A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture
The exclusion volume approximation in the Poisson-Boltzmann theory is analysed at the mean field level for an ion-dipole mixture against a plane, uniformly charged hard wall. An earlier treatment is extended to take account of the deviation of the exclusion volume term from the uncharged wall –...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture / C.W. Outhwaite, S. Lamperski // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 739-748. — Бібліогр.: 28 назв. — англ. |
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irk-123456789-1205362017-06-13T03:02:51Z A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture Outhwaite, C.W. Lamperski, S. The exclusion volume approximation in the Poisson-Boltzmann theory is analysed at the mean field level for an ion-dipole mixture against a plane, uniformly charged hard wall. An earlier treatment is extended to take account of the deviation of the exclusion volume term from the uncharged wall – uncharged hard sphere distribution function. Preliminary numerical results are presented for a 1:1 electrolyte at c = 1.0 mole/dm³ with unequal ion and dipole sizes. Наближення виключеного об’єму в теорії Пуассона-Больцмана аналізується на рівні середнього поля для іонно-дипольної суміші біля плоскої, однорідно зарядженої твердої стінки. Попередній розгляд є розширений до врахування відхилення члена виключеного об’єму від функції розподілу незаряджена стінка – незаряджена тверда сфера. Попередні числові результати є представлені для електроліту 1:1 при c=1.0 моль/дм³ з нерівними розмірами іонів і диполів. 2001 Article A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture / C.W. Outhwaite, S. Lamperski // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 739-748. — Бібліогр.: 28 назв. — англ. 1607-324X PACS: 68.08.-p DOI:10.5488/CMP.4.4.739 http://dspace.nbuv.gov.ua/handle/123456789/120536 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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English |
| description |
The exclusion volume approximation in the Poisson-Boltzmann theory is
analysed at the mean field level for an ion-dipole mixture against a plane,
uniformly charged hard wall. An earlier treatment is extended to take account
of the deviation of the exclusion volume term from the uncharged
wall – uncharged hard sphere distribution function. Preliminary numerical
results are presented for a 1:1 electrolyte at c = 1.0 mole/dm³ with unequal
ion and dipole sizes. |
| format |
Article |
| author |
Outhwaite, C.W. Lamperski, S. |
| spellingShingle |
Outhwaite, C.W. Lamperski, S. A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture Condensed Matter Physics |
| author_facet |
Outhwaite, C.W. Lamperski, S. |
| author_sort |
Outhwaite, C.W. |
| title |
A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture |
| title_short |
A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture |
| title_full |
A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture |
| title_fullStr |
A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture |
| title_full_unstemmed |
A treatment of the exclusion volume term in the inhomogeneous Poisson-Boltzmann theory for an ion-dipole mixture |
| title_sort |
treatment of the exclusion volume term in the inhomogeneous poisson-boltzmann theory for an ion-dipole mixture |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120536 |
| citation_txt |
A treatment of the exclusion volume
term in the inhomogeneous
Poisson-Boltzmann theory for an
ion-dipole mixture / C.W. Outhwaite, S. Lamperski // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 739-748. — Бібліогр.: 28 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT outhwaitecw atreatmentoftheexclusionvolumetermintheinhomogeneouspoissonboltzmanntheoryforaniondipolemixture AT lamperskis atreatmentoftheexclusionvolumetermintheinhomogeneouspoissonboltzmanntheoryforaniondipolemixture AT outhwaitecw treatmentoftheexclusionvolumetermintheinhomogeneouspoissonboltzmanntheoryforaniondipolemixture AT lamperskis treatmentoftheexclusionvolumetermintheinhomogeneouspoissonboltzmanntheoryforaniondipolemixture |
| first_indexed |
2025-07-08T18:05:37Z |
| last_indexed |
2025-07-08T18:05:37Z |
| _version_ |
1837102998787981312 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 4(28), pp. 739–748
A treatment of the exclusion volume
term in the inhomogeneous
Poisson-Boltzmann theory for an
ion-dipole mixture
C.W.Outhwaite 1 , S.Lamperski 2
1 Department of Applied Mathematics, University of Sheffield,
Sheffield, S3 7RH, UK
2 Department of Physical Chemistry, Faculty of Chemistry,
A.Mickiewicz University,
Grunwaldzka 6, 60–780 Poznan, Poland
Received August 8, 2001
The exclusion volume approximation in the Poisson-Boltzmann theory is
analysed at the mean field level for an ion-dipole mixture against a plane,
uniformly charged hard wall. An earlier treatment is extended to take ac-
count of the deviation of the exclusion volume term from the uncharged
wall – uncharged hard sphere distribution function. Preliminary numerical
results are presented for a 1:1 electrolyte at c = 1.0mole/dm3 with unequal
ion and dipole sizes.
Key words: excluded volume term, Poisson-Boltzmann, ion-dipole
mixture, electric double layer
PACS: 68.08.-p
1. Introduction
The electric double layer, in the neighbourhood of a charged surface in an elec-
trolyte solution, plays a key role in understanding the behaviour of many physical,
chemical and biological phenomena. The classical treatment of the electrical dou-
ble layer is based on the mean field Poisson-Boltzmann (PB) equation where the
electrolyte solution is modelled by a system of point ions moving in a dielectric
continuum. Many attempts have been made to improve both the electrolyte model
and the theory of this classical approach. A basic reference electrolyte model is the
primitive model (PM) where the ions in the dielectric are given finite size by being
represented as charged hard spheres. More realistic models treat the solvent on an
equal footing with the solute, the solvent having dipoles and higher order multipoles.
c© C.W.Outhwaite, S.Lamperski 739
C.W.Outhwaite, S.Lamperski
Theoretical improvements have been based on improved or modified PB equations,
integral equations and simulation [1,2].
We consider here a PM electrolyte or an ion-dipole mixture against a uniformly
charged plane hard wall. The emphasis will be upon examining the influence of
ion or solvent size upon the structure of the electric double layer within a mean
electrostatic potential framework. Various attempts have been made to account for
the steric effect of the ions and the solvent. The earliest work was that of Stern [3]
who introduced the idea of an inner compact layer to correct the Gouy-Chapman
[4,5] theory which overestimated ionic adsorption. Stern’s approach has led to many
different models and theories for the structure of the solution next to an electrode
[6–8], but there are difficulties in reconciling many of these approaches with an
overall consistent statistical mechanical picture of the double layer. An important
related aspect is a realistic model of the electrode in assessing its influence on ionic
distribution and adsorption [9,10].
The first formal statistical mechanical analysis detailing the deficiences of the
PB equation was due to Kirkwood [11]. He showed that the two approximations
in the theory were the neglect of the exclusion volume term and of the fluctuation
potential. The exclusion volume term arises from the discharged ion in the formal
charging process, and in the double layer it plays a major role in controlling the
surface concentration of the ions. The various treatments and relative importance of
the two approximations within the mean electrostatic framework for the restricted
PM (RPM) are given in [1,12]. There has been a recent resurgence of interest in
incorporating steric effects into the PB equation in a simple fashion. These attempts
are based on lattice model [13–15] and functional [16–18] approaches. The derivation
in [13,18] of these improved PB equations invokes solvent exclusion volume effects
and the new theories impose an upper limit on the surface concentration of the
counterions. No treatment is made of fluctuation effects and the solvent is solely
considered as neutral hard spheres. The underlying electrolyte model, when the
solvent is uncharged hard spheres moving in a dielectric continuum, is the solvent
primitive model (SPM). The modified Poisson-Boltzmann (MPB) theory [1,12,19–
21] treats both the exclusion volume and the fluctuation potential term and formally
prevents unphysical counterion surface coverage at high surface charge. By neglecting
the fluctuation term, we show here how the MPB exclusion volume term reduces
to expressions analogous to the recent improved PB results. Preliminary results are
also presented for the ion-dipole mixture with different sizes which extends earlier
work [22] on a corrected PB theory incorporating exclusion volume terms.
2. Theory
The electric double layer is modelled by a mixture of charged hard spheres and
dipolar hard spheres moving in a medium of background relative electric permittivity
ε∞ in the neighbourhood of a uniformly charged plane hard wall. At a normal
distance x1 from the electrode into the solution, Poisson’s equation for the mean
740
Exclusion volume term in electric double layer
electrostatic potential ψ = ψ(1) is
d2ψ
dx21
= −
1
ε0ε∞
∑
k
eknkgk +
1
ε0ε∞
dP
dx1
, (1)
where
P = nd
∫
µ · x̂1 gd dω. (2)
Here P = P (1) is the polarization with dω the dipole rotational elements, ek the
charge on an ion of species k, µ the dipole moment, nt the bulk number density
of species t and gk, gd the singlet distribution functions for an ion k or dipole d
at x1 respectively. The labels s and t are for either ions or dipoles while k and d
signify ions and dipoles respectively. For both the PM and SPM, P vanishes and
ε∞ is replaced by the bulk relative electric permittivity (bulk dielectric constant) ε.
However, in the PM, exclusion volume effects only arise through ion size while the
SPM also incorporates hard sphere solvent effects.
Neglecting the fluctuation term [23],
gk = ζk exp(−βekψ), (3)
gd = (ζd/4π) exp(−βµ · ∇ψ), (4)
where ζk = gk(ek = 0), ζd = gd(µ = 0) are the exclusion volume terms. For high
values of the surface charge σ, the exclusion volume terms will impose a limit on
the surface concentration of the ions and solvent. The mean field approximation is
given by replacing ζk, ζd by the appropriate unit step functions.
To treat the exclusion volume terms, consider the exact equation (2.11) of [24],
ln gs = −βφs +
∫
1
0
∫
∑
t
nt cst
∂gt
∂λ
d2 dλ, (5)
where φs is the interaction between the particle s and the field due to the electrode,
cst = cst(1, 2|λ) is the nonuniform direct correlation function and λ is the coupling
parameter for the electrode field. Although developed for the PM, equation (5)
also holds for the hard sphere ion-dipole mixture when d2 includes the rotational
elements for t running over the dipole species.
The exclusion volume term from equation (5) is
ln ζs = −βφs(s = 0) +
∫
1
0
∫
∑
t
nt cst(s = 0)
∂gt
∂λ
d2 dλ, (6)
where s = 0 means that the ion or dipole at x1 is discharged.
A formal decomposition of cst into a short and long range part means we expect
the long range term to vanish and cst(s = 0) to be of short range, independent of λ.
Also φ(s = 0) is simply the hard sphere – hard wall interaction. So integrating with
respect to λ,
ln ζs =
∫
∑
t
nt cst(s = 0)[gt − 1]d2, (7)
741
C.W.Outhwaite, S.Lamperski
with ζs = 0 when x1 is less than the distance of closest approach of s to the electrode.
Now putting the surface charge zero in equation (7), and then subtracting the result
from equation (7), gives
ln(ζs/ζ
0
s ) =
∫
∑
t
nt
{
[cst(s = 0)gt − c0stgt(σ = 0)] + [c0st − cst(s = 0)]
}
d2, (8)
where ζ0s = ζs(σ = 0) and c0st indicates that in cst both the molecule s at x1 and
surface charge are discharged. Setting cst(s = 0) = c0st, which is true for the bulk,
gives
ln(ζs/ζ
0
s ) =
∫
∑
t
ntc
0
st[gt − gt(σ = 0)] d2. (9)
Equations similar to equation (9) can be derived from the other standard statistical
mechanical theories [1, p. 188–189], although now c0st takes its bulk value rather
than the inhomogeneous value in equation (9). Given a value for c0st, such as the
bulk Percus-Yevick (PY) value for a mixture of uncharged hard spheres [25], the
equations (1), (3), (4), (9) form a consistent set of equations for ψ. We shall not
consider the full solution here but only analyse how the exclusion volume term given
by equation (9) may alter the PB equation. A solution of the system for ζs given
by the PY uncharged hard sphere singlet distribution function has been treated
elsewhere [22].
Alternatively an equation analogous to equation (8) for the PM can be derived
from the BBGY hierarchy [19]
ln(ζs/ζ
0
s ) =
∫
∞
x1
∑
t
nt
∫
S12
x̂ · r̂12 [gtgst(es = 0)
− gt(σ = 0)gst(es = σ = 0)] dS2dx2. (10)
We first show how equation (7) in conjunction with equations (3), (4) relates to the
earlier work of Borukhov et al. [14,15]. Consider a single electrolyte with charges
ei, ej , with all the ions and dipoles having the same diameter R and c st(s = 0) given
by the bulk uncharged direct correlation function. In this special case we write ζ
and c0 for ζs and c
0
st respectively as they are the same for all the species.
Equation (7) then simplifies to
ln ζ =
∫
∑
k
nk c
0[ζ exp(−βekψ)− 1] d2
+
∫
(nd/4π) c
0[ζ exp(−βµ · ∇ψ)− 1]d2. (11)
We now assume that the exclusion volume term only deviates slightly from its bulk
value of unity, and write ζ = 1+ δ where |δ| < 1. So expanding to the first order in
δ and approximating δ in the integrand by its mean value δ at x1, gives
δ = δX +X −Xb, (12)
742
Exclusion volume term in electric double layer
where
X = niFi + njFj + ndGd, (13)
Xb =
∑
s
ns
∫
c0 d2, (14)
Fk =
∫
c0 exp(−βekψ) d2, (15)
Gd =
∫
c0D d2, (16)
D = (1/4π)
∫
exp(−βµ · ∇ψ) dω = sinhΘ/Θ, (17)
Θ = µβdψ/dx2. (18)
Equating δ to δ and solving equation (12) gives
ζ =
1−Xb
1−X
, (19)
which is implicit in the RPM work of Bell and Levine [26]. To see if (19) may predict
a finite counterion gi for a large surface charge, we use the approximation c0 =
−1 +O(n), r < R, and zero elsewhere, and so for x1 > R/2, Fk ∼ −v exp(−βekψ),
Gd ∼ −v exp(−βµ ·∇ψ), where v = 4πR3/3 or the truncated sphere for R/2 < x1 <
R. With these approximations
ζ =
1 + v
∑
s ns
1 + v [
∑
k nk exp(−βekψ) + nd exp(−βµ · ∇ψ)]
, (20)
thus for |βeiψ| ≫ 1 with −β(µ · ∇ψ − eiψ) ≪ 0
gi →
1
vni
(
1 + v
∑
s
ns
)
, (21)
which is finite. Multiplying numerator and denominator of equation (20) by 1 +X b
and neglecting terms of order n2, then putting nd = 0 or µ = 0 gives the special
cases of [14]. The corrected PB equation is given by equations (1), (3), (4), (19)
and when nd = 0 we are working with the PM so that the correction arises solely
from the ion sizes, and when µ = 0 the correction is within the SPM and comes
from the solvent size. As stressed by Kralji-Iglic and Iglic [13], Borukhov et al. [14],
an exclusion volume term imposes a limit on the surface ion concentration. The
present formulation of the MPB theory [1,20] for the RPM utilizes approximations
to equation (10) and would be expected to prevent unphysical counterion adsorption.
As yet there are no MPB numerical results for high surface charge.
A failure of result (19) or (20) is that when the ions and dipole are discharged the
exclusion volume term is unity, as in the PB theory, rather than the uncharged hard
sphere-hard wall distribution function. Indeed this uncharged distribution function
743
C.W.Outhwaite, S.Lamperski
must contain important steric effects at high electrolyte concentrations. To overcome
this deficiency we generalize the derivation of (19) by writing ζs = ζ0s (1 + δs) where
we assume |δs| is small. So, substituting in equation (9) for ζs, using (3), (4) for the
singlet distribution functions, expanding to first order in δs, and approximating δs
under the integral sign by its mean value δ s at x1 gives
δs =
∑
k
δknkFsk + δdndGsd +Xs −X0
s , (22)
where
Xs =
∑
k
nkFsk + ndGsd, (23)
X0
s = Xs(σ = 0), (24)
Fsk =
∫
c0skζ
0
k exp(−βekψ) d2, (25)
Gsd =
∫
c0sdζ
0
dD d2. (26)
If we approximate δs by δs then equation (22) becomes a system of linear equations
which can readily be solved for δs. When σ → 0, δs → 0 and so as required ζs → ζ0s .
For a single electrolyte with equal size ions of charge e i and ej, simplifications occur
as then for c0ii = c0ij = c0jj, Fii = Fji, Fjj = Fij and Gid = Gjd. Furthermore if the
dipoles are also the same size as the ions, Fdi = Fii, Fdj = Fjj, Gid = Gjd = Gdd. In
particular for a z : z electrolyte for equisized ions and dipoles,
ζ = ζ0
1− n(F 0
ii + F 0
jj)− ndG
0
dd
1− n(Fii + Fjj)− ndGdd
, (27)
where ni = nj = n; and for the 1 : z case with equal sizes,
ζ = ζ0
1− n(zF 0
ii + F 0
jj)− ndG
0
dd
1− n(zFii + Fjj)− ndGdd
, (28)
where ni = znj = zn and ζ = ζs, ζ
0 = ζ0s . As before these two results are analogous
to those of Borukhov et al.. In any investigation of the influence of the exclusion
volume term given by equation (9), we would expect equation (22) to provide a
reasonable first approximation to ζs for the numerical solution of the corrected PB
equation, or the corresponding MPB theory in the RPM. In (22) the ζ 0s and c0st can
be estimated by any appropriate uncharged hard wall – hard sphere distribution
function and bulk hard sphere direct correlation function respectively.
3. Results and discussion
We now present some preliminary results for the unequal size ion-dipole mixture
based on equation (9) for ζk and equation (22) for ζd. From equation (22), with
744
Exclusion volume term in electric double layer
δk = 0, we obtain
ζd = ζ0d
1 +
∑
k nk(Fdk − F 0
dk)− ndG
0
dd
1− ndGdd
. (29)
For consistency the dipole exclusion volume term should also be calculated from
equation (9). The present numerical procedure needs to be adapted to solve this
consistent situation, but we expect equation (29) to be a good approximation to
ζd for the considered parameter range. The exclusion volume term for zero surface
charge ζ0s , and the uncharged bulk direct correlation function, are approximated by
the corresponding Percus-Yevick values for uncharged hard sphere mixtures [25].
The three component PY g0s has been given elsewhere [22] and the PY cst can be
derived from Baxter’s work [27], namely
rcij =
{
−αir − βi + 2π
∑
s nsIs, r < Rij
0, r > Rij ,
(30)
where
Is = Ib = −
R2
i
24
[
12β2
s + 12αsβs(r +Rs) + α2
s(2Rir + 6Rsr + 3R2
s)
]
, r < Sji , (31)
Is = Ib + (Sij + r)2
{
β2
s
2
+
αsβs
2
(Ris +Rjs) +
α2
s
24
[
r(2Sij − r) + 3(Rij +Rs)
2
]
}
,
Rij > r > Sji , (32)
and
αi =
1− ξ3 + 3Riξ2
(1− ξ3)2
, βi = −
3R2
i ξ2
2(1− ξ3)2
, ξj =
π
6
∑
i
niR
3
iR
j
i . (33)
Here Ri is the diameter of i with Rij = (Ri +Rj)/2, and Sij = (Ri − Rj)/2.
The numerical solution of Poisson’s equation (1), in conjunction with equations
(3), (4) and exclusion volumes given by equation (9) for the ions and (29) for the
dipoles, was obtained by a modification of the quasi-linearization technique of [22].
Calculations were performed for a 1:1 electrolyte at concentration c = 1.0mole/dm3
with R+ = 3.6 Å, R− = 3.0 Å while the solvent parameters were Rd = 2.4 Å, µ =
1.84D and concentration cd = 45mole/dm3. The background permittivity ε∞, which
reflects the atomic and electronic polarization of solute and solvent, was taken to be
2 and temperature T = 298.15K.
In the figures, comparisons are made with the case when the exclusion volume
term is simply given by the PY uncharged singlet distribution function g 0
s , this
situation being denoted by g∗s . Figure 1 indicates a large reduction in the value of
ζk from the PY g0k at σ = 0.1C/m2 in the vicinity of the electrode. In contrast
there is only a small reduction in ζd, so now it is the largest, not the smallest,
exclusion volume term near the electrode. The small reduction in ζ d is significant as
it indicates a correlation with the surface charge which will become more important
as σ increases. A similar overall picture occurs for σ = −0.1C/m2.
The variation of the contact values of ζs and gs for σ varying from −0.15 to
0.15C/m2 are given in figures 2–4. For the ions ζk is less than the PY g0k, except
745
C.W.Outhwaite, S.Lamperski
x
*
10
10
/ m
0
2
4
6
ζ
s
,
g
s
0
0
1
2
3
d
-
+
σ
/ Cm
-2
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
ζ
,
g
0
10
20
30
ζ
-
g
-
0
(PY)
g
-
*
g
-
Figure 1. Exclusion volume term ζs
(solid line) and PY g0s (dashed line)
at σ = 0.1C/m2. Other parameters as
in text. The solvent, anion and cation
functions are denoted by d, −, + re-
spectively.
Figure 2. Contact value of the an-
ion exclusion volume term and singlet
function for |σ| < 0.15C/m2. The sin-
glet functions g−, g
∗
−
are calculated us-
ing ζ− and PY g0
−
respectively.
σ
/ Cm
-2
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
ζ
,
g
0
10
20
30
ζ
+
g
+
0
(PY)
g
+
*
g
+
σ
/ Cm
-2
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
ζ
,
g
1.8
2.0
2.2
2.4
2.6
2.8
ζ
d
g
d
0
(PY)
g
d
*
g
d
Figure 3. Contact value of the cation
exclusion volume term and singlet
function for |σ| < 0.15C/m2. The sin-
glet functions g+, g
∗
+ are calculated us-
ing ζ+ and PY g0+ respectively.
Figure 4. Contact value of the sol-
vent exclusion volume term and singlet
function for |σ| < 0.15C/m2. The sin-
glet functions gd, g
∗
d are calculated us-
ing ζd and PY g0d respectively.
746
Exclusion volume term in electric double layer
σ
/ Cm
-2
-0.10
-0.05
0.00
0.05
0.10
C
/
F
m
-2
0.122
0.124
0.126
0.128
0.130
0.132
0.134
0.136
0.138
0.140
Figure 5. The differential capacitance C.
The dashed line is the capacitance when
the exclusion volume term is given by the
PY g0s .
when σ = 0. This in turn reduces the
contact value of gk for σ 6= 0 from g∗k.
However for the solvent when σ > 0, the
contact value of gd is larger than that of
g∗d in contrast to its lower value when
σ < 0. This is a reflection of the dif-
ferent molecular sizes, with the dipole
being the smallest. At these low surface
charges the larger counterion (cation for
σ < 0) reduce the adsorption of the sol-
vent molecules relative to the g0d, and the
smaller counterion (anion for σ > 0) in-
crease the solvent adsorption. The effect
is clearly seen in figure 5 for the differen-
tial capacitance. Rather than an asym-
metric ‘U’ shape from the g∗s as found in
[22], the capacitance begins to adopt a
shape reminiscent of group 2 of Parson’s
classification [28].
The absolute values of the capacitance are too low, and this can be attributed to
many factors, two of the most probable being the poor representation of the solvent
and the neglect of the fluctuation term in the distribution functions. However it ap-
pears that the formulation of the exclusion volume terms presented here which takes
into account surface charge correlations, married with treating solute and solvent
molecules on an equal footing, may prove fruitful. Work is in progress studying the
implication of various parameter variations.
References
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Трактування члена виключеного об’єму в
неоднорідній теорії Пуассона-Больцмана для
іонно-дипольної суміші
К.В.Аутвайт 1 , С.Ламперскі 2
1 Відділення прикладної математики, Шефілдський університет,
Шефілд, Великобританія
2 Інститут фізичної хімії, хімічний факультет, Університет А.Міцкевича,
Познань, Польща
Отримано 8 серпня 2001 р.
Наближення виключеного об’єму в теорії Пуассона-Больцмана ана-
лізується на рівні середнього поля для іонно-дипольної суміші біля
плоскої, однорідно зарядженої твердої стінки. Попередній розгляд є
розширений до врахування відхилення члена виключеного об’єму від
функції розподілу незаряджена стінка – незаряджена тверда сфера.
Попередні числові результати є представлені для електроліту 1:1 при
c=1.0 моль/дм 3 з нерівними розмірами іонів і диполів.
Ключові слова: член виключеного об’єму, Пуассон-Больцман,
іонно-дипольна суміш, електричний подвійний шар
PACS: 68.08.-p
748
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