Crystallization Beyond Crystals: The Mathematics of Non-Periodic Ordered Materials
Abstract
The proposed research takes a fresh look at the longstanding, celebrated mathematical problem known as the ???crystallization problem???. The context of this work is the recent discovery of a host of materials and nanostructureshaving a completely different structure than that hypothesized to be the solution of this problem. These structures include 2D materials (such as graphene and the dichalcogenides), the icosahedral or helical structures of viruses, and a variety of non periodic structures that arise from simulation as minimizers or relative minimizers of problems in the calculus of variations. A disproportionate number of these structures are ???Objective Structures??? [Jam06, DJ12, BEJ15, DJ10, JMNV17, JFJ15], and this therefore provides an attractive framework for the study of generalized crystallization. The work involves a close coupling between the development of models of molecular interaction that are appropriatefor these emerging structures and the development of new rigorous mathematical methods in the calculus of variations to deduce structure. Recognizing that the pathway may also influence the nature of a final structure, the research also encompasses the development of dynamic models and rigorous methods for the study of self-assembly. The working hypothesis of this research is that the crystallization problem ??? prove, based on the use of certain central force potentials that the face-centered cubic (FCC) structure is the energy minimizing structure ??? is formulated too narrowly, and has little relevance to recent profound developments in nanoscience. The team will formulate new mathematical models of molecular forces and new dynamic theories for self-assembly, and will investigate the relation between these interactions and final structure. The team will also develop new rigorous mathematical methods for the prediction of structure from the associated problems in the calculus of variations. Methods of simplification and qualitative methods will be developed in the context of this more general description of molecular interaction forces.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 27, 2018
- Source ID
- N000141812766
Entities
People
- Richard G James
Organizations
- Office of Naval Research
- Regents of the University of Minnesota
- United States Navy