Extended space charge near non-ideally selective membranes and nanochannels
aa r X i v : . [ c ond - m a t . s o f t ] M a r APS/123-QED
Extended space charge near non-ideally selective membranes and nanochannels
Jarrod Schiffbauer, Neta Leibowitz, and Gilad Yossifon Faculty of Mechanical Engineering, Micro- and Nanofluidics Laboratory,Technion - Israel Institute of Technology - Technion City 32000, Israel (Dated: September 13, 2018)We demonstrate the role of selectivity variation in the structure of the non-equilibrium extendedspace-charge using 1D analytic and 2D numerical Poisson-Nernst-Planck models for the electro-diffusive transport of a symmetric electrolyte. This provides a deeper understanding of the underly-ing mechanism behind a previously-observed maximum in the resistance-voltage curve for a shallowmicro-nanochannel interface device [Schiffbauer, Liel, Leibowitz, Park, and Yossifon, submitted toPhys. Rev. E. ]. The current study helps to establish a connection between parameters such asthe geometry and nanochannel surface-charge and the control of selectivity and resistance in theover-limiting current regime.
PACS numbers: 47.61.Fg, 47.57.jd, 82.39.Wj, 82.45.Yz
I. INTRODUCTION
The current-voltage response of fabricated micro-nanochannel devices and ion-selective membranes arein many respects similar. Both exhibit ion-selectivetransport and produce concentration polarization (CP)in adjacent solution resulting in a near limiting sat-uration of the DC current response. At sufficientlyhigh voltages, both exhibit an over-limiting current(OLC) which may be associated with electro-convectivevortices [1–4] among other mechanisms [5–8]. However,there are a number of differences between ion-selectivemembranes, such as Nafion, and fabricated nanochanneldevices. The geometry of pores within a membrane areoften irregularly shaped, highly tortuous, and poorlyconnected [9], and the fixed charge density quite high.One critical difference is that, while membranes such asNafion typically exhibit near ideal selectivity across awide range of parameters [10], nanochannel selectivitycan vary appreciably. The similarity between membranesand nanochannels implies that a non-ideal membranemodel can serve as a simple model for electro-diffusivetransport through a nanochannel.Previous studies have shown variation in selectivityin the under-limiting regime [11] and demonstrated therole of non-ideality in rectification due to diffusion-layerasymmetry [12]. Other studies concern the role ofa non-equilibrium extended space charge (ESC) ontransport of competing counter-ions through an idealmembrane [13], or consider space-charge dynamics fora binary electrolyte through a non-ideal membrane inthe over-limiting current (OLC) regime [14]. However,these previous studies omit the first-order correction tothe salt-concentration in the space charge regime andits coupling to the selectivity. Here we demonstrate thenecessity of including these terms in capturing the effectsof changing selectivity on the resistance maximum. Thisis expected to have consequences for the observabilityof the resistance maximum in real systems, where otherOLC mechanisms can enhance or compete with the ESCdriven resistance maximum [15]. Furthermore, it may have implications for the control of selectivity and ESCstructure in practical applications, such as bio-molecularsensing, analyte pre-concentration, or electrokineticdesalination.
II. THEORY
In the following, we consider one-dimensional steady-state electro-diffusive transport of a symmetric, binaryelectrolyte through a system consisting of two solutionlayers flanking a non-ideally cation-selective membrane(see Fig. 1). We omit effects such as surface conductionand fluid-flow for simplicity. In particular, we focus ourattention on the modulation of charge-selectivity in theover-limiting regime and its effect on the previously ob-served resistance maximum [15]. Using the steady-state(non-dimensional) Poisson-Nernst-Planck (PNP) equa-tions as a starting point, the ion concentration field isgiven by, D ( x ) (cid:20) − dc ± dx ∓ c ± ( x ) dφdx (cid:21) = j ± (1)for each species ± , and the Poisson equation describesthe electric potential − ǫ d φdx = c + ( x ) − c − ( x ) − N ( x ) (2)where the (negative) fixed charge density is N ( x ) = N for x ∈ {− , } and and 0 elsewhere. Lengths have beenscaled by the membrane half-thickness, ℓ , concentrationby the bulk concentration, c o , fixed at the ends, and theelectric potential by the thermal voltage, kT /e = 0 . D ( x ) = 1 in the electrolyte, but in general may have adifferent value, D m , in the membrane interior. The ap-plied field, hence electric current, is directed towards theright (positive) so the ESC sub-layer forms at the x = − FIG. 1: Basic geometry for the 1D model. Regions a and d arerespectively the depleted and enriched quasi-electroneutraldiffusion layers, region c is charge-selective, and region b ap-proximates the width of the ESC. length, ǫ = p εε o kT /e Lc o is taken as a small parameter, ǫ ≪
1. Its appearance in Eqn. 2 multiplying the highestderivative of the potential introduces the possibility ofboundary layers and singular perturbation methods aretypically used to obtain an analytic solution to the fullproblem [16].The solutions to the 1D problem are well-knownin various approximations. Here we review results rel-evant to the non-ideal case. The problem is re-cast interms of c = ( c + + c − ) / J + = ( j + + j − ) /
2, and charge flux density J − = ( j + − j − ) /
2. The two flux densities are related bythe selectivity factor G = J − /J + and the electric currentdensity is I = 2 J − . While the full solution requires ei-ther singular perturbation theory or numerical methods,a useful first approximation may be obtained by takingthe leading-order solution, or equivalently, setting ǫ = 0,which is tantamount to imposing local electro-neutrality(LEN). These locally electro-neutral solutions for the saltconcentration in the 1D problem are bi-linear functionsof the salt-flux density and the position, c l ( x ) = 1 − J + ( L + x ) , (3)in the depleted (left) side and c r ( x ) = J + ( L − x ) + 1 (4)in the enriched (right) side. The potentials are given bythe integrals over the inverse concentration, or φ ( x ) = G ln [1 − J + ( L + x )] (5)for the depleted region and φ ( x ) = G ln [ J + ( L − x ) + 1] − V (6)for the potential in the enriched region. The difficultywith the LEN approximation can be seen clearly inEqns. 3 and 5. As the current approaches a value cor-responding to the salt-flux density J + ,lim = 1 / ( L − x = − x ∈ {− , } , we assumelocal quasi-electro-neutrality to hold to arbitrarily highcurrents. So Eqn. 2 reduces to c + = c − + N . The ques-tion of practical limits on this condition, especially in thecontext of the micro-nanochannel interface, is beyond thescope of the present study. The governing equations forthe salt concentration and potential in the membrane in-terior are obtained from Eqns. 1 and 2, dc m dx + N dφ m dx = − J + D m (7)and dφ m dx = − J − D m c − m . (8)These are integrated to yield the following transcendentalequation for the concentration profile,[ c m ( x ) − c m ( − N G ln h c m ( x ) − NG c m ( − − NG i = − J + D m ( x + 1) , (9)and φ m ( x ) = − J − D m F ( x ) + φ m ( −
1) (10)for the electric potential inside the ion selective region,where the integral F ( x ) = Z x − c − m ( x ′ ) dx ′ (11)is defined for convenience.For salt-flux densities lower than the limiting value, theresponse of the system may be obtained from the bound-ary conditions at x = ± L by assuming continuity of theelectrochemical potentials across the quasi-equilibriumelectric double layers (EDLs) between each region, asfirst proposed by Kirkwood [18]. Strictly speaking, thisis an assumption of local, homogenous thermodynamicequilibrium between two physically distinct phases, sep-arated by a defining, infinitesimally thin Gibbs dividingplane. Because the EDLs are quite thin and the struc-ture appears to maintain a quasi-equilibrium characterunder current, this approximation may be employed innon-equilibrium scenarios. The extent of the validity ofthe internal quasi-equilibrium structure has been exam-ined thoroughly and shown to be maintained well intothe non-equilibrium regime [16]. The system retains aquasi-equilibrium EDL as the innermost sub-layer, withthe ESC developing as an adjacent structure [14]. Thusone should always be able to impose, at least approxi-mately, a continuity condition to jump across this innerEDL boundary, resulting in a Donnan-like equilibriumbetween the interior of the ESC (adjacent to the thinEDL) and the membrane interior. We employ this as-sumption herein.For the full solution, we follow the usual procedurefor obtaining a master equation for the scaled E-field, E = − ǫ∂φ/∂x (see for instance Ref [16].) The main re-sults are the following equation for the scaled E-field, ǫ d Edx − E J + E ( x − x o ) = − ǫGJ + (12)and the equation for the cation concentration, c + ( x ) = ǫ E ′ + E − J + ( x − x o ) (13)and the salt concentration, c ( x ) = E − J + ( x − x o ) (14)Thus the concentrations may be calculated once a so-lution for the E-field is obtained. Note the integrationconstant x o . It can be associated with the intersection ofthe extrapolated depleted linear salt concentration profileto zero concentration. At and above the limiting current,this point moves into the region x ∈ {− L, − } , yieldingan estimate of the edge position of the ESC.Asymptotic expansions for the scaled E-field are givenby the following for regions sufficiently far ( ≫ ǫ / ) fromeither side of the point x = x o : E ( x ) = − ǫG ( x − x o ) + ǫ (4 G − G )4 J + ( x − x o ) + O ( ǫ )2 p J + ( x − x o ) + ǫG x − x O ) + O ( ǫ ) (15)The first expansion is valid for regions to the left of x o ,corresponding to the outer edge of the ESC and (at nottoo high a voltage) a quasi-LEN diffusion layer qualita-tively similar to Eqn. 3. The second expansion is validwithin the ESC itself and can be integrated to find thevoltage drop across the ESC to leading order,∆ V ESC = − p J + ǫ ( − − x o ) / (16)Note that while this gives a large ( O ( ǫ − )) contributionto the voltage drop, hence dominating the response interms of voltage, the lowest-order non-vanishing contri-bution to the concentration in the ESC arises from thefirst-order correction. It was shown in [19] that this ap-proach yields a well-defined minimum in the marginalstability of the ESC. Because this contribution affects thestructure and resistance of the ESC and is also coupledto the selectivity of the membrane, we choose to employit calculating the ESC-membrane Donnan condition.A rigorous treatment of the problem (see Ref. [16])involves defining appropriate boundary layer variablesand careful consideration of the behavior of the (trans-formed) parameter x o across the full range of applicableapplied voltages in the limit ǫ →
0. Here we adopt a lessrigorous approach, similar to that employed in [19], to demonstrate the coupling between ESC structure, non-ideal selectivity, and the Ohmic-to-OLC transitional ESCresistance maximum [15]. The idea is akin to solution-patching with a first-order correction and demandingthat Eqns. 3 and 15 yield the same value for the con-centration at our estimated ESC edge, x = x o .Transforming to the boundary layer variables, E =(2 J + ǫ ) / GF and x − x o = (2 J + ) − / ǫ / z , the salt con-centration at the ESC edge is given by c ( x o ) = ( ǫ J + ) / G F (0)4 (17)so that the ESC edge position can be approximated by, x o = 1 J + − L − (2 ǫ ) / G F (0)4 J − / . (18)This requires the value of the E-field at x o , or in termsof the boundary layer variables, the value of F (0), where F is the solution of the transformed Painleve equation, F ′′ − G F + zF = − F ( z ) = − z z ≪ √ zG z ≫ G . The solution for higher values isobtained by employing a (finite-difference) Taylor expan-sion of F ( z ; G ) in δG . F ( z ) is seen to decrease in the ESCwith decreasing selectivity (see Fig. 2). To use this resultconveniently in subsequent calculations, the value of F (0)as a function of G is obtained using a fourth-order poly-nomial interpolation of the results. By evaluating thevalue of x o at the limiting salt flux density, it is shown inFig. 3 that decreasing the selectivity increases the ESCthickness and that longer diffusion length to membranethickness ratios are more sensitive to changes in the se-lectivity. A more pertinent question is how the selectivitychanges couple to the current density itself. This requiresuse of Eqn. 9 to obtain the dependence of selectivity onapplied current. A similar analysis was carried out in [11]up to the limiting current. Here we extend those consid-erations to the over-limiting case.As stated previously, we impose continuity of theelectrochemical potentials, µ ± ( x ) = ln c ( x ) ± φ ( x ) ,be-tween the ESC at x = − x = − c m,ul ( −
1) = q [1 − J + ( L − + N / H L G = = = - -
10 0 10 20012345 zF
FIG. 2: Numerical solution to modified Paineleve equation(Eqn. 19) for several values of G (solid black). Large-z asymp-totes (Eqn. 20) are shown also (dashed black.) The insetshows the interpolated value of F (0) as a function of G , usedin subsequent calculations. L = = - - - - - - - x o FIG. 3: Estimate of the ESC width at the limiting salt fluxdensity as a function of selectivity for ǫ = 0 . and in the over-limiting regime, using the concentration c ( x o ) of Eqn. 17, c m,ol ( −
1) = s N − ǫ J + x o (cid:18) G − (cid:19) (22)and the usual expression for the concentration at the en-riched side, c m (1) = q [ J + ( L −
1) + 1] + N / O ( ǫ √ / ). Were we to neglect this contribution (such as the approach taken in [14]), thequalitative picture would not change.The current-voltage relationship for the over-limitingcase can be obtained simply by modifying the under-limiting (LEN) model [11, 12], V = ln h c m (1)+ N/ c m,ul ( − N/ i − ( G + 1) ln h c r (1) c ℓ ( − i − GJ + D F (1) (24)The above equation, like the internal membrane concen-trations in Eqns. 21 and 22, is obtained from the con-tinuity of electrochemical potentials. However, upon in-spection, it is clear that it may be thought of as a sum ofvoltage drops across the layers in the system. While it isgenerally true that one cannot arbitrarily add responsecoefficients in a multi-layer system [20], one can alwaysadd voltage drops themselves, provided such drops canactually be resolved in a straightforward way. Thus, onemay insert the ESC voltage drop, Eqn. 16 into the sum.It is worth emphasizing that Kirkwood’s argument re-garding the continuity of electrochemical potentials be-tween phases only applies between the membrane inte-rior and the interior of the ESC and across the EDLin the enriched side. Here, the EDLs are in fact thin,quasi-equilibrium boundary layers separating physicallydistinct phases. However this is not the case at the ar-tificial boundary, implied by x o . This is a mathemati-cal artifice corresponding to the poles in the asymptoticexpansions, and not a Gibbs plane between physicallydistinct phases. So it is sufficient to demand continu-ity of concentration, as we did to obtain the estimate of x o . However, it is necessary to modify the concentrationat the edge of the quasi-electro-neutral DL and the DLlength accordingly. Doing so, one obtains the followingapproximate form for the current-voltage relationship inthe over-limiting regime, V = ln h c m (1)+ N/ c m,ol ( − N/ i − ( G + 1) ln h c r (1)1 − J + ( L + x o ) i − GJ + D F (1) + ∆ V ESC . (25) III. RESULTS
The dependence of the selectivity on salt flux densitymay be extracted directly from Eqn. 9 by substitutingthe concentration at x = −
1, using Eqn. 21 or 22 andthat at x = 1 Eqn. 23, for the appropriate regime. Thisis shown in Fig. 4. The ratio of the absolute value of co-to counter-ion flux density is plotted against overall saltflux density for a range of parameters, with an increase inthe ratio corresponding to loss of selectivity. The rangeof salt flux density is the same for all three cases, fromzero to five times the limiting value. The selectivity de-creases as the salt-flux is increased because the differencein co- and counter-ion electrochemical potential acrossthe charge-selective region increases. This corresponds N = = = = J + È j - È (cid:144) È j + È H a L L = = = = J + È j - È (cid:144) È j + È H b L D m = D m = D m = D m = J + È j - È (cid:144) È j + È H c L FIG. 4: Selectivity variation with salt flux density in termsof ratio of co- to counter-ion flux density for various systemparameters where the standard case is N = 10, L = 3, and D m = 1. Each plot shows a comparison of the standard toother cases; (a) varying fixed charge density N , (b) varyingdiffusion length L , and (c) varying membrane diffusivity, D m . to both an increase in conductivity and increasing co-ion concentration within the charge-selective region. Inagreement with previous studies [11], the change in theunder-limiting regime is rather modest. However, sincethe enriched concentration has no imposed upper bound,this difference can be expected to increase as J + exceeds J + ,lim .In Fig. 4a, the change in selectivity is shown for several values of the fixed charge density. Unsurpris-ingly, as the charge density increases–corresponding toincreased restriction of the co-ion flux–both the over-all selectivity and change in selectivity decrease. Thecases in Figs 4b and c bear some relationship to the ef-fects of field-focusing in the sense that increased field-focusing corresponds to both increasing flux density inthe focused region and effective shortening of the diffu-sion length [21]. For different diffusion lengths L , thetotal change in selectivity across the range of salt fluxdensity, from J + = 0 to J + = 5 J + ,lim , is the same for all L . However, the overall selectivity itself decreases withincreasing L because the concentration at the enrichedboundary is higher for the same salt flux density. Thesame is essentially true of varying D m ; the total changein selectivity is independent of the effective diffusivity inthe charge-selective region. However, while the limitingsalt flux density in the diffusion layers is the same for allcases, a decrease in effective diffusivity may be thoughtof as an increase in effective salt flux density inside thecharge-selective region. So a comparably small changein concentration at the enriched side, corresponding to acomparably small total change in electrochemical poten-tial across the membrane, leads to lower overall selectiv-ity without shifting the voltage.To validate our approach, we compare the results ofthe above model with numerical calculations made us-ing COMSOL software, shown in Fig. 5. The agreementbetween the approximate analytic model and the numer-ical model is quite good, diverging appreciably only atthe highest applied voltages where the analytic modelunder-predicts the current density by a few percent. Thetwo models yield essentially identical selectivity responseacross the entire voltage range, showing that the bulk ofthe selectivity change occurs at higher voltages.Concerning the resistance maximum, both the maxi-mum value of the resistance and the voltage at which itoccurs are related to the changes in selectivity, and thustied to the other parameters in the system. The changein selectivity itself does not require an accurate value ofthe concentration in the ESC; similar change would beobtained by simply equating the over-limiting concentra-tion to zero. However, the very existence of the maxi-mum itself arises from the fact that not only is the con-centration non-zero, but increases with current beyondthe limiting value (see Eqn. 17). The basic mechanismbehind the maximum may be understood as follows: Ini-tially, the increase in the extent of the depleted region,as indicated by x o , causes a rapid increase in the ESCresistance. However, as the ESC concentration increases,the effective conductivity of the depleted region increasesboth within the ESC and at the edge of the quasi-electro-neutral region, giving rise to the maximum.By considering the variation of the concentration c ( x o ) (Eqn. 17) along with selectivity variation one canunderstand the results shown in Fig. 6. The parameterscorrespond to the same cases as shown in Fig. 4. Bothtotal resistance and ESC resistance are shown for each H a L È j - (cid:144) j + È
10 100 1000 70000.20.30.40.5 V È j - (cid:144) j + È H b L FIG. 5: Comparison of theoretical model (dashed black) toCOMSOL calculations (solid blue) for a 1D PNP model, bothusing the standard parameter set N = 10, L = 3, and D m = 1.(a) Current-voltage response is shown in the top figure andchanges in selectivity in the bottom (b). Inset in top figureshows discrepancy between approximate model and COMSOLgrows with voltage. case. For the range of parameters considered here, theresistance is almost totally dominated by the ESC resis-tance.Fig. 6a shows the resistance-voltage curves for differ-ent values of the fixed charge density, N . For large val-ues of N , the selectivity is high and, as seen in Fig. 4a,changes little with applied voltage. The co-ion concen-tration inside the charge-selective region remains low andthe counter-ion concentration is governed primarily bythe value of N itself. Thus G remains very close to theideal value, G = 1 for all voltages and the ESC con-centration increases slowly with voltage, being governedprimarily by the slow increase of J / . So the overallresistance is high and the slow change in concentrationimply a maximum resistance at higher voltage than theother cases. For smaller values of N , the change in G (equivalently, the ratio | j − /j + | ) is larger, resulting in in-creasing G faster than the decrease in F (0). Hence thereis a larger increase in the ESC concentration and lowerresistance occurring at a lower voltage for smaller N .The dependence of the resistance-voltage curve on dif- N = = = = H a L L = = = = H b L D m = D m = D m = D m = H c L FIG. 6: Resistance vs. voltage curves, for cases matchingfigure 4. The ‘standard’ case (solid red) has N = 10, L =3, and D m . The total resistance (dashed black) and ESCresistance (solid blue or red) are shown for each case. fusion length L is more indicative of the changes in thelimiting salt flux density and in the enriched concentra-tion than the increased resistance of the longer diffusionlayers. This can be expected to differ from a true 2D/3Dcase, where field-focusing plays a strong role [21], or a sit-uation in which non-uniform fluid flow can alter the de-pleted concentration profile [6, 7]. The maximum occursat higher voltages for longer diffusion lengths because thetotal change in selectivity occurs over a smaller range ofsalt flux density while the actual value of the salt flux FIG. 7: COMSOL geometry sketch, note that L here doesnot correspond directly to that in the 1D model. density is lower for longer L (see the discussion associ-ated with Fig. 4), corresponding to a smaller increase inESC concentration for a given change in total appliedvoltage. Thus it takes a larger overall voltage to reachthe resistance maximum. The reasoning for changing dif-fusivity is somewhat similar in the sense that higher ef-fective salt flux in the membrane for smaller values of D m corresponds to an overall higher selectivity, hence lowerco-ion concentration and flux. However, because it doesnot correspond to an actual modification of the concen-trations or electric potentials adjacent to the membrane,the voltage does not shift.To gain some insight into the influence of field-focusing on the structure of the ESC, selectivity varia-tion, and dc response, a 2D numerical solution for thePNP equations is obtained using COMSOL. The geome-try is shown in Fig. 7. A uniform volumetric charge den-sity, N = 10, is specified in the constricted ‘nanochan-nel’ region. No surface charge is specified on the channelwalls, so that we may study the effects of field-focusingon the ESC in isolation from surface-conduction. Fig. 8ashows the current-voltage characteristics in the main fig-ure, which are qualitatively similar to those obtained foreither decreasing diffusion-length L or decreasing D m inthe 1D model. However, note the selectivity response(inset) shows that the increased field-focusing results inboth higher selectivity initially, and a greater overallchange in selectivity with increasing salt flux (equiva-lently, voltage.) It is important to keep in mind that thenumerical calculations are done under potentiostatic con-ditions, as opposed to the galvanostatic conditions em-ployed in analytic model. Therefore, the voltage rangeis kept fixed and the salt-flux and current densities aredetermined in response to the voltage. Thus the samevoltage range produces a change in selectivity over asmaller range of salt-flux density for an increasing de-gree of field-focusing. The resistance-voltage curve inFig. 8b further illustrates the difference. While the valueof the resistance increases, in this case corresponding toa smaller nanochannel height for a fixed microchannelheight, the resistance maximum shifts to a higher volt-age. This stands in contrast to both decreasing D m anddecreasing L , where by analogy we would expect the in-creased degree of field focusing to correspond to either noshift in the voltage of the maximum, or a shift to lower J + È j - (cid:144) j + È H i LH ii LH iii L H a L H i L H ii L H iii L H b L FIG. 8: Comsol cases L = 2, N = 10, bulk diffusivity ev-erywhere and (i) H = h = 1 corresponding to the standardcase of the analytic model, (ii) H = 1, h = 1 /
2, (iii) H = 1, h = 1 /
4. (a) I-V with selectivity ratio inset, (b) R-V voltages.To better understand what is happening in the 2Dcase, we consider the evolution of the 2D ESC withvoltage. The centerline salt concentration near thenanochannel entrance is plotted in Fig. 9 for voltagesbelow and above the maximum. Note that, as one mightexpect, increasing field-focusing corresponds to a smallerESC at the same voltage. However, the ESC concentra-tion undergoes less of an increase with voltage when thedegree of field-focusing is higher. While this seems at firstcounter-intuitive, it is worth noting that the range in saltflux density for the same voltage range is reduced consid-erably. Thus, at a given voltage, a lower salt flux densitydrives the change in ESC salt concentration leading to aslower increase in ESC concentration with increased fieldfocusing.
IV. CONCLUSIONS
We have presented theoretical and numerical resultsconcerning the connection between variation of selectiv-ity, ESC structure, and dc over-limiting resistance of a H i L H ii L H iii
L H i L H ii L H iii L - - - - - FIG. 9: Depleted salt concentration profile for numerical 2Dcase, evaluated on centerline near nanochannel entrance with L = 2, N = 10, bulk diffusivity everywhere. Solid linesevaluated for V = 1000 and dashed lines for V = 7000 (i) H = h = 1 corresponding to the standard case of the analyticmodel, (ii) H = 1, h = 1 /
2, (iii) H = 1, h = 1 / Acknowledgments
We wish to acknowledge Israel Science Foundation,grant number 2015240, the Stephen and Nancy GrandWater Research Institute, grant number 2017720, forfinancial support. [1] I. Rubinstein and B. Zaltzman, Phys. Rev. E, , 2238(2000)[2] S.M. Rubinstein, G. Manukyan, A. Staicu, I. Rubinstein,B. Zaltzman, R.G.H. Lammertink, F. Mugele, and M.Wessling, Phys. Rev. Lett. , 236101 (2008)[3] G. Yossifon and H-C. Chang, Phys. Rev. Lett., ,254501 (2008)[4] E. Demekhin, V. Shelistov and S. Polyanskikh, PhysicalReveiw E. , 036318 (2011)[5] E.V. Dydek, B. Zaltzman, I. Rubinstein, D.S. Deng, A.Mani, and M.Z. Bazant, Phys. Rev. Lett., , 118301(2011)[6] A. Yaroschuk, E. Zholkoskiy, S. Pogodin, and V. Baulin,Langmuir, ,11710-11721 (2011)[7] C.P. Nielsen and H. Bruus, arXiv:1408.4610v1[physics.flu-dyn] (2014)[8] S.Nam, I. Cho, J. Heo, G. Lim, M.Z. Bazant, G. Sung,S.J. Kim, Phys. Rev. Lett., , (2015)[9] D.J. Gargas, D. A. Bussian, and S. K. Buratto, NanoLett., ,2184-2187 (2005)[10] J.J. Krol, M. Wessling, and H. Strathmann, J. MembraneSci., , 155-164 (1999)[11] R. abu-Rjal, V. Chinaryan, M. Z. Bazant, I. Rubinstein,and B. Zaltzman, Phys. Rev. E, , 012302 (2014) [12] J. Schiffbauer, Doctoral dissertation (2011).[13] V.I. Zabolotsky, J.A. Manzanares, V.V. Nikonenko, andK.A. Lebedev, Desalination, , 387-392 (2002)[14] I. Rubinstein and B. Zaltzman, Phys. Rev. E., , 061502(2010)[15] J.Schiffbauer., U. Liel, N. Leibowitz, S. Park, and G.Yossifon manuscript submitted to Phys. Rev. E. [16] B. Zaltzman and I. Rubinstein, Journal of Fluid Mech., , 173 (2007)[17] I. Rubinstein and L. Shtilman, J. Chem. Soc., FaradayTrans. 2 , 231 (1979)[18] J.G. Kirkwood, “Ion Transport Across Membranes”, ed.Clarke, Acad. Press, N.Y., p. 119. (1954)[19] I. Rubinstein and B. Zaltzman, Phys. Rev. E., , 032501(2003)[20] O. Kedem and A. Katchalsky, Trans. Faraday Soc., ,1941-1953 (1963)[21] G. Yossifon, P. Mushenheim, Y.-C. Chang, and H.-C.Chang, Phys. Rev. E. , 046301 (2010).[22] A. Mani and M.Z. Bazant, Phys. Rev. E, , 061504(2011)[23] I. Rubinstein and B. Zaltzman, Phy. Rev. Lett.,114