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Page 8 of 37 Ye et al. J Mater Inf 2023;3:15 https://dx.doi.org/10.20517/jmi.2023.08

The molar Gibbs energies of Fe Tb , Fe Tb , Fe Tb, Fe Dy , Fe Dy , Fe Dy, Fe Er , Fe Er , Fe Er, Fe Lu ,
23
23
2
17
6
2
6
3
23
6
23
17
3
3
6
2
17
Fe Lu, Fe Y , Fe Y, and Fe Y can be expressed based on the Neumann-Kopp rule as:
3
23 6
2
3
in which b and q represent the stoichiometric numbers, J and K represent the optimized parameters.
- and , as shown in Equation (6) and Equation (7), are the magnetic contribution to
the molar Gibbs energy, and the formula is expressed by Equation (3). T and β of the Fe-RE intermetallic
C
0
compounds used in the present work can be found in Tables 1-5, respectively.
Thermodynamic calculation
The parameters of all stable phases in the Fe-RE (RE = Tb, Dy, Er, Lu, and Y) binary systems were
optimized using the PARROT module of Thermo-Calc software considering the experimental
[136]
information (e.g., phase equilibria, enthalpy of mixing, enthalpy of formation, and heat capacity) reported
in the literature. As for the Fe-RE binary systems, the experimental data are relatively limited because of the
experimental difficulties. However, the systematic trends were observed for the thermodynamic
characteristics of the RE-B , RE-Mn [137,138] , and RE-Ni binary systems with the increasing of RE atomic
[139]
[24]
numbers. The Fe-RE binary systems also show a similar systematic trend, which was considered in the
present calculations. For example, because the experimental enthalpy of mixing of liquid Fe-Er alloys was
not reported in the literature, which was estimated by Konar et al. using the formula
[19]
[ΔH mix,Fe − Er = ΔH mix,Fe − Dy + 0.5(ΔH mix,Fe − Lu − ΔH mix,Fe − Dy )], where ΔH mix,Fe − Lu and ΔH mix,Fe − Dy were obtained from
the measured experimental results .
[98]
The thermodynamic parameters were optimized finally, as shown in Tables 1-5, respectively. The enthalpy
of formation (including the reported experimental results and the calculated results by the CALPHAD
method and first-principles calculation) and the crystal structure data of the Fe-RE intermetallic
compounds are shown in the Supplementary Materials.
CALCULATED RESULTS
Fe-Tb and Fe-Dy
Thermodynamic parameters of liquid phase and Fe Tb and Fe Dy were assessed in consideration of the
2
2
experimental results and the calculations . In this work, the calculated Fe-Tb binary system is compared
[101]
[27]
with the measured data [92,93,95] and the calculations , as shown in Figure 1. The temperatures of invariant
[104]
reactions in the Fe-Tb binary system are calculated as listed in Table 6. The present calculations agree with
the reported calculations [27,104] , and the optimized parameters are shown in Table 1.
The presently calculated enthalpy of mixing and partial enthalpy of mixing in liquid Fe-Tb alloys at 1,833 K
are accordant well with the calculations , as shown in Figure 2 and Figure 3. The enthalpies of formation
[27]
of the Fe-Tb intermetallic compounds are calculated at 298 K in Figure 4, which is in good accordance with
the calculations [19,27,104,140] and the experimental data [54,102] . The presently calculated enthalpy of formation of
Fe Tb is -9.005 kJ/mol-atom. As presented in Figure 5, the calculated thermodynamic properties [e.g., heat
2
capacity, the entropy of formation, and enthalpy difference of Fe Tb] are in accordance with the
data and show no artificial break point at 120 K in the heat capacity curve.
[101]