Application of a density functional approach to nonuniform ionic fluids: the effect of association
In the present paper we discuss a density functional approach for nonuniform ionic fluids, which takes into account the existence of ion pairs. The theory is based on a fundamental measure theory of hard-spheres, the theory of Gillespie et al., which leads to a more accurate description of the e...
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irk-123456789-1190092017-06-04T03:04:13Z Application of a density functional approach to nonuniform ionic fluids: the effect of association Reszko-Zygmunt, J. Sokołowski, S. In the present paper we discuss a density functional approach for nonuniform ionic fluids, which takes into account the existence of ion pairs. The theory is based on a fundamental measure theory of hard-spheres, the theory of Gillespie et al., which leads to a more accurate description of the electrostatic part of the grand potential as well as on Wertheim’s association theory. The results of model calculations indicate that the inclusion of the associative term in the grand potential leads to the structure of the double layer, which differs from the structure evaluated by neglecting the association. These differences are important at low temperatures only У цій роботі ми обговорюємо метод функціоналу густини для неоднорідних іонних флюїдів, який враховує існування іонних пар. Підхід базується як на теорії фундаментальної міри твердих сфер, теорії Гіллеспі та ін., яка веде до більш точного опису електростатичної частини великого термодинамічного потенціалу, так і на асоціативній теорії Вертхейма. Результати модельних обчислень вказують на те, що включення асоціативного члена у великий термодинамічний потенціал веде до структури подвійного шару, яка відрізняється від структури, коли нехтується асоціацією. Ця різниця є важливою тільки при низьких температурах. 2004 Article Application of a density functional approach to nonuniform ionic fluids: the effect of association / J. Reszko-Zygmunt, S. Sokołowski // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 793–804. — Бібліогр.: 44 назв. — англ. 1607-324X PACS: 68.43-h, 68.43.De, 71.15.Mb DOI:10.5488/CMP.7.4.793 http://dspace.nbuv.gov.ua/handle/123456789/119009 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In the present paper we discuss a density functional approach for nonuniform
ionic fluids, which takes into account the existence of ion pairs. The
theory is based on a fundamental measure theory of hard-spheres, the
theory of Gillespie et al., which leads to a more accurate description of the
electrostatic part of the grand potential as well as on Wertheim’s association
theory. The results of model calculations indicate that the inclusion
of the associative term in the grand potential leads to the structure of the
double layer, which differs from the structure evaluated by neglecting the
association. These differences are important at low temperatures only |
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Article |
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Reszko-Zygmunt, J. Sokołowski, S. |
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Reszko-Zygmunt, J. Sokołowski, S. Application of a density functional approach to nonuniform ionic fluids: the effect of association Condensed Matter Physics |
| author_facet |
Reszko-Zygmunt, J. Sokołowski, S. |
| author_sort |
Reszko-Zygmunt, J. |
| title |
Application of a density functional approach to nonuniform ionic fluids: the effect of association |
| title_short |
Application of a density functional approach to nonuniform ionic fluids: the effect of association |
| title_full |
Application of a density functional approach to nonuniform ionic fluids: the effect of association |
| title_fullStr |
Application of a density functional approach to nonuniform ionic fluids: the effect of association |
| title_full_unstemmed |
Application of a density functional approach to nonuniform ionic fluids: the effect of association |
| title_sort |
application of a density functional approach to nonuniform ionic fluids: the effect of association |
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Інститут фізики конденсованих систем НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/119009 |
| citation_txt |
Application of a density functional approach to nonuniform ionic fluids: the effect of association / J. Reszko-Zygmunt, S. Sokołowski // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 793–804. — Бібліогр.: 44 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT reszkozygmuntj applicationofadensityfunctionalapproachtononuniformionicfluidstheeffectofassociation AT sokołowskis applicationofadensityfunctionalapproachtononuniformionicfluidstheeffectofassociation |
| first_indexed |
2025-07-08T15:05:02Z |
| last_indexed |
2025-07-08T15:05:02Z |
| _version_ |
1837091630915518464 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 4(40), pp. 793–804
Application of a density functional
approach to nonuniform ionic fluids:
the effect of association
J.Reszko-Zygmunt∗, S.Sokołowski†
Faculty of Chemistry, Maria Curie-Skłodowska University,
20–031 Lublin, Poland
Received August 18, 2004, in final form October 1, 2004
In the present paper we discuss a density functional approach for nonuni-
form ionic fluids, which takes into account the existence of ion pairs. The
theory is based on a fundamental measure theory of hard-spheres, the
theory of Gillespie et al., which leads to a more accurate description of the
electrostatic part of the grand potential as well as on Wertheim’s associ-
ation theory. The results of model calculations indicate that the inclusion
of the associative term in the grand potential leads to the structure of the
double layer, which differs from the structure evaluated by neglecting the
association. These differences are important at low temperatures only.
Key words: adsorption, associating fluids, density functional theory
PACS: 68.43-h, 68.43.De, 71.15.Mb
1. Introduction
The theoretical study of electrolytes and in particular association phenomena
have been of great interest over the recent years. Ion association has a significant
effect on physical and chemical characteristic of electrolytes. In recent years several
theoretical methods have been proposed and discussed to take the phenomena of
ion association into account [1].
The properties of an electrochemical interface within the so-called restricted pri-
mitive model (RPM) and several theoretical approaches have included the RPM
model into theories aimed at describing the uniform, as well as the nonuniform
electrolytes and molten salts. In particular, these systems have been intensively in-
vestigated using computer simulations [2–5] and integral equations theories [6–9].
1Contribution for the collection of papers of the NATO ARW Workshop “Ionic Soft Matter:
Novel trends in theory and applications” April, 2004, Lviv, UA
∗jreszko@hermes.umcs.lublin.pl
†stefan@zool.umcs.lublin.pl
c© J.Reszko-Zygmunt, S.Sokołowski 793
J.Reszko-Zygmunt, S.Sokołowski
However, the most successful approaches for determining the thermodynamic and
structural properties of nonuniform RPM fluids are those based on recent density
functional formulations. Density functional theory (DF) determines the structure
and thermodynamic properties of a system minimizing the grand potential functi-
onal. Usually, the construction of the grand potential is based on a perturbation
expansion around a hard-sphere reference system. One of the most accurate DF
theories for simple hard-sphere fluid, which is known in the literature as the funda-
mental measure theory, was developed by Rosenfeld [10]. In subsequent years there
were several papers published devoted to the application of DF approaches to the
description of nonuniform RPM fluids at adsorbing surfaces [11–16].
In many cases the DF approach is quite accurate in predicting the structure of
nonuniform electrolytes. According to common DF treatment the Helmholtz free
energy, F , is divided into the ideal term, Fid, the contributions resulting from hard-
sphere interactions, FHS, and the electrostatic contribution, Fel. Of course, the ideal
term is known exactly. However, in order to determine all remaining contributions we
need to introduce additional approximations. Various formulations of the functional
FHS exist [17], including Rosenfeld’s [10] and the improved formulations based on
it [18–20]. However, the contributions to the free energy resulting from electrostatic
interactions have been evaluated using a perturbational approach with respect to a
bulk uniform fluid. It is obvious that not all the systems are capable of perturbing
around a bulk fluid. If local densities vary in large amounts within the system, then
the bulk-fluid perturbation ansatz is probably not the best choice.
Recently Gillespie et al. [21,22] have drawn attention to this problem. They pre-
sented an approximate electrostatic excess free energy functional for charged hard
sphere fluids. Their method is based on a perturbation expansion concerning a suit-
ably chosen position-dependent reference fluid. This method accurately reproduces
the results of Monte Carlo simulations [21,22].
Real bulk electrolyte solutions consist of free ions and ionic pairs and even more
complex ionic aggregates. Therefore the description of thermodynamic properties of
electrolyte solutions calls for applying the associative concept of Bjerrum [23] and
Ebeling and Grigo [24] to the RPM. In fact, the ideas of Ebeling and Grigo have
already been used to improve the equation of state calculations of bulk fluids [1,25–
29]. In accordance with this approach, the density of ions is divided into the density
of free and bound parts, which are connected by the mass action law.
Associative concept of nonuniform ionic fluids can be realized by two differ-
ent methods. According to the first approach only unbounded ions are taken into
account, i.e. the total fluid density is modified by subtracting the concentration
of associated ions, which is obtained from the mass action law [30]. The second
method starts from the associative mean spherical approximation [27] that is based
on Wertheim’s association theory [31] of inhomogeneous chemically associating flu-
ids [32]. The incorporation of association into integral equation theory of uniform,
as well as nonuniform electrolytes has been reported in [33–36].
Recently, Yu and Wu have focused attention on the case of fluids interacting
via short-range associative forces [32]. They have proposed an extension of the fun-
794
Ionic associating fluids at surfaces
damental measure density functional theory [10] to the case of fluids interacting
via short-range associative forces [32]. Their theory has been successfully applied to
the description of nonuniform non-ionic fluids, exhibiting close-loop demixing phase
diagrams in bulk systems [37,38].
In this paper we report the study on the behavior of nonuniform associating
ionic fluids in contact with a charged surface. We propose a density functional ap-
proach, which combines the ideas of Gillespie et al [21,22] and Yu and Wu [32].
To our best knowledge, the association effects in nonuniform electrolytes have been
considered so far within the framework of the associative mean spherical approx-
imation only, cf. [34,35] and the present work is the first attempt to incorporate
the association of ions into density-functional approach. This work is also the first
step towards the development of a theory which would predict phase transitions in
confined electrolytes.
2. Theory
The model system in this study consists of a two-component mixture of ions
(a generalization to n-component mixture is straightforward). The electrolyte is
assumed to be a fluid of charged hard spheres, all of the same diameter d and of
valences zi, i = 1, 2. The fluid is in contact with a charged hard wall. The interaction
between the ions of species i and the wall is given by
ui(z) = vi(z) + wi(z), (1)
where wi(z) and vi(z) are the electrostatic and the non-electrostatic (van der Waals)
parts of the external potential, respectively. The van der Waals interaction poten-
tial is
vi(z) =
{
∞, z < d/2,
0, otherwise,
(2)
where z is the distance from the surface. The electrostatic interaction between an
ion and the surface is given by
wi(z) =
4πqzie
2
ε
z. (3)
In the above e is the electron charge, zi is the valency of ions of i-th type, q is the
surface charge density and ε is the dielectric constant.
In the present paper we apply the theory, which adapts the approach of Gillespie
et al. [21,22]. The first step toward the development of the grand potential of the
system as a functional of the local density, ρi(r), is its decomposition into: ideal (id),
excess hard-sphere (hs), electrostatic (el), and associative (as) terms:
Ω [{ρi(r)}] = Ωid[{ρi(r)}] + Ωhs[{ρi(r)}] + Ωel[{ρi(r)}] + Ωas[{ρi(r)}]. (4)
The term Ωid is
Ωid[{ρi}]/kT =
∑
i=1,2
∫
drρi(z)
[
ln ρi(z) − 1 + zieΨ(r)
1
kT
+
1
kT
{ui(r) − µi}
]
, (5)
795
J.Reszko-Zygmunt, S.Sokołowski
where e is the electron charge and Ψ(r) is the electrostatic potential. Similarly to
[21,22] the last equation also involves non-ideal contributions to the configurational
chemical potential µi. The electrostatic potential Ψ(z) is obtained by integrating
[39] Poisson’s equation:
∇2Ψ(z) = −4π e
ε
∑
i
ziρi(z). (6)
The hard-sphere component of the grand potential is calculated from the version
of the fundamental measure theory [10], proposed by Roth et al. [40] and by Yu and
Wu [41]
Ωhs[{ρi}] =
∫
drfhs[{nα}] (7)
with
fhs = −n0 ln(1− n3) +
n1n2 − n1 · n2
1 − n3
+ n3
2
(
1 − ξ2
)3 n3 + (1 − n3)
2 ln(1 − n3)
36πn2
3(1 − n3)2
. (8)
In the above ξ(r) = |n2(r)|/n2(r) and nα, α = 0, 1, 2, 3, denote the weighted den-
sities defined as sums of the spatial convolutions of the average densities and the
corresponding weight functions
nα(r) =
∫
dr′ρi(r
′)wα(r − r
′) (9)
with the weight functions wα(r) given in [10,18–22].
The electrostatic part of the grand potential is approximated by [21,22]:
Ωel[{ρi}] ≈ Ωel
[{
ρref
i
}]
+
∑
i=1,2
∫
drµes.i
({
ρref
i
})
∆ρi(r)
− kT
2
∑
i,j=1,2
∫
drdr′cref
ij (r, r′)∆ρi(r)∆ρj(r
′) +
1
2
∑
i=1,2
zie
∫
drΨ(r)ρi(r), (10)
where ∆ρi(r) = ρi(r) − ρref
i (r) and ρref
i (r) is a suitably chosen position-dependent
reference system density. cref
ij (r, r′) is a part of the reference system two-particle
direct correlation function, which results from electrostatic interactions.
According to Gillespie et al. [21,22] the reference system density is determined
by defining an auxiliary system. This system locally has the same ionic strength as
the system under study and locally satisfies the condition of electroneutrality. For
an 1:1 electrolyte the density of this system equals to ρ̄i(r) = [ρ1(r) + ρ2(r)]/2. The
reference system density is evaluated by averaging the densities ρ̄i(r) over a sphere
of the radius Rf
ρref
i (r) =
∫
dr′ρ̄i(r
′)w(r, r′), (11)
with the weight function
w(r, r′) =
θ (|r − r
′| − Rf )
4πR3
f/3
. (12)
796
Ionic associating fluids at surfaces
The radius Rf is position-dependent and for the 1:1 electrolyte it equals to Rf =
d/2 + 1/2Γ(r), where Γ = (
√
1 + 2dκ − 1)/2d, κ2 = (4π/T ∗)
∑
i z
2
i ρ
ref
i and T ∗ is
the usual reduced temperature, T ∗ = kTεd/e2. The definitions ρ̄i and Rf for an
arbitrary (multicomponent) system are given in [21,22].
The evaluation of the radius Rf requires the knowledge of the reference system
density and vice versa. Therefore, their determination must be carried out using a
self-consistent procedure. Gillespie et al. have proposed an application of an additi-
onal iterational method, which is described in detail in [21,22].
The short-range electrostatic part of the two-particle direct correlation function,
resulting from the MSA approximation
kTc
(el)
ij (r) +
zizje
2
εr
= −zizje
2
2εr
[
(λi + λj) r
λiλj
− r2
2λiλj
− (λi − λj)
2
2λiλj
]
(13)
is the complement of the electrostatic part of grand potential term. In the above
equation λi = d/2 + 1/2Γ is the capacitance length of ion species (obviously, in the
system in question λi = λj).
The last term of the grand potential arises from association. The associative
contribution is approximated by using Wertheim’s treatment [31]. This approach,
which has been widely used [37,38,42] to describe association in simple nonuniform
fluids leads to the expression [41]
Ωas [{ρ(r)}] =
∫
drfas(r) =
∑
i=1,2
∫
dr
[
ρref
i (r) ln α(r) + ρref
i (r)(0.5 − α(r)/2)
]
. (14)
In the above the dissociation constant is
α =
√
1 + 2K
∑
i=1,2 ρref
i − 1
K
∑
i=1,2 ρref
i
. (15)
The density of ions, ρb, is divided into the density of free ions, ρb0 and bounded
pairs, ρbb, which are connected by the mass action law [30]
1 − α
α
= ρbK, (16)
where α = ρb0/ρb, ρb = ρb0+ρbb is the degree of dissociation and K is the dissociation
constant. The dissociation constant depends on density and temperature and at a
given temperature it can be separated into two terms: K = K0(T )Kγ(T, ρb), i.e.
into the density-independent and the density dependent terms. From the Ebeling’s
theory [3,24,29] we obtain the density-independent term
K0(T
∗) = 8πd3
∑
m>2
(T ∗)−2m
(2m)!(2m − 3)
. (17)
The density-dependent term, however, is given by
Kγ(T, ρref
i ) = ghs(d) + gel,MSA(d), (18)
797
J.Reszko-Zygmunt, S.Sokołowski
with ghs(d) being the contact value of the hard-sphere radial distribution functi-
on between unlike ions. This part has been evaluated from the Carnahan-Starling
equation of state (cf. [1]). The electrostatic part of the radial distribution function
between unlike ions, gel, arises from the MSA approach
gel,MSA(d) =
4Γ2{ρref
i }
∑
ρiz2
i
. (19)
In the theory presented above two sets of averaged densities are applied: the
set {nα} – to calculate the hard-sphere contribution to the grand potential and the
densities {ρref
i } – to calculate the electrostatic contribution. The association is the
result of electrostatic interactions. Thus, to be consistent, the contact values of the
radial distribution function have been evaluated for the set of densities {ρref
i }.
The derivative of Ωas is
δΩas
δρi(r)
=
∫
dr′f ′
as
[{
ρref
i (r′)
}] δρref
i (r′)
δρi(r)
. (20)
The expressions for the functional derivatives of Ωhs and Ωel are given in [10,18,
21]. Requiring the grand potential to be at a minimum, δΩ/δρi(r) = 0, we obtain
equilibrium density profiles.
0 0.5 1 1.5 2 2.5 3
z/d
0
0.5
1
1.5
ρ i* (z
)
1
2
ρ
bi
*
=0.04
T
*
=0.15
q
*
=0.00765
a
0.5 1.5 2.5 3.5 4.5
z/d
0
2
4
6
8
10
ρ i* (z
)
b
ρ
bi
*
=0.04623
T
*
=0.59492
q
*
=0.70
Figure 1. (a) A comparison of the counterion (1) and co-ion (2) density profiles at
a charged wall resulting from the present theory (dashed and solid lines, respec-
tively), from the theory of Boda et al. [44] (dotted and dash – dotted lines) and
from Monte Carlo simulations (filled and empty circles) [44]. (b) A comparison of
theoretical predictions with simulation data of Torrie and Valleau [4,5]. All the
parameters are given in the figure.
The knowledge of one-dimensional profiles ρi(z) allow us to calculate the adsorp-
tion isotherms
Ai =
∫ ∞
0
dz [ρi(z) − ρb,i] (21)
798
Ionic associating fluids at surfaces
and the surface charge density
q = −
∑
i=1,2
zie
∫
ρi(z)dz. (22)
Our calculations were proceeded on a grid of the size of 0.02d. To evaluate the
density profiles we have employed Piccard iterational procedure [43], as described
in [21,22]. The input quantity to our calculations was the value of the electrostatic
potential at the wall Ψ∗
0 = Ψ(z = 0)(4πed/ε) and the value of the charge at the
wall was calculated from equation (22). All the calculations were continued until the
maximum difference between two consecutive profiles was smaller than 10−7 percent.
Our calculations are carried out for symmetric 1:1 nonuniform mixtures.
0 2 4 6 8 10
z/d
0
0.02
0.04
0.06
0.08
0.1
ρ*
i(z
)
T*=0.09
ρ*
bi
=0.05
0.03
0.001
0.01
Ψ*
0
=0.1
a
0 2 4 6 8 10
z/d
0
0.02
0.04
0.06
0.08
0.1
ρ∗
i(z
) ρ∗
bi
=0.05
0.03
0.001
0.01
T*=0.3
Ψ*
0
=0.1
b
0 2 4 6 8 10 12
z/d
0
0.02
0.04
0.06
0.08
0.1
0.12
ρ*
i(z
)
ρ*
bi
=0.05
0.03
0.001
0.01
T*=0.09
Ψ*
0
=0.2
c
0 2 4 6 8 10 12
z/d
0
0.02
0.04
0.06
0.08
0.1
ρ*
i(z
) ρ*
bi
=0.05
0.03
0.001
0.01
T*=0.3
Ψ*
0
=0.2
d
Figure 2. Density profiles (part a) obtained for counterions (solid lines) and co-
ions (dashed) for bulk density, ρ∗bi = 0.05 at T ∗ = 0.09 and Ψ∗
0 = 0.1. Parts (b),
(c) and (d) show the same as in part (a), but for different parameters listed in
the figure.
3. Results and Discussion
In order to test the method we first perform some comparisons of the structural
properties of the system with computer simulation data. Figure 1a presents a com-
799
J.Reszko-Zygmunt, S.Sokołowski
parison of theoretical density profiles of ions at ρb = 0.04 and T ∗ = 0.15, in contact
with a charged wall q∗ = qd2/e = 0.00765 with Monte Carlo simulation data [44]
and with previous calculations based on an earlier version of the density functional
theory, described in detail in [44]. We observe that the present theory provides a
better agreement with simulations for this thermodynamic state than the previous
version of the DF theory [44]. Next comparisons are given in figure 1b. Here we
show the results for the density profiles of ions in contact with a single charged
wall at T ∗ = 0.59492. The Monte Carlo data were taken from the works by Torrie
and Valleau [4,5]. In general, the present theory describes the distributions of ions
reasonably well. However, the temperature is quite high in this case and under such
conditions the present approach reduces to that of Gillespie et al. [21,22].
Since the association of ions plays an important role at low temperature only
we concentrate our discussion on the results obtained at the reduced temperatures
lower than 0.5.
0 5 10 15
z/d
0
0.2
0.4
0.6
0.8
1
0.05
0.03
ρ∗
bi
=0.001
0.01
T*=0.09 Ψ*
0
=0.1
α(z)
a
0 5 10 15
z/d
0.885
0.935
0.985
ρ*
bi
=0.001
0.03
0.05
0.01
T*=0.3
Ψ∗
0
=0.1
α(z)
b
Figure 3. Dependence of α(z) for counterions (solid) and co-ions (dashed) on the
distance from wall. The curves in part (b) are for the same bulk densities as the
relevant curves in part (a). All the parameters are given in the figure.
Figure 2 illustrates how the temperature and the value of electrostatic potential
at the wall Ψ∗
0 effect the structure of the solid-fluid interface. Here we have plotted
the density profiles of co-ions and counterions. The effect of the association on the
density profiles of counterions is smaller than on the profiles of co-ions. Figures
2a and 2c present the profiles obtained at a relatively low reduced temperature
T ∗ = 0.09, whereas figures 2b and 2d – at higher temperature, T ∗ = 0.3. An increase
of the temperature changes the structure of the solid-fluid interface. At a higher
temperature we can observe “the smoothing” of the shape of the obtained profiles.
As we can see in figures 2a (and 2c, respectively) low temperature causes a high
concentration of the ions of a given type and causes “a sucking” of the ions of an
opposite sign. A competition between repulsive ion-wall force and a tendency to
form ionic pairs results in the appearance of the local density peak of co-ions at a
distance of z∗ = z/d ≈ 0.8 from the wall. Since the first maximum of the counterions
800
Ionic associating fluids at surfaces
concentration is at z/d = 0.5, the shift of the location of the first local density peaks
for counter- and co-ions suggests that the ionic pairs do not assume orientation with
their axes parallel to the surface, but rather the associates are tilted with respect
to the surface. An increase of the value of the electrostatic potential leads to the
growth of concentrations of counterions at the wall and causes the second distinct
local density peak of counterions. This is observed in figures 2a and 2c.
Figures 3a and 3b present the fraction of unbounded ions, α, for the systems in
figure 2a and figure 2b respectively. We can observe that a decrease of temperature
causes an increase of the association. In both cases, directly at the wall the fraction of
unbounded co-ions is high, whereas the fraction of unbounded counterions is rather
small. This is the result of a high wall-ion repulsion, which causes “the absence” of
co-ions in the nearest vicinity of the surface.
0 0.01 0.02 0.03 0.04 0.05 0.06
ρ∗
bi
−0.065
−0.045
−0.025
−0.005
0.015
0.035
A
id
2
T*=0.5
T*=0.1
T*=0.09
Ψ*
0
=0.1
T*=0.15
a
0 0.01 0.02 0.03 0.04 0.05 0.06
ρ∗
bi
−0.05
0
0.05
0.1
A
id
2
Ψ*
0
=0.2
Ψ∗
0
=0.15
T*=0.09
Ψ∗
0
=0.1
Ψ∗
0
=0.125
Ψ∗
0
=0.05
b
0 0.01 0.02 0.03 0.04 0.05 0.06
ρ∗
bi
−0.064
−0.044
−0.024
−0.004
0.016
A
id
2
T*=0.09
T*=0.1
Ψ*
0
=0.1
c
Figure 4. Adsorption isotherms (a) obtained for Ψ∗
0 = 0.1. Part (b) presents
isotherms obtained at low temperature for different values of Ψ∗
0. Part (c)
shows comparison results evaluated by “switching on” (solid) and “switching
off” (dashed) association, respectively. All the parameters are given in the figure.
To complete the description of the studied system we present examples of the
adsorption isotherm. Figure 4a shows the isotherms for the same value of electro-
801
J.Reszko-Zygmunt, S.Sokołowski
static potential at the wall Ψ∗
0 but obtained at different reduced temperatures. We
can see only small differences for the isotherm obtained at temperature higher than
T ∗ = 0.15. The illustration of the effect of the value of electrostatic potential on
the adsorption is plotted in figure 4b. As we expected, an increase of Ψ∗
0 causes the
increase of adsorption of the system investigated. In figure 4c we compare isotherms
obtained for Ψ∗
0 = 0.1 at low temperatures with those obtained for the system ne-
glecting association. At lower temperature, T ∗ = 0.09 the isotherm obtained with
the association “switched on” and “switched off” differ significantly. The differences
decrease with an increase of the temperature.
In this paper we have applied the density functional theory to the study of
nonuniform associating ionic fluids. The obtained results indicate that at low tem-
peratures the inclusion of the associative term in the grand potential leads to the
structure of solid-fluid interface. This structure differs from the structure evaluated
for the system neglecting association, but the differences become negligibly small at
high temperatures.
Acknowledgement
The EU is kindly acknowledged for partially funding this work as a TOK project,
under contract number 509249.
References
1. Jiang J., Blum L., Bernard O., Prausnitz J.M., Sandler S.I. // J. Chem. Phys., 2002,
vol. 116, p. 7977.
2. Boda D., Chan K.-Y., Henderson D. // J. Chem. Phys., 1998, vol. 109, p. 7362.
3. Boda D., Henderson D., Chan K.-Y., Wasan D. T. // Chem. Phys. Lett., 1999, vol. 308,
p. 473.
4. Torrie G.M., Valleau J.P. // J. Chem. Phys. 1980, vol. 73, p. 5807.
5. Torrie G.M., Valleau J.P. // J. Chem. Phys. 1982, vol. 86, p. 3251.
6. Lozada-Cassou M. Fundamentals of Inhomogeneous Fluids, (Henderson D., ed.), chap-
ter 8. New York, Marcel Dekker, 1992.
7. Lozada-Cassou M., Olivares W., Sulbar’an B. // Physical Review E, 1996, vol. 53,
p. 522.
8. Greberg H., Kjellander R. // Mol. Phys., 1994, vol. 83, p. 789.
9. Henderson D., Boda D., Wasan D.T. // Chem. Phys. Lett., 2000, vol. 325, p. 6553
10. Rosenfeld Y. // Phys Rev. Lett., 1989, vol. 63, p. 980; J. Chem. Phys. 1993, vol. 98,
p. 8126.
11. Boda D., Henderson D., Rowley R., Soko lowski S. // J. Chem. Phys., 1999, vol. 111,
p. 9382
12. Boda D., Henderson D., Patrykiejew A., Soko lowski S. // J. Chem. Phys., 2000,
vol. 113, p. 802
13. Henderson D., Bryk P., Soko lowski S., Wasan D. T. // Phys. Rev. E, 2000, vol. 61,
p. 3896
802
Ionic associating fluids at surfaces
14. Boda D., Henderson D., Patrykiejew A., Soko lowski S. // J. Coll. Interf. Sci., 2001,
vol. 239 p. 432
15. Mier-y-Teran L., Suh S.H., White H.S., Davis H.T. // J. Chem. Phys., 1990, vol. 92,
p. 5087.
16. Tang Z., Mier-y-Teran L., Davis H.T., Scriven L.E., White H.S. // Molec. Phys., 1990,
vol. 71, p. 369.
17. Evans R. Fundamentals of Inhomogeneous Fluids, (Henderson D., ed.), chapter 4. New
York, Marcel Dekker, 1992.
18. Rosenfeld Y., Schmidt M., L’owen H., Tarazona P. // Phys. Rev. E., 1997, vol. 55,
p. 4245.
19. Tarazona P. // Phys. Rev. Lett., 2000, vol. 84, p. 694.
20. Cuesta J.A., Mart́ınez-Ratón Y., Tarazona P. // J. Phys. Condens. Matter, 2002,
vol. 14, 11965.
21. Gillespie D., Nonner W., Eisenberg R.S. // J. Phys. Condens. Matter, 2002, vol. 14,
p. 12129.
22. Gillespie D., Nonner W., Eisenberg R.S. // Phys. Rev. E, 2003, vol. 68, p. 031503.
23. Bjerrum N. K. // K. Dan. Vidensk. Selsk. Mat. Fys. Medd., 1926, vol. 7, p. 1.
24. Ebeling W., Grigo M. // Ann. Phys. (Leipzig), 1980, vol. 37, p. 21.
25. Zhou Y., Yeh S., Stell G. // J. Chem. Phys., 1995, vol. 102, p. 5785.
26. Stell G., Zhou Y.Q. // J. Chem. Phys., 1989, vol. 91, p. 3618.
27. Holovko M.F., Kalyuzhnyi Yu.V. // Mol. Phys., 1991, vol. 73, p. 1145.
28. Holovko M.F., Vakarin E.V. // Mol. Phys., 1996, vol. 87, p. 123.
29. Blum L., Bernard G. // J. Stat. Phys. 1995, vol. 79, p. 569.
30. Barthel J., Kienke H., Kunz W. Physical Chemistry of Electrolyte Solutions – Modern
Aspects. Darmstadt, Steinkopff, New York, Springer, 1998.
31. Wertheim M.S. // J. Stat. Phys., 1984, vol. 35 19, p. 35.
32. Yu Y.-X., Wu J. // J. Chem. Phys., 2002, vol. 116, p. 7094.
33. Holovko M.F., Kalyuzhnyi Yu.V. // Mol. Phys., 1991, vol. 73, p. 1145.
34. Holovko M.F., Vakarin E.V. // Mol. Phys., 1996, vol. 87, p. 123.
35. Holovko M.F., Kapko V., Henderson D., Boda D. // Chem. Phys. Letters, 2001,
vol. 341, p. 363.
36. Kalyuzhnyi Yu. V., Vlachy V., Holovko M.F. // J. Stat. Phys., 2000, vol. 100, p. 243.
37. Bucior K., Patrykiejew A., Soko lowski S., Pizio O. // Mol. Phys., 2003, vol. 101,
p. 1477.
38. Patrykiejew A., Pizio O., Soko lowski S., Pusztai P. // Mol Phys., 2003, vol. 101,
p. 2219.
39. Henderson D., Bryk P., Soko lowski S., Wasan D.T. // Physical Rev. E, 2000, vol. 4,
p. 3896.
40. Roth R., Evans R., Lang A., Kahl G. // J. Phys. Condens. Matter, 2002, vol. 14,
p. 12063.
41. Yu Y.-X., Wu J. // J. Chem. Phys., 2002, vol. 117, p. 10156.
42. Borówko M., Pizio O., Soko lowski S. Computational Methods in Surface and Colloid
Science, (Borówko M., ed.). New York, Marcel Dekker, 2000.
43. Piccard S. // Sur Ensembles Distan, 1939, vol. 13, p. 1.
44. Boda D., Henderson D., Mier-y-Teran L., Soko lowski S. // J. Phys.: Condensed Mat-
ter, 2002, vol. 14, p. 11945.
803
J.Reszko-Zygmunt, S.Sokołowski
Застосування методу функціоналу густини до
неоднорідних іонних флюїдів: ефект асоціації
Я.Режко-Зигмунт, С.Соколовський
Хімічний факультет, університет М. Кюрі-Склодовської, Люблін,
Польща
Отримано 18 серпня 2004 р., в остаточному вигляді –
1 жовтня 2004 р.
У цій роботі ми обговорюємо метод функціоналу густини для
неоднорідних іонних флюїдів, який враховує існування іонних пар.
Підхід базується як на теорії фундаментальної міри твердих сфер,
теорії Гіллеспі та ін., яка веде до більш точного опису електрос-
татичної частини великого термодинамічного потенціалу, так і на
асоціативній теорії Вертхейма. Результати модельних обчислень
вказують на те, що включення асоціативного члена у великий
термодинамічний потенціал веде до структури подвійного шару, яка
відрізняється від структури, коли нехтується асоціацією. Ця різниця
є важливою тільки при низьких температурах.
Ключові слова: адсорбція, асоціативні флюїди, метод функціоналу
густини
PACS: 68.43-h, 68.43.De, 71.15.Mb
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