Study of liquid crystal space groups using controlled tilting with cryogenic transmission electron microscopy.

We developed a method that enables differentiation between liquid crystalline-phase particles corresponding to different space groups. It consists of controlled tilting of the specimen to observe different orientations of the same particle using cryogenic transmission electron microscopy. This leads to the visualization of lattice planes (or reflections) that are present for a given structure and absent for the other one(s) and that give information on liquid crystalline structures and their space groups. In particular, we show that we can unambiguously distinguish among particles having the inverted micellar cubic (space group Fd(3)m, 227), the inverted bicontinuous gyroid (space group Ia(3)d, 230), the inverted bicontinuous diamond (space group Pn(3)m, 224), and the inverted bicontinuous primitive cubic structure (space group Im(3)m, 229).