Spatially localized self-assembly driven by electrically charged phase separation

Self-assembly driven by phase separation coupled to Coulombic interactions is fundamental to a wide range of applications, examples of which include soft matter lithography via di-block copolymers, membrane design using polyelectrolytes, and renewable energy applications based on complex nano-materials, such as ionic liquids. The most common mean field framework for these problems is the nonlocal Cahn-Hilliard, such as the Ohta-Kaw asaki model. Unlike the common investigations of spatially extended patterns, the focus here is on the emergence of spatially localized states in both the classical and the extended Ohta-Kawasaki model. The latter also accounts for (i) asymmetries in long-range Coulomb interactions that are manifested by differences in the dielectric response, and (ii) asymmetric short-range interactions that correspond to differences in the chemical potential between two materials or phases. It is shown that in one space dimension there is a multiplicity of coexisting localized solutions, which organize in the homoclinic snaking structure. These, however, appear in a vertical structure as in dissipative systems, and not slanted as in conserved models with uniquely defined chemical potential (Lagrange multiplier), e.g., the conserved Swift-Hohenberg model. Differences between the cases and mechanism of localized solution selection are discussed. In addition, an analysis of two-dimensional extension is performed and distinct secondary instability mechanisms (related to extended and localized modes) of localized stripes are discussed with respect to model parameters and domain size. Finally, implications to localized hexagonal patterns are also made. The insights provide an efficient mechanistic framework to design and control localized self-assembly that might be a plausible strategy for low cost of nanoelectronic applications, i.e., a rather simple nanoscale fabrication of isolated morphologies.