Variational Modeling of Molecular Geometries

Wider research context: Driven by their fascinating electronic and mechanical properties, research on low-dimensional materials (such as graphene) is exponentially growing. New findings are emerging at an always increasing pace, ranging from fundamental concepts to applications. In contrast to the wealth of experimental and numerical evidence currently available, rigorous mathematical results on local and global crystalline geometries are scant and the study of the emergence of different scales within molecular structures is still in its infancy. Objectives: We focus on the variational modeling of molecular geometries within the frame of Molecular Mechanics: effective configurations are identified as minimizers of classical configurational potentials. The project aims at obtaining new mathematical understanding of molecular geometries and at investigating the emergence of scale effects across scales. Approach: Ranging from the nano to the macroscale, we address crystallization for molecular compounds, the description of local molecular features including defects and rigidity, the occurrence of global geometric characteristics such as flatness in 3d and stratification, and the passage from discrete to continuum theories. Grounded on variational methods for atomistic models, the methodology will integrate techniques from discrete mathematics and stochastics as well. Innovation: The project targets a number of hot research fronts in Materials Science from the rigorous mathematical standpoint. Compared with simulations, the theoretical approach bears the advantage of being system-size independent, a crucial asset for investigating effects across scales. Researchers involved: The new international research team between Münster and Vienna will be coordinated by Manuel Friedrich and Ulisse Stefanelli and will benefit from a network of local and international collaborators, including experimental and computational groups. Keywords: Variational methods, Molecular Mechanics, crystallization, local and global geometries, scale effects, discrete to continuum.

Driven by their fascinating electronic and mechanical properties, research on low-dimensional materials (such as graphene) is exponentially growing. New findings are emerging at an always increasing pace, ranging from fundamental concepts to applications. In contrast to the wealth of experimental and numerical evidence currently available, rigorous mathematical results on local and global crystalline geometries are scant and the study of the emergence of different scales within molecular structures is still in its infancy. In this project, we have focused on the variational modeling of molecular geometries within the frame of Molecular Mechanics in order to deepen the mathematical understanding of molecular ge- ometries and to investigate the emergence of scale effects across scales. Ranging from the nano to the macroscale, we have addressed crystallization for molecular compounds, the description of local molecular structures including defects, the occurrence of global geometric characteristics such as nonflatness in 3d, and the passage from discrete-to-continuum theories. The methodology was mainly based on variational techniques for atomistic models, but integrated also methods from graph theory, combinatorics, and discrete differential geometry. The innovative idea of the project consisted in tackling challenging problems in Materials Science from a rigorous mathematical standpoint. Compared with simulations, the theoretical approach bears the advantage of being system-size independent, a crucial asset for investigating effects across scales.