2025/09/06 by Manuel Friedrich, Friedrich, Manuel, Leonard Kreutz +3
Earth and Planetary Sciences · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #nanoparticles nucleation surface interactions
paper · pdf · doi:10.48550/arxiv.2509.05642
openalex publication_date 2025/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We address finite crystallization in two dimensions in the presence of a flat crystalline substrate. Particles interact through short-range two- and three-body potentials favoring local square-lattice arrangements. An additional interaction term of relative strength β>0 couples the particles and the substrate. Our first main result proves crystallization for all β>0, corresponding to the onset of discrete Winterbottom configurations. The proof relies on a stratification technique from [31], characterizing the topology of the bond graph of minimizing configurations. Our second main result concerns fluctuations estimates for β∈ (0,1). We obtain bounds on the distance between distinct minimizers with the same number N of particles, showing a sharp scaling law N3/4 when β is rational, and N1/3 when β is irrational and algebraic. This reveals a genuine substrate-driven effect on fluctuation laws. As a corollary, we derive a discrete-to-continuum convergence of minimizers towards the Winterbottom equilibrium shape in the large-particle limit.