Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer
The density functional and modified Poisson-Boltzmann descriptions of a spherical (electric) double layer are compared and contrasted vis-a-vis existing Monte Carlo simulation data (for small ion diameter 4.25 · 10⁻¹⁰ m) from the literature for a range of physical parameters such as macroion surf...
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irk-123456789-1195492017-06-08T03:03:42Z Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer Bhuiyan, L.B. Outhwaite, C.W. The density functional and modified Poisson-Boltzmann descriptions of a spherical (electric) double layer are compared and contrasted vis-a-vis existing Monte Carlo simulation data (for small ion diameter 4.25 · 10⁻¹⁰ m) from the literature for a range of physical parameters such as macroion surface charge, macroion radius, valencies of the small ions, and electrolyte concentration. Overall, the theoretical predictions are seen to be remarkably consistent between themselves, being also in very good agreement with the simulations. Some modified Poisson-Boltzmann results for the zeta potential at small ion diameters of 3 and 2 · 10⁻¹⁰ m are also reported. Теорія функціоналу густини та модифікована теорія Пуассона-Больцмана для сферичного електронного подвійного шару порівнюються з даними комп’ютерного Монте Карло експерименту (для малих іонів з розмірами 4.25 · 10⁻¹⁰ m), які є опубліковані в літературі для широкого набору фізичних параметрів, таких як поверхневий заряд макроіонів, радіус макроіонів, валентності малих іонів та концентрація електроліту. Загалом, теоретичні передбачення досить добре узгоджуються між собою і з комп’ютерним експериментом. Окремі результати модифікованої теорії Пуассона-Больцмана для зета потенціалу, коли розміри малих іонів є 3 та 2 · 10⁻¹⁰ m, також представлені у роботі. 2005 Article Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer / L.B. Bhuiyan, C.W. Outhwaite // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 287–302. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.20.Qg, 82.35.RS, 82.45.Gj DOI:10.5488/CMP.8.2.287 http://dspace.nbuv.gov.ua/handle/123456789/119549 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 functional and modified Poisson-Boltzmann descriptions of
a spherical (electric) double layer are compared and contrasted vis-a-vis
existing Monte Carlo simulation data (for small ion diameter 4.25 · 10⁻¹⁰ m)
from the literature for a range of physical parameters such as macroion surface
charge, macroion radius, valencies of the small ions, and electrolyte
concentration. Overall, the theoretical predictions are seen to be remarkably
consistent between themselves, being also in very good agreement
with the simulations. Some modified Poisson-Boltzmann results for the zeta
potential at small ion diameters of 3 and 2 · 10⁻¹⁰ m are also reported. |
| format |
Article |
| author |
Bhuiyan, L.B. Outhwaite, C.W. |
| spellingShingle |
Bhuiyan, L.B. Outhwaite, C.W. Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer Condensed Matter Physics |
| author_facet |
Bhuiyan, L.B. Outhwaite, C.W. |
| author_sort |
Bhuiyan, L.B. |
| title |
Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer |
| title_short |
Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer |
| title_full |
Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer |
| title_fullStr |
Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer |
| title_full_unstemmed |
Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer |
| title_sort |
comparison of density functional and modified poisson-boltzmann structural properties for a spherical double layer |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119549 |
| citation_txt |
Comparison of density functional and modified Poisson-Boltzmann structural properties for a spherical double layer / L.B. Bhuiyan, C.W. Outhwaite // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 287–302. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bhuiyanlb comparisonofdensityfunctionalandmodifiedpoissonboltzmannstructuralpropertiesforasphericaldoublelayer AT outhwaitecw comparisonofdensityfunctionalandmodifiedpoissonboltzmannstructuralpropertiesforasphericaldoublelayer |
| first_indexed |
2025-07-08T16:09:26Z |
| last_indexed |
2025-07-08T16:09:26Z |
| _version_ |
1837095681828847616 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 287–302
Comparison of density functional and
modified Poisson-Boltzmann structural
properties for a spherical double layer
L.B.Bhuiyan 1 , C.W.Outhwaite 2
1 Laboratory of Theoretical Physics,
Department of Physics, Box 23343,
University of Puerto Rico,
San Juan, Puerto Rico 00931–3343
2 Department of Applied Mathematics,
School of Mathematics,
The University of Sheffield,
Sheffield S3 7RH, UK
Received December 24, 2004
The density functional and modified Poisson-Boltzmann descriptions of
a spherical (electric) double layer are compared and contrasted vis-a-vis
existing Monte Carlo simulation data (for small ion diameter 4.25 · 10−10 m)
from the literature for a range of physical parameters such as macroion sur-
face charge, macroion radius, valencies of the small ions, and electrolyte
concentration. Overall, the theoretical predictions are seen to be remark-
ably consistent between themselves, being also in very good agreement
with the simulations. Some modified Poisson-Boltzmann results for the ze-
ta potential at small ion diameters of 3 and 2 · 10−10 m are also reported.
Key words: spherical double layer, structure, density functional theory,
modified Poisson-Boltzmann
PACS: 05.20.Jj, 61.20.Qg, 82.35.RS, 82.45.Gj
It is a pleasure to dedicate
this paper to Douglas J. Henderson
on the occasion of his 70th Birthday
1. Introduction
The accumulation of an ionic charge cloud in the vicinity of an electrode in an
electrolyte is commonly referred to as an electric double layer with the geometry of
the electrode determining the geometry of the double layer. The microscopic charac-
terization of the charge cloud has wide ranging significance in important biological,
c© L.B.Bhuiyan, C.W.Outhwaite 287
L.B.Bhuiyan, C.W.Outhwaite
industrial, and technological processes. Such relevance has made the double layer
phenomenon a major research area in colloid and polyelectrolyte science for over
three decades, see the recent reviews [1–6].
Recently there has been a resurgence of interest in the use of the density functi-
onal theory (DFT) as a theoretical probe to study the properties of electric double
layers in all geometries. For example, Boda et al. [7,8] have applied the theory to
a planar double layer (PDL) containing a restricted primitive model (RPM) elec-
trolyte (equisized rigid ions moving in a continuum dielectric) and have obtained
structural and thermodynamic results for different ion sizes. The application to the
cylindrical (electric) double layer (CDL) has been considered by Patra and Yethiraj
[9,10]. Yu et al. [11] have employed the technique to spherical double layers (SDL).
In another study Valisko et al. [12] have further treated the PDL containing a pri-
mitive model electrolyte but now with unequal ionic radii. In many cases the DFT
results from these studies have been compared with the corresponding Monte Carlo
(MC) simulation data. A consensus that emerges from these comparisons is that
the DFT is a viable approach to the electric double layer theory. It ought to be
mentioned here that the DFT was originally applied to the PDL in the early 1990s
by various groups [13–16].
It is natural to wonder how the DFT compares with other more established
theories of the electric double layer, viz., integral equation methods and potential
based theories. To this end some work has already been reported. In an earlier
paper [17] we have compared the DFT results of Boda et al. [7,8] for a RPM planar
double layer at ionic diameter of a = 4.25 ·10−10 m with the corresponding modified
Poisson-Boltzmann (MPB) results. With few exceptions the two theories were seen
to be on par with each other in reproducing the MC simulation data of Boda et al.
[7,8] for the model systems. More recently, Patra and Bhuiyan [18] have compared
the DFT and MPB results for a CDL where the calculations were done at physical
parameters for a double stranded DNA. Notwithstanding the fact that MC results
for the CDL are presently rather sketchy in the literature, the two theories again
showed notable consistency in their predictions.
The above findings have encouraged us to extend such comparisons to a double
layer involving spherical symmetry. In this paper we will compare the structural
descriptions of a SDL arising out of the DFT and the MPB theory. As indicated
previously, the DFT for the SDL has recently been solved by Yu et al. [11], the
calculations having been made at the ionic diameter of a = 4.25·10−10 m. These DFT
results will be utilized in the present study, and in the rest of the paper “DFT” will
refer to this version. The MC simulations at this ionic diameter were obtained earlier
by Degreve et al. [19] and Degreve and Lozada-Cassou [20]. González-Továr et al.
[21] have reported simulation and hypernetted chain/mean spherical approximation
(HNC/MSA) integral equation results at higher ionic diameters.
As the name suggests the modified Poisson-Boltzmann formalism arose from
modifications to the classical Poisson-Boltzmann (PB) theory to account for (i) the
interionic correlations, and (ii) the ionic exclusion volume effects, both of which are
ordinarily neglected in the classical mean field theory. The MPB theory was first
288
Structural properties for a spherical double layer
applied to the SDL by Outhwaite and Bhuiyan [22,23]. Some zeta potential results
from this study were subsequently used by Degreve and co-workers [19,20] in their
MC studies for comparison purposes. Indeed the MPB zeta potentials compared
rather well with the simulations and the HNC/MSA results of González-Továr and
Lozada-Cassou [24].
2. Model and theory
The SDL model utilized in this study consists of an isolated spherical macroion
of radius R and uniform surface charge density σ, bathed in a RPM electrolyte, the
model being equivalent to a macroion at infinite dilution. The common diameter of
the small, simple ions is a, while the mean number density and charge of an ion
of species s are ρs and es = zs|e| (zs is the valency and e the electron charge),
respectively.
The development of the MPB theory for the SDL has been described in detail
in reference [23]. We will therefore outline here only the principal equations.
The mean electrostatic potential ψ(r) satisfies Poisson’s equation
∇2ψ(r) = −
1
ε0εr
∑
s
esρsgs(r), (1)
where r is the radial distance from the macroion centre, and ε0, εr are the vacuum
and relative permittivities, respectively. In spherical symmetry, the macroion-small
ion radial distribution function gs(r) reads
gs(r) = ξs(r) exp
[
−
βe2s
8πε0εra
(F − F0) − βesL(u)
]
, (2)
where
L(u) =
1
2r
[
F (u(r + a) + u(r − a)) −
(F − 1)
a
∫ r+a
r−a
u(s)ds
]
, (3)
F =
4{4 +
(
κ
r
)
[(r + a)2 −
(
R + a
2
)2
]}−1, R + a
2
6 r 6 R + 3a
2
,
(1 + κa)−1, r > R + 3a
2
,
(4)
κ2 =
e2β
ε0εr
∑
s
z2
sρsgs(r), (5)
F0 = lim
r→∞
F = (1 + κ0a)
−1 , (6)
κ2
0 =
e2β
ε0εr
∑
s
z2
sρs (7)
with u(r) = rψ(r) and β = 1/(kBT ) (kB the Boltzmann’s constant and T the
absolute temperature). The exclusion volume term ξs(r) is approximated using the
Bogoliubov-Born-Green-Yvon hierarchy
289
L.B.Bhuiyan, C.W.Outhwaite
Figure 1. Macroion-small ion radial distribution function gi(r) in a spherical double
layer for different RPM electrolytes. The symbols represent the MC data, while the solid,
dashed, and dotted lines represent the MPB, DFT, and PB results, respectively. For the
1:1 electrolyte c = 1 mol/dm3, σ = 0.102 C/m2, and for the 2:2 and 1:2 electrolytes
c = 0.5 mol/dm3, σ = 0.204 C/m2. The macroion radius is R = 15 · 10−10 m in all cases.
DFT results from reference [11] and MC data from references [19,20].
290
Structural properties for a spherical double layer
ξs(r) = H (r − (R + a/2)) exp
[
π
∫
∞
r
∑
t
ρt
∫ r+a
max(R+a/2,r−a)
(
X
x
)2
(
X2 − x2 − a2
)
gt(x) exp {−βetφ(x,X)}dXdx
]
, (8)
and
φ(x,X) =
F
4πa
∫
V
∇2ψdV (9)
is the fluctuation potential evaluated on the surface of the exclusion volume V of
the discharged ion.
ψ
Figure 2. Macroion-small ion radial distribution function gi(r) and the mean elec-
trostatic potential ψ(r) in a spherical double layer with a 1:1 RPM electrolyte at
c = 1 mol/dm3, and (a) σ = 0.102 C/m2, and (b) σ = 0.204 C/m2. The macroion
radius is R = 15 · 10−10 m. The meaning of the curves as in figure 1. DFT results from
reference [11].
291
L.B.Bhuiyan, C.W.Outhwaite
The equations (1)–(9) are collectively known as the MPB equation for the SDL.
The classical PB equation is obtained upon taking ξ(r) = H(r − R), F = F0, and
a → 0. The MPB and the PB equations were solved numerically using a quasi-
linearization iterative procedure [25,26] for a range of R, σ, zs, a, and electrolyte
concentration c.
ψ
Figure 3. Macroion-small ion radial distribution function gi(r) and the mean elec-
trostatic potential ψ(r) in a spherical double layer with a 2:1 RPM electrolyte at
c = 0.5 mol/dm3, and (a) σ = 0.102 C/m2, and (b) σ = 0.204 C/m2. The macroion
radius is R = 15 · 10−10 m. The meaning of the curves as in figure 1. DFT results from
reference [11].
3. Results and discussion
The physical parameters used in the present work follow closely those used in
the earlier MC [19,20] and DFT [11] calculations. Thus unless otherwise mentioned
292
Structural properties for a spherical double layer
Figure 4. Zeta potential in a spherical double layer for a 1:1, 2:1/1:2, and 2:2 RPM
electrolyte at different concentrations. For 1:1 the macroion radius is (a) R = 15·10−10 m,
and (b) R = 5 · 10−10 m, while for 2:1/1:2 and 2:2 electrolytes R = 15 · 10−10 m. The
meaning of the symbol and curves as in figure 1.
293
L.B.Bhuiyan, C.W.Outhwaite
all of the MPB and PB results pertain to a = 4.25 ·10−10 m and a water-like solvent
(εr = 78.5) at room temperature T = 298 K.
Figure 5. Variation of the zeta potential for 2:1 and 1:2 electrolytes as a function of
inverse macroion radius at a fixed macroion surface charge density of σ = 0.204 C/m2.
The meaning of the symbol and curves as in figure 1. DFT results from reference [11]
and MC data from reference [20].
The singlet distribution functions of the simple ions around the spherical colloid
of radius R = 15 · 10−10 m are shown in figure 1 for 1:1, 1:2 and 2:2 electrolytes.
The 1:1 electrolyte has σ = 0.102 C/m2 with c = 1 mol/dm3 while the divalent
electrolytes have σ = 0.204 C/m2 with c = 0.5 mol/dm3. Comparison with the
MC results indicates that both the MPB and DFT theories accurately reproduce
the simulation results. For the 1:2 and 2:2 cases both the theories predict a non-
monotonic behaviour in the distribution functions indicative of overscreening. This
non- monotonic behaviour cannot be produced by the standard PB theory. Figures 2
and 3 illustrate the ionic singlet distribution functions and the mean electrostatic
potential around the colloid for 1:1 and 2:1 electrolytes at c = 1 and 0.5 mol/dm3,
294
Structural properties for a spherical double layer
respectively, with σ = 0.102 and 0.204 C/m2. Note that there are misprints in the
captions of figures 4 and 6 in reference [11] where it should read 0.204 C/m2 rather
than 0.306 C/m2 [27]. Again the MPB and DFT theories are in close agreement,
with both theories having a shallow minimum in the mean electrostatic potential
(see insets). As the surface charge is increased we expect the steric effect of the
counterions to become important and a second layer of counterions to form at r ∼
R+3a/2. This is seen in the planar electric double layer in both theory and simulation
[28–30]. Provided R is not too small this layering effect is predicted by the DFT
theory (figures 4–6 of Yu et al. [11]) but not by the MPB theory due to the breakdown
of the numerical procedure at high σ. It is expected that the MPB theory will
predict counterion layering on using an alternative numerical technique. In practice,
however, such high surface charges are not observed in typical colloidal suspensions.
Figure 6. Charge inversion in a spherical double layer with a 2:2 RPM electrolyte at
c = 1.25 mol/dm3 and a = 4 · 10−10 m. The macroion radius is R = 10 · 10−10 m and
contains 20 elementary charges. The meaning of the symbol and curves as in figure 1.
DFT results from reference [11] and MC data from reference [31].
In figure 4 we plot the zeta potential ζ = ψ(R + a/2) as a function of surface
charge σ for 1:1, 2:1/1:2, and 2:2 electrolytes at various electrolyte concentrations.
Again we use the colloid radius R = 15 · 10−10 m with also R = 5 · 10−10 m for the
1:1 case. There is good agreement between the MPB, DFT, and simulation results
with all the three electrolyte cases, although the MPB is closer to the MC values
for divalent counterions at c = 0.005 mol/dm3. The PB theory gives the correct
qualitative response for 1:1 electrolytes but fails to predict the maxima and minima
for the 2:2 and 2:1 electrolytes, respectively. Figure 5 illustrates the variation of ζ
with a/R for 1:2 and 2:1 electrolytes at σ = 0.204 C/m2 with c = 0.005 and 0.5
mol/dm3. The MPB results for 1:1 and 2:2 electrolytes with the same parameters
are very similar to the 2:1 and 1:2 graphs, respectively, in figure 5 and hence are
not shown. Increasing R for divalent counterions leads to a maximum in the ζ plot
295
L.B.Bhuiyan, C.W.Outhwaite
which cannot be predicted by the PB theory. The recent HNC/MSA calculations
of González-Továr et al. [21] further illustrate the severe limitations of using the
PB zeta potential to analyse electrophoretic mobility experiments. These authors
studied the behaviour of ζ at varying ionic radii and surface charge for fixed R and
c and found that for both 1:1 and 2:2 electrolytes there is a change in curvature sign
as the ion size is increased.
Figure 7. Charge inversion in a spherical double layer for 1:1 and 2:1 RPM electrolytes
at (a) c = 0.05 mol/dm3, (b) c = 1 mol/dm3, and (c) c = 1.5 mol/dm3. The macroion
radius is R = 15 ·10−10 m and contains 18 elementary charges. The meaning of the curves
as in figure 1. DFT results from reference [11].
The non-monotonic behaviour of gi(r) and ψ(r) seen in figures 1–3 indicate that
the local density of the counterions is such that the colloidal charge is overcompen-
sated giving rise to overcharging or overscreening. To illustrate this overscreening
296
Structural properties for a spherical double layer
we consider the integrated charge distribution [11]
P (r) = Z + 4π
∑
s
zsρs
∫ r
0
x2gs(x)dx. (10)
Figure 8. Zeta potential in a spherical double layer for a 1:1 RPM electrolyte at different
concentrations with small ion diameter (a) a = 3 · 10−10 m, and (b) a = 2 · 10−10 m. The
macroion radius is R = 15 · 10−10 m. The meaning of the curves as in figure 1.
Figure 6 compares the theoretical predictions with the MC results of Terao and
Nakayama [31] for a 2:2 electrolyte with R = 10 · 10−10 m, a = 4 · 10−10 m and
macroion charge Z|e| = −20|e|. The damped oscillatory behaviour of the charge
density profile is clearly seen with the MPB and DFT giving excellent agreement
with the simulations, while the PB is qualitatively different, showing a monotonic
behaviour. The graphs in figure 7 give P (r) for 1:1 and 2:1 electrolytes respectively
at 3 different concentrations with Z = −18 and R = 15 · 10−10 m.
297
L.B.Bhuiyan, C.W.Outhwaite
At the lowest concentration of 0.05 mol/dm3 the integrated charge is mono-
tonic, but when the concentration is increased to the higher concentrations of 1
and 1.5 mol/dm3 a damped oscillatory behaviour sets in. As expected the divalent
counterion charge inversion response is more pronounced than that of the univalent
counterion at the same concentration.
Finally, in figures 8–10 we present some MPB and PB zeta potential results when
the small ion diameter is 3 ·10−10 and 2 ·10−10 m, respectively. (The macroion radius
has been kept at R = 15 · 10−10 m.) A motivation for these calculations is the fact
Figure 9. Zeta potential in a spherical double layer for 2:1, 1:2 RPM electrolytes at
different concentrations with small ion diameter (a) a = 3 · 10−10 m, and (b) a = 2 ·
10−10 m. The macroion radius is R = 15 · 10−10 m. The meaning of the curves as in
figure 1.
that in their DFT and MC work on the PDL at these diameters, Boda et al. [7,8]
had noted an increasing discrepancy between the classical PB and DFT (or MC)
diffuse layer potentials with decreasing ion size. The behaviour was confirmed by
298
Structural properties for a spherical double layer
the MPB calculations in the PDL [17], and by the DFT and MPB calculations in
the CDL [18]. In the present case the trend is clearly visible at the higher absolute
values of the surface charge for each of the 1:1, 2:1/1:2, and 2:2 situations with the
deviation between the PB and the MPB ζ increasing as the ion size decreases from
3 · 10−10 m to 2 · 10−10 m. For divalent counterions the MPB ζ continues to show a
maximum (or a minimum), while the PB ζ remains monotonic as in the planar or
cylindrical geometries [17,18].
Figure 10. Zeta potential in a spherical double layer for a 2:2 RPM electrolyte at
different concentrations with small ion diameter (a) a = 3 · 10−10 m, and (b) a = 2 ·
10−10 m. The macroion radius is R = 15 · 10−10 m. The meaning of the curves as in
figure 1.
299
L.B.Bhuiyan, C.W.Outhwaite
4. Concluding remarks
This paper has focussed on a comparative study of the structure of an electric
double layer in spherical symmetry for a range of physical parameters using the
density functional and modified Poisson-Boltzmann theories. Although quite distinct
in their origins, the theories show a remarkable consistency overall for the cases
studied, and their further predictions closely follow the simulation data.
The DFT and MPB approaches to the electric double layer have now been com-
pared for planar [17], cylindrical [18], and spherical geometries. The accumulated
evidence clearly suggests that the aforementioned consistency is global, extending
to all the three symmetries, and that under ordinary laboratory conditions, viz.,
σ 6 0.3 C/m2 and electrolyte solution regimes, the two theories are comparable in
their predictive properties.
Acknowledgements
Support of a National Foundation Grant 0137273 is gratefully acknowledged.
We are grateful to Dr. Yang-Xin Yu for sending us the numerical values of the DFT
results utilized in this work.
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L.B.Bhuiyan, C.W.Outhwaite
Застосування теорії функціоналу густини та
модифікованої теорії Пуассона-Больцмана
дослідження структурних властивостей
сферичного подвійного шару
Л.Б.Бхуян 1 , К.У.Аутвайт 2
1 Лабораторія теоретичної фізики,
університет Пуерто Ріко,
Пуерто Ріко
2 Інститут математики,
університет Шеффілда,
Шеффілд, Великобританія
Отримано 24 грудня 2004 р.
Теорія функціоналу густини та модифікована теорія Пуассона-
Больцмана для сферичного електронного подвійного шару порівню-
ються з даними комп’ютерного Монте Карло експерименту (для
малих іонів з розмірами 4.25 · 10−10 m), які є опубліковані в літерат-
урі для широкого набору фізичних параметрів, таких як поверхневий
заряд макроіонів, радіус макроіонів, валентності малих іонів та кон-
центрація електроліту. Загалом, теоретичні передбачення досить
добре узгоджуються між собою і з комп’ютерним експериментом.
Окремі результати модифікованої теорії Пуассона-Больцмана для
зета потенціалу, коли розміри малих іонів є 3 та 2 · 10−10 m, також
представлені у роботі.
Ключові слова: сферичний подвійний шар, структура, теорія
функціоналу густини, модифікований Пуассон-Больцман
PACS: 05.20.Jj, 61.20.Qg, 82.35.RS, 82.45.Gj
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