Thermodynamic Modeling of the Al–Co–Pd Ternary System, Aluminum Rich Corner

The aluminum-rich corner of the Al–Co–Pd ternary system was thermodynamically modeled by the CALPHAD method in the present study. The ternary system is a complex system with many ternary phases (W, V, F, U, Y2, C2). All ternary phases, except phase U, were modeled as stoichiometric compounds. The order–disorder model was used to describe the BCC–B2 and BCC-A2 phases. Solubility of the third element in binary intermetallic phases (Al5Co2, Al3Co, Al9Co2, Al13Co4, Al3Pd and Al3Pd2) was modeled. The experimental results collected from the literature were used in the optimization of the thermodynamic parameters. A good agreement between the experimental results and the calculations was achieved.


Introduction
Thermodynamic assessment for the Al-Co-Pd ternary system has not yet been published. However, the system is of interest for the possible use of the Al-Pd based materials, for example, for catalysts [1] or as coatings with good oxidation resistance, low adhesion and high hardness [2][3][4] and also in electronics as semiconductors [5,6]. In addition, the system is also interesting from a scientific point of view because it contains a large number of complicated intermetallics, quasicrystals and quasi-crystalline approximants [7,8].
The present work is focused on the modeling of the Al-Co-Pd ternary system in an Al-rich corner by the Calphad method using experimental results collected from the literature.

Co-Pd
The phase diagram of the Co-Pd binary system is presented in Figure 2. The diagram was calculated based on the thermodynamic data from study [17]. Part of the diagram at temperatures below 1400 K was not shown in the mentioned study [17] and consistent experimental results concerning the miscibility gap in the FCC phase at low temperatures have not been found in the literature. The diagram presented in Figure 2 was calculated using the mentioned thermodynamic data [17]. Liquid, Co-rich hcp solid solution and FCC solid solution with complete solubility are equilibrium phases present in the system.

Al-Pd
Thermodynamic data for the Al-Pd binary subsystem can be found in papers Li et al. [18] and Duriska et al. [19]. The AlPd phase (having the BCC-B2 crystallographic structure) with a wide homogeneity range was described by a sublattice asymmetrical model (Al%, Pd)(Pd%, Va) in both studies [18,19]. Extrapolation of the Al-Pd binary system to the ternary Al-Pd-Co system requires compatibility of models in corresponding binary subsystems. In the Al-Co binary system [16], the BCC-B2 phase corresponding to the AlPd phase in Al-Pd system is described by the order-disorder model. Therefore, the order-disorder model was used for the BCC phases also in an Al-Pd system. The ordered BCC-B2 phase (AlPd) is the stable phase and disordered BCC-A2 is an unstable phase in the Al-Pd system. The low-temperature modification of the AlPd phase (α-AlPd) has not been modeled in previously published assessments [18,19]. BCC-B2 is modeled as stable also at low temperatures. As a result of incomplete experimental information on the existence of a low-temperature modification of this phase and completely missing information on the possible existence of a low-temperature phase in the Al-Co system, the BCC-B2 phase was modeled as equilibrium also at low temperatures similarly as in study [18].
The phase diagram of the system is shown in Figure 3. There are seven other intermetallic phases (Al 4 Pd, Al 3 Pd (ε), Al 21 Pd 8 , Al 3 Pd 2 , Al 3 Pd 5 , Al 2 Pd 5 and AlPd 2 ), liquid and FCC solid solutions in the diagram.

Thermodynamic Models
Thermodynamic modeling by the Calphad method was performed by Thermo-Calc software (Thermo-Calc Software AB, Solna, Sweden) [23]. Parameters for pure elements were taken from Dinsdale [24].

Thermodynamic Models for FCC Solid Solution
The Gibbs energy of the FCC solid solution phase in the system is described using the two-sublattice model (Al,Co,Pd) 1 (Va) 1 . This two-sublattice model with the vacancies in the second sublattice was used to ensure the consistency with the model used for this phase in large metal-based databases (e.g., for steels), where the interstitial elements in the FCC phase play an important role. As there are no interstitial elements in this system, y Va is always 1 and the model is generally equal to the model for liquid (see below).
The Gibbs energy for the FCC phase is expressed as: In Equation (1), y represents the site fraction of component i in the relevant sublattice. G 0 i:Va is the Gibbs energy of pure element i in the phase.
All values of G are given relative to the Stable Element Reference state (SER) that is defined as the stable state of the element under standard conditions (298.15 K and 10 5 Pa). Interaction parameters L are expressed by a Redlich-Kister-Muggianu polynomial [25]: The temperature dependence of the L v parameter is expressed as follows: G mag in Equation (1) is the magnetic contribution to the Gibbs energy. Its value is calculated according to the model of Hillert and Jarl [26].

Thermodynamic Models for Liquid and Hcp Solid Solution
The liquid phase is described by a single sublattice. Its Gibbs energy is described as follows: where x i is a mole fraction of the component i.
The hcp phase is also described using single sublattice model as in the case of the liquid phase.

Thermodynamic Model for Stoichiometric Phases
The Al 4 Pd, Al 21 Pd 8 , Al 3 Pd 5 phases are described as stoichiometric phases with two sublattices (Al) a (Pd) c . The Gibbs energy per mole of the phase Al a Pd c is expressed by the Gibbs energy of formation relative to the chosen reference state as follows: G f can be given by the following expression where m and n are the parameters to be evaluated in the present work. These parameters correspond to the enthalpy and entropy of formation of a given stoichiometric phase, respectively, with respect to SER. The Al 3 Pd (ε), Al 3 Co, Al 5 Co 2 , Al 13 Co 4 , Al 9 Co 2 phases are also described using the model with two sublattices, but the solubility of the third element is taken into account. The Gibbs energy per mole of the formula Al a (Co,Pd) c is given as follows: Al:i + aRTy I Al lny I Al + cRT ∑ i y I I i lny I I i + y I Al y I I Co y I I Pd L Al:Co,Pd i = Co, Pd The W, V, F, Y2, C2 ternary phases are described as stoichiometric phases with a three sublattices model with formula The U ternary phase is described by a three sublattices model with (Al) 0.704 (Pd) 0.113 (Co,Pd) 0.183 formula. The model used for this phase is an extension of the model described by Equation (7).

Thermodynamic Model for Intermediate Phases with Homogeneity Ranges
There are several phases with a homogeneity range in the Al-Pd binary system. The AlPd 2 phase is described with a two-sublattice model with formula (Al%,Pd) 1 (Al,Pd%) 2 . The symbol % denotes a major component in the corresponding sublattice. Gibbs energy is given as follows: The Al 3 Pd 2 phase is described with two-sublattice models with formula (Al%,Pd) 3 (Al,Co,Pd%) 2 . The Gibbs energy is expressed as: The Al 2 Pd 5 phase is also described with a two-sublattice model; however, the first sublattice is occupied only by aluminum. The formula of the Al 2 Pd 5 phase is (Al) 2 (Al,Pd%) 5

Thermodynamic Model for BCC Ordered and Disordered Phases
The BCC-B2 phase is an ordered phase described with a two-sublattice model (Al,Co, Pd,Va) 0.5 (Al,Co,Pd,Va) 0.5 based on the model used in [16]. The model of the phase is based on a BCC-A2 solid solution model with formula (Al,Co,Pd,Va) 1 . The BCC-A2 is an unstable phase in the investigated Al-Pd-Co system; however, it is modeled to achieve consistency with other systems when the system is extrapolated to the higher systems. The single function describing the Gibbs energy of the ordered and disordered phase together is given by the following equation: Term G dis BCC is the Gibbs energy of the disordered BCC-A2 phase and can be calculated by a solution model similarly to the hcp-phase or liquid. ∆G ord BCC is the ordering contribution. The sublattices of the BCC-B2 phase model are crystallographically equivalent to each other, therefore G 0 i:j = G 0 j:i and L v i,j:k = L v k:,i,j and L v i,j:k,l = L v k,l:,i,j i,j,k,l = Al,Co,Pd,Va.

Al-Pd Binary System
The Al-Pd binary phase diagram is based on an assessment created by Li et al. [18]. However, in contrast to [18], the intermediate AlPd phase with a wide homogeneity range is modeled as the ordered BCC-B2 phase to achieve compatibility of description with other binary subsystems of the studied ternary system. For this purpose, also the BCC-A2 phase, unstable in the Al-Pd binary system, was modeled. This change of the model resulted in the necessary reassessment of the system and also parameters for Al 3 Pd 2 and Al 3 Pd 5 Al 21 Pd 8 intermediate phases had to be slightly modified. Parameters for Al 21 Pd 8 were modified with the aim of achieving better agreement with experimental results [21], which show that the Al 3 Pd phase forms by peritectic reaction liquid+Al 3 Pd 2 ↔ Al 3 Pd at 1062 K. On the contrary, according to assessment [18] (Figure 4a), the Al 3 Pd phase formed as a result of different (liquid+Al 21 Pd 8 ↔ Al 3 Pd) reactions based on the study [27]. According to Grushko [21], reaction liquid + Al 21 Pd 8 ↔ Al 3 Pd at 1055 K that was suggested in study [27] is considered as quite uncertain because only a weak kink associated with this reaction was observed in a DTA plot of the study [27]. The result of this reassessment is shown in Figure 3 and Figure 4b.

Al-Co-Pd Ternary System
In addition to a large number of binary phases, the Al-Co-Pd system also contains several stable ternary phases (W, V, F, U, Y, C2).
The homogeneity range of the W, V, C2 and Y phases is generally narrow. Therefore, they were modeled as stoichiometric phases. The F phase has a very narrow homogeneity range at 1323 and 1063 K, and slightly wider at 1273 and 1213 K according to Yurechko et al. [10]. The composition range of this phase at 1273 K is from 71.6Al-17.2Co-11.2Pd to 71.8Al-18.8Co-9.4Pd, and at 1213 K from 71.3Al-12.4Co-16.3Pd to 72.3Al-8.8Co-18.9Pd. Because the range of composition is relatively narrow, the phase was also modeled as a stoichiometric phase. For the U phase, the model described by formula (Al) 0.704 (Pd) 0.113 (Co,Pd) 0.183 was used. This model takes into account the homogeneity region of the phase. The experimentally determined width of the homogeneity region of the U-phase is about 5 at.% with a constant amount of aluminum [10].
The model for Al 13 Co 4 covers all phases of the Al 13 Co 4 family (M-Al 13 Co 4 , O-Al 13 Co 4 and Y-Al 13 Co 4 ) in the binary system and is suitable for the Al-Co binary system (Figure 1). On the other hand, the Y-phase observed in the Al-Co-Pd system by Yurechko et al. [10] has a similar structure to the high-temperature binary phase Y-Al 13 Co 4 in the Al-Co system [28]. However, no continuous range between the ternary Y phase and the binary Y-Al 13 Co 4 was observed in the Al-Co-Pd system [10]. Therefore, in the presented calculations, Y is presented as a ternary phase, and a different model was used. This phase was modeled as a stoichiometric phase from approximately 1063 to 1213 K based on the experimental results from Yurechko et al. [10]. The phase was presented as a separate ternary phase also in other experimental studies [11][12][13][14]. Figure 5a-d shows the isothermal sections of the Al-Co-Pd system at various temperatures. The calculated three-phase equilibria are in agreement with the experimentally determined results according to Yurechko et al. [10]. Only the experimentally determined three-phase equilibrium Al 9 Co 2 +M-Al 13 Co 4 +O-Al 13 Co 4 at 1213 K ( Table 2)    At 1063 K (Figure 5d), the Al 3 Pd+Al 3 Pd 2 +liquid equilibrium was calculated. The equilibrium is not presented in a phase diagram according to Yurechko et al. [10]. Yurechko presented homogeneity range of the Al 3 Pd phase from binary system (Al 3 Pd) to 73.7Al-11.1Pd-15.2Co at this temperature. However, the Al 3 Pd phase forms by peritectic reaction liquid + Al 3 Pd 2 ↔ Al 3 Pd at 1062 K in the binary system according to Grushko [21], which is in accordance with the presented calculations.
Al 3 Pd phase represents phases of ε-family (ε6, ε28 known from the binary system and ε16 ε22 ε34 observed in the ternary system [8]). The Al 3 Pd model was taken from the work of Li et al. [18] and was extended to the ternary system by the addition of Co into the model. The homogeneity range of the phase increases with decreasing temperature (Figure 5), which is in accordance with the experimental results from [10,13]. Nevertheless, such a wide range of homogeneity as in [10] was not achieved at lower temperatures. The modeled isothermal sections of the Al-rich corner of the Al-Co-Pd phase diagram ( Figure 5) take into account information about the formation of the ternary phases in the system. According to [10], the F, V and W ternary phases solidified from a liquid above 1323 K. V and W decompose between 1273-1213 and 1323-1273 K, respectively. F is observed at all studied temperatures. U melts between 1273 and 1323 K. The C2 is formed from a solid phase. The C2 was observed at 1063 K. The calculations are consistent with these experimental findings.
Better agreement between the experimentally determined maximum solubility of the third element in binary phases (Pd in Al 5 Co 2 , Al 3 Co, Al 13 Co 4 , Al 9 Co 2 ; Co in Al 3 Pd, Al 3 Pd 2 ), and the calculation was obtained for lower temperatures. The calculated solubility of Co is 4 and 15 at.% in Al 3 Pd 2 and Al 3 Pd, respectively, at 1063 K. Yurechko et al. [10] observed the same values of solubility in these phases. The calculated solubility of Pd is 1.7, 2.7, 0.45, and 5 in Al 9 Co 2 , Al 13 Co 4 , Al 3 Co and Al 5 Co 2 , respectively. Experimentally determined values are 2.6, 2.7, 1.6, and 3 at.% in the same phases, respectively [10]. Slightly larger differences between experimental and calculated values of solubility are in Al-Co binary phases at 1323 K, but due to the relatively small Pd solubility in these phases (about 3 at.% [10]), this does not have a significant effect on the character of the phase equilibria.
The liquidus surface prediction of the system was calculated and is shown in Figure 6. The calculations are in good agreement with liquidus lines presented in the isothermal sections in study [10] for higher temperatures. The agreement is slightly worse close to the Al-rich corner. Here the reason can be partially explained by observed disagreement between the calculated and experimental isotherms very close to the binary Al-Co and Al-Pd subsystems, which influences the shape of the isotherms in the ternary region as well. For example, in the Al-Pd system, the compositions of the binary system for the melting temperatures 1323, 1273 and 1213 K extrapolated from the experimental isotherms are very close to each other; nevertheless, they still reasonably correspond to the values obtained from the theoretical assessment of the binary Al-Pd system. On the other hand, the extrapolated position of the experimental isotherm at 1063 K is much closer to the Al corner than the calculated one. Similar disagreement can be found for the Al-Co binary especially close to the Al-rich corner. The discrepancies existing in binary subsystems indicate some doubts about the precision of experimentally established isotherms. Figure 6. Calculated liquidus surface prediction of the Al-Co-Pd system. Dotted lines represent liquidus lines from isothermal sections from study [10].
The thermodynamic parameters assessed in this work are given in Table 3. The parameters for phases of the Al-Co-Pd system from the region below 50 at.% of aluminum were not modified because experimental results are missing in the literature for this part of the diagram. Al 3 Pd 5 has been modified in connection with the modification of the Al-Pd binary diagram as mentioned above. Table 3. Thermodynamic parameters of the Al-Pd system.

Summary and Conclusions
The work deals with modeling of an Al-rich corner of the Al-Co-Pd phase diagram by the CALPHAD method using available experimental literature data. The achieved results can be summarized as follows: • The assessment of the Al-Pd binary system was modified. Intermediate AlPd phase with a wide homogeneity range in the binary system was modeled as an ordered BCC-B2 phase using order-disorder model to achieve compatibility of the description of the phases with other binary subsystems of the studied Al-Co-Pd ternary system. For this purpose, also the BCC-A2 phase, unstable in the Al-Pd binary system, was modeled. Parameters for Al 3 Pd 2 , Al 3 Pd 5 and Al 21 Pd 8 intermediate phases were slightly modified. • Thermodynamic assessment of Al-Co-Pd was performed. All ternary phases, except the U phase, were modeled as stoichiometric phases. The U phase was described by the (Al) 0.704 (Pd) 0.113 (Co,Pd) 0.183 formula. The solubility of the third element in binary phases (Al 5 Co 2 , Al 3 Co, Al 13 Co 4 , Al 9 Co 2 , Al 3 Pd and Al 3 Pd 2 ) was modeled. In addition, liquidus surface prediction was calculated. The binary phases from the region under 50 at.% Al were not modified.

•
The calculated phase diagram is in good agreement with experimental results from the literature. Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study (including TDB file) are available on request from the corresponding author.