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Research Article  |  Open Access  |  11 Jul 2023

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

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J Mater Inf 2023;3:15.
10.20517/jmi.2023.08 |  © The Author(s) 2023.
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Abstract

This study focuses on investigating phase equilibria and thermodynamic stability of intermetallic compounds made of the transition metal Fe and rare earth (RE) elements. By using the CALPHAD method and reliable experimental information from the literature, the binary systems of Fe-Y, Fe-Er, and Fe-Lu were reassessed. To improve our previous calculations of Fe-RE (RE = Tb and Dy) binary systems, the Gibbs energy expressions of intermetallic compounds Fe2Tb and Fe2Dy were modified to avoid artificial breaks in their heat capacity curves. Thermodynamic parameters obtained are self-consistent, and the Gibbs energies of the Fe-RE (RE = Tb, Dy, Er, Lu, and Y) phases were accurately expressed to reappear available both thermodynamic data and phase equilibria. This work was further combined with the previous calculations of the Fe-RE (RE = La, Ce, Pr, Nd, Sm, Gd, Ho, and Tm) systems to discuss thermodynamic characteristics and phase equilibria of Fe-RE binary systems in detail. A trend was noticed for the change of thermodynamic properties and phase equilibria of the Fe-RE binary systems with RE atomic number. Generally, as the RE atomic number increases, the formation temperatures of the Fe-RE intermetallic compounds increase gradually, and the enthalpy of mixing of liquid Fe-RE (apart from Fe-Y and Fe-Ce) alloys and the enthalpy of formation of the Fe-RE (apart from Fe-Y, Fe-Ce, Fe-Gd, and Fe-Dy) intermetallic compounds become increasingly negative. The results provide a thorough set of thermodynamic parameters of thirteen Fe-RE binary systems, which could serve as a sound basis for developing a thermodynamic database of Fe-RE-based alloy systems.

Keywords

Fe-RE binary systems, Phase equilibria, Thermodynamic

INTRODUCTION

The intermetallic compounds made of the transition metal Fe and rare earth (RE) elements have been studied extensively because of their excellent physical properties[1-3]. For example, Fe17RE2 compounds exhibit large negative thermal expansion below the Curie temperature, which has numerous high-precision technology and positioning instrument applications[4-6]. Fe2RE compounds, such as Terfenol-D plates (i.e., Fe2TbxDy1-x compound), enable efficient energy and information conversion between electromagnetic and mechanical energy[7-9]. Furthermore, Fe2RE compounds have maximum strain characteristics among giant magnetostrictive materials[10,11]. Additionally, the intermetallic compounds made of RE elements, the transition metal Fe, and a third light element B (e.g., Nd2Fe14B, Dy2Fe14B, and Tb2Fe14B) have excellent magnetic properties as permanent magnets, widely used in various industries, including electronic information, electrical machine, and medical equipment[12-17]. The reliable phase diagrams and thermodynamic properties of the Fe-RE binary systems are essential to better understand the effect of RE metals on the phase formation of the Fe-RE intermetallic compounds.

Konar et al. executed a systematic calculation of the Fe-RE binary systems by using a modified quasichemical model to describe the liquid phase considering the reported experimental results[18,19]. Although the calculated thermodynamic properties and phase equilibria of the Fe-RE binary systems were in agreement with the experimental information, Konar et al. did not consider the heat capacity of Fe2RE and Fe17RE2 phases (e.g., Fe2Tb, Fe2Dy, Fe2Er, Fe2Lu, Fe17Lu2, and Fe17Y2) at low temperature (below 298 K) in their calculations[19]. Moreover, to ensure model compatibility inside the thermodynamic database in the Fe-RE-based systems. The description of the liquid phase reported by Konar et al.[18,19] needs to be revised to match thermodynamic calculations of the B-RE binary systems in our previous work[20-24]. In our earlier work, we also assessed the Fe-RE (RE = La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, and Tm) binary systems[25-29]. In addition, the Fe-RE (RE = Pm, Eu, and Yb) binary systems have not been assessed because of a lack of the experimental data, while the Fe-Y binary system was investigated through the experimental determination and thermodynamic calculations[19].

This study focuses on the Fe-RE (RE = Er, Lu, and Y) binary systems by the CALPHAD method in consideration of experimental results and previous calculations. To improve our previous calculations of the Fe-RE (Tb and Dy) binary systems[27], the Gibbs energies of intermetallic compounds Fe2Tb and Fe2Dy were modified to eliminate the artificial break point in their heat capacity curves that appear in the earlier work. By combining our present evaluations with previous optimizations[25-29], we provide a comprehensive discussion of phase equilibria and thermodynamic characteristics of the Fe-RE binary systems.

LITERATURE INFORMATION

Fe-La and Fe-Ce

Several studies have experimentally investigated the Fe-La binary phase diagram[30-33]. These studies identified three peritectic reactions and one eutectic reaction, but stable intermetallic compounds were not observed. In addition, the thermodynamic properties in liquid Fe-La alloys were measured[34,35].

The Fe-Ce binary system was experimentally investigated by different researchers[36-41]. Fe17Ce2 and Fe2Ce compounds were found to form through peritectic reactions. In terms of thermochemical properties of liquid Fe-Ce alloys, the enthalpy of mixing and the partial enthalpy of mixing of Fe and Ce were measured experimentally by different researchers[35,42-43], while the enthalpy of formation of Fe17Ce2 and Fe2Ce was measured[44-45].

Thermodynamic calculation of the Fe-La system was performed by several researchers[18,46], and that of the Fe-Ce system was performed[18,47]. In our previous work, the Fe-La and Fe-Ce binary systems were reassessed in consideration of new measured experimental results[29]. The calculated results of these two binary systems[29] are consistent well with the experimental information.

Fe-Pr and Fe-Nd

The Fe-Pr binary system has been experimentally studied[48-50]. According to the review[51,52], Fe17Pr2 is the only stable phase. There are two peritectic reactions, one peritectoid reaction, and one eutectic reaction. The thermodynamic properties in liquid Fe-Pr alloys were determined[53], while the enthalpy of formation of Fe17Pr2 was reported in Refs[44,54].

Several researchers investigated the Fe-Nd binary system[55-59]. These experimental results[55-59] show that Fe17Nd2 is the only stable phase. However, the researchers[60,61] found that Fe17Nd5 is also a stable intermetallic compound, and then the Fe-Nd phase diagram was revised accordingly. Okamoto[62,63] reviewed the Fe-Nd binary system. The thermodynamic properties of liquid Fe-Pr alloys were determined[53], while the enthalpy of formation of Fe17Nd2 was reported[44,54].

The Fe-Pr binary system was calculated[18,64-66], while the Fe-Nd system was calculated by several authors[18,60,67-68] considering these two stable intermetallic compounds, Fe17Nd2 and Fe17Nd5. In our previous work, the Fe-Pr and Fe-Nd binary systems were re-optimized[25] considering new determined experimental results. The calculated results of these two binary systems[25] are in accordance with the experimental information.

Fe-Sm and Fe-Gd

The Fe-Sm binary system has been determined by several authors[69]. Three intermetallic compounds, Fe17Sm2, Fe3Sm, and Fe2Sm, are formed (L + fcc-Fe ↔ Fe17Sm2, L + Fe17Sm2 ↔ Fe3Sm, L + Fe3Sm ↔ Fe2Sm). Using the high-temperature calorimetric method, the enthalpy of mixing and the partial enthalpy of mixing at 1,829 K were determined[70], while the enthalpy of formation of Fe17Sm2 was measured[44].

The Fe-Gd binary system has been studied experimentally[71-78]. According to these measurements, four Fe17Gd2, Fe23Gd6, Fe3Gd, and Fe2Gd intermetallic compounds are formed by the peritectic transformation. The thermodynamic properties of liquid Fe-Gd alloys were measured[53,79]. The enthalpy of formation of Fe2Gd was reported[44,80]. The enthalpy of formation of Fe17Gd2, Fe3Gd, and Fe2Gd was determined[81].

The Fe-Sm binary system was calculated[18,82-83], and the Fe-Gd binary system was evaluated[19,84-86]. The Fe-Sm and Fe-Gd systems were reassessed in our previous work[26], and the calculations are in accordance with the experimental information.

Fe-Ho and Fe-Tm

Roe et al. investigated the Fe-Ho binary system[87]. Fe17Ho2, Fe23Ho6, and Fe2Ho are formed by congruent melting, and Fe3Ho forms through the peritectic transformation. Four eutectic reactions are separately L ↔ Fe17Ho2 + fcc-Fe, L ↔ Fe17Ho2 + Fe23Ho6, L ↔ Fe2Ho + Fe3Ho, and L ↔ hcp-Ho + Fe2Ho. Besides, the enthalpy of formation of Fe2Ho was investigated[44], while those of Fe17Ho2 and Fe23Ho6 were reported[54].

Kolesnichenko et al. determined the Fe-Tm binary system[88]. Considering the review of Massalski[89], four phases exist in this system. Fe2Tm is formed congruently from liquid phase at 1573 K, while Fe17Tm2, Fe23Tm6, and Fe3Tm form by three peritectic transformation. The enthalpy of formation of Fe17Tm2 and Fe2Tm was studied[44,54].

The Fe-Tm and Fe-Ho binary systems were evaluated in Refs[19,90,91]. A reassessment of these two binary systems was performed in our previous study[28], and the assessments are consistent well with the experimental information.

Fe-Tb and Fe-Dy

The Fe-Tb binary system has been studied[92,93]. However, the determined melting temperatures[93] show the large experimental errors (± 50 K and about 3 at.%). Based on the experiment results[92], Okamoto[94] reported the Fe-Tb binary system. Four Fe17Tb2, Fe23Tb6, Fe3Tb, and Fe2Tb intermetallic compounds are produced via the peritectic transformation at 1,585 K, 1,549 K, 1,485 K, and 1,460 K, respectively, and Fe17Tb2 is produced at 1,589 K by the congruent melting. A eutectic reaction (L ↔ Fe17Tb2+bcc-Fe) takes place according to the DTA results[95] at 1,574 K.

The Fe-Dy binary system has been determined[96]. Four Fe17Dy2, Fe23Dy6, Fe3Dy, and Fe2Dy intermetallic compounds were found[96-98]. According to the literature[96,99], there are two congruent transformations (L ↔ Fe17Dy2 and L ↔ Fe3Dy), two peritectic transformations (L + Fe17Dy2 ↔ Fe23Dy6 and L + Fe3Dy ↔ Fe2Dy), and three eutectic transformations (L ↔ Fe17Dy2 + fcc-Fe,L ↔ Fe23Dy6 + Fe3Dy, and L ↔ Fe2Dy + hcp-Dy).

Ivanov et al. measured the enthalpy of mixing and the partial enthalpy of mixing of Fe, Tb, and Dy at 1,833 K[53]. The activities of Dy in Fe-Dy alloys between 1,273 K and 1,573 K were reported[100]. The heat capacity, the enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$ and the entropy of formation $$ \left(S_{T}^{0}\right) $$ of Fe2Tb and Fe2Dy at low temperature (below 300 K) were measured[101]. Furthermore, the enthalpy of formation of Fe17Tb2 and Fe17Dy2 was measured[54]. The enthalpy of formation of Fe2Tb, Fe17Tb2, Fe2Dy, and Fe17Dy2 was recently measured to be -5.5 ± 2.4, -2.1 ± 3.1, -1.6 ± 2.9, and -5.3 ± 1.7 kJ/mol-atom[102], respectively. Similarly, the enthalpy of formation of Fe2Dy, Fe3Dy, and Fe17Dy2 was reported[103]. Using ab initio calculation, the enthalpy of formation of Fe2Dy was reported to be -7.7 kJ/mol-atom[44].

The Fe-Tb and Fe-Dy binary systems were calculated by several authors[19,83,104,105]. In our previous work, a thermodynamic reassessment of the Fe-RE (RE = Tb and Dy) systems was conducted[27]. However, the heat capacity curves of Fe2Tb and Fe2Dy show artificial break points at 120 K due to the unreasonable expressions of their Gibbs energies. Therefore, the Gibbs energies of Fe2Tb and Fe2Dy were reassessed in this work based on the previous calculations[27].

Fe-Er

Buschow and Goot[106] investigated the Fe-Er binary phase diagram, and found Fe17Er2, Fe23Er6, Fe3Er, and Fe2Er. Meyer[107] studied the Fe-Er binary system, but the melting temperatures of these intermetallic compounds determined by Meyer[107] are much lower (about 70 K) than the results[106]. Subsequently, Kolesnikov et al.[108] re-determined the Fe-Er binary system and confirmed the reported results[106]. Furthermore, the experimental results[106-108] were reviewed by Okamoto[109]. The experimental results[106-108] show two eutectic transformations (L ↔ Fe17Er2 + Fe23Er6 at 1,588 K and L ↔ Fe2Er + hcp-Er at 1,188 K). Fe17Er2, Fe23Er6, and Fe3Er are produced by three peritectic transformations (L + fcc-Fe ↔ Fe17Er2 at 1628 K, L + Fe3Er ↔ Fe23Er6 at 1,603 K, and L + Fe2Er ↔ Fe3Er at 1,618 K), while Fe2Er melts congruently at 1,633 K. The solubility in terminal solid solution phases (hcp-Er, fcc-Fe, and bcc-Fe) was not found in the literature.

The heat capacity of Fe2Er at low temperature (8-300 K) was measured by Germano and Butera[101] using an adiabatic calorimeter, and thermodynamic functions $$ \left[\left(H_{T}^{0}-H_{0}^{0}\right)\right. $$ and $$ \left.S_{T}^{0}\right] $$ were obtained. The enthalpies of formation of the Fe-Er intermetallic compounds at different temperatures were measured[44,54,103]. Using indirect solution calorimetry, the enthalpies of formation of Fe2Er and Fe3Er were reported[103] at 1,100 K. The enthalpy of formation of Fe17Er2 was studied[54], while that of Fe2Er was determined[44].

The Fe-Er binary system was calculated by several researchers[19,110-112]. It was pointed out that the magnetism of the Fe-Er intermetallic compounds was not considered in the calculation[110]. Considering the experimental heat capacity of Fe2Er[101], the Fe-Er binary system was re-evaluated[111]. Although the calculations[111] are consistent with the results[106], the obtained heat capacity of Fe2Er at low temperature (below 30 K) displays a noticeable deviation from the experimental results[101]. Besides, Fe17Er2 and Fe3Er are decomposed at low temperatures (below 600 K)[112], which are contradictory to the experimental information. Therefore, the Fe-Er binary system is re-assessed in this study.

Fe-Lu

The Fe-Lu binary system has been studied[88], and four Fe17Lu2, Fe23Lu6, Fe3Lu, and Fe2Lu intermetallic compounds were found. There are three peritectic reactions (1,593 K, 1,563 K, and 1,583 K), which are corresponding to the formation of Fe17Lu2, Fe23Lu6, and Fe3Lu, respectively. The congruent reaction at 1,618 K is corresponding to the formation of Fe2Lu. Two eutectic transformations are L ↔ Fe2Lu + hcp-Lu at 1,243 K and 75 at.% Lu, and L ↔ Fe23Lu6 + Fe17Lu2 at 1,548 K and 17.8 at.% Lu, respectively. The solubility in terminal solid solution was not measured in the literature.

The thermodynamic properties in liquid Fe-Lu alloys were determined at 1,950 K[53]. The heat capacity, the enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$, and the entropy of formation $$ \left(S_{T}^{0}\right) $$ of Fe2Lu at low temperature (10-300 K) were measured[101]. Tereshina and Andreev[113] investigated the heat capacity of Fe17Lu2 at low temperature. The enthalpy of formation of Fe2Lu was studied[54], while that of Fe2Lu was reported[44].

The Fe-Lu binary system has been calculated[90]. However, the calculated thermodynamic properties including the partial enthalpy of mixing and the enthalpy of mixing are inconsistent with new measured data[53]. Thus, the Fe-Er binary system needs to be reassessed.

Fe-Y

The Fe-Y binary system has been investigated experimentally[114,115]. Due to the limitation of the purity of raw materials and experimental equipment, the experimental results[114] are not acceptable. Domagala et al. determined the eutectic reaction (L ↔ hcp-Y + Fe2Y at 1,173 K) by metallography and X-ray diffraction[115]. The intermetallic compounds Fe17Y2, Fe23Y6, and Fe3Y are produced by congruent reactions at 1,673 ± 25 K, 1,573 K, and 1,608 K, respectively. Gscheidner[116] and Kubaschewski[117] assessed the Fe-Y system in consideration of the experimental information[115]. The most recent evaluation of the Fe-Y system was carried out[118]. However, this evaluation focused only on the information of intermetallic compounds. Hellawell[119] reported that the solubility of 1 at.% Y reduces the transformation temperature [fcc(γ-Fe)/bcc(δ-Fe)], but Kubaschewski[117] still considered that the effect of Y on the transformation [fcc(γ-Fe)/bcc(δ-Fe)] was not known. According to Buschow[120], Fe17Y2 has two modifications, the rhombohedral Th2Znl7 and the hexagonal Th2Ni17. However, the temperature of this structure transformation was not reported, and thus Fe17Y2 was treated as a single phase in this work. It was found that the maximum solubility of Fe in hcp-Y is 1.5 at.% at 1173 K, but that of Y in fcc-Fe is not more than 0.6 at.% at 1,623 K[115].

The thermodynamic properties in liquid Fe-Y alloys at 1,873 K were measured[121,122]. The experimental results[121] show that the minimum enthalpy of mixing at 1,873 K is -8.44 kJ/mol at around 47 at.% Y. The measured partial enthalpy of mixing of Y in liquid Fe-Y alloys in the Fe-rich part[122] is much less negative than the measured results[121].

The Gibbs energy of formation of Fe17Y2 was measured[54], and then the enthalpy of formation of Fe17Y2 was deduced. The Gibbs energies of formation of Fe17Y2, Fe23Y6, Fe3Y, and Fe2Y were studied[123] between 893 K and 1,271 K.

Dariel et al. studied experimentally the specific heat and the Curie temperature of Fe2Y by using differential scanning calorimetry[124]. However, Dariel et al. did not report the specific heat data but only gave the Curie temperature of Fe2Y to be 535 K[124]. The heat capacity of Fe17Y2 was reported[125] at low temperature (below 300 K). The activities of Fe and Y were determined[126] at 1,473 K and 1,573 K.

Thermodynamic optimization of the Fe-Y binary system was performed[127], and the calculated results are in accordance with the reported data[115]. However, the calculated thermodynamic properties show significant deviations from the reported data[121], and Fe23Y6 is produced by the peritectic reaction rather than the congruent reaction, which was determined by Domagala et al.[115]. Later, several authors[128-130] re-optimized the Fe-Y binary system. Unfortunately, the relevant thermodynamic parameters were not published[128-130]. Combining the reported experimental data and the calculations, Kardelass et al. used two different models to optimize the Fe-Y binary system[131]. Combining with the reported results[114-119] and thermodynamic calculations[127-131], Saenko et al. optimized the new thermodynamic parameters of this binary system[132]. The calculations[132] are reasonably accordant with the experimental results. However, the calculations[131,132] show considerable homogeneity ranges of Y6Fe23 and YFe2, which are not confirmed by the reported experimental results. The calculations[133] show that the temperature of L ↔ hcp-Y + Fe2Y is inconsistent with the reported data[115], and the partial enthalpy of mixing of Fe is inconsistent with the data[121]. Besides, the measured heat capacity of Fe17Y2[125] was not considered in the calculations[133] at low temperature (10-300 K). Therefore, the Fe-Y binary system needs to be reassessed.

THERMODYNAMIC CALCULATION

Thermodynamic model

The Gibbs energy of solution phase φ (including liquid, fcc, bcc, and hcp) in the Fe-Tb, Fe-Dy, Fe-Er, Fe-Lu, and Fe-Y binary systems is described as:

$$ G_{\mathrm{m}}^{\varphi}={ }^{0} G_{\mathrm{Fe}}^{\varphi} \mathrm{x}_{\mathrm{Fe}}+{ }^{0} G_{\mathrm{RE}}^{\varphi} \mathrm{x}_{\mathrm{RE}}+\mathrm{R} T \sum_{\mathrm{i}=\mathrm{RE}, \mathrm{Fe}} \mathrm{x}_{\mathrm{i}} \ln \mathrm{x}_{\mathrm{i}}+\mathrm{x}_{\mathrm{RE}} \mathrm{x}_{\mathrm{Fe}} \sum_{\mathrm{n}=0}{ }^{\mathrm{n}} L_{\mathrm{RE}, \mathrm{Fe}}^{\varphi}\left(\mathrm{x}_{\mathrm{RE}}-\mathrm{x}_{\mathrm{Fe}}\right)^{\mathrm{n}}+{ }^{m a g} G_{\mathrm{m}}^{\varphi} $$

$$ { }^{\mathrm{n}} L_{\mathrm{RE}, \mathrm{Fe}}^{\varphi}=A_{n}+B_{n} T $$

$$ { }^{m a g} G_{\mathrm{m}}^{\varphi}=\mathrm{R} T \ln \left(\beta_{0}+1\right) g(\tau) $$

Where $$ { }^{0} G_{\mathrm{Fe}}^{\varphi} $$ and $$ { }^{0} G_{\mathrm{RE}}^{\varphi} $$ present the Gibbs energy of pure Fe and RE in the structure of the solution phase φ, and their values are obtained from the database[134]. xRE and xFE are mole fractions of Fe and RE. R is the universal gas constant, and T is the absolute temperature in Kelvin. The interaction parameters $$ { }^{\mathrm{n}} L_{\mathrm{RE}, \mathrm{Fe}}^{\varphi} $$ are presented by two constants An and Bn, which are optimized. $$ { }^{m a g} G_{m}^{\varphi} $$ present the magnetic contribution to the molar Gibbs energy of the of the solution phase φ. τ is expressed as $$ \tau=T / T_{C}^{\varphi} $$, which $$ T_{C}^{\varphi} $$ is the Curie temperature. Based on the proposed equation[135], g(τ) is expressed as:

$$ g(\tau)=\left\{\begin{array}{ll} 1-\frac{1}{M}\left[\frac{79 \tau^{-1}}{140 p}+\frac{474}{497}\left(\frac{1}{p}-1\right)\left(\frac{\tau^{3}}{6}+\frac{\tau^{9}}{135}+\frac{\tau^{15}}{600}\right)\right] & (\tau>1) \\ -\frac{1}{M}\left[\frac{\tau^{-5}}{10}+\frac{\tau^{-15}}{315}+\frac{\tau^{-25}}{1500}\right] & (\tau<1) \end{array}\right. $$

$$ M=\frac{518}{1125}+\frac{11692}{15975}\left(\frac{1}{p}-1\right) \\ $$

in which p is a constant dependent on the structure of the solution phase φ (0.4 for the bcc phase and 0.28 for other phases).

Fe17RE2, Fe23RE6, Fe3RE, and Fe2RE, are considered to be stoichiometric compounds in the Fe-RE (RE = Tb, Dy, Er, Lu, and Y) binary systems due to the lack of their composition range data. The heat capacities of Fe2Tb, Fe2Er, Fe2Dy, Fe17Lu2, Fe2Lu, and Fe17Y2 were experimentally determined by Germano and Butera[99], Tereshina et al.[113] and Mandal et al.[125]. In this work, according to available experimental data on heat capacities, the Gibbs energies of Fe2Tb, Fe2Dy, Fe2Er, Fe2Lu, Fe17Lu2, and Fe17Y2 were described by using a thermodynamic model, which can establish well the heat capacities of intermetallic compounds from 0 K. This model was employed in our previous assessments of Fe2Ce and Fe17Ce2[29]. The molar Gibbs energies of these intermetallic compounds FefREg are presented as:

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{\mathrm{f}} \mathrm{RE}_{\mathrm{g}}}=A+B T+C T^{2}+3 \mathrm{R}\left\{\frac{D}{2}+T \ln \left[1-\exp \left(-\frac{D}{T}\right)\right]\right\}+{ }^{m a g} G_{m}^{\mathrm{Fe}_{\mathrm{f}} \mathrm{RE}_{\mathrm{g}}} \\ $$

Where f and g are the stoichiometric numbers, and the parameters, A, B, C, and D, are to be assessed.

The molar Gibbs energies of Fe17Tb2, Fe23Tb6, Fe3Tb, Fe17Dy2, Fe23Dy6, Fe3Dy, Fe17Er2, Fe23Er6, Fe3Er, Fe23Lu6, Fe3Lu, Fe23Y6, Fe3Y, and Fe2Y can be expressed based on the Neumann-Kopp rule as:

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{\mathrm{q}} \mathrm{RE}_{\mathrm{b}}}=\frac{b}{b+q}{ }^{0} G_{\mathrm{RE}}^{\varphi}+\frac{q}{b+q}{ }^{0} G_{\mathrm{Fe}}^{b c c}+J+K T+{ }^{m a g} G_{m}^{\mathrm{Fe}_{\mathrm{q}} \mathrm{RE}_{\mathrm{b}}} \\ $$

in which b and q represent the stoichiometric numbers, J and K represent the optimized parameters. $$ { }^{m a g} G_{\mathrm{m}} \mathrm{Fe}_{\mathrm{f}} \mathrm{RE}_{\mathrm{g}} $$ and $$ { }^{m a g} G_{\mathrm{m}}^{\mathrm{Fe}_{\mathrm{q}} \mathrm{RE}_{\mathrm{b}}} $$, 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). TC and β0 of the Fe-RE intermetallic compounds used in the present work can be found in Tables 1-5, respectively.

Table 1

Thermodynamic parameters in the Fe-Tb binary system

PhaseThermodynamic parameters*Reference
Liquid

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Tb}}^{\mathrm{Liq}}=-20115.512-3.2 T $$

$$ { }^{1} L_{\mathrm{Fe}, \mathrm{Tb}}^{\mathrm{Liq}}=-3332.5193-11.1 T $$

$$ { }^{2} L_{\mathrm{Fe}, \mathrm{Tb}}^{\mathrm{Liq}}=660.3-8.1 T $$

This work
bcc-(Fe/Tb)

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Tb}}^{\mathrm{bcc}}=58000 $$

[27]
hcp-(Tb)

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Tb}}^{\mathrm{hcp}}=150000 $$

[27]
fcc-(Fe)

$$ { }^{0} G_{\mathrm{Tb}}^{\mathrm{fcc}}={ }^{0} G_{\mathrm{Tb}}^{\mathrm{hcp}}+5000 $$

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Tb}}^{\mathrm{fc}}=75000 $$

[27]
Fe17Tb2

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{17}\mathrm{~Tb}_{2}}=-10343+1.668 T+0.8947^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.1053^{0} G_{\mathrm{Tb}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{17}\mathrm{~Tb}_{2}}=408, \beta_{0}^{\mathrm{Fe}_{17} \mathrm{~Tb}_{2}}=0.167 $$

This work

[2]
Fe23Tb6

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{2}\mathrm{~Tb}_{6}}=-12501+0.355 T+0.7931^{0} G_{\mathrm{Fe}}^{\text {bcc }}+0.2069^{0} G_{\mathrm{Tb}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{23}\mathrm{~Tb}_{6}}=574, \beta_{0}^{\mathrm{Fe}_{23}\mathrm{~Tb}_{6}}=0.010 $$

This work

[2]
Fe3Tb

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{3}\mathrm{~Tb}}=-12008-0.277 T+0.75^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.25^{0} G_{\mathrm{Tb}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{3}\mathrm{~Tb}}=652, \beta_{0}^{\mathrm{Fe}_{3}\mathrm{~Tb}}=0.426 $$

This work

[2]
Fe2Tb

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{2}\mathrm{~Tb}}=-14540-4.332 T-0.0074 T^{2} + 3 \mathrm{R}\left\{\frac{185}{2}+T \ln \left[1-\exp \left(-\frac{185}{T}\right)\right]\right\} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{2}\mathrm{Tb}}=704, \beta_{0}^{\mathrm{Fe}_{2}\mathrm{~Tb}}=0.762 $$

This work

[2]
Table 2

Thermodynamic parameters in the Fe-Dy binary system

PhaseThermodynamic parameters*Reference
Liquid

$$ { }^{0} L_{\mathrm{Dy}, \mathrm{Fe}}^{\mathrm{Liq}}=-32524.791+10.7 T $$

$$ { }^{1} L_{\mathrm{Dy}, \mathrm{Fe}}^{\mathrm{Liq}}=14150.219-1.17 T $$

$$ { }^{2} L_{\mathrm{Dy}, \mathrm{Fe}}^{\mathrm{Liq}}=422.595-8.2 T $$

This work
bcc-(Fe/Dy)

$$ { }^{0} L_{\mathrm{Dy}, \mathrm{Fe}}^{\mathrm{bcc}}=38000 $$

[27]
hcp-(Dy)

$$ { }^{0} L_{\mathrm{Dy}, \mathrm{Fe}}^{\mathrm{hcp}}=100000 $$

[27]
fcc-(Fe)

$$ { }^{0} G_{\mathrm{Dy}}^{\mathrm{fcc}}={ }^{0} G_{\mathrm{Dy}}^{\mathrm{hcp}}+5000 $$

$$ { }^{0} L_{\mathrm{Dy}, \mathrm{Fe}}^{\mathrm{fcc}}=38000 $$

[27]
Fe17Dy2

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{17}\mathrm{Dy}_{2}}=-12882+3.744 T+0.8947^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.1053^{0} G_{\mathrm{Dy}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{17} \mathrm{Dy}_{2}}=371, \beta_{0}^{\mathrm{Fe}_{17} \mathrm{Dy}_{2}}=0.161 $$

This work

[2]
Fe23Dy6

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{23} \mathrm{Dy}_{6}}=-15700+3.909 T+0.7931^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.2069^{0} G_{\mathrm{Dy}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{23} \mathrm{Dy}_{6}}=534, \beta_{0}^{\mathrm{Fe}_{23} \mathrm{Dy}_{6}}=0.100 $$

This work

[2]
Fe3Dy

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{3} \mathrm{Dy}}=-15506+3.101 T+0.75^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.25^{0} G_{\mathrm{Dy}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{3} \mathrm{Lu}}=606, \beta_{0}^{\mathrm{Fe}_{3} \mathrm{Lu}}=0.493 $$

This work

[2]
Fe2Dy

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{2} \mathrm{Dy}}=-17860-5.419 T-0.00672 T^{2} + 3 \mathrm{R}\left\{\frac{210}{2}+T \ln \left[1-\exp \left(-\frac{210}{T}\right)\right]\right\} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{2} \mathrm{Dy}}=635, \beta_{0}^{\mathrm{Fe}_{2} \mathrm{Dy}}=0.890 $$

This work

[2]
Table 3

Thermodynamic parameters in the Fe-Er binary system

PhaseThermodynamic parameters*Reference
Liquid

$$ { }^{0} L_{\mathrm{Er}, \mathrm{Fe}}^{\mathrm{Liq}}=-37100+12.1 T $$

$$ { }^{1} L_{\mathrm{Er}, \mathrm{Fe}}^{\mathrm{Liq}}=15300-15.2 T $$

$$ { }^{2} L_{\mathrm{Er}, \mathrm{Fe}}^{\mathrm{Liq}}=-3500+2.8 T $$

This work
bcc-(Fe)

$$ { }^{0} L_{\mathrm{Er}, \mathrm{Fe}}^{\mathrm{bcc}}=75000 $$

This work
hcp-(Er)

$$ { }^{0} L_{\mathrm{Er}, \mathrm{Fe}}^{\mathrm{hcp}}=80000 $$

This work
fcc-(Fe)$$ ^{0} G_{\mathrm{Fe}}^{\mathrm{fcc}} $$ cited from SGTE database[134]
Fe17Er2

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{17} \mathrm{Er}_{2}}=-11321+4.4546 T+0.8947^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.1053^{0} G_{\mathrm{Er}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{17} \mathrm{Er}_{2}}=305, \beta_{0}^{\mathrm{Fe}_{17} \mathrm{Er}_{2}}=0.165 $$

This work

[2]
Fe23Er6

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{23} \mathrm{Er}_{6}}=-14250+4.4056 T+0.7931^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.2069^{0} G_{\mathrm{Er}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{23} \mathrm{Er}_{6}}=494, \beta_{0}^{\mathrm{Fe}_{23} \mathrm{Er}_{6}}=0.071 $$

This work

[2]
Fe3Er

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{3} \mathrm{Er}}=-14347+3.6854 T+0.75^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.25^{0} G_{\mathrm{Er}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{3} \mathrm{Er}}=552, \beta_{0}^{\mathrm{Fe}_{3} \mathrm{Er}}=0.452 $$

This work

[2]
Fe2Er

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{2} \mathrm{Er}}=-17458-6.4717 T+0.00538 \mathrm{~T}^{2} + 3 \mathrm{R}\left\{\frac{210}{2}+T \ln \left[1-\exp \left(-\frac{210}{T}\right)\right]\right\} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{2} \mathrm{Er}}=587, \beta_{0}^{\mathrm{Fe}_{2} \mathrm{Er}}=0.802 $$

This work

[2]
Table 4

Thermodynamic parameters in the Fe-Lu binary system

PhaseThermodynamic parameters*Reference
Liquid

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Lu}}^{\mathrm{Liq}}=-44158.73+14.99 T $$

$$ { }^{1} L_{\mathrm{Fe}, \mathrm{Lu}}^{\mathrm{Liq}}=-19539.28+9.78 T $$

$$ { }^{2} L_{\mathrm{Fe}, \mathrm{Lu}}^{\mathrm{Liq}}=-1356.5296+3.17 T $$

This work
bcc-(Fe)$$ ^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}} $$ cited from SGTE database[134]
hcp-(Lu)

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Lu}}^{\mathrm{hcp}}=50000 $$

This work
fcc-(Fe)$$ ^{0} G_{\mathrm{Fe}}^{\mathrm{fcc}} $$ cited from SGTE database[134]
Fe17Lu2

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{17} \mathrm{Lu}_{2}}=-11769-2.856 T-0.002 T^{2} +3 \mathrm{R}\left\{\frac{240}{2}+T \ln \left[1-\exp \left(-\frac{240}{T}\right)\right]\right\} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{17} \mathrm{Lu}_{2}}=268, \beta_{0}^{\mathrm{Fe}_{17} \mathrm{Lu}_{2}}=0.206 $$

This work

[2]
Fe23Lu6

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{23} \mathrm{Lu}_{6}}=-15608+4.769 T+0.7931^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.2069^{0} G_{\mathrm{Lu}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{23} \mathrm{Lu}_{6}}=481, \beta_{0}^{\mathrm{Fe}_{23} \mathrm{Lu}_{6}}=0.140 $$

This work

[2]
Fe3Lu

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{3} \mathrm{Lu}}=-15802+4.039 T+0.75^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.25^{0} G_{\mathrm{Lu}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{3} \mathrm{Lu}}=606, \beta_{0}^{\mathrm{Fe}_{3} \mathrm{Lu}}=0.493 $$

This work

[2]
Fe2Lu

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{2} \mathrm{Lu}}=-18167-4.61 T+0.00313 T^{2} +3 \mathrm{R}\left\{\frac{230}{2}+T \ln \left[1-\exp \left(-\frac{230}{T}\right)\right]\right\} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{2} \mathrm{Lu}}=596, \beta_{0}^{\mathrm{Fe}_{2} \mathrm{Lu}}=0.578 $$

This work

[2]
Table 5

Thermodynamic parameters in the Fe-Y binary system

PhaseThermodynamic parameters*Reference
Liquid

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Y}}^{\mathrm{Liq}}=-33500+17.5 T $$

$$ { }^{1} L_{\mathrm{Fe}, \mathrm{Y}}^{\mathrm{Liq}}=-2800+1.2 T $$

$$ { }^{2} L_{\mathrm{Fe}, \mathrm{Y}}^{\mathrm{Liq}}=9000-4.0 T $$

This work
bcc-(Fe/Y)

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Y}}^{\mathrm{bcc}}=63000 $$

This work
hcp-(Y)

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Y}}^{\mathrm{hcp}}=34000 $$

This work
fcc-(Fe)

$$ { }^{0} G_{\mathrm{Y}}^{\mathrm{fcc}}={ }^{0} G_{\mathrm{Y}}^{\mathrm{hcp}}+5000 $$

$$ { }^{0} L_{\mathrm{Fe}, \mathrm{Y}}^{\mathrm{fcc}}=60000 $$

[142]

This work
Fe17Y2

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{17} \mathrm{Y}_{2}}=-11250-1.538 T+0.0061 T^{2} +3 \mathrm{R}\left\{\frac{270}{2}+T \ln \left[1-\exp \left(-\frac{270}{T}\right)\right]\right\} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{17} \mathrm{Y}_{2}}=324, \beta_{0}^{\mathrm{Fe}_{17} \mathrm{Y}_{2}}=0.204 $$

This work

[132]
Fe23Y6

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{23} \mathrm{Y}_{6}}=-11910+3.583 T+0.7931^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.2069^{0} G_{\mathrm{Y}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{23} \mathrm{Y}_{6}}=481, \beta_{0}^{\mathrm{Fe}_{23} \mathrm{Y}_{6}}=0.139 $$

This work

[132]
Fe3Y

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{3} \mathrm{Y}}=-11500+2.784 T+0.75^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.25^{0} G_{\mathrm{Y}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{3} \mathrm{Y}}=569, \beta_{0}^{\mathrm{Fe}_{3} \mathrm{Y}}=0.581 $$

This work

[132]
Fe2Y

$$ G_{\mathrm{m}}^{\mathrm{Fe}_{2} \mathrm{Y}}=-11885+3.068 T+0.6667^{0} G_{\mathrm{Fe}}^{\mathrm{bcc}}+0.3333^{0} G_{\mathrm{Y}}^{\mathrm{hcp}} $$

$$ T_{\mathrm{C}}^{\mathrm{Fe}_{2} \mathrm{Y}}=542, \beta_{0}^{\mathrm{Fe}_{2} \mathrm{Y}}=0.574 $$

This work

[132]

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[136] considering the experimental 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[24], RE-Mn[137,138], and RE-Ni[139] binary systems with the increasing of RE atomic 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.[19] using the formula Hmix,Fe − Er = ΔHmix,Fe − Dy + 0.5(ΔHmix,Fe − Lu − ΔHmix,Fe − Dy)], where ΔHmix,Fe − Lu and ΔHmix,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 Fe2Tb and Fe2Dy were assessed in consideration of the experimental results[101] and the calculations[27]. In this work, the calculated Fe-Tb binary system is compared with the measured data[92,93,95] and the calculations[104], as shown in Figure 1. The temperatures of invariant 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.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 1. Calculated Fe-Tb binary phase diagram with (A) the calculations[104] and (B) the experimental data[92,93,95].

Table 6

Invariant reactions of the Fe-Tb binary system

ReactionsReaction typeTemperature (K)Composition $$ \left(x_{T b}^{L}\right) $$Reference
bcc(δ-Fe) ↔ fcc(γ-Fe) + LMetatectic1667.0
1665.1
1664.0
1666.9
0.120
0.080
0.100
0.050
[94] (exp.)
[104] (cal.)
[27] (cal.)
This work (cal.)
Fe17Tb2 + fcc(γ-Fe) ↔ bcc(α-Fe)Peritectoid1185.0
1184.8
1185.0
1184.8



[94] (exp.)
[104] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe17Tb2 + fcc(γ-Fe)Eutectic1574.0
1579.0
1576.0
1575.0
0.075
0.082
0.075
0.076
[95] (exp.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe17Tb2Congruent1589.0
1585.0
1586.0
1587.5
0.105
0.105
0.105
0.105
[95] (exp.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe17Tb2 + Fe23Tb6Eutectic1549.0
1547.0
1542.7
1546.0
1546.0
1546.5
0.216

0.212
0.190
0.208
0.167
[92] (exp.)
[95] (exp.)
[104] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe23Tb6Congruent1547.0
1561.1
0.207
0.207
[19] (cal.)
This work (cal.)
L+ Fe23Tb6 ↔ Fe3TbPeritectic1485.0
1473.1
1481.0
1485.0
1486.0
0.349
0.361
0.388
0.340
0.319
[92] (exp.)
[104] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L + Fe3Tb ↔ Fe2TbPeritectic1460.0
1456.6
1459.0
1460.0
1462.7
0.412
0.403
0.424
0.405
0.348
[92] (exp.)
[104] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe2Tb + hcp-TbEutectic1120.0
1123.0
1122.0
1120.0
1117.1
0.720
0.687
0.700
0.690
0.675
[92] (exp.)
[104] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)

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[27], as shown in Figure 2 and Figure 3. The enthalpies of formation 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 Fe2Tb is -9.005 kJ/mol-atom. As presented in Figure 5, the calculated thermodynamic properties [e.g., heat capacity, the entropy of formation, and enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$ of Fe2Tb] are in accordance with the data[101] and show no artificial break point at 120 K in the heat capacity curve.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 2. The calculated enthalpy of mixing of liquid Fe-Tb alloys at 1,833 K with the calculations[104] and the experimental data[53].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 3. Calculated partial enthalpy of Fe (A) and Tb (B) in liquid Fe-Tb alloys at 1,833 K with the calculation[104] and the experimental data[53].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 4. Calculated enthalpies of formation of the Fe-Tb intermetallic compounds at 298 K with the experimental results[54,102] and the calculations[19,27,104,140].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 5. The calculated (A) heat capacity, (B) the entropy of formation, and (C) the enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$ of Fe2Tb with the calculations[27] and the experimental data[101].

Figure 6 compares the presently calculated Fe-Dy binary system with the data[96] and the calculations[104,105]. The calculated temperatures of the invariant reactions in Fe-Dy binary system were seen in Table 7. The presently calculated results agree well with the reported results[96] and the calculations[104,105].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 6. Calculated Fe-Dy binary phase diagram with (A) the calculations[104,105] and (B) the experimental data[96].

Table 7

Invariant reactions of the Fe-Dy binary system

ReactionsReaction typeTemperature (K)Composition $$ \left(x_{D y}^{L}\right) $$Reference
bcc(δ-Fe) ↔ fcc(γ-Fe) + LMetatectic1663.0
1667.9
1667.5
1662.0
1664.0
0.085
0.070
0.060
0.060
0.055
[99] (exp.)
[104] (cal.)
[105] (cal.)
[27] (cal.)
This work (cal.)
Fe17Dy2 + fcc(γ-Fe) ↔ bcc(α-Fe)Peritectoid1183.0
1184.8
1184.8
1185.0
1184.8




[99] (exp.)
[104] (cal.)
[105] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe17Dy2 + fcc(γ-Fe)Eutectic1633.0
1641.7
1630.7
1639.0
1638.0
1634.0
1630.0
0.090
0.083
0.078

0.074
0.080
0.066
[96] (exp.)
[104] (cal.)
[105] (cal.)
[83] (cal.)
[19] (cal.)
[27] (cal.)
This work(cal.)
L ↔ Fe17Dy2Congruent1648.0
1644.8
1637.5
1640.0
1649.0
1647.0
1651.0
0.105
0.105
0.105
0.105
0.105
0.105
0.105
[96] (exp.)
[104] (cal.)
[105] (cal.)
[83] (cal.)
[19] (cal.)
[27] (cal.)
This work(cal.)
L + Fe17Dy2 ↔ Fe23Dy6Peritectic1563.0
1561.9
1574.0
1565.0
1567.4
0.212
0.218
0.210
0.212
0.209
[96] (exp.)
[105] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe23Dy6 + Fe3DyEutectic1558.0
1561.0
1573.0
1557.0
1567.3
0.225
0.225
0.215
0.221
0.212
[96] (exp.)
[105] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe3DyCongruent1578.0
1581.8
1563.5
1580.0
1580.0
1579.0
1576.4
0.250
0.250
0.250
0.250
0.250
0.250
0.250
[96] (exp.)
[104] (cal.)
[105] (cal.)
[83] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L + Fe3Dy ↔ Fe2DyPeritectic1543.0
1542.9
1535.9
1545.0
1548.0
1547.0
1545.5
0.401
0.363
0.335

0.339
0.390
0.335
[96] (exp.)
[104] (cal.)
[105] (cal.)
[83] (cal.)
[19] (cal.)
[27] (cal.)
This work (cal.)
L ↔ Fe2Dy + hcp-DyEutectic1163.0
1162.1
1165.3
1164.0
1161.0
1161.0
1160.0
0.715
0.716
0.717
0.730
0.691
0.710
0.696
[96] (exp.)
[104] (cal.)
[105] (cal.)
[83] (cal.)
[19] (cal.)
[27] (cal.)
This work(cal.)

The presently calculated partial enthalpy of mixing and enthalpy of mixing in liquid Fe-Dy alloys are consistent with the calculations[27], as given in Figure 7 and Figure 8, respectively. In Figure 9, the enthalpies of formation of the Fe-Dy intermetallic compounds in the present calculation at 298 K agree well with the calculations[19,27,44,83,104,105,141] and the experimental data[54,102,103]. The enthalpy of formation of Fe2Dy is calculated as -12.327 kJ/mol-atom. Additionally, the calculated thermodynamic properties [e.g., heat capacity, entropy of formation and enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$] of Fe2Dy, as shown in Figure 10, are much better accordance with the results[101] and show no artificial break point in the heat capacity curve.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 7. The calculated enthalpy of mixing of liquid Fe-Dy alloys at 1,850 K with the experimental results[53] and the calculations[104,105].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 8. Calculated partial enthalpy of mixing of Fe (A) and Dy (B) in liquid Fe-Dy alloys at 1,850 K with the experimental data[53] and the calculations[104,105].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 9. Calculated enthalpy of formation of the Fe-Dy intermetallic compounds at 298 K with the experimental data[54,102,103] and the calculations[19,27,44,83,104,105,141].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 10. The calculated (A) heat capacity, (B) the entropy of formation, and (C) the enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$ of Fe2Dy with the experimental data[101] and the calculations[27].

Fe-Er

Figure 11 displays the presently calculated Fe-Er binary system with the earlier calculations[110-112] and the experimental results[106-108]. As can be seen, Fe17Er2, Fe23Er6, Fe3Er, and Fe2Er are stable at room temperature, but some compounds are unstable at low temperature in the literature[112]. The presently calculated liquidus and solidus in Figure 11(B) agree with the experimental data[106-108]. Table 8 compares the calculated results of invariant reactions together with the experimental results. The temperatures and compositions of two eutectic reactions, L ↔ Fe17Er2 + Fe23Er6 and L ↔ Fe2Er + hcp-Er, are calculated to be 1,601.8 K and 18.2 at.% Er and 1,185.6 K and 66.9 at.% Er, respectively. A congruent reaction of Fe2Er (L ↔ Fe2Er) happens at 1640.4 K, and those of four peritectic reactions (L + bcc-Fe ↔ fcc-Fe, L + fcc-Fe ↔ Fe17Er2, L + Fe3Er ↔ Fe23Er6, and L + Fe2Er ↔ Fe3Er) are calculated to be 1,663.2 K, 1,625.1 K, 1,601.8 K, and 1,615.0 K, respectively, agreeing well with the results[106-108].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 11. Calculated Fe-Er binary phase diagram with (A) the calculations[110-112] and (B) the experimental data[106-108].

Table 8

Invariant reactions of the Fe-Er binary system

ReactionsReaction typeTemperature (K)Composition $$ \left(\boldsymbol{x}_{E r}^{L}\right) $$Reference
bcc(δ-Fe) ↔ fcc(γ-Fe) + LMetatectic1658.0
1667.5
1658.0
1658.0
1661.5
0.100
0.081
0.082
0.089
0.084
[106] (exp.)
[110] (cal.)
[111] (cal.)
[112] (cal.)
This work (cal.)
Fe17Er2 + fcc(γ-Fe) ↔ bcc(α-Fe)Peritectoid1184.8This work
L + fcc(γ-Fe) ↔ Fe17Er2Peritectic1628.0
1616.0
1619.0
1626.0
1630.0
1625.1
0.104
0.105
0.105
0.108
0.106
0.106
[106] (exp.)
[110] (cal.)
[111] (cal.)
[112] (cal.)
[19] (cal.)
This work (cal.)
L ↔ Fe17Er2 + Fe23Er6Eutectic1588.0
1599.0
1600.0
1600.5
1581.0
1601.8
0.165
0.174
0.181
0.178
0.171
0.182
[106] (exp.)
[110] (cal.)
[111] (cal.)
[112] (cal.)
[19] (cal.)
This work (cal.)
L+ Fe3Er ↔ Fe23Er6Peritectic1603.0
1601.0
1600.0
1602.7
1595.0
1602.7
0.197
0.191
0.182
0.190
0.202
0.189
[106] (exp.)
[110] (cal.)
[111] (cal.)
[112] (cal.)
[19] (cal.)
This work (cal.)
L + Fe2Er ↔ Fe3ErPeritectic1618.0
1611.0
1611.0
1615.7
1612.0
1615.0
0.248
0.240
0.235
0.250
0.279
0.250
[106] (exp.)
[110] (cal.)
[111] (cal.)
[112] (cal.)
[19] (cal.)
This work (cal.)
L ↔ Fe2ErCongruent1633.0
1632.0
1633.0
1636.0
1633.0
1640.4
0.333
0.333
0.333
0.333
0.333
0.333
[106] (exp.)
[110] (cal.)
[111] (cal.)
[112] (cal.)
[19] (cal.)
This work (cal.)
L ↔ Fe2Er + hcp-ErEutectic1188.0
1185.0
1189.0
1187.9
1189.0
1185.6
0.700
0.692
0.719
0.687
0.684
0.669
[106] (exp.)
[110] (cal.)
[111] (cal.)
[112] (cal.)
[19] (cal.)
This work (cal.)

The available experimental enthalpy of mixing was not reported. The enthalpy of mixing is calculated at 1833 K and compared with the calculations[19,110-112] in Figure 12. Figure 13 is the presently calculated enthalpy of formation of the Fe-Er intermetallic compounds together with the results[44,54,103] and the calculations[19,110-112,140,141]. The presently calculated enthalpy of formation of Fe17Er2, Fe23Er6, Fe3Er, and Fe2Er are -3.255, -7.216, -8.881, and -11.611 kJ/mol-atom, respectively, which are accordant with the experimental results[54,103]. The presently calculated enthalpy of formation of Fe2Er is more negative than the measured results[44]. On the other hand, it should be pointed out that the measured data[44] are significantly less negative than that measured data[103]. The heat capacity, entropy of formation, and enthalpy difference of Fe2Er are calculated to be compared with the experimental results[101] and the calculations[110-112] in Figure 14A-c, respectively. The presently calculated heat capacity of Fe2Er is much better consistent with the experimental data[101] than the earlier calculations[110-112].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 12. The calculated enthalpy of mixing of liquid Fe-Er alloys at 1,833 K with the calculations[19,110-112].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 13. Calculated enthalpies of formation of the Fe-Er intermetallic compounds at 298 K with the experimental data[44,54,103] and the calculations[19,110-112,140,141].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 14. The calculated (A) heat capacity, (B) the entropy of formation and (C) the enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$ of Fe2Er with the experimental data[101] and the calculated results[110-112].

Fe-Lu

Figure 15 presents the calculated Fe-Lu binary system, the calculations[90], and the experimental results[88]. It is observed in Figure 15A that Fe17Lu2 decomposes at low temperature according to the calculations[90], while Fe17Lu2, Fe23Lu6, Fe3Lu, and Fe2Lu are stable down to room temperature in this work. As displayed in Figure 15B, the present calculations agree with the results[88]. Table 9 displays the calculated results of invariant reactions with the measured data[88]. The results of two eutectic reactions, L ↔ Fe17Lu2 + Fe23Lu6 and L ↔ Fe2Lu + hcp-Lu, are calculated to be 1,559.9 K and 18.4 at.% Lu and 1,240.3 K and 64.2 at.% Lu, respectively. The congruent melting of Fe2Lu (L ↔ Fe2Lu) is calculated to be 1,619.6 K, while those of four peritectic reactions, L + bcc-Fe ↔ fcc-Fe, L + fcc-Fe ↔ Fe17Lu2, L + Fe3Lu ↔ Fe23Lu6, and L + Fe2Lu ↔ Fe3Lu, are calculated to be 1,667.5 K, 1,590.1 K, 1,560.9 K, and 1,580.4 K, respectively. These present calculations results are in accordance with the reported results[88].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 15. Calculated Fe-Lu binary phase diagram with (A) the calculations[90] and (B) the experimental data[88].

Table 9

Invariant reactions of the Fe-Lu binary system

ReactionsReaction typeTemperature (K)Composition $$ \left(x_{L u}^{L}\right) $$Reference
L + bcc(δ-Fe) ↔ fcc(γ-Fe)Peritectic1667.0
1667.0
1667.5


0.070
[88] (exp.)
[90] (cal.)
This work (cal.)
Fe17Lu2 + fcc(γ-Fe) ↔ bcc(α-Fe)Peritectoid1185.0
1185.0
1184.8


[88] (exp.)
[90] (cal.)
This work (cal.)
L + fcc(γ-Fe) ↔ Fe17Lu2Peritectic1593.0
1593.0
1602.0
1590.1
0.130
0.090
0.111
0.105
[88] (exp.)
[90] (cal.)
[19] (cal.)
This work(cal.)
L ↔ Fe17Lu2 + Fe23Lu6Eutectic1548.0
1549.0
1546.0
1559.9
0.180
0.158
0.172
0.184
[88] (exp.)
[90] (cal.)
[19] (cal.)
This work (cal.)
L + Fe3Lu ↔ Fe23Lu6Peritectic1563.0
1560.0
1560.0
1560.9
0.196
0.183
0.202
0.188
[88] (exp.)
[90] (cal.)
[19] (cal.)
This work (cal.)
L + Fe2Lu ↔ Fe3LuPeritectic1583.0
1581.0
1580.0
1580.4
0.225
0.229
0.333
0.241
[88] (exp.)
[90] (cal.)
[19] (cal.)
This work (cal.)
L ↔ Fe2LuCongruent1618.0
1618.0
1620.0
1619.6
0.333
0.333
0.333
0.333
[88] (exp.)
[90] (cal.)
[19] (cal.)
This work (cal.)
L ↔ Fe2Lu + hcp-LuEutectic1243.0
1237.0
1249.0
1240.3
0.750
0.741
0.670
0.642
[88] (exp.)
[90] (cal.)
[19] (cal.)
This work(cal.)

Figure 16 shows the calculated enthalpy of mixing of liquid Fe-Lu alloys at 1,950 K with the measured results[53] and the calculations[90]. A significant deviation between the calculations[90] and the experimental data[53] is found, while the present calculations agree with the experimental data[53]. In Figure 17, the partial enthalpy of mixing of Fe and Lu in liquid Fe-Lu alloys is calculated and compared with the measured results[53] and the calculations[90]. It is obvious that our calculations are accordant well with the measured data[53], while the calculations[90] show large deviations.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 16. The calculated enthalpy of mixing of liquid Fe-Lu alloys at 1,950 K with the experimental data[53] and the calculations[90].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 17. Calculated partial enthalpy of Fe (A) and Lu (B) in liquid Fe-Lu alloys at 1,950 K with the experimental data[53] and the calculations[90].

Figure 18 depicts the calculated enthalpy of formation of the Fe-Lu compounds at 298 K with the measured results[44,54] and the calculations[19,90,140,141]. The enthalpy of formation of Fe17Lu2, Fe23Lu6, Fe3Lu, and Fe2Lu are calculated to be -3.69, -8.762, -10.615, and -11.957 kJ/mol-atom, respectively. The presently calculated enthalpy of formation of Fe17Lu2 is in accordance with the measured data[54], while the calculated that of Fe2Lu is slightly lower than the measured data[44]. It is worth mentioning that the measured enthalpy of formation of Fe2RE in the Fe-RE binary systems by the authors[44] is all less negative. Further experiments are still needed to determine the enthalpy of formation of the Fe-Lu intermetallic compounds.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 18. Calculated enthalpies of formation of the Fe-Lu intermetallic compounds at 298 K with the experimental data[44,54] and the calculations[19,90,140,141].

Figure 19 shows the presently calculated heat capacity of Fe17Lu2 along with the measured results[113] and the calculations[90]. The calculated heat capacity of Fe17Lu2 agrees with the results[113], while the results calculated by Kardellass et al. are significantly different[90]. Figure 20 is the calculated the heat capacity, the entropy of formation, and the enthalpy difference of Fe2Lu with the measured results[101] and the calculations[90]. Comparing with the previous calculations[90], the heat capacity of Fe2Lu calculated in this study is in better accordance with the results[101].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 19. Calculated heat capacity of Fe17Lu2 with the experimental data[113] and the calculations[90].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 20. The calculated (A) heat capacity, (B) the entropy of formation, and (C) the enthalpy difference $$ \left(H_{T}^{0}-H_{0}^{0}\right) $$ of Fe2Lu with the experimental data[101] and the calculations[90].

Fe-Y

Figure 21 shows the calculated Fe-Y binary system along with the earlier calculations[127,131-133] and the experimental results[115]. It is evident in Figure 21A that the previous calculations[127] show that Fe23Y6 and Fe2Y are not stable at low temperature, while four intermetallic compounds (including Fe23Y6 and Fe2Y) are stable down to room temperature in this study. As displayed in Figure 21B, the present calculations agree with the measured results[115]. Table 10 summarizes the present calculations of the invariant reactions together with the calculations[127,131-133], and the optimized parameters in this work are shown in Table 5. Four eutectic reactions, L ↔ Fe17Y2 + fcc-Fe, L ↔ Fe17Y2 + Fe23Y6, L ↔ Fe23Y6 + Fe3Y, and L ↔ Fe2Y + hcp-Y, are calculated to be 1,647.3 K and 8.8 at.% Y, 1,602.7 K and 19.7 at.% Y, 1,602.4 K and 22.1 at.% Y, and 1,163.5 K and 65.6 at.% Y, respectively. The temperatures of the congruent reaction of Fe17Y2, Fe23Y6, and Fe3Y are calculated to be 1,620.1 K, 1,603.0 K, and 1,605.5 K, respectively, while two peritectic reactions, L + Fe3Y ↔ Fe2Y and L + bcc-Fe ↔ fcc-Fe, are calculated to be 1,666.8 K and 1,163.5 K, respectively, which are in accordance with the results[115].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 21. Calculated Fe-Y binary phase diagram with (A) the calculations[127,131-133] and (B) the experimental data[115].

Table 10

Invariant reactions of the Fe-Y binary system

ReactionsReaction typeTemperature (K)Composition $$ \left(\boldsymbol{X}_{Y}^{L}\right) $$Reference
bcc(δ-Fe) ↔ fcc(γ-Fe) + LMetatectic1673±25
1667.0
1665.0
1725.0
1663.0
1664.0
1660.0
1665.8

0.058
0.072
0.039
0.048
0.048
0.060
0.077
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
fcc(γ-Fe) ↔ bcc(α-Fe) + Fe17Y2Eutectoid1185.0
1185.0
1185.0
1185.0
1185.0
1184.8





[115] (exp)
[132]
[133]
[131] (PTD) (cal.)
[131] (ETD) (cal.)
This work
L ↔ Fe17Y2+ fcc(γ-Fe)Eutectic1623±25
1623.0
1641.0
1657.0
1633.0
1663.0
1635.0
1647.3
0.082
0.074
0.084
0.069
0.056
0.072
0.070
0.088
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L↔ Fe17Y2Congruent1673±25
1673.0
1644.0
1679.0
1676.0
1672.0
1650.0
1650.1
0.105
0.105
0.105
0.105
0.105
0.105
0.105
0.105
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L ↔ Fe17Y2 + Fe23Y6Eutectic~1553.0
1553.0
1605.0
1645.0
1605.0
1638.0
1613.0
1602.7
0.129
0.164
0.197
0.159
0.182
0.179
0.173
0.197
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L ↔ Fe23Y6Congruent~1573.0
1606.0
1667.0
1606.0
1641.0
1623.0
1603.0

0.201
0.207
0.207
0.207
0.207
0.207
[115] (exp.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L ↔ Fe23Y6 + Fe3YEutectic~1523.0
1573.0
1604.0
1667.0
1605.0
1640.0
1619.0
1602.4
0.237
0.186
0.237
0.210
0.207
0.216
0.207
0.221
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L ↔ Fe3YCongruent~1608.0
1623.0
1605.0
1681.0
1623.0
1647.0
1621.0
1605.5
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L + Fe3Y ↔ Fe2YPeritectic1398±25
1398.0
1396.0
1397.0
1409.0
1397.0
1416.0
1421.0
~0.424
0.421
0.525
0.475
0.443
0.482
0.467
0.490
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L ↔ Fe2Y + hcp-YEutectic1173±10
1173.0
1201.0
1156.0
1146.0
1118.0
1120.0
1163.5
~0.660
0.651
0.661
0.601
0.612
0.637
0.640
0.656
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)
L + bcc-Y ↔ hcp-YPeritectic1758.0
1752.0
1756.0
1752.0
1750.0
1749.0
1754.0
1757.1

0.979
0.983
0.978
0.979
0.976
0.980
0.981
[115] (exp.)
[127] (cal.)
[132] (cal.)
[19] (cal.)
[131] (PTD) (cal.)
[131] (ETD) (cal.)
[133] (cal.)
This work(cal.)

As shown in Figure 22, the enthalpy of mixing of liquid Fe-Y alloys is calculated at 1,873 K and compared with the determined results[121] and the calculations[127,131-133]. The calculations[127,131] are a visible deviation from the experimental results[121], while the present calculations are in accordance with the results[121]. Figure 23 depicts the partial enthalpy of mixing of Fe and Y calculated by this study at 1,873 K with the measured data[121,122] and the calculations[127,131-133]. It was found that the present calculations are accordant with the results[121], while the previous calculations[127,131-133] show obvious differences. Figure 24 shows the calculated enthalpy of formation of the Fe-Y intermetallic compounds at 298 K with the measured data[123] and the calculations[127,131-133]. The enthalpies of formation of Fe17Y2, Fe23Y6, Fe3Y, and Fe2Y are calculated to be -2.994, -5.061, -6.417, and -7.446 kJ/mol-atom, respectively. It was found that the calculated enthalpy of formation of Fe2Y is the most negative. Generally, the enthalpy of formation of Fe2RE is the most negative in Fe-RE binary systems. It is still necessary to measure the enthalpy of formation of the Fe-Y compounds in further experiments.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 22. The calculated enthalpy of mixing of liquid Fe-Y alloys at 1,873 K with the experimental data[121] and the calculations[127,131-133].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 23. Calculated partial enthalpy of (A) Fe and (B) Y in liquid Fe-Y alloys at 1,873 K with the experimental data[121,122] and the calculations[127,131-133].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 24. Calculated enthalpies of formation of the Fe-Y intermetallic compounds at 298 K with the experimental data[123] and the calculations[127,131-133].

Figure 25 depicts the calculated heat capacity of Fe17Y2 with the reported results[125]. The calculated heat capacity of Fe17Y2 at low temperature (below 298 K) is accordant well with the results[125], while Konor et al. did not consider the experimental heat capacity of Fe17Y2[19]. Figure 26 shows the calculated activities of Fe and Y at 1,473 K and 1,573 K with the determined results[126]. It is clear that the calculated activities of Fe and Y in the Fe-rich region are much better consistent with the reported results[126] than those of Fe and Y in the Y-rich region, which could be resulted from the oxidation of Fe-Y alloys.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 25. The calculated heat capacity of Fe17Y2 with the experimental data[125].

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 26. The calculated activities of Fe and Y at (A) 1,473 K and (B) 1,573 K with the experimental data[126].

DISCUSSION

RE metals show similar physicochemical properties because of their unique and similar electronic configurations. The alloys made of RE metals and transition metals are expected to show a trend in phase equilibria and thermodynamic properties of the RE-TM (transition metals) binary systems as the RE atomic number increases. Our previous work[25,26,28,29] shows that the calculated phase equilibria and thermodynamic properties of some Fe-RE binary systems are accordant well with the experimental data, as displayed in Figure 27. In the following section, phase equilibria and thermodynamic properties of thirteen Fe-RE binary systems were analyzed systematically.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 27. The calculated Fe-RE binary phase diagrams. (A) Fe-La[29]; (B) Fe-Ce[29]; (C) Fe-Pr[25]; (D) Fe-Nd[25]; (E) Fe-Sm[26]; (F) Fe-Gd[26]; (G) Fe-Ho[28]; (H) Fe-Tm[28]

Phase equilibria

In Figure 28, the Fe-RE binary phase diagrams demonstrate that the number of stable intermetallic compounds in generally increases from 0 to 4 as the RE atomic number increases. With the increasing of the RE atomic number, Fe17RE2, Fe17RE5, Fe3RE, Fe2RE, and Fe23RE6 appear successively in the Fe-RE (RE = Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y) binary phase diagrams, as given in Table 11. Notably, Fe17RE2, Fe23RE6, Fe3RE, and Fe2RE are stable in the Fe-RE (RE = Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y) binary systems. However, Fe23RE6 does not exist in the Fe-RE (RE = La, Ce, Pr, Nd, and Sm) binary systems. In addition, Fe17RE5 only exists in the Fe-Nd binary system, and Fe17RE2 is stable in the Fe-RE (apart from Fe-La) binary systems.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 28. The reaction temperatures of the Fe-RE (RE = La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu and Y) intermetallic compounds. (A) Fe17RE2; (B) Fe23RE6; (C) Fe3RE; (d) Fe2RE.

Table 11

Temperatures of formation reaction for Fe-RE intermetallic compounds

SystemsFe17RE2Fe23RE6Fe17RE5Fe3REFe2RE
Fe-La
Fe-Ce1344.6P1205.6P
Fe-Pr1383.0P
Fe-Nd1490.0P1071.0P
Fe-Sm1557.0P1287.0P1174.0P
Fe-Gd1604.0P1556.0P1429.0P1354.5P
Fe-Tb1587.5C1561.1C1486.0P1462.7P
Fe-Dy1651.0C1567.4P1576.4C1545.5P
Fe-Ho1615.6C1605.2C1567.6P1565.0C
Fe-Er1625.1P1602.7P1615.0P1640.4C
Fe-Tm1573.4P1544.6P1552.5P1572.6C
Fe-Lu1590.1P1560.9P1580.4P1619.6C
Fe-Y1650.1C1603.0C1605.5C1421.0P

The types and temperatures of the invariant reactions for the formation of the Fe-RE (e.g., Fe17RE2, Fe3RE, Fe2RE, and Fe23RE6) intermetallic compounds are demonstrated in Figure 28 and Table 11. As can be easily seen in the Fe-RE (RE = Ce, Pr, Nd, Sm, Gd, Er, Tm, and Lu) binary systems, Fe17RE2 is produced through the peritectic reaction, but by the congruent reaction in the Fe-RE (RE = Tb, Dy, Ho, and Y) binary systems; Fe23RE6 is produced by the congruent reaction in the Fe-RE (RE = Tb, Ho, and Y) binary systems, but by peritectic reaction in the Fe-RE (RE = Gd, Dy, Er, Tm, Lu, and Y) binary systems; Fe3RE is produced by the congruent reaction in the Fe-Dy and Fe-Y binary systems, but by the peritectic reaction in the Fe-RE (RE = Sm, Gd, Tb, Ho, Er, Tm, and Lu) binary systems; Fe2RE is produced through the peritectic reaction in the Fe-RE (RE = Ce, Sm, Gd, Tb, Dy, and Y) binary systems, but through the congruent reaction in the Fe-RE (RE=Ho, Er, Tm, and Lu) binary systems. The melting points of RE metals increase gradually with the RE atomic number increases, and the reaction temperatures and types for the intermetallic compounds show a similar trend. In general, the reaction temperatures of Fe17RE2, Fe3RE, and Fe2RE in the Fe-RE (RE = Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, and Lu) binary systems become higher with the increase of the RE atomic number. The reaction temperature of Fe23RE6 also displays this trend in the Fe-RE (RE = Gd, Tb, Dy, Ho, and Er) binary systems, but it is not clear in the Fe-Tm and Fe-Lu binary systems.

Thermodynamic properties

Figure 29 depicts the calculated enthalpy of mixing of liquid Fe-RE alloys. It was found that the enthalpy of mixing of liquid Fe-La alloys is positive, while that of liquid Fe-Pr alloys is positive and negative over different composition regions. Meanwhile, the calculated enthalpy of mixing in liquid Fe-RE (RE = Ce, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu, and Y) alloys is all negative. The minimum enthalpy of mixing of liquid Fe-RE (RE = Ce, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, and Lu) alloys is located in the composition range of 30 at.%-50 at.% RE. In general, the enthalpy of mixing of liquid Fe-RE (apart from Fe-Ce and Fe-Y) alloys is calculated and displayed a tendency that the higher the RE atomic number, the more negative the enthalpy of mixing. The similar irregularities of the enthalpy of mixing of liquid Fe-Ce and Fe-Y alloys in the Fe-RE binary systems were also observed in the B-RE[24] and Ni-RE[139] binary systems. Due to a lack of the experimental data on the enthalpy of mixing of liquid Fe-Er alloys, the thermodynamic calculation of the Fe-Er binary system was performed by taking this regular trend into account in the present study.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 29. The calculated enthalpies of mixing of liquid Fe-RE (RE = La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu and Y) alloys.

Figure 30 displays the calculated enthalpy of formation of the Fe-RE intermetallic compounds at 298 K (e.g., Fe17RE2, Fe17RE5, Fe3RE, Fe2RE, and Fe23RE6). The enthalpy of formation Fe2RE is the most negative and shows a trend that the enthalpy of formation of Fe2RE becomes increasingly negative with the RE atomic number increases in the Fe-RE (apart from Fe-Y, Fe-Gd, and Fe-Dy) binary systems. It is noted that the enthalpy of formation of the Fe-Gd intermetallic compounds is more negative than those of the Fe-Tb intermetallic compounds. In particular, the enthalpy of formation of the Fe-Y intermetallic compounds is between that of the Fe-Sm intermetallic compounds and that of the Fe-Gd intermetallic compounds. There are similar irregularities that also appear in the RE-B[24], RE-Mn[137,138], and RE-Ni[139] binary systems. Generally, the enthalpy of formation of the Fe-RE intermetallic compounds become increasingly negative with an increasing of the RE atomic number. It indicates stronger bond in Fe-RE (apart from Fe-Y) binary systems as a consequence of the reducing atomic radius with the increasing of the RE atomic number.

Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems

Figure 30. The calculated enthalpies of formation of the Fe-RE (RE = La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu and Y) intermetallic compounds at 298 K.

SUMMARY

Three Fe-Er, Fe-Lu, and Fe-Y binary systems were reassessed through the CALPHAD method in the present work according to the reliable experimental results and the earlier calculations in the literature. An improvement to our previous assessments of the Fe-Tb and Fe-Dy binary systems was obtained by revising the expressions of the Gibbs energies of Fe2Tb and Fe2Dy and thus eliminating the artificial break in their heat capacity curves that appears in the earlier assessments. A set of available thermodynamic parameters for describing the Gibbs energies of all the phases in the Fe-RE (RE = Tb, Dy, Er, Lu, and Y) binary systems are used to calculate thermodynamic properties and phase equilibria of these five binary systems accurately.

Combining with the present assessments of the Fe-RE (RE = Tb, Dy, Er, Lu, and Y) binary systems and the previous calculations of the Fe-RE (RE = La, Ce, Pr, Nd, Sm, Gd, Ho, and Tm) binary systems, phase equilibria, and thermodynamic characteristics of thirteen Fe-RE binary systems were discussed systematically. A tend was demonstrated for the change of phase equilibria and thermodynamic properties of the Fe-RE alloys with the RE atomic number. Generally, as the increase of the RE atomic number, the formation temperatures of the Fe-RE intermetallic compounds increase gradually, and the enthalpy of mixing of liquid Fe-RE (apart from Fe-Ce and Fe-Y) alloys as same as the enthalpy of formation of the Fe-RE (apart from Fe-Y, Fe-Ce, Fe-Gd, Fe-Dy) intermetallic compounds become increasingly negative. In conclusion, the self-consistent set of thermodynamic parameters of the Fe-RE binary systems was finally obtained in this study, which would provide a solid basis for developing a thermodynamic database of Fe-RE-based alloy systems.

DECLARATIONS

Author’s contribution

Thermodynamic calculation and writing: Ye H, Rong M, Wang J

Data analysis and discussion: Ye H, Chen Q, Wang J

Performed literature survey and discussion: Yao Q, Rao G, Zhou H

Availability of data and materials

Supplementary Materials are available from the Journal of Materials Informatics or from the authors.

Financial support and sponsorship

This work was supported financially by Guangxi Natural Science Foundation (2020GXNSFFA297004), the National Natural Science Foundation of China (51971069, 51761008), Guangxi Key Laboratory of Information Materials (211007-Z) and Engineering Research Center of Electronic Information Materials and Devices (EIMD-AA202004), and Guilin University of Electronic Technology, China. The authors thank the support from the foundation for Guangxi Bagui scholars.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2023.

Supplementary Materials

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Cite This Article

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Ye H, Rong M, Chen Q, Yao Q, Wang J, Rao G, Zhou H. Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems. J Mater Inf 2023;3:15. http://dx.doi.org/10.20517/jmi.2023.08

AMA Style

Ye H, Rong M, Chen Q, Yao Q, Wang J, Rao G, Zhou H. Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems. Journal of Materials Informatics. 2023; 3(3): 15. http://dx.doi.org/10.20517/jmi.2023.08

Chicago/Turabian Style

Ye, Hongjian, Maohua Rong, Qing Chen, Qingrong Yao, Jiang Wang, Guanghui Rao, Huaiying Zhou. 2023. "Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems" Journal of Materials Informatics. 3, no.3: 15. http://dx.doi.org/10.20517/jmi.2023.08

ACS Style

Ye, H.; Rong M.; Chen Q.; Yao Q.; Wang J.; Rao G.; Zhou H. Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems. J. Mater. Inf. 2023, 3, 15. http://dx.doi.org/10.20517/jmi.2023.08

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