Ion size effects in a primitive level model of the diffuse double layer
An analytical expression is developed for the potential drop across the diffuse layer φ^d in terms of a cubic polynomial in the corresponding estimate in the Gouy-Chapman approximation ϕ^dGC . The coefficients of this polynomial are defined in terms of the hard sphere volume fraction η and the M...
Збережено в:
| Дата: | 2004 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2004
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/119016 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Ion size effects in a primitive level model of the diffuse double layer / W.R. Fawcett, T.G. Smagala// Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 709–718. — Бібліогр.: 7 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-119016 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1190162017-06-04T03:03:43Z Ion size effects in a primitive level model of the diffuse double layer Fawcett, W.R. Smagala, T.G. An analytical expression is developed for the potential drop across the diffuse layer φ^d in terms of a cubic polynomial in the corresponding estimate in the Gouy-Chapman approximation ϕ^dGC . The coefficients of this polynomial are defined in terms of the hard sphere volume fraction η and the MSA dimensionless reciprocal distance parameter Γ . The resulting expression is shown to describe the Monte-Carlo estimates of φ^d obtained in a primitive level simulation of diffuse layer properties Отримано аналітичний вираз для зниження потенціалу через дифузійний шар φ^d у термінах кубічного поліному в рамках наближення Гуї-Чепмена ϕ^d GC. Коефіцієнти поліному є визначені в термінах об’ємної густини твердих сфер η і безрозмірного параметра оберненої відстані Γ (середньосферичне наближення). Показано, що остаточний вираз описує оцінки для φ^d у примітивній моделі дифузійного шару, отримані методом Монте-Карло. 2004 Article Ion size effects in a primitive level model of the diffuse double layer / W.R. Fawcett, T.G. Smagala// Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 709–718. — Бібліогр.: 7 назв. — англ. 1607-324X DOI:10.5488/CMP.7.4.709 PACS: 05.20.Jj, 05.10.Zn http://dspace.nbuv.gov.ua/handle/123456789/119016 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
An analytical expression is developed for the potential drop across the diffuse
layer φ^d in terms of a cubic polynomial in the corresponding estimate
in the Gouy-Chapman approximation ϕ^dGC . The coefficients of this polynomial
are defined in terms of the hard sphere volume fraction η and the
MSA dimensionless reciprocal distance parameter Γ . The resulting expression
is shown to describe the Monte-Carlo estimates of φ^d obtained
in a primitive level simulation of diffuse layer properties |
| format |
Article |
| author |
Fawcett, W.R. Smagala, T.G. |
| spellingShingle |
Fawcett, W.R. Smagala, T.G. Ion size effects in a primitive level model of the diffuse double layer Condensed Matter Physics |
| author_facet |
Fawcett, W.R. Smagala, T.G. |
| author_sort |
Fawcett, W.R. |
| title |
Ion size effects in a primitive level model of the diffuse double layer |
| title_short |
Ion size effects in a primitive level model of the diffuse double layer |
| title_full |
Ion size effects in a primitive level model of the diffuse double layer |
| title_fullStr |
Ion size effects in a primitive level model of the diffuse double layer |
| title_full_unstemmed |
Ion size effects in a primitive level model of the diffuse double layer |
| title_sort |
ion size effects in a primitive level model of the diffuse double layer |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119016 |
| citation_txt |
Ion size effects in a primitive level model of the diffuse double layer / W.R. Fawcett, T.G. Smagala// Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 709–718. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT fawcettwr ionsizeeffectsinaprimitivelevelmodelofthediffusedoublelayer AT smagalatg ionsizeeffectsinaprimitivelevelmodelofthediffusedoublelayer |
| first_indexed |
2025-07-08T15:05:41Z |
| last_indexed |
2025-07-08T15:05:41Z |
| _version_ |
1837091671685201920 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 4(40), pp. 709–718
Ion size effects in a primitive level
model of the diffuse double layer
W.R.Fawcett, T.G.Smagala
Department of Chemistry University of California Davis, CA 95616
Received August 20, 2004, in final form November 15, 2004
An analytical expression is developed for the potential drop across the dif-
fuse layer φd in terms of a cubic polynomial in the corresponding estimate
in the Gouy-Chapman approximation ϕd
GC
. The coefficients of this poly-
nomial are defined in terms of the hard sphere volume fraction η and the
MSA dimensionless reciprocal distance parameter Γ . The resulting ex-
pression is shown to describe the Monte-Carlo estimates of φd obtained
in a primitive level simulation of diffuse layer properties.
Key words: double layer theory, diffuse double layer, ion size effects
PACS: 05.20.Jj, 05.10.Zn
1. Introduction
We have made considerable effort in recent years to develop an analytical model
for the diffuse double layer which takes into consideration the effects of finite ion size
[1–3]. Our approach is based on the equations from the hypernetted chain approxi-
mation (HNCA) as a solution of the Ornstein-Zernike integral equation for a charged
wall in an electrolyte solution [4]. The HNCA keeps the non-linear character of the
double layer problem but it does not have an analytical solution. Thus, the estimates
of the wall-ion correlation functions must be obtained by iteration. Henderson and
Blum [5] described an approximate way of solving the HNC equations in which the
first estimates of the wall-ion correlation functions are assumed to be equal to those
given by the Gouy-Chapman (GC) theory. As is well known, this theory ignores the
effects of ion size. The integrals needed to evaluate the same correlation functions
in the HNCA are then evaluated using the GC estimates. These results are then
assumed to be sufficiently good estimates of the wall-ion correlation functions with
consideration of ion size.
The above approach which is referred to as the Henderson-Blum (HB) approach
[5] is clearly approximate but a good beginning in a more complete solution of the
problem. In order to assess the results of the HB approach one makes use of Monte
Carlo (MC) data also obtained for a primitive level system. In the present problem,
c© W.R.Fawcett, T.G.Smagala 709
W.R.Fawcett, T.G.Smagala
“primitive” refers to the fact that the molecular nature of the solvent is ignored and
it is treated as a dielectric continuum with the permittivity of the pure solvent. In
addition the dielectric properties of the wall are assumed to be the same as those
of the solvent. This means that the formation of images in the wall, which occurs
when it is a medium with a different permittivity such as a metal, are ignored. MC
data obtained at the primitive level have been published recently for 1–1 electrolytes
assuming realistic sizes for the component ions [6,7].
In the present paper our previous work[1–3] is extended on the basis of the equa-
tions of the HNCA, and the appropriate MC data are used to obtain an analytical
expression for the potential drop across the diffuse layer for a 1–1 electrolyte.
2. Theory
In the GC theory for the diffuse double layer, the Poisson-Boltzmann equation
is solved assuming a dielectric continuum and ions which are point charges with
no volume. The potential drop across the diffuse layer ϕd is calculated from the
following equation in the case of a 1–1 electrolyte:
Θ2
GCce
[
exp(−ϕd) + exp(ϕd) − 2
]
= σ2
m , (1)
where ϕd is the dimensionless value of the potential drop, σm, the charge density
on the wall, ΘGC, the GC constant, and ce, the electrolyte concentration. The GC
constant is given by
ΘGC = (2000RTεsε0)
1/2 , (2)
where εs is the relative permittivity of the pure solvent, ε0, the permittivity of
free space, and the electrolyte concentration is expressed in M. The dimensionless
potential is related to the potential in volts by the equation
ϕ =
Fφ
RT
. (3)
The value of ϕd is now used to calculate the correlation functions for the ions
with the wall. At the outer Helmholz plane (oHp), the correlation function for the
cation is given by
gw+ = exp(−ϕd) (4)
and that for the anion,
gw−
= exp(ϕd). (5)
It follows that the wall-sum correlation function is
gws =
exp(ϕd) + exp(−ϕd)
2
= cosh ϕd. (6)
Combining equations (1) and (6), it follows that
gws = cosh ϕd =
σ2
m
2Θ2
GC
ce
+ 1 =
E2
2
+ 1. (7)
710
Ion size effects in a primitive level model of the diffuse double layer
Here,
E =
σm
ΘGCc
1/2
e
(8)
is the dimensionless electrical field at the oHp.
Now, these equations are written in the HNCA. The wall-cation correlation func-
tion at the oHp is
gw+ = Ξ0 exp(−Φ0) = Ξ0 exp(−ϕd − H0) (9)
and that for the anion,
gw−
= Ξ0 exp(Φ0) = Ξ0 exp(ϕd + H0). (10)
Ξ0 is a function which depends on the volume fraction η and is calculated as an
integral of the wall-sum correlation function. The ionic volume fraction for a 1–1
electrolyte with a concentration ce in M is given by
η =
2000NLπceσ
3
6
. (11)
NL is the Avogadro constant, and σ, the ion diameter in meters. On the other hand,
H0 is a function which depends on the reciprocal distance parameter Γ defined in
the mean spherical approximation (MSA). The dimensionless expression for Γ is
Γ =
(1 + 2σκ)1/2
2
−
1
2
, (12)
where κ is the Debye-Hückel reciprocal distance given by
κ =
(
2000F 2ce
ε0εsRT
)1/2
. (13)
Here εs is the relative permittivity of the solvent at 25˚C (78.46 for water) and ε0,
the permittivity of free space. The function H0 is calculated as an integral of the
wall-difference correlation function. Exact expressions for Ξ0 and H0 were given in
earlier papers[1,2]. The wall-sum correlation function in the HNCA is
gws = Ξ0 cosh
(
ϕd + H0
)
=
E2
2
+ Ξ0. (14)
When the ionic diameter goes to zero, Ξ0 goes to unity and H0, to zero. Then the
HNC expression for gws reduces to that in the GC theory (equation (7)).
Explicit expressions for ϕd are now derived. In the GC theory,
ϕd
GC = cosh−1
(
E2
2
+ 1
)
(15)
and in the HNCA,
ϕd = cosh−1
(
E2
2Ξ0
+ 1
)
− H0 , (16)
711
W.R.Fawcett, T.G.Smagala
where the value obtained in the GC theory is now designated as ϕd
GC. The strategy
used here is to expand the hyperbolic functions as power series in E.
In order to proceed further, the dependence of the functions Ξ0 and H0 on E
must be established. These functions were estimated following the method suggested
by Henderson and Blum [5]. Accordingly, the integrals defining Ξ0 and H0 were
calculated using the wall-ion correlation functions given by the GC theory, but
limiting the field E to values in the range −3 6 E 6 3. The results for Ξ0 could be
fit to the equation
Ξ0 = a0sech(a1E), (17)
where
a0 =
(1 + η + η2 − η3)
(1 − η)3
(18)
and a1 is also a function of η. In the case of H0, the dependence on E is given by
H0 = b1E + b3E
3 , (19)
where
b1 = Γ2 (20)
and b3 is also a function of Γ. More details about these functions are given later in
this paper.
3. Results and discussion
MC simulations of the diffuse double layer were carried out for the restricted
primitive model for five electrolyte concentrations and three ion sizes. The ion di-
ameters chosen (200, 300 and 400 pm) cover the range found for simple monoatomic
ions. The electrolyte concentrations, namely, 0.1, 0.2, 0.5, 1.0, and 2.0 M cover the
range where significant departure from the GC theory is expected. Finally, data
were obtained for charge densities between 5 and 40 µC cm−2 at increments of
5 µC cm−2. Details about the procedure followed for the MC calculations are given
in earlier papers [6,7].
MC data obtained for an electrolyte concentration of 0.1 M are shown in figure 1.
For smaller ion diameters (200 and 300 pm) the value of ϕd
MC is always less than the
GC estimate, ϕd
GC. However, for a diameter of 400 pm, the MC data fall below the
GC estimates for smaller electrode charge densities but then begin to increase with
respect to ϕd
GC, eventually becoming greater than ϕd
GC. In all cases, the MC data
could be fit to the equation
ϕd
MC = d1ϕ
d
GC + d3(ϕ
d
GC)3. (21)
Results at a concentration of 0.5 M are shown in figure 2. The MC data are qualita-
tively similar to those at 0.1 M except that the values of ϕd
MC and ϕd
GC are quanti-
tatively smaller for a given value of electrode charge density σm. Finally, results at
2.0 M are shown in figure 3. In this case the value of ϕd
MC is always less than that
712
Ion size effects in a primitive level model of the diffuse double layer
Figure 1. Plots of the potential drop across the diffuse layer ϕd according to
the MC simulation (filled symbols) and the present model (open symbols) for an
electrolyte concentration of 0.1 M and for three ion sizes: 200 pm (O, H), 300 pm
(◦, •) and 400 pm (M, N). The results at 300 and 400 pm are shifted vertically by
2 and 4 units, respectively, for the sake of clarity.
Figure 2. As in figure 1 but for an electrolyte concentration of 0.5 M.
713
W.R.Fawcett, T.G.Smagala
of ϕd
GC. However, the curvature of the plot of ϕd
MC against ϕd
GC remains positive for
an ionic diameter of 400 pm.
In order to relate the MC results to the calculations based on the integral equa-
tion approach, equations (15) and (16) must be expanded as an infinite series in the
dimensionless field E. This leads to the result
ϕd = ϕd
GC + c1E + c3E
3 + c5E
5 + · · · . (22)
It is also possible to express ϕd as an infinite series in ϕd
GC which gives the equation
ϕd = d1ϕ
d
GC + d3(ϕ
d
GC)3 + d5(ϕ
d
GC)5 + · · · . (23)
The first two coefficients in this series are
d1 =
1
a
1/2
0
− b1 (24)
and
d3 =
1
24
[
1
a
1/2
0
+
6a2
1
a
1/2
0
−
1
a
3/2
0
− b1 − 24b3
]
. (25)
Figure 3. As in figure 1 but for an electrolyte concentration of 2.0 M.
If the present approach is to agree with the MC data then only the first two
coefficients in equation (23) are significant, and equation (23) becomes the same
as equation (21). Furthermore, we assume that the results of the HB approach give
sufficiently good estimates of a0 and b1 and accept the values given by equations (18)
and (20). This assumption is easily checked by comparing the values of d1 determined
714
Ion size effects in a primitive level model of the diffuse double layer
Figure 4. Plots of d1 obtained by fitting equation (18) to MC data obtained at
5 electrolyte concentrations and three ion sizes against a
−1/2
0 − Γ2; results are
shown for 200 pm (N), 300 pm (•), and 400 pm (H).
in a least squares fit of equation (21) to the MC data with values estimated on
the basis of equation (24). This comparison is shown in figure 4 where the value
of d1 estimated from the MC data is plotted against a
−1/2
0 − Γ2 for five different
concentrations, and for three values of the ionic diameter. There is a tendency for
the estimates of d1 obtained from MC results to be too high but the maximum
deviation between the value obtained from the MC data and equation (24) is never
greater than 10 percent. Thus, it is reasonable to assume that d1 is given by equation
(24) to a very good approximation.
The MC data were then refitted to equation (21) in a one parameter least squares
fit in which the values of d1were forced to be those given by equation (24). The
resulting values of d3 vary with both ce and σ but no clear trend is apparent. This is
due to the fact that d3 is a complex function of the four parameters, a0, a1, c1, and
c3. Two of these parameters, a0 and c1, determine the linear coefficient d1and can
be accepted as correct. However, the contributions of a1 and c3 must be reevaluated
if the curvature coefficient d3 is to be correctly estimated. Therefore, it is reasonable
to evaluate a new curvature coefficient e3 defined as
e3 = d3 −
1
24 a
1/2
0
+
1
24 a
3/2
0
+
c1
24
=
a2
1
4a1/2
− c3 . (26)
Values of e3 are plotted against electrolyte concentration ce in figure 5. Clearly, e3
increases with increase in electrolyte concentration but the change for σ = 200 pm is
very small. In addition e3 becomes positive for higher values of ce and σ. This means
that the contribution from the term in a1 must dominate under these conditions.
715
W.R.Fawcett, T.G.Smagala
Figure 5. Plots of e3 obtained from the MC results using equation (23) against
the electrolyte concentration; results are shown for 200 pm (N), 300 pm (•), and
400 pm (H).
In order to optimize the values of the curvature parameter the MC data for e3
were fit to the equation
e3 = α1f1(η) + α2f2(Γ), (27)
where f1(η) is a function of the volume fraction η, f2(Γ), a function of the reciprocal
distance parameter Γ, and α1 and α2 are numerical constants. Guided by the HB
results various simple functions involving both integer and fractional powers of η
and Γ were tested. The best fit to the MC results was obtained for the relationship
e3 =
η1/2
4a
1/2
0
−
Γ
8(1 + Γ)
. (28)
Thus, we find that
a1 = η1/4 (29)
and
c3 =
Γ
8(1 + Γ)
. (30)
These results are very different from those obtained from the analysis of the functions
Ξ0 and H0estimated using the HB approach. This means that the HB approach can
only be used in the limit of extremely small fields. Finally, the equation used to
estimate ϕd is
ϕd = d1ϕ
d
GC + d3(ϕ
d
GC)3, (31)
where the parameters d1 and d3 are given by
d1 =
1
a
1/2
0
− Γ2 (32)
716
Ion size effects in a primitive level model of the diffuse double layer
and
d3 =
1
24a
1/2
0
+
η1/2
4a
1/2
0
−
1
24a
3/2
0
−
Γ2
24
−
Γ
8(1 + Γ)
. (33)
The results obtained using equations (31)–(33) are also shown in figures 1 to 3.
It is clear that the revised results fit the MC data very well. Moreover, the revised
estimates of ϕd are easily estimated from the GC value ϕd
GC once the volume fraction
η and the reciprocal distance parameter Γ have been calculated. Thus, the present
approach gives simple analytical equations which can be used to estimate the diffuse
layer potential with consideration of ion size effects. In the case of 1–1 electrolytes,
the correction to the GC estimates of ϕd are important at high concentrations and
electrode charge densities. The failure of the GC theory is much more important for
2–1, 1–2, and 2–2 electrolytes. Application of the present approach to these systems
will be considered in a future paper.
Acknowledgements
The authors are grateful to Dr. Dezső Boda for making available his MC program.
This research was supported by a grant from the National Science Foundation,
Washington (CHE 0133758).
References
1. Fawcett W.R., Henderson D.J. // J. Phys. Chem. B, 2000, vol. 104, p. 6837.
2. Fawcett W.R. // Russ. J. Electrochem., 2002, vol. 38, p. 2.
3. Andreu R., Fawcett W.R. // J. Electroanal. Chem., 2003, vol. 552, p. 105.
4. Henderson D.J., Abraham F.F., Barker J.A. // Mol. Phys., 1976, vol. 31, p. 1291.
5. Henderson D.J., Blum L. // J. Electroanal. Chem., 1980, vol. 111, p. 217.
6. Boda D., Fawcett W.R., Henderson D.J., Sokolowski S. // J. Chem. Phys., 2002,
vol. 116, p. 7170.
7. Boda D., Henderson D.J., Plaschko P., Fawcett W.R. // Molec. Simulation, 2004,
vol. 30, p. 137.
717
W.R.Fawcett, T.G.Smagala
Ефекти розміру іона в примітивній моделі
дифузійного подвійного шару
В.Р.Фоцетт, Т.Г.Смагала
Відділення хімії, університет Каліфорнії, Давіс, США
Отримано 20 серпня 2004 р., в остаточному вигляді –
15 листопада 2004 р.
Отримано аналітичний вираз для зниження потенціалу через диф-
узійний шар φd у термінах кубічного поліному в рамках наближення
Гуї-Чепмена ϕd
GC
. Коефіцієнти поліному є визначені в термінах
об’ємної густини твердих сфер η і безрозмірного параметра
оберненої відстані Γ (середньосферичне наближення). Показано,
що остаточний вираз описує оцінки для φd у примітивній моделі
дифузійного шару, отримані методом Монте-Карло.
Ключові слова: теорія подвійного шару, подвійний дифузний шар,
ефекти іонного розміру
PACS: 05.20.Jj, 05.10.Zn
718
|