Scaling in charged fluids: beyond simple ions
The analytical solution of the Mean Spherical Approximation for a quite general class of interactions is always a function of a reduced set of scaling matrices Γχ. We extend this result to systems with multipolar interactions: We show that for the ion-dipole mixture the thermodynamic excess...
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| Zitieren: | Scaling in charged fluids: beyond simple ions / L. Blum // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 611-620. — Бібліогр.: 26 назв. — англ. |
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irk-123456789-1205192017-06-13T03:04:55Z Scaling in charged fluids: beyond simple ions Blum, L. The analytical solution of the Mean Spherical Approximation for a quite general class of interactions is always a function of a reduced set of scaling matrices Γχ. We extend this result to systems with multipolar interactions: We show that for the ion-dipole mixture the thermodynamic excess functions are a functional of this matrix. The result for the entropy is S = −{kV/3π}(F[Γα])α∈χ where F is an algebraic functional of the scaling matrices of irreducible representations χ of the closure of the OrnsteinZernike. The result is also true for arbitrary electrostatic multipolar interactions. Аналітичний розв’язок середньосферичного наближення для достатньо загального класу взаємодій є завжди функцією редукованого набору скейлінгових матриць. Ми розширюємо цей результат на випадок систем з мультипольними взаємодіями. Ми показуємо, що для іонно-дипольної суміші термодинамічні надлишкові функції є функціями цієї матриці. Результат для ентропії є S = −{kV /3π}(F[Γα])α∈χ, де F – алгебраїчний функціонал скейлінгових матриць незвідних представлень замикання Орнштейна-Церніке. Результат дійсний також і для довільних електростатичних мультипольних взаємодій. 2001 Article Scaling in charged fluids: beyond simple ions / L. Blum // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 611-620. — Бібліогр.: 26 назв. — англ. 1607-324X PACS: 61.20.Gy DOI:10.5488/CMP.4.4.611 http://dspace.nbuv.gov.ua/handle/123456789/120519 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
The analytical solution of the Mean Spherical Approximation for a quite
general class of interactions is always a function of a reduced set of scaling
matrices Γχ. We extend this result to systems with multipolar interactions:
We show that for the ion-dipole mixture the thermodynamic excess
functions are a functional of this matrix. The result for the entropy is
S = −{kV/3π}(F[Γα])α∈χ where F is an algebraic functional of the scaling
matrices of irreducible representations χ of the closure of the OrnsteinZernike.
The result is also true for arbitrary electrostatic multipolar interactions. |
| format |
Article |
| author |
Blum, L. |
| spellingShingle |
Blum, L. Scaling in charged fluids: beyond simple ions Condensed Matter Physics |
| author_facet |
Blum, L. |
| author_sort |
Blum, L. |
| title |
Scaling in charged fluids: beyond simple ions |
| title_short |
Scaling in charged fluids: beyond simple ions |
| title_full |
Scaling in charged fluids: beyond simple ions |
| title_fullStr |
Scaling in charged fluids: beyond simple ions |
| title_full_unstemmed |
Scaling in charged fluids: beyond simple ions |
| title_sort |
scaling in charged fluids: beyond simple ions |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120519 |
| citation_txt |
Scaling in charged fluids: beyond
simple ions / L. Blum // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 611-620. — Бібліогр.: 26 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bluml scalinginchargedfluidsbeyondsimpleions |
| first_indexed |
2025-07-08T18:01:00Z |
| last_indexed |
2025-07-08T18:01:00Z |
| _version_ |
1837102701740032000 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 4(28), pp. 611–620
Scaling in charged fluids: beyond
simple ions
L.Blum
Department of Physics P.O. Box 23343, University of Puerto Rico,
Rio Piedras, PR 00931-3343
Received September 28, 2001
The analytical solution of the Mean Spherical Approximation for a quite
general class of interactions is always a function of a reduced set of scal-
ing matrices Γχ . We extend this result to systems with multipolar inter-
actions: We show that for the ion-dipole mixture the thermodynamic ex-
cess functions are a functional of this matrix. The result for the entropy is
S = −{kV /3π}(F [Γα])α∈χ where F is an algebraic functional of the scal-
ing matrices of irreducible representations χ of the closure of the Ornstein-
Zernike. The result is also true for arbitrary electrostatic multipolar interac-
tions.
Key words: Coulomb systems, mean spherical approximation, entropy,
ion-dipole mixtures
PACS: 61.20.Gy
1. Introduction
It is my pleasure to contribute to this issue dedicated to J.-P.Badiali a true
scientist and a gentleman.
The remarkable simplicity of the mean spherical approximation (MSA) [1–4] and
its extensions using Yukawa closures [5–7] can be summarized by the fact that the
entropy for a wide class of systems has a very simple functional form. The MSA
[8] is the solution of the linearized Poisson-Boltzmann equation, just as the Debye-
Hückel (DH) theory. It shares with the DH theory the remarkable simplicity of a
one parameter description (the screening length κ) of all the thermodynamic and
structural properties of rather diverse systems. The MSA shares this feature with
the DH theory [9]. The major difference is that in the MSA the excluded volume of
all the ions is treated exactly.
The MSA is attractive for chemists examining the thermodynamic properties of
real electrolyte solutions not only because the model gives simple analytical formulas,
but also satisfies exact asymptotic relations, such as the large charge, large density
limits of Onsager [10–13] and more recently, the large charge small density limits
c© L.Blum 611
L.Blum
implied in the Wertheim-Ornstein-Zernike equation. Very recently [5,6] we have been
able to extend the MSA closure analytical solutions to any arbitrary closure that
can be expanded in damped exponentials (Yukawa functions), and to obtain explicit,
analytical forms of the excess thermodynamic functions in terms of a matrix of
scaling parameters (the EMSSAP, or Equivalent Mean Spherical Scaling Approach
[7]).
For systems with Coulomb and screened Coulomb interactions in a variety of
mean spherical approximations (MSA) it is known that the solution of the Orn-
stein Zernike (OZ) equation is given in terms of a single screening parameter Γ.
This includes the ‘primitive’ model of electrolytes,in which the solvent is a contin-
uum dielectric, but also models in which the solvent is a dipolar hard sphere, and
much more recently the YUKAGUA model of water that has the correct tetrahedral
structure. The MSA can be deduced from a variational principle in which the energy
is obtained from simple electrostatic considerations and the entropy is a universal
function. For the primitive model it is
∆S = −kV
Γ3
3π
,
where Γ is the MSA screening parameter and in general it will be of the form
∆S = S(Γ),
which is independent of the form of the cavity in this approximation and
Γ .
is now the scaling matrix. We have shown that in all known cases the scaling matrix
Γ is obtained from the variational principle
∂A
∂Γ
= 0 (1)
Ionic solutions are mixtures of charged particles, the ions, and the neutral solvent
particles, most commonly water, which has an asymmetric charge distribution, a
large electric dipole and higher electric moments. Because of the special nature of
these forces the charge distribution around a given ion and the thermodynamics does
satisfy a series of conditions or sum rules. One remarkable property of mixtures of
classical charged particles is that because of the very long range of the electrostatic
forces, they must create a neutralizing atmosphere of counterions, which shields
perfectly any charge or fixed charge distribution. Otherwise the partition function,
and therefore all the thermodynamic functions, will be divergent [14]. The size of
the region where this charge shielding occurs depends not only on the electrostatics,
but also on all the other interactions of the system. For spherical ions this means:
1. The internal energy E of the ions is always the sum of the energies of capac-
itors. For spherical ions the capacitor is a spherical capacitor, and the exact
612
Scaling in charged fluids
form of the energy is
∆E = −e2
ε
∑
i
ρizi
z∗i
1/Γi + σi
, (2)
where z∗i is the effective charge, β = 1/kT is the usual Boltzmann thermal
factor, ε is the dielectric constant, e is the elementary charge, and ions i
have charge, diameter and density ezi, σi, ρi, respectively. For the continuum
dielectric primitive model Γ i = Γ for all i.
2. The Onsagerian limits. When the ionic concentration goes to infinity and at
the same time the charge diverges, then the limiting energy is bounded by
∆E = −e2
ε
∑
i
ρizi
z∗i
σi
, (3)
obtained by setting Γi → ∞
3. A further exact limit is the DH limiting law, which simply requires that for
all ions in the system
2Γi → κ with κ2 =
4πβe2
ε
m
∑
j=1
ρjz
2
j . (4)
4. Finally in systems that are strongly associating in the limit of total association
the above equation still holds. This means that if component 1 forms a n-mer
the DH limiting law must satisfy
κ2 =
4πβe2
ε
m
∑
j=2
ρjz
2
j + ρ1(nz1)
2
. (5)
This limiting law is not satisfied by any closure of the regular Ornstein-Zernike
equation, but only for closures of the Wertheim-Ornstein-Zernike equation
[15,16].
1.1. Charge-charge interactions
For the primitive model of ionic solutions in the general case [9] the parameter
Γ is determined from the equation
4πe2
ǫWkBT
∑
i
ρiz
2
i
(1 + Γ)2
= 4Γ2, (6)
where the ionic charge is zie and number density ρi = Ni/V , where Ni is the number
of ions and V is the volume of the system. We have
κ2
(1 + Γ)2
= 4Γ2, (7)
613
L.Blum
where κ is defined by equation (5). The known analytical MSSA solutions for dimers
and polymers satisfy a ‘universality’ principle for the excess entropy
∆S(MSA) = −kV
Γ3
3π
. (8)
Then Γ is determined in every case by the simple variational equation (1)
∂[β∆E(Γ) + Γ3/(3π)]
∂Γ
= 0. (9)
This equation is also obtained by solving the MSA using the standard procedure.
1.2. Dipole-dipole interactions
For a system of hard spheres with a permanent dipole moment µs the MSA result
can be expressed in terms of a single parameter λ. Following Wertheim [17], we have
d22 =
λ2(λ+ 2)2
9
(
1− 1
ǫW
)
, (10)
where
d22 =
4πρsµ
2
s
3kBT
(11)
and ρs is the solvent number density. Furthermore, the MSA dielectric constant ǫW
of the solvent is given by
ǫW =
λ2(λ+ 1)4
16
. (12)
As has been often done in the literature, the parameter λ can be computed direct-
ly from the dielectric constant ǫW using the above cubic equation. This parametriza-
tion defines an effective polarization parameter. Just as in the case of the ions, there
is a physically meaningful way of interpreting the MSA for point dipoles using the
variational principle (1). The dipolar system can be represented by a collection of
dipolar spheres [12].
A dipolar sphere in a dielectric continuum
λ =
ǫin
ǫout
,
b2
6
= geffk ,
where b2 is the dipole-dipole energy parameter defined below equation (23) and g eff
k
is the effective Kirkwood parameter for this system. From here we calculate the
induced dipole
Xd =
3d2
λ+ 2
= d2β6
and the excess energy
βE = d2Xd .
614
Scaling in charged fluids
So that the closure equation can be rewritten as
9d22
(λ+ 2)2
= λ2 − 16
(λ+ 1)4
. (13)
This corresponds to exactly equation (1) in the form
∂βE
V ∂λ
= − π
V k
∂S
∂λ
= λ2 − 16
(λ+ 1)4
, (14)
which can be integrated to yield
− π
kV
S =
1
3
[
λ3 + 2
(
2
λ+ 1
)3
]
− 1. (15)
Now if we define the scaling lengths for the irrep χ = 0
Γ0 = λ
and for χ = ±1
Γ1 =
2
λ+ 1
,
then
− π
V k
S =
1
3
[
(Γ0)
3 + 2 (Γ1)
3
]
− 1. (16)
Notice that they satisfy the Wertheim ‘density’ of the irreps since they are obtained
by setting
ρ1 = −(1/2)ρ0 .
Furthermore, observe that because of the structure of the equations the natural
assignment is
Xd =
3
λ+ 2
d2 = β6d2 . (17)
1.3. Charge-dipole interactions
We summarize the results of the previous work [18–24]. We use the invariant
expansion formalism [25], in which the total pair correlation h(12) is expanded in
terms of rotational invariants
h(12) = ĥ000(r12)+ĥ011(r12)Φ̂
011+ĥ101(r12)Φ̂
101+ĥ110(r12)Φ̂
110+ĥ112(r12)Φ̂
112, (18)
where ĥmnℓ(r12) is the coefficient of the invariant expansion, which depend only on
the distance r12 between spheres 1 and 2. The rotational invariants Φ̂mnℓ depend
only on the mutual orientations of the molecules. For the present case the relevant
correlation functions are
• ion-ion:
hii(r) = (1/2)
[
ĥ000
++(r)− ĥ000
+−
(r)
]
; (19)
615
L.Blum
• ion-dipole:
hin(r) = (1/2)
[
ĥ011
+n (r)− ĥ011
−n (r)
]
(r̂ · µ̂); (20)
• dipole-dipole:
hnn(r) = −
√
3ĥ110
nn (r)µ̂1 · µ̂2
+
√
15
2
ĥ112
nn (r) [3(r̂ · µ̂1)(r̂ · µ̂2)− µ̂1 · µ̂2] , (21)
where µ̂ is the unit vector in the direction of µ. The solution of the MSA is given in
terms of the ‘energy’ parameters
• ion-ion:
b0 = 2πρi
∫
∞
0
drhii(r)r; (22)
• ion-dipole:
b1 = 2π
√
ρiρs
3
∫
∞
0
drhin(r); (23)
• dipole-dipole:
b2 = 3πρs
√
2
15
∫
∞
0
dr
ĥ112(r)
r
, (24)
which, as will be shown below are proportional to the ion-ion, ion-dipole and dipole-
dipole excess internal energy [21]. In the MSA they are functions of the ion charge
and the solvent dipole moment, through the parameters
d20 =
4πe2
kBT
∑
j
ρjz
2
j (25)
and d22 is defined by equation (11) These parameters are required to satisfy the
following equations [18]
a21 + a22 = d20 , (26)
a1K10 − a2[1−K11] = d0d2, (27)
K2
10 + [(1−K11)]
2 = y21 + d22 . (28)
where
D = 1 + B1 (29)
with
B1 =
b21
4β2
6
=
b21(λ+ 2)2
36
, (30)
β6 = 1− b2
6
.
616
Scaling in charged fluids
A simple set of equations is obtained when we use the proper scaling lengths [22,23].
In terms of the excess energy parameters b0, b1 and b2 of equations (22–24):
b0 =
−Γ
1 + Γ
+ B1DΛ , (31)
b1 =
√
B1
6
2 + λ
(32)
and
b2 = 6
λ− 1
2 + λ
. (33)
The entropy can be computed using [21,26]
S
kV
=
β
V
[E −A], (34)
which leads to
S
kV
=
1
12π
(
b0d0
2 − 4 b1d0d2 − 6b2d
2
2 + 2 q′
2
+Q′
dd
2
+ 2Q′
id
2
+ [Q′
ii]
2
)
. (35)
The contact pair correlations are [21–23]
• ion-ion
Q′
ii =
2
D{[Γ]2 + B1}; (36)
• ion-dipole
Q′
id =
2
√
B1
D {[1 + Γ]Dλg − 1]}; (37)
• dipole-dipole
Q′
dd =
2
D{[λ2 − 1 + B1[D2
λg − 1]} (χ = 0),
q′ =
(λ− 1)(λ+ 3)
(λ+ 1)2
(χ = 1). (38)
We also get
a1 =
2
DΓ(1 + Γ), (39)
a2 = −2
√
B1
D (1 + Γ)Dλg (40)
with
Dλg = 1 + [Γ + λ], (41)
Ddd = 1 +
3Γ
2 + λ
, (42)
617
L.Blum
DΛ =
1
1 + Γ
+
{
1
2 + λ
}
, (43)
Df =
β6
2(1 + Γ)
1 +
(
b1(2 + λ)
6
)2
, (44)
B1 =
(
b1(2 + λ)
6
)2
. (45)
We have
1−K11 =
2 + λ
3D {λ+ [1 + Γ]B1DλgDΛ} (46)
and
K10 =
(2 + λ)
√B1
3D (1 + Γ)
[
(1 + Γ)DΛ −
{
1
2 + λ
}]
. (47)
Our main result is the new expression for the MSA excess entropy
S = −
(
kV
3π
)
{s0 + s1} (48)
with
s0 =
Γ3
T
√
√
√
√
{
1− 16D
Γ2
T (1 + λ)4
}{
1 +
16[(D(1− 2Γ2
T ) + λ2 − 1]
Γ4
T (1 + λ)4
}
, (49)
s1 =
[
32
(1 + λ)3
(
1− D
2(1 + λ)
)
− 3
]
(50)
and
ΓT =
√
Γ2 + λ2 +
2B1 (1 + Γ) (1 + λ)
D . (51)
The excess pressure can also be computed [26]. The expression is [22]
P/kBT = S/V kB . (52)
Then,
G = E (53)
still holds.
It is easy to see that this expression yields the correct asymptotic results when
either the ions or the dipoles are turned off. Another interesting limit is that of very
large dielectric constant. Then we get
S =
(
kV
3π
)
{Γ3
T}. (54)
A full discussion of these results will be done in a future publication.
618
Scaling in charged fluids
2. Acknowledgements
The author thanks the National Science Foundation for support through grant
NSF–CHE–95–13558 and to the Department of Energy for grant DOE–EPSCoR
grant DE–FCO2–91ER75674.
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619
L.Blum
Скейлінг у заряджених флюїдах: поза простими
іонами
Л.Блюм
Фізичний факультет, університет Пуерто Ріко,
Ріо П’єдрас, PR 00931–3343
Отримано 28 вересня 2001 р.
Аналітичний розв’язок середньосферичного наближення для до-
статньо загального класу взаємодій є завжди функцією редукова-
ного набору скейлінгових матриць. Ми розширюємо цей резуль-
тат на випадок систем з мультипольними взаємодіями. Ми пока-
зуємо, що для іонно-дипольної суміші термодинамічні надлишко-
ві функції є функціями цієї матриці. Результат для ентропії є S =
−{kV /3π}(F [Γα])α∈χ , де F – алгебраїчний функціонал скейлінгових
матриць незвідних представлень замикання Орнштейна-Церніке.
Результат дійсний також і для довільних електростатичних мультипо-
льних взаємодій.
Ключові слова: кулонівські системи, середньосферичне
наближення, ентропія, іонно-дипольні суміші
PACS: 61.20.Gy
620
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