Focal conic domains energy

M. Kleman and O.D. Lavrentovich, Curvature energy of a focal conic domain with arbitrary eccentricity, Phys. Rev. E, 61, 1574-1578 (2000). 

This paper is written as an introductory review for the unlearnt. It is constructed as follows. The first part deals with some generalities of the geometry of Focal Conic Domains (FCD s). The second one is devoted to an analytical description of the sheets which constitute the domains. This is the starting point of the calculation of the energy of different types of FCD s, which is made in the third part. We complete the discussion with some experimental observations in lyotropic systems (fourth part). The fifth part is devoted to the problem of space filling we are considering it in the case of FCD of the first and second species. The last part is devoted to the question of the nucleation and growth of FCD s. 

The vertices of two focal conics are singular, since they correspond to a reverse in the orientation of the materialized conic sheets of Dupin s cyclides. These points often disjoin into pairs of lower energy (see Fig. 31 d and Fig. 11 in Bouligand [53]). At a transition from a smectic A to a smectic C phase disclination lines appear in focal domains, linking the two conics, and are of the type shown in Fig. 21a and b [118]. Their origin is easily understood from Figs. 14 and 25 in Bouligand and Kleman [57]. 

The total energy F scales like a(l-e ) and becomes in principle vanishingly small when e 1, i.e. for the so-called parabolic focal domains (PFD), where the two conjugate conics are two parabolae. But note that K(e ) increases without limit for e 1, and that the... 

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