Chemistry Reference
In-Depth Information
Hence, (
60
) reduces to:
2
2
g
¼
g
0
ðfÞþk
1
r
f þ k
2
rðÞ
þ
...
(63)
Integrating over a volume
V
of the system, the total free energy of this volume is:
Z
dV g
0
ðfÞþk
1
r
2
2
G
¼
f þ k
2
rðÞ
þ
...
(64)
Application of the divergence theorem, results in:
Z
dV k
1
r
Z
Z
¼
2
dV
2
f
ð
d k
1
=
d f
Þ rðÞ
þ
ð
k
1
rf
n
Þ
dS
(65)
S
where
S
is an external surface with a normal vector n. Since one is not concerned
with effects at the external surface, by choosing a boundary of integration in (
65
)
such that
rf
·n
¼
0 at the boundary, the surface integral vanishes. Using (
65
)to
2
eliminate the term
r
f
from (
64
) one obtains:
Z
dV g
0
ðfÞþk rðÞ
2
G
¼
þ
...
(66)
where:
0
0
þ
2
g
@f@r
2
g
@ rf
d f þ k
2
¼
@
@
k ¼
d k
1
=
(67)
2
2
f
j
j
)
2
is the additional positive contribution to the free energy, which arises from
the local composition gradient. The coefficient of the square gradient term is related
to the inhomogeneous fluid structure [
220
,
221
]. It is essentially the second moment
of the Ornstein Zernike direct correlation function,
C
(
s
,
k
(
rf
f
), of a uniform fluid of
composition
f
. The relationship is:
Z
s
4
C
4
p
kT
6
kðfÞ¼
ð
s
; fÞ
ds
(68)
C
(
s
,
f
) depends on the range of correlation and is a function of the composition
f
of
the system.
Following the derivation of Cahn Hilliard, the total free energy for the case of a
one-dimensional composition gradient and a flat interface of area
A
becomes:
"
#
dx
A
þ1
1
2
d
dx
G
¼
g
0
ðfÞþk
(69)
