Chemistry Reference
In-Depth Information
zero-order moments. Coming back to the morphology of liquids, we see that the cor-
relation length describes the amount of correlation inside disorder, but it is clearly
insufficient to describe morphology. Let us see how we can bring up the concept of
morphology through the study of simple liquid models.
7.2.2 c
orrelaTions
and
F
lucTuaTions
in
a
s
imPle
l
liquid
wiTh
r
ePulsive
i
inTeracTions
To make things simple, let us consider the following pair interaction between spheri-
cal particles of diameter σ, with
a
> 0,
+∞
r
<
σ
ur
()
=
σ
(7.26)
r
a
exp(
−
)
r
≥
λ
We can use the analytical expression for the DCF from the Percus-Yevick (PY)
theory (Hansen and McDonald 2006) to compute the direct correlations inside the
core region
r
< σ, and just use Equation 7.6 outside the core. This is in the spirit
of the mean spherical approximation (MSA) (Hansen and McDonald 2006). This
approximation reproduces a structure factor in relatively good agreement with that
of the hard spheres (HS) as computed from the PY approximation, as can be seen
in Figure 7.1, which itself is in good agreement with the computers for dense fluids.
The resulting first three moments of the DCF can be computed following Equation
7.8. These moments are shown in Figure 7.2. The zero-order moment
c
0
is always
negative while the second moment
c
2
is always positive when
a
> 0, so the dis-
cussion above on the possibility of an unstable fluid phase does not apply for this
2.5
2
1.5
1
0.5
0
0
5
10
15
20
k
σ
FIGURE 7.1
Structure factor for the hard sphere fluid versus wave vector times the diameter
of the sphere σ. Full curve for PY approximation, dotted curve for the model used in the text
with
a
= 1 and κ = 1, dashed curve for
a
= 1 and κ = 2.
