Chemistry Reference
In-Depth Information
One sees that there is only one correlation length and this is equally true for any
number of components. Following Equation 7.17, the decay of the pair correlations
has the same Yukawa form as in Equation 7.10,
A
r
−
r
()
ij
lim
hr=
exp
(7.21)
ij
ξ
The isothermal compressibility of a mixture is given by the expression
ˆ
()
S
0
κ
T
=
(7.22)
∑
ˆ
ij
xx
S
ij
ij
,
an expression that trivially differs from that found in Hansen and McDonald because
of the definition of the structure factor we have adopted above, a choice that keeps
compatibility with the form of the OZ equation (Hansen and McDonald 2006). By
using the OZ equation, we find an equivalent expression in terms of the DCFs,
1
*
κ
=
(7.23)
∑
T
ˆ
()
1
−
ρ
xxc
0
ijij
which is identical to that reported in Hansen and McDonald (2006).
In particular,
we see that for a mixture, the isothermal compressibility is not related to the
c
instability of the isotropic phase. The latter is governed by the determinant
∙
I
-
C
(0)
∙
= 1/
∙
S
(0)
∙
, which is also related to the fluctuation of the number of particles through
the relations:
ˆ
S=
NN
−
NN
i
j
i
j
()
(7.2 4)
ij
NN
i
j
We also see that the correlation length in Equation 7.19 diverges with the divergence
of the correlations through the term
∙
S
(0)
∙
. Another important link with thermody-
namics is provided by the Kirkwood-Buff theory (Kirkwood and Buff 1951),
∂
∂
βµ
ρ
i
()
ˆ
ρ
xx
=
δρ
−
x xc
0
(7.25)
ij
ij
ijij
j
TV x
, {}
′
Equation 7.9 and Equation 7.19 reveal an important feature that is completely
missed in any fluctuation theory, that the correlation length is related to the sec-
ond moment of the DCF (Fisher 1964), while all fluctuation theories concern only
