Shapes, contact angles, and line tensions of dropl

arXiv:cond-mat/0002427v1[cond-mat.soft]28Feb2000Shapes,contactangles,andlinetensionsofdropletsoncylindersC.BauerandS.DietrichFachbereichPhysik,BergischeUniversit¨atWuppertal,D-42097Wuppertal,GermanyAbstractUsinganinterfacedisplacementmodelwecalculatetheshapesofnanometer-sizeliquiddropletsonhomogeneouscylindricalsurfaces.Wedetermineeffec-tivecontactanglesandlinetensions,thelatterdefinedasexcessfreeenergiesperunitlengthassociatedwiththetwocontactlinesattheendsofthedroplet.Thedependencesofthesequantitiesonthecylinderradiusandonthevolumeofthedropletsareanalyzed.68.45.Gd,68.10.-m,82.65.DpTypesetusingREVTEX1I.INTRODUCTIONThewettingpropertiesofafiberinliquidmatrices(e.g.,dyemixtures,polymermelts,ormoltenresins)playanimportantroleinthetextileindustryandinthefabricationofhigh-performance,fiber-reinforcedcompositematerials.Sincecontactanglesofliquiddropletsonsolidsubstratesprovideavaluablecharacterizationofsuchwettingpropertiestherearenumerousexperimentalandtheoreticalstudiesoftheshapeandthespreadingofdropletsdepositedonacylindricalsubstrate(see,e.g.,Refs.[1–15]).Themorphologyofliquiddropsonafiberisparticularlyinterestinginsofarasonaplanarsubstratethereisonlyone,sphericalcaplikedropletshape,whereasonacylindricalsubstratedropletsmayexhibittwo,topologicallydifferentshapes,a“clamshell”-anda“barrel”-typeone,dependingonthedropletvolume,thecontactangle,andthecylinderradius[1–3].Theaforementionedstudiesdealwiththickfibersandlargedrops,i.e.,thelengthscalesareμmandlarger.Inthisrangethefluidstructuresaredeterminedbymacroscopicpropertiesalone,i.e.,volumeoftheliquid,surfacetensionσoftheliquidvaporinterface,Young’scontactangleθ∞,andradiusRofthecylinder.However,withthediscoveryofnanotubestheinterestinsuchfluidstructureshasshiftedtomuchsmallerscales.Thereareseveralapplicationsforwhichthesesmallsolid-fluidstruc-turesareveryimportant.(i)Forfabricatingvaluablecompositematerialsinvolvingnano-tubestheirwettingbytheliquidhostmatrixisnecessarytocoupletheinherentstrengthofthenanotubestothematrix,reinforcingmaterialsorfillersforplasticsandceramics[16].(ii)Nanotubescanbeusedassupportsforheterogeneouscatalysisorastemplatesforcreatingsmallwiresortubularstructuresbycoatingthemwithmetalsormetaloxidesintheliquidstate[17]orbyattachinginorganicandorganicmoietiestothenanotubesurfaces[18].(iii)Inordertousenanotubesas“nanostraws”potentialcandidatesforexploitingsuchcapil-laritymustbescreenedbyfirstseeingiftheliquidwetstheoutsideofnanotubes[19].Theperformanceofthenanotubesascatalysts,adsorbants,anddeodorantscanvarydepend-ingonwhethertheyarecomposedofcarbon,boronnitride,oroxides(SiO2,Al2O3,V2O5,MoO3,TiO2)[20].Thisvarietydemonstrates,thatthesubstratepotentialofthesetubescanberegardedasatunableparameter.(iv)Byusingnanotubesasnanotweezers[21]itmightbepossibletograbandmanipulatesmallliquiddrops.Forthisapplicationthesubstratemustbenonwettable.Thesesmallscalesarecomparablewiththerangeofthesubstratepotentialofthecylin-dersandofthemolecularforcesbetweenthefluidparticlesadsorbingonthem.Thusthedropletsformundertheactionoftheso-calledeffectiveinterfacepotentialωwhichaccountsfortheneteffectofthecompetitionbetweentheforcesamongthefluidparticlesandthesubstratepotential[22].Accordinglythecalculationofthecorrespondingdeformeddropletshapesrequiresamoredetailedtheoreticaldescriptionwhichtakestheeffectiveinterfacepotentialintoaccount.Toourknowledgethereisonlyone,recentpublicationinwhichthiseffectofωonthedropletshapeonfibershasbeenanalyzed[23].Itisthepurposeofourstudyheretorefineandtoextendthisanalysisinvariousdirections.IftheradiusRofthefiberreducestoafewnm,asitisthecasefornanotubes,theeffectiveinterfacepotentialitselfwilldependonRandthusdeviatefromthatofthecorrespondingsemi-infiniteplanarsubstrateusedinRef.[23].Accordinglywepresentasystematicanalysisofthedependenceoftheshapeofthedropletsandtheirsuitablydefinedcontactangleson2bothRandthedropletvolume.ThisenablesustodescribesystematicallythecrossoverinshapeandcontactanglebetweenthoseofdropletsonacylinderandonthelimitingcaseR→∞ofaplanarsubstrate.Weremarkonhowthestructureoftheeffectiveinterfacepotential,dependingonwhetheritleadstofirst-orderorcontinuouswettingtransitions,influencesthemorphologyofthedroplets.Weconfineouranalysistobarrel-typedropletsandestimatetheirmetastabilityagainstroll-uptotheclamshellconfiguration.Finallywestudytwotypesoflinetensions.Thefirstoneconcernsthelinetensionofthree-phasecontactbetweenliquid,vapor,andsubstrateemergingattheendsofmacroscopiclylargedropsonfiberswhichreducestothefamiliarlinetensionofthestraightthree-phasecontactlineonaplanarsubstrate.Thesecondexcessfreeenergyconcernstheeffectivelinetensionassociatedwiththecircularshapeofthethree-phasecontactlineonaplanarsubstrateasfunctionofthedropletvolume.Theseresultsarerelevantforunderstandinghowtoextractlinetensionsfromcontactanglemeasurements.Weareencouragedtopresentourrefinedanalysesbyrecentexperimentaladvancestodeterminedropletshapessuchasmicroscopicinterferometry[24],ellipsometricmi-croscopy[25],scanningpolarizationforcemicroscopy[26],andtapping-modescanningforcemicroscopy[27]