Improved atomistic Monte Carlo models based on ab-initio-trained neural networks: Application to FeCu and FeCr alloys

We significantly improve the physical models underlying atomistic Monte Carlo (MC) simulations, through the use of ab initio fitted high-dimensional neural network potentials (NNPs). In this way, we can incorporate energetics derived from density functional theory (DFT) in MC, and avoid using empirical potentials that are very challenging to design for complex alloys. We take significant steps forward from a recent work where artificial neural networks (ANNs), exclusively trained on DFT vacancy migration energies, were used to perform kinetic MC simulations of Cu precipitation in Fe. Here, a more extensive transfer of knowledge from DFT to our cohesive model is achieved via the fitting of NNPs, aimed at accurately mimicking the most important aspects of the ab initio predictions. Rigid-lattice potentials are designed to monitor the evolution during the simulation of the system energy, thus taking care of the thermodynamic aspects of the model. In addition, other ANNs are designed to evaluate the activation energies associated with the MC events (migration towards first-nearest-neighbor positions of single point defects), thereby providing an accurate kinetic modeling. Because our methodology inherently requires the calculation of a substantial amount of reference data, we design as well lattice-free potentials, aimed at replacing the very costly DFT method with an approximate, yet accurate and considerably more computationally efficient, potential. The binary FeCu and FeCr alloys are taken as sample applications considering the extensive literature covering these systems.

Figure 1

Radial shaping factors used in this work, i.e., bcc Fe-based alloys with $a_0=2.831\AA$. Dashed curves show the two terms in Eq. (6), and vertical arrows the distances associated with shells of nearest neighbors in bcc. Top: Cutting interaction between the 3nn and the 4nn distances: $R_{\mathrm{c}}=1.75a_0=4.954\AA$, $\alpha =18$, and $\beta =15$. Bottom: Cutting interaction between the 2nn and the 3nn distances: $R_{\mathrm{c}}=1.6a_0=4.530\AA$, $\alpha =10$, and $\beta =6.5$.

Figure 2

Atomic configurations extracted from DFT-NEB calculations in this work. Each cubic box with 8 atoms (gray circles) pictorially depicts a DFT calculation supercell ($5a_0\times 5a_0\times 5a_0$) for the case of vacancy migration (dashed circle).

Figure 3

Comparison of the (left) end-state energy differences and (right) migration energies found by NEB using DFT or the EAM potentials.

Figure 4

Accuracy of prediction by the rigid-lattice NNPs of the supercell formation energy [defined in Eq. (15)]. $\delta$ is the mean absolute error of prediction, reported in Table 2.

Figure 5

Phase diagrams predicted with the rigid-lattice NNPs obtained in this work. Bottom: FeCu case. The dashed line shows a calculation with the ATAT method for the bcc/bcc coherent phase in Ref. [53]. The prediction with the EAM potential is taken from Ref. [58]. Top: FeCr case. The dashed line is a review of experimental evidence by Bonny et al. in Ref. [56].

Figure 6

ANN quality of prediction for the activation energy associated with vacancy migration in FeCr alloys. Only points from the reference set are shown. The residual mean absolute error of prediction is 48.3 meV ($R^2=0.90$) for the jumping Fe case, and 32.9 meV ($R^2=0.86$) for the jumping Cr case.

Figure 7

Prediction with AKMC of a thermal annealing experiment on an Fe 20% Cr alloy at $500{}^{\circ }\mathrm{C}$. Both results obtained in this work (labeled AKMC-DFT) and in Ref. [24] (labeled AKMC-SEAM) are shown. APT experimental data are taken from Novy [61, 62], Jacquet [63], and Bley [64].

Figure 8

Accuracy of prediction achieved by the lattice-free NNPs designed for the FeCu and FeCr systems. Reference values are taken from the databases of configurations described in Sec. 2, and formation energies are defined in Eq. (15). The figures for the latter are split into the vacancy migration events and the SIA migration events. $\delta$ is the mean absolute error of prediction, reported in Table 2.

Figure 9

Comparison of the end-state energy differences (left) and NEB migration energies (right), found using DFT or the NNPs. $\delta$ is the mean absolute error of prediction, reported in Table 2.

Figure 10

ANN quality of prediction for the migration energy associated with single-SIA migration in FeCr alloys. Target values were obtained with the NNP. The residual mean absolute error of prediction is 39.1 meV ($R^2=0.89$).