Numerical Simulation of the Electrical Double Layer Based on the Poisson-Boltzmann Models for AC Electroosmosis Flows

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1 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble Numecal Smulaton of the lectcal Double Laye Based on the Posson-Boltzmann Models fo AC lectoosmoss Flows Pascale Pham (1), Mattheu Howoth (1), Anne Planat-Chéten (1) and Sedat Tadu (2) (1) CA/LTI - Dépatement des mcotechnologes pou la Bologe et la Santé (2) LGI, UMR 5519, B.P. 53 X 3841 Genoble Cedex Fance coespondng autho: 17 Avenue des Matys, 3854 Genoble cedex 9, Fance, pascale.pham@cea.f Abstact: In ths pape, the analytcal valdaton of Posson-Boltzmann () equaton computed wth Comsol Multphyscs, n the case of a polazed suface n contact wth the electolyte [1]-[2], s fst pesented. Comsol Multphyscs algothms easly handle the hghly nonlnea aspect of the equaton. The lmtatons of the model, that consdes ons as pontlke chages, ae outlned. To account fo the stec effects of the on cowdng at the chaged suface, the Modfed Posson-Boltzmann model, poposed by Klc et al. [3], s analysed fo symmetc electolytes. The M equaton s then coupled to the complex AC electoknetc and the Nave-Stokes equatons to smulate the AC electoosmoss flow obseved nsde an ntedgtated electodes mcosystem [4]-[6]. Keywods: numecal smulaton, Posson- Boltzmann, Fnte lement Method, AC electoknetcs. 1. Intoducton The lectcal Double Laye () epesents the nteface between a sold suface (polazed electode) and an electolyte. The chaged suface attacts neaby counteons and epels coons pesent n the soluton. In mcosystems, the same electostatc phenomenon s also pesent aound chaged nanopatcles (bomolecules, latex beads ) mmesed nto an electolyte: they expeence electostatc nteactons whch gve se to a counteon cloud. The o the counteon cloud s lkely to eact to the appled electc felds and can stongly nfluence vaous electcal phenomena such as delectophoess, electophoess of polyelectolytes (DNA, potens, ) o AC electoknetc flows. RC ccut models ae wdely used by electochemsts fo epesentng the. Howeve, n mcosystems whee appled electc felds can be vey stong because of the vey small dmensons, ths appoxmaton fals [3] [7]. Despte the explosve gowth of multscale modelng fo mcofludcs, whee the contnuum s usually coupled to Molecula Dynamcs technques, we nvestgated hee the use of coupled contnuum models, based on the Posson-Boltzmann () equaton. Fo us, t s nteestng to epesent the usng the Comsol Multphyscs softwae applcaton because ts stong couplng to macoscopc equatons (Nave-Stokes n ou case) s possble. 2. Theoy 2.1 The electcal double laye In ths pape, we consde that electodes ae deally polazable.e. that no electon tansfe (electochemcal) eactons occu at the electode. The model whch gave se to the tem 'electcal double laye' was fst put fowad n the 185's by Helmholtz. In ode fo the nteface to eman neutal, the chage held on a polazed electode s balanced by the edstbuton of ons close to the electode suface. In Helmholtz's vew of ths egon, the attacted ons ae assumed to appoach the electode suface wth a dstance assumed to be lmted to the sze of the on: the oveall esult s two layes of chage (the double laye) and a lnea potental dop whch s confned to ths egon only. A late model put fowad by Gouy and Chapman supposed that ons ae able to move n soluton and so the electostatc nteactons ae n competton wth Bownan moton. The esult s stll a egon close to the electode suface contanng an excess of one type of on but now the potental dop s exponental and occus ove the egon called the dffuse laye:

2 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble suface potental ψ Fgue 1. The Gouy-Chapman epesentaton of the used by the equaton. The most common epesentaton of the lectcal Double Laye () s due to Sten (1924): the s composed of two layes (see Fgue 2). The nne laye (called the compact laye) whch s n contact wth the electode and whee ons ae absobed on to the suface due to hgh electostatc nteactons. Outsde the compact laye, thee s the dffuse double laye: suface potental V zeta potental ζ dffuse laye compact laye dffuse laye Fgue 2. The Sten epesentaton of the composed of the compact laye and the dffuse laye. The vaaton of the electcal potental V thu the (ed lne) s epesented fo the case of a postvely chaged suface. The potental at the nteface between the compact and the dffuse laye s called the zeta potental ζ whch can be detemned fom electoknetcs measuements. 2.2 The Posson-Boltzmann equaton bulk () ψ = bulk () The Posson-Boltzmann () theoy s based on the Gouy-Chapman epesentaton [3]. The dffuse laye s consdeed to be dectly n contact wth the chaged suface who s potental o chage s known (see Fgue 1). z We wll see n ths secton that the theoy pedcts that the suface potental deceases exponentally n the. Ths s the sceenng phenomenon of the suface chages by the counteons. Because the equaton has lmtatons (see secton 3.2), we voluntay name the electc potental used n the equaton by ψ nstead of V used n the Sten epesentaton (Fgue 2). In the equaton, ons ae supposed to be pontlke chages, the onc soluton s supposed to be a dlute soluton (so the ons do not nteact wth each othe) and the solvent (wate) s consdeed as a contnuum delectc of pemttvty ε = ε ε. The chages of the suface nduce an electc potental ψ (V) n the electolyte whch acts on each spece of ons. ach on concentaton dstbuton c (ons/m 3 ) s gven by the Boltzmann dstbuton whee electostatc (z eψ) and themal () eneges balance each othe: c z e ψ = c e (1) c = n c s the on concentaton n the bulk n beng the numbe of ons n the electolyte ( fomula, c s the bulk concentaton), T s the tempeatue (K) and k the Boltzmann constant ( J/K). e s the poton chage ( C) and z s the on chage numbe. Fo convenence, concentatons can be expessed n 3 Mola unt (M = mole/l): M = 1 c N A whee N A s the Avogado s numbe ( ). ach on dstbuton coesponds to a volume fee chage dstbuton q such that: q = z e c (2) In etun, the total fee chage densty q = q = z e c (3) acts on the potental dstbuton thu the Posson equaton whch lnks the electc potental ψ to ts souces (q):. ε ψ = (4) ( ) q Combnng (4) wth (1) gves se to the non lnea Posson-Bolztmann () equaton:

3 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble.( ε ψ) = z ec e z eψ (5) The bounday condtons assocated to the equaton ae the classcal ones used n electostatcs (see Fgue 3). On the electode, the potental ψ coesponds to the followng suface chage densty [8]: σ chaged suface: ψ = ψ o σ ψ = D.n = ε n Fgue 3. Bounday condtons assocated to the equaton fo a 2D sem-nfnte electolyte n contact wth a flat chaged suface. (6) In the patcula case of a bnay symmetc electolyte (fo example KCl o MgSO 4, z z = z c = c c ), the equaton + =, + = becomes the Gouy-Chapman () equaton [1] [3]:. z eψ ( ε ψ) = 2ze c snh (7) 2.3 The Debye-Huckel theoy: the lneazed equaton The lneazaton of the equaton s obtaned unde the assumpton that the electostatc enegy s small compaed to the themal enegy: ψ << ψ T = (8) z e At oom tempeatue (298 K), fo monovalent ons (z = 1) ψ T ~ 26 mv, fo dvalent ons, ψ T ~ 13 mv. Unde assumpton (8), equaton (1) can be lneazed:. ψ = ε n electolyte: equaton z e c ( ε ψ) = ψ bulk (): ψ ( ) = nsulaton: ψ = n S (9) Ths equaton admts the followng soluton: z κ 1 ψ (z) = ζe (1) When movng away fom the polazed electode, the potental deceases exponentally 1 wth a chaactestc length κ called the Debye length: κ 1 = ε z e c (11) The Debye length (λ D ) s wdely used to estmate the thckness because ts smple fomula depends only on the electolyte chaactestcs. In ths pape, we always consde the case of an aqueous electolyte (ε = 78.5) at ambent tempeatue (298 K). 3. Numecal smulaton of the and the M equatons Usng the Debye-Hückel theoy s qute estctve fo mcosystems because appled electode potentals ae often much geate than ψ T. The equaton s hghly non lnea and ou fst concen s evaluatng how Comsol Multphyscs and the Fnte lement Method can handle ths dffculty. 3.1 Analytcal valdaton of the equaton The equaton s mplemented n Comsol Multphyscs as a PD equaton. The valdaton of the numecal model s made by the compason of numecal solutons fom and equatons (5) and (7) wth analytcal solutons n the case of bnay symmetc semnfnte electolytes n contact wth a flat polazed suface [1]: 4 zeψ ψ z) = actan h tanh( ) ze 4 e κ ( z (12) Tests ae pefomed on the geomety of Fgue 3 fo two dffeent types of electolytes (1:1 and 2:2), vaable bulk concentatons, vaable suface potentals ψ. In the two followng fgues, the cuves wee dawn fo suface potentals ψ of +5 mv and +1V and bulk concentatons c of.1m and.1m.

4 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble The numecal soluton of both the CG equaton (5) and the equaton (7) s n a good ageement wth the analytc soluton (12) KCl +.1V M 5 mv analytc thckness (nm) Debye Huckel M bulk concentaton (M) Fgue 4. Compason of numecal (black =, blue = ) and analytcal (ed) electc potentals fo a 1:1 electolyte at bulk concentatons of.1 M and.1 M. ψ = +5mV. s the dstance fom the electode suface V.1 M analytc Fgue 5. Compason of numecal (black =, blue = ) and analytcal (ed) electc potentals fo a 1:1 electolyte at bulk concentaton of.1 M and ψ = +1V. Fgue 6 compaes the Debye length wth the wdth ( L ) computed fom the and the solutons accodng to the bulk concentaton of a KCl electolyte, fo the appled voltage +.1V. As expected, the Debye length undeestmates the wdth and the eo commtted when usng (11) s qute mpotant. It nceases wth the bulk concentaton. Fgue 6. The wdth L accodng to the bulk concentaton fo KCl, at +.1V: fom soluton (blue), soluton (black) and Debye length fomula (11) (ed). 3.2 Lmtatons of the equaton valdty One could expect that the equaton (and the equaton fo bnay symmetc electolytes), when fully solved n the non lnea egme (ψ > ψ T ), would gve a good estmaton of the. Howeve, even at lage appled potentals, the and the CG equatons have lmted applcablty. One of the assumptons made n the equaton s that ons ae pontlke chages. Ths means that the ons ae consdeed to have no sze. The consequence s that the equaton can pedct an nfnte concentaton of counte-ons nea the chaged suface, whch s not ealstc. Fo example, fo the aqueous electolyte (Na +, Cl - ), at a bulk concentaton of 1 mm, ambent tempeatue and ψ = +1V, the suface chage calculated fom expesson (6) coesponds to a concentaton of sphecal counte-ons (Cl - ) of ons/m 3 hence M! Ths would mean that the chlode on adus s m whch s 1 tmes smalle than the eal value. Hee we use the numecal model to detemne the aea n whch the equaton s vald, the lmt beng gven by the stec effect whch coesponds to a maxmum concentaton eached at the chaged suface due to the hydated on cowdng.

5 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble hydodynamc adus (nm) [9] maxmum concentaton (ons/m 3 ) fo a face cented cubc packng [1] maxmum concentaton (M) ) fo a face cented cubc packng maxmum concentaton (M) fo Klc M model [3] Na + K + Mg 2+ Cl - SO Table 1. xamples of hydodynamc adus and maxmum concentaton epesentng the stec lmt fo seveal ons. In Table 1, the stec lmt s estmated fom the face cented cubc sphee packng model [1] fo whch the packng densty s.74 and fom the Klc model [3] whee each on of damete a s supposed to occupy a volume equals to a 3. Usng the stec lmt values of Table 1 (fo the face cented cubc model), Fgue 7 epots the equaton valdty domans n tems of the appled voltage ψ and the bulk concentaton foa KCl and MgCl 2. These calculatons show that the valdty doman beyond the lnea appoxmaton s estcted to voltages of seveal hundeds of mv. suface potental (mv) Cl n MgCl2 K + n KCl Cl n KCl Mg 2 + n MgCl2 3.3 The Modfed Posson-Boltzmann (M) equaton Recently, an equaton takng nto account the stec effects of the ons has been poposed by Klc et al [3]. It s called the Modfed Posson- Boltzmann (M) equaton. In the M equaton, the Boltzmann dstbuton pat of the equaton s modfed. The modfed Boltzmann dstbuton s gven by the followng expesson: c e c = νsnh z eψ z eψ 2 (13) whee ν s the packng paamete such as ν = 2 a 3 c and a s the effectve on sze. We consde hee a as the damete of the hydated on, see Table 1. Fo a bnay symmetc z:z electolyte, the M equaton can be wtten as follows [3]:. ( ε ψ) = zec 1+ z eψ 2snh 2 2 z eψ 2νsnh 2 (14) The M equaton (14) can be genealzed to non symmetc electolytes by combnng (13) wth (4). In Fgue 8, fomula (13) s plotted vesus the appled voltage ψ and compaed to the dstbuton (1). The dstbuton pedcts a contnuous ncease of the concentaton of the ons at the suface wth the suface potental. Wth the M dstbuton, the concentaton of each on satuates and cannot exceed the stec lmt gven by a bulk concentaton (M) Fgue 7. Valdty domans of the equaton fo KCl and MgCl 2 electolytes.

6 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble suface concentaton (% of max) stec lmt M.1M Fgue 8. Compason of the suface concentaton (% of the maxmum concentaton gven by the stec lmt) fo the dstbuton (blue) and the M dstbuton (ed) accodng to the postve appled voltage. The anon s Cl - and ts bulk concentaton s.1m. On Fgue 9, the M equaton has been solved fo quas-lnea condtons (+.1V,.1M fo KCl): as expected, the M soluton and the soluton ae dentcal M.1V, Cl.1M Fgue 9. Valdaton of the M equaton on a quaslnea case (KCl electolyte, +.1V,.1 M) by compason wth and solutons. Fo hghe voltages (+1V, see Fgue 1), the M and solutons do not ovelap anymoe because the equaton valdty fals. The M equaton pedcts an wdth much bgge (~.2 nm) than the one gven by the soluton (<<.1 nm). The cowdng effect at the chaged suface epels the counteons nto the dffuse laye and povdes a much lage wdth than what the equaton s pedctng M 1V, Cl.1M zoom zoom M 1V, Cl.1M Fgue 1., M and solutons fo a KCl electolyte of bulk concentaton.1m and a hgh suface potental (+1V). The lowe fgue s a zoom of the uppe one nea the chaged suface. Next fgue plots the chlode concentaton pofle coespondng to the pevous potental pofle: nea the chaged suface, the cuve clealy shows that the M model lmts the concentaton to ts maxmum value (117 M fo Cl -, see Table 1): Cl concentaton (M) V,.1M Fgue 11. The Cl - concentaton pofle gven by the M equaton fo a KCl electolyte of bulk concentaton.1m and a hgh appled suface potental (+1V). 3.4 Gettng convegence dung the and the M equaton computaton The pevous numecal esults show that the hghe the electode potental ψ s, the hghe the non lneaty of the poblem s. To obtan a good convegence of the soluton, seveal tcks ae used. Fst, the mesh s hghly efned nea the electode suface whee gadents ae vey steep. The one dmensonal chaacte of the soluton allows the use of quadlateal elements wth low qualty: the mapped mesh has the advantage to educe dastcally the total numbe of Fnte lements and so the soluton tme and the memoy equements. Second, pevous solutons obtaned fom the soluton and/o

7 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble wth lowe appled voltages wee used as ntal condton by selectng the estat button. 4. Couplng the M equaton wth the Nave-Stokes equaton fo AC electoosmoss AC electoosmoss s the flud flow nduced above a chaged suface by the dft of the moble chages by the electc feld. The convecton of the fee chages n the can be neglected n compason wth the dft [11] leadng to a weak couplng between the electcal stess and the flud flow. In most papes whch deal wth ac electoosmoss modelng, the thn double laye appoxmaton unde the lnea egme s used fo the [4]-[6]. The s estmated fom the Debye-Hückel theoy and s not ncluded nsde the computaton doman. The electc feld nsde the bulk s computed wth the ac electoknetc equaton (see (16)) connected to the thu a Neumman bounday condton (18). The flud moton s obtaned fom the Nave-Stokes equaton (2) whee the electcal stess acts as a slp velocty mposed as a bounday condton on the electode suface. Ths slp velocty s estmated fom empcal paametes (the capactance of the compact and the dffuse layes) and the tangental component of the electc feld gven by (16). Ou goal hee s to take off these empcal paametes fom the numecal model. Ths supposes that the s fully epesented nsde the computaton doman fo the flud moton. The electcal volume foce actng nsde the on the flud s not tansfomed nto a slp velocty. The numecal dffculty hee s the multscale couplng that has to be pefomed: the, whch s tens of nm wde, has to be ncluded n a mcosystem whch sze eaches 1 mm. 4.1 Ac electoomoss equatons In ou numecal model, we use the M equaton to estmate moe pecsely the featues: the wdth fo the electc feld calculaton nsde the bulk and the fee chage densty fo the velocty feld. The s assumed to be a capactance pe unt aea C (C/m 2 ) such that: C ε = (15) L whee L s the wdth computed fom the M equaton (14). Unde AC voltages of angula fequency ω, the electc feld nsde the bulk (.e. outsde the ) s gven by the ac complex electoknetc equaton fo eal delectcs [11]-[12]:. σ + ωε V =. σ V (16) ( ( ) ) ( ) whee σ s the bulk conductvty (S/m), V the complex electcal potental of eal pat Re(V) = V and σ the complex conductvty. On nsulated sufaces n contact wth the electolyte, the bounday condton assocated to (16) s of Neumann homogeneous type: Re V = n σ (17) wth n beng the oute nomal. Above the chaged electodes, whch ae assumed pefectly polazable (no electochemcal eactons), the bulk s n contact wth the. quaton (16) s connected to the M equaton at ths nteface whee the consevaton of the nomal cuent densty gves: Re whee : V n σ = ω C ψ (18) ψ s the potental dop acoss the ψ = ψ V (19) and C s gven by (15). The tme-aveaged flud flow s obtaned fom the Nave-Stokes equaton whee effects fom the Joule heatng ae supposed to be neglgble: 2 p η v + ρ v. v = F (2) m ( ) ρ m s the mass densty of the flud (1 kg/m 3 fo wate) and η s ts dynamc vscosty (1-3 kg/m/s fo wate). Because Reynolds numbes ae vey low n mcosystems [12], the neta tem s geneally vey low n (2).

8 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble F s the tme-aveaged electcal foce due to the nteacton of the ac electc feld wth the fee chages of the. Unde the assumpton that the flud pemttvty s unfom (whch s not geneally the case fo hgh voltages [7]): F 1 = q Re( ) (21) 2 whee q s the fee chage densty nsde the defned by (3) and computed fom the souce tem of the M equaton (14). As the M equaton nvolves only the dffuse laye of the, the bounday above the electodes fo equaton (2) epesents the nteface between the compact laye and the dffuse laye: the bounday condton of type slp/symmety, whch s equvalent to the nonpemeablty condton, s used at ths nteface (see fgue 12): v.n = (22) Ths condton s also used on othe boundaes because the vetcal ones ae symmety planes and the uppe one s supposed to be a fee suface. 4.2 Numecal settngs The 2D ntedgtated electode mcosystem studed by Geen and al. [4] s consdeed hee: the electode wdth s 5 µm fo a gap of 25 µm. The electolyte thckness above the electode plane s about 1 mm. The electolyte s a KCl soluton of conductvty 2.1 ms/m (electolyte A): as the authos don t specfy the coespondng bulk concentaton, we use the Kohlaush s law to estmate t [9]: M. The dagam of Fgue 12 summazes the way couplngs between equatons ae done n ou AC electoosmoss model: the M equaton (blue) s solved on a 1D geomety, the wdth L s computed fom the 1D potental by usng an ntegaton couplng vaable: t s used n fomula (18) fo the bounday condton above electodes of the AC complex electoknetc equaton (geen). The fee chage densty (souce tem of (14)) s extuded fom the 1D geomety nto the 2D geomety fo the solvng of the Nave-Stokes equaton (ed). L +ψ -ψ Fgue 12. quatons couplng fo the AC electoosmoss model. The electodes shown as black hashed aeas do not take pat of the doman computaton. 4.3 Numecal esults (18) AC complex electoknetc equaton (17) (19) usng L half electodes M equaton (14) +ψ -ψ q- Nave-Stokes equaton (21) (22) -q +q q+ The followng fgues gve examples of numecal esults obtaned wth the AC electoosmoss model descbed n ths pape, fo the case ψ = ±.1V. On Fgue 13, the electc feld and the potental s epesented accodng to the fequency. By lookng at the maxmum value of the potental, we can see that, fo 1 Hz, not all the maxmum potental nsde the bulk s only.4v compaed to the electode value whch s.1v: a bg pat of the electcal potental dop occus nsde the. On the contay, the does not nfluence anymoe the potental dstbuton nsde the bulk when fequency eaches 1 khz.

9 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble Fgue 13. Isopotentals (V) and electc feld vectos nsde the bulk fo ψ = ±.1V, 1 Hz (uppe) and 1 khz (lowe). The next Fgue shows the coespondng velocty feld (whte) and ts magntude (sovalues) fo the 2 fequences. Maxmum velocty s eached at the electode suface (ed aeas), nea the gap. The flud flow dstbuton s vayng wth fequency but seems n good ageement wth the Geen s obsevatons [4]. Fgue 14. Velocty magntude (m/s) and velocty vectos nsde the bulk fo ψ = ±.1V, 1 Hz (uppe) and 1 khz (lowe). The maxmum velocty values obtaned fo these two confguatons (.13 m/s at 1 Hz and.32 m/s at 1 khz) seem to be vey hgh when compang them to Geen s measuements [4] (about hundeds of µm/s). Ths s maybe due to the bounday condton type we selected fo the computaton of the flud moton (slp condton). Obvously, a no slp condton should dmnsh the maxmum velocty (.27 m/s at 1 khz), as shown by the followng fgue: Fgue 15. Velocty feld when usng a no slp bounday condton at the nteface between the dffuse laye and the compact laye (1 khz).

10 xcept fom the Poceedngs of the COMSOL Uses Confeence 27 Genoble 5. Conclusons In ths pape, we popose a numecal model mplemented nto Comsol Multphyscs fo the modelng of the ac electoomoss phenomena. In ths model, no empcal paametes ae necessay to compute the velocty feld. The analytcal valdaton of equaton and ts compason wth the M equaton poposed by Klc et al. [3] at low voltages show that Comsol Multphyscs algothms easly handle the hghly nonlnea aspect of these equatons. The weak couplng between the 1D M equaton wth the 2D complex AC electoknetc and the Nave-Stokes equatons has been computed on the ntedgtated electodes mcosystem studed by Geen [4]. The fst numecal esults seem to be n a good ageement wth Geen s esults. Nave-Stokes convegence could be mpoved by usng a nonpmtve set of vaables (steam functon and votcty) nstead of the velocty and the pessue [13]. A moe detaled analyss of the Geen s ntedgtated electodes mcosystem must be contnued to fully valdate the model.. 6. Refeences [7] M.Z. Bazant, K. Thonton, A. Adja, Dffuse-chage dynamcs n electochemcal systems, Physcal evew, (24). [8]. Body, lectomagnétsme, théoes et applcatons, ISBN , 199. [9] P.W. Atkns, Physcal Chemsty, 5 th edton, ISBN , [1] Conway, J. H. and Sloane, N. J. A. Sphee Packngs, Lattces, and Goups, 2nd ed. New Yok: Spnge-Velag, [11] P. Pham, A.S. Laea, R. Blanc, F. Revol- Cavale, I. Texe, F. Peaut, Numecal desgn of a 3D mcosystem fo DNA delectophoess: the pyamdal mcodevce, Jounal of lectostatcs, 65 (27) [12] A. Castellanos, A. Ramos, A. Gonzales, N G Geen, H. Mogan, lectohydodynamcs and delectophoess n Mcosystems: scalng laws, J. Phys. D: Appl. Phys. 36 (23) [13] P. Pham, J.L. Achad, P. Massé, J. Bethe Modélsaton d un écoulement Maangon dans une goutte en équlbe avec sa vapeu, Jounal La Houlle Blanche, vol 5, n 8, 23. [1] L. Renaud, tudes de systèmes mcofludques : applcaton à l électophoèse su puces polymèes, Thèse de Doctoat, Unvesté Claude Benad Lyon1, jullet 24. [2] M.Z. Bazant, K. Thonton, A. Adja, Dffuse-chage dynamcs n electochemcal systems, Physcal evew 7, 2156 (24). [3] M.S. Klc, M.Z. Bazant, Stec effects n the dynamcs of electolytes at lage appled voltages. I. Double-laye chagng, Physcal Revew 75, 2152 (27). [4] N. G. Geen, A. Ramos, A. Gonzales, H. Mogan, A. Castellanos, Flud flow nduced by nonunfom ac felds n electolytes on mcoelectodes. III Obsevaton of steamlnes and numecal smulaton, Physcal evew, (22). [5], A. Ramos, H. Mogan, N G Geen, A. Castellanos, AC lectoknetcs: a evew of foces n mcostuctues, J. Phys. D: Appl. Phys. 31 (1998) [6] S. Tadu, The electcal double laye effect on the mcochannel flow stablty and heat tansfe, Supelattces and Mcostuctues 35 (24)