ASYMPTOTIC ANALYSIS ON DIELECTRIC BOUNDARY EFFECTS OF MODIFIED POISSON NERNST PLANCK EQUATIONS

Transcription

1 ASYMPTOTIC ANALYSIS ON DIELECTRIC BOUNDARY EFFECTS OF MODIFIED POISSON NERNST PLANCK EQUATIONS LIJIE JI, PEI LIU, ZHENLI XU, AND SHENGGAO ZHOU Abstract. The charge transport in an environment with inhomogeneous dielectric permittivity is ubiquitous in many areas such as electrochemical energy devices and biophysical systems. We theoretically study the equilibrium and dynamics of electrolytes between two blocking electrodes based on a modified Poisson Nernst Planck model with the dielectric boundary effect. Matched asymptotic analysis shows that a two-layer interfacial structure exists in the vicinity of the interfaces when the dielectric self-energy correction to the potential mean-force is relatively weak. For this two-layer structured solution, the dielectric effect plays the dominate role in the first layer, while the solution in the second layer is mainly determined by the classical Poisson Boltzmann equation. When the dielectric self energy becomes stronger, there is only one interfacial layer which is governed by the modified Poisson Boltzmann equation with the dielectric self-energy correction in the Boltzmann factor. We perform a systematic investigation for symmetric and asymmetric electrolytes on ionic concentrations, electrostatic potential, diffuse charges, differential capacitance, and charge inversion phenomenon, to show the effects of the dielectric inhomogeneity on the solutions near interfaces. Key words. Poisson Nernst Planck equations; Dielectric interfaces; Matched asymptotic expansion; Boundary layers AMS subject classifications. 8C, 8D5, 35Q9. Introduction. The ion transport and distribution in an aqueous solution near interfaces is fundamental to a wide variety of electrochemical applications and biological processes [3, 6, 34]. The ion transport in solutions is usually described through the Poisson Nernst Planck (PNP) theory based on a mean-field approximation. The Nernst Planck (NP) equations model diffusion of ions under the concentration gradient and the electrostatic potential. The Poisson s equation governs the electrostatic potential with the charge density stemming from transporting ions. The classical PNP theory has been successful in many applications [9,,6,7,7,3,3,47], but the theory may fail to accurately predict dynamics and equilibrium distributions of ions in many scenarios when the steric effect, the ion-ion correlation or the dielectric boundary effect plays the role in the system, since it ignores these features due to the mean-field nature. For example, in the presence of dielectric interfaces, the dielectric self energy, which is the interaction energy between an ion and the inhomogeneous dielectric medium, plays an important role in the surface tension of air/water interfaces [44] and other soft materials [58] even for electrolytes at the weak-coupling regime. It has been reported that the PNP theory overestimates the effective channel pore size due to the point-charge approximation of ions and the ignorance of the dielectric self energy that is a substantial energy barrier to ion permeation through a narrow channel [, 54]. The validity of the mean-field approximation in the Poisson- Boltzmann (PB) and PNP theory has been tested by comparing the results with those of Brownian dynamics simulations on narrow ion channels whose pore radii are less than the Debye length [4, 5, 4]. Considerable differences on the concentration School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 4, China. Department of Mathematics, The Penn State University, University Park, PA 68, USA. School of Mathematical Sciences, Institute of Natural Sciences, and Key Lab of Scientific and Engineering Computing (Ministry of Education), Shanghai Jiao Tong University, Shanghai 4, China (xuzl@sjtu.edu.cn). Department of Mathematics and Mathematical Center for Interdiscipline Research, Soochow University, Shizi Street, Suzhou 56, Jiangsu, China(sgzhou@suda.edu.cn).

2 L. Ji, P. Liu, Z. Xu and S. Zhou profiles and ionic conductance demonstrate that the mean-field approximation breaks down for narrow ion channels [4]. Interestingly, significant qualitative improvements can be achieved by incorporating a dielectric self-energy correction to remove artificial shielding effects in the mean-field approximation [5]. Many modified versions of the PNP theory have been put forward in literature to take into account the ignored effects beyond the mean-field approximation. The steric effect of ions are considered by incorporating an excess free energy of solvent entropy[,9,3,38,39], Lennard-Jones interaction kernel[8], or modified fundamental measure theory [46, 56]. The ionic correlations can be described by a fourth-order partial differential equation with a correlation length [8, 35, 49], which can be also viewed as introducing an effective inhomogeneous dielectric permittivity as function of the Laplacian operator. Alternatively, ionic correlations can be taken into account by incorporating the self energy of solvated ions that is obtained by solving the diagonal of Green s function from a generalized Debye-Hückel equation [4, 4, 45]. The effect due to dielectric inhomogeneity can be accounted for under the framework of the self-energy-modified model [53, 55]. Recently, Liu et al. [36] proposed a modified PNP (mpnp) model to consider the Coulombic correlations in variable dielectric media, where the excess free energy from the Coulombic correlation and dielectric effect is obtained by a Debye charging process and is further supplemented with asymptotic expansions to deal with the difficulty arising from finite ionic sizes. In addition to the dielectric inhomogeneity across the boundary, the dielectric coefficient inside electrolytes could depend on local ionic concentrations as well as the electric field [7,,,4,5,33,57],which can play important role in many physical systems. Mathematical analysis based on singular perturbation methods for the understanding of charge-diffusion properties has made much progress in recent decades [, 4 6,9,9,8,9,37,48,5]. In these analysis work, the Debye length is often assumed to be much smaller than the characteristic length scale of the geometries, thus one has a small perturbation parameter for asymptotic expansions, ǫ, which is the ratio between two lengths. The method of matched asymptotic expansions (MAE) has been used to obtain the singular perturbation solutions to the steady-state PNP e- quations [5], to investigate the diffuse-charge dynamics in electrochemical systems with time-dependent applied voltages [9], and to analyze the current-voltage relations for electrochemical thin films with Faradaic reactions [6, 3]. It is also used to study the impact of steric effects on the double-layer charging [8] and the dynamics of electrolytes at large applied voltages [9]. Recently, Wang et al. [5] have studied the PNP equations using the matched asymptotic analysis, and discussed the existence and uniqueness of the solution to the PNP equations with multiple ionic species. This paper employs the method of MAE to investigate the dielectric boundary effect on the charge dynamics of electrolyte solutions between two blocking electrodes. The dielectric self energy is described through the WKB approximation of the generalized Debye-Hückel equation for the system with two parallel dielectric interfaces. An additional parameter that is the ratio between the Bjerrum length and the separation of electrodes is introduced to represent the strength of dielectric boundary effect, and hence two small parameters are present in the modified PNP system. It is observed that the boundary-layer structure strongly depends on the magnitude of the two parameters. We study the parameters in different regimes and perform the matched asymptotic analysis for the leading order solutions. Remarkably, a two-layer structure in the vicinity of electrodes is observed when the dielectric boundary effect is weak. When the dielectric self energy is comparable to the direct ion-electrode

3 Asymptotic analysis on mpnp equations 3 V - V γz - e ε B ε W z - e ε B γz - e x= - D x= x= D Fig... Schematic illustration of the model system. Voltage is imposed on two electrodes located at x = ±D. The dielectric coefficient of electrodes is ε B and that of electrolyte solutions is ε W. The charges outside the electrolyte region represent the induced image charges of a point source due to the dielectric inhomogeneity, which contribute a dielectric self energy under the WKB approximation. interaction, the asymptotic analysis reveals that there is only one layer for inner solution whose governing equations are the modified Poisson Boltzmann equation with the self-energy correction to the potential of mean force in the Boltzmann factor. Results on symmetric and asymmetric electrolytes are systematically investigated, which demonstrate the impact of dielectric boundary effect on the ionic concentrations, electrostatic potential, charge density, diffuse charges, and differential capacitance. Of much interest is that the dielectric boundary effect, no matter attractive or depletive, is able to induce charge inversion for asymmetric electrolytes. The rest ofthe paperis organizedasfollows. In section, we describe the physical setup and the modified PNP equations with dielectric boundary effect. In section 3, singular perturbation solutions by the method of MAE are obtained. In section 4, The asymptotic results on symmetric and asymmetric electrolytes are analyzed and discussed. Finally, concluding remarks are drawn in section 5.. The mpnp model. Consider a binary electrolyte confined between two parallel planar electrodes of separation D; see Fig.. for a schematic view. The valences of the two ion species are denoted by z ± with z < and z >. Their ionic concentration distributions c ± are homogeneous in the y-o-z plane. We describe the dielectric coefficient as a piecewise constant function, { ε ε ε ε(x) = W, x < D, ε ε B otherwise, (.) where ε is the vacuum dielectric constant, ε W is the relative dielectric coefficient of the solvent, and ε B is the relative dielectric coefficient of the electrodes. Since ions have finite sizes, the electrolyte region [ L, L] is actually smaller than [ D, D], where L = D/(ξ) and ξ is a constant to represent the inaccessible layer at the electrodes. Under the setup, one can write the electrostatic free energy per unit area with taking into account the effect of the dielectric boundaries as F free = F mf F ex, where the

4 4 L. Ji, P. Liu, Z. Xu and S. Zhou mean-field free energy reads F mf = ε ε L Φ dxk B T c i (logc i )dx, (.) L i=± with constants e,k B and T being the elementary charge, the Boltzmann constant and the temperature, respectively. The mean electric potential Φ is determined by Poisson s equation with proper boundary conditions imposed at x = ±D, ε ε Φ = i=±z i ec i. (.3) Here c ± are zero for x > L. The excess free energy F ex is given by the Debye charging process [36], L ( ) N F ex = dx dλ λ c i (x)u i (x;λ), (.4) L where U i (x;λ) is the self energy of a particle at the charging state λ, defined by the nonsingular part of the self Green s function, which is governed by the generalized Debye-Hückel equation in three dimensions [4, 5], ε ε(x) G λ (r,r )λ I(x)G λ (r,r ) = δ(r r ), U i (x;λ) = z ie lim [G r λ(r,r ) /(4πε W r r (.5) )]. r Here, I = i=± z i e c i / is the local ionic strength. It is noted that, the Green s function is defined over three-dimensional space with r = (x,y,z), while the self energy is homogeneous in yz coordinates thus U i (x;λ) is just a function of x and λ. We employ the WKB approximation [,5,55] for Eq. (.5). In the WKB approximation, the Green s function is approximated by the screened Coulomb potential from all the image charges of the point source due to the two dielectric interfaces. S- ince the screening length of our system is much smaller than the separation D, the contribution from reflected image charges is neglected and the WKB approximation for the self energy is expressed as, U i (x;λ) = γz i e 8πε ε W i= [ e (Dx)λκ(x) (Dx) ] e (D x)λκ(x), (.6) (D x) where κ(x) = i=± z i e c i /(ε ε W k B T) is the inverse of the local Debye screening length, and γ = (ε W ε B )/(ε W ε B ) describes the dielectric ratio. It is noted that the inaccessible region described by the constant ξ is essential to avoid the singularity of the self energy. Plugging Eq. (.6) into the free energy functional (.4), one can see that the free energy integration on λ can be calculated analytically. A simple calculation gives, µ ex i = δf ex /δc i = U(x;), namely, the excess chemical potential is coincidentially the self energy at the full charging state, λ =. Using the variational approach and the Fick s law leads us to the NP equations, c i (x,t) t J i =, i = ±, (.7)

5 Asymptotic analysis on mpnp equations 5 and the flux density J i = D i [ c i βc i (z i eφu i (x;))], which is coupled with the Poisson s equation (.3) for the electric potential and the self-energy equation (.6) at the full charging state to close an mpnp system. Here β = /k B T, and D ± are the diffusion constants. We assume D = D = D. Eq. (.7) describes the fact that the transport of ions is governed by the diffusion arising from the concentration gradient and the advection arising from the electric potential and self-energy gradient. We study the asymptotic solution to the mpnp equations. In the analysis, initial ionic concentrations are assumed to be uniform and charge neutral, c ± (x,) = c ±,b = z c with c being a characteristic constant. Let l = ε ε W /βe c be a length scale which is proportional to the Debye length, and l B = e /(4πε ε W k B T) be the Bjerrum length. One shall introduce the dimensionless quantities x = x/l, t = td /l L, c ± = c ± /c, ε = ε/ε W, Φ = βeφ, and u = βu i (x;)/zi. Now one has the dimensionless mpnp equations. For simplicity, the tildes over all variables can be dropped and the mpnp system reads c i t = ǫ x [ x c i c i x (z i Φ zi u )], i = ±, (.8) ǫ x Φ = z c z c, (.9) which is subject to the initial-boundary conditions: c ± (x,) = z, Φ(±,t) = V ±, (.) J ± (±,t) =. Here u(x) = γq e (ξx)κ(x) ξx γq e (ξ x)κ(x) ξ x, κ(x) = zc (x)z c (x)/ǫ, and J ± = x c ( i c i x zi Φ z i u). We assume that ǫ = l /L and q = l B /L are two small parameters. As q goes to zero, the PDE system (.8-.9) degenerates to the classical PNP equations. 3. Asymptotic solutions. In this section, the method of MAE is used to solve the initial-boundary value problem of the mpnp equations. We aim to find the leading order expansion in terms of the small parameters ǫ and q. In the analysis, we focus on electrolyte solutions such that the Bjerrum length is much smaller or comparable to the Debye length (q = o(ǫ) or q = O(ǫ)), otherwise it corresponds to a strong-correlated Coulomb system which cannot be correctly described by the modified PNP equations. The case of q = o(ǫ) corresponds to systems in many areas when the surface charge can be very weak or the interface-interface interaction is long-ranged [44,5,58]. The small quality ξ is assumed to be at the same order as q such that the self energy does not blow up. Throughout the paper, we use a bar accent to represent the outer solution, and a hat accent to represent the inner solution of the singular perturbation problem. 3.. Outer solution. When ǫ, Eq. (.8) becomes, c ±, t =, (3.) whichimpliestheleadingorderconcentration: c ±, (x,t) = c ±, (x,) = z. Hereand afterwards, the subscript represents the leading term in the asymptotic expansion,

6 6 L. Ji, P. Liu, Z. Xu and S. Zhou e.g., c ± = c ±, O(ǫ). Summing two equations in Eq. (.8) together and using (.9), one obtains, ǫ 4 x Φ = [ ǫ3 t x 4Φǫ (z x c z c ) Φ x z3 c z c 3 ] u. (3.) x So, the leading order term of the potential Φ satisfies, [ (z x c, z c, ) Φ ] =. (3.3) x Here one has used the fact that κ = O(/ǫ) and thus the spatial derivative of the self energy vanishes in the outer region ( < x < ), and because of it the outer solution is the same as that of the classical PNP equations [9]. Since c ±, are constants, Φ is a linear function of x. One can express it by Φ (x,t) = j (t)x A(t), where j (t) is the leading term of the Faradaic current density in the outer region and its initial value is given by j () = (V V )/. It follows from boundary conditions that A() = (V V )/. To summarize, the outer solution reads { c±, (x,t) = z, (3.4) Φ (x,t) = j (t)xa(t). Note that one shall determine values of j (t) and A(t) given their initial data by using the time-dependent matching. This will be discussed later. 3.. Inner solution. Since the outer solution shown above does not satisfy the boundary conditions (.), there will be a boundary layer near each electrode, i.e., the inner solution. We study the inner solution near the left interface x = and that near the right interface can be obtained similarly. To this end, a variable transformation y = (x )/ǫ α is used, where ǫ α is the thickness of the boundary layer and the parameter α > is to be determined. The corresponding inner solution is denoted by ĉ ± (y,t) and Φ(y,t). After the transformation, the mpnp equations become ǫ α ĉ i ( ) t = ĉ i y y z Φ iĉ i y û z iĉi, i = ±, y ǫ α Φ y = z ĉ z ĉ, (3.5) where ( ) e (ǫ α yξ)κ y)κ û = qγ ǫ α e (4ξ ǫα y ξ 4ξ ǫ α. (3.6) y To analyze the influence of the dielectric self energy, one assumes q = ηǫ θ, where η is a positive constant. The parameter θ is discussed in the following two cases: θ = and θ >. Since θ < corresponds to strong coupling systems for which the Coulomb correlation becomes significant, this work does not analyze this regime.

7 Asymptotic analysis on mpnp equations Case of θ =. Consider the Nernst-Planck equation for the ionic concentration of species i in Eq. (3.5). In the case of θ =, we have ( ) ( ) ĉ i t = ĉi O(ǫ ), = O(ǫ ), and Φ z i ĉ i = O(ǫ α ). (3.7) y y y y ( The order of the term related to the dielectric self energy, y z i ĉ i y û ), depends on the value of α, which can be determined by the following. If < α <, one has û = o(ǫ α ). The leading-order term in the NP equation is y (z i ĉ i, y Φ ). The zero-flux boundary condition implies that either the cation concentration is zero or the potential is a constant. Clearly both cases cannot match the outer solution. If α >, one has û = O(min(ǫ α,ǫ/ξ))( where ξ is smaller than ǫ, and then the leading-order term in the NP equation is y z i ĉ i y û ) /. Again, the zero-flux boundary condition implies either zero cation concentration or constant û. The later case implies that the dielectric self energy does not affect the solution, and the former case cannot match the outer solution. When α =, one has û = O(ǫ ). All the terms in the right side of the NP equation have the same orders, and the leading asymptotics satisfy y ( ĉ i, y z iĉ i, Φ y z iĉi, û y ) =. (3.8) The zero-flux boundary condition implies that the function in the parentheses is zero and can be integrated with respect to y to obtain ĉ i, (y,t) = a i (t)e zi Φ (y,t) z iû. (3.9) Here coefficients a i (t), i = ±, are functions of time t to be determined by the timedependent matching later. Using the asymptotic matching for the ion concentrations and potential in the inner solution and the values in outer solution, i.e., yields, lim x c ±,(x,t) = lim y ĉ±,(y,t), (3.) lim Φ (x,t) = lim x y Φ (y,t), (3.) a ± (t) = z e z±φ(,t). (3.) Hence, the ionic concentrations in the boundary layer can be implicitly expressed as ĉ ±, (y,t) = z e [z± ϕ(y,t) z ±û(y,t)], (3.3) with ϕ (y,t) = Φ (y,t) Φ (,t). In summary, the inner solution of the mpnp equations (3.5) is described by ϕ y = z ĉ, z ĉ,, ĉ ±, (y,t) = z e [z± ϕ(y,t) z ±û(y,t)], (3.4) û = qγ e ǫyξ ǫ zĉ,z ĉ, ǫyξ,

8 8 L. Ji, P. Liu, Z. Xu and S. Zhou with boundary conditions ϕ (,t) = and ϕ (,t) = V j (t) A(t). Notice that the equations resemble the Poisson Boltzmann equation but with dielectric self energy as a correction to the mean potential energy in the Boltzmann factor [5] Case of θ >. One can easily obtain the orders of the left and the first two terms in the right sides of (3.5), similar to the case of θ =. But the order of the self-energy contribution is different. To analyze the order of y (ĉ i y û), one shall consider the following cases. If α > θ, then û = O(min(ǫ θ α,ǫ θ /ξ)) where ξ is smaller than ǫ θ. Thus, the term from the dielectric self energy is the leading-order term in (3.5). The zeroflux boundary conditions lead to z iĉi y û =. This implies that either the cation concentration is zero or û is a constant, both resulting in nonphysical solutions. If < α < θ, then û = O(ǫ θ α ). The orderanalysisshowsthat yy ĉ i isthe leading-order term in (3.5). Due to the zero-flux boundary condition, the ionic concentrations have to be constant. Matching the concentrations between the inner solution and the outer solution, one finds that the concentrations are constant in the whole domain, which is unphysical or a trivial solution. If < α <, then û = o(ǫ θ α ) and y ( z i ĉ i, y Φ ) is the leading-order term. The zero-flux boundary conditions imply that either the ionic concentrations are zero or the potential is a constant, both of which cannot match the outer solution and boundary conditions. When α =, then û = O(ǫ θ ), and y ( y ĉ i, z i ĉ i, y Φ ) are the leadingorder terms. When α = θ, then û = O(ǫ ), and y ( y ĉ i, z iĉi, y û ) are the leading-order terms. These two cases lead to physical solutions. Actually, a two-layer structure in the boundary layer can be obtained by matching the inner and outer solutions, which will be discussed below. The first layer. When α = θ, the width of the boundary layer is of order O(ǫ θ ), and one uses the variable change w = (x)/ǫ θ and denotes the solution in this layer by ĉ I (w,t), ĉi (w,t), and Φ I (w,t). In Eq. (3.5), taking the leading-order terms and using the zero-flux boundary condition, one has Integrating with respect to w, one gets ĉ I i, w z iĉ I û I i, =. (3.5) w ĉ I ±, (w,t) = b ±(t)e z ±ûi, (3.6) where b ± (t) are to be determined functions of t. By the expression of the self energy, one has û I = qγe ǫ θ wξ ǫ z ĉ I, z ĉi, ǫ θ wξ qγ ǫ θ, as ǫ. (3.7) wξ Since θ >, one has ǫ θ = o(ǫ ). From the leading-order term of Poisson s equation, one finds that ww ΦI =. Thus, the potential Φ I is a linear function, Φ I = d(t)w V, where d(t) is a function of t. In order to match with the solution in other regions, one must have d(t) = and Φ I = V to avoid the singularity. The leading terms of the ion concentrations are given by qγ ĉ I ±, = b ±(t)e z ± 4ǫ θ w4ξ, (3.8)

9 Asymptotic analysis on mpnp equations 9 where b ± (t) are to be determined by asymptotic matching. It is noted that the dielectric self energy contributes to the distributions of the two species in the same way in the exponential factor, no matter the sign of ionic valences is positive or negative. This is in agreement with the fact that the self energy of an ion due to a dielectric interface is quadratic to the ion charge. We remark that the small thickness of the first layer will lead to great challenge if grid-based numerical approximation is used in order to resolve the boundary layer, and this can be treated using a renormalized boundary condition [7] to replace the contribution of this layer, considering that the integrated charge can be calculated from the asymptotic solution. The second layer. When α =, the width of the boundary layer is of order O(ǫ) and the variable transform y = (x )/ǫ is used. The solution in this layer is denoted by ĉ II ± (y,t) and Φ II (y,t). Taking the leading-order terms in (3.5), one has ĉ II i, y z iĉ II Φ II i, y = B, (3.9) where B is a constant independent of y. Matching the flux densities in the inner solution onto the outer solution yields ( ) ĉ II i, lim y ǫ y z iĉ II Φ II ( i, y O(ǫ) ci, = lim x x c i, x (z iφ ) z i u ). z± Φ II (3.) Clearly, one finds B =. Integrating Eq. (3.9) gives ĉ II ±, = f ±(t)e where f ± (t) are functions of t and can be determined by matching the ion concentrations and electric potential, i.e., y ĉii lim ±, = lim ±,, x (3.) lim Φ II y = lim, x (3.) and consequently, f ± (t) = z e z±φ(,t). Let ϕ II (y,t) := Φ II (y,t) Φ(,t). One obtains the following equations, ĉ II ±, = z e z± ϕii (y,t), II ϕ y = z ĉ II, z ĉ II,, (3.3) Here ϕ II (y,t) represents the leading-order term of ϕ II (y,t). It is noted that the dielectric boundary effect does not come into play in the solution of the second layer. The boundary conditions of the Poisson s equation for the potential drop are given by the potential matchings between the first, the second and the outer layers, namely, ϕ II (,t) = ζl (t) V Φ (,t), and ϕ II (,t) =. By matching the ionic concentrations in the first and second layers, lim w ĉi ±, = lim ±,, one obtains b ± (t) = z e z±ζl (t). Finally, one reaches the solution in the y ĉii first layer, which is given by, ĉ I ±, = z e z±ζl (t) z qγ ± 4ǫ θ w4ξ, (3.4) Φ I = V.

10 L. Ji, P. Liu, Z. Xu and S. Zhou In contrast to the θ = case, this θ > case has the dielectric boundary effect only in the first layer, which is of width O(ǫ θ ). It is noted that the leading contribution of the potential in the first layer is a constant and the potential drop is ζ L (t) which is an O(ǫ) term. It can be observed that the dielectric self energy has a strong impact on ionic concentrations through the Boltzmann factor. The whole set of the asymptotic solution in the first layer, second layer, and outer area will be determined, once A(t) andthepotentialdropζ L (t)issolvedbythetime-dependentmatching; cf. Section3.4. At the equilibrium state, these quantities are constant and the asymptotic solution is then determined by the above equations Uniformly asymptotic solutions. We have obtained the outer and the inner solutions, and consequently the asymptotic solution can be written into a form of uniformly valid approximations by summing up the inner and outer solutions and subtracting the overlaps. Since there are two electrodes for the physical system, the solution is split into two parts where the solution in the right half plane is obtained by simply repeating the procedure exerting to the left one. We use the superscript L and R to distinguish the left and right asymptotic solutions. These two parts coincide in the middle between two electrodes, once the appropriate j (t) and A(t) are determined by the time-dependent matching. For the case of θ =, the uniformly valid solutions can be written as ( ) ( ) x x c ± (x,t) = ĉ L ±,,t ĉ R ±,,t z O(ǫ), ( ǫ ) ( ǫ ) Φ(x,t) = Φ x L,t ǫ Φ (3.5) x R,t j ǫ (t) A(t)O(ǫ), where A(t) = [ Φ L (,t) Φ R (,t)]/. It is noted that in this case the outer solution for the ionic concentrations coincides with the overlap solution, and the uniform solution happens to be the inner solution. In addition, the two boundary layers at the left and right electrodes also coincide with the same outer solutions. For the case of θ >, each boundary layer has two overlaps since the inner solution has two layers. The uniformly valid solutions can be written as ( ) ( ) ( ) ( ) x x x x c ± (x,t) = ĉ L,I ±, ǫ θ,t ĉ L,II ±,,t ĉ R,I ±, ǫ ǫ θ,t ĉ R,II ±,,t [ ] ǫ z e z±ζl (t) e z±ζr (t) O(ǫ), ( ) ( ) L,II x R,II x Φ(x,t) = Φ,t Φ,t j ǫ ǫ (t)x A(t)O(ǫ), (3.6) where A(t) = [ Φ L,II R,II (,t) Φ (, t)]/. In the solution for the ionic concentrations, the first four terms represent the first- and second-layer inner solutions on both sides, and the fifth term represents the three overlap solutions. For the electric potential, the potential drop on the first layer is small, and thus there is only one overlap solution Time-dependent matching for mpnp equations. One shall study the time evolution of the above asymptotic solutions for concentrations and electrostatic potential by determining the time-dependent coefficients j(t), A(t), ζ L (t) and ζ R (t). We perform a time-dependent matching for the asymptotic solutions by following Bazant et al. [9]. We discuss the case of θ >, which has a two-layer structure in the boundary layer. The same procedure can be readily applied to the case of θ =

11 Asymptotic analysis on mpnp equations without much difference, which results in the same solution since the first layer for case θ > has neglectable contribution. Consider the dynamics of the total diffuse charge which is the net charge in the half space defined by Q(t) = ρ(x)dx with ρ = z c z c. Since the outer solution is electrically neutral up to order O(ǫ) and the net charge in the first layer is ρ I (w,t) O(ǫ θ ) with θ >, one can only consider the leading asymptotic expansion in the second layer, Q. By Eq. (3.3), one has Q = = [ z ĉ II, (y,t)z ĉ II, (y,t)] dy II ϕ y dy = ϕii y, (3.7) y= where one has used the relation lim y ϕ II (y,t) = lim y Φ II (y,t) = to obtain the y y last equality. Taking the time derivative of the leading-order term of the total diffuse charge and using the NP equations in Eq.(3.5) yields, d Q dt = ( = lim y ǫ ĉ II, z t [ ρ II z ĉ II, t ) dy y ( zĉ II, z ĉ II ) Φ II, y [ ρ = lim x x ( z z z z ) Φ x z3 ĉii, z3 ĉii, ] û II y ], (3.8) where the zero-flux boundary condition is used. In Eq. (3.8), the electroneutrality condition ρ = leads us to, d Q dζ L dζ L dt = ( z z z z ) j (t), (3.9) where j (t) = x Φ is the current and is independent of x as the outer solution Φ is linear function of x, and ζ L is the potential drop defined below Eq. (3.3). Define the differential capacitance, C(ζ L ) = d Q /dζ L, and recall ζ L (t) = V Φ (,t) = V j (t) A(t). Eq. (3.9) can be written as, C(ζ L ) dζl = ( z dt z zz )( ζ L ) A(t) V. (3.3) Similarly, for the boundary layer on the right, we have, C(ζ R ) dζr dt = ( z z z z ) j (t) = ( z z z z )( A(t)ζ R (t) V ), (3.3) where ζ R (t) = V Φ (,t) = V j (t) A(t). By the definitions of ζ L and ζ R, one has A(t) = V V ζ L ζ R. Eqs. (3.3) and (3.3) form a closed system of ODEs for the potential drops, C(ζ L ) dζl = ( z dt z z ) V V ζ L ζ R z, C(ζ R ) dζr = ( z dt z zz ) V V ζ R ζ L (3.3),

12 L. Ji, P. Liu, Z. Xu and S. Zhou with initial conditions: ζ L () = ζ R () =. In particular, for a symmetric salt system, the potentialdropsatthe twoelectrodessatisfythatζ L (t)ζ R (t) anda VV and these two ordinary equations are the same. In this case, we only need to solve one of them. 4. Results and discussion. In this section, we present results obtained by the asymptotic approximations ( ), in comparison to numerical solutions solved by a second-order finite difference method with small grid sizes. We first report the equilibrium state solutions to the mpnp system in Section 4. and 4. for symmetric and asymmetric (:) electrolytes, respectively. The results on the dynamics are presented in the Section Cation Concentration γ= 9/ γ= γ= -/3 γ= - (a).5.5 d Anion Concnetration γ= 9/ γ=. γ= -/3 γ= - (b).5.5 d Potential Distribution (c) d γ= 9/ γ= γ= -/3 γ= - Charge Density d γ= 9/ γ= γ= -/3 γ= - (d) Fig. 4.. Ionic concentrations, electrostatic potential, and charge density for symmetric salt with ǫ = q =.,θ =,d = (x)/ǫ and various dielectric ratios. The asymptotic approximations are shown in lines and the numerical solutions are shown with triangle symbols. 4.. Symmetric salt. Consider a symmetric monovalent electrolyte, i.e., z = z =, with dielectric constant ε W = 8. The dielectric boundary effect is studied by investigatingthe solution with variousvalues ofthe dielectric coefficient ε B. Unless otherwise stated, we take ǫ =. and ξ = q/5 in the calculations. We first consider the case of θ = for which the inner solution has a one-layer structure. We take q =., V = V = V =.5, and ε B = 4,8,6and, where ε B = corresponds to metallic electrodes; correspondingly, the dielectric ratios are γ = 9/,, /3 and, respectively. The dimensionless V =.5 corresponds to a boundary voltage of.9 mv. Fig. 4. displays the equilibrium profiles of the cation and anion concentrations, the electrostatic potential, and the charge density

13 Asymptotic analysis on mpnp equations 3 as function of d = (x )/ǫ which is the distance to the left electrode rescaled by ǫ. Clearly, one can see that the dielectric self energy due to a low-dielectric electrode (γ = 9/) depletes both counterions and coions. For γ =, there is no dielectric boundary effect, and counterions are attracted to the electrode and coions are repelled from the electrode. In contrast, the dielectric self energy exerts attraction on both counterions and coions for a high-dielectric electrode (the γ < cases). More significant attraction can be observed as the dielectric ratio gets larger. Total Diffuse Charge γ= 9/ γ= γ= -/3 γ= - (a) V Differential Capacitance..5 γ= 9/ γ= γ= -/3 γ= - (b) V Fig. 4.. Total diffuse charge and differential capacitance against applied voltages. Using the same parameters, we also investigate the effect of dielectric self energy on the total diffuse charge Q = (z c z c )dx and the differential capacitance C(V) = dq/dv. As shown in Fig. 4., the total diffuse charge increases monotonically as the applied voltage gets larger. Also, it is observed that the dielectric self energy suppresses the total diffuse charge when γ >, and enhances the total diffuse charge when γ <. The dielectric ratio has a considerable impact on the differential capacitance as well. The capacitance increases linearly for weak applied voltages. In addition, the dielectric self energy with larger dielectric ratio has more significant attraction to ions and therefore increases the capacitance pronouncedly. There is great agreement between the asymptotic approximations and numerical solutions, validating the effectiveness of our asymptotic analysis. Cation Concentration d γ= 9/ γ= γ= -/3 γ= -... (a) Charge Density d γ= 9/ γ= γ= -/3 γ= - (b) Fig Cation concentration and charge density for symmetric salt with ǫ =. and q = 6 for various dielectric ratios. d = (x)/ǫ. The numerical solution corresponds to the case of q =. According to our asymptotic analysis, there is a two-layer structure in the bound-

14 4 L. Ji, P. Liu, Z. Xu and S. Zhou ary layer for the case of θ >, and the first layer close to the electrode is rather thin with width O(ǫ θ ). In order to show the deck layers more intuitively, we use the same parameters as the one-layer case but taking q = 6. As illustrated in Fig. 4.3, there is a sharp transition on ionic concentrations, when the distance to the electrode is less than.ǫ. The zoom-in inset clearly demonstrates that the dielectric self energy exerts remarkable attraction or depletion on ionic distributions in the first layer, depending on the value of dielectric ratios. Right next to the first layer, the second layer cannot see the dielectric boundary effect at all, and ionic concentrations follow the classical Boltzmann distribution. In this case, the dielectric boundary effect is so weak that it only alters ionic distributions in the first layer and is completely screened by ions in the first layer. From the profile of the charge density, we also find that the dielectric boundary affects both counterions and coions in the first layer. Such results confirm and further explain the asymptotic analysis on the two-layer structure of the boundary layer. It should be noted that the numerical solutions are obtained with taking q = because it is computationally prohibitive to resolve the first thin layer with a finite-difference grid. The agreement outside the first layer can be observed for such a small parameter, showing that the expansion with respect to q is a regular perturbation for the solution outside the first layer. The embedded figures demonstrate that the dielectric self energy will provide strong dielectric-dependent interaction to ions close to the surface, which cannot be ignored if one aims to understand interface properties. Cation Concentration γ= 9/ γ= γ= -/3 γ= - Cation Concnetration.5.5 γ= 9/ γ= γ= -/3 γ= - (a) (b) d L d R Anion Concentration γ= 9/ γ= γ= -/3 γ= - (c).5.5 Anion Concentration γ= 9/ γ= γ= -/3 γ= -.5 (d) d L d R Fig Concentrations of divalent cations (upper panel) and monovalent anions (lower panel) close to the electrodes with different dielectric ratios. d L = (x)/ǫ,d R = ( x)/ǫ, and the applied voltage V =.5.

15 Asymptotic analysis on mpnp equations Asymmetric salt. We use our model to investigate the dielectric boundary effect on the equilibrium state of an asymmetric electrolyte, which consists of divalent cations and monovalent anions, i.e., z = and z =. In the calculations, we take V = V =.5, ǫ =., and four different γ as in previous examples. The parameter q =. is taken such that the asymptotic solution of θ = is used. We will not show results of θ > since the phenomenon due to the effect of the first layer is similar to that of the symmetric salt. As displayed in Fig. 4.4, the profiles of ionic concentration are monotone when no dielectric boundary effect is taken into account (γ = case). In contrast, ions distribute in a totally different fashion when the dielectric self energy comes into play. The equilibrium ionic distribution is resulted from competitions between the dielectric contribution and the direct ion-electrode electrostatic interaction. Notice that the direct ion-electrode interaction is rather weak with V =.5. The dielectric boundary effect dominates in the vicinity of electrodes, and depletes cations when γ >. As the distance to the electrode, d, gets larger, the attraction from the electrode starts to prevail over the dielectric boundary effect, giving rise to a slightly larger cation concentration than that of the bulk. As γ <, the divalent cations feel strong attraction near the electrodes and their concentrations drop quickly to a value smaller than that of the bulk as the distance becomes large. This is attributed to the conservation of cations in the system. Ionic concentrations have similar distributions close to the anode. Although the dielectric boundary effect is much weaker for monovalent anions, it still dominates over the direct ion-electrode electrostatic interactions. As illustrated in the lower panel of Fig. 4.4, anions feel attraction and depletion when γ < and γ >, respectively. To further understand the effect of the dielectric self energy, we plot in Fig. 4.5 the charge density and electrostatic potential close to electrodes. We observe charge inversion [3, 5] for both attractive and depletive dielectric boundary effects. When γ >, the electrostatic potential increases from V at the cathode to a value greater than V in the bulk and decreases to V at the anode. On the contrary, the electrostatic potential increases to a smaller constant potential and then attains V when γ <. Such a behavior can be explained by the distributions of total charge densities that are shown in lower panel of Fig Again, we can see remarkable impacts of the dielectric self energy on the charge distribution. To characterize the impact of dielectric boundary effects on the differential capacitance, we perform numerical simulations with varying applied voltages and compute the differential capacitance of the whole system, which is treated as a series connection of two capacitors at electrodes. From Fig. 4.6, we can see that the attractive dielectric boundary effect promotes the diffusion of charges, and less charge is accumulated at the electrode when the dielectric boundary effect is depletive. Also, the capacitance becomes larger when the attractive dielectric boundary effect is taken into account. The capacitance grows nonlinearly as the applied voltage gets larger. Furthermore, the total diffuse charge and differential capacitance are much larger and grow faster for the case of asymmetric salts, in comparison with the case of symmetric salts, cf. Figs. 4. and 4.6. Again, the comparison between asymptotic approximations and numerical solutions evidences that the asymptotic analysis works well for asymmetric electrolytes as well Dynamics of ions. In this section, we first study the ion concentrations and potential distributions at different times. We use the following parameters in the calculations: V =.5,V =.5,ǫ = q =.,ξ = q/5, and γ = 9/. Figs. 4.7

16 6 L. Ji, P. Liu, Z. Xu and S. Zhou.. Potential Distribution γ= 9/ γ= γ= -/3 γ= - (a) Potential Distribution γ= 9/ γ= γ= -/3 γ= - (b) d L d R 3 Charge Density γ= 9/ γ= γ= -/3 γ= (c) Charge Density γ= 9/ γ= γ= -/3 γ= - (d) d L d R Fig Electrostatic potential and charge density close to the electrodes with different dielectric ratios. d L = (x)/ǫ and d R = ( x)/ǫ. Total Diffuse Charge γ= 9/ γ= γ= -/3 γ= - (a) V Differential Capacitance γ= 9/ γ= γ= -/3 γ= - (b) V Fig Total diffuse charge and differential capacitance against applied voltages at the anode with different dielectric ratios. The salt consists of divalent cations and monovalent anions. and 4.8 present the asymptotic approximations obtained by (3.3) and the numerical solutions solved with finite difference methods for : and : salts, respectively. In Fig. 4.7, one observes that the cation concentration has a depletion layer in the vicinity of the electrode and gradually forms a hump near the electrode. With accumulated cations, the potential gets screened as time evolves, but keeps being a linear function in the outer layer. For the asymmetric salt, from the potential distribution plotted in Fig. 4.8, one sees that, as time evolves, the potential are quickly screened by the

17 Asymptotic analysis on mpnp equations 7 Cation Concentration..8.6 (a) x t=. t=.5 t= t= 3 Potential Distribution (b) x t=. t=.5 t= t= 3 Fig Cation concentration and potential distribution for symmetric electrolytes (:) at t =.,.5, and 3. Asymptotic approximations are shown in lines and numerical solutions are shown with triangle symbols..5.5 Potential Distribution t =.5 t = t = Potential Distribution t =.5 t = t = (a) x (b) x Charge Density - t =.5 t = t = Charge Density - t =.5 t = t = (c) x (d) x Fig Cation concentration and potential distribution for asymmetric electrolytes (:) at t =.5, and. Asymptotic approximations are shown in lines and numerical solutions are shown with triangle symbols. accumulated net positive charge density near the cathode. At large time t when the solution approaches the equilibrium state, one can see charge inversion phenomenon near the cathode, with the potential being positive. In the vicinity of the anode, the potential also gets screened by negative charge density. However, one cannot see charge inversion close to the anode with monovalent anions. Both figures show that the asymptotic approximations agree very well with the numerical solutions for

18 L. Ji, P. Liu, Z. Xu and S. Zhou different time snapshots, validating that the time-dependent matching between the outer layer and inner layers in asymptotic analysis are accurate t t x (a) NUM (b) ASY x. Fig The evolution of cation concentration with oscillatory applied voltages. (a) Asymptotic solution; (b) Numerical solution. It is also significant to study the response of the boundary layer on variable surface voltages. For the purpose, the mpnp solution for an monovalent electrolyte with applied voltages ±V = ±.5 sin(t) is calculated by both the asymptotic and the numerical approximations. The parameters take ǫ =.,q =.4 and γ = 9/. The multiple-layer structure of the electric double layer has been reported in the literature [6, 3, 43] for systems with large applied voltages, for which the steric effect between ions plays important role. Here we consider weak oscillatory voltages to investigate the dielectric boundary effect as higher voltages screen the effect. The numerical solutions displayed in Fig. 4.9 reveal that the weak oscillatory voltages do not alter the layer structure in the EDL. Since q is much smaller than ǫ, a two-layer structure is present, where a thin layer can be observed near the electrode, followed by a thick diffusion layer. The asymptotic solution agrees well with the numerical approximation, showing that the asymptotic expansion is also accurate for the time dynamics as well as the steady-state solution. We remark that in order to see how the dielectric self energy affects the structure with a dense electrolyte or large applied voltage, the steric effect [43] should be incorporated into the model such that the excess chemical potential includes both effects. 5. Concluding remarks. In this paper, we have performed an asymptotic expansion analysis to understand the dielectric boundary effect on the charge dynamics of electrolyte solutions between two blocking electrodes based on a modified PNP model with the dielectric self energy included in the potential of mean force. Asymptotic solutions have shown that there is a two-layer structure close to electrodes when the dielectric boundary effect is weak. The dielectric self energy only comes into play in the first layer, while the second layer behaves like the classical PNP solution, without any correction from the dielectric boundary effect. When the dielectric boundary

19 Asymptotic analysis on mpnp equations 9 effect is relatively strong, the asymptotic analysis have demonstrated that there is only one layer in which the inner solution satisfies modified Poisson Boltzmann equations with the dielectric self energy appearing in the Boltzmann factor. Systematic investigations on the ionic concentrations, electrostatic potential, charge density, diffuse charges, differential capacitance, and charge inversion have deepened our understanding of the dielectric boundary effect on electrostatic phenomena near interfaces. We now discuss several issues and possible further refinements of the present work. Since the electrolytes under consideration are dilute and applied voltages are weak, the steric effect and ion-ion correlations have not been taken into account in our current model. However, they could play important roles if we consider concentrated electrolytes or stronger applied voltages. Steric effects could be included via adding solvent entropic contributions to the total free energy [8, 9, 35]. Alternatively, a fourth-order Poisson s equation could be applied to describe the effect of ion-ion correlations [, 8, 35]. Ionic correlations induce overscreening when the applied voltage is small; while, the steric effect becomes dominant in determining the structure of electric double layer when the applied voltage is strong. It remains for future work to elaborate the interplay among steric effects, ion-ion correlations, and dielectric boundary effects in variable dielectric environment. Also, it is of great interest to apply the refined mpnp model to study ionic conductance in ion channel problems. It has been known that, in addition to the steric effects and ion-ion correlations, the dielectric effects play key roles in ion permeation through a thin channel [5,]. Nevertheless, not much effort has been put in this direction using self-consistent PNP models. We have considered two blocking electrodes in our current development. The charge dynamics and layer structure close to electrodes will have a totally different picture if we consider electrodes with Faradaic reactions with nonlinear Butler-Volmer kinetics. With more refined treatments of electrodes, the present model will be a promising tool in predicting electrochemical properties near electrodes of different materials. Acknowledgments. The authors thank the anonymous reviewers for their useful comments and suggestions. L. Ji and Z. Xu acknowledge the support from grants NSFC 5736 and and HPC center of Shanghai Jiao Tong University. S. Zhou acknowledges the support from grants NSFC 636, NSFC 77365, Natural Science Foundation of Jiangsu Province BK63, and Q4745. REFERENCES [] N. Abaid, R. Eisenberg, and W. Liu, Asymptotic expansions of I-V relations via a Poisson Nernst Planck system, Phys. Rev. E, 7 (8), pp [] A.Yochelis, Transition from non-monotonic to monotonic electrical diffuse layers: impact of confinement on ionic liquids, Phys.Chem.Chem.Phys., 6 (4), pp [3] P. Balbuena and Y. Wang, Lithium ion batteries: solid electrolyte interphase, Imperial College Press, London, 4. [4] V. Barcilon, D. Chen, and R. Eisenberg, Ion flow through narrow membrane channels: Part II, SIAM J. Appl. Math., 5 (99), pp [5] V. Barcilon, D. Chen, R. Eisenberg, and J. Jerome, Qualitative properties of steadystate Poisson Nernst Planck systems: Perturbation and simulation study, SIAM J. Appl. Math., 57 (997), pp [6] M. Z. Bazant, K. T. Chu, and B. J. Bayly, Current-voltage relations for electrochemical thin films, SIAM J. Appl. Math., 65 (6), pp [7] M. Z. Bazant, M. S. Kilic, B. D. Storey, and A. Ajdari, Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions, Adv. Colloid Interface Sci., 5 (9), pp

20 L. Ji, P. Liu, Z. Xu and S. Zhou [8] M. Z. Bazant, B. D. Storey, and A. A. Kornyshev, Double layer in ionic liquids: Overscreening versus crowding, Phys. Rev. Lett., 6 (), p. 46. [9] M. Z. Bazant, K. Thornton, and A. Ajdari, Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 7 (4), p. 56. [] D. Ben-Yaakov, D. Andelman, and R. Podgornik, Dielectric decrement as a source of ion-specific effects, J. Chem. Phys., 34 (), p [] F. P. Buff and F. H. Stillinger, Statistical mechanical theory of double-layer structure and properties, J. Chem. Phys., 39 (963), pp [] W. Cheng, C.-X. Wang, W.-Z. Chen, Y.-W. Xu, and Y.-Y. Shi, Investigating the dielectric effects of channel pore water on the electrostatic barriers of the permeation ion by the finite difference Poisson-Boltzmann method, Eur. Biophys. J., 7 (998), pp. 5. [3] K. T. Chu and M. Z. Bazant, Electrochemical thin films at and above the classical limiting current, SIAM J. Appl. Math., 65 (5), pp [4] B. Corry, S. Kuyucak, and S.-H. Chung, Tests of continuum theories as models of ion channels. II. Poisson Nernst Planck theory versus Brownian Dynamics, Biophys. J., 78 (), pp [5] B. Corry, S. Kuyucak, and S.-H. Chung, Dielectric self-energy in Poisson Boltzmann and Poisson Nernst Planck models of ion channels, Biophysical J., 84 (3), pp [6] R. Eisenberg, From structure to function in open ionic channels, J. Membrane Biol., 7 (999), pp. 4. [7] R. Eisenberg, Crowded charges in ion channels, Adv. Chem. Phys., 48 (), pp [8] R. Eisenberg, Y. Hyon, and C. Liu, Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids, J. Chem. Phys., 33 (), p. 44. [9] R. Eisenberg and W. Liu, Poisson Nernst Planck systems for ion channels with permanent charges, SIAM J. Math. Anal., 38 (7), pp [] N. Gavish and K. Promislow, On the structure of generalized Poisson Boltzmann equations, Euro. J. Appl. Math., 7 (6), pp [] P. Graf, A. Nitzan, M. G. Kurnikova, and R. D. Coalson, A dynamic lattice Monte Carlo model of ion transport in inhomogeneous dielectric environments: method and implementation, J. Phys. Chem. B, 4 (), pp [] D. C. Grahame, The electrical double layer and the theory of electrocapillarity, Chem. Rev., 3 (947), pp [3] A. Y. Grosberg, T. T. Nguyen, and B. I. Shklovskii, The physics of charge inversion in chemical and biological systems, Rev. Mod. Phys., 74 (), pp [4] X. Guan, M. Ma, Z. Gan, Z. Xu, and B. Li, Hybrid Monte Carlo and continuum modeling of electrolyte with concentration-induced dielectric variations, Phys. Rev. E, 94 (6), p [5] M. M. Hatlo, R. V. Roij, and L. Lue, The electric double layer at high surface potentials: The influence of excess ion polarizability, Europhys. Lett., 97 (), p. 8. [6] B. Hille, Ion channels of excitable membranes, Sinauer Associates, 3rd ed.,. [7] P. C. Jordan, R. J. Bacquet, J. A. McCammon, and P. Tran, How electrolyte shielding influences the electrical potential in transmembrane ion channels, Biophys. J., 55 (989), pp [8] M. S. Kilic, M. Z. Bazant, and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. I. double-layer charging, Physical Review E, 75 (7), p. 5. [9] M. S. Kilic, M. Z. Bazant, and A. Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson Nernst Planck equations, Phys. Rev. E, 75 (7), p. 53. [3] M. G. Kurnikova, R. D. Coalson, P. Graf, and A. Nitzan, A lattice relaxation algorithm for three-dimensional Poisson Nernst Planck theory with application to ion transport through the gramicidin a channel, Biophys. J., 76 (999), pp [3] D. G. Levitt, Strong electrolyte continum theory solution for equilibrium profiles, diffusion limitation, and conductance in charged ion channels, Biophys. J., 48 (985), pp [3] B. Li, P. Liu, Z. Xu, and S. Zhou, Ionic size effects: generalized Boltzmann distributions, counterion stratification, and modified debye length, Nonlinearity, 6 (3), p [33] B. Li, J. Wen, and S. Zhou, Mean-field theory and computation of electrostatics with ionic concentration dependent dielectrics, Commun. Math. Sci., 4() (6), pp [34] D. Li, Electrokinetics in microfluidics, vol., Academic Press, 4. [35] J.-L. Liu and R. Eisenberg, Correlated ions in a calcium channel model: A Poisson-Fermi theory, J. Phys. Chem. B, 7 (3), pp [36] P. Liu, X. Ji, and Z. Xu, Modified Poisson Nernst Planck model with accurate coulomb