Towards an analytical theory for charged hard spheres
Ion mixtures require an exclusion core to avoid collapse. The Debye Hueckel (DH) theory, where ions are point charges, is accurate only in the limit of infinite dilution. The mean spherical approximation (MSA) is the embedding of hard cores into DH, and is valid for higher densities. The properties...
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Інститут фізики конденсованих систем НАН України
2007
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| Цитувати: | Towards an analytical theory for charged hard spheres / L. Blum, D.V.P. Veloz // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 381-385. — Бібліогр.: 27 назв. — англ. |
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irk-123456789-1187032017-06-01T03:03:23Z Towards an analytical theory for charged hard spheres Blum, L. Veloz, D.V.P. Ion mixtures require an exclusion core to avoid collapse. The Debye Hueckel (DH) theory, where ions are point charges, is accurate only in the limit of infinite dilution. The mean spherical approximation (MSA) is the embedding of hard cores into DH, and is valid for higher densities. The properties of any ionic mixture can be represented by the single screening parameter Γ which for the equal ionic size restricted model is obtained from the Debye parameter κ. This Γ representation, the binding mean spherical approximation (BIMSA), is also valid for complex / associating systems, such as the general n-polyelectrolytes. The BIMSA is the only theory that satisfies the infinite dilution limit of the DH theory for any chain length. Furthermore, the contact pair distribution function calculated from our theory agrees with the Monte Carlo of Bresme ea.(Phys. Rev. E, 1995, 51, 289). Щоб уникнути колапсу, iоннi сумiшi вимагають включення в теорiю твердого кору. Теорiя Дебая-Гюккеля (ДГ), в рамках якої iони вважаються точковими зарядами, є точною тiльки в границi нескiнченного розведення. Середньо-сферичне наближення, яке враховує твердий кор, є справедливим для бiльших густин. Властивостi будь-якої iонної сумiшi можуть бути представленi за допомогою одного параметра Γ, який для примiтивної моделi iонiв однакового розмiру отримується з параметра Дебая . Це Γ-представлення (BIMSA) має мiсце також для складних/асоцiйованих систем, таких як загальнi n-полiелектролiти. BIMSA є єдиною теорiєю, яка задовiльняє границi нескiнченого розведення теорiї ДГ для довiльної довжини ланцюжка. Крiм того, парна функцiя розподiлу на контактi, обчислена в рамках нашої теорiї, узгоджується з результатами Монте Карло Бресме та iн.(Phys. Rev. E, 1995, 51, 289). 2007 Article Towards an analytical theory for charged hard spheres / L. Blum, D.V.P. Veloz // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 381-385. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 61.20.Gy DOI:10.5488/CMP.10.3.381 http://dspace.nbuv.gov.ua/handle/123456789/118703 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Ion mixtures require an exclusion core to avoid collapse. The Debye Hueckel (DH) theory, where ions are point charges, is accurate only in the limit of infinite dilution. The mean spherical approximation (MSA) is the embedding of hard cores into DH, and is valid for higher densities. The properties of any ionic mixture can be represented by the single screening parameter Γ which for the equal ionic size restricted model is obtained
from the Debye parameter κ. This Γ representation, the binding mean spherical approximation (BIMSA), is also valid for complex / associating systems, such as the general n-polyelectrolytes. The BIMSA is the only theory that satisfies the infinite dilution limit of the DH theory for any chain length. Furthermore, the contact pair distribution function calculated from our theory agrees with the Monte Carlo of Bresme ea.(Phys. Rev. E, 1995, 51, 289). |
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Article |
| author |
Blum, L. Veloz, D.V.P. |
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Blum, L. Veloz, D.V.P. Towards an analytical theory for charged hard spheres Condensed Matter Physics |
| author_facet |
Blum, L. Veloz, D.V.P. |
| author_sort |
Blum, L. |
| title |
Towards an analytical theory for charged hard spheres |
| title_short |
Towards an analytical theory for charged hard spheres |
| title_full |
Towards an analytical theory for charged hard spheres |
| title_fullStr |
Towards an analytical theory for charged hard spheres |
| title_full_unstemmed |
Towards an analytical theory for charged hard spheres |
| title_sort |
towards an analytical theory for charged hard spheres |
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Інститут фізики конденсованих систем НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118703 |
| citation_txt |
Towards an analytical theory for charged hard spheres / L. Blum, D.V.P. Veloz // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 381-385. — Бібліогр.: 27 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT bluml towardsananalyticaltheoryforchargedhardspheres AT velozdvp towardsananalyticaltheoryforchargedhardspheres |
| first_indexed |
2025-07-08T14:29:18Z |
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2025-07-08T14:29:18Z |
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1837089403165474816 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 3(51), pp. 381–385
Towards an analytical theory for charged hard spheres
L.Blum1, D.V.P.Veloz2
1 Department of Mathematics, Hill Center, Busch Campus, Rutgers University, Piscataway, N.J. 08854
2 Department of Physics, Faculty of Natural Sciences, University of Puerto Rico, Rio Piedras, Puerto Rico
00931
Received July 23, 2007
Ion mixtures require an exclusion core to avoid collapse. The Debye Hueckel (DH) theory, where ions are
point charges, is accurate only in the limit of infinite dilution. The mean spherical approximation (MSA) is the
embedding of hard cores into DH, and is valid for higher densities. The properties of any ionic mixture can be
represented by the single screening parameter Γ which for the equal ionic size restricted model is obtained
from the Debye parameter κ. This Γ representation, the binding mean spherical approximation (BIMSA), is
also valid for complex / associating systems, such as the general n-polyelectrolytes. The BIMSA is the only
theory that satisfies the infinite dilution limit of the DH theory for any chain length. Furthermore, the contact
pair distribution function calculated from our theory agrees with the Monte Carlo of Bresme ea.(Phys. Rev. E,
1995, 51, 289).
Key words: electrolytes, MSA, ESMSA, contact pair distribution function
PACS: 61.20.Gy
1. Introduction
Stable ionic mixtures require an exclusion core to avoid the collapse of the system. For this
reason the Debye Hueckel (DH) [1]theory, in which the ions are point particles, is accurate only
in the limit of infinite dilution. The MSA [2–4] that is an embedding of the DH theory into the
hard-core Ornstein Zernike (OZ) equation is valid for high densities, and is even asymptotically
exact at infinite density [5]. In the MSA the properties of any ionic mixture can be represented
by a single screening parameter Γ [6] which in the simplest equal ionic size restricted model is
obtained from the Debye screening parameter κ:
Γ =
1
2σ
{√
1 + 2κσ − 1
}
, (1)
κ is the inverse of the Debye screening length defined by
κ2 =
4πβe2
ε
∑
k
ρkz2
k , (2)
where zk is the electrovalence, ρk is the density of component k, β is the Boltzmann thermal factor,
σk is the hard core diameter, and ε is the dielectric constant. The one parameter representation
[7] is valid for a number of complex and associating systems such as dimers, and even polymers,
where as in the Debye-Hueckel theory, the thermodynamic properties depend on a single screening
parameter Γ: for the dimerization of equally charged hard ions we get
κ2
α + Γσ
1 + Γσ
= 4Γ2(1 + Γσ)2. (3)
After a detailed analysis [8–10] we can generalize equation (3) in the form
κ2F(n, α) = 4Γ2(1 + Γσ)2. (4)
c© L.Blum, D.V.P.Veloz 381
L.Blum, D.V.P.Veloz
For dimers
F(2, α) =
α + Γσ
1 + Γσ
. (5)
In the limit of total association α = 0, and we recover the DH limiting law with the charge of the
polyelectrolyte, for any n > 3. Then after a lengthy calculation
lim
ρ→0
F(n, 0) ∼ n2; correct DH limit,
lim
ρ→∞
F(n, 0) ∼ n; high density limit. (6)
The criticality of ionic systems, initiated by M.E. Fisher and collaborators is still a subject of
current interest. [11–13]. In the original discussion of this problem Fisher and Levin [11] used the
DH [1] theory in combination with Bjerrum association [14]. It is clear that to get a mathemati-
cally well defined system all the ions need to have an exclusion core. Most of the subsequent work
is directed at the inclusion of the excluded core in the nonlinear Coulomb and the association
problem. In our own previous work [15] we used different combinations of the binding mean spheri-
cal approximation (BIMSA) [9,10] and various treatments of ion association, all derived from the
work of Bjerrum [14]. The best agreement with computer simulations was obtained from the SIS-
BIMSA treatment of the association constant (Jiang, Blum, Bernard, Prausnitz and Sandler [15].
In the SIS approximation of Stell et al [16] the chemical association constant is computed from the
contact pair correlation function. A discussion of the merits of the different approximations was
recently given by Aqua, Banerjee and Fisher [12]). The real problem here is that, as it has been
shown by Wertheim for the associating systems, the classical virial expansion does not converge
[17,18] and a new MSA (the BIMSA), based on the Wertheim-Ornstein-Zernike equation (WOZ)
has to be formulated. As has been shown elsewhere [9,10], for dense systems, the scaling solution
and the remarkably simple thermodynamics of the MSA applies to the new theories but with a
renormalized screening constant, ΓN . In the ESMSA the low density limits are also included [21].
The real issue is that we need an internally consistent procedure and therefore there is an open
(big) question about their reliability and accuracy that we will try to answer.
2. Theory
Ever since the DH [1] theory was formulated, almost a century ago, there has been a steady
effort in finding the ways to improve and extend the range of its multiple applications. Most
of the theoretical effort has been dedicated to the inclusion of the hard exclusion core and the
nonlinearity of the Poisson-Boltzmann equation. Now we know [5] that the MSA is asymptotically
exact at very high densities. At lower densities the association limits yield exact conditions that
are satisfied by closures of the Wertheim-Ornstein-Zernike equation: the exact DH limit at infinite
dilution must contain the charges of the associated ions. As has been pointed out in this work,
since the parameter space of these systems is large, we need to have an accurate, fully analytical
theory for the properties of these systems. The general problem was discussed in the past by several
authors (see for example [19]). In the recent work we have proposed a new theory, the ESMSA
which could be such a theory, since there is only one screening parameter Γ, as in the MSA, but
also the degree of dissociation, ionic diameters, effective dielectric constants enter in the calculation
through the contact pair correlation function.
The most interesting feature of the MSA even for complex systems, is that, as in the Debye-
Hueckel theory, the thermodynamic properties depend on a single screening (scaling) parameter Γ:
4[Γ]2(1 + Γσ)2 = κ2
(α + Γσ)
(1 + Γσ)
(7)
κ is the inverse of the Debye screening length defined by
κ2 =
4πβe2
ε
∑
k
ρkz2
k (8)
382
Towards an analytical theory for charged hard spheres
and α is the degree of dissociation. The remarkable property of this equation is that it yields the
correct asymptotic limits for zero density as well as for very large density [5], where
Γ =
1
2σ
{
√
1 + 2κσ − 1} ∼
{
√
κ
2σ
}
. (9)
In the case of associating (polymerizing) ions the BIMSA for the restricted primitive model yields
a very simple expression ΓB. This result applies to Polyelectrolytes and accounts for the remark-
able agreement with the simulations of Orkulas et al. [20]. The question arises which is the best
interpolation scheme between these two regimes? One possible answer can be obtained by testing
different combinations as was done in our previous work [15]: However, it is clear that one cannot
separate the sources of errors in the analysis of criticality. Slill, a more illuminating perspective can
be obtained considering the simulations by Bresme et al [22], as we will see below. The usefulness
of the MSA resides in the fact that the scaling solution is formally valid for the general mixture of
the ions of arbitrary charge and size.
It has been shown by Rosenfeld and Blum (Y. Rosenfeld and L. Blum, [5] that the MSA is
asymptotically exact at a very large density. However, the point remains that at low temperature
ANY theory based on the normal Ornstein Zernike (OZ) equation will fail, because it will not be
capable of reaching the full association limits. This is corrected by the Wertheim Ornstein Zernike
(WOZ) [18] equation, which has been shown to work remarkably well even for the limit of infinitely
long polyelectrolyte chains [8].
For many applications it is important to have an analytical, but at the same time an accurate
theory of electrolytes. This is not an easy task to carry out, but recent advances have made
this an attainable goal. One important ingredient of this theory is presumably the contact pair
correlation function (PCF). In this communication we evaluate a simple functional form of the
contact PCF, using the Monte Carlo simulations by Bresme et al. [22]. We find that the best
analytical representation is obtained from this formula. It is increasingly erroneous at high charges.
One possible reason for this failure is the fact that the simulations do not satisfy the PST, as is
suggested by some preliminary simulation runs. Another source of errors is the neglect of the field
parameter η.
3. The contact pair distribution function
In the present contribution we show that the contact pair distribution function for a hard ion
mixture can be represented explicitly by a functional of Γ [9,10,21]:
g00
ij (σij) = gHS
ij (σij)e
−
βe2
σijε0
X01
i X01
j , X01
i =
zi − ησ2
i
1 + Γσi
' zi
1 + Γσi
(10)
η is a mean field parameter which depends on many body interactions, such as α, usually very
small, and ε0 is an effective dielectric constant. The hard core contact correlation is from the
Carnahan-Starling approximation [24]
gHS
ij (σ) =
1 − π
12
∑
i ρiσ
3
(1 − π
6
∑
i ρiσ3)3
(11)
but in our case it can be taken to be 1, since our system is very dilute.
Indeed the proper discussion of this system is to treat it as a mixture of ions and dipoles for
the associated part [27]. For our restricted equal diameter problem we get
σij = σi = σj = σ. (12)
The simplest approximation is to take η = 0, which means that we neglect higher order correlation
effects [10]. Then from equation (10) we get
g00
ij (σ) = e
−
zizjβe2
εσ(1+Γσ)2 . (13)
383
L.Blum, D.V.P.Veloz
Figure 1. Comparison of the contact pair distribution function for g
ESMSA−BIMSA
+−
. The squares
are the Monte Carlo results by Bresme et al. [22]. The triangles are the INV-C theory of Haymet
et al [25]. The circles are the ESMSA-BIMSA with η = 0 and an effective dielectric constant of
ε0 = 1.17 (which also depends on the association constant α).
The results of the calculation are shown in figure 1. The agreement for βe2
σ
up to 16 is quite good,
and much better than the HNC theory [26] which does not easily converge or the INV-C of Duh
and Haymet [25] which is shown in figure 1. However, in our theory the error in the last point is
sizable, and we have no explanation for this at the time, other than the suspicion that this may
be a consequence of the failure to satisfy the PST [23].
Acknowledgements
The authors are indebted to the NSF-PREM program for financial support. We wish a very
happy 60th birthday to Prof. Fumio Hirata.
384
Towards an analytical theory for charged hard spheres
References
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До аналiтичної теорiї заряджених твердих сфер
Л.Блюм1, Д.В.П.Велоз2
1 Математичний факультет, Унiверситет Рутжер, Пiскатавей, США
2 Фiзичний факультет, Унiверситет Пуерто-Рiко, Пуерто-Рiко, США
Отримано 23 липня 2007 р.
Щоб уникнути колапсу, iоннi сумiшi вимагають включення в теорiю твердого кору. Теорiя Дебая-
Гюккеля (ДГ), в рамках якої iони вважаються точковими зарядами, є точною тiльки в границi нескiн-
ченного розведення. Середньо-сферичне наближення, яке враховує твердий кор, є справедливим
для бiльших густин. Властивостi будь-якої iонної сумiшi можуть бути представленi за допомогою
одного параметра Γ, який для примiтивної моделi iонiв однакового розмiру отримується з парамет-
ра Дебая κ. Це Γ-представлення (BIMSA) має мiсце також для складних/асоцiйованих систем, таких
як загальнi n-полiелектролiти. BIMSA є єдиною теорiєю, яка задовiльняє границi нескiнченого роз-
ведення теорiї ДГ для довiльної довжини ланцюжка. Крiм того, парна функцiя розподiлу на контактi,
обчислена в рамках нашої теорiї, узгоджується з результатами Монте Карло Бресме та iн.(Phys.
Rev. E, 1995, 51, 289).
Ключовi слова: електролiти, середньо-сферичне наближення, парна функцiя розподiлу на
контактi
PACS: 61.20.Gy
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