The Wulff Construction of Icosahedral Quasicrystals

Quasicrystals are fascinating materials with their long-range aperiodicity and forbidden rotational symmetry forcing solid-state physicists to reconsider the traditional crystallography. This breakthrough originated from investigations into the growth morphology and atomic structure of rapidly solidified alloys. The emergence of pentagonal symmetry in quasicrystals revealed a new kind of order that could be the origin of solid formation, leading to the exploration of theoretical and computational approaches to understand the formation mechanisms. Density functional theory (DFT) is often used to simulate the crystals, but quasicrystals are long-range aperiodic and cannot be simulated under periodic boundary conditions. Here, we present a new technique called the 'sphere-cut method' to construct the thermodynamic Wulff shape of quasicrystals by calculating the surface energies in DFT. We compute the surface energies of finite-sized hemispheres from cutting the very center of scooped nanospheres, and then construct the geometries connecting all local minima of the γ–plot. Using well-trained machine learning interatomic potentials from pre-calculated DFT data, we calculate the surface energies and construct the Wulff shapes of Tsai-type ScZn and YbCd icosahedral quasicrystals, whose structures have previously been resolved with atomistic resolution. The Wulff geometry shows a polyhedron with aperiodic units, which would explain the nature of pentagonal morphologies as it grows.