Design of continuous transport of droplet by contact-boiling regime - Wang et al. - Unknown - Unknown

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SupportingInformationDesignofcontinuoustransportofdropletbycontact-boilingregimeShanlinWang1,*,XiaofengZhao1,3,XianWu1,QingyuZhang2,*,YuanchengTeng1,RajeevAhuja3,YoufaZhang41.StateKeyLaboratoryforEnvironment-FriendlyEnergyMaterials,SchoolofMaterialsScienceandEngineering,SouthwestUniversityofScienceandTechnology,Mianyang621010,P.R.China;2.ShagangSchoolofIronandSteel,SoochowUniversity,Suzhou215137,P.R.China;3.CondensedMatterTheoryGroup,MaterialsTheoryDivision,DepartmentofPhysicsandAstronomy,UppsalaUniversity,75120Uppsala,Sweden;4.JiangsuKeyLaboratoryofAdvancedMetallicMaterials,SchoolofMaterialsScienceandEngineering,SoutheastUniversity,Nanjing211189,P.R.China;*Correspondingauthor.Email:wangshanlin@swust.edu.cn(S.W.);qingyu.zhang@suda.edu.cn(Q.Z.);CONTENTSupplementarytextS1SupplementaryFiguresS1~S9LegendsforMoviesS1~S7S1

1SupplementarytextS1.NumericalSimulationThebehaviorofanethanoldropletimpactingonahotmicro-holearraysurfaceissimulatedbyusingatwo-dimensionalnine-velocity(D2Q9)thermalmultiphaselatticeBoltzmann(LB)model(1-3).IntheLBmodel,itisassumedthatthemultiphaseflowsarecomposedofmassivefluidparticles,andtheflowphenomenaarerepresentedbytheevolutionoftheparticledistributionfunctions.Usingamulti-relaxation-time(MRT)collisionoperator,theLBequation,describingtheprocessesofparticlestreamingandcollision,iswrittenas1eq'fixeit,ttfix,tMΛMfjfjFitij(S1)i,j0,1,...,8wherefandfeqaretheparticledistributionfunctionandtheequilibriumdistributionfunction,iirespectively.x,t,andΔtaretheposition,time,andtimestep,respectively.IntheD2Q9model,theformsoftheorthogonaltransformationmatrix,M,andthediscretevelocities,ei,aregivenby111111111411112222422221111010101111M020201111(S2)001011111002021111011110000000001111010101111ec(S3)i001011111wherec=Δx/ΔtisthelatticespeedandΔxisthelatticespacing.InEq.(S1),M-1istheinversematrixofM,andF'istheforcingterm.Thediagonalmatrix,Λ,isicomposedoftherelaxationtimesandexpressedas111111111Λdiag,e,,j,q,j,q,,(S4)whereτνandτedeterminethedynamicviscosityandthebulkviscosity,respectively.Inthepresentwork,therelaxationtimesaresettobeτρ=τj=1.0,τe=τζ=1.25,τq=10/11andτν=1.1,respectively.S2

2Thedistributionfunctionandthecorrespondingequilibriumdistributionfunctioninthemomentspaceareobtainedbym=Mfandmeq=Mfeq.meqaregivenbyeq2222Tm[1,23u,13u,u,u,u,u,uu,uu](S5)xxyyxyxy8whereρisthemacroscopicdensityobtainedbyfi.u=[ux,uy]Trepresentsthemacroscopici0velocityandiscalculatedby8ueifiFt2(S6)i0whereFisthetotalforceactingonthefluidparticles.UsingthetransformationmatrixM,theLBequationinthemomentspaceiswrittenas*eqΛmmΛmmtIS(S7)2whereIistheunittensor,andSistheforcingterminthemomentspace.TherelationshipbetweenSand?′is(I-Λ/2)S=M?′.Thestreamingprocessisimplementedinthevelocityspace,whichiscalculatedby*fixeit,ttfix,t(S8)wheref*=M-1m*isthetransformationfromthemomentspacetothevelocityspace.TheforcingtermSinEq.(S7)isexpressedas22S[0,6uF12Fct(e0.5),6uF2212Fct(0.5),Fx,Fx,Fy,Fy,(S9)T2uxFxuyFy,uxFyuyFx]wheretheparameter,ε,fortuningthemechanicalstabilityconditionisselectedtobeε=0.09inthepresentwork.Thetotalforce,F,includesthegravitationalforce,Fg,thefluid-fluidcohesionforce,Fc,andthefluid-solidadhesionforce,Fads.FgiscalculatedbyFg(x,t)(x,t)g,whereg=[0,-g]isthegravitationalaccelerationandgissettobe1×10-5.FciscalculatedbyS3

38Fc(x,t)Gcx,twi(xeit,t)ei(S10)i0whereGcisthefluid-fluidinteractionstrength.Fadsiscalculatedby82Fads(x,t)Gadsx,twis(xeit,t)ei(S11)i0wheres(x+eiΔt)isanindicatorfunctionforindicatingasolid(s=1)orafluid(s=0)neighboringnode,respectively.Gadsisthefluid-solidadhesionparameter.InEqs.(S10)and(S11),theweightcoefficientswiaregivenbywi=4/3for|ei|2=0,wi=1/3for|ei|2=1,andwi=1/12for|ei|2=2,respectively.Thepseudopotentialfunction,ψ,isdefinedby22PcG(S12)EOSscwhereGc=-1isadoptedtoensurethattheterminsidethesquarerootispositive.PEOSistheprescribednon-idealequationofstate.Inthepresentstudy,thePeng-Robinsonequationofstateisemployed(4),whichiswrittenas2RTa(T)P(S13)EOS221b12bbwhereTisthetemperatureofeachnode.(T)isafunctionofT,whichiswrittenas22(T)1(1TT)(0.374641.52260.26992).ωistheacentricfactorofethanol,whichcisfixedtobeω=0.644(5).AccordingtotheworkofLietal.(1-3)andYuan&Schaefer(4),the22parametersaandbaredefinedasa0.45724RTcpcandb0.0778RTcpc,respectively.Inaddition,wesettheparametersa,b,andRtobea=2/49,b=2/21,andR=1,whichareidenticaltotheworkofYuan&Schaefer(4).Asaresult,bysolvingtheequationsofaandb,thevalueofthecriticaltemperatureTcisobtainedasTc=0.073.InEqs.(S10-S13),thetemperatureofeachnodeislinkedtotheequationofstateintheLBmodel.Thegoverningequationforcalculatingtemperaturefieldiswrittenas(1-3)T1TPEOSTuTu(S14)tccTvvS4

4wherecvisthespecificheatcapacityatconstantvolume.λ(ρ)isthedensity-dependentthermalconductivityandexpressedasλ(ρ)=ρcvχ,whereχ=0.35(1/τν-0.5)/3.ThemacroscopicvelocityuiscalculatedfromEq.(S6).Thetermsontheright-handsideofEq.(S14)representtheheatandtemperaturevariationscausedbyheatconduction,fluidconvection,andliquid-gasphasechange,respectively.Inthepresentwork,Eq.(S14)issolvedbyusingthefourth-orderRunge-Kuttaalgorithmforthetimediscretizationandthecentraldifferencealgorithmforthespatialdiscretization.Throughthecalculationsofflowandtemperaturefields,the2-DthermalmultiphaselatticeLBmodelincorporatesthemechanismsofliquid-gasphasechange,heattransfer,anddropletmovement.Intheflowfieldcalculation,theperiodicboundaryconditionisappliedonallwallsofthe2-Ddomain,andabounce-backruleisemployedonthefluid-solidboundaries.Inthetemperaturefieldcalculation,aNeumann(zero-flux)boundaryconditionisadoptedonthenorth,west,andeastwalls,whileaDirichletboundaryconditionisappliedonthefluid-solidboundaries.Inaddition,thequantitiesinthesimulationsaretakenastheLBunits,i.e.lu(latticeunit)forthespatialspacingandts(timestep)forthetime.SupplementaryFigureS6presentstheschematicillustrationofanethanoldropletaboveahotmicro-holearraysurfaceatthestartoftheLBcalculation.Thediameteroftheinitiallystaticdropletis100lu,andtheinitialheightofthedropletis200lu.Themicro-holearraysarecharacterizedbydiameter,separationdistanceandheight,whicharesettobe3,3,6luinthe2-Dsimulation.Themicro-holearraysurfaceishotterthanthedropletandatmospherebysettingatemperaturedifferenceof0.42Tc.SincethecriticaltemperatureofethanolisTc=240.7Cinthephysicalworld(5),thetemperaturedifferenceintheLBsimulationcorrespondsto101.1C.AstheLBsimulationproceeds,theethanoldropletfallsdownwardsundertheeffectofgravity.Whenthedropletcontactsthehotmicro-holearraysurface,thelowerpartofthedropletvaporizesspontaneously.Coherentlyinfluencedbytheeffectsofpressureunderneaththedropletandgravity,thedropletundergoescontinuousreboundingandcomplexmorphologicalchanges.Inthepresentstudy,theheightofthedropletcentroidiscalculatedbyyLiyii,wherethesubscriptiindicateseachnodeofiLiLS5

5theliquidphase,andyiistheverticalcoordinate.Thepressureunderneaththedropletanddroplettemperatureareobtainedbyaveragingthepressuresandtemperaturesofthecorrespondingnodesunderneathandwithinthedroplet,respectively.References[1]LiQ.;KangQ.J.;FrancoisM.M.;HeY.L.;LuoK.H.LatticeBoltzmannmodelingofboilingheattransfer:Theboilingcurveandtheeffectsofwettability.InternationalJournalofHeatandMassTransfer2015,85,787-796.[2]LiQ.;YuY.;ZhouP.;YanH.J.Enhancementofboilingheattransferusinghydrophilic-hydrophobicmixedsurfaces:AlatticeBoltzmannstudy.AppliedThermalEngineering2018,132,490-499.[3]LiQ.;KangQ.J.;FrancoisM.M.;HuA.J.LatticeBoltzmannmodelingofself-propelledLeidenfrostdropletsonratchetsurfaces.SoftMatter2016,12,302-312.[4]YuanP.;SchaeferL.EquationsofstateinalatticeBoltzmannmodel.PhysicsofFluids2006,18,042101.[5]ReidR.C.;PrausnitzJ.M.;PolingB.E.ThePropertiesofGases&Liquids.McGraw-HillInc.(1987).S6

6SupplementaryFigureS1LiquidHV=6μLLSide-viewH0=10mmDLSiliconwaferMicron-hole-arraysHeatingstageFigureS1.Schematicfortheimpactingtest.Whentest,thesiliconwaferwithmicro-holearrayswasfirstlygluedontheheatingstagebythermalgrease.Then,startingupthecalefactioncircuitanditssurfacetemperaturewasmaintainedat110℃.6μLabsoluteethanolwillgoingintofreefallfrom10mmafterextrudedinquantitativeinfusiondeviceandsubsequentlycrashingintothemicron-hole-arrayssurface.Thebouncingprocesseswereobservedbystereomicroscopeandrecordedbytime-lapsephotography.S7

7SupplementaryFigureS20ms3.2ms15.6ms17.8ms36.2ms39.2ms51.2ms54.0ms63.4ms67.8ms79.6ms82.0ms91.4ms95.8ms104.8ms108.8ms122.0ms127.0ms135.0ms143.2ms158.2ms161.8ms168.2ms178.8ms1mmFigureS2.OverallprocessofcontinuouslyreboundingonMHAssurfaceatTs=110C.Selectedsnapshotsofhigh-speedcamerashowthecontinuoustouching,spreading,shrinkingandreboundingbehaviorofethanoldropletonMHAssurfaceatTs=110C.(MoredetailsseeMovieS1)S8

8SupplementaryFigureS30ms3.2ms15.6ms17.8ms45°2mmFigureS3.EthanoldropletimpactingonplanesurfaceatTs=110C.Selectedsnapshotsofhigh-speedcamerashowthetouching,spreadingandshrinkingbehaviorofethanoldropletonpolishedsiliconwaferatatTs=110C.Afterstabilization,wecouldmeasuretheintrinsiccontactangleofethanoldropletonsiliconwaferisabout45°.(MoredetailsseeMovieS1)S9

9SupplementaryFigureS4a6.8ms43.6ms71.6ms100.0ms130.0ms168.4msbEthanolVapourMicrobubbleMicrobubbleextensionformationconnectionVapourSiliconFigureS4.Bubbleformationmechanismbycontact-boilingregime.a)Selectedsnapshotsofhigh-speedcameraperformthatvaporbubblesexistwidelyinsolid-liquidinterfaceineachcyclewhentheethanoldropletcontactswiththeMHAssurfaceatTs=110C.b)Schematicoftheprocessofformation,extensionandconnectionofvaporbubbleonMHAssurface.S10

10SupplementaryFigureS5112)aPk110,P(erus107.95kPase108rpruopa106V104-1.0-0.50.00.51.0Reboundheightdifference(h,mm)FigureS5.MotioncharacteristicofethanoldropletonMHAssurfaceatTs=110C.Lineardiagrambetweenvaporpressure(P)andreboundheightdifference(∆h)bytheoreticalcalculationusinganassumptionthatthemassandvolumelossesfromcollisionandevaporationarenegligible.Thereinto,thevaporpressuretomaintainthemomentumconservation(∆h=0)isabout107.95kPa.S11

11SupplementaryFigureS6DropletGasDropletGasDLsimplifyintoHHotMHAsFigureS6.SketchesoflatticeBoltzmannmodel.Adropletwithinitiallydiameterof100luwasreleasedandfallingtohotmicro-holearrayssurfacefromtheinitialheightof200lu,andbasicparametersofdiameter(D),separationdistance(L)andheight(H)weresetto3,3and5lu,respectively.S12

12SupplementaryFigureS720060005000150)2lu4000()eu100m(ltluh3000ovigetH50lep2000orD01000-500012345674Time(10ts)FigureS7.Motionparametersofethanoldropletinthelaterperiod.Thedecaybehaviorofthecentroidheightandvolumeofethanoldropletwerepredictedinthelaterperiodofcontinuousreboundingonhotmicro-holearrayssurfacebyLBsimulation.S13

13SupplementaryFigureS8concentriccirclesDropletC1l1C1l2l1FigureS8.SchematicillustrationofthedecreasingdirectionoftheLaplacepressure.WedesignasimplemathematicalmodeltounderstandthatthedecreasingdirectionoftheLaplacepressureisconsistentwiththeradiusincreasingdirectionontheconcentricmicroridgessurface.Firstly,wecouldobtainarclengthl1withchordlengthofC1bysuperimposingtwoequalradiicirclesontheleftofthecontactprojectionbetweendropletandconcentricmicroridgessurface.Here,weassumethatthedropletwasfullyplacedontherightofthecenteroftheconcentriccircles.Then,wecanfindauniqueequalchordlengthofC1ontherightofthecontactprojection.Thus,wecanacquirearclengthl2byincreasetheradiusuntiltheintersectingchordlengthisC1.Accordingtothenegativecorrelationbetweenradiusandarclengthundertheequallengthchordcondition,wecandrawtheconclusionthatarclengthl1isgreaterthanarclengthl2.Basedonthis,wecancrudeextrapolatethatthereversethrustfromvaporpressureontheleftofthecontactprojectionisgreaterthantherightofoneduetothepositivecorrelationbetweenpressureandcontactarea.Therefore,theradiusincreasingdirectionoftheconcentriccirclesisthedecreasingdirectionoftheLaplacepressure.S14

14SupplementaryFigureS90ms10ms20ms30ms40ms50ms60ms70ms80ms90msTs=25℃Ts=80℃Ts=90℃Ts=100℃Ts=110℃Ts=120℃FigureS9.ThemigrationpatternsofwaterdropletonCMRssurfaceatTs=25-120C.Scalebar,2mm.S15

15SupplementaryFigureS1016EthanolonCMRsWateronCMRsEthanolontraditionalratchets12)/sWateronPlanarratchetsm8c(ThisworkWaterontraditionalratchetsv4DryiceontraditionalratchetsR134aontraditionalratchets00100200300400T(℃)FigureS10.Comparisonoftheterminalvelocity(v)andthetemperaturedifference(ΔT)ondifferentsurfaces.ΔTwasdefinedasthedifferencebetweensurfacetemperature(Ts)andliquidboilingpoint(Tp).Thedataonthetraditionalandplanarratchetswerereferredfrombelowreferences.[1]LinkeH.;Alema´nB.J.;MellingL.D.;TaorminaM.J.;FrancisM.J.;Dow-HygelundC.C.;NarayananV.;TaylorR.P.;StoutA.Self-propelledLeidenfrostdroplets.PhysicalReviewLetters2006,96,154502.[2]LagubeauG.;MerrerM.;ClanetC.;QuéréD.Leidenfrostonaratchet.NaturePhysics2011,7,395-398.[3]LiJ.;ZhouX.;ZhangY.;HaoC.;ZhaoF.;LiM.;TangH.;YeW.;WangZ.RectificationofMobileLeidenfrostDropletsbyPlanarRatchets.Small2019,1901751.S16

16LegendsforMoviesMovieS1.ThemovementpatternsofethanoldropletonMHAsandplanesurfacesatTs=110C.MovieS2.Thecontinuousdropletreboundingbehavioranditspressureandtemperatureevolutionsonheatedmiro-holearrayssurfacebyLBmodel.MovieS3.ThecontinuousreboundingbehaviorofethanoldropletonMHAs-BandNHAssurfacesatTs=110C.MovieS4.ThemovementpatternsofethanoldropletonCMRssurfaceatdifferentTs.MovieS5.ThedirectionalmovementofwaterdropletonCMRssurfaceatdifferentTs.MovieS6.WaterdropletwasalwaystransportedalongthedirectionoftheincreaseinradiusonCMRssurfaceatTs=180C.MovieS7.ThemovementpatternofwaterdropletonCMRssurfaceatTs=180C.S17