Thermotropic Behavior
Limit Between LC Conjugated Polymers and LC Conjugated Oligomers
Thermotropic LC conjugated polymers are crystalline becoming an ordered liquid
with heating. It is important to consider the limit between LC hairy-rod polymers and
LC oligomers. This consideration means the phase behavior as a function of
molecular weight, M n . Depending on M n , the polymers manifest at least Nem and
Hex phases. The phase equilibrium of the system can be understood from an analysis
of the corresponding free energies. The free energy of the Nem phase is given as
(Stepanyan et al. 2003)
F N % k B TVcln
f
e
þ k B TVcln
4π
Ω N
,
(1)
where the translational and orientational entropy are represented by the first and
second terms. Here k B T is a Boltzmann factor, V the volume of the sample, c the
concentration of the molecules, f the volume fraction of the backbone, and e the
Euler number. The quantity Ω N describes the degree of overall (uniaxial) alignment:
The smaller it is, the more aligned is the system.
The free energy of the ordered Hex phase does not involve translational entropy
but the interaction between ordered molecules which stems from the inhomogeneous
distribution of the side chain ends starts playing an important role. For hairy rods, it
reads as
F lattice % k B TV
v
v 0 l
2
K l u
,
(2)
where v 0 is the volume of one repeat unit of the hairy rod. From Eq. (2), the free
energy of the Hex phase is
F H % k B TVcln
4π
Ω H
À k B TV
v
v 0 l
2
K l u
:
(3)
Due to incompressibility, the concentration c = M u /(v 0 M n ) is directly related to
M n . Eqs. (1) and (3) imply that the higher the M n , the more favorable the Hex
packing. The value of M n separating Nem and Hex phases is
M n
Ã
% M u
l
2
K l u
v
ln
eΩ N
f Ω H
:
(4)
To elucidate further on the temperature dependency of the phase equilibrium, one has
to consider the temperature variation of the solid angles accessible for rotational degrees
of freedom,Ω N , and Ω H . For both phases, the direction of each main-chain segment can
fluctuate in the solid angle as (Khokhlov 1991; Semenov and Khokhlov 1988)
11 Liquid Crystalline Conjugated Polymers
321
Limit Between LC Conjugated Polymers and LC Conjugated Oligomers
Thermotropic LC conjugated polymers are crystalline becoming an ordered liquid
with heating. It is important to consider the limit between LC hairy-rod polymers and
LC oligomers. This consideration means the phase behavior as a function of
molecular weight, M n . Depending on M n , the polymers manifest at least Nem and
Hex phases. The phase equilibrium of the system can be understood from an analysis
of the corresponding free energies. The free energy of the Nem phase is given as
(Stepanyan et al. 2003)
F N % k B TVcln
f
e
þ k B TVcln
4π
Ω N
,
(1)
where the translational and orientational entropy are represented by the first and
second terms. Here k B T is a Boltzmann factor, V the volume of the sample, c the
concentration of the molecules, f the volume fraction of the backbone, and e the
Euler number. The quantity Ω N describes the degree of overall (uniaxial) alignment:
The smaller it is, the more aligned is the system.
The free energy of the ordered Hex phase does not involve translational entropy
but the interaction between ordered molecules which stems from the inhomogeneous
distribution of the side chain ends starts playing an important role. For hairy rods, it
reads as
F lattice % k B TV
v
v 0 l
2
K l u
,
(2)
where v 0 is the volume of one repeat unit of the hairy rod. From Eq. (2), the free
energy of the Hex phase is
F H % k B TVcln
4π
Ω H
À k B TV
v
v 0 l
2
K l u
:
(3)
Due to incompressibility, the concentration c = M u /(v 0 M n ) is directly related to
M n . Eqs. (1) and (3) imply that the higher the M n , the more favorable the Hex
packing. The value of M n separating Nem and Hex phases is
M n
Ã
% M u
l
2
K l u
v
ln
eΩ N
f Ω H
:
(4)
To elucidate further on the temperature dependency of the phase equilibrium, one has
to consider the temperature variation of the solid angles accessible for rotational degrees
of freedom,Ω N , and Ω H . For both phases, the direction of each main-chain segment can
fluctuate in the solid angle as (Khokhlov 1991; Semenov and Khokhlov 1988)
11 Liquid Crystalline Conjugated Polymers
321