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Dynamics and Rheology of Particulate Suspensions Undergoing Steady and Unsteady Shear Flows

Permanent Link: http://ufdc.ufl.edu/UFE0041243/00001

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Title: Dynamics and Rheology of Particulate Suspensions Undergoing Steady and Unsteady Shear Flows
Physical Description: 1 online resource (100 p.)
Language: english
Creator: Park, Hyun-Ok
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

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Subjects / Keywords: 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: Suspension rheology has applications in fields from cutting edge nano-technology to biotechnology as well as existing industrial processes. Suspension systems exhibit interesting behaviors under flows such as shear-thinning and shear-thickening, but the dynamics of many systems are not well-understood. In this work, I introduce two examples of well-defined, even simple, systems where the dynamics and the rheology surprisingly remain mysterious. In addition, the rheology of suspensions of single-walled carbon nanotubes is studied. The experiments are complimented by Stokesian dynamics simulations for the non-colloidal suspensions of spheres in unsteady shear flow. The rheology of rigid rod suspensions in steady shear flow is experimentally investigated. The work presented in this dissertation provides a significant contribution towards generating a more comprehensive view of rheology of particulate suspensions undergoing steady and unsteady flows. Moreover, the work demonstrates the potential for rheology to be used as a quantitative tool, rather than simply a qualitative one. First, the dynamics of a suspension of rigid rods in steady shear flow is studied. Theories predict a steady shear viscosity that is independent of shear rate for a non-colloidal suspension of rods. However, unexpected shear-thinning behaviors are observed, although a well-defined suspension system is used. The shear thinning behaviors become stronger with increasing volume fraction and aspect ratio of the rods. Possible mechanisms such as flocculation are explored. Through direct comparison of processed spheres dispersed in an identical suspending liquid as the rod suspension, flocculation mechanisms are ruled out. Based upon a recent study of Park (2009), residual weak Brownian torque is argued to cause net migration toward the center in a torsional flow, resulting in the observed shear thinning behaviors. Secondly, the stability and rheology of the single walled carbon nanotube (SWNT) suspensions prepared by interfacial trapping method is examined and compared to a conventional method of ultracentrifugation. Since the rheological properties are sensitive to the suspension microstructure, the change of rheological properties, such as viscosity, can be employed as a systematic standard of stability. The steady shear viscosities have been measured and compared as a function of shear rate and aging time of the suspension. Also, the visual states of the suspension have been observed. The rheology of the SWNT suspensions depends on the preparation of surfactant solution. Also, the interfacial trapping method generated similar behaviors to the SWNT suspension prepared by the ultracentrifugation method. Finally, the dynamics of non-colloidal spheres in oscillatory shear flow is studied by experiment and simulation. Two distinct scales are observed for the development of the rheology in time. At small total strains, a rapid decay of the apparent elastic component of the viscosity is observed, while the apparent viscous component of the viscosity and the complex viscosity remain constant. However, the evolution of the complex viscosity is observed over large total strains, indicating microstructural changes over long times. Also, this suspension system shows a non-monotonic dependence of viscosity on strain amplitude. Stokesian dynamics simulations are used to correlate the rheology and microstructures.
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 Hyun-Ok Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Butler, Jason E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041243:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041243/00001

Material Information

Title: Dynamics and Rheology of Particulate Suspensions Undergoing Steady and Unsteady Shear Flows
Physical Description: 1 online resource (100 p.)
Language: english
Creator: Park, Hyun-Ok
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: 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: Suspension rheology has applications in fields from cutting edge nano-technology to biotechnology as well as existing industrial processes. Suspension systems exhibit interesting behaviors under flows such as shear-thinning and shear-thickening, but the dynamics of many systems are not well-understood. In this work, I introduce two examples of well-defined, even simple, systems where the dynamics and the rheology surprisingly remain mysterious. In addition, the rheology of suspensions of single-walled carbon nanotubes is studied. The experiments are complimented by Stokesian dynamics simulations for the non-colloidal suspensions of spheres in unsteady shear flow. The rheology of rigid rod suspensions in steady shear flow is experimentally investigated. The work presented in this dissertation provides a significant contribution towards generating a more comprehensive view of rheology of particulate suspensions undergoing steady and unsteady flows. Moreover, the work demonstrates the potential for rheology to be used as a quantitative tool, rather than simply a qualitative one. First, the dynamics of a suspension of rigid rods in steady shear flow is studied. Theories predict a steady shear viscosity that is independent of shear rate for a non-colloidal suspension of rods. However, unexpected shear-thinning behaviors are observed, although a well-defined suspension system is used. The shear thinning behaviors become stronger with increasing volume fraction and aspect ratio of the rods. Possible mechanisms such as flocculation are explored. Through direct comparison of processed spheres dispersed in an identical suspending liquid as the rod suspension, flocculation mechanisms are ruled out. Based upon a recent study of Park (2009), residual weak Brownian torque is argued to cause net migration toward the center in a torsional flow, resulting in the observed shear thinning behaviors. Secondly, the stability and rheology of the single walled carbon nanotube (SWNT) suspensions prepared by interfacial trapping method is examined and compared to a conventional method of ultracentrifugation. Since the rheological properties are sensitive to the suspension microstructure, the change of rheological properties, such as viscosity, can be employed as a systematic standard of stability. The steady shear viscosities have been measured and compared as a function of shear rate and aging time of the suspension. Also, the visual states of the suspension have been observed. The rheology of the SWNT suspensions depends on the preparation of surfactant solution. Also, the interfacial trapping method generated similar behaviors to the SWNT suspension prepared by the ultracentrifugation method. Finally, the dynamics of non-colloidal spheres in oscillatory shear flow is studied by experiment and simulation. Two distinct scales are observed for the development of the rheology in time. At small total strains, a rapid decay of the apparent elastic component of the viscosity is observed, while the apparent viscous component of the viscosity and the complex viscosity remain constant. However, the evolution of the complex viscosity is observed over large total strains, indicating microstructural changes over long times. Also, this suspension system shows a non-monotonic dependence of viscosity on strain amplitude. Stokesian dynamics simulations are used to correlate the rheology and microstructures.
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 Hyun-Ok Park.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Butler, Jason E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041243:00001


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D YNAMICSANDRHEOLOGYOFPARTICULATESUSPENSIONSUNDERGOING STEADYANDUNSTEADYSHEARFLOWS By HYUN-OKPARK ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

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c r 2010 Hyun-OkPark 2

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T omyhusband,YuKyoum 3

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A CKNOWLEDGMENTS Thisdissertationbenetedtremendouslyfrommycommittee.Iamtrulyhonored thatIhavelearnedfromthebestcommitteemembers.Firstofall,Iwouldliketothank myadvisor,Dr.JasonE.Butler.Iamtrulyindebtedforhisadvice,encouragement, andtremendoussupportheprovidedoveryears.Icannotndwordstoexpressmy wholeheartedappreciationforhim.Next,IextendthankstoDr.KirkJ.ZieglerandDr. Z.HughFanfortheirhelpfuladviceanddirectionsandDr.AnujChauhanwhohasbeen reallyencouragingandsupportiveinnumerousways. IamalsoverygratefulthatIhaveworkedwiththebestcolleagues,O.BerkUsta, JoontaekPark,RahulKekre,andCarlosSilvera-Batista.Iextendspecialthanksto JonathanBricker,whohasbeenfriendly,supportiveandhelpfulasaseniorlabmember. Finally,Ideeplythankmyfamilyandfriends.Iwouldliketoproposeanaltoast toYuKyoumKim,myhusband,friend,andpartner.Icouldnothaveovercomeallthe hurdlesIhaveencounteredthroughoutmydoctoralstudieswithouthissupportandlove. 4

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T ABLEOFCONTENTS page A CKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................7 LISTOFFIGURES .....................................8 ABSTRACT .........................................11 CHAPTER 1RHEOLOGYOFPARTICULATESUSPENSIONS .................13 1.1Introduction ...................................13 1.2SuspensionsofSphericalParticles ......................15 1.3SuspensionsofRigidRodParticles ......................18 2RHEOLOGYOFSEMI-DILUTESUSPENSIONSOFRIGIDPOLYSTYRENE ELLIPSOIDSATHIGHPECLETNUMBERES ...................20 2.1Introduction ...................................20 2.2Experiments ..................................21 2.3Results .....................................25 2.4Discussion ...................................32 2.4.1RateDependentRheology .......................32 2.4.2Scalingsof r atHigh Pe r .......................37 2.5Conclusions ...................................39 3RHEOLOGYANDSTABILITYOFSINGLE-WALLEDCARBONNANOTUBES .41 3.1Introduction ...................................41 3.2Experiments ..................................44 3.3ResultsandDiscussion ............................47 3.3.1SDS-DispersedSWNTSuspensions .................48 3.3.2RheologyofGumArabicSurfactantSolution .............49 3.3.3GA-DispersedSWNTSuspensionsasaControl ...........51 3.3.3.1GA-dispersedSWNTsuspensionspreparedbyinterfacial trapping ............................53 3.3.3.2GA-dispersedSWNTsuspensionpreparedbyultracentrifugation process ............................56 3.4Conclusions ...................................58 4RHEOLOGYOFOSCILLATINGSUSPENSIONSOFNON-COLLOIDALSPHERES ATSMALLANDLARGETOTALSTRAINS .....................61 4.1Introduction ...................................61 4.2Experiments ..................................62 5

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4.3 ResultsandDiscussion ............................64 4.3.1StressResponsesatSmall N .....................64 4.3.2ApparentViscositiesatSmall N ....................67 4.3.3ApparentViscositiesatLarge N ....................70 4.4Conclusions ...................................74 5STOKESIANDYNAMICSSIMULATIONSFOROSCILLATINGSUSPENSIONS OFNON-COLLOIDALSPHERESATSMALLTOTALSTRAINS .........76 5.1Introduction ...................................76 5.2RheologySimulations:StokesianDynamicsSimulations .........76 5.3PreliminaryResultsandDiscussion .....................79 6CONCLUSIONS ...................................83 6.1Accomplishments ................................83 6.2SuggestedFutureWork ............................85 APPENDIX:TheStressofaSlenderBodyinDiluteSuspension ...........88 REFERENCES .......................................92 BIOGRAPHICALSKETCH ................................100 6

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LIST OFTABLES T able page 2-1 Summaryofthecharacteristiclengthscales,volumeperparticle,andsurface areaperparticleofpolystyreneparticleswithaspectratiosof L = d =1,4,and7. 24 3-1AgingeffectoftheGA-dispersedSWNTsuspensionpreparedbyultracentrifugation with =0.017mg/mL. .................................59 A-1ComparisonoftheviscositiesoftheSDS-dispersedSWNTsuspensionsprepared byultracentrifugation. ................................90 A-2ComparisonoftheviscositiesoftheGA-dispersedSWNTsuspensionsasa control. ........................................90 A-3ComparisonoftheviscositiesoftheGA-dispersedSWNTsuspensionsprepared byinterfacialtrapping. ................................90 7

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LIST OFFIGURES Figure page 1-1 Scanningelectronmicrographofbloodcellsandanimageoftobaccomosaic virus ..........................................14 1-2ComparisonofwovenKevlarfabricwithoutandwithimpregnationofashear thickeningsuspensionafteraballistictest .....................15 1-3Anelectrorheologicalsuspensionwithoutanelectriceldandwithanelectric eld ..........................................16 1-4Thesteadystatespatialandorientationaldistributionofbersat r =0andat r> 0 .........................................18 2-1Scanningelectronmicrographsofpolystyrenespheresbeforeprocessingand ellipsoidswithaverageaspectratiosof 4 and 7. ...............21 2-2Normalizednumberdistributionsforellipsoidswithanaverageaspectratioof 4 and 7. .....................................25 2-3Relativeviscosityasafunctionoftimeandshearrateforpolystyreneellipsoids suspendedinapolyalkyleneglycol/water/KClmixture. ..............26 2-4Relativeviscosityasafunctionoftimeandvolumefractionforpolystyrene ellipsoidswithanaspectratioof4suspendedinapolyalkyleneglycol/water/KCl mixture. ........................................27 2-5Relativeviscosityasafunctionoftimeandvolumefractionforpolystyrene ellipsoidswithanaspectratioof7suspendedinapolyalkyleneglycol/water/KCl mixture. ........................................28 2-6Relativeviscosityasafunctionofshearrateandvolumefractionforpolystyrene ellipsoidswithanaspectratioof4suspendedinapolyalkyleneglycol/water/KCl mixture. ........................................29 2-7Relativeviscosityasafunctionofshearrateandvolumefractionforpolystyrene ellipsoidswithanaspectratioof7suspendedinapolyalkyleneglycol/water/KCl mixture. ........................................30 2-8Comparisonoftherelativeviscositymeasuredinthecone-and-plateandparallel-plate geometry .......................................31 2-9SEMimagesofprocessedpolystyrenespheresbeforeandafterwashing. ...34 2-10Relativeviscosityasafunctionofshearrateandaspectratiofordifferentsuspension systems. .......................................36 8

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2-11 Steadystaterelativeviscosityat Pe r =10 6 asafunctionofvolumefraction andaspectratio. ...................................38 2-12Steadystaterelativeviscosityat Pe r =10 4 and 10 6 asafunctionofdimensionless numberdensityandaspectratio. ..........................39 3-1TEMimageofaSWNTbundle. ...........................42 3-2OverallprocessofremovingSWNTbundlesfromaqueoussuspensionsvia liquid-liquidinterfaces. ................................45 3-3TheSDS-dispersedSWNTsuspensionpreparedbytheultracentrifugation process. ........................................46 3-4SteadyshearviscositiesoftheSDS-dispersedSWNTsuspensionasafunction ofconcentrationandshearrate. ..........................47 3-5RheologicalbehaviorsofGAsolutions .......................48 3-6PicturesofGAsolution. ...............................49 3-7Acontrolsample ...................................50 3-8Steadyshearviscositiesofthecontrolsamples ..................51 3-9Aftersteadysheartest,aggregatesofSWNTsareformedinthecontrolsample. 52 3-10Steadyshearviscositiesofacontrolsample ....................53 3-11Steadyshearviscositiesofacontrolsampleareplottedasafunctionofaging timeandshearrate. .................................55 3-12TheGA-dispersedSWNTsuspensionpreparedbyinterfacialtrappingprocess with c =0.026mg/mL. .................................56 3-13Steadyshearviscositiesareplottedasafunctionofshearrateandconcentration fortheGA-dispersedSWNTsuspensionpreparedbytheinterfacialtrapping technique. .......................................57 3-14DistributionofthediameterofSWNTsintheGA-dispersedSWNTsuspension preparedbytheinterfacialtrappingprocess. ....................58 3-15SteadyshearvaluesoftheshearviscositiesoftheGA-dispersedSWNTsuspensions producedafterhomogenizationandultrasonication(control),interfacialtrapping, andultracentrifugation. ...............................59 4-1Stressresponsesofthesuspensionsduringthe15 th oscillation( N =15). ...65 4-2Theapparentvaluesof 00 and 0 plottedasafunctionofnumberofoscillations forthesuspensionofvolumefraction =0.4. ...................66 9

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4-3 Theapparentvaluesof 00 and 0 plottedasafunctionofnumberofoscillations N forthesuspensionofvolumefraction =0.2. ..................68 4-4Comparisonofpreshearedsuspensionstosuspensionshavinganinitially randomcongurationoftheparticles. .......................69 4-5Apparentcomplexviscosityversustotalstrain ...................71 4-6Comparisonofthecomplexviscositiesasafunctionof A= H ...........72 4-7Thecomplexviscosityatlargetotalstrainsasafunctionofstrainamplitude A= H forvolumefractions rangingfrom0.20to0.50. ..............73 5-1Theperiodiccellusedinthesimulations. .....................77 5-2Thestrain-stressresponse .............................79 5-3Theapparentvaluesof 00 and 0 plottedasafunctionofnumberofoscillations forthesuspensionofvolumefraction =0.4. ...................80 5-4Comparisonofsuspensionswithdifferentinitialcongurations. .........81 A-1ComparisonofviscositiesoftheSDS-dispersedSWNTsuspensions. .....91 10

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Abstr actofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DYNAMICSANDRHEOLOGYOFPARTICULATESUSPENSIONSUNDERGOING STEADYANDUNSTEADYSHEARFLOWS By Hyun-OkPark May2010 Chair:JasonE.Butler Major:ChemicalEngineering Suspensionrheologyhasapplicationsineldsfromcuttingedgenano-technology tobiotechnologyaswellasexistingindustrialprocesses.Suspensionsystemsexhibit interestingbehaviorsunderowssuchasshear-thinningandshear-thickening,but thedynamicsofmanysystemsarenotwell-understood.Inthiswork,Iintroduce twoexamplesofwell-dened,evensimple,systemswherethedynamicsandthe rheologysurprisinglyremainmysterious.Inaddition,therheologyofsuspensions ofsingle-walledcarbonnanotubesisstudied.Theexperimentsarecomplimented byStokesiandynamicssimulationsforthenon-colloidalsuspensionsofspheresin unsteadyshearow.Therheologyofrigidrodsuspensionsinsteadyshearowis experimentallyinvestigated. Theworkpresentedinthisdissertationprovidesasignicantcontributiontowards generatingamorecomprehensiveviewofrheologyofparticulatesuspensions undergoingsteadyandunsteadyows.Moreover,theworkdemonstratesthepotential forrheologytobeusedasaquantitativetool,ratherthansimplyaqualitativeone. First,thedynamicsofasuspensionofrigidrodsinsteadyshearowisstudied. Theoriespredictasteadyshearviscositythatisindependentofshearratefora non-colloidalsuspensionofrods.However,unexpectedshear-thinningbehaviors areobserved,althoughawell-denedsuspensionsystemisused.Theshearthinning behaviorsbecomestrongerwithincreasingvolumefractionandaspectratiooftherods. 11

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P ossiblemechanismssuchasocculationareexplored.Throughdirectcomparisonof processedspheresdispersedinanidenticalsuspendingliquidastherodsuspension, occulationmechanismsareruledout.BaseduponarecentstudyofPark(2009), residualweakBrowniantorqueisarguedtocausenetmigrationtowardthecenterina torsionalow,resultingintheobservedshearthinningbehaviors. Secondly,thestabilityandrheologyofthesinglewalledcarbonnanotube(SWNT) suspensionspreparedbyinterfacialtrappingmethodisexaminedandcomparedto aconventionalmethodofultracentrifugation.Sincetherheologicalpropertiesare sensitivetothesuspensionmicrostructure,thechangeofrheologicalproperties,such asviscosity,canbeemployedasasystematicstandardofstability.Thesteadyshear viscositieshavebeenmeasuredandcomparedasafunctionofshearrateandaging timeofthesuspension.Also,thevisualstatesofthesuspensionhavebeenobserved. TherheologyoftheSWNTsuspensionsdependsonthepreparationofsurfactant solution.Also,theinterfacialtrappingmethodgeneratedsimilarbehaviorstotheSWNT suspensionpreparedbytheultracentrifugationmethod. Finally,thedynamicsofnon-colloidalspheresinoscillatoryshearowisstudiedby experimentandsimulation.Twodistinctscalesareobservedforthedevelopmentofthe rheologyintime.Atsmalltotalstrains,arapiddecayof 00 isobserved,while 0 and remainconstant.However,theevolutionofthecomplexviscosityisobservedoverlarge totalstrains,indicatingmicrostructuralchangesoverlongtimes.Also,thissuspension systemshowsanon-monotonicdependenceofviscosityonstrainamplitude.Stokesian dynamicssimulationsareusedtocorrelatetherheologyandmicrostructures. 12

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CHAPTER 1 RHEOLOGYOFPARTICULATESUSPENSIONS 1.1Introduction Particulatesuspensionsareubiquitous,withexamplesinbiologicalsystemssuch asblood[ 19],householdgoodssuchaspaints,andindustrialprocessingsuchaswaste slurries(seeFig. 1-1).Thereforeunderstandingthedynamicsofparticulatesuspensions isimportantforefcientcontrolandprocessing.Althoughmanystudieshavebeen conducted,thedynamicsstillremainmysteriousandcontinuetosurpriseus. Therheologyofparticulatesuspensionsdifferswidelywithrespecttotheindividual typesofparticlesandsuspendingliquidsandinteractionsamongthem.Ingeneral, particleadditiontothesuspendingliquidenhancesthesuspensionviscosity[ 8, 38, 39]. ForadilutesuspensionofrigidspheresinaNewtonianuid,Einstein[ 38 ]predicteda suspensionviscosityof = o (1+2.5), (1) where issuspensionviscosity, o istheviscosityofthesuspendingliquid,and is thevolumefraction.Also,BatchelorandGreen[ 8]obtainedthebulkstressformula expressedbyorder 2 inasuspensionofforce-freesphericalparticlesinNewtonianuid aswellasthebulkstressinasuspensionofnon-sphericalforce-freeparticles[ 6 ]. Somesuspensionsexhibitshearratedependentrheologysuchasshearthinning andshearthickening.Forsuspensionsofcolloidalparticles,Brownianforcescontribute totheparticlestressaswellashydrodynamicforces[ 43, 47].WhenBrownianmotion dominates,thesuspensionremainswell-dispersedandtherheologyshowsshear thinningbehavior[ 43].Anexamplefromeverydaylifeispaint,asuspensionofcolloidal silicaspheresdispersedinanorganicsolventthatshearthinsuponapplicationwitha roller[ 26].Colloidalinteractionsalsocauseshearthinningbehaviors[ 47].However,at highervolumefractionsandshearstresses,shearthickeningbehavioroccurs[ 43 47]. Oneexcitingapplicationofshearthickeningpropertiesisfabricbodyarmor[ 60, 100]. 13

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Figure 1-1.Scanningelectronmicrographofbloodcells[ 19](left)andanimageof tobaccomosaicvirus[41 ](right). Leeetal.[ 60]testedtheballisticpenetrationperformanceofthewovenKevlarfabric, whichisimpregnatedwithashearthickeningcolloidalsuspensionconsistingofsilica particlesdispersedinethyleneglycol.TheresultsshowthattheKevlarfabrichashighly enhancedballisticresistancewithoutanylossofmaterialexibility(seeFig. 1-2). Inaddition,colloidalinteractionsamongparticlessuchascharge[23]orattractive particleswillcontributetotheparticlestress[ 106].Forexample,electrorheological(ER) suspensionschangephasefromliquidtosoliduponapplyinganexternalelectriceld [101].ERsuspensionsconsistofpolarizableparticlesdispersedinanon-conducting suspendingmedium.Underelectricelds,particlesformdipolesresultinchaining ofparticlesparalleltotheelectricelddirection.Consequently,theviscositiesare enhancedbyseveralordersofmagnitudeandthesuspensionscanstopowing.Figure 1-3 showstheformationofchainstructuresofparticlesfromahomogeneousinitial state.Thischangeisrapidandreversibleviaanelectriceld.Theseuniqueproperties ofERuidshavepotentialapplicationsindampingdevices,brakes,actuators,andmore [101]. Thesuspensionrheologyalsodependsonsuspendingliquidproperties.Chanand Powell[ 20]comparedsuspensionsdispersedinNewtonianuidsandinnon-Newtonian uids,respectively.Theyfoundthesuspensionsinnon-Newtonianuidsdemonstratean 14

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Figure 1-2.ComparisonofwovenKevlarfabricwithout(left)andwith(right) impregnationofashearthickeningsuspensionafteraballistictest[60]. onsetofnon-Newtonianbehaviorsofstressrelaxationandstressgrowthatlowershear ratesthansuspensionsinNewtonianuids. Theeffectofparticleshapeontherheologyisreadilyapparent.Forexample, sedimentationofrodsinNewtoniansolventsresultinginformationofclustersofrods [18 ],whilesphericalparticlessedimentwithoutformingclusters.Duetotherelatively simpleshapeofthespheres,manyrheologicalstudiesonsuspensionsofsphereshave beenperformed.Unlikethewell-knowndynamicsofsphericalsuspensions,rheology ofsuspensionsofrigidrodsislessunderstoodanddisparitiesbetweentheoriesand experimentsexist. Inthiswork,Iwillintroducetwosimpleexamplesofwell-denedsuspension systems.SuspensionsofneutrallybuoyantrigidrodsinaNewtonianuidarestudiedin Chapter 2 andasuspensionofneutrallybuoyant,non-colloidalspheresisinvestigatedin Chapter 4 .However,wewillseethedynamicsarestillsurprisingevenforthosesimple cases. 1.2SuspensionsofSphericalParticles Therheologyofviscoussuspensionsofnon-colloidalandneutrallybuoyantspheres [28 ]hasbeenstudiedprimarilyunderconditionsofsteadyshearthroughexperiments [20 44 102]aswellassimulation[ 11, 93, 94].UnderconditionsofhighPecletnumber 15

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Figure 1-3.Anelectrorheologicalsuspensionwithoutanelectriceld(left)andwithan electriceld(right).TheERuidconsistsofcornstarchparticlesdispersed incornoil.Thevolumefractionofthesuspensionis =0.02andtheapplied electriceldstrengthisE=600V. (> 10 3 )andlowReynoldsnumber( < 10 )Tj /T3_1 7.97 Tf 6.59 0 Td (3 ),theeffectiveviscosityatsteadystateis expectedtobeafunctionofvolumefractiononly.Forinstance,inadilutesuspensionof spheresinaNewtonianuid,therelativeviscosityisexpressedby = o (1+2.5 + B 2 + O ( 3 )), (1) whereB=7.6forBrowniansuspensionsinisotropicstructuresatinnitePecletnumber [8].OneempiricalformulasatisedforhigherconcentrationlimitsisderivedbyKrieger andDougherty[ 57], = o (1 )Tj /T1_1 11.955 Tf 11.96 0 Td (= m ) )Tj /T3_1 7.97 Tf ([ ] m (1) wherethemaximumpackingfractionis m Tomeasurerheologicalpropertiesofconcentratedsuspensionsaccurately, shear-inducedmigrationoftheparticlesshouldbeavoided.Theshear-induced migrationinconcentratedsuspensionshasreceivedconsiderableinterestsince Gadala-MariaandAcrivos[ 44 ]andLeightonandAcrivos[ 61 ].Theyobservedviscosity decreasesinaCouettegeometryafterlongperiodsofsteadyshear.Theyarguedthat theviscositydecreasearisesfromshear-inducedmigrationoftheparticles.Shear inducedmigrationisthephenomenoninwhichbulkmigrationofparticlesoccursfrom theshearedgap,withhighshearrates,totheuidreservoir,withlowshearrates. 16

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P ossiblefactorscausingparticlemigrationareaconcentrationgradient,viscosity gradient,andashearrategradient.Phillipsetal.[ 81 ]developeda`diffusiveux'model basedonparticleinteractionanddiffusivescalinglawsproposedbyLeightonand Acrivos[ 61].Thismodelsuccessfullydescribedsteadystateandtransientprolesof particleconcentrationsinCouetteowandPoiseuilleow,butfailstoexplainqualitative featuresofconcentrationprolesmeasuredinoscillatoryows[ 17].Anothermodel calledthe`suspensionbalance'modelwasproposedbyNottandBrady[ 74].The migrationoccursasaresultofsatisfactionofconservationofmass,momentum,and energy.Theyintroducedaconceptofthesuspension`temperature',ameasureof particlevelocityuctuations.Thismodelworkswellforsteadyshearowsinachannel. Ontheotherhand,inoscillatorypressuredrivenow,theparticlemigration dependsontheappliedstrainamplitude[ 17 68 108].Withlargestrains,theparticles migratetothecenterofthetubeorthechannelsimilartothecaseofsteadyshearows. However,atsmallstrainamplitudes,particlesmigratetowardsthewalls.Yapicietal. [108]proposedastructure-tensorbasedmodel,wheremigrationtothewallsatlow strainamplitudestakesplacecombinedwithorderedmicrostructuresofthesuspension. Thereareavarietyoftheargumentsconcerningthereversibilityofparticlepositions inoscillatoryshearow.TrajectoriesoftwoisolatedsmoothspheresinStokesow areknowntobesymmetricandreversible.However,ifparticlesurfacesarerough, particletrajectoriescanbeirreversible[ 29].Inaddition,three-bodyinteractionsofthe sphereshavebeencitedasapossiblesourceofchaoticbehavior[ 12, 34].Inoscillatory shearow,Pineetal.[ 83]claimthepresenceofathresholdofstrainamplitudesfor irreversibility.Theyarguethatatsmallstrainamplitudesbelowthethreshold,particle trajectoriesarereversiblewhileabovethethresholdparticledynamicsbecomes irreversible.Yet,somestudiesreportedirreversibilityofnoncolloidalsuspensions subjecttooscillatoryshearows.Forexample,BrickerandButler[ 14],Gadala-Maria andAcrivos[ 44],GondretandPetit[ 46]observedviscositychangesoverlargetotal 17

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Figure 1-4.Thesteadystatespatialandorientationaldistributionofbersat r =0(left) andat r> 0(right). strainsevenatsmallstrainamplitudes.BrickerandButler[ 14 ]foundthecorresponding microstructuresofthesuspensionsviaStokesiandynamicssimulations.Inaddition,a netmigrationoftheparticlesatlowstrainamplitudewasalsoreportedbyButlerand Bonnecaze[ 17]andMorris[68].Inthiswork,toelucidatethereversibilityissue,rheology ofoscillatingsuspensionsofnoncolloidalspheresisstudiedexperimentallyatsmalland largestrainsinChapter 4. 1.3SuspensionsofRigidRodParticles Thedynamicsofrigidrodsuspensionsisnotwell-understoodandhasdistinguishable featurescomparedtosuspensionsofspheres.Forexample,inbersuspensions, theorientationdistributionofthebersisimportantaswellasspatialconguration ofthebers.Anotherexampleisinstabilitiesofsuspensionsofrigidrodsduring sedimentationasmentionedbefore[ 18 ].Alsoduetotheparticleshape,rodshave anisotropicdiffusivities,whilesphereshaveisotropicdiffusivities[ 70 ]. Fornon-Browniansuspensionsofrigidrods,dynamicsofrigidrodsuspensionsin thediluteregimecanbewellexplainedbyJefferyorbit[ 55].Insemi-diluteconcentration regime,rheologyshowsdiscrepanciesbetweentheoriesandexperiments.Modelsand simulationspredictshearindependentbehaviorofsteadyshearviscosity[ 6, 30, 91].As showninFig. 1-4,rodparticleshaveanisotropicdistributionatinitialstate( r =0).Under steadyshearow,rodstendtoalignparalleltotheowdirectionhavingsteadyshear 18

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viscosities independenttoshearrates.However,experimentsontherheologyofrigid bersshowshearthinningbehaviorsofnon-colloidalbersuspensionsinsteadyshear ows[ 22 45 90 ].Possiblesourcesoftheshearthinningbehaviorareproposed.For example,densitymismatchbetweenparticlesandsuspendinguidscancauseshear thinningbehaviors[ 45 ].Flexibilityoftheberscanalsocausetheshearthinningby formingentanglementsatlowshearratesandbyreleasingentanglementsathighshear rates[ 9].Suchnon-idealcharacteristicspreventfromclarifyingwhethertheshearrate dependenceexistsinawell-denedrodsuspensionsornot. InChapter 2,therheologyofwell-denedrodsuspensionsathighPeisstudied experimentally.Furthermore,themechanismsforshearthinningbehaviorsare discussed.Moreover,thestabilityofsingle-walledcarbonnanotube(SWNT)suspensions isstudiedinChapter 3 usingrheologyasacharacteristicmeasureofSWNTsuspensions. 19

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CHAPTER 2 RHEOLOGYOFSEMI-DILUTESUSPENSIONSOFRIGIDPOLYSTYRENE ELLIPSOIDSATHIGHPECLETNUMBERES 2.1Introduction Experimentsontherheologyofsemi-dilutesuspensionsofrigidbersatlarge rotationalPecletnumbersconsistentlyshowadecreaseinviscositywithincreasing shearrate[ 22, 45].Distinctlydifferentbehaviorispredictedbytheoriesandsimulations. Theoriesevaluatingthestressgeneratedinasemi-dilutesuspensionofrigidnon-Brownian bers[ 7 30, 91 ]donotpredictadependenceofthesuspensionviscosityonshearrate. Likewise,simulationsonsemi-dilutesuspensionsofbers[ 84, 107]ndnodependence ontheshearrate,evenwhenallowingmechanicalcontactbetweenbers[ 99 ].Though thetheoriesandsimulationsareperformedinthelimitofinniterotationalPeclet number,theresultsareexpectedtoqualitativelypredicttheresultsofexperimentsat sufcientlylargerotationalPecletnumbers. Toaccountfortheshearratedependenceobservedinsomespecicexperiments, explanationssuchasberexibility[ 9]andberadhesion[ 22]havebeenoffered. However,theseeminglyuniversalnatureoftheshearthinningphenomenonin experimentsremainsunclear.Interpretationofresultsfromexperimentsarefurther complicatedbydeviationsfrommodelsystems,whichcaneffecttheobservedrheology. Forexample,suspensionsaretypicallycomposedofheavyberssuspendedinalight liquid[ 45],resultinginbersedimentation.Furthermore,berstypicallyhavelarge lengthscaleswhichcanresultincomplicationsfromberbreakageifusingfragile materials[ 90]andboundaryeffects[ 31 ].Clearly,astudyoftherheologyofwell-dened bersuspensionsisneeded. Inthefollowingsections,wereportontherheologyofpolystyreneellipsoid suspensionsinthesemi-diluteconcentrationregime.Therheologyofsuspensions ofsphereswiththesamematerialpropertiesastheellipsoidsarealsoevaluatedand compared.Weobserveshearthinningbehaviorinsuspensionsofrigidellipsoids 20

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Figure 2-1.ScanningelectronmicrographsofpolystyrenespheresbeforeprocessingA), andellipsoidswithaverageaspectratiosof 4 B)and 7 C).Thescalebar ineachimageis2 m,withtheexceptionofC),whichhasascalebarof1 m. atrotationalPecletnumbersgreaterthan10 3 ,whereasthesuspensionsofspheres exhibitnothinningoverthesamerangeofshearrates.InSec. 2.2 ,adescriptionofthe fabricationofthebersisprovidedalongwithacharacterizationoftheparticlesizeand distribution.ResultsarepresentedinSec. 2.3 fortwodifferentparticleaspectratios.A discussionofpossiblemechanismsfortheratedependentrheologyalongwithscalings ofthesteadystateviscosityisgiveninSec. 2.4.Conclusionsarepresentedinthelast section. 2.2Experiments PolystyreneellipsoidsweremanufacturedusingthemethodofNagyandKeller[71 ]. Monodispersepolystyrenespheres(Polysciences,Inc.)withameandiameterof1.06 0.02 mwereaddedtoamixtureof5% byweightpolyvinylalcohol(MPBiomedicals, Inc.)inwater.Themixturewasdriedtomakethinlmswhichweredeformeduniaxially atatemperatureof190 Ctoadesireddrawratio.Ellipsoidalparticleswereobtained 21

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b ydissolvingthestretchedlmsina30 % byvolumemixtureofisopropanolinwater. Tolimittheamountofresidualpolyvinylalcoholfromtheparticlesurface,theparticles wererepeatedlywashedandthencentrifuged.Theissueoftheroleofresidualpolyvinyl alcoholontheobservedrheologyisdiscussedindetailinSec. 2.4.Amoredetailed descriptionoftheprocedureispresentedelsewhere[ 49 71].Thisparticularmethod allowsfortheproductionofellipsoidswithaspectratiosandsizedistributionswhichare highlycontrollable[ 49 ]. Tostudytheeffectofvaryingtheellipsoidaspectratioontherheology,different drawratioswereused,resultinginellipsoidswithaverageaspectratios, L =d (where L isthelengthofthelongaxisand d theshortaxis),of4.17 0.81and7.14 1.49. ScanningelectronmicrographsofthespheresandresultingellipsoidsareshowninFig. 2-1.NormalizednumberdistributionsoftheparticlesaregiveninFig. 2-2,andaverage valuesoftheparticlevolumeandsurfaceareaforeachaspectratioaretabulatedin Table 2-1.Alllengthscalesareobtainedfromscanningelectronmicroscopyandthe valuesrepresentaveragesover50randomparticles. ThesuspendingliquidconsistedofUCON50-HB-5100oil(DowChemicals) blendedwith10 % byvolumedistilledwater;potassiumchloridewasaddedatatotal concentrationof1mM.Thecomponentswerechoseninparttomatchthedensityof theparticlesandsuspendingliquid.Themeasureddensityofthesuspendingliquid was1.06g/cm 3 andthereportedparticledensitywas1.056g/cm 3 ,thusbuoyancy effectsarenegligibleoverthetimescaleoftheexperiments.Theadditionofpotassium chloridewasalsousedtohelpstabilizetheparticles;thetotalconcentrationof1mM saltcorrespondstoamaximumintheelectrophoreticmobility[ 50].Thesuspending liquidslightlyshearthins,exhibitingadecreaseinviscosityfrom1.93Pasto1.75Pas overtherangeofshearrates0.1s )Tj /T3_2 7.97 Tf (1 r 400s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 .Althoughthedifferenceoverthe shearraterangeislessthan10%,therelativeviscosity, r ,reportedintheremainder 22

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of thepaperisdenedastheeffectiveviscosityofthesuspensionatagivenshearrate normalizedbytheviscosityofthesuspendingliquidatthesameshearrate. Allsuspensionswerepreparedbygentlyhandmixingtheellipsoidsinsmall incrementsuntilahomogeneousstatewasreached.Thesuspensionswereplaced undervacuumpriortotestingtoeliminateanyairbubblesentrainedwithinthe suspension.Inthesemi-diluteregime,denedby1 < nL 3 < L= d (wherenisthenumber densityofellipsoids)[ 33],theinterparticlespacingissuchthattheellipsoidsareunable torotatefreelywithoutbeingimpededbyneighboringellipsoids.Therheologywas studiedforconcentrationswithinthesemi-diluteconcentrationregimecorresponding to nL 3 =1.00to4.10forellipsoidswithanaspectratioof4,and nL 3 =1.13to6.95for ellipsoidswithanaspectratioof7. Themaximumparticle-basedReynoldsnumberforthesuspensionsystemsisRe =10 )Tj /T3_1 7.97 Tf 6.59 0 Td (6 ,sotheeffectsofinertiaareminimal.TheminimumrotationalPecletnumber, Pe r = r=D r ,calculatedbasedontherotationaldiffusion, D r ,ofaprolatespheroid[ 13 ],is 10 3 .ThePecletnumberbasedonthetranslationaldiffusionofaprolatespheroid, D t isdenedas Pe t = r L 2 = D t .SincetherotationalPecletnumberrepresentsthesmaller ofthetwoPecletnumbers, Pe r isusedintheremainderofthepapertodenetheow strength.Adiscussionofotherpossiblenon-hydrodynamiceffectsisgiveninSec. 2.4 Experimentswereperformedusinga50mmdiameterparallel-plategeometry.The gapbetweenplateswassetto500 m,resultinginaminimumgap-to-particlelength ratioof125.Additionalexperimentsperformedatalargergapof1000 m( H =L =250, whereHisthegapheight)werestatisticallyequivalent.Intheparallel-plategeometry, therateofstrainvariesintheradialdirection,resultingininhomogeneousow.Since themicrostructureofbersuspensionsisstraindependent,additionalexperiments wereperformedusinga50mmdiametercone-and-plategeometry,whichmaintainsa constantshearrateintheradialdirection.Forallcone-and-platemeasurements,the 23

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T able2-1.Summaryofthecharacteristiclengthscales,volumeperparticle,andsurfaceareaperparticleofpolystyrene particleswithaspectratiosof L= d = 1,4,and7. Aspect Ratio d L Volume/particleSurfaceArea/particle 1 1.06 0.02 m 1.06 0.02 m 0.63 0.03 m 3 3.55 0.13 m 2 4.17 0.810.65 0.06 m 2.71 0.38 m 0.61 0.11 m 3 4.48 0.59 m 2 7.14 1.490.55 0.07 m 3.87 0.62 m 0.63 0.17 m 3 5.33 1.03 m 2 24

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Figure 2-2.Normalizednumberdistributionsforellipsoidswithanaverageaspectratio of 4 (solid)and 7 (dashed).Ineachcase,particlenumbersare normalizedbytheirrespectivepeakvalues.Eachdistributioniscalculated from50randomparticles. coneanglewas0.04radians,andthegapwassetto45.7 m,resultinginaminimum gap-to-particlelengthratioof12attheapexoftheconeand270attheedge. TherheometerusedinallexperimentswasanARESLS-1straincontrolled rheometer(TAInstruments).Thesteadyshearrheologywasinvestigatedforarange ofberconcentrationsandaspectratios.Toensurethateachtestbeganfromasimilar initialstate,thesuspensionswerepreshearedatashearrateof r =100s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 for300 seconds,whichwassufcienttoenablethesuspensiontoreachasteadystate.The temperaturewasmaintainedat25 Candthetemperatureuctuationwaslessthan 0.05 Cduringatypicalexperiment. 2.3Results Start-upexperimentswereperformedintheparallel-plategeometrytoevaluate thetransientbehaviorofellipsoidsuspensions.Figure 2-3 showstherelativeviscosity asafunctionoftimeandshearrateforsuspensionsofellipsoidswith L =d =4.The particlevolumefraction, ,of0.103fallswithinthesemi-diluteconcentrationregime. 25

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Figure 2-3.Relativeviscosityasafunctionoftimeandshearrateforpolystyrene ellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.The ellipsoidshaveanaverageaspectratioof4andthevolumefractionis0.103. Foreachshearrate,thesuspensionsarepreshearedat r =100s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 for300s. Experimentswereperformedusinga50mmparallel-plategeometrywitha gapof500 m. Eachstart-upexperimentwasperformedimmediatelyfollowingapreshearat r =100 s )Tj /T3_1 7.97 Tf (1 for300s.Forsmallershearrates,thetransientresponseoftherelativeviscosityis moredramatic.Forexample,at r =100s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 ,therelativeviscosityisalreadyatsteady stateandnoadditionalchangesareobservedoverthetimescaleoftheexperiment.At lowershearrates, r increasescontinuouslyanddoesnotreachsteadystateoverthe durationoftheexperiment.Furthermore,althoughtheviscositiesatt=0areequivalent, theviscosityat r =0.1s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 increasesto r 5.5after300swhereasat r =1s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 ,the increaseismuchmoremodest( r 3.5)overthesametimeperiod.Theresultsare qualitativelysimilartoexperimentsonsuspensionsoflargeaspectratio( L= d 40)bers [22 32 ],whichshowanincreaseinviscositywithtimeatlowshearrates. Figure 2-4 showstherelativeviscosityat r =0.251s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 plottedasafunctionoftime andconcentrationforsuspensionsofberswith L = d =4.Theresultsareplottedfor threedifferentvolumefractionsandrepresentthetimeevolutionfollowingapreshearat 26

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Figure 2-4.Relativeviscosityasafunctionoftimeandvolumefractionforpolystyrene ellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.The ellipsoidshaveanaverageaspectratioof4andallvolumefractionslie withinthesemi-diluteconcentrationregime.Foreachvolumefraction,the suspensionsareshearedatarateof r =0.251 s )Tj /T3_1 7.97 Tf (1 immediatelyfollowinga preshearat r =100 s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 .Experimentswereperformedusinga50mm parallel-plategeometrywithagapof500 m. r =100s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 for300s.Atallthreevolumefractions,therelativeviscosityincreaseswith time.Forthevolumefractionsstudied,nosteadystateisobservedthrought=300s. Additionalexperimentswereconductedoverlargertimescalesforsuspensionswith = 0.079;nosteadystatewasobservedandtheviscositycontinuedtoincreasethrought= 2000s. Todeterminetheeffectofellipsoidaspectratioontherheology,suspensionsof ellipsoidswithanaverageaspectratioof L= d =7werestudied.Therelativeviscosityat r =0.1s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 ,plottedasafunctionoftimeandconcentrationforsuspensionsofellipsoids with L =d =7,isshowninFig. 2-5.Theresultsareplottedforthreedifferentvolume fractionsandrepresentthetimeevolutionfollowingapreshearat r =100s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 .Similarto thesuspensionscontainingellipsoidswithasmalleraspectratio(Fig. 2-4),therelative viscosityincreaseswithtime,withtheexceptionofthelowestvolumefraction, =0.012, 27

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Figure 2-5.Relativeviscosityasafunctionoftimeandvolumefractionforpolystyrene ellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.The ellipsoidshaveanaverageaspectratioof7andallvolumefractionslie withinthesemi-diluteconcentrationregime.Foreachvolumefraction,the suspensionsareshearedatashearrateof r =0.1s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 immediatelyfollowing apreshearat r =100s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 .Experimentswereperformedusinga50mm parallel-plategeometrywithagapof500 m. whichshowsasteadyvalueoverthedurationoftheexperiment.Comparisonofthe transientresponseofsuspensionsofellipsoidswithdifferentaspectratiosrevealsa dependenceon L= d .Specically,theviscosityincreaseovert=300sisgreaterfor suspensionswithellipsoidsofhigheraspectratio.Forexample,at 0.055,therelative viscosityforsuspensionsofellipsoidswith L= d =7increasesby32 % overatotaltimeof 300s,whereasforsuspensionsofellipsoidswith L= d =4therelativeviscosityincreases byonly21 % overthesametime. Thedependenceoftherelativeviscosityonshearrate(and Pe r )isplottedinFig. 2-6 forsuspensionsofberswith L= d =4.Ratesweepswereperformedusingthe parallel-plategeometryforsuspensionsatsixdifferentvolumefractionsspanningthe semi-diluteconcentrationregime.Ateachshearrate,thevalueoftherelativeviscosityis recordedafter360secondsofshear,whichwaschosentoreducethetotalexperiment 28

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Figure 2-6.Relativeviscosityasafunctionofshearrateandvolumefractionfor polystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture. Eachdatapointrepresentstheapparentviscosityafter360secondsof shearateachcorrespondingshearrate.Theellipsoidshaveanaverage aspectratioof4andallvolumefractionsliewithinthesemi-dilute concentrationregime.Experimentswereperformedusinga50mm parallel-plategeometrywithagapof500 m. timeandconsequently,tominimizepossibleevaporationofthesuspendingliquid. Sincetherelativeviscosityateachshearrateissensitivetothetimeoverwhichthe suspensionissheared,notallvaluesrepresentsteadystateviscositiesafter360s. Thus,opensymbolsrepresentsteadystatevaluesoftherelativeviscosityandclosed symbolsrepresentvaluesthathavenotattainedasteadystatevalueaftert=360s. Forallvolumefractionsstudied,thesuspensionsshowshearthinningbehavior. Thedependenceoftherelativeviscosityonshearratebecomesweakerasthevolume fractiondecreases.Forexample,atthehighestvolumefraction( =0.127),thevalue oftherelativeviscositydecreases350%overthreedecadesofshear,whileatthe lowestvolumefraction( =0.031),therelativeviscositydecreasesbyonly43 % over thesamerangeofshearrates.Below r =1s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 ,thevaluesoftherelativeviscosities 29

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Figure 2-7.Relativeviscosityasafunctionofshearrateandvolumefractionfor polystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture. Eachdatapointrepresentstheapparentviscosityafter360secondsof shearateachcorrespondingshearrate.Theellipsoidshaveanaverage aspectratioof7andallvolumefractionsliewithinthesemi-dilute concentrationregime.Experimentswereperformedusinga50mm parallel-plategeometrywithagapof500 m. reportedinFig. 2-6 ceasetorepresentsteadystatevalues.However,eveninshearrate rangeswherethevaluesoftheviscosityareatsteadystate(forexample,10s )Tj /T3_0 7.97 Tf 6.59 0 Td (1 r 251s )Tj /T3_0 7.97 Tf 6.59 0 Td (1 ),shearthinningisapparent.Moreover,ratesweepsfromhightolow r were performedandshowthattherheologyisreversible.Althoughratedependentrheology isnotexpectedforsuspensionsofbersathighPecletnumbers[ 80 ],shearthinning similartothatobservedhereisconsistentlyobservedinexperiments[ 45 ]. Figure 2-7 comparestherelativeviscositiesat L= d =7forvedifferentvolume fractionsspanningthesemi-diluteconcentrationregime.Thevaluesoftherelative viscosityateachshearratearemeasuredastheywereinFig. 2-6.Opensymbols representsteadystatevaluesoftherelativeviscosityandclosedsymbolsrepresent viscositieswhichhavenotreachedsteadystate.Therangeofshearratesstudiedis 30

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Figure 2-8.Comparisonoftherelativeviscositymeasuredinthecone-and-plate(dotted line)andparallel-plate(solidline)geometryforsuspensionsofpolystyrene ellipsoidswith L = d =4 inapolyalkyleneglycol/water/KClmixture.The relativeviscosityisplottedasafunctionofshearrateandvolumefraction. Eachdatapointrepresentstheapparentviscosityafter360secondsof shearateachcorrespondingshearrate.Thecone-and-plategeometryhad adiameterof50mmandaconeangleof 0.04 radians.Thegeometrygap wassetat45.7 m. shiftedtoenablecomparisonatsimilar Pe r forthetwosystemswithdifferentaspect ratios.Similartosuspensionscontainingellipsoidswith L= d =4,higheraspectratio ellipsoidsshowshearthinningbehaviorforallvolumefractions.Again,theshearrate dependenceoftherelativeviscositybecomeslessnoticeableastheconcentration decreases.Forexample,atthelowestvolumefractionstudiedforeitheraspectratio( = 0.012),theviscosityisnearlyindependentoftheshearrateoverthreedecadesofshear. Sincetherheologyofbersuspensionsdependsonthespatialcongurationand orientationofbers,whichisstraindependent,theparallel-plategeometrymaybe unsuitableforbersuspensionrheologybecausethestrainvarieswithradialposition [31 ].Todeterminetheeffectofgeometryontherheology,additionalexperimentswere 31

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perf ormedinacone-and-platexturewhichmaintainsaconstantstrainintheradial direction.Figure 2-8 showstherelativeviscosityinthecone-and-plategeometryplotted asafunctionofshearrateforsuspensionsofberswith L= d =4.Althoughexperiments wereperformedovertheentirerangeofvolumefractionspresentedinFig. 2-6,only threearereportedandcomparedtoexperimentsintheparallel-plategeometry.For allvolumefractionsstudied,thevaluesoftherelativeviscosityforsuspensionsin thecone-and-plategeometryareconsistentlyhigherthanthoseintheparallel-plate geometry.Thedifferencesaremostsignicantatthelowestshearrates,wherethe reportedviscositieshavenotyetattainedsteadystate.Despitethesedifferences, thequalitativebehaviorofoursystemisinsensitivetothegeometryused.Duetothe smallergap-to-particlelengthratioattheapexinthecone-and-plategeometry,the rheologywassubjecttostressjumps,possiblyduetoparticlejammingattheapexofthe cone.Anexampleofparticlejammingoccursfor =0.079,wheretherelativeviscosity anomalouslyincreasesat r =1.58s )Tj /T3_2 7.97 Tf (1 2.4Discussion Inthissection,wediscusspossiblemechanismsfortheratedependenceofthe relativeviscosities.Wealsoinvestigatescalingsofthesteadystateviscosityinthe semi-diluteconcentrationregimeandcomparetoresultsfromsimulations. 2.4.1RateDependentRheology FibersuspensionsatlargerotationalPecletnumberspredominantlyexhibitshear thinningbehavior[ 22, 45, 90]similartothatobservedinthepresentwork.However, theoriesandsimulationsforrigidrodsdonotpredictratedependentrheologyfor rigidrodsinthelimitsoflowReynoldsnumberandhigh Pe r ,wherehydrodynamic interactionsdominate[ 80].Inanattempttoelucidatetheshearthinningphenomenonin ourexperiments,weconsiderpossiblehydrodynamiceffects,suchasberexibility[ 9 ] andparticleorcolloidalinteractions,suchasberadhesion[ 22 ]. 32

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Iner tia,shear-inducedalignment,andberdeformationorbreakageduetoshear areunlikelysourcesoftheobservedshearthinning.Althoughthereisaniteamountof inertia,theReynoldsnumberistoolow( Re =10 )Tj /T3_1 7.97 Tf (6 )toimpacttherheology.Similarly,the Pecletnumbersaretoolargetoobserveanysignicantreductioninstresscausedby shearalignmentofwell-dispersedBrownianrods[ 48 ].Fiberexibility,whichcanresult inratedependentrheology[ 9],isalsonegligible.Scanningelectronmicroscopyimages oftheellipsoidsrevealnobentordeformedparticles.Shear-induceddeformationofthe particlesisalsounlikelysincethemaximumstressgeneratedduringtheexperimentsis signicantlylessthantheminimumstressrequiredtodeformpolystyrene[ 42 ]byafactor of10 3 Boundaryeffectsareofpossibleconcernwithbersuspensions.Forexample, models[ 78]andsimulations[88]indicatethatindividual,rigidbersundergoing rectilinearowathigh Pe r inthevicinityofawallundergoanetmigrationawayfrom thewalls,creatingadepletionlayerthatincreaseswithowstrength.However,the depletionlayerpredictedfordilutesystemsissignicantlysmallerthanthegapsize usedintheexperimentsandthereductioninviscositywithincreasingshearrateis consequentlyverysmall.Additionally,experimentsperformedatdifferentgapwidths indicatednochangesintheeffectiveviscositiesofthesuspensions,regardlessof concentration. Shearthinninginoursystemcouldbecausedbyhydrodynamicforcesbreaking mechanicalbridgesbetweenparticlesformedbyhardened,residualPVAspanning particlesurfaces.Forexample,Fig. 2-9 showsscanningelectronmicrographsof polystyrenesphereswhichweresubjectedtothesameproceduredescribedinSec. 2.2, butnotstretched.Priortowashingtheprocessedparticles,thePVAmatrixspansgaps betweenparticlesandappearstomechanicallybindparticlestooneanother.Afterthe washingcycleshowever,theprocessedparticlesresembleplainpolystyrenespheres withnoevidenceofPVAbridgingparticlesurfaces.Uponbreakingofanyphysicalbonds 33

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Figure 2-9.SEMimagesofprocessedpolystyrenespheresbeforeA)andafterB) washing.Ineachimage,thescalebarrepresents1 m. betweenparticles,therheologyofasuspensionofparticlessimilartothoseinFig. 2-9A wouldundergoapermanentchange.Consequently,thereversibilityoftheviscosity withrespecttoshearrate,atleastforthoseexperimentsonellipsoidsthatreachsteady state,arguesstronglyagainstthismechanism.However,polyvinylalcoholbindsstrongly topolystyreneandcanaffectparticlesurfaceproperties;infact,adsorbedPVAsterically stabilizessuspensionsofpolystyreneparticles[ 21, 73 ]. Comparisonsweremadebetweensuspensionsofplainpolystyrenespheres(as receivedfromthemanufacturer)andprocessedpolystyrenespheresfollowingthe washingcyclestofurthertestanyaffectsofresidualPVA.Bothsuspensionswere preparedatavolumefractionof =0.07,andtherelativeviscosityasafunction ofshearrateisplottedinFig. 2-10 .Forsuspensionscontainingplainpolystyrene spheres,therelativeviscosityisconstantoverthreedecadesofshearrate.Suspensions containingprocessedspheresshowanearlyconstantrelativeviscosityoverthesame rangeofshearrates,withslightshear-thinningdetectablefor r< 1 s )Tj /T3_1 7.97 Tf (1 ,afterwhichthe relativeviscosityremainsconstantandstatisticallyequivalenttotherelativeviscosityfor suspensionscontainingplainpolystyrenespheres.Therheologymaybeaffectedbythe presenceofPVAontheparticlesurface,butthequantitativedifferenceissmallandonly apparentatthelowestshearrates. 34

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Chaouche andKoch[ 22]relatedtheshear-thinningobservedinsuspensions ofnylonrodstothebreak-upofocscausedbycolloidalinteractions,despitethe relativelylargeparticlesize(alengthof0.5mm)andhighPecletnumbersusedintheir experiments.Experimentsonsuspensionofrodsthatareclearlywithinthecolloidal range(sub-micronlengths)showavarietyofbehaviors.Recently,measurementsof calciumcarbonaterodssuspendedathighconcentrations( > 0.25 )inpoly(ethylene glycol)showedshearthickeningatPecletnumbersuptoapproximately 10 4 [36, 37 ]. Morerelevanttothepresentstudyaremeasurementsshowingshearthinningfor dilutesuspensionsofhematiterodshavinglengthssmallerthan200nm[ 96 ].These measurementscoveredasimilarrangeofaspectratiosandconcentrationsasthe presentstudy,butwereperformedatlowerrotationalPecletnumbersoflessthan5. SolomonandBoger[ 96]attributedthesignicantshear-thinningintheirelectrostatically stabilizedsuspensionofrodstoelectroviscouseffects. Itiswelldocumentedthatelectrostaticinteractionscancauseratedependent rheologyinsuspensionsofpolystyrenespheres[ 86].Generallyhowever,variations duetoelectrostaticinteractionswouldbeexpectedforvolumefractionshigherand shearrateslowerthanthoseusedhere.TheDebyelengthinthepresentsystemis approximately )Tj /T3_1 7.97 Tf 6.59 0 Td (1 3 nm,resultinginasmallincreaseintheeffectivevolumefraction, whichisdiluteinthecaseofthesuspensionsofspheresstudiedhere.Consequently, increasesinzeroshearviscosityduetoparticleinteractions(secondaryelectroviscous effect)wouldbesmall[ 85 ].Furthermore,electroviscouseffectsareexpectedtovanish athighshearrateswherehydrodynamicforcesdominate[ 59 87 ].Evidenceofthelack ofelectroviscouseffectsforthespheresappearsinFig. 2-10,whichshowsastable viscositywithincreasingshearrateforspheres. Althoughsimilarargumentswouldbeexpectedtoholdforthesuspensionsof ellipsoids,adirectcomparisonshowninFig. 2-10 demonstratesthattherheologyof suspensionsofspheresandellipsoidshavingidenticalmaterialpropertiesandvolume 35

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Figure 2-10.Relativeviscosityasafunctionofshearrateandaspectratiofordifferent suspensionsystems.Theresultsarefrompolystyreneparticlessuspended inapolyalkyleneglycol/water/KClmixtureatavolumefractionof 0.07. Experimentswereperformedusinga50mmparallel-plategeometrywitha gapof500 m.Thevaluesrepresentaveragesovertwoindividualruns. Errorbarsarespeciedwhentheerrorislargerthanthesizeofthesymbol. fractionsqualitativelydiffer.Argumentshavebeenmadethatparticleshapecanimpact electrostaticeffects[ 50 ].Furthermore,boundaryelementstudiessuggestthatalthough thesecondaryelectroviscouseffectisindependentofparticleshape,theprimary electroviscouseffectdramaticallydifferswithparticleaspectratioforweaklycharged prolatespheroidssuchasthepolystyreneparticlesusedinthisstudy[ 2].Theoretical studiesoftheprimaryelectroviscouseffectonrodshavebeenperformedforDebye lengthssignicantlylargerthanthediameter[ 23 92],howevermakingacomparisonto oursystem,where d > 400,isdifcult. Anal,andperhapsmorelikely,possibilityisthattheshearthinninginellipsoids arisesfromshear-induced(orthokinetic)aggregation.Visualinspectionofoursystems indicatesthatthesuspensionsremainstablewhenatrestoveraperiodofatleastone month.Beyondelectrostaticbarriers,stabilityofthesuspensionisalsoassistedby thepresenceofresidualamountsofPVAadsorbedontheparticlesurfaces,though 36

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whether ornotsufcientamountsofPVAremaininsolutiontocreateaneffectivesteric layerisunclear.However,stabilizedsuspensionsofparticlesareknowntosometimes formaggregatesduringshear[ 86].Boththedecreaseineffectiveviscositywithshear rateandtheslowapproachtoasteadystatevaluefortheellipsoidsatlowratesof shearfollowingthepreshearareconsistentwiththisproposedmechanism.Incontrast, thesignicantdifferencesbetweenidenticalsuspensionsofspheresandellipsoids seeninFig. 2-10 wouldseemtoargueagainstthismechanism.Onepossibilityisthat thespheresdoaggregateduringshearwithoutgeneratingameasurabledifference inviscosity,whereasaggregationoftheellipsoidsdoesalterthemeasurement. Forexample,thestructureandsizeofaggregatesformedbyanisotropicparticles candifferfromaggregatesofspheres[ 65];asignicantdifferenceinthestructureof aggregatescouldberesponsibleforthecontrastingmeasurement,thoughdirectoptical observationsduringshearareneededtoverifytheoccurrenceofanyshear-induced aggregation. 2.4.2Scalingsof r atHigh Pe r AtthehighestPecletnumbers,therelativeviscositiesapproachasteadystatefor allconcentrationsandaspectratios.Thesehighestvaluesof Pe r presumablyrepresent thelimitatwhichanynon-hydrodynamiceffects,asdiscussedintheprevioussection, ceasetoaffectthemeasurements.Figure 2-11 showstherelativeviscosityat Pe r =10 6 asafunctionofvolumefractionandaspectratio.Forbothaspectratios,theviscosity increaseswithvolumefraction;theincreaseismuchfasterforlargeraspectratios.For example,at 0.07thesteadystaterelativeviscosityfor L= d =7isapproximately one-and-a-halftimesgreaterthanthatfor L = d =4. Thedependenceofthesteadystateviscosityonaspectratioagreesqualitatively withresultsfromexperiments[ 32]andsimulations[24].Computationalresults[24 ]for dispersionsofspheroidswithaspectratiossimilartothosestudiedhereareplotted alongwithourexperimentalresultsinFig. 2-11.Boththecalculationsandmeasurement 37

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Figure 2-11.Steadystaterelativeviscosityat Pe r =10 6 asafunctionofvolumefraction andaspectratio.Theresultsarefrompolystyreneellipsoidssuspendedin apolyalkyleneglycol/water/KClmixture.Theexperimentalresultsare comparedwithresultsfromsimulationsofisotropicsuspensionsof ellipsoidalparticles[ClaeysandBrady(1993)]. ofrelativeviscositiesexhibitalineardependenceonvolumefraction.Quantitatively, theresultsagreereasonablywellforthesmalleraspectratio,consideringthatthe calculationswereperformedforstatisticallyhomogeneoussuspensionswithanisotropic orientationdistributionofspheroids.However,theexperimentsshowamuchstronger dependenceoftherelativeviscosityonaspectratiocomparedtothesimulationresults ofClaeysandBrady[ 24 ].Thedifferencesmayarisebecausethecomputationalresults ofClaeysandBrady[ 24 ]areforforce-freeellipsoids,whereastheellipsoidsinthe experimentsmaybeinuencedbynon-hydrodynamiceffects,asdiscussedinthe previoussection. Plottingtheresultsat Pe r = 10 6 versusdimensionlessnumberdensity, nL 3 ,also producesalinearrelationshipfortheexperimentswith L = d =7and4asshowninFig. 2-12.Onthisscale,thecalculationsofClaeysandBrady[24 ]predictamuchstronger dependenceoftherelativeviscosityonaspectratio.However,theexperimentalvalues oftherelativeviscosityat Pe r = 10 6 collapsetowithinasmallrangewhenplottedversus 38

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Figure 2-12.Steadystaterelativeviscosityat Pe r =10 4 and 10 6 asafunctionof dimensionlessnumberdensityandaspectratio.Theresultsarefrom polystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KCl mixture. nL 3 ratherthan ;thisindicatesthattherelativeviscositycanbepredictedfromthe numberdensityforsemi-dilutesuspensionsofsmallaspectratioellipsoidsathigh Pe r Figure 2-12 alsoshowstherelativeviscosityat Pe r = 10 4 asafunctionof nL 3 ForthelowerrotationalPecletnumbers,theresultsforthetwoaspectratiosdonot overlapwhenplottedversus nL 3 andtheviscositynolongerincreaseslinearlywith nL 3 .Thesignicantdifferencesindicatethattherheologyispossiblyaffectedby non-hydrodynamicinteractionsatlowervaluesof Pe r 2.5Conclusions Therheologyofwellcharacterizedsuspensionsofrigidpolystyreneellipsoidsat rotationalPecletnumbersexceeding10 3 wasevaluated.Polystyreneellipsoidshaving twodifferentaspectratioswerefabricatedusingthemethodofNagyandKeller[ 71 ]. SimilartopreviousexperimentsonbersuspensionsatlargePecletnumbers[ 45 ], ellipsoidsuspensionsexhibitshearthinningbehaviorforbothaspectratiosoverall concentrationsspanningthesemi-diluteconcentrationregime.Theshearthinning 39

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beha viorisreversibleandindependentofthegeometryused.Distinctlydifferent behaviorispredictedbytheoriesandsimulations,whichshownodependenceofthe viscosityonshearrateinthelimitofinnite Pe r Possiblemechanismsfortheshearthinningbehaviorwerediscussedandevaluated throughdirectcomparisonoftherheologyofsuspensionsoftheellipsoidalparticleswith suspensionsofsphericalparticles,processedinanidenticalfashion.ForlargePeclet numbers,thedependenceoftherelativeviscosityonvolumefractionisqualitatively similartosimulationsofisotropicsuspensionsofspheroidswithsimilaraspectratios [24 ].Additionally,atthelargestPecletnumbersstudied,therelativeviscosityscales linearlywith nL 3 ,regardlessoftheellipsoidaspectratio. 40

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CHAPTER 3 RHEOLOGYANDSTABILITYOFSINGLE-WALLEDCARBONNANOTUBES 3.1Introduction Theuniqueelectricalandmechanicalpropertiesofsingle-walledcarbonnanotubes (SWNTs)havemanypotentialapplicationsindiverseeldssuchasbiomedical, electronics,andnanotechnology.Althoughmuchresearchhasbeenperformedsince thediscoveryofcarbonnanotubesin1991[ 53],developmentofapplicationshasbeen limitedpartiallyduetostrongvanderWaalsattractionamongtubes,whichresultsin bundledSWNTs(seeFig. 3-1).ThebundledSWNTshavediminishedmechanical andelectricalpropertiescomparedtoindividualSWNTs[ 103].Consequently,many attemptsatdispersingandstabilizingindividualSWNTsinsolutionhavebeenmade. Amongthem,ionicandnon-ionicsurfactantsarewidelyusedtodisperseandstabilize individualSWNTs[ 5, 103].Moreover,differentlevelsofprocessingarecombinedto assistwithgenerationofsurfactant-stabilizedsuspensionsofSWNTs.Forinstance, sonicationisappliedtobreakapartlargebundlesofSWNTs[ 54 79].Centrifugation alsohelpstoseparateindividualSWNTsfromthebundledSWNTsremaininginthe suspension[ 3 67].Additionally,arelativelynewprocessthatreliesoninterfacial trappingtoseparateindividualSWNTsfrombundleshasbeenintroduced[ 104, 105 ]. InthisworkIwillexaminefourSWNTsuspensionsystemsaccordingtotheapplied processesandsurfactantsfordispersingandstabilizingindividualSWNTs:thesodium dodecylsulfate(SDS)-dispersedSWNTsuspensionpreparedbycentrifugation,the GumArabic(GA)-dispersedSWNTsuspensionpreparedbythehomogenization andsonicationprocess(acontrol),theinterfacialtrappingprocesswithGA,andthe centrifugationprocesswithGA. ThemostcommonmethodusedfordispersingSWNTsemploysanionicsurfactant togetherwithphysicalagitationofthesuspension[ 54, 67, 79].Thesurfactantcommonly usedissodiumdodecylsulfate(SDS),ananionicsurfactantknownasagooddispersing 41

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Figure 3-1.TEMimageofaSWNTbundle.Theimageistakenfromworkperformedat theLeibnizInstituteforSolidStateandMaterialsResearch,Dresden[40]. agentforSWNTsintheaqueousmedium[ 67, 103].TheindividualSWNTsare stabilizedintheaqueousmediumbyelectrostaticrepulsionofnegativelycharged SDS,whichadsorbsonthesurfaceofSWNTs.TobreakupbundlesofSWNTs, high-shearhomogenizationfollowedbyultrasonicationisused.Ultracentrifugationis performedtoremovebundlesthatstillremaininthesuspension[ 75].However,dueto thetime-consumingnatureofultracentrifugation,alternativewaysofremovingbundles fromSWNTsuspensionsareneededtoeconomicallygeneratelarge-scaledispersions. ThesurfactantGumArabichasalsobeenusedasadispersingagent.Itisawater solublepolysaccharidehavingalargemolecularweightof 250,000 g = mol .Thelong chainsofGumArabicadsorbedontheSWNTsstabilizeindividualtubesthroughsteric hinderance.AlthoughsomestudiesarguethatthesuspensionofSWNTsdispersedby GumArabicshowsgoodstabilityforseveralmonths[ 5],Iobservedinstabilitiesofthe SWNTsuspensionsduringexperiments.TherheologicalbehavioroftheGumArabic solution,evenintheabsenceofSWNTs,iscomplexandcontroversial[ 89, 109]. Recently,Wangetal.[ 105]introducedaninterfacialtrappingmethodemploying aPickeringemulsionsystem[ 82]asasimplealternativetoultracentrifugation.This 42

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technique takesadvantagesofsurfaceenergydifferencesbetweenbundlesand individualSWNTsinthemixtureoforganicsolventandanaqueoussuspension ofSWNTs.Thebundleshavealargerpotentialofmeanforceinteractionwiththe interfacethananindividualSWNT,resultinginselectiveseparationofbundlesfrom thesuspension.Separationisachievedbyextractingtheaqueousphasecontaining individualSWNTs.Inaddition,thePickeringsystemscanbeusedforlarge-scale separationandhasbetterseparationefcienciesthancentrifugation[ 51]. Despiteeffortstogeneratewell-dispersedandstablesuspensionsofindividual SWNTs,makinggoodSWNTsuspensionsremainschallenging.Moreover,establishment ofmethodscharacterizingthestabilityisanurgentproblem.Forinstance,previous evaluationsofthestabilityofindividualSWNTsinthesuspensionareproblematic; stabilityofsuspensionsofSWNTsdispersedinSDSsolutionhavebeenroughlyjudged onthebasisofwhetherornotocsformaftermorethan24hours[ 75].Thestabilityof suspensionsofSWNTsdispersedinGumArabicsolutionwasevaluated[ 5 54, 76]in thesamemanner.Moreaccurateandsystematicmeasurementsofstabilitywillaidin thedevelopmentandevaluationofsurfactant-dispersedSWNTsuspensions. Inthiswork,thestabilityandrheologyoftheSWNTsuspensionspreparedby interfacialtrappingmethodwillbeexaminedandcomparedtoaconventionalmethod ofultracentrifugation.Acontrolsampleundergoeshigh-shearhomogenizationfollowed byultrasonication.Sincetherheologicalpropertiesaresensitivetothesuspension microstructure,thechangeofrheologicalproperties,suchasviscosity,canbeemployed asasystematicstandardofstability.Eventhoughthelowtorquevalueoftheaqueous suspensionsofSWNTsmakestherheologicalmeasurementdifcult,therheological measurementshavemerits;theyarerelativelyfast,preparation-insensitive,andan objectivemethodincomparisonwithopticalmethodssuchasAFMwhicharesensitive tosamplepreparationprocessesandonlymeasurethestateofSWNTswithinthe selectedeldofview[ 5, 52]. 43

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I haveexaminedtheviscositiesofthesuspensionsundergoingsteadyshear owfortheSWNTssuspensionshavingbeenstabilizedwithdifferentsurfactantsand preparationmethods.Inthiswork,theconcentrationofthesuspensionofSWNTs andthediameterofSWNTs[ 35]areopticallymeasuredbyUV-visabsorbanceand uorescencespectra,respectively.Thesteadyshearviscositieshavebeenmeasured andcomparedasafunctionofshearrateandagingtimeofthesuspension.Also,the visualstatesofthesuspensionhavebeenobserved. 3.2Experiments SWNTsuspensionsarepreparedwithaninitialmassofrawSWNTs(RiceHPR 145.1)andmixedwith200mLofasurfactantsolutionof1wt % togiveaninitial concentrationof0.03-0.2mg/mL[ 5].Sodiumdodecylsulfate(SDS),ananionic surfactant,andGumArabic(GA),ahetero-polysaccharide,areused.Thesuspensions experiencehigh-shearhomogenization(IKAT-25Ultra-Turrax)for1.5hoursand ultrasonication(MisonixS3000)for10minutes.Thisdenesthecontrolsample. AcontrolsampleisfurtherprocessedtoremovethebundledSWNTsfromthe suspensions.ToprepareultracentrifugedsuspensionsofSWNTs,controlsamplesare ultracentrifugedatspeedsof26,000rpm(BeckmanCoulterOptimaL-80K)for4hours toremovenanotubebundles.SincethebundledSWNTssettlefaster,thesupernatantof thecentrifugedsuspensioncontainingindividualSWNTsiscollected. Figure 3-2 illustratestheoverallprocessoftheinterfacialtrappingmethod:toluene (Acros,99 %)isaddedtotheaqueousSWNTsuspensionstoformatwo-phasesystem. Thetwo-phasesystemisthenshakenvigorouslyfor30secondstoincreaseinterfacial area.Phaseseparationoccurswithin1-2minutes,howeverthesolutionsareallowed tosettlefor30-60minutestoensurethatsteadystateisachievedbeforeperforming spectroscopymeasurements.Allexperimentswereconductedatanorganictoaqueous SWNTsuspensionvolumeratioof0.1. 44

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Figure 3-2.OverallprocessofremovingSWNTbundlesfromaqueoussuspensionsvia liquid-liquidinterfaces.Priortointerfacialtrapping,therawSWNTsarerst homogenizedwithahighshearmixerandultrasonicated,resultingina mixtureofindividuallysuspendedSWNTsandSWNTbundles.A)Tolueneis addedtotheaqueousphaseandmixedtoformemulsions;B)SWNT bundlesaretrappedattheemulsioninterfaces;C)Creamingand coalescenceofemulsionsseparatesthebundledSWNTs;andD)SWNT bundlesareremovedfromthebulkaqueousuid. 45

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Figure 3-3.TheSDS-dispersedSWNTsuspensionpreparedbytheultracentrifugation process.Theconcentrationofthesuspensionis0.026mg/mL.No aggregatesareobservable. TherheologyofsuspensionsofSWNTsisexaminedusinganARESLS-1,astrain controlledrheometer(TAInstruments).Aconeandplategeometrywithadiameterof50 mmandanangleof0.04radiansisusedandallexperimentsareperformedatT=20 C. Thesteadyshearviscosityismeasuredatshearratesbetween r =1 s )Tj /T3_2 7.97 Tf (1 and100 s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 ForthestabilitytestoftheSWNTsuspensions,thesuspensionsareleftundisturbedfor daysandthenstirredimmediatelybeforethetest.Thesteadyshearviscositiesat r =10 or20 s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 and r =100 s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 arecomparedasafunctionoftheageofthesamples. TheconcentrationandthediameterofSWNTsinthesuspensionsareoptically evaluatedbyvis-NIRabsorbancespectrausinganAppliedNanoFluorescence Nanospectrolyzer(Houson,TX)withexcitationfrom662and784nmdiodelasers. Measuredabsorbanceat763nminthevis-NIRspectrumisconvertedtoconcentration (mg/L)bytheBeer-Lambertlaw, A = b c (3) 46

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Figure 3-4.SteadyshearviscositiesoftheSDS-dispersedSWNTsuspensionasa functionofconcentrationandshearrate. whereAisabsorbance, istheextinctioncoefcient,bisthepathlengthofthelight, andcistheconcentration.Inthiswork,theextinctioncoefcientis0.043L/mg cm at =763nm[66]andthepathlengthis1cm.Theconcentrationsareconvertedto volumefractionusingadensityof1.45g/cm 3 ,whichisthedensityfor(7,8)SWNTs [79 ].Fluorescencespectraaresensitivetothetypeandtheaggregationstatesof SWNTs[ 75].MetallicSWNTsdonotuorescebecausethereisnobandgap.Moreover, bundledSWNTsperturbtheelectronicstructureofthetube[ 75]andconsequently,only semiconductingandindividualSWNTsuoresce.Therefore,theuorescencespectra canbeusedtoidentifythe(n,m)typeandthediameterofsemiconductingSWNTs[ 4]. 3.3ResultsandDiscussion SincethemotivationistodeterminestabilityoftheSWNTsuspensions,Irst characterizewell-dispersedsystemsandthencharacterizeasystemhavingaggregates. 47

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Figure 3-5.A)ComplexrheologicalbehaviorsofGAsolutionintheabsenceofSWNTs withoutcentrifugation.B)NewtonianbehaviorofcentrifugedGAsolutionin theabsenceofSWNTs. SDS-dispersedSWNTsuspensionpreparedbyultracentrifugationisthemost widelyused[ 5].Asacontrolsystematlowerlimitofdispersity,GA-dispersedSWNT suspensionpreparedbyhigh-shearhomogenizationfollowedbyultrasonicationis examined.Afterthecharacterizationoftheextremecases,Ifocusoncharacterizing theSWNTsuspensionpreparedbythemethodofinterfacialtrapping.Additionally,the GA-dispersedSWNTsuspensionpreparedbyultracentrifugationisexaminedtoproduce faircomparisontotheGA-dispersedSWNTsuspensionpreparedbytheinterfacial trappingtechniquebyusingsamesurfactantofGA. 3.3.1SDS-DispersedSWNTSuspensions TomonitorthestabilityofSWNTsuspensionsrheologically,rstIexaminedthe well-dispersedSWNTsuspension.Ultracentrifugationisconventionallyusedtogenerate well-dispersedsuspensionsbyremovingbundlesfromthesuspensionsinspiteof thelargetimerequirement.TheSDS-dispersedSWNTsuspensionpreparedby thecentrifugationprocessappearshomogeneouswithoutanyvisibleevidenceof aggregatesasshowninFig. 3-3. Rheologicalproperties,suchasviscosity,havebeenmeasuredfortwoconcentrations of c =0.019mg/mLand0.026mg/mLundersteadyshearow.Theviscositiesof thesuspensionsaresmallduetothelowviscosityofthesuspendingmediumand 48

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Figure 3-6.PicturesofGAsolution.GAsolution,whichhasnotbeencentrifuged,is hazyandsedimentisseenandthecentrifugedGAsolutionisclearwithout sediment(right). lowconcentrationoftheSWNTs.Still,weakshear-thinningbehaviors(Fig. 3-4)are observable.ThesuspensionofSWNTswith c =0.019mg/mLhaslargererrorbars(8% at r =10 s )Tj /T3_2 7.97 Tf (1 )thanthosefor0.026mg/mL(2 % at r =10 s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 )becausetheviscosityis closertothelowerlimitofthetorquedetectablebytherheometer. Toexplainshear-thinningofSWNTsuspensions,asimpletheoreticalcalculation oftheshearstressandtheeffectiveviscosityofthesuspensionispresentedin Appendix.ThecalculationmodelstheSWNTsasslenderbodiesandBrownianmotion isconsidered.Thetheorypredictstheaverageviscosityofthesuspensions,butfails topredicttheshear-dependentbehavioroverthisrangeofshearrates.Themodel indicatesthatotherfactors,beyondsimplealignmentprocesses,areaffectingthe rheologyofSWNTsuspensions. Totestthestabilityofthissuspension,steadyshearviscositiesaremeasuredas functionsofthetimeofaging.Theviscositydecreasesonlyby5 % at10 s )Tj /T3_2 7.97 Tf (1 after14 daysandthereisnoobservableaggregatesofSWNTs.So,thisSDS-dispersedSWNT suspensionpreparedbyultracentrifugationcanbesaidtobestableforatleasttwo weeks. 3.3.2RheologyofGumArabicSurfactantSolution IexaminedthestabilityoftheGA-dispersedSWNTsuspensionspreparedby interfacialtrappingbyperformingrheologicaltestssimilartothoseintheprevious 49

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Figure 3-7.Acontrolsample:theGA-dispersedSWNTsuspensionpreparedbyhigh shearhomogenizationandultrasonication.Theconcentrationofthe suspensionis0.028mg/mL.TheblackdotsareaggregatesofSWNTs. section.IfoundthattherheologyoftheSWNTsuspensionschangedwitheach experiment. Therheologyof1wt% GAaqueoussolutionwithoutSWNTswasstudiedto elucidatethecauseofthevariability.AsshowninFig. 3-5 A,theGAsolutionshows shear-thickeningbehavioraswellasNewtonianbehaviordependingonthesample. Thesevariousrheologicalbehaviorsareconsistentwithotherworks[ 89, 109].The complexanduncertainstructuresoftheGAsurfactantgeneratedifferentrheological behaviorsfromNewtoniantoshear-thinningaswellasshear-thickening.Ialsoobserved thattheGAsolutionformssedimentandaggregatesinsolutionwithtime(seeFig. 3-6, left),whileitisclearwithoutsedimentaftercentrifugation(seeFig. 3-6,right).Tolimit theaffectsofthecomplexpropertiesofGAsolutionontheresults,theGAsurfactant solutioniscentrifuged.Aftercentrifugation,theGAsolutionexhibitsaNewtonian behaviorandnovariabilitybetweensamplesasshowninFig. 3-5 B.Therefore,the centrifugedGAsolutionisusedtoprepareGA-dispersedSWNTsuspensions. 50

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Figure 3-8.Steadyshearviscositiesofthecontrolsamples,GA-dispersedSWNT suspensionspreparedbyhomogenizationandultrasonication,with c =0.041 mg/mL,0.028mg/mL,and0.014mg/mL. 3.3.3GA-DispersedSWNTSuspensionsasaControl Asacontrolsamplewithpoordispersity,theGA-dispersedSWNTsuspension processedbyhighshearhomogenizationandultrasonicationisexamined.Thevisual observationoflargeaggregatesandsedimentofSWNTsinthesuspensionsupports assertionsthatfurtherprocessingisneededtogeneratewell-dispersedsuspensions (seeFig. 3-7). Therheologicalpropertiesofthecontrolsamplesaremeasuredundersteadyshear ow.Threeconcentrationsof c =0.041mg/mL,0.028mg/mL,and0.014mg/mLare tested.AsshowninFig. 3-8,thesuspensionshowsshear-thinningbehaviorbetween shearratesof5 s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 and100 s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 ,whichisenhancedwithconcentration.Thesuspension of c =0.041mg/mLexhibitsthestrongestshear-thinningtendency,buttheerroris 50%. 51

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Figure 3-9.Aftersteadysheartest,aggregatesofSWNTsareformedinthecontrol sample,theGA-dispersedSWNTsuspensionpreparedbyhomogenization andultrasonication,with =0.041mg/mL.Parallelplategeometrywith diameterof50mmandgapof0.5mmisused. AftershearingtheSWNTsuspension,Iobservedaninterestingphenomenon. Largeocswithanirregularshapewerefoundinthesuspension(seeFig. 3-9).The ocsareobservedinboththeconeandplategeometryandparallelplategeometry. Theshear-thinningmightbeattributedtotheformationofaggregatesduringshearing. However,theoriginandmechanismforformationofaggregatesduringshearing oftheSWNTsuspensionsisunclear.AlthoughMaetal.[ 62, 63 ]arguedthatthe formationofaggregatestructures,named`helicalbands',mightarisefrommechanical aggregationduetocontactandwallconnement,differentsystemconditionsmake directcomparisondifcult. Also,theshear-thinningbehaviorisirreversibleasshowninFig. 3-10.Thechange inmicrostructuresoftheSWNTsislikelythecauseofirreversibility.Oncetheyform aggregatesandbundleswhichhavestrongvanderWaalsforce[ 5 ],themicrostructures 52

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Figure 3-10.Steadyshearviscositiesofacontrolsample,theGA-dispersedSWNT suspensionpreparedbyhomogenizationandultrasonication,with c =0.041 mg/mL.Thesuspensionwasagedfor18days. donotreturntotheinitialstateuponloweringtheshearrates.Therefore,therheological behaviorshowsirreversibility. Timestabilityofthecontrolsampleisstudied.Thesuspensionshavebeenaged andthenstirredrightbeforethetest.Thechangeofviscositiesisplottedasafunction ofagingtimeandshearrateinFig. 3-11.Figure 3-11 demonstratesthatthecontrol suspensionshavepoorstability.Forinstance,theviscositydecreasedby11 % in3days andby26 % in44daysatashearrateof20 s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 .Moreover,manySWNTshavesettled visiblytothebottom. 3.3.3.1GA-dispersedSWNTsuspensionspreparedbyinterfacialtrapping TheinterfacialtrappingtechniqueisbasedonthePickeringeffect,whichpredicts theadsorptionofcolloidalparticlesatinterfaces[ 82, 104].Wangetal.[ 104]argued 53

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that selectivepartitioningofSWNTbundles,asopposedtoindividualSWNTs,onthe interfacecanbeexplainedbyexaminingthefreeenergeyassociatedwiththeinterfacial adsorption.Themodelforfreeenergychange, E,uponinsertingthenanotubeatthe interfaceis E =2RLr ow )Tj /T1_1 11.955 Tf 5.48 -9.69 Td [( cos = 180 )Tj /T3_1 11.955 Tf 11.95 0 Td (sin (3) whereRisaparticleradius,Lisalength, isacontactangle,and r ow istheinterfacial tensionoftheoil-waterinterface.Since EisproportionaltoRandL,aSWNTbundle ontheinterfacelowersthefreeenergymorethananindividualSWNT,resultingin selectiveextractionofbundles. TheGA-dispersedSWNTsuspensionspreparedbyinterfacialtrappingprocess havesomeaggregatesofSWNTsinthesolutionasshowninFig. 3-12.Theamountof aggregatesinthesolutionismorethanintheSDS-dispersedSWNTsuspension,butfar lessthaninthecontrolsuspension(seeFig. 3-7). Thesteadyshearviscositiesareplottedasafunctionofshearrateandconcentration inFig. 3-13.ThesuspensionsofSWNTsshowshear-thinningbehaviorsforall concentrations.Theshear-thinningbehaviorisirreversiblesimilartothecontrolsample. Theviscositiesdonotincreaselinearlywithconcentrationinthisregime.Thegroup ofthesuspensionswiththeconcentrationmorethan0.018mg/mLshowssimilar viscositiesandthesuspensionwith c =0.009mg/mLhaslowerviscosities.Afterthe shearratesweeptest,someaggregatesofSWNTsareobservableasinthecontrol sample.ThereforethesamerationaleofBrownianmotionandformationofaggregates canbeappliedtoexplaintheshear-thinningbehavioroftheGA-dispersedSWNT suspensions. Forthesameseparationprocessoftheinterfacialtechnique,whenIusetheGA solutionwithoutcentrifugation,theGA-dispersedsuspensionsofSWNTshavealarge agingeffect.Theviscosityofthesuspensiondecreasesby11 % in2days.Ontheother hand,whenthecentrifugedGAsolutionisusedtoproducetheGA-dispersedSWNT 54

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Figure 3-11.Steadyshearviscositiesofacontrolsample,theGA-dispersedSWNT suspensionpreparedbyhomogenizationandultrasonication,with =0.041 mg/mLareplottedasafunctionofagingtimeandshearrate.The viscositiesofthesuspensionarereducedintime. suspensions,theGA-dispersedSWNTsuspensionshaveasmalleragingeffect.For instance,thecentrifugedGA-dispersedsuspensionsofSWNTsexhibita3 % decrease inviscositiesin15days.Therefore,thecentrifugedGAsolutionisusedtoproduce theGA-dispersedSWNTsuspensionpreparedbytheinterfacialtrappingmethod.The GA-dispersedSWNTsuspensionpreparedbyinterfacialtrappingprocessisstablein termsofviscositychange,butstillhasinstabilityregardingtheformationofaggregates andsedimentintime.Also,theinstabilitycanbeattributedtotheGAsurfactanteffect basedontheexperimentalresultforthecomplexGAsolutionasdescribedabove. TheestimateddiameterdistributionofSWNTsisshowninFig. 3-14 fromuorescence spectra.AsshowninFig. 3-14,theSWNTsexistasamixtureofvariouskindofSWNTs anditisalmostimpossibletoseparateeachtypeofSWNTs.Theaveragediameter 55

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Figure 3-12.TheGA-dispersedSWNTsuspensionpreparedbyinterfacialtrapping processwith c =0.026mg/mL.Here,centrifugedGAsolutionisusedfor makingthesuspensionofSWNTs. oftheSWNTsis0.97nm.ThevaluesofdiameterofSWNTsmatchwiththereported values[ 58]of0.8 1.4nmmanufacturedbyHiPcoprocess[16]. 3.3.3.2GA-dispersedSWNTsuspensionpreparedbyultracentrifugationprocess Intheprevioussection,theGA-dispersedSWNTsuspensionpreparedbyanew separationmethodofinterfacialtrappingprocesswascomparedtotheSDS-dispersed SWNTsuspensionpreparedbyultracentrifugationprocess,whichisconsideredtobea well-dispersedandstablesystem. Figure 3-15 comparesthesteadyvaluesoftheshearviscositiesforthethree differentSWNTsuspensions:acontrolsamplewith c =0.020,aGA-dispersedSWNT suspensionbytheinterfacialtrappingprocesswith c =0.026,andaGA-dispersedSWNT suspensionbyultracentrifugationwith c =0.025.Theviscosityofallsystemsnearly matchesthehighestshearrateof100 s )Tj /T3_1 7.97 Tf (1 ,demonstratingthatthesmalldifferencein concentrationacrossthethreesystemshasalimitedimpactontheresults.Differences intherheologyareapparentatlowershearrates.Exceptforthelowestshearrate, 56

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Figure 3-13.Steadyshearviscositiesareplottedasafunctionofshearrateand concentrationfortheGA-dispersedSWNTsuspensionpreparedbythe interfacialtrappingtechnique. thecontrolsamplehasanearlyconstantviscosityasafunctionofshearrateoverthe testedrangeofconcentration.Theultracentrifugationandtheinterfacialtrappingmethod producesamplesthatshearthin.Thedecreaseinviscositywithincreasingshearrate ismorepronouncedfortheultracentrifugedsample,butagreeswithinmeasurement errorwiththeresultfromthesamplepreparedusinginterfacialtrapping.Thefavorable comparisonimpliesthatinterfacialtrappingmethodproducesasuspensionofSWNTs similarinowcharacteristics,andhencedispersionquality,totheultracentrifugation method.Moreover,thevisualstateofthecentrifugedSWNTsuspensionissimilartothe GA-dispersedSWNTsuspensionbytheinterfacialtrappingprocess,butthereareless aggregatesofSWNTs. 57

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Figure 3-14.DistributionofthediameterofSWNTsintheGA-dispersedSWNT suspensionpreparedbytheinterfacialtrappingprocess.Theaverage diameteris0.092nm. FortimestabilityoftheGA-dispersedSWNTsuspensionpreparedbyultracentrifugation, theviscositychangeisinsignicantwithrespecttotheerrorbars(23 % at10 s )Tj /T3_2 7.97 Tf (1 )as shownintable 3-1.Absorbance,orescence,andRamanspectroscopyofeachSWNT suspensionsystemwerealsomeasuredandcomparedbycolleagues[ 104].Wang etal.[ 104]arguedthatincreaseduorescenceintensityanddecreasedabsorbancein comparisontothecontrolsamplesindicateselectivebundleremovalfromtheSWNT suspensionsthroughtheinterfacialtrappingmethod. 3.4Conclusions TherheologicalbehaviorsoftheSWNTssuspendedinaqueoussolutions wasexamined.First,IfoundthatGAsolutionaffectedtherheologyoftheSWNT suspensionsdependingonwhetherthecentrifugedGAsolutionisusedornot.Contrary 58

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Figure 3-15.SteadyshearvaluesoftheshearviscositiesoftheGA-dispersedSWNT suspensionsproducedafterhomogenizationandultrasonication(control), interfacialtrapping,andultracentrifugation.Theconcentrationofthe control,interfacialtrappingsuspension,andultracentrifugedsamplesare 0.020,0.026,and0.025mg/mL,respectively. Table3-1.AgingeffectoftheGA-dispersedSWNTsuspensionpreparedby ultracentrifugationwith =0.017mg/mL. Viscosity 1 st da y 47 th dayViscositychange( %) at 10 s )Tj /T3_3 7.97 Tf 6.59 0 Td (1 0.022 0.005P0.022 0.003P 3 % at80 s )Tj /T3_3 7.97 Tf 6.59 0 Td (1 0.018 0.003P0.019 0.001P 4 % to theargumentthatGAsurfactantproducesastableSWNTsuspension[ 5],the GAsolutionitselfshowscomplexrheologyandinstabilityintime.Toregulatethe complicatedrheologyofGAsolution,centrifugationisappliedbeforeaddingSWNTsin thesuspensions. Secondly,aggregatesofSWNTsareobservedinGA-dispersedSWNTsuspensions, whileSDS-dispersedSWNTsuspensionbyultracentrifugationhasnoobservable 59

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agg regates.However,forthesuspensionswithasamesurfactantofGA,theextent oftheaggregateoftheSWNTsuspensionpreparedbyinterfacialtrappingmethod decreasestotheleveloftheSWNTsuspensionbyultracentrifugation. Rheologically,allSWNTsuspensionsexhibitshear-thinningbehavior.The theoreticalcalculationaccountingBrownianforcecannotpredicttheshear-dependent behaviorovertherangeofshearrate,althoughitpredictsviscositiesclosetothemean valueofviscositiesobtainedbyexperiments(seeAppendix).Moreover,Iobservedthat shearowpromotestheformationofSWNTaggregatesandsimilarobservationshave alsobeenreportedbyothers[ 63 64 ].Thissupportsthehypothesisthatfactorsbesides Brownianmotion,suchasformationofaggregates,playaroleinSWNTsuspension rheology.Similarly,Maetal.[ 62]suggestedanaggregation =orientationmodeltoexplain theshear-thinningandaggregateformation. Forstabilitytest,allGA-dispersedSWNTsuspensionsformsedimentand aggregatesintime,althoughtheviscositychangecanbesmall.SDS-dispersed SWNTsuspensionpreparedbyultracentrifugationshowsgoodstabilitybasedon bothnoformationofaggregatesandsmallviscositychange.Toevaluateanewprocess ofinterfacialtrappingmethodexcludingasurfactanteffect,GA-dispersedSWNT suspensionsarecompared.Asaresult,thenewseparationprocessofinterfacial trappingproducesacomparablesuspensiontoultracentrifugationintermsofamount ofaggregatesandtimestabilityofviscositywhenthesameGAsurfactantisused. Besides,thisnewtechniqueprovidesasimpleroutetoachieveeconomiclarge-scale dispersionofSWNTs.However,thereisstillapossibilityofproducingmorestable suspensionwithoutaggregatesbyusingbettersurfactant. 60

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CHAPTER 4 RHEOLOGYOFOSCILLATINGSUSPENSIONSOFNON-COLLOIDALSPHERESAT SMALLANDLARGETOTALSTRAINS 4.1Introduction Therheologyofviscoussuspensionsofnon-colloidalandneutrallybuoyantspheres hasbeenstudiedprimarilyunderconditionsofsteadyshearthroughexperimentation [20 44 102]aswellassimulation[ 10, 93, 94]andisthesubjectofrecentreviews [69 98 ].InthisregimeofinterestwherethePecletnumberislarge( > 10 3 )andthe Reynoldsnumbersmall( < 10 )Tj /T3_1 7.97 Tf 6.59 0 Td (3 ),theeffectiveviscosityatsteadystateisexpected tobeafunctionofvolumefraction,butnottherateofshear[ 57, 98]owingtothe presumedlackofdependenceofthestructureofthesuspensionuponanythingother thanhydrodynamicforces.Theshearviscositiescanbesuccessfullydescribedusing correlationssuchasthatofKriegerandDaugherty[ 57]aswellasmoreadvanced descriptions[ 97]thatattempttopredictthemicrostructureandrelateittothemore generalrheologicalproperties.However,theaccuratemeasurementoftherheological propertiesofsuchasuspensionunderconcentratedconditionsrequiresavoidance ofshear-inducedmigration[ 61 ],whichcanalterthedistributionofparticleswithinthe testinggeometryandfrustratetheaccurateevaluationoftheexperimentalresults. Relativelyfewstudieshavebeenmadeofconcentratedsuspensionsinoscillatory shearows.Investigationshaveconrmedthattheoscillatoryresponsecanbe nonlinear[ 12, 14, 44, 72]withtheexactresultsdependinguponconcentration, andstrainamplitude, A= H .Gadala-MariaandAcrivos[ 44]andBreedveldetal.[12 ] reportedasinusoidalandlinearresponseatsmallamplitudesofstrainasopposedto aclearlynonlinearresponseatastrainamplitudeof A= H 1,whereasBrickerand Butler[ 14 ]reportedthattheresponseatsmallamplitudesisstillslightlynonlinear.At strainamplitudesgreatlyinexcessofone,bothBreedveldetal.[ 12]andBrickerand Butler[ 14 ]foundthattheresponsereturnedtoaformclosertolinearthanthatobserved at A=H 1.BothGadala-MariaandAcrivos[ 44 ]andBrickerandButler[ 14 ]found 61

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the rheologytobeindependentof,oronlyweaklydependenton,frequencyoverlarge ranges,whereasBreedveldetal.[ 12]reportedadependence. Theuseofdifferentsuspensionsystemsandmeasurementgeometriesmay beinpartresponsibleforanydiscrepanciesbetweendifferentinvestigations.Also, differentstudieshavereportedresultsfromdifferentpointsintheexperiments:atvery shorttimes[ 12],averagedovertherstfewoscillations[ 14 ],andafterafewhundred oscillations[ 44].Thepointofmeasurementmaymattersincetherheologycanchange rapidlyoverthersttentotwentyoscillationsdependinguponthespecicconditions, asreportedhere.Furthermore,therheologicalresponsecancontinuetochangeover manythousandsofoscillations[ 14, 15, 46 ]andthenalvalueofthecomplexviscosity candependuponthestrainamplitudeinanon-monotonicfashion[ 14]. Inthispaper,experimentalcharacterizationoftheoscillatoryrheologyofa suspensionofnon-colloidalspheresinaNewtonianuidexpandstheexperimental measurementstoawiderrangeofconditionsthaninpreviousworks.Wequantifythe deviationfromalinearresponseandndthatalthoughtheresponsemayappearcloser toasinusoidalformatsmallamplitudesofoscillation,theresultsaremorenonlinear thanatastrainamplitudeof A= H 1 .Ignoringthenon-linearities,thestressresponses areseparatedintocomponentsin-phaseandout-of-phasewiththestraintorevealthe existenceoftwoscales.Detailedstudiesatsmalltotalstrainsdemonstratealargeand rapiddecayinthestoragemodulusoverthersttentotwentyoscillationsforsome combinationsofthevolumefractionandamplitudeofoscillation.Thedynamicviscosity attainssteadystateonamuchslowerscale,requiringmanythousandsofoscillations, andthecomplexviscosityatlargetotalstrainshasanon-monotonicdependenceonthe strainamplitudeforvolumefractionsinexcessof0.2. 4.2Experiments Polystyrenebeadswereusedintheexperiments.Characterizationoftheparticle sizeanddistribution,asmeasuredbyaCoultercounterandveriedbyscanning 62

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electron microscopy,returnedameandiameterof 99 6 m.Thesesphericalparticles weresuspendedinpolyalkyleneglycoloilatvolumefractionsrangingfrom =0.2to 0.5.PolyalkyleneglycoloilisNewtonianwithaviscosityof18.6Pandadensityof1.052 g/cm 3 thatcloselymatchesthatofthepolystyrenespheres.Testsdemonstratedthe stabilityofthissuspensionsystemoverthelongtimesrequiredfortheexperiments[ 14]. MeasurementswereperformedwithanARESLS-1straincontrolledrheometer(TA Instruments)usingbothaparallelplateandCouettegeometry.Theparallelplateshave adiameterof50mmandtheinnercylinderoftheCouettecellhasadiameterof32mm; thegap H wasmaintainedat1mmforallexperimentsinbothgeometries.Allvalues reportedfortheparallelplates,includingshearrate,strainamplitude,totalstrain,and viscosities,arebasedupontheconditionsevaluatedattheedgeoftheplates.When comparedonthisbasis,themeasurementsoftheoscillatoryresponseofthesuspension inthetwogeometriesgivequalitatively,andsometimesquantitatively,similarresultsas reportedhereandinapreviousstudy[ 14]. Thesuspensionswerepreshearedat24 s )Tj /T3_1 7.97 Tf 6.59 0 Td (1 foradurationof120to180spriorto eachoscillatorymeasurementtoensureaconsistent,initialconguration.Thestrainof 2880to4320doesnotcauseasignicantmigrationofparticlesfromtheshearinggapof theCouettedevicetothereservoirbelowthecup[ 61].Fortheparallelplategeometry, nomigrationbehaviorisobservablefromtherheologicaldataoverlargeperiodsof steadyshear[ 14]andatotalstrainof2880givesaviscositythatnearlymatchesthe steadyvalue.Oscillatorymeasurementswereinitiatedatafrequencyof1.59cycles/s immediatelyfollowingpreshear.Thestrainamplitudeoftheoscillations, A= H ,was variedbetween0.05and5.0.Thetemperatureofthesuspensionwasmaintainedat 25.00 0.05 o Cduringtherheologicalmeasurements. Theparticle-basedReynoldsnumberis 10 )Tj /T3_1 7.97 Tf (5 atmostandtheminimumPeclet numberis 10 8 fortheseexperiments,indicatingthatinertialeffectsandthermaldiffusion arenegligible.Othercolloidalforces,suchaselectrostaticinteractions,areexpectedto 63

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be ofnoimportanceduetotherelativelylargesizeoftheparticles.Comparisonsfrom measurementsatfrequenciesof0.159cycles/swithmeasurementsat1.59cycles/s indicatethattheresultsareindependentoftimewithinthisrangeofparameters,which isconsistentwiththeexpectationthatshearviscositydependsprimarilyuponthetotal strainandstrainamplitudeforthissystem.Allresultsareconsequentlyreportedin termsofthevolumefraction ,thestrainamplitude A=H ,andeitherthenumberof oscillations N ortotalstrain r t =4AN = H 4.3ResultsandDiscussion Examplesofthedetailedstressresponsesatrelativelysmallvaluesofthe totalstrainarereportedrstandcomparedtolinearresponses.Notingthatthe out-of-phaseandin-phasecomponentsoftheapparentlinearresponseschangeon clearlydifferentiatedscales,thedataatsmallvaluesof N arepresentedinSec. 4.3.2 beforepresentingtheresultsforlarge N inSec. 4.3.3. 4.3.1StressResponsesatSmall N Figure 4-1 showstheoutputstressresponsetothesinusoidalstraininputatthree amplitudesofoscillationduringthe15 th oscillationforsuspensionsofvolumefraction =0.2and0.4.EachsetofdatashowninFig. 4-1 representsanaverageoveratleast fourexperiments;theexperimentswerehighlyreproducible,asreectedbyastandard deviationinthedatathatisnolargerthanthesizeofthesymbolsasappearinginthe gure.Thedataiscomparedtoasinusoidalcurvehavingafrequencyidenticaltothat oftheinputstrainthatbesttstheexperimentaloutputsonthebasisofthesquared error;assuch,thecurverepresentsthelinearresponsethatbestapproximatesthe experimentaldata. Comparingthedataandthettothedataillustratesthatthestressescandeviate fromlinearresponsetheory,withthelargestdifferencesoccurringforthemore concentratedsuspensionandatthesmalleramplitudesofoscillation.Thissummary ofthedataappearinginFig. 4-1 isbaseduponquanticationsofthedeviationofthe 64

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Figure 4-1.Stressresponsesofthesuspensionsduringthe15 th oscillation(N =15). Thelinesrepresentthelinearresponsesthatbestapproximatethe experimentaldata,shownassymbols.Circlesrepresentexperimentaldata measuredintheCouettegeometryandtrianglescorrespondto measurementsfromtheparallelplategeometry.GraphslabeledfromA)to C)areforthesuspensionwith =0.2andgraphslabeledfromD)toF)are forthesuspensionwith =0.4. datafromthettothedataasmeasuredbythecorrelationcoefcient, R 2 ;thereported valuesof R 2 arefromthetstotheCouettegeometry. Resultsfromthesuspensionwith =0.2correspondcloselytoalinearresponse forthehigheramplitudesof A=H =1.0and5.0asindicatedbycorrelationcoefcientsof 0.9983and0.9996,respectively.Thecomparisonispoorerat A=H =0.05,asapparent fromavisualcomparisonoftheoutputstressandtheapproximate,linearresponseand asreectedbytherelativelypoorvalueof R 2 =0.9864. Thesuspensionwith =0.4alsoappearstocorrespondcloselytoalinear responseatthelargestamplitudeof A= H =5.0,thoughthedifferencesthatcanbe 65

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Figure 4-2.Theapparentvaluesof 00 and 0 plottedasafunctionofnumberof oscillations, N ,fortherst12cyclesofoscillation.Resultsareshownforthe suspensionofvolumefraction =0.4forfouramplitudesofoscillation, A= H asmeasuredwithintheparallelplategeometry(toptwo)andtheCouette geometry(bottomtwo). seeninFig. 4-1 fordataproducedintheCouettegeometryresultinasmallercorrelation coefcientof R 2 =0.9974thanthecorrespondingresultfor =0.2.Theattened,or broadened,stressresponsefor =0.4and A=H =1.0isobviouslynotsinusoidal.Yet, thecorrelationcoefcientof R 2 =0.9910islargerthanthatatthesmalleramplitudeof A= H =0.05wheretheresponseappearstobemoresinusoidaland R 2 =0.9825. ComparisonsbetweendatacollectedfromtheCouetteandparallelplategeometries appearwithinFig. 4-1.Quantitativedifferencesbetweenthetwosetsofresultsexist, withthelargestdiscrepancytakingplaceforthemoreconcentratedsuspension( =0.4) oscillatedatanamplitudeof A= H =5.0.However,thequalitativebehaviorissimilarin allcases.Forexample,theslightlyasymmetricshapeoftheresponseat A=H =0.05, 66

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f or =0.2and0.4,isseeninbothgeometries.Evenat A=H =5.0and =0.4,wherethe differencesbetweenmeasurementsfromtheCouetteandparallelplategeometriesare largest,therelativedifferencesbetweenthedataandtheapproximate,linearresponse followsimilartrends. Studies[12, 44]ofsimilarsystemshavereportedthedistortionofoutputstress signalsforconcentratedsuspensionsthatismaximuminthevicinityof A= H 1 and closertolinearatsmalleramplitudes.However,quanticationsofthedeviationsfrom linearitywerenotreportedinpreviousstudies[ 12, 14, 44].Thequanticationsofthe deviationfromalinearresponseperformedhereindicateamorenonlinearresultfor thesmalleramplitudesof A=H =0.05thanat A= H =1.0,althoughtheresponseat0.05 visuallyappearsclosertoasinusoidalformthanat A=H =1.0.Thedifferenceinthe conclusionsconcerningthelinearityoftheresponsescouldbeduetothisfact.The differentconclusionsalsocouldarisefromthedependenceoftherheologyonthe specicchoiceofparticlesandsuspendinguidsinadditiontothedifferentconditions forthemeasurements. 4.3.2ApparentViscositiesatSmall N Fromtheassumptionofalinearresponse,thetstothedatasimilartothoseshown inFig. 4-1 canbedecomposedintoalossmodulus G 00 andstoragemodulus G 0 that aredividedbythefrequencytogivetheassociatedviscositiesof 0 and 00 .Datafrom theexperimentsisreportedintermsofdynamicviscosities( 0 G 00 =! ),out-of-phase componentofthecomplexviscosities( 00 G 0 =! ),andcomplexviscosities( = 0 )Tj /T3_0 11.955 Tf 10.39 0 Td (i 00 ) fromthispointforward.Thesequantitiesarecalculatedfromtheclassicaltheoryofsmall amplitudeoscillatoryshear[ 70],sovaluesreportedhereshouldbeinterpretedas apparentquantitiesthatdonotstrictlycorrespondtotheclassicaldenitionsbasedupon linearresponsetheory[ 12, 14]. Figure 4-2 showstheevolutionoftheapparentviscosities 0 and 00 forthe suspensionofvolumefraction =0.4overtherst12oscillationsfollowingcessation 67

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Figure 4-3.Theapparentvaluesof 00 and 0 plottedasafunctionofnumberof oscillations N fortherst20cyclesofoscillation.Resultsareshownforthe suspensionofvolumefraction =0.2forfourvaluesoftheamplitudesof oscillationasmeasuredwithintheparallelplategeometry. ofthepreshear.Eachsetofdataisaveragedfromfourexperimentsandthetest resultsarerepeatable.Forinstance,themaximumerrorof 00 is4% bothintheparallel plateandCouettegeometry.Overtheserstfewoscillations,thedynamicviscosities, 0 ,remainnearlyconstantforallamplitudesofoscillation.Since 0 isthedominant contributiontothecomplexviscosity, alsomaintainsanapproximatelyconstantvalue. Behaviorssimilartothecaseof =0.4for 0 areobservedforthesuspensionswith = 0.2,0.3and0.5;Fig. 4-3 showstheresultfor =0.20. Incontrastto 0 ,Fig. 4-2 showsthatthequalitativebehaviorof 00 varieswiththe amplitudeofoscillation.Forthehighestandlowestamplitudesof A =H =0.05and5.0, theapparentvalueof 00 declinesonlyslightlyovertherst12oscillations,whereas thevaluedecreasesbyanorderofmagnitudeormorefor A=H =0.5and1.0.The qualitativebehaviorof 00 alsochangeswithvolumefraction.Figure 4-3 showsthatthe valuedeclinesfortheamplitudeof5.0inadditionto A= H =0.5and1.0forthecaseof =0.20.Also,thereductionin 00 for =0.2islessthanthatobservedfor =0.4. Figure 4-2 includesacomparisonbetweendatacollectedfromtheCouetteand parallelplategeometries.Theresultsfromthetwogeometriesagreequalitatively, thoughquantitativedifferencesexist.Similartothedataspecicto N =15(seeFig. 4-1), thelargestquantitativedifferenceisatthehighestamplitudeof A= H =5.0,wherethe 68

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Figure 4-4.Theapparentvaluesof 00 and 0 forpreshearedsuspensionsarecompared tothoseforsuspensionshavinganinitiallyrandomcongurationofthe particles.Resultsareshownforthesuspensionofvolumefraction =0.4 shearedintheparallelplategeometry. valueof 0 is 10% higherintheparallelplategeometryasopposedtotheCouette geometry.Also,thereductionintheapparentvalueof 00 for A= H =0.5and1.0isnot aslargewhenmeasuredintheCouettedeviceasopposedtotheparallelplatedevice. Similar,comparableresultsbetweenthedatafromthetwogeometrieswasfoundforall volumefractionsandamplitudesofoscillationstudied. Thedeclinein 00 canspananorderofmagnitude,asisthecasefor =0.4atstrain amplitudesof A= H =0.5and1.0.Anorderofmagnitudereductionintheapparentvalue of 00 wasalsoreportedforthespeciccaseof =0.3and A=H =1.2byCort e etal. [27 ],thereductionoccurredrapidlyuponinitiatingtheoscillatoryshearfollowingashort periodofsteadyshearasseenhereinFigs. 4-2 and 4-3.Therapiddecayof 00 might beassumedtoberelatedtothemicrostructuredevelopedduringthepreshear.Testing demonstratesotherwise,asmadeclearinFig. 4-4.Despitewhetherstartingfromthe preshearedorrandomstate,thequalitativebehaviorremainssimilar. Giventhepresumedabsenceofanycolloidalforces,thechangesin 00 must representalterationsintherelativearrangement,ormicrostructure,oftheparticles insuspension.Cortetal.arguedthattherapidattainmentofthesteadyvalueof 00 representsthecompletionofthereorganizationofthesuspensionintoareversible, absorbingstate[ 27 ].Theconclusionwasbolsteredbydirectmeasurementsofparticle 69

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displacements andamodelthatconsideredparticlecollisions.Asnotedinthenext sectionhowever,therheologycontinuestochangeslowlyanddoesnotattainasteady stateuntilverylargevaluesofthetotalstrain,ornumberofoscillations N 4.3.3ApparentViscositiesatLarge N Whiletheapparentvalueof 00 remainsconstantofcontinuestodeclinetoan immeasurablysmallvalueaftertherst10-20oscillations,thevaluesof 0 and changeoverlargetotalstrainsforthemoreconcentratedsuspensions.Figure 4-5 shows theevolutionwithtotalstrainofthecomplexviscositiesofthesuspensionatavolume fractionof =0.3forfouramplitudesofoscillation.Thecomplexviscositieshavebeen normalizedbythecomplexviscositiesat N =15todepictthechangesatlarge N more clearly.Forstrainamplitudesof A= H =0.05and0.5,theviscositiesincreaseoveratotal strainthatisverylarge;thesuspensionsshowdecreasesintheviscositiesforthestrain amplitudesof A= H =1.0and5.0.Reachingthesteadyvalueofthecomplexviscosities requiresalargetotalstrainof r t 10 4 forthesmalleststrainamplitudeof A= H =0.05. Similarexperimentsoverlargetotalstrainswereperformedalsoforvolumefractions of =0.2to0.5.Theconcentratedsuspensions( 0.3)exhibitedchangesinthe rheologyoverlargetotalstrains,althoughtheyshoweddifferenttransientbehaviors dependinguponthevolumefractionandappliedstrainamplitude.Forinstance,the viscositiesofthesuspensionwith =0.3increasewith r t at A= H =0.05and0.5(see Fig. 4-5).However,theviscositiesfor =0.4decreasewith r t at A= H =0.5asseenin Fig. 4-6.Figure 4-6 alsoshowscomparisonsofthecomplexviscositiesat N =15and theresultatlarge r t forthreeadditionalamplitudesofoscillation.Forthesuspensionof =0.4,thecomplexviscositiesdecreasewith r t for A=H 0.1 after N =15. Thesuspensionsexperienceoscillatoryshearforlargetotalstrainsbeforeasteady valueoftheviscosityisattained,asseeninFig. 4-5.Thisraisesthepossibilitythatthe observationisduetoashear-inducedmigrationorasmallnon-hydrodynamiceffect, suchasaslightmismatchindensitiesbetweentheuidandparticles.Toinvestigate 70

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Figure 4-5.Apparentcomplexviscosityversustotalstrainforstrainamplitudesranging from0.05to5.0withavolumefractionof =0.3measuredinparallelplates. Thecomplexviscosityhasbeennormalizedbythecomplexviscosityat N = 15. thesepossibilities,therheologicalbehaviorsofthesuspensionswereexaminedinthe parallelplateandCouettegeometryandfoundtobeinqualitativeagreementandas alsofoundinapreviousstudylimitedtothevolumefractionof =0.4[14].Though thesimilaritybetweenmeasurementresultsinthetwogeometriesfailstoeliminate thepossibilityofeitheroftheseproblems,theagreementofthemeasurementsseems unlikelyifeitherprocessisoccurring.Furtherevidencethatneithersedimentationnor shear-inducedmigrationcausesthechangeinviscosityoverlarge N isprovidedby simulations[ 15 ]thatpredictsimilarchangesofthecomplexviscosityoverlargetotal strains. Recentstudies[15, 27, 83 ]investigateductuationsinparticlepositionsatthe microscopiclevelinoscillatoryshearbytrackinglabeledparticles.Theyfoundastrong declineinuctuationsuponloweringthestrainamplitude,wherethepointoftransition dependsuponthevolumefractionofthesuspension.Althoughrheologycannotgive 71

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Figure 4-6.Comparisonofthecomplexviscositiesasafunctionof A= H for =0.4 evaluatedat N =15(circles)andatlargetotalstrainsapproachingthe steadystatevalues( N ss ,triangles).Thesemeasurementsarefromthe Couettegeometryandthecomplexviscosityhasbeennormalizedbythe viscosityforsteadyshear, ss ,atthesamevolumefractionof =0.4. adirectmeasureofmicroscopicuctuationsinparticlepositions,themacroscopic rheologicalmeasurementsaresensitivetotherelativearrangement,ormicrostructure, oftheparticles.Consequentlythechangesinrheology,suchasthoseshowninFig. 4-5, indicatetheexistenceofalterationsinthemicrostructure.Thechangespersistforlarge totalstrainsandareobservableevenatthesmallestamplitudesofoscillationstudied. Figure 4-7 showsthecomplexviscosityatlargetotalstrains, ss ,asafunctionof bothvolumefractionandstrainamplitude.Eachdatapointfor ss representsavalue averagedfromatleastfoursetsofexperiments;thevalueshavebeennormalizedby themeasuredvaluesoftheviscosityforsteadyshear( ss ),whichcloselyagreeswith establishedcorrelations[ 57 ].DatashowninFig. 4-7 for =0.2and0.3wastakenfrom measurementsintheparallelplategeometry,whereasmeasurementsshownfor = 0.4and0.5weregeneratedintheCouettegeometry.NotethattheresultsshowninFig. 4-7 havebeenveriedtobequalitativelyindependentofthegeometryforthisrangeof 72

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Figure 4-7.Thecomplexviscosityatlargetotalstrainsasafunctionofstrainamplitude A=H forvolumefractions rangingfrom0.20to0.50.Thecomplex viscositiesateachvolumefractionhavebeennormalizedbythesteady shearviscosity, ss ,attheidenticalvolumefraction. parameters.Forthesuspensionof =0.2,thecomplexviscosityatsteadystatedoes notexhibitacleardependenceonstrainamplitude.Thecomplexviscosityofthemore concentratedsuspensions( 0.3)showaweakdependenceontheamplitudeof oscillationthatincreaseswithvolumefraction.At =0.3,thedifferencesbetween atdifferentamplitudesofoscillationsareonlyafewpercent,butaresignicantasseen bycomparingvaluesandrelativeerrorsasshowninFig. 4-7.Thedependenceof on A= H isstronglyevidentforthehighervolumefractionsof0.40and0.50. Figure 4-7 showsthatthevalueofthecomplexviscosityatlargetotalstrains dependsonthestrainamplitudeforvolumefractionshigherthan0.2,presumably duetotherearrangementoftheparticlemicrostructureratherthanamacroscopic demixingofthesuspensionasarguedabove.Remarkably,thecomplexviscositiesare non-monotonicforthemoreconcentratedsuspensions,havingaminimumvaluein 73

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the vicinityof A=H 1 thatcanbeassmallas ss = 2.Thistrendisconsistentwiththe previousmeasurements[ 14]thatwerelimitedto =0.4. Furthermore,simulationsofamonolayerofparticlesatanarealfractionof0.6, whichcorrespondsto =0.4,producedcomplexviscositiessimilartothoseobserved herethatcorrelatedwithchangesinthemicrostructure[ 15 ].Forexample,thehigh viscositiesatlowstrainamplitudessuchas A= H =0.05arerelatedtothestructure ofalocallyorderedcrystal.Theminimumviscositiesat A= H =1.0correspondtoa microstructuresoforderedlayersofparticlesalignedintheowdirection.Forthehigher strainamplitudessuchas A= H =5.0,themicrostructureissimilartothatseenunder steadyshearconditions.Similarchangesofthemicrostructureuponchanging A=H are probablyresponsibleforthenon-monotonicviscositiesseeninFig. 4-7 forothervolume fractions. 4.4Conclusions Therheologyofanon-colloidalsuspensionofspheressubjecttooscillatorymotion hasbeenstudied.Performingdetailedmeasurementsatrelativelysmalltotalstrains andoverlargetotalstrainsdemonstratesthattherheologyofthesuspensionvaries overtwodistinctscalesdependinguponthevolumefractionandstrainamplitude.Inthe rstregime,thesuspensionsdemonstratearapiddeclineinthevalueof 00 depending uponthestrainamplitudeasapparentinFigs. 4-2 and 4-3.Measurementsstartingfrom bothpreshearedandrandomcongurationindicatethatthisbehaviorisqualitatively independentoftheinitialcondition. Inthesecondregime,therheology,dependinguponthevolumefractionand amplitudeofoscillation,developsoveraverylargetotalstrain,ornumberofoscillations. Argumentshavebeenmadethattherheologychangesduetoaslowlydeveloping microstructure,ratherthanamacroscopicmigrationmechanism.Furthermore,the steadyvalueoftheapparent,complexviscosityisanon-monotonicfunctionofthestrain 74

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amplitude forvolumefractionsinexcessof0.20.Forthesesuspensions,thecomplex viscosityisminimumatastrainamplitudeofapproximatelyone. Extendingtherangeofvolumefractionsandtotalstrainsbeyondprevious investigations,aswellasamoredetailedstudyoftheresponseatsmalltotalstrain, hasprovidedamorecomprehensiveviewoftherheology.Theresultspresented hereshouldenableimprovedmodelsofsuspensionrheology,informcomparisons withsimulations,andaidinthedevelopmentofimprovedmodelsforthedynamicsof concentratedsuspensionsinowsmorecomplexthansimpleshear. 75

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CHAPTER 5 STOKESIANDYNAMICSSIMULATIONSFOROSCILLATINGSUSPENSIONSOF NON-COLLOIDALSPHERESATSMALLTOTALSTRAINS 5.1Introduction Therheologyofconcentratedsuspensionsinoscillatoryshearowsisnotwell understoodasarguedinChap. 4 .Thendingsfromexperimentsconrmthatthe oscillatoryresponsecanbenonlinear,ashaveotherstudies[ 12, 14 44 72 ].The oscillatoryrheologyofconcentratedsuspensionsofnon-colloidalspheresobserved atlargetotalstrainscanbeexplainedbytheslowevolutionofthemicrostructures withstain,whichisveriedbyStokesiandynamicssimulations[ 15].However,the rheologyobservedatsmalltotalstrainsneedsfurtherinvestigation:thestress responsesdecaywithtimeatsmallstrainsandthenseemtoreachanewsteady state.Thecorrespondingviscosity( 00 )oftheapparentelasticcomponentsshowsa rapiddecaydependinguponstrainamplitude,whilethedynamicviscosity( 0 ),relatedto theapparentviscouscomponents,remainsnearlyconstantatsmalltotalstrains. Inthiswork,Stokesiandynamicssimulationsareperformedtoelucidatethe mechanismatsmalltotalstrains,orshort-timerheology.Specically,wefocusonthe originsofelasticityatasmallnumberofoscillations.AbriefexplanationoftheStokesian dynamicssimulationsfollowsinSec. 5.2 andpreliminaryresultsfromthesimulations arepresentedinSec. 5.3. 5.2RheologySimulations:StokesianDynamicsSimulations TheStokesiandynamicsmethodisusedtosimulateoscillatoryshearowsof suspensionsofnon-colloidalspheres.DetailsofthismethodarepresentedinBricker andButler[ 15].Thehydrodynamicforce, F ,andstresslets, S ,ontheparticlesare relatedtothevelocities, U ,andtherateofstrain, E ,bytheresistancetensor, R 0 B @ F S 1 C A = R 0 B @ U)]TJ /T1_2 11.955 Tf 21.29 0 Td (< u > E 1 C A (5) 76

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Figure 5-1.Acartoonoftheperiodiccellusedinthesimulations,whichisnottoscale. Wallparticlesarewhiteandbulkparticlesaredark.Alayerofclearuidis introducedunderthelowerwallduetotheperiodicboundaryconditions. where < u > isthemeanvelocityoftheentiredomain. IntheStokesiandynamicssimulations,theresistancetensorisexpressedbya combinationoffar-eldandnear-eldcomponents, R =(M 1 ) )Tj /T3_2 7.97 Tf 6.59 0 Td (1 + R 2 b )-222(R 1 2b (5) where M 1 isthefar-eldmobilitytensor, R 2b isthenear-eldtwobodyresistance tensor,and R 1 2b isthefar-eldcomponentof R 2 b ,whichissubtractedtopreventdouble counting. ShearowisinducedbymovingwallsinthesamemannerasNottandBrady [74 ]andSinghandNott[ 95].Thewallsaremodeledbyamonolayerofastringofthe spheresasillustratedinFigure 5-1.Thewallparticleshaveanimposedvelocityonlyin thex-directionandthevelocitycomponentsinthey-andz-directionsaresettozero. Theimposedwallvelocityis U w x = U max cos(! t ), (5) where U max isthemaximumwallvelocity,whichissettooneand isthefrequencyof theoscillationsandtistime.Theupperwallmoveswith U x ( t ) andthelowerwallmoves 77

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with )Tj /T3_0 11.955 Tf (U x ( t ) Duetotheperiodicboundarycondition,alayerofclearuidisplacedat thebottomofthecell.Sincethemovingwallsarethedrivingforceoftheshearow,the meanvelocityoftheentiredomain, < u > issettozeroandtherearenoexternalbody forces.Weemploytheconstraintinthesimulations, X N w F w x =0, (5) where N w isthenumberofwallparticlesandthesummationisoverupperandbottom walls.Thisconstraintisimposedtoensurethattheowispureshear. Topreventparticleoverlapping,ashortrangerepulsiveforceisemployed[15 74 95], F f = F o e )Tj /T1_4 7.97 Tf [( 1 )Tj /T3_0 11.955 Tf 11.96 0 Td (e )Tj /T1_4 7.97 Tf 6.59 0 Td ( r f jr f j (5) where F f is theforceexertedbysphere f onsphere .Theparameters F o and are themagnitudeandtherangeoftheforce,respectively.Theseparationdistancebetween thesurfacesofthespheresis and r f istheunitvectorconnectingthecentersof thespheres.Theparametersof F o and aresetto F o =1.0x10 )Tj /T3_5 7.97 Tf (4 and =100in SinghandNott[ 95].However,wevarytheparameterstoinvestigatetheeffectofthe magnitudeandtherangeoftheforceonshort-timerheology. Thefar-eldinteractionsareupdatedevery10timestepsasintheworkofNottand Brady[ 74 ]andBrickerandButler[ 15 ]whilethenear-eldlubricationinteractionsare updatedateverytimestep.Inthisstudy,thenumberofwallparticlesis14andthegap issetto30.Theparticlediameterissetto2andthearealfractionof = 0.60istested. ThepreliminaryresultsreportedinSec. 5.3 areanaverageoverseveralruns. Theshearstresscanbeevaluatedfromtheforcesonthewallparticles[ 94]orthe stressletsactingonthesuspendedparticles[ 11].Theshearstressfromtheforceson thewallparticlesisevaluatedby yx = N w X i =1 F x A y (5) 78

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Figure 5-2.Theoutputstressforthesuspensionofthevolumefraction =0.4atstrain amplitudeof A= H =1.0andafrequencyof =0.1Hz. whereasthestressfromthestressletsofthesuspendedparticlescanbeevaluatedby yx = N s V (< S H yx > + < S P yx > ), (5) where N s is thenumberofthesuspendedparticlesandthesuperscript H and P denotes thehydrodynamicportionandtheshort-rangerepulsionportiondenedinEquation 5.Althoughequation 5 representstheparticlecontributiontothestress,equation 5 representsthetotalstress.Inthiswork,onlytheparticlecontributiontothestress willbereported.Therefore,thetotalstressinEquation 5 ismodiedtotheparticle contributionbysubtractingtheportionofthewallforce[ 15 ]. 5.3PreliminaryResultsandDiscussion SimilartoexperimentalresultsinChapter 4,thestrain-stressresponseisplotted asafunctionofsimulationtimeinFig. 5-2.Theappliedstrainamplitudeis1.0with afrequencyof =0.1Hz.AsseeninFig. 5-2,theoutputstressshowsanon-linear responseandtheshearstressdecayswithtimeandseemstoapproachasteady response.Thisstressdecayatshorttimesissimilartotheexperimentalresults.The 79

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Figure 5-3.Theapparentvaluesof 00 and 0 plottedasafunctionofnumberof oscillations, N ,fortherst12cyclesofoscillation.Resultsareshownforthe suspensionofvolumefraction =0.4foramplitudesof A=H =1.0. linerepresentstheviscouscomponentofthestress.Thedeviationfromthettingline indicatesnon-linearresponsesoftheoutputstressatagivenstrainamplitude. UsingthesamemethodsasdescribedinChap. 4,theapparentvaluesof 00 and 0 areevaluatedbasedontheoutputstressesfromFig. 5-2.Figure 5-3 showsthe apparentvaluesof 00 and 0 asafunctionofnumberofoscillationsforthesuspension ofarealfraction = 0.60,correspondingtothevolumefraction =0.4,atastrain amplitudeof A= H =1.0.Here,theconditionsof F o =1.0x10 )Tj /T3_2 7.97 Tf (4 and =0.01areapplied fortheshortrangerepulsiveforce.AsseeninFig. 5-3,the 00 exhibitsadecreaseand 0 remainsalmoststeadyliketheexperimentalresults(seeChap. 4,Fig. 4-2).Therefore, ourStokesiandynamicssimulationssuccessfullyreproducetheexperimentalresultsand supportstheobservationofanelasticcomponentatasmallnumberofoscillations. Moreover,wefoundthatthesimulationalresultsaresensitivetotheshortrange repulsiveforce.Whenweappliedtheconditionof F o =1.0x10 )Tj /T3_2 7.97 Tf 6.59 0 Td (4 and =100likein SinghandNott[ 95]andBrickerandButler[15],thesimulationresultsshownodecayof 00 .However,decreasingtherangeofrepulsiveforcefrom =100to =0.01,wewere 80

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Figure 5-4.Theapparentvaluesof 00 and 0 forsuspensionshavingdifferentinitial congurations;preshearedandrandomconguration. abletoobtainthedropin 00 .Thismayindicatethattheshortrangerepulsion,meant tomimicacontactforce,canbethesourceoftheelasticityinoscillatingsuspensions. However,morestudiesareneededtoclarifytheoriginoftheelasticity;forinstance,the effectofthemagnitudeofshortrangerepulsiveforce, F o ,needsinvestigating. Figure 5-4 comparesthesuspensionwithrandominitialcongurationstothe preshearedsuspension.Thearealfractionofsuspensionsis0.6andthestrain amplitudeis1.0.Theapplied F o and is1.0x10 )Tj /T3_2 7.97 Tf 6.59 0 Td (4 and0.01,respectively.Results showthattheapparentvaluesof 00 ofthesuspensionwitharandominitialconguration doesnotshowadecay,whereasthepreshearedsuspensionexhibitsthedecreasein 00 asshowninFig. 5-4.However,inChap. 4,weexperimentallyobservedthatpresheared suspensionsandthesuspensionshavingrandomcongurationsshowsimilarbehaviors of 00 .Thedisagreementbetweenexperimentsandsimulationscouldbeexplained bythefactthattheinitialcongurationinexperimentalconditionsmaynotberandom duetotheeffectofloadingthesuspensionintheapparatus.However,wecanhave trulyrandominitialcongurationsinsimulations.Therefore,themicrostructuresduring preshearingsuspensionsmightaffecttheelasticityatshorttimes. 81

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Fur thersimulationswillprovidemoreinformationontheshortrangeinteractions withoutloadingeffects,shear-inducedmigration,andsedimentationasplausiblecauses oftheobservations.Moreover,thecorrelationbetweenmicrostructuresandrheologyfor short-timeoscillationsusingStokesiandynamicssimulationswillbeperformed. 82

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CHAPTER 6 CONCLUSIONS Theworkpresentedinthisdissertationsignicantlyadvancesknowledgeofthe dynamicsofparticulatesuspensions.Applicationsofparticulatesuspensionsare numerousinexistingandemergingtechnologiesasassertedinChapter 1 .Therefore, studiesmustbeconductedtobetterunderstandthedynamicsofsuspensionssothat efcientcontrolandprocessingcanbeachieved.Thisworkfocusedontwosimple examplesofsuspensionsystems:neutrallybuoyantsuspensionsofrigidrodsand non-colloidalspheres.Theaccomplishmentsandcontributionsofthisdissertationare presentedinSec. 6.1 andsuggestionsforfutureworkfollowinSec. 6.2. 6.1Accomplishments Thisworkdemonstratedtheeffectofparticleaspectratioonrheology,independent ofotherpropertiesofthesuspension,viadirectcomparisonofthedynamicsof suspensionsofspheresandellipsoids.Theresultsprovideclearevidencethatthe changeoftheparticleshapealterstherheology qualitatively.Moreover,theresults suggestthatafundamentalmechanismthatinuencestherheologicalcharacterization ofsuspensionsofbersatlargePecletnumbershasyettoberevealed. Thekeyadvantageofourexperimentalsystemsisthatweareabletocomparethe effectiveviscositiesofsuspensionsofspherestothoseofrigidbersandcanattribute thedifferencessolelytothechangeinshapeoftheparticle.Thisinnovationisenabled byusingidenticalprocessingmethodsforboththesuspendinguidandtheparticle manufacture;asaresult,thespheresandellipsoidshavethesamevolume,material composition,andsurfaceproperties.Themeasurementandcomparisonoftherheology ofthesuspensionsreturnanintriguingresult:alteringtheparticleshapefromspherical toellipsoidalproducesaratedependentrheologyandthesheardependencebecomes strongeruponincreasingtheaspectratio,whereastheviscosityisindependentofthe rateofshearforthesuspensionofspheres.Theexperimentalsystemandconsequent 83

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compar isonmakesitpossibletoconcludethattheobservedratedependentrheology ofthebersuspensionsprobablyisnotcausedbyunwantedeffectssuchasparticle aggregationduetomechanicaladhesionorcolloidalinteractionsthatwouldcauseshear thinninginthesuspensionofspheresaswellasellipsoids. Theexperimentsclearlyhighlightthediscrepancybetweentheoriesandexperiments fortheratedependentrheologyofbersuspensions.Theoriesinasemi-dilute suspensionofrigid,non-Brownianbersdonotpredictadependenceofthesuspension viscosityonshearrate[ 7 30, 91].Likewise,simulationsndnodependenceonthe shearrate[ 84, 107].However,experimentsontherheologyofsemi-dilutesuspensions ofrigidbersathighPecletnumbersconsistentlyshowadecreaseinviscositywith increasingshearrate[ 22 45 ].Withwell-controlledbersuspensions,wealsoobserved shear-thinningbehaviorsofbersuspensionsasdescribedinChapter 2 Toaddressthisdiscrepancy,amechanismthatpredictsanetmigrationresulting fromthecompetitionbetweenhydrodynamicforcesandtheweakthermalrotationat high,butnotinnite,Pecletnumbersisproposed.Themechanismcanaccountforshear thinninginthesuspensionofellipsoids,butpredictsnoshearthinningforthespheres, accordingtoarecentstudy[ 77]. Theidenticationoftwodifferentstrainregimesinoscillatoryshearowsfor suspensionsofneutrallybuoyant,non-colloidalspheresrepresentsanothermajor contributionoftheworkpresentedinthisdissertation.Longtimechangesofthe viscosity( 0 )occuroveralargenumberofoscillationsduetotheslowdevelopment ofmicrostructureswithstrain.Ontheotherhand,afastdecayof 00 isobserved,while 0 and remainnearlyconstantoverthecorresponding,smallnumberofoscillations.The ndingsprovideimportantinputstomodeltheoscillatoryshearowsofthesesystems aswellastomodelmorecomplexowsofsuspensions.Forinstance,preliminary modelingindicatesthatthefastdecayoftheelasticcomponentofthestressresponseis 84

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related tothenatureoftheshort-rangeinteractionsbetweenthesphereswhichmay,in part,beresponsiblefortheirreversiblebehaviorofsuchsuspensions. DatagiveninChapter 4 alsoextendconclusionsregardingtheexistenceofa non-monotonicviscositydependenceuponthestrainamplitudetoawiderrange ofvolumefractionsthanpreviouslyinvestigated.Thenon-monotonicdependence oftheviscosityonthestrainamplitudesisobservedforvolumefractionsfrom =0.30to =0.50,conrmingpreviousndingsthatwerelimitedto =0.4[ 14]. Thenon-monotonicbehaviorindicatesthepresenceofaqualitativechangeinthe steady-statemicrostructureofthesuspensionthatdependsupontheamplitudeofthe oscillatorystrain. TheworkpresentedinChapter 3 representsanewendeavortoexploitrheology asatoolforcharacterizingthestabilityofsuspensionsofsinglewallcarbonnanotubes (SWNTs).DispersingandstabilizingSWNTsinsolutionremainsoneofthechallenging issuesthatcurrentlylimitsapplicationsofSWNTs.Multiplemethodsofdispersion andstabilizationhavebeendevelopedthatincludedifferentlevelsofprocessingas wellasthetheuseofsurfactants.Thesuccessofthevariousmethodshavebeen roughlyassessedby,forexample,makingvisualinspectionofwhetherornotocs formaftermorethan24hours[ 5 54 75 76 ].InChapter 3 ,thestabilityoftheSWNT suspensionsaresystematicallyinvestigatedbymeasuringviscositychangeswithtime. Althoughrheologicalmeasurementwaslimitedbythelowtorquevaluegeneratedbythe aqueoussuspensionsofSWNTs,themethodwasusedsuccessfullytotestthestability ofSWNTsuspensionspreparedbyanovelmethodofinterfacialtrapping.Additionally, therheologicalworkresultedintheproductionofenhancedstabilityforsuspensionsof SWNTsthatutilizeGumArabicsurfactant. 6.2SuggestedFutureWork Suggestionsforfutureworkareaimedatstrengtheningandsupportingthendings discussedinthisdissertation.Imagingofparticledynamics,forexample,provide 85

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a directroutetoconrmmigrationmechanisms,aggregation,andpossiblyeven microstructures.Additionalsimulationscanalsobeusedtofurthercorrelatemeasured rheologiestounderlyingdynamicsandmicrostructuresofthesuspensions. First,imagingofrigidrodsinowswillassistindiscriminatingbetweenmechanisms thatmayberesponsiblefortheratedependentrheologyofbersuspensionsinthe semi-diluteregimeasdescribedinChapter 2.Specically,thepredictionofanet migrationofabersintorsionalowsduetothecompetitionbetweenhydrodynamic forcesandthermalforces[ 77]canbeveriedthroughdirectobservation.Fluorescence microscopycouldbeusedtoidentifythedistributionofparticlesintheradialdirectionfor anappropriatelyconstructedanddesignedsystem.Moreover,confocalmicroscopy couldbeusefulforinvestigatingthecross-streammigrationforrodsundergoing steadyshearows[ 78]andinoscillatingshearowsinmicrouidicchannels.For thislattertask,preliminarytestingwithaconfocalmicroscopehasbeenperformedfor adilutesuspensionofpolystyreneellipsoidsowingthroughamicrochannel.However, limitationssuchaslowresolutionanddifcultiesinidentifyingparticlesduetotheir anisotropicshapeneedstobeovercome.Also,preliminarytestsshowedthatthe particledistributionwasinuencedbygravityandthatdoubletsandtripletsofrodsexist withinthesuspension.Therefore,weneedtodevelopasuspensionsystemofindividual, rigidbershavingamatchedrefractiveindexaswellasamatcheddensitytogenerate morestablesuspensions. Secondly,imagingconcentratedsuspensionsofsphereswillclarifythedynamicsof thesuspensionsatsmallnumberofoscillationsaswellasconrmthestructure-rheology correlationsuggestedbyStokesiandynamicssimulations;suchndingswould greatlystrengthentheconclusionsoftheworkpresentedinChapter 4.Forimaging ofnon-colloidalsuspensionsofspheres,amagneticresonanceimaging(MRI)technique canbeutilized.MRIhastheadvantageofbeingapplicabletohighvolumefractionsand largeparticlesystems,whilecommonimagingmethodssuchasscatteringandoptical 86

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or uorescencemicroscopesarelimitedtodiluteconcentrationsand/orsmallparticle sizes. Asidefromimaging,Stokesiandynamicssimulationsarebeingconductedtofurther investigatetheoriginoftherapiddeclineof 00 forsuspensionsofnon-colloidalspheres overasmallnumberofoscillationsasintroducedinChapter 5.Preliminaryresults reproducedqualitativelysimilarbehaviorsof 00 over N =20.Furthersimulationswill provideadditionalinsightintotheshort-rangeinteractionswhileeliminatingloading effects,shear-inducedmigration,andsedimentationasplausiblecausesofthe observations.SimilarlytoBrickerandButler[ 15 ],wecouldcorrelatemicrostructures torheologyforshort-timeoscillationsusingStokesiandynamicssimulations,ifinfactthe simulationsreturnanaccuratepredictionofthemeasuredrheology. 87

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APPENDIX: TheStressofaSlenderBodyinDiluteSuspension ThePecletnumber, Pe ,isadimensionlessgroupphysicallyrepresentingaratioof shearforcetoBrownianforceactingontherod.ThePecletnumbercanbedenedby thetranslationalparameters, Pe t ortherotationalones, Pe r ,whicharedenedas Pe t = r D t P e r = r D r where r is theshearrate, D t isthetranslationaldiffusivity,and D r istherotational diffusivity.Sincetherotationaldiffusivityislargerthanthetranslationaldiffusivityforthe slender-bodymodel, L 2 D ro =D to =9[ 25 ]. TherotationalPecletnumberforaslender-bodyisexpressedas Pe r = rL 3 3 kTln (2 r ) where is thesolventviscosity,Listheparticlelength,kistheBoltzmannconstant,Tis temperature,andristheaspectratiooftheparticle( L =d ,wheredisthediameterof theparticle)[ 33]. Ifwelettheparticlelengthandtheaspectratiobe500,thenforthesuspensions ofSWNTsinaqueoussolutions, Pe r isintherangeof0.05 0.8for r =10 100 s )Tj /T3_3 7.97 Tf (1 Since Pe r isapproaching1,theremightbeatransitionfortheorientationdistributions ofparticlesresultinginhighviscosity.Theshearforcetendstoalignparticlesparallelto theshearowdirectiongeneratinglowviscosity.Therefore,thetransitionfromasystem dominatedbyBrownianforcestoshearforcesmaycausetheshear-thinningbehavior. TheshearstressofaSWNTinaqueoussuspensionsistheoreticallycalculated usingslender-bodytheory[ 6, 56 ]consideringBrownianmotion.ASWNTisassumed tobearigidslender-bodyandthediluteconcentrationregimeisassumedtoenable asimplecalculationalthoughoursuspensionsareintheregimeofsemi-dilute concentration.AssumptionsofaNewtoniansuspendingliquidandStokesoware appliedandthesuspensionsareforce-freeandtorque-free. 88

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T opredictmacroscopicproperties,astatisticalensembleaverageisperformed.The averageshearstressofaslender-bodyundergoingsimpleshearowisexpressedas h ij i = 1 V Z v ij dV = 1 V Z v f f ij dV + 1 V N X =0 Z v p ij dV The lefttermisthestressoftheuidphaseandtherighttermisthestressofthe particlephase.ForaNewtonianuid, f ij = )Tj /T3_2 11.955 Tf (P f ij +2E f ij Theuidvolume, V f ,isexpressedas V f = V )Tj /T1_3 11.955 Tf 12.35 8.97 Td (P N V p ,where E ij istherateofstrain tensor,denedas 1 2 [(ru ) +( ru ) T ] foranincompressibleuid.Usingtheserelationships,theaveragestresscanbedivided intouidandparticlephases.Thestressduetothesuspendinguidis, h f ij i = 1 V Z v f f ij dV = )Tj /T3_1 11.955 Tf 12.36 8.09 Td (1 V ( Z v f PdV ) ij + 1 V Z v f 2E ij dV The stressduetotheparticleis, h p ij i = 1 V N X =1 Z v p ij dV = 1 V N X =0 Z v p @ ( ij x j ) @ x k )Tj /T1_3 11.955 Tf 11.95 16.28 Td (Z v p x j @ ik @ x k dV F ortheparticlesthatareforce-free,theequationreducesto h p ij i = 1 V N X =1 Z S ij x j n k dS = 1 V N X =1 Z S f i x j dS = 1 V N X =1 ij dV where f i is theforcedensityand S ij isthestresslet[ 56].Also, 2 Z V p E ij dV = Z V p ( @ u i @ x j + @ u j @ x i ) dV = Z S ( u i n j + u j n i ) dS Combining theseresultstogether,theaveragestressofthesuspensionisgivenby h ij i = )Tj /T1_2 11.955 Tf 12.62 0 Td (< P > ij +2< E ij > + 1 V N X =1 S ij 89

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T ableA-1.ComparisonoftheviscositiesoftheSDS-dispersedSWNTsuspensions preparedbyultracentrifugation. v ol%cal.viscosityexp.viscositydifference( %) 0.00260.01215 P0.01313P 7% 0.00190.01184P0.01242P 5% T ableA-2.ComparisonoftheviscositiesoftheGA-dispersedSWNTsuspensionsasa control. v ol%cal.viscosityexp.viscositydifference( %) 0.00410.01980 P0.02731P27% 0.00280.01892P0.02059P 8% Here ,thestressletforaslender-bodyisexpressedintermsoftheparticleorientation vector, ~ p S ij = L 3 6ln (2r ) ( < p i p j p k p l > )Tj /T3_0 11.955 Tf 10.49 8.09 Td (1 3 ij < p k p l > ) < E ij >, where < p i p j > = Z P ( ~ p t )p i p j d ~ p < p i p j p k p l >= Z P ( ~ p t ) p i p j p k p l d ~ p The probabilitydensity, P ,istheprobabilityofndingaparticlewithorientation ~ p attime t.Therefore,theshearstressofthesuspensioncontainingBrownianmotionundergoing steadyshearowisexpressedas < 12 >=2< E 12 > +nkT [3 < p 1 p 2 > ]+ nL 3 6 ln (2 r ) [2E 12 < p 1 p 2 p 1 p 2 >]. where 1and2arethex-andy-directions,respectively. OthermembersinourgroupperformedsimulationsfordynamicsofBrownian rodsundersteadyshearow[ 78].Therodsarerigidandslender-bodytheorywas applied.Theyobtainedthesecondandthefourthmomentoftheorientationvectorsas functionsof Pe r .Usingthesimulationresults,weobtainedtheinnerproductvaluesfor TableA-3.ComparisonoftheviscositiesoftheGA-dispersedSWNTsuspensions preparedbyinterfacialtrapping. v ol%cal.viscosityexp.viscositydifference( %) 0.00260.01878 P0.01903P 1% 0.00190.01830P0.01942P 6% 90

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Figure A-1.ComparisonofviscositiesoftheSDS-dispersedSWNTsuspensions. theorientationvectorsofSWNTs.ThetheoreticalandexperimentalvaluesofSWNT suspensionsarecomparedinTables A-1, A-2 ,and A-3,wheretheviscosityvaluesare obtainedat r =10 s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 .AsshowninTables A-1, A-2,and A-3,theviscosityvaluesfrom theoreticalcalculationarewithin7 % oftheexperimentalresultsfortheSDS-dispersed SWNTsuspensionspreparedbyultracentrifugationandfortheGA-dispersedSWNT suspensionspreparedbytheinterfacialtrappingmethod.However,qualitativelythe theorydoesnotshowshear-thinningbehaviorovertherangeof10 100 s )Tj /T3_2 7.97 Tf 6.59 0 Td (1 asshown inFig. A-1.TheresultimpliesthatotherfactorsbesidesBrownianmotionplayarolein therheologyoftheSWNTsuspensions.Basedontheobservation,theshear-dependent behaviormightbeattributedtotheformationofaggregates. 91

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BIOGRAPHICAL SKETCH Hyun-OkParkwasborninOnyang,Korea,toNowonParkandJungsikShin. Hyun-OkattendedOnyangOnchonElementaryschoolandgraduatedwithtophonors. Hyun-OkattendedOnyangWomen'smiddleschool,wastheclasschairfortwoyears, andalsograduatedwithtophonors.Forhighschool,Hyun-OkcommutedtoBokja Women'shighschoolinthecityofChon-an.There,Hyun-OkbecameaCatholic. Hyun-Okgraduatedfromthehighschoolwithtophonors,too.Forcollege,Hyun-Ok enteredKoreaUniversitylocatedinSeoulin1996.Hyun-Okmajoredinchemical engineeringandthencontinuedherstudiesatgraduateschoolinSeoulNational University.Hyun-OkreceivedanM.S.degreeatSeoulNationalUniversityin2003. Hyun-OkworkedforDCChemicalCompanyR &DCenterasaresearcherfrom2003to 2004.Hyun-OkmarriedYuKyoumKimin2004andcametotheUnitedStates.Hyun-Ok enteredtheUniversityofFloridain2005andjoinedDr.Butler'sresearchgroupfor herdoctoraldegree.Hyun-Ok'sresearchfocusedonthedynamicsandrheologyof suspensionsystems.NowHyun-Oklooksforwardtothenextchapterinherlifeasshe takestheknowledgeandexperiencethatshehasgainedinearningaPh.D.inchemical engineeringintoacademia. 100