Thermodynamic Modeling of Formation Enthalpies of Amorphous and Crystalline Phases in Zr, Nd, and Ce-Substituted Fe-Si Systems

Abstract

The alloys that crystallize in a tetragonal ThMn₁₂-type (space group I4/mmm) structure and are based on Fe and rare earth elements are believed to have a potential to plug the performance gap between ferrite and Nd-based magnets. Nevertheless, the progress is hindered by their poor structural stability, compared with other phases competing during the synthesis process, e.g., Th₂Zn₁₇-type. In this work, the enthalpies of the formation (and other thermodynamic parameters) of various phases in (Zr, Nd, Ce)-Fe-Si systems were calculated, with paramount focus on the Fe-rich compositions. We compared and discussed the stability range and stabilization routes for amorphous phases, solid solutions, and intermetallics. The beneficial influence of Zr and Si on the crystallization of intermetallic compounds was confirmed, simultaneously being valid for other phases. Among all of the analyzed Fe-rich phases, the lowest values for enthalpy of the formation of the amorphous phase and solid solution were determined for ZrFe₁₀Si₂ (−17.5 and −18.2 kJ/mol, respectively). Moreover, substitution by elements with a large atomic radius is indicated as a method for the introduction of topological disorder, giving possibility for the synthesis of metastable phases (even amorphous) and the utilization of more sophisticated synthesis routes in the future.

1. Introduction

High performance permanent magnets are currently mostly based on rare earth elements (REE). One can easily refer to the Sm- or Nd-based compounds [1,2,3,4,5] with the highest energy product BH_|max|. The main shortcomings of the Nd₂Fe₁₄B (space group P4₂/mnm) compound is its low Curie temperature (312 °C [6]) and strong deterioration of its anisotropy field with temperature. Therefore, to enhance the anisotropy field and coercivity of modern 2:14:1 magnets, substantial amounts of Dy and other critical REE are used [7,8]. The criticality of such heavy REE forced the development in the field of REE free or at least REE lean permanent magnets. Summarizing, we can distinguish two important branches of research lately, namely: (i) decrease in REE content with the preservation of magnets’ magnetic performance and the (ii) search for novel REE-free materials. When considering the first approach, the tetragonal ThMn₁₂-type phase [9] (space group I4/mmm) is one of the candidates. It is believed to fill the gap between cheap ferrite and expensive 2:14:1 magnets [10]. One of the main issues in the case of materials based on the ThMn₁₂-type phase is their structural stability. Unfortunately, the ThMn₁₂-type phase is less stable for the most REE containing Fe-based compounds than, e.g., hexagonal 2:17-type phase (space group P6₃/mmc) [11]. Therefore, additional stabilizing elements are necessary to change the energy landscape and to increase the probability of 1:12-type phase formation. There are various types of substitution/doping mechanisms decreasing the relative enthalpy of the formation of 1:12-type phase and simultaneously allowing for its synthesis [12]. Silicon was mentioned as a stabilizer many years ago by Buschow [13] and de Mooij [14], and its role was later confirmed by other authors [15,16,17,18,19]. The formation of the 1:12-type phase was recently confirmed for the ZrFe₁₀Si₂ compound [20] and in some other systems, where Zr was substituted with REE [21]. In this paper, we focus on the formation enthalpy calculations of Fe-Si-based alloys with various substitutions (Zr, Ce, and Nd), with an emphasis put on the Fe-rich compositions.

The main aim of the paper was to determine the structural stability of various phases in (Ce, Zr, and Nd)-Fe-Si systems. We compared the enthalpy of the formation of hypothetical amorphous phases, solid solutions, and intermetallic compounds of chosen stoichiometry to answer more detailed questions, including the following: (i) Is it possible to synthesize metallic glass (to have an amorphous precursor for further crystallization of 1:12-type phase) with particular compositions and how broad is the glass forming range? (ii) Is it possible to synthesize NdFe₁₂ or NdFe₁₀Si₂ (not synthesized up to date) through the formation of off-stoichiometric alloys (to prepare structurally disordered precursors for further treatment or to restrain the formation of competing 2:17-type phase). To determine the enthalpies of formation, mismatch entropy, and glass forming ability parameter, we utilized a thermodynamic approach based on the semi-empirical Miedema’s model [22,23,24,25], which has been proven to be reliable in these types of calculations and matchless when dealing with the use of computing power. It should also be noted here that other thermodynamical models such as the cBΩ model [26,27,28] have also been applied to magnetic materials [29]. Later, a regular solution model was proposed by Takeuchi and Inoue [30]. To take into account the effect of the additive element on the properties of the binary alloy, asymmetric methods have been proposed [31], which in turn, in many cases, generate inaccuracy by the experience-based choice of asymmetric constituents. Therefore, the asymmetric geometric model proposed by Y. Ouyang et al. [32], independent on the choice of asymmetric element, was successfully implemented before [25,33,34,35] and is utilized in this paper.

2. Methods

The thermodynamic properties are calculated on the basis of Miedema’s semi-empirical model [23,24], also called the macroscopic-atom model. The atoms are taken in simplified form, as building blocks with the boundaries of the Wigner–Seitz cell. The enthalpy is the main thermodynamic quantity that is discussed throughout the paper and its variation is triggered by the contact of two blocks of atoms of different elements. Such an alloying process is thermodynamically described by three quantities that differentiate each element from the periodic table: (i) molar volume (V), (ii) potential similar to work function of an electron (φ), and (iii) density at the boundary of the Wigner–Seitz cell (n_ws). The interfacial enthalpy ΔH^inter(A in B) for solving one mole of A atoms in an excess of B is defined as follows:

$$\mathrm{∆}{H}^{inter}\left(A in B\right)=\frac{V_A^{2/3}}{\frac{1}{2}\left(\frac{1}{n_{wsA}^{1/3}}+\frac{1}{n_{wsB}^{1/3}}\right)}\left\{-P{\left(\mathrm{∆}\phi \right)}^2+Q{\left(\mathrm{∆}{n}_{ws}^{1/3}\right)}^2\right\},$$

(1)

where P and Q are empirical constants that depend on the alloying elements. To determine the possibility of the formation of various phases, one should compare their formation enthalpies. The formation enthalpy of the amorphous phase is expressed as the following:

$$\mathrm{\Delta }{H}^{am}=\mathrm{\Delta }{H}^{chem}+\mathrm{\Delta }{H}^{topo},$$

(2)

where ΔH^chem is the chemical contribution equal to the following:

$$\mathrm{∆}{H}^{chem}=c_A{c}_B\left(c_B^s\mathrm{∆}{H}^{inter}\left(A in B\right)+c_A^s\mathrm{∆}{H}^{inter}\left(B in A\right)\right),$$

(3)

while c_A, c_B and c^s_A(= $\frac{c_A{V}_A^{\frac{2}{3}}}{c_A{V}_A^{\frac{2}{3}}+c_B{V}_B^{\frac{2}{3}}}$), c^s_B are the volume and surface fractions of each atom type. ΔH^topo is a topological contribution reflecting the disorder in amorphous state and strongly dependent on the melting temperature.

$$\mathrm{\Delta }{H}^{topo}=3.5\left(c_A{T}_{m,A}+c_B{T}_{m,B}\right)\cdot {10}^{-3}.$$

(4)

The formation enthalpy of the solid solution is defined as follows:

$$\mathrm{\Delta }{H}^{ss}=\mathrm{\Delta }{H}^{chem}+\mathrm{\Delta }{H}^{elast}+\mathrm{\Delta }{H}^{struct}.$$

(5)

The first of the contributions unique for a solid solution is the elastic misfit enthalpy Equation (6), connected with the atom size mismatch:

$$\mathrm{∆}{H}^{elast}\left(A in B\right)=\frac{2K_A{G}_B{\left(W_A-W_B\right)}^2}{4G_B{W}_A+3K_A{W}_B},$$

(6)

where:

$$W_A=V_A+\alpha \frac{\left({\phi }_A-{\phi }_B\right)}{n_{ws}^A},\mathrm{and}\alpha =1.5\frac{V_A^{\frac{2}{3}}}{\frac{1}{n_{ws}^A{}^{\frac{1}{3}}}+\frac{1}{n_{ws}^B{}^{\frac{1}{3}}}}.$$

(7)

K is a bulk modulus, G is a shear modulus, and W_A Equation (7) and W_B are the corrected molar volumes (because of electron transfer from B to A and from A to B). The structural contribution ΔH^struct originates from the valence and the crystal structure of the solvent and the solute atoms. As a structural term can be only roughly estimated, according to Bakker [24], even for binary transition metal systems, we omitted it from the analysis. Moreover, it should be underlined that the structural contribution for the analyzed transition metal and rare earth element substitutions should not provide much energy difference, according to the structural enthalpies of binaries [24], especially when focussing on Fe-rich compositions. The structural enthalpy for solving one mole of Zr (four valence electrons) in Fe (eight valence electrons) is equal to 0 kJ/mol, while for solving Ce or Nd (both ~ 3 valence electrons) in Fe it is equal to 2 kJ/mol. The emphasis was put on the calculations of enthalpies of formation of amorphous phases and solid solutions, complementary to the vast number of literature data on intermetallic compounds, so as to compare the formation enthalpies of all phases that compete during the solidification process. The thermodynamic quantities determined for three sub-binary systems must be extrapolated to obtain the values for each ternary system. A geometric model was chosen for this purpose, as discussed in the introduction section. Apart from enthalpy of formation, the effect of atomic radius differences was determined by mismatch entropy (S_σ/k_B) calculations, on the basis of the relation given by Mansoori et al. [36], while the glass forming ability was additionally estimated using the glass forming ability parameter ΔP_HS [22], defined as the product of chemical enthalpy and normalized mismatch entropy ($\mathrm{∆}{H}^{chem}\mathrm{∆}\frac{S_{\sigma }}{k_B}$). A more detailed description of the utilized method can be found in [25]. Basic parameters used for the calculations are listed in

Table 1. Basic parameters used in the calculations of enthalpies of formation [24,37].

Chemical Element	φ (V)	nws (Arb. Units)	V (10−6 m3/mol)	K (1010 Pa)	G (1010 Pa)	d (Å)
Ce	3.18	1.69	21.62	2.395	1.197	3.64
Fe	4.93	5.55	7.09	16.83	8.152	2.56
Nd	3.19	1.73	20.58	3.268	1.452	3.64
Si	4.70	3.38	8.60	9.888	3.973	2.34
Zr	3.45	2.80	14.00	8.335	3.414	3.20

.

3. Results and Discussion

The enthalpies of formation of various phases in (Ce, Zr, Nd)-Fe-Si systems were calculated. The emphasis in the discussion is put on the compositions with a high Fe content, due to the arguments put forward in the introduction. This computational paper is intended to provide a keystone for further experimental works on the stabilization of 1:12-type hard magnetic phase. Substituting elements were chosen on the basis of a vast number of references with the intention to improve the structural stability of the 1:12-type phase in comparison with the other competing phases, bearing in mind that the superior aim is to retain the hard magnetic properties. We also check the possibility of the formation of amorphous phase in order to assess the potential of (Ce, Zr, and Nd)-Fe-Si alloys for a more sophisticated synthesis route i.e., vitrification by rapid quenching and further nanocrystallization. It can be beneficial when designing soft/hard magnetic exchange spring magnet composites, comprising of a soft magnetic Fe matrix and hard magnetic nanocrystals [3].

The contour maps of the enthalpies of formation of amorphous phase in Zr-Fe-Si, Ce-Fe-Si, and Nd-Fe-Si (Figure 1) are qualitatively similar. The most negative values are reached for Fe-free compositions. Large negative interfacial enthalpies between (Zr, Ce, and Nd) and Si are responsible for such behavior. Bearing in mind that the large atomic radii difference of alloy constituents promotes glassy state formation, the significant difference in (Zr, Ce, and Nd) and Si atomic radii, which are equal to 1.6–1.82 Å and 1.17 Å [37], respectively, is also fundamental here. The Fe atomic radius with its intermediate value of 1.28 Å plays a minor role. This gives us a first impression of the limited or even destructive role of Fe in the process of the formation of structural disorder in the Fe-rich alloys. The enthalpies of the formation of the amorphous phase in the Zr-Fe-Si system are more negative than those for Ce- and Nd-containing alloys. The most negative values of $\mathrm{∆}{H}^{am}$, obtained for Zr-Si, Ce-Si, and Nd-Si, are equal to −72.1 kJ/mol, −56.2 kJ/mol, and −53.0 kJ/mol for Zr₄₇Si₅₃, Ce₄₃Si₅₇, and Nd₄₃Si₅₇, respectively. A similar role for Zr has been already shown for (Mn, Zr)-Co-Ge alloys [38], where the enthalpies of formation of Zr-containing amorphous phases are much more negative, mainly due to the strongly negative values of the interfacial enthalpies of Zr-Co and Zr-Ge binaries. As can be seen in Figure 1, that a simultaneous decrease in Zr and Si content leads to a gradual increase in $\mathrm{∆}{H}^{am}$. Positive values for $\mathrm{∆}{H}^{am}$ are reached for a small number of, mainly Fe-rich, Zr-Fe-Si compositions. For Ce-Fe-Si and Nd-Fe-Si systems, this range is extended to all Ce- and Nd-rich alloys (most compositions with Ce and Nd content extending 95 at.%). Focusing on compositions of interest, namely (Zr, Ce, Nd)Fe_12−xSi_x (0 ≤ x ≤ 2) and (Zr, Ce, Nd)₂Fe_17-xSi_x (0 ≤ x ≤ 2), crucial differences are visible again when comparing Zr-containing and Ce- and Nd-containing alloys. For nominal compositions of the ThMn₁₂-type phase, so ZrFe₁₂, CeFe₁₂, and NdFe₁₂, we found $\mathrm{∆}{H}^{am}$ = −2.0 kJ/mol, 7.2 kJ/mol, and 6.5 kJ/mol, respectively. For Si-substituted compositions, which are believed to have a lower enthalpy of formation of the ThMn₁₂-type phase, namely ZrFe₁₀Si₂, CeFe₁₀Si₂, and NdFe₁₀Si₂, we found $\mathrm{∆}{H}^{am}$ = −17.5 kJ/mol, −8.9 kJ/mol, and −9.3 kJ/mol, respectively. At first, CeFe₁₂ and NdFe₁₂ compositions will unlikely form an amorphous phase, as their enthalpies of formation are highly positive, whereas the enthalpy of formation of the amorphous ZrFe₁₂ phase is slightly negative and theoretically exhibits some driving force for vitrification. Substitution of Fe by Si strongly influences enthalpy values, which are negative also for Ce and Nd. The most negative value of $\mathrm{∆}{H}^{am}$ among the analyzed compositions was found for the ZrFe₁₀Si₂ alloy, confirming the beneficial influence of the Zr and Si substitutions. One should bear in mind that kinetic parameters were not taken into account in the calculations and the formation of the amorphous phase would be more probable with specific methods (e.g., for high cooling rates). Moreover, the determined $\mathrm{∆}{H}^{am}$ values did not include any information on other phases (intermetallics and solid solution) that were competing during solidification process. To have a full picture, one needs to compare all of them. Therefore, another paragraph is devoted to the formation enthalpy of solid solutions.

The contour maps of the enthalpies of formation of solid solutions in Zr-Fe-Si, Ce-Fe-Si, and Nd-Fe-Si (Figure 2) are similar to the respective contour maps calculated for the amorphous phases (presented in Figure 1). The most negative values were reached for low Fe-content and for Zr-containing alloys. When we take a closer look at (Zr, Ce, Nd)Fe_12−xSi_x (0 ≤ x ≤ 2) and (Zr, Ce, Nd)₂Fe_17-xSi_x (0 ≤ x ≤ 2) compositions, rare earth alloys exhibited higher values of $\mathrm{∆}{H}^{ss}$ (as already reported also for $\mathrm{∆}{H}^{am}$). The enthalpy of formation of solid solution $\mathrm{∆}{H}^{ss}$ was equal to −2.8 kJ/mol, 5.9 kJ/mol, and 5.9 kJ/mol, for ZrFe₁₂, CeFe₁₂, and NdFe₁₂, respectively. Si-substituted compositions exhibited lower values of $\mathrm{∆}{H}^{ss}$ than Si-free counterparts mentioned above. $\mathrm{∆}{H}^{ss}$ is equal to −18.2 kJ/mol, −10.3 kJ/mol, and -10.0 kJ/mol, for ZrFe₁₀Si₂, CeFe₁₀Si₂, and NdFe₁₀Si₂, respectively. All of the $\mathrm{∆}{H}^{am}$ and $\mathrm{∆}{H}^{ss}$ values listed above are gathered in

Table 2. Enthalpies of the formation of amorphous phase $\mathrm{∆}{H}^{am}$, of solid solution $\mathrm{∆}{H}^{ss}$, their difference $\mathrm{∆}{H}^{am-ss}$, normalized mismatch entropy S_σ/k_B, and glass forming ability parameter ∆P_HS calculated for the chosen binary and ternary 1:12-type and 2:17-type compositions.

	ΔHam (kJ/mol)	ΔHss (kJ/mol)	ΔHam-ss (kJ/mol)	Sσ/kB	ΔPHS (kJ/mol)
ZrFe12	−2.0	−2.8	0.8	0.11	−0.9
CeFe12	7.2	5.9	1.3	0.31	0.3
NdFe12	6.5	5.9	0.6	0.31	0.1
ZrFe10Si2	−17.5	−18.2	0.7	0.14	−3.5
CeFe10Si2	−8.9	−10.3	1.4	0.36	−5.8
NdFe10Si2	−9.3	−10.0	0.7	0.36	−6.0
Zr2Fe17	−4.6	−3.6	−1.0	0.14	−1.6
Ce2Fe17	7.4	7.8	0.4	0.40	0.6
Nd2Fe17	6.5	7.8	−1.3	0.40	0.1
Zr2Fe15Si2	−16.6	−15.7	−0.9	0.17	−4.0
Ce2Fe15Si2	−5.1	−5.0	−0.1	0.44	−5.3
Nd2Fe15Si2	−5.7	−4.7	−1.0	0.44	−5.6

along with the data for 2:17-type phases (believed to be the main competing phases). The $\mathrm{∆}{H}^{am}$ and $\mathrm{∆}{H}^{ss}$ values calculated for Zr₂Fe₁₇ were more negative than those determined for the ZrFe₁₂ phase. Inverse dependence was observed for Ce and Nd substitutions, where both enthalpies were higher for 2:17-type phases. Si substitution plays a crucial role in all cases, reducing the enthalpy values significantly. To have a clear comparison of the $\mathrm{∆}{H}^{am}$ and $\mathrm{∆}{H}^{ss}$, the difference between both values (ΔH^am-ss) is presented in Figure 3 in the form of a contour map. This gives us a more reliable way to show the phase formation preferences for each composition. The ΔH^am-ss data specific for 1:12 and 2:17 compositions are shown in Table 2. The positive values point out compositions for which the formation of the solid solution is thermodynamically preferred over an amorphous phase. This is the case for all 1:12-type compositions gathered in Table 2. In turn, for all 2:17-type phases, except Ce₂Fe₁₇, the amorphous phase is preferred. If we take into account the influence of the structural enthalpy term (exact values discussed in the Calculations section), this preference should even be slightly strengthened. The most negative value of ΔH^am-ss = −1.3 kJ/mol was calculated for the Nd₂Fe₁₇ composition. To have a more overarching view, one should focus on the general trends visible in Figure 3. To improve the glass forming ability, or to at least destabilize the solid solution and to introduce structural disorder, the content of the elements with a large atomic radius, and preferably with a large negative heat of mixing with Fe (e.g., Zr), should be further increased at the expense of the Fe and Si content. The change in ΔH^am-ss sign (for about 15 at.% of Zr, Ce, and Nd) marks the change in phase preference (Figure 3).

To extend the analysis, the composition dependence of the normalized mismatch entropy was also calculated (Figure S1). Most of the compositions exhibited S_σ/k_B values higher than 0.1. This is the minimum value for the achievement of considerable glass forming ability [30]. Normalized mismatch entropy reflects the effect of atomic radii mismatch between the main constituents and, as expected due to the large atomic radius difference between Zr, Ce, Nd, and Si, reaches the highest values for Fe-free compositions. The lowest value of S_σ/k_B among the alloys of interest (Table 2) was determined for ZrFe₁₂ and was equal to 0.11. The highest mismatch reflected in the S_σ/k_B value of 0.44 was reached for Ce₂Fe₁₅Si₂ and Nd₂Fe₁₅Si₂. As expected, the results indicate a beneficial effect of the presence of largely mismatched atomic radii elements on the possibility of the formation of a disordered structure.

To combine both, the enthalpy of formation and the mismatch entropy, the glass forming ability parameter ∆P_HS (Figure S2) was also calculated as a product of S_σ/k_B and chemical enthalpy (basic term in the expression for enthalpy of formation for both, amorphous phase and solid solution). The obtained results were comparable to those of the formation enthalpy of the amorphous phase. The most negative values indicating the highest glass forming ability were reached again for the Fe-free compositions. Large negative interfacial enthalpies and an atomic radii difference between (Zr, Ce, and Nd) and Si are believed to be crucial here, as already stated in the case of $\mathrm{∆}{H}^{am}$. However, ∆P_HS reached the most negative values for the 1:12-type phases, −5.8 kJ/mol for CeFe₁₀Si₂ and −6.0 kJ/mol for NdFe₁₀Si₂. To conclude the first part of our research, one should expect that the substitution of Fe simultaneously by the elements with larger and smaller atomic radii, as in the case of (Zr, Ce, Nd) and Si substitutions, will promote the vitrification process. The minimum content of Zr, Ce, and Nd (equal to 15 at.%), determined on the basis of $\mathrm{∆}{H}^{am-ss}$, should be taken as an estimation only, as kinetic factors of synthesis process can be decisive. We can be more precise when comparing systems with various substitutions for similar compositions, and the data gathered in Table 2 can be treated as a guide. Nevertheless, qualitative and quantitative agreement of the calculated enthalpies with existing data can be found. For example, when one compares the $\mathrm{∆}{H}^{am-ss}$ contour map of the Nd-Fe-Si system with the experimental data reported in [39] (Nd_90-xFe_xSi₁₀ amorphous alloys were successfully synthesized in the range of 20 ≤ x ≤ 50), it is evident that the very broad glass forming range confirmed here for the low Si content converged well with the literature.

Planning future experimental works, we should compare the calculated values of the enthalpies of formation of amorphous alloys and solid solutions to those of intermetallic compounds apparently also competing during the synthesis process. There is experimental evidence on the formation of the phase of interest (ThMn₁₂-type) in (Zr, Ce)Fe₁₀Si₂ and similar systems. Sakurada et al. [40] showed that the ThMn₁₂-type phase crystallized (along with relatively small amount of α-Fe) in Nd_1-xZr_xFe₁₀Si₂ (x = 0.25, 0.5 and 0.75), indicating the atomic radius of the rare earth site element as a controlling factor. Moreover, it has been reported that the changes in the cooling rate in a rapid quenching process influenced the phase constitution. The ThMn₁₂-type structure has been stabilized in the samples quenched with a roll velocity of 10 m/s for Sm, Gd, and Er substitutions, while for 30 m/s, Zr and Nd also favored its formation [40]. Other studies have confirmed the stability of the tetragonal structure in (REE)Fe₁₀Si₂ for rare earth site elements having an atomic radius equal or smaller than 0.181 nm [20,41,42]. CeFe₁₀Si₂ and NdFe₁₀Si₂ preferably crystallize in the Th₂Zn₁₇-type structure [21]. The stabilization of the 1:12 structure in such alloys can be obtained through the substitution of larger (REE) atoms with smaller Zr [21,43,44]. To date, there is no direct evidence on the stability of 1:12-type phase in the alloys containing a combination of Nd, Zr, and Ce atoms. Nevertheless, it has been reported by Gabay and Hadjipanayis [18] for (Sm, Zr, and Ce)Fe₁₀Si₂ that the combination of three rare earth site substitutions resulted in the stabilization of a 1:12 structure with an improved hard magnetic performance (in comparison with stable alloys with the mixture of two elements on rare earth site) and simultaneously with a low content of critical rare earth elements.

A more exact comparison requires the consideration of the values of the enthalpies of formation of amorphous phases, solid solutions, and intermetallics. Because of the coexistence of many various types of phases, e.g., tetragonal 1:12-type (space group I4/mmm) or hexagonal (space group P6₃/mmc) and rhombohedral (space group R3m) 2:17-type phases in the region of interest in the phase diagram, and the limitations of Miedema’s model-based semi-empirical approach, the enthalpies of formation of intermetallic compounds were taken from theoretical works based on more sophisticated methods, e.g., ab initio calculations. The general conclusion, which can be drawn from the literature data, is the improvement of 1:12-type phase stability by the substitution of Fe by various stabilizing elements, as reported for SmFe₁₂ and SmFe₁₁(Co, Ti, V, and Ga) [45]. The lowering of the enthalpy of formation for the substituted SmFe₁₁(Co, Ti, V, Ga) samples resembled the variation of enthalpies of formation of solid solutions reported in the present paper (Table 2). While the energy of formation for SmFe₁₂ was approximately equal to 2 kJ/mol, it decreased for Co, Ti, V, and Ga substitutions and reached a value lower than -5 kJ/mol for SmFe₁₁Ti [45]. The recent study by Landa et al. [46] showed that nickel can also play a role as a stabilizer in the 1:12-type structure. Comparing these values with the formation energies reported in the paper by Harashima et al. [11] shows us a general picture about the relative stability of various intermetallic phases. At first, the value of the formation energy of SmFe₁₂ was determined to be equal to approx. 0.2 eV. Recalculation of this value to kJ/mol provided a much higher value than the 2 kJ/mol calculated in [45]. Nevertheless, the overall picture remained consistent. The formation energy of 1:12-type phase increased with increasing the atomic radius on the rare earth site. The energy of formation of ZrFe₁₂ was strongly negative (approx. −0.25 eV). It also remained negative for Lu, Tm, Er, Ho, and Dy. A further increase in the atomic radius made the formation of the 1:12-type phase unfavorable. For NdFe₁₂, the formation energy was highly positive and doubled the value determined for SmFe₁₂. When comparing these values with the enthalpies of formation reported in Table 2, it is clear that the energy of formation of intermetallics always had a higher absolute value. It means that in the case of compositions of interest, the formation of intermetallic phases was preferred in respect to the solid solution and amorphous phase. Nevertheless, one has to bear in mind that any deviation from ideal stoichiometry and growing chemical disorder could reverse this situation. Finally, we compared the formation energy values for the competing 2:17-type and 1:12-type phases. As opposed to the enthalpies of formation of solid solutions (Table 2), the energies of formation of rhombohedral Th₂Zn₁₇ and hexagonal Th₂Ni₁₇ were 0.04–0.16 eV lower than the energy of formation of tetragonal ThMn₁₂ phase for all of the analyzed rare earth site substitutions (Hf, Zr, Sc, Lu, Tm, Er, Ho, Dy, Y, Gd, Sm, Nd, Pr, and La) [11]. In conclusion, the utilization of Zr as the main rare earth site substitution had the highest impact on the lowering of the enthalpy of formation of all types of phases, intermetallics (as already reported in [11]), solid solutions, and amorphous. A further decrease is expected when substituting elements with a smaller atomic radius (e.g., Si, as discussed in the present paper) on Fe sites. The situation was slightly different when comparing various stable intermetallic phases containing the same chemical elements. In this case, the crystalline structure and hence varying interatomic distances played a more significant role. Therefore, the discussion that excludes mentioned factors is rather incomplete, but some general trends are still valid. The complex Zr-Fe-Si system with a vast number of stable intermetallic phases can be treated as an example [47]. Intermetallics with a relatively low content of Si and Zr (in comparison to Fe) are characterized by enthalpy of formation values equal to approximately −52 kJ/mol and −56 kJ/mol for Fe₁₆Si₇Zr₆ and Fe₃SiZr₂, respectively [48]. A decreasing content of Fe (Fe₄Si₇Zr₄) leads to a more negative formation enthalpy equal to approx. −68 kJ/mol [48]. We observed a similar trend for the formation enthalpy of a solid solution in the Zr-Fe-Si system (Figure 2), with slightly less negative values of enthalpy compared to the intermetallics. The enthalpy changes from −28 kJ/mol for Fe₃SiZr₂, through −31 kJ/mol for Fe₁₆Si₇Zr₆, to −48 kJ/mol for the Fe₄Si₇Zr₄ composition. In general, intermetallic phases are favored for stoichiometric (1:12 and 2:17) compositions, but it is believed that the further introduction of chemical disorder on rare earth and Fe sites can result in the formation of a solid solution (as in the case of high entropy alloys) or even amorphous phase. In this instance, the energy convex hull [12] can be used to determine the relative stability of the phases. The formation of a glassy structure can be also favored by the utilization of non-equilibrium synthesis methods.

4. Conclusions

In the present paper, we focused on the calculations of formation enthalpy of (Zr, Ce, and Nd)-Fe-Si systems. The most effort was put into the comparison of the formation enthalpies of various phases and the characterization of the mutual dependences between them for the Fe-rich compositions. As it was already elucidated, the ThMn₁₂-type phase was, for most Fe-based compounds containing rare earth elements, less stable than other intermetallic compounds with a similar stoichiometry [11]. The formation of the 1:12-type phase has been lately confirmed, e.g., in ZrFe₁₀Si₂ compound [12], which was a starting point for our research. The beneficial influence of Zr was confirmed for intermetallic compounds, but was also apparent when comparing the enthalpy of formation of solid solution $\mathrm{∆}{H}^{ss}$, which was equal to −2.8 kJ/mol, 5.9 kJ/mol, and 5.9 kJ/mol for ZrFe₁₂, CeFe₁₂, and NdFe₁₂, respectively. Further lowering of the $\mathrm{∆}{H}^{ss}$ was observed after the substitution of Fe by Si; ZrFe₁₀Si₂, CeFe₁₀Si₂, and NdFe₁₀Si₂ exhibited lower values of $\mathrm{∆}{H}^{ss}$ (−18.2 kJ/mol, −10.3 kJ/mol, and −10.0 kJ/mol, respectively) than their Si-free counterparts. The substitution of Nd and Ce had a similar impact on the formation enthalpies for all of the analyzed phases. Both elements were not favorable when aiming for the improvement of stability, but could be substituted to some extent (needed from the point of view of optimization of magnetic properties). The most negative value for $\mathrm{∆}{H}^{am}$ among the analysed compositions was reached for the ZrFe₁₀Si₂ alloy, confirming advantageous impact of Zr and Si atoms. In turn, the glass forming ability parameter was the lowest for the compositions containing Nd and Ce, indicating the great importance of large atomic radii substitutions on the formation of topological disorder. Intermetallic phases were favored for the analyzed stoichiometric compositions (1:12 and 2:17), but the substituting effect could be crucial for the formation of chemical and topological disorder and for the stabilization of solid solutions (growing entropy) or amorphous phase.