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Thin Liquid Films Theoretical and Mathematical Physics The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals thattheseriesisasuitablepublicationplatformforboththemathematicalandthethe- oretical physicist. The wider scope of the series is reflected by the composition of the editorialboard,comprisingbothphysicistsandmathematicians. Thebooks,writteninadidacticstyleandcontainingacertainamountofelementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminarsatgraduatelevel,butalsoforscientistsenteringafieldofresearch. EditorialBoard W.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Heidelberg, Germany P.Chrusciel,GravitationalPhysics,UniversityofVienna,Vienna,Austria J.-P. Eckmann, Département de Physique Théorique, Université de Genève, Geneva, Switzerland H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria A.Kupiainen,DepartmentofMathematics,UniversityofHelsinki,Helsinki,Finland H. Löwen, Institute of Theoretical Physics, Heinrich-Heine-University of Düsseldorf, Düsseldorf,Germany M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,USA N.A.Nekrasov,IHÉS,Bures-sur-Yvette,France M.Ohya,TokyoUniversityofScience,Noda,Japan M. Salmhofer, Institute of Theoretical Physics, University of Heidelberg, Heidelberg, Germany S.Smirnov,MathematicsSection,UniversityofGeneva,Geneva,Switzerland L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,StonyBrook,USA J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria Forfurthervolumes: www.springer.com/series/720 Ralf Blossey Thin Liquid Films Dewetting and Polymer Flow RalfBlossey CNRSUSR3078 InstitutdeRechercheInterdisciplinaire Villeneuved’AscqCedex France ISSN1864-5879 ISSN1864-5887(electronic) TheoreticalandMathematicalPhysics ISBN978-94-007-4454-7 ISBN978-94-007-4455-4(eBook) DOI10.1007/978-94-007-4455-4 SpringerDordrechtHeidelbergNewYorkLondon LibraryofCongressControlNumber:2012938859 ©SpringerScience+BusinessMediaDordrecht2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Two in one. This AFM picture displays two types of thin films instabilities at the sametime:dewettingholesthatopenupinathinpolymerfilmandaninstabilityof thefilmrimalongitsrecedingedge.ThisinstabilityissimilartotheclassicRayleigh instabilityofaliquidcolumn.Inthecaseofathinfilmthepolymeraccumulatesat the rim which upon becoming unstable produces a characteristic pattern which is reminiscentofacarpetfringe. Courtesy:KarinJacobs Howdoesitcomethatpapersonwetting phenomenaand othersoftmattertopicsare oftensoconfusing,whileincondensedmatter physicseverythingisso clear? VincentSenez, sometimein2010 Preface Thisbookisatreatiseonthethermodynamicanddynamicpropertiesofthinliquid filmsatsolidsurfacesand,inparticular,theirruptureinstabilities.Forthequantita- tivestudyofthesephenomena,polymerthinfilmshavenproventobeaninvaluable experimentalmodelsystem. Whatisathinliquidfilm?Forthepurposeofthisbook,thinfilmsare(polymeric) liquids at surfaces whose properties are controlled by interfacial forces—capillary andintermolecular,likevanderWaalsforces.Gravitydoesnotplayaroleforthem. Someresearchersprefertocallsuchfilmsultrathin. Whatisitthatmakesthinfilminstabilitiesspecialandinteresting,warrantinga whole book? There are several answers to this. Firstly, thin polymeric films have an important range of applications, and with the increase in the number of tech- nologiesavailabletoproduceandtostudythem,thisrangeislikelytoexpand.An understandingoftheirinstabilitiesisthereforeofpracticalrelevanceforthedesign ofsuchfilms. Secondly, thin liquid films are an interdisciplinary research topic. Interdisci- plinary research is surely not an end to itself, but in this case it leads to a fairly heterogeneous community of theoretical and experimental physicists, engineers, physical chemists, mathematicians and others working on the topic. It justifies at- temptingtowriteatextwhichaimsatacoherent,theoreticalpresentationofthefield whichresearchersacrosstheirspecialisedcommunitiesmightbeinterestedin.Itis insomesensearesponsetoV.Senez’question:insolidstatephysicsthecommu- nityhasmuchmoreconvergedtoacommonconceptualunderstanding,sincepeople fromacommonscientificbackgroundworkinit.Butthereismore:thewettingor soft matterfieldisdominatedbyanenormousdiversityofphenomenaandmostly experimentalwork(andseeminglysimpletheoreticalexplanations),apartfromthe theory of wetting phase transitions, which has a rigorous grounding in statistical physics. Thin liquid films are an interesting laboratory for a theorist to confront a well-establishedtheory,hydrodynamics,withitslimits.Liquidsatsurfacestakeno- ticeofthesurfacetheyareplacedupon,andthisisreflectedintheirdynamics.And thepolymers,whenconfinedtothinfilms,canimprintmolecularpropertiesonthe filmdynamics. ix x Preface In the end, of course, we have only really learnt something about Nature when thetheorieshavebeenconfrontedwithreality.Here,again,liesatremendousadvan- tageinthecaseofthinpolymericfilmsduetothemodernexperimentaltechniques withwhichtheycanbemadeandstudied.Thisthereforeisafieldinwhichahighly fruitfulexchangeandcollaborationispossiblebetweenexperimentalistsandtheo- rists. The material in the book is arranged in two Parts. Part I covers the basics of wettinganddewettingphenomena,andisofinteresttoresearchersworkinginthe field also outside of polymeric systems. It can be read as a brief introduction into thetheoryofwettingphasetransitions.PartIIdelvesexclusivelyintopolymericthin films, their mathematical description, and the confrontation with experiment. The expositionofthisbookistheoreticalormathematicalinthesensethatwithineach chapterandeachsection,calculationsarepresentedatagreatlevelofdetail,butno proofsinanystrictmathematicalsensearegiven.Foranexperimentalscientist,the bookshouldserveasareferenceandguidetowhatisthecurrentconsensusofthe theoreticalunderpinningsofthefieldofthinfilmdynamics. ThefieldofwettinganddewettingowesagreatdebttoPierre-GillesdeGennes andhiscollaboratorsandstudentswhoweresoinfluentialforthefieldofSoftMat- ter Physics. Their work has produced deep insights which are, at the same time, presentedinamathematically‘light’andelegantfashion,oftenmakinguseofscal- ingarguments.Fortheuntrained,thisapproachisnotalwayseasytofollow.There is no point in trying to replace it with tedious technicalities, and this is not what is intended here. The present book attempts to bridge between the ‘light’ and the ‘rigorous’,alwayswiththeambitiontoenhanceinsightandunderstanding—andto notletgotheeleganceofthetheory. Thisbookowesagreatdealtomycollaboratorsanddiscussionpartnersoverthe years.IhopetheyallwillfindthatitalsoreflectswhatIlearntfromthem. Lille RalfBlossey May2012 Contents PartI Dewetting 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 TheLandau-Levich-DerjaguinProblem . . . . . . . . . . . . . . . 3 1.2 DewettingofaLiquidFilmofaBinaryMixture . . . . . . . . . . 6 2 StatisticalMechanicsofThinFilms . . . . . . . . . . . . . . . . . . . 9 2.1 CapillarityandSurfaceTension . . . . . . . . . . . . . . . . . . . 9 2.2 ForcesActingatInterfaces . . . . . . . . . . . . . . . . . . . . . 15 2.3 WettingandDewetting:TheWettingPhaseDiagram . . . . . . . . 19 2.4 TheHamiltonianandtheLineTension . . . . . . . . . . . . . . . 22 2.4.1 TheEffectiveInterfaceHamiltonian. . . . . . . . . . . . . 22 2.4.2 LineTensionI:TheModifiedYoungEquation . . . . . . . 22 2.4.3 LineTensionII:TheElasticityoftheContactLine . . . . . 26 2.5 CharacterizingMetastableThinFilms . . . . . . . . . . . . . . . . 30 2.5.1 WettingbyNucleation:TheCriticalDroplet . . . . . . . . 32 2.5.2 DewettingbyNucleation:TheCriticalHole . . . . . . . . 35 2.5.3 MappingtoaDynamicalSystem . . . . . . . . . . . . . . 38 2.5.4 ScalingBehaviouroftheCriticalHole . . . . . . . . . . . 41 2.5.5 UndercoolingaThickFilm:APhysicalInterpretation,and SomeTheory. . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6 DewettingforanUnstableFilm:SpinodalDecomposition . . . . . 45 2.7 DensityFunctionalTheory . . . . . . . . . . . . . . . . . . . . . 46 3 FromClassicalLiquidstoPolymers. . . . . . . . . . . . . . . . . . . 49 3.1 ClassicalLiquids&SoapFilms . . . . . . . . . . . . . . . . . . . 49 3.2 LiquidHelium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 WettinginSuperconductors . . . . . . . . . . . . . . . . . . . . . 52 3.4 DewettingofPolymerThinFilms . . . . . . . . . . . . . . . . . . 53 xi xii Contents PartII PolymerFlow 4 HydrodynamicsofThinViscousFilms . . . . . . . . . . . . . . . . . 59 4.1 TheThin-FilmProblem . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 LubricationApproximationI . . . . . . . . . . . . . . . . . . . . 62 4.2.1 LubricationScaling . . . . . . . . . . . . . . . . . . . . . 63 4.2.2 ChoiceofScaleRatios . . . . . . . . . . . . . . . . . . . . 64 4.3 MathematicalPropertiesoftheThin-FilmEquation . . . . . . . . 65 4.4 ThinFilmRuptureinViscousThinFilms . . . . . . . . . . . . . . 67 4.5 ThermalFluctuationsintheFilm . . . . . . . . . . . . . . . . . . 71 4.6 LubricationApproximationII . . . . . . . . . . . . . . . . . . . . 74 4.7 DewettingHoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.7.1 HoleGrowth . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.7.2 TheDewettingRim . . . . . . . . . . . . . . . . . . . . . 79 4.7.3 DewettingRim:AsymptoticAnalysisintheNo-SlipCase . 80 4.7.4 DewettingRimsinSlippingFilms . . . . . . . . . . . . . 83 4.7.5 RimInstabilities . . . . . . . . . . . . . . . . . . . . . . . 85 5 ViscoelasticThinFilms . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1 ViscoelasticFlow . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.1.1 RateofStrainTensor . . . . . . . . . . . . . . . . . . . . 90 5.1.2 TheJeffreysModel . . . . . . . . . . . . . . . . . . . . . 91 5.2 LubricationApproximationIII . . . . . . . . . . . . . . . . . . . 93 5.2.1 WeakSlip . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2.2 StrongSlip . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2.3 DispersionRelations . . . . . . . . . . . . . . . . . . . . . 99 5.2.4 DewettingRims . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 BeyondtheJeffreysModel . . . . . . . . . . . . . . . . . . . . . 102 5.3.1 TheCorotationalJeffreysModel . . . . . . . . . . . . . . 102 5.3.2 ConfrontationwithExperiment:APhenomenological ModificationoftheJeffreysModel . . . . . . . . . . . . . 106 5.4 MicroscopicsofPolymerFilms . . . . . . . . . . . . . . . . . . . 109 5.4.1 MicroscopicsoftheSlipBoundaryCondition . . . . . . . 109 5.4.2 PolymersareGlasses:T ofThinFilms . . . . . . . . . . . 113 g 6 ConclusionsandOutlook . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1 FiniteandStructuredGeometries . . . . . . . . . . . . . . . . . . 118 6.2 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3 MetallicFilms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4 PolymersUnderExternalFields . . . . . . . . . . . . . . . . . . . 123 6.5 ActivePolarGels . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.6 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 AppendixA PolymericThinFilms . . . . . . . . . . . . . . . . . . . . . 127 A.1 TheSubstratesandtheSurfaceCoatings . . . . . . . . . . . . . . 128 A.2 PreparationandMeasurementofThinFilms . . . . . . . . . . . . 128 AppendixB MinkowskiMeasures . . . . . . . . . . . . . . . . . . . . . 129