New features appear in the kinetics of phase ordering and phase separation in systems where long-range repulsive interactions (LRRI) compete with the shortrange attractive interactions considered earlier. Competing interactions can lead to the emergence of modulated phases, where a particular symmetry, wavelength, and amplitude are selected (Seul and Andelman, 1995). Both in equilibrium and nonequilibrium systems such modulated phases have domain structures with various shapes, patterns, and morphologies. Figure 13.1 shows some domain structures seen in systems displaying modulated phases. Modulated phases in materials are important in technological applications (Park et al., 1997; Black et al., 2000). An understanding of such phases is crucial in order to be able to design materials with specific properties and control their morphology.

Many systems in nature can be modeled through the inclusion of long-range interactions. Examples of such systems are uniaxial ferromagnetic films, ferromagnetic surface layers, ferrofluid films, ferroelectrics, Langmuir (lipid) monolayers, block copolymers, and cholesteric liquid crystals. A uniaxial ferromagnetic film in the presence of an external magnetic field can be modeled by augmenting the standard scalar order parameter model A with an additional long-range interaction arising from the parallel orientation of magnetic dipoles (Roland and Desai, 1990). This repulsive interaction competes with the attractive domain wall energy. An external magnetic field makes the film's magnetization a nonconserved quantity so that a description based on model A is appropriate. Block copolymers and Langmuir monolayers are examples of conserved order parameter (model B) systems where the connectivity between the covalently bonded blocks of the polymer chains results in an effective LRRI (Sagui and Desai, 1994).