We studied millimeter-size aqueous sessile drops in an ambient oil environment in a classical electrowetting conguration with a wire shaped electrode placed at a variable height above the substrate. Within a certain range of height and above a certain threshold voltage the drop oscillates periodically between two morphologies where it is either attached to the wire or detached from it. We determine the range of control parameters wire height and voltage in which oscillations occur and explain it by a simple capillary model. Furthermore, we analyze the dynamics of the oscillations using high speed video microscopy and numerical uid dynamics modeling. We develop a one-dimensional harmonic oscillator model that describes the dependence of the drop oscillations on the relevant intrinsic (surface tension, viscosity, density) and extrinsic (wire height, voltage) parameters.

Self-excited drop oscillations in electrowetting tenEghristophe fret hilips eserh vortories iindhovenD rofF rolstln RD STSTee iindE hoven @xvA 1 wihel wF tF heré hilips eserh vortories iindhovenD rofF rolstln RD STSTee iindE hoven @xvA prieder wugele niversity of wenteD y fox PIUD USHHei inshede @xvA his is the uthor version of the mnusript pulished s Self-excited drop oscillations in electrowettingD tg fret et lF vngmuir PHHU @doiForgGIHFIHPIGlHTPIRWfA 1 Introduction wiromnipultion of liquid droplets is the sis of digitl miro)uidi sysE tems in whih droplets re the elementry units tht rry iologil or hemil ontentF 1 o e funtionlD the miro)uidi system onsidered does not only hve to provide severl elementry hndling opertions on the unitD like produtionD tutionD splitting nd fusionY it should lso e dpted to the prtiulr iohemil onstrints of the liquid usedD redue rossEontmintion etween smples nd promote mixing t smll slesF emongst the vrious wys to tute liquid t the mirosleD eletrowetE ting 2,3 is prtiulrly promising tehniqueF sts e0ieny nd verstility hs een demonstrted y vrious groupsD not only for funtionl lEonEEhip devies 1,49 ut lso for tunle lensesD 10,11 displysD 12 nd optil swithesF 13 hespite the lrge numer of pplitionsD modeling of eletrowetting hs so fr minly een restrited to stti situtions E exept for few reent ontriE utionsF sn the present pper we disuss the ehvior of drops in lssil eletrowetting on(gurtion with wire shped eletrode held t vrile distne d ove the sustrteF he eletrode is rely immersed into the dropD suh tht it dethes from the wire upon pplying su0iently high voltgeF efter the dethmentD no more voltge is pplied to the drop nd hene it relxes k towrds its shpe t zero voltgeF pon doing soD it retthes to the eletrodes whereupon it spreds ginD therey strting yle of periodi drop osilltionsF he present work is n extension of our previous work on osillting pillry ridgesF 14,15 sn previous ommuniE tionD we showed tht the relxtion of the drop towrds its zero voltge shpe P n e ontrolled either eletrilly if the drop retins (nite hrge fter the dethmentD or purely hydrodynmilly if it is dishrged during the pinh o' from the wireF 16 hile the stti stility digrm to e desried elow pplies in oth sesD for the dynmi prt we will fous in the present pper on hydrodynmilly ontrolled osilltionsF he pper is orgnized s followsX fter rief introdution into the sis of eletrowetting t the end of this setionD we desrie the experimentl setup in setion ssF usequentlyD we present the experimentl dt nd simple stti pillry model to desrie the rnge of stility nd instility of oth the tthed nd the dethed morphology in the prmeter spe spnned y the sustrteEwire distne d nd the pplied voltge U @setion sssAF sn setion s we nlyze the dynmi ehvior of the osillting drops y ompring high speed video mirosopy imges nd numeril @volumeE ofE)uidA simultionsF sn the disussion setion @AD we mp the dynmis of the osillting drops onto oneEdimensionl dmped hrmoni osilltorF his model turns out to ontin ll the relevnt ingredients to understnd the si physis of the osilltionsF vet us now rie)y review the sis of eletrowettingF hen drop is deE posited on surfe the energies of the vrious interfes determine the vlue of ontt ngle 17 θ Y F hen voltge is pplied etween ondutive liquid nd n eletrode on sustrteD the solidEliquid interfil energy dereses qudrtilly with the pplied voltgeD s shown y vippmnn in IVUSF 18 sn reent yersD eletrowetting on dieletriD vrint of vippmnn9s originl 4eletropillrity e'et4 hs eome the stndrd on(gurtion for eletroE stti tution of sessile drops @see pigF IAF sn this on(gurtion thin dieletri oting seprtes the eletrode from the liquid nd thus prevents eletril urrents from )owingF nder these onditionsD pplying voltge etween the drop nd the eletrode gives rise to eletrostti hrging of the prllel plte pitor formed y the @perfetly ondutiveA drop nd the eletrode on the sustrteF he resulting eletrostti energy gin of the sysE tem is proportionl to the solidEliquid interfil reF reneD eletrowetting on dieletri E lthough physilly eing n eletrostti e'et E leds to n eectively redued solidEliquid interfil energy γ sl,ef f (U ) = γ sl,0 -cU 2 /2D where c = ε 0 ε r /T is the pitne per unit re etween the drop nd the sustrte eletrodeF ε 0 ε r is the dieletri permittivity of the insulting lyerD T its thiknessF es onsequeneD the pprent ontt ngle θ(U ) of the drop depends on oth oung9s ngle nd the pplied voltge U vi the eletrowetting equtionX 2

cos θ = cos θ Y + c 2γ U 2 = cos θ Y + U U L 2 @IA Q
where γ the liquidGvpor surfe tensionF U L = c/2γ is the hrteristi voltge sle of the prolemF iletrowetting is thus wy to tively ontrol the wettility of surfe y voltgeF sn the following we will denote the voltgeEdependent ontt ngle s the pprent ontt ngle nd the ontt ngle t zero voltge s oung9s ngleF pigure IX iletrowetting setEup with shemti droplet morphology in the onneted stteF he drop pprent ontt ngle @mesured y the sideE views of the sessile dropsA θ is modulted y the rms vlue of the pplied voltge U 0 F he distne d is mehnilly djustleF he urrent is meE sured vi the resistor R m F 2 Material and Methods e used typil eletrowetting setEup mde of ondutive sustrteD n insulting lyerD drop of ondutive liquid @typil volume V ≈ 1µlA nd pltinum wire immersed in the dropF he wire @rdius r e a SH or IPS µmA is positioned t ontrollle distne d ove the sustrteF en eg voltge U a H FFF IHH V rms with frequeny f a IFFFPH krz is pplied etween the wire nd the solid sustrteF he solid sustrte is ondutive n + rseni doped silion wfer @ondutivity IES mmA on top of whih silion oxide insulting lyer hs een thermlly grown up to thikness T a IFIS ± HFIS µmF en dditionl hydrophoi lyer @y monolyerA hs een deposited on the surfe from the liquid phse using stndrd protoolsF 19 he liquids used re mixtures of wterD glyerol nd xgl in vrious ompositions whih llows to tune the visosity η @P FFF UHmsA nd ondutivity σ @HFI FFF IH mGmA without signi(nt modi(tion of its surfe tension γ ≈ QV mxGmF he whole setEup is immersed in silion oil th @ker silione oil eu R SD visosity η = S msA in order to @iA prevent evportion on the time of the experimentsD @iiA redue ontt ngle hysteresis nd @iiiA redue e'et of grvityF oung9s ontt ngle θ Y of the solution under oilD mesured using side view of sessile drop of wterD is ISS¦on the y lyer nd out IRH¦for glyerol with few degrees @`S¦A hysteresis in oth sesF he reltive density of the liquids in the silione oil is ρ a HFPEHFQ IH 3 kgGm 3 F en yhmi resistor @R m a IH kΩA ws inluded in series with the drop @see pigFIAF he urrent through the iruit is monitored y mesuring the voltge ross the resistneF he jump of the urrent mplitude from zero vlue in the disonneted stte to nonEzero vlue in the onneted stte provides n eletril mesurement of the drop osilltion frequenyF sn dditionD sideE views of the drop re reorded with highEspeed mer @hotron pstm ultim SIPA t frme rte of PHHH fps nd with n opening time of the shutter in the rnge PH E RH µsF he pprent ontt ngle θ ws determined from side view imges of the pigure PX iletrowetting urve for wter drop in oil showing the dependene of the pprent ontt ngle with the pplied voltgeF he eletrowetting eqution I holds elow SH F rere U L a SH F sessile drop using ustom written softwreF θ dereses with inresing voltge in greements with eqF I up to SH 2 @see pigF PAF he hysteresis etween inresing nd deresing voltge is smller thn S • F 3 Morphology diagram he experiments were performed s followsF e drop with rdius smller thn the pillry length ws deposited on the sustrteF he t wire ws 2 Above 50 V contact angle saturation set in, as typically observed in electrowetting. 2 S ove the drop t zero voltgeF here re two possile morphologies for droplet deposited on surfeX it n e either tthed to the wire or deE thed from itF he stility of oth morphologies depends on wireEsustrte distne d nd on the pplied voltge U F 22 e determined the stility limE its of oth morphologies y vrying d nd U systemtillyF he result is shown in pigF QD where we resled the siss nd the ordinte y U L nd y V 1/3 D respetivelyF et lrge seprtion @region eAD the droplet is lwys dethed from the wireD wheres t smll seprtion @region fAD it is lwys tthedF et low voltgeD there is region @gA in whih the morphology is history dependentF sf region g is entered from ove @iFeF from region eAD the drop is dethed from the wireD wheres upon entering from elow @region fA it is tthedF sn region g oth morphologies re thus metstleF he upper oundry of region gD d max (U 0 ) @solid line nd lk irlesAD repreE sents the stility limit of the tthed stteD wheres the lower oundryD d min @dshed lineAD represents the stility limit of the dethed phseF his is the wellEknown phenomenon of pillry hysteresisF pon inresing U the width of region g deresesF et the threshold voltge U T D the stility limits d min (U ) nd d max (U ) ross nd ontinue s smooth ontinuous funtions of U F ling the wire t distne d suh tht d min < d < d max for U > U T @region hA gives rise to the following sitution situtionX d is oth ove the stility limit of the tthed morphology nd elow the stility limit of the dethed morphologiesF rene oth morphologies re unstleF es onsequene the system osilltes periodilly etween the tthed nd the dethed stte in this region @tringles in pigF QAF o hrterize this dynmi phse digrm in more detilD we determined the threshold voltge for drop osilltions for vrious drop volumes V @V ∼ HFIP F F F SFQ µv A nd for two di'erent rdii r e of the eletrode @SHµm nd IPSµmAF foth inresing V nd deresing r e were found to redue U T @see inset RAF he rw dt ould e ollpsed onto mster urve y resling r e y V 1/3 D see pigFRF o understnd this morphologil digrm nd the ehvior of the threshE old voltgeD we developed simple pillry modelF he stility limit of the dethed phse is rther oviousX s soon s

d < h(U = 0), @PA where h(θ(U )) = 3V π 1/3 1 -cos θ(U ) 2 -3 cos θ(U ) + 3 cos 3 θ(U ) T A B C D pigure QX hynmi phse digrm for drop of V ≈ R µv with U L = 50
F sn region eD onneted sttes re unstle @the lk dots represent the experimentl stility limit of the onneted morphologyFAY in region f only onneted sttes re stleY g is the region of pillry hysteresis nd h is the osilltion region in whih osilltions re stle @tringle upA or unstle @tringle downAF he limits of the regions re in greement with the model @solid lineX stility limit of tthed morphologies d max Y dshed lineX height of the drop with oung9s ngle h(U = 0)/V 1/3 AF pigure RX hependene of the threshold ontt ngle θ T with resled drop sizeF himonds hve een otined with r e a SH µm nd rosses with r e = IPS µmF he full line orresponds to iqF QD the dshed line is otined y model sed on stility of unduloidsF 22 U is the height of the drop t voltge U D the wire touhes the drop nd thus gets immersedF por drops tht re ompletely dishrged fter dethment this ondition thus gives rise to voltgeEindependent horizontl line tht depends only on the ontt ngle if we resle ll lengths y V 1/3 @see pigFQAF ht determines the stility limit of the tthed morphologyc es long s the drop is su0iently lrge ompred to the wireD we my regrd the tthed drop s spheril p of @resledA height h(U ) = h(U )/V 1/3 with tiny pillry nek of negligile volume tthed to itF he stility limit of the tthed morphology is thus determined y the mximum length of tht nekF sing yleigh9s lssil rgument we expet tht the nek eomes unstle when its height is of the order of its irumferene Pπr e 31 @see eFgF ottom right snpshot of pigure UAF his leds to mximum height of the eletrodeX

dmax (θ) = h(θ) + kr e @QA
where k is (t prmeterD whih we expet to e of order PπF sing the experimentl eletrowetting urve @pigFPA to onvert the pplied voltge into ontt ngleD we otin k ≈ 5 upon (tting the eqFQ to the experimentl dt in pigFQF yviouslyD the mximum eletrode height dereses with deE resing ontt ngleD iFeF with inresing voltgeF es onsequeneD there is E t low voltge E rnge of eletrode heights for whih oth morphologies re @metAstleF et high voltgeD howeverD there is rnge of d vlues where neither morphology is stleF his rnge of droplet osilltions is determined y the riterion

dmax (U ) < d < h(0)
@RA prom this eqution it is lso ler tht the threshold of voltge for drop osE illtions is given y dmax (θ(U T )) = h(0)F he solid line in pigFR shows the solution of this eqution for U T @using kaSAF st produes good greement with the experimentl dt for lrge volumesD V 1/3 /r e ∼ 10 nd lrger ut fils t smller volumesF he reson for this filure is tht the deomposition of the tthed drop into spheril p nd pillry nek of negligile volume is no longer justi(ed for smll dropsF sn tht rnge the stility of the ext solutions of the pillry equtionD the generl xisymmetri surfes of onstnt mens urvture @heluny surfesA 27 hs to e nlyzed s desried reentlyF 22 ithout ny (t prmeter it gives rise to the dshed line in pigFRD whih reprodues the experimentl results lso for smll volE umesF his nlysis lso shows tht the relevnt length sle for the pinh o' prolem is tully the eletrode rdius rther tht the drop sizeD whih is oviously of no relevneF V sn the followingD we fous on the dynmis of drop osilltionsF essume tht we immerse the wire deeply into the drop nd set voltge U bU T F hen we inrese d in qusistti proess to enter the osilltory regionF pon rossing d max (U )D the drop reks o' the eletrodeF wo di'erent kinds of drop ehvior re then oservedD depending on the ondutivity nd the eg frequeny of the pplied voltgeF 16 et low frequeny @or t hg voltgeA ndGor high ondutivityD the drop dethes without hnging the ontt ngle sustntilly @on vergeAF e thus end up with spheril p with ontt ngle θ ∼ θ(U )F yn time sle of mny seonds up to minutes @depending on the qulity of the insulting lyerA θ slowly relxes k towrds θ Y F et some point the pex of the drop rehes the height d nd the drop touhes the eletrode ginF et this momentD the ontt ngle suddenly drops k to θ(U )D the drop spreds gin until it dethes one more from the wire nd the entire yles egins ginF his gives rise to highly 9symmetri9 osilltions in whih the drop spends most of its time in the dethed stte nd only short time in the tthed oneF he explntion of this dynmis is very simpleX one the drop is dethed from the wireD its ontt ngle is determined y its hrge following

cos θ L (q) = cos θ Y + q 2 2CA sl γ @SA
where C = cA sl is the totl pitne etween the drop nd the sustrte eletrode @A sl X solidEliquid interfil reAF por hg voltgeD the hrge is simply given y q = Q max = CU D whih mens tht the ontt ngle does not hnge upon dethment of the drop from the eletrodeF yn longer time slesD the drop slowly dishrges vi unvoidle reep urrentsD giving rise to the oserved slow relxtion of the ontt ngle nd the overll drop shpeF et high eg frequeny @ndGor t low ondutivityAD the ehvior is omE pletely di'erentF sn this regimeD the ontt ngle inreses immeditely more of less to θ Y upon dethment from the wireF es onsequeneD the drop shpe evolves k towrds spheril p with θ = θ Y on time sle of order milliseonds rther thn seonds to minutesF huring this proessD it touhes the wire ginD whereupon the ontt ngle swithes k to θ(U ) nd the yle strts ginF his gives rise very fst drop osilltions with frequeny rnging from ∼IHrz to ∼IHHrzD depending on the ext drop size nd the visosityF hese fst osilltions re thus muh more symmetri with the frtion of time spent in the tthed stteD t on D nd the dethed W stteD t of f D oth eing of the order of tens of milliseondsF hetilsD howeverD still depend on oth d nd U D s will e disussed elowF prom eqFS it is ovious tht the rpid inrese in ontt ngle in the fst osilltory regime implies tht the drop is essentilly dishrged fter the dethment proessF e desried in detil in previous pulition 16 tht suh dishrge n indeed tke ple during the lst moments of pinh o' of the pillry nekF sn tht work we showed tht the hrge in the (nl stte is determined y lne of three ftorsX @iA the eg frequenyD @iiA the gE time onstnt formed y the ohmi resistne of the pillry nek @whih diverges lgerilly upon pinh o'A nd the pitne etween the drop nd the sustrte eletrodeD nd @iiiA hrteristi hydrodynmi time sle τ of the pillry pinh o'D whih depends on the oilEwter interfil tension nd the visosityF ell these ftors n e omined in dimensionE less prmeter α = 2πf (R 0 Ct µ 0 ) 1/(1+µ) F sn this expression IGP`µ`QGP is the exponent of the nek resistne divergene nd R 0 = 1/σr e is the hrE teristi resistne of the nekF por α 1 the drop hrge vries rndomly etween ±Q max = CU D for α 1D howeverD it goes to zero 3 sn the followingD we will fous on fst osilltions with α 1F pigFS shows the eletril urrent through the iruitD whih displys nonEzero vlue when the drop is in ontt with the eletrode during drop osilltionsF pigFS@A orresponds to prmeter vlues of d nd U deep inside osilltory regionF ysilltions re very periodi nd the frequeny is wellEde(nedF he solute frequeny depends on oth prmetersF por instneD for ∼R µv drop of visosity IS ms frequeny of the osilltions is inresing from PT rz t QU to RP rz t SH with only few perent vrition when d is vried in the whole rnge of the osillting regimeF purthermoreD the frequeny lso depends on the drop size nd on the visositiesF roweverD no signi(nt vritions hve een oserved with the eletril ondutivity @within the rnge tht gurntees omplete drop dishrge upon dethmentAF he mesurement of the eletril urrent llows to mesure the time spent in the tthed stte nd in the dethed stteF ritions of d do 'et the reltive time spent in the tthed nd in the dethed stteD respetivelyF sn pigF T we disply the frtion of time α i = t i /(t on + t of f )) with iaonD of f spent in either stte s funtion of dF es nturlly expetedD the time spent in the on @respF o'A stte inreses @respF deresesA with deresing @respF inresingA dF glose to the oundry of the osilltory region the osilltion frequeny pigure TX ime spent in the on stte s funtion of distne for Q voltges in the osillting regimeF ixperimentl dt @+A nd omprison with the model @full lineAF he prmeters of the model re θ =IHHD WH nd VH degrees @top to ottomAD Q = 0.5D V =QFUT µvD θ Y = 140 degrees in resonle greement with the experimentl vluesF is not wellEde(nedF pigF S@A shows tht the osilltions re rther irregulr with periods of slowly relxing drops @vi reep urrentsY see oveA interE hnging with periods of fst osilltionsF II eprt from the prmeters d nd U D whih ontrol lso the morphologil digrm in pigF QD the osilltion dynmis depend lso on the visosity of the drop nd on its volumeF wo typil dynmis of osilltions hve een oservedF por lrge volumes nd smll visosities the drop shpe displys lrge mplitude deformtion while for smll volumes nd lrge visosities the drop remins more or less spheril p shpedF sn oth sesD the pE prent ontt ngle of the drop is dynmi dvning ontt ngle lose to the eletrowetting ngle θ(U ) in the onneted stte nd reeding ontt ngle lose to oung9s ngle θ Y in the disonneted stteF he swith from θ(U ) to θ Y nd k ours on time sles muh fster thn the osilltion periodF ixmples @snpshots from high speed video lipsA of the two types of osilltions re displyed in pigF U@A nd V@AF sn view of the fst swith of the ontt ngleD we onluded tht the osilltion dynmis re in ft ompletely determined y the hydrodynmi response of the drop to this instntneous hnge of the oundry onditionF o test this ideD we implemented numeril simultion sheme of drop osilltions using the volume of )uid @ypA method with the ommeril )uid dynmis pkge gphEegiC @pigF U @A nd V @AAF sn the odeD the drop dynmis re driven entirely y the swithing of the ontt ngle etween the indited vlues s the drop either dethes from the wire or retthes to it 4 F es eomes pprent from the (guresD the qulittive fetures of the dynmis re rther well reprodued E lthough the solute osilltion frequeny is not perfetly reprodued @see elowAF le I shows the vrition of the times spent in the tthed nd the dethed sttesD respetivelyD s well s the osilltion frequenies for series of prmeter vluesF hese results on(rm on semiquntittive level the vlidity of the pE proh tken hereF he osilltion dynmis re entirely determined y the hydrodynmi response of the drop to vritions of the ontt ngleF ileE tri (elds do not in)uene the drop osilltions diretlyF hey enter only indiretly vi the indued vrition of the ontt ngleF hile the overE ll drop shpes re orretly reprodued y the numeril lultionsD quntittive omprison of the frequenies shows slight mismthD proE ly used y the di'erene of the prmeters etween experiments nd simultions or more fundmentlly y the physil desription of the ontt line motionF foth the experimentl results s well s the numeril lultions show 4 Numerical methods such as VOF are known to have diculties in capturing accurately singularities such as during capillary pinch o. However, since the pinch o occurs on a time scale much smaller than the period of the drop oscillation we do not expect such errors to play an important role for the overall dynamics. pigure VX righEspeed snpshots of the osilltions of I µv queous drop in oil nd orresponding numeril simultions @ypD gphEegiC pkgeAD visous regime η ≈ VH msF @AX d = 1.10 mmD F = 24 rzD time etween two frmes S msF@A d = 1.10 mmD F = 34 rzD time etween two frmes QFU ms drop surfe pro(leD whih re lredy present in the stti situtionD 26 or dynmi spets of the entrpment of thin (lms of the mient oilD 20 lso 'et the drop dynmisF uh e'etsD howeverD require dditionl simulE tions with more sophistited numeril tools thn urrently ville to usF ithin the sope of the present nlysisD we ontent ourselves with the onE lusion tht the qulittive spets of the osilltion dynmis for oth low visosity inertiEdominted drops s well s for visosityEdominted drops re well reproduedF

IR d @mmA θ deg HFWH IFHH IFIH IFIS IFIU IFPH SH t on ms RH t of f ms U TS t on ms ∞ SI IR IQ IQ W t of f ms T V W IH IQ US t on ms ∞ PS IT IR t of f ms V W IH VS t on ms ∞ QP t of f ms IH
le IX xumeril simultions resultsX time in the onneted stte @t on A or disonneted stte @t of f AF 5 Discussion fy hnging the ontt ngle ruptly etween low nd high vlue upon reking the pillry nek or upon retthing to the eletrodeD we exite osilltions of the dropF he spetrum of eigenfrequenies of free drops ws investigted y vmF 21 he evolution of the drop shpe suggests tht sevE erl eigenmodes re exited simultneouslyD in prtiulr for the low visosity seF ther thn nlyzing these eigenmodes in detil we will nlyse the osillting drops s simple dmped hrmoni osilltor in order to provide physil understnding of how the osilltion dynmis depend on @iA pE rmeters hrterizing the drop @density nd sizeD visosityD surfe tensionA nd @iiA on the externl ontrol prmeters d nd U F o perform the mppingD we follow model similr to the one desried y ykumur et al. 30 for ouning drops on superhydrophoi surfesF he deformtions indued y the fst hnges of the ontt ngle re lned either y visous dissiption or y inerti or omintion of othF he drop shpe is desried in rdil oordinte y funtion R = R 0 + R 1 (φ, t) with R 0 the drop rdius t rest @zero voltgeAD R 1 timeEdependent perturtion nd φ the ngle with respet to the vertil @see pigF WAF sn order to determine the timeEsle of the drop dynmis we onsider the xvierEtokes eqution expressing the lne of elertion y pillry fores nd visous dissiptionX du dt

+ u∇u = η ρ ∇ 2 u - 1 ρ ∇p @TA
where the speed u ∼ dR/dtF prom dimensionl perspetiveD we (nd u ∼ IS pigure WX keth of the pillry prolem onsideredF he relxtion of the perturtion R 1 is lned y visous dissiption or inertiF h is the height of the pex of the drop @see iqF V nd WA R 1 /τ where τ is the timeEsle of the motionD du/dt ∼ R 1 /τ 2 nd u∇u ∼ R 2 1 /(R 0 τ 2 )F rene the nonEliner term u∇u n e negleted s long s the mplitude R 1 of the perturtions is smller thn the drop size R 0 F p is the pillry pressure linked to the urvture of the interfe ∇p ∼ γ/R 0 •R 1 /R 2 0 F herefore iqF T n e rewritten s dmped hrmoni osilltor eqution for R 1 X

R1 + 1 Q Ṙ1 + R 1 = 0
@UA with dimensionless time t = tω where ω 2 0 = γ/(ρR 3 0 ) is the eigenfrequeny of the undmped systemF he dots indite time derivtivesD s usulF @es lredy notedD we neglet the presene of higher eigenmodes in this nlysisFA he qulity ftor of the osilltor Q = (ργR 0 ) 1/2 /η is given y the inverse of the yhnesorge numer Oh = η/(ργR 0 ) 1/2 F he ltter is the hrteristi numer ppering whenever visosityD inerti nd pillrity ompete with eh otherD s for instne in drop impt prolems 23 nd pillry rekupF 24 prom eqFU it is thus ler tht the dependene of the osilltion dynmis of the drop prmeters is represented y the two prmeters ω 0 nd Oh in this homogeneous equtionF hepending on the vlue of Oh di'erent regimes re otinedF sn the extreme se of invisid )uidsD Oh 1 @ Q ∼ ∞AD the system ehves s spring without dmpingF he frequeny of the osilltions is therefore given y the only remining timeEsleX

F c ∼ ω 0 = γ ρR 3 0 1/2
por the drop shown in pigF S we hve Oh ≈ HFHID iFeF the system ehves s wekly dmped osilltorF his explins the overshoot ove the stti IT height nd the other inerti driven deformtions of the drop surfeF sn the se of n extremely visous )uid @Oh 1AD inerti is of minor importneF ueeping only the visous term the frequeny of the osilltions is now given y the relxtion time of the systemX

F v ∼ γ ηR 0
por the dt shown in pigF V we hve Oh ≈ IF hese osilltion thus fll into the intermedite regime inertiD where pillrity nd visous dmping re eqully importntF hile ω 0 nd Oh hrterize the intrinsi response of the dropD the experimentl ontrol prmeters d nd U will determine the strength nd the timeEdependene of the externl foresD iFeF the inhomogeneities to e dded in eqF UF o understnd this spet it is onvenient to rewrite eqF U fousing on the dynmis of the drop pex hrterized y h(t)D iFeF R(φ = π)F ḧ + Oh × ḣ + h = h eq @VA where

h eq = h 0 for h(t) < h c h(U ) for h(t) > d @WA
with h c = d -kr e is the ritil height where the pillry nek reks @fF iqF QAF por d -kr e < h(t) < dD the vlue of h eq n dopt either vlue depending on the historyD iFeF h eq = h(U ) if the drop ws previously tthedD nd h eq = h 0 if the drop ws previously dethedF his explins the role of the externl ontrol prmetersX the lrger U D the lrger the di'erene etween h 0 nd h(U ) nd thus the stronger the externl driving fore ting on the osilltorF d ontrols the timing when externl fores re swithed etween the two vluesF sn pigFIH we disply the results of the dynmis of the drop pex for visous drop of volume V = QFUT µv with Q = 0.5F he drop is initlly onneted to the wire pled t d = IFS mm t taHF sn this seD h eq = h(U ) nd the drop pex egins to relx exponentilly towrds this equilirium vlue @dshed line in pigF IHAF pon doing soD the pillry nek reks t t 1 whereupon the equilirium vlues swithes to h eq = h 0 F rene the drop egins to relx towrds this new equilirium vlue @dotted nd dshed line in pigF IHA until it rehes h = d t t = t 2 D whereupon h eq swithes k to h(U )D etF he pex height thus osilltes periodilly etween d nd h c @full line in pigF IHD s oserved experimentllyF he time the systems spends in either the tthed @t on A or the dethed @t of f A morphology nd thus the IU showing tht the model predits orretly the vritions of the experimentl pointsF he model however overestimtes the drop frequeny y ftor of ∼P whih might e used y dditionl dissiptions @t the ontt line or in the surrounding )uidA tht re not onsidered hereF sf we inlude the e'et of inerti into this pitureD the min qulittive di'erene is tht we expet the vlue of h(t) no longer to e on(ned etween d nd h c F therD dynmi overshoots nd undershoots will ourD s oserved in oth the experiments nd the numeril simultions @see pigFUAF sn greement with the experimentsD these overshoots lso explin tht drop osilltions @one strtedA n e oserved t wire positions d > h 0 for low visosity dropsF he model gives lso qulittive explntion for the irregulr urstE like osilltions oserved lose to the edge of the osilltory regionF sn most experiments there is still smll rndom residul hrge hrge q on the drop fter the dethmentD with mximum vlue is given y Q max F 16 his IV redues the equilirium height of the drop from h 0 to h(q)D s n e derived from eqFSF sf h 0 < d < h(Q max ) the rndomly piked vlue of q determines whether the drop n reh the wire within the hydrodynmilly governed timeEsle of the fst osilltions or notF por d just ove h(Q max D this is the se fter most dethment eventsD ut sometimes q is su0iently lose to Q max suh tht the osilltions re loked for ertin timeF 6 Conclusions sn summry we were le to quntify the geometri nd eletri onditions required for the genertion of selfEexited drop osilltions in eletrowettingF e simple geometri pillry model ws developed to identify the regions within the twoEdimensionl prmeter spe d -U where drop osilltions ourD inluding the threshold voltge U T F o model the osilltion dynmE isD the drop ws mpped onto dmped hrmoni osilltor in whih the intrinsi )uid prmeters determine the response of the droplet to n externl fore tht is swithed periodilly etween two vlues s the drop tthes to the wire nd dethes from itF he timing of the externl fore is set y the externl ontrol prmeters d nd U F prom more generl perspetiveD the suess of the modeling presented in this work orroortes two sttements tht pply to more or less ny eletrowetting experimentX @iA the hrterE isti hydrodynmi response times of drops re determined y the lne of three intrinsi )uid dynmi properties of the dropsX surfe tensionD visE osityD nd densityF he reltive importne of these three ontriutions n e red from the yhnesorge numer Oh = η/(ργR 0 ) 1/2 F @iiA vritions of the ontt ngle re instntneous on the time sle of the typil hydroE dynmi response times of the dropsF nder usul experimentl onditionsD there is thus seprtion of time sleF es onsequeneD the drop dynmis n e modeled y n instntneous vrition of the lol ontt ngle to the voltge pplied to the eletrode@sA in the systemD followed y purely hydrodynmi response of the liquid on longer timesF e thnk ente xikopoulosD do ur't nd wnfred rörger for tehE nil ssistneF e re lso indeted to lf eemnnD tephn rermingE hus nd wrtin frinkmnn for fruitful disussions nd omments on the mnusriptF his work ws supported y the qermn iene poundtion within the priority progrm etting nd truture pormtion t snterfes nd y the snstitute of wehnisD roess nd gontrol wente @swegAF tFEgF fret knowledges support y wrie gurie sndustry rost pellowship sEIWWWEVHHHRF IW

3

  For intermediate values of α the charge can adopt a random value between zero and an α-dependent maximum value pigure SX iletril urrent inditing the drop G eletrode onnetionF he urrent is normlized y its mximl vlueF @A deep in the osillting regime @U = SH D d = IFTW mmA the osilltions re stleF @A lose to oundry @U = RH D d = IFUQ mmAD the osilltions re errtiF

  snpshots of the osilltions of I µv queous drop in oil nd orresponding numeril simultions @ypD gphEegiC pkgeAD inertil regime η ≈ P msF @AX d = 1.26 mmD F = 72 rzD time etween two frmes IFS E P msF @AX numeril lultion d = 1.21 mmD F = 93 rzD time etween two frmes IFR msF tht the osilltion frequeny depends strongly on the ontt ngleF huring the osilltionsD one should oviously tke into ount the speedEdependent dynmi ontt ngle rther thn the stti vlueD whih we used hereF purE thermoreD it is to e expeted tht eletri (eldEindued distortions of the IQ

  pigure IHX hynmis of drop pexF olution of the model for θ Y =IRH deE greesD θ = VH degreesD Q a HFSD V = QFUT µvF he dshed line orresponds to the spreding without dethmentD the dshedEndEdotted line to the reE lxtion to drop with oung9s ontt ngle without retthment nd the full line the osilltionsF ih time the interfe rosses the gry linesD the ondition of equilirium is swithedF he overE nd underE shoots re results of inertiF osilltion period is strongly 'eted y oth d nd U F por instne it is ovious tht t on should inrese upon reduing d @nd thus h c AF sn ftD we expet t on to diverge s h c pprohes h(U )F imilrlyD t of f is expeted to inrese nd diverge s d is inresed nd pprohes h 0 F e omprison of the model with the experimentl mesurements of t on is displyed on pigF T

		2
		1,8
					h(U=0)
	h [mm]	1,4 1,6			d
		1,2			d -k r e
					h(U)
		0 1	20	t ω	40

Present address: ISIS-ULP, 8 allée Gaspard Monge, 67083 Strasbourg, (F) I

IF ollkD wF qFY henderovD eF hFY pirD F fF Lab on a Chip 2002, 2, WTEIHIF PF wugeleD pFY fretD tFEgF J. Phys. Cond. Matter 2005, 17, UHSEUURF QF hrhuerD eF eFY roinD F wF Annu. Rev. Fluid Mech. 2005, 37, RPSE SSF RF ghoD F uFY woonD rFY uimD gF tF J. Microelectromechanical systems 2003, 12, UHEVHF SF rinivsnD FY mulD F uY pirD F fF Lab Chip 2004, 4, QIHEQISF TF huoisD FY wrhndD qY pouilletD FY ferthierD tY houkiD FY rsE sineD pY qmouthD FY ultierD wF Anal. Chem. 2006, 78, RWHWERWIUF UF heelerD eF FY woonD rY uimD gF tFY vooD tF eFY qrrellD F vF Anal. Chem 2004, 76, RVQQERVQVF VF oonD tF FY qrrellD F vF Anal. Chem 2003, 75, SHWUESIHPF WF ikD FY mulD F uFY pirD F fF Lab Chip 2003, 3, PSQEPSWF IHF fergeD fFY eseuxD tF Eur. Phys. J. 2000, E, ISWEITQF IIF rendriksD fFY esD F uF wF FY endersD gFY ukkerD F Optical Review 2005, 12, PSSEPSWF IPF ryesD F eFY peenstrD fF tF Nature 2003, 425, QVQEQVSF IQF urupenkinD FY ngD FY whD F Appl. Phys. Lett. 2003, 82, QITEQIVF IRF ulingnerD eFY rerminghusD FY wugeleD pF Appl. Phys. Lett 2003, 81, RIVUERIVWF ISF ulingnerD eFY fuehrleD tFY wugeleD pF Langmuir 2004, 20, TUUHETUUUF ITF fretD tFEgFY wugeleD pF Phys. Rev. Letters 2006, 96, IHTIHTF IUF oungD F Philos. Trans. R. Soc. London 1805, 95, TSEVUF IVF vippmnnD wF qF Ann. Chim. Phys 1875, 5, RWRF IWF givD tF J. Am. Chem. Soc. 1980, 102,F PH PHF tiuD eFY wugeleD pF Phys. Rev. Letters 2006, 97, ITUVHIF PIF vmD rF Dover Publications; 6th Edn. 1993, F PPF fretD tFEgFY frinkmnnD wF Phys. Rev. Letters 2006, 94, IRTIHTF PQF hi0noD FY oninD eF eF Phys. Fluids 1997, 9, QIUPEQIVTF PRF iggersD tF Rev. Mod. Phys. 1997, 69, VTSEWPWF PSF fergeD fF C. R. Acad. Sci. III 1993, 317, ISUF PTF fuehrleD tFY rerminghusD FY wugeleD pF Phys. Rev. Lett. 2003, 91, TIHIF PUF helunyD gF J. math. pures et appliquées 1841, 6, QHWEQIWF PVF wugeleD pFY ulingnerD eFY fuehrleD tFY teinhuserD hFY rerminghusD F J. Phys. Cond. Matter 2005, 17, SSWESUTF PWF wugeleD pFY fretD tFEgFY teinhuserD hF Appl. Phys. Letters 2006, 88, PHRIHTF QHF ykumurD uFY ghevyD pFY ihrdD hFY uéréD hFY glnetD gF Europhys. Lett 2003, 62, PQUEPRQF QIF yleighD vF Proc. R. Soc. London Ser 1879, A 29,F PI