Chemistry Reference
In-Depth Information
where
a
I
is a measure of the effective interfacial thickness and:
r
0
d r
A
=
2
b
a
I
¼
j
x¼
0
¼
(38)
dx
1
=
2
ð
6
wÞ
The interfacial tension was calculated as:
Z
Z
þ1
1
k
B
T
r
0
g ¼
d l
dxwr
A
ð
x
;
lwÞr
B
ð
x
;
lwÞ
(39)
0
1
where
r
i
(
x
;
lw
) is determined from the self-consistent equations by replacing
w
with
lw
. Using (
37
), one gets:
1
=
2
6
g ¼
r
0
bk
B
T
(40)
The theory was originally compared to three polymer pairs, namely PS/PMMA;
PMMA/poly(
n
-butyl methacrylate), PnBMA; and PnBMA/poly(vinyl acetate),
PVA. The calculated interfacial tension agreed exactly with the experimental
value for PnBMA/PVA; it compared well for PMMA/PnBMA and differed by
50% for PS/PMMA. Helfand and Tagami suggested that, if
w
is too large, then
the characteristic interfacial thickness is too small for the mean-field theory to be
appropriate. The theory has been widely used to estimate the interfacial tension in
many different polymer polymer systems with acceptable success.
However, the theory cannot be used if the asymmetry between A and B is too
severe. Helfand and Sapse [
29
] refined the theory of Helfand and Tagami so as to
remove the restrictive approximation of property symmetry of the two polymers.
For a Gaussian random walk in a mean field, they obtained:
"
#
2
b
A
þ b
B
2
1
6
ð
b
A
b
B
Þ
1
=
2
g ¼
k
B
Ta
þ
(41)
b
A
þ b
B
a
is the mixing parameter,
a ¼ w
(
r
0A
r
0B
)
1/2
and
b
2
i
6. It was assumed that
there was no volume change upon mixing and that the isothermal compressibility
was small and independent of composition. The theory makes reasonable predic-
tions, which are slightly improved when nonlocal interactions are considered.
Inclusion of these nonlocal interactions gave:
¼ r
0
i
b
i
=
1
=
2
"
b
A
þb
B
2
!
!
#
2
2
1
6
ð
b
A
b
B
Þ
1
18
c
2
b
A
þb
B
2
5
ð
b
A
b
B
Þ
2
g ¼
k
B
Ta
þ
þ
a
þ
...
3
b
A
þb
B
ð
b
A
þb
B
Þ
(42)
