<%BANNER%>

Dynamic Simulations of Suspensions of Rod-Like Polymers and Colloids


PAGE 4

Iwouldliketothankeveryonewhohelpedmeinthisendeavor.Firstandforemostismyadvisor,JasonButler,whoneverdidgiveuponme,butinsteadpushedmetoworkharderandproducebetterwork.Ithankhimforhispatienceandhisdedicationtobeingagoodmentorandadvisor.Iwanttothankallofthepeopleinmylabgroupfortheirhelpinmanywayswithmyresearchandlifeingeneralingraduateschool.IthankBerkUstaandJonathanBrickerfortheirhelpinreadingandeditingmypapers,listeningtomypracticetalks,helpingmeunderstanddierentsoftwarepackages,andjustbeingtheretobounceideasoof.IalsothankJoontaekParkforhisinsight.Iwouldliketothankmyfriendswhohelpedmeinmanywayswithdierentaspectsofmygraduateschoolexperience.IwanttothankChristineDuboisandLeahPolkowskispecicallyfortheirhelpineditingmycandidacyproposalandlisteningtomypracticetalks.Last,butnotleast,Iwanttothankmyroommateswhohelpedkeepmesanethroughoutmytimeingraduateschool:KenBrown,ZachPulkin,TimCobb,andAmauryGarcia.Ialsowanttothankmyotherfriendswhowereinstrumentalinkeepingmegroundedduringthistime:SwapnaMony,LaurenPauly,andCourtneyMaibach. iv

PAGE 5

page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. viii LISTOFFIGURES ................................ ix ABSTRACT .................................... xii 1BROWNIANFIBERSYSTEMS:PASTEXPERIMENTS,THEORIES,ANDSIMULATIONS ........................... 1 1.1Introduction .............................. 1 1.2ExperimentalSystemsofRigidRods ................ 4 1.3CharacterizationofSuspensionsofRigidRods ........... 7 1.4ExistingTheories ........................... 10 2BROWNIANDYNAMICSSIMULATIONSOFASINGLEROD .... 13 2.1HydrodynamicModelsofRigidRods ................ 13 2.1.1TheSlender-BodyModel ................... 13 2.1.2TheRigid-DumbbellModel .................. 17 2.1.3AGeneralApproach ...................... 22 2.2BrownianForcesandTorques .................... 26 2.3DiscretizedEquationsofMotion ................... 27 2.3.1TheCorrectedEulerMethod ................. 27 2.3.2TheMidpointMethod ..................... 35 2.4SimulationsofDiluteSuspensionsofBrownianFibers ....... 36 2.4.1TestingtheBrownianMotion ................. 37 2.4.2ErrorintheCorrectedEulerMethod ............ 40 2.4.3ComparisonofCorrectedEulerMethodtoAnotherEstab-lishedMethod ........................ 43 2.5Conclusion ............................... 46 3DYNAMICSIMULATIONSOFCONCENTRATEDSUSPENSIONSOFRIGIDFIBERS:RELATIONSHIPBETWEENSHORT-TIMEDIFFUSIVITIESANDTHELONG-TIMEROTATIONALDIFFUSION 47 3.1Introduction .............................. 47 3.2SimulationMethod .......................... 49 3.2.1GoverningEquations ..................... 50 v

PAGE 6

.... 55 3.2.3NumericalIntegrationoftheGoverningEquations ..... 56 3.3Results ................................. 60 3.3.1RotationalDiusivities .................... 60 3.3.2TranslationalDiusivities ................... 64 3.4Discussion ............................... 67 3.4.1DependenceofRotationalDiusivitiesonFiberModel ... 67 3.4.2RelationwithExistingTheories ............... 72 3.4.3RotationalDiusivityunderLimitingConditions ...... 76 3.4.4ReinterpretationofComparisonwithExperiments ..... 79 3.4.5TranslationalDiusivities ................... 80 3.5Conclusions .............................. 82 4DYNAMICSIMULATIONSOFCONCENTRATEDSUSPENSIONSOFSEMI-RIGIDFIBERS:EFFECTOFBENDINGONTHERO-TATIONALDIFFUSIVITY ........................ 85 4.1Introduction .............................. 85 4.2SimulationMethod .......................... 86 4.2.1GoverningEquations ..................... 87 4.2.2EvaluationofBrownianForces ................ 88 4.2.3EvaluationofExcludedVolumeandBendingForces .... 89 4.2.4EvaluationofConstraintandCorrectionForces ....... 90 4.2.5DiusivityCalculations .................... 92 4.3Results ................................. 94 4.3.1SimulationsofIndividualRods ................ 94 4.3.2RotationalDiusivitiesatHighConcentrations ....... 97 4.3.3TranslationalDiusivitiesatHighConcentrations ..... 100 4.4Discussion ............................... 101 4.5Conclusions .............................. 107 5IMPROVEDCOMPUTATIONALPERFORMANCEOFBROWNIANDYNAMICSSIMULATIONSWITHHYDRODYNAMICINTERAC-TIONSTHROUGHPARALLELIMPLEMENTATIONUSINGTHEMESSAGEPASSINGINTERFACE ................... 109 5.1Introduction .............................. 109 5.2CalculatingtheMulti-bodyHydrodynamicInteractions ...... 110 5.3ImplementingaParallelAlgorithmusingtheMessagePassingIn-terface(MPI) ............................ 114 5.3.1ParallelCalculationoftheMobilityMatrix ......... 115 5.3.2DecompositionoftheGrandMobilityMatrix ........ 115 5.3.3PLAPACK:AGeneralApproachtotheParallelCholeskyDecomposition ........................ 117 5.4PerformanceoftheParallelCholeskyDecomposition ........ 123 5.5Discussion ............................... 129 vi

PAGE 7

............................... 132 6CONCLUSION ................................ 134 REFERENCES ................................... 138 BIOGRAPHICALSKETCH ............................ 147 vii

PAGE 8

Table page 3{1ValuesoftheparametersandusedinEquation( 3.21 ) ....... 55 3{2Valuesfortheexponentforthescalingoftherotationaldiusivity 65 viii

PAGE 9

Figure page 1{1Simulationresultsofsphero-cylindersofvariousaspectratios(A) .. 2 1{2ThesimulationresultsofWilliamsandPhilipse[ 1 ] ........... 2 1{3Dierentconcentrationregimesofsuspensionsofrigidbers. ..... 3 1{4Picturesoffabricsmadeupofcarbonbers(a),Kevlarbers(b),andacompositeofboth(c) ...................... 4 1{5Anelectronmicrographpictureofthetobaccomosaicvirus ...... 5 1{6TransmissionelectronmicroscopeimagesofCdSenanorods ...... 6 1{7Carbonnanotubecompositeribbon ................... 7 1{8Transmissionelectronmicroscopeimageoftheboehmiterods ..... 9 1{9Imageofapolystyreneprobesphereinasuspensionofsilicacoatedboehmiterodstakenbyatransmissionelectronmicroscope ..... 10 2{1Diagramshowingtheeectofthedriftcorrectionupontheorienta-tionvectorpi(t) ............................ 35 2{2Plotofthespheresweptoutbytheorientationvector ......... 37 2{3PlotoftheBrowniandisplacementsoftheorientationwithonecom-ponent(p(3))lockedwiththez-axis ................. 38 2{4Plotoftheautocorrelationfunctionofthecenterofmassdisplace-mentsovertime ............................. 40 2{5Plotoftheautocorrelationfunctionoftheorientationovertime ... 41 2{6Long-timerotationaldiusivitiescalculatedusingeithertheEulermethodorthealgorithmofLowen[ 2 ] ................. 42 2{7Thepercentageerrorinthelong-timerotationaldiusivities ..... 45 3{1Summaryofthecharacteristicsandvariablesdescribingtheslender-bodyandrigid-dumbbellmodels ................... 50 3{2TherotationalcorrelationfunctionatnL3=150,L2DR0=DT0=9,andDk0=D?0=2forrodswithA=50 ................ 61 ix

PAGE 10

......... 62 3{4Resultsareshownfortherotationaldiusivitiesversusconcentrationfortheslender-bodymodel ....................... 62 3{5Rotationaldiusivitiesversusnumberdensityfortheslender-bodyandrigid-dumbbellmodelswithA=50 ............... 64 3{6Exponentforthescalingoftherotationaldiusivity ......... 65 3{7Translationaldiusivitiesasafunctionofnumberdensity ....... 66 3{8Averagesquaredisplacementsofthecenterofmassversuselapsedtime 68 3{9Linearextrapolationoftherotationaldiusivities ........... 71 3{10DirectcomparisonofthesimulationresultsofDoietal.[ 3 ]withtheslender-bodymodel ........................... 72 3{11RotationaldiusivitiesoverarangeofL2DR0=Dk0witheitheriso-tropicoranisotropiccenterofmassdiusivitiesforrodswithA=25 74 3{12RotationaldiusivitiesoverarangeofL2DR0=Dk0witheitheriso-tropicoranisotropiccenterofmassdiusivitiesforrodswithA=50 75 3{13ComparisonofrotationaldiusivitiesforrigidberswithA=25withperpendiculardiusivityremoved ................ 78 3{14Rotationaldiusivitiesfromsimulationsusingtheslender-bodyandtherigid-dumbbellmodelswithA=50comparedtothediusivi-tiesofPBLG .............................. 79 3{15PerpendiculardiusivitiesforberswithA=25incomparisontothereptationmodelofSzamel[ 4 ] ................... 83 4{1Thetwomodelsofthesemi-rigidrodsusedinthesimulations .... 87 4{2Probabilitydistributionofp(;1)ip(;2)i 95 4{3ValuesoftheratioofthedilutevaluesofthediusivitiesL2DR0=DT0resultingfromtheKvaluechosen ................... 96 4{4Plotoftheorientationautocorrelationfunction1 2lnDp()i(t+)p()i(t)Eovertime 97 4{5Rotationaldiusivitiesofrigidslender-bodiesandthree-beadtrumbells 98 4{6Rotationaldiusivitiesoftherigid-rodmodelsincomparisontothesemi-rigidrodmodels .......................... 99 x

PAGE 11

... 100 4{8RatioofthePersistencelengths(P)fortheslender-bodydimerandthree-beadtrimermodelsoverthetotalrodlength(L)atK=10 103 4{9RotationaldiusivityasafunctionofthestinessparameterK 104 4{10Rotationaldiusivitiesoftheslender-bodydimerwithK=10incomparisontosimulationsofarigid-dumbbell[ 5 ] .......... 107 5{1ParallelCholeskydecompositionsofa20002000matrixwithvary-ingb ................ 125 5{2ParallelCholeskydecompositionsofa50005000matrixwithvary-ingb ................ 126 5{3ParallelCholeskydecompositionsofa1000010000matrixwithvaryingb ............. 126 5{4TimetoperformtheparallelCholeskydecompositionasafunctionofthematrixsize,withrunsperformedwithdierentnumbersofprocessors ................................ 127 5{5Timerequiredtodecomposedierentsizematricesasafunctionofthenumberofprocessorsused ..................... 128 5{6TimetoperformtheparallelCholeskydecompositionasafunctionofthenumberofbers .......................... 130 xi

PAGE 12

Simulationspresentedinthisdissertationadvanceknowledgeofthedynamicsofsuspensionsofrigidandsemi-rigidBrownianbers.Thisworkresolvescom-petingclaimsconcerningthepower-lawscalingfortheconcentrationdependenceoftherotationaldiusivities.Thepower-lawscalingstatesthattherotationaldiusivityDRscalesasDR=DR0(nL3),whereDR0istherotationaldiusivityofasingleberininnitedilution,nisthenumberdensity(numberofbersperunitvolume),andListheberlength.Thechoiceofhydrodynamicmodel,withanintrinsicratiooftherotationaltotranslationaldiusivitiesatinnitedilutionL2DR0=DT0,setsthevalueoftheexponentinthescaling.Theaspectratioofthebersalsoaectsthescalings,withstrongvariationsforratioslessthanfty;ratiosofftyorhighercanconsideredinnitelythin.Ananalysisofthenumericalintegrationmethodwasperformed,resultinginanewalgorithmwithlesserrorandhighereciency. Addingexibilitydelaysthenumberdensityatwhichbersbecomesigni-cantlyhinderedbytheirneighborsandentertheregimewhereastrongdecreaseintherotationaldiusivityoccurs.Oncewithinthissemi-diluteregime,the xii

PAGE 13

Includinghydrodynamicinteractionsintothesimulationswillprovidefurtherinsightsintothedynamicsofbersuspensions.Investigationsofaparallelcompu-tationofthepairinteractionsandCholeskydecompositionindicatethatsimulatingsystemsofoveronehundredbersisfeasible. xiii

PAGE 14

1.1 Introduction Manyinorganicandpolymericcolloidaldispersionsconsistofrigidandnon-sphericalparticles,suchasrod-likemacromolecules.Thesepolymersinteractthroughinter-particlecolloidalandhydrodynamicforcesinacontinuumuid.ThemacromoleculesalsoexperienceBrownianforceswhicharisefromthethermaluctuationsoftheuid.Incolloidaldispersions,ingeneral,oneisfacedwithalargespreadofwellseparatedtimescales[ 6 7 ].Thesetimescalesrangefromthesmallesttimescaleassociatedwiththerandommotionoftheuidmoleculestothemuchlargertimescalesassociatedwithdiusionofparticlesintheuid.Asaresult,thedynamicbehaviorofacolloidaldispersionisusuallysimpliedintoamodeloflargemacromoleculesor\Brownian"particlessuspendedinacontinuumuid. Theunderstandingofthedynamicbehaviorofconcentratedpolymersolutionsandmeltsremainsamajorchallengeofmodernpolymerphysics.Thecomplexitiesareduetouniquefeaturesofpolymermolecules.Polymermoleculespossessalargenumberofdegreesoffreedomassociatedwiththeirintermolecularrotationalandvibrationalmotioninadditiontothetranslationaldegreesoffreedomoftheircenterofmass.Theinterplaybetweenmacroscopicandintramolecularmotion,aswellasthepreservationofthechainconnectivity,resultsintheentanglementeect.Thiseectisbelievedtodominatethedynamicbehaviorofthesesystems[ 6 ]. Brownianbers,whetherrigidorexible,producephasesinsuspensionwhicharenotseeninsuspensionsofspheres.Rigid,rod-likepolymersdierfrom 1

PAGE 15

Simulationresultsofsphero-cylindersofvariousaspectratios(A)pro-ducedbythesimulationsofWilliamsandPhilipse[ 1 ].ThepicturesareforA=0(upper-left),A=0:4(upper-right),A=2(lower-left),andA=40(lower-right). ThesimulationresultsofWilliamsandPhilipse[ 1 ]areplottedwiththevolumefraction()asafunctionoftheparticleaspectratio(A).Thesolidlineisatheoreticallimitoftherandompackingasgivenbyequation(1)intheworkofWilliamsandPhilipse[ 1 ].

PAGE 16

exiblepolymersinmanyrespects,yettheyarethesimplestpolymerswhereentanglementsoccur[ 6 ].WilliamsandPhilipse[ 1 ]performedsimulationsofparticleswithsphericalendcapsandvaryinglengths.Theythencalculatedthemaximumobtainablevolumefractionatwhichtheorientationdistributionremainsrandom.Figure 1{1 showsgraphicallythatastheaspectratioA(berlengthoveritsradius)increasesfromA=0(sphere)toA=40,moreemptyspaceisseenaroundtheparticles.AgraphofthemaximumrandompackingvolumefractionasafunctionoftheaspectratioisshowninFigure 1{2 [ 8 ].Theseresultsshowthatastheparticleaspectratioincreases,thevolumefractionofthemaximumrandompackingdecreases.Particlesofhighaspectratiowillhinderthemotionofthesurroundingparticlesatmuchlowerconcentrationsthanspheres,resultinginanentanglementeect,eventhoughtheyareperfectlyrigid. TheconcentrationregimesinwhichWilliamsandPhilipse[ 1 ]performedsimulationsisknownasthesemi-diluteandconcentratedisotropicregimes.Suspensionswithhigherconcentrationsofberscanbemade;howevertheyenterintoanewregimewheretheretheorientationdistributionofthebersisnolongerrandom.Astheconcentrationisincreasedfromtherandom,orconcentrated,isotropicregime,thebersformanematicphasewherethereisorderingintheorientationsofthebers,butnotinthecenterofmasslocations. Dierentconcentrationregimesofsuspensionsofrigidbers.Figurea)isthediluteregime,gureb)istheconcentratedisotropicregime,Figurec)isthenematicconcentrationregime,andFigured)isthesmecticconcentrationregime.

PAGE 17

Iftheconcentrationisincreasedfromthenematicregime,theberswillenterintothesmecticregime,wherethereisorderingofboththeorientationandcenterofmassofthebers.ThesedierentconcentrationregimesareshownschematicallyinFigure 1{3 1.2 ExperimentalSystemsofRigidRods Apartfromtheirimportanceasthesimplestsystemsofentangledpolymermolecules[ 6 ],nondilutesolutionsofrigid,rod-likepolymershaveconsiderableinterestontheirown.RigidpolymersandBrownianbersarefoundinexistingtechnologiesandexcitingnewareasofresearch.Manyrigid-rodpolymersexhibithightensilemodulusandstrength.Thesematerialsalsohavealowdensity,whichmakethemidealforuseincreatinglightweight,high-strengthbersandlms.Consequently,thesepolymers,aswellascompositesmadewiththesepolymers,arewidelyusedashigh-performancematerialsinconsumer,aerospace,andelectronicsapplications[ 10 ].Oneofthemostwidelyrecognizedexamplesispoly(1,4-phenyleneeterephtalamide),soldunderthetradenameKevlar.ImpactandcompressiontestsofKevlarandcarbonberfabrics[ 9 ]investigatingthetensilestrengthandstinessofthesecompositematerialsshowthatthecombinationofbothhighstrengthbersinonecompositeproducesanevenstrongermaterial. Picturesoffabricsmadeupofcarbonbers(a),Kevlarbers(b),andacompositeofboth(c)aftertheyhavebeensubjectedtoimpactandcompressiontestingintheworkofGustinetal.[ 9 ].

PAGE 18

Anelectronmicrographpictureofthetobaccomosaicvirusillustrat-ingitsrod-likecharacteristics[ 11 ]. Figure 1{4 showstheresultsofthetestswithfabricsmadeentirelyofcarbonbers(a),Kevlarbers(b),andacompositeofboth(c). Withintherealmofbiology,manymacromoleculesresemblerigidbers.Shortlengthsofbiopolymers,suchasDNA[ 12 13 14 ],actin[ 15 ],andcollagen[ 16 17 ],canbemodeledasrigidbers.Xanthamgum,ahelicalpolysaccharidewhichroughlyformsacylinder,isusedtoenhanceviscosityofmanyfoodproducts.Othertypesofpolysaccharides,peptides,andpolynucleotidesalsoformrigidlinearstructures.Evensomesimplemicro-organismshaveanelongated,ellipsoidalconguration,andforsucientlylowforcesmaintainarigidstructure.Theseincludefdbacteriophagesandtobaccomosaicviruses[ 18 19 20 ].Anelectronmicrographofthetobaccomosaicvirusshowingitsrod-likeshapeisseeninFigure 1{5 [ 11 ].

PAGE 19

TransmissionelectronmicroscopeimagesofCdSenanorodsproducedinexperimentsbyHuynhetal.[ 21 ]inordertoimprovetheeciencyofhybridsolarcells.TherodsonthelefthaveanaspectrationofA5andtheonesontheleftareA10. Intherealmofmanmade,ornon-biological,Brownianbers,thereexistnanotubesandnanorods.Thescienceofthesematerialsisthesubjectofanimmenseamountofresearch,resultinginexcitingtechnologicalapplications.Processingofthebersoftenoccursinsolution,wheretheeectsofdiusivityandowuponthemicrostructurebecomerelevant.InoneexperimentperformedbyHuynhetal.[ 21 ],semiconductorCdSenanorodshavebeencastintoalmtomakesolarcells.Bycontrollingthediameterandlengthofthenanorods,aswellasthemicrostructure,Huynhetal.[ 21 ]wereabletoimprovetheperformanceofthesolarcells.Apictureofthesenanorodstakenbyatransmissionelectronmicroscope(TEM)isseeninFigure 1{6 Inasecondexample,Vigoloetal.[ 22 ]disbursedindividualcarbonnanotubesinasolutionandthenalignedthemusingaoweldtocreateexiblemacroscopicbersorribbons.Theseribbonsaremuchlongerthantheindividualnanotubes(ontheordersofcentimetersorlonger),andhaveveryhightensilestrength.Apictureofoneoftheseribbonswhichwastakenbyopticalmicroscopyisshown

PAGE 20

CarbonnanotubecompositeribbonproducedthroughtheexperimentsofVigoloetal.[ 22 ]. inFigure 1{7 .Byunderstandingthedynamicpropertiesofthecarbonnanotubesinsolution,theyieldoftheseribboncompositescouldbeimproved,aswellastheirmaterialproperties.Manyoftheexperimentsnotedusedcharacterizationtechniquestostudythemotionofthebersinsuspension,whichwillbediscussedinthefollowingsection. 1.3 CharacterizationofSuspensionsofRigidRods MaterialsscientistsandengineersemployanumberofexperimentaltechniquestocharacterizethepropertiesofrigidpolymersandBrownianberssuspendedinsolution.AcommonmeasurementusedtocapturethemotionofBrownianbersinsuspensionisdynamiclightscattering(DLS)[ 17 20 23 24 25 26 ].Thecongurationofdierentlightscatteringset-upsmayvary,butthegeneralconceptisthesame.Alightsourceisfocusedonasuspensionofrodswhereitisscatteredasithitstherods.Thescatteredlightismeasuredbyadevicethatanalyzestheamountofscatteredlight.Dynamiclightscatteringhasafewdrawbacks.Oneof

PAGE 21

thesedrawbacksisthatthedynamicsofindividualrodsandbersareverydiculttocalculateatanythingotherthandiluteconcentrations.Athighconcentrationsthereisaneedforwelldenedtracerparticles,whicharediculttoobtaininBrowniansystems. AwaytoavoidthedicultiesofrequiringtracerparticleswhenusingDLSathighconcentrationsisbyperformingabirefringencemeasurement.Withbirefrin-gencestudies,thesuspensionisaligned,eitherbyauidow(owbirefringence),orbyavoltagedierenceacrosstheuid(electricbirefringence).Theoworelectriceldisthenturnedoandthesuspensionisallowedtorelaxbackfromanalignedtoanisotropicstate.Therotationaldiusivitiesarethencalculatedbasedupontherateofchangeoftheintensityofabeamoflightpassingthroughthesuspension.Thisformofmeasurementcanproducecalculationsforthelong-timeparticlerotationaldiusivities,whichwouldhavebeenextremelydiculttocal-culatewithDLSalone.Phalankornkuletal.[ 25 ]usedthiscombinationofelectricbirefringenceandDLStomeasuretherotationaldiusivitiesofpoly(-benzyl--L-glutamate)(PBLG)overawiderangeofconcentrations.Otherbirefringencedevicesworksimilarly,exceptthattheymeasureachangeintheamountofelectriccurrentowingthroughthesampleasthesamplerelaxestomeasuretherotationaldiusivities. AnotherwayofmeasuringthediusivepropertiesofBrownianbersincludestheuseofuorescencerecoveryafterphotobleaching.Inthistechnique,theparti-clesarecoatedwithordyedbyaspecicchemicalthatwillcausethemtouoresceundergivenconditions.Theuorescentparticlescanthenbevisualizedandthediusivepropertiesmeasuredandcalculated.Asanexample,vanBruggenetal.[ 27 28 ]usedasuspensionofboehmiterodswhichwerecoatedwithuorescentcolloidalsilica.Theparticleswerethenvisualizedusingatransmissionelectronmicroscope;animageoftheparticlesisshowninFigure 1{8

PAGE 22

TransmissionelectronmicroscopeimageoftheboehmiterodswhichwhereuorescedintheworkofvanBruggenetal.[ 27 28 ]. AnotherformofmeasuringpropertiesofBrownianbersuspensionsisthroughmicro-ornano-rheology,dependingonthelengthscalesofthebersinthesuspension.Theideabehindtheseconceptsisthatpropertiessuchasviscositycanbemeasuredlocallyinasuspension,andfromthat,othersuspensionpropertiescanbedetermined.OneexampleisgivenbyKluijtmansetal.[ 29 ]whereuorescenttracerspheresareobservedsedimentingthroughasuspensionofsilica-coatedboehmiterods.Bycalculatingthediusivepropertiesofthespheresatdierentpointsinthesuspension,theeectiveviscosityofthesuspensionatthoselocationscanbecalculated.SimilarexperimentswereperformedbyTracyandPecora[ 30 ]andKangetal.[ 31 ],exceptthatdynamiclightscatteringwasusedtotrackthetracerspheresinsteadofuorescencerecoveryafterphotobleaching. Heldenetal.[ 32 ]performedexperimentswheretotalinternalreectionmicroscopywasusedincombinationwithaprobespheretocalculatetheentropicforcesinducedbyrigidrodsinasuspension.Inthisexperiment,apolystyrene

PAGE 23

ImageofapolystyreneprobesphereinasuspensionofsilicacoatedboehmiterodstakenbyatransmissionelectronmicroscopeintheexperimentperformedbyHeldenetal.[ 32 ]. sphereinasuspensionofsilicacoatedboehmiterodsispressedclosetoasilicacoatedglasswall.Theentropicforceinducedbytherodswasthenmeasuredasafunctionoftheconcentration.ApictureoftheprobeandrodsisshowninFigure 1{9 1.4 ExistingTheories Tointerprettheresultsofmeasurementsofdiusivitiesandrheology,re-searchersrelyontheoriesofrigidberswhichusuallyprovidequalitativescalinglaws,ratherthanquantitativepredictions.Thesetheoriesaccuratelycapturesomeobservations,butthescalinglawsareoftenlimited.ThelimitationscanincluderestrictionsonparameterssuchastheconcentrationregimeandPecletnumber.Thoughapredictionexistsforonerangeofparameters,theremaynotbeacomplementarytheoryforotherranges.

PAGE 24

Theoriesexistwhichattempttopredictmacroscopicproperties,suchasviscosity,ofasuspensionofrigidbers.Thesetheorieswerenecessarybecauseithasbeenobservedthatthebehaviorofrigidrodsuspensionsisqualitativelydierentfromthebehaviorofsuspensionsofsphericalparticles,whichhavebeenstudiedextensively[ 1 33 ].DoiandEdwards[ 6 ]predictedarelationshipbetweenthelowshearviscosityandtherotationaldiusivitiesofasuspensionofrigidrodsinthesemi-diluteconcentrationregimeof whereisthesuspensionviscosity,0isthesolventviscosity,nisthenumberdensity(numberofrodsperunitvolume),kBTisthethermalenergy,DRisthelong-timerotationaldiusivity,andDR0istherotationaldiusivityofasinglerodininnitedilution.Thesemi-diluteregimeisdenedas(1=L3n1=dL2),whereListherodlength,anddisthediameteroftherod[ 6 7 ]. Severaltheorieshavebeenproposedwhichattempttopredictthelong-timerotationaldiusivity(DR)ofasuspensionofrodsasafunctionofthedimensionlessnumberdensity(nL3).Doi[ 7 ]developeda\tube"theoryforinnitelythin,rigidrodswhichpredictsthattherotationaldiusivitiesscaleasDR=DR0(nL3)2fornumberdensitieswithinthesemi-diluteregime.Thistheorystatesthatasthesuspensionofrodsbecomesmoreconcentrated,itwillenterintoaregimewheretherodssurroundingaparticularrodwillforma\cage,"or\tube,"whichwillhinderthatrodfromrotatingordiusinginthedirectionperpendiculartoitscentralaxis.Thisproberodis,however,freetodiusealongthedirectionparalleltoitscentralaxis,becauseitisinnitelythin;thusthereisnothingtohinderitsdiusioninthatdirection.Theproberodwillthendiusehalfitslengthoutofthe\tube,"atwhichpointitwillbeabletorotateandenterintoanother\tube."SimulationsperformedbyDoietal.[ 3 ]ofinnitelythinbers,inwhich

PAGE 25

thehydrodynamicinteractionsbetweentherodswereignoredandcrossingofthecenterlinesoftherodswaspreventedthroughareectionrule,conrmedthescalingof(nL3)2. SimulationsperformedbyFixman[ 34 35 ]foundadierentscalingforthelong-timerotationaldiusivitiesofDR(nL3)1forinnitelythinrodsinthesemi-diluteconcentrationregime.SimilartothesimulationsofDoietal.[ 3 ],adynamictechniquewasimplemented,butshort-rangepotentialswereusedtopreventtherodsfromoverlapping.Fixman[ 34 35 ]proposedanalternativetheoryfoundedupontheconceptof\cooperativerotation"asthemechanismfortherotationaldiusionofthebersinordertoaccountforthesimulationresults.Thistheorystatesthatsincetherodsthatmakeupthe\cage,"or\tube,"inthetheoryofDoi[ 7 ]arealsodiusingandrotating,the\tube"isnotastaticentity.The\tube"willinfactdiuse,rotate,andbreakuponatimescalewhichisfasterthanthecenterofmassdiusionoftheproberod.The\tube"motion,aswellasitsbreakingupandreforming,resultsinhigherrotationaldiusivitiesinthesemi-diluteconcentrationregime. Eversincethepublicationofthesetwocompetingtheories,therehasbeenanongoingcontroversyinthescienticliteratureoverwhichtheoryisinfactvalid[ 36 37 38 39 ].ThiscontroversyisresolvedbytheworkreportedinChapter 3 .Thedierenceinthescalingoftherotationaldiusivitiescomesfromthemannerinwhichthebersaremodeledinthetheoriesandsimulations.Questionshavealsobeenraisedaboutthelimitationsofthesetheoriesandtheirusefulnessininterpretingexperimentalresultsfromrealpolymerrodsystems.Extensionshavebeenmadetothe\tube"theoryofDoi[ 7 ]inordertoaccountforexibility[ 23 ],polydispersity[ 40 ],andhydrodynamicinteractions[ 41 ].Theeectofaddingaslightdegreeofexibilityonthesetheorieswillbediscussedinchapter 4 ,andtheinclusionofhydrodynamicinteractionsisdiscussedinchapter 5

PAGE 26

2.1 HydrodynamicModelsofRigidRods Inthepastithasbeenassumedthatthehydrodynamicmodelusedtodescribepolymerbersinsolutionhasnoeectonthedynamicsofthebersystem.Dierentmodelsfortherodscomefromdierentdescriptionsofthedistributionofthehydrodynamicresistance,orfriction,alongtherod.Batchelor[ 42 ]andCox[ 43 44 ]formulatedadescriptionofhighaspectratio(lengthoverdiameter)bersusingslender-bodytheory.Otherhydrodynamicmodelsincludebead-rodmodelssuchasdumbbells(twobeads)andtrumbells(threebeads)todescribethebers.Researchers[ 2 4 8 38 39 45 46 47 ]haveassumedthatthedierencesbetweenthemodelswoulddisappearoncetherelevantdimensionshavebeenscaledaway.Chapter 3 showsthatthisassumptionisinvalid,astheratiooftheshorttimerotationaldiusivitiestotheaverageshorttimetranslationaldiusivitiesL2DR0=DT0isdierentforeachmodelandhasalargeimpactonthedynamics,whereListherodlength,DR0istherotationaldiusivityatinnitedilution,andDT0istheaveragecenterofmassdiusivityatinnitedilution. 2.1.1 TheSlender-BodyModel Batchelor[ 42 ]andCox[ 43 44 ]usedslender-bodytheorytodescribearigidberasalinedistributionofStokeslets.AStokesletrepresentstheeectofaforceappliedtoapointinauidinStokesow(zeroReynoldsnumberow).Thevelocityatapointxiinaninniteuidcausedbyaforcefjisgivenby 8ij 13

PAGE 27

whereistheuidviscosity.Ifthevelocityisthenintegratedoverthelengthoftheber(L)withtheoriginatthecenteroftheber,thevelocitybecomes 8"ZL=2L=2ijfj(s0) wheresisthelengthalongthemajorberaxis,ristheberradius,piisaunitvectoralignedwiththemajoraxis,andp?iisaunitvectorperpendiculartopi.Astheaspectratio(berlengthdividedbydiameter)increasesabove10,theintegralsasymptoticallyapproachasolutionof and wheredistheberdiameter.SubstitutingthesolutionsfromEquations( 2.3 )and( 2.4 )intoEquation( 2.2 )gives _xi+s_piu0i(s)=ln(2L=d) 4ij+pipjfj(s);(2.5) wherethevelocityoftheberhasbeensplitintoacenterofmassvelocity_xiandarotationalvelocity_pi,andu0i(s)isthedisturbancevelocitycausedbyowintheuidaswellasanyotherbers. Tosolveforthecenterofmassandrotationalvelocities,additionalstepsmustbeperformed.Equation( 2.5 )isintegratedovertheberlengthtogive 4ij+pipjFj+

PAGE 28

_xiZL=2L=2ds+_piZL=2L=2sdsZL=2L=2u0i(s)ds=ln(2L=d) 4ij+pipjFj+ 4ij+pipjFj+ 4Lij+pipjFj:(2.6) Thetotalforceactingontheberisgivenby Asimilarcalculationisdonetosolvefortherotationalvelocities,whereacrossproductistakenofEquation( 2.5 )withspiandthentheresultisintegratedovertheberlengthasshown ln(2L=d) 4ijkpjTk=ZL=2L=2ijkspj_xk+s_pku0k(s)ds+ 4ijkpjTk=ijkpj_xkZL=2L=2sds+ijkpj_pkZL=2L=2s2dsZL=2L=2sijkpju0k(s)ds+ 4ijkpjTk=ijkpj_xk(0)+ijkpj_pkL3 Thetorqueactingontheberisgiveby

PAGE 29

ThecrossproductofEquation( 2.8 )isthenperformedagainwithpiandtheresultis _pi=12 ThevectoridentityofijkklmpjplAm=(ijpipj)AjforanyarbitraryvectorAi[ 48 ]hasbeenusedalongwithdeningaweightedforcewhere Forsimulationsperformedwithoutmulti-bodyhydrodynamicinteractions,thetranslationalandrotationalvelocitiesfromEquations( 2.6 )and( 2.10 )become _xi=ln(2L=d) 4Lij+pipjFj:(2.12) and _pi=3ln(2L=d) ThecenterofmassandrotationalmobilitiesmatricescanthenbeobtaineddirectionfromEquations( 2.12 )and( 2.13 )togive 4Lij+pipj(2.14) whereMTijisthecenterofmassmobilitymatrix,andMRijistherotationalmobilitymatrix.Thecenterofmassmobilitymatrixcanbeseparatedintomobilitymatricesparallelandperpendiculartothecentralaxisoftherod.Thesemobilitiesare 2Lpipj(2.16) 4Lijpipj;(2.17)

PAGE 30

whereMkijisthemobilityparalleltothecentralrodaxis,andM?ijisthemobilityperpendiculartothecentralrodaxis.Itcanbeseenthatthemagnitudeoftheparallelmobilityisequaltotwicethatoftheperpendicularmobility. Thediusivitiesoftheslender-bodymodelcanbecalculatedfromthemagni-tudeofthemobilities, 2L(2.18) 4L(2.19) wherekBTistheBoltzmanntemperature,andDk0,D?0,andDR0aretheparallel,perpendicular,androtationaldiusivitiesoftherodsatinnitedilution.TheratioofDk0=D?0=2isthemaximumtheoreticalvalueforaslender-bodymodel,whichassumesthattheslender-bodyisinnitelythin.Theslender-bodymodelhasaninherentratiooftherotationalandaveragecenterofmassdiusivitiesatinnitedilutionwhichcannotbeneglectedthroughscalingarguments.Thisratiois 3kBTln(2L=d) 2L+2kBTln(2L=d) 4L=9;(2.21) whereDT0istheaveragecenterofmassdiusivityatinnitedilutiongivenbyDT0=1 3Dk0+2D?0.ThisratioofL2DR0=DT0playsasignicantroleinthedynamicsofconcentratedsuspensionsofrods,whichisdiscussedinChapters 3 and 4 2.1.2 TheRigid-DumbbellModel Arigiddumbbellcanbemodeledastwobeadswithcentersx(1)iandx(2)iwhichareseparatedbyaxeddistanceL.Theequationsforthecenterofmassvelocitiesforthetwobeadsundertheassumptionthatinertiacanbeneglectedsincethisis

PAGE 31

inStokesforarethengiveby _x(1)i=1 6a"F(1)iT@g @x(1)i#and_x(2)i=1 6a"F(2)iT@g @x(2)i#;(2.22) whereaisthediameterofthebead,F(1)iandF(2)iistheforceoneachbead,andTisthetensionalongthelinebetweenthecentersofthebeads.InordertoenforcethexedlengthofLbetweenthebeadsonthedumbbell,aconstraintgisdenedsuchthat ThederivativesoftheconstraintswithrespecttothecenterofmasspositionsofeachbeadgiveninEquations( 2.22 )are @x(1)i=2x(2)ix(1)iand@g @x(2)i=2x(2)ix(1)i:(2.24) Anotherrequirementfortherigiddumbbellisthattherelativevelocitiesalongtheconstraininglinebetweenthebeadsmustbezero, 0=_x(2)i_x(1)ix(2)ix(1)i:(2.25) SubstitutingthecenterofmassvelocitiesfromEquation( 2.22 )intoEquation( 2.25 )gives 0=1 6aF(2)iF(1)ix(2)ix(1)i+1 6aT@g @x(1)iT@g @x(2)i!x(2)ix(1)i:(2.26) ThederivativesoftheconstraintsfromEquations( 2.24 )aresubstitutedintoEquation( 2.26 )tosolveforthetensionT, @x(2)i@g @x(1)i!x(2)ix(1)i+

PAGE 32

4L2F(2)iF(1)ix(2)ix(1)i:(2.27) Thecenterofmassvelocityoftherigiddumbbell,_xi,isdenedastheaverageofthecenterofmassvelocitiesofeachbead, _xi=1 2_x(1)i+_x(2)i:(2.28) SubstitutingthebeadvelocitiesfromEquations( 2.22 ),aswellastheconstraintderivativesfromEquations( 2.24 ),andthevalueforthetensionfromEquation( 2.27 )intotheequationfortherigiddumbbellcenterofmassgives _xi=1 2"1 6aF(1)i+F(2)iT @x(2)i+@g @x(1)i!#+ 12aF(1)i+F(2)i+ 12aijFj; whereFjisthetotalforceactingonthedumbbell.TheijterminEquation( 2.29 )istheidentitymatrix,whichshowsthattherigiddumbbellhasanisotropiccenterofmassmobility(Dk0=D?0=1)atinnitedilution.

PAGE 33

Therotationalvelocityoftherigiddumbbellisdenedas _pi=1 BysubstitutinginthecenterofmassvelocitiesofeachbeadfromEquations( 2.22 )intoEquation( 2.30 )aswellasthevaluesfortheconstraintderivativesinEquations( 2.24 ),andthetensionfromEquation( 2.25 ),therotationalvelocityoftherigiddumbbellcanbecalculated, _pi=1 6aF(2)iF(1)iT @x(2)i@g @x(1)i!#+ 6aLF(2)iF(1)i4 4L2F(2)jF(1)jx(2)ix(1)ix(2)jx(1)j+ 6aLF(2)iF(1)i1 6aL0@ijx(2)ix(1)i 6aLijpipjF(2)jF(1)j;(2.31) wheretheorientationpiisdenedasthevectorbetweenthetwobeadsdividedbythedistancebetweenthebeads,thusmakingitaunitvector.ItisseenthattheforcesactingoneachbeadarestillbeingusedinEquation( 2.31 ).Theseforcescanbeconvertedtoanoveralltorque(orweightedforce)actingontherigiddumbbellbymultiplyingtherotationalvelocitybythedistancefromthecenterofthedumbbellatwhichtheforcesareacting(L=2).Inordernottochangethevalue

PAGE 34

oftherotationalvelocities,theequationisalsodividedbyL=2whichgives _pi=1 6aLijpipjF(2)jF(1)jL 3aL2ijpipj~F(2)j~F(1)j+ 3aL2ijpipj~Fj; where~Fjisthetotalweightedforceactingontherigiddumbbell. ThecenterofmassandrotationalmobilitiesmatricesfortherigiddumbbellareobtainedfromEquations( 2.29 )and( 2.32 )where 12aij(2.33) 3aL2ijpipj:(2.34) Theaveragecenterofmassmobilitymatrixisisotropic,meaningthattheparallelandperpendicularmobilitymatricesareequaltoMTij.Therotationalandcenterofmassdiusivitiesarethengivenby Theratiooftheshorttimerotationalandtranslationaldiusivitiesfortherigiddumbbellmodelis

PAGE 35

2.1.3 AGeneralApproach Otherhydrodynamicmodelsexistwhichalsocouldbeusedtodescribethebersinthesimulations.Brenner[ 49 ]describedthreesuchmodels,whichincludeaprolatespheroid,asymmetricaldoublecone,andacircularcylinder.Otherhydrodynamicmodelsalsoexist,andeachmodelcanbepreciselycalculated.TheratioofDk0=D?0forthesemodelsrangebetween1and2andencompasstheslender-bodyanddumbbellmodels.Suchahighlyprecisemodelthoughisnotrequiredforthehighaspectratiosthatwillbestudiedinthesesimulations. AmoregeneralapproachcanbetakeninderivingthehydrodynamicmodelsusedinthesimulationsofBrownianbers,whichincorporatesboththeslender-bodyandrigid-dumbbellmodels.ThisapproachbeginswiththeLangevinequa-tionsofmotionforeachrodwithinertiaincluded, wheremisthemassoftherod,Misthemomentofinertia,tisthetime,and_xiandiarethecenterofmassandangularvelocityvectorsoftherod.ThehydrodynamicforcesactingontherodareTij_xjandRi,whereTijisthecenterofmassresistancematrix,andRistherotationalresistance.Rotationaboutthemajoraxisofthehighaspectratiorodsisignored,consequentlytherotationalresistanceRinEquation 2.39 canbewrittenasascalar. TheBrownianforces,Fi,andtorques,Ti,ontherodaregivenbytheuctua-tiondissipationtheorem, and

PAGE 36

where(tt0)istheDiracdeltafunction.Forthepurposeofmakingdiscretetimestepsoflengtht,theDiracdeltafunctionisapproximatedas1=tfortimeswithinthesametimestepandzerootherwise.Equation( 2.39 )canberewrittenintermsoftheorientationvector(directioncosine)pi,whichisstrictlyaunitvector,andtheweightedBrownianforces~Fi, wheretherotationalvelocity_piisrelatedtotheangularvelocityaccordingto i=ijkpj_pkand@i Thetorqueisdenedintermsofaweightedforce, Theconditionthattheuctuation-dissipationtheoremmustbesatisedisusedtodeterminetheweightedforces.Theweightedforcesareofzeromean, Thevariancesoftheforcesarenotcorrelatedandhavetheform Todeterminethevariances,Equation( 2.42 )isintegratedovershorttimesttogiveasolutionfortherotationalvelocities, _pi(t)=Zt01 M(t)d:(2.47)

PAGE 37

Theautocorrelationfunctionfortherotationalvelocitiesisthengivenby[ 50 ] M(t)eR M(t0)dd0+ M(t)eR M(t0)dd0: Undertheassumptionofanequipartitionofenergyast!1,thevarianceoftherotationalvelocitiesis[ 50 ] Mijpipj:(2.49) EquatingEquations( 2.48 )and( 2.49 )demonstratesthat andthevarianceintheweightedforcesis ThevarianceintheweightedforcesinEquation( 2.51 )isproportionalto2kBTtimestheresistancematrix,whichappearsinEquation( 2.42 )asRijpipj.Therotationalmobilitymatrixisthentheinverseoftheresistancematrix, Notethatinvertingtheresistanceseemstorequireinversionofthesingularmatrix(ijpipj).Theappropriateviewisthataninverseisnotactuallyperformed,butratheraprojectionisperformedwhichextractsonlythosecomponentsonbothsidesoftheequationwhichareperpendiculartopi.

PAGE 38

Forexample,intheabsenceofinertia(@_pi=@t=0),therotationalvelocityisgivenbytheproductofthemobilityandweightedforces, _pi=1 wheretheprojectionof~Fjby(ijpipj)ensuresthattherotationalvelocity_piisperpendiculartopi.Insteadofsolvingthemobilityproblemwheretheweightedforcesareknownandtherotationalvelocityisdesired,onemightbeinterestedinsolvingtheresistanceproblemwheretherotationalvelocityisknownandtheweightedforcesareunknown.Inthiscase,theequationbecomes ~Fi=Rijpipj_pj;(2.54) wheretheprojectionof_pjbythematrix(ijpipj)ensuresthattheweightedforces~Fihavenocomponentsparalleltotheorientationvector.Theabovemobilityandresistanceproblemsindicatethatapseudo-inverseofthe(ijpipj)matrixcanbedenedby thoughcaremustbetakentonotviolatetheexactrelationshipbetweenEquations( 2.53 )and( 2.54 )of Theexpectedshort-timediusivitymatricesarenotalteredbychangingthedescriptionoftherodsfromangularvelocitiesandtorquestorotationalvelocitiesandweightedforces;theyaregivenbythethermalenergytimesthemobility,orinverseresistancematrices,

PAGE 39

Usingthisinformation,theshort-timediusivitycanbewrittenas whereasexpectedtherotationaldiusivityataninstantintimeisperpendiculartotheorientationvectorpi.Furthermore,thematrix(ijpipj)canbefactoredtogiveanexpression whereAikAjk=(ijpipj).ThisisanecessarystepinformingthediscreteBrowniandisplacementspioftheorientationsasseeninthefollowingsection. 2.2 BrownianForcesandTorques SolvingfortheBrownianforces(andtorques)fromEquations( 2.40 )and( 2.51 )gives t1=2AijWTjand~F(br)i=2 t1=2BijWRj;(2.61) whereWTjandWRjarevectorsofrandomnumberswithlengthsof3generatedbytheran2subroutine[ 51 ].Therandomnumbershavethepropertiesofzeromeanandunitvariance, Auniformdistributionsucesforthesimulations,sincearstorderintegrationtechniquewillbeused[ 52 ].ThematricesAijandBijareconstructedinsuchawaythat TheevaluationofAijandBijistraditionallydonebyCholeskydecomposition[ 51 ],althoughotherapproachesfordeterminingAijandBijareavailable[ 53 54 ].

PAGE 40

2.3 DiscretizedEquationsofMotion Oncetherotationalandcenterofmassvelocityequationsareknownforthevarioushydrodynamicmodels,thevelocitiesmustbediscretizedintimetosimulatethemotionofthebers.Toperformadynamicsimulation,thedierentialequationmustbeintegrated.Sincethegoverningdierentialequationsarestochastic,theusualmethodsofnumericalintegrationfordeterministicequationsmustbereconsidered[ 55 ]. 2.3.1 TheCorrectedEulerMethod OptionsforthenumericalintegrationincludeamodiedEulermethod,wherethedivergenceoftheshorttimediusiontensormustbeevaluated,multipliedbythetimestep,andaddedtotheBrowniandisplacement.Thismodicationisneededtocorrectforthespatialvariationsofthemobilityduringthetimestepwhichcausesameanerroratsecondorderwhichisnon-zero[ 56 ].ThismethodwasusedinthecalculationsofBrowniansuspensionsofspheresbyPhungetal.[ 57 ]andFossandBrady[ 58 ].Unfortunately,numericalevaluationofthespatialderivativesofthemobilitycanbeacostlyexercise. RatherthanintegratingEquations( 2.38 )and( 2.42 )todeterminethetimedependentcenterofmassandrotationalvelocities,andthendisplacements,ErmakandMcCammon[ 56 ]demonstratedthattheLangevinequationswithstochasticforcingscanbeintegrateddirectlytogivethedisplacements,butthatacorrectionisgenerallyneeded.ThisanalysisissimilarlyrepeatedherefortheLangevinequationsshowninEquations( 2.38 )and( 2.42 ).Eliminatingtheinertialtermsandsolvingfortherotationalvelocitygives _pi(t)=MRij(t)~Fj(t) (2.64)

PAGE 41

_pi=1 Aswritten,thisequationhasadimensionofthree,despitethefactthatonlytwocomponentsof_pi(t)andpi(t)areindependent.However,ateachinstantoftime,themultiplyingmatrix(ijpi(t)pj(t))appearinginEquation( 2.65 )extractsonlythecomponentsof~Fj(t)whichareperpendiculartopi(t).Therefore,_pi(t)isguaranteedtobeperpendiculartotheorientationandonlytwoprincipalcomponentsofpi(t)arealtered. IntegratingEquation( 2.65 )directlyusingastraightforwardEulerscheme,however,failstoproducethecorrectrotationaldiusivityduetoadriftintheaveragerotationalvelocity(h_pi(t)i6=0)atrstorderinthetimestept,unlessextremelysmalltimestepsareused.ThisnonzerodriftinthevelocityhasoriginsintheeliminationoftheinertialtermsfromtheLangevinequation.ErmakandMcCammon[ 56 ]developedanalgorithminwhichtheEulermethodcanbemodiedtoproducetheproperdiusivity;thisalgorithmiscommonlyusedtosolvestochasticLangevinequations[ 59 55 52 ]. Derivationofthedriftvelocity Toderivethedrifttermneededtocorrecttherstorderdiscretealgorithm,theformalismusedbyGrassiaetal.[ 60 ]isimplemented.Inthisscheme,theorientations,rotationalvelocities,andmobilityareexpanded, _pi(t)=_p(1)i(t)+_p(2)i(t)+:::(2.67) wherethesuperscriptsindicatetheorderoftheapproximation.SimilarlytoGrassiaetal.[ 60 ],p(1)i(t)representsthestandardlinearBrownianrotation,whichiscalculatedbyreplacingRij(t)withRp(0)i.Thetermp(2)i(t)isasmallnonlinear

PAGE 42

correctionwhicharisesfromthesmallvariationinthefrictionbetweenRjp(0)iandRij(t).SubstitutingtheseexpansionsintoEquation( 2.65 )givesexpressionsfortherotationalvelocitiesatrstorder, _p(1)i(t)=MRijp(0)i~Fj(t);(2.69) whereitcanbeseenthat_p(1)i(t)(andconsequentlyp(1)i(t))isproportionaltop _p(2)i(t)=0@p(1)j(t)@MRij where_p(2)i(t)(andp(2)i(t))isproportionaltotsincebothp(1)i(t)and~Fj(t)areoforderp Averaging_p(1)i(t)gives sinceMRijp(0)iisconstantandD~Fi(t)EiszeroasspeciedinEquation( 2.47 ).However,themeanrotationalvelocityatsecondorder, isgenerallynon-zerosincetherstorderdisplacementsandweightedforces~Fi(t)arecorrelated.

PAGE 43

Todeterminethecorrelation,thedisplacementsatrstorderarecalculatedbyintegratingEquation( 2.69 )overtime, wherep(1)i(0)istheknownpositionattimezeroandtislimitedtoshorttimes.Multiplyingp(1)i(t)by~Fi(t)andaveraginggives, sinceR(tt0)dt0=1=2andMRijp(0)iRkjp(0)i=(ijpi(t)pj(t))insteadofij. SubstitutingthecorrelationfromEquation( 2.74 )intoEquation( 2.72 )givesthemeandriftvelocityatsecondorder, whichisnon-zero.Thedriftvelocitycanbesimpliedusingtheexplicitexpressionforthemobility,

PAGE 44

Rjkpj(t)pk(t)@ @pi(t)jkpj(t)pk(t)+ Rjkpj(t)pk(t)@jk Rjkpj(t)pk(t)ijpi(t)pj(t)pk(t)pj(t)ikpi(t)pk(t)+ R"jkpj(t)pk(t)jkpj(t)pk(t)pi(t)pi(t)jkpj(t)pk(t)jkpj(t)pk(t)#: Inderivingthislastequation,usehasbeenmadeoftheidentity whichstatesthattheorientationvectorpi(t)cannotchangeindirectionsparalleltopi(t).CompletingthemultipliesinEquation( 2.76 )gives R"jkpj(t)pk(t)jkpj(t)pk(t)pi(t)pi(t)jkpj(t)pk(t)jkpj(t)pk(t)#+ Rh2pi(t)i+

PAGE 45

Rpi(t): ThisisequivalenttothenegativeofthedivergenceoftherotationaldiusivitymatrixDRij, @pi(t)"kBT Rjkpj(t)pk(t)#+ Rpi(t)+ whichcanalsobeseendirectlyinEquation( 2.75 ). Thenumericaldiscretization AnaiveimplementationofanEulermethodtointegrateEquation( 2.65 )producesunnecessarilylargeerrorsbecausethedriftvelocityD_p(2)i(t)Easderivedinthelastsectionisnon-zero.However,thissystematicerrorcanbecorrectedbyaddingD_p(2)i(t)EtotheEuleralgorithmateachstepintheEuleralgorithmtoproducediusivitieswhichareaccuratetoorder(t)intime.Thisgivestheexpressionof wherepiisthestochasticdisplacementperpendiculartotherodaxis. Usingtheexpressionforthedivergenceoftherotationaldiusivityasappear-inginEquation( 2.78 ),thediscretealgorithmcanbewrittenas Rpi(t)t+pi:(2.81)

PAGE 46

Thecorrectionof2kBT Rpi(t)tindicatesthattherodlengthmustbeshortenedasmallamountatthebeginningofeachtimestep.Thisdoesnotmeanthattheorientationvectorgrowsshorterwithtime.Infact,thecorrectioncanbeinterpretedinasimple,physicalmanner:thiscorrectionmaintainspiasaunitvectoronaverageatordert.Afteratimestepoft,theaverageexpectedvaluehpipiiequalsoneplusanerroroforder(t)2.Thisisprovenbymultiplyingthediscreteequationforpi(t+t)byitselfandaveragingovermultiplerealizations, Rpi(t)t+pipi(t)2kBT Rpi(t)t+pi+ Rt+2kBT R2(t)2+D(pi)(pi)E: IngoingfromthersttosecondlineinEquation( 2.82 ),thevectorpiisassumedtobeaunitvectorattimetandthestochasticdisplacementofpiisperpendiculartopi, Thecorrelateddisplacementsh(pi)(pi)iarecalculatedusingEquations( 3.26 )and( 3.28 )fromChapter 3

PAGE 47

Rijpipjijpipj(t)+ R(t); whichistheexpectedresultfortheshorttimedisplacements[ 6 ].SubstitutingtheresultfromEquation( 2.84 )intoEquation( 2.82 )gives R2(t)2:(2.85) Theaveragelengthoftherodisexactly1toO(t)intime.OnlyatO(t)2doestheaveragerodlengthincrease.Withoutthecorrection,thelengthoftherodwouldextendlinearlywiththetimestep.Moreimportantly,thediusivitywillcontainanunnecessarilylargeerrorwithoutproperapplicationofthecorrection,thoughitshouldbenotedthattherotationaldiusivitycanbemadearbitrarilyclosetothecorrectresultbymakingvanishinglysmalltimesteps. Figure 2{1 demonstratestheeectofthecorrectiongraphicallyforarepre-sentativedisplacement.Withoutapplyingacorrection,thenewpositionfortheorientation, isguaranteedtowalkothesurfaceoftheunitsphereasshowninthegure.Applyingthecorrectionlowersqiintheoriginaldirectionoftheorientationpi(t)togive Rtpi(t):(2.87) ForthespeciccaseshowninFigure 2{1 ,thecorrectionmovesqifromapositionoutsidetheconstrainingspheresurfacetoapositioninsidetheunitsphere.Dependingonthestochasticvalueofpiattimet,thevalueofpi(t+t)canfalleitherinsideoroutsideoftheunitsphereafterthediscretetimestep.The

PAGE 48

Diagramshowingtheeectofthedriftcorrectionupontheorientationvectorpi(t)afteradiscretetimestept.Lengthscalesinthegurehavebeenexaggeratedtomakethequalitativepointsclear. correctionissuchthattheleadingerroroftiseliminated,andtheunitvectormaintainsitsaveragelengthexceptforasmallerroroforder(t)2. 2.3.2 TheMidpointMethod Asanalternativenumericalintegrationmethod,Grassiaetal.[ 60 ]andGrassiaandHinch[ 61 ]advocatethemidpointmethoddevelopedbyFixman[ 62 ].Anadvantageofthemidpointmethodisthatthespacialderivativesofthediusivitiesarenotrequired,whichcanbeanintensecalculationforcomplexsystems.LikethemethoddueofErmakandMcCammon[ 56 ],themidpointmethodcorrectsforthedriftinthemobilityoveratimestep.Thismidpointmethodissimilartotheclassicalmidpointmethodforsolvingdeterministicequationsexceptfortwoissues.TheBrownianforcesofEquation( 2.61 )mustbeheldconstantovertheentiretimestep,thoughotherforcespresentinthesimulationarereevaluatedatthemidpointasusual.Also,themethodisrstorderinaccuracy,ratherthan

PAGE 49

second;forstochasticproblems,themidpointmethoddoesnotnecessarilyimprovetheaccuracyofthesolution. Usingtheequationsfortherotationalandtranslationalvelocitiesoftherigid-dumbbellmodelasexampleinusingthemidpointmethodgivesthefollowingdiscretizedequationsforthehalf-step 12aijFjt 3aL2ijpi(t)pj(t)~Fjt wherexiandpiarethehalf-stepcenterofmasspositionsandorientations.TheBrownianforcesarekeptconstantacrosstheentiretimestep,butallotherforcesarerecalculatedusingthehalf-stepvaluesofthepositionsandorientations.Thefullstepequationsarethencompletedusingthehalf-stepvaluesaswellasthenewlycalculatedforcesandtorques, 12aijFjt(2.90) 3aL2ijpipj~Fjt:(2.91) 2.4 SimulationsofDiluteSuspensionsofBrownianFibers BeforesimulationsofsuspensionsofBrownianberscanbeperformed,thevalidityandaccuracyofthenumericaldiscretizationandintegrationmethodsmustbeconrmed.Oncethemobilityforthecollectionofhydrodynamicallyinteractingrodsiscalculated,theBrownianforcesandtorquescanbeaddedtothesimulationmethod.Fortheseinitialsimulations,multi-bodyhydrodynamicinteractionsareignored,andsothemobilitiesarethoseseeninEquations( 2.14 )to( 2.17 ),andEquations( 2.33 )and( 2.34 ).

PAGE 50

Plotofthespheresweptoutbytheorientationvector,wherep(1),p(2),andp(3)arethethreeCartesiandirectionsofpi.Thetimestepusedwas1105,andatotalof1106timestepsweretaken. 2.4.1 TestingtheBrownianMotion ThereareseveralqualitativeandquantitativetestswhichcanbeusedtoverifythevalidityoftheBrownianmotionproducedbythedierentnumericalintegrationmethodsusedintheprevioussections.TheBrownianforcesandtorquescalcu-latedinsection 2.2 dependonthesizeofthetimestept.TherandomnumbergeneratorusedtoproducetheBrownianforcescreatesauniformdistributionofforcesandtorques.TheBrowniantorquesshouldproducedisplacementsintheorientationpiwhichsweepsoutaspherearoundtheorigin.Thethreeindividualcomponentsoftheorientation(p(1),p(2),andp(3))arethenplottedinFigure 2{2 .Itcanbeseenthatafter1106timesteps,thesphereisevenlysweptout. Anothertestofthedisplacementsoftheorientationsistolockoneofthethreeindicesofpiinplace.TheothertwocomponentsoftheorientationarethenallowedtosteponcealongthesurfaceofthespheresweptoutbytheorientationvectorseeninFigure 2{2 .Afterthisinitialstep,thecomponentsareresetback

PAGE 51

PlotoftheBrowniandisplacementsoftheorientationwithonecom-ponent(p(3))lockedwiththez-axis.Timestepsgreaterthan1104donotproduceBrowniandisplacementsadequatemaximumdisplace-ments. totheirinitialpositions.TheBrowniandisplacementsproducedwillmapoutasquareintwodimensionswherethefreecomponentsofpiareequallyprobabletobedisplacedfromtheorigin.Themagnitudeofthedisplacementisdependentonthetimestepsizet,whichisseeninequation( 2.61 ).AnadequatetimestepsizeneededtoaccuratelyresolvetheBrownianmotionofthesystemwillproduceaclearlydenedsquareinthetwocomponentsofpiwhicharenotlocked.Thedisplacementsshouldequallyllinasquareintheplainperpendiculartothelockedindex.ThesizeofthesquareisdependentonthemagnitudeoftheBrownianforcesandtorques.Thesedisplacements,whichdependonthetimestepsize,areshowninFigure 2{3 ,whereitisseenthatfortimestepssmallerthan1104donotadequatelyllouttheedgesofthesquares.Allofthesesimulationswereperformedwithatotalof1106timesteps.

PAGE 52

Theprevioustwotestsarequalitative,providingnoinformationaboutthevalidityofthemagnitudeoftheBrownianforcesandtorqueswhichproducethedisplacementsinthesimulations.Themagnitudesoftheforcesandtorquesaretestedbycalculatingthediusivityvaluesofthecenterofmassandrotationbyusingtheautocorrelationfunctionsofthecenterofmassandrotationaldisplace-ments.Theaveragecenterofmassdiusivityiscalculatedfromtheautocorrelationfunctionas forlargevaluesof,whichareshowninFigure 2{4 .Atsmallvaluesofthedierenceinthediusivitiesparallelandperpendiculartothecentralrodaxiscanbeseen,where and wherexki(t+)xki(t)andx?i(t+)x?i(t)arethedisplacementsparallelandperpendiculartothecentralrodaxisovertime.Afterlongenoughtimesthemotionoftherodsbecomede-correlatedfromtheoriginalorientationsandsotheparallelDkandperpendicularD?diusivitiesbecomethesameastheaveragecenterofmassdiusivityDT.Figure 2{4 showsthattheMidpointandcorrectedEulermethodsproduceidenticalresults,withinthelimitsofnumericalaccuracy. Therotationaldiusivitycanbecalculatedinasimilarmannerusingtheautocorrelationfunctionsoftheorientationswhere fortimescalesofDRgreaterthanorequalto1.TheplotofthisfunctionisdisplayedinFigure 2{5 .Itisdiculttoobtainanaccuraterotationaldiusivity

PAGE 53

Plotoftheautocorrelationfunctionofthecenterofmassdisplace-mentsovertimeforboththecorrectedEulerandMidpointmethods,whichproduceindistinguishableresults.ForshorttimesDk0=2D?0,butastime()increasesbothconvergetothevalueofDT.Thelong-timecenterofmassdiusivityDTisequaltotheshorttimediusivityforasinglerodininnitedilution. fromthisautocorrelationfunction,asdoinganumericaltofanexponentiallineisdicult.However,takingthenaturallogoftherotationalautocorrelationfunction[ 38 39 ] 2lnDpi(t+)pi(t)E=2DR(2.96) producesalinewiththeslopeequaltoDR.ThistransformationisseenintheinsetofFigure 2{5 .OnceagaintheMidpointandcorrectedEulermethodsproduceidenticalresultsatshorttimes.Theseresultscanbeimprovedsothattheyareidenticalfortheentiretimeregimeshownwithfurthertimeaveraging. 2.4.2 ErrorintheCorrectedEulerMethod IthasbeenmentionedpreviouslythattheEulermethodusedincludesacorrectionforthestochasticnatureoftheproblemwhichimprovestheaccuracyofthesimulations.Aninvestigationintowhatthesimulationresultswouldbewithoutthecorrectionhasbeenconductedandtheresultswillbeshown.Simulationswere

PAGE 54

Plotoftheautocorrelationfunctionoftheorientationovertime.BoththecorrectedEulermethodandtheMidpointmethodproducesimilarresults.Therotationaldiusivityismoreeasilytakenfromatoftheinsetdata,whichislinear. performedatdierenttimesstepsforasinglerodinanunboundeduidusingthenumericaldiscretizationofEquation( 2.81 ).Figure 2{6 showstheresultsofthesimulations,whereinthelong-timerotationaldiusivitieswerecalculatedusingEquation( 3.33 ).Errorsinthevaluesoftherotationaldiusivitieswereestimatedbycalculatingthestandarddeviationoftheaveragerotationaldiusivityfromanensembleoftendierentsimulationsoveratotaldimensionlesstimeperiodof1104.Figure 2{6 alsoshowsacomparisontoanestablishednumericalmethod[ 2 ]whichisdescribedindetailinthefollowingsection.Therenormalizationofpi(t+t)isdiscussedinmoredetailinthefollowingsection,buttosummarize,insomecasestheorientationvectorpimustberenormalizedbacktoaunitvectoraftereachtimestepistakentoavoidunnecessarilylargeerrors. Theresultsdemonstratethatthediusivityiscorrectforasucientlysmalltimestep,eventhoughtheorientationvectorpiwasnotrenormalizedafter

PAGE 55

Long-timerotationaldiusivitiescalculatedusingeithertheEulermethodorthealgorithmofLowen[ 2 ].Thesquaresaresimulationsus-ingtheEulermethodwiththedriftcorrectionandnorenormalizationofpi(t+t).ThediamondsaresimulationsusingtheEulermethodwiththedriftcorrectionwherepi(t+t)isrenormalizedaftereachtimestep.ThetrianglesareforsimulationsusingtheEulermethodwithoutthecorrection,andthecirclesareforsimulationsusingthealgorithmofLowen[ 2 ].BoththeEulermethodwithoutthedriftcor-rectionandthealgorithmofLowen[ 2 ]requiredrenormalizationaftereachtimestep. eachtimestep.ThisisinkeepingwiththecalculationoftherodlengthgiveninEquation( 2.85 ),whichshowsthattherodlengthisproperlymaintainedtorstorderinthetimestep.However,thesimulationswhichwereperformedinChapter 3 employedtheEulermethodwiththedriftcorrectionandpi(t+t)wasrenormalizedaftereachtimestep.Theorientationisrenormalizedtoaccountforthedeterministicforcingduetotheexcludedvolumeforceactingonarod;thedriftcorrectiononlycorrectsthestochasticportionofthealgorithm.Figure 2{6 demonstratesthattherenormalizationdoesnotchangetheresults. Failingtoapplythedriftcorrectionproducesnumericalresultswhicharesignicantlylessaccurate,thoughFigure 2{6 showsthattherotationaldiusivity

PAGE 56

isproducedwithinnumericalaccuracyfortimesstepssmallerthant=4105;thealgorithmwiththecorrectionproducestheexpectedresultfortimestepsaslargeast=3104.Renormalizationofpi(t+t)isrequiredaftereachtimestepwhenusingtheEulermethodwithoutthecorrectionregardlessofthetimestepduetotheerrorofordertinthelengthoftherod,hpi(t+t)pi(t+t)i1+O(t). 2.4.3 ComparisonofCorrectedEulerMethodtoAnotherEstablishedMethod Lowen[ 2 ]employedasimilarnumericaldiscretization,butnocorrectionwasincluded.Ingeneral,acorrectionexistsfortherstorderEulermethodisthedivergenceofdiusivityisnon-zero.However,thedivergenceofthediusivityifformallyzeroforLowen's[ 2 ]formulationsinceequationsintheformoforientationsandtorqueswereused,ratherthanorientationsandweightedforcesasdonehere. ThediscreteequationforthechangeintheorientationoveratimestepusedbyLowen[ 2 ]is wherex1andx2arerandomnumberswithzeromeanandvarianceof2kBT Rt,ande(1)i(t)ande(2)i(t)aretwounitvectorsperpendiculartopi(t)andtooneanother.UsingasimilartechniquetothatusedinEquation( 2.82 )wherepi(t+t)ismultipliedbyitselfandthenaveragedovermultipleiterations,theexpectederrorinthelengthoftherodafteronetimestepis

PAGE 57

Rt: ComparingtheresultsofthisequationtothatofEquation( 2.85 )indicatesthattheEulermethodwiththedriftcorrectionismoreaccurate.Inordertomaintainpi(t)asaunitvectorwhenusingthenumericalalgorithmofEquation( 2.97 ),pi(t+t)mustberenormalizedaftereachtimestep. SimulationswereperformedusingthealgorithmofEquation( 2.97 )withthesameparametersasthesimulationsusingtheEulermethod.Figure 2{6 showstheresultsincomparisontotheEulermethodwithandwithoutthecorrection.TheresultsofthediscretizationusedbyLowen[ 2 ]andtheEulermethodwithoutthecorrectionareinterestinglyequivalent.Thevaluesoftherotationaldiusivityareaccurateforallthemethodsfortimestepssmallerthan1105.ItisevidentthatcorrectvaluesoftherotationaldiusivitiesarestillcalculatedusingtheEulermethodwiththecorrectionfortimestepsupto3104,while4105isthelargesttimestepwhichstillonaverageproducescorrectvaluesfortherotationaldiusivityforthealgorithmusedbyLowen[ 2 ]. TheEulermethodwiththecorrectionismoreaccuratethanthealgorithmdenedbyLowen[ 2 ]asseeninthecomparisonofthenumericalerrorsinFigure 2{7 .Sincethecomputationsareforthesimplecaseofasinglerodwheretheshort-timeandlong-timediusivitiesareequivalent,amoredetailedanalysisoftherelativeerrorsofthealgorithmscanbeperformed.Theexpectedvalueoftheshort-timerotationaldiusivitycanbedirectlycalculatedasafunctionofthetimestepfrom 4tDpi(t+t)pi(t)2E;(2.99) whereeitherthealgorithmofEquation( 2.81 )or( 2.97 )canbeusedforpi(t+t).Inthecaseswhereanormalizationoftheorientationvectoriscarriedoutafterthemoveoftimestept,Equation( 2.99 )mustbemodiedbydividingpi(t+t)by

PAGE 58

Thepercentageerrorinthelong-timerotationaldiusivitiescalcu-latedusingtheEulermethodwiththecorrectionandthealgorithmofLowen[ 2 ].Thedottedlineisthetheoreticalerrorintheexpectedshort-timediusivitiesforsimulationsusingthealgorithmofLowen[ 2 ],whilethedashedlineisthetheoreticalerrorusingtheEulermethodwiththedriftcorrection.ThesymbolsarethesameasinFigure 2{6 .Therelativeerrorbetweenthetwomethodsisseenintheinset,wherethetheoreticalerrorofthealgorithmofLowen[ 2 ]isdividedbythetheoreticalerroroftheEulermethodwiththedriftcorrection. themagnitudeofpi(t+t), 4t*pi(t+t) Figure 2{7 showsthattheerrorinthelong-timerotationaldiusivitiesisaccuratelypredictedbytheerrorvaluescalculatedfromtheexpectedshort-timerotationaldiusivitiesasgivenbyEquation( 2.100 ).ThisconrmsthattheerrorinthecorrectedEulermethodissmallerthantheerrorresultingfromthealgorithmgivenbyLowen[ 2 ].Theinsertingure 2{7 showsthattherelativeerrorbetweenthetwoalgorithmsascalculatedusingEquation( 2.100 )approachesaconstant

PAGE 59

valueasthetimestepgoestozero.TheerrorintheEulermethodwiththecorrectionisatleastthreetimessmallerthanthealgorithmusedbyLowen[ 2 ]foralltimesteps. 2.5 Conclusion TheadvantagesofusingthecorrectedEuleralgorithmincludetheincreasedaccuracyandeciency.ThoughtheEulermethodwiththecorrectionismoreaccuratethanthealgorithmofLowen[ 2 ],thedierenceinaccuracyisnegligibleforthesmalltimesstepsusedinthesimulationsoftheconcentratedrodsystems.Sim-ulationshavebeenperformedusingthealgorithmofLowen[ 2 ]atconcentrationsofnL3=70and150fortheslender-bodymodelwithaspectratiosof25and50,andtheresultsarestatisticallyequivalenttothoseinwhichtheEulermethodwiththedriftcorrectionwasused.Sincetheaccuracyofthesimulationsisessentiallyequivalentforeithermethod,numericaleciencyisthenextconsiderationwhenchoosingbetweenthetwonumericalmethodsforthisparticularapplication.ThealgorithmusedbyLowen[ 2 ]requiresdeterminingtwounitvectorsthatareper-pendiculartopi(t)andtooneanotherateverytimestep.TheseextracalculationsmakethemethodlessecientthantheEulermethodwiththecorrection;forourimplementations,theEulermethodwiththecorrection(Equation( 2.81 ))wasfoundtobe26%fasterthanthealgorithmgivenbyEquation( 2.97 ).

PAGE 60

3.1 Introduction Rigidpolymersarewidelyusedashighperformanceplastics[ 10 ]andexamplesofBrownianberscanbefoundintheformofmacromoleculesofbiologicalorigin[ 19 ]andinnanotechnologyintheformofnanotubesandnanorods[ 63 64 ].RigidBrownianrodsarealsothesimplestcolloidalsystemdemonstratingtheeectsofentanglement[ 6 ],thereforethestudyofBrownianbershasthepotentialtoilluminatefundamentalissuesinthegeneralareaofpolymerphysics.Consequently,numeroustheoretical,computational,andexperimentalresultshavebeenpublishedonthedynamicsandrheologyofBrownianrods. Conictingtheoriesfortherotationaldiusivity(DR)inthelimitofinnitelythinrodsinconcentratedsuspensionspredictscalingsofDR=DR0(nL3)witheither=2or1,wherenisthenumberdensity,Listhelengthofthebers,andDR0istherotationaldiusivityofanisolatedberinaninniteuid.Doi[ 7 ]developeda\tube"theorywhichpredictsrotationaldiusivitiesscalingas(nL3)2fornumberdensitieswithinthesemi-diluteregime(1=L3n1=dL2)[ 7 6 ],wheredisthediameteroftheber.Extensionsofthetheoryhavebeenmadeforexibility[ 23 ],polydispersity[ 40 ],andhydrodynamicinteractions[ 41 ].SimulationsperformedbyDoietal.[ 3 ]ofinnitelythinbers,inwhichthehydrodynamicinteractionsbetweentherodswereignoredandcrossingofthecenterlinesoftherodswaspreventedthroughareectionrule,conrmedthescalingof(nL3)2.SimulationsperformedbyFixman[ 34 35 ]foundadierent 47

PAGE 61

scalingofDR(nL3)1fortherotationaldiusivityathighconcentrationsinthelimitofinnitelythinrods.SimilartoDoietal.[ 3 ],adynamicsimulationtechniquewasimplemented,butshort-rangepotentialspreventedtherodsfromoverlapping.Simulationsperformedbyotherresearchersconrmedascalingexponentthatwascloseto=1fortherotationaldiusivity[ 38 39 ].Fixman[ 34 35 ]proposedanalternativetheoryfoundedupontheconceptof\cooperativerotation"asthemechanismfortherotationaldiusionoftheberstoaccountforthesimulationresults. Browniandynamicssimulationspresentedheredemonstratethatthediscrep-anciesbetweenprevioussimulationresultsarisefromthechoiceofmodelfortherods.ThedimensionlessratioofL2DR0=DT0determinesthescalingofDR=DR0,whereDT0istheaveragecenterofmassdiusivityforaberinsolutionatinnitedilution.Itisalsoobservedthatthethicknessoftherodsplaysaroleinthecal-culatedscalingfortherotationaldiusivity,evenatrelativelyhighvaluesfortheaspectratios(A).AsL2DR0=DT0variesbetween4(rigid-dumbbellmodel)and9(slender-bodymodel)forrodswithanaspectratioof50,theexponenttransitionsbetweenapproximately1and2.Forrodswithanaspectratioof25,theratioL2DR0=DT0mustbevariedbetween1and9inordertoproduceascalingthatrangesbetweenapproximately=1and2.TheexponentremainsnearlyconstantforL2DR0=DT09,regardlessofwhichrodthicknesswasused.Intherangeof0L2DR0=DT04,theexponentdecreasessignicantlyforallaspectratiosstudied.Thesendingspresentedheredemonstratethattherotationaldiusivityathighconcentrationsisnotexclusivelycontrolledbytheexcludedvolume. TheBrowniandynamicsmethodusedtodemonstratethedependenceofthescalingexponentupontheratioofshort-timediusivitiesispresentedinSec. 3.2 .Themethodignoreshydrodynamicinteractionsbetweentherodsandusesshort

PAGE 62

rangepotentialstoenforceexcludedvolumefortherods.TheratioL2DR0=DT0wasvariedbetween1and9forvaryingaspectratios(berlengthoverdiameter),withthelimitingcasesofL2DR0=DT0=0andL2DR0=DT0=1alsobeingstudied.TheremainingthreesectionscontaintheresultsoftherotationaldiusivitystudiesinSec. 3.3.1 andtheresultsofthetranslationaldiusivitystudiesinSec. 3.3.2 .AdiscussionoftheresultsandhowtheycomparetotheoriesandexperimentscanbefoundinSec. 3.4 .TheconclusionsreachedthroughthisworkaresummarizedinSec. 3.5 3.2 SimulationMethod Theunderlyingphysicalmodelfortheshort-timediusivitiesoftheindividualrodsgreatlyinuencesthebehavioroftherotationaldiusivitiesforconcentratedsystemsascalculatedusingsimulations.ResearchershaveselecteddierentmodelsfortherodswhichhavedierentvaluesofthecriticalparameterL2DR0=DT0.Thisdierencearisesfromthemannerinwhichthehydrodynamicresistanceisdistributedalongthelengthofthebers.Doietal.[ 3 ]usedaslender-bodymodelfortheindividualberswhenperformingsimulationsandfoundascalingofDR=DR0(nL3)2forinnitelythinrods.Theslender-bodymodel[ 42 43 44 ]yieldsaratioofL2DR0=DT0=9.Additionally,thismodelhasananisotropiccenterofmassmobility,whereforaninnitelythinrodatinnitedilution,themobilityinthedirectionparalleltothecentralaxisoftherodistwicethemobilityinthedirectionperpendiculartothecentralaxis.Otherauthors[ 34 35 ]foundascalingofDR=DR0(nL3)1whenusingarigid-dumbbellmodelwhichhasaratioofL2DR0=DT0=4.Therigid-dumbbellmodel,inwhichthespheresdonotinteracthydrodynamically,hasanisotropiccenterofmassmobility.Figure 3{1 summarizestheimportantcharacteristicsofthetwomodels. Toinvestigatetheroleplayedbytheratioofdiusivitiesatinnitedilution,simulationswereperformedforvaluesoftheparameterL2DR0=DT0between1and

PAGE 63

Summaryofthecharacteristicsandvariablesdescribingtheslender-bodyandrigid-dumbbellmodels.Thevectorxidenesthepositionofthecenterofmassoftheberandtheunitvectorpidenestheori-entation.TheberlengthisL,disthehydrodynamicdiameteroftheberintheslender-bodymodel,andaisthehydrodynamicdiameterofthesphereintherigid-dumbbellmodel. 9forrodsofvaryingaspectratios.Inaddition,simulationswereperformedforrodswithnoBrowniancontributiontotheircenterofmassmotion(L2DR0=DT0!1)andforrodswithnoBrowniancontributiontotheirrotation(L2DR0=DT0=0).ForeachvalueofL2DR0=DT0,simulationswereperformedforisotropicmodelswitharatioDk0=D?0=1andanisotropicmodelswitharatioDk0=D?0=2,withtheexceptionofthesimulationswithanaspectratio(A)of500.ThediusivitiesDk0andD?0arethediusivitiesatinnitedilutionofthecenterofmassinthedirectionsparallelandperpendiculartothecentralrodaxis. 3.2.1 GoverningEquations ThemotionofeachrodinthesuspensionisgivenbytheLangevinequations

PAGE 64

wheremisthemassoftherod,Misthemomentofinertia,tisthetime,andx()iand()iarethecenterofmassandangularvelocityvectorsofrod.Thehydrodynamicforcesactingontherodare(;T)ijx()jandR()i,where(;T)ijisthecenterofmassresistancematrixandRistherotationalresistance.Rotationaboutthemajoraxisofthehighaspectratiorodsisignored,consequentlytherotationalresistanceRinEquation( 3.2 )canbewrittenasascalar. andavarianceof and asrequiredbytheuctuation-dissipationtheorem.TheparameterkBTisthethermalenergy,ijistheidentitymatrix,and(tt0)istheDiracdeltafunction.Thebracketshiindicateanensembleaverage.TheBrownianforcesandtorquesareuncorrelatedintimeandtherearenocorrelationsbetweenrodsbecausemulti-bodyhydrodynamicinteractionsareignoredwithinthesesimulations. Ratherthanworkingwithanglesandangularvelocitiestodescribetherods,thesimulationspresentedhereuserodorientationsandrotationalvelocitiesashasbeendonebyotherresearchers[ 2 65 ].Inaddition,thetorqueisrewrittenintermsoftheweightedforcedistributionactingontherodsasexplainedinthefollowing.ToconvertEquation( 3.2 )intoaformincludingonlyrotational

PAGE 65

velocities,accelerations,andweightedforces,therelationships ()i=ijkp()j_p()kand@()i areused,wheretheorientationvectorp()iisstrictlyaunitvectoralignedwiththemajoraxisofrodand_p()iistherotationalvelocity.Theorientationvectorisequivalenttothedirectioncosinefortherod.Forrodsofhighaspectratio,thetotaltorqueactingontherodscanbeapproximatedasthecrossproductoftheorientationvectorandtherstmomentoftheforcedistributionintegratedalongthecenterlineoftherod[ 43 44 ], wherefi(s)isthelineforcedensityatapositionsalongtheprimaryaxisoftherodandsrangesfromL=2toL=2,withLbeingtherodlength.Deningtheweightedforce~F()iactingonrodas ~F()i=ZL=2L=2sfi(s)ds(3.8) allowsthetorquetobewrittenas[ 66 ] PerformingthecrossproductofEquation( 3.2 )withtherodorientationp()iandusingEquations( 3.6 )and( 3.9 ),theLangevinequationforangularmomentumcanbewrittenas Makinguseofthevectoridentityijkp()jklmp()lam=ijp()ip()jaj[ 48 ],whereajisanarbitraryvectorquantity,allowsEquation( 3.10 )tobewritteninthe

PAGE 66

form where(;R)ijistheresistancematrix(;R)ij=Rijp()ip()j.SimilarlytotheBrowniantorquesappearinginEquations( 3.3 )and( 3.4 ),theweightedforces~F()imustsatisfytheuctuationdissipationtheorem, and ThevarianceintheweightedforcesinEquation( 3.13 )isproportionalto2kBTtimestheresistancematrix(;R)ijwhichappearsinEquation( 3.11 );aderivationforthevarianceoftheweightedforcesisgivenintheChapter 2 Equations( 3.1 )and( 3.11 ),togetherwiththeBrownianforcingsinEquations( 3.3 ),( 3.4 ),and( 3.13 ),describethemotionofahighaspectratiorodwithdiusivitiesof whereD(;T)ijandD(;R)ijarethediusionmatricesforthecenterofmassandrotation.Theinverseofthematrixijp()ip()jappearsinthecalculationoftherotationaldiusivity,howeverperformingatraditionalinverseofthismatrixisnotpossiblebecauseitissingular.Understandingthatthissingularityarisesfromtheprojectionperpendiculartotherodaxiswhichensuresthat_p()ihasnocomponentsinthedirectionofp()i(therodisinextensible)allowsthedenitionofanappropriatepseudo-inverse,ijp()ip()j1=ijp()ip()j(seeChapter 2 forfurtherexplanation).Thecenterofmassandrotationaldiusivitiescorrespond

PAGE 67

tothemodelchosentorepresenttheresistancesoftherods.Fortherigid-dumbbellmodel[ 6 ],theseresistancesresultinthefollowingshort-timediusivitiesfortherodssystemsbeingsimulated: whereistheviscosityofthesuspendinguid,aisthehydrodynamicdiameterofeachsphereontherigid-dumbbell,Listherodlength,Dkisthediusioninthedirectionparalleltothemajorrodaxis,D?isthediusioninthedirectionperpendiculartothemajorrodaxis,andDRistherotationaldiusion.Theresistancesfortheslender-bodymodel[ 42 43 44 ]giveshort-timediusivitiesof 2L(3.18) 4L(3.19) wheredisthehydrodynamicdiameteroftherod.Forbothmodels,theaveragecenterofmassdiusivitycanbewrittenasDT=1 3Dk+2D?. Themobilitymatrices(invertedresistances)fortherodmodelscanbewrittenintheuniedform wheretheparametersandareconstantsshowninTable 3{1 .WritingthemobilitiesinthisformenablestheuseofasinglesetofLangevinequations(oneforthecenterofmassandonefortherotation)forthepurposesofcodingwhich

PAGE 68

Table3{1. ValuesoftheparametersandusedinEquation( 3.21 )inthischap-ter.ThevalueofchangesdependingontheisotropyofthemodeltokeeptheaveragecenterofmassdiusivityatinnitedilutionDT0con-stant.DT0,Dk0,D?0,andDR0arethediusivitiescalculatedforthemodelsforasinglerodatinnitedilution. ModelDk 12a014Rigid-dumbbell(anisotropic)1 16a124Slender-body(isotropic)ln(2A) 3L019Slender-body(anisotropic)ln(2A) 4L129 ModelDT0Dk0D?0DR0 3LkBTln(2A) 3LkBTln(2A) 3L3kBTln(2A) 3LkBTln(2A) 2LkBTln(2A) 4L3kBTln(2A) 3.2.2 EvaluationofExcludedVolumeForcesandTorques Ashort-rangerepulsivepotentialmaintainstheexcludedvolumeofthebers.Thisforcecontributestoboththerotationanddisplacementofthecenterofmassandactsontheneighboringbersatthepointsofclosestapproach[ 67 ].The

PAGE 69

forcesbetweenaninteractingpairofbersareequalinmagnitudeandoppositeindirection.Multiplerepulsiveforcesmayactonasingleber,dependingontheseparationdistancebetweenthesurroundingbers,whicharethenaddedtotheLangevinequationsinthesimulation(seeEquations( 3.1 ),( 3.2 ),and( 3.11 )).Theformoftherepulsivepotentialis where"andaretheparametersthatdeterminethemagnitudeandrangeofthepotential,sin()istheanglebetweenthepairofinteractingbers(jijkp()jp()kj),andhistheclosestdistancebetweenthecenterlinesofapairofbers[ 67 ].ForthesimulationswhereA=50,theparametersusedwere"=(50=3)kBTand=0:03855L,whichwerethesameasusedbyBitsanisetal.[ 38 39 ].ForthesimulationsofA=500,theparameterthatsetstherange()wasdividedby10.ForthesimulationswithA=25,thesameparametersasforthesimulationsofA=50wereused,buthwascalculatedfromadistanceof0:01Lfromthebercenterlines,thusdoublingthediameterandmakinganeectiveaspectratioof25. Dividingbytheanglebetweentherodsposesaproblemforparallelbers.Astheanglebetweentheinteractingrodsapproacheszero,therepulsiveforceincreasestoinnity.Usingsmalltime-stepspartiallycompensatesforthisproblem.Atthehighestconcentration,themaximumdisplacementproducedbytheparallelrodsisnogreaterthan2%oftherodlength,or1=5oftheinteractiondistanceoftherepulsiveforce.Therepulsiveforceactsalongthenormalvectorbetweenthepointofclosestapproachofapairofrods[ 67 ]. 3.2.3 NumericalIntegrationoftheGoverningEquations Tosimulatethemotionoftherods,Equations( 3.1 )and( 3.11 )mustbeintegratedintime.However,theinterestofthisstudyisinresolvingthediusivemotion,sothereisnoneedtocalculatethevelocities.ErmakandMcCammon

PAGE 70

[ 56 ]showedthatthetimedependentpositionscanbecalculatedusingamodiedEulermethodwithoutcomputingthevelocitiesexplicitly[ 56 55 66 ].ApplyingthismethoddirectlytoEquations( 3.1 )and( 3.11 )producesexpressionsfortheroddisplacements wheretisthediscretetimesteptakeninthesimulation,andx()iandp()iarethestochasticdisplacements.Thestochasticdisplacementsaregivenby x()i=(;T)ij1F(;br)j=B()ijw(;T)j(3.25) p()i=(;R)ij1~F(;br)j=C()ijw(;R)j;(3.26) wherew(;T)iandw(;R)iarerandomvectorsoflengththreewithpropertiesof Therandomnumbersaregeneratedusing\ran2,"asubroutinepresentedbyPressetal.[ 51 ].AuniformdistributionfortherandomnumbersissucientbecausethenumericalalgorithmisaccurateonlytoO(t)[ 52 ].ThematricesB()ijandC()ijarerelatedtothemobilities,where TheappearanceofthedivergenceoftherotationaldiusivityinEquation( 3.24 ), @p()jhkBT1Rijp()ip()ji=2kBT1Rp()i;(3.29) correctsforthedriftintherotationalvelocitiesandisadirectconsequenceofusingorientationsandweightedforcesinEquation( 3.11 )(seeChapter 2 forderivation).

PAGE 71

Thisdriftcorrectionmaintainsp()i(t)asaunitvectoronaverageatordert.Afteratimestepoft,theaverageexpectedvalueofDp()i(t+t)p()i(t+t)Eequals Withinthealgorithm,p()i(t+t)isrenormalizedaftereachtimesteptoaccountforthesmallO(t)2error.Failuretoincludethedivergenceoftherotationaldif-fusivityinEquation( 3.24 )resultsinanunnecessarilylargeerrorintherotationaldiusivity.NosimilarcorrectionexistswhencalculatingthedivergenceofthecenterofmassdiusivityD(;T)ij, @x()jhkBTij+p()ip()ji=0:(3.31) Simulationswerealsoperformedusingthemidpointmethod[ 62 60 ],whichproducedresultsthatwereequivalenttothoseproducedintheintegrationschemeshownabove. Atlongtimesinasuspensionofrods,thediusivitiesarealteredbythepresenceoftheforcesusedtomaintaintheexcludedvolume.Theselong-timediusivitiesarecalculatedfromthesimulationresults.Theaveragecenterofmassdiusivityiscalculatedbytheaveragesquareddisplacementsofthebers, forlargevaluesof.Therotationaldiusivityiscalculatedinasimilarmanner,where fortimescalesofDR0greaterthanorequalto1.ThealgorithminEquations( 3.23 )-( 3.29 )resultsintheexpecteddiusivitiesaslistedinEquations( 3.15 )-( 3.20 )forthebersinthelimitofinnitedilution(DR0andDT0)[ 6 ].

PAGE 72

Periodicboundaryconditionswereusedtoapproximateanunboundedsuspension.Acubicboxwithsidesoflength2:1LwasusedforallsimulationsexceptthosewithanumberdensityofnL3=5,forwhichaboxoflengthof2:63timestheberlengthwasusedinordertohavemorebersoverwhichtoaverage.Thesizeofthesimulationboxandthedesirednumberdensitydictatedthenumberofparticlesineachsimulation.ForthenumberdensitiesofnL3=5to150,thesimulationscontainedbetweenN=91and1389bers.ThedimensionlesstimestepusedinthesimulationsofA=25andA=50was5107forconcentrationsuptonL3=50;smallertimestepsoft=5108wererequiredforhigherconcentrations.ForthesimulationsofA=500,whichwereonlyperformedfortheconcentrationsofnL3=70tonL3=150,usedadimensionlesstimestepof5109.Thetimestepwasmadedimensionlessbydividingby4L3=kBTln(2A)fortheslender-bodymodel,andby6aL2=kBTfortherigid-dumbbellmodel. Oneparticulardicultyinsimulatingarandomdispersionofbersathighconcentrationsiscreatingtheinitialdistribution.ForberswithA=50,themaximumrandompackingoccursatavolumefractionofapproximately10%[ 1 ].ThisvolumefractioncorrespondstoanumberdensityofaboutnL3=345.Forberswithanaspectratioof25,themaximumnumberdensityforrandompackingisaboutnL3=172.ForconcentrationslowerthannL3=100,theprocessofplacingbersatrandominthebox,testingforoverlappingbers,andthenstartingthesimulationwasstraightforward.Forhigherconcentrations,aninitiallyorderedcongurationwasspeciedanddatacollectionbeganonlyafterthesuspensionbecamerandom.Therandomnessoftheorientationswasconrmedbycheckingtheorientationdistributionfunctionfororderingacrosstheentiresystemaswellasforlocalordering.

PAGE 73

3.3 Results 3.3.1 RotationalDiusivities Figure 3{2 showsthecorrelationfunctionfortheorientationasafunctionoftimeforsimulationsatanumberdensityofnL3=150forrodswithanaspectratioof50,L2DR0=DT0=9,andDk0=D?0=2.AlltheinputpropertiesofDR0,DT0,andDk0=D?0wereconrmedcorrectforthissimulation,andallothersimulations,byexaminingthediusivitiesatshorttimes.Forexample,therotationaldiusivityequalsthediusivityofaberatinnitedilutionatveryshorttimesasseenintheinsetofgure 3{2 .Thebersarenothinderedbythesurroundingbersattheshortesttimescales,andthereforerotateandtranslateasifatinnitedilution.Atlongertimes,theexcludedvolumeinteractionsbetweenthebersreducethefreedomofmotion,andhencethediusivity,whichisgivenbytheslopeofthelinesrelating1 2lnDp()i(t)p()i(t+)EandasindicatedinEquation( 3.33 ).FornL3=150,therotationaldiusivityisreducedby97%incomparisontothedilutevalueofthediusivity,andthecoecientofdetermination(R2)betweenthelineartandthesimulationdatafortimeslargerthan=0:01isgreaterthan0:99. Astheconcentrationoftherodsincreases,theberssurroundingatestberstronglyhindertherotationandsignicantlyreducetherotationaldiusivity.PlotsoftherotationaldiusivityasafunctionofnumberdensityappearinFigure 3{3 forboththeslender-bodymodel(L2DR0=DT0=9andDk0=D?0=2)andtherigid-dumbbellmodel(L2DR0=DT0=4andDk0=D?0=1)forrodshavinganaspectratioof50.Errorsinthediusivityvalueswereestimatedbycalculatingthestandarddeviationoftheaveragediusivityfromanensembleofthreedierentsimulationsoveratotaldimensionlesstimeperiodof1.Themagnitudeoftheerrorisgreatestatthelowerconcentrations,wheretherearefewerrodsoverwhichtoaverage.Atthelowestconcentration(nL3=5),theerrorisabout5-6%ofthemeanvalue.Astheconcentrationofthesystemincreases,theerrordecreasestoabout3-4%at

PAGE 74

TherotationalcorrelationfunctionatnL3=150,L2DR0=DT0=9,andDk0=D?0=2forrodswithA=50.Theinsetshowsthatthediusivityatshorttimesisequivalenttotherotationaldiusivityatinnitedilution.Atlongertimes,excluded-volumeinteractionsbetweenthebersreducesthediusivitytoDR.Forthismodel,thetimehasunitsof4L3=kBTln(2A). theconcentrationofnL3=150forallaspectratios.Theerrorbarsinthegurearesmallerthanthesymbolsusedforthedatapoints.Themaximumestimatederrorforallthesimulationsisnogreaterthan6%. Convergenceofthevalueswithrespecttothetimestepwasconrmedforallsimulations.ThelowervaluesofL2DR0=DT0requireasmallertime-stepinordertoachieveconvergentresults.Astheaspectratioincreases,asmallertime-stepisalsoneededtoachieveconvergentresults.Atimestepoft=5107wassucientforconcentrationsuptoanL3=50anduptoA=50.ForthesimulationsofnL370andforA=25andA=50,atime-stepoft=5108wasrequiredtoachieveconvergentresultsatthehighestconcentrations.ToconvergethesimulationsofA=500,atime-stepoft=5109wasrequired.ThistimestepwasusedforallsimulationsforconcentrationsnL370forA=500.

PAGE 75

Therotationaldiusivitiesversusthenumberdensityareshownfortheslender-bodymodel(L2DR0=DT0=9andDk0=D?0=2)andtherigid-dumbbellmodel(L2DR0=DT0=4andDk0=D?0=1),bothforrodswithA=50.ThenumericalresultsofDoiandEdwards[ 6 ]andBitsanisetal.[ 38 39 ]areplottedforcomparison,aswellasthe(nL3)1and(nL3)2scalings. Resultsareshownfortherotationaldiusivitiesversusconcentrationfortheslender-bodymodel(L2DR0=DT0=9)withA=25,A=50,andA=500.Theresultsfortherigid-dumbbellmodel(L2DR0=DT0=4)forA=50andA=500areshown,aswellastheresultsforanalternativemodelfortherodswhereL2DR0=DT0=1andA=25.

PAGE 76

SimulationsofmodelswithvaryingvaluesofL2DR0=DT0wereperformedtodeterminethedependenceofthescalingexponentfortherotationaldiusivityscaling,DR=DR0=(nL3),withintheconcentrationregimeofnL3=70to150.Simulationswerealsoperformedusingaspectratiosfortherodsof25,50,and500tostudytheeectthattherodthicknessplaysinthescalingoftherotationaldiusivities.Figure 3{3 comparesthesimulationsresultsobtainedherewiththoseofBitsanisetal.[ 38 39 ]andDoietal.[ 3 ].Therotationaldiusivitiesfortherigid-dumbbellmodelmorecloselyfollowthescalingof=1,whereastheslender-bodymodelhasascalingcloserto=2athighconcentrations.AtofthedatawhichincludespointswithconcentrationsinexcessofnL3=50indicatesascalingof=1:89fortheslender-bodies(anisotropic)withA=50and=1:13fortherigid-dumbbells(isotropic)ofthesameaspectratio.ForthesimulationsofrodswithA=25,theratioL2DR0=DT0=1wasrequiredtoreproducetheapproximate=1scaling,asseeninFigure 3{4 .Simulationswithanaspectratioof500werealsoperformedfortheslender-bodymodel(L2DR0=DT0=9andDk0=D?0=2)andtherigid-dumbbellmodel(L2DR0=DT0=4andDk0=D?0=1)forthehigherconcentrationregime.Thescalingsthatwerecalculatedfromthesesimulationswere=1:01fortherigid-dumbbellmodel,and=1:91fortheslender-bodymodel.WithintherangeofnL3=70to150,thepower-lawtshaveacoecientofdetermination(R2)greaterthan0:92forallcases.TheratioofDk0=D?0hasasmallimpactontherotationaldiusivitiesasseeninFigure 3{5 ,regardlessofwhetherL2DR0=DT0equals4or9.ChangingthevalueofL2DR0=DT0between4and9howeverisseentoalternotonlythevaluesoftherotationaldiusivities,butalsothescalingsatthehigherconcentrations. Adecreaseofabout1=2foroccursbetweenL2DR0=DT0=1and0fortherodswithanaspectratioof25,andadecreaseofabout2=3fortherodswithA=50.Thisdecreaseindicatesarapidchangeovertherange.Setting

PAGE 77

Rotationaldiusivitiesversusnumberdensityfortheslender-bodyandrigid-dumbbellmodelswithA=50.SettingtheratioofDk0=D?0equalto1(isotropic)or2(anisotropic)causessmallchangesinthebehavioroftherotationaldiusivity. 3{6 arelistedinTable 3{2 3.3.2 TranslationalDiusivities TheaveragediusivitiesofthecenterofmassweredeterminedfromthesimulationdatausingEquation( 3.32 ).Figure 3{7 showsthatadierenceexistsbetweenthediusivitiescalculatedfromtheslender-bodyandrigid-dumbbell

PAGE 78

ExponentforthescalingoftherotationaldiusivityasgivenbyDR=DR0=(nL3)fortheconcentrationrangeof70nL3150.Theexponenttransitionsbetweentheapproximatevaluesof1and2astheratioofrotationaldiusivityDR0totranslationaldiusivityDT0increasesfrom1to9forrodswithA=25,andfrom4to9forrodswithA=50andA=500.TherodthicknessplaysalargerroleforvaluesofL2DR0=DT08,butmakesaminimaldierenceforlargervalues.ThepointsforA=500showthecalculatedscalingfortherigid-dumbbellmodel(L2DR0=DT0=4andDk0=D?0=1)andtheslender-bodymodel(L2DR0=DT0=9andDk0=D?0=2). Table3{2. Valuesfortheexponentforthescalingoftherotationaldiusiv-ityasgivenbyDR=DR0=(nL3)plottedinFigure 3{6 .Thevaluesareformodelswitheitherisotropic(Dk0=D?0=1)oranisotropic(Dk0=D?0=2)centerofmassmobilities.Asstatedinthetext,thepower-lawttingsareintendedasageneralguidelineforthescalingofDR=DR0athighconcentrations.Apurelypower-lawscalingfortherotationaldiusivityofrigid-rodsystemsathighconcentrationshasnotbeenrigorouslyjustied[ 68 ].

PAGE 79

Translationaldiusivitiesasafunctionofnumberdensityfortheslender-bodymodel(L2DR0=DT0=9andDk0=D?0=2)andtherigid-dumbbellmodel(L2DR0=DT0=4andDk0=D?0=1)withA=25.Theaveragecenter-of-massdiusivityDT,diusivityparallelDk,anddiusivityperpendicularD?totherodsarenormalizedbytheparalleldiusivityatinnitedilutionDk0. modelsuptoanumberdensityofnL330forrodswithaspectratiosof25.Forthesimulationsofrodswithaspectratiosof50,thequalitativedierenceextendstoanumberdensityofnL340.Beyondthisconcentration,thediusivityDT=Dk0continuestodecreaseforboththeslender-bodymodelandtherigid-dumbbellmodeltoavalueofapproximately0:4atnL3=150.Aswiththedatafortherotationaldiusivities,errorswereestimatedbycalculatingthestandarddeviationofthemeandiusivityforanensembleofthreedierentsimulationsoveratotaldimensionlesstimeperiodof1;theerrorsweredeterminedtobenomorethan5%ofthemeanvalueovertheentireconcentrationregime. Examiningthecomponentsofthediusivitiesdemonstratesthatthehindranceofthemotionperpendiculartotherodsisresponsibleforthereductionintheaveragediusivitywithconcentration;theparalleldiusivitiesremainvirtuallyconstantovertheentirerangeofconcentrationsasseeninFigure 3{7 ,evenfor

PAGE 80

therelativelythickrodswithA=25.Tocalculatethediusivitiesintheparallelandperpendiculardirections,theberdisplacementsareprojectedontotheberorientationattimet.Figure 3{8 showsthattheparalleldiusivityatinnitedilution,Dk0,isrecoveredatshorttimes,asistheperpendicularcomponent,D?0.Forlarge,thetimerateofchangeofthesquaredisplacementsinthedirectionsperpendicularandparalleltotheinitialorientationsoftherodsareequivalenttotheaveragediusivity,DA.Thisconvergenceofthevaluesoverlongtimesarisesfromthegraduallossinthecorrelationoftheinstantaneousandinitialorientationstogetherwithcouplingofthetranslationalmotionofaberwiththeinstantaneousorientationoftheber.Theaveragesquaredisplacementsattheintermediatetimeof=0:1denethediusivitiesDkandD?asindicatedinFigure 3{8 .Thevalueof0:1waschosensothatthediusivitiescorrespondtothosecalculatedbyBitsanisetal.[ 38 39 ]asseeninFigure 3{7 ;thetranslationaldiusivitiescalculatedfortherigid-dumbbellmodelfallwithintheerrorrangesofthesepreviousresults. 3.4 Discussion Inthefollowingsections,thedependenceofthepower-lawscalingfortherotationaldiusivityontheratioL2DR0=DT0andtheaspectratioAisexamined.AcomparisonismadeinSec. 3.4.2 betweenthesimulationresultsandsomeexistingtheoriesfortherotationaldiusion.AdiscussioninSec. 3.4.3 ofthreedierentlimitingconditionsforthemodeloftherodsprovidessomeinsightintotheroleplayedbytheratiooftheshort-timediusivities.Thecalculationsoftherotationaldiusivitiesarealsocomparedtoexperiments(Sec. 3.4.4 )andthetranslationaldiusivitiesarediscussedinSec. 3.4.5 3.4.1 DependenceofRotationalDiusivitiesonFiberModel Thedependenceofrotationaldiusivityonnumberdensitydependsstronglyuponthehydrodynamicmodeldescribingtheindividualbers,regardlessoftheaspectratio.Forexample,withintheconcentrationrangenL3between70and

PAGE 81

Averagesquaredisplacementsofthecenterofmassversuselapsedtime.Theaveragesquaredisplacementatatimeof=0:1,pro-jectedalongtheparallelandperpendiculardirectionat=0:0,denesthediusivitiesDkandD?.ThedataherecorrespondstosimulationswithnL3=30,L2DR0=DT0=9,andDk0=D?0=2forrodswithA=50.TheunitsoftimearethesameasthoseinFigure 3{2 150,ttingthedependenceofrotationaldiusivityonnumberdensityusingapowerlawscalingofDR=DR0(nL3)givesverydierentresultsforthescalingexponentdependinguponthehydrodynamicmodeldescribingtheindividualbersasseeninFigure 3{6 andTable 3{2 .Inkeepingwiththeliteratureonthesubjectofrotationaldiusivityofrigidrods,thepower-lawtforthedatawithintheconcentrationregimehasbeenusedasaconvenientmethodfordescribingthescalingbehaviorwithinthesemi-diluteconcentrationregime.However,theassumptionthatDR=DR0conformstoapower-lawwithrespecttonL3isnotrigorouslyjustied[ 68 ].Infact,theagreementbetweenthesimulationdataandthepower-lawtsarenotparticularlyconvincingasnotedbythelowvaluesofcorrelationbetweenthedataandthepower-lawscalingsasreportedinSec. 3.3.1 ThekeydierenceinthemodelswhichleadstothevariationintheobservedvalueforthescalingsisL2DR0=DT0,thedimensionlessratiooftherotational

PAGE 82

andtranslationaldiusivitiesforanindividualrodatinnitedilution.Therodthickness,asseeninFigure 3{6 ,alsohasasignicantimpactonforsmallervaluesofL2DR0=DT0.Forvalueslessthanabout8,thethicknessoftherodplaysaroleindeterminingtheexponentialscaling.Inordertoproducethe=1scaling,avalueofL2DR0=DT0=1isneededforrodswithA=25,whileavalueofL2DR0=DT0=4isneededtoproducethe=1scalingforrodswithA=50.ForlargevaluesofL2DR0=DT0,Figure 3{6 showsthatthereisessentiallynodierenceinthescalingforaspectratiosofatleast25orlarger.Simulationswerealsoperformedusinganaspectratioof500fortherigid-dumbbellmodel(L2DR0=DT0=4andDk0=D?0=1)andtheslender-bodymodel(L2DR0=DT0=9andDk0=D?0=2)giveresultssimilartothesimulationswithaspectratiosof50.Fortherigid-dumbbellmodelascalingof=1:01wascalculated,whilea=1:91scalingwascalculatedfortheslender-bodymodel.Thisshowsthatforsimulationsofrodswithnitethicknesses,thereisaqualitativetransitioninthescalingbehaviorduetothehydrodynamicmodelfortherods. Anotherpossiblesourceforthedierenceinthescalingsistheisotropyofthemodels.However,Figure 3{5 demonstratesthattheratioofDk0=D?0hasonlyasmalleectontherotationaldiusivitiesiftheoverallcenterofmassdiusivityDT0isheldconstant.Thesimulationofgeneralmodelsotherthantheslender-bodyandrigid-dumbbellalsodemonstratethattheexponentisafunctionofL2DR0=DT0,asplottedinFigure 3{6 .Theexponentisnotasingle-valuedfunctionofL2DR0=Dk0.SettingD?0equaltoDk0inEquation( 3.32 )resultsintherelationL2DR0=Dk0=L2DR0=DT0fortheisotropicmodel.Fortheanisotropicmodel,L2DR0=Dk0equals2L2DR0=3DT0becauseD?0=1 2Dk0.PlottingtheexponentsversustheratioL2DR0=Dk0consequentlyresultsintwodistinctcurvesfortheaspectratiosstudied.

PAGE 83

Sincetheorieshavebeenlargelybasedupontheresultsofsimulations,thisqualitativedierenceinthescalingbehaviorhasledtosignicantconfusionoverthesubjectofrotationaldiusioninconcentratedsystemsofBrownianbers.SimulationsbyDoietal.[ 3 ]usingtheslender-bodymodelwithinnitelythinrods,resultsofwhichareshowninFigure 3{3 ,conrmedthetubetheory[ 7 ]whichresultsinascalingtheorywith=2.Fixman[ 34 35 ]performedsimulationsandobtainedascalingof=1inthelimitofthinrods,directlycontrastingwiththetubetheoryproposedbyDoi[ 7 ].Consequently,Fixman[ 34 35 ]proposedanalternativetheoryfortherotationaldiusivitiesofBrownianrodsatsemi-diluteconcentrations.RoughlythesamesimulationresultsofFixman[ 34 35 ]werelaterconrmedbythesimulationsofBitsanisetal.[ 38 39 ].ThedierentresultsofDoietal.[ 3 ]andtheseotherauthorsstemfromthedierentmodelsusedtodescribetherods.Fixman[ 34 35 ]usedathree-beadmodelinhissimulationsofrodsofvanishingdiameter,statingthattheratioofL2DR0=DT0equals6(seefootnote7intheworkbyFixman[ 35 ]).Bitsanisetal.[ 38 39 ]usedarigid-dumbbellmodelwithanaspectratioof50toconrmthe1scaling;adirectcomparisonismadeinFigure 3{3 betweenthoseresultsandthesimulationsperformedaspartofthiswork. Fixman[ 34 35 ]andBitsanisetal.[ 38 39 ]dismissedthesimulationresultsofDoietal.[ 3 ]asbeinginerror,statingthatDoietal.[ 3 ]probablyusedatimesteptoolargetoaccuratelyresolvethedynamicsofthesystem.However,thesimulationspresentedhereindicatethatthesimulationsofDoietal.[ 3 ]wereessentiallycorrect.ThedataofDoietal.[ 3 ]andoursimulationsatA=50areshowninFigure 3{3 andplotsofrotationaldiusivitiesasafunctionofconcentrationareshowninFigure 3{4 forallthreeaspectratiosof25,50,and500simulated.Doietal.[ 3 ]simulatedaninniteaspectratio,whichwouldseemtopreventmakingadirectcomparisonoftheresults.However,ouranalysisindicates

PAGE 84

Linearextrapolationoftherotationaldiusivitiescalculatedusingtheslender-bodymodeltothelimitofaninnitelythinrodusingthevaluescalculatedfromtheaspectratiosofA=50and500. thattheslender-bodymodelwithA=500producesresultsnearlyequivalenttotheslender-bodymodelwithaninniteaspectratio,asarguedhere. TheresultsforA=50and500varyonlyby9.6%onaverage.Recallingthatthevaluesforeachpointareknownonlywithinapproximately6%indicatesthattherotationaldiusivitiesfor50and500arenearlythesamewithinstatisticalaccuracy.Furthermore,alinearextrapolationismadeusingthevaluesatA=50and500toestimateameanvalueoftherotationaldiusivityforthelimitA=1.TheextrapolationisshowninFigure 3{9 andtheextrapolatedvaluesareplottedasafunctionofnL3inFigure 3{10 .ThedierencesbetweentheextrapolatedvaluesandthevaluesatA=500dierbyanaverageamountofonly1:1%,anerrorwhichiswellbelowthecapabilityofresolvingusingtheBrowniandynamicssimulations.Inotherwords,theresultsforA=500(fortheslender-body)areessentiallyequivalenttoA=1fortheconcentrationregimeofinterest.ThedirectcomparisonoftheextrapolatedvalueswiththeresultsofDoietal.[ 3 ]showninFigure 3{10 showaclosecorrespondence,especiallywhenconsideringtheerrorintheA=1resultsofatleast6%(duetotheadditionalerrorsarisingfrom

PAGE 85

DirectcomparisonofthesimulationresultsofDoietal.[ 3 ]withtheslender-bodymodelusingthelimitingvaluesofinniteaspectratioasappearinginFigure 3{9 .ThetheoryofFixman[ 35 ]giveninEquation( 3.39 )forA=1andL2DR0=DT0=9isalsoshowninthegraph. theextrapolation)andtheerrorsreportedbyDoietal.[ 3 ]ofapproximately5%.Attheleast,thesimulationsofDoietal.[ 3 ]accuratelycapturedascalingofapproximately(nL3)2fortheslender-bodymodelwithintheconcentrationregimeofinterest,inpartjustifyingthepredictionofthetubetheorywhichisdiscussedinthenextsection. 3.4.2 RelationwithExistingTheories Existingtheoriesfortherotationaldiusivityofbersofhighaspectratiodonotaccuratelyaccountforthedependenceofthescalingexponentupontheratioofshort-timediusivities.Forexample,Doi[ 7 ]andDoiandEdwards[ 69 ]arguedthatinnitelythinbersathighconcentrationbecomecagedinstatictubesformedbythesurroundingbers.Rotationcanthenoccuronlywhentherodescapesthecagebydiusingalongthecentralaxisoftheber,whereupontheberwillrotateupon

PAGE 86

enteringanewtube.Forthisscaling,therotationaldiusivityisgivenby L21 whereaisthediameterofthetubeconstrainingtherotationanddistheaveragetimenecessaryforthepolymertoleaveatube.Thetubediameterisestimatedtoscaleas andthetimeforarodtoleaveatubeisproportionaltothetimeneededforarodtodiuseoneparticlelengthalongtheprimaryaxis, CombiningEquations( 3.34 )-( 3.36 )anddividingbytherotationaldiusivityatinnitedilutionresultsinthepredictionofthetubetheoryforDR,[ 7 69 ] Asdiscussedintheprevioussection,thesimulationsofDoietal.[ 3 ]accuratelyprovidedevidenceforthisscalinglaw,buttheresultof=2islimitedtoonlycertainconditionsasshowninFigure 3{6 .Thisshortcomingarisesfromtheassumptionthattherotationalmotioniscontrolledexclusivelybytheexcludedvolumeathighconcentrationsandaspectratios;theratioofdiusivitiesappearsonlyasamultiplicativefactoroforderone[ 7 6 ]. Asidefrompredictingaconstantpower-lawscalingof=2,thescalingofEquation( 3.37 )impliesthattheratioofDR=DR0shouldbeproportionaltoDk0=L2DR0ataxednumberdensity.Figure 3{11 showsthattherotationaldiusivityasafunctionoftheratiooftheshort-timediusivitiesdemonstratesthattherelationshipisnon-linear.TheanisotropicdiusivityvaluesareosetfromtheisotropicvaluesbycomparingtoDk0=L2DR0insteadofDT0=L2DR0.Figure

PAGE 87

RotationaldiusivitiesoverarangeofL2DR0=Dk0witheitheriso-tropicoranisotropiccenterofmassdiusivitiesforrodswithA=25.Forrodswithanisotropicmobility,DT0=Dk0,whileforrodsananisotropicmobility,DT0=3 4Dk0.Thesolidsymbolsarefortherodswithanisotropicmobilities,whiletheopensymbolsarefortherodswithisotropicmobilities. 3{11 showstheresultsforrodswithA=25,andthedeviationfromalinearbehaviorisseentobecomelesspronouncedastheconcentrationincreases,butisstillclearlynon-linearevenfornL3=150.Figure 3{12 showsthesameresultsforthesimulationsofrodswithA=50,andthebehaviordeviatesevenmorefromalinearrelationshipthanwiththethickerrods. Mosttheoreticalworkhasconrmedthe=2scalingfortherotationaldiusivityinthelimitofhighconcentrations.AnumberofthestudieshavesoughttorenetheconceptofcagesasoriginallyputforthbyDoi[ 7 ].Forexample,KeepandPecora[ 68 ]developedatheorywhichreducestothe2scalingonlyforconcentrationsgreaterthenL3=500.Forlowerconcentrations,thistheorypredictsascalingwhichisamixtureof=1and2.Teraokaetal.[ 36 ]andTeraokaandHayakawa[ 37 ]alsoconrmedtherotationalscalingof=2inthe

PAGE 88

RotationaldiusivitiesoverarangeofL2DR0=Dk0witheitheriso-tropicoranisotropiccenterofmassdiusivitiesforrodswithA=50.Thelledinsymbolsarefortherodswithanisotropicmobilities,whiletheclearsymbolsarefortherodswithisotropicmobilities. limitofhighconcentrationforverythinbers,asseenintheirprediction Thisrelationshipindicatesarotationaldiusivitywhichdependsupontheshort-timediusivitiesinanon-trivialmannerandprovidesagoodtforthedatageneratedfortheslender-bodymodelovertheentireconcentrationrangestudied.Theagreementofthispredictionwiththesimulationsresultsusingotherhydro-dynamicmodelsfortherods,suchastherigid-dumbbell,ishoweverpoor.SatoandTeramato[ 70 ]latermodiedEquation( 3.38 )toaccountforthepresumedreductioninlongitudinaldiusivitycausedbythethicknessoftherods.However,Dkremainsessentiallyconstantevenforrelativelythickrodsofaspectratio25asseeninFigure 3{7 ,atleastforconcentrationsthroughnL3=150. AnalternativetothecagetheoriescanalsobeseenintheworkofFixman[ 34 35 ],whereitisarguedthatthestatictubeofDoiandEdwards[ 6 ]wouldmove,

PAGE 89

rotate,break-upandreformonatime-scalecomparabletothereptationtime,orthetimeneededtoescapethecage,ofL2=Dk0.Thistheorypredictsarotationaldiusivityof wheretheparameters(f),(f),andfarenotarbitraryconstants,butaredeterminedthroughtheanalysis(seeEquation(4)intheworkofFixman[ 35 ])andareweakfunctionsofthenumberdensity.Forsuspensionsofbersofnitethicknessandatsucientlylargeconcentrations,thetheoryapproachesthescalingofDR=DR0(nL3)2.ThescalingofDR=DR0(nL3)1,oftenquotedintheliterature[ 38 39 4 2 45 46 47 8 ]whendiscussingthetheoryofFixman[ 35 ],correspondstotheduallimitofinniteaspectratioandhighconcentration.Thetheoryiscomparedtotheresultsofthesimulationsusingtheslender-bodymodelwithinFigure 3{10 ,whereAwassetto1andL2DR0=DT0to9withinEquation( 3.39 ).Thediusivitiespredictedbythetheoryarelargerthanthoseresultingfromthesimulationsundertheseconditions.WithintherangeofnL3=70to150overwhichthesimulationdatahasbeenconsistentlytusingthepower-law,thebestttothetheorycorrespondsto=0:69.Thelimitingvalueof=1isnotapproacheduntiltheconcentrationreachesatleasttheunrealisticallylargevalueofnL3=1000. 3.4.3 RotationalDiusivityunderLimitingConditions Thesimulationresultsdemonstratethatthecageisdynamicandcannotbetreatedasastaticgeometricconstraintontherotationoftracerrods,atleastfortherangeofaspectratiosandconcentrationregimestudiedhere.Forexample,removingtheBrownianforcesappliedtothecenterofmassoftherodssetstheshort-timediusivityDT0tozero.Thecentersoftherodsarestillmobile,however,andtheresultsshowthatasignicanttranslationaldiusivityathigh

PAGE 90

concentrationsiscausedbytheexcludedvolumeoftherods.Thismotionofthecentersoftherodsisinducedbythecollisionsbetweentherotatingrodsandisconsistentwithdiusioninthatthesquaredisplacementsgrowlinearlyintime.Ascomparedtoresultsfromtheslender-bodymodel,DR=DR0isreducedby75%forrodswithaspectratiosof50,andby70%forrodswithaspectratiosof25.Theconstantscalingof=2formodelswithvaluesofL2DR0=DT0greaterthan9indicatesthattherelaxationofrotationalconstraintsthroughthereptationmechanismisnotveryimportant.Similarstudiesinwhichthecenterofmassmotionoftherodswasfrozenwereperformedforhighaspectratiorods[ 35 ]andforlowaspectratiospheroids[ 2 ],buttheresultswerenotclearlystated. Therotationcausedbycollisionsbetweenrodsisamechanismwhichhasbeenignoredinprevioustreatments.AstheratioofL2DR0=DT0goestozero(byremovingtheBrowniantorquescontributingtotheberrotationwhilestillallowingtherodstorotatefreely),Figure 3{6 showsthattherotationaldiusivityscalesas=0:2fortheconcentrationrangebetweennL3of70and150forrodswithA=50,and=0:5forrodswithA=25.Thevaluesfortherotationaldiusivityaresmallerinthiscase,butarenotvanishinglysmall.Atnl3=70,thediusivityis12%ofthevalueofDRfortheslender-bodymodelwithA=50.Astheconcentrationincreasestonl3=150,thediusivityforL2DR0=DT0=0is43%ofthevalueofDRcalculatedfortheslender-bodymodel.FortherodswithA=25,thedierencegoesfrom20%to48%overthesameconcentrationregime.Therefore,rotationsinducedbycollisionsmakeasignicantcontributiontotherotationaldiusivity. ThedependenceoftherotationaldiusivityonDT0ratherthanDk0isindirectcontrasttomostexistingtheories,particularlythetubetheory[ 7 6 ].There-fore,athirdlimitingcasewasexploredinwhichtheshort-timediusivityintheperpendiculardirection,D?0,wassetequaltozerobyremovingtheperpendicular

PAGE 91

ComparisonofrotationaldiusivitiesforrigidberswithA=25withperpendiculardiusivityremovedandbersmaintainingtheperpendicularcontributiontothediusivity.TheresultsshownhereareforaconcentrationofnL3=90. componentsoftheBrownianforcesactingonthecenterofmassoftherods.Figure 3{13 showsthattheperpendiculardiusivitydoeshaveasignicantimpactontherotationaldiusivities,evenathighconcentrations.RodswithA=25wereusedinthiscomparison,asthethicknessmoreclearlyshowstheeectoftheperpendiculardiusivitybytherodbeingmoreconnedandthuslesslikelytodiuseinthatdirection.AsseenfromthedataatnL3=90ofrodswithA=25,notincludingD?0hasasmallerimpactforlowvaluesofL2DR0=DT0thanforhighervalues.Adierenceof11to20%inDR=DR0existforL2DR0DT0=1,whereasthedierenceisapproximately31to36%forL2DR0=DT0=9.Theperpendiculardiusivityindeeddoesplayamorepivotalroleinthedynamicsthanassumedinthetubethe-ory,atleastfortheconcentrationrangestudiedhere,andsupportstheclaimthatcooperativemotionbetweenatracerrodandconningrodsmustbeconsideredinthetheoriesofrotationaldynamics.

PAGE 92

Rotationaldiusivitiesfromsimulationsusingtheslender-bodyandtherigid-dumbbellmodelswithA=50comparedtothediusivitiesofPBLGmeasuredbyMorietal.[ 71 ]. 3.4.4 ReinterpretationofComparisonwithExperiments Figure 3{14 comparestheresultsfortherotationaldiusivitycalculatedfromtheslender-bodyandrigid-dumbbellmodelsforrodswithaspectratiosof50totheexperimentalmeasurementsofpoly(-benzyl-L-glutamate)(PBLG)inm-cresol[ 71 ]fornumberdensitiesuptonL3=160.ThissamecomparisonwasmadebyBitsanisetal.[ 38 39 ]togetherwiththeconclusionthattheBrowniandynamicssimulationswerefaithfullyreproducingtheexperimentallymeasuredvalues.However,thegoodagreementbetweenthesimulationsofrigid-dumbbellsandtheexperimentsarisesfromafortuitouschoiceoftherigid-dumbbellmodeltorepresenttheshort-timediusivitiesoftheindividualrodscomposingthesuspension;anyothermodelproducesqualitativelydierentresults.Yet,thereisnobasisforarguingthattherigid-dumbbellissomehowspecial,becausemoreaccuratemodelsareavailablesuchastheslender-bodyforhighaspectrodsandofellipsoidalspheroids[ 72 73 13 ]ofhighaspectratio.

PAGE 93

ThefactthatDR=DR0dependsupontheratioofshort-timediusivitiesinanon-trivialmanner,aswellasthedependenceontherodthicknessforsomemod-els,bringsadditionalcomplicationstotheproblem.Beyondchoosingthecorrectmodelfortherods,theeectsofthelongrangehydrodynamicinteractions,whichaltertheshort-timediusivitiesdependinguponconcentration,mustbeincluded.ForsuspensionsofBrownianspheres,previousworkindicatesthattheratiooflong-timeandshort-timediusivitiesremainnearlyconstantatanyconcentrationregardlessofthelevelofapproximationatwhichhydrodynamicinteractionsarecalculated[ 74 ].Sucharelationremainstobedemonstratedforrigidrods,butthisrelationwouldhavetodependinpartupontheratioofshort-timediusivities.Nocalculationofthelong-timediusivitiesforconcentratedsystemsofBrownianrodswithhydrodynamicinteractionsexist,thoughcalculationsofshort-timemobilitiesareavailableforellipsoidalparticles[ 75 ]andmethodsforincludinghydrodynamicinteractionsinsuspensionsofslender-bodiesexist[ 76 77 66 ]. Additionally,recentcriticismsintheexperimentalliteraturehaveraisedquestionsastowhetherthesimulationsofrigidbersshouldbecomparedtotheresultsofPBLGmeasurementsinthestudiesperformedbyMorietal.[ 71 ].PBLGisasti,helicalhomopolypeptidewhichmayormaynotberigid[ 27 28 ].CushandRusso[ 20 ]claimthatexibility,especiallyattheendsofthePBLGbers,causeadownturninthepredictionforthetranslationaldiusivities,whichisdiscussedinthefollowingsection.FlexibilityinthePBLGberscouldcauseanincreaseintherotationaldiusivity,astheberswouldmoreeasilyescapethetubesandrotate[ 23 ]. 3.4.5 TranslationalDiusivities Theratioofshort-timediusivities,L2DR0=DT0,doesnotaectthelong-timetranslationaldiusivitiesasisthecaseforthelong-timerotationaldiusivities.Comparisonsoftheslender-bodyandrigid-dumbbellmodelsshowninFigure 3{7

PAGE 94

demonstratethatthetranslationaldiusivitiesareindistinguishablefornumberdensitiesnL3greaterthan50.Clearly,forconcentrationsinexcessof50,theapproximate18to1factorfortheparalleltoperpendiculardiusivityarisespurelyfromtheeectsofexcludedvolumeandincreasingconcentration;neitherthedegreeofisotropyinthehydrodynamicmodeloftherodsnorthethicknessoftherods,atleastforaspectratioslargerthan25,hasanynoticeableeectupontheseresults.ThetranslationaldiusivitiescalculatedbyBitsanisetal.[ 38 39 ]usingtherigid-dumbbellmodelarealsoplottedinFigure 3{7 .Themodelsindicatethatthediusivityinthedirectionparalleltotherodorientationremainsessentiallyconstantovertheconcentrationrangestudied.ThisisincontrasttothendingsofBitsanisetal.[ 38 39 ]thatshowedadecreaseinDk=Dk0atthehighestconcentrations.However,Bitsanisetal.[ 38 39 ]reportederrorboundswhichoverlapwiththedataforboththerigid-dumbbellandslender-bodymodelsascalculatedhere. Theassumptionsconcerningtheisotropyofthemodelimpactsthesimulationresultsfortheaveragecenterofmassandperpendiculardiusivitiesatlowconcentrations.Forbersofhighaspectratio,themodelsshouldbeanisotropic[ 6 38 ],suggestingthattherigid-dumbbellmodelisqualitativelyincorrectatlowconcentrationsunlesshydrodynamicsinteractionsbetweenthebeadsistakenintoaccount.Ontheotherhand,theratioofDk0=D?0=2representsatheoreticallimitforaninnitelythinber[ 42 43 44 ].Usingmodelsofellipsoidalparticlesdemonstratesthatthedegreeofisotropyliesbetween1and2,andthattheaspectratioofanellipsoidmustbeveryhigh[ 78 ]beforereachingthelimitingvalueof2.Foranellipsoidalparticle[ 79 80 ]withA=50,theratioofDk0=D?0equals1:61. Aswiththerotationaldiusivities,multiplepredictionsofthetranslationaldiusivitieshavebeenmadeandarecomparedtotheresultsofthesimulations.Szamel[ 4 ]developedareptationmodelfortheperpendiculardiusivitiesof

PAGE 95

innitelythinbers,whichpredictsD?=Dk0(nL3)2fornumberdensitieslargerthannL3=100,whereisanumericalconstant.Figure 3{15 showsthatthisscalingisaccuratefortheconcentrationshigherthannL3=100.Fittingthesimulationdataoverthissamerangeofconcentrationgivesanearlyidenticalscalingexponentof1:99fortheslender-bodymodeland1:96fortherigid-dumbbellmodelforrodswithA=25.AscalingofD?=Dk0(nL3)1moreaccuratelycapturesthediusivitiesformid-rangenumberdensities,downtoanumberdensityofnL3=20.TeraokaandHayakawa[ 81 ]proposedascaling,similartotheoneappearinginEquation( 3.38 )forDR,fortheaveragecenterofmassdiusivity, where1=2isanumericalconstant.Experimentshavebeenperformedmeasuringtheaveragecenterofmassdiusivitiesasafunctionofincreasingconcentrations[ 24 27 28 20 ].ThetubetheoryofDoi[ 7 ],throughtheassumptionofD?goingtozero,impliesthatathighconcentrations,theaveragecenterofmassdiusivityoftheslender-bodiesshouldbereducedby50%fromthedilutevalue,whileDTforrigid-dumbbellsshouldbereducedbyonly33%ofthevalueforadilutesolution.Experimentswithtobaccomosaicvirus(TMV)demonstratethatDTisreducedby60%athighconcentrations[ 20 ].CushandRusso[ 20 ]claimthatTMVisabettermodelsystemforstudyingthetranslationaldiusivityofrigidrodsthansolutionofPBLG,whichexhibitsslightexibilitiesneartheendsofthemacromolecule.Figure 3{7 showsthatDT,ascalculatedfromtheBrowniandynamicssimulations,isindeedreducedby60%. 3.5 Conclusions Thefunctionalrelationshipbetweenrotationaldiusivityandnumberdensitydependsstronglyuponthehydrodynamicmodeldescribingtheindividualbers,regardlessoftheaspectratio.Forexample,withintheconcentrationrangenL3

PAGE 96

PerpendiculardiusivitiesforberswithA=25incomparisontothereptationmodelofSzamel[ 4 ].Formid-rangenumberdensities,thetheoryofD?=Dk0(nL3)1followsthedatawell,whileathigherconcentrations,D?=Dk0(nL3)2isamoreappropriatetheory. between70and150,ttingthedependenceofrotationaldiusivityonnumberdensityusingapowerlawscalingofDR=DR0(nL3)givesverydierentresultsforthescalingexponentdependinguponthehydrodynamicmodeldescribingtheindividualbers.TheBrowniandynamicscalculationsperformedheredemonstratethatmodelingtherodsusingeitherslender-bodiesorrigid-dumbbellsresultsin=1:89or=1:13respectivelyforberswithaspectratiosofA=50.ForA=500,whichhasbeenarguedtocloselycorrespondtothelimitA=1,=1:91fortheslender-bodymodeland=1:01fortherigid-dumbbellmodel.Thekeyparameterinthemodelscausingthedierenceistheratioofshort-timediusivities,L2DR0=DT0.Theexcludedvolumeoftherod,orrodthickness,alsosignicantlyaectstherotationaldiusivities,butthelargercontributionisseentocomefromtheratiooftheshort-timediusivities,atleastforaspectratioslargerthan25.Thisqualitativedierenceintheresultshasgeneratedsignicantconfusionoverthesubjectofrotationaldiusioninconcentratedsystemsof

PAGE 97

Brownianbers.Thendingspointtotheneedforimprovedsimulationswhichincludehydrodynamicinteractions,whichwillaccountforthealterationsinshort-timediusivitiesasafunctionofconcentration,aswellasimprovedtheorieswhichaccuratelyaccountforthedependenceofthelong-timerotationaldiusionupontheratioofshort-timediusivities.

PAGE 98

4.1 Introduction ItwasdemonstratedinChapter 3 thatthechoiceofmodelusedtorepresentindividualrodswithinaconcentratedsuspensionofrigidrodswithexcludedvolumeinteractionssubstantiallyimpactstherotationaldynamics.Thekeydierenceisthedistributionofthehydrodynamicresistancealongtherodswhichalterstheratiooftheshort-timediusivities,L2DR0=DT0,whereListhetotallengthoftherod,andDR0andDT0aretherotationalandaveragecenterofmassdiusivitiesofasinglerodatinnitedilution.Therotationaldiusivityhasbeentheorizedtohaveapower-lawscalinginthesemi-diluteregimeofDR=DR0(nL3)[ 3 34 35 ].CobbandButler[ 5 ]showedthatcanbeaslowas2inagreementwiththesimulationsandtheoriesofDoi[ 3 ],canequal1inagreementwithotherworkbyFixman[ 34 35 ]andBitsanisetal.[ 38 39 ],andrangesfromanyvaluebetweenapproximately0:2and2dependingontheratioofL2DR0=DT0forthemodelusedinthesimulation. Smalldeviationsfromperfectrigidityhavealsobeencitedashavingasig-nicantimpactonthedynamicsofrodsuspensions[ 6 23 20 ].Experimentsonbersofpoly(-benzyl-L-glutamate)(PBLG)suspendedinm-cresol[ 71 ]havebeencitedmanytimesasamodelsystemforsuspensionsofrigidrods[ 38 39 19 25 ].However,recentcriticismshaveraisedquestionsaboutwhethertheresultsoftheexperimentsperformedbyMorietal.[ 71 ]shouldbecomparedtotheresultsofsimulationsofrigidrodssincePBLGmaynotbecompletelyrigid[ 27 28 20 ]. 85

PAGE 99

CushandRusso[ 20 ]claimthatexibilityinthePBLGpolymer,especiallyneartheendsofthebers,causesadierenceintheresultsincomparisontorigidrods. Thesimulationsperformedhereusedtwosimplemodelstoinvestigatethedualroleofslightbendingandchoiceofmodelonthedynamicsofconcentratedsystemsofsemi-rigidrods.Thesmalldeviationsfromrigiditydelaytheonsetofsemi-dilutebehaviorinsuspensionsofrods,whilethechoiceofmodelresultsindierentpower-lawscalingsinthesemi-dilutetoconcentratedregime.Themethodsusedtoperformsimulationsarediscussedinsection 4.2 .Theresultsofthesimulationsarethenpresentedinsection 4.3 ,andthediscussionandconclusionsbasedontheseresultsareinsections 4.4 and 4.5 4.2 SimulationMethod Inthesimulationsperformedhere,twodierentmodelswereusedfortherodsasshowninFig. 4{1 .Therstmodelisadimercomposedoftwoslender-bodies[ 82 83 ],andthesecondisatrimercomposedofthreebeads[ 84 ].Bothmodelsarecomposedoftworigidsegmentsconnectedbyconstraints,similartothe\once-brokenrod"model[ 85 ].Abendingforceisappliedwhichhindersthebendingoftherods.Forahighbendingforce,theslender-bodydimerbehavesasarigidslender-body,havingavalueofL2DR0=DT0=9andDk0=D?0=2[ 42 43 44 ],whereDk0isthediusivityinthedirectionparalleltothecentralrodaxisatinnitedilution,andD?0isthediusivityinthedirectionperpendiculartothecentralrodaxisatinnitedilution.Thethree-beadtrimerbehavesasarigidtrimerwhenahighbendingforceisapplied,havingL2DR0=DT0=6andDk0=D?0=1. Thecenterofmasspositionx()iofeachrodiscalculatedbyaveragingthecenterofmasspositionsofeachsegmentoftherod.Theslender-bodydimermodeliscomposedoftwosegmentseachwithacenterofmass,x(;1)iandx(;2)i.Thethree-beadtrimermodeliscomposedofthreebeadswithcentersatx(;1)i,x(;2)i,andx(;3)i.Theunitvectorp()idenestheoverallorientationofrod,whichis

PAGE 100

Thetwomodelsofthesemi-rigidrodsusedinthesimulations,wherelisthelengthofeachindividualsegmentoftherod,disthediameteroftheslender-bodydimermodel,aisthediameterofeachbeadinthethree-beadtrimermodel.Alsoincludedaretheequationsforcalculat-ingtheoverallcenterofmasspositionx()iandorientationp()iofrodbasedonthepositionsandorientationsofeachcomponent. calculatedbyaveragingthetwoindividualsegmentorientations,p(;1)iandp(;2)i.Forthethree-beadtrimermodel,theorientationofeachofthetwosegmentsiscalculatedbytakingthedierencebetweenthecenterofmasspositionsofthetwobeadsonthesegment(beads1and2forsegment1andbeads2and3forsegment2)dividedbythemagnitudeofthedistance(l)betweenthem. 4.2.1 GoverningEquations ThedynamicsofeachrodisgivenbytheLangevinequationswithoutinertia, where()ijisthehydrodynamicresistancematrixofrodand_r()iisavectorcontainingthecenterofmassandrotationalvelocitiesofrod.Fortheslender-bodydimermodelthevector_r()jisdenedas _r()j=h_x(;1)i;_x(;2)i;_p(;1)i;_p(;2)ii;(4.2)

PAGE 101

wherejhasalengthof12,ihasalengthof3,_x(;1)iand_x(;2)iarethecenterofmassvelocitiesofsegments1and2ofrod,and_p(;1)iand_p(;2)iaretherotationalvelocitiesofeachofthetwosegments.Forthethree-beadtrimermodelthevector_r()jisdenedas _r()j=h_x(;1)i;_x(;2)i;_x(;3)ii;(4.3) wherejhasalengthof9,and_x(;1)i,_x(;2)i,and_x(;3)iarethecenterofmassvelocitiesofthethreebeads.ThetermsF(;Br)i,F(;Ev)i,F(;Bend)i,F(;Const)i,andF(;Corr)iaretheBrownian,excludedvolume,bending,constraint,andcorrectionforcesandtorquesactingoneachsegmentofrod.Eachforcevectoristhendenedfortheslender-bodymodelas andforthethree-beaddimermodelas wheretheindicesiandjhavethesamevaluesasthoseforthevelocityvectorsinEquations( 4.2 )and( 4.3 ),andtheindicatesthetypeofforceortorquebeingapplied(Br,Ev,Bend,Const,orCorrasinEquation( 4.1 )).Theforcesonthecentersofmassoftheslender-bodiesandthebeadsofthethree-beaddimermodelsaregivenbyFi,andthetorques(weightedforces[ 5 ])aregivenby~Fi. 4.2.2 EvaluationofBrownianForces TheuctuatingBrownianforcessatisfyingtheuctuationdissipationtheoremhaveameanofzero, andavarianceof

PAGE 102

wherekBTisthethermalenergyand(tt0)istheDiracdeltafunction.ThesesimulationsuseunprojectedBrownianforces[ 86 82 83 ],unlikeotherapproaches[ 84 ]wheretheBrownianforcesareinitiallyprojectedorthogonaltotheconstraints.TheBrownianforcesaregivenby wherew()jisarandomvectoroflength12fortheslender-bodydimermodeland9forthethree-beadtrimermodelforeachrod.Thevectorw()jhasthepropertiesof ThematrixB()ijisrelatedtotheresistancematrixthroughtherelation wheretheDiracdeltafunctionisapproximatedby1=tforthepurposeofmakingdiscretetimesteps[ 52 ]. 4.2.3 EvaluationofExcludedVolumeandBendingForces Ashort-rangerepulsivepotentialmaintainstheexcludedvolumeoftherods.Thisforceactsonneighboringrodsatthepointsofclosestapproach[ 67 5 ],contributingtoboththerotationanddisplacementofthecenterofmass.Theforcesbetweenaninteractingpairofrodsegmentsareequalinmagnitudeandoppositeindirection,actingalongthecommonnormalvectorbetweenthesegments.Multiplerepulsiveforcesmayactonasinglesegmentofarod,dependingontheseparationdistancebetweenthesurroundingbers.Therepulsiveforceiscalculatedbetweeneachsegmentofonerodandthetwosegmentsofeveryotherrodinthesystem.Theformoftherepulsivepotentialis[ 38 39 ]

PAGE 103

where"andaretheparametersthatdeterminethemagnitudeandrangeofthepotential,sin()istheanglebetweenthepairofinteractingbers(jijkp()jp()kj),andhistheclosestdistancebetweenthecenterlinesofapairofbersascal-culatedintheworkofFrenkelandMaguire[ 67 ].Theexcludedvolumeforces,F(;Ev)i,usedinEquation( 4.1 )arecalculatedfromthepotentialinEquation( 4.11 )bytakingthederivativeofthepotentialwithrespecttotheseparationdis-tanceh.Allsimulationspresentedhereusedaneectiveaspectratioof50,where"=(50=3)kBTand=0:0771L.Theseparametersfortheexcludedvolumepotentialresultinashort-rangeforcewhichisapproximatelyzeroafteradistanceofabout20%oftherodlength.Theexcludedvolumepotentialbecomessingularwhentherodsareperfectlyparallel,andsocaremustbetakeninthesimulationstoavoidunrealisticresults[ 5 ]. Therigidityoftherodsiscontrolledbyabendingforce.Theformofthebendingpotentialis[ 87 ] wherethestinessparameterKsetsthemagnitudeofthepotential.Thebendingforces,F(;Bend)i,arecalculatedbytakingthederivativeofthepotentialwithrespecttotheconguration.Forthesimulationsperformedinthisstudy,thedimensionlessparameterKwassetequalto300forrodswithlittlebendingand10forsimulationsofslightlymoreexiblerods. 4.2.4 EvaluationofConstraintandCorrectionForces Rigidconstraintsareplacedontherodssothatthetworodsegmentsre-mainedattached.Theconstraintequationsaredenedas

PAGE 104

fortheslender-bodydimermodeland forthethree-beadtrimermodel.Theconstraintsfortheslender-bodymodelstatethatthepositiveendofsegment1ofrodmustcoincidewiththenegativeendofsegment2.Theconstraintsforthethree-beadtrimermodelrequirethatneighboringbeadsremainseparatedbyaxeddistancelatalltimes.Therstconstraint(a=1)ofthethree-beadtrimermodelactsbetweenbeads1and2ofrodandthesecondconstraint(a=2)actsbetweenbeads2and3. Theconstraintforceisgivenby[ 84 88 86 ] wheretheforceistheproductofthevectorofLagrangemultipliers()jandthederivativeoftheconstraintsg()j.TheLagrangemultipliersarecalculatedusing[ 84 82 ] _r()i@g()j whichrequiresthattherodvelocitiesremainperpendiculartotheconstraints. Oncetheforcesactingontherodsarecalculated,Equation( 4.1 )isintegratedintimeusingthemidpointmethod[ 62 60 ].NaiveapplicationofrigidconstraintsusingunmodiedBrownianforcesasdenedinsection 4.2.2 inconjunctionwiththemidpointmethodproducesaprobabilitydistributionfunctionwhichdoesnotgenerallyagreewithkinetictheory.Anadditionalforcegivenby[ 86 82 83 ] isappliedtocorrectthedynamicssothattheintegrationprocedureproducesadistributionconsistentwiththekinetictheoryforaexiblepolymermodel

PAGE 105

employingstispringforces.ThecorrectionforcedenedinEquation( 4.17 )canbeevaluatedanalyticallyasafunctionoftheinternalcongurationforthemodelsappearinginFig. 4{1 4.2.5 DiusivityCalculations Thesimulationsperformedinthisstudyignorehydrodynamicinteractionsbetweenrodsandrodsegments.Underthissimplifyingconditions,themobilitymatrixfortheslender-bodydimermodelisgivenby[ 42 43 44 ] wherethemobility()mn1hasbeendividedbythequantityln(2A)=4l,whereAistheaspectratio(rodsegmentlengthldividedbyroddiameterd)andistheviscosityofthesuspendinguid.Theindicesmandndeningthematrixsizerangefrom1to12andtheindicesiandjdeningthesub-blocksrangefrom1to3.Bythedenitionof_r()imadeinEquation( 4.2 ),thersttwotermslyingalongthediagonalcorrespondtothecenterofmassmobilitiesofsegments1and2oftheslender-bodydimermodelandthelasttwoblockscorrespondtotherotationalmobilities.Themobilitymatrixforthethree-beadtrimermodelisdenedas where()mn1hasbeenmadedimensionlessbydividingbythequantity1=6a,whereaisthediameterofeachbead.Forthethree-beadtrimermodeltheindicesmandndeningthematrixsizerangefrom1to9andtheindicesiandjarethe

PAGE 106

sameaswiththeslender-bodydimermodel.Theupperleftsub-blockcorrespondstothecenterofmassmobilityofbead1,thecentersub-blockcorrespondstothemobilityofbead2,andthelowerrightsub-blockcorrespondstothemobilityofbead3. Astheconcentrationofthesemi-rigidrodsincreases,themobilityofeachrodisreducedduetotheexcludedvolumeinteractionswiththesurroundingrods.Consequently,thelong-timediusivityisaltered.Thelong-timediusivitiesarecalculatedfromthepositionsandorientationsoftherodsovertime.Theaveragecenterofmassdiusivityiscalculatedby forlargevaluesof.Thebracketshiindicateanensembleaverageoverthetimeandrods.TheaveragecenterofmasspositioniscalculatedasseeninFig. 4{1 foreachmodel. Tocharacterizetherotationaldynamicsofthesemi-rigidrods,arotationaldiusioncoecient,DR,isdenedintermsoftheorientationoftheend-to-endvectorfortherod[ 6 23 ], wherep()iistheaverageofp(;1)iandp(;2)i(seeFig. 4{1 ).TheoverallorientationisrenormalizedbeforethediusivitiesarecalculatedinEquation( 4.21 )toensurethatp()iisaunitvector. Periodicboundaryconditionswereusedinthesimulationstoapproximateanunboundedsuspension.Acubicboxoflength3:0LwasusedforthesimulationsfordimensionlessnumberdensitiesnL3from50to150,whereListhetotallengthoftherod.Aboxwithsidesoflength2:25LwasusedfortheconcentrationsofnL3=200and250.Thenumberofrodssimulatedwasdeterminedbythesizeofthebox

PAGE 107

andthedesirednumberdensity.Thevarioussimulationsusedbetween169and506rodsandusedadimensionlesstime-stepof1108foratotaldimensionlesssimulationtimeof1:0.Thetime-stepwasmadedimensionlessbydividingby4L3=kBTln(2A)fortheslender-bodydimermodel,andby6aL2=kBTforthethree-beadtrimermodel.Thetime-stepwasdeterminedtobesmallenoughtoresolvetheexcludedvolumeforces.Atthehighestconcentrations,thedisplacementcausedbytherepulsiveforcesisnomorethan2%oftherodlength,or1=10oftheinteractiondistanceoftherepulsiveforce.ThevaluesofthediusivitiescalculatedfromthesimulationsweredeterminedtobeconvergentintimebydividingthetimestepbytwoforthecaseofnL3=150forbothmodelsateitherKvalueandtheresultswereequivalent.Theeectofthesizeofthesimulationboxwasalsodeterminedtonotbeafactorintheresultsofthesimulationsbyincreasingthesizeofthesimulationboxby50%attheconcentrationsofnL3=50and150forbothmodels. 4.3 Results Thefollowingsectionspresenttheresultsofthesimulations.First,simulationswereperformedofindividualrodstoquantifytheeectofthebendingforces.TheresultsofsimulationsofconcentratedsuspensionsarethenpresentedwithastudyofthebehaviorofDR=DR0asafunctionofthechoiceofmodelandthedegreeofrigidityoftherods.Theresultsoftheeectofbendingonthetranslationaldiusivitiesarealsopresented. 4.3.1 SimulationsofIndividualRods Todeterminetheeectofbendingonindividualrods,studieswereperformedinwhichthemagnitudeofthebendingforcewasvaried.Withnoappliedbendingforce(K=0),theprobabilitydistributionofp(;1)ip(;2)iisuniformasseeninFig. 4{2 .Theprobabilitydistributionofp(;1)ip(;2)iiscalculatedbydividingtherangeof1to1into200equalsizedbins,countingthenumberoftimesthattheangle

PAGE 108

Probabilitydistributionofp(;1)ip(;2)ibasedonthevalueofKusedforthebendingforcewiththeslender-bodydimermodel.Theval-uesofKusedinthesimulationswere10,100and300.Theinsetisamagnicationoftheareawherep(;1)ip(;2)iiscloseto1. betweentherodsfallsintoeachbin,andthennormalizingthecountineachbinbythelargestcount.Thevalueofp(;1)ip(;2)i=0correspondstoarodbentat90degrees,whileavalueofp(;1)ip(;2)i=1correspondstoastraightrod.TheprobabilitydistributionseeninFig. 4{2 approachesthatofarigidrodasthevalueofKincreases.TheinsetinFig. 4{2 showsthequickdecayoftheprobabilitydistributiontozerofortherodswithK=100andK=300.Asexpected,simulationsconrmedthattheprobabilitydistributionsfortheslender-bodydimerandthethree-beadtrimermodelsmatch. ThevalueofKaectstheratioofthediusivitiesatinnitedilution(L2DR0=DT0)fortherodmodels,asshowninFig. 4{3 .Thevaluesofthera-tiosforeachsemi-rigidrodmodelhavebeennormalizedbythevaluesforthecorrespondingdilute,rigid-rodmodels.ForK=5,theratioL2DR0=DT0fortheslender-bodydimermodelis18:6%higherthantheratioforarigidslender-body

PAGE 109

ValuesoftheratioofthedilutevaluesofthediusivitiesL2DR0=DT0resultingfromtheKvaluechosenforthemagnitudeofthebendingforcebetweenthesegmentsofrod.ThevaluesofL2DR0=DT0forthesemi-rigidrodswerenormalizedbytherigid-rodvaluesforeachmodel. model.Similarly,thenormalizedvalueoftheratioforthethree-beadtrimermodelis9:1%higherthantheratioforarigidtrumbell.AsthevalueofKincreasesfrom5,thevaluesoftheratiosforthetworodmodelsapproachthevaluesforrigidrods.Thedecreaseintheratioforthesemi-rigidrodmodelscomesfromadecreaseintherotationaldiusivityastheKvalueincreases.ThevaluesofthecenterofmassdiusivitiesremainconstantregardlessofthevalueforKsincehydrodynamicinteractionsbetweentherodsegmentsareignored. Forstudiesofconcentrationeects,twodierentvaluesofKwerechosen.Therst,K=300,waschosentoproducerodswhichwerenearlyrigid.Thesecondvalue,K=10,waschosentoproducerodswithasmalldegreeofexibility.ThevalueofK=10resultsinarodthatisstillfairlyrigid,withp(;1)ip(;2)irarelyfallingbelow0:6.

PAGE 110

Plotoftheorientationautocorrelationfunction1 2lnDp()i(t+)p()i(t)Eovertimefortheslender-bodydimermodelataconcentrationofnL3=50andK=300.TheinsetofthegraphisamagnicationatshorttimesshowingthatDR0isrecovered. 4.3.2 RotationalDiusivitiesatHighConcentrations Systemswithhigherconcentrationswerestudiedusingthemodelsiden-tiedintheprevioussection.Figure 4{4 showsaplotoftheautocorrelation0:5lnDp()i(t+)p()i(t)Easafunctionoftimeforasuspensionofslender-bodydimersataconcentrationofnL3=50andK=300.Therotationaldiusivityiscalculatedfromtheslopeofthelineafterthesystemhascometoequilibrium.ForthesimulationswithK=300thedilutevalueoftherotationaldiusivityisrecoveredatshorttimes.Thelong-timerotationaldiusivityisthencalculatedfromtheslopeofthegraphatlongertimes. Figure 4{5 showsthevaluesoftherotationaldiusivitiesforthemodelswithK=300aswellasforcompletelyrigidrods.ThesimulationsoftherigidrodswereperformedusingthealgorithmdescribedbyCobbandButler[ 5 ].Therotationaldiusivitiesforeachmodelarenormalizedbytherotationaldiusivitiesofthatspecicmodelatinnitedilution.Apower-lawscalingofDR=DR0(nL3)isused

PAGE 111

Rotationaldiusivitiesofrigidslender-bodiesandthree-beadtrum-bellsincomparisontotheslender-bodydimerandthree-beadtrimermodelswithK=300.Thediusivitiesarenormalizedbytheirrespec-tivedilutevaluesandareplottedasafunctionofthenumberdensitynL3. tocomparetheresultsofthesimulationsobtainedinthisstudywiththepublishedtheories[ 3 6 34 35 23 ].Thescalingcalculatedfortheslender-bodydimermodelis1:95,andthescalingforthethree-beadtrimermodelis1:36overtheconcentrationsofnL3=70to150.Thesevaluesforthepower-lawscalingareclosetothoseofrigidrods[ 5 ].Therelativevaluesoftherotationaldiusivitiesareconsistentlyhigherforthesemi-rigidrods,whichshowsthatevensmalldeviationsfromrigidmodelscauseadierence.Theaveragedierencebetweenthediusivityvaluesis9:8%overtherangeofconcentrationsstudied. Overthesamerangeofconcentrations,thevaluesoftherotationaldiusivitiesforthemoreexiblerodswithK=10diersignicantlyfromthediusivitiesofrigidrods.Figure 4{6 showsthatovertherangeofnL3=70to150,thevaluesofDR=DR0arehigherforthesimulationswiththesemi-rigidrods.Thepower-lawscalingsoverthisrangeofconcentrationsfortherodswithK=10are1:40fortheslender-bodydimermodeland1:01forthethree-beadtrimermodel.

PAGE 112

Rotationaldiusivitiesoftherigid-rodmodelsincomparisontothesemi-rigidrodmodelswithK=10.Thescalingofthesemi-rigidrodmodelsatconcentrationsgreaterthannL3=130approachthatoftherigidrods. Thepower-lawtisalsonotparticularlygoodoverthisconcentrationregime,withthecoecientofdetermination(R2)havingavalueofabout0:92forbothmodels.Iftheconcentrationrangeoverwhichthepower-lawscalingiscalculatedischangedtonL3=130to250,thenthescalingsapproachthevaluesforrigidrodswhere1:91fortheslender-bodydimermodeland1:35forthethree-beadtrimermodel.Thepower-lawtovertheseconcentrationsimprovestoavalueofR2ofabout0:97fortheslender-bodydimermodeland0:96forthethree-beaddimermodel. Thepercentageerrorinthediusivitiescalculatedfromthesimulationresultswaslargestatthelowestconcentrations,wheretherewerefewerrodsoverwhichtoaverage.However,theerrorneverexceeded5%ofthemeanvalueforanyofthediusivityvaluescalculated.TheerrorbarsinFigs. 4{5 and 4{6 aresmallerthanthesymbols.

PAGE 113

Translationaldiusivitiesforthesemi-rigidmodelswithK=10incomparisontothediusivitiesforrigidrods.ThevalueofDkforthesemi-rigidmodelsisreducedbyabout20%fromthedilutevalue,andsolowerstheoverallcenterofmassdiusivity. 4.3.3 TranslationalDiusivitiesatHighConcentrations TheaveragediusivitiesofthecenterofmassfortherodsdisplayedinFig. 4{7 werecalculatedusingEquation( 4.20 ).Thedisplacementsparallelandperpendiculartothecentralrodaxiswerecalculatedbyprojectingtherodcenterofmassdisplacementalongtheorientationovertime[ 38 39 5 ].ThediusivitiesoftherodswithK=300areindistinguishablefromtherigidrodvaluesatallconcentrationssimulated.ThevaluesofDk=Dk0arequalitativelydierentformodelswithK=10ascomparedtotherigid-rodmodelsatconcentrationsgreaterthannL3=50.ThevaluesofDT=Dk0arehigherforthesemi-rigidrodswithK=10uptoconcentrationsofnL3=70.ThevaluesofDT=Dk0arelowerforthesemi-rigidrodsatconcentrationsgreaterthannL3=130.ThevaluesofD?=Dk0arehigherforthesemi-rigidrodsuntilaboutnL3=110.

PAGE 114

Examiningthedierencebetweenthesemi-rigidandrigidrodsshowsthatthediusivityparalleltothecentralrodaxis(Dk=Dk0)isreducedtoabout0:8atthehighestconcentrationsfortherodmodelswithK=10,whereasDk=Dk0wasvirtuallyunchangedfortherigid-rodmodels.ThisdecreaseinDk=Dk0resultsinareductionoftheoverallcenterofmassdiusivityathighconcentrationsfromabout0:4fortherigidrodsto0:3.Thediusivityperpendiculartothecentralrodaxisisalmostnonexistentatthesehighconcentrations,whereD?0=Dk0isreducedbyabout97%fortheslender-bodydimermodeland98%forthethree-beadtrimermodel.Theperpendiculardiusivitiesfortheslender-bodydimermodelatlowconcentrationsuptonL3=90aresmallerthanthecorrespondingdiusivitiesforthethree-beaddimermodelforboththerigidandsemi-rigidrodmodels.Thisdierencearisesfromtheanisotropicversusisotropiccenterofmassdiusivitiesfortheslender-bodydimerandthree-beadtrimermodels,respectively.Thepercentageerrorinthecenterofmassdiusivitycalculationswasalsowithinthe5%range,similarlytotheerrorintherotationaldiusivities,withtheerrorbarsbeingsmallerthanthesymbolsinFig. 4{7 4.4 Discussion Incomparingthesimulationresultsofthesemi-rigidtotherigidpolymers,twomainfeaturesareapparent.First,adelayintheonsetofthesemi-diluteregimeisobservedforthemoreexiblerods.Second,thesesemi-rigidrodsproducethesamepower-lawscalingfortherotationaldiusivitiesastheircorrespondingrigid-rodmodelsatsucientlyhighconcentrations.Theoriginofthesetwomainfeaturesofthedynamicsarediscussedwithinthecontextofexistingtheoriesandrecentsimulationresults. ThetheoriesofDoiandEdwards[ 6 ]predicttheonsetofsemi-dilutebehaviorinsystemsofrigidrodsatlowerconcentrationsthanobservedinexperiments[ 89 90 ].Thesemi-diluteconcentrationregimeisgenerallydenedastheconcentration

PAGE 115

atwhichrodrotationsbecomesucientlyhinderedtocauseamarkeddecreaseintherotationaldiusivities[ 89 ].KeepandPecora[ 23 ]arguethatexibilityinpolymersrequiresmoreblockingpoints(andsomorerods)beforetherodsbecomecaged[ 3 ]andenterthesemi-diluteregime.Theendofaexiblerodislesshinderedthanarigidrod,andthereforerotatesmorequickly,eventhoughthetrailingportionmaystillbetrapped.ThetheoryofKeepandPecora[ 23 ]statesthatbelowacriticalconcentration,therodsmaybeconsideredrigidwhencalculatingtherotationaldynamics.Oncethesystemexceedsthiscriticalconcentration,eectscausedbytheexibilityoftherodareseen,includingadelayintheonsetofthesemi-diluteconcentrationregime.Thecriticalconcentrationisgivenby[ 23 ] 15+"418 152+768 L#1=2;(4.22) wherePisthepersistencelength,Listhetotallengthoftherod,and(nL3)critistheconcentrationatwhichexibilityeectsmustbeconsideredforpolymerrodsystems. Thepersistencelengthisdenedby[ 23 ] wheresisthedistancealongthecontourlengthoftherod.Forthesimulationsperformedhereofsemi-rigidrodsmadeupoftworigidsegments,thevalueofsisthedistancealongthelengthoftherodsegmentsbetweenthecenters(s=l).Inthesesimulationstherodshavealengthof2l(L=2l).AccordingtothepredictionofKeepandPecora[ 23 ],thiscriticalconcentrationiswheretherotationaldynamicsofthesemi-rigidrodswilldivergefromthedynamicsofrigidrods.Equation( 4.22 )predictsthecriticalconcentrationquitewellforthesimulationsofsemi-rigidrodswithK=10.UsingthecalculationfromEquation

PAGE 116

RatioofthePersistencelengths(P)fortheslender-bodydimerandthree-beadtrimermodelsoverthetotalrodlength(L)atK=10.Thesemi-rigidrodsstraightenonaverageastheconcentrationincreases. ( 4.22 )alongwiththepersistencelengthsforthemodelsatinnitedilutionseeninFig. 4{8 producesacriticalconcentrationof(nL3)crit40.Figure 4{6 showstherotationaldiusivitiesfortherigid-rodmodelsincomparisontothesemi-rigidrodmodels.TherotationaldiusivitiesforconcentrationslessthannL3=30areequivalent,withintheerrorofthesimulations.ForconcentrationsgreaterthannL3=40therotationaldiusivitiesforthemodelswithK=10arehigherthanthatoftherigidrods.Thischangeinthedynamicsfromrigidtosemi-rigidcanbeseeninaplotofthediusivityvaluesasafunctionofthestinessparameterKusedinthesimulations.Figure 4{9 showsthatfortheslender-bodydimermodelatlowconcentrations,thevaluesofDR=DR0donotchangesignicantlywithdierentvaluesofK,orfortherigidmodels.Astheconcentrationincreases,therigidmodelsandmodelswithhighbendingstinesshavemuchlowerrotationaldiusivityvaluesthanthemodelswithK=10.Therotationaldiusivityvaluesforthethree-beadtrimermodelproducearesultwhichisqualitativelysimilartothatofFig. 4{9

PAGE 117

RotationaldiusivityasafunctionofthestinessparameterKandconcentrationforsimulationsusingtheslender-bodydimermodel. Thesecondmajorobservationofthedynamicsofrodsuspensionsisthesim-ilarityinthepower-lawscalingbetweentherigidandsemi-rigidrodmodels.TherotationaldiusivitiesseeninFig. 4{6 showthatastheconcentrationincreasesuptonL3=250,thepower-lawscalingsofDR=DR0approachthevaluesforrigidrods.Thiswouldseemtoindicatethatthesemi-rigidrodsbehaveasfullyrigidrodsatsucientlyhighconcentrations.Tostudytheonsetofrigid-rodlikebehaviorasafunctionofconcentration,thepersistencelengthoftherodswascalculatedfromthesimulationdataasafunctionofconcentration.Therigidityoftherodisrelatedtotheratioofthepersistencelengthtothelengthoftherod(P=L).AcompletelystraightrodwillhavearatioofP=L=1.Figure 4{8 showsthepersistencelengthratiosofthesemi-rigidrodswithK=10ascalculatedinEquation( 4.23 ).Thepersistencelengthratiosforthesesemi-rigidrodsvarybetweenP=L=3:9,corre-spondingtoanaverageanglebetweentherodsofabout39degreesfromastraightconformation,andP=L=10:7,correspondingtoanangleofabout24degrees,overtherangeofconcentrationsstudied.AvalueofP=L=10(averageangleofabout

PAGE 118

25degreesfromrigid),isconsideredtobeverysti[ 23 ].Figure 4{8 alsoshowsthatthepersistencelengthincreaseswithincreasingconcentration.Thisincreasedemonstratesthattherodsspendmoretimeinastraight,orrigidconformationastheconcentrationincreases.Oncethecongurationoftherodsbecomesstraightenoughduetocaging,therotationaldiusivitybeginstodecreasewithnearlythesamepower-lawscalingasthecorrespondingrigidmodel.Thepower-lawscalingfortherotationaldiusivitiesofthesemi-rigidslender-bodydimermodelwithK=10atconcentrationshigherthannL3=110isalmostidenticaltothepower-lawscal-ingoftherigid-slenderbodymodelatconcentrationsgreaterthannL3=70.Thesameistrueforthepower-lawscalingofthesemi-rigidthree-beadtrimermodelincomparisontotherigidthree-beadtrumbellmodeloverthesameconcentrationregime. Thepower-lawscalingvaluesaredierentforeachhydrodynamicmodel,asexpectedfromrecentsimulationwork[ 5 ].Theoriesforthescalingoftherotationaldiusivitiesofrodsystemsinsolutionhaveexistedinthescienticliteratureformanyyears,withtheearliesttheoriesbasedonsystemsofrigidrodswithhighaspectratios[ 3 34 35 ].Thepower-lawscalingofDR=DR0(nL3)2predictedbythetheoriesandsimulationsofDoiandEdwards[ 6 ]havebeenreportedtobeincorrectbasedonsimulationresults[ 34 35 ].Disagreementsconcerningtheobservedpower-lawscalinghavebeenshowntooriginatewiththechoiceofhydrodynamicmodelusedwithinthesimulations[ 5 ].Thepower-lawscalingpredictedbyDoiandEdwards[ 6 ]is,however,observedinmanyexperiments[ 23 ]. Asanoteaboutthescalings,theuseofapower-lawscalingtocharacterizetherotationaldynamicsofpolymerrodsystemsathighconcentrationshasnotbeenrigorouslyjustied[ 68 5 ].Infact,theagreementbetweenthesimulationdataandthepower-lawtsarenotparticularlyconvincing;relativelyfewpointsareusedtocalculatetheexponent,andthecorrelationisnotparticularlygood.

PAGE 119

However,inkeepingwiththeliteratureonthesubjectofrotationaldiusivityofpolymerrods,apower-lawtofthesimulationdataisusedtoprovideaconvenientwayofcollapsingallofthedatafortherotationaldiusivitiesinthesemi-diluteconcentrationregimeintoasinglenumber,whichcanbethenreadilycomparedtoexistingtheories. Therotationaldiusivitiesfortheslender-bodydimerrodswithK=10areshowninFig. 4{10 incomparisontotheexperimentalndingsofMorietal.[ 71 ],aswellassimulationsofrigid-dumbbellsbyCobbandButler[ 5 ].Someresearchershaveclaimedthatthe=1scalingmeasuredintheexperimentsofMorietal.[ 71 ]overthisintermediateconcentrationregimeshowthattherigid-dumbbellmodelshouldbeusedinsimulationsofpolymerrodsbecauseitreproducestheexperimentalresultswell[ 38 39 ].TheresultsinFig. 4{10 showthatsimulationsofsemi-rigidslender-bodydimerswillalsoreproducetherotationaldiusivityresultsofMori[ 71 ]overthisrangeofconcentrations.Consequently,thedegreeofexibilityandhydrodynamicmodelcannotbeuniquelydeterminedonthesolebasisofacomparisonbetweenthemeasuredandcalculatedrotationaldiusivitiesforthisintermediaterangeofconcentrations.Bycomparingwithexperimentsathigherconcentrations,distinguishingbetweenthetwoeectsofexibilityandhydrodynamicmodelshouldbepossible. Flexibilityalsonoticeablyimpactsthecenterofmassdiusionofthesemi-rigidrods.DoiandEdwards[ 6 ]predictthattheconcentrationdependenceoftheaveragecenterofmassdiusivitydecreasesforincreasinglyexiblerods.Thedropintheaveragecenterofmassdiusivity,asseeninFig. 4{7 ,conrmstheaccuracyoftheprediction.Thisdecreaseintheparalleldiusivityiscausedbythefactthatwhentherodisbent,themotionparalleltothecentralaxesofeachsegmentdoesnottranslatecompletelyintothemotionalongtheoverallorientationoftherodas

PAGE 120

Rotationaldiusivitiesoftheslender-bodydimerwithK=10incomparisontosimulationsofarigid-dumbbell[ 5 ],andtherotationaldiusivitiesmeasuredbyMorietal.[ 71 ]insystemsofPBLG. awhole.Abentrodwilltravelalongitsparalleldirectionthroughatubelessfreelythanarigidrod. 4.5 Conclusions Simulationsofmodelsincorporatingasmalldegreeofexibilitycanprovideasimple,yetvaluableinsightintothedynamicsofrodsuspensions.Thedegreeofexibilityandthetypeofhydrodynamicmodelchosenhaveadualeectontherotationaldiusivitiesofrodsystems.Theexibilityoftherodsbecomesimportantaboveacriticalconcentration,whereitcausesadelayintheonsetofthesemi-diluteconcentrationregime.Thedegreeofexibilitydoesnothoweverchangethevalueofinthepower-lawscalingofDR=DR0(nL3)athighconcentrationsforthatspecichydrodynamicmodel.Insteadtheexibilitysimplyincreasestheconcentrationatwhichthescalingisobserved.Thechoiceofhydrodynamicmodelusedtorepresenttherodsinthesimulationshasasignicantimpactonthevalueofinthepower-lawscalingathighconcentrations.

PAGE 121

Dierentmicro-mechanicalmodelshavebeenshowntobeabletoreproduceexperimentalresults.Therodssimulatedherearenotcompletelyrealisticincomparisontoactualexperimentalpolymerrodssincetheimportanteectofmulti-bodyhydrodynamicinteractionshavenotbeentakenintoaccount.Thehydrodynamicinteractionswillaltertheshort-timedynamicsoftherodsand,consequently,thelong-timediusivities.Thesimulationresultscanbeusedasatooltoobtainqualitativecomparisonstoexperimentalsystemsandalsoasareferencetocomparetosimulationsinwhichmulti-bodyhydrodynamicinteractionsareincluded.

PAGE 122

5.1 Introduction TheBrowniandynamicssimulationsinchapters 3 and 4 wereperformedwithoutincludinghydrodynamicinteractions[ 5 91 ].Theomissionofthehydro-dynamicinteractionsfromtheseprevioussimulationsintroducesaalterationintothedynamicsofthesystemsstudied.Themotionofaparticleinauidcausesadisturbanceinthatuidwhichaectsthemotionofthesurroundingparticles,andthemotionofthesurroundingparticleslikewiseaectsthetestparticle.Thisdisturbancecanplayasignicantroleintheoveralldynamicsofthesuspension.Itisknownthattheinclusionofhydrodynamicinteractionsintoasimulationwillchangethevaluesoftheshort-timediusivitiesinasuspensionofbersdependingontheconcentration.Itistheorizedthatiftheshort-timediusivitiesofasuspen-sionofberscanbecalculated,thenthosevaluescouldbeusedintheBrownianDynamicssimulationsfromchapter 3 tocalculateaccuratelong-timediusivities. Includinghydrodynamicinteractionsisnotatrivialproblem.Thecomputa-tionalcostofincludinghydrodynamicinteractionsintothesimulationslimitthenumberofbersthatcanbesimulatedwithinareasonableamountoftime.Onewayofreducingthecomputationalcost,andsoincreasingtheperformanceofthesimulations,istoperformthesimulationsinparallelacrossanetworkedclusterofcomputerprocessors.Themethodofincorporatingthehydrodynamicinteractionsintothesimulationswillbediscussed,aswellasanimplementationofaparallel 109

PAGE 123

codeforcalculatingthedynamicsofthesystem.Apreliminaryanalysisoftheperformancegainsispresented. 5.2 CalculatingtheMulti-bodyHydrodynamicInteractions Usingtheslender-bodymodel[ 42 43 44 ]asanexampleforincludinghydro-dynamicinteraction,theequationofmotionforparticleisgivenby _x()i+s_p()iu0i(s)=ln(2A) 4ij+p()ip()jf()j(s);(5.1) whereu0i(s)isthedisturbancevelocityonrodcausedbythesumoftheimposedvelocityeldontheuidandtheuidvelocitiescausedbyrodsotherthan.ThemotionoftherodsdependontheaspectratioA(rodlengthoverdiameter)andthedistributionofforcesactingalongtherodlengthf()j(s). Asshowninchapter 2 ,thecenterofmassvelocities_x()icanbecalculatedbyintegratingEquation( 5.1 )overtherodlengthgiving _x()i=1 4Lij+p()ip()jF()j;(5.2) wherex()i+sp()iisanypointalongthecenteraxisofrod,andF()jisthetotalforceactingonrodgivenby Therotationalvelocities_p()iarecalculatedinasimilarmannerwherethecrossproductofEquation( 5.1 )istakenwithsp()i,integratedovertherodlength,andthencrossedwithsp()iagaintoobtain _p()i=12 wherethetorqueT()ionrodisrelatedtotheweightedforce~F()iby[ 5 ]

PAGE 124

Equations( 5.2 ),( 5.2 ),and( 5.4 )aremadedimensionlesswithacharacteristiclengthlc,timetc,andforcefcgivenby L;(5.6) togive _x()i+s_p()iu0i(s)=ij+p()ip()jf()j(s);(5.7) _x()i=ZL=2L=2u0ix()i+sp()ids+ij+p()ip()jF()j;(5.8) and _p()i=12ijp()ip()jZL=2L=2su0jx()i+sp()ids+12ijp()ip()j~F()j:(5.9) Thedisturbancevelocityu0ix()i+sp()iactingonrodisgivenbythesumoftheimposedvelocityeldontheuidu(1)iandthedisturbancevelocitiescausedbyalltheotherrodsinthesuspensionv()i, whereNisthenumberofrodsinthesuspension.Ifthereisnoimposedvelocityeldonthesystemthenu(1)iissettozero. TheoriginalanalysisofBatchelor[ 42 ]isusedtocalculatecontributionsofv()i.Theslender-bodyapproximationwasderivedfromaboundaryintegralequationwherethesurfaceintegraliscalculatedasanintegralalongthecenterlineoftheparticle.Theerrorcausedbythisapproximationissmalliftheparticleisofhighaspectratio.Thus,thevelocitydisturbancegeneratedbyparticleatapositionxicanbeapproximatedas 2ln(2A)Z1=21=2ij

PAGE 125

wherey()i=xix()i+sp()i,andy()isthemagnitudeofy()i.Becauseperiodicboundaryconditionsareusedtoapproximateanunboundedsuspension,thefundamentalsolutionoftheStokesequationsforaperiodicarrayofpointforcesasderivedbyHasimoto[ 92 ]replacestheOseentensorinEquation( 5.11 ).AlsonotethattheaspectratiodoesnotappearinanyofthedimensionlessequationsuntilEquation( 5.11 ).Whennotincludinghydrodynamicinteractions,theaspectratiocanbescaledoutofthefar-eldrelations.Whenincludinghydrodynamicinteractions,thefar-eldinteractionsdependupontheaspectratiooftheparticleinanon-trivialmanner. Thoughconceptuallystraightforward,adirectnumericalsolutionofthepreviousequationsforthemotionoftheindividualbersisprohibitivelycomputa-tionallyexpensive.Forexample,MackaplowandShaqfeh[ 93 ]discretizedeachberandsolvethesetofequationsdirectlyfortheforcedistributionandvelocitydistur-banceoneachber.Themethodwasfoundtobetoocostlytouseinsimulatingthedynamicsofasuspensionofnon-colloidalbers,soasimpliedpoint-particlemethodwasdevelopedtocarryoutdynamicsimulations. AmoreecientapproachofsolvingfortheforcedistributionwasutilizedbyHarlenetal.[ 77 ],andsubsequentlybyButlerandShaqfeh[ 66 ].Inthismethod,Equations( 5.7 ),( 5.8 ),and( 5.9 )aresolvedfortheforcedistributionintermsofthetotalforce,torque,anddisturbancevelocity.ThedisturbancevelocityisthenexpandedintermsofaLegendrepolynomialinthebercoordinates[ 66 ].Retainingonlythersttwotermsintheexpansionresultsinthelinearizedforcedistribution, Inthisexpression,thenethydrodynamicforce,F()i,andtorqueT()jmustbalancethesumoftheforcesandtorquesactingontheber,suchasthosearisingfromBrownianmotion,repulsiveforces,orotherinteractionsthatthebersmay

PAGE 126

experience.ThethirdterminEquation( 5.12 )includesthestressletS()forber,where 2Z1=21=2sp()iu()ix()i+sp()ids:(5.13) Sincetheparticlesaremodeledasslenderbodies,theypossessonlyonenitedimensioninthehydrodynamicmodel,andthestressletisthereforeascalarquantityassociatedwitheachrodwhicharisesfromtheinabilityofanindividualrodtostretchorcompressalongitsmajoraxis. ThestressletinEquations( 5.12 )and( 5.13 )incorporatesmulti-bodyhydro-dynamicinteractionsintothecalculationofthemobilitymatrix.Equation( 5.13 )isusedtorelatethevalueofS()foreachrodtotheforcesandtorquesactingonalltheotherrods.Todothis,thedisturbancevelocitiesintheequationforthestressletisevaluatedusingEquation( 5.10 )withtheforcedistributiongivenbyEquation( 5.12 ).Alinearsetofequationsforthestressletswhichdependsupontheinstantaneouscongurationofalltheparticleswithinthesuspensionresults,whichgivesrisetothemulti-bodyapproximationforthemobilitymatrix[ 66 ].TheexplicitexpressionforthestressletsaresubstitutedintothelinearizedforcedensityofEquation( 5.12 )andareusedtogetherwithEquations( 5.10 )and( 5.11 )tosolveforthedisturbancevelocity.ThedisturbancevelocitycanthenbeusedinEquations( 5.8 )and( 5.9 )tocalculatethemotionoftherods. Thus,therelationshipbetweenthestressletsandtheforcesandtorquescanbeincorporateddirectlyintotheformulationtogive _ri=MijFj(5.14) where_riisavectorcontainingthevelocitiesofalltherodsinthesuspensionandFiisavectorcontainingtheforcesandtorques: _ri=_x()j_p()jandFi=F()jT()j:(5.15)

PAGE 127

ThevectorsinEquations( 5.15 )areconstructedsuchthati=3(1)+jifthevariableis_xorFandi=3(N+1)+jfor_pandT;thisinsuresthatthemobilitymatrixMijhastheappropriatestructure.Themobilitymatrixisaninstantaneousfunctionofonlythepositionsandorientationsri,where Aswritten,themobilitymatrixincorporatesanapproximationtomulti-bodyhydrodynamicinteractions.However,themobilitymatrixcanbeconstructedwithlowerlevelsofapproximation.Forexample,leavingoutthestressletinEquation( 5.12 )givesanapproximationtothemobilitymatrixwhichonlyincludestwobodyinteractionssincethematrixisformedbypairwiseadditionoftheforcesandtorques.Itshouldbenoted,though,thattheinverseofthismobilitymatrixgivesaresistancematrixwhichdoesrepresentamulti-bodyapproximation.Theversionwiththestresslets,however,wouldbemoreaccurate[ 94 ].CompletelyignoringthedisturbancevelocityinEquations( 5.8 )and( 5.9 )resultsinamobilitymatrixwhichfailstoincorporatehydrodynamicinteractions.Therefore,bychangingthelevelofapproximation,theimpactofthehydrodynamicinteractionscanbedirectlydetermined. 5.3 ImplementingaParallelAlgorithmusingtheMessagePassingInterface(MPI) Asmentionedpreviously,onereasonfornotincludingthemulti-bodyhydro-dynamicinteractionsinthesimulationsinchapters 3 and 4 isthecomputationalexpenseofcalculatingtheseinteractionsforsuspensionswithmanyparticles.Usingasingleprocessor,thelimitofnumberofrodsinthesimulationsisontheorderof100rods.Asseeninthepreviouschapters,simulationswereperformedwithasmanyas1500rods.Inordertoperformsimulationswithsucientlyhighcon-centrationswherethemulti-bodyinteractionsarecalculated,aparallelalgorithmmustbeimplemented.TheMessagePassingInterface(MPI)[ 95 96 ]standardwas

PAGE 128

developedtoprovideauniedandportablestandardforparallelprogramming.TheMPIstandardprovidesaninterfacethroughwhichnodesinacomputerclustershareinformationandthereforeworktogethertosolveasingleproblem. 5.3.1 ParallelCalculationoftheMobilityMatrix FormingthegrandmobilitymatrixMijseeninEquation( 5.14 )istherstplacewhereaparallelimplementationcouldpotentiallyspeedupthecalculationandallowforsimulationsoflargersuspensions.Thisisaratherstraightforwardcalculationsincetheindividualentriesdependonlyuponindividualpairsofparticles,wherethepositionsandorientationsofthebersarebroadcasttothedierentprocessorsinthecluster.Eachprocessorwillthenbeassignedasetofrods,whichthencalculatestheinteractionsonthosebersaccordingtoEquation( 5.10 ).Thetimethataprocessortakestocalculatetheinteractionsonthesetofbersassignedtoitismuchlongerthanthecommunicationtimeacrossthenetworkofnodesinthecluster,andsotherateofcalculatingtheinteractionsshouldscalewiththenumberofprocessorsdoingthecalculations. Oncethegrandmobilitymatrixhasbeenformed,therepulsiveorlubricationinteractionsarecalculated.Theseshort-rangeinteractionskeeptherodsfromoverlappingwhentheyareincloseproximity.TheseinteractionsarethenaddedbackintothegrandmobilitymatrixMijbeforeitisdecomposed.Therepulsiveorlubricationinteractionsareallpairinteractions,andcanbeimplementedinaparallelcodeinasimilarmannertowhenthegrandmobilitymatrixwasinitiallyformed.Thecalculationtimefortheseinteractionswouldbesimilartothatoftheformationofthegrandmobilitymatrix. 5.3.2 DecompositionoftheGrandMobilityMatrix Thesecondpossibleimplementationofaparallelalgorithmintothesimula-tionsismorecomplicated.OncethegrandmobilitymatrixMijisformed,and

PAGE 129

thelubricationinteractionsareaddedintoit,thenitmustbedecomposedtocal-culatethedisplacementsoftherods.AsmentionedinSection 2.2 ofChapter 2 ,aCholeskydecomposition[ 51 ]isusedtodecomposethemobilitymatrixforserialimplementationsofthecode.ConversionofaserialCholeskydecompositioncodetoaparallelcodeisnotatrivialmatter. InCholeskydecomposition,thematrixAijmustbesymmetricandpositivedenite.Asymmetricmatrixisdenedas[ 51 ] andispositivedeniteif forallvectorsvi.TheCholeskydecompositionbuildsalowertriangularmatrixLijsuchthat Thisfactorizationisalsoreferredtoas\takingthesquarerootofthematrixAij"[ 51 ].Thelowertriangularmatrixcanthenbecalculatedby and whereNisthesizeofthematrix.TheCholeskydecompositionrequiresN3=6calculationsforeachtimethematrixisdecomposed,whichcanbecomeverytimechallengingasthesuspensionsbecomelarger.Asanexample,asuspensionof1000rodswouldcreateagrandmobilitymatrixwhichwouldhaveadimensionof60006000(eachrodhas3componentsforthecenterofmassand3componentsfortheorientation),requiring3:61010operationstodecomposethematrix.

PAGE 130

5.3.3 PLAPACK:AGeneralApproachtotheParallelCholeskyDecomposition MultipleimplementationsofaparallelCholeskydecompositionhaveappearedinthescienticliterature[ 97 98 99 100 101 ].However,manyoftheseimplemen-tationsaredesignedspecicallyforthecomputerarchitecturethattheresearcherswereusingandarenotportabletoothersystems.Forexample,EdwardsandTomlin[ 97 ]designedaparallelalgorithmfortheIntelhypercubeparallelcomputersystem,whileZhengandChang[ 98 ]developedaparallelCholeskydecompositionforacomputerclusterwhereeveryprocessorsharesacommonbankofmemory. AportablesolutionforimplementingaparallelCholeskydecompositionhasbeendevelopedaspartofalargerlinearalgebrapackagecalled\PLAPACK"[ 102 ].ThePLAPACKinfrastructurewasdevelopedtoprovideasimplewaytodistributedataacrossaparallelnetworkofprocessorswhichcouldbeeasilyportedfromonesystemarchitecturetoanother.PLAPACKtakesadvantageofthefactthatdierentcomputervendorshavedevelopedcomputationalkernelsknownastheBasicLinearAlgebraSubprograms(BLAS).Thesesetofkernelscontainbasiclinearalgebraoperationssuchasvectordotproducts,matrix-vectormultiplication,andmatrix-matrixmultiplication.TheseBLASlibrariesareoptimizedbythecomputervendorstotakeadvantageofthespecicarchitectureinthemachinesbeingusedandsoproduceahigherperformance.TheBLASlibrariesthatwillbeusedinthisimplementationaretheGotoBLASlibraries[ 103 ].ThedierentlinearalgebraoperationsincludedintheBLASlibrariesaregroupedintothreecategories: 104 ]:vector-vectoroperations.ThereisatotalofO(n)calculationsperformedonO(n)amountofdata,wherenisthelengthofthevectors. 105 ]:matrix-vectoroperations.ThereisatotalofO(n2)computationsperformedonO(n2)amountofdata,wherenisthesizeofthematrix.

PAGE 131

106 ]:matrix-matrixoperations.ThereisatotalofO(n3)calculationsdoneonO(n2)amountofdata. Aproblemthatoccursinmanyparallelcomputingenvironmentsisthatittakesamuchlongertimeforthecomputertoaccessitsmainmemorythanitdoesfortheprocessortodocalculations,whichcancauseaproblemwhenprocessorshavetowaitfortheaccesstothememorybeforecontinuingtodoothercalculations.Level-3BLASistargetedasthesolutiontothisproblembecauseofthehigherorderofcalculationsrequiredforalessamountofdata.PLAPACKalsousestheSingleProgramMultipleData(SPMD)paradigm,wheretheidenticalprogramwillrunonallthedierentprocessorsinthecomputernetwork,butthepaththroughtheprogramisdeterminedbytheindexofeachindividualprocessor.ThisparadigmwaschosensothatitwouldtseamlesslywithintheMPIinfrastructure. Choleskydecomposition:Theright-lookingvariant ThesimplestformoftheCholeskydecompositionistheright-lookingvariant,whichwasdescribedintheprevioussection.Theimplementationofthatformofthedecompositioncanbedescribedbypartitioningthematricesas where11and11arescalars,a21andl21arevectorsoflengthn1,andA22andL22arematricesofsize(n1)(n1).TheindicatesthesymmetricpartofAij,whichisnotoperatedon.ThematrixAijisthengivenby

PAGE 132

fromwhichthefollowingequationsarederived ThealgorithmforcomputingtheCholeskydecompositionisthengivenby 1. PartitionAijasinEquation( 5.22 ) 2. ContinuerecursivelywithA22 5.22 )aresolvedforwiththe\="sign,butinsteadofcreatinganewmatrix,thevaluesarestoredintheoriginalAijmatrixassigniedwiththe\"sign.ThePLAPACKsubroutineoverwritestheoriginalmatrixAijwiththeCholeskyfactorizedlowertriangularmatrixLij.Thealgorithmwillcontinueoperatingoneachsubsequentbottom-rightsubmatrixinAijuntilitreachestheendofthematrix.Thisstrictlyserialversiondoesnotconvertverywelltoaparallelversion.PLAPACKusesthedierentlevelsofBLAStoincreasetheperformanceinparallel. Level-1andlevel-2BLASimplementations ToincreasetheperformanceofaparallelimplementationoftheCholeskydecomposition,increasedlevelsoftheBLASlibrariesareused.Whenconsideringhowtoimprovethealgorithmshownintheprevioussection,itisclearthatthe

PAGE 133

square-rootoperationinstep2performedisanoperationonascalarvalue.Theoperationsperformedontherestofthecolumna21l21=a21=11instep3oftheserialalgorithmisavectoroperation,andsoLevel-1BLAScanbeusedtoimprovetheperformance.TheBLASroutineusedforthisoperationistheDSCALsubroutine,wheretheDsigniesadouble-precisionoperation.Inordertoconvertstep4intheserialalgorithmtoavector-vector(Level-1BLAS)operation,A22ispartitionedbycolumnsandl21ispartitionedbyelements andsotheupdateofA22instep4ofthealgorithmbecomes Inthisupdate,amultipleofl21issubtractedfromeachcolumnofA22inanoperationcalledDAXPY,whichstandsforascalarA timesavectorX P lusavectorY inD oubleprecision.SinceonlythelowertriangularpartofA22isneededbecauseAijissymmetric,onlypartofthecolumnisupdatedhere. Theperformanceofthealgorithminparallelcanbefurtherimprovedbyintro-ducingmatrix-vectorlinearalgebrasubroutinesfromtheLevel-2BLASlibraries.ThedierencecomesinthepartitioningoftheA22matrixintosubmatrices,insteadofsimplycolumnsofvectors,

PAGE 134

whereqisthenumberofcolumnsofA22includedinthesubmatrix. Level-3BLASimplementation OptimizingtheserialCholeskydecompositionalgorithmwithLevel-1andLevel-2BLASlibrariesdoesinfactimprovetheperformancewhenruninaparallelcomputingenvironment.However,asnotedbefore,thenumberoftimesrequiredtoaccessthecomputermemoryisonthesameorderasthenumberofcalculationsperformedbytheprocessor(O(n)forLevel-1BLASandO(n2)forLevel-2BLAS).Moderncomputerprocessorsperformcalculationsmuchfasterthantheprocessofaccessingthecomputer'smemory.ThismeansthatthereisalimittotheamountthattheperformanceoftheCholeskydecompositioncanbeimprovedbyonlyusingLevel-1andLevel-2BLAS. ByimplementingLevel-3BLAS,whichrequiresO(n3)operationstobeperformedonO(n2)data,theperformanceofthesimulationscanbeimproved.ByfurtherdividingthematrixAijintosubmatrices,whichwaspartiallydoneintheuseoftheLevel-2BLASimplementation,itistheoreticallypossibletoachievenearpeakperformance[ 102 ].ThematricesAijandLijarepartitionedas whereA11andL11aresubmatricesofsizebb,whereb<
PAGE 135

Thefollowingequationsarederivedfromtheaboveoperation ThealgorithmfortheLevel-3BLASimplementationoftheCholeskydecompositionisthen 1. PartitionAijasinEquation( 5.31 ) 2. ContinuerecursivelywithA22. Conceptually,theLevel-3BLASimplementationisnotsignicantlydierentfromtheLevel-1BLASimplementation.TheimprovementintheperformancecomesfromthefactthattheLevel-1BLASCholeskydecompositionisnowperformedinitsentiretyoneachindividualprocessorwhichisassignedaportionofthematrixAij. TheLevel-3BLASalgorithmcanbeconrmedconceptuallybypartitioningAijas whereATLandLTLarekk,and\TL,"\BL,"and\BR"standfor\Top-Left,"\Bottom-Left,"and\Bottom-Right,"respectively.Aswasshownbefore,

PAGE 136

andtheequationsderivedare Itisreadilyseenthataftergoingthrougheachstepinthealgorithm,theremainingdecompositionthatneedstobedoneisinthesubmatrixABR,whereATisoverwrittenbyLTL,ABLisoverwrittenbyLBL,andABRisoverwrittenbyABRLBLLTBL. 5.4 PerformanceoftheParallelCholeskyDecomposition OncethePLAPACKinfrastructurewassuccessfullyinstalled,theparallelCholeskydecompositionwastested.Thetestsimulationswereperformedonaclusterof8nodes,eachhavingtwoPentiumRIIIprocessors.ThersttestperformedwastoverifythattheCholeskydecompositionwasproducingthecorrectresults.Inordertoverifytheresults,thepropertyfromEquation( 5.19 )thatthedecomposedmatrixLijmultipliedbyitstransposeLkjwillreturntheoriginalmatrixAij.ThedierenceinthemagnitudeoftheoriginalmatrixAijandtheonecalculatedfromEquation( 5.19 )wasthendetermined,andwasfoundtobesmallerthan11014.Thiserroriswithinboundsofthemachineprecision. OnceitwasdeterminedthattheparallelCholeskydecompositionwasindeedproducingcorrectresults,itwasthennecessarytotestitsperformancerunningparallelcomputations.Aparallelprogramisgenerallycomposedoftwoparts.Therstispassingdatabetweenprocessors,andthesecondistheactualcalculationsthattheprocessorsperform.Thetimeittakesforaprocessortoperformacalculationismuchfasterthanthetimeittakesforthecomputermemoryto

PAGE 137

beaccessedandtopasstheinformationtoanotherprocessor.Theidealparallelprogramwillminimizetheamountoftimespentinsendinginformationbetweencomputers.InthePLAPACKCholeskydecomposition,therearetwofactorswhichareusedtotunetheprogramsothatitwillrunmoreecientlyinparallel. AsdiscussedinSection 5.3.3 aboutaLevel-3BLASimplementationoftheparallelcode,thematrixcanbesubdividedintoblocksofsizebtoimprovetheeciencyofthecode.Thesizeofbismuchsmallerthanthesizeofthematrixn,butlargeenoughtominimizethenumberoftimesthattheprocessorsmustpassinformation.Inthisimplementation,bisfurthersubdividedintotwofactors:adistributionblocksize(b Simulationswereperformedwithdierentmatrixsizeson16processorstodeterminetheoptimalblockingparameters.Therstimplementationwaswitha20002000matrix,theretheparametersb 5{1 .Inordertoidentifytheoptimalblockingvalues,thesamedataisplottedintwodierentways.Inthegraphontheleft,thetimetodecomposethematrixisplottedasafunctionofthesizeofb

PAGE 138

ParallelCholeskydecompositionsofa20002000matrixwithvaryingb theblockingparametersfromFigure 5{1 areb Similarsimulationswereperformedwith50005000,and1000010000matriceson16processors,whichareshowninFigures 5{2 and 5{3 respectively.Forthematrixwheren=5000,theoptimalblockingparametersb ThenextsetoftestsperformedweretostudytheeectofthenumberofprocessorsonthetimetodecomposethematrixAij.Thevaluesoftheblocking

PAGE 139

ParallelCholeskydecompositionsofa50005000matrixwithvaryingb ParallelCholeskydecompositionsofa1000010000matrixwithvaryingb

PAGE 140

TimetoperformtheparallelCholeskydecompositionasafunctionofthematrixsize,withrunsperformedwithdierentnumbersofproces-sors.Constantvaluesofb parameterswerekeptthesameforalloftherunstofocusontheeectoftheprocessors.Thevaluesoftheblockingparameterschosenwereb 5{4 asafunctionofthematrixsize,witheachlinebeingasetofsimulationrunsonadierentnumberofprocessors.Forthesetofparametersusedinthesesimulationsitappearsthatitisactuallyfastertodecomposethematrixonasingleprocessorifthematrixissmallerthann=500.Thissizecanmostlikelybecutinhalformorewithamoredetailedstudyofwhatsizeblockingparameterstouseformatricesinthatsizerange.Formatriceslargerthann=500,theadvantageofhavingmultipleprocessorsworkingontheproblembecomesreadilyapparent. TheinsetinFigure 5{4 showsthetimetodecomposethematricesonalineartimescale,whichmakesthedierenceintimeeasiertosee.Fora1000010000

PAGE 141

Timerequiredtodecomposedierentsizematricesasafunctionofthenumberofprocessorsused.Multipleprocessorsbecomeasignicantadvantageoncethematrixislargeenough. matrix,thedecompositiontimewith16processorsisaboutfourtimessmallerthanforasingleprocessor.Thistimeadvantagecanalsomostlikelyimprovedwithamoredetailedstudyofthesizeoftheblockingparametersusedinthedecomposition.Figure 5{5 showsaplotofthedecompositiontimeasafunctionofthenumberofprocessorsforvaryingmatrixsizesaboven=1000.Thisgurefurtherillustratesthatasthematrixsizeincreases,thevalueofdoingtheCholeskydecompositioninparallelalsoincreases.Figure 5{5 alsoshowsthatthereappearstobeapointatwhichtheadditionofmoreprocessorstotheproblemdoesnotseemtohelpincreasethespeedofcalculationsignicantly,ascanbeseeninthedierenceinthecalculationtimebetween8processorsand16processors.Improvementsinthecalculationtimecanbemadebyusingmorehighlyoptimizeddistributionblocksizes,andbyimprovementstotheactualparallelcode,butiftheydonotsignicantlyimprovetheprocessingtimebetween8and16processors,thenitwouldbemoreecienttohavetwodierentsimulationsrunningon8processorsratherthanoneon16.

PAGE 142

5.5 Discussion TheultimateobjectiveofperformingtheseparallelcomputationsistoimprovetheperformanceofacomputersimulationofBrownianbersinwhichhydrody-namicinteractionshavebeenincluded.Thesecomputersimulationscanbebrokendownintofourmaincategories: 1. FormationofthemobilitymatrixAij. 2. Calculationofrepulsive/lubricationinteractions. 3. CalculationoftheberdisplacementsthroughtheCholeskydecomposition. 4. Allothercalculationsinvolvedinadvancingthesimulation. Theformationofthemobilitymatrixintherstmaincategoryinthesesimulationsisdonebythemethoddescribedinsection 5.2 .Thetimerequiredtoperformthecalculationsoftheinteractionsbetweenthebersinthesimulationgoesas 2;(5.41) whereTformisthetimetoformthemobilitymatrixAij,Nisthenumberofbersinthesimulation,andC1isaconstantbasedupontheprocessorcalculationspeed.Assuggestedinsection 5.3.1 ,thisprocessshouldscalewellwiththenumberofprocessorsinaparallelimplementation,andsoEquation( 5.41 )wouldgoas 2P;(5.42) wherePisthenumberofprocessors. Thesecondcategoryinthesimulationsisthecalculationoftheshortrangerepulsiveorlubricationinteractionsbetweenthebers.Theseinteractionskeepthebersfromcrossingwhentheyareincloseproximityandareaddedbackintothemobilitymatrixoncetheyarecalculated.Becausethesearepairinteractions,the

PAGE 143

TimetoperformtheparallelCholeskydecompositionasafunctionofthenumberofbers,whereeachbercontributes6ntothematrixsize.Constantvaluesofb calculationtimeshouldbesimilartothatofformingthemobilitymatrix, 2P;(5.43) whereTlubisthecalculationtimefortheinteractions,andC2isaconstantsimilartoC1inEquations( 5.41 )and( 5.42 ). ThethirdcategoryinthesimulationsisthetimerequiredtoperformtheCholeskydecompositionofthemobilitymatrixAijonceithasbeenformedintherstcategoryandhadthelubricationorrepulsiveinteractionsaddedintoit.ThedatafromFigure 5{4 canbepresentedasafunctionofthenumberofbersinthesimulation,aseachberwillcontribute6ntothelengthofthemobilitymatrix.TheresultingplotofthecalculationtimeasafunctionofthenumberofbersinthesimulationisgiveninFigure 5{6

PAGE 144

Thefourthandnalcategoryinthesimulationsisthegroupingofalltheothercalculationsperformedinthesimulationtoadvancethesimulationintime.Thetimerequiredforthesecalculationsgoesas whereC3isaconstantsimilartotheotherconstantsC1andC2.AddingEquations( 5.42 ),( 5.43 ),and( 5.44 )alongwiththetimetodecomposethemobilitymatrixTdecompfromFigure 5{6 givesthetotaltimetocalculateonedisplacementofthebers 2P+C2N(N1) 2P+TdecompON3+C3N; whereTsimisthetotalsimulationtimeforasingledisplacement.AlloftheO(N)calculationscanbeessentiallyignoredforlargesimulationssincetheO(N2)andO(N3)processeswilldominate.Thetotalsimulationtimethenbecomes whereCisasumoftheconstantsC1andC2.Forsimulationsathighconcen-trations,theCholeskydecompositionportionofthecodewillclearlydominatethecalculationtime,unlesstheconstantCislarge.Anyimprovementintheper-formanceofthecalculationofthepairinteractionswillonlyhelptoimprovetheoveralleciencyandperformanceoftheparallelcode,andsoshouldbeincluded. ThesimulationsperformedinChapter 3 atthehighestconcentrationsrequiredthatthesimulationswererunforatotaldimensionlesstimeof1.Ifadimensionlesstimestepof5106isused,thisresultsinasimulationwhichmustperforma

PAGE 145

totalof200;000steps.InordertocomparetheresultsofsimulationswhichincludehydrodynamicinteractionswiththoseinChapter 3 ,adimensionlessnumberdensityofnL3=150mustbesimulated,wherenisthenumberdensity(numberofbersperunitvolume),andListheberlength.Thesmallestperiodicboxsizewhatmaybeusedinthesimulationsis1:1Linsize,whichcorrespondstoasimulationwith200bers.Figure 5{6 showsthatforsimulationswithunder100bersthereisnoadvantagetoperformingtheCholeskydecompositioninparallel.Forsimulationswithmorethan100bers,parallelcomputationsbegintoprovideanincreaseintheoverallperformance. Asapreliminarycalculation,ifthetwoO(N2)calculationsareassumedtorequirethesameorderofmagnitudeoftimetoperformforasuspensionof200bers,thenthetotalsimulationtimeforasinglestepwouldbeontheorderofabout2to3secondsasseeninFigure 5{6 .If200;000stepsaretaken,thismeansthatthesimulationwouldrequireatotalofbetween4and7daystoperform.Usingthisanalysis,thelargestsuspensionthatcouldbereasonablysimulatedwouldhavetotakenolongerthanabout13secondsperstep,whichwouldresultinatotalsimulationtimeof30days.IfthetimetoperformtheCholeskydecompositionis1=3ofthetotaltime,thena4:3secondtimewouldcorrespondtoamaximumnumberofbersimulatedofjustover400forasingleprocessor,andalmost800for8ormoreprocessors. 5.6 Conclusion IthasbeenshownthatbyperformingtheCholeskydecompositionofamatrixinaparallelenvironmentforsucientlylargematricesyieldsaperformanceincrease.Thesizeoftheblocksofdatasenttothedierentprocessorshasasignicantimpactontheoveralltimeneededtoperformthedecomposition,where,foranoptimallysizeddistributionblock,thecalculationtimecanimprovebyovertwoordersofmagnitude.Theseblocksizesarespecictothearchitecture

PAGE 146

beingusedtodecomposethematrix(processortype,amountofmemory,speedofcommunication)aswellasthematrixsize,andsomustbeindividuallydeterminedonthatarchitectureforwhateversizematrixisbeingusedforthesimulations. TheresultspresentedhereshowthatitisprobablynotworthittouseaparallelimplementationoftheCholeskydecompositionforsuspensionswithfewerthan100bers,althoughimplementationofaparallelcodetocalculatethepairinteractionsmightlowerthisvalue.Forsimulationswithmorethan100berstherearesubstantialtimesavingsfromusingaparallelcode.Withfurtherdevelopment,simulationsoflargesuspensionofBrownianberswithhydrodynamicinteractionsisadenitepossibility.

PAGE 147

Theworkpresentedinthisdissertationsignicantlyadvancesknowledgeofthedynamicsofsuspensionsofrigidandsemi-rigidBrownianbers.Theimportanceofthesesystemsishighlightedbytheirwidespreaduseinexistingandemergingtechnologies,withexamples(seeChapter 1 )includingcarbonnanorodsandnanotubes[ 22 ],compositematerials[ 9 ],andviruses[ 11 ]andothermacromoleculesofbiologicalorigin.Theequilibriumpropertiesanddynamicsofsuspensionsofbershavebeenstudiedforovertwodecades,butarestillnotclearlyunderstood.Experimentaltechniquesforcharacterizingthesesuspensionsexist,suchasdynamiclightscattering[ 20 23 ]andelectricbirefringence[ 25 ],yetinterpretingthesemeasurementsremainsproblematic.Theproblemsincludenotonlyinterpretingspecicmeasuredvalues,butalsothescalingbehavior. TherstmajorcontributionoftheworkpresentedinthisdissertationregardsthescalingbehavioroftherotationaldiusivityDRwithrespecttothedimen-sionlessnumberdensitynL3.Competingtheoriesonthescalingfortherotationaldiusivitieshaveexistedintheliteratureforthepasttwentyyears.Doi[ 7 ]initiallyproposeda\tube"theorypredictingascalingof forinnitelythinrodsatsucientlyhighconcentrations,whereDR0istheshort-timevalueoftherotationaldiusivity.Baseduponaphysicalpictureofaproberodconnedwithinastaticcageofneighboringbers,thescalingwasconrmedthroughcomputersimulationsbyDoi[ 3 ].LatersimulationsbyFixman[ 34 35 ] 134

PAGE 148

producedascalingof andFixmanconsequentlyproposedanalternativetheorybasedupona\coopera-tiverotation"mechanism.Asjusticationforthediscrepancy,Fixman[ 35 ]statedthatthesimulationsofDoi[ 3 ]usedatimestepthatwastoolargetoaccuratelycapturethedynamicsofthesystem.Therehasbeenanongoingdisagreementintheliteratureaboutwhichscalingiscorrect,asdiscussedinChapter 3 Theworkpresentedinthisdissertationresolvesthiscontroversy.Bothpower-lawscalingsof1and2canbereproducedwithinaccuratesimulationsaspresentedinChapter 3 .Simulationsofrigidberswithdierenthydrodynamicmodels,orratiosofshort-timerotationaltotranslationaldiusivities(L2DR0=DT0),producepower-lawscalingswithdierentexponents, rangingbetween0:25to2.Doi[ 7 ]andFixman[ 34 35 ]chosedierentvaluesofL2DR0=DT0,consequentlythesourceoftheirdiscrepancywasthefailuretoconsiderthisimportantratio,ratherthananumericalproblemasassertedbyFixman[ 35 ].Theaspectratio,A,alsoplaysapivotalroleinthescalings,withastrongvariationbetweenthevaluesofforrodswithAlessthan50;aspectratiosofA50orhighercanbeassumedinnitelythin. Aspartofperformingsimulations,adetailedanalysisofthenumericalintegrationwasperformedandimprovementsonexistingmethodswerefound.AnEulermethodwithanItocorrection[ 56 ]waschosenasthenumericalintegrationmethodforthesimulationsofrigidBrownianbers,andtheerrorwasanalyzedinChapter 2 andcomparedtothatofanestablishednumericaltechniqueusedbyLowen[ 2 ].Thisanalysisproducedanewnumericalalgorithmwithsignicantlylesserrorwhileatthesametimebeingmoreecient.

PAGE 149

Theworkpresentedinthisdissertationalsoadvancestheunderstandingofsystemsofsemi-rigidrods.Comparisonsoftheoriestoexperimentalresultsiscomplicatedbythefactthatmostrodpolymersandcolloidsarenotperfectlyrigid[ 6 23 20 ].Althoughtheoriesexist[ 23 ]forchangesduetodeviationfromrigidity,untilnownosimulationshavebeenperformedcomparingthescalingresultsofbothrigidandsemi-rigidrods.Theadditionofexibilitycausesadelayinthenumberdensityatwhichtherodsbecomesignicantlyhinderedbytheirneighborsandenterthesemi-diluteconcentrationregimewhereastrongdecreaseintherotationaldiusivityoccurs(seeChapter 4 ).Oncewithinthesemi-diluteconcentrationregime,thepower-lawscalingsofthesemi-rigidrodscloselymatchedthoseoftherigidrodswiththesamecorrespondinghydrodynamicmodelandaspectratio. Animportantconclusionresultsfromcomparingsimulationsofrigidandsemi-rigidrodsuspensionstoexperimentalresults:dierentmicro-mechanicalmodelscanproduceresultsoverlimitedconcentrationregimeswhichareindis-tinguishablewithinmeasurementcapabilities.AnexamplewasgiveninFigure 4{10 showingacomparisonofrotationaldiusivitiesmeasuredinexperimentsonPBLG[ 71 ]tosimulationresultsusingarigidmodelandasemi-rigidmodel.Overtheconcentrationrangeshown,therotationaldiusivitiesfromthetwomodelsreproducetheexperimentalresults.Thisresultsuggeststhatpropermodelscannotbedistinguishedbasedsolelyontheirrotationaldiusivities,soothermeasures,suchascenterofmassdiusivities,mustbeusedtoidentifytheappropriatemodel. Asidefromtheissueofrigidversussemi-rigidrodsisthequestionofwhichhydrodynamicmodelisbestforcomparingtoexperimentalresultsonrodsystems.Forthinrodsatdiluteconcentrations,theslenderbodymodelisthemostaccuratemodeltouseinsimulationssincethehydrodynamicresistanceisdistributedcontinuouslyacrosstherod,ratherthanlocalizedatarbitrarypointsalongtherodaswithbead-rodmodels.Atsemi-diluteconcentrations,theshort-timediusivities

PAGE 150

arenotequivalenttothedilutediusivitiesasemployedintheBrownianDynamicssimulationsinChapters 3 and 4 .Infact,theshort-timediusivityisconcentrationdependentduetohydrodynamicinteractionsbetweentherods.Inthelimitofinniteaspectratio,hydrodynamicinteractionshavenoeectonthesuspensionofrods.Consequentlyforthistheoreticalsituation,thescalingof=2asoriginallyproposedbyDoi[ 7 ]iscorrect.Forrealrodsystemswithlarge,butnotinnite,aspectratios,hydrodynamicinteractionshaveameasurableeectonthedynamics.Thiseectcomesfromtheratiooftheincreasedresistanceduetothepresenceoftheotherrodstothehydrodynamicresistanceofasinglerodwhichscalesasln(2A);thisfactordecaysveryslowlywithincreasingA,consequentlyincludinghydrodynamicinteractionsisnecessary. Thus,hydrodynamicinteractionswillbeaddedtofurtherimproveontheworkpresentedinthisdissertation.AnalgorithmforincludinghydrodynamicinteractionshasbeenoutlinedinChapter 5 .Thoughconceptuallystraightforward,thekeyproblemtorealizingasimulationwithhydrodynamicinteractionsisthecomputationalexpense.Therefore,ananalysisofthecomputationalloadinaparallelenvironmenthasbeenmadeandthecostsandbenetshavebeencomparedtothoseofperformingthesimulationsinaserialenvironment.Themajortasksareformationofthemobilitymatrix,whichshouldscalewiththenumberofprocessors,andthedecompositionofthemobilitymatrix.Forthelattertask,aparallelCholeskydecompositionwassuccessfullyimplemented.SimulationsoflargesuspensionsofBrownianberswithhydrodynamicinteractionsarefeasible;theresultsofthesesimulationswillprovidefurtherandimprovedinsightsintothedynamicsofrod-likeparticlesinsuspension.

PAGE 151

[1] S.R.WilliamsandA.P.Philipse,\RandomPackingsofSpheresandSpherocylindersSimulatedbyMechanicalContraction,"Phys.Rev.E.,vol.67,pp.051301,2003. [2] H.Lowen,\BrownianDynamicsofHardSpherocylinders,"Phys.Rev.E,vol.50,pp.1232{1243,1994. [3] M.Doi,I.Yamamoto,andF.Kano,\Monte-CarloSimulationoftheDynamicsofThinRodlikePolymersinConcentrated-Solution,"J.Phys.Soc.Jpn.,vol.53,pp.3000{3003,1984. [4] G.Szamel,\ReptationasaDynamicMean-FieldTheory:StudyofaSimpleModelofRodlikePolymers,"Phys.Rev.Lett.,vol.70,pp.3744{3747,1993. [5] P.D.CobbandJ.E.Butler,\SimulationsofConcentratedSuspensionsofRigidFibers:RelationshipBetweenShort-TimeDiusivitiesandtheLong-TimeRotationalDiusion,"J.Chem.Phys.,vol.123,pp.054908,2005. [6] M.DoiandS.F.Edwards,TheTheoryofPolymerDynamics,OxfordUniversityPress,NewYork,NY,1986. [7] M.Doi,\RotationalRelaxation-TimeofRod-LikeMacromoleculesinConcentratedSolution,"J.Physique,vol.36,pp.607{611,1975. [8] A.M.WierengaandA.P.Philipse,\Low-shearViscosityofIsotropicDispersionsof(Brownian)RodsandFibres;AReviewofTheoryandExperiments,"Coll.Surf.A:Physicochem.Eng.Aspects,vol.137,pp.355{372,1998. [9] J.Gustin,A.Joneson,M.Mahinfalah,andJ.Stone,\LowVelocityImpactofCombinationKevlar/CarbonFiberSandwichComposites,"Comp.Struct.,vol.69,pp.396{406,2005. [10] W.W.Adams,R.K.Eby,andD.E.McLemore,TheMaterialsScienceandEngineeringofRigid-RodPolymers,MaterialsResearchSociety:Pittsburgh,Pittsburgh,PA,1989. [11] \Tobaccomosaicvirus,"Wikipedia,thefreeencyclopedia(http://en.wikipedia.org/wiki/Tobacco mosaic virus),March2006. 138

PAGE 152

[12] J.Newman,H.L.Swinney,S.A.Berkowitz,andL.A.Day,\HydrodynamicPropertiesandMolecularWeightoffdBacteriophageDNA,"Biochemistry,vol.13,pp.4832{4838,1974. [13] M.M.Tirado,C.L.Martinez,andJ.G.Delatorre,\ComparisonofTheoriesfortheTranslationalandRotationalDiusion-CoecientsofRod-LikeMacromolecules-ApplicationtoShortDNAFragments,"J.Chem.Phys.,vol.81,pp.2047{2052,1984. [14] L.Wang,M.M.Garner,andH.Yu,\Self-DiusionandCooperativeDiusionofaRodlikeDNAFragment,"Macromolecules,vol.24,pp.2368{2376,1991. [15] F.G.Schmidt,F.Ziemann,andE.Sackmann,\ShearFieldMappinginActinNetworksbyusingMagneticTweezers,"Euro.Biophys.J.Biophys.Lett.,vol.24,pp.348{353,1996. [16] Y.R.OhandO.O.Park,\TransientFlowBirefringenceofCalfSkinCollagenSolutions,"J.Chem.Eng.Japan,vol.25,pp.243{249,1992. [17] K.ClaireandR.Pecora,\TranslationalandRotationalDynamicsofCollageninDiluteSolution,"J.Phys.Chem.B,vol.101,pp.746{753,1997. [18] F.C.Chen,G.Koopmans,R.L.Wiseman,L.A.Day,andH.L.Swinney,\DimensionsofXfVirusfromitsRotationalandTranslationalDiusionCoecients,"Biochemistry,vol.19,pp.1373{1376,1980. [19] M.A.TracyandR.Pecora,\DynamicsofRigidandSemirigidRodlikePolymers,"Ann.Rev.Phys.Chem.,vol.43,pp.525{557,1992. [20] R.C.CushandP.S.Russo,\Self-DiusionofaRodlikeVirusintheIsotropicPhase,"Macromolecules,vol.35,pp.8659{8662,2002. [21] W.U.Huynh,J.J.Dittmer,andA.P.Alivasatos,\HybridNanorod-PolymerSolarCells,"Science,vol.295,pp.2425{2427,2002. [22] B.Vigolo,A.Penicaud,C.Coulon,C.Sauder,R.Pailler,C.Journet,P.Bernier,andP.Poulin,\MacroscopicFibersandRibbonsofOrientedCarbonNanotubes,"Science,vol.290,pp.1331{1334,2000. [23] G.T.KeepandR.Pecora,\DynamicsofRodlikeMacromoleculesinNondiluteSolutions-Poly(Normal-alkylIsocyanates),"Macromolecules,vol.21,pp.817{829,1988. [24] Z.Bu,P.S.Russo,D.L.Tipton,andI.I.Negulescu,\Self-DiusionofRodlikePolymersinIsotropicSolutions,"Macromolecules,vol.27,pp.6871{6882,1994.

PAGE 153

[25] J.K.Phalakornkul,A.P.Gast,andR.Pecora,\RotationalandTranslationalDynamicsofRodlikePolymers:ACombinedTransientElectricBirefrin-genceandDynamicLightScatteringStudy,"Macromolecules,vol.32,pp.3122{3135,1999. [26] F.Pignon,A.Magnin,andJ.Piau,\TheOrientationDynamicsofRigidRodSuspensionsunderExtensionalFlow,"J.Rheol.,vol.47,pp.371{388,2003. [27] M.P.B.vanBruggen,H.N.W.Lekkerkerker,andJ.K.G.Dhont,\Long-TimeTranslationalSelf-DiusioninIsotropicDispersionsofColloidalRods,"Phys.Rev.E,vol.56,pp.4394{4403,1997. [28] M.P.B.vanBruggen,H.N.W.Lekkerkerker,G.Maret,andJ.K.G.Dhont,\Long-TimeTranslationalSelf-DiusioninIsotropicandNematicDispersionsofColloidalRods,"Phys.Rev.E,vol.58,pp.7668{7677,1998. [29] S.G.J.M.Kluijtmans,G.H.Koenderink,andA.P.Philipse,\Self-DiusionandSedimentationofTracerSpheresin(Semi)DiluteDispersionsofRigidColloidalRods,"Phys.Rev.E,vol.61,pp.626{636,2000. [30] M.A.TracyandR.Pecora,\Synthesis,Characterization,andDynamicsofaRod/SphereCompositeLiquid,"Macromolecules,vol.25,pp.337{349,1992. [31] K.Kang,J.Gapinski,M.P.Lettinga,J.Buitenhuis,G.Meier,M.Ratajczyk,J.K.G.Dhont,andA.Patkowski,\DiusionofSpheresinCrowdedSuspensionsofRods,"J.Chem.Phys.,vol.122,pp.044905,2005. [32] L.Helden,R.Roth,G.H.Koenderink,P.Leiderer,andC.Bechinger,\DirectMeasurementofEntropicForcesInducedbyRigidRods,"Phys.Rev.Lett.,vol.90,pp.048301,2003. [33] B.HerzhaftandE.Guazzelli,\ExperimentalStudyoftheSedimentationofDiluteandSemi-DiluteSuspensionsofFibres,"J.FluidMech.,vol.384,pp.133{158,1999. [34] M.Fixman,\EntanglementsofSemidilutePolymerRods,"Phys.Rev.Lett.,vol.54,pp.337{339,1985. [35] M.Fixman,\DynamicsofSemidilutePolymerRods-AnAlternativetoCages,"Phys.Rev.Lett.,vol.55,pp.2429{2432,1985. [36] I.Teraoka,N.Ookubo,andR.Hayakawa,\MolecularTheoryontheEntanglementEectofRodlikePolymers,"Phys.Rev.Lett.,vol.55,pp.2712{2715,1985. [37] I.TeraokaandR.Hayakawa,\TheoryofDynamicsofEntangledRod-LikePolymersbyuseofaMean-FieldGreenFunctionFormulation.II.RotationalDiusion,"J.Chem.Phys.,vol.91,pp.2643{2648,1989.

PAGE 154

[38] I.Bitsanis,H.T.Davis,andM.Tirrell,\BrownianDynamicsofNondiluteSolutionsofRodlikePolymers.1.LowConcentrations,"Macromolecules,vol.21,pp.2824{2835,1988. [39] I.Bitsanis,H.T.Davis,andM.Tirrell,\BrownianDynamicsofNondiluteSolutionsofRodlikePolymers.2.HighConcentrations,"Macromolecules,vol.23,pp.1157{1165,1990. [40] G.MarrucciandN.Grizzuti,\TheFree-EnergyFunctionoftheDoi-EdwardsTheory-AnalysisoftheInstabilitiesinStress-Relaxation,"J.ofRheol.,vol.27,pp.433{450,1983. [41] J.K.Moscicki,\OntheRotationalDiusionofRod-LikeMacromoleculesinLypotropicMesomorphicPhases,"Mol.Phys.,vol.51,pp.919{933,1984. [42] G.K.Batchelor,\Slender-BodyTheoryforParticlesofArbitraryCross-SectioninStokesFlow,"J.FluidMech.,vol.44,pp.419{440,1970. [43] R.G.Cox,\TheMotionofLongSlenderBodiesinaViscousFluid.Part1.GeneralTheory,"J.FluidMech.,vol.44,pp.791{810,1970. [44] R.G.Cox,\TheMotionofLongSlenderBodiesinaViscousFluid.Part2.ShearFlow,"J.FluidMech.,vol.45,pp.625{657,1971. [45] J.SeilsandR.Pecora,\Dynamicsofa2311BasePairSuperhelicalDNAinDiluteandSemidiluteSolutions,"Macromolecules,vol.28,pp.661{673,1995. [46] A.ImmaneniandA.J.Mchugh,\TheEectofConcentration,Temperature,andMolecularWeightontheDynamicsofRigid-RodMoleculesinSemi-DiluteSolutions,"J.Poly.Sci.:PartB:Poly.Phys.,vol.36,pp.181{190,1998. [47] G.Petekidis,D.Vlassopoulos,andG.Fytas,\DynamicsofHairy-RodPolymers:SemidiluteRegime,"Macromolecules,vol.31,pp.1406{1417,1998. [48] R.Aris,Vectors,Tensors,andtheBasicEquationofFluidMechanics,DoverPublicationsInc.,NewYork,NY,1962. [49] H.Brenner,\RheologyofaDiluteSuspensionofAxisymmetricBrownianParticles,"Int.J.MultiphaseFlow,vol.1,pp.195{341,1974. [50] D.A.McQuarriee,StatisticalMechanics,UniversityScienceBooks,Sausalito,CA,2000. [51] W.H.Press,S.A.Teukolsky,W.T.Vetterling,andB.P.Flannery,Numer-icalRecipesinFORTRAN:TheArtofScienticComputing,CambridgeUniversityPress,NewYork,NY,1992.

PAGE 155

[52] H.C.Ottinger,\VelocityFieldinNondrainingPolymerChains,"RheologicaActa,vol.35,pp.134{138,1996. [53] R.M.Jendrejack,M.D.Graham,andJ.J.dePablo,\HydrodynamicInter-actionsinLongChainPolymers:ApplicationoftheChebyshevPolynomialApproximationinStochasticSimulations,"J.Chem.Phys.,vol.113,pp.2894{2900,2000. [54] M.Fixman,\ImplicitAlgorithmForBrownianDynamicsofPolymers,"Macromolecules,vol.19,pp.1195{1204,1986. [55] P.E.KloedenandE.Platen,\Higher-OrderImplicitStrongNumericalSchemesforStochasticDierential-Equations,"J.Stat.Phys.,vol.66,pp.283{314,1992. [56] D.L.ErmakandJ.A.McCammon,\BrownianDynamicswithHydrody-namicInteractions,"J.Chem.Phys.,vol.69,pp.1352{1360,1978. [57] T.N.Phung,J.F.Brady,andG.Bossis,\StokesianDynamicsSimulationofBrownianSuspensions,"J.FluidMech.,vol.313,pp.181{207,1996. [58] D.R.FossandJ.F.Brady,\Self-DiusioninShearedSuspensionsbyDynamicSimulation,"J.FluidMech.,vol.401,pp.243{274,1999. [59] B.Oksendal,StochasticDierentialEquations,Springer,Dover,DE,1985. [60] P.S.Grassia,E.J.Hinch,andL.C.Nitsche,\ComputerSimulationsofBrownianMotionofComplexSystems,"J.FluidMech.,vol.282,pp.373{403,1995. [61] P.GrassiaandE.J.Hinch,\ComputerSimulationsofPolymerChainRelaxationviaBrownianMotion,"J.FluidMech.,vol.308,pp.255{288,1996. [62] M.Fixman,\SimulationofPolymerDynamics.1.GeneralTheory,"J.Chem.Phys.,vol.69,pp.1527{1537,1978. [63] P.Jiang,J.F.Berone,andV.L.Colvin,\ALost-WaxApproachtoMonodisperseDolloidsandTheirCrystals,"Science,vol.291,pp.453{457,2001. [64] V.M.CepakandC.R.Martin,\PreparationofPolymericMicro-andNanostructuresUsingaTemplate-BasedDepositionMethod,"Chem.Mater.,vol.11,pp.1363{1367,1999. [65] T.Kirchho,H.Lowen,andR.Klein,\DynamicalCorrelationsinSus-pensionsofChargedRodlikeMacromolecules,"Phys.Rev.E,vol.53,pp.5011{5023,1996.

PAGE 156

[66] J.E.ButlerandE.S.G.Shaqfeh,\DynamicSimulationsoftheInhomoge-neousSedimentationofRigidFibres,"J.FluidMech.,vol.468,pp.205{237,2002. [67] D.FrenkelandJ.F.Maguire,\Molecular-DynamicsStudyoftheDynamicalPropertiesofanAssemblyofInnitelyThinHard-Rods,"Mol.Phys.,vol.49,pp.503{541,1983. [68] G.T.KeepandR.Pecora,\ReevaluationoftheDynamic-ModelforRotationalDiusionofThin,RigidRodsinSemidilutePolymerRods,"Phys.Rev.Lett.,vol.18,pp.1167{1173,1985. [69] M.DoiandS.F.Edwards,\DynamicsofRod-likeMacromoleculesinConcentrated-Solution.1,"J.Chem.Soc.,FaradayTrans.,vol.74,pp.560{570,1978. [70] T.SatoandA.Teramoto,\DynamicsofSti-ChainPolymersinIsotropicSolution:Zero-ShearViscosityofRodlikePolymers,"Macromolecules,vol.24,pp.193{196,1991. [71] Y.Mori,N.Ookubo,R.Hayakawa,andY.Wada,\RelaxationalBehaviorofBirefringenceofAqueousCarboxymethylcelluloseunderanAlternatingElectricFieldatFrequenciesRangingfrom0.1Hzto100kHz,"J.Polym.Sci.Polym.Phys.Ed.,vol.20,pp.2111,1982. [72] S.Broersma,\RotationalDiusionConstantofaCylindricalParticle,"J.Chem.Phys.,vol.32,pp.1626{1631,1960. [73] S.Broersma,\ViscousForceConstantforaClosedCylinder,"J.Chem.Phys.,vol.32,pp.1632{1635,1960. [74] M.Medinanoyola,\Long-TimeSelf-DiusioninConcentratedColloidalDispersions,"Phys.Rev.Lett.,vol.60,pp.2705{2708,1988. [75] I.L.ClaeysandJ.F.Brady,\SuspensionsofProlateSpheroidsinStokesFlow.Part2.StatisticallyHomogeneousDispersions,"J.FluidMech.,vol.251,pp.443{477,1993. [76] M.B.MackaplowandE.S.G.Shaqfeh,\ANumericalStudyoftheRheo-logicalPropertiesofSuspensionsofRigid,Non-BrownianFibres,"J.FluidMech.,vol.329,pp.155{186,1996. [77] O.G.Harlen,R.R.Sundararajakumar,andD.L.Koch,\NumericalSimulationsofaSphereSettlingThroughaSuspensionofNeutrallyBuoyantFibres,"J.FluidMech.,vol.388,pp.355{388,1999. [78] M.B.Mackaplow,AStudyoftheTransportPropertiesandSedimentationCharacteristicsofFiberSuspensions,Ph.D.dissertation,StanfordUniversity,October1995.

PAGE 157

[79] J.HappelandH.Brenner,LowReynoldsNumberHydrodynamics,PrenticeHall,Stoneham,MA,1965. [80] A.Oberbeck,\UeberStationareFlussigkeitsbewegungenmitBerucksicthi-gungderInnerenReibung(OnSteadyStateFluidFlowandtheCalculationoftheDrag),"J.ReineAngew.Math.,vol.81,pp.62{80,1876. [81] I.TeraokaandR.Hayakawa,\TheoryofDynamicsofEntangledRod-LikePolymersbyuseofaMean-FieldGreenFunctionFormulation.I.TransverseDiusion,"J.Chem.Phys.,vol.89,pp.6989{6995,1988. [82] J.E.ButlerandE.S.G.Shaqfeh,\BrownianDynamicsSimulationsofaFlexiblePolymerChainWhichIncludesContinuousResistanceandMultibodyHydrodynamicInteractions,"J.Chem.Phys.,vol.122,pp.014901,2005. [83] J.E.Butler,\BrownianDynamicsofMicro-MechanicalModelswithRigidConstraintsBetweenSubunitswithCongurationallyDependentMassandMobilityMatrices,"J.FluidMech.(Submitted),2005. [84] E.J.Hinch,\BrownianMotionwithStiBondsandRigidConstraints,"J.FluidMech.,vol.271,pp.219{234,1994. [85] H.YuandW.H.Stockmayer,\IntrinsicViscosityofaOnce-BrokenRod,"J.Chem.Phys,vol.47,pp.1369{1373,1967. [86] D.C.Morse,\TheoryofConstrainedBrownianMotion,"Adv.Chem.Phys.,vol.128,pp.65{189,2004. [87] M.Pasquali,V.Shankar,andD.C.Morse,\ViscoelasticityofDiluteSolutionsofSemiexiblePolymers,"Phys.Rev.E,vol.64,pp.020802,2001. [88] E.KlavenessandA.Elgsaeter,\BrownianDynamicsofBead-Rod-Nugget-SpringPolymerChainswithHydrodynamicInteractions,"J.Chem.Phys.,vol.110,pp.11608,1999. [89] T.Odjik,\OntheStatisticsandDynamicsofConnedorEntangledStiPolymers,"Macromolecules,vol.16,pp.1340{1344,1983. [90] D.Thirumalai,\RotationalRelaxationofaCylinderandaSemirigidMoleculeinConcentratedSolutions,"J.Chem.Phys,vol.98,pp.9265{9269,1994. [91] P.D.CobbandJ.E.Butler,\SimulationsofConcentratedSuspensionsofSemi-RigidFibers:EectofFlexibilityontheRotationalDiusivity,"Macromolecules,vol.39,pp.886{892,2006.

PAGE 158

[92] H.Hasimoto,\OnthePeriodicFundamentalSolutionsoftheStokesEquationsandtheirApplicationtoViscousFlowPastaCubicArrayofSpheres,"J.FluidMech.,vol.5,pp.317{328,1959. [93] M.B.MackaplowandE.S.G.Shaqfeh,\ANumericalStudyoftheSed-imentationofFibreSuspensions,"J.FluidMech.,vol.376,pp.149{182,1998. [94] L.Durlofsky,J.F.Brady,andG.Bossis,\DynamicSimulationofHydro-dynamicallyInteractingParticles,"J.FluidMech.,vol.180,pp.21{49,1987. [95] W.Gropp,E.Lusk,andA.Skjellum,UsingMPI,TheMITPress,London,England,1994. [96] M.Snir,S.W.Otto,S.Huss-Lederman,D.W.Walker,andJ.J.Dongarra,MPI:TheCompleteReference,TheMITPress,London,England,1996. [97] J.EdwardsandJ.A.Tomlin,\ParallelCholeskyFactorisation,"inFifthAustralianSupercomputingConference,1992,pp.105{114. [98] D.ZhengandT.Y.P.Chang,\ParallelCholeskyMethodonMIMDwithSharedMemory,"Comp.Struct.,vol.56,pp.25{38,1995. [99] A.V.Gerbessiotis,\PracticalConsiderationsofParallelSimulationsandArchitectureIndependentParallelAlgorithmDesign,"J.Par.Dist.Comp.,vol.53,pp.1{25,1998. [100] P.P.ChuandE.E.Santos,\ParallelLoopingAlgorithmsforCholeskyDecomposition,"Par.Dist.Comp.andSys.,vol.342,pp.419{424,2001. [101] E.E.SantosandP.P.Chu,\EcientandOptimalParallelAlgorithmsforCholeskyDecomposition,"J.Math.Mod.andAlg.,vol.2,pp.217{234,2003. [102] R.A.vandeGeijn,UsingPLAPACK,TheMITPress,London,England,1997. [103] \GotoBLAS,"TexasAdvancedComputingCenter:SoftwareandTools(http://www.tacc.utexas.edu/resources/software/),March2006. [104] C.L.Lawson,R.J.Hanson,D.R.Kincaid,andF.T.Krogh,\BasicLinearAlgebraSubprogramsforFORTRANusage,"ACMTrans.Math.Soft.,vol.5,pp.308{323,1979. [105] J.J.Dongarra,J.DuCroz,S.Hammarling,andR.J.Hanson,\AnExtendedSetofFORTRANBasicLinearAlgebraSubprograms,"ACMTrans.Math.Soft.,vol.14,pp.1{17,1988.

PAGE 159

[106] J.J.Dongarra,J.DuCroz,S.Hammarling,andI.Du,\ASetofLevel3BasicLinearAlgebraSubprograms,"ACMTrans.Math.Soft.,vol.16,pp.1{17,1990.

PAGE 160

PhilipDanielCobbwasborninthecityofTegucigalpa,Honduras,onApril16,1978,toStephenandHelenCobb,whowereChristianmissionariesthere.PhilipattendedAcademiaPinaresfromkindergartenthroughsecondgrade.TheCobbfamilythenmovedtoQuito,Ecuador.Philipwashome-schooledfromthirdgradethroughsixthgrade.PhilipwasthenacceptedintoAllianceAcademy,whereheattendedfromseventhgradethroughtwelfthgrade.PhilipgraduatedthirdinhisclassofthirtyfourstudentsandchosetoattendcollegeatFloridaInstituteofTechnologyinMelbourne,Florida.PhilipgraduatedfromFloridaInstituteofTechnologywithabachelorofsci-encedegreeinchemicalengineeringwithhighesthonors.WhilehewasatFloridaInstituteofTechnology,hewasheavilyinvolvedwithInterVarsityChristianFellow-ship,TauKappaEpsilonnationalsocialfraternity,andtheStudentAmbassadors.PhilipwasalsoaResidentAdvisorforhisnalyearatFloridaTech.Philipalsohadtheopportunitytoworkforfoursemestersinaco-oppositionatNASA,KennedySpaceCenter,whereheperformedheattransferstudiesandhelpeddesignandautomatetestingequipment.UpongraduatingfromFloridaTech,Philipcontinuedhiseducationingrad-uateschoolattheUniversityofFlorida.PhilipworkedinDr.JasonButler'sresearchgroup,wherehisresearchfocusedonperformingcomputersimulationsofrigidandsemi-rigidpolymerbersuspensions.NowPhiliplooksforwardtothenextchapterinhislifeashetakestheknowledgeandexperiencethathehasgainedinearningaPh.D.inchemicalengineeringintotheworkplace. 147


xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20110218_AAAACY INGEST_TIME 2011-02-18T18:21:28Z PACKAGE UFE0013662_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 8711878 DFID F20110218_AABXQG ORIGIN DEPOSITOR PATH cobb_p_Page_156.tif GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
dcd8ce272547eaec8e1460e302d962d4
SHA-1
dab636b3874f7558f28f98015fe8195350f38fb2
F20110218_AABXPR cobb_p_Page_128.tif
9d2e07798861f7e5f7011466a7297479
30dc36e56dfe959addca0a4a21484ac8f28e4551
F20110218_AABXQH cobb_p_Page_157.tif
90f4b1b3104fa50416cba699a2939d83
3bc715b1c014c27457d7ee13d1b3f803dbf7bca3
F20110218_AABXPS cobb_p_Page_130.tif
c8eba3c30412a4cee4f012bdcd9aee36
d315c6a6f18883e77eacf71008ac584153318f7c
F20110218_AABXPT cobb_p_Page_132.tif
0f0c191f252ddc2bad7039ffa5d94b29
ecaf31db71af978d6356325d1761d72b288cda9a
F20110218_AABXQI cobb_p_Page_158.tif
ac994b85a1adbe0260947daad945f92f
fe742fbfd8cafd4663dfccaae716d5ebeea53804
F20110218_AABXPU cobb_p_Page_133.tif
a6932c6fcc52d52122a8ab1ea9355192
b8a7fb0050aea228c51b4801b2d0a75afa01a89f
F20110218_AABXQJ cobb_p_Page_160.tif
28aeed22d9cb98c9468b52647711f3dd
7aa235e26ddf58cbf1a547f80e08a0b590306ae4
F20110218_AABXPV cobb_p_Page_137.tif
c6cddc5dbb8f3dd966e26dd42cb32c5b
370e733ade7658715fb5de8ab693757acb88427b
451 F20110218_AABXQK cobb_p_Page_001.txt
ca8b9f645b41588cf574c7b42da63225
ebbc7636b69344fa1cba78e2d6ea911219a6679e
F20110218_AABXPW cobb_p_Page_138.tif
ea9a03708a0b2ca44a52554fcce6265d
496f7976191c4b8951aab6b01bd6be21125a8e52
1822 F20110218_AABXRA cobb_p_Page_040.txt
303fb6d88e8dc3b54d0e028ce382de25
7eafa05748f47f03a801a3512f81cb9f86215941
109 F20110218_AABXQL cobb_p_Page_002.txt
edc0ef6756833a5e6894c8b317f4ded7
4e5ce9dca81cefbad2aab3ad5c7e7ed7257a61ae
F20110218_AABXPX cobb_p_Page_139.tif
b72227c27839cb63be60651c898daeef
43cd452efd2dc343d2ea54bc81279ddc0bf8b64d
2009 F20110218_AABXRB cobb_p_Page_041.txt
35cb8c4fb8635e4849e341294228955f
1ea1b1dcadfa2b33a7c1e8ea060cf03c2c53f9ce
1318 F20110218_AABXQM cobb_p_Page_003.txt
7379107b5ec924329f2789058103ee71
de8e8bc482a1edbd13ecb2cbb6c05922c8863c15
F20110218_AABXPY cobb_p_Page_141.tif
8f00af1716220a7d710baf959c1ea1db
1c09e4805541c485968e6f7c317a183eff748825
1202 F20110218_AABXRC cobb_p_Page_042.txt
34b886b788e96f7e2c965f68884cb2e2
b472908a77402558f7fdab280d20a6a6a2f8d02c
275 F20110218_AABXQN cobb_p_Page_007.txt
d37ba7e23a3d5bcf1bc1c6627f3d884f
645c3b32c79afa8222a836e9b86d3ec490a3334a
F20110218_AABXPZ cobb_p_Page_142.tif
71b8d2aec7130a09998425a1a293c660
da80c6b9a5d04698fbacad4bc5bb2df53a6ef30b
1433 F20110218_AABXRD cobb_p_Page_043.txt
12ca17a54bc1692337449610f6776c5a
e90005cbd730a945590f81757e7f3effc111db6a
1511 F20110218_AABXQO cobb_p_Page_011.txt
5e2fd5f0b71914d7aa381436593b65c8
823476cc5051353c3492add8c77e4a29cd71559d
1371 F20110218_AABXRE cobb_p_Page_046.txt
e34bff9b17270626272e57013f15287f
89ce3235b0ad65ab5f1e4b68786ec61f56af7a4b
1877 F20110218_AABXQP cobb_p_Page_012.txt
d98c7d1af9ccee227965e010a991b339
165998fda4fa9a8a4d5d4678db4536b0d9690be3
1938 F20110218_AABXRF cobb_p_Page_049.txt
f15ec4490f817ef81219e80a220ef4eb
66ea14bb6c82886cbff42d63d4d02feb58d9cf74
1924 F20110218_AABXQQ cobb_p_Page_014.txt
959e66b6c83af72be2b9810f3358e6d1
40f67ce0f4b8031d649df301dabc4938de27b231
2033 F20110218_AABXRG cobb_p_Page_051.txt
4069598068e644828812f7707103e9df
ebb012c4a63a258de8d6136f21cbf970dd616416
750 F20110218_AABXQR cobb_p_Page_015.txt
a87a22aa21d0d9c741fe418c8d68c987
30fa3f3ecdab6c96d4fb6e5f04b1dece7a9aafa7
2125 F20110218_AABXRH cobb_p_Page_054.txt
1fe43e5197481f753074879f88d851cc
e4ea71003eceb8b3857a92f0fc6b436056d1a80f
1141 F20110218_AABXQS cobb_p_Page_023.txt
25e3546e31668d463847a90e9dd39c70
b872afa19d779b468feccd063b3b8f037e27d21e
1505 F20110218_AABXRI cobb_p_Page_059.txt
e1450260eaaf3077d5f9c0eeb5803851
ae9132c6644d15af58128738b111a412f0bde2ed
2287 F20110218_AABXQT cobb_p_Page_024.txt
eb4a6aee4b7eb300d146e21b938a38e9
9ea546674c877336415e1ffca4dad3f1bfe08026
2179 F20110218_AABXQU cobb_p_Page_025.txt
940f3f0e979b6167f6cd4fa0daff21a4
a3be073dd2325b8c9bbd14014d8a9490ae4a97b8
1940 F20110218_AABXRJ cobb_p_Page_060.txt
6b7949b63224a4f3b140a2b7d4a62447
91909c2ce48b5516ebaaeaf64742cbfb72f0d23b
1920 F20110218_AABXQV cobb_p_Page_027.txt
1ba36e4eff4546e15684e6f9633a9e35
76193720af46cce997ba84ddcc9f06e418123c50
2147 F20110218_AABXRK cobb_p_Page_061.txt
6de3f36d16cdf45258179cad030d5fa1
3b624944fe5a835c6143ddc2c49cd86570b0b41f
1544 F20110218_AABXQW cobb_p_Page_034.txt
b8e4e3cb362513e47372391547111bd8
22ced0a7dae430528dbb9b08919f76a1d90d32e7
2171 F20110218_AABXRL cobb_p_Page_062.txt
296e230e19e9015fcb98a8734d2f0341
e311baef1667711489e5ff64ce343abdc1c6535e
1893 F20110218_AABXQX cobb_p_Page_035.txt
88839b9af8725da433fba57807e46f22
0005b3957e0a9e0cc4fa6f4deeff133886e00cee
1738 F20110218_AABXSA cobb_p_Page_085.txt
92b053423d7d2a23682252b86f606d13
86eb1852fa117a7d1820993e1617f5cefc1bc4f5
1880 F20110218_AABXRM cobb_p_Page_064.txt
659579b8007a4e4491ab65d8656099b2
27b23786598b82c1f8ef7661bf7ada04c4250947
1776 F20110218_AABXQY cobb_p_Page_038.txt
c5277469451596f8e056265a6681a6ef
afce3787ba37adb4838a8ff8090d6d446ea7caf6
2123 F20110218_AABXSB cobb_p_Page_086.txt
24d599315e889ac3b15ed673110cb49f
2b0a61811afd695ff0d71543efcdb8c0285eff17
1931 F20110218_AABXRN cobb_p_Page_065.txt
9c2bed644f34d0fd326e3b5c6d08cc3b
4a12b82467fae9860e947bceef5fd7bb8c45de38
1536 F20110218_AABXQZ cobb_p_Page_039.txt
59cd96572a53ac2e1b69474a6e4e8870
202ccedf703002d569861a273f376cc59ef5a9d7
1855 F20110218_AABXSC cobb_p_Page_087.txt
8062997b31e97d62464e2639f4e2da46
a66429a26e7266cff4ba08b73fd5931459f39a21
2263 F20110218_AABXRO cobb_p_Page_067.txt
dcab3478c789cc3887f9bb839ab571e3
3754dfebe9873096faaafcfba63f8ae3ea324e38
2160 F20110218_AABXSD cobb_p_Page_089.txt
f71f29b0a9aeabe334f9870ad46c2cfd
4c54663da79b06f42a6b51ed31408344cc0ec0b0
2801 F20110218_AABXRP cobb_p_Page_068.txt
89c05e050dc757dce479c7fa61446dfb
67131aa889a571c02f55df6a556e4f4ccd7c9017
1835 F20110218_AABXSE cobb_p_Page_091.txt
3b397b8d0acf74a480e717ebfc3d3a95
6eabb0880e95e702e0bd72a57c086fe8c0ceb18c
2200 F20110218_AABXRQ cobb_p_Page_069.txt
85193e2de6da41b0d9a7a9f1c2b8a121
05b019cfd19867b6fa8895601d9462937155c854
2189 F20110218_AABXSF cobb_p_Page_093.txt
5a615ffca86325ae79f95b24b778b7c8
24a019e3542f4a2509e22c09b13ecb43c271f1de
1866 F20110218_AABXRR cobb_p_Page_070.txt
26bc08469bb008fe8f529ff8adcd3888
149d7e9858a4920df903cb340749f04fb7edce95
1917 F20110218_AABXSG cobb_p_Page_096.txt
ad031946cc1ac81a882fab5fe7b58b1a
a61e012963659d01b2d5869956cec1771876020f
2262 F20110218_AABXRS cobb_p_Page_073.txt
d5696043503fe9263d976b058c71477b
00beedb43fc0c976c386bee5dc8ec808e98632e2
1769 F20110218_AABXSH cobb_p_Page_101.txt
b4f1c9d430eeb4e4e287a9e80f6e95b0
349a277ac01ad41c66f52cb6f7e7a8b19520586a
1944 F20110218_AABXSI cobb_p_Page_102.txt
22ffc75db23f131cbd561838c706c4e8
f66f81399e11dcacc61c9daf174439d4da52da3d
2246 F20110218_AABXRT cobb_p_Page_076.txt
aaab789ad43a1a7719cf04f847383c25
1c2af0cbe5e80e292bee7204d1b955af05413d53
2058 F20110218_AABXSJ cobb_p_Page_104.txt
768aea1e5770bc34aaed841b17344f91
be5899030347e37db3672f8d80ef6857a698f5f3
2194 F20110218_AABXRU cobb_p_Page_078.txt
77962221990218d508de4357317de938
1ef5ff011467df1325812c8cfd7f175d40a13307
2006 F20110218_AABXRV cobb_p_Page_079.txt
9e1cb88d3482ecce22489896b449e806
72d29d898ad0619f0520de837455cacb8184f433
1986 F20110218_AABXSK cobb_p_Page_105.txt
9aac7daefd97a63f3a4c6e0a1e42ace3
93a36f08082210010d6bf96e8abee037c0657f38
2252 F20110218_AABXRW cobb_p_Page_080.txt
89b25a830550ebff0edac4abfa66caf2
2ffcf7bf42e0fcc3569d40509d08f20555bef500
2148 F20110218_AABXTA cobb_p_Page_137.txt
0ac19351d5600038320b7ab5ba5d0df7
0422abaaa99fd518c04e81b83a675be84698b833
1850 F20110218_AABXSL cobb_p_Page_106.txt
cf0c6adcdc0d83970267beda3a027f50
8c3cfeca48f4e5e6106af4a45959e77394938662
1961 F20110218_AABXRX cobb_p_Page_081.txt
517892a5fbc858e64d000cf82a4010fa
8b42ce55766269e231ed9a954071d1719ea829f6
2327 F20110218_AABXTB cobb_p_Page_140.txt
f4ab79a172846caa23c2c0a2e09849b9
1ecda469275f7c0723833921bf39a2d69ce8cc33
1706 F20110218_AABXSM cobb_p_Page_109.txt
a9a37d77d4543952047bfde290f7e883
2a817febecc2339426ba96e06fa5ca4acc29f051
2235 F20110218_AABXRY cobb_p_Page_083.txt
8c46a428f8faf051e02bf9df9b942f2d
06bd17e791dbdf5a034f2f96fe7e915567d44b52
2162 F20110218_AABXTC cobb_p_Page_145.txt
2f7bcda88cc8e7b65d71fa537fd9bab1
cb0aea8f7569d9380abe663810544673cbbde307
2003 F20110218_AABXSN cobb_p_Page_110.txt
7332974579e5d00ca3c344ec5afc7a5f
9a8915c2473b3f1b09f35bb751a57e87e07213f5
2142 F20110218_AABXRZ cobb_p_Page_084.txt
281fdd8f715c8903c56ca5b69fcb99fa
18aa38c521ba6f5f54a55ccd1a8dc0b9980fb5b3
808 F20110218_AABXTD cobb_p_Page_146.txt
bc3dadb4090e3e935a65f47044c97705
1433c3f74cf9c1bf1abd9d2816e1b9c637611b82
F20110218_AABXSO cobb_p_Page_112.txt
d5e5a191acef3947d8f9a06bf526a37f
23b9f8b98ae16cf3932cebe009a012667cde4c0c
1891 F20110218_AABXTE cobb_p_Page_147.txt
8e60fe4dcc497b8bf875e42348b8eb50
3992b73eeceee7544f5b7eb20a990af1f11c0dd2
1975 F20110218_AABXSP cobb_p_Page_113.txt
ef18005987e7a8dd10b193df34338ff1
6dfdaa82abfb091c6d9ac2931d15b30ced2b4a32
2295 F20110218_AABXTF cobb_p_Page_149.txt
301020b8efb792534c3bd1309f91497a
8f72bcb76c24ffcd078f7c7ecebcfad3e9d8b6bc
1925 F20110218_AABXSQ cobb_p_Page_116.txt
ab4843d7b95b07301006ed91b2f1629a
1a17d4f5eec6477d4a30a7c8eb6c41027e0f7944
1945 F20110218_AABXTG cobb_p_Page_151.txt
7cf4e3d66c48a39fc7f534a598b19457
4c588fcc7624b2654b4fcddeda8a252e332ce4af
2214 F20110218_AABXSR cobb_p_Page_118.txt
1d5484b33011149096b6715328a757b7
002c8aaec88cde5519cff9922335ee2f895dc864
2389 F20110218_AABXTH cobb_p_Page_152.txt
d9a8af5f6a4cffff9e9874eff2d460e8
fd2dcb3505aad17c48df53ca4779d128b940288c
723 F20110218_AABXSS cobb_p_Page_121.txt
cf6bdb72e80598369a11132513295dce
f77a45d454a2332e6cb422eb8786ba77ef00f490
2535 F20110218_AABXTI cobb_p_Page_153.txt
65335b9995619da4986702711600720b
1a3d1818ea343762d9000a84e002a084c07b0ded
2064 F20110218_AABXST cobb_p_Page_126.txt
67498eccc17b80d615329eaaa1db4939
07b90183fd4d31a345d41825ce2c4948694e20bc
2348 F20110218_AABXTJ cobb_p_Page_154.txt
8f16d6bf9e4f79c6835ec5d5fbf256b0
8f1d7a16f3b4f5be8f60cfe96a0face153c98cca
2068 F20110218_AABXSU cobb_p_Page_128.txt
3bc57b1edd8ffaa391ef90bb35581b44
10a45290161df3f1f9966d96b5621bd9d4b55fdf
2247 F20110218_AABXTK cobb_p_Page_155.txt
edd8c6db41fc1078b63d09712a640c76
3e236a2e560a47dc8b62d918af97ce7980eebd65
1884 F20110218_AABXSV cobb_p_Page_129.txt
5776b230b7501fdef49696f6aed169f2
141169ad7d8372bfde643147af49a72e4d4a040f
1709 F20110218_AABXSW cobb_p_Page_132.txt
942ea8598d99239564af43de4ef45843
78b8f9f74a92b84a60e226d150cc4f7e6602594e
2289 F20110218_AABXTL cobb_p_Page_157.txt
8394cf40d8dcbc2c6450ddb631e5c0ee
638c76478411edbf063075a7e1860096e1a87b62
F20110218_AABXSX cobb_p_Page_133.txt
95c6fe8d80465c159df1a557488304c8
3b2a659354dd9e9acb20b863f59fc2772ef2d900
31379 F20110218_AABXUA cobb_p_Page_029.pro
506e95ba3f3b3d40159378e8ff1a26a3
42b27fbd9c8905b2b6cbeaa8b81a822ff05a2643
236 F20110218_AABXTM cobb_p_Page_159.txt
b80f83cbb2f989fd4c91594bcb478150
0f4e69fdcd28d6654ee0b19fc098120712a00607
1427 F20110218_AABXSY cobb_p_Page_135.txt
2a51a5c44241bc3b15a6fcc95d02cda9
15933cc2fe36e5aa9aa9701c7294dae93be1da15
31682 F20110218_AABXUB cobb_p_Page_031.pro
c8b7f613dc65e9e222a202e9c9add30a
0df4e2e1f49901e84ead4e82c1eaf032260e3c4e
1146 F20110218_AABXTN cobb_p_Page_002.pro
360453c45cb5117f85339e91aecb5de1
75e29b769c17d853355fd922649568c84e048070
2021 F20110218_AABXSZ cobb_p_Page_136.txt
7483d4770925664f43c91febfca8a3b5
b63a0f84e57eee36cb8cb359edc3c2bfb76dd803
30711 F20110218_AABXUC cobb_p_Page_036.pro
560f54a5fc87c5593c539634aced7eae
152ecc3cd96002c6d421f5fbdba45f1778723ec7
32429 F20110218_AABXTO cobb_p_Page_003.pro
64cbf0915815241900097daa0fd06193
3dd675088e5321e7a7615fb57bd8e3992c796f88
29512 F20110218_AABXUD cobb_p_Page_037.pro
a4c004b89d09c58463b7885531d94ba5
1d153aa4f02deaed00470ef7e954edeec14c2bf2
71662 F20110218_AABXTP cobb_p_Page_006.pro
372345d7d51e8c8697363b20da7b1245
7e8f8e60d358eefb110eb92edc4df314ff4a2c1a
38475 F20110218_AABXUE cobb_p_Page_038.pro
7dde589f32d87795fb5e55afd91198d4
75de33d27cdc03f66df675cb994f8e615d1a546d
5903 F20110218_AABXTQ cobb_p_Page_008.pro
4a2f306b8dda9fb165aa5fa5fcb37b79
5b8b03ac2ea4d7ec1b0d91098029747ca4acde96
46898 F20110218_AABXUF cobb_p_Page_041.pro
ecebf2344b3daf9045782cf88d42b2b5
659bd8371f157ffea64773a82958875442573cf4
28673 F20110218_AABYAA cobb_p_Page_087.QC.jpg
824f3b8a99d43c6725469e75c5e11866
97feb9849ea349c662d76eb951a18b9b714fd0a3
51935 F20110218_AABXTR cobb_p_Page_009.pro
350bc9296c5ef495cd9936e220dd03ec
81ee646b5ef9c0d4ce652062b179f951dcc491d1
24843 F20110218_AABXUG cobb_p_Page_043.pro
32d93c970c5a3b401c2eb3b31d945982
1b94578db374f168157eb3d7aebbf8768ab82ae6
104925 F20110218_AABYAB cobb_p_Page_089.jpg
0be28921921c7349f9d3a73717a87c96
493ee6fab11f8ff0d395b886427cf4f805ec755e
60970 F20110218_AABXTS cobb_p_Page_010.pro
42a90e0c697d39f6670d9c9b5b7c48f4
28c141a3bcc538c72b19c602a57bf0633fc69ac9
18991 F20110218_AABXUH cobb_p_Page_044.pro
4ef6b9987eb822852b1300f61c4628c9
338418ca8ef87bbb908d376f800dd47c16133926
113721 F20110218_AABYAC cobb_p_Page_090.jpg
4e5e57f70fe605abd3fac9f3616cf1c6
642976587b2e22df390288d46a837def5b4dc321
21530 F20110218_AABXTT cobb_p_Page_013.pro
a9d89bebbfbd7401260c2d6ca51c3f6d
dbbb73bb05a8d12e01fdc972400e13ea7fe401dc
29410 F20110218_AABXUI cobb_p_Page_045.pro
026369c49162949b4c2ea9601fe3b249
f11d18c760f3a9f35e32f87a5caa8d76a256a76f
35337 F20110218_AABYAD cobb_p_Page_090.QC.jpg
7a091ddf3310acd4383799f096118fba
735f1855b11ae3a0e2c0c0ffd459d6f415f8af97
16174 F20110218_AABXTU cobb_p_Page_015.pro
f0aff4cb3bf325b65712f1ceaf11d4e3
69b9505f6cb66c125f51352fd148e5fe232e58e3
41771 F20110218_AABXUJ cobb_p_Page_047.pro
3b69014edc928c0518df42299a246747
14713345ea5ed5c0e216eb4df9e6fad3e357fb33
26347 F20110218_AABYAE cobb_p_Page_091.QC.jpg
f8ea085fdd09d0893686673385232690
0fea90bf70752ac33ad83526a192bb911bc49a8a
41176 F20110218_AABXTV cobb_p_Page_017.pro
390cb2ead1024fa18e2893a8620b116e
2ad20ebe6e2200aa5a966114decf1377b3151f94
39643 F20110218_AABXUK cobb_p_Page_049.pro
017c3aa02d4ce940a932e6d1a61a09a7
55ac80f0774904491b3624e8af5b96f1679b6b97
83891 F20110218_AABYAF cobb_p_Page_092.jpg
d0d08d4defd83ca206dbb7bac073edc8
43eb9f761f05ad7cc8b669a13f01e5a9bad8dd7a
55510 F20110218_AABXTW cobb_p_Page_021.pro
48d5667620806a03aea61c3cd305f798
d72ab5ade1f63b8d39cf27271df49d0e58fcb0ca
48941 F20110218_AABXUL cobb_p_Page_050.pro
98506d3f22fb0f3035d5b387c2b46372
a53fb43dd3fe57a2c2fe32bfb6d4517840d97d94
113370 F20110218_AABYAG cobb_p_Page_093.jpg
5c6d0df42cdfdbfdc77797a273fdfbae
d762ee943de756a2a8420a856683eab6fdcf8de6
33887 F20110218_AABXTX cobb_p_Page_022.pro
add53eb8feb5d0c616463001700650a2
c797ce3f1b8ed8d654a324c06f38eb9843ff8185
39356 F20110218_AABXVA cobb_p_Page_077.pro
4670d128f8968bd682f79733e71caf79
b9ff8f673c6fc84132f9e92513546b7f8b59f009
114291 F20110218_AABYAH cobb_p_Page_094.jpg
4c6b58307f5296099dae56cb5591c5c0
4fbbf123b66627613a236cf3cb8ec14a569dfba4
55410 F20110218_AABXTY cobb_p_Page_025.pro
c504a863493e8f4b4395fa6d1a69759b
2420879a96097f3c93aa39a148e284961eeb09a3
47768 F20110218_AABXVB cobb_p_Page_078.pro
f2f18c1a78c317b3577b187d1af9473d
a371c6c8cb53fdb2838b8f913237f4bea0743df9
39775 F20110218_AABXUM cobb_p_Page_053.pro
288ed983ca381c6936f0e3716623693e
c1f6190ab58f41350415f6f9f39bfd8e82c3ddbb
34678 F20110218_AABYAI cobb_p_Page_094.QC.jpg
a510372e83e0dc1e0890744d45b39a40
752276cb3bfb88b9b8b46e3741dabc253b59b6b1
26337 F20110218_AABXTZ cobb_p_Page_028.pro
782db66f444c4453faac6c274cc4c7f0
218352cf2ecaa23d207db6484b2196a4d9e220bb
43058 F20110218_AABXVC cobb_p_Page_079.pro
27c9405c664ab56153e66a41acd9bda1
2d4012f02ce2c0c7dad0e6176b6bb80322f413a3
42983 F20110218_AABXUN cobb_p_Page_054.pro
6747fde6be409ea57bbce46c2c9eb438
c0df4246a7258f0fc2514a86b147e53c0c941813
106624 F20110218_AABYAJ cobb_p_Page_095.jpg
8503230b88547dc065cf6896fb6824b5
21a1606202c7d11af69fb6fa3d09758770c70ded
39031 F20110218_AABXVD cobb_p_Page_087.pro
0b4815b102e594818affe6430d835608
f531e47a9460a8a911a744b878f0e0e68473628f
41924 F20110218_AABXUO cobb_p_Page_055.pro
bc0e3cb2721bf0a557776f97e3a75595
2b978406a80e7422d3ce454593811f8f8c07dbb5
23931 F20110218_AABYAK cobb_p_Page_097.jpg
d40ee266da990d3270590a3dd235fbad
bd24872b6c3bc23594355a6a88a69cf28ff9ee13
53991 F20110218_AABXVE cobb_p_Page_089.pro
3c63efa5f1ca772108db5e58fdb47ffd
380234110ef10ed961f0f73d2f30479bf82e81f9
35341 F20110218_AABXUP cobb_p_Page_058.pro
7752380e82c356bddd6a0d5b95569184
613641e62656e9d0a72379738776c601344764b3
7162 F20110218_AABYAL cobb_p_Page_097.QC.jpg
653a599a3c2669886350baa9cddfa4ea
ba850c3618d3e1654b8dd38b599737f9f3296047
56576 F20110218_AABXVF cobb_p_Page_090.pro
8c66dbac08b3dcfed406fb02df9cfd22
25cf7f7fd6f4ff5d4c4287cfa48f540fffd828b7
36903 F20110218_AABXUQ cobb_p_Page_059.pro
2f67fb153069652a0cc64f6378992a64
0dd6e9b849e74f7ce6a9a015fcd9edecf4e9da1e
28464 F20110218_AABYBA cobb_p_Page_111.QC.jpg
8cd273ab1f7601b303b00db1a9d2f34e
00deb97a20b46dc9946a0b059e3cf45bc6ad0f39
29848 F20110218_AABYAM cobb_p_Page_098.QC.jpg
cb00977641c22c6506f73adabd1116b0
0c12270b3fd842a10649c6c481a26302abfbdef2
912671 F20110218_AABWSE cobb_p_Page_086.jp2
04bf835c4acf8d263cd0da37afba7d74
cb28d1db467bc88f92c1a15be3b6e93a3de197b4
39017 F20110218_AABXVG cobb_p_Page_091.pro
248f87352fa7f31c00384628cec23506
a9bd67162261258821469e2e1eebedb732d4367e
54442 F20110218_AABXUR cobb_p_Page_061.pro
a1143adb0a8100d282a81e95ab3d158b
b4bce2d9fe11633aad4180c3e01649fe7fc2f40b
24834 F20110218_AABYBB cobb_p_Page_112.QC.jpg
11a32622edbc6e2e89ea5cb498cab126
451d3448f38605173895fbdd001bce1a97d752c1
114051 F20110218_AABYAN cobb_p_Page_099.jpg
3ff3d3664d55b3491feb5358cd7570fb
c96c9307864af84900217b91c322add83068a570
1313 F20110218_AABWSF cobb_p_Page_037.txt
be5e79e722780bb4647fbb4730d642f3
0aad4306cb973a56eaa0c168e4e125659d198769
52289 F20110218_AABXVH cobb_p_Page_095.pro
ee1636add3ba9be93c115a3c65e1580a
d6df1bd9c0566e6d19da8c679ad3d3d302e80bb2
54453 F20110218_AABXUS cobb_p_Page_062.pro
97fbd149118f22d3696623cf2c3e4427
abc69db113bf049d6b5038f80f6fbd4f52cebbaa
110763 F20110218_AABYBC cobb_p_Page_114.jpg
20ce84880777467b15793b60dd8ec3bd
e68b6a31c5ecf5a85e1e585f6479cf26bd02df77
28967 F20110218_AABYAO cobb_p_Page_100.QC.jpg
7b576cb6421ecf5528a9d78d9563a889
990014370f800ca06892212c99c9d48a229c5a9c
972115 F20110218_AABWSG cobb_p_Page_136.jp2
34a3161d97c469f0a8d5fa7235cffe86
4ec0dd176dbe92eaa0f6d05ce3bdb67c39560c0d
9998 F20110218_AABXVI cobb_p_Page_097.pro
c7e06936139d7c0bbfaf6ec1acb1b7de
75fbb4b74c0c475d90587a3e4a37546340051c69
37835 F20110218_AABXUT cobb_p_Page_063.pro
2322a2eb4fc220e61a1c6d88e091d99b
282a62d3eebe6d06cea72f4f3ce7e6cb77c50a67
34019 F20110218_AABYBD cobb_p_Page_114.QC.jpg
6b11ef8cd70b458e11c82ce181a48e7b
aa7b6f563836783acf2e95da488ed68788ce245d
24987 F20110218_AABYAP cobb_p_Page_101.QC.jpg
7ffdd5311bb3d8468ad491e5a3b2603b
899fdc46e8d9529e2e45a28393c36e980ed3e14c
32744 F20110218_AABWSH cobb_p_Page_006.QC.jpg
562e041926c57c8d83812ab9fd27db31
ea29355b26181989871f310cb812299518b65c90
49910 F20110218_AABXVJ cobb_p_Page_100.pro
3086ae66a982708246a54277386beb51
783e69cc0d5b8c3504c592a6e8155c02fb0372ae
37886 F20110218_AABXUU cobb_p_Page_065.pro
3e003d0d03177afe38b4c6d908ca0bba
32f659ba868a8446809bd9fd33826ae00c70c2c6
86853 F20110218_AABYBE cobb_p_Page_116.jpg
1f08cea9c7886b2089cb240c63979621
8ea7f5d27f08b19e5e04a5476913db6727f5defd
85702 F20110218_AABYAQ cobb_p_Page_102.jpg
b64868049633514a805a04741fd75fe4
9de856ce94b659b3a3a911d8d0aa7360e07dcbe9
F20110218_AABWSI cobb_p_Page_121.tif
d410cc19628ef48ba144173f9201576a
4facccd0d04526cedbe15f38537d2a76072c8c5c
38980 F20110218_AABXVK cobb_p_Page_101.pro
ebb9cffa81df2d9ca7b6fe22942df32b
3517d9476bb027ae9bae5f9fbb4638daa9635457
43320 F20110218_AABXUV cobb_p_Page_066.pro
987290e54f4b890c446c9d47868d0136
87869d7fcbd472a76edd8532d87274e4f5a04016
28248 F20110218_AABYBF cobb_p_Page_117.QC.jpg
cac9bff1744f51cd8a78c0f6b2f0034e
ae87a71cba4554750bf3f04a87e8c911e773529d
90746 F20110218_AABYAR cobb_p_Page_103.jpg
41728729689ac8f5f12b9cb86cd14f38
84c61367279b05404d233e6d212c2cf05b97c652
8135 F20110218_AABWSJ cobb_p_Page_021thm.jpg
6a8992a18b7bd17eb339d5a5d983f9f2
9cf4ff9780ce39d726bf4156de93ab7e4bb67e67
42397 F20110218_AABXVL cobb_p_Page_102.pro
b1509924ae73f720f04916391fbaa2e2
6912d96cb0f7ea1ecb0d7dde184ab2dfce684bed
42121 F20110218_AABXUW cobb_p_Page_067.pro
5e37d15cf2519664c8955dcb4faab298
689fbb974203451225ed9284dceff6f5c89f6015
108736 F20110218_AABYBG cobb_p_Page_119.jpg
e3fb360f7ac26416b9bdfe8c43987ca1
7532aa6167ac15710d3436928f2840d952e2731b
28662 F20110218_AABYAS cobb_p_Page_103.QC.jpg
f8d1b8aa3eb9fdaf8e302414c2ac1e8b
8d6aa733cee147aa6fc793718fd16c5f43894831
41311 F20110218_AABXWA cobb_p_Page_131.pro
df08d6e0cd91f274d547d5b3f62c5009
214e202a436a3603077b3e26e56002e75cbf1209
42213 F20110218_AABWSK cobb_p_Page_096.pro
7d22cf4a3f4819861efa8e9ff8dc7936
6ae305d77b4fee4993892977ae824f7e1fff7665
46012 F20110218_AABXVM cobb_p_Page_103.pro
655d8202282cb395e7258e3eec4e860a
84af61b2688b1b27e2e1c6fe299e4e968dc4ad11
52693 F20110218_AABXUX cobb_p_Page_069.pro
3b3d27c872d14bb3a7a4d5dc92701633
e3b3ac112ec37aad97acd67b60e510574d25c839
97593 F20110218_AABYBH cobb_p_Page_122.jpg
2ebd9ea318ec7b3f4208b36378e3badc
5210ca9b85de382da1c5a927c2d2e3b50e833350
27480 F20110218_AABYAT cobb_p_Page_105.QC.jpg
ee806bf1faef43a0616e459686e0c21c
bcb767f4209ccea3b0c993f9899821f19bb0ebae
37022 F20110218_AABXWB cobb_p_Page_132.pro
e433652bb4da33c7c7150856ee4d70c0
c2dfbce7edbfd9b976ccfda0fe783886b8af9feb
51110 F20110218_AABXUY cobb_p_Page_072.pro
4d5cf70b880d524ebc973f89100ac6d1
67b62bcd81e7d9299949ca447cc1b7f30aa06efc
30191 F20110218_AABYBI cobb_p_Page_122.QC.jpg
ed45ada8a5ec354500ff93a05917f5c0
8b19f9ed6c898127f6d7875890c628f55c1caeb7
92423 F20110218_AABYAU cobb_p_Page_106.jpg
f030896091f2a469f5883326f33910d3
46ba5eea146e023afd513aecd4939087375d7c91
52596 F20110218_AABWTA cobb_p_Page_044.jpg
e7dd47440018d55ed46ead305c4386b7
6c2e4525ba48dc928b61096c69564f16e9005df7
42257 F20110218_AABXWC cobb_p_Page_133.pro
080a47beb850c245be655bb3762a1d5f
356abe31cb6639ebb92a69177223e47532001933
85495 F20110218_AABWSL cobb_p_Page_086.jpg
fc2be73909d5f07bd9d2d88c4e771f44
40247338fbeeba35e9458a2f78dd76fd39a5f861
43840 F20110218_AABXVN cobb_p_Page_104.pro
c18f38906da6a4babec3c47cd4bdd284
94bef47042d0c051096b3a346f174b8e9256d3ed
30873 F20110218_AABXUZ cobb_p_Page_075.pro
47b5a51b423318dc3a5819cbc7a12e37
2026ecf2f7f6eb05636be36bf706404c2f468122
87749 F20110218_AABYBJ cobb_p_Page_123.jpg
3d359f5c4acc9946531fcc949c8f9191
a1ab706571b596a46139b22a61ff9cf3b8f07a75
28133 F20110218_AABYAV cobb_p_Page_106.QC.jpg
273540b4975a336c702a2398f08d185a
d65f71301abb067fdc5bbd655eea81041fd9cec2
23400 F20110218_AABWTB cobb_p_Page_004.QC.jpg
04de99c2bbeadc896d267d0cb3d3e98a
5473ac39a8c8d66618667ac89a7731c9fb62e49b
31156 F20110218_AABXWD cobb_p_Page_135.pro
d0589f42a56d51c81ebae35569fc10a2
333ec492c8a0bac033a4d9f583fb43eac70e3bd9
20056 F20110218_AABWSM cobb_p_Page_045.QC.jpg
5c7c83c3aa606f7866787fec403fa3ba
d482d6acdf6c9360ce564937e92a11f7b7178c18
36141 F20110218_AABXVO cobb_p_Page_109.pro
865799832cae9d82e49ffcb935d6fab1
272adb1ea8c45a1f74af2dce0ba2984f4bb00fd2
77479 F20110218_AABYBK cobb_p_Page_124.jpg
93bb1dea449d9b2b7c15bc4af9cb75aa
04cdbb04668d7c5a5ac18e40c59b08ee880d2157
32962 F20110218_AABYAW cobb_p_Page_107.QC.jpg
df08c27eba6f30b59c6a13c031b13843
8f8172ec9afb8b11a9f3a9b80b7baf913558762e
4868 F20110218_AABWTC cobb_p_Page_032thm.jpg
90ccb041a571708a0909f7ee4420bb19
a77a3d92b6b5cf4570b28ccf00a29504b8eb5fb3
54559 F20110218_AABXWE cobb_p_Page_137.pro
1c34bd3d30daf91c850e938fc3a0ce48
6968b7d1a5c5c91f541e982245b731368a88b012
F20110218_AABWSN cobb_p_Page_062.tif
1ce99711c26c100cef669be6af729871
3a94936729b8dbba74e82cb0c4eb963b02d8351e
37463 F20110218_AABXVP cobb_p_Page_112.pro
00a315d5a61df11901c43a2fcb10ee7f
14323b138185bc21d326c6e3995af4a3793fa5b1
109526 F20110218_AABYBL cobb_p_Page_125.jpg
429c6e17374cffe70a112a79c90d8b3a
e7cf9f12046fefc3b0a6b9908d05784de56122c9
1904 F20110218_AABWTD cobb_p_Page_098.txt
bef4b37d640d322e42d40f6b17a6fd49
e31156adb3725498f1fbf3251495047c72c26d23
45316 F20110218_AABXWF cobb_p_Page_138.pro
efad06b142801bde1fd635c82c85be58
aeddb5e8915d714b8e89c7eb70dde49965a625f4
1282 F20110218_AABWSO cobb_p_Page_020.txt
167b7810f5fb10bf6fea48c0383d5460
3be5f5fb3414a7db055fb17d44b5cb5f43d1385b
40384 F20110218_AABXVQ cobb_p_Page_113.pro
f811edd30bb70cd354f7d392de3f5687
29dc0cff11db89b66648c62b3cddf9d452cea91f
110279 F20110218_AABYCA cobb_p_Page_137.jpg
8935edf7dd6775242695322b4935392d
e528cc5b19e333ae4cf294b521e3235a939d4477
34107 F20110218_AABYBM cobb_p_Page_125.QC.jpg
6c92ce2960bd33c4718e2903665fe917
87dddc37954051fda13d62aed7ad398797b4b41b
78212 F20110218_AABYAX cobb_p_Page_109.jpg
ec645c46489314784ffb56fb22156d4b
b92aed691bd7b02dfc13b823aafe5c4b411a6b55
48144 F20110218_AABWTE cobb_p_Page_115.pro
ba8374dbccbb99be2b81f9146c34ab12
5c98c7144f510cf9048520fb5a38e0fc0d1a189d
46542 F20110218_AABXWG cobb_p_Page_140.pro
1303434956e144ce1b33e53fcecf820e
850c6afb1783f37103c99c385d57216dd665e87c
28467 F20110218_AABWSP cobb_p_Page_141.QC.jpg
d1541984b03959308a9733bb75c029fb
e4698d1105dec947cbe2a8c8076bdd0ea2cd5b93
41420 F20110218_AABXVR cobb_p_Page_117.pro
d60b2540a0efc69f95b471fba4cba254
0423133d8ee49d199960b4f5f7a32c26cb45f8ac
103218 F20110218_AABYCB cobb_p_Page_138.jpg
572c7bb26e58da6812ebdfbf00a509e2
a38b145b16c4227c3c062845c6eece64b2dcfe02
29309 F20110218_AABYBN cobb_p_Page_126.QC.jpg
f2ed398b13f0a873a78d42e738761f9f
0b078672b1a243ed8bc4bcf42f69cb785d6abdd9
28365 F20110218_AABYAY cobb_p_Page_110.QC.jpg
cd8ad9c3ada6b43605184f338c9ca7ed
72d07ab4883172fca3ae33cd4ae1162eae5b0ab4
6056 F20110218_AABWTF cobb_p_Page_112thm.jpg
6ebbb14835df988f17a056d3dbd8fd9e
943a37763c70760dd847327ff766be9efa508bb3
42802 F20110218_AABXWH cobb_p_Page_144.pro
cc07c69d5569fd772d87f48b2f781951
85d222b64b4d51845f4c7d7bc356bd66a114da78
2261 F20110218_AABWSQ cobb_p_Page_158.txt
8b98ebe69d32867c7c74c49e9e6b77c8
479426c7aca9bb02fff078cbc3fa45936ab1c074
56382 F20110218_AABXVS cobb_p_Page_118.pro
82dd9a11d79da89f72a53599b9a335bf
75337acb0ac1c0ab2a48df2a06837d7b4f87e1c8
31638 F20110218_AABYCC cobb_p_Page_138.QC.jpg
cbcc48e29749f67a7f1e50da4bbd1fa6
a1e540d2264e3b937c1fa58b635e5a2b8dd79e90
107716 F20110218_AABYBO cobb_p_Page_127.jpg
99b7563280f9fac7f73306efa08dd9b2
eb172705b54fbefe0e60b502d931b489279b3da5
96275 F20110218_AABYAZ cobb_p_Page_111.jpg
6774bf5b1aae768b73b08318e78a351a
8d4b8f72297d2853c6493d46577f196f3152f407
54195 F20110218_AABXWI cobb_p_Page_145.pro
0a3f9b96f8ddce4ec23fb151c09e2d17
10016aa7fa1872e5379b6ad84059e8dcfa39008e
20393 F20110218_AABWSR cobb_p_Page_029.QC.jpg
d1e1cb1305a4a88e588cb1e13499dae3
0905790d44d61369947bee71f6382912d3773a26
54099 F20110218_AABXVT cobb_p_Page_119.pro
496b9856a037445550f745360d30598b
1295289e42980e73ffedee2441836114978f1871
36280 F20110218_AABWTG cobb_p_Page_076.QC.jpg
762ad38ccf1a6ea98f809ab30090d6a6
8716d96d2eff526fea137ad89c2e037592d53a9b
89400 F20110218_AABYCD cobb_p_Page_139.jpg
61afdfacaa876d06c07a3eb7fd7793de
918e407998e2744e4c1bcaedc1b7be6fce281c5f
33548 F20110218_AABYBP cobb_p_Page_127.QC.jpg
245e4c2b58f8c242d259536f51663afc
6c32ba5647bbc3cb8b5e24637ff83b8aaec9ca6a
20186 F20110218_AABXWJ cobb_p_Page_146.pro
7d9ea0697aff39c752a1c21dbab03e3f
d29bf91f7e5c5bbb96ad6c6170c7a7113688bc1d
965428 F20110218_AABWSS cobb_p_Page_103.jp2
fd08e5429f2566e2a0d3389c5060735e
a7f8ecd686fab81e89c7fd19772d8f75fc31fb75
35359 F20110218_AABXVU cobb_p_Page_120.pro
e7e1512a5ca76b44ac5e3e7a79f8e094
74c19cf88d657793fdd449a670cb8359fadf1849
7870 F20110218_AABWTH cobb_p_Page_079thm.jpg
b12146b613b194ad0d8737168bb54f1b
6cf18fd6b74101214ddada30024f196efea7a2eb
93525 F20110218_AABYCE cobb_p_Page_141.jpg
e8aab87b84ba9e2ef93bc6ccc4f42dad
7cc3ad91e25a12cb4ad5bd2cc64707616d2e12c3
33123 F20110218_AABYBQ cobb_p_Page_128.QC.jpg
fc13c9d179397ee3f937c5242d15ccff
7c9dffd98d28334799dcefac67cfdd5379364ad3
44094 F20110218_AABXWK cobb_p_Page_147.pro
91bf25c9c5b09e6e10d68e48c6a9c3c4
acffc14970b8b17d5f90a3286bef94e2bf398042
F20110218_AABWST cobb_p_Page_081.tif
5b864ed3a2213719022128e94de1fe44
071c32c53dfaf52009d932862aeeac998a474142
17077 F20110218_AABXVV cobb_p_Page_121.pro
88be38b6962fa03e9ca301e742e34cd3
bdc2ac9b99b89dbf93d4fe6147d46ff60061692d
861373 F20110218_AABWTI cobb_p_Page_049.jp2
f88e0db08afb0e7ea99b06688996f3f4
5f94cc33bdcc84884a97a98f96426957537b4447
26827 F20110218_AABYCF cobb_p_Page_142.QC.jpg
2bda6ba4e1736498f0cef81979c89ced
e1e4d64ef96cd2ff447458e329007588a2cfb13c
78637 F20110218_AABYBR cobb_p_Page_129.jpg
10c324521beb7ec2992d9493200a0be3
62701539ac8b80a5c79a99b01b32ef34a516ba5e
49852 F20110218_AABXWL cobb_p_Page_148.pro
3eb1e1d923368bd3bd1d5f802ecc5683
c207c15b85b8b2aeb16ab9e2a2a7a1fdbf3f1626
44662 F20110218_AABWSU cobb_p_Page_086.pro
4eda8ac927e045571a387021637d0af5
d941a304d2321dcb6449d90d8387afd8d189a1ab
46620 F20110218_AABXVW cobb_p_Page_122.pro
610e7d07e397d0ffac99d6cebdf04c72
8f1edc04b7a63a39a0c54262e4c2fff56bcf5ff6
7281 F20110218_AABWTJ cobb_p_Page_064thm.jpg
6d06f52998da3ba2632a37274539f7a2
1cd00d3fbf71e3759eccd75cbd4ee6d9bdf344af
77850 F20110218_AABYCG cobb_p_Page_143.jpg
63e5591ff65ce541fbfb23eef5c4effa
3434fc2c7d54ae9ed1bd8fd495683139a342610c
24564 F20110218_AABYBS cobb_p_Page_129.QC.jpg
667d06a4869b144ebff5699c5e9df4c5
37da2eb5dc2877516018565f5f8a7af7c74143ce
4443 F20110218_AABXXA cobb_p_Page_008.QC.jpg
d8e36670e7d09087c121a7f9d7fc39c4
35ab8937a3a889a139b6f02348e0857b32093afa
58388 F20110218_AABXWM cobb_p_Page_149.pro
102f7784a0adb641a4b1a3a3364f6b30
ae68f24ffe895c497e02b04b9037f45d55eb137e
6104 F20110218_AABWSV cobb_p_Page_075thm.jpg
cf6d45df2629cf7fc02eaa543308868a
4876085311b620b21af84771ac28c5229bf3f034
47096 F20110218_AABXVX cobb_p_Page_126.pro
71ea3c21b13e02a55e6ad779a965caea
c9a0a85adfe9a630f8714336bb54d26ee792a895
2023 F20110218_AABWTK cobb_p_Page_072.txt
3697d8bd9d81c437a3bd55477519c29e
2a09df495ef526744c4b9fe5b2950cd6f3451d3f
23873 F20110218_AABYCH cobb_p_Page_143.QC.jpg
352314f5e047d5b1fae47bd838b53f5e
396693f8971f2a171b173d11e16d5bb68b8d81db
111336 F20110218_AABYBT cobb_p_Page_130.jpg
d824ddd47929edb2bfb66d8b348a077d
4953aea1c9ed16800814dfd5083cde98aa02cb78
26488 F20110218_AABXXB cobb_p_Page_009.QC.jpg
3152292b3025f85fbb4102cafcf79890
0475673068a8ee5313e98ff02292e09e99ac6841
59290 F20110218_AABXWN cobb_p_Page_152.pro
484b66618f59dbc2a8f270425d5006e7
3545bfcaa3fa50185a7be90e5696a9e7547bf5dc
109584 F20110218_AABWSW cobb_p_Page_051.jpg
5f11d60ec4881057a2b50bd25e59a11f
bb1244a9677dd393d9382e7aaad5a44fa762775a
52211 F20110218_AABXVY cobb_p_Page_128.pro
cc98ebec555c59e60a246072762c0fc4
af62942886ea9910366609d0fd8536240dd2041b
8217 F20110218_AABWTL cobb_p_Page_149thm.jpg
098dd5abafcf86f8798a174fad4749c4
eb807313d932c86ad173d5ae60d56f0886fb0976
84587 F20110218_AABYCI cobb_p_Page_144.jpg
324b5e6e8e59ca9412b7a5ed00b4ae6c
a36f8cf06e8fc15110a8b2ae9411d186d173d8c4
34398 F20110218_AABYBU cobb_p_Page_130.QC.jpg
0160c1fb6b49f93ec24dfbc7458da992
8841701cc47be9d81c7bc1dbdf37f6d88e802618
105125 F20110218_AABXXC cobb_p_Page_010.jpg
6c30ceec4b0d7ebd2c87b19de2314123
df9aa61f1bfc7f6c480295641ecf270a79f394dc
87465 F20110218_AABWSX cobb_p_Page_009.jpg
c25384739acccae7f1d72a341de270ed
8f995449ccbaf0cc1537aa96a7beb8e283935479
39665 F20110218_AABXVZ cobb_p_Page_129.pro
a78fb2b5231e61938a466fd855e66b3a
c7e0aa11be39912c8b44eaa81c09829c1dc3b977
2192 F20110218_AABWUA cobb_p_Page_114.txt
027146be5f5459853d7a53394c221fe2
172b03905bc8c7d4693820af9a1a8d1d6f11fa20
33237 F20110218_AABYCJ cobb_p_Page_145.QC.jpg
e4b1236970633f914151eda079b132d5
0579edf883a7c3ae730d4475130bc4c82903527a
90056 F20110218_AABYBV cobb_p_Page_131.jpg
7aa4b6c001c123d78a033592b133a338
4b304b7b14a1a2c6df7253487fe8210d528d42a5
18678 F20110218_AABXXD cobb_p_Page_011.QC.jpg
35b87fefd22ad25f8299e6e28821414a
3b582a05f15de33f7d8ec0749981123db10ef7e3
63099 F20110218_AABXWO cobb_p_Page_153.pro
17ffb119f517e8cb80797991c683ec64
cf0e4a7fc1c88fa840e976338e93d9d4e128b63e
55536 F20110218_AABWSY cobb_p_Page_099.pro
01b068b5f0cd3aa3292b8726b67f3f64
3211f9b9b93d22824b5ef2898b74af0aff82ed9d
F20110218_AABWUB cobb_p_Page_103.tif
f0af50562dd8f913fe43f90743941ad4
120c4c112f4881b1c8292eb7a7ccb2d680dcb178
F20110218_AABWTM cobb_p_Page_082.tif
f6d6d4e3c1bba773a0e6482730b4f979
46b33c815ad61ef67a220e1cc08d101019122e8f
99264 F20110218_AABYCK cobb_p_Page_148.jpg
4d66a6ca566ec4c7b49e557c74da7c25
5596592599c0107a474ffa429dabf9ecdf4f1f80
27113 F20110218_AABYBW cobb_p_Page_133.QC.jpg
285d18e687617748c44b5d664e48e2a2
3abf391b623449ad5317b40d4d201ffa6f4a9821
88727 F20110218_AABXXE cobb_p_Page_012.jpg
322b6281593939056f8b5c45fd96913a
35d2477e78da13e8a47a26aa50207b9ee8feb87f
58337 F20110218_AABXWP cobb_p_Page_154.pro
2cf9fd4bf130f980b5b4127f07981779
9031bfd4f6ff34d57206e5ec6467510446843350
53336 F20110218_AABWSZ cobb_p_Page_130.pro
003866eb7adac6a716bd12f6c6ff01e5
b691da40bdf6ef97e7f6d5bfd78a51e8b5c09ec5
823216 F20110218_AABWUC cobb_p_Page_067.jp2
431887b6d5d4dc46305973b23745c5ab
76b1576c55fa88c5b2cb8154850c665df6f7e0af
2516 F20110218_AABWTN cobb_p_Page_010.txt
fbe1c38891357303250714247aa389f3
a5e5a89cad96f7611e943d7eca9a60848442df29
117477 F20110218_AABYCL cobb_p_Page_149.jpg
05df1a660bc749060eaf5f024dd725ff
e004280433ed33ba157aa8a3145fdd5e16726e29
67092 F20110218_AABYBX cobb_p_Page_135.jpg
ef4bb3c42a5bdeeb6ef80768436cb63a
e7f9153cf5b128ee23648b20fc99f9b28a20baef
46853 F20110218_AABXXF cobb_p_Page_013.jpg
1cd69477877a8186e8d18876f4a2ef2c
46521faafc64ec68918d5f92c4e0943eb0f99daa
44694 F20110218_AABXWQ cobb_p_Page_160.pro
71f14bfee8b0314d2291efb8d5f182f1
6c3d39ee9a6633abdf70b62305cefc73763d15be
935070 F20110218_AABWUD cobb_p_Page_052.jp2
bc6e8965c454fd8a7814031630926f1e
acbc69ec6d57c92ba185a11901b42433df5e9484
F20110218_AABWTO cobb_p_Page_049.tif
215cac4d499d8348df074e6ab8bd9add
431a1bebd3c490fad31534206425f005506fe3e6
760262 F20110218_AABYDA cobb_p_Page_004.jp2
8fd98ca1c9c89455faf10b5dc4330ed2
a47b55d9a26c379458ce7a65d0ad43bf38862622
101078 F20110218_AABYCM cobb_p_Page_150.jpg
49ca3a447cb6aaef134e696e78738e70
d738d9433db61b189076a2d9689a2987dcc9679a
14212 F20110218_AABXXG cobb_p_Page_013.QC.jpg
2295204da5bdceb0d68ed9571320302d
d25ea23f6822e48a3fdd47fb8b876599a6ebb5e0
26485 F20110218_AABXWR cobb_p_Page_001.jpg
44007deb1506385f8c4bc86ddc1d8244
1871ff151f1e61303a0f60c2c1e458ff711254c4
1071060 F20110218_AABWUE cobb_p_Page_138.jp2
5098fe3adc353da9bded1ca197e85278
cf833202840eddeb60db21140df9249038730f58
7984 F20110218_AABWTP cobb_p_Page_089thm.jpg
1b49be584a6529f04356135bb321b2aa
ab3bfc402deacf65dcc3983dda7d4dc5aca13187
93471 F20110218_AABYDB cobb_p_Page_007.jp2
fcdc8063a57881a6f53bcfd86ddf385a
1022a702269b8e0ece25bc89c017c285573d4e2d
31184 F20110218_AABYCN cobb_p_Page_150.QC.jpg
7700265c924d2265422db5a369e11abb
5a99d65cfe255a9dc82436cb7c2d982f3f6656c4
94257 F20110218_AABYBY cobb_p_Page_136.jpg
62b3422fad6544bb57e816c77d2d1112
590a0d5c1025f8f7feb1e1fed83421b333a0a814
93728 F20110218_AABXXH cobb_p_Page_014.jpg
2f21afea9d54cbd52227b290f5385207
2b71855ebc201624a65fc2cd384bcf56ff4ce225
4681 F20110218_AABXWS cobb_p_Page_002.jpg
b954277d4cdb789a90c420b68fd7b22d
60c757d1155091a4bb1d6aceacd3fe7bd800be53
46641 F20110218_AABWUF cobb_p_Page_108.pro
0253e76b6d15b77a83d9d1dea7b16a13
7e2e0538e0410babd97d640329314ecebebcde24
6467 F20110218_AABWTQ cobb_p_Page_006thm.jpg
cada9c29e7f1cf01feaaa940e8e9fcd5
624e6444f2890eda19b58656325f1840f4993bb3
107296 F20110218_AABYDC cobb_p_Page_008.jp2
163bce8f8dba323afc32748299bc905f
88ea7d3e2cc6fba6f130a71a1b7c112ee61cfca3
105858 F20110218_AABYCO cobb_p_Page_151.jpg
54e1d563f26e1b0eba8323a541103222
5228cd76302fff31c935dd59a419090400480135
30495 F20110218_AABYBZ cobb_p_Page_136.QC.jpg
623ba2d634b5f70f7eb42405c48fb3f0
5ed89c1224b5527370a3c74636b298adc8b68391
29518 F20110218_AABXXI cobb_p_Page_014.QC.jpg
240fe1659a882dc9e8f068a513ca115b
5506b7b13103b39af01acc44a10737ab5b77713f
1489 F20110218_AABXWT cobb_p_Page_002.QC.jpg
1fe036c15304b1ce3977075c7261dbf3
2cdcb52daac42c51247cbdaf1b7ee58867b67244
1381 F20110218_AABXAA cobb_p_Page_022.txt
e23c9f2ff087481b99c462a4229ceea1
263b1dda8bf6570ef3d40aeabcfbaf18e46decf0
2221 F20110218_AABWUG cobb_p_Page_127.txt
3bffbcc0124fdf0f4e6500a1e7004c8b
b9fbe63092099abcabd27958e866b29293702a33
827 F20110218_AABWTR cobb_p_Page_044.txt
2f8df69ca9233f0dfc5933d712a55426
fedf90594b551eb12c55d384c8e32babd4bfd926
1037991 F20110218_AABYDD cobb_p_Page_010.jp2
cbabf439c4f2079c802fbdadac95356c
c12a748598fe2a58fd294ffe7f82d80e09377f5e
118534 F20110218_AABYCP cobb_p_Page_152.jpg
0db62ad38045224c0221a11251a8f95d
024f03d420f0cee58bd2dae2e0b4b3a0294ec866
36060 F20110218_AABXXJ cobb_p_Page_016.QC.jpg
48b29d1778b9b2de3ac79c1b040910ed
cca80081c4bb7d71ceaac0278108ca9ca56baa0b
68204 F20110218_AABXWU cobb_p_Page_003.jpg
5f160a521e862e43e224eca1afcb081f
29c528dac99adebf5d35687abe85725a18c88b27
7462 F20110218_AABXAB cobb_p_Page_072thm.jpg
f667f34a60f73a69ec1f15ac0aa3af7c
5a0b103e7c8059e2904c33f8d952d6a7181c88bc
8387 F20110218_AABWUH cobb_p_Page_076thm.jpg
53492c45e77ca2222baff09cde3905ea
fd266a84bb154c29af353d24556849b75a4c870f
2421 F20110218_AABWTS cobb_p_Page_156.txt
7236d136add1e3279e11826d2e390121
d0a0d694697b9d1633924d8654c8866404fe69fd
637977 F20110218_AABYDE cobb_p_Page_011.jp2
b507ed63f36776b7ab3a18ed6998c716
adbc387c87ddba8495ffd178fb1210597f8b87b1
123616 F20110218_AABYCQ cobb_p_Page_153.jpg
95dd5258894f5875d5be3415caf8551e
b1cee765cb1cecfca403e1a12a93d0a5b76a5e47
111446 F20110218_AABXXK cobb_p_Page_017.jpg
bd65b1c86879df1b90af204f59fe00af
7e1620e62d72d96e5e5eaadfc39217c522641bad
21114 F20110218_AABXWV cobb_p_Page_003.QC.jpg
9c336beb80b0be6f75da48224c881d9b
929f1c32a7030a648961880293873c5ed2e538c8
27346 F20110218_AABXAC cobb_p_Page_078.QC.jpg
6f01f391e172ca20071f222b6b0db64a
cc6445b2c1d6aaff44daf0ac6234d5247676f7bf
6007 F20110218_AABWUI cobb_p_Page_132thm.jpg
0ece639bd998f296ad2524d9cc150de5
959f3297a41e29254ee26f2aae105e9ab235c1fe
31190 F20110218_AABWTT cobb_p_Page_148.QC.jpg
9d1ed12ae6c4e3d30651fbcc12fe3242
eb5a251c3a3aa4025dea3d0db46d9055fc9b08b9
479670 F20110218_AABYDF cobb_p_Page_013.jp2
73ba129cd8c69fa6d50670a5063637ff
fa7e15a58816fd1f6d1526331d6cd730dd31df09
34046 F20110218_AABYCR cobb_p_Page_154.QC.jpg
7d372a5808451a33e4d7dad089464596
5c7f6db09524d98ff940af7309d899ac36c721ee
33405 F20110218_AABXXL cobb_p_Page_017.QC.jpg
97a6c1f172ad3573da8aed835e3eb73a
efc6d15ad7c02107f64fe19f0d70516ed9d45621
26105 F20110218_AABXWW cobb_p_Page_005.QC.jpg
680129c8e55276a332fb752f7279943c
2dd5536eb086adaa7f1c11330a3780d10d98bee0
90094 F20110218_AABWUJ cobb_p_Page_030.jpg
975e54a3eeb73b24d97eaf5a5fdbfe78
eabdbae826d7d06fc872a4a54ae2c6eb52b2c2bb
62046 F20110218_AABWTU cobb_p_Page_011.jpg
baa26b1dad63b19b918868d1c75a1caf
7ad41dbfe393c2c1497bc41bf44c55fd622d76cb
F20110218_AABXAD cobb_p_Page_079.tif
a53dbcb10f12873b69ceacd39bf4ab79
127b6702be8cd22486c4eefea64d8ab907e6c6c6
1016281 F20110218_AABYDG cobb_p_Page_014.jp2
badb4f2644c58cbf594b9ce1b9751756
f6ef1f399057be1a709a7bb0e92409ea0206775b
117550 F20110218_AABYCS cobb_p_Page_155.jpg
e69d54074688313632bcb208699a52d3
77991aa3dc06c888e0193807e347e666040201fc
17407 F20110218_AABXYA cobb_p_Page_034.QC.jpg
41039f7b17a13dac0dbbe30a1a3e41f5
bdbee7fd61cca1b821a2b314afc5216f37d03fc5
96333 F20110218_AABXXM cobb_p_Page_018.jpg
5f692f90d4717fb5dcdcc9ca43fd9d45
0b7d4da9f143028ee446ea1d937a2227b3c4c59c
121329 F20110218_AABXWX cobb_p_Page_006.jpg
dde9f199bd49b265755c223b00c36a3d
443be8c8112bd355130d9eeef1439632bb07d5bd
76520 F20110218_AABWUK cobb_p_Page_027.jpg
21503ea8160d7c4847d30a2c6970de3a
265a06fc3a1d87335368db05a70732edbb6ae54a
1087845 F20110218_AABWTV cobb_p_Page_118.jp2
9ad58748fad4c353e97d14da4063be40
12b61d71c9a38e7d10ce42eedf8e777424dc4d0a
91629 F20110218_AABXAE cobb_p_Page_005.jpg
20c213c2bff3d0f78d6cc8d0dea2b9e0
a3c3de7ec4d0b0be4051d1303efb7721a72e2856
1087890 F20110218_AABYDH cobb_p_Page_016.jp2
0b25619ba348cf5efeb44782bc6cc510
0fe73fc76982d920c8959013a2bd54374665aa27
118465 F20110218_AABYCT cobb_p_Page_157.jpg
b52c0618226eb2730578213d34538c55
795d9dd25f3c1f6eed5ddb8d7a9a6b4185d20a93
89426 F20110218_AABXYB cobb_p_Page_035.jpg
ac2cc8879087596c8c9084053076d33b
af38981e3067db4bfbfb07153f8d66df7993b9dc
28142 F20110218_AABXXN cobb_p_Page_018.QC.jpg
6e9f31a2bae3b265903bdc15b0b1ca65
7013437db144e77e7851a731653250f9b884ec58
4804 F20110218_AABXWY cobb_p_Page_007.QC.jpg
7d019d8df37855fb8336b0703ff769ec
3cbfe3d5cd4656fffb566bef76ecdaa74d7bfb88
41150 F20110218_AABWUL cobb_p_Page_116.pro
d40803bc28db059858792e5e706caa71
e21fe46115400adb2d0d064d6208b4be03c696a4
1860 F20110218_AABWTW cobb_p_Page_026.txt
fd97ec53024143086db2aa186a5289fd
9fd12f06a0cf3df734edeb765a21f5d4e2782fbb
F20110218_AABXAF cobb_p_Page_013.tif
1e4ce5e16327a4184fd006a6f1b4da12
b1f7ab7714e5fce726c4edb94a9ba42626cf600b
1087899 F20110218_AABYDI cobb_p_Page_017.jp2
1510414149cba15ea96f669e49c67031
bcd78143ca3b361721ab54ef5da47bb3c1e84be8
32437 F20110218_AABYCU cobb_p_Page_157.QC.jpg
61dad889ed04156552600f6b09795b25
4681d124a88c309cbeb2a8866304d06250510311
28735 F20110218_AABXYC cobb_p_Page_035.QC.jpg
11a44e247ac103b3f0c1bbf676735451
bdc05c7a7891bee59ca22347658a3e010b390913
110934 F20110218_AABXXO cobb_p_Page_019.jpg
3494db304bef60ae2a3f2cd3b9ad1111
4991a626c012c185cab86b2f55c8d368a84cb809
13251 F20110218_AABXWZ cobb_p_Page_008.jpg
2678b1f13a473b97577711aa0ea3e3a2
abcc8cdc74307659471384a9e39bc8ec2fff0af6
1671 F20110218_AABWVA cobb_p_Page_045.txt
537d029e03118d492916ad53af565d02
5e824ed8fc7010d67d569822b082b657fa94aa64
F20110218_AABWUM cobb_p_Page_070.tif
480bbe77bc7a2b6b3b55ad162526eb36
4435ef03153c5fa3ffed46fc3b7be0fcb1feaace
55727 F20110218_AABWTX cobb_p_Page_155.pro
ed556d1aae1802c7a27c2c05a321c6d2
527ba30c60b7b73f4b5c1d26f30409194c771017
25286 F20110218_AABXAG cobb_p_Page_034.pro
9c1d2283881c819f1a231d64521e4a60
f651ec3f6e0a8368f3ad890503610480458d478e
1087895 F20110218_AABYDJ cobb_p_Page_018.jp2
0abcf19901041e40faa5f3e44418cdd2
a8d9e9bab0192192dbd8277a1a35fbc33ee6ad8d
3348 F20110218_AABYCV cobb_p_Page_159.QC.jpg
0963dd1b39794efa2dfc4ea22d2331c1
2bc235b8bfa84825d069350c479e7a5b927fbe87
65313 F20110218_AABXYD cobb_p_Page_036.jpg
555b1959cc08d80f266049341f8f342f
7b9c6338b022dfe17911add55d6ccb4b4774c2e8
6903 F20110218_AABWVB cobb_p_Page_056thm.jpg
2cb499074cba35167fb88e8b0339bd1e
f5bd46014b237cd362058d1a1fab8430bb9722bb
F20110218_AABWTY cobb_p_Page_071.tif
cff288865abd25d68db65f586dc2d641
a4230dd53257e81257c91368767b7f00e1053c5a
43646 F20110218_AABXAH cobb_p_Page_026.pro
51da1ddaf4e3883c09e6e1a9ee0fd48e
37dc107fee117eed13f4bc976869138ddd600dbc
1087798 F20110218_AABYDK cobb_p_Page_019.jp2
f7c16253f6033c6d34a2f1aea3bd0171
cbb10fa1b4a30c959d6c592c86863309b1ca3d35
94146 F20110218_AABYCW cobb_p_Page_160.jpg
fa70b30552e91eefb551deb6f91b81fe
17dcbeb502ac97d4a1d7dd3641722d5ccf26501d
70817 F20110218_AABXYE cobb_p_Page_037.jpg
568d8155a1e35933f429646c330add80
0fbbb2e12cc12d677041fa1bf65e0139857a0890
34509 F20110218_AABXXP cobb_p_Page_019.QC.jpg
5887010009e068a9d22401867f1609b4
83c1e69bc2965379291a1581adc61671056a27f3
28812 F20110218_AABWVC cobb_p_Page_140.QC.jpg
0d06304df012d5a06ea59ad175643974
dc2b6098919b33fdb9ba78252fd990fd5ca595bf
F20110218_AABWUN cobb_p_Page_064.tif
39e22a0e80cddb80c4012ef2d85ad14e
5fd3000569ae311b181e336adc3421f59e5d54e5
F20110218_AABWTZ cobb_p_Page_053.tif
59d6569383e9a34c16b21af2a1d43b43
7aba2b1aaeb670486b96be487642591d60549d45
26758 F20110218_AABXAI cobb_p_Page_047.QC.jpg
4336502f67d2c9cab132f4dd503e2fbf
0fa44c83408e90147a0a3b30eb9a316faa77c3e4
983959 F20110218_AABYDL cobb_p_Page_020.jp2
bdcf867b56075f064bec974ad5e0321f
c7da78a1eb13d31a29356c5f3a7b49b93c965041
29513 F20110218_AABYCX cobb_p_Page_160.QC.jpg
ca81c0613b63a6e5183f7254d47330d3
f4eaa8f078b2738b3e130df5c54fb7c2c8cdde9f
21863 F20110218_AABXYF cobb_p_Page_037.QC.jpg
b4388ff89a28c3125fa735f089ee92b0
d55b2faa55487be095c5c4346eb161eed25c9f4c
26858 F20110218_AABXXQ cobb_p_Page_020.QC.jpg
16ddeca40c65024088f46a818bb1dd67
5c33154ebbc9aa2a4d2f41ce798fad8f94bd5347
1061592 F20110218_AABWVD cobb_p_Page_148.jp2
c0c1aa2fa39fe2e201d5e35d251c1385
1c7fa42e83736dc490b9c9076b2262435b008aa9
7819 F20110218_AABWUO cobb_p_Page_145thm.jpg
4a95383f6f1fae0c6a5a5ee3fea1f15f
ea1564c82f9d6f7a6ded275cddd59f07b79a7ab6
F20110218_AABXAJ cobb_p_Page_124.tif
c41584e1357d3b979902fdb2af09376a
4783e569dd7be7870f07ae9132e18ebddd7d62a5
649487 F20110218_AABYEA cobb_p_Page_045.jp2
19bdb291dce3f10891f6bb8611f325a9
f2d76e04227505acb679f0ade10b6d5c0c083e5f
1087898 F20110218_AABYDM cobb_p_Page_021.jp2
f5a87e9f9917a55e4c874194c3ee0c33
37540748465ec42a194c2accdcdd9567635308a9
236259 F20110218_AABYCY cobb_p_Page_001.jp2
6dfcca140e67390f0208a5a08ab0cee3
c7cb47bc94ab36860b71d4a403311f5eb8dc8a5d
25359 F20110218_AABXYG cobb_p_Page_038.QC.jpg
5afc058d54da8c26393610de4c91f76a
65811c922ac1a485c39d70d9a8ccbada796cf9a1
33504 F20110218_AABXXR cobb_p_Page_021.QC.jpg
6fb149f601b7041ec01cef4c41a609fc
cdb040b672e2018ce4239da67f3391cfffb3ad36
27111 F20110218_AABWVE cobb_p_Page_104.QC.jpg
3af1f759438fffa60ae6b71058ac7053
52c64e85b18d56072335898c4eb16afb55757e9e
F20110218_AABWUP cobb_p_Page_083.tif
5e05b76b42281a847ba249a56b46bd19
d335f834bb446c709e6b5d08455b29576a8748e3
88767 F20110218_AABXAK cobb_p_Page_053.jpg
df50078d1042068e32a20d0568aabb5f
c261d8447c525952724ed17dfb55921694a2f3a2
685925 F20110218_AABYEB cobb_p_Page_046.jp2
9b9bc7896f9c8e0a4cb5a2e7a031bcaa
6e70f400721f61b2cd8b1ebaa16c92884612738f
1042369 F20110218_AABYDN cobb_p_Page_023.jp2
b713902d1443f10069dc49b8dc497076
a09496e7baa4cb7a26b17aced842bc4b58f5bfe1
58539 F20110218_AABXYH cobb_p_Page_042.jpg
285adbfb3eb2c2155256246152b1ae4f
dc8626a5ecbab3181599a2c9de43370149c60d73
129334 F20110218_AABXXS cobb_p_Page_022.jpg
9fc22d7be655e9216a9c7a7e7390c7b0
ff1c1612c52b2fa2c5b56df0760a2f1f11d26547
1960 F20110218_AABWVF cobb_p_Page_117.txt
2397f95dc88e96b7000b0af72e956f2f
ea3e8aa2c94c1c852d69a37ef7d773db6c198fb4
F20110218_AABWUQ cobb_p_Page_065.tif
332b92e06f94e0329ad6c41eb85d9677
ed9f558f717b1e7d70cd46f083377d5be567bd80
8216 F20110218_AABXAL cobb_p_Page_118thm.jpg
374825edde230237ccd8770d2865e936
3ea947aed193fc7af258a06368f98a9f0d65aeb8
884380 F20110218_AABYEC cobb_p_Page_047.jp2
4e8456a1534f2ab2eb092923260e3955
bbaf0404cedabbf453f609f3bd30e815ff69af06
1087880 F20110218_AABYDO cobb_p_Page_024.jp2
9d7bf23764215656e408302023b126be
e46a4814229475f10a93f25a71bcc5be186f54d4
708819 F20110218_AABYCZ cobb_p_Page_003.jp2
df561c2eea4d167f0bc40d274fa16273
0747b7023c8d3e9699692f8dd314c8bed815939e
17381 F20110218_AABXYI cobb_p_Page_043.QC.jpg
3fd16131dde5a93301038dd7ff0e9c0b
b144a031c41bfbfe7f8a7c033c46758eef8d0b8e
40146 F20110218_AABXXT cobb_p_Page_022.QC.jpg
da4ea0a9b25951c788df318a083eb66c
f31b7c97c0efbd0e889a9919331d5b062caf0d78
47249 F20110218_AABWVG cobb_p_Page_016.pro
4e642e874b1ebc1896c2a547d26a30c8
46b369d5d7efbe4fe05877f0f83fe0a2f30f0199
F20110218_AABWUR cobb_p_Page_134.tif
e8ae74317e8809824d15941b31a85576
e652f2e0f8215ddb50e6ce28a80564cc2455bcf0
32780 F20110218_AABXBA cobb_p_Page_155.QC.jpg
5daa0c3e90cd18eb869f436b3753ad66
b257a2a01d962b84af88f96ee8177223413ea0ac
1415 F20110218_AABXAM cobb_p_Page_033.txt
669d65494d2b559bdd28afd33ede56f6
c91fd47b8cd3db21258cc613fcb74c2c3776745e
815690 F20110218_AABYED cobb_p_Page_048.jp2
316b614cdf3d2eacdcb5647b2c07bed9
7e4346984297c8d197cf3677b0e12d800a1db87f
796576 F20110218_AABYDP cobb_p_Page_027.jp2
4c82ac3b80f265bc93aa6af416f9968f
c0edb0b06b1b9cf161e09e6e9976e0b8744f1916
16631 F20110218_AABXYJ cobb_p_Page_044.QC.jpg
e40b41a8adf85b79e5f1e44979db7a0d
10cca7f78bae62b1b3bd5f94089ed72e5b605738
82051 F20110218_AABXXU cobb_p_Page_023.jpg
29dde62765653d1a5b88a082ac16af8b
446a6a9cdfc4e6388dc7770883401b3324d19c7c
86899 F20110218_AABWVH cobb_p_Page_091.jpg
2d029a1bfb40e7b0e6789c0ad7234d32
bc80568c617e0649e0a6f0bfe0ad70798d362b5c
F20110218_AABWUS cobb_p_Page_103.txt
c9d2c7a264f6094838e94cda19f353d3
7a877e94543f84372cefa05b58b357b8c4573f3f
5448 F20110218_AABXBB cobb_p_Page_036thm.jpg
0f30f4a0c9a77762edba5daff50a4696
5f8545386471663ca4e3a644d92354228c3203c8
95491 F20110218_AABXAN cobb_p_Page_081.jpg
766e13d15a024b5c04a125bc38f193d0
b261ee03b7f27126827db765378bdbb700c1805a
F20110218_AABYEE cobb_p_Page_051.jp2
a883334255548760aeda4ac684bd70cd
8547fa62438c508f59401b89d85e424feb1db5b5
552714 F20110218_AABYDQ cobb_p_Page_028.jp2
23d7d52120afbd5010a35ef39ac1f8e2
acc1cfecac5f7794028c30537c64f2cc17f4491c
84774 F20110218_AABXYK cobb_p_Page_047.jpg
e57859a795e2ac7a2e11cf8e380da48b
fd740340840ae0f029f968017e14bf9b70937bc6
108441 F20110218_AABXXV cobb_p_Page_024.jpg
db948a1228c0edec231c2497c0c0fcef
75afecd3a0fa549f4b833b076d1bd095dffd9c64
57665 F20110218_AABWVI cobb_p_Page_068.pro
49f1c82029c9309339ab5e4f7e6c32bc
b050e855be360dddbe1a7d8e9ddf008007898b22
1032226 F20110218_AABWUT cobb_p_Page_079.jp2
6abd069d1f7e9362541611fff67a029c
c7c3b346ef618865de92efad10833fa03a67e8cd
F20110218_AABXBC cobb_p_Page_072.jp2
06c26448e93cf557b9187299759d44e7
c86e23577ff2347f148e8e81da3ab77ba6d144de
1087884 F20110218_AABXAO cobb_p_Page_095.jp2
c31a5eee3d32fcb0928a9c66d9763ef4
39bb3c826c8604d7441c0d3b6af9aad3f159d8e9
886883 F20110218_AABYEF cobb_p_Page_054.jp2
fddd581857b069693e9e945e467abeaa
50a0dc7cede368edb6f9eac88f8de794459af796
920733 F20110218_AABYDR cobb_p_Page_030.jp2
2fc6b1361c40d5cbfec8b71a5d3059fe
556ad0ee27fdbbc3504b8123a505a0c867d5215d
25849 F20110218_AABXYL cobb_p_Page_049.QC.jpg
8c99bd4a4069b017e3eabeb6108ebf27
563bd9e310d0e8cc9adbf6c7303ffd885b82aaac
33833 F20110218_AABXXW cobb_p_Page_025.QC.jpg
b72b4713a15a30b0a2a0241b010f4ec4
ac705338fcfb8448c89d3e44f0b5d2eae0c02541
108646 F20110218_AABWVJ cobb_p_Page_128.jpg
59314c19e9ef965ff7d19c5ecec1706d
bfaafc13df143a327fd1362403c326806751f6ea
34618 F20110218_AABWUU cobb_p_Page_039.pro
e83e61e697e70c31d50c7f5d9af47eea
593de261c5a4a2e1af789699ade72ed0a1630b12
8592 F20110218_AABXBD cobb_p_Page_016thm.jpg
ff1028d8a552999d9f6dbcd5fe7738f5
dc7b7c03a06989226fbada35995008c7d4a4d4bb
6722 F20110218_AABXAP cobb_p_Page_098thm.jpg
8ade1557e3c1b33f75acad25c5b973c3
5c86954c10c7cd38444b11ef89f1e78017327a98
1087882 F20110218_AABYEG cobb_p_Page_057.jp2
4b6ac44ff810cb6156e0e639c44a2c7a
73e0506679c23653fcf48bf548454f7d95c16254
531584 F20110218_AABYDS cobb_p_Page_032.jp2
aafca2222ec2a79435ee49854ddc485c
43e7dbcc1f87c4cb577fc98d64ce99733e97846d
110624 F20110218_AABXZA cobb_p_Page_061.jpg
9b4792879e80723dd4faf391191cc4f3
a0ced9a6d4f6051346f38a2b133a15b0f92cd857
97977 F20110218_AABXYM cobb_p_Page_050.jpg
2b08196e28c19a0ac9d8dbe21a3f5888
047b474e52796c90f622fbb9dd053a20c506f7e0
18088 F20110218_AABXXX cobb_p_Page_028.QC.jpg
aad2c24d355d43b9c15cd34cb736c5c5
eea5439375a0b9c1ef667c425704e57959fd68a6
6802 F20110218_AABWVK cobb_p_Page_110thm.jpg
293e5f1832867cbee66926203c69e4e6
2c65983dbcac96cba7f815014cff6965a9851377
1561 F20110218_AABWUV cobb_p_Page_019.txt
99af56410d3357b6dc3f1d69e1f76910
a8f91ea128ac947fbe058359d492850f477e9d25
F20110218_AABXBE cobb_p_Page_111.tif
31b35e14df3e618ff2ca667078817ed8
9b339ba068573a4647afc1d1c9383a15ea82f28c
55398 F20110218_AABXAQ cobb_p_Page_034.jpg
5225152961c8d864f9082f6843b46527
9d6fc470273d095de4763e9036b08f50349749a0
878475 F20110218_AABYEH cobb_p_Page_058.jp2
3b563a0a6a8bbf408ffa96de2f91123c
e7deb342612d641addf2fff0ad5f0ddca45b29d2
563895 F20110218_AABYDT cobb_p_Page_034.jp2
cd250524b4701cc33d35216427377726
8d336a366555e1e6d286718210e9ff9b3b276733
34336 F20110218_AABXZB cobb_p_Page_061.QC.jpg
98c1752d43ee5c1e9566445da57df351
09be2e9ea327e5c4c07b4043ae43ab3667472af0
29790 F20110218_AABXYN cobb_p_Page_050.QC.jpg
896e4a02542262454edb99bb1fbdd5e5
f443f8cb14c7882695bcb5689343afd56c25c12b
67627 F20110218_AABXXY cobb_p_Page_029.jpg
22b407a5c3d4147559f93522123386ba
69b656e7b0ef7f4722a56ae726f8e6bde34c2dd0
6501 F20110218_AABWVL cobb_p_Page_091thm.jpg
dbde378cc4e5bc923dd5f93047904939
4553c99ab703f2bf0b7f757771a217200c494c89
81496 F20110218_AABWUW cobb_p_Page_112.jpg
5e0b6765a0c771aa26c56b9a9f90e8e2
253f3fc384f38923066caefc538503154c19fccf
7862 F20110218_AABXBF cobb_p_Page_130thm.jpg
ac8f733f474ee48e4ddef090f2934d02
24498e6b61a90f150e7a8698212414f9d2e7a683
F20110218_AABXAR cobb_p_Page_089.tif
9e7f4ac002d90a563d49f7106ac1a0ea
1b0b6a23648c9b61cc7b82a348a75f895a884d9e
1054971 F20110218_AABYEI cobb_p_Page_060.jp2
7470f5112eda6715c315c210f3fc3158
9d7d255f0cb6b9a750107d450d383cdb0084192b
720523 F20110218_AABYDU cobb_p_Page_037.jp2
f75a3c54a2427a85f646e0f89c067c7f
8bf18540692bfe6ac1ce5dc2b70cdbab2f18e6d5
110759 F20110218_AABXZC cobb_p_Page_062.jpg
5f7e11f8966365023458e65f3b7e7f24
28959aa486277b97f59f17ce43ec593f47560c3f
27950 F20110218_AABXYO cobb_p_Page_052.QC.jpg
d5fa8d6cb1c170ca8ac5d30005ae2711
9b9330733451aee7b1091efdeabd40c38bfe8219
54257 F20110218_AABXXZ cobb_p_Page_032.jpg
0594e7ce9766ed26ec669325f6b472a4
badb7a5d37a9d90894166863c12e18fcebba65fd
F20110218_AABWVM cobb_p_Page_033.tif
b5d24cf20dedb14eb2c835812ad87d61
c2ffb2c6156642d81035d24eb6426a95c99a094a
F20110218_AABWUX cobb_p_Page_136.tif
1375f4376879267608166780ee36ba56
777030d7751ca57d377b037a454b84be467f7f26
F20110218_AABXBG cobb_p_Page_073.tif
1a8da21d9b4f9c04732c7e850479b29b
b86555b39eb8b63c1ef3f481f2f19ac317f6b60b
43977 F20110218_AABWWA cobb_p_Page_040.pro
e0ebf8e0cf129a2ab88918f42366de0e
73a3ae482bb76096c9ce16b22349b4ec376bc958
F20110218_AABXAS cobb_p_Page_016.txt
bf4bfef5fcdb4a7a2d01b840821cf64a
23256b9a340569a66613f6a2a909ac9982efd01c
1087888 F20110218_AABYEJ cobb_p_Page_062.jp2
4d573c18f445865c2bc51f2bf213472b
e9679ec01bb0530a1561a451d9a98335c73b1912
777019 F20110218_AABYDV cobb_p_Page_039.jp2
f2db6d57e6db8ce2db9e1610ec82a0ff
2217e004ede6c75a22d16520a758ecd48e14329e
91609 F20110218_AABXZD cobb_p_Page_064.jpg
3f8528ad18554c60fc27fa11ddf17310
2351b9e296e0be65149cb374f3d84c3d3e872d12
86472 F20110218_AABXYP cobb_p_Page_054.jpg
dbaa572eada2dd4f530b98ec8358e188
d9b4ee24636bd8a99490c901d3b14bb624de7f56
44155 F20110218_AABWVN cobb_p_Page_084.pro
cb8c97011b7418fcdcdf84233653f473
2ddc34ee3eb8d36f087c2eeb71115727b5c4b082
F20110218_AABWUY cobb_p_Page_129.tif
e540493635c5f43f5842bde6f4ca0fdd
540dbeb87ac11b72ff9fcca6ffa80f51339a5fc3
79834 F20110218_AABXBH cobb_p_Page_015.jpg
a5eed3e8f66838e4080256ccc900f726
6fafae6c8f25012590642a0852787458975e7853
F20110218_AABWWB cobb_p_Page_125.txt
8f4ae1481aee3989bd860ccbc48b7036
a5c0aac51061976c11e5721ccf3afbc53849d35e
56407 F20110218_AABXAT cobb_p_Page_080.pro
b0f765ad4c943449dac72d3eb63089b2
02f176b1a72637c2fd2f5be938659fe11a984630
919739 F20110218_AABYEK cobb_p_Page_063.jp2
dbd2c90eaac589fe125c2e1de936debe
813f00743ef722554dbd7761d10c6f6d605c4acf
1008032 F20110218_AABYDW cobb_p_Page_041.jp2
8d61de8009b98c73dca3965026a6735f
7dac365711aa32d0f900501bcf813f0a67a826d1
78821 F20110218_AABXZE cobb_p_Page_065.jpg
ed190042dbb0e6478c1ed801b9eb434f
de893ff7e60c49a7ccbbc440203ff3cb5ed437a7
F20110218_AABWUZ cobb_p_Page_043.tif
efdc1c220de2dcd88ea3f45a16804ca7
62a13270fcc89de703dd848411421cf0d6a075ea
F20110218_AABXBI cobb_p_Page_031.tif
014dcf9e4183ccac7b9dfb679a13691c
62a7636e4907dc43865f5367a5f6cac31e78deca
6773 F20110218_AABWWC cobb_p_Page_134thm.jpg
aeeed5dfdd91474afdd341b9581ffb88
8b0fc20cf553a0971f9e51e48d8ddffbb84eed14
F20110218_AABXAU cobb_p_Page_145.tif
9f6d06da12d3707b8e867eee36a7b21a
8afa6df4e498bb049e499667d0a3e62ae83fe718
981066 F20110218_AABYEL cobb_p_Page_064.jp2
9a11f968fc147ddae28b402befdff962
68f0fe492373d1609fd8233f85f2d5d3a63d9c36
609831 F20110218_AABYDX cobb_p_Page_042.jp2
0660ad12ca57f2d3b1c55600ec735f2a
44756a911707bdd9a66d5d61584fda05716a07aa
24804 F20110218_AABXZF cobb_p_Page_065.QC.jpg
6ae04414ac2b33205e116bf7a6dee25b
106b9e35bb5ff43e905529e2bf9ea7ba7602357b
27316 F20110218_AABXYQ cobb_p_Page_054.QC.jpg
1547ac09cd1c30556de918a39366c501
e72d682cf35b88387e8997b1693a50b9de0eb125
F20110218_AABWVO cobb_p_Page_009.tif
5fd4e468f0a873fc7bda8dcb4a175930
d79c1455b5f3fc7875d6b0fbfe1d0477ee6575ce
F20110218_AABXBJ cobb_p_Page_148.txt
bf5bbad4eb87340264c966635bcf92ad
6d77d19ddc5ff77ff600993d82e85ca8c8ca1a47
F20110218_AABWWD cobb_p_Page_084.tif
26f25d0b437fb1f2d2bb00c4bffe6773
4749c61e39df8743d7b7edbbc24647ab4fe6a78d
28732 F20110218_AABXAV cobb_p_Page_064.QC.jpg
bad1508e751ed2b07b1141f6cd050035
ec842df80f40f4f6536116f2613a8eb9f402ae65
1087871 F20110218_AABYFA cobb_p_Page_094.jp2
0e7db7e879b3fa1cebf0b8f50b8571a2
1af5ecd91bd1c76ab6951ba0e951542b46d579fd
830650 F20110218_AABYEM cobb_p_Page_065.jp2
1e241267599db5b4d7dc70fe28e3f631
745f6abd43b51ac5050e13a6cbb4788e0efc9f33
581971 F20110218_AABYDY cobb_p_Page_043.jp2
d425a36a6b8db7fa7877d6635874d689
06484ce2adb7cba65b6de7ee12326f5c9d1b6eab
80152 F20110218_AABXZG cobb_p_Page_067.jpg
22b06a0b62371743576ba145aa40903f
6a4fb17bf423668f07314a419869c23b84ed85b1
90149 F20110218_AABXYR cobb_p_Page_055.jpg
80b272f1de5ffef5106efac6b3ebe302
e303a4f36fddb8e3f11907e9257c126b5e5cb5f2
28280 F20110218_AABWVP cobb_p_Page_151.QC.jpg
33669027638c6d716968a2226102c775
63a9cf00096a764aa71193802f7e929698fa0c32
45547 F20110218_AABXBK cobb_p_Page_098.pro
07022675a89ecad58ed94edb4cbd6179
190aad0fa35eff6c2f194eb4a4934ddd5b2d7db4
F20110218_AABWWE cobb_p_Page_112.tif
8d5d507009e686579432708930dfe18c
77ce8404a0bed1998b08f5121f89137f4f6b87c5
867438 F20110218_AABXAW cobb_p_Page_074.jp2
576f6ead44c05173a81a4e583a12a921
7a02d79336481fefc90c26065d2d642004d5810e
216669 F20110218_AABYFB cobb_p_Page_097.jp2
e3d92d6206e3e5859664668418822366
f4dbf187d147287f8d9447f467eb4e089d21ab76
1087830 F20110218_AABYEN cobb_p_Page_068.jp2
77837e62b25002898cf67ca985398b6a
135f9b1b718841c56dae61c27bc328510da2b73e
522938 F20110218_AABYDZ cobb_p_Page_044.jp2
caf688f6e786b678dc8ea00ec18ce290
dd0f7cc4ca2972df8843d32b4625ce81ce0be828
25011 F20110218_AABXZH cobb_p_Page_067.QC.jpg
2fcc62ab101208d8ebd29580b144e10b
cf1236843aacf09785ca5672429dd3d76d82dec5
26368 F20110218_AABXYS cobb_p_Page_055.QC.jpg
9449af97d4c1138f127ac8bfc43635c6
470c30e75c3b5e6be78a1d35a37c9183095319b2
4792 F20110218_AABWVQ cobb_p_Page_159.pro
7ad7359ae11a7a053c873ef65abebc0d
a66cb20200e2606a02fef80670b557f069ae349c
53313 F20110218_AABXBL cobb_p_Page_082.pro
4531446685357ad6426bae5e42c82254
e86fb7a239d0b57ac8acd84004f0d1cf9427d9a8
6580 F20110218_AABWWF cobb_p_Page_026thm.jpg
7c8765b0aac33574e8f1f8c0aba8c949
bb2478ebf37d33f127ea99b3103fe7811f0c7c71
1752 F20110218_AABXAX cobb_p_Page_029.txt
5c1bd813f24a69459eab83b5c14f9c94
1f9615adc66dcb70108357f8740a4ba1b507a5ac
823603 F20110218_AABYFC cobb_p_Page_101.jp2
ebf023adf483a111c53a4d81f7178d70
bc068335d8bfa0c158d617bdd95c9ae4dd713520
1087900 F20110218_AABYEO cobb_p_Page_069.jp2
7d82a1ba7fcaf01da723d1e4acee35ad
89993acd2c59cec034fadac141e6a09409b4bc39
108504 F20110218_AABXZI cobb_p_Page_068.jpg
c8b2a6917ed1552d9fb71f34a510182f
b1af600e43a36e0d53f44e749e5c29d9f05e994d
87556 F20110218_AABXYT cobb_p_Page_056.jpg
9b3d7827da8875ac7843f037157273e3
faae84a09e07d12f47f2b8c81f924b757abbfd7a
1087891 F20110218_AABWVR cobb_p_Page_093.jp2
1ab2de169c68edbf2ac4a9f450e1ce04
9805367d9d4c7147c007ec270523f67fad901638
68817 F20110218_AABXCA cobb_p_Page_031.jpg
a78ec3537a09145be24b6dc860fed7af
27e9ba734db99067c9da900b95c8f6d5863f59db
1087808 F20110218_AABXBM cobb_p_Page_006.jp2
9a79a27b4bab968aadda85e697a2345d
f612a1405bdd29316bab60272f3c967a84352689
6610 F20110218_AABWWG cobb_p_Page_058thm.jpg
776d7b6e5369402ac94b4574bf03cd74
56dd0e92e27c97bcc56ae09e5907c0e3537299d3
898216 F20110218_AABYFD cobb_p_Page_102.jp2
9ec95de4538ef874771e8c97cef84db0
fbfe6d83d29dc67ba09f76a104589d711e1602ce
917756 F20110218_AABYEP cobb_p_Page_070.jp2
2ccb25f22e6d3c08a879d2866bd86e5c
48553e3348a23b4e920d3e1647905d85c0e64c7e
32502 F20110218_AABXZJ cobb_p_Page_069.QC.jpg
39e08d9d65814e3feae76649808e1b19
d2e8a7bf01c24cf7981bcc2a6d29083aeabf090e
27161 F20110218_AABXYU cobb_p_Page_056.QC.jpg
8047adb680e839e3f140fdd226a826a9
aa312016ce24fb043626dfed27fbe4973f1c7949
26042 F20110218_AABWVS cobb_p_Page_042.pro
0c6af23538d554ccf01ef63e65c9ef02
ac71df4f751d135910d9e5c21b8df1d9617d18c1
91071 F20110218_AABXCB cobb_p_Page_084.jpg
2395679bcc2193a5e4925c17b7db7b8d
0e33d65945f0c16505e0c88ba5a43d145d30e5a8
1787 F20110218_AABXBN cobb_p_Page_071.txt
56752be4977ddad76dc2d382da2b74ba
aad2671e052cac563950c0cc795e44cb92e8abfa
90676 F20110218_AABWWH cobb_p_Page_088.jpg
d49246bd0c84af4d91eddd307b576010
6285c622a2a490c6251a2b2fbbfce4aff8782281
47734 F20110218_AABXAY cobb_p_Page_151.pro
41104d8c7cee9fe197911ec31415e18b
0c37a411d641c54e61d035dffb1616a0dbe5ea4e
920407 F20110218_AABYFE cobb_p_Page_104.jp2
c108b7b29ba4ab74826225017efde533
976cbd792e0cec64b6ff4781b7c0ed982c5d50f8
914739 F20110218_AABYEQ cobb_p_Page_071.jp2
2c53eaae32c9c9e266fdeb2f8617afd2
26e9602da56edda65531d9b455fcafba0f743286
87137 F20110218_AABXZK cobb_p_Page_070.jpg
4d2e8d01672b0a9aa864a7cfc67d1ac2
7855ba13300fe681aa398cc0f158d55c0fd55062
104018 F20110218_AABXYV cobb_p_Page_057.jpg
b0d563dafb11c7d8ad6827beb53030b2
eee126129c0553e8ede4f465c580eaa3875efe86
23013 F20110218_AABWVT cobb_p_Page_135.QC.jpg
501a121d214f8e78eee4fcec9729c7d9
01accc21f570ce739d086723cd0a20026802a011
7177 F20110218_AABXCC cobb_p_Page_071thm.jpg
d4aebf61a75ff3edb524e41eac2ba228
fabf19737e82d7ad7fb8ce65b4a7a1a69bc9d21e
26791 F20110218_AABXBO cobb_p_Page_018.pro
8f34108f89f6c8283b858ac6499bda0b
c72e51ed1eee466e3c815a520ec1e15eaa31b669
39598 F20110218_AABWWI cobb_p_Page_123.pro
ac2d8840d6ab7a845e651a143d0a3734
c05306a236bd02ed528aaa9c2465acbf1b447ada
21147 F20110218_AABXAZ cobb_p_Page_031.QC.jpg
c51bd18b9a63cf586db2d06e88b31a8d
3cd187643ca7eeada7005aa9aa560489f3af4000
966825 F20110218_AABYFF cobb_p_Page_106.jp2
216495c572f74cbdd3779877308e4951
d0335fc877115d38818ba2ae6b41110786fb7a09
F20110218_AABYER cobb_p_Page_076.jp2
c1752f2dd9f3b7589cfbbc11a10d4d80
53bb7e133f0340891cf2e2ce79edbf7d2d524e1f
27716 F20110218_AABXZL cobb_p_Page_070.QC.jpg
27119778f45dbf80fba7bdb2964a3952
e6d087ef24da28217c814471fb469b8264248d22
32987 F20110218_AABXYW cobb_p_Page_057.QC.jpg
eaf1ae776e743b7fbc109c06e55d4530
0bc7c38d97c2acbaabdebf446d010bb1d32aa728
2091 F20110218_AABWVU cobb_p_Page_057.txt
b7eb3527ae3c4b40c74b35a85898ca2b
d92998aba29eba5c4617616f3c237acabb5671e1
86971 F20110218_AABXCD cobb_p_Page_104.jpg
cfdd6cc9b63836bccecb5e4eb4b63c04
54ebaff956859d932f7685665577b2c41ac05f2b
23799 F20110218_AABXBP cobb_p_Page_085.QC.jpg
e9543206b587477ba5dd6b3e2c1456f0
b08b340bd2acc6c89ca91684d22339e36588dc53
27736 F20110218_AABWWJ cobb_p_Page_066.QC.jpg
6b357e8aef2ba6e49368832cc1d12d7b
07396ff4f0daba6a98d8381e8c7a6f5ab9768ba1
909450 F20110218_AABYFG cobb_p_Page_110.jp2
2cb1a2f667d5c64e575409b14fc68acd
a93ff5f3128d1cc7a7d4aca81ca93187ccafae3c
F20110218_AABYES cobb_p_Page_080.jp2
6c4a04c9bd7385e0f1cd763351c2c0fa
1a03be819f8d2d72191ba7376ee9e1abf012f721
28242 F20110218_AABXZM cobb_p_Page_071.QC.jpg
295606990758f03cabef6101e96dbfe8
3dbc8a1ece715af0cfc352aea0a1ff6cfb9e8a4e
86279 F20110218_AABXYX cobb_p_Page_058.jpg
d2a12b282b2e537896f2ec9fdaefb83f
4773f1414ce32155a16dd0c35e0302b9590452b2
7959 F20110218_AABWVV cobb_p_Page_137thm.jpg
e04c18c83e77fb4db17f3651c0d3be72
4d6dfd6d19dc3e5acd3b2488b41a5a5a369478ac
2129 F20110218_AABXCE cobb_p_Page_119.txt
d33689bd9526ed7bdc783e24f834d3be
27c3abbb9f1cc01a9c4084dbbbc468304ab6a4e9
1800 F20110218_AABXBQ cobb_p_Page_097thm.jpg
699f02abd8e35961ecf6e4f52fce5c14
447de1ca61e5e1583fcfeee41231edc8324e2701
34869 F20110218_AABWWK cobb_p_Page_153.QC.jpg
706ef382f995358868046cb7af7b5ab5
626fcdc7b1420ed2e60941e1846b5608d8f335ec
959635 F20110218_AABYFH cobb_p_Page_113.jp2
1b26d2803f47368f1a622989d91d9497
c3a6a6985cc9716fd22764b059b52c3557adedd0
1087870 F20110218_AABYET cobb_p_Page_082.jp2
9f8ad190f13c7706c53d8dc9341d0996
34d781a4b9ab0ba84ba430afc6eefad99bfc633d
104397 F20110218_AABXZN cobb_p_Page_072.jpg
b6b3830fb4d8b68e4278813a22d105f2
1f8b730c654eeeb53f79c0484469334606b69dc3
25907 F20110218_AABXYY cobb_p_Page_058.QC.jpg
854413b0afa9104f0168692dc93bcd1b
22892bdb54d9761ced2d8bba4e3911ac1dc08469
92731 F20110218_AABWVW cobb_p_Page_096.jpg
973d7d4cc9534d7c75cbe90ac4eae3f8
21266b731864feea7b6e081c2acef711f67f2b6b
36393 F20110218_AABXCF cobb_p_Page_085.pro
077dc04804cd36569b45d83a260bbd41
a056a4e28db9bbb3f3bfe35c517c4eee528921aa
29667 F20110218_AABXBR cobb_p_Page_041.QC.jpg
500a9d64cd0d7094d5e0e5e8d5a66a2a
ff465b08816d2d4323027efa9c0fc12314e6f3d8
76200 F20110218_AABWWL cobb_p_Page_132.jpg
a6a0542561e3fc2b1a19c4cc3ecf487b
bc39856ab9cc7d4496a9c1f06c77745ee20fdc71
F20110218_AABYFI cobb_p_Page_114.jp2
699e03b2fd64912e7a3731044bce5b03
043ce261ce433ce066d6d83c5fc01119ac7418bf
960005 F20110218_AABYEU cobb_p_Page_084.jp2
558a24d21bd6249f2fcd6a3795314720
f44dca2fe48619399c5d18117bb60b36dcd5546d
112947 F20110218_AABXZO cobb_p_Page_073.jpg
b5aa48a9f896b8619bce707ea52ddf15
0e8565c83fcd2f1929415c9e43d80dafcea97b2b
100009 F20110218_AABXYZ cobb_p_Page_060.jpg
ae6ac99742a32a99e4910a1697abc48e
3afe94e5fce9e1f458d2d237fe2e4faecb80d638
1087868 F20110218_AABWVX cobb_p_Page_050.jp2
c1cc53bddfc41280ffa365adbc74606f
d85b0837cc222dc9aab85010e854a2ed43660189
51295 F20110218_AABXCG cobb_p_Page_057.pro
93865c2cd3ae02d11d9cfebb3d45ab84
804953ce41f5efe296157ce92eefda45cf82273e
767340 F20110218_AABWXA cobb_p_Page_132.jp2
88f87e48c3b8aae11f028f506efe876d
8d583716aa2ad8af646ee890f494c7eac5a74878
1791 F20110218_AABXBS cobb_p_Page_131.txt
b3613457f862ed657a64a266c45d6e9d
4a529331e2a58530f4f5ed10802f227bd8c7aba1
56501 F20110218_AABWWM cobb_p_Page_043.jpg
5525f2ffac04006a0d3f1e7c2eae1b96
fbec8926b0a366240cf5627e7427e7c616248ce2
1026956 F20110218_AABYFJ cobb_p_Page_115.jp2
212711411741c3580b305a90cc9566ba
25a31c77de2c596438391b41d142ab5879eb31a3
786459 F20110218_AABYEV cobb_p_Page_085.jp2
04e5de15a197de9edf13a4100dbe2b96
b090bd7d9c9465f9a3a0bc2708f306e942241582
84186 F20110218_AABXZP cobb_p_Page_074.jpg
d88bbfa1cc116eda04fe6a102e598401
9dacce407b6b7438419d29ae8c84c09546abb25d
45079 F20110218_AABWVY cobb_p_Page_146.jpg
418bbb35038a99415107aa8ee997dd7f
b75a8aaa0b2de40f9dff9b2170db7078b77226d4
803634 F20110218_AABXCH cobb_p_Page_059.jp2
ae995072a6b90993e116254a1e01ae80
44df01ecb57c3ef3f85cc07b5eaa1ef61c556b67
1087896 F20110218_AABWXB cobb_p_Page_025.jp2
1d58738872e1fcfb3fc037bbe3e56d71
bccb29c6c19ac566b22128effe13a737f5e0df20
36870 F20110218_AABXBT cobb_p_Page_011.pro
832a9f9946196a990ec96c3be3e7aa9b
c4de7d25580a131891c4f4095c8987fcd9316824
40470 F20110218_AABWWN cobb_p_Page_074.pro
e412fd0c24ad2fa9a16b424898d04dd0
064059184a63e5c7d67e540e1e92849eafa30420
886518 F20110218_AABYFK cobb_p_Page_116.jp2
9a763781f0440423226badfce327d006
f54b94e34529380d47b6cfc39287775b14e59ad6
1087875 F20110218_AABYEW cobb_p_Page_089.jp2
8a510b9bd79120c43dea47bc7cb189be
9e51c5ddff171d1749702de8ef3c842476a8d8fa
114819 F20110218_AABXZQ cobb_p_Page_076.jpg
10c3e7258a2720ccce7a3d9bd6623d13
6c4cfa203628154b653e611fd2e46b10b95932eb
F20110218_AABWVZ cobb_p_Page_152.tif
03a2b8b8ed216dcddc5490ddbfbf95ad
53a224e54cf2888cdb5327d06dabad03b0ad8811
F20110218_AABXCI cobb_p_Page_119.tif
b4943e1df581a1fc26b61763dfd70716
488e5b9c4e95f0bb115454391124a8fa6a6c08db
F20110218_AABWXC cobb_p_Page_073.jp2
95f7769fb42dbfbefbdf025124582feb
d1090904f0c712d4bbc3bcc608bfff4b93a25d55
8070 F20110218_AABXBU cobb_p_Page_095thm.jpg
cce4f529789ae2314ab2da8b6e2662eb
d555be06ac278d87885a007dd895a091decbc4ca
F20110218_AABWWO cobb_p_Page_125.tif
b978556b5c1a640532afd4a68ea295c4
4db280a809e9ae6ac5d29faf8b7d94798785e926
936605 F20110218_AABYFL cobb_p_Page_117.jp2
251d1071c62bd4be8e0a3ea0090639ad
14f138c92d12d139318509efc76463063d7bc660
1087863 F20110218_AABYEX cobb_p_Page_090.jp2
dbd4366eec5ec8586dc3ebe57fc3dc60
1626e5935c3e1df536137fc95b54fcc94419b5a6
8242 F20110218_AABXCJ cobb_p_Page_001.QC.jpg
fec40943e92d1b2ebdca975d92e26b30
7a2ffb0546496a78cece989893240d515d60b256
1810 F20110218_AABWXD cobb_p_Page_124.txt
1914c3cc8848c2542b1752a8cff19ce4
19b30c302852dc38d975342574f1ff18178577a2
6315 F20110218_AABXBV cobb_p_Page_143thm.jpg
dc2723122930c2a657c7b4e322b30e99
d7439667cd83a6a47854b7f80d38b6070aa32993
1087865 F20110218_AABYGA cobb_p_Page_137.jp2
8b0534b741c715c7b57e95cd3e48c0db
82e290904bc7971a2fc5134e7c816f9dc864be74
F20110218_AABYFM cobb_p_Page_119.jp2
d0372f9014a1aeef1344633abcb6db5f
af6bd6783d0f7b4c8c7f9888bd5eb99da6717157
888738 F20110218_AABYEY cobb_p_Page_091.jp2
f61f93193c58d9b8cc249400d718becf
0ba05bc7a7a4c223215fbb4f24a13d7d125a2e17
83038 F20110218_AABXZR cobb_p_Page_077.jpg
e3d75a441d00221df973fe83f4257634
9dd3e508e78d61c3fea587881caec2cdfdf8e9d6
7411 F20110218_AABXCK cobb_p_Page_126thm.jpg
e7ec8f1c27e0caafef67546315f1be34
f0513c798fb5447085d07413e5c7594adfd82af7
1581 F20110218_AABWXE cobb_p_Page_134.txt
2f66c6942efd71047ea1b9d3ed158ffe
068b5c012198466a7a863c5bd4b3f1267ff8cb18
F20110218_AABXBW cobb_p_Page_023.tif
b549dcf269189a17d4e1ff5543a68093
69de62ccf8bb0d5264a278b4819cbc1d5506832f
24421 F20110218_AABWWP cobb_p_Page_015.QC.jpg
54d2528f953657f597f4a4488be99ff4
481b8e27d8b5d0d060c9ac15211b5c025bae1fb6
895055 F20110218_AABYGB cobb_p_Page_139.jp2
b730096dd34318e549ce4d00bffec526
722912a032a4dd3fb6134c95ed5f45ecb43f64e9
784468 F20110218_AABYFN cobb_p_Page_120.jp2
9bd648cddd567a9079a3245dfd6d8740
3985a65a5eb96f3c0178fe4a7f49992fde764d82
851019 F20110218_AABYEZ cobb_p_Page_092.jp2
bfe1614df0ccc663d95e5487ff03df27
ba3d10e0373eedfb23055df83c39b79b3e49aa50
27181 F20110218_AABXZS cobb_p_Page_077.QC.jpg
869d335e7c57cbbf75ecc81650843464
8b50dab394a89e0847f87cfd7d2ec07eeb5d7cb3
F20110218_AABXCL cobb_p_Page_034.tif
0337797bdd454d4ac3c847905eb86d02
acab7ce8a181c2c9d121b458331e252e77b61ca9
6575 F20110218_AABWXF cobb_p_Page_101thm.jpg
38dab5e2daf79fd6179170273aaa88b7
fbbcd29b787cd9788de1513f98abf7b42431f757
41869 F20110218_AABXBX cobb_p_Page_070.pro
19c222c9a1bd42641ff396ae614e324a
bf65f49a6d49907c6471a89eadad6d42f3927ac6
47094 F20110218_AABWWQ cobb_p_Page_060.pro
48f273fd08b2e298fc340a8abc7061de
cbc20e55dc5609807065a6a24e30a31e00796e8b
994671 F20110218_AABYGC cobb_p_Page_141.jp2
f248d0fab82a5b5a01bd2f581f177c6c
3d57fcbeb06b6f0d830a86f47f31cca6e892e426
376714 F20110218_AABYFO cobb_p_Page_121.jp2
b4299e8ea130d0cde65c935f1ca36847
1faf06d5ce2061a30faf5fddd15c3cb040b12954
99867 F20110218_AABXZT cobb_p_Page_079.jpg
1fb9cfc9f628c46ccb0dc9f6e25e0af3
937081b4f75cd0c6ba47fb76eaf4d02f1e51b4ef
20773 F20110218_AABXCM cobb_p_Page_036.QC.jpg
737c44b5544692c511a06d5bd93f9509
29d1dc0b5624b5daf22ea7a1d8669697cacbc90d
F20110218_AABWXG cobb_p_Page_131.tif
66dd183b1e6a2e62d1b1470c6f70b605
50c31e7d4a5efdc59ba557e49b8a11b0590381c6
974316 F20110218_AABXBY cobb_p_Page_081.jp2
6e69ab267aed84dd94c56500e2ab3374
e6fccfe96da255620747d5490ad10e6d411fd6bf
439 F20110218_AABWWR cobb_p_Page_097.txt
7e2628c876dd32ee6f6d80e2fe7c2bd4
ba963f4ad9508b128fc87f60e9ed80985cff6359
27637 F20110218_AABXDA cobb_p_Page_139.QC.jpg
74d612edf308630db24b7aff29c13795
5e8bd7da3f5b9e4b2010c0646910a098e8abdcc7
879086 F20110218_AABYGD cobb_p_Page_142.jp2
400475e6a8d9244f79a15d36869560ce
ab3f8f57d1d4c9e8d5ca411877c525274a2036b9
1057310 F20110218_AABYFP cobb_p_Page_122.jp2
6aecdac9c9ba703d6f2d3b91a1fc9840
6d83be0e74baa6ace3f29f71eb99413490dad219
30004 F20110218_AABXZU cobb_p_Page_079.QC.jpg
56fc211b76c216279abfd7c659da372a
5899aad42a956b26110f2f87d8eb5a30d36e6018
78816 F20110218_AABXCN cobb_p_Page_048.jpg
0d9c4acbd74961321d40d36c4aa0b17f
af4cb88a31eb9410df410d6b0df9855d09fc9b95
24838 F20110218_AABWXH cobb_p_Page_027.QC.jpg
e12bf502291e0e1f0218868c418b1515
f71cee21b19e7be9fe044a2be75b7487b452fa4f
87144 F20110218_AABWWS cobb_p_Page_071.jpg
533159890f8371f40e4be2447ce01cb5
3b8bef2821fb9b77f64f6d7a87038316f1254e52
7187 F20110218_AABXDB cobb_p_Page_113thm.jpg
8533a2bc292ce42d690d28f9a0633eed
229b7623d18d5ff97c884f386cede6dd9880c54c
837074 F20110218_AABYGE cobb_p_Page_143.jp2
f13c912e8828b3068918f546eeb69274
1e77f9bbd044538db8eb59638b69f6478634bfce
899524 F20110218_AABYFQ cobb_p_Page_123.jp2
f9d42467467ae948ce8815f7019c4d15
c5ba51adb61736691c42c405c3c25d7f2664707e
112231 F20110218_AABXZV cobb_p_Page_080.jpg
fbddb9135e7527e2e77be30138e70368
0bdcedfc5b9a9d6d5f2edc10a1c85b9dda872e93
73023 F20110218_AABXCO cobb_p_Page_075.jpg
99943e162c087ed59f93dd3b97b8e75e
b31528fff097736651c4120327d60d44b7a945cf
961636 F20110218_AABWXI cobb_p_Page_100.jp2
aae3266bc19c6eb2e9fb302b5af53f69
b37a26bf3ce0100532edec44d32747d78d60e4ae
683507 F20110218_AABXBZ cobb_p_Page_029.jp2
5e23178be47f611dd79b41fdac8a8cd2
9abc39ca3fca625929a609e1018c3c94c8cf1232
1464 F20110218_AABWWT cobb_p_Page_075.txt
2280f65b1796fae64ac5e6bc5d78c041
abfd1c599ff6f5e48a10a658bdacb2575503f5eb
F20110218_AABXDC cobb_p_Page_067.tif
832fb385fd883c5bb92160f88e35d889
2331dd03ec44f1977d425a1d47754f46a312509c
F20110218_AABYGF cobb_p_Page_145.jp2
7688becc6c771894813957605aa0e6ad
2ce50c4c9972180689b48d99d1af71984129ae69
794110 F20110218_AABYFR cobb_p_Page_124.jp2
9a404f1704b22ae7b58eac148c08f6a9
1f535359fce1f89b48dded0ae43b06d0f65b56f9
34557 F20110218_AABXZW cobb_p_Page_080.QC.jpg
2bf79b68a84356f6274fffdc477c0a9b
e69751ed6fad35d02dcf1e59b445fcf3960d4263
45967 F20110218_AABXCP cobb_p_Page_136.pro
131fb70fae99de747add057a749c71d0
1e1d3df70f123986241667e8a72d676b4325a64c
8162 F20110218_AABWXJ cobb_p_Page_138thm.jpg
61b6329fe7eed9c8e4f740c0e7813edf
95d41750892c5642ddd42099be492c999b29ab3a
94867 F20110218_AABWWU cobb_p_Page_041.jpg
fc7c328a0ff48b9c8af43436509e7570
6f28a1afda0963b18e5dacd558200c07f4153173
8134 F20110218_AABXDD cobb_p_Page_154thm.jpg
c7e8117d50d16c880b9c6d29bc95691a
1c81675b64e0bf5572e8510168fa69a0e49c4a56
442148 F20110218_AABYGG cobb_p_Page_146.jp2
6a7d2ee445204fb171202f2ad35d91a0
f085f6c5c8b408aadb66d8a8e66f447cfab87389
F20110218_AABYFS cobb_p_Page_125.jp2
29640a04b05b7223c50c861c569b20d9
48519b9473c648b176edf095a5db94aea0a5af39
28599 F20110218_AABXZX cobb_p_Page_081.QC.jpg
a445ac58a93e9828efd11265e859389c
8f79ecc792b489a4899f0de5515366b66636164a
957438 F20110218_AABXCQ cobb_p_Page_026.jp2
ba860b452e19e6d3c7609363fe48997b
7a0935c49e4722a16ac2c5214792c62f79cca380
1351 F20110218_AABWXK cobb_p_Page_139.txt
b73d76363d42a477d720e28df819ba5b
c5cd30df0a9cddd82ec80a7c62dfae32caa25ea2
F20110218_AABWWV cobb_p_Page_114.tif
a2137dd4ec81be75d7dc408520430c20
ef8da38799ceedfd2b3c868fc44ffe5c0ce8f1cb
878048 F20110218_AABXDE cobb_p_Page_108.jp2
d68145671a1d983dd864e20dd6338801
4c27bb7d3d71cf0001b596bb8b6b8db994ffa87c
952640 F20110218_AABYGH cobb_p_Page_147.jp2
3a9cce4eab470522991db37b814162d6
ce2a804ace784aea3c18fcc9dfa0a308244ae68c
993669 F20110218_AABYFT cobb_p_Page_126.jp2
5bab043c45ca4e0982608fd994f3016e
370140418a243c0ba36a82ee189121e01fef03c4
113710 F20110218_AABXZY cobb_p_Page_083.jpg
14071993c0cbb498a74fb1a193760f08
1072fd0f574699e700db906efbdf0c0f944330ec
26907 F20110218_AABXCR cobb_p_Page_012.QC.jpg
4820570838c13fa46885df5f72087afe
64078ce20bb08089517cadf7c5bec297561a1b44
1087886 F20110218_AABWXL cobb_p_Page_130.jp2
47a9edf2ac227022bcdb5eaaf8c9087e
032bc5b748cf52397a27e0d84ea30eb57d5c9f1c
82474 F20110218_AABWWW cobb_p_Page_142.jpg
9d0f9f9b3c8c9cff4a945d1d921a2f1f
30b8bc7614523aff5d7cc871aa2e180c60fb9817
6300 F20110218_AABXDF cobb_p_Page_120thm.jpg
d21e3e80c57c53efd170632136b54fde
cffbf45e6713d3b5d553273ce9f2e18d606542ee
1087901 F20110218_AABYGI cobb_p_Page_152.jp2
87c37c44c40919ae051c83c2616fe481
e41c43aa396726faa6ae475ec9bbb45642882856
F20110218_AABYFU cobb_p_Page_127.jp2
82de7f2768d74ca42ba3a7854673dedf
a215d756fe1ead58c28d74227ad46d9cb5c2192f
27597 F20110218_AABXZZ cobb_p_Page_086.QC.jpg
4547cb7a4fd6420e6a381df667a4dda7
d462856ee6167a52ec55f094c9d1205edb0051c6
1875 F20110218_AABWYA cobb_p_Page_047.txt
c0b13b66ba501b8d026b8e148127d255
d3b37d43dccabbea5f1ea1eac91e473ed9574100
5288 F20110218_AABXCS cobb_p_Page_004thm.jpg
032d894531745c28d50a0ca71dfe74df
1251e29a5c7703e1341dc2d4af715f04be352fa3
F20110218_AABWXM cobb_p_Page_004.tif
6f1b728d0c7d58e53950e75a15a372d1
3078fea0070555b8a723e297c6b0914342cc1926
116811 F20110218_AABWWX cobb_p_Page_156.jpg
1866c7e96fe9124523391b50608f9a27
3e69b3cbd35c4e352334a6b4ce6884c69d617b54
7195 F20110218_AABXDG cobb_p_Page_084thm.jpg
71dc3da5db8b6d5831683c817892a764
135d109c4c63fbe66fcc9d8f75661441b6882217
F20110218_AABYGJ cobb_p_Page_153.jp2
4b98c930925eb44af85536327b2961cc
dca1daea44e8f1e404bdfa67d0714bd6b2c438fb
1087881 F20110218_AABYFV cobb_p_Page_128.jp2
1b79e82ea72aa46796ad8df0b829dac9
f39f6a507048a0c696346615be1b24265ac34a11
79300 F20110218_AABWYB cobb_p_Page_038.jpg
fc5395497692954ac68c6e89c935fd38
adfa4e5ea674e129e1e7e38f51925e6c03941812
F20110218_AABXCT cobb_p_Page_094.txt
9e730e29a9b2394962e6e9aef6b2b543
ce7823cf32870bce73d76620adc303366b902dda
57216 F20110218_AABWXN cobb_p_Page_094.pro
aed4cdfb19a49b8f1d353f417fb67862
6171c003a3144d3cd1ccd9861ae58253d237efef
31401 F20110218_AABWWY cobb_p_Page_115.QC.jpg
db79c86662495e355e3423351ef3be6e
4ca3f10989102b99f75c49b0445883efbfda4922
F20110218_AABXDH cobb_p_Page_012.tif
685253c828c77ebc0bfade1fba812133
02a8564de47d9de9b8f7164b0c15c0b6042c7852
F20110218_AABYGK cobb_p_Page_155.jp2
d1071e4846f94a954c4159c8567f1c32
1729d79c6b0d5ae18dc2169765efbb326644b366
819277 F20110218_AABYFW cobb_p_Page_129.jp2
2343c747bd889de65b945c2f5865cf93
2a250067658a38eae992b9d420fc61ebab178a07
F20110218_AABWYC cobb_p_Page_088.tif
b8ce09636cfd1f5e9e99f02db8135192
e67a0d1bc54fad302aceff0b7647298085b8add5
93208 F20110218_AABXCU cobb_p_Page_087.jpg
4b892e80b57565e58772bc5adb549fd6
50820b9c6cc85034718b67d1d2b2e8da367beb0c
88679 F20110218_AABWXO cobb_p_Page_052.jpg
f1cbce9c808cf86cb3dc71cce1f898a5
18a4ec6d17d37d9e67ea976f5de635e5b451bba4
F20110218_AABWWZ cobb_p_Page_086.tif
7fa9a844a8e52bad5a6a80315fd8d449
ffdce11330c180bcdee2e8b5eee0811055dbf808
F20110218_AABXDI cobb_p_Page_140.tif
d42aa13648c9ee2d4c71a58935473cc0
250785496e7af749e3a3160e9dc576644bb7a481
F20110218_AABYGL cobb_p_Page_157.jp2
dd273f8ea2622755337c194d56c9a19e
af8fb0116ca30c8cd31916aef94f34be0aa7a2ea
913837 F20110218_AABYFX cobb_p_Page_131.jp2
b33fa387731c70e9851f9121364a8e79
cb351456fccc772fb54b8e07028cba03ee2f4a75
8271 F20110218_AABXCV cobb_p_Page_017thm.jpg
308727b079d078226fb39d1969488023
aa4867da77b16b7de5e531fa3fbc3469f64652a0
89878 F20110218_AABWXP cobb_p_Page_066.jpg
414df98e3ba31096ec299b87d9505269
cf30f3f46421c1fa0c8b27255c41c3957eae223c
56718 F20110218_AABXDJ cobb_p_Page_157.pro
d7175a0e9ad007f1fb856561026593e0
7fd82da8a86169d98ef45938f46c5c42ae0f23a5
F20110218_AABWYD cobb_p_Page_108.jpg
1c86e1417cb4dad706ce416ac218fd6c
1ff59badcd626d51286ad441070b7d61ea6b5b3e
5714 F20110218_AABYHA cobb_p_Page_029thm.jpg
7024370c32e5fd3b23a32d6db17c7654
22144e8e51084f11788728b32605bda2f42308de
F20110218_AABYGM cobb_p_Page_158.jp2
ebb01bfe67a1ac85a95b2db19bc3ae6f
64cf0358b2789b14a40131998059d9a30fef06a6
859140 F20110218_AABYFY cobb_p_Page_133.jp2
5756613adea87472499373512a932610
eeca9398fbe0d19bf59b358d03e525d284c9acbb
91949 F20110218_AABXCW cobb_p_Page_110.jpg
f8730afcc3c02dda9507e8bb5be4e407
42ddd7e5d945e00650634bc8e5589f8bc841cf14
54445 F20110218_AABXDK cobb_p_Page_028.jpg
0ebf6923c1b075cd172602e5d5948e68
013d17f593ee06f72b3d42d1b57ebc03dd3c6eb1
2483 F20110218_AABWYE cobb_p_Page_138.txt
60680aee6be9d671c607854b45c74163
34c92c8baf3a7a6c4649bf27875c75d8ffe34ccd
6278 F20110218_AABYHB cobb_p_Page_033thm.jpg
ca20e7d1f1d775b4207210406d2629a0
5b4cddbb236abc03aa880ae957fd348c973cf8e8
994542 F20110218_AABYGN cobb_p_Page_160.jp2
8053ea7940a919cb6931ef804ad07fde
87f615dc6718e2d4050e0e3a3ae02e6c2b8aaf16
674186 F20110218_AABYFZ cobb_p_Page_135.jp2
4f85454db7ca323cfd06f9d17c36bf37
407a12c3b726547c803c7e4ae978520aa7c63557
6969 F20110218_AABXCX cobb_p_Page_140thm.jpg
76e3dc0bde69a057937ab0424131968d
d0ccdd513fff48cdd56d80cf7334a6764d4e5a5b
42808 F20110218_AABWXQ cobb_p_Page_071.pro
b6bf7afaec63f1999e64619b6edfcd14
64459990f46750194ae54996825d482c59aab1d2
F20110218_AABXDL cobb_p_Page_026.tif
035426dd7d45eca85387fb30cb6bda7a
77122b93a808d9b48c20553e32f47e469cf499b5
1087893 F20110218_AABWYF cobb_p_Page_022.jp2
f74b1f5750f2d54337f294618b238370
750306bc320f00c50e10e69f839d3ca758ec2eaa
4987 F20110218_AABYHC cobb_p_Page_034thm.jpg
650f27cb96ee291276a68f0af4c8a578
ab3f8f99f1607949aa43f1a767151f272909dde9
600 F20110218_AABYGO cobb_p_Page_002thm.jpg
2a197e3bac18fda82c4e9a731a443a04
2ea1f5e69eb7e82ed941c490636e458aefb7945d
1905 F20110218_AABXCY cobb_p_Page_144.txt
dceacfccdf92b9e217913e672f067498
f066a6ed3d4ddb024bcae5a28f29e07a45451572
876762 F20110218_AABWXR cobb_p_Page_009.jp2
1d0a36b9181094e97551d1ebab488cd0
b651021fd48b017a9e93a4bca53883561f7ebe7b
F20110218_AABXEA cobb_p_Page_096.tif
922b8b5cfe7bd2a0b4ca7f8ed2e981bd
4572f9c52017c5484341a0559ed2b2d969ec51c9
F20110218_AABXDM cobb_p_Page_021.tif
2ee12e56a41bdebc4f81aa680efa5808
da5f2e0cfc41256afb0dc89aab04917345bfc1a2
6963 F20110218_AABWYG cobb_p_Page_105thm.jpg
9d67268475086e34dfceb0f1f9c30862
a41b8c508a50d7ba18ef7beb5854161796378fec
6616 F20110218_AABYHD cobb_p_Page_035thm.jpg
85af93b72fdbf1bb728c6f49d30d6f16
97f5d0650bbd49a5f99b60a482cfd0e455021f18
5536 F20110218_AABYGP cobb_p_Page_003thm.jpg
9546a089dd4aad8430274fa87cec63d1
bd1bb3e02637babec34242dd1cd9a45b2f0cc63a
7891 F20110218_AABXCZ cobb_p_Page_024thm.jpg
b2ab57b109026e6075234dd6daa2d118
bc5a10a05f695e261e0a322692e8800a50344b05
863908 F20110218_AABWXS cobb_p_Page_134.jp2
54d154f3e6bb5ebe5e09c4b5be5f8039
098bd038eb017e8ccd4710bf3b04ea36b9addfeb
26032 F20110218_AABXEB cobb_p_Page_092.QC.jpg
517f19ce24fb35f8dc4d86fe280774d1
361f037894e1efa57c9c0bd64bd5bc1be939503f
91549 F20110218_AABXDN cobb_p_Page_026.jpg
8efe48a1bb8dd64fc4ffc32725d0cb6d
1c0d658295bd06534093db7d7d49a673f9dc0e64
26518 F20110218_AABWYH cobb_p_Page_023.pro
3c8e85cdabcd88aa3685d7abf71c0d78
af5b09cdc528f2008249ff5c98a37006a93b3907
6070 F20110218_AABYHE cobb_p_Page_037thm.jpg
ce0a498fd6bfa1adb2dec8b0bcb5ad68
8aeccc6c4fbe9c106e64f5230b35ce0f4ceb4a61
5982 F20110218_AABYGQ cobb_p_Page_009thm.jpg
fea27449d97965ea37da05fe5cc4e084
5b8caf044899e8e87e57dee624a4680f72ca54be
22643 F20110218_AABWXT cobb_p_Page_075.QC.jpg
abd99d317fbaeb0d8fe93c7b0d43e37a
ec97d0ae982ad27a4a851cfd8dfd0284bcf70d08
7693 F20110218_AABXEC cobb_p_Page_128thm.jpg
4140224659db6666ed62808ce3a10888
93428922c5c4547a95eec0c8f7376fa1bf2e4a6e
935469 F20110218_AABXDO cobb_p_Page_066.jp2
3ceedf2b13d3c40709d09e1b66419d91
23bd62d790d0d6f5519c94414f12ee287f265280
1916 F20110218_AABWYI cobb_p_Page_142.txt
74962c44e3c372d8cfb2ff5e316e744f
3afcd4ddf2032c59ffac86a4d122c441ac9852b5
6459 F20110218_AABYHF cobb_p_Page_039thm.jpg
16610c32d1a1e965b533c747a36780ff
d20cd07f231138c5716c387653d346482b5623f1
6787 F20110218_AABYGR cobb_p_Page_010thm.jpg
66637000363d3bf85ada07de3e97a306
33d4c6e923c549e9c0bb62763bfa8415d8790c57
F20110218_AABWXU cobb_p_Page_018.tif
83fdfb91936e08afe8b1302b12449ed4
4b781bb27426c8e1e067b5c1f25d70770e64e329
1948 F20110218_AABXED cobb_p_Page_122.txt
7baaafcd7b9b152b1055b3b6cb4e2b5b
24d05ff9237738b480730f4f6c29a1fb7b8cadc0
45617 F20110218_AABXDP cobb_p_Page_030.pro
b2c673e55f02eb982bf0a243e7dfbd59
25b6a903ca7acc70b0975b0ea0552b1cfab89bf4
54722 F20110218_AABWYJ cobb_p_Page_114.pro
6c91d4a6163c4471a231fd9c5db8cc5a
95ace61edc3d0787743eef12a410d6b5129804c8
6659 F20110218_AABYHG cobb_p_Page_040thm.jpg
a769fcb36e5cabe4e6a6edbd78029168
fc7016bcc9d9f388c71b928153da36d7b0dc77fe
4221 F20110218_AABYGS cobb_p_Page_011thm.jpg
69c76410a30ffe0f4de0db6ce97bb7bc
f9a27290c185fd8a9171f9f33c9b213632fe82b7
F20110218_AABWXV cobb_p_Page_118.tif
331bbd0bf61f2dba25cf4cd33f3f2cd4
77b74670d608321b4bf4e38a3d1a4217e0774ec3
1966 F20110218_AABXEE cobb_p_Page_143.txt
adc2b0480b88ba9d7e9747bbd837f562
5bdf893f3bbb1d933c130ff3f0278af543d97f13
F20110218_AABXDQ cobb_p_Page_054.tif
7db67c1e4895a8f74396228c1e8146d4
f14884015ac6674731145a8dd4ec372e866989a7
F20110218_AABWYK cobb_p_Page_144.QC.jpg
250478e57c210523d70fe42bfc101fff
c0359557fb0dcd36b335621aa13c4af0f0ec2b30
7053 F20110218_AABYHH cobb_p_Page_041thm.jpg
704922700b146e79688c330381b57a50
de316d3a1eb3330724e5af7d9b49869faf0f9db5
3699 F20110218_AABYGT cobb_p_Page_013thm.jpg
2f1ef1fdac756d141e76b316439c93af
0a29fad5516d404d6cd5a62a76c595fc4e29a01f
931916 F20110218_AABWXW cobb_p_Page_012.jp2
dd918fbe7dd8ecd03245a275d6a64c40
c6a5e1761ab5038cb4fead85546f846546b09cf9
806650 F20110218_AABXEF cobb_p_Page_112.jp2
f1de6ff210492303e67dd22118148793
b14f6ac7c3f3ea8d7c6c688bfdb4117dded87e96
6044 F20110218_AABXDR cobb_p_Page_142thm.jpg
ae21717b584cd1d1b8866100a7de134c
97b9ad27388d3dfcaabbc485ffe0fcbcd5f92454
2226 F20110218_AABWYL cobb_p_Page_090.txt
2e3113b14eee7394d98a14431309b910
9a4ff6a8b234c1281e6e271fa8b707c1c9d1bfce
5204 F20110218_AABYHI cobb_p_Page_042thm.jpg
72f5a06db1c090da16b54672aadf2e87
3cf2e8218616b11645ba6ba023f2e515d926696d
6715 F20110218_AABYGU cobb_p_Page_015thm.jpg
8b1d3dff61eee7a9ead07bd5231898d8
16a3b9d9321ba8832b6d0ca75d04c1edfb3192e1
91053 F20110218_AABWXX cobb_p_Page_040.jpg
3f17de757af3b8bd84d6cf02d7362f2e
ff4df7e34a8d7456daf3fff4d08465e9d32bda24
61939 F20110218_AABXEG cobb_p_Page_045.jpg
5b6318bf26efbe3737581e821dd45543
c1ddf0f4cd193ec7aa66c9998b37f70f2e102177
114117 F20110218_AABWZA cobb_p_Page_118.jpg
634a4ebb260c47a54b7b6e851f877a5c
61ee64c41ccdb283bac743e915e46c7fe77a647f
F20110218_AABXDS cobb_p_Page_110.tif
3ad357de1cf5de765504c08c115d8dcd
18530887ee2fbad0ff255531a14da99297f746c0
F20110218_AABWYM cobb_p_Page_016.tif
f53f9d875c78f2e4e0bcbcdffd94400d
aabf6a479eee2ff32d4ddc8ff9ad7433b3ba9959
5126 F20110218_AABYHJ cobb_p_Page_043thm.jpg
19f755f9127468b1c66036ee68cbd964
af9cafd6647569b9b37a36c5362bfd25a103f5b2
6692 F20110218_AABYGV cobb_p_Page_018thm.jpg
7562f4005adbfce787c39a3888bf07a9
4b09273327f0125455df5d375b1627aa063ff0ba
55122 F20110218_AABWXY cobb_p_Page_024.pro
800b6c2d9fd809b3d1cae890f74907da
5d8270c31c8b08088c6e0e7da1a778bdc72d3491
F20110218_AABXEH cobb_p_Page_149.jp2
4d1a08c68ed818b8e81f13a05776f38a
afad6e5c05944cdac1ce81354a97b30b267d284f
28691 F20110218_AABWZB cobb_p_Page_096.QC.jpg
a0a13fb4435fb6e0f1f6533887388bd8
e1724dc3226f05adec7c986f1433d393c7ad9043
F20110218_AABXDT cobb_p_Page_055.tif
e45d97c6c17fdee6ae1f1e74d3e516fa
24e638dd3783b927e6cd8e04c6d1a7ada73a4eb3
1087867 F20110218_AABWYN cobb_p_Page_156.jp2
656680a50d69f9febd99b06a903c8801
d843f2c34ca3c1e310fc72d68aff87d585c9e107
5266 F20110218_AABYHK cobb_p_Page_045thm.jpg
75889a02928f9e143251d8d326fe6616
19a47f165843319599f619d420448b91a301daa0
8885 F20110218_AABYGW cobb_p_Page_019thm.jpg
d0c2eb878c5fb33c02723371bf29e83f
f2e05863efe56b5fff6ac438e28b5913c25ac34b
98072 F20110218_AABWXZ cobb_p_Page_115.jpg
d34ba15257cf778142da392f1db4041d
d969aad90c1257d0e8281f3e72d0aca66ae85814
30895 F20110218_AABXEI cobb_p_Page_051.QC.jpg
e348b061279da37a7b66f32c09438d0d
d438f4101975469c918cc0d3e1d592bdfbb8fe5b
66825 F20110218_AABWZC cobb_p_Page_046.jpg
eb4a783bedf857a9c12d8e55a44aaac4
bd62274140691640c3a10ab1386ad34b93062384
29654 F20110218_AABXDU cobb_p_Page_046.pro
4f5fe0144a8f3b5eb7f104762c8f80dd
1e3289e5be9524a855b06072511e41a1eaa4df53
1968 F20110218_AABWYO cobb_p_Page_150.txt
f34bcaca610f99bfbb8270b72d111f09
fb0816021b227dfd3d0734e3da7b4a49ca827929
5809 F20110218_AABYHL cobb_p_Page_046thm.jpg
6446a2f39c0bb723b1579047d6562f97
eba81e65906d5a8ca0a12dc58f03e4f9e5b5d8ad
10010 F20110218_AABYGX cobb_p_Page_022thm.jpg
6e5b9053577163959dfb48d23bde8456
f6fd517a37d4426154eaaeb7b30119f98ef84ab7
6918 F20110218_AABXEJ cobb_p_Page_160thm.jpg
72729891933c7d0318ca60e3e90d937e
7ecc7659c4bc95e762f5537ba5dda9bbc683c4d5
12199 F20110218_AABWZD cobb_p_Page_159.jpg
a495dd3466f50a027cd00076fb90d5d7
4c31dfc0ef8083103cc713a3238acfbc094e4a76
28671 F20110218_AABXDV cobb_p_Page_063.QC.jpg
00d020f29f33e7454f95f14434c22187
d82e9d48f9c00c18f450895cc777f3ab329f037b
3119 F20110218_AABWYP cobb_p_Page_006.txt
514464f3a8297bf50ed013770eceaba9
9948fb3943dcee1032e14ba191036354e4df6e79
7490 F20110218_AABYIA cobb_p_Page_068thm.jpg
2a8fc69db657599a644c22800398a318
1abd2919f2907d29ebc938dd622cbc3a1e89c7a1
6835 F20110218_AABYHM cobb_p_Page_047thm.jpg
e69cdd86092dbfe95010ea32672ddad6
d0d19afe53cce208df35d580c1fe89fd6a032fb5
6492 F20110218_AABYGY cobb_p_Page_023thm.jpg
9e7d40287ed8bebdc44d14c452c722db
5137134f4b49af2283bb8e526c0c63c4739f7a27
60111 F20110218_AABXEK cobb_p_Page_156.pro
21c1bb1b7b41cc257e123b2895cf1742
c836262254f0dbecc9c040be3756dd58b0b58817
53251 F20110218_AABWZE cobb_p_Page_127.pro
8e49a976f57e7a3cb6089845aefa1592
e2f646d8b9a1845ae14234a0272d4859d9b9e2ec
39336 F20110218_AABXDW cobb_p_Page_088.pro
0f26c7ddd3b580c7b685b1aec86ef16e
e6d633d8bfe27470050b37b37570c4f95648057a
93868 F20110218_AABWYQ cobb_p_Page_140.jpg
91a416d4a245fad72bf729b6d4344130
ae5fa220cf88f9bfb63be0d2bd4aac8a118d6428
7791 F20110218_AABYIB cobb_p_Page_069thm.jpg
9c201b1dcb27cbb61912e9bbf1bf98a6
3c1919bf21e569f7c20fa621d65650036a896385
F20110218_AABYHN cobb_p_Page_048thm.jpg
ff848ed8c3c717040239df0593218549
cce238e106bbfb1db257d9999c10e91316b64557
8123 F20110218_AABYGZ cobb_p_Page_025thm.jpg
945b6c013d2c13ef40673296c274738d
45825f835baf0f090c99caada38ec72ae50b52e1
93075 F20110218_AABXEL cobb_p_Page_126.jpg
9fae40848ba18678fb35a69ef43f2010
e36c463af9a6abf806ec35c1b232f1da2c5b1c90
1856 F20110218_AABWZF cobb_p_Page_052.txt
018e675c3e05cf79e9c852fe50297d68
cc9ab9a4c47c2c7b055ad7ee66e77c6fdc6752f2
1907 F20110218_AABXDX cobb_p_Page_088.txt
d3b26bf767ae079338e61ce1b77d3b93
2de2b0ce01ec7e62695927298030e1d4c4011889
7908 F20110218_AABYIC cobb_p_Page_073thm.jpg
16fd5a29a533bc1b90ff3f5ef85f2f9f
562d517afba5acec21daadca8e264b1eb3d62f68
7126 F20110218_AABYHO cobb_p_Page_050thm.jpg
f3f2ffad98377cdc74c96f5b3a4a360b
505de5b47f43919fafc586508f3d45f959270741
2182 F20110218_AABXEM cobb_p_Page_021.txt
90b83676f82ae7b69be0c92103c6c0b5
bde9d691ac43c55fbe39ce017ce8e8150a72904d
113757 F20110218_AABWZG cobb_p_Page_016.jpg
5cda02b34c3db7bc1684955d70634546
81e7c1e2170ac11f9e6fe872b0926ee93d6786c1
F20110218_AABXDY cobb_p_Page_116.tif
d5077311a13518e479943829ad2f2a37
081a88849e70e8dd735598876c894e3e64f4bc59
83503 F20110218_AABWYR cobb_p_Page_133.jpg
e60d1c8083f9edd81243d62d399b02e9
13ebd2e9333e0d31f35881a40f14c7891bd8f848
6961 F20110218_AABXFA cobb_p_Page_117thm.jpg
af496a45a309dde5822280032d1e887a
788915b912a390b0125eb8f247d79830cea4586c
7051 F20110218_AABYID cobb_p_Page_077thm.jpg
eacedad8f0850cc3c0154b61fd094cab
8dcf4d019dd250456dad33416770919aead3c6a3
7328 F20110218_AABYHP cobb_p_Page_051thm.jpg
10140bd6d3dc8951e583d5a63dcd773e
1ae1319072d325570a6bcd89301b787fb8b7b0a2
2136 F20110218_AABXEN cobb_p_Page_130.txt
147cb326a588e3b337a510b38ba04937
7782d8e01fad6e909fc4971422ed686f02ea6904
F20110218_AABWZH cobb_p_Page_122.tif
f06988b2563a93fe64bfe66c5f5cbe67
8756fe60cf1144003d4f87aa42a03b86ef8474d2
78822 F20110218_AABXDZ cobb_p_Page_120.jpg
f8813aa3e9f3b048ede00592b61f8ec4
bbf301a2e1419b8de6aa4d5658dd8ad77c706357
108825 F20110218_AABWYS cobb_p_Page_082.jpg
66c4f30b7e2adeb0e28fa43360883e9a
5358776c4d880c9f2a47b7197e482831e9bfdd78
F20110218_AABXFB cobb_p_Page_092.tif
d53f755500c3b55416e67938250853f2
d4f2e03c9eb437ab22814adbb84e59b7e82bd86f
6742 F20110218_AABYIE cobb_p_Page_078thm.jpg
95bd67a139d046cd52d7a64e7e9cd763
425b316f2d0edbf452675d65d3988ae5efb25d74
7282 F20110218_AABYHQ cobb_p_Page_052thm.jpg
b2a7ec0dd6a8652e1292d84340d9ea56
45e074d498f59edd5622d10faaa13cf009aa36b8
907115 F20110218_AABXEO cobb_p_Page_056.jp2
b45217aa86648efb80ef43c5c8713482
f1ef7862fc57f5afca81baf4a31534760ec4e6dd
56735 F20110218_AABWZI cobb_p_Page_073.pro
ded813303c2518f1c0d69ddc94b11663
48eece7edd718e867b0681d5328605cb97ca242a
34029 F20110218_AABWYT cobb_p_Page_095.QC.jpg
8209900768811ebc97c502bbc65726ff
030fd29db842cf0eeeef9c99685be7089ad3bd6a
94058 F20110218_AABXFC cobb_p_Page_100.jpg
dbf709715f1d9f9058331eb5683cf8fd
8a7a3c46294034218fe074652cc6675ea082a680
8173 F20110218_AABYIF cobb_p_Page_080thm.jpg
4dc89eb3df7376ffc95b1dfa5357f1f4
c9843972e18465000b65c28ee32b1cd4acf59af0
6719 F20110218_AABYHR cobb_p_Page_053thm.jpg
d4b1d9d434d8b7243ee3d551ac1f31ee
3c151271112d9c0f022637557909bd4e85ada58b
6521 F20110218_AABXEP cobb_p_Page_116thm.jpg
2e26443738ac757aaf8bf19a21a37c1a
4c85ce17d8833c9bc2dd232123841fa54ca64adb
F20110218_AABWZJ cobb_p_Page_104.tif
f6525bc56e42798b70276f47721e5592
8d7e43a7462037441c86bc5a0427929ee7f7cfd4
41862 F20110218_AABWYU cobb_p_Page_051.pro
35fe6f14475fbfa4f661c15788d052df
fea01af03e41d86f87c6128c3347c618c0ad6345
6386 F20110218_AABXFD cobb_p_Page_055thm.jpg
3e97af3aeb26775cdb0c5363eeb8e36a
fd4263f8430506eb4c2b0f2f01a248c315b0e625
6894 F20110218_AABYIG cobb_p_Page_081thm.jpg
b500f34eb09f584631893787c4395494
d17a2003135295d8c661275f66124dfa6765d173
6676 F20110218_AABYHS cobb_p_Page_054thm.jpg
76b5d3450201f36e24b6f1718fb6dea1
5433d3e9ca57548547e1041e970302411a95ceec
106341 F20110218_AABXEQ cobb_p_Page_069.jpg
3d8f8c2cf88f7a4c1b5325bcf2627d7a
c6f09a317427b615759390a7555db5c31cc9e568
79544 F20110218_AABWZK cobb_p_Page_085.jpg
4151d2e427c723624340ec9a09432860
1f513e25a35312b60328ef0a10b42393cb62dfa8
7857 F20110218_AABWYV cobb_p_Page_082thm.jpg
1fdbdc62458b3dd6d98a113ffe3fdc0f
317603bfd08f66cee29b4efc963dbec566213c5d
F20110218_AABXFE cobb_p_Page_024.tif
a6effd2d000ab0f0fb04c999955cc156
6f2165df85f512c8c07395cbd37203d1d22bd027
8129 F20110218_AABYIH cobb_p_Page_083thm.jpg
f4c43956b2ff1c2cdeb4aa3143a31928
90b9d71e434158956291452e93fb4bef486aa697
7846 F20110218_AABYHT cobb_p_Page_057thm.jpg
69c40153581586a411109cb104d92f26
d79c7a21997205bb51659c10f23863ebea470087
7622 F20110218_AABXER cobb_p_Page_139thm.jpg
06a5f0d5496406df0f4cece10b7056a4
6efdb2950780c0f1ff84eb589a0a0fab168f8b28
2101 F20110218_AABWZL cobb_p_Page_107.txt
e65848d198a2e79fb79444e3ee30e73f
22e12456a23bb004f2b3cf3748e62dbebb1bd531
110366 F20110218_AABWYW cobb_p_Page_025.jpg
9d8a446681833b67afee5ffd757e384d
01fb9fc2ef8350bd8672f104fe37443dc2f15062
960452 F20110218_AABXFF cobb_p_Page_111.jp2
cf8a63f707be8876159f28314e1aacfb
bf015c12f498a7b9b067783d475107d168e50cdf
5941 F20110218_AABYII cobb_p_Page_085thm.jpg
679441b206344573a97231fdf4146f0b
bd7e09c4b4dc67975624b9700850175319bee987
5717 F20110218_AABYHU cobb_p_Page_059thm.jpg
8997dc7eb33c9fe14c065dc26706354e
09536931a7bac8814f7022f89192e55f12807eb4
31171 F20110218_AABXES cobb_p_Page_020.pro
73d55a4f0607dd350a0536c7337f485a
2acf9abb5b38b19e28181724dae97b1743c717d8
42700 F20110218_AABWZM cobb_p_Page_012.pro
cce1a9f2f51fa8baf2fc03cd66155e4f
bba40bac16ae20cf441e53769296b773ff8fe77d
34531 F20110218_AABWYX cobb_p_Page_062.QC.jpg
36b64ed6c0b8daaccb341c5d2813d777
629cd8e126dc73582806a17e47872f5046bdc18c
45415 F20110218_AABXFG cobb_p_Page_064.pro
83dd3252774bd6a4654b9f1d560a5705
8ba56d04597e1e14777981b6d5dbab3c1e6af083
7033 F20110218_AABYIJ cobb_p_Page_088thm.jpg
24ca6fa5b0aafdc78765e0363abe5b68
d7176246c7fdd3695266d5a53a0d800dc4acb40e
6916 F20110218_AABYHV cobb_p_Page_060thm.jpg
eb63f56453379c82c2191467e142a607
84f5ca78b0dcfdaec9202a20daad7b6554213d81
85471 F20110218_AABXET cobb_p_Page_020.jpg
05a22f7825b522f12a0495ed7aa57d13
826d61a8a0b9468ea8c65f22de1a817ef3e42893
1087851 F20110218_AABWZN cobb_p_Page_061.jp2
30a9f77684681d9ec8107a7bd3ff6df0
f2fda822ea43d0556d8ea8ee2aaee5ededa33d7d
6450 F20110218_AABWYY cobb_p_Page_074thm.jpg
270086c3de1c162b438648ac489f0165
1b7fe918d3d6048dbdfe121c6a20978860f3e299
F20110218_AABXFH cobb_p_Page_047.tif
6940ea0bb009a0ae800e49d4c0335290
f56e6f4ddbfe156434c39da77e5c587fae747b82
8317 F20110218_AABYIK cobb_p_Page_090thm.jpg
abcc496ea216263811598fdef55bd2b7
d8402d4b4fc37dcc433e365b38ab3f1471737688
8137 F20110218_AABYHW cobb_p_Page_061thm.jpg
1e0bed71db14b9f9c8dd221e323b5135
71d0d946e978decdd8ea3566860100bd116b515e
35443 F20110218_AABXEU cobb_p_Page_099.QC.jpg
450f83a6173e1ecb303e31c8c840d5a0
fa6af9b9b5e3f90158c42e011b8213cd28e4eed0
2212 F20110218_AABWZO cobb_p_Page_099.txt
7d50659060e340fc3ebbbba73d7c7418
db41b8e81ead702bcb838ce8f439377452d3b312
946341 F20110218_AABWYZ cobb_p_Page_055.jp2
d74d0665b0d15713764f20b7538126ce
fe6346c95ef918f3ff416b249855e9ca82bf35f1
7961 F20110218_AABXFI cobb_p_Page_157thm.jpg
2bd85ca2ebbf2490a18ea626dee99f6d
b422bee67e0d3c3288a17684b6fc5fd64de34041
6951 F20110218_AABYIL cobb_p_Page_096thm.jpg
c0c63b7d5fb56a53ea62f227f940e1b2
6d2295f9bf3cbe172c734a5ffa7f42be754550c7
7856 F20110218_AABYHX cobb_p_Page_062thm.jpg
8f769d2923fc904a5891a89cd5000825
a8694a219cc09239d24c4716ea0e0873d0733efc
14357 F20110218_AABXEV cobb_p_Page_007.jpg
7cb876a4af3ed32c46c6654e78c54fa3
11191d9fde197b8d023708dbe8507ee68f21b2b5
117276 F20110218_AABWZP cobb_p_Page_158.jpg
ee78739b1d80b1620e7aab04fe73a129
ccc2c836294971e97bc0b6383bfdc8424f9eb5f4
1688 F20110218_AABXFJ cobb_p_Page_017.txt
3084be6e50c717ba102abb93e9753535
20402725fcbf8642dd8ddf109473ecbd56d5b534
3311 F20110218_AABYJA cobb_p_Page_146thm.jpg
a85dbb321a0615ec99328e6d148e48a6
9377ab580402218046eaefa9d3c505920e1061c9
8205 F20110218_AABYIM cobb_p_Page_099thm.jpg
8f6864909b8e87c6ec31c482982d2ac0
8bf41997ce7ae642e1a69a4482c8e60fa7142bf0
7496 F20110218_AABYHY cobb_p_Page_063thm.jpg
3af2b710103cb76c94ee3bf535109ea6
217e2c00e3edcff7212d294b267f75ec38689e6d
F20110218_AABXEW cobb_p_Page_042.tif
62ecd11ce4f7f225f764d92cc36500c2
3a2e7f0c117c2cd2346ead2aa2b95b6cb6172ea4
6897 F20110218_AABWZQ cobb_p_Page_131thm.jpg
db4143938ff67fa022104dbc34e4ba5f
e763ec374f963b9f517aa410c43c35860d5da436
6751 F20110218_AABXFK cobb_p_Page_144thm.jpg
cfeb92532bc04feb1cc36aba74b3e12a
9fb3b993bf018a8122b5abf80c3aef8498eb59dd
6474 F20110218_AABYJB cobb_p_Page_147thm.jpg
d89a59fe8047fcd29b8f6b0a65a3e8d4
63ac24cb90a280f2ed280bf65342295bfcce999f
7500 F20110218_AABYIN cobb_p_Page_100thm.jpg
69844194853dcbc1d4c8261bb2bc2b18
106e7c6220c7e690c1c19198afc1292ef047399e
6561 F20110218_AABYHZ cobb_p_Page_065thm.jpg
aa75aa73afbe7f8fee23ccaeff85af88
aec64e4bb6c93794c5d82adde426b1a4a782cd67
6661 F20110218_AABXEX cobb_p_Page_012thm.jpg
8fab43f410f26348564cbe1887ad3f42
f416e08b10398c663732272ad9a7ef97c2535e6e
2296 F20110218_AABWZR cobb_p_Page_141.txt
1d56519a9b3a6b4bd0a97708d9f39b8f
8906b78f6dce910b39f74a8c5c43df13f46257c1
1805 F20110218_AABXFL cobb_p_Page_160.txt
e78f70a17d461bfb6e526047f2d9ec76
d2dcd60c67302e7c9cfe2587f3892de7c112c6d7
7519 F20110218_AABYJC cobb_p_Page_148thm.jpg
78c5c7ad68fb6cee413b54ec09494e80
91b04ac6d148be1a7f7194a9d6a1271c0a3ae983
6631 F20110218_AABYIO cobb_p_Page_102thm.jpg
44497f63ae2fdde4b7744264d07daf15
5f60931777209af4d9698c5c258fe5044280ac26
1715 F20110218_AABXEY cobb_p_Page_056.txt
597f0003f2e0b5ea4a7a19e6bb1dad83
1f063aeccbd04c8c10bc4045b504b180ab9e334c
7034 F20110218_AABXGA cobb_p_Page_066thm.jpg
2cf2884d0c4171067e56cee485333abe
49f4fb481293eaebe001d3333080197de1e735ed
33024 F20110218_AABXFM cobb_p_Page_158.QC.jpg
72363e4bf14a8261163d86a2acae3415
403d959ca46dd5616bedae86c2942eec2a145fcf
6869 F20110218_AABYJD cobb_p_Page_151thm.jpg
6b0878c6f57fbd5af113912ddef37026
85ff124bb4d64a8c1806a01eb80b1b0ebe5dfe8a
6898 F20110218_AABYIP cobb_p_Page_103thm.jpg
aad3ff552811c27c6ec3c174b9e566e6
c9545b8743ea182ab70bde07b7518c9c460131e0
6509 F20110218_AABXEZ cobb_p_Page_049thm.jpg
e05e6540b1ad40380bcfd44fc4d01f94
00128ec9d89b61b3a07755be6f41334f196de52f
33861 F20110218_AABWZS cobb_p_Page_082.QC.jpg
325d4da3de8c6030560fa766113e7f1d
c839786b6c7a5d83025755edd0aad3fffea1ea20
28949 F20110218_AABXGB cobb_p_Page_026.QC.jpg
0c2ea777c37a84ea8d36f091d36f6505
ad74e4b467e36675bc49fa4cd05a018efe5dd11f
919077 F20110218_AABXFN cobb_p_Page_088.jp2
f3d4bc00fdabf71f03aa5075c1e1e1cf
c75dbd90b4ad2812b039a8c784ec7d27625ba2b7
8107 F20110218_AABYJE cobb_p_Page_152thm.jpg
f077fb69c50ca16c10a9f755c0f639aa
552dfcbb0bdad60385a2fcfe7ddf694ed12a3a8d
6769 F20110218_AABYIQ cobb_p_Page_104thm.jpg
9322ef89362453e3d121bc15eba7183a
cad43490e0016aec4f74fcbba826a8b40745b144
1987 F20110218_AABWZT cobb_p_Page_115.txt
aa6bd449069bb0810b1b21542e1093d1
475800b943d67db123c2510b78eec267c03e3957
1006675 F20110218_AABXGC cobb_p_Page_140.jp2
37f6e54add28396c28ba082262b4325a
1419d3d6a81dbfda5f56ffc59f8ba5f6d13bf88e
71516 F20110218_AABXFO cobb_p_Page_033.jpg
aa4c0da810b5aec39e80ec7755fcdfd2
690e673db97881053b21eed958f4e5d75c23b0ae
8476 F20110218_AABYJF cobb_p_Page_153thm.jpg
6b743cb255811f8b202a17409c0360b6
ceab5ef4575ac66dbcf76333d442428769d15dd0
7205 F20110218_AABYIR cobb_p_Page_106thm.jpg
ba1ae491a8a453364fea1f447129683a
b8442bbf6e1b1555dde692f69ff0f7c37dfa8514
939098 F20110218_AABWZU cobb_p_Page_096.jp2
647dbdc886fd598885fe364c5a7cd780
75bc11e46fe8cb9da56c358ea1ece130d174204b
42926 F20110218_AABXGD cobb_p_Page_081.pro
bd7bc1185086c61e2f7fe684f3611c61
93b28b6efe3f711117fd0cf58733048d7b89803e
28547 F20110218_AABXFP cobb_p_Page_131.QC.jpg
4e50f6108408e072c3832b3e23c421f6
da2571c0e3629b04b49e7713a8dd7615864ebcc4
F20110218_AABYJG cobb_p_Page_155thm.jpg
310d9a70a10f6a49dbeb1f7329ab9acc
b5ea92e6b709ce016c4d64e324a060b6f7dce12b
6750 F20110218_AABYIS cobb_p_Page_108thm.jpg
000078f8b6d7c2193a422384cd15e959
39dfc8b986b53730b63656e4653f683e53c8cb1a
F20110218_AABWZV cobb_p_Page_154.jp2
d67ffb06bc5f241da660ccbb0f78f1a6
e7ca64597f957652f81b83fbf9c62f39e86868f6
813686 F20110218_AABXGE cobb_p_Page_109.jp2
c4e21d1c09c06481f0b9f69c33acca0c
a44fb346498afa6f42042ec664f4f60f9b32a93c
2016 F20110218_AABXFQ cobb_p_Page_001thm.jpg
2016c609d0a0a60825624e8863126581
7355bad690cb00e9ea82fbd98f1a30068d9b41c3
8441 F20110218_AABYJH cobb_p_Page_156thm.jpg
8e1f6c042917be1e721dd489fc238655
142a30e3437243786dfcd2215970037e49d664bd
7233 F20110218_AABYIT cobb_p_Page_111thm.jpg
a60e733375a707f0c8405584c8dbb682
09acb4cf5bd17c11bc01a265660213c9c8b9cbc9
F20110218_AABWZW cobb_p_Page_135.tif
2a6607228a044512d7fb6f89466e71a2
76ba633f78a5edd2d5dbc3f2659dab50efe29b27
1011956 F20110218_AABXGF cobb_p_Page_015.jp2
c58948694a4621b956c60ecbc218a05d
1d19fff759246a280a7a7f894fd35a760ac50ee9
1075714 F20110218_AABXFR cobb_p_Page_150.jp2
2f32c7788da7ca5b00ced5da7d4fb82c
0afe21ab372d114644fa2afd1cff14d62476bc6c
1175 F20110218_AABYJI cobb_p_Page_159thm.jpg
d2ed40043a2d6d538fba226711f16d35
8c778dd52984c31579d53960a40b78714c562581
7669 F20110218_AABYIU cobb_p_Page_115thm.jpg
7f8be2c87aeb85bcf14f954168550488
a9345fe576021feac70ecf98f3a76d4027ad6dc3
90141 F20110218_AABWZX cobb_p_Page_147.jpg
fc36101f25be0f3690b8ce8633cf4542
34edd8ef5a8156b314832c99baed9739b377e9ab
6705 F20110218_AABXGG cobb_p_Page_030thm.jpg
80be06289a80dcc7a0c4f2272e9916e5
7ee382d540b14fb4d7b469eb09f06b4146460eb5
6755 F20110218_AABXFS cobb_p_Page_087thm.jpg
60e2ce937cb83393ca3d365e9b81cbba
f3f5c6fade0a9ddbd4c0e0d3ba3f2948bf38093b
2285595 F20110218_AABYJJ cobb_p.pdf
53b5b2c60fe64a6d5e38d73afb108617
ad0ab95cbfb06a74f55e98252771e2d512ea74e9
7697 F20110218_AABYIV cobb_p_Page_119thm.jpg
cd5e822aa44c60bd370a28bdfccb3d47
31bfa64c0130d71a0f13ed0a2b6e4a289e55c1bd
77809 F20110218_AABWZY cobb_p_Page_059.jpg
2f91187ce8a248059b8d18c6f5b07105
6def4df4019e6acbdccb1d40caa9006ce0a0ab48
5875 F20110218_AABXGH cobb_p_Page_135thm.jpg
21d98ef21b370fbd3c67e0e92a09426b
1363cbfb89aa6c1018823e95e9f2e226fc1bb73e
28765 F20110218_AABXFT cobb_p_Page_113.QC.jpg
4145496127804ed65d7412c4a954f4fc
6100119531cfb14155fb870aba9ff7fc6fea03bc
186419 F20110218_AABYJK UFE0013662_00001.mets FULL
467cd34467640d11fe3c6065f2c04486
5e6ea57f27caa8dac510a1fc612342fa9d1edcd1
F20110218_AABYIW cobb_p_Page_122thm.jpg
cb29c23ad4a443d611a6eba786023384
fc48f89f91d61fc41d76b05d6546a7292555b481
41425 F20110218_AABWZZ cobb_p_Page_056.pro
f8a48035793018a5d5442d8cfcb8278a
91a38a0ff5647d0731e740dd8163eeca7df6d1db
7017 F20110218_AABXGI cobb_p_Page_141thm.jpg
58a96513c82796eb99617fef4c1da312
3b0825f0412b4582316f224a362c48ee05086b3a
F20110218_AABXFU cobb_p_Page_151.tif
dfb937a1ecb3ba16caa5d2ddfe21dfb3
e0577ee3761bfb7668e3b980647c77b5b1d57c0f
7211 F20110218_AABYIX cobb_p_Page_123thm.jpg
8528840dd50137b0a372fbb229aa6fed
6101ddab2d19a723e35bdcb9a592c97d2851a312
32524 F20110218_AABXGJ cobb_p_Page_068.QC.jpg
a65b7fa3702f3ecf25b5eec3a89a2c52
f2725e4d54086aa43fd4496f2fcd411baa52c6ea
34637 F20110218_AABXFV cobb_p_Page_137.QC.jpg
423ad024d83288d29120e9a2beb0b4cb
ea6a217596bc18bc7e04717245965a79617e933e
7634 F20110218_AABYIY cobb_p_Page_127thm.jpg
25dad4baeb02a38891e8f747c6380618
b3199a8483ab0c866b2cc9be6e064c98791d19d4
2169 F20110218_AABXGK cobb_p_Page_050.txt
9d1f6c13334dce8b98ae0cff173c9e00
d79b0745752ffba9bcd5b9352aea5a255fc020a6
34998 F20110218_AABXFW cobb_p_Page_073.QC.jpg
54d699a424523fec9620ca9cb2a387b4
a4b87c1c87e44b3663ebad4e736db26237f35269
7005 F20110218_AABYIZ cobb_p_Page_136thm.jpg
5488e882e20c676ccc24b04a03182f79
55f8e2c596352560502dc35dd41ab38c448c75a8
F20110218_AABXGL cobb_p_Page_159.tif
1a9e5cb7cea48811907e8c3828e7c107
2ffdb39c1cad0a9d62313e48fb0c3aea76c0373a
F20110218_AABXFX cobb_p_Page_019.tif
184d30259340cc8ae9eab967397ed8f6
a0e4bab32222bf6fa9417cd972a9e226fc93b6e5
106419 F20110218_AABXHA cobb_p_Page_107.jpg
9fd4103fcfb349c340479d4d5a1bf520
2c25996898293b4cb44a4a1ed885b74e14f32218
44028 F20110218_AABXGM cobb_p_Page_052.pro
b36148c74fa8d67b72195bdae89cb0b6
2382c12126bc57cea968ad4cee043537a53d04e8
25502 F20110218_AABXFY cobb_p_Page_048.QC.jpg
24ae968f66b88f6ae7131fba51e9b4ea
fcd28491682cf3bb730f845d16afd110ebfe97d2
79499 F20110218_AABXHB cobb_p_Page_101.jpg
9c3463864eed56b30dc0ccbad12098c8
573dd333074db0d38def59a7fd8116eff60ab769
F20110218_AABXGN cobb_p_Page_074.tif
f7f2e6a5ebbb183e7c5d2061e128a3be
1f3df66210758786f08bb24c27868f467803f592
6505 F20110218_AABXFZ cobb_p_Page_067thm.jpg
d54ce64292eb6bd5d96a6a4f6d877bfc
41eb52bb538a1f8a7316f81fe3827ac64ae10a0d
83906 F20110218_AABXHC cobb_p_Page_049.jpg
77f37bebafa19ecb24fa5f623034eed1
8184189642b0cd91004e09ce659a911194fc3bdb
7820 F20110218_AABXGO cobb_p_Page_107thm.jpg
b98afa92a5ae4013309365acf36c521d
796f46d027b83953014d1c1517a6d381588f97c1
34774 F20110218_AABXHD cobb_p_Page_083.QC.jpg
3cb1bc8c6a61beed691830fa15d94cb2
2d12546e00f2db3a3f5d90e2fa4ee00a9868ad94
100441 F20110218_AABXGP cobb_p_Page_159.jp2
b040ad360b90eed38eb818ba6abad014
3e5193c56cc5be03f169df20fbd91978058238a1
1087832 F20110218_AABXHE cobb_p_Page_083.jp2
ebec501c62b34f51dd8d160f4a65b96e
8794ae61a0f94e3a13d3f22967f36868878a542d
2134 F20110218_AABXGQ cobb_p_Page_095.txt
fa50e62c757df4cc4f7e2b6115e7ec35
1b969297199011636a530f7d2cf51915d1e785a8
25422 F20110218_AABXHF cobb_p_Page_023.QC.jpg
9be6881e807bfcde58314c957c5e6dd2
a0cc63da3ada94c87dd63c51a91fc9d1774a149d
27027 F20110218_AABXGR cobb_p_Page_123.QC.jpg
ae4b872dba20e3e2972416ba69ddc945
ad5eba38b13e6046e761d8a64dec622708bf2f49
F20110218_AABXHG cobb_p_Page_146.tif
5bfa1965726ed2af8fde774d35f08859
3fa20e782c2bb554b4708a8ad9b1a0d06f713791
883539 F20110218_AABXGS cobb_p_Page_144.jp2
ff861f094c7e95d2e298c3ee1b44e2ac
df53c86fdbbcbf629c74e7daa77f49c579858888
96873 F20110218_AABXHH cobb_p_Page_098.jpg
6dd72da934a69dd9592c81ea72c11a16
15ba52b5682402e8ff5cbd0f66e66148db688c20
822647 F20110218_AABXGT cobb_p_Page_005.jp2
3f0ecb3caf3b5f5570f7ee78c17f7817
61a2e7a690bc6b38425f6505e3914ee2899d023b
6909 F20110218_AABXHI cobb_p_Page_020thm.jpg
a18ece1046fba5a7d3e62a64737ef780
0d3172b977963219f344e2214cbdc7d3c94eee2d
F20110218_AABXGU cobb_p_Page_101.tif
f7ba8d592a2611e0a4b1150275c62fbc
38fdefe0a9dabad05e8a927a2e8f676a85311e06
26022 F20110218_AABXHJ cobb_p_Page_102.QC.jpg
c838a31c14b6ee0eebfb3f3f5df3c1b4
48401a7977523070f461e65732807f6085a0ee0c
33502 F20110218_AABXGV cobb_p_Page_152.QC.jpg
b016beb3b157bdb57dea0e36ff613d3e
2de54936d01f57d46bd9c7440bb0521b05dece7c
30462 F20110218_AABXHK cobb_p_Page_010.QC.jpg
8f4beb72a47bf53db5ed10e9258ed2d8
b3b861153c8a9a331003ce8fee28d2433112861b
35891 F20110218_AABXGW cobb_p_Page_149.QC.jpg
25fe7c0b4406f6d3452c55263bc0b090
78906ba63ccce251df908f6d8b1294e34fadb7dd
2102 F20110218_AABXHL cobb_p_Page_082.txt
7f1984bd5b7202bfb7a1cd1ae3e5f08d
f2531cb925b44e96b868b7692e5d0a32e5aa388b
6703 F20110218_AABXGX cobb_p_Page_007.pro
158ee55e758bcf587afff8a2b3e77fe2
f091f9022b1966e9db9d5317f4ea6915d36e7b42
730898 F20110218_AABXHM cobb_p_Page_033.jp2
cc40c91918c6f4077bd6d427ae82ebfb
737960b42b65b025fce240da3a8a4b5ccbcb9839
34364 F20110218_AABXGY cobb_p_Page_124.pro
8fc88c3b03a42cd6adebee1c85c5ce12
64e83a18590a361f668cb1664215abceeeb481ab
109889 F20110218_AABXIA cobb_p_Page_021.jpg
14962ce70423397a120cd473ebd569d2
e3a5c124a1eff47917e15f111d2bb256b38c1d5d
40455 F20110218_AABXHN cobb_p_Page_027.pro
ecd81c6b4c794aa83940492d263a7d72
3a5a2c6aca66953d8d255a7ac3aac26942c7d5d4
37512 F20110218_AABXGZ cobb_p_Page_143.pro
59210bd6c7194941790204fc70ee1485
88d3d0e13ff48ce320960eba5e45b8adb712551b
2039 F20110218_AABXIB cobb_p_Page_111.txt
105f4df4824a829b4e9d953bff789bb3
9e0f77733f1fe1a6194fb07cd308979d9237aebe
24639 F20110218_AABXHO cobb_p_Page_109.QC.jpg
8f4d10c3e0a817a3677e9fa46e8d5495
034fe34e06aabd62b31f3878e4e872060259c5f9
683853 F20110218_AABXIC cobb_p_Page_075.jp2
a32c02414a933ec1a0377b196e034208
2e0c882c96f065228287508a65a90483e17f93a2
1928 F20110218_AABXHP cobb_p_Page_123.txt
a1e2e19254067bf4188924c1f8bdfd49
5a06aa532bdc1a8596fe916f965ba849e7abbd76
33164 F20110218_AABXID cobb_p_Page_024.QC.jpg
beb448277946fcf7b683c2a54ab46aed
6905d596c0caddb0ec137bf44eb97f323dd6912a
F20110218_AABXHQ cobb_p_Page_011.tif
7addf0bc5768eb5bea27156fcc799226
bca168df80a348b5429400b114462ee8fd69accc
1061540 F20110218_AABXIE cobb_p_Page_151.jp2
3111d7afe20bdb072a6e5188bde7d188
ad7e75cc8d3074275d3e7a2783ad685c37e2cf24
107802 F20110218_AABXHR cobb_p_Page_145.jpg
dcb3215f2e84f386fd442755e8b667f4
f6ff3dc63794e32de4df41e6beb88fe2f3e59517
30824 F20110218_AABXIF cobb_p_Page_060.QC.jpg
3524157ab49eb3a94b34befa0cc95d36
7477e451cad361faa07dce073776affeb0648aef
F20110218_AABXHS cobb_p_Page_091.tif
7159d2475eb988cd8706cace73ce0265
1499ce03ce12ee56ac92e68dae2aec4e0e053201
4957 F20110218_AABXIG cobb_p_Page_044thm.jpg
9c8358f520ce66aa22a18782f09282e8
59242e745756daa6ce46beedb00463bb04e34cfa
1892 F20110218_AABXHT cobb_p_Page_053.txt
6c9a007127ae35371e541e32c043a991
6bcb10aa371f59c5913c9e8e820519280ce44513
24381 F20110218_AABXIH cobb_p_Page_032.pro
da25e0a1ed5308eeca6b48259d99703a
d77fbb654047ec0b8166687147d837686563738a
88499 F20110218_AABXHU cobb_p_Page_117.jpg
8bf46bbe868292cf3b6ab3362a5216e0
a51914874c301dd591e55eb7c7c1bcf16614c482
26473 F20110218_AABXII cobb_p_Page_116.QC.jpg
317c2968eea670070b80c0484ed5e7f5
e2ceed535ff733dca0d473e6e583065dd9d10394
34352 F20110218_AABXHV cobb_p_Page_093.QC.jpg
2a013a77ef68b4ed416177254d82074f
72ef8f3e00545548c2f93c0b1de73dc95e5a4761
7168 F20110218_AABXIJ cobb_p_Page_150thm.jpg
8817ca8c59d70246899bc8b8daa8252c
ac045eff892a8ffb1c34a2b19ac9c50fdebe22d4
5503 F20110218_AABXHW cobb_p_Page_028thm.jpg
35886a222ee478bffd0233b25d86e617
27501aa692a00bcaad16b63bc3a57810d76528d7
973355 F20110218_AABXIK cobb_p_Page_040.jp2
8a36e4bf406e52256ef3dd9b76c45eb5
013eee00c4c20d450695df8873da117a37f27dd5
F20110218_AABXHX cobb_p_Page_126.tif
38b5a6b744f5bd3943e121dda43ebfbd
d98854f61cb47150027f5783131a4028e0eb8690
74912 F20110218_AABXJA cobb_p_Page_039.jpg
f9dfd226ae842a7421dbf6a29f6ccac1
b8c2bb6cf47412490b6fd37fe45f68ad82388e05
6370 F20110218_AABXIL cobb_p_Page_027thm.jpg
ca7e8f0b19dba15a6d82a9461f91da3b
7028c862e11dba37b5c7fe8c9807baf45fc6de91
8088 F20110218_AABXHY cobb_p_Page_094thm.jpg
f44ba095847cab22e6c092515d7c4e0b
703eb8626daf5735e6e2c575095979309ad06283
45134 F20110218_AABXIM cobb_p_Page_106.pro
a4be3c1621863d8ce8469523a84c12f3
3c18bc785c599a180359cc91523ca9db4600e11c
7845 F20110218_AABXJB cobb_p_Page_001.pro
6065e58fccc6da03030ee126a439f3c7
7b61a20534fc27cdcc1a5f890aae33cd17a5d360
18107 F20110218_AABXIN cobb_p_Page_042.QC.jpg
389c0277df225f3e2fff88a83a226129
06714a94087578f496c6761f9fc7ede2c6e7efbb
F20110218_AABXHZ cobb_p_Page_149.tif
6c4ab7271f4a4dfa813a26666d370ba6
a671651e905055d04659d019e49c6fd736a1e742
27746 F20110218_AABXJC cobb_p_Page_147.QC.jpg
f1e71af3694650aeb7a4852c8b65fdd2
c9e4ba978a802aa5705c58aa507cb74694bf9359
22821 F20110218_AABXIO cobb_p_Page_039.QC.jpg
426b08db650d6fa3299279955867c814
bdc80c16aae1b55d03c4ee2777c048a64cbcad53
33835 F20110218_AABXJD cobb_p_Page_004.pro
76622e9be1b413b058095c3495a2ebd5
05705a324ee2c03bb6e21a8fa6585bae08802781
1472 F20110218_AABXIP cobb_p_Page_058.txt
272c1c264e4e5c40ac8fcf5cb2cdbfe0
323f86819495d2372679f94aed741cd183c98e66
319 F20110218_AABXJE cobb_p_Page_008.txt
b7551e22232d8517a59ce9892f8b8df3
105f565cdf1ce13e37ecdafc520ecde67a74bf3a
41788 F20110218_AABXIQ cobb_p_Page_142.pro
a1a637a243fc95f461b6a48fbdbcbc05
5dbcc41e74fc78357f708eaaa5da417113220e05
1740 F20110218_AABXJF cobb_p_Page_063.txt
5ca01ef89ddc72dbbaa230156ebfc69d
8a600a5650545e98edbd01fefe0cebf16e5a4eb2
8157 F20110218_AABXIR cobb_p_Page_158thm.jpg
82dc9727b39c2567ec5d633e9e0182b1
e0e7dad8d686fd5073b8644ea425eb4437432a3b
1465 F20110218_AABXJG cobb_p_Page_028.txt
ac718f6e889be466365d71b207cf7eb8
f88347be659ebadb58307e2840aae1c3a13e4ad8
7436 F20110218_AABXIS cobb_p_Page_070thm.jpg
c979b9495b7c9a67288768d94b753fa3
f898f27a2cbe5f7cb80c80677f358ee9ac652008
F20110218_AABXJH cobb_p_Page_105.tif
e25cc85649390985109641db73c78d52
e089fb5d72b6ea766551efc7b649d1e11607653e
1210 F20110218_AABXIT cobb_p_Page_008thm.jpg
fcdd77f465ade2b77ce5b64736eeafce
3de87c1735d24d80d05257f5a03925d5d571599b
20525 F20110218_AABXJI cobb_p_Page_046.QC.jpg
44e20de8cab0decd8f76262644c7cdca
c3e26bd1e2b1c0de749d41b65606a9a84f5aa3e7
666224 F20110218_AABXIU cobb_p_Page_036.jp2
c09fbfa8f07a868d0ddd141ee640bcd2
4035785942ae0ecdb553e72eba32fd247b87214d
24408 F20110218_AABXJJ cobb_p_Page_124.QC.jpg
02e496b43e65723d0fafbffa3add3fa4
803bb382d299edf8229ccd7c1a7c786e21e0e25d
42390 F20110218_AABXIV cobb_p_Page_110.pro
2e4d76d1378846b3806e0f8835f48da0
ca8fc64820d7f46f1cca09eac1fdb05a09701aa4
1879 F20110218_AABXJK cobb_p_Page_066.txt
6b25c9bd68c2c47817177583ffb29002
d0da5502138007e88524c65316dbd8fb88707aec
35017 F20110218_AABXIW cobb_p_Page_118.QC.jpg
e3656dd1cfe6216e4b57a5b7e16b080b
9af5f25240246af4604c248d36e230f7bc0b0487
948033 F20110218_AABXJL cobb_p_Page_035.jp2
c7a3253b302d67e9eec03f81565d103d
7af45e2a4117be66e1c8c79b75569bef6ec095f6
1448 F20110218_AABXIX cobb_p_Page_036.txt
8c33c5a4fc295c5d1f32662715c0ff34
39528cb1a1bc8758903c4d66caadfbcd90e73786
13848 F20110218_AABXKA cobb_p_Page_146.QC.jpg
0ae92a943015d21d103be973e1b33fdc
355f4e300bf7fbe04fe1f73ed4c00245570d75b4
33434 F20110218_AABXJM cobb_p_Page_156.QC.jpg
1dd88d9bac68e336f22e7a28ed287fba
7c3e77b5f20602ef76488dd6e4fd334ba0c4668c
914124 F20110218_AABXIY cobb_p_Page_053.jp2
7ad60bf09b43081aca2ebea9e624bb0f
920e9f6418b070c7819542979c3e350096ff8b48
982326 F20110218_AABXKB cobb_p_Page_078.jp2
74aa7ac9a83d72d1a1e2089d45bf4a94
e88cd012ff496c03a13a189e29caf6c406d09b8e
85995 F20110218_AABXJN cobb_p_Page_134.jpg
767db51b1f5d2e343efdead4465b5a76
5932488f83da789ebfb984edef4dd74fc3ed87e6
2799 F20110218_AABXIZ cobb_p_Page_121thm.jpg
67a270e8b6c2b48733e5ae1f007b59e3
845efa2f0bd0c3884e7eb745c0b442865ca3ab34
F20110218_AABXJO cobb_p_Page_061.tif
36efdc71c789d46d745b410a831b1b12
8b47e37a224ed15f79c28635e23d9444861793a2
6143 F20110218_AABXKC cobb_p_Page_109thm.jpg
849940f17b345e2716a9cb2f1cfa48ba
7c6a2f9655996e14478aa18e8cd681bc687c9a6e
F20110218_AABXJP cobb_p_Page_093.pro
f8b5930a633f75f51eaf56dc2d2ba4e3
e6ec8d54458bf06bac2cf397d6e96ad7a732902c
2265 F20110218_AABXKD cobb_p_Page_009.txt
1fbed1ce1da192b09c9dc8609625a4ae
1ef480843e5328cb4b7379f1bdb0e9c3894ceec8
17296 F20110218_AABXJQ cobb_p_Page_032.QC.jpg
1eb45215d1fb8db7cac253b7e99b6eed
75d4d0830d388348078ce74ff77ab890b2838c7f
91933 F20110218_AABXKE cobb_p_Page_063.jpg
dac82880c85dcb36cb1528991df6412f
9b6134dcb722d0647a9afb0d8831c4a7bed79c2b
2416 F20110218_AABXJR cobb_p_Page_108.txt
80d0cc60d2de0d630c8c66feb49de420
dff0da0c1d008fad36cf96d7dae34b90395531ff
F20110218_AABXKF cobb_p_Page_153.tif
bb675506d34aeb18225021ec337a4c0b
8ecf0f04b561d9c1a4fc577e9e5ea8b98028e23b
F20110218_AABXJS cobb_p_Page_030.txt
c66b7d8745c3c9bfa8596dc5df9043d8
a40624d5d4e4be551c4c2ecde7a91f123d20822e
F20110218_AABXKG cobb_p_Page_015.tif
0ba74c727766426d0e7e7812284f88d8
db94cf56a2df94046aa2956c4353b82adfe5d2a8
38183 F20110218_AABXJT cobb_p_Page_019.pro
abfe28429e49c16fafe8de96687d5a52
88f58d8afb7bc11f9c658fb7ff9a48d4cb15768e
11688 F20110218_AABXKH cobb_p_Page_121.QC.jpg
d3fcc65209fd710cdc50db17782a3343
848bc6f420cf35e3abaf900bfc8e53acacaf1e14
F20110218_AABXJU cobb_p_Page_133thm.jpg
07c038319a7692e65a725754951680fe
7c4c74d8312f741f7b6d49bb33fabb3a334b4ffc
31728 F20110218_AABXKI cobb_p_Page_033.pro
a282d00c7bd8663f65abb169b7c90ba5
19488eeb153431a1da091f801fc8cf4cf6b55cbf
1015481 F20110218_AABXJV cobb_p_Page_098.jp2
0bc266dba4461dfe8e9e5eb15e4b2276
22cf555b6b7d61d7998bbaff53ce2dc451873a10
1406 F20110218_AABXKJ cobb_p_Page_004.txt
f2441b5dae275ddba79b9094f5c56ff9
11d2efcc26a4615c042c1851bd1c363631e12143
2029 F20110218_AABXJW cobb_p_Page_092.txt
92ee77e0ce6393579afb0d0273cf1f80
92d44240ae827213d59a9d7f330038b2f81228fc
45642 F20110218_AABXKK cobb_p_Page_141.pro
926f6bae57377b1a3d97d7fe98d763f0
efca501efa9cc84cc10cf9d2ba2d028cfd1262ed
46073 F20110218_AABXJX cobb_p_Page_014.pro
c33299dcdbcebc2603e77af73ae4f49c
838a876d94ee00eb8569f7222546b2b3e8af003f
F20110218_AABXLA cobb_p_Page_037.tif
ea9b87092407814641457b2bae90e443
8418e857b285422d1d17406fbf906eda571f2dd9
F20110218_AABXKL cobb_p_Page_068.tif
b6f29c77f0d5cfb9c327e0c885c478bb
9611beb023ac639ae643a16e535109edfdf757c0
6355 F20110218_AABXJY cobb_p_Page_031thm.jpg
6c020f7f961c198e7b4c3a408c2ae770
9fb9da3ab46615653c6700e0eff84d1f886ba3bf
7928 F20110218_AABXLB cobb_p_Page_114thm.jpg
ffa3a37dbfeaadb9f414c8040a952612
0dcff1a53600b103961b575ce1e5d227397a5ede
32539 F20110218_AABXKM cobb_p_Page_072.QC.jpg
7dec1c23dcf6a7b97cee6b7be2b8ecc8
379d0ff01fed121b4b9d35fca6080502a10af525
5470 F20110218_AABXJZ cobb_p_Page_005thm.jpg
8f59e5ab3297167b0e00ff43f604a679
9b4ecc57026bfdc317c4fffeb5da2ba23f305392
916199 F20110218_AABXLC cobb_p_Page_087.jp2
e5f447bc0ed1410d69a504e67eeb01a8
c73a269de8d334b197fb9544a0e48404b033234d
98385 F20110218_AABXKN cobb_p_Page_078.jpg
10bbc38a285f1af4e49ed19a9cad2fbc
0c8d83b78f7d9b9e56979d3d82cdd541149dc1c4
6576 F20110218_AABXKO cobb_p_Page_129thm.jpg
4f50b7244a9b10230977b2948c7bce9b
2727226053daed09d6d603f65fe93e01baeee905
861359 F20110218_AABXLD cobb_p_Page_077.jp2
f56f326292173a46be538c32d97a3f47
2f0764ba4f49c8dda4c1a760c8ee96c41505ec68
F20110218_AABXKP cobb_p_Page_020.tif
230ad30fa70da84af3c7f635aefa8135
fc262455c142dda41340dce346ad076ebf77bb91
26932 F20110218_AABXLE cobb_p_Page_108.QC.jpg
717f39ab1dde297ae94e26854c88b27c
0dad4c2a34314a23343579d7259613597f826d3a
30223 F20110218_AABXKQ cobb_p_Page_139.pro
fb7f043aad30182886d0c74d5a8bff89
ff3999665432547f91b8d4149d00cf0f9a034095
23784 F20110218_AABXLF cobb_p_Page_132.QC.jpg
f66ab01ce3b54079bd80b0c8dc955ab0
07a80f1eb975b126b68d9a5aa33dcead647544ad
1070 F20110218_AABXKR cobb_p_Page_007thm.jpg
93a3a2f0216f6db79990d5fbe0cbdd3e
d3a274344ea68e95fe5df8ab0ede5b78259dee08
7031 F20110218_AABXLG cobb_p_Page_014thm.jpg
a19a8d29697c5d875088dcc8cedfecea
3ff45e58efb031c26b2b868a3c6b41db3b2d4d96
55457 F20110218_AABXKS cobb_p_Page_158.pro
30d68fe2fc550f2cad9d74ca989526e8
d9afaf08e79916ca5f6f517b7d4087b2982d2155
F20110218_AABXLH cobb_p_Page_077.tif
1fce78983729e076cbf29325e93c4403
576b68b2f032b200e0fe8199a0ab17a2f35778b6
24569 F20110218_AABXKT cobb_p_Page_120.QC.jpg
ed65e35221d74971e1a6925ca2cbf417
d3d3e9dbf9af2d81c37ab3bad36bc209b39cd595
61231 F20110218_AABXLI cobb_p_Page_048.pro
2ef7e562c845aac1e94907d14e02baf3
5b75b166bac7880077d6bbd0046c09acc8a81a27
1888 F20110218_AABXKU cobb_p_Page_074.txt
a8e96c103a54ff2ad8c7fb05d90e7112
d3f584ef8fbb7915eef48ccf24ce8066831d5193
F20110218_AABXLJ cobb_p_Page_097.tif
69620a477d7808b5d52aedbfac5147c5
fa6ee192c63fd40f63c9365c4d343426cba2b7e6
24299 F20110218_AABXKV cobb_p_Page_059.QC.jpg
54f09c9e2f23ce318f57659358a8781e
dc57bd0f4b802dd5a33f335b98def0a7325cf5bd
50395 F20110218_AABXLK cobb_p_Page_005.pro
4c8a86f9422caa4236d3f83cb7f41e0f
f949d4f518c93052da73b5da8b0b73210f84c86b
56919 F20110218_AABXKW cobb_p_Page_083.pro
78e8f7f2107ca76a38068f5e25b550f5
52b2e6d31146e87ea00f14aedbbeea1693a33f98
26820 F20110218_AABXMA cobb_p_Page_053.QC.jpg
e4944c89af45d134bad7d357dce86f83
aa8c7f012ad3108538d68ca4e5cd4a805a0c292f
33264 F20110218_AABXLL cobb_p_Page_089.QC.jpg
42e425ff91af7b275c2d87475aa82428
51a17dabb51cf0bc0fe1b215e3e7b20b396bbdd0
823636 F20110218_AABXKX cobb_p_Page_038.jp2
1a015976cd43e26651ea1efc9b2350cd
af8186db152f3f5423a2a1345d8eda7bae3a124a
888811 F20110218_AABXMB cobb_p_Page_105.jp2
08afdbb8def01328b780576d1afc99be
edfee21222146ab0a01114c75db334a70bdff0b6
F20110218_AABXLM cobb_p_Page_028.tif
d4b131dff8cca244b44e2644e98e646f
f2ce635c8ebd6da015ce58dc1a30356471db4082
33692 F20110218_AABXKY cobb_p_Page_119.QC.jpg
d5c63906626bc385c273d4517b8c1758
f7fa4a43b663420cf587484cc89540cd892005b7
93148 F20110218_AABXMC cobb_p_Page_113.jpg
1c5741d412f3e07d0eda4ba4035376fe
bd17af71e5a2557bfc26add611bd4ef60482b636
29321 F20110218_AABXLN cobb_p_Page_084.QC.jpg
7a949ae82d24a180cead1779dd772106
a68077f611adfd6a601280a9112387148246ad61
42781 F20110218_AABXKZ cobb_p_Page_035.pro
ea011984f2659fb09c1e39c898bee5e5
90e9b47c82b321212b1f3f57a1bfefe385fc75b5
25888 F20110218_AABXMD cobb_p_Page_002.jp2
6bed5e752f9672769c044432aff6e77e
a67c1c9e3a1f1daee95c31ab142c4c97d0d71531
695938 F20110218_AABXLO cobb_p_Page_031.jp2
a616e6a69c2f625accb515e2d16ffbd2
b2901134f189e7c05c514c5579e09a1121fa3817
49723 F20110218_AABXLP cobb_p_Page_150.pro
09505e45b9d44d501ff42435bd511ca8
c06764e942a8b5ac24426b040b7204d85cc12bab
1356 F20110218_AABXME cobb_p_Page_032.txt
c3f6d0c41d70778133634546d092c9e4
156a765bb2933b7d9492be5f960d65899483b576
F20110218_AABXLQ cobb_p_Page_055.txt
193bc617209a3102f3e14e4d96f7f013
2ced3fd4fe470a89c36f130a3f98432ffdfba27c
F20110218_AABXMF cobb_p_Page_058.tif
c7a204118db57e8b62683bcce34bb03b
21f392bd810a1c5eea2a1035d58a425ca7854945
72714 F20110218_AABXLR cobb_p_Page_004.jpg
145780a4dd2daedbf4c4fc58cecf2beb
3fa75dc59675f62fd0e9cba9c857fe04c8b3ce2d
44997 F20110218_AABXMG cobb_p_Page_111.pro
cdbe286ccea3684b6f4d2fd572a87540
fa7259a61b5501e26b30ac5c85939276c9d2019a
2615 F20110218_AABXLS cobb_p_Page_048.txt
0d3e2cac55d93b5201fa12ecb160f559
d55e526e5483330ab010c8ec0dd8c0367a72d3b7
7181 F20110218_AABXMH cobb_p_Page_086thm.jpg
3622fc9277b1a48fa51192f3ad2810b0
51799fb45e15f18627fc6e7e94be5b99aeeabc14
28829 F20110218_AABXLT cobb_p_Page_088.QC.jpg
d98cc1ac2c73fdf873d6d0e0eeea6a9d
c25bafdf2509c484d913cf4a568f92487cb599b2
F20110218_AABXMI cobb_p_Page_057.tif
eb8a9124120197adc7c284d504f33cb4
8d4b31eac7bdb20d99e2d927ac38c08b4e8f1706
27242 F20110218_AABXLU cobb_p_Page_134.QC.jpg
aff3a9645226ad623cbab6c11bf29dbc
6c03d3476c4b20fd8e1dafde4bdf8e7aea02848e
52553 F20110218_AABXMJ cobb_p_Page_107.pro
62288d645f4b15dc0c3e861a10bad6c0
e7e0d4c8ae9b97e6080e8f05085c81ab105ee9b4
1087860 F20110218_AABXLV cobb_p_Page_099.jp2
1c23831ef261290dbdd9eb91918ab1c1
263d97f7a507faeb7056647c18f5e1f2c25073e1
1497 F20110218_AABXMK cobb_p_Page_031.txt
82f2e86242c8502820ee4a75912c9a46
bc20a6ef30d0d2af806f5e37cdcea0c05174a750
6342 F20110218_AABXLW cobb_p_Page_038thm.jpg
618f6fd0d177804a41a4a22d95447f9a
14aae94c1fc8317d1c303805b54e0ab4077ca54b
F20110218_AABXML cobb_p_Page_059.tif
4f61e915052fe0d13f5b81210fcf42af
fe6495428676bc08d28d63c914288c347552269b
42624 F20110218_AABXLX cobb_p_Page_105.pro
6e2961561200e9587ec41e3d2c5a5a9e
b2736ca9ed3f24a4041156a832c54dc94277f517
122617 F20110218_AABXNA cobb_p_Page_154.jpg
81d7ed3d34c4d28ee5ccdab6477fe79a
0d5c9e34ebb51792bd2b30b71423b2148ebfcfce
28110 F20110218_AABXMM cobb_p_Page_030.QC.jpg
41f1dca774491e3e3186e53b11f303b6
ea592f1d77b8aa5e9b8b713dc472202ca2c11f6e
23626 F20110218_AABXLY cobb_p_Page_033.QC.jpg
56012bc8966897023532dc9e86a3f2e5
78f30ee7561a495c134104e34ecdf62b58e7d5b6
F20110218_AABXNB cobb_p_Page_078.tif
7ee960612d6b51c7be4f4939604e8729
9fb3ad29443897f9cdcf043dae9118209ebc0659
2014 F20110218_AABXMN cobb_p_Page_077.txt
626da1b8a68037f7d179ff1b32acf6ee
d593cc8459b30d93b0c8f53b38d98b34c1e9a4b7
53586 F20110218_AABXLZ cobb_p_Page_125.pro
0dd93fa00b3c6b2e31fda4679a04e740
b5ec182daefd52b3d803544ef8472c94c0ba3652
6367 F20110218_AABXNC cobb_p_Page_124thm.jpg
535cfba6ff97221bf3cedd03ae11a0d9
a1ea9e0139a5e26f0c8431f73c8da79eccc6c829
25470 F20110218_AABXMO cobb_p_Page_074.QC.jpg
0bde9fb48cafe9a49a1f5263863ddd59
587ab631407c64ca6b9e5efa9802b78b3ebc40da
862 F20110218_AABXND cobb_p_Page_013.txt
15210155fab74c06f8de5028486c2041
1616eaa3680395ee77d95888760951a38a30a58e
86466 F20110218_AABXMP cobb_p_Page_105.jpg
5dbab48b6175f76cef6387f1ff7ad9a5
1d537fc5a96e14b4465eef4744cb707d3493e600
F20110218_AABXNE cobb_p_Page_150.tif
cb035c82e1dd0a5c2a7fef9f5299084f
5deb61f27d0964580e1ab40ed955b266bba6de21
F20110218_AABXMQ cobb_p_Page_006.tif
f47ea60049c8d5c7aed320b181e19cc7
5495a42cd973c453a1098485ec22ad5bd87ed469
F20110218_AABXMR cobb_p_Page_030.tif
8832ca20529403d5768971812c04840f
62c1adb3aedd35668387b8d131f6f4307513d9dd
F20110218_AABXNF cobb_p_Page_005.txt
40f74b556bf87a4d59ad957b81f47aff
6432596365629c90e00ea4fe31d3e56a4c0b7377
28192 F20110218_AABXMS cobb_p_Page_040.QC.jpg
3392f0ae2d78960bb0aa3f103c79e9fe
fd9676632e088d33490a4c7986f25c8707cbc536
8059 F20110218_AABXNG cobb_p_Page_093thm.jpg
1c094cad1f6dbd9119d9919c6152cf8b
98d8695bb118886f6b33b3801919f3ececab4fc8
38640 F20110218_AABXMT cobb_p_Page_121.jpg
46b8922d03dc38775ef2713b1e5d82ed
fa45e9b891f17396ebf3b367b22546cd2f3928fe
57122 F20110218_AABXNH cobb_p_Page_076.pro
07ebaaf98edc9abd066c26667ab5137d
d9e38222de2c55703c5e07283f078195d9441d57
F20110218_AABXMU cobb_p_Page_039.tif
1b3a8947b83829774aa3cdc4a6918f82
5704ac6bb82a1fa18d930feb2074267b7563c418
7951 F20110218_AABXNI cobb_p_Page_125thm.jpg
2d9f46628973529cf4d6e8e487f1db2c
e2d9eebe211154ed35ac05ed6ef70c3f0d1eebcb
1724 F20110218_AABXMV cobb_p_Page_120.txt
9833c187cf5dd2ddc04c7d69f8eef3d1
a60a866d3567701d876a870307b445c47de3b7a1
36032 F20110218_AABXNJ cobb_p_Page_134.pro
122ed6249640b2c11c32982673bdbb3d
bbc5fd6e73b0fae867cab2c09cba3e6e23bf97de
38536 F20110218_AABXNK cobb_p_Page_092.pro
2dd200d1ea0e3bb4ef8be20091ea7f42
fcd61a17e509efbd696846113d68788d8a45387c
F20110218_AABXMW cobb_p_Page_107.jp2
2895c6e21b73069ebe4d003fb2b853de
9ef2f25b19d8c386a7750ad72a5ab6147309e300
F20110218_AABXOA cobb_p_Page_029.tif
9dbe24df1d7ac1e7490d1d5fa2cde401
00bacae0465ada835877f525236c7aa8e7796f0b
255699 F20110218_AABXNL UFE0013662_00001.xml
da5390cd239201fb2b7aaaafe16efe3e
3e414d35703cfbd570634a9330383c260ad00b19
2074 F20110218_AABXMX cobb_p_Page_100.txt
2ed6d98a89a4281b2e8ebd7cf8f7a658
deccd09bbca7b64b58549f74633a6a1eb9b5a793
F20110218_AABXOB cobb_p_Page_032.tif
72a2d7894786f5811fa530d6bd0f8dea
2603ffcf5024f8bd4789fa83e61760050104ffb4
1148 F20110218_AABXMY cobb_p_Page_018.txt
bf73d1ada68a739cdcdfb91014adc101
7f1457b6a24b052080333438c6887ad3681bdaab
F20110218_AABXOC cobb_p_Page_035.tif
c670351260c233de30834ed9086a7d41
4f3d4bbaeabb47189b2f752c40f44591fdaed37a
6256 F20110218_AABXMZ cobb_p_Page_092thm.jpg
7d5c33c57ba2ef111a248d438217cd3c
ca4e69340cd5c566f02b322534e1771851afb796
F20110218_AABXOD cobb_p_Page_036.tif
0dd071ea666af49a3707d9e917978f1c
2df32349e108860e199aed1f4c430c54b81238cb
F20110218_AABXNO cobb_p_Page_001.tif
7057172bf1884d0971b984adb29b8e3e
e9b29de2896cd198d148ed37126a9de73788f7aa
F20110218_AABXOE cobb_p_Page_038.tif
0ffd1439ed20adcc96d69e8b4ff29374
de73964cf72c6274c2f251dbddde02e0acc9549c
F20110218_AABXNP cobb_p_Page_002.tif
ebec3f0624d52d0a849adf8e09095aa3
57d6909f55a39640769ccd36fcd011a9c0f5a99a
F20110218_AABXOF cobb_p_Page_040.tif
863d0b72860c61b9aa1b5b6152c068a8
45357a95cfe67d00ecfb0fe3d9979344c76b9321
F20110218_AABXNQ cobb_p_Page_003.tif
945cdcf384f18586cddfd6b63b9e4c78
25c154148440c56ae26f0510e68f42ea37f90fab
F20110218_AABXNR cobb_p_Page_005.tif
4d63572a3e151ac6c2e5a24b1dc21169
532be0e42288658b019a068bd8f57bde2fbb801c
F20110218_AABXOG cobb_p_Page_041.tif
81778f2e00728c7bff4eb0f5dace77ab
7b82db565575304aa454d0052a0236b1bb342769
F20110218_AABXNS cobb_p_Page_007.tif
838c03d029692b5f994cda82710c4e64
8f3b7a3bc9427e8ea983f28fb698156310b7ea9e
F20110218_AABXOH cobb_p_Page_044.tif
0a3a24919532630f19efe9087a62a111
dbabbc0a7a9893eddb456b48b3194818bdacb0db
F20110218_AABXNT cobb_p_Page_008.tif
452f43688cb977485a8c04edf66ec5bd
638d024b4c85be6c399ff90c826fe44c1d60ab24
F20110218_AABXOI cobb_p_Page_045.tif
7753ab5b4447c1d3592228264d21b689
ab300304d4db41d7305a6dbe6c10f887046e244c
F20110218_AABXNU cobb_p_Page_010.tif
d032afa55caa89eb05ae84fe7270d820
94bc2dd590e7154db0516fd04db74132579f8cb6
F20110218_AABXOJ cobb_p_Page_046.tif
6cca07da530d7920f2753f09dc1f816f
641717caadbc8297cb7d0ad1deaf6595e21808d8
F20110218_AABXNV cobb_p_Page_014.tif
3e785ca3410e0ee58527beb9c6485c21
1a8cf130cda7212f748ff3575b074d6a014c932c
F20110218_AABXOK cobb_p_Page_048.tif
2516cac4dc2c40ca4eddf2f7a379ddca
9e28390f36dd45047a345907a590fdf98eca1844
F20110218_AABXNW cobb_p_Page_017.tif
bc4e4b634bb1cfb23d2a072f85e8b0cf
fbf94b580be279e72bff8171951657f1417ed3ec
F20110218_AABXOL cobb_p_Page_050.tif
a31bcd5ba3992ff41c6969692b73d631
fa523db321a1dbcb0fb865b664414d943da01446
F20110218_AABXNX cobb_p_Page_022.tif
03b632c0340ad652896898f3e4934c3a
bf628df2f1eb38afd7c1f80f149f42570e871893
F20110218_AABXPA cobb_p_Page_093.tif
55a0225861e4d432532bbddabe97a53c
30ed2054d680212ed365589494ba9b5a4804b151
F20110218_AABXOM cobb_p_Page_051.tif
dd2852db309adbbb43bf1e678fc57613
35f54f52a3a1370a8099f96ac1d1de577bc8f6e0
F20110218_AABXNY cobb_p_Page_025.tif
487657ca8fd7492db1b36a0c89ce1c63
7b8393f53b429118866a44b49753861cb8e10f9f
F20110218_AABXPB cobb_p_Page_094.tif
20ac1a5d14d944f9a91bd609b96359e5
8128fa913a4db86156da548c7e662d7c1e293f72
F20110218_AABXON cobb_p_Page_052.tif
0c26b96626454929ce5f6e68cb6861c9
4779d98a61f4afb9614021b6979728c1d23d7f71
F20110218_AABXNZ cobb_p_Page_027.tif
3ea7baf4d8aee9b3f260d846a9c95922
aa3c9e066d1492cbb64022c4a3afbc93830d2bb7
F20110218_AABXPC cobb_p_Page_095.tif
1c644a4abb7e1382c8e207ad9b85539f
886ff730882148e28dd65c4dcfc514545e2dc868
F20110218_AABXOO cobb_p_Page_056.tif
e1ba4c3bba7a18600a1ab46f42a6e906
3b868766de4249e15cb2fe34984f5d1b42ad331d
F20110218_AABXPD cobb_p_Page_098.tif
1c719e695718b0c4ddfcb4b5759d5c60
d07b13730cf5aa3e7efaf1234ceca30ef0148be9
F20110218_AABXOP cobb_p_Page_060.tif
7bb74431096dc4944f5e0a765b06a13c
efdc9e0898b4fbfc9232a002d43f93b9135174b3
F20110218_AABXPE cobb_p_Page_099.tif
6df1b71753561a02f0bc812c64b6c666
6193fdb869a69b7cb7ddffe150979db3d8f3162d
F20110218_AABXOQ cobb_p_Page_063.tif
c4d2418d54c8a44ff0a99896d0c6320b
bedc9374364453e17d40747bcb82550375cd0532
F20110218_AABXPF cobb_p_Page_100.tif
12b6ffc27b3c7c55374b1ffe6c6639d6
53034fea56440a309c142309b5dcad5c085d8b1d
F20110218_AABXOR cobb_p_Page_066.tif
b5ac67f7ee40365082d0cdc150c3726a
4b176da0c1605a79c97030392f685d1204f26771
F20110218_AABXPG cobb_p_Page_102.tif
ecd21b57f5f68cf6b0ecb4f0654e7243
cd4d976deba2881301233c4a3af52338c8664e2d
F20110218_AABXOS cobb_p_Page_069.tif
d51445e782c81e2f03876dc0661ed282
10d31acea8c86d92c37bdfcd394dd240f36dcbe7
F20110218_AABXOT cobb_p_Page_072.tif
0f6a0f2c1e697df2afe98eef9048d1f8
679cbd4d7d9a77ebbda0a06cca27c57d9de9657c
F20110218_AABXPH cobb_p_Page_106.tif
fc8a79c8220f0c12d93829c25afec598
f902f675328f9f3998a4e7d91b8bee4350822c05
F20110218_AABXOU cobb_p_Page_075.tif
d6a9e0140628371130ba4ddccd0f1c16
1fe65cc257be69bcd9961d2496042e7465973da5
F20110218_AABXPI cobb_p_Page_107.tif
a2c33db43b037b22f41b639d49c48f2e
3356b4bf2a7160591fa9ca159112d6455349a7e8
F20110218_AABXOV cobb_p_Page_076.tif
826a740612223a1859344de334afdad0
8ddb2274918af7f1d1787e13a2808a54763a6470
F20110218_AABXPJ cobb_p_Page_108.tif
878ca823b8c4c9970833c43217ae53e3
e4bb23758222c640dae99dc56c99ad635542cb8d
F20110218_AABXOW cobb_p_Page_080.tif
a7ef3484538260e12e638916078f0a3a
384bb337c4e9aa1721608516887a4dd949e114a1
F20110218_AABXPK cobb_p_Page_109.tif
0c53b3e1ddf409dc9d56d548a6d60ff5
7d6e53fcfce7d288c87018f758c8e4fdcf002f2f
F20110218_AABXOX cobb_p_Page_085.tif
8a126a71d59bb822fac20ead391c5fb2
26e613d57b4c78e8f009f6bddf784d05423e5888
F20110218_AABXQA cobb_p_Page_143.tif
b89e4fa34b85142dea6a08761ef40699
bf8f2f62b7df2ef55bf24319b438d91be0d92264
F20110218_AABXPL cobb_p_Page_113.tif
3f641158f7ebb323073889be855fbb09
770a6bb1025928ab6e2ec1940c6153ce44a0bffd
F20110218_AABXOY cobb_p_Page_087.tif
731bbbb0d74a345093c7348e203e6f32
784b138673073e77612ad59b0a36e9ed0c01a90e
F20110218_AABXQB cobb_p_Page_144.tif
e6dde19fd5434c75a6df0170c99082aa
25765d5c63a1c90a63d762dee31a1135a3f4d01e
F20110218_AABXPM cobb_p_Page_115.tif
0cd04bc71e53ff835b32348b6d19ca27
7da8f7318744bdcd298202642351393e592b6772
F20110218_AABXOZ cobb_p_Page_090.tif
37cf4797a7a27ba2ab3de2e347da552f
358c62ecc2ba76d5e1a5a5bf4e2bef2a2e6cfd7a
F20110218_AABXQC cobb_p_Page_147.tif
74dad906dcee568926a721eadf354149
57f6e2dd8b91f7330054b4d67f7c8e2a3dad9249
F20110218_AABXPN cobb_p_Page_117.tif
7fb7053dd6fd0487ab21579b647388d2
e10b1b6cbb1f78f04a3bfd2b040086245d5dcdb0
F20110218_AABXQD cobb_p_Page_148.tif
a59ae664083e67e3db1b084a5f82b097
0c47ece62bc4cca65c8837a3d69ef0b0d84d65d2
F20110218_AABXPO cobb_p_Page_120.tif
c36c70a9b7217601d570f27a10c4d485
a8b8ee6a01c3054365ca7bb1a78bc97628d6102d
F20110218_AABXQE cobb_p_Page_154.tif
652b99fb884b80ec98e751e67c016c28
f31951aebeb1472b19d39e065c226ee3705c05e9
F20110218_AABXPP cobb_p_Page_123.tif
6e9fa91cf0b6bd59d55b0a6772b70280
75717da24037a40e452c5f8a6d7f354e592a44a3
F20110218_AABXQF cobb_p_Page_155.tif
8e6626eb8342b89bd4fd0dc5633ed8d6
b2e25f6551774c915aa217dd1082a346c06d28b1
F20110218_AABXPQ cobb_p_Page_127.tif
eaf135650d4ff49912c5a7899deabc3f
cd005073a70c89d742789102d706eaf638e60095


Permanent Link: http://ufdc.ufl.edu/UFE0013662/00001

Material Information

Title: Dynamic Simulations of Suspensions of Rod-Like Polymers and Colloids
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013662:00001

Permanent Link: http://ufdc.ufl.edu/UFE0013662/00001

Material Information

Title: Dynamic Simulations of Suspensions of Rod-Like Polymers and Colloids
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013662:00001


This item has the following downloads:


Full Text











DYNAMIC SIMULATIONS OF SUSPENSIONS OF ROD-LIKE POLYMERS
AND COLLOIDS















By

PHILIP D. COBB


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Philip D. Cobb
















I dedicate this work first and foremost to God, my heavenly Father, who gave

me the strength, perseverance, and ability to do the work and understand the

problems that have presented themselves. I consider myself to be extremely blessed

for the life that I have had, and all of that is only possible through the love of Jesus

C'!i 1-I in my life.

I also dedicate this work to my parents, Steve and Helen Cobb for their love

and support while I have been in college. My life has been extremely blessed to

have two wonderful, loving parents who's love and support of anything I do has

never been in doubt. I thank God for having them in my life.

Lastly, I also dedicate this work to my grandparents, William (Grandpa Bill)

and Peggy (N\ i I) Lassiter, who have blessed me more than can be measured while

I have been in college. From setting me up with the things I needed when I started

my undergraduate education, to picking me up and providing a place for me to stay

during breaks and vacations, to just being there to talk with me and share their

wisdom, they have been a blessing in my life. I would give anything to have Nana

here todiv to share this moment in my life with; I know she would have enjo, d1

this time so much.















ACKNOWLEDGMENTS

I would like to thank everyone who helped me in this endeavor. First and

foremost is my advisor, Jason Butler, who never did give up on me, but instead

pushed me to work harder and produce better work. I thank him for his patience

and his dedication to being a good mentor and advisor.

I want to thank all of the people in my lab group for their help in many v--i-

with my research and life in general in graduate school. I thank Berk Usta and

Jonathan Bricker for their help in reading and editing my papers, listening to my

practice talks, helping me understand different software packages, and just being

there to bounce ideas off of. I also thank Joontaek Park for his insight.

I would like to thank my friends who helped me in many v--iv with different

aspects of my graduate school experience. I want to thank Christine Dubois and

Leah Polkowski specifically for their help in editing my candidacy proposal and

listening to my practice talks.

Last, but not least, I want to thank my roommates who helped keep me sane

throughout my time in graduate school: Ken Brown, Zach Pulkin, Tim Cobb, and

Amaury Garcia. I also want to thank my other friends who were instrumental in

keeping me grounded during this time: Swapna Mony, Lauren Pauly, and Courtney

Maibach.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF TABLES ................... .......... viii

LIST OF FIGURES ..................... ......... ix

ABSTRACT .. .. .. ... .. .. .. .. .. .. .. .. .. .. ... xii

1 BROWNIAN FIBER SYSTETS8: PAST EXPERIMENTS, THEORIES,
AND SIMULATIONS ............. ........ 1

1.1 Introduction ............... .... .......... 1
1.2 Experimental Systems of Rigid Rods ........ ......... 4
1.3 ('C! i'':terization of Suspensions of Rigid Rods ...... ..... 7
1.4 Existing Theories .. .......... ... ..... 10

2 BROWNIAN DYNAMICS SIMULATIONS OF A SINGLE ROD .... 13

2.1 Hydrodynamic Models of Rigid Rods ...... .......... 13
2.1.1 The Slender-Body Model ........ ........... 13
2.1.2 The Rigid-Dumbbell Model .............. 17
2.1.3 A General Approach ......... ............. 22
2.2 Brownian Forces and Torques ................. 26
2.3 Discretized Equations of Motion .............. .. 27
2.3.1 The Corrected Euler Method ..... . . ..... 27
2.3.2 The Midpoint Method. .............. .. 35
2.4 Simulations of Dilute Suspensions of Brownian Fibers ...... ..36
2.4.1 Testing the Brownian Motion ..... . . ..... 37
2.4.2 Error in the Corrected Euler Method . . ... 40
2.4.3 Comparison of Corrected Euler Method to Another Estab-
lished M ethod ............... ...... 43
2.5 Conclusion ............... ......... .. 46

3 DYNAMIC SIMULATIONS OF CONCENTRATED SUSPENSIONS
OF RIGID FIBERS: RELATIONSHIP BETWEEN SHORT-TIME
DIFFUSIVITIES AND THE LONG-TIME ROTATIONAL DIFFUSION 47

3.1 Introduction .................. ........... .. 47
3.2 Simulation Method .................. ....... .. 49
3.2.1 Governing Equations ................ .. .. 50









3.2.2 Evaluation of Excluded Volume Forces and Torques . 55
3.2.3 Numerical Integration of the Governing Equations . 56
3.3 Results . . . . . . .... 60
3.3.1 Rotational Diffusivities ............ . .. .. 60
3.3.2 Translational Diffusivities .................. .. 64
3.4 Discussion ......... ...... ............... 67
3.4.1 Dependence of Rotational Diffusivities on Fiber Model 67
3.4.2 Relation with Existing Theories . . ..... 72
3.4.3 Rotational Diffusivity under Limiting Conditions . 76
3.4.4 Reinterpretation of Comparison with Experiments . 79
3.4.5 Translational Diffusivities .................. .. 80
3.5 Conclusions .................. ........... .. 82

4 DYNAMIC SIMULATIONS OF CONCENTRATED SUSPENSIONS
OF SEMI-RIGID FIBERS: EFFECT OF BENDING ON THE RO-
TATIONAL DIFFUSIVITY .................. .... .. 85

4.1 Introduction ............... ......... .. 85
4.2 Simulation Method ............... ...... .. 86
4.2.1 Governing Equations ..... ........... .... 87
4.2.2 Evaluation of Brownian Forces . . ...... 88
4.2.3 Evaluation of Excluded Volume and Bending Forces . 89
4.2.4 Evaluation of Constraint and Correction Forces ...... 90
4.2.5 Diffusivity Calculations ............. .. .. 92
4.3 Results ...... ... .. .... ......... 94
4.3.1 Simulations of Individual Rods . . 94
4.3.2 Rotational Diffusivities at High Concentrations ...... ..97
4.3.3 Translational Diffusivities at High Concentrations ..... 100
4.4 Discussion .................. ............ .. 101
4.5 Conclusions .................. ........... 107

5 IMPROVED COMPUTATIONAL PERFORMANCE OF BROWNIAN
DYNAMICS SIMULATIONS WITH HYDRODYNAMIC INTERAC-
TIONS THROUGH PARALLEL IMPLEMENTATION USING THE
MESSAGE PASSING INTERFACE .................. 109

5.1 Introduction ................... ............ 109
5.2 Calculating the Multi-body Hydrodynamic Interactions ..... 110
5.3 Implementing a Parallel Algorithm using the Message Passing In-
terface (\ PI) .............. ... .. ....... 114
5.3.1 Parallel Calculation of the Mobility Matrix ......... 115
5.3.2 Decomposition of the Grand Mobility Matrix ........ 115
5.3.3 PLAPACK: A General Approach to the Parallel C'!..,! -l:y
Decomposition. .................. ..... .. 117
5.4 Performance of the Parallel C('! i!. -l:y Decomposition . ... 123
5.5 Discussion . . . . . . ..... 129









5.6 Conclusion ... .. .. .. .. .. ... .. . . ... 132

6 CONCLUSION ............... .......... .. 134

REFERENCES .................. ................ 138

BIOGRAPHICAL SKETCH ............... ......... 147















LIST OF TABLES
Table page

3-1 Values of the parameters X and K used in Equation (3.21) ...... ..55

3-2 Values for the exponent v for the scaling of the rotational diffusivity 65















LIST OF FIGURES
Figure page

1-1 Simulation results of sphero-cylinders of various aspect ratios (A) 2

1-2 The simulation results of Williams and Philipse [1] . . .. 2

1-3 Different concentration regimes of suspensions of rigid fibers. . 3

1-4 Pictures of fabrics made up of carbon fibers (a), Kevlar fibers (b),
and a composite of both (c) ................ ...... 4

1-5 An electron micrograph picture of the tobacco mosaic virus . 5

1-6 Transmission electron microscope images of CdSe nanorods . 6

1-7 Carbon nanotube composite ribbon .............. 7

1-8 Transmission electron microscope image of the boehmite rods . 9

1-9 Image of a polystyrene probe sphere in a suspension of silica coated
boehmite rods taken by a transmission electron microscope ..... .. 10

2-1 Diagram showing the effect of the drift correction upon the orienta-
tion vector pi (t) .................. ......... .. 35

2-2 Plot of the sphere swept out by the orientation vector . ... 37

2-3 Plot of the Brownian displacements of the orientation with one com-
ponent (p (3)) locked with the z-axis ............... 38

2-4 Plot of the autocorrelation function of the center of mass displace-
ments over time .................. .......... .. 40

2-5 Plot of the autocorrelation function of the orientation over time 41

2-6 Long-time rotational diffusivities calculated using either the Euler
method or the algorithm of Lowen [2] .. . 42

2-7 The percentage error in the long-time rotational diffusivities . 45

3-1 Summary of the characteristics and variables describing the slender-
body and rigid-dumbbell models .............. .. .. 50

3-2 The rotational correlation function at nL3 = 150, L2DRo/DTo = 9,
and Dlo/D10 = 2 for rods with A = 50 ............... .61









3-3 The rotational diffusivities versus the number density . ... 62

3-4 Results are shown for the rotational diffusivities versus concentration
for the slender-body model .................. .. 62

3-5 Rotational diffusivities versus number density for the slender-body
and rigid-dumbbell models with A = 50 . ..... 64

3-6 Exponent v for the scaling of the rotational diffusivity . ... 65

3-7 Translational diffusivities as a function of number density ...... ..66

3-8 Average square displacements of the center of mass versus elapsed
tim e r . . . .. .... . .. . ... 68

3-9 Linear extrapolation of the rotational diffusivities . ... 71

3-10 Direct comparison of the simulation results of Doi et al. [3] with the
slender-body model .................. ........ .. 72

3-11 Rotational diffusivities over a range of L2DRo/Dlo with either iso-
tropic or anisotropic center of mass diffusivities for rods with A = 25 74

3-12 Rotational diffusivities over a range of L2DRo/Dlo with either iso-
tropic or anisotropic center of mass diffusivities for rods with A = 50 75

3-13 Comparison of rotational diffusivities for rigid fibers with A = 25
with perpendicular diffE'-i-, il- removed ............... .. 78

3-14 Rotational diffusivities from simulations using the slender-body and
the rigid-dumbbell models with A = 50 compared to the diffusivi-
ties of PBLG .................. ........... .. 79

3-15 Perpendicular diffusivities for fibers with A = 25 in comparison to
the reputation model of Szamel [4] ................... 83

4-1 The two models of the semi-rigid rods used in the simulations . 87

4-2 Probability distribution of pi ,.1) ..'2 ..... . 95

4-3 Values of the ratio of the dilute values of the diffusivities L2DRo/DTo
resulting from the K value chosen .................. .. 96

4-4 Plot of the orientation autocorrelation function -'lni () + )a)t
over tim e r .................. ............ .. 97

4-5 Rotational diffusivities of rigid slender-bodies and three-bead trumbells 98

4-6 Rotational diffusivities of the rigid-rod models in comparison to the
semi-rigid rod models .................. ....... .. 99









4-7 Translational diffusivities for the semi-rigid models with K = 10 100

4-8 Ratio of the Persistence lengths (P) for the slender-body dimer and
three-bead trimer models over the total rod length (L) at K = 10 103

4-9 Rotational diF'ii-i- il i as a function of the stiffness parameter K .. 104

4-10 Rotational diffusivities of the slender-body dimer with K = 10 in
comparison to simulations of a rigid-dumbbell [5] . ... 107

5-1 Parallel C'!i .! -l:y decompositions of a 2000 x 2000 matrix with vary-
ing b_dist and b_alg using 16 processors . . 125

5-2 Parallel ('! iI. -l:y decompositions of a 5000 x 5000 matrix with vary-
ing b_dist and b_alg using 16 processors . . 126

5-3 Parallel C'!i.!. -l:y decompositions of a 10000 x 10000 matrix with
varying b_dist and b_alg using 16 processors . . 126

5-4 Time to perform the parallel C'!i. !. -l:y decomposition as a function
of the matrix size, with runs performed with different numbers of
processors . . . . . . .. 127

5-5 Time required to decompose different size matrices as a function of
the number of processors used .................. .. 128

5-6 Time to perform the parallel C'!i. !. -l:y decomposition as a function of
the number of fibers .................. ....... 130















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

DYNAMIC SIMULATIONS OF SUSPENSIONS OF ROD-LIKE POLYMERS
AND COLLOIDS

By

Philip D. Cobb

May 2006

C'!I iir: Jason E. Butler
M, i r Department: C'! im i d Engineering

Simulations presented in this dissertation advance knowledge of the dynamics

of suspensions of rigid and semi-rigid Brownian fibers. This work resolves com-

peting claims concerning the power-law scaling for the concentration dependence

of the rotational diffusivities. The power-law scaling states that the rotational

diffusivity DR scales as DR/DRo ~ (nL3)", where DRo is the rotational diffusivity

of a single fiber in infinite dilution, n is the number density (number of fibers per

unit volume), and L is the fiber length. The choice of hydrodynamic model, with

an intrinsic ratio of the rotational to translational diffusivities at infinite dilution

L2DRo/DTo, sets the value of the exponent v in the scaling. The aspect ratio of

the fibers also affects the scalings, with strong variations for ratios less than fifty;

ratios of fifty or higher can considered infinitely thin. An ,ia 1,i--i; of the numerical

integration method was performed, resulting in a new algorithm with less error and

higher efficiency.

Adding flexibility d.-41iv the number density at which fibers become signifi-

cantly hindered by their neighbors and enter the regime where a strong decrease

in the rotational diffE -iv. ili, occurs. Once within this semi-dilute regime, the









power-law scalings of the semi-rigid fibers closely match those of rigid fibers with

corresponding hydrodynamic models and aspect ratios. Comparing simulations

of rigid and semi-rigid fibers to experimental results demonstrates that different

micro-mechanical models can produce results which are indistinguishable within

measurement capabilities. Consequently, proper models cannot be distinguished

based solely on their rotational diffusivities, so other measures are needed to

identify the appropriate model.

Including hydrodynamic interactions into the simulations will provide further

insights into the dynamics of fiber suspensions. Investigations of a parallel compu-

tation of the pair interactions and C(!i. .1 ly decomposition indicate that simulating

systems of over one hundred fibers is feasible.















CHAPTER 1
BROWNIAN FIBER SYSTEMS: PAST EXPERIMENTS, THEORIES, AND
SIMULATIONS

1.1 Introduction

Many inorganic and polymeric colloidal dispersions consist of rigid and non-

spherical particles, such as rod-like macromolecules. These polymers interact

through inter-particle colloidal and hydrodynamic forces in a continuum fluid.

The macromolecules also experience Brownian forces which arise from the thermal

fluctuations of the fluid. In colloidal dispersions, in general, one is faced with a

large spread of well separated time scales [6, 7]. These timescales range from the

smallest timescale associated with the random motion of the fluid molecules to

the much larger time scales associated with diffusion of particles in the fluid. As

a result, the dynamic behavior of a colloidal dispersion is usually simplified into a

model of large macromolecules or "Brc.v iin particles suspended in a continuum

fluid.

The understanding of the dynamic behavior of concentrated polymer solutions

and melts remains a i, i" challenge of modern polymer physics. The complexities

are due to unique features of polymer molecules. Polymer molecules possess a

large number of degrees of freedom associated with their intermolecular rotational

and vibrational motion in addition to the translational degrees of freedom of their

center of mass. The interplay between macroscopic and intramolecular motion, as

well as the preservation of the chain connectivity, results in the entanglement effect.

This effect is believed to dominate the dynamic behavior of these systems [6].

Brownian fibers, whether rigid or flexible, produce phases in suspension

which are not seen in suspensions of spheres. Rigid, rod-like polymers differ from






























Figure 1-1. Simulation results of sphero-cylinders of various aspect ratios (A) pro-
duced by the simulations of Williams and Philipse [1]. The pictures are
for A = 0 (upper-left), A = 0.4 (upper-right), A = 2 (lower-left), and
A = 40 (lower-right).


0 20 40 60 80 100 120 140 160
A


Figure 1-2. The simulation results of Williams and Philipse [1] are plotted with
the volume fraction (0) as a function of the particle aspect ratio (A).
The solid line is a theoretical limit of the random packing as given by
equation (1) in the work of Williams and Philipse [1].









flexible polymers in many respects, yet they are the simplest polymers where

entanglements occur [6]. Williams and Philipse [1] performed simulations of

particles with spherical end caps and varying lengths. They then calculated the

maximum obtainable volume fraction at which the orientation distribution remains

random. Figure 1-1 shows graphically that as the aspect ratio A (fiber length over

its radius) increases from A = 0 (sphere) to A = 40, more empty space is seen

around the particles. A graph of the maximum random packing volume fraction

as a function of the aspect ratio is shown in Figure 1-2 [8]. These results show

that as the particle aspect ratio increases, the volume fraction of the maximum

random packing decreases. Particles of high aspect ratio will hinder the motion of

the surrounding particles at much lower concentrations than spheres, resulting in

an entanglement effect, even though they are perfectly rigid.

The concentration regimes in which Williams and Philipse [1] performed

simulations is known as the semi-dilute and concentrated isotropic regimes.

Suspensions with higher concentrations of fibers can be made; however they

enter into a new regime where there the orientation distribution of the fibers

is no longer random. As the concentration is increased from the random, or

concentrated, isotropic regime, the fibers form a nematic phase where there is

ordering in the orientations of the fibers, but not in the center of mass locations.









a) b) c) d)

Figure 1-3. Different concentration regimes of suspensions of rigid fibers. Figure
a) is the dilute regime, figure b) is the concentrated isotropic regime,
Figure c) is the nematic concentration regime, and Figure d) is the
smectic concentration regime.









If the concentration is increased from the nematic regime, the fibers will enter into

the smectic regime, where there is ordering of both the orientation and center of

mass of the fibers. These different concentration regimes are shown schematically in

Figure 1 3.

1.2 Experimental Systems of Rigid Rods

Apart from their importance as the simplest systems of entangled polymer

molecules [6], nondilute solutions of rigid, rod-like polymers have considerable

interest on their own. Rigid polymers and Brownian fibers are found in existing

technologies and exciting new areas of research. A1 i ., rigid-rod polymers exhibit

high tensile modulus and strength. These materials also have a low density,

which make them ideal for use in creating lightweight, high-strength fibers and

films. Consequently, these polymers, as well as composites made with these

polymers, are widely used as high-performance materials in consumer, aerospace,

and electronics applications [10]. One of the most widely recognized examples is

poly(1,4-phenyleneeterephtalamide), sold under the trade name Kevlar. Impact and

compression tests of Kevlar and carbon fiber fabrics [9] investigating the tensile

strength and stiffness of these composite materials show that the combination of

both high strength fibers in one composite produces an even stronger material.












Figure 1-4. Pictures of fabrics made up of carbon fibers (a), Kevlar fibers (b), and
a composite of both (c) after they have been subjected to impact and
compression testing in the work of Gustin et al. [9].







5























Figure 1-5. An electron micrograph picture of the tobacco mosaic virus illustrat-
ing its rod-like characteristics [11].


Figure 1-4 shows the results of the tests with fabrics made entirely of carbon fibers

(a), Kevlar fibers (b), and a composite of both (c).

Within the realm of biology, many macromolecules resemble rigid fibers.

Short lengths of biopolymers, such as DNA [12, 13, 14], actin [15], and collagen

[16, 17], can be modeled as rigid fibers. Xantham gum, a helical polysaccharide

which roughly forms a cylinder, is used to enhance viscosity of many food products.

Other types of polysaccharides, peptides, and polynucleotides also form rigid

linear structures. Even some simple micro-organisms have an elongated, ellipsoidal

configuration, and for sufficiently low forces maintain a rigid structure. These

include fd bacteriophages and tobacco mosaic viruses [18, 19, 20]. An electron

micrograph of the tobacco mosaic virus showing its rod-like shape is seen in Figure

1-5 [11].


























Figure 1-6. Transmission electron microscope images of CdSe nanorods produced
in experiments by Huynh et al. [21] in order to improve the efficiency of
hybrid solar cells. The rods on the left have an aspect ration of A ~ 5
and the ones on the left are A ~ 10.


In the realm of manmade, or non-biological, Brownian fibers, there exist

nanotubes and nanorods. The science of these materials is the subject of an

immense amount of research, resulting in exciting technological applications.

Processing of the fibers often occurs in solution, where the effects of di'ff -ivi iF,

and flow upon the microstructure become relevant. In one experiment performed

by Huynh et al. [21], semiconductor CdSe nanorods have been cast into a film to

make solar cells. By controlling the diameter and length of the nanorods, as well as

the microstructure, Huynh et al. [21] were able to improve the performance of the

solar cells. A picture of these nanorods taken by a transmission electron microscope

(TEM) is seen in Figure 1-6.

In a second example, Vigolo et al. [22] disbursed individual carbon nanotubes

in a solution and then aligned them using a flow field to create flexible macroscopic

fibers or ribbons. These ribbons are much longer than the individual nanotubes

(on the orders of centimeters or longer), and have very high tensile strength. A

picture of one of these ribbons which was taken by optical microscopy is shown






























Figure 1-7. Carbon nanotube composite ribbon produced through the experiments
of Vigolo et al. [22].


in Figure 1-7. By understanding the dynamic properties of the carbon nanotubes

in solution, the yield of these ribbon composites could be improved, as well as

their material properties. Many of the experiments noted used characterization

techniques to study the motion of the fibers in suspension, which will be discussed

in the following section.

1.3 Characterization of Suspensions of Rigid Rods

Materials scientists and engineers employ a number of experimental techniques

to characterize the properties of rigid polymers and Brownian fibers suspended

in solution. A common measurement used to capture the motion of Brownian

fibers in suspension is dynamic light scattering (DLS) [17, 20, 23, 24, 25, 26]. The

configuration of different light scattering set-ups may vary, but the general concept

is the same. A light source is focused on a suspension of rods where it is scattered

as it hits the rods. The scattered light is measured by a device that analyzes the

amount of scattered light. Dynamic light scattering has a few drawbacks. One of









these drawbacks is that the dynamics of individual rods and fibers are very difficult

to calculate at anything other than dilute concentrations. At high concentrations

there is a need for well defined tracer particles, which are difficult to obtain in

Brownian systems.

A way to avoid the difficulties of requiring tracer particles when using DLS at

high concentrations is by performing a birefringence measurement. With birefrin-

gence studies, the suspension is aligned, either by a fluid flow (flow birefringence),

or by a voltage difference across the fluid (electric birefringence). The flow or

electric field is then turned off and the suspension is allowed to relax back from an

aligned to an isotropic state. The rotational diffusivities are then calculated based

upon the rate of change of the intensity of a beam of light passing through the

suspension. This form of measurement can produce calculations for the long-time

particle rotational diffusivities, which would have been extremely difficult to cal-

culate with DLS alone. Phalankornkul et al. [25] used this combination of electric

birefringence and DLS to measure the rotational diffusivities of poly(7-benzyl-a-

L-glutamate) (PBLG) over a wide range of concentrations. Other birefringence

devices work similarly, except that they measure a change in the amount of electric

current flowing through the sample as the sample relaxes to measure the rotational

diffusivities.

Another way of measuring the diffusive properties of Brownian fibers includes

the use of fluorescence recovery after photobleaching. In this technique, the parti-

cles are coated with or dyed by a specific chemical that will cause them to fluoresce

under given conditions. The fluorescent particles can then be visualized and the

diffusive properties measured and calculated. As an example, van Br-i.--. -ni et al.

[27, 28] used a suspension of boehmite rods which were coated with fluorescent

colloidal silica. The particles were then visualized using a transmission electron

microscope; an image of the particles is shown in Figure 1-8.






























Figure 1-8. Transmission electron microscope image of the boehmite rods which
where fluoresced in the work of van Br--i.- -ni et al. [27, 28].


Another form of measuring properties of Brownian fiber suspensions is

through micro- or nano-rheology, depending on the length scales of the fibers in the

suspension. The idea behind these concepts is that properties such as viscosity can

be measured locally in a suspension, and from that, other suspension properties can

be determined. One example is given by Kluijtmans ct al. [29] where fluorescent

tracer spheres are observed sedimenting through a suspension of silica-coated

boehmite rods. By calculating the diffusive properties of the spheres at different

points in the suspension, the effective viscosity of the suspension at those locations

can be calculated. Similar experiments were performed by Tracy and Pecora [30]

and Kang et al. [31], except that dynamic light scattering was used to track the

tracer spheres instead of fluorescence recovery after photobleaching.

Helden et al. [32] performed experiments where total internal reflection

microscopy was used in combination with a probe sphere to calculate the entropic

forces induced by rigid rods in a suspension. In this experiment, a polystyrene































Figure 1-9. Image of a pc.l- I i-, probe sphere in a suspension of silica coated
boehmite rods taken by a transmission electron microscope in the
experiment performed by Helden et al. [32].


sphere in a suspension of silica coated boehmite rods is pressed close to a silica

coated glass wall. The entropic force induced by the rods was then measured as a

function of the concentration. A picture of the probe and rods is shown in Figure

1-9.

1.4 Existing Theories

To interpret the results of measurements of diffusivities and rheology, re-

searchers rely on theories of rigid fibers which usually provide qualitative scaling

laws, rather than quantitative predictions. These theories accurately capture

some observations, but the scaling laws are often limited. The limitations can

include restrictions on parameters such as the concentration regime and Peclet

number. Though a prediction exists for one range of parameters, there may not be

a complementary theory for other ranges.









Theories exist which attempt to predict macroscopic properties, such as

viscosity, of a suspension of rigid fibers. These theories were necessary because

it has been observed that the behavior of rigid rod suspensions is qualitatively

different from the behavior of suspensions of spherical particles, which have been

studied extensively [1, 33]. Doi and Edwards [6] predicted a relationship between

the low shear viscosity and the rotational diffusivities of a suspension of rigid rods

in the semi-dilute concentration regime of

UnkT nkBT
r = r10+o + (1.1)
30DRo 1ODR'

where rl is the suspension viscosity, rlo is the solvent viscosity, n is the number

density (number of rods per unit volume), kBT is the thermal energy, DR is the

long-time rotational diffusivity, and DR0 is the rotational diffusivity of a single rod

in infinite dilution. The semi-dilute regime is defined as (1/L3 < n < 1/dL2), where

L is the rod length, and d is the diameter of the rod [6, 7].

Several theories have been proposed which attempt to predict the long-

time rotational diffusivity (DR) of a suspension of rods as a function of the

dimensionless number density (nL3). Doi [7] developed a "tube" theory for

infinitely thin, rigid rods which predicts that the rotational diffusivities scale as

DR/DRo ~ (nL3)-2 for number densities within the semi-dilute regime. This theory

states that as the suspension of rods becomes more concentrated, it will enter into

a regime where the rods surrounding a particular rod will form a "cage," or "tube,"

which will hinder that rod from rotating or diffusing in the direction perpendicular

to its central axis. This probe rod is, however, free to diffuse along the direction

parallel to its central axis, because it is infinitely thin; thus there is nothing to

hinder its diffusion in that direction. The probe rod will then diffuse half its length

out of the "tube," at which point it will be able to rotate and enter into another

"tube." Simulations performed by Doi et al. [3] of infinitely thin fibers, in which









the hydrodynamic interactions between the rods were ignored and crossing of

the centerlines of the rods was prevented through a reflection rule, confirmed the

scaling of (nL3)-2

Simulations performed by Fixman [34, 35] found a different scaling for the

long-time rotational diffusivities of DR ~ (nL3)-1 for infinitely thin rods in the

semi-dilute concentration regime. Similar to the simulations of Doi et al. [3], a

dynamic technique was implemented, but short-range potentials were used to

prevent the rods from overlapping. Fixman [34, 35] proposed an alternative theory

founded upon the concept of "cooperative rotation" as the mechanism for the

rotational diffusion of the fibers in order to account for the simulation results.

This theory states that since the rods that make up the "cage," or "tube," in the

theory of Doi [7] are also diffusing and rol ,I hli the "tube" is not a static entity.

The "tube" will in fact diffuse, rotate, and break up on a time scale which is faster

than the center of mass diffusion of the probe rod. The "tube" motion, as well

as its breaking up and reforming, results in higher rotational diffusivities in the

semi-dilute concentration regime.

Ever since the publication of these two competing theories, there has been an

ongoing controversy in the scientific literature over which theory is in fact valid

[36, 37, 38, 39]. This controversy is resolved by the work reported in C'! lpter 3.

The difference in the scaling of the rotational diffusivities comes from the manner

in which the fibers are modeled in the theories and simulations. Questions have

also been raised about the limitations of these theories and their usefulness in

interpreting experimental results from real polymer rod systems. Extensions have

been made to the "tube" theory of Doi [7] in order to account for flexibility [23],

polydispersity [40], and hydrodynamic interactions [41]. The effect of adding a

slight degree of flexibility on these theories will be discussed in chapter 4, and the

inclusion of hydrodynamic interactions is discussed in chapter 5.














CHAPTER 2
BROWNIAN DYNAMICS SIMULATIONS OF A SINGLE ROD

2.1 Hydrodynamic Models of Rigid Rods

In the past it has been assumed that the hydrodynamic model used to describe

polymer fibers in solution has no effect on the dynamics of the fiber system.

Different models for the rods come from different descriptions of the distribution

of the hydrodynamic resistance, or friction, along the rod. Batchelor [42] and

Cox [43, 44] formulated a description of high aspect ratio (length over diameter)

fibers using slender-body theory. Other hydrodynamic models include bead-rod

models such as dumbbells (two beads) and trumbells (three beads) to describe the

fibers. Researchers [2, 4, 8, 38, 39, 45, 46, 47] have assumed that the differences

between the models would disappear once the relevant dimensions have been scaled

away. C'! lpter 3 shows that this assumption is invalid, as the ratio of the short

time rotational diffusivities to the average short time translational diffusivities

L2DRo/DTo is different for each model and has a large impact on the dynamics,

where L is the rod length, DRo is the rotational diffusivity at infinite dilution, and

DTo is the average center of mass diffusivity at infinite dilution.

2.1.1 The Slender-Body Model

Batchelor [42] and Cox [43, 44] used slender-body theory to describe a rigid

fiber as a line distribution of Stokeslets. A Stokeslet represents the effect of a

force applied to a point in a fluid in Stokes flow (zero Reynolds number flow). The

velocity at a point xi in an infinite fluid caused by a force fj is given by

i T- 12 + (i;6i3/2 fi (2.1)
87/1 ({x 1/2 (xx 2)








where p is the fluid viscosity. If the velocity is then integrated over the length of
the fiber (L) with the origin at the center of the fiber, the velocity becomes

1 L/2 /
us (s) 8= [ ( f(S) 1ds'
87/ -L/2 T((S- S )2 1+ 2)1/2
+ L/2 ((S ') + drp) ( s')p + rp) f ) ds' (2.2)
-L/2 ((s- )2 + /2

where s is the length along the i1 ii' Pr fiber axis, r is the fiber radius, pi is a unit
vector aligned with the i1 ii' 'r axis, and pi is a unit vector perpendicular to pi. As
the aspect ratio (fiber length divided by diameter) increases above 10, the integrals
.-i-','i.i. ically approach a solution of

L/2 J 12 ds' 2 In (2L/d) 6f (2.3)
-L/2 ((s- '2 + r2) 12

and

/ L/2 pp, rpf) f (
2 -- sp ') S ) 3j2 ds' 2 In (2L/d) pipjfj, (2.4)
-L/2 (( 2 + r2)

where d is the fiber diameter. Substituting the solutions from Equations (2.3) and
(2.4) into Equation (2.2) gives

In (2L/d) /
xi + spi ui (s) = ( + paj) fj (s), (2.5)

where the velocity of the fiber has been split into a center of mass velocity 'i and
a rotational velocity pi, and u' (s) is the disturbance velocity caused by flow in the
fluid as well as any other fibers.
To solve for the center of mass and rotational velocities, additional steps must
be performed. Equation (2.5) is integrated over the fiber length to give

SL/2 ln (2L/d) ( Pi+ Y
(2 i +4 spi ui (s)) ds = + pipy
J-L/2 47/ V/









-L/2 L/2


X-(L/2 + -/(0

-j (L) + pj (0) -


L/2 / In (2L/d) ( )
sds fLu (s) ds = -- 6ij +ppy J-j





J-L/2 47/1


1 JL/2 u, (n (2L/d) ( i
xT = aut (s) ds + nt fb + pri si.
L -L/2 47rpL V
The total force acting on the fiber is given by


(2.6)




(2.7)


/L/2
Fi L/fi (s) ds.
J-L/2


A similar calculation is done to solve for the rotational velocities, where a cross

product is taken of Equation (2.5) with spi and then the result is integrated over

the fiber length as shown

In (2L/d) L/2
nL ijkPjk ijkSj k+ u, () ds
47/r f-L/2


In (2L/d)
4 ijkjT


-L/2
CijkPj k \
J-L/2


In (2L/d)
4171


sds + CijkPjPk / 2ds


jL3
CijkPj k (0) + CijkfjPk 1


SL/2
-L/2


SL/2
L/2


12 SL/2
QijkPjPk 1 2ijkSPjUk
L J-L/2

The torque acting on the fiber is give by


/ L/2
-L/2


s) ds +


31n(2L/d)
37pL3 ijkPjTk
;rl-.


eijkSpjfk (s) ds


SCijkpjUk (s) ds


SCijkPjUk (s) ds


(2.8)


(2.9)









The cross product of Equation (2.8) is then performed again with pi and the result
is
12 L/2 31n (2L/d)
i (ij Pipj) () ds + pL3 (ij pipj) Fj, (2.10)
Ll J-L/2 7/1
The vector identity of EijkCklimjPlAm = (6ij pipj) Aj for any arbitrary vector Ai
[48] has been used along with defining a weighted force where

Ti= ePijk k. (2.11)

For simulations performed without multi-body hydrodynamic interactions, the
translational and rotational velocities from Equations (2.6) and (2.10) become

In (2L/d) (2.2)
x 4 =(2.12)

and
3 In (2L/d) (2.13)
7ipL3
The center of mass and rotational mobilities matrices can then be obtained
direction from Equations (2.12) and (2.13) to give

ln (2L/d) ( + pi) (2.14)
S47pL

4 R -3-3In 6i pip (2.15)

where MA4 is the center of mass mobility matrix, and MR is the rotational
mobility matrix. The center of mass mobility matrix can be separated into mobility
matrices parallel and perpendicular to the central axis of the rod. These mobilities
are
M4H In (2L/d) / (
Mr = (pip) (216)

M274 L PL
47xpL









where Mi1 is the mobility parallel to the central rod axis, and Mt is the mobility

perpendicular to the central rod axis. It can be seen that the magnitude of the

parallel mobility is equal to twice that of the perpendicular mobility.

The diffusivities of the slender-body model can be calculated from the magni-

tude of the mobilities,

D0 kBTM kBT n (2Ld) (2.18)
D1lo = kfiTM (2.18)
27pL

D kBT3M kBTln (2L/d) (2.
DIo = klBTM1 = (2.19)
47pL
DRo kBTMR 3kBT In (2L/d)
Do = kL3 (2.20)

where kBT is the Boltzmann temperature, and Dil0, DIo, and DR0 are the parallel,

perpendicular, and rotational diffusivities of the rods at infinite dilution. The ratio

of DIIo/DIo = 2 is the maximum theoretical value for a slender-body model, which

assumes that the slender-body is infinitely thin. The slender-body model has an

inherent ratio of the rotational and average center of mass diffusivities at infinite

dilution which cannot be neglected through scaling arguments. This ratio is

2 L2 (3kBT n(2L/d)
L L3 ) 9, (2.21)
DT 1 (kBTln(2L/d) + 2( kTn(2L/d)))
3\ 27FxL 4xpL ))

where DTo is the average center of mass diffTi-i',lii, at infinite dilution given by

DTo = (DI|o + 2D0o). This ratio of L2DRo/Dro p'1 i', a significant role in the

dynamics of concentrated suspensions of rods, which is discussed in C'!i pters 3 and

4.

2.1.2 The Rigid-Dumbbell Model

A rigid dumbbell can be modeled as two beads with centers x1) and x 2) which

are separated by a fixed distance L. The equations for the center of mass velocities

for the two beads under the assumption that inertia can be neglected since this is









in Stokes for are then give by


(1) 1 F(1) T and x(2) 1 (2) T 9g
6(pa x1) [ 67pa [ (2)


(2.22)


where a is the diameter of the bead, Fi() and Fi2) is the force on each bead, and T

is the tension along the line between the centers of the beads. In order to enforce

the fixed length of L between the beads on the dumbbell, a constraint g is defined

such that


gO (2) 1))2


(2.23)


The derivatives of the constraints with respect to the center of mass positions of

each bead given in Equations (2.22) are

0g 9 z (2 ) ()Og ( _
g= -2 (x) 1) and =2 2 (x2) ). (2.24)


Another requirement for the rigid dumbbell is that the relative velocities along the

constraining line between the beads must be zero,


(O= (2 1) (2) (1))


(2.25)


Substituting the center of mass velocities from Equation (2.22) into Equation (2.25)

gives

0 F(2) F)(X) ) + (T ag T X) (2.26)
67rta 67a ax 2
\ i )
The derivatives of the constraints from Equations (2.24) are substituted into

Equation (2.26) to solve for the tension T,


T X (2) 2 (1)
\9xi i'9


(F(2) Fi1)) (X(2) (1)
i i)









i(2) F(1) ( (2) (1) ( 2) (1) (2) (1)




F(2) F( (X(2) X() 2 4T2


S4 (2 l) F(1) (1). (2. 7)
T- 4L2 i-F-^ (2.27)

The center of mass velocity of the rigid dumbbell, xi, is defined as the average of

the center of mass velocities of each bead,


S() + 2)). (2.28)

Substituting the bead velocities from Equations (2.22), as well as the constraint

derivatives from Equations (2.24), and the value for the tension from Equation

(2.27) into the equation for the rigid dumbbell center of mass gives

r 1 +1 (1 ) T ( ag
S 2 67 ra i 67rla (2) + a x()




Xi F i(1) + Fi(2))
127rx/a




x 127i i (2.29)

where Tj is the total force acting on the dumbbell. The 6ij term in Equation (2.29)

is the identity matrix, which shows that the rigid dumbbell has an isotropic center

of mass mobility (DIIo/Do = 1) at infinite dilution.








The rotational velocity of the rigid dumbbell is defined as

P 1 ( -2) 1)) (2.30)

By substituting in the center of mass velocities of each bead from Equations
(2.22) into Equation (2.30) as well as the values for the constraint derivatives in
Equations (2.24), and the tension from Equation (2.25), the rotational velocity of
the rigid dumbbell can be calculated,

1 1 ( (2) (1) 9
4 L 6 Fa ) 67wtpa (2) a (1)



A4 T \(Fi(2) -Fi(1) 4 (F(2) F1) (2) _x\ (1) (2) _xl))]




S( ( 2) t ) (X(2) (1) (2) (2) ( 1) )





67paL L L ) (t



S- \ Pip F) ( (2) F1), (2.31)

where the orientation pi is defined as the vector between the two beads divided
by the distance between the beads, thus making it a unit vector. It is seen that
the forces acting on each bead are still being used in Equation (2.31). These
forces can be converted to an overall torque (or weighted force) acting on the rigid
dumbbell by multiplying the rotational velocity by the distance from the center of
the dumbbell at which the forces are acting (L/2). In order not to change the value









of the rotational velocities, the equation is also divided by L/2 which gives







tp 3 paL2 Pi (2) P> 1)


11


P 3i -r l-aL- 2 -, (2.32)
37/1qaL2 (

where Fj is the total weighted force acting on the rigid dumbbell.

The center of mass and rotational mobilities matrices for the rigid dumbbell

are obtained from Equations (2.29) and (2.32) where


MT (6(*) (2.33)


3 paL2 (6 p (2.34)
"" 3paL2
The average center of mass mobility matrix is isotropic, meaning that the parallel

and perpendicular mobility matrices are equal to MA The rotational and center of

mass diffusivities are then given by

Do = D = DT =T kBTMT = BT (2.35)
127rpa

kBT
DRo = kBTM=R kBT. (2.36)
3/tpaL2
The ratio of the short time rotational and translational diffusivities for the rigid

dumbbell model is
L2DR L2 ( k13T
D k = 4. (2.37)
D12 kBT
S12Fpa









2.1.3 A General Approach

Other hydrodynamic models exist which also could be used to describe the

fibers in the simulations. Brenner [49] described three such models, which include

a prolate spheroid, a symmetrical double cone, and a circular cylinder. Other

hydrodynamic models also exist, and each model can be precisely calculated.

The ratio of DIIo/DIo for these models range between 1 and 2 and encompass the

slender-body and dumbbell models. Such a highly precise model though is not

required for the high aspect ratios that will be studied in these simulations.

A more general approach can be taken in deriving the hydrodynamic models

used in the simulations of Brownian fibers, which incorporates both the slender-

body and rigid-dumbbell models. This approach begins with the Langevin equa-

tions of motion for each rod with inertia included,

at
ma-O i (2.38)

M a _RQ + T, (2.39)

where m is the mass of the rod, M is the moment of inertia, t is the time, and

xi and fi are the center of mass and angular velocity vectors of the rod. The

hydrodynamic forces acting on the rod are -iLxj and -RfSi, where _-T is the

center of mass resistance matrix, and R is the rotational resistance. Rotation

about the 1i ii' ',r axis of the high aspect ratio rods is ignored, consequently the

rotational resistance R in Equation 2.39 can be written as a scalar.

The Brownian forces, i, and torques, 7, on the rod are given by the fluctua-

tion dissipation theorem,


( (t)) 0 and F(t)F(t')) = 2kBTE(t t') (2.40)

and

( (t)= 0 and z(t);(t')) 2kBTR6sj(t t'), (2.41)








where 6(t t') is the Dirac delta function. For the purpose of making discrete
time steps of length At, the Dirac delta function is approximated as 1/At for times
within the same time step and zero otherwise. Equation (2.39) can be rewritten in
terms of the orientation vector (direction cosine) pi, which is strictly a unit vector,
and the weighted Brownian forces i,

M(bJ, i- Pj)9 = (6 PpJp )j + 6 pPj)Fj, (2.42)

where the rotational velocity pi is related to the angular velocity according to

Qi = CijkPjPk and t = ijkPj (2.43)

The torque is defined in terms of a weighted force,

T= ijkpjFk ijkPj sfk (s) ds, (2.44)
-L/2
The condition that the fluctuation-dissipation theorem must be satisfied is used to
determine the weighted forces. The weighted forces are of zero mean,


(K (t)) -0. (2.45)

The variances of the forces are not correlated and have the form


(J (t) Fy (t')) = Gi6 (t t') (2.46)

To determine the variances, Equation (2.42) is integrated over short times t to give
a solution for the rotational velocities,

MP () = 6ij, Pipj ) (7-) e----d. (2.47)








The autocorrelation function for the rotational velocities is then given by [50]


(P (t) p () t' tJ 6i ( iPkk Gk (r ) i ( T-TRdr. (2.48) T


,t >t 1


Under the assumption of an equipartition of energy as t oo, the variance of the
rotational velocities is [50]

K()tPt' kB() 6i PipJ (2.49)

Equating Equations (2.48) and (2.49) demonstrates that

Gy 2kIBTR (,J pp) (2.50)

and the variance in the weighted forces is


( (t) xt'l)= 2kBT R(6, ppS) 6(t t'). (2.51)

The variance in the weighted forces in Equation (2.51) is proportional to 2kBT
times the resistance matrix, which appears in Equation (2.42) as R (t j ipj).
The rotational mobility matrix is then the inverse of the resistance matrix,

MAf j (V R6 -(t PP 6) -j pp (2.52)

Note that inverting the resistance seems to require inversion of the singular matrix
(6ij -pipj). The appropriate view is that an inverse is not actually performed, but
rather a projection is performed which extracts only those components on both
sides of the equation which are perpendicular to pi.








For example, in the absence of inertia (ap/ilt = 0), the rotational velocity is
given by the product of the mobility and weighted forces,

pi = ( Pipj) j, (2.53)

where the projection of 1Tj by (6ij pipy) ensures that the rotational velocity pi is
perpendicular to pi. Instead of solving the mobility problem where the weighted
forces are known and the rotational velocity is desired, one might be interested
in solving the resistance problem where the rotational velocity is known and the
weighted forces are unknown. In this case, the equation becomes

F = (6,j -PiP )Pi, (2.54)

where the projection of ij by the matrix (6ij pipj) ensures that the weighted
forces Fi have no components parallel to the orientation vector. The above mobility
and resistance problems indicate that a pseudo-inverse of the (6ij pipj) matrix
can be defined by

(ij Pipj) (6ij PiPj, (2.55)

though care must be taken to not violate the exact relationship between Equations
(2.53) and (2.54) of

(ij pipJ)FJ = R (ij PiPj) P. (2.56)
The expected short-time diffusivity matrices are not altered by changing the
description of the rods from angular velocities and torques to rotational velocities
and weighted forces; they are given by the thermal energy times the mobility, or
inverse resistance matrices,

Dr kBT ) kTMA4 (2.57)

DR kBT () k1 TM (2.58)
ii B1 kBTM









Using this information, the short-time diffusivity can be written as

D^ = kBTQ( p (2.59)

where as expected the rotational diffusivity at an instant in time is perpendicular
to the orientation vector pi. Furthermore, the matrix (6ij pipj) can be factored to
give an expression

A, = (6ij- pip), (2.60)

where AikAik (6ij -pipj). This is a necessary step in forming the discrete
Brownian displacements Api of the orientations as seen in the following section.
2.2 Brownian Forces and Torques

Solving for the Brownian forces (and torques) from Equations (2.40) and (2.51)
gives
2F) ( ) AJ W and T(br) ( 2i ByJf, (2.61)

where WT and WV are vectors of random numbers with lengths of 3 generated by
the ran2 subroutine [51]. The random numbers have the properties of zero mean
and unit variance,


(Wf) 0 (W ) = 0, (W Wff) -= ij and (WWf) = 6. (2.62)

A uniform distribution suffices for the simulations, since a first order integration
technique will be used [52]. The matrices Aij and Bij are constructed in such a way
that

AkAik j kBT and B'ikjk kBT. (2.63)

The evaluation of Aij and Bij is traditionally done by Cholesky decomposition [51],
although other approaches for determining Aij and Bij are available [53, 54].









2.3 Discretized Equations of Motion

Once the rotational and center of mass velocity equations are known for the

various hydrodynamic models, the velocities must be discretized in time to simulate

the motion of the fibers. To perform a dynamic simulation, the differential equation

must be integrated. Since the governing differential equations are stochastic,

the usual methods of numerical integration for deterministic equations must be

reconsidered [55].

2.3.1 The Corrected Euler Method

Options for the numerical integration include a modified Euler method, where

the divergence of the short time diffusion tensor must be evaluated, multiplied

by the time step, and added to the Brownian displacement. This modification is

needed to correct for the spatial variations of the mobility during the time step

which causes a mean error at second order which is non-zero [56]. This method

was used in the calculations of Brownian suspensions of spheres by Phung et al.

[57] and Foss and Brady [58]. Unfortunately, numerical evaluation of the spatial
derivatives of the mobility can be a costly exercise.

Rather than integrating Equations (2.38) and (2.42) to determine the time

dependent center of mass and rotational velocities, and then displacements, Ermak

and McCammon [56] demonstrated that the Langevin equations with stochastic

forcing can be integrated directly to give the displacements, but that a correction

is generally needed. This analysis is similarly repeated here for the Langevin

equations shown in Equations (2.38) and (2.42). Eliminating the inertial terms and

solving for the rotational velocity gives


(t) = M (t) j (t) (2.64)









p = bi pi (t) pj (t) ) "(t). (2.65)


As written, this equation has a dimension of three, despite the fact that only
two components of pi (t) and pi (t) are independent. However, at each instant
of time, the multiplying matrix (6ij -pi (t)pj (t)) appearing in Equation (2.65)
extracts only the components of Fj (t) which are perpendicular to pi (t). Therefore,

pi (t) is guaranteed to be perpendicular to the orientation and only two principal
components of pi (t) are altered.
Integrating Equation (2.65) directly using a straightforward Euler scheme,
however, fails to produce the correct rotational diffusivity due to a drift in the
average rotational velocity ((pi (t)) / 0) at first order in the time step At, unless
extremely small time steps are used. This nonzero drift in the velocity has origins
in the elimination of the inertial terms from the Langevin equation. Ermak and
McCammon [56] developed an algorithm in which the Euler method can be
modified to produce the proper diffusivity; this algorithm is commonly used to
solve stochastic Langevin equations [59, 55, 52].

Derivation of the drift velocity

To derive the drift term needed to correct the first order discrete algorithm,
the formalism used by Grassia et al. [60] is implemented. In this scheme, the
orientations, rotational velocities, and mobility are expanded,

p ( 0)+ (t) + 2) (t) + ... (2.66)

S(t) (t) + (2) (t) + ... (2.67)

Mj(t) M ( ( +PO (t) + + (2.68)
Pi (t ) (0)
where the superscripts indicate the order of the approximation. Similarly to
Grassia et al. [60], pi (t) represents the standard linear Brownian rotation, which

is calculated by replacing $ (t) with p 10). The term p2) (t) is a small nonlinear







correction which arises from the small variation in the friction between (i. ... and
R (t). Substituting these expansions into Equation (2.65) gives expressions for the
rotational velocities at first order,

p() (t) = M o (t), (2.69)

where it can be seen that pil) (t) (and consequently pl) (t)) is proportional to V t
since Fj (t) is of order vAt. At second order,

(2) () -o' M (t) (2.70)

where p2) (t) (and pj) (t)) is proportional to At since both pl) (t) and fj (t) are of
order A/Yt.
Averaging p1' (t) gives

.(1) (t)) K/M ,, ...j (t)\ M K. j ((t)) 0, (2.71)

since M o) is constant and (i (t) is zero as specified in Equation (2.47).
However, the mean rotational velocity at second order,

(2) 1 ((O5 ,.))
(( (t) a M (t)


(2) (t)) I (t )-'k ())t (2.72)


is generally non-zero since the first order displacements and weighted forces Fi (t)
are correlated.








To determine the correlation, the displacements at first order are calculated by
integrating Equation (2.69) over time,
t/ t
..(1) (t(1) __ (1)
p~)(t) = () d+p) (0)
Jt'= 0



p) (t) M o (t')dt' + )(0), (2.73)
Jt'= Pi
where pl) (0) is the known position at time zero and t is limited to short times.
Multiplying pl) (t) by F(t) and averaging gives,
m)() R f Jk (tt/ t'





(pm Rt) BT M( (- t1)k (2dt)
pi Pt=0




Jt'=0



p (() (t)) ) kBT (6ij, pi (t) pj (t)), (2.74)

since 6(t- t')dt' = 1/2 and M ,,,, (6i pi (t) pj (t)) instead of 6ij.
Substituting the correlation from Equation (2.74) into Equation (2.72) gives
the mean drift velocity at second order,

p) (t) -} k --- (t) Pk (t) ) (2.75)
)Pi (t)

which is non-zero. The drift velocity can be simplified using the explicit expression
for the mobility,

() (t) kT, ) kt).
PiPi(pi (t)








S(6jk Ft ) Pk


api (t0


py (t)
api (t) t


k t)pk (t) 6ij pi





(,k Pi (t) P (t)) (j- p

P ()jk (t) Pk (t) (jk


pj (t)


-(t) P k (t) )P


(t)pk (t)) t)


- (t)Pk (t))] (2.76)


In deriving this last equation, use has been made of the identity


ap, (t)
Opj (t)


(2.77)


which states that the orientation vector pi (t) can not change in directions parallel
to pi (t). Completing the multiplies in Equation (2.76) gives


(jk P (t)Pk (t))(k -



(t) M= 2i (t)Pk (t)) (6


(t) (t)p (t))

- P Pk(t))


S.(2)
Pi


KP2' (t))


kBT (jk
-^[R


K (2)())


(2)





.(2 )


kBT (j


kBT (6


kBT


KP2 (t))


kBT
R


j (t)Pk (t))


6ij Pi (t) pj ,









(p2) 2k (t)p (2.78)

This is equivalent to the negative of the divergence of the rotational di'iiff-i, il

matrix DR

ODRk k9 T
ap, (t) p () ( -)




aOD9 2ksT
-pi (t)




d' (2) \t)
S= p M (t) (2.79)

which can also be seen directly in Equation (2.75).

The numerical discretization

A naive implementation of an Euler method to integrate Equation (2.65)

produces unnecessarily large errors because the drift velocity (p2 (t) a)s derived
in the last section is non-zero. However, this systematic error can be corrected

by adding ( 2) (t) to the Euler algorithm at each step in the Euler algorithm

to produce diffusivities which are accurate to order (At) in time. This gives the

expression of
OD.
Pi (t + At) Pi (t) + + A ,-t+ (2.80)
aOp
where Api is the stochastic displacement perpendicular to the rod axis.

Using the expression for the divergence of the rotational diffusivity as appear-

ing in Equation (2.78), the discrete algorithm can be written as

2kBT
p, (t + At) = pi (t)- pi (t) At + Api. (2.81)
CR








The correction of -2kBTpi (t) At indicates that the rod length must be shortened
a small amount at the beginning of each time step. This does not mean that
the orientation vector grows shorter with time. In fact, the correction can be
interpreted in a simple, physical manner: this correction maintains pi as a unit
vector on average at order At. After a time step of At, the average expected value
(pip) equals one plus an error of order (At)2. This is proven by multiplying the
discrete equation for pi (t + At) by itself and averaging over multiple realizations,

i (t + At) pi (t + At)) ((p (t) 2-BT (t) At + Ap

( 2kBT
(p (t) P (t) At + Ap2)




(p, (t + At)p, (t + At)) 4kT At+ T )(t)2+ p + )( (2.82)

In going from the first to second line in Equation (2.82), the vector pi is assumed to
be a unit vector at time t and the stochastic displacement of pi is perpendicular to
Pi,
pi (Api) = 0. (2.83)

The correlated displacements ((Api) (Api)) are calculated using Equations (3.26)
and (3.28) from C'!i plter 3,

(Api) (Api)) (C ,.) (CiW )




(Api) (Api)) (C! {C,) KPwIR)














S 4kBT
((Api) (Ai)) 2kT ( T Pat)( P )(At)




(Ap) (Ap)) 4k (At), (2.84)

which is the expected result for the short time displacements [6]. Substituting the

result from Equation (2.84) into Equation (2.82) gives


S(t + At) (t + At) 1 + (2kT (At)2. (2.85)

The average length of the rod is exactly 1 to O (At) in time. Only at O (At)2 does

the average rod length increase. Without the correction, the length of the rod

would extend linearly with the time step. More importantly, the diffusivity will
contain an unnecessarily large error without proper application of the correction,

though it should be noted that the rotational diffusivity can be made arbitrarily

close to the correct result by making vanishingly small time steps.
Figure 2 1 demonstrates the effect of the correction graphically for a repre-

sentative displacement. Without applying a correction, the new position for the

orientation,

q, (t + At) = pi (t) + (Api) (2.86)

is guaranteed to walk off the surface of the unit sphere as shown in the figure.

Applying the correction lowers qi in the original direction of the orientation pi (t) to
give
2kBT
Pi (t + At) =q, Atpi(t). (2.87)

For the specific case shown in Figure 2 1, the correction moves qi from a position

outside the constraining sphere surface to a position inside the unit sphere.

Depending on the stochastic value of Api at time t, the value of pi (t + At) can
fall either inside or outside of the unit sphere after the discrete time step. The









unit sphere displacement Api


/ Cli(t)

pi(t) -2kBTlpiAt



pp(t+At)









vector pi (t) after a discrete time step At. Length scales in the figure
have been < r-,ted to make the qualitative points clear.

correction is such that the leading error of At is eliminated, and the unit vector

maintains its average length except for a small error of order (At)2

2.3.2 The Midpoint Method

As an alternative numerical integration method, Grassia et al. [60] and Grassia

and Hinch [61] advocate the midpoint method developed by Fixman [62]. An

advantage of the midpoint method is that the spacial derivatives of the diffusivities

are not required, which can be an intense calculation for complex systems. Like

the method due of Ermak and McCammon [56], the midpoint method corrects

for the drift in the mobility over a time step. This midpoint method is similar to

the classical midpoint method for solving deterministic equations except for two

issues. The Brownian forces of Equation (2.61) must be held constant over the
/







Figure 2 1. Diagram showing the effect of the drift correction upon the orientation











entirvector p (t) after a discrete time step, though other forces presentscales in the simulation are reevaluated at
the midpoint as usualbeen Also, the method is first order in accuracy, rather than
correction is such that the leading error of At is eliminated, and the unit vector
maintains its average length except for a small error of order (At)2.
2.3.2 The Midpoint Method

As an alternative numerical integration method, Grassia et al. [60] and Grassia

and Hinch [61] advocate the midpoint method developed by Fixman [62]. An
advantage of the midpoint method is that the spacial derivatives of the diffusivities

are not required, which can be an intense calculation for complex systems. Like
the method due of Ermak and McCammon [56], the midpoint method corrects

for the drift in the mobility over a time step. This midpoint method is similar to

the classical midpoint method for solving deterministic equations except for two
issues. The Brownian forces of Equation (2.61) must be held constant over the

entire time step, though other forces present in the simulation are reevaluated at
the midpoint as usual. Also, the method is first order in accuracy, rather than









second; for stochastic problems, the midpoint method does not necessarily improve

the accuracy of the solution.

Using the equations for the rotational and translational velocities of the rigid-

dumbbell model as example in using the midpoint method gives the following

discretized equations for the half-step

1 At)
x = x) (t) + 12Q 4 Y (2.88)


i + pi (t) p(t) (2.89)

where x* and p* are the half-step center of mass positions and orientations. The

Brownian forces are kept constant across the entire time step, but all other forces

are recalculated using the half-step values of the positions and orientations. The

full step equations are then completed using the half-step values as well as the

newly calculated forces and torques,

1
xi (t + At) = xi (t) + -- s6 At (2.90)
127rpa
1 /
p (t + At) = Pi (t) + 2 (6^ pp y At. (2.91)
37paL2 \ j

2.4 Simulations of Dilute Suspensions of Brownian Fibers

Before simulations of suspensions of Brownian fibers can be performed, the

validity and accuracy of the numerical discretization and integration methods must

be confirmed. Once the mobility for the collection of hydrodynamically interacting

rods is calculated, the Brownian forces and torques can be added to the simulation

method. For these initial simulations, multi-body hydrodynamic interactions are

ignored, and so the mobilities are those seen in Equations (2.14) to (2.17), and

Equations (2.33) and (2.34).









p(3)


0.8 -
0.6 .
0.4 -
0.2 -
-0.2












Figure 2 2. Plot of the sphere swept out by the orientation vector, where P (1),
p (2), and P (3) are the three Cartesian directions of pi. The time step
used was 1 x 10-5, and a total of 1 x 106 time steps were taken.
2.4.1 Testing the Brownian Motion
-0.8 -


.6














There are several qualitative and quantitative tests which can be used to verify
p(1) O 0. 1 -1






Figure 2. Plovalidity of the Browniansphere swept out by the orientation vector, where patio
methods used in the p (3)revious sections. The Brownian directions of pand torques calcu-time step
lated in section 2.2 depe, and on the sizetotal of the time step At. The random number

generator used to produce the Brownian Motionrces creates a unirm distribution of
There are several qualitative and quantitative tests which can be used to verify

the validity of the Brownian motion produced by the different numerical integration

methods used in the previous sections. The Brownian forces and torques calcu-

lated in section 2.2 depend on the size of the time step At. The random number

generator used to produce the Brownian forces creates a uniform distribution of

forces and torques. The Brownian torques should produce displacements in the

orientation pi which sweeps out a sphere around the origin. The three individual

components of the orientation (p (1), p (2), and p (3)) are then plotted in Figure

2-2. It can be seen that after 1 x 106 time steps, the sphere is evenly swept out.

Another test of the displacements of the orientations is to lock one of the

three indices of pi in place. The other two components of the orientation are then

allowed to step once along the surface of the sphere swept out by the orientation

vector seen in Figure 2-2. After this initial step, the components are reset back








38


At= Ix03 At= xl0-4
0.3 0 -



C., 0 0-



-0 .3 I -0 .1 I i I
-0.3 0 0.3 -0.1 0 0.1
p(1) p(1)
At= lxl05 At= x106
0.03 0.01







-0.03- i -0.01- i I i
-0.03 0 0.03 -0.01 0 0.01
p(l) p(l)

Figure 2-3. Plot of the Brownian displacements of the orientation with one com-
ponent (p (3)) locked with the z-axis. Time steps greater than 1 x 10-4
do not produce Brownian displacements adequate maximum displace-
ments.


to their initial positions. The Brownian displacements produced will map out a

square in two dimensions where the free components of pi are equally probable

to be displaced from the origin. The magnitude of the displacement is dependent

on the time step size At, which is seen in equation (2.61). An adequate time

step size needed to accurately resolve the Brownian motion of the system will

produce a clearly defined square in the two components of pi which are not locked.

The displacements should equally fill in a square in the plain perpendicular to

the locked index. The size of the square is dependent on the magnitude of the

Brownian forces and torques. These displacements, which depend on the time step

size, are shown in Figure 2-3, where it is seen that for time steps smaller than

1 x 10-4 do not adequately fill out the edges of the squares. All of these simulations

were performed with a total of 1 x 106 time steps.








The previous two tests are qualitative, providing no information about the
validity of the magnitude of the Brownian forces and torques which produce the
displacements in the simulations. The magnitudes of the forces and torques are
tested by calculating the diffusivity values of the center of mass and rotation by
using the autocorrelation functions of the center of mass and rotational displace-
ments. The average center of mass diffusivity is calculated from the autocorrelation
function as

((xi (t + ') x ) (t) 6DrTr (2.92)

for large values of T, which are shown in Figure 2-4. At small values of r the
difference in the diffusivities parallel and perpendicular to the central rod axis can
be seen, where

(( (t +, ) -x (t) 2D (2.93)

and

(x (t+ ) X- (t)) 2 2D 7, (2.94)

where x (t + r) x(t)) and (x (t + r) x (t)) are the displacements parallel
and perpendicular to the central rod axis over time T. After long enough times
the motion of the rods become de-correlated from the original orientations and so
the parallel D11 and perpendicular D1 diffusivities become the same as the average
center of mass diffusivity DT. Figure 2-4 shows that the Midpoint and corrected
Euler methods produce identical results, within the limits of numerical accuracy.
The rotational diffusivity can be calculated in a similar manner using the
autocorrelation functions of the orientations where

pi (t + r) p, (t) ) 2e -2D (2.95)

for time scales of TDR greater than or equal to 1. The plot of this function is
di-p il, 'it in Figure 2-5. It is difficult to obtain an accurate rotational diffusivity









0.08-
Corrected Euler Method
Midpoint Method
A DT
0.06- D .



0.04- D
-.. 0


0.02 .
V .



0 0.01 0.02 0.03 0.04 0.05

Figure 2-4. Plot of the autocorrelation function of the center of mass displace-
ments over time for both the corrected Euler and Midpoint methods,
which produce indistinguishable results. For short times D10 = 2Dl0,
but as time (r) increases both converge to the value of DT. The long-
time center of mass diffusivity DT is equal to the short time diffusivity
for a single rod in infinite dilution.


from this autocorrelation function, as doing a numerical fit of an exponential line is

difficult. However, taking the natural log of the rotational autocorrelation function

[38, 39]
SIn (p (t + 7) p (t) 2DRT (2.96)


produces a line with the slope equal to DR. This transformation is seen in the

inset of Figure 2-5. Once again the Midpoint and corrected Euler methods produce

identical results at short times. These results can be improved so that they are

identical for the entire time regime shown with further time averaging.

2.4.2 Error in the Corrected Euler Method

It has been mentioned previously that the Euler method used includes a

correction for the stochastic nature of the problem which improves the accuracy of

the simulations. An investigation into what the simulation results would be without

the correction has been conducted and the results will be shown. Simulations were









1--
-- Euler Method 1
..-- Midpoint Method /
O0.8 /
(-2DRt) 0.8 -
0.8 e

A \ 0.4 /
0.6-
S0.2
\ | | | | |

S0.4 0.02 0.04 00 0.06 0.08 0.1



V

0-
0.2-






-0.2
0 0.1 0.2 0.3 0.4 0.5


Figure 2-5. Plot of the autocorrelation function of the orientation over time. Both
the corrected Euler method and the Midpoint method produce similar
results. The rotational diffusivity is more easily taken from a fit of the
inset data, which is linear.


performed at different times steps for a single rod in an unbounded fluid using the

numerical discretization of Equation (2.81). Figure 2-6 shows the results of the

simulations, wherein the long-time rotational diffusivities were calculated using

Equation (3.33). Errors in the values of the rotational diffusivities were estimated

by calculating the standard deviation of the average rotational diffusivity from

an ensemble of ten different simulations over a total dimensionless time period of

1 x 104. Figure 2-6 also shows a comparison to an established numerical method

[2] which is described in detail in the following section. The renormalization of

pi (t + At) is discussed in more detail in the following section, but to summarize,

in some cases the orientation vector pi must be renormalized back to a unit vector

after each time step is taken to avoid unnecessarily large errors.

The results demonstrate that the diffusivity is correct for a sufficiently small

time step, even though the orientation vector pi was not renormalized after









1.02




1.00




0.98


Figure 2-6.


0.96 I 1 .11111 I I 11111
10-5 10-4 10-3
At
Long-time rotational diffusivities calculated using either the Euler
method or the algorithm of Lowen [2]. The squares are simulations us-
ing the Euler method with the drift correction and no renormalization
of pi (t + At). The diamonds are simulations using the Euler method
with the drift correction where pi (t + At) is renormalized after each
time step. The triangles are for simulations using the Euler method
without the correction, and the circles are for simulations using the
algorithm of Lowen [2]. Both the Euler method without the drift cor-
rection and the algorithm of Lowen [2] required renormalization after
each time step.


each time step. This is in keeping with the calculation of the rod length given

in Equation (2.85), which shows that the rod length is properly maintained to

first order in the time step. However, the simulations which were performed in

C'i Ilpter 3 employ, ,1 the Euler method with the drift correction and pi (t + At)

was renormalized after each time step. The orientation is renormalized to account

for the deterministic forcing due to the excluded volume force acting on a rod; the

drift correction only corrects the stochastic portion of the algorithm. Figure 2-6

demonstrates that the renormalization does not change the results.

Failing to apply the drift correction produces numerical results which are

significantly less accurate, though Figure 2-6 shows that the rotational diffusivity








is produced within numerical accuracy for times steps smaller than At = 4 x 10-5;
the algorithm with the correction produces the expected result for time steps as
large as At = 3 x 10-4. Renormalization of pi (t + At) is required after each time
step when using the Euler method without the correction regardless of the time
step due to the error of order At in the length of the rod, (pi (t + At) pi (t + At)) ~
1 + (At).
2.4.3 Comparison of Corrected Euler Method to Another Established Method
Lowen [2] employ, a similar numerical discretization, but no correction was
included. In general, a correction exists for the first order Euler method is the
divergence of diffusivity is non-zero. However, the divergence of the diffusivity if
formally zero for L6wen's [2] formulation since equations in the form of orientations
and torques were used, rather than orientations and weighted forces as done here.
The discrete equation for the change in the orientation over a time step used
by Lowen [2] is
p (t + At) = pi (t) + x (t) + Lc (t) (2.97)

where x1 and x2 are random numbers with zero mean and variance of 2kBTAt, and
e) (t) and e) (t) are two unit vectors perpendicular to pi (t) and to one another.
Using a similar technique to that used in Equation (2.82) where pi (t + At) is
multiplied by itself and then averaged over multiple iterations, the expected error
in the length of the rod after one time step is

Pi (t + At) Pi (t + At) K Pi (t) + x I1) (t) + x2 (t))
i (t) + X 1 (t) (+ X2 t))




(Pi (t + At)pi (t + At) 1+ (xixi) + (x2X2)









(p (t + At) p (t + At) 1 4kBTAt. (2.98)

Comparing the results of this equation to that of Equation (2.85) indicates that the

Euler method with the drift correction is more accurate. In order to maintain pi (t)

as a unit vector when using the numerical algorithm of Equation (2.97), pi (t + At)

must be renormalized after each time step.

Simulations were performed using the algorithm of Equation (2.97) with the

same parameters as the simulations using the Euler method. Figure 2-6 shows the

results in comparison to the Euler method with and without the correction. The

results of the discretization used by Lowen [2] and the Euler method without the

correction are interestingly equivalent. The values of the rotational diffusivity are

accurate for all the methods for time steps smaller than 1 x 10-5. It is evident

that correct values of the rotational diffusivities are still calculated using the Euler

method with the correction for time steps up to 3 x 10-4, while 4 x 10-5 is the

largest time step which still on average produces correct values for the rotational

diffusivity for the algorithm used by L6wen [2].

The Euler method with the correction is more accurate than the algorithm

defined by L6wen [2] as seen in the comparison of the numerical errors in Figure

2-7. Since the computations are for the simple case of a single rod where the

short-time and long-time diffusivities are equivalent, a more detailed analysis of

the relative errors of the algorithms can be performed. The expected value of the

short-time rotational diffusivity can be directly calculated as a function of the time

step from

DR (Pi (t+At)- p(t)) 2 (2.99)

where either the algorithm of Equation (2.81) or (2.97) can be used for pi (t + At).

In the cases where a normalization of the orientation vector is carried out after the

move of time step At, Equation (2.99) must be modified by dividing pi (t + At) by


















W2



1



0


10-5 10-4
At


Figure 2-7.


The percentage error in the long-time rotational diffusivities calcu-
lated using the Euler method with the correction and the algorithm
of Lowen [2]. The dotted line is the theoretical error in the expected
short-time diffusivities for simulations using the algorithm of Lowen [2],
while the dashed line is the theoretical error using the Euler method
with the drift correction. The symbols are the same as in Figure 2-6.
The relative error between the two methods is seen in the inset, where
the theoretical error of the algorithm of Lowen [2] is divided by the
theoretical error of the Euler method with the drift correction.


the magnitude of pi (t + At),


DR- t i (t + At)
4At pi (t + At0


(2.100)


Figure 2-7 shows that the error in the long-time rotational diffusivities is

accurately predicted by the error values calculated from the expected short-time

rotational diffusivities as given by Equation (2.100). This confirms that the error in

the corrected Euler method is smaller than the error resulting from the algorithm

given by Lowen [2]. The insert in figure 2-7 shows that the relative error between

the two algorithms as calculated using Equation (2.100) approaches a constant


10-6 105 10-4
At


10-3 10-









value as the time step goes to zero. The error in the Euler method with the

correction is at least three times smaller than the algorithm used by Lowen [2] for

all time steps.

2.5 Conclusion

The advantages of using the corrected Euler algorithm include the increased

accuracy and efficiency. Though the Euler method with the correction is more

accurate than the algorithm of Lowen [2], the difference in accuracy is negligible for

the small times steps used in the simulations of the concentrated rod systems. Sim-

ulations have been performed using the algorithm of Lowen [2] at concentrations

of nL3 = 70 and 150 for the slender-body model with aspect ratios of 25 and 50,

and the results are statistically equivalent to those in which the Euler method with

the drift correction was used. Since the accuracy of the simulations is essentially

equivalent for either method, numerical efficiency is the next consideration when

choosing between the two numerical methods for this particular application. The

algorithm used by Lowen [2] requires determining two unit vectors that are per-

pendicular to pi (t) and to one another at every time step. These extra calculations

make the method less efficient than the Euler method with the correction; for our

implementations, the Euler method with the correction (Equation (2.81)) was

found to be 2 '.- faster than the algorithm given by Equation (2.97).














CHAPTER 3
DYNAMIC SIMULATIONS OF CONCENTRATED SUSPENSIONS OF RIGID
FIBERS: RELATIONSHIP BETWEEN SHORT-TIME DIFFUSIVITIES AND
THE LONG-TIME ROTATIONAL DIFFUSION

3.1 Introduction

Rigid polymers are widely used as high performance plastics [10] and examples

of Brownian fibers can be found in the form of macromolecules of biological origin

[19] and in nanotechnology in the form of nanotubes and nanorods [63, 64]. Rigid

Brownian rods are also the simplest colloidal system demonstrating the effects

of entanglement [6], therefore the study of Brownian fibers has the potential to

illuminate fundamental issues in the general area of polymer physics. Consequently,

numerous theoretical, computational, and experimental results have been published

on the dynamics and rheology of Brownian rods.

Conflicting theories for the rotational diffusivity (DR) in the limit of infinitely

thin rods in concentrated suspensions predict scalings of DR/DRo ~ (nL3) with

either v = -2 or -1, where n is the number density, L is the length of the fibers,

and DRo is the rotational diffusivity of an isolated fiber in an infinite fluid. Doi

[7] developed a "tube" theory which predicts rotational diffusivities scaling as

(nL3)-2 for number densities within the semi-dilute regime (1/L3 < n < 1/dL2)

[7, 6], where d is the diameter of the fiber. Extensions of the theory have been

made for flexibility [23], polydispersity [40], and hydrodynamic interactions [41].

Simulations performed by Doi et al. [3] of infinitely thin fibers, in which the

hydrodynamic interactions between the rods were ignored and crossing of the

centerlines of the rods was prevented through a reflection rule, confirmed the

scaling of (nL3)-2. Simulations performed by Fixman [34, 35] found a different









scaling of DR (nL3)-1 for the rotational diffusivity at high concentrations in

the limit of infinitely thin rods. Similar to Doi et al. [3], a dynamic simulation

technique was implemented, but short-range potentials prevented the rods from

overlapping. Simulations performed by other researchers confirmed a scaling

exponent that was close to v = -1 for the rotational diffusivity [38, 39]. Fixman

[34, 35] proposed an alternative theory founded upon the concept of "cooperative

rotation" as the mechanism for the rotational diffusion of the fibers to account for

the simulation results.

Brownian dynamics simulations presented here demonstrate that the discrep-

ancies between previous simulation results arise from the choice of model for the

rods. The dimensionless ratio of L2DRo/DTo determines the scaling of DR/DRo,

where DTo is the average center of mass diff'i-i -ili-v for a fiber in solution at infinite

dilution. It is also observed that the thickness of the rods pl ,i- a role in the cal-

culated scaling for the rotational diffusivity, even at relatively high values for the

aspect ratios (A). As L2DRO/DTO varies between 4 (rigid-dumbbell model) and 9

(slender-body model) for rods with an aspect ratio of 50, the exponent v transitions

between approximately -1 and -2. For rods with an aspect ratio of 25, the ratio

L2DRo/DTo must be varied between 1 and 9 in order to produce a scaling that

ranges between approximately v = -1 and -2. The exponent v remains nearly

constant for L2DRo/DTO > 9, regardless of which rod thickness was used. In the

range of 0 < L2DRo/DTo < 4, the exponent v decreases significantly for all aspect

ratios studied. These findings presented here demonstrate that the rotational

diffusivity at high concentrations is not exclusively controlled by the excluded

volume.

The Brownian dynamics method used to demonstrate the dependence of the

scaling exponent upon the ratio of short-time diffusivities is presented in Sec. 3.2.

The method ignores hydrodynamic interactions between the rods and uses short









range potentials to enforce excluded volume for the rods. The ratio L2DRo/DTo

was varied between 1 and 9 for varying aspect ratios (fiber length over diameter),

with the limiting cases of L2DRO/Do = 0 and L2DRo/DTo = o also being studied.

The remaining three sections contain the results of the rotational diffusivity studies

in Sec. 3.3.1 and the results of the translational diffusivity studies in Sec. 3.3.2. A

discussion of the results and how they compare to theories and experiments can be

found in Sec. 3.4. The conclusions reached through this work are summarized in

Sec. 3.5.

3.2 Simulation Method

The underlying physical model for the short-time diffusivities of the individual

rods greatly influences the behavior of the rotational diffusivities for concentrated

systems as calculated using simulations. Researchers have selected different models

for the rods which have different values of the critical parameter L2DRo/DTo.

This difference arises from the manner in which the hydrodynamic resistance

is distributed along the length of the fibers. Doi et al. [3] used a slender-body

model for the individual fibers when performing simulations and found a scaling of

DR/DRo ~ (nL3)-2 for infinitely thin rods. The slender-body model [42, 43, 44]

yields a ratio of L2DR/DTO = 9. Additionally, this model has an anisotropic center

of mass mobility, where for an infinitely thin rod at infinite dilution, the mobility

in the direction parallel to the central axis of the rod is twice the mobility in the

direction perpendicular to the central axis. Other authors [34, 35] found a scaling

of DR/DRo ~ (nL3)-1 when using a rigid-dumbbell model which has a ratio of

L2DRo/DT = 4. The rigid-dumbbell model, in which the spheres do not interact

hydrodynamically, has an isotropic center of mass mobility. Figure 3-1 summarizes

the important characteristics of the two models.

To investigate the role pl i, d by the ratio of diffusivities at infinite dilution,

simulations were performed for values of the parameter L2DRo/DTo between 1 and









Slender Body







Characteristics:
DRO/DTO = 9
DI /D _= 2
H4-o,=


Rigid Dumbbell

(eQ (CL)
.X.




Characteristics:
L2DRO/DTO= 4
D/DL= 1


Figure 3-1. Summary of the characteristics and variables describing the slender-
body and rigid-dumbbell models. The vector xi defines the position of
the center of mass of the fiber and the unit vector pi defines the ori-
entation. The fiber length is L, d is the hydrodynamic diameter of the
fiber in the slender-body model, and a is the hydrodynamic diameter of
the sphere in the rigid-dumbbell model.

9 for rods of varying aspect ratios. In addition, simulations were performed for rods
with no Brownian contribution to their center of mass motion (L2DRO/DTO oo)
and for rods with no Brownian contribution to their rotation (L2DR/DTo = 0).
For each value of L2DRo/DT, simulations were performed for isotropic models
with a ratio DIIo/D1 o = 1 and anisotropic models with a ratio DIIo/D1 o = 2, with
the exception of the simulations with an aspect ratio (A) of 500. The diffusivities
DI0l and DI0 are the diffusivities at infinite dilution of the center of mass in the
directions parallel and perpendicular to the central rod axis.
3.2.1 Governing Equations
The motion of each rod in the suspension is given by the Langevin equations

S -aT (a) (aT(a,br) (a,ev)
m- X] i + b) (3.1)

MQ" Q (aQ) + 7a-br) + (a,ev), (3.2)
a t









where m is the mass of the rod, M is the moment of inertia, t is the time, and

x a) and Qf ) are the center of mass and angular velocity vectors of rod a. The

hydrodynamic forces acting on the rod are -aIT)a x-) and -Raf6 ), where -(,T)

is the center of mass resistance matrix and R is the rotational resistance. Rotation

about the 1i ii' ',r axis of the high aspect ratio rods is ignored, consequently the

rotational resistance R in Equation (3.2) can be written as a scalar.

Fa"br) and T((abr) are the fluctuating Brownian force and torque vectors acting

on rod a, and F ',( ) and T(a'7 ) are the excluded volume force and torque vectors.

The fluctuating Brownian forces and torques acting on the rods have a mean of

zero,

F(abr) (t) O and ((abr) (t)) 0, (3.3)

and a variance of

(abr) ( ab) (t')) 2kTTJ (t t') (3.4)

and

(7(abr) (t) T(abr) (t') = 2ksTRn J6 (t tl), (3.5)

as required by the fluctuation-dissipation theorem. The parameter kBT is the

thermal energy, 6ij is the identity matrix, and 6 (t t) is the Dirac delta function.

The brackets (.) indicate an ensemble average. The Brownian forces and torques

are uncorrelated in time and there are no correlations between rods because

multi-body hydrodynamic interactions are ignored within these simulations.

Rather than working with angles and angular velocities to describe the rods,

the simulations presented here use rod orientations and rotational velocities as

has been done by other researchers [2, 65]. In addition, the torque is rewritten

in terms of the weighted force distribution acting on the rods as explained in

the following. To convert Equation (3.2) into a form including only rotational









velocities, accelerations, and weighted forces, the relationships

S(a) (a) .(a) and (a) ap (3.6)


are used, where the orientation vector p ) is strictly a unit vector aligned with the

i i r axis of rod a and pYa) is the rotational velocity. The orientation vector is

equivalent to the direction cosine for the rod a. For rods of high aspect ratio, the

total torque acting on the rods can be approximated as the cross product of the

orientation vector and the first moment of the force distribution integrated along

the centerline of the rod [43, 44],

S) L/2
~(aj) kP sf(s) ds, (3.7)
J-L/2

where fi(s) is the line force density at a position s along the primary axis of the

rod and s ranges from -L/2 to L/2, with L being the rod length. Defining the

weighted force ') acting on rod a as

-L/2
a) / (s) ds (3.8)
J-L/2

allows the torque to be written as [66]



Performing the cross product of Equation (3.2) with the rod orientation pa)

and using Equations (3.6) and (3.9), the Langevin equation for angular momentum

can be written as

M Jkp a) ( klm1) (a) Rm\ijk (ca) kl(a) (a))


ijkPj) (Cklm a) + Jabr) (c) kl\mP,(a) av)) (3.10)

(a) (a) (a) ri
Making use of the vector identity 6ijk ) ((kimPi a) (6 P [48],

where aj is an arbitrary vector quantity, allows Equation (3.10) to be written in the








form
M( )p( R)(a)
M (6i = -Pj +
(t (a)p(a)) .(abr) + (ij P p()) ,
(aR) -(aR) (a)

where ija is the resistance matrix ( ,R) 6i p a))). Similarly to the
Brownian torques appearing in Equations (3.3) and (3.4), the weighted forces '(a)
must satisfy the fluctuation dissipation theorem,

Kj(a,br) (t) 0 (3.12)

and
abr ) a,br) (t') 2kBT R -( p (a) (a)) 6 (t t'). (3.13)

The variance in the weighted forces in Equation (3.13) is proportional to 2kBT
times the resistance matrix ( R)) which appears in Equation (3.11); a derivation
for the variance of the weighted forces is given in the C'! Ilpter 2.
Equations (3.1) and (3.11), together with the Brownian forcing in Equations
(3.3), (3.4), and (3.13), describe the motion of a high aspect ratio rod with
diffusivities of

D(aT) kBT (~;)' and D R) = kBT (i, p a)pa) (3.14)

where D(aT) and D('R) are the diffusion matrices for the center of mass and
rotation. The inverse of the matrix (6ij p a)) appears in the calculation of
the rotational diffusivity, however performing a traditional inverse of this matrix
is not possible because it is singular. Understanding that this singularity arises
from the projection perpendicular to the rod axis which ensures that p a) has no
components in the direction of p() (the rod is inextensible) allows the definition of
i (a ) (a)) -1 (P a ) (a )') '- r 2
an appropriate pseudo-inverse, (ij p a) j (= (see C pter 2
for further explanation). The center of mass and rotational diffusivities correspond









to the model chosen to represent the resistances of the rods. For the rigid-dumbbell

model [6], these resistances result in the following short-time diffusivities for the

rods systems being simulated:

kBT
D = kBTH 1 l (3.15)
127rpa

D1 kBTT1 kBT (3.16)
12r/ia
kBT
DR = kBTR1 kB (3.17)
3rplaL2'
where p is the viscosity of the suspending fluid, a is the hydrodynamic diameter

of each sphere on the rigid-dumbbell, L is the rod length, Dll is the diffusion in

the direction parallel to the 1 ii' ', rod axis, DI is the diffusion in the direction

perpendicular to the 1 ii, r rod axis, and DR is the rotational diffusion. The

resistances for the slender-body model [42, 43, 44] give short-time diffusivities of

SkT In (2L/d) (.
D =kT = -1 (3.18)2

k kT1n (2L/d)
DI = kBTI = ln(2L/d) (3.19)

DR BT 3kT n (2L/d)
DR = kB L3 (3.20)

where d is the hydrodynamic diameter of the rod. For both models, the average

center of mass diffusivity can be written as DT = (DII + 2DI).

The mobility matrices (inverted resistances) for the rod models can be written

in the unified form


ij X ["ij + /Pi PjI and ij I1 0 ij R Pi Pj (. t)
(aoT)) 1 ( + (a) (a) and R)) (6 P (a) (a))( (321)


where the parameters X and K are constants shown in Table 3-1. Writing the

mobilities in this form enables the use of a single set of Langevin equations (one

for the center of mass and one for the rotation) for the purposes of coding which







55

Table 3-1. Values of the parameters X and K used in Equation (3.21) in this chap-
ter. The value of X changes depending on the isotropy of the model to
keep the average center of mass diff' I-i.ili at infinite dilution DTo con-
stant. DTo, DI0o, DL, and DRo are the diffusivities calculated for the
models for a single rod at infinite dilution.

D LZDRo
Model x D L2DT
Rigid-dumbbell isotropicc) 0 1 4
12 2 4
Rigid-dumbbell (anisotropic) 1 2 4
Slender-body isotropicc) ln2) 0 1 9
Slender-body (anisotropic) n(2) 1 2 9


Model DTo D0lo DI0 DRo
Rigid-dumbbell isotropicc) kB kB kB' kB-
S127pa 127rpa 127Tpa 37paL2
Rigid-dumbbell (anisotropic) kBT kBT kBT kBT
12-pa 8O pa 167pa 3 paL2
Slender-body isotropicc) kBTln(2A) kBTln(2A) kBTln(2A) 3kBTln(2A)
3enpL 3wpL 3wpL wpL3
Slender-body (anisotropic) kBTln(2A) kBTln(2A) kBTln(2A) 3kBTln(2A)
lender-L 2wL 4wFL wFL3



encompass a wide range of models, including the slender-body model and the

rigid-dumbbell model. Examples of these models include a slender-body model

where the average center of mass diffusivity at infinite dilution (DTo) remains

constant, but the parallel and perpendicular diffusivities are changed so that the

overall diffT -i- ili becomes isotropic (Dllo = Do). This model is referred to as the

:-i.1 ~i"p' slender-body" model. An ,in- ropic rigid-dumbbell" model can also be

developed, where the center of mass diffusivity remains constant at infinite dilution

and the diffT -i- ili is made anisotropic (DI0o = 2Do0). In a similar way, the ratio

L2DRo/DTo can be changed to produce models for the rods with values of this ratio

other than 4 (rigid-dumbbell model) and 9 (slender-body model).

3.2.2 Evaluation of Excluded Volume Forces and Torques

A short-range repulsive potential maintains the excluded volume of the fibers.

This force contributes to both the rotation and displacement of the center of mass

and acts on the neighboring fibers at the points of closest approach [67]. The









forces between an interacting pair of fibers are equal in magnitude and opposite

in direction. Multiple repulsive forces may act on a single fiber, depending on the

separation distance between the surrounding fibers, which are then added to the

Langevin equations in the simulation (see Equations (3.1), (3.2), and (3.11)). The

form of the repulsive potential is

e-(h/<7)2
U = ( (3.22)
U sin (0)3.22)


where E and a are the parameters that determine the magnitude and range of the

potential, sin (0) is the angle between the pair of interacting fibers (|.t.p ) Pa)

and h is the closest distance between the center lines of a pair of fibers [67]. For

the simulations where A = 50, the parameters used were E = (50/3) kBT and

a = 0.03855L, which were the same as used by Bitsanis et al. [38, 39]. For the

simulations of A = 500, the parameter that sets the range (a) was divided by 10.

For the simulations with A = 25, the same parameters as for the simulations of

A = 50 were used, but h was calculated from a distance of 0.01L from the fiber

center lines, thus doubling the diameter and making an effective aspect ratio of 25.

Dividing by the angle between the rods poses a problem for parallel fibers. As

the angle between the interacting rods approaches zero, the repulsive force increases

to infinity. Using small time-steps partially compensates for this problem. At the

highest concentration, the maximum displacement produced by the parallel rods

is no greater than S''. of the rod length, or 1/5 of the interaction distance of the

repulsive force. The repulsive force acts along the normal vector between the point

of closest approach of a pair of rods [67].

3.2.3 Numerical Integration of the Governing Equations

To simulate the motion of the rods, Equations (3.1) and (3.11) must be

integrated in time. However, the interest of this study is in resolving the diffusive

motion, so there is no need to calculate the velocities. Ermak and McCammon









[56] showed that the time dependent positions can be calculated using a modified
Euler method without computing the velocities explicitly [56, 55, 66]. Applying
this method directly to Equations (3.1) and (3.11) produces expressions for the rod
displacements

xa) (t + At) x(a) (t) + ( T))- Fa,ev)At + A a) (3.23)


p.) (t t) (a) (t) +(( ')( R))1- +ev) DI} At+Api (3.24)
Aj + 9p} (a)

where At is the discrete time step taken in the simulation, and Ax ) and Ap a) are
the stochastic displacements. The stochastic displacements are given by

Ax ) ( ,T)) 1 'a,br) B-') WT) (3.25)

Ap ) ,R) (a,br) C (a) ( aR) (3.26)

where r' ) and ,1,(a,') are random vectors of length three with properties of

(aT) ,(a,R) ,,(a,T)W ) i, and (a,R) (aR) 6j. (3.27)


The random numbers are generated using i ,i 2," a subroutine presented by Press
et al. [51]. A uniform distribution for the random numbers is sufficient because the
numerical algorithm is accurate only to O (At) [52]. The matrices B() and C( are
related to the mobilities, where

3(a) L(a) 2kBTAt /C(a,'T)) and C (a) C (a) (Ta \ (a'R) (3.28)
ik = 2kBTAt 7, and ik C 2kBTAt (.j (3.28)

The appearance of the divergence of the rotational diffusivity in Equation (3.24),
SOD') () 1(a)
,) kB- ) R \ ( -P p P)j -2kBTR Pi (3.29)
Opj O'pj
corrects for the drift in the rotational velocities and is a direct consequence of using
orientations and weighted forces in Equation (3.11) (see ('! plter 2 for derivation).








This drift correction maintains p (t) as a unit vector on average at order At.
After a time step of At, the average expected value of (pi) (t + At)p (t + At))
equals

(a) (t + At) ) (t + At)) 1 + (2kBTQ )2 (At)2. (3.30)

Within the algorithm, p a) (t + At) is renormalized after each time step to account
for the small O (At)2 error. Failure to include the divergence of the rotational dif-
fii-i.-ii i- in Equation (3.24) results in an unnecessarily large error in the rotational
diffusivity. No similar correction exists when calculating the divergence of the
center of mass diffusivity D 'T),


a x $ j a x $ L k B T X ( + ( a ) ( a))p 1 0 ( 3 3 1 )
oxj oxj .

Simulations were also performed using the midpoint method [62, 60], which
produced results that were equivalent to those produced in the integration scheme
shown above.
At long times in a suspension of rods, the diffusivities are altered by the
presence of the forces used to maintain the excluded volume. These long-time
diffusivities are calculated from the simulation results. The average center of mass
diffusivity is calculated by the average squared displacements of the fibers,

(xa) (t + T) i () ( 6DTT, (3.32)

for large values of r. The rotational diff' -i-. i,, is calculated in a similar manner,
where

(pa) (t + T(a) (t)) e-2DT, (3.33)

for timescales of TDRo greater than or equal to 1. The algorithm in Equations
(3.23) (3.29) results in the expected diffusivities as listed in Equations (3.15) -
(3.20) for the fibers in the limit of infinite dilution (DRo and DTo) [6].









Periodic boundary conditions were used to approximate an unbounded

suspension. A cubic box with sides of length 2.1L was used for all simulations

except those with a number density of nL = 5, for which a box of length of 2.63

times the fiber length was used in order to have more fibers over which to average.

The size of the simulation box and the desired number density dictated the number

of particles in each simulation. For the number densities of nL3 = 5 to 150, the

simulations contained between N = 91 and 1389 fibers. The dimensionless time

step used in the simulations of A = 25 and A = 50 was 5 x 10-7 for concentrations

up to nL" = 50; smaller time steps of At = 5 x 10-8 were required for higher

concentrations. For the simulations of A =500, which were only performed for

the concentrations of nL3 = 70 to nL3 = 150, used a dimensionless time step of

5 x 10-9. The time step was made dimensionless by dividing by 47rpL3/kBTln (2A)

for the slender-body model, and by 67paL2/kBT for the rigid-dumbbell model.

One particular difficulty in simulating a random dispersion of fibers at high

concentrations is creating the initial distribution. For fibers with A = 50, the

maximum random packing occurs at a volume fraction of approximately 10' -

[1]. This volume fraction corresponds to a number density of about nL3 = 345.

For fibers with an aspect ratio of 25, the maximum number density for random

packing is about nL3 = 172. For concentrations lower than nL3 = 100, the

process of placing fibers at random in the box, testing for overlapping fibers, and

then starting the simulation was straightforward. For higher concentrations, an

initially ordered configuration was specified and data collection began only after the

suspension became random. The randomness of the orientations was confirmed by

checking the orientation distribution function for ordering across the entire system

as well as for local ordering.









3.3 Results

3.3.1 Rotational Diffusivities

Figure 3-2 shows the correlation function for the orientation as a function of

time for simulations at a number density of nL3 = 150 for rods with an aspect ratio

of 50, L2DRo/DTO = 9, and Dllo/Do = 2. All the input properties of DRo, DTo,

and Dllo/DIo were confirmed correct for this simulation, and all other simulations,

by examining the diffusivities at short times. For example, the rotational diffusivity

equals the diffusivity of a fiber at infinite dilution at very short times as seen in

the inset of figure 3-2. The fibers are not hindered by the surrounding fibers at

the shortest timescales, and therefore rotate and translate as if at infinite dilution.

At longer times, the excluded volume interactions between the fibers reduce the

freedom of motion, and hence the diffusivity, which is given by the slope of the

lines relating -- In (p ( t) p) (t + r) and r as indicated in Equation (3.33). For

nL = 150, the rotational diffE'-i.- il is reduced by 97'. in comparison to the dilute

value of the diffusivity, and the coefficient of determination (R2) between the linear

fit and the simulation data for times larger than r = 0.01 is greater than 0.99.

As the concentration of the rods increases, the fibers surrounding a test fiber

strongly hinder the rotation and significantly reduce the rotational diffusivity. Plots

of the rotational diffusivity as a function of number density appear in Figure 3-3

for both the slender-body model (L2DR/DTo = 9 and DIIo/D1o = 2) and the rigid-

dumbbell model (L2DRo/DTO = 4 and DIIo/D1o = 1) for rods having an aspect ratio

of 50. Errors in the diffusivity values were estimated by calculating the standard

deviation of the average diffusivity from an ensemble of three different simulations

over a total dimensionless time period of 1. The magnitude of the error is greatest

at the lower concentrations, where there are fewer rods over which to average. At

the lowest concentration (nL3 = 5), the error is about 5 i '. of the mean value.

As the concentration of the system increases, the error decreases to about 3 !' at









0.03 0001oo
00008- DRO

A 0.025- 00006-
r0 00004-
+
0.02- -
a 0 00001 00002 00003 00004 00005
0.015-


V 0.01 DRO DR


0.005-



0.00 0.01 0.02 0.03 0.04 0.05


Figure 3-2. The rotational correlation function at nL3 = 150, L2DRo/DT = 9,
and DIo/DIo 2 for rods with A 50. The inset shows that the
diffusivity at short times is equivalent to the rotational diffusivity at
infinite dilution. At longer times, excluded-volume interactions between
the fibers reduces the diffusivity to DR. For this model, the time 7 has
units of 47pL3/kBT In (2A).


the concentration of nL3 = 150 for all aspect ratios. The error bars in the figure are

smaller than the symbols used for the data points. The maximum estimated error

for all the simulations is no greater than I'.

Convergence of the values with respect to the time step was confirmed for all

simulations. The lower values of L2DRo/DTo require a smaller time-step in order to

achieve convergent results. As the aspect ratio increases, a smaller time-step is also

needed to achieve convergent results. A time step of At = 5 x 10-' was sufficient

for concentrations up to a nL3 = 50 and up to A = 50. For the simulations

of nL3 > 70 and for A = 25 and A = 50, a time-step of At = 5 x 10-8 was

required to achieve convergent results at the highest concentrations. To converge

the simulations of A = 500, a time-step of At = 5 x 10-9 was required. This time

step was used for all simulations for concentrations nL3 > 70 for A = 500.


















Bitsanis et al13,14
Rigid-dumbbell Mod

Doi etal. 0
Slender-body Model


1.00-







S0.10-







0.01-


S(nL3 -1
EU 'l(nL)







(nL3 -2
"\


100


Figure 3-3. The rotational diffusivities versus the number density are shown for
the slender-body model (L2DRo/DTO 9 and DIIo/Do 2) and
the rigid-dumbbell model (L2DRo/DTo 4 and DIIo/D1o 1), both
for rods with A 50. The numerical results of Doi and Edwards [6]


and Bitsanis et al.
(nL3)-1 and (nL3)


[38, 39] are
-2 scaling.


plotted for comparison, as well as the


SL2DRO/DTO

D LDRO/DTO

A L2DRO/DTO

0 L2DRO/DTO
SL2DRO/DTO

O L2DRO/DTO


1, A=

4, A=

4, A=

9, A=

9, A=

9, A=


I I I I I I I I


E(.L3 -1\

2550 EL -

50 \
"\ D
500

25\ E6

50
500 (L3 -2
500 (nL) [


100


Figure 3-4. Results are shown for the rotational diffusivities versus concentration
for the slender-body model (L2DRo/DT 9) with A 25, A 50, and
A 500. The results for the rigid-dumbbell model (L2DRo/Do 4)
for A = 50 and A = 500 are shown, as well as the results for an
alternative model for the rods where L2DRo/Do = 1 and A = 25.


10


1.00-


0.10-


d Q
8
e, ,









Simulations of models with varying values of L2DRO/DTo were performed to

determine the dependence of the scaling exponent v for the rotational diffusivity

scaling, DR/DRo = (nL3), within the concentration regime of nL3 = 70 to 150.

Simulations were also performed using aspect ratios for the rods of 25, 50, and

500 to study the effect that the rod thickness p1 ii- in the scaling of the rotational

diffusivities. Figure 3-3 compares the simulations results obtained here with those

of Bitsanis et al. [38, 39] and Doi et al. [3]. The rotational diffusivities for the rigid-

dumbbell model more closely follow the scaling of v = -1, whereas the slender-

body model has a scaling closer to v = -2 at high concentrations. A fit of the data

which includes points with concentrations in excess of nL3 = 50 indicates a scaling

of v = -1.89 for the slender-bodies (anisotropic) with A = 50 and v -1.13 for

the rigid-dumbbells isotropicc) of the same aspect ratio. For the simulations of rods

with A = 25, the ratio L2DRo/DT = 1 was required to reproduce the approximate

v = -1 scaling, as seen in Figure 3-4. Simulations with an aspect ratio of 500 were

also performed for the slender-body model (L2DRo/DT = 9 and DIIo/D1o = 2)

and the rigid-dumbbell model (L2DRo/Do = 4 and DIIo/D1 o = 1) for the higher

concentration regime. The scalings that were calculated from these simulations

were v = -1.01 for the rigid-dumbbell model, and v -1.91 for the slender-body

model. Within the range of nL3 = 70 to 150, the power-law fits have a coefficient

of determination (R2) greater than 0.92 for all cases. The ratio of DIIo/DIo has

a small impact on the rotational diffusivities as seen in Figure 3-5, regardless of

whether L2DRo/DTo equals 4 or 9. ('!CI ,iig the value of L2DRo/DTo between 4

and 9 however is seen to alter not only the values of the rotational diffusivities, but

also the scalings at the higher concentrations.

A decrease of about 1/2 for v occurs between L2DRo/D, = 1 and 0 for

the rods with an aspect ratio of 25, and a decrease of about 2/3 for the rods

with A = 50. This decrease indicates a rapid change over the range. Setting









1.00 -



(nL3)
[-H W%

L2DRO/DTO = 4, Isotropic
0.10 2
O LDRO/DTO = 4, Anisotropic '
2 i
L DR /DT = 9, Isotropic \
2
OL DRo/DTO 9, Anisotropic


0.0 1 ... i .. ..'
1 10 100
nL3

Figure 3-5. Rotational diffusivities versus number density for the slender-body and
rigid-dumbbell models with A 50. Setting the ratio of DIIo/D1 o equal
to 1 isotropicc) or 2 (anisotropic) causes small changes in the behavior
of the rotational diffusivity.


L2DRO/DT = 0 was achieved by imposing the condition DR0 = 0. This is

the limiting case where rotations occur only due to random collisions between

neighboring rods. Setting DR0 = 0 prohibits direct comparison of DR/DRo to the

other values of L2DRo/DTO because DR/DRo cannot be calculated. The power-law

scaling can however be calculated and compared to the scaling calculated from

other values of L2DRo/DTO. Setting Do = 0 allows calculation of the other limit

where translation of the particles arises purely from the collisions of the rotating

rods. In this limit of L2DRo/DT -- oc, the exponent v goes to -2 for rods with

A = 25 or A = 50. The values for scalings plotted in figure 3-6 are listed in Table

3-2.

3.3.2 Translational Diffusivities

The average diffusivities of the center of mass were determined from the

simulation data using Equation (3.32). Figure 3-7 shows that a difference exists

between the diffusivities calculated from the slender-body and rigid-dumbbell













'S

""


OOA
cmA
<0 A
AAA
< A
> A


L I I I I I I I
0 1 2 3 4 5 6 7
01234567
2
LDRo/DTo


0.00-

-0.25 '

-0.501

-0.75-

-1.00-

-1.25-

-1.50-

-1.75-

-2.00-


Figure 3-6. Exponent v for the scaling of the rotational (iffT L-i, ili, as given by
DR/DRo (nL3)" for the concentration range of 70 < nL3 < 150. The
exponent transitions between the approximate values of -1 and -2 as
the ratio of rotational diffusivity DR0 to translational diffusivity DTO
increases from 1 to 9 for rods with A = 25, and from 4 to 9 for rods
with A 50 and A 500. The rod thickness p1 '. l a larger role for
values of L2DRo/DTO < 8, but makes a minimal difference for larger
values. The points for A = 500 show the calculated scaling for the
rigid-dumbbell model (L2DRo/DT 4 and DIo/DIo = 1) and the
slender-body model (L2DRo/DT 9 and DIIo/Do 2).

Table 3-2. Values for the exponent v for the scaling of the rotational diffusiv-
ity as given by DR/DRo = (nL3) plotted in Figure 3-6. The values
are for models with either isotropic (DI|o/Do = 1) or anisotropic
(D0Io/D1o = 2) center of mass mobilities. As stated in the text, the
power-law fittings are intended as a general guideline for the scaling
of DR/DRo at high concentrations. A purely power-law scaling for the
rotational diffusivity of rigid-rod systems at high concentrations has not
been rigorously justified [68].


A D
1
25 2
1
50 2


0
-0.48
-0.53
-0.22
-0.22


1
-1.08
-1.08
-0.64
-0.63


500


3
1.34
1.36


L2DRRo/DT
4 5
-1.78
-1.79


1.13
1.15
1.01


6 7
-1.83
-1.86


1.37
1.35


9
1.86
1.90
1.85
1.89


2.03
-2.04
-2.02
-2.04


= 25 Isotropic
= 25 Anisotropic
= 50 Isotropic
= 50 Anisotropic
= 500 Isotropic
= 500 Anisotropic





8 9 10

8 9 10 00
















S* 13,14
A A\x Bitsanis etal.
0.4- --- ..
-.\\ center of mass
0.2 ..
perpendicular
0.0 I
0 50 100 150
nL3

Figure 3-7. Translational diffusivities as a function of number density for the
slender-body model (L2DRO/DTo 9 and DIIo/Do = 2) and the rigid-
dumbbell model (L2DRo/DTo 4 and DIIo/Do 1) with A 25.
The average center-of-mass diffusivity DT, diffusivity parallel DII, and
diffusivity perpendicular DI to the rods are normalized by the parallel
diffusivity at infinite dilution Dlo0.


models up to a number density of nL3 m 30 for rods with aspect ratios of 25. For

the simulations of rods with aspect ratios of 50, the qualitative difference extends

to a number density of nL3 w 40. Beyond this concentration, the diffusivity

DT/DIIo continues to decrease for both the slender-body model and the rigid-

dumbbell model to a value of approximately 0.4 at nL3 = 150. As with the data

for the rotational diffusivities, errors were estimated by calculating the standard

deviation of the mean diffusivity for an ensemble of three different simulations over

a total dimensionless time period of 1; the errors were determined to be no more

than 5'. of the mean value over the entire concentration regime.

Examining the components of the diffusivities demonstrates that the hindrance

of the motion perpendicular to the rods is responsible for the reduction in the

average diffusivity with concentration; the parallel diffusivities remain virtually

constant over the entire range of concentrations as seen in Figure 3-7, even for









the relatively thick rods with A = 25. To calculate the diffusivities in the parallel

and perpendicular directions, the fiber displacements are projected onto the fiber

orientation at time t. Figure 3-8 shows that the parallel diffusivity at infinite

dilution, Dll0, is recovered at short times, as is the perpendicular component, DI0.

For large T, the time rate of change of the square displacements in the directions

perpendicular and parallel to the initial orientations of the rods are equivalent to

the average dif'l -i-l ii, DA. This convergence of the values over long times arises

from the gradual loss in the correlation of the instantaneous and initial orientations

together with coupling of the translational motion of a fiber with the instantaneous

orientation of the fiber. The average square displacements at the intermediate time

of T = 0.1 define the diffusivities Dll and DI as indicated in Figure 3-8. The value

of 0.1 was chosen so that the diffusivities correspond to those calculated by Bitsanis

et al. [38, 39] as seen in Figure 3-7; the translational diffusivities calculated for the

rigid-dumbbell model fall within the error ranges of these previous results.

3.4 Discussion

In the following sections, the dependence of the power-law scaling for the

rotational diffusivity on the ratio L2DRo/DTo and the aspect ratio A is examined.

A comparison is made in Sec. 3.4.2 between the simulation results and some

existing theories for the rotational diffusion. A discussion in Sec. 3.4.3 of three

different limiting conditions for the model of the rods provides some insight into

the role p1 i- -1 by the ratio of the short-time diffusivities. The calculations of

the rotational diffusivities are also compared to experiments (Sec. 3.4.4) and the

translational diffusivities are discussed in Sec. 3.4.5.

3.4.1 Dependence of Rotational Diffusivities on Fiber Model

The dependence of rotational difTI -i.-lii on number density depends strongly

upon the hydrodynamic model describing the individual fibers, regardless of the

aspect ratio. For example, within the concentration range nL3 between 70 and









0.08

0.07

0.06 D1O / DT

S0.05- /

0.04-

0.03-

"0.02- / /

0.01

0.00
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Figure 3-8. Average square displacements of the center of mass versus elapsed
time 7. The average square displacement at a time of 7 = 0.1, pro-
jected along the parallel and perpendicular direction at T = 0.0, defines
the diffusivities D11 and D The data here corresponds to simulations
with nL3 30, L2DRo/DT 9, and DIIo/Do 2 for rods with
A 50. The units of time 7 are the same as those in Figure 3-2.


150, fitting the dependence of rotational diffusivity on number density using a

power law scaling of DR/DRo ~ (nL3)" gives very different results for the scaling

exponent v depending upon the hydrodynamic model describing the individual

fibers as seen in Figure 3-6 and Table 3-2. In keeping with the literature on the

subject of rotational diffusivity of rigid rods, the power-law fit for the data within

the concentration regime has been used as a convenient method for describing

the scaling behavior within the semi-dilute concentration regime. However, the

assumption that DR/DRo conforms to a power-law with respect to nL3 is not

rigorously justified [68]. In fact, the agreement between the simulation data and

the power-law fits are not particularly convincing as noted by the low values of

correlation between the data and the power-law scalings as reported in Sec. 3.3.1.

The key difference in the models which leads to the variation in the observed

value for the scalings v is L2DRo/DTo, the dimensionless ratio of the rotational









and translational diffusivities for an individual rod at infinite dilution. The rod

thickness, as seen in Figure 3-6, also has a significant impact on v for smaller

values of L2DRo/DTo. For values less than about 8, the thickness of the rod pl i,

a role in determining the exponential scaling. In order to produce the v = -1

scaling, a value of L2DR/DTO = 1 is needed for rods with A = 25, while a

value of L2DRo/DT = 4 is needed to produce the v = -1 scaling for rods with

A = 50. For large values of L2DRo/DTy, Figure 3-6 shows that there is essentially

no difference in the scaling for aspect ratios of at least 25 or larger. Simulations

were also performed using an aspect ratio of 500 for the rigid-dumbbell model

(L2DR/DTo = 4 and DIIo/Do = 1) and the slender-body model (L2DRo/DO = 9

and DIIo/DIo = 2) give results similar to the simulations with aspect ratios of

50. For the rigid-dumbbell model a scaling of v = -1.01 was calculated, while a

v = -1.91 scaling was calculated for the slender-body model. This shows that for

simulations of rods with finite thicknesses, there is a qualitative transition in the

scaling behavior due to the hydrodynamic model for the rods.

Another possible source for the difference in the scalings is the isotropy of the

models. However, Figure 3-5 demonstrates that the ratio of DIIo/DIo has only a

small effect on the rotational diffusivities if the overall center of mass diffusivity

DTo is held constant. The simulation of general models other than the slender-

body and rigid-dumbbell also demonstrate that the exponent v is a function

of L2DRo/DTo, as plotted in Figure 3-6. The exponent v is not a single-valued

function of L2DRo/DIIo. Setting DI0 equal to D0lo in Equation (3.32) results in the

relation L2DRo/DII = L2DRo/DT for the isotropic model. For the anisotropic

model, L2DRo/DIlo equals 2L2DR0/3DTo because DI0 = DII0. Plotting the

exponents v versus the ratio L2DRo/DIIo consequently results in two distinct curves

for the aspect ratios studied.









Since theories have been largely based upon the results of simulations, this

qualitative difference in the scaling behavior has led to significant confusion over

the subject of rotational diffusion in concentrated systems of Brownian fibers.

Simulations by Doi et al. [3] using the slender-body model with infinitely thin

rods, results of which are shown in Figure 3-3, confirmed the tube theory [7] which

results in a scaling theory with v = -2. Fixman [34, 35] performed simulations and

obtained a scaling of v = -1 in the limit of thin rods, directly contrasting with

the tube theory proposed by Doi [7]. Consequently, Fixman [34, 35] proposed an

alternative theory for the rotational diffusivities of Brownian rods at semi-dilute

concentrations. Roughly the same simulation results of Fixman [34, 35] were later

confirmed by the simulations of Bitsanis et al. [38, 39]. The different results of Doi

et al. [3] and these other authors stem from the different models used to describe

the rods. Fixman [34, 35] used a three-bead model in his simulations of rods of

vanishing diameter, stating that the ratio of L2DRo/DTO equals 6 (see footnote 7 in

the work by Fixman [35]). Bitsanis et al. [38, 39] used a rigid-dumbbell model with

an aspect ratio of 50 to confirm the v -1 l- ii:.. a direct comparison is made

in Figure 3-3 between those results and the simulations performed as part of this

work.

Fixman [34, 35] and Bitsanis et al. [38, 39] dismissed the simulation results

of Doi et al. [3] as being in error, stating that Doi et al. [3] probably used a time

step too large to accurately resolve the dynamics of the system. However, the

simulations presented here indicate that the simulations of Doi et al. [3] were

essentially correct. The data of Doi et al. [3] and our simulations at A = 50

are shown in Figure 3-3 and plots of rotational diffusivities as a function of

concentration are shown in Figure 3-4 for all three aspect ratios of 25, 50, and 500

simulated. Doi et al. [3] simulated an infinite aspect ratio, which would seem to

prevent making a direct comparison of the results. However, our analysis indicates









02 3
SnL3 = 70
nL3 = 90
*-nL3 = 110
o 0-- nL3 = 130
SnL3 = 150

01-







0 001 002 003 004
1/A

Figure 3-9. Linear extrapolation of the rotational diffusivities calculated using
the slender-body model to the limit of an infinitely thin rod using the
values calculated from the aspect ratios of A = 50 and 500.


that the slender-body model with A = 500 produces results nearly equivalent to the

slender-body model with an infinite aspect ratio, as argued here.

The results for A = 50 and 500 vary only by 9.1 '. on average. Recalling that

the values for each point are known only within approximately i' indicates that

the rotational diffusivities for 50 and 500 are nearly the same within statistical

accuracy. Furthermore, a linear extrapolation is made using the values at A = 50

and 500 to estimate a mean value of the rotational diffusivity for the limit A = xo.

The extrapolation is shown in Figure 3-9 and the extrapolated values are plotted

as a function of nL3 in Figure 3-10. The differences between the extrapolated

values and the values at A = 500 differ by an average amount of only 1.1 an

error which is well below the capability of resolving using the Brownian dynamics

simulations. In other words, the results for A = 500 (for the slender-body) are

essentially equivalent to A = oo for the concentration regime of interest. The direct

comparison of the extrapolated values with the results of Doi et al. [3] shown in

Figure 3-10 show a close correspondence, especially when considering the error

in the A = oo results of at least I.' (due to the additional errors arising from









1.00 -------






0.10 -

o Doi etal. [10]
Slender-body Model (A oo) \
-- Fixman [12] (A = oo)(nL
:(nL3 -2
0.01 .. .
11 10 100
nL3
Figure 3 10. Direct comparison of the simulation results of Doi et al. [3] with the
slender-body model using the limiting values of infinite aspect ratio as
appearing in Figure 3 9. The theory of Fixman [35] given in Equation
(3.39) for A oc and L2DRo/DTo 9 is also shown in the graph.


the extrapolation) and the errors reported by Doi et al. [3] of approximately 5' .

At the least, the simulations of Doi et al. [3] accurately captured a scaling of

approximately (nL3)-2 for the slender-body model within the concentration regime

of interest, in part juL-I Fii-i; the prediction of the tube theory which is discussed in

the next section.

3.4.2 Relation with Existing Theories

Existing theories for the rotational diffusivity of fibers of high aspect ratio do

not accurately account for the dependence of the scaling exponent upon the ratio of

short-time diffusivities. For example, Doi [7] and Doi and Edwards [69] argued that

infinitely thin fibers at high concentration become caged in static tubes formed by

the surrounding fibers. Rotation can then occur only when the rod escapes the cage

by diffusing along the central axis of the fiber, whereupon the fiber will rotate upon









entering a new tube. For this scaling, the rotational diffusivity is given by


DR1 -, (3.34)


where a is the diameter of the tube constraining the rotation and Td is the average

time necessary for the polymer to leave a tube. The tube diameter is estimated to

scale as
1
a nL- (3.35)
nL2

and the time for a rod to leave a tube is proportional to the time needed for a rod

to diffuse one particle length along the primary axis,


d --. (3.36)
Dll0

Combining Equations (3.34) (3.36) and dividing by the rotational diffusivity at

infinite dilution results in the prediction of the tube theory for DR,[7, 69]

DR D10 (1 2 (
DRo L2DRo nL (3.

As discussed in the previous section, the simulations of Doi et al. [3] accurately

provided evidence for this scaling law, but the result of v = -2 is limited to

only certain conditions as shown in Figure 3-6. This shortcoming arises from the

assumption that the rotational motion is controlled exclusively by the excluded

volume at high concentrations and aspect ratios; the ratio of diffusivities appears

only as a multiplicative factor of order one [7, 6].

Aside from predicting a constant power-law scaling of v = -2, the scaling

of Equation (3.37) implies that the ratio of DR/DRo should be proportional to

DIIo/L2DRo at a fixed number density. Figure 3-11 shows that the rotational

diffusivity as a function of the ratio of the short-time diffusivities demonstrates

that the relationship is non-linear. The anisotropic diffusivity values are offset from

the isotropic values by comparing to DIIo/L2DRo instead of DTo/L2DRo. Figure









0.50- O nL = 50
WAnL = 70
0.40 nL3= 110
SnL = 150

o 0.30


0.20 -


0.10


0.00 -
0 0.2 0.4 0.6 0.8 1
DK/L2D
D110 RO

Figure 3-11. Rotational diffusivities over a range of L2DRo,/DI with either iso-
tropic or anisotropic center of mass diffusivities for rods with A 25.
For rods with an isotropic mobility, DTo = Dii0, while for rods an
anisotropic mobility, DT, = 3D0,. The solid symbols are for the rods
with anisotropic mobilities, while the open symbols are for the rods
with isotropic mobilities.


3-11 shows the results for rods with A = 25, and the deviation from a linear

behavior is seen to become less pronounced as the concentration increases, but is

still clearly non-linear even for nL3 = 150. Figure 3-12 shows the same results for

the simulations of rods with A = 50, and the behavior deviates even more from a

linear relationship than with the thicker rods.

Most theoretical work has confirmed the v = -2 scaling for the rotational

diffusivity in the limit of high concentrations. A number of the studies have sought

to refine the concept of cages as originally put forth by Doi [7]. For example,

Keep and Pecora [68] developed a theory which reduces to the -2 scaling only

for concentrations greater the nL3 = 500. For lower concentrations, this theory

predicts a scaling which is a mixture of v = -1 and -2. Teraoka et al. [36] and

Teraoka and Hayakawa [37] also confirmed the rotational scaling of v = -2 in the









00nL3 = 50
0.60- nL3= 70
W nL3 = 110
0.50- nL3= 1
AA nL3 = 150
a 0.40

9 0.30

0.20

0.10

0.00--
0 0.2 0.4 0.6 0.8 1
DII/L2D
D110 RO

Figure 3-12. Rotational diffusivities over a range of L2DRO/DIlo with either iso-
tropic or anisotropic center of mass diffusivities for rods with A 50.
The filled in symbols are for the rods with anisotropic mobilities,
while the clear symbols are for the rods with isotropic mobilities.


limit of high concentration for very thin fibers, as seen in their prediction


DR= 1 + 0.011 2DR (3.38)3 -2


This relationship indicates a rotational diffusivity which depends upon the short-

time diffusivities in a non-trivial manner and provides a good fit for the data

generated for the slender-body model over the entire concentration range studied.

The agreement of this prediction with the simulations results using other hydro-

dynamic models for the rods, such as the rigid-dumbbell, is however poor. Sato

and Teramato [70] later modified Equation (3.38) to account for the presumed

reduction in longitudinal diffusivity caused by the thickness of the rods. However,

DII remains essentially constant even for relatively thick rods of aspect ratio 25 as

seen in Figure 3-7, at least for concentrations through nL3 = 150.

An alternative to the cage theories can also be seen in the work of Fixman

[34, 35], where it is argued that the static tube of Doi and Edwards [6] would move,









rotate, break-up and reform on a time-scale comparable to the reputation time, or
the time needed to escape the cage, of L2/DIIo. This theory predicts a rotational

diffusivity of

DR +aM L2DRO (nL3) + 3(f) L2DRO (nL3)2 (3.39)
= 1 + ( f) D AL 1 o {nL3)2 (3.39)
DRO DT A DTo

where the parameters a (f), 3 (f), and f are not arbitrary constants, but are
determined through the analysis (see Equation (4) in the work of Fixman [35])

and are weak functions of the number density. For suspensions of fibers of finite
thickness and at sufficiently large concentrations, the theory approaches the scaling
of DR/DRo ~ (nL3)-2. The scaling of DR/DRo ~ (nL3)-1, often quoted in the

literature [38, 39, 4, 2, 45, 46, 47, 8] when discussing the theory of Fixman [35],

corresponds to the dual limit of infinite aspect ratio and high concentration. The
theory is compared to the results of the simulations using the slender-body model
within Figure 3-10, where A was set to oo and L2DRo/DTO to 9 within Equation

(3.39). The diffusivities predicted by the theory are larger than those resulting
from the simulations under these conditions. Within the range of nL3 = 70 to 150

over which the simulation data has been consistently fit using the power-law, the
best fit to the theory corresponds to v = -0.69. The limiting value of v = -1

is not approached until the concentration reaches at least the unrealistically large
value of nL3 = 1000.
3.4.3 Rotational Diffusivity under Limiting Conditions

The simulation results demonstrate that the cage is dynamic and can not

be treated as a static geometric constraint on the rotation of tracer rods, at
least for the range of aspect ratios and concentration regime studied here. For

example, removing the Brownian forces applied to the center of mass of the rods

sets the short-time diffusivity DTo to zero. The centers of the rods are still mobile,
however, and the results show that a significant translational diffusivity at high









concentrations is caused by the excluded volume of the rods. This motion of the

centers of the rods is induced by the collisions between the rotating rods and is

consistent with diffusion in that the square displacements grow linearly in time.

As compared to results from the slender-body model, DR/DRo is reduced by 7.' .

for rods with aspect ratios of 50, and by 711' for rods with aspect ratios of 25.

The constant scaling of v = -2 for models with values of L2DRo/DTo greater

than 9 indicates that the relaxation of rotational constraints through the reputation

mechanism is not very important. Similar studies in which the center of mass

motion of the rods was frozen were performed for high aspect ratio rods [35] and

for low aspect ratio spheroids [2], but the results were not clearly stated.

The rotation caused by collisions between rods is a mechanism which has

been ignored in previous treatments. As the ratio of L2DRo/DTo goes to zero

(by removing the Brownian torques contributing to the fiber rotation while still

allowing the rods to rotate freely), Figure 3-6 shows that the rotational diffusivity

scales as v = -0.2 for the concentration range between nL3 of 70 and 150 for rods

with A = 50, and v = -0.5 for rods with A = 25. The values for the rotational

diffusivity are smaller in this case, but are not vanishingly small. At nl3 = 70, the

diffusivity is 12'"- of the value of DR for the slender-body model with A = 50. As

the concentration increases to nl3 = 150, the diffusivity for L2DRo/Do = 0 is

! :'. of the value of DR calculated for the slender-body model. For the rods with

A = 25, the difference goes from 211' to ,!*'- over the same concentration regime.

Therefore, rotations induced by collisions make a significant contribution to the

rotational diffusivity.

The dependence of the rotational diffusivity on DTo rather than Dll0 is in

direct contrast to most existing theories, particularly the tube theory [7, 6]. There-

fore, a third limiting case was explored in which the short-time diffusivity in the

perpendicular direction, DIo, was set equal to zero by removing the perpendicular









0.4
W- Isotropic No D
w-E Isotropic D Included
0.3 00 Anisotropic No D
0 0E Anisotropic D Included

S0.2



0.1



0.0-
1 2 3 4 5 6 7 8 9

L DR /DTO

Figure 3-13. Comparison of rotational diffusivities for rigid fibers with A = 25
with perpendicular diffusivity removed and fibers maintaining the
perpendicular contribution to the diffusivity. The results shown here
are for a concentration of nL3 90.


components of the Brownian forces acting on the center of mass of the rods. Figure

3-13 shows that the perpendicular diffusivity does have a significant impact on the

rotational diffusivities, even at high concentrations. Rods with A = 25 were used in

this comparison, as the thickness more clearly shows the effect of the perpendicular

diffusivity by the rod being more confined and thus less likely to diffuse in that

direction. As seen from the data at nL3 = 90 of rods with A = 25, not including

DI has a smaller impact for low values of L2DRo/DTO than for higher values. A

difference of 11 to 211', in DR/DRo exist for L2DRoDTO = 1, whereas the difference

is approximately 31 to :', for L2DRo/DT = 9. The perpendicular diffusivity

indeed does p1 i, a more pivotal role in the dynamics than assumed in the tube the-

ory, at least for the concentration range studied here, and supports the claim that

cooperative motion between a tracer rod and confining rods must be considered in

the theories of rotational dynamics.







79

1.004 see go


AS
A A

A
A 4P
Q 0.10- Rigid-dumbbell Model A
Q A Slender-body Model A
Mori etal. [49] A
A


0.01-
1.0 10.0 100.0
nL3
Figure 3-14. Rotational diffusivities from simulations using the slender-body and
the rigid-dumbbell models with A = 50 compared to the diffusivities
of PBLG measured by Mori et al. [71].


3.4.4 Reinterpretation of Comparison with Experiments

Figure 3-14 compares the results for the rotational diffusivity calculated from

the slender-body and rigid-dumbbell models for rods with aspect ratios of 50 to the

experimental measurements of poly(7-benzyl-L-glutamate) (PBLG) in m-cresol [71]

for number densities up to nL3 = 160. This same comparison was made by Bitsanis

et al. [38, 39] together with the conclusion that the Brownian dynamics simulations

were faithfully reproducing the experimentally measured values. However, the good

agreement between the simulations of rigid-dumbbells and the experiments arises

from a fortuitous choice of the rigid-dumbbell model to represent the short-time

diffusivities of the individual rods composing the suspension; any other model

produces qualitatively different results. Yet, there is no basis for arguing that the

rigid-dumbbell is somehow special, because more accurate models are available such

as the slender-body for high aspect rods and of ellipsoidal spheroids [72, 73, 13] of

high aspect ratio.









The fact that DR/DRo depends upon the ratio of short-time diffusivities in a

non-trivial manner, as well as the dependence on the rod thickness for some mod-

els, brings additional complications to the problem. Beyond choosing the correct

model for the rods, the effects of the long range hydrodynamic interactions, which

alter the short-time diffusivities depending upon concentration, must be included.

For suspensions of Brownian spheres, previous work indicates that the ratio of

long-time and short-time diffusivities remain nearly constant at any concentration

regardless of the level of approximation at which hydrodynamic interactions are

calculated [74]. Such a relation remains to be demonstrated for rigid rods, but this

relation would have to depend in part upon the ratio of short-time diffusivities. No

calculation of the long-time diffusivities for concentrated systems of Brownian rods

with hydrodynamic interactions exist, though calculations of short-time mobilities

are available for ellipsoidal particles [75] and methods for including hydrodynamic

interactions in suspensions of slender-bodies exist [76, 77, 66].

Additionally, recent criticisms in the experimental literature have raised

questions as to whether the simulations of rigid fibers should be compared to the

results of PBLG measurements in the studies performed by Mori et al. [71]. PBLG

is a stiff, helical homopolypeptide which may or may not be rigid [27, 28]. Cush

and Russo [20] claim that flexibility, especially at the ends of the PBLG fibers,

cause a downturn in the prediction for the translational diffusivities, which is

discussed in the following section. Flexibility in the PBLG fibers could cause an

increase in the rotational diffusivity, as the fibers would more easily escape the

tubes and rotate [23].

3.4.5 Translational Diffusivities

The ratio of short-time diffusivities, L2DRo/DTo, does not affect the long-time

translational diffusivities as is the case for the long-time rotational diffusivities.

Comparisons of the slender-body and rigid-dumbbell models shown in Figure 3-7









demonstrate that the translational diffusivities are indistinguishable for number

densities nL3 greater than 50. Clearly, for concentrations in excess of 50, the

approximate 18 to 1 factor for the parallel to perpendicular diffusivity arises

purely from the effects of excluded volume and increasing concentration; neither

the degree of isotropy in the hydrodynamic model of the rods nor the thickness

of the rods, at least for aspect ratios larger than 25, has any noticeable effect

upon these results. The translational diffusivities calculated by Bitsanis et al.

[38, 39] using the rigid-dumbbell model are also plotted in Figure 3-7. The models

indicate that the diffusivity in the direction parallel to the rod orientation remains

essentially constant over the concentration range studied. This is in contrast to

the findings of Bitsanis et al. [38, 39] that showed a decrease in D\\/DI0o at the

highest concentrations. However, Bitsanis et al. [38, 39] reported error bounds

which overlap with the data for both the rigid-dumbbell and slender-body models

as calculated here.

The assumptions concerning the isotropy of the model impacts the simulation

results for the average center of mass and perpendicular diffusivities at low

concentrations. For fibers of high aspect ratio, the models should be anisotropic

[6, 38], sl...-- -ii-; that the rigid-dumbbell model is qualitatively incorrect at low

concentrations unless hydrodynamics interactions between the beads is taken into

account. On the other hand, the ratio of Dlo/DLo = 2 represents a theoretical

limit for an infinitely thin fiber [42, 43, 44]. Using models of ellipsoidal particles

demonstrates that the degree of isotropy lies between 1 and 2, and that the aspect

ratio of an ellipsoid must be very high [78] before reaching the limiting value of 2.

For an ellipsoidal particle [79, 80] with A = 50, the ratio of Dllo/DLo equals 1.61.

As with the rotational diffusivities, multiple predictions of the translational

diffusivities have been made and are compared to the results of the simulations.

Szamel [4] developed a reputation model for the perpendicular diffusivities of









infinitely thin fibers, which predicts DI/D\I, ~ 3 (nL3)-2 for number densities

larger than nL = 100, where 3 is a numerical constant. Figure 3-15 shows that

this scaling is accurate for the concentrations higher than nL = 100. Fitting

the simulation data over this same range of concentration gives a nearly identical

scaling exponent of -1.99 for the slender-body model and -1.96 for the rigid-

dumbbell model for rods with A = 25. A scaling of DI/D0ll ~ (nL3)-1 more

accurately captures the diffusivities for mid-range number densities, down to a

number density of nL = 20. Teraoka and Hayakawa [81] proposed a ,liiI-

similar to the one appearing in Equation (3.38) for DR, for the average center of

mass diffusivity,

DT/DTr, [1 + 7-/2 (nL3)]-2 (3.40)

where 7-1/2 is a numerical constant. Experiments have been performed measuring

the average center of mass diffusivities as a function of increasing concentrations

[24, 27, 28, 20]. The tube theory of Doi [7], through the assumption of D- going

to zero, implies that at high concentrations, the average center of mass diffusivity

of the slender-bodies should be reduced by 50'. from the dilute value, while DT for

rigid-dumbbells should be reduced by only : :'. of the value for a dilute solution.

Experiments with tobacco mosaic virus (T\ IV) demonstrate that DT is reduced by
1.11' at high concentrations [20]. Cush and Russo [20] claim that T\ iV is a better

model system for studying the translational diffusivity of rigid rods than solution

of PBLG, which exhibits slight flexibilities near the ends of the macromolecule.

Figure 3-7 shows that DT, as calculated from the Brownian dynamics simulations,

is indeed reduced by ,i I' .

3.5 Conclusions

The functional relationship between rotational diffusivity and number density

depends strongly upon the hydrodynamic model describing the individual fibers,

regardless of the aspect ratio. For example, within the concentration range nL3









1.0


0.8- slender-body model
S ~m rigid-dumbbell model
S(3 -1
S flA (nL)








0.0-
0 50 100 150
nL3
Figure 3-15. Perpendicular diffusivities for fibers with A = 25 in comparison to
the reputation model of Szamel [4]. For mid-range number densities,
the theory of DI/Dlo0 ~ (nL3)-1 follows the data well, while at higher
concentrations, DI/DIIo ~r (nL3)-2 is a more appropriate theory.


between 70 and 150, fitting the dependence of rotational diffusivity on number

density using a power law scaling of DR/DRo ~ (nL3)" gives very different results

for the scaling exponent v depending upon the hydrodynamic model describing the

individual fibers. The Brownian dynamics calculations performed here demonstrate

that modeling the rods using either slender-bodies or rigid-dumbbells results in

v = -1.89 or v = -1.13 respectively for fibers with aspect ratios of A = 50.

For A = 500, which has been argued to closely correspond to the limit A = oo,

v = -1.91 for the slender-body model and v = -1.01 for the rigid-dumbbell model.

The key parameter in the models causing the difference is the ratio of short-time

diffusivities, L2DRo/DTo. The excluded volume of the rod, or rod thickness, also

significantly affects the rotational diffusivities, but the larger contribution is seen

to come from the ratio of the short-time diffusivities, at least for aspect ratios

larger than 25. This qualitative difference in the results has generated significant

confusion over the subject of rotational diffusion in concentrated systems of







84

Brownian fibers. The findings point to the need for improved simulations which

include hydrodynamic interactions, which will account for the alterations in short-

time diffusivities as a function of concentration, as well as improved theories which

accurately account for the dependence of the long-time rotational diffusion upon

the ratio of short-time diffusivities.














CHAPTER 4
DYNAMIC SIMULATIONS OF CONCENTRATED SUSPENSIONS OF
SEMI-RIGID FIBERS: EFFECT OF BENDING ON THE ROTATIONAL
DIFFUSIVITY

4.1 Introduction

It was demonstrated in Chapter 3 that the choice of model used to represent

individual rods within a concentrated suspension of rigid rods with excluded

volume interactions substantially impacts the rotational dynamics. The key

difference is the distribution of the hydrodynamic resistance along the rods which

alters the ratio of the short-time diffusivities, L2DR/DTo, where L is the total

length of the rod, and DRo and DTo are the rotational and average center of

mass diffusivities of a single rod at infinite dilution. The rotational diffusivity

has been theorized to have a power-law scaling in the semi-dilute regime of

DR/DRo ~ (nL3)" [3, 34, 35]. Cobb and Butler [5] showed that v can be as low

as -2 in agreement with the simulations and theories of Doi [3], can equal -1 in

agreement with other work by Fixman [34, 35] and Bitsanis et al. [38, 39], and

ranges from any value between approximately -0.2 and -2 depending on the ratio

of L2DRo/DTo for the model used in the simulation.

Small deviations from perfect rigidity have also been cited as having a sig-

nificant impact on the dynamics of rod suspensions [6, 23, 20]. Experiments on

fibers of poly(7-benzyl-L-glutamate) (PBLG) suspended in m-cresol [71] have been

cited many times as a model system for suspensions of rigid rods [38, 39, 19, 25].

However, recent criticisms have raised questions about whether the results of the

experiments performed by Mori et al. [71] should be compared to the results of

simulations of rigid rods since PBLG may not be completely rigid [27, 28, 20].









Cush and Russo [20] claim that flexibility in the PBLG polymer, especially near

the ends of the fibers, causes a difference in the results in comparison to rigid rods.

The simulations performed here used two simple models to investigate the dual

role of slight bending and choice of model on the dynamics of concentrated systems

of semi-rigid rods. The small deviations from rigidity delay the onset of semi-dilute

behavior in suspensions of rods, while the choice of model results in different

power-law scalings in the semi-dilute to concentrated regime. The methods used

to perform simulations are discussed in section 4.2. The results of the simulations

are then presented in section 4.3, and the discussion and conclusions based on these

results are in sections 4.4 and 4.5.

4.2 Simulation Method

In the simulations performed here, two different models were used for the rods

as shown in Fig. 4-1. The first model is a dimer composed of two slender-bodies

[82, 83], and the second is a trimer composed of three beads [84]. Both models are

composed of two rigid segments connected by constraints, similar to the "once-

broken rod" model [85]. A bending force is applied which hinders the bending

of the rods. For a high bending force, the slender-body dimer behaves as a rigid

slender-body, having a value of L2DR/DTo = 9 and DIo/DIo = 2 [42, 43, 44],

where D0lo is the diffusivity in the direction parallel to the central rod axis at

infinite dilution, and DI0 is the diffusivity in the direction perpendicular to the

central rod axis at infinite dilution. The three-bead trimer behaves as a rigid trimer

when a high bending force is applied, having L2DRo/DTo = 6 and DIo/DIo = 1.

The center of mass position x &) of each rod is calculated by averaging the

center of mass positions of each segment of the rod. The slender-body dimer model

is composed of two segments each with a center of mass, ,1) and x 2). The

three-bead trimer model is composed of three beads with centers at x i1, x ,)

and x$ ,3 The unit vector pi) defines the overall orientation of rod a, which is










Three-Bead Trimer


( ,i) (a,2) (,1) ,2)
1P P

S (u,2)

;"1 a,1) (a,2) d X(,l) X (a,3)
Pi Pi


(a) (a,l) (a,2)
xi = (x. + x. )/2

pi = (U + p )/2
<.>. x < )


Figure 4-1.


(ci) (iU1) (c,2) (ci,3)
(a) = (x(a) + 2)x. + x )/3
pi() = (pi( ) pi(,2)
Pi (Pi +Pi )/2
i(al) (a,2) (a,)) (a,2) (al)
Pi =(xi -xi )/( xi -x I
(a,2) (a,3) (a,2) (a,3) (a,2)
i (xi -xi )/( -x.i


The two models of the semi-rigid rods used in the simulations, where
1 is the length of each individual segment of the rod, d is the diameter
of the slender-body dimer model, a is the diameter of each bead in the
three-bead trimer model. Also included are the equations for calculat-
ing the overall center of mass position x(a) and orientation p(-) of rod a
based on the positions and orientations of each component.


calculated by averaging the two individual segment orientations, p a,1) and p a,2)

For the three-bead trimer model, the orientation of each of the two segments is

calculated by taking the difference between the center of mass positions of the two

beads on the segment (beads 1 and 2 for segment 1 and beads 2 and 3 for segment

2) divided by the magnitude of the distance (1) between them.

4.2.1 Governing Equations

The dynamics of each rod is given by the Langevin equations without inertia,


a) (a) F(a,Br) + (a,Ev) + i(a,Bend) + (a,Const) + (a,Corr)
Sij rj i+ F, i F,


(4.1)


where /) is the hydrodynamic resistance matrix of rod a and r(a) is a vector

containing the center of mass and rotational velocities of rod a. For the slender-

body dimer model the vector r a) is defined as

(a) (a,1) (a,2) .((a,1) .(a,2)]
-' i xi Pi Pi 1 ) (4.2)


Slender-Body Dimer