Chemistry Reference
In-Depth Information
approach predicts higher transition temperatures and less latent entropy at
transition. One reason is the simplicity of the discrete orientation approxi-
mation in the transfer matrix approach. The difference may also result from
the fact that the Maier-Saupe method is actually specified for small molec-
ular mass liquid crystals and does not take into account that connections of
consecutive segments reduce the system entropy. Experimentally, the N-I
transition temperature increases as the degree of polymerization increases.
The higher
T
c
prediction for a freely-jointed-rod chain of liquid crystalline
polymers is expected from the same argument. Meanwhile, the latent
entropy at transition is reasonably less than that of the Maier-Saupe results.
For the elastically-jointed-rod chain, the nematic mean field energy is
expressed as
U
n
=
−
i
pSP
2
(
n
i
)
,
(2.81)
where
p
is the Maier-Saupe coe
cient of one monomer and
n
i
denotes
the orientation of the
i
-th segment. In addition, there is an elastic energy
associated with the bending of the successive rods,
U
B
=
i
q
2
(
n
i
+1
−
n
i
)
2
,
(2.82)
where,
q
is the bend elasticity.
Thus the Hamiltonian of a polymer is expressed as
H
i
=
i
2
(
n
i
+1
−
n
i
)
2
.
−
pSP
2
(
n
i
)+
q
(2.83)
In principle,
n
i
can be any orientation in three-dimensional space. For
simplicity, we limit the orientation of each segment to six directions,
i.e
.,
the three orthogonal axes in a cubic lattice. The six orientations are:
n
=1
for pointing up,
n
=
1 for pointing down and
n
= 0 for four transverse
directions, rightward, leftward, forwards and backward, the degeneracy
being four, as shown in Figure 2.16.
Split the Maier-Saupe potential into two terms and express the partition
function as
Z
=
n
i
−
exp
a
n
i
)
2
,
b
2
(
n
i
+1
−
−
2
[
P
2
(
n
i
+1
)+
P
2
(
n
i
)]
(2.84)
i
where
a
=
βpS
and
b
=
βq
.
b
is actually the ratio of persistence length
to repeated unit length,
l/l
0
. We assume that
b
is constant. The summa-
tion
is made over all possible orientations of each segment while the
