Effective zero-thickness model for a conductive membrane driven by an electric field

Transcription

1 Effective zero-thickness odel for a conductive ebrane driven by an electric field Falko Ziebert, 1 Martin Z. Bazant, and David Lacoste 1 1 Laboratoire de Physico-Chiie Théorique, UMR CNRS Gulliver 783, ESPCI, 1 rue Vauquelin, F-7531 Paris, France Departent of Cheical Engineering and Departent of Matheatics, Massachusetts Institute of Technology, Cabridge, Massachusetts 139, USA Received Noveber 9; revised anuscript received 11 January 1; published 11 March 1 The behavior of a conductive ebrane in a static dc electric field is investigated theoretically. An effective zero-thickness odel is constructed based on a Robin-type boundary condition for the electric potential at the ebrane, originally developed for electrocheical systes. Within such a fraework, corrections to the elastic oduli of the ebrane are obtained, which arise fro charge accuulation in the Debye layers due to capacitive effects and electric currents through the ebrane and can lead to an undulation instability of the ebrane. The fluid flow surrounding the ebrane is also calculated, which clarifies issues regarding these flows sharing any siilarities with flows produced by induced charge electro-ososis ICEO. Nonequilibriu steady states of the ebrane and of the fluid can be effectively described by this ethod. It is both sipler, due to the zero thickness approxiation which is widely used in the literature on fluid ebranes, and ore general than previous approaches. The predictions of this odel are copared to recent experients on supported ebranes in an electric field. DOI: 1.113/PhysRevE PACS nuber s : b, 8.39.Wj, 5.7.Np I. INTRODUCTION Bilayer ebranes fored fro phospholipid olecules are an essential coponent of the ebranes of cells. The echanical properties of equilibriu ebranes are characterized by two elastic oduli, the surface tension and the curvature odulus 1, which typically depend on the electrostatic properties of the ebranes. Understanding how these properties are odified when the ebrane is driven out of equilibriu is a proble of considerable iportance to the physics of living cells. A ebrane can be driven out of equilibriu in any ways, for instance by ion concentration gradients or electric fields, either applied externally or generated internally. The external application of electric fields on lipid fils is used to produce artificial vesicles by electroforation, as well as to create holes in the ebrane by electroporation 3. Both processes are iportant for biotechnological applications, they are widely used experientally although they are still rather poorly understood. The generation of ion concentration gradients by internal eans is controlled in biological cells by ebrane-bound ion pups and channels, which play key roles in any areas of biology 4. The nonequilibriu fluctuations of ebranes including ion channels and pups were first analyzed in Refs. 5,6 by eans of an hydrodynaic theory. Artificially ade active ebranes inspired by these ideas were then studied experientally 7 9. Several theoretical studies followed, ainly otivated by the question of how to odel nonequilibriu effects produced by protein conforation changes One liitation of existing active ebrane odels is that they do not describe electrostatic effects associated with ion transport in details. In previous papers by our group 14,15, we have addressed this liitation by studying a theoretical odel for a ebrane with a finite conductivity transverse to the ebrane plane due for instance to ion channels or pups using electrokinetic equations Our work copleents Ref. 19, where the correction to the elastic oduli of a ebrane in a dc electric field were calculated using an approach purely based on electrostatics no currents. It is also inspired by Refs.,1. where siilar probles were considered using electrokinetic equations. In contrast to these studies, our approach focuses on the nonequilibriu case, where electrokinetic corrections to the elastic oduli arise due to currents through the ebrane. In particular, a negative correction to the surface tension arises due to capacitive effects, also called Lippan tension. This negative tension leads to instabilities as can be understood fro the high-salt liit 3. A first experiental proof of the destabilizing effect of the electric field on a stack of lipid ebranes was brought by x-ray scattering studies 4. Recently, the lowering of the tension due to electrostatic or electrokinetic effects has been observed experientally with supported ebranes subjected to an ac electric field 5 and in active ebranes 9. The resulting flow fields around the undulating ebrane are interpreted within the fraework of induced charge electro-ososis ICEO 18,6,7. Siilar flow patterns within vesicles subject to ac electric fields have been observed experientally and analyzed theoretically in Ref. 8. The deforation of lipid vesicles in alternating fields in various ediu conditions has been odeled theoretically in Refs. 9,3. All these studies show that lipid ebranes in electric fields present a rich panel of possible behaviors 31,8. This paper extends previous work 14,15, by providing an effective zero-thickness ebrane odel that contains both capacitive effects and ionic currents. In a first attept 14, a zero-thickness ebrane odel has been proposed with the boundary condition BC of zero electric field at the ebrane. Although the shape of the potential was acceptable, the charge distribution had the wrong sign and the elastic oduli were orders of agnitude too sall. A odel with finite ebrane thickness and dielectric constant has thus been considered in Ref. 15, leading to correct signs of the /1/81 3 / The Aerican Physical Society

2 ZIEBERT, BAZANT, AND LACOSTE charge distribution and correct orders of agnitude of the elastic oduli. However, this odel needed approxiations and finally nuerical evaluations. In view of this, we present here an iproved zero-thickness odel, by using the ore realistic BC of a dielectric interface sustaining Faradaic currents 3. Although this Robin-type BC has been introduced in Ref. 15, its consequences were not developed. In particular this odel leads to siple analytical expressions for the corrections to the elastic constants of the ebrane. The odel clearly captures both nonequilibriu effects due to ion currents and equilibriu effects, of capacitive nature. We also calculate the flow field around the ebrane, which has in fact the opposite sign as copared to the one of Ref. 15 for the zero-thickness case, and is thus siilar to standard ICEO flow fields. The presented effective zero-thickness odel for a dc-field driven conductive ebrane is siple enough to be the starting point of ore refined further studies. The work is organized as follows: in Sec. II we describe the equations for the charges in the electrolyte. A special ephasis is put on the boundary conditions which is the crucial point here. Then the base state solution corresponding to a flat ebrane is calculated in Sec. III. In Sec. IV we calculate the leading order contributions to the electric and ion density fields for a spatially odulated ebrane height and analyze the corresponding hydrodynaic flows around the ebrane. Using the boundary conditions for the stress tensor at the ebrane which includes Maxwell and hydrodynaic stresses, we calculate in Sec. V the growth rate of ebrane fluctuations. In Sec. VI, our results are discussed and copared to previous calculations and to related experients. II. MODEL EQUATIONS We consider a steady dc current driven by a voltage V between two electrodes at a fixed distance L, applied to an initially flat ebrane located initially at z=. The ebrane is ebedded in an electrolyte of onovalent ions with densities n + and n. The ebrane has channels for both ion species but is itself neutral no fixed charges at the ebrane. The channels/pups are assued to be hoogeneously distributed in the ebrane and enter only in the effective conductance G, as introduced below. For the effect of nonunifor distributions of channels/pups in ebranes we refer to Refs. 11,33. A point in the ebrane is characterized by its Monge representation valid in the liit of sall undulations by introducing a height function h r, where r is a two-diensional in-plane vector. In the electrolyte, the governing equation for the electric potential is Poisson s equation = 1 en+ en. Here e is the eleentary charge and is the dielectric constant of the electrolyte. For the sake of siplicity, we assue a syetric 1:1 electrolyte, so that far away fro the ebrane n + =n =n. We also assue that the total syste is electrically neutral. 1 The densities of the ion species are assued to obey the Poisson-Nernst-Planck equations for a dilute solution t n + j =, with ionic current densities j = D n n e, k B T 3 where k B T is the theral energy. We have assued that both ion types have the sae diffusion coefficient D, and neglected various corrections for concentrated solutions 34. We consider a steady-state situation and use the Debye- Hückel approxiation by linearizing the concentrations n =n + n, leading to = e n+ n, n en k B =. T 5 For syetric binary electrolytes, it is useful to introduce half of the charge density 3,35, = e n+ n. 6 The su of the two ionic concentrations, n + + n, turns out not to be a relevant variable since it is decoupled fro the field at sall applied voltages 35. Moreover, since we have considered a steady state and a syetric situation, there is no net particle current. We also define j = j+ j = D + e n k B T, 7 which represents half of the electric current density, and we obtain the equations =, + e n k B T =. 9 Cobining Eqs. 8 and 9 leads to =, where = e n 1 k B T and 1 = D is the Debye length that defines the characteristic length scale for charge relaxation in the electrolyte. Boundary conditions At the electrodes located at z= L, we externally ipose the voltage leading to z = L = V This BC is oversiplified for real electrodes, since it neglects interfacial polarization across the double layers pass

3 EFFECTIVE ZERO-THICKNESS MODEL FOR A ing Faradaic currents 3,36, which akes the voltage iposed across the electrolyte, outside the double layers, different fro the applied voltage. Since we focus on the ebrane dynaics, however, electrode polarization is inconsequential, and the voltage V in the odel siply serves as a eans to apply a steady dc current, which could be directly easured or iposed in experients testing our theory. We assue in the following that the distance between the electrodes is uch larger then the Debye length, L D = 1. In that case, the bulk electrolyte is quasineutral, n + =n =n, with negligible charge density copared to the total salt concentration, z = L =. 1 Since the conductivity of a quasineutral electrolyte is constant, the applied unifor current is equivalent to an applied electric field far fro the ebrane. As we will see, the BC at the ebrane is crucial to recover the correct physical behavior. Let n be the vector noral to the ebrane. In the siple zero-thickness odel proposed in Ref. 14 the Neuann BC, n z=h =, 13 was used for the potential, corresponding to a vanishing noral coponent of the electric field at the ebrane. Thus, the dielectric isatch between the electrolyte and the ebrane was accounted for only approxiatively. When copared to the full finite thickness calculation, the agreeent was poor. To address this issue, a ore general Robintype BC was introduced 15 n z=h + = n z=h = h + h, 14 where = d 15 is a length scale that contains the ebrane thickness d and the ratio of the dielectric constant of the electrolyte,, and of the ebrane,. This BC with one side held at constant potential was originally developed for electrodes sustaining Faradaic current 3,36 38 or charging capacitively 18,35. In that context the analog of our ebrane is a Stern onolayer of solvent olecules or a thin dielectric coating, such as a native oxide, on a etallic surface, and is denoted S. Note that this BC has also been used in Ref. 1. The odified boundary condition 14 introduces a new diensionless paraeter 3, = = = D /d = C D. 16 C For a blocking or ideally polarizable surface, which does not pass noral current and only allows capacitive charging of the double layer, this paraeter controls the relative iportance of the capacitance of the surface here, the ebrane C = /d copared to that of the diffuse part of the double layer, C D =. The BC Eq. 14 then iplies that these capacitances are effectively coupled in series in an equivalent-circuit representation of the double layer +ebrane surface 35. For a surface sustaining noral current, either by electron-transfer reactions at an electrode or by ionic flux through a ebrane, the situation is ore coplicated. It can be shown that the sae electrostatic BC Eq. 14 reains valid for a thin dielectric layer, as long as it has zero total free charge 34, which is typical for ebranes containing a high density of fixed counter charge. However, the sae paraeter no longer plays the role of a capacitance ratio. Instead, it controls the effect of diffuse charge on the noral current, the so-called Frukin correction to reaction kinetics in electrocheistry, reviewed in Ref. 36. Two distinct regies were first identified in Ref. 3 in the context of electrolytic cells and recently extended to galvanic cells 36 : i the Helholtz liit 1, where ost of the double layer voltage is dropped across the surface or ebrane and the diffuse-layer has no effect on the current, and ii the Gouy-Chapan liit 1 where the diffuse layer carries all of the voltage and thus deterines the current. In the Helholtz liit, the Robin BC Eq. 14 reduces to the Neuann BC Eq. 13 used in Ref. 15, so that paper analyzed the liit where the diffuse charge is sall and has little effect on the current. In this paper we consider the general case of finite. III. BASE STATE The base state of the proble is a flat ebrane. The electric field, assued to be perfectly aligned in z direction, is then perpendicular to the ebrane. In the bulk fluid, the syste is copletely characterized by the electrostatic potential z or by the field E z =E z z = z, by the steady-state ion distribution z, as well as by the pressure P z. Inside the ebrane, an internal electrostatic potential z and field E is present. Equations 8 and 9 are readily solved leading to the charge distribution and to the potential z = z = e z ; z e z ; z, 17 j D z L e z + V ; z j D z + L + e z V ; Here j = j is the electric current density and z. 18 = z =+ z = 1 z= 19 represents the jup in the charge density across the ebrane. We have introduced the notation f z=a = f z = a + f z = a, by which we denote the jup of the field f at position z=a. Note that the jup in the charge density at the ebrane

4 ZIEBERT, BAZANT, AND LACOSTE can be interpreted in ters of a surface dipole localized on the ebrane. The existence of this surface dipole is the physical reason for the discontinuity of the potential at the ebrane. At the ebrane, the Robin-type BC reads z z=+ = z z= = z=. 1 Although the ebrane has a zero thickness in this odel, one can still define an internal field E and an internal potential z. This is done by keeping a finite thickness d at first, and then take the liit d, see Ref. 15 for details. Continuity of the potential at the ebrane boundaries then iplies a constant internal field E = z= /d and the internal potential z = z= z/d. Using Eq. 18, we obtain E = 1 d j L D + V. Now one has to deterine and j. Using Eq. 1, one obtains the jup in the charge density V j D L + =. 3 + Two rearks on this derivation are in order: first, Eq. 3 illustrates that the asyetry of the charge distribution results either fro the accuulation of charges due to the applied voltage a capacitive effect proportional to V, present also for nonconductive ebranes or due to ionic currents across the ebrane a nonequilibriu effect proportional to j present only for conductive ebranes. Second, the expressions for z, z, derived above are independent of the response of the ion channels and thus reain unchanged if a nonlinear ion channel response is used. Only the expression for the current j, that enters as a paraeter, will be affected by such a nonlinearity. For precisely that reason, the general for of the base state does not depend on the echanis that has created the asyetry of charge distribution or the ionic currents external due to an applied field or internal due to pups. To deterine the current density j at the ebrane position, we use a linear response approach for nonlinear ionic response, see for instance Refs. 4,1,39 j z= = G e z=, 4 where is the cheical potential per particle and G is the ebrane conductance per unit surface across the ebrane, not in plane. We assued equal G for both ion species. In the bulk one has j = edn k B T z, which leads to the usual expression for the electro- cheical potential, =k B T +e. By equating en j = j z= = e G k BT en z= + e z=, we finally arrive at the siple expression 5 j = j = GV D GL This relation is consistent with the usual electric circuit representation of ion channels in a ebrane that is surrounded by an electrolyte 4,14. As far as the sign convention of the currents is concerned, the cathode toward where the cations drift is located at z= L/ and the anode at z =L/. Thus j is positive, in accordance with the usual convention for transport of positive charges fro the anode to the cathode. Insertion of j into yields = V 1 D G 1+ D GL +. 7 Note that this derivation is general and holds for any value of =. The jup in the charge distribution at the ebrane is positive equivalent to a surface dipole oriented in the +z direction,, for poorly conductive ebranes when D/ G. This corresponds well to the case of biological ebranes see Sec. VI, which are typically uch less conductive than the surrounding electrolyte. In this case, positive charges pile up at the side of the positive electrode and due to syetry, negative charges will do the sae on the other side, leading to. For highly conductive ebranes, this piling up effect does not arise anyore, because positive charges are able to cross the ebrane easily. This results in a negative jup of the charge distribution at the ebrane equivalent to a surface dipole oriented in the z direction when D/ G. The threshold on ebrane conductance per unit area is given by the conductance of a layer of electrolyte of thickness equal to the ebrane thickness ties the ratio of dielectric constants, i.e., G c = e n D/ k B Td. This threshold G c is uch too high to be accessible with biological ebranes even in the best conditions of low salt. However, with artificial ebranes of very high conductivity, this effect ay be observable. We now understand why the zero thickness odel with the siple BC, Eq. 13, is not realistic for biological ebranes. Indeed, it corresponds to =, iplying. We note that by perforing the liit in Eq. 7, one regains the results for the zero-thickness odel of Ref. 15 with BC Eq. 13. To coplete the description of the base state, we have to consider the total stress tensor ij = P ij + i v j + j v i + E i E j 1 ije 8 at the ebrane. It contains the pressure, the viscous stresses in the fluid and the Maxwell stress due to the electrostatic field. We denote by the viscosity of the electrolyte and by v its velocity field. The electric field is given by E =

5 EFFECTIVE ZERO-THICKNESS MODEL FOR A In the base state, where the ebrane is flat and the electric field is oriented in z direction, fro = we get z P = z z = z. By using Eqs. 17 and 18 and iposing P z =, 3 this is readily solved leading to P z = 4 j D e z + e z, 9 and siilarly with z z for z. For the stress we thus get zz, z = zz, z = 4 D j. 3 Note that the stress is constant and is due to the current density j, with no contributions fro the induced charges. At the ebrane, the stress is balanced. In addition to the part of the stress tensor due to the electric field in the electrolyte, there also is the part due to the electric field inside the ebrane, already entioned above. For the force balance in the base state however, this contribution vanishes is thus not iportant. IV. LEADING ORDER CONTRIBUTION OF MEMBRANE FLUCTUATIONS In the following, we derive the corrections to the base state to first order in the ebrane height h r. Fro such a calculation, we obtain the growth rate of ebrane fluctuations by iposing the BC for the noral stress at the ebrane. Fro this growth rate, we then can identify electrostatic and electrokinetic corrections to the elastic oduli of the ebrane. A. Electrostatic proble We use here the quasistatic approach 15,19 by assuing that ebrane fluctuations are uch slower than the characteristic diffusion tie D = 1 of the ions to diffuse on a D Debye length. With the definition of the Fourier transfor f k,z = dr e ik r f r,z, we expand the electric field and the charge density as k,z = z + 1 k,z, k,z = z + 1 k,z, 31 3 where k lies in the plane defined by the ebrane and z, z are the base state solutions given by Eqs. 17 and 18. Let us introduce l, the inverse characteristic length scale of the electrostatic potential near the slightly undulated ebrane, defined by l = k As shown in Appendix A, first-order corrections to the potential and charge density read 1 k,z = h k e lz, l 1 k,z = h k e lz, 34 l for z and a syetric expression i.e., with e lz for z. B. Linear hydrodynaic flow When the ebrane starts to undulate with sall aplitude h r, a flow is induced in the surrounding electrolyte. Using again the quasistatic assuption and low Reynolds nuber, this flow is governed by incopressibility and the Stokes equation, v =, p + v + f =, 35 where f is a body force density due to the electric field. Introducing the triad 1,4 of unit vectors kˆ,nˆ,tˆ with kˆ =k /k, nˆ =ẑ and tˆ=kˆ nˆ,weget z v z + ik v =, ik p + z k v + f =, z p + z k v z + f z =, z k v t + f t =. 39 Two forces drive the flow: one is given by the coupling to the ebrane, enters via the BC, and is discussed below. The second one is the bulk force due to the electric field acting on the charge distribution and reads f=qe=. Note that one has to use the total charge density, Q=. To leading order in the ebrane height, this driving force is f= 1 1 +O h with coponents f = z ik 1 k,z, f z = z z 1 k,z 1 k,z z z, 4 and f t =. Because of the latter, Eq. 39 is decoupled and trivial. To solve for v z, v, and p we proceed as follows: using incopressibility for the perpendicular fluid velocity, one solves for the pressure. Insertion into Eq. 38 then yields a single equation for v z, for z z k v z = k 1 z As the driving ter on the r.h.s. is a constant ties e lz due to 1, by iposing v z z = the solution is of the for v z = B + Cz e k z + Fe lz. 4 The coefficient F is deterined by the driving force f, but two ore BCs are needed. At the ebrane, continuity of the noral velocity iposes v z + =v z = t h r =sh k where we have introduced the growth rate for e

6 ZIEBERT, BAZANT, AND LACOSTE brane fluctuations, s, fro the teporal Fourier representation h t e st. Note that s is also a function of k. The continuity of the tangential velocity, v + =v =, together with the incopressibility iplies a second BC for v z, naely z v z += z v z =. With the notation = 4 j D 4, 43 which quantifies the aplitude of the ICEO flow, see below, the velocity and pressure fields in the doain z read v z z = h k s 1+k z e k z k k z k l k z e k z k l l e lz, 44 v z = h k ik sze k z 1 k z + k z l e k z + e lz, 45 p z = h k k s k k l e k z + e lz + 4 l e l+ z. 46 The solutions for z can be obtained by syetry operations: v z z is obtained by perforing the irror operation with respect to the plane defined by the ebrane, z z, in the forula for v z z. Siilarly v z and p z are obtained by doing this operation on v z and p z, respectively. In the absence of electric effects, ==j, one gets the typical flow induced by a ebrane bending ode 41,4. An additional flow field due to the ebrane currents arises that has the for of an ICEO flow 18. A detailed discussion of this effect is postponed to Sec. VI, but we stress that the additional flow is purely due to ebrane conductivity since j G, and is a nonequilibriu effect. For nonconductive ebranes this flow vanishes but there still is charge accuulation in the Debye layers, leading via the pressure field, Eq. 46, to corrections to surface tension and bending rigidity proportional to. V. GROWTH RATE OF MEMBRANE FLUCTUATIONS To discuss the dynaics of the ebrane, we still have to deterine the growth rate s of the ebrane. The elastic properties of the ebrane are described by the standard Helfrich free energy F H = 1 d r h + K h, 47 where is the bare surface tension and K the bare bending odulus of the ebrane. Force balance on the ebrane iplies that the restoring force due to the ebrane elasticity is equal to the discontinuity of the noral-noral coponent of the stress tensor defined in Eq. 8 zz,1 z= = F H h r = k K k 4 h k. 48 We should stress that the coupled electrostaticshydrodynaics proble under investigation can not be forulated only in ters of bulk forces, i.e., f and the divergence of a stress tensor, because the hydrodynaic and Maxwell stress tensors enter the BC Eq. 48 explicitly. For this reason, the force localized on the ebrane surface is a priori unknown, i.e., ust be deterined by BCs for the velocity and the stress. Equation 48 deterines the growth rate s=s k entering the noral stress difference. Details of the evaluation of zz,1 z= can be found in Appendix B. After isolating s and expanding in wave nuber k, the growth rate s k finally has the for k s k = k k K + K k The electrostatic corrections to the surface tension, = +, and to the bending odulus, K= K + K have been decoposed into outside contributions due to the Debye layer, index and inside contributions due to the voltage drop at the ebrane, index. They are given by = j 4 D, 5 K = 3 5 for the contributions due to the Debye layers and by = E d, d K = E 1 d E 3, 53 for the contributions due to the field inside the ebrane, cf. Eq.. InEq. 49, we also obtain a purely nonequilibriu correction = 4 j 5 D =. 54 It corresponds to a ter proportional to k 3 in the effective free energy of the ebrane, which is forbidden for an equilibriu ebrane but allowed in nonequilibriu. This particular contribution arises due to the electro-osotic flows around the ebrane cf. Sec. VI C, as can be shown fro a siple calculation using the Helholtz-Soluchowski equation for the electro-osotic slip velocity near the ebrane 15. We should ention that an independent check of Eqs. 5 and 5 is provided by a direct integration of the lateral

7 EFFECTIVE ZERO-THICKNESS MODEL FOR A pressure profile as shown in Appendix C. This route avoids the consideration of hydrodynaics but is liited to the calculation of the surface tension correction. Σ -.1 a) VI. DISCUSSION Let us discuss the corrections to the ebrane oduli, Eqs and copare the to previous theoretical work on this question. Two particular liits have been considered before: first, the high-salt liit of a conductive ebrane. In this case, one finds that the correction to the surface tension is only due to the inside field, since V and d, in agreeent with Ref. 3. The corrections to the bending odulus were not considered in that reference, but can be obtained fro the present calculation as K and K 1d, siilar to the nonconduc- V tive case. In this liit, we get a siple expression for the ratio K / = d /1, which is thus independent of the applied voltage and proportional to the square of the ebrane thickness. While this siple dependence on the thickness is restricted to the high-salt liit where d is the only relevant length scale, the independence of voltage is general: the ratio K/ is always independent of the applied voltage, since, j, and E are proportional to V. This is a consequence of our use of a linear ohic response for the ebrane, linear electrostatics Debye-Hückel approxiation for the electrolyte and of the assuption that the ebrane does not carry any fixed charges. A dependence of the ratio on applied voltage, if it could be observed experientally, would iply the violation of one of these assuptions. The second situation considered in the literature is the case of a nonconductive ebrane j =, with an arbitrary aount of salt 19. For G= j =, the su of the two corrections to the surface tension given by Eqs. 5 and 5 can be expressed as = V 1+ d d 1+ d/, 55 which agrees with the result of 19 note that there, the ebrane thickness was defined as d. For the total bending odulus correction, we obtain 3 K = dv 1+ 3 d/ d/ d/ This forula is of the sae for as the one given in 19, but the nuerical prefactors of the finite salt correction ters second and third ter in the upper bracket differ slightly. This is ost probably due to a difference in the boundary conditions which leads to differences in the expression of the potential inside the ebrane, given by Eqs. B9 and B1. Nevertheless, the high-salt liit is correct and the overall shape of K is well captured, as shown in the next section. Thus, our general results, Eqs. 5 53, extend the K b) κ FIG. 1. Panel a shows the electrostatic corrections to the surface tension in units of N 1 and panel b those to the bending odulus units J as a function of units 1 in the nonconductive case, G=. Dashed lines: contributions due to the Debye layer and K respectively. Dash-dotted lines: contributions due to the field inside the ebrane and K respectively. Solid lines: su of both corrections. The figure was ade with the paraeters given in the text and V=1 V, L=1. work of Abjörnsson et al. to the case of a conductive ebrane. A. Effect of salt and ebrane conductivity With the expressions for the jup of the charge density at the ebrane, Eq. 7, for the current density, Eq. 6, and for the internal field, Eq., we can discuss the corrections to the ebrane elastic constants given by Eqs in detail. In particular we obtain the dependence of these elastic oduli on the ionic strength of the electrolyte, and on the ion conductance of the ebrane per unit area, G. We have used the following paraeters: dielectric constants are =8 for the electrolyte and = for the ebrane. The ebrane thickness is typically d =5 n leading to = d= n. The diffusion coefficient of ions is of the order of D=1 9 s 1, and the viscosity =1 3 Pa s. Figure 1 a displays separately the contributions to the surface tension as a function of the inverse Debye length. The dashed line represents the contribution fro the Debye layers, the dash-dotted line represents the contribution fro the field inside the ebrane and the solid line is the su of both contributions. The value of the inverse Debye length, varies fro D =1 for pure water to = D.3 n for 1M NaCl. Figure 1 b shows the respective contributions to the ebrane bending odulus. In this figure we have assued that the ebrane is nonconductive G=. As shown in Fig. 1, the contributions fro the Debye layers doinate for low salt for and for K. For high salt, the contributions fro the ebrane doinate, and approach the liiting values discussed above. In this case of

8 ZIEBERT, BAZANT, AND LACOSTE Σ a) Σ a) K b) κ FIG.. Electrostatic corrections to the surface tension panel a and to the bending odulus panel b as a function of in a slightly conductive case G=.1 S. Dashed lines: contributions due to the Debye layer. Dash-dotted lines: contributions due to the field inside the ebrane. Solid line: su of both corrections. Paraeters as in previous figure except for G. zero conductivity, both the Debye and the inside contribution to the surface tension are always negative, and there is good agreeent with previous treatents for nonconductive ebranes 15,19. Figure displays the corrections to the elastic coefficients in the case where a finite ebrane current j is present, induced by a sall conductance per unit area G =.1 S, for otherwise unchanged paraeters. We find that this rather sall conductance has already a large effect on both oduli: first, the effect of the Debye layers on the surface tension is suppressed and the contribution fro the ebrane is doinating. Second, the overall contribution gets relevant for higher salt than in the nonconductive case. The effect on the bending odulus is even ore significant: although the Debye contribution is still doinating for about 1 8 1, it is uch saller in aplitude than in the nonconductive case around 1 J copared to 1 17 Jat and furtherore becoes nononotonous 15. The effect of ebrane conductance is highlighted in Fig. 3, where the total contributions to surface tension panel a and bending odulus panel b are shown as a function of the inverse Debye length for conductances per unit area in the range G=.1 1 S, and otherwise unchanged paraeters. We find that, except for the high-salt liit, the bending odulus correction tends to be reduced by increasing ebrane conductance. To give soe nubers, for a nonconductive ebrane G= and V=1 V, the jup in the charge density for = is = e. 3 Already a sall value of the conductance per unit area G =.1 S halves the charge density to =8.6 1 e e s and creates the current density j = , or A. Conductances of biological ebranes can be as high as G=1 S, which is the value for a squid axon, corresponding to a density of potassiu channels of K b) κ FIG. 3. Electrostatic corrections to the surface tension panel a and to the bending odulus panel b as a function of for different ebrane conductivities. Paraeters are as in the two previous figures except for G: solid line G=; dashed line G=.1; dashdotted line G=.1; dash-two-dots line G=1; dotted line G=1 in units of S. We also note that the distance between the electrodes L, i.e., the confineent, is a relevant paraeter and influences the shape of Figs. 1 and. Here we have used a acroscopic distance L=1, corresponding to the experients entioned below. If L was instead of the order of icrons, the suppression of the bending odulus correction due to ebrane conductivity would be uch less pronounced and the corresponding figure would becoe siilar to the one given in Ref. 15. Moreover, for high enough ebrane conductivity, the Debye layer contribution to the surface tension can becoe positive, i.e., stabilizing. In fact, the sign of is governed by a factor D G. In a way siilar as discussed in Sec. III concerning the sign of, for G D D +GL there is a sign change, rendering the correction positive for sall. However, the denoinator containing the distance L between the electrodes suppresses this effect for acroscopic distances. It can be seen only if L is sall, e.g., L=1 as used in Ref. 15. We note that the icron scale is particularly relevant to experients with cell ebranes subitted to electric fields. It is also relevant to experients that one could propose to test these ideas using icrofluidics devices. B. Mebrane instability Since the corrections to the ebrane surface tension are typically negative with the exception entioned above, they can overcoe the bare surface tension. At this point, an instability toward ebrane undulations sets in 3. Our theory is able to go beyond previous odeling of this instability still for early stages of the instability, which were liited to the high-salt liit and did not include electrostatic corrections to the bending odulus or hydrodynaic effects associated with the odulus. The linear growth rate of the

9 EFFECTIVE ZERO-THICKNESS MODEL FOR A τ D s(k ) V(k ) a) b) k ebrane fluctuations is given by Eq. 49 and has the for s k = 1 4 k k 1 4 K k 3, k 57 where we have written siply k for k and introduced the effective surface tension and odulus, = +, K =K + K. Figure 4 a shows this growth rate, or dispersion relation, in rescaled units where we scaled the wave vector by, k =k/, and the tie by the typical tie for ions to diffuse a Debye length, D = 1. The paraeters are the sae as in the D previous sections for a nonconductive ebrane, i.e., G =. The control paraeter is the external voltage V. Figure 4 a shows the growth rate for three different levels of the voltage: the dashed line is for V=.7 V, which lies below the threshold of the instability, all wave nubers are daped and the ebrane is stable. The solid and the dashdotted line correspond to V=.75 V and V=.8 V and are above threshold. A certain window of wave nubers k, k ax V has positive growth rates and the ebrane is thus unstable. This window gets larger with increasing voltage. The linear growth will be doinated by the axiu of the growth rate defining the fastest growing wave nuber k fg. Given Eq. 57, one easily calculates k fg = 3K K, k ax = K K, 59 for. The sae inforation given by the dispersion relation can be expressed by the so-called neutral curve which is shown in Fig. 4 b. This curve, given by the solid line, separates the negative below fro the positive above growth rates in the control paraeter-wave nuber plane. If the voltage is 1 s< FIG. 4. a The renoralized growth rate or dispersion relation, D s, as a function of the rescaled wave nuber k =k / for three voltages: V=.7 V dashed line, V=.75 V solid line, V =.8 V dash-dotted line. b The neutral curve solid line separating the regions of s and s, and the fastest growing wave nuber k fg dashed line in the plane voltage vs rescaled wave nuber k =k /. Paraeters as previously except: no conductivity, G=; = ; =1 N 1 ; K =1k B T. below the section of the neutral curve with the voltage axis, the syste is stable. Otherwise a certain band of wave nubers is unstable. The position of the fastest growing ode k fg is given by the dashed line. Since we have the dispersion relation in analytical for, in principle one has forulas for all relevant observables like the threshold voltage V c. In ters of the syste paraeters, however, they are quite lengthy. The threshold voltage is given by the change of sign of the leading order contribution in s k. In the nonconductive case it has the siple for V c G = = d + + d. 6 Since both and j in case of G are proportional to the voltage, as expected the critical voltage scales like V c. In the liit of sall ebrane conductance, one gets to leading order using that L is acroscopic V c = V c G = 1+ 4GL D. 61 Thus ebrane conductance increases the voltage value needed to cross the instability. In the liit of high salt,, one regains the known result V c = d/. The typical wavelength of the ebrane undulations above threshold i.e., the one of the fastest growing ode for paraeters as in Fig. 4 is of order = D, so several ties the Debye length. C. ICEO flows We now discuss the for of the fluid flows which arise near the ebrane when it is driven by ionic currents. Figure 5 c shows the flow field for a high ebrane conductivity and low salt, in the regie where the ebrane is unstable due to the electrostatic correction to the surface tension and thus starts to undulate. This figure was generated by selecting the fastest growing wave nuber k fg, defined in Sec. VI B, and using the respective axiu growth rate s k fg. Since this wave nuber has the fastest growth rate in the linear regie, it will doinate the initial behavior. The shape of the ebrane undulation is represented as the black solid curves in all plots of Fig. 5. The resulting flow, shown in Fig. 5 c is a superposition of two distinct flows: first, the typical flow associated to a ebrane bending ode 41,4 as shown in Fig. 5 a. This contribution corresponds to the ters proportional to the growth rate s in Eqs. 44 and 45. Second, the flow associated with the reaining ters in Eqs. 44 and 45, proportional to. This contribution yields the typical counter-rotating vortices of an ICEO flow 18, as shown in Fig. 5 b. Clearly, the superposition of these two flow contributions, as shown in Fig. 5 c, results in a parallel flow close to the ebrane, in contrast to the usual bending ode flow given by Fig. 5 a. Since the jup of the charge density for biological ebranes and j by definition, the induced flow occurs for this case in the sae direction as in standard ICEO flows. Note that an inverse ICEO flow was obtained in 15, due to the opposite sign of obtained with the siple but

10 ZIEBERT, BAZANT, AND LACOSTE z z z x FIG. 5. Representation of the flows around the ebrane beyond the instability threshold. The orientation of the electric field is toward negative values of z. Panel a shows the flow generated by the ebrane bending instability ters proportional to s in Eqs. 44 and 45. Panel b shows the ICEO flow ters proportional to in Eqs. 44 and 45. Finally, panel c shows the actual flow, which is the superposition of the forer two and results in a strong flow near the ebrane, oriented parallel to the surface. Both axes are scaled by the Debye length 1. Paraeters are as in previous figures except V=3.165 V, =1 7 1, G=1 S and L =1. unrealistic BC Eq. 13. Also, the situation of ICEO flows is less general as suggested earlier: for ost paraeters odest conductivities, not too low salt the flow generated by ebrane bending is usually doinating and hides the sall ICEO contribution. This is due to the fact that the forer is proportional to s which has contributions, while the ICEO flow is j. Thus, to see the situation given by Fig. 5, a high ebrane conductance G is needed. Second, one needs low salt, since otherwise the ebrane instability is shifted to very high voltages. Also, since for acroscopic electrode distances L of order illieter and high conductance the voltage needed to induce the instability is very high, we have used a icroscopic electrode distance L a b c =1. While it ight still be possible to see these flows for higher salt and acroscopic electrode separations, such situations will be clearly far beyond the Debye-Hückel approxiation used so far. The ICEO flows near the ebrane could also becoe relevant once the syste has reached a steady state. Indeed in the case of lipid vesicles for instance, nonlinear effects associated with the conservation of the nuber of lipids on the vesicle 3 guarantee a saturation of the ebrane fluctuations for not too high voltages that ight lead to vesicle rupture, as copared to the case of the planar ebrane considered here. Since the ebrane fluctuations are confined by nonlinear effects and becoe quasistationary in the long tie liit, the syste can reach a well defined nonequilibriu steady state. In this nonequilibriu steady state, fluid flows will persist due to ionic currents going through the ebrane, after the transient flow associated with the ebrane bending ode has disappeared. D. Applications of the odel to experients Recently, Lecuyer et al. 5 have investigated a pair of nearby ebrane bilayers in an electric field by neutron reflectivity. The first bilayer was close to the botto electrode and used to protect the second one fro interacting with the wall. Since the bare values of the elastic oduli were known fro x-ray off-specular experients for a siilar syste 43.5 N 1 and K 4k B T, the surface tension correction was extracted fro the data under the assuption that the bending odulus is not affected by the field. The experients were perfored in an ac electric field at several frequencies. For the lowest frequency 1 Hz, the electrostatic correction to the surface tension was obtained to be 3 N 1. In the experientally probed regie of low salt D O was used as the electrolyte, the electrostatic corrections to the elastic oduli depend rather sensitively on both the aount of salt and on the ionic conductance of the ebrane, as discussed in Sec. VI A. Moreover, in the above experient, the correction to the bending odulus was not easured. Thus we restrain ourselves to a coparison of orders of agnitude only. The distance between the electrodes was about L=1, while the voltage was in the 1 5 V range. With the other paraeters as used above, and assuing that the ebrane is nonconductive, G=, for = and V=1 V, our odel yields N 1 and K 19k B T. The odel thus successfully accounts for the order of agnitude of the electrostatic correction to the surface tension. However, it also shows that the bending odulus increases about five ties. In order to obtain an experiental test of the odel, it would be interesting to easure the correction to the surface tension and to the bending odulus siultaneously. We would also like suggest to carry out experients in which the applied electric field or the ionic strength would be varied. Another interesting possibility would be to study ebranes of different conductivities or thicknesses in an applied electric field. A second field of application of the odel are active ebranes, which are artificial lipid vesicles containing

11 EFFECTIVE ZERO-THICKNESS MODEL FOR A ionic pups such as bacteriorhodopsin 7 9. In these experients, no external electric field is applied. Instead the pups are activated by light to transport protons across the ebrane. In Ref. 9, a lowering of the ebrane tension produced by the activity of the pups has been reported, which could be due to an accuulation of charges near the ebrane, as discussed here. The specificity of that experient is that this charge accuulation would result fro the activity of the pups rather than fro an applied electric field. However, it is difficult to ake a precise coparison between the experients and the present theory, because only the correction to the surface tension is accurately easured and any aspects of the transport of ions are unknown. Nevertheless, if we assue that the passive state of that experient corresponds to a nonconductive ebrane G= and the active state to a ebrane with G=1 S, and if we use a typical transebrane potential of the order of 5 V, we get the sae order of agnitude for the observed tension lowering, N 1, if we account for the rather high aount of salt with We also find that there is no easurable difference for the bending odulus between the active and passive state, as observed experientally. The odel predicts that a current density of j 1 A arises when the pups are active, which corresponds to an overall current of 1pA on a vesicle of size 1. To better copare to the odel, again it would be desirable to have experients in varying conditions ionic strength and conductance of the ebrane, for instance. Another interesting possibility would be to easure the ebrane current and the transebrane potential in the course of the experient, for instance using patch-clap techniques. when the potential satisfies everywhere the condition For the experients on supported ebranes discussed above, one finds that the charge accuulation on the ebrane is too large for this approxiation to hold, since one has e 4k B T. It would thus be relevant to solve the nonlinear Poisson-Nernst-Planck equations and otherwise proceed siilarly as proposed in this work, in order to describe the behavior of ebranes surrounded by high charge densities. Furtherore, it would be interesting to investigate a odel suitable for sall syste sizes, since uch of the results of this paper are based on the assuption that the syste size L is uch larger than all other length scales in the proble, as well as for ore realistic boundary conditions at the electrodes. We would like to thank Thierry Charitat and Pierre Sens for fruitful discussions and Luis Dinis for a careful reading of the anuscript. F.Z. acknowledges financial support fro the Geran Science Foundation DFG, M.Z.B. support fro the U.S. National Science Foundation under Contract No. DMS and D.L. fro the Indo-French Center for the Prootion of Advanced Research under Grant No. 35 and ANR for funding. APPENDIX A: ELECTROSTATICS TO FIRST ORDER IN h(r ) Fro the expansion of the potential given in Eq. 3, and using the equation of ion conservation and Poisson s equation, one obtains the following equations to first order in ebrane height: z k 1 k,z + 1 k,z =, A1 VII. CONCLUSIONS AND PERSPECTIVE This paper offers a route to describe capacitive effects near a conductive lipid ebrane while keeping the siplicity of the zero thickness approxiation on which ost of the literature on lipid ebranes is based. These capacitive effects are the ain player in the corrections to the elastic oduli of ebranes driven by an electric field or by internal pups or ionic channels. The present theory goes beyond available descriptions by including nonequilibriu effects which arise due to ionic ebrane currents. These ionic currents have a siilar for as the ICEO flows studied in the context of icrofluidics and can odify the fluid flows around the ebrane fro usual bending doinated flow. Our approach is sufficiently siple to be the starting point for further generalizations, which could include various nonlinear effects: nonlinear elastic ters associated with the ebrane or the cytoskeleton in case of a biological ebrane, nonlinear current-voltage response of the channels. Also density fluctuations of the ion channels and the experientally sipler case of an ac electric field should be investigated. Further theoretical work is also needed to extend the odel to higher voltage where the Debye-Hückel approxiation breaks down. In fact, this approxiation only holds e 4k B T z k 1 k,z + 1 k,z =, A and for the particle currents at the ebrane one has the condition D z 1 + = 1 z=h G 1 z= + 1 z=. A3 Since we assued L D, we can use 1 k, = 1 k, = far fro the ebrane, and Eqs. A and A3 and the BCs at infinity are satisfied by choosing 1 = 1 and n + 1 = n 1 iplying 1 =en + 1. It follows that to first order in the height, one has a zero flux condition at the ebrane. Accordingly, the zeroth order solution enters in the equations for the first-order solution 1 only via the boundary conditions. It thus reains to solve the Poisson equation A1, z k 1 =. A4 Introducing l =k + one easily gets 1 =A e lz for z and z respectively. To deterine the constants A, one expands