2018 Impact factor 1.686
Soft Matter and Biological Physics
Eur. Phys. J. E 2, 47-57

Grain boundaries and the law of corresponding cones in smectics

M. Kléman1 - O.D. Lavrentovich2

1 Laboratoire de Minéralogie-Cristallographie (UMR 7590), Universités de Paris-VI & de Paris-VII, Case 115; 4 place Jussieu 75252 Paris cédex 05, France
2 Chemical Physics Interdisciplinary Program and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
odl@lci.kent.edu

Received 21 June 1999 and Received in final form 10 September 1999

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 $\omega=0$ made of toric focal conic domains, to any $\omega\not=0$ TiGB. Finally we present a realistic model of the energy of the $\omega\not=0$ 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.

PACS
61.30.Jf Defects in liquid crystals - 61.72.Lk Linear defects: dislocation disclinations - 61.72.Mm Grain and twin boundaries

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