Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles
The density, charge, polarization and potential profiles of a simple model of an associative electrolyte are studied in an associative mean spherical approximation (AMSA). The limits of the full association and complete disassociation are considered.
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| Дата: | 2001 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут фізики конденсованих систем НАН України
2001
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles / V.I. Kapko, M.F. Holovko // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 201-208. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1204232017-06-13T03:04:47Z Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles Kapko, V.I. Holovko, M.F. The density, charge, polarization and potential profiles of a simple model of an associative electrolyte are studied in an associative mean spherical approximation (AMSA). The limits of the full association and complete disassociation are considered. Профілі густини, заряду, поляризації і потенціалу простої моделі асоціативного електроліту вивчаються в асоціативному середньо-сферичному наближенні (АССН). Розглядаються границі повної асоціації і дисоціації. 2001 Article Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles / V.I. Kapko, M.F. Holovko // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 201-208. — Бібліогр.: 9 назв. — англ. 1607-324X PACS: 05.20.-y DOI:10.5488/CMP.4.2.201 http://dspace.nbuv.gov.ua/handle/123456789/120423 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The density, charge, polarization and potential profiles of a simple model
of an associative electrolyte are studied in an associative mean spherical
approximation (AMSA). The limits of the full association and complete disassociation
are considered. |
| format |
Article |
| author |
Kapko, V.I. Holovko, M.F. |
| spellingShingle |
Kapko, V.I. Holovko, M.F. Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles Condensed Matter Physics |
| author_facet |
Kapko, V.I. Holovko, M.F. |
| author_sort |
Kapko, V.I. |
| title |
Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles |
| title_short |
Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles |
| title_full |
Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles |
| title_fullStr |
Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles |
| title_full_unstemmed |
Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles |
| title_sort |
associative electrolyte solution near the charge hard wall. density, charge, polarization and potential profiles |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120423 |
| citation_txt |
Associative electrolyte solution near the charge hard wall. Density, charge, polarization and potential profiles / V.I. Kapko, M.F. Holovko // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 201-208. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kapkovi associativeelectrolytesolutionnearthechargehardwalldensitychargepolarizationandpotentialprofiles AT holovkomf associativeelectrolytesolutionnearthechargehardwalldensitychargepolarizationandpotentialprofiles |
| first_indexed |
2025-07-08T17:51:30Z |
| last_indexed |
2025-07-08T17:51:30Z |
| _version_ |
1837102103196073984 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 201–208
Associative electrolyte solution near
the charge hard wall. Density, charge,
polarization and potential profiles
V.I.Kapko, M.F.Holovko
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received August 26, 2000
The density, charge, polarization and potential profiles of a simple model
of an associative electrolyte are studied in an associative mean spherical
approximation (AMSA). The limits of the full association and complete dis-
association are considered.
Key words: electrolyte-electrode interface, associative mean spherical
approximation
PACS: 05.20.-y
A set of important results for the electrode-electrolyte interface description can
be obtained on the basis of models and methods, which have been developed and
tested previously for bulk systems with electrostatic interactions and then general-
ized for inhomogeneous systems. The Henderson-Abraham-Barker (HAB) approach
[1] enables us to describe the interface properties through the bulk properties of
fluids. The simplest ion-dipole model for an electrolyte in the mean spherical ap-
proximation (MSA) [2] has been used to investigate the electrolyte properties near
the charged wall by Blum and Henderson [3]. Recently the associative mean spherical
approximation (AMSA) was solved [4], within the frameworks of which the analyti-
cal solution of the ion-dipole mixture against the charge hard wall has been obtained
[5]. The purpose of this note is to discuss the influence of interionic association on
the density, charge, polarization and potential profiles.
Our system consists of a mixture of charged hard spheres and dipolar hard
spheres near the charged hard wall. Let us call ρ+,ρ−,ρd (ρ+ = ρ− = ρi/2) the
number density of the cations, anions and solvent, respectively. The diameters of
all particles are equal to σ. The modules of the charges of the cations and anions
are also equal (Z+e = −Z−e=q). The oppositely charged ions can form the neutral
pairs due to the attractive sites placed on the surface of each ion.
The associative version of the wall-particle Ornstein-Zernike (OZ) equation can
c© V.I.Kapko, M.F.Holovko 201
V.I.Kapko, M.F.Holovko
be written as
hβ
y (z,Ω2) = cβy (z,Ω2) +
∑
x
∑
γδ
∫
d3hγ
x(z
′,Ω3)ρ
γδ
x Cδβ
xy (32) (1)
together with MSA closure relations
hβ
y (z,Ω2) = −δβ0, r12 < σ/2
cβy (z,Ω2) = −δβ0
uel
y (z)
kT
, r12 > σ/2, (2)
where hβ
y (z,Ω2), c
β
y (z,Ω2) are the vectors of the pair and direct wall-particle correla-
tion functions, z is the distance of the particle from the wall surface, Ω2 orientation
of dipolar moment. Subscripts x and y point at the sort of the particle and super-
scripts α, β, γ and δ at the degree of bonding (0 for bonded and 1 for unbonded
particle). Cαβ
xy (12) is the matrix of the direct correlation functions for the bulk phase.
d3 means integration over the positions ~r3 and possible orientations of the dipole.
The wall particle interaction is for the ion
uel
i (z) = −eE z (3)
and for the dipole
uel
s (z) = −(~µ ~E), (4)
where ~E is the bare (unscreened) electric field, which is connected with the surface
charge density on the wall qs by
E = 4π qs . (5)
Matrix of density is defined as [6,7] ρ00x = ρx, ρ01x = ρ10x = ρ0x, ρ11x = 0, where ρx
is the total density of particles of sort x and ρ0
x is density of unbonded particles of
sort x, which are connected by self-consistent relation [8]:
ρi = ρ0i + 2π(ρ0i )
2σ3Bg00+−
(σ+). (6)
The analytical solution can be represented in the form of the two integral equa-
tions:
g(S)βy (z)−
∑
x
∑
γδ
ρ∗x
z
∫
0
g(S)γx (z − r)Rγδ
x Q(S)δβ
xy (r)dr = Kβ
y (0), (7)
√
ρ∗yh
(D)β
y (z)−
∑
x
∑
γδ
√
ρ∗x
z
∫
0
h(D)γ
x (z − r)Rγδ
x Q(D)δβ
xy (r)dr = −F β
y (z). (8)
The expressions for Q̂(S)(r), Q̂(D)(r), Kβ
y (0) and F β
y (z) can be found in [5]. The
density profiles are connected with the functions g
(S)β
y (z) and h
(D)β
y (z) by
g±(z) =
1
∑
β=0
(
g
(S)β
i (z)± h
(D)β
i (z)
)
, (9)
gd(z,Θ) = g
(S)0
d (z) +
√
3h
(D)0
d (z) cos(Θ) (10)
202
Associative electrolyte solution near the charge hard wall
with Θ the angle between the dipolar moment and the normal to the wall. It must
be pointed out that the factor ρ0i /ρi is included in g
(S)1
i (z) and h
(D)1
i (z) in contrast to
Wertheim’s definition [6,7]. This avoids the uncertainty in the case of full association
(ρ0i = 0).
We solve numerically the integral equations (7) and (8) by the well-known Per-
ram’s method [9].
The profiles in figures 1–4, 7–8 have been calculated by parameters
η = π/6× (ρiσ
3 + ρdσ
3) = 0.3178,
ci = ρi/(ρi + ρd) = 0.0115,
q∗ 2 =
Z2
i e
2
σkT
= 40,
µ∗ 2 =
µ2
σ3kT
= 2.5,
E∗ =
√
σ3
kT
E = 1.5,
which correspond to the Debye screening region. The ionic densities in the curves
plotted in the figures 4 and 5 are ci = 0.11 and ci = 0.5 which correspond the short-
range ionic ordering and the short density ordering near the wall, respectively. We
consider the three cases of the ionic association in all figures: (a) corresponds to the
case of the complete dissociation (the ionic monomer fraction α = ρ0
i /ρi = 1); (b)
corresponds to α = 0.5; (c) corresponds to the case of the fully ionic dimerization
α = 0.
The charge profiles at the low ionic concentration are plotted in figure 4. The
charge profile at α = 0.5 is close to the one for the free ions. The curve for the fully
dimerized ions is less than the charge profiles for (a) and (b) cases. At z = 1.5σ the
curve (c) has a sharp maximum which indicates an orthogonal configuration of the
dimers.
Figure 5 corresponds to the middle ionic concentration c i = 0.11. For the fully
dimerized ions (c) the charge profile sign change is observed at the first maximum
region. At z < 0.8σ all the curves coincide. For bigger distances (z > 2σ) – after the
first minimum the charge profiles decay fast.
In figure 6 the charge profiles for the comparatively high concentration c i = 0.5
are shown. Under such conditions the dimerization process exerts no influence on
the charge profiles. At z > 1.5σ we observe some lag between (c)-(b), (b)-(a), (c)-(a)
profiles.
The polarization profiles (figure 7) possess the oscillative behaviour which cor-
responds to a solvent discrete structure. At high distances (z > 2.5σ) the higher
degree of the ionic association leads to the increase of a solvent polarization.
The potential profiles are plotted in figure 8. They also show the oscillative
behaviour and decay at high z. The association leads to potential profiles increasing
especially for low α.
203
V.I.Kapko, M.F.Holovko
-1
0
1
2
0.5 1.5 2.5 3.5
g+(z)
z/σ
a
b
c
Figure 1. The density profiles of cations at ci = 0.0115.
1
2
3
0.5 1.5 2.5 3.5
g−(z)
z/σ
a
b
c
Figure 2. The density profiles of anions at ci = 0.0115.
204
Associative electrolyte solution near the charge hard wall
0
1
2
3
0.5 1.5 2.5 3.5
g
(S)
d (z)
z/σ
a,b,c
Figure 3. The S-part of solvent density profiles at ci = 0.0115.
-2
-1.5
-1
-0.5
0
0.5 1 1.5 2 2.5 3 3.5
q∗(z)
z/σ
a
b
c
Figure 4. The charge profiles at ci = 0.0115.
205
V.I.Kapko, M.F.Holovko
-0.3
-0.2
-0.1
0
0.1
0.5 1 1.5 2 2.5 3 3.5
q∗(z)
z/σ
a
b
c
Figure 5. The charge profiles at ci = 0.11.
-0.3
-0.2
-0.1
0
0.1
0.5 1 1.5 2 2.5 3 3.5
q∗(z)
z/σ
a
b
c
✯
Figure 6. The charge profiles at ci = 0.5.
206
Associative electrolyte solution near the charge hard wall
0
0.05
0.1
0.15
0.5 1.5 2.5 3.5 4.5 5.5
P ∗(z)
z/σ
a
b
c
Figure 7. The polarization profiles at ci = 0.0115.
-0.2
0
0.2
0.4
0.6
0.5 1.5 2.5 3.5
Ψ∗(z)
z/σ
a
b
c
Figure 8. The potential profiles at ci = 0.0115.
207
V.I.Kapko, M.F.Holovko
References
1. Henderson D., Abraham F.F., Barker J.A. // Molec. Phys., 1976, vol. 31, p. 129.
2. Blum L. // J. Chem. Phys., 1974, vol. 61, p. 2129.
3. Blum L., Henderson D. // J. Chem. Phys., 1981, vol. 74, p. 1902.
4. Holovko M.F., Kapko V.I. // Condens. Matter Phys., 1998, vol. 1, p. 239.
5. Holovko M.F., Kapko V.I. (in preparation).
6. Wertheim M.S. // J. Stat. Phys., 1984, vol. 35, p. 19.
7. Wertheim M.S. // J. Stat. Phys., 1984, vol. 35, p. 35.
8. Holovko M.F., Kalyuzhnyi Yu.V. // Mol. Phys., 1991, vol. 73, p. 1145.
9. Perram J.W. // Mol. Phys., 1975, vol. 30, p. 1505.
Асоціативний електроліт біля зарядженої твердої
стінки. Профілі густини, заряду, поляризації і
потенціалу
В.І.Капко, М.Ф.Головко
Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 26 серпня 2000 р.
Профілі густини, заряду, поляризації і потенціалу простої моделі асо-
ціативного електроліту вивчаються в асоціативному середньо-сфе-
ричному наближенні (АССН). Розглядаються границі повної асоціації
і дисоціації.
Ключові слова: інтерфейс електроліт-електрод, асоціативне
середньо-сферичне наближення
PACS: 05.20.-y
208
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