Chemistry Reference
In-Depth Information
A
r
−
r
()
lim
hr =
exp
(7.10)
ξ
The asymptotic relations (Equations 7.9 and 7.10) are exact for any one-component
liquid with spherical interactions, except when the Taylor expansion (Equation 7.7)
fails to hold, which is near a critical point (Fisher 1964). At the critical point, the
DCF becomes a nonanalytic function, and the correlations develop a nonexponential
algebraic decay in 1/
r
1+η
where η ≈ 0.041 is a critical exponent (Fisher 1964). In all
that follows, we will stay away from criticality, hence the OZ forms, Equations 7.9
and 7.10, are essentially exact. When the critical point is approached, the isothermal
compressibility diverges, and the analysis above shows that the correlation length
diverges as the square root of the compressibility. The divergence of ξ leads to a
Coulomb decay of the pair correlation in Equation 7.10, but the correct exponent is
slightly faster than pure Coulomb decay.
7.2.1.2 Liquid State Theory for Binary Mixtures of Simple Liquids
The formalism described above applies equally to any mixtures of simple liquids.
The form of the decay is very similar to that obtained above, except that the prefactor
A,
and the correlation length ξ, are complicated expressions. The important point is
the following: while the prefactor depends on the species pairs, the correlation length
is the same for all species pairs. In other words, all pair correlation functions decay
in an identical fashion. We will demonstrate this in a very general case in Section
7.3. Here, we examine the case of isotropic interactions and binary mixtures. If we
label the two species as 1 and 2, we have 3 independent interactions:
u
11
(
r
)
and
u
22
(
r
)
between like species 1 and 2, respectively, and
u
12
(
r
) =
u
21
(
r
) between unlike species.
Similarly, there will be three RDFs:
g
11
(
r
),
g
22
(
r
), and
g
12
(
r
), and three DCFs:
c
11
(
r
),
c
22
(
r
), and
c
12
(
r
). All these functions are related to each other through the two same
exact equations, the OZ equation written in the Fourier space as
∑
ρ
ˆ
ˆ
()
()
() ()
ˆ
ˆ
hk=c
k+
hkck
(7.11)
ij
ij
n n
nj
and the closure equations,
(
)
()
()
()
−
()
+
()
gr=
exp
−
β
u
r+hr cr br
(7.12)
ij
ij
ij
ij
ij
The OZ equation contains the partial number densities that are defined as ρ
i
=
N
i
/
V
,
where
N
i
is the number of molecules of species
i
within the volume
V
. One can also
define the mole fraction of species
i
through ρ
i
=
x
i
ρ. The OZ equation can be conve-
niently cast into the following matrix form,
(
)
=
ˆ
ˆ
SI CI
−
(7.13)
