On the contact values of the density profiles in an electric double layer using density functional theory
A recently proposed local second contact value theorem [Henderson D., Boda D., J. Electroanal. Chem., 2005, 582, 16] for the charge profile of an electric double layer is used in conjunction with the existing Monte Carlo data from the literature to assess the contact behavior of the electrode-ion di...
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irk-123456789-1202862017-06-12T03:03:05Z On the contact values of the density profiles in an electric double layer using density functional theory Bhuiyan, L.B. Henderson, D. Sokołowski, S. A recently proposed local second contact value theorem [Henderson D., Boda D., J. Electroanal. Chem., 2005, 582, 16] for the charge profile of an electric double layer is used in conjunction with the existing Monte Carlo data from the literature to assess the contact behavior of the electrode-ion distributions predicted by the density functional theory. The results for the contact values of the co- and counterion distributions and their product are obtained for the symmetric valency, restricted primitive model planar double layer for a range of electrolyte concentrations and temperatures. Overall, the theoretical results satisfy the second contact value theorem reasonably well, the agreement with the simulations being semi-quantitative or better. The product of the co- and counterion contact values as a function of the electrode surface charge density is qualitative with the simulations with increasing deviations at higher concentrations. Нещодавно запропоновану теорему про локальне друге контактне значення [Henderson D., Boda D., J. Electroanal. Chem., 2005, 582, 16] для профiлю заряду електричного подвiйного шару поєднано з iснуючими в лiтературi даними Монте Карло з метою оцiнки контактної поведiнки електрод-iонних розподiлiв, передбачених теорiєю функцiоналу густини. Результати для контактних значень розподiлiв ко- i протиiонiв та їхнього добутку отримано для випадку симетричної валентностi в рамках обмеженої примiтивної моделi плоского подвiйного шару для низки концентрацiй i температур електролiту. В цiлому, теоретичнi результати досить добре задовольняють теорему про друге контактне значення, узгодження iз симуляцiями – напiвкiлькiсне або краще. Добуток ко- i протиiонних контактних значень як функцiя густини заряду поверхнi електрода якiсно узгоджується з симуляцiями, але вiдхилення мiж обома зростає при вищих концентрацiях. 2012 Article On the contact values of the density profiles in an electric double layer using density functional theory / L.B. Bhuiyan, D. Henderson, S. Sokołowski // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23801:1-10. — Бібліогр.: 40 назв. — англ. 1607-324X PACS: 82.45.Fk, 61.20.Qg, 82.45.Gj DOI:10.5488/CMP.15.23801 arXiv:1207.3274 http://dspace.nbuv.gov.ua/handle/123456789/120286 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A recently proposed local second contact value theorem [Henderson D., Boda D., J. Electroanal. Chem., 2005, 582, 16] for the charge profile of an electric double layer is used in conjunction with the existing Monte Carlo data from the literature to assess the contact behavior of the electrode-ion distributions predicted by the density functional theory. The results for the contact values of the co- and counterion distributions and their product are obtained for the symmetric valency, restricted primitive model planar double layer for a range of electrolyte concentrations and temperatures. Overall, the theoretical results satisfy the second contact value theorem reasonably well, the agreement with the simulations being semi-quantitative or better. The product of the co- and counterion contact values as a function of the electrode surface charge density is qualitative with the simulations with increasing deviations at higher concentrations. |
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Bhuiyan, L.B. Henderson, D. Sokołowski, S. |
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Bhuiyan, L.B. Henderson, D. Sokołowski, S. On the contact values of the density profiles in an electric double layer using density functional theory Condensed Matter Physics |
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Bhuiyan, L.B. Henderson, D. Sokołowski, S. |
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Bhuiyan, L.B. |
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On the contact values of the density profiles in an electric double layer using density functional theory |
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On the contact values of the density profiles in an electric double layer using density functional theory |
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On the contact values of the density profiles in an electric double layer using density functional theory |
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On the contact values of the density profiles in an electric double layer using density functional theory |
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On the contact values of the density profiles in an electric double layer using density functional theory |
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on the contact values of the density profiles in an electric double layer using density functional theory |
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On the contact values of the density profiles in an electric double layer using density functional theory / L.B. Bhuiyan, D. Henderson, S. Sokołowski // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23801:1-10. — Бібліогр.: 40 назв. — англ. |
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Condensed Matter Physics |
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| first_indexed |
2025-07-08T17:36:01Z |
| last_indexed |
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| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 2, 23801: 1–10
DOI: 10.5488/CMP.15.23801
http://www.icmp.lviv.ua/journal
On the contact values of the density profiles
in an electric double layer using density
functional theory∗
L.B. Bhuiyan1 , D. Henderson2, S. Sokołowski3
1 Laboratory of Theoretical Physics, Department of Physics, University of Puerto Rico,
Box 70377, San Juan, Puerto Rico 00936-8377, USA
2 Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah 84602-5700, USA
3 Department for the Modelling of Physico-Chemical Processes, Faculty of Chemistry, MCSU,
20031 Lublin, Poland
Received September 18, 2011, in final form October 18, 2011
A recently proposed local second contact value theorem [Henderson D., Boda D., J. Electroanal. Chem., 2005,
582, 16] for the charge profile of an electric double layer is used in conjunction with the existing Monte Carlo
data from the literature to assess the contact behavior of the electrode-ion distributions predicted by the density
functional theory. The results for the contact values of the co- and counterion distributions and their product
are obtained for the symmetric valency, restricted primitive model planar double layer for a range of electrolyte
concentrations and temperatures. Overall, the theoretical results satisfy the second contact value theorem rea-
sonably well, the agreement with the simulations being semi-quantitative or better. The product of the co- and
counterion contact values as a function of the electrode surface charge density is qualitative with the simula-
tions with increasing deviations at higher concentrations.
Key words: electric double layer, restricted primitive model, density profiles
PACS: 82.45.Fk, 61.20.Qg, 82.45.Gj
1. Introduction
One of the more interesting recent developments in the electric double layer research has been the
advancement of contact value theorems involving the charge profile in a primitive model (PM) planar
double layer (charged hard spheres moving in a dielectric continuum next to a planar electrode) (see, for
example, references [1–4]). Such exact conditions, or sum rules as they are often called, are important per
se in statistical mechanics since they permit unambiguous assessment of various approximate theories
and hence aid in theoretical development.
The most famous contact theorem in the double layer literature is the one formulated by Henderson
and Blum [5], and Henderson, Blum, and Lebowitz (HBL) [6] over thirty years ago. It is a condition on the
contact value of the total density profile in a planar double layer, and for a symmetric valency restricted
primitive model (RPM) (equisized ions in the PM) planar double layer – the model system of interest in
this paper, the HBL relation reads
gsum(d/2) = [gco(d/2)+ gctr(d/2)]/2 = a +
b2
2
. (1)
∗It is a pleasure to dedicate this paper to Dr. Orest Pizio on the occasion of his 60th Birthday. Douglas Henderson recalls with
fondness his first meeting with “Don Oresto” in Telavi in the Republic of Georgia in the mid 1980s, where Orest and Myroslav
Holovko invited him to visit Lviv. During this visit, Orest showed him the city sights, including Stefan Banach’s grave and the
Scottish Cafe, where Banach and his colleagues formulated many important theorems in functional analysis.
© L.B. Bhuiyan, D. Henderson, S. Sokołowski, 2012 23801-1
http://dx.doi.org/10.5488/CMP.15.23801
http://www.icmp.lviv.ua/journal
L.B. Bhuiyan, D. Henderson, S. Sokołowski
Here gco, gctr are the co- and counterion singlet distribution functions, d is the common ionic diameter,
and a = p/(ρkBT ) is the bulk osmotic coefficient with p being the bulk pressure, kB the Boltzmann con-
stant, and T the absolute temperature. The quantity b = zeσ/(ǫ0ǫrkBTκ) is a dimensionless parameter
where z, e , and σ are, respectively, the absolute value of the ionic valency, the magnitude of the elemen-
tary charge, and the uniform surface charge density on the electrode, ǫ0 is the vacuum permittivity, and
ǫr is the relative permittivity of the continuum solvent. Also, κ=
√
z2e2ρ/(ǫ0ǫrkBT ) is the Debye-Hückel
parameter (inverse Debye screening length) with ρ =
∑
i ρi where ρi is the mean number density of the
i th ionic species. In the RPM case the contact distance, that is, the distance of closest approach of an ion
to the electrode, occurs at d/2, where dco = dctr = d . Note that gsum(x) (x is the perpendicular distance
from the electrode into the solution) is related to the total density profile
ρ(x) =
∑
i
ρi (x) =
∑
i
ρi gi (x) = ρgsum(x), (2)
where ρi (x) is the singlet density profile of the i th species. It is of interest that the second term in equa-
tion (1) is just the Maxwell stress. Although equation (1) was obtained from statistical mechanics it is
consistent with Maxwell’s equations.
Equation (1) is a local expression and the consequent ease of its use has made the HBL contact con-
dition very appealing in double layer research over the years. For example, the classical Gouy-Chapman-
Stern (GCS) [7–9] theory of the double layer satisfies equation (1) but with a = 1, the ideal gas value. Thus,
for an electrolyte with osmotic coefficient substantially different from unity, the GCS theory can lead to
appreciable errors especially at low surface charges.
Sum rules such as equation (1) are useful not only for assessing theories but also for the insight they
provide. For example, because the coion contact value becomes small at a large surface charge and the
counterion contact value becomes large, according to this equation, the latter contact value increases as
square of the surface charge density. On the other hand, the local electroneutrality condition (also a sum-
rule) requires that the area of the charge profile be equal but opposite in sign to the electrode charge
density. As a result this area increases linearly with the electrode charge density. Consequently, at large
electrode charge, the oscillations and charge inversions in the charge profile should diminish but cannot
disappear altogether relative to the contact value.
Analogous relations for the contact value of the total charge profile in the double layer – the theme
of the present paper, have been relatively recent. A formal, rigorous relation was derived by Holovko et
al. [1, 10] and Holovko and di Caprio [2] using the Bogoliuobov-Born-Green-Yvon hierarchy. For symmet-
ric valency RPM planar double layer their expression is as follows:
gdiff(d/2) =−zeβ
∞
∫
d/2
dxgsum(x)
dψ(x)
dx
, (3)
where β = 1/(kBT ), ψ(x) is the mean electrostatic potential, and gdiff(x) = [gctr(x) − gco(x)]/2. Again,
gdiff(x) is now related to the total charge profile, viz.,
q(x) = e
∑
i
ziρi gi (x) = zeρgdiff(x), (4)
with zi being the valency of the ionic species i and z = zco = −zctr. The definition of gdiff(x) is, for con-
venience only, designed to make this quantity positive in general. Since the use of this equation implies a
knowledge of ψ(x), and gi (x) throughout the double layer, the expression is non-local.
Independently, Henderson and Boda (HB) [3] have proposed an approximate, local expression for
gdiff(d/2) at low electrode charges from empirical considerations, viz.,
gdiff(d/2) = ab +O(b3
). (5)
To date a formal, analytic connection between equations (3) and (5) remains obscure, although in a later
paper Holovko et al. [11] have outlined a very approximate connection. In a series of papers Henderson
and Bhuiyan [4] and Bhuiyan and co-workers [12–15] have tested equation (3) against exact Monte Carlo
(MC) simulation data for a spectrum of physical states including asymmetric electrolytes [13–15] and
23801-2
Contact values of profiles using density functional theory
found the equation to be remarkably consistent with the simulations. Theoretical support came from an
application of the modified Poisson-Boltzmann (MPB) equation, which was found to satisfy equation (3)
to a very good degree [12, 13]. Bhuiyan and Henderson [16] have also compared the two relations (equa-
tions (2) and (3)) numerically using simulations and the conclusion was that although exact, equation (2)
is difficult to implement numerically because of its non-local nature. We note here that an approximate,
non-local relation for gdiff(d/2) has also been suggested by Henderson and Bhuiyan [17]. Following con-
vention, we will call equation (1) the first contact value theorem, while equations (3) and (5) represent
two versions of the second contact value theorem.
Another interesting recent result that also concerns us in this paper is the behavior of the product of
the co- and counterion contact values f = gco(d/2)gctr(d/2) in the RPM planar double layer. The classical
GCS result for this quantity is strictly unity under all circumstances and thus constitutes a basis for the
classical theory. For example, the value of the counterion contact gctr(d/2) is as high as the reciprocal
of the coion contact gco(d/2). However, the corresponding simulation results [18] dramatically alter the
classical picture. The product f is seen to be not only different from unity, but also that its characteristics
as a function of the electrode charge change with the salt concentration. At low concentrations there is a
maximum before f becomes vanishingly small at high electrode charge. As the concentration increases,
the height of the maximum decreases and at sufficiently high concentrations the maximum disappears
completely with f decreasing monotonously. Again, theoretical support for such a behavior of f came
from the MPB [18] and although the hypernetted chain/mean spherical approximation theory does not
show a maximum, the product f does become very small when the electrode charge is large [19].
In this study we propose to utilize the HB second contact value theorem and the existing MC simula-
tion results from the literature for f to assess the density functional theory (DFT) of the planar double
layer. The DFT has been one of the more successful theories of the electric double layer phenomenon and
compares favorably with the MPB across planar, cylindrical, and spherical geometries (see for example,
references [20–22]). Early applications of the DFT to the planar double layer were made by Tang et al. [23]
andMier y Teran et al. [24]. Later Rosenfeld’s [25] techniques were utilized byMier y Teran et al. [26] and
Boda et al. [27–30]. For even recent publications on application of the DFT to the planar double layer, we
refer the interested reader to the works by Gillespie et al. [31, 32], Valiskó et al. [33], Wang et al. [34], Yu et
al. [35], and Pizio et al. [36]. Since there is more than one version of the DFT for the planar double layer,
in the next section we will briefly outline the DFT method used in this paper. Results will be shown in
section 3, and some conclusions drawn in section 4.
2. Model and methods
2.1. Molecular model
As indicated in the previous section, themodel double layer system consists of a binary, symmetric va-
lency RPM next to a non-penetrable, non-polarizable, uniformly charged planar electrode with a surface
charge density of σ. Since for a given salt concentration, solvent dielectric constant, and temperature, b
has a linear dependence on σ, it is often convenient to specify σ in terms of b.
The ion-ion interaction potential in the Hamiltonian is thus
ui j (r )=
{
∞ r < d ,
e2zi z j /(4πǫ0ǫrr ) r > d ,
(6)
where r is the distance between a pair of ions. We also assume that the dielectric constant, ǫr, is uniform
throughout the entire system. The bare interaction between an ion of species i and the wall is given by
ui (x) = vi (x)+wi (x), (7)
where vi (x) and wi (x) are the non-electrostatic and electrostatic (Coulombic) parts of the ion-wall poten-
tial. The non-electrostatic contribution is a hard-wall potential
ui (x) =
{
∞ x < d/2,
0 x > d/2.
(8)
23801-3
L.B. Bhuiyan, D. Henderson, S. Sokołowski
The electrostatic part wi (x) is given by
wi (x) =−
σzi e
ǫ0ǫr
x, >
d
2
. (9)
2.2. Density functional theory
The essence of the density functional theory (see for example, reference [37]) is that an expression
for the grand potential, Ω, as a functional of the singlet density profiles, ρi (x), of each of the species i ,
is initially constructed. At equilibrium the grand potential is minimal with respect to variations in the
density profiles, viz.,
δΩ
δρi (x)
= 0. (10)
This condition is then used to calculate the density profiles and other relevant quantities like the free
energy.
In the density functional theory the grand potential of an inhomogeneous fluid can be written in the
form [30, 36]
Ω= F ({ρi })+
1
2
∑
i=co,ctr
ezi
∫
ρi (x)ψ(x)dr+
∑
i=co,ctr
∫
[ui (x)−µi ]dr, (11)
where µi denotes the chemical potential of species i . The free energy functions F ({ρi }) is decomposed into
ideal (id), hard-sphere (hs), and electrostatic (el) terms as follows F ({ρi }) = Fid({ρi })+Fhs({ρi })+Fel({ρi }).
The ideal term is known exactly
Fid({ρi }) =
∑
i=co,ctr
∫
dr[ρi (x) lnρi (x)−ρi (x)]. (12)
For the hard-sphere term, however, we apply the expression resulting from a recent version of the Fun-
damental Measure Theory [38], with the free energy consisting of the terms dependent on scalar and
vector weighted densities, for details see reference [36].
Following Pizio et al. [36] electrostatic contribution to the free energy, Fel({ρi }), is represented by
Fel({ρi })=
∫
dr fel({ρ̄i (x)}), (13)
where {ρ̄i (x)} denotes a set of suitably defined inhomogeneous average densities of a reference fluid. One
of the simplest possible choices of fel({ρ̄i (x)}) is to apply the expression resulting from the MSA equation
of state evaluated via the energy route, namely [39]
fel({ρ̄i (x)})/kT =−
d
T ∗
∑
i=co,ctr
z2
i ρ̄i (x)
Γ
1+Γd
+
Γ
3
3π
. (14)
For a symmetric valency situation as in the present case the reduced temperature is T ∗ = 4πkBT ǫ0ǫrd
e2z2
Moreover,
Γ=
(p
1+2κd −1
)
/2d . (15)
The inverse Debye screening length κ can be cast in terms of T ∗
κ
2 = (4πd/T ∗
)
∑
i
z2
i ρ̄i (x). (16)
The last three expressions above correspond to an electroneutral fluid, so that the construction of the
averaged densities ρ̄i (x) at the electroneutrality condition is satisfied. In our approach we follow the
development proposed by Gillespie et al. [31, 32] described briefly below.
Let us define the weighted densities ρ̃i (x) as
ρ̃i (x) =
∫
ρi (x′
)W (|r−r
′|)dr
′
, (17)
23801-4
Contact values of profiles using density functional theory
where W (|r−r
′|) is a weight function. Gillespie et al. made the assumption, viz.,
W (|r−r
′|) =
θ(|r−r
′|)−R f (r
′)
(4π/3)R3
f
(r
′)
, (18)
where θ(|r− r
′|) is the step-function. The radius of the sphere over which averaging is performed, R f , is
approximated by the “capacitance” radius, that is, by the ion radius plus the screening length
R f (r)=
d
2
+
1
2Γ({ρ̄i (x)})
. (19)
In addition, Gillespie et al. [31, 32] required that the fluid with the densities {ρ̄i (x)} have the same ionic
strength as the system with weighted densities, {ρ̃i (x)}. Consequently, in the case of a symmetric 1:1
electrolyte the averaged densities {ρ̄i (x)} are given by
ρ̄1(x) = ρ̄2(x) =
ρ̃1(x)+ ρ̃2(x)
2
. (20)
Because equations (15) and (17)–(20) are coupled, the evaluation of R f requires an iteration procedure.
This iteration loop has to be carried out in addition to the main iteration procedure for evaluating the
density profiles.
The mean electrostatic potential ψ(x) is determined by the Poisson equation
d2ψ(x)
dx2
=−
e
ǫ0ǫr
∑
i
ziρi (x). (21)
The integration of the Poisson equation is carried out subject to the boundary conditions limz→∞ψ(x) = 0
and limx→∞ψ′(x) = 0.
Having specified all the contributions to the free energy functional, the requisite density profiles can
be obtained by minimizing the grand potential (cf. equation (11)).
All the details of our approach can be found in reference [36].
3. Results and discussion
The DFT equations have been solved numerically using the established methods (see for example,
references [23, 36, 40]. We will also present the classical GCS results for comparison purposes, which for
the RPM case can be obtained analytically. It is convenient to discuss the results in terms of universal
reduced parameters such as the reduced density ρ∗ =
∑
i ρi d3
i
and the reduced temperature T ∗ defined
earlier. Calculations were done at two different reduced temperatures, T ∗ = 0.150 and 0.595, respectively,
and at each reduced temperature a number of physical states were treated. The value of the ionic diam-
eter was kept at d = 4.25 ×10−10 m throughout. Although a 1:1 valency system was used in the actual
calculations, in view of universality of T ∗ this becomes a moot point since for a given T ∗ a 1:1 system
at T is equivalent to a 2:2 system at 4T . For example, in the specific case of T ∗ = 0.15, a 1:1 valency case
corresponds to ∼75 K, while a 2:2 valency case corresponds to ∼300 K.
In implementing the HB contact condition, Henderson and Bhuiyan [4] found it convenient to recast
equation (5) in the form
lim
b→0
(
gdiff(d/2,b)
b
)
= a, (22)
for symmetrical valency electrolytes. We have followed the procedure here. We note though that a
straightforward linear plot of equation (5) with a as the slope has also been done [16]. In figures 1–6
we present the results for gdiff(d/2,b)/b and the contact product f = gco(d/2)gctr(d/2) as functions b
for T ∗ = 0.15 at ρ∗ = 0.02 (c = 0.216 mol/dm3), 0.03 (c = 0.324 mol/dm3), 0.05 (c = 0.541 mol/dm3), 0.10
(c = 1.08 mol/dm3), 0.20 (c = 2.16 mol/dm3), and 0.25 (c = 2.70 mol/dm3), respectively. The lone filled cir-
cle on the vertical axis in the upper panel of a figure corresponds to the osmotic coefficient a, which is
evaluated from the simulations at b = 0 using equation (1). Noticeable immediately from the figures is
23801-5
L.B. Bhuiyan, D. Henderson, S. Sokołowski
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
* = 0.02, T* = 0.15
g co
(d
/2
) g
ct
r(d
/2
)
b
0 2 4 6 8 10
0
2
4
6
* = 0.02, T* = 0.15
g di
ff(d
/2
,b
)/b
b
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
* = 0.03, T* = 0.15
g
co
(d
/2
)g
ct
r(d
/2
)
b
0 2 4 6 8 10
0
2
4
6
* = 0.03, T* = 0.15
g di
ff(d
/2
,b
)/b
b
Figure 1. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.02 (c = 0.216 mol/dm3) and T∗ =
0.15. The symbols represent MC data, while the solid
line represents the DFT results, and the dash-dotted
line the GCS results. The filled circle on the vertical
axis in the upper panel is gsum(d/2,b = 0) = a =
0.597. MC data from reference [18].
Figure 2. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.03 (c = 0.324 mol/dm3) and T∗
= 0.15. The filled circle on the vertical axis in the
upper panel is gsum(d/2,b = 0) = a = 0.606. The rest
of symbols and notation as in figure 1. MC data from
reference [18].
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
* = 0.05, T* = 0.15
g
co
(d
/2
)g
ct
r(d
/2
)
b
0 2 4 6 8 10
0
2
4
6
* = 0.05, T* = 0.15
g di
ff(d
/2
,b
)/b
b
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
* = 0.10, T* = 0.15
g co
(d
/2
) g
ct
r(d
/2
)
b
0 2 4 6 8 10
0
2
4
6
* = 0.10, T* = 0.15
g di
ff(d
/2
,b
)/b
b
Figure 3. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.05 (c = 0.541 mol/dm3) and T∗
= 0.15. The filled circle on the vertical axis in the
upper panel is gsum(d/2,b = 0) = a = 0.627. The rest
of symbols and notation as in figure 1. MC data from
reference [18].
Figure 4. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.10 (c = 1.08 mol/dm3) and T∗
= 0.15. The filled circle on the vertical axis in the
upper panel is gsum(d/2,b = 0) = a = 0.684. The rest
of symbols and notation as in figure 1. MC data from
reference [18].
23801-6
Contact values of profiles using density functional theory
the trend that the DFT gdiff(d/2,b)/b (upper panels of the figures) follows the corresponding simulation
results very closely for not too high b for the range of concentration treated. Only a very slight discrep-
ancy is seen at b = 0, which is a consequence of the fact that the DFT does not satisfy the HBL first contact
theorem exactly. The classical GCS theory satisfies equation (5) but with a = 1, the ideal gas value. This is
clear from the figures and for ρ∗ = 0.02, 0.03, 0.05, and 0.10 (figures 1–4), where a is somewhat less than
unity, the GCS theory leads to deviations from the MC data. The results at a different temperature T ∗ =
0.595 and at ρ3 = 0.00925 (c = 0.1 mol/dm3) and ρ3 = 0.0925 (c = 1.0 mol/dm3) are shown in figures 7 and 8,
respectively. Here too the trends shown by the DFT gdiff(d/2,b)/b and their agreement with the corre-
sponding simulations are similar to that seen in figures 1–6. We note that the HNC satisfies equation (1)
with the first term being a function of the hard sphere compressibility, and equation (5) with a = 1 in the
first term. For contact values, it is little better than the GCS theory.
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
* = 0.20, T* = 0.15
g
co
(d
/2
)g
ct
r(d
/2
)
b
0 2 4 6
0
2
4
6
* = 0.20, T* = 0.15
g di
ff(d
/2
,b
)/b
b
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
* = 0.25, T* = 0.15
g co
(d
/2
)g
ct
r(d
/2
)
b
0 2 4 6 8 10
0
2
4
6
* = 0.25, T* = 0.15
g di
ff(d
/2
,b
)/b
b
Figure 5. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.20 (c = 2.16 mol/dm3) and T∗
= 0.15. The filled circle on the vertical axis in the
upper panel is gsum(d/2,b = 0) = a = 0.934. The rest
of symbols and notation as in figure 1. MC data from
reference [18].
Figure 6. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.25 (c = 2.70 mol/dm3) and T∗
= 0.15. The filled circle on the vertical axis in the
upper panel is gsum(d/2,b = 0) = a = 1.09. The rest
of symbols and notation as in figure 1. MC data from
reference [4].
The behavior of the contact product function f is displayed in the lower panel of the figures. Overall
the characteristics of the DFT plots are in qualitative agreement with the simulations. At ρ∗ É 0.10, the
MC data show a maximum with the height of the maximum decreasing as ρ∗ increases. The DFT result is
qualitative and there is a distinct maximum at ρ∗ = 0.03 (figure 2), 0.05 (figure 3), and 0.10 (figure 4), and
at ρ∗ = 0.02, there is the hint of a maximum. Further, the maximum in the DFT curves tends to occur at a
greater value of b than that in the MC. In figures 5 (ρ∗ = 0.20) and 7 (ρ∗ = 0.00925) the plots are initially
flat, while in figures 6 (ρ∗ = 0.25) and 8 (ρ∗ = 0.0925) the initial slope of f is negative. In all of these figures
the DFT continues to be qualitative with the simulations. The characteristics of the initial slope of f as
the salt concentration increases can be understood from the following. From equations (1) and (5) one
has for low b
gco(d/2)gctr(d/2) = a2 + (a −a2
)b2
, (23)
(see for example, equation (18) of reference [18]). This equation is exact in the limit b →0. It is easy to
see at b = 0 that the value of f depends on the value of a. Furthermore, the initial slope of f is negative
for a < 1 (figures 1–4), the initial slope is approximately zero and the plots are initially flat when a ∼ 1
(figures 5 and 7), and the initial slope is negative when a > 1 (figures 6 and 8). Note again that since in the
GCS theory a = 1, the right hand side of equation (19) is unity and the initial slope is zero being consistent
with the observations.
23801-7
L.B. Bhuiyan, D. Henderson, S. Sokołowski
0 1 2 3 4 5
0.0
1.5
3.0
4.5
0 1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
b
* = 0.00925, T* = 0.595
g di
ff(d
/2
,b
)/b
* = 0.00925, T* = 0.595
g co
(d
/2
) g
ct
r(d
/2
)
b
0 1 2 3 4 5
0.0
1.5
3.0
4.5
b
* = 0.0925, T* = 0.595
g di
ff(d
/2
,b
)/b
0 1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
* = 0.0925, T* = 0.595
b
g co
(d
/2
) g
ct
r(d
/2
)
Figure 7. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.00925 (c = 0.1 mol/dm3) and T∗
= 0.595. The filled circle on the vertical axis in the
upper panel is gsum(d/2,b = 0) = a = 0.947. The
rest of symbols and notation as in Figure 1. MC data
from reference [12].
Figure 8. gdiff(d/2,b)/b (upper panel) and
gco(d/2)gctr(d/2) (lower panel) as functions of
b in a RPM planar double layer for symmetric
valencies at ρ∗ = 0.0925 (c = 1 mol/dm3) and T∗
= 0.595. The filled circle on the vertical axis in the
upper panel is gsum(d/2,b = 0) = a = 1.31. The rest
of symbols and notation as in Figure 1. MC data
from reference [12].
An important property of the simulation data is that the contact product f tends to very small values
at large values of b. As the surface charge increases the coion population near the electrode is depleted,
while the counterion population increases. However, the latter also induces packing problems that in-
hibit distant counterions from migrating too close to the electrode surface. All these lead to the observed
behavior of f . In the GCS theory however, the decrease in gco is always proportional to the increase in
gctr so that classically f = 1 consistently. The DFT f generally follows the MC trend in figures 2–8. Al-
though in figure 1 the lack of DFT data beyond b = 8 implies that one cannot be definitive, in view of the
results in the rest of the figures, it is a fair conjecture that here also the contact product will assume small
values at still higher values of b.
4. Conclusions
In this paper we have examined the predictions of a density functional theory of the planar electric
double layer with regards to (i) the HB second contact value theorem, and (ii) the behavior of the product
of the DFT contact values of the co- and counterion distributions vis-a-vis exact MC simulation data from
the literature. The principal finding regarding (i) is that generally the DFT follows the MC results very
closely over the range of concentrations and temperatures studied. There is only just a hint of discrepancy
at b = 0, which is probably tied to the approximation used for the hard-sphere term in the free energy
functional used to construct the grand potential. By contrast, the GCS results show greater deviations
from the simulations, especially at lower concentrations when the MC a is less than unity. It is of interest
to note that the degree to which the DFT satisfies the HB contact condition is very similar to what some of
us have observed with the MPB theory [12, 13] with both of the approaches showing slight deviations at b
= 0. This is not surprising since none of the theories satisfies the HBL first contact value theorem exactly.
With respect to (ii) above, on the other hand, our calculations indicate that the DFT is broadly in
qualitative agreement with the characteristics of the simulations, with the product f of the contact values
tending to small values with increasing surface charge on the electrode. A maximum in f as a function
23801-8
Contact values of profiles using density functional theory
of b seen at lower concentrations although this occurs as, what might be termed, a delayed maximum .
Importantly though, the behavior of the the initial slope of f as the electrolyte concentration increases
follows the MC trend. Admittedly, however, there is a quantitative discrepancy between the DFT results
and the MC data beyond c ∼ 1 mol/dm3. This is understandable in view of the fact that the product of the
contact values of the distributions can be a rather more sensitive quantity than their difference so that a
slight error in either of the contact values tends to become magnified in the contact product.
The version of density functional that is employed in this paper gives good results for the contact
values but is less satisfactory in predicting oscillatory profiles. In contrast, other versions of density func-
tional theory [35,40] are better at producing oscillatory profiles but are less successful for contact values.
There is more to be done in the development of a fully satisfactory density functional theory.
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http://dx.doi.org/10.1063/1.2137707
http://dx.doi.org/10.1063/1.2909973
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http://dx.doi.org/10.1080/08927020701461247
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http://dx.doi.org/10.1063/1.2873466
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http://dx.doi.org/10.1135/cccc20080558
http://dx.doi.org/10.1016/j.jelechem.2006.10.010
http://dx.doi.org/10.1039/b316098j
http://dx.doi.org/10.1080/00268979000101851
http://dx.doi.org/10.1080/00268979100100581
http://dx.doi.org/10.1063/1.464569
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Контактнi значення профiлiв густини в електричному
подвiйному шарi, використовуючи теорiю
функцiоналу густини
Л.Б. Бгуян1, Д. Гендерсон2, С. Соколовскi3
1 Лабораторiя теоретичної фiзики, фiзичний факультет, Унiверситет Пуерто Рiко, США
2 Факультет хiмiї i бiохiмiї, Унiверситет Брiгема Янга, Прово, США
3 Вiддiл моделювання фiзико-хiмiчних процесiв, хiмiчний факультет,
Унiверситет iм. Марiї Складовської-Кюрi, Люблiн, Польща
Нещодавно запропоновану теорему про локальне друге контактне значення [Henderson D., Boda D., J.
Electroanal. Chem., 2005, 582, 16] для профiлю заряду електричного подвiйного шару поєднано з iснуючи-
ми в лiтературi даними Монте Карло з метою оцiнки контактної поведiнки електрод-iонних розподiлiв,
передбачених теорiєю функцiоналу густини. Результати для контактних значень розподiлiв ко- i протиiо-
нiв та їхнього добутку отримано для випадку симетричної валентностi в рамках обмеженої примiтивної
моделi плоского подвiйного шару для низки концентрацiй i температур електролiту. В цiлому, теоретичнi
результати досить добре задовольняють теорему про друге контактне значення, узгодження iз симуляцi-
ями – напiвкiлькiсне або краще. Добуток ко- i протиiонних контактних значень як функцiя густини заряду
поверхнi електрода якiсно узгоджується з симуляцiями, але вiдхилення мiж обома зростає при вищих
концентрацiях.
Ключовi слова: електричний подвiйний шар, обмежена примiтивна модель, профiлi густини
23801-10
http://dx.doi.org/10.1088/0953-8984/14/46/317
http://dx.doi.org/10.1021/jp073703c
http://dx.doi.org/10.1063/1.2137710
http://dx.doi.org/10.1063/1.1818677
http://dx.doi.org/10.1063/1.1520530
http://dx.doi.org/10.1063/1.1468638
http://dx.doi.org/10.1103/PhysRevE.47.4088
Introduction
Model and methods
Molecular model
Density functional theory
Results and discussion
Conclusions
|