Chemistry Reference
In-Depth Information
where
v
is the mean molar volume,
r
is the degree of polymerization, and
k
is the
square gradient coefficient, which is considered a constant given by
k ¼ wrb
2
/6. To
estimate
r
in the interfacial region, he chose the expression
wr
¼
2.093
T
0
/
T
, with
T
0
being an adjustable parameter. Thus, (
74
) leads to:
"
#
1
=
2
2
464
RT
v
2
T
0
T
733
T
T
T
0
1
=
g ¼
0
:
bw
1
0
:
T
0
0
:
267
(78)
However, Kammer incorrectly assumed that the interfacial tension is the free
energy of mixing per unit area, instead of the correct expression that defines
interfacial tension as the
excess
free energy per unit interfacial area. Although
interfacial tension is predicted to decrease with temperature, the results are not
accurate fundamentally and the derivations should be recalculated.
Poser and Sanchez [
229
] used the generalized density gradient theory of inter-
faces [
216
] in conjunction with the compressible lattice fluid model of Sanchez and
Lacombe [
237 240
] to approximate the interfacial tension and thickness between
two immiscible high molecular weight polymer liquids. The theory is not expected
to apply near the critical point, where the lattice fluid theory incorrectly describes
the coexistence curve, or for highly polar polymers. Furthermore, the theory
neglects intramolecular correlation effects present in long polymer chains, as well
as changes in the configurational entropy at the interface. Due to the fact that the
calculated phase diagrams, using the lattice fluid model, are extremely sensitive to
the values of the two interaction parameters inherent in the model, and the assump-
tion that the entropy in the interfacial region is independent of concentration
gradients, Poser and Sanchez suggested that “
in its present form, the theory is
being pushed to its limits when applied to a polymer polymer interface.
”
The resultant equations yield predictions comparable to those of Helfand and
Sapse [
29
]. Formally the two theories look quite similar. Conceptually, however,
they are quite different. Gradient effects arise only from energetic considerations in
the Poser Sanchez theory, whereas they arise from the intrinsic connectivity of the
polymer chain in the theory of Helfand Sapse. In the simplest version of the
Helfand Sapse theory, compressibility effects are ignored whereas they play an
important role in the Poser Sanchez formulation. Poser and Sanchez suggested that
a proper theory for polymeric interfaces should not ignore the compressible nature
of polymer liquids (even though it is very small), nor can it ignore the intrinsic
connectivity of a polymer chain.
Anastasiadis et al. have also developed a theory for polymer polymer interfacial
tension [
20
,
122
], based upon the generalized square-gradient theory of Cahn and
Hilliard [
216
] in conjunction with the Flory-Huggins theory of the free energy of
mixing [
206
]. The free energy is calculated as:
D
g
0
ðfÞ
k
B
T
¼
f
r
A
u
A
ln
f þ
1
f
r
B
u
B
w
u
A
fð
ln
ð
1
fÞþ
1
fÞ
(79)
