How smectic-A and smectic-C liquid crystals resolve confinement-induced frustration in spherical shells

The layered structure of smectic liquid crystals cannot develop unobstructed when confined to spherical shells with layers extending in the radial direction, since the available cross section area increases from the inside to the outside of the shell yet the number and thickness of layers must be constant. For smectic-A (SmA) liquid crystals, with the layer normal m parallel to the director n, the frustration breaks up the texture into spherical lune domains with twist deformations of alternating sense, overlaid with a herringbone-like secondary modulation and mediated via localized bend regions where the boundary conditions are violated. The SmC phase has more degrees of freedom to resolve the frustration thanks to its non-zero tilt angle tau between n and m, but its response to tangential shell confinement was never studied. We show experimentally and theoretically that the lunes in shells undergoing a SmA-SmC transition become twice as wide and half as many and they lose the secondary modulation, adopting a configuration with no layer twist but uniform layer bend if tau reaches a large enough value. Our study expands our understanding of how smectics respond to spherical confinement and it opens new soft matter research opportunities, given the rich diversity of phases with SmC-like symmetry, including chiral and spontaneously polarized phases.