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interaction seems to require a nonspherical symmetry breaking interaction. Water
is a good example of such a nonspherical interaction that contains the potential for
creating morphology. Aqueous mixtures are microheterogeneous because of water
(Dixit et al. 2002; Allison et al. 2005). Similarly, polymers are highly nonspheri-
cal objects and their inclusion equally breaks the spherical symmetry of colloidal
particles. The STL of nonspherical interactions is therefore required if one wants to
further explore domain formation in realistic fluctuating systems.
7.3 STATISTICAL THEORY OF MOLECULAR LIQUIDS
Here, we formulate the molecular version of the STL for an arbitrary number of com-
ponents, but the practical results will be restricted to binary mixtures.
7.3.1 a s
urvey
oF
The
F
ormalism
We consider a binary mixture made of two species of nonspherical molecules, and
we will consider that each molecule is made of spherical sites, which is a convenient
approximation for most realistic systems. Two such molecules will then interact through
the sum of all their site-site interactions terms. We will consider the generic interaction
where each site-site interaction is the sum of a Lennard-Jones (LJ) interaction and a
Coulomb interaction if the sites bear partial charges localized at their centers. Each of
the molecular site
i
a
belonging to species
a
is then characterized by its diameter σ
i
a
, its
LJ energy parameter ε
a
, and its partial charge
q
i
a
. The total interaction between any two
molecules, labeled 1 and 2, is then
12
6
−
σ
σ
ij
qq
r
∑∑
ε
(
)
()
ij
i
a
j
b
ij
u
1,2
=ur=
4
ab
ab
−
(7.31)
ij
ij
ij
ab
ab
ab
r
r
ij
ij
ab
ab
ab
where
r
i
a
,
j
b
is the distance between the two sites
i
and
j
on molecules of species
a
and
b
, respectively. The first term is the LJ term while the second is the Coulomb inter-
action term. The LJ energy and size parameters are usually defined empirically by
the following rules, the arithmetic mean is called the
Lorentz rule
σ
i
a
,
j
b
= (σ
i
a
+ σ
j
b
)/2
and the geometrical mean is called the
Berthelot rule
ε
=
. The interaction
in Equation 7.31 can also be written in terms of the distance
r
=
r
2
-
r
1
between the
center of masses of any two molecules 1 and 2, as well as their respective orienta-
tions through the unit vectors, Ω
1
and Ω
2
, each describing molecular orientations
through the set of Euler angles (θ
i
, β
i
, φ
i
) defined in the lab fixed frame (Gray and
Gubbins 1984). Since any site
i
of a given molecule can be related to the center of
mass at position
r
i
through the relation
r
i
a
=
r
i
-
I
i
a
(Ω), where the last term is the
position of the site in the reference frame attached to the molecule, one can indeed
rewrite Equation 7.31 as
ε ε
i
a
j
b
i
a
j
b
u
(, )
12
=
u
( ,
r
ΩΩΩ
,
)
=
ur
( ,
ΩΩΩ
,
,
r
/
r
)
(7.32)
12
12
