<%BANNER%>

Polymer-Induced Forces at Interfaces


PAGE 4

IoermyprayersandpranamstotheAlmightyParabrahmanandParashakti;itisentirelyduetoDivineGracethatIhavebeenabletoaccomplishthismeagerwork.Mydeepestrespect,gratitudeandpranamsarealsoduetomyGuruAmmawhoseunerringandeternalguidanceandblessingshaveshapedmeineveryway,shapeandform.IwouldliketooermysincerethankstoDr.AnthonyJ.C.Laddforhiswhole-heartedsupport,guidanceandpatience.TheyearsofmydoctoralworkhavebeensomeofthemostchallengingyearsandIhavegrownasapersoninwaysIneverevenimag-ined.Dr.Laddwasverypatientwithme,wasalwaysavailableforhelpandguidanceinacademicandpersonalmatters,andmotivatedmewhensomepersonalproblemstriedtooverwhelmme.Intimeswherebeinganacademicianhasincreasinglybecomeajobofseekingfundsandmanagingasmallcompanyconsistingofgraduatestudents,whentrueteacher-studentinteractionsarehardtocomeby,Dr.Laddhasshowedmehowonecanstillndtimeandwaystomentorstudentsandeducatetheminissuesconcerningtheirresearchandbeyond.Ishallremembermyinteractionswithhiminshapingmyfutureinteractionswithcolleaguesandstudentsinmycareer.IwouldalsoliketothankDr.RichardDickinsonandDr.DanielPurichfortheirsupportandguidance.Whileworkingwiththem,Ilearnedthevalueofcollaborationandfreeexchangeofideas.Bothofthemwereverysupportiveduringtimesofpersonalcrisis,forwhichtheyhavemysinceregratitude.IwouldliketothankDr.RajRajagopalan,Dr.JorgeJimenez,Dr.JasondeJoannis,Ms.TianyCloud,Ms.PatriciaSocias,Mr.SheltonWrightandMr.MichaelSmithforinteractionsandcollaborationsontheworkonpolymer-inducedforcesexaminedusingmean-eldtheories.Dr.RajalsopaidformyconferencepresentationsandIwouldliketoacknowledgehissupport.MycolleaguesinthelabofDr.Laddmademystayoneofpleasure.Theysup-portedmeinallmyworkandweenjoyeddiscussingeverythingunderthesunand iv

PAGE 5

v

PAGE 6

Page ACKNOWLEDGMENTS ................................ iv LISTOFTABLES .................................... ix LISTOFFIGURES ................................... x ABSTRACT ....................................... xiii CHAPTER 1INTRODUCTION ................................. 1 1.1FlexibleandSemiexiblePolymers ................... 1 1.1.1FreelyJointedChains:AModelofFlexiblePolymers ..... 1 1.1.2FreelyRotatingChains ...................... 3 1.1.3WormlikeChains:AModelofSemiexiblePolymers ...... 4 1.1.4PhysicalInterpretationofPersistenceLength .......... 5 1.2ScopeofthePresentWork ........................ 6 1.3OrganizationofthePresentWork .................... 9 2ADSORPTIONOFFLEXIBLEPOLYMERSATINTERFACES:ALIT-ERATUREREVIEW ............................... 10 2.1Introduction ................................ 10 2.2TheoriesofPolymerAdsorption:Overview ............... 12 2.2.1Density-FunctionalTheories:Cahn-deGennesApproach .... 12 2.2.2Free-Energy-Functional(FEF)/ScalingApproach ........ 15 2.2.3AnalyticalMean-FieldTheories .................. 16 2.2.4NumericalMean-FieldTheory ................... 19 2.3ComputationalStudiesofPolymerAdsorption ............. 19 2.4ExperimentalStudiesofPolymerAdsorption .............. 20 2.5EstimationofPolymer-InducedForces:ValidityoftheTheoreticalandComputationalResults ........................ 21 3POLYMER-INDUCEDFORCESFROMNUMERICALMEAN-FIELDTHE-ORIES ....................................... 25 3.1Introduction ................................ 25 3.1.1BackgroundInformation ...................... 25 3.1.2OrganizationoftheChapter .................... 27 3.2SystemandSurroundings ......................... 28 3.3ThermodynamicsofInhomogeneousSystems .............. 30 3.4FreeEnergyfromLatticeNumericalMean-FieldTheory ........ 32 3.4.1System,Preliminaries,andPartitionFunction ......... 32 3.4.2CongurationalEntropy ...................... 35 3.4.3EquilibriumDistribution ..................... 37 vi

PAGE 7

............................. 40 3.5InhomogeneousContinuumDescriptionoftheFilm .......... 42 3.5.1ContinuumFormulationoftheFundamentalEquation ..... 42 3.5.2EstimationofDisjoiningPressure,FilmTensionandInterfa-cialTension ............................. 44 3.6FilmTension,InterfacialTensionandForceofCompressionfromLat-ticeModels ................................ 46 3.6.1FullEquilibrium .......................... 46 3.6.2RestrictedEquilibrium ....................... 49 3.7ValidityoftheApproach ......................... 50 3.8ResultsandDiscussion .......................... 50 3.8.1FullEquilibrium .......................... 51 3.8.2RestrictedEquilibrium ....................... 52 3.8.3ImplicationsontheMean-FieldPredictions ........... 56 3.9ConcludingRemarks ............................ 58 4EFFECTSOFPOLYMER-LAYERANISOTROPYONTHEINTERAC-TIONBETWEENADSORBEDLAYERS ................... 61 4.1Introduction ................................ 61 4.2Self-ConsistentAnisotropicMean-FieldTheory(SCAFT) ....... 63 4.2.1PreliminariesandNotations .................... 63 4.2.2AnisotropicMeanField ...................... 66 4.2.3StatisticalWeightsandCompositionRule ............ 67 4.2.4StructureoftheAdsorbedLayer ................. 69 4.2.5EstimationofInteractionForces ................. 73 4.3Resultsanddiscussion ........................... 73 4.3.1StructureoftheAdsorbedLayers ................. 74 4.3.2InteractionForces ......................... 76 4.4ConcludingRemarks ............................ 83 5BENDINGOFSEMIFLEXIBLEPOLYMERS ................. 86 5.1Introduction ................................ 86 5.2WormlikeChainsasSlenderElasticRods ................ 86 5.3BendingofSlenderElasticRods ..................... 87 5.3.1SpecialCases ........................... 88 5.3.2BoundaryConditions ....................... 89 5.3.3EquationofPureBendingofaRodofCircularCrossSection 89 5.4Resultsanddiscussion ........................... 90 5.4.1Double-HingedCase ........................ 91 5.4.2Clamped-HingedCase ....................... 92 6F-ACTINASASEMIFLEXIBLEELASTICROD:MOTILITYOFListe-riamonocytogenesPROPELLEDBYACTINFILAMENTS ......... 98 6.1Introduction ................................ 98 6.2TheListeria-ActinSystem ........................ 98 6.3BiophysicalModeloftheMotilityofListeria:Actoclampin ...... 102 6.4ResultsandDiscussion .......................... 105 6.4.1Actin-BasedMotilityofListeria 105 6.4.2LongLength-ScaleRotationofListeria 107 vii

PAGE 8

.................... 112 7.1ProblemsAddressed ............................ 112 7.2Conclusions ................................. 112 7.2.1InteractingPolymerLayers .................... 112 7.2.2BendingofSemiexiblePolymers:F-ActinPropelledMotilityofListeria 113 7.3FutureWork ................................ 114 7.3.1InteractingPolymerLayers .................... 114 7.3.2ModelingofSemiexiblePolymers:F-ActinPropelledMotil-ityofListeria 116 APPENDIX ASTRUCTUREOFTHEADSORBEDPOLYMERLAYERSUSINGSELF-CONSISTENTANISOTROPICMEAN-FIELDTHEORY(SCAFT):AD-DITIONALRESULTS .............................. 119 BSEGMENTDENSITYDISTRIBUTIONS:OVERALL,TAIL,BRIDGE,ANDLOOPDENSITIES ............................. 126 REFERENCES ...................................... 130 BIOGRAPHICALSKETCH ............................... 136 viii

PAGE 9

Table Page B{1SegmentDensityDistributionfor=0:75,H=a=5:0,r=200,s=1:0kT 127 B{2SegmentDensityDistributionfor=0:25,H=a=5:0,r=200,s=1:0kT 128 B{3SegmentDensityDistributionfor=1:25,H=a=6:0,r=200,s=1:0kT 129 ix

PAGE 10

Figure Page 1{1Thefreelyjointedchainmodelofapolymerona2-Dlattice ........ 2 1{2Thewormlikechainmodelofapolymer ................... 4 1{3Twointeractingphysisorbedlinear,exible,homopolymerlayerscon-nedbyat,parallel,adsorbingsurfaces ................... 7 1{4Anillustrationoftheactoclampinmodel:Anensembleofelongatingclampedlamentsundercompressionortensionpropellingthemotilesurface ..................................... 8 3{1TwointeractingadsorbedlayersseparatedbyadistanceH:Denitionofthesystem ................................... 29 3{2Filmtensionandinterfacialtension2asafunctionofsurfacesepara-tion(H/a)underfullequilibriumconditions.Theresultsareshownforanadsorptionenergys=0:5kT,chainlengthr=100,bulkconcentra-tionb=0:05,andgoodsolventconditions,=0:0 ............ 52 3{3Averagedensityofpolymersegments(t/H)intheinterfaceasafunc-tionofsurfaceseparation(H/a)underfullequilibriumconditions.s=0:5kT,r=100,b=0:05,and=0:0 .................... 53 3{4Forceperunitareafasafunctionofsurfaceseparation(H/a)underfullequilibriumconditionsforr=100and=0:0.Thenegativederiva-tiveofexcessgrandcanonicalfreeenergyisalsoshownforcomparison.(a)s=0:5kT,b=0:05;(b)s=1:0kT,b=0:005 ........... 54 3{5InteractionpotentialWbetweenthesurfacesinfullequilibriumwithso-lutionsofvaryingbulkconcentrations.s=0:5kT,r=100,and=0:0 55 3{6Correctcalculationofexcesssemigrandfreeenergyunderrestrictedequi-libriumconditions.Theresultsshownhereareforasurfacecoverage=0:75,s=1:0kT,r=200,and=0:0 .................. 56 3{7Forceperunitareaasafunctionofsurfaceseparation(H/a)underre-strictedequilibriumconditions.=0:5,s=1:0kT,r=200,and=0:0 57 3{8Thevariationofdeviatoricstresspersegmentuponcompression,fortwosurfacecoverages=0:5and=0:75.Theresultsareshownfors=1:0kT,r=200,and=0:0,underrestrictedequilibrium .......... 58 3{9InteractionpotentialWbetweensurfacesinrestrictedequilibriumforcoverages=0:5and=0:75.Theresultsshownherearefors=1:0kT,r=200,and=0:0 ............................ 59 x

PAGE 11

......... 60 4{1Bondorientationsandanisotropicmeaneld ................ 65 4{2Overallsegmentdensitydistribution:Comparisonoflatticemean-eldresults(SF1,SF2,SCAFT)withlatticeMonteCarlosimulations ..... 74 4{3Numberandsizedistributionofbridges,asafunctionofH=a,foracon-stantadsorptionenergy,s=1:0kTand=0:75 ............. 76 4{4Forceperunitareabetweenthesurfacesfasafunctionofsurfacesepa-rationH=a 79 4{5ForceperunitareafataxedseparationofH=a=4:5asafunctionofsurfacecoverage ............................... 80 4{6Numberandsizedistributionofbridges,asafunctionofrescaledsur-facecoverage=0,foraconstantrescaledadsorptionenergy,sc=0:74,andH=a=5:0 .............................. 81 4{7ForceperbridgefbrasafunctionofforaxedwallseparationH=a=4:5 ........................................ 82 5{1Shapesofadouble-hingedrodundercompressionforforcesgreaterthanthecriticalforce ................................ 93 5{2Forceonthedouble-hingedrodasafunctionofendposition,correspond-ingtotheshapesshowninFigure5-1 .................... 94 5{3Possiblelamentendpositionsforalamentclampedattheoriginandorientedalongthez-axisforvariousaxialandcompressiveforces.Notetheexistenceofmultiplesolutionsforeachforce:Fy>0 .......... 96 5{4Possiblelamentendpositionsforalamentclampedattheoriginandorientedalongthez-axisforvariousaxialandcompressiveforces:Fy<0 97 5{5Shapeofalamentclampedattheoriginandorientedatanangle45degreesfromthez-axisundercompression .................. 97 6{1Shapeofalamentundertension.Fz=19:2pN .............. 106 6{2Motilityofthesurfaceduetoasinglelament:SurfacepositionandFil-amentendposition .............................. 107 6{3Motilityofthesurfaceduetoasinglelament:Force ........... 108 6{4Trajectoryofthemotilesurfacepropelledbyvelaments ......... 109 A{1Averagenumberofbridges,nbr,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0 .......... 119 A{2Averagesizeofbridges,lbr,asafunctionofsurfacecoverageforacon-stantadsorptionenergy,s=1:0kTandH=a=5:0 ............ 120 xi

PAGE 12

.................. 120 A{4Averagesizeofloops,llo,asafunctionofH=aforaconstantadsorptionenergy,s=1:0kTand=0:75 ....................... 121 A{5Averagenumberofloops,nlo,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0 .......... 121 A{6Averagesizeofloops,llo,asafunctionofsurfacecoverageforacon-stantadsorptionenergy,s=1:0kTandH=a=5:0 ............ 122 A{7Averagenumberoftails,nta,asafunctionofH=aforaconstantadsorp-tionenergy,s=1:0kTand=0:75 .................... 122 A{8Averagesizeoftails,lta,asafunctionofH=aforaconstantadsorptionenergy,s=1:0kTand=0:75 ....................... 123 A{9Averagenumberoftails,nta,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0 .......... 123 A{10Averagesizeoftails,lta,asafunctionofsurfacecoverageforacon-stantadsorptionenergy,s=1:0kTandH=a=5:0 ............ 124 A{11Averagenumberofloops,nlo,asafunctionofrescaledsurfacecoverage=0,foraconstantrescaledadsorptionenergy,sc=0:74,andH=a=5:0 ................................... 124 A{12Averagesizeofloops,llo,asafunctionofrescaledsurfacecoverage=0,foraconstantrescaledadsorptionenergy,sc=0:74,andH=a=5:0 125 xii

PAGE 13

Thisdissertationconcernsstudiesofforcesgeneratedbyconnedandphysisorbedexiblepolymersusinglatticemean-eldtheories,andthosegeneratedbyconnedandclampedsemiexiblepolymersmodeledasslenderelasticrods. Latticemean-eldtheorieshavebeenusedinunderstandingandpredictingthebehaviorofpolymericinterfacialsystems.Inordertoecientlytailorsuchsystemsforvariousapplicationsofinterest,onehastounderstandtheforcesgeneratedintheinterfaceduetothepolymermolecules.Thepresentworkexaminestheabilitiesandlimitationsoflatticemean-eldtheoriesinpredictingthestructureofphysisorbedpoly-merlayersandtheresultantforces.Withinthelatticemean-eldtheory,adenitionofnormalforceofcompressionasthenegativederivativeofthepartition-function-basedexcessfreeenergywithsurfaceseparationgivesmisleadingresultsbecausethetheorydoesnotexplicitlyaccountforthenormalstressesinvolvedinthesystem.Correctexpressionsfornormalandtangentialforcesareobtainedfromacontinuum-mechanics-basedformulation.PreliminarycomparisonswithlatticeMonteCarlosimulationsshowthatmean-eldtheoriesfailtopredictsignicantattractiveforceswhenthesurfacesareundersaturated,asonewouldexpect.Thecorrectionstotheexcludedvolume(non-reversalchains)andthemean-eld(anisotropiceld)approximationsimprovethepredictionsoflayerstructure,butnottheforces. xiii

PAGE 14

xiv

PAGE 15

Thisdissertationconcernsstudiesofforcesgeneratedbyconnedandphysisorbedexiblepolymersusinglatticemean-eldtheories,andthosegeneratedbyconnedandclampedsemiexiblepolymersmodeledasslenderelasticrods.Inthischapter,werstdeneandoutlinemodelsofexibleandsemiexiblepolymersandidentifythecontributionsofthepresentwork.Finally,wediscusstheorganizationoftherestofthedissertation. 1 ],Yamakawa[ 2 ]). 1

PAGE 16

Figure1{1: Thefreelyjointedchainmodelofthepolymerona2-Dlattice.Thelledcirclesarethesegmentsandthethicklinesarethebonds. thepolymerlyingonthelatticepointsarecalledsegmentsandtherodsconnectingthesegmentsarecalledbonds.Letbbethelengthofeachbond. Figure1-1showsatwo-dimensionallattice.Duetorandomthermalmotionofthesurroundingmedium,thepolymerchainhastheexibilitytotakeondierentcongurations(allonthe2-Dlattice).Thesimplestpossiblemodelthenwouldbewhenthereisnocorrelationbetweenthedirectionsthatdierentbondstakeandthatalldirectionshavethesameprobability.Thisistheso-calledfreely-jointed-chainmodel(alsocalledrandom-coilmodelorrandom-ight-chainmodel).Inthiscase,thecongurationofapolymerwillbethesameasarandomwalkonthelattice. Theend-to-endvectorRjoiningoneendofthepolymertotheotherisameasureofthesizeofthepolymer.IfthepolymerismadeofNbonds,withrnthevectorofthenthbond,wehave TheaveragevalueofR,hRi=0,sincetheprobabilityoftheend-to-endvectorbeingRisthesameasitbeingR.Therefore,onecalculatesR2,andexpressesthe

PAGE 17

sizeofthepolymerbytherootmeansquare(rms)valueofR.Thisisgivenby Sinceforafreelyjointedchainthereisnocorrelationbetweendierentbondvectors,thesecondtermontherighthandsideofEquation( 1-2 )iszero.Therefore,forafreelyjointedchain,theend-to-enddistanceofthechainisgivenby 2Nb23=2exp3R2 ThisprobabilitydistributionfunctionisaGreen'sfunctionwhichsatisesthediusionequationassociatedwiththerandomprocess(position)r(s)ofaBrownianparticlewithL=Nbregardedastime. 1-2 ),onethengets ForlargeN,thisresultapproaches

PAGE 18

Figure1{2: Thewormlikechainmodelofapolymer.ThegureshowsaninstantaneouscongurationofthecontinuouschainoflengthLwithanend-to-endradiusR. Fromtheseresults,itispossibletoderiveanexpressionfortheaverageprojectionoftheend-to-endvectorRontheinitialtangentofthechainu1=r1=b. AsNtendstoinnity,cosNtendstozerofor0<<=2,andthereforethisresultreducesto Thequantityontheleft-handsideofEquation( 1-8 )providesanoperationaldenitionofanimportantpropertyofapolymerchain,namely,thepersistencelength.ItisimportanttonotethatthisdenitionofpersistencelengthisvalidonlyforlargeN.Persistencelength,asdenedabove,measuresthedistancefromorigintillthechain`remembers'theinitialdirectionu1.(OnlyuntilthenwillthedotproductinthesummationofEquation( 1-7 )benon-zero.) Forafreelyjointedchain,therearenorestrictionsin,i.e.,hr1rni=0(ifn6=1).ThereforeitiseasytoseethatinEquation( 1-7 ),onlythersttermofthesummationwillsurvive,andthepersistencelengthissimplyb.Ontheotherhand,forafreelyrotatingchain,thepersistencelengthisalwaysgreaterthanb.

PAGE 19

polymer.FromEquation( 1-8 ),onecanwrite Lp=1L NLp;(1-9) whereL=Nbisthetotallengthofthechain.Thewormlikechain,illustratedinFigure1-2isdenedasalimitingcontinuouschainformedfromthisdiscretechainbylettingN!1,b!0,and!0undertherestrictionthatLremainconstant.Therefore,ifwenotethat NLpN=expL Lp;(1-10) Equation( 1-7 )gives Lp;(1-11) which,forlargeL,reducestoLp.Further,fromEquation( 1-5 ),onecanwrite Lp;(1-12) deningthehR2iforwormlikechainsintermsofthepersistencelengthLp.ForlargeL(LLp),hR2i!2LLp,andforlargeLp(LpL),hR2i!L2.Intheintermediateregime,whereL>Lp,theformula canbeusedwithlessthan5%errorwhenL3Lp. whereu(s)=dr(s)=dsisthetangentvectorats,thearclength.Further,uiuj=cosij,whereijistheanglebetweenthepointsiandjalongthechain.Theextenttowhichthechainisexibleisdeterminedbythecorrelationhuiuji=hcosiji.Ingeneral,hcos(s)idenotesthemeancosineoftheanglebetweenchainsegments

PAGE 20

separatedbythecontourlengths.Thisfunctionpossessesthepropertyofso-calledmultiplicativity:ifthechainhastwoneighboringsectionswithlengthssands0,then Thefunctionhavingthispropertyisanexponential,i.e., Lp(1-16) wherethepre-exponentialfactorisunity,becausecos(s=0)=1,andLpisaconstantforeachgivenpolymer.ThisconstantisthebasiccharacteristicofpolymerexibilityandEquation( 1-16 )istheexactdenitionofpersistencelength. ThephysicalinterpretationofpersistencelengthisreadilydrawnfromEqua-tion( 1-16 ).WhensLp,Equation( 1-16 )giveshcos(s)i1.Hencetheangle(s)uctuatesaroundzero.ThissimplymeansthatchainsegmentsthatarecloserthanLphavenearlythesamedirection.FortheoppositecasesLp,hcos(s)i0.Thismeans,(s)canbeanythingfrom0to360withequalprobability.Sothechaindirectiongetscompletely`forgotten'atlengthsmuchgreaterthanLp. 3 4 5 6 7 8 9 10 11 12 13 ]haveappearedintheliteraturethathaveindicatedtheimportanceofunderstandingpolymer-inducedforcesintailoringpolymerlayersforspecicapplications.Inadditiontosubstantialexperimentalstudiesonpolymeradsorptionatinterfaces,anumberoftheoriesofpolymeradsorptionhavebeendevel-opedoverthepastfourdecades.Amodelcasethathasbeeninvestigatedwidelyistheadsorptionoflinear,exiblepolymersontooneortwouniformatsurfaces[ 14 ].Thestructureoftheadsorbedpolymerlayerisdescribedintermsoftrains,loops,andtails,

PAGE 21

Figure1{3: Twointeractingphysisorbedlinear,exible,homopolymerlayersconnedbyat,parallel,adsorbingsurfacesseparatedbyadistanceH.Theadsorbedchainconformationsaredescribedintermsofloops,trains,tails,andbridges. andadditionallyintermsofbridgesinthecaseoftwointeractinglayers,asillustratedinFigure1-3.Theprimarypurposeoftheoreticalattemptsinthisareaistodeveloppredictiveequationsforthestructureofthelayersandtorelatethestructuretotheforces. Thepresentworkdevelopstheappropriatethermodynamicsofinteractingpolymerlayersandacorrectformulationforestimatingpolymer-inducedforcesinthemodelcasedescribedabove.Whilethemethodofestimatingpolymer-inducedforceshasbeendiscussedinsomepreviousworks[ 15 16 ],therearethreecontributionsfromthepresentwork:

PAGE 22

Figure1{4: Anillustrationoftheactoclampinmodel:Anensembleofelongatingclampedlamentsundercompressionortensionpropellingthemotilesurface.Re-producedfromDickinsonandPurich[ 18 ]. 17 ]. 18 ],treatsthemotilesystemasananity-modulated,processivemotorcomplexwhich,whilesimultaneouslyclampedtotheelongatingendoftheactinlamentattachedtothemotilesurface,facilitateslamentelongationandusestheresultingforcestopropelthemotilesurface(seeFigure1-4foranillustration). ThepresentworkmodelsF-ActinasaslenderelasticrodandestimatestheforcesgeneratedinthepropulsionofListeriabyactoclampinmotors[ 18 19 ].ThestepwisemotilityobservedexperimentallybyKuoandMcGrath[ 20 ]andpredictedusingthe

PAGE 23

actoclampinmechanismbyDickinsonandPurich[ 18 ]isreproduced.Anestimateofthetorquesgeneratedinthesystemismadetoshowhowthelamentelongationandclamptranslocationresultsinlong-lengthscale,right-handedhelicaltrajectoriesofthemotilesurfaceandtheattachedactintails. 2 presentsabriefsum-maryofthetheoreticalandcomputationalstudiesconcerningtheadsorptionofexiblepolymersatinterfaces.InChapter 3 ,wediscussthethermodynamicsofinhomogeneoussystems,developtheformalismtoevaluatetangentialandnormalforcesininteractingphysisorbedpolymerlayers,andestimatetheseforcesusingthelattice,numericalmean-eldtheoryofScheutjensandFleer.Theeectsofincorporatingbondcorrelationsandpolymer-layeranisotropyintothemean-eldonthestructureofthepolymerlayersandtheinteractionforcesareexaminedforgoodsolventsunderrestrictedequilibriumconditionsinChapter 4 .AnanalysisofthebendingofsemiexiblepolymersmodeledaselasticrodsispresentedinChapter 5 ,wheretwocases(arodhingedatbothendsandarodclampedatoneendandhingedattheotherend)areconsidered.ModelingF-Actinasaslenderelasticrodandemployingtheactoclampinmodelofforcegener-ation,Chapter 6 examinesthemotilityofListeriamonocytogenespropelledbyactinlaments.Finally,Chapter 7 summarizesthekeyresultsandidentiesproblemsoffutureinterestthatevolveoutofthepresentwork.

PAGE 24

12 ].Athoroughunderstandingofthestructureofthesephysisorbedpolymerlayersalongwiththeforcesgeneratedbytheselayersisessentialfortailoringpolymerinterfacesforspecicapplications.Experimentalstudiesinthisdisciplinehavefocusedon: Theyhavecontributedsignicantlytotheunderstandingofpolymeradsorption.How-ever,experimentsarestilllimitedintheirabilitytodiscernthenerdetailsofthelayerstructure(e.g.,thearrangementsofthepolymersegmentsonthesurface-asloops,tails,trains,seeFigure1-3),andthecontributionsofthevariousinteractionstowardsthenetpolymer-inducedforce.Asaresult,theneedfortheoreticalguidelinesthatrelatetheconformationsofadsorbedpolymerswiththeresultingforcesisindispensable.Inthiscontext,amodelcasethathasbeenwidelyinvestigatedistheadsorptionoflinear,exible,monodispersehomopolymerspresentinamonomericsolventontooneortwouniformatsurfaces[ 14 ].Inthiscase,thepolymerlayerstructureisdescribedintermsofloops,tailsandtrains(andbridgesincaseoftwosurfacesnotveryfarapart). 10

PAGE 25

Theconformationaldetailsoftheadsorbedchainsarecrucialindeterminingwhethertheinteractionbetweenthesurfacesisattractiveorrepulsive. Theinteractionsbetweenpolymersegmentsandthesolventmoleculescanleadtotheclassicationofthesolventsasgoodsolventsinwhichthesegment-solventinteractionisclosetothesamemagnitudeormorefavoredthanthesegment-segmentandsolvent-solventinteractions(thepolymerchainspreadsout),orpoorsolventsinwhichthesegment-solventinteractionsaresignicantlylessfavoredthanthesegment-segmentandsolvent-solventinteractions(thepolymerchaintendstocollapse).TheseinteractionsarecharacterizedbytheFlory-Hugginsparameter.Forthecase=0,thesegmentsandsolventmoleculeshavenospecicpreferenceoveroneanother.Inthepresentwork,allcalculationsareconsideredforthecase=0. Thepresentworkconsidersforcesarisingduetotheconnementofpolymersbytwoatsurfaces.Dependingonthecharacteristictimeofcompressionofthephysisorbedlayers,onecandistinguishthreepossibilitiesinsuchasystem.

PAGE 26

14 ]basedontheso-calledmagnetanalogy(analogywithhighlyuctuatingmagneticsystemsnearthecriticaltemperature),theresultsofthedensity-functionalapproachhavebeenmodied,givingrisetotheso-calledscaling/free-energy-functional(FEF)theory.Hereweprovidethesalientfeaturesoftheguidelinesonthestructureofadsorbedpolymerlayersandpolymer-inducedforcesprovidedbytheseapproaches. 21 22 ].Inthisapproach,afreeenergyfunctional(FEF)isderivedforpolymerlayersusingCahnandHilliard's[ 23 ]ideastoobtainthefreeenergyofanonuniformsystem. Thefreeenergyisexpressedasasumofasurfaceinteractiontermaccountingfortheattractive/repulsiveinteractionsbetweenthesurfaceandthepolymersegmentsandafunctionalaccountingforthenonuniformconcentrationthatdevelopsbetweenthesurfaceandthebulk.Thetypicalsurfaceinteractionparameterisexpressedasalinearfunctionoftheconcentrationofthesegmentsatthesurface,aboundarycondition,andthebindingenergybetweenthesegmentsandthesurface.Aslongasthebinding

PAGE 27

energyforachainissignicantlyhigherthanthethermalenergybutweakwhenconsideredpersegment(trueforlongchains),thelinearapproximationsuces. Thefunctionaltermcontainsa`local'freeenergycontributiongivenbythevirialexpansionoftheFlory-Hugginsfreeenergyofinteractionbetweenthesegmentsandthesolvent,anda`nonlocal'termaccountingfortheentropicconstraintsinplacingthechainsduetothepresenceofotherchainsandothersegments(excludedvolume)aswellastheenergetictermsaccountingformechanicalandchemicalequilibriumwiththebulk,dependingonthetypeofequilibriumthatexistsbetweentheadsorbedchainsandthebulk.Theentropiccontributionisafunctionofnotonlythesegmentdensitybutalsoofitsgradientsinthesystem,withthedensityanditsgradientstreatedasindependentvariables. Thenumberofvirialcoecients(andtheirvalues)consideredintheFlory-Hugginsvirialexpansiondependsonthequalityofthesolvent.Forinstance,forgoodsolventsitissucienttokeeponlythesecondvirialcoecientv[ 21 ]. 24 ].Forpoorsolventsboththesecondandthirdvirialcoecientsareused(inthiscasevhasanegativevalue)[ 25 26 ]. ThedetailsofthederivationofthefreeenergyarepresentedinFleeretal.[ 12 ],andaresimilaritiesbetweenthisapproachandthemean-eldapproacharediscussedtherein. Thesegmentdensityproleasafunctionofdistancefromthesurfacebyminimiz-ingtheFEFwithrespecttoposition.Thefreeenergyfunctionalisderivedinterms

PAGE 28

ofasingleorderparameter(z),whichisrelatedtothesegmentvolumefractionas(z)=2(z),wherezisthenormaldistancefromthesurface. Beforeproceedingtosummarizesomeusefulresults,thefollowingcommentsareinorder.DeGennes[ 21 22 ]referstotheaboveformulationasamean-eldtheory.Thelabel`meaneld'inthiscontextdiersfromthemean-eldapproximationusedinthenumericalaswellasanalyticalmean-eldtheorieswediscusslater.Thereisclearlyamean-eldassumptioninthevirialexpansionforthe`local'contributionofthefreeenergyfunctional.Nevertheless,fundamentallytheCahn-deGennesformulationisadensity-functionalapproachwiththefunctionalbeingobtainedusingsomemean-eldapproximations.Further,ithasbeshownbyJimenez[ 27 ]thatundermarginalsolventconditions,theCahn-deGennestheoryisequivalenttotheso-calledGroundStateDominanceApproximation(GSDA),asimpliedsolutionofamean-eldformulation(seeSection 2.2.3 )inwhichtheadsorptionisdescribedbytheconformationsofasinglepolymerchaininthepresenceofanexternaleld[ 28 ].ThisformulationleadstoaSchrodinger-likediusionequation,whichcanbesolvedintermsofaneigenfunctionexpansion.Theleadingtermofthissolutioncorrespondstothechainsinthelowestenergystate(the`groundstate'),i.e.theadsorbedchains. ThefollowingaretheimportantpredictionsbytheCahn-deGennesapproach: 2.2.2

PAGE 29

22 ]determinedthattheinteractionforce(andthefreeenergy)forthecaseofrestrictedequilibriumvanishedforallseparationdistancesbetweenthetwosurfaces. Asameanstoexaminingforcesunderrestrictedequilibriumconditions,deGennesemployedtheself-similarityargumentsofscalinginbulkpolymersolutions(basedonthe`magnetanalogy')inmodifyingtheFEF.Thisstudyresultedinasetofworkswhicharepresentedunderthenextsubsection. 22 ]modiedtheFEFusingrenormalizedscalingarguments,whereinthelocalandnonlocaltermsoftheFEFaremodiedbasedonthescalingresultsofthecorrelationlengthandtheosmoticpressureinsemidilutesolutions.Onceagain,thetheoryonlyconsiderschainswithinnitemolecularweight(N!1).Inthecentralregion,thepowerlawhasacoecientof4=3,adecaylesspronouncedthantheonepredictedfrommean-eldarguments(2).Underfullequilibriumconditions,bothscaling/FEFandtheCahn-deGennesapproach,alongwiththeGSDA(discussedinSection 2.2.3 )showthesameattractionbetweenthesurfaces.Underrestrictedequilibriumconditions,arepulsiveforceisobservedatalldistancesforsaturatedsurfaces.Furtherattemptsinthisapproachincludegeneralizationtovarioussolventconditions[ 26 29 30 ]andstudyofinteractionsbetweenundersaturatedlayers[ 31 ]. TheFEF-ScalingapproachhassincebeenextendedbyRossiandPincus[ 31 32 ]tothecaseofundersaturatedpolymerlayers.Theirresultsshowthat,formoderateundersaturations,i.e.,0:5<=0<1:0,theforcesbecomelessrepulsive(ascomparedtothesaturatedcase).Forlargeundersaturations,i.e.,=0<0:4,anattractiveforceoccurs.

PAGE 30

Thenextsubsectionsummarizesthekeyaspectsofanalyticalmean-eldap-proachestoexaminethestructureofthepolymerlayer.ThisapproachisbasedontheEdwardsequation,whichexploitstheanalogybetweentheconformationofapolymerchainandthediusionequation.Anumericalanalogofthesameapproachisthelat-tice,mean-eldtheoryofScheutjensandFleer[ 33 34 ].AnanalysisofthistheoryistheprimaryfocusofthisworkandispresentedinChapters 3 and 4 28 ],G(z;n)satisesthefollowingSchrodinger-likeequation Theinteractionsbetweenthesegmentsandthewallcanbeintroducedintwodierentways.Onepossibilityistoincludeanadditivedeltafunction(atz=0)intheexternaleldU(z).Theotheroptionistoaccountfortheinteractionbyusingan 2-1 ),sometimesreferredtoasthepropagationequa-tion,isdiscussedinChapter 3 .Thenumericalsolutionofthelatticeversionisforchainsofarbitrarylength,whereastheanalyticalsolutionstotheEdwardsequationshowninthisandthefollowingsectionaddressthelimitofinnitechainlength.

PAGE 31

eectiveboundarycondition: wherebisacharacteristiclength,knownastheextrapolationlength,associatedwiththestrengthoftheadsorptionenergy.Ifthelattermethodisused,theexternaleldU(z)containsonlythemean-eldexcluded-volumepotentialactingonasegment.Inthecaseofapolymerlayerundermarginal-solventconditions,U(z)isgivenby wherevisthesecondvirialorexcluded-volumecoecientand(z)thesegmentconcentrationatadistancezfromthesurface. Self-consistencyrequiresU(z)toberelatedtothestatisticalweightG(z;n):Thisrelationisgivenbytheso-calledcompositionrule TheevaluationoftheconstantCdependsonwhetheroneisconsideringthecaseofapolymerlayerinequilibriumwithabulksolution,i.e.,fullequilibrium[ 35 ],orthecaseofrestrictedequilibrium[ 36 ].Ground-StateDominanceApproximation 2-1 )maybeobtainedasaneigenfunctionexpansion.Theleadingtermintheeigenfunctionexpansioncorrespondstochainsinthelowestenergystate,thegroundstate,whichcorrespondstoadsorbedchains.Theapproximationwhereallbuttheleadingtermintheeigenfunctionexpansionareneglectedistheso-calledground-statedominanceapproximation.Thisapproximation,whichinessenceneglectstheeectoftailsintheadsorbedlayer,isequivalenttothefree-energyfunctionalapproachpresentedearlier(Section 2.2.1 ),asshownby 3

PAGE 32

Jimenez[ 27 ].AswiththecaseoftheCahn-deGennesapproach,theforcesbetweentwosaturatedlayersunderfullequilibriumconditionsarealwaysattractiveandthoseunderrestrictedequilibriumconditionsareclosetozero.BeyondGround-StateDominance:Two-Order-ParameterTheory 35 36 37 38 ]considerthetailcontributionsbydistinguishingbetweentheadsorbedandfreechainsindeningstatisticalweightsanddeningseparatesegmentdensitiesforloopsandtails.Theoverallsegmentdensityproleisobtainedsimplyaddingtheloopdensityandthetaildensity.Inordertodistinguishloopandtailcontributions,theydeneasecondorderparameterandanotherpartialdierentialequationforthatorderparameter.Detailsonthederivationsfordierentconditionscanbefoundinreferences[ 35 36 37 38 ].Weshallfocushereonthethreecharacteris-ticlengthsthatresultfromtheaboveanalysis. Theinterval(b;)correspondstothecentralregiondenedbydeGennes(seeSec-tion 2.2.1 ).Withinthisregion,zdelineatestheloop-dominatedregionfromthetail-dominatedportionofthelayer. Herewemakeuseofthesecharacteristiclengthstopresenttheresultsobtainedforadsorptionfromdiluteandsemi-diluteconditions.Theloopdensityproleforthesecasesisgivenby and

PAGE 33

whereasthetaildensityproleisgivenby and Ascanbeseenfromtheaboveequations,theoveralldensityprolearisingfromthetwo-order-parametertheoryleadstothesamescalingrelationobtainedfromtheCahn-deGennesapproach,i.e., Thestrengthoftheapproachsummarizedhereisthatitallowsforthedeterminationofthetail-dominantandloop-dominantregionsandforthedeterminationofasymptoticlawsforthedensityofloops,tailsandfreechains. Thetwo-order-parametertheoryprovidesanalyticalinsightsintothenatureoftail-inducedeectssuchastail-inducedrepulsionbetweenlayers[ 36 ].Thistheorypredictsadecreaseinrepulsionforverysmallundersaturation,andattractionforany(=0)<0:98. 33 34 39 40 41 ].Thepresentworkwillprovideapreliminaryexaminationofthecapabilitiesandlimitationsofthenumericalmean-eldtheoryinprovidingquantitativeguidelinesonthestructureoftheadsorbedlayersandthepolymer-inducedforces.Adetaileddescriptionofthetheoryitselfisprovidedinthenextchapter.Theequivalenceofthistheorywiththeanalyticalself-consistentmean-eldtheoryofSemenovandcoworkersisdiscussedin[ 37 42 ].

PAGE 34

manipulatingthesimulationsystem,theaccuracyofthemean-eldpredictionsandthevalidityoftheapproximationsinthemean-eldtheoriescanbetestedinatargetedfashion.AnecientMonteCarloalgorithmcanprovideallthestructuralfeaturesofthepolymerlayersbetweenthesurfaces.DeJoannis[ 43 ]hastakentherststepinthisdirectionbyacarefulexaminationofthestructureofphysisorbedlayersforawiderangeofadsorptionenergiesandmolecularweights.Jimenez[ 27 ]hastakentherststepincorrelatingthestructureofthepolymerlayerwiththepolymer-inducedforcesusinglatticeMonteCarlosimulationsandtheso-calledcontact-distributionmethod(CDM,[ 44 ])toevaluateforcesbetweennite/semi-inniteobjectsinthepresenceofpolymerchains[ 45 ]. However,asdiscussedinSection 2.5 andChapter 3 ,thereisacaveatastowhetherthischangeinHelmholtzfreeenergycalculatedasaboveisindeedthenormalforce.Evenundergoodsolventconditions,wheresomeconcernsregardingsolventequilibriumarenolongercrucial,theveracityoftheresults[ 27 ]obtainedusingCDMisstillanissue.Therearenoeasywaysoftestingthevalidityofthesecalculationswithinalattice-basedformalismunlessonecancalculatetheosmoticpressuresinthelm.However,comparisonswithMonteCarloresultsofsimilarbutcontinuoussystemswouldshedsomelightinthisregard.ThesecalculationsarebeyondthescopeofthepresentworkbutaremerelysuggestedhereasanecessaryexercisebeforeonecanventuretofurtheruseCDMasamethodofestimatingforcesininterfaciallatticesystems.

PAGE 35

giveinformationonthemicrostructureofthelayer,andthosefocusingonforcemeasurements.Wepresentabriefdiscussionofsomeofthesehere. 46 ],andevanescent-wave-induceduorescence[ 47 ]aresomeofthetechniquesthathavebeenusedtomeasurethelayerthicknessorthetotalamountofadsorbedpolymers.Theseexperimentshaveconrmedthatthethicknessoftheadsorbedlayerisoftheorderofthecoilsizeandhavebeenusedextensivelyinstudiesofadsorptionkineticsorcompetitiveadsorption. 48 49 ].Theresultsobtainedwiththesetechniquesareinqualitativeagreementwiththeoreticalpredictions. 13 50 ].Theanalysisoftheexperimentaldatashowsqualitativeagreementwiththeories,butagain,nodenitecomparisonswiththetheoriesarepossibleduetothelargenumberofuncontrolledfactorsthattypicallyexistinexperiments.Forexample,ithasbeenobservedexperimentallythatpolymerchainsdonotadsorbuniformlyonsurfacesbutform`islands'[ 51 52 ].Therefore,itisdiculttouse`average'measurementssuchastheonesobtainedfromSFAandcomparethoseforceswiththeoreticalpredictions.Atomicforcemicroscopy(AFM)couldbeusedasa`local'probeinthisrespect,butonemustpayattentiontotheinuenceoftheshapeandsizeoftheAFMtipanditsanity(orlackthereof)tothepolymerchainsinordertointerpretthemeasurementsmeaningfully.Moreover,currenttheoriesmostlyignorelateralsurfaceinhomogeneitiesandareinadequateforpredictingtheforceofcompressionbynite-sizedobjects. 53 ],asrequiredofthe

PAGE 36

termsintheintegratedequationofstateofasystem.Otherreasonsforthisargument,specictothedenitionofpartitionfunctionwithinthelatticemean-eldformalism,areconsideredinChapter 3 Ahomogeneousfunctionisafunctionwithmultiplicativescalingbehavior.Iftheargumentismultipliedbysomefactor,theresultismultipliedbythepowerofthisfactor.Letf:V!WbeafunctionbetweentwovectorspacesoveraeldF.Wesaythatfishomogeneousofdegreekiftheequation holdsforallFandvV.Afunctionf(x)=f(x1;x2;:::;xn)thatishomogeneousofdegreekhaspartialderivativesofdegreek1.Furthermore,itsatisesEuler'shomogeneousfunctiontheorem,whichstatesthat Inclassicalthermodynamics,wewritedierentialequationsofstate Inthisequation,theextensivevariablessuchasentropy(S),volume(V)andamountofsubstance(N)arehomogeneousfunctionsofrstdegree,whereastheintensivevariablessuchastemperature(T),pressure(P)andchemicalpotential()areho-mogeneousfunctionsofzerothdegree.Therefore,usingEuler'shomogeneousfunctiontheorem,onecanwritetheintegratedequationofstateas 2-12 )isanalogoustoEquation( 2-11 ),whenk=1.Thisistrueforthefreeenergyasitisanextensivevariable.

PAGE 37

withtheGibbs-Duhemequationbeing Classicalthermodynamicsrequiresthattheappropriatesetofmacroscopicvari-ablestodescribetheenergyofasystem(usinganequationofstate)beasetcompris-ingofatleastoneextensivevariableandtherestbeingeitherextensivevariablesortheircorrespondingintensivevariable(suchasS-T,V-P,andN-).ForphysisorbedpolymersconnedbetweentwosurfacesofareaAeach,separatedbyadistancehandinequilibriumwithabulksolution,onecannotarbitrarilydenetheexcessfreeenergyFexatconstanttemperature,bulkosmoticpressure,andbulkchemicalpotentialsas wheredenotesthetotalsurfacetension,andfdenotestheforce.Thisisbecause,whileAandarehomogeneousfunctionsofdegreeoneandzerorespectively,handfAarenot.Inotherwords,whiledoublingthesurfaceareadoublesthefreeenergywithoutchangingthesurfacetension,freeenergyandsurfaceseparationdonothavesuchalinearrelationship.Thisresultsinthefactthatonecannotintegratea`dierentialequationofstate'suchasEquation( 2-15 ).Further,onecannotwriteaGibbs-DuhemequationforthesystemconsistentwithEquation( 2-15 ).However,whenFexisdenedappropriately,amodiedversionofEquation( 2-15 )isvalid,withcertainrestrictions.ThisisthefundamentalproblemconcerningthethermodynamicsofnonuniformsystemsandisaddressedinChapter 3 Itappearsthatalltheworksreviewedinthischapterthatconcerntheestimationofpolymer-inducednormalforcesofcompression,beitthedensity-functionalapproachormean-eldapproachorthecomputersimulationsofJimenez,maynothaveconsid-eredtheaboveargumentindeningtheirforcesintheinterface.Basedonthepresentstudyandtheworksonthermodynamicsofnonuniformsystems(seediscussioninSection 3.3 ),itappearsthatadenitionofnormalforceofcompressionas

PAGE 38

isfundamentallyincorrectunlessthepartitionfunctionexplicitlyaccountsforthereversiblework-modeinvolvedinthenormalcompressionofthelayers.Intheaboveequation,Asisthesurfaceareaofeachoftheadsorbingsurfaces,Histhesurfaceseparation,andexistheexcessgrandcanonicalfreeenergyderivedfromthepartitionfunction.Moreover,thereareadditionalrestrictionsastotheappropriatechemicalpotentialsthatareconservedinthisequilibrium(seeSection 3.6 forfurtherdiscussion).TheseissuesbringtoquestionthevalidityofthesimulationresultsofJimenez[ 27 ]andthecomputationalmean-eldresultsofScheutjensandFleer[ 34 ],andSemenovandcoworkers[ 36 38 ].Whethertheresultsofthescalingtheories(whichdonotobtainforcesfromthepartitionfunctionofthesystem)arevalidisaquestionthatneedsfurtherexploration. Inthenextchapter,wediscussthethermodynamicsofinteractingpolymerlayersanddevelopexpressionsforevaluatingforcesofcompressionoftwophysisorbedpolymerlayersconnedbyat,parallelsurfacesunderfullandrestrictedequilibriumconditionsusingthelattice,numericalmean-eldtheoryofScheutjensandFleer.Wepresenttherstcorrectresultsofforcesofcompressionusingthemean-eldtheory,andthenexaminethecapabilitiesandlimitationsofthistheoryanditsvariousimprovementsinprovidingquantitativeguidelinesforrelatingthelayerstructuretotheinteractionforces.

PAGE 39

34 ].Weshowthat,withinaone-dimensionalmean-eldapproximation,thenormalforceofcompressionisnotequaltothenegativederivativeofthefreeenergywithrespecttothenormaldistanceHbetweenthetwoadsorbingsurfacesundergoodsolventconditions.ThisisbecausethepartitionfunctionfailstoexplicitlyaccountforthemechanicalworkinmaintainingthedistanceHbetweenthesurfaces.Therefore,inordertoobtainexpressionsfortangentialandnormalstressesintheinterface,weconsideracontinuumanalysisoftheinterfaceandthenadopttheresultsforalatticesystem.Usingtheaboveformalism,weexaminethelatticemean-eldpredictionsofinteractionforces,interfacialtension,andinteractionpotentialsofpolymerlayersingoodsolventsunderfullandrestrictedequilibriumconditionsoverarangeofsurfacecoverages. 54 ]thattheforceactingbetweenthetwoplatescanbeobtainedeitherfromthefreeenergyorfromtheosmoticstressesinthesystem.Employingtheformerapproachinobtainingtheforcesbetweeninteractingpolymerlayers,boththelatticeandthecontinuum-formulationanalyticaltheoriesderiveexpressionsforexcessfreeenergyofthesolutionbetweenthesurfaces,whichisthedierencebetweenthefreeenergiesofthesysteminthepresenceandabsenceofthesurfaces.Notethatintheabsenceofthesurfaces,thesystemishomogeneouswith 25

PAGE 40

thesamecompositionasthatofthebulksolutionwithwhichitisin(full)equilibrium.Therefore,theexcessgrandcanonicalfreeenergyisdenedas ex(H)=(H)b:(3-1) Thefreeenergyofinteractionbetweenthelayersisdenedasthechangeinfreeenergywhenthesurfacesaremovedreversiblyfroman`innite'separationdistancetoagivenseparationdistanceH, int=ex(H)ex(1):(3-2) Theforcebetweenthetwosurfacesisthendenedasthederivativeoftheinteractionfreeenergywithrespecttoseparationdistance, Theaboverelationimpliesthattheworkperformedforthecompressionofthepolymerlayers,namely,dW=fdH,iscontainedinthethermodynamicpotentialderivedusingtheappropriatestatisticalmechanicalformulationfortheconformationsofthechainsintheinterface,andthattheappropriatechemicalpotentialsofthepolymerchainsandthesolventmoleculesareinvariantuponcompression(seeSection 3.6 forfurtherdiscussion).However,thisanalysisbreaksdowninthecontextoflatticeandanalyticalmean-eldtheoriesbecauseoftheinabilityofthepartitionfunctiontoexplicitlybuildintheworkofcompressionarisingfromtheattractiveorrepulsiveforcesthatmightexistbetweentheconningsurfaces.Inparticular,thefreeenergyformulatedusingthelatticemean-eldtheoriesonlyaccountsforthetangentialstressesintheinterfacerelativetothebulkstress,asshowninthischapter. Inalatticemean-elddescriptionofpolymerlayers(fordetails,seeSection 3.4 andScheutjensandFleer[ 33 34 ]),onewritesthegrandcanonicalpartitionfunctionandobtainstheequilibriumconformationofthepolymerchainsandsolventmoleculesbyminimizingthecorrespondinggrandcanonicalpotential.Theequilibriumconformationisthenrelatedtothedensitydistributioninthesystem.Thisdensitydistributiondependsonthecompositionofthebulksolutionwithwhichthepolymerlayersareinequilibrium.Fromthedensitydistribution,onecanobtaintheexcessfreeenergyofthe

PAGE 41

layersrelativetothatofthebulksolution,atagivenseparationdistanceandareaofthesurfaces.InSections 3.4 and 3.6 ,weshowthatthisexcessfreeenergyisequaltotheso-calledlmtension,whichistheenergyrequiredtocreateathinpolymerlm(consistingoftwointeractingpolymerlayers)ofagivenarea2AsandmaintainitatathicknessH. Inordertoobtainthecorrectestimateoftheinteractionforcesbetweenthelayers,weturntoacontinuumanalysisofthepolymerlayersalongthelinesofEvans[ 15 ]andPloehn[ 16 55 ].Theforceofcompression,thelmtensionandtheinterfacialtensionofthelayersarethenobtainedbyrelatingtheresultingnormalandtangentialstresses(respectively)tothethermodynamicpotentialgivenbytheworkfunctionbasedoncontinuummechanics.Thepolymer-inducedforceofcompressionperunitareaisshowntobeequaltothedisjoiningpressureasdenedinthetheoryofthinliquidlm(seeforinstance,Babak[ 56 ]),i.e.,theVolterraderivativeoftheexcessgrandcanonicalpotentialofthepolymerlmwithrespecttothesurfaceseparation.Thedisjoiningpressureisthedierencebetweentheosmoticpressureinthemidpointandthatinthebulk,whenthepolymerlayersareinfullequilibriumwiththebulksolution,analogoustotheresultinelectricaldoublelayers[ 54 ].Thecaseofthepolymerlayersbeinginrestrictedequilibrium(totalamountofpolymerbetweenthesurfacesisxed)isalsodiscussed.TheformulationpresentedhereisanalogoustotheonespresentedbyEvans[ 15 ]andPloehn[ 16 55 ],butissimplerandmoredirect.Moreover,whereasEvans[ 15 ]andPloehn[ 16 ]haveexaminedthepredictionsofanalyticalmean-eldtheoriesbasedonEdwards'equation,therehasbeennoanalysisofthecorrectpredictionsofforceofinteractionbythelattice,numericalmean-eldtheoryofScheutjensandFleer[ 34 ],tothebestofourknowledge.Thisformsthefocusoftherestofthestudy.Wepresenttherstcorrectresultsofpolymer-inducedforcesbetweentwoplanarsurfacesusinglatticenumericalmean-eldtheory. 3.2 ,thesys-temunderconsiderationisdened.Section 3.3 discussesthegeneralthermodynamics

PAGE 42

ofinhomogeneoussystems,anddenestheappropriatepartitionfunctions,thermo-dynamicpotentials,andthecriterionforequilibrium.Usingthelatticemean-eldtheoryofScheutjensandFleer[ 34 ],theexpressionforgrandcanonicalpotentialandsemi-grandpotentialarederivedunderfullequilibriumandrestrictedequilibrium,respectively,inSection 3.4 .Expressionsfortheforceofcompressionandinterfacialten-sionareobtainedinSection 3.5 ,usingtheprinciplesofcontinuummechanics.Followingthis,inSection 3.6 ,wedeveloptheexpressionsfornormalandtangentialstressesforthelatticemodelandshowthattheexcessgrandcanonicalpotentialderivedinSection 3.4 isindeedequaltothelmtension.InSection 3.8 ,weillustratethepredictionsofthelatticemean-eldtheoryforafewsamplecases.Thediscrepanciesobservedbe-tweentheresultsbasedonthecorrectformulationderivedinthisworkandsomeofthepreviousresultspublishedintherecentliterature,basedontheincorrectformulation,(Jimenezetal.[ 57 ])arediscussed.

PAGE 43

TwointeractingadsorbedlayersseparatedbyadistanceH.NotethatthedividingsurfacesarenotGibbs'surfaces.Here,thesuperscriptorphasebdenotesbulkconditions,thesuperscriptintdenotestheinterface,thesubscriptmdenotestheconditionsatthemidpointoftheinterface(z=H=2ina1-Dsystem),Preferstopres-sure,denotestheosmoticpressure,existheexchangechemicalpotential(denedinSection 3.6 ),andisthelmtension. solidsurfacesandthepolymersolution,andformaintainingthenormaldistanceofseparationHbetweenthetwosurfaces.Theformerworkisusuallyreportedintermsoftheinterfacialtension,whilethelatterworkistypicallyrepresentedintermsofthedisjoiningpressureofthethinlmdorthenormalforceofcompressionofthepolymerlayersdAswhereInmodelingthelm,thedividingsurfacesaredenedtobeatthez-planesofcontactbetweenthesurfacesandthesolution.ThereforethedividingsurfacesarenotGibbs'surfacesandthesurfaceseparationHisanindependentvariable. Withthispicture,onecanunderstandwhythedisjoiningpressurecannotbewrittenasthenegativederivativeoftheexcessgrandcanonicalpotentialderivedfromapartitionfunction.Duetothepresenceoftheparallelconningsurfaces,theensembleaveragepropertiesinthelmareinhomogeneousonlyinonedimension.(Thisisdenotedasthez-direction.)Mechanicalstabilitydemandsthatthenormalstressbeuniformeverywhereinthelmwhilethetangentialstressisuniformineach

PAGE 44

3.4 and 3.6 ,theexcessgrandcanonicalpotentialinsuchaformulationisindeedequaltothelmtension.Whilethetworeversiblemechanicalworkmodesarepresentinthelmtension,thenormalstressinthelmisnotexplicitlypresent.Thisisthereasonthatonecannotdenethenormalforcetobethenegativederivativeofapartition-function-basedthermodynamicpotentialwithrespecttotheseparationdistance. 58 ] Thegrandcanonicalpartitionfunctionofageneralinhomogeneoussystemwithintheassumptionoflocalequilibriumisgivenby [V;;] ref=XNZdNexp[ZVdr(r)^"(r)+cX=1(r)^n(r)];(3-4) wheredenotesthetermstoensureindistinguishabilityofidenticalsystemcongura-tions,Ndenotesthesetofallpossibleparticlenumbers,Ndenotesthephasespace

PAGE 45

variables,^"(r)and^n(r)denotetheenergydensityeld,andthenumberdensitiesofcomponents(=1;2;:::;c),and(r)and(r)denotethecorrespondingLagrangemultiplierelds.ThepartitionfunctionisafunctionaloftheseLagrangemultiplierelds. Itfollowsfromthedenitionofthegrandcanonicalpartitionfunctionthat ref and ref Here"(r)=h^"(r)iandn(r)=h^n(r)aretheensembleaveragesoftheenergydensityeldandthenumberdensitiesrespectively.Thegrandcanonicalpartitionfunctionleadstothedenitionoftheappropriategrandcanonicalpotentialofthesystem kT=ln[V;;] ref=ZVdr(r):(3-7) Here(r)isthe`densityofln[V;;] refandisathermodynamiceld. Ifoneweretowritethefundamentalequationoftheentropyofthisinhomogeneoussystem,onewouldhave where Thevariationsofentropyanditsdensitywouldturnouttobe where withthecorrespondingGibbs-Duhemequationas

PAGE 46

wheretheintegrandisnotequaltozeroateachpointinr.Thisisbecausethethermo-dynamiceld(r)isanonlocalfunctionaloftheLagrangemultiplierelds(r)and(r). Inthecontextofinteractionbetweenpolymerlayers,itisthisnonlocaldependenceofthe`grandcanonicalpotentialdensity'thataccountsfortheworkofcompressionarisingfromtheattractiveorrepulsiveforcesbetweenthetwosurfaces,aswellastheworkrequiredinmaintainingtheareaofthesurfacesexposedtothesolution.However,intheproblemofinteractingpolymerlayersconnedbytwoparallelsolidsurfaces,theintegralinEquation( 3-12 )one-dimensional,i.e.,inz-directionalone.WeshallusethisinderivingthecriterionforthermodynamicequilibriuminthelminSection 3.6.1 3.4.1System,Preliminaries,andPartitionFunction

PAGE 47

surfaces),thenitfollowsthat, Wedenethevolumefractionsofpolymerandsolventatanygivenlayeras, suchthat, and, Whenthesolutioninthelmisopentoabulksolutionwithrespecttonumbers,thenumbersofpolymerchainsandsolventmoleculesbetweenthesurfacevary.Theappropriategrandcanonicalpartitionfunctionisgivenbythesumoftheappropri-atelyweightedcanonicalpartitionfunctions(thecanonicalpartitionfunctionQbeingdenedforaxednumberofsolventmoleculesandaxednumberofpolymerchainsinadenedsetofconformationscorrespondingtoagivenenergyofthesystem)fordierentvaluesofpossiblenumberofpolymerchains(andhencedierentnumbersofsolventmolecules). =8><>:Palln0s(fncg;V;A;T)expUsurf where,fncgisasetofpermissibleconformationsofthepolymerchainsinthelmsuchthat,

PAGE 48

Theinteractionenergiesaregivenby and TheequilibriumdistributionofthepolymersegmentsinthelatticeisdeterminedbymaximizingthegrandcanonicalpartitionfunctioninEquation( 3-17 ).Adetailedderivationoftheinteractionenergies,congurationalentropyandtheequilibriumdistributionaregivenintheoriginalScheutjens-Fleerpapersxxandisnotrepeatedhere.Instead,wemerelypresenttheimportantequationsinthetheorythatpertaintotheestimationofthegrandcanonicalpotential. Itistobenotedthatthechainstatisticsisdenedintermsofconcentrationsofchainconformationsandnotintermsofconcentrationsofindividualsegments.Achainisthentreatedasconnectedsegments.Achainconformationischaracterizedbydeningthelayernumbersinwhicheachofthesuccessivesegmentsarepresent.Aconformationcanthenbedenotedas indicatingthattherstsegmentisinlayeri,thesecondsegmentisinlayerjandsoon.Thisimpliesthatmanydierentactualarrangementsofthesegments(inthelm)cancorrespondtoadenedconformation.Ifasegmentsisplacedinlayerz=iandsegments+1isplacedinlayerz=j,thenthenumberofdierentallowedplacementsofs+1relativetosisD0ifj=i,andD1ifj=i1.Itthenfollowsthatadimerwithconformation(1;i)(2;j)canassumeLDjidierentpositions.Atrimerwiththeconformation(1;i)(2;j)(3;k)canassumeLD2jikjpositions,ifbackfoldingofthechainisallowed.Apartialcorrectiontothebackfoldingofsegmentswillbeappliedlateron.Usingsimilarlogic,thenumberofwaysofarrangingrsegmentsofachaininconformationcgiveninequation( 3-21 ),inanemptylatticeisLDr1jikj:::ml,

PAGE 49

whichcanbewrittenasL!cDr1,where where, Further,ifoneconsidersonlythenumberofarrangementsofapartofthepolymerchain,then,thenotation!c(s;t)isused,wherethepartofthechainconsideredisbetweensegmentssandt,includingbothofthem.Similarly,thesummationPc(s;t)wouldaccountonlyforallthepossibleconformationsofthatpartofthechain.Itisevidentthat!c=!c(1;r),andthatPc()=Pc(1;r)().Thedenitionofstatisticsofthechainintermsofconformationsandintermsofindividualsegmentsareinterchangeableandarerelatedbythefollowingrelation: where,rz;cisthenumberofsegmentsofachaininconformationcinlayerz.Thenota-tionusedtoindicatethelayernumbercorrespondingtothesegmentsofconformationcisthesubscriptk(s;c),kbeingthelayernumber. Finally,wedenethefollowingreferencesystem:

PAGE 50

ofthesystem.Incalculatingthedegeneracy,theBragg-Williamsapproximationofrandommixingwithineachlayerisused,i.e.,thepolymersegmentsineachlayerareconsideredtoberandomlydistributedovertheLlatticesitesinthatlayer.AsdiscussedinSection 3.4.1 ,thenumberofwaysofplacingachaininaconformationcinanemptylatticeisL!cDr1.Ifthelatticeispartiallyoccupied,thenachaincanonlybeplacediftheappropriatelayers(asdenedintheconformation,e.g.,equation( 3-21 ))havevacantsites.Therefore,wehavetoapplyrcorrectionfactorstothecombinatoricsinordertopartiallyaccountfortheexcludedvolume,onefactorforeachsegmentofthechain.Thiscorrectionfactor,inthesimplestcase,istheprobabilitythatagivenlayerhasatleastonefreesite(usuallycalledthevacancyprobability).Ifthenumberofoccupiedsitesinlayerzatanygiveninstantis(z),thenthevacancyprobabilityisgivenby(1(z)=L). Therefore,thenumberofpossibilitiesofplacingachaininconformationcisgivenbyL!cDr1Qrs=11k(s;c)=L=!c(D=L)r1Qrs=1(Lk(s;c)),wherek(s;c)isthenumberofpreviouslyoccupiedsegmentsinthelayerkwherethesegmentsofthechaininconformationcisplaced. Therefore,thenumberofarrangements!ofplacingtherstchainofconformationc(ofthencchains)inanemptylatticeisgivenby Placingncchainsinconformationcwouldgiverisetothefactor!ncc(D=L)(r1)nc,whilethemultiplicationextendsto(z)=ncrz;c1.Placingalln=Pcncchainswouldleadtothenumberofarrangements NowtheremainingLn(z)=n0(z)sitesineachoftheMlayershavetobelledwithn0solventmolecules.Thenumberofpossibilitiesofarrangingn0(z)numberofsolventmoleculesinlayerzisgivensimplybyQL1(z)=n(z)(L(z)).Uponsimplication,

PAGE 51

theexpressionforcongurationalentropyisfoundtobe =(L!)M(D=L)(r1)nYc!ncc Thefactorialsnc!andn0(z)!correctforindistinguishabilityofthencchainsineachconformationcandofthesolventmoleculesineachlayerz. Thecongurationalentropyofthereferencesystem+canbederivedusingsimilararguments.Inthebulk,thedistinctionbetweenlayersisirrelevant.Sincethenumberof(equivalent)latticesitesinthebulkpolymerisnr,thefactor(L!)Minequation( 3-27 )isreplacedby(nr)!,andthefactorL(r1)nby(nr)(r1)n.Further,inthebulk,allconformationsareequallyprobable.Hence,nc=n;and!c=1.Alsosincetherearenosolventmolecules,n0(z)=0.Therefore,equation( 3-27 )reducesto +=(nr!) ThisexpressionisalsoderivedbyFlory[ 1 ].Therefore,thisformulationisconsistentwithearliertheories. 3-17 )).Atequilibrium,theappropriatefreeenergyofthesystemtakesitsminimumvalue,i.e.,thepartitionfunctionisatitsmaximumvalue.Thissituationcorrespondstothemostprobablesetofconformations(withacorrespondingnumberofchainsneq:andnumberofsolventmoleculesn0eq:),whichwillbeobtainedinthissection.Ifweneglectuctuationsoncethesystemhasattainedequilibrium,thenthenumberofchainsandsolventmoleculesarexed.Thismeansthatthesumoveralln'sinequation( 3-17 )isreplacedbythemaximumterm.Toobtaintheequilibriumdistribution,i.e.,thenumberofchainsndinconformationdintheequilibriumsituation,thetermswithinthesuminequation( 3-17 )aredierentiatedwithrespecttondandsettozero.RealizingthatPzn0(z)=MLrPcnc,wehave @ndV;A;T;fnc6=ndg+0=kT=0:(3-29)

PAGE 52

Thisdierentiationcorrespondstoaddingonechaininconformationdfrombulkandplacingrz;dsegmentsineachlayerz;andremovingrsolventmoleculesfromtheappropriatelayersinordertomaintainconstantvolume(i.e.,allsitesareoccupied).Thisconservationoftotalvolumecanbeexpressedas Thismaximizationissubjecttoanadditionalconstraintthateachlayerinthelatticehastobecompletelylledwitheitherpolymersegmentsorsolventmolecules.Thisconstraintisimplicitlyincluded,becauseinperformingthedierentiation,n0(z)=LPcrz;cnc=Ln(z)isused.However,formulticomponentsystems,thisconstraintshouldbeexplicitlyincludedwithanappropriateLagrangemultiplier. Performingsimplealgebraicmanipulationsonequations( 3-27 )and( 3-28 ),andusingStirling'sapproximationofthelogarithmofafactorial,oneobtains ln +=MLlnLXcnclnnc Upondierentiationandsimplication,theequilibriumsetofconformationsarethenobtainedas lnnd Dening lnC=r1+lnr+(0) wewrite, lnnd where, lnGz=s(1;z+M;z)+h(z)i0(z)+ln0(z):(3-35)

PAGE 53

Equation( 3-35 )isadiscretizedversionoftheEdwardsequation,discussedinChapter 2 Thenumberofchainsinaparticularequilibrium-conformationdisproportionalto!d,aproductof(r1)step-weights,i.e.,'s,whichdeterminetheconformationasisevidentfromEquation( 3-21 ).Further,ndisalsoproportionaltorweightingfactors(Gz's),asshowninEquation( 3-34 ).EachsegmentinthechaincontributesaweightingfactorGz,whichisaBoltzmann-typefactoraccountingforthechangeinfreeenergywhenasolventmoleculeinlayerzisreplacedbyasegment.ItconsistsofacontributionfortheexchangeadsorptionenergyskT,whenthelayerzisadjacenttoasurface(z=1orz=M);afactorfortheexchangeinteractionenergybetweensegmentsandsolventmolecules(h(z)ih0(z)i),forreplacingasolventmoleculebyasegment;andafactorforthelocalentropykln0(z)ofthesolventmolecule.Forthesimplestcaser=1(monomer),wehave,fromEquation( 3-34 ),(z)=n(z)=L=KGzexp(0)=kT,whereKisaconstant.SincethevolumefractionvarieslinearlywiththeweightingfactorGzforamonomer,Gzisalsocalledmonomerweightingfactor.AnotherwayofviewingGzisasfollows:Themean-eldexperiencedbyasegmentinthepresenceofpolymerchainsandsolventmoleculesisgivenby where, suchthat, Here,u0(z)isthehard-coreorexcludedvolumeinteractioninthelayer.

PAGE 54

TheunnormalizedprobabilityofndingthesthpolymersegmentofachaininthelayerzisgivenbythefunctionG(z;s) whereisthefractionofnearestneighborsitesinthelayerz0ofthelattice.ThisequationisanalogoustothepropagationequationdiscussedinSection 2.2.3 inChapter 2 .Itstatesthatthesthsegmentofachaincanbeinagivensiteinlayerzifandonlyifthe(s1)thsegmentcanbeinoneofitsnearestneighbors.Inenforcingthisconstraintofconnectivity,amean-eldassumptionthatdoesnotdistinguishthesiteswithineachlayerisused.Inotherwords,connectivityisenforcedlayer-by-layeronly.Thisconstraintpartiallyaccountsforconnectivityandexcludedvolume. Further,analogoustothecompositionrulediscussedinSection 2.2.3 inChapter 2 ,thesegmentvolumefraction(z)isgivenbythecompositionrule G(z)XsG(z;s)G(z;rs+1);(3-42) whereCisanormalizationconstant.Forapolymerlayerinequilibriumwithabulksolution,C=b=N,andforrestrictedequilibrium,C==G(N),whereG(N)istheunnormalizedprobabilityofndingtheendsegmentofthechainanywhereinthesystem. ref:(3-43) Thelogarithmofthegrandcanonicalpartitionfunctionisgivenby ref=ln +Usurf ThebulkchemicalpotentialsareestimatedusingFlory-Hugginstheory kT=b1

PAGE 55

and Substitutingthevarioustermsin( 3-44 ),oneobtainstheexcessgrandcanonicalfreeenergytobe kT=LXz(11 whichisthesameas kT=LXz(z)b;(3-48) whereistheosmoticpressure. Theaboverelationoftheexcessgrandcanonicalfreeenergyisanalogoustothewell-knownKirkwood-Buequation[ 16 ].AsmentionedinSection 3.2 ,byconsideringtheexcessosmoticpressurewithreferencetothebulk(reservoir)osmoticpressure,thispotentialaccountsfortheworkdoneincreatingthethinlm(interfacialtension,2L)andmaintainingitatthegiventhicknessM(disjoiningpressure-dAsM).Inotherwords,thispotentialisindeedthelmtension.ThisisconsistentwiththeargumentpresentedattheendofSection 3.3 Whenthesystemisinrestrictedequilibrium,thelmisopenwithrespecttosolventbutnotwithrespecttopolymer.Inthiscase,thesemi-grandpotentialissimply kT+Xcnc:(3-49) Thereforewegettheexpressionforsemi-grandpotentialas wherethetotalamountofpolymerinthegapt=Pz(z);thesolventdensitydistribution0(z)=1(z),andG(r)istheunnormalizedprobabilityofndingachainofrmonomeric(persistence)unitsinthegapregion.WeshallfurtherdiscussthispotentialanditsinterpretationinSection 3.8

PAGE 56

3.2 and 3.3 ,oneconcludesthattherearetworeversiblemechanicalwork-modesinvolvedintheproblemofinteractingpolymerlayers,namelytheinterfacialtensionandthedisjoiningpressured.However,toobtainusefulpredictiveequationsforthedisjoiningpressure,onehastoexaminethestressesinthesystem.Forthispurpose,weemployaquasi-thermodynamicframeworkbasedontheprinciplesofcontinuummechanics.Withinthisframework,ineachmicroscopicvolume,thesystemisconsideredasaninhomogeneouscontinuousmediumwherethefundamentalthermodynamicrelationspresentedinSection 3.3 arevalid.Thishypothesisholdswhenthecorrelationlengthinthesystemisgreaterthantherangeofintermolecularforces. 59 ]forthepolymerlayersandthereservoirofhomogeneousbulksolutionthelayersareinequilibriumwith. dt=(r:q)+tr(T:rv)r:Xim;biji!;(3-51) whered() Notethatheretheosmoticstressesaredenedwithreferencetothereservoirbulkpressureratherthantheas-yet-undeterminednormalstressinthelm.

PAGE 57

dt+1 where^Sistheentropyperunitmass.SubtractingEquation( 3-53 )from( 3-51 )givesusthetotalHelmholtzfreeenergybalance. dt=Pb(r:v)r:Ximiji!+tr(:rv):(3-54) Uponfurthersimplications,conversionintomolarbasisandintegrationoverthevolumeofthesystem,equation( 3-54 )becomes dt=PbdV dt+Xibidni Forhomogeneoussystems,theosmoticstresstensorvanishesuniformly,i.e.,=0.ThereforeEquation( 3-55 )reducesto whichisthewell-knownfundamentalequationatconstanttemperature. Itisimportanttonotethattheabovefundamentalequationofthesystem,Equation( 3-55 ),isforboththelmaswellasforthereservoir.Thelasttermaccountsforthenetmechanicalworkperformedinmaintainingthearea,2As,ofcontactbetweenthesurfacesandthesolutionaswellasmaintainingthetwosurfacesatagivendistanceH.Thistermcanbebrokendownintotwocontributions-onefromthelmandtheotherfromthereservoir-arisingfromtheexchangeofpolymersandsolventmoleculesbetweentheinterfaceandthereservoiruponequilibration.Thereforeitisevidentthatthevariationoftheexcessgrandcanonicalpotentialofthelmatconstanttemperatureisgivenby whereVIdenotesthevolumeofthelm.Thisvariationcorrespondstothemechanicalworkcontributionduetoaxisymmetriccompressionorexpansionofthelayersat

PAGE 58

constanttemperature.Equation( 3-57 )isanalogousthevariationofthegrandcanonicalpotentialdenedinEquation( 3-7 )inSection 3.3 Purecompressionisirrotational,sorvissimplytherateofdeformationtensorDwithcomponents wherethedilationsiarethestretchedlengthsoflinesegmentsintheprincipaldirectionsi=x;y;zwithinitiallengthsofunity(intheundeformedstate). wherekand?arethetangentialandnormalstressesrespectively.(Theunitvectorineachdirectionisdenotedbye.)SubstitutionoftheaboveequationinEquation( 3-57 )thenleadsto ItfollowsfromEquation( 3-60 )that whichdenestheexcesstangentialstress(withrespecttothereservoirbulkpressure)asthelmtension.Inevaluatingtheabovederivativeatconstanttemperature,itistobenotedthatthecorrespondingGibbs-DuhemequationgivenbyEquation( 3-12 )mustalsobesatised.Itisnotnecessarythatthechemicalpotentialsofeachspecies 3-57 )isperformedbymappingthevolumeVItoaninvariantreferencevolumeVI;RusingthetransformationVI=lVVI;R,wherelVlxlylz.TheabovetransformationalsoenablesonetoevaluatethevariationsoffunctionalsaccordingtotheequalityRVI;R(lVf)dVI;R=RVI;RflVdVI;R=RVIfdV.

PAGE 59

beconstantinthelm.Weshalldiscussthenecessarycriteriaforequilibriuminthenextsectionandmakefurtherobservationsinthisregard. Moreover,fromEquation( 3-60 )onealsohas whichshowsthatthedisjoiningpressureissimplythenegativeofthecenter-linenormalosmoticstress?;m.Hereandelsewherewedenotethemidpoint(H=2)betweenthetwosolidsurfacesbythesubscriptm,i.e.,Hm=H=2. Theaboveexpressionsforanddcanalsobeexpressedintermsoftheisotropicanddeviatoriccomponentsofthetensor.Sincetheosmoticstresstensorcanbewrittenintermsoftheisotropicanddeviatoricstressesinthesystemas 2exex1 2eyey+ezez(3-63) withtheisotropiccomponentgivenby 3tr=2 3k+1 3?(3-64) andthedeviatoriccomponentgivenby 3(?k);(3-65) onehas,forthelmtension, and,forthedisjoiningpressure, Theinterfacialtensioncanthenbeobtainedfromthedenitionoflmtension,=2dH.

PAGE 60

3-66 )and( 3-67 )forthecaseoflatticemean-eldapproximation.Thepurposeofthissectionisdeveloptheexpressionsnecessarytoobtaintheosmoticstresstensor(andhencetheisotropicanddeviatoricstresses)withinthelatticeformalismsothattheforceofinteraction,thelmtensionandtheinterfacialtensioncanbeobtainedfromlatticemodels.Althoughtheresultspresentedherecanbegeneralizedtoanyvariationsofthelatticetheory,weshallprimarilyconsidertheminthecontextoftheScheutjens-Fleertheoryasithasbeenusedextensivelyintheliterature. Herethelasttermisconsistentwiththefactthattheinhomogeneityofthermodynamicpropertiesinthelmisone-dimensional(seeearlierdiscussionsonthethermodynamicsofinhomogeneoussystems).P(z)=Pb+b(z),consistentwiththedenitionofthestresstensorinEquation( 3-52 ),i(z)arethechemicalpotentialsofspeciesievaluatedforahomogeneoussolutionatlocalcompositioni(z),andi(z)areposition-dependentnonlocaleldswhichactonindividualcomponents.Theeldsiarechosensuchthati(z)isanequilibriumdistributionconsistentwiththecongurationalconstraintsimposedbytheconnectivityofpolymerchains.Thetotalamountofeachspeciesinthesystemisgivenby

PAGE 61

ThecriterionforequilibriumisdeterminedbyminimizingthetotalHelmholtzenergyofthelmandthereservoiratconstantT,ni,As,andH.Thereforewehave sincethetermsVbPbandnbibiarezero.NotethatthersttwotermsinEqua-tion( 3-70 )arenon-zerobecausetheyaccountforthechangeintheHelmholtzenergyinthereservoirduetotheexchangeofmaterialandenergywiththelm.Sincethetotalamountofeachspeciesinthesystemisconstant,wehave UsingtheaboveresultandusingpartialmolarvolumesithersttwotermsofEquation( 3-70 )canberewrittenas ThereforeEquation( 3-70 )becomes WeidentifytheleadingtermontherighthandsideofEquation( 3-73 )tobetheGibbs-Duhemequationforthelm,analogoustoEquation( 3-12 )andhencethistermvanishes,leaving Forequilibrium,wehaveA0forallvariationsatconstantT,ni,As,andH.Thiscriteriongivesustheappropriatenonlocalself-consistenteldsi(z)as wherewehaveincorporatedtheosmoticstresses.Itisevidentfromtheaboveexpres-sionthatforthesolventmolecules,(z)=0,sincetheyhavenocongurationaldegreesoffreedom.Theseeldsalsovanishinthehomogeneousbulkphase.Further,wecan

PAGE 62

rewriteEquation( 3-75 )as Thisisanalogoustoageneralized`membrane'equilibriumasdiscussedbyLyklema[ 60 ].Alternatively,thecriterionforequilibriumcanberepresentedastheconstancyoftheexchangechemicalpotentials iiii inthelmandthebulk.Whilstoneobtainspolymer-inducedforcesfromfreeenergiesinagrandcanonicalensemble,itisessentialthatthesetransferchemicalchemicalsareheldconstant. SubstitutingEquation( 3-75 )into( 3-68 )andsimplifying,oneobtainstheexcessgrandcanonicalpotentialofthelmas equivalenttotheresult,Equation( 3-48 ),obtainedinSection 3.4 andanalogoustotheKirkwood-Buformula.Forcompressionunderfullequilibriumdeviatoricstressesareabsent.Therefore,Equations( 3-64 )and( 3-66 )implythat ItthenfollowsthatthenormalstressinthelmissimplytheosmoticpressurematthemidpointHmandthatthedisjoiningpressureortheforceofcompressionperunitareaunderfullequilibriumconditionsis Thisresultisinaccordancewiththatforelectrostaticinteractionsbetweenchargedsurfaces.Whereasintheelectrostaticinteractionsbetweentwoparallelplatesonehasanelectricpotentialgradientbalancedbytheosmoticpressure,inthecaseofinteractingpolymerlayers,onehasaconcentrationgradientarisingfromentropicandenergeticinteractionsofthepolymersegmentsandthesolvent,whichisbalancedbytheosmoticpressure.Forasymmetricsystem(wheretheadsorptionenergyper

PAGE 63

segmentsisthesameforbothsurfacesandthesurfaceshavesameareaAsexposedtothelmandareotherwiseidentical),symmetrydemandsthattheconcentrationgradientvanishesatthemidpoint,leavingthenormalstresseverywhereinthelmtobethemidpointosmoticpressure.Thus,underfullequilibriumconditions,thenormalforceofcompressionofthelmissimplythedierencebetweenthenormalstressinthelmandthatinthereservoir.Inourcalculations,weonlyconsidersymmetricsystems. Consistentwiththedenitionoflmtension,theinterfacialtensionisgivenby 2=Xz[m(z)]:(3-81) 61 ] Thelmtensionforthecaseofrestrictedequilibriumthereforebecomes ComparisonofEquation( 3-66 )withEquation( 3-83 )showsthattheexpressionforthedeviatoricstressinthelmis AstheisotropicstressisstillgivenbyEquation( 3-79 ),thedisjoiningpressureforthecaseofrestrictedequilibriumisgivenby Thenormalstresseverywhereinthelmisthereforem2^gmm.

PAGE 64

Theinterfacialtensionunderrestrictedequilibriumconditionsdoesnotfollowfromthedenitionoflmtensionbecausethedeviatoriccorrectionsdierforthelmtensionandthedisjoiningpressure,aswasshowninSection 3.5 .Theinterfacialtensionissimplydenedwithreferencetothenormalstressinthelmandisgivenby 2=Xz[m(z)2^gmm^g(z)(z)]:(3-86) 3.4 ,asthestatistical-mechanicalmodeltoestimatetheforcesofcompressionofpolymerlayers.

PAGE 65

34 57 ].Theforcesarepredominantlyrepulsive,asopposedtothemonotonicallyattractivebehaviorpredictedintheliteratureforinteractionsunderfullequilibrium.Ourresultsagreequalitativelywiththeanalyticalmean-eldpredictionsofPloehn[ 16 ]. Finally,inFigure3-5weshowtheinteractionpotentialsWfordierentbulkconcentrationsforanadsorptionenergyofs=0:5kTpersegmentforachainof100segments.Theinteractionpotentialisobtainedsimplybyintegratingtheforceprole wherehisusedasadummyvariableforthedistancebetweenthetwosurfaces.Thisinteractionpotentialisclearlynotequaltotheexcessgrandcanonicalpotential(interfacialtension)showninFigure3-2.

PAGE 66

Figure3{2: Filmtensionandinterfacialtension2asafunctionofsurfacesepara-tion(H/a)underfullequilibriumconditions.Theresultsareshownforanadsorptionenergys=0:5kT,chainlengthr=100,bulkconcentrationb=0:05,andgoodsolventconditions,=0:0. 34 ].Oncethesegmentdensitiesareknown,onecanevaluatethesemi-grandcanonicalfreeenergyrelativetoan`eective'bulksolutionwithwhichthelayersareinfullequilibrium.Again,thenegativederivativeofthisfreeenergyhasbeeninterpretedintheliteratureastheforceofcompression.Inthissection,weshallrstdeveloptheequationsforthedeviatoricstressesandtheforceofcompressionwithinthelatticemean-eldformulationbycorrectlyincorporatingtheresultsofthegeneralizedmembraneequilibriumdescribedearlier.

PAGE 67

Figure3{3: Averagedensityofpolymersegments(t/H)intheinterfaceasafunctionofsurfaceseparation(H/a)underfullequilibriumconditions.s=0:5kT,r=100,b=0:05,and=0:0. AsobtainedinSection 3.4 ,thesemi-grandfreeenergyisgivenby(alsoseeEquation24in[ 34 ]), ThesecondterminEquation( 3-88 )isanentropicmixingtermwhilethethirdtermcontainscontributionsfromenthalpicandentropicinteractions.Therstandthefourthtermspartiallyaccountforthedeviatoricstressesthatariseinthesystemduetotheconnementofpolymerinthegap.ScheutjensandFleer[ 34 ],inpage1886oftheirmanuscript,denotethelasttermas\asmallattractivetermaccountingfortheosmoticpressureofthesolutionoutsidetheplates"andneglectthetermintheircalculations.Weinterpretthistermasanosmoticpressurecontributionarisingduetotheconnementofpolymereitheroutsideorinsidethesurfacesdependingupontheamountofpolymerinthegap.Therefore,therecouldbeconditionswherethistermissignicant.ThisisdemonstratedinFigure3-6,whichshowsthenegativederivativeofthesemi-grandfreeenergywithandwithoutthelastterm.Inwhatfollows,wehaverecalculatedthemean-eldresultsinthatworkbyincludingtheaboveterm.

PAGE 68

Figure3{4: Forceperunitareafasafunctionofsurfaceseparation(H/a)underfullequilibriumconditionsforr=100and=0:0.Thenegativederivativeofexcessgrandcanonicalfreeenergyisalsoshownforcomparison.(a)s=0:5kT,b=0:05;(b)s=1:0kT,b=0:005. Aconsequenceoftheapproximationofrestrictedequilibriumasanequivalentfullequilibriumproblemisthat,forapolymersegment,Equation( 3-89 )becomes wherebeffandbeffarethechemicalpotentialofthepolymerchainandtheos-moticpressurecorrespondingtothe`eective'bulksolution.Thus,theeective-full-equilibriumapproximationleadstog,aneectiveadditionalpotentialorstressneededpersegment,thatisafunctionofsurfaceseparationbutindependentofz g=bbeff+bbeff;(3-91) fromwhichitfollowsthattheinterfacialtensionunderrestrictedequilibriumconditionsis 2=MXz=1b(z)g(z):(3-92) Correspondingly,forthedisjoiningpressure,Equation( 3-85 )isreplacedby ThechemicalpotentialsandtheosmoticpressuresinEquations( 3-91 )and( 3-93 )areevaluatedwithintheFlory-HugginsapproximationforeachlayerusingEquations

PAGE 69

Figure3{5: InteractionpotentialWbetweenthesurfacesinfullequilibriumwithsolu-tionsofvaryingbulkconcentrations.s=0:5kT,r=100,and=0:0. ( 3-45 )and( 3-46 ).(ItfollowsfromEquation( 3-91 )thatforfullequilibrium,forwhichbeff=bandbeff=b,g=0.) Figure3-7showsthat,undercertainconditions,theforcesofcompressionpredictedbyEquation( 3-85 )canbequalitativelydierentfromthederivativeofsemi-grandfreeenergy.Ingeneral,weobservethatthisdierenceisquantitativeforextremelyundersaturatedlayersandforlayersclosetosaturationcoverageandbeyond.Intheintermediaterange,qualitativedierences(i.e.,repulsioninsteadofattraction)couldresult.Toindicatetheeectsofthedeviatoricstressesinthesystem,weshowthevariationoftheeectivedeviatoricstresspersegmentfortwosurfacecoverages=0:5and0:75(Figure3-8).Theformercasecorrespondstoastarvedlayerwhereinthesurfacecoverageislessthanthesaturationcoverage0 3

PAGE 70

Figure3{6: Correctcalculationofexcesssemigrandfreeenergyunderrestrictedequilib-riumconditions.Theresultsshownhereareforasurfacecoverage=0:75,s=1:0kT,r=200,and=0:0. Herethepolymerchainsareconstrainedfromleavingthegap,resultinginanincreasedrepulsion.TheappropriateinteractionpotentialscorrespondingtotheabovecasesareshowninFigure3-9. 57 ]alreadypublishedintheliteraturebasedontheincorrectthermodynamicformulationofdeningtheforcetobethenegativederivativeofexcesssemi-grandfreeenergy.Aplotoftheforceperunitareaasafunctionofsurfacecoverage,forasurfaceseparationofH=a=4:5,whereaisthelatticespacing,showninFigure3-10areinforcestheaboveobservations.Fromourresults,itisseenthatSF2predictshigherrepulsiveforcesthanSF1,probablybecausetheeliminationofbackfoldingthroughSF2servestodecreasetheentropyofthechains,makingthemmore`rigid'andhardertocompress.ThedeviationofSF1fromSF2isparticularlypronouncedathighercoverages.Further,uponenlargingthelow-coverageregionofFigure3-10a,oneobservesthatSF2predictsslightlymoreattractiveforcesthanSF1.Thisattraction,observedatlowcoverages,shouldcorrespondtothatofidealchains(SF1)andnon-reversalchains(SF2).Itisto

PAGE 71

Figure3{7: Forceperunitareaasafunctionofsurfaceseparation(H/a)underre-strictedequilibriumconditions.=0:5,s=1:0kT,r=200,and=0:0. benotedthatdespitetherathersignicantlyattractiveeectivedeviatoricstresspersegmentatlowcoverages(asshowninFigure3-8),thetotalforceisonlyveryweaklyattractive. Ithasbeenobserved[ 57 ]that,comparedtoMonteCarlosimulationswithself-avoiding-walkchains,SF1andSF2underestimatethesurfacesaturationcoverage,becausethechainstatisticspermitoverlappingofthepolymersegments.Forthisreason,weexaminetheforceasafunctionofreducedsurfacecoverage(=0);seeFigure3-10b.TherescalingpreservestheobservationsalreadymadeinthecontextofFigure3-10a.Inaddition,thecorrectedforcecalculationsshowthattheforcesatverylowcoveragesaretoosmalltoprovideanymeaningfulcomparisonwiththelinearityofforceversuscoveragepredictedbyscalingarguments,whicharebasedonananalysisofthesystemasacollectionofisolatedbridges[ 62 63 ].TheearlierresultsbasedontheSF2formulation[ 57 ]appeartoshowanagreementwiththescalingargumentsforcoveragesuptoabout0:4;butthoseresultsdonotcorrespondtotheforceofinteraction,asalreadynoted.

PAGE 72

Figure3{8: Thevariationofdeviatoricstresspersegmentuponcompression,fortwosurfacecoverages=0:5and=0:75.Theresultsareshownfors=1:0kT,r=200,and=0:0,underrestrictedequilibrium. Thecrossovercoverages 32 ],c;r0:4).Inthiscontext,itisinterestingtocontrastthenumericalmean-eldresultswiththetwo-order-parameteranalyticalmean-eldtheoryofSemenovetal.[ 36 ],whichpredictsattractionforany=0.0:98,forallvaluesofH=a.Duetoreasonsasyetunclear,theanalyticalmean-eldresults[ 36 ]showsurprisinglystrongattractioncomparedtothenumericalmean-eldtheory.

PAGE 73

Figure3{9: InteractionpotentialWbetweensurfacesinrestrictedequilibriumforcov-erages=0:5and=0:75.Theresultsshownherearefors=1:0kT,r=200,and=0:0. adsorbingsurfaces.Therefore,onehastobecarefulindeningtheinterfacialtensionandthenormalforceofcompression. Further,wehavepresentedtherstcorrectresultsfortheinteractionforcesbetweenadsorbedpolymerlayersbasedonthelattice-basednumericalmean-eldtheory.Itisshownthatqualitativedierencesbetweenthecorrectresultsandtheearlierincorrectformulationcouldoccurunderbothfullandrestrictedequilibriumconditions.TheforceofcompressionoftheadsorbedlayersinfullequilibriumwithahomogeneousbulksolutionisneithermonotonicallyattractiveasseenfromtheresultsofScheutjensandFleer[ 34 ],norisitmonotonicallyrepulsive,asclaimedbyPloehn[ 16 ].Underrestrictedequilibriumconditions,additionaldeviatoricstressesdevelopinthesystem.Dependingonwhetherthepolymermoleculesareconnedinthegaporoutsidethegap,thesestressescauseadditionalrepulsionorattractionrespectively.Itappearsfromthesepreliminaryresultsandfurtherresults[ 17 ]showninChapter 4 thatmean-eldtheorieshighlyunderestimatetheattractiveforcesatconditionswhenbridgingeectsareexpectedtodominate.Theeliminationofbackfoldinginthechainstatisticshardlyimprovesthepredictionsoftheforceofcompression.Theresultspresentedherearesignicantlydierentcomparedtothe

PAGE 74

Figure3{10: Forceperunitareafasafunctionofsurfacecoverageunderrestrictedequilibriumconditionsforaxedseparationof(H/a)=4.5.(a)s=1:0kT,r=200,and=0:0;(b)Eectsofrescalingthesurfacecoveragebynormalizingwiththesaturationcoverage0ontheforcefor(H/a)=4.5,s=1:0kT,r=200,and=0:0. predictionsofthetwo-order-parametertheory,eventhoughtheassumptionsinvolvedinthetheoriesaresimilarinnature.

PAGE 75

2 ,wereviewedvarioustheoretical(scaling,analyticalandnumericalmean-eldtheories)andcomputationalapproachestotheproblemofphysisorptionoflinear,exiblehomopolymersontooneortwouniform,at,parallel,impenetrablesolidsurfaces.Wethendevelopedathermodynamicformalismforexaminingtheforcesofinteractionoftheselayersandapplieditusingthelatticenumericalmean-eldtheorydevelopedbyScheutjensandFleer[ 34 ]withandwithoutbackfoldinginChapter 3 .BoththeseMarkovapproximationsconsideranisotropicmeaneld.Furtherimprovementinthetheoryaroseduetotheintroductionofanisotropyinthemeaneldbypartiallyaccountingfortheeectsofbondorientationsontheequilibriumpropertiesofthesystem.Thisself-consistentanisotropicmean-eldtheory(SCAFT)wasrstproposedbyLeermakersandScheutjens[ 40 ]tostudyphasetransitionsinlipidbilayermembranes,inordertoaccountfortheanisotropicorientationalinteractionsbetweenthelipid-likemoleculesinamembrane.Thetheorywasabletosuccessfullypredictthecriticalphasebehaviorofthemembraneobservedinexperiments.Thisworkalsoprovidedanelegantderivationofthetheoryfromthebasicprinciplesofstatisticalthermodynamics.Morerecently,vanderLindenetal.[ 41 ]extendedtheScheutjensandFleertheorytosemiexiblepolymers,inwhichbondcorrelationswereincorporated.Inboththeseworks,theauthorsstudiedonlytheoverallsegmentdensitiesandsomeofthebroadstructuralfeatures.Thefocusintheformerworkwastodevelopatheoryformembranesbasedonstatisticalthermodynamicsandthatofthelatterworkwastoformulatealatticemean-eldtheoryforsemiexiblepolymers(e.g.,`wormlike'chains).Fleeretal.[ 42 ]haveattemptedtorelatethenumericalmean-eldformalismandthetwo-order-parametertheoryinanattempttoobtainclosed-formsolutionsthatreproducethenumericalmean-eldresults. 61

PAGE 76

Thenumericalmean-eldtheoryofScheutjensandFleeris,presently,theonlytheoryfromwhichonecanobtainquantitativeguidelinesontheadsorptionofshortchains(ofnitemolecularweight).Itisthereforeimportanttoascertainthereliabilityofthistheoryinpredictingthestructuraldetailsoftheadsorbedlayer,andtheforcesofcompressionoftwopolymerlayers.Comparisonshavebeenmadewithexperimentsintheliterature(e.g.,Fleeretal.[ 12 ]).However,manyoftheintricatestructuraldetailsoftheadsorbedlayerareeitherinaccessibleornoteasilyaccessiblethroughexperiments,suchasspecicinformationonloops,tails,andbridges.Moreover,experimentsinvolvemanyparametersthatcannotbepreciselycontrolledordonotfactorintomosttheories,suchassurfaceroughnessandpolydispersityofthepolymer.Anotherveryimportantissueincomparisonswithexperimentsistheambiguityinrelatingatheoreticalstateofthesystem(asdenedbythenumberofchains,numberofKuhnunits,atheoreticaladsorptionenergy,andatheoreticalsegment-solventinteraction()parameter)withappropriateexperimentalconditions.Theseconsiderationslimittheextentofanysuchcomparisontooneofaqualitativenature.Inviewofthese,whileitmaybeinstructiveandimportanttoassessthevalidityofatheoryagainstexperiments,asimple,directcomparisonwouldnotonlybelimitedbutalso,undermostconditions,bemisleading.Ontheotherhand,computersimulationsare`exact'withintheirapproximations(e.g.,latticeapproximationinlatticeMCsimulations),andcomparisonsagainstlattice-basedsimulationswouldenableonetoassessthelimitationsofthemean-eldapproximationinthecontextoflatticemean-eldtheories.We,therefore,useMonteCarloresults[ 27 ]ofrealisticchains(self-avoidingwalks)withinthelatticeformalismasareference,toexaminethepredictionsofthemean-eldtheories. Inthischapter,weconsidertheeectsofanisotropyinthelayerontheresultingmean-eldpredictionsofthestructureofthelayeraswellastheforcesarisingfromtheinteractionsbetweentwolayers.Specically,weintroduceanisotropyintheSF2formulation(ScheutjensandFleernumericalmean-eldtheorywithsecond-orderMarkovchains).Wealsoprovidepreliminarycomparisonsoftheanisotropicmean-eldpredictionsofthestructureoftheadsorbedlayersandtheforcesofcompressionof

PAGE 77

thelayersagainstresultsfromrigorouscomputersimulations[ 57 ].Asummaryoftheanisotropicmean-eldformulationisprovidedinthenextsection.(Furtherdetailsareavailableelsewhere;seeFleeretal.[ 12 ];LeermakersandScheutjens[ 40 ].)Wethendiscusstheimprovementsinthepredictionsduetotheintroductionofanisotropyandcommentonthelimitationsofmean-eldtheoriesasseenfromthesecomparisons.Oneofourobjectivesinthenextsectionistopresentaclearandeasilyunderstandableanisotropicmean-eldformulationandtoprovideexpressionsforthevariousstructuralfeaturesoftheadsorbedpolymerlayer. 4.2.1PreliminariesandNotations

PAGE 78

Wedenethevolumefractionsofpolymerandsolventatanygivenlayeras, suchthat, and, Wenowpresenttherelevantequationsforthecaseofrestrictedequilibrium.Thepolymerchainsaremodeledasstep-weightedrandomwalksinasimplecubiclattice.Weconsiderasecond-order-Markovchainstatisticsinwhichimmediatestep-reversals(backfoldingofsegments)aredisallowed.Thedimensionlessmean-eldpotentialu(z)thatapolymersegmentexperiencesinlayerzisgivenas wheresiistheSilberbergadsorptionenergyparameterforthepolymer/solventpaironthesurfaceatlayerzandijisthestandardKroneckerdeltafunction.Forasymmetricsystem,s1=sM.Thetermuint(z)accountsfortheenergeticinteractionsbetweenthepolymersegmentsandsolventmoleculeswithintheBragg-Williamsrandommixingapproximation. Here,istheFlory-Hugginssegment-solventinteractionparameterthatdecidesthesolventquality.Inthepresentwork,weexamineonlygoodsolventconditions,forwhich=0.Thenear-neighboraverageofvolumefractionofthesegmentsinlayerzisgivenby, which,foracubiclatticewithonlythenearest-neighborinteractions,becomesh(z)i=1(z1)+0(z)+1(z+1):

PAGE 79

TheLagrangeparameteru0(z)inEquation( 4-5 )partiallyaccountsfortheexcludedvolumeofthesegmentandsolventinagivenlayer.ThisisusuallybasedontheBragg-Williamsrandom-mixingapproximation. Figure4{1: Bondorientationsandanisotropicmeaneld.(a)Thenotionofbondori-entations.(i)Atypicalpolymerchaininalattice.(ii)Segmentsandbonds.(iii)BondOrientations.Intheillustrationabove,theorientationofthebondbetweensegment(s1)andsegments(in(ii))isk=1(see(iii)).Thiswayofrepresentingbondorien-tationthusindirectlyspeciesthepositionoftheprevious,e.g.,(s1)th,segment.Inacubiclatticetherearesixpossibleorientations.Thesesixorientationscanbethoughtofasthreepairsof`opposite'orientations.Forinstance,k=1andk=3are`opposite'orientations.Consecutivesegmentswith`opposite'orientationswillcauseabackfoldedconformation;(b)Conformationsofconsecutivebonds.(i)Straight(ii)Perpendicular(iii)Backfolded;(c)Schematicrepresentationoftheanisotropicmeaneldinasquarelattice.IsotropicMeanField:Probabilityofplacinganewsegmentinlayer2=3=6.AnisotropicMeanField:Probabilityofplacinganewsegmentinlayer2=3=5.

PAGE 80

theideaofanisotropicmean-eld,letusconsiderapolymerchainasasetofsegmentsconnectedbybondsofdenedorientations.Thenotionofbondorientationsisillus-tratedinFigure4-1a.Betweenanythreeconsecutivesegments,threeconformationscanbeidentiednamely,astraightconformation,aperpendicularconformationandabackfoldedconformation.TheseconformationsareillustratedinFigure4-1b.Asecond-orderMarkovchainstatisticsdoesnotallowthephysicallyunrealisticbackfoldedconformation.Inthecaseofisotropicmeaneld,theexcludedvolumeisaccountedforbyjustrequiringthattheprobabilityofhavingasegmentinalayeristhefractionofemptysitesinthatlayer.(ThisisthestandardBragg-Williamsrandommixingapproximation.)Therefore,allthebondorientationsareequallylikely(withinthere-strictionsimposedbythechainstatistics).Inananisotropicmeaneld,theorientationofapolymersegmentinagivenlayerdependsontheorientationsofitsneighboringsegments(theneighboringbonds),whichapriorilimittheprobabilityofhavingthesegmentinthatlayer.ThisisillustratedinFigure4-1c.Thisthereforemeansthateachorientationhastobeweightedappropriately.Thisintroducesabiasinthestep-weightsoftherandomwalk,makingthemeaneldanisotropic Inanisotropicmeaneld,theLagrangeparameteru0(z)isgivenbythefractionof`empty'latticesitesinthelayerz,whichisequivalenttothevolumefractionofsolvents(for`good'solvents). Incaseofanisotropicmeaneld,thereareapriorimore`empty'sitesavailableinthelayerzonceweaccountforthefactthatthosesegmentsinlayerzwiththesamebondorientationandthoseinlayerz1withacomplementaryorientationwillnot`block'thegivenbondorientation.Let(z;k)bethefractionofbondswiththoseorientations

PAGE 81

thatwillnotblockabondwithanorientationk. (z;k)=Xsf(z;s;k)+(z0;s;k0)g(4-9) Here,kandk0are`opposite'bondorientations(seeFigure4-1(c)),andz0=zorz1,asthecasemaybe.(z;s;k)isthevolumefractionofsthsegments(inr-mers)withabondorientationk,inlayerz.Therefore,1(z;k)isthemaximumavailablefractionof`empty'sitesinlayerzforasegmenttooccupywithabondorientationk. Thiscorrectionfactorisgivenas, 1(z;k)(4-10) where(z;k)isthefractionofbondswiththoseorientationsthatwillnotblockabondwithanorientationk.Notethatincaseofisotropicmeaneld,thecorrectionfactorisunity. denestheso-calledfree/monomersegmentdistributionfunction.Itisevidentlytheunnormalizedprobabilityofa`monomer'segmentinlayerz.Wenowdenetheunnormalizedprobability

PAGE 82

Foracubiclattice,wethereforewrite, 6(4-12) Thecorrectionsfortheanisotropyduetobondorientationsshouldbeintroducedfromdimerson.Wenowwritethestatisticalweightsfordimers. Usingsecond-orderMarkovstatistics,wecanfurtherwritethestatisticalweightsofas-meras Tointroduceashort-handnotation,wewritetheaboveequationsas NotethatthiswayofevaluatingthestatisticalweightsautomaticallyensureschainconnectivityasshownbyScheutjensandFleer[ 33 ].Wenowproceedtoevaluatethesegmentvolumefractions(whichwecallsegmentdensities)usingtheso-calledcompositionrule (5=6)Gm(z);ifs6=1orr (4-18) Here,Cisanormalizationconstanttoaccountforthefactthatthestatisticalweightsarenotnormalized.Forthecaseofrestrictedequilibrium,itisdenedas LG(r)(4-19) wheretisthetotalamountofpolymerbetweentheplatesandG(r)istheso-calledend-segmentdistribution,i.e.,thestatisticalweightofndingar-meranywhere

PAGE 83

betweenthetwosurfaces.Theformerisdenedas L(4-20) and,thelatterisdenedas Aself-consistentsolutionisobtainedbyassumingadensityproleandisotropicmean-eld(asinitialguess)andusinganiterativeproceduretoevaluatethevolumefractions. Thereforewehave, Sincet=CrG(r),wedene, 34 ]here.

PAGE 84

andfurther, foreachofthegroup(representedby*).Nowweproceedtodenerecursiverelation-shipsforthestatisticalweightsoffree,adsorbedandbridgedchainssimilartothewaywedenedthestatisticalweightsbefore. subjecttotheconditions (4-28) (4-29) 5Xl6=4fGa1(2;s1;l)+Gf(2;s1;l)g subjecttotheconditions Inequation( 4-31 ),thetermGm(1)g(1;2)Pl6=4Gf(2;s1;l)wouldoccurintherecursiveexpressionforGf(1;s;2)accordingtothenotation.However,itactuallycorrespondstoconformationsofs-merswithonlytheendsegmentadsorbed.HenceitisaddedtoGa1(1;s;2).

PAGE 85

(4-32) (4-33) 5Xl6=4fGb1(2;s1;l)+Gb2(2;s1;l)+Ga2(2;s1;l)g subjecttotheconditions Again,thetermsGm(1)g(1;2)Pl6=4Gb2(2;s1;l)andGm(1)g(1;2)Pl6=4Ga2(2;s1;l)wouldoccurintherecursiveexpressionsforGb2(1;s;2)andGa2(1;s;2)respectively.Howevertheyactuallycorrespondtobridgedchainswiththelastchainendatsurface`1'.HencetheyareaddedtoGb1(1;s;2): Sinceachainhastwoends,theprefactor2isaddedtotheequation. Oncewehavethestatisticalweightsoffree,adsorbedandbridgedchains,wecanalsoestimatetheaveragenumberandsizesofloops,tails,trainsandbridges.DetailsoftheseareprovidedinScheutjensandFleer'spaper[ 34 ].Togiveanillustration,weconsidertheaveragenumberandsizeofloops.Inaloop,boththeendsareadsorbedonthesamesurface.Therefore,weconsiderthestatisticalweightofanadsorbedr-merhavingthesegmentsinlayer`2'andthesegments+1inlayer`1'(adsorbed),withther-meradsorbedatleastoncebeforesegments.ThisisequaltotheproductGa1(2;rs+1;4)Pk6=4Ga1(2;s;k).Wenormalizethestatisticalweightwiththestatisticalweightofanadsorbedr-mertoobtaintheprobabilityofndinganadsorbedr-merhavingthesegmentsinlayer`2'andthesegments+1inlayer`1'(adsorbed),

PAGE 86

withther-meradsorbedatleastoncebeforesegments.Thiswillbeequaltothefractionofadsorbedchainswithloopsendinginsegments.Summingoverallthepossiblevaluesofsgivestheaveragenumberofloopsperadsorbedchain,nl.Theaveragenumberofloopsperadsorbedchain(adsorbedonsurface`1')na1;listhengivenby (4-37) Theaveragesizeoftheloopsinthechainsadsorbedonsurface`1'la1;listhengivenby wherea1;listhefractionofloops,a1istheamountofpolymeradsorbedonthesurface`1',andaregivenby Loopsareformedonsurface`1'bychainsingroups`2'and`4',i.e.,bychainsadsorbedonsurface`1'withorwithoutformingbridges.Therefore,anexpressioncanbewrittenfortheaveragenumberofloopsonsurface`1'perbridgingchain,nb1;lasexplainedinScheutjensandFleer'spaper[ 34 ].TheaveragesizeofsuchloopscanbecalculatedusinganexpressionsimilartoEquation( 4-39 ).Theaveragenumberofloopsonsurface`1'perunitarea,nl,isthengivenby Lfna1;lfa1+nb1;lfb1g(4-42) wherefa1isthefractionofchainsadsorbedonsurface`1',fb1isthefractionofbridgingchainswiththelastchainendleavingfromsurface`1',andaregivenby (4-43)

PAGE 87

Theaveragesizeofloopsadsorbedonsurface`1'l1;listhenobtainedasaweightedsumofthetwoaveragesizesla1;landlb1;l,andisgivenby 3 .Also,abriefsummaryofthemethodofestimatingpolymer-inducedforcesusingMonteCarlosimulationswaspresentedinChapter 2 57 ].WeshallcomparethepredictionsofthestructureofadsorbedlayerandtheinteractionforcesbySCAFTandSF2withsomesimulationresults.ThesimulationresultsusedinthefollowingdiscussionarebasedonalatticeMonteCarlotechniqueinwhichwemodelthepolymerchainsasself-avoidingwalks(SAW's)andsamplethechainstatisticsusingamodicationofthecongurationalbiasalgorithmofSiepmannandFrenkel[ 64 ]duetodeJoannis[ 43 ].Weuseperiodicboundaryconditionsinthexandydirectionsandconsidertwoimpenetrable,adsorbingsurfacesconningthelayersinthez-direction.Thesurface-segmentinteractionisconsideredonlyintheadjacentlayer,asshowninEquation( 4-5 ).Allresultshavebeengeneratedforchainsof200segments,inagoodsolventwitha

PAGE 88

simple-cubiclatticeofsizeL=a=40inthexandydirections Inthiswork,wefocusontheintroductionofanisotropyinthemean-eldtheoryandattempttounderstandwhatimprovementitoers.Anattemptismadetorelatetheforcebetweenthetwoadsorbingsurfacestothestructureoftheadsorbedlayers. Figure4{2: Overallsegmentdensitydistribution:Comparisonoflatticemean-eldresults(SF1,SF2,SCAFT)withlatticeMonteCarlosimulations.(a)H=a=5:0.(b)H=a=40:0.

PAGE 89

intheappendix(seeTablesB-2andB-3)fordierentsurfaceseparationsandsurfacecoverages. Basedontheaboveresults,rstoneobservesthat,forH=a=5:0,whilethepredictionsoftheoverallsegmentdensityofSF1agreewellwiththesimulations,thoseofSF2dierconsistently.SinceSF2preventssegmentbackfolding,itpermitsalowernumberofallowableconformationsnearthesurfacethanSF1,whichresultsinreducedsegmentdensitiesnearthesurface(wheretheconcentrationishigh)andhighervalueofdensitiesawayfromthesurface.ThisresultsinSF2predictinghighersegmentdensitiesforthebridges,loopsandtailsandlowerdensitiesfortrainsascomparedtoSF1andsimulations.Introductionofanisotropysignicantlyimprovesthepredictions.SCAFTpartiallycorrectsSF2intheappropriatemanner.TheoverallsegmentdensityprolesshowasurprisingagreementwiththesimulationsforH=a=5:0.However,forH=a=40:0,itisclearthatthecorrectionsduetoanisotropyareinsucienttocapturetheinteractionsbetweenthesegmentsaccurately.EvenSCAFTpredictsanorderofmagnitudehighersegmentdensitiesnearthecenter. Ingeneral,thecorrectionduetoanisotropyinthemeaneldistwo-fold:

PAGE 90

Figure4{3: Numberandsizedistributionofbridges,asafunctionofH=a,foracon-stantadsorptionenergy,s=1:0kTand=0:75(a)Averagenumberofbridges,nbr,(b)Averagesizeofbridges,lbr. SCAFTalsopredictslowersegmentdensitiesoftailsandloops

PAGE 91

uponcompression.Whilebridgesgenerallycontributetoanattractiveinteractionbe-tweenthesurfaces,interactionsamongtails,bridgesandloopsresultinstericrepulsion.Itisofinteresttoexaminetheeectsofbondcorrelationsonthebalancebetweenthetwocompetingfactorsindeterminingtheinteractionbetweentwoadsorbedlayers.Further,asnotedearlier,anevaluationoftheperformanceofmean-eldapproximationswouldbeincompletewithoutanexaminationoftheforcesofcompressionduetotheinteractionbetweentwolayers.Here,weshallconsidertheeectoftheintroductionofanisotropyinthemeaneldonpolymer-inducedforces. Atlowsurfacecoverages,oneessentiallyencountersasingle-chainregime.There-forethesystemcanbeanalyzedasacollectionofsinglechains.Whenthesurfacesareclosetoeachother,atstrongadsorptionconditions,thechainsformbridgesbetweenthesurfaces.Thesebridgesarelooselystretched,andasarstapproximation,canbeconsideredtobehaveasHookeansprings[ 63 ] 62 ],witheachtethergovernedbythePincuslawofelasticity[ 65 ].Inthissingle-chainregime,theinteractionbetweenthetwosurfacesispredominantlyattractive,withtheexceptionofverylowseparations,wherestericinteractionsbetweenthesegmentsofthechainlowertheattraction.Asthesurfacecoverageincreases,thechainsarepackedcloserandcloserthatbeyondtheproximalregionsofthelayers,thesystemisinasemi-diluteregime,andstericinteractionsbetweendierentsectionsofachainandbetweenchainsbecomeimportant.Undersuchconditions,theforceofinteractionbetweenthesurfacescrossesoverfromattractiontorepulsion.Suchbehaviorhasalsobeenqualitativelyobservedinexperiments(seeforinstance,Fleeretal.[ 12 ]).

PAGE 92

ScheutjensandFleer[ 34 ]haveexaminedthisprobleminthelatticemean-eldapproximation,andhavebeenabletoreproducethebridging-attraction-and-steric-repulsionforceprolequalitatively.Theyusedtherst-orderMarkovapproximationtomodelthechainsinthelattice(SF1).Whatissurprisingabouttheirresultsisthatthemean-eldtheoryappearstoprovidereasonablepredictionsinthesingle-chainregime,whichiscounterintuitive.InChapter 3 wehavediscussedthecorrectcalculationofforcesofcompressionusinglatticemean-eldtheory,andhavereinterpretedtheScheutjens-Fleerresults.Here,weshowthatevenwiththeintroductionofanisotropyinthemeaneld,thelatticemean-eldtheoryfailstocaptureanysignicantbridgingattraction.Weshallrestrictourdiscussionstotheeectsofpolymerlayeranisotropyonthepredictionsofforceofcompressionofthepolymerlayers. Werstexaminetheforceperunitarea(theexcessnormalstressinthesystem)asafunctionofsurfaceseparationH=a,forsurfacecoverages=0:25through1:25;andforanadsorptionenergys=1:0kT,inFigures4-4athrough4-4f.Atlowcoverages,fortheextentofbridgingobserved(seeforinstanceFigures4-3aand4-3bandFigureA1inAppendixA),mean-eldtheoriespredictnearlynegligibleattractiveforces.Further,themean-eldresults,evenconsideringbond-orientationeects,aresignicantlymorerepulsivethanpredictedbythesimulations.Weshallrevisittheseobservationstowardstheendofthissectionanddrawfurtherconclusions.Aninterestingobservationtobemadefromtheseguresisthatathighcoverages,SCAFTandSF2predictarepulsiveforcewithapowerlawbehavior(shownonlyfor=1:25),theexponentbeingindependentofthecoverage.Theexponentsare3:67and3forSCAFTandSF2,respectively. Tofurtherillustrateandunderstandthelimitationsofmean-eldtheories,weexaminetheforceasafunctionofforaxedsurfaceseparationH=a=4:5,showninFigure4-5a.TheresultsofSF1(withthecorrectcalculationofforces)arealsoincludedforcomparison.Itisclearthatatlowcoverages,thereisseldomanyattractionpredictedbyanyofthemean-eldformulations.Ontheotherhand,simulationsshowthatatlowcoverages,wherethesurfaceconsistsofisolatedpolymerchains,thereisalinearincreaseintheattractiveforcewithacorrespondingincreaseinthecoverage.

PAGE 93

Figure4{4: (a)=0:25.(b)=0:5.(c)=0:75.(d)=1:0.(e)=1:125.(f)=1:25.

PAGE 94

Figure4{5: ForceperunitareafataxedseparationofH=a=4:5,(a)asafunctionofsurfacecoverage,foradsorptionenergys=1:0kT,(b)asafunctionofrescaledsurfacecoverage=0,forrescaledadsorptionenergysc=0:74. Athighcoverages,weobservethatSF2predictshigherrepulsionthanSCAFT.Thisiscounterintuitive,sincetheintroductionofanisotropyservestoincreasethe`rigidity'ofthechains.Forthisreason,onewouldexpectapolymerlayerinananisotropicmean-eldtobelesscompressiblethanoneinisotropicmean-eld.However,mean-eldtheoriesunderestimatethesaturationcoveragebecausetheexcluded-volumeinteractionsareimplementedinamean-eldsenseandthatthechainstatisticspermitoverlapandcrossover.Itisfoundthatthesaturationcoveragesforsimulations[ 66 ],SF1,SF2andSCAFTare,respectively,1.2,0.7,0.72and0.92.ThisexplainswhySF2predictslesscompressiblelayersthanSCAFT,contrarytointuition. Moreover,duetothedierencesinthechainstatisticsandthemean-eldassump-tions,onecanexpectthatthecriticaladsorptionenergy 66 ].

PAGE 95

Figure4{6: Numberandsizedistributionofbridges,asafunctionofrescaledsurfacecoverage=0,foraconstantrescaledadsorptionenergy,sc=0:74,andH=a=5:0.(a)Averagenumberofbridges,nbr,(b)Averagesizeofbridges,lbr. Ourcalculationsshowthatthecriticaladsorptionenergiesare,respectively,0.18,0.22,0.21and0.26.Duetothedierencesin0andc,itisinstructivetoexaminetheresultsforthesamerelativesurfacesaturationandrelativeadsorptionenergies.There-fore,weplottheforceperunitareaasafunctionofrescaledcoverage=0foraxedH=aandarescaledadsorptionenergy(sc)inFigure4-5b.TheresultsaresimilartothoseshowninFigure4-5a,exceptthatathighcoverages,SCAFTisnowseentopredicthigherrepulsiveforces,aswouldbeexpected.Theaveragenumberandsizeofbridges,loopsandtailsareplottedasafunctionof=0foragiven(sc).Figures4-6aand4-6bshowthenumberandsizeofbridges.Thenumberandsizeofloopsandtailsareavailableinthesupplementalinformation(seeFiguresA411throughA414). Anenlargedviewoftheresultsatlowcoverages,illustratedinFigures4-4a,4-4b,4-5a,4-5b,showsthatSCAFTandSF2dopredictsomeattraction,eventhoughquantitativelynegligiblecomparedtothecorrespondingsimulationresults.Asstatedearlier,inahighlyundersaturatedregime,thechainconformationsaredictatedessentiallybysingle-chainstatistics.Theassumptionofmean-eldisexpectedtobehighlyinaccuratehere.Forchainsinalattice,constructedwitharst-orderMarkovstatistics,withrandommixingapproximationinthelateraldimensions,thefollowingexpressionhasbeenused[ 67 ]toestimatetheradiusofgyrationRg,expressedasa

PAGE 96

Figure4{7: ForceperbridgefbrasafunctionofforaxedwallseparationH=a=4:5. multipleofthelengthofastepinthelattice, whereristhechainlength,andDisthelatticecoordinationnumber.Forachainlengthof200segmentsinacubiclattice,theradiusofgyrationisabout5.7latticeunits.Basedontheestimatesofrmsthicknessoftheadsorbedlayer,onecanestimatethatsingle-chainstatisticsprevailsuptoasurfacecoverageof0:5.Therefore,theobservedattractionthencorrespondstothatofanidealnon-reversalchain(SF2),andaweightedidealnon-reversalrandomwalk(SCAFT).Wenotedearlierthatatlowcoveragesthesystemisexpectedtobehaveasacollectionofsinglebridgeswithnegligibleinteractionsamongbridges.Therefore,themagnitudeoftheattractiveforceisexpectedtoincreaselinearlywithsurfacecoverage,asseeninthesimulationresults.Thenumberofbridgesinthisregimedecreasesexponentiallyandthelength

PAGE 97

ofthebridgesincreaseslinearly 65 ])inthestronglystretchedregion,weplottheforceperbridgeasafunctionofinFigure4-7.Atverylowcoverages,onedoesobserveaqualitativelysimilarbehavior,thoughthemagnitudeoftheforceismuchlessthanthePincuslawestimateandthepredictionsfromthesimulations. Theprecedingobservationsindicatethatthemean-eldtheoriesareseverelylimitedintheirabilitytoprovideareasonablequantitativeestimateoftheattractionbetweeninteractingpolymerlayersatlowcoverages,underrestrictedequilibrium.Theypredictqualitativelyexpectedbehaviorathighercoverages,wherestericinteractionsaresignicant.Theattempttoimprovemean-eldpredictionsbytheintroductionofanisotropyseemstoworkinthepredictionofstructuraldetailsoftheadsorbedlayer,butisclearlywantinginthepredictionofforces.Toobtainanysignicantimprovementinthepredictionsofthetheoriesofpolymeradsorption,onemayhavetogobeyondtheconstraintsofthemeaneld.Further,inadequatechainstatistics(whichoneoftenresortstoinordertoretainmanageablepropagationrelations)introducefurtherlimitationsinthetheoreticalpredictions.Eventhoughtheeliminationofbackfoldingleadstoasignicantimprovement,asshowninthisworkandbyJimenezetal.[ 57 ]andSimonandPloehn[ 68 ],thisismerelythesimplestoftherenementsthatmaybenecessary.Acarefulexaminationisneededtostudythelimitationsduetochainstatistics. 62 ],whoshowthattheelas-ticityofsuchtethersisnonlinearandisgovernedbythePincuslawofelasticity[ 65 ].

PAGE 98

hasbeentopresentatleastapreliminaryexaminationofwhatimprovementscanbeexpectedthroughtheintroductionofanisotropy. Theresultsshowthatintroductionofanisotropydoesimprovethesegmentdensitydistributionclosetotheadsorbingsurface.However,farawayfromthesurface,mean-eldtheoriesconsistentlypredicthigherdensitiesthansimulations.Itisalsofoundthatmean-eldtheoriesfailtoprovideusefulresultsatlowercoveragesontheforcesofinteraction.Alinearforceprole,qualitativelyconsistentwiththescalingpredictionsofJietal.[ 62 ],isobservedatlowsurfacecoverages,eventhoughtherearequantitativediscrepancies.Boththemean-eldformulationspredictthecrossoverfromnetattractiontorepulsiontooccuratsimilarcoverages(=00:5). Intheabovecomparisons,wehaveusedthestructuralpropertiesandinteractionforcescomputedusingMonteCarlosimulationsasareference.However,acaveatisinorderinthisrespect,namelythatthereisnoclearwaytocomparerigorouslythemean-eldtheorieswithexactcalculations(e.g.,simulations).Inordertobeabletomakeanyconclusivecommentsonthereliabilityofmean-eldtheoriesevenforthesimpleproblemofhomopolymeradsorption,onehastoconsidermanyissues.First,anagreementbetweenamean-eldformulation(say,SF1)andsimulationsforonlycertainproperties(say,segmentdensities)doesnotnecessarilyimplythattheformulationwillbeequallygoodforallproperties.Secondly,thefactthatimprovingthechainstatisticsleadstoimprovedsegmentdensitiesincertainregions(aswenotedintheprevioussection)doesnotguaranteethatsimilarimprovementscanbeexpectedintheoverallperformance.Thatis,agreementincertainrespectscouldoccurbecauseoffortuitouscancellationoferrors,whiletheerrorscouldcompoundeachotherinotherrespects.Thirdly,itisunclearcurrentlyastowhatistherightway(or,thebestway)tocomparemean-eldcalculationswithsimulations.Intuitively,itappearsreasonabletocomparetheresultsforthesamerescaledcoverage,=0,andrescaledadsorptionenergy,(sc).However,whethersucharescalingissucient(or,evenvalid)remainsunclear.Thisissuedeservesfurtherattention,sincetheinterpretationscouldbemisleadingiftherightcomparisonsarenotmade.Ontheotherhand,despitesuchuncertainties,theuseofsimulationsasbenchmarksremainstheonlyoptiontoexamine

PAGE 99

thereliabilityofthemean-eldtheories,sinceadirectcomparisonwithexperimentsonlyintroducesadditionaluncertainties.

PAGE 100

1 (seeSection 1.1.3 ),wediscussedthewormlikechainmodelforsemiexiblepolymers.Herewestatetheconnectionbetweenthewormlike-chainmodel,amicroscopicdescriptionofsemiexiblepolymers,andamacroscopiccontinuumdescriptionbasedonthetheoryofelasticity,aselasticrods.Fordetailedderivations,theinterestedreaderisreferredtoChapter3ofYamakawa[ 2 ].Theprobabilitydistributionofndingtheendsegment(correspondingtos=L)ofawormlikechainatpositionRgiventhattherstsegmentisatoriginandhasaspeciedorientationu0forL>0istheGreen'sfunction.Onceagain,r(s)andu(s)mayberegardedasMarkovrandomprocessesonthepropertimescaleofs(orL).ThisGreen'sfunctionsatisesaSchrodinger-likeequationanalogoustothebehaviorofarigidelectricdipoleinanelectriceld.Thesolutionsforsuchequationsareknown.The`Lagrangian'forthisproblemisgivenby where 86

PAGE 101

with 5-1 ),thersttermontheright-handsideisthe`kineticenergy'oftherigiddipoleandthesecondtermisthenegativeofits`potentialenergy'.Equation( 5-1 )maybegivenastatisticalmechanicalinterpretation.TheUlgivenbyEquation( 5-2 )isjustthebendingenergyperunitlengthofaslenderelasticrodwithbendingforceconstantorbendingstiness(exuralrigidity),andthereforethetotalpotentialenergyUofthewormlikechainasaslenderelasticrodisgivenby Thisisafamiliarequationintheelasticityofstislenderrods.Foranisotropicandhomogeneousslenderelasticrod,theexuralrigidityissimplyEI,whereEistheYoung'smodulusandIisthesecondmomentofinertiaofthecross-sectionoftherod.Havinggivenabasisforthewormlikechainmodelforapolymerasaninextensibleslenderrod,weshallnextdiscusstheelasticbehavioroftheserodsundertheactionofexternalforces.InEquation( 5-4 ),theproportionalityconstantisthebendingstinessorexuralrigidity.FromEquation( 5-3 ),thepersistencelengthofthewormlikechain,Lp,isdirectlyproportionaltotheexuralrigidityofthechain,andthereforeisameasureofthechain'sresistancetobendingbythermalforces. 69 ].Consideraninnitesimalelementboundedbytwoadjoiningcross-sectionsoftherod.Letthetwoadjoiningcross-sectionsformtheupperandlowerendsoftheelement;aforceF+dFactsontheupperend,andaforceFactsonthelowerend;thesumoftheseisthedierentialdF.LetKbetheexternalforceontherodperunitlength.Then,theexternalforceactingontheelementdsisKds.Atequilibrium,theresultantforcedF+Kdsmustequalzero.Therefore,forcebalancegives

PAGE 102

Asecondequationisobtainedfromtheconditionthatthetotalmomentoftheforces(torques)onthiselementmustbezero.LetMbethemomentoftheforcesonthecross-section.Thisisthemomentaboutapoint(theorigin)O0whichliesintheplaneofthecross-sectionofthelowerend.ConsideranotherpointOlyingintheplaneoftheupperendoftheelement.TheforcesonthisendgiverisetoamomentM+dM.ThemomentaboutOoftheforcesonthelowerendhastwocomponents:themomentMofthoseforcesabouttheoriginO0intheplaneofthelowerend,andthemomentaboutOofthetotalforceFonthatend,givenbydsF,wheredsisthevectoroftheelementoflengthoftherodbetweenO0andO.ThemomentduetotheexternalforcesKisofahigherorderofsmallnessandhenceisneglected.ThusthetotalmomentactingontheelementconsideredisdM+(dsF)andthishastobezeroforequilibrium.Dividingthistotalmomentbyds,notingthatds=ds=ttheunitvectortangentialtotherod(regardedasaline),andtakingthelimitds!0,wehavethetorquebalance Equations( 5-5 )and( 5-6 )formacompletesetofequilibriumequationsforarodbentinanymanner. 5-5 )becomes ThevalueoftheconstantisfoundfromthefactthatthedierenceF2F1oftheforcesattwopoints1and2isthesumofalltheexternalforcesappliedtothesegmentoftherodbetweenthetwopoints,i.e.,F2F1=PK,where2isfurtherfromthepoints=0than1.Inparticular,ifonlyoneconcentratedforcefactsontherod,andisappliedatthefreeend,thenF=constant=fatallpointsontherod.Equation( 5-6 )isalsosimplied.Ifwenotethatt=ds=ds=dr=ds,whereristhepositionvectorfromanylab-xedframetothepointconsidered,andthattheforceFisindependentofthearclength,Equation( 5-6 )canbeintegratedtogive

PAGE 103

69 ].Forpurebendingina2-Dplanewherethetangentoftherodisalongthez-axisandthebendingoccursinthezy-plane,themomentofbendingisproportionaltothelocalradiusofcurvature.Thenormalvectorisgivenbytheunitvectoralongdt=ds=(1=R)n=(1=R)^y,whereRisthelocalradiusofcurvature.Themomentactsalongthex-axis,whichisthebinormalaxisofaFrenetframe(denedbythetangentt,normaln,andbinormalb=tn)denedatanypointoftherodelement.Forarodofcircularcross-section,themomentsofinertiaIx=Iy=I.Ingeneral,foracircularrodsubjecttopurebending(notwist),theconstitutiverelation

PAGE 104

oflinearelasticityisgivenby Therefore,combiningthetorquebalance(Equation( 5-6 ))andtheconstitutiverelationgives Equation( 5-10 )istheequationforpurebendingofacircularrod. Inbothcases,theendsoftherodsarespecied.Whereastheinitialorientationoftherodisspeciedintheformercase,thelattercasehasazeromomentattheinitialend(aswellastheotherend).TheanalysisthatfollowsisverysimilartothoseinLandauandLifshitz[ 69 ]andinLove[ 70 ]. Wetaketherodtobebentinzy-plane,therefore,theforceactingontherodisconstrainedtolieinthezy-planeaswell.Introducingtheanglebetweenthetangenttoftherodandthez-axis,onehasdy=ds=sinanddz=ds=cos,wherey,zarethecoordinatesofapointontherod,and,F=Fy^y+Fz^z.Dening=tan1(Fy=Fz),onecanrotatetheaxissuchthatthenewz-axisisparalleltoF.Thischangeofframesimpliestheanalysisconsiderably.Wenowhave,intherotatedframe,F=f^yandt=sin^y+cos^z,wheref=p 5-10 )becomes ds2+fsin=0(5-11) inanon-dimensionalformwiths=s=Landf=fL2=EI.Firstintegrationgives 1 2d ds2=fcos+C(5-12)

PAGE 105

Theboundaryconditionsintherotatedframefortheproblemsofinterestarefollows: End1: ClampedEnd:s=0;=;y=0;z=0; HingedEnd:s=0;d End2: HingedEnd:s=1;d Further,thereisaconstraintinthetotallengthoftherodgivenbytheincompleteellipticintegraloftherstkindF sin0=2;sin0 Theshapeoftherodisgivenby and sin0=2;sin0 sin0=2;sin0 whereEistheincompleteellipticintegralofthesecondkind.Therefore,foraknownforce(f,),onegetstheappropriate0fromthelengthconstraintandsolvesfortheshapeofthelament.Conversely,ifonexestheendpositionsyN=y(s=1;=0)andzN=z(s=1;=0),oneshouldbeabletogettheforce(f,)requiredtoxtheendsoftherodattheoriginand(yN,zN)withtheendsbeingclampedorhingedasthecasemaybe. 70 ].Thissimpliestheaboveequations.Thelengthconstraintbecomesacompleteellipticintegraloftherstkind,givenby

PAGE 106

Theshapeoftherodissymmetricwithrespecttothecenterwhere=0.Theendpointsintherotatedframearegivenbythecompleteellipticintegralofthesecondkind andyN=0.ForagivenzN,thecorrespondingnon-dimensionalforcesand0areobtainedbyminimizingthefunction withrespectto0.Itisimportanttonotethatthesesolutionsexistonlybeyondacritical(non-dimensional)forcegivenforthedouble-hingedcaseby whenthelinearshapeceasestobestable.Figures5-1athrough5-1fshowtheshapesoftherodforvariouszN,analogoustotheinexionalelasticapredictedbyLove[ 70 ].Thecorrespondingnon-dimensionalforcesareshownasafunctionoftheendpositionzNinFigure5-2.Itcanbeseenthatastherodisbentfurther,atverylargeforces,therodformsaloop. Sincethebendingforceactsalongthelineconnectingthetwoendsoftherod,onenotesthat,foranygivensetofendpositions(0,0)and(yN,zN)oftherod,thereisauniquevalueofthebendingforce(f,)whereisgivenby andfisgivenbyEquation( 5-16 ).

PAGE 107

Figure5{1: Shapesofadouble-hingedrodundercompressionforforcesgreaterthanthecriticalforce.(a)f=fc=2.(b)f=10:68.(c)f=13:82.(d)f=16:36.(e)f=21:55.(f)f=400:0.

PAGE 108

Figure5{2: Forceonthedouble-hingedrodasafunctionofendposition,correspondingtotheshapesshowninFigure5-1. higher(second)ordersolution,wheretherodhastwoinexions-oneatthehingedend(s=1)andanothersomewhereinthemiddleoftherod(si;0
PAGE 109

discussedinthedouble-hingedcaseexistonlybeyondanon-zerocriticalbendingforce,uponxingtheorientationofatleastoneendoftherod,thelinearshapebecomesunstableforanynon-negligiblevalueofthebendingforce.Thereforetherodbendsunderanyforcegreaterthanzero.Itisseenthatthereareeitherone,twoorthreesolutions,dependingonthestrengthoftheyandthezcomponentsoftheforce.Thisisseenbybracketingthezerosofthefunction sin0=2;sin0 wheremistheorderofthesolution.Calculationswereperformedforvariousvaluesofthenon-dimensionalforcesFy(50:0
PAGE 110

Possiblelamentendpositionsforalamentclampedattheoriginandorientedalongthez-axisforvariousaxialandcompressiveforces.Notetheexistenceofmultiplesolutionsforeachforce.Fy>0. changeintheendposition,andthiscouldbeareasonwhyweobservesofewpossibleendpositions.Ifoneweretoconsideranexperimentofathin,elasticrodbentwithclamped-hingedconditions,onemayvisualizethatallpossible(y,z)positionsastheend-positionoftherod.Thequestioniswhetherallowingtherodtohaveasmallshearorallowingthemomenttotendtozeroratherthanenforceittobezeroatthehingedendwouldallowsignicantlymoresolutions.Onewaytotestthiswouldbetoapplyaxedbutnon-zeromomentatthehingedendandstudythesolutions.

PAGE 111

Possiblelamentendpositionsforalamentclampedattheoriginandorientedalongthez-axisforvariousaxialandcompressiveforces.Notetheexistenceofmultiplesolutionsforeachforce.Fy<0. Shapeofalamentclampedattheoriginandorientedatanangle45de-greesfromthez-axisundercompression.ThesolutionsshownarerstordersolutionsofEquation( 5-24 ).

PAGE 112

71 72 73 74 75 ].Thereisconsensusintheliteraturethatthepush-pullforcesinvolvedinthepropulsionaregen-eratedbyactinlamentelongation.Inordertounderstandtheunderlyingbiophysicalmechanismthatgeneratesthenecessaryforcesfromthebiochemicalprocessoflamentelongation,anumberofmodelshavebeenproposed[ 18 19 76 77 78 79 80 ].Inthenextsection,wesummarizethebiochemistryofactin-basedmotilitywithinthecontextofthemotilityofListeria.Thissectionalsodiscussesthevariousmodelsproposedtoexplainactin-basedmotility,savethe`actoclampin'modelproposedbyDickinsonandPurich[ 18 ].Section 6.3 discussestheactoclampinmodelinsomedetail.WethenmodeltheactinlamentsasslenderelasticrodsandseektoreproducetheresultsofDickinsonandPurich[ 18 ]withinthecontextoftheactoclampinmodel.Wethenseektoshow,usingaback-of-the-envelopeestimate,howactoclampinmodelcanaccountforthetwisttorquesgenerateduponmonomeradditionandhowthisextensionexplainssomeoftheexperimentallyobservedfeaturesofthemotilityofListeria,i.e.,thelong-lengthscaleright-handedhelicaltrajectories[ 81 82 ]. 98

PAGE 113

subunitsareconnectedbya2.7-nmaxialrise.Thispolymerizationprocess,theso-calledtreadmilling,isahighly-regulatedsteady-stateprocessmaintainedbytherate-limitingdepolymerizationofthelamentatoneendevenasitpolymerizesontheotherendatacontrolledrate,withthelengthofthelamentremainingapproximatelyconstant.Thepredominantlypolymerizingendofthepolarlamentiscalledthebarbed(+)end(thebarbedendofthemonomerisexposed),whilethepredominantlydepolymerizingendiscalledthepointed()end.MonomericG-actinattachedtoATPassemblesnearthebarbedend,associatedwithanirreversiblehydrolysisoftheboundATP.Thedepolymerizedactin-ADPexchangesitsADPforanATPandisreadyforassemblyonceagain. Anumberofproteinscontroltreadmilling.Cappingproteinsthatcapthebarbedendsraisethetreadmillingofuncappedlaments,whereasthosethatcapthepointedendsstabilizethelaments.ADF(actindepolymerizingfactor),suchascolin,facilitatedepolymerizationatthepointedend.ColinboundwithG-actin-ADPalsoinhibitsADP/ATPexchangeandsoinhibitspolymerization.Prolinregulatesactinpolymerizationbyforminga1:1complexwithG-actin.ThiscomplexisknowntopromoteADP/ATPexchange,therebyincreasingthelocalconcentrationofG-actin-ATP,theformthatpolymerizes.Prolinmayalsosequesteractinmonomers.Arp2=3isacomplexthatnucleatesactinpolymerizationandfacilitateslamentbranching.Itcontainstwoactin-relatedproteinsArp2andArp3,alongwithveothersmallerproteins.Whenactivated,theArp2=3complexbindstothesideofanexistinglamentandandnucleatesassemblyofanewactinlament,resultinginaY-shapedbranch.Gelsolin,activatedbyCa2+ionsinthecell,seversthebarbedendofthelamentandcapsit,preventinglamentregrowth.Thisresultsintheformationofmany,shorteractinlaments.Moredetailsofthetreadmillingcouldbefoundin[ 71 72 73 74 ]. Inthecontextofactin-basedmotility,thepathogenListeriamonocytogenesisofgreatinterestbecauseofitsremarkableabilitytoreproducethesalientfeaturesofin-tracellularactinmotilitybyusinghost-cellcytoskeletalcomponents.TheproteinActA,presentonthesurfaceofListeria,promotesmotilitysingle-handedlybystimulatingArp

PAGE 114

2=3-mediatedlamentnucleationandbylocalizingandbindingVASP(Vasodilator-StimulatedPhosphoprotein),whichinturnmediatesprolin-actin-ATPadditiontothebarbedendsoftheelongatinglaments[ 83 ].Kuhneletal.[ 84 ]havestudiedthecrystalstructureofVASPandhaveidentieda45-residue-longtetramerizationdomainthatformsaright-handed-helicalcoiled-coilstructure,whichcanbindprolin-actincomplexandF-actin.Therefore,usingActA,Listeriacaneectivelyassembleandengagethecytoskeletalcomponentsdiscussedaboveandreproduceactin-basedmotility.Insummary,ActAonthesurfaceofthepathogenlocalizesandbindsVASP,whichinturnbindsthebarbed-endoftheelongatingactinlamentandprolin-actin-ATPthatpolymerizesonthebarbedend.WherethehydrolysisofATPtsinthissequencehasnotbeenestablished,butitisknownthathydrolysisprovides20kTperATP(andhencepermonomer).Oneofthekeyinterestsinthissystemandothersystemsofactin-basedmotilityistounderstandingoftheunderlyingbiophysicalmechanismthatgeneratesthenecessaryforcesformotility.Beforewesummarizethemodelingeortstodeterminethismechanism,werstdiscussoneoftheerstwhilecontroversialsuppositionthattheactinlamentisboundtothemotilesurfacebecausesuchabindingplacessignicantrequirementsonthebiophysicalmechanismofforcegeneration. Thethesisthattheactinlamentistetheredtothemotilesurfacehasbeenconrmedbyanumberofexperimentalobservations.ThiswasobservedwithListeriabyKuoandMcGrath[ 20 ],whoshowedusingahigh-resolutiontrajectoryanalysisthattheuctuationsofthebacteriumwasafewordersofmagnitudelessthanthoseofthefreepathogens.Theyfurthershowedthatthemotilityoccurredinactinmonomer-sizedstepswithanarrowstep-sizedistribution(5:10:1nm).Cameronetal.[ 85 ]showedthatsucientlysmall(50nm)ActA-coatedbeadsgrewlongactin-lamenttailsconsistingofasinglelament.Suchlongsinglelament-tailscouldnotformifthelamentwerenotalwaysattachedduringelongation,becauseifthelamenthaddissociatedfromthebeadsurfaceforthe0:01s1requiredtoaddanactinmonomer,thebeadwouldhavediusedarootmeansquareddisplacement250nm

PAGE 115

fromthelamentend(DickinsonandPurich 77 ]usedanopticaltraptomeasuretheforcerequiredtoseparatethebacterialcellfromtheactintail,whichturnedouttobesucient(>10pN)towarrantatleasttransientattachmentofthetailtothecellsurface.RobbinsandTheriot[ 81 ]andZeileetal.[ 82 ]haveshownrecentlythatthetrajectoriesofListeriaandthecomettailsattachedtothebacterialsurfacearelong-length-scaleright-handedhelices.Thisshowstheexistenceoftorquesonthesurfaceandthetails,whichmustobviouslybeattachedtoeachother.Finally,theworkofKuhneletal.[ 84 ]hasprovidedastructuralbasisfortheattachmentofactinlamentstothemotilesurface.RobbinsandTheriot[ 81 ]havearguedagainstVASPbridgingActAtolamentsonthebasisoftheirobservationsofslowrotationofafewweaklymotilebacteriawithanActAvariantwithouttheVASPbindingregions.However,itappearsthatVASPbindingtothismutantActAhasnotbeencompletelyruledoutduetothepresenceoftwoadditionalpotentialVASP-bindingsites(DickinsonandPurich,personalcommunication).Further,SouthwickandPurich[ 86 ]havedemonstratedthatthemicroinjectionofasyntheticVASP-bindingpeptide,whichblocksActAbinding,completelyinhibitsListeriamotility. Threesignicantmodelshavebeenproposedthatseektoexplainforcegenerationinthesesystems.Inelasticpropulsion-typemodels[ 76 77 ],thepushingforcesaregen-eratedbyanetworkofgrowinglamentsonthecurvedmotilesurface.Circumferentialstressesmountuponbarbed-endgrowthuntilacriticalpointisreachedwherelocalgelfractureoccursandtheobjectisthrustforward.Thesecondclassofmodelsaretheso-calledBrownianratchet-typemodels[ 78 79 80 ],wheretheforceisgeneratedbytheBrownianmotionofthepolymerizinglaments.Monomeradditiontakesplaceduringthemonomer-sizedexcursionsofthebarbedendsfromthemotilesurface.Inconsistentwiththerecentexperimentaladvancesthathaveestablishedlament-surfaceattachmentandtheroleofArp2=3complexinlamentnucleation,MogilnerandOs-ter[ 80 ]haveamendedtheirelasticBrownianratchetmodeltoatetheredratchetmodel,whichemploystwotypesoflaments-thosethataredetached,undergoelongationand

PAGE 116

generateforce,andthosethatundergonucleationbysurface-boundArp2=3complex,whichisalsotetheredtotheir`mother'lamentsalreadylinkedtothecytoskeletalnetwork.Suchamechanismgeneratespush-pullforcesuntiltheArp2=3unitsdetachfromthemotilesurface,atwhichpointtheybecome`working'lamentsthatgenerateforcestopropelthesurface.Thereareanumberofissueswiththismodel,includingthefactthatitdoesnotconsistentlyaccountforthestructureofVASPdeterminedbyKuhneletal.[ 84 ],andthatitisunabletogeneratetwisttorquesresponsiblefortheobservedlong-lengthscalerotationofListeria.Thethirdclassofmodels,proposedbyDickinsonandPurich[ 18 19 ]considermotilityeectedbyanity-modulatedprocessive`actoclampin'motors.Abriefdiscussionoftheactoclampinmodelisprovidedinthenextsection. 19 ]performedanenergyinventoryoflamentelongationandtreadmillingandconcludedthefollowing: 87 88 ]).

PAGE 117

bythelamentend-trackingmotorwhilealsostronglytetheringtheelongatinglamentendtothemotilesurface. Dickinsonetal.[ 19 ]proposedtwomechanismsbywhichend-trackingmotorscouldfacilitateelongationandforcegenerationofanactinlament.Inbothmechanisms,themotorcomplexisboundtothesurface.Asspeculatedintheprevioussection,apossiblemotorcomplexinthemotilityofListeriaistheActA-VASPcomplex.BothmechanismsinvolvemonomeradditionfollowedbyATPhydrolysisandmotortranslocation.ThekeydierenceinthetwomechanismsisthatinonemechanismthemotoralsofacilitatesthetransferofanATP-G-actin(withorwithoutprolin)tothelamentinadditiontolamentelongationandmotility. TodeterminewhethertheactoclampinmodelcangeneratethestepwiseprogressionandsmalluctuationsofListeriaobservedbyKuoandMcGrath[ 20 ],DickinsonandPurich[ 18 ]examinedthemotilityofthesurfacebyrelatingthesurfaceend-positiontothecompressiveandtensile(push-pull)forcesappliedonthesurfacebythelamentsmodeledasHookeansprings.Theseforcesaregivenby whereTandCarethelamentstinessesundertensionandcompressionrespec-tively,thesubscriptireferstotheithlament,z0representsthelamentendposition,andzsrepresentsthepositionofthemotilesurface,withthedirectionofelongationandmotilitybeingassumedtobealongthez-axis.Themotortranslocationintheithlamentcorrespondstoashiftinz0;ibyadistanced.Themeantotaltimerequiredforeachcycle,inwhichthemotoradvances5:4nm(duetotheadditionofmonomerstobothprotolaments)consistsofthetimeTmrequiredformonomeradditionandthetimerequiredforshiftingandredockingofthemotortothenewterminalmonomer.Tmisabout0:005to0:1s,determinedfromthelamentelongationrates(typically0:05to1m/s).Sincemotortranslocationisanity-modulatedthroughATPhydrolysis,thetimeisdeterminedbyassumingaatenergylandscapebetweentheshallowenergywellatthehydrolyzedsiteandaninnitelydeepwellcorrespondingtoirreversiblebindingatthenon-hydrolyzedsite.Thetranslocationisthentreated

PAGE 118

asaone-dimensionaldiusionofthefreelamentendoveradistanceof5:4nm.Thisdiusionisfacilitatedbythetensileforceonatautlamentoropposedbytheexuralforceonacompressedlament.AschematicillustrationoftheactoclampinmodelandforcegenerationinthemodelisprovidedinFigure1-4.ThedetailsareavailableinDickinsonandPurich[ 18 ]. Neglectingtheinertiaofthemotilesurface,themotionofthemotilesurfaceisgovernedbythestochasticdierentialequation where istheinstantaneousdeterministicvelocityofthemotilesurface,fisthedragcoe-cientassociatedwitheachlament, Nf=Df=N(6-4) istheeectivediusivityofthesurface,Dfisthelamentdiusivity),anddWtisanincrementintheWienerprocess,denedsuchthat (6-5) DickinsonandPurich[ 18 ]numericallyintegratedequation( 6-2 )withatimeincrementtasaboutone-tenthofthecharacteristicrelaxationtimeofasingletrailinglament,about0:5sfor80laments.Theinitialequilibriumendpositionsofthelamentswererandomlydistributedovera60-nmrange.Monomeradditionwasperformedwhenauniformpseudorandomnumbergeneratorontheinterval[0;1]successfullyyieldedvalueslessthant=(Tm+).TheyusedthevaluesofT=60pN/nm,C=0:15pN/nm,andthediusivityofthelaments,Df=4106nm2/s.

PAGE 119

6.4.1Actin-BasedMotilityofListeria 5 .Thestretchingforcesoftherodsundertensionweremodeledbyassumingthattheresultantdeectionsaresmall.Thesmalldeectionassumptionentailsthatthedirectionofthetangentvectorvariesslowlyalongitslength,i.e.,theradiusofcurvatureofthebentrodislargeeverywherecomparedwiththerodlength.Thisistypicallysatisedifthetransversedeectionissmallcomparedtothelengthoftherod,anassumptionmostlytrueforlamentsundertension.Filamentscanbeundertensioniftheyarestraightandstretchedbeyondtheircontourlengthsoriftheyarebentandsucientlypulledawayfromtheirinitialaxisoforientationthattheyarestretchedbeyondtheircontourlengths.Sinceactinlamentsaresti(Young'smodulusE=1:8109N/m2[ 89 ]),thetotalstrainmaybeassumedtobeasmallfractionofthecontourlength,evenwhentheyarebentandpulledawayfromtheirinitialorientations.Further,thelamentsareassumedtobeincompressibleandthereforetheminimumlengthofthelamentsistheirrestlength(undertheabsenceofanystress). Withinthesmall-deectionapproximation(validforactinlamentsfordeectionslessthan=4),thelamentshapey(z)andtheforceFzaregivenbytheequations, dz4kzd2y dz2=0;(6-7) where withLbeingthelengthofthelament(chosentobe210nm,approximatelythreehelicalrepeatsofthelamentandIisthemomentofinertiaofthelamentgivenby

PAGE 120

Shapeofalamentundertension.Fz=19:2pN. rbeingtheouterradiusoftheactinlament(equalto8nm),assumingthelamenttobeahomogeneouscylinder.Theboundaryconditionsaresimilartothoseofcom-pressedlaments-hingedatbothends. TheshapeofalamentundertensionisshowninFigure6-1.Thiscalculationcorrespondedtoatensileforceof19:2pN.Asimpletestofthefeasibilityofexaminingactin-basedmotilityusingactoclampinmodelwiththelamentsmodeledaselasticrodswasconductedbyconsideringsingle-lamentelongation.HerethelamentisalwaysundertensionandthecodeistestedtoensurethatthelamentgrowsforeveryTmseconds,consistentwiththeactoclampinmodelequationsdiscussedintheprevioussection.Inallourcalculations,thediusionofthesurface(thedWtterm)hasbeenneglected.TheresultsareshowninFigures6-2and6-3.Itisseenthatthestepwisemotionofthemotilesurfaceisreproduced.ThiscalculationwasrepeatedforvelamentsandthetrajectoryofthemotilesurfaceisshowninFigure6-4.OnecanseequalitativeagreementwiththeresultsofDickinsonandPurich[ 18 ],wheretheyhavemodeledthelamentsaslinearsprings,allelseidenticalbetweenthetwocalculations.

PAGE 121

Motilityofthesurfaceduetoasinglelament:SurfacepositionandFila-mentendposition. Onceagain,itisimportanttonotethatinallthesecalculations,theeectsofdiusionofthesurface(thesecondterminEquation( 6-2 ))wereneglected. 81 ]rstreportedthatListeriamonocytogenesrotatesarounditslongaxisasitispropelledbyactinpolymerization,bytrackingtheuorescenceofmicrospheresnon-specicallycoupledtothebacterialsurface.Theyobservedanaveragebacterialspeedof0:09m=s.Zeileetal.[ 82 ]studiedthemotilityofListeriainbrainextractssupplementedwithrhodamine-actinandobservedthattheresultingrockettailsresistdepolymerization,andthereforefacilitateunambiguousobservationoftheListeriatrajectories.Therockettailsshowedconclusiveevidenceofright-handedtrajectorieswithaperiodicityofthe4to40m,nearly100to1000timeslongerthanthehelicalperiodicityoftheright-handedactinlaments(70nm). Intheactoclampinmodel,astheanity-modulatedmotorprocessivelytracksalongtheelongatinglament,eachcycleofmonomeradditionandmotortranslocationshouldgeneratetranslationalandtorsionalforces.Inthefollowingback-of-the-envelope

PAGE 122

Motilityofthesurfaceduetoasinglelament:Force. calculation,weshowthatwithintheframeworkofanity-modulatedclamped-lamentelongationmodel,onecanexplaintheaboveexperimentalobservations. Basedonthehelicalperiodicityofactinlaments,wenotethatthetwiststoredineachmonomerunitis2=13.Further,theaveragelengthofalamentfromthesurfacetothecytoskeletalnetworkisaboutthreehelicalrepeats,whichistakentobeL210nm.Tsudaetal.[ 89 ]haveestimatedthetorsionalrigidityofasingleactinlamenttobeR81026Nm2.Thetwisttorqueoneachlamentuponmonomeradditionisgivenby, Theenergyrequiredtostorethetwist2=13isgivenby, 2R;f2 whichisnearlyhalftheenergyobtainedfromthehydrolysisofanATPmoleculeattheATPandADPconcentrationsinthesystem.Thisleadsustoconcludethatitismuchhardertotwistanactinlamentthanitis,forinstancetobendit.(Tocompare,the

PAGE 123

Trajectoryofthemotilesurfacepropelledbyvelaments. energycostofbendinganactinlamentoflength210nmby10nmisapproximately1:4pNnm,whenoneendisclampedandthefreeendisdisplaced.)Therefore,itisreasonabletoassumethatinthemotorcomplex,theclampismorelikelytostorethetwist.Therealityisprobablysomewhereinbetween.However,weproceedwiththeassumptionthatthetwisttorquegeneratedduetomonomeradditionisappliedontothemotorandthatthemotorcanbemodeledasanelasticspring.WefurthertakethelengthofthemotortobeLc10nmandthetwiststinessofthemotortobeR1000pNnm2.Thetwisttorqueisthen, Theenergycostforthisisoftheorderof10pNnm,whichisamorereasonablevalue.Now,inordertobalancethenettorquegeneratedbymanysuchmotors,themotilesurfacerotatesbyananglewhichwecall.Anyrotationofthemotilesurfacewouldcauseallthemotorstorotateaswell.Sincethemotorsareclampedtothesurface,theirpositionsaredeterminedinpolarcoordinatesbyasetof(ri,i),wherethesubscriptsilabelthemotors.Uponrotationofthesurfacebyanangle,thepositions

PAGE 124

ofthemotorsonthesurfacechangefrom to (6-15) Thereforethechangeinpositionsaregivenby, xi=ri[cos(i+)cosi] (6-17) yi=ri[sin(i+)sini]: Theresultantforcesonthemotorscanbemodeledaselasticsprings,givingriseto, Thenetforcecanberesolvedintocomponentsas, (6-20) where^isaneectivespringconstantforeachmotor,Nisthetotalnumberofmotorsattachedtothesurface,andCandSaremeanx-andy-positionsofthemotors,givenby Thetorqueduetothisrotationactsalongtheaxisofrotation,i.e.,thez-axis.Foreachmotor,thistorqueisgivenby

PAGE 125

Thereforethenettorqueactingonthesurfaceisgivenby 2^Na2sin;(6-25) whereaistheradiusofthemotilesurface,andthemeansquarepositionofthemotorsonthesurfaceisgivenby 2Na2:(6-26) Torquebalanceonthesurfaceis whichgives sin107 Beforeweproceedwithourestimates,someobservationsaredue: 82 ] Theangleofrotationofthesurface,andhencethepitchoftheright-handedhelicaltrajectory,dependsonthevalueoftheeectivespringconstantofthemotor,^.Weexpectthisvaluetobelowerthanthecompressivestinessofanactinlament(0:15pN/nm).Takinganupperboundof0:15pN/nm,weestimatethatthesurfacewillrotatebyanangleof6:67104radiansforevery5:4nmlateraldisplacement.Thismeans,forarotationof2radians,thelateraldisplacementisabout50m,whichisclosetotheupperboundoftheperiodicityobservedbyZeileetal.[ 82 ] Thekeyconclusionfromtheaboveestimateisthattheactoclampinmodelcanindeedpredictthelong-lengthscalerotationofListeriamonocytogenesduringactin-basedmotility.Amoredetailedanalysisofthissystemisleftforfuturework.

PAGE 126

7.2.1InteractingPolymerLayers 112

PAGE 127

thelayers,andthelatterformulationparticularlyhasbothqualitativeandquantitativeagreementwiththeresultsofthelatticesimulations. 34 ]inthesingle-chainregime.Partialcorrectionsofexcludedvolume(SF2)andanisotropiceld(SCAFT)slightlyimprovethemean-eldformulationatlowsurfacecoverages,buttheattractiveforcesbetweenthesurfacesarenotqualitativelysignicant. 69 ]andLove[ 70 ]. where,EIistheexuralrigidityoftherodoflengthL.TheshapesoftherodsandthecorrespondingforcesareconsistentwiththeresultsinLove[ 70 ].Foranysetofendpositionsoftherod,onecanobtainauniqueforcevectorthattherodissubjectedto.

PAGE 128

andassumesabentshapeconsistentwiththeshapesdictatedbytheappropriateellipticintegrals.Foraknownforcevector,therearemanypossibleendpositionsoftherod,andthereversemappingisnotunique.Further,forreasonsunclearasyet,notallendpositionsseemtobeavailableforthehinged-endpositionofarodclampedattheorigin,whenevaluatingthesolutionsasellipticintegrals. 18 ]wheretheymodelactinlamentsaselasticsprings.Theseresultsarealsoconsistentwiththeexperimentallyobserved5:4nmstepsofListeria[ 20 ].

PAGE 129

chains.Whenthesurfaceisadsorbing,poorsolventqualitycausesstrongeradsorption,therebykeepingthechainsatandtogether.Therefore,onewouldexpectthepolymerlayerstonotinteractuntiltheyaremuchclosertoeachotherthanobservedingoodsolvents.Further,onewouldexpectlesserattractiveforcesintheundersaturatedregimeduetothelesserdensityofbridgesinthesystem.Athighercoverages,thecompetitionbetweenbridgingattractionandstericrepulsionwouldbetitledtowardsthelatterbythepoorsolventquality.Itwouldbeinterestingtoseeifthemean-eldformulationscancapturethisexpectedbehaviorunderpoorsolventconditions.Further,itwouldbeinterestingtoseewhether,ifatall,mean-eldformulationswouldpredictattractiveforcesbetweenthesurfaces. 27 ],whereinthecontactdistributionmethodisusedtoobtainfreeenergydierences.Thepresentworkhasmadesomecomparisonswiththoseresultsdespitetheambiguityastowhetherthesefreeenergydierencesindeedcorrespondtothenormalforcesofcompressionofthelayers.Anindependentinvestigationofthisusing,sayMDsimulations,hastobedonetoanswertheabovequestionbeforetheycanbeusedasbenchmarks. 90 ]).Itwouldbeinterestingtondifthetwo-dimensionalmeaneldleadstobetterpredictionsintheundersaturated

PAGE 130

regime.Further,itwouldbeinterestingtoseeif,inthetwo-dimensionalmean-eldformulation,theexcessfreeenergyderivedfromthepartitionfunction,isstillequaltothelmtension.Thiswoulddetermineiftheinabilitytowritethenormalforceofcompressionasthenegativederivativeofthefreeenergywithrespecttosurfaceseparationisuniquetoaone-dimensionalmean-eldapproximation. 42 ]haveshowntheequivalenceofthetwo-orderparametertheoryandtheone-dimensional,latticemean-eldformulationofScheutjensandFleer,onecouldalsoexploremeansofimprovingthemean-eldapproximationsandchainexcludedvolumestatisticswithintheanalyticalmean-eldformulations. 20 77 81 82 84 85 ],theactinlamentsthatpropelListeriamotilityareattachedatalltimetothemotilesurface,andareclampedtothecytoskeletalnetworkastheygrowthroughbranching.Theattachmenttothemotilesurfaceisachievedthroughthepresenceofa`clamp'(believedtobetheVASPcomplexoroneofitsanalogs).Whiletheendsegmentortheprevious

PAGE 131

segmentstaysattachedtothe`clamp',itisreasonabletoassumeatleastapartialexibilitywithrespecttoitsorientation.Therefore,theclamped-hingedcasediscussedinChapter 5 (seeSection 5.4.2 )ishighlyrelevantinthecontextofmodelingListeriamotility.Inexaminingthiscase,weobservedtheexistenceofmultiplesolutionsfortheshapeoftherodforagivenforceandnotedthatmorethanoneofthemcouldbestable.AthoroughstabilityanalysisofthesystemisessentialtobetterunderstandhowonemayusethesecalculationsinthecontextofListeriamotility.Further,itwasobservedthatforagivenforceandgivenclamped-endposition,notallendpositionsoftherodsatisfythezero-momentconditionofthehingedend.Inotherwords,therewasasignicantdicultyinmappingaknownforcevectorwiththehinged-endpositioninthetwo-dimensionalplane.Theseissuesfurthercomplicatethesolutionoftheinverseproblem,namely,predictingtheforcealamentisunderforagivensetofclamped-andhinged-endpositions.ItisthisinverseproblemthatonehastoapplyinthecontextofListeriamotility. 81 ]aswellasbyZeileetal.[ 82 ]thattheforcesandtorquesgeneratedintheactin-basedpropulsionofListeriamonocytogenescausethebacteriumandhencethecomettailattachedtoittotracearight-handedhelicaltrajectories.Themostlogicaloriginofthesetorquesandtheresultanthelicityinthetailsistheinherenttwistintheactinlaments. Actinlamentshaveaninherenttwistwitheachmonomercontributingatwistangleof2=13radians.(Thelamenttwistsby2radiansforeverythirteenmonomers.)Asactinlamentsgrow,theenergyexpendedforstoringatwistof2=13radiansinalamentduringtreadmillingisabout45pNnm.Thisindicatesthatitismuchhardertotwistthelamentsthantobendit.Thereforewemayconcludethatthetwistisnotstoredinthelaments.Allthetwisttorquemaybereasonablyassumedtobeappliedtotheclamp,whichmaybemodeledasatwistspring.Basedonthisassumption,wepresentedanestimatethatexplained,withintheframeworkoftheanity-modulatedactoclampinmotormodel,theexperimentallyobservedperiodicity(pitch)ofthehelixanditsright-handedness.OneofthemostessentialextensionsofthepresentworkistoincorporatethistorquebalanceinthebiophysicalmodeldiscussedinChapter 6 .Thiswouldaccountforthetwistgeneratedinherentlyinthesystemandshouldhopefullypredicthelicaltrajectoriesofthemotilesurfacewithreasonablepitches. 6 ,onealsohastobalancetheradialforcesgeneratedduetotherotationofthesurface,shownintheestimateattheendofChapter 6 .Further,theeectsoftherelativepositionsofthelamentsonthemotilesurfacemustalsobeaccountedfor.Whilemostlamentscanbereasonablyassumedtobeattachedtooneendofthebacterial

PAGE 132

surface(ratherthanthesides),therearesomelamentsontheouteredgeofthetailthatareindeedattachedtothesidesofthesurface.Whiletheselamentsarelesslikelytobeundertension,theyaremostaccessibletoactinmonomersandthereforeshouldhavenon-negligibleprobabilitiesofgrowth.Suchgrowthscancausetumbling-likemotionofthesurfaceandcouldcontributetosomeofthetighterhelicalcomettailsobservedbyZeileetal.[ 82 ](Theobservedpitcheswouldbeclosertothebacteriallength;SeepanelsJandKinFigure1.)Aninvestigationofthesurfacetrajectoriesundertheseconditionsisimportant. Anotherimportantconsiderationisthedistributionofmotorsonthemotilesurfaceisanotherfactorthatcouldplayasignicantroleindeterminingthetrajectoryofthesurface.Incasesofnon-uniformdistributionofthemotorsonasurface,theshiftinthecenterof`gravity'fromtheaxisofthesurfacecouldresultinapersistentdisplacementofthemotilesurfaceoitsaxis,resultinginalongerlength-scaleperiodicityorcurvaturesuperimposedovertheobservedhelicalpitch.Atleastoneobservedtrajectory(panelAinFigure1ofZeileetal.[ 82 ])showsalongercurvaturesuperimposedonthehelicaltrajectoryofthebacterium.Unfortunately,motordistributiononmotilesurfacesisnotalwaysreadilyaccessiblethroughexperimentscurrently.Thecausesoftheso-calledsymmetrybreakinginanumberofactin-basedmotilitysystemsmaywellbehere.However,acomputationalexaminationoftheeectsofthedistributionofmotorsonthesurfaceisanachievableextensionofthepresentwork.

PAGE 133

ThisappendixcontainsadditionalresultsonthestructureoftheadsorbedpolymerlayersasevaluatedfromSCAFTandSF2(numberandsizeofbridges,loops,andtailsasafunctionofsurfacecoverageandsurfaceseparation(H/a)). FigureA{1: Averagenumberofbridges,nbr,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0. 119

PAGE 134

FigureA{2: Averagesizeofbridges,lbr,asafunctionofsurfacecoverageforacon-stantadsorptionenergy,s=1:0kTandH=a=5:0. FigureA{3: Averagenumberofloops,nlo,asafunctionofH=aforaconstantadsorp-tionenergy,s=1:0kTand=0:75

PAGE 135

FigureA{4: Averagesizeofloops,llo,asafunctionofH=aforaconstantadsorptionenergy,s=1:0kTand=0:75. FigureA{5: Averagenumberofloops,nlo,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0.

PAGE 136

FigureA{6: Averagesizeofloops,llo,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0. FigureA{7: Averagenumberoftails,nta,asafunctionofH=aforaconstantadsorp-tionenergy,s=1:0kTand=0:75.

PAGE 137

FigureA{8: Averagesizeoftails,lta,asafunctionofH=aforaconstantadsorptionenergy,s=1:0kTand=0:75. FigureA{9: Averagenumberoftails,nta,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0.

PAGE 138

FigureA{10: Averagesizeoftails,lta,asafunctionofsurfacecoverageforaconstantadsorptionenergy,s=1:0kTandH=a=5:0. FigureA{11: Averagenumberofloops,nlo,asafunctionofrescaledsurfacecoverage=0,foraconstantrescaledadsorptionenergy,sc=0:74,andH=a=5:0.

PAGE 139

FigureA{12: Averagesizeofloops,llo,asafunctionofrescaledsurfacecoverage=0,foraconstantrescaledadsorptionenergy,sc=0:74,andH=a=5:0.

PAGE 140

Thisappendixtabulatesthesegmentdensitydistributions(overalldensity,taildensity,bridgedensity,andloopdensity)ofcompressedphysisorbedpolymerlayersobtainedusingnon-reversalchainsinisotropic(SF2)andanisotropic(SCAFT)meanelds,alongwithcorrespondinglatticeMonteCarlosimulationresultsobtainedbyJimenezetal.[ 57 ]Notethatthedensityoftrainsissimplythedensityofsegmentsintheendlayers. 126

PAGE 141

TableB{1: SegmentDensityDistributionfor=0:75,H=a=5:0,r=200,s=1:0kT. OverallSegmentDensity TailSegmentDensity BridgeSegmentDensity LoopSegmentDensity Simulations SCAFT SF2 SCAFT SF2 SCAFT SF2 SCAFT SF2 0.543 0.542 0.485 0 0 0 0 0 0 0 0 0 2 0.160 0.160 0.199 0.0071 0.0073 0.0086 0.050 0.040 0.055 0.104 0.114 0.136 3 0.093 0.090 0.132 0.0061 0.0065 0.0087 0.054 0.045 0.065 0.033 0.038 0.058 4 0.160 0.160 0.199 0.0071 0.0073 0.0086 0.049 0.040 0.055 0.104 0.114 0.136 5 0.544 0.542 0.485 0 0 0 0 0 0 0 0 0

PAGE 142

TableB{2: SegmentDensityDistributionfor=0:25,H=a=5:0,r=200,s=1:0kT. OverallSegmentDensity TailSegmentDensity BridgeSegmentDensity LoopSegmentDensity Simulations SCAFT SF2 SCAFT SF2 SCAFT SF2 SCAFT SF2 0.213 0.217 0.207 0 0 0 0 0 0 0 0 0 2 0.0299 0.029 0.037 0.0013 0.0012 0.0014 0.008 0.004 0.006 0.021 0.024 0.030 3 0.012 0.008 0.012 0.007 0.006 0.008 0.008 0.004 0.006 0.003 0.003 0.005 4 0.0310 0.029 0.037 0.0013 0.0012 0.0014 0.008 0.004 0.006 0.021 0.024 0.030 5 0.212 0.217 0.207 0 0 0 0 0 0 0 0 0

PAGE 143

TableB{3: SegmentDensityDistributionfor=1:25,H=a=6:0,r=200,s=1:0kT. OverallSegmentDensity TailSegmentDensity BridgeSegmentDensity LoopSegmentDensity Simulations SCAFT SF2 SCAFT SF2 SCAFT SF2 SCAFT SF2 0.682 0.619 0 0 0 0 0 0 0 0 0 0.328 0.329 0.351 0.0199 0.0209 0.0219 0.089 0.074 0.085 0.219 0.234 0.244 0.242 0.239 0.280 0.0228 0.0261 0.0292 0.116 0.102 0.121 0.103 0.111 0.130 0.240 0.239 0.280 0.0228 0.0261 0.0292 0.115 0.102 0.121 0.102 0.111 0.130 0.328 0.329 0.351 0.0197 0.0209 0.0219 0.089 0.074 0.085 0.219 0.234 0.244 0.682 0.682 0.619 0 0 0 0 0 0 0 0 0

PAGE 144

[1] M.Doi.IntroductiontoPolymerPhysics.OxfordSciencePublications,Oxford,1997. [2] H.Yamakawa.HelicalWormlikeChainsinPolymerSolutions.Springer-Verlag,Berlin,1997. [3] K.U.Dee,M.L.Shuler,andA.H.Wood.Inducingsingle-cellsuspensionofBTI-TN5B1-4insectcells.1.Theuseofsulfatedpolyanionstopreventcellaggregationandenhancerecombinantproteinproduction.BiotechnologyandBioengineering,54:191{205,1997. [4] K.U.Dee,A.H.Wood,andM.L.Shuler.Inducingsingle-cellsuspensionofBTI-TN5B1-4insectcells.2.Theeectofsulfatedpolyanionsonbaculovirusinfection.BiotechnologyandBioengineering,54:206{220,1997. [5] E.EvansandD.Needham.Attractionbetweenlipidbilayermembranesinconcentratedsolutionsofnonadsorbingpolymers:Comparisonofmean-eldtheorywithmeasurementsofadhesionenergy.Macromolecules,21:1822{1831,1988. [6] T.L.Kuhl,Y.Guo,J.L.Alderfer,A.D.Berman,D.Leckband,J.N.Israelachvili,andS.W.Hui.Directmeasurementofpolyethyleneglycolinduceddepletionattractionbetweenlipidbilayers.Langmuir,12:3003{3014,1996. [7] T.L.Kuhl,A.D.Berman,S.W.Hui,andJ.N.Israelachvili.Part1.Directmeasurementofdepletionattractionandthinlmviscositybetweenlipidbilayersinaqueouspolyethyleneglycolsolutions.Macromolecules,31:8250{8257,1998. [8] T.L.Kuhl,A.D.Berman,S.W.Hui,andJ.N.Israelachvili.Part2.Crossoverfromdepletionattractiontoadsorption:Polyethyleneglycolinducedelectrostaticrepulsionbetweenlipidbilayers.Macromolecules,31:8258{8263,1998. [9] S.W.Hui,T.L.Kuhl,Y.Q.Guo,andJ.N.Israelachvili.Useofpoly(ethyleneglycol)tocontrolcellaggregationandfusion.ColloidsandSurfaces:B,14:213{222,1999. [10] A.Razatos,Y.L.Ong,F.Boulay,D.L.Elbert,J.A.Hubbell,M.M.Sharma,andG.Georgiou.Forcemeasurementsbetweenbacteriaandpoly(ethyleneglycol)-coatedsurfaces.Langmuir,16:9155{9158,2000. [11] H.W.HuandS.Granick.Universalandsystem-specicfeaturesofsurfaceforcesbetweenadsorbedpolystyreneinanear-solvent.Macromolecules,23:613{623,1990. [12] G.J.Fleer,M.A.CohenStuart,J.M.H.M.Scheutjens,T.Cosgrove,andB.Vincent.PolymersatInterfaces.ChapmanandHall,London,1993. 130

PAGE 145

[13] J.KleinandG.Rossi.Analysisoftheexperimentalimplicationsofthescalingtheoryofpolymeradsorption.Macromolecules,31:1979{1988,1998. [14] P.G.deGennes.ScalingConceptsinPolymerPhysics.CornellUniversityPress,Ithaca,NewYork,1979. [15] E.Evans.Forcesbetweensurfacesthatconneapolymersolution:Derivationfromself-consistenteldtheories.Macromolecules,22:2277{2286,1989. [16] H.J.Ploehn.Compressionofpolymerinterphases.Macromolecules,27:1627{1636,1994. [17] M.Rangarajan,J.Jimenez,andR.Rajagopalan.Eectsofpolymerlayeranisotropyontheinteractionbetweenadsorbedlayers.Macromolecules,35:6020{6031,2002. [18] R.B.DickinsonandD.L.Purich.Clamped-lamentelongationmodelforactin-basedmotors.BiophysicalJournal,82:605{617,2002. [19] R.B.Dickinson,L.Caro,andD.L.Purich.Forcegenerationbycytoskeletallamentend-trackingproteins.BiophysicalJournal,87:2838{2854,2004. [20] S.C.KuoandJ.L.McGrath.StepsanductuationsofListeriamonocytogenesduringactin-basedmotility.Nature,407:1026{1029,2000. [21] P.G.deGennes.Polymersolutionsnearaninterface.1.Adsorptionanddepletionlayers.Macromolecules,14:1637{1644,1981. [22] P.G.deGennes.Polymersataninterface.2.Interactionbetweentwoplatescarryingadsorbedpolymerlayers.Macromolecules,15:492{500,1982. [23] J.W.CahnandJ.E.Hilliard.Freeenergyofanonuniformsystem.1.Interfacialfreeenergy.TheJournalofChemicalPhysics,28:258{267,1958. [24] K.Ingersent,J.Klein,andP.A.Pincus.Forcesbetweensurfaceswithadsorbedpolymers.3.solvent.Calculationsandcomparisonwithexperiment.Macro-molecules,23:548{560,1990. [25] J.KleinandP.A.Pincus.Interactionbetweensurfaceswithadsorbedpolymers:Poorsolvents.Macromolecules,15:1129{1135,1982. [26] K.Ingersent,J.Klein,andP.A.Pincus.Interactionsbetweensurfaceswithadsorbedpolymers:Poorsolvent.2.Calculationsandcomparisonwithexperiment.Macromolecules,19:1374{1381,1986. [27] J.Jimenez.TestingtheTheoreticalFoundationsofPolymerLayersandInterac-tions.PhDthesis,UniversityofFlorida,2000. [28] S.F.Edwards.Thestatisticalmechanicsofpolymerswithexcludedvolume.ProceedingsofthePhysicalSociety,85:613{624,1965. [29] I.CarmesinandJ.Noolandi.First-orderwettingtransitionsofpolymermixturesincontactwithawall.Macromolecules,22:1689{1697,1989.

PAGE 146

[30] A.JohnerandJ.F.Joanny.Polymeradsorptioninapoorsolvent.JournaldePhysiqueII,1:181{194,1991. [31] G.RossiandP.A.Pincus.Interactionsbetweenundersaturated-polymeradsorbedsurfaces.EurophysicsLetters,5:641{646,1988. [32] G.RossiandP.A.Pincus.Propertiesofpolymerlayersadsorbedonsurfacesundernon-equilibriumconditions.Macromolecules,22:276{283,1989. [33] J.M.H.M.ScheutjensandG.J.Fleer.Statisticaltheoryoftheadsorptionofinteractingchainmolecules.1.Partitionfunction,segmentdensitydistribution,andadsorptionisotherms.TheJournalofChemicalPhysics,83:1619{1635,1979. [34] J.M.H.M.ScheutjensandG.J.Fleer.Interactionbetweentwoadsorbedpolymerlayers.Macromolecules,18:1882{1900,1985. [35] A.N.Semenov,J.Bonet-Avalos,A.Johner,andJ.F.Joanny.Adsorptionofpolymersolutionsontoaatsurface.Macromolecules,29:2179{2196,1996. [36] A.N.Semenov,J.F.Joanny,A.Johner,andJ.Bonet-Avalos.Interactionbetweentwoadsorbingplates:Theeectofpolymerchainends.Macromolecules,30:1479{1489,1997. [37] A.Johner,J.Bonet-Avalos,C.C.vanderLinden,A.N.Semenov,andJ.F.Joanny.Adsorptionofneutralpolymers:Interpretationofthenumericalself-consistenteldresults.Macromolecules,29:3629{3638,1996. [38] J.Bonet-Avalos,J.F.Joanny,A.Johner,andA.N.Semenov.Equilibriuminteractionbetweenadsorbedpolymerlayers.EurophysicsLetters,35:97{102,1996. [39] F.A.M.Leermakers,J.M.H.M.Scheutjens,andR.J.Gaylord.Modelingtheamorphousphaseofameltcrystallized,semi-crystallinepolymer:Segmentdistribution,chainstinessanddeformation.Polymer,25:1577{1588,1984. [40] F.A.M.LeermakersandJ.M.H.M.Scheutjens.Statisticalthermodynamicsofassociationcolloids.1.Lipidbilayermembranes.TheJournalofChemicalPhysics,89:3264{3274,1988. [41] C.C.vanderLinden,F.A.M.Leermakers,andG.J.Fleer.Adsorptionofsemiexiblepolymers.Macromolecules,29:1172{1178,1996. [42] G.J.Fleer,J.vanMale,andA.Johner.Analyticalapproximationtothescheutjens-eertheoryforpolymeradsorptionfromdilutesolution.1.Trains,loops,andtailsintermsofproximalanddistallengths.Macromolecules,32:825{844,1999. [43] J.deJoannis.EquilibriumPropertiesofPolymerSolutionsAtSurfaces:MonteCarloSimulations.PhDthesis,UniversityofFlorida,2000. [44] J.JimenezandR.Rajagopalan.Anewsimulationmethodforthedeterminationofforcesinpolymer/colloidsystems.TheEuropeanPhysicalJournalB,5:237{243,1998.

PAGE 147

[45] J.JimenezandR.Rajagopalan.InteractionbetweenagraftedpolymerchainandanAFMtip:Scalinglaws,forces,andevidenceofconformationaltransition.Langmuir,14:2598{2601,1998. [46] P.G.deGennes.PhysicalBasisofCell-CellAdhesion,chapterModelPolymersatInterfaces.CRCPress:BocaRaton,Florida,1988. [47] Z.FuandM.M.Santore.Kineticsofcompetitiveadsorptionofpeochainswithdierentmolecularweights.Macromolecules,31:7014{7022,1998. [48] P.Auroy,Y.Mir,andL.Auvray.Localstructureanddensityproleofpolymerbrushes.PhysicalReviewLetters,69:93{95,1992. [49] R.Levicky,N.Koneripalli,andM.Tirrell.Concentrationprolesindenselytetheredpolymerbrushes.Macromolecules,31:3731{3734,1998. [50] S.S.PatelandM.Tirrell.Measurementofforcesbetweensurfacesinpolymeruids.AnnualReviewofPhysicalChemistry,40:597{635,1989. [51] G.HuberandT.A.Vilgis.Polymeradsorptiononheterogeneoussurfaces.TheEuropeanPhysicalJournalB,5:217{223,1998. [52] K.SumithraandA.Baumgaertner.Polymeradsorptiononplanarrandomsurfaces.TheJournalofChemicalPhysics,109:1540{1544,1998. [53] Wikipediacontributors.Homogeneousfunction:Wikipedia,Thefreeencyclopedia.http://en.wikipedia.org,2006.[Online;accessedJune-2006]. [54] E.J.W.VerweyandJ.Th.G.Overbeek.TheoryoftheStabilityofLyophobicColloids.DoverPublications,NewYork,2000. [55] H.J.Ploehn.Freeenergybalanceforcompressionofpolymerinterphases.Macromolecules,35:5331{5333,2002. [56] V.G.Babak.Thermodynamicsofplane-parallelliquidlms.ColloidsandSurfaces:A,142:135{153,1998. [57] J.Jimenez,J.deJoannis,I.Bitsanis,andR.Rajagopalan.Interactionbetweenundersaturatedpolymerlayers:Computersimulationsandnumericalmean-eldcalculations.Macromolecules,33:8512{8519,2000. [58] E.Wajnryb,A.R.Altenberger,andJ.S.Dahler.AdvancesinChemicalPhysics,volumeXCI,chapterThePhenomenologicalandStatisticalThermodynamicsofNonuniformSystems.JohnWileyandSons,NewYork,1995. [59] L.E.Malvern.IntroductiontotheMechanicsofaContinuousMedium.PrenticeHall,EnglewoodClis,NewJersey,1997. [60] J.Lyklema.FundamentalsofInterfaceandColloidsScience,Vol.1.Fundamentals.Academic,London,1991. [61] H.J.Ploehn.Structureofadsorbedpolymerlayers:Molecularvolumeeects.Macromolecules,27:1617{1626,1994.

PAGE 148

[62] H.Ji,D.Hone,P.Pincus,andG.Rossi.Polymerbridgingbetweentwoparallelplates.Macromolecules,23:698{707,1990. [63] J.Jimenez,J.deJoannis,I.Bitsanis,andR.Rajagopalan.Bridgingbyasinglepolymerchain.Macromolecules,33:7157{7164,2000. [64] J.I.SiepmannandD.Frenkel.CongurationalbiasMonteCarlo:Anewsamplingschemeforexiblechains.MolecularPhysics,75:59{70,1992. [65] P.Pincus.Bridgingbyasinglepolymerchain.Macromolecules,9:386{388,1976. [66] J.deJoannis,C.W.Park,J.Thomatos,andI.Bitsanis.Homopolymerphysisorp-tion:AMonteCarlostudy.Langmuir,17:69{77,2001. [67] J.M.H.M.ScheutjensandG.J.Fleer.Statistical-theoryoftheadsorptionofinteractingchainmolecules.2.Train,loop,andtailsizedistribution.JournalofPhysicalChemistry,84:178{190,1980. [68] P.P.SimonandH.J.Ploehn.Backfoldingcorrectionsforfreelyjointedchainsinself-consistenteldlatticemodels.Macromolecules,31:5880{5891,1998. [69] L.D.LandauandE.M.Lifshitz.TheoryofElasticity.ButterworthHeinemann,Oxford,1986. [70] A.E.H.Love.ATreatiseontheMathematicalTheoryofElasticity.DoverPublications,NewYork,1944. [71] D.Pantaloni,C.LeClainche,andM.F.Carlier.Mechanismofactin-basedmotility.Science,292:1502{1506,2001. [72] T.D.PollardandG.G.Borisy.Cellularmotilitydrivenbyassemblyanddisassem-blyofactinlaments.Cell,112:453{465,2003. [73] M.F.Carlier,C.LeClainche,S.Wiesner,andD.Pantaloni.Actin-basedmotility:Frommoleculestomovement.BioEssays,25:336{345,2003. [74] G.G.BorisyandT.M.Svitkina.Actinmachinery:Pushingtheenvelope.CurrentOpinioninCellBiology,12:104{112,2000. [75] A.Mogilner.Ontheedge:Modelingprotrusion.CurrentOpinioninCellBiology,18:32{39,2006. [76] F.Gerbal,P.Chaikin,Y.Rabin,andJ.Prost.AnelasticanalysisofListeriamonocytogenespropulsion.BiophysicalJournal,79:2259{2275,2000. [77] V.Noireaux,R.M.Golsteyn,E.Friederich,J.Prost,C.Antony,D.Louvard,andC.Sykes.Growinganactingelonsphericalsurfaces.BiophysicalJournal,78:1643{1654,2000. [78] C.S.Peskin,G.M.Odell,andG.F.Oster.Cellularmotionsandthermaluctuations:Thebrownianratchet.BiophysicalJournal,65:316{324,1993. [79] A.MogilnerandG.F.Oster.Cellmotilitydrivenbyactinpolymerization.BiophysicalJournal,71:3030{3045,1996.

PAGE 149

[80] A.MogilnerandG.F.Oster.Forcegenerationbyactinpolymerization:Theelasticratchetandtetheredlaments.BiophysicalJournal,84:1591{1605,2003. [81] J.R.RobbinsandJ.A.Theriot.Listeriamonocytogenesrotatesarounditslongaxisduringactin-basedmotility.CurrentBiology,13:R754{R756,2003. [82] W.L.Zeile,F.Zhang,R.B.Dickinson,andD.L.Purich.Listeria'sright-handedhelicalrocket-tailtrajectories:Mechanisticimplicationsforforcegenerationinactin-basedmotility.CellMotilityandtheCytoskeleton,60:121{128,2005. [83] L.A.Cameron,M.J.Footer,A.vanOudenaarden,andJ.A.Theriot.MotilityofActAprotein-coatedmicrospheresdrivenbyactinpolymerization.ProceedingsoftheNaturalAcademyofSciencesoftheUSA,96:4908{4913,1999. [84] K.Kuhnel,T.Jarchau,E.Wolf,I.Schlichting,U.Walter,A.Wittinghofer,andS.V.Strelkov.TheVASPtetramerizationdomainisaright-handedcoiledcoilbasedona15-residuerepeat.ProceedingsoftheNaturalAcademyofSciencesoftheUSA,101:17027{17032,2004. [85] L.A.Cameron,T.M.Svitkina,D.Vignjevic,J.A.Theriot,andG.G.Borisy.Dendriticorganizationofactincomettails.CurrentBiology,11:130{135,2001. [86] F.S.SouthwickandD.L.Purich.ArrestofListeriamovementinhostcellsbyabacterialActAanalogue:Implicationsforactin-basedmotility.ProceedingsoftheNaturalAcademyofSciencesoftheUSA,91:5168{5172,1994. [87] A.Upadhyaya,J.R.Chabot,A.Andreeva,A.Samadani,andA.vanOude-naarden.Probingpolymerizationforcesbyusingactin-propelledlipidvesicles.ProceedingsoftheNaturalAcademyofSciencesoftheUSA,100:4521{4526,2003. [88] P.A.Giardini,J.R.Fletcher,andJ.A.Theriot.Compressionforcesgeneratedbyactincomettailsonlipidvesicles.ProceedingsoftheNaturalAcademyofSciencesoftheUSA,100:6943{6948,2003. [89] Y.Tsuda,H.Yasutake,A.Ishijima,andT.Yanagida.Torsionalrigidityofsingleactinlamentsandactin-actinbondbreakingforceundertorsionmeasureddirectlybyinvitromicromanipulation.ProceedingsoftheNaturalAcademyofSciencesoftheUSA,93:12937{12942,1996. [90] T.D.CloudandR.Rajagopalan.Stericinteractionsbetweenniteobjectsandend-graftedpolymerchains:Atwo-dimensionalmean-eldanalysis.JournalofColloidandInterfaceScience,266:304{313,2003.

PAGE 150

MuraliRangarajanwasborninMadurai,TamilNadu,India,andspenthischildhoodinTamilNaduandKarnatakastates.Heobtainedhisbachelor'sdegreeinchemicalandelectrochemicalEngineeringintheCentralElectrochemicalResearchInstitute(CECRI),Karaikudi,India,in1997,duringwhichtimeheco-wroteathesisontheelectrolysisofpotassiumsulfateintosulfuricacidandpotassiumhydroxideusingaNaonTMmembraneelectrolyzer.HethenjoinedtheIndianInstituteofTechnology(IIT)Kanpur,India,asagraduatestudentandobtainedhismaster'sdegreeinchemicalengineering.Heworkedwithateamofgraduatestudentsonthedevelopmentofonlinesensorofthepropertiesoftheproductsofacrudedistillationunitandanonlineoptimizerofthecrudedistillationunit.ThisprojectwasajointworkwithIndianOilCorporation,MathuraRenery,andthesensorswereeventuallyinstalledthere.UponnishinghisstudiesatIITKanpur,MuralijoinedtheDepartmentofChemicalEngineeringattheUniversityofFlorida,Gainesville,in1999.Heworkedonlatticenumericalmean-eldtheoriesforpolymeradsorption,bendingofstipolymerchains,andtheactin-basedmotilityofListeriamonocytogenesduringthecourseofhisdoctoralstudiestill2005.HetaughtchemistryfortwosemestersattheSantaFeCommunityCollegein2005-2006,andhassincejoinedtheCleanEnergyResearchCenterattheUniversityofSouthFlorida,Tampa,asaVisitingResearchScholarinMay2006. 136


Permanent Link: http://ufdc.ufl.edu/UFE0015689/00001

Material Information

Title: Polymer-Induced Forces at Interfaces
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: UFE0015689:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015689/00001

Material Information

Title: Polymer-Induced Forces at Interfaces
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: UFE0015689:00001


This item has the following downloads:


Full Text











POLYMER-INDUCED FORCES AT INTERFACES


By

MURALI RANGARAJAN




















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

Murali Rangarajan















This dissertation is dedicated with all my love, prayers and humble pranams to

AMMA, the source and soul of my life.















ACKNOWLEDGMENTS

I offer my prayers and pranams to the Almighty Parabrahman and Parashakti; it is

entirely due to Divine Grace that I have been able to accomplish this meager work. My

deepest respect, gratitude and pranams are also due to my Guru Amma whose unerring

and eternal guidance and blessings have shaped me in every way, shape and form.

I would like to offer my sincere thanks to Dr. Anthony J. C. Ladd for his whole-

hearted support, guidance and patience. The years of my doctoral work have been some

of the most challenging years and I have grown as a person in v--~ I never even imag-

ined. Dr. Ladd was very patient with me, was ahv--li- available for help and guidance

in academic and personal matters, and motivated me when some personal problems

tried to overwhelm me. In times where being an academician has increasingly become

a job of seeking funds and managing a small company consisting of graduate students,

when true teacher-student interactions are hard to come by, Dr. Ladd has showed me

how one can still find time and v--,v- to mentor students and educate them in issues

concerning their research and beyond. I shall remember my interactions with him in

shaping my future interactions with colleagues and students in my career.

I would also like to thank Dr. Richard Dickinson and Dr. Daniel Purich for their

support and guidance. While working with them, I learned the value of collaboration

and free exchange of ideas. Both of them were very supportive during times of personal

crisis, for which they have my sincere gratitude.

I would like to thank Dr. Raj R i, I palan, Dr. Jorge Jimenez, Dr. Jason de

Joannis, Ms. Tiffany Cloud, Ms. Patricia Socias, Mr. Shelton Wright and Mr. Michael

Smith for interactions and collaborations on the work on polymer-induced forces

examined using mean-field theories. Dr. Raj also paid for my conference presentations

and I would like to acknowledge his support.

My colleagues in the lab of Dr. Ladd made my stay one of pleasure. They sup-

ported me in all my work and we enjol, 1 discussing everything under the sun and









beyond! I would like to thank my colleagues and friends Dr. Jonghoon Lee, Dr. Piotr

Symczak, Dr. Nhan-Quyen Nguyen, Mr. Byoungjin Chun, Mr. Berk Usta, Mr. Gaurav

Misra and Dr. Dazhi Yu. I would particularly like to thank Byoungjin who was been

helpful in more v--,i-< than I can count. Every time I ever had a question, particularly

on C programming or LaTeX, he unhesitatingly helped me in any way he could.

I would like to thank the Department of C'!I. i,, il Engineering and the University

of Florida Alumni Graduate Fellowship for funds.

I had a great blessing of finding amazing friends, my family away from home,

during my stay in Gainesville. They are simply far too many to list. I sincerely

hope I have told each and every one of them enough times that I love them deeply

and that all my prayers and best wishes go for them. The Maitri family, as we have

called ourselves, has been the source of my emotional and spiritual strength. Growing

together with them has been such a blessing and pleasure; I pray to God that this love

and atmosphere of growth will continue to exist for the rest of our lives.

Last but definitely not the least, I would like to thank my parents for their love

and support. Their love has been a constant source of strength. They taught me to

never give up and alv- i- strive to better myself and succeed. For everything they have

ever done for me, mere words will not do any justice. My prv .---i are for them to find

happiness and peace in every moment of their life.















TABLE OF CONTENTS
Page

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

LIST OF TABLES ...................... ............ ix

LIST OF FIGURES ........... ................. ...... x

ABSTRACT . . . . . . ... . xiii

CHAPTER

1 INTRODUCTION ................... ............. 1

1.1 Flexible and Semiflexible Polymers .. ............... 1
1.1.1 Freely Jointed ('! Ch,- A Model of Flexible Polymers ..... 1
1.1.2 Freely Rotating C'!i ,i ..................... 3
1.1.3 Wormlike ('!i ,ii- A Model of Semiflexible Polymers ..... 4
1.1.4 Physical Interpretation of Persistence Length . . .. 5
1.2 Scope of the Present Work ............... ... 6
1.3 Organization of the Present Work ............... . 9

2 ADSORPTION OF FLEXIBLE POLYMERS AT INTERFACES: A LIT-
ERATURE REVIEW ..... ........... ... ........ 10

2.1 Introduction ............. . . ... 10
2.2 Theories of Polymer Adsorption: Overview . . 12
2.2.1 Density-Functional Theories: Cahn-de Gennes Approach . 12
2.2.2 Free-Energy-Functional (FEF)/Scaling Approach . ... 15
2.2.3 Analytical Mean-Field Theories ..... . . ..... 16
2.2.4 Numerical Mean-Field Theory .................. .. 19
2.3 Computational Studies of Polymer Adsorption . . 19
2.4 Experimental Studies of Polymer Adsorption . . .. 20
2.5 Estimation of Polymer-Induced Forces: Validity of the Theoretical
and Computational Results .................. ... 21

3 POLYMER-INDUCED FORCES FROM NUMERICAL MEAN-FIELD THE-
ORIES ................... ............ ...... 25

3.1 Introduction .................. ............. .. 25
3.1.1 Background Information ................. . .. 25
3.1.2 Organization of the C!i lpter ........... . .. 27
3.2 System and Surroundings .................. ..... 28
3.3 Thermodynamics of Inhomogeneous Systems . . .... 30
3.4 Free Energy from Lattice Numerical Mean-Field Theory . ... 32
3.4.1 System, Preliminaries, and Partition Function . ... 32
3.4.2 Configurational Entropy ................. . .. 35
3.4.3 Equilibrium Distribution ............... . .. 37









3.4.4 Free Energy ..... . . ..... ........... 40
3.5 Inhomogeneous Continuum Description of the Film . ... 42
3.5.1 Continuum Formulation of the Fundamental Equation . 42
3.5.2 Estimation of Di-, i iiii.;- Pressure, Film Tension and Interfa-
cial Tension ............ . . .... 44
3.6 Film Tension, Interfacial Tension and Force of Compression from Lat-
tice M odels .................. ............. .. 46
3.6.1 Full Equilibrium .................. ..... .. 46
3.6.2 Restricted Equilibrium ................ .... .. 49
3.7 Validity of the Approach .................. ..... .. 50
3.8 Results and Discussion .................. ..... .. 50
3.8.1 Full Equilibrium .................. ..... .. 51
3.8.2 Restricted Equilibrium .................. ... .. 52
3.8.3 Implications on the Mean-Field Predictions . .... 56
3.9 Concluding Remarks .................. ......... .. 58

4 EFFECTS OF POLYMER-LAYER ANISOTROPY ON THE INTERAC-
TION BETWEEN ADSORBED LAYERS ................ ..61

4.1 Introduction ..... ...... ... ......... ... .... 61
4.2 Self-Consistent Anisotropic Mean-Field Theory (SCAFT) ...... ..63
4.2.1 Preliminaries and Notations .............. .. 63
4.2.2 Anisotropic Mean Field ............ ... . .. 66
4.2.3 Statistical Weights and Composition Rule . . ... 67
4.2.4 Structure of the Adsorbed L -iv.r ................ .. 69
4.2.5 Estimation of Interaction Forces ................ .. 73
4.3 Results and discussion .................. ...... .. .. 73
4.3.1 Structure of the Adsorbed L- rs . . . 74
4.3.2 Interaction Forces .................. ..... .. 76
4.4 Concluding Remarks .................. ......... 83

5 BENDING OF SEMIFLEXIBLE POLYMERS ................ .. 86

5.1 Introduction ............ . . . .... 86
5.2 Wormlike C('! ,ii- as Slender Elastic Rods . . ..... 86
5.3 Bending of Slender Elastic Rods ............... . .. 87
5.3.1 Special Cases .................. ........ .. 88
5.3.2 Boundary Conditions .... . . ..... 89
5.3.3 Equation of Pure Bending of a Rod of Circular Cross Section 89
5.4 Results and discussion .... ........... ...... .. 90
5.4.1 Double-Hinged Case ................ .. .. 91
5.4.2 Clamped-Hinged Case ............... . .. 92

6 F-ACTIN AS A SEMIFLEXIBLE ELASTIC ROD: MOTILITY OF Liste-
ria I,,,.:., ',.,/, ,:, PROPELLED BY ACTIN FILAMENTS . ... 98

6.1 Introduction .................. ............. .. 98
6.2 The Listeria-Actin System . . ..... ... 98
6.3 Biophysical Model of the Motility of Listeria: Actoclampin ...... 102
6.4 Results and Discussion ..... ........ . .. 105
6.4.1 Actin-Based Motility of Listeria . . 105
6.4.2 Long Length-Scale Rotation of Listeria . . .. 107

vii









7 CONCLUSIONS AND FUTURE WORK ................... .112

7.1 Problems Addressed ........... . . ... 112
7.2 Conclusions .. .. .. .. .. .. .. ... .. .. ... .. .. 112
7.2.1 Interacting Polymer Layers . . . .. 112
7.2.2 Bending of Semiflexible Polymers: F-Actin Propelled Motility
of Listeria ....... ........ ....... ....... 113
7.3 Future Work .................. ...... ....... 114
7.3.1 Interacting Polymer Layers . . . .. 114
7.3.2 Modeling of Semiflexible Polymers: F-Actin Propelled Motil-
ity of Listeria .................. ......... 116

APPENDIX

A STRUCTURE OF THE ADSORBED POLYMER LAYERS USING SELF-
CONSISTENT ANISOTROPIC MEAN-FIELD THEORY (SCAFT): AD-
DITIONAL RESULTS .................. ........... .. 119

B SEGMENT DENSITY DISTRIBUTIONS: OVERALL, TAIL, BRIDGE,
AND LOOP DENSITIES .................. .......... 126

REFERENCES ................... ..... .... ........ 130

BIOGRAPHICAL SKETCH ................... ......... 136















LIST OF TABLES
Table Page

B-1 Segment Density Distribution for F = 0.75, H/a = 5.0, r = 200, = 1.0
kT . . . . .. . . . . . 127

B-2 Segment Density Distribution for F = 0.25, H/a = 5.0, r = 200, = 1.0
kT . . . . .. . . .... .. . 128

B-3 Segment Density Distribution for F = 1.25, H/a = 6.0, r = 200, X, 1.0
kT . . . . .. . . .... .. . 129















LIST OF FIGURES
Figure Page

1-1 The freely jointed chain model of a polymer on a 2-D lattice . . 2

1-2 The wormlike chain model of a polymer ................. 4

1-3 Two interacting physisorbed linear, flexible, homopolymer 1 i-rS- con-
fined by flat, parallel, adsorbing surfaces ..... . . .. .. 7

1-4 An illustration of the actoclampin model: An ensemble of elongating
clamped filaments under compression or tension propelling the motile
surface ............................... ........ 8

3-1 Two interacting adsorbed lr ,-i s separated by a distance H: Definition of
the system . . . . . .. . . 29

3-2 Film tension 7 and interfacial tension 2a as a function of surface separa-
tion (H/a) under full equilibrium conditions. The results are shown for
an adsorption energy X, 0.5 kT, chain length r-100, bulk concentra-
tion (b 0.05, and good solvent conditions, X 0.0 . . ... 52

3-3 Average density of polymer segments (8t/H) in the interface as a func-
tion of surface separation (H/a) under full equilibrium conditions. X,
0.5 kT, r-100, ob 0.05, and X 0.0 ................ 53

3-4 Force per unit area f as a function of surface separation (H/a) under
full equilibrium conditions for r=100 and X = 0.0. The negative deriva-
tive of excess grand canonical free energy is also shown for comparison.
(a), 0.5 kT, Ob 0.05; (b), 1.0 kT, b 0.005 . .... 54

3-5 Interaction potential W between the surfaces in full equilibrium with so-
lutions of varying bulk concentrations. X, = 0.5 kT, r 100, and X = 0.0 55

3-6 Correct calculation of excess semigrand free energy under restricted equi-
librium conditions. The results shown here are for a surface coverage
7 0.75, X, 1.0 kT, r 200, and X = 0.0 . . ...... 56

3-7 Force per unit area as a function of surface separation (H/a) under re-
stricted equilibrium conditions. 7 = 0.5, X, = 1.0 kT, r=200, and X = 0.0 57

3-8 The variation of deviatoric stress per segment upon compression, for two
surface coverages 7 = 0.5 and 7 = 0.75. The results are shown for X,
1.0 kT, r 200, and X = 0.0, under restricted equilibrium . ... 58

3-9 Interaction potential W between surfaces in restricted equilibrium for
coverages 7 = 0.5 and 7 = 0.75. The results shown here are for X, = 1.0
kT, r 200, and X = 0.0 .................. ......... .. 59









3-10 Force per unit area f as a function of surface coverage 7 under restricted
equilibrium conditions for a fixed separation of (H/a) 4.5 . ... 60

4-1 Bond orientations and anisotropic mean field ............... ..65

4-2 Overall segment density distribution: Comparison of lattice mean-field
results (SF1, SF2, SCAFT) with lattice Monte Carlo simulations . 74

4-3 Number and size distribution of bridges, as a function of H/a, for a con-
stant adsorption energy, X, = 1.0 kT and F = 0.75 ............ ..76

4-4 Force per unit area between the surfaces f as a function of surface sepa-
ration H/a ................... ... ........ 79

4-5 Force per unit area f at a fixed separation of H/a = 4.5 as a function of
surface coverage F .................. ...... ..... 80

4-6 Number and size distribution of bridges, as a function of rescaled sur-
face coverage F/Fo, for a constant rescaled adsorption energy, X, Xc =
0.74, and H /a = 5.0 .................. ........... .. 81

4-7 Force per bridge fbr as a function of F for a fixed wall separation H/a
4.5 . . . . . . . .. . .. 82

5-1 Shapes of a double-hinged rod under compression for forces greater than
the critical force .. ... .. .. .. .. .. .. .. .. .. .... .. 93

5-2 Force on the double-hinged rod as a function of end position, correspond-
ing to the shapes shown in Figure 5-1 ................ 94

5-3 Possible filament end positions for a filament clamped at the origin and
oriented along the z-axis for various axial and compressive forces. Note
the existence of multiple solutions for each force: Fy > 0 . ... 96

5-4 Possible filament end positions for a filament clamped at the origin and
oriented along the z-axis for various axial and compressive forces: Fy < 0 97

5-5 Shape of a filament clamped at the origin and oriented at an angle -45
degrees from the z-axis under compression ................. ..97

6-1 Shape of a filament under tension. F, = 19.2 pN . ..... 106

6-2 Motility of the surface due to a single filament: Surface position and Fil-
ament end position ............... .......... 107

6-3 Motility of the surface due to a single filament: Force . . ... 108

6-4 Trajectory of the motile surface propelled by five filaments . ... 109

A-1 Average number of bridges, nbr, as a function of surface coverage F for a
constant adsorption energy, X, = 1.0 kT and H/a = 5.0 . ... 119

A-2 Average size of bridges, lbr, as a function of surface coverage F for a con-
stant adsorption energy, X, = 1.0 kT and H/a = 5.0 . . ... 120









A-3 Average number of loops, nlo, as a function of H/a for a constant ad-
sorption energy, X, 1.0 kT and F = 0.75 ................. ..120

A-4 Average size of loops, lio, as a function of H/a for a constant adsorption
energy, X, 1.0 kT and F = 0.75 ................... ...... 121

A-5 Average number of loops, nlo, as a function of surface coverage F for a
constant adsorption energy, X, = 1.0 kT and H/a = 5.0 . ... 121

A-6 Average size of loops, lio, as a function of surface coverage F for a con-
stant adsorption energy, X, = 1.0 kT and H/a = 5.0 . . .... 122

A-7 Average number of tails, nta, as a function of H/a for a constant adsorp-
tion energy, X, = 1.0 kT and F = 0.75 ................. ..122

A-8 Average size of tails, Ita, as a function of H/a for a constant adsorption
energy, X, 1.0 kT and F = 0.75 ................... ...... 123

A-9 Average number of tails, nta, as a function of surface coverage F for a
constant adsorption energy, X, = 1.0 kT and H/a = 5.0 . ... 123

A-10 Average size of tails, Ita, as a function of surface coverage F for a con-
stant adsorption energy, X, = 1.0 kT and H/a = 5.0 . . .... 124

A-11 Average number of loops, nlo, as a function of rescaled surface coverage
F/F0, for a constant rescaled adsorption energy, X, Xc = 0.74, and
H/a = 5.0 ................. ......... ....... 124

A-12 Average size of loops, lio, as a function of rescaled surface coverage F/F0,
for a constant rescaled adsorption energy, X, Xc = 0.74, and H/a = 5.0 125















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

POLYMER-INDUCED FORCES AT INTERFACES

By

Murali Rangarajan

August 2006

Chair: Anthony J. C. Ladd
Major Department: Chemical Engineering

This dissertation concerns studies of forces generated by confined and physisorbed

flexible polymers using lattice mean-field theories, and those generated by confined and

clamped semiflexible polymers modeled as slender elastic rods.

Lattice mean-field theories have been used in understanding and predicting the

behavior of polymeric interfacial systems. In order to efficiently tailor such systems

for various applications of interest, one has to understand the forces generated in the

interface due to the polymer molecules. The present work examines the abilities and

limitations of lattice mean-field theories in predicting the structure of physisorbed poly-

mer lV-, -i and the resultant forces. Within the lattice mean-field theory, a definition of

normal force of compression as the negative derivative of the partition-function-based

excess free energy with surface separation gives misleading results because the theory

does not explicitly account for the normal stresses involved in the system. Correct

expressions for normal and tangential forces are obtained from a continuum-mechanics-

based formulation. Preliminary comparisons with lattice Monte Carlo simulations show

that mean-field theories fail to predict significant attractive forces when the surfaces

are undersaturated, as one would expect. The corrections to the excluded volume

(non-reversal chains) and the mean-field (anisotropic field) approximations improve the

predictions of lv.- r structure, but not the forces.









Bending of semiflexible polymer chains (elastic rods) is considered for two bound-

ary conditions where the chain is hinged on both ends and where the chain is

clamped on one end and hinged on the other. For the former case, the compressive

forces and chain shapes obtained are consistent with the inflexional elastica pub-

lished by Love. For the latter, multiple and higher-order solutions are observed for the

hinged-end position for a given force. Preliminary studies are conducted on actin-based

motility of Listeria morn.'. ;//. ',,i ,' i by treating actin filaments as elastic rods, using the

actoclampin model. The results show qualitative agreement with calculations where the

filaments are modeled as Hookean springs. The feasibility of the actoclampin model to

address long length-scale rotation of Listeria during actin-based motility is addressed.















CHAPTER 1
INTRODUCTION

This dissertation concerns studies of forces generated by confined and physisorbed

flexible polymers using lattice mean-field theories, and those generated by confined

and clamped semiflexible polymers modeled as slender elastic rods. In this chapter, we

first define and outline models of flexible and semiflexible polymers and identify the

contributions of the present work. Finally, we discuss the organization of the rest of the

dissertation.

1.1 Flexible and Semiflexible Polymers

A polymer is a large molecule consisting of many small, simple chemical units

called monomers, joined together by a chemical reaction. They can be classified as

flexible or semiflexible polymers on the basis of a microscopic property (persistence

length) or a macroscopic one (flexural rigidity) that determines polymer flexibility. In

this section, we derive microscopic models of flexible and semiflexible polymers and

relate persistence length and flexural rigidity. These microscopic models consider a

single polymer chain under the action of thermal forces and define various quantitative

properties of the chain. It is important to note that the size of a polymer molecule is

such that it is impossible to neglect the effect of thermal forces on the molecule, as one

often does for macroscopic objects. For a detailed description of polymer models and

the concepts involved, the interested reader is referred to classic texts (e.g., Doi [1],

Yamakawa [2]).

1.1.1 Freely Jointed Chains: A Model of Flexible Polymers

A polymer molecule has many internal degrees of freedom. For example, one may

imagine the rotational, vibrational, and torsional freedoms about -C C- bonds in

polyethylene molecules. It is because of this high degree of flexibility that one often

models a polymer rod as a long, flexible piece of string. The simplest mathematical

model of a polymer chain assumes the chain to follow a regular lattice. The portions of



































Figure 1-1: The freely jointed chain model of the polymer on a 2-D lattice. The filled
circles are the segments and the thick lines are the bonds.


the polymer lying on the lattice points are called -, jir, ',1 and the rods connecting the

segments are called bonds. Let b be the length of each bond.

Figure 1-1 shows a two-dimensional lattice. Due to random thermal motion of

the surrounding medium, the polymer chain has the flexibility to take on different

configurations (all on the 2-D lattice). The simplest possible model then would be

when there is no correlation between the directions that different bonds take and

that all directions have the same probability. This is the so-called freely-jointed-chain

model (also called random-coil model or random-flight-chain model). In this case, the

configuration of a polymer will be the same as a random walk on the lattice.

The end-to-end vector R joining one end of the polymer to the other is a measure

of the size of the polymer. If the polymer is made of N bonds, with r, the vector of the

nth bond, we have
N
R r,. (1-1)
n=l
The average value of R, (R) = 0, since the probability of the end-to-end vector

being R is the same as it being -R. Therefore, one calculates R2, and expresses the









size of the polymer by the root mean square (rms) value of R. This is given by
M N
(R2 E E(r. rm) Nb2 + 2 ( E rm). (1-2)
n-l rl 1 nmrn

Since for a freely jointed chain there is no correlation between different bond

vectors, the second term on the right hand side of Equation (1-2) is zero. Therefore, for

a freely jointed chain, the end-to-end distance of the chain is given by


(R2)= Nb2 (1-3)

i.e., the size of the polymer is proportional to N1/2. Further, it can be shown that if

the initial segment of a freely jointed chain is fixed at origin of a Cartesian coordinate

system, and if P(R) is the probability distribution of finding the Nth segment at a

position R, for large N and for R > b, the probability distribution function is

Gaussian.

P(R, N) -( 3 3/Vb 2 (_32. (1-4)
27Nb2 2Nb2
This probability distribution function is a Green's function which satisfies the diffusion

equation associated with the random process (position) r(s) of a Brownian particle with

L = Nb regarded as time.

1.1.2 Freely Rotating Chains

In this model, the bonds have rotational freedom but the bond angles are fixed at

r 0, where r 0 is the angle between any nth and (n + 1)th bonds (0 < 0 < r/2).

Therefore (r, rn+) = b2cos0. It can easily be shown that for any n and m (n < m),

(r, r,) b2cosm-"O. From the right hand side of Equation (1-2), one then gets

(R2) Nb2 1 + -cos 2b2cosO ( -cos .O) (1-5)
1 cosO (1 -cosO)2

For large N, this result approaches


(R2) Nb21 + (1-6)
1 cosO











u(s)


Figure 1-2: The wormlike chain model of a polymer. The figure shows an instantaneous
configuration of the continuous chain of length L with an end-to-end radius R.

From these results, it is possible to derive an expression for the average projection of

the end-to-end vector R on the initial tangent of the chain ul = rl/b.

N 1 cosNO
(R ui) b b-1 (ri r) cos (1-7)
n=l

As N tends to infinity, cosNO tends to zero for 0 < 0 < r/2, and therefore this

result reduces to
b
LtUN, (R ul) = L. (1-8)
1 cosO

The quantity on the left-hand side of Equation (1-8) provides an operational

definition of an important property of a polymer chain, namely, the persistence length.

It is important to note that this definition of persistence length is valid only for large

N. Persistence length, as defined above, measures the distance from origin till the

chain 'remembers' the initial direction ul. (Only until then will the dot product in the

summation of Equation (1-7) be non-zero.)

For a freely jointed chain, there are no restrictions in 0, i.e., (ri r,) = 0 (if n / 1).

Therefore it is easy to see that in Equation (1-7), only the first term of the summation

will survive, and the persistence length is simply b. On the other hand, for a freely

rotating chain, the persistence length is ah--iv- greater than b.

1.1.3 Wormlike Chains: A Model of Semiflexible Polymers

As discussed above, a freely rotating chain restricts the flexibility of the polymer

chain by restricting the bond angles, and hence is the simplest model of a semiflexible









polymer. From Equation (1-8), one can write

b L
cosO = L (1-9)
L, N L,

where L = Nb is the total length of the chain. The wormlike chain, illustrated in Figure
1-2 is defined as a limiting continuous chain formed from this discrete chain by letting
N oo, b 0, and 0 0 under the restriction that L remain constant. Therefore, if

we note that

LtN ,oo,-ocosNO = LtN-D 1 = exp (1-10)

Equation (1-7) gives

LtNo(R -U) =L, p(- exp -- (1-11)

which, for large L, reduces to Lp. Further, from Equation (1-5), one can write


(R2) 2LLp 1 (1 exp (1-12)

defining the (R2) for wormlike chains in terms of the persistence length Lp. For large L

(L > Lp), (R2) 2LLp, and for large Lp (Lp > L), (R2) L2. In the intermediate
regime, where L > Lp, the formula


(R2) = 2LLp (1-13)

can be used with less than 5'. error when L > 3Lp.
1.1.4 Physical Interpretation of Persistence Length

In the discrete freely rotating chain model, we saw that (rn rn+1) = b2cos0.
However, for a continuous wormlike chain,
/L
R r(L)- r(0) u(s)ds, (1-14)

where u(s) = dr(s)/ds is the tangent vector at s, the arc length. Further, u. uj
(. -0 ., where Oij is the angle between the points i and j along the chain. The extent

to which the chain is flexible is determined by the correlation (ui uj) =((. .).
In general, (cosO(s)) denotes the mean cosine of the angle between chain segments









separated by the contour length s. This function possesses the property of so-called

multiplicativity: if the chain has two neighboring sections with lengths s and s', then


(cos0(s + s')) = (cose(s))(cose(s')). (1-15)

The function having this property is an exponential, i.e.,


(cosO(s)) = exp -- (1-16)

where the pre-exponential factor is unity, because cos0(s = 0) = 1, and Lp is a constant

for each given polymer. This constant is the basic characteristic of polymer flexibility

and Equation (1-16) is the exact definition of persistence length.

The physical interpretation of persistence length is readily drawn from Equa-

tion (1-16). When s < L,, Equation (1-16) gives (cos0(s)) t 1. Hence the angle

O(s) fluctuates around zero. This simply means that chain segments that are closer

than Lp have nearly the same direction. For the opposite case s > Lp, (cosO(s)) } 0.

This means, O(s) can be anything from 0 to 3600 with equal probability. So the chain

direction gets completely 'forgotten' at lengths much greater than Lp.

1.2 Scope of the Present Work

The present work focuses on two problems of interest:

Forces generated by confined, adsorbed flexible polymers: Mean-

field studies: Adsorption of flexible polymers is of great importance in various

processes of technological and biological relevance (e.g., cell .r-'regation, inhibition

of viral replication, bacterial adhesion to surfaces, colloidal/nanoparticle interactions,

adhesion, lubrication, and bilayer membranes). A number of experimental stud-

ies [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] have appeared in the literature that have indicated

the importance of understanding polymer-induced forces in tailoring polymer 1lV. r-

for specific applications. In addition to substantial experimental studies on polymer

adsorption at interfaces, a number of theories of polymer adsorption have been devel-

oped over the past four decades. A model case that has been investigated widely is the

adsorption of linear, flexible polymers onto one or two uniform flat surfaces [14]. The

structure of the adsorbed polymer 1-v.-r is described in terms of trains, loops, and tails,










Tall \ j r- -- ; -: | I
ail Adsorbed chains Free chain



\ "1 .............. Bridge


I s ^


..... \ \

IH .
Z.. ..........
Loop Train



z=O z=H

Figure 1-3: Two interacting physisorbed linear, flexible, homopolymer 1lr, -i confined
by flat, parallel, adsorbing surfaces separated by a distance H. The adsorbed chain
conformations are described in terms of loops, trains, tails, and bridges.

and additionally in terms of bridges in the case of two interacting 1-,-is, as illustrated

in Figure 1-3. The primary purpose of theoretical attempts in this area is to develop

predictive equations for the structure of the l.-s-i~ and to relate the structure to the

forces.

The present work develops the appropriate thermodynamics of interacting polymer

1i.. r- and a correct formulation for estimating polymer-induced forces in the model

case described above. While the method of estimating polymer-induced forces has been

discussed in some previous works [15, 16], there are three contributions from the present

work:


A complete, thorough, and general discussion of estimating interfacial tension and
normal force of compression of interacting physisorbed polymer l. r-i confined by
two flat, parallel, solid surfaces,


The first correct results of forces between two planar physisorbed polymer l~i. rs
in good solvents using lattice mean-field theories, and,




























Figure 1-4: An illustration of the actoclampin model: An ensemble of elongating
clamped filaments under compression or tension propelling the motile surface. Re-
produced from Dickinson and Purich [18].


An examination of the effects of polymer- i .-r anisotropy (bond correlations) on
the structure of the polymer 1.- r- and the interaction forces under restricted
equilibrium conditions [17].

Biophysical modeling of the motility of Listeria monocytogenes pro-

pelled by actin filaments: Actin-based motility is seen in cell crawling using

filipodial/lamellipodial extensions, intracellular propulsion of organelles, and the motil-

ity of microbial pathogens such as Listeria and S'i,. //.I The push-pull forces involved

in the propulsion are generated by actin filament elongation. In order to understand

the underlying '1.i,1l1, ,! i. mechanism that generates the necessary forces from the

biochemical process of filament elongation, a number of models have been proposed.

One of the models, proposed by Dickinson and Purich [18], treats the motile system as

an affinity-modulated, processive motor complex which, while simultaneously clamped

to the elongating end of the actin filament attached to the motile surface, facilitates

filament elongation and uses the resulting forces to propel the motile surface (see Figure

1-4 for an illustration).

The present work models F-Actin as a slender elastic rod and estimates the forces

generated in the propulsion of Listeria by actoclampin motors [18, 19]. The stepwise

motility observed experimentally by Kuo and McGrath [20] and predicted using the









actoclampin mechanism by Dickinson and Purich [18] is reproduced. An estimate of the

torques generated in the system is made to show how the filament elongation and clamp

translocation results in long-length scale, right-handed helical trajectories of the motile

surface and the attached actin tails.

1.3 Organization of the Present Work

The rest of the dissertation is organized as follows: C'!i lpter 2 presents a brief sum-

mary of the theoretical and computational studies concerning the adsorption of flexible

polymers at interfaces. In ('! Ilpter 3, we discuss the thermodynamics of inhomogeneous

systems, develop the formalism to evaluate tangential and normal forces in interacting

physisorbed polymer 1l-V.-i and estimate these forces using the lattice, numerical mean-

field theory of Schecl i,' .- and Fleer. The effects of incorporating bond correlations and

polymer-!i-,- r anisotropy into the mean-field on the structure of the polymer 1-v.. ri

and the interaction forces are examined for good solvents under restricted equilibrium

conditions in ('! Ilpter 4. An analysis of the bending of semiflexible polymers modeled

as elastic rods is presented in Chapter 5, where two cases (a rod hinged at both ends

and a rod clamped at one end and hinged at the other end) are considered. Modeling

F-Actin as a slender elastic rod and employing the actoclampin model of force gener-

ation, C'! Ilpter 6 examines the motility of Listeria mon'... '. u', propelled by actin

filaments. Finally, C'! Ilpter 7 summarizes the key results and identifies problems of

future interest that evolve out of the present work.















CHAPTER 2
ADSORPTION OF FLEXIBLE POLYMERS AT INTERFACES: A LITERATURE
REVIEW

2.1 Introduction

Flexible polymers physisorbed at interfaces are interesting from a fundamental

point of view and in terms of their numerous applications of technological relevance

(e.g., lubrication, adhesion, self-assembled drug-delivery vehicles, stabilization or

flocculation of colloidal dispersions, bil I -r membranes, and prevention of protein

adsorption) [12]. A thorough understanding of the structure of these physisorbed

polymer lI--irs along with the forces generated by these rlV- -i~ is essential for tailoring

polymer interfaces for specific applications. Experimental studies in this discipline have

focused on:


Characterizing the surfaces (e.g., active sites on surfaces)


Probe the structural details of the l1- -r (e.g., segment density distribution and
average thickness), and,


Measure the net polymer-induced forces arising from steric, electrostatic, hydro-
dynamic, and other interactions.

They have contributed significantly to the understanding of polymer adsorption. How-

ever, experiments are still limited in their ability to discern the finer details of the l1v.-r

structure (e.g., the arrangements of the polymer segments on the surface as loops,

tails, trains, see Figure 1-3), and the contributions of the various interactions towards

the net polymer-induced force. As a result, the need for theoretical guidelines that

relate the conformations of adsorbed polymers with the resulting forces is indispensable.

In this context, a model case that has been widely investigated is the adsorption of

linear, flexible, monodisperse homopolymers present in a monomeric solvent onto one or

two uniform flat surfaces [14]. In this case, the polymer liv r structure is described in

terms of loops, tails and trains (and bridges in case of two surfaces not very far apart).









The conformational details of the adsorbed chains are crucial in determining whether

the interaction between the surfaces is attractive or repulsive.

The interactions between polymer segments and the solvent molecules can lead

to the classification of the solvents as good solvents in which the segment-solvent

interaction is close to the same magnitude or more favored than the segment-segment

and solvent-solvent interactions (the polymer chain spreads out), or poor solvents in

which the segment-solvent interactions are significantly less favored than the segment-

segment and solvent-solvent interactions (the polymer chain tends to collapse). These

interactions are characterized by the Flory-HIl-.-ii-; parameter X. For the case X = 0,

the segments and solvent molecules have no specific preference over one another. In the

present work, all calculations are considered for the case X = 0.

The present work considers forces arising due to the confinement of polymers

by two flat surfaces. Depending on the characteristic time of compression of the

physisorbed 1ri-.;-i, one can distinguish three possibilities in such a system.


When the characteristic time of compression of the li.-r is larger than that of
desorption, the chemical potentials of the polymer chains and solvent molecules
are the same as the corresponding ones in the 'surrounding' bulk solution. This
situation is referred to as full equilibrium situation. In order to maintain the
constant chemical potential, the total amount of polymer between the surfaces
varies as the system is compressed.


The solvent is in equilibrium with the bulk solution whereas the total amount of
polymer in the system is fixed. This would correspond to a case when the charac-
teristic time for desorption of polymer chains is larger than the characteristic time
of compression. If the characteristic time for rearrangement of the conformations
is significantly smaller than the time of compression, the solvent molecules 'drain'
out while the polymer chains (whose total mass is fixed) between the surfaces
rearrange themselves into new 'equilibrium' conformations. This situation is
referred to as restricted equilibrium.


The third possibility would correspond to the case in which the polymer chains
take longer time to rearrange themselves than the time taken to compress the
system. Then, the chains would not have reached equilibrium with the bulk
solution. This is the non-equilibrium case.









2.2 Theories of Polymer Adsorption: Overview

The theories of polymer adsorption are broadly based on a density-functional

approach or a mean-field approach. In a density-functional approach, the free energy of

the system is written as a density functional1 and it is required that the equilibrium

segment density distribution corresponds to the free energy minimum. In the mean-

field approach, the equilibrium segment density distribution is determined by the

conformation of a single polymer chain in the presence of an external mean field,

appropriately defined to account for the rn i, ,v- body interactions in the system. This

aside, using the scaling results proposed by de Gennes [14] based on the so-called

magnet analogy (analogy with highly fluctuating magnetic systems near the critical

temperature), the results of the density-functional approach have been modified, giving

rise to the so-called scaling/free-energy-functional (FEF) theory. Here we provide

the salient features of the guidelines on the structure of adsorbed polymer l-. -?i and

polymer-induced forces provided by these approaches.

2.2.1 Density-Functional Theories: Cahn-de Gennes Approach

The first successful attempt at describing polymer adsorption may be termed the

Cahn-de Gennes density functional approach [21, 22]. In this approach, a free energy

functional (FEF) is derived for polymer 1-.. -i~ using Cahn and Hilliard's [23] ideas to

obtain the free energy of a nonuniform system.

The free energy is expressed as a sum of a surface interaction term accounting for

the attractive/repulsive interactions between the surface and the polymer segments and

a functional accounting for the nonuniform concentration that develops between the

surface and the bulk. The typical surface interaction parameter is expressed as a linear

function of the concentration of the segments at the surface, a boundary condition,

and the binding energy between the segments and the surface. As long as the binding



1 In simple terms, a functional is a function of a variable which itself is a function of
position and/or time. In this case, the free energy of the system is expressed as a func-
tion of the segment density, which is a function of position. Thus a functional is derived
for free energy.









energy for a chain is significantly higher than the thermal energy but weak when

considered per segment (true for long chains), the linear approximation suffices.

The functional term contains a 'local' free energy contribution given by the virial

expansion of the Flory-H l-.iii' free energy of interaction between the segments and

the solvent, and a nonlocall' term accounting for the entropic constraints in placing the

chains due to the presence of other chains and other segments (excluded volume) as

well as the energetic terms accounting for mechanical and chemical equilibrium with

the bulk, depending on the type of equilibrium that exists between the adsorbed chains

and the bulk. The entropic contribution is a function of not only the segment density

but also of its ,j,,,l. l,/' in the system, with the density and its gradients treated as

independent variables.

The number of virial coefficients (and their values) considered in the Flory-flrI-'-ii_ -

virial expansion depends on the quality of the solvent. For instance, for good solvents it

is sufficient to keep only the second virial coefficient v[21].2 For theta conditions v = 0

and hence only the third virial coefficient remains [24]. For poor solvents both the

second and third virial coefficients are used (in this case v has a negative value)[25, 26].

The details of the derivation of the free energy are presented in Fleer et al. [12],

and are similarities between this approach and the mean-field approach are discussed

therein.

The segment density profile as a function of distance from the surface by minimiz-

ing the FEF with respect to position. The free energy functional is derived in terms



2 The condition where only the second virial coefficient is important (and a mean-
field approximation is appropriate) is also referred to as i,,irj.:,,l solvent condition. The
assumption that only the second virial coefficient is important implies that only pair
correlations are significant and that three-body and higher-order correlations are neg-
ligible. On the other hand, mean-field approximation is appropriate when the spatial
fluctuations in the segment concentration are small. Therefore, this definition implicitly
assumes that the segment concentration is large enough so that its spatial fluctuations
are small. So, the marginal solvent regime is a region between the semi-dilute and the
concentrated regimes, often encountered in experimental systems and most of the re-
sults in the present work.









of a single order parameter b(z), which is related to the segment volume fraction as

O(z) = (z), where z is the normal distance from the surface.

Before proceeding to summarize some useful results, the following comments are in

order. De Gennes [21, 22] refers to the above formulation as a mean-field theory. The

label 'mean field' in this context differs from the mean-field approximation used in the

numerical as well as analytical mean-field theories we discuss later. There is clearly a

mean-field assumption in the virial expansion for the 'local' contribution of the free

energy functional. Nevertheless, fundamentally the Cahn-de Gennes formulation is a

density-functional approach with the functional being obtained using some mean-field

approximations. Further, it has be shown by Jimenez [27] that under marginal solvent

conditions, the Cahn-de Gennes theory is equivalent to the so-called Ground State

Dominance Approximation (GSDA), a simplified solution of a mean-field formulation

(see Section 2.2.3) in which the adsorption is described by the conformations of a single

polymer chain in the presence of an external field [28]. This formulation leads to a

Schr6dinger-like diffusion equation, which can be solved in terms of an eigenfunction

expansion. The leading term of this solution corresponds to the chains in the lowest

energy state (the 'ground state'), i.e. the adsorbed chains.

The following are the important predictions by the Cahn-de Gennes approach:


This approach ignores the chain end effects and considers chains of infinite
molecular weight (N -- oc). Hence, it does not account for the presence of tails
and their effects in the structure and forces.


The existence of three regions in space, namely, the proximal, central and the
distal regions, in the concentration profiles is observed. The proximal region is
dominated by short-range surface interaction parameter. The concentration profile
in the central region is universal. The adsorption liv-r here has the local structure
of a semi-dilute solution and the correlation length of concentration fluctuations
corresponds to the thickness of the adsorbed li-r. In the distal region, the
concentration profile decays exponentially to the bulk value.


In the central regime, the segment concentration follows the power law decay as
a function of distance from the surface and is independent of the bulk concentra-
tion. Under marginal solvent conditions, the concentration profile has an inverse
square dependence on the position from the surface. This result will be contrasted
with the scaling results in Section 2.2.2.









The force between the two walls is alv--,v- attractive for an undersaturated l1.- r
(i.e., (F/Fo) < 1.0, where, F is the surface coverage, and Fo is the saturation
surface coverage, in equivalents of rni_..,1l i, rs) under full equilibrium conditions,
with a power law dependence on the surface separation, h, f ~ -h-3. These
conditions are hard to reproduce in experiments because the chains are usually so
strongly adsorbed that they cannot desorb or diffuse much within the timescale
of compression. The experimental results, therefore, mostly indicate repulsive
forces. Restricted equilibrium conditions, where the total amount of polymers
between the surfaces is fixed, are probably closer to experiments. However, using
the Cahn-de Gennes approach, de Gennes[22] determined that the interaction
force (and the free energy) for the case of restricted equilibrium vanished for all
separation distances between the two surfaces.

As a means to examining forces under restricted equilibrium conditions, de Gennes

employ, .1 the self-similarity arguments of scaling in bulk polymer solutions (based on

the 'magnet analogy') in modifying the FEF. This study resulted in a set of works

which are presented under the next subsection.

2.2.2 Free-Energy-Functional (FEF)/Scaling Approach

In this approach, de Gennes [22] modified the FEF using renormalized scaling

arguments, wherein the local and nonlocal terms of the FEF are modified based on the

scaling results of the correlation length and the osmotic pressure in semidilute solutions.

Once again, the theory only considers chains with infinite molecular weight (N -- o0).

In the central region, the power law has a coefficient of -4/3, a decay less pronounced

than the one predicted from mean-field arguments (-2). Under full equilibrium

conditions, both scaling/FEF and the Cahn-de Gennes approach, along with the

GSDA (discussed in Section 2.2.3) show the same attraction between the surfaces.

Under restricted equilibrium conditions, a repulsive force is observed at all distances

for saturated surfaces. Further attempts in this approach include generalization to

various solvent conditions [26, 29, 30] and study of interactions between undersaturated

1..- s [31].

The FEF-Scaling approach has since been extended by Rossi and Pincus [31, 32]

to the case of undersaturated polymer 1-., r-. Their results show that, for moderate

undersaturations, i.e., 0.5 < F/Fo < 1.0, the forces become less repulsive (as compared

to the saturated case). For large undersaturations, i.e., F/Fo < 0.4, an attractive force

occurs.









The next subsection summarizes the key aspects of analytical mean-field ap-

proaches to examine the structure of the polymer lI-.-r. This approach is based on the

Edwards equation, which exploits the analogy between the conformation of a polymer

chain and the diffusion equation. A numerical analog of the same approach is the lat-

tice, mean-field theory of Schecl i' .-' and Fleer [33, 34]. An analysis of this theory is the

primary focus of this work and is presented in ('! Ilpters 3 and 4.

2.2.3 Analytical Mean-Field Theories

The Edwards Equation

The adsorption of polymers at interfaces can be described, in a mean-field treat-

ment, by the conformations of a single polymer chain in the presence of an appropri-

ately defined external field U(r) (r being the spatial position vector), which in turn

depends on the polymer conformations or segment density distribution. In the case of

adsorption on a flat surface, with the assumption of lateral homogeneity in the system,

the external field U(r) is only a function of the normal distance z from the surface.

Such a type of problem requires a self-consistent approach in which the external field

U(r) is dependent on the local concentration along the z direction, while the con-

centration is itself a function of the external field. The mathematical description of

the polymer conformation is based on the statistical weight, G(z, n), of a chain that

contains n segments, with the end-segment being located at a distance z from the

surface. As first shown by Edwards [28], G(z, n) satisfies the following Schr6dinger-like

equation3 known as the Edwards equation:

G(z, n) a2 G(z, n) U (z
S 6 9z2 U(z)G(z, n). (2-1)
On 6 OZ2

The interactions between the segments and the wall can be introduced in two

different v--iv-. One possibility is to include an additive delta function (at z = 0) in

the external field U(z). The other option is to account for the interaction by using an



3 A lattice version of Equation (2-1), sometimes referred to as the propagation equa-
tion, is discussed in ('!i lpter 3. The numerical solution of the lattice version is for
chains of arbitrary length, whereas the analytical solutions to the Edwards equation
shown in this and the following section address the limit of infinite chain length.









effective boundary condition:

G(z, n) G(O,n)
(2-2)
On 0 b

where b is a characteristic length, known as the extrapolation length, associated with the

strength of the adsorption energy. If the latter method is used, the external field U(z)

contains only the mean-field excluded-volume potential acting on a segment. In the case

of a polymer lv,-,r under marginal-solvent conditions, U(z) is given by


U(N) =v(), (2-3)


where v is the second virial or excluded-volume coefficient and Q(z) the segment

concentration at a distance z from the surface.

Self-consistency requires U(z) to be related to the statistical weight G(z, n). This

relation is given by the so-called composition rule ,


() = C j G(z, n m)G(z, m)dm (2-4)

The evaluation of the constant C depends on whether one is considering the case of a

polymer -1v- r in equilibrium with a bulk solution, i.e., full equilibrium [35], or the case

of restricted equilibrium [36].

Ground-State Dominance Approximation

Analytical solutions of Equation (2-1) may be obtained as an eigenfunction

expansion. The leading term in the eigenfunction expansion corresponds to chains

in the lowest energy state, the i,..-;;,./ state, which corresponds to adsorbed chains.

The approximation where all but the leading term in the eigenfunction expansion are

neglected is the so-called p.i.'ni,,l-tate dominance approximation. This approximation,

which in essence neglects the effect of tails in the adsorbed l~V,-r, is equivalent to

the free-energy functional approach presented earlier (Section 2.2.1), as shown by



4 At this time we will not attempt to describe the physical meaning of the propaga-
tion equation or the composition rule. This can be done more easily when one considers
the corresponding lattice equations discussed in Chapter 3.









Jimenez [27]. As with the case of the Cahn-de Gennes approach, the forces between two

saturated lv. --i~ under full equilibrium conditions are .li. --,- attractive and those under

restricted equilibrium conditions are close to zero.

Beyond Ground-State Dominance: Two-Order-Parameter Theory

Going beyond the ground-state dominance approximation facilitates the char-

acterization of the contribution of tails to the segment density profile. Semenov and

coworkers [35, 36, 37, 38] consider the tail contributions by distinguishing between the

adsorbed and free chains in defining statistical weights and defining separate segment

densities for loops and tails. The overall segment density profile is obtained simply

adding the loop density and the tail density. In order to distinguish loop and tail

contributions, they define a second order parameter and another partial differential

equation for that order parameter. Details on the derivations for different conditions

can be found in references [35, 36, 37, 38]. We shall focus here on the three characteris-

tic lengths that result from the above analysis.


The first length is the extrapolation length which is related to the strength of the
adsorptive potential and defines the extension of the proximal region.


The second is the characteristic length z*, which represents the crossover of the
loop and tail density profiles.


Finally, the third length of importance is related to the cutoff distance that
defines where the segment density starts to decay exponentially, and therefore is
also referred to as the I7.,;. thickness.

The interval (b, A) corresponds to the central region defined by de Gennes (see Sec-

tion 2.2.1). Within this region, z* delineates the loop-dominated region from the

tail-dominated portion of the 1lV.-r.

Here we make use of these characteristic lengths to present the results obtained for

adsorption from dilute and semi-dilute conditions. The loop density profile for these

cases is given by
2
d forb < (2-5)

and


S~ z-8 for z* < z < A,


(2-6)









whereas the tail density profile is given by


t ~ 3 n ) for b < z < z* (2-7)

and
20
t ~ for z* < z < A. (2-8)

As can be seen from the above equations, the overall density profile arising from

the two-order-parameter theory leads to the same scaling relation obtained from the

Cahn-de Gennes approach, i.e.,

Total -2. (2-9)

The strength of the approach summarized here is that it allows for the determination of

the tail-dominant and loop-dominant regions and for the determination of i-mptotic

laws for the density of loops, tails and free chains.

The two-order-parameter theory provides analytical insights into the nature of

tail-induced effects such as tail-induced repulsion between 1-.- ri [36]. This theory

predicts a decrease in repulsion for very small undersaturation, and attraction for any

(F/Fo) < 0.98.

2.2.4 Numerical Mean-Field Theory

The primary work in the numerical mean-field theory has been by the group of

Fleer (cite Fleer's book). Fleer and coworkers have developed a lattice-based numerical

mean-field theory and have improved upon some of its limitations [33, 34, 39, 40, 41].

The present work will provide a preliminary examination of the capabilities and

limitations of the numerical mean-field theory in providing quantitative guidelines

on the structure of the adsorbed lv-- ris and the polymer-induced forces. A detailed

description of the theory itself is provided in the next chapter. The equivalence of this

theory with the analytical self-consistent mean-field theory of Semenov and coworkers is

discussed in [37, 42].

2.3 Computational Studies of Polymer Adsorption

Computer simulations provide the most convenient way to test these theories and

the various approximations that go into them almost individually. They provide rigor-

ous, exact solutions of the problem within a given set of assumptions. By judiciously









manipulating the simulation system, the accuracy of the mean-field predictions and the

validity of the approximations in the mean-field theories can be tested in a targeted

fashion. An efficient Monte Carlo algorithm can provide all the structural features of

the polymer 11-V,-ir between the surfaces. De Joannis [43] has taken the first step in

this direction by a careful examination of the structure of physisorbed 1.v.--i~ for a wide

range of adsorption energies and molecular weights. Jimenez [27] has taken the first

step in correlating the structure of the polymer i-v.-r with the polymer-induced forces

using lattice Monte Carlo simulations and the so-called contact-distribution method

(CDM, [44]) to evaluate forces between finite/semi-infinite objects in the presence of

polymer chains [45].

However, as discussed in Section 2.5 and C! Ilpter 3, there is a caveat as to whether

this change in Helmholtz free energy calculated as above is indeed the normal force.

Even under good solvent conditions, where some concerns regarding solvent equilibrium

are no longer crucial, the veracity of the results [27] obtained using CDM is still an

issue. There are no easy v--~i- of testing the validity of these calculations within a

lattice-based formalism unless one can calculate the osmotic pressures in the film.

However, comparisons with Monte Carlo results of similar but continuous systems

would shed some light in this regard. These calculations are beyond the scope of the

present work but are merely sir.-.- I. here as a necessary exercise before one can

venture to further use CDM as a method of estimating forces in interfacial lattice

systems.

2.4 Experimental Studies of Polymer Adsorption

In the previous subsections we have summarized some of the theoretical predictions

for the density profile and the force of compression of physisorbed polymer 1i-,--r. A

number of experimental techniques (e.g., hydrodynamic measurements, ellipsometry,

evanescent-wave-induced fluorescence, small-angle neutron scattering, surface force

apparatus, and atomic force microscopy) have been used to study the structure

of physisorbed and grafted 1lv. -ir and the resulting polymer-induced forces. The

experimental efforts made so far can be classified broadly into three categories, namely,

those which focus on the amount of adsorbed polymers (,i/. ., rl features), those which









give information on the microstructure of the l--r, and those focusing on force

measurements. We present a brief discussion of some of these here.


Hydrodynamic measurements, ellipsometry [46], and evanescent-wave-induced
fluorescence [47] are some of the techniques that have been used to measure the
l-1v-r thickness or the total amount of adsorbed polymers. These experiments have
confirmed that the thickness of the adsorbed l-1v-r is of the order of the coil size
and have been used extensively in studies of adsorption kinetics or competitive
adsorption.


Neutron scattering techniques, based on either small-angle scattering or reflec-
tivity experiments, have been used to probe the density profile of adsorbed and
end-grafted lv-, r-i [48, 49]. The results obtained with these techniques are in
qualitative agreement with theoretical predictions.


The third group of experiments have studied the interaction force between
physisorbed polymer l--ir-. Most of the force experiments have made use of
the SFA and have measured the forces under good, theta and below-theta
conditions [13, 50]. The analysis of the experimental data shows qualitative
agreement with theories, but again, no definite comparisons with the theories
are possible due to the large number of uncontrolled factors that typically
exist in experiments. For example, it has been observed experimentally that
polymer chains do not adsorb uniformly on surfaces but form 'islands' [51, 52].
Therefore, it is difficult to use 'average' measurements such as the ones obtained
from SFA and compare those forces with theoretical predictions. Atomic force
microscopy (AFM) could be used as a 'local' probe in this respect, but one
must 1p .i attention to the influence of the shape and size of the AFM tip and
its affinity (or lack thereof) to the polymer chains in order to interpret the
measurements meaningfully. Moreover, current theories mostly ignore lateral
surface inhomogeneities and are inadequate for predicting the force of compression
by finite-sized objects.

2.5 Estimation of Polymer-Induced Forces: Validity of the Theoretical and
Computational Results

It is important to note that in all the theoretical and computational results

discussed in this chapter pertaining to the calculation of the force of compression of

two physisorbed polymer lI-,-i, the force is defined as the negative of the derivative

of the free energy with respect to surface separation, under constant surface areas,

temperature, and other appropriate 'extensive' variables. The present work contests

the validity of such a definition within the lattice mean-field formalism, especially

because the force-surface separation pair of 'intensive/extensive' pair of variables are

not positive homogeneous functions of the zeroth/first degree [53], as required of the









terms in the integrated equation of state of a system. Other reasons for this argument,

specific to the definition of partition function within the lattice mean-field formalism,

are considered in Chapter 3.

A homogeneous function is a function with multiplicative scaling behavior. If the

argument is multiplied by some factor, the result is multiplied by the power of this

factor. Let f : V W be a function between two vector spaces over a field F. We -i

that f is homogeneous of degree k if the equation


f(av) -= kf(v) (2-10)

holds for all a c F and v c V. A function f(x) = f(xx, ..., Xn) that is homogeneous

of degree k has partial derivatives of degree k 1. Furthermore, it satisfies Euler's

homogeneous function theorem, which states that


x'ixf (x)- kf(x) (2-11)
i=1

In classical thermodynamics, we write differential equations of state5 of a macro-

scopic system as, for instance,


dF = -SdT PdV + IdN. (2-12)


In this equation, the extensive variables such as entropy (S), volume (V) and amount

of substance (N) are homogeneous functions of first degree, whereas the intensive

variables such as temperature (T), pressure (P) and chemical potential (p) are ho-

mogeneous functions of zeroth degree. Therefore, using Euler's homogeneous function

theorem, one can write the integrated equation of state as


F -PV + pN, (2-13)



5 Note that Equation (2-12) is analogous to Equation (2-11), when k = 1. This is
true for the free energy as it is an extensive variable.









with the Gibbs-Duhem equation being


SdT + VdP Ndp = 0. (2-14)


Classical thermodynamics requires that the appropriate set of macroscopic vari-

ables to describe the energy of a system (using an equation of state) be a set compris-

ing of at least one extensive variable and the rest being either extensive variables or

their corresponding intensive variable (such as S-T, V-P, and N-/i). For physisorbed

polymers confined between two surfaces of area A each, separated by a distance h and

in equilibrium with a bulk solution, one cannot arbitrarily define the excess free energy

F" at constant temperature, bulk osmotic pressure, and bulk chemical potentials as


dF" = adA fAdh, (2-15)


where a denotes the total surface tension, and f denotes the force. This is because,

while A and a are homogeneous functions of degree one and zero respectively, h and fA

are not. In other words, while doubling the surface area doubles the free energy without

changing the surface tension, free energy and surface separation do not have such a

linear relationship. This results in the fact that one cannot integrate a 'differential

equation of state' such as Equation (2-15). Further, one cannot write a Gibbs-Duhem

equation for the system consistent with Equation (2-15). However, when F" is defined

appropriately, a modified version of Equation (2-15) is valid, with certain restrictions.

This is the fundamental problem concerning the thermodynamics of nonuniform

systems and is addressed in C'! plter 3.

It appears that all the works reviewed in this chapter that concern the estimation

of polymer-induced normal forces of compression, be it the density-functional approach

or mean-field approach or the computer simulations of Jimenez, may not have consid-

ered the above argument in defining their forces in the interface. Based on the present

study and the works on thermodynamics of nonuniform systems (see discussion in

Section 3.3), it appears that a definition of normal force of compression as

1 ,ex
f t M -) (2-16)
A, OH / ,Vpi,As









is fundamentally incorrect unless the partition function 'il'/. ./il accounts for the

reversible work-mode involved in the normal compression of the 1 i,-r-. In the above

equation, A8 is the surface area of each of the adsorbing surfaces, H is the surface

separation, and Q"x is the excess grand canonical free energy derived from the partition

function. Moreover, there are additional restrictions as to the appropriate chemical

potentials that are conserved in this equilibrium (see Section 3.6 for further discussion).

These issues bring to question the validity of the simulation results of Jimenez [27] and

the computational mean-field results of Scheul i' I,' and Fleer [34], and Semenov and

coworkers [36, 38]. Whether the results of the scaling theories (which do not obtain

forces from the partition function of the system) are valid is a question that needs

further exploration.

In the next chapter, we discuss the thermodynamics of interacting polymer

1 .i. -i and develop expressions for evaluating forces of compression of two physisorbed

polymer 1--. rs confined by flat, parallel surfaces under full and restricted equilibrium

conditions using the lattice, numerical mean-field theory of Sd!, n i' 11P' and Fleer.

We present the first correct results of forces of compression using the mean-field

theory, and then examine the capabilities and limitations of this theory and its various

improvements in providing quantitative guidelines for relating the l ,-rv structure to the

interaction forces.















CHAPTER 3
POLYMER-INDUCED FORCES FROM NUMERICAL MEAN-FIELD THEORIES

3.1 Introduction

In this chapter, we develop the thermodynamic formalism for estimating tangential

and normal stresses in interacting polymer lv. rs within a 1-D lattice, numerical mean-

field theory [34]. We show that, within a one-dimensional mean-field approximation, the

normal force of compression is not equal to the negative derivative of the free energy

with respect to the normal distance H between the two adsorbing surfaces under good

solvent conditions. This is because the partition function fails to explicitly account for

the mechanical work in maintaining the distance H between the surfaces. Therefore,

in order to obtain expressions for tangential and normal stresses in the interface, we

consider a continuum analysis of the interface and then adopt the results for a lattice

system. Using the above formalism, we examine the lattice mean-field predictions of

interaction forces, interfacial tension, and interaction potentials of polymer 1. rlS in

good solvents under full and restricted equilibrium conditions over a range of surface

coverages.

3.1.1 Background Information

Consider the problem of two interacting polymer li- r- physisorbed on flat, par-

allel, impenetrable surfaces placed at a fixed but arbitrary separation distance H (i.e.,

the dividing surfaces are not Gibbs' surfaces, as is usually defined in thermodynamics

of thin films). In the context of interacting electrical double 1-., ri, it has been shown

by D. i ii 1-ii Langmuir, Verwey and Overbeek [54] that the force acting between the

two plates can be obtained either from the free energy or from the osmotic stresses in

the system. Employing the former approach in obtaining the forces between interacting

polymer l~V. rs, both the lattice and the continuum-formulation analytical theories

derive expressions for excess free energy of the solution between the surfaces, which is

the difference between the free energies of the system in the presence and absence of

the surfaces. Note that in the absence of the surfaces, the system is homogeneous with

25









the same composition as that of the bulk solution with which it is in (full) equilibrium.
Therefore, the excess grand canonical free energy is defined as

` (H) = Q(H) (3-1)

The free energy of interaction between the lz. --i~ is defined as the change in free energy

when the surfaces are moved reversibly from an 'infinite' separation distance to a given
separation distance H,
nt Qex(H) Qe(oo) (3-2)

The force between the two surfaces is then defined as the derivative of the interaction

free energy with respect to separation distance,

1 81 (aQe'
f t (3-3)
+ T,Vjpi,As U / T,V1pi,As

The above relation implies that the work performed for the compression of the polymer
lv-. -is, namely, dW = -fdH, is contained in the thermodynamic potential derived

using the appropriate statistical mechanical formulation for the conformations of the
chains in the interface, and that the appropriate chemical potentials of the polymer

chains and the solvent molecules are invariant upon compression (see Section 3.6 for
further discussion). However, this analysis breaks down in the context of lattice and

analytical mean-field theories because of the inability of the partition function to

explicitly build in the work of compression arising from the attractive or repulsive
forces that might exist between the confining surfaces. In particular, the free energy
formulated using the lattice mean-field theories only accounts for the tangential stresses

in the interface relative to the bulk stress, as shown in this chapter.
In a lattice mean-field description of polymer l-v.-i s (for details, see Section 3.4 and

Scheul i, -' and Fleer [33, 34]), one writes the grand canonical partition function and

obtains the equilibrium conformation of the polymer chains and solvent molecules by
minimizing the corresponding grand canonical potential. The equilibrium conformation

is then related to the density distribution in the system. This density distribution
depends on the composition of the bulk solution with which the polymer 1 i-rs- are in

equilibrium. From the density distribution, one can obtain the excess free energy of the









1-.-ris relative to that of the bulk solution, at a given separation distance and area of

the surfaces. In Sections 3.4 and 3.6, we show that this excess free energy is equal to

the so-called film tension, which is the energy required to create a thin polymer film

(consisting of two interacting polymer lI.-i ) of a given area 2A, and maintain it at a

thickness H.

In order to obtain the correct estimate of the interaction forces between the l1.1rs,

we turn to a continuum analysis of the polymer l.r-i~ along the lines of Evans [15]

and Ploehn [16, 55]. The force of compression, the film tension and the interfacial

tension of the 1.ir-i are then obtained by relating the resulting normal and tangential

stresses (respectively) to the thermodynamic potential given by the work function

based on continuum mechanics. The polymer-induced force of compression per unit

area is shown to be equal to the di-i. ,ii:iii-; pressure as defined in the theory of thin

liquid film (see for instance, Babak [56]), i.e., the Volterra derivative of the excess

grand canonical potential of the polymer film with respect to the surface separation.

The di-i. ,ili.i; ; pressure is the difference between the osmotic pressure in the midpoint

and that in the bulk, when the polymer 1..r-i are in full equilibrium with the bulk

solution, analogous to the result in electrical double l.r1-i [54]. The case of the polymer

1.. -ir being in restricted equilibrium (total amount of polymer between the surfaces

is fixed) is also discussed. The formulation presented here is analogous to the ones

presented by Evans [15] and Ploehn [16, 55], but is simpler and more direct. Moreover,

whereas Evans [15] and Ploehn [16] have examined the predictions of analytical

mean-field theories based on Edwards' equation, there has been no analysis of the

correct predictions of force of interaction by the lattice, numerical mean-field theory of

Scheul i' ,' and Fleer [34], to the best of our knowledge. This forms the focus of the rest

of the study. We present the first correct results of polymer-induced forces between two

planar surfaces using lattice numerical mean-field theory.

3.1.2 Organization of the Chapter

The remaining part of this chapter is organized as follows: In Section 3.2, the sys-

tem under consideration is defined. Section 3.3 discusses the general thermodynamics









of inhomogeneous systems, and defines the appropriate partition functions, thermo-

dynamic potentials, and the criterion for equilibrium. Using the lattice mean-field

theory of Schel i,' and Fleer [34], the expression for grand canonical potential and

semi-grand potential are derived under full equilibrium and restricted equilibrium,

respectively, in Section 3.4. Expressions for the force of compression and interfacial ten-

sion are obtained in Section 3.5, using the principles of continuum mechanics. Following

this, in Section 3.6, we develop the expressions for normal and tangential stresses for

the lattice model and show that the excess grand canonical potential derived in Section

3.4 is indeed equal to the film tension. In Section 3.8, we illustrate the predictions of

the lattice mean-field theory for a few sample cases. The discrepancies observed be-

tween the results based on the correct formulation derived in this work and some of the

previous results published in the recent literature, based on the incorrect formulation,

(Jimenez et al. [57]) are discussed.

3.2 System and Surroundings

The system under consideration (see Figure 3-1) is a monodisperse, homopolymer

solution in a monomeric good solvent that is confined between two parallel, solid,

impenetrable, adsorbing surfaces. The polymer molecules physisorb on the surfaces

to form two polymer l1v. rs, with the distance between the adsorbing surfaces denoted

by H. We denote this region between the two surfaces as a thin or a thick film. The

solution in the film is exposed to and is in equilibrium with an infinite reservoir of a

homogeneous bulk solution whose properties are denoted with a superscript or subscript

b. When the two lI-. -ir are non-interacting, i.e., the force between the two surfaces is

zero, the film consists of two physisorbed polymer interfaces and a homogeneous bulk in

the middle with the same composition as that of the phase b and is considered a thick

film. However, when the two solid surfaces are close enough so that there is no uniform

'bulk' phase in the middle, the polymer solution between the surfaces are said to form

a thin film. The 'film-excess' properties are defined with reference to the homogeneous

bulk phase b. The interfacial tension is then defined with respect to the normal stress in

the film. In order to confine the thin film of polymer solution between the two surfaces,

mechanical work has to be performed for maintaining the area of contact between the











Solid Surfaces-,



Intel
La





ex,int.--.
ijex,int = Cjex~b


r b



l-
Reservoir; Phase b






Reservoir; Phase b


Figure 3-1: Two interacting adsorbed 'l.- r- separated by a distance H. Note that the
dividing surfaces are not Gibbs' surfaces. Here, the superscript or phase b denotes
bulk conditions, the superscript int denotes the interface, the subscript m denotes the
conditions at the midpoint of the interface (z = H/2 in a 1-D system), P refers to pres-
sure, II denotes the osmotic pressure, p / is the exchange chemical potential (defined in
Section 3.6), and 7 is the film tension.

solid surfaces and the polymer solution, and for maintaining the normal distance of
separation H between the two surfaces. The former work is usually reported in terms of
the interfacial tension a, while the latter work is typically represented in terms of the
disjoining pressure of the thin film TTd or the normal force of compression of the polymer
1-. -is wT AswhereIn modeling the film, the dividing surfaces are defined to be at the z-
planes of contact between the surfaces and the solution. Therefore the dividing surfaces
are not Gibbs' surfaces and the surface separation H is an independent variable.
With this picture, one can understand why the disjoining pressure cannot be
written as the negative derivative of the excess grand canonical potential derived
from a partition function. Due to the presence of the parallel confining surfaces, the
ensemble average properties in the film are inhomogeneous only in one dimension.
(This is denoted as the z-direction.) Mechanical stability demands that the normal
stress be uniform everywhere in the film while the tangential stress is uniform in each









/;,-plane. The normal force required to maintain the film is simply the excess normal

stress in the film relative to the reservoir while the interfacial tension is simply the

integral of the tangential stress relative to the normal stress in the film. However,

this normal stress in the film is not known apriori when the film is thin, i.e., when

the normal force is non-zero. What one does know is the pressure of the bulk fluid in

the reservoir. Therefore, one writes the partition function of the thin film (within the

mean-field approximation in this study) with reference to the bulk reservoir phase. In

other words, the mechanical work required to maintain the surface separation of a thin

film by compressing the polymer l V.--i~ is not explicitly accounted for in the partition

function. As will be shown in Sections 3.4 and 3.6, the excess grand canonical potential

in such a formulation is indeed equal to the film tension 7. While the two reversible

mechanical work modes are present in the film tension, the normal stress in the film

is not explicitly present. This is the reason that one cannot define the normal force to

be the negative derivative of a partition-function-based thermodynamic potential with

respect to the separation distance.

3.3 Thermodynamics of Inhomogeneous Systems

The appropriate statistical mechanical ensemble for the problem of interacting

polymer l1iv-,rs under full equilibrium conditions is the grand canonical ensemble.

However, the corresponding grand canonical potential is not equal to PV because the

system under consideration has a nonuniform spatial distribution of species, and hence

nonuniform pressure and chemical potential distributions. In this section, we review

the thermodynamics of a general inhomogeneous system and identify some key results

therein that are relevant to our discussion. For further details, the interested reader is

referred to an elegant review by Wajnryb et al. [58]

The grand canonical partition function of a general inhomogeneous system within

the assumption of local equilibrium is given by

S[V; dFN p[- drf(r)(r) + pp (r)Zd,(r)], (3-4)
2ref N. a=-

where ( denotes the terms to ensure indistinguishability of identical system configura-

tions, Na denotes the set of all possible particle numbers, FN, denotes the phase space










variables, (r) and h,(r) denote the energy density field, and the number densities of

components (a = 1, 2, ..., c), and P3(r) and po(r) denote the corresponding Lagrange

multiplier fields. The partition function is a functional of these Lagrange multiplier

fields.

It follows from the definition of the grand canonical partition function that

[Sln V;,p]
-- F(r) (3-5)


and
^ ef I
--- n -(r). (3-6)


Here E(r) = ((r)) and no(r) = (r(r) are the ensemble averages of the energy density

field and the number densities respectively. The grand canonical partition function

leads to the definition of the appropriate grand canonical potential of the system

-- In ([V; ) P dr (r). (3-7)
kT eef V

Here b(r) is the 'density of l n and is a thermodynamic field.

If one were to write the fundamental equation of the entropy of this inhomogeneous

system, one would have

S= j dra(r), (3-8)

where
c
u(r) k- [3(r)(r) + (r (r)na(r) + (r)]. (3-9)
a=1
The variations of entropy and its density would turn out to be


6 = dr6s(r), (3-10)


where
c
6s k[O3(r) (r) + Pa(ri%, ,(r)] (3-11)
a=l1
with the corresponding Gibbs-Duhem equation as

cr C
Sdr[ (r)6/3(r) + n(r)P(r) (r) (r)] 0 (3-12)
Sa=1









where the integrand is not equal to zero at each point in r. This is because the thermo-

dynamic field Q(r) is a nonlocal functional of the Lagrange multiplier fields P(r) and

pa(r).

In the context of interaction between polymer 1-_. -i~, it is this nonlocal dependence

of the 'grand canonical potential density' that accounts for the work of compression

arising from the attractive or repulsive forces between the two surfaces, as well as the

work required in maintaining the area of the surfaces exposed to the solution. However,

in the problem of interacting polymer 1.. -ir confined by two parallel solid surfaces, the

integral in Equation (3-12) one-dimensional, i.e., in z-direction alone. We shall use this

in deriving the criterion for thermodynamic equilibrium in the film in Section 3.6.1.

3.4 Free Energy from Lattice Numerical Mean-Field Theory

3.4.1 System, Preliminaries, and Partition Function

Consider a lattice system confined between two flat, parallel, impenetrable surfaces

and filled with polymer segments (linear, monodisperse chains of finite length r) and

solvent molecules (good solvent). We only consider fluctuations in a single direction

normal to the surface, which we define as the z-direction. Each segment or solvent

molecule occupies one lattice site. The lattice adjoins the two hard/adsorbing surfaces

and is divided into M 1--.- rs of sites parallel to the adsorbing surfaces, in the z-

direction, i.e., H = 1,M, where lz, the length of each lattice site in the z-direction is

taken to be unity. Each lI-v-r contains L lattice sites. The surfaces therefore correspond

to z = 0 and z = M + 1, and adsorption takes place in z = 1 and z = M. If D is

the coordination number of the lattice, then, a lattice site in any 1-. r z has D nearest

neighbors, of which a fraction A/,-z is in l-v-r z'. The fraction of nearest neighbors that

lie in the same l-1 .-r, then, is A0, that in the .,.i i:ent l-1v.-r is A1 and so on. Therefore,

A can be viewed as the fraction of the nearest neighbors of a lattice site. In the case

of a first-order Markov chain statistics, A's are the step-weights of the random walk.

Furthermore, in a simple cubic lattice, there are six nearest neighbors to a lattice site

in a 1,- -r z, of which four are in 1-. r z and one each in the lI-r-i z 1. Therefore, for

a cubic lattice, A1 = 1/6, Ao = 4/6, and, A = 0, | j I> 1. If there are n polymer

chains and no solvent molecules in the film (defined as the region enclosed by the two









surfaces), then it follows that,

nr + no = ML. (3-13)

We define the volume fractions of polymer and solvent at any given l v-r as,


()= ); 0(z) n(Z (3-14)
L L

such that,

n(z) + nO(z) = L; (z) = 1 (z) (3-15)

and,

n(J) n r; nO(z) = n. (3-16)
z z
When the solution in the film is open to a bulk solution with respect to numbers,

the numbers of polymer chains and solvent molecules between the surface vary. The

appropriate grand canonical partition function E is given by the sum of the appropri-

ately weighted canonical partition functions (the canonical partition function Q being

defined for a fixed number of solvent molecules and a fixed number of polymer chains

in a defined set of conformations corresponding to a given energy of the system) for

different values of possible number of polymer chains (and hence different numbers of

solvent molecules).


a uns ({nc} V, A, T) exp ( exp (Us -so1
= (3-17)
exp ( n (z) exp (zc) (3-17
ex kT ex kT

where, {nc} is a set of permissible conformations of the polymer chains in the film such

that,

n' = n, (3-18)
c
f is the configurational entropy of arranging the polymer segments and the solvent

molecules in one of the n~ conformations within mean field, V is the volume of the

system, A is the surface area, T is the temperature, p0 is the chemical potential

of the solvent molecules corresponding to the bulk solvent concentration, and p is

the chemical potential of the polymer segment corresponding to the bulk polymer

concentration.









The interaction energies are given by


Usrf = -kTxs(n(1) + n(M)) (3-19)


and

Use_ = kTx n(z) ((z)). (3-20)

The equilibrium distribution of the polymer segments in the lattice is determined

by maximizing the grand canonical partition function in Equation (3-17). A detailed

derivation of the interaction energies, configurational entropy and the equilibrium

distribution are given in the original Schecl i. i,--Fleer papersxx and is not repeated

here. Instead, we merely present the important equations in the theory that pertain to

the estimation of the grand canonical potential.

It is to be noted that the chain statistics is defined in terms of concentrations

of chain conformations and not in terms of concentrations of individual segments. A

chain is then treated as connected segments. A chain conformation is characterized

by defining the l-1 v-r numbers in which each of the successive segments are present. A

conformation can then be denoted as


Conformation c : (1, i)(2,j)(3, k)...(r 1,)(r, m), (3-21)


indicating that the first segment is in lIV--r i, the second segment is in liv1-r j and so

on. This implies that many different actual arrangements of the segments (in the film)

can correspond to a defined conformation. If a segment s is placed in l rv-.r z i and

segment s + 1 is placed in 1l- .r z j, then the number of different allowed placements

of s + 1 relative to s is DAo if j = i, and DA1I if j = i 1. It then follows that a dimer

with conformation (1, i) (2,j) can assume LDAj_ different positions. A trimer with

the conformation (1, i) (2,j) (3, k) can assume LD2 Xjj k-j positions, if backfolding of

the chain is allowed. A partial correction to the backfolding of segments will be applied

later on. Using similar logic, the number of v- iv- of arranging r segments of a chain in

conformation c given in equation (3-21), in an empty lattice is LDr-1Aj_iAk-j...A -l,









which can be written as L~cDr-1, where


Lc = (As,+1)c (3-22)


where,


s,s+ A0o, if s=s+ (3-23)
Ai, otherwise

Further, if one considers only the number of arrangements of a part of the polymer

chain, then, the notation wc(s, t) is used, where the part of the chain considered is

between segments s and t, including both of them. Similarly, the summation C:(,t)

would account only for all the possible conformations of that part of the chain. It is

evident that c = wc(1, r), and that c() = Zc(1,r)(). The definition of statistics of the

chain in terms of conformations and in terms of individual segments are interchangeable

and are related by the following relation:


n(z) Y rzn^ Y r^ r, (3-24)
c z

where, r,,c is the number of segments of a chain in conformation c in 1lv. r z. The nota-

tion used to indicate the l -r number corresponding to the segment s of conformation

c is the subscript k(s, c), k being the l,-v-r number.

Finally, we define the following reference system:


nr amorphous polymer segments forming a pure polymer liquid (n chains each
with r segments), occupying nr lattice sites


Two flat, parallel, impenetrable surfaces, each with L sites on them, with no
mL, sites between them, enclosing no solvent molecules


Every site is filled with either a polymer segment or a solvent molecule.

3.4.2 Configurational Entropy

The configurational entropy term f ({n} V, A, T) is the number of v--,v of placing

the n chains of r segments each and n solvent molecules between the two surfaces,

corresponding to the energy given by the two exponential terms, i.e., the degeneracy









of the system. In calculating the degeneracy, the Bragg-Williams approximation of

random mixing within each 1'.- r is used, i.e., the polymer segments in each 1'i.r

are considered to be randomly distributed over the L lattice sites in that l, r. As

discussed in Section 3.4.1, the number of v-~ i of placing a chain in a conformation c

in an empty lattice is LcD'r-1. If the lattice is partially occupied, then a chain can

only be placed if the appropriate l-., r- (as defined in the conformation, e.g., equation

(3-21)) have vacant sites. Therefore, we have to apply r correction factors to the

combinatorics in order to partially account for the excluded volume, one factor for each

segment of the chain. This correction factor, in the simplest case, is the probability

that a given 1-, -r has at least one free site (usually called the vacancy probability). If

the number of occupied sites in l1-v r z at any given instant is v(z), then the vacancy

probability is given by (1 v(z)/L).

Therefore, the number of possibilities of placing a chain in conformation c is given

by LcDr'-1 [ 1 (1 vk(s,c)/L) = uc(D/L)r1 H 1(L k(s,c)), where Vk(s,c) is the

number of previously occupied segments in the 1l.,r k where the segment s of the chain

in conformation c is placed.

Therefore, the number of arrangements u of placing the first chain of conformation

c (of the n, chains) in an empty lattice is given by

M rzc-1
= c(D/L)r-1 (L v)). (3-25)
z=1 v(z)=0

Placing nc chains in conformation c would give rise to the factor wc"(D/L)(r-1)"0, while

the multiplication extends to v(z) = ncrz,c 1. Placing all n = e n~ chains would lead

to the number of arrangements

( ) M n(z)-1
w(n) (D/L)(r-1)n ( ( nf J (L- v)). (3-26)
c z=l v(z) 0

Now the remaining L n(z) = no(z) sites in each of the M 1li.-i~s have to be filled

with no solvent molecules. The number of possibilities of arranging no(z) number of

solvent molecules in l--r z is given simply vby Y z)=n(z)(L~ v(z)). Upon- simplification,
solvent molecules in lv z is given simply by 11,(,) -( < vz)). Upon simplification,









the expression for configurational entropy is found to be

1c, 1
(1-(7!) nKI (171 nj)!) (3-27)

The factorials nc! and no(z)! correct for indistinguishability of the rc chains in each

conformation c and of the solvent molecules in each 1.-r z.

The configurational entropy of the reference system f+ can be derived using

similar arguments. In the bulk, the distinction between 1-v. rs is irrelevant. Since the

number of (equivalent) lattice sites in the bulk polymer is nr, the factor (L!)M in

equation (3-27) is replaced by (nr)!, and the factor L(r-1)" by (nr)(r-1)". Further, in the

bulk, all conformations are equally probable. Hence, nc = n, and Uw 1. Also since

there are no solvent molecules, no(z) = 0. Therefore, equation (3-27) reduces to


Q (ur) (D/nr) 1)n (3-28)
n!

This expression is also derived by Flory [1]. Therefore, this formulation is consistent

with earlier theories.

3.4.3 Equilibrium Distribution

The grand canonical partition function of the system is given by Equation (3-17)).

At equilibrium, the appropriate free energy of the system takes its minimum value,

i.e., the partition function E is at its maximum value. This situation corresponds to

the most probable set of conformations (with a corresponding number of chains nfq.

and number of solvent molecules n%,), which will be obtained in this section. If we

neglect fluctuations once the system has attained equilibrium, then the number of

chains and solvent molecules are fixed. This means that the sum over all n's in equation

(3-17) is replaced by the maximum term. To obtain the equilibrium distribution, i.e.,

the number of chains nd in conformation d in the equilibrium situation, the terms

within the sum in equation (3-17) are differentiated with respect to nd and set to zero.

Realizing that Ez no(z) = ML r E n~, we have


nQ I+ (p ,) /kT= 0. (3-29)
8nd VA,T,{nc-nd}









This differentiation corresponds to adding one chain in conformation d from bulk

and placing r,,d segments in each 1lv. r z, and removing r solvent molecules from the

appropriate l -i~s in order to maintain constant volume (i.e., all sites are occupied).

This conservation of total volume can be expressed as

O(noa(z)) (az) (3-30)
rz-a =^ (3-30)
8Od ) V,A,T,{n nd} ( Ond ) V,A,T,{n nd}*

This maximization is subject to an additional constraint that each liv.r in the lattice

has to be completely filled with either polymer segments or solvent molecules. This

constraint is implicitly included, because in performing the differentiation, no(z) =

L c rzcc = L n(z) is used. However, for multicomponent systems, this constraint

should be explicitly included with an appropriate Lagrange multiplier.

Performing simple algebraic manipulations on equations (3-27) and (3-28), and

using Stirling's approximation of the logarithm of a factorial, one obtains

In = MLInL ncn ( c) no(z) nno(z) nr -(r 1)nlnL. (3-31)

Upon differentiation and simplification, the equilibrium set of conformations are then

obtained as

In lnd+r- 1+lnr+ ( +
L kT
Sr,,d{ Xs (61,, + 6MK) + X KK(z)) ) + ln0(z)} (3-32)

Defining

In C = r 1 + Inr + k (3-33)
kT
we write,

In = n wld + In C + r,d n G,
L
z
M
or, C dfJ G, (334)
z=1
where,


In G, xs (l,z + 6M,z) +X ((n z)) (0())) + n0(z).


(3-35)









Equation (3-35) is a discretized version of the Edwards equation, discussed in C'! plter

2.

The number of chains in a particular equilibrium-conformation d is proportional

to WUd, a product of (r 1) step-weights, i.e., A's, which determine the conformation

as is evident from Equation (3-21). Further, nd is also proportional to r weighting

factors (G,'s), as shown in Equation (3-34). Each segment in the chain contributes a

weighting factor Gz, which is a Boltzmann-type factor accounting for the change in

free energy when a solvent molecule in liv, r z is replaced by a segment. It consists

of a contribution for the exchange adsorption energy -Xy kT, when the li-v-T z is

.,.li i,:ent to a surface (z = 1 or z = M); a factor for the exchange interaction energy

between segments and solvent molecules -X ((O(z)) (o(z))), for replacing a solvent

molecule by a segment; and a factor for the local entropy -k ln (z) of the solvent

molecule. For the simplest case r = 1 (monomer), we have, from Equation (3-34),

((z) = n(z)/L = KG, exp(p po)/kT, where K is a constant. Since the volume

fraction varies linearly with the weighting factor Gz for a monomer, Gz is also called

monomer weighting factor. Another way of viewing Gz is as follows: The mean-field

experienced by a segment in the presence of polymer chains and solvent molecules is

given by

u(z) = Usrf(z) + Useg-,so(z) + u'(z), (3-36)

where,


Usurf(z) = -kTX, (61, + ,z) (3-37)

use-so 5(z) -kTX (((z)) (())) and, (3-38)

u'(z) = -kTInfo(z), (3-39)

such that,

G() exp ( (3-40)

Here, uW(z) is the hard-core or excluded volume interaction in the I'iV'T.









The unnormalized probability of finding the sth polymer segment of a chain in the

1,-.-r z is given by the function G(z, s)


G(z, s) = G(z) A,, G(z', s 1), (3-41)


where A is the fraction of nearest neighbor sites in the 1I-vr z' of the lattice. This

equation is analogous to the propagation equation discussed in Section 2.2.3 in C'! Ilpter

2. It states that the sth segment of a chain can be in a given site in l-1v--r z if and

only if the (s 1)th segment can be in one of its nearest neighbors. In enforcing this

constraint of connectivity, a mean-field assumption that does not distinguish the sites

within each 1iv-r is used. In other words, connectivity is enforced 1l-- -b.v--1vr only.

This constraint partially accounts for connectivity and excluded volume.

Further, analogous to the composition rule discussed in Section 2.2.3 in C'! Ilpter 2,

the segment volume fraction Q(z) is given by the composition rule


(z) = 0 G(zs)G(zr s + 1), (3-42)

where C is a normalization constant. For a polymer liv.-r in equilibrium with a bulk

solution, C = b/N, and for restricted equilibrium, C = F/G(N), where G(N) is

the unnormalized probability of finding the end segment of the chain anywhere in the

system.

3.4.4 Free Energy

The free energy of interaction between the two l., r-i is obtained from the grand

canonical partition function as

S= -kTIn ). (3-43)
-ref

The logarithm of the grand canonical partition function is given by

( Usurf Useg-sol Y0 Z no () P Y, nc4
'( f In + + I (3-44)
Yre kT kT kT kT

The bulk chemical potentials are estimated using Flory-HtI-I-i_ -; theory


Si- 1 + ',. + x,2 (3-45)
kT r









and

= b 1 )+ In o + x (3-46)

Substituting the various terms in (3-44), one obtains the excess grand canonical

free energy to be

k L ( )(() b + )) (3-47)
KU z ( Obo ()/

which is the same as

T L (1(z) fIb) (3-48)
z
where II is the osmotic pressure.

The above relation of the excess grand canonical free energy is analogous to the

well-known Kirkwood-Buff equation [16]. As mentioned in Section 3.2, by considering

the excess osmotic pressure with reference to the bulk (reservoir) osmotic pressure, this

potential accounts for the work done in creating the thin film interfaciall tension, 2Lcr)

and maintaining it at the given thickness M (d -i. Piii.-; pressure -TTdA. 11). In other

words, this potential is indeed the film tension 7. This is consistent with the argument

presented at the end of Section 3.3.

When the system is in restricted equilibrium, the film is open with respect to

solvent but not with respect to polymer. In this case, the semi-grand potential is simply



+ (3-49)
kT kT
Therefore we get the expression for semi-grand potential as

Fe (H) n t lnO(z) + X () ((z))- (H ) (3-50)


where the total amount of polymer in the gap Ot = Y, O(z), the solvent density

distribution O(z) = 1 (z), and G(r) is the unnormalized probability of finding a

chain of r monomeric (persistence) units in the gap region. We shall further discuss this

potential and its interpretation in Section 3.8.









3.5 Inhomogeneous Continuum Description of the Film

Based on the arguments in Sections 3.2 and 3.3, one concludes that there are two

reversible mechanical work-modes involved in the problem of interacting polymer lV1.-is,

namely the interfacial tension a and the di-i. iiii;.-: pressure TTd. However, to obtain

useful predictive equations for the disjoining pressure, one has to examine the stresses

in the system. For this purpose, we employ a quasi-thermodynamic framework based

on the principles of continuum mechanics. Within this framework, in each microscopic

volume, the system is considered as an inhomogeneous continuous medium where

the fundamental thermodynamic relations presented in Section 3.3 are valid. This

hypothesis holds when the correlation length in the system is greater than the range of

intermolecular forces.

3.5.1 Continuum Formulation of the Fundamental Equation

We begin by writing the internal energy and entropy balance equations [59] for

the polymer lV-. -ir and the reservoir of homogeneous bulk solution the 1v. r s are in

equilibrium with.

Internal energy balance: The rate of change of internal energy per unit mass of

a material element of a multicomponent fluid is given by

p dU (V.q) + tr(T.Vv) V. ,bi (3-51)
P dt


where denotes the material derivative, q is the heat flux vector, tr denotes trace

of the tensor, T is the stress tensor, ,m'b is the chemical potential of species i defined

on a mass basis, and ji is the corresponding mass flux. The first term on the right

corresponds to energy conduction, the second term corresponds to the mechanical

work-modes, while the last term corresponds to energy change due to material diffusion.

The stress tensor T consists of the isotropic bulk pressure pb and the osmotic stress

tensor T and can be written as

T -pbl + T. (3-52)

Note that here the osmotic stresses are defined with reference to the reservoir bulk

pressure rather than the ..-- vet-undetermined normal stress in the film.









Entropy inequality: For slow and reversible compression, the second law of

thermodynamics becomes an equality. At constant temperature, it reads


p + V.q 0, (3-53)
dt T

where S is the entropy per unit mass. Subtracting Equation (3-53) from (3-51) gives us

the total Helmholtz free energy balance.

Helmholtz free energy balance:

dA t (3-54)
St V.V- t Vv). (3-54)

Upon further simplifications, conversion into molar basis and integration over the

volume of the system, equation (3-54) becomes

dA t dV + b dni + tr(Vv)dV. (3-55)
at dt 't .I

For homogeneous systems, the osmotic stress tensor vanishes uniformly, i.e., r = 0.

Therefore Equation (3-55) reduces to

6hom _pb + _/t6, (3-56)


which is the well-known fundamental equation at constant temperature.

It is important to note that the above fundamental equation of the system,

Equation (3-55), is for both the film as well as for the reservoir. The last term accounts

for the net mechanical work performed in maintaining the area, 2A,, of contact between

the surfaces and the solution as well as maintaining the two surfaces at a given distance

H. This term can be broken down into two contributions one from the film and the

other from the reservoir arising from the exchange of polymers and solvent molecules

between the interface and the reservoir upon equilibration. Therefore it is evident that

the variation of the excess grand canonical potential of the film at constant temperature

is given by

69 e= tr (T 6v)dV, (3-57)
JvI
where V' denotes the volume of the film. This variation corresponds to the mechanical

work contribution due to axisymmetric compression or expansion of the lV. r? S at









constant temperature. Equation (3-57) is analogous the variation of the grand canonical

potential defined in Equation (3-7) in Section 3.3.

Pure compression is irrotational, so Vv is simply the rate of deformation tensor D

with components

Dii i = y, z. (3-58)
Ai
where the dilations Ai are the stretched lengths of line segments in the principal

directions i = x, y, z with initial lengths of unity (in the undeformed state).

3.5.2 Estimation of Disjoining Pressure, Film Tension and Interfacial
Tension

For axisymmetric compression or expansion of two flat, parallel l. -is, the osmotic

stress tensor assumes the following form


7= TH (exe, + eyey) + Te ee, (3-59)


where Tr and Tr are the tangential and normal stresses respectively. (The unit vector in

each direction is denoted by e.) Substitution of the above equation in Equation (3-57)

then leads tol


6ex= 6 T dV rdV ( T-) A dV. (3-60)

It follows from Equation (3-60) that

( Jqex\ jH
-- TH = r dz, (3-61)
6A T,H JO

which defines the excess tangential stress (with respect to the reservoir bulk pressure)

as the film tension. In evaluating the above derivative at constant temperature, it is

to be noted that the corresponding Gibbs-Duhem equation given by Equation (3-12)

must also be satisfied. It is not necessary that the chemical potentials of each species



1 Since the volume V1 is not constant during deformation, the evaluation of integral
in Equation (3-57) is performed by mapping the volume V1 to an invariant reference
volume VI'R using the transformation V' lV6VIR, where lv =- Ilyl,. The above
transformation also enables one to evaluate the variations of functionals according to
the equality fv,, 6(lvf)dV'R 6 (fV,,, flvdV',) 6 (fs fdV).









be constant in the film. We shall discuss the necessary criteria for equilibrium in the

next section and make further observations in this regard.

Moreover, from Equation (3-60) one also has

= -TdA8s As,-,m (3-62)
6H /T,A,

which shows that the disjoining pressure is simply the negative of the center-line normal

osmotic stress T ,m. Here and elsewhere we denote the midpoint (H/2) between the two

solid surfaces by the subscript m, i.e., Hm = H/2.

The above expressions for 7 and 7rd can also be expressed in terms of the isotropic

and deviatoric components of the tensor T. Since the osmotic stress tensor can be

written in terms of the isotropic and deviatoric stresses in the system as


7 = iso + Tdev ( xe ee ee + ee, (3-63)

with the isotropic component given by

1 2 1
Tso -trT -T 2| + -T (3-64)

and the deviatoric component given by

2
Tdev -= r(T 7), (3-65)
3

one has, for the film tension,


j7= H iso- -)dz, (3-66)

and, for the disjoining pressure,


TTd = [Tiso + Tdev]Hm (3-67)

The interfacial tension a can then be obtained from the definition of film tension,

7 2o- TdH.









3.6 Film Tension, Interfacial Tension and Force of Compression from
Lattice Models

We now reduce the general expressions presented above for the film tension and

the disjoining pressure in Equations (3-66) and (3-67) for the case of lattice mean-field

approximation. The purpose of this section is develop the expressions necessary to

obtain the osmotic stress tensor (and hence the isotropic and deviatoric stresses) within

the lattice formalism so that the force of interaction, the film tension and the interfacial

tension can be obtained from lattice models. Although the results presented here can

be generalized to any variations of the lattice theory, we shall primarily consider them

in the context of the Scheui i' j--Fleer theory as it has been used extensively in the

literature.

3.6.1 Full Equilibrium

We first consider the case when the polymer 1-i r~ are in full equilibrium with the

bulk phase b, on a simple cubic lattice of size L x L x H, with each lattice element

having unit length in each direction (i.e., the area A8 = L2, and z increases in unit

steps). The total Helmholtz energy of the system (film + the reservoir) is then given by


A -PbV + jPir + As (p() ez)) W(z) Pz) (3-68)
i z I
Here the last term is consistent with the fact that the inhomogeneity of thermodynamic

properties in the film is one-dimensional (see earlier discussions on the thermodynamics

of inhomogeneous systems). P(z) = pb + Hb H(), consistent with the definition of the

stress tensor in Equation (3-52), pi(z) are the chemical potentials of species i evaluated

for a homogeneous solution at local composition Qi(z), and ci(z) are position-dependent

nonlocal fields which act on individual components. The fields ci are chosen such that

(i(z) is an equilibrium distribution consistent with the configurational constraints

imposed by the connectivity of polymer chains. The total amount of each species in the

system is given by


ni = ni + A i(z).


(3-69)









The criterion for equilibrium is determined by minimizing the total Helmholtz energy of

the film and the reservoir at constant T, ni, As, and H. Therefore we have

6A = -Pb6Vb + pn\ + A 6 ((z) ) )C (z ) P(z) (3-70)
i z I

since the terms Vb6Pb and n6pfi are zero. Note that the first two terms in Equa-

tion (3-70) are non-zero because they account for the change in the Helmholtz energy in

the reservoir due to the exchange of material and energy with the film. Since the total

amount of each species in the system is constant, we have

6bn -As6Z{(z). (3-71)

Using the above result and using partial molar volumes vi the first two terms of

Equation (3-70) can be rewritten as

-pbVb + P 6n = -A (pb Pb) i(z). (3-72)
i z i
Therefore Equation (3-70) becomes


6A A, Y, C 6pi(z) 6e(z) u6P(z) O(z) (3-73)
6A = < (3-73)
S+ E ~i(z) C(z) v P(z) 1p + VPb 6bi(z)

We identify the leading term on the right hand side of Equation (3-73) to be the

Gibbs-Duhem equation for the film, analogous to Equation (3-12) and hence this term

vanishes, leaving

6A A Pi(z) C(Z) P (P(z) Pb) i(Z) (3-74)

For equilibrium, we have 6A = 0 for all variations at constant T, ni, As and H. This
criterion gives us the appropriate nonlocal self-consistent fields ci(z) as


C(z) P(z) + 7, (n(z) Pb) (3-75)

where we have incorporated the osmotic stresses. It is evident from the above expres-

sion that for the solvent molecules, c0(z) = 0, since they have no configurational degrees
of freedom. These fields also vanish in the homogeneous bulk phase. Further, we can









rewrite Equation (3-75) as


(z) Ce(z) + vn(z) = p + vb. (3-76)

This is analogous to a generalized 'membrane' equilibrium as discussed by Lyklema [60].

Alternatively, the criterion for equilibrium can be represented as the constancy of the

exchange chemical potentials

Pii ii ci (3-77)

in the film and the bulk. Whilst one obtains polymer-induced forces from free energies

in a grand canonical ensemble, it is essential that these transfer chemical chemicals are

held constant.

Substituting Equation (3-75) into (3-68) and simplifying, one obtains the excess

grand canonical potential of the film as

I 7A, =A, [f b- n(z)], (3-78)


equivalent to the result, Equation (3-48), obtained in Section 3.4 and analogous to the

Kirkwood-Buff formula. For compression under full equilibrium deviatoric stresses are

absent. Therefore, Equations (3-64) and (3-66) imply that


riso(z) = b (z). (3-79)

It then follows that the normal stress in the film is simply the osmotic pressure UnI at

the midpoint Hm and that the ld-i- .iiiii.-; pressure or the force of compression per unit

area under full equilibrium conditions is


TTd = lm nb. (3-80)

This result is in accordance with that for electrostatic interactions between charged

surfaces. Whereas in the electrostatic interactions between two parallel plates one

has an electric potential gradient balanced by the osmotic pressure, in the case of

interacting polymer liv.i, one has a concentration gradient arising from entropic

and energetic interactions of the polymer segments and the solvent, which is balanced

by the osmotic pressure. For a symmetric system (where the adsorption energy per









segment X, is the same for both surfaces and the surfaces have same area A8 exposed

to the film and are otherwise identical), symmetry demands that the concentration

gradient vanishes at the midpoint, leaving the normal stress everywhere in the film to

be the midpoint osmotic pressure. Thus, under full equilibrium conditions, the normal

force of compression of the film is simply the difference between the normal stress in the

film and that in the reservoir. In our calculations, we only consider symmetric systems.

Consistent with the definition of film tension, the interfacial tension is given by


2aj- [In II(z)] (3-81)
z

3.6.2 Restricted Equilibrium

The case of more practical interest, commonly referred to as restricted equilibrium,

arises when the total amount of polymer is fixed in the system during compression.

This constraint on the total amount of polymer is imposed by introducing a Lagrange

parameter g(z) that accounts for the additional potential (per segment in z) needed to

maintain a fixed number of polymer chains in the system. Through a similar analysis as

in the previous section, the criterion for equilibrium in the film can be shown to be [61]


i(z) Ce(z) + Van(z) + g4(z) = P + Vab. (3-82)

The film tension for the case of restricted equilibrium therefore becomes

r' 7A A= > [nLb n(z) g(z) )] (3-83)


Comparison of Equation (3-66) with Equation (3-83) shows that the expression for the

deviatoric stress in the film is


Tdev(z) = 2g(z)O(z). (3-84)

As the isotropic stress is still given by Equation (3-79), the dli-i 1111:-; pressure for the

case of restricted equilibrium is given by


7d Il, IIb 2 ., (3-85)


The normal stress everywhere in the film is therefore 'ln 2.), .









The interfacial tension under restricted equilibrium conditions does not follow

from the definition of film tension because the deviatoric corrections differ for the film

tension and the di]-i 1ii11ii-; pressure, as was shown in Section 3.5. The interfacial tension

is simply defined with reference to the normal stress in the film and is given by


2a = [II- I() 2 g(z)(z). (3-86)


3.7 Validity of the Approach

The approach needed to calculate the force of axisymmetric normal compression

of a binary system of interacting polymer liv--ir at constant interfacial area and the

work needed in maintaining the area of the surfaces exposed to the polymer 1., ris at

a constant surface separation in lattice systems interfaciall tension) were described

above. The traditional meaning of the interfacial tension as the work needed in lateral

stretching or compression of the interface at constant surface separation does not

apply to the lattice system. All the results derived in these sections follow from the

thermodynamics of inhomogeneous systems, principles of continuum mechanics and

the assumptions involved therein. Further all these relations are independent of the

statistical-mechanical model one uses to calculate the composition of components in the

film, osmotic pressure and appropriate chemical potentials. In what follows, we shall

focus on the numerical mean-field theory of Sd!. il i' 1-' and Fleer, a summary of which

was presented in Section 3.4, as the statistical-mechanical model to estimate the forces

of compression of polymer I iT-.

3.8 Results and Discussion

We begin here with the some results for the force of compression between two

physisorbed polymer 1 i,-r-i using a lattice (numerical) mean-field theory. All calcu-

lations reported here are performed for first-order Markovian chains in a monomeric

good solvents (X = 0.0), on a simple-cubic lattice under full or restricted equilibrium

conditions. Since our calculations are performed on a lattice, the results are normalized

per lattice site in a l ivr rather than per unit volume, as shown earlier. The deviatoric

stress is therefore calculated on a per segment basis. Forces and segment densities are

reported per unit area (i.e., per the number of lattice sites in a l-iv-).









3.8.1 Full Equilibrium

Figure 3-2 shows the variation of film tension and interfacial tension as a function

of surface separation under full equilibrium conditions. The decrease in the film

tension upon compression indicates that the polymer molecules in the film prefer to

adsorb on the surfaces even for an adsorption energy2 of X = 0.5 kT. This is also

evident from the fact that upon compression, polymer chains do not leave the film,

resulting in an increase in the average density of the polymer segments (Ot/H) in the

film (Figure 3-3). This makes it harder to compress the l'r -iS, which is evident from

the strong osmotic repulsive forces obtained in Figures 3-4a and 3-4b as well as in

the sharp increase in the interfacial tension in Figure 3-2. The trend is very similar

for stronger adsorption energies. The force profiles shown in Figures 3-4a and 3-4b

reveal qualitative discrepancies between our results and the negative derivative of the

excess grand canonical potential (as obtained from the mean-field theory), which has

been interpreted in the literature as the force of interaction [34, 57]. The forces are

predominantly repulsive, as opposed to the monotonically attractive behavior predicted

in the literature for interactions under full equilibrium. Our results agree qualitatively

with the analytical mean-field predictions of Ploehn [16].

Finally, in Figure 3-5 we show the interaction potentials W for different bulk

concentrations for an adsorption energy of X, = 0.5 kT per segment for a chain of 100

segments. The interaction potential is obtained simply by integrating the force profile

rH
W(H)= f d( h)dh, (3-87)
infty

where h is used as a dummy variable for the distance between the two surfaces. This

interaction potential is clearly not equal to the excess grand canonical potential

interfaciall tension) shown in Figure 3-2.



2 The critical adsorption energy Xc for the lattice model is estimated to be 0.18 kT.
The critical adsorption energy represents the magnitude of X, at which the entropic
repulsion between the walls and the polymer chains cancel any wall-segment affinity.
Adsorption occurs only for X, > Xc.










A.04
0
C -0.05 Film Tension, y
----- Interfacial Tension, 2a
C -0.06 =0.05
Sa. =0.5 k T
S -0.07 v r =100


-0.08

E -0.09
ir.
-0. .1.1.1.1.1.1.1.1..
6 10 14 18 22 26 30
Surface Separation, (H/a)

Figure 3-2: Film tension 7 and interfacial tension 2a as a function of surface separa-
tion (H/a) under full equilibrium conditions. The results are shown for an adsorption
energy X, 0.5 kT, chain length r-100, bulk concentration Qb 0.05, and good solvent
conditions, X = 0.0.


3.8.2 Restricted Equilibrium

Under conditions of restricted equilibrium, the gap region is open with respect to

the solvent but closed with respect to the polymer. In this case, the usual practice is

to approximate the restricted equilibrium problem to an equivalent full equilibrium

problem with an additional constraint that the total amount of polymer is invariant

upon compression. By appropriately minimizing the partition function for a mixture

of polymer chains and solvent molecules in the gap region, the segment density distri-

bution between the gap is obtained [34]. Once the segment densities are known, one

can evaluate the semi-grand canonical free energy relative to an 'effective' bulk solution

with which the l~V. rs are in full equilibrium. Again, the negative derivative of this

free energy has been interpreted in the literature as the force of compression. In this

section, we shall first develop the equations for the deviatoric stresses and the force of

compression within the lattice mean-field formulation by correctly incorporating the

results of the generalized membrane equilibrium described earlier.









0.25 .. ... . .... ....

4 -+-Average number of segments
02 per unit volume
0.20



S0.15
E b = 0.05
; X =0.5
Sr = 100
0.10 -



0.05 ,
0 5 10 15 20 25 30 35
Surface Separation, (H/a)

Figure 3-3: Average density of polymer segments (8t/H) in the interface as a function
of surface separation (H/a) under full equilibrium conditions. X, = 0.5 kT, r 100,
Qb 0.05, and x 0.0.


As obtained in Section 3.4, the semi-grand free energy is given by (also see

Equation 24 in [34]),

F ) n ( )ln0 (z) + x (z) ((z)) (-O) (3-88)


The second term in Equation (3-88) is an entropic mixing term while the third term

contains contributions from enthalpic and entropic interactions. The first and the

fourth terms partially account for the deviatoric stresses that arise in the system due

to the confinement of polymer in the gap. Schel i,' and Fleer [34], in page 1886

of their manuscript, denote the last term as "a small attractive term accounting for

the osmotic pressure of the solution outside the pl! Ii and neglect the term in their

calculations. We interpret this term as an osmotic pressure contribution arising due to

the confinement of polymer either outside or inside the surfaces depending upon the

amount of polymer in the gap. Therefore, there could be conditions where this term is

significant. This is demonstrated in Figure 3-6, which shows the negative derivative of

the semi-grand free energy with and without the last term. In what follows, we have

recalculated the mean-field results in that work by including the above term.


i(z) eC(z) + ,nl(z) + g (z) = + v_,b.


(3-89)










0.0002 0.015

S0.0001 0.010 --Force
&y----/H
o.oooo : --"- S -- ... : *-.-. --' 0.005 -- --- \

S-0.0001 m 0.000
S--Force
0 -0.0002 .05 -By/H 0.005
Sb =0.05 X =1
-0.0003 5= 0 -0.010 ,r =100
L.. r =100
-0.0004 ....1 .. .... I .. .. i -0.015 I I
0 5 10 15 20 25 30 0 5 10 15 20
H/a Surface Separation, (H/a)

Figure 3-4: Force per unit area f as a function of surface separation (H/a) under
full equilibrium conditions for r 100 and X = 0.0. The negative derivative of excess
grand canonical free energy is also shown for comparison. (a)X, = 0.5 kT, b 0.05;
(b), 1.0 kT, Qb 0.005.


A consequence of the approximation of restricted equilibrium as an equivalent full

equilibrium problem is that, for a polymer segment, Equation (3-89) becomes


p9(z) (z) + vfI(z) = ,bff + VIlff =b + vfb g, (3-90)


where t fbf and II f are the chemical potential of the polymer chain and the os-

motic pressure corresponding to the 'effective' bulk solution. Thus, the effective-full-

equilibrium approximation leads to g, an effective additional potential or stress needed

per -. iir that is a function of surface separation but independent of z

g ( b b

(pb ) I II f) (3-91)

from which it follows that the interfacial tension under restricted equilibrium conditions

is
M
2a = [fIb () g(z)] (3-92)
z=1
Correspondingly, for the d(-i- .iPii-: pressure, Equation (3-85) is replaced by


f = In nIb 2,/. (3-93)


The chemical potentials and the osmotic pressures in Equations (3-91) and (3-93)

are evaluated within the Flory-fli-,-ii' : approximation for each li-r using Equations










0.0020 i


^ 0.0015 \ -0.00005
= 0.00005
b --= 0.005
0.0010 ... b 0.05
X 0.5
Sr =100
0.0005 -
ciP

S0.0000


-0.0005 *
5 10 15 20 25
Surface Separation, (H/a)

Figure 3-5: Interaction potential W between the surfaces in full equilibrium with solu-
tions of varying bulk concentrations. X, = 0.5 kT, r 100, and X = 0.0.


(3-45) and (3-46). (It follows from Equation (3-91) that for full equilibrium, for which

b =b and I1b i b, g 0.)

Figure 3-7 shows that, under certain conditions, the forces of compression predicted

by Equation (3-85) can be qualitatively different from the derivative of semi-grand

free energy. In general, we observe that this difference is quantitative for extremely

undersaturated 1l-vr-i and for 1-< -irs close to saturation coverage and beyond. In the

intermediate range, qualitative differences (i.e., repulsion instead of attraction) could

result. To indicate the effects of the deviatoric stresses in the system, we show the

variation of the effective deviatoric stress per segment for two surface coverages F = 0.5

and 0.75 (Figure 3-8). The former case corresponds to a starved 1- v-r wherein the

surface coverage is less than the saturation coverage F03 Here the polymer chains

are constrained from entering the gap region, resulting in an attraction. On the other

hand, the latter case corresponds to a slightly oversaturated regime (F > Fo = 0.7).




3 The saturation coverage oF is defined as the limiting value of F corresponding to
b 0. For short chains, Fo is obtained from the adsorption isotherm (a plot of F as
a function of Qb) from an extrapolation of the equilibrium coverage F beyond the initial
rise to b = 0.










0.012
-*--*Without solvent exchange term
0.010
-*-- With solvent exchange term
0.008 I = 0.75
r = 200
0.006

0.004

0.002



-0.002 ---
3 4 5 6 7 8 9 10 11
Surface Separation, (H/a)

Figure 3-6: Correct calculation of excess semigrand free energy under restricted equilib-
rium conditions. The results shown here are for a surface coverage 7 = 0.75, X, = 1.0
kT, r 200, and X = 0.0.


Here the polymer chains are constrained from leaving the gap, resulting in an increased

repulsion. The appropriate interaction potentials corresponding to the above cases are

shown in Figure 3-9.

3.8.3 Implications on the Mean-Field Predictions

We now turn to the implications of the new force calculations on the capabilities

and limitations of the lattice mean-field theories. Here, we reinterpret the results [57]

already published in the literature based on the incorrect thermodynamic formulation

of defining the force to be the negative derivative of excess semi-grand free energy.

A plot of the force per unit area as a function of surface coverage F, for a surface

separation of H/a = 4.5, where a is the lattice -Ip 'i- shown in Figure 3-10a reinforces

the above observations. From our results, it is seen that SF2 predicts higher repulsive

forces than SF1, probably because the elimination of backfolding through SF2 serves to

decrease the entropy of the chains, making them more 'rigid' and harder to compress.

The deviation of SFi from SF2 is particularly pronounced at higher coverages. Further,

upon enlarging the low-coverage region of Figure 3-10a, one observes that SF2 predicts

slightly more attractive forces than SF1. This attraction, observed at low coverages,

should correspond to that of ideal chains (SF1) and non-reversal chains (SF2). It is to










0.015 i

U" 0.010

0.005

0.000 ,-- ^ ... ...

-0.005
S- r0o.5
S -0.010 r -= 200
I
-0.015 --8---8F/18H
o0 I -"-- Force
L -0.020 -

-0.025 I I I I
2 4 6 8 10
Surface Separation, H/a


Figure 3-7: Force per unit area as a function of surface separation (H/a) under re-
stricted equilibrium conditions. 7 0.5, X, 1.0 kT, r=200, and X = 0.0.


be noted that despite the rather significantly attractive effective deviatoric stress per

segment at low coverages (as shown in Figure 3-8), the total force is only very weakly

attractive.

It has been observed [57] that, compared to Monte Carlo simulations with self-

avoiding-walk chains, SF1 and SF2 underestimate the surface saturation coverage,

because the chain statistics permit overlapping of the polymer segments. For this

reason, we examine the force as a function of reduced surface coverage (F/Fo); see

Figure 3-10b. The rescaling preserves the observations already made in the context of

Figure 3-10a. In addition, the corrected force calculations show that the forces at very

low coverages are too small to provide any meaningful comparison with the linearity of

force versus coverage predicted by scaling arguments, which are based on an analysis

of the system as a collection of isolated bridges [62, 63]. The earlier results based on

the SF2 formulation [57] appear to show an agreement with the scaling arguments for

coverages up to about F ~ 0.4; but those results do not correspond to the force of

interaction, as already noted.









0.020

0.010
t-S
g: o.ooo ---- *----ee --
S 0.000

S-0.010
a.

4 -0.050
-0 .040 r=5
-0040 r=200
.S X = 1.0
4 4.050

-0.060
0 5 10 15 20 25
Surface Separation, (H/a)

Figure 3-8: The variation of deviatoric stress per segment upon compression, for two
surface coverages 7 = 0.5 and 7 = 0.75. The results are shown for sX = 1.0 kT, r 200,
and x = 0.0, under restricted equilibrium.


The crossover coverages4 obtained from the corrected calculations are significantly

different from those obtained from the previous calculations. The corrected reduced

crossover coverages are close to the prediction of the scaling/free energy functional

theory (Rossi and Pincus [32], Fc,r ~ 0.4). In this context, it is interesting to contrast

the numerical mean-field results with the two-order-parameter analytical mean-field

theory of Semenov et al. [36], which predicts attraction for I,':u F/Fo < 0.98, for all

values of H/a. Due to reasons as yet unclear, the analytical mean-field results [36] show

surprisingly strong attraction compared to the numerical mean-field theory.

3.9 Concluding Remarks

We have developed the thermodynamics of interacting polymer l1i. r-i and have

derived the appropriate expressions for polymer-induced force of compression between

two surfaces, within the framework of quasi-thermodynamics. The main conclusion

here is that one-dimensional mean-field approximations do not i, /'/., ./li account for

the mechanical work-mode involved in maintaining the separation distance between the




4 Crossover coverage corresponds to the surface coverage at which the interaction
between the surfaces changes from attractive to repulsive.









0.05 r -- -- -- -- | -- -- | -- -- | -- --
0.05

0.04 -- 0.75-- ------


0.03 r II200




0.01 0






Surface Separation, (H/a)

Figure 3-9: Interaction potential W between surfaces in restricted equilibrium for cov-
erages 7 = 0.5 and 7 = 0.75. The results shown here are for X, = 1.0 kT, r=200, and
X 0.0.


adsorbing surfaces. Therefore, one has to be careful in defining the interfacial tension

and the normal force of compression.

Further, we have presented the first correct results for the interaction forces

between adsorbed polymer lIV'Trs based on the lattice-based numerical mean-field

theory. It is shown that qualitative differences between the correct results and the

earlier incorrect formulation could occur under both full and restricted equilibrium

conditions. The force of compression of the adsorbed lv.r-i~ in full equilibrium with

a homogeneous bulk solution is neither monotonically attractive as seen from the

results of Schecl, i,' 1 and Fleer [34], nor is it monotonically repulsive, as claimed by

Ploehn [16]. Under restricted equilibrium conditions, additional deviatoric stresses

develop in the system. Depending on whether the polymer molecules are confined

in the gap or outside the gap, these stresses cause additional repulsion or attraction

respectively. It appears from these preliminary results and further results [17] shown

in C'!h ipter 4 that mean-field theories highly underestimate the attractive forces

at conditions when bridging effects are expected to dominate. The elimination of

backfolding in the chain statistics hardly improves the predictions of the force of

compression. The results presented here are significantly different compared to the







60

0.04 ....0.04
M 0.03 Force, SF / 0.03 Force, SF -

0.02 Force, SF / 0.02 Fce,SF /
/ 1Force, / S
0.01 / I 0.01


-0.01 8-F/H, S -0. 01 -
S0.00 -0.00

t. SF H/a =4.5
-0.02 H/a = 4.5 -0.02 8F/8H, SF r r=200
L -SF/SH, SFi r = 200 -8F/8H, SF2
-0.03 I I -0.03 1 i 1
0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2
Surface Coverage, r Reduced Surface Coverage, F/FT

Figure 3-10: Force per unit area f as a function of surface coverage 7 under restricted
equilibrium conditions for a fixed separation of (H/a) 4.5. (a)Xs = 1.0 kT, r 200,
and x = 0.0; (b) Effects of rescaling the surface coverage 7 by normalizing with the
saturation coverage o7 on the force for (H/a) 4.5, X, = 1.0 kT, r 200, and X = 0.0.


predictions of the two-order-parameter theory, even though the assumptions involved in


the theories are similar in nature.















CHAPTER 4
EFFECTS OF POLYMER-LAYER ANISOTROPY ON THE INTERACTION
BETWEEN ADSORBED LAYERS

4.1 Introduction

In C!i ipter 2, we reviewed various theoretical (scaling, analytical and numerical

mean-field theories) and computational approaches to the problem of physisorption

of linear, flexible homopolymers onto one or two uniform, flat, parallel, impenetrable

solid surfaces. We then developed a thermodynamic formalism for examining the

forces of interaction of these lv. r- and applied it using the lattice numerical mean-

field theory developed by Sd<, ul i, 1. and Fleer [34] with and without backfolding in

Chapter 3. Both these Markov approximations consider an isotropic mean field. Further

improvement in the theory arose due to the introduction of anisotropy in the mean

field by partially accounting for the effects of bond orientations on the equilibrium

properties of the system. This self-consistent anisotropic mean-field theory (SCAFT)

was first proposed by Leermakers and Scheul i' 1-' [40] to study phase transitions in lipid

'ili-i r membranes, in order to account for the anisotropic orientational interactions

between the lipid-like molecules in a membrane. The theory was able to successfully

predict the critical phase behavior of the membrane observed in experiments. This work

also provided an elegant derivation of the theory from the basic principles of statistical

thermodynamics. More recently, van der Linden et al. [41] extended the Scheul i' I-' and

Fleer theory to semiflexible polymers, in which bond correlations were incorporated.

In both these works, the authors studied only the overall segment densities and some

of the broad structural features. The focus in the former work was to develop a theory

for membranes based on statistical thermodynamics and that of the latter work was

to formulate a lattice mean-field theory for semiflexible polymers (e.g., 'wormlike'

chains). Fleer et al. [42] have attempted to relate the numerical mean-field formalism

and the two-order-parameter theory in an attempt to obtain closed-form solutions that

reproduce the numerical mean-field results.









The numerical mean-field theory of Scheul i' I-' and Fleer is, presently, the only

theory from which one can obtain quantitative guidelines on the adsorption of short

chains (of finite molecular weight). It is therefore important to ascertain the reliability

of this theory in predicting the structural details of the adsorbed 1~- .-r, and the forces

of compression of two polymer l1 --ir-. Comparisons have been made with experiments

in the literature (e.g., Fleer et al. [12]). However, many of the intricate structural

details of the adsorbed l v-r are either inaccessible or not easily accessible through

experiments, such as specific information on loops, tails, and bridges. Moreover,

experiments involve many parameters that cannot be precisely controlled or do not

factor into most theories, such as surface roughness and polydispersity of the polymer.

Another very important issue in comparisons with experiments is the ambiguity

in relating a theoretical state of the system (as defined by the number of chains,

number of Kuhn units, a theoretical adsorption energy, and a theoretical segment-

solvent interaction (X) parameter) with appropriate experimental conditions. These

considerations limit the extent of any such comparison to one of a qualitative nature.

In view of these, while it may be instructive and important to assess the validity of a

theory against experiments, a simple, direct comparison would not only be limited but

also, under most conditions, be misleading. On the other hand, computer simulations

are 'exact' within their approximations (e.g., lattice approximation in lattice MC

simulations), and comparisons against lattice-based simulations would enable one

to assess the limitations of the mean-field approximation in the context of lattice

mean-field theories. We, therefore, use Monte Carlo results [27] of realistic chains (self-

avoiding walks) within the lattice formalism as a reference, to examine the predictions

of the mean-field theories.

In this chapter, we consider the effects of anisotropy in the 1v.-r on the resulting

mean-field predictions of the structure of the lIv-r as well as the forces arising from

the interactions between two l --i r-. Specifically, we introduce anisotropy in the SF2

formulation (S(!. u lP i -' and Fleer numerical mean-field theory with second-order

Markov chains). We also provide preliminary comparisons of the anisotropic mean-field

predictions of the structure of the adsorbed lV--ir and the forces of compression of









the Ii. -i against results from rigorous computer simulations [57]. A summary of the

anisotropic mean-field formulation is provided in the next section. (Further details are

available elsewhere; see Fleer et al. [12]; Leermakers and Sdt, l i. 11,' [40].) We then

discuss the improvements in the predictions due to the introduction of anisotropy and

comment on the limitations of mean-field theories as seen from these comparisons. One

of our objectives in the next section is to present a clear and easily understandable

anisotropic mean-field formulation and to provide expressions for the various structural

features of the adsorbed polymer l1.-r.

4.2 Self-Consistent Anisotropic Mean-Field Theory (SCAFT)

4.2.1 Preliminaries and Notations

Consider a lattice system confined between two flat, parallel, impenetrable surfaces

and filled with polymer segments (chains of finite length, r) and solvent molecules. We

only consider fluctuations in a single direction normal to the surface, which we define

as the z-direction. Each segment or solvent molecule occupies one lattice site. The

lattice adjoins one or two hard/adsorbing surfaces and is divided into M 1lIv-i~s of sites

parallel to the adsorbing surfaces, in the z-direction. Each lI--r contains L lattice sites.

The surfaces therefore correspond to z = 0 and z = M + 1, and adsorption takes

place in z = 1 and z = M. If D is the coordination number of the lattice, then, a

lattice site in any 1i.-r z has D nearest neighbors, of which a fraction A/,-z is in 1-V-.

z'. The fraction of nearest neighbors that lie in the same 1.iVr, then, is A0, that in the

.,li ,:ent 1i-v.r is A1 and so on. Therefore, A can be viewed as the fraction of the nearest

neighbors of a lattice site. In the case of a first-order Markov chain statistics, A's are

the step-weights of the random walk. Furthermore, in a simple cubic lattice, there are

six nearest neighbors to a lattice site in a l-.- r z, of which four are in 1- r z and one

each in the lI'. rs z 1. Therefore, for a cubic lattice, A i = 1/6, Ao = 4/6, and,

Aj = 0, | j I> 1. If there are n polymer chains and no solvent molecules between the

surfaces, then it follows that,


nr + n = ML.


(4-1)









We define the volume fractions of polymer and solvent at any given 1.,-r as,


~) z) ;() O) (4-2)
L L

such that,

n(z) + nO(z) = L; (z) = 1 0(z) (4-3)

and,

n(z) nr; n(z) = n. (4-4)
z z
We now present the relevant equations for the case of restricted equilibrium. The

polymer chains are modeled as step-weighted random walks in a simple cubic lattice.

We consider a second-order-Markov chain statistics in which immediate step-reversals

(backfolding of segments) are disallowed. The dimensionless mean-field potential u(z)

that a polymer segment experiences in 1 i-r z is given as


u(z) = -Xs1 61z XsM 6Mz + uint(Z) + u'(z), (4-5)


where Xsi is the Silberberg adsorption energy parameter for the polymer/ solvent

pair on the surface at l-i .r z and %ij is the standard Kronecker delta function. For a

symmetric system, Xs X=sM. The term uint(z) accounts for the energetic interactions

between the polymer segments and solvent molecules within the Bragg-Williams

random mixing approximation.


int(z) =x [(z())- o(z))]. (4-6)


Here, X is the Flory-Hu'--I-ii: segment-solvent interaction parameter that decides the

solvent quality. In the present work, we examine only good solvent conditions, for which

X = 0. The near-neighbor average of volume fraction of the segments in 1i-v.r z is given

by,
M
(A)) A (), (4-7)
S1
which, for a cubic lattice with only the nearest-neighbor interactions, becomes


(O(z)) = A-10(z 1) + AoQ(z) + A i(z + 1).










The Lagrange parameter u'(z) in Equation (4-5) partially accounts for the excluded

volume of the segment and solvent in a given 1 r, v. This is usually based on the

Bragg-Williams random-mixing approximation.


k=4


I
I

Layer z


I
I
Layer z+1


k=4


1 2 3 4 5
0
U


Empty Sites
Filled Sites
New segment
Segments whose bond
orientation will not
'block' the placing of
the new segment ofthe
chain in layer 2.


Figure 4-1: Bond orientations and anisotropic mean field. (a) The notion of bond ori-
entations. (i) A typical polymer chain in a lattice. (ii) Segments and bonds. (iii) Bond
Orientations. In the illustration above, the orientation of the bond between segment
(s 1) and segment s (in (ii)) is k 1 (see (iii)). This way of representing bond orien-
tation thus indirectly specifies the position of the previous, e.g., (s )th, segment. In
a cubic lattice there are six possible orientations. These six orientations can be thought
of as three pairs of 'opposite' orientations. For instance, k = 1 and k = 3 are 'opposite'
orientations. Consecutive segments with 'opposite' orientations will cause a backfolded
conformation; (b) Conformations of consecutive bonds. (i) Straight (ii) Perpendicular
(iii) Backfolded; (c) Schematic representation of the anisotropic mean field in a square
lattice. Isotropic Mean Field: Probability of placing a new segment in 1.v-r 2 = 3/6.
Anisotropic Mean Field: Probability of placing a new segment in 1.v-r 2 = 3/5.


4.2.2 Anisotropic Mean Field

Anisotropic mean-field theory attempts to improve upon the Bragg-Williams ap-

proximation in order to better account for the excluded volume. To better understand


k=4

k=2
(iii)


7- I









the idea of anisotropic mean-field, let us consider a polymer chain as a set of segments

connected by bonds of defined orientations. The notion of bond orientations is illus-

trated in Figure 4-la. Between any three consecutive segments, three conformations

can be identified namely, a straight conformation, a perpendicular conformation and a

backfolded conformation. These conformations are illustrated in Figure 4-lb. A second-

order Markov chain statistics does not allow the physically unrealistic backfolded

conformation. In the case of isotropic mean field, the excluded volume is accounted

for by just requiring that the probability of having a segment in a 1v-,vr is the fraction

of empty sites in that 1~ ,- (This is the standard Bragg-Williams random mixing

approximation.) Therefore, all the bond orientations are equally likely (within the re-

strictions imposed by the chain statistics). In an anisotropic mean field, the orientation

of a polymer segment in a given 1l-v-r depends on the orientations of its neighboring

segments (the neighboring bonds), which apriori limit the probability of having the

segment in that 1 I -r. This is illustrated in Figure 4-1c. This therefore means that each

orientation has to be weighted appropriately. This introduces a bias in the step-weights

of the random walk, making the mean field anisotropic 1 .This idea is explained further

below.

In an isotropic mean field, the Lagrange parameter u'(z) is given by the fraction of

'empty' lattice sites in the 1 i,-v z, which is equivalent to the volume fraction of solvents

(for 'good' solvents).

u'(z) = n(l (z)) (4-8)

In case of anisotropic mean field, there are apriori more 'empty' sites available in the

1l-.-r z once we account for the fact that those segments in li--r z with the same bond

orientation and those in 1l,-r z 1 with a complementary orientation will not 'block'

the given bond orientation. Let Q(z, k) be the fraction of bonds with those orientations



1 Notice that the correction to the mean field is independent of the chain statistics
used (in this case, a second-order Markov statistics.)









that will not block a bond with an orientation k.


Q(z, k) = j (z, s, k) + (z', s, k')} (4-9)
s

Here, k and k' are 'opposite' bond orientations (see Figure 4-1(c)), and z' = z or z 1,

as the case may be. Q(z, s, k) is the volume fraction of sth segments (in r-mers) with a

bond orientation k, in l-v- r z. Therefore, 1 Q(z, k) is the maximum available fraction

of 'empty' sites in -1v.-r z for a segment to occupy with a bond orientation k.

This correction factor is given as,

1
g(z, k) (4-10)
1 Q(z, k)

where Q(z, k) is the fraction of bonds with those orientations that will not block a bond

with an orientation k. Note that in case of isotropic mean field, the correction factor is

unity.

4.2.3 Statistical Weights and Composition Rule

The Boltzmann factor

GCzm(z) exp (uz (4-11)
\ kT /

defines the so-called free/monomer segment distribution function. It is evidently

the unnormalized probability of a 'monomer' segment in liv.r z. We now define the

unnormalized probability2 G(z, s, k), of a s-mer, with the end-segment located in l1v-r

z, with a bond orientation k (as per the definition of bond orientations in figure 4-1(a)).

Since we model the chain using Markov statistics, the probability of a s-mer can be

obtained by summing the probabilities of adding a monomer to the different possible

orientations of a (s 1)-mer that will give rise to the desired conformation (defined by

G(z, s, k)), thereby resulting in a recursive relation for evaluating the statistical weights.

To be consistent with the notation, we need to define the monomer statistical weights

as G(z, 1, k) even though bond orientations do not have a physical meaning in this case.


2 We call this the statistical weight for obvious reasons.









For a cubic lattice, we therefore write,


G(z, k)= (z) (4-12)
6

The corrections for the anisotropy due to bond orientations should be introduced from

dimers on. We now write the statistical weights for dimers.
6
G(z, 2, k) = Gm(z) g(z, k)- G(z', 1,1) (4-13)
= 1

Using second-order Markov statistics, we can further write the statistical weights of a

s-mer as

G(z, s, k) = Gm(z) g(z, k) G(z', 1,1) (4-14)
l1k'
To introduce a short-hand notation, we write the above equations as


G(z, s, k) = Gm(z) g(z, k) (G(z', s 1,1)) (4-15)


Note that this way of evaluating the statistical weights automatically ensures chain

connectivity as shown by Sd. i, l i: and Fleer [33]. We now proceed to evaluate

the segment volume fractions (which we call segment densities) using the so-called

composition rule

G(z, s, k) ,,, G(z, r -s + 1,l)
(4, s, k) C if s / 1 or r (4-16)
(5/6) Gm(z)
G(z,s,k)y 6=1G(z,r +1,1)
(z, s, k) = C if s = 1 or r (4-17)
Gm, (z)
r 6
i(z) (z s, k) (4-18)
s=1 k=1

Here, C is a normalization constant to account for the fact that the statistical weights

are not normalized. For the case of restricted equilibrium, it is defined as


C = (4-19)
r G(r) L G(r)

where Ot is the total amount of polymer between the plates and G(r) is the so-called

end-segment distribution, i.e., the statistical weight of finding a r-mer anywhere









between the two surfaces. The former is defined as


Ot= z=n (4-20)
z

and, the latter is defined as

M 6
G(r) = G(z, r, k) (4-21)
z=1 k 1

A self-consistent solution is obtained by assuming a density profile and isotropic

mean-field (as initial guess) and using an iterative procedure to evaluate the volume

fractions.

4.2.4 Structure of the Adsorbed Layer

Once a self-consistent solution is obtained for the segment densities, one can easily

obtain details of the structure of the adsorbed li-r and the free energy of the system.

To do this, we see that the polymer chains between the two surfaces (0t) are in one of

the five following groups 3


The chains are free, 0f.


The chains are adsorbed only to surface '1', 0f.


The chains are adsorbed only to surface '2', 0'.


The chains form bridges, with the last chain end leaving from surface '1', 0 .


The chains form bridges, with the last chain end leaving from surface '2', 20.

Therefore we have,

Ot = f + Oa + O + O + O (4-22)

Since t CrG(r), we define,


G(r)) Cf(r) () + GI(r) + G0(r) + G (r) (4-23)


3 We use the approach of Fleer [34] here.









and further,
M 6
G*(r) Z=G*(z,r,k) (4-24)
z=1 k 1
for each of the group (represented by *). Now we proceed to define recursive relation-

ships for the statistical weights of free, adsorbed and bridged chains similar to the way

we defined the statistical weights before.

Free chains: Since free chains cannot have a segment in 1-_. ri 1 or M, we have,


Gf(z, 1, k)


SG(z, k),

0,


S< z < M

z 1, M


Gf(z, s, k)


subject to the conditions


(4-25)


(4-26)




(4-27)


Gf(z, s, k)= 0, for all s, z 1, M.


Adsorbed chains: We define similar expressions for adsorbed chains as for free

chains. For instance, the statistical weight of a chain adsorbed to surface '1' is given by


GI(z, 1, k)


{ G(z, k),

0,


z 1, M

otherwise


(4-28)


G(z, s, k)

GI(1, s,2) -


SGm(z) g(z,k) (G(z',s

G.()5l4, 2)


1,1)), if z / 1, k (/ 2

, 1) + G(2, s-l 1)}


(4-29)

(4-30)


subject to the conditions


G (M, s, k) 0; G (1,s, 4)- 0, for all s, k.


(4-31)


In equation (4-31), the term Gm(1) g(1, 2) Y14 Gf(2, s 1,1) would occur in the

recursive expression for Gf(1, s, 2) according to the notation. However, it actually

corresponds to conformations of s-mers with only the end segment adsorbed. Hence it

is added to Ga(1, s, 2).


Gm(z) g(z, k) (G'(z', s 1, 1))









Bridged chains: The recursive relations for bridged chains are very similar to

those for the adsorbed chains. For instance, the statistical weight of a chain forming a

bridge with the last chain end leaving surface '1' is given by


G1(z,, k) = 0, z> 1 (4-32)

zs, k) G(z) gz, k) Gz', 1,1), if z k / 2 (4-33)

G(1,s,2) Gm(1)5g(1, 2) { (2, ,1) + G(2, -1,1) + G(2, -1,}34)
1,4

subject to the conditions


G (M, s, k) 0; G (1, s, 4) 0, for all s, k. (4-35)


Again, the terms Gm(1) g(1,2) Y14 G (2, s-1,1) and G,(1) g(1,2) :14 G(2, s-1,1)

would occur in the recursive expressions for G (1, s, 2) and Gj(1, s, 2) respectively.

However they actually correspond to bridged chains with the last chain end at surface

'1'. Hence they are added to Gb(1, s, 2).

Now the segment densities of adsorbed chains and bridged chains are found by

writing similar composition rules. As an example, the volume fraction of the segments

which form tails from surface '1' is given by

zr 62b Ga(z,s,k)Zl Y kG(z,r -S+1,)
(z) 2 C m (4-36)
s=1 k=l G )
Since a chain has two ends, the prefactor 2 is added to the equation.

Once we have the statistical weights of free, adsorbed and bridged chains, we can

also estimate the average number and sizes of loops, tails, trains and bridges. Details

of these are provided in Schel iP' '-u and Fleer's paper [34]. To give an illustration, we

consider the average number and size of loops. In a loop, both the ends are adsorbed

on the same surface. Therefore, we consider the statistical weight of an adsorbed r-

mer having the -.g,,, ,.I s in 7.';., '2' and the -gn, ,.. s + 1 in 7.';., '1' adsorbedd),

with the r-mer adsorbed at least once before -yi,, ,., s. This is equal to the product

Gal (2, r s + 1, 4) >kt4 Gal (2, s, k). We normalize the statistical weight with the

statistical weight of an adsorbed r-mer to obtain the probability of finding an adsorbed

r-mer having the segment s in li--r '2' and the segment s + 1 in 1-, r '1' adsorbedd),









with the r-mer adsorbed at least once before segment s. This will be equal to the

fraction of adsorbed chains with loops ending in -'i,,n ,,i s. Summing over all the

possible values of s gives the average number of loops per adsorbed chain, ni. The

average number of loops per adsorbed chain adsorbedd on surface '1') n,l is then given

by

G Z3 a{(2,r -s + 1,4) EZkcG(2, s, k)}
Gn' -a(r) G.(2) (4-37)
M 6
G ) = G z, r, k). (4-38)
z=1 k 1
The average size of the loops in the chains adsorbed on surface '1' la, is then given by

(a
,1 (4-39)
'1,l

where (,1 is the fraction of loops, O0 is the amount of polymer adsorbed on the surface

'1', and are given by


', = a (4-40)
0" ,tr ,(z) + ,ta(z)}. (4-41)

Loops are formed on surface '1' by chains in groups '2' and '4', i.e., by chains

adsorbed on surface '1' with or without forming bridges. Therefore, an expression can

be written for the average number of loops on surface '1' per bridging chain, n, as

explained in Schel i' .1 and Fleer's paper [34]. The average size of such loops can be

calculated using an expression similar to Equation (4-39). The average number of loops

on surface '1' per unit area, ni, is then given by

n= Lnil + nb,l f} (4-42)

where f is the fraction of chains adsorbed on surface '1', fb is the fraction of bridging

chains with the last chain end leaving from surface '1', and are given by


0 (4-43)

OCb
fb (4-44)
fl >7 z









The average size of loops adsorbed on surface '1' 11,1 is then obtained as a weighted sum

of the two average sizes l", and l/,, and is given by

a a a+ lb b ,b
S+ nl fb (4-45)
nif + bI 11


4.2.5 Estimation of Interaction Forces

A detailed derivation of the thermodynamics OF interacting polymer li.- rs and

the method of calculation of polymer-induced forces was presented in C'! lpter 3. Also,

a brief summary of the method of estimating polymer-induced forces using Monte Carlo

simulations was presented in C'! lpter 2.

4.3 Results and discussion

We shall now examine what changes or improvements one observes concerning

the structure of the adsorbed l. r-i and the forces resulting from the interaction

between two adsorbing (polymer-bearing) surfaces when anisotropy is introduced in the

mean-field formalism. Our primary focus here will be on the results of SCAFT (with

non-reversal chain statistics analogous to SF2) relative to its isotropic version, SF2.

The structure and isotropic stresses in the system can also be obtained exactly using

simulations, as we have discussed elsewhere [57]. We shall compare the predictions of

the structure of adsorbed 1I-, -r and the interaction forces by SCAFT and SF2 with

some simulation results. The simulation results used in the following discussion are

based on a lattice Monte Carlo technique in which we model the polymer chains as

self-avoiding walks (SAW's) and sample the chain statistics using a modification of the

configurational bias algorithm of Siepmann and Frenkel [64] due to de Joannis [43].

We use periodic boundary conditions in the x and y directions and consider two

impenetrable, adsorbing surfaces confining the Il- -i~ in the z-direction. The surface-

segment interaction is considered only in the .,.i i:'ent Ii- Vr, as shown in Equation (4-5).

All results have been generated for chains of 200 segments, in a good solvent with a










simple-cubic lattice of size L/a = 40 in the x and y directions 4 where a is the lattice

spacing.

In this work, we focus on the introduction of anisotropy in the mean-field theory

and attempt to understand what improvement it offers. An attempt is made to relate

the force between the two adsorbing surfaces to the structure of the adsorbed 1lv r-.



0.5 C 1 -*-SCAFT
S* SF --S10--F
0.4 SCAFT --SF
V Simulations
0.2 0 SCAFT --
*(z) 0.3 4(z) 10o2

S= 0.75 .. 10-' =I.0 v
0.1 r = 200 r=200 v v
s = 1.0 s =1.0
0.0 I I I 10 T"" "
0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40
z/a z/a

Figure 4-2: Overall segment density distribution: Comparison of lattice mean-field
results (SF1, SF2, SCAFT) with lattice Monte Carlo simulations. (a) H/a = 5.0. (b)
H/a = 40.0.


4.3.1 Structure of the Adsorbed Layers

We first focus on the overall segment density distribution. Segment densities are

reported, following the normal convention, as the fraction of a lattice li- r occupied

by segments (i.e., area fraction). As an example, Figure 2 presents the overall segment

density for the case of surface coverage F = 0.75 and H/a = 5.0 (Figure 4-2a) and

for F = 1.0 and H/a = 40 (Figure 2b). The predictions of SF1 (i.e., based on the

first-order Markov statistics for the chains) are also shown in Figures 4-2a and 4-2b.

The corresponding segment densities of loops, tails, trains and bridges for F = 0.75 and

H/a = 5.0 are given in the appendix (Table B-l). Some more results on the overall

segment densities and segment densities of loops, tails, trains and bridges are available




4 The numerical mean-field theories considered here are one-dimensional, i.e., the x
and y directions extend to infinity, and variations are considered only in the z-direction.
Hence in the mean-field context, the lattice size of (L/a)= 40 specified in the x and y
directions simply implies that 1600 lattice sites are considered per lI--r (normal to the
x y plane.









in the appendix (see Tables B-2 and B-3) for different surface separations and surface

coverages.

Based on the above results, first one observes that, for H/a = 5.0, while the

predictions of the overall segment density of SF1 agree well with the simulations, those

of SF2 differ consistently. Since SF2 prevents segment backfolding, it permits a lower

number of allowable conformations near the surface than SF1, which results in reduced

segment densities near the surface (where the concentration is high) and higher value

of densities away from the surface. This results in SF2 predicting higher segment

densities for the bridges, loops and tails and lower densities for trains as compared to

SF1 and simulations. Introduction of anisotropy significantly improves the predictions.

SCAFT partially corrects SF2 in the appropriate manner. The overall segment density

profiles show a surprising agreement with the simulations for H/a = 5.0. However, for

H/a = 40.0, it is clear that the corrections due to anisotropy are insufficient to capture

the interactions between the segments accurately. Even SCAFT predicts an order of

magnitude higher segment densities near the center.

In general, the correction due to anisotropy in the mean field is two-fold:


Near the surface: In typical situations near an adsorbing surface one would
expect a significant density of trains under strongly adsorbing conditions (i.e.,
(z = 1)) in which all segments are oriented in the same direction (i.e., parallel
to the surface). Introduction of anisotropy will cause more bonds to orient along
the surface, thus giving rise to an increase in the number of segments forming
trains. This is confirmed by the increase of segment densities of trains (relative
to SF2) observed in Figures 4-2a and 4-2b. This improvement also gives rise to
an excellent agreement between SCAFT and simulations in the overall segment
density near the surface.


Far from the surface: Anisotropy also leads to an under-prediction of the
segment density of bridges compared to SF2 and simulations. Far from the
adsorbing surfaces, the bond orientations are almost random and hence the
introduction of anisotropy (ordering) only serves to increase the 'rigidity' of the
chain, i.e., decrease the number of allowable conformations for that part of the
chain, resulting in a lower density compared to isotropic mean-field predictions.
Further, restricted equilibrium requires that the total amount of polymer chains
between the surfaces remain constant. Therefore, an increase in the segment
density near the surface will have to be accompanied by a corresponding decrease
away from the surface.











030
8 V Simulations = 0.75 Simulations s
SCAFT r=200 *-SCAFT,
1.' SFF X=I. 2.i
_. 2 1 ----SF A
O 20 r 0.75 ,-
e r=200 ."
C 0

o ; '
0) 10


2 4 6 8 10 12 02 4 6 8 10 12
H/a H/a
(a) (b)

Figure 4-3: Number and size distribution of bridges, as a function of H/a, for a con-
stant adsorption energy, X = 1.0 kT and F = 0.75 (a) Average number of bridges, nbr,
(b) Average size of bridges, lbr.


SCAFT also predicts lower segment densities of tails and loops 5 relative to SF2

but higher values compared to the simulations. Again, one observes that the correction

due to anisotropy leads to an improvement but the results still differ from those of the

simulations possibly because of the mean-field approximation in the introduction of

anisotropy. The results obtained for the number and size of bridges, loops and tails

confirm the above observations. Sample results are shown in Figures 4-3a and 4-3b

for the number and size of bridges (for F = 0.75). Further results for the number and

size of bridges, loops and tails as functions of F and H/a are available in Appendix A

(figures A 1 through A 12). SCAFT predicts lower number of bridges than SF2 and

simulations, but bridges of larger average size than the simulations. This difference is

pronounced at larger separation distances. The net result is an underprediction of the

segment densities contributed by bridges.

4.3.2 Interaction Forces

One of the central issues that is not clearly understood is the relationship between

the structure of the adsorbed li. r and the resulting forces of interaction between two

adsorbed 1i- rv. Evidently, the work done in compressing two lV. -is (and hence the

interaction force) depends on the rearrangement of the bridges, loops, tails and trains


5 This could explain why SCAFT predicts lower repulsion than SF2 in Figures 4-5a.









upon compression. While bridges generally contribute to an attractive interaction be-

tween the surfaces, interactions among tails, bridges and loops result in steric repulsion.

It is of interest to examine the effects of bond correlations on the balance between the

two competing factors in determining the interaction between two adsorbed l1,- rs.

Further, as noted earlier, an evaluation of the performance of mean-field approximations

would be incomplete without an examination of the forces of compression due to the

interaction between two 1i-, r. Here, we shall consider the effect of the introduction of

anisotropy in the mean field on polymer-induced forces.

At low surface coverages, one essentially encounters a single-chain regime. There-

fore the system can be analyzed as a collection of single chains. When the surfaces are

close to each other, at strong adsorption conditions, the chains form bridges between

the surfaces. These bridges are loosely stretched, and as a first approximation, can be

considered to behave as Hookean springs L;]'' As the surface separation increases,

the bridges become strongly stretched, and then one can consider the system to be a

collection of non-linear elastic tethers [62], with each tether governed by the Pincus law

of elasticity [65]. In this single-chain regime, the interaction between the two surfaces

is predominantly attractive, with the exception of very low separations, where steric

interactions between the segments of the chain lower the attraction. As the surface

coverage increases, the chains are packed closer and closer that beyond the proximal

regions of the l.. r-i, the system is in a semi-dilute regime, and steric interactions

between different sections of a chain and between chains become important. Under such

conditions, the force of interaction between the surfaces crosses over from attraction to

repulsion. Such behavior has also been qualitatively observed in experiments (see for

instance, Fleer et al. [12]).



6 In this work, the distribution of bridges and snapshots of simulations are shown
for a single chain of 1000 segments, to illustrate the importance of bridge-bridge steric
interactions, especially at weak adsorption energies, and how such interactions p1 l a
vital role in systems with multiple chains.









Schenll i' i1 and Fleer [34] have examined this problem in the lattice mean-field

approximation, and have been able to reproduce the bridging-attraction-and-steric-

repulsion force profile qualitatively. They used the first-order Markov approximation to

model the chains in the lattice (SF1). What is surprising about their results is that the

mean-field theory appears to provide reasonable predictions in the single-chain regime,

which is counterintuitive. In C'!i pter 3 we have discussed the correct calculation

of forces of compression using lattice mean-field theory, and have reinterpreted the

Scheul, i. i--Fleer results. Here, we show that even with the introduction of anisotropy

in the mean field, the lattice mean-field theory fails to capture any significant bridging

attraction. We shall restrict our discussions to the effects of polymer 1?v- r anisotropy

on the predictions of force of compression of the polymer 1.-r -.

We first examine the force per unit area (the excess normal stress in the system)

as a function of surface separation H/a, for surface coverages F = 0.25 through

1.25, and for an adsorption energy X = 1.0 kT, in Figures 4-4a through 4-4f. At

low coverages, for the extent of bridging observed (see for instance Figures 4-3a and

4-3b and Figure A 1 in Appendix A), mean-field theories predict nearly negligible

attractive forces. Further, the mean-field results, even considering bond-orientation

effects, are significantly more repulsive than predicted by the simulations. We shall

revisit these observations towards the end of this section and draw further conclusions.

An interesting observation to be made from these figures is that at high coverages,

SCAFT and SF2 predict a repulsive force with a power law behavior (shown only for

F = 1.25), the exponent being independent of the coverage. The exponents are -3.67

and -3 for SCAFT and SF2, respectively.

To further illustrate and understand the limitations of mean-field theories, we

examine the force as a function of F for a fixed surface separation H/a = 4.5, shown

in Figure 4-5a. The results of SF1 (with the correct calculation of forces) are also

included for comparison. It is clear that at low coverages, there is seldom any attraction

predicted by any of the mean-field formulations. On the other hand, simulations show

that at low coverages, where the surface consists of isolated polymer chains, there is

a linear increase in the attractive force with a corresponding increase in the coverage.






















S2/





-*-SCAFT, r=0.25
--- SF2, r = 0.25
-C--Simulations, = 0.25
r =200
; X, = 1.0


2 3 4 5 6 7 8 9 1C


4 6


10 12


0










S SCAFT, F = 0.5
-4- SF2, F = 0.5
--U--Simulations, F = 0.5
Sr = 200
Xs =1.0


2 3 4 5 6 7 8 9 10

H/a



-- SCAFT, F = 1.0
-- SF r = 1.0
--*--Simulations, r =1.0
S r = 200
1 Xs = 1.0








e#.-...m --




2 4 6 8 10 12 14 16

H/a



SCAFT, F =1.25
0 0 SF2, =1.25
0* Simulations, F = 1.25

S0o r=200
0 = 1.0
o
0 0





"- '\o


. i .


2 4 6 8 10 12 14


Figure 4-4:
F= 1.25.


(a) F


0.25. (b) r


0.5. (c) F 0.75. (d)r 1.0. (e) F


0.25


-0.05


0.35


1.125. (f)










0.25 0.6 .

0.2 SCAFT 0.5 --,SCAFT
0.2 -SF
SF I *- *SF
S-- --Simulations 0.4 /a=
0. 15 04~H/a =5.0
H/a=4.5 r =200
r= 200 0.3 =0.74
0.1 X. I
: / 02
5 / 0.2
0.05 a




-0.05 -0 .1 .. .. .
0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2
r ric
F F/F
(a) (b)

Figure 4-5: Force per unit area f at a fixed separation of H/a = 4.5, (a) as a function
of surface coverage F, for adsorption energy X, = 1.0 kT, (b) as a function of rescaled
surface coverage F/Fo, for rescaled adsorption energy X, Xc = 0.74.


At high coverages, we observe that SF2 predicts higher repulsion than SCAFT. This

is counterintuitive, since the introduction of anisotropy serves to increase the 'rigidity'

of the chains. For this reason, one would expect a polymer 1-v.-r in an anisotropic

mean-field to be less compressible than one in isotropic mean-field. However, mean-

field theories underestimate the saturation coverage because the excluded-volume

interactions are implemented in a mean-field sense and that the chain statistics permit

overlap and crossover. It is found that the saturation coverages for simulations [66],

SF1, SF2 and SCAFT are, respectively, 1.2, 0.7, 0.72 and 0.92. This explains why SF2

predicts less compressible l--, rs than SCAFT, contrary to intuition.

Moreover, due to the differences in the chain statistics and the mean-field assump-

tions, one can expect that the critical adsorption energy 7 Xc will be different for the

different approximations, namely, SF1, SF2 and SCAFT, and SAW-simulations [66].




7 The critical adsorption energy Xc is the effective adsorption energy for which the
energetic attraction of the segments to the surface is compensated exactly by the en-
tropic repulsion arising from the conformational restrictions imposed by the presence of
surface. Therefore, at Xs = Xc, the overall segment density profile is flat. The critical
adsorption energy is estimated as the adsorption energy at which the surface excess 0"
(= (z( ) wb), where Qb is the bulk concentration) is zero.








81

0.05 9.5

SSCAFT o 9 0 SCAFT a
S 0.04 SF SF SF
S8.5 *
H/ a=5.0 H/a = 5.0 [
0 r=200 r =200
0.03 -X.= 0.74 m = 0.74
E 2 .N
E El 7.5
Z 0.02 -
0 75
0) 6.5
Q 0.01


0 5.5 .
0 0.5 1 1.5 2 0 0.5 1 1.5 2
F/F FlF
0 0
(a) (b)

Figure 4-6: Number and size distribution of bridges, as a function of rescaled surface
coverage P/F0, for a constant rescaled adsorption energy, X, Xc= 0.74, and H/a = 5.0.
(a) Average number of bridges, nbr, (b) Average size of bridges, lbr.


Our calculations show that the critical adsorption energies are, respectively, 0.18, 0.22,

0.21 and 0.26. Due to the differences in oF and Xc, it is instructive to examine the

results for the same relative surface saturation and relative adsorption energies. There-

fore, we plot the force per unit area as a function of rescaled coverage F/Fo for a fixed

H/a and a rescaled adsorption energy (Xs Xc) in Figure 4-5b. The results are similar

to those shown in Figure 4-5a, except that at high coverages, SCAFT is now seen to

predict higher repulsive forces, as would be expected. The average number and size of

bridges, loops and tails are plotted as a function of F/Fo for a given (X, Xc). Figures

4-6a and 4-6b show the number and size of bridges. The number and size of loops

and tails are available in the supplemental information (see Figures A4 11 through

A4 14).

An enlarged view of the results at low coverages, illustrated in Figures 4-4a, 4-

4b, 4-5a, 4-5b, shows that SCAFT and SF2 do predict some attraction, even though

quantitatively negligible compared to the corresponding simulation results. As stated

earlier, in a highly undersaturated regime, the chain conformations are dictated

essentially by single-chain statistics. The assumption of mean-field is expected to be

highly inaccurate here. For chains in a lattice, constructed with a first-order Markov

statistics, with random mixing approximation in the lateral dimensions, the following

expression has been used [67] to estimate the radius of gyration R,, expressed as a








82

10 -

-- SCAFT
8 -- SF I
-- Pincus Law
--S--Simulations /
6 P
H/a = 4.5
r=200

4 -0s = 1.0
-- 4


2


o. '0 -- ..0


-2 .- --,
-2 -- I --- I --- I --- I --- I -- I -
0 0.2 0.4 0.6 0.8 1 1.2 1.4

F

Figure 4-7: Force per bridge fb, as a function of F for a fixed wall separation
H/a = 4.5.


multiple of the length of a step in the lattice,


R (r/6) ( + D-) (1- D-), (4-46)


where r is the chain length, and D is the lattice coordination number. For a chain

length of 200 segments in a cubic lattice, the radius of gyration is about 5.7 lattice

units. Based on the estimates of rms thickness of the adsorbed l. -r, one can estimate

that single-chain statistics prevails up to a surface coverage of F r 0.5. Therefore,

the observed attraction then corresponds to that of an ideal non-reversal chain (SF2),

and a weighted ideal non-reversal random walk (SCAFT). We noted earlier that at

low coverages the system is expected to behave as a collection of single bridges with

negligible interactions among bridges. Therefore, the magnitude of the attractive

force is expected to increase linearly with surface coverage, as seen in the simulation

results. The number of bridges in this regime decreases exponentially and the length









of the bridges increases linearly 8 with H/a. In order to see if the system behaves as

a collection of elastic tethers (governed by Pincus law of elasticity [65]) in the strongly

stretched region, we plot the force per bridge as a function of F in Figure 4-7. At very

low coverages, one does observe a qualitatively similar behavior, though the magnitude

of the force is much less than the Pincus law estimate and the predictions from the

simulations.

The preceding observations indicate that the mean-field theories are severely

limited in their ability to provide a reasonable quantitative estimate of the attraction

between interacting polymer l1.- r- at low coverages, under restricted equilibrium. They

predict qualitatively expected behavior at higher coverages, where steric interactions

are significant. The attempt to improve mean-field predictions by the introduction of

anisotropy seems to work in the prediction of structural details of the adsorbed l.,-r,

but is clearly wanting in the prediction of forces. To obtain any significant improvement

in the predictions of the theories of polymer adsorption, one may have to go beyond

the constraints of the mean field. Further, inadequate chain statistics (which one

often resorts to in order to retain manageable propagation relations) introduce further

limitations in the theoretical predictions. Even though the elimination of backfolding

leads to a significant improvement, as shown in this work and by Jimenez et al. [57]

and Simon and Ploehn [68], this is merely the simplest of the refinements that may

be necessary. A careful examination is needed to study the limitations due to chain

statistics.

4.4 Concluding Remarks

In summary, we have presented an examination of the effects of the orientational

anisotropy of polymer chains in a physisorbed 1 v.-r on mean-field predictions of struc-

ture of the li--rv and the resulting forces of interactions between two lI-i r-. In addition

to chain statistics, anisotropic effects are expected to be among the most important

elements of mean-field theories having an impact on the predictions. Our objective here



s This type of behavior has been -ii--. -1. .1 by Ji et al. [62], who show that the elas-
ticity of such tethers is nonlinear and is governed by the Pincus law of elasticity [65].









has been to present at least a preliminary examination of what improvements can be

expected through the introduction of anisotropy.

The results show that introduction of anisotropy does improve the segment

density distribution close to the adsorbing surface. However, far away from the surface,

mean-field theories consistently predict higher densities than simulations. It is also

found that mean-field theories fail to provide useful results at lower coverages on the

forces of interaction. A linear force profile, qualitatively consistent with the scaling

predictions of Ji et al. [62], is observed at low surface coverages, even though there are

quantitative discrepancies. Both the mean-field formulations predict the crossover from

net attraction to repulsion to occur at similar coverages (F/Fo ~ 0.5).

In the above comparisons, we have used the structural properties and interaction

forces computed using Monte Carlo simulations as a reference. However, a caveat is

in order in this respect, namely that there is no clear way to compare rigorously the

mean-field theories with exact calculations (e.g., simulations). In order to be able to

make any conclusive comments on the reliability of mean-field theories even for the

simple problem of homopolymer adsorption, one has to consider many issues. First, an

agreement between a mean-field formulation (- i-, SFi) and simulations for only certain

properties (F~-, segment densities) does not necessarily imply that the formulation

will be equally good for all properties. Secondly, the fact that improving the chain

statistics leads to improved segment densities in certain regions (as we noted in the

previous section) does not guarantee that similar improvements can be expected in

the overall performance. That is, agreement in certain respects could occur because of

fortuitous cancellation of errors, while the errors could compound each other in other

respects. Thirdly, it is unclear currently as to what is the right way (or, the best way)

to compare mean-field calculations with simulations. Intuitively, it appears reasonable

to compare the results for the same rescaled coverage, F/F0, and rescaled adsorption

energy, (X, Xc). However, whether such a rescaling is sufficient (or, even valid)

remains unclear. This issue deserves further attention, since the interpretations could

be misleading if the right comparisons are not made. On the other hand, despite such

uncertainties, the use of simulations as benchmarks remains the only option to examine







85

the reliability of the mean-field theories, since a direct comparison with experiments

only introduces additional uncertainties.















CHAPTER 5
BENDING OF SEMIFLEXIBLE POLYMERS

5.1 Introduction

This chapter concerns the problem of bending of a single semiflexible polymer

modeled as an incompressible elastic rod, under two different conditions:


one end of the rod is clamped with respect to both position and orientation while
the other end is clamped with respect to position but has a flexible orientation
(the I ,riI,, -i,:,,,: problem), and


both ends of the rod are clamped with respect to position but have flexible
orientations (the double-hinge problem).

5.2 Wormlike Chains as Slender Elastic Rods

In C'!i pter 1 (see Section 1.1.3), we discussed the wormlike chain model for

semiflexible polymers. Here we state the connection between the wormlike-chain model,

a microscopic description of semiflexible polymers, and a macroscopic continuum

description based on the theory of elasticity, as elastic rods. For detailed derivations,

the interested reader is referred to C'!i pter 3 of Yamakawa [2]. The probability

distribution of finding the end segment (corresponding to s L) of a wormlike chain

at position R given that the first segment is at origin and has a specified orientation

uo for L > 0 is the Green's function. Once again, r(s) and u(s) may be regarded as

Markov random processes on the proper time scale of s (or L). This Green's function

satisfies a Schrodinger-like equation analogous to the behavior of a rigid electric dipole

in an electric field. The solutions for such equations are known. The 'Lagrangian' for

this problem is given by

L c +k u, (5-1)
kT

where

U [u(s)]2, (5-2)
2