Variational Modeling of Molecular Geometries

• Applicant: Professor Dr. Manuel Friedrich
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 427980274

• Final Report Year: 2024

Final Report Abstract

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 geometries 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.

Publications

• Crystallization in a One-Dimensional Periodic Landscape. Journal of Statistical Physics, 179(2), 485-501.
  Friedrich, Manuel & Stefanelli, Ulisse
  
• Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy. Archive for Rational Mechanics and Analysis, 240(2), 627-698.
  Friedrich, Manuel; Kreutz, Leonard & Schmidt, Bernd
  
• Lattice ground states for embedded-atom models in 2D and 3D. Letters in Mathematical Physics, 111(4).
  Bétermin, Laurent; Friedrich, Manuel & Stefanelli, Ulisse
  
• Stability of Z2 configurations in 3D. Nonlinearity 34, 12:8392–8413.
  Bétermin, Laurent; Friedrich, Manuel & Stefanelli, Ulisse
  
• From atomistic systems to linearized continuum models for elastic materials with voids. Nonlinearity, 36(1), 679-733.
  Friedrich, Manuel; Kreutz, Leonard & Zemas, Konstantinos
  
• Tilings with Nonflat Squares: A Characterization. Milan Journal of Mathematics, 90(1), 131-175.
  Friedrich, Manuel; Seitz, Manuel & Stefanelli, Ulisse
  
• A Proof of Finite Crystallization via Stratification. Journal of Statistical Physics, 190(12).
  Friedrich, Manuel & Kreutz, Leonard
  
• The double-bubble problem on the square lattice. Interfaces and Free Boundaries, Mathematical Analysis, Computation and Applications, 26(1), 79-134.
  Friedrich, Manuel; Górny, Wojciech & Stefanelli, Ulisse
  
• Nonlocal-to-local limit in linearized viscoelasticity. Communications in Applied and Industrial Mathematics, 15(1), 1-26.
  Friedrich, Manuel; Seitz, Manuel & Stefanelli, Ulisse
  
• A characterization of ℓ1 double bubbles with general interface interaction. Advances in Calculus of Variations, 18(3), 609-637.
  Friedrich, Manuel; Górny, Wojciech & Stefanelli, Ulisse
  
• Discrete-to-continuum linearization in atomistic dynamics. Discrete and Continuous Dynamical Systems, 45(3), 847-874.
  Friedrich, Manuel; Seitz, Manuel & Stefanelli, Ulisse
  
• The ℓ1 Double-Bubble Problem in Three Dimensions. The Journal of Geometric Analysis, 35(10).
  Friedrich, Manuel; Górny, Wojciech & Stefanelli, Ulisse