3
r  (Å)
0
1
2
3
β V~
12
SR
(r)
β V 12
SR
(r)
β V~
33
SR
(r)
4 6 8 10
0
1
2
3 β V~
13
SR
(r)
4 6 8 10
β V~
23
SR
(r)
(a)
(b) (c)
(d)
FIG. 3: Eﬀective pair potentials derived for MSA3 and
BIMSA3. (a) Cation anion (dashed line: without taking the
pair into account), (b) pair cation, (c) pair anion, and (d) pair
pair. The internal potential of the pair β ˜Vint(r) is set equal
to βV eﬀ
ij (r) for distances less than 4 ˚
A.
trapolating the original potential at the barrier separat-
ing pairs from free ions (as shown in Fig.
3). We assume
that the interaction potential is averaged over the rota-
tional degrees of freedom of the CIP and thus pairwise
additive. Hereafter, the quantities referring to such a
three-component model are written with a tilda symbol.
The short-range potentials involving the pair can be de-
rived, in the inﬁnite dilution limit, from an average of
the contributing ion interactions. In Fourier space,
˜V SR
3i (k) = ˜w(k/ 2)
[
V SR
1i + V SR
2i
]
(k), i = 1 , 2 (2a)
˜V SR
33 (k) = ˜w(k/ 2)2[
V SR
11 + V SR
22 + 2V SR
12
]
(k) (2b)
where ˜w(r) is the pair probability distribution
˜w(r) = K− 1
0 e− β ˜Vint(r) (2c)
˜Vint(r) is the internal part of the pair potential (see
Fig.
3), and K0 is the association constant, deﬁned as:
K0 =
∫ ∞
0
dr 4πr2e− β ˜Vint(r) = 0 . 43 L . mol− 1 (3)
The excess free-energy density of the original system
βf ex
v is that of the three component mixture β ˜fex
v plus a
correction term
βf ex
v = β ˜fex
v − ˜ρ3 ln K0, (4)
which is due to the change in standard chemical potential
between the two component and three component mod-
els. It should be noted that the fraction of pairs is now an
additional parameter in the minimization scheme, which
serves to ensure chemical equilibrium. Within this rep-
resentation, the pair can be modeled as a hard sphere
(MSA3) or as a dumbbell-like CIP (BIMSA3) [4]. Since
0 0.5 1 1.5
c
1/2
  (mol.L
-1
)
1/2
-1.5
-1
-0.5
0
β f v
ex
  (mol.L
-1
)
MC
MSA2
MSA3
BIMSA3
DHLL
Exp
0 0.5 1
0.1
0.2
Pair Fraction
FIG. 4: (Color online) Excess free-energy density βf ex
v as
a function of the square root of the concentration √ c. (dia-
mond) MC simulations, (dot dashed) MSA2, (dashed) MSA3,
(solid) BIMSA3, (dot) DHLL, and (cross) experiments. The
inset gives the fraction of pairs (MSA3, BIMSA3) as a func-
tion of√
c.
we have no additional information, we consider only sym-
metric dumbbells. Furthermore, since analytic expres-
sions for the RDF within BIMSA are not known, we ap-
proximate the dumbbell as a hard sphere when comput-
ing the perturbation term (this is not necessary for the
reference term, since an expression for the free energy
is available). Let˜σc be the diameter of the cation (an-
ion) within the dumbbell, the diameter of the hard sphere
representing this dumbbell is taken to be˜σ3 = 4
√
2
π ˜σc[21].
Using these two reference systems, the three-
component MSA3 and BIMSA3, we obtain results in
much better agreement with the MC simulations, as
shown in Fig.
4. The diameters obtained for species 1,
2, and 3 are 3.65, 4.79, and 5.76 ˚
A for MSA3 and 3.69,
4.75 and 6.19 ˚
A for BIMSA3. The free ion diameters are
similar for MSA2, MSA3, and BIMSA3. The pair diam-
eter is smaller when modeled as a hard sphere (MSA3)
than when modeled as a dumbbell (BIMSA3). At high
concentration (about 1 mol l− 1), the MSA3 overestimates
the free energy, because the excluded volume repulsion
becomes too important for the pairs to be represented as
hard spheres. The BIMSA3 model is the closest to the
MC simulation results. It is worth noting that even at
the lowest concentration considered, the fraction of pairs
(shown in the insert of Fig.
4), although less then 5%,
has a non-negligible eﬀect on the thermodynamics of the
system.
This procedure also provides an accurate description of
the structure over the whole range of concentrations. A
development similar to the one that leads to Eq. (
2) de-
rives the average unpaired RDF from the corresponding
paired quantities: