Thermodynamic Modeling of the Au-Ge-X (X = In, Sb, Si, Zn) Ternary Systems

In this study, the CALPHAD approach was employed to model the thermodynamics of the Au-Ge-X (X = In, Sb, Si, Zn) ternary systems, leveraging experimental phase equilibria data and previous assessments of related binary subsystems. The solution phases were modeled as substitutional solutions, and their excess Gibbs energies were expressed using the Redlich–Kister polynomial. Owing to the unavailability of experimental data, the solubility of the third elements in the Au-In, Au-Sb, and Au-Zn binary intermetallic compounds was excluded from consideration. Additionally, stable ternary intermetallic compounds were not reported in the literature and, thus, were not taken into account in the present thermodynamic calculations. Calculations of liquidus projections, isothermal sections, and vertical sections for these ternary systems have been performed, aligning with existing experimental findings. These thermodynamic parameters form a vital basis for creating a comprehensive thermodynamic database for Au-Ge-based alloys, which is essential for the design and development of new high-temperature Pb-free solders.


Introduction
The Pb-5wt.% Sn solder is commonly employed as a high-temperature solder in electronic packaging within the modern electronics industry.The environmental and health risks associated with lead necessitate research into high-temperature solders devoid of lead, aiming to supplant conventional solders that have high lead content.[1][2][3].
Gold-based alloys, such as Au-Sn, Au-Sb, and Au-Ge eutectic alloys, are recognized for their exceptional corrosion resistance, high electrical and mechanical strength, and superior thermal conductivity.These attributes render them highly advantageous for electronic packaging applications [4][5][6][7][8].Au-Ge-based alloys have especially attracted much attention as high-temperature Pb-free solders [4,9].To mitigate the costs of Au-based alloys, alloying elements, such as Ag, Bi, Cu, Ga, Ge, In, Sb and Zn, can be introduced to partially replace Au.In order to comprehensively understand the impact of alloying elements, a thorough understanding of the reliable phase equilibria and thermodynamic properties of Au-based alloys is essential.This knowledge is critical for the development of advanced high-temperature Pb-free solders based on Au, contributing to significant advancements in electronic packaging materials.

The Au-Ge-Sb Ternary System
The Au-Sb system, a pivotal component of the Au-Ge-Sb system, has been extensively investigated by numerous researchers [19, [29][30][31][32].Liu et al. [31] conducted an optimization of the Au-Sb system, taking into account the existing experimental data.The phase diagram and thermodynamic data previously published align well with the recent calculations [30].The optimization of the Ge-Sb system was initially conducted by Wang et al. [15] and Chewvalier et al. [27].Due to the oversight regarding the solubility of Ge in the rhombohedral Sb phase in these earlier studies, Liu et al. [21] conducted a re-optimization of the Ge-Sb system.The re-optimized results [21] correlate closely with the experimental observations.Consequently, the refined thermodynamic data for the Ge-Sb system from Liu et al. [21], along with the Au-Sb system data from Liu et al. [31], have been incorporated into the current calculations for the Au-Ge-Sb system.
Zwingmann [33] conducted a study on the phase equilibria of the Au-Ge-Sb system; however, the thermodynamic properties of this system have not been reported in the literature.Prince et al. [34] conducted a review of the Au-Ge-Sb system as part of their compilation of Au-based alloy phase diagrams.According to the experimental results [33], the Au-Ge-Sb system contains two invariant reactions: the peritectic reaction (U), liquid + rhombohedral (Sb) ↔ diamond (Ge) + AuSb 2 at 703 K and the eutectic reaction (E), liquid ↔ diamond (Ge) + fcc (Au) + AuSb 2 at 561 K. Three vertical sections of AuSb 2 -Ge, Ge 0.1 Sb 0.9 -Au, and 15 at.%Ge were measured by Zwingmann [33].Zwingmann [33] did not observe the presence of stable ternary intermetallic compounds in the Au-Ge-Sb system.
The Au-Ge-Sb system was optimized by Wang et al. [15].Afterward, Liu et al. [21] updated the thermodynamic parameters of the Ge-Sb system.Liu et al. [21] took into account the solubility of Ge in rhombohedral(Sb) during the optimization of the Ge-Sb system.The thermodynamic assessment of the Au-Ge-Sb system by Wang et al. [15] was found to be in good agreement with the experimental data [33].However, updated thermodynamic parameters for the Ge-Sb systems were provided by Liu et al. [21], indicating a necessity to re-optimize the Au-Ge-Sb system based on the findings reported in reference [33].

The Au-Ge-Si Ternary System
Meng et al. [20] conducted optimization studies on the Au-Si system, while Olesinski and Abbaschian [35] and Bergman et al. [36] focused on reviewing and optimizing the Ge-Si system, respectively.The optimization results for the Au-Si and Ge-Si systems from references [20,36] were well-aligned with the experimental findings and were, therefore, integrated into the current modeling of the Au-Ge-Si system.
While the literature does not detail the thermodynamic properties of the Au-Ge-Si system, Predel et al. [37] have explored its phase equilibria using thermal and metallographic analyses.They identified four vertical sections, namely Au-Ge 0.20 Si 0.80 , Au-Ge 0.40 Si 0.60 , Au-Ge 0.60 Si 0.40 , and Au-Ge 0.80 Si 0.75 , and constructed the liquidus projection for this system.The experimental findings from Predel et al. [37] have been incorporated into the ongoing calculations for the Au-Ge-Si system.

The Au-Ge-Zn Ternary System
The Au-Zn system was assessed by Okamoto and Massalski and Krachler et al. [38,39].Liu et al. [40] provided an updated thermodynamic assessment of the Au-Zn system.The thermodynamic calculation of the Ge-Zn system was performed by Chewvalier et al. [27].The calculated results of the Au-Zn and Ge-Zn systems by Liu et al. [40] and Chewvalier et al. [27] are in accordance with experimental data and were consequently utilized in the present calculation of the Au-Ge-Zn system.
As for the phase equilibria of the Au-Ge-Zn system, the vertical section of AuZn-Ge was investigated by Butt et al. [41] using differential thermal analysis (DTA) and the metallographic technique.Their experimental vertical section shows a pseudo-binary eutectic characteristic, and the eutectic composition and temperature were determined to be 12.2 ± 0.2 at.%Ge and 946 K, respectively.At the eutectic temperature, the solubility of Ge in the AuZn alloy was determined to be 1.3 atomic percent Ge, as reported in reference [41].These experimental findings [41] were utilized in the current calculations.

Calculated Results and Discussion
Leveraging the lattice stabilities compiled by Dinsdale [24], the optimization of thermodynamic parameters for the Au-Ge-X (X = In, Sb, Si, Zn) systems was performed using the PARROT module within the Thermo-Calc ® software 2024a, as developed by Sundman et al. [44].The final results, detailing the thermodynamic parameters for all phases within these systems, are comprehensively presented across Tables 2-5.
Figure 3 compares the calculated four vertical sections of AuIn-Ge, Au 41.5 In 58.5 -Ge, AuIn 2 -Ge, and Au 20 In 80 -Ge with the experimental results [28].As can be seen, these four vertical sections display a pseudo-binary eutectic characteristic.In Figure 3a, the calculated transition temperature and the composition of the liquid phase for the AuIn-Ge vertical section are 764 K and 3.0 at.%Ge, which are consistent with the experimental results (761 K and 2.0 at.%Ge) [28], respectively.

The Au-Ge-Sb System
Figure 4 displays the liquidus projection for the Au-Ge-Sb system calculated in this study, alongside comparisons to the projections made by Wang et al. [15].Figure 5 depicts the reaction scheme derived from the current calculations for the same system.Additionally, Table 7 lists the temperatures and compositions for the invariant reactions within the Au-Ge-Sb system, validated against both experimental findings [33] and previous calculations [15].As shown in Figure 4, the liquidus projection of the Au-Ge-Sb system includes one E-type invariant reaction (E1: Liquid ↔ diamond(Ge) + fcc(Au) + AuSb2) and one U- Specifically, in panel (a) of Figure 3, the transition temperature and the composition of the liquid phase for the AuIn-Ge section were determined to be 764 K and 3.0 atomic percent Ge, aligning closely with the experimental values of 761 K and 2.0 atomic percent Ge [28].Panels (b), (c), and (d) of Figure 3 show that the results for the Au 41.5 In 58.5 -Ge, AuIn 2 -Ge, and Au 20 In 80 -Ge sections also correspond well with the experimental findings [28].The calculated liquidus in these sections within the Au-Ge-In system demonstrates good agreement with the documented experimental results [28].

The Au-Ge-Sb System
Figure 4 displays the liquidus projection for the Au-Ge-Sb system calculated in this study, alongside comparisons to the projections made by Wang et al. [15].Figure 5 depicts the reaction scheme derived from the current calculations for the same system.Additionally, Table 7 lists the temperatures and compositions for the invariant reactions within the Au-Ge-Sb system, validated against both experimental findings [33] and previous calculations [15].As shown in Figure 4, the liquidus projection of the Au-Ge-Sb system includes one Etype invariant reaction (E 1 : Liquid ↔ diamond(Ge) + fcc(Au) + AuSb 2 ) and one U-type invariant reaction (U 1 : Liquid + rhombohedra(Sb) ↔ diamond(Ge) + AuSb 2 ).In this work, the temperatures of these two invariant reactions were calculated to be 561 K and 703 K, which are consistent well with the experimental data (561 K and 703 K) measured by Zwingmann [33].The calculated results of this work are much better than the results (560 K and 702 K) of Wang et al. [15].Moreover, the compositions of the liquid phase for these two invariant reactions were calculated to be 64.1 at.%Au-16.6 at.%Ge and 38.0 at.%Au-13.7 at.%Ge, which are consistent with the experimental data (68.0 at.%Au-15.0 at.%Ge and 35.0 at.%Au-14.0 at.%Ge) [33] and the calculated results [15].6 presents the calculated vertical sections of Au-Ge0.1Sb0.9,AuSb2-Ge, and 15 at.%Ge, compared against both the experimental findings [33] and the previously calculated results [15].Panels (a) and (c) of Figure 6 reveal that the phase-transition temperatures and liquidus lines for the Au-Ge0.1Sb0.9 and 15 atomic percent Ge sections align closely with the experimental data [33].For the AuSb2-Ge section, the calculated liquidus corresponds with the calculated data [15] but exhibits a minor variance from the experimental observations [33].

The Au-Ge-Si Ternary System
The calculated liquidus projection of the Au-Ge-Si system is shown in Figure 7; the Au-Ge-Si system presents a monotropic eutectic reaction including the liquid phase, fcc(Au) and diamond(Ge, Si).The monovariant curve from the Au-Si binary eutectic point e1 to the Au-Ge binary eutectic point e2 passes smoothly through a minimum point (599 K and 13.8 at.%Ge-11.7 at.%Si), which is consistent with the reported data (599 K and 8.8 at.%Ge-13.2 at.%Si) by Predel et al. [37].
Comparing the experimental data and the calculated results of the four vertical sections of Au-Ge0.8Si0.2,Au-Ge0.6Si0.4,Au-Ge0.4Si0.6, and Au-Ge0.2Si0.8, the calculated liquidus temperatures and phase-transition temperatures for the four vertical sections match the experimental data well, as shown in Figure 8.All the four vertical sections present a pseudo-binary eutectic characteristic [37].

The Au-Ge-Si Ternary System
The calculated liquidus projection of the Au-Ge-Si system is shown in Figure 7; the Au-Ge-Si system presents a monotropic eutectic reaction including the liquid phase, fcc(Au) and diamond(Ge, Si).The monovariant curve from the Au-Si binary eutectic point e 1 to the Au-Ge binary eutectic point e 2 passes smoothly through a minimum point (599 K and 13.8 at.%Ge-11.7 at.%Si), which is consistent with the reported data (599 K and 8.8 at.%Ge-13.2 at.%Si) by Predel et al. [37].Comparing the experimental data and the calculated results of the four vertical sections of Au-Ge 0.8 Si 0.2 , Au-Ge 0.6 Si 0.4 , Au-Ge 0.4 Si 0.6 , and Au-Ge 0.2 Si 0.8 , the calculated liquidus temperatures and phase-transition temperatures for the four vertical sections match the experimental data well, as shown in Figure 8.All the four vertical sections present a pseudo-binary eutectic characteristic [37].

The Au-Ge-Zn Ternary System
Figure 9 illustrates the reaction scheme of this system as obtained in this study, and Figure 10 shows the liquidus projection of the Au-Ge-Zn system that was calculated.Table 8 provides a summary of the temperatures and compositions of the invariant reactions.

The Au-Ge-Zn Ternary System
Figure 9 illustrates the reaction scheme of this system as obtained in this study, and Figure 10 shows the liquidus projection of the Au-Ge-Zn system that was calculated.Table 8 provides a summary of the temperatures and compositions of the invariant reactions.It is obvious that the liquidus projection of the Au-Ge-Zn system contains one E-type invariant reaction (E1: Liquid ↔ β2-AuZn + diamond(Ge) + γ1-AuZn3 at 920 K) and four U-type invariant reactions (U1: Liquid + γ1-AuZn3 ↔ diamond(Ge) + γ3-AuZn3 at 841

The Au-Ge-Zn Ternary System
Figure 9 illustrates the reaction scheme of this system as obtained in this study, and Figure 10 shows the liquidus projection of the Au-Ge-Zn system that was calculated.Table 8 provides a summary of the temperatures and compositions of the invariant reactions.It is obvious that the liquidus projection of the Au-Ge-Zn system contains one E-type invariant reaction (E1: Liquid ↔ β2-AuZn + diamond(Ge) + γ1-AuZn3 at 920 K) and four U-type invariant reactions (U1: Liquid + γ1-AuZn3 ↔ diamond(Ge) + γ3-AuZn3 at 841  It is obvious that the liquidus projection of the Au-Ge-Zn system contains one E-type invariant reaction (E 1 : Liquid ↔ β 2 -AuZn + diamond(Ge) + γ 1 -AuZn 3 at 920 K) and four U-type invariant reactions (U 1 : Liquid + γ 1 -AuZn 3 ↔ diamond(Ge) + γ 3 -AuZn 3 at 841 K, U 2 : Liquid + β 2 -AuZn ↔ diamond(Ge) + fcc (Au) at 802 K, U 3 : Liquid + γ 3 -AuZn 3 ↔ diamond(Ge) + ε 1 -Au 3 Zn 17 at 747 K, U 4 : Liquid + ε 1 -Au 3 Zn 17 ↔ diamond(Ge) + hcp (Zn) at 693 K).Moreover, the liquidus projection of the Au-Ge-Zn system illustrates that the presence of two maximum points (m 1 : 944 K and 44.0 at.%Au-9.4 at.%Ge; m 2 : 928 K and 31.9 at.%Au-4.1 at.%Ge) that are located on the monovariant curves of E 1 -U 2 and E 1 -U 1 .Because of the absence of experimental information on the liquidus projection of the Au-Ge-Zn system, additional experiments need to be performed to determine these invariant reactions and maximum points in the monovariant curves, which would be used to confirm the present calculations.diamond(Ge) + ε1-Au3Zn17 at 747 K, U4: Liquid + ε1-Au3Zn17 ↔ diamond(Ge) + hcp (Zn) at 693 K).Moreover, the liquidus projection of the Au-Ge-Zn system illustrates that the presence of two maximum points (m1: 944 K and 44.0 at.%Au-9.4 at.%Ge; m2: 928 K and 31.9 at.%Au-4.1 at.%Ge) that are located on the monovariant curves of E1-U2 and E1-U1.Because of the absence of experimental information on the liquidus projection of the Au-Ge-Zn system, additional experiments need to be performed to determine these invariant reactions and maximum points in the monovariant curves, which would be used to confirm the present calculations.

Conclusions
This study employed the CALPHAD approach, integrating it with data from the literature reviews and previous binary subsystem evaluations to conduct thermodynamic modeling of the Au-Ge-X (X = In, Sb, Si, Zn) systems.The comparison of the calculated liquidus projections and vertical sections of the Au-Ge-X (X = In, Sb, Si, Zn) systems demonstrates that the present calculations are consistent with the reported experimental data.Finally, a set of available and self-consistent thermodynamic parameters of the Au-

Conclusions
This study employed the CALPHAD approach, integrating it with data from the literature reviews and previous binary subsystem evaluations to conduct thermodynamic modeling of the Au-Ge-X (X = In, Sb, Si, Zn) systems.The comparison of the calculated liquidus projections and vertical sections of the Au-Ge-X (X = In, Sb, Si, Zn) systems demonstrates that the present calculations are consistent with the reported experimental data.Finally, a set of available and self-consistent thermodynamic parameters of the Au-Ge-X (X = In, Sb, Si, Zn) systems was obtained.These parameters enable the prediction of phase equilibria and thermodynamic behaviors of the Au-Ge-X alloys.Such data are crucial for the development of a thermodynamic database for multi-component Au-based alloys, enhancing the exploration of innovative high-temperature lead-free solders for electronic applications.

Materials 2024 , 17 Figure 1 .
Figure 1.The calculated liquidus projection of the Au-Ge-In system in this study.

Figure 1 .
Figure 1.The calculated liquidus projection of the Au-Ge-In system in this study.

Figure 1 .
Figure 1.The calculated liquidus projection of the Au-Ge-In system in this study.

Figure 2 .
Figure 2. Reaction scheme of the Au-Ge-In system.

Figure 2 .
Figure 2. Reaction scheme of the Au-Ge-In system.

Figure 4 .
Figure 4. Calculated liquidus projection of the Au-Ge-Sb system in this study with the calculated results by Wang et al. [15].

Figure 5 .
Figure 5. Reaction scheme of the Au-Ge-Sb system.

Figure 4 .
Figure 4. Calculated liquidus projection of the Au-Ge-Sb system in this study with the calculated results by Wang et al. [15].

Figure 4 .
Figure 4. Calculated liquidus projection of the Au-Ge-Sb system in this study with the calculated results by Wang et al. [15].

Figure 5 .
Figure 5. Reaction scheme of the Au-Ge-Sb system.

Figure 5 .
Figure 5. Reaction scheme of the Au-Ge-Sb system.

Figure 6
Figure 6 presents the calculated vertical sections of Au-Ge 0.1 Sb 0.9 , AuSb 2 -Ge, and 15 at.%Ge, compared against both the experimental findings [33] and the previously calculated results [15].Panels (a) and (c) of Figure 6 reveal that the phase-transition temperatures and liquidus lines for the Au-Ge 0.1 Sb 0.9 and 15 atomic percent Ge sections align closely with the experimental data [33].For the AuSb 2 -Ge section, the calculated liquidus corresponds with the calculated data [15] but exhibits a minor variance from the experimental observations [33].Materials 2024, 17, x FOR PEER REVIEW 12 of 17
For in the images, please m the caption.

Materials 2024 , 17 Figure 7 .
Figure 7. Calculated liquidus projection of the Au-Ge-Si system in this study.

Figure 7 .
Figure 7. Calculated liquidus projection of the Au-Ge-Si system in this study.

Figure 7 .
Figure 7. Calculated liquidus projection of the Au-Ge-Si system in this study.

Figure 9 .
Figure 9. Reaction scheme of the Au-Ge-Zn system.

Figure 10 .
Figure 10.Calculated liquidus projection of the Au-Ge-Zn system in this study.

Figure 9 .
Figure 9. Reaction scheme of the Au-Ge-Zn system.

Figure 9 .
Figure 9. Reaction scheme of the Au-Ge-Zn system.

Figure 10 .
Figure 10.Calculated liquidus projection of the Au-Ge-Zn system in this study.

Figure 10 .
Figure 10.Calculated liquidus projection of the Au-Ge-Zn system in this study.

Figure 11
illustrates the computed vertical section of the AuZn-Ge system alongside the documented experimental outcomes [41].This section exhibits characteristics of a pseudo-binary eutectic system.The study determined the eutectic reaction's temperature and composition at 943 K and 10.4 atomic percent Ge, aligning closely with the experimental measurements of 946 K and 12.2 atomic percent Ge [41].Notably, the alignment of the calculated liquidus for this section with the experimental data [41] demonstrates the model's accuracy.Materials 2024, 17, x FOR PEER REVIEW 15 of 17

LFigure 11
Figure 11 illustrates the computed vertical section of the AuZn-Ge system alongside the documented experimental outcomes [41].This section exhibits characteristics of a pseudo-binary eutectic system.The study determined the eutectic reaction's temperature and composition at 943 K and 10.4 atomic percent Ge, aligning closely with the experimental measurements of 946 K and 12.2 atomic percent Ge [41].Notably, the alignment of the calculated liquidus for this section with the experimental data [41] demonstrates the model's accuracy.

Figure 11 .
Figure 11.Calculated vertical sections of the Au-Ge-Zn system with the experimental results [41].
in the images, pl the caption.

Figure 11 .
Figure 11.Calculated vertical sections of the Au-Ge-Zn system with the experimental results [41].

Table 1 .
Crystallographic data of the solid solution phases in the Au-Ge-X (X = In, Sb, Si, Zn) systems.

Table 3 .
Thermodynamic parameters for the Au-Ge-Sb system.

Table 4 .
Thermodynamic parameters for the Au-Ge-Si system.

Table 6 .
Invariant reactions in the Au-Ge-In system.

Table 7 .
Invariant reactions in the Au-Ge-Sb system.

Table 7 .
Invariant reactions in the Au-Ge-Sb system.

Table 7 .
Invariant reactions in the Au-Ge-Sb system.

Table 8 .
Invariant reactions in the Au-Ge-Zn system.

Table 8 .
Invariant reactions in the Au-Ge-Zn system.