Chemistry Reference
In-Depth Information
whereas the interfacial width is given by:
"
#
1
=
2
2
5
4
2
wr
þ
1
4
2
wr
a
I
¼
a
I
1
1
(58a)
2/3
over a wide
It is interesting that (57) shows an almost linear dependence on (
wr
)
range of 2.5
< wr
<
20, consistent with experimental observations for intermediate
molecular weights.
The asymptotic expressions for very high molecular weights are:
35
2
wr
g
g
1
1
1
:
(57b)
5
8
2
wr
a
I
a
I1
1
þ
(58b)
The coefficient 5/8
0.625 in (
58b
) is very close to that (ln2
0.693) of Broseta
in (
87
) (to be discussed later), whereas the coefficient of 1.35 in (
57b
) is about
50% larger than that (
2
/12
p
0.82) in (
86
) and about twice as large as the value of
ln2
0.693 in (
56
).
The respective equations in the weak segregation limit (WSL) are:
h
i
3
=
2
2
3
w
r
2
g
g
1
p
;
ð
Þ
1
WSL
(57c)
1
=
2
2
3
2
wr
p
a
I
a
I1
1
;
ð
WSL
Þ
(58c)
3.2.3 Square-Gradient Approach
A conceptually different approach to the calculation of interfacial tensions is the
use of the generalized square-gradient approach as embodied in the work of Cahn
and Hilliard [
216
]. The Cahn Hilliard theory provides a means for relating a
particular equation of state, based on a specific statistical mechanical model, to
surface and interfacial properties. The local free energy,
g
, in a region of nonuni-
form composition will depend on the local composition as well as the composition
of the immediate environment. Thus,
g
can be expressed in terms of an expansion in
the local composition and the local composition derivatives. Use of an appropriate
free energy expression derived from statistical mechanics permits calculation of the
surface or interfacial tension.
