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arXiv:cond-mat/0411688v3[cond-mat.soft]25Jun2007The‘Cheerioseffect’DominicVellaandL.Mahadevan∗DivisionofEngineeringandAppliedSciences,HarvardUniversity,PierceHall,29OxfordStreet,CambridgeMA02138(Dated:February2,2008)Objectsthatfloatattheinterfacebetweenaliquidandagasinteractbecauseofinterfacialdeformationandtheeffectofgravity.Wehighlightthecrucialroleofbuoyancyinthisinteraction,which,forsmallparticles,prevailsoverthecapillarysuctionthatisoftenassumedtobethedominanteffect.Weemphasizethispointusingasimpleclassroomdemonstration,andthenderivethephysicalconditionsleadingtomutualattractionorrepulsion.Wealsoquantifytheforceofinteractioninsomeparticularinstancesandpresentasimpledynamicalmodelofthisinteraction.Theresultsobtainedfromthismodelarethenvalidatedbycomparisontoexperimentalresultsforthemutualattractionoftwoidenticalsphericalparticles.Weconcludebylookingatsomeoftheapplicationsoftheeffectthatcanbefoundinthenaturalandmanmadeworlds.I.INTRODUCTIONBubblestrappedattheinterfacebetweenaliquidandagasrarelyrest.Overatimescaleofseveralsecondstominutes,long-livedbubblesmovetowardsoneanotherand,whencontained,tendtodrifttowardstheexteriorwalls-asshowninfig.1.Thescepticalreadermayread-ilyverifytheseclaimsbypouringthemselvesaglassofsparklingwater(or,iftheyprefer,wine)andfollowingthemotionofthosebubblesatthesurface-particularlythoseattheperipheryoftheglass.Thisphenomenonhasevenbeenaffectionatelydubbedthe‘Cheerioseffect’af-tertheobservationthatbreakfastcerealsfloatinginmilkoftenclumptogetherorsticktothewallsofthebreakfastbowl1.Inthisarticle,webringtogetheranumberofperspec-tivesontheCheerioseffectgatheredfromtheliteratureandourownexperienceathome,inthekitchen,andinthelaboratory.Weshowhowsimplephysicalideasleadtoanunderstandingnotonlyoftheattractionit-self,butalsoofitsdynamicalconsequences.Despitebe-ingasubjectwithenormouspotentialforsimple,reliablepartytricks,thetechnologicalimplicationsoftheChee-rioseffectarefarfromfrivolous.Indeed,muchresearchiscurrentlybeingundertakentoinvestigatethepossibilityFIG.1:Bubblesfloatingonwaterinapetridish.Thebubblesareobservedtoaggregatebeforemovingtothewallofthecontainer.Aftersufficientlylongtimes,theislandofbubblesinthecentrealsomigratestothewall.ofusingsurfacetensiontoinducethe‘self-assembly’ofsmall-scalestructures2.Understandingthewayinwhichparticlesaggregateataninterfaceandthencebeingabletocontroltheformoftheaggregateaswellasthedynam-icsofitsformationmayonedayenablemuchsimplifiedmanufactureofcomponentsofmicro-electromechanicalsystems(MEMS).Forfloatingobjectsinequilibriumormotion,wemustconsiderthebalanceoflinearmomentumbothintheflotationplaneaswellasoutoftheplane,andinadditionthebalanceofangularmomentuminallthreedirections.Manyofthemisconceptionsinthefieldarisefromconsid-eringonlysomebutnotallofthesebalanceequations.Inparticular,neglectingtheconditionofverticalforcebalanceleadstoanunderestimationoftheimportanceoftheparticle’sbuoyancyindeterminingthenatureoftheinteraction.Webeginwithadiscussionofthephysicalmechanismthatleadstotheobservedattractioninmostinstancesandillustratetheroleofparticlebuoyancybymeansofasimpleexperiment.Wethenfocusonaseriesofsim-pleexamplesthatallowustoquantifythemagnitudeoftheattractiveforce.Thefirstoftheseisinspiredbyanoversimplifiedphysicalpicturethatisoftenperceivedasthedominanteffect.Byconsideringcarefullytheverticalforcebalancethatmustbesatisfiedforparticlestofloat,wethenshowthatitisthisthatisdominantforsmallparticlesandsubsequentlyproposeasimpledynamicalmodelfortheattractionoftwosphericalparticles.Fi-nally,weshowthatconsiderationoftheremainingequi-libriumcondition,thatoftorquebalance,canleadtoamphiphilicstripsandconcludewithsomepossiblebio-logicalimplicationsofourobservations.II.THEPHYSICALORIGINOFATTRACTIONThemechanismbehindtheapparentattractionbe-tweenbubblesorbetweenabubbleandthewallofaglassiseasytounderstandbyconsideringthegeometryoftheinterfaceatwhichthebubblesaretrapped.Forsimplic-ity,weconsiderthelattercase(schematicallyillustrated2FIG.2:Schematicofasinglebubbleclosetoawall,alongwiththedefinitionofthecontactangle.infig.2),althoughtheexplanationoftheclusteringofmanybubblesissimilar.Here,theair-waterinterfaceissignificantlydistortedbythepresenceofthewall(thewell-knownmeniscuseffect)and,sincethebubbleispos-itivelybuoyant,thereisanetupwardforce,Fg,onthebubble.Becauseitisconstrainedtolieattheinterface,however,thebubblecannotsimplyriseverticallyandin-steaddoesthenextbestthingbymovingupwardsalongthemeniscus.Asglassisingeneralwetting(thecontactangleθ,definedinfig.2,satisfiesθπ/2),inmovingupwardsalongthemeniscusthebubblealsomovesclosertothewall.Viewedfromabove,itappearsasifthereisanexoticattractiveforceactingbetweenthewallandthebubblewheninfactthebubbleismerelyobeyinggravityandmovinginaconstrainedpathimposedbythepresenceofthewall.Now,asinglebubblewilldeformtheinterfacejustasthepresenceofawalldoes,thoughthistimeforadiffer-entreasonandtoalesserextent.Inthebubble’scase,itcanonlyremainattheinterfacebecausethebuoy-ancyforce(thatistendingtopushthebubbleoutoftheliquid)iscounterbalancedbythesurfacetensi