Nature Materials | 2021
Arresting grain boundaries with topology
Abstract
It has been long known that the mechanical properties of polycrystalline metals and alloys can be controlled by engineering the nature of their grain boundaries. For example, the Hall–Petch effect, discovered in the 1950s, is a strengthening that arises as grain size decreases, due to the pinning of dislocations at the grain boundaries. It typically leads to a maximal yield strength when grain sizes are in the range of tens of nanometres1. But what nanocrystalline metals gain in strength, they may lose in ductility as dislocations become immobile. Grain-boundary effects can also cause embrittlement due to the segregation of impurities such as hydrogen to those locations2. There’s good reason, then, to seek ways of manipulating grain boundaries to advantage. But it’s challenging. The boundaries are high-energy locations in the (poly)crystal lattice, and will be eliminated by atomic rearrangements if a metal is heated, typically through a gradual coarsening process in which the grains grow larger, somewhat like the coarsening of the network of bubble junctions in a foam as it drains. In some nanocrystalline metals such coarsening can occur even at room temperature. But Li et al. recently found that the network of grain boundaries in nanocrystalline copper can relax to a structure that becomes unusually stable — in fact, metastable — against further rearrangement and coarsening, because the boundaries form a labyrinthine structure with minimal surface area3. Here the analogy with soap films becomes particularly apt. The tendency of soap films to minimize their surface area when constrained to particular geometries has been long known: Leonhard Euler and Joseph Lagrange studied such ‘minimal surfaces’ mathematically in the eighteenth century. One of the striking properties of such surfaces is that they also acquire zero mean curvature at every point: the positive (hill-like) and negative (bowl-like) curvatures balance one another. In 1834, mathematician Heinrich Scherk showed that it is possible to construct such minimal surfaces that extend forever, without boundaries, by a periodic repetition of identical curved units. A few decades later, Hermann Schwarz found that one of the building blocks of periodic minimal surfaces may be the saddle-like shape of a soap film stretched across a tetrahedral framework. This shape can be assembled into three-dimensional structures with cubic symmetry (a P-surface), diamond-lattice symmetry (D-surface), or a more complex form called a gyroid (G-surface)4. All divide three-dimensional space into interpenetrating networks, and they have been reported in biological membranes and copolymer phases. Li et al. found in molecular dynamics simulations of nanocrystalline copper that an initial orderly array of packed polyhedral grains — a ‘Kelvin’ polycrystal resembling the ‘ideal foam’ posited by Lord Kelvin5 — will anneal into the Schwarz D-surface, either thermally or through imposed strain at low temperatures. This accounted for the thermal stability of nanograins against coarsening seen previously by the researchers after plastic deformation of copper and nickel6. Once in a Schwarz-like configuration, the grain-boundary network has ‘nowhere to go’: its surface area and mean curvature are minimal, and so it is topologically stable. At the same time, the initial grain boundaries evolve into a three-dimensional ordered network of ‘twinned’ boundaries that share a symmetry axis, which also confers mechanical stability. The Schwarz surface and twinned-boundary network interlock and reinforce each other’s stability: the grains can’t coarsen, nor the twin boundaries migrate. As a result, the material’s yield strength approaches the ideal value for the single-crystal metal. The researchers now report more detailed atomistic simulations of the formation of such ‘Schwarz crystals’ — grain-boundary networks corresponding to both the Pand D-surfaces — in polycrystalline metals7, revealing exactly how they evolve from Kelvin crystals. They also describe an apparent D-surface morphology seen experimentally by transmission electron microscopy in a region of polycrystalline copper. As well as stabilizing the grain-boundary structure, the formation of a Schwarz crystal can suppress atomic diffusion. Xu et al. report that thermal diffusion of magnesium, which is usually highly mobile in this setting, in aluminium– magnesium alloys is effectively prevented after strain-induced formation of such a microstructure, thereby preventing phase separation and precipitation of the Al3Mg2 phase from the Mg-supersaturated aluminium nanograins8. Here again, then, engineering such a robust grain-boundary structure can boost the stability of the material itself. ❐