Configurational thermodynamics of Fe-Ni alloys at
Earth’s core conditions:
Supplementary material
M. Ekholm, A. S. Mikhaylushkin, S. I. Simak, B. Johansson, I. A. Abrikosov
1. Equations of state
In Tables 1, 2, and 3 we account for volumes and compositions used in total energy calculations. Figure 1 shows our equations of state (EOS) calculated within the EMTO-CPA and PAW-VCA methods. In Figure 2 we compare the EOS for Fe and FeNi alloys with experimental data.
2. Effective Cluster Interactions
We have calculated effective cluster interactions in the fcc- and hcp-based Fe-Ni system, using the (screened) generalised perturbation method (GPM), (Ruban and Abrikosov, 2008). In Fig. 3(a) we show GPM pair interactions in fcc-Fe0
.875Ni0.125. In the low temperature limit, we find several long-ranged
interactions. These are found as far away as in the 10th coordination shell. However, at elevated temperature, interactions beyond the 4th coordination shell are diminished. Comparing with the high pressure interactions, we find the pair-interactions to be increased in strength. Although interactions beyond the second coordination shell are all weakened with temperature, several long-ranged interactions of non-negligible magnitude remain even at 5000 K.
In Fig. 3(b) we may observe how a significant increase in Ni concentration to 50 % greatly enhances several of the interactions in the first few shells but decreases almost all of the long-ranged interactions. Electronic temperature and pressure have the same effects also in this case. Fig. 4(a) shows effective pair interactions in hcp-Fe Ni . Also here we find non-negligible
long-150 200 250 300 350 6.5 7 7.5 8 8.5 9 P [ GPa ] V [ Å 3 / atom ] PAW−VCA EMTO−CPA Fe Fe 0.80Ni0.20
(a) fcc-Fe and fcc-Fe0.8Ni0.2.
150 200 250 300 350 6.5 7 7.5 8 8.5 9 PAW−VCA EMTO−CPA P [ GPa ] V [ Å 3 / atom ] Fe Fe 0.80Ni0.20 Fe 0.75Ni0.25
(b) hcp-Fe, hcp-Fe0.80Ni0.20, and hcp-Fe0.75Ni0.25.
Figure 1: Calculated EOS for fcc- (a) and hcp- (b) Fe and FeNi alloys using the EMTO-CPA and PAW-VCA methods, and electronic temperature of 5000 K
100 150 200 250 300 350 6.5 7 7.5 8 8.5 9 P [ GPa ] V [ Å 3 / atom ] EMTO Fe
Exp. Fe (Mao et al., 1990)
(a) Calculated EOS for hcp-Fe and comparison with experimen-tal data. 100 150 200 250 300 350 6.5 7 7.5 8 8.5 9 P [ GPa ] V [ Å 3 / atom ] EMTO−CPA Fe 0.75Ni0.25 EMTO−CPA Fe 0.875Ni0.125 Exp. Fe 0.8Ni0.2 (Mao et al., 1990)
(b) Calculated EOS for hcp-Fe0.75Ni0.25 and hcp-Fe0.875Ni0.125,
with comparison to experimental data.
Figure 2: Calculated EOS (lines) for hcp-Fe (a) and hcp-FeNi (b) using the EMTO-CPA method at T=0 K, with comparison to experimental data by Mao et al. (1990) (red circles).
−2 0 2 V 2 [mRy] P, T = 0 GPa, 1 K P, T = 0 GPa, 5000 K 0.5 1 1.5 2 2.5 3 3.5 −2 0 2 V 2 [mRy] distance [ a ] P, T = 400 GPa, 1 K P, T = 400 GPa, 5000 K (a) fcc-Fe0.875Ni0.125 −6 −4 −2 0 2 V 2 [mRy] P, T = 0 GPa, 1 K P, T = 0 GPa, 5000 K 0.5 1 1.5 2 2.5 3 3.5 −6 −4 −2 0 2 V 2 [mRy] distance [ a ] P, T = 400 GPa, 1 K P, T = 400 GPa, 5000 K (b) fcc-Fe0.50Ni0.50
Figure 3: Effective pair interactions in fcc-FeNi as a function of distance, given in terms of the lattice constant.
−6 −4 −2 0 2 V 2 [mRy] P, T = 0 GPa, 1 K P, T = 0 GPa, 5000 K 1 1.5 2 2.5 3 −6 −4 −2 0 2 distance [ a ] V 2 [mRy] P, T = 400 GPa, 1 K P, T = 400 GPa, 5000 K (a) hcp-Fe0.875Ni0.125 −10 −5 0 V 2 [mRy] P, T = 0 GPa, 1 K P, T = 0 GPa, 5000 K 1 1.5 2 2.5 3 −10 −5 0 V 2 [mRy] distance [ a ] P, T = 400 GPa, 1 K P, T = 400 GPa, 5000 K (b) hcp-Fe0.50Ni0.50
Figure 4: Effective pair interactions in hcp-FeNi as a function of distance, given in terms of the lattice constant.
and electronic temperature. Increasing Ni concentration to 50 % has the same effect as in the case of fcc, shown in Fig. 4(b).
The diminishing effect of increased temperature on the long-ranged pair interactions can be explained by Fermi surface smearing. Adding other tem-perature effects, such as ion vibrations, would lead to further strong smear-ing of the electronic structure, as reported for the case of Mo (Asker et al., 2008). From this we may conclude that the range of interactions should be even shorter if all effects at elevated temperature were to be included. It is therefore reasonable to assume that for many applications, the interatomic interactions in FeNi alloy at the conditions of the Earth’s core can be con-sidered relatively short-ranged — up to 3rd or 4th nearest neighbour shell. 3. Density of Free Energy States
In Figure 5 we show the density of states, g(E), at high pressure (360 GPa) and temperature, corresponding to the density of free energy states, relative to the Fermi level, EF, for the fcc (a) and hcp (b) phases of Fe and
Fe0
.8Ni0.2. Its shape can be correlated with the results presented in Section
3.3 of the main text, which reveal that adding Ni to Fe has opposite effects on the relative fcc-hcp stability at moderate temperature (below 4000 K) and very high temperature (above 4000 K). In particular, below 4000 K, increase of Ni content favours the fcc phase, whereas above 4000 K it favours the hcp phase.
Phase stability of transition metals and their alloys is known to be sensi-tive to the shape of g(E) around EF (Massalki, 1996) If EF is situated in a
local minimum of g(E), the phase appears to be relatively more stable. Con-versely, a high density of states in the vicinity of EF indicates destabilisation
of the phase.
Beginning at the moderate temperature of 1000 K, we see in Figure 5 (b) that both hcp-Fe and hcp-Fe0
.8Ni0.2 show a local minimum at the Fermi level.
The impact of adding Ni to Fe is only a small change in the occupancy of the states, which shifts the Fermi level within the local minimum towards a large peak. For fcc-Fe at the same conditions, the Fermi level is seen in Figure 5 (a) to be located close to the top of a peak, and adding Ni shifts the Fermi level towards a local minimum, which translates into relative stabilisation of fcc over hcp.
-9 -6 -3 0 3 E - E F [eV] 0.0 0.5 1.0 1.5 g(E) [eV -1 ] (a) fcc (b) hcp 0.0 0.5 1.0 1.5 Fe 1000 K Fe 6600 K Fe0.8Ni0.2 1000 K Fe0.8Ni0.2 6600 K
Figure 5: Density of states in FeNi alloys at finite temperature and high pressure (P=360 GPa), shown relative to the Fermi level energy (EF).
files are still quite different. In hcp-Fe, the Fermi level is still close to a pronounced local minimum. With increase of Ni content, the Fermi level is slightly shifted away from the local minimum. In contrast, the Fermi level of fcc-Fe is situated in a very shallow local minimum. Adding Ni results in a redistribution of the states, which changes the profile of g, and the mini-mum disappears altogether. This means that the fcc phase is unfavourable compared to hcp at these conditions.
The above discussion helps us to rationalise the underlying cause of the different roles played by Ni at low and high temperature in the stabilisation of hcp or fcc crystal structures. It provides an important starting point for a future, more quantitative, investigation of this issue.
References
Asker, C., Belonoshko, A. B., Mikhaylushkin, A. S., Abrikosov, I. A., 2008. First-principles solution to the problem of Mo lattice stability. Physical Review B (Condensed Matter and Materials Physics) 77 (22), 220102. Mao, H., Y. Wu, L. Chen, J. Shu, A. Jephcoat, 1990. Static Compression
of Iron to 300 GPa and Fe0
.8Ni0.2 Alloy to 260 GPa: Implications for
Composition of the Core. Journal of Geophysical Research 95, 21737. Massalki, T. B., 1996. Structure and stability of alloys. In: Cahn, R. W.,
Haasen, P. (Eds.), Physical Metallurgy, 4th Edition. Vol. 1.
Ruban, A. V., Abrikosov, I. A., 2008. Configurational thermodynamics of alloys from first principles: effective cluster interactions. Rep. Prog. Phys. 71, 046501.
Table 1: Volumes (in ˚A3
) used for total energy calculations of fcc-Fe1−xNixalloys.
x= 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 6.25537 6.25537 6.34265 6.34265 6.34265 6.08321 6.08321 6.16889 6.25537 6.25537 6.25537 6.60936 6.60936 6.6999 6.6999 6.6999 6.43074 6.43074 6.51964 6.60936 6.60936 6.60936 6.97647 6.97647 7.07032 7.07032 7.07032 6.79126 6.79126 6.88345 6.97647 6.97647 6.97647 7.35692 7.35692 7.45415 7.45415 7.45415 7.16501 7.16501 7.26054 7.35692 7.35692 7.35692 7.75096 7.75096 7.85162 7.85162 7.85162 7.55223 7.55223 7.65116 7.75096 7.75096 7.75096 8.15882 8.15882 8.26298 8.26298 8.26298 7.95315 7.95315 8.05555 8.15882 8.15882 8.15882 8.58075 8.58075 8.68846 8.68846 8.68846 8.36801 8.36801 8.47394 8.58075 8.58075 8.58075 9.01698 9.01698 9.1283 9.1283 9.1283 8.79706 8.79706 8.90657 9.01698 9.01698 9.01698 9.46774 9.46774 9.58274 9.58274 9.58274 9.24053 9.24053 9.35368 9.46774 9.46774 9.46774 9.93329 9.93329 10.052 10.052 10.052 9.69866 9.69866 9.81551 9.93329 10.6598 9.93329 10.4139 10.9097 10.5364 10.5364 10.5364 10.1717 10.1717 10.2923 10.4139 11.421 10.4139 9
Table 2: Volumes (in ˚A3
) used for total energy calculations of hcp-Fe1−xNixalloys.
x= 0.0 0.125 0.25 0.50 0.70 1.0 6.16889 6.34265 6.16889 6.16889 6.34265 6.34265 6.88345 6.6999 6.51964 6.51964 6.51964 6.6999 7.26054 7.07032 6.88345 6.88345 6.88345 7.07032 8.05555 7.45415 7.26054 7.26054 7.26054 7.45415 8.47394 7.85162 7.65116 7.65116 7.65116 7.85162 9.35368 8.26298 8.05555 8.05555 8.05555 8.26298 10.2923 8.68846 8.47394 8.47394 9.58274 8.68846 10.7843 9.1283 8.68846 8.90657 10.052 9.1283 9.46774 9.1283 9.35368 10.5364 9.58274 9.93329 9.58274 10.2923 11.2917 10.052 10.4139 10.052 11.421 10.5364 10.2923 11.036 10.7843
Table 3: Volumes (in ˚A3
) used for total energy calculations of ordered fcc-based FeNi alloys.
FeNi3 FeNi FeNi3
6.25537 6.16889 6.16889 6.43074 6.51964 6.51964 7.16501 6.88345 6.88345 7.55223 7.26054 7.26054 8.15882 7.65116 7.65116 8.90657 8.05555 8.05555 9.58274 8.47394 8.47394 10.1717 8.90657 8.90657 10.5364 9.35368 9.35368 9.81551 9.81551 10.2923 10.2923 10.7843 10.7843
