HUNGARIAN JOURNAL OF
INDUSTRY AND CHEMISTRY

Vol. 43(2) pp. 55-66 (2015)
hjic.mk.uni-pannon.hu

DOI: 10.1515/hjic-2015-0010

SOME ANALYTIC EXPRESSIONS FOR THE CAPACITANCE AND
PROFILES OF THE ELECTRIC DOUBLE LAYER FORMED
BY IONS NEAR AN ELECTRODE

DOUGLAS HENDERSON ∗

Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah 84602, USA

The electric double layer, which is of practical importance, is described. Two theories that yield analytic results, the
venerable Poisson-Boltzmann or Gouy-Chapman-Stern theory and the more recent mean spherical approximation, are
discussed. The Gouy-Chapman-Stern theory fails to account for the size of the ions nor for correlations amoung the
ions. The mean spherical approximation overcomes, to some extent, these deficiencies but is applicable only for small
electrode charge. A hybrid description that overcomes some of these problems is presented. While not perfect, it gives
results for the differential capacitance that are typical of those of an ionic liquid. In particular, the differential capacitance
can pass from having a double hump at low concentrations to a single hump at high concentrations.

Keywords: Electric double layer, capacitance, Gouy-Chapman-Stern theory, mean spherical approx-
imation, density functional theory, computer simulation

1. Introduction

A double layer (DL) or an electric double layer (EDL) is
formed when charged particles are attracted to a charged
surface. The most obvious case is an electrolyte near a
charged electrode (as in a battery). However, DNA can
play a role that is analogous to the electrode. Ions can
be attracted to membranes. A membrane can be thought
of as a pseudo electrode. Ions are absorbed (often selec-
tively) into physiological channels in membranes. Such
channels permit the transport of nutrients into the cell and
the removal of waste from the cell and are essential to
the functioning of cells and life. The reader’s attention is
drawn to some recent reviews of EDLs [1–3].

It is the case of an electrolyte near a charged flat sur-
face that is considered here. This is the simplest case; it
is an interesting and important application of statistical
mechanical theory. The theory of the DL is important to
our understanding of batteries. It can be used in the analy-
sis of experimental electrochemical data and in analytical
chemistry.

In the model DL that is presented here, the electrode
is approximated as a smooth flat charged surface located
at x = 0. This surface is impenetrable and the ions are
confined to the region x > 0. The charge of the elec-
trode is located on the surface. There is no charge inside
the electrode (x < 0). The electric field does not pene-
trate the surface. The electrode is a classical metal. Ob-
viously, this is an approximation but there has been very

∗Author for correspondence: doug@chem.byu.edu

little work that takes into account the electronic structure
of the electrode. The electrode charge is presumed to be
uniform; the charge density of the electrode is σ and has
the units of C/m2. Ions in the electrolyte near an elec-
trode that have a charge opposite to that of the electrode
are attracted to the electrode and form a layer whose net
charge is equal in magnitude, but opposite in electric sign,
to the charge of the electrode. The electrode and the at-
tracted charge are together called an EDL. The charge in
the EDL of the electrolyte can be spread over an extended
region, usually called the diffuse layer, and need not con-
sist solely of counterions whose charge is opposite to the
electrode charge. The counterions can bring some coions
with them. There may be regions of alternating charge
where the coions predominate. However, the net charge
of the attracted charged region in the electrolyte is equal
in magnitude but opposite in sign to that of the electrode.
Otherwise, the electric field would not vanish far from the
electrode.

For simplicity, the model electrolyte that is employed
here is a fluid of charged hard spheres of diameter d. In
this study, the electrolyte is assumed to be binary. For ad-
ditional simplicity, the ions are assumed in this article to
be symmetric both in the magnitude of their charge and
diameter. The value of the charge of an ion of species i
is zie, where zi is the ion valence and has the sign of
the ion charge. The magnitude of the elementary charge
is e. Because the ions are symmetric, |zi| = z. In the
bulk, the density of the ions of species i is ρi = Ni/v,
where Ni is the number of ions of species i in the bulk



56 HENDERSON

and v is the volume of the system. Electrical neutrality
requires that N1 = N2 or ρ1 = ρ2 or

∑
ziρi = 0. The

solvent (usually water) of the electrolyte is characterized
by a dielectric constant, �. Any change of the dielectric
constant with a change of ion concentration is ignored.
This model electrolyte is appropriately called the primi-
tive model (PM). In the particular case considered here,
where the ions all have the same diameter, this model is
called the restricted primitive model (RPM).

In this model, the interaction between a pair of ions,
whose centers are separated by the distance r, is given by

uij(r) =



∞ for r < d
zizje

2

4π�0�r
for r ≥ d

, (1)

where �0 is the permittivity of free space, and the interac-
tion of an ion with the surface is given by

uwi(x) =

{
∞ for x < d/2
−
σziex

�0�
for x ≥ d/2 , (2)

where x is the distance between the center of the ion and
the surface.

Our task is to determine the density profile, ρi(x), of
the ions, or equivalently, gi(x) = ρi(x)/ρi. Note that
ρi(∞) = ρi, so that gi(∞) = 1. Once, the gi(x) are
known, the charge profile (C/m2), for x > d/2, is given
by

q(x) = e
∑
i

ziρihi(x), (3)

where hi(x) = gi(x) − 1. In writing Eq. 3, the global
charge neutrality condition

∑
ziρi = 0 has been invoked.

The charge density on the electrode is given by

σ = −e
∑
i

ziρi

∫ ∞
d/2

hi(t)dt. (4)

There is no point including the region 0 < t < d/2 in the
integral since

∑
hi(t) = 0 in this region. The potential

profile (in Volts) is given by

φ(x) = −
e

��0

∑
i

ziρi

∫ ∞
x

(t−x)hi(t)dt. (5)

In particular, the potential (Volts) of the electrode is given
by

V = φ(0) = −
e

��0

∑
i

ziρi

∫ ∞
0

thi(t)dt. (6)

Note that these equations satisfy Poisson’s equation

d2φ(x)

dx2
= −

q(x)

��0
. (7)

Indeed, Eqs. 3 and 5 are obtained by integrating Poisson’s
equation. An alternative procedure for computing the po-
tential profile has been proposed by Boda and Gillespie
[4] for simulation purposes.

It is often convenient to use dimensionless, or re-
duced, values that are denoted by an asterisk. For a sys-
tem whose temperature (K) is T , the reduced temper-
ature is T∗ = 4π��0dkT/z2e2. The reduced density
is ρ∗i = ρid

3, the reduced electrode charge density is
σ∗ = σd2/e, and the reduced potential is φ∗ = βeφ,
where β = 1/kT , with k being the Boltzmann constant
(the gas constant per particle).

2. Poisson–Boltzmann or
Gouy–Chapman–Stern theory:
comparison with simulations

The classic theory of the EDL was developed by Gouy
[5], Chapman [6], and Stern [7] (GCS) a century ago. The
theory is based on Poisson’s equation together with the
Boltzmann formula,

gi(x) =

{
0 x < d/2
exp[−βzieφ(x)] x ≥ d/2

. (8)

In electrostatics, Poisson’s equation is exact and is equiv-
alent to one of Maxwell’s equations. The Boltzmann for-
mula is approximate and neglects ion size and correla-
tions between the ions. Eq. 8 states that gi(x) for the
coions is the reciprocal of gi(x) for the counterions. This
is not true, in general [8].

Equation 8, when inserted into Poisson’s equation,
yields what may be called the Poisson-Boltzmann (PB)
or GCS approximation. This approximation is also em-
ployed in the Debye-Hückel (DH) theory for bulk elec-
trolytes that was developed some years later. However,
because of the three dimensional geometry of the DH the-
ory, the nonlinear PB equation cannot be solved analyti-
cally and the PB equations in the DH theory are usually
linearized. In the GCS theory, the resultant PB equation
is a nonlinear second order differential equation. As has
been pointed out, such equations generally do not yield
analytic solutions. However, for the one dimensional ge-
ometry of the planar DL that is considered here, an ana-
lytic solution is possible in the case of the GCS theory.

The resulting PB/GCS potential is

βzeφ(x)

2
= ln

{
1 +

b/2

1 +
√

1 + b2/4
exp[−κy]

}

− ln

{
1 −

b/2

1 +
√

1 + b2/4
exp[−κy]

}
, (9)

where y = x−d/2 > 0 and

b =
βzeσ

��0κ
, (10)

where κ is the Debye screening parameter that is given
by

κ =

√
βz2e2ρ

��0
(11)

Hungarian Journal of Industry and Chemistry



DOUBLE LAYERS NEAR AN ELECTRODE 57

with ρ =
∑
ρi. The parameter b is another dimension-

less measure of the electrode charge density. However, it
is not as fundamental a quantity as σ∗ since it arises from
a theory. The parameter κ is a screening parameter; it is
an inverse measure of the distance over which the pro-
files reach their asymptotic values within the GCS and
DH theories.

In the GCS theory the relationship between the poten-
tial difference and electrode charge density is given by

sinh

[
βzeφd/2

2

]
=
b

2
, (12)

where φd/2 = φ(d/2) is often called the diffuse layer
potential. Some relations that are equivalent to Eq. 12 are

cosh

[
βzeφd/2

2

]
=
√

1 + b2/4, (13)

tanh

[
βzeφd/2

2

]
=

b/2√
1 + b2/4

, (14)

and

tanh

[
βzeφd/2

4

]
=

b/2

1 +
√

1 + b2/4
. (15)

The equivalence of Eqs. 12–15 is a result of identities
among the hyperbolic functions.

Thus, Eq. 9 can be written as

βzeφ(x)

2
= ln

{
1 + tanh

[
βzeφd/2

4

]
exp(−κy)

}
− ln

{
1 − tanh

[
βzeφd/2

4

]
exp(−κy)

}
. (16)

Alternative forms of Eqs. 9 and 16 are

tanh

[
βzeφ(x)

4

]
= tanh

[
βzeφd/2

4

]
exp(−κy) (17)

or

tanh

[
βzeφ(x)

4

]
=

b/2

1 +
√

1 + b2/4
exp(−κy). (18)

In the GCS theory, the potential difference across the
EDL is

V = −
ze

��0

∑
i

ρi

∫ ∞
0

thi(t)dt =
σd

2��0
+ φd/2, (19)

where φd/2 is given by Eq. 12. Thus, the capacitance,
C = σ/V , of the EDL is

1

C
=

d

2��0
+

2 sinh−1(b/2)

��0κb
(20)

and the differential capacitance, Cd = ∂σ/∂V , of the
EDL is given by

1

Cd
=

d

2��0
+

1

��0κ
√

1 + b2/4
. (21)

Equations 20 and 21 are formally identical to a diffuse
layer capacitor with capacitance,

Cdl = ��0κ
b/2

sinh−1(b/2)
, (22)

or differential capacitance

Cdld = ��0κ
√

1 + b2/4 (23)

in series with an inner–layer parallel plate capacitor with
capacitance (or differential capacitance),

Cil = Cild =
2��0
d
. (24)

At contact,

gi(d/2) = exp[βzieφd/2] = 1 +
b2

2
−
zi
z
b

√
1 +

b2

4
,

(25)
so that gsum(d/2) =

1
2
[g1(d/2) +g2(d/2)] is, in the GCS

theory, given by

gsum(d/2) = 1 +
b2

2
. (26)

This is to be compared with the exact result (for the re-
stricted PM) due to Henderson and Blum [9] and Hender-
son, Blum, and Lebowitz [10],

gsum(d/2) =
p

ρkT
+
b2

2
, (27)

where p is the osmotic pressure of the electrolyte. The
second term in the above equation is just the Maxwell
electrostatic stress. Thus, Eq. 27 is just a force balance
condition where the momentum transfer to the electrode
is equal to the sum of the osmotic term and the Maxwell
stress. The GCS theory deals with the electrostatic term
correctly but replaces the osmotic pressure with the ideal
gas result p = ρkT because of the neglect of the ion
diameters.

For comparison with the mean spherical approxima-
tion (MSA), which is a linear response theory that will
be considered in the next section, it is worthwhile to give
the linearized GCS theory results, obtained for the case
of small electrode charge. In this case,

gi(x) =

{
0 for x < d/2
1 −βzieφ(x) for x ≥ d/2

, (28)

or

gi(x) =




0 for x < d/2

1 +
βzieσ

��0κ
exp (−κy) for x ≥ d/2

.

(29)
The potential profile in the diffuse layer is given by

φ(x) = φd/2 exp(−κy) (30)

with φd/2 given by

φd/2 =
σ

��0κ
. (31)

43(2) pp. 55-66 (2015) DOI: 10.1515/hjic-2015-0010



58 HENDERSON

0.00 0.01 0.02 0.03 0.04

(C
d

dl
)
-1

 [cm
2
/µF]

0.02

0.03

0.04

0.05

0.06

0.07
C

d

-1
 [

c
m

2
/µ

F
]

-8

-4

-2

0

2

4

8

Figure 1: Experimental values of the inverse differential
capacitance, Cd, of an aqueous solution of NaH2PO4 at 25
◦C near a hanging drop mercury electrode as a function of
the inverse diffuse layer capacitance, Cdld , obtained from
Eq. 23. The points are the experimental results of Parsons
and Zobel. The light straight lines of unit slope give the
results of the GCS theory but with the experimental in-
ner layer capacitance obtained empiricially. The numbers
at the low concentration end of the lines give the elec-
trode charges in units of µCcm−2. The heavy solid curve
gives the results of the MSA using a dipolar hard sphere
model for the solvent together with an estimate of the con-
tribution of the electronic structure of the electrode and is
intended only as an aid to the eye. This figure has been
reproduced, with permission, from Ref. [1].

The potential difference across the EDL is

V =
σd

2��0
+

σ

��0κ
(32)

and the capacitance (and differential capacitance) of the
EDL is given by

1

C
=

d

2��0
+

1

��0κ
. (33)

In the GCS theory, in the limit of large κ (high concentra-
tion) or large b (high electrode charge), the diffuse layer
capacitance is large and, as a result, the inner layer capac-
itance dominates, due to the reciprocal or series additivity
of Eqs. 20 and 21. Hence, in the GCS theory, C = 2��0/d
is the limiting (maximum) value of C or Cd at large con-
centrations or large electrode charges. Further, the differ-
ential capacitance at low concentrations looks something
like a parabola but flattens out at large σ. At high concen-
trations, the differential capacitance is constant. Any ad-
ditional shape in the experimental differential capacitance
is added by an empirical fit of the inner layer capacitance
to the experimental results. However, the diffuse layer ca-
pacitance is presumed to be given adequately by the GCS
theory.

Parsons and Zobel (PZ) [11] have plotted their exper-
imental results for the inverse of the differential capaci-
tance as a function of the inverse of the diffuse layer dif-

ferential capacitance, given by Eq. 23. Such a plot is of-
ten called a Parsons-Zobel plot. If the GCS theory were
correct, this should result in a straight line. The extrapola-
tion of the straight line to 1/Cdld = 0 (high concentration
and/or high electrode charge density) should, if the GCS
theory were correct, yield the reciprocal of the inner layer
capacitance. As is seen in Fig. 1, at first sight the exper-
imental results of PZ (the points) do seem to follow a
straight line and, conventionally, are presumed to provide
an experimental verification of the GCS theory. In Fig.
1, the light straight lines are the GCS results. The solid
curve is the result of the MSA that has not yet been dis-
cussed. For the moment the solid line can be considered
to be an aid to the eye in following the trend of the experi-
mental results. In the conventional GCS picture, the inner
layer capacitance might not be given by Eq. 24 but might
differ because of the presumed effect of the presence and
nature of the solvent molecules and the electronic struc-
ture of the electrode that are beyond the GCS theory. Pos-
sible solvent effects might be a lower dielectric constant
due to the alignment of the solvent molecules because of
the strength of the electrode charge. The important point
is that, in the GCS theory, such solvent effects are pre-
sumed to be confined only to the inner layer. The GCS
theory is conventionally considered to provide an ade-
quate description of the diffuse layer where the ions are
present. Also, it is thought to provide a description of the
EDL when combined with some treatment of the solvent
molecules in the inner layer, or even an empirical fit. In-
deed, the extrapolation of the straight lines to 1/Cdld = 0
is one method of obtaining presumed “experimental” val-
ues of Cild .

However, a careful examination of the PZ experimen-
tal results in Fig. 1 indicates that the experimental dif-
ferential capacitance does not follow Eq. 21 at high con-
centrations (the left side of the figure) but rises above the
extrapolated intercept, possibly without limit. Until re-
cently, most experimentalists have ignored this point and
did not concern themselves with this issue because ions
are not soluble in water when their concentration is large
and it is difficult to obtain results with other solvents.
Additionally, experimental results are difficult to obtain
at high electrode charges for conventional electrolytes.
However, as we shall see, DLs in ionic liquids can be
formed at high concentrations and the deficiencies of the
GCS theory become quite apparent.

Given that experiments on aqueous systems are dif-
ficult in regimes where problems with the GCS theory
become apparent, it is useful to consider computer sim-
ulations. One simulation technique is the Monte Carlo
(MC) method. Until recently, it has been the most com-
mon simulation tool in DL studies. In MC simulations the
ions undergo a random walk and the profiles and other
properties of interest are obtained by averages over this
random walk. A simple random walk would take forever
before useful results could be obtained. However, mean-
ingful results can be obtained by means of a biased ran-
dom walk that confines the ions to regions in which they

Hungarian Journal of Industry and Chemistry



DOUBLE LAYERS NEAR AN ELECTRODE 59

0 5 10 15
b

0

2

4

6

βe
φ(

d/
2)

a

0.0 0.1 0.2

σd
2
/e

0

2

4

6

βe
φ(

d/
2)

b

Figure 2: Diffuse layer potential of a 1:1 electrolyte (d =
3 Å) at room temperature as a function of b (part a) and
σ (part b). The curves are, from top to bottom, for 0.01,
0.1, and 1 M solutions. The symbols give the simulation
results. The solid curve in part a gives the GCS results.
The dotted lines connect the MC results for easier visual-
ization. The lines in part b give the GCS results. Part a is
reproduced, with permission, from Ref. [1].

have a high probability of residing. The simulation cell
consists of a parallelopiped with a charged wall (the elec-
trode) at x = 0 and another wall (charged or uncharged)
at x = L, where L is so large that the two walls do not
interfere. Periodic boundary conditions are used in the
other two directions. The size of the cell is chosen to be
large enough that electrostatic screening eliminates the
effects of the periodic image cells. The number of ions of
each species is chosen so that the system is electroneu-
tral. The first use of MC simulations for the study of the
EDL was that of Torrie and Valleau [12, 13]. After their
seminal studies, there was a hiatus in simulation studies
of the EDL. However in recent years, there has been a
renewed interest in simulations of the EDL that includes
the work of Bhuiyan et al. [8], Boda et al. [14, 15], and
Lamperski et al. [16, 17].

Another simulation technique is the molecular dy-
namics (MD) method in which the equations of motion
are solved and the properties of the system of interest are
obtained by averaging over the positions and velocities
of the ions. A simulation cell that is similar to that used

0

1

2

3

4

5

g i
(x

)

a

0 1 2 3 4 5
x/d

0

1

2

3

4

5

g i
(x

)
b

Figure 3: Normalized density profiles, gi(x), of a 1:1 elec-
trolyte (d = 3 Å) at 1 M and room temperature for the
state for which the MC values are σd2/e = 0.1685 and
βeφ(d/2) = 2.6. The points give the simulation results
and the curves give the GCS results. The comparison is
made at the same charge density (part a) and the same dif-
fuse layer potential (part b).

in MC simulations is employed. In recent years, there has
been an interest in MD simulations of the EDL, especially
for ionic liquids. Some representative studies are those of
Vatamanu et al. [18, 19], Hu et al. [20], and Feng [21] et
al..

A comparison with simulation gives an unambigu-
ous test of the GCS theory since uncertainties resulting
from empirical fits of the diffuse layer capacitance can-
not arise. The GCS theory and simulations both use the
same model and interaction parameters that are defined
in Eqs. 1 and 2. Additionally, simulations and theory
give results for the density profiles, gi(x), that cannot be
obtained by present experimental methods. The simula-
tions plotted in Figs. 2–5 are those of Boda et al. [22].
In Fig. 2a, the electrostatic potential βeφ(d/2) for a 1:1
electrolyte is plotted as a function of b for three con-
centrations (0.01M, 0.1M, and 1M). If the GCS theory
were correct, these curves would be identical and inde-
pendent of concentration. Hence, there can be only one
GCS curve in Fig. 2a. As is seen, φ(d/2) as a function
of b actually decreases with increasing concentration. In
Fig. 2b, φ(d/2) is plotted as a function of σd2/e. The
CGS curves are greater than the simulation results, espe-

43(2) pp. 55-66 (2015) DOI: 10.1515/hjic-2015-0010



60 HENDERSON

-0.2 -0.1 0.0 0.1 0.2

σd
2
/e

-4

-2

0

2

4

6
βe

φ(
d/

2)

Figure 4: Diffuse layer potential of a 2:1 electrolyte (d =
3 Å) at room temperature as a function of σ. The curves
are, from top to bottom, for 0.01, 0.1, and 1 M for positive
σ and the reverse for negative σ. The symbols give the
simulation results. The curves give the GCS results.

cially as the concentration increases. The density profiles
for a 1:1 electrolyte are plotted in Fig. 3. The compari-
son is made at the same value of σ in part a and the same
value of φ(d/2) in part b. In principle, there is no reason
to choose whether the comparison should be made at the
same σ, the same φ(d/2), or the same φ(0). It was natural
for Torrie and Valleau to make their comparisons at the
same σ because σ is the input variable in their method.
However, φ is the natural variable in the GCS theory. In
any case, the GCS theory looks best when σ is used as
the input variable. Torrie and Valleau overstated things
when they said that the GCS theory was reasonable for a
1:1 electrolyte. Their statement is most applicable if the
comparison is made at the same value of σ. The value
of the counterion profile would be in poor agreement at
x = d/2 if φ(d/2) or φ(0) were used as the input vari-
able. A similar comparison is made for a 2:1 electrolyte
in Figs. 4 and 5. The agreement of the GCS theory with
simulations is much poorer. The electrostatic interactions
are stronger because of the presence of the divalent ions.
When the divalent ions are the counterions, the potential,
φ(d/2) has a maximum and then decreases with increas-
ing electrode charge. This is not seen in the GCS results
which are monotonic. Further, the simulation profiles are
not monotonic whereas the GCS results are monotonic.
The simulation profiles have oscillations. The EDL can
consist of regions where counterions or coions predomi-
nate. When the coions predominate, this phenomenon is
known as charge inversion. Of course, the net charge in
the diffuse layer is still equal in magnitude, but opposite
in sign, to that of the electrode charge. This is required
to screen the electrode charge and potential far from the
electrode.

Generally, experimentalists have been content to ig-
nore the discrepancies in the results of the GCS theory
and state that these differences are unimportant since they
occur at high electrode charges or high concentrations

0

1

2

3

4

5

g i
(x

)

a

0 1 2 3 4 5
x/d

0

1

2

3

g i
(x

)
b

Figure 5: Normalized density profiles, gi(x) of a 2:1 elec-
trolyte (d = 3 Å) at 1 M and room temperature for the
state for which the MC values are σd2/e = −0.1685 and
βeφ(d/2) = −0.15. The points give the simulation re-
sults and curves give the GCS results. The comparison is
made at the same charge density (part a) and the same dif-
fuse layer potential (part b).

or for high valence electrolytes or nonaqueous systems,
where experimental results are difficult to obtain. How-
ever, this is short-sighted. As scientists, one of our goals
is to understand what is happening. This cannot be done
with an inaccurate theory even with curve fitting. In the
remainder of this article, attention is directed to more ac-
curate, but still analytic, theories.

3. Mean spherical approximation

The mean spherical approximation (MSA) is a natural
extension of the linearized GCS theory in which the
size of the ions is taken into account. It was first ap-
plied to the EDL by Blum [23]. The GCS is usually ob-
tained by means of the solution of a differential equa-
tion whereas the MSA is obtained from the solution of an
integral equation. At first sight, the connection between
the GCS and MSA theories is unclear. However, Hen-
derson and Blum [24] demonstrated that the GCS the-
ory could also be obtained from an integral equation. In
fact, the linearized GCS integral equation is just the MSA
integral equation with the effect of ion size ignored. Ac-
tually, Henderson and Blum proved a more general re-
sult. They showed that the GCS theory followed from the

Hungarian Journal of Industry and Chemistry



DOUBLE LAYERS NEAR AN ELECTRODE 61

hypernetted–chain approximation (HNCA) when ion size
was neglected. The MSA can be regarded as a linearized
version of the HNCA and, because of this, the stated rela-
tion of the linearized GCS theory to the MSA follows. In
this article, the HNCA is not considered because it does
not yield analytic results and has severe problems when
σ is large [25]. Also, the emphasis in this article is upon
analytic, or at least explicit, results that can be valuable
in practical calculations.

The MSA result that is analogous to Eq. 29 is

gi(x) =




0 for x < d/2

g0(x) −
βzieσ

��0κ
f(y) for x ≥ d/2

,

(34)
where g0(x) is the Percus-Yevick (PY) profile for hard
spheres near a hard surface. Earlier, Henderson, Abra-
ham, and Barker (HAB) [26] obtained an integral equa-
tion for g0(x). The second term gives the electrostatic
part of the profile for charged hard spheres near a charged
hard surface. Blum [23] did not obtain a result for f(y)
but he did obtain an analytic result for the Laplace trans-
form of f(y),∫ ∞

0

exp(−sy)f(y)dy =

=
s

s2 + 2(Γσ)s + 2(Γσ)2(1 − exp[−s])
, (35)

where 2Γ is a renormalized screening parameter that is
related to κ by κ = 2Γ(1 + Γσ) or 2Γσ =

√
1 + 2κσ-

1. Note that for small κ (small concentrations), 2Γσ =
κσ−(κσ)2/2+···. Thus, the MSA screening parameter is
smaller than the GCS screening parameter. This suggests
that the MSA EDL is wider than that of the GCS theory.
This agrees with the simulation results. The notation of
Blum has been followed. However, it might have been
preferable if he had incorporated the factor of 2 into the
definition of Γ so that Γ became κ at low concentrations.

Note that at low concentrations, the Laplace transform
of f(y) becomes 1/s(1 + κs). This means that

f(y) = exp(−κy), (36)

in the limit of low concentrations. Also, g0(x) = 1 in this
limit. Thus, at low concentrations, the MSA becomes the
GCS theory. Blum did not invert the Laplace transform of
f(y). However, he did obtain the contact value of f(y) by
examining the Laplace transform of f(y) at large s. He
showed that f(0) = 1. Using the earlier result of HAB
for g0(d/2), the contact value of gi(x) is

gi(d/2) =
1 + 2η

(1 −η)2
−
βzieσ

��0
, (37)

where η = πρd3/6. The MSA contact value is an im-
provement over the GCS result that contains only the
ideal gas term. However, the osmotic pressure should
have both a hard sphere term and an electrostatic term.
Additionally, the hard sphere term is accurate only for
low values of ρ.

Equation 37 does not contain the quadratic term b2 of
Eq. 27. This is because the MSA is a linearized theory. A
better expression for the osmotic term is

p

ρkT
=

1 + η + η2 −η3

(1 −η)3
−

Γ3

3πρ
, (38)

where Γ is the renormalized screening parameter that has
been defined above. This result is obtained from the ap-
plication of the MSA to bulk electrolytes. The MSA, as
is the case for most theories, is not fully self-consistent.
Henderson et al. [27] have compared this expression with
their simulations (see their Fig. 1) and found it to be very
accurate. Despite these problems, the MSA contact value
given in Eq. 37 does represent an advance.

By expansion of the Laplace transform of f(y), it is
easy to show that the MSA EDL satisfies electroneutral-
ity. That is, the charge in the EDL is equal in magnitude,
but opposite in sign, to the electrode charge. Again, by
expanding the Laplace transform, the MSA expressions
for the total and diffuse layer potentials of the EDL are
found to be

V =
σ

��0(2Γ)
(39)

and
φd/2 =

σ

��0κ
[1 − (Γd)2]. (40)

Thus, in the MSA, the capacitance (and differential ca-
pacitance) of the EDL is

1

C
=

1

��0(2Γ)
. (41)

Expanding the expression that defines Γ,

2Γ = κ−κ2d/2 + κ3d2/2 + · · ·. (42)

Therefore,

1

C
=

1

��0κ
+

d

2��0
−
κd2

4��0
+ · · · . (43)

The MSA capacitance does not reach a maximum at
2��0/d but continues to increase, as is indicated in Fig.
1. The MSA (solid) curve in Fig. 1 was not calculated
from Eq. 41 but from a more sophisticated version of the
MSA, that is not discussed in detail here, which includes
the contribution resulting from explicit solvent molecules
and the electronic structure of the metal [28–30]. How-
ever, the results of Eq. 41 are qualitatively similar to the
more sophisticated results. In this paper, the solid curve
serves to guide the eye. The inner layer capacitance con-
tinues to be 2��0/d but it is simply the electrode charge
divided by potential difference across the inner layer and
not a ‘catch all’ for the deficiencies of the GCS theory.
The diffuse capacitance is the electrode charge divided
by the potential difference from the distance of closest
approach to the bulk electrolyte and contains correction
terms to the GCS theory. This is the reverse of the usual
interpretation of the GCS theory where the GCS expres-
sions are assumed to be accurate for the diffuse layer

43(2) pp. 55-66 (2015) DOI: 10.1515/hjic-2015-0010



62 HENDERSON

0.14 0.16 0.18 0.2 0.22 0.24

ρ*

1.4

1.6

1.8

2

2.2

2.4

C
/F

m
-2

MC

MSA

GCS

Figure 6: Double layer capacitance, C, as a function of
the reduced density, ρ∗ = ρd3, at the reduced temperature
T∗ = 0.08. The circles are the MC data of Henderson
et al. [27] and the lines are the MSA and GCS results.
The line through the circles is given as a guide to the eye.
Because the MSA is a linearized theory, the MC, MSA,
and GCS capacitances are for σ = 0.

capacitance and everything else is ‘lumped’ into the in-
ner layer capacitance. In the more sophisticated version
MSA, the contributions due to the solvent molecules ap-
pear in both the diffuse and inner layer potentials. The
solvent molecule profile is as diffuse as that of the ions.
The effect of the molecular nature of the solvent is not
confined to the inner layer.

The initial slope of the φ(d/2) vs. σ curves in Figs. 2
and 4 is just the inverse of the diffuse portion of the ca-
pacitance. It is seen that the initial slope of the MC curves
is well described by the GCS theory at low concentrations
but increasingly falls below the GCS initial slope (the in-
verse of the differential capacitance) with increasing con-
centration as was seen in the experimental results in Fig.
1. Henderson et al. [27] compared the GCS and MSA
differential capacitances with their simulation results for
a broad range of densities and at a temperature that was
meant to be qualitatively representative of an ionic liquid.
As is seen in Figs. 6 and 7, the MSA results are consid-
erably improved over the GCS results. The comparison
with the MC results is made for a small value of σ be-
cause the MSA is a linearized theory that is applicable
only for small σ.

Although an analytic expression for f(y) is not avail-
able, Henderson and Smith [31] were able to obtain a
zonal expansion for f(y). They showed that

f(x) =

∞∑
n=1

fn(z)u(z), (44)

where z = t−n + 1, t = x/d, u(z) is the Heaviside step
function that is zero for z < 0 and one for z ≥ 0 and

fn(z) = exp(−µ)
µn

(n− 1)!
[jn−2(µ) − jn−1(µ)] (45)

with µ = (Γd)z. The function jm(µ) is the spherical

x / d

0.5 1.0 1.5 2.0 2.5

0

1

2

3

4

g
(x

/d
)

0

1

2

3

4

5

counterions

co-ions

counterions

co-ions

(a)

(b)

Figure 7: The electrode-ion normalized density profiles,
gi(x/d), at the reduced density ρ∗ = ρd3 = 0.5 for the
reduced temperature T∗ = 0.8 and surface charge den-
sity, σ = 0.05 C/m2, is small enough that the MSA is
applicable. The circles are the MC results of Henderson
et al. [26] and the dashed line gives the MSA result. The
line through the circles is given as a guide to the eye. This
figure is reproduced, with permission, from Ref. [26].

Bessel function that is easily calculated using the recur-
rence formula for this function. Hundreds of jm(µ) can
be calculated without difficulty, even with a laptop com-
puter.

Henderson and Smith [31] also obtained a zonal ex-
pansion for g0(x). Their result is

g0(x) =

∞∑
n=1

gn0 (z)u(z), (46)

where z is again given by z = t − n + 1 with t = x/d.
The expressions for the gn0 (x) are rather complex. How-
ever, Henderson and Smith gave results for n ≤ 5. The
formulae for g0(x) and f(x) are not quite analytic since
they involve infinite series. However, these results are ex-
plicit and easily used. Fortran programs to obtain g0(x)
and f(x) are given in Supplementary Material. Note that
the program for g0(x) consists of two parts. One part cal-
culates those parameters that depend only on the state of
the electrolyte and the other subroutine in each code cal-
culates profiles for a given x. The user should resist the
temptation to combine the two parts into one. McQuarrie
[32] did this in an appendix to his excellent book and pro-
duced an inefficient, and probably incorrect, code that he
referred to as ‘Henderson’s code’. Fortunately, his code

Hungarian Journal of Industry and Chemistry



DOUBLE LAYERS NEAR AN ELECTRODE 63

is illegible in the later printing of his book. If the reader
does combine the codes, the reader is on his/her own and
should not refer to the combined, or otherwise modified,
code as ‘Henderson’s code’.

The functions g0(x) and f(x) are oscillatory, in ac-
cord with the simulations. They are improvements, qual-
itative and quantitative, to the monotonic functions of the
GCS theory.

4. A useful hybrid description

As has been mentioned, the deficiencies of the GCS
theory could, until recently, be dismissed as appearing
mainly under conditions that are of limited experimental
interest. However, there has been considerable recent in-
terest in EDLs formed by ionic liquids. Ionic liquids can
be thought of as room–temperature molten salts. Because
there is no solvent, the ions do not become insoluble in
some solvent and experimental results can be obtained at
high concentrations. The fact that they exist at room tem-
perature is a great experimental convenience. Kornyshev
[33] has drawn attention to these electrochemical systems
and aptly suggested that they provide a paradigm change
in electrochemistry. He modestly ends the title of his im-
portant paper with a question mark. An exclamation mark
might have been more appropriate. As well as exposing
the deficiencies of the GCS theory, EDLs in ionic liq-
uids are important in green technologies, the design of
novel energy storage devices, such as high-tech batter-
ies and super-capacitors [34]. Ionic liquid DLs have at-
tracted recent experimental [35, 36] and theoretical inter-
est [18–21, 37–39]. The differential capacitance, as de-
termined by MC simulations, of a simple model [38] of
an ionic salt in which T∗ = 0.8 and d = 8 Å is given
in Fig. 8 for ρ∗ = 0.04, 0.14 and 0.24. At low concen-
trations, the differential capacitance is parabolic-like, as
the GCS theory suggests. However, Cd does not become
flat at large electrode charges. At higher concentrations,
Cd at small electrode charges continues to increase with
increasing concentration. This has been seen in Fig. 6.
At high electrode charges, the capacitance decreases. The
nature of this decrease seems to be independent of the
concentration. This is similar to the GCS theory except
that the capacitance is not flat at high electrode charges
but decreases. The decrease is due to the fact that the ions
are not point charges but occupy space and cannot sit on
top of each other. The diffuse layer must become thicker
and the capacitance decreases as the electrode charge in-
creases. This is not because the distance of closest ap-
proach of the ions increases. Strong secondary peaks in
the counterion profile appear [37]. The beginnings of this
trend were first observed by Torrie and Valleau [12] and
seem to be quite universal.

The GCS theory satisfies Eq. 27 at high electrode
charges but fails at low electrode charges whereas the
MSA gives reasonable results at small electrode charges.
This implies that a repair of the GCS so that it gives the
MSA results in the regime of the low electrode charges

-2 -1 0 1 2
σ∗

3

4

5

6

7

8

9

C
df

*

0.24

0.04

0.14

Figure 8: Differential capacitance, C∗df = Cdd/4π�0, ob-
tained from MC simulation for the EDL of a model ionic
liquid with d = 4 Å and T∗ = 0.8. The curves are, from
bottom to top, for ρ∗ = 0.04, 0.14, and 0.24. This figure
has been reproduced, with permission, from Ref. [40].

but leaves the high electrode charge part unchanged
might be useful. Likely, there is no way to accomplish this
in a fundamental way. Additionally, there are probably
several semi-empirical ways in which this could be done.
Henderson and Lamperski [40] have presented one pro-
cedure. Because it is not based on any fundamental ideas,
it is not a theory. It would be more appropriate to refer to
their procedure as a description. They proposed that the
differential capacitance for a symmetric salt could use-
fully be written as

1

Cd
=

d′

2��0
+

d′

2��0
√

1 + b2/4

(
1

Γd′
− 1
)
. (47)

The parameter d′ is an adjustable parameter and repre-
sents the effective thickness of the diffuse layer. At small
b (small electrode charge), Eq. 47 yields

1

Cd
=

1

2��0
, (48)

which is the MSA result. At large b (large electrode
charge), Eq. 47 yields

1

Cd
=

d′

2��0
. (49)

The results of Eq. 47, using d′ = 2d for the system that
Lamperski et al. simulated, were given by Henderson and
Lamperski. Qualitatively, the results are very similar to
the simulation results shown in Fig. 8. Better agreement
could be obtained by making d′ increase with electrode
charge. Figure 2 of Henderson and Lamperski suggests
that Cd is proportional to 1/σ∗ at large σ∗ (electrode
charge) with the proportionality constant being indepen-
dent of concentration. This behavior was first predicted
by Kornyshev [33] on the basis of a lattice theory and
seems to be universal. As well as the simulations of Hen-
derson and Lamperski, it has been seen experimentally

43(2) pp. 55-66 (2015) DOI: 10.1515/hjic-2015-0010



64 HENDERSON

-1.5 -1 -0.5 0 0.5 1 1.5

σ*

4

6

8

10

C
*
d

0.24

0.14

0.04

Figure 9: Differential capacitance, C∗d = Cdd/4π�0, ob-
tained from the hybrid description of the EDL of a model
ionic liquid with d = 4 Å and T∗ = 0.8. The curves
are, from bottom to top, for ρ∗ = 0.04, 0.14, and 0.24.
The solid and broken curves give the results of the hybrid
approach and GCS theory, respectively.

by Islam et al. [35]. Something of the nature of

d′ = d1 + d2|σ∗| (50)

would give the desired decrease of Cd at large σ∗. Equa-
tion 50 is sensible because it is consistent with the diffuse
layer becoming thicker as the electrode charge increases.
The results of this ansatz with d1 = 2d and d2 = d are
given in Fig. 9. The results are similar to the simulation
results in Fig. 8. This hybrid approach is capable of yield-
ing a capacitance with a double hump at low concentra-
tions and a single hump at high concentrations. This be-
havior is predicted by simulations and all the good theo-
ries of the DL of ionic liquids.

A hybrid treatment of the profiles, gi(x), is possible.
One could start with the MSA expression for gi(x) and
add to this gGCSi (x; b) − g

GCS
i (x; b = 0). However, it

must be realized that the MSA expressions for the pro-
files are less accurate than the MSA expressions for the
potential and capacitance. The potential and capacitance
are integrals and tend to average out any inaccuracies in
the profiles.

5. Conclusion

The study of the electric DL is an important application
of statistical mechanics that is of experimental and ap-
plied interest. The GCS theory is popular with exper-
imentalists because it is intuitively simple and easy to
use in the routine analysis of experiments. However, the
GCS theory has deficiencies. Its use leads to the idea that
any problems with the GCS theory can be ‘swept under
the carpet’ by placing all of these problems into an em-
pirical treatment of the inner layer. In reality, the defi-
ciencies of the GCS theory lie with the GCS treatment
of the diffuse layer. Admittedly, it is difficult to observe
this in aqueous systems. However, it is not impossible.

The departure from linearity in the Parsons-Zobel plot
(Fig. 1) is real and should not be ignored. The important
field of ionic liquid electrochemistry requires something
more adequate than the GCS theory. The best theories of
the EDL are the modified Poisson-Boltzmann theory [41]
and the density functional theory [42]. However, both the-
ories are numerical and require an iterative numerical so-
lution of a fairly large set of equations and may not be
appealing in an experimental analysis. A hybrid descrip-
tion, such as that explored here, preserves the advantage
of an analytic treatment of the capacitance and is no more
cumbersome than the GCS theory.

Supplementary Information

Fortran programs to calculate g0(x) and f(x−d/2) using
MSA. The codes can be downloaded free of charge from
http://tinyurl.com/hjic-2015-0010-suppl.

Acknowledgement

Professor Dezső Boda assisted with the preparation
of this paper. Professor Lutful Bari Bhuiyan read the
manuscript prior to submission and suggested several im-
portant modifications. The author is grateful to both col-
leagues for their continuing wise advice.

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