Stabilization of a modulated phases in the presence of long range interactions

Many systems exhibit a phase where the order parameter is spatially modulated. These phases are the result of frustration caused by the competition between interaction forces with opposite effects. They form a disordered phase at high temperature and an ordered phase at low temperature in which the order parameter is spatially modulated. In all models with local interactions, the ordered phases disappear in the strong segregation regime (low temperature). It is expected however that these phases should persist in the case of long range interactions, which can't be correctly described by a Ginzburg-Landau type model with only a finite number of spatial derivatives of the order parameter. An alternative approach is to study the dynamics of the phase transition or pattern formation. While, in the usual process of Ostwald ripening, succession of doubling of the domain size leads to a total segregation, or macro -segregation, C. Misbah and P. Politi have shown that long-range interactions could cause an interruption of this coalescence process, stabilizing a pattern that then remains in a micro structured state or super-cristal. We show that this is the case for a modified Cahn-Hilliard dynamics due to Oono which includes a non local term and which is particularly well suited to describe systems with a modulated phase.