214
10 Importance of Molecular Crystals
h
k
LT
LT
b* HT
a* HT
Fig. 10.7 Relation between reciprocal lattices of the high-temperature (HT) basic structure and the
low-temperature (LT) superstructure. Crossing points of grids, lattice points of the superstructure;
black dots, lattice points of the HT structure; dotted arrows, reciprocal lattice vectors of the HT
structure; orange arrows, a wave vector of the modulation. Successive shifts by the orange arrow
from lattice points of the HT structure completely cover all lattice points of the LT structure.
Prepared based on J. Chem. Phys., 146, 074503 (2017) [83]
coverage is possible if new reciprocal lattice points appear as a set of double-lined
(2 × j, j ≥ 2) clusters. In this case, the multiplicity is (2 j + 1). Although there is
another choice of the modulation wave vector in Fig. 10.7 because of the minimum
multiplicity, no arbitrariness remains in cases with j > 2.
The consideration in the previous paragraph lapses into a dream without a real
counterpart. Indeed, the formation of superstructures with odd multiplicity is uncommon in non-hexagonal cases. However, there exists a compound that undergoes a
phase transition accompanying the formation of a superstructure with the multiplicity 5 [83]. Since the transition belongs to the minimum case, the identification of the
real example itself seems consistent with the consideration. It suggests that the quest
for exotic superstructure deserves to pursue.
10.3.5 Molecular Aggregation in a Gyroid Phase
The Gyroid phase (briefly introduced in Sect. 4.3.1) is a bicontinuous cubic phase
most widely observed [84] in soft matter ranging from polymers with a cell constant
in µm-scale [85] to thermotropic and lyotropic liquid crystals with that in the nmscale [86, 87]. There are two complementary descriptions, which pay emphases on
one of the wall (surface) and the jungle gyms. Fig. 4.1a depicted the Gyroid phase
in the latter spirit. The two subspaces divided by the G (gyroid) surface (shown in
Fig. 10.8), which has extensively been studied in differential geometry, are physically equivalent but have opposing handedness. Among the Gyroid phase formed in
10 Importance of Molecular Crystals
h
k
LT
LT
b* HT
a* HT
Fig. 10.7 Relation between reciprocal lattices of the high-temperature (HT) basic structure and the
low-temperature (LT) superstructure. Crossing points of grids, lattice points of the superstructure;
black dots, lattice points of the HT structure; dotted arrows, reciprocal lattice vectors of the HT
structure; orange arrows, a wave vector of the modulation. Successive shifts by the orange arrow
from lattice points of the HT structure completely cover all lattice points of the LT structure.
Prepared based on J. Chem. Phys., 146, 074503 (2017) [83]
coverage is possible if new reciprocal lattice points appear as a set of double-lined
(2 × j, j ≥ 2) clusters. In this case, the multiplicity is (2 j + 1). Although there is
another choice of the modulation wave vector in Fig. 10.7 because of the minimum
multiplicity, no arbitrariness remains in cases with j > 2.
The consideration in the previous paragraph lapses into a dream without a real
counterpart. Indeed, the formation of superstructures with odd multiplicity is uncommon in non-hexagonal cases. However, there exists a compound that undergoes a
phase transition accompanying the formation of a superstructure with the multiplicity 5 [83]. Since the transition belongs to the minimum case, the identification of the
real example itself seems consistent with the consideration. It suggests that the quest
for exotic superstructure deserves to pursue.
10.3.5 Molecular Aggregation in a Gyroid Phase
The Gyroid phase (briefly introduced in Sect. 4.3.1) is a bicontinuous cubic phase
most widely observed [84] in soft matter ranging from polymers with a cell constant
in µm-scale [85] to thermotropic and lyotropic liquid crystals with that in the nmscale [86, 87]. There are two complementary descriptions, which pay emphases on
one of the wall (surface) and the jungle gyms. Fig. 4.1a depicted the Gyroid phase
in the latter spirit. The two subspaces divided by the G (gyroid) surface (shown in
Fig. 10.8), which has extensively been studied in differential geometry, are physically equivalent but have opposing handedness. Among the Gyroid phase formed in