154
7 Liquid Crystals
s =
1
q(β)
π/2
0
P 2 (cos θ) exp [βzs P 2 (cos θ)] sin θdθ
(7.14)
=
1
q(β)
1
0
P 2 (cos θ) exp [βzs P 2 (cos θ)] d(cos θ).
=
1
q(β)
1
0
P 2 (x) exp [βzs P 2 (x)] dx
with
q(β) =
1
0
exp [βzs P 2 (x)] dx,
(7.15)
where β = 1/k B T is an inverse temperature.
The number of states appearing in the model is always infinite due to the classical
nature of the model except for the perfect order θ i = 0 for all i, in contrast to lattice
problems treated in Chap. 6 assuming discrete states for each molecule. Therefore,
the entropy cannot be evaluated in this problem by counting the possible number of
states based on the Boltzmann principle. Alternatively, Eq. 7.15 is regarded as the
partition function (state sum) for a single molecule in a mean-field −zs P 2 (cos θ).
It is, however, essential to know that the internal energy E/N derived naïvely from
this partition function differs from that derived above (Eq. 7.13). This difference is
due to the double count of the interaction, as is usually the case in any mean-field
treatments. The corrected free energy is given by
F
N
= −β
−1 ln q(β) +
z
2
s
2
.
(7.16)
We can not solve the self-consistency equation analytically, but find solutions
numerically. The Maier–Saupe theory predicts a first-order transition between the
disordered state (isotropic liquid) and the nematic liquid crystal, in accordance with
the symmetry consideration (Sect. 7.1.2.1). The comparison of free energies of the
nematic phase and disordered state (with s = 0) is necessary. The free energy of the
disordered state is F = −k B T ln 1 = 0 by Eq. 7.15. The Maier–Saupe theory predicts
a first-order transition at T trs = 0.2201zV 0 /k B . Figure 7.5 shows the temperature
dependence of the nematic order parameter s. Since the intermolecular interaction
prefers the parallel arrangement, the perfect nematic order appears at the absolute
zero.
The Maier–Saupe theory explains the formation of only the nematic phase. To
treat other phases, at least, the introduction of a length scale is necessary. Indeed, a
simple introduction of the molecular length explains the formation of layered liquid
crystal, SmA phase, as shown by McMillan [18].
7 Liquid Crystals
s =
1
q(β)
π/2
0
P 2 (cos θ) exp [βzs P 2 (cos θ)] sin θdθ
(7.14)
=
1
q(β)
1
0
P 2 (cos θ) exp [βzs P 2 (cos θ)] d(cos θ).
=
1
q(β)
1
0
P 2 (x) exp [βzs P 2 (x)] dx
with
q(β) =
1
0
exp [βzs P 2 (x)] dx,
(7.15)
where β = 1/k B T is an inverse temperature.
The number of states appearing in the model is always infinite due to the classical
nature of the model except for the perfect order θ i = 0 for all i, in contrast to lattice
problems treated in Chap. 6 assuming discrete states for each molecule. Therefore,
the entropy cannot be evaluated in this problem by counting the possible number of
states based on the Boltzmann principle. Alternatively, Eq. 7.15 is regarded as the
partition function (state sum) for a single molecule in a mean-field −zs P 2 (cos θ).
It is, however, essential to know that the internal energy E/N derived naïvely from
this partition function differs from that derived above (Eq. 7.13). This difference is
due to the double count of the interaction, as is usually the case in any mean-field
treatments. The corrected free energy is given by
F
N
= −β
−1 ln q(β) +
z
2
s
2
.
(7.16)
We can not solve the self-consistency equation analytically, but find solutions
numerically. The Maier–Saupe theory predicts a first-order transition between the
disordered state (isotropic liquid) and the nematic liquid crystal, in accordance with
the symmetry consideration (Sect. 7.1.2.1). The comparison of free energies of the
nematic phase and disordered state (with s = 0) is necessary. The free energy of the
disordered state is F = −k B T ln 1 = 0 by Eq. 7.15. The Maier–Saupe theory predicts
a first-order transition at T trs = 0.2201zV 0 /k B . Figure 7.5 shows the temperature
dependence of the nematic order parameter s. Since the intermolecular interaction
prefers the parallel arrangement, the perfect nematic order appears at the absolute
zero.
The Maier–Saupe theory explains the formation of only the nematic phase. To
treat other phases, at least, the introduction of a length scale is necessary. Indeed, a
simple introduction of the molecular length explains the formation of layered liquid
crystal, SmA phase, as shown by McMillan [18].