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 permittivityis ubiquitous in many areas such as electrochemical energy devices and biophysical systems. Wetheoretically study the equilibrium and dynamics of electrolytes between two blocking electrodesbased on a modified Poisson–Nernst–Planck model with the dielectric boundary effect. Matchedasymptotic analysis shows that a two-layer interfacial structure exists in the vicinity of the interfaceswhen the dielectric self-energy correction to the potential mean-force is relatively weak. For thistwo-layer structured solution, the dielectric effect plays the dominate role in the first layer, while thesolution in the second layer is mainly determined by the classical Poisson–Boltzmann equation. Whenthe dielectric self energy becomes stronger, there is only one interfacial layer which is governed bythe modified Poisson–Boltzmann equation with the dielectric self-energy correction in the Boltzmannfactor. We perform a systematic investigation for symmetric and asymmetric electrolytes on ionicconcentrations, electrostatic potential, diffuse charges, differential capacitance, and charge inversionphenomenon, to show the effects of the dielectric inhomogeneity on the solutions near interfaces.

Key words. Poisson–Nernst–Planck equations; Dielectric interfaces; Matched asymptotic ex-pansion; Boundary layers

AMS subject classifications. 82C21, 82D15, 35Q92

1. Introduction. The ion transport and distribution in an aqueous solution nearinterfaces is fundamental to a wide variety of electrochemical applications and biolog-ical processes [3, 26, 34]. The ion transport in solutions is usually described throughthe Poisson–Nernst–Planck (PNP) theory based on a mean-field approximation. TheNernst–Planck (NP) equations model diffusion of ions under the concentration gradi-ent and the electrostatic potential. The Poisson’s equation governs the electrostaticpotential with the charge density stemming from transporting ions. The classicalPNP theory has been successful in many applications [9, 12, 16, 17, 27, 30, 31, 47], butthe theory may fail to accurately predict dynamics and equilibrium distributions ofions in many scenarios when the steric effect, the ion-ion correlation or the dielectricboundary effect plays the role in the system, since it ignores these features due to themean-field nature. For example, in the presence of dielectric interfaces, the dielectricself energy, which is the interaction energy between an ion and the inhomogeneousdielectric medium, plays an important role in the surface tension of air/water in-terfaces [44] and other soft materials [58] even for electrolytes at the weak-couplingregime. It has been reported that the PNP theory overestimates the effective chan-nel pore size due to the point-charge approximation of ions and the ignorance of thedielectric self energy that is a substantial energy barrier to ion permeation through anarrow channel [21,54]. The validity of the mean-field approximation in the Poisson-Boltzmann (PB) and PNP theory has been tested by comparing the results withthose of Brownian dynamics simulations on narrow ion channels whose pore radii areless than the Debye length [14, 15, 40]. Considerable differences on the concentration

∗School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China.†Department of Mathematics, The Penn State University, University Park, PA 16802, 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 200240,China (xuzl@sjtu.edu.cn).

§Department of Mathematics and Mathematical Center for Interdiscipline Research, SoochowUniversity, 1 Shizi Street, Suzhou 215006, Jiangsu, China(sgzhou@suda.edu.cn).

1

2 L. Ji, P. Liu, Z. Xu and S. Zhou

profiles and ionic conductance demonstrate that the mean-field approximation breaksdown for narrow ion channels [14]. Interestingly, significant qualitative improvementscan be achieved by incorporating a dielectric self-energy correction to remove artificialshielding effects in the mean-field approximation [15].

Many modified versions of the PNP theory have been put forward in literatureto take into account the ignored effects beyond the mean-field approximation. Thesteric effect of ions are considered by incorporating an excess free energy of solvententropy [20,29,32,38,39], Lennard-Jones interaction kernel [18], or modified fundamen-tal measure theory [46,56]. The ionic correlations can be described by a fourth-orderpartial differential equation with a correlation length [8, 35, 49], which can be alsoviewed as introducing an effective inhomogeneous dielectric permittivity as functionof the Laplacian operator. Alternatively, ionic correlations can be taken into accoun-t by incorporating the self energy of solvated ions that is obtained by solving thediagonal of Green’s function from a generalized Debye-Huckel equation [41, 42, 45].The effect due to dielectric inhomogeneity can be accounted for under the frame-work of the self-energy-modified model [53, 55]. Recently, Liu et al. [36] proposeda modified PNP (mPNP) model to consider the Coulombic correlations in variabledielectric media, where the excess free energy from the Coulombic correlation anddielectric effect is obtained by a Debye charging process and is further supplementedwith asymptotic expansions to deal with the difficulty arising from finite ionic sizes. Inaddition to the dielectric inhomogeneity across the boundary, the dielectric coefficientinside electrolytes could depend on local ionic concentrations as well as the electricfield [7, 10, 22, 24, 25, 33, 57],which can play important role in many physical systems.

Mathematical analysis based on singular perturbation methods for the under-standing of charge-diffusion properties has made much progress in recent decades [1,4–6,9,19,28,29,37,48,52]. In these analysis work, the Debye length is often assumedto be much smaller than the characteristic length scale of the geometries, thus onehas a small perturbation parameter for asymptotic expansions, ǫ, which is the ratiobetween two lengths. The method of matched asymptotic expansions (MAE) hasbeen used to obtain the singular perturbation solutions to the steady-state PNP e-quations [5], to investigate the diffuse-charge dynamics in electrochemical systemswith time-dependent applied voltages [9], and to analyze the current-voltage relationsfor electrochemical thin films with Faradaic reactions [6, 13]. It is also used to studythe impact of steric effects on the double-layer charging [28] and the dynamics ofelectrolytes at large applied voltages [29]. Recently, Wang et al. [52] have studied thePNP equations using the matched asymptotic analysis, and discussed the existenceand uniqueness of the solution to the PNP equations with multiple ionic species.

This paper employs the method of MAE to investigate the dielectric boundaryeffect on the charge dynamics of electrolyte solutions between two blocking electrodes.The dielectric self energy is described through the WKB approximation of the gen-eralized Debye-Huckel equation for the system with two parallel dielectric interfaces.An additional parameter that is the ratio between the Bjerrum length and the sep-aration of electrodes is introduced to represent the strength of dielectric boundaryeffect, and hence two small parameters are present in the modified PNP system. Itis observed that the boundary-layer structure strongly depends on the magnitude ofthe two parameters. We study the parameters in different regimes and perform thematched asymptotic analysis for the leading order solutions. Remarkably, a two-layerstructure in the vicinity of electrodes is observed when the dielectric boundary effectis weak. When the dielectric self energy is comparable to the direct ion-electrode

Asymptotic analysis on mPNP equations 3

x= - D x= + Dx= 0

--

V - V+

εW

εBεB

γz-eγz-e

-

-

-

-

-

-

-

--

- -

--

--

z-e+

++

+

+

+

++

+

+

+

+

++

+

++

+

+ -

Fig. 2.1. Schematic illustration of the model system. Voltage is imposed on two electrodeslocated at x = ±D. The dielectric coefficient of electrodes is εB and that of electrolyte solutionsis εW. The charges outside the electrolyte region represent the induced image charges of a pointsource due to the dielectric inhomogeneity, which contribute a dielectric self energy under the WKBapproximation.

interaction, the asymptotic analysis reveals that there is only one layer for inner so-lution whose governing equations are the modified Poisson–Boltzmann equation withthe self-energy correction to the potential of mean force in the Boltzmann factor. Re-sults on symmetric and asymmetric electrolytes are systematically investigated, whichdemonstrate the impact of dielectric boundary effect on the ionic concentrations, elec-trostatic potential, charge density, diffuse charges, and differential capacitance. Ofmuch interest is that the dielectric boundary effect, no matter attractive or depletive,is able to induce charge inversion for asymmetric electrolytes.

The rest of the paper is organized as follows. In section 2, we describe the physicalsetup 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 anddiscussed. Finally, concluding remarks are drawn in section 5.

2. The mPNP model. Consider a binary electrolyte confined between twoparallel planar electrodes of separation 2 ·D; see Fig. 2.1 for a schematic view. Thevalences of the two ion species are denoted by z± with z− < 0 and z+ > 0. Their ionicconcentration distributions c± are homogeneous in the y-O-z plane. We describe thedielectric coefficient as a piecewise constant function,

ε0ε(x) =

{ε0εW , |x| < D,ε0εB otherwise,

(2.1)

where ε0 is the vacuum dielectric constant, εW is the relative dielectric coefficient ofthe solvent, and εB is the relative dielectric coefficient of the electrodes. Since ionshave finite sizes, the electrolyte region [−L,L] is actually smaller than [−D,D], whereL = D/(ξ+1) 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 takinginto account the effect of the dielectric boundaries as F free = Fmf + F ex, where the

4 L. Ji, P. Liu, Z. Xu and S. Zhou

mean-field free energy reads

Fmf =

∫ ∞

−∞

ε0ε

2|∇Φ|2 dx+ kBT

∫ L

−L

∑

i=±

ci(log ci − 1)dx, (2.2)

with constants e, kB and T being the elementary charge, the Boltzmann constantand the temperature, respectively. The mean electric potential Φ is determined byPoisson’s equation with proper boundary conditions imposed at x = ±D,

−ε0∇ · ε∇Φ =∑

i=±

zieci. (2.3)

Here c± are zero for |x| > L. The excess free energy F ex is given by the Debyecharging process [36],

F ex =

∫ L

−L

dx

∫ 1

0

dλ

(2λ

N∑

i=1

ci(x)Ui(x;λ)

), (2.4)

where Ui(x;λ) is the self energy of a particle at the charging state λ, defined by thenonsingular part of the self Green’s function, which is governed by the generalizedDebye-Huckel equation in three dimensions [41, 51],

−ε0∇ · ε(x)∇Gλ(r, r

′) + 2λ2I(x)Gλ(r, r′) = δ(r− r′),

Ui(x;λ) =1

2z2i e

2 limr′→r

[Gλ(r, r′)− 1/(4πεW |r− r′|)] .

(2.5)

Here, I =∑

i=± z2i e2ci/2 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 selfenergy is homogeneous in yz coordinates thus Ui(x;λ) is just a function of x and λ.

We employ the WKB approximation [11, 50, 55] for Eq. (2.5). In the WKB ap-proximation, the Green’s function is approximated by the screened Coulomb potentialfrom 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, thecontribution from reflected image charges is neglected and the WKB approximationfor the self energy is expressed as,

Ui(x;λ) =γz2i e

2

8πε0εW

[e−2(D+x)λκ(x)

2(D + x)+

e−2(D−x)λκ(x)

2(D − x)

], (2.6)

where κ(x) =√∑

i=± z2i e2ci/(ε0εWkBT ) is the inverse of the local Debye screening

length, and γ = (εW − εB)/(εW + εB) describes the dielectric ratio. It is noted thatthe inaccessible region described by the constant ξ is essential to avoid the singularityof the self energy.

Plugging Eq. (2.6) into the free energy functional (2.4), one can see that the freeenergy integration on λ can be calculated analytically. A simple calculation gives,µexi = δF ex/δci = U(x; 1), namely, the excess chemical potential is coincidentially the

self energy at the full charging state, λ = 1. Using the variational approach and theFick’s law leads us to the NP equations,

∂ci(x, t)

∂t+∇ · Ji = 0, i = ±, (2.7)

Asymptotic analysis on mPNP equations 5

and the flux density Ji = −Di[∇ci+βci∇(zieΦ+Ui(x; 1))], which is coupled with thePoisson’s equation (2.3) for the electric potential and the self-energy equation (2.6) atthe full charging state to close an mPNP system. Here β = 1/kBT , and D± are thediffusion constants. We assume D+ = D− = D0. Eq. (2.7) describes the fact that thetransport of ions is governed by the diffusion arising from the concentration gradientand the advection arising from the electric potential and self-energy gradient.

We study the asymptotic solution to the mPNP equations. In the analysis, initialionic concentrations are assumed to be uniform and charge neutral, c±(x, 0) = c±,b =

|z∓|c0 with c0 being a characteristic constant. Let ℓ0 =√ε0εW /βe2c0 be a length

scale which is proportional to the Debye length, and ℓB = e2/(4πε0εWkBT ) be theBjerrum length. One shall introduce the dimensionless quantities x = x/L, t =

tD0/ℓ0L, c± = c±/c0, ε = ε/εW , Φ = βeΦ, and u = 2βUi(x; 1)/z2i . Now one has

the dimensionless mPNP equations. For simplicity, the tildes over all variables canbe dropped and the mPNP system reads

∂ci∂t

= ǫ∂

∂x

[∂

∂xci + ci

∂

∂x

(ziΦ+

1

2z2i u

)], i = ±, (2.8)

−ǫ2∂2

∂x2Φ = z+c+ + z−c−, (2.9)

which is subject to the initial-boundary conditions:

c±(x, 0) = |z∓|,Φ(±1, t) = V±,

J±(±1, t) = 0.

(2.10)

Here u(x) = γq e−(2+2ξ+2x)κ(x)

2+2ξ+2x + γq e−(2+2ξ−2x)κ(x)

2+2ξ−2x , κ(x) =√z2+c+(x) + z2−c−(x)/ǫ, and

J± = ∂∂x

ci + ci∂∂x

(ziΦ+ 1

2z2i u). We assume that ǫ = ℓ0/L and q = ℓB/L are two

small parameters. As q goes to zero, the PDE system (2.8-2.9) degenerates to theclassical PNP equations.

3. Asymptotic solutions. In this section, the method of MAE is used to solvethe initial-boundary value problem of the mPNP equations. We aim to find theleading 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 orcomparable to the Debye length (q = o(ǫ) or q = O(ǫ)), otherwise it correspondsto a strong-correlated Coulomb system which cannot be correctly described by themodified PNP equations. The case of q = o(ǫ) corresponds to systems in many areaswhen the surface charge can be very weak or the interface-interface interaction islong-ranged [44, 51, 58]. The small quality ξ is assumed to be at the same order asq such that the self energy does not blow up. Throughout the paper, we use a baraccent to represent the outer solution, and a hat accent to represent the inner solutionof the singular perturbation problem.

3.1. Outer solution. When ǫ → 0, Eq. (2.8) becomes,

∂c±,0

∂t= 0, (3.1)

which implies the leading order concentration: c±,0(x, t) = c±,0(x, 0) = |z∓|. Here andafterwards, the subscript “0” represents the leading term in the asymptotic expansion,

6 L. Ji, P. Liu, Z. Xu and S. Zhou

e.g., c± = c±,0 +O(ǫ). Summing two equations in Eq. (2.8) together and using (2.9),one obtains,

−ǫ2∂2

∂x2

∂

∂tΦ = −ǫ3

∂4

∂x4Φ+ ǫ

∂

∂x

[(z2+c+ + z2−c−)

∂Φ

∂x+

z3+c+ + z3−c−

2

∂u

∂x

]. (3.2)

So, the leading order term of the potential Φ0 satisfies,

∂

∂x

[(z2+c+,0 + z2−c−,0)

∂Φ0

∂x

]= 0. (3.3)

Here one has used the fact that κ = O(1/ǫ) and thus the spatial derivative of the selfenergy vanishes in the outer region (−1 < x < 1), and because of it the outer solutionis the same as that of the classical PNP equations [9].

Since c±,0 are constants, Φ0 is a linear function of x. One can express it byΦ0(x, t) = j0(t)x + A(t), where j0(t) is the leading term of the Faradaic currentdensity in the outer region and its initial value is given by j0(0) = (V+ − V−)/2. Itfollows from boundary conditions that A(0) = (V−+V+)/2. To summarize, the outersolution reads

{c±,0(x, t) = |z∓|,Φ0(x, t) = j0(t)x+A(t).

(3.4)

Note that one shall determine values of j0(t) and A(t) given their initial data by usingthe time-dependent matching. This will be discussed later.

3.2. Inner solution. Since the outer solution shown above does not satisfy theboundary conditions (2.10), there will be a boundary layer near each electrode, i.e.,the inner solution.

We study the inner solution near the left interface x = −1 and that near theright interface can be obtained similarly. To this end, a variable transformationy = (x + 1)/ǫα is used, where ǫα is the thickness of the boundary layer and theparameter α > 0 is to be determined. The corresponding inner solution is denoted byc±(y, t) and Φ(y, t). After the transformation, the mPNP equations become

ǫ2α−1 ∂ci∂t

=∂

∂y

(∂ci∂y

+ zici∂Φ

∂y+

1

2z2i ci

∂u

∂y

), i = ±,

−ǫ2−2α ∂2Φ

∂y2= z+c+ + z−c−,

(3.5)

where

u = qγ

(e−(2ǫαy+2ξ)κ

2ǫαy + 2ξ+

e−(4+2ξ−2ǫαy)κ

4 + 2ξ − 2ǫαy

). (3.6)

To analyze the influence of the dielectric self energy, one assumes q = ηǫθ, whereη is a positive constant. The parameter θ is discussed in the following two cases:θ = 1 and θ > 1. Since θ < 1 corresponds to strong coupling systems for which theCoulomb correlation becomes significant, this work does not analyze this regime.

Asymptotic analysis on mPNP equations 7

3.2.1. Case of θ = 1. Consider the Nernst-Planck equation for the ionic con-centration of species i in Eq. (3.5). In the case of θ = 1, we have

∂ci∂t

= O(ǫ0),∂

∂y

(∂ci∂y

)= O(ǫ0), and

∂

∂y

(zici

∂Φ

∂y

)= O(ǫ2α−2). (3.7)

The order of the term related to the dielectric self energy, ∂y(z2i ci∂yu

), depends on

the value of α, which can be determined by the following.If 0 < α < 1, one has u = o(ǫ1−α). The leading-order term in the NP equation

is ∂y(zici,0∂yΦ0). The zero-flux boundary condition implies that either the cationconcentration is zero or the potential is a constant. Clearly both cases cannot matchthe outer solution. If α > 1, one has u = O(min(ǫ1−α, ǫ/ξ)) where ξ is smaller thanǫ, and then the leading-order term in the NP equation is ∂y

(z2i ci∂yu

)/2. Again, the

zero-flux boundary condition implies either zero cation concentration or constant u0.The later case implies that the dielectric self energy does not affect the solution, andthe former case cannot match the outer solution.

When α = 1, one has u = O(ǫ0). All the terms in the right side of the NPequation have the same orders, and the leading asymptotics satisfy

∂

∂y

(∂ci,0∂y

+ zici,0∂Φ0

∂y+

1

2z2i ci,0

∂u0

∂y

)= 0. (3.8)

The zero-flux boundary condition implies that the function in the parentheses is zeroand can be integrated with respect to y to obtain

ci,0(y, t) = ai(t)e−ziΦ0(y,t)−

12 z

2i u0 . (3.9)

Here coefficients ai(t), i = ±, are functions of time t to be determined by the time-dependent matching later. Using the asymptotic matching for the ion concentrationsand potential in the inner solution and the values in outer solution, i.e.,

limx→−1

c±,0(x, t) = limy→∞

c±,0(y, t), (3.10)

limx→−1

Φ0(x, t) = limy→∞

Φ0(y, t), (3.11)

yields,

a±(t) = |z∓|ez±Φ0(−1,t). (3.12)

Hence, the ionic concentrations in the boundary layer can be implicitly expressed as

c±,0(y, t) = |z∓|e−[z±ϕ0(y,t)+12 z

2±u0(y,t)], (3.13)

with ϕ0(y, t) = Φ0(y, t) − Φ0(−1, t). In summary, the inner solution of the mPNPequations (3.5) is described by

− ∂2ϕ0

∂y2= z+c+,0 + z−c−,0,

c±,0(y, t) = |z∓|e−[z±ϕ0(y,t)+12 z

2±u0(y,t)],

u0 = qγe−

2ǫy+2ξǫ

√z+c+,0+z−c−,0

2ǫy + 2ξ,

(3.14)

8 L. Ji, P. Liu, Z. Xu and S. Zhou

with boundary conditions ϕ0(∞, t) = 0 and ϕ0(0, t) = V− + j0(t) − A(t). Noticethat the equations resemble the Poisson–Boltzmann equation but with dielectric selfenergy as a correction to the mean potential energy in the Boltzmann factor [51].

3.2.2. Case of θ > 1. One can easily obtain the orders of the left and the firsttwo terms in the right sides of (3.5), similar to the case of θ = 1. But the order ofthe self-energy contribution is different. To analyze the order of ∂y(ci∂yu), one shallconsider the following cases.

If α > θ, then u = O(min(ǫθ−α, ǫθ/ξ)) where ξ is smaller than ǫθ. Thus, theterm from the dielectric self energy is the leading-order term in (3.5). The zero-flux boundary conditions lead to z2i ci∂yu0 = 0. This implies that either the cationconcentration is zero or u0 is a constant, both resulting in nonphysical solutions. If1 < α < θ, then u = O(ǫθ−α). The order analysis shows that ∂yy ci is the leading-orderterm in (3.5). Due to the zero-flux boundary condition, the ionic concentrations haveto be constant. Matching the concentrations between the inner solution and the outersolution, one finds that the concentrations are constant in the whole domain, which is

unphysical or a trivial solution. If 0 < α < 1, then u = o(ǫθ−α) and ∂y

(zici,0∂yΦ0

)is

the leading-order term. The zero-flux boundary conditions imply that either the ionicconcentrations are zero or the potential is a constant, both of which cannot matchthe outer solution and boundary conditions.

When α = 1, then u = O(ǫθ−1), and ∂y

(∂y ci,0 + zici,0∂yΦ0

)are the leading-

order terms. When α = θ, then u = O(ǫ0), and ∂y(∂y ci,0 + 12z

2i ci,0∂yu0) are the

leading-order terms. These two cases lead to physical solutions. Actually, a two-layerstructure in the boundary layer can be obtained by matching the inner and outersolutions, which will be discussed below.

The first layer. When α = θ, the width of the boundary layer is of order O(ǫθ),and one uses the variable change w = (x+1)/ǫθ and denotes the solution in this layer

by cI+(w, t), cI−(w, t), and ΦI(w, t). In Eq. (3.5), taking the leading-order terms and

using the zero-flux boundary condition, one has

∂cIi,0∂w

+1

2z2i c

Ii,0

∂uI0

∂w= 0. (3.15)

Integrating with respect to w, one gets

cI±,0(w, t) = b±(t)e− 1

2 z2±uI

0 , (3.16)

where b±(t) are to be determined functions of t. By the expression of the self energy,one has

uI0 = qγ

e−2ǫθw+2ξ

ǫ

√z2+cI+,0+z2

−cI−,0

2ǫθw + 2ξ∼ qγ

2ǫθw + 2ξ, as ǫ → 0. (3.17)

Since θ > 1, one has ǫ−2+2θ = o(ǫ0). From the leading-order term of Poisson’s

equation, one finds that ∂wwΦI0 = 0. Thus, the potential ΦI

0 is a linear function,

ΦI0 = 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) = 0 and ΦI0 = V− to avoid the singularity. The

leading terms of the ion concentrations are given by

cI±,0 = b±(t)e−z2

±

qγ

4ǫθw+4ξ , (3.18)

Asymptotic analysis on mPNP equations 9

where b±(t) are to be determined by asymptotic matching. It is noted that thedielectric self energy contributes to the distributions of the two species in the same wayin 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 dielectricinterface is quadratic to the ion charge. We remark that the small thickness of thefirst layer will lead to great challenge if grid-based numerical approximation is usedin order to resolve the boundary layer, and this can be treated using a renormalizedboundary condition [7] to replace the contribution of this layer, considering that theintegrated charge can be calculated from the asymptotic solution.

The second layer. When α = 1, the width of the boundary layer is of orderO(ǫ) and the variable transform y = (x + 1)/ǫ is used. The solution in this layer is

denoted by cII± (y, t) and ΦII(y, t). Taking the leading-order terms in (3.5), one has

∂cIIi,0∂y

+ zicIIi,0

∂ΦII0

∂y= B, (3.19)

where B is a constant independent of y. Matching the flux densities in the innersolution onto the outer solution yields

limy→∞

1

ǫ

(∂cIIi,0∂y

+ zicIIi,0

∂ΦII0

∂y+O(ǫ)

)= lim

x→−1

(∂ci,0∂x

+ ci,0∂

∂x(ziΦ0 +

1

2z2i u0)

).

(3.20)

Clearly, one finds B = 0. Integrating Eq. (3.19) gives cII±,0 = f±(t)e−z±ΦII

0 wheref±(t) are functions of t and can be determined by matching the ion concentrationsand electric potential, i.e.,

limy→∞

cII±,0 = limx→−1

c±,0, (3.21)

limy→∞

ΦII0 = lim

x→−1Φ0, (3.22)

and consequently, f±(t) = |z∓|ez±Φ0(−1,t). Let ϕII(y, t) := ΦII(y, t) − Φ(−1, t). Oneobtains the following equations,

cII±,0 = |z∓|e−z±ϕII0 (y,t),

−∂2ϕII0

∂y2= z+c

II+,0 + z−c

II−,0,

(3.23)

Here ϕII0 (y, t) represents the leading-order term of ϕII(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 givenby the potential matchings between the first, the second and the outer layers, namely,ϕII0 (0, t) = ζL(t) , V− − Φ0(−1, t), and ϕII

0 (∞, t) = 0.By matching the ionic concentrations in the first and second layers, lim

w→∞cI±,0 =

limy→0

cII±,0, one obtains b±(t) = |z∓|e−z±ζL(t). Finally, one reaches the solution in the

first layer, which is given by,

cI±,0 = |z∓|e−z±ζL(t)−z2±

qγ

4ǫθw+4ξ ,

ΦI0 = V−.

(3.24)

10 L. Ji, P. Liu, Z. Xu and S. Zhou

In contrast to the θ = 1 case, this θ > 1 case has the dielectric boundary effect onlyin the first layer, which is of width O(ǫθ). It is noted that the leading contribution ofthe potential in the first layer is a constant and the potential drop is ζL(t) which is anO(ǫ) term. It can be observed that the dielectric self energy has a strong impact onionic concentrations through the Boltzmann factor. The whole set of the asymptoticsolution in the first layer, second layer, and outer area will be determined, once A(t)and the potential drop ζL(t) is solved by the time-dependent matching; cf. Section 3.4.At the equilibrium state, these quantities are constant and the asymptotic solution isthen determined by the above equations.

3.3. Uniformly asymptotic solutions. We have obtained the outer and theinner solutions, and consequently the asymptotic solution can be written into a formof uniformly valid approximations by summing up the inner and outer solutions andsubtracting the overlaps. Since there are two electrodes for the physical system, thesolution is split into two parts where the solution in the right half plane is obtainedby 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 partscoincide in the middle between two electrodes, once the appropriate j0(t) and A(t)are determined by the time-dependent matching.

For the case of θ = 1, the uniformly valid solutions can be written as

c±(x, t) = cL±,0

(1 + x

ǫ, t

)+ cR±,0

(1− x

ǫ, t

)− |z∓|+O(ǫ),

Φ(x, t) = ΦL0

(1 + x

ǫ, t

)+ ΦR

0

(1− x

ǫ, t

)+ 2j0(t)−A(t) +O(ǫ),

(3.25)

where A(t) = [ΦL0 (∞, t) + ΦR

0 (∞, t)]/2. It is noted that in this case the outer solu-tion for the ionic concentrations coincides with the overlap solution, and the uniformsolution happens to be the inner solution. In addition, the two boundary layers atthe left and right electrodes also coincide with the same outer solutions.

For the case of θ > 1, each boundary layer has two overlaps since the innersolution has two layers. The uniformly valid solutions can be written as

c±(x, t) = cL,I±,0

(1 + x

ǫθ, t

)+ cL,II

±,0

(1 + x

ǫ, t

)+ cR,I

±,0

(1− x

ǫθ, t

)+ cR,II

±,0

(1− x

ǫ, t

)

− |z∓|[1 + e−z±ζL(t) + e−z±ζR(t)

]+O(ǫ),

Φ(x, t) = ΦL,II0

(x+ 1

ǫ, t

)+ ΦR,II

0

(1− x

ǫ, t

)+ 2j0(t)x −A(t) +O(ǫ),

(3.26)

where A(t) = [ΦL,II0 (∞, t) + ΦR,II

0 (∞, t)]/2. In the solution for the ionic concen-trations, the first four terms represent the first- and second-layer inner solutions onboth sides, and the fifth term represents the three overlap solutions. For the electricpotential, the potential drop on the first layer is small, and thus there is only oneoverlap solution.

3.4. Time-dependent matching for mPNP equations. One shall study thetime evolution of the above asymptotic solutions for concentrations and electrostaticpotential by determining the time-dependent coefficients j(t), A(t), ζL(t) and ζR(t).We perform a time-dependent matching for the asymptotic solutions by followingBazant et al. [9]. We discuss the case of θ > 1, which has a two-layer structure inthe boundary layer. The same procedure can be readily applied to the case of θ = 1

Asymptotic analysis on mPNP equations 11

without much difference, which results in the same solution since the first layer forcase θ > 1 has neglectable contribution.

Consider the dynamics of the total diffuse charge which is the net charge in the

half space defined by Q(t) =∫ 0

−1 ρ(x)dx with ρ = z+c+ + z−c−. Since the outersolution is electrically neutral up to order O(ǫ) and the net charge in the first layeris ρI(w, t) ∼ O(ǫ2θ−2) with θ > 1, one can only consider the leading asymptotic

expansion in the second layer, Q0.By Eq. (3.23), one has

Q0 =

∫ ∞

0

[z+c

II+,0(y, t) + z−c

II−,0(y, t)

]dy

=

∫ ∞

0

−∂2ϕII0

∂y2dy =

∂ϕII0

∂y

∣∣∣∣y=0

, (3.27)

where one has used the relation limy→∞

∂yϕII0 (y, t) = lim

y→∞∂yΦ

II0 (y, t) = 0 to obtain the

last equality. Taking the time derivative of the leading-order term of the total diffusecharge and using the NP equations in Eq.(3.5) yields,

dQ0

dt=

∫ ∞

0

(z+

∂cII+,0

∂t+ z−

∂cII−,0

∂t

)dy

= limy→∞

1

ǫ

[∂ρII0∂y

+(z2+c

II+,0 + z2−c

II−,0

) ∂ΦII0

∂y+

z3+cII+,0 + z3−c

II−,0

2

∂uII0

∂y

]

= limx→−1

[∂ρ0∂x

+(z2−z+ − z2+z−

) ∂Φ0

∂x

], (3.28)

where the zero-flux boundary condition is used. In Eq. (3.28), the electroneutralitycondition ρ0 = 0 leads us to,

dQ0

dζLdζL

dt=(z2−z+ − z2+z−

)j0(t), (3.29)

where j0(t) = ∂xΦ0 is the current and is independent of x as the outer solution Φ0

is linear function of x, and ζL is the potential drop defined below Eq. (3.23). Define

the differential capacitance, C(ζL) = −dQ0/dζL, and recall ζL(t) = V−−Φ0(−1, t) =

V− + j0(t)−A(t). Eq. (3.29) can be written as,

−C(ζL)dζL

dt=(z2−z+ − z2+z−

) (ζL +A(t)− V−

). (3.30)

Similarly, for the boundary layer on the right, we have,

− C(ζR)dζR

dt= −

(z2−z+ − z2+z−

)j0(t)

=(z2−z+ − z2+z−

) (A(t) + ζR(t)− V+

), (3.31)

where ζR(t) = V+ − Φ0(1, t) = V+ − j0(t) − A(t). By the definitions of ζL and ζR,one has 2A(t) = V+ + V− − ζL − ζR. Eqs. (3.30) and (3.31) form a closed system ofODEs for the potential drops,

−C(ζL)dζL

dt=(z2−z+ − z2+z−

) V+ − V− + ζL − ζR

2,

−C(ζR)dζR

dt=(z2−z+ − z2+z−

) V− − V+ + ζR − ζL

2,

(3.32)

12 L. Ji, P. Liu, Z. Xu and S. Zhou

with initial conditions: ζL(0) = ζR(0) = 0. In particular, for a symmetric salt system,

the potential drops at the two electrodes satisfy that ζL(t)+ζR(t) ≡ 0 and A ≡ V++V−

2and these two ordinary equations are the same. In this case, we only need to solveone of them.

4. Results and discussion. In this section, we present results obtained by theasymptotic approximations (3.25-3.26), in comparison to numerical solutions solvedby a second-order finite difference method with small grid sizes. We first report theequilibrium state solutions to the mPNP system in Section 4.1 and 4.2 for symmetricand asymmetric (2:1) electrolytes, respectively. The results on the dynamics arepresented in the Section 4.3.

d

Cha

rge

Den

sity

0 0.5 1 1.5 20

0.5

1

1.5

2

γ= 19/21γ= 0γ= -1/3γ= -1Num.

(d)

d

Ani

on C

oncn

etra

tion

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

γ= 19/21γ= 0γ= -1/3γ= -1Num. (b)

d

Cat

ion

Con

cent

ratio

n

0 0.5 1 1.5 20

1

2

3

γ= 19/21γ= 0γ= -1/3γ= -1Num. (a)

d

Pot

entia

l Dis

trib

utio

n

0 0.5 1 1.5 2-0.5

-0.4

-0.3

-0.2

-0.1

0

γ= 19/21γ= 0γ= -1/3γ= -1Num.

(c)

Fig. 4.1. Ionic concentrations, electrostatic potential, and charge density for symmetric saltwith ǫ = q = 0.02, θ = 1, d = (x+1)/ǫ and various dielectric ratios. The asymptotic approximationsare shown in lines and the numerical solutions are shown with triangle symbols.

4.1. Symmetric salt. Consider a symmetric monovalent electrolyte, i.e., z+ =−z− = 1, with dielectric constant εW = 80. The dielectric boundary effect is studiedby investigating the solution with various values of the dielectric coefficient εB. Unlessotherwise stated, we take ǫ = 0.02 and ξ = q/5 in the calculations.

We first consider the case of θ = 1 for which the inner solution has a one-layerstructure. We take q = 0.02, V+ = −V− = V = 0.5, and εB = 4, 80, 160 and ∞, whereεB = ∞ corresponds to metallic electrodes; correspondingly, the dielectric ratios areγ = 19/21, 0,−1/3 and −1, respectively. The dimensionless V = 0.5 corresponds toa boundary voltage of ∼ 12.9 mV . Fig. 4.1 displays the equilibrium profiles of thecation and anion concentrations, the electrostatic potential, and the charge density

Asymptotic analysis on mPNP equations 13

as function of d = (x + 1)/ǫ which is the distance to the left electrode rescaled byǫ. Clearly, one can see that the dielectric self energy due to a low-dielectric electrode(γ = 19/21) depletes both counterions and coions. For γ = 0, there is no dielectricboundary effect, and counterions are attracted to the electrode and coions are re-pelled from the electrode. In contrast, the dielectric self energy exerts attraction onboth counterions and coions for a high-dielectric electrode (the γ < 0 cases). Moresignificant attraction can be observed as the dielectric ratio gets larger.

V

Diff

eren

tial C

apac

itanc

e

0 0.2 0.4 0.6 0.8 10.01

0.015

0.02γ= 19/21γ= 0γ= -1/3γ= -1Num.

(b)

V

Tot

al D

iffus

e C

harg

e

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04γ= 19/21γ= 0γ= -1/3γ= -1Num.

(a)

Fig. 4.2. 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 =∫ 0

−1(z+c+ + z−c−)dx and the differential capacitance

C(V ) = dQ/dV . As shown in Fig. 4.2, the total diffuse charge increases monoton-ically as the applied voltage gets larger. Also, it is observed that the dielectric selfenergy suppresses the total diffuse charge when γ > 0, and enhances the total diffusecharge when γ < 0. The dielectric ratio has a considerable impact on the differentialcapacitance as well. The capacitance increases linearly for weak applied voltages.In addition, the dielectric self energy with larger dielectric ratio has more signifi-cant attraction to ions and therefore increases the capacitance pronouncedly. Thereis great agreement between the asymptotic approximations and numerical solutions,validating the effectiveness of our asymptotic analysis.

d

Cat

ion

Con

cent

ratio

n

0 0.5 1 1.5 2

1

1.5

2

2.5

(a)

d

Cha

rge

Den

sity

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

(b)

0 0.001 0.0021.2

1.4

1.6

1.8

2

2.2 γ= 19/21

γ= 0

γ= -1/3

γ= -1

Num.

0 0.0005 0.001 0.00150.8

1

1.2

1.4γ= 19/21

γ= 0

γ= -1/3

γ= -1

Num.

Fig. 4.3. Cation concentration and charge density for symmetric salt with ǫ = 0.02 and q = 106

for various dielectric ratios. d = (x+1)/ǫ. The numerical solution corresponds to the case of q = 0.

According to our asymptotic analysis, there is a two-layer structure in the bound-

14 L. Ji, P. Liu, Z. Xu and S. Zhou

ary layer for the case of θ > 1, and the first layer close to the electrode is rather thinwith width O(ǫθ). In order to show the deck layers more intuitively, we use the sameparameters as the one-layer case but taking q = 10−6. As illustrated in Fig. 4.3, thereis a sharp transition on ionic concentrations, when the distance to the electrode isless than 0.002ǫ. The zoom-in inset clearly demonstrates that the dielectric self ener-gy 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 secondlayer cannot see the dielectric boundary effect at all, and ionic concentrations followthe classical Boltzmann distribution. In this case, the dielectric boundary effect is soweak that it only alters ionic distributions in the first layer and is completely screenedby ions in the first layer. From the profile of the charge density, we also find that thedielectric boundary affects both counterions and coions in the first layer. Such resultsconfirm and further explain the asymptotic analysis on the two-layer structure of theboundary layer. It should be noted that the numerical solutions are obtained withtaking q = 0 because it is computationally prohibitive to resolve the first thin layerwith a finite-difference grid. The agreement outside the first layer can be observed forsuch a small parameter, showing that the expansion with respect to q is a regular per-turbation for the solution outside the first layer. The embedded figures demonstratethat the dielectric self energy will provide strong dielectric-dependent interaction toions close to the surface, which cannot be ignored if one aims to understand interfaceproperties.

dL

Cat

ion

Con

cent

ratio

n

0 0.5 1 1.5 20

1

2γ= 19/21γ= 0γ= -1/3γ= -1Num.

(a)

dR

Cat

ion

Con

cnet

ratio

n

00.511.520

0.5

1

1.5

2γ= 19/21γ= 0γ= -1/3γ= -1Num.

(b)

dR

Ani

on C

once

ntra

tion

00.511.520

0.5

1

1.5

2

2.5

3

3.5γ= 19/21γ= 0γ= -1/3γ= -1Num.

(d)

dL

Ani

on C

once

ntra

tion

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3γ= 19/21γ= 0γ= -1/3γ= -1Num.

(c)

Fig. 4.4. Concentrations of divalent cations (upper panel) and monovalent anions (lower panel)close to the electrodes with different dielectric ratios. dL = (x+1)/ǫ, dR = (1−x)/ǫ, and the appliedvoltage V = 0.05.

Asymptotic analysis on mPNP equations 15

4.2. Asymmetric salt. We use our model to investigate the dielectric boundaryeffect on the equilibrium state of an asymmetric electrolyte, which consists of divalentcations and monovalent anions, i.e., z+ = 2 and z− = −1. In the calculations, wetake V+ = −V− = 0.05, ǫ = 0.02, and four different γ as in previous examples. Theparameter q = 0.02 is taken such that the asymptotic solution of θ = 1 is used. Wewill not show results of θ > 1 since the phenomenon due to the effect of the first layeris similar to that of the symmetric salt.

As displayed in Fig. 4.4, the profiles of ionic concentration are monotone whenno dielectric boundary effect is taken into account (γ = 0 case). In contrast, ionsdistribute in a totally different fashion when the dielectric self energy comes into play.The equilibrium ionic distribution is resulted from competitions between the dielectriccontribution and the direct ion-electrode electrostatic interaction. Notice that the di-rect ion-electrode interaction is rather weak with V = 0.05. The dielectric boundaryeffect dominates in the vicinity of electrodes, and depletes cations when γ > 0. Asthe distance to the electrode, d, gets larger, the attraction from the electrode startsto prevail over the dielectric boundary effect, giving rise to a slightly larger cationconcentration than that of the bulk. As γ < 0, the divalent cations feel strong attrac-tion near the electrodes and their concentrations drop quickly to a value smaller thanthat of the bulk as the distance becomes large. This is attributed to the conservationof cations in the system. Ionic concentrations have similar distributions close to theanode. Although the dielectric boundary effect is much weaker for monovalent anions,it still dominates over the direct ion-electrode electrostatic interactions. As illustratedin the lower panel of Fig. 4.4, anions feel attraction and depletion when γ < 0 andγ > 0, respectively.

To further understand the effect of the dielectric self energy, we plot in Fig. 4.5the charge density and electrostatic potential close to electrodes. We observe chargeinversion [23, 50] for both attractive and depletive dielectric boundary effects. Whenγ > 0, the electrostatic potential increases from V− at the cathode to a value greaterthan V+ in the bulk and decreases to V+ at the anode. On the contrary, the electro-static potential increases to a smaller constant potential and then attains V+ whenγ < 0. Such a behavior can be explained by the distributions of total charge densitiesthat are shown in lower panel of Fig. 4.5. Again, we can see remarkable impacts ofthe dielectric self energy on the charge distribution.

To characterize the impact of dielectric boundary effects on the differential capac-itance, we perform numerical simulations with varying applied voltages and computethe differential capacitance of the whole system, which is treated as a series connectionof two capacitors at electrodes. From Fig. 4.6, we can see that the attractive dielectricboundary effect promotes the diffusion of charges, and less charge is accumulated atthe electrode when the dielectric boundary effect is depletive. Also, the capacitancebecomes 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 fasterfor the case of asymmetric salts, in comparison with the case of symmetric salts, cf.Figs. 4.2 and 4.6. Again, the comparison between asymptotic approximations andnumerical solutions evidences that the asymptotic analysis works well for asymmetricelectrolytes as well.

4.3. Dynamics of ions. In this section, we first study the ion concentrationsand potential distributions at different times. We use the following parameters in thecalculations: V− = −0.5, V+ = 0.5, ǫ = q = 0.02, ξ = q/5, and γ = 19/21. Figs. 4.7

16 L. Ji, P. Liu, Z. Xu and S. Zhou

dL

Cha

rge

Den

sity

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5

2

2.5

3

γ= 19/21γ= 0γ= -1/3γ= -1Num.

(c)

dR

Cha

rge

Den

sity

00.511.52-1

-0.5

0

0.5

1

1.5

2

γ= 19/21γ= 0γ= -1/3γ= -1Num.

(d)

dR

Pot

entia

l Dis

trib

utio

n00.511.52-0.15

-0.1

-0.05

0

0.05

0.1

γ= 19/21γ= 0γ= -1/3γ= -1Num. (b)

dL

Pot

entia

l Dis

trib

utio

n

0 0.5 1 1.5 2-0.15

-0.1

-0.05

0

0.05

0.1

γ= 19/21γ= 0γ= -1/3γ= -1Num.

(a)

Fig. 4.5. Electrostatic potential and charge density close to the electrodes with differentdielectric ratios. dL = (x+ 1)/ǫ and dR = (1 − x)/ǫ.

V

Diff

eren

tial C

apac

itanc

e

0 0.2 0.4 0.6 0.8 10.02

0.022

0.024

0.026

0.028

0.03

0.032γ= 19/21γ= 0γ= -1/3γ= -1Num.

(b)

V

Tot

al D

iffus

e C

harg

e

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06γ= 19/21γ= 0γ= -1/3γ= -1Num.

(a)

Fig. 4.6. Total diffuse charge and differential capacitance against applied voltages at the anodewith different dielectric ratios. The salt consists of divalent cations and monovalent anions.

and 4.8 present the asymptotic approximations obtained by (3.32) and the numericalsolutions solved with finite difference methods for 1:1 and 2:1 salts, respectively. InFig. 4.7, one observes that the cation concentration has a depletion layer in the vicinityof the electrode and gradually forms a hump near the electrode. With accumulatedcations, the potential gets screened as time evolves, but keeps being a linear functionin the outer layer. For the asymmetric salt, from the potential distribution plottedin Fig. 4.8, one sees that, as time evolves, the potential are quickly screened by the

Asymptotic analysis on mPNP equations 17

xP

oten

tial D

istr

ibut

ion

-1 -0.8 -0.6 -0.4 -0.2 0-0.5

-0.4

-0.3

-0.2

-0.1

0

t= 0.1t=0.5t= 1t= 3Num.

(b)

x

Cat

ion

Con

cent

ratio

n

-1 -0.98 -0.96 -0.94 -0.92 -0.90.4

0.6

0.8

1

1.2

t= 0.1t= 0.5t= 1t= 3Num.

(a)

Fig. 4.7. Cation concentration and potential distribution for symmetric electrolytes (1:1) att = 0.1, 0.5, 1 and 3. Asymptotic approximations are shown in lines and numerical solutions areshown with triangle symbols.

x

Cha

rge

Den

sity

-1 -0.98 -0.96-2

-1

0

1

2

t = 0.5t = 1t = 2Num.

(c)

x

Cha

rge

Den

sity

0.96 0.98 1-2

-1

0

1

2

t = 0.5t =1t =2Num.

(d)

x

Pot

entia

l Dis

trib

utio

n

0 0.2 0.4 0.6 0.8 1-0.5

-0.25

0

0.25

0.5

t = 0.5t = 1t = 2Num.

(b)

x

Pot

entia

l Dis

trib

utio

n

-1 -0.8 -0.6 -0.4 -0.2 0-0.5

-0.25

0

0.25

0.5

t = 0.5t = 1t = 2Num.

(a)

Fig. 4.8. Cation concentration and potential distribution for asymmetric electrolytes (2:1) att = 0.5, 1 and 2. Asymptotic approximations are shown in lines and numerical solutions are shownwith triangle symbols.

accumulated net positive charge density near the cathode. At large time t when thesolution approaches the equilibrium state, one can see charge inversion phenomenonnear the cathode, with the potential being positive. In the vicinity of the anode,the potential also gets screened by negative charge density. However, one cannotsee charge inversion close to the anode with monovalent anions. Both figures showthat 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 theouter layer and inner layers in asymptotic analysis are accurate.

0.4

0.4

0.5

0.5

0.6

0.6

0.70.7

0.7

0.7

0.7

0.7

0.80.8

0.8

0.80.8

0.8

0.80.8

0.8

0.80.8

0.8

0.8

0.90.9

0.9

0.9

0.90.90.9

0.9

0.90.9 0.9

0.9 0.9 0.9

111

1

11

1

111

1

1 1 1

1.11.11.1

1.1

1.1 1.1 1.1

1.11.11.1

1.1 1.1 1.1

1.21.2

1.2

1.21.21.2

1.2

1.2 1.21.2

1.21.21.2

1.3

1.3

1.3

1.3

1.3

1.3

1.3

1.3

0 0.1 0.2 0.3 0.4 0.5

x

0.01

2.01

4.01

6.01

8.01

10.01

12.01

13

t

0.3

0.3

0.40.4

0.4

0.5

0.5

0.5

0.5

0.6

0.6

0.6

0.6

0.7

0.7

0.7

0.7

0.7

0.7

0.80.8

0.8

0.8

0.80.8

0.8

0.80.8

0.8

0.80.8

0.8

0.90.9

0.90.9

0.90.90.9

0.9

0.90.9 0.9

0.9 0.9

111

1

11 1

111

1

1 1 1

1.11.1

1.1

1.1 1.1 1.1

1.11.11.1

1.1 1.1 1.1

1.21.2

1.21.21.2

1.2

1.21.2

1.21.21.2

1.2

1.3

1.3

1.3

1.3

1.3

1.3

1.3

0 0.1 0.2 0.3 0.4 0.5

x

0.01

2.01

4.01

6.01

8.01

10.01

12.01

13

t

(a) NUM (b) ASY

Fig. 4.9. The evolution of cation concentration with oscillatory applied voltages. (a) Asymp-totic solution; (b) Numerical solution.

It is also significant to study the response of the boundary layer on variablesurface voltages. For the purpose, the mPNP solution for an monovalent electrolytewith applied voltages ±V = ±0.5 sin(t) is calculated by both the asymptotic and thenumerical approximations. The parameters take ǫ = 0.02, q = 0.0004 and γ = 19/21.The multiple-layer structure of the electric double layer has been reported in theliterature [6,13,43] for systems with large applied voltages, for which the steric effectbetween ions plays important role. Here we consider weak oscillatory voltages toinvestigate the dielectric boundary effect as higher voltages screen the effect. Thenumerical solutions displayed in Fig. 4.9 reveal that the weak oscillatory voltages donot alter the layer structure in the EDL. Since q is much smaller than ǫ, a two-layerstructure is present, where a thin layer can be observed near the electrode, followedby a thick diffusion layer. The asymptotic solution agrees well with the numericalapproximation, showing that the asymptotic expansion is also accurate for the timedynamics as well as the steady-state solution. We remark that in order to see howthe dielectric self energy affects the structure with a dense electrolyte or large appliedvoltage, the steric effect [43] should be incorporated into the model such that theexcess chemical potential includes both effects.

5. Concluding remarks. In this paper, we have performed an asymptotic ex-pansion analysis to understand the dielectric boundary effect on the charge dynamicsof electrolyte solutions between two blocking electrodes based on a modified PNPmodel with the dielectric self energy included in the potential of mean force. Asymp-totic solutions have shown that there is a two-layer structure close to electrodes whenthe dielectric boundary effect is weak. The dielectric self energy only comes into playin the first layer, while the second layer behaves like the classical PNP solution, with-out any correction from the dielectric boundary effect. When the dielectric boundary

Asymptotic analysis on mPNP equations 19

effect is relatively strong, the asymptotic analysis have demonstrated that there is onlyone layer in which the inner solution satisfies modified Poisson–Boltzmann equationswith the dielectric self energy appearing in the Boltzmann factor. Systematic inves-tigations on the ionic concentrations, electrostatic potential, charge density, diffusecharges, differential capacitance, and charge inversion have deepened our understand-ing of the dielectric boundary effect on electrostatic phenomena near interfaces.

We now discuss several issues and possible further refinements of the presentwork. Since the electrolytes under consideration are dilute and applied voltages areweak, the steric effect and ion-ion correlations have not been taken into account in ourcurrent model. However, they could play important roles if we consider concentratedelectrolytes or stronger applied voltages. Steric effects could be included via addingsolvent entropic contributions to the total free energy [28, 29, 35]. Alternatively, afourth-order Poisson’s equation could be applied to describe the effect of ion-ion cor-relations [2, 8, 35]. Ionic correlations induce overscreening when the applied voltageis small; while, the steric effect becomes dominant in determining the structure ofelectric double layer when the applied voltage is strong. It remains for future workto elaborate the interplay among steric effects, ion-ion correlations, and dielectricboundary effects in variable dielectric environment.

Also, it is of great interest to apply the refined mPNP model to study ionic con-ductance in ion channel problems. It has been known that, in addition to the stericeffects and ion-ion correlations, the dielectric effects play key roles in ion permeationthrough a thin channel [15, 21]. Nevertheless, not much effort has been put in thisdirection using self-consistent PNP models. We have considered two blocking elec-trodes in our current development. The charge dynamics and layer structure close toelectrodes will have a totally different picture if we consider electrodes with Faradaicreactions with nonlinear Butler-Volmer kinetics. With more refined treatments ofelectrodes, the present model will be a promising tool in predicting electrochemicalproperties near electrodes of different materials.

Acknowledgments. The authors thank the anonymous reviewers for their use-ful comments and suggestions. L. Ji and Z. Xu acknowledge the support from grantsNSFC 11571236 and 21773165 and HPC center of Shanghai Jiao Tong University.S. Zhou acknowledges the support from grants NSFC 11601361, NSFC 21773165,Natural Science Foundation of Jiangsu Province BK20160302, and Q410700415.

REFERENCES

[1] N. Abaid, R. Eisenberg, and W. Liu, Asymptotic expansions of I-V relations via a Poisson–Nernst–Planck system, Phys. Rev. E, 7 (2008), pp. 1507–1526.

[2] A.Yochelis, Transition from non-monotonic to monotonic electrical diffuse layers: impact ofconfinement on ionic liquids, Phys.Chem.Chem.Phys., 16 (2014), pp. 2836–2841.

[3] P. Balbuena and Y. Wang, Lithium ion batteries: solid electrolyte interphase, Imperial Col-lege Press, London, 2004.

[4] V. Barcilon, D. Chen, and R. Eisenberg, Ion flow through narrow membrane channels:Part II, SIAM J. Appl. Math., 52 (1992), pp. 1405–1425.

[5] V. Barcilon, D. Chen, R. Eisenberg, and J. Jerome, Qualitative properties of steady-state Poisson–Nernst–Planck systems: Perturbation and simulation study, SIAM J. Appl.Math., 57 (1997), pp. 631–648.

[6] M. Z. Bazant, K. T. Chu, and B. J. Bayly, Current-voltage relations for electrochemicalthin films, SIAM J. Appl. Math., 65 (2006), pp. 1463–1484.

[7] M. Z. Bazant, M. S. Kilic, B. D. Storey, and A. Ajdari, Towards an understandingof induced-charge electrokinetics at large applied voltages in concentrated solutions, Adv.Colloid Interface Sci., 152 (2009), pp. 48–88.

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: Over-screening versus crowding, Phys. Rev. Lett., 106 (2011), p. 046102.

[9] M. Z. Bazant, K. Thornton, and A. Ajdari, Diffuse-charge dynamics in electrochemicalsystems, Phys. Rev. E, 70 (2004), p. 021506.

[10] D. Ben-Yaakov, D. Andelman, and R. Podgornik, Dielectric decrement as a source ofion-specific effects, J. Chem. Phys., 134 (2011), p. 074705.

[11] F. P. Buff and F. H. Stillinger, Statistical mechanical theory of double-layer structure andproperties, J. Chem. Phys., 39 (1963), pp. 1911–1923.

[12] W. Cheng, C.-X. Wang, W.-Z. Chen, Y.-W. Xu, and Y.-Y. Shi, Investigating the dielectriceffects of channel pore water on the electrostatic barriers of the permeation ion by thefinite difference Poisson-Boltzmann method, Eur. Biophys. J., 27 (1998), pp. 105–112.

[13] K. T. Chu and M. Z. Bazant, Electrochemical thin films at and above the classical limitingcurrent, SIAM J. Appl. Math., 65 (2005), pp. 1485–1505.

[14] B. Corry, S. Kuyucak, and S.-H. Chung, Tests of continuum theories as models of ionchannels. II. Poisson–Nernst–Planck theory versus Brownian Dynamics, Biophys. J., 78(2000), pp. 2364–2381.

[15] B. Corry, S. Kuyucak, and S.-H. Chung, Dielectric self-energy in Poisson–Boltzmann andPoisson–Nernst–Planck models of ion channels, Biophysical J., 84 (2003), pp. 2658–2669.

[16] R. Eisenberg, From structure to function in open ionic channels, J. Membrane Biol., 171(1999), pp. 1–24.

[17] R. Eisenberg, Crowded charges in ion channels, Adv. Chem. Phys., 148 (2011), pp. 77–223.[18] R. Eisenberg, Y. Hyon, and C. Liu, Energy variational analysis of ions in water and chan-

nels: Field theory for primitive models of complex ionic fluids, J. Chem. Phys., 133 (2010),p. 104104.

[19] R. Eisenberg and W. Liu, Poisson–Nernst–Planck systems for ion channels with permanentcharges, SIAM J. Math. Anal., 38 (2007), pp. 1932–1966.

[20] N. Gavish and K. Promislow, On the structure of generalized Poisson–Boltzmann equations,Euro. J. Appl. Math., 27 (2016), pp. 667–685.

[21] P. Graf, A. Nitzan, M. G. Kurnikova, and R. D. Coalson, A dynamic lattice Monte Carlomodel of ion transport in inhomogeneous dielectric environments: method and implemen-tation, J. Phys. Chem. B, 104 (2000), pp. 12324–12338.

[22] D. C. Grahame, The electrical double layer and the theory of electrocapillarity, Chem. Rev.,32 (1947), pp. 441–501.

[23] A. Y. Grosberg, T. T. Nguyen, and B. I. Shklovskii, The physics of charge inversion inchemical and biological systems, Rev. Mod. Phys., 74 (2002), pp. 329–345.

[24] X. Guan, M. Ma, Z. Gan, Z. Xu, and B. Li, Hybrid Monte Carlo and continuum modelingof electrolyte with concentration-induced dielectric variations, Phys. Rev. E, 94 (2016),p. 053312.

[25] 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 (2012), p. 28010.

[26] B. Hille, Ion channels of excitable membranes, Sinauer Associates, 3rd ed., 2001.[27] 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 (1989),pp. 1041–1052.

[28] M. S. Kilic, M. Z. Bazant, and A. Ajdari, Steric effects in the dynamics of electrolytes atlarge applied voltages. I. double-layer charging, Physical Review E, 75 (2007), p. 021502.

[29] M. S. Kilic, M. Z. Bazant, and A. Ajdari, Steric effects in the dynamics of electrolytesat large applied voltages. II. Modified Poisson–Nernst–Planck equations, Phys. Rev. E, 75(2007), p. 021503.

[30] M. G. Kurnikova, R. D. Coalson, P. Graf, and A. Nitzan, A lattice relaxation algorithm forthree-dimensional Poisson–Nernst–Planck theory with application to ion transport throughthe gramicidin a channel, Biophys. J., 76 (1999), pp. 642–656.

[31] D. G. Levitt, Strong electrolyte continum theory solution for equilibrium profiles, diffusionlimitation, and conductance in charged ion channels, Biophys. J., 48 (1985), pp. 19–31.

[32] B. Li, P. Liu, Z. Xu, and S. Zhou, Ionic size effects: generalized Boltzmann distributions,counterion stratification, and modified debye length, Nonlinearity, 26 (2013), p. 2899.

[33] B. Li, J. Wen, and S. Zhou, Mean-field theory and computation of electrostatics with ionicconcentration dependent dielectrics, Commun. Math. Sci., 14(1) (2016), pp. 249–271.

[34] D. Li, Electrokinetics in microfluidics, vol. 2, Academic Press, 2004.[35] J.-L. Liu and R. Eisenberg, Correlated ions in a calcium channel model: A Poisson-Fermi

theory, J. Phys. Chem. B, 117 (2013), pp. 12051–12058.[36] P. Liu, X. Ji, and Z. Xu, Modified Poisson–Nernst–Planck model with accurate coulomb

Asymptotic analysis on mPNP equations 21

correlation in variable media, SIAM J. Appl. Math., 78 (2018), pp. 226–245.[37] W. Liu, Geometric singular perturbation approach to steady-state Poisson–Nernst–Planck sys-

tems, SIAM J. Appl. Math., 65 (2005), pp. 754–766.[38] B. Lu and Y. Zhou, Poisson–Nernst–Planck equations for simulating biomolecular diffusion-

reaction processes II: size effects on ionic distributions and diffusion-reaction rates, Bio-phys. J., 100 (2011), pp. 2475–2485.

[39] G. Minton and L. Lue, The influence of excluded volume and excess ion polarisability on thecapacitance of the electric double layer, Mol. Phys., 114 (2016), pp. 2477–2491.

[40] G. Moy, B. Corry, S. Kuyucak, and S.-H. Chung, Tests of continuum theories as modelsof ion channels. I. Poisson–Boltzmann theory versus Brownian Dynamics, Biophys. J., 78(2000), pp. 2349–2363.

[41] R. R. Netz and H. Orland, Beyond Poisson–Boltzmann: Fluctuation effects and correlationfunctions, Eur. Phys. J. E, 1 (2000), pp. 203–214.

[42] R. R. Netz and H. Orland, Variational charge renormalization in charged systems, Eur.Phys. J. E, 11 (2003), pp. 301–311.

[43] L. H. Olesen, M. Z. Bazant, and H. Bruus, Strongly nonlinear dynamics of electrolytes inlarge ac voltages, Phys. Rev. E, 82 (2010), p. 011501.

[44] L. Onsager and N. N. T. Samaras, The surface tension of Debye-Huckel electrolytes, J.Chem. Phys., 2 (1934), pp. 528–536.

[45] R. Podgornik, Electrostatic correlation forces between surfaces with surface specific ionicinteractions, J. Chem. Phys., 91 (1989), pp. 5840–5849.

[46] R. Roth, R. Evans, A. Lang, and G. Kahl, Fundamental measure theory for hard-spheremixtures revisited: the white bear version, J. Phys.: Condens. Matter, 14 (2002), p. 12063.

[47] R. Schoch, J. Han, and P. Renaud, Transport phenomena in nanofluidics, Rev. Mod. Phys.,80 (2008), pp. 839–883.

[48] A. Singer, D. Gillespie, J. Norbury, and R. Eisenberg, Singular perturbation analysis ofthe steady-state Poisson–Nernst–Planck system: Applications to ion channels, Euro. J.Appl. Math., 19 (2008), pp. 541–560.

[49] B. D. Storey and M. Z. Bazant, Effects of electrostatic correlations on electrokinetic phe-nomena, Phys. Rev. E, 86 (2012), p. 056303.

[50] R. Wang and Z.-G. Wang, Effects of image charges on double layer structure and forces, J.Chem. Phys., 139 (2013), p. 124702.

[51] R. Wang and Z. G. Wang, On the theoretical description of weakly charged surfaces, J. Chem.Phys., 142 (2015), p. 104705.

[52] X. Wang, D. He, J. Wylie, and H. Huang, Singular perturbation solutions of steady-statePoisson–Nernst–Planck systems, Phys. Rev. E, 89 (2014), p. 022722.

[53] Z. G. Wang, Fluctuation in electrolyte solutions: The self energy, Phys. Rev. E, 81 (2010),p. 021501.

[54] P. Weetman, S. Goldman, and C. G. Gray, Use of the Poisson–Boltzmann equation to esti-mate The electrostatic free energy barrier for dielectric models of biological ion channels,J. Phys. Chem. B, 101 (1997), pp. 6073–6078.

[55] Z. Xu, M. Ma, and P. Liu, Self-energy-modified Poisson-Nernst-Planck equations: WKBapproximation and finite-difference approaches, Phys. Rev. E, 90 (2014), p. 013307.

[56] Y.-X. Yu and J.-Z. Wu, Structures of hard-sphere fluids from a modified fundamental-measuretheory, J. Chem. Phys., 117 (2002), pp. 10156–10164.

[57] H. Zhao and S. Zhai, The influence of dielectric decrement on electrokinetics, J. Fluid Mech.,724 (2013), pp. 69–94.

[58] J. W. Zwanikken and M. O. de la Cruz, Tunable soft structure in charged fluids confinedby dielectric interfaces, Proc. Nat. Acad. Sci. USA, 110 (2013), pp. 5301–5308.