Grain boundaries and the law of corresponding cones in smectics

Abstract:

Focal Conic Domains (FCDs) in smectic phases often assemble according to a particular rule, experimentally discovered by G. Friedel, the law of corresponding cones (l.c.c.). This paper reports various results relating to this type of association. First we show that a l.c.c. contact between 2 focal conic domains has a vanishing energy, yielding metastable local equilibrium. Then we use some projective properties of conic sections to extend the celebrated Apollonian tiling, which describes a tilt grain boundary (TiGB) of vanishing disorientation made of toric focal conic domains, to any TiGB. Finally we present a realistic model of the energy of the TiGB, which we compare to the energy of a TiGB split into dislocations, and to the energy of a curvature wall. This model explains why FCD tilings show macroscopic zones not filled with FCDs.