Solidification Behavior of Dy-Tb-Fe Alloys through Experimental Study and Thermodynamic Calculation

Tan, Cong; Wei, Qi; Cheng, Weifeng; Liu, Xingyu; Bai, Yuchen; Wang, Jiang

doi:10.3390/ma16134697

Open AccessArticle

Solidification Behavior of Dy-Tb-Fe Alloys through Experimental Study and Thermodynamic Calculation

by

Cong Tan

¹,

Qi Wei

¹,

Weifeng Cheng

¹,

Xingyu Liu

¹,

Yuchen Bai

¹ and

Jiang Wang

^1,2,*

¹

Guangxi Key Laboratory of Information Materials, School of Materials Science and Engineering, Guilin University of Electronic Technology, Guilin 541004, China

²

Engineering Research Center of Electronic Information Materials and Devices, Ministry of Education, Guilin University of Electronic Technology, Guilin 541004, China

^*

Author to whom correspondence should be addressed.

Materials 2023, 16(13), 4697; https://doi.org/10.3390/ma16134697

Submission received: 3 June 2023 / Revised: 24 June 2023 / Accepted: 27 June 2023 / Published: 29 June 2023

(This article belongs to the Special Issue Metallic Materials: Microstructure, Phase Equilibria and Thermodynamics)

Download

Browse Figures

Review Reports Versions Notes

Abstract

:

In this work, the solidification microstructure and phase transitions of Dy-Tb-Fe alloy samples were studied by using scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS), X-ray diffraction (XRD) and differential thermal analysis (DTA). No stable ternary compound was detected in the present experiments. The phase transformation temperatures of eight Dy-Tb-Fe alloy samples were measured. Based on the experimental results determined in this work and reported in the literature, the phase equilibria of the Dy-Tb-Fe system was calculated using the CALPHAD method. The calculated vertical sections are consistent with the experimental results determined in this work and reported in the literature. Furthermore, in combination with the experimental solidification microstructure, the solidification behavior of Dy-Tb-Fe alloy samples was analyzed through the thermodynamic calculation with the Gulliver–Scheil non-equilibrium model. The simulated results agree well with the experimental results. This indicates that the reasonable thermodynamic parameters of the Dy-Tb-Fe system were finally obtained.

Keywords:

Dy-Tb-Fe; solidification microstructure; phase transition; thermodynamic calculation

1. Introduction

Nd-Fe-B permanent magnets with excellent magnetic properties have been widely used in various industrial fields such as wind turbines, electric vehicles, and aerospace [1,2,3,4,5]. With the increase in the operation temperature, the magnetic properties (e.g., coercivity and remanence) of Nd-Fe-B permanent magnets decrease [6,7,8,9]. In order to ensure the sufficient coercivity of Nd-Fe-B permanent magnets at the operation temperature, the addition of heavy rare earth elements (Dy, Tb, etc.) to Nd-Fe-B permanent magnets to partially substitute Nd is an effective method because the magnetocrystalline anisotropy fields of Dy₂Fe₁₄B and Tb₂Fe₁₄B are much higher than that of Nd₂Fe₁₄B, which would result in the great improvement of the coercivity for the permanent magnets [10,11,12]. To reduce the content of heavy rare earth metals Dy and Tb in Nd-Fe-B permanent magnets, the grain boundary diffusion process (GBDP) was developed recently [13]. During the GBDP, first, Dy and Tb can diffuse along the grain boundary phase into the interior of the magnet; then, the partial Nd on the surface of Nd₂Fe₁₄B grains is replaced by Dy and Tb to form a core–shell microstructure containing Nd₂Fe₁₄B and (Dy, Tb)₂ Fe₁₄B. The substitution of Nd by Dy and Tb dilutes the ferromagnetism of the grain boundary phase and further weakens the coupling between neighboring grains, which enhances the coercivity of the magnets after the grain boundary diffusion process [14,15]. In order to gain a deeper understanding of the grain boundary diffusion process, the thermodynamics and kinetics of Nd-Fe-B permanent magnets containing Tb and Dy elements are fundamental to enhance the overall performance of Nd-Fe-B magnets [16]. In particular, the solidification process of Nd-Fe-B-based alloys is significantly important to control microstructure and improve magnetic properties of Nd-Fe-B permanent magnets [17]. Solidification is a process in which the liquid phase decreases gradually and the solid phase is formed by peritectic, eutectic and other reactions during the cooling process until the liquid phase disappears completely [18]. Using thermodynamic calculations, the Scheil–Gulliver model can be employed to simulate the non-equilibrium solidification process of as-cast alloys [19]. In our previous work, the RE-Fe (RE = Pr, Nd, Sm, Gd, Dy, Tb, Ho, Tm, Lu, Y) [20,21,22,23,24], RE-B (RE = La, Ce, Pr, Nd, Sm, Gd, Dy, Tb, Ho, Tm, Lu, Y) [25,26,27], and RE₁-RE₂ binary systems [28] were calculated, and then thermodynamic calculations of the RE₁-RE₂-Fe (e.g., La-Ce-Fe and Ce-Nd-Fe [29], La-Pr-Fe and Ce-Pr-Fe [30]) ternary systems and the Nd-Fe-B ternary system [31] were performed. Furthermore, the development of a thermodynamic database of Nd-Dy-Tb-Fe-B magnets is in progress in our group.

In this work, the solidification microstructure and phase transitions of Dy-Tb-Fe as-cast alloy samples were investigated experimentally, and then phase equilibria of this ternary system was calculated using the CALPHAD method. Finally, the solidification behaviors of Dy-Tb-Fe as-cast alloy samples were investigated using the Scheil–Gulliver non-equilibrium model. It could demonstrate a solid foundation for the microstructure design of high-performance Nd-Fe-B permanent magnet materials with Dy and Tb elements.

2. Literature Information

2.1. Binary Systems

The thermodynamic database of Dy-Fe and Tb-Fe systems in the high-temperature range (above 800 K) was established by Landin et al. [32], but the contribution of magnetism to the Gibbs energy and experimental heat capacity of all intermetallic compounds were not taken into account. Rong et al. [22] optimized the Dy-Fe and Tb-Fe systems using the CALPHAD method considering the magnetic contribution and experimental heat capacity of the intermetallic compounds. The calculation results including phase relationship and thermodynamic properties are in good agreement with the experimental results. Recently, Ye et al. [24] re-conducted thermodynamic calculations for Dy-Fe and Tb-Fe systems to achieve compatibility with the RE-Fe thermodynamic database. The results of the Dy-Fe and Tb-Fe systems optimized by Ye et al. [24] were used in the present calculation of the Dy-Tb-Fe system.

The Dy-Tb phase diagram was not calculated in the reported literature up to now. Gschneidner et al. [33,34] reported that Dy and Tb are completely miscible and measured lattice parameters of the Dy-Tb alloys in the systematic review of lanthanide binary systems. According to the reported data [33,34], the Dy-Tb phase diagram was drawn by Moffatt [35] considering the continuous solid solution phases formed from β-Dy, β-Tb, α-Dy and α-Tb phases due to their same crystal structures [36]. Therefore, all the phases including liquid phase, bcc (β-Dy, β-Tb) and hcp (α-Dy, α-Tb) in the Dy-Tb system were described by using the ideal solution model. Figure 1 shows the calculated Dy-Tb phase diagram in this work.

2.2. Ternary System

The Dy_0.73Tb_0.27-Fe vertical section of the Dy-Tb-Fe system was measured by Westwood et al. [37] with DTA, X-ray diffraction and metallography. Landin et al. [32] directly extrapolated the Dy-Tb-Fe system based on the experimental results of Westwood et al. and Abell et al. [38], including the Dy_0.73Tb_0.27-Fe vertical section and liquidus projection. Although the values of the calculated Dy_0.73Tb_0.27-Fe vertical section are consistent with the experimental data [37], Ye et al. [24] have recently reassessed the calculations of the Dy-Fe and Tb-Fe systems. Therefore, thermodynamic calculations for the Dy-Tb-Fe system are still necessary.

3. Experimental Procedure

Eight Dy-Tb-Fe alloy samples with compositions of

x_{D y} : x_{T b} = 1 : 1

were prepared from bulk Dy, Tb and Fe (99.99% purity, China New Metal Material Technology Company, Ltd., Beijing, China). Each alloy sample (about 4 g) was melted 3–5 times in a vacuum arc melting furnace filled with argon gas to ensure uniform composition. The alloy samples were cooled in a copper crucible with cooling water, and thus the solidification process of the alloy samples was non-equilibrium due to the fast cooling rate.

The as-cast alloy samples were prepared by standard metallographic procedure. The morphology and phase composition of the alloy samples were measured by scanning electron microscope with energy dispersive spectroscopy (SEM-EDS, FEI 450G, FEI Company, Hillsboro, OR, USA). The compositions of each phase in the alloy samples were measured four times by EDS, and the standard deviations of the measured composition data were determined. After the alloy sample was ground into powder in anhydrous ethanol, the phase structures of the formed phases in the alloy samples were analyzed by X-ray powder diffraction (XRD, PLXcel 3D, Cu K_α radiation). The phase transition temperatures of the alloy samples were measured by differential thermal analysis (DTA, TA Instruments SDT/Q-600) using high-purity Al₂O₃ crucibles in a flowing argon atmosphere. Considering that rare earth metals are prone to oxidation, a heating/cooling rate of 20 K/min was used in DTA measurement.

4. Thermodynamic Calculation

4.1. Solution Phases

The solution phase φ including liquid, fcc, bcc and hcp is described by using the substitutional solution model. The molar Gibbs energy of the solution phase φ can be expressed as follows:

G_{m}^{φ} = \sum_{i = D y, T b, F e}^{} {x_{i}}^{} {}^{0}{G_{i}^{φ}} + R T \sum_{i = D y, T b, F e}^{} {x_{i}}^{} l n {x_{i}}^{} + {}^{m a g}{G_{m}^{φ}} + {}^{e x}{G_{m}^{φ}},

(1)

{}^{e x}{G_{m}^{φ} =} {x_{D y}}^{} {x_{T b}}^{} \sum_{j = 0}^{} {}^{j}{L_{D y, T b}^{φ} {(x_{D y} {- x}_{T b})}^{j}} + {x_{D y}}^{} {x_{F e}}^{} \sum_{j = 0}^{} {}^{j}{L_{D y, F e}^{φ} {(x_{D y} {- x}_{F e})}^{j}}, + {x_{T b}}^{} {x_{F e}}^{} \sum_{j = 0}^{} {}^{j}{L_{T b, F e}^{φ} {(x_{T b} {- x}_{F e})}^{j}} + {x_{D y}}^{} {x_{T b}}^{} {x_{F e}}^{} L_{D y, T b, F e}^{φ},

(2)

{}^{m a g}{G_{m}^{φ}} = R T l n (β_{0} + 1) g (τ),

(3)

L_{D y, T b, F e}^{φ} = x_{D y} {}^{0}{L_{D y, T b, F e}^{φ} +} x_{T b} {}^{1}{L_{D y, T b, F e}^{φ} +} x_{F e} {}^{2}{L_{D y, T b, F e}^{φ}},

(4)

where

{x_{i}}^{}

is the mole fraction of element i (i = Dy, Tb, Fe) and

{}^{0}{G_{i}^{φ}}

means the molar Gibbs energy of phase φ for element i (i = Dy, Tb, Fe); these values refer to the SGTE database [39]. R is the gas constant and T is the absolute temperature (Kelvin).

{}^{m a g}{G_{m}^{φ}}

is the magnetic contribution to Gibbs energy of the magnetic phase. In Equation (3),

τ = T / T_{c}^{φ}

, and

T_{c}^{φ}

is the Curie temperature of the phase φ.

β_{0}

is the Bohr magnetrons.

{}^{j}{L_{D y, F e}^{φ}}

and

{}^{j}{L_{T b, F e}^{φ}}

are the interaction parameters and are taken from the Dy-Fe and Tb-Fe systems evaluated by Ye et al. [24], respectively.

{}^{0}{L_{D y, T b, F e}^{φ}}

,

{}^{1}{L_{D y, T b, F e}^{φ}}

and

{}^{2}{L_{D y, T b, F e}^{φ}}

are the ternary interaction parameters to be evaluated.

4.2. Intermetallic Compounds

In the Dy-Tb-Fe system, a continuous solid solution is formed because of the same crystal structure of DyFe₂ and TbFe₂ [40,41,42]. Similarly, DyFe₃ and TbFe₃, Dy₆Fe₂₃ and Tb₆Fe₂₃, Dy₂Fe₁₇ and Tb₂Fe₁₇ also form a continuous solid solution in the Dy-Tb-Fe system. Therefore, these intermetallic compounds are modeled by (Dy, Tb)_0.3333Fe_0.66667, (Dy, Tb)_0.25Fe_0.75, (Dy, Tb)_0.2069Fe_0.7931 and (Dy, Tb)_0.1053Fe_0.8947, which are named REFe₂, REFe₃, RE₆Fe₂₃ and RE₂Fe₁₇ (RE = Dy, Tb). Their molar Gibbs energies can be expressed as

G_{m}^{{R E F e}_{2}} = Y_{D y}^{} {G_{D y : F e}^{{R E F e}_{2}} +} {Y_{T b}^{} {G_{T b : F e}^{{R E F e}_{2}} +}}_{}^{} 0.3333 R T (Y_{D y}^{} l n Y_{D y}^{} + Y_{T b}^{} l n Y_{T b}^{}) + Y_{D y}^{} Y_{T b}^{} {}^{j}{L_{D y, T b : F e}^{{R E F e}_{2}} {(x_{D y} {- x}_{T b})}^{j}} + {}^{m a g}{G_{m}^{{R E F e}_{2}}},

(5)

{}^{j}{L_{D y, T b : F e}^{{R E F e}_{2}}} = A_{1} + B_{1} T,

(6)

G_{m}^{{R E F e}_{3}} = Y_{D y}^{} {G_{D y : F e}^{{R E F e}_{3}} +} {Y_{T b}^{} {G_{T b : F e}^{{R E F e}_{3}} +}}_{}^{} 0.25 R T (Y_{D y}^{} l n Y_{D y}^{} + Y_{T b}^{} l n Y_{T b}^{}) + Y_{D y}^{} Y_{T b}^{} {}^{j}{L_{D y, T b : F e}^{{R E F e}_{3}} {(x_{D y} {- x}_{T b})}^{j}} + {}^{m a g}{G_{m}^{{R E F e}_{3}}},

(7)

{}^{j}{L_{D y, T b : F e}^{{R E F e}_{3}}} = A_{2} + B_{2} T,

(8)

{G_{m}^{{R E}_{6} {F e}_{23}}} = Y_{D y}^{} {G_{D y : F e}^{{R E}_{6} {F e}_{23}} +} {Y_{T b}^{} {G_{T b : F e}^{{R E}_{6} {F e}_{23}} +}}_{}^{} 0.2069 R T (Y_{D y}^{} l n Y_{D y}^{} + Y_{T b}^{} l n Y_{T b}^{}) + Y_{D y}^{} Y_{T b}^{} {}^{j}{L_{D y, T b : F e}^{{R E}_{6} {F e}_{23}} {(x_{D y} {- x}_{T b})}^{j}} + {}^{m a g}{G_{m}^{{R E}_{6} {F e}_{23}}},

(9)

{}^{j}{L_{D y, T b : F e}^{{R E}_{6} {F e}_{23}}} = A_{3} + B_{3} T,

(10)

{G_{m}^{{R E}_{2} {F e}_{17}}} = Y_{D y}^{} {G_{D y : F e}^{{R E}_{2} {F e}_{17}} +} {Y_{T b}^{} {G_{T b : F e}^{{R E}_{2} {F e}_{17}} +}}_{}^{} 0.1053 R T (Y_{D y}^{} l n Y_{D y}^{} + Y_{T b}^{} l n Y_{T b}^{}) + Y_{D y}^{} Y_{T b}^{} {}^{j}{L_{D y, T b : F e}^{{R E}_{2} {F e}_{17}} {(x_{D y} {- x}_{T b})}^{j}} + {}^{m a g}{G_{m}^{{R E}_{2} {F e}_{17}}},

(11)

{}^{j}{L_{D y, T b : F e}^{{R E}_{2} {F e}_{17}}} = A_{4} + B_{4} T,

(12)

in which

{}^{j}{L_{D y : F e}^{{R E F e}_{2}}}

,

{}^{j}{L_{T b : F e}^{{R E F e}_{2}}}

,

{}^{j}{L_{D y : F e}^{{R E F e}_{3}}}

,

{}^{j}{L_{T b : F e}^{{R E F e}_{3}}}

,

{}^{j}{L_{D y : F e}^{{R E}_{6} {F e}_{23}}}

,

{}^{j}{L_{T b : F e}^{{R E}_{6} {F e}_{23}},} {}^{j}{L_{D y : F e}^{{R E}_{2} {F e}_{17}}} a n d {}^{j}{L_{T b : F e}^{{R E}_{2} {F e}_{17}}}

are interaction parameters taken from the Dy-Fe and Tb-Fe systems assessed by Ye et al. [24].

{}^{j}{L_{D y, T b : F e}^{{R E F e}_{2}}}

,

{}^{j}{L_{D y, T b : F e}^{{R E F e}_{3}}}

,

{}^{j}{L_{D y, T b : F e}^{{R E}_{6} {F e}_{23}}}

and

{}^{j}{L_{D y, T b : F e}^{{R E}_{2} {F e}_{17}}}

are the interaction parameters to be optimized.

5. Results and Discussion

The microstructure and phase transitions of eight Dy-Tb-Fe as-cast alloy samples were determined in this work. The phase compositions and phase transition temperatures of the alloy samples measured by EDS, XRD and DTA are shown in Table 1.

5.1. Solidification Microstructure

Figure 2 contains the BSE micrograph and XRD patterns of Dy_42.5Tb_42.5Fe₁₅ alloy sample. In Figure 2a, the microstructure of this sample shows the formation of two phases, and the composition of the gray phase was measured by EDS to be 17.88 at.% Dy, 13.91 at.% Tb and 68.21 at.% Fe, while that of the light gray phase was determined to be 46.77 at.% Dy, 47.74 at.% Tb and 5.49 at.% Fe. According to the results determined by EDS in Table 1, the gray phase and the light gray phase were identified to be (Dy, Tb) Fe₂ and hcp (Dy, Tb), respectively, which was same as those of the XRD patterns in Figure 2b. Moreover, there is a large amount of the hcp (Dy, Tb) phase in Figure 2b, and the background of the diffraction pattern is too high and the spectral peak is not smooth, which is the result of the internal stress or preferred orientation generated during the preparation of metal powders. In addition, the microstructure characteristics of the Dy_42.5Tb_42.5Fe₁₅ alloy sample indicate that the light gray hcp (Dy, Tb) phase was formed first from the liquid phase during the solidification process.

Figure 3 shows the BSE and XRD images of Dy₂₅Tb₂₅Fe₅₀ and Dy₂₀Tb₂₀Fe₆₀ alloy samples. In Figure 3a,c, Dy₂₅Tb₂₅Fe₅₀ and Dy₂₀Tb₂₀Fe₆₀ alloy samples are composed of two phases, while their XRD patterns demonstrate the formation of the (Dy, Tb) Fe₂ and hcp (Dy, Tb) phase in Figure 3b,d. According to EDS results, there are two phases present in these two samples, with (Dy, Tb) Fe₂ in dark gray and hcp (Dy, Tb) in light gray. It means that the SEM-EDS results of these two samples are consistent with their XRD results. Meanwhile, the formation of the primary phase (Dy, Tb) Fe₂ and similar eutectic microstructure including (Dy, Tb) Fe₂ and hcp (Dy, Tb) was observed from the BSE micrographs of these two as-cast alloy samples.

As shown in Figure 4a,c, Dy₁₅Tb₁₅Fe₇₀ and Dy_13.5Tb_13.5Fe₇₃ alloy samples present the microstructure of two phases. Based on the composition measurements in Table 1, these two phases are identified to be (Dy, Tb) Fe₂ and (Dy, Tb) Fe₃, which are in good agreement with the XRD patterns in Figure 4b,d. Similarly, the microstructure characteristics of Dy₁₅Tb₁₅Fe₇₀ and Dy_13.5Tb_13.5Fe₇₃ alloy samples show that the (Dy, Tb) Fe₃ phase is the primary phase during this solidification process.

Figure 5 displays the BSE micrographs and XRD patterns of Dy₁₂Tb₁₂Fe₇₆ and Dy_10.5Tb_10.5Fe₇₉ alloy samples. In Figure 5a,c, three different phases were formed in these two samples. As given in Table 1, the experimental results obtained by EDS indicate that the light gray phase, the gray phase, and the dark black phase are (Dy, Tb) Fe₃, (Dy, Tb)₆ Fe₂₃ and (Dy, Tb)₂ Fe₁₇, respectively, which are same as those of the XRD patterns in Figure 5b,d. The microstructures of Dy₁₂Tb₁₂Fe₇₆ and Dy_10.5Tb_10.5Fe₇₉ alloy samples suggest that the (Dy, Tb)₆Fe₂₃ phase as the primary phase was formed.

Figure 6 presents the BSE and XRD images of the Dy₉Tb₉Fe₈₂ alloy sample. Based on the microstructure and XRD patterns in Figure 6a,b with the phase compositions measured by EDS in Table 1, the formation of the (Dy, Tb)₆ Fe₂₃ phase and the (Dy, Tb)₂ Fe₁₇ phase was found. Moreover, (Dy, Tb)₂ Fe₁₇ was formed as the primary phase in the Dy₉Tb₉Fe₈₂ alloy sample.

Based on the experimental results of eight Dy-Tb-Fe alloy samples with the compositions of

x_{D y} : x_{T b} = 1 : 1

determined by SEM-EDS and XRD, it was noted that the stable ternary intermetallic compound was not determined in this work. In addition, the EDS results measured in Table 1 illustrate that the solubilities of Dy in TbFe₂, TbFe₃, Tb₆Fe₂₃ and Tb₂Fe₁₇ as well as those of Tb in DyFe₂, DyFe₃, Dy₆Fe₂₃ and Dy₂Fe₁₇ are different. This indicates that (Dy, Tb) Fe₂, (Dy, Tb) Fe₃, (Dy, Tb)₆Fe₂₃ and (Dy, Tb)₂ Fe₁₇ all form continuous solid solution phase in the Dy-Tb-Fe system, which was also reported by Westwood et al. [37].

5.2. Phase Transition

Figure 7 shows the thermal analysis curve of Dy-Tb-Fe alloy samples in this work. Based on the thermal analysis results, the transition temperatures of Dy-Tb-Fe alloy samples were analyzed, and the results are listed in Table 1. In Figure 7a, the thermal curve of the Dy_42.5Tb_42.5Fe₁₅ alloy sample displays three peaks at 1030 K, 1057 K and 1435 K. Combined with Figure 2a, the peaks at 1030 K and 1057 K are generated due to the formation of the hcp (Dy, Tb) phase and the (Dy, Tb) Fe₂ phase, while the third peak at 1435 K is corresponding to the formation of the hcp (Dy, Tb) phase as the primary phase. The thermal curve of the Dy₂₅Tb₂₅Fe₅₀ alloy sample in Figure 7b presents three signal peaks at 1121 K, 1131 K and 1459 K, respectively, which correspond to the formation of the (Dy, Tb) Fe₂ and the hcp (Dy, Tb) phase. Nevertheless, the thermal curve of the Dy₂₀Tb₂₀Fe₆₀ alloy sample in Figure 7c shows only one endothermic peak at 1490 K, corresponding to the formation of (Dy, Tb) Fe₂, although the solidification microstructures of both Dy₂₅Tb₂₅Fe₅₀ and Dy₂₀Tb₂₀Fe₆₀ alloy samples contain the (Dy, Tb) Fe₂ and hcp (Dy, Tb) phases. In Figure 7d,e, the thermal analysis curves of Dy₁₅Tb₁₅Fe₇₀ and Dy_13.5Tb_13.5Fe₇₃ alloy samples indicate two peaks at 1494/1475 K and 1501/1505 K, corresponding to the formation of (Dy, Tb) Fe₂ and (Dy, Tb) Fe₃. Similarly, the thermal analysis curve of the Dy₁₂Tb₁₂Fe₇₆ alloy sample in Figure 7f indicates three signal peaks at 1482 K, 1495 K and 1555 K, corresponding to the formation of (Dy, Tb) Fe₃, (Dy, Tb)₆ Fe₂₃ and (Dy, Tb)₂ Fe₁₇. However, two peaks at 1497 K and 1562 K were observed in the DTA results of the Dy_10.5Tb_10.5Fe₇₉ alloy sample in Figure 7g, although the solidification microstructures of both Dy₁₂Tb₁₂Fe₇₆ and Dy_10.5Tb_10.5Fe₇₉ alloy samples consist of (Dy, Tb) Fe₃, (Dy, Tb)₆ Fe₂₃ and (Dy, Tb)₂ Fe₁₇. As shown in Figure 7h, two endothermic peaks at 1523 K and 1560 K were observed in the Dy₉Tb₉Fe₈₂ alloy sample, which correspond to the formation of (Dy, Tb)₆ Fe₂₃ and (Dy, Tb)₂ Fe₁₇. In addition, the oxidation peaks due to the easy oxidation of Dy-Tb-Fe alloy samples at high temperatures were also observed in Figure 7f,h.

5.3. Thermodynamic Calculation

Based on the experimental data determined in this work and reported by Westwood et al. [37] as well as previous evaluations of Dy-Fe, Tb-Fe, and Tb-Dy systems, the Dy-Tb-Fe system was calculated. The thermodynamic parameters of the Dy-Tb-Fe system obtained are listed in Table 2.

Figure 8 shows the calculated liquidus projection of the Dy-Tb-Fe system. It was found that no invariant reactions are existent in this ternary system. The calculated liquidus projection agrees with the experimental results of the primary phase determined in the experiment. Figure 9 shows the vertical sections of Tb_0.50Dy_0.50-Fe and Tb_0.27Dy_0.73-Fe calculated based on the experimental data determined in this work and reported by Westwood et al. [37]. The calculation results differ slightly from the experimental results and are still accepted within the experimental errors considering the oxidation of Dy-Tb-Fe alloy samples in the thermal analysis measurements at high temperature.

In order to gain a deeper understanding of the phase transformation of Dy-Tb-Fe alloy samples during the non-equilibrium solidification process, the thermodynamic parameters of the Dy-Tb-Fe system obtained in this work were employed to simulate the solidification process of the as-cast alloy samples using the Gulliver–Scheil model. Figure 10 displays the solidification process of four alloy samples (Dy_42.5Tb_42.5Fe₁₅, Dy₁₅Tb₁₅Fe₇₀, Dy_10.5Tb_10.5Fe₇₉, and Dy₉Tb₉Fe₈₂) simulated using the Gulliver–Scheil module. The simulated solidification path of the Dy_42.5Tb_42.5Fe₁₅ alloy sample in Figure 10a is as follows: L → hcp (Dy, Tb) + L → hcp (Dy, Tb) +L + (Dy, Tb) Fe₂. The calculated results show that the solidification structure of the Dy_42.5Tb_42.5Fe₁₅ alloy sample consists of a hcp (Dy, Tb) phase and (Dy, Tb) Fe₂, which is consistent with the microstructure observation results shown in Figure 2a. Figure 10b shows the simulated solidification path of the Dy₁₅Tb₁₅Fe₇₀ alloy sample: L → (Dy, Tb) Fe₃ + L → (Dy, Tb) Fe₃ + L + (Dy, Tb) Fe₂. It indicates that the microstructure of the Dy₁₅Tb₁₅Fe₇₀ alloy sample consists of (Dy, Tb) Fe₃ and (Dy, Tb) Fe₂. The simulation results are the same as the SEM results shown in Figure 4a. Similarly, Figure 10c shows the simulated solidification path of the Dy_10.5Tb_10.5Fe₇₉ alloy sample: L → L + (Dy, Tb)₆Fe₂₃ → L + (Dy, Tb)₆ Fe₂₃ + (Dy, Tb) Fe₃, while the simulated solidification path of the Dy₉Tb₉Fe₈₂ alloy sample in Figure 10d is as follows: L → L + (Dy, Tb)₂Fe₁₇ → L + (Dy, Tb)₂Fe₁₇ + (Dy, Tb)₆Fe₂₃, which is consistent with the microstructure observation results shown in Figure 5c and Figure 6a. This indicates that reliable thermodynamic data of the Dy-Tb-Fe system was obtained in this work, which can be employed to reproduce well the solidification processes of Dy-Tb-Fe alloy samples using the Gulliver–Scheil module.

6. Conclusions

This work investigates the solidification behavior of the Dy-Tb-Fe system using experimental measurements and thermodynamic calculations. The conclusions drawn are as follows:

The phase transition temperatures and phase compositions of eight Dy-Tb-Fe alloy samples were determined. Based on the experimental results determined in this work and reported in the literature, the thermodynamic calculation of the Dy-Tb-Fe system was performed using the CALPHAD method. The calculated vertical section and liquidus projection are consistent with the experimental results.
The solidification behaviors of several Dy-Tb-Fe alloy samples were simulated by using the Gulliver–Scheil non-equilibrium model with the obtained thermodynamic parameters. The simulation results were compared with the solidification structure of the experimental samples, and they were in good agreement with the experimental results. This means that the thermodynamic parameters of the Dy-Tb-Fe system optimized in this work are reliable and will provide reference for the microstructure design of high-performance Nd-Dy-Tb-Fe-B magnets.

Author Contributions

C.T. and J.W. conceived and designed the calculations and the experiments; C.T. and Q.W. performed the calculations and the experiments; W.C., X.L. and Y.B. analyzed the data; C.T. and J.W. wrote the paper. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Guangxi Natural Science Foundation (2020GXNSFFA297004), 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), Guilin University of Electronic Technology, China.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

All the data that support the findings of this study are included within the article.

Acknowledgments

The authors thank the support from the foundation for Guangxi Bagui scholars.

Conflicts of Interest

The authors declare no conflict of interest.

References

Poudyal, N.; Liu, J.P. Advances in nanostructured permanent magnets research. J. Phys. D Appl. Phys. 2012, 46, 043001. [Google Scholar] [CrossRef]
Coey, J.M.D. Perspective and prospects for rare earth permanent magnets. Engineering 2020, 6, 119–131. [Google Scholar] [CrossRef]
Nakamura, H. The current and future status of rare earth permanent magnets. Scr. Mater. 2018, 154, 273–276. [Google Scholar] [CrossRef]
Gutfleisch, O.; Willard, M.A.; Brück, E.; Chen, C.H.; Sankar, S.G.; Liu, J.P. Magnetic materials and devices for the 21st century: Stronger, lighter, and more energy efficient. Adv. Mater. 2011, 23, 821–842. [Google Scholar] [CrossRef]
Sugimoto, S. Current status and recent topics of rare-earth permanent magnets. J. Phys. D Appl. Phys. 2011, 44, 064001. [Google Scholar] [CrossRef]
Davies, B.E.; Mottram, R.S.; Harris, I.R. Recent developments in the sintering of NdFeB. Mater. Chem. Phys. 2001, 67, 272–281. [Google Scholar] [CrossRef]
Li, W.F.; Ohkubo, T.; Hono, K.; Sagawa, M. The origin of coercivity decrease in fine grained Nd-Fe-B sintered magnets. J. Magn. Magn. Mater. 2009, 321, 1100–1105. [Google Scholar] [CrossRef]
Li, J.; Sepehri-Amin, H.; Sasaki, T.; Ohkubo, T.; Hono, K. Most frequently asked questions about the coercivity of Nd-Fe-B permanent magnets. Sci. Technol. Adv. Mater. 2021, 22, 386–403. [Google Scholar] [CrossRef] [PubMed]
Chen, F.; Zhang, T.; Zhao, Y.; Wang, X.; Jiang, C.; Chen, J.; Zhao, W. A novel strategy to design and fabricate Nd-Fe-B magnets. J. Alloys Compd. 2021, 867, 159102. [Google Scholar] [CrossRef]
Trench, A.; Sykes, J.P. Rare earth permanent magnets and their place in the future economy. Engineering 2020, 6, 115–118. [Google Scholar] [CrossRef]
Löewe, K.; Brombacher, C.; Katter, M.; Gutfleisch, O. Temperature-dependent Dy diffusion processes in Nd-Fe-B permanent magnets. Acta Mater. 2015, 83, 248–255. [Google Scholar] [CrossRef]
Pan, M.; Zhang, P.; Li, X.; Ge, H.; Wu, Q.; Jiao, Z.; Liu, T. Effect of Terbium addition on the coercivity of the sintered NdFeB magnets. J. Rare Earths 2010, 28, 399–402. [Google Scholar] [CrossRef]
Li, W.; Yang, L.; Zhang, Q.; Xu, C.; Zhu, Q.; Song, Z.; Zheng, B.; Hu, F.; Jiang, J. Effect of the grain boundary Tb/Dy diffused microstructure on the magnetic properties of sintered Nd-Fe-B magnets. J. Magn. Magn. Mater. 2020, 502, 166491. [Google Scholar] [CrossRef]
Divinski, S.V.; Bokstein, B.S. Recent advances and unsolved problems of grain boundary diffusion. Defect Diffus. Forum 2011, 1212, 309–310. [Google Scholar] [CrossRef]
Chen, F. Recent progress of grain boundary diffusion process of Nd-Fe-B magnets. J. Magn. Magn. Mater. 2020, 514, 167227. [Google Scholar] [CrossRef]
Li, W.; Zhang, Q.; Zhu, Q.; Xiao, S.; Xu, C.; Yang, L.; Zheng, B.; Mao, S.; Song, Z. Formation of anti-shell/core structure of heavy rare earth elements (Tb, Dy) in sintered Nd-Fe-B magnet after grain boundary diffusion process. Scr. Mater. 2019, 163, 40–43. [Google Scholar] [CrossRef]
Liu, Z.; He, J.; Ramanujan, R.V. Significant progress of grain boundary diffusion process for cost-effective rare earth permanent magnets: A review. Mater. Des. 2021, 209, 110004. [Google Scholar] [CrossRef]
Luo, Q.; Guo, Y.; Liu, B.; Feng, Y.; Zhang, J.; Li, Q.; Chou, K. Thermodynamics and kinetics of phase transformation in rare earth–magnesium alloys: A critical review. J. Mater. Sci. Technol. 2020, 44, 171–190. [Google Scholar] [CrossRef]
Schaffnit, P.; Stallybrass, C.; Konrad, J.; Stein, F.; Weinberg, M. A Scheil–Gulliver model dedicated to the solidification of steel. Calphad 2015, 48, 184–188. [Google Scholar] [CrossRef]
Chen, T.L.; Wang, J.; Rong, M.H.; Rao, G.H.; Zhou, H.Y. Experimental investigation and thermodynamic assessment of the Fe-Pr and Fe-Nd binary systems. Calphad 2016, 55, 270–280. [Google Scholar] [CrossRef]
Chen, X.L.; Wang, J.; Chen, T.L.; Lin, X.D.; Rong, M.H.; Rao, G.H.; Zhou, H.Y. Thermodynamic reassessment of the Fe-Gd and Fe-Sm binary systems. Calphad 2017, 58, 151–159. [Google Scholar] [CrossRef]
Rong, M.H.; Chen, X.L.; Wang, J.; Rao, G.H.; Zhou, H.Y. Thermodynamic re-assessment of the Fe-Dy and Fe-Tb binary systems. Calphad 2017, 59, 154–163. [Google Scholar] [CrossRef]
Xu, L.; Wang, J.; Li, S.; Chen, X.L.; Rong, M.H.; Rao, G.H.; Zhou, H.Y. Thermodynamic reassessment of the Fe-Tm and Fe-Ho binary systems. Calphad 2019, 66, 101646. [Google Scholar] [CrossRef]
Ye, H.J.; Rong, M.H.; Yao, Q.R.; Wang, J.; Chen, Q.; Rao, G.H.; Zhou, H.Y. Assessment of phase equilibria and thermodynamic properties in the Fe-RE (RE = rare earth metals) binary systems. J. Mater. Inform. 2023, 3, 14. [Google Scholar]
Li, S.; Rong, M.H.; Xu, L.; Wei, Q.; Wang, J.; Rao, G.H.; Zhou, H.Y. Thermodynamic assessment of the RE-B (RE = Ce, Dy, Lu) binary systems. Calphad 2020, 68, 101740. [Google Scholar] [CrossRef]
Wei, Q.; Rong, M.H.; Li, S.; Su, D.; Yao, Q.R.; Wang, J.; Rao, G.H.; Zhou, H.Y. Thermodynamic assessment of the RE-B (RE = Ho, Er, Tm) binary systems. Calphad 2020, 70, 101796. [Google Scholar] [CrossRef]
Wei, Q.; Rong, M.H.; Li, S.; Su, D.; Rao, Q.R.; Wang, J.; Chen, Q.; Rao, G.H.; Zhou, H.Y. Thermodynamic calculation of phase equilibria of rare earth metals with boron binary systems. Int. J. Mater. Res. 2022, 113, 400–418. [Google Scholar] [CrossRef]
Yang, K.C.; Wang, J.; Yao, Q.R.; Lu, Z.; Rong, M.H.; Zhou, H.Y.; Rao, G.H. Phase diagrams of permanent magnet alloys: Binary rare earth alloy systems. J. Rare Earths 2019, 37, 1040–1046. [Google Scholar] [CrossRef]
Su, D.; Yang, K.C.; Rong, M.H.; Liu, P.P.; Yao, Q.R.; Wang, J.; Chen, Q.; Rao, G.H.; Zhou, H.Y. Phase transition and solidification microstructure of Ce-La-Fe and Ce-Nd-Fe alloys: Experimental investigation and thermodynamic calculation. Calphad 2023, 80, 102506. [Google Scholar] [CrossRef]
Su, D.; Rong, M.H.; Yang, K.C.; Yao, Q.R.; Wang, J.; Rao, G.H.; Zhou, H.Y. Thermodynamic calculation and solidification behavior of the La-Pr-Fe and Ce-Pr-Fe ternary systems. Calphad 2021, 74, 102285. [Google Scholar] [CrossRef]
Chen, T.L.; Wang, J.; Guo, C.P.; Li, C.R.; Du, Z.M.; Rao, G.H.; Zhou, H.Y. Thermodynamic description of the Nd-Fe-B ternary system. Calphad 2019, 66, 101627. [Google Scholar] [CrossRef]
Landin, S.; Ågren, J. Thermodynamic assessment of Fe-Tb and Fe-Dy phase diagrams and prediction of Fe-Tb-Dy phase diagram. J. Alloys Compd. 1994, 207, 449–453. [Google Scholar] [CrossRef]
Gschneidner, K.A.; Calderwood, F.W. Use of systematics for the evaluation of rare-earth phase diagrams and crystallographic data. Bull. Alloy Phase Diagr. 1983, 4, 129–131. [Google Scholar] [CrossRef]
Gschneidner, K.A.; Calderwood, F.W. Intra rare earth binary alloys: Phase relationships, lattice parameters and systematics. Handb. Phys. Chem. Rare Earths 1986, 8, 1–161. [Google Scholar] [CrossRef]
Moffatt, W.G. Binary Phase Diagrams Handbook; General Electric Comp: Schenectady, NY, USA, 1987. [Google Scholar]
Raghavan, V. Dy-Fe-Tb (dysprosium-iron-terbium). J. Phase Equilibria Diffus. 2004, 25, 169. [Google Scholar] [CrossRef]
Westwood, P.; Abell, J.S.; Pitman, K.C. Phase relationships in the Tb-Dy-Fe ternary system. J. Appl. Phys. 1990, 67, 4998–5000. [Google Scholar] [CrossRef]
Abell, J.S.; Lord, D.G. Microstructural studies of ternary rare earth-iron alloys. J. Less Common Met. 1986, 126, 107–112. [Google Scholar] [CrossRef]
Dinsdale, A.T. SGTE data for pure elements. Calphad 1991, 15, 317–425. [Google Scholar] [CrossRef]
Van der Goot, A.S.; Buschow, K.H.J. The dysprosium-iron system: Structural and magnetic properties of dysprosium-iron compounds. J. Less-Common Met. 1970, 21, 151–157. [Google Scholar] [CrossRef]
Okamoto, H. Dy-Fe (dysprosium-iron). J. Phase Equilibria 1996, 17, 80–81. [Google Scholar] [CrossRef]
Dariel, M.P.; Holthuis, J.T.; Pickus, M.R. The terbium-iron phase diagram. J. Less-Common Met. 1976, 45, 91–101. [Google Scholar] [CrossRef] [Green Version]

Figure 1. The calculated Dy-Tb binary phase diagram in this work.

Figure 2. (a) Back-scattered electron (BSE) micrograph and (b) XRD patterns of Dy_42.5Tb_42.5Fe₁₅ as-cast alloy sample.

Figure 3. Back-scattered electron (BSE) micrographs and XRD patterns of Dy₂₅Tb₂₅Fe₅₀ and Dy₂₀Tb₂₀Fe₆₀ as-cast alloy samples. (a,b) Dy₂₅Tb₂₅Fe₅₀, (c,d) Dy₂₀Tb₂₀Fe₆₀.

Figure 4. Back-scattered electron (BSE) micrographs and XRD patterns of Dy₁₅Tb₁₅Fe₇₀ and Dy_13.5Tb_13.5Fe₇₃ as-cast alloy samples. (a,b) Dy₁₅Tb₁₅Fe₇₀, (c,d) Dy_13.5Tb_13.5Fe₇₃.

Figure 5. Back-scattered electron (BSE) micrographs and XRD patterns of Dy₁₂Tb₁₂Fe₇₆ and Dy_10.5Tb_10.5Fe₇₉ as-cast alloy samples. (a,b) Dy₁₂Tb₁₂Fe₇₆, (c,d) Dy_10.5Tb_10.5Fe₇₉.

Figure 6. (a) Back-scattered electron (BSE) micrographs and (b) XRD patterns of Dy₉Tb₉Fe₈₂ as-cast alloy sample.

Figure 7. Thermal analysis curves of Dy-Tb-Fe alloy samples measured by DTA at heating rate of 20 K/min. (a) Dy_42.5Tb_42.5Fe₁₅, (b) Dy₂₅Tb₂₅Fe₅₀, (c) Dy₂₀Tb₂₀Fe₆₀, (d) Dy₁₅Tb₁₅Fe₇₀, (e) Dy_13.5Tb_13.5Fe₇₃, (f) Dy₁₂Tb₁₂Fe₇₆, (g) Dy_10.5Tb_10.5Fe₇₉, (h) Dy₉Tb₉Fe₈₂.

Figure 8. The calculated liquidus projection of the Dy-Tb-Fe system with the experimental results determined in this work.

Figure 9. The calculated vertical sections under the different composition conditions with the experimental data measured in this work and reported by Westwood et al. [37]. (a)

x_{D y} : x_{T b} = 1 : 1

, (b)

x_{D y} : x_{T b} = 0.73 : 0.27

.

Figure 9. The calculated vertical sections under the different composition conditions with the experimental data measured in this work and reported by Westwood et al. [37]. (a)

x_{D y} : x_{T b} = 1 : 1

, (b)

x_{D y} : x_{T b} = 0.73 : 0.27

.

Figure 10. Calculated Scheil–Gulliver solidification curves of the representative Dy-Tb-Fe as-cast alloy samples. (a) Dy_42.5Tb_42.5Fe₁₅, (b) Dy₁₅Tb₁₅Fe₇₀, (c) Dy_10.5Tb_10.5Fe₇₉, (d) Dy₉Tb₉Fe₈₂.

Table 1. Phase compositions and identified phases of Dy-Tb-Fe alloy samples.

Nominal Composition (at.%)	SEM-EDS Results (at.%)			Identified Phases by EDS and XRD	Phase Transitions (K)	Solidification Process
Nominal Composition (at.%)	Dy	Tb	Fe	Identified Phases by EDS and XRD	Phase Transitions (K)	Solidification Process
1#-Dy_42.5Tb_42.5Fe₁₅	46.77 ± 0.2	47.74 ± 0.6	5.49 ± 0.4	hcp (Dy, Tb)	1030, 1057, 1435	L → hcp (Dy, Tb) L → hcp (Dy, Tb) + (Dy, Tb) Fe₂
1#-Dy_42.5Tb_42.5Fe₁₅	17.88 ± 0.6	13.91 ± 0.2	68.21 ± 0.4	(Dy, Tb) Fe₂	1030, 1057, 1435	L → hcp (Dy, Tb) L → hcp (Dy, Tb) + (Dy, Tb) Fe₂
2#-Dy₂₅Tb₂₅Fe₅₀	17.51 ± 0.7	15.46 ± 0.3	67.03 ± 0.6	(Dy, Tb) Fe₂	1121, 1131, 1459	L → (Dy, Tb) Fe₂ L → hcp (Dy, Tb) + (Dy, Tb) Fe₂
2#-Dy₂₅Tb₂₅Fe₅₀	40.64 ± 0.2	52.09 ± 0.1	7.27 ± 0.2	hcp (Dy, Tb)	1121, 1131, 1459
3#-Dy₂₀Tb₂₀Fe₆₀	17.33 ± 0.7	15.96 ± 0.1	66.71 ± 0.8	(Dy, Tb) Fe₂	1490
3#-Dy₂₀Tb₂₀Fe₆₀	37.29 ± 0.4	55.31 ± 0.8	7.40 ± 1.0	hcp (Dy, Tb)	1490
4#-Dy₁₅Tb₁₅Fe₇₀	11.45 ± 0.6	11.81 ± 0.5	76.74 ± 0.6	(Dy, Tb) Fe₃	1494, 1501	L → (Dy, Tb) Fe₃ L → (Dy, Tb) Fe₂ + (Dy, Tb) Fe₃
4#-Dy₁₅Tb₁₅Fe₇₀	14.57 ± 0.1	17.95 ± 0.1	67.48 ± 0.1	(Dy, Tb) Fe₂	1494, 1501
5#-Dy_13.5Tb_13.5Fe₇₃	10.34 ± 0.3	12.91 ± 0.2	76.75 ± 0.1	(Dy, Tb) Fe₃	1475, 1505
5#-Dy_13.5Tb_13.5Fe₇₃	13.07 ± 0.3	17.41 ± 0.3	69.52 ± 0.6	(Dy, Tb) Fe₂	1475, 1505
6#-Dy₁₂Tb₁₂Fe₇₆	3.96 ± 0.3	6.37 ± 0.5	89.67 ± 0.4	(Dy, Tb)₂ Fe₁₇	1482, 1495, 1555	L → (Dy, Tb)₆ Fe₂₃ L→ (Dy, Tb) Fe₃ + (Dy, Tb)₆ Fe₂₃
	9.05 ± 0.1	11.73 ± 0.3	79.22 ± 0.3	(Dy, Tb)₆ Fe₂₃
	11.01 ± 0.2	12.95 ± 0.3	76.04 ± 0.4	(Dy, Tb) Fe₃
7#-Dy_10.5Tb_10.5Fe₇₉	3.54 ± 0.1	6.16 ± 0.5	90.30 ± 0.4	(Dy, Tb)₂ Fe₁₇	1497, 1562
	9.14 ± 0.1	11.48 ± 0.2	79.38 ± 0.2	(Dy, Tb)₆ Fe₂₃
	10.10 ± 0.2	13.33 ± 0.2	76.57 ± 0.4	(Dy, Tb) Fe₃
8#-Dy₉Tb₉Fe₈₂	3.54 ± 0.1	5.18 ± 0.3	91.28 ± 0.4	(Dy, Tb)₂ Fe₁₇	1523, 1560	L → (Dy, Tb)₂ Fe₁₇ L→ (Dy, Tb)₂ Fe₁₇ + (Dy, Tb)₆ Fe₂₃
8#-Dy₉Tb₉Fe₈₂	10.21 ± 0.1	11.87 ± 0.4	77.92 ± 0.5	(Dy, Tb)₆ Fe₂₃	1523, 1560	L → (Dy, Tb)₂ Fe₁₇ L→ (Dy, Tb)₂ Fe₁₇ + (Dy, Tb)₆ Fe₂₃

Table 2. Thermodynamic parameters of the Dy-Tb-Fe system.

Phase	Thermodynamic Parameters	Reference
Liquid	${}^{0}{L_{F e, D y}^{L i q u i d} = - 32,524.791 + 10.7 T}$	[24]
	${}^{1}{L_{F e, D y}^{L i q u i d} = 14,150.219 - 1.17 T}$	[24]
	${}^{2}{L_{F e, D y}^{L i q u i d} = 422.595 - 8.2 T}$	[24]
	${}^{0}{L_{F e, T b}^{L i q u i d} = - 20,115.512 - 3.2 T}$	[24]
	${}^{1}{L_{F e, T b}^{L i q u i d} = - 3332.5193 - 11.1 T}$	[24]
	${}^{2}{L_{F e, T b}^{L i q u i d} = 660.3 - 8.1 T}$	[24]
	${}^{0}{L_{D y, T b, F e}^{L i q u i d} = 6000}$	This work
bcc (α-Fe, δ-Fe, β-Dy, β-Tb)	${}^{0}{L_{F e, D y}^{b c c} = 38,000}$	[24]
bcc (α-Fe, δ-Fe, β-Dy, β-Tb)	${}^{0}{L_{F e, T b}^{b c c} = 58,000}$	[24]
fcc (γ-Fe)	${}^{0}{L_{F e, D y}^{f c c} = 38,000}$	[24]
	${}^{0}{G_{D y}^{f c c}} = {}^{0}{G_{D y}^{h c p}} + 5000$	[24]
	${}^{0}{L_{F e, T b}^{f c c} = 75,000}$	[24]
	${}^{0}{G_{T b}^{f c c}} = {}^{0}{G_{T b}^{h c p}} + 5000$	[24]
hcp (α-Dy, α-Tb)	${}^{0}{L_{F e, D y}^{h c p}} = 100,000$	[24]
hcp (α-Dy, α-Tb)	${}^{0}{L_{F e, T b}^{h c p}} = 150,000$	[24]
Fe₁₇RE₂ (Fe₁₇Dy₂, Fe₁₇Tb₂)	${G_{m}^{{F e}_{17} {D y}_{2}}} = - 12,882 + 3.477 T + 0.1053 {{}^{0}G}_{D y}^{h c p} + 0.8947 {{}^{0}G}_{F e}^{b c c}$	[24]
	${T_{c}^{{F e}_{17} {D y}_{2}}} = 371, {β_{0}^{{F e}_{17} {D y}_{2}}} = 0.161$	[24]
	${G_{m}^{{F e}_{17} {T b}_{2}}} = - 10,343 + 1.668 T + 0.1053 {{}^{0}G}_{T b}^{h c p} + 0.8947 {{}^{0}G}_{F e}^{b c c}$	[24]
	${T_{c}^{{F e}_{17} {T b}_{2}}} = 408, {β_{0}^{{F e}_{17} {T b}_{2}}} = 0.167$	[24]
	${}^{0}{L_{D y, T b : F e}^{{F e}_{17} {R E}_{2}}} = - 2000$	This work
Fe₂₃RE₆ (Fe₂₃Dy₆, Fe₂₃Tb₆)	${G_{m}^{{F e}_{23} {D y}_{6}}} = - 15,700 + 3.909 T + 0.2069 {{}^{0}G}_{D y}^{h c p} + 0.7931 {{}^{0}G}_{F e}^{b c c}$	[24]
	${T_{c}^{{F e}_{23} {D y}_{6}}} = 534, {β_{0}^{{F e}_{23} {D y}_{6}}} = 0.1$	[24]
	${G_{m}^{{F e}_{23} {T b}_{6}}} = - 12,501 + 0.355 T + 0.2069 {{}^{0}G}_{T b}^{h c p} + 0.7931 {{}^{0}G}_{F e}^{b c c}$	[24]
	${T_{c}^{{F e}_{23} {T b}_{6}}} = 574, {β_{0}^{{F e}_{23} {T b}_{6}}} = 0.01$	[24]
	${}^{0}{L_{D y, T b : F e}^{{F e}_{23} {R E}_{6}}} = - 2000$	This work
Fe₃RE (Fe₃Dy, Fe₃Tb)	${G_{m}^{{F e}_{3} {D y}_{}}} = - 15,506 + 3.101 T + 0.25 {{}^{0}G}_{D y}^{h c p} + 0.75 {{}^{0}G}_{F e}^{b c c}$	[24]
	${T_{c}^{{F e}_{3} {D y}_{}}} = 606, {β_{0}^{{F e}_{3} {D y}_{}}} = 0.493$	[24]
	${G_{m}^{{F e}_{3} {T b}_{}}} = - 12,008 - 0.277 T + 0.25 {{}^{0}G}_{T b}^{h c p} + 0.75 {{}^{0}G}_{F e}^{b c c}$	[24]
	${T_{c}^{{F e}_{3} {T b}_{}}} = 652, {β_{0}^{{F e}_{3} {T b}_{}}} = 0.426$	[24]
	${}^{0}{L_{D y, T b : F e}^{{F e}_{3} {R E}_{}}} = - 3000$	This work
Fe₂RE (Fe₂Dy, Fe₂Tb)	${G_{m}^{{F e}_{2} {D y}_{}}} = - 17,860 - 5.419 T - 0.00672 T^{2} + 3 R \{105 + T l n [1 - e x p (- \frac{210}{T})]\}$	[24]
	${T_{c}^{{F e}_{2} {D y}_{}}} = 635, {β_{0}^{{F e}_{2} {D y}_{}}} = 0.89$	[24]
	${G_{m}^{{F e}_{2} {T b}_{}}} = - 14,540 - 4.332 T - 0.0074 T^{2} + 3 R \{\frac{185}{2} + T l n [1 - e x p (- \frac{185}{T})]\}$	[24]
	${T_{c}^{{F e}_{2} {T b}_{}}} = 704, {β_{0}^{{F e}_{2} {T b}_{}}} = 0.762$	[24]
	${}^{0}{L_{D y, T b : F e}^{{F e}_{2} {R E}_{}}} = - 4000$	This work

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Share and Cite

MDPI and ACS Style

Tan, C.; Wei, Q.; Cheng, W.; Liu, X.; Bai, Y.; Wang, J. Solidification Behavior of Dy-Tb-Fe Alloys through Experimental Study and Thermodynamic Calculation. Materials 2023, 16, 4697. https://doi.org/10.3390/ma16134697

AMA Style

Tan C, Wei Q, Cheng W, Liu X, Bai Y, Wang J. Solidification Behavior of Dy-Tb-Fe Alloys through Experimental Study and Thermodynamic Calculation. Materials. 2023; 16(13):4697. https://doi.org/10.3390/ma16134697

Chicago/Turabian Style

Tan, Cong, Qi Wei, Weifeng Cheng, Xingyu Liu, Yuchen Bai, and Jiang Wang. 2023. "Solidification Behavior of Dy-Tb-Fe Alloys through Experimental Study and Thermodynamic Calculation" Materials 16, no. 13: 4697. https://doi.org/10.3390/ma16134697

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Menu

Solidification Behavior of Dy-Tb-Fe Alloys through Experimental Study and Thermodynamic Calculation

Abstract

1. Introduction

2. Literature Information

2.1. Binary Systems

2.2. Ternary System

3. Experimental Procedure

4. Thermodynamic Calculation

4.1. Solution Phases

4.2. Intermetallic Compounds

5. Results and Discussion

5.1. Solidification Microstructure

5.2. Phase Transition

5.3. Thermodynamic Calculation

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI