FRG: Modeling and Computation of Objective Structures in Materials Science and Biology

Luskin

DMS-0757355

The investigators in this FRG project develop and study the

theory of objective structures. By definition, these are

structures composed of identical molecules having the property

that corresponding atoms in each molecule see precisely the same

environment up to an orthogonal transformation. Objective

structures generalize classical crystal structures, and include

many of the most intensely studied structures in science today,

including carbon nanotubes, buckyballs, viral capsids and other

parts (necks, tails, baseplates), many common proteins, bilayers,

and many other nanostructures now being synthesized, especially

via the process of self-assembly. The investigators exploit the

symmetries of these structures to develop computational numerical

methods for molecular dynamics, a mathematical theory for the

self-assembly of objective structures, a quasicontinuum numerical

method for defective objective structures, and simpler

first-principles calculations of the energy of these structures.

Nanostructures are becoming increasingly important in a

variety of scientific and technological applications. Objective

structures are the building blocks of nanostructures, both

organic and inorganic. A comprehensive and unified mathematical

treatment of such structures has the potential to lead to the

discovery of new structures with unusual forms of ferromagnetism

and ferroelectricity and unexpected transport properties. The

detailed investigation of the self-assembly of objective

structures can lead to new methods of synthesis of such

structures, especially methods that produce nanostructures of

desired dimensions and molecular arrangement. These, in turn,

could lead to new strategies to combat viral infections, and new

methods for the templated growth of particular nanostructures

such as carbon nanotubes. The quasicontinuum mathematical

methods deliver a general strategy for the systematic

investigation of the process of nucleation and growth of defects

in nanostructures. These methods are expected to give a

systematic new tool for the discovery of exceptionally strong

molecular structures.