Chemistry Reference
In-Depth Information
As polymers enter from the isotropic phase into the nematic phase,
−
∆
2
jumps from zero to a finite value of about 7. The corresponding ratio of
the mean square root of the end-to-end distance component parallel to the
director is about 3.
2.5.3.
Order parameter
In terms of the Spheroidal wave function, the order parameter may be
re-expressed as
S
=
1
0
[
Sp
0
(
z
)]
2
P
2
(
z
)
dz.
(2.117)
Sp
0
(
z
) is a function of ∆
2
, hence,
S
and
T
. Thus, the equation is
self-consistent.
The nematic-isotropic transition occurs when the free energies of the
nematic and the isotropic phases are equal,
i.e
.,
S
2
(
T
)
2
λ
0
+
=0
,
(2.118)
where
λ
0
is the eigenvalue of the ground state of the Spheroidal wave
function.
The
S
vs.
T
relationship (Wang & Warner, 1986) is similar to the we
ll-
known Maier-Sauipe shape, the temperature scale being
T
=
k
B
T/
√
νε
instead of
T
=
k
B
T/p
. The order parameter decreases as
T
increases
until the critical temperature
T
c
=0
.
388. The order parameter discontinu-
ally jumps from 0.356 to zero. Because
T
is the reduced temperature the
transition temperature
T
c
is a function of the geometric mean of
ν
and
ε
.
The greater
v
, the higher
T
c
. The more
ε
, the more stable the nematic
phase. These conclusions are consistent with experiments. It was found that
the critical order parameter for the semi-flexible liquid crytalline polymers
ranges from 0.3 to 0.45.
2.5.4.
Dependence of N-I transition on polymer chain length
It is worth pointing out that after expanding the free energy in the Landau-
de Gennes form and taking into account the entropic contribution of the
free ends of the chain (contrary to circle polymers) and the
n
= 2 term in
