Theory of domain patterns in systems with long-range interactions of   Coulomb type
  We develop a theory of the domain patterns in systems with competing short-range attractive interactions and long range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derived an interfacial representation of the pattern's free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interaction's coupling constant. We also derived integrodifferential equations describing the stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variation of the interfacial free energy. We showed that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We showed that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration. 
C. B. Muratov