithcontourlengthLC=10.6m.Inaddition,thebeadsexperienceashortrangerepulsionwithanearbywall,basedonthelossofcongurationalentropyofanidealchain, 5 )-( 6 )isusedinboththeLBandBDsimulations. 5.2.2 withtheadditionofthelinkbounce-backrule[ 72 ]todescribetheno-slipconditionatstationaryormovingwallslocatedaty=0andy=H.PeriodicboundaryconditionsareappliedinthexandzdirectionswitharepeatlengthofL.TocomparewithBrowniandynamics,wherethesystemisunconnedindirectionsparalleltothewalls,werequireLH. 112

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14 ]andL>2Hhasbeenfoundtobesufcienttoapproximateasinglepolymerboundedbetweenparallelplates[ 2 ].Inthisworkweusedarangeofvalues2L=H4toeliminateartifactsfromtheperiodicboundaryconditions.ThekeyparameterscharacterizinganinertialsimulationofashearowarethePecletorWeissenbergnumber(R2G=D)andtheReynoldsnumber(R2G=).Theradiusofgyrationofthepolymer,RG,isthecharacteristiclength,Disthepolymerdiffusivity,andistheshearrate.AlargePecletnumbercanbeachievedwhilekeepingtheReynoldsnumbersmalliftheratioSc==Dissufcientlylarge.Inpreviousworkonglobularpolymers[ 88 ]wefoundnoevidenceofinertialeffectsiftheSchmidtnumberwaslargerthan4050.Inordertoallowforhydrodynamicinteractionstodevelopbetweensegmentsofchainsthatareextendedbytheow,wehereuselargerSchmidtnumbers,fromSc300toSc2500.Singlechainpropertiesareinsensitivetothemassofthebead,theuidviscosity,andtheinterpolationmethod;theroleoftheseparametersinaconnedgeometrywasfoundtobenegligible.Thecrucialparameterinobtainingaccurateandconvergentresultsforthedynamicpropertiesisthegrid-resolution,x.Weusedx=2bandx=1.33b,correspondingtoaroot-mean-squarebondlengthhr2i1=20.87xandhr2i1=21.30x.Theeffectivetime-stepisgivenbyt=b2=D0,whereD0=T=(6a)isthemonomerdiffusioncoefcient.Thehydrodynamicradiusa=0.36bischosentomatchreportedvaluesforDNA[ 49 ]. 5.2.1 .Usuallyinsuchsimulations,aregularizedGreen'sfunctionisusedtoensurethatthemobilitymatrixremainspositive-denite.Here,wedescribeanimplementationofaregularizedandsymmetricmobilitymatrixforparticlesconnedbetweentwoparallelno-slipboundaries,withthegeometryillustratedinFig. 6-1 113

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Illustrationofthecongurationofbeadsiandjbetweentwoparallelplates. Theow-eldbetweentwoparallelplatesduetoapointforceFis 14 ].ThisGreen'sfunctioncanbedecomposedintothefree-spaceGreen'sfunction,G1(rr0),correctionsduetoaplanarboundary,GW(r,r0;Y)[ 12 ](Yisthelocationofthewall),andcorrectionsduetomultiplereectionsbetweentheboundariesGR(r,r0jH), 6-1 )duetotheforceactingonallthebeadsis 114

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10 ].TheRPtensorincludingthenear-eldvaluesofrij=rjriis 6a8>>>>>>><>>>>>>>:C1I+C2rijrij 4a rij+1 2a3 4a rij3 2a3 32rij 32rij Thus,themobilitymatrixofapairofconnedparticlesis 6 ),removestheneedforaspecicshort-rangefunctionfortheHIwiththewall.ExpressionsfortheregularizedGreen'sfunctionforasinglewall,LiLjGW(ri,rj;Y),aregiveninRef.[ 13 ].Thecontributionfromhigherreections,LiLjGR,iscalculatedbynumericallyapplyingtheFaxenoperatorstothereectiontermGR,isolatedfromtheLiron-Mochonsolution,Eq.( 6 ),forpointparticles.ExplicitexpressionsforGRareavailableintheliterature[ 89 90 ].ThecomputationtimeforLiLjGRisseveralordersofmagnitudelargerthanthesuperposedmobilitymatrix, 89 ],iscomputedateachtimestep,whilecontributionsfromthemultiplereectionsare 115

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B .TheinterpolatedGreen'sfunctionisnotdivergencefree.Therefore,theLangevinequationdescribingthemotionofthepolymerisintegratedusingFixman'smid-pointalgorithm[ 91 ], 91 ],because,inconjunctionwiththeexcludedvolumeinteraction,Eq.( 6 ),andaboundedrandomnumbergenerator,thecenterofmassofthebeadnevercrossesawall.Theupperbound,zm,oftherandomnumbergeneratorcanbedeterminedfromthemaximumbeaddisplacement yp 116

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DecayofthenormalizedcorrelationofRE(t),X1(t),R2E(t),Rmax(t),andxy(t),Eq.( 6 ).Theslopeofthelinearlydecayingregionsareusedtodeterminetheirrelaxationtimes.TimetismadedimensionlessusingtheRouserelaxationtime,C. viscousrelaxationtime,,obtainedfromtheautocorrelationofthepolymerstress,xy,andthelongestconformationalrelaxationtime,C,obtainedfromtheautocorrelationoftherstnormalmode,X1,ortheend-to-endvector,RE.WithintheRouseandZimmapproximations,C=2.SinglemoleculeexperimentswithDNA[ 32 ]suggestthatcanbeobtainedbyttingtheasymptoticdecayofthemaximumvisiblestretch,hRmax(t)ihRmax(1)i,toasingleexponential.FromBDsimulationsofasinglepolymermoleculeatinnitedilution,wecomputedthenormalizedautocorrelationfunctions, 117

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6-2 byasingleexponential.OurresultsconrmthatX1REandalsothatR2ERMaxxy.ContrarytotheRouseandZimmapproximations,wendC=2.6.ThedeviationfromRouseandZimmtheoryisprimarilyduetotheexcludedvolumeinteractionbetweenthepolymerbeads.ForanidealharmonicchainC=2,butCcanincreaseto5whenincludingexcludedvolumeandniteextension[ 92 ].Inourpresentsimulations,weuse=0.38CasthecharacteristicrelaxationtimedeningtheWeissenbergnumber. 1 3 36 ].Thepolymerchainstretchesandorientsitselfinthepresenceofthelocalshearrateandthedisturbanceeldreectedfromano-slipboundarythendrivesitawayfromthewall.Thesteady-statedistributionofthepolymeracrossthechannelisthusameasureofthehydrodynamicinteractionsinthesystem.Figure 6-3 showsthecenterofmassdistributionobtainedfromLBandBDsimulationsatfourdifferentvaluesoftheWeissenbergnumber.Toobtainstatisticallyprecisedata,eachproleisaveragedoveratimeofapproximately1000H;hereH=H2=D=3000t0isthetimerequiredbythepolymerchaintodiffuseacrossthechannelofwidthH=10RG,andt0=6a=.ProlesfromBDsimulationsweregeneratedusingboththesuperposedGreen'sfunction,Eq.( 6 ),andthefullGreen'sfunction,Eq.( 6 ).Theeffectofmultiplereectionsissignicantathighershearrates;forinstanceatWi=76thepeakinthedistributionisincreasedby33%comparedtothesuperpositionapproximation.ResultsfromLBsimulationsateachWinumbershowanidenticaldepletionlayertotheBDresultsobtainedfromthecompleteGreen'sfunction. 118

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Centerofmassdistributionofaconnedpolymerdrivenbyuniformshearow.ResultsforLBandBDsimulationsarecomparedforfourvaluesoftheWeissenbergnumber,Wi=;thechannelwidthisH=10RG 6-3 usedatime-stept4104t0,aperiodicboxlengthL=3H,andagridresolutionx=2b.Typically,xb[ 65 ]isrequiredtoaccuratelycapturehydrodynamicinteractionswithinthepolymer,buthereacoarsergridresolutionissufcienttoobtainthecorrectdistributionofthecenterofmass,whichisdominatedbyHIonthelengthscaleofthepolymer.Themeanstretchofthepolymermoleculeisgivenbyitsend-to-endvectorandtheresultsobtainedfromthetwomethodsforWi=9.5areplottedinFig. 6-4 ;comparisonsatotherWishowsimilartrends.ResultsfromLBsimulationsindicateadependenceontheeffectivetime-stepsize,t,andthegrid-resolution,x.Byreducingthetime-step 119

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Distributionoftheend-to-endvectoracrossthechannelforasingleconnedpolymerchaindrivenbyuniformshearowiscomparedforLBandBDsimulationsforWi=9.5.LBresultsareobtainedfortwovaluesofgridresolution:x=b=2.00and1.33andthreevaluesofthetime-step. 120

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Distributionofthecenterofmassforaconnedpolymerinapressure-drivenow.ResultsforLBandBDsimulationsarecomparedforfourvaluesoftheWeissenbergnumber,Wi=. 93 ],resultinginoff-centerpeaksandacentraldipinthepolymerconcentrationprole.ThedistributionprolesfromLBandBDareagaininquantitativeagreement,asshowninFig. 6-5 .ThemaximumdeviationsoccuratWi=38;3%attheoff-centerpeaksand4%atthecentraldip.AtthehighestWeissenbergnumber,Wi=76,theow-rateandtemperaturewerereducedbyafactorof4.TheincreaseinSchmidtnumberisneededtoallowthehydrodynamicinteractionstopropagateoverthefullyextendedchain. 121

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9 ],whichincorrectlysuggestaqualitativedisagreementbetweenLBandBDsimulationsofaconnedpolymerinapressure-drivenow.PolymerextensionismoresensitivetothesmallscaleHI.Thepolymerisstretchedbytheshear,andthetimetakenforadisturbanceintheow-eldtopropagatetoneighboringbeadsislarger.Convergentresultsforthepolymerstretchrequireatwotofourtimessmallertime-stepthancomparablesimulationsofacollapsedpolymer.Inadditionanergridisneededtoresolvethebalancebetweenthetensioninthepolymer,theshear,andtheBrownianforce.Asxandtbecomesmall,convergentresultsareobtainedforthepolymerstretchaswell.BrowniandynamicssimulationsdependonanaccurateGreen'sfunctionfornite-sizebeadsbetweentwoparallelplates.ThemobilitymatrixcomputedusingEq.( 6 )issymmetricandpositive-deniteforallaccessiblecongurationsofthepolymer.ThesuperposedGreen'sfunctionoffersacomputationallyfasteralternativeforlowervaluesofWi;theerrorsintheconcentrationprolearelessthan5%.However,forWi40,thesuperposedsolutionisinerrorbymorethan25%andreectionsarenecessarytoproperlyaccountfortheHI. 122

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21 50 ].ThepresenceofHIinducedbyanelectriceldalsoexplainsunansweredobservationscross-streammigrationofDNAdrivenbyacombinationofelectriceldandpressuregradient[ 5 8 ].TodemonstratetheeffectsofelectrophoreticallyinducedHIinrealsystems,Iemployedkinetictheoryofanelasticdumbbelltoinvestigatecross-streammigrationwhendrivenbyacombinationofowandexternalforce.Theoreticalcalculations,that 123

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5 8 ]despitereducedconformationaldegreesoffreedomofthedumbbell.Toimprovethepredictionsforlateralmigration,amulti-beadmodelofDNAwhichincludeslong-rangeelectrohydrodynamiceldswassimulatedusingtheBrownian-dynamicstechnique.Remarkably,theresultsfromBDsimulationsagreequantitativelywithexperiments[ 5 8 ]withoutusinganyadjustableparameters.Inagreementwiththeexperiments,thesimulationsalsocapturethecontour-lengthdependenceofcross-streammigration.Thismodeladvancespreviouscoarse-grainedDNAmodelsbyincludingcounterionswithinameaneldapproximation.Long-rangeowdisturbancesduringelectrophoresisofthisblobarethecumulativeeffectofallmonomerswithinablobwithanassumptionthateachmonomerislocatedattheblob'scenter.Animportantaspect,alsohighlightedbythesimulations,isthatthefar-eldowdisturbancesaredominantwhentheDebyelayerissignicantlylargerthanthemonomerradius(whichismuchsmallerthanthehydrodynamicradiusoftheblob).ThesaltconcentrationinthesolutiondeterminesthesizeoftheDebyelayerandisidentied 124

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21 ]demonstratesthedependanceofelectrophoreticmobilityofDNAonitsinstantaneousconformation;theirexperimentsshowthatastretchedDNAmoleculehashigherelectrophoreticvelocitythanacoiledmolecule.Second,apolyelectrolytemoleculeispredictedtoswelltoasmuchastwiceitssizeinthepresenceoflargeelectriceldsduetointernaldispersionofitssubunits.Measuringthestaticanddynamicpropertiesofasinglemoleculeisacumbersomeexperiment.Insteadthepresenceofanoptimumelectriceldcorrespondingtomaximumcross-streammigrationataxedowratewouldconrmtheswellingphenomena.IndeedpreliminaryexperimentalresultspresentedinthisdissertationsuggestthattheextentofmigrationofDNAincreasesnonlinearlyastheelectriceldisincreasedandattainssaturation.Experimentsatevenhigherelectriceldsarerequiredtoconrmthedecreaseinmigration.Thisthesisalsoenhancesourcondenceinnumericalsimulationsofneutralpolymersbyperformingbenchmarkcalculationsthatcomparethestaticanddynamicpropertiesof-DNA.Twocontrastingmethods,theimplicitsolventBrownian-dynamicsandexplicitsolventlattice-Boltzmann,areshowntoquantitativelyagreeforanidenticalpolymermodelthroughasystematicinvestigationofallmodelparameters.Comparisonforstaticanddynamicspropertiesofasinglechaininisolationandcross-streammigrationinstronglyconnedchannelsclarifythemisconceptionsbasedontheresultsofRef.[ 9 ].Theirresultsincorrectlyimplyinability,ofatleastoneofthesetwo 125

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9 ],isreinforcedbyourresults.Themostimportantcontributionofthisdissertationisthatitoffersanewmechanismtocontrolpolyelectrolytemotioninfreesolution.Currently,separationofDNAstrandsaccordingtolengthisimplementedusinggelelectrophoresis[ 94 95 ]basedonthenotionthatnofar-eldhydrodynamiceldsexistduringDNAelectrophoresis.ThisnotionhasbeensuccessfullychallengedinthisdissertationandwarrantsfurtherinvestigationasameanstoDNAseparation.TheabilitytodevelopamicrouidicdevicethatusesacombinationofelectriceldandowtoseparateDNAwouldhavenumerousadvantagessuchasreductioninpostprocessingofDNAandomissionofsievingagents[ 96 97 ].ThelackofanyspecializedequipmentrequiredforDNAseparationdevicewouldhavefar-reachingconsequences.Otherpotentialapplications[ 98 100 ]basedoncross-streammigrationofDNAcouldincludeDNAbiosensorofDNAbiochipswhichmonitorinteractionsbetweenanalyatesandDNAonindividualhybridizationspotslocatedonthesurfaceofthedevice.TheabilitytodirectDNAmoleculestowardsthewallbyapplyinganelectriceldinreversedirectiontoowwouldprovideimprovedcontrolandsensitivityinthesemicrodeviceswithoutsignicantlyincreasing[ 101 ]theircomplexity.Theresultsinthisthesisprovideastrongfoundationforfutureworkintheeldofpolyelectrolytedynamics.Theshort-termgoalsbasedonthisstudymustfocuson 126

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127

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@rG(r,rc)r=rcq @r0G(rc,r0)r0=rc,(A)wheretheminussignisassociatedwithbead1andtheplussignwithbead2.ThewallcontributionstothehydrodynamicmobilitiesdenedinEqs.( 2 )-( 2 )arethencalculatedtolinearorderinq: @r0G(rc,r0)r0=rc, @rG(r,rc)r=rc. Inwhatfollows,weonlyconsidertheGreen'sfunction,G0,foraplanarwalllocatedaty=0.TheGreen'sfunctionfortwowalls,Eq.( 2 ),isthenobtainedbysuperpositionG=G0+Gh.IfwedenethevectorRfromtheimagesource,locatedatr02y0ey,totheeldpointr, 128

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8 8 8RyRy Thefunctionsg()is+1forthecomponents=xand=z,and1when=y.Whenthesourceandeldpointsarebothatthecenterofmassofthedumbbell,R=2ycey,theStokeslet,dipole,anddoubletare: 16yc(+yy), 64y3c(+yy), 32y2c(+yy), wherethefollowingrelationswereusedtosimplifytheKroneckerdeltafunctions: CombiningtheresultsinEqs.( A )-( A ),weobtainthecontributiontotheKirkwoodmobility,DK=kBT,fromthewallath=0, 32yc(I+eyey).(A)ThegradientsofG0canbecalculatedusingthetranslationalinvarianceoftheGreen'sfunctionwithrespecttothelocationoftheeldpointandtheimagesource 129

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A )), @r=@ @R=r, @r0=sg()r. Theremainingnon-zeromobilitymatrices,GandGarethen, TheexpressionforG0containstwoadditionalterms,8ycqyPD(2ycey)+4qySD(2ycey),buttheyexactlycancel,ascanbeseenfromEqs.( A )and( A ).FinallyweneedtocalculatethegradientsoftheStokeslet,dipole,anddoubletterms,againusingtheresultsofEqs.( A )and( A ): 32y2c(y+y+y3yyy), 128y4c(y+yyyyy), 32y3c(y+y2y). TheseresultscanbecombinedtogivetheexpressionsforG0andG0: 32y2c[(I+eyey)qyeyq+qey], 32y2c[(I+eyey)qy+eyqqey]. Aftersuperposingthecontributionfromtheupperwall,weobtaintheexpressionsforGandGgiveninEqs.( 2 )and( 2 ).InfactGandGarethematrixtransposeofoneanother.TheresultsforGKandGagreewiththeresultsofMaandGraham.[ 36 ] 130

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6 ),takesseveralordersofmagnitudelongerthanthesuperpositionapproximation,Eq.( 6 ).Thus,thecontributionduetomultiplereectionsbetweentheplates,Rij,=LiLjGR(ri,rjjH),ispretabulatedandinterpolatedduringthesimulations;notethattheindexesiandjnumberthebeadswhileandarethecomponents.Thelook-uptableisconstructedonaCartesiangridwithNy+1gridpointsinthedirectionperpendiculartothewall(y)andNx+1gridpointsinthedirectionsparalleltothewall(xandz).Thegrid-spacingsarey=H=Nyandx,z=LC=NxwhereLCisthecontourlengthofthepolymer.Themobilitymatrix,Rij,,isafunctionofthefourdimensionalvectorRij=[yi,rij],andeachelementinthelook-uptablecanbeidentiedbyasetoffourintegerindexes,=(ks,kx,ky,kz): 131

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Illustrationofsymmetrybasedonthecenterofthechannel.Thedisturbanceintheow-eldatrijrelativetoasourcelocatedatyiisidenticaltothedisturbanceatrklrelativetoasourcelocatedatyk=Hyi. Thecomputationalcostofconstructingalook-uptablewith(2Nx+1)2(Ny+1)2elementscanbesignicantly(1=32)reducedbyexploitingthesymmetriesinRij,.First,thesymmetryofthemobilitymatrixisused, B-1 ,duetoasourcelocatedatyiisidenticaltotheow-eldatrlduetoasourcelocatedatyk=Hyi,providedthatrkl=rij.Thus, B-2 .OnaCartesiangridthereareeightequivalentlocationsindicatedintheTopViewofFig. B-2 .TheelementsoftheHImatrixattheselocationscanbeobtainedbyswapping 132

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Illustrationofcylindricalsymmetryinthesystem.Thesourcebeadislocatedatyi.Thedisturbanceeldatthereceiver(solidlineintopview)isrelatedtotheindicatedlocations(dashedlines)throughsimplelinearoperationsgivenbyEqs.( B )and( B ). elementsandsigns: 133

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6 .Asparsergrid(orxy)canbeusedinthedirectionparalleltothewallbecausethepolymerisextendedinthatdirectionbytheshear. 134

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RahulKekrewasborninMumbai,Indiain1984toJyotsnaandAnoopMukundKekre.HeattendedtheAtomicEnergyCentralSchoollocatedinthetownshipoftheCenterforAdvancedTechnology,Indorefrom1989-2001.HeisarecipientoftheNationalTalentSearchExamination1999scholarshipofferedbyGovernmentofIndiaandwasselectedfortheRegionalroundofthePhysicsOlympiadconductedbytheIndianAssociationofPhysicsTeachers.RahuljoinedtheChemicalEngineeringdepartmentatIndianInstituteofTechnology,Bombay(IITB)in2001.Hegraduatedwithabachelor'sdegreein2005withastronginterestinmathematicalmodeling.Aftercompletinghisdegree,RahulworkedasaSoftwareEngineeratHeadstrongPrivateLimitedlocatedinBangaloreforoneyear.In2006,RahulacceptedanofferfromChemicalEngineeringdepartmentatUniversityofFloridatopursueaPhDdegree.HejoinedtheresearchgroupledbyProfessorTonyLaddandProfessorJasonButlerandhisresearchprojectfocusedonthedynamicbehaviorofpolyelectrolytesduringelectrophoresis.HewasawardedtheGraduateAlumniFellowshipbytheChemicalEngineeringdepartmenttopursuehisDoctoraldegree.RahulplanstojointheIntelfacilitylocatedatHillsboro,Oregonafternishinghisgraduatestudies. 140