Chemistry Reference
In-Depth Information
number of the individual particles, to obtain equations for the mean field acting on
the polymers. They used this general theory to calculate the interfacial tension for
polymer/polymer/solvent systems [
214
], where good agreement was obtained with
experiment [
215
] for the PS/PBD/styrene ternary system. Their final results, how-
ever, are in integral form, which requires numerical integration. The application of
the theory to the ternary systems polymer A/polymer B/diblock copolymer AB will
be presented in Sect.
4.3.1
.
Helfand and coworkers [
30
] responded to the experimental interest on the molec-
ular weight effects on interfacial tension (see Sect.
3.1
) by solving the equations
they had derived earlier [
27
,
28
,
199
,
208
] for the case of finite molecular weights;
these equations were solved only in the infinite molecular weight limit earlier
[
27
,
28
]. The leading correction to the interfacial tension, which is of order
r
1
(where
r
is the degree of polymerization of the two polymers in a symmetric system),
is solely due to the placement entropy, i.e., it originates from the gain in translational
entropy for finite chains, which can penetrate slightly more into the other phase.
The interfacial tension for a symmetric system (polymers A and B with the same
properties when pure) of large but finite molecular weights is, thus, calculated as:
ln 2
2
wr
g ¼ g
1
1
(56)
The leading correction to the concentration profile is also of the order of
r
1
and
is due to the entropic attraction of the chain ends to the interfacial region and the
necessary readjustment of the remainder of the molecule. The authors gave a
nonanalytic expression for the interfacial width. The concentration correction
does not contribute to the interfacial tension at leading order because the free
energy is calculated within a mean field approximation, where any change in the
concentration can affect it in the second order, producing in this case a correction to
the interfacial tension of the order of
r
2
.
Tang and Freed [
32
] used density functional theory to investigate the effects of
molecular weight on polymer polymer interfacial tension. They considered possi-
ble reasons for the discrepancy between the theories available at that time and the
experimental investigations on interfacial tension and concentration profiles across
the interface. They postulated that certain approximations in the density functional
previously used might be appropriate only in certain limited domains and, conse-
quently, that higher order contributions to free energy functionals could contribute
significantly to interfacial properties. Moreover, they considered the possible com-
position dependence of the Flory-Huggins interaction parameter. Tang and Freed
calculated the interfacial tension for a symmetric blend for the entire two-phase
region (from the weak to the strong segregation regime); it is given as:
"
#
3
=
2
2
g ¼ g
1
1
0
:
90
2
2
wr
wr
0
:
10
(57a)
