Chemistry Reference
In-Depth Information
2
being the second moment
of the direct correlation function [
199
]. The interfacial thickness is predicted to be:
where
c
is a measure of the range of nonlocality, with
c
1
=
2
2
A
2
B
2
b
þ b
a
I
¼
(43)
2
a
Tagami [
200
,
201
] extended the theories of Helfand and coworkers to the case of
compressible nonsymmetric polymer mixtures. A slight decrease in the predicted
interfacial tension was found, due to the presence of finite compressibility of the
polymers. This tendency was particularly apparent in the case of nearly symmetric
polymer pairs, when the intersegmental interactions are of nonlocal nature. The
results reduce to the results of Helfand and Sapse in the appropriate limits. How-
ever, the resulting equations are much too complicated, although the results do not
differ significantly from those predicted by (
41
).
The difficulty in applying the above-mentioned theories is the paucity of accu-
rate data for the physical parameters required by the theories. In particular, data for
w
or
a
are not generally available, and the Hildebrand regular solution theory
expression:
2
ð
d
1
d
2
Þ
a ¼
(44)
kT
has frequently been used, where
d
i
is the solubility parameter of the i-th constituent.
The fact that solubility parameters are normally available at only one temperature
necessitates the additional assumption that they are temperature independent. Use
of this expression for
a
yields interfacial tensions of reasonable magnitude, but
gives the wrong sign for the temperature coefficient. Indeed, substitution of (
44
)in
(
40
)or(
41
) results in an effective
T
1/2
dependence, whereas a linear decrease with
temperature is experimentally observed. However, a proper temperature depen-
dence can be obtained if a small entropic term is added to the expression for
a
[
19
];
an apparent interaction density parameter of the form
a ¼ a
H
/
T
+
a
S
gives a good
agreement between theory and experiment.
The Gaussian random coil model is appropriate when the scale of inhomogeneity
(e.g., the interfacial thickness) is large compared with the length of a bond,
b
, and
the range of interactions,
c
. To handle the case where this is not true, a lattice model
has been proposed by Helfand [
202 205
], in the spirit of the Flory-Huggins
approach [
206
]. For infinite molecular weights, he obtained:
h
i
k
B
T
a
ðwm
1
2
1
=
2
1
=
2
arctan
1
=
2
g ¼
Þ
1
þð
1
þ wÞw
w
(45)
where a is the cross-sectional area of a lattice cell and
m
is a lattice constant, defined
such that the number of nearest neighbors of a cell in the same layer parallel to the
interface is
z
(1
2
m
) and in each of the adjacent layers is
zm
, where
z
is the number
