Effect of disjoining pressure on the drainage and rela - PDF document

Effectofdisjoiningpressureonthedrainageandrelaxationdynamicsofliquid“lmswithmobileinterfacesSoniaS.Tabakova,KrassimirD.DanovLaboratoryofPhysicochemicalHydrodynamics,InstituteofMechanics,BulgarianAcademyofScience,1113So“a,BulgariaLaboratoryofChemicalPhysicsandEngineering,FacultyofChemistry,UniversityofSo“a,1164So“a,BulgariaDepartmentofMechanics,TU-So“a,BranchPlovdiv,4000Plovdiv,Bulgariaarticleinfo Correspondingauthor.Fax:+35929625643.E-mailaddress:(K.D.Danov). JournalofColloidandInterfaceScience336(2009)273…284 ContentslistsavailableatScienceDirectJournalofColloidandInterfaceScience low.Thekinematicboundaryconditionatthe“lmsurface,seeEq.below,introducesanadditionalfunction,thelocal“lm.Hence,thekinematicboundaryconditiondoesnotde-creasethenumberofunknownparameters.Todeterminethem,thedynamicboundaryconditionattheinterfacemustbetakenintoaccount.Inlubricationapproximationthenormalstressboundaryconditionrelates=1,2),,and,seeEq.low.However,thetangentialstressboundarycondition,Eq.isautomaticallyful“lledbytheleadingordersolutionoftheprob-lem.Forthatreason,thesecondordertermsshouldbeconsidered.Thisapproachisknownintheliteratureastheextendedlubrica-tionapproachŽapproachŽ….Forsomerecentpracticalapplicationsitisveryimportanttoproducethinliquid“lmsfromsurfactant-freephasesorfromphaseswithaverysmallamountofsurfaceactivecomponents.Similarproblemsarisewhenthe“lmsarestabilizedbynanome-ter-sizedparticles.Inthesecases:(i)thelateraldimensionsaremuchlargerthanthelocal“lmthickness;(ii)the“lmdrainageoc-cursinadynamicregimeatarbitraryvaluesoftheReynoldsnum-ber.Modelingofthedynamicsofsuchsystemsispossibleusingtheextendedlubricationapproachapproach…,formerlyappliedtostudylinearandnonlinearstabilityof“lms.Thisapproachhasalsoproveditswideapplicabilityformodelingthedrainageandstabil-ityofevaporating“lms“lms…andformodelingthesolidi“cationoffreethin“lmsattachedtoaframeandpulledfromameltmelt.OriginallythesemodelsaccountforthevanderWaalsinteractionsbetweeninterfaces.Recentexperimentalandtheoreticalinvestiga-tionsmanifesttheimportanceofanumberofothercomponentsofthedisjoiningpressure.Experimentalmeasurementsshowthat,eveninpureliquid,thegasbubbleshave-potentialof65mVmV,whichforthexylene,dodecane,hexadecane,andper”uorom-ethyldecalindropletschangesfrom100mVto20mV,depend-ingontheelectrolyteconcentrationandpHofthesolutionsolution.Thechargeaccumulationonthesurfacesisduetothespontaneousadsorptionofhydroxylions,whichcauseselectrostaticrepulsionbetweenthe“lminterfaces.Manyexperimentswithemulsionandfoam“lms“lms…suggestthatthelongrangehydrophobicattractionforcescanbeconsiderablylargerthanthevanderWaalsinteractions,andmustbeincludedasadisjoiningpressurecompo-nent.Atclosedistancesbetween“lminterfacesstrongstericinter-actionsbetweenadsorbedparticlescanstabilize“lmdrainageandprevent“lmrupturerupture….Theaimofthepresentstudyistotakeintoaccounttheroleofdifferenttypesofintermolecularinter-actionsforthe“lmdrainageandstabilityindynamicconditions,usingtheextendedlubricationapproach.Thepaperisorganizedasfollows.InSection,wedescribetheextendedlubricationapproachappliedfor“lmswithtangentiallymobilesurfacesintheframeworkofthedisjoiningpressureap-proach.Thedimensionlessnumbersappearinginthemodelequa-tionsforone-dimensionalsymmetric“lmsareintroducedin.Theytakeintoaccountthecontributionofinertia,inter-facialtension,vanderWaals,hydrophobic,electrostaticandstericforces.Therein,theboundaryconditionsandpossibleregimesofthe“lmdrainageandrelaxationarediscussed.TheeffectoftheintermolecularforcesandthemagnitudesoftheReynoldsandWe-bernumberson“lmdrainagearestudiedinSectionandtheirin”uenceon“lmrelaxation…inSection.ThegeneralconclusionsaregiveninSection2.Modelingofthedynamicsofthinliquidfree“lms2.1.ConservationofmassandmomentuminthebulkphaseWeconsiderasymmetricthinfoam“lmwithalocalthicknessFig.1).Thebulkphaseinthe“lmisdescribedasanincompress-iblehomogeneousliquidwithvelocityvector,pressuredynamicviscosityanddensity.TheCartesiancoordinatesystem,,isplacedintheplaneof“lmsymme-=0,where=1,2)arethelateralcoordinatesandtheverticalcoordinate(Fig.1).Thecharacteristiclateralvelocity,thelaterallengthscaleis,andthenaturalscaleoftime,.The“lmthickness,,ismeasuredwiththeverticallength,andisassumedtobesosmallthattheratio,canbeconsideredasasmallparameterintheproblem.Thedimen-sionlesstime,,thelateralcoordinatesandcomponentsofthevelocityvector,=1,2),arede“nedasfollows UTa;xa Xaa;andua Thedimensionless“lmthickness,,theverticalcoordinate,,andtheverticalcomponentofthevelocity,,areintroducedthroughthefollowingrelationships Hb;x3 X3b;andu3 accountsforthefactsthat:(i)inthecaseofthinliquid“lmstheverticalcomponentofthevelocityismuchsmallerthantherespectivelateralcomponents;(ii)thecontinuityequationmustbeinvariantlyexpressedintermsofdimensionlessquantities ouboxbþ InEq.andbelowwewillusetheEinsteinsummationconven-tion,wheretheGreeklettersubscripts()indicatesumsre-latedtothelateralcomponentsdenotedwithindexes1and2.ForNewtonian”uidsthemomentumbalanceisdescribedbytheNavier…Stokesequation.Forfoam“lmswithfullymobilesur-faces(seeSection)aviscousscale,,isappropriateformea-suringthepressure.Thedimensionlesspressure,,andtheReynoldsnumber,Re,areintroducedbythede“nitions andRe Usingtheaboveexpressions,Eqs.(2.1),(2.2)and(2.4),weobtainthedimensionlessformofthemomentumbalanceequationinthelateraldirections(=1,2) ouaotþ ooxbðubuaÞþ oox3ðu3uaÞ¼ opoxaþ o2uaoxboxbþ 1e2 o2uaox23:ð2:5Þ Fig.1.Sketchofasymmetricthinliquid“lmwithalocal“lmthickness.TheverticalcoordinateaxisoftheCartesiancoordinatesystem,,isperpendic-ulartothe“lmmiddleplane,=0.Thecharacteristiclaterallength,,ismuchlargerthanthecharacteristicverticallength,.Thepositionoftheupper“lmsurfaceisde“nedas/2andthepositionofthelowerone…as/2.Theuppersurfacenormalandtangentvectorsare,and,respectively.S.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 TherespectivedimensionlessformoftheverticalprojectionoftheNavier…Stokesequationreads ou3otþ ooxbðubu3Þþ oox3ðu3u3Þ¼ 1e2 opox3þ o2u3oxboxbþ 1e2 TheReynoldsnumber,Re,appearinginEqs.(2.5)and(2.6)andde-“nedbyEq.,isassumednottobeverylarge,i.e.ReForexample,forwater“lms(=10=10Pas)withaninitialthickness=1mmandaframelaterallength=1cm=0.1),thelateralvelocity,,atwhichRe==1isequalto1mm/s,thusinthatcasethemodelwouldbeapplicableforsuctionvelocitiessmallerthan1mm/s.Theeffectoftheinertiaonthe“lmdrainageisdiscussedinSectionFig.3Becauseofthesymmetryoftheconsideredproblematthemiddleplane=0.TakingintoaccountEq.oneobtainstheleadingordersolutionofEqs.(2.5)and(2.6)[22] wherethefunctionsdependontime,,andlateralcoordi-.AftersubstitutingEq.intothecontinuityequa-tion,Eq.,integratingtheobtainedresultwithrespecttousingthesymmetrycondition=0atthemiddleplane,weobtaintheleadingorderexpressionfortheverticalvelocitycomponent: Theunknownfunctions,andaredeterminedfromthekine-maticanddynamicboundaryconditionsatthe“lminterfaceusinganappropriateasymptoticprocedure(seeSectionAppendixA2.2.KinematicanddynamicboundaryconditionsattheinterfaceUsingtheabovede“nitions,Eqs.(2.1)and(2.2),onepresentsthekinematicboundaryconditionatthe“lmsurfaceinasimpledimensionlessform ohotþub ohoxb¼2u3atx3¼ Throughsubstitutingtheleadingordersolutions,Eqs.(2.7)and,intoEq.weobtaintherespectiveexpressiondescribingtheconservationofmassinthecontinuous“lmphase ohotþ Insurfactant-free“lmstheinterfacialtension,,isconstantandthesurfaceshearanddilatationalviscositiesarenegligible.There-fore,inthiscase,thebulkviscousfrictionforceatthe“lmsurfaceiszerozero.Theleadingorderofthetangentialstressboundarycon-ditionat/2iscalculatedinAppendixA,seeEq..AftercombiningtheobtainedresultwiththeintegratedformofEq.AppendixA,wederivetheasymptoticformofthemomen-tumbalanceequationinthe“lm: ootðhwaÞþ ooxbðhwbwaÞ¼h ooxaqþ owboxbþ ooxbh owboxaþ owaoxbþ2 istheKroneckerdelta.ItisimportanttoclearlyindicateforwhichsystemsthemomentumbalanceEq.,originallyderivedforpureliquids,isapplicable.(i)Ifthesolutioncontainsonlyindifferentelectrolytes(salts),thenthechangeofthesurfacetensionwiththeionicstrengthissmall,theGibbselasticityissmallandtheMarangonistresscanbeneglectedneglected.TheelectrostaticandvanderWaalscomponentsofthedisjoiningpressureappearatsmall“lmthicknessandcontrolthe“lmdrainageandstability.(ii)Inthecaseofsmallamountoflowmolecularweightsurfac-tants,thesurfactantrelaxationtimeiscomparabletothecharacteristictimeof“lmdrainageandtheMarangonieffectbecomesevident.Thiseffectsuppressesthesurfacemobilityanddecreasestherateof“lmthinningthinning.TheaccountoftheMarangonistressmakestherespectiveana-logueofEq.muchmorecomplex.(iii)Inthecaseoflargeamountoflowmolecularweightsurfac-tants(closeorabovethecriticalmicelleconcentration)thesurfactantrelaxationtimedecreasesconsiderablyanditisoftheorderoforsmallerthanseveralmicrosecondsmicroseconds.Inthesesystemsthesurfacetensiondoesnotchangeduetothe“lmdrainageandtheMarangonistresscanbeneglected.Fromtheliteratureitisknownthatlowmolecularweightsurfactantshaveverysmallvaluesofsurfaceviscos-viscos-andtherefore,theroleofinterfacialrheologyisalsonegligible.ForsuchsystemsEq.isvalid.Inaddi-tiontotheclassicalcomponentsofthedisjoiningpressure,thesteric,oscillatory,etc.non-DLVOsurfaceforcesshouldbeconsidered.(iv)Filmsstabilizedbyproteinsorpolymershavesigni“cantval-uesofthesurfacedilatationalandshearviscosities,whichconsiderablyincreasetheeffectofthesurfacerheologyonthedrainageandrelaxationdynamics.Forsuchsystemsourmodelisnotapplicable.Forsmall“lmthicknesses(below200nm)thesurfaceforcesofintermolecularoriginmustbeaccountedfor.OnetypeofsuchforcesisthevanderWaalsattractionforce,whichischaracter-izedbytheHamakerconstant,constant,.ManyauthorsincludethevanderWaalsinteractionsasabulkpotentialforceintheNavier…Stokesequation.Thisapproachisknownintheliteratureasthebodyforceapproachapproach….Thebodyforceapproachap-pliedforionicsolutionsleadstoverycomplexmathematicalcal….Manyothertypesofsurfaceforcesaretheoreticallyandexperimentallyinvestigatedintheliterature(thestericrepulsion,thehydrophobicattraction,theoscillatorystructuralforces,etc.).Inmanycasestheoriginoftheobtainedinteractionsisnotclearlyunderstoodunderstood.Forthatreason,amoreconvenientwaytoaccountfortheroleofthesurfaceforcesistousethesocalleddisjoiningpressureapproachapproach.InthisapproachthebulkforcedensityintheNavier…Stokesequationisomittedandallintermolecularinteractionsinthe“lmaretakenintoconsiderationbyintroducinganaddi-tionaldisjoiningpressureterm,,inthenormalstressboundarycondition.Thedisjoiningpressuredependsonlyonthelocal“lm,andthephysicochemicalpropertiesofthesolutions.Thespeci“cexpressionsforthedifferentcomponentsofgiveninSection.Thenormalcomponentofthebulkforceact-ingonthe“lmsurfaceisthencompensatedbythe“lmcapillaryanddisjoiningpressuresandthepressureinthegasphase,.InAppendixAwederivetheasymptoticformofthenormalstressboundarycondition.TheobtainedresultreadsS.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 qþ owboxbþ e2Ca wherethecapillarynumber,Ca,thedimensionlessdisjoiningpres-,andthepressureinthegasphase,,arede“nedas gUr;pdp 2Pa2rb;andpg SubstitutingEq.intoEq.wearrivetothe“nalequationofthemodel: ootðhwaÞþ ooxbðhwbwaÞ¼ eh2Ca ooxa o2hoxboxbþpdp !þ ooxbh owboxaþ owaoxbþ2 Notethatthetypicalvaluesofthecapillarynumberaresmall.Thustheparameter/CainEq.isoftheorderoforlargerthanErneuxandDavisDavisshowedthatthemodelisanalogoustoarespectiveapproachingasdynamics.Eqs.(2.10)and(2.14)spondtothetwo-dimensionalviscous-”owequationsifweiden-tifythethicknessŽasthedensityŽofthecompressiblegas,thethicknessŽastheshearviscosityŽ.The“rsttermintheright-handsideofEq.canbeconsideredasthepressure)Ž,whichadmitsthefollowingcomplexequationofÞ¼ eq2Ca o2qoxaoxbþ e4Ca2Z1qn dpdpðnÞdndnþ oqoxc generalizedforanarbitraryfunction).Forsmallvaluesofthe“lmthickness,,thedensityŽinEq.isverysmallandthesoundspeedŽbecomesequalto eq2Ca showsthatifd�0thesoundspeedŽisimaginary,whichcorrespondstoinstability.Incontrast,ifd0thesys-temhasarealsoundspeedŽ,correspondingtowavepropagation.Thisfactiswellknownintheliterature,i.e.d&#x-241;0destabilizesandd0stabilizesthedrainageofthinliquid“lms“lms.3.One-dimensionalthinliquid“lmsattachedtoframesWeconsiderthesimplecaseofone-dimensionalthinliquid“lmattachedtoaframe(Fig.2).Theparametersoftheframeare:width;left-andright-hand-sideheights,respectively.The”owratesfrombothframechannelsarecharacterizedbytheFig.2),the“lmsurfacesaresymmetricwithrespecttothemiddleplaneandhave“xedcontactlinesattheframeborders.Atinitialtime,=0,the“lmthicknessisgivenbythefunction),whichdeterminestheinitialliquidvolumecaughtintheframe.Themodelequations,Eqs.(2.10)and(2.14)describingthedynamicsofsuch“lms,aresimpli“edtothe ohotþ ooxðhwÞ¼0;ð3:1Þ owotþw owox¼ 1We o3hox3þ 4Re o2wox2þ 4Re owox olnhoxþ oox isthedimensionlessvolumeoftheliquidcaughtinthe“lm,perunitchannellength;,andarethedimensionlesslat-eralcoordinate,thicknessandvelocity,respectively;andtheclassi-calWebernumberisrescaledwiththesmallparameterasfollows 2eCaRe¼2 qa2U2rb¼ Atinitialtime,=0,theliquidisatrest,the“lmpro“leisaknown),andtheliquidvolumeis.Therefore,theinitialconditionsfornumericalcalculationofEqs.(3.1)and(3.2)The“lmsurfaceshave“xedcontactlineswiththeframe,whichas-signsthefollowingboundaryconditionsforthe“lmthickness:The”owratesfrombothsidesoftheframeareknown(Fig.2);theyde“netheboundaryconditionsforvelocityisthecharacteristiclateralvelocity(seeSection).Theminussigninthe“rstequationofEq.accountsforthedirec-tionofthe”ow(Fig.2).InSectionwewillstudytheprocessof“lmdrainagetothepointofminimumpossible“lmthickness,Inthiscasethe”owratesarenon-zeroduringthewholeprocess.Inthe“lmthinsundertheactionofthesuctionvelocities,,foratime,,necessarytoreachagivenvolume,.Atsuctionisstoppedandthe“lmisallowedtorelaxtoits“nalshape.Inthiscasetheright-handsidesofbothequationsinEq.aresettozerofor,whichdoesnotchangethenumericalschemede-scribedinAppendixBThedisjoiningpressureappearinginEq.canbepresentedasasumofthevanderWaalsandhydrophobicattraction,,elec-trostaticrepulsion,,stericrepulsion,,andotherinteractions.Hereweconsideronlythe“rstthreecomponents:.Therespectivedimensionlesscomponentsofthedisjoiningpressurearedenotedas,and.ThevanderWaalsandhydrophobiccomponentsofthedisjoiningpressurearecalculatedbytheexpression),whereaneffectiveconstantconstant.ItisimportanttonotethatrecentexperimentssuggestthehydrophobicinteractionsareinmanycasesstrongerthanthevanderWaalsinteractioninteraction….TheauthorsshowthatthehydrophobicinteractionsaredescribedbythesameexpressionasthatforthevanderWaalscomponentofthedisjoiningpressure,butwithmuchlargerinteractionconstant,,thantheHamakerconstant,.Forthatreasontheeffectiveisusedbelow.FromEqs.(2.13)and(3.3) Fig.2.Sketchofone-dimensionalsymmetricthinliquid“lmattachedtoaframewithaleft-hand-sideheight,right-hand-sideheight,andwidth2.Theinitial“lmthicknessis),the”owratesattheleftandrightbordersoftheframearecharacterizedbyvelocities,respectively.S.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 theattractivecomponentofthedisjoiningpressureispresentedinthefollowingform: p1We¼ B1h3andB1 isthedimensionlessinteractionconstant.Inthecaseof1:1electrolytestheelectrostaticcomponentofthedisjoiningpressure,,iscalculatedfromtheexpression1),whereisthethermalenergy,istheionicstrengthofthesolution,andisthedimensionlesselectricpo-tentialatthemiddleplaneplane.FromEqs.(2.13)and(3.3)weob-taintherelationshipbetweentheelectrostaticcomponentofthedimensionlessdisjoiningpressureandthedimensionlessionic p2We¼B2ðcoshUm1ÞandB2 ItiswellknownintheliteratureliteraturethatthesolutionofthePois-son…Boltzmannequationfortheelectricpotentialinthe“lmleadstothetranscendentalequation, 1jbZUsUm whichrelatestheelectricpotentialatthemiddleplane,,withthesurfaceelectricpotential,,thelocal“lmthickness,,andtheDebyescreeninglength,.Foragivensystemthesurfaceelectricpotentialchangesinsigni“cantlyduringthe“lmdrainage.Onlysmallvariationsofarepossibleforverythin“lmswhenthelocal“lmthicknessissmallerthan10nmnm.Belowwewillas-sumethat:(i)theconstantvaluesofareknown;(ii)thenumericalsolutionofEq.providesthedependence);(iii)Eq.).VarioussimplerapproximationsofEqs.(3.8)and(3.9)areknownintheliteratureliterature.Theseapproximationsarevalidfor1or1andcannotbeappliedforthewhole“lmpro“lewheresigni“cantly.Thepowerlawexpressionsforthestericcomponentofthedis-joiningpressure,widelyusedintheliteratureliteratureforproteinandpolymersolutions,arenotconsistentwithourmodel,asseeninthediscussionafterEq..Inthecaseoflowmolecularweightsurfactants,theadsorbedmolecules,forwhichourmodelcanbeapplied,oftenpossessbulkyhydrophilicheads,e.g.,whenthepolarpartoftheamphiphileconsistsofoxyethylenechainschains….Asthe“lmthins,thetwosurfacespackedwithsurfactantmoleculescloselyapproachoneanotherandthehydrophilicheadsmaybegintooverlap.Letbethecharacteristicthresholddistanceatwhichthestericinteractionbecomesactive.Therepulsiveforcewillbedescribedbymeansofamodelexpressionfortherespec-tivedisjoiningpressurecomponent,),whereistheinteractionconstant.Thevalidityofthisasymptoticequationisdiscussedbymanyauthorsauthors.FromEqs.(2.13)and(3.3)oneobtainstheexpressionsforandfortheinteractionconstant, p3We¼B3exp hbdandB3 Thenonlinearsystemofequations,Eqs.(3.1)and(3.2),withini-tialandboundaryconditions,Eqs.,canbesolvednumericallybyappropriateconservative“nitedifferenceschemeonstaggeredgridsanditerativealgorithmsalgorithms.Here,wedevelopafasternumericalmethodbrie”ydescribedinAppendixB.Wechosearegularspaceandtimegridwithalengthstep,,andtimestep,.Theprecisionofthenumericalscheme()is),whichguaranteeshighaccuracyandef“ciencyofthealgorithm.Allcalculationsbelowareperformedwith=10.Thesesmallvaluesofmakeitpossibletocalcu-latetheformulatedproblemuptotheminimum“lmthickness,,oftheorderoforlargerthan10.Moreprecisely,thevaluedependsontheprecisionofthedifferenceformulausedforthe“rstderivativeofdisjoiningpressure,Eq..Inthecaseofacentraldifferenceschemewithprecision)therelativeerrorofthederivativecalculatedat=10is3.4%.Thiserrorde-creaseswiththeincreaseof:for=20therelativeerroris0.84%;for=30itis0.37%,etc.4.Drainageofone-dimensionalthinliquid“lms4.1.NodisjoiningpressureWhenthedisjoiningpressurehasnoeffectonthe“lmdynam-ics,themodel,givenbyEqs.(3.1)and(3.2),containstwodimen-sionlessnumbers,ReandWe,de“nedbyEqs.(2.4)and(3.3)Thesenumberschangeinadifferentwayforgivenphysicochemi-calparametersoftheliquidphasedependingonthevelocity,andontheratio.Forexample,ifforagivencon“gurationRe=1andWe=1(solidlinesinFig.3a),thenthe“vetimesin-creaseofthevelocity,,leadstoRe=5andWe=25(dashedlinesFig.3a),whilethe“vetimesdecreaseofreducesthesenum-berstoRe=0.2andWe=0.04(solidlinesinFig.3b).Fora“xedva-lueofRetheWebernumberincreaseswiththeincreaseof Fig.3.Timeevolutionofthe“lmpro“lesinthecaseofsymmetricboundary,andaninitialcondition)=1:(a)Re=1andWe=1(solidlines),Re=1andWe=10(dash-dottedlines),andRe=5andWe=25(dashedlines);(b)Re=0.2andWe=0.04(solidlines)andRe=1andWe=0.1(dashedlines).S.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 ,andWedecreaseswiththeincreaseofdensity,frameheight,,andsurfacetension,,seeEq..Notethatthepossiblechangesofthesurfacetensionoffoam“lmsarefrom72mN/m(surfactant-freesystems)to25mN/m(closeorabovethecriticalmicelleconcentration),whiletheotherparameters()canvaryinordersofmagnitude.Theexamplesfortheevo-lutionofthe“lmpro“le,),inthecaseofRe=1,We=10andRe=1,We=0.1areillustratedinFig.3a(dash-dottedlines)andFig.3b(dashedlines),respectively.Forsymmetricboundaryconditions,,andaconstantvalueoftheinitial“lmthickness,)=1,theinitial“lmvolumeis=2andthedimensionlesstimeneededforthecom-pletesuctionofthisvolumeisequalto1.ForRe=1andWe=1the“lmdrainsupto=0.7withoutdimpleformationandwithavirtually“xedlengthofitsplane-parallelregion(Fig.3a…solidlines).Subsequentlythe“lmringspreadstotheborders,apro-nounceddimpleisformed,andat=0.9the“lmrupturesbecausethemenisciatthe“lmperipherytouchŽtheframeborders.There-fore,theruptureisnotaresultofreachingtheminimalpossible.Withthe“vetimesincreaseofthe”owrateawellpronounceddimpleisformedatanearlystage,thedimplegrows,the“lmringspreads,andat=0.6the“lmrupturesbeforereach-Fig.3a…dashedlines).Atthe“nalmomentthedimpleheightat=0hasquitealargevalueof0.6,thusthedimplecovers40%oftheinitial“lmvolume.IncreasingtheWebernumber(Re=1andWe=10)resultsinthe“lmsurfacesbecomingmoredeformable(Fig.3a…dash-dottedlines).Inthiscasetheevolution)issimilartothecasewiththetentimessmallerWebernumber:thethicknessesofthetwo“lmsat=0areclosetoeachother;the“lmringforWe=10isinitiallylargerthanthatforWe=1,butatthe“nalstageboth“lmpro“lesarevirtuallyidenti-cal.Therefore,theWebernumbercanbeconsideredasacharacter-isticofthedeformabilityof“lmsurfaces.Fromapracticalviewpointitisveryimportanttoproduce“lmswiththicknessasuniformaspossibleandwithalargeplane-par-allelarea.Onewayistoreducethe”owrate:asitisshowninFig.3b(solidlines),the“vetimesdecreaseofthe”owrate(Re=0.2,We=0.04)leadstoaregular“lmthinningwithacontin-uousincreaseoftheplane-parallelregionupto=0.9whenthe“lmthicknessat=0is4.25andthe“lmringisplacedapproximatelyat0.8.Anotherwayistodecreasethedeforma-bilityof“lmsurfaces:forRe=1thetentimesdecreaseoftheWe-bernumber,We=0.1,stabilizestheprocessof“lmdrainageFig.3b…dashedlines).Such“lmshaveregularpro“les,),dur-ingthewholeprocess.Atmoment=0.9theirthicknessat=0is,i.e.smallerthanthatforthecaseofRe=0.2andWe=0.04,andthe“lmringisplacedapproximatelyatthesameInthecaseofunsymmetricalboundaryconditionstheproduc-tionof“lmswithlargeplane-parallelareasisamorecomplicatedtask.Onecanexpectthattheincreaseofthesurfacetensionwillhelptosolvethisproblem.OurcalculationsforRe=1,We=0.1,)=1,andwithoutaliquid”owfromtheleft-hand-sideboundary(Fig.4a)showthatinthiscaseadimpledoesnotform,i.e.alargeamountofliquidisleftcapturedclosetotheleft-hand-sideboundary.Becauseofthetwicesmaller”owrate(onlyfromoneside)thedimensionlesstimenecessaryforthecompletesuctionoftheinitialvolumeisequalto2.The“lmreachesitsmin-imalthicknessslightlyafter=1.4withacomplexshapewithoutplane-parallelregions.Theminimal“lmthicknessislocatedat=0.75.Theprocessof“lmdrainageisregular,withoutwavesandinstabilities.WehaveperformedcalculationsforthesamecasedecreasingtheWebernumbertoWe=0.01.The“lmpro“leatthemomentofreachingofisplottedinFig.4a(dashedline),whereitcanbeseenthatislocatedat=0.38.Thefollowingconclusionscanbedrawn:these“lmsreachtheminimalthicknessatapproximatelythesametime;thedecreaseofdeformabilityof“lmsurfacesleadstomoresymmetric“lmthicknessesbuttheplane-parallelareasofthe“lmsareverysmallforunsymmetricalboundaryconditions.Toextendtheplane-parallel“lmregionwehavedecreased“vetimestheheightoftheleft-hand-sideborderassumingthat=0.2andhaveusedthesameconditionsasinFig.4awithasmal-lervalueoftheWebernumber,We=0.01(Fig.4b).Startingwithaninitialcondition,)=0.6+0.4,correspondingtoaninitialdimensionlessvolume,=1.2,wehaveobtainedthattheevolu-tionofsuch“lmsisregularuntiltheminimal“lmthickness,isreachedat=0.9and=0.23.Therefore,theconsiderablede-creaseofthedeformabilityof“lmsurfacesandtheframeheight,,isnotsuf“cienttoproducelargeplane-parallel“lmareas.ThecalculationsillustratedinFigs.3and4showthatthemostef“cientwaytoaccomplishthistechnologicaltaskistousesymmetric”owratesandsymmetricgeometricalparametersoftheframesasmuchaspossible.4.2.EffectofvanderWaalsandhydrophobicattractionThevanderWaalsandhydrophobicattractions,accountedforbythedisjoiningpressurecomponent,arealwaysactiveandhavethelongestrangecomparedtotheothercomponentsofTheroleofthiskindofattractionforcesontheevolutionofthe“lmpro“lesinthecaseofsymmetricboundaryconditionsforRe=1, Fig.4.Plotsof)for“lmdrainageintheabsenceofa”owfromtheleft-hand-sideborder:(a)andinitialcondition)=1atRe=1,We=0.1,till=1.4(solidlines)andatRe=1,We=0.01,=1.4(dashedline);(b)Re=1,We=0.01,=0.2,andinitialcondition)=0.6+0.4S.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 We=1,and=0.1isillustratedinFig.5a(solidlines).Thecom-parisonbetweenFigs.3aand5a(solidlines)showsthatuptotime=0.7thedisjoiningpressuredoesnotconsiderablyaffectthepro-cessofthe“lmdrainage.Whenthelocal“lmthicknessdecreasestoacertaincriticalvalue,theattractionforcesbecomesigni“cantandtheareaofthe“lmringthinsfaster.Attime=0.818the“lmatposition=0.984andbreaks.The“naldimplecov-ers1.82%oftheinitialliquidvolume.Oneexpectsthatthedecreaseofthe”owratealwaysmakesthe“lmdrainagemoreregular(seeFig.3).Whenkeepingalltheotherparametersconstantanddecreasing“vetimesthe”owrate,thevalueofincreases25times,seeEq..TheobtainednumericalresultsforRe=0.2,We=0.04,and=2.5areplottedinFig.5a(dashedlines).ThecomparisonoftheseresultswiththoseplottedinFig.3b(solidlines)inthecasewithoutdisjoiningpressuretakenintoaccountshowsthatboth“lmsfollowthesameregularbehavioruptotime=0.4.Atsubsequentmomentstheattractionmolecularforcessig-ni“cantlyacceleratethe“lmthinninginthecentralzone,(comparetherespectivepro“lesfor=0.5,0.55,0.58,and0.591).=0.591atypicalinstabilityappearsandthe“lmbreaks.ThiseffectisknownintheliteratureaspimpleŽformationasopposedtothecaseofthe“lmbreakageattheringring.4.3.EffectofelectrostaticandstericrepulsionOnepossibilitytoreducethedestabilizingroleoftheattractivecomponentsofthedisjoiningpressureistouseionicsurfactantsandthusincreasetheelectrostaticcomponentof.Itisprovenintheliteraturethatevenwithoutaddedsurfactantsthesurfacepotentialisdifferentfromzero:itsoriginforsuchsystemsisthesurfacechargeduetothespontaneousadsorptionofhydroxylionsions.Inthecaseoflargevaluesoftheasymptoticexpressionfortheelectrostaticcomponentofthedisjoiningpressure,=644.Therefore,amorecon-venientdimensionlessparameter,characterizingthemagnitudeofthisforceis/4),seeEq..InFig.3(dashedlines)wehaveillustratedthatatRe=1thedecreaseoftheWebernumbertoWe=0.1leadstoaregulardrainageofsuch“lms.Ifwetakeintoaccounttheactionoftheattractivecompo-nentofthedisjoiningpressurebyassumingthat=1andcalcu-latethe“lmevolution,weobtainthesolidlinesinFig.5b.Itisclearlyseenthatafter=0.4thepro“les)accelerateintime,=0.537apimpleinstabilityappearsatthecentralzoneandsubsequentlythe“lmbreaks.Notethatthe“lminFig.3bobtainedatthesameprocessparametersbutwithouttheactionofhasawideplane-parallelregionat=0.9.InordertohaveanideaaboutthemagnitudeofonecalculatesitsvalueforwatersolutionwithoutsurfactantsatpH=9androomtemperature:=45We).TakingthevaluesWe=0.1,m,and=5mmweob-tainthat=1.1andtherespectivevalueofis10.3.Experimentsfor“lmdrainageandequilibriumthicknessofsimilarsystemsarereportedinRef.Ref..Theincreaseofthesurfacepoten-tialandionicstrengthincreases.The“lm,whichisunstablewithoutaccountingfortheelectrostaticrepulsion,be-comesmorestablefor=10=10,seeFig.5b(dashedlines).ThecomparisonbetweenFigs.3band5b(dashedlines)showsthatinthepresenceofdisjoiningpressuretheelectrostaticrepulsioncounterbalancestheattractionforcesinawideplane-parallelcentralzone.Nevertheless,atthelateststageof“lmdrain-ageawellpronounced“lmringat=0.864isformedandsubse-quentlythe“lmbreaksdown.Thedecreaseofthesurfacepotential,,suppressestheelec-trostaticinteractions.Forexample,ifwetaketheabovesystemandchangepHto5whilekeepingalltheotherparameterscon-stants,thevalueofdecreasestentimes.Thesmallerrepulsiveforceisnotsuf“cienttopreventthepimpleformation,seeFig.6(solidline).ThecomparisonbetweenFigs.5band6(solidlines)illustratesthattheelectrostaticcomponentofdeceleratestheappearanceofsuchkindofinstabilities.The“lm(Fig.6,solidline)thinsregularlyforalongertime=0.65comparedtothatwithout,whichthinsinasimilarmannerupto=0.52(Fig.5b,solidline).Nevertheless,at=0.682(Fig.6,solidlines)theattractivecomponentsofthedisjoiningpressureprevailandthe“lmrup-turesatitscenterline.Inaddition,theshorterrangerepulsiveforcecanbeincluded,namelytheeffectofstericinteractionscanbeinvestigated.Asarulerule…forsuchforcesismuchlarger,exceptforthecasesoflargeionicstrengths,andtheinter-actionconstant,,isverylarge,seeEq..InFig.6lines)wecalculatethe“lmevolutionadditionallyaccountingforthestericinteractionsat=20and.Inthiscasethedrainageprocessstabilizesandregular“lmthinningisob-servedupto=0.9.Themechanismofthe“lmrupturechangestoaringformationatpositionandtime,=0.916and=0.921,respectively,andalargeplane-parallelcentralzonewiththickness0.064isformed.Figs.5band6showthatthedynamiceffectscan-notbesuppressedbytheactionoftherepulsivedisjoiningpressure.5.Relaxationofone-dimensionalthinliquid“lmsIfonewantstoproducea“lmwithagivenvolume,,the”owratesarestoppedaftertime,.Dependingonthemagnitudesof Fig.5.Roleofdisjoiningpressureonthedrainageof“lmswithsymmetricboundaryconditionsandinitialcondition)=1:(a)Re=1,We=1,=0.1(solidlines)andRe=0.2,We=0.04,=2.5(dashedlines);(b)Re=1,We=0.1,(solidlines)andadditionally=10,and=10(dashedlines).S.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 interactionforcesandregimesof“lmdrainagethe“lmsurfaceswillrelaxtothesteadystate“lmpro“le,),oreventuallythe“lmmayrupturebeforereaching)(seebelow).Inthecaseofthick“lmswhentheroleofthedisjoiningpressurecanbene-glected,thestaticvariantofEq.,simpli“estod=0.Thesolutionfor“lmswithsymmetricboundarycondi-(±1)=1,anda“xedvolume,,is 3vf24þ Therefore,theminimumpossibledimensionlessvolume(minimumstaticequilibriumvolume)is2/3andif2/3theLaplaceequationofcapillarityhasnosolution,thusnostaticequilibriumexists.Therefore,ifonestopsthe”owratesat&#x-248;2/3inthecaseshownFig.3therespective“lmswillbreak.Whenthedisjoiningpres-sureistakenintoaccount,thestaticvariantofEq.hasnoana-lyticalsolutionand)isobtainedasalimitof)aftercompleterelaxation.The“rststepinourstudyistoobtainthecharacteristicrelaxa-tiontimeandthetypeof“lmpro“lerelaxationinthesimplestcase,withoutthedisjoiningpressureeffectbeingtakenintoac-count.WeconsidertwocasesforRe=1,We=1,andvolume=1,largerthantheminimumpossibleone,2/3:initial“lmpro-“leis)=1(”atsurfaces)and=0.5,andparabolicinitialpro-)=(5+3)/8with=1.5and=0.25(Fig.7).Thedrainageofthese“lmsisillustratedinFig.3a(solidlines)andFig.7a(dashedlines);therespectiverelaxationcurvesshowasim-ilarbehavior.ThesolidlinesinFig.7ashowtheevolutionof“lm),duringtheprocessofrelaxationfor.Thestableregularapproachtotheequilibriumstate)=(1+3)/4iswellillustrated.Thechangesofthethicknessdifference,withtimeforbothsetsofinitialconditionsareshowninFig.7Oneseesthatinthesecasestheinitialconditionsdonotaffectthetypeofrelaxationforallvaluesofthelateralcoordinate.IntheinsetofFig.7bwepresentthechangesofthicknessdifferenceinthecentralzone,(0),inalogarithmicscaleasafunc-tionof.Theexcellentparallelstraightlineswithaslopeof1.5provesthattherelaxationobeysanexponentialrule,),forbothkindsofinitialconditions.Therefore,theelasticforcearisingfromtheactionofthesurfacetensionisstrongenoughtostabilizesuch“lms.Theattractioncomponentofthedisjoiningpressuredestabi-lizesthe“lmsnotonlyinthe“lmdrainageprocess(seeFig.5butalsoduringthe“lmrelaxation.Ifwetakethe“lmsgiveninFig.5aandstopthe”owratesat=0.45,whentheliquidvolume=1.1islargerthanthatforthe“lmsinFig.7,the“lmscannotreachthesteadystateandthereforerupture,asseenonFig.8.InthecaseofRe=1,We=1,and=0.1thechangeof)for0.45withatimestep0.1isshowninFig.8.Thepimpletypeinstabilityappearsat=2.245andsubsequentlythe“lmrupturesinitscentralzone.TheinsetofFig.8demonstratesthedecreaseof,0)withtimeforthis“lm(solidline)andforthe“lmwiththesamephysicochemicalparameters,butwith5timessmallersuc-tionvelocity(dashedline).Oneseesthatinbothcasestheinstabil-ityappearsat=0andforsmaller”owratesthedimensionlessrupturetimeof0.707forRe=0.2islowerthanthatof2.245forRe=1.Takingintoaccountthede“nition,Eq.,oneconcludesthatduetothe5timessmaller”owratetherealrupturetime,forRe=0.2becomeslonger(50.707=3.535)thanthatforRe=1.Thisfactcanbeexplainedbythelongertimeneededtosup-pressthelargerlocalsurfacevelocitiesafterstoppingthe“lmdrainageatlarger”owrates.Asdiscussedabove,theproductionofplane-parallel“lmswithsmallvolumes,2/3,andthicknessesisimpossiblewithoutusingadditives,whichincreasetherepulsivecomponentsofthedisjoiningpressure.However,theattractivedisjoiningpressureal-waysappearsandcannotbeexcludedfromconsideration.ThedashedlinesinFigs.5band6representregular“lmdrainagein Fig.6.Roleofelectrostaticandstericcomponentsofthedisjoiningpressureforthedrainageof“lmswithsymmetricboundaryconditionsandinitialcondition)=1forRe=1,We=0.1,=1,=10=10(solidlines),and=20(dashedlines). Fig.7.Relaxationof“lmsforRe=1,We=1,and=1:(a)Local“lmthickness,),withinitialpro“le,)=(5+3)/8,andinitialvolume,=1.5,=0.25(solidlines).For�0.35solidlinescorrespondtotimestep0.2;(b)thickness),linesplottedwithtimestep0.1for)=1,=2,=0.5(solidlines)andforcase(a)(dashedlines).Theinsetshowsthedifference(0)intime,,forthetwocasesafterthe“lmsuctionisstopped.S.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 thepresenceof.Wetookthesesystemsandstoppedthe”owratesattwodifferenttimes,at=0.7when=0.6,andat=0.85whenthevolumeissmaller,=0.3.Forbothsystemsandtimesthe“nal“lmpro“les,),areshowninFig.9.Oneseesthatthe“lmscontainlargealmostplane-parallelcentralzonesandsteepmenisciaroundthem.These“lmsarestablebecausetherepulsivecomponentsofthedisjoiningpressure,,prevailtherespectivevanderWaalsandhydrophobiccomponents,TheinsetofFig.9illustratesthetypeofrelaxationofthese“lms…inallcasesthe“lmthickness,,0),foroscillatesaroundtheequilibriumthickness,(0),buttheamplitudesofoscillationsexponentiallydecreasewithtime.Byapplyinglinearstabilityanal-ysis,analogoustothatgiveninRef.Ref.,forthedisturbancesofthe“lmsurfacesinacentralplane-parallelzoneoneprovesthatthecriticalwavelengthsforthecasesillustratedinFig.9aremuchlar-gerthanthelateraldimensionoftheframe.Therefore,the“nal“lmthicknessescorrespondtoastableequilibrium.Theseconclusionsareimportantforpracticalapplicationsbe-causetheygiveonepossiblewayforacontrollableproductionoflargethin“lms.6.SummaryandconclusionsAmodeldescribingthedrainageandrelaxationdynamicsofsymmetricthinliquid“lmswithtangentiallymobilesurfacesisdeveloped.Theobtainedequations,Eqs.(2.10)…(2.12),areapplica-bleforlaminar”owswhenthe“lmthicknessesaremuchsmallerthanthecharacteristiclateraldimensions.Themodeldescribesthe“lmpro“le,surfacevelocityandpressureforarbitraryvaluesoftheReynoldsandcapillarynumber,accountingfordifferentcompo-nentsofthedisjoiningpressure.Inthecaseofone-dimensional“lmattachedtoaframetheproblemissimpli“edinSectionandthevanderWaals,hydrophobic,electrostaticandstericinter-actionsbetween“lmsurfacesareincludedinthedisjoiningpres-sure.Afastalgorithmfornumericalsolutionofthisproblem,whichhasasecondorderprecisionintimeandspace,isdevelopedAppendixBandappliedinSections4and5Inthecaseofunsymmetricalboundaryconditions(different”owratesorframeheights)the“lmsdrainwithcomplexshapesoftheirsurfaces,thereforetheformationoflargeplane-parallelareasbycontrollingthe”owratesisdif“cult(Fig.4).For“lmswithsymmet-ricalboundaryconditionstheincreaseof”owratesleadstoapro-nounceddimpleformationwitharingclosetotheframe(Fig.3Theprocessof“lmdrainagebecomesmoreregularwithalmostnodimpleformationbydecreasingthe”owrateorthedeformabilityofthe“lmsurfaces,i.e.forsmallervaluesoftheReynoldsandWebernumbers(Fig.3b).Theattractivedisjoiningpressurefavorsthe“lmbreakageintwodifferentways:forlarger”owratesthedimplerup-turesatitsring;forsmaller”owratesthe“lmbreaksinitscentralzone(Fig.5).TheelectrostaticandstericrepulsivecomponentsofthedisjoiningpressurecansuppressthevanderWaalsandhydro-phobicattraction,makingtheprocessof“lmdrainageregular,withawidecentralplane-parallelzone(Figs.5and6).Thetypeof“lmdrainagecanbeef“cientlycontrolledbyvaryingthemagnitudesofdifferentcomponentsofthedisjoiningpressure.Whenthe”owratesarestopped,the“lmsurfaceseitherrelaxtotheir“nalequilibriumpro“le,orthe“lmruptures.ForagivenliquidvolumetheLaplaceequationofcapillaritydescribesthesteady“lm).Withoutaccountingfortherepulsivecomponentsofthedisjoiningpressure,)hasaparabolicorpimpleformwith-outplane-parallelareas.The“lmscanbeeitherstablewithanexpo-nentialdecayofsurfacecorrugations(Fig.7)orunstableifattractivesurfaceforcesarepresent(Fig.8).Theonlypossiblewaytoproduce“lmswithsmallvolumesandthicknessesistocontrolthemagni-tudesofelectrostaticandstericinteractions(Fig.9).These“lmsarestable:theirthicknessesoscillatearoundtheequilibriumpro-),andtheamplitudesofsurfacecorrugationsexponentiallydecreasewithtime.Theequilibriumpro“lescontainlargealmostplane-parallelcentralzonesandsteepmenisciclosetotheframes.ThemodeldescribedinSectioncanbeappliedtostudythedrainageandrelaxationdynamicsofcircular“lms.Howeverinthiscasethe“lmsthinundertheactionofaconstantpressurediffer-ence,whichchangestheboundaryconditions.Thechangeofgeometryandboundaryconditionsmustbetakenintoaccountinthemodelequationsandnumericalscheme.Thisproblemisanaimforourfuturepublication.AcknowledgmentThisstudyispartiallyfundedbyprojectINZ01/01024oftheNa-tionalScienceFundofBulgaria(programIntegratedResearchCentersintheUniversitiesŽ). Fig.8.Relaxationof“lmsurfacesinthecaseofsymmetricboundaryconditionsforRe=1,We=1,=0.1,and=0.45.Solidlinesareplottedwithtimestep0.1andthemomentbeforethe“lmruptureis2.245.Theinsetcomparestherelaxationof,0)forthis“lm(solidline)withthecaseofrelaxationofthesame“lmbutwith“vetimessmaller”owrateRe=0.2,We=0.04,=2.5,and=0.45(dashedline). Fig.9.Equilibrium“lmthicknessesforRe=1,We=0.1,and=1correspondingto=0.7and=0.85.Solidlinesareplottedfor=10=10withoutaccountingforthestericinteractions(seeFig.5b)anddashedlines…for=10=10,,and=20(seeFig.6).Theinsetillustratestherespectiverelaxationsof,0)afterstoppingthe”owrates.S.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 AppendixA.Derivationoftheleadingorderofthedynamicboundaryconditionsatthe“lmsurfaceToobtainthedensityoftheforce,,actingfromthebulk”uidontotheupper“lminterfaceoneneedstocalculatethenormalprojectionofthebulkstresstensor,,atthesurface.Wewillusethefollowingnotations:=1,2,3)aretheunitvectorsoftheCartesiancoordinatesystems=1,2)arethetangen-tialvectorsattheupper“lmsurface;istherespectivesurfacenormalvector.Therefore, 12 oHoXae31þ 1 oHoXa2"#1=2ða¼1;2Þ;ðA:1Þn¼ 1 oHoXbebþe31þ 1 oHoX12þ 1 ForincompressibleNewtonian”uidsthestresstensor,,isgivenbythede“nition oUioXkþ istheKroneckerdelta.FromEqs.onederivestheexpressionfortheforcedensity, Pkb2 oHoXbþPk31þ 1 oHoX12þ 1 whereallparametersarecalculatedat/2and=1,2,3.FromEqs.(A.1)and(A.4)weobtaintheformulaforthetangen-tialcomponentoftheforce: P332 oHoXa Pab2 oHoXb P3b4 oHoXb oHoXa1þ 1 oHoX12þ 1 oHoX22"#1=21þ 1 oHoXa2"#1=2atX3¼ Thetangentialstressboundarycondition(seeSection)forfreesurfacesreads=0.Substitutingthede“nitionofthestresstensor,Eq.,intoEq.onereducesthetangentialstressboundaryconditiontothefollowingrelationship oUaoX3þ oU3oXaþ4 oU3oX3 oHoXa2 oUaoXbþ oUboXa oHoXb oU3oXbþ oUboX3 oHoXb 0at FromEqs.(2.1),(2.2)and(A.6)weobtainthedimensionlessformofthisboundarycondition: 1e2 ouaox3¼ ou3oxa ou3ox3 ohoxaþ 1 ouaoxbþ ouboxa ohoxbþ 1e2 ou3oxbþ oubox3 ohoxb ohoxaatx3¼ Usingtheleadingordersolutionoftheconsideredproblem,Eqs.(2.7)and(2.8),wesimplifytheright-handsideofEq.totheasymptoticexpression: 1e2 ouaox3x3¼h=2¼ h2 o2wboxaoxbþ ohoxa owboxbþ 12 owaoxbþ owboxa SubstitutingEq.intoEq.wereduceittotheasymptoticrelationship: 1e2 o2uaox23¼ ooxaq owboxb o2waoxboxbþRe owaotþ ooxbðwbwaÞþ IntegratingEq.withrespecttofrom0to/2andaccountingforthekinematicboundarycondition,Eq.,onederives 1e2 ouaox3x3¼h=2¼ h2 oqoxa h2 o2wboxaoxb h2 o2waoxboxbþRe oot h2waþ ooxb Therefore,theright-handsidesofEqs.(A.8)and(A.10)mustbeequalandaftersimpletransformationswearrivetotheresultgivenbyEq.inSectionFromEqs.(A.2)and(A.4)onecalculatestheexpressionforthenormalcomponentoftheforcedensity: Pab4 oHoXa oHoXb oHoXbPb3þP331þ 12 oHoX12þ 1 whereallparametersarecalculatedat/2.Bysubstitutingthede“nitionsofthestresstensor,Eq.,andofthedimensionlessparameters,Eqs.(2.1),(2.2)and(2.4),inEq.wederivetherelationship gUapþ gUa2 ouboxbþ ohoxb oubox3þe2 ohoxb ou3oxb e24 ouaoxbþ ouboxa ohoxa ohoxb1þe2 1 ohox12þe2 1 writtenatFromtheleadingordersolution,Eq.,thetangentialstressboundarycondition,Eq.,andEq.oneobtains: gUaqþ Thenormalstressboundaryconditionreads(seeSectionSection)Fs\bnþPcþP¼Pg;ðA:14ÞwherePgisthepressureinthegasphase,isthe“lmcapillarypressure,andisthedisjoiningpressure.FromEqs.(A.13)andtheleadingorderformofthenormalstressboundarycondi-tionisreducedto owboxbþ rb2gUa o2hoxboxbþ 2Pa2rb Byde“ningthecapillarynumberandthedimensionlessdisjoiningandgaspressuresbyEq.wetransformEq.intothe“-nalresultgivenbyEq.AppendixB.NumericalschemeToconstructthenumericalschemewedenotethevaluesofallparametersattimewiththesuperscript(-)andthosecalculatedattimewiththesuperscript(+),whereisthelocalnumer-icaltimestep.Forexample,aretakenattimelevelS.S.Tabakova,K.D.Danov/JournalofColloidandInterfaceScience336(2009)273…284 …atthesubsequenttimelevel.Therespec-tivetimedifferences,,arede“nedas.Eqs.(3.1)and(3.2)areconsideredatmo-/2andthetimederivativesappearingthereinareapproximatedwithprecision DhDt¼RhjtþDt=2þOðDt2Þand wheretheoperators,,arede“nedbytheexpressions ooxðhwÞ;ðB:2ÞRw oox w22þ 1We o3hox3þ oox pdpWeþ 4Re o2wox2þ 4Re owox UsingtheCrank-Nicolsonmethoddtheright-handsidesoftheequationsinEq.arereplacedwiththemeanvaluesofoperatorsattimelevelswithaccuracy),i.e. DhDt RðþÞhRðÞ2¼RðÞþOðDt2Þand DwDt InordertoapplytheNewtonmethodforlinearizationtionwecalculatethevariationsofwithsecondor-derprecision)usingtheabovede“nitions,Eqs.(B.2)and.Aftersimpletransformationsonearrivestotherelationships DhDtþ 12 oox½hðÞDwþwðÞDh\t¼RðÞh;ðB:5Þ DwDtþ oox wðÞDw2 12We o3Dhox3 oox ddhð pdpWeÞðÞ Dh2() 2Re o2Dwox2 2Re olnhðÞox oDwox 2Re owðÞox oox TosolvenumericallyEqs.(B.5)and(B.6)weusearegulargridforthespacecoordinate,,de“nedas=0,1,2,...isthenumberofintervalswithalength=2/.Theval-uesofthefunctionsinthenodesofthegridaredenotedwiththe.Forexample,thecentraldifferenceinspacewithpreci-)appliedtoEq. wðÞk14DxDhk1 hðÞ14DxDwk1þ DhkDtþ wðÞþ14DxDhkþ1þ hðÞþ14DxDwkþ1¼RðÞh;k;ðB:7ÞwhereRðÞ;k¼ =1,2,Therespectivedifferenceschemewithprecision)appliedtoEq. lnhðÞþ1lnhðÞ12ReDx2 wðÞ14Dx 2ReDx2"#Dwk1þ wðÞkþ1wðÞ12ReDx2hðÞ1þ 14Dx ddh pdpWeðÞk1()Dhk1þ 1Dtþ 4ReDx2Dwk 12WeDkðDhÞþ wðÞkþ14Dx 2ReDx2 lnhðÞþ1lnhðÞ12ReDx2"#Dwkþ1þ 12WeDx3 14Dx ddh pdpWeðÞkþ1 wðÞkþ1wðÞ12ReDx2hðÞkþ1()Dhkþ1¼RðÞ;k;ðB:9ÞwhereRðÞ;k¼ wðÞ1wðÞ1wðÞþ1wðÞþ14Dxþ 1WeDk½hðÞ\tþ pðÞdp;kþ1pðÞ;k12WeDxþ 4Re wðÞþ12wðÞþwðÞ1Dx2þ isthedifferenceoperatorrepresentingthethirdderivativewithrespecttothespacecoordinateBecauseofthethirdderivativewithrespectto,thedifferenceoperator,,inthenodes=2,3,2hasdifferentformthanthosewrittenforthenodes1and1.For=2,3,2weusethefollowingcentraldifferenceexpressionfor isde“nedbytherelationship iscalculatedusingthevaluesofthefunctioninthenodes1,andthroughequation 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