Many systems exhibit a phase where the order parameter is spatially modulated. These patterns can be the result of a frustration caused by the competition between interaction forces with opposite effects.In all models with local interactions, these 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, Misbah and Politi have shown that long-range interactions could cause an interruption of this coalescence process, stabilizing a pattern which then remains in a micro-structured state or super-crystal. We show that this is the case for a modified Cahn–Hilliard dynamics due to Oono which includes a nonlocal term and which is particularly well suited to describe systems with a modulated phase.
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The authors would like to thank Dr. Chaouqi Misbah (LIPhy, Grenoble) for fruitful discussions and an invitation in Grenoble where part of this work was done.
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