The effect of ionic size on polyion-small ion distributions in a cylindrical double layer
The structure of an electrolyte surrounding an isolated, cylindrical polyion – the cylindrical double layer – is studied using a density functional approach and the modified Poisson Boltzmann theory. The polyion is modelled as an infinitely long, rigid, and impenetrable charged cylinder, while t...
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| Zitieren: | The effect of ionic size on polyion-small ion distributions in a cylindrical double layer / C.N. Patra, L.B. Bhuiyan // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 425–446. — Бібліогр.: 70 назв. — англ. |
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irk-123456789-1196562017-06-08T03:04:04Z The effect of ionic size on polyion-small ion distributions in a cylindrical double layer Patra, C.N. Bhuiyan, L.B. The structure of an electrolyte surrounding an isolated, cylindrical polyion – the cylindrical double layer – is studied using a density functional approach and the modified Poisson Boltzmann theory. The polyion is modelled as an infinitely long, rigid, and impenetrable charged cylinder, while the electrolyte consists of rigid ions moving in a dielectric continuum. The results for the zeta potential, polyion-small ion distribution, and the mean electrostatic potential are obtained for a wide range of physical conditions including three different ionic diameters of 2, 3, and 4·10⁻¹⁰ m. The zeta potentials show a maximum or a minimum with respect to the polyion surface charge density for a divalent counterion. The polyion-ion distributions and the mean electrostatic potential profiles show considerable variations with the concentration of the electrolyte, the valency of the ions constituting the electrolyte, and the ionic size. The theories are seen to be generally consistent with each other overall, and are capable of predicting the charge inversion phenomenon – a feature, which is completely absent in the classical Poisson-Boltzmann theory. Moreover, the theories reproduce well some Monte Carlo results (for ion distributions) from the literature. Структура електроліту навколо ізольованого циліндричного полііона – циліндричного подвійного шару – досліджується методом функціоналу густини та модифікованою теорією Пуассона-Больцмана. Полііон моделюється як нескінченно довгий, жорсткий і непроникний заряджений циліндр, тоді як електроліт складається з жорстких іонів, які рухаються у неперервному діелектричному середовищі. Результати для зеті-потенціалу, розподілу полііон-іон та середнього електростатичного потенціалу отримані для широкого діапазону фізичних умов, включаючи три різні діаметри: 2, 3 та 4·10⁻¹⁰ m. У випадку дво-валентних контраіонів, зета-потенціали демонструють максимум або мінімум відносно густини поверхневого заряду полііона. Профілі розподілів полііон-іон та середнього електростатичного потенціалу демонструють суттєві зміни як функції концентрації електроліту, валентності іонів електроліту та іонних розмірів. Обидва теоретичні підходи в загальному є узгоджені між собою і обидва передбачають явище зарядової інверсії, яке є відсутнє в рамках звичайної теорії Пуассона-Больцмана. Більше того, обидві теорії добре відтворюють окремі результати Монте Карло (для іонних розподілів), які є відомі з літератури. 2005 Article The effect of ionic size on polyion-small ion distributions in a cylindrical double layer / C.N. Patra, L.B. Bhuiyan // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 425–446. — Бібліогр.: 70 назв. — англ. 1607-324X PACS: 82.70.Dd; 64.10.+h DOI:10.5488/CMP.8.2.425 http://dspace.nbuv.gov.ua/handle/123456789/119656 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 |
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English |
| description |
The structure of an electrolyte surrounding an isolated, cylindrical polyion
– the cylindrical double layer – is studied using a density functional
approach and the modified Poisson Boltzmann theory. The polyion is modelled
as an infinitely long, rigid, and impenetrable charged cylinder, while
the electrolyte consists of rigid ions moving in a dielectric continuum. The
results for the zeta potential, polyion-small ion distribution, and the mean
electrostatic potential are obtained for a wide range of physical conditions
including three different ionic diameters of 2, 3, and 4·10⁻¹⁰ m. The zeta
potentials show a maximum or a minimum with respect to the polyion surface
charge density for a divalent counterion. The polyion-ion distributions
and the mean electrostatic potential profiles show considerable variations
with the concentration of the electrolyte, the valency of the ions constituting
the electrolyte, and the ionic size. The theories are seen to be generally
consistent with each other overall, and are capable of predicting the charge
inversion phenomenon – a feature, which is completely absent in the classical
Poisson-Boltzmann theory. Moreover, the theories reproduce well some
Monte Carlo results (for ion distributions) from the literature. |
| format |
Article |
| author |
Patra, C.N. Bhuiyan, L.B. |
| spellingShingle |
Patra, C.N. Bhuiyan, L.B. The effect of ionic size on polyion-small ion distributions in a cylindrical double layer Condensed Matter Physics |
| author_facet |
Patra, C.N. Bhuiyan, L.B. |
| author_sort |
Patra, C.N. |
| title |
The effect of ionic size on polyion-small ion distributions in a cylindrical double layer |
| title_short |
The effect of ionic size on polyion-small ion distributions in a cylindrical double layer |
| title_full |
The effect of ionic size on polyion-small ion distributions in a cylindrical double layer |
| title_fullStr |
The effect of ionic size on polyion-small ion distributions in a cylindrical double layer |
| title_full_unstemmed |
The effect of ionic size on polyion-small ion distributions in a cylindrical double layer |
| title_sort |
effect of ionic size on polyion-small ion distributions in a cylindrical double layer |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119656 |
| citation_txt |
The effect of ionic size on polyion-small ion distributions in a cylindrical double layer / C.N. Patra, L.B. Bhuiyan // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 425–446. — Бібліогр.: 70 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT patracn theeffectofionicsizeonpolyionsmalliondistributionsinacylindricaldoublelayer AT bhuiyanlb theeffectofionicsizeonpolyionsmalliondistributionsinacylindricaldoublelayer AT patracn effectofionicsizeonpolyionsmalliondistributionsinacylindricaldoublelayer AT bhuiyanlb effectofionicsizeonpolyionsmalliondistributionsinacylindricaldoublelayer |
| first_indexed |
2025-07-08T16:20:47Z |
| last_indexed |
2025-07-08T16:20:47Z |
| _version_ |
1837096396510986240 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 425–446
The effect of ionic size on polyion-small
ion distributions in a cylindrical double
layer
C.N.Patra 1,3 , L.B.Bhuiyan 2
1 Theoretical Chemistry Section, RC & CD Division, Chemistry Group,
Bhabha Atomic Research Centre, Mumbai 400 085, India
2 Laboratory of Theoretical Physics, Department of Physics, Box 23343,
University of Puerto Rico, San Juan, Puerto Rico 00931–3343
3 Department of Materials Science and Engineering University of Utah
122 S. Central Campus Drive Salt Lake City, Utah 84112–0560
Received July 12, 2004
The structure of an electrolyte surrounding an isolated, cylindrical polyi-
on – the cylindrical double layer – is studied using a density functional
approach and the modified Poisson Boltzmann theory. The polyion is mod-
elled as an infinitely long, rigid, and impenetrable charged cylinder, while
the electrolyte consists of rigid ions moving in a dielectric continuum. The
results for the zeta potential, polyion-small ion distribution, and the mean
electrostatic potential are obtained for a wide range of physical conditions
including three different ionic diameters of 2, 3, and 4·10−10 m. The zeta
potentials show a maximum or a minimum with respect to the polyion sur-
face charge density for a divalent counterion. The polyion-ion distributions
and the mean electrostatic potential profiles show considerable variations
with the concentration of the electrolyte, the valency of the ions constituti-
ng the electrolyte, and the ionic size. The theories are seen to be generally
consistent with each other overall, and are capable of predicting the charge
inversion phenomenon – a feature, which is completely absent in the classi-
cal Poisson-Boltzmann theory. Moreover, the theories reproduce well some
Monte Carlo results (for ion distributions) from the literature.
Key words: cylindrical double layer, structure, density functional theory,
modified Poisson-Boltzmann
PACS: 82.70.Dd; 64.10.+h
It is a pleasure to dedicate this paper
to Dr. Douglas J. Henderson
on the occasion of his 70th Birthday
c© C.N.Patra, L.B.Bhuiyan 425
C.N.Patra, L.B.Bhuiyan
1. Introduction
The study of the interactions of a polyion with the ionic atmosphere surrounding
it is quite important in diverse areas such as colloidal systems [1], industrial polyelec-
trolytes [2], and biologically significant systems like DNA [3–5]. The structure and
thermodynamics of all these systems manifest large salt dependencies. A detailed
knowledge of the spatial distribution of small ions in the vicinity of a polyelectrolyte
is therefore fundamental to a microscopic understanding of these polyelectrolyte
solutions.
The polyion together with its charge cloud constitute what is known as an electric
double layer in the literature with the geometry of the polyion shaping the double
layer geometry. In the planar symmetric situation one has the well known planar
double layer (PDL), which has been extensively studied over the past three decades
using formal statistical mechanical methods (see for example, reviews [6–9]). In
contrast, the double layer in other geometries, for instance the cylindrical double
layer (CDL) and the spherical double layer (SDL), has received comparatively less
attention.
Theoretical advances in the last few decades have made possible the calculation of
the ionic distribution and mean electrostatic potential profiles, and consequently the
characterization of various other thermodynamic properties of the polyion – small
ion systems. In almost all of these studies, the polyion is modelled as an infinitely
long, rigid charged cylinder, while the small, mobile ions are treated as charged hard
spheres in a continuum dielectric. The situation when the polyion is at infinite diluti-
on, viz., the primitive model cylindrical double layer system, and the situation when
the polyion has a finite concentration, viz., the cylindrical cell model, have both been
treated. The theoretical works are mainly based on four distinct approaches. The first
one is the counterion condensation (CC) theory of Manning (see for example, refer-
ence [10]), which has been quite extensively applied in several biological problems
[11] including analyses of competitive binding equilibria [12]. The second approach
corresponds to methods based on the classical Poisson-Boltzmann (PB) theory which
has been applied in different forms, viz. linear [13], nonlinear [14,15], and the cell
model [16,17]. The third approach is based on more rigorous formal statistical me-
chanical methods which includes functional expansions [18], liquid structure integral
equations [19–26], the modified Poisson-Boltzmann theory (MPB) (including the cell
model) [27–33], and the density functional theory (DFT) [34,35], all of which have
been found to be successful in varying degrees in predicting the structural and dy-
namical information of such systems. The fourth one involves Monte Carlo (MC)
simulations (again including the cell model) [30,32,33,36–42], which gives interesting
insights into the exact energetics, density distributions, and colligative properties of
such systems. The qualitative and quantitative comparisons of these approaches have
been attempted from time to time (see for example, references [24,40,43]), but it is
beyond the scope of the present paper. We intend to concentrate more on some of
the specific issues such as the effect of ionic size on double layer structure, involving
the application of the DFT and MPB theories to the cylindrical double layer.
426
Ionic distributions in cylindrical double layers
The DFT is a relatively new analytical approach to the electric double layer
phenomenon and has been found to be useful in some earlier structural studies
of the PDL [44–47]. In recent years these techniques have been utilized in more
exhaustive studies of the PDL system by Boda et al. [48–51] and Patra and Ghosh
[52], while Yu et al. [53] have applied the methods to the SDL. The studies have
confirmed the usefulness of the density functional approach for these systems. Such
detailed studies for the CDL, however, lacking in the literature. An earlier DFT
study of the CDL was limited, viz., the zeta potentials were not calculated [34,35].
Furthermore, it is tempting to see how the approach compares with earlier and more
established theories of the CDL such as the MPB, which has already seen successful
applications to the cylindrical geometry [27–29]. The DFT technique applied in this
work is a partially perturbative one [34,54] where the hard sphere contribution to
the first order correlation function is evaluated from the weighted density approach
(WDA) of Denton and Ashcroft [55] and the electrical contribution is approximated
by a quadratic expansion with respect to the corresponding bulk electrolyte [56].
Henceforth in this paper the term “density functional theory” or its acronym DFT,
will refer to this version.
The MPB theory belongs to the family of potential based theories in the statis-
tical mechanics of charged fluids. It has evolved from the attempts to improve upon
the classical PB theory by incorporating
(i) inter-ionic correlations through a fluctuation potential term, and
(ii) ionic exclusion volume effects – features that are missing in the classical for-
mulation.
For a cylindrical double layer situation the MPB theory was first derived by Out-
hwaite in the mid-1980s [27]. Subsequently the theory was applied to a primitive
model CDL by Bhuiyan and Outhwaite [28,29]. In these applications the value of
the ionic diameter was held fixed at d = 4.25 · 10−10 m. It is important to point out
that with few exceptions, most of the double layer studies in planar, cylindrical, and
spherical symmetries have been at this value of the ionic diameter. The very recent
MC and DFT works by Boda et al. [50,51] on PDL have for the first time examined
the dependency of double layer properties on ionic size with diameters d = 2 · 10−10
and 3·10−10 m being used besides the habitual d = 4.25·10−10 m. An interesting out-
come is that the classical Gouy-Chapman-Stern theory of the PDL [57–59] becomes
increasingly less accurate as the ionic size decreases. In a later detailed comparative
study by Bhuiyan and Outhwaite [60] of the DFT results versus the corresponding
MPB predictions vis-a-vis the MC, the two theories were found to be remarkably
consistent over a broad range of physical parameters including different ionic sizes.
In this paper we will explore the DFT and MPB theories for a primitive model
CDL with a particular emphasis on the effect of variation of ionic size on structural
properties of the double layer. It is further noted that no comparison of the DFT
and the MPB theory for the CDL exists in the polyelectrolyte literature.
427
C.N.Patra, L.B.Bhuiyan
2. Theoretical formulation
2.1. Molecular model
The polyion is modelled as an infinite, isolated, rigid cylinder bearing a uniform
axial charge density given by
ξ =
βe2
εb
, (1)
where e is the proton charge, b is the length per unit charge (inverse of linear charge
density), β = (kBT )−1, kB is the Boltzmann’s constant, T is the temperature, and
ε is the dielectric constant of the pure solvent modelled as a uniform dielectric
continuum, which is taken to be ε = 78.358 (characterizing water). Throughout this
work we set T = 298.15 K, ξ = 4.2, and b = 1.7 · 10−10 m, which are the accepted
values for a double-stranded DNA [43]. The small ions (with α denoting the species)
are modelled using the restricted primitive model (RPM), viz. they are charged hard
spheres of equal diameter, d = 2, 3, or 4 · 10−10 m., and charge qα = Zαe, with Zα
being the ionic valency. The polyion was ascribed a radius R = 8 · 10−10 m so that
the distance of closest approach of an ion varied from 9 · 10−10 m to 10 · 10−10 m. In
another case, we have taken R = 20·10−10 m, which should resemble the distributions
corresponding to that in a planar double layer.
2.2. Density functional theory
In density functional theory [61], one starts with an expression for the grand
potential, Ω, as a functional of the singlet density profiles, ρα(r), of each of the
species, α. At equilibrium the grand potential is minimal with respect to variations
in the density profiles, viz.,
δΩ
δρα(r)
= 0 (2)
for each α and this condition is used to determine the density profiles and the free
energy. Without going into details, we present here the relevant equations for the
nonuniform density distribution of small ions
ρα(r) = ρ0
α exp
{
−βqαψ(r) + c(1) hs
α (r; [{ρα}]) − c̃(1) hs
α ([{ρ0
α}])
+ c(1) el
α (r; [{ρα}]) − c̃(1) el
α ([{ρ0
α}])
}
(3)
in the region r > (R + d/2). In the above expression, ψ(r) represents the mean
electrostatic potential and the quantities c
(1)hs
α (r; [{ρα}]) and c
(1)el
α (r; [{ρα}]) denote,
respectively, the hard sphere and electrical contributions to the first order corre-
lation function. The electrostatic potential ψ(r) is obtained from a solution of the
corresponding PB equation in cylindrical geometry, and is given as
ψ(r) =
4π
ε
∫
∞
r
dr′ r′ ln
( r
r′
)
∑
α
qαρα(r′). (4)
428
Ionic distributions in cylindrical double layers
In the absence of explicit expressions, c
(1) hs
α (r; [{ρα}]) and c
(1) el
α (r; [{ρα}]) have been
approximated as
c(1) hs
α (r; [{ρα}]) = c̃(1) hs
α (ρ̄(α)(r)), (5)
c(1)elα (r, [{ρα}]) = c̃(1)elα ([{ρ0
α}]) +
∑
β
∫
dr′ r′ c̃
(2) el
αβ
(
r, r′;
[
{ρ0
α}
]) (
ρβ(r′) − ρ0
β
)
,(6)
where “tilde” (∼) refers to the uniform fluid, with ρ0
α being the mean ionic num-
ber density of the αth species. The quantity ρ̄(α)(r) is the total weighted density,
which has been calculated using the WDA scheme of Denton and Ashcroft [55]. The
uniform second order correlation functions, c̃
(2) hs
αβ (r; [{ρ0
α}]) and c̃
(2) el
αβ (r; [{ρ0
α}]) are
taken from the analytical expressions [62] within the mean spherical approximations
for a uniform mixture of charged hard spheres. Thus the density functional theory
is completely specified through equations (3)–(6) for calculation of the ionic dis-
tribution and the mean electrostatic potential profiles. It ought to be pointed out
that the DFT reduces to the nonlinear PB theory [15] once the correlation between
the small ions is neglected, i. e., by setting c
(1) hs
α (r; [{ρα}]) = c̃
(1) hs
α ([{ρ0
α}]) = 0 and
c
(1) el
α (r; [{ρα}]) = c̃
(1) el
α ([{ρ0
α}]) = 0.
2.3. Modified poisson Boltzmann theory
As indicated in the Introduction, the MPB equation appropriate for the cyli-
ndrical geometry as applied to the present polyion-ion system was formulated by
Outhwaite [27]. Without delving into the details, which have been described in ref-
erences [28,29], for completeness we outline here the relevant equations for the ionic
density distributions.
In cylindrical symmetry, the MPB polyion-ion singlet density distribution reads
ρα(r) = ρ0
αξα(r) exp
[
−βq
2
α
2εd
(F − F0) −
βqαF
2
√
r
{u(r + d) + u(r − d)}
+
βqα(F − 1)
2d
√
r
∫ r+d
r−d
u(R)dR
]
, (7)
where, u(r) =
√
rψ(r). In equation (7), F and F0 are given by
F =
{
{(1 + κd) − (κd/π)S}−1 , R + d/2 6 r 6 R + 3d/2 ,
1/(1 + κd), r > R + 3d/2,
(8)
F0 = lim
r→∞
F = (1 + κ0d)
−1, (9)
with
S =
∫ π/2
θ0
sin θ cos−1
{
c− cos2 θ
(2r/d) sin θ
}
dθ, (10)
θ0 = sin−1
[
r − (R + d/2)
d
]
, (11)
429
C.N.Patra, L.B.Bhuiyan
c = 1 −
(
R
d
+
1
2
)2
+
(r
d
)2
, (12)
and κ and κ0 are the local and bulk Debye-Hückel screening parameters
κ2 =
4πβ
ε
∑
α
q2
αρα(r), (13)
κ0 = lim
r→∞
κ. (14)
The exclusion volume term ξα(r) is approximated using the Bogoliubov-Born-Green-
Yvon hierarchy for a planar symmetry [63]
ξα(r) =
ρα(r|qα = 0)
ρ0
α
= Θ
(
r −
(
R +
d
2
))
× exp
[
2π
∫
∞
r
∑
γ
∫ r+d
max(R+d/2,r−d)
(x− y)ργ(x) exp {−βqγφ(y, x)}dxdy
]
, (15)
φ(y, x) =
F
4πd
∫
V
∇2ψdV, (16)
where Θ(x) is the Heaviside function and φ(y, x) is the fluctuation potential eval-
uated on the exclusion surface of the discharged ion. This approximate ξα(r) was
found to be reasonable in the earlier applications [28,29], and as we will see lat-
er, is adequate for the physical parameters apropos of DNA utilized in the present
work. It should also be noted at this point that MPB theory reduces to the classical
nonlinear PB theory upon taking ξα(r) = 1, F = F0 and d → 0 [28]. We remark
however, that the PB theory used here is in the Stern [59] spirit, that is, the distance
of closest approach of a small ion to the polyion continues to be R + d/2 so that
ξα(r) = Θ(r − (R + d/2)).
3. Results and discussion
The DFT and MPB equations have been solved using numerical iterative meth-
ods, the details of which can be found elsewhere in the literature (see for example,
references [44,47] for DFT and references [64–66] for MPB). We have also obtained
numerical solution of the PB theory for the cylindrical polyion-ion system in question
for comparison purposes. Some of the physical parameters utilized in the calculations
have been mentioned earlier in sub-section 2.1. We have treated 1:1, 2:1/1:2, and 2:2
valency electrolyte systems in the concentration range 0.01 mol/dm3 – 2 mol/dm3,
while the surface charge density on the polyion varied from 0 to 0.3 C/m2. At the
ionic diameter d = 4 · 10−10 m some calculations have also been performed at a
higher polyion radius R = 20 · 10−10 m.
We first discuss the zeta potential profiles for a cylindrical double layer system
as a function of polyion surface charge density σ, bulk salt concentration c, ionic
valency Zα, and ionic size (diameter) d. The zeta potential is defined as the mean
430
Ionic distributions in cylindrical double layers
Figure 1. Zeta potentials for a 1:1 electrolyte as a function of surface charge
density of the polyion at different electrolyte concentrations (in Molar (mol/dm3))
with the ionic diameter (a) d = 2 · 10−10 m, (b) d = 3 · 10−10 m, and (c) d =
4 ·10−10 m. The dot-dashed, solid, and dashed curves are predictions of the DFT,
MPB, and PB theories, respectively. The legend a and b in the bottom sub-figure
corresponds to two values of polyion radius, R = 8·10−10 m and R = 20·10−10 m,
respectively.
431
C.N.Patra, L.B.Bhuiyan
Figure 2. Zeta potentials for a 2:1/1:2 electrolyte as a function of surface charge
density of the polyion at different electrolyte concentrations (in Molar (mol/dm3))
with the ionic diameter (a) d = 2 · 10−10 m, (b) d = 3 · 10−10 m, and (c) d =
4 · 10−10 m. Notation as for figure 1.
432
Ionic distributions in cylindrical double layers
Figure 3. Zeta potentials for a 2:2 electrolyte as a function of surface charge
density of the polyion at different electrolyte concentrations (in Molar (mol/dm3))
with the ionic diameter (a) d = 2 · 10−10 m, (b) d = 3 · 10−10 m, and (c) d =
4 · 10−10 m. Notation as for figure 1.
433
C.N.Patra, L.B.Bhuiyan
electrostatic potential ψ(r) at the closest approach between a small ion and the
polyion and is given as
ζ = ψ(R + d/2) =
4π
ε
∫
∞
R+d/2
dr′r′ ln
(
R + d/2
r′
)
∑
α
qαρα(r′). (17)
The DFT, MPB, and PB zeta potentials are presented as a function of σ in
figures 1–3 for 1:1, 2:1/1:2, and 2:2 valency electrolytes, respectively. The sub-figures
a-c in each figure correspond to d = 2, 3, and 4 · 10−10 m, respectively, while in each
sub-figure the results for different c are shown. Some general trends of the results are
clear from the figures. Firstly, at low concentrations, the dependence of ζ on ionic
diameter is rather weak. For instance, the curves at c = 0.01 mol/dm3 corresponding
to different d are quite similar, both qualitatively and quantitatively. Secondly, the
magnitude of ζ decreases with salt concentration for all ionic valencies and all ionic
diameters. Thirdly, the classical mean field ζ generally tends to overestimate the
corresponding statistical mechanical results. Fourthly, the deviation between the
PB ζ and the DFT, MPB ζ increases as the ionic size decreases. These trends are
consistent with similar observations for the PDL [50,51,60]. The relative magnitude
of the deviation mentioned above is however, the least for 1:1 systems (cf. figure 1)
and increasing for asymmetric and higher valency systems (cf. figures 2 and 3). This
is of course a well known feature of the mean field result that the theory is more useful
for monovalent systems where the effect of the neglected inter-ionic correlations is
relatively smaller than in the presence of multivalent ions. In figure 1c (1:1 salt,
d = 4 · 10−10 m) the zeta potential corresponding to the bigger polyion radius
R = 20 ·10−10 m (curves b) is seen to lie above that for R = 8 ·10−10 m (curves a) for
all the theories. This is expected since for higher R the system approaches the PDL.
We note further that the MPB results for d = 4 · 10−10 m are reminiscent of results
seen in the earlier MPB application [28,29] with very similar physical parameters,
viz., d = 4.25 · 10−10 m. One of the more noteworthy features of figure 1 is the
consistency of the DFT and MPB predictions for all ionic sizes and all polyion radii,
the agreement between the two theories increasing as d increases from 2 · 10−10 m
through 3 ·10−10 m to 4 ·10−10 m. At the highest d treated here, the DFT and MPB
curves are virtually indistinguishable at c = 1 mol/dm3. A similar level of agreement
between the two approaches has been reported for the PDL [60].
Turning now to 2:1/1:2 electrolytes in figure 2, we notice immediately that the
behaviour of the zeta potential for 2:1 valency systems (monovalent counterions) is
very similar to that in 1:1 valency systems. This is a well established result in the
double layer literature, which testifies to the fact that it is the electrode-counterion
interaction that shapes the system characteristics. In particular, the DFT and MPB
results again reveal consistency although the differences between them for the two
smaller ionic diameters at the highest concentration are comparatively larger than
that for the earlier 1:1 system. The situation, however, changes for divalent coun-
terions (1:2 valency systems) when the zeta potential is seen to pass through a
minimum. This is true at almost all the concentrations and at all the ionic diame-
ters studied here. Although the theoretical predictions now reveal more deviations
434
Ionic distributions in cylindrical double layers
between them, the overall agreement between the two theories again improves as
the ionic diameter increases. In the absence of exact simulation data, it is difficult,
however, to make any definitive statement about the relative accuracy of either the
DFT or the MPB theory. The PB theory does not capture the above effects and is
not even in qualitative correspondence with the DFT and MPB.
Figure 4. Concentration profiles (lower panel) of counterions and coions around a
polyion of radius R = 8·10−10 m and surface charge density σ = 0.187 C/m2 for a
0.016 mol/dm3 1:1 electrolyte. The plots in the upper panel are the corresponding
reduced mean electrostatic potentials. The symbols are the MC simulations, and
the dot-dashed, solid, and dashed curves are predictions of the DFT, MPB, and
PB theories, respectively. MC data from reference [42].
As we move on to 2:2 electrolytes as shown in figure 3, the DFT and MPB zeta
potential shows a maximum and then starts decreasing. Expectedly, the 2:2 results
435
C.N.Patra, L.B.Bhuiyan
now show similarities with the 1:2 results and similar considerations as for the earlier
seen latter results apply. Such maxima or minima have been previously predicted
theoretically through simulations for the PDL [6,50,51,60] and the SDL [53,67–69],
and have also been observed earlier for the CDL [19,22,28,29]. The maximum or
minimum in the ζ for an electrolyte with a divalent counterion implies that the
differential capacitance can become infinite before becoming negative [50].
Figure 5. Concentration profiles (lower panel) of counterions and coions around a
polyion of radius R = 8·10−10 m and surface charge density σ = 0.187 C/m2 for a
0.016 mol/dm3 2:1 electrolyte. The plots in the upper panel are the corresponding
reduced mean electrostatic potentials. Notation as for figure 4. MC data from
reference [42].
In the next two figures (figures 4 and 5) we compare the theoretical results with
some published MC data [42] for ionic density distributions (in the form of local
436
Ionic distributions in cylindrical double layers
concentration profiles concα(r)[= cρα(r)/ρ0
α]) for 1:1 and 2:1 salts at c = 0.016
mol/dm3 and σ = 0.187 C/m2. Figure 4 shows the results for a 1:1 salt where
the reduced mean electrostatic potential ψ∗(r) [= βeψ(r)] is also given in the top
panel. The DFT and MPB local concentration profiles reproduce the MC data quite
accurately. Note also that these profiles show little structure. Not surprisingly, for
such a low concentration 1:1 system the PB results are also in good agreement with
the simulations. The predicted ψ∗(r)s also show similar consistency. A slightly more
structure begins to appear in the polyion-ion distributions for a 2:1 valency system
shown in figure 5. Again the DFT and MPB results generally agree well with the
simulation data, with the MPB co-ion profile tending to follow the MC data more
closely. The PB results now show a greater deviation than in the 1:1 case and predict
a more diffuse, thicker double layer.
Figure 6. Concentration profiles (lower panels) and the reduced mean electro-
static potential profiles (upper panels) of a 1 mol/dm3 1:1 electrolyte around a
polyion of radius R = 8 · 10−10 m and surface charge density σ = 0.187 C/m2.
The dot-dashed, solid, and dashed curves are predictions of the DFT, MPB, and
PB theories, respectively. The legend a, b, and c corresponds to the three values
of the ionic diameters, d = 4, 3, and 2 · 10−10 m, respectively.
To investigate the effect of ionic size on the double layer structure, we present,
in figures 6 and 7, the local concentration and potential profiles for 1:1 and 2:1
electrolytes, respectively, at different ionic diameters. These calculations are at c =
1 mol/dm3, and in each figure the panels a, b, and c correspond to d = 4, 3, and
437
C.N.Patra, L.B.Bhuiyan
2 · 10−10 m , respectively. A glance at the figures across the panels from left to right
reveals the principal effect of ionic size on structure – the double layer becomes
thicker and diffuse as the ionic diameter decreases. For example, in figure 6a, the
concentration profiles reach their bulk values at around r ∼ 5d from the polyion
axis. In figures 6b, 6c this occurs at r ∼ 7d and r > 8d, respectively. For 2:1 salts,
these numbers are r ∼ 5d for d = 4 ·10−10 m (figure 7a), r ∼ 6.5d for d = 3 ·10−10 m
(figure 7b), and r ∼ 8d for d = 2 · 10−10 m (figure 7c). This feature is corroborated
by the corresponding potential profiles shown. Decreasing the size of the ions leads
to smaller accumulation near the polyion, which in turn yields a more diffuse double
layer. The DFT and MPB ψ∗(r) for the 2:1 salt systems at d = 4, and 3 · 10−10 m
Figure 7. Concentration profiles (lower panels) and the reduced mean electro-
static potential profiles (upper panels) of 1 mol/dm3 2:1 electrolyte around a
polyion of radius R = 8 · 10−10 m and surface charge density σ = 0.187 C/m2.
Notation as for figure 6.
shown in figures 7a and 7b dip below zero near the polyion surface. This is indicative
of polyion overcharging, that is, more counterions accumulate in the vicinity of the
polyion than is necessary to screen the surface charge. This has been observed in
earlier double layer studies involving, besides the DFT and MPB, other theoretical
and numerical methods [22,26,24]. We refer the reader to a recent excellent review
on macroion overcharging by Quesada-Perez et al. [70] and references therein. At
a given electrolyte concentration overscreening is also a function of ionic size as it
is not observed at d = 2 · 10−10 m and c = 1 mol/dm3. This is not to say that
438
Ionic distributions in cylindrical double layers
overscreening or charge inversion does not occur for a 2:1 salt at this value of d or
for that matter for a 1:1 salt. Indeed there is evidence in the literature to suggest
that overscreening in these systems is possible at sufficiently high concentrations
[28,34,46]. The DFT and MPB results again show remarkable consistency for both
1:1 and 2:1 salts at all ionic diameters. As expected, the PB profiles are closer to their
statistical mechanical counterparts in 1:1 systems but deviate more in 2:1 systems.
Further, in the latter case the PB ψ∗(r) are monotonic in variance to the DFT and
MPB ψ∗(r) and do not show any overcharging effect. Evidently the phenomenon
has origins in the inter-ionic correlations and cannot be accounted for within the
traditional PB framework unless counterion binding is invoked [6,70].
Figure 8. Concentration profiles (lower panel) and the reduced mean electrostatic
potential profiles (upper panel) of 1 mol/dm3 1:2 electrolyte around a polyion of
radius R = 8 ·10−10 m and surface charge density σ = 0.187 C/m2 with the value
of the ionic diameter d = 4 · 10−10 m. Notation as for figure 6.
Increased electrostatic attraction between the polyion and a divalent counterion
can play a major role in characterizing the polyion-ion distributions and mean elec-
439
C.N.Patra, L.B.Bhuiyan
trostatic potential profiles. This is clearly noticeable in figures 8 and 9, which show
the structure of a double layer containing 1:2 and 2:2 electrolytes, respectively, at
c = 1 mol/dm3 and d = 4 · 10−10 m. The conspicuous structural features here are
the pronounced oscillations in the polyion-ion distributions and the appearance of
a substantial well in the mean potential around the distance where the local con-
centrations of the ions cross each other. These would imply a rather large charge
inversion. This is examined in some details in figures 10 and 11 where the 1:2 and
2:2 ψ∗(r) are shown for the three ionic sizes. Once again we notice that a decreasing
ionic size leads to a thickening of the double layer. Parallely, the magnitude of the
potential minimum also decreases. It is relevant to note that the comparative be-
haviour of the DFT and the MPB theory seen earlier [60] extends to these situations.
Figure 9. Concentration profiles (lower panel) and the mean electrostatic poten-
tial profiles (upper panel) of 1 mol/dm3 2:2 electrolyte around a polyion of radius
R = 8 · 10−10 m and surface charge density σ = 0.187 C/m2 with the value of the
ionic diameter d = 4 · 10−10 m. Notation as for figure 6.
440
Ionic distributions in cylindrical double layers
Figure 10. Mean electrostatic potential profiles of 1 mol/dm3 1:2 electrolyte
around a polyion of radius R = 8 · 10−10 m and surface charge density σ = 0.187
C/m2 with ionic diameter (a) d = 2 · 10−10 m, (b) d = 3 · 10−10 m, and (c)
d = 4 · 10−10 m. Notation as for figure 6.
441
C.N.Patra, L.B.Bhuiyan
Figure 11. Mean electrostatic potential profiles of 1 mol/dm3 2:2 electrolyte
around a polyion of radius R = 8 · 10−10 m and surface charge density σ = 0.187
C/m2 with value ionic diameter (a) d = 2 · 10−10 m, (b) d = 3 · 10−10 m, and (c)
d = 4 · 10−10 m. Notation as for figure 6.
442
Ionic distributions in cylindrical double layers
Although the predicted results are generally consistent over the range of the ionic
size studied in this work, the agreement diminishes a little at the lower diameters.
The PB theory on the other hand deviates substantially from the formal theories in
all cases with the deviation increasing with decreasing ionic size.
4. Concluding remarks
The density functional and the modified Poisson-Boltzmann theories have been
employed in an in-depth study of the equilibrium properties of a primitive model
cylindrical double layer with a particular focus on the effect of ionic size on double
layer structure. The principal findings are (i) the double layer tends to become more
diffuse, thicker as the ion-size decreases, and (ii) the classical Poisson-Boltzmann de-
scription of structure becomes progressively poorer with decreasing ion-size. Indeed
the PB results are not even qualitatively similar to the statistical mechanical results
in some instances, for example, at higher concentrations and/or when asymmetric
or higher valency electrolytes are present. These results re-affirm similar conclusions
reached earlier for the corresponding planar double layers. [50,51] Other relevant
results include (i) the appearance of a maximum or a minimum in zeta potentials
in a double layer containing electrolytes with multivalent counterions, and (ii) the
phenomenon of polyion overcharging in certain cases. The former clearly indicates a
change in sign in the differential capacitance in presence of multivalent counterions.
These features are tied to the inter-ionic correlations since such features cannot be
captured by the PB theory.
The relative behaviour of the DFT and MPB approaches is interesting. Over-
all the theoretical predictions are reasonably consistent over the range of physical
parameters studied. Similar behaviour was observed for these theories in their appli-
cation to the PDL [50,51]. Although in contrast to the planar situation the limited
availability of simulation data at the present parameters makes a more detailed
assessment of the theories difficult, the general consistency of the polyion – small
ion distribution functions and the mean electrostatic potentials out of the two very
different theories adds to the reliability of the results. Another detailed simulation
study of the CDL would be quite useful nonetheless, and such a study is planned
for the future.
Acknowledgements
We are very grateful to Professor C.W. Outhwaite for critically going through
the manuscript, his comments and suggestions, which have helped greatly in im-
proving the paper. Support of a National Foundation Grant 0137273 is grateful-
ly acknowledged. C.N.P. acknowledges Prof. A. Yethiraj for guiding him through
the initial stage of the work on DFT of CDL. C.N.P thanks Dr. S.K. Ghosh and
Dr. T. Mukherjee for their interest and encouragement. L.B.B. acknowledges an
institutional grant through FIPI, University of Puerto Rico.
443
C.N.Patra, L.B.Bhuiyan
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C.N.Patra, L.B.Bhuiyan
Вплив розміру іона на полііон-іон розподіли в
циліндричному подвійному шарі
К.Н.Патра 1,3 , Л.Б.Бхуян 2
1 Атомний центр досліджень, Бгабга, Мамбаі, Індія
2 Університет Пуерто Ріко, Пуерто Ріко
3 Факультет матеріалознавства та інженерії, університет Юти,
Солт Лейк Сіті, Юта, США
Отримано 12 липня 2004 р.
Структура електроліту навколо ізольованого циліндричного полііона
– циліндричного подвійного шару – досліджується методом функц-
іоналу густини та модифікованою теорією Пуассона-Больцмана.
Полііон моделюється як нескінченно довгий, жорсткий і непроникний
заряджений циліндр, тоді як електроліт складається з жорстких
іонів, які рухаються у неперервному діелектричному середовищі.
Результати для зеті-потенціалу, розподілу полііон-іон та середнього
електростатичного потенціалу отримані для широкого діапазону
фізичних умов, включаючи три різні діаметри: 2, 3 та 4·10−10 m.
У випадку дво-валентних контраіонів, зета-потенціали демонстр-
ують максимум або мінімум відносно густини поверхневого зар-
яду полііона. Профілі розподілів полііон-іон та середнього елек-
тростатичного потенціалу демонструють суттєві зміни як функц-
ії концентрації електроліту, валентності іонів електроліту та іонних
розмірів. Обидва теоретичні підходи в загальному є узгоджені між
собою і обидва передбачають явище зарядової інверсії, яке є відс-
утнє в рамках звичайної теорії Пуассона-Больцмана. Більше того,
обидві теорії добре відтворюють окремі результати Монте Карло
(для іонних розподілів), які є відомі з літератури.
Ключові слова: циліндричний подвійний шар, структура, теорія
функціоналу густини, модифікований Пуассон-Больцман
PACS: 82.70.Dd; 64.10.+h
446
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