6.2 Plastic Crystals and Liquid Crystals
135
T OL =
zw
4k B
ν.
(6.42)
In an intermediate region of ν, the melting occurs to the isotropic liquid in a single
step. The locating this boundary requires numerical minimization of the Helmholtz
energy concerning ξ and σ , and numerical comparison of the Helmholtz energies
of two phases because the transition is of the first-order. Practically, the following
simultaneous equations are to be solved
ln
ξ
1 − ξ
=
zw
2k B T
− σ (1 − σ )
z
w
k B T
(2ξ − 1)
(6.43)
ln
σ
1 − σ
=
z
w
k B T
(1 − 2ξ + 2ξ
2
)(2σ − 1)
(6.44)
and, the corresponding Helmholtz energy is compared with that with ξ =
1
2
and
σ =
1
2
.
Although transitions predicted assuming constant volume are mostly of secondorder, they might become of first-order under constant pressure conditions, as shown
in the previous section for simple melting. Indeed, assuming the same v
−4 dependence of energy penalties (w and w
) as that assumed by Lennard-Jones and Devonshire [29]. Pople and Karasz [30, 31] indicated that phase transitions became of the
first-order in most cases.
Some issues can be pointed out to be examined in the model by Pople and Karasz.
As we see in Sect. 6.1, plastic crystals appear in systems of globular molecules. In
such cases, the number of possible molecular orientations is much larger than the two
assumed in the model. Amzel and Becka [32] later made an extension concerning the
available orientations. Another variation came from the side of ordered liquids (liquid
crystals). The same dependences of energy penalties on volume imply their ratio, ν,
remains the same upon the volume variation. Chandrasekhar et al. [33, 34] claimed
that the volume dependence of the orientational penalty, w
, is weaker. Assuming
w
∝ v
−3 , they reported an improved phase diagram under constant pressure. Even
taking these modifications into account, the parameter ν cannot practically be related
to the anisotropy of molecular shape, except ν = 0 for the completely globular case.
In this respect, models are to be treated as not quantitative but qualitative ones to
understand the general trend of melting behavior depending on molecular anisotropy.
A variety of partial melting can be imagined for either orientational and positional
order. Plural crystalline phases of methane (CH 4 ) [35–37] and varieties of liquid
crystals discussed in Sect. 7.1.2 are such examples.
Realistic models based on molecular details were constructed for systems consisting of simple molecules such as diatomic molecules [38] and methanes [36,
39]. Due to their small moments of inertia, proper consideration of possible quantum effects are sometimes necessary for good descriptions. Especially in cases of
methanes, quantum effects bring a drastic difference in physical properties among
135
T OL =
zw
4k B
ν.
(6.42)
In an intermediate region of ν, the melting occurs to the isotropic liquid in a single
step. The locating this boundary requires numerical minimization of the Helmholtz
energy concerning ξ and σ , and numerical comparison of the Helmholtz energies
of two phases because the transition is of the first-order. Practically, the following
simultaneous equations are to be solved
ln
ξ
1 − ξ
=
zw
2k B T
− σ (1 − σ )
z
w
k B T
(2ξ − 1)
(6.43)
ln
σ
1 − σ
=
z
w
k B T
(1 − 2ξ + 2ξ
2
)(2σ − 1)
(6.44)
and, the corresponding Helmholtz energy is compared with that with ξ =
1
2
and
σ =
1
2
.
Although transitions predicted assuming constant volume are mostly of secondorder, they might become of first-order under constant pressure conditions, as shown
in the previous section for simple melting. Indeed, assuming the same v
−4 dependence of energy penalties (w and w
) as that assumed by Lennard-Jones and Devonshire [29]. Pople and Karasz [30, 31] indicated that phase transitions became of the
first-order in most cases.
Some issues can be pointed out to be examined in the model by Pople and Karasz.
As we see in Sect. 6.1, plastic crystals appear in systems of globular molecules. In
such cases, the number of possible molecular orientations is much larger than the two
assumed in the model. Amzel and Becka [32] later made an extension concerning the
available orientations. Another variation came from the side of ordered liquids (liquid
crystals). The same dependences of energy penalties on volume imply their ratio, ν,
remains the same upon the volume variation. Chandrasekhar et al. [33, 34] claimed
that the volume dependence of the orientational penalty, w
, is weaker. Assuming
w
∝ v
−3 , they reported an improved phase diagram under constant pressure. Even
taking these modifications into account, the parameter ν cannot practically be related
to the anisotropy of molecular shape, except ν = 0 for the completely globular case.
In this respect, models are to be treated as not quantitative but qualitative ones to
understand the general trend of melting behavior depending on molecular anisotropy.
A variety of partial melting can be imagined for either orientational and positional
order. Plural crystalline phases of methane (CH 4 ) [35–37] and varieties of liquid
crystals discussed in Sect. 7.1.2 are such examples.
Realistic models based on molecular details were constructed for systems consisting of simple molecules such as diatomic molecules [38] and methanes [36,
39]. Due to their small moments of inertia, proper consideration of possible quantum effects are sometimes necessary for good descriptions. Especially in cases of
methanes, quantum effects bring a drastic difference in physical properties among