Evaluation of overpotentials on graphite and liquid Bi–Mg electrodes by current interruption

Abstract

Our group proposes a new titanium smelting process via Bi–Ti alloy. This process is comprised of reduction of TiCl₄ by Bi–Mg liquid alloy, separation of Ti from Bi by distillation, and Mg electrolysis. In this study, the Mg electrolysis with Bi–Mg liquid alloy cathode was investigated. We firstly measured the IR-corrected polarization curves on graphite and Bi–Mg liquid alloys by the current interruption method. The results indicated that the Bi–Mg alloy cathode can reduce the electricity consumption of the Mg electrolysis. In addition, from the relaxation curves on graphite and Bi–Mg alloys, the concentration overpotential on the Bi–Mg alloy is mainly due to mass transfer of Mg from the electrode/molten salt interface to the liquid alloy bulk. At current densities higher than 300 mA cm⁻², Mg-rich solid phases such as Bi₂Mg₃ and/or pure solid Mg are assumed to be deposited on the Bi–Mg liquid alloy cathode. Finally, we estimated the electricity consumption of the Mg electrolysis in the new smelting process based on the measured overpotentials, assuming that Bi–Mg liquid alloy cathode is stirred sufficiently and a low current density, 275 mA cm⁻², is applied. Under these conditions, the total electricity consumption of the Mg electrolysis in the new process will be lower than that in the Kroll process when the anode–cathode distance is smaller than 8 cm.

Graphic abstract

IR-corrected polarization curves of (a) Mg²⁺ reduction on graphite and Bi–Mg liquid alloys and (b) Cl₂ evolution on graphite in MgCl₂–NaCl–KCl at 550 °C were measured by the current interruption method, and electricity consumption of Mg electrolysis in the new Ti smelting process was estimated from these results.

1 Introduction

Titanium (Ti) is commercially produced by the Kroll process [1] in which titanium tetrachloride (TiCl₄) is reduced to porous Ti metal by magnesium (Mg). However, the productivity is low, and new smelting processes have been investigated, e.g., FFC-Cambridge process [2], OS process [3], Armstrong process [4], Subhalide reduction process [5], PRP [6], and HAMR process [7]. Our group proposes a new Ti smelting process via bismuth (Bi)–Ti liquid alloy [8,9,10,11,12]. Figure 1 is a schematic diagram of the new process for Ti. In the reduction cell, TiCl₄ is reduced by Bi–Mg alloy, and Bi–10 mol% Ti liquid alloy is produced at 900 °C. In the next segregation cell, the alloy is cooled to 500 °C, and a Ti-rich compound, Bi₉Ti₈, is precipitated in liquid Bi. Bi₉Ti₈ is then separated from liquid Bi and distilled to produce Ti powder in the distillation cell. Magnesium chloride (MgCl₂), a by-product in the reduction cell, is electrolyzed to produce Mg and chlorine gas (Cl₂) with a Bi–Mg liquid alloy cathode. The addition of Bi into the reduction cell makes it possible to transfer Ti in the liquid alloy state continuously. In addition, the feed rate of TiCl₄ can be improved because Bi also works as a coolant [9].

Fig. 1

Schematic of the proposed new smelting process

In the Kroll process, Mg and Cl₂ are produced by electrolysis of MgCl₂ with graphite cathode and anode. The reactions are described as follows:

$$ {\text{Cathode: Mg}}^{{ 2 { + }}} {\text{ + 2 e }} \to {\text{ Mg (}}l ) $$

(1)

$$ {\text{Anode: }}2{\text{ Cl}}^{ - } \to {\text{ Cl}}_{2} (g ) { } + \, 2{\text{ e}} $$

(2)

Because the Mg electrolysis accounts for 70–80% [13] of the total energy consumed in the Kroll process, it is quite important to improve the efficiency of Mg electrolysis. For this reason, many studies on Mg deposition in molten salts have been made [14,15,16,17,18,19]. In the new process, Bi–Mg liquid alloy is used as the cathode of Mg electrolysis, and Mg²⁺ ion is reduced at a liquid Bi cathode by the following reaction.

$$ {\text{Cathode: Mg}}^{{ 2 { + }}} {\text{ + 2 e }} \to {\text{ Mg (in Bi)}} $$

(3)

The liquid Bi cathode may reduce the electricity consumption of the Mg electrolysis due to the small activity coefficient of Mg in Bi. For example, the activity coefficient of Mg is 1.81 × 10⁻³ in Bi–25 mol% Mg alloy at 550 °C [20]. Figure 2 shows the activity of Mg in Bi–Mg alloy and corresponding oxidation–reduction potential of Mg in the alloy relative to pure liquid Mg. Considering the small activity of Mg in the alloy, the decomposition voltage of MgCl₂ is reduced by 12% in comparison with that of the Mg electrolysis. The calculation of the decomposition voltage is described in Appendix. The Bi–Mg alloy can be directly used for the reduction of TiCl₄ in the reduction cell. In addition, Bi-rich Bi–Mg alloys have lower melting points than pure Mg as seen in Fig. 3 [21]. Thus, the electrolysis can be operated at a lower temperature than that of the Mg electrolysis in the Kroll process, which prolongs the lifetime of the electrolytic cell. Furthermore, the liquid Bi cathode may improve the current efficiency of the electrolysis. In the electrolytic cell of the Kroll process, liquid Mg droplets move upward with Cl₂ gas bubbles in the flowing electrolyte. Then, it is reported that a back reaction, recombination of Mg and Cl₂, occurs and decreases the current efficiency [22]. Since the density of Bi is much higher than that of electrolyte, Bi–Mg alloy can be easily separated from Cl₂ bubbles.

Fig. 2

Reported activity of Mg in Bi–Mg alloy [20] and potential of Bi–Mg alloy versus Mg²⁺/Mg (s) at 550 °C. Fitted curve is determined assuming a subregular solution model (interaction parameter, $\varOmega_{\text{sub}}^{\text{liq}}$/kJ mol⁻¹ = − 256.6 $x_{\text{Mg}}^{2}$ − 42.16 $x_{\text{Mg}}$ − 48.65 where x_Mg is the mole fraction of Mg in Bi–Mg alloy). The standard state of Mg is converted from solid to liquid by the Gibbs energy of melting of Mg at 550 °C, 907.6 J mol⁻¹ [35]

Fig. 3

Bi–Mg phase diagram [21]

Liquid alloy electrodes, such as Mg–lead (Pb) alloy cathode for Mg electrolysis [23, 24] and Mg–antimony (Sb) alloy electrode for liquid metal batteries [25, 26], were investigated. However, the electrochemical behavior of Bi–Mg alloy electrode has not been researched sufficiently. In the present study, the overpotentials on graphite anode and Bi–Mg alloy cathode were measured by the current interruption method, and the electricity consumption of the Mg electrolysis in the new process was estimated.

2 Experimental

2.1 Preparation of electrolyte and electrodes

MgCl₂ (97%, Wako Pure Chemical Industries, Ltd.), NaCl (99.5%, Nakalai Tesque, Inc.), and KCl (99.5%, Nakalai Tesque, Inc.) were mixed in a glove box with argon (Ar) atmosphere at the eutectic composition of MgCl₂:NaCl:KCl = 50:30:20/mol% (m.p. 396 °C [27]). The mixed salt was dried under vacuum at higher temperatures than 226 °C for more than 54 h. The salt was filled in a nickel (Ni) crucible (OD 8.2 cm × ID 7.8 cm) and heated to 550 °C under Ar gas flow. After the salt was melted, pre-electrolysis with graphite anode and cathode was carried out at a cell voltage of 1.2–1.3 V. Before pre-electrolysis, small anodic and cathodic peaks were confirmed between 1.1 and 1.7 V versus Mg²⁺/Mg in cyclic voltammogram. These peaks could be attributed to reaction of impurities derived from water. After the pre-electrolysis for 10 h, the residue current became smaller than 1 mA cm⁻². In the present study, Mg was electrochemically deposited on a Mo wire in the molten salt and used as reference electrode (RE). It was confirmed that the potential difference between Mg deposited on the Mo wire and pure Mg plate was smaller than 1.5 mV for more than 100 min.

Figure 4 shows the experimental apparatus. Bi–Mg alloy electrodes were prepared from pure Bi shot (99.999%, Kamioka Mining & Smelting Co., Ltd.). A magnesia (MgO) crucible (OD 1.4 cm × ID 1.0 cm, 99.6%, Nikkato Co., Ltd.) was fixed to a stainless steel tube (S. S. tube) as shown in Fig. 4. About 6 g of Bi shot was put into the MgO crucible, and Mo wire covered by a MgO tube (OD 2 mm × ID 1 mm, 99.6%, Nikkato Co., Ltd.) was inserted into the Bi shot as an electrical lead. After the Bi electrode was immersed into the molten salt, Bi–x mol% Mg alloy electrodes (x = 5, 10, 15, 20, and 25) were prepared by cathodic Mg deposition and anodic Mg dissolution on the molten Bi. The concentrations of Mg in the alloys were controlled by the electrical charge of the electrolysis. After all the electrochemical measurements, the Bi–25 mol% Mg alloy electrode was analyzed by energy-dispersion X-ray spectroscopy (EDX, JXA-8530F, JEOL Ltd.) and inductively coupled plasma atomic emission spectroscopy (ICP-AES, SPS3520UV, SII Nano Technology Inc.). Other metals such as sodium and potassium were not detected in the alloy from the results of EDX, and the concentration of Mg was confirmed to be 25.1 mol% by ICP-AES.

Fig. 4

Apparatus for electrochemical measurements

2.2 Current interruption measurement

The current interruption method was used to measure the overpotential of Cl₂ evolution and Mg deposition. The counter electrode (CE) was a pure Mg plate or a graphite plate, and Mg deposited on Mo wire was used as RE. Galvanostatic electrolysis at various current densities was carried out for 2 or 250 s with a potentiogalvanostat (Solartron 1287, Solartron Analysis), and current interruption was carried out. After the constant current was interrupted, the relaxation of potential of working electrode (WE) versus RE was recorded with a high-speed data recorder (USB-6251 or USB-6341, National Instruments) at a sampling rate of 1.25 × 10⁶ or 2 × 10⁶ Hz until 2.0 ms just after the current interruption. From 0.05 to 600 s, the potential of WE versus RE was also recorded with a data recorder (Agilent 34972A, Agilent Technologies) at a sampling rate of 10 or 20 Hz. After 2 s electrolysis, the potential of the Bi–Mg alloy easily relaxed to the initial potential because of little change in the Mg concentration in the alloy. Thus, the next measurement was carried out immediately. In contrast, the change was not small at 250 s electrolysis. Thus, Mg in the Bi–Mg alloy was anodically dissolved before the next measurement. The same current interruption measurement was carried out for pure solid Mg deposition and Cl₂ evolution on a graphite rod electrode. Mg deposited on the graphite electrode was completely dissolved by applying an anodic potential before the next measurement.

3 Results

3.1 Cl₂ evolution on graphite

Figure 5 shows relaxation curves on the graphite electrode where the evolution of Cl₂ occurs by reaction (2). In Fig. 5, the time of the last data point before the current interruption is defined as 0 s. During the galvanostatic electrolysis (time, t ≤ 0), the potential of WE versus RE is represented as:

Fig. 5

Relaxation curves on graphite in MgCl₂–NaCl–KCl eutectic electrolyte in aµs time range, b ms time range

$$ \phi_{\text{WE}} (I \ne 0, \, t \le 0) \, - \phi_{\text{RE}} = \, \Delta E_{\text{eq}} \, + \, \eta_{\text{r}} \, + \, \eta_{\text{c}} \, + \, V_{\text{IR}} $$

(4)

where ϕ_WE (I ≠ 0, t ≤ 0) is the potential of WE during electrolysis, ϕ_RE is the potential of RE, η_r is the reaction overpotential, η_c is the concentration overpotential, and V_IR is the IR drop due to the ohmic resistance of the electrolyte, assuming the resistance of metallic parts is negligible. ΔE_eq is the difference between the electrode potentials of WE and RE where the following reactions (5) and (6) are assumed to be at equilibrium, respectively.

$$ 2 {\text{Cl}}^{-} \rightleftharpoons {\text{Cl}}_{ 2}\; \left( g, {\text{1 atm}} \right) {\text{ + 2 e}} $$

(5)

$$ {\text{Mg}}^{{ 2 { + }}} {\text{ + 2 e }} \rightleftharpoons {\text{Mg }}\left( {s} \right) $$

(6)

In this case, ΔE_eq is equal to the decomposition voltage of MgCl₂ in eutectic MgCl₂–NaCl–KCl. The activity of MgCl₂ in the present molten salt at 550 °C is calculated to be 0.182 (standard state: pure liquid Mg) with a subregular solution model and the reported thermodynamic data, and the corresponding decomposition voltage is 2.67 V (see Appendix). As seen in Fig. 5a, unexpected oscillation of the potential was observed for 5 µs after the current interruption. This unexpected oscillation will be attributed to the instruments or inductance of lead wires, and we did not use the initial 5 μs data for analysis. The IR drop, V_IR, immediately becomes zero at the current interruption measurements. Obvious relaxation is not observed from 5 to 20 μs, and the relaxation of the reaction and concentration overpotentials are assumed to be much slower than 20 μs. We can therefore take the average of the data points from 5 to 20 μs as ΔE_eq + η_r + η_c. Figure 6 shows the IR-corrected polarization curve of Cl₂ evolution on the graphite. As the data are already IR-corrected, the potential corresponds to the overpotential. The current density increases exponentially with overpotential. The behavior in this study is similar to the reported polarization curve at 500 °C [28]. Moreover, the overpotential in this study is larger than the reported values. If we assume that the small temperature difference and the concentration difference in the molten salts do not affect, the reasons are probably attributed to difference in roughness of graphite electrode surface, coverage ratio of the electrode by Cl₂ bubbles on electrode, and so on.

Fig. 6

IR-corrected polarization curves of Cl₂ evolution on graphite. Open circles denote data in the present study at 550 °C. The electrode potential of Cl₂/Cl⁻ was calculated to be 2.67 V versus Mg²⁺/Mg from the reported thermodynamic data (see Appendix). Open triangles denote the reported polarization behavior at steady-state in MgCl₂–KCl at 500 °C [28]. The concentration of MgCl₂ in the literature is unknown but is assumed to be between 28 and 58 mol% from the MgCl₂–KCl phase diagram [36]

3.2 Mg²⁺ reduction on Bi–Mg alloy

Figures 7 and 8 show the relaxation curves on the Bi–Mg alloy after 2 s and 250 s electrolysis, respectively. On the Bi–Mg alloy electrode, reaction (3) is expected to occur. The sum of the overpotentials was determined in the same way as for the graphite electrode. ΔE_eq in Eq. (4) was determined by the open circuit potential of the Bi–Mg alloy before the electrolysis. Figures 7c and 8c show that the potential relaxes slowly and there is a plateau region after electrolysis at current densities higher than 300 mA cm⁻².

Fig. 7

Relaxation curves on Bi–15 mol% Mg alloy after 2 s electrolysis in MgCl₂–NaCl–KCl eutectic electrolyte in aμs time range, b ms time range, c s time range. The last data point before the current interruption is defined as 0 s

Fig. 8

Relaxation curves on Bi–5 mol% Mg alloy after 250 s electrolysis in MgCl₂–NaCl–KCl eutectic electrolyte in aμs time range, b ms time range, c s time range. The last data point before the current interruption is defined as 0 s

4 Discussion

4.1 The concentration overpotentials on Bi–Mg alloy

We don’t have the data between 2.0 ms and 0.05 s after the current interruption. However, the plateau region suggests that the relaxation of the reaction overpotential completes in 0.1 s after the current interruption, and the slow relaxation after 0.1 s should be attributed to the concentration overpotential. On the Bi–Mg alloy, the concentration overpotential consists of two parts and is represented as:

$$ \eta_{\text{c}} = \eta_{\text{c}}^{\text{electrolyte}}+\eta_{\text{c}}^{\text{alloy}} $$

(7)

where $\eta_{\text{c}}^{\text{electrolyte}}$ is the concentration overpotential caused by mass transfer of Mg²⁺ in the electrolyte, and $\eta_{\text{c}}^{\text{alloy}}$ is caused by mass transfer of Mg from the electrode/molten salt interface to the liquid alloy bulk.

As a simple case, Mg deposition on a graphite rod electrode was investigated. The relaxation curves on graphite after 2 s electrolysis are shown in Fig. 9. In Fig. 9c, no plateau was observed on the relaxation curves on graphite. In addition, the concentration overpotential on graphite were much smaller than those on Bi–Mg alloy shown in Fig. 7c. For example, the values at 0.1 s on graphite and Bi–Mg alloy were about − 17 mV and − 164 mV at 1000 mA cm⁻², respectively. If the concentration overpotential in the electrolyte on Bi–Mg alloy, $\eta_{\text{c}}^{\text{electrolyte}}$, behaves in the same way as the concentration overpotential on graphite, the large overpotential on Bi–Mg alloy is not due to the concentration overpotential in electrolyte, but mainly due to the concentration overpotential in the alloy, $\eta_{\text{c}}^{\text{alloy}}$. The slow relaxation after 250 s electrolysis in Fig. 8c is also considered to be due to the concentration overpotential in the alloy. Considering the activity of Mg in Bi–Mg alloy shown in Fig. 2, the potential of liquid Bi–Mg alloy is more positive than 0.24 V versus Mg²⁺/Mg. At current densities higher than 300 mA cm⁻², each relaxation curve shows a plateau at a more negative potential than 0.24 V as shown in Figs. 7c and 8c. Therefore, Mg-rich solid phases such as Bi₂Mg₃ and pure Mg were probably deposited on the surface of Bi–Mg alloy at current densities higher than 300 mA cm⁻². This situation is schematically illustrated in Fig. 10. At current densities lower than 275 mA cm⁻², the Bi–Mg alloy should be in a single liquid phase as shown in Fig. 10a. After the electrolysis, the potential relaxes as the concentration gradient of Mg decreases by the convection and the diffusion in the liquid alloy. In contrast, at current densities higher than 300 mA cm⁻², solid phases are deposited on the surface as shown in Fig. 10b. The Mg concentration in the liquid phase at the surface does not change until the solid phases disappear. Thus, the solid phases make the relaxation of potential slower.

Fig. 9

Relaxation curves on graphite after 2 s electrolysis in MgCl₂–NaCl–KCl eutectic electrolyte in aμs time range, b ms time range, c s time range. The last data point before the current interruption is defined as 0 s

Fig. 10

Schematic of Mg concentration in Bi–Mg alloy after electrolysis

4.2 Polarization curves on Bi–Mg alloy and graphite

The IR-corrected polarization curves on Bi–Mg alloy and graphite are shown in Fig. 11. From Fig. 11, the reduction of Mg²⁺ on Bi–Mg alloy starts at more positive potentials than that on graphite. This result indicates that the electricity consumption of Mg electrolysis can be reduced by Bi–Mg alloy. However, in a prolonged electrolysis at higher current densities than 300 mA cm⁻², the overpotential on Bi–Mg alloy becomes so large that the potential becomes almost 0 V versus Mg²⁺/Mg. In this situation, solid phases may form at the interface. Thus, to reduce electricity consumption, it is important to keep the concentration overpotential small by stirring the alloy or applying a low current density.

Fig. 11

IR-corrected polarization curves on Bi–Mg alloy and graphite in MgCl₂–NaCl–KCl eutectic electrolyte at 550 °C. Dashed line is the IR-free polarization curve of Mg (l) deposition on Mg (l) electrode in MgCl₂–NaCl equimolar electrolyte at 665 °C calculated using the Butler-Volmer equation of the symmetry coefficient is 0.5 and reported exchange current density of 3.65 A cm⁻² [17]

In Fig. 11, the dashed line is a polarization curve of molten Mg deposition on molten Mg electrode in MgCl₂–NaCl equimolar electrolyte at 665 °C in literature [17]. There is a large difference between the values in the literature and the present study. We do not consider that the small temperature difference and the different state of deposited Mg between liquid and solid can account for this difference. The difference of electrolysis time, about 0.1 ms in the literature [16] and 2 s in this study, might have affected the results.

4.3 Estimation of the industrial electricity consumption

We can estimate the electricity consumption of Mg electrolysis using the data obtained in this study. Figure 12 shows the results of estimation. The operating temperature in the new process is tentatively fixed at 550 °C. Inlet and outlet Mg concentrations in alloy are assumed to be 5 and 25 mol%, respectively. Theoretical electricity consumption, W_ΔG, and the enthalpy change of the cell reaction, W_ΔH, are calculated from the reported thermodynamic data as shown in Appendix. The current density is set at 275 mA cm⁻², above which solid phases would form on the alloy cathode. The values of the overpotentials obtained in the 2 s electrolysis experiment are used in this estimation, assuming that the same concentration overpotential would be achieved in industrial operation by stirring the alloy. The electricity consumed by the IR drop due to the electrolyte, W_IR, is estimated from the reported conductivity of MgCl₂–NaCl–KCl eutectic electrolyte, 1.18 S cm⁻¹, at 475 °C [29]. The current efficiency is assumed to be 100%. As for the Mg electrolysis in the Kroll process, the reported operation temperature and current density are 660 °C [22] and 600 mA cm⁻² [13], respectively. The total electricity consumption is reported to be 10155 kWh/ton-Ti [13], and the theoretical electricity consumption, W_ΔG, is calculated from thermodynamic data. From Fig. 12, when the anode–cathode distance is smaller than 8 cm in the new process, the total electricity consumption will be lower than that in the Kroll process. For isothermal operation at 550 °C without heat supply to the electrolytic cell, the total energy cannot be lower than W_ΔH. Therefore, the anode–cathode distance in the new process should be larger than 1 cm. We also have to consider heat dissipation from the bath in industrial electrolysis, so a longer distance is appropriate.

Fig. 12

Estimated electricity consumption of Mg electrolysis. W_ΔG is the theoretical electricity consumption. W_η-anode and W_η-cathode are the electricity consumed by overpotentials on the anode and cathode, respectively. W_IR is the electricity consumed by the IR drop attributed to the electrolyte. W_ΔH is the enthalpy change of the cell reaction of the Mg electrolysis

5 Conclusions

In this study, the IR-corrected polarization curves on graphite and Bi–Mg liquid alloy are obtained by the current interruption method. At current densities higher than 300 mA cm⁻², solid phases such as Bi₂Mg₃ and/or pure solid Mg are expected to be deposited on the Bi–Mg alloy cathode. It is found that the concentration overpotential on the Bi–Mg alloy is mainly due to mass transfer of Mg from the electrode/molten salt interface to the liquid bulk. Consequently, stirring the Bi–Mg alloy is needed for electrolysis with smaller overpotentials. We then estimated the electricity consumption of the Mg electrolysis in the new smelting process based on the measured overpotentials, assuming sufficient stirring of alloy cathode and a low current density. The total electricity consumption of the Mg electrolysis in the new process will be lower than that in the Kroll process when the anode–cathode distance is smaller than 8 cm.

Appendix 1

1.1 Calculation of the theoretical electricity consumption of Mg electrolysis

Data used for the thermodynamic calculation are given in Table 1. The activities of MgCl₂ in Table 1 are calculated by a subregular solution model, in which the excess molar Gibbs energy of ternary molten salt, G^ex, is represented as Eq. (8).

Table 1 Data used for the calculation of theoretical electricity consumption of the Mg electrolysis in the Kroll process and the new process

$$ G^{\text{ex}} \, = \, \varOmega_{12} x_{1} x_{2} \, + \, \varOmega_{23} x_{2} x_{3} \, + \varOmega_{31} x_{3} x_{1} $$

(8)

In Eq. (8), x₁, x₂, and x₃ are the mole fractions of components 1, 2, and 3 in the ternary molten salts, respectively, and Ω_ij are interaction parameters of binary molten salts composed of i and j. The compositions of the molten salts are shown in Table 1. We defined the components 1 and 2 as MgCl₂ and NaCl, respectively. The component 3 is defined as CaCl₂ or KCl in the Kroll process or new process, respectively. The activity coefficient of MgCl₂, γ₁, can be calculated from the following equation.

$$ RT{ \ln }\gamma_{1} { = }G^{\text{ex}} \, - x_{ 2} \left( {\frac{{\partial G^{\text{ex}} }}{{\partial x_{2} }}} \right)_{{P, T, x_{ 3} }} - x_{ 3} \left( {\frac{{\partial G^{\text{ex}} }}{{\partial x_{3} }}} \right)_{{P, T, x_{ 2} }} $$

(9)

In Eq. (9), R is the gas constant, T is the absolute temperature, P is the pressure. Then, interaction parameters of the binary salts, Ω_ij, are collected as below.

For MgCl₂–NaCl, NaCl–CaCl₂, and KCl–MgCl₂ binary salts, data points of excess molar Gibbs energies for binary salts, $G_{ij}^{\text{ex}}$, are reported in literature [30]. Then, Ω_ij are obtained by Eq. (10) and fitted by Eq. (11) which is a Redlich–Kister polynomial [31].

$$ \varOmega_{ij} { = }\frac{{G_{ij}^{\text{ex}} }}{{y_{i} \, y_{j} }} $$

(10)

$$ \varOmega_{ij} { = }\sum\limits_{{n{ = 0}}}^{m} {L_{n}^{ij} } \left( {y_{i} \, - \, y_{j} } \right)^{n}\ (m \, \le { 3)} $$

(11)

In Eqs. (10) and (11), y_i and y_j are mole fractions of i and j in the binary salts, and $L_{n}^{ij}$ should be function of temperature. However, we assumed that $L_{n}^{ij}$ are independent of temperature and adopted the data above 700 °C. Then, Ω_ij are function of the mole fractions. Figure 13 shows Ω_ij versus y_i − y_j and fitted curves by Eq. (11). The fitting parameters, ${{L}}_{{n}}^{{ij}}$, are shown in Table 2. As for CaCl₂–MgCl₂ [32] and NaCl–KCl [33] molten salts, interaction parameters are reported as Eqs. (12) and (13), respectively.

Fig. 13

Interaction parameters of MgCl₂–NaCl, NaCl–CaCl₂, and KCl–MgCl₂ molten salts calculated from the reported data. Solid lines are fitted curves by Eq. (11)

Table 2 Parameters used for the calculation

$$ {\text{CaCl}}_{2}{-}{\text{MgCl}}_{2}{:}\ \varOmega /{\text{kJ}}\,{\text{mol}}^{{ - 1}} = 9.87 - 15.1{\text{y}}_{{{\text{CaCl}}_{2} }} + 5.44{\text{y}}_{{{\text{CaCl}}_{2} }}^{2} $$

(12)

$$ {\text{NaCl}}{-}{\text{KCl}}\!\!\!:\varOmega /{\text{kJ mol}}^{{{-1}}} = -2.05 - 0.272y_{\text{NaCl}} $$

(13)

Using these data, parameters $L_{n}^{ij}$ are obtained as shown in Table 2.

Finally, each Ω_ij expressed as Eq. (11) is substituted into Eqs. (8) and (9) under the condition that y_i − y_j is equal to x_i − x_j, which is called a Muggianu-type approximation [34].