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TherModynaMIc properTIeS predIcTIon of Mg-al-Zn MelTS BaSed on The aToM and Molecule coexISTence Theory

A developed and verified thermodynamic model based on the atom and molecule coexistence theory (AMCT) is employed to predict activities relative to pure liquids in standard state in Mg-Al, Mg-Zn, Al-Zn and Mg-Al-Zn melts through the calculated mass action concentrations of structural units, i.e., N i . According to AMCT, N i can be extrapolated and calculated by the chemical equilibrium constant of a structural molecule, i.e., K i , in the Mg-Al-Zn ternary system and binary subsystems. In this paper, the standard Gibbs free energy function, for reported activities and mixing thermodynamic properties in Mg-Al, Mg-Zn and Al-Zn melts, was regressed and optimized. The results showed that K i and N i were deduced by Gibbs free energy function at the studied temperature. The results of calculating thermodynamic properties in the full composition range for liquid Mg-Al-Zn from 880 to 1100 K, as well as Mg-Al from 923 to 1073 K, Mg-Zn from 880 to 973 K and Al-Zn from 1000 to 1073 K, are presented in the paper by coupling with N i and AMCT. An excellent agreement is noticed between the calculated values of this study and measured thermodynamic data from the references, suggesting that the AMCT can be well applied to describe and predict the activities of the Mg-Al-Zn system


Introduction
Magnesium, aluminum and zinc alloys have been the promising materials in various fields, including the lightweight metal, functional materials in electronics and protective coatings [1][2][3]. Since the various applications of Mg-Al-Zn, previous researchers had directed tremendous attention to the Mg-Al-Zn system, especially to the thermodynamic behavior of the molten system. During the investigation of thermodynamic properties in liquid Mg-Al-Zn ternary system including Mg-Al [3][4][5][6], Mg-Zn [8][9][10] and Al-Zn [11][12][13] subsystems, some thermodynamic models and empirical formulas, listed in Table 1, were established and introduced to describe thermodynamic reaction abilities. Nevertheless, one of the potential problems for thermodynamic models is that the artificial parameters are employed in prediction formulas, triggering controversial results in the prediction models, so that the above mentioned prediction models may not work well. Moreover, very limited number of studies have been devoted to the thermodynamic properties in Mg-Al-Zn alloys. Even though Živković et al. [4] calculated the activities between 900 to 1200 K in the Mg-Al-Zn system by the general solution model, thermodynamic data for comparisons is not sufficient. Liang et al. [5] assessed thermodynamics by the crystal structure and cubic symmetry information, but within a limited temperature range in Mg-Al-Zn melts. Therefore, a thermodynamic prediction model without artificial parameters should be employed to investigate thermodynamic properties of Mg-Al-Zn system with more comparisons and wider temperature intervals in the present paper.
A thermodynamic model without artificial parameters, based on the atom and molecule coexistence theory, can be applied to predict the activities through the calculated mass action concentrations in binary solutions and binary melts [14]. The atom and molecule coexistence theory, i.e.,

J. Min. Metall. Sect. B-Metall. 55 (2) B (2019) 135 -145
AMCT, has been developed and verified through comparisons between the measured activities relative to pure liquids as standard state and mass action concentrations of structural units, i.e., N i , indicating that the mass action concentrations can be applied to substitute the measured activities of corresponding elements relative to pure liquids as standard state [15][16][17][18]. Therefore, in order to investigate the thermodynamic properties of Mg-Al-Zn in the full composition range more effectively, AMCT can be well applied to discuss the reaction abilities above the liquid temperature.
Considering great practical significance, the current research aims to contribute to the thermodynamic study of Mg-Al-Zn ternary melts and its binary subsystems on the basis of AMCT in the full composition range from 880 K to 1100 K by calculating N i . Since N i can be deduced and calculated according to the obtained in Mg-Al-Zn ternary melts and binary subsystems, the process of regression and modification for the chemical equilibrium constant and standard molar Gibbs free energy function of structural molecules, i.e., and, in Mg-Al, Mg-Zn and Al-Zn binary melts will be presented in the paper. What's more, once and were obtained, thermodynamic properties at the given temperature in the full composition range can be extrapolated and calculated through AMCT. In order to test the accuracy of thermodynamic properties predicted by AMCT, the calculation thermodynamics results are compared with the reference data in the Mg-Al-Zn ternary system and binary subsystems in this paper.

hypotheses
According to the AMCT [15][16][17][18], Zhang et al. [16,17] argued that both atoms and their intermetallic compound molecules, i.e., structural characteristics, can coexist in melts. The main points of AMCT [15][16][17][18] can be briefly summarized as: 1) Structural units are composed of both atoms and molecules according to phase diagram at elevated temperature; 2) The atoms would participate in chemical equilibrium reactions with intermetallic molecules, such as: (1) where presents the chemical equilibrium constant of A x B y ; a i the activity of the given structural unit; N i the mass action concertation of the structural unit; 3) The chemical reaction above obeys the mass action law; 4) The mass action concentrations can be applied to substitute the measured activities of corresponding elements relative to pure liquids as standard state.
Given assumptions and main points, Zhang et al.  [16,17] introduced formulas and conclusions of mixing thermodynamic properties presented as follows. (2) where is the integral molar Gibbs energy of mixing; the integral molar enthalpy of mixing; the integral molar entropy of mixing; the sum mass action concentration of component elements; T the temperature; R the gas constant; , and are the Gibbs energy, enthalpy and entropy of the structural molecule i, respectively.

results and discussion
In order to investigate thermodynamic properties in Mg-Al-Zn melts based on AMCT, the information of structural units in the Mg-Al-Zn ternary system should be deduced by Mg-Al, Mg-Zn and Al-Zn binary subsystems. Zhang et al. [16,17] have studied activities and mixing thermodynamics in the molten Mg-Al at 1073 K and Al-Zn at 1000 K. However, due to the calculation results built on a given temperature without optimization and the lack of enough measured data for comparison [16][17] for Mg-Al and Al-Zn, it is necessary to investigate and confirm the Mg-Al, Al-Zn system, as well as Mg-Zn melts with more data. The present paper would reassess the Mg-Al, Al-Zn systems and evaluate the Mg-Zn, Mg-Al-Zn by verifying the results, both measured and calculated in the mentioned melts. Hence, combined with reported activities, mixing thermodynamics and chemical reaction isotherm, the results of calculations would be more reliable through regression and modification in the present paper.

Mg-Al binary system
According to AMCT [15][16][17][18], the mass action concentration, i.e., N i can be calculated, once could be obtained. In order to describe N i at other temperature, all the of structural molecules at the same temperature should be known. However, very limited amounts of experimental activities data have been found in the literature for Mg-Al system, so that can't be regressed directly based on AMCT [15][16][17][18]. Fortunately, Bhatt et al. [19] measured the activity and mixing parameter of liquid Al-Mg alloys by vapor pressure measurements at 1000 K, while Lu, Tiwari, and Hultgren et al. [20][21][22] applied EMF method to measure the activities and mixing thermodynamics measurement at 1073 K. Considering the lack of enough measured activities with respect to the wider temperature and concentration ranges in liquid Mg-Al, optimization should be applied to those measured data, so that the function between and T could be obtained. According to the previous works of Zhang et al. [16,17], two kinds of atoms, Mg and Al, as well as three kinds of structural molecules, i.e., Mg 2 Al, Mg 17 Al 12 and MgAl 2 , coexisting in the Mg-Al system at 1073 K, are included in the present study. In addition, once the activities and the mixing thermodynamics from literature [19][20][21][22] are substituted into Eqs. (6)-(8), the thermodynamic parameters, i.e., , , and summarized as Table 2, could be regressed.
During the calculation process, the measured activities of Mg and Al are assumed to be substituted by the mass action concentration, so that Eqs. (5) - (9) are the functions of , , with N Mg and N Al . With the regressing of Eqs. (5) -(9) by the MATLAB software , , , as well as can be optimized and obtained as it is shown in Table 2. Moreover, the standard molar Gibbs free energy of structural molecules in Mg-Al melts can be expressed as: The , at the studied temperature, can be obtained from Eqs. (10) - (12), so that N i at the same temperature can be calculated based on the AMCT [15][16][17][18]. In this paper, in order to describe N i at more investigated temperature, Eqs. (10) -(12) were extrapolated to calculate and N i from 923 to 1073 K. Furthermore, comparisons of calculated quantities by AMCT with the reported activities of Mg or Al and mixing thermodynamic properties [19][20][21][22] from 923 to 1073 K are presented in Figure 1.
The results of comparisons between calculated mass action concentrations of Mg(N Mg ), Al(N Al ) as well as structural molecules , , by AMCT and measured activities of Mg (a Mg ), Al (a Al ) relative to pure liquids as standard state in the full composition range of Mg-Al melts are shown in Figure 1(a), (c) and (e) from 923 to 1073 K, respectively. Furthermore, Figure 1(b) and (d) present the comparison of mixing thermodynamics between calculated and reported of Mg-Al melts at 1000 K and 1073 K, respectively. As can be seen in Figure 1, calculated values have an excellent agreement with the measured data, obviously, indicating the calculated mass action concentration of N Mg and N Al can be successfully applied to substitute the measured a Mg and a Al in Mg-Al melts in the full composition range from 923 to 1073K.
where Eq. (13) is the regression for primary , according to AMCT [15][16][17][18]; i in Eq. (14) represents structural molecules, i.e., MgZn 2 , Mg 2 Zn 11 , Mg 4 Zn 7 and MgZn, respectively. Compared to Section 3.1, there are more activities available in Mg-Zn, enabling and to be modified by the activities without mixing thermodynamics. Therefore, through the primary and the chemical reaction isotherm, i.e., Eqs. (13) - (14), the standard molar Gibbs free energy of structural molecules of MgZn 2 , Mg 2 Zn 11 , Mg 4 Zn 7 and MgZn can be described as follows, so that can be worked out and optimized at studied temperatures.
Therefore, coupled with AMCT [15][16][17][18], the modified equilibrium constants and mass action concentration can be deduced by the extrapolation of Eqs. (15) - (18). The modified in Mg-Zn system from 880 to 973 K by Eqs. (15) -(18) is listed in Table  3, while the mass action concentrations and comparison activities reported are shown in Figure 2. Figure 2 shows the comparison between calculated mass action concentration of structural units, such as, N Mg , N Zn , N MgZn , , , N MgZn , and measured activities of Mg (a Mg )or Zn (a Zn ) relative to pure liquids as standard state in the full composition range of Mg-Zn melts from 880 to 1073 K, respectively. It is obvious that the calculated mass action concentrations of N Mg and N Zn nearly equal to the measured a Mg and a Zn in Mg-Zn binary melts in Figure 2(a) to (e), suggesting that thermodynamic properties in the Mg-Zn melts can be predicted and calculated by AMCT at investigated temperature.

Al-Zn binary system
Balanović et al. [11] measured and calculated the activity of Al and Zn at 1000 K, and Wasiur et al. [23] summarized the activities of Al-Zn at 1073 K, showing that both Al and Zn have a positive deviation with Raoult's law. Zhang et al. [16] argued that this kind of melts can be defined as the heterogeneous system and introduced the calculation process. In addition, Zhang et al. [16] had investigated thermodynamic properties of Al-Zn at 1000 K, and argued that there are three structural characteristics, Al, Zn and AlZn, coexisting in Al-Zn melts. Similar as optimization in Section 3.1, through measured data of activities at 1000 K and 1073 K as well as mixing thermodynamics data at 1000 K, this paper has regressed and modified the and . Through the liner regression by the MATLAB software, , as well as can be optimized, with thermodynamic data substituted into Eqs.
Where Eq. (19) is the regression for primary , according to AMCT [16]; and a and b are the molar fraction of component elements Al and Zn, respectively.
On the basis of , the standard Gibbs free energy function of structural molecules in Al-Zn melts can be expressed as: (23) Therefore, according to obtained by Eq. (23), the thermodynamic properties of Al-Zn at 1000 K and 1073 K can be calculated. The results of calculating mass action concentration in the full composition range of Al-Zn binary melts at different temperature are presented in Figure 3. Figure 3 (a) and (c) show the comparisons between calculated mass action concentration of Al(N Al ), Zn(N Zn ) as well as N AlZn and measured activities of Al, Zn (a Mg , a Al ) relative to pure liquids as standard state in the full composition range of Al-Zn melts from 1000 to 1073 K, respectively. Figure 3(b) shows the comparison between calculated and reported of mixing thermodynamics. As can be seen in Figure 3, calculated values have an excellent agreement with the measured data, obviously, indicating the calculated mass action concentration of N Zn and N Al can be successfully applied to represent the measured a Mg and a Al in Al-Zn binary melts in the full composition range at a temperature from 1000 K to 1073 K.

Prediction in Mg-Al-Zn ternary system
Comparisons of calculated thermodynamic properties illustrated excellent consistency with the reported experimental data of Mg-Al, Mg-Zn, Al-Zn in Figures 1-3, respectively, suggesting the revised and Gibbs free energy function worked well. In addition, the standard Gibbs free energy functions are assumed to never change at above liquid temperature, so that at other investigated temperatures can be calculated with Eqs.  [16] that the homogeneous system together with the heterogeneous system could be subject to the heterogeneous system, suggesting that the Mg-Al-Zn ternary system would be subject to the Al-Zn calculated process. Hence, according to AMCT [16] and aforementioned results in Mg-Al, Mg-Zn, Al-Zn systems, the structural units in Mg-Al-   (23), mass action concentrations of components, i.e., N Al , N Mg , N Zn , can be calculated at the studied temperature in Mg-Al-Zn system from 880 to 1100 K. In order to describe the thermodynamic properties of Mg-Al-Zn system at details, thermodynamic calculation would be carried out in two sections: 1) the results of thermodynamic calculations in the Mg-Al-Zn melts determined for nine sections with the constant molar ratios of two components at different temperature are given in the following figures, i.e., as Figure 4, the Al corner sections, Mg :Zn=1:3, 1:1 ,3:1; as Figure 5, the Mg corner sections, Al : Zn: =1:3, 1:1 ,3:1 as well as Figure 6, the Zn corner sections, Al :Mg =1:3, 1:1 ,3:1; 2) the iso-activity diagrams at studied temperatures are introduced for Mg, Al and Zn component elements shown in Figure 7.
Thermodynamic properties of Mg-Al-Zn melts in the full composition range from 880 K to 1100 K are investigated in the paper by them with N i . The N i obtained in Section 3.1-3.4 has a good agreement with the reported activities data from literature. Therefore, the mass action concentrations of structural units based on the AMCT can well represent and describe the activities relative to pure liquids as standard state in Mg-Al, Mg-Zn and Al-Zn binary melts as well as the Mg-Al-Zn ternary melts.
The activities in Mg-Al-Zn has been shown above, therefore, according to AMCT, the mixing Gibbs free energy can be predicted and calculated by Eqs.    (30) The mixing Gibbs free energy was calculated same as for Figures 4~6, i.e. keeping the constant molar ratios of two components at different temperature. The results were presented as Figure 8, Figure 9, and Figure 10.
The standard molar mixing Gibbs free energy of Mg-Al-Zn melts change of composition from 880 K to 1100 K was further deduced. With the aid of AMCT, the thermodynamic properties of Mg-Al-Zn can be calculated and predicted well.

conclusion
According to the current research results on the alloy melt structure, it is generally believed that there     are short-range and medium-range ordered structures in the alloy melt [28], with various intermetallic compounds existing in the form of clusters or associates. Based on the above, Zhang et al. [17,18] proposed the atom and molecule coexistence theory (AMCT) and the hypothesis. Therefore, the prediction of thermodynamic properties of Mg-Al-Zn melts has been carried out based on the AMCT. Critical evaluations and optimizations of the constituent subsystems Mg-Al, Mg-Zn and Al-Zn have been present.
1) The chemical reaction equilibrium constant and standard molar Gibbs free energy function of structural molecules, i.e., and , are regressed and optimized in Mg-Al melts with the activities and mixing thermodynamics at 1000 K and 1073 K, due to the lack of enough activities data. Moreover, the standard molar Gibbs free energy of structural molecules in Mg-Al melts can be expressed as: ; ; ; 2) The in Mg-Zn binary system is regressed with sufficient activities data from 880 to 973 K. Therefore, coupling with chemical reaction isotherm and primary regressed , the standard molar Gibbs free energy of structural molecules in Mg-Zn melts can be deduced as: ; ; ; .
3) With measured data of activities at 1000 K and 1073 K as well as mixing thermodynamics data at 1000 K, and in Al-Zn have been regressed and optimized in this paper. What's more, the standard Gibbs free energy function of structural molecules in Al-Zn melts can be given as: . 4) Mass action concentrations of components, i.e., N Al , N Mg , N Zn , can be calculated at studied temperatures in Mg-Al-Zn system from 880 to 1100 K, through AMCT and the obtained standard Gibbs free energy functions in subsystems. Once and were obtained, thermodynamic properties at given temperature in the full composition range can be extrapolated and calculated with AMCT.