Snowflake growth provides a fascinating example of spontaneous pattern formation in nature. Attempts to understand this phenomenon have led to important insights in non-equilibrium dynamics observed in various active scientific fields, ranging from pattern formation in physical and chemical systems, to self-assembly problems in biology. Yet, very few models currently succeed in reproducing the diversity of snowflake forms in three dimensions, and the link between model parameters and thermodynamic quantities is not established. Here, we report a modified phase field model that describes the subtlety of the ice vapour phase transition, through anisotropic water molecules attachment and condensation, surface diffusion, and strong anisotropic surface tension, that guarantee the anisotropy, faceting and dendritic growth of snowflakes. We demonstrate that this model reproduces the growth dynamics of the most challenging morphologies of snowflakes from the Nakaya diagram. We find that the growth dynamics of snow crystals matches the selection theory, consistently with previous experimental observations.

This article is devoted to the study of the tip shape of dendritic crystals grown from a supercooled liquid. The recently developed theory (Alexandrov & Galenko 2020 Phil. Trans. R. Soc. A 378 , 20190243. ( doi:10.1098/rsta.2019.0243 )), which defines the shape function of dendrites, was tested against computational simulations and experimental data. For a detailed comparison, we performed calculations using two computational methods (phase-field and enthalpy-based methods), and also made a comparison with experimental data from various research groups. As a result, it is shown that the recently found shape function describes the tip region of dendritic crystals (at the crystal vertex and some distance from it) well. This article is part of the theme issue ‘Transport phenomena in complex systems (part 1)’.

Microstructure patterning using the ion beam mixing process results from the competition between thermal diffusion and ballistic disordering induced by impinging ions. Although microstructure patterning under irradiation is now qualitatively understood, so far, no study could quantitatively estimate irradiation conditions leading to patterning. In this work, a new multiscale approach based on phase field was developed to simulate the microstructure evolution, and the occurrence of patterning due to ion irradiation in a silver-copper alloy, from atomic to microstructural scale. For that purpose, an efficient numerical scheme was developed to simulate the microstructure dynamics, within the framework of phase field. Equilibrium parameters of AgCu were computed using a mixed Monte Carlo-molecular dynamics approach. Ballistic effects induced by krypton ion irradiation, and point defect recreation leading to irradiation enhanced diffusion, were estimated using the binary collision approximation framework. As a result, we predicted the range of temperatures and irradiation fluxes leading to the formation of patterned microstructures in the AgCu alloy under krypton ion irradiation. Results were summarized in a pseudo phase diagram in the temperature-flux plane. Our model was in good agreement with a previous diffraction experimental study.