Reconstruct Morphological Phases from Nonlocal Geometric Systems

George Washington University, Washington DC

Investigators

Abstract

Pattern formation is an orderly outcome of self-organization principles. Morphological phases are complex geometric structures that form as structured patterns in physical and biological systems. Examples include morphological phases in block copolymers, animal coats, and skin pigmentation in cell development. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. In addition, pattern formation provides a longer ranging confinement of the locally self-enhancing process. Studying pattern formation reveals the mechanical, optical, electrical, ionic, barrier, and other properties of these systems. In this project, the investigator studies morphological phases in biological and physical systems having multiple constituents. Often these phases have been observed in experiments where no current theory or models predict them. The investigator aims for a rigorous analysis that not only predicts the existence of these structures but also their particular properties. Morphological phases are important in modern materials such as block copolymers, which are commercially used as thermoplastic elastomers: wine bottle stoppers, outer coverings for optical fibre cables, adhesives, bitumen modifiers, or in artificial organ technology. Graduate and undergraduate students participate in the work of the project. The investigator studies morphological phases in physical and biological systems. Systems of particular interest are the Ohta-Kawasaki, FitzHugh-Nagumo, Gierer-Meinhardt, and tri- and tetrablock copolymers. Typically a morphological phase consists of micro-domains separated by narrow interfaces. Often, narrow-interface morphological phases can be approximated by their sharp interface limits, where constituents are fully separated by interfaces with no thickness. In the sharp interface case this project investigates assemblies of tori, Hopf links, double bubbles, and triple bubbles. Analytical methods are developed to use sharp interface morphological phases as skeletons to reconstruct morphological phases with narrow interfaces. Through renormalization and an infinite-dimensional reduction technique, where Coulomb forces are treated as long-range interactions, free energies of lower-dimensional structures are derived. Graduate and undergraduate students participate in the work of the project.