Grain boundaries and the law of corresponding cones in smectics

Focal Conic Domains (FCDs) in smectic phases often assemble according to a particular rule, experimentally discovered by G. Friedel, the law of corres ponding 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 c onic domains has a vanishing energy, yielding metastable local equilibrium. Then we use some projective properties of conic sections to extend the cel ebrated Apollonian tiling, which describes a tilt grain boundary (TiGB) of vanishing disorientation omega = 0 made of toric focal conic domains, to an y omega not equal 0 TiGB. Finally are present a realistic model of the ener gy of the omega not equal 0 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.