Chemistry Reference
In-Depth Information
are indicated by the subscript
w
and may differ markedly in their numerical values
from w. The expression for the residual Gibbs energy of dilution is also given an
index as a reminder that weight fractions were used to calculate its combinatorial
part. Despite the practical advantages of
w
w, we stay with volume fractions for all
subsequent considerations, because they account at least partly for the differences
in the free volume of the components and because most of the published thermo-
dynamic information uses this composition variable.
One of the consequences of composition-dependent interaction parameters lies
in the necessity to distinguish between different parameters, depending on the
particular method by which they are determined. The Flory Huggins interaction
parameter w relates to the integral interaction parameter
g
as:
Þ
@
g
@'
w
¼
g
ð
1
'
(9)
The expression analogous to (
5
), referring to the solvent, reads for the polymer
(index P):
D
m
P
RT
¼
N
G
P
RT
¼
D
2
ln
' þ
ð
1
N
Þ
ð
1
'
Þþ
x
N
1
ð
'
Þ
(10)
This relation defines the differential interaction parameter x in terms of the chemi-
cal potential of the polymer and is calculated from
g
by means of:
þ '
@
g
@'
x
¼
g
(11)
Out of the three types of interaction parameters, it is almost exclusively w that is
of relevance for the thermodynamic description of binary and ternary polymer-
containing liquids, as will be described in the section on experimental methods
(Sect.
3
). The integral interaction
g
parameter is practically inaccessible, and the
parameter x, referring to the polymer, suffers from the difficulties associated with
the formation of perfect polymer crystals, because it is based on their equilibria
with saturated polymer solutions.
Measured Flory Huggins interaction parameters soon demonstrated the neces-
sity to treat w as composition-dependent. A simple mathematical description con-
sists of the following series expansion:
2
¼
w
o
þ
w
1
' þ
w
2
'
...
w
(12)
A more sophisticated approach [
21
] accounts for the differences in the molecular
surfaces of solvent molecules and polymer segments (of equal volume) and
formulates w(
'
) as:
