Electrolytes at charged interfaces: Pair integral equation approximations for model 2–2 electrolytes

Abstract

The structure and thermodynamics for model 2–2 electrolytes at a charged interface have been determined by the so-called “pair” approximation of integral equation theory. In addition to Coulombic interactions, the potential models for the ion–ion and ion–wall interactions employ “soft” continuous potentials rather than “hard”-sphere or “hard”-wall potentials. The solvent is modeled as a structureless dielectric continuum at 25 °C. The structure is calculated using the inhomogeneous Ornstein–Zernike relation, together with the hypernetted chain closure and two choices for the functional relationship between the singlet and pair correlation functions. Both the interfacial density profile and the inhomogeneous pair correlation functions are calculated. Some thermodynamic properties of these systems are also evaluated. The results of the pair approximation are compared with the so-called “singlet” approximation, selected computer simulation results, Gouy–Chapman–Stern predictions, and experimental data. While qualitative agreement is generally found between the two levels of integral equation approximation, measurable quantitative improvements exist for both structural and thermodynamic predictions in the pair approximation. © 2001 American Institute of Physics.

Full text

Electrolytes at charged interfaces: Pair integral equation approximations
for model 2–2 electrolytes
Andrew C. Eaton
School of Chemistry, University of Sydney, NSW 2006 Australia
A. D. J. Haymet
a)
Department of Chemistry, University of Houston, Houston, Texas 77204
共Received 19 December 2000; accepted 5 April 2001兲
The structure and thermodynamics for model 2–2 electrolytes at a charged interface have been
determined by the so-called ‘‘pair’’ approximation of integral equation theory. In addition to
Coulombic interactions, the potential models for the ion–ion and ion–wall interactions employ
‘‘soft’’ continuous potentials rather than ‘‘hard’’-sphere or ‘‘hard’’-wall potentials. The solvent is
modeled as a structureless dielectric continuum at 25°C. The structure is calculated using the
inhomogeneous Ornstein–Zernike relation, together with the hypernetted chain closure and two
choices for the functional relationship between the singlet and pair correlation functions. Both the
interfacial density profile and the inhomogeneous pair correlation functions are calculated. Some
thermodynamic properties of these systems are also evaluated. The results of the pair approximation
are compared with the so-called ‘‘singlet’’ approximation, selected computer simulation results,
Gouy–Chapman–Stern predictions, and experimental data. While qualitative agreement is generally
found between the two levels of integral equation approximation, measurable quantitative
improvements exist for both structural and thermodynamic predictions in the pair approximation.
© 2001 American Institute of Physics. 关DOI: 10.1063/1.1375141兴
I. THE ELECTRICAL DOUBLE LAYER
The region where an electrolyte solution meets a charged
solid surface, resulting in a charge separation at the bound-
ary, is often referred to as the electrical double layer,
1–5
since
order is induced in the liquids next to the charged surface.
This phenomenon occurs in many naturally-occurring and
electrochemical systems, such as colloids, micelles, mem-
branes and the electrode–electrolyte interface, and can have
profound effects on the chemistry and physics of such
systems.
6–11
In this paper we solve the ‘‘pair’’ approximation for the
electrical double layer for symmetric models of 2–2 electro-
lytes, using ‘‘soft’’ continuous potentials. In this approach,
we solve directly for the inhomogeneous ion-wall and ion–
ion correlations functions. The aqueous solvent is approxi-
mated by a structureless dielectric continuum of fixed dielec-
tric constant. We require as input only the structure of bulk
electrolytes from work by Duh and Haymet.
12
We calculate
concentration profiles, inhomogeneous correlation functions,
and a range of thermodynamic quantities, in order to make
comparison with the ‘‘singlet’’ approximation for the same
electrolyte model reported earlier by us.
13
The calculation of the structure of an electrolyte near a
planar surface, both charged and uncharged, is the principal
topic considered in this paper. To show the value of this
paper, we display, in Fig. 1, the normalized density profiles
calculated using AHNC the pair approximation discussed be-
low are compared with simulation data based on the charged
hard-sphere/charged hard-wall 共CHS/CHW兲 potential model.
Allowing for the different potential functions used, this com-
parison shows good agreement. It is worth noting once again
the dramatic effect the surface has on the structure, since all
of the functions shown are unity in the absence of the sur-
face. Absent from this figure are any data from the singlet
approximation, which calculates the density profile for an
electrolyte but assumes that ion–ion correlations are unaf-
fected by the surface. For this case, the singlet approximation
is unable to provide a solution. Hence the need for the kind
of pair approximations discussed in this paper.
The pair approximation, as applied to electrolytes next to
charged surfaces, was first detailed by Henderson and
Plischke
14–16
for an electrolyte next to a single surface, and
by Kjellander and co-workers
17–21
for an electrolyte between
two surfaces. More recently, the pair approximation has been
further applied in wider contexts by Kjellander and
co-workers.
22–26
Work on the charged surface/electrolyte in-
terface has largely used the charged hard-sphere/charged
hard-wall potential model.
An excellent review of theoretical contributions to the
understanding of the electrical double layer up to 1995 has
been presented by Attard.
27
A compact and informative deri-
vation of both pair and singlet approximations has also been
presented recently by our group.
28
This paper is organized as follows: The equations used
in this work are presented in Sec. II, with numerical details
in Sec. III. Our structural and thermodynamic predictions are
collected in Secs. IV and V. In Sec. VI we present our dis-
cussion and conclusions.
a兲
Author to whom correspondence should be addressed.
JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 24 22 JUNE 2001
109380021-9606/2001/114(24)/10938/10/$18.00 © 2001 American Institute of Physics
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

II. SUMMARY OF PAIR EQUATIONS AND INPUT
The pair approximation that we have solved can be sum-
marized in terms of a definition and two equations: 共i兲 the
inhomogeneous Ornstein–Zernike 共OZ兲 relation, which de-
fines the direct correlation function c(r) from the total cor-
relation function h(r); 共ii兲 the hypernetted chain 共HNC兲 clo-
sure, an approximate relation between the direct and total
correlation functions; and 共iii兲 a density equation, a func-
tional relationship between the inhomogeneous correlation
functions and the singlet density profile. We emphasize that
the exact OZ relation serves purely as a definition of the
direct correlation function.
In previous works, the Lovett–Mou–Buff–Wertheim
共LMBW兲 equation,
29
Triezenberg–Zwanzig 共TZ兲 equation
30
共which is equivalent to the LMBW equation兲, or the Yvon–
Born–Green 共YBG兲 equation
31
have been employed as the
functional relation between the pair and singlet correlation
functions. We have used the LMBW equation in this work.
Additionally, we also use an alternative relation called the
AHNC approximation. This equation has been previously
derived in a somewhat lengthy manner by Kjellander and
co-workers.
17
More recently, our group has shown that it
may be obtained by minimization of an approximate free
energy functional for the grand potential.
28
The equations for
the pair approximations that we have solved in this work are
summarized below.
A. The inhomogeneous OZ relation
For a fluid of
␯
species next to an infinite planar solid
surface with an external potential applied normal to the sur-
face, denoted henceforth as the charged ‘‘wall,’’ the inhomo-
geneous OZ relation between the total and direct correlation
functions is
h
wij
共
z
1
,z
2
,s
12
兲
⫽ c
wij
共
z
1
,z
2
,s
12
兲
⫹
兺
k⫽1
␯
冕
0
⬁
dz
3
␳
wk
共
z
3
兲
⫻
冕
⫺ ⬁
⬁
dx
3
冕
⫺ ⬁
⬁
dy
3
c
wik
共
z
1
,z
3
,s
13
兲
⫻ h
wkj
共
z
3
,z
2
,s
23
兲
. 共1兲
Figure 2 depicts the cylindrical geometry used in Eq. 共1兲.
The wall lies in the xy-plane. Particles 1 and 2 are at per-
pendicular distances z
1
and z
2
from the wall and are sepa-
rated by a distance R
12
⫽ R. Since this system is cylindrically
symmetric, it is useful to define the length
s⫽ s
12
⫽
冑
共
x
1
⫺ x
2
兲
2
⫹
共
y
1
⫺ y
2
兲
2
. 共2兲
By Pythagoras’ theorem
s
2
⫽ R
2
⫺
共
z
1
⫺ z
2
兲
2
. 共3兲
The inhomogeneous pair correlation function between
the wall w and species i and j is
g
wij
共
z
1
,z
2
,s
12
兲
⫽ h
wij
共
z
1
,z
2
,s
12
兲
⫹ 1. 共4兲
The bulk number density of species i and number density of
species i at a perpendicular distance z from the surface are
␳
i
B
and
␳
wi
(z), respectively. This is one of the system of
equations solved numerically in this work. This equation has
been called the OZ2 relation in the literature of inhomoge-
neous fluids.
16
We choose to retain the 共redundant兲 subscript
w in our equation to explicitly distinguish the inhomoge-
neous correlation functions from their bulk counterparts
g
ij
(r) and c
ij
(r).
Using the 2D Fourier 共Hankel兲 transformation
32,33
de-
fined by
f
ˆ
共
k
兲
⫽ 2
␲
冕
0
⬁
dssf
共
s
兲
J
0
共
ks
兲
,
共5兲
f
共
s
兲
⫽
1
2
␲
冕
0
⬁
dkkf
ˆ
共
k
兲
J
0
共
ks
兲
,
Equation 共1兲 can be written in Fourier space as
h
ˆ
wij
共
z
1
,z
2
,k
兲
⫽ cˆ
wij
共
z
1
,z
2
,k
兲
⫹
兺
k⫽1
v
冕
0
⬁
dz
3
␳
wk
共
z
3
兲
cˆ
wik
共
z
1
,z
3
,k
兲
⫻ h
ˆ
wkj
共
z
3
,z
2
,k
兲
共6兲
FIG. 1. 共a兲 Cationic g
w⫹
(z) and 共b兲 anionic g
w⫺
(z) normalized concentra-
tion profiles for a 0.05 M 2–2 electrolyte at a surface charge density of
⫺ 8.65
␮
Ccm
⫺2
: pair using AHNC approximation 共solid line兲; pair using
LMBW approximation 共dashed line兲; and the MC results 共circles兲 of Torrie
and Valleau 共Ref. 46兲, using a CHS/CHW potential.
FIG. 2. The geometry used in the pair approximation. The charged wall lies
in the xy-plane. Particles 1 and 2 are at perpendicular distances z
1
and z
2
from the flat surface and are separated by a distance R
12
. We define s
12
using the relation s
12
2
⫽ (x
1
⫺ x
2
)
2
⫹ (y
1
⫺ y
2
)
2
, where x
i
and y
i
are x and y
coordinates of particle i.
10939J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 Electrolytes at charged interfaces
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

where J
0
is the zeroth order Bessel function. We now only
need to perform the one-dimensional integrals in Eq. 共6兲.
Due to the long-range nature of direct and total inhomo-
geneous correlation functions, an arbitrary decomposition of
the direct and total correlation functions into short-range
parts 共denoted by the superscript ‘‘s’’兲 and long-range parts,
q and
␸
, respectively, must be introduced to renormalize the
equations. The decomposition is
h⫽ h
s
⫹ q,
c⫽ c
s
⫹
␸
, 共7兲
␶
⫽ h
s
⫺ c
s
.
Both
␸
and q are usually chosen to have convenient analyti-
cal forms that yield well-behaved, short-ranged forms for c
s
and h
s
. We introduce
␶
as a convenient variable to be used in
conjunction with c
s
to solve the pair approximation. Two
choices of
␸
and q will be discussed in the next section. The
numerical details for solving the Ornstein–Zernike equation
are described in Ref. 34. Note that we do not include the
asymptotic form discussed by Ulander and Kjellander,
21
which may be important at very high concentrations.
B. The HNC closure
The HNC closure may be derived by discarding high
order entropy contributions from the functional expansion of
the free energy.
28
The HNC closure in real space, also pre-
sented as Eq. 共17兲 by Bush et al.,
28
is
h
wij
共
z
1
,z
2
,s
12
兲
⫹ 1⫽ exp
关
⫺
␤
u
ij
共
R
12
兲
⫹ h
wij
共
z
1
,z
2
,s
12
兲
⫺ c
wij
共
z
1
,z
2
,s
12
兲
兴
, 共8兲
where u
ij
(R
12
) is the ion–ion potential for the interparticle
separation R
12
,
␤
⫺ 1
⫽ kT, k is the Boltzmann constant, and
T is the absolute temperature. Using the definitions in Eq.
共7兲, Eq. 共8兲 can be written as
c
wij
s
共
z
1
,z
2
,s
12
兲
⫽ exp
关
⫺
␤
u
ij
共
R
12
兲
⫹
␶
wij
共
z
1
,z
2
,s
12
兲
⫹ q
wij
共
z
1
,z
2
,s
12
兲
⫺
␸
wij
共
z
1
,z
2
,s
12
兲
兴
⫺ 1⫺
␶
wij
共
z
1
,z
2
,s
12
兲
⫺ q
wij
共
z
1
,z
2
,s
12
兲
. 共9兲
Two analytical choices of
␸
and q have been used in the
literature by Henderson and Plischke,
14
and by Kjellander
and co-workers.
19
A natural choice for
␸
is setting it equal to
the negative 共dimensionless兲 Coulombic part of the ion–ion
potential.
19
The Coulombic part of the potential has the form,
⌽
ij
共
R
12
兲
⫽
q
i
q
j
␧R
12
, 共10兲
where q
i
is the charge on species i, and the solvent is treated
as a structureless dielectric continuum of dielectric constant
␧⫽ 4
␲
␧
r
␧
0
, with ␧
0
the permittivity of free space and ␧
r
⫽ 78.358 is the relative permittivity chosen to model water at
the temperature 25 °C and density 0.997 gcm
⫺3
. In this work
the wall is assumed to have the same relative permittivity as
the solvent, and, thus, no image effects are considered.
In order to avoid numerical difficulties arising from
this choice of the long-range potential for small values of
兩
z
1
⫺ z
2
兩
, Henderson
15
proposed a slightly altered form of
␸
,
␸
wij
共
z
1
,z
2
,s
12
兲
⫽⫺
␤
⌽
ij
共
z
1
,z
2
,s
12
兲
⫻
共
1⫺ e
⫺ ␭[s
12
2
⫹ (z
1
⫺ z
2
)
2
]
1/2
兲
3
, 共11兲
where ␭ is chosen such that, for the most distant point R
N
from the wall analyzed, e
⫺ ␭[R
N
2
⫹ (z
1
⫺ z
2
)
2
]
1/2
is arbitrarily
small. Similarly, q is defined as
q
wij
共
z
1
,z
2
,s
12
兲
⫽
␸
wij
共
z
1
,z
2
,s
12
兲
• e
⫺
␬
[s
12
2
⫹ (z
1
⫺ z
2
)
2
]
1/2
,
共12兲
where the inverse Debye length
␬
⫺ 1
is given by
␬
2
⫽ 4
␲
␤
兺
i⫽1
v
␳
i
B
(q
i
2
/␧d), and d is the hydrated ion size. We
note that the Hankel transforms of 1/
关
s
2
⫹ b
2
兴
1/2
and
exp(⫺a
关
s
2
⫹b
2
兴
1/2
)/
关
s
2
⫹ b
2
兴
1/2
are 2
␲
exp(⫺
兩
b
兩
k)/k and
2
␲
exp(⫺b
关
k
2
⫹a
2
兴
1/2
)/
关
k
2
⫹ a
2
兴
1/2
, respectively. We have
adopted the decomposition scheme of Henderson
15
for the
results reported in this paper. Results based upon the decom-
position scheme of Kjellander
19
are discussed in Ref. 34.
C. Density equations: The AHNC approximation
The OZ and HNC equations make up only two of the
three equations required to solve a fully self-consistent pair
approximation. A third equation may be derived by minimi-
zation of an approximate free energy functional for the grand
potential, known as the AHNC approximation.
28
One version of the AHNC density equation, obtained
from Bush et al.,
28
is
ln
冋
␳
wi
共
z
1
兲
␳
i
B
册
⫽⫺
␤
q
i
␺
共
z
1
兲
⫺
␤␾
wi
LJ
共
z
1
兲
⫺
1
2
关
1⫹ c
wii
共
z
1
,z
1
,0
兲
兴
⫹
␮
i
(ex)
⫹
兺
j⫽ 1
v
冕
0
⬁
dz
2
␳
wj
共
z
2
兲
冋
␤
q
i
q
j
兩
z
1
⫺ z
2
兩
2␧
0
␧
r
⫹ cˆ
wij
*
共
z
1
,z
2
,0
兲
⫺
␲
冕
0
⬁
ds
12
s
12
h
wij
2
共
z
1
,z
2
,s
12
兲
册
, 共13兲
where
␮
i
(ex)
⫽
1
2
共
1⫹ c
ii
共
0
兲兲
⫺ 2
␲
兺
j⫽ 1
v
␳
j
B
冕
0
⬁
drr
2
⫻
冉
c
ij
*
共
r
兲
⫺
1
2
关
h
ij
共
r
兲
兴
2
冊
, 共14兲
c
ij
*
共
r
兲
⫽ c
ij
共
r
兲
⫹
␤
q
i
q
j
4
␲
␧
0
␧
r
r
, 共15兲
c
wij
*
共
z
1
,z
2
,s
12
兲
⫽ c
wij
共
z
1
,z
2
,s
12
兲
⫹
␤
q
i
q
j
4
␲
␧
0
␧
r
共
s
12
2
⫹
共
z
1
⫺ z
2
兲
2
兲
1/2
, 共16兲
10940 J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 A. C. Eaton and A. D. J. Haymet
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

q
i
␺
共
z
1
兲
⫽⫺
␲␴
兩
z
1
兩
2␧
0
␧
r
⫹
q
i
␺
0
2
⫺ q
i
兺
j⫽ 1
v
冕
0
⬁
dz
2
q
j
␳
wj
共
z
2
兲
兩
z
1
⫺ z
2
兩
2␧
0
␧
r
, 共17兲
and
␺
0
⬅
␺
共
0
兲
⫽⫺
␲
␧
0
␧
r
兺
j⫽ 1
v
冕
0
⬁
dz
2
q
j
z
2
␳
wj
共
z
2
兲
. 共18兲
Defining t
wi
(z)⫽ ln
关
g
wi
(z)
兴
⫹
␤␾
wi
LJ
(z), Eq. 共13兲 can be written
with t
wi
(z) as the subject,
t
wi
共
z
兲
⫽⫺
1
2
␤
q
i
␺
0
⫺
1
2
关
1⫹ c
wii
共
z
1
,z
1
,0
兲
兴
⫹
␮
i
(ex)
⫹
兺
j⫽ 1
v
冕
0
⬁
dz
2
␳
wj
共
z
2
兲
冋
␤
q
i
q
j
兩
z
1
⫺ z
2
兩
2␧
0
␧
r
⫹ cˆ
wij
*
共
z
1
,z
2
,0
兲
⫺
␲
冕
0
⬁
ds
12
s
12
h
wij
2
共
z
1
,z
2
,s
12
兲
册
.
共19兲
D. Density equations: The LMBW equation
The Lovett–Mou–Buff–Wertheim 共LMBW兲 equation is
a popular choice
15,22
to complement the OZ and HNC equa-
tions, but it does not lead to a consistent set of equations in
the sense of being a minimum of an approximate free energy
surface.
28
This equation is derived from balancing the gradi-
ent of the density with the force of the external potential and
the forces within the double layer.
35,36,27
There are a number
of formally equivalent versions of the LMBW equation.
28
The version we have used in numerical work is
⳵
共
ln
␳
wi
共
z
1
兲兲
⳵
z
1
⫽
⫺
␤
⳵
u
wi
共
z
1
兲
⳵
z
1
⫹
兺
j⫽ 1
v
冕
0
⬁
dz
2
⳵␳
wj
共
z
2
兲
⳵
z
2
⫻
冋
␤
q
i
q
j
兩
z
1
⫺ z
2
兩
2␧
0
␧
r
⫹ cˆ
wij
*
共
z
1
,z
2
,0
兲
册
. 共20兲
Defining u
wi
(z)⫽
␾
wi
LJ
(z)⫹ ⌽
wi
(z), where ⌽
wi
(z)
⫽⫺
␲␴
兩
z
兩
/2␧
0
␧
r
, with t
wi
(z) as the subject, Eq. 共20兲 be-
comes
⳵
t
wi
共
z
1
兲
⳵
z
1
⫽
⫺
␤
⳵
⌽
wi
共
z
1
兲
⳵
z
1
⫹
兺
j⫽ 1
v
冕
0
⬁
dz
2
⳵␳
wj
共
z
2
兲
⳵
z
2
⫻
冋
␤
q
i
q
j
兩
z
1
⫺ z
2
兩
2␧
0
␧
r
⫹ cˆ
wij
*
共
z
1
,z
2
,0
兲
册
. 共21兲
E. Model potentials
The model two-component bulk electrolyte used in our
study was introduced by Rossky and Friedman and
co-workers,
37,38
and is the same as that used earlier by us to
study the singlet approximation.
13
This model ionic solution
is that of equal-sized, spherically symmetric, ‘‘soft’’ ions,
interacting via the potential energy,
u
ij
共
r
兲
⫽
kB
d
ij
冉
d
ij
r
冊
9
⫹
q
i
q
j
⑀
r
. 共22兲
The parameters are chosen to be B⫽5377.75
兩
q
i
q
j
兩
Å K, and
d
ij
⫽ 2.8428 Å, the same as those used by Rossky and
co-workers.
37
This potential for a 2–2 electrolyte is dis-
played in Fig. 3共a兲共solid lines兲.
The short-range wall–ion potential employed in these
calculations is a Lennard-Jones 共LJ兲 9-3 potential, obtained
by integrating a LJ 12-6 potential over the half-space of wall
atoms, as described by Steele,
39
␾
wi
共
z
兲
⫽
2
3
␲
⑀
wi
再
2
15
冉
d
wi
z
冊
9
⫺
冉
d
wi
z
冊
3
冎
, 共23兲
where
⑀
wi
is the depth of the LJ 12-6 potential energy well
and d
wi
is the LJ 12-6 distance parameter. The value of d
wi
is chosen to be 4.2 Å, the hydrated ion size d, which corre-
sponds to the location of the minimum in the bulk potential
energy. The 9-3 well depth
␾
min
is adjusted by the parameter
⑀
wi
. This parameter is chosen so that, at zero surface poten-
tial, the normalized concentration profile is approximately
equal to one for all z⬎ d/2. Consequently, the value of the LJ
9-3 reduced well depth
␾
min
/kT is chosen to be 0.372 84 for
all calculations presented. Our short-range wall–ion potential
is shown in Fig. 3共b兲共solid line兲. In the double layer litera-
ture, the majority of calculations that we compare with, par-
ticularly any Monte Carlo 共MC兲 data we present, use a
‘‘hard-wall’’ potential, as shown in Fig. 3共b兲共dashed line兲.
F. Surface thermodynamics
Once ionic concentration profiles have been determined,
thermodynamic quantities can be calculated. Direct compari-
son of the profiles with experimental data is not yet possible,
but it is possible to compare some thermodynamic quantities.
We collect here the formulas for the surface thermodynamics
properties for a two component electrolyte near a wall at a
surface potential
␺
0
and a surface charge density
␴
.
We define the normalized concentration profile g
wi
(z)
by
FIG. 3. 共a兲 Bulk dimensionless potential energy u
ij
(r)/kT for the model
2–2 electrolyte 共solid lines兲 at T⫽25 °C. The dashed line is a restricted
primitive model 共RPM兲 potential. 共b兲 Short-range LJ 9-3 wall–ion dimen-
sionless potential energy
␾
wi
(z)/kT 共solid line兲 at T⫽ 25 °C. The dashed
line is a ‘‘hard-wall’’ potential. Note the definition of
␾
min
and that the
vertical scale in 共b兲 is one order of magnitude less than in 共a兲.
10941J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 Electrolytes at charged interfaces
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

␳
wi
共
z
兲
/
␳
i
B
⫽ g
wi
共
z
兲
⫽
关
1⫹ h
wi
共
z
兲
兴
, 共24兲
where h
wi
(z) is the singlet wall–ion total correlation func-
tion. The cationic ⌫
⫹
and anionic ⌫
⫺
surface excess can be
expressed as integrals over the respective singlet total corre-
lation functions h
w⫹
(z) and h
w⫺
(z),
⌫
i
⫽
冕
0
⬁
dzh
wi
共
z
兲
, i⫽⫹,⫺ . 共25兲
The surface charge density
␴
, used as input for our cal-
culations, can be calculated across the interfacial region,
␴
⫽⫺
兺
i⫽⫹,⫺
q
i
␳
i
B
⌫
i
. 共26兲
Combining Eqs. 共17兲, 共18兲, and 共26兲, the mean electrostatic
potential may be rewritten as
␺
共
z
兲
⫽
␲
␧
0
␧
r
兺
i⫽1
v
q
i
冕
z
⬁
dt
共
z⫺ t
兲
␳
wi
共
t
兲
. 共27兲
Electrocapillary curves are plots of the surface free energy
change ⌬
␥
as the charge on the electrode is increased. The
change in surface free energy is calculated from the equation,
⌬
␥
⫽⫺
冕
0
␺
0
d
␺
␴
共
␺
兲
. 共28兲
The electrical double layer is comprised of separated layers
of charge, analogous to a conventional electrical capacitor. It
is a region where charge can be stored by varying the surface
potential. The differential capacitance C is defined to be
C⫽
d
␴
d
␺
0
, 共29兲
and has been calculated by performing a numerical differen-
tiation on a plot of
␴
against
␺
0
.
G. Contact theorem
The contact theorem provides a consistency check for
the accuracy of approximate theories. For a charged hard-
wall, the contact theorem connects the values of the ionic
densities at contact
␳
wi
(d/2) with the bulk osmotic pressure
p and the surface charge density
␴
. For a soft short-range
wall–ion potential the ‘‘contact’’ theorem for an exact
theory is
40,41
kT
兺
i⫽⫹,⫺
␳
wi
共
0
兲
⫽ p⫹
2
␲␴
2
⑀
⫹
兺
i⫽⫹,⫺
冕
0
⬁
dz
␳
wi
共
z
兲
d
␾
wi
共
z
兲
dz
. 共30兲
For the HNC approximation, an alternative contact condition
may be derived,
42,43
kT
兺
i⫽⫹,⫺
␳
wi
共
0
兲
⫽
1
2
冉
kT
兺
i⫽⫹,⫺
␳
i
B
⫹
␹
⫺ 1
冊
⫹
2
␲␴
2
⑀
⫹
兺
i⫽⫹,⫺
冕
0
⬁
dz
␳
wi
共
z
兲
d
␾
wi
共
z
兲
dz
, 共31兲
where the isothermal compressibility
␹
is calculated using
kT
兺
i⫽⫹,⫺
␳
i
B
⫺
␹
⫺ 1
⫽ kT
兺
i,j⫽⫹,⫺
␳
i
B
␳
j
B
冕
drc
ij
*
共
r
兲
. 共32兲
Henderson
15
stated that pair calculations using the LMBW
equation ‘‘appear to satisfy’’ Eq. 共31兲. Recalling that the
right-hand side of either contact theorem is necessarily zero
for our potential, we have found numerically that the AHNC
approximation satisfies the exact contact theorem as it
must.
28
In its dimensionless form, numerically we find it
holds to within 10
⫺ 1
for the majority of our calculations. The
pair approximation with the LMBW equation does not, for
the majority of calculations, satisfy the dimensionless form
of either Eq. 共30兲 or 共31兲 to within 10
⫺ 1
.
III. NUMERICAL CONSIDERATIONS
The pair approximation must be solved numerically. The
input to our calculation is: 共i兲 the short-range potential en-
ergy between the wall and the ions
␾
wi
(z); 共ii兲 the short-
range ion–ion potential energy
␾
ij
(r); 共iii兲 the temperature
T; 共iv兲 the bulk electrolyte concentration c
i
; 共v兲 the structure
of the bulk model electrolyte, contained in the direct and
total correlation functions, c
ij
(r) and h
ij
(r), respectively,
for all distinct pairs of species, ij⫽⫹⫹, ⫹⫺, and ⫺⫺,as
a function of distance r; and 共vi兲 the surface charge density
␴
.
The equations are solved iteratively using a solver NK-
SOL; the advantages of using such a solver are discussed by
Booth.
44
For all calculations, the temperature T⫽ 25°C. The
bulk correlation functions c
ij
(r) and h
ij
(r) are based on cal-
culations made by Duh and Haymet,
12
using the HNC clo-
sure. Discrete grids are used for the normal and radial direc-
tions. Integrals were calculated using the trapezoidal method
of integration.
In the z-direction, a nonlinear grid z
i
was used, where
each grid point is defined by z
i
⫽ exp
关
(i⫺b)*
␦
Z
兴
⫹a, where a
共real兲, b 共integer兲, and
␦
Z 共real兲 were chosen to ensure that
density profiles are numerically zero at the first point z
1
and
within a few percent of the bulk density at a cutoff, z
cut
,
from the wall. The grid in the z-direction was broken into
two regions. The first section represented the grid between
the wall and an arbitrarily chosen cutoff, z
cut
, beyond which
the density of ions was assumed to be that of the bulk. A
second region was selected from z
cut
to z
max
, a point arbi-
trarily chosen to represent infinity in integrals calculated in
the z-direction. Any inhomogeneous correlation function
which involves an ion located between z
cut
and z
max
is as-
sumed to have a value of the bulk correlation function of the
same ion–ion separation R. In general, a total of approxi-
mately 50 to 80 points were used in the z-direction, with
about 45–75 used between the wall and z
cut
.
The Hankel transforms expressed by Eq. 共5兲 can be
approximated
45
by
10942 J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 A. C. Eaton and A. D. J. Haymet
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

f
ˆ
共
k
m
兲
⫽
4
␲
k
N
2
兺
j⫽ 1
N
f
共
s
j
兲
J
0
共
k
m
s
j
兲
J
1
2
共
k
M
s
j
兲
,
共33兲
f
共
s
j
兲
⫽
1
␲
s
N
2
兺
m⫽1
N
f
ˆ
共
k
m
兲
J
0
共
k
m
s
j
兲
J
1
2
共
k
m
s
N
兲
,
where J
1
is the first order Bessel function. The points in the
radial direction (k
1
¯ k
N
and s
1
¯ s
N
) are determined by
the conditions that J
0
(k
m
s
N
)⫽ J
0
(k
N
s
j
)⫽ 0 and s
N
is suit-
ably chosen so that all functions f(s
i
) approach zero as i
approaches N. For the majority of calculations, N⫽ 90, with
s
N
⫽ 30 Å. The numerical details of the pair approximation
calculations are further discussed in Ref. 34.
IV. RESULTS: ION CONCENTRATION PROFILES
A. Comparison of AHNC approximation and
simulations
To our knowledge, no simulation data exist for the real-
istic ‘‘soft’’ potentials we have studied with pair and singlet
theory. As a guide, we have chosen to make comparisons
with the Monte Carlo 共MC兲 data of Torrie and Valleau.
46,47
In their simulations, Torrie and Valleau use a different set of
potentials than us, namely charged hard-spheres 关e.g., the
potential in Fig. 3共a兲, dashed lines兴 for the model electrolyte,
and a short-range hard-wall potential, as shown in Fig. 3共b兲
共dashed line兲.
Figure 4 shows the cationic and anionic normalized con-
centration profiles for 0.5 M 2–2 electrolyte at a surface
charge density of ⫺ 21.29
␮
Ccm
⫺2
. The normalized concen-
tration profiles for the singlet approximation have been in-
cluded for comparison. The corresponding surface potential
for each of the integral equation approximations is listed
below in brackets. The circles are the digitized MC computer
simulation results. The solid lines are the pair results using
the AHNC (⫺ 99 mV) approximation. The long-dashed lines
are the singlet approximation (⫺ 100 mV).
Taking into consideration the differences in the potential
functions, the approximate integral equation theories predict
qualitatively the same behavior as the MC data. The pair
approximations show quantitative agreement with MC data,
particularly for the anionic profile, and a measurable im-
provement over the singlet approximation.
For divalent electrolytes at lower concentrations it is no
longer possible to obtain singlet solutions at the surface
charge densities used in the MC simulations. In Fig. 1 above
we compare the AHNC approximation 共solid lines,
⫺ 69.6 mV) with the MC results 共circles兲 for 2–2 electrolytes
at 0.05 M and a surface charge density of ⫺8.65
␮
Ccm
⫺2
.
We also show the pair using the LMBW approximation
共dashed lines, ⫺ 70.3 mV) for comparison. Allowing for the
differences in potential models, the pair approximations pre-
dict qualitatively the same behavior as the MC data.
B. Comparison of AHNC and singlet approximations
We now consider comparisons of normalized concentra-
tion profiles for the singlet approximation, using our ‘‘soft’’
potential model, and the pair approximation using the AHNC
density equation. In Fig. 5, comparisons for cationic and an-
ionic normalized concentration profiles are made between
pair results using the AHNC approximation 共solid lines兲 and
the singlet approximation 共dashed lines兲, for 2–2 electrolytes
at 0.5 M with surface charge densities of 0, ⫺10 and
⫺ 30
␮
Ccm
⫺2
. A surface charge density of ⫺30
␮
Ccm
⫺2
represents the upper limit for obtaining solutions for the sin-
glet approximation.
With no surface charge on the wall, the profiles of the
singlet and pair approximations are similar, with the pair
approximation result showing slightly less structure. At a
surface charge density of ⫺ 10
␮
Ccm
⫺2
, there is very little
difference in the two profiles, though, as the magnitude of
the surface charge increases, the singlet approximation pre-
dicts significantly more structure than the pair approxima-
tion, with secondary peaks developing in both the cationic
and anionic profiles at a surface charge density of
⫺ 30
␮
Ccm
⫺2
. While both integral equations carry approxi-
mations, and hence conclusions drawn from differences in
their results must be made with some caution, the differences
in density profiles would seem to be a consequence of the
combination of approximations in the singlet plus HNC
approximation,
28
namely that the inhomogeneous correlation
FIG. 4. 共a兲 Cationic g
w⫹
(z) and 共b兲 anionic g
w⫺
(z) normalized concentra-
tion profiles for a 0.5 M 2–2 electrolyte at a surface charge density of
⫺ 21.29
␮
Ccm
⫺2
: pair using AHNC approximation 共solid line兲; singlet
共long-dashed line兲; and the MC results 共circles兲 of Torrie and Valleau 共Ref.
46兲, using a CHS/CHW potential.
FIG. 5. 共a兲 Cationic g
w⫹
(z) and 共b兲 anionic g
w⫺
(z) normalized concentra-
tion profiles for a 0.5 M 2–2 electrolyte at surface charge densities of 0,
⫺ 10, and ⫺ 30
␮
Ccm
⫺2
: pair using AHNC approximation 共solid line兲;and
singlet approximation 共long-dashed line兲.
10943J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 Electrolytes at charged interfaces
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

functions are equivalent to their bulk counterparts. The sin-
glet approximation exaggerates the density profiles since the
charged liquid is not allowed to relax from its bulk structure
even very close to the charged surface. Indeed, for all con-
centrations studied, the general trend is that, for increasing
surface charge, the pair approximation predicts less ionic
density structure than the singlet approximation.
C. Results: Concentration dependence
We now study the effect of changing the concentration
for 2–2 electrolytes, with a surface charge density of
⫺ 30
␮
Ccm
⫺2
. We have obtained solutions for concentra-
tions over 3 orders of magnitude in concentration, at 25 °C.
The values of the Debye length
␬
⫺ 1
and the calculated sur-
face potential are indicated in brackets: 0.005 M 共21.49 Å,
⫺ 162.7 mV), 0.05 M 共6.80 Å, ⫺ 149.6 mV), 0.5 M 共2.15 Å,
⫺ 130.5 mV) and 5.0 M 共0.68 Å, ⫺ 92.6 mV). In Fig. 6, the
cationic and anionic normalized density profiles are shown.
The major feature of the counterion profiles is the rapidly
increasing peak as the concentration is decreased. The an-
ionic profile does not show the same rapidly increasing peak
height. Despite the cationic peak height rapidly rising, the
surface potential only increases by moderate amounts. For
the 5.0 M concentration, strong oscillations are seen in the
density profiles. As the concentrations is lowered to 0.5 M,
the profiles only oscillate weakly. No oscillations are seen
for concentrations below 0.5 M.
V. RESULTS: ELECTROSTATICS AND SURFACE
THERMODYNAMICS
Although we have shown that significant differences be-
tween the pair and singlet approximations can occur in the
structure of the double layer, the impact of this finding would
be somewhat lessened were these differences not to translate
into noticeable changes in the electrostatics and thermody-
namics of the double layer. The quantities in this section
currently represent the strongest links between theory and
experiment, and their evaluation and comparison is vital to
an understanding of the double layer and theoretical models
that are applied to it.
A. Mean electrostatic potential
The surface charge density
␴
is used as input for the pair
calculations. For a 0.5 M 2–2 electrolyte, the dependence of
surface potential on surface charge density is shown in Fig.
7. The pair results using the AHNC 共solid line兲 and LMBW
共dashed line兲 approximations are compared with the singlet
approximation 共long-dashed line兲. There is no simulation or
experimental data for comparison to our knowledge.
Comparing the pair 共AHNC兲 and singlet approximations,
it can be seen that there are two distinct regions. As the
charge is increased from zero to about 10
␮
Ccm
⫺2
, there is
little difference among any of the integral equation approxi-
mations. After 10
␮
Ccm
⫺2
, the singlet approximation pre-
dicts substantially less surface potential for the same surface
charge density.
The two pair approximations show similar behavior.
While the curve for the LMBW approximation predicts
slightly higher surface potentials than for the AHNC ap-
proximation, they remain similar over the entire range. An
interesting feature of both pair approximations is the linear
nature of the functions at surface charge densities greater
than 10
␮
Ccm
⫺2
.
Figure 8 shows the change in the mean electrostatic po-
tential of the AHNC approximation, as the surface charge
density is increased by 10
␮
Ccm
⫺2
from ⫺ 10 to
⫺ 80
␮
Ccm
⫺2
, indicated by alternating solid and dashed
lines. This corresponds to an increase in the surface potential
from ⫺ 54.6 to ⫺ 311.8 mV. At a low charge on the wall, the
mean electrostatic potential has a very shallow minimum at a
z value near 1.4d. As the wall charge increases to
⫺ 40
␮
Ccm
⫺2
, the minimum becomes deeper and shifts to
about z⫽ 0.9d. As the charge on the wall continues to in-
crease up to ⫺ 80
␮
Ccm
⫺2
, there is very little change in the
overall functional form of the mean electrostatic potential.
B. Ionic surface excess
Figure 9 shows the cationic surface excess charge den-
sity as a function of surface charge density. The solid line
FIG. 6. 共a兲 Cationic g
w⫹
(z) and 共b兲 anionic g
w⫺
(z) normalized concentra-
tion profiles for a wall with a surface charge density of ⫺ 30
␮
Ccm
⫺2
for a
5.0 M 2–2 electrolyte 共dotted–dashed line兲; 0.5 M 2–2 electrolyte 共solid
line兲; 0.05 M 2–2 electrolyte 共long-dashed line兲; and 0.005 M 2–2 electro-
lyte 共dashed line兲.
FIG. 7. The surface potential as a function of surface charge density for a
0.5 M 2–2 electrolyte: pair using the AHNC approximation 共solid line兲; pair
using the LMBW approximation 共dashed line兲; and the singlet approxima-
tion 共long-dashed line兲.
10944 J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 A. C. Eaton and A. D. J. Haymet
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

represents the pair approximation with the AHNC density
equation for a 0.5 M 2–2 electrolyte. The long-dashed line
shows the singlet approximation. The circles represent the
experimental data of Harrison, Randles and Schriffrin
48
for a
0.5 M aqueous solution of magnesium sulfate at 25°C, ob-
tained from electrocapillary measurements on a mercury-
aqueous electrolyte interface. To our knowledge, no simula-
tion data exist for relevant ‘‘soft’’ potentials. Merely for a
simple comparison, the Gouy–Chapman–Stern approxima-
tion 共GCS兲共for hard potentials兲
13
and is the short-dashed
line. The results of the LMBW approximation show no dis-
cernible difference from those of the AHNC approximation,
so they are not included here.
For a negative electrode, the curve of the AHNC ap-
proximation agrees well with both the experimental data and
the curves of the other theoretical approaches. At positive
surface charges, the results from the AHNC approximation
are quantitatively and qualitatively different from the exist-
ing experimental data. The pair approximation predicts little
change in the cationic surface excess charge density as the
wall becomes more positively charged. This is a direct result
of the lack of change in the coion density profile as the wall
gains charge, whereas the experimental results indicate sub-
stantial increases in the coion density for these surface
charges. The pair approximation for our ‘‘soft’’ potentials
does qualitatively predict similar behavior to the GCS ap-
proximation 共for hard potentials兲. The AHNC approximation
results also differ markedly from the results of the singlet
approximation, which may seem in this case to predict the
behavior of the experimental data better, possibly due to a
cancellation of errors. This unexpected result is probably an
indication of the limitations of the models that are being
used. Our models fail to account for the nature of the solvent
and the important role that it plays. The size of the magne-
sium ion relative to the sulfate ion may also be a factor in the
differences that have been seen, and the nature of the solid
surface, including any image interactions, cannot be ignored.
These will be explored in future work, together with connec-
tions to the work of Greberg and Kjellander.
26
C. Electrocapillary curves
The change in surface free energy resulting from charg-
ing the interface can be calculated by the numerical integra-
tion of the surface charge density as a function of the surface
potential, as shown in Eq. 共28兲 above. In Fig. 10, the surface
free energy is plotted as a function of surface potential for
0.5 M 2–2 electrolyte, using the AHNC approximation. For
comparison, the 0.5 M 2–2 electrolyte predictions for the
singlet approximation are shown. The digitized experimental
data 共circles兲 of Grahame
49
for NaCl at 18 °C is only in-
cluded to show the order of magnitude of this quantity in the
absence of experimental data for a 0.5 M 2–2 electrolyte.
Figure 10 indicates that the predictions of the pair approxi-
mation are in good agreement with those of the singlet ap-
proximation. This indicates that the surface free energy is
relatively insensitive to the differences in the fluid structure
near the solid surface.
D. Differential capacitance
Given the charge separation between the interface and
the electrolyte solution, the double layer may be viewed as a
FIG. 8. The change in the mean electrostatic potential, as a function of
distance from the wall, as the surface charge density is increased from ⫺ 10
to ⫺80
␮
Ccm
⫺2
, in increments of ⫺10
␮
Ccm
⫺2
, for the AHNC approxi-
mation.
FIG. 9. The cationic surface excess charge density as a function of surface
charge density, using the AHNC approximation 共solid line兲, for a 0.5 M 2–2
electrolyte. Also shown are the singlet approximation 共long-dashed line兲,
and the Gouy–Chapman–Stern approximation 共GCS兲共short-dashed line兲.
For comparison, the experimental data of Harrison, Randles and Schriffrin
共circles兲 are shown for a 0.5 M aqueous solution of magnesium sulfate at
25 °C.
FIG. 10. The surface free energy is plotted as a function of surface potential
for a 0.5 M 2–2 electrolyte, using the AHNC approximation 共solid line兲. For
comparison, the 0.5 M 2–2 electrolyte predictions for the singlet approxi-
mation 共long-dashed line兲 are shown. The digitized experimental data
共circles兲 of Grahame 共Ref. 49兲 for the 1–1 electrolyte NaCl at 18 °C is
included merely to show the order of magnitude of this quantity in the
absence of experimental data for a 0.5 M 2–2 electrolyte.
10945J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 Electrolytes at charged interfaces
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

parallel plate capacitor. This phenomenon has drawn much
interest recently.
50,51
The differential capacitance is calcu-
lated using Eq. 共29兲. Figure 11 shows the differential capaci-
tance as a function of the surface charge density for a 0.5 M
2–2 electrolyte for the pair 共AHNC兲 approximation 共solid
line兲. Also shown are our earlier results the singlet approxi-
mation 共long-dashed line兲. No simulation data are available
for soft potentials. To provide some basis for comparison, we
include hard potential based GCS predictions 共dotted–
dashed line兲 and digitized data from Teran et al.
52
共short-
dashed line兲, calculated for the RPM/charged hard wall
model within the hypernetted chain/mean spherical approxi-
mation singlet approximation. The results of the latter two
are discussed at greater length in our previous paper.
13
Note
that nothing in this figure alters the fact that the pair approxi-
mation is believed to have the best accuracy, the singlet next
in accuracy, and the GCS last. The predictions of the LMBW
approximations and the AHNC approximations do not differ
noticeably and so the LMBW results have not been included
here.
It can be seen that for soft potentials, the pair and singlet
approximation vary in their functional form, with qualita-
tively different behavior. The singlet approximation predicts
a rapidly increasing capacitance as the wall is charged. Con-
versely, the pair approximation shows a slowly increasing
capacitance as the interface becomes more charged, almost
to the point of being constant at higher charges. The capaci-
tance is dependent on the surface potentials of the pair and
singlet approximations as a function of the surface charge
density, but, unlike the electrocapillary curves, is very sensi-
tive to the differences in the density profiles of the pair and
singlet approximations, most particularly, the lack of change
in the coion density profile of the pair approximation.
VI. CONCLUSIONS
In this paper, we have calculated the structure, electro-
statics and thermodynamics of electrolytes using two pair
approximations, the LMBW and AHNC approximations.
The density profiles and thermodynamics produced by both
approximations were found to be similar for all electrolyte
concentrations considered. The pair approximations predict
ionic density profiles that are qualitatively, and frequently
quantitatively, in good agreement with MC simulation data,
whereas the singlet approximation generally predicts concen-
tration profiles with more structure than the pair approxima-
tion as the surface charge is increased.
Integral equations provide some interesting insights into
electrostatics and thermodynamics. For example, the surface
free energy is found to be relatively unaffected by the differ-
ences in the density profiles predicted by the different inte-
gral equations. The differential capacitance is extremely sen-
sitive to the type of integral equation used. The electrostatic
potentials show some differences between the singlet and
pair approximations. The ionic surface excess plot showed a
noticeable difference in the behavior of the pair and singlet
approximations. The experimental data for this quantity was
significantly different from the predictions of both integral
equations. This highlights some of the shortcomings of the
simple models we have used, in particular, the neglect of
discrete solvent molecules, the simplification of same ionic
radii and the lack of a realistic solid surface. These are not
new considerations, but are being revisited with renewed
vigor after some neglect.
53–58
The successful application of
the pair approximations to electrolytes next to a charged sur-
face, opens the door for their use on more complex systems,
such as those with polarizable potentials like the central
force models of water.
59
ACKNOWLEDGMENTS
This research was supported in part by the Australian
Research Council 共ARC兲共Grant No. A29530010兲, and in
part by the Welch Foundation 共Grant No. E1429兲.Weac-
knowledge gratefully many helpful conversations on this
topic with Dr. Stephan Marc
˘
elja 共ANU兲 and Dr. Roland
Kjellander 共Gothenburg兲. Computational aspects of this re-
search have also been supported by ANUSF and the
NSWCPC. A.D.J.H. thanks Dr. Grant Goodyear and Dr. Jo-
han Ulander for a critical reading of the manuscript.
1
H. Helmholtz, Ann. Phys. 共Leipzig兲 7, 337 共1879兲.
2
J. S. Rowlinson, ‘‘Development of theories of inhomogeneous fluids,’’ in
Fundamentals of Inhomogeneous Fluids, edited by D. Henderson 共Marcel
Dekker, New York, 1992兲, Chap. 1.
3
G. Gouy, J. Phys. 共Paris兲, Colloq. 9,457共1910兲.
4
D. L. Chapman, Philos. Mag. 25, 475 共1913兲.
5
O. Stern, Z. Elektrochem. Angew. Phys. Chem. 30, 508 共1924兲.
6
N. T. Skipper, M. V. Smalley, G. D. Williams, A. K. Soper, and C. H.
Thompson, J. Chem. Phys. 99, 14201 共1995兲.
7
J. Israelachvili and H. Wennerstro
¨
m, Nature 共London兲 379,219共1996兲.
8
J. P. Quirk and S. Marc
ˇ
elja, Langmuir 13, 6241 共1997兲.
9
M. Ullner, C. E. Woodward, and B. Jo
¨
nsson, J. Chem. Phys. 105, 2056
共1996兲.
10
G. Toikka and R. A. Hayes, J. Colloid Interface Sci. 191,102共1997兲.
11
M. J. Weaver, J. Phys. Chem. 100, 13079 共1996兲.
12
D. M. Duh and A. D. J. Haymet, J. Chem. Phys. 97, 7716 共1992兲.
13
M. J. Booth, A. C. Eaton, and A. D. J. Haymet, J. Chem. Phys. 103,417
共1995兲.
14
M. Plischke and D. Henderson, J. Chem. Phys. 88, 2712 共1988兲.
15
D. Henderson and M. Plischke, J. Phys. Chem. 92, 7177 共1988兲.
16
D. Henderson, ‘‘Integral equation theories for inhomogeneous fluids,’’ in
Fundamentals of Inhomogeneous Fluids, edited by D. Henderson 共Marcel
Dekker, New York, 1992兲, Chap. 4.
17
R. Kjellander and S. Marc
˘
elja, J. Chem. Phys. 82, 2122 共1985兲.
18
R. Kjellander and S. Marc
˘
elja, Chem. Phys. Lett. 127,402共1986兲.
FIG. 11. The differential capacitance as a function of the surface charge
density for a 0.5 M 2–2 electrolyte for the pair 共AHNC兲 approximation
共solid line兲. Also shown is the singlet approximation 共long-dashed line兲,
GCS predictions 共dotted–dash line兲 and digitized data from Teran et al.
共Ref. 52兲共short-dashed line兲, using a CHS/CHW potential with a singlet
approximation.
10946 J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 A. C. Eaton and A. D. J. Haymet
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

19
R. Kjellander, J. Chem. Phys. 88, 7129 共1988兲, and references therein.
20
R. Kjellander, T. Åkesson, B. Jo
¨
nsson, and S. Marc
ˇ
elja, J. Chem. Phys.
97,1424共1992兲.
21
R. Kjellander and J. Ulander, J. Phys. 共Paris兲, Colloq. 10, 431 共2000兲.
22
H. Greberg and R. Kjellander, Mol. Phys. 83,789共1994兲.
23
J. Ennis, S. Marc
ˇ
elja, and R. Kjellander, Electrochim. Acta 41, 2115
共1996兲.
24
H. Greberg, R. Kjellander, and T. Åkesson, Mol. Phys. 87, 407 共1996兲.
25
H. Greberg, R. Kjellander, and T. Åkesson, Mol. Phys. 92,35共1997兲.
26
R. Kjellander and H. Greberg, J. Electroanal. Chem. 462, 273 共1999兲.
27
P. Attard, Advances in Chemical Physics, edited by I. Prigogine and S. A.
Rice 共Wiley, New York, 1996兲, Vol. XCII.
28
M. R. Bush, M. J. Booth, A. Haymet, and A. Schlijper, Mol. Phys. 95, 601
共1998兲.
29
J. Quintana, D. Henderson, and M. Plischke, J. Phys. Chem. 93,4304
共1989兲.
30
D. G. Triezenberg and R. Zwanzig, Phys. Rev. Lett. 28, 1183 共1972兲.
31
J. P. Hansen and I. R. McDonald, Theory of Simple Liquids 共Academic,
London, 1986兲.
32
R. V. Churchill, Operational Mathematics, 3rd ed. 共McGraw–Hill, New
York, 1972兲.
33
H. Bateman, Tables of Integral Transforms, edited by A. Erde
`
lyi
共McGraw–Hill, New York, 1954兲, Vol. II.
34
A. C. Eaton, ‘‘Electrolytes at Charged Interfaces Integral Equation Ap-
proach,’’ Ph.D. thesis, School of Chemistry, University of Sydney, Aus-
tralia, 2000.
35
R. Lovett, C. Y. Mou, and F. P. Buff, J. Chem. Phys. 65, 570 共1976兲.
36
M. S. Wertheim, J. Chem. Phys. 65, 2377 共1976兲.
37
P. J. Rossky, J. B. Duduwicz, B. L. Tembe, and H. L. Friedman, J. Chem.
Phys. 73, 3372 共1980兲.
38
R. Bacquet and P. J. Rossky, J. Chem. Phys. 79, 1419 共1983兲.
39
W. A. Steele, The Interaction of Gases with Solid Surfaces 共Pergamon,
Oxford, 1974兲.
40
D. Henderson, L. Blum, and J. L. Lebowitz, J. Electroanal. Chem. 102,
315 共1979兲.
41
S. L. Carnie and D. Y. C. Chan, J. Chem. Phys. 74, 1293 共1981兲.
42
S. L. Carnie, D. Y. C. Chan, D. J. Mitchell, and B. W. Ninham, J. Chem.
Phys. 74, 1472 共1981兲.
43
L. Blum and D. Henderson, ‘‘Statistical mechanics of electrolytes at in-
terfaces,’’ in Fundamentals of Inhomogeneous Fluids, edited by D. Hend-
erson 共Marcel Dekker, New York, 1992兲, Chap. 6.
44
M. J. Booth, A. G. Schlijper, L. E. Scales, and A. D. J. Haymet, Comput.
Phys. Commun. 119, 122 共1999兲.
45
F. Lado, J. Chem. Phys. 49,3092共1968兲.
46
G. M. Torrie and J. P. Valleau, J. Phys. Chem. 86,3251共1982兲.
47
S. L. Carnie and G. M. Torrie, Adv. Chem. Phys. 56,141共1984兲.
48
J. A. Harrison, J. E. B. Randles, and D. J. Schiffrin, J. Electroanal. Chem.
25,197共1970兲.
49
D. C. Grahame, Chem. Rev. 41,441共1947兲. The concentration of NaCl is
not specified. It appears to be a 1.0 M solution.
50
D. Boda, D. Henderson, and K.-Y. Chan, J. Chem. Phys. 110,5346
共1999兲.
51
D. Boda and D. Henderson, J. Chem. Phys. 112, 8934 共2000兲.
52
L. Mier y Tera
´
n, E. Dı
´
az-Herrera, M. Lozada-Cassou, and D. Henderson,
J. Phys. Chem. 92, 6408 共1988兲.
53
G. To
´
th, E. Spohr, and K. Heinzinger, Chem. Phys. 200,347共1995兲.
54
D. Boda, K.-Y. Chan, and D. Henderson, J. Chem. Phys. 109,7362
共1998兲.
55
J. Forsman, B. Jo
¨
nsson, and C. E. Woodward, J. Phys. Chem. 100, 15005
共1996兲.
56
M. F. Toney et al., Nature 共London兲 368, 444 共1994兲.
57
E. Dı
´
az-Herrera and F. Forstmann, J. Chem. Phys. 102, 9005 共1995兲.
58
M. F. Toney et al., Surf. Sci. 335, 326 共1995兲.
59
M. J. Booth, D.-M. Duh, and A. D. J. Haymet, J. Chem. Phys. 101, 7925
共1994兲, and references therein.
10947J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 Electrolytes at charged interfaces
Downloaded 16 Jan 2004 to 129.171.128.66. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp