72
4 Molecular Crystals
spherical symmetry around the molecular center of gravity. The spherical symmetry means that the resulting crystal is a “plastic crystal” (see Chap. 6), where the
molecules are isotropically disordered in their orientation. The formation of such
plastic crystals is physically unrealistic for highly anisotropic molecules, as shown
by simple theories [16, 17] and computer simulations [18, 19]. The lowest indexes
are, however, clearly favored by nature because the higher the indexes, the longer
the unit cell dimension that requires an effectively long-ranged interaction.
As described in Sect. 4.1.2, the normalization of the free energy of the symmetric
phase, i.e., the isotropic liquid, is mandatory for meaningful comparison. Expansion
coefficients should remain unchanged irrespective of indexing methods because the
identical wavevector is characterizing the most significant density fluctuation. The
transition temperature is primarily determined by the number of the wavevectors we
take into account. The number is 12 for {1, 1, 0}: (1, 1, 0), (1, −1, 0), (−1, 1, 0)
, (−1, −1, 0) and their cyclic permutations. Similarly, they are 24, 12, and 48 for
{2, 1, 1}, {2, 2, 0}, and {3, 2, 1}, respectively.
The first two groups in the list of indexes are known to be crucially important for
an exotic structure called the Gyroid phase, the basic structure of which is shown
in Fig. 4.1a. For the Gyroid phase, which belongs to the space group I a3d, the two
lowest surviving indexes of diffractions are {2, 1, 1} and {2, 2, 0} when irradiated
by a suitable ray. Needless to say, {2, 1, 1} is more important because only {2, 2, 0}
merely leads to the same structure as that in the BCC case, although the importance
of {2, 2, 0} in enhancing the stability of the Gyroid phase was pointed out [20]. It
is notable here that not only both {2, 1, 1} and {2, 2, 0} form equilateral triangles
separately, but also the two wavevectors (2, 1, 1) and (2, 2, 0), and their equivalents
can form isosceles triangles like (2, 1, 1), (−2, 1, 1) and (0, −2, −2). Two groups of
wavevectors cooperatively enhance the stability of the Gyroid phase. The mismatch
in length is the same as that in the FCC case.
It is noted that the present discussion is free of microscopic details of systems,
and, consequently, is equally applicable even to other soft matter than liquid crystals
consisting of small anisotropic molecules. The Gyroid phase is characterized by
a triply periodic minimal surface (TPMS) called gyroid [21]. The gyroid surface
forms a family with other TPMS, called P and D surfaces. These TPMSs are related
to each other by a mathematical transformation called the Bonnet transformation,
which keeps the local geometry at all points on the surfaces [21]. This property
implies that the structures are degenerate in energy if the energy is a function of local
properties on surfaces. In reality, however, the physical appearance of structures
characterized by the P and D surfaces has been infrequent in contrast to the Gyroid
phase. Similar analyses of mesophases characterized by P or D surfaces indicate that
such stabilization is only operative in the Gyroid phase [15].
A structure closely related to higher indexes (Fig. 4.1b ) is also known [15]. The
most important index is {3, 2, 1} in this case. The wavevector having the nearest length is {4, 0, 0}. This wavevector can form isosceles triangles as (3, 2, 1),
(−3, 2, −1), and (0, −4, 0). The mismatch is calculated as (4 −
√
14)/4 ≈ 0.06,
the smallest in all combinations considered. Because of this good matching, the
wavelength dependence of the expansion coefficients may be ignored. Under some
4 Molecular Crystals
spherical symmetry around the molecular center of gravity. The spherical symmetry means that the resulting crystal is a “plastic crystal” (see Chap. 6), where the
molecules are isotropically disordered in their orientation. The formation of such
plastic crystals is physically unrealistic for highly anisotropic molecules, as shown
by simple theories [16, 17] and computer simulations [18, 19]. The lowest indexes
are, however, clearly favored by nature because the higher the indexes, the longer
the unit cell dimension that requires an effectively long-ranged interaction.
As described in Sect. 4.1.2, the normalization of the free energy of the symmetric
phase, i.e., the isotropic liquid, is mandatory for meaningful comparison. Expansion
coefficients should remain unchanged irrespective of indexing methods because the
identical wavevector is characterizing the most significant density fluctuation. The
transition temperature is primarily determined by the number of the wavevectors we
take into account. The number is 12 for {1, 1, 0}: (1, 1, 0), (1, −1, 0), (−1, 1, 0)
, (−1, −1, 0) and their cyclic permutations. Similarly, they are 24, 12, and 48 for
{2, 1, 1}, {2, 2, 0}, and {3, 2, 1}, respectively.
The first two groups in the list of indexes are known to be crucially important for
an exotic structure called the Gyroid phase, the basic structure of which is shown
in Fig. 4.1a. For the Gyroid phase, which belongs to the space group I a3d, the two
lowest surviving indexes of diffractions are {2, 1, 1} and {2, 2, 0} when irradiated
by a suitable ray. Needless to say, {2, 1, 1} is more important because only {2, 2, 0}
merely leads to the same structure as that in the BCC case, although the importance
of {2, 2, 0} in enhancing the stability of the Gyroid phase was pointed out [20]. It
is notable here that not only both {2, 1, 1} and {2, 2, 0} form equilateral triangles
separately, but also the two wavevectors (2, 1, 1) and (2, 2, 0), and their equivalents
can form isosceles triangles like (2, 1, 1), (−2, 1, 1) and (0, −2, −2). Two groups of
wavevectors cooperatively enhance the stability of the Gyroid phase. The mismatch
in length is the same as that in the FCC case.
It is noted that the present discussion is free of microscopic details of systems,
and, consequently, is equally applicable even to other soft matter than liquid crystals
consisting of small anisotropic molecules. The Gyroid phase is characterized by
a triply periodic minimal surface (TPMS) called gyroid [21]. The gyroid surface
forms a family with other TPMS, called P and D surfaces. These TPMSs are related
to each other by a mathematical transformation called the Bonnet transformation,
which keeps the local geometry at all points on the surfaces [21]. This property
implies that the structures are degenerate in energy if the energy is a function of local
properties on surfaces. In reality, however, the physical appearance of structures
characterized by the P and D surfaces has been infrequent in contrast to the Gyroid
phase. Similar analyses of mesophases characterized by P or D surfaces indicate that
such stabilization is only operative in the Gyroid phase [15].
A structure closely related to higher indexes (Fig. 4.1b ) is also known [15]. The
most important index is {3, 2, 1} in this case. The wavevector having the nearest length is {4, 0, 0}. This wavevector can form isosceles triangles as (3, 2, 1),
(−3, 2, −1), and (0, −4, 0). The mismatch is calculated as (4 −
√
14)/4 ≈ 0.06,
the smallest in all combinations considered. Because of this good matching, the
wavelength dependence of the expansion coefficients may be ignored. Under some