Mahadevan a Division of Engineering and Applied Sciences Harvard University Pierce Hall 29 Oxford Street Cambridge Massachusetts 02138 Received 22 November 2004 accepted 25 February 2005 Objects that 64258oat at the interface between a liquid a ID: 66529
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TheCheerioseffectDominicVellaandL.MahadevanDivisionofEngineeringandAppliedSciences,HarvardUniversity,PierceHall,29OxfordStreet,Cambridge,Massachusetts02138Received22November2004;accepted25February2005Objectsthatßoatattheinterfacebetweenaliquidandagasinteractbecauseofinterfacialdeformationandtheeffectofgravity.Wehighlightthecrucialroleofbuoyancyinthisinteraction,which,forsmallparticles,prevailsoverthecapillarysuctionthatoftenisassumedtobethe duetosurfacetensionwhichisproportionaltothecurvatureoftheinterfaceisequaltothehydrostaticpressurediffer-encecausedbythedeformationoftheinterfaceseeRef.4,p.65forathoroughdiscussion.Forsmallinterfacialdeßec-tions,thisbalancemaybewrittenas: isthesurfacetensioncoefÞcientoftheliquidÐgasisthedensityoftheliquid,andistheaccelera-tionduetogravity.Equationistobesolvedwiththeboundaryconditionsthatthecontactangles,,aregivenateachoftheplatesandthedeßectionoftheinterfaceshoulddecayfarawayfromtheplates.Intheregionslabeled1,2,3inFig.4,thesolutionofEq.,where isthecapillarylength,whichdeÞnesthelengthscaleoverwhichinterac-tionsoccur.Theconditions)give,which,combinedwiththecontactanglecondi-,givetheinterfaceshapeoutsidetheplatesas:Thecontactangleconditions,,givetheinterfaceshapebetweenthetwoplates: Lccot1coshdx Lccot2coshx/Lc BecauseoftheinterfacialdeformationgivenbyEqs.,theplatesarenowsubjectedtoacapillarypressurethatresultsinahorizontalforceontheplates.Thereisnoresult-anthorizontalsurfacetensionforcebecauseitscomponentsoneithersideofaplatecancelexactly.Thisforcemayacteithertobringthemtogetherortopullthemapartdependingonthecontactangles.Thevalueofthehorizontalforceperunitlength,,canbecalculatedbyintegratingthehydrostaticpressurealongeachofthewettedsidesofoneoftheplatessaytheoneontheleftinFig.4andtakingthedifferenceasfollows: sothatwehave: 2cot1coshd/Lccot22 wherethesignconventionissuchthat0correspondstoattractionbetweentheplates.Typicallythisforceiseitherattractiveforallplateseparationsorrepulsiveatlargesepa-rationsandattractiveatshortseparationswithanunstableequilibriumatanintermediatedistance.Anexampleofaforce-displacementcurveinthelattercaseisshowninFig.5.canbeusedtoshowthatrepulsionispossibleonlyifcot0,thatis,ifoneplateiswettingandtheothernonwetting.Wenotethatifcot,thefact0asimpliesthatrepulsioncanoccuronlyifhasamaximumvaluesomewhere,becauseas.Asimplecalculationshowsthattheonlyturningpointofthefunction(cotcursat,wherecosh,whichonlyhasarealsolutionifcot0.Whencot,theshortrangeattractiondoesnotexist,andinsteadthereisrepulsionatdisplacements.However,theresultthatrepulsioncanonlyoccurwhencot0stillThisresultshowsthatverticalplatesataliquidÐgasinter-facewillattractiftheyhavelikemenisciandotherwisere-pel,aswesawinSec.IIwithßoatingobjects.However,thephysicalmechanismhereissubtlydifferent.Wecannolongerargueintermsofoneplatefollowingthemeniscusimposedbytheotherbecausetheseplatesdonotßoat,mean-ingthatthereisnoanalogueoftheconditionofverticalforcebalanceinthiscase.Instead,wemustconsidertheeffectsofhydrostaticpressurewhichresultfromthedeformationoftheinterface,asexplainedinRef.8followingearlierargumentsbyKelvinandTait.WegivehereanabbreviatedversionoftheirargumentintermsoftheconÞgurationsshowninFigs.and6inwhichplatesoflikewettabilityareatthe Fig.4.ThegeometryoftwoinÞniteplatesinasemi-inÞniteßuid.Theplaneshavecontactanglesandareatahorizontaldistance Fig.5.Atypicalforce-separationcurveforawettingandnonwettingplateandaliquidÐgasinterface.Here,/3and/4,andweobserverepulsionatlargeseparationsandattractionatshortrange.Am.J.Phys.,Vol.73,No.9,September2005D.VellaandL.Mahadevan ancyforceduetothedisplacedbulkßuid.TheÞrstoftheseforcesiseasilyseentobegivenby.Thesecondisgivenbytheweightofthewaterthatwouldoccupytheareabetweenthewettedregionofthesphereandtheundisturbedinterface,whichisshownasthehatchedareainFig.7.TounderstandphysicallythisgeneralizationofArchimedesÕprinciple,noticethattheliquidhasnoknowl-edgeofthegeometryoftheobjectthatisattheinterfaceoutsideofitswettedperimeter.TheliquidmustthereforeproduceanupwardforceequaltowhatitwouldprovidetoanobjectÞllingtheentirehatchedregion,whichweknowfromtheusualArchimedesresultistheweightofthedis-placedliquidthatwouldÞllthisvolume.Forelegantrigor-ousderivationsofthisresult,seeRefs.10or11.Thisvol-umecanbecalculatedbysplittingitintoacircularcylinderofradiusandheight,andasphericalcapof)andbase.Theseconsiderationsgivethebuoyancyforce Rsin c2 3cos c1 Thebalanceoftheverticalforcesmaynowbewrittenex-plicitlyas: 32Rsin czc 1zc2gR3zc Rsin c2 3cos c1 Ifwesubstituteandkeeponlythosetermslinearin,weobtainanexpressionforcuratetoÞrstorderintheBondnumber 31 2cos1 Asaconsistencycheck,observethat0when/2and1/2,whichweexpect,becauseinthiscasetheArchimedesbuoyancyaloneisenoughtobal-ancetheweightofthespherewithoutanyinterfacialdefor-containstwodimensionlessparameters,.TheBondnumber,,isthemostimportantdimensionlessparameterinthissystem.Itgivesameasureoftherelativeimportanceoftheeffectsofgravityandsurfacetension:largecorrespondstolargeparticlesorsmallsur-facetensioncoefÞcientÑinbothcasesthesurfacetensionisinconsequential.TheexpressionfortheslopeoftheinterfaceinthevicinityofthesphericalparticlegiveninEq.validforcorrespondingtoaradiusof1mmorsmallerforasphereatanairÐwaterinterface,inwhichcasesurfacetensionisveryimportant.Theotherdimensionlessparameter,,canbethoughtofasaantweightoftheparticleoncetheArchimedesforcehasbeensubtracted.Thisinterpretationarisesnaturallyfromtheverticalforcebalancecondition,becausethere-sultantweightoftheobjectinthelinearizedapproximationis2TocalculatetheinteractionenergyusingtheNicolsonap-proximation,wealsomustcalculatetheinterfacialdisplace-mentcausedbyanisolatedßoatingsphere,whichisdeter-minedbythehydrostaticbalance,thecoordinateinvariantstatementofEq..Withtheassump-tionofcylindricalsymmetry,thisgeneralizationofEq. rd r drh withtheboundaryconditionsthat0as.EquationhasasolutionintermsofmodiÞedBesselfunctionsoftheÞrstkindandoforder wherewehaveusedtheasymptoticresultsee,forexample,Ref.121tosimplifytheprefactor.B.TwointeractingparticlesHavingcalculatedtheeffectiveweightofasphereatadeformedinterfaceas2asdeÞnedinEq.aswellastheinterfacialdeformationcausedbythepres-enceofasinglesphere,wearenowinapositiontocalculatetheenergyofinteractionbetweentwospheres.Toleadingorderin,thisenergyistheproductoftheresultantweightofonesphereanditsverticaldisplacementduetothepres-enceofanotherspherewithitscenterahorizontaldistanceaway.Wemaythereforewritetheenergy,),as: andfromEq.,theforceofinteractionisgivenby,or: Lc.13 Problem1.Repeatthepreviouscalculationfortwocylin-dersofinÞnitelengthlyinghorizontallyandparalleltooneanotherataninterface.FirstconsidertheinterfacialproÞlecausedbyanisolatedcylinderandshowthatitisgivenby1.Nextusethelinearizedverticalforcebalanceandthegeometricalrelationshipshowthat: 2D11 sin2 Theresultantweightoftheobjectcanbefoundfromtheforcebalancetobe.WiththiseffectiveweightandtheinterfacialproÞleinEq.,calculatetheenergyofperunitlength),betweentwocylinderswithcenterÐcenterseparation.Showthat dl 2B2C2expl deÞnedasinEq.Am.J.Phys.,Vol.73,No.9,September2005D.VellaandL.Mahadevan