Double layer for hard spheres with an off-center charge
Simulations for the density and potential profiles of the ions in the planar electrical double layer of a model electrolyte or an ionic liquid are reported. The ions of a real electrolyte or an ionic liquid are usually not spheres; in ionic liquids, the cations are molecular ions. In the past, this...
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| Zitieren: | Double layer for hard spheres with an off-center charge / W. Silvestre-Alcantara, L.B. Bhuiyan, S. Lamperski, M. Kaja, D. Henderson // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13603: 1–11. — Бібліогр.: 41 назв. — англ. |
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irk-123456789-1557922019-06-18T01:27:28Z Double layer for hard spheres with an off-center charge Silvestre-Alcantara, W. Bhuiyan, L.B. Lamperski, S. Kaja, M. Henderson, D. Simulations for the density and potential profiles of the ions in the planar electrical double layer of a model electrolyte or an ionic liquid are reported. The ions of a real electrolyte or an ionic liquid are usually not spheres; in ionic liquids, the cations are molecular ions. In the past, this asymmetry has been modelled by considering spheres that are asymmetric in size and/or valence (viz., the primitive model) or by dimer cations that are formed by tangentially touching spheres. In this paper we consider spherical ions that are asymmetric in size and mimic the asymmetrical shape through an off-center charge that is located away from the center of the cation spheres, while the anion charge is at the center of anion spheres. The various singlet density and potential profiles are compared to (i) the dimer situation, that is, the constituent spheres of the dimer cation are tangentially tethered, and (ii) the standard primitive model. The results reveal the double layer structure to be substantially impacted especially when the cation is the counterion. As well as being of intrinsic interest, this off-center charge model may be useful for theories that consider spherical models and introduce the off-center charge as a perturbation. Виконано симуляцiї профiлiв густини i потенцiалу iонiв у плоскому електричному подвiйному шарi модельного електролiту або iонної рiдини. Iони електролiту або iонної рiдини є зазвичай несферичними; у iонних рiдинах катiони є молекулярними iонами. У недалекому минулому цю асиметрiю моделювали розглядаючи або сфери, якi були асиметричними за розмiром i/чи валентнiстю (примiтивна модель), або димернi катiони, якi утворенi за допомогою тангенцiйно дотичних сфер. Тут ми розглядаємо сферичнi iони, якi є асиметричними за розмiром та моделюють асиметричну форму за допомогою заряду, який розташований не в центрi катiонних сфер, в той час як заряд анiона помiщено в центрi анiонних сфер. Рiзнi одночастинковi профiлi густини i потенцiалу порiвнюються з (i) ситуацiєю з димерами, коли складовi сфери димерного катiона тангенцiйно прив’язанi i з (ii) стандартною примiтивною моделлю. Результати показують, що структура подвiйного шару зазнає значного впливу, особливо коли катiон є контрiоном. Будучи цiкавою сама по собi, ця модель нецентрального заряду може стати корисною для теорiй, якi розглядають сферичнi моделi та вводять нецентральний заряд як збурення. 2016 Article Double layer for hard spheres with an off-center charge / W. Silvestre-Alcantara, L.B. Bhuiyan, S. Lamperski, M. Kaja, D. Henderson // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13603: 1–11. — Бібліогр.: 41 назв. — англ. 1607-324X DOI:10.5488/CMP.19.13603 arXiv:1603.02172 PACS: 61.20.Qj, 82.45.Fk, 82.45.Gj, 82.45.Jn http://dspace.nbuv.gov.ua/handle/123456789/155792 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Simulations for the density and potential profiles of the ions in the planar electrical double layer of a model electrolyte or an ionic liquid are reported. The ions of a real electrolyte or an ionic liquid are usually not spheres; in ionic liquids, the cations are molecular ions. In the past, this asymmetry has been modelled by considering spheres that are asymmetric in size and/or valence (viz., the primitive model) or by dimer cations that are formed by tangentially touching spheres. In this paper we consider spherical ions that are asymmetric in size and mimic the asymmetrical shape through an off-center charge that is located away from the center of the cation spheres, while the anion charge is at the center of anion spheres. The various singlet density and potential profiles are compared to (i) the dimer situation, that is, the constituent spheres of the dimer cation are tangentially tethered, and (ii) the standard primitive model. The results reveal the double layer structure to be substantially impacted especially when the cation is the counterion. As well as being of intrinsic interest, this off-center charge model may be useful for theories that consider spherical models and introduce the off-center charge as a perturbation. |
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Silvestre-Alcantara, W. Bhuiyan, L.B. Lamperski, S. Kaja, M. Henderson, D. |
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Silvestre-Alcantara, W. Bhuiyan, L.B. Lamperski, S. Kaja, M. Henderson, D. Double layer for hard spheres with an off-center charge Condensed Matter Physics |
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Silvestre-Alcantara, W. Bhuiyan, L.B. Lamperski, S. Kaja, M. Henderson, D. |
| author_sort |
Silvestre-Alcantara, W. |
| title |
Double layer for hard spheres with an off-center charge |
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Double layer for hard spheres with an off-center charge |
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Double layer for hard spheres with an off-center charge |
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Double layer for hard spheres with an off-center charge |
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Double layer for hard spheres with an off-center charge |
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double layer for hard spheres with an off-center charge |
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Інститут фізики конденсованих систем НАН України |
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2016 |
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Double layer for hard spheres with an off-center charge / W. Silvestre-Alcantara, L.B. Bhuiyan, S. Lamperski, M. Kaja, D. Henderson // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13603: 1–11. — Бібліогр.: 41 назв. — англ. |
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Condensed Matter Physics |
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2025-07-14T08:01:39Z |
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2025-07-14T08:01:39Z |
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1837608573721378816 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 1, 13603: 1–11
DOI: 10.5488/CMP.19.13603
http://www.icmp.lviv.ua/journal
Double layer for hard spheres with an off-center
charge
W. Silvestre-Alcantara1, L.B. Bhuiyan1∗, S. Lamperski2, M. Kaja2, D. Henderson3
1 Department of Physics and Laboratory for Theoretical Physics, University of Puerto Rico,
San Juan, Puerto Rico 00936-8377, USA
2 Department of Physical Chemistry, Adam Mickiewicz University in Poznań,
Umultowska 89b, 61-614 Poznań, Poland
3 Department of Chemistry and Biochemistry, Brigham Young University, Provo UT 84602-5700, USA
Received November 9, 2015, in final form December 16, 2015
Simulations for the density and potential profiles of the ions in the planar electrical double layer of a model elec-
trolyte or an ionic liquid are reported. The ions of a real electrolyte or an ionic liquid are usually not spheres;
in ionic liquids, the cations are molecular ions. In the past, this asymmetry has been modelled by consider-
ing spheres that are asymmetric in size and/or valence (viz., the primitive model) or by dimer cations that are
formed by tangentially touching spheres. In this paper we consider spherical ions that are asymmetric in size
and mimic the asymmetrical shape through an off-center charge that is located away from the center of the
cation spheres, while the anion charge is at the center of anion spheres. The various singlet density and po-
tential profiles are compared to (i) the dimer situation, that is, the constituent spheres of the dimer cation are
tangentially tethered, and (ii) the standard primitive model. The results reveal the double layer structure to be
substantially impacted especially when the cation is the counterion. As well as being of intrinsic interest, this
off-center charge model may be useful for theories that consider spherical models and introduce the off-center
charge as a perturbation.
Key words: electrical double layer, simulations, density functional theory, off-center charged spheres
PACS: 61.20.Qj, 82.45.Fk, 82.45.Gj, 82.45.Jn
This article is dedicated to our colleague and friend, Stefan Sokołowski, in commemoration of his 65th
birthday. DH first met and collaborated with “Don Esteban” in Mexico City but had admired his work long
before that. Stefan has been our good friend and frequent collaborator since that time. We wish him a
happy birthday and continued good health and productivity.
1. Introduction
The electrical double layer (EDL) has long fascinated experimentalists and theorists. An EDL is formed
when the charged particles in a Coulomb fluid are attracted (or adsorbed) by a charged surface or elec-
trode. The charge within or on the charged surface forms one layer and the charge of the adsorbed fluid
forms the second layer that is of opposite sign to the charge of the surface but is equal in magnitude to
the electrode charge; hence, the name double layer. In reality, these two layers of the EDL often consist
of sublayers, so the term double layer is not quite accurate. Generally, the structure of the (usually metal)
electrode is ignored and the electrode is considered to be a classical metal with the (electrode) charge be-
ing distributed uniformly on the electrode surface. The fluid layer, on the other hand, can be thought of
as an extended diffuse layer whose net charge is equal in magnitude but opposite in sign to the electrode
charge. The fluid layer may consist of sublayers, sometimes with alternating charge.
∗E-mail: beena@beena.uprrp.edu
©W. Silvestre-Alcantara, L.B. Bhuiyan, S. Lamperski, M. Kaja, D. Henderson, 2016 13603-1
http://dx.doi.org/10.5488/CMP.19.13603
http://www.icmp.lviv.ua/journal
W. Silvestre-Alcantara et al.
The EDL has been of great practical importance in electrochemistry and analytical chemistry. In re-
cent years, the EDL has been found to be promising for new batteries, fuel cells, and supercapacitors [1]
as well as for studies in biology [2]. The selectivity of a physiological channel can be thought of as an
EDL problem with the channel playing the role of an electrode and the adsorbed ions playing the role
of the diffuse layer [3]. The adsorption of ions by DNA is another example of an EDL (see for example,
reference [4]).
The importance of the EDL is attested to in some recent reviews [2, 5, 6]. Despite this importance,
the acceptance of the recent theoretical developments has been somewhat hampered by the fact that
deviations from the predictions of the classical and often inadequate Poisson-Boltzmann (PB) theory of
Gouy [7], Chapman [8], and Stern [9] (GCS) occur at high concentrations and/or high electrode charge and
can be difficult to observe in aqueous systems. The popularity of the GCS theory stems from its intuitive
simplicity and ease of use. Thus, it might be useful in quick and qualitative analysis of experimental
results but it is unsatisfactory if one wishes to understand these results at a fundamental level. In passing,
we note that the classical theory for bulk electrolytes that parallels the GCS theory is the more well-known
Debye-Hückel (DH) theory (see for example, reference [10]). Historically, the GCS theory predates the DH
theory, however, unlike the former theory, the latter theory is a linearized version of the bulk PB theory.
Until recently, a broad interest in the theory of the EDL has not been helped by the fact that it is diffi-
cult to achieve experimentally the electrode charges and/or high electrolyte concentrations with aqueous
electrolytes where the deficiencies of the GCS theory are most apparent. By contrast, computer simula-
tions provide a means of testing theories for such difficult situations. They are the gold standard against
which theories can be compared since the theory and simulation are based on the same assumed model
for the inter-particle interactions. For a given model of the electrode and ions, simulations provide exact
results, apart from statistical uncertainties, against which a theory may be compared. Torrie and Valleau
[11] found in their seminal simulations that the capacitance near the point of zero charge (pzc) and at
high concentration was greater than what was predicted by the GCS theory. Blum [12], and Henderson,
Saavedra-Barrera and Lozada-Cassou [13] showed that the sophisticated mean spherical approximation
(MSA) and hypernetted chain theory (HNC) also exhibited this behavior. The MSA and HNC are closely
related; the MSA is a linearized version of the HNC. A careful examination [5, 14] of experimental results
confirms this behavior but the effect is small. In their simulations, Torrie and Valleau also found that at
high electrode charge the capacitance is smaller than the GCS predictions. One of us recalls a meeting at
which a prominent experimentalist ridiculed this result as being outside experimental range. However,
the effect is still real. Ions occupy space and their charge cannot be crowded indefinitely. They cannot
continue to form a monolayer on the electrode as the electrode charge increases. Regrettably, the MSA
and the HNC fail to predict this latter effect. The MSA is a linear response theory andmakes no prediction
about the behavior of the capacitance at large electrode charge, while the HNC proves inadequate and
predicts a greater capacitance at a large charge [15].
The modified Poisson-Boltzmann theory (MPB) [16] also predicts an increasing capacitance with an
increasing concentration at small electrode charge and seems consistent with the continued decline in the
capacitance as a function of electrode charge at high electrode charge. Although the numerical solutions
of the MPB equations lack convergence at sufficiently high electrode charge, within the range of the
surface charge for which solutions exist, the capacitance curve passes from a double hump curve at
small concentrations to a single hump curve at high concentrations consistent with simulations [17, 18].
In these works [17, 18], the MC and MPB were applied to a restricted primitive model (RPM) (charged
hard spheres with a common diameter). The density functional theory (DFT)[19] seems to be satisfactory
in a similar vein. Here, too the DFT applied to the RPM double layer gave very good agreement with the
corresponding MC simulation results for the capacitance [20]. The capacitance at small electrode charge
continues to increase with an increasing concentration but decreases with an increasing electrode charge
at large electrode charge, and shows the aforementioned double hump to single hump transition.
Interest in the EDL at high concentrations and electrode charge is increasing because EDLs can be
formed in ionic liquids. Ionic liquids are organic electrolytes. Due to the fact that a solvent is not present,
ionic liquids are, in essence, room temperature molten salts. They can exist at high concentrations and
can support higher electrode charges. As a result, they are ideal for the testing of modern theories of
the EDL. As Kornyshev has aptly stated [20], an ionic electrolyte gives a new paradigm for EDL studies.
As Kornyshev observes, ionic liquid capacitance curves can show double and single hump shapes. There
13603-2
Off-center charges
have been a variety of recent experimental studies in which the capacitance exhibits both single and
double hump shapes [21–25] as the ion concentration is varied.
Summarizing the situation, the differential capacitance of the EDL of an ionic liquid at small electrode
charge increases, apparently without limit, and the differential capacitance, at large electrode charge de-
creases as the electrode charge increases in magnitude. This latter effect is due to the fact that the ions
cannot be adsorbed into a monolayer as the GCS theory suggests. These two features have the effect of
causing the capacitance to pass from a double hump shape with a minimum at or near the pzc at low con-
centrations to a single hump shape at higher concentrations. By contrast, the GCS theory predicts only a
minimum in the capacitance that gradually fills in with an increasing concentration with the capacitance
gradually becoming independent of the concentration and electrode beyond some high concentration
result.
Ionic liquids, in the real world, are formed from asymmetric ions. We have considered EDLs in a wide
range of models that approximate ionic liquids [26]. We have modelled the asymmetry of ionic liquids by
means of size asymmetry [27], through the use of dimers formed by tangentially touching spheres [27–34],
and dimers of fused spheres [35]. Another method of introducing asymmetry is to keep spherical ions but
allow the charge of one species of the ions to be off-center. We do this by considering the charged sphere
of a dimer cation to be smaller than the diameter of the uncharged sphere of the dimer and fusing the
smaller charged sphere of the dimer cation so that it should be entirely within the larger neutral sphere.
This results in an electrolyte consisting of spheres but with one species (the anions) having a charge at
the sphere center and the other species (the cations) having an off-center charge. In this paper, we will
report some results for the density and potential profiles of this off-center ion model double layer. It is to
be hoped that as well as its intrinsic interest, this model of off-center charges might lend itself to theories
that use an expansion in powers of the degree to which the charge is off-center. In the event of such
theories being developed, our simulations will be of value for testing these theories.
In addition to our work using spheres, there is another body of work that uses simulations with fairly
realistic complex ionic liquids [36–39]. The investigators using complex ionic liquid models seek speci-
ficity whereas we seek generality. Such complex ionic liquid models do not lend themselves to theory. At
present, only simulations are possible for these complex models. Both approaches have their merit.
2. Model; simulation method
We employ a fluid electrolyte model that consists of anions and cations that are represented by
charged hard spheres in a uniform dielectric background. The charged spheres have a charge whose
magnitude is equal to the proton charge e and have an equal diameter, d = 4.25×10−10 m. This model
is similar to that of the restricted primitive model electrolyte but we allow the charge of the cations to
be off-center, while the charge of the anions continues to be located at the sphere center. The electrode
is considered to be a non-polarizable, hard planar wall of uniform (surface) charge density, σ, that is lo-
cated on the electrode surface. To yield cation spheres with an off-center charge, we use dimer cations
consisting of an uncharged hard sphere and a smaller charged hard sphere, which can fuse into each
other. Thus, when the charged sphere is encapsulated entirely within the uncharged sphere, we get a
cation with an off-center charge. The anion is a charged sphere whose diameter is equal to that of the
uncharged or neutral sphere of the dimer. Specifically, if d+, d−, and d0 are the diameters of the positive,
negative, and neutral spheres, respectively, then we have, d− = d0 = d and d+ = d/2.
In the Hamiltonian, the various interaction potentials occurring in the system are as given below. The
interaction between the spheres is
ui j (r )=
{
∞, r < dc
i j
,
Zi Z j e2
4πǫ0ǫr r
, r Ê dc
i j
,
(1)
while the interaction of a sphere with the electrode is given by
ws (x) =
{
∞, x < dc
w s ,
−
σZs ex
ǫ0ǫr
, x Ê dc
w s .
(2)
13603-3
W. Silvestre-Alcantara et al.
Here, Zs is the valency of particle species s, ǫ0 and ǫr are the vacuum and relative permittivities, r is the
distance between the centers of two spheres, and x is the perpendicular distance of a sphere from the
electrode plane.
The quantity dc
i j
is the distance of closest approach between two particles of type i and j , respectively,
while dc
w s is the distance of closest approach of a particle of species s to the electrode. In the present study,
the following three cases may be distinguished:
(i) for the dimer cation case (no fusion), we get as before (see for example, reference [34]) dc
i j
= (di +
d j )/2 and dc
w s = ds /2, ds being the diameter of a particle of type s,
(ii) for the off-centered charged cation case, we have dc
++
∈ [d/2,3d/2], dc
+0
(= dc
+−
) ∈ [3d/4,5d/4], and
dc
i j
= d where i j = 00,0−,−−. Also, dc
w+
∈ [d/4,3d/4] and dc
w0
= dc
w−
= d/2, and finally
(iii) when the smaller positively charged sphere is completely fused inside the larger neutral sphere
such that the centers of the two spheres are coincident, we have dc
i j
= d and dc
w s = d/2.
Figure 1. (Color online) Schematic diagram of an off-
center cation containing RPM. The distance s+0 is be-
tween the centers of the positively charged sphere
and the neutral sphere. From top to bottom: dimer
model (no fusion), the off-center charge RPM, and
standard RPM, respectively.
The degree to which the charge of the cations
is off-center is controlled by the separation of, or
distance between, the centers of the spheres in the
dimer, s+0. Thus, when there is no fusion, and
the spheres in the dimer are merely tangentially
touching, the separation between them is s+0 =
3d/4. We have s+0 = d/4 when the positively
charged sphere is completely inside the neutral
sphere and an off-center charged cation results.
This is the case of most interest in the present
work. The s+0 = 0when the centers of the positive
and neutral spheres coincide, and the standard
RPM ensues. It ought to be mentioned, however,
that although the sizes of the cation and the an-
ion are different, viz., d/2 and d , respectively, the
fact that the positive sphere is symmetrically em-
bedded within the neutral sphere, makes the com-
posite sphere an effective hard sphere cation of
the same size as the hard sphere anion, and hence
the RPM. For the diameters considered here, the
off-center charge lies between the center of the
neutral sphere and a distance of d/4 from its cen-
ter (i.e., mid-way to the surface of the neutral
sphere). Results for an off-center charge at a dis-
tance greater than mid-way to the surface, d/4,
could be obtained by using a smaller value for the
diameter of the positive sphere. However, we ex-
pect that a value of s+0 that is less than, or equal
to, d/4 would be of most interest, at least in a the-
ory in which the degree to which the charge is off-
center was treated as a perturbation. The relation between s+0 and the degree to which the small charged
sphere is fused into the larger uncharged sphere is shown in figure 1.
The MC simulations were performed in the canonical (NV T ) ensemble following the standard Metro-
polis algorithm methods. The details of the simulations are the same as in our previous publications [28–
35] and will not be repeated here. When s+0 = 3d/4 and s+0 = 0, that is the dimer and RPM situations,
respectively, the simulations were checked against independent simulations of the dimer and RPM double
layers using our previous programs. These results are useful tests of our present simulations and were
found to be consistent with our earlier results within statistical errors.
13603-4
Off-center charges
3. Results
We consider a system of anions, cations, and uncharged hard spheres with a background relative
permittivity of ǫr = 78.5 at a temperature of T = 298.15 K. The hard sphere anions and the neutral spheres
have a diameter of d = 4.25×10−10 m, while the hard sphere cations are smaller (half the anion or the
uncharged sphere diameters). The electrolyte concentration considered here is c = 1 mol/dm3. In this
paper we will restrict ourselves to the symmetric valency cases of monovalent and divalent charged ions,
viz., Z+ =−Z− = Z = 1 or 2, for the 1:1 or 2:2 cases, respectively.
In our numerical calculations, it is convenient to use reduced or dimensionless units that are de-
noted by an asterisk. However, in reporting results we will also give the equivalent physical units. The
reduced temperature is T ∗
= 4πǫ0ǫr dkBT /Z 2e2, whose values in the present study are T ∗
= 0.595 for
the monovalent case (Z = 1), and T ∗
= 0.149 for the divalent case (Z = 2), respectively. The reciprocal of
the T ∗ is the plasma coupling constant Γ of the literature, viz., Γ= 1/T ∗. The bulk reduced density of the
free particles of species s is defined as ρ∗
s = ρs d3 with ρs being the corresponding bulk number density.
Thus, for c = 1 mol/dm3, we have ρ∗
+
= 0.00578, ρ∗
−
= 0.0462, and ρ∗
0
= 0.0462, respectively. The reduced
surface charge density on the electrode is defined σ∗
= σd2/e , while the reduced electrode potential is
ψ
∗
= eψ/kBT , where ψ is the electrode potential in volts.
The double layer structure is described principally by the electrode–particle singlet distribution func-
tion gs(x) = ρ∗
s (x)/ρ∗
s = ρs (x)/ρs , where ρ
∗
s (x) (or ρs (x)) is the local value of the corresponding quantity.
In addition to reporting the values for the gs(x) of the particles of species s, we will also report the values
of the reduced potential, ψ∗(x). The potential profile ψ(x) is a weighted integral of the singlet distribu-
tions, viz.,
ψ(x) =
e
ǫ0ǫr
∑
s
Zsρs
∞
∫
x
dx′
(x − x′
)gs(x′
). (3)
It is a useful indicator of the overall charge distribution in the system besides being the relevant quantity
in characterizing the capacitance behaviour of the double layer. All of the profiles are calculated as func-
tions of the perpendicular distance, x from the electrode, which is located at x = 0. The reduced unit for
x is, of course, x/d .
In figures 2–4, results are presented for Z = 1 for σ =±0.1 C/m2, which corresponds to σ∗
= ±0.113,
whereas the results shown in figures 5–7 are for these same values of σ, or σ∗, but for Z = 2. Our main
interest in this work is the case of spheres with an off-center charge, s+0 = d/4. These are shown in the
middle panels of the density profiles in figures 2, 3, 5, 6, and in both panels of the potential profiles in
figures 4 and 7. Results are also given for s+0 = 3d/4 , that is, a dimer consisting of tangentially touching
spheres, and s+0 = 0, which is the well studied RPM. The results for these two cases are not new, but the
fact that these results represent the starting point and the end point in our quest for an off-center charged
ionmodel, is a valuable internal consistency test of the new code. Furthermore, comparison of the results
for various values of s+0, especially as the degree of fusion increases, makes the effect of fusing the cation
into the neutral sphere physically more apparent.
Considering first figure 2, the case of a negative surface charge, we notice in general that being coun-
terions, the cations are attracted to the negatively charged electrode and bring the neutral spheres with
them. On the other hand, the anions are the coions and are repelled from the electrode. In the no fusion
situation, s0+ = 3d/4 (top panel), the profile of the neutral sphere of the dimer has a peak for x slightly
less that d . The charged head of the dimer can approach the electrode more closely than can the neutral
tail of the dimer since the positive sphere has a smaller diameter than the neutral one and the dimer can
rotate. At the other extreme we have the case of complete fusion such that s+0 = 0 (bottom panel), that
is, the RPM case, and the distributions of the positive and neutral spheres become identical because their
centers are coincident so that the neutral spheres have effectively disappeared. The middle panel of this
figure corresponds to s+0 = d/4 when the cation has just disappeared completely into the neutral sphere
leading to a RPM with off-center charged cations, the central theme of the present study. The profiles
for the center of this off-center charged sphere has a peak for x somewhat greater than d/2, whereas
the profile of the charge of this off-center charged sphere has its peak for x nearer 3d/4. These profiles
also indicate that the distance of the charge of the off-center sphere from the electrode plane can be less
than the corresponding distance of the center of the sphere itself since the sphere can rotate to bring
13603-5
W. Silvestre-Alcantara et al.
Figure 2. (Color online) Electrode–particle singlet
distributions gs for a 1:1 valency electrolyte sys-
tem at concentration c = 1 mol/dm3, reduced elec-
trode surface charge density σ∗
= −0.113 (σ =
−0.1 C/m2), and for s+0 = 3d/4 (top panel), s+0 =
d/4 (middle panel), and s+0 = 0 (bottom panel).
Figure 3. (Color online) Electrode–particle singlet
distributions gs for a 1:1 valency electrolyte sys-
tem at concentration c = 1 mol/dm3 and reduced
electrode surface charge density σ∗
= +0.113 (σ =
+0.1 C/m2. Rest of the notation and legend as in fig-
ure 2.
the charge center closer to the wall. The profiles of the monomer anions are only slightly affected by the
value of s+0; this is because the anion population near the negatively charged electrode is relatively less.
This profile seems to be monotonous for all cases.
The case of a positively charged electrode is considered in figure 3. The striking feature of the results
here is the qualitative similarity of the various profiles across the panels. This can be understood from
the fact that now it is the negatively charged monomer spheres that are the counterions and are attracted
to the electrode, while the positively charged spheres and their attached neutral spheres are repelled.
13603-6
Off-center charges
Figure 4. (Color online) Reduced mean electrostatic potential ψ∗(x/d) for a 1:1 valency electrolyte
system at concentration c = 1 mol/dm3 with reduced electrode surface charge density σ∗
= −0.113
(σ = −0.1 C/m2) (upper panel), and σ
∗
= +0.113 (σ = +0.1 C/m2) (lower panel). The three plots in each
panel correspond to (i) solid circles , s+0 = 3d/4), (ii) open squares, s+0 = d/4), and (iii) stars, s+0 = 0.
The latter feature, in turn, implies that the internal geometry of the cation has little influence on the
double layer structure in the immediate vicinity of the electrode. The profiles are thus more akin to that
for a PM and having relatively less features than when the electrode is negatively charged. Analogous
observations weremade in some of our previous studies (see for example, reference [34]). It is interesting
that the profile of the counterion has a small peak near the contact. In the course of these calculations
we have also examined the cases with a higher surface charge density where this peak tended to become
less conspicuous so that the peakmay well be related to hard core effects. At any rate, it should have little
affect on the potential profile since the potential profile is a first moment. The potential profile is also
strongly influenced by the charges further from the electrode. The profiles for the positive and neutral
spheres of the dimers are very nearly indistinguishable and are monotonous. These particles reside far
away from the electrode and seemingly behave as a single entity. It is noted again that in the bottom
panel s+0 = 0 for the RPM case, the g+ and g0 are identical as expected.
The 1:1 mean electrostatic potential profiles for the negatively and positively charged walls are illus-
trated in figure 4. In either situation the s+0 dependence is generally weak. For the monovalent case and
hence relatively low coupling, the geometry of the ions gets masked in taking the weighted average of the
gs to get theψ
∗. We further note that the neutral species spheres have no bearing on the potential. These
potential profiles are also monotonous.
The results for Z = 2, displayed in figures 5–7, show the effect of higher valency on the structure. The
singlet distributions reveal oscillations, not seen with the 1:1 case, and the double layer is more compact.
For a negative electrode (figure 5), for the no fusion dimer case (top panel) and the off-center charge fused
sphere (middle panel) case, we note again that the positive spheres can approach the electrode more
closely than the neutral spheres for reasons stated earlier. In figure 6 (positive electrode), the profiles
13603-7
W. Silvestre-Alcantara et al.
Figure 5. (Color online) Electrode–particle singlet
distributions gs for a 2:2 valency electrolyte sys-
tem at concentration c = 1 mol/dm3 and reduced
electrode surface charge density σ∗
= −0.113 (σ =
−0.1 C/m2). Rest of the notation and legend as in
figure 2.
Figure 6. (Color online) Electrode–particle singlet
distributions gs for a 2:2 valency electrolyte sys-
tem at concentration c = 1 mol/dm3 and reduced
electrode surface charge density σ∗
= +0.113 (σ =
+0.1 C/m2). Rest of the notation and legend as in
figure 2.
for positively charged and neutral ends of the dimer (top panel) or the profiles for the center of the off-
center charge sphere and the charge-center of this sphere (middle panel) are very nearly the same as
seen earlier in the 1:1 case. The general shape of the profiles is determined by the nature of counterions,
which are spheres in all cases. However, rotation of the dimer coions is clearly evident. The dimer rotates
so that the charged end is further from the electrode. This is still true for the off-center charged sphere
but to a lesser extent. In the bottom panels (RPM case) of these two figures, the g+ and g0 are identical as
before for the 1:1 case.
13603-8
Off-center charges
Figure 7. (Color online) Reduced mean electrostatic potential ψ∗(x/d) for a 2:2 valency electrolyte
system at concentration c = 1 mol/dm3 with reduced electrode surface charge density σ∗
= −0.113
(σ =−0.1 C/m2) (upper panel), and σ
∗
= +0.113 (σ =+0.1 C/m2) (lower panel). Rest of the notation and
legend as figure 4.
The ψ∗ profiles depicted in figure 7 are of interest. Unlike the monovalent cases in figure 5, the po-
tential profiles for the divalent ions are no longer monotonous. Near the electrode, there is a broad max-
imum or a shallow minimum inψ∗ for the negative and positive surface charges, respectively. This is the
phenomenon of charge inversion or overscreening, which refers to the physical situation where a layer
of counterions predominates near the electrode and over-compensates the charge on the electrode. This
is a common occurrence when divalent ions are present. In the present case, this feature manifests for
all s+0.
4. Summary
In this paper we have explored through MC simulations an asymmetric variation of the RPM pla-
nar double layer where asymmetry is imparted to the electrolyte by having an off-center charge for the
cations. This is a logical extension of the familiar RPM. In the model introduced here, the repulsive hard,
non-electrostatic, part of the inter-ionic interaction is spherical and central, and is represented by a hard
sphere potential, which inmanyways is the most difficult part of the interaction but which has been stud-
ied extensively and is now quite well understood. However, there is no reason to expect that the center
of charge of an ion should be located at the center of mass of the ion. This would be especially true for the
ions in an ionic liquid. Oneway of deriving a theory is to accept the hard sphere portion of a charged hard
sphere fluid as the starting point and to introduce the electrostatic contribution as a perturbation. For ex-
ample, it is possible to obtain the mean spherical approximation by means of a ‘ring sum’ of electrostatic
terms. It may well be possible to formulate an analogous theory when the electrostatic interaction has a
13603-9
W. Silvestre-Alcantara et al.
center of charge that differs from the geometric center, or center of mass, of the ion, especially when the
difference between the center of charge and center of mass is small enough that it can be regarded as a
perturbation.
Another pertinent question in line with the above concerns the generalization of the Henderson-
Blum-Lebowitz (HBL) contact value theorem for PM double layers [40,41] to include the present off-center
charge RPM double layer. Such an extension will certainly aid in the theoretical development as has
been the case with the HBL sum rule in the double layer literature. But an expression for the osmotic
pressure for a bulk charged hard sphere fluid with an off-center charge does not yet exist so that again a
perturbation approach would be preferable. We hope to take up the issue in the near future.
A cation with an off-center charge was conceived from a dimer cation formed by a tangentially teth-
ered pair of a large hard sphere and a smaller charged hard sphere such that the latter can fuse entirely
into the uncharged sphere. The interesting changes that occur as a result in the structural pattern of the
double layer relative to that of the standard RPM makes the off-center charge RPM promising for the
future.
Variations of the present off-center charge model can be contemplated. For instance, one could have
off-center charges for both the anion and the cation. Still further asymmetry could be incorporated by
having in addition, asymmetric ionic valencies and/or sizes. Work on such models is in progress.
Acknowledgements
Monika Kaja and Stanisław Lamperski acknowledge the financial support from the Faculty of Chem-
istry, Adam Micklewicz University in Poznań.
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Подвiйний електричний шар у моделi твердих сфер
iз нецентральним зарядом
В. Сильвестр-Алькантара1, Л.Б. Буян1, С. Ламперский2, M. Кайя2, Д. Гендерсон3
1 Фiзичний факультет i лабораторiя теоретичної фiзики, Унiверситет Пуерто Рiко, Пуерто Рiко, США
2 Факультет фiзичної хiмiї, Унiверситет Адама Мiцкевича в Познанi, Познань, Польща
3 Факультет хiмiї та бiохiмiї, Унiверситет Бригама Янга, Прово, США
Виконано симуляцiї профiлiв густини i потенцiалу iонiв у плоскому електричному подвiйному шарi мо-
дельного електролiту або iонної рiдини. Iони електролiту або iонної рiдини є зазвичай несферичними;
у iонних рiдинах катiони є молекулярними iонами. У недалекому минулому цю асиметрiю моделювали
розглядаючи або сфери, якi були асиметричними за розмiром i/чи валентнiстю (примiтивна модель), або
димернi катiони, якi утворенi за допомогою тангенцiйно дотичних сфер. Тут ми розглядаємо сферичнi
iони, якi є асиметричними за розмiром та моделюють асиметричну форму за допомогою заряду, який
розташований не в центрi катiонних сфер, в той час як заряд анiона помiщено в центрi анiонних сфер.
Рiзнi одночастинковi профiлi густини i потенцiалу порiвнюються з (i) ситуацiєю з димерами, коли складовi
сфери димерного катiона тангенцiйно прив’язанi i з (ii) стандартною примiтивною моделлю. Результати
показують, що структура подвiйного шару зазнає значного впливу, особливо коли катiон є контрiоном.
Будучи цiкавою сама по собi, ця модель нецентрального заряду може стати корисною для теорiй, якi
розглядають сферичнi моделi та вводять нецентральний заряд як збурення.
Ключовi слова: електричний подвiйний шар, моделювання, теорiя функцiоналу густини, сфери з
нецентральним зарядом
13603-11
http://dx.doi.org/10.1021/jp212082k
http://dx.doi.org/10.1021/jp304362y
http://dx.doi.org/10.1063/1.4799886
http://dx.doi.org/10.1063/1.4817325
http://dx.doi.org/10.1080/00268976.2014.968651
http://dx.doi.org/10.1016/j.jcis.2014.11.070
http://dx.doi.org/10.1080/00268976.2015.1083132
http://dx.doi.org/10.1021/ja104273r
http://dx.doi.org/10.1021/jp301399b
http://dx.doi.org/10.1039/C3CP51218E
http://dx.doi.org/10.1021/ct200914j
http://dx.doi.org/10.1063/1.436535
http://dx.doi.org/10.1016/S0022-0728(79)80459-3
Introduction
Model; simulation method
Results
Summary
|