Critical Systematic Evaluation and Thermodynamic Optimization of the Fe-RE System: RE = Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y

Abstract

As the second part of the thermodynamic study of binary Fe-RE system, critical evaluations and optimizations of all available phase diagrams and thermodynamic data for the Fe-heavy RE (heavy RE = Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y) systems were conducted to obtain reliable thermodynamic functions of all the phases in the systems. In the thermodynamic modeling of the heavy RE systems, systematic variations in the phase diagrams and thermodynamic properties such as the enthalpy of mixing in the liquid state and enthalpy of formation of solid compounds with the atomic number of lanthanide series were observed. These systematic trends were incorporated in the optimization of the Fe-heavy RE system to resolve inconsistencies between available experimental data and to estimate unknown thermodynamic properties. The systematic trends in thermodynamic properties of solid and liquid phases and phase diagram of the entire Fe-RE systems were summarized.

1 Introduction

Rare earth (RE) elements are used as important alloying elements for the applications to permanent magnets, electronics, and power industries. The addition of heavy-RE elements enhances the magnetic, electronic, optical and mechanical properties of alloys. Compounds of heavy-RE elements with other metals like Co, Fe, Ni etc. have unique properties of optical,[1] magnetic,[2] magnetocaloric effect,[3,4,5] magnetostriction[6] and electrical conductivity,[7] and electrocatalytic properties.[8,9] These properties make them valuable to the electronic, appliance, green technology, weapon and medical device manufacturing industry.[10,11] In addition, several studies[12,13,14] related to the use of RE elements in steel metallurgy as strong deoxidants, microstructure refining alloys, and inclusion modifiers have been performed.

Accurate thermodynamic properties and phase diagrams of alloys are indispensable for the development of alloy design and processing. In particular, CALPHAD type thermodynamic databases are widely used in such applications. The thermodynamic database can be developed through ‘thermodynamic optimization’ process involving critical and simultaneous evaluation of all thermodynamic and phase equilibrium data available from the literature. During optimization, the thermodynamic properties such as activity, enthalpy and entropy are considered along with phase diagram information. The discrepancy even between different types of experimental data is evaluated to comply with the thermodynamic principles. After a critical evaluation of the available experimental data, a self-consistent set of Gibbs energy equations for all phases in the given system was obtained to reproduce all reliable experimental data. This set of Gibbs equations can be used along with computer software containing the Gibbs energy minimizing routine like FactSage[15] to back calculate any phase equilibria, phase diagram and thermodynamic properties useful for new materials development and processing.

As part of the wide research program for the recycling of RE elements from waste RE magnets and electronic components,[16,17] and also to introduce RE elements in new Mg alloy development,[18] thermodynamic optimization of many binary and ternary RE elements containing systems was performed. In particular, to obtain an accurate thermodynamic description of binary RE-ME (ME = Fe, Sn, Mn, Si, Pb, Al, Mg, and Zn) systems, systematic and critical assessments of the thermodynamic and phase diagram data were performed. For example, we have already performed thermodynamic optimization for RE-Sn,[19,20] Mn,[21,22,23] Si,[24,25] Pb,[26] Al,[27,28] Mg,[29] and Zn.[30]

In our previous study[31] the critical evaluation and optimization of the Fe-light RE systems, where RE = La, Ce, Pr, Nd, and Sm, was performed. As the second part of the thermodynamic modeling study of the whole Fe-RE systems, the critical optimization of the Fe-heavy RE systems where RE = Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y is presented in this study. As pointed out in many binary RE alloy systems,[16,21,23,25] the systematic trends in thermodynamic properties and phase diagram information are used to resolve existing inconsistency between experimental data and predict the unknown thermodynamic and phase diagram data.

2 Thermodynamic Models

The thermodynamic models for liquid and solid phases are the same as described in the previous work[31] for the Fe-light RE system. The modified quasichemical model (MQM) was used to describe the liquid solution in all binary systems. The model has been described in detail by Pelton et al.[32,33] which accounts for short-range ordering and gives a realistic thermodynamic description of liquid phase compared to that obtained from conventional simple random-mixing Bragg–Williams model.[32,33] The Gibbs energies of all pure elements were taken from SGTE database version 5.0.[34] The Gibbs energies of stoichiometric compounds were optimized based on available thermodynamic data such as heat capacity, enthalpy and entropy of formation at 298 K. If the heat capacities of stoichiometric compounds were not available, they were typically estimated using the Neumann–Kopp (N–K) rule.[35] If the entropies of the compounds were unknown, typically N–K rule was applied as the first approximation. In some instances, they were slightly adjusted based on the periodic trend along the RE series to reproduce the phase diagram of the system. Iron and most of the compounds in Fe-RE binary system have magnetic transitions involving the Néel or Curie temperature. The magnetic contribution (G _Mag ) to the Gibbs energy of phase was described following an empirical relationship suggested by Inden[36] and modified by Hillert and Jarl.[37] The solid solutions, Fe-rich solution and RE-rich solution, appearing in a binary system were described by the Compound Energy Formalism (CEF)[38] with one sublattice (substitutional site) model which can be occupied by Fe and RE atom. The details of thermodynamic models can be found in the previous study[31] for the Fe-light RE systems.

3 Critical Evaluation and Thermodynamic Optimization

The crystal structures of the intermetallic phases considered in the present assessment are summarized in Table 1. The optimized thermodynamic model parameters of solid and liquid phases are presented in Table 2.

Table 1 Summary of the crystal structures of the compounds in the heavy Fe-RE system

Table 2 Optimized model parameters for the Fe-RE system (J mol⁻¹ or J mol⁻¹ K⁻¹)

3.1 The Fe-Gd (Iron-Gadolinium) System

The thermodynamic assessment of Fe-Gd system was performed by Liu et al.,[39] Zinkevich et al.[40] and Konar.[41] Kubaschewski[42] and Okamoto[43] also reviewed this system. In all assessments, four intermetallic compounds (Gd₂Fe₁₇, Gd₆Fe₂₃, GdFe₃ and GdFe₂), bcc and fcc Fe solutions, bcc and hcp Gd solutions, and liquid (Liq) phase were considered.

The optimized calculated phase diagram of the Fe-Gd system is shown in Fig. 1 along with the experimental data.[44,45,46,47,48,49,50] The invariant reactions are summarized in Table 3. The phase diagram of the Fe-Gd system was first investigated by Novy et al.[50] Seven intermetallic phases, Gd₂Fe₃, GdFe₂, GdFe₃, Gd₂Fe₇, GdFe₄, GdFe₅ and Gd₂Fe₁₇ along with five invariant reactions were reported by differential thermal analysis (DTA) technique and optical pyrometry. It can be said that this is very preliminary phase diagram of this system. Contrary to Novy et al.,[50] Copeland et al.[44] suggested three intermetallic phases GdFe₉, GdFe₃, and GdFe₂ with peritectic melting using thermal analysis (TA), optical microscopy (OM) and x-ray diffraction (XRD) phase analysis. The solidus and liquidus temperatures were determined by TA and visually by optical pyrometry. Savitskii et al.[45] investigated the phase diagram over the entire composition range with TA and equilibration technique followed by OM and XRD phase analyses and reported two compounds Gd₂Fe₁₇ and GdFe₂ with peritectic melting. An eutectic reaction between Gd-rich phase and GdFe₂ was also reported at 0.72 mol fraction Gd (X_Gd) and 830 ± 7 °C. Later, Savitskii[46] reported four intermetallic phases Gd₂Fe₁₇, GdFe₄, GdFe₃, and GdFe₂ where all showed peritectic melting behaviors. In the review by Kubaschewski,[42] the phase diagram of the Fe-Gd system was proposed based on the experimental results of Copeland et al.[44] with the modification of peritectic melting of Gd₆Fe₂₃ based on Savitskii et al.[46] and Burov et al.[48] Most recently, Atiq et al.[49] investigated the phase transformations in the Fe-rich side (less than 0.32 X_Gd) which were determined by DTA and diffusion couple technique. For the determination of phases in the diffusion couples, energy dispersive x-ray analysis (EDAX) was used. Atiq et al.[49] suggested a polymorphic transformation of the Gd₂Fe₁₇ phase from rhombohedral (Th₂Zn₁₇ type) to hexagonal (Th₂Ni₁₇ type) structures at around 1215 ± 2 °C.

Fig. 1

The optimized phase diagram of the Fe-Gd system with experimental data (a) overall composition range,[44,46,47,48,49,50] (b) in the Fe-rich side,[46,48] and (c) in the Gd-rich side[46]

Table 3 Calculated invariant reactions in the Fe-Gd system along with the experimental data

In the present optimization, four intermetallic phases GdFe₂, GdFe₃, Gd₆Fe₂₃ and Gd₂Fe₁₇, and bcc, fcc and hcp Fe-Gd solutions were considered as solid phases. The GdFe₉ and GdFe₄ reported by Copeland et al.[44] and Savitskii[46] should be corresponding to Gd₂Fe₁₇ and Gd₆Fe₂₃, respectively, according to more updated crystallographic information. The liquidus and solidus data by Savitskii et al.,[45,46] Burov et al.,[48] and Atiq et al.[49] were mainly considered in the present modeling. The polymorphic transformation of the Gd₂Fe₁₇ phase reported by Atiq et al.[49] was not considered in the present study due to lack of information about the enthalpy of transformation. All the liquidus and peritectic transition temperatures in the Fe-Gd system were well reproduced in the present optimization within the experimental error. The peritectic reaction of Liq + Gd₂Fe₁₇ → Gd₆Fe₂₃ is calculated at 0.281 X_Gd in this study, which is slightly higher than the experimental value 0.28 X_Gd from Savitskii,[46] while the previous assessments calculated at about 0.235[39] and 0.210 X_Gd.[40] The other invariant reaction compositions are all calculated within experimental error ranges.

The solubility of Gd in bcc Fe and that of Fe in hcp Gd was reported by Savitskii et al.[46] and Burov et al.[48] as presented in Fig. 1(b) and (c). Savitskii et al.[46] reported the maximum solubility of Gd in bcc Fe to be 0.001 mol fraction of Gd at 800 °C and solubility of Fe in hcp Gd to be 0.006 mol fraction of Fe at 700 °C, respectively, using XRD measurement of quenched samples. Burov et al.[48] performed dilatometric analysis and determined the bcc Fe to fcc Fe transition and presented less than 0.001 mol fraction solubility of Gd in Fe in their diagram. On the contrary, Copeland et al.[44] reported no solubility of the solid phases from their optical metallographic observation of the sample. In this study, the data from Savitskii et al.[46] and Burov et al.[48] were accepted, and a regular solution parameter for a bcc Fe-Gd solution was determined to reproduce the solubility of Gd in bcc Fe. It should be noted that the eutectoid transition of bcc Fe → fcc Fe in the Fe-rich side determined by Atiq et al.[49] was also well reproduced as shown in Fig. 1(a). A temperature-dependent parameter for fcc Fe solution was necessary to reproduce the metatectic transition of bcc Fe → fcc Fe + Liq at 1381 °C, compared to 1389 ± 5 °C[49] as shown in Fig. 1(a). A temperature-dependent parameter for hcp Gd solution was also needed to reproduce the rapidly increasing solubility of Fe in hcp Gd solution by Savitskii et al.[46]

The integral and the partial enthalpy of mixing of the Fe-Gd liquid were reported by Nikolaenko and Nosova[51] at 1850 K and Ivanov et al.[52] at 1833 K as presented in Fig. 2. The experiments were conducted using a high flux high-temperature isoperibolic calorimeter (SC) under high purity[51] Ar and He[52] gas. Nikolaenko and Nosova[51] used yttria-coated alumina crucibles, whereas in the recent study by Ivanov et al.[52] the partial enthalpy of mixing of the Fe-rich and the Gd-rich compositions was measured using zirconia crucibles and Mo crucibles, respectively. The integral enthalpy of mixing was calculated at 1833 K and presented in Fig. 2(a). Even though the two sets of experimental data show negative enthalpy of mixing, a large discrepancy in terms of the location of minima of enthalpy of mixing and partial enthalpies of Fe and Gd is observed. It is rather unusual to have minima of enthalpy of mixing where no solid compound exists. In addition, the results of Ivanov et al.[52] are consistent with the systematic trend of enthalpy of mixing of all other binary liquid Fe-RE solutions, which will be discussed later in section 4. Therefore, the present model parameters of liquid solution were optimized to reproduce the more recent enthalpy data by Ivanov et al.[52] in Fig. 2 and the phase diagram in Fig. 1, simultaneously.

Fig. 2

Enthalpy of mixing of the Fe-Gd liquid at 1850 K with the experimental data[51,52] and assessment.[39] (a) Integral enthalpy of mixing and (b) partial enthalpy of mixing

The $\Delta H_{298}^{o}$ of GdFe₂, GdFe₃, and Gd₂Fe₁₇ were measured by Colinet et al.[53,54] and of GdFe₂ by Meschel et al.,[55] as presented in Fig. 3. Colinet et al.[53,54] prepared samples with Gd (99.9 wt.%) and Fe (99.99 wt.%) and conducted experiments by solution calorimetry in molten Al (SC-Al) at about 1000 K and reported the $\Delta H_{298}^{o}$ for GdFe₂, GdFe₃, and Gd₂Fe₁₇ to be −11.6, −9.3 and −2.3 kJ mol-atom⁻¹, respectively. Meschel et al.[55] performed direct synthesis calorimetry (DSCa) of GdFe₂ at 1100 °C (1373 K) which contained up to 5 wt.% Gd₂Fe₁₇ as an impurity. But it should be noted that the melting of GdFe₂ occurs at 1074 °C (1347 K) according to the phase diagram in Fig. 1, and therefore the measurement of Meschel et al.[55] is less reliable. Deodhar and Ficalora[56] also reported the $\Delta H_{298}^{o}$ of the GdFe₂ compound from their DTA experiments. The reported values were not reliable due to several kinetic limitations of the formation reaction, along with reported complex reaction mechanisms and C _p assumptions.

Fig. 3

Enthalpy of formation at 298 K ($\Delta H_{298}^{o}$) of the intermetallic Fe-Gd compounds compared with the experimental data[53,54]

Figure 4 shows the C _p of the GdFe₂ compound measured by Germano et al.[57] using adiabatic calorimetry (AC) from 15 to 300 K. In the present study, high-temperature C _p was firstly obtained using N–K rule and then modified to fit the low-temperature C _p data by increasing the temperature independent C _p term by 12.31 J mol⁻¹ K⁻¹. The magnetic properties (Bohr magnetons per mole and Curie temperature) of the intermetallic phases were incorporated into the present modeling from Segal and Wallace.[58] As there are no heat capacity data reported for other compounds, the C _p of GdFe₃, Gd₆Fe₂₃ and Gd₂Fe₁₇ were estimated from the N–K rule using the C _p function of GdFe₂ and Fe, and the magnetic properties were taken into account. The $S_{298}^{o}$ of GdFe₂ derived from the above low-temperature C _p data is 115.6 J mol⁻¹ K⁻¹. It should also be noted that Meschel et al.[55] measured the heat content change (H ₁₃₇₃  − H ₂₉₈ ) of GdFe₂ to be 50.8 ± 1.3 kJ mol-atom⁻¹. The calculated value of 40.04 kJ mol-atom⁻¹ from the optimized heat capacity in Fig. 4 is quite different from the data of Meschel et al.[55]

Fig. 4

The optimized C _p of GdFe₂ along with the experimental data[57]

Although the phase diagram of the Fe-Gd system is reasonably well determined, the experimental thermodynamic properties of solid phases are insufficient to constrain the Gibbs energies. Based on the phase diagram data, liquid enthalpy data, and limited thermodynamic data of solid compounds, the model parameters of all phases were optimized. Five parameters including two small temperature dependent terms were required to describe the Gibbs energy of liquid phase. The enthalpies of formation of solid compounds by Colinet et al.[53,54] were mostly well reproduced within the experimental error limits. However, it was very difficult to exactly reproduce the $S_{298}^{o}$ of GdFe₂, 115.6 J mol⁻¹ K⁻¹ as derived from low-temperature C _p data. The optimized value from the present study is 109.38 J mol⁻¹ K⁻¹, which is well constrained by the experimental data of $\Delta H_{298}^{o}$ and C _p of GdFe₂ and the thermodynamic properties of the liquid phase in conjunction with the phase diagram. All model parameters obtained from the present study are summarized in Table 2.

In all the previous assessments by Liu et al.,[39] Zinkevich et al.[40] and Konar,[41] the liquid enthalpy was optimized using the data by Nikolaenko and Nosova[51] as the experiments of Ivanov et al.[52] were not available at the time of the assessments. Liu et al.[39] assessed the $\Delta H_{298}^{o}$ for Gd₂Fe₁₇, GdFe₃, and GdFe₂ to be −3.74, −8.29 and −9.47 kJ mol-atom⁻¹, respectively, which are consistent with the results of Colinet et al.[53,54] However, Gibbs energy of eight compounds (Gd₂Fe₁₇, Gd₆Fe₂₃, GdFe₃, GdFe₂, GdFe₅, Gd₂Fe₇, Gd₃Fe and Gd₄Fe₃) were defined. Moreover, the $S_{298}^{o}$ of GdFe₂ was determined to be 143.86 J mol⁻¹ K⁻¹ which is much higher than the experimental data of 115.6 J mol⁻¹ K⁻¹.[57] The phase equilibria below 900 K were not shown. In the assessment by Zinkevich et al.,[40] the $\Delta H_{298}^{o}$ of Gd₂Fe₁₇, GdFe₃, and GdFe₂, were −3.66, −4.91 and −4.59 kJ mol-atom⁻¹, respectively, which significantly deviate from the results of Colinet et al.[53,54] The C _p of GdFe₂ by Germano et al.[57] was not considered, and the $S_{298}^{o}$ for GdFe₂ was modeled to be 128.5 J mol⁻¹ K⁻¹ deviating from experimental 115.6 J mol⁻¹ K⁻¹.[57] In addition, their liquid Gibbs energy function was modeled with several large temperature-dependent parameters. In both the assessments by Liu et al.[39] and Zinkevich et al.,[40] all the $\Delta H_{298}^{o}$ and $S_{298}^{o}$ of intermetallic compounds were not concomitantly optimized, and therefore several intermetallic phases were calculated to be unstable at room temperature.

3.2 The Fe-Tb (Iron-Terbium) System

The Fe-Tb system was reviewed by Okamoto[59] and assessed by Landin and Ågren[60] and Konar.[41] Landin and Ågren[60] and Konar[41] suggested four stable intermetallic phases Tb₂Fe₁₇, Tb₆Fe₂₃, TbFe₃, and TbFe₂, which is characteristic of the Fe-heavy RE systems, in addition to bcc Fe, fcc Fe, hcp Tb, bcc Tb solution and liquid phase, based on the experimental study of Dariel et al.[61]

The phase diagram of the Fe-Tb system is shown in Fig. 5 along with experimental data by Dariel et al.,[61] Orlova et al.[62] and Chen et al.[63] Dariel et al.[61] used both TA and quenching experiments followed by XRD, OM, and electron probe microanalysis (EPMA) for phase analysis. Orlova et al.[62] determined the melting point of the compounds by a pyrometer with an error of ± 50 °C followed by compositional chemical analysis. Overall results of Orlova et al.[62] are not reliable as shown in Fig. 5. Dariel et al.[61] reported that Tb₂Fe₁₇, TbFe₃, and TbFe₂ formed peritectically at 1312, 1212, and 1187 °C, respectively, and also reported the eutectic reaction Liq → TbFe₂ + hcp Tb at 847 °C. Recently, Chen et al.[63] re-examined the phase diagram in the composition range of 0.045 to 0.1625 mol fraction Tb carefully using DTA (see Fig. 5(b)). The DTA results for Tb₂Fe₁₇ compound show a congruent melting at 1316 °C and a eutectic reaction of Liq → Tb₂Fe₁₇ + bcc Fe at 1301 °C with 0.075 mol fraction Tb. While Dariel et al.[61] reported two polymorphic forms of Tb₂Fe₁₇ (α-Tb₂Fe₁₇ with Th₂Zn₁₇-type rhombohedral in Tb-rich side and β-Tb₂Fe₁₇ with Th₂Ni₁₇-type hexagonal in Fe-rich side), Chen et al.[63] reported the presence of only one polymorph of Tb₂Fe₁₇ (Th₂Ni₁₇ structure) from DTA study. According to Chen et al.,[63] Tb₂Fe₁₇ was observed at the solidification temperature with no homogeneity range. As part of the phase diagram study of the Fe-Pt-Tb system, Gu et al.[64] also confirmed the existence of a stoichiometric Tb₂Fe₁₇ with Th₂Zn₁₇ type rhombohedral structure at 900 °C (1173 K) with XRD. In the present study, a single stoichiometric phase of Tb₂Fe₁₇ was considered without polymorphic transition. Dariel et al.[61] reported limited solid solution of Tb in bcc Fe at 880 °C to be 0.0007 ± 0.0003 and 0.001 ± 0.0005 mol fraction of Tb by XRD lattice parameter measurements and EPMA analysis, respectively. No measurable solubility of Tb in fcc Fe was observed by Dariel et al.[61] The solubility of Tb in bcc Fe is depicted in Fig. 5(c). Overall, the phase diagram study by Dariel et al.[61] is reliable except the melting behavior of Tb₂Fe₁₇ which was further properly examined by Chen et al.[63]

Fig. 5

The optimized phase diagram of the Fe-Tb system with the experimental data (a) overall composition range,[61,62,63,64] (b) in the high-temperature region[61,62,63] (c) in the Fe-rich side[61]

Figure 6(a) and (b) show the integral and partial enthalpies of mixing of liquid Fe and Tb measured by Ivanov et al.[52] using a SC at 1833 K. Ivanov et al.[52] measured the partial enthalpy of mixing of the Fe-rich alloys using zirconia crucible and Tb-rich compositions using molybdenum crucible. A smoothed equation for integral enthalpy of mixing was obtained with a minimum at 0.45 mol fraction Tb with a value of −5.06 ± 0.43 kJ mol⁻¹. The present optimization reproduced these experimental data accurately. At the time of previous assessments by Landin and Ågren,[60] and Konar,[41] no such experimental enthalpy of mixing data were available, hence the estimated enthalpies are quite different from the present result as shown in Fig. 6(a).

Fig. 6

Enthalpy of mixing of the Fe-Tb liquid at 1833 K with the experimental data[52] and assessments.[41,60] (a) Integral enthalpy of mixing and (b) partial enthalpy of mixing

The $\Delta H_{298}^{o}$ of Tb₂Fe₁₇ and TbFe₂ measured by Gozzi et al.[65] and Meschel et al.[55] are presented in Fig. 7. Gozzi et al.[65] reported the $\Delta H_{298}^{o}$ for the Tb₂Fe₁₇ phase derived from the Gibbs energy of formation determined using galvanic cells with CaF₂ electrolyte. Meschel et al.[55] determined the formation enthalpies of Tb₂Fe₁₇ and TbFe₂ by DSCa, and also reported the predicted values from Miedema method. The experimental data and estimated data from Miedema method are consistent within the experimental error limits. The experimental value of $\Delta H_{298}^{o}$ for Tb₂Fe₁₇ was measured to be −2.1 ± 3.1 and −3.3 kJ mol-atom⁻¹ by Meschel et al.[55] and Gozzi et al.,[65] respectively, which is reproduced as −2.45 kJ mol-atom⁻¹ in the present optimization. The experimental $\Delta H_{298}^{o}$ value of TbFe₂ by Meschel et al. was −5.5 ± 2.4 kJ mol-atom⁻¹ and the present optimized value is about 1.5 kJ mol-atom⁻¹ more negative than the experimental error range of their data.

Fig. 7

Enthalpy of formation at 298 K ($\Delta H_{298}^{o}$) of the intermetallic Fe-Tb compounds compared with the experimental data[55,65]

Figure 8 shows the experimental low-temperature C _p of TbFe₂ measured by Germano et al.[57] using AC from 4.2 K to 300 K and the C _p curve from the present study. The $S_{298}^{o}$ of TbFe₂ was determined to be 122.7 J mol⁻¹ K⁻¹, from the experimental low-temperature C _p . The estimated high-temperature C _p of TbFe₂ presented in Fig. 8 was firstly derived using the N–K rule and the temperature independent term of C _p was modified by 9.09 J mol-K to fit the low-temperature C _p data. The magnetic moment and the Curie temperature of the compounds were obtained from Buschow.[66] The C _p of the other compounds TbFe₃, Tb₆Fe₂₃, and Tb₂Fe₁₇ were also estimated by the N–K rule from the C _p of hcp Tb and bcc Fe, and the magnetic properties reported by Buschow[66] were taken into account.

Fig. 8

The optimized C _p of TbFe₂ along with the experimental data[57]

The present optimization of the Fe-Tb system was carried out by following a similar procedure like the Fe-Gd system. The enthalpy of the liquid phase and the Gibbs energy of TbFe₂ were firstly optimized together with the phase diagram information in order to reproduce the enthalpy of mixing by Ivanov et al.[52] and the thermodynamic properties of TbFe₂ compound. Then, the thermodynamic properties of $\Delta H_{298}^{o}$ and $S_{298}^{o}$ of TbFe₃, Tb₆Fe₂₃ and Tb₂Fe₁₇ and the bcc solution parameters were optimized to reproduce their phase stabilities. Six model parameters including temperature-dependent terms were required to describe the Gibbs energy of liquid phase. The regular solution parameter with temperature dependent term was determined to reproduce the solubility of Tb in bcc Fe. The parameters for fcc and hcp solutions were determined to reproduce the associated invariant reaction temperatures. In the literature, no experimental homogeneity range data for these solutions are available. In general, the phase diagram data and the enthalpy of formation data by Meschel et al.[55] were well reproduced. The optimized $S_{298}^{o}$ of TbFe₂ (116.8 J mol⁻¹ K⁻¹) is slightly smaller than the experimentally determined value (122.7 J mol⁻¹ K⁻¹). In the present study, the congruent melting of Tb₂Fe₁₇ at 1312 °C is calculated which is consistent with the detailed experimental study by Chen et al.[63] (1316 °C), while Dariel et al.[61] proposed a peritectic melting of this compound. It should be also noted that the congruent melting of the RE₂Fe₁₇ compound was observed in the Fe-Dy, and Fe-Ho system (see next sections). Although Dariel et al.[61] reported the incongruent melting of the Tb₆Fe₂₃ at 1276 °C, it is optimized to be congruent melting at 1274 °C in the present study. According to the liquidus of Tb₂Fe₁₇ by Chen et al.[63] (see Fig. 6(b)), it is very difficult to get the incongruent melting of Tb₆Fe₂₃. So, the present study is more consistent with the experimental results by Chen et al. All other invariant reactions in the Fe-Tb system are in good agreement with experimental data, as summarized in Table 4.

Table 4 Calculated invariant reactions in the Fe-Tb system along with the experimental data

3.3 The Fe-Dy (Iron-Dysprosium) System

The Fe-Dy system was reviewed by Okamoto[67] and thermodynamically assessed by Landin and Ågren[60] and Konar et al.[68] Four intermetallic phases, DyFe₂, DyFe₃, Dy₆Fe₂₃, and Dy₂Fe₁₇ were considered as stable intermetallic compounds. In the present study, the previous model parameters of Konar et al.[68] were slightly modified to reproduce new experimental data by Nagai et al.[69] for activity and Ivanov et al.[52] for liquid enthalpy.

The optimized phase diagram of the Fe-Dy system is shown in Fig. 9 along with the experimental data of Van der Goot and Buschow.[70] Van der Goot and Buschow investigated the phase diagram over the whole composition range of the Fe-Dy system by means of TA and quenching experiment with OM and XRD phase analysis. Based on the experimental micrographs, they suggested that DyFe₃ and Dy₂Fe₁₇ melted congruently while Dy₆Fe₂₃ and DyFe₂ melted peritectically. The mutual solubilities of Fe and Dy have not been studied, but the transition temperature of bcc and fcc Fe phase indicates the very small amount of the solubility of Dy in Fe if it exists.

Fig. 9

The optimized phase diagram of the Fe-Dy system with the experimental data[70]

Figure 10 shows the integral and partial enthalpies of mixing of the Fe-Dy liquid phase measured by Ivanov et al.[52] using a SC. Ivanov et al.[52] measured the partial enthalpy of mixing of Fe-rich alloys using zirconia crucible and Dy-rich compositions using molybdenum crucible at 1833 K. A smoothed equation for integral enthalpy of mixing was obtained with a minimum at 0.35 mol fraction Dy with a value of −8.80 ± 1.33 kJ mol⁻¹. The previous assessment results from Konar et al.[68] are slightly higher than experimental data of Ivanov et al.[52]

Fig. 10

Enthalpy of mixing of the Fe-Dy liquid at 1850 K with the experimental data[52] and assessments[68] (a) Integral enthalpy of mixing and (b) partial enthalpy of mixing

In Fig. 11, the $\Delta H_{298}^{o}$ determined by the present assessment is depicted along with experimental data reported by Norgren et al.,[71] Gozzi et al.[65] and Meschel et al.[55] The $\Delta H_{298}^{o}$ of DyFe₂, DyFe₃ and Dy₂Fe₁₇ were measured by Norgren et al.[71] using SC-Al at 1100 K. The reported $\Delta H_{298}^{o}$ of compounds become more negative with increasing Dy content. Gozzi et al.[65] and Meschel et al.[55] reported $\Delta H_{298}^{o}$ of Dy₂Fe₁₇ and DyFe₂ from EMF measurements and DSCa, respectively, and also conducted empirical Miedema calculations, as shown in Fig. 11. The experimental $\Delta H_{298}^{o}$ value of Meschel et al.[55] of DyFe₂ is more positive than the value of Dy₂Fe₁₇, which is a completely opposite trend from Norgren et al.[71] As seen in all Fe-RE system, the optimized $\Delta H_{298}^{o}$ of compounds become more negative with increasing RE content. Therefore, the result of Norgren et al.[71] was considered more accurate in the present optimization.

Fig. 11

Enthalpy of formation at 298 K ($\Delta H_{298}^{o}$) of the intermetallic Fe-Dy compounds compared with the experimental data[55,65,71]

The low-temperature C _p of DyFe₂ was measured by Germano et al.[57] using an AC as presented in Fig. 12. Germano et al. calculated the $S_{298}^{o}$ of DyFe₂ from the heat capacity data to be 124.8 J mol⁻¹ K⁻¹. Like the previous systems, the high-temperature C _p for DyFe₂ phase was formulated based on the N–K rule with small adjustment as tabulated in Table 2. As there are no data available, the C _p of DyFe₃, Dy₆Fe₂₃ and Dy₂Fe₁₇ was estimated using the N–K rule. The magnetic contributions to the C _p of all intermetallic compounds were taken into account using the compilation data by Buschow.[66]

Fig. 12

The optimized C _p of DyFe₂ along with the experimental data[57]

Nagai et al.[69] measured the thermodynamic properties of several compositions in the temperature range of 1273 to 1573 K using the Knudsen effusion mass spectrometry (KEMS). Several alloys were prepared at 0.07, 0.17, 0.23, 0.28, 0.57, 0.74, and 0.89 mol fraction of Dy using electron beam melting technique with reagent grade Dy and electrolytic Fe. The chemical compositions of the alloys are verified by inductively coupled plasma atomic emission spectroscopy (ICP-AES) and phases were identified by XRD before KEMS experiment. The ion currents were detected for Dy with the isotopes, ¹⁵⁸Dy, ¹⁶⁰Dy, ¹⁶¹Dy, ¹⁶²Dy, ¹⁶³Dy, and ¹⁶⁴Dy. However, the ion currents due to ⁵⁴Fe, ⁵⁶Fe, ⁵⁷Fe and ⁵⁸Fe isotopes were not detected, as the vapor pressures in equilibrium with the alloys were assumed to be low. The activities of Dy of different alloys derived from KEMS measurement are summarized in Fig. 13.

Fig. 13

The temperature dependence of the activity of Dy of 7, 17, 23, 28, 57, 74 and 89 at.% Dy Fe-Dy alloys with the experimental data[69]

In the present optimization, all the available experimental data of thermodynamic properties and phase diagram was simultaneously considered. The enthalpy of mixing data for the liquid phase is well reproduced in Fig. 10. The Gibbs energy of DyFe₂ can be relatively well defined from $\Delta H_{298}^{o}$, $S_{298}^{o}$ and C _p data in Fig. 11 and 12. The optimized $\Delta H_{298}^{o}$ is in good agreement with the experimental data by Norgren et al.,[71] and $S_{298}^{o}$ of DyFe₂ was optimized to be 118.10 J mol⁻¹ K⁻¹ which is slightly lower than the value 124.8 J mol⁻¹ K⁻¹ derived from low-temperature heat capacity data by Germano et al.[57] The $\Delta H_{298}^{o}$ and $S_{298}^{o}$ of DyFe₃, Dy₆Fe₂₃ and Dy₂Fe₁₇ were optimized to reproduce their melting point and the phase diagram. All invariant reactions of the Fe-Dy system are summarized in Table 5. In general, the calculated invariant reactions are consistent with experimental data by Van der Goot and Buschow[70] within ± 10 °C, except the invariant reactions related to Dy₆Fe₂₃; the largest difference is calculated for the eutectic reaction Liq → Dy₆Fe₂₃ + DyFe₃ which is calculated to be 15 °C higher than the experimental data. The optimized activity data of Dy in Fig. 13 is in reasonable agreement with the measured data by Nagai et al.[69] In fact, the activity data of Dy in the alloy up to 0.28 mol fraction of Dy below 1573 K are directly determined from the Gibbs energies of intermetallic phases, and the calculated values are in reasonable agreement with the experimental data, which proves that the optimized thermodynamic properties of solid compounds are reasonable. Considering the difficulty in obtaining accurate Dy activity as pointed out by Nagai et al.[69] themselves, more weight was put on the phase diagram data in the present optimization. It should be mentioned that similar incoherencies of the data by Nagai et al.[72] were also pointed out in our previous assessment for the Fe-La system.[31]

Table 5 Calculated invariant reactions in the Fe-Dy system along with the experimental data

The Fe-Dy system was thermodynamically assessed by Landin and Ågren,[60] Konar[41] and Konar et al.[68] In the assessment done by Landin and Ågren,[60] the $\Delta H_{298}^{o}$ of the compounds were not constrained by the experimental data of Norgren et al.[71] Moreover, their optimization was viable only above 527 °C (800 K), and the magnetic contributions to the Gibbs energies of the compounds were not considered. Landin and Ågren[60] also calculated the formation of Dy₆Fe₂₃ by a peritectoid reaction rather than an experimental peritectic reaction.[70] In the all previous thermodynamic assessments, the integral enthalpy of mixing was calculated to be less negative than the experimental data as shown in Fig. 10(a).

3.4 The Fe-Ho (Iron-Holmium) System

The Fe-Ho system was reviewed by Kubaschewski[42] and Okamoto[73] and assessed by Kardellass et al.[74] and Konar.[41] Four stable intermetallic compounds, HoFe₂, HoFe₃, Ho₆Fe₂₃, and Ho₂Fe₁₇ exist in the Fe-Ho system.

The optimized phase diagram of the Fe-Ho system in the present study is presented in Fig. 14 along with experimental data. The phase diagram was investigated by Roe and O’Keefe[75] up to 0.77 mol fraction Ho. They established the phase diagram by DTA and quenching experiments followed by XRD and metallographic phase analysis. HoFe₂ and HoFe₃ were reported to melt peritectically and Ho₆Fe₂₃ and Ho₂Fe₁₇ were reported to melt congruently. The eutectic reaction of Liq → HoFe₂ + hcp Ho was reported at 0.63 mol fraction of Ho and 875 °C. The other two eutectic reactions reported by Roe and O’Keefe[75] are Liq → Ho₂Fe₁₇ + Ho₆Fe₂₃ at 0.178 mol fraction of Ho and 1284 °C, and Liq → fcc Fe + Ho₂Fe₁₇ at 0.083 mol fraction of Ho and 1338 °C. In fact, the nature of melting of HoFe₂ was very difficult to be determined in the experiment by Roe and O’Keefe[75] due to the close melting temperatures of HoFe₂ and HoFe₃. Their experimental DTA results seem to be more favorable to the congruent melting of HoFe₂. In an earlier study by O’Keefe et al.,[76] HoFe₂ was reported to melt at 1335 ± 15 °C. Considering the melting temperatures of HoFe₃ and Ho₂Fe₁₇ in the same study, congruent melting of HoFe₂ can be reasonably concluded. However, this result was not taken into account by Roe and O’Keefe[75] later. In the present study, the congruent melting of HoFe₂ was assumed considering the melting behavior of REFe₂ compound changing from peritectic (incongruent) to congruent melting with increasing atomic number of RE. For example, ErFe₂, LuFe₂, and TmFe₂ in the succeeding Fe-heavy RE systems melt congruently. The congruent melting behavior of Ho₂Fe₁₇ and Ho₆Fe₂₃ concur with those in the Fe-Tb and Fe-Dy systems.

Fig. 14

The optimized phase diagram of the Fe-Ho system with the experimental data[75]

Unfortunately, no enthalpy for liquid phase was experimentally determined. In order to constrain the model parameters of liquid phase reasonably, the enthalpy of mixing is necessary. Therefore, in the present study, the enthalpy of mixing of liquid Fe-Ho solution was estimated from the systematic trend in the Fe-RE system. As can be seen in the enthalpy of mixing data through all Fe-RE systems, there is a systematic change in the enthalpy of mixing; with increasing atomic number of RE, it becomes more negative and the location of the minima slowly moves towards the Fe-rich composition. Therefore, the mixing enthalpy of liquid Fe-Ho solution could be estimated to be an average of the experimental mixing enthalpy data of the Fe-Dy (just prior to Ho) and Fe-Er (just next to Ho) solution. Unfortunately, the experimental data of liquid Fe-Er solution is unavailable too. Thus, in the present study, the enthalpy of mixing of liquid Fe-Ho solution was estimated from the Fe-Dy and Fe-Lu solution as ΔH _mix,Fe-Ho = ΔH _mix,Fe-Dy + 0.25 (ΔH _mix,Fe-Lu − ΔH _mix,Fe-Dy), and the result is shown in Fig. 15. The factor of 0.25 was determined by neglecting Yb (in between Tm and Lu) which does not follow the general periodic trend of the physico-chemical properties of RE elements. Petiffor[77,78,79] proposed the phenomenological coordinate, called Mendeleev number (M) for each element in the periodic table where M is based on the size, electronegativity, valence and the bond orbitals, and predicts the nature and stoichiometry of compound formation. The M of Yb and Eu are quite different from other RE elements. According to the prediction from Miedema’s method, the formation enthalpies of Fe-Yb and Fe-Eu compounds are positive while those of other Fe-heavy RE systems are negative. This means that Yb and Eu are not following the general trend of RE series. Thus, the choice of the present estimation scheme of the enthalpy of mixing of the Fe-Ho liquid solution is in accordance with the results of the Mendeleev number (M) and Miedema’s prediction.

Fig. 15

Integral enthalpy of mixing of the Fe-Ho liquid at 1833 K with assessments[41,74]

The $\Delta H_{298}^{o}$ of Ho₂Fe₁₇ and HoFe₂ compounds were measured by Gozzi et al.[65] and Meschel et al.,[55] respectively, as shown in Fig. 16. Gozzi et al. derived $\Delta H_{298}^{o}$ of the Ho₂Fe₁₇ phase (−6.7 kJ mol-atom⁻¹) from the galvanic cells with CaF₂ electrolyte. Meschel et al. measured the $\Delta H_{298}^{o}$ of HoFe₂ using DSCa. The $\Delta H_{298}^{o}$ of HoFe₂ is even positive than that of Ho₂Fe₁₇. Moreover, the $\Delta H_{298}^{o}$ values for all the REFe₂ compounds by Meschel et al. were found to be systematically more positive than other experimental data and optimized data, and are inconsistent with the minima of the enthalpy of mixing. So the result by Meschel et al. was treated as less reliable.

Fig. 16

Enthalpy of formation at 298 K ($\Delta H_{298}^{o}$) of the intermetallic Fe-Ho compounds compared to the experimental data[55,65]

Figure 17 shows the low-temperature C _p of HoFe₂ measured by Germano et al.[57] using AC. Germano et al.[57] also derived the $S_{298}^{o}$ (127.4 J mol⁻¹ K⁻¹) from the C _p data. The high-temperature C _p for HoFe₂ phase was estimated based on the N–K rule with small adjustment as tabulated in Table 2 to reproduce the experimental C _p at 300 K. The heat content (H_1373K − H_298K) of HoFe₂ measured by Meschel et al.[55] was 37.4 ± 1.5 kJ mol-atom⁻¹ compared to the calculated value 39.56 kJ mol-atom⁻¹ in the present study. Thus, the C _p of HoFe₂ shown in Fig. 17 is considered to be reliable. The C _p of HoFe₃, Ho₆Fe₂₃, and Ho₂Fe₁₇ were also estimated from the N–K rule. The magnetic properties (Bohr magnetons and Curie temperatures) of the compounds were obtained from the compilation of Buschow.[66]

Fig. 17

The optimized C _p of HoFe₂ along with the experimental data[57]

Five model parameters including two small temperature dependent terms were determined to describe the thermodynamic behavior of liquid Fe-Ho phase. As mentioned above, the estimated enthalpy of mixing was used to constrain the model parameters of liquid phase first. Then, the thermodynamic data of solid HoFe₂ and other compounds were simultaneously considered to optimize the model parameters of all phases in the system to reproduce the phase diagram. The optimized $S_{298}^{o}$ of HoFe₂ is 122.10 J mol⁻¹ K⁻¹ which is slightly smaller than the experimental data of 127.4 J mol⁻¹ K⁻¹.[57] The $S_{298}^{o}$ of other compounds were slightly adjusted if necessary from those derived from the N–K rule. It should be noted that the congruent melting behavior of HoFe₂ was considered instead of peritectic melting as discussed above. All other invariant reactions are well reproduced. The invariant reactions are summarized in Table 6. No measurable mutual solubilities of Fe and Ho in both fcc, bcc and hcp solid solutions were considered in the present optimization.

Table 6 Calculated invariant reactions with the experimental data in the Fe-Ho system

The previous assessment conducted by Konar[41] has limitation due to the lack of thermodynamic data for the Fe-Ho liquid, similar to the assessment for the Fe-Dy system. In a later optimization by Kardellass et al.,[74] a significantly large temperature dependent term was used to assess the Gibbs energy of the liquid phase of the Fe-Ho system. As shown in Fig. 15, the minimum of the enthalpy of mixing calculated from the study by Kardellass et al. is −11.5 kJ mol⁻¹ at 0.5 mol fraction of Ho, which is noticeably different from the estimated data. The assessed $\Delta H_{298}^{o}$ of Ho₂Fe₁₇ (−0.295 kJ mol-atom⁻¹) and HoFe₂ (−2.65 kJ mol-atom⁻¹) are much smaller than the optimized values in the present study. In fact, their $\Delta H_{298}^{o}$ of HoFe₂ is close to that measured by Meschel et al.[55] (−2.6 ± 3.3 kJ mol-atom⁻¹). On the other hand, the $S_{298}^{o}$ of HoFe₂ optimized by Kardellass et al.[74] was 149.54 J mol⁻¹ K⁻¹ which is significantly larger than the experimental data of 127.4 J mol⁻¹ K⁻¹.[57]

3.5 The Fe-Er (Iron-Erbium) System

The Fe-Er system was assessed by Konar[41] and Zhou et al.[80] where four stable compounds ErFe₂, ErFe₃, Er₆Fe₂₃, and Er₂Fe₁₇ were considered.

The optimized phase diagram of the Fe-Er system is presented in Fig. 18 along with experimental data. Meyer[81] used both DTA and quenching experiments followed by XRD and metallographic (OM) phase analysis. It was noted that when Er amount exceeded 0.4 mol fraction, alloy samples reacted with the alumina crucible at elevated temperatures. In general, the melting points of the compounds reported by Meyer[81] were significantly lower than other studies. As stated by Buschow and Van Der Goot,[82] this could result from the reduction of the crucible materials (Al₂O₃) by liquid Er during the experiment. Therefore, Buschow and Van Der Goot[82] used a larger amount of sample than Meyer[81] for the DTA experiment using Al₂O₃ crucible to minimize the composition change. The presences of four compounds in the binary system were confirmed by XRD and OM. ErFe₂ was reported to have a congruent melting at 1360 °C, whereas ErFe₃, Er₆Fe₂₃, and Er₁₇Fe₂ were reported to have peritectic meltings at 1345, 1330, and 1355 °C, respectively. Koleshnikov et al.[83] conducted experiments for the Fe-Er system using DTA and reported liquidus similar to the results of Buschow and Van der Goot[82] except for ErFe₃ and ErFe₂. They claimed that ErFe₃ melted congruently and ErFe₂ melted incongruently, which is contradictory to the results of Buschow and Van der Goot.[82] No mutual solubility of Fe and Er was reported in any investigation.

Fig. 18

The optimized phase diagram of the Fe-Er system with the experimental data[81,82,83]

No experimental enthalpy of mixing for liquid Fe-Er system is available. The optimized enthalpy of mixing of the Fe-Er liquid at 1833 K from this study is depicted in Fig. 19 along with the Miedema prediction published by Zhou.[80] Similar to the Fe-Ho system, the enthalpy of mixing in liquid Fe-Er solution was estimated based on the periodic trend of the Fe- heavy RE system, as presented in Fig. 19: ΔH _mix,Fe-Er = ΔH _mix,Fe-Dy + 0.5(ΔH _mix,Fe-Lu − ΔH _mix,Fe-Dy), based on the experimental results of Fe-Dy and Fe-Lu.[52] The predicted value for the present system from the Miedema method[80] is similar to the experimental value of the Fe-Tb system (see Fig. 6), and the value is much positive than the estimated value based on the systematic trend of the enthalpy of mixing in the Fe-RE system.

Fig. 19

Integral enthalpy of mixing in Fe-Er liquid at 1833 K with assessments[41,80]

The $\Delta H_{298}^{o}$ of compounds was measured by Norgren et al.,[71] Gozzi et al.[65] and Meschel et al.[55] These results are plotted in Fig. 20 along with the previous optimization by Zhou et al.[80] Norgren et al.[71] measured the $\Delta H_{298}^{o}$ of ErFe₂ and ErFe₃ to be −12.5 ± 1.4 and −7.9 ± 1.4 in kJ mol-atom⁻¹, respectively, using the SC-Al at 1100 K. Gozzi et al.[65] conducted Miedema calculation and EMF measurements and reported the $\Delta H_{298}^{o}$ for Er₂Fe₁₇. Meschel et al.[55] performed DSCa of the ErFe₂ in a BN crucible.

Fig. 20

Enthalpy of formation at 298 K ($\Delta H_{298}^{o}$) of the intermetallic Fe-Er compounds compared to the experimental data[55,65,71]

The low-temperature C _p of ErFe₂ was reported by Germano et al.[57] using AC in the temperature range of 15-300 K, as shown in Fig. 21. The $S_{298}^{o}$ of ErFe₂ was calculated to be 133.79 J mol⁻¹ K⁻¹. The high-temperature C _p was estimated using the N–K rule with a small modification of the temperature-independent term to have continuity with experimental low-temperature C _p . Meschel et al.[55] also reported enthalpy change (H ₁₃₇₃ − H ₂₉₈) to be 34.8 ± 1.7 kJ mol-atom⁻¹ as part of their calorimetric experiment, which is calculated to be 37.52 kJ mol-atom⁻¹ from the present evaluated C _p . Similarly, the C _p functions for ErFe₃, Er₆Fe₂₃ and Er₂Fe₁₇ were estimated from the N–K rule. The magnetic properties of all compounds were taken from literature.[66,67,78]

Fig. 21

The optimized C _p of ErFe₂ along with the experimental data[57]

In the present optimization, the estimated enthalpy of mixing of the liquid phase in Fig. 19 and $\Delta H_{298}^{o}$ of compounds by Norgren et al.[71] were considered along with phase diagram data to optimize the model parameters of all phases. The liquid was described by using seven parameters including two temperature dependent terms. Enthalpies of formation of the compounds ErFe₂ and ErFe₃ were optimized to be −11.0 and −8.26 kJ mol-atom⁻¹, respectively, compared to the results (−12.2 ± 1.2 and −8.2 ± 1.2 kJ mol-atom⁻¹, respectively) by Norgren et al. The optimized $S_{298}^{o}$ of ErFe₂ is 128.27 J mol⁻¹ K⁻¹ which is slightly lower than the experimentally determined value of 133.79 J mol⁻¹ K⁻¹. The $S_{298}^{o}$ of other compounds were optimized by the slight modification from the values obtained from the N–K rule. The overall experimental phase diagram data was well reproduced in the present optimization. The optimized invariant reactions are compared with experimental data in Table 7. All invariant reactions in this study are in good agreement with the results from Buschow and Van der Goot[82] except ErFe₃. There is inconsistency in the melting behavior of ErFe₃: incongruent melting by Buschow and Van der Goot[82] and congruent melting by Koleshnikov et al.[83] It was very difficult to reproduce the incongruent melting of ErFe₃ with simultaneous reproduction of all other data. It should be also noted that X ⁵_Fe-Fe parameter in the liquid phase was introduced to reproduce the peritectic melting of Er₂Fe₁₇.

Table 7 Calculated invariant reactions with the experimental data in the Fe-Er system

The previous assessment by Zhou et al.[80] put the emphasis on the phase diagram data by Buschow and Van der Goot.[82] However, as depicted in Fig. 19, the enthalpy of mixing of Fe-Er liquid assessed by Zhou et al. has the minimum enthalpy of mixing in the Er-rich side, which is most probably wrong from the systematic change in the enthalpy of mixing of the Fe-RE system. They used a large temperature dependent term in liquid parameters, and the $\Delta H_{298}^{o}$ of Er₂Fe₁₇ was too negative (about −10 kJ mol-atom⁻¹) compared to the experimental data, as shown in Fig. 20, and the assessed value of $S_{298}^{o}$ for ErFe₂ is 149.53 J mol⁻¹ K⁻¹ which is much larger than the experimental value[57] of 133.79 J mol⁻¹ K⁻¹.

3.6 The Fe-Tm (Iron-Thulium) System

The system was reviewed by Kubaschewski[42] and Okamoto,[84] and thermodynamically assessed by Konar[41] and Kardellass et al.[85] Four stable compounds TmFe₂, TmFe₃, Tm₆Fe₂₃, and Tm₂Fe₁₇ were considered in this system.

The thermodynamic and magnetic properties of the solid and liquid phases are presented in Table 2 and the invariant reactions are presented in Table 8. The phase diagram of the Fe-Tm system is shown in Fig. 22. Kolesnichenko et al.[86] investigated this phase diagram using TA followed by XRD and OM phase analysis but did not mention the detailed experimental procedure in the paper. TmFe₂ and Tm₆Fe₂₃ with cubic structure and TmFe₃ and Tm₂Fe₁₇ with hexagonal structure were found as intermetallic compounds. TmFe₂ was reported to melt congruently at 1300 °C, while TmFe₃ and Tm₆Fe₂₃ melted incongruently. Tm₂Fe₁₇ was assumed to melt incongruently without strong experimental evidence. The mutual solubility of Fe and Tm has not been investigated, but it is expected to be nearly zero as can be seen in neighboring Fe-RE system such as the Fe-Er and Fe-Lu systems.

Table 8 Calculated invariant reactions with the experimental data in the Fe-Tm system

Fig. 22

The optimized phase diagram of the Fe-Tm system with the experimental data[86]

No enthalpy of mixing was experimentally investigated. In the present study, the enthalpy of mixing was estimated from the experimental data of the adjacent Fe-Ho and Fe-Er systems: ΔH _mix,Fe-Tm = ΔH _mix,Fe-Dy + 0.75 (ΔH _mix,Fe-Lu − ΔH _mix,Fe-Dy), as presented in Fig. 23. It should be noted that the calculated enthalpy of mixing from the previous assessment by Kardellass et al.[85] show the unmixing trend, which is most probably incorrect considering the enthalpy trend of the Fe-RE system.

Fig. 23

Integral enthalpy of mixing in Fe-Tm liquid at 1833 K with assessments[85]

Meschel et al.[55] measured the $\Delta H_{298}^{o}$ of TmFe₂ to be −2.2 ± 2.8 kJ mol-atom⁻¹ using the DSCa, and the result is plotted in Fig. 24. The low-temperature C _p of TmFe₂ was measured by Germano et al.[57] using AC. The $S_{298}^{o}$ of TmFe₂ derived from the C _p data is 127.55 J mol⁻¹ K⁻¹. The high-temperature C _p was estimated using the N–K rule with a slight adjustment of temperature independent term to make the continuity from the experimentally measured low-temperature C _p . The magnetic contributions to heat capacity were taken from Buschow.[66] The C _p of TmFe₂ is presented in Fig. 25. Meschel et al.[55] also measured the heat content change, H ₁₃₇₃ − H ₂₉₈ to be 44.0 ± 1.8 kJ mol-atom⁻¹. According to the evaluated C _p data in Fig. 25, the heat content is calculated to be 36.45 kJ mol-atom⁻¹. The heat capacities of other compounds were also estimated using the N–K rule with the consideration of the magnetic transition,[66] and $\Delta H_{298}^{o}$ and $S_{298}^{o}$ of the compounds were optimized to reproduce the phase diagram data.

Fig. 24

Enthalpy of formation at 298 K ($\Delta H_{298}^{o}$) of the intermetallic Fe-Tm compounds compared to the experimental data[55]

Fig. 25

The optimized C _p of TmFe₂ along with the experimental data[57]

In the present optimization, the estimated enthalpy of mixing and experimental phase diagram data was mainly taken into account to obtain the model parameters of liquid phase first. The Gibbs energy of solid compounds was optimized to reproduce their melting temperature. The optimized $\Delta H_{298}^{o}$ of TmFe₂ is −11.23 kJ mol-atom⁻¹ which is more negative than the experimental value of −2.2 ± 2.8 kJ mol-atom⁻¹ determined by Meschel et al.[55] However, it should be noted that the $\Delta H_{298}^{o}$ of REFe₂ measured by Meschel et al.[55] in all other Fe-RE systems tend to be more positive than other experimental data and optimized results in the present study, and also considering the congruent melting of this compound, the $\Delta H_{298}^{o}$ of this compound should be the most negative among all intermetallic phases in this system, as in all other Fe-heavy RE systems. Therefore, it is hard to believe the data of Meschel et al. The optimized $S_{298}^{o}$ of TmFe₂ is 126.72 J mol⁻¹ K⁻¹, consistent with the experimental value of 127.55 J mol⁻¹ K⁻¹. All the phase diagram results are well reproduced by the present optimization and the optimized invariant reactions are summarized in Table 8.

The previous thermodynamic assessment by Kardellass et al.[85] was based on phase diagram experiments by Kolesnichenko et al.[86] The excess Gibbs energy of liquid phase was modeled using six parameters including three temperature dependent terms. As shown in Fig. 23, the assessed enthalpy of mixing had a strange unmixing tendency, which is less plausible. Their high-temperature C _p function for TmFe₂ shows discontinuity from the low-temperature C _p data determined by Germano et al.[57] The optimized $S_{298}^{o}$ of TmFe₂ is 133.83 J mol⁻¹ K⁻¹, which is much larger than experimental value of 127.56 J mol⁻¹ K⁻¹.[57]

3.7 The Fe-Lu (Iron-Lutetium) System

The binary Fe-Lu system was reviewed by Kubaschewski[42] and Okamoto,[84] and assessed by Konar[41] and Kardellass et al.[85] The Fe-Lu system has characteristic four intermetallic phases LuFe₂, LuFe₃, Lu₆Fe₂₃, and Lu₂Fe₁₇. The thermodynamic and magnetic properties of the solid and liquid phases are presented in Table 2 and the invariant reactions are presented in Table 9.

Table 9 Calculated invariant reactions with the experimental data in the Fe-Lu system

The optimized Fe-Lu phase diagram is presented in Fig. 26. Kolesnichenko et al.[86] investigated this system using TA followed by XRD and OM. Four compounds Lu₂Fe₁₇, Lu₆Fe₂₃, LuFe₃, and LuFe₂ and bcc Fe, fcc Fe and hcp Lu phases were experimentally found in this system. According to Kolesnichenko et al.,[86] LuFe₂ melted congruently at 1345 °C and the remaining compounds LuFe₃, Lu₆Fe₂₃, and Lu₂Fe₁₇ were formed by peritectic reactions at 1310, 1290, and 1328 °C, respectively. Two eutectic reactions were observed at 970 °C and 0.73 mol fraction Lu and at 1275 °C, and 0.18 mol fraction Lu. No mutual solubility of Fe and Lu has been determined experimentally.

Fig. 26

The optimized phase diagram of the Fe-Lu system with the experimental data[86]

Figure 27 shows the integral and partial enthalpies of liquid measured by Ivanov et al.[52] Ivanov et al.[52] measured the partial enthalpy of mixing using a SC of Fe-rich alloys using zirconia crucible, and Lu-rich alloys using molybdenum crucible at 1950 K. The derived integral enthalpy of mixing shows a minimum at 0.4 mol fraction Lu with a value of −11.56 ± 0.57 kJ mol⁻¹. The Gibbs energy of the liquid phase in the present study is optimized based on these experimental data.

Fig. 27

Enthalpy of mixing in the Fe-Lu system at 1950 K along with the experimental data[52] and assessments[85] (a) Integral enthalpy of mixing and (b) partial enthalpy of mixing

Figure 28 shows the $\Delta H_{298}^{o}$ of the compounds in the Fe-Lu system along with the data from Gozzi et al.[65] and Meschel et al.[55] Gozzi et al.[65] conducted Miedema calculation and EMF measurements and reported the $\Delta H_{298}^{o}$ for Lu₂Fe₁₇. The $\Delta H_{298}^{o}$ of LuFe₂ measured by Meschel et al.[55] using DSCa was −3.6 ± 3.1 kJ mol-atom⁻¹ which is even lower than that of Lu₂Fe₁₇ determined by Gozzi et al. Such a small $\Delta H_{298}^{o}$ for LuFe₂ is less plausible as discussed above in the previous Fe-Tm system. The low-temperature C _p of LuFe₂ and Lu₂Fe₁₇ were measured by Germano et al.[57] using AC and Tereshina and Andreev[87] using a Physical Property Measurement System (PPMS) magnetometer device, respectively, as shown in Fig. 29. The high-temperature C _p of both phases were estimated using the N–K rule with a small modification of temperature independent term to have continuity with low-temperature C _p data. The magnetic contributions of both phases were taken from Buschow.[66] As part of their calorimetric measurement, Meschel et al.[55] measured the heat content change H ₁₃₇₃  − H ₂₉₈ of LuFe₂. The calculated value from the heat capacity is 35.49 kJ mol-atom⁻¹ consistent with experimental value, 32.9 ± 3.1 kJ mol-atom⁻¹. The $S_{298}^{o}$ of LuFe₂ and Lu₂Fe₁₇ calculated from the low-temperature C _p in Fig. 29 are 107.41 J mol⁻¹ K⁻¹ and 647.74 J mol⁻¹ K⁻¹, respectively. The heat capacities of the other two phases, Lu₆Fe₂₃ and LuFe₃ were estimated using the N–K rule, and their $\Delta H_{298}^{o}$ and $S_{298}^{o}$ were optimized to reproduce the phase diagram data as close as possible.

Fig. 28

Enthalpy of formation at 298 K ($\Delta H_{298}^{o}$) of the intermetallic Fe-Lu compounds compared to the experimental data[55]

Fig. 29

The optimized C _p of LuFe₂ and Lu₂Fe₁₇ along with the experimental data[57,87]

Seven model parameters including two small temperature dependent terms were used to describe the Gibbs energy of liquid phase. The liquid enthalpy data and phase diagram data were simultaneously used to evaluate the Gibbs energy of liquid phase and Gibbs energies of solid compounds. The optimized $S_{298}^{o}$ of LuFe₂ and Lu₂Fe₁₇ are 106.6 J mol⁻¹ K⁻¹ and 646 J mol⁻¹ K⁻¹, respectively, which are close to the experimental values of 107.41 J mol⁻¹ K⁻¹[57] and 647.74 J mol⁻¹ K⁻¹,[87] respectively. The optimized $\Delta H_{298}^{o}$ of LuFe₂ is −11.77 kJ mol-atom⁻¹ which is much negative than experimental data by Meschel et al.[55] Similar problem of the data for REFe₂ by Meschel et al. were pointed out above in the other Fe-RE systems. All invariant reactions by Kolesnichenko et al.[86] were reproduced within ±6 °C in the present optimized diagram except the melting behavior of LuFe₃. It was very difficult to reproduce incongruently melting of LuFe₃ as reported by Kolesnichenko et al. and it is calculated to melt congruently in the present study. Similar difficulty was witnessed in the Fe-Er and Fe-Tm systems too. The mutual solubilities of Fe and Lu for the terminal fcc, bcc and hcp solid solutions were assumed to be negligible in the present optimization.

The assessed phase diagram by Kardellass et al.[85] can reproduce all experimental invariant reactions. Like the previous Fe-Tm system, however, the critical weakness of the assessment by Kardellass et al.[85] is the unrealistic enthalpy of mixing of liquid Fe-Lu solution as shown in Fig. 27, compared to the experimental data. The C _p functions for LuFe₂ and Lu₂Fe₁₇ optimized in their study were also inconsistent with experimental C _p data. In their assessment, a homogeneity range was introduced to the compounds Lu₂Fe₁₇ and Lu₆Fe₂₃ without strong experimental evidence.

3.8 The Fe-Y (Iron-Yttrium) System

The Fe-Y system has been reviewed by Gschneidner[88] and Zhang et al.[89] and assessed by Du et al.,[90] Konar,[41] Lu et al.[91] and Kardellass et al.[92] The crystal structures of the intermetallic phases considered in the present assessment are tabulated in Table 1. The optimized thermodynamic and magnetic parameters of the solid and liquid phases are listed in Table 2 and the invariant reactions are summarized in Table 10.

Table 10 Calculated invariant reactions with the experimental data in the Fe-Y system

The optimized binary phase diagram of the Fe-Y system is presented in Fig. 30. The binary phase diagram was investigated by Farkas and Bauer,[93] Domagala et al.,[94] and Nagai et al.[72] Farkas and Bauer[93] studied the Fe-rich side (up to 30 wt.% Y) by TA. The presence of YFe₅ and YFe₄ was analyzed by XRD and OM. The phase equilibria and the transition temperatures were also measured by Farkas and Bauer. But due to the high heating/cooling rate of 400 °C min⁻¹ in the TA, severe undercooling was introduced. Therefore, their data cannot be used in the present study. Domagala et al.[94] used visual melting observation and quenching experiment followed by OM and XRD phase analysis. A eutectic reaction (Liq → hcp Y + YFe₂) was found at 0.65 mol fraction Y at 900 °C. With the help of XRD analysis, the polymorphic transition from hcp Y to bcc Y at 1478.5 °C and melting point of Y was revealed. Domagala et al. reported congruent meltings of YFe₃, YFe₄, and YFe₉, but the incongruent melting of YFe₂. As part of the thermodynamic study by Nagai et al.[72] using KEMS, the phase transition was identified from the measured activity change at 1200-1300 °C. The mutual solid solubility of Fe and Y is shown in Fig. 30(b) and (c). Domagala et al.[94] measured the solubility by XRD measurement and reported the terminal solubilities of 0.015 mol fraction of Fe in Y at 900 °C, and 0.066 mol fraction solubility of Y in Fe at 1350 °C. Li and Xing[95] conducted EPMA to determine the solubility of Y in bcc Fe from 600 to 880 °C.

Fig. 30

The optimized phase diagram of the Fe-Y system with the experimental data (a) Overall composition range,[72,93,94] (b) in the Fe-rich side,[94,95] and (c) in Y rich side[94]

Ryss et al.[96] and Sudavtsova et al.[97] measured the partial enthalpy of mixing of liquid Fe-Y solution using SC at 1873 and 1870 K, respectively, and derived the integral enthalpy of mixing, as shown in Fig. 31(a) and (b). Ryss et al. suggested that the minimum of the integral enthalpy of mixing is about −8.144 kJ mol⁻¹ at 0.47 mol fraction Y. Sudavtsova et al. conducted SC in the Fe-rich side of the Fe-Y liquid under He atmosphere. The partial enthalpy of mixing of Y in the Fe-rich side determined by Sudavtsova et al. is more positive than that of Ryss et al. In the present optimization, the enthalpy of mixing data by Ryss et al. was more weighted than the results of Sudavtsova et al.

Fig. 31

Enthalpy of mixing in the Fe-Y system at 1873 K along with experimental data[96,97] and assessments[41,90,92] (a) Integral enthalpy of mixing and the previous optimization and (b) partial enthalpy of mixing

The $\Delta H_{298}^{o}$ of the compounds were estimated by Van Mal et al.[98] and Watson and Bennet.[99] Van Mal et al. predicted the $\Delta H_{298}^{o}$ of the compounds based on Miedema’s theory.[98] Watson and Bennet developed a simple electron band theory model which can predict the $\Delta H_{298}^{o}$ of the intermetallic phases of metals with d and/or f bands. Gozzi et al.[65] reported the $\Delta H_{298}^{o}$ of the Y₂Fe₁₇ phase, derived from the Gibbs energy of formation determined using galvanic cells with CaF₂ electrolyte, to be −8.7 kJ mol-atom⁻¹. As seen in the above Fe-RE systems, the $\Delta H_{298}^{o}$ of RE₂Fe₁₇ measured by Gozzi et al.[65] is typically much negative than the experimental data and also optimized values in the present study. As no other experimental $\Delta H_{298}^{o}$ are available in the literature, the enthalpy was modeled based on the Gibbs energy of formation of the Fe-Y compounds measured by Subramaniam and Smith,[100] as depicted in Fig. 32(a) and (b). Subramaniam and Smith[100] employed EMF technique with solid electrolyte to determine the Gibbs free energy of four intermetallic phases over the temperature range between 620 and 998 °C. Their results of the Gibbs energy of formation and derived enthalpy and entropy of formation from the elements at 973 K are plotted in Fig. 32.

Fig. 32

Thermodynamic properties of the Fe-Y compounds with experimental data[100] and assessments,[41,90,92] (a) Gibbs energy of formation of compounds at 973 K, (b) Enthalpy of formation at 973 K and (c) Entropy of formation of compounds at 973 K

The C _p of YFe₂ from 300 to 600 K was measured by Dariel et al.[101] to determine Curie temperature using DSC. However, actual C _p data were not given in the paper except for the Curie temperature at 535 K. The low-temperature C _p of Y₂Fe₁₇ was measured by Mandal et al.[102] with a PPMS device, as presented in Fig. 33. The $S_{298}^{o}$ derived from this result is 660.71 J mol⁻¹ K⁻¹. In the present study, the high-temperature C _p function of Y₂Fe₁₇ was obtained by the N–K rule with a slight modification of the temperature independent term to have continuity with the low-temperature C _p . It should be noted that the ΔS ₉₇₃ calculated from the C _p of Y₂Fe₁₇ in Fig. 33 is inconsistent with the results of Subramaniam and Smith[100] in Fig. 32(c). This casts doubt on the accuracy of the EMF results of Subramaniam and Smith. The C _p functions of other compounds were also obtained using the N–K rule. The magnetic contributions to C _p functions obtained by several researchers[103,104,105,106] are listed in Table 2.

Fig. 33

Low-temperature heat capacity of Y₂Fe₁₇[102]

The activities of Fe and Y at 1473 and 1573 K were derived by Nagai et al.[72] from their KEMS experiment, as shown in Fig. 34. Owing to the high affinity of Y containing alloys to oxygen, they were stored under spindle oil. Oxygen amount of alloys in pre and post experiment was minimal, as determined by inert gas fusion-infrared absorptiometry. The KEMS measurements were carried out for the samples across the entire composition in the temperature range of 1473-1573 K. The ion currents for Y were determined with the help of ⁸⁹Y isotope. The more abundant ⁵⁴Fe and ⁵⁶Fe were chosen for Fe detection among the possible ⁵⁴Fe, ⁵⁶Fe, ⁵⁷Fe and ⁵⁸Fe isotopes. As can be seen in Fig. 34, the activity of Y shows a large negative deviation from ideal solution behavior in Y rich side, which seems to be too negative compared to the rest of liquid Fe-RE solution. This will be discussed again below.

Fig. 34

The activity of Fe and Y at (a) 1473 K and (b) 1573 K with the experimental data[72]

In the present study, the enthalpy of the liquid phase, Gibbs energy of solid compounds and low-temperature C _p ($S_{298}^{o}$) of Y₂Fe₁₇ were simultaneously considered along with phase diagram information to optimize the thermodynamic parameters of solid and liquid phases. The enthalpy of mixing and partial enthalpy of liquid phase were well reproduced. The optimized $S_{298}^{o}$ value of Y₂Fe₁₇ in the present work is 623.0 J mol⁻¹ K⁻¹, which is within 5% error from experimental value of 660.71 J mol⁻¹ K⁻¹ from low-temperature C _p .[102] It should be noted that if the experimental $S_{298}^{o}$ (660.71 J mol⁻¹ K⁻¹) is used, a larger error will be obtained in the entropy of formation in Fig. 32(c). The calculated formation Gibbs energies, enthalpies, and entropies of compounds at 983 K from the present study are plotted in Fig. 32 along with the experimental data of Subramaniam and Smith.[100] The calculated Gibbs energy data from the present study is slightly lower than the experimental values. But the previous assessments[90,91,92,107] show a much larger deviation from the experimental data. The enthalpy of formation data also shows similar results. In the case of the entropies of formation, the calculated values from the present study are positive as opposed to the negative experimental data. As discussed above, the experimental entropy of formation of Y₂Fe₁₇ at 298 K is already positive,[102] and the entropy of formation at 973 K would most probably be positive as calculated from the present data.

The activities of Fe and Y at both 1573 and 1473 K are substantially underestimated. This can be due to an error in the ion current measurements for Fe and Y, and high oxygen contamination of Y. The activities of Y and Fe are also well reproduced at 1473 and 1573 K in the Fe-rich region, while the activity of Y in the Y-rich region is more positive than the experimental results of Nagai et al.[72] In order to reproduce the activity of Y of Nagai et al.[72] in the Y-rich region, more negative Gibbs energy of liquid is required, which results in much lower eutectic temperature for Liq → YFe₂ + hcp Y. In the present study, more weight was given to the phase diagram than the KEMS data by Nagai et al.[72] All phase diagram data from Domagala et al.[94] was well reproduced, as shown in Fig. 30(a), except the congruent melting of Y₆Fe₂₃. As mentioned earlier, there is no critical evidence of peritectic melting of this compound. The mutual solubilities of Fe and Y for the terminal fcc, bcc, and hcp solid solutions were reproduced by adding one regular solution model parameter. In the case of the fcc solution, the temperature dependent term was also necessary to reproduce the steep solubility as reported by Domagala et al.,[94] as shown in Fig. 30(b) and (c). All invariant reactions in the binary system are listed in Table 10.

The Fe-Y system has been assessed by Du et al.,[90] Lu et al.[91] and Kardellass et al.[92,107] Du et al.[90] assessed the thermodynamic properties of the liquid and solid phases, where the Gibbs energy of compounds was significantly different from the experimental data in Fig. 32. The Gibbs energy of formation data reported by Subramaniam and Smith[100] was only reproduced for Y₂Fe₁₇ and not for Y₆Fe₂₃, YFe₃, and YFe₂. Lu et al.[91] reassessed the modeling conducted by Du et al.[90] All the intermetallic phases were considered to be stoichiometric phases. Neither the thermodynamic parameters for the liquid or solid phases, nor the invariant reactions were given in their paper, so it is hard to evaluate their work. The phase diagram depicted is identical to that assessed from the present study, except no terminal solid solutions were considered. Kardellass et al.[92,107] presented a phase diagram with considerable homogeneity range in the Y₆Fe₂₃ and YFe₂ phase without any experimental evidence. Several investigators[108,109] reported the presence of a stoichiometric phase, without any solid solubility of Fe in Y₆Fe₂₃. Kardellass et al.[92,107] did not consider the terminal solid solutions of Y and Fe in their assessment.

4 Discussion and Systematic Analysis

Certainly, there are systematic trends in the thermochemical properties and phase diagrams of the Fe-RE systems. The trends observed in the enthalpy and entropy of liquid phase, the enthalpy and entropy of formation of solid phases, and the phase diagram information in the Fe-RE system will be discussed below.

The optimized enthalpy of mixing of the liquid Fe-RE solution at 1873 K is presented in Fig. 35, which manifests a periodic trend. The enthalpy of mixing for all liquid Fe-RE alloys shows maximum or minimum values at around 0.4 mol fraction RE. It varies systematically from 5.5 to −12 kJ mol⁻¹ with increasing atomic number of RE. This systematic change in the mixing enthalpy was taken into account to estimate the enthalpy of mixing of the Fe-Ho, Fe-Er, and Fe-Tm solution, where no experimental data are available. The position of minimum enthalpies of mixing is near 0.4 mol fraction of RE. This asymmetric nature of mixing enthalpy may result from the size difference between Fe and RE: RE has a larger atomic size (r_La = 188 pm and r_Lu = 173 pm) than Fe (r_Fe = 140 pm).[58] It should be noted that the Fe-Y system shows a bit different behavior than other RE systems. A similar trend was observed in the Mn-RE systems[21] where enthalpy of mixing varies from 8 to −8 kJ mol⁻¹. In other binary RE systems, the enthalpies of mixing are already very negative, so it is hard to observe the clear trend in enthalpy of mixing data: for example, the Si-RE alloys (−65 to −75 kJ mol⁻¹),[24] Pb-RE alloys (−40 to −60 kJ mol⁻¹),[26] Al-RE systems (−30 to −40 kJ mol⁻¹)[27] and Zn-RE systems (−20 to −32 kJ mol⁻¹).[30]

Fig. 35

Comparison of integral enthalpies of liquid Fe-RE at 1873 K

The optimized $\Delta H_{298}^{o}$ and $\Delta S_{298}^{o}$ of the binary Fe-RE systems from the respective elements are presented in Fig. 36 and 37, respectively, which also show definitive similarity in trend. The REFe₂ compound shows the most negative $\Delta H_{298}^{o}$ value of all Fe-RE systems, and also most negative $\Delta S_{298}^{o}$. It should be also noted that the $\Delta H_{298}^{o}$ values measured by Meschel et al.[55] are mostly more positive than the optimized values in the present study. Therefore, more investigation into these thermodynamic properties is necessary for future.

Fig. 36

Comparison of enthalpies of formation of Fe-RE compounds at 298 K. (based on one mole of Fe and RE)

Fig. 37

Comparison of entropies of formation of Fe-RE compounds at 298 K. (based on one mole of Fe and RE) from elemental references

Similarly, a systematic trend is observed in the melting temperatures of the intermetallic compounds. The melting temperatures of the Fe-RE compounds and the eutectic temperature (eutectic reaction: Liq → REFe₂ + RE) is shown in Fig. 38. The melting point of all the compounds is increasing gradually with increasing atomic number of RE. The increase becomes flattened out from Fe-Ho system. The temperature of the eutectic reaction ‘Liq → REFe₂ + RE’ shows a clear increasing trend with increasing atomic number of RE. A similar trend was previously discussed in the Si-RE system.[24,25]

Fig. 38

Systematic changes in the melting temperatures of Fe-RE compounds (a) RE₂Fe₁₇ and RE₆Fe₂₃, (b) REFe₃ and REFe₂ (c) RE melting point and the eutectic temperatures between REFe₂ and RE

A collection of all the optimized phase diagrams of the Fe-RE systems obtained in the present study and by Konar et al.,[31] is shown in Fig. 39. As can be seen in the collection, no intermetallic phase forms in the Fe-La system, and then RE₂Fe₁₇, REFe₃, and REFe₂ are gradually forming with increasing atomic number of RE. Additionally, RE₆Fe₂₃ compound forms in the heavy RE systems. The melting behavior of intermetallic compounds also changes from peritectic to congruent melting with increasing atomic number of RE. This means that the enthalpy of formation (Gibbs energy of formation) of intermetallic Fe-RE phase becomes more stable (negative) with increasing atomic number of RE, as also seen in the enthalpy of mixing in liquid Fe-RE solution. In the case of the Fe-Er, -Tm, and -Lu systems, however, we found that it is necessary to add positive excess parameters in the Fe-rich liquid solution to reproduce the peritectic melting behavior of RE₂Fe₁₇ compounds.

Fig. 39

A collection of the optimized phase diagrams of the Fe-RE systems

The enthalpy of formation is one of the important thermodynamic properties of solid compounds that determine their thermal stabilities. Amongst the semi-empirical approaches to predict the standard enthalpy of formation ($\Delta H_{298}^{o}$), Miedema model[98,110,111] improved by Zhang et al.[112,113,114] using statistical approach (ab initio throughput data-mining (HTDM) method) was used in this study for the estimation of enthalpies of formation all the Fe-RE compounds. The semi-empirical nature of this Miedema model comes from the element specific constants such as electronic configuration, electron densities, and atomic radius. In Fig. 40, the optimized enthalpies of formation of all Fe-RE compounds in the present study are compared to those predicted from the Miedema model. In Fig. 40, only the intermetallic compounds in the Fe-heavy RE system was compared as Fe-light RE system has only a few intermetallic compounds. It should be noted that the Miedema results for the compounds in the Fe-Gd and Fe-Y systems would be less reliable because Zhang et al.[111,112,113] contain the same ionic radius and similar electron density parameter for Gd and Y. Similar incongruences were also observed in the ionic radius and electron density parameter of Ho.

Fig. 40

Comparison of the enthalpy of formation of Fe-RE compounds at 298 K between the optimized results (solid symbols) and predicted results using the Miedema method (empty symbols)[112,113,114]

In the previous studies[23,28] on RE containing systems, it was also demonstrated that the prediction from the Miedema model was less accurate for RE containing compounds. Although there are significant differences in the results from present optimization and the Miedema model, the general trend in the formation enthalpy is the same. The enthalpy of formation (per mol-atom) of a given Fe-RE system decreases in the order of RE₂Fe₁₇, RE₆Fe₂₃, REFe₃, and REFe₂. That is, the minimum of enthalpy of formation in solid state occurs at REFe₂. In the case of RE₂Fe₁₇, the enthalpies of formation are not significantly changed with the atomic number of RE, while those of REFe₂ are noticeably decreasing with increasing atomic number of RE.

In the present analysis, the general trends of the enthalpies in the solid and liquid state are plotted with the variation of the atomic number of RE. The physical and chemical properties of RE show the systematic changes with the atomic number of RE. For example, the electronegativity and density of RE increase with increasing atomic number of RE. Therefore, the general systematic trends discussed in this study can be also valid with electronegativity and density of RE.

In general, there is a lack of experimental information on the homogeneity range of terminal Fe and RE solid solution. Although it is expected to be small, such solubility should be further investigated for the complete understanding of the phase diagram of the Fe-RE system.

5 Conclusions

A critical evaluation and optimization of all available phase diagram and thermodynamic data of the Fe-RE (RE = Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y) systems was conducted to obtain consistent thermodynamic functions of all the phases in each system. The Gibbs energy functions for all stoichiometric compounds were obtained with consideration of their magnetic properties. The liquid Fe-RE phase was described using the MQM and the solid solution phases such as bcc, fcc, and hcp solutions were described using the one-sublattice Compound Energy Formalism.

In the thermodynamic modeling, it is found that the Fe-RE systems show systematic trends in the phase diagrams and thermodynamic properties. In general, the formation enthalpy of intermetallic compounds and enthalpy of mixing of liquid phase become more negative with an increasing atomic number of RE. The melting point of the intermetallic phases increases gradually with increasing atomic number of RE. Such trend was used to estimate unknown thermodynamic properties of several Fe-RE systems, and clarify the melting behavior of compounds. In conclusion, in the present study, a more accurate thermodynamic description of the Fe-RE system is achieved through simultaneous thermodynamic optimization of all Fe-RE systems.