Crystal Ordering and Dynamics of Interacting Particles

Many materials present a crystal structure at low temperature. The microscopic ordering influences on the macroscopic scale, resulting in different electromechanical and optical properties. The understanding of the crystal formation phenomenon from first principles is still open. Large-systems crystallization is expected to be well-described in terms of minimization of suitable configurational potential taking into account particles interactions. The proof of periodicity of minimizers for configurational potentials is however presently unavailable in a majority of relevant cases. The project aims at obtaining novel rigorous crystallization results. Moving beyond the current state of the art, we aim at deepening the understanding of pattern formation in materials. Moreover, we will be focusing on explicit energy and surface-tension estimation, macroscopic geometry of ground states and their stability with respect to perturbations. Finally, we are interested in describing the evolution towards ordered equilibria in evolutive systems, exploiting the analogy with other paradigmatic problems of collective behavior of individuals, from biology, kinetics, and population dynamics. The goal of the project is to fast-forward the study of the mathematics of crystallization for complex materials and describe the emergence of nanostructures. By progressively enriching the choice of the representative interaction model, we aim at addressing three-dimensional structures and at obtaining more general, rigorous description of crystal pattern. On the other hand, we would like to investigate geometric and mechanical properties of carbon allotropes, fullerenes, carbon nanotubes. The ultimate objective of the proposed program is to contribute to a better understanding of the basic principles of crystallization, toward application for the new emerging technologies.

The project is concerned with a mathematical description of the typical crystalline structure of solid matter, with special focus on carbon nanostructures. At zero temperature, interatomic interactions can be expected to be ruled solely by the geometry of atom configurations, and the mutual attraction-repulsion effects between the particles is described by means of classical potentials, in the framework of Molecular Mechanics. Crystalline structures may or may not arise as ground states for the configurational energy accounting for particles interactions and typically depending on interatomic distances and bond angles. The main result that we obtained is a stability result for carbon rolled-up configurations such as carbon nanotubes. These structures have paramount relevance for modern engineering applications. We have provided a systematic mathematical description of their geometry, along with some modeling of their mechanical properties.Competing effects of aggregation, spreading and ordering of particle distributions also appear in many mathematical models involving evolution, both in a discrete or continuous description, ranging from population dynamics to motion of gas particles or biological agents. A second main point of the project has been the analysis of interaction driven evolution, i.e., of the dynamics of particle densities toward optimal states of specific interaction energies. In this respect, new theoretical results in the mathematical research fields of evolution equations and calculus of variations have been obtained.