Abstract
The crystal surface free energy is one of the most fundamental aspects influencing the morphology and performance of nanoscale materials across a diverse range of applications such as catalysis, drug delivery, and semiconductor technology. Despite this importance, its direct measurement, particularly for arbitrary high-index crystal facets, remains challenging due to the vast number of potential orientations and computational limitations. Here, we present a universal empirical model for estimating the surface energy of arbitrary Miller planes in high-symmetry crystals, including diamond cubic (DC), face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP) structures. The model uses stereographic projection of surface energy anisotropy and partitioning of the given plane into triangular zones defined by (low index) singular planes, enabling surface energy estimation with practical accuracy within 5% error in the whole orientation space. This approach offers significant computational efficiency while retaining sufficient precision for applications in crystal engineering. We demonstrate the model's extensive applicability even for oxide nanocrystals: the shape evolution of NiO nanocrystals under varying oxygen vacancy conditions is computed, achieving excellent agreement with experiments. The proposed framework provides a cost-effective, accessible starting point for surface energy calculations and hence serves as a bridge to more intensive computational methods, such as density functional theory (DFT) or molecular dynamics, supporting broader adoption across materials design workflows.