Free Energy Criterion for Thermal Stability of Schwarz Nanocrystals

The discovery of Schwarz nanocrystals (SCs)─characterized by interpenetrating networks of minimal surface grain boundaries (GBs) stabilized by coherent twin boundaries (CTBs)─has pushed the boundaries of nanoscience and nanotechnology to length scales of only a few nanometers. However, the physical mechanisms governing the thermal stability remain unresolved. Here we establish a free energy criterion for thermal stability of SC using large-scale thermodynamic integration, benchmarking SCs against other nanostructures such as Voronoi and Kelvin nanocrystals. Surprisingly, the free energy of minimal surface GBs in SCs is higher than that of conventional GBs in the Voronoi nanocrystal. Therefore, the competition between decreasing the GB volume fraction and increasing the local GB energy imposes a fundamental size limit. Kinetic stabilization of SCs is accommodated by the topological interlock between the CTBs and GBs. The free energy-based criterion for stability of SCs provides a rule for selecting Schwarz-like nanostructures stable at extreme conditions.