|M.Sc Student||Versano Idan|
|Subject||Localized States of Non-local Cahn Hilliard Equations|
|Department||Department of Applied Mathematics||Supervisor||Professor Nir Gavish|
The non-local Cahn-Hilliard (Ohta-Kawasaki) models describe spatio-temporal behavior driven by competing short-range repulsive forces and long-range attractive interactions (e.g., Coulomb interactions). These models are often employed in the study of di-block copolymers and renewable energy applications based on complex nano-materials such as ionic liquids and polyelectrolyte membranes. In this work, we study the asymmetric Ohta-Kawasaki equation, an extended model which accounts for possible differences in dielectric response and free energy between the two phase materials.
We study families of distinct solutions to this model using perturbation and numerical continuation methods. Specifically, we focus on spatially localized states in one spatial dimension. We show that the asymmetric Ohta-Kawasaki equation has multiple stable localized solutions. Unlike other conservative gradient models, such as the conserved Swift-Hohenberg equation, these localized solutions form a non-slanted homoclinic snaking structure.
We also show that unlike other conserved models, a finite wavenumber instability may still arise as a result of a coupling to electro-neutrality demand. We also investigate localized states in two spatial dimensions and study the dependence of stable periodic (stripe- like) solutions on the domain size. We reveal that stable localized stripes may arise in the asymmetric Ohta-Kawasaki equation when limiting the domain size.