Multiscale Modeling of Vacancy-Cluster Interactions and Solute Clustering Kinetics in Multicomponent Alloys

Abstract

Prediction of solute clustering kinetics in aged multicomponent alloys requires a quantitative understanding of complex vacancy-cluster interactions across multiple scales. Here, we develop an integrated computational framework combining on-lattice kinetic Monte Carlo (KMC) simulations, absorbing Markov chain models, and mesoscale cluster dynamics (CD) to investigate these interactions in Al-Mg-Zn alloys. The Markov chain model yields vacancy escape times from solute clusters and identifies a two-stage behavior of the vacancy-cluster binding energy. These binding energies are used to estimate residual vacancy concentrations in the Al matrix after quenching, which serve as critical inputs to CD simulations to predict long-term cluster evolution kinetics during natural aging. Our results quantitatively demonstrate the significant impact of quench rate on natural aging kinetics. Results provide insights to guide alloy chemistry, quench rates, and aging time at finite temperature to control the evolution of solute clusters and eventual precipitates in aged multicomponent alloys.

Figures (9)

• Figure 1: Time evolution of the average energy per atom $E$ of two typical KMC quenching simulationsxi2024kinetic. The curve above is at a constant temperature of 300 K from the first step, representing an infinitely fast (instantaneous) cooling simulation from a solid solution. The curve below is from a water-quenching simulation. The change in the simulation temperature is denoted by the colors defined in the right color contour bar. Snapshots illustrate the configurations before and after quenching from these simulations. The length of the cubic simulation supercells in the simulation snapshots is 12 nm. Clusters here are defined as solute atoms (Mg and Zn) that are connected as their 1$^{\text{st}}$ nearest neighbors. Al atoms surrounded by 12 clustered solute atoms as their first nearest neighbors are also counted. In the snapshots, only clusters with more than 10 atoms are presented.
• Figure 2: Analyses of local cluster energetics from SA simulations based on surrogate models trained on first-principles data xi2022mechanismxi2024kinetic. (a)-(c) An example solute cluster showing its internal structures: (a) spatial distribution of elemental species, (b) local vacancy binding energy at each site, $E^{\text{site}}_{\text{bind}}$, and (c) local average vacancy migration barrier at each site, $E^{\text{site}}_{\text{mig}}$. (d) Probability density plots obtained using kernel density estimation (KDE) for the local vacancy migration barrier associated with each first-nearest-neighbor bond, $E^{\text{bond}}_{\text{mig}}$, in this example cluster. Three distributions are shown, corresponding to different migration event types defined in the inset schematic: yellow indicates migration events within the cluster interior; light blue denotes vacancy jumps from the cluster surface to its first nearest-neighbor (NN) shell; dark blue represents vacancy jumps from the NN shell to the surrounding matrix. Dashed colored lines indicate average values for each event type, and the black dotted line marks the migration barrier in pure Al, $E_{\text{mig}}^{\text{Al}} = 0.58$ eV. (e) Schematic illustration of a conceptual "vacancy prison" in the potential energy landscape (PEL) of vacancy migration around a solute cluster. The dashed line traces local energy minima, while the solid curved line connects both minima and transition states. Curved double-headed arrows represent vacancy migration.
• Figure 3: Workflow for constructing an absorbing Markov chain model to analyze vacancy-cluster interactions and predict vacancy escape times. (a) Atomic configuration of an example solute cluster, showing atomic arrangements and elemental distributions, with colors indicating Al (blue), Mg (green), and Zn (orange). (b) Graph representation of the cluster, including the cluster atoms and an additional layer of neighboring atoms. Nodes correspond to lattice sites occupied by atoms or vacancies, and edges represent bonds between nearest neighbors. (c) Absorbing Markov chain (bidirected graph) constructed from the graph in (b), where nodes are colored by local vacancy binding energies, $E_{\text{bind}}^{\text{site}}$, and edges are shaded according to the migration barriers between connected sites, $E_{\text{mig}}^{\text{bond}}$. The outermost layer denotes absorbing states, representing sites where vacancies are considered to have escaped from the cluster region.
• Figure 4: KMC simulations for the verification of the absorbing Markov chain model described by Eq. \ref{['eq:exit_time']}. (a) Illustration of the KMC strategy used to determine the vacancy escape event from a solute cluster. Red open circles represent vacancies, while blue, green, and orange circles indicate Al, Mg, and Zn atoms, respectively. White stars mark the first nearest-neighbor shell surrounding the cluster, which is used to define the escape condition. (b) Comparison of vacancy escape times, $t_{\text{esc}}$, obtained from the analytical solution of Eq. \ref{['eq:exit_time']} and from direct KMC simulations for solute clusters generated by SA simulations with varying cluster sizes, $N^{\text{clu}}$.
• Figure 5: Analysis for vacancy trapping ability of solute clusters using the Markov chain model. (a) Vacancy escape time $t_{\text{esc}}$ calculated by Eq. \ref{['eq:exit_time']} at 300 K of solute clusters in terms of cluster size $N^{\text{clu}}$. Multiple cluster configurations with the same $N^{\text{clu}}$ value were analyzed to statistically represent the trapping behavior. (b) A conceptual figure to explain the single-energy-basin approximation for a solute cluster. (c) Effective vacancy binding energy of a cluster $\tilde{E}^{\text{clu}}_{\text{bind}}$ defined in Eq. \ref{['eq:effective_binding_energy']} in terms of cluster size $N^{\text{clu}}$. The dashed line represents a numerical fit.