Phase Equilibria and Thermodynamic Properties of Selected Compounds in the Ag-Ga-Te-AgBr System

Abstract

The equilibrium T − x space of the Ag-Ga-Te-AgBr system in the part Ag₂Te-GaTe-Te-AgBr-Ag₂Te below 600 K has been divided into separate phase regions using the electromotive force (EMF) method. Accurate experimental data were obtained using the following electrochemical cells (ECs): (−) IE | NE | SSE | R{Ag⁺} | PE | IE (+), where IE is the inert electrode (graphite powder), NE is the negative electrode (silver powder), SSE is the solid-state electrolyte (glassy Ag₃GeS₃Br), PE is the positive electrode, R{Ag⁺} is the region of PE that is contact in with SSE. At the stage of cell preparation, PE is a non-equilibrium phase mixture of the well-mixed powdered compounds Ag₂Te, GaTe, Ga₂Te₃, AgBr, and tellurium, taken in ratios corresponding to two or three different points of interest for each of the phase regions. The equilibrium set of phases was formed in the R{Ag⁺} region at 600 K for 48 h with the participation of the Ag⁺ ions. Silver cations, displaced for thermodynamic reasons from the NE to the PE of ECs, acted as catalysts, i.e., small nucleation centers of equilibrium phases. The spatial position of the established phase regions relative to the position of silver was used to express the overall reactions of synthesis of the binary Ga₂Te₅, Ga₇Te₁₀, Ga₃Te₄, ternary AgGa₅Te₈, and quaternary Ag₃Ga₁₀Te₁₆Br, Ag₃Ga₂Te₄Br, Ag₂₇Ga₂Te₁₂Br₉ compounds in the PE of ECs. The values of the standard thermodynamic functions (Gibbs energies, enthalpies, and entropies) of these compounds were determined based on the temperature dependencies of the EMF of the ECs.

1 Introduction

Over the past few decades, silver-containing chalcogenides and chalcohalides have attracted significant attention among scientists and engineers due to their potentially environmentally sound application in solid-state ionics, and non-linear optical devices, photovoltaic absorbers, as well as thermoelectric materials (TMs).^{[1,2,3,4,5,6]} Moreover, multinary chalcogenides and chalcohalides have been considered interesting scientific objects due to the diversity of their crystal structures and physicochemical properties.^{[7,8,9,10,11,12]}

The TMs offer various opportunities in power generation by directly converting waste heat to electricity. Moreover, TMs devices are quiet in operation, do not release emissions, and are components of environmentally friendly technologies.^[13,14,15] The typical binary TMs based on bismuth and lead tellurides have a narrow application due to their low efficiency and environmental reasons.^[16,17] The criterion for choosing compounds for application in the thermoelectric devices is a dimensionless thermoelectric figure of merit parameter ZT = (S²σ)T/k (where S is the Seebeck coefficient, σ is the electrical conductivity, k is the thermal conductivity, and T is the absolute temperature).^[18] In crystals, which simultaneously have a strong covalent bond that provides high electronic conductivity, and disordered mobile ions (like a "quasi-liquid"), there is a possibility of independent optimization of both factors.^[19,20,21] The focus on tellurium as the heaviest non-radioactive chalcogen is due to the possibility of decreasing the thermal conductivity coefficient.^[22] Optimization of synthesis of new TMs is impossible without a comprehensive analysis of the thermodynamic properties of intermediate phases and construction of equilibrium phase diagrams.

According to different literature reports,^{[23,24,25,26,27,28,29,30,31]} most of the binary, ternary, and quaternary compounds of the Ag-Ga-Te-AgBr system are located in the Ag₂Te-GaTe-Te-AgBr-Ag₂Te part. In particular, the GaTe-Te part consists of the following compounds Ga₃Te₄, Ga₇Te₁₀, Ga₂Te₃, and Ga₂Te₅; section Ag₂Te-Ga₂Te₃ contains Ag₉GaTe₆, AgGaTe₂, and AgGa₅Te₈; section Ag₂Te-AgBr contains Ag₃TeBr; and section AgBr-Ga₂Te₃ contains AgGa₂Te₃Br. According to Blachnik and Klose,^[29] compounds GaTe and Ga₂Te₃ melt congruently at 1108 and 1071 K, respectively; Ga₃Te₄ is formed at 1057 K by the peritectic reaction of the melt and Ga₂Te₃; Ga₂Te₅ exists in a limited temperature range of 681-757 K. A trigonal Ga₇Te₁₀ was synthesized from pure components at 1020 K.^[27] To our knowledge, there is no information on the thermal and thermodynamic stability of the Ga₇Te₁₀ at 300 K. Kramer et al.^[24] found that Ag₉GaTe₆ melts incongruently at 978 K and underdoes a phase transition in the range of 275-303 K. The Ag₃TeBr compound is formed at 710 K by the peritectic reaction of Ag₂Te with the melt.^[23] The crystal structure of the AgGa₂Te₃Br compound has been indexed to the tetragonal system, space group I-4, a = 0.62977(3) nm and c = 1.19473(7) nm.^[25] The ternary compounds Ag₉GaTe₆, AgGaTe₂, and AgGa₅Te₈ of the quasi-binary system Ag₂Te-Ga₂Te₃ belong to the class of thermoelectric materials.^[28,30,31]

The effect of replacing part of the gallium cations of the compound p-Ag₉GaTe₆ according to the scheme Ag₉Ga_1-δM_δTe₆ (M = Cd, Zn, Mg, Nb; δ = 0.05) on the ZT values is given in (Ref 31). In the case of Cd doping, achieving the thermoelectric figure of merit parameter of ZT≈0.6 at 600 K. Such method of increasing the ZT value is ineffective in the case of a thermodynamically unbalanced state of the doping component in the crystal lattice of the compound. The action of external factors, such as changes in temperature, pressure, radiation, etc. will contribute to the migration of impurities at the grain boundary, which will lead to a decrease in the ZT value of the sample over time. It is possible to avoid a decrease in the ZT value during the operation of a doped thermocouple by producing it in the form of an equilibrium solid solution based on a quaternary compound. As an example, for doping of the Ag₉GaTe₆, AgGaTe₂, and AgGa₅Te₈, it is possible to use quaternary compounds of the Ag₂Te-Ga₂Te₃-AgBr region. This region is part of the Ag-Ga-Te-AgBr system, where the formation of quaternary compounds Ag₃Ga₁₀Te₁₆Br, Ag₃Ga₂Te₄Br, and Ag₂₇Ga₂Te₁₂Br₉ is possible at the intersection points of the AgGa₅Te₈-AgBr, AgGaTe₂-AgBr, Ag₉GaTe₆-AgBr composition lines with the Ga₂Te₃-Ag₃TeBr composition line. There are no previous reports on quaternary compounds of mentioned compositions. The thermodynamic conditions for the formation of quaternary phases likely correspond to the temperature values T < 600 K, although there are kinetic obstacles to such a process. Kinetic obstacles to the synthesis of phases from pure substances and binary compounds can be overcome with the participation of a catalyst, for example Ag⁺ ions, as small nucleation centers of equilibrium phases.^[32,33]

The purpose of this work was to establish the phase equilibria of the Ag-Ga-Te-AgBr system in the part Ag₂Te-GaTe-Te-AgBr-Ag₂Te below 600 K and to determine the standard thermodynamic properties of compounds by the electromotive force (EMF) method.

2 Experimental Procedure

The high-purity substances Ag (>99.9 wt.%, Alfa Aesar, Germany), Ga, Te (>99.99 wt.%, Alfa Aesar, Germany), and binary compound AgBr (>99.99 wt.%, Alfa Aesar, Germany) were used to synthesize the binary and ternary compounds. Evacuated melts of the Ag₂Te, GaTe, and Ga₂Te₃ compounds, cooled to room temperature, were crushed to a particle size of ~ 1⋅10⁻⁶ m for preparation of the positive electrodes (PE) of electrochemical cells (ECs). Melts of the Ga₇Te₁₀, AgGaTe₂, and AgGa₅Te₈ compounds cooled to a temperature of 630 K were annealed for two weeks, followed by cooling to room temperature with the furnace turned off. The phase composition of the synthesized compounds was analyzed by an x-ray diffraction (XRD) technique. The STOE STADI P diffractometer equipped with a linear position-sensitive detector PSD, in a Guinier geometry (transmission mode, CuKα₁ radiation, a bent Ge(111) monochromator, and 2 θ/ω scan mode) was used for these investigations. The following software programs STOE WinXPOW,^[34] PowderCell,^[35] FullProf,^[36] and crystal structure databases^[37,38] were applied to analyze the obtained results.

The modified EMF method^[39,40] was used both to establish the phase equilibria in the GaTe-AgGa₅Te₈-Te region below 600 K and to determine the thermodynamic parameters of compounds. For these investigations, a certain number of ECs were assembled:

$$ \left( - \right)\;{\text{IE}}\;\left| {\;{\text{NE}}\;} \right|\,{\text{SSE}}\;\left| {\;\text{R}\left\{ {{\text{Ag}}^{ + } } \right\}\;} \right|\;{\text{PE}}\;|\;{\text{IE}}\;\left( + \right) $$

where IE is the inert electrode (graphite powder), NE is the negative electrode (silver powder), SSE is the solid-state electrolyte (glassy Ag₃GeS₃Br^[41]), and R{Ag⁺} is the region of PE that is in contact with SSE. At the stage of cell preparation, PE is the non-equilibrium phase mixture of the well-mixed powdered binary compounds Ag₂Te, GaTe, Ga₂Te₃, AgBr, and pure substance tellurium. Compositions of these mixtures covered the entire composition space of the Ag₂Te-GaTe-Te-AgBr-Ag₂Te system. An equilibrium set of phases was formed in the R{Ag⁺} region at 600 K after 48 h. The Ag⁺ ions, displaced for thermodynamic reasons from the NE to the PE electrodes of the ECs, acted as catalysts, i.e., small nucleation centers of equilibrium phases.^[32,33]

The experiments were performed in a resistance furnace as previously described in literature.^[42] To assemble the ECs, a fluoroplastic base with a hole with a diameter of 2 mm was used. The powder components of ECs were pressed at pressure 10⁸ Pa into the hole under a load of (2.0 ± 0.1) tons to a density of ρ = (0.93 ± 0.02)⋅ρ₀, where ρ₀ is the experimentally determined density of cast samples. The assembled cells were placed in a quartz tube with nozzles for the purging of argon gas.^[43,44] The argon gas flow had a direction from the NE to PE of ECs at the rate of (10.0 ± 0.2) cm³⋅min⁻¹. The temperature of ECs was maintained by an electronic thermostat with ± 0.5 K accuracy. A Picotest M3500A digital voltmeter with an input impedance of > 10¹² Ohms was used to measure the EMF (E) values of the cells (accuracy ± 0.3 mV) at different temperatures. The reproducibility of the E versus T dependences of ECs in heating-cooling cycles was a criterion for completing the formation of the equilibrium set of phases in the R{Ag⁺} region.

3 Results and Discussion

3.1 Phase Equilibria and Thermodynamic Properties of Selected Compounds of the Ag-Ga-Te System in the GaTe-AgGa₅Te₈-Te Part Below 600 K

Charoenphakdee et al.^[30] reported that the AgGa₅Te₈ compound was obtained by cooling the melt to a temperature of 873 K and subsequent homogenization for 1 week. In our research, the AgGa₅Te₈ compound was not obtained by cooling the melt to 630 K followed by annealing for two weeks. The diffraction pattern of the synthesized powder sample is shown in Fig. 1.

Fig. 1.

X-ray powder diffraction pattern of the sample with nominal composition AgGa₅Te₈, obtained by cooling the melt to 630 K. Detected composition of the sample and identified phase (with structure type and space group indicated) are shown in the upper right corner

The presence of a solid solution of Ag in Ga₂Te₃, i.e. Ag_xGa₂Te₃ (structure type (ST) ZnS, space group (SG) F–43 m) and a minor admixture of an unidentified phase have been established by the XRD method. The refined unit-cell parameter a = 0.59786(5) nm is greater than a ~ (0.587-0.590) nm for Ga₂Te₃^[37,38] (JCPDS cards No. 35-1490, 79-2443, 89-7202), thus indicating the formation of a solid solution.

A metastable, for kinetic reasons, combination of the mentioned phases was confirmed by an attempt to implement the reaction AgGaTe₂ + 2Ga₂Te₃ = AgGa₅Te₈. For this reason, a well-mixed blend of the AgGaTe₂ and 2Ga₂Te₃ compounds was evacuated, kept for two weeks at 630 K, and cooled to room temperature. According to the XRD results presented in Fig. 2, the orthorhombic AgGa₅Te₈ compound was also not obtained. The sample contained characteristic peaks of the phases Ag_xGa₂Te₃ (ST ZnS, SG F–43m), AgGaTe₂ (ST CuFeS₂, SG I–42d), and pure Te (ST Se, SG P3₁21)^[37] (JCPDS Cards No. 45-1278, 75-0116, 80-2200, 65-2748, 36-1452). Thus, results of the XRD have shown that the orthorhombic AgGa₅Te₈ compound decomposes at a certain value of the temperature in the range of 630-873 K.

Fig. 2.

X-ray powder diffraction pattern of the sample with nominal composition AgGa₅Te₈, obtained by solid-state synthesis mixture of the AgGaTe₂ and 2Ga₂Te₃ compounds. Detected compositions of the sample and identified phases (with structure type and space group indicated) are shown in the upper right corner

The XRD pattern of a sample with a nominal composition Ga₇Te₁₀, synthesized at 1020 K followed by annealing at 630 K for 2 weeks is shown in Fig. 3. The main phase Ga₂Te₃ (ST ZnS, SG F-43m, refined unit-cell parameter a = 0.5897(7) nm) and the additional phase Ga₇Te₁₀ with rhombohedral structure were identified^[37] (JCPDS Cards No. 35-1490, 79-2443, 89-7202, 85-0007). Hence, the Ga₂Te₃ phase is a decomposition product of the high-temperature modification of the Ga₇Te₁₀ phase.

Fig. 3.

X-ray powder diffraction pattern of the sample with nominal composition Ga₇Te₁₀. Compositions of the sample and identified phases (with structure type and space group indicated) are shown in the upper right corner

Thus, from the analysis of the XRD patterns Fig. 1, 2 it follows that the stable at 873 K compound AgGa₅Te₈, decomposes at a certain value of the temperature in the range of 630-873 K. However, in this work, the existence of the AgGa₅Te₈ compound below 600 K was established. The existence of a low-temperature modification of AgGa₅Te₈ is based on the results of division of the composition space of the GaTe-AgGa₅Te₈-Te system by the EMF method (Fig. 4), calculations the values of thermodynamic functions of binary and ternary compounds with their further use in studies of the properties of quaternary phases. Other examples of the existence of silver-based ternary and quaternary compounds in two temperature ranges are described in elsewhere literature.^[45,46,47]

Fig. 4.

Division of the composition space of the GaTe-AgGa₅Te₈-Te system at T < 600 K. Red dots indicate compositions of the PE of ECs and EMF (mV) values of the ECs at 400.4 K

The division of the composition space of the Ag-Ga-Te system in the GaTe-AgGa₅Te₈-Te part below 600 K into separate three-phase regions was carried out on the basic rules of the EMF method:^[48,49,50]

• 1) within a specific phase region, the EMF value of the cell does not depend on the composition of the PE;

• 2) ECs with PE of different phase regions are characterized by different EMF values at T = const (see Table 1 and Fig. 4);

• 3) the three-phase region further away from the figurative point of Ag is characterized by a higher EMF value at a specific temperature.

Table 1 Measured values of temperature (T) and EMF (E) of the ECs with PE of different phase regions at pressure P = 10⁵ Pa

The spatial position of three-phase regions Ga₂Te₅-AgGa₅Te₈-Te (I), Ga₂Te₃-AgGa₅Te₈-Ga₂Te₅ (II), Ga₇Te₁₀-AgGa₅Te₈-Ga₂Te₃ (III), and Ga₃Te₄-AgGa₅Te₈-Ga₇Te₁₀ (IV) relative to the silver point was used to establish the overall potential-determining reactions:

$$ {\text{2Ag}} + {\text{5Ga}}_{{2}} {\text{Te}}_{{5}} = {\text{2AgGa}}_{{5}} {\text{Te}}_{{8}} + {\text{9Te}}, $$

(R1)

$$ {\text{4Ag}} + {\text{9Ga}}_{{2}} {\text{Te}}_{{3}} + {\text{Ga}}_{{2}} {\text{Te}}_{{5}} = {\text{4AgGa}}_{{5}} {\text{Te}}_{{8}} , $$

(R2)

$$ {\text{Ag}} + {\text{6Ga}}_{{2}} {\text{Te}}_{{3}} = {\text{AgGa}}_{{5}} {\text{Te}}_{{8}} + {\text{Ga}}_{{7}} {\text{Te}}_{{{1}0}} , $$

(R3)

$$ {\text{Ag}} + {\text{2Ga}}_{{7}} {\text{Te}}_{{{1}0}} = {\text{AgGa}}_{{5}} {\text{Te}}_{{8}} + {\text{3Ga}}_{{3}} {\text{Te}}_{{4}} . $$

(R4)

Reactions (R1)-(R4) took place in the PE of ECs, the phase mixtures correspond to phase regions (I)-(IV), respectively. The binary and ternary compounds in reactions (R1)-(R4) are written assuming that the phases have constant composition, which was confirmed by further calculations of thermodynamic functions of quaternary compounds (see Section 3.2). According to reactions (R1)-(R4), the ratios of binary compounds and pure tellurium for assembling the PE of ECs were established. In particular, the compounds AgGa₅Te₈, Ga₂Te₅, Ga₇Te_10, and Ga₃Te₄ are present in the PE with the following ratios of mixtures of the binary compounds and the pure substance Te: 0.5Ag₂Te + 2.5Ga₂Te₃, Ga₂Te₃ + 2Te, GaTe + 3Ga₂Te₃, and GaTe + Ga₂Te₃, respectively.

The E versus T relations were obtained by the least squares method^[51,52] and are presented in the form of Eq 1:

$$E=a+bT\pm {k}_{{{St}}}\sqrt{\left(\frac{{u}_{E}^{2}}{n}+{u}_{b}^{2}{\left(T-\overline{T }\right)}^{2}\right)},$$

(1)

where $a$ and $b$ are coefficients of linear equation, ${k}_{{{St}}}$ is the Student’s parameter,^[53] $n$ is the number of experimental pairs ${E}_{i}$ and ${T}_{i}$, ${u}_{E}^{2}$ and ${u}_{b}^{2}$ are the statistical dispersions of the E and $b$ quantities, respectively.

Listed in Table 1 are the experimental values of E and T that were used to calculate the coefficients and statistical dispersions of Eq 1 in the phase regions (I)-(IV). The results of the calculations are listed in Table 2.

Table 2 The coefficients and statistical dispersions of E versus T dependencies of the ECs

The Gibbs energies (${\Delta }_{{\text{r}}}G$), enthalpies (${\Delta }_{{\text{r}}}H$), and entropies (${\Delta }_{{\text{r}}}S$) of the reactions (R1)-(R4) were calculated by the following thermodynamic equations:

$${\Delta }_{{\text{r}}}G=-n \mathrm{F }E,$$

(2)

$${\Delta }_{{\text{r}}}H=-n\mathrm{ F }\left[E-\left(\frac{{{d}}E}{{{d}}T}\right) T\right],$$

(3)

$${\Delta }_{{\text{r}}}S=n\mathrm{ F }\left({{d}}E/{{d}}T\right).$$

(4)

where $n$ is the number of electrons involved in the reactions (R1)-(R4), F is the Faraday’s constant, and E is the EMF of the ECs.

The values of the thermodynamic functions of reactions (R1)-(R4) in the standard state (T = 298 K and P = 10⁵ Pa) were calculated according to Eq 2-4 and are listed in the Table 3.

Table 3 The values of standard thermodynamic functions of reactions (R1)-(R4). Standard uncertainties for ${\Delta }_{{\text{r}}}G^\circ$, ${\Delta }_{{\text{r}}}H^\circ$, and ${\Delta }_{{\text{r}}}S^\circ$ are also listed

The Gibbs energies of reactions (R1) and (R2) are related to the standard Gibbs energies of compounds by Eq 5 and 6:

$${\Delta }_{{\text{r}}({\text{R}}1)}G^\circ =2{\Delta }_{{\text{f}}}{G}_{{{\text{AgGa}}}_{5}{{\text{Te}}}_{8}}^{\circ }-5{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ },$$

(5)

$${\Delta }_{{\text{r}}({\text{R}}2)}{G}^{\circ }=4{\Delta }_{{\text{f}}}{G}_{{{\text{AgGa}}}_{5}{{\text{Te}}}_{8}}^{\circ }-9{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{3}}^{\circ }-{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ }.$$

(6)

By solving the system of Eq 5 and 6 one obtains:

$${\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ }=\frac{{\Delta }_{{\text{r}}({\text{R}}2)}{G}^{\circ }-{2\Delta }_{{\text{r}}({\text{R}}1)}{G}^{\circ }}{9}+{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{3}}^{\circ }.$$

(7)

Equations for determining the enthalpy of formation and entropy of the Ga₂Te₅ compound were similarly obtained:

$${\Delta }_{{\text{f}}}{H}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ }=\frac{{\Delta }_{{\text{r}}({\text{R}}2)}{H}^{\circ }-{2\Delta }_{{\text{r}}({\text{R}}1)}{H}^{\circ }}{9}+{\Delta }_{{\text{f}}}{H}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{3}}^{\circ },$$

(8)

$${S}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ }=\frac{{\Delta }_{{\text{r}}({\text{R}}2)}{S}^{\circ }-{2\Delta }_{{\text{r}}({\text{R}}1)}{S}^{\circ }}{9}+2{S}_{{\text{Te}}}^{\circ }+{S}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{3}}^{\circ }.$$

(9)

Reactions to determine the standard thermodynamic properties ${\Delta }_{{\text{f}}}{G}^{\circ }$, ${\Delta }_{{\text{f}}}{H}^{\circ }$, and ${S}^{\circ }$ of the AgGa₅Te₈, Ga₇Te₁₀, and Ga₃Te₄ compounds were written in a similar way using reactions (R2)-(R4) with the corresponding stoichiometric numbers.

For the first time, the standard thermodynamic values for compounds of the GaTe-AgGa₅Te₈-Te system were determined using Eq 7-9 and the thermodynamic data of pure substances (Ag, Ga, Te) and the binary compound Ga₂Te₃.^[54] The results of the calculations are listed in Table 4.

Table 4 The values of standard (T = 298 K and P = 10⁵ Pa) thermodynamic properties of compounds of the GaTe-AgGa₅Te₈-Te system

The temperature dependences of the Gibbs energies of the formation of compounds of the GaTe-AgGa₅Te₈-Te system are described by Eq 10-13:

$$\frac{{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}}{\left(\mathrm{kJ }{{\text{mol}}}^{-1}\right)}= -\left(276.5\pm 4.6\right)+\left(18.1\pm 0.3\right)\times \frac{{10}^{-3}T}{{\text{K}}},$$

(10)

$$\frac{{\Delta }_{{\text{f}}}{G}_{{{\text{AgGa}}}_{5}{{\text{Te}}}_{8}}}{\left(\mathrm{kJ }{{\text{mol}}}^{-1}\right)}= -\left(711.3\pm 8.8\right)+\left(30.9\pm 0.4\right)\times \frac{{10}^{-3}T}{{\text{K}}},$$

(11)

$$\frac{{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{7}{{\text{Te}}}_{10}}}{\left(\mathrm{kJ }{{\text{mol}}}^{-1}\right)}= -\left(958.4\pm 11.7\right)+\left(58.6\pm 0.7\right)\times \frac{{10}^{-3}T}{{\text{K}}},$$

(12)

$$\frac{{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{3}{{\text{Te}}}_{4}}}{\left(\mathrm{kJ }{{\text{mol}}}^{-1}\right)}= -\left(408.6\pm 7.9\right)+\left(25.4\pm 0.4\right)\times \frac{{10}^{-3}T}{{\text{K}}}.$$

(13)

The obtained ${\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ }$, ${\Delta }_{{\text{f}}}{G}_{{{\text{AgGa}}}_{5}{{\text{Te}}}_{8}}^{\circ }$, ${\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{7}{{\text{Te}}}_{10}}^{\circ }$, and ${\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{3}{{\text{Te}}}_{4}}^{\circ }$ values do not exclude the hypothetical reactions of the synthesis of compounds under standard conditions:

$$ {\text{Ga}}_{{2}} {\text{Te}}_{{3}} + {\text{2Te}} = {\text{Ga}}_{{2}} {\text{Te}}_{{5}} , $$

(R5)

$$ {\text{Ag}}_{{2}} {\text{Te}} + {\text{5Ga}}_{{2}} {\text{Te}}_{{3}} = {\text{2AgGa}}_{{5}} {\text{Te}}_{{8}} , $$

(R6)

$$ {\text{GaTe}} + {\text{3Ga}}_{{2}} {\text{Te}}_{{3}} = {\text{Ga}}_{{7}} {\text{Te}}_{{{1}0}} , $$

(R7)

$$ {\text{GaTe}} + {\text{Ga}}_{{2}} {\text{Te}}_{{3}} = {\text{Ga}}_{{3}} {\text{Te}}_{{4}} . $$

(R8)

The calculated values of Gibbs energies of the reaction (R5)-(R8) ${\Delta }_{{\text{r}}({\text{R}}5)}{G}^{\circ }=-1.2$ $\mathrm{kJ }{{\text{mol}}}^{-1}$, ${\Delta }_{{\text{r}}({\text{R}}6)}{G}^{\circ }=-19.6$ $\mathrm{kJ }{{\text{mol}}}^{-1}$, ${\Delta }_{{\text{r}}({\text{R}}7)}{G}^{\circ }=-9.3$ $\mathrm{kJ }{{\text{mol}}}^{-1}$, and ${\Delta }_{{\text{r}}({\text{R}}8)}{G}^{\circ }=-9.2$ $\mathrm{kJ }{{\text{mol}}}^{-1}$ are negative. Thus, the standard thermodynamic values of compounds presented in Table 4 do not contradict the thermodynamics laws.

3.2 Phase Equilibria and Thermodynamic Properties of Quaternary Compounds of the Ag-Ga-Te-AgBr System Below 600 K

The division of composition space of the Ag-Ga-Te-AgBr system in the Ag₂Te-GaTe-Te-AgBr-Ag₂Te part is shown in Fig. 5(a)-(e). The triangulation is performed according to:

• 1) the results of the E versus T dependencies of the ECs with PE of different phase regions;

• 2) the established phase composition of the GaTe-AgGa₅Te₈-Te phase region (see Section 3.1); and

• 3) the existence of quaternary compounds in the AgTe-Ga₂Te₃-AgBr system.

Fig. 5.

Divisions of composition space of the Ag-Ga-Te-AgBr system in the Ag₂Te-GaTe-Te-AgBr-Ag₂Te part on separate phase regions below 600 K

The locations of individual tetrahedra Ga₂Te₃-Ga₂Te₅-AgBr-Ag₃Ga₁₀Te₁₆Br (phase region (V)), Ga₂Te₃-AgBr-Ag₃Ga₁₀Te₁₆Br-Ga₇Te₁₀ (VI), Ga₇Te₁₀-AgBr-Ag₃Ga₁₀Te₁₆Br-Ga₃Te₄ (VII), Ag₃Ga₁₀Te₁₆Br-Ga₂Te₅-AgBr-Ag₃Ga₂Te₄Br (VIII), Ag₃Ga₁₀Te₁₆Br-AgBr-Ag₃Ga₂Te₄Br-GaTe (IX), Ag₃Ga₂Te₄Br-AgBr-Ag₂₇Ga₂Te₁₂Br₉-GaTe (X), and Ga₂Te₅-AgBr-Ag₃Ga₂Te₄Br-Ag₂₇Ga₂Te₁₂Br₉ (XI) of the Ag₂Te-GaTe-Te-AgBr-Ag₂Te part are shown in Fig. 5(b)-(e). All of these tetrahedra contain quaternary compounds. The spatial position of four-phase regions (V)-(XI) relative to the silver point was used to establish the overall potential-determining reactions of synthesis of the quaternary compounds. The common planes between two tetrahedra are illustrated as shaded areas in Fig. 5(b), (d) and (e).

The reactions for these individual phase regions can be described as follows

According to Fig. 5(b), reactions (R9) and (R10) for the phase regions (V) and (VI), respectively, can be expressed as:

$$ {\text{4Ag }} + {\text{ 9Ga}}_{{2}} {\text{Te}}_{{3}} + {\text{ Ga}}_{{2}} {\text{Te}}_{{5}} + {\text{ AgBr }} = {\text{ 2Ag}}_{{3}} {\text{Ga}}_{{{1}0}} {\text{Te}}_{{{16}}} {\text{Br}}, $$

(R9)

$$ {\text{2Ag}} + {\text{12Ga}}_{{2}} {\text{Te}}_{{3}} + {\text{AgBr}} = {\text{Ag}}_{{3}} {\text{Ga}}_{{{1}0}} {\text{Te}}_{{{16}}} {\text{Br}} + {\text{2Ga}}_{{7}} {\text{Te}}_{{{1}0}} , $$

(R10)

the common plane for these phase regions is Ag₃Ga₁₀Te₁₆Br-Ga₂Te₃-AgBr;

for phase region (VII), Fig. 5(c):

$$ {\text{2Ag}} + {\text{4Ga}}_{{7}} {\text{Te}}_{{{1}0}} + {\text{AgBr}} = {\text{Ag}}_{{3}} {\text{Ga}}_{{{1}0}} {\text{Te}}_{{{16}}} {\text{Br}} + {\text{6Ga}}_{{3}} {\text{Te}}_{{4}} ; $$

(R11)

for the phase regions (VIII) and (IX), Fig. 5(d):

$$ {\text{16Ag}} + {\text{4Ga}}_{{2}} {\text{Te}}_{{5}} + {\text{8AgBr}} + {\text{Ag}}_{{3}} {\text{Ga}}_{{{1}0}} {\text{Te}}_{{{16}}} {\text{Br}} = {\text{9Ag}}_{{3}} {\text{Ga}}_{{2}} {\text{Te}}_{{4}} {\text{Br}}, $$

(R12)

$$ {\text{4Ag}} + {\text{Ag}}_{{3}} {\text{Ga}}_{{{1}0}} {\text{Te}}_{{{16}}} {\text{Br}} + {\text{2AgBr}} = {\text{3Ag}}_{{3}} {\text{Ga}}_{{2}} {\text{Te}}_{{4}} {\text{Br}} + {\text{4GaTe}}, $$

(R13)

the common plane is Ag₃Ga₁₀Te₁₆Br-3Ag₃Ga₂Te₄Br-AgBr;

for the phase regions (X) and (XI), Fig. 5(e):

$$ {\text{8Ag}} + {\text{5Ag}}_{{3}} {\text{Ga}}_{{2}} {\text{Te}}_{{4}} {\text{Br}} + {\text{4AgBr}} = {\text{Ag}}_{{{27}}} {\text{Ga}}_{{2}} {\text{Te}}_{{{12}}} {\text{Br}}_{{9}} + {\text{8GaTe}}, $$

(R14)

$$ {\text{32Ag}} + {\text{8Ga}}_{{2}} {\text{Te}}_{{5}} + {\text{16AgBr}} = {\text{Ag}}_{{{27}}} {\text{Ga}}_{{2}} {\text{Te}}_{{{12}}} {\text{Br}}_{{9}} + {\text{7Ag}}_{{3}} {\text{Ga}}_{{2}} {\text{Te}}_{{4}} {\text{Br}}, $$

(R15)

the common plane is Ag₃Ga₂Te₄Br-AgBr-Ag₂₇Ga₂Te₁₂Br₉.

According to these reactions, ratios of binary compounds in the PE of ECs were calculated. Quaternary compounds are presented in heterophase mixtures of PE by the following binary phases: Ag₃Ga₁₀Te₁₆Br → Ag₂Te, 5Ga₂Te₃, and AgBr; Ag₃Ga₂Te₄Br → Ag₂Te, Ga₂Te₃, and AgBr; Ag₂₇Ga₂Te₁₂Br₉ → 9Ag₂Te, Ga₂Te₃, and 9AgBr.

Experimental quantities of temperature and EMF of the ECs with PE of the phase regions (V)-(XI) are listed in Table 5.

Table 5 Measured values of temperature (T) and EMF (E) of the ECs with PE of different phase regions at pressure P = 10⁵ Pa

The experimental values of T and E listed in Table 5 were used to calculate the coefficients and statistical dispersions of Eq 1 for the phase regions (V)-(XI). The results of the calculations are listed in Table 6.

Table 6 The coefficients and statistical dispersions of E versus T dependencies of the ECs

The values of the thermodynamic functions of reactions (R9)-(R15) in the standard state (T = 298 K and P = 10⁵ Pa) were calculated according to Eq 2-4 and are listed in Table 7.

Table 7 The values of standard thermodynamic properties of the reactions (R9)-(R15)

The standard thermodynamic properties of the quaternary compounds of the Ag-Ga-Te-AgBr system were calculated in a similar way to the ternary and binary compounds of the Ag-Ga-Te system (see Section 3.1). The results of the calculations are listed in Table 8.

Table 8 The values of standard (T = 298 K and P = 10⁵ Pa) thermodynamic properties of selected compounds of the Ag-Ga-Te-AgBr system

The determined Gibbs energy values of the quaternary compounds Ag₃Ga₁₀Te₁₆Br in the phase regions (V), (VI), (VII) and Ag₃Ga₂Te₄Br in regions (VIII), (IX) are the same within the experimental errors. The standard Gibbs energy of formation of the Ag₂₇Ga₂Te₁₂Br₉ compound calculated using (R14) ${\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{27}{{\text{Ga}}}_{2}{{\text{Te}}}_{12}{{\text{Br}}}_{9}, ({\text{R}}14)}^{\circ }=-1629.9\mathrm{ kJ }{{\text{mol}}}^{-1}$ differs significantly from the value obtained from (R15) ${\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{27}{{\text{Ga}}}_{2}{{\text{Te}}}_{12}{{\text{Br}}}_{9}, ({\text{R}}15)}^{\circ }=-1492.5\mathrm{ kJ }{{\text{mol}}}^{-1}$. To analyze the obtained values from a thermodynamic point of view, the Gibbs energy of a hypothetical reaction

$$ {\text{9Ag}}_{{2}} {\text{Te}} + {\text{Ga}}_{{2}} {\text{Te}}_{{5}} + {\text{9AgBr}} = {\text{Ag}}_{{{27}}} {\text{Ga}}_{{2}} {\text{Te}}_{{{12}}} {\text{Br}}_{{9}} , $$

(R16)

of formation of the quaternary compound from the binary compounds was calculated by Eq 14:

$${\Delta }_{{\text{r}}({\text{R}}16)}{G}^{\circ }={\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{27}{{\text{Ga}}}_{2}{{\text{Te}}}_{12}{{\text{Br}}}_{2}}^{\circ }-9{\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{2}{\text{Te}}}^{\circ }-{\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ }-9{\Delta }_{{\text{f}}}{G}_{{\text{AgBr}}}^{\circ }.$$

(14)

Considering data from Ref.^[54] ${\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{2}{\text{Te}}}^{\circ }=-41.558 \mathrm{kJ }{{\text{mol}}}^{-1}$, ${\Delta }_{{\text{f}}}{G}_{{{\text{Ga}}}_{2}{{\text{Te}}}_{5}}^{\circ }=-271.019 \mathrm{kJ }{{\text{mol}}}^{-1}$, ${\Delta }_{{\text{f}}}{G}_{{\text{AgBr}}}^{\circ }=-97.095 \mathrm{kJ }{{\text{mol}}}^{-1}$ it was found that ${\Delta }_{{\text{r}}({\text{R}}16)}{G}^{\circ }<0$ using the value of ${\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{27}{{\text{Ga}}}_{2}{{\text{Te}}}_{12}{{\text{Br}}}_{9}, ({\text{R}}14)}^{\circ }$ and ${\Delta }_{{\text{r}}({\text{R}}16)}{G}^{\circ }>0$ using the value of ${\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{27}{{\text{Ga}}}_{2}{{\text{Te}}}_{12}{{\text{Br}}}_{9}, ({\text{R}}15)}^{\circ }$. Consequently, the ${\Delta }_{{\text{f}}}{G}_{{{\text{Ag}}}_{27}{{\text{Ga}}}_{2}{{\text{Te}}}_{12}{{\text{Br}}}_{9}, ({\text{R}}15)}^{\circ }$ value contradicts the thermodynamics laws. This means that in PE of ECs the mixture of compounds formed with the participation of the catalyst Ag⁺ does not correspond to that expressed in (R15). it is likely that the Ga₂Te₅ compound in the set of phases in (R15) is not formed for thermodynamic reasons. The experimentally established participation of the Ga₂Te₅ in the divisions of the GaTe-AgGa₅Te₈-Te region below 600 K indicates the possibility of its existence as a thermodynamically stable compound in the aggregate of three phases of PE—reactions (R1) and (R2), as well as four phases of PE—reaction (R9). Similar properties were established for the metastable phases Ag₅Te₂Br of the Ag-Te-Br system and Ag₂NiSnS₄ of the Ag-Ni-Sn-S system.^[39,40]

4 Conclusions

The EMF method was applied to divide the T − x space of the Ag-Ga-Te-AgBr system in the Ag₂Te-GaTe-Te-AgBr-Ag₂Te part below 600 K into three- and four-phase regions with the participation of stable Ga₃Te₄, Ga₇Te₁₀, AgGa₅Te₈, Ag₃Ga₁₀Te₁₆Br, Ag₃Ga₂Te₄Br, Ag₂₇Ga₂Te₁₂Br₉ and metastable Ga₂Te₅ compounds. The AgGa₂Te₃Br compound did not form below 600 K. The synthesis of the equilibrium set of phases using the calculated amounts of binary compounds Ag₂Te, GaTe, Ga₂Te₃, AgBr, and pure Te was carried out in the positive electrodes of electrochemical cells with the participation of the Ag⁺ catalysts. The standard thermodynamic functions (Gibbs energies, enthalpies, and entropies) for the compounds Ga₂Te₅, Ga₇Te₁₀, Ga₃Te₄, AgGa₅Te₈, Ag₃Ga₁₀Te₁₆Br, Ag₃Ga₂Te₄Br, and Ag₂₇Ga₂Te₁₂Br₉ were determined by the EMF method, for the first time. The obtained values of the Gibbs energies of formation of the compounds are in accordance with the laws of thermodynamics. Based on x-ray diffraction phase analysis and EMF measurements of the electrochemical cells, it was concluded that compounds Ga₇Te₁₀ and AgGa₅Te₈ exist in two temperature ranges. The established existence of two-phase equilibria between the compounds along the cross-sections AgGa₅Te₈-Ag₃Ga₁₀Te₁₆Br, AgGaTe₂-Ag₃Ga₂Te₄Br, and Ag₉GaTe₆-Ag₂₇Ga₂Te₁₂Br₉ allows the formation of solid solutions based on them. The formation of solid solutions is an effective strategy for increasing the thermoelectric figure of merit ZT parameters of different materials.