Theories of Liquid Crystalline Polymers - Liquid Crystalline Polymers - page 82

Figure 2.12. Critical volume fraction vs. axial ratio: comparisons of Onsager and
Flory theories with experimental data. (Modified from Ciferri, 1991. Reproduced by
permission of John Wiley & Sons, Inc.)

while Flory predicts that the critical volume fraction is

1 .

8
x

2
x

φ I = − (2.63)

These two predictions are depicted in Figure 2.12 together with some
experimental data. In this figure, the dashed lines are the expectations
from the Onsager and Flory theories. The points are for PBLG (molec-
ular diameter D =15 . 6 A in m-cresol (Hermans, 1962), marked “ ”
and in dioxane, “ ” (Robinnov et al ., 1958). In addition, the data of
PBA ( D =5A in DMAC (+3% LiCl) ( • ) (Bruzone, 1986) and in H 2 SO 4 ( )
(Kwolek et al., 1977) and schizophylian ( D =16 . 7 A) dissolved in H 2 O( )
(Itou et al ., 1985) were compared in Figure 2.12.
It is shown that the predictions of the Onsager theory are smaller than
that of the experimental data. This discrepancy is due to the second virial
approximation. Adopting higher terms of virial coe cients may improve
the predictions.
Meanwhile, Flory predicts a higher critical volume fraction. It is thought
that the Flory lattice model is comparatively simple, even though the
addition of the soft interactions does not change this shortcoming.
The real liquid crystalline polymers exhibit finite flexibility. This kind of
polymer was studied extensively by Khokhlov and his co-workers. Assume
that the semiflexible chain has the total contour length L , and Kuhn length l
and diameter D , and L D . Analogous to the Onsager model, the

free energy of the polymer in solution is composed of the conformational

l