  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 non-local Cahn-Hilliard (a.k.a. Ohta-Kawasaki) framework.
In this work, we study 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 phases. It is shown that in one space dimension (1D) there is a
multiplicity of coexisting localized solutions, which organize in the
homoclinic snaking structure, bearing similarity to dissipative systems. In
addition, an analysis of 2D extension is performed and distinct 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 nano
electronic applications, i.e., a rather simple nano scale fabrication of
isolated morphologies.

Analysis of PDEs;Pattern Formation and Solitons