Frequency Effects on Periodical Potential and Ion Concentration

Applied Mathematical Sciences, Vol. 5, 2011, no. 33, 1631 - 1647

Frequency Effects on Periodical Potential and Ion Concentration Oscillation in Microchannel with an AC Electric Field Perpendicular to Wall * Yao Zhang, Jiankang Wu (1) and Bo Chen Mechanics Department, Huazhong University of Science and Technology Wuhan National Laboratory of Optoelectronics, Wuhan 430074, China

Abstract Without any approximation, this paper presents a numerical analysis of full Poisson-Nernst-Planck equation to investigate periodical potential and ion transport in microchannel under an AC electric field perpendicular to a wall in broad frequency range. Numerical solutions indicate that behavior of periodical potential and ion concentration oscillation strongly depend on frequency. The study found the follows: (1) In low frequency regions, amplitude of potential and ion concentration oscillations retains constants that are independent of frequency. With respect to applied potential, phase angles are close to zero. Electrolyte solution is fully charged in the microchannel. (2) In moderate frequency regions, amplitude of potential increases, and the phase angle first increases and then decreases, with the frequency increase. The amplitude of ion concentration oscillation decreases, but phase angle increases, with the frequency increase. A maximum phase angle is found at a moderate frequency. Electrolyte solution in the microchannel is partially charged. (3) In high frequency regions, amplitude of potential approaches a steady maximum that is independent of frequency, phase angle of potential decreases to zero, amplitude of ion concentration oscillation approaches zero, and phase angle reaches a steady maximum. Ion concentration is almost uniform in the microchannel. Induced charge density vanishes, and potential distribution shows a linear profile in the microchannel. The electrolyte solution in the microchannel can not be charged. Moreover, a double-frequency Fourier component of ion concentration oscillation is found by spectral analysis.

Supported by National Science Foundation of China, (1)

Corresponding author : Jiankang Wu,

Professor,

No. 10872076,

No. 50805059

email: [email protected]

1632

Yao Zhang, Jiankang Wu and Bo Chen

Keywords: Poisson-Nernst-Planck equation, Microchannel, AC electric field， Ion concentration oscillation

INTRODUCTION Microfluidics is a research frontier of combinative fields associated with fluid mechanics, biology, chemistry, electricity, and heat transfer. Electroosmotic flows (EOF) has become an increasingly efficient fluid transport mechanism in microfluidic systems. Magnitude, direction, and flow profile of EOF in a microchannel can effectively be controlled by applying electric fields perpendicular to a solid wall [1,2,3,4]. Such electric fields can be introduced with the aid of electrodes that are electrically insulated from a liquid and embedded beneath solid-liquid interface. An applied electric field can be steady or transient when used for different applications. In studying the effects of transverse voltage bias on wall potentials, capacitor–resistor boundary conditions were used in most of the reported analyses [5,6,7,8,9]. Recently, increased attention was given to new applications of AC electrokinetic mechanism to pump liquids in microchannels (ACEO) [10,11,12]. AC electrokinetic phenomena can also be applied in various fields of microfluidics [13,14,15,16]. Periodical traveling-wave electric field can be used to create electroosmotic flow in microchannels [17,18,19,20,21]. Critical issue on AC electroosmotic phenomena is the periodical behavior of induced potential and ion transport in microchannels. Asymptotic solution of P-N-P equations describing potential drop and ion transport near a polarized electrode under AC were therefore presented [22, 23], the approximate solutions of which were described in terms of triple-layer structure. Transient electrokinetic response from abrupt application in electric fields on symmetric binary electrolyte between parallel-plate blocking electrodes was analyzed by matched asymptotic expansions [24]. The electrohydrodynamics of the Debye layer in an aqueous binary solution was studied by first-order perturbation theory in applying a modulated AC voltage on a planar insulating wall [25]. Analytic solutions that are in good agreement with the linear theory were presented. Transient currents of electrolytes with high ionic strength in response to a voltage step were investigated [26]. Frequency dependance of the double-layer impedance of metal ion-amalgam electrodes with charge-transfer reaction [27], and at a rough surface [28] were studied. These theoretical works provide fundamental insight on AC electrokinetic phenomena in microfluidics, but involve approximations to some degree. Moreover, most of these works used blocking electrodes that are directly brought to contact with aqueous solutions. In some cases, it is desirable to coat electrode surfaces with thin dielectric layers to prevent Faradaic reaction. The applied potential on the electrodes can be quite different from the induced channel wall potential, both in amplitude and phase angles, due to the presence of the dielectric layer. The induced

Frequency effects on periodical potential

1633

potential in the microchennal strongly depends on the dielectric layer. The effects of the electric layer properties and thickness on periodical potential and ion concentration oscillation are also not so clear When applying an AC electric field, electrolyte ions are periodically attracted to the wall and then repulsed from the wall, thus resulting in ion concentration oscillations, as well as induced periodical charge density and electric potential in microchannels. The relationships of induced potential, ion concentration oscillation, and applied electrode potential are somehow unclear in low to high frequency AC cases. Without any approximation, an exact theoretical solution on the AC electrokinetic phenomena seems extremely difficult to achieve because of the inherently strong non-linearity of the full P-N-P equation. Numerical analysis could effectively explore the physical nature of the AC electrokinetic phenomena in microfluidics. Motivated by this argument, we attempted to perform numerical analysis to investigate AC electrokinetic behavior without any approximation of P-N-P equations. Specifically, this study focused on the effects of frequency on periodical behaviors of induced potential and ion transport in a microchannel under an AC field that was perpendicular to the wall and insulated from the solution by a thin electric layer. The dielectric layer and liquid phase were set up as a continuous domain unlike the multi-layer model. The effects of the electric layer on periodical potential and ion concentration oscillation are detailedly analysed including the amplitude and the phase lag.

1. Governing equations and boundary conditions

An uniform microchannel with transverse AC electric field is shown in Fig. (1）

Electrode Dielectric layer

Channel wall

x

ψ L = ψ 0 cos(ω t )

y

ψ R = −ψ 0 cos(ω t )

E

δ

Fig. 1

W

Systematic sketch of an uniform microchannel with transverse AC electric field

1634


where δ is thickness of the dielectric layer, W is channel width, ψ L , ψ R are the AC potentials applied on the left and the right electrodes. ψ 0 is amplitude of the applied AC electrode potential, and ω is angular frequency. E is the periodical electric field in the microchannel. Generally and in most practical cases, ion transport resulting from flow convection can be neglected within the double layer. The following assumptions were made in this study: (1) The wall potential is below steric limitation [29, 30] and the steric effects of finite ionic size will not be taken into account. (2) The thickness of the dielectric layer δ has the same order of magnitude as with the Debye length, and there is no Faradaic current on the channel wall. (3) The thickness of the stern layer is much smaller than Debye length, and the potential of slip plane (Zeta potential) is approximately equal to wall potential. (4) A symmetry electrolyte solution has been adopted as the working fluid in this study. The diffusivities of the cation and the anion are approximately the same. Electric potential ψ and charge density ρ e are governed by the following Poisson equations:

∇ 2ψ = −

ρe ， ρe = ∑ e ( zi ni ) ε rε 0 i

∇ 2ψ = 0

in liquid region

(1)

in dielectric layer

(2)

where zi is i-type ion valence, e is elementary charge, ε r is the permittivity of the solution, ε 0 is the dielectric constants in the vacuum, and ni is i-type ion number concentration. The ion number concentration is governed by the Nernst-Planck equation:

∂ni + ∇ ⋅ J i = 0, ∂t

J i = − Di∇ni −

Di zi ni e ∇ψ kbT

in liquid phase

(3)

where J i is i-type ion number flux, Di is i-type ion diffusivity, kb is Boltzmann constant, and T is absolute temperatute. For generality, the variables are nondimensionalized as follows [31]: ∂ ∂ (4) i+ j x = kx, y = ky , t = k 2 Dt , ψ = ψ ζ 0 , n = n n0 , ∇ = ∂x ∂y where

ζ 0 = kbT ( ze) ≈ 25 mV , k −1 = λD =

εε 0 kbT 2n0 e 2 z 2

is the Debye length of

the electric double layer, and n0 is the ion concentration of the bulk solution. The dimensionless forms of the equations in the dialectic layer (1)-(3) are written as follows:


1635

∇ 2ψ = 0

(5)

For symmetry ( 1:1) electrolyte solution dimensionless Poisson equation is written as

∇ 2ψ = −0.5(n+ − n− )

in liquid phase

(6)

The dimensionless Nernst-Planck equation is written as ∂n+ + ∇ ⋅ (−∇n+ − n+∇ψ ) = 0 in liquid phase (7) ∂t ∂n− (8) + ∇ ⋅ (−∇n− + n−∇ψ ) = 0 in liquid phase ∂t where n+ = n+ n0 , n− = n− n0 are dimensionless number concentrations of cation and anion, respectively.

2. Set up of the numerical computation The periodical potentials and ion concentration oscillation in this study are essentially one-dimensional problem due to uniformity in channel length. One-dimensional numerical model was used in this study to solve equations (5) to (8) in the united region of the dielectric layer and electrolyte solution using the commercial software, COMSOL Multiphysics, based on the finite element method. Boundary conditions are illustrated in Fig. (2 ） and described as follows: ( J± )y = 0 ( J± ) = 0 y

y

ψ L = ψ 0 cos(ω t )

ψ R = −ψ 0 cos(ω t )

Fig. 2 The boundary conditions of P-N-P equations in microchannel with AC electric field Where ψ L = ψ 0 cos(ω t ) on the left electrode surface and ψ R = −ψ 0 cos(ω t ) on the right electrode surface, where ψ 0 = ψ 0 ζ 0 is the dimensionless amplitude of the AC electrode potential, ω is angular frequency, and normal ion flux is ( J ± ) y = 0 on solid wall. The match condition of the potential on ∂ψ d ∂ψ s . Here, β = ε d ε s is the = ∂y ∂y ratio of the dielectric constants of the dielectric layer to the solution. Initial potential was set to zero and initial ion concentrations were specified at liquid-solid interface is ψ s = ψ d ,

β

1636


n+ = n− = 1 . Time step was set at Δt = T 100 , with T as the period of AC electric field. Computational parameters were specified as follows: W = 20λD , δ =（2 ~ 4）λD , Bulk concentration of KCL solution was n0 = 10−6 M , The diffusivity of K + , CL− was D = 2 × 10−9 m 2 / s , while Debye length was λD = 304nm . Permittivities of dielectric layer and solution are ε d = 2.0，ε s = 80.0 The grids close to the liquid-solid interface were sufficiently refined. A series of grids with different element numbers was tested to ensure that the numerical solutions were independent of the grids. It was found that numerical solutions with (1000 ~ 1500) elements were in fairly good agreement, as shown in Fig. (3). Computation was carried out until a stable and repeatable periodical numerical solution was achieved. All numerical solutions in this present work were recorded after 10 periods from initial time.

3 Non-linear periodical behavior of the ion concentration oscillation in the microchannel When a periodical electric field was applied on electrodes the wall ion concentrations are shown in Fig. (4), in which the electrode potential amplitude

ψ 0 = 150 , the frequency is f =

ω = 103 Hz . The transient ion concentration 2π

profile in the microchannel is shown in Fig. (3). Induced wall ion concentration oscillation is also shown in Fig. (4).

Fig.3

The transient ion concentration profile in microchannel Numerical solution of 1200 elements Numerical solution of 1500 elements


Fig. 4

1637

Periodical wall ion concentration fluctuation

n+ (t)

n− (t)

It can be observed that the wall ion concentration has an asymmetrical periodical oscillation with an average of 1.0, rather than a pure cos(ωt ) behavior. One possible reason may be the non-linear ion response to periodical electric forcing. Cation concentration oscillation is the same as in the anion concentration, but in a reverse phase. Ion concentration oscillation lags behind the electrode potential. Ion concentration oscillation can generally be expressed as n(t ) = A ⋅ P(ωt + α ) (9) Equation (9) is a periodical function with amplitude A and phase angle α . P is a symbol of periodical function. The maximum nmax , minimum nmin , and the amplitude of the ion concentration oscillation A = nmax − nmin are shown in Figs. (5) and (6).

Fig. 5 The maximum and minimum of the wall ion concentration fluctuation versus frequency

nmax

nmin

1638

A


Fig. 6 The amplitude of the wall ion concentration fluctuation versus frequency

It can be observed that nmax , nmin , and A do not vary with frequency in low frequency regions (about f < 103 Hz in the current case). As frequency increases, maximum nmax decreases while minimum nmin increases in the moderate frequency region (about 103 Hz < f < 106 Hz ). Ion concentration approaches bulk ion concentration n0 in a high frequency region (about f > 106 Hz ), the amplitude of the ion concentration oscillation approaches to zero. With decreases of the thickness of dielectric layer, the amplitude of the ion concentration increases, as shown in Fig. (6). Similar behavior is found for the phase angle of the ion concentration oscillation. Phase angle does not vary in low frequency (about f < 102 Hz ), increases with frequency in moderate frequency (about 102 Hz < f < 105 Hz ), and reaches a steady maximum at high frequency (about f > 105 Hz ), as shown in Fig. (7).

α0 Fig. 7 The phase angle of the wall ion concentration fluctuation versus frequency


1639

Moreover, it was found that phase angle of ion concentration oscillation is independent to the thickness of the dielectric layer. Conservation of ion ∂ni ∂J = − i in present study, where J i is periodical transport is expressed as ∂t ∂y ion number flux with dominant frequency ω . Approximately, one may obtain ∂ni 1 ≈ Q(ωt ) , and ni (t ) ≈ ∫ Q(ωt )dt ≈ R(ωt ) + c ∂t ω

(10)

where Q, R are symbols of periodical functions. Amplitude of ion concentration oscillation decreases as frequency increases. In low frequency （ f < 103 Hz ）time scale （ T > 1 f ≈ 10−3 s ）of external electric forcing is larger than the charging time （ λDW D ≈ 10−3 s ）of the electrolyte solution in microchannel. Hence, electrolyte solution in the microchannel can be fully charged. Electrolyte solution can not be fully charged in moderate frequency (about 103 Hz < f < 106 Hz ). In high frequency (about f > 106 Hz ), the time scale of electric forcing is much smaller than charging time, the ions cannot be attracted to wall and resident. The amplitude of ion concentration oscillation approaches zero even though the transient wall potential can be large, as will be seen in the next section. As a result, the charging of electrolyte solution in the microchannel becomes totally impossible in high frequency. Transient ion concentration profiles in the microchannel, with wall ion concentration at its maximum, are shown in Fig. (8). It can be seen that ion concentration oscillation is significant at low frequency but trivial at high frequency. The region of effective ion concentration oscillation shrinks with increasing frequency. By spectral analysis of the wall ion concentration oscillation, a high order Fourier component of 2ω can be observed, as shown in Figs. (9) and (10), which may have resulted from the non-linearity of the P-N-P equation. The ion flux given by electric migration is J i = − ( Di zi e kbT ) ni ∇ψ , wherein ni , ψ have dominant frequency ω ; thus, a high order Fourier component of 2ω can be created in J i and ni . However, the spectral density of the high order component of 2ω is much smaller than that of the dominant component of ω . Ion concentration oscillation may also include higher components with smaller amplitudes, which may be less significant.

1640

n+ ( y )


The

Fig. 8 transient ion concentration profiles in microchannel

Fig. 9 Spectral density of wall ion concentration fluctuation with dominant frequency f 0 = 102 Hz

Fig. 10 Spectral density of wall ion concentration fluctuation with dominant frequency f0 = 104 Hz

4 Periodical behavior of the electric potential in the microchannel


1641

When a periodical electric field ψ L = ψ 0 cos(ωt ) and ψ R = −ψ 0 cos(ωt ) , is applied on electrodes insulated from the solution by the dielectric layer, electric potentials in the microchannel are induced. Electrode potential amplitude is

ψ 0 = 150 while frequency is f =

ω = 103 Hz . It can be expected that the 2π

induced wall potential lags behind the electrode potential, and the amplitude is smaller than in the electrode potential. This can be expressed as ζ w = ζ a cos(ωt + β ) , where ζ a , β are amplitude and phase angles, respectively. It was found that both amplitude and phase angle of the wall potential are strongly dependent on frequency, as shown in Figs. (11) and (12) , wherein electrode potential amplitude ψ 0 is fixed.

ζa Fig. 11 Amplitude of the wall periodical potential versus frequency

β0 Fig. 12 Phase angle of the wall periodical potential versus frequency

It can also be observed that the amplitude of the wall potential is a constant independent of frequency in low frequency; it increases with increasing

1642


frequency in moderate frequency, and approaches a steady maximum in high frequency. Moreover, as the thickness of dielectric layer decreases, amplitude increases. The phase angle of the wall potential is zero in low frequency. In moderate frequency the phase angle first increases and then decreases with increasing frequency. A maximum phase angle can be observed. The phase angle returns to zero in high frequency. Similar to ion concentration oscillation, the thickness of the dielectric layer does not affect the phase angle of the potential in the microchannel. The transient potential profiles in the microchannel in the first half of the period are shown in Fig. (13) for f = 102 Hz , in Fig.(14) for f = 103 Hz , and in Fig. (15) for f = 104 Hz .

Fig. 13 The transient profile of the electric potential in microchannel

f = 102 Hz

Fig. 14 The transient profile of the electric potential in microchannel

f = 103 Hz


1643

Fig.15 The transient profile of the electric potential in microchannel

f = 104 Hz

It can be seen that potential profile rapidly varies only in a region close to the wall in low frequency. The potential-varying region gradually extends to the bulk fluid and potential profiles are straightened as frequency increases. The potential profile evolution in the microchannel with increasing frequency is shown in Fig. (16), wherein wall potentials are at their maximum. The potential is almost a linear variation in high frequency.

Fig. 16 Profile evolution of periodical potential in microchannel with frequency when wall potentials are maximum

5

Conclusion

A numerical analysis based on the full Poisson-Nernst-Planck equation was carried out to investigate the frequency behavior of the periodical potential and

1644


the ion concentration oscillation in a microchannel under an AC electric field perpendicular to the wall. It was found that electric frequency plays a significant role in the periodical charging of electrolyte solutions in microchannels. Primary conclusion is summarized as follows (1) In low frequency regions, the characteristic time of electric forcing is larger than the charging time of electrolyte solution. Adequate time can be accorded to fully charge electrolyte solutions in microchannels. Periodical charging of electrolyte solutions can be equivalent to steady charging, except for periodicial veration in time. The potential amplitude of periodical charging is equal to the potentials of steady charging. The phase angle of periodical charging relative to electric forcing is almost zero. This is referred to as the full charging frequency region. (2) In moderate frequency region, the characteristic time of electric forcing is somewhat less than charging time. Only a small portion of ions can be attracted to the wall. The electrolyte solution in the microchannel cannot be fully charged. The amplitude of the potential increases with increasing frequency. The amplitude of ion concentration oscillation decreases, phase angle of the potential first increases and then decreases, and the phase angle of ion oscillation increase. This is referred to as the partial charging frequency region. (3) In high frequency region, the characteristic time of electric forcing is much less than charging time. Charging of the electrolyte solution in the microchannel is totally impossible. There is no surplus counter-ion on the wall, even though the transient wall potential could be high. Ion concentration is uniformly equal to bulk ion concentration. Induced charge density is almost zero and the potential shows a linear profile in the microchannel. The periodical charging of an electrolyte solution in high frequency can be equivalent to steady charging of a dielectric medium in terms of average time. This is referred to as the non-charging frequency region. (4) Ion concentration oscillation includes a high-order double-frequency Fourier component.

References [1] C. S. Lee, William C. Blanchard, C.T Wu, Direct control of the electroosmosis in capillary zone electrophoresis by using an external electric field, Anal. Chem. 62 (1990), 1550-1552. [2] J. S. Buch, P. C. Wang, D. L. DeVoe, C. S. Lee, Field-effect flow control in a polydimethylsiloxane-based microfluidic system, Electrophoresis. 22 (2001), 3902-3907. [3] Richard B. M. Schasfoort, Field-Effect Flow Control for Microfabricated Fluidic Networks, Science. 286 (1999), 942-945.


1645

[4] M. A. Hayes, A. G. Ewing, Electroosmotic flow control and monitoring with an applied radial voltage for capillary zone electrophoresis, Anal. Chem. 64 (1992), 512-516. [5] Ajdari, A. Pumping liquids using asymmetric electrode arrays, Phys Rev. E 61(2000) R45-R48 [6] N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, Fluid flow induced by nonuniform ac electric fields in electrolytes on microelectrodes. III. Observation of streamlines and numerical simulation, Phys Rev. E 66 (2002), 026305. [7] John Paul Urbanski, b, Jeremy A. Levitan, Damian N. Burch, Todd Thorsen and Martin Z. Bazant, The effect of step height on the performance of three-dimensional ac electro-osmotic microfluidic pumps, J. Colloid Interface Sci. 309 (2007) ,332-341 [8] A. Ramos, H. Morgan, N. G. Green, A. González and A. Castellanos, (2005), Pumping of liquids with traveling-wave electroosmosis, J. Appl. Phys. 97 (2005) 084906 [9] D.N. Petsev, Theory of transport in nanofluidic channels with moderately thin electrical double layers: Effect of the wall potential modulation on solutions of symmetric and asymmetric electrolytes, J. Chem. Phys. 123 (2005), 244907 [10]

A. Ramos, H Morgan, N.G. Green, A. Castellanos, AC electric-field-induced fluid flow in microelectrodes, J. Colloidal Interface Sci. 217 (1999), 420-422

[11] M. Z. Bazant, AC electroosmotic flow, Encyclopedia of Microfluidics and Nanofluidics, Vol. part 14, Spriger, (2008), 1461-1470. [12] M. Z. Bazant and Y. Ben, Theoretical prediction of fast 3D AC electro-osmotic pumps, Lab Chip, 6 (2006),1455-1461. [13] M. Mpholo, C. G. Smith, and A. B. D. Brown, Low voltage plug flow pumping using anisotropic electrode arrays, Sens. Actuators B 92 (2003), 262-268. [14] V. Studer, A. Pepin, Y. Chen, and A. Ajdari, Fabrication of microfluidic devices for AC electrokinetic fluid pump, Microelectron. Eng. 61-62 (2002), 915.

1646


[15] M. Z. Bazant. and. T. M. Squires, Induced-Charge Electrokinetic Phenomena: Theory and Microfluidic Applications, Phys. Rev. Lett. 92 (2004), 066101. [16] T. M. Squires and M. Z. Bazant, Breaking symmetries in induced-charge electro-osmosis and electrophor, J. Fluid Mech. 560 (2005), 65-101. [17] B. P. Cahill, L. J. Heyderman, J. Gobrecht, A. Stemmer, Electroosmotic streaming on application of traveling wave electric fields, Physical Review, E 70 (2004), 036305. [18] A. Ramos, H. Morgan, N. G. Green, A. Gonzalez, A. Castellanos, Pumping of liquids with traveling wave electroosmosis, J. Appli. Phys. 97 (2005), 084906. [19] P. Garcia-Sanchez, A. Ramos, N. G. Green, H. Morgan, Experiments on AC electrokinetic pumping of liquids using arrays of microelectrodes, IEEE Transactions on Dielectrics and Electrical Insulation 13 (2006) ,670-667. [20] J. P. Urbanski, T. Thorsen, J. A. Levitan and M. Z. Bazant, Fast AC electro-osmotic micropumps with nonplanar electrodes, APPLIED PHYSICS LETTERS 89 (2006), 143508 [21] A. Ramos, A. Gonzalez, P. Garcia-Sanchez and A. Castellanos, A linear analysis of the effect of faradaic current on traveling wave electroosmosis, J. Colloidal Interface Sci. 309 (2007), 323-331. [22] Y. K. Suh1, and S. Kang, Asymptotic analysis of ion transport in a nonlinear regime around polarized electrodes under AC , Phys. Rev. E 77 (2008), 031504. [23] Suh Y. K. and Kang S., Numerical prediction of AC electro-osmotic flows around polarized electrodes, Phys. Rev. E 79 (2009), 046309. [24] Martin Z. Bazant, Katsuyo Thornton, and Armand Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E 70 (2004), 021506. [25] N. A. Mortensen, L. H. Olesen, L. Belmon, and H. Bruus, Electro-hydrodynamics of binary electrolytes driven by modulated surface potentials, Phys. Rev. E 71 (2005), 056306. [26] Matthias Marescaux, Filip Beunis, Filip Strubbe, Bart Verboven, and Kristiaan Neyts, Impact of diffusion layers in strong electrolytes on the transient current, Phys. Rev. E 79 (2009), 011502.


1647

[27] P. Delahay, G. G. Susbielles, Double-Layer Impedance of Electrodes with charge-transfer reaction, The Journal of Physical Chemistry, 70 (10) (1966), 3150-3157, [28] T. S. Halsey, Frequency dependence of double-layer impedance at a rough surface Phsycal Review A, 35(8) (1987), 3512-3521, [29] M. S. Kilic, M. Z Bazant, A, Ajdari,. Steric effects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging, Phys. Rev. E. 75 (2007), 021502. [30] M. S. Kilic, M. Z. Bazant, A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations, Phys. Rev. E 75 (2007), 021503. [31] Ali Mansouri, Carl Scheuerman, Subir Bhattacharjee, Daniel Y. Kwok and Larry W. Kostiuk, Transient streaming potential in a finite length microchannel, Journal of Colloid and Interface Science 292 (2005), 567–580. Received: September, 2010