Generalization of the mixed-space cluster expansion method for arbitrary lattices 

Editorial Summary: Mixed-Space Cluster Expansion: A New Perspective Across Real and Reciprocal Space

In modern materials science, the occupancy distribution of atoms on the parent lattice is crucial, which not only determines the thermodynamic properties of alloys, but also influences the transition process from disorder to order, which in turn relates to the phase stability and other properties of materials. The cluster expansion (CE) method combined with density functional theory (DFT) is able to accurately describe these phenomena, and although an infinite number of clusters is theoretically required, in practice, simple clusters are sufficient for high-precision energy parameterization. For the accurate modeling of long-range strain interactions, Laks et al. proposed the mixed-space cluster expansion (MSCE) method, which makes it possible to characterize both short- and long-range interactions in complex alloy systems by taking into account both real and inverse space. A team lead by Dr. Kang Wang and Dr. Bi-Cheng Zhou from Department of Materials Science and Engineering, University of Virginia, USA, proposed a mixed-space cluster expansion (MSCE) method, which is generalized to systems with multiple sublattices by formulating compatible reciprocal space interactions and combined with a crystal-symmetry-agnostic algorithm for the calculation of constituent strain energy. The MSCE method owes the high accuracy, compared with r-space CE, to three aspects. (1) The long-ranged limit of the constituent strain energy (CSE) due to size-mismatch are explicitly incorporated using the k-space formalism. (2) The attenuation of the long-ranged interactions accommodates for medium-ranged structures, which is the case for the majority of the structures in the training set. (3) The regularization of r-space effective cluster interactions (ECIs) allow us to include much larger number of clusters, which enhances the fitting capability. This generalized approach is demonstrated in a hypothetical HCP system and Mg-Zn alloys. In Mg-Zn system, the preferred orderings of Zn atoms are identified with Zn-rods and local C14 MgZn2 arranged along [0001]. Noteworthy, C14 MgZn2 rods here agree with the peak-age precipitates β₁’ rods along [0001]_α with the orientation relationship in Mg-Zn system. In the current MSCE and MC, all the fully relaxed HCP orderings in DFT are included in the training set, which enables the energy predicted by MSCE to be fully incorporate the relaxations from HCP lattice sites. In the calculations with only moderately relaxed structures (with d < 0.1), many overly relaxed structures are excluded in r-space CE. Such calculation reflects the energies of orderings very close to HCP lattice sites and coherent orderings resembling GP zones were revealed. However, such potential GP zones were not found in the current MSCE and MC simulations. In Mg-Zn system, experimental evidences of GP zones were reported, which, however, is deemed insufficient. Comparison of the calculations using r-space CE and MSCE offers a possible explanation. When Zn atoms start to aggregate in local regions of Mg matrix, lattice distortion will be involved due to the lattice mismatch between Mg and Zn atoms. When such distortions are small and the atoms sit close to the ideal HCP lattice sites, local arrangements of Zn atoms resembling GP zones can be found. When the local Zn concentration gets high and local lattice distortion becomes severe, the local Zn-rich regions prefer to transform to more stable structures, such as C14 or C15 MgZn2. Considering the lattice becomes less rigid when the temperature is elevated, the GP zones are more likely to be found at low aging temperatures in the samples with low overall Zn concentrations. This may explain why concrete evidence of GP zones in Mg-Zn is elusive even with modern microscopy.