Numerical simulation of spreading drops

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Legendre, Dominique and Maglio, Marco

Numerical simulation of

spreading drops.

(2013) Colloids and Surfaces A: Physicochemical

and Engineering Aspects, vol. 432 . pp. 29-37. ISSN 0927-7757

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Numerical

simulation

of

spreading

drops

Dominique

Legendre

,

Marco

Maglio

InstitutdeMécaniquedesFluidesdeToulouse(IMFT),UniversitédeToulouse,CNRS-INPT-UPS,France

g

r

a

p

h

i

c

a

l

a

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•Numerialsimulationofspreadingdropsonawettablesurface.

•Wediscusstheeffectofthemainparameters(density,viscosity,surfacetension,dropradiusandwettability).

•Powerlawevolutionsobservedinexperiments(t1_/2,t1_{/10)arerecoveredanddiscussed.}

•Thecombinedeffectofviscosityandwettabilityhasasignificanteffectonthespreading.

Keywords: Dropspreading Numericalsimulation

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Weconsideraliquiddropthatspreadsonawettablesurface.Differenttimeevolutionshavebeen observedforthebaseradiusrdependingoftherelativeroleplayedbyinertia,viscosity,surface ten-sionandthewettingcondition.Numericalsimulationswereperformedtodiscusstherelativeeffectof theseparametersonthespreadingdescribedbytheevolutionofthebaseradiusr(t)andthespreading timetS.Differentpowerlawevolutionsr(t)∝tnhavebeenobservedwhenvaryingtheparameters.Atthe

earlystageofthespreading,thepowerlawt1/2_{(n=1/2)isobservedaslongascapillarityisbalancedby}

inertiaatthecontactline.Whenincreasingtheviscositycontribution,theexponentnisfoundtoincrease despitetheincreaseofthespreadingtime.Theeffectofthesurfacewettabilityisobservedforliquids moreviscousthanwater.Forasmallcontactangle,thepowerlawt1/2_{isthenfollowedbythefamous}

Tannerlawt1/10_{oncethedropshapehasreachedasphericalcap.}

1. Introduction

When a liquid drop contacts a wettable surface, the liquid spreadsoverthesolidtominimizethetotalsurfaceenergy.The firstmomentsofspreadingtendtoberapid(somemilli-seconds for millimetersized waterdrops) whileit takes a muchlonger timeforviscousliquidstocompletelyspread.Fromtheliterature

[22,17,9,1,6,3,2,24,7],itappearsthatseveraltimeevolutionscanbe observedforthebaseradiusdependingifinertia,viscosityand/or

∗ Correspondingauthors.Tel.:+33534322818;fax:+33534322991. E-mailaddress:legendre@imft.fr(D.Legendre).

gravityareinvolved.Inaddition,theeffectofthesubstrate wett-abilityondropspreadingisnotyetclearlyunderstood.

Theevolutionofthewettedsurfaceisreportedusingeitherthe baseradiusr(t)orthewettedsurfaceA(t)=r2_(t).The

experimen-talresultsareusuallypresentedinordertoprovideapowerlaw evolutionundertheform:

r(t)∼tn_,orA(t)∼t2n_{, (1)}

Themostpopularlawfordropspreadingiscertainlytheso-called Tanner law[23,22] withn=1/10. Thisevolution can be simply obtainedbyconsideringthattheshapeofthedropisathin spheri-calcapandthatthecontactanglefollowstheCox-Voinovrelation. Suchevolution hasbeenfoundtobeinagreementwithseveral

experiments(see[3]).TheTannerlawconcernsspreading situa-tionsinwhichsurfacetensionisbalancedbyviscosity.Varyingthe relativedifferentcontributions(inertia,gravity,viscosity,surface tension)differentevolutionshavebeenreported:n=0.1,n=0.14, n=0.16,n=1/8,n=1/7,n=1/2[17,9,3].Thet1/2_{powerlawresults}

fromabalancebetweenthecapillarypressureandtheinertial pres-sure.Such evolution hasbeenobserved in severalexperiments

[1,2,24].LaviandMarmur[15]performedexperimentsfor differ-entliquidsonacoatedsiliconwaferandreportedthefollowing exponentialevolution: A(t) Af = 1−exp





whereAfisthefinalvalueofthewettedareaandisthe

dimen-sionlesstime=t/V1/3_{.LaviandMarmur[15]reportsvaluesfor}

mintherangem=0.618−0.964.Theinterestofthisrelationisto proposeapowerlawoftheformtm_{inthelimitoflargefinalwetted}

surfaceAf.Asitwillbeshowninthesimulationsreportedinthis

paper,theviscosityhasanimportanteffectonthedrop spread-ing.Forlowviscousliquids(aswater),apowerlawoftheform(1)

canbeobservedduringalmostalltheevolution.Whenincreasing theviscosity,theslopeoftheevolutionchangesovertime, show-inganexponentialdependencyoftheform(2).Thewettabilityhas alsoanimpactonthedropspreading.Thedistinctionbetweenthe differentwettingstatesisusuallymadebyconsideringthe equi-libriumspreadingcoefficientSeq=(cosS−1),whichrepresents

thesurfacefreeenergyrelativetoitsvalueforcompletewetting. Whenadropisdepositedona planesurface,theinitialcontact anglei≈180◦ madebytheinterfacewiththewallislargerthan

equilibriumS.AninitialspreadingcoefficientSi=(cosi−1)can

beintroducedtocharacterizetheinitialenergyofthesystem.The differenceS=Seq−Si=cosS−cosiindicatesthenatureofthe

spreading.SinceS<0,thedropspreadsuntilthecontactangle stabilizesattheequilibrium angleS.Neitherdifferent

theoret-icalmodelsnorexperimentalobservationshave showna direct effectofS=cosS−cosionthedropspreading.Furthermore,the

impactofthesurfacewettabilityonthedropspreadingisstillunder controversy.Bianceetal.[1]observedthescalinglawr(t)_∼t1/2_in

theirexperimentsofwaterdropsspreadingonacompletely wet-tingsurface.Birdetal.[2]performedexperimentswithdroplets ofdifferentwater-glycerolmixtures(1.0,3.7,and10.7cP)on sili-conwafersurfacesofvariablewettability(fromS=0◦toS=180◦).

Thespreadingradiusisfoundtofollowapower-lawevolutionof theform(1)wheretheexponentndependsontheequilibrium contactangle.Typicallyitvariesfromn=1/2forperfectwetting surfaceston=1/4fornonwettingsurfaces.Chenetal.[7]observed spreadingofwaterdroponhydrophobicglasswithn=0.3.Onthe contrary,therecentexperimentsofWinkelsetal.[24]pointtoward aspreadingregimethatisindependentofwettabilitywithn=1/2. Theexperiments weremade withwater dropsonthree differ-entsubstrateswithdifferentequilibriumcontactanglesS:clean

glass(almostperfectlywetting,S∼0◦),coatedglass(S∼65◦),and

Teflon-coatedglass(S∼115◦).

Theaimofthisworkistodiscusstheeffectofthemain param-eters controllingthe drop spreading:viscosity, density, surface tension,dropvolumeandsurfacewettability.Thetimeevolution r(t)ofthebaseradiusisanalyzedaswellasthetimetSforthe

com-pletespreadingofthedrop.Eachparameterhasbeenvariedina largerangewhilekeepingconstanttheotherparameters.

2. Numericalprocedure

2.1. NavierStokessolver

Thenumericalsimulationsreportedinthisworkareperformed withthevolumeoffluid(VoF)solverdevelopedintheJADIMcode

[4,12].Theone-fluidsystemofequationsisobtainedbyintroducing theone-fluidfunctionCusedtolocalizeoneofthetwophases.In thisstudy,wedefineCasC=1inthedrop(phase1),andC=0inthe surroundingair(phase2).Theone-fluidfunctionCmakespossible thedefinitionoftheonefluidvariablesU=CU1+(1−C)U2forthe

velocity,P=CP1+(1−C)P2 forthepressure,=C1+(1−C)2 for

thedensityand=C1+(1−C)2fortheviscosity.Theposition

oftheinterfaceisthengivenbythetransportequation:

∂

∂

∇

The two fluids areassumed tobe Newtonian and incompress-iblewithnophasechange.Underisothermalconditionandinthe absenceofanysurfactant,thesurfacetensionisconstantand uni-format theinterfacebetweenthetwo fluids.In suchcase,the velocityfieldUandthepressurePsatisfytheclassicalone-fluid formulationoftheNavier–Stokesequations:

∇



∂

∂

∇



∇

∇

where istheviscous stresstensor, gis thegravity.F isthe

capillarycontribution:

F=

∇

whereisthesurfacetension,ndenotesbyarbitrarychoicethe unitnormaloftheinterfacegoingoutfromthedropandıIisthe

Diracdistributionassociatedtotheinterface.

ThesystemofEqs.(3)–(6)isdiscretizedusingthefinite vol-umemethod.Timeadvancementisachievedthroughathird-order Runge-Kuttamethodfortheviscousstresses.Incompressibilityis satisfiedattheendofeachtimestepthoughaprojectionmethod. Theoverallalgorithmissecond-orderaccurateinbothtimeand space. The volume fraction C and the pressure P are volume-centeredandthevelocitycomponentsareface-centered.Dueto thediscretizationofC,itresultsanumericalthicknessofthe inter-face,thecellscutbytheinterfacecorrespondingto0<C<1.The specificaspectofourapproachcomparedtotheclassicalVoFor LevelSetmethods[19–21]concernsthetechniqueusedtocontrol thestiffnessoftheinterface.Inourapproachnointerface recon-structionorredistancingtechniquesareintroduced.Theinterface locationandstiffnessarebothcontrolledbyanaccuratetransport algorithmbasedonFCT(Flux-Corrected-Transport)schemes[25]. Thismethodleadstoaninterfacethicknessofaboutthreegridcells duetotheimplementationofaspecificprocedureforthevelocity usedtotransportC,inflowregionofstrongstrainandshear[4].The numericaldescriptionofthesurfacetensionisoneofthecrucial pointswhenconsideringsystemswherecapillaryeffectscontrol theinterfaceshape.Thisinterfacialforceissolvedusingtheclassical CSF(continuumsurfaceforce)modeldevelopedby[5].Aclassical problemofthisformulationisthegenerationofspuriouscurrents

[13,18].Inordertodecreasespuriouscurrentsintensity,aclassical methodintroducedby[5]wasemployed,whichconsistsin cal-culatingthesurfacecurvaturefromasmootheddensitygradient, whilethediscretizationofthedeltafunctionusesanun-smoothed density.ThespuriouscurrentsinourcodeJADIMhavebeen char-acterizedby[12].Theirmaximummagnitudeevolvesas0.004/, inagreementwithothercodesusingtheBrackbill’sformulation. 2.2. Numericalmodelingofthecontactangle

Themodeling usedinthis workaims torespectthephysics atthecontactline.Itisbasedontwomainconsiderationsatthe macroscopicscale:

Fig.1.Statementoftheproblem.Initialandfinaldropshapecharacterizedbythe baseradiusr0andrf,respectively.TheradialandverticalsizeareLr=Lz=3R0.his

theinitialdistancebetweenthedropcenterandthewall.

-Thewallconditionseenbythefluidisanonslipcondition. -There is a direct relation between the interfacevelocity and

theshapeoftheinterfacecharacterizedbytheapparentcontact angle.

TheobjectiveistosolvethephysicattheapparentscaleLwhere thecontactangleistheapparentordynamiccontactangle.Atthis scalethefluidobeysanonslipconditionfortheresolutionofthe momentumequation:

UW =0 (7)

HydrodynamicsatasmallerscalethanLhasanimportantrolesince itlinkstheapparentangletothestaticangle.Asub-gridmodeling ofsucheffectneedstobeintroducedinthesimulation.Thisisdone byusingtherelationderivedbyCox[8]inordertolinktheapparent angletothestaticangle:

g(W)=g(S)+CaLn





(8) whereCa=Ucl/isthecontactlinecapillarynumber.Thefirst

Vol-umeofFluidnodebeinglocatedat/2fromthewall,itfollowsthat L=/2.Consideringthevalueoftheobservedsliplength[14],we haveusedafixedvalue=10−9mforthesliplengthin(4)inthe simulationsreportedinthiswork.Notethatexperimental observa-tionsgivealsoarecommendationforthechoiceofthemacroscopic lengthLconsistentwiththedynamicmodelusedinournumerical modeling.Indeedoneshouldhave/2=L≈10−4−10−5m.

Thecomparisonbetweenourmodelingwithothervariant mod-elings,i.e.staticcontactangleversusdynamiccontactangle,noslip conditionversusslipcondition,ispresentedin[16].Agridandtime convergencehasbeenperformedanddiscussedforthedifferent possiblemodels.Thesecomparisonsclearlystressthestrongeffect ofthenumericalmodelingusedforthesimulationofdynamic con-tactlinesasillustratedinthenextsection,wherethesimulations arecomparedwithexperimentsforwaterandmoreviscousliquids. 2.3. Comparisonwithexperimentsforwaterandviscousdrops

We consider a spherical droplet of volume V=4R3 0/3

depositedonanhorizontalwallthatentersincontactwithan hori-zontalwettablesurfaceattimet=0.Allthenumericalexperiments reportedinthisworkaredescribedbyFig.1.Thecomputational domainisasquaredomainofequalradialandverticalextension

10−2 10−1 100 101 102 103 0 0.2 0.4 0.6 0.8 1 τ r / r f

Fig.2.Comparisonwiththeexperimentsforviscousdrop[15].Thenormalized radiusr/rfisreportedversusthenormalizedtime=t/V1/3.Contactanglemodels

usedforthesimulations:—Dynamicmodel,−−−staticmodel

Lr=Lz=3R0.Theaxisofsymmetrycorrespondstothewest

bound-ary,thewettingconditionsgivenbyeq.(7-4)areimposedonthe southboundaryandclassicalwallconditions(noslip)areimposed onthetwootherboundaries(eastandnorth).Theinitial condi-tioncorrespondstoadropinitiallylocatedatthedistanceh≤R0

fromthewall. Theeffectofthis initialconditionas wellasthe gridandtimeconvergenceshavebeenperformed[16]by consid-eringdifferentinitiallocationsofthedropcenter(h=0.95,0.98, 0.99,0.995,0.998R0),differentgrids(correspondingto32,64,128

and167nodesperradius)anddifferenttimesteps(t=10−5 to t=10−8s).

Squalaneandwaterdropsareconsideredforcomparisonwith theexperimentsofLavi&Marmur [15]Winkelsetal.and[24], respectively.

We first consider the experiments of [15]. They have con-sidered the spreading of squalane drops with R0=1mm. The

fluidpropertiesare=809kg/m3_{,=0.034Pasand=0.032N/m.}

Thecomparisonbetweenoursimulationsandtheexperimentsis showninFig.2.Thenormalizedradiusr/rfisreportedversusthe

normalizedtime=t/V1/3_{.Thedynamicmodeliscomparedwith}

thestaticmodel(aconstantcontactangleSbeingimposed

dur-ingallthesimulation).Asignificantdifferenceisobservedbetween thedynamicmodel(4)andthestaticmodel(W=S).Thedynamic

modelsgiveasatisfactoryagreementwiththeexperimentsof[15]. Wenowconsiderthespreadingofwaterdropsandwe com-pare our simulations with the experiments performed by [24]

formillimeter-sizedwaterdropletsinair(R0=0.5mm).Wehave

selectedthecasescorrespondingtoverydifferentcontactangles S=115◦andS≈0◦.

Basedonpreliminarytests[16],thecomparisonwiththe exper-imentsperformedby[24]areperformedwithagridcorresponding to128nodesperradiusR0andatimestept=10−7s.Theinitial

positionish=0.998R0.ThecomparisonisshowninFig.3wherethe

normalizedcontactlineradialpositionr/R0isshownasfunctionof

thetimenormalizedbythecapillary-inertialtimet=(R3₀/)1/2.

Theagreementisverysatisfactoryforthetwocasesconsidered.In particular,thepowerlawt1/2_{isclearlyreproducedbythe}

simu-lations.Consequently,oursimulationsdonotshowanoticeable effect ofthesurface wettabilityonthe powerlawevolution in agreementwith[24]butinoppositionwith[2].Below,we con-siderindetailthispointwithadditionalsimulationsperformedfor afluidtentimesmoreviscousthanwater.Notealsothatour sim-ulationrevealstheoscillationofthecontactlineattheendofthe

32 D.Legendre,M.Maglio/ColloidsandSurfacesA:Physicochem.Eng.Aspects432 (2013) 29–37 10−3 10−2 10−1 100 101 10−2 10−1 100 101 1 2 t / t_ρ r / R0

Fig.3.Evolutionofthecontactlineradialpositionr/R0asfunctionofthe

nor-malizedtimet/(R3 0/)

1/2

.Experiments[24]:()S=115o,(♦)S≈0◦.Numerical

simulations:−−−S=115◦,—S=5◦.

spreading.Suchoscillationarenotreportedintheexperimentsof

[24]andwillbediscussedinthefollowing.

3. Results

Weconsidertheeffectofthemainparameterscontrollingdrop spreading: fluid properties (liquid density, liquid viscosity and surfacetension),thedropsizeandthewettingpropertiesofthe sur-face.Eachparameterisvariedindependentlyoverawiderangeof variation.Allthesimulationshavebeenperformedconsideringair asthesurroundingfluid.Asaconsequencethedensityandviscosity ratiovalues(i.e.2/1and2/1)arenotconstantbuttheyalways

remainmuchsmallerthanunitysothatthereisnoexpectedeffect ofboththeviscosityandthedensityoftheaironthespreading. 3.1. Inertial-capillaryregimeofspreading

Wefirstconsiderlow-viscousspreading.Thedensity,the sur-facetensionandthedropradiusarevariedindependentlywhile theviscosityis setfixedtotheviscosityofwater =0.001Pas.

Fig. 4 shows the evolution of the contact line radius when varying the density from =100kg/m3 _{to =4000kg/m}3 _with

theotherparameterskeptconstant:=0.072N/m,=0.001Pas, R0=0.5mmandS=65◦.Whenthedensityisincreasedthe

spread-ingtimegrows(thecurvesaretranslatedtotheright)sincethe inertiaoftheliquidtobeacceleratedbythemovingcontactline ismore important. We notice that thesame powerlaw evolu-tiont1/2_{isobservedforalltherange ofdensityconsidered.The}

t1/2 _{powerlaw resultsfrom acapillary-inertial balance.Indeed,}

whenthesphericaldropofradiusR0contactsthesolid,thebalance

betweenthecapillarypressure∼R0/r2andtheinertialpressure

∼(dr/dt)2_{givesafterintegrationthefollowingevolution:}

r(t)∼







t1/2 ₍₉₎

Fig.4(bottom)confirmsthatthedropspreadingiscontrolled byabalancebetweencapillary andinertia,sincetheuseofthe timenormalizedusingthecapillary-inertiacharacteristictimet=

(R3 0/)

1/2

makespossibleagoodcollapseofthedifferentcurves.

Fig.4alsorevealscontactlineoscillationsatboththeearlyand thefinalstageofspreading.IndeedasshowninFig.5,theinterface issignificantlydeformedbythepropagationof capillarywaves.

Fig. 4.Effect of the liquid densityon the dropspreading for=0.072N/m, =0.001Pas, S=65◦ and R0=0.5mm. The normalized contact line radius

r*_=(r−r

0)/(rf−r0)isreportedasafunctionof(top)thedimensionaltimetand

(bottom)thenormalizedtimet/(R3 0/)

1/2

.

Theappearanceofacapillarywavethattravelsfromthecontact linetothetopofthedroplethasalsobeenreportedinthe simula-tionsreportedby[11]usingadiffuseinterfacemethod.Thisfigure reportstheevolutionoftheshapeofawaterdropforS=65◦.The

sequenceispresentedlinebylineforthetimesgiveninthelegend. Thetimestepsarenonregularinordertoshowparticularevents observedduringthespreading.Atthebeginningofthespreading, onlythecontactlineismoving.Thetopofthedropstartstomove duetothecapillarywaveinducedbytheinitialimpulsegenerated atthecontactline.Surprisinglythiscapillarywaveinducesthe ele-vationofthetopofthedrop.Asaconsequence,combinedwiththe displacementofthecontactline,thedropiselongatedintotwo orthogonaldirectionsresultinginthegenerationofapinchevent (see2ndimage(t=2.52ms),line3).Inthiscase,thepinch-offdoes notoccursresultinginthepropagationofarelativelylarge ampli-tudewave.Thiscapillarywavedisplacesagainthetopofthedrop inthedirectionoppositetothewall.Thisgeneratesasecondpinch thatresultsintheejectionofasmallsatellitedroplet(see2ndimage (t=3.34ms),line5).Then, thesatellitedropletimpactsand coa-lesceswiththedropgeneratingagainwaveoscillations,explaining thefinaloscillationsobservedforthecontactlineinFig.4.Such regimeofdropletejectionfollowingthedepositionofdropsonto

Fig.5. Shapeevolutionofawaterdrop(radialsection)forS=65◦.Thesequenceisshownlinebylineforthetimest=0.06,0.36,0.56,1.00,1.40,1.66,1.86,2.52,2.70,2.86,

2.98,3.12,3.28,3.34,3.52,3.60,4.80and5.80ms.

a solid substratehasbeen identified inexperiments and direct numericalsimulationsby[10]forsmallercontactangles(S≤65◦).

Weshowherethatthedropletejectioncanalsobeobservedfor largercontactangles.

WeillustrateinFig.6theeffectofthedensityonthedroplet ejec-tion.Thesimulationsareshownforthreedensities=700kg/m3_,

=1000kg/m3 _{and=4000kg/m}3_{.Theimagescorrespondingto}

thetwosuccessivepinch-offsandtothemaximumheightofthe satellitedropletareshown.For=700kg/m3_{twopincheventsare}

observedbutnodropletejectionisgeneratedbythesurface oscil-lation,whilefor=4000kg/m3_{thedropletisejectedatthefirst}

pinchevent.

The effect of thesurface tensionis now reported in Fig. 7. The surfacetensionis variedfrom=0.007N/m to=0.2N/m. The other parameters are fixed to =103_kg/m3_{, =0.001Pas,}

R0=0.5mmandS=65◦.Thespreading timeis decreasedwhen

increasingthesurfacetension,sincetheinitialimpulse transmit-tedtothedropisincreased.Wealsoobservethatthepowerlaw

Fig.6.Shapeevolutionofawaterdrop(radialsection)showingthepincheventsforS=65◦.Firstline:=700kg/m3–fromlefttoright:t=2.16,2.64ms.Secondline:

=1000kg/m3_{–fromlefttoright:t=2.52,3.28,3.34ms.Thirdline:=4000kg/m}3_{–fromlefttoright:t=4.56,4.94ms.}

evolutiont1/2_{isnotsignificantlyaffectedbythechangeofthe}

sur-facetension. Againtheuseofthenormalizedtime t/(R3 0/)

1/2

makespossiblethecollapseofthecurvesandclearlyrevealsthe capillaryoscillationsthatimpactthecontactlinespreading.

WereportinFig.8thespreadingtimetSforthecontactline

evo-lutionshowninFigs.4and7.Thespreadingtimesdeducedfrom simulationsforwaterdropswithradiiR0=0.5mmandR0=5mm

arealsoreported.tSisshownasafunctionofthecharacteristic

capillary-inertialtimet=(R3₀/)1/2.tSisdefinedasthetime

nec-essarytoreach98%ofthefinalequilibriumradiusrfofthecontact

line.Theplotconfirmsthatspreadingisclearlycontrolledbythe balancebetweeninertiaandcapillarity.Thespreadingtimecanbe estimatedas: tS≈1.7





WereportinFig.9theeffectoftheviscosityonthedrop spread-ing.Theviscosityisvariedfrom=0.001Pasto=0.07Pasfor =103_kg/m3_{,=0.072N/m,R}

0=0.5mmandS=65◦.The

spread-ingtimeisenlargedwhenincreasingtheviscosity(thecurvesbeing translatedtotheright)andtheoscillationsobservedattheendof thespreadingduetothepropagationofcapillarywavearedamped. Despitetheincreaseofthespreadingtime,theslopeofthe evolu-tionattheearlystageofthespreadingincreasesfromt1/2_(water)

tot2/3_{forthelargestviscosityconsidered.Butthedurationofthe}

powerlawregimeisreducedbecausetheendofthespreadingis characterizedbyanexponentialevolutionoftheformgivenby(2). Forcomparison,theTannerlawt1/10_{isalsoshowninthefigure.}

Thereisnoclearevidencethatthistypeofevolutionisobservedin oursimulationsduringthefinalstageofthespreading.Infact,the staticcontactangleS=65◦istoolargetoobservesucharegime.

Forliquidsmoreviscousthanwater,weexpectthattheevolutionis controlledbyaviscous–capillarybalance.Theviscousstressnear thecontact linecanbeestimatedas∼(dr/dt)/where isthe characteristiclengthcontrollingtheshearstressatthecontactline. Assuminganevolutionoftheform=R1−p₀ rp_{,thebalancebetween}

theviscousstressandthecapillarypressure∼R0/r2 gives,after

integration,theobservedt2/3_evolution:

r(t)∼



for p=3/2 corresponding to ∼R−1/20 r3/2. The corresponding

viscous–capillarycharacteristictime t=R0/ is usedinFig.4

(bottom)tonormalizethetime.Foraviscositylargerthan0.01Pas, asatisfactorycollapseofthecurvesisobservedindicatingthe tran-sitiontoaspreading regimecontrolledbythebalancebetween capillarityandviscosity.

TheevolutionofthespreadingtimetSisreportedinFig.10as

afunctionoftheviscous–capillarycharacteristictimet=R0/.

Twodifferentbehaviorsare clearlyobserved. For low viscosity, tSisfoundtoevolveast1/5(i.e.1/5sincealltheother

param-etersarenotvarying).We havenoexplanationoftheparticular evolution tS∝1/5 revealed by the simulation for low-viscous

drops. Forlargerviscositiesan expectedlinearevolution tS∝t

isobserved.Thelinearbehaviorresultsfromthelinearityofthe governingequations(Stokesequation)atlowReynoldsnumber. Theviscous–capillaryspreadingtimecanbeestimatedusingthe relation

tS≈55

R0

Fig.7. Effectofthe surfacetension onthedropspreading for =103_kg/m3_,

=0.001Pas, R0=0.5mm and S=65◦. The normalized contact line radius

r*_=(r−r

0)/(rf−r0)isreportedasafunctionof(top)thedimensionaltimetand

(bottom)thenormalizedtimet/(R3 0/)

1/2

.

3.3. Contactangleeffect

WefinallydiscusstheeffectofthestaticcontactangleSon

dropspreading.Figs.3,4and7showthatthet1/2_{powerlawisnot}

affectedbythesurfacewettabilityatthebeginningofthe spread-ing,whenconsideringafluidhavingthesameviscosityaswater foralargerangeofcontactangles(S=5◦,65◦and115◦),in

agree-mentwiththeexperimentsreportedby[24]forwaterdrops.The wettingconditionsconsideredby[24](S=5◦,65◦ and115◦)are

nowsimulatedforafluidwithaviscositytentimeslargerthanthat ofwater(=0.01Pas),theotherparametersbeingkeptthesame asforwater(=0.072N/m,=103_kg/m3_)andR

0=0.5mm.

Addi-tionalwettingconditions(S=140◦andS=160◦)havealsobeen

considered.

Fig.11clearlyrevealsthatthewettingconditionhasnowa sig-nificanteffectonthespreadingrate.Whenthecontactangleis increasedtheslopeseemstodecreaseinagreementwiththe exper-imentsof[2].However,iftheradiusrisnormalizedusingtheradius reachedattheendoftheinertial-capillaryregime,allthecurves collapseonasinglecurveshowingat1/2_{evolutionindependenton}

thecontactangleSinagreementwiththeexperimentsof[24].

Fig.8.SpreadingtimetSversusthecapillary-inertialtimet=(R30/) 1/2_when

varyingtheliquiddensity,thesurfacetensionandthedropradius.(◦)is vary-ingfrom100kg/m3_to4000kg/m3_{for=0.001Pas,=0.072N/m,R}

0=0.5mmand

S=65◦.()isvaryingfrom0.007N/mto0.2N/mfor=103kg/m3,=0.001Pas,

R0=0.5mmandS=65◦.( )R0isvaryingfrom0.5mmto5mmfor=103kg/m3,

=0.001Pas,=0.072N/mandS=65◦.Thecontinuouslinerepresentstherelation

tS=1.7t

Fig.9. Effectof theliquidviscosityonthe dropspreadingfor =0.072N/m, =103_kg/m3_,R

0=0.5mmandS=65◦.Thenormalizedradiusr*=(r−r0)/(rf−r0)is

Fig.10.SpreadingtimetSversustheviscous–capillarycharacteristictimet=R0/

for=0.072N/m,=103_kg/m3_,R

0=0.5mmandS=65◦.(—)tS∝t1/5,(−−−)

tS∝t.

Fig.11.EffectofthecontactanglesSonthetimeevolutionofthedropbaseradius

rfor=0.01Pas,=0.072N/m,=103_kg/m3_andR

0=0.5mm.Thedropradial

sec-tionscorrespondingtoa,bandcareshowninFig.12.

ForthesmallestcontactangleS=5◦,weobserveafterthet1/2

evolutionacleart1/10_{evolutioncharacteristicoftheTannerLaw.A}

similarbehaviorhasbeenreportedinthesimulationsof[11]where aregimeclosetotheTanner’slawisobservedwhenincreasing theviscositywhilearegimeclosetot1/2_{isobservedattheearly}

stageofthespreading.Fig.12showstheshapeofthedropduring

thet1/2_{evolution,duringthet}1/10_{evolutionandatthetransition}

betweenthesetworegimes.Thecorrespondingtimesaredepicted bybulletsinFig.11.Asphericalcapisclearlyidentifiedduringthe t1/10_evolution.

Inthelimitofaverythinsphericalcap,thecontactangleW,

thedropvolumeVandthecontactlineradiusrarelinkedbythe geometricalrelationW≈4V/r3.Inthelimitofasmalldynamic

contactanglewithS≈0,theCoxrelation(4)canbeexpressedas:

_W3 =9 

dr dtln(L/)

Combiningthesetworelationsandintegratingintime,theradius ofthecontactlineisfoundtofollowthefamousTannerlaw:

r(t)∼R0





Forspreadingdrops,suchevolutionisthusobservedforsmall contactanglesoncethedrophasreachedasphericalcapshape asshowninFig.12c.InthelimitofS→0,thebaseradiusofthe

sphericalcapcanbeexpressedasrf≈R0−1/3_S anditfollowsan

estimationofthespreadingtime:

tS≈t−10/3_S (14)

For the simulations reported in Fig. 11, relation (14) gives tS≈0.2swhichisacorrectorderofmagnitudeforS=5◦.The

corre-spondingviscous–capillaryspreadingtimegivenbyrelation(12)is twoordersofmagnitudesmaller≈0.0038s,butinagreementwith theevolutionreportedforthelargercontactangles(S=65◦,115◦,

140◦and160◦).

4. Conclusion

Wehaveconsideredthespreadingofadroponahorizontal wet-tablesurfacebynumericalsimulation.Inournumericalmodelingof movingcontactlines,theapparentcontactangleisgivenbythe[8]

relationwhileazerosliplengthisimposedforthemomentum cal-culation.Thesimulationshavebeensatisfactorilycomparedwith theexperimentsof[15]forsqualaneandwiththeexperimentsof

[24]forwater.Theeffectsofthemainparameterscontrollingthe dropspreadinghavebeenstudiedseparately.Forviscosityofthe orderofthatofwater,thet1/2_{powerlawhasbeenrecoveredat}

theearlystageofthespreadingforallthecontactangles consid-eredinagreementwiththeexperimentsof[24].Whenincreasing theviscosity,theearlystageofthespreading is still character-izedbya powerlawevolutionbutwithanincreasingexponent inagreementwiththeexperimentsof[2].Boththeviscosityand thecontactanglearethenshowntoimpactthelineexpansion.In particular,consideringthespreadingofaviscousdroponahighly wettablesubstrate(S=5◦),weobservetwosuccessivepowerlaw

evolutions.Thecapillary-inertialregimer∝t1/2_{isfollowedbythe}

Fig.12.Dropradialsectioncorrespondingtothetimesa,bandcshowninFig.11(S=5◦).(a)inthemiddleofthet1/2evolution(t=4×10−4s),(b)thetransitionbetween

Tannerr∝t1/10_{evolutiononcethedroplethasreacheda}

spheri-calcapshape.Oursimulationsshowthatthecombinedeffectof theviscosityandthecontactangleplaysanimportantroleinthe spreadingofadrop.Theymakeconsistentthedifferent experimen-talobservations(aprioricontradictory)concerningtheeffectofthe wettingconditions.

enddocument

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