Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory
The modified Poisson-Boltzmann theory of the restricted primitive model double layer is revisited and recast in a fresh, slightly broader perspective. Derivation of relevant equations follow the techniques utilized in the earlier MPB4 and MPB5 formulations and clarifies the relationship between th...
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irk-123456789-1570152019-06-20T01:28:46Z Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory Bhuiyan, L.B. Outhwaite, C.W. The modified Poisson-Boltzmann theory of the restricted primitive model double layer is revisited and recast in a fresh, slightly broader perspective. Derivation of relevant equations follow the techniques utilized in the earlier MPB4 and MPB5 formulations and clarifies the relationship between these. The MPB4, MPB5, and a new formulation of the theory are employed in an analysis of the structure and charge reversal phenomenon in asymmetric 2:1/1:2 valence electrolytes. Furthermore, polarization induced surface charge amplification is studied in 3:1/1:3 systems. The results are compared to the corresponding Monte Carlo simulations. The theories are seen to predict the “exact” simulation data to varying degrees of accuracy ranging from qualitative to almost quantitative. The results from a new version of the theory are found to be of comparable accuracy as the MPB5 results in many situations. However, in some cases involving low electrolyte concentrations, theoretical artifacts in the form of un-physical “shoulders” in the singlet ionic distribution functions are observed. В роботi переглядається i представляється у новому, бiльш широкому свiтлi модифiкована теорiя Пуасона-Больцмана (МТПБ) обмеженої примiтивної моделi подвiйного шару. На основi попереднiх формулювань МТПБ4 i МТПБ5 виводяться вiдповiднi рiвняння, якi дозволяють краще зрозумiти зв’язок мiж цими формулюваннями. МТПБ4, МТПБ5 i нове формулювання теорiї застосовуються для аналiзу структури i явища змiни знаку заряду в електролiтах з асиметричною валентнiстю 2:1/1:2. Вивчається також явище iндукованого поляризацiєю збiльшення поверхневої густини заряду у системах 3:1/1:3. Проводиться порiвняння з даними моделювання методом Монте-Карло. Показано, що наведенi теорiї вiдтворюють данi моделювання з рiзним ступенем точностi вiд якiсного до майже кiлькiсного. Результати нового варiанту теорiї в багатьох випадках практично не вiдрiзняються вiд результатiв МТПБ5. Однак при певних умовах, зокрема при низьких концентрацiях електролiтiв, спостерiгаються теоретичнi артефакти у формi нефiзичних “плечей” в унарних функцiях розподiлу iонiв. 2017 Article Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory / L.B. Bhuiyan, C.W. Outhwaite // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33801: 1–15. — Бібліогр.: 65 назв. — англ. 1607-324X PACS: 82.45.Fk, 61.20.Qg, 82.45.Gj DOI:10.5488/CMP.20.33801 arXiv:1710.01537 http://dspace.nbuv.gov.ua/handle/123456789/157015 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 modified Poisson-Boltzmann theory of the restricted primitive model double layer is revisited and recast
in a fresh, slightly broader perspective. Derivation of relevant equations follow the techniques utilized in the
earlier MPB4 and MPB5 formulations and clarifies the relationship between these. The MPB4, MPB5, and a
new formulation of the theory are employed in an analysis of the structure and charge reversal phenomenon
in asymmetric 2:1/1:2 valence electrolytes. Furthermore, polarization induced surface charge amplification is
studied in 3:1/1:3 systems. The results are compared to the corresponding Monte Carlo simulations. The theories are seen to predict the “exact” simulation data to varying degrees of accuracy ranging from qualitative to
almost quantitative. The results from a new version of the theory are found to be of comparable accuracy as the
MPB5 results in many situations. However, in some cases involving low electrolyte concentrations, theoretical
artifacts in the form of un-physical “shoulders” in the singlet ionic distribution functions are observed. |
| format |
Article |
| author |
Bhuiyan, L.B. Outhwaite, C.W. |
| spellingShingle |
Bhuiyan, L.B. Outhwaite, C.W. Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory Condensed Matter Physics |
| author_facet |
Bhuiyan, L.B. Outhwaite, C.W. |
| author_sort |
Bhuiyan, L.B. |
| title |
Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory |
| title_short |
Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory |
| title_full |
Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory |
| title_fullStr |
Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory |
| title_full_unstemmed |
Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory |
| title_sort |
charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified poisson-boltzmann theory |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157015 |
| citation_txt |
Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory / L.B. Bhuiyan, C.W. Outhwaite // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33801: 1–15. — Бібліогр.: 65 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bhuiyanlb chargereversalandsurfacechargeamplificationinasymmetricvalencerestrictedprimitivemodelplanarelectricdoublelayersinthemodifiedpoissonboltzmanntheory AT outhwaitecw chargereversalandsurfacechargeamplificationinasymmetricvalencerestrictedprimitivemodelplanarelectricdoublelayersinthemodifiedpoissonboltzmanntheory |
| first_indexed |
2025-07-14T09:21:41Z |
| last_indexed |
2025-07-14T09:21:41Z |
| _version_ |
1837613608494694400 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 3, 33801: 1–15
DOI: 10.5488/CMP.20.33801
http://www.icmp.lviv.ua/journal
Charge reversal and surface charge amplification in
asymmetric valence restricted primitive model
planar electric double layers in the modified
Poisson-Boltzmann theory
∗
L.B. Bhuiyan1, C.W. Outhwaite2
1 Laboratory of Theoretical Physics, Department of Physics, Box 70377, University of Puerto Rico,
San Juan, Puerto Rico 00936-8377, USA
2 Department of Applied Mathematics, University of Sheffield, Sheffield S3 7RH, UK
Received April 27, 2017, in final form May 31, 2017
The modified Poisson-Boltzmann theory of the restricted primitive model double layer is revisited and recast
in a fresh, slightly broader perspective. Derivation of relevant equations follow the techniques utilized in the
earlier MPB4 and MPB5 formulations and clarifies the relationship between these. The MPB4, MPB5, and a
new formulation of the theory are employed in an analysis of the structure and charge reversal phenomenon
in asymmetric 2:1/1:2 valence electrolytes. Furthermore, polarization induced surface charge amplification is
studied in 3:1/1:3 systems. The results are compared to the corresponding Monte Carlo simulations. The theo-
ries are seen to predict the “exact” simulation data to varying degrees of accuracy ranging from qualitative to
almost quantitative. The results from a new version of the theory are found to be of comparable accuracy as the
MPB5 results in many situations. However, in some cases involving low electrolyte concentrations, theoretical
artifacts in the form of un-physical “shoulders” in the singlet ionic distribution functions are observed.
Key words: electric double layer, imaging, restricted primitive model, modified Poisson-Boltzmann theory,
Monte Carlo simulations
PACS: 82.45.Fk, 61.20.Qg, 82.45.Gj
1. Introduction
Electric double layers play a key role in a wide variety of phenomena, in the fields diverse as colloids,
physical and biological systems, and electrochemistry [1–8]. Recent years have seen an increasing
relevance of double layers in DNA sequencing [9, 10], drug delivery [11, 12], and green technology,
viz., designing novel energy storage devices as in super-capacitors (see for example, reference [13]).
An electric double layer appears when a system of charged particles in a fluid is in the neighborhood
of a charged surface. Various models and theories have been developed depending upon the physical
system being treated. A simple model of an electrolyte next to a plane charged surface is a basic model
in many areas. Gouy and Chapman [14, 15] analysed this model, based on the Poisson-Boltzmann (PB)
equation, where the electrolyte was treated as a system of point ions moving in a medium of constant
permittivity. Later on, Stern [16] proposed to lower the permittivity in the region immediately next to
the planar electrode (Stern layer) in order to bring the computed double layer capacitance in line with
the experiments. At not too high surface charge and electrolyte concentration for univalent ions, the
Gouy-Chapman-Stern (GCS) theory describes reasonably well the diffuse part of the electric double
layer. However, double layer research over the last four decades involving formal statistical mechanical
∗It is a pleasure to dedicate this paper to the memory of Dr. J.P. Badiali.
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33801-1
https://doi.org/10.5488/CMP.20.33801
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
L.B. Bhuiyan, C.W. Outhwaite
theories, simulations, and experiments has shown the shortcomings of the theory in capturing many
physical phenomena (see for example, the review [8]). As a case in point, the GCS theory cannot explain
the two phenomena that are the focus of this work, namely, charge reversal (CR) (or charge inversion
or overcharging) and surface charge amplification (SCA). In CR, the local density of the counterions
overcompensate the surface charge, while in SCA, in the absence of specific ion adsorption, there is an
apparent increase in the surface charge due to an excess of coions.
CR was indicated in the simulation work of Torrie and Valleau [17, 18], Snook and van Megen [19]
and in the modified Poisson-Boltzmann (MPB) and hypernetted-chain/mean spherical approximation
(HNC/MSA) theories [20]. The important feature of these works was allowing the ions to have size
and accounting for the approximations in the PB equation. There is now a large body of theoretical and
experimental work on charge reversal in colloidal systems due to its importance in both equilibrium
and non-equilibrium situations [1, 2, 21–29]. Surface charge amplification is more counterintuitive than
CR. It can occur when the ions have asymmetry in size and charge [30–34]. Imaging can also promote
SCA, even when the ions are of equal size [35–37]. Here, we discuss the MPB theory with imaging for
equisized ions with 1:2/2:1, 1:3/3:1 electrolytes.
2. Model and theory
The electric double layer is modelled by a restricted primitive model electrolyte in the neighborhood
of a uniformly charged plane electrode. The solute molecules are hard spheres of diameter a with point
charges at their center moving in a medium of constant relative permittivity εr , while the electrode has a
constant relative permittivity of εw. The analysis presented here builds upon the earlier formulations of
MPB4 [38, 39] andMPB5 [40] versions of the theory and clarifies the relation between them. The present
approach is a bit more general since the MPB5 situation will be seen to follow as a special limiting case.
The connections to the MPB4 are also explored.
At a perpendicular distance x from the electrode into the solution, the mean electrostatic potential
ψ(x) satisfies Poisson’s equation
d2ψ
dx2 = −
1
ε0εr
∑
s
esnsgs(x), (1)
where the sum is over the ion species, ε0 is the vacuum permittivity, es is the charge of an ion of species s,
ns and gs(x) are the mean number density and singlet distribution function for an ion of species s,
respectively. Kirkwood’s charging process [41] for an ion i at x1 gives
gi(x1) = ζi(x1) exp
[
− βeiψ(x1) − β
ei∫
0
lim
r→0
(φ∗i − φ
∗
ib)dei
]
, (2)
where
φ∗i = φi(1; 2) −
ei
4πε0εrr
(3)
with φ∗
ib being the bulk value of φ∗i . Here, ζi(x1) = gi(x1/ei = 0) is the exclusion volume term,
φi = φi(1; 2) is the fluctuation potential at r2 for an ion at r1, r = r12, and β = 1/(kBT) (kB is the
Boltzmann’s constant and T is the absolute temperature).
The fluctuation potential can be shown to satisfy the following set of differential equations [40]
∇2φi = 0, x < a/2, (4)
∇2φi = −
d2φi
dx2 , r < a, (5)
∇2φi = κ
2(x)φi , x > a/2, r > a, (6)
where x = x2 and κ(x) is the local Debye-Hückel parameter defined by
κ2(x) =
β
ε0εr
∑
s
e2
snsgs(x). (7)
33801-2
Charge reversal and surface charge amplification using the modified Poisson-Boltzmann theory
Equations (4) and (5) are exact, while equation (6) involves two approximations. These are Loeb’s
closure [42], which gives a closed system of equations for φi , and the linearization of the equation
involving Loeb’s closure. The boundary conditions associated with equations (4)–(6) are φi and ∂φi/∂n
continuous (n denotes normal) at x = a/2 and r = a, while at the electrode φi is continuous but
εr∂φi/∂x+ = εw∂φi/∂x−. Approximate solutions of equations (4)–(6) have been considered for both
point ions [43] and finite size ions [40]. The point ion solution suggests an appropriate solution in the
regions x < 0, 0 < x < a/2. From the analysis of reference [43], the limiting case of having no distance
of closest approach and k → 0 gives
φi =
ei
4πε0εrr
(1 + f ), x < 0, (8)
φi =
ei
4πε0εr
(
1
r
+
f
r∗
)
, x > 0, (9)
where f = (εr − εw)/(εr + εw) and r∗ is the distance of the image of ei in the electrode from the field point.
These two results suggest that an appropriate approximation for the ion size result for the equation (4) is
φi = C
(
1
r
+
f
r
)
, x < 0, (10)
φi = C
(
1
r
+
f
r∗
)
, 0 < x < a/2, (11)
which were used in reference [40].
Previous work [38, 40, 44–47] has shown that a good approximation to the solution of equation (6) is
φi = B
exp(−κr)
r
+ B∗
exp(−κr∗)
r∗
, x > a/2, r > a, (12)
where κ = κ(x1). The point ion result, with a distance of closest approach, indicates a complex situation
such that B, B∗, C can contain the imaging factor f .
The use of the approximate solutions (10)–(12) mean that the boundary conditions cannot be satisfied.
An approximate fitting of the boundary conditions was considered in [40], giving theMPB5 theory. Here,
we adopt the approach of reference [38]. The application of Green’s second identity to φ∗i and 1/r in the
exclusion sphere V , where V is a truncated sphere if a/2 < x < 3a/2, gives, on using equation (5)
4π lim
r→0
φ∗i =
∫
V
(
1
r
−
1
a
)
∇2ψdV +
1
a2
∫
SI
φidS −
1
a
∫
SW
∂φi
∂n
dS
+
∫
SW
[
1
r
∂φi
∂n
− φi
∂
∂n
(
1
r
)]
dS −
ei
aε0εr
. (13)
Here, SW and SI are respectively the wall and double layer surface portions of the truncated sphere. The
integrals over SW vanish for x > 3a/2, and SI is then the complete spherical surface. For reference,
the values of the integrals appearing in equation (13), using the approximations for φi , are given in the
appendix.
To determine the relations between B, B∗ and C we use Gauss’s theorem for ion i∫
S
∂φi
∂n
dS = −
ei
ε0εr
−
∫
V
∇2ψdV . (14)
Thus, combining the expressions resulting from the evaluation of equations (13), (14)with equation (2)
33801-3
L.B. Bhuiyan, C.W. Outhwaite
gives
gi(x) = ζi(x) exp
{
−
1
2
βei
[ (
ei
4πε0εr
)
(F − F0) + Fψ(x + a) + Fψ(x − a)
−
(F − 1)
a
x+a∫
x−a
ψ(X)dX
]}
, (15)
where for a/2 < x < 3a/2
F =
A
D
, (16)
A = (2x + a)e−y +
C
B
(3a − 2x) +
C
B
a f
x
(a − 2x + s) +
a
κx
B∗
B
[
e−κs − e−κ(2x+a)
]
, (17)
D = (1 + y)(2x + a)e−y +
C
B
(3a − 2x) +
C
B
f (s − 2a) −
a
κx
B∗
B
[
(1 + y)e−κ(2x+a)
−
1
a
(a + κsa − κsx)e−κs
]
(18)
and for x > 3a/2
F =
1 + 1
2κx
(
B∗
B
)
e−κ(2x−a) sinh y
1 + y − 1
2κx
(
B∗
B
)
e−κ(2x−a)(y cosh y − sinh y)
, (19)
where F0 = limx→∞ F, y = κa and s =
√
a2 + 2xa.
At x = 3a/2, we have that F is continuous but ∂F/∂x is discontinuous in general. The expression for
F depends upon the ratios C/B, B∗/B for a/2 < x < 3a/2 and on the single ratio B∗/B for x > 3a/2.
Putting C = B exp(−y), B∗ = B f exp(−y + κs) in a/2 < x < 3a/2 and B∗ = B f exp y in x > 3a/2 gives
the MPB5 theory [40]. Early calculations of F did not consider the exclusion region 0 < x < a/2 for φi
so that F is defined in the two regions a/2 < x < a and x > a, for example the MPB4 theory [38, 39]. In
these earlier works, the value of F differs from that of the MPB5 when x > 3a/2. On putting B∗ = f B
in equation (19), the F of MPB4 for x > a is derived. This suggests an alternative F, analogous to that
for MPB4, which is given by C = B in equations (17), (18) and B∗ = f B in equations (17)–(19). There
are many possibilities of choosing the ratios C/B, B∗/B, depending upon the chosen criterion. However,
a better approach may be to improve upon the analysis of equations (4), (5) and the nonlinear version of
equation (6).
Various expressions have been used for evaluating the exclusion volume term ζi(x) in equation (2)
[38, 39, 48]. The most accurate exclusion volume terms are those based on the BBGY expansion [39] or
charging up the wall and introducing the non-uniform direct correlation function [48]. Here, we will use
the BBGY approach, where for x > a/2
ζi(x) = exp
−2π
x∫
∞
∑
s
nsζis(a)
y+a∫
max( a2 , y−a)
(X − y)gs(X) exp[−βeiΦ(y, X)]dXdy
(20)
with ζi(x) = 0 for x < a/2. The contact value ζi j(a) between ions i and j is that proposed by Fischer [49]
while Φ(y, X) = φ(1; 2/es = 0/r12 = a).
2.1. Monte Carlo simulations
The Monte Carlo simulations were performed in the canonical (N,V,T) ensemble using the standard
and widely used Metropolis algorithm [50, 51]. The techniques were the same as those used in some of
our earlier works (see for example, references [52, 53]). The MC cell was a rectangular parallelepiped of
dimensions Lx , Ly , and Lz , with Ly = Lz . One of the y-z faces was a charged wall, viz., the electrode
33801-4
Charge reversal and surface charge amplification using the modified Poisson-Boltzmann theory
with a uniformly distributed surface charge at x = 0, while the other at x = Lx was a neutral wall. The
surface charge density on the electrode was determined from
σ = −
(N+ |Z+ | − N− |Z− |)e
L2
y
, (21)
where N+ and N− are the number of cations and anions, respectively. It is to be noted that with the above
definition of σ, the local electro-neutrality at the wall is built in. In order to simulate the semi-infinite
system, conventional minimum image techniques along with periodic boundary conditions in the y and
z directions were implemented [51]. The effect of the long range nature of the Coulomb interactions was
treated by the parallel charge sheets method pioneered by Torrie and Valleau [17]. This procedure was
later examined and improved upon by Boda et al. [54], who also confirmed the validity of the method for
charge hard spheres.
The situation when the permittivity of the medium of the charged wall differs from that of the
electrolyte solution has also been treated by Torrie at al. [55] and their procedure adopted here. These
authors regarded the polarizability of the electrode as a classic electrostatic image problem, viz., the total
surface charge density σ has a contribution from the fictitious image charge − fσ due to the polarization
of the electrode. In passing we would like to note that the Torrie and co-workers did their simulations in
the grand canonical (µ,V,T) ensemble unlike the canonical ensemble in the present case.
In the canonical ensemble, the bulk target concentration can only be achieved by a trial-and-error
method of adjusting the cell length Lx and/or the number of particles. We used both of these methods
where necessary and imposed a tolerance of less than 2% error in reproducing the bulk concentration.
Typically, around 400 particles were simulated and approximately 108 configurations sampled out of
which the first 107 (10% of the total number of configurations) were used for system equilibration before
statistics were taken.
3. Results
The properties of the diffuse double layer are found by solving the Poisson equation, equations (1) and
(15), for the mean electrostatic potential and hence from equation (15) the singlet distribution function.
Three different values of F were treated in detail, these corresponding to those of MPB4, MPB5, and the
new F given by C = B, B∗ = f B in equations (16)–(19), respectively. A quasi-linearisation technique
was used [38, 56] which has been successfully employed in past investigations. The parameters chosen
were those used by Wang [37] who simulated charge asymmetric 1:3 and 3:1 RPM electrolytes with
imaging. For the RPM electrolyte, we treat 1:2, 2:1 and some 1:3, 3:1 valences with a = 4 · 10−10 m,
εr = 80, T = 298 K while the wall is characterized by εw = 2, 80 or ∞. When εw , εr , imaging occurs
which is repulsive for εw < εr and attractive for εw > εr . The value εw = 2, which is relevant for a medium
such as silica, while εw = ∞ represents a metallic electrode.
We start by looking in detail at the predictions of the MPB5 theory for 2:1 electrolytes with and
without imaging. Figures 1 and 2 treat the repulsive ( f = 0.9513) and attractive ( f = −1) cases,
respectively, at electrolyte concentration c = 0.24 mol/dm3 and surface charge density σ = −0.02 C/m2.
Displayed are the singlet distribution functions gs(x/a), the dimensionless mean electrostatic potential
ψ∗(x/a) = |e|βψ(x/a) and the integrated charge density q(x/a) defined by
q(x) = σ +
∑
s
nses
x∫
0
gs(y)dy. (22)
The integrated charge gives a measure of the total charge within a distance x of the electrolyte and enables
a succinct illustration of the two phenomena CR and SCA.
For no imaging, the simulated singlet distribution functions apparently display a monotonous be-
haviour of the co and counter ions, with a corresponding monotonous behaviour of the mean electrostatic
potential and the integrated charge. The MPB5 theory has a small shoulder in the counterion gs(x) at
x ∼ 3a/2 arising from the discontinuity of ∂F/∂x at x = 3a/2. This is an unfortunate feature at low
33801-5
L.B. Bhuiyan, C.W. Outhwaite
Figure 1. (Color online) The MPB5 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 2:1 electrolyte.
The symbols represent MC data, while the lines
represent the MPB results. The notations “co” and
“cntr” stand for “coion” and “counterion”, respec-
tively. Legend as given in the figure.
Figure 2. (Color online) The MPB5 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 2:1 electrolyte. No-
tation as in figure 1 and legend as given in the figure.
concentrations which has been observed earlier [40]. The influence of this feature is not observed in ψ∗(x)
since ψ∗(x) and its derivative are continuous everywhere. The principal effect of imaging in the distribu-
tion functions for the repulsive situation is that the divalent counterion is attracted for large x but reduces
and becomes small near the wall, while a slight reduction is seen in the monovalent coion. Conversely
for the attractive situation, the coion profile increases to a value greater than 1 near the electrode, and
the counterion profile rapidly increases in the neighborhood of the electrode. These imaging effects are
reflected in the reduction (repulsive case) and increase (attractive case) over the no imaging situation, of
the mean electrostatic potential and integrated charge. A very small CR is seen when f = −1. The MPB5
accurately predicts ψ∗(x) and q(x), and apart from the shoulder feature, the gs(x) are quantitatively or
qualitatively correct.
At the higher concentration c = 1 mol/dm3 and surface charge σ = −0.2 C/m2, there is a clear change
in the electrolyte properties, viz., figures 3, 4. The striking feature is the obvious damped oscillatory
behaviour of various functions for all three situations ( f = 0.9513, f = 0, f = −1). The onset of this
oscillatory behaviour mainly depends, at a small surface charge, upon the value of y0 = κ0a, where κ0 is
33801-6
Charge reversal and surface charge amplification using the modified Poisson-Boltzmann theory
Figure 3. (Color online) The MPB5 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 2:1 electrolyte. No-
tation as in figure 1 and legend as given in the figure.
Figure 4. (Color online) The MPB5 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 2:1 electrolyte. No-
tation as in figure 1 and legend as given in the figure.
the bulk Debye-Hückel constant. A damped oscillatory behaviour in ψ∗(x) is predicted in the linear MPB
theory for y0 > 1.2412 [57] for symmetric valences and for y0 > 1.15 [45] for 2:1/1:2 valences. However,
the latter is probably too high since in simulation and the HNC, the gs(x) value for 1:2 homogeneous
electrolytes implies y0 ≈ 0.76 [58]. Here, y0 = 2.26, while the previous concentration of 0.24 mol/dm3
gave y0 = 1.11, so that a qualitative change in the behaviour between the two concentrations is predicted
by the MPB theories. With no imaging, the initial maximum in ψ∗(x) is at x ∼ a with CR starting at
about the same value of x and having a maximum at x ∼ 5a/4. Both the repulsive and attractive cases
are qualitatively similar to that for no imaging, showing small but different deviations from f = 0. For
example, with the mean potential, the first maximum is shifted to a larger x for the repulsive case, but
closer to the electrode with a larger maximum for the attractive case. At this higher concentration, there is
no shoulder in theMPB5 gs(x) at x ∼ 3a/2, and theMPB5 gives an accurate picture of all three functions.
Clearly, the effect of discontinuity in ∂F/∂x at x = 3a/2 gets masked at high concentrations. The MPB4
theory seems a possible alternative to the MPB5 theory as it does not have this discontinuity at x = 3a/2.
Figures 5 and 6 show the MPB4 for the 2:1 attractive cases. Clearly, the theory is inadequate, especially
for the lower concentration. The main inaccuracy is the unphysical upturn in the coion distribution
function close to the wall. A further possible alternative theory is that given by C = B, B∗ = f B in
33801-7
L.B. Bhuiyan, C.W. Outhwaite
Figure 5. (Color online) The MPB4 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 2:1 electrolyte. No-
tation as in figure 1 and legend as given in the figure.
Figure 6. (Color online) The MPB4 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 2:1 electrolyte. No-
tation as in figure 1 and legend as given in the figure.
equations (16)–(19). Looking again at the 2:1 attractive cases, figures 7 and 8, this theory incorrectly
predicts (i) an unphysical rise in the coion gs(x) near the wall, (ii) a shoulder in the counterion gs(x)
for no imaging at the lower concentration. Apart from the unphysical behaviour (i), the theory based on
the new F gives a good representation of the mean potential and integrated charge at c = 1 mol/dm3.
Unfortunately, neither the MPB4 nor that based on the new F is an improvement on the MPB5.
Figures 9 and 10 consider the prediction of the MPB5 theory for the 1:2 electrolyte where now the
coion is divalent. Counterion attraction with the wall plays a dominant role in shaping the structural
properties of the electric double layer and hence many of the 1:2 double layer properties are similar
to those of the 1:1 case, while the 2:1 properties resemble that for the 2:2 situation (see for example,
reference [59]). Thus, it comes as no surprise that the predicted structure for the 1:2 cases are relatively
closer to the simulations than that seen earlier for the 2:1 cases. Only attractive imaging is treated in
figures 9, 10 since any deviation from no imaging tends to be greater than those for the repulsive case. At
a low concentration, figure 9, the coion is attracted near the wall, and since it is divalent, it has a greater
contact value than in the monovalent case in figure 2. The difference in the valence of the counterion
means that the deviations from no imaging for the mean potential and an integrated charge are less and
oppositely directed. At a higher concentration, figure 10, the imaging has little effect. There is still a
33801-8
Charge reversal and surface charge amplification using the modified Poisson-Boltzmann theory
Figure 7. (Color online) The MPB (new F)
electrode-ion singlet distribution functions gs(x/a)
(lower panel), the mean electrostatic potential
ψ∗(x/a) (middle panel), and the integrated charge
q(x/a) (upper panel) as functions of x/a for a 2:1
electrolyte. Notation as in figure 1 and legend as
given in the figure.
Figure 8. (Color online) The MPB5 and MPB
(new F) electrode-ion singlet distribution functions
gs(x/a) (lower panel), the mean electrostatic po-
tential ψ∗(x/a) (middle panel), and the integrated
charge q(x/a) (upper panel) as functions of x/a for
a 2:1 electrolyte. Notation as in figure 1 and legend
as given in the figure.
damped oscillatory feature in the functions, with CR occurring, but it is not as pronounced as for the
divalent counterion. Overall, the MPB5 theory accurately predicts the 1:2 double layer, apart from the
behaviour of the coion in the neighborhood of x = 3a/2. The alternative MPB4 and new F theories
perform well in the 1:2 case.
Lastly, we briefly look at the MPB theory for a 1:3 and 3:1 electrolyte with the imaging at c =
0.24 mol/dm3 and σ = −0.02 C/m2. Replacing the divalent coion by a trivalent ion leads to CR
( f = 0.9513, f = 0) and also to SCA when there is an attractive imaging ( f = −1), viz., figures 11, 12.
Alternatively, when the trivalent ion is the counterion there is again CR and SCA, but now SCA occurs
when there is a repulsive imaging and CR for an attractive imaging, viz., figure 13. MC simulations [37]
indicate that theMPB5 prediction of SCA for no imaging is incorrect. The CR is expected as y0 = 1.5636,
while the counterintuitive SCA arises solely through polarization forces. In the 1:3 situation, the attraction
of the coion image overcomes the repulsion of the surface charge, while for the 3:1 electrolyte, a much
larger repulsive force acting on the counterion is sufficient to neutralise the screening in the neighborhood
of the wall. SCA does not occur for either 1:2 or 2:1 electrolytes at these parameters. Higher valences
provide a stringent test for theories. The discrepancies in gs(x)mean that the MPB5 and the new F theory
predict both CR and SCA with varying degrees of success.
33801-9
L.B. Bhuiyan, C.W. Outhwaite
Figure 9. (Color online) The MPB5 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 1:2 electrolyte. No-
tation as in figure 1 and legend as given in the figure.
Figure 10. (Color online) The MPB5 electrode-ion
singlet distribution functions gs(x/a) (lower panel),
the mean electrostatic potential ψ∗(x/a) (middle
panel), and the integrated charge q(x/a) (upper
panel) as functions of x/a for a 1:2 electrolyte. No-
tation as in figure 1 and legend as given in the figure.
4. Conclusions
In this paper, we have looked again at the formulation of a modified Poisson-Boltzmann equation for
the planar electric double layer formed by a restricted primitive model electrolyte. The derivation of the
MPB equation follows the pattern and techniques utilized in some of our earlier works [38, 39] but is a
bit more general. Our focus was on obtaining a general expression for the quantity F, a key ingredient
in the theory, which incorporates important aspects of inter-ionic correlations. We have shown that an
earlier version of the theory, viz., MPB5 [40], follows as a special case of the generalized F. Although
the MPB4 version does not follow directly, the relationship between these two versions can be clearly
assessed. With such features built in, the theory is more flexible and it is likely that the MPB equation
will become more adaptable to particular physical situations.
The MPB theory does not satisfy exactly the contact condition [60–62] which holds for no imaging.
Previous work [63–65] has shown that the MPB5 theory fairly accurately satisfies the contact condition
for 1:1 electrolytes and gives a good agreement for 1:2/2:1 and 2:2 electrolytes. A similar conclusion
holds in general when the new F is used, but the MPB4 theory can be poor. Adapting the MPB approach
to satisfy the contact condition would be useful for inner consistency. However, for the present univalent
and divalent RPM electrolytes, the MPB5 theory has shown to be in good overall agreement with both
structural and thermodynamic simulation properties [59].
33801-10
Charge reversal and surface charge amplification using the modified Poisson-Boltzmann theory
Figure 11. (Color online) The MPB5 and MPB (new F) electrode-ion singlet distribution functions
gs(x/a) (lower panel) and the mean electrostatic potential ψ∗(x/a) (upper panel) as functions of x/a
for a 1:3 electrolyte. Notation as in figure 1 and legend as given in the figure. The MC data are from
reference [37].
Figure 12. (Color online) The MPB5 and MPB (new
F) integrated charge q(x/a) as a function of x/a for
a 1:3 electrolyte. The symbols represent MC data,
while the lines represent the MPB results. Legend
as given in the figure. The MC data are from refer-
ence [37].
Figure 13. (Color online) The MPB5 integrated
charge q(x/a) as a function of x/a for a 3:1 elec-
trolyte. The symbols represent MC data, while the
lines represent the MPB results. Legend as given in
the figure. The MC data are from reference [37].
We have studied the effect of imaging on 1:2, 2:1 valence electrolytes with the emphasis on the
phenomenon of charge reversal. Also briefly treated were 1:3, 3:1 electrolytes where, besides the charge
inversion, polarization can produce the counterintuitive surface charge amplification. The conclusions
are mainly twofold:
(a) structural and charge reversal behaviour are generally well described by the various MPB theories
33801-11
L.B. Bhuiyan, C.W. Outhwaite
compared to MC simulations. For the 1:3, 3:1 systems, the theories are poor relative to the 1:2, 2:1 cases,
which is not unexpected at the higher multi-valence situations. Furthermore, the theoretical predictions
tend to be closer to the simulation data formonovalent counterions rather than formultivalent counterions,
a feature that is well known in the double layer literature.
(b) The new formulation of F allows for different choices to be made for F. One of the new F’s
analyzed gives reasonable results that in some situations are at a par with MPB5. However, there are
issues, the main one being the appearance of an nonphysical shoulder in the singlet distribution functions
at lower concentrations, similar to that seen in MPB5.
The shoulders are artifacts of the theory,which come fromdiscontinuities in the slope of the fluctuation
potential φ at x = 3a/2. In the analysis outlined, although φ is continuous across the planar surface of
the truncated exclusion sphere (cf. section 2), the normal derivative is not continuous. The continuity
of the normal derivative may well be achieved by an appropriate choice of B, B∗, C in equations 16,
19. It is to be seen if such choices in the present approach will give a simple means of eliminating the
shoulder in the distribution function. Further analytic work along the lines of MPB5 may be difficult, so
that eventually the best approach may well be a numerical evaluation of the fluctuation potential.
Acknowledgements
We would like to thank Dr. Z.-Y. Wang of the School of Optoelectronic Information, Chongqing
University of Technology, People’s Republic of China, for making available to us the numerical data
from his Monte Carlo simulations.
Appendix
Evaluation of the integrals in equations (13), (14) using the expressions (11) and (12) for the fluctuation
potential φi in a/2 < x < 3a/2∫
SI
φdS = π
{
B(2x + a)e−y +
aB∗
κx
[
e−κs − e−κ(2x+a)
]}
, (23)
∫
SW
∂φ
∂n
dS = πC
[
2x − 3a
a
+
2 f
s
(
s − x −
a
2
)]
, (24)
∫
SW
[
1
r
∂φ
∂n
− φ
∂
∂n
(
1
r
)]
dS =
πC f
xas
(
a2 − 2x2 + ax + as
)
, (25)
∫
SI
∂φ
∂n
dS = π
{
−
B
a
(1 + y)(2x + a)e−y +
(
B∗
κx
) [
(1 + y)e−κ(2x+a) −
(
1
a
)
(a + ys − κsx)e−κs
]}
(26)
and in x > 3a/2 ∫
S
∂φ
∂n
dS = 2π
[
−2B(1 + y)e−y +
(
B∗e−2κx
κx
)
(y cosh y − sinh y)
]
, (27)∫
S
φdS = 2πa
[
2Be−y +
(
B∗e−2κx
κx
)
sinh y
]
. (28)
The mean electrostatic potential integrals are∫
V
∇2ψdV = 2πa
[
ψ(x + a) + ψ(x − a) −
1
a
x+a∫
x−a
ψ(X)dX
]
, (29)
33801-12
Charge reversal and surface charge amplification using the modified Poisson-Boltzmann theory∫
V
∇2ψ
r
dV = 2π [ψ(x + a) + ψ(x − a) − 2ψ(x)] , (30)
which for a truncated sphere are to be evaluated in conjunction with the linear result in 0 < x < a/2,
ψ(x) = ψ(0) + x
[
dψ
dx
]
x=0
. (31)
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Charge reversal and surface charge amplification using the modified Poisson-Boltzmann theory
Змiна знаку i збiльшення поверхневої густини заряду у
валентно-асиметричнiй обмеженiй примiтивнiй моделi
планарних електричних подвiйних шарiв в рамках
модифiкованої теорiї Пуасона-Больцмана
Л.Б. Буян1, К.У. Аутвейт2
1 Лабораторiя теоретичної фiзики, вiддiл фiзики, А/с 70377, Унiверситет Пуерто-Рiко, Сан Хуан, Пуерто-Рiко
2 Вiддiл прикладної математики, унiверситет Шеффiлда,Шеффiлд S3 7RH, Велика Британiя
В роботi переглядається i представляється у новому, бiльш широкому свiтлi модифiкована теорiя
Пуасона-Больцмана (МТПБ) обмеженої примiтивної моделi подвiйного шару. На основi попереднiх фор-
мулювань МТПБ4 i МТПБ5 виводяться вiдповiднi рiвняння, якi дозволяють краще зрозумiти зв’язок мiж
цими формулюваннями. МТПБ4, МТПБ5 i нове формулювання теорiї застосовуються для аналiзу структу-
ри i явища змiни знаку заряду в електролiтах з асиметричною валентнiстю 2:1/1:2. Вивчається також яви-
ще iндукованого поляризацiєю збiльшення поверхневої густини заряду у системах 3:1/1:3. Проводиться
порiвняння з даними моделювання методом Монте-Карло. Показано, що наведенi теорiї вiдтворюють
данi моделювання з рiзним ступенем точностi вiд якiсного до майже кiлькiсного. Результати нового ва-
рiанту теорiї в багатьох випадках практично не вiдрiзняються вiд результатiв МТПБ5. Однак при певних
умовах, зокрема при низьких концентрацiях електролiтiв, спостерiгаються теоретичнi артефакти у формi
нефiзичних “плечей” в унарних функцiях розподiлу iонiв.
Ключовi слова: електричний подвiйний шар, вiдображення, обмежена примiтивна модель,
модифiкована теорiя Пуасона-Больцмана, моделювання Монте-Карло
33801-15
Introduction
Model and theory
Monte Carlo simulations
Results
Conclusions
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