Phase stability of ternary fcc and bcc Fe-Cr-Ni alloys

The phase stability of fcc and bcc magnetic binary Fe-Cr, Fe-Ni, and Cr-Ni alloys, and ternary Fe-Cr-Ni alloys is investigated using a combination of density functional theory (DFT), cluster expansion (CE), and magnetic cluster expansion (MCE) approaches. Energies, magnetic moments, and volumes of more than 500 alloy structures have been evaluated using DFT, and the predicted most stable configurations are compared with experimental observations. Deviations from the Vegard law in fcc Fe-Cr-Ni alloys, resulting from the nonlinear variation of atomic magnetic moments as functions of alloy composition, are observed. The accuracy of the CE model is assessed against the DFT data, where for ternary Fe-Cr-Ni alloys the cross-validation error is found to be less than 12 meV/atom. A set of cluster interaction parameters is defined for each alloy, where it is used for predicting new ordered alloy structures. The fcc ${\mathrm{Fe}}_2\mathrm{CrNi}$ phase with ${\mathrm{Cu}}_2\mathrm{NiZn}$-like crystal structure is predicted to be the global ground state of ternary Fe-Cr-Ni alloys, with the lowest chemical ordering temperature of 650 K. DFT-based Monte Carlo (MC) simulations are applied to the investigation of order-disorder transitions in Fe-Cr-Ni alloys. The enthalpies of formation of ternary alloys predicted by MC simulations at 1600 K, combined with magnetic correction derived from MCE, are in excellent agreement with experimental values measured at 1565 K. The relative stability of fcc and bcc phases is assessed by comparing the free energies of alloy formation. The evaluation of the free energies involved the application of a dedicated algorithm for computing the configurational entropies of the alloys. Chemical order is analyzed, as a function of temperature and composition, in terms of the Warren-Cowley short-range order (SRO) parameters and effective chemical pairwise interactions. In addition to compositions close to binary intermetallic phases ${\mathrm{CrNi}}_2$, FeNi, ${\mathrm{FeNi}}_3$, and ${\mathrm{FeNi}}_8$, pronounced chemical order is found in fcc alloys near the center of the ternary alloy composition triangle. The calculated SRO parameters compare favorably with experimental data on binary and ternary alloys. Finite temperature magnetic properties of fcc Fe-Cr-Ni alloys are investigated using an MCE Hamiltonian parameterized using a DFT database of energies and magnetic moments computed for a large number of alloy configurations. MCE simulations show that the ordered ternary ${\mathrm{Fe}}_2\mathrm{CrNi}$ alloy phase remains magnetic up to 850–900 K due to the strong antiferromagnetic coupling between (Fe,Ni) and Cr atoms in the ternary Fe-Cr-Ni matrix.

Figure 1

Enthalpies of mixing [(a) and (b)], volumes per atom [(c) and (d)], and magnetic moments [(e) and (f)] of Fe-Ni structures on fcc [(a), (c), and (e)] and bcc [(b), (d), and (f)] lattices, calculated using DFT. Experimental data are taken from Refs. [60, 65, 66]. GS refers to the ground state on fcc [(a) and (c)] or bcc [(b) and (d)] crystal lattices; ST is the most stable structure and magnetic configuration for the corresponding alloy composition.

Figure 2

Enthalpies of mixing [(a) and (b)], volumes per atom [(c) and (d)], and magnetic moments [(e) and (f)] predicted by DFT for fcc [(a), (c), and (e)] and bcc [(b), (d), and (f)] Fe-Cr alloys. Experimental data are taken from Refs. [78, 79, 80]. GS: ground states of alloys on fcc [(a) and (c)] or bcc [(b) and (d)] crystal lattices, ST: the most stable structure for a given composition.

Figure 3

Enthalpies of mixing [(a) and (b)], volumes per atom [(c) and (d)], and magnetic moments [(e) and (f)] calculated using DFT for Cr-Ni alloys on fcc [(a), (c), and (e)] and bcc [(b), (d), and (f)] lattices. Experimental data are taken from Ref. [65]. GS: ground states of alloys on fcc [(a) and (c)] or bcc [(b) and (d)] lattices, ST: the most stable structure found for a given alloy composition.

Figure 4

Effective cluster interactions (ECIs) derived using CE method for fcc Fe-Ni (a), bcc Fe-Ni (b), fcc Fe-Cr (c), bcc Fe-Cr (d), fcc Cr-Ni (e), and bcc Cr-Ni (f) alloys.

Figure 5

Enthalpies of formation of Fe-Ni (a), Fe-Cr (b), and Cr-Ni (c) binary structures.

Figure 6

Enthalpies of formation predicted by DFT (filled circles) and CE (open circles) for ternary Fe-Cr-Ni alloys at 0 K. Only the most stable structures for each composition are shown. Blue and red circles show computed formation enthalpies of fcc and bcc Fe-Cr-Ni ternary alloys. Blue and red surfaces show convex hulls for fcc and bcc crystal structures, respectively. Black solid line corresponds to the intersection between fcc and bcc convex hulls. Cross-validation errors between DFT and CE are 10.2 and 11.2 meV/atom for fcc and bcc ternary alloys, respectively.

Figure 7

Effective cluster interactions obtained using the CE method for fcc (a) and bcc (b) Fe-Cr-Ni ternary alloys.

Figure 8

Volumes (in Å${}^3$) of stable fcc [(a)and (c)] and bcc [(b) and (d)] ordered structures predicted by DFT calculations at 0 K for various alloy compositions. Filled and open circles in (a) and (b) correspond to DFT data above and below the interpolated values, represented by the respective surfaces. (c) and (d) are the orthogonal projections of (a) and (b).

Figure 9

Magnetic moment per atom (in ${\mu }_B$) in the most stable fcc [(a)and (c)] and bcc [(b) and (d)] ordered alloy structures predicted by DFT calculations at 0 K for each alloy composition. Filled and open circles in (a) and (b) correspond to DFT data above and below the interpolated values represented by the respective surfaces. (c) and (d) are the orthogonal projections of (a) and (b). (e) The average atomic magnetic moment of fcc and bcc structures is discontinuous across the fcc-bcc phase transition line, shown in the figure as solid black line.

Figure 10

Magnetic moments (in ${\mu }_B$) of Fe [(a) and (b)], Cr [(c) and (d)], and Ni [(e) and (f)] on fcc [(a), (c), and (e)] and bcc [(b), (d), and (f)] lattices.

Figure 11

Enthalpies of formation (in eV/atom) of fcc (a) and bcc (b) alloys calculated using MC simulations at 300 K.

Figure 12

Difference between enthalpies of formation [(a) and (c)] and free energies of formation [(b and (d)] of fcc and bcc alloys, in eV/atom units, predicted using MC simulations for 300 K [(a) and (b)], 600 K [(c) and (d)], and 900 K [(e) and (f)]. Black solid lines separate the Ni-rich region of stability of fcc alloys and the region of stability of bcc alloys predicted using the enthalpy and free-energy criteria.

Figure 13

Difference between enthalpies of formation [(a) and (c)] and free energies of formation [(b) and (d)] of fcc and bcc alloys, in eV/atom units, calculated using MC simulations at 1600 K without [(a) and (b)] and with magnetic correction applied to the formation enthalpies (c) and free energies of formation (d). Black solid lines separate the Ni-rich region of stability of fcc alloys and the region of stability of bcc alloys predicted using the enthalpy and free energy criteria.

Figure 14

(a) Enthalpies of formation (in eV/atom) for the most stable crystal structures of Fe-Cr-Ni alloys computed using MC simulations at 1600 K with magnetic correction applied, compared to experimental data (b) from Refs. [91, 92].

Figure 15

Enthalpies of mixing and short-range order parameters (SRO) computed as functions of temperature for Fe-Cr, Cr-Ni and Fe-Ni atomic pairs in the first coordination shell of fcc ${\mathrm{Fe}}_{50}{\mathrm{Cr}}_{25}{\mathrm{Ni}}_{25}$ alloy.

Figure 16

Order-disorder temperatures of fcc (a) and bcc (b) Fe-Cr-Ni alloys computed using Monte Carlo simulations. Order-disorder temperatures for pure elements are assumed to be 0 K.

Figure 17

(a) Short-range order parameters as functions of temperature, calculated for Fe-Cr, Cr-Ni, and Fe-Ni pairs occupying two coordination shells in ${\mathrm{Fe}}_{0.56}{\mathrm{Cr}}_{0.21}{\mathrm{Ni}}_{0.23}$ alloy, compared with experimental values from Ref. [24]; (b) 1NN SRO between Ni and average (Fe,Cr) atoms in ${\mathrm{Fe}}_{42.5}{\mathrm{Cr}}_{7.5}{\mathrm{Ni}}_{50}$, ${\mathrm{Fe}}_{38}{\mathrm{Cr}}_{14}{\mathrm{Ni}}_{48}$, and ${\mathrm{Fe}}_{34}{\mathrm{Cr}}_{20}{\mathrm{Ni}}_{46}$ calculated at 600, 900, and 1300 K, and compared with experimental data taken from Ref. [25] and presented as a function of Cr content; (Fe,Cr)-Ni indicates average SRO between Ni and average (Fe,Cr) atoms obtained from Eq. (24).

Figure 18

Effective interactions between different pairs of atoms: Fe-Cr [(a) and (b)], Fe-Ni [(c) and (d)], and Cr-Ni [(e) and (f)] on fcc [(a), (c), and (e)] and bcc [(b), (d), and (f)] lattices in ternary Fe-Cr-Ni and binary alloys.

Figure 19

(a) Configurational specific heat $C_{\mathrm{conf}}$, configurational entropy $S_{\mathrm{conf}}$, (b) the enthalpy of formation $H_{\mathrm{form}}$, the product of temperature and configurational entropy $-T{S}_{\mathrm{conf}}$ and the free energy of formation $F_{\mathrm{form}}$ of fcc and bcc ${\mathrm{Fe}}_2\mathrm{CrNi}$ alloys. The black dotted line is the entropy of ideal random solid solution of ${\mathrm{Fe}}_2\mathrm{CrNi}$ equal to $1.04k_B$.

Figure 20

Enthalpies of formation (in eV/atom) [(a) and (b)], the product of temperature and configurational entropies $T{S}_{\mathrm{conf}}$ (in eV/atom units) [(c) and (d)] and free energies of formation (in eV/atom units) [(e) and (f)] at 900 K computed using MC simulations for Fe-Cr-Ni alloys on fcc [(a), (c), and (e)] and bcc [(b), (d), and (f)] lattices.

Figure 21

Total magnetic moment per atom in (${\mathrm{Fe}}_{0.5}{\mathrm{Ni}}_{0.5}$)${}_{1-x}{\mathrm{Cr}}_x$ alloy as a function of chromium content $x$, predicted by MCE. DFT results (Sec. 3e) are shown by red circles.

Figure 22

Temperature dependence of the total magnetic moment of ordered ${\mathrm{Fe}}_2\mathrm{CrNi}$ alloy and magnetic moments of atoms forming the alloy.