# Asymmetric electrostatic properties of an electric double layer: a
generalized Poisson-Boltzmann approach taking into account non-uniform size
effects and water polarization

Jun-Sik Sin<EMAIL_ADDRESS>Song-Jin Im Kwang-Il Kim Department of
Physics, Kim Il Sung University, Daesong District, Pyongyang, DPR Korea

###### Abstract

We theoretically study electrostatic properties of electric double layer using
a generalized Poisson-Boltzmann approach taking into account the orientational
ordering of water dipoles and the excluded volume effect of water molecules as
well as those of positive and negative ions with different sizes in
electrolyte solution.

Our approach enables one to predict that the number densities of water
molecules, counterions and coions and the permittivity of electrolyte solution
close to a charged surface, asymmetrically vary depending on both of sign and
magnitude of the surface charge density and the volume of counterion. We treat
several phenomena in more detail. Firstly, an increase in the volume of
counterions and an increase in the surface charge density can cause the
position of the minimum number density of water molecules to be farther from
the charged surface. Secondly, width of the range of voltage in which the
properties at the charged surface symmetrically vary decreases with increasing
bulk salt concentration. In addition, we show that the excluded volume effect
of water molecules and the orientational ordering of water dipoles can lead to
early onset and lowering of the maximum of electric capacitance according to
surface voltage. Our approach and results can be applied to describing
electrostatic properties of biological membranes and electric double layer
capacitor for which excluded volume effects of water molecules and ions with
different sizes may be important.

###### keywords:

Electric double layer, Excluded volume effect, Poisson-Boltzmann equation,
Differential capacitance, Orientational ordering of water PACS: 82.45.Gj,
82.60.Lf, 66.10.-x, 61.20.Q

##  1 Introduction

The concept of electric double layer was first presented by Herman von
Helmholtz[1] before more than one century. Since then many researchers have
developed realistic theories of electric double layer to solve various
problems of biology, medicine, colloid science and electrochemistry such as
the binding of charged ligands to the membrane surface, the interactions of
vesicles with the membrane, osteoblast attachment to biomaterials, fundamental
nucleic acid processes, RNA folding, and differential capacitance of electric
double layer capacitor [2, 3, 4, 5].

To correctly represent electrostatic properties of electric double layer, a
number of computational approaches such as Monte Carlo method and numerical
solutions of integral equations[6, 7, 8, 9] were introduced, but they involve
more complicated calculations than for the Poisson-Boltzmann (PB) approach.

However, the original PB approach proposed by Gouy and Chapmann [10, 11]
doesn’t consider the finite volumes of ions in electrolyte and it is known
that the approach heavily overestimates the ionic concentrations close to
charged surfaces in electrolyte. As an early attempt to eliminate such a
shortcoming of the original PB approach, Stern [12] considered the finite size
effect of ions by combining the Helmholtz model with the Gouy-Chapmann model.
To include volume effects of ions directly into the PB approach, Bikerman[13]
empirically modified Boltzmann distribution by correcting ion concentrations
for the volume excluded by all ions.

In [14, 15, 16, 17, 18, 19], the authors considered finite volumes of ions and
water molecules within lattice statistical mechanics approach. The assumption
of the same size of ions and water molecules in electrolyte has been common to
their work. Although this assumption does properly work in many situations,
ionic transport across narrow channels and ionic adsorption in objects of
subnanometer size cannot be described by means of the assumption. For example,
when the size of a negative ion is larger than one of a positive ion, negative
ionic transport across narrow channels of biological membranes or pores of
electrode of electric double layer capacitor will hardly proceed.

Modified PB approaches using lattice statistics [20, 21, 22] were developed
for considering the difference in sizes of positive and negative ions. For
this purpose, in [20], a lattice in which one cell can contain several ions
was used. In [21], the authors semiempirically extended a modified PB equation
to the case of an asymmetric salt. Recently, in [22], the authors assumed that
the lattice cell size was an integer number of times smaller than a linear
dimension of ion.

In fact, the Booth model [23, 24] is well-known for generalization of the
Onsager-Kirkwood-Fröhlich permittivity model [25, 26] in the saturation regime
of orientational ordering of water dipoles, but the model doesn’t consider the
sizes of both ions and water molecules in electrolyte solution.

In [27, 28, 29, 30, 32], the authors took into account the excluded volume
effect of water molecules and the orientational ordering of water dipoles
together with the excluded volume effect of ions in the modification of the PB
approach using lattice statistics. In particular, the authors described that
the permittivity of an electrolyte solution near a strongly charged surface
may be heavily decreased by orientational ordering of water dipoles and
depletion of water molecules [28, 29, 30]. In their approach, each particle
occupies one cell of lattice based on the assumption that the ions and water
molecules have the same excluded volume, which allowed the PB equations to
have analytical and intuitive solutions. However, in cases where effects of
difference in excluded volumes of water molecules and ions may be important,
for example, where the electric capacitance is asymmetric due to the
difference in sizes of positive and negative ions [22], a more general
approach taking into account the effects is needed.

In this paper we will incorporate not only the asymmetric size effect of ions
in electrolyte but also both the orientational ordering of water dipoles and
excluded volume effect of water molecules into the Poisson-Boltzmann approach.
In a word, our approach generalizes that of [28] to include non-uniform ionic
sizes. We introduce a lattice statistics where more than one cell can be
occupied by each ion as in [22] and also by each water molecule for
considering effects of different excluded volumes of ions and water molecules.
We show that electrostatic properties of electrolyte solution close to a
charged surface aren’t symmetric in positive and negative surface charge
densities of the charged surface. We study effects of the volume of counterion
and sign and magnitude of the surface voltage on electrostatic properties of
the electrolyte solution. Finally, an early onset and lowering of the maximum
of electric capacitance are predicted.

##  2 The generalized Poisson-Boltzmann approach

We consider an electrolyte solution composed of multivalent ions and water
molecules in contact with a charged planar surface, where a positive ion has
charge $+ze_{0}$ and a negative ion has charge $-ze_{0}$. The total free
energy $F$ can be written in terms of the local electrostatic potential
$\psi\left(r\right)$ and the number densities of ions $c_{+}\left(r\right)$,
$c_{-}\left(r\right)$ and water molecules
$c_{w}\left(r\right)=\left<\rho\left(\omega,r\right)\right>_{\omega}$.

$\displaystyle F=\int{d{\bf r}}\left(-\frac{\varepsilon_{0}\varepsilon
E^{2}}{2}+e_{0}z\psi\left(c_{+}-c_{-}\right)+\left<\rho\left(\omega\right)\gamma{p_{0}}E\cos\omega\right>_{\omega}-\mu_{+}c_{+}-\mu_{-}c_{-}-\left<\mu_{w}\left(\omega\right)\rho\left(\omega\right)\right>_{\omega}-Ts\right),$
(1)

where $\left<f\left(\omega\right)\right>_{\omega}=\int
f\left(\omega\right)2\pi\sin\left(\omega\right)d\omega$ in which $\omega$ is
the angle between the vector p and the normal to the charged surface. Here p
is the dipole moment of water molecules and E is the electric field strength.
The first term is the self energy of the electrostatic field, where
$\varepsilon$ equals $n^{2}$ and $n=1.33$ is the refractive index of water.
The next term corresponds to the electrostatic energy of the ions in the
electrolyte solution, where $e_{0}$ is the elementary charge. The third one
represents the electrostatic energy of water dipoles[28], where
$\gamma=\left(2+n^{2}\right)/2$, $p_{0}=\left|{\bf p}\right|$ and
$E=\left|{\bf E}\right|$. The next three terms are responsible for coupling
the system to a bulk reservoir, where $\mu_{+,-}$ are the chemical potentials
of positive ions and negative ions and $\mu_{w}\left(\omega\right)$ is the
chemical potential of water dipoles with orientational angle $\omega$. $T$ is
the temperature and $s$ is the entropy density.

Consider a unit volume of the electrolyte solution. The entropy density is the
logarithm of the number of translational and orientational arrangements of
non-interacting $c_{+}$ positive ions, $c_{-}$ negative ions and
$\rho\left(\omega_{i}\right)\Delta\Omega_{i}\left(i=1\cdots N\right)$ water
molecules, where
$\Delta\Omega_{i}=2\pi\sin\left(\omega_{i}\right)\Delta\omega$ is an element
of a solid angle and $\Delta\omega=\pi/N$. The positive ion, negative ion and
water molecule occupy volumes of $V_{+},V_{-}$ and $V_{w}$, respectively.

Within a lattice statistics approach each particle in the solution occupies
more than one cell of a lattice as in [22]. Considering translational
arrangements of ions and orientational ordering of water dipoles, the number
of arrangements can be calculated as follows. As in [22], we first place
$c_{+}$ positive ions of the volume $V_{+}$ and then $c_{-}$ negative ones of
the volume $V_{-}$ in the lattice. Finally, taking into account the
orientational ordering of water dipoles, we put in
$\rho\left(\omega_{i}\right)\left(i=0,1,...\right)$ water molecules of the
volume $V_{w}$ in the lattice. The number of arrangements $W$ is written as

$W=\frac{c_{s}\left(c_{s}-1\cdot
v_{+}\right)\cdots\left(c_{s}-\left(c_{+}-1\right)v_{+}\right)}{c_{+}!}\frac{\left(c_{s}-c_{+}v_{\pm}\right)\cdots\left(c_{s}-c_{+}v_{\pm}-\left(c_{-}-1\right)v_{-}\right)}{c_{-}!}\frac{\left(c_{s}-c_{+}v_{+}-c_{-}v{-}\right)\cdots
v_{w}}{\lim_{N\rightarrow\infty}\prod^{N}_{i=1}\rho\left(\omega_{i}\right)\Delta\Omega_{i}!},$
(2)

where $v_{+,-,w}=V_{+,-,w}/a^{3}$ are the numbers of cells that the positive
ion, negative ion and water molecule occupy, respectively. $c_{s}=1/a^{3}$ is
the number of cells per unit volume and $a$ denotes the linear dimension of
one cell.

From the standpoint of physics the entropy density should be symmetric in $+$
and $-$. For this purpose, we assume that the positive(negative) ion excludes
$v_{\pm}=\left(v_{+}+v_{-}\right)/2$ for the negative(positive) ion.

Expanding the logarithms of factorials using Stirling formula, we obtain the
expression for the entropy density, $s=k_{B}\ln W$,

$\displaystyle\frac{s}{k_{B}}=\ln
W=-c_{+}\ln{a^{3}}-c_{-}\ln{a^{3}}-\left(\frac{1-c_{+}V_{+}-c_{-}V_{-}}{V_{w}}\right)\ln{a^{3}}-c_{+}\ln
c_{+}-c_{-}\ln
c_{-}-\left(\frac{1}{V_{+}}-c_{+}\right)\ln\left(1-c_{+}V_{+}\right)$
$\displaystyle+\left(\frac{1}{V_{-}}-\frac{c_{+}V_{\pm}}{V_{-}}\right)\ln\left(1-c_{+}V_{\pm}\right)-\left(\frac{1}{V_{-}}-\frac{c_{+}V_{\pm}}{V_{-}}-c_{-}\right)\ln\left(1-c_{+}V_{\pm}-c_{-}V_{-}\right)$
$\displaystyle-\left(\frac{1}{V_{w}}-\frac{c_{+}V_{+}}{V_{w}}-\frac{c_{-}V_{-}}{V_{w}}\right)\ln\left(1-c_{+}V_{+}-c_{-}V_{-}\right)-\lim_{N\rightarrow\infty}\sum^{N}_{i=1}\left[\rho\left(\omega_{i}\right)\Delta\Omega_{i}\ln\Delta\Omega_{i}+\rho\left(\omega_{i}\right)\Delta\Omega_{i}\ln\rho\left(\omega_{i}\right)-\rho\left(\omega_{i}\right)\Delta\Omega_{i}\right],$
(3)

where $k_{B}$ is the Boltzmann constant. Assuming that
$c_{i}V_{j}\left(i=+,-;j=+,-,\pm\right)$ are small, the above expression gets
simpler:

$\displaystyle\frac{s}{k_{B}}=-c_{+}\ln{c_{+}a^{3}}-c_{-}\ln{c_{-}a^{3}}-\left(\frac{1-c_{+}V_{+}-c_{-}V_{-}}{V_{w}}\right)\ln{a^{3}}+c_{+}\left(1-\frac{V_{+}}{V_{w}}\right)+c_{-}\left(1-\frac{V_{-}}{V_{w}}\right)$
$\displaystyle-\frac{c_{+}^{2}V_{+}}{2}-\frac{c_{-}^{2}V_{-}}{2}+\frac{\left(c_{+}V_{+}\right)^{2}}{2V_{w}}+\frac{\left(c_{-}V_{-}\right)^{2}}{2V_{w}}-c_{+}c_{-}V_{\pm}+c_{+}c_{-}\frac{V_{+}V_{-}}{V_{w}}-\lim_{N\rightarrow\infty}\sum^{N}_{i=1}\left[\rho\left(\omega_{i}\right)\ln\rho\left(\omega_{i}\right)+\rho\left(\omega_{i}\right)\ln\Delta\Omega_{i}\right]\Delta\Omega_{i}.$
(4)

Like the modification of the Poisson-Boltzmann approach of [22], the
expression for the entropy density is symmetric in negative and positive ions.

All lattice cells should be occupied by either ions or water molecules[16, 32,
33, 34, 35], therefore

$\displaystyle c_{s}=c_{+}v_{+}+c_{-}v_{-}+c_{w}v_{w}.$ (5)

In order to find the free energy in equilibrium taking into account Eq. (5),
we will use the method of undetermined multipliers. The Lagrangian of the
electrolyte solution is

$\displaystyle L=F-\int\lambda\left({\bf
r}\right)\left(1-c_{+}V_{+}-c_{-}V_{-}-c_{w}V_{w}\right)d{\bf r},$ (6)

where $\lambda$ is a local Lagrange parameter. Once the Lagrangian is
established, the Euler$-$Lagrange equations are obtained and solved with
respect to the functions $c_{+},c_{-}$ and $\rho\left(\omega\right)$. The
variation of the Lagrangian with respect to $c_{+}$ yields an equation from
which we can get the number density of positive ions in the electrolyte
solution:

$\displaystyle\frac{\delta L}{\delta c_{+}}=e_{0}z\psi-\mu_{+}+k_{B}T\left(\ln
c_{+}a^{3}+c_{+}V_{+}\left(1-\frac{V_{+}}{V_{w}}\right)+c_{-}V_{\pm}-c_{-}\frac{V_{+}V_{-}}{V_{w}}\right)+\lambda
V_{+}=0.$ (7)

Using the boundary conditions $\psi\left(x\rightarrow\infty\right)=0$ and
$c_{+,-}\left(x\rightarrow\infty\right)=c_{0}$ and
$\lambda\left(x\rightarrow\infty\right)=\lambda_{0}$, we get the chemical
potential for positive ions from Eq.(7):

$\displaystyle\mu_{+}=k_{B}T\left(\ln
c_{0}a^{3}+c_{0}V_{+}\left(1-\frac{V_{+}}{V_{w}}\right)+c_{0}V_{\pm}-c_{0}\frac{V_{+}V_{-}}{V_{w}}\right)+\lambda_{0}V_{+}.$
(8)

Inserting Eq.(8) into Eq.(7), we obtain $c_{+}$ by exponentiation:

$\displaystyle
c_{+}=c_{0}\exp\left(-\frac{e_{0}z\psi}{k_{B}T}\right)\exp\left(-\left[\left(c_{+}-c_{0}\right)V_{+}\left(1-\frac{V_{+}}{V_{w}}\right)+\left(c_{-}-c_{0}\right)\left(V_{\pm}-\frac{V_{+}V_{-}}{V_{w}}\right)\right]\right)\exp\left(\lambda
V_{+}\right),$ (9)

where $\left(\lambda-\lambda_{0}\right)/k_{B}T\rightarrow\lambda$ for
simplicity. Like the derivation of Eq.(9), the expressions for $c_{-}$ and
$\rho\left(\omega\right)$ are simply obtained:

$\displaystyle
c_{-}=c_{0}\exp\left(\frac{e_{0}z\psi}{k_{B}T}\right)\exp\left(-\left[\left(c_{-}-c_{0}\right)V_{-}\left(1-\frac{V_{-}}{V_{w}}\right)+\left(c_{+}-c_{0}\right)\left(V_{\pm}-\frac{V_{+}V_{-}}{V_{w}}\right)\right]\right)\exp\left(\lambda
V_{-}\right),$ (10)
$\displaystyle\rho\left(\omega\right)=\rho_{0}\exp\left(-\frac{\gamma
p_{0}E\cos\omega}{k_{B}T}\right)\exp\left(\lambda V_{w}\right).$ (11)

In general, within our approach the number densities of ions and water
molecules are obtained implicitly not explicitly.

In the case when the ions and water molecules have the same size and the
surface charge density is negative, that is, when $V_{+}=V_{-}=V_{w}$ and
$\sigma<0$, we can recover all basic equations of [28]. When we neglect
orientational ordering of water dipoles, our approach is identical to that of
[22].

The Euler$-$Lagrange equation for $\psi\left(r\right)$ yields the Poisson-
Boltzmann equation

$\displaystyle\nabla\left(\varepsilon_{0}\varepsilon_{r}\nabla\psi\right)=-e_{0}z\left(c_{+}-c_{-}\right),$
(12)

where

$\displaystyle\varepsilon_{r}\equiv n^{2}+\frac{{\bf P}}{\varepsilon_{0}{\bf
E}}.$ (13)

Here, ${\bf P}$ is the polarization vector due to a total orientation of
point-like water dipoles. From the planar symmetry of this problem, one can
see that the electric field strength is perpendicular to the charged surface
and have the same magnitude at all points equidistant from the charged
surface. The $x$ axis points in the direction of the bulk solution and is
perpendicular to the charged surface. Consequently, along the $x$ axis E and P
have only an $x$ component and P is given as [28]

$\displaystyle{\bf
P}\left(x\right)=c_{w}\left(x\right)\left(\frac{2+n^{2}}{3}\right)p_{0}\mathcal{L}\left(\gamma{p_{0}}E\beta\right)\hat{{\bf
e}},$ (14)

where a function $\mathcal{L}\left(u\right)=\coth\left(u\right)-1/u$ is the
Langevin function, $\hat{{\bf e}}={{\bf E}/E}$ and
$\beta=1/\left(k_{B}T\right)$ .

Differentiation of Eqs.(9)-(11) with respect to the distance from the charged
surface provides linear algebraic equations in terms of
$d{c_{+}}/dx,d{c_{-}}/dx,d{c_{w}}/dx,d{\lambda}/dx$:

$\displaystyle\frac{d{c_{+}}}{dx}=c_{+}\left[-\frac{d\Phi}{dx}-V_{+}\left(1-\frac{V_{+}}{V_{w}}\right)\frac{d{c_{+}}}{dx}-\left(V_{\pm}-\frac{V_{+}V_{-}}{V_{w}}\right)\frac{d{c_{-}}}{dx}+V_{+}\frac{d\lambda}{dx}\right],$
(15)
$\displaystyle\frac{d{c_{-}}}{dx}=c_{-}\left[\frac{d\Phi}{dx}-V_{-}\left(1-\frac{V_{-}}{V_{w}}\right)\frac{d{c_{-}}}{dx}-\left(V_{\pm}-\frac{V_{+}V_{-}}{V_{w}}\right)\frac{d{c_{+}}}{dx}+V_{-}\frac{d\lambda}{dx}\right],$
(16)
$\displaystyle\frac{d{c_{w}}}{dx}=c_{w}\left[\mathcal{L}\left(\gamma{p_{0}}E\beta\right)\left(\gamma{p_{0}}\beta\right)\frac{dE}{dx}+V_{w}\frac{d\lambda}{dx}\right],$
(17)

where $\Phi\left(x\right)=e_{0}z\psi\left(x\right)/k_{B}T$.

Solving the system of Eqs.(5),(15)-(17) for
$d{c_{+}}/dx,d{c_{-}}/dx,d{c_{w}}/dx,d{\lambda}/dx$ results in the following
coupled differential equations:

$\displaystyle\frac{d{c_{+}}}{dx}=\frac{c_{+}}{D}\left(c_{-}V_{-}^{2}+c_{-}V_{+}V_{-}-c_{-}c_{w}V_{+}V_{-}V_{w}-c_{-}c_{w}V_{-}^{2}V_{w}+c_{-}c_{w}V_{-}V_{w}^{2}+c_{-}c_{w}V_{\pm}V_{w}^{2}+c_{w}V_{w}^{2}\right)\frac{d\Phi}{dx}$
$\displaystyle+\frac{c_{+}c_{w}V_{w}}{D}\left(-c_{-}V_{\pm}V_{-}+V_{+}+c_{-}V_{+}V_{-}\right){\mathcal{L}}\left(\gamma{p_{0}}E\beta\right)\left(\gamma{p_{0}}\beta\right)\frac{dE}{dx},$
(18)
$\displaystyle\frac{d{c_{-}}}{dx}=\frac{c_{-}}{D}\left(-c_{+}V_{+}^{2}-c_{+}V_{+}V_{-}+c_{+}c_{w}V_{+}V_{-}V_{w}+c_{+}c_{w}V_{+}^{2}V_{w}-c_{+}c_{w}V_{+}V_{w}^{2}-c_{+}c_{w}V_{\pm}V_{w}^{2}-c_{w}V_{w}^{2}\right)\frac{d\Phi}{dx}$
$\displaystyle+\frac{c_{-}c_{w}V_{w}}{D}\left(-c_{+}V_{\pm}V_{+}+V_{-}+c_{+}V_{+}V_{-}\right)\mathcal{L}\left(\gamma{p_{0}}E\beta\right)\left(\gamma{p_{0}}\beta\right)\frac{dE}{dx},$
(19)
$\displaystyle\frac{d{c_{w}}}{dx}=-\frac{c_{w}V_{w}}{D}\left[c_{+}V_{+}-c_{-}V_{-}+c_{+}c_{-}V_{\pm}\left(V_{+}-V_{-}\right)\right]\frac{d\Phi}{dx}$
$\displaystyle-\frac{c_{w}}{D}\left[c_{+}V_{+}^{2}+c_{-}V_{-}^{2}+c_{+}c_{-}V_{+}V_{-}\left(V_{+}+V_{-}-2V_{\pm}\right)\right]\mathcal{L}\left(\gamma{p_{0}}E\beta\right)\left(\gamma{p_{0}}\beta\right)\frac{dE}{dx},$
(20)

where

$\displaystyle
D=-c_{+}V_{+}^{2}\left(1-c_{w}V_{w}\right)\left(1+c_{-}V_{-}\right)-c_{-}V_{-}^{2}\left(1-c_{w}V_{w}\right)\left(1+c_{+}V_{+}\right)$
$\displaystyle-
c_{w}V_{w}^{2}\left[\left(1+c_{+}V_{+}\right)\left(1+c_{-}V_{-}\right)-c_{+}c_{-}V_{\pm}^{2}\right]+2c_{+}c_{-}\left(1-c_{w}V_{w}\right)V_{+}V_{-}V_{\pm},$
(21)

The electrostatic potential and the number densities of the ions and water
molecules are obtained by solving Eqs.(12),(18)-(21). In many cases of
biological and chemical systems, effective volumes of positive ions, negative
ions and water molecules in electrolyte solutions differ from each other, so
that our approach will be useful for actual cases.

##  3 Results and Discussion

Under the boundary conditions $\psi\left(x\rightarrow\infty\right)=0$ and
$E\left(x=0\right)=\sigma/\left(\varepsilon_{0}\varepsilon_{r}\left(x=0\right)\right)$,
we combine Eq.(12) and Eqs.(18-21) and solve these differential equations for
$c_{+},c_{-},c_{w},\psi$ by using the fourth order Runge-Kutta method. As in
[28], the water dipole moment $p_{0}$ should be 3.1 D(Debye is $3.336\times
10^{-30}$C/m) so that far away from the charged surface the relative
permittivity of electrolyte solution reaches 78[27, 28]. In calculations, we
choose $c_{0w}/N_{A}=55mol/l$ for the number density of water molecules in the
bulk electrolyte solution [27, 28].

Figure 1: (Color online) The number densities of water molecules, coions and
counterions as functions of the distance from the charged planar surface.
Dipole moment of water $p_{0}=3.1D$, monovalent bulk salt concentration
$c_{0}/N_{A}=0.15mol/l$ and temperature $T=300K$. (a) The number density of
water molecules as a function of the distance from the charged planar surface.
The case of $V_{-}=V_{+}=V_{w}=0.03nm^{3}$ is compared with the case of
$V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$. Here the surface charge density is
$\sigma=+0.2C/m^{2}$. (b). The number density of water molecules as a function
of the distance from the charged planar surface. For
$V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$, the case of $\sigma=+0.2C/m^{2}$ is
compared with the case of $\sigma=-0.2C/m^{2}$. (c). For the case of
$V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$, the number densities of counterions
(squares) and coions(diamonds). For the case of
$V_{-}=V_{+}=V_{w}=0.03nm^{3}$, the number densities of counterions (circles)
and coions(triangles). Here the surface charge density is
$\sigma=+0.2C/m^{2}$. (d). For the case of $\sigma=+0.2C/m^{2}$, the number
densities of counterions (squares) and coions(diamonds). For the case of
$\sigma=-0.2C/m^{2}$, the number densities of counterions (circles) and
coions(triangles). Here $V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$. Figure 2: (Color
online) The magnitude of electric field strength (a) and the magnitude of the
first derivative of electric field strength (b) according to the distance from
the charged surface for the cases of $V_{-}=V_{+}=V_{w}=0.03nm^{3}$(circles),
$V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$ (solid line)and
$V_{+}=2.21V_{-},V_{-}=0.15nm^{3}$(dashed line). Here surface charge density
is $\sigma=+0.2C/m^{2}$ and other parameters are the same as in Fig. 1(a).

Fig. 1(a) compares the calculated number densities of water molecules for
$V_{-}=V_{+}=V_{w}=0.03nm^{3}$ used in [30] and
$V_{+}=0.15nm^{3},V_{-}=2.21V_{+},V_{w}=0.03nm^{3}$ (of our case), where the
surface charge density is $\sigma=+0.2C/m^{2}$. We note that in [22] the
asymmetric parameters $V_{+}=0.15nm^{3},V_{-}=2.21V_{+}$ are used, however the
authors didn’t consider the excluded volume effect of water molecules. Here,
monovalent bulk salt concentration is taken to be $c_{0}/N_{A}=0.15mol/\l$. In
the case when ions and water molecules are of equal size the number density of
water molecules is monotonously decreased towards the charged surface in
agreement with [30]. For our case when ions and water molecules aren’t of the
same size, the number density of water molecules first decreases with
increasing distance from the charged surface and reaches a minimum. Then this
density increases and reaches the bulk value of water 55mol/l. Such a
behaviour of the number density of water molecules in the vicinity of the
charged surface was also predicted and shown to be a consequence of a balance
between the counterion Boltzman factor
$c_{0}\exp\left(e_{0}|\phi\left(x\right)|\beta\right)$ and rotationally
averaged water Boltzmann factor $c_{0w}\left<\exp\left(-\gamma
p_{0}E\beta\cos\omega\right)\right>_{\omega}$ in [30]. However, it is noticed
that in our case the surface charge density for this characteristic behaviour
is smaller than that of [30]. This is attributed to the excluded volume of
counterions larger than that in [30]. Fig. 1(b) represents the number
densities of water molecules for surface charge densities of equal magnitude
and opposite sign, where $V_{+}=0.15nm^{3},V_{-}=2.21V_{+},V_{w}=0.03nm^{3}$.
It is shown that the position of the minimum for $\sigma=+0.2c/m^{2}$ is
farther from the charged surface than that for $\sigma=-0.2c/m^{2}$. Beyond
the positions of the minima, the number density of water molecules for
$\sigma=+0.2c/m^{2}$ is smaller than that for $\sigma=-0.2c/m^{2}$. We note
that the effective volume of counterion for $\sigma=+0.2c/m^{2}$ is larger
than that for $\sigma=-0.2c/m^{2}$, which results in surface-charge-sign-
dependent $c_{w}$ as mentioned above.

Fig. 1(c) and Fig. 1(d) show the number densities of ions for the cases of
Fig. 1(a) and Fig. 1(b), respectively. Fig. 1(c) and Fig. 1(d) indicate that
the number density of coions is much less than the number density of
counterions near the charged surface.

As shown in Fig. 1(b), it is interesting that an increase in effective volume
of counterion causes the position of minimum number density of water molecules
to be farther from the charged surface. Eq.(20) provides a way to understand
physical meaning of the fact. The value of the right term of Eq.(20) is equal
to zero at the position of the minimum number density of water molecules.
Considering that the number density of coions near the charged surface is much
smaller than that of counterions as shown in Fig. 1(c) and Fig. 1(d), Eq.(22)
is obtained.

$\displaystyle\frac{V_{w}}{V_{-}}=\frac{\gamma{p_{0}}}{e_{0}z}\frac{\mathcal{L}\left(\gamma{p_{0}}E\beta)\right)}{E}\left|\frac{dE}{dx}\right|,$
(22)

where without loss of generality, we assume that the charged surface has a
positive charge. But it is difficult to analyze this equation directly because
all the physical quantities(i.e. the electric field strength, the number
densities of water molecules and ions and the permittivity of the electrolyte
solution) are subtly linked together. The electric field strength and the
first derivative of the electric field strength are the key to understanding
the above equation. Fig. 2(a) and Fig. 2(b) show the magnitudes of the
electric field strength and first derivative of electric field strength
according to the distance from the charged surface, respectively, where the
surface charge density is $\sigma=+0.2C/m^{2}$. Other parameters are the same
as in Fig. 1(a). Fig. 2(a) and Fig. 2(b) obviously illustrate the fact that
the larger the volume of counterion, the weaker the screening property of
counterion: an increase in volume of counterion causes both an increase in the
electric field strength and a decrease in the first derivative of electric
field strength. We can easily know that the function
$\mathcal{L}\left(x\right)/x$ in Eq.(13) decreases with increasing value of
$x$. From Eq. (22) and the above facts, it is deduced that the position shift
of minimum water density is attributed to difference in screening properties
of counterions due to the difference in volumes of counterions.

Figure 3: (Color online)The number density of water molecules according to the
distance from the charged surface for different surface charge densities:
$\sigma=+0.1C/m^{2},+0.2C/m^{2},+0.3C/m^{2},+0.4C/m^{2}$. Here
$V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$. Other parameters are the same as in Fig.
1(a).

Fig. (3) shows the variation of number density of water molecules with the
distance from the charged surface for different surface charge densities. Fig.
(3) indicates that the minimum number density of water molecules doesn’t
depend on the surface charge density and that the larger the surface charge
density, the more distant the position of the minimum value from the charged
surface. In fact, an increase in surface charge density of the charged surface
causes increases in magnitude for both the electric field strength and the
first derivative of electric field strength in the electrolyte solution.
Consequently, the position at which Eq.(22) is satisfied is farther from the
charged surface than that for the original surface charge density.

Figure 4: (Color online) (a) The variation of the permittivity of the
electrolyte solution with the distance from the charged surface for both
surface charge densities $\sigma=+0.2C/m^{2}$ and $\sigma=-0.2C/m^{2}$. (b)
The orientational distribution function of cos($\theta$) where $\theta$ is the
angle between a water dipole and the vector normal to the electrode surface
(pointing into the bulk). For the case of $\sigma=+0.2C/m^{2}$ the curves at
the charged surface($x=0$)(dotted line) and $x=1nm$(dashed line) are compared
with corresponding ones at the charged surface($x=0$)(solid line) and
$x=1nm$(circles) for the case of $\sigma=-0.2C/m^{2}$. Other parameters are
the same as in Fig. 1(b).

Fig. 4(a) represents the permittivities of the electrolyte solution for
$\sigma=+0.2c/m^{2}$ and $\sigma=-0.2c/m^{2}$. Other parameters are the same
as in Fig. 1(b). Fig. 4(a) shows that the permittivity for the larger size of
counterion is smaller than that for the smaller size of counterion. The reason
for it is explained by two effects. On one hand, the number density of water
molecules for the larger size of counterion is small compared to that for the
smaller size of counterion as shown in Fig. 1(b). On the other hand, the
increase in electric field strength due to difference in volumes of
counterion(as shown in Fig. 2(a)) also causes a decrease of permittivity of
electrolyte solution since the function $\mathcal{L}\left(x\right)/x$ in
Eq.(13) decreases with increasing value of $x$. By using Eqs.(13), (14)
together with the above facts, it is found that the permittivity of the
electrolyte solution for the larger size of counterion becomes smaller than
that for the smaller size of counterion.

Figure 5: (Color online) Permittivity (a), number density of water molecules
(b), number densities of coions and counterions (c) and magnitude of electric
field strength (d) in the electrolyte solution at charged planar surface
according to magnitude of the surface voltage. Physical quantity for
$c_{0}/N_{A}=0.15mol/l$ and positive voltage (dashed line), Physical quantity
for $c_{0}/N_{A}=0.15mol/l$ and negative voltage(dash-dotted line), Physical
quantity for $c_{0}/N_{A}=0.01mol/l$ and positive voltage(solid line),
Physical quantity for $c_{0}/N_{A}=0.01mol/l$ and negative voltage(dotted
line). Here $V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$. Other parameters are the same
as in Fig. 1(a).

In order to understand the dependence of the permittivity on the distance from
the charged surface, it is helpful to look at orientational distribution
function $p\left(\Omega\right)$. According to the definition of orientational
distribution function, $p\left(\Omega\right)d{\Omega}$ is the probability of
finding a molecule with solid angles $\Omega=\left(\theta,\phi\right)$. In the
case of planar surface, $p\left(\Omega\right)$ is independent of the azimuthal
angle $\phi$. Since $p\left(\Omega\right)$ should be normalized,
$\int^{\pi}_{0}p\left(\cos\left(\theta\right)\right)\sin\left(\theta\right){d}\theta=1$.
The orientational distribution function is obtained by applying Eq.(11) to the
normalization condition.

$\displaystyle p\left(\cos\left(\theta\right)\right)=\frac{\gamma
p_{0}E}{k_{B}T}\cdot\frac{\exp\left(-\frac{\gamma
p_{0}E\cos\left(\theta\right)}{k_{B}T}\right)}{\left(\exp\left(\frac{\gamma
p_{0}E}{k_{B}T}\right)-\exp\left(-\frac{\gamma p_{0}E}{k_{B}T}\right)\right)}$
(23)

Fig. 4(b) shows the orientational distribution functions
$p\left(\cos\left(\theta\right)\right)$ at different distances($x=0$ and
$x=1nm$) from the charged surface for $\sigma=+0.2c/m^{2}$ and
$\sigma=-0.2c/m^{2}$. Other parameters are the same as in Fig. 4(a). Due to
simplicity of our mean-field approach, Fig.4(b) differs from the results of
[37, 38, 39]. However, Fig. 4(b) well represents the fact that at the
positively charged surface the population of water dipoles pointing into the
charged surface($\cos\left(\theta\right)<0$) is depleted but at the negatively
charged surface a large population of water dipoles points into the charged
surface. Since a large population of water molecules is highly oriented under
a large electric field, more polarization is hardly provided and the
permittivity of electrolyte solution should therefore decrease. In [36], the
authors reported both the electric field strength dependence and spatial
variation of relative permittivity for different solvent composition. The
behaviours of permittivities for our and their approaches are similar. Unlike
the situation in our approach, the authors assumed that possible directions
for the solvent dipoles are perpendicular to the electrode surface: the
dipoles moments pointing toward the surface or opposite to the surface.

Fig. 5(a), Fig. 5(b), Fig. 5(c) and Fig. 5(d) show the permittivity, the
number density of water molecules, the number densities of coions and
counterions, and the magnitude of the electric field strength at the charged
surface as functions of surface voltage for different bulk salt concentrations
($c_{0}/N_{A}=0.01mol/l,0.15mol/l$), respectively. Other parameters are the
same as in Fig. 1(a).

Let’s consider the case of $c_{0}/N_{A}=0.01mol/l$. For $|U|<0.1V$, the
symmetric shape of the permittivity curve of the electrolyte is due to the
small number density of counterions near the charged surface and the
negligible excluded volume effect of counterions as shown in Fig. 5(c). For
$|U|>0.1V$, the permittivity doesn’t symmetrically vary with $U$ due to the
different excluded volume effects of positive and negative ions. In the region
of $0.1<|U|<0.2V$, the permittivity of the electrolyte solutions for the
larger size of counterion is larger than that for the smaller size of
counterion. Conversely, for $|U|>0.2V$, the permittivity for the smaller size
of counterion is higher than for the larger size of counterion. The behaviour
of the permittivity for $|U|>0.2V$ is attributed to the following two facts.
Firstly, Fig. 5(b) shows that at the charged surface, the number density of
water molecules is a non-monotonic function according to the surface charge
density. This fact is understood by the formation of the minimum number
density of water molecules as shown in Fig. 1(a). Secondly, Fig. 5(d) shows
that an increase in magnitude of the surface voltage at the charged surface is
accompanied by an increase in magnitude of the electric field strength at the
position. Finally, the magnitudes of both quantities for the larger size of
counterion are smaller than that for the smaller size of counterion as shown
in Fig. 5(b) and Fig. 5(d). The behaviour of the permittivity for $|U|>0.2V$
is explained by using Eqs.(13), (14), together with these three facts.

Fig. 5(a), (b), (c) and (d) show that the higher the bulk salt concentration,
the narrower the voltage region where the permittivity is symmetric with
respect to the surface voltage. This is explained by the fact that at the
higher bulk salt concentration, the number density of counterions at the
neutral surface($\sigma=0$) is larger than that for the smaller bulk salt
concentration. Fig. 5(a) also indicates that at high voltages the effect due
to difference in bulk salt concentrations is negligible and the permittivity
doesn’t depend on the bulk salt concentration. This is attributed to the fact
that at high voltages, the counterionic densities at the charged surface are
equal to each other, regardless of the bulk salt concentration of the
electrolyte solution as shown in Fig. 5(c).

Figure 6: (Color online) (a) The differential capacitance according to surface
voltage for our approach( $V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$), Popovic’s
approach( $V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$) and Gongadze’s
approach($V_{-}=V_{+}=V_{w}$). Here $c_{0}/N_{A}=0.01mol/l$. (b) The
differential capacitance according to surface voltage for our approach. The
concentrations of the electrolyte solution are 0.01mol/l,0.05mol/l, 0.15mol/l,
1mol/l, respectively. Here $V_{-}=2.21V_{+},V_{+}=0.15nm^{3}$. Other
parameters are the same as in Fig. 1(a).

Fig. 6(a) shows the voltage dependences of differential capacitance calculated
by using our approach , Popovic’s approach[22] and Gongadze’s approach[28].
Fig. 6(b) represents the voltage dependences of differential capacitance
obtained by our approach for the following bulk salt concentrations;
$c_{0}/N_{A}=0.01mol/l,0.05mol/l,0.15mol/l,1mol/l$. In Fig. 6(a) and Fig.
6(b), other parameters are the same as in Fig. 1(a). As shown in Fig. 6(a) and
Fig. 6(b) all the differential capacitances aren’t monotonic functions of
surface voltage but show the same behavior, i.e. they first increase at low
voltages, then have maxima at intermediate voltages and slowly decrease toward
zero at higher voltages.

In the low-voltage region, counterions can come closer to the charged surface
easily by a removal of water molecules and the permittivity of electrolyte
solution hardly changes. In the high-voltage region, the packing and excluded
volume effects become more important and the charge storage is more difficult,
leading to a decrease of capacitance. Moreover, our approach predicts that
water polarization lowers the permittivity of electrolyte solution at
intermediate and high voltages and therefore enhances the decrease of
differential capacitance.

Our approach provides the results which significantly differ from those in
[22]. In particular, we note that the magnitude of the maximum differential
capacitance for our approach is about half of that in [22]. In the case of
Popovic’s approach [22], the change of permittivity of the electrolyte
solution due to the orientational ordering and excluded volume effect of water
molecules is not taken into account which results in the overestimated
magnitude of the maximum differential capacitance. In the case of our
approach, the voltage for the maximum differential capacitance is slightly
smaller than that for Popovic’s approach. This is attributed to the fact that
the excluded volume effect of water molecules prevents accumulation of ions
close to the charged surface and causes the early onset of saturation of
counterionic density.

Gongadze’s approach [28] accounts for the above-mentioned change of
permittivity of electrolyte solution. However, Gongadze’s approach is based on
the assumption that all kinds of ions have the same size as water molecules
have. As shown in Fig. 6(a), the maximum differential capacitance for
Gongadze’s approach is larger than that for our approach. The reason for the
discrepancy in the maximum capacitance between the two cases is that the
larger size of counterions causes the early onset of saturation of
counterionic density.

At high voltages the differential capacitance for our approach is identical
with one for Gongadze’s approach as shown in Fig. 6(a). The differential
capacitance is defined as follows,

$\displaystyle c=d\sigma/dU,$ (24)

where $U=\psi\left(x=0\right)$ and
$\sigma=\epsilon_{0}\epsilon_{r}\left(x=0\right)E\left(x=0\right)$. The larger
size of counterion results in both the smaller permittivity of electrolyte
solution and the larger magnitude of electric field strength as shown in Fig.
5(a) and Fig. 5(d). These two effects cancel out each other at high voltages,
so that the variation of surface charge density with the surface voltage
doesn’t depend on volumes of ions.

In the high voltage region, the differential capacitance for our approach
varies with the surface voltage, regardless of the bulk salt concentration, as
shown in Fig. 6(b). This fact is explained by using Eq. (24), together with
the fact that in the high voltage region, the permittivity and the magnitude
of the electric field strength doesn’t depend on the bulk salt concentration
as shown in Fig. 5(a) and Fig. 5(d).

Finally, we note that the differential capacitances for these three approaches
are quite similar at low voltages because the electrolyte permittivity at the
low voltages doesn’t exhibit significant change as shown in Fig. 5(a) and the
excluded volume effect of ions is negligible.

Although our differential capacitance curves represent some characteristics of
experimental results of [31, 40, 41], the comparison with experimental data is
rather misleading. According to the Stern model [12], the total capacitance is
given by an inner layer capacitance and a diffuse layer capacitance in series.
For all practical cases the inner layer capacitance, which is due to the
effect of electrode material and its coupling with the solution, is comparable
to or smaller than the diffuse layer capacitance. Therefore, the diffuse layer
capacitance is significantly higher than the total capacitance in practice.
Because our theory predicts only the diffuse layer capacitance, our calculated
capacitance is still higher than the experimental values. In order to
accurately determine properties of electric double layer, quantum mechanics
should be used and it requires a lot of computational efforts as in [42, 43,
44, 45]. However, if we are interested in the properties of electric double
layer related to only the electrolyte solution, our approach can predict the
properties with a modest accuracy. Actually, our differential capacitance
curves show behaviours similar to the results of density functional theory
such as [8, 46].

##  4 Conclusions

We have presented the generalized Poisson-Boltzmann approach based on a
lattice statistics taking into account the excluded volume effects of both
ions and water molecules and the orientational ordering of water dipoles. Our
approach takes into account also the influence of the sign of surface charge
density on electrostatic properties near the charged surface.

It is predicted that the position of the minimum number density of water
molecules can be more distant from the charged surface by an increase in
volume of counterions and an increase in the surface charge density.

The width of the voltage region in which electrostatic properties of
electrolyte solution symmetrically vary with surface voltage, decreases with
increasing bulk salt concentration.

Additionally, the asymmetric, double-humped shape of differential capacitance
curve compared to ones of [22, 28] shows early onset and lowering of the
maximum of capacitance according to surface voltage which are attributed to
not only the excluded volume effect of ions and water molecules but also the
saturated orientational ordering of water dipoles.

Reference

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