Formation Mechanism of Triply Periodic Minimal Surface as a Thermodynamic Equilibrium Structure

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Thermodynamic and structural study was conducted on thermotropic and lyotropic liquid crystals paying special attention on why so complex structures closely related to triply periodic minimal surfaces (TPMS) are spontaneously formed. Smaller heat capacity than neighboring liquid crystalline phases is common to TPMS-related cubic phases both in thermotropic and lyotropic systems. This trend can be understood by considering the suppressed fluctuation due to three-dimensional connectivity of TPMS. The analysis of the phase transition entropy between lamellar (smectic) and cubic phases suggests the possible construction of "thermodynamics" on the basis of "surfaces", where a unit cell of cubic phase would play a role of a basic unit. Appearance of plural cubic phases depending on the alkyl chain length was proved through putting a new example, BABH series. On the basis of the alkyl chain length dependence of the reflection intensities from a cubic phase having Ia3d symmetry (gyroid phase), the aggregation structure at the molecular level was, for the first time, revealed. The ends of alkyl chain of BABH molecules are located on the gyroid while the molecular cores aggregated to form a "jungle gym" consisting of rod-like aggregates. This aggregation mode survives even if the volume of the alkyl chain becomes larger than that of the molecular core. This is in a strong contrast to the cases of block copolymer and lyotropic liquid crystals, where the aggregation mode changes into the inverse phase. In the region where their volumes are roughly equal to each other, the other cubic phase with Im3m symmetry appears commonly in BABH and ANBC. This suggests that the clarification of the molecular aggregation mode in this Im3m phase and of its stability would be a key to solve the issue how different are low-weight thermotropic system from lyotropic system and block copolymers, in both of which physical surface can easily be imagined.