Inhibitory Long Range Interaction in Pattern Forming Physical and Biological Systems

George Washington University, Washington DC

Investigators

Abstract

Inhibitory systems, such as block copolymers, are known for their structured patterns of morphological phases. Studying these patterns reveals the mechanical, optical, electrical, ionic, barrier and other properties for applications. For instance, block copolymers are commercially used as thermoplastic elastomers: outer coverings for optical fiber cables, adhesives, bitumen modifiers, or in artificial organ technology. The mathematical theory to be studied by the investigator will offer insights into the nature of these materials and ultimately inform the building of industrial devices. This project will also provide laboratory experience and research mentoring opportunities for undergraduate and graduate students. Inhibitory interaction appears in geometric variational problems with connections to phenomena in other branches of mathematics: bubble clusters in geometric measure theory, elliptic curves in analytic number theory, and sphere packing in geometry. The negative fractional and the negative fractional screened Laplace operators provide accurate descriptions of inhibitory interaction and are relevant to other mathematical theories. The first task in this project is to identify the ranges for the fractional exponents in which these operators possess inhibition properties. The second is to construct building blocks for periodic structures in inhibitory systems, such as the non-standard double bubble, and to use them to find periodic stationary points for the whole space. A related question is to find an optimal lattice that gives rise to a periodic configuration with the least free energy density. Building blocks can also be used to forge near periodic stationary assemblies on a general bounded domain. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.