A Turing instability in the solid state: void lattices in irradiated metals

Turing (or double-diffusive) instabilities describe pattern formation in reaction-diffusion systems, and were proposed in 1952 as a potential mechanism behind pattern formation in nature, such as leopard spots and zebra stripes. Because the mechanism requires the reacting species to have significantly different diffusion rates, only a few liquid phase chemical reaction systems exhibiting the phenomenon have been discovered. In solids the situation is markedly different, since species such as impurities or other defects typically have diffusivities $\propto\!\exp\left( -E/k_{\rm B} T\right)$, where $E$ is the migration barrier and $T$ is the temperature. This often leads to diffusion rates differing by several orders of magnitude. Here we use a simple, minimal model to show that an important class of emergent patterns in solids, namely void superlattices in irradiated metals, could also be explained by the Turing mechanism. Analytical results are confirmed by phase field simulations. The model (Cahn-Hilliard equations for interstitial and vacancy concentrations, coupled by creation and annihilation terms) is generic, and the mechanism could also be responsible for the patterns and structure observed in many solid state systems.