Chemistry Reference
In-Depth Information
Because of the well-documented composition dependencies of the individual
binary interaction parameters, an unmindful use of (
56
) would lead to totally
unrealistic results. This feature requires twofold adaption.
First of all, it is necessary to account for the fact that the contribution of a certain
binary contact to the total Gibbs energy of mixing depends on its particular
molecular environment, which in the general case also contains the third compo-
nent. We can allow for that circumstance by multiplying
g
ij
with the factor (
'
j
þ '
i
)
¼
1.
Secondly, we need to specify whether the composition dependencies of the
g
ij
parameters are formulated in terms of
(1
'
k
)
'
i
or of
'
j
, because the resulting mathemati-
cal expressions are not identical.
In order to enable a straightforward application of the new approach to the most
interesting ternary systems (polymer solutions in mixed solvents and solutions of a
polymer blend in a common solvent), it is expedient to express the binary interac-
tion parameters for polymer solvent systems (
26
) and that for polymer blends (
49
)
in the same form. This requirement is met by the relation:
a
ij
g
ij
¼
z
ij
d
ij
þ
e
ij
'
j
(57)
1
1
n
ij
n
ij
'
j
For polymer solutions:
d
¼
1
and
e
¼
ð
1
l
Þ
(58)
whereas the corresponding equation for a polymer blend (the composition depen-
dence being expressed in terms of
'
B
) reads:
2
a
þ
b
b
a
d
¼
and
e
¼
(59)
3
3
By means of (
57
) and the required modification formulated at the beginning of this
section, one obtains the following expression for the reduced residual segment
molar Gibbs energy of mixing of ternary systems, if one neglects ternary interac-
tions (
t
ijk
¼
0) for the time being:
G
res
RT
D
a
12
¼
Þ
z
12
d
12
þ
ð
e
12
'
2
1
ð
'
3
Þ
Þ
'
1
'
2
ð
1
n
12
Þ
ð
1
n
12
'
2
1
ð
'
3
Þ
a
23
þ
Þ
z
23
d
23
þ
ð
e
23
'
3
1
ð
'
1
Þ
Þ
'
2
'
3
ð
1
n
23
Þ
ð
1
n
23
'
3
1
ð
'
1
Þ
a
31
þ
Þ
z
31
d
31
þ
ð
e
31
'
1
1
ð
'
2
Þ
Þ
'
3
'
1
ð
1
n
31
Þ
ð
1
n
31
'
1
1
ð
'
2
Þ
(60)
