Journal of Molecular Structure (Theochem) 493 (1999) 241–247
www.elsevier.nl/locate/theochem
The solvation shell in ionic solutions: variational mean spherical
scaling approximation
L. Blum a,b,*, E.S. Velázquez b,c
a
Department of Physics, University of Puerto Rico, P.O. Box 23343, Ro Piedras, PR 00931, USA
Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA
c
Department of Physics, University of Puerto Rico, Mayaguez Campus, P.O. Box 9016, Mayaguez, PR 00681-9016, USA
b
Received 3 December 1998; received in revised form 16 February 1999; accepted 2 July 1999
Abstract
The non-specific solvent effects in neutral dipolar (or better yet, effective dipolar) solvents were discussed in the literature
with success. However the situation is not so clear when the solvent is an electrolyte, where the reaction field approach is no
longer valid. In the present communication we present a theory based on the Mean Spherical Scaling Approximation in which
the system is encapsulated in a ionic cloud that is represented by an equivalent capacitor of shape determined by precise
prescriptions derived from exact relations and sum rules. The construction of the equivalent capacitor depends on the geometry
of the system, on the nature of the solvent and on the nature and concentration of the ions. The treatment proposed here satisfies
exact relations like the Stillinger–Lovett sum rules, the perfect screening sum rules, and the high and low coupling conditions
and constitutes an interpolation scheme between exact high density and low density limits in it simplest form. The calculation is
illustrated with an example of an ellipsoidal cavity. q 1999 Elsevier Science B.V. All rights reserved.
Keywords: Ionic solvation; Solvation shells; Variational mean spherical approximation; Scaling theory
1. Introduction
The computational description of molecular
systems and reactions in solution was given a fair
amount of attention [1–9] and a number of very interesting approaches were developed. In most of them
the system is encapsulated in a region, which may or
may not include some of the solvent (water) molecules, and the exterior is treated as a continuum
dielectric. Different kinds of boundary conditions
were used ranging from metallic to pure dielectric.
In our present contribution we discuss only systems
in electrolytic solutions. We would like to show that
* Corresponding author. Tel.: 1 1-809-764-0000.
E-mail address: l_blum@upr1.upr.clu.edu (L. Blum)
the Mean Spherical Approximation (MSA) or
more properly the Mean Spherical Scaling
Approximation (MSSA) [10] and its recent
improvements provide a procedure that is coherent
and simple. Just as the Debye–Huckel (DH)
theory [11,12], the MSSA is a linearized Poisson–Boltzmann approximation for electrolytes,
but unlike the DH, the MSSA takes into account
the excluded volume of all the ions, those in the
center and those in the ionic cloud. The MSSA
also admits analytical solutions for the primitive
model (continuum dielectric solvent) [13], the iondipole model [14,15], the ion-YUKAGUA model
[16] and more recently for inhomogeneous
confined systems. All of these solutions are
expressed in terms of a screening parameter (or
0166-1280/99/$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S0166-128 0(99)00245-6
242
L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247
more generally a matrix) that is related to the DH
screening parameter k . The MSSA satisfies a
number of exact conditions:
1. Stillinger–Lovett sum rule [17,18] This rule is
obeyed by all approximations that are symmetric
and have ions with excluded volume. It is violated
by the DH theory and all Poisson–Boltzmann
theories that take the ions in the ionic atmosphere
as points.
2. Perfect screening sum rule [19] 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 do 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 [19]. 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. This also means
that for any approximation the internal energy of
the ionic system can be expressed as the sum of
grounded metal capacitors surrounding each
charge density. If this is a spherical ion then the
capacitor is a spherical capacitor, and then the
exact form of the energy is
e2 X
zpi
ri zi
;
DE ˆ 2
1 i
1=Gi 1 si
zpi
1†
is the effective charge, we will use
where
b ˆ 1=kT, the usual Boltzmann thermal factor, 1
is the dielectric constant, e is the elementary
charge, and ions i have charge, diameter and
density ezi , si , ri , respectively. For the continuum
dielectric primitive model Gi ˆ for alli. Here g ù
rs2 …ez†2 =1kT is the plasma coupling parameter.
3. High r -High g : Onsager limits [20] The
Onsagerian limit is obtained by setting Gi ! ∞ in
our previous expression. When the ionic
concentration goes to infinity and at the same
time the charge diverges, then the limiting energy
for the spherical ion is bounded by
DE ˆ 2
e2 X
zp
ri zi i :
1 i
si
2†
4. Low r -Low g : DH limits This exact limit
requires that the screening parameter satisfy
2G ! k with k2 ˆ
m
4pbe2 X
r z2 :
1 jˆ1 j j
3†
It also means that the excess entropy must be of
the form
DS…MSA† ˆ 2k
G3
:
3p
4†
All the known analytical MSSA solutions for the
dimers and the polymers satisfy this relation at
infinite dilution and at any given concentration.
We conjecture that there is an ‘universality’ principle for the excess entropy for any arbitrarily
shaped system, and this is the basis of our approximation.
5. Low r -High g : Binding limits [21] The Binding
Mean Spherical Approximation (BIMSA) obtained
by solving the Wertheim form of the Ornstein–
Zernike (OZ) equation satisfies exactly the correct
DH limits for both the associated and dissociated
dimers or n-mers of any composition. This means
that if component 1 forms a n-mer the DH limiting
law must satisfy
k2 ˆ
m
4pbe2 X
‰ rj z2j 1 r1 …nz1 †2 Š:
1
jˆ2
5†
This limiting law is not satisfied by any closure of
the regular OZ equation, but only for closures of
the Werteim–Ornstein–Zernike (WOZ) equation
[22–26].
Treating the solvent effects in quantum mechanics
is often a challenge because of the difficulties in defining precisely the limits of the solvent ‘cell’ (or
capsule) in which the reaction takes place. The cavity
size and shape is not defined in most cases, and the
continuum region is also difficult to assert [27–30].
When the solvent is an ionic solution, however, a very
L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247
clear set of prescriptions can be formulated. This is so
because of the very special nature of the Coulomb
forces in classical mechanics, which forces perfect
screening of the unbalanced charges in the system
[19]. In our treatment we use the MSA and the
BIMSA in which the cavity size is defined by a variational expression, and the cavity shape is defined from
geometrical excluded volume considerations. All of
these are ionic strength dependent parameters, and
satisfy the required asymptotic limiting expressions
[13,31,32].
For systems with Coulomb and screened Coulomb
interactions in a variety of mean spherical approximations it is known that the solution of the OZ equation
is given in terms of a single screening parameter G .
This includes the ‘primitive’ model of electrolytes, in
which the solvent is a continuum 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
DS ˆ 2k
G3
;
3p
where G is MSA the screening parameter and in
general it will be of the form
The internal energy for a given object is written as
DE…G†. Then G is determined by
2‰bDE…G† 1 G3 =…3p†Š
ˆ 0:
2G
2A
ˆ 0:
2G
The analytical solution of the MSSA can also be
derived from a variational principle in which the
screening parameter is obtained from the minimization of a trial function [13]. The general principle was
formulated by Baxter [33] for the Percus-Yevick
approximation, by Chandler [34] for the MSA, and
by one of us for the soft MSA (SMSA) [35,36].
We extend the variational approach for the screening parameters [13,37] to arbitrarily shaped objects.
6†
In Section 2 we review previous results of the
MSSA obtained for dimers from the WOZ equation.
In Section 3 we discuss a variational form, the Variational Mean Spherical Scaling Approximation
(VMSSA) that will satisfy all exact sum rules
described above.
2. The BIMSA for dimers
In this section we review our previous results for
dimers [38,39]. Consider the case of dimers formed in
a solution containing m species of ions of diameters s i
and charges zi. The dimers are formed by species 1
and 2, and the rest of the ions do not associate.
Now we remark that the excess internal energy is a
function of the set of the aNi xNi :
DE ˆ
3
e2 X zk X
rk ‰
aN xN 2 zk Š
1 k
sk Nˆ0 k k
or
DE ˆ 2
DS ˆ S…G†;
which is independent of the form of the cavity in this
approximation, and G is now the scaling matrix. We
propose in this article that for a general form of the
cavity, the scaling matrix G is obtained from the variational principle
243
1
e2 X
GT zk 1 hT sk
rk zk
1 k
1 1 G T sk
3
e2 X zk X
rk
aN xN ;
1 k
sk Nˆ0 k k
7†
with
hT ˆ
2
X
p X
r k sk
aNk xNk :
2D k
Nˆ0
8†
Now we will consider the case of the charged chain
constituted by the particles 1; 2; …; n with z1 ˆ ^z2 ˆ
… ˆ zn and s1 ˆ s2 ˆ … ˆ sn . In this case, from
Eq. (7) we deduce [21]
"
!
e2
z1 G
12a
1
rz
2
DE ˆ 2
1 1 1 1 1 Gs
2s …1 1 Gs†2
#
X 2 G
ri zi
:
9†
1
1 1 Gs
i
244
L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247
The closure relation obtained directly from the boundary condition of the WOZ is
"
!
G2
be 2
z1
z2 …1 2 a†
ˆ
r1 z1
1
p
1
1 1 Gs†2
1 1 Gs†3
#
X 2
1
;
10†
ri zi
1
1 1 Gs†2
i
where the degree of dissociation is 0 , a , 1 and it
is obtained from the dimerization equilibrium relation
Ka ˆ r1
1 2 a†2
:
a
11†
It can be verified that Eq. (10) does satisfy the variational relation (6). It can also be verified that conditions 3 and 4 are satisfied for all values of a . To our
knowledge, no other theory does this. However condition 2 is not satisfied correctly in the limit of perfect
association, a ˆ 0, because one always gets the result
of independent particles.
an incomplete elliptic integral of the first kind [42]
Cˆ
2l
;
F…fum†
15†
where
F…fum† ˆ
lˆ
1
2
Zf
0
q
c20 2 a20 ;
16†
17†
a0
;
c0
18†
c20 2 b20
:
c20 2 a20
19†
f ˆ cos21
mˆ
1 2 m sin 2 u†21=2 du;
For our model capacitor
C…G† ˆ
2l…G†
;
F…f…G†um…G††
20†
where l…G†, f…G†, and m…G† are obtained by letting
3. The variational mean spherical scaling
approximation for ellipsoids
a0 ! a…G† ˆ a0 1
s0
1
1
2
2G
21†
We consider an ellipsoidal shell of semiaxes a0 ,
b0 , c0 centered at the origin of a Cartesian coordinate system. The distance of closest approach along
the x; y; z axes of a sphere of radius s 0 are a0 1 s 0 =2,
b0 1 s 0 =2, c0 1 s 0 =2, respectively. Based on the spherical capacitor model for hard spheres, we take the
excess internal energy to be
b0 ! b…G† ˆ b0 1
s0
1
1
2
2G
22†
c0 ! c…G† ˆ c0 1
s0
1
1
2
2G
23†
DE ˆ 2
e2 X 2 1
;
rz
21 i i i C…G†
12†
where C…G† is the capacitance of the model capacitor.
The capacitance of an ellipsoid of semiaxes a0 ,
b0 , c0 is [40,41]
C ˆ Z∞
0
2
R21
j dj
;
where
q
Rj ˆ …j 2 a20 †…j 2 b20 †…j 2 c20 †:
13†
in Eqs. (16)–(19). Note that the excess internal energy
has the expected form in the limit G ! ∞, since
lim C…G† ˆ C∞ ;
G!∞
24†
where C∞ is the capacitance of a conductor shaped as
the boundary, i.e. an ellipsoid of semiaxes a0 s 0 =2,
b0 1 s 0 =2, c0 1 s 0 =2 and therefore
lim DE ˆ 2
G!∞
e2 X 2 1
rz
:
21 i i i C ∞
25†
The excess entropy of the system is
DS ˆ 2k
14†
This result can be conveniently rewritten in terms of
G3
;
3p
26†
and thus we find the excess Helmholtz free energy
DA ˆ DE 2 TDS:
27†
L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247
Fig. 1. Concentration vs. screening parameter for oblate (O) and
prolate (P) ellipsoids. The ratio of the largest semiaxis to the smallest is 5 in both cases.
The closure relation obeyed by the VMSSA can be
written in the form
d‰DAŠ ˆ 0;
28†
Fig. 3. Excess Helmholtz free energy versus screening parameter
for oblate (O) and prolate (P) ellipsoids. The ratio of the largest
semiaxis to the smallest is 5 in both cases.
and from Eq. (20), we obtain
2
1
2
C…G† ˆ C…G†
l…G† 2G l…G†
2G
!
1
2
F…f…G†um…G†† :
2
F…f…G†um…G†† 2G
from which
2
G2
DE ˆ 2
:
pb
2G
245
31†
29†
Furthermore,
Using Eq. (12) we have
2
e X 2 1
2
Eˆ
C…G†;
rz
2G D
21 i i i C 2 …G† 2G
2
30†
1 2
1
;
l…G† ˆ 2
l…G† 2G
2Gm…G†
32†
where
m…G† ˆ 1 1 G…a0 1 c0 †;
33†
and
2
1 2f
F…fum† ˆ
1
2G
n…G† 2G
F…fum†
E…fum†
1
2m
2m…1 2 m†
!
sin 2f
2m
;
2
4…1 2 m†n…G† 2G
2
34†
Fig. 2. Excess internal energy versus screening parameter for oblate
(O) and prolate (P) ellipsoids. The ratio of the largest semiaxis to the
smallest is 5 in both cases.
where the dependence of f…G† and m…G† on G has not
been explicitly written for compactness, E…fum† is the
incomplete elliptic integral of the second kind, and
q
n…G† ˆ 1 2 m…G† sin2 f…G†:
35†
246
L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247
It is straightforward to find
2
c 2 a0 †1=2
f…G† ˆ 1=2 0
;
2G
G …1 1 2Gc0 †m1=2 …G†
2
b 2 a0 †…c0 2 b0 †
m…G† ˆ 0
:
2G
c0 2 a0 †m2 …G†
36†
37†
Since we can write
k
;
DE ˆ 2
8pbC…G†
38†
the closure relation (6) takes the form
2
C…G† ˆ 28G2 C 2 …G†:
2G
39†
Fig. 1 shows the dependence of the concentration on
the screening parameter, while Figs. 2 and 3 show the
excess internal energy and the excess Helmholtz free
energy respectively. In each of these figures two
curves are shown corresponding to a prolate ellipsoid
with a0 ˆ b0 ˆ 1; c0 ˆ 5 and to an oblate ellipsoid
with a0 ˆ 1=5, b0 ˆ c0 ˆ 1.
4. Discussion
We have already shown that the excess internal
energy (12) satisfies the Onsager limit (condition 3).
We will now show that the same expression provides
the correct form in the limit of low coupling, i.e. we
should obtain
2
e X 2
DE ˆ 2
rz ;
lim
G!0 2G
1 i i i
2
40†
which implies that we have to show
lim 2
G!0
1
2
C…G† ˆ 2:
C 2 …G† 2G
41†
We first obtain the following series expansions for
G!0
p
c 0 2 a0
l…G† ˆ
1 O…G1=2 †;
42†
2G1=2
p
f…G† ˆ 2 c0 2 a0 G1=2 1 O…G3=2 †
c 2 b0
1 O…G†:
m…G† ˆ 0
c 0 2 a0
F…uum0 † ˆ u 1
m0 u3
1 O…u5 †;
6
45†
to obtain
C…G† ˆ
2
k2
Since f…G† ! 0 and m…G† ! m0 , where m0 is a
fixed value, we need the following series, valid for
u!0
1
1 O…1†:
2G
46†
To obtain the limit of the derivative of C…G† we use
Eq. (31). The first term in parenthesis is, using Eqs.
(32) and (33)
1 2
1
1 O…1†:
l…G† ˆ 2
l…G† 2G
2G
47†
The limit of the second term in parenthesis is obtained
from Eq. (34). First using Eqs. (35), (36) and (43) we
obtain
n…G† ˆ 1 1 O…G†
so that the first term of Eq. (34) is
p
c 0 2 a0
1 2
f…G† ˆ
1 O…1†:
n…G† 2G
G1=2
48†
49†
For the reason stated above when finding the limit of
F…f…G†um…G††, we need
E…u um0 † ˆ u 2
m0 u3
1 O…u5 †:
6
50†
Furthermore, since all three terms inside the parenthesis in Eq. (34) are of order f , and consequently of
order G 1/2, and since from Eq. (37) we see that the
limit of the derivative of m…G†k is of order 1, we
conclude that the first term of Eq. (34), which is of
order G 21/2 is the leading term and therefore
p
c 0 2 a0
2
F…f…G†um…G†† ˆ
1 O…1†:
51†
2G
G1=2
Substitution in Eq. (31) leads to
2
1
C…G† ˆ 2 2 1 O…1†;
2G
2G
52†
which combined with Eq. (46) gives
43†
lim 2
G!0
44†
1
C2 …G†
2
C…G† ˆ 2;
2G
as we expected.
53†
L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247
Acknowledgements
This research was supported in part by the National
Science Foundation under Grant No. PHY-94-07194.
L.B. thanks the NSF for support through grants PHY94-07194, CHE-95-13558 and EPSCoR OSR-9452893. We also thank J. Given, J. Hubbard, O.
Bernard, and M. Zerner for very helpful discussions.
References
[1] D. Rinaldi, J.L. Rivail, Theor. Chim. Acta. 32 (1973) 57.
[2] O. Tapia, O. Goscinsky, Mol. Phys. 29 (1975) 1653.
[3] K. Sharp, A. Jean-Charles, B. Honig, J. Phys. Chem. 96 (1992)
3822.
[4] N. Roesch, M. Zerner, J. Phys. Chem. 98 (1994) 5817.
[5] J. Tomasi, M. Persico, Chem. Rev. 94 (1994) 2027.
[6] C.J. Cramer, D.G. Truhlar, in: D.B. Boyd, K.B. Lipkowitz
(Eds.), Reviews on Computational Chemistry, 6, VCH, New
York, 1995.
[7] A. Klammt, J. Phys. Chem. 99 (1995) 2224.
[8] J.L. Rivail, R. Rinaldi, in: J. Leszcynski (Ed.), Computational
Chemistry–Review of Current Trends, World Scientific,
Singapore, 1995, pp. 139–174.
[9] M. Cossi, et al., Phys. Lett. 286 (1998) 253.
[10] L. Blum, Mol. Phys. 30 (1975) 1529.
[11] P. Debye, E. Huckel, Phys. Z. 24 (1932) 185.
[12] D.A. McQuarrie, Statistical Mechanics, Harper & Row, New
York, 1971, Chap. 15.
[13] L. Blum, Y. Rosenfeld, J. Stat. Phys. (J.K. Percus Festschrift)
63 (1991) 1177.
[14] L. Blum, Chem. Phys. Lett. 26 (1974) 200.
[15] L. Blum, F. Vericat, W.R. Fawcett, J. Chem. Phys. 96 (1992)
3039.
[16] L. Blum, F. Vericat, L. Degreve, Physica A 365 (1999) 396.
247
[17] F.H. Stillinger, R. Lovett, J. Chem. Phys. 48 (1968) 3858.
[18] F.H. Stillinger, R. Lovett, J. Chem. Phys. 49 (1968) 1991.
[19] L. Blum, Ch. Gruber, J.L. Lebowitz, Ph.A. Martin, Phys. Rev.
Lett. 48 (1982) 1769.
[20] Y. Rosenfeld, L. Blum, J. Chem. Phys. 85 (1986) 1556.
[21] O. Bernard, L. Blum, in: G. Kalman, M. Rommel, A. Budaev
(Eds.), Proceedings of the International Conference on
Strongly Coupled Plasmas, Plenum, New York, 1998.
[22] M.S. Wertheim, J. Stat. Phys. 35 (1984) 19–35.
[23] M.S. Wertheim, J. Stat. Phys. 42 (1986) 459–477.
[24] M.S. Wertheim, J. Chem. Phys. 85 (1985) 2929.
[25] M.S. Wertheim, J. Chem. Phys. 87 (1987) 7323.
[26] M.S. Wertheim, J. Chem. Phys. 88 (1988) 1214.
[27] M. Karelson, A.R. Katritzky, M. Szafran, M.C. Zerner, J. Org.
Chem. 54 (1989) 6030.
[28] M. Karelson, M.C. Zerner, J. Phys. Chem. 96 (1992) 6949.
[29] M. Karelson, T. Tamm, M.C. Zerner, J. Phys. Chem. 97
(1993) 11 901.
[30] M. Roesch, M.C. Zerner, J. Phys. Chem. 98 (1994) 5817.
[31] L. Onsager, J. Phys. Chem. 43 (1939) 189.
[32] Y. Rosenfeld, L. Blum, J. Chem. Phys. 85 (1986) 1556.
[33] R.J. Baxter, J. Chem. Phys. 52 (1970) 4559.
[34] H.C. Andersen, D. Chandler, J. Chem. Phys. 57 (1972) 1918.
[35] L. Blum, A.H. Narten, J. Chem. Phys. 56 (1972) 5197.
[36] A.H. Narten, L. Blum, R.H. Fowler, J. Chem. Phys. 60 (1974)
3378.
[37] E.S. Velázquez, L. Blum, J. Mol. Liquids 73-74 (1997) 75.
[38] O. Bernard, L. Blum, J. Chem. Phys. 104 (1996) 4746.
[39] L. Blum, Yu.V. Kalyuzhnyi, O. Bernard, J.N. Herrera, J. Phys.
Condens. Matter 8 (1996) A143.
[40] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New
York, 1941.
[41] W.R. Smythe, Static and Dynamic Electricity, McGraw-Hill,
New York, 1950.
[42] T.H. Shumpert, D.J. Galloway, IEEE Trans, Antennas &
Propag. AP-25 (1977) 284.