Short-range order and phase stability of CrCoNi explored with machine learning potentials

We present an analysis of temperature-dependent atomic short-range ordering and phase stability of the face-centered cubic CrCoNi medium-entropy alloy employing a combination of ab initio calculations and on-lattice machine learning interatomic potentials. Temperature-dependent properties are studied with canonical Monte Carlo simulations. At around 975 K a phase transition into an ordered $\mathrm{Cr}{(\mathrm{Ni},\mathrm{Co})}_2$ phase (${\mathrm{MoPt}}_2$-type) is found. This hitherto not reported state has an ordering energy twice as large than the ordered structures previously suggested. We show that magnetism is not responsible for the observed chemical ordering.

Figure 1

The temperature dependence of specific-heat capacity $C_V$ based on Monte Carlo simulation for an equiatomic FCC CrCoNi alloy with supercell size (a) $3\times 3\times 3$, i.e., 108 atoms (brown and red correspond to MC simulation results based on initially trained and retrained ensemble of 10 LRP potentials, respectively) and (b) for $6\times 6\times 6$ and $12\times 12\times 12$, i.e., 864 and 6912 atoms, respectively. The solid lines represent the averaged heat capacity over an ensemble of 10 LRPs, and the shaded area corresponds to the standard deviation of the LRPs ensemble.

Figure 2

In (a) the calculated temperature dependence of the mean internal energies from MC simulations for an equiatomic FCC CrCoNi alloy has been compared for different supercell sizes: $3\times 3\times 3$ (108 atoms), $6\times 6\times 6$ (864 atoms), and $12\times 12\times 12$ (6912 atoms) in the case of a cubic supercell. In the inset, the temperature-dependent internal energies from MC simulation on an orthorhombic supercell have been compared to that of a cubic one for $6\times 6\times 6$ supercell size. The structures containing 864 atoms (in the case of a cubic supercell) correspond to the (b) lower energy Co and Ni separated ordered structure at 50 K (referred to as ${\mathrm{CrCo}}_2+{\mathrm{CrNi}}_2$), (c) ordered layered structure at 510 K (referred to as $\mathrm{Cr}{(\mathrm{Co},\mathrm{Ni})}_2$), and (d) random solid solution (referred to as random) at very high temperature (left to right), respectively. The structures in (e) and (f) are the snapshots taken at 50 K and 510 K from MC simulation on the orthorhombic supercell with supercell size $6\times 6\times 6$ (1296 atoms) and are equivalent to ${\mathrm{CrCo}}_2+{\mathrm{CrNi}}_2$ and $\mathrm{Cr}{(\mathrm{Co},\mathrm{Ni})}_2$ structures, respectively. Energies in (a) are referenced to the energy of the random solid solution.

Figure 3

Temperature-dependent Warren-Cowley SRO parameters for different atom pairs in the first coordination shell in an equiatomic CrCoNi alloy. The parameters from the MC simulations are averaged over an ensemble of 10 LRPs, and the light shaded areas show the standard deviation (uncertainty level) of the utilized LRPs. The insets show the qualitatively different nature of the low- (continuous) and high-temperature (discontinuous) transitions.

Figure 4

DFT calculated energy differences as compared to a fully random state for the two ordered states, ${\mathrm{CrCo}}_2+{\mathrm{CrNi}}_2$ and $\mathrm{Cr}{(\mathrm{Co},\mathrm{Ni})}_2$, found in the present work [see also Figs. 2 and 2]. The calculations include ionic relaxations (red) as well as volume and cell shape relaxations, i.e., full relaxation (green). The impact of electronic excitations (light red and light green) is shown as well. The DFT calculated energy differences found in previous literature results (blue shades) by Pei et al. [47], Tamm et al. [43], and Walsh et al. [49] between the random state and their proposed ordered state also have been added and compared to our thermodynamic equilibrium computed ordered configurations.

Figure 5

The arrangement of constituent atoms in (a) the here obtained ordered structure (in the case of cubic supercell) of CrCoNi at a temperature of 510 K compared with (c) the ground state structure as suggested by Pei et al. [47]. In the case of (a), for clear visualization of our obtained $\mathrm{Cr}{(\mathrm{Co},\mathrm{Ni})}_2$ structure, only a certain part of the 864-atom cell with a smaller boundary is shown. Another orthorhombic primitive cell (also see Appendix pp3) as considered in the case of orthorhombic supercell for $\mathrm{Cr}{(\mathrm{Co},\mathrm{Ni})}_2$ structure also has been added in (b) for better visualization. The half-colored spheres with yellow and red represent the 50% occupancy probability of Co and Ni atoms for those sites.

Figure 6

The alignment of local magnetic moments in (a) our here proposed ordered $\mathrm{Cr}{(\mathrm{Co},\mathrm{Ni})}_2$ structure of CrCoNi compared with (b) the one proposed in [47], both calculated with spin-polarized DFT for a 108-atom cell. Blue, yellow, and red spheres represent Cr, Co, and Ni atoms, respectively. Visualization is performed with the ovito software [79].

Figure 7

The calculated temperature dependence of mean internal energies obtained from magnetic and nonmagnetic LRP-based MC simulations for an equiatomic CrCoNi alloy with supercell size $12\times 12\times 12$. The energy of high-temperature magnetic random solid solution has been included as reference (gray line).

Figure 8

In (a) the calculated temperature dependence of the mean internal energies from heating and cooling MC simulations for an equiatomic FCC CrCoNi alloy are compared for different supercell sizes: $3\times 3\times 3$ (108 atoms), $6\times 6\times 6$ (864 atoms), and $12\times 12\times 12$ (6912 atoms) with $2\times {10}^5$ number of MC steps. In the inset, the observed hysteresis in the case of a $6\times 6\times 6$ (864 atoms) supercell is shown near the high-temperature phase transition. In (b) the convergence of the internal energy with the number of MC steps is shown. A temperature of 945 K, which is in the hysteresis region [marked by a square in the inset of (a)], is used in the simulations. The calculations are performed in a $6\times 6\times 6$ supercell and taken for each of the fitted 10 LRPs. Energies are referenced to the energy of the random solid solution.

Figure 9

The moment distribution for the three different atom types Cr, Co, and Ni in the three different structures: [(a)–(c)] random, [(d)–(f)] ordered $\mathrm{Cr}{(\mathrm{Co},\mathrm{Ni})}_2$, and [(g)–(i)] ordered ${\mathrm{CrCo}}_2$ $+$ ${\mathrm{CrNi}}_2$ structures.