Thermodynamic Reassessment of the Na-Cu and NaK Binary Systems

The Na-Cu and Na-K systems are of significant interest due to the use of liquid sodium and melt of sodium and potassium in the nuclear industry as a cooling agent in nuclear reactors. In the present work, thermodynamic modeling of phase equilibria in the Na-Cu and Na-K systems is carried out, based on the available published experimental data. This modeling was done using the “FactSage” software package (version 7.0). The set of Redlich-Kister equation parameters was obtained, which allows one to describe the dependence of Gibbs energy from composition and temperature for solutions that can be formed in the studied systems. Phase diagrams (T-x diagrams) of the investigated systems were calculated.


Introduction
Liquid sodium is widely used as a cooling agent in nuclear reactors [1].In some cases, it is considered as appropriate to use an alloy of Na-K.Because of the properties of such alloys, they can be used as cooling agents for other technical devices.
Active alkaline metals, which are used as cooling agents, should circulate in a closed system.Some important parts of this closed system are made of copper (a cheap material with a high thermal conductivity).This is why the study of the Cu-Na-K system is of interest.
Presently, phase equilibrium in the Cu-Na-K system has never been the subject of full thermodynamic modeling.This kind of simulation requires a preliminary thermodynamic description of phase equilibria in binary systems containing Cu, Na, and K. Experimental data about the Cu-K system are uncommon in the literature.It is just noted that this system does not form any intermetallic compounds [2].At the same time, there is a large amount of experimental data about the Na-Cu and Na-K systems.The aim of this work is to model the phase equilibria in the Na-Cu and Na-K systems based on the available experimental data by means of a widely used thermodynamic model common to both systems.

Thermodynamic Modeling
Software package "FactSage" (version 7.0, "Thermfact/CRCT", Montreal, QC, Canada and "GTT-Technologies", Aachen, Germany) [3][4][5] is used in the simulations.Calculation of phase diagrams Metals 2018, 8, 629; doi:10.3390/met8080629www.mdpi.com/journal/metals is carried out using the "Phase Diagram" module.The "Compound" module is used to creating the individual substances base.The "Solution" module (model #1) is used to calculate the thermodynamic parameters of the solutions.The possibility of miscibility gaps formation is taken into account during the phase diagrams calculation.All calculations are performed for a fixed total system pressure of 1 bar.The temperature dependences of Gibbs energy for solid (BCC and FCC phases separately) and liquid simple substances (sodium, copper, and potassium) are taken (with minor necessary corrections) from work [6].For calculation of Na 2 K Gibbs energy, the Equation ( 1) is used: This Equation ( 1) is obtained according to FTlite (FACT Light Metal Alloy) Database for FactSage 7.0.
For the simulation of the solutions (Na-Cu melt and Na-K melt, as well as the solid solutions with crystal lattices BCC and FCC), the Redlich-Kister polynomials [7] are used (Equations ( 2) and ( 3)): in which L is the interaction parameter, which is expressed as the following form: Parameters, describing the solutions in the studied systems, are given in the following Table 1.Values of the most part of parameters (excluding the parameters for solid solutions in the Na-Cu system, taken from [8]) were determined in this study.
Table 1.The model parameters used to describe the solutions.

Results and Discussion
In Figure 1, some results of experimental study on the solubility of copper in liquid sodium are presented, according to the data given in [9].In Figures 2-4, a general view of the calculated phase diagram of the Na-Cu system and a zoom of its fragments are shown.
In Figure 5, results of the phase diagram calculation for the Na-K system are shown.The experimental data, used for the selection of model parameters, are also presented here.It allows one to confirm the accuracy of the present modeling.
The Na-Cu system has been studied experimentally in several works [10][11][12][13][14].However, the attention of researchers has been focused mainly on the study of copper solubility in liquid sodium.A critical review of all works performed by 1986 was presented by A. Pelton [9].In this review, the author plots the phase diagram for the Na-Cu system, in which most of it is made schematically, without relying on experimental data.
In [8], thermodynamic modeling and calculation of phase diagrams for the Na-Cu system are performed, and they were included in the review of H. Okamoto [15].The authors of [8] used Redlich-Kister polynomials to build the model.Unfortunately, the authors of [8] did not justify their choice of model parameters for solid solutions (experimental data [10][11][12][13][14], to which they refer, do not contain the necessary information).However, any additional information of this kind of data is absent in the literature.Hence, for our calculation we used the values presented in [8].
Metals 2018, 8, x FOR PEER REVIEW 3 of 7 contain the necessary information).However, any additional information of this kind of data is absent in the literature.Hence, for our calculation we used the values presented in [8].contain the necessary information).However, any additional information of this kind of data is absent in the literature.Hence, for our calculation we used the values presented in [8].         A. Pelton [9] correctly pointed out that at low temperatures and at low concentrations of copper in sodium, the values are too scattered.For this reason, it is incorrect to take these data for the equation development, which one can be used for extrapolation at much higher temperatures.Therefore, in the present work only data obtained for relatively high temperatures (above 723 K) were used for the solubility approximation.In Figure 1, used experimental points are shown as black and unused as white.The result of approximation (with R 2 = 0.8995) is presented by the Equation (4): in which x Cu is the mole fraction of copper in the sodium-based melt.1-data from [16], 2-data from [17] according to [16], 3-data from [18], 4-data from [19], and 5data from [20].
A. Pelton [9] correctly pointed out that at low temperatures and at low concentrations of copper in sodium, the values are too scattered.For this reason, it is incorrect to take these data for the equation development, which one can be used for extrapolation at much higher temperatures.Therefore, in the present work only data obtained for relatively high temperatures (above 723 K) were used for the solubility approximation.In Figure 1, used experimental points are shown as black and unused as white.The result of approximation (with R 2 = 0.8995) is presented by the Equation (4): in which xCu is the mole fraction of copper in the sodium-based melt.The continuous line, shown in Figure 1, is the result of our calculation for the liquidus line in the Na-Cu system at the temperatures 500-1111 K.It includes the compositions, in which sodium-based melt is in equilibrium with a copper-based solid solution.The parameters used for this calculation (presented in the Table 1) are selected in such a way that the liquidus line overlaps the line, calculated by the equation, obtained in approximation.The liquidus lines calculated by using a set of parameters from [8] and from FTlite Database are shown on the same figure.Comparison of the calculation results with the experimental data justified the proposed set of parameters.
During the modeling of the Na-K system, experimental data of works [16][17][18][19][20] are used.The results of these works are in a good agreement.In particular, all researchers confirm that the system produces compound Na2K, which is incongruent decomposed at a temperature of about 280 K. Earlier, the thermodynamic description of the Na-K system was performed in [21] within the modeling of the Na-K-S system, and the phase diagram of this system was calculated.A modified 1-data from [16], 2-data from [17] according to [16], 3-data from [18], 4-data from [19], and 5-data from [20].
The continuous line, shown in Figure 1, is the result of our calculation for the liquidus line in the Na-Cu system at the temperatures 500-1111 K.It includes the compositions, in which sodium-based melt is in equilibrium with a copper-based solid solution.The parameters used for this calculation (presented in the Table 1) are selected in such a way that the liquidus line overlaps the line, calculated by the equation, obtained in approximation.The liquidus lines calculated by using a set of parameters from [8] and from FTlite Database are shown on the same figure.Comparison of the calculation results with the experimental data justified the proposed set of parameters.
During the modeling of the Na-K system, experimental data of works [16][17][18][19][20] are used.The results of these works are in a good agreement.In particular, all researchers confirm that the system produces compound Na 2 K, which is incongruent decomposed at a temperature of about 280 K. Earlier, the thermodynamic description of the Na-K system was performed in [21] within the modeling of the Na-K-S system, and the phase diagram of this system was calculated.A modified quasi-chemical model [22] was used to describe the melt.It prevents us from using the obtained parameters of the Na-K description for the study of more complex systems by more common approaches.In this work, the model parameters allowing one to describe phase equilibria in this system by means of Redlich-Kister polynomials were obtained.

Figure 2 .
Figure 2. General view of the calculated phase diagram of the Na-Cu system.

Figure 2 .
Figure 2. General view of the calculated phase diagram of the Na-Cu system.Figure 2. General view of the calculated phase diagram of the Na-Cu system.

Figure 2 .
Figure 2. General view of the calculated phase diagram of the Na-Cu system.Figure 2. General view of the calculated phase diagram of the Na-Cu system.

Figure 3 .
Figure 3. Calculated phase diagram of the Na-Cu system.Zoom for the area with low copper concentration.

Figure 4 .
Figure 4. Calculated phase diagram of the Na-Cu system.Zoom for the area with low sodium concentration.

Figure 3 .
Figure 3. Calculated phase diagram of the Na-Cu system.Zoom for the area with low copper concentration.

Figure 3 .
Figure 3. Calculated phase diagram of the Na-Cu system.Zoom for the area with low copper concentration.

Figure 4 .
Figure 4. Calculated phase diagram of the Na-Cu system.Zoom for the area with low sodium concentration.

Figure 4 .
Figure 4. Calculated phase diagram of the Na-Cu system.Zoom for the area with low sodium concentration.