Solute-defect interactions in Al-Mg alloys from diffusive variational Gaussian calculations

Resolving atomic-scale defect topologies and energetics with accurate atomistic interaction models provides access to the nonlinear phenomena inherent at atomic length and time scales. Coarse graining the dynamics of such simulations to look at the migration of, e.g., solute atoms, while retaining the rich atomic-scale detail required to properly describe defects, is a particular challenge. In this paper, we present an adaptation of the recently developed “diffusive molecular dynamics” model to describe the energetics and kinetics of binary alloys on diffusive time scales. The potential of the technique is illustrated by applying it to the classic problems of solute segregation to a planar boundary (stacking fault) and edge dislocation in the Al-Mg system. Our approach provides fully dynamical solutions in situations with an evolving energy landscape in a computationally efficient way, where atomistic kinetic Monte Carlo simulations are difficult or impractical to perform.

Figure 1

Schematics of the main approximations used in the DMD approach for binary alloys. Solvent and solute atoms (shown in different colors) undergoing thermal vibrations around the same position ${\mathbf{X}}_i$ in MD (left) are represented in DMD by superposition of Gaussians (right) for each atomic species at this atomic site.

Figure 2

(a) Enthalpy and (b) free energy of mixing versus Mg solute concentration $c_0$ given by VG calculations for Al-Mg solid solution (with fcc Mg) at different temperatures. Here, MS data correspond to properties at $T=0$ and the calphad data at $T=800$ K.

Figure 3

Temperature dependence of (a) lattice constant and (b) elastic constants of Al-Mg alloy for different compositions obtained by DMD method.

Figure 4

(a) Plot of solute concentration normalized by the initial concentration $c_0$ versus the distance to SF, $d_{\text{SF}}$, at $T=600$ K. (b) Atomistic representation of solute distribution near the SF defect depicted by highest solute concentration.

Figure 5

Time evolution of (a) the solute concentration of atoms located in the SF planes and (b) free-energy difference between the current and initial states per SF area for $c_0=1\%$, 5%, and 10% at $T=600$ K from NVT DMD calculations.

Figure 6

Contour plots of solute concentration near the split edge dislocation from DMD calculation (left) and from the hydrostatic pressure field existing prior to segregation (right) for (a) $c_0=1\%$ and (b) $5\%$ at $T=600$ K. (c) Contour plots of the difference between these results and (d) magnified snapshots of the concentration field illustrating the details of the solute distribution for 1% Mg (left) and 5% Mg (right).

Figure 7

Contour plots of hydrostatic pressure fields measured in GPa near the split edge dislocation in (a) initially disordered Al-Mg solid solution, DMD relaxed structures in the (b) NVT and (c) $\mu \mathrm{VT}$ ensembles for $c_0=5\%$ at $T=600$ K.

Figure 8

Time evolution of the solute concentration in the vicinity of the dislocation from (a) NVT and (b) $\mu \mathrm{VT}$ DMD calculations for $c_0=5\%$ at $T=600$ K. (c) Comparison of the solute segregation kinetics in the tensile part of the core region [denoted by white circle with radius $R_c$ in (d)] between different ensembles and continuum model predictions. (d) Atomic representation of solute distribution near the split edge dislocation core. The circle of radius $R_c=13\text{Å}$ ($\sim \mathrm{distance}$ between partials) indicates the core region.

Figure 9

(a) Two snapshots from DMD simulations (times indicated above the figures) for solute segregation near the split edge dislocation for $c_0=10\%$ at $T=600$ K. The unit cell of a fully ordered precipitate obtained from cooling this system to a lower temperature is shown at the bottom right. (b) Time evolution of the normalized excess solute concentration (blue symbols) in the region indicated by the white circle in (a) and the free-energy difference (green symbols) between the current and initial states.