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We develop a theory of the domain patterns in systems with competingshort-range attractive interactions and long range repulsive Coulombinteractions. We take an energetic approach, in which patterns are consideredas critical points of a mean-field free energy functional. Close to themicrophase separation transition, this functional takes on a universal form,allowing to treat a number of diverse physical situations within a unifiedframework. We use asymptotic analysis to study domain patterns with sharpinterfaces. We derived an interfacial representation of the pattern's freeenergy which remains valid in the fluctuating system, with a suitablerenormalization of the Coulomb interaction's coupling constant. We also derivedintegrodifferential equations describing the stationary domain patterns ofarbitrary shapes and their thermodynamic stability, coming from the first andsecond variation of the interfacial free energy. We showed that the lengthscale of a stable domain pattern must obey a certain scaling law with thestrength of the Coulomb interaction. We analyzed existence and stability oflocalized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patternsin two dimensions. We showed that these patterns are metastable in certainranges of the parameters and that they can undergo morphological instabilitiesleading to the formation of more complex patterns. We discuss nucleation of thedomain patterns by thermal fluctuations and pattern formation scenarios forvarious thermal quenches. We argue that self-induced disorder is an intrinsicproperty of the domain patterns in the systems under consideration.Comment: 59 pages (RevTeX 4), 9 figures (postscript), to be published in the  Phys. Rev.

Theory of domain patterns in systems with long-range interactions of Coulomb type