Phase Equilibria and Thermodynamic Properties of Selected Compounds in the Ag–Ga–S–AgBr System for Modern Application in Energy Conversion Devices

Abstract

The phase equilibria of the Ag–Ga–S–AgBr system in the part GaS–Ga₂S₅–AgBr–Ag₂S below 600 K were investigated by the modified electromotive force (EMF) method using the Ag⁺ catalysts as small nucleation centers of equilibrium phases. Division of the GaS–Ga₂S₅–AgBr–Ag₂S was carried out with the participation of the following compounds Ag₂S, GaS, Ga₂S₃, AgBr, Ag₉GaS₆, AgGaS₂, Ag₃SBr, Ag₃Ga₂S₄Br, and Ag₂₇Ga₂S₁₂Br₉. Reactions were performed by applying electrochemical cells (ECs) with the structure: (−) 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 Ag⁺ diffusion into PE. The measured EMF and temperature values of ECs were used to determine the standard thermodynamic functions of the compounds Ag₃Ga₂S₄Br and Ag₂₇Ga₂S₁₂Br.

Introduction

To date, establishing the phase composition of the equilibrium T-x space of multi-component inorganic systems at T ≤ 600 K, when there are kinetic obstacles to achieving a state of thermodynamic equilibrium, remains relevant. The effect on samples of such external factors as long-term annealing during temperature and pressure variations is ineffective in many cases. The possibility of overcoming such kinetic obstacles was established in Refs. [1, 2]. For this purpose, the silver ions Ag⁺ were used as catalysts, i.e., small nucleation centers of equilibrium phases.

The concentration tetrahedra of the Ag–Ga–X–Y (X = S, Se, Te; Y = Cl, Br, I) in part of the quasi-ternary Ag₂X–Ga₂X₃–AgY systems are characterized by the presence of semiconductor compounds of the formula composition AgGa₂X₃Y (structure type CuIn₂Te₃Cl, space group I–4) [3]. Quaternary compounds decompose upon annealing at 600 K [4].

For the case X = S and Y = Br, the quasi-ternary Ag₂S–Ga₂S₃–AgBr system, in addition to the quaternary compound AgGa₂S₃Br, is characterized by the following ternary phases Ag₉GaS₆, AgGaS₂ (quasi-binary system Ag₂S–Ga₂S₃) and Ag₃SBr (quasi-binary system Ag₂S–AgBr) [5, 6]. The solid-state phase equilibria in the Ag–Ga–S system and thermodynamic properties of ternary phases were reported in Ref. [7]. The argyrodite family compound Ag₉GaS₆ is a promising thermoelectric material with the figure of merit parameter ZT ~ 0.6 and has intrinsic ultralow lattice thermal conductivity [8]. Moreover, Ag₉GaS₆ has a high silver ionic conductivity [9, 10]. The AgGaS₂ belongs to the chalcopyrite-structured ternary semiconductor compounds with a direct band gap of (2.48–2.75) eV. This compound has a high transparency in the mid-IR range and can be used as a commercial material for photovoltaic and nonlinear optical applications as well as a promising candidate for X-ray dosimetry [11,12,13]. The Ag₃SBr compound belongs to the class of superionic materials [14]. Thus, the multi-component compounds and solid solutions based on phases of the Ag–Ga–S system have been considered interesting scientific objects due to the diversity of their crystal structures and physicochemical properties [15,16,17,18,19]. However, these compounds no longer fully meet all the requirements of a new generation of devices for modern applications. For example, the band gap value and weak absorption in the visible light region limit the use of AgGaS₂ as absorber material for photovoltaic solar cells. Recently, the photo-electrochemical cells based on the AgGaS₂ compound showed an efficiency of 5.85% [20]. Optimization technology for the synthesis of new materials and improving their technical characteristics is impossible without a comprehensive analysis of the thermodynamic properties of intermediate phases and construction equilibrium phase diagrams.

The points of intersection of the cross-sections AgGaS₂–AgBr and Ag₉GaS₆–AgBr of the quasi-ternary system with the tie-line Ga₂S₃–Ag₃SBr are places of potential formation of quaternary compounds Ag₃Ga₂S₄Br and Ag₂₇Ga₂S₁₂Br₉. There are no previous reports on quaternary compounds of mentioned composition. The thermodynamic conditions for the formation of quaternary phases likely correspond to the temperature values T < 600 K, where there are kinetic obstacles to such a process.

The purpose of this work was to establish by the electromotive force (EMF) method the phase composition of the Ga₂S₃–Ag₃SBr cross-section of the Ag₂S–Ga₂S₃–AgBr system below 600 K and to determine the values of the standard thermodynamic functions of the quaternary compounds in the system. The two-phase equilibrium between compounds of the AgGaS₂–Ag₃Ga₂S₄Br and Ag₉GaS₆–Ag₂₇Ga₂S₁₂Br₉ cross-sections can be used to vary the nonlinear optical properties of the phases in the way of forming solid solutions on a mutual basis.

Experimental

The high-purity substances Ag (> 99.9 wt%, Alfa Aesar, Germany), Ga, and S (> 99.99 wt%, Alfa Aesar, Germany) were used to synthesize the binary compounds Ag₂S, GaS, and Ga₂S₃. Melts of the Ag₂S, GaS, and Ga₂S₃ compounds in an inert atmosphere were cooled to room temperature, then crushed to a particle size of ~ 1 × 10^–6 m for preparation of the positive electrodes (PE) of electrochemical cells (ECs) [21, 22].

The modified EMF method [1, 2] was used both to establish the phase equilibria in the GaS–Ga₂S₅–AgBr–Ag₂S part of the Ag–Ga–S–AgBr concentration tetrahedron below 600 K and to determine the thermodynamic parameters of compounds. For these investigations, a certain number of ECs were assembled:

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

where IE is the inert electrode (graphite powder), NE is the negative electrode (silver powder), SSE is the solid-state electrolyte (glassy Ag₃GeS₃Br [23]), and R{Ag⁺} is the region of PE that contacts with SSE. At the stage of cell preparation, PE is the non-equilibrium phase mixture of the well-mixed powdered binary compounds Ag₂S, GaS, Ga₂S₃, and AgBr (99.5 wt%, Alfa Aesar, Germany). Compositions of these mixtures covered the entire concentration space of the GaS–Ga₂S₅–AgBr–Ag₂S region. An equilibrium set of phases was formed in the R{Ag⁺} region at 600 K for 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 [21, 22].

The experiments were performed in a resistance furnace described in Ref. [24, 25]. 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 [26, 27]. The argon gas 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 [28].

Results and Discussion

The division of the concentration tetrahedron Ag–Ga–S–AgBr into separate four-phase regions in the GaS–Ga₂S₅–AgBr–Ag₂S part below 600 K is shown in Fig. 1. The division was carried out based on the experimental results of the E versus T relations of the ECs with PE of different phase regions and taking into account the basic rules of the EMF method [29,30,31]:

Fig. 1

Spatial position of tetrahedra GaS–Ga₂S₃–AgBr–Ag₃Ga₂S₄Br (left) and GaS–AgBr–Ag₂₇Ga₂S₁₂Br₉–Ag₃Ga₂S₄Br (right) in the concentration space of the Ag–Ga–S–AgBr system

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

• (2) ECs with PE of different phase regions are characterized by different EMF values at T = const, Table 1;

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

• (3) the four-phase region further away from the figurative point of Ag is characterized by a higher EMF value at a specific temperature, Fig. 2.

  Fig. 2

  

  E versus T dependencies of the ECs with PE of the phase regions (I) and (II)

The spatial position of the established four-phase regions GaS–Ga₂S₃–AgBr–Ag₃Ga₂S₄Br (phase region (I)) and GaS–AgBr–Ag₂₇Ga₂S₁₂Br₉–Ag₃Ga₂S₄Br (phase region (II)) relative to the silver point was used to establish the overall potential-determining reactions:

$$ {\text{2Ag}} + {\text{2Ga}}_{{2}} {\text{S}}_{{3}} + {\text{AgBr}} = {\text{Ag}}_{{3}} {\text{Ga}}_{{2}} {\text{S}}_{{4}} {\text{Br}} + {\text{2GaS}}, $$

(R1)

$$ {\text{8Ag}} + {\text{5Ag}}_{{3}} {\text{Ga}}_{{2}} {\text{S}}_{{4}} {\text{Br}} + {\text{4AgBr}} = {\text{Ag}}_{{{27}}} {\text{Ga}}_{{2}} {\text{S}}_{{{12}}} {\text{Br}}_{{9}} + {\text{8GaS}}. $$

(R2)

Reactions (R1) and (R2) were carried out in the PE of ECs, and the phase mixtures correspond to phase regions (I) and (II), respectively. According to reactions (R1) and (R2), the ratios of binary compounds for assembling the PE of ECs were established. In particular, the compounds Ag₃Ga₂S₄Br and Ag₂₇Ga₂S₁₂Br₉ are present in the PE compositions in the following ratios of mixtures of the binary compounds: Ag₂S:Ga₂S₃:AgBr = 1:1:1 and Ag₂S:Ga₂S₃:AgBr = 9:1:9, respectively.

From the data analysis of Fig. 2, it follows that the E versus T dependencies of the ECs in the phase regions (I) and (II) are linear. Therefore, the results of EMF measurements processed by the least squares method [32] can be presented in the form of Eq. (1):

$$ E = a + bT \equiv \overline{E} + b\left( {T - \overline{T}} \right), $$

(1)

where $\overline{E} = \frac{{\sum E_{i} }}{n}$, $\overline{T} = \frac{{\sum T_{i} }}{n}$ ($E_{i}$ is the EMF of the cell at temperature $T_{i}$; $n$ is a number of experimental pairs $E_{i}$ and $T_{i}$).

Coefficients $a$ and $b$ were calculated by the following Eqs. (2) and (3):

$$ a = \overline{E} - b\overline{T}, $$

(2)

$$ b = \frac{{\sum \left[ {\left( {E_{i} - \overline{E}} \right)\left( {T_{i} - \overline{T}} \right)} \right]}}{{\sum \left( {T_{i} - \overline{T}} \right)^{2} }}. $$

(3)

The statistical dispersions of the measurement uncertainties consisted of the calculation variances of experimental values of EMF $E$ ($u_{E}^{2}$), coefficients $b$ ($u_{b}^{2}$) and $a$ ($u_{a}^{2}$), as well as dispersions of the calculated by Eq. (1) EMF values $\tilde{E}$ ($u_{{\tilde{E}}}^{2}$):

$$ u_{E}^{2} = \frac{{\sum \left( {E_{i} - \tilde{E}_{i} } \right)^{2} }}{n - 2}, $$

(4)

$$ u_{b}^{2} \left( T \right) = \frac{{u_{E}^{2} }}{{\sum \left( {T_{i} - \overline{T}} \right)^{2} }}, $$

(5)

$$ u_{a}^{2} \left( T \right) = \frac{{u_{E}^{2} }}{n} + \frac{{u_{E}^{2} \overline{T}^{2} }}{{\sum \left( {T_{i} - \overline{T}} \right)^{2} }}, $$

(6)

$$ u_{{\tilde{E}}}^{2} \left( T \right) = \frac{{u_{E}^{2} }}{n} + u_{b}^{2} \left( {T - \overline{T}} \right)^{2} . $$

(7)

Uncertainties ($\Delta_{i}$) of the corresponding quantities can be calculated by the Eq. (8):

$$ \Delta_{i} = k_{St} u_{i} $$

(8)

where $k_{St}$ is the Student’s coefficient, and $u_{i}$ is the standard deviation. At the confidence level of 95% and $n = 12$, the Student’s coefficient is equal $k_{St} = 2.179$ [32].

According to [33, 34], the final equation of the E versus T dependences together with the statistical dispersions can be expressed as:

$$ E = a + bT \pm k_{St} \sqrt {\left( {\frac{{u_{E}^{2} }}{n} + u_{b}^{2} \left( {T - \overline{T}} \right)^{2} } \right)} . $$

(9)

An example of calculating the coefficients of Eq. (9) for the phase region (I) is given in Table 2.

Table 2 Coefficients of the E versus T dependence of the cell with PE of the phase region (I)

Analogously to the phase region (I), coefficients E versus T dependence of the cell with PE of the phase region (II) were calculated. The results of the calculations are listed in Table 3.

Table 3 Coefficients and statistical dispersions of the E versus T dependencies of the ECs in the phase regions (I) and (II)

The Gibbs energies ($\Delta_{{\text{r}}} G$), enthalpies ($\Delta_{{\text{r}}} H$), and entropies ($\Delta_{{\text{r}}} S$) of the reactions (R1) and (R2) were calculated by the following thermodynamic equations:

$$ \Delta_{{\text{r}}} G = - z\,FE, $$

(10)

$$ \Delta_{{\text{r}}} H = - z\,F\left[ {E - \left( {{\text{d}}E/{\text{d}}T} \right) T} \right], $$

(11)

$$ \Delta_{{\text{r}}} S = z\,F\left( {{\text{d}}E/{\text{d}}T} \right). $$

(12)

where $z$ is the number of electrons involved in the reactions (R1) and (R2), F is the Faraday’s constant, and E is the EMF of the ECs.

The values of the thermodynamic functions of reactions (R1) and (R2) in the standard state (T = 298 K and P = 10⁵ Pa) were calculated according to Eqs. (10)–(12) and are listed in Table 4.

Table 4 Values of standard thermodynamic function of the reactions (R1) and (R2)

The Gibbs energy, enthalpy, and entropy of the reaction (R1) are related to the Gibbs energy, enthalpy, and entropy of the compounds Ga₂S₃, AgBr, Ag₃Ga₂S₄Br, GaS, and pure substance Ag by the following equations:

$$ \Delta_{{{\text{r}}\left( {{\text{R}}1} \right)}} G^{\circ } = \Delta_{{\text{f}}} G_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}}^{\circ } + 2\Delta_{{\text{f}}} G_{{{\text{GaS}}}}^{\circ } - 2\Delta_{{\text{f}}} G_{{{\text{Ga}}_{2} {\text{S}}_{3} }}^{\circ } - \Delta_{{\text{f}}} G_{{{\text{AgBr}}}}^{\circ } , $$

(13)

$$ \Delta_{{{\text{r}}\left( {{\text{R}}1} \right)}} H^{\circ } = \Delta_{{\text{f}}} H_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}}^{\circ } + 2\Delta_{{\text{f}}} H_{{{\text{GaS}}}}^{\circ } - 2\Delta_{{\text{f}}} H_{{{\text{Ga}}_{2} {\text{S}}_{3} }}^{\circ } - \Delta_{{\text{f}}} H_{{{\text{AgBr}}}}^{\circ } , $$

(14)

$$ \Delta_{{{\text{r}}\left( {{\text{R}}1} \right)}} S^{\circ } = S_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}}^{\circ } + 2S_{{{\text{GaS}}}}^{\circ } - 2S_{{{\text{Ag}}}}^{\circ } - 2S_{{{\text{Ga}}_{2} {\text{S}}_{3} }}^{\circ } - S_{{{\text{AgBr}}}}^{\circ } . $$

(15)

It follows from Eqs. (13)–(15) that:

$$ \Delta_{{\text{f}}} G_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}}^{\circ } = 2\Delta_{{\text{f}}} G_{{{\text{Ga}}_{2} {\text{S}}_{3} }}^{\circ } + \Delta_{{\text{f}}} G_{{{\text{AgBr}}}}^{\circ } - 2\Delta_{{\text{f}}} G_{{{\text{GaS}}}}^{\circ } + \Delta_{{{\text{r}}\left( {{\text{R}}1} \right)}} G^{\circ } , $$

(16)

$$ \Delta_{{\text{f}}} H_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}}^{\circ } = 2\Delta_{{\text{f}}} H_{{{\text{Ga}}_{2} {\text{S}}_{3} }}^{\circ } + \Delta_{{\text{f}}} H_{{{\text{AgBr}}}}^{\circ } - 2\Delta_{{\text{f}}} H_{{{\text{GaS}}}}^{\circ } + \Delta_{{{\text{r}}\left( {{\text{R}}1} \right)}} H^{\circ } , $$

(17)

$$ S_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}}^{\circ } = 2S_{{{\text{Ag}}}}^{\circ } + 2S_{{{\text{Ga}}_{2} {\text{S}}_{3} }}^{\circ } + S_{{{\text{AgBr}}}}^{\circ } - 2S_{{{\text{GaS}}}}^{\circ } + \Delta_{{{\text{r}}\left( {{\text{R}}1} \right)}} S^{\circ } . $$

(18)

Reactions to determine the standard thermodynamic properties $\Delta_{{\text{f}}} G^{\circ }$, $\Delta_{{\text{f}}} H^{\circ }$, and $S^{\circ }$ of the Ag₂₇Ga₂S₁₂Br₉ compound were written in similarity using (R2) with the corresponding stoichiometric numbers.

For the first time, the standard thermodynamic quantities of the quaternary compounds of the Ag–Ga–S–AgBr system were determined using Eqs. (16)–(18) and thermodynamic data of pure substances (Ag, Ga, S, Br₂) and the binary compound GaS, Ga₂S₃, AgBr [35]. The results of the calculations are listed in Table 5.

Table 5 Values of standard (T = 298 K and P = 10⁵ Pa) thermodynamic properties of selected compounds of the Ag–Ga–S–AgBr system

The temperature dependences of the Gibbs energies of formations of the quaternary compounds of the Ag–Ga–S–AgBr system are described by the following equations:

$$ \Delta_{{\text{f}}} G_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}} /\left( {{\text{kJ mol}}^{ - 1} } \right) = - \left( {722.0 \pm 10.2} \right) - \left( {33.4 \pm 0.4} \right) \times 10^{ - 3} \,{\text{T}}/{\text{K}}, $$

(19)

$$ \Delta_{{\text{f}}} G_{{{\text{Ag}}_{27} {\text{Ga}}_{2} {\text{S}}_{12} {\text{Br}}_{9} }} /\left( {{\text{kJ mol}}^{ - 1} } \right) = - \left( {2443.1 \pm 31.7} \right) - \left( {377.3 \pm 4.9} \right) \times 10^{ - 3} \,{\text{T}}/{\text{K}}. $$

(20)

Included in Table 5 values of $\Delta_{{\text{f}}} G_{{{\text{Ag}}_{3} {\text{Ga}}_{2} {\text{S}}_{4} {\text{Br}}}}^{\circ }$ and $\Delta_{{\text{f}}} G_{{{\text{Ag}}_{27} {\text{Ga}}_{2} {\text{S}}_{12} {\text{Br}}_{9} }}^{\circ }$ do not contradict the hypothetical reactions of the synthesis of quaternary compounds from binary phases under standard conditions:

$$ {\text{Ag}}_{{2}} {\text{S}} + {\text{Ga}}_{{2}} {\text{S}}_{{3}} + {\text{AgBr}} = {\text{Ag}}_{{3}} {\text{Ga}}_{{2}} {\text{S}}_{{4}} {\text{Br}}, $$

(R3)

$$ {\text{9Ag}}_{{2}} {\text{S}} + {\text{Ga}}_{{2}} {\text{S}}_{{3}} + {\text{9AgBr}} = {\text{Ag}}_{{{27}}} {\text{Ga}}_{{2}} {\text{S}}_{{{12}}} {\text{Br}}_{{9}} . $$

(R4)

Calculated values of the Gibbs energies of reactions (R3) and (R4) are equal, respectively: $\Delta_{{{\text{r}}\left( {{\text{R}}3} \right)}} G^{\circ } = - 40.4\;{\text{kJ mol}}^{ - 1}$ and $\Delta_{{{\text{r}}\left( {{\text{R}}4} \right)}} G^{\circ } = - 811.3\;{\text{kJ mol}}^{ - 1}$.

Conclusions

The phase space of the Ag–Ga–S–AgBr system in the GaS–Ga₂S₅–AgBr–Ag₂S part is characterized by the binary (Ag₂S, GaS, Ga₂S₃, AgBr), ternary (Ag₉GaS₆, AgGaS₂, Ag₃SBr), and quaternary (Ag₃Ga₂S₄Br, Ag₂₇Ga₂S₁₂Br₉) compounds. Quaternary compounds are components of the concentration tetrahedra GaS–Ga₂S₃–AgBr–Ag₃Ga₂S₄Br and GaS–AgBr–Ag₂₇Ga₂S₁₂Br₉–Ag₃Ga₂S₄Br. The spatial position of the established tetrahedra relative to the silver point was used to establish the overall potential-determining reactions of the synthesis of compounds. The synthesis of quaternary compounds was carried out from the calculated amounts of binary phases in the positive electrodes of the cells with the participation of the Ag⁺ catalyst. For the first time, the values of standard thermodynamic functions (Gibbs energies, enthalpies, and entropies) of quaternary compounds were calculated based on the temperature dependences of the EMF of electrochemical cells. The variation of the composition of ternary and quaternary compounds within the homogeneity regions opens wide possibilities for changing their physicochemical properties.