Thermodynamic Evaluation and Optimization of the Ag-As-S system

A critical evaluation of all available phase diagram and thermodynamic data has been performed for Ag-As, Ag-S, As-S and Ag-As-S systems, including a detailed review of the gaseous species involved in the As-S system. Thermodynamic assessments over the whole composition range for these four systems are presented using the CALPHAD method. To predict thermodynamic properties and phase equilibria, the Modified Quasichemical Model for short range ordering was used for the liquid phases, and the Compound Energy Formalism was used for the solid solutions. For the As-S binary system, natural logarithm terms for the temperature dependence of the excess Gibbs energy of the liquid solution have been used. This led to a significant improvement compared with the previous assessment of this system. The optimization of Ag-As-S systems is in good agreement with the existing experimental data.


Introduction
Over the last decades, the demand for various metals has increased rapidly leading to a growth in their production. The high demand is accompanied by a gradual decline in high-grade ores treated in smelters and an evident rise in complex ores containing high arsenic contents. [1][2][3] This is concerning since arsenic is a toxic impurity that contaminates the surrounding environment and increases the risk of developing health problems. [4] To help the industry develop solutions to overcome the arsenic emission problem, the thermodynamic interaction between arsenic and other elements contained in ores entering the smelting process must be understood. Many transition metal sulfides are found in complex ores, particularly Ag 2 S. which is most often associated with lead ores and can also be found in small amounts in copper ores. [5] Ag 2 S is important for many operations like the manufacturing of photovoltaic cells and it can also be utilized as an electrode material for various applications. [6,7] To study complex systems contained in sulfide ores, it is necessary to develop thermodynamic models of compounds and mixtures for Ag-As-S systems and their associated databases. In this work, the available experimental data of Ag-As, Ag-S, As-S and Ag-As-S systems are critically evaluated, and a set of model parameters is optimized for each system to predict thermodynamic properties and phase equilibria. The Modified Quasichemical Model in the Pair Approximation suggested by Pelton et al. [8,9] was used for the liquid phases, and the Compound Energy Formalism (CEF) suggested by Sundman and Å gren [10] was used for solid solutions. All calculations and optimizations in this work were performed with the FactSage thermochemical software. [11,12] The obtained results are compared with previous evaluations of Ag-As and Ag-S systems by Shi et al. [13] and As-S system by Prostakova et al. [14] .
2 Literature Review

Experimental Determination of Ag-As Binary Phase Diagram
The Ag-As binary system has a single intermediate hcp phase. The experimental studies measuring phase diagram data of this binary system are presented in Table 1. Details of these studies are also discussed and reviewed in this section. Two other studies, by Descamps in 1878 [15] and by Friedrich and Leroux in 1906 [16] were considered obsolete by Hansen and Anderko [17] and were not considered in his compiled phase diagram. These two studies were also not examined in this work.

Ag-As System
Solid solubility: According to the phase diagram proposed by Heike and Leroux, [18] the maximum solubility of arsenic in silver is constant at 7 at.% As at all temperatures below 868 K. Measurements made by Owen and Rowlands [20] and Owen and Morris [21] show that there is a variation in the maximum solubility of As in Ag as a function of temperature, between approximately 4.5 at.% As at 573 K to 8.8 at.% As at 868 K. The results of Eade and Hume-Rothery [22] confirm the variation in maximum solubility as a function of temperature. However, the proposed maximum solubility is about 0.3 at.% As lower than the values of Owen and Rowlands [20] at 573 K and about 1 at.% As lower at higher temperatures. For the solid solubility of Ag in As, it has been concluded by Heike and Leroux [18] and Broderick and Ehret [19] that the solubility of silver in arsenic is negligeable. Eutectic: According to Heike and Leroux, [18] the eutectic reaction takes place at a temperature of 813 K and a composition of 25.3 at.% As to form solid As and an hcp phase (n). Liquidus: Liquidus curves on the silver-rich side have been determined by Heike and Leroux [18] and Eade and Hume-Rothery [22] and they are in good agreement. The liquidus curve on the arsenic-rich side has also been measured by Heike and Leroux [18] under pressure.

Hcp Phase (nÞ
The Ag-As system contains a single hcp phase (nÞ. The phase is stable from 647 K (n Ag ? As) to 868 K (Liquid ? Ag n) according to Heike and Leroux, [18] with a likely composition of 10.5 at.% As. However, Broderick and Ehret [19] reported a eutectoid temperature greater than 663 K, which does not agree with the temperature measured by Heike and Leroux. [18] The eutectoid temperature according to Eade and Hume-Rothery [22] takes place between 716 and 723 K and the peritectic temperature between 852 and 858 K, in contrast with the results of Heike and Leroux. [18] At 813 K, King and Massalski [23] measured the compositions of 10.5 and 12.5 at.% As as the limits of the n phase. Hall [24] could not accurately measure the limits of the n phase but proposed that it corresponds to a formula of Ag 9 As. Hall [24] was also able to find that the phase is not stable below 729 K, contrary to the temperature reported by Heike & Leroux. [18] No thermodynamic information has been reported for this phase.

Experimental Determination of Thermodynamic Properties of Ag-As Binary System
For temperatures between 791.8 K and 932.1 K, Lüdecke et al. [25] used the pseudo-isopiestic method to measure the vapor pressure above Ag-As alloys for compositions up to 8 at.% As. The data were then used to determine the activity of As in solid Ag-As alloys, with respect to solid As as the standard state. It was concluded that As and As 3 molecules are not present in the vapor. Additionally, it was observed that the vapor above the very low-concentrated alloys consists mainly of As 2 molecules, and as the alloy Table 1 Ag-As phase diagram data reported in the literature References Experimental methods and results [18] TA and microstructural analysis: Determination of the phase diagram for the entire composition range [19] X-ray diffraction: Determination of the solid solubility of As in Ag and the stability compositions of the hcp solid solution (nÞ [20] X-ray diffraction: Determination of the solid solubility of As in Ag [21] X-ray diffraction: Determination of the solid solubility of As in Ag [22] TA, microstructure analysis and x-ray diffraction: Determination of the phase diagram up to a composition of 20 at.% As [23] X-ray diffraction: Determination of the stability compositions of the hcp solid solution (nÞ [24] X-ray diffraction and microscopic method: Determination of the stability compositions of the hcp solid solution (nÞ composition in As increases, As 4 molecules become increasingly dominant in the vapor. Table 2 Ag-S phase diagram data reported in the literature References Experimental methods and results [28] TA and metallography: Determination of the phase diagram up to a composition of 33 at.% S [33] TA: Determination of the phase diagram up to a composition of 33 at.% S [32] TA and microphotography: Determination of the phase diagram up to a composition of 33 at.% S [27] TA and DTA: Determination of the phase diagram for the entire composition range [34] Gaz-metal equilibria: Determination of the liquidus and the phase boundary of the miscibility gap between compositions of 5 and 31 at.% S [36] Emf: Determination of the solubility of Ag in Ag 2 S between 433 K and 573 K [29] Tracer method with the radio isotope 35 S: Determination of the solid solubility of S in Ag [30] Microscopic investigations, conductivity measurements and sulfide formation temperature measurements: Determination of the solid solubility of S in Ag [37] H 2 S=H 2 method and dew point method: Determination of the homogeneity range of Ag 2 S [38] Emf: Determination of the homogeneity range of Ag 2 S betweeen 296 K and 449 K [39] Emf: Determination of the homogeneity range of Ag 2 S between 466 K and 823 K [31] Coulometric method: Determination of the solid solubility of S in Ag [40] Dew point method: Determination of the solubility of S in Ag 2 S between 503 K and 742 K [41] Electrochemical investigation: Determination of the solubility limit of Ag 2 S at 723 K [35] Emf: Determination of the eutectic temperature, the monotectic temperature and the phase boundary of the miscibility gap up to a composition of 32 at.% was stated by Jeannot et al. [26] , no indication of Ag 4 S was found. [32] 2.4.1 Ag 2 S-S Subsystem Solid solubility: No information has been reported regarding the solid solubility of silver in sulfur. It is most likely to be negligeable according to. [42] Monotectic: The Ag 2 S-S subsystem forms a miscibility gap and the monotectic temperature is 1013 K according to Kracek [27] and 1017 K according to Rau. [37] There has been no experimental determination of the phase boundary of the miscibility gap. Eutectic: The eutectic reaction that forms S and Ag 2 S occurs at a temperature of 392 K according to

Ag 2 S Compound
The natural form of Ag 2 S is acanthite with a monoclinic structure. A reversible polymorphic transformation takes place to form argentite which has a body centered cubic structure (bcc). [43] In mineralogy, argentite refers to a pseudomorph of acanthite after argentite. [44] A third polymorphic transformation takes place to form a face-centered cubic phase (fcc). [45] The transformation temperatures reported in the literature for are shown in Table 3. The transformation temperatures of Ag 2 S vary with composition, especially at higher temperatures, which indicates its non-stoichiometry. According to the experiments performed by Kracek, [27] the Ag 2 S compound undergoes the first polymorphic transformation at a temperature of 450:8 AE 0:7 K with excess sulfur and 449:3 AE 0:5 K with excess silver. The second polymorphic transformation varies largely with composition: 895 AE 3 K with excess sulfur and 859 AE 3 K with excess silver. [27] Ag 2 S melts congruently and the reported melting temperatures are also presented in Table 3. According to Hansen and Anderko, [17] the lower values of the melting point reported can be explained by the insufficient protection against oxidation or a small excess of silver or sulfur. A narrow Ag 2 S homogeneity range is suggested, and the different results are in very good agreement. Wagner [36] determined the solubility of Ag in Ag 2 S by meausring the electomotive force (emf) of the cell Ag=AgI=Ag 2 S/Pt for temperatures  from 433 to 573 K. Using the H 2 S=H 2 and the dew point methods, Rau [37] constructed a defect model from S 2 fugacity measurements and calculated the complete homogeneity range of Ag 2 S. Bonnecaze et al. [38,39] also determined the homogeneity range from 296 to 449 K and 466 K to 823 K, and the non stoichiometry was controled by coulometry using the cell Ag=RbAg 4 I 5 =Ag 2þe S=Pt: Ditman and Kulikova [40] determined the excess sulfur that Ag 2 S can solubilize by dew point method at temperatures of 503 AE 10 K and 743 AE 25 K. Finally, Reye [41] determined by an electrochemical investigation the solubility limit of Ag 2 S at a temperature of 723 K. The various results agree in that the low temperature form has nearly an ideal stoichiometric composition (Fig. 3). The heat of formation of Ag 2 S determined using a variety of techniques are listed in Table 4 and are in good agreement. Certain values were obtained by evaluation of the third law of galvanic cell data and equilibrium data for the following reactions: Several studies have been conducted to measure the heat capacity of Ag 2 S phases, and some results do not agree. Calorimetry 448 (175) 3.98 AE 0.04 [28] TA 448 (175) … [33] TA 448 (175) … [47] TA 448 (175) … [48] XRD 453 (180) … [49] P-T studies, Clausius-Clapeyron 449 (176) 6.38 AE 2 [50] Adiabatic calorimetry 450 (177)  Drop calorimetry 859 (586) 0.50 AE 0.54 [53] Adiabatic calorimetry 865 (592) 0.784 AE 0.005 [55] DTA 867 (594) … [57] DSC and dilatometry 863 (590)AE 1 1.2 AE 0.3 [58] XRD 850 (577)-860 (587) … [59] Adiabatic TA 1098 (825) … [47] TA 1107 (834) … [61] Freezing-point-depression 1111 (838) 14.23 [52] Drop calorimetry 1103 (830) 7.87 AE 0.6 [62] Adiabatic calorimetry 1113 (840)AE 2 7.66 AE 0.23 [53] Adiabatic calorimetry 1109 (836) … This work … 1117 (898) 15.7 Over the range 323 K to 673 K, Perrott and Fletcher [50] measured the heat capacity of the stoichiometric Ag 2 S using adiabatic calorimetry. They also measured the heat capacity of non-stoichiometric Ag 2 S containing 1 mol % excess silver or 1 mol % excess sulfur. The absolute accuracy of their measurements was estimated as AE 10%. As can be seen from Fig. 4, the data measured for the stoichiometric compound differ greatly from the other measurements. The main difference comes from an additional transition reported at 623 K. This was explained by the partial disordering of the stoichiometric compound, analogous to that observed for AgI. Gr nvold and Westrum [53] rejected this explanation and suggested that this non reproducible transition might be associated with the presence of another phase in their sample. The heat capacity of Ag 2 S was also measured using calorimetry by Bellati and Lussana [46] for temperatures from 320 to 490 K, by Gultyaev and Petrov [63] from 80 to 300 K, by Walsh et al. [64] from 13 to 296 K, and by Jost and Kubaschewski [65] from 200 to 300 K. Okazaki and Takano [59] also measured the heat capacity using adiabatic scanning calorimetry from 350 to 1200 K. Thompson and Flengas [52] also determined the heat capacity from 448 to 997 K using drop calorimetry. Gr nvold and Westrum [53] performed the measurements from 5 to 1000 K using adiabatic-shield calorimetry with a sample with stoichiometric composition and a sample with Ag 2 S 1:0526 composition. The results of both samples agree and only the values obtained for the stoichiometric composition are shown in Fig. 4. Gusev and Sadovniko [57] measured the heat capacity for a coarse-crystalline sample and for nanoparticles. The difference in particle size resulted in different [34]   heat capacity values. A good agreement was obtained between the results of Thompson and Flengas 1971, [52] Grønvold and Westrum, [53] and Gusev and Sadovnikov, [57] showing a slight decrease over the range of 460 K to 620 K. The experimental data of Grønvold and Westrum [53] are the most reliable according to Sadonikov and Gusev. [66] The standard entropy obtained from low temperature heat capacity measurements and second law evaluation of galvanic cell measurements are presented in Table 4. According to the mass spectroscopic studies of Glazov and Korenchuk [76] and Dutchak et al., [77] Ag 2 S thermally decomposes according to the following reaction: Vapor pressure measurements were also performed using the Knudsen effusion method. Piacente et al. [78] also measured the vapor pressure using torsion and Knudsen effusion methods and found no significant difference using cells of different materials (graphite, pyrophyllite and quartz). Smith and Smeltzer [79] carried sulfidation tests of iron using Ag 2 S and reported a P S 2 value for a temperature of 1023 K. The vapor pressure data are presented in Fig. 2.

Experimental determination of thermodynamic properties of Ag-S binary system
Rosenqvist [34] determined the partial pressure of S 2 by measuring the ratio H 2 S=H 2 of the gas in equilibrium with Ag-S alloys. The measurements were performed for alloys with compositions up to 33.3 at.% S, between 773 and 1573 K, according to the following equilibrium: The composition of the gas mixture was determined from its density obtained by magnetic balancing. Fukatsu and Kozuka [35] determined the emf for the following cell: Pt; AirjZrO 2 ðþCaOÞ|Ag-S, SO 2 ; Ir; Kanthal, and from the measured oxygen partial pressures, the partial pressure of S 2 was calculated between 867 and 1538 K, for compositions up to 32 at.% S. Yazawa, et al. [80] also measured the partial pressure of sulfur over Ag-S liquid alloys up to 33 at.% S. The measurements were performed between 1273 [50], Different runs for the stoichiometric Ag 2 S [50], Excess Ag [59], Experimental [57], Coarse [57], Nanopowder  and 1473 K by bubbling nitrogen through the melt, and the sulphur content was determined by chemical analysis. The different measurements of the partial pressure of S 2 are in good agreement.

Experimental Determination of As-S Binary Phase Diagram
The As-S binary system contains three stoichiometric intermediate compounds stable at ambient pressure: As 2 S 3 , As 4 S 4 et As 4 S 3 . Phase diagram data for this binary system are challenging to measure. For compositions lower than 45 at.% As, the measurements are hard to record due to the glass formation tendency of the alloys. And for compositions higher than 45 at.% As, the thermodynamically stable As 4 S 3 tends to not crystallize from synthetic melts, creating a metastable system. The experimental studies measuring phase diagram data are summarized in Table 5 and discussed in detail in this section.

S-As 2 S 3 Subsystem
As-S alloys with compositions lower than 45 at.% As are highly viscous according to the experiments of Jonker. [81] Since the liquid-solid equilibrium is reached very slowly, it was not possible to measure the liquidus curve of the S-As 2 S 3 subsystem using thermal analysis. Therefore, Jonker [81] estimated the liquidus curve by comparing the viscosities of several alloys with that of As 2 S 3 at its melting temperature. Thus, at temperatures between 473 and 673 K, the viscosities of pure sulfur and alloys at the compositions of 5, 10, 20 and 28.6 at.% As were evaluated quantitatively. It was possible to estimate two points of the liquidus curve. Jonker [81] also analytically determined the vapor curve in equilibrium with boiling liquid of known compositions and identified an azeotropic point at the composition of As 2 S 3 . It was concluded that the vapor coexisting with the liquid is rich in sulfur. Blachnik et al. [87] also failed to measure the liquidus curve due to the high alloy viscosities in this subsystem.

As 2 S 3 -As 4 S 4 Subsystem
According to measurements by Jonker, [81] a eutectic reaction takes place at a temperature of 565 K to form As 2 S 3 and As 4 S 4 at a composition around 43 at.% As. It was also concluded that the vapor coexisting with the liquid is rich in arsenic. However, according to Fedorova and Epova, [88] As 4 S 4 is incongruent and decomposes at temperatures above 569 ± 2 K. A monotectic reaction occurs at a temperature of 576 ± 2 K. According to the suggested phase diagram of Blachnik et al., [87] As 2 S 3 and As 4 S 4 form by a eutectic reaction at a temperature of 573 K for a composition around of 43 at.% As.

As 4 S 4 -As Subsystem
According to the work of Jonker, [81] an invariant equilibrium between the gaseous phase, the liquid phase and solid arsenic exists at an estimated temperature of 807 K at 52.2 at.% As. Since the experiments were performed at atmospheric pressure, no liquid phase was formed for alloys very rich in arsenic due to the sublimation. For this subsystem, the vapor is richer in arsenic relative to the liquid in equilibrium. Clark [82] estimated that at a temperature of 873 K, the saturated liquid contains 60 ± 0.2 at.% As. He also confirms that the vapor phase is richer in arsenic compared to the liquid in equilibrium. According to the results obtained by Barton, [83] a monotectic reaction takes place at a temperature of 1070 ± 5 K. The miscibility gap of two liquids forms between the compositions of 83 ± 3 and 97 ± 3 at.% As. According to Ceolin et al. [85] , the monotectic reaction involves a vapor phase in addition to solid arsenic and the two liquids. The reaction takes place at a temperature of 1056 ± 1 K, with an upper critical temperature of 1210 K. The metallographic examination Table 5 As-S phase diagram data reported in the literature

References
Experimental methods and results [81] TA, analytical and quantitative analysis: Determination of the phase diagram for the entire composition range at atmospheric pressure [82] TA: Determination of the liquidus temperature of an alloy of the sub-system As 2 S 2 -As at unspecified pressures above atmospheric pressure [83] DTA: Determination of the phase diagram between compositions of 80 to 100 at.% As [84] DTA, XRF and electron microscopy: Determination of the phase diagram up to a composition of 50 at.% As, with crystallized alloys from a pressure of 70 kbar and a temperature of 673 K [85] DTA: Determination of the phase diagram between compositions of 50 to 100 at.% As [86] TA: Determination of the phase diagram up to a composition of 40 at.% As [87] DSC: Determination of the metastable phase diagram up to a composition of 40 at.% As High temperature XRD: Proposition of a stable phase diagram for the entire composition range confirms the presence of the miscibility gap. The monotectic reaction takes place at a temperature of 1048 ± 3 K according to the experiments of Fedorova and Epova. [86] Also, the formation of As 4 S 3 was reported for the first time by Fedorova and Epova. [86] The compound is incongruent and stable only at temperatures below 404 ± 3 K. The eutectic reaction between As 4 S 4 and As takes place at a temperature of 452 ± 4 K and at a composition around 56 at.% As. The limits of the miscibility gap could not be determined by Blachnik et al. [87] due to technical difficulties. The liquidus curves of Blachnik et al. [87] and Ceolin et al. [85] agree, unlike the curve measured by Barton [83] found at lower arsenic compositions. The phase diagram measured by Blachnik et al. [87] does not show the formation of the thermodynamically stable As 4 S 3 . The metastable eutectic temperature forming As 4 S 4 and As is 515 K, while the eutectic temperature forming As 4 S 3 and As 4 S 4 in the suggested stable phase diagram is 473 ± 4 K.

As 2 S 3 Compound
The natural crystalline form of As 2 S 3 is orpiment. At high pressures, the crystalline form undergoes two polymorphic transformations according to the experiments of Timofeeva et al. [84] , at 463 K and 483 K. At atmospheric pressure As 2 S 3 occurs only in one form. [89] The synthesized As 2 S 3 is usually amorphous and almost impossible to crystallize. Also, As 2 S 3 melt is a strong glass-forming liquid at ambient pressure, with a viscosity of 8000 Pa:s near the melting temperature. [90] Pure As 2 S 3 crystals cannot be prepared from As 2 S 3 melts. However, the amorphization of As 2 S 3 crystals is relatively easy. The optically induced transition of crystalline As 2 S 3 to the amorphous state studied by Frumar et al. [91] results in a glass transition temperature T g of 453 K. Slightly higher T g values have been reported for synthetic samples. [89,[92][93][94] The temperatures and heats of fusion for crystalline As 2 S 3 are presented in Table 6. Espeau et al. [89] measured and compared the melting temperatures obtained from synthetic and natural samples. According to the DSC results, with a heating rate of 5 K. min À1 , the average melting temperature obtained for the synthetic samples is 601 K, while the average for the natural samples of two different sources is 596 K and 591 K. Even though elemental analyzes showed that the synthetic sample and the natural samples have the same composition, Espeau et al. [89] suggests that the difference may be due to the higher purity of the synthetic sample. The average heat of fusion DH fusion ð Þ for the synthetic samples is 26.7 kJ/mol, and for natural samples from the different sources are 29.8 kJ/mol and 27.4 kJ/mol.
Since the synthesized As 2 S 3 is usually glassy and difficult to crystallize, several studies have used natural orpiment to determine its thermodynamic properties. Č ernošek et al. [98] successfully synthesized high purity As 2 S 3 crystals for the first time in 1999 and questioned the purity of the natural samples used in the previous studies. As can be seen from Table 7, there is a large discrepancy between the suggested heat of formation values by the different sources, ranging from -83 kJ/mol to -169 kJ/mol. Britzke et al. [99] obtained the first enthalpy of formation value of -145 kJ/mol by combustion calorimetry in oxygen. From the data of Britzke et al. [99] , Wagman et al. [100] recalculated a value of -169 kJ/mol, and Mills [101] a value of 159 ± 42 kJ/mol. Neverov et al. [102] calculated two enthalpy of formation values from the reactions of arsenic sulfide with an alkaline solution of NaOH saturated with bromine as well as with an alkaline solution of NaOH saturated with chlorine. The values

References
Experimental methods Temperatures, in K (in°C) DH Fusion kJ/mol As 2 S 3ðmP20Þ ! As 2 S 3ðliquidÞ [81] TA 583 (310) … [24] TA 590 (317) [95] DTA 583 (310)AE 5 … [84] DTA 583 (310 ÞAE 10 … [96] DSC 588 (315)AE 5 28.7 AE1; 3 [86] DTA 591 (318) … [87] DSC 581 (308) 30.05 [97] DTA 591 (318) … [98] DSC 580 (307) 19.19 [89] DSC obtained are -138 ± 12 kJ/mol and -126 ± 10 kJ/mol respectively. Barton [83] does not agree with such negative values of DH formation of orpiment and estimated a value of -96.23 kJ/mol from his equilibrium study on As-Fe-S alloys. From the combustion energy of the glassy As 2 S 3 measured in fluorine by Johnson et al. [103] , a value of DH formation of the crystalline compound of -91.6 ± 4, 8 kJ/mol was derived, agreeing with the value of Barton. [83] This is also close to the value of -88.7 ± 3.8 kJ/mol recalculated by O'Hare. [104] Using high temperature calorimetry, a value of -83 ± 3.8 kJ/mol was obtained by Bryndzia and Kleppa. [105] Babanly et al. [106] calculated a value of -94.09 ± 2.11 kJ/mol from emf measurements, with a liquid electrolyte d KBr in glycol with 0.01% mass of Br 3 As and solid arsenic as reference electrode. Nordstrom and Archer [107] evaluated and fitted the different enthalpy of formation values reported in the literature with a weighted least-squares regression and suggested a value of 85.8 kJ/mol. Enthalpy of formation values for amorphous As 2 S 3 are also shown in Table 7. The heat capacity of crystalline As 2 S 3 was first measured at low temperatures by Romanovsky and Tarasov [108] using adiabatic calorimetry between 65 and 300 K. The heat capacity was also measured at higher temperatures by Blachnik et al. [87] using DSC between 300 and 650 K. For vitreous As 2 S 3 ; heat capacities were also measured at low temperatures by Hattori et al., [109] Kageyama et al., [110] Tarasov and Zhdanov, [111] Haggerty et al. [112] and Schnaus et al. [113] and are in reasonable agreement. Only the heat capacity data for the crystallin As 2 S 3 are presented in Fig. 4. The standard entropy calculated from these data are presented in Table 7.
The compound As 2 S 3 sublimates congruently, and the experiments determining the composition of its vapor are not in agreement. The electron diffraction study results suggested by Lu and Donohue [119] show that As 4 S 6 molecules are probably the only species present in the vapor state. The interpretation of the results of this study is very doubtful according to Rogstad. [120] Based on their mass spectrometric analyses, Janai and Rudman [121] also suggest that the compound evaporates as As 4 S 6 based on the appearance of all its possible fragments in the spectra. The composition of the vapor in equilibrium with As 2 S 3 was also studied by Ban and Knox [122] using mass spectrometry with laser-induced vaporization. It was concluded that vaporization of As 2 S 3 forms S 2 and As 3 S n species (n = 1, 2, 3, 4), with abundance of As 3 S and As 3 S 4 species. According to the mass spectrometry by Faure et al. [123] , the vaporization of As 2 S 3 forms gaseous AsS and S 2 The mass spectrum was obtained by scanning the ion accelerating voltage at a constant magnetic field, instead of the usual practice of varying the magnetic field at a constant ion accelerating voltage. This method used to obtain the Oxygen bomb calorimetry -145.6 … … [108] Adiabatic calorimetry … 163.18 116.2 [83] High-temperature equilibria -96.2 163.2 … [103] Fluorine combustion calorimetry -91.6 AE 4.8 … … [87] DSC … … 114.9 [114] Solubility measurements -86.5 … … [100] Recalculated from [99] -169 … … [105] High-temperature calorimetry -83 AE 3.8 … … [102] Calorimetry -138 AE 12 … … [102] Calorimetry -126 AE 10 … … [115] cited in [102] … -146 … … [116] cited in [102] … -126 … … [104] Recalculated from [99] -88.7 AE 3.8 … … [117] … -91.6 AE 5 … Amorphous compound [103] Fluorine combustion calorimetry -69.6 AE 4.2 … … [118] Solubility measurements -88.1 AE 5 … … [107] Weighted least-squares regression -66.9 200 76.8 spectrum does not allow the identification of species with higher masses according to Brittain et al. [124] . Moreover, according to Chakraborti and Lynch, [125] the presence of AsS species in their results can be associated with the destruction of larger molecules due to the high ionization potentials used. The high temperature mass spectrometry by Pashinkin et al. [126] reveals the following gaseous specie: As 2 S 2 ; As 3 S 3 and As 4 S 4 . Since the measurements were made with ionizing electron energies of the order of 50 eV, the results also reflect the presence of products associated with the destruction of larger molecules according to Brittain et al. [124] and Chakraborti and Lynch. [125] High temperature mass spectrometry performed by Brittain et al. [124] was obtained with an ionizing potential of 15 eV. The results show that the vaporization of the compound As 2 S 3 mainly produces the following gaseous species: As 4 S 4 , As 4 S 3 , S 2 and a minor amount of As 4 S 5 . It was also concluded that the vapor does not contain AsS; As 2 S 2 and As 3 S 3 . Thus, the process of vaporization is mainly done according to the following reactions: The ion intensity data indicate that reaction 5 predominates over reaction 6. Also, the following reaction contributes slightly to the process: According to the results obtained by Brunetti et al. [127] , reactions 5 and 6 take place independently of temperature. Several experiments determined the vapor pressures of As 2 S 3 . Figure 5 shows data from each experiment. Using the Knudsen-Langmuir technique, Hsiao and Schlechten [128] determined the vapor pressures for temperatures between 453 and 603 K. The vapor pressure values were labeled by Hsiao and Schlechten [128] as ''apparent'' because of the uncertainty as to how much is due to evaporation and how much to dissociation. Chakraborti and Lynch [125] rejected the values measured by Hsiao and Schlechten. [128] Isakova and Nesterov [129] determined the  When processing experimental data, it was assumed that only high dosage As 2 S 3 molecules are in the vapor phase.
Since the obtained vapor pressures measured are very close to those obtained for As 4 S 4 ; Chakraborti and Lynch [125] suggested the presence of a small amount of As 4 S 4 in the solid sample used by Isakova and Nesterov. [129] Isakova [130] determined the vapor pressures using flux and boiling point methods between 623 and 773 K. Ustyugov et al. [131] determined the vapor pressures with the static method between 729 and 966 K using a quartz membrane manometer. Faure et al. [123] determined the vapor pressures with the Knudsen method between 497 and 569 K, and Novoselova and Pashinkin [132] expressed the vapor pressure as a function of temperature. Gospodinov and Pashinki [133] determined the vapor pressures with the Knudsen method between 476 and 536 K, and the Langmuir method between 470 and 546 K. Brunetti et al. [127] determined the total vapor pressures with the torsion-effusion method between 501 and 583 K. The boiling temperature of the compound As 2 S 3 is around 980 K according to Jonker [81] and it corresponds to an azeotropic point.

As 4 S 4 Compound
As 4 S 4 Is found in nature in a crystalline form called realgar and it can easily change to orpiment with light exposure. [120] The formula of realgar is also given by AsS or As 2 S 2 in some references. Since the mineral consists of As 4 S 4 rings, [134] the formula considered in this work is As 4 S 4 . Realgar undergoes a polymorphic transformation to form pararealgar. Since both forms are monoclinic, [135] the transformation enthalpy is very small. The experiments performed by Rolland [136] show that the transformation temperature for the stoichiometric compound occurs at 525 K and it varies between 512 and 536 K in the presence of elemental As and As 2 S 3 ; respectively. The compound has a very narrow homogeneity range, [136] and it is still not experimentally investigated. The measured transformation temperatures are shown in Table 8. Hall [24] defined three distinguishable optical properties of As 4 S 4 , and more recently, a third form was reported by Bindi et al., [137] analogous to the new mineral bonazziite found in Khaidarkan deposit in Kyrgystan. The As 4 S 4 compound is reported as congruent, except for the measurements of Fedorova et al. [138] that show an incongruent compound. The measured melting temperatures are shown in Table 8. The heat of formation values of As 4 S 4 reported in the literature are presented in Table 9. Britzke et al. [99]
Several experiments studied the composition of the vapor in equilibrium with As 4 S 4 . By determining the vapor density at several temperatures, Szarvasy and Messinger [149] concluded that the vapor contains mainly As 4 S 4 molecules at temperatures below 823 K, and at higher temperatures, the molecules dissociate into As 2 S 2 . At temperatures above 1473 K, Szarvasy and Messinger [149] suggested that another dissociation might occur. Castro et al. [150] confirm that the gaseous As 4 S 4 dissociates in two stages at high temperatures to As 2 S 2 and AsS: According to the results of the mass spectrometry performed by Munir et al. [151] , the vaporization of synthetic and natural realgar produces a vapor that mainly contains As 4 S 4 species at high temperatures. The following species were also found in minor quantities: As 4 S 3 , As 4 and As 2 , and their presence decreases with increasing temperature. Therefore, it was concluded that the process of vaporization takes place according to the following reaction: This agrees with the results of Brittain et al. [124] who used mass spectrometry and mass-torsion effusion. It was concluded that at least 90% of the vapor produced following the sublimation of realgar comprises the gaseous As 4 S 4 species. Also, it was determined that the vapor pressure of the two forms, a-As 4 S 4 and b-As 4 S 4 , are similar. Various experiments determined the vapor pressures of As 4 S 4 . The measured data are shown in Fig. 5. Using spoon gauge, Strathdee and Pidgeon [152] performed vapor pressure measurements for temperatures between 573 and 773 K. Ustyugov et al. [131] also determined the vapor pressures for temperatures between 663 and 838 K using the same technique. The data obtained by Ustyugov et al. [131] have a systematic error caused by the non-uniformity of the thermal field according to Novoselova and Pashinkin. [132] This is due the large sizes of the membrane chambers. High-temperature calorimetry -108.8 … … [102] Calorimetry -236 … … [104] Recalculated from [99] -127.5 AE5:3 … … [117] … -138.0 AE6:8 … … [87] DSC … … 190 [143] DSC … … 189 [107] Weighted Munir et al. [151] performed vapor pressure measurements using torsion-effusion for temperatures between 452 and 534 K and using torsion-Langmuir between 411 and 465 K. Godovikov et al. [144] also performed vapor pressure measurements using a quartz membrane manometer for temperatures between 614 and 814 K. Their pressures are lower than those measured by Strathdee and Pidgeon [152] and Ustyugov et al. [131] Finally, measurements were performed by Pashinkin et al. [153] between 470 and 580 K using the Knudsen method, and the results are in good agreement with those by Munir et al. [151] .

As 4 S 3 Compound
The natural form of As 4 S 3 is dimorphite. There is few thermodynamic information in the literature for this compound and the data available are inconsistent across the sources. As 4 S 3 undergoes a polymorphic transformation and the measured transformation temperatures are shown in Table 10. The orthorombic structure of the synthetic forms has been determined by Whitfield, [154,155] and has been confirmed for the natural forms by Gavezzoti et al. [156] . Since both forms are orthorombic, the transformation enthalpy is very small. [87] Another modification from crystalline to plastic-crystalline has been found by Blachnik et al. [87] and Chattopadhyay et al. [157] . It is probably of tetragonal symmetry similar to the one found in P 4 S 3 and it possibly only occurs under non-equilibrium conditions according to Blachnik et al. [87] . The compound is reported as congruent and the measured melting temperatures are shown in Table 11. Fedorova and Epova [86] measured a much lower melting point of 404 AE 3 K which is incongruent with no polymorphic transformation. Based on emf measurements, Babanly et al. [106] calculated a DH formation value of -125.5 kJ/mol for As 4 S 3 . Using the second law of thermodynamics, Pashinkin et al. [158] calculated a value of -111.1 kJ/mol. A lower value of -82.4 kJ/mol from Isabaev et al. [159] was cited by Prostakova et al. [14] without reporting the measurement method. The heat capacity of As 4 S 3 was first measured by Blachnik et al. [87] using DSC for temperatures between 130 and 600 K. These measurements are uncertain because the sample has undergone partial evaporation according to Chattopadhyay et al. [143] Measurements were also performed using DSC by Chattopadhyay et al. [143] between 100 and 400 K. Pashinkin et al. [158] approximated the heat capacity of As 4 S 3 by structural analogy to P 4 S 3 . As shown in Fig. 4, the heat capacity values from each source agree DSC … … 166.67 [143] DSC … … 165.4 [159] cited in [14] … only in a limited temperature range. The standard thermodynamic properties of As 4 S 3 are presented in Table 11. The vaporization behavior of solid As 4 S 3 is still not well established in the literature. As reported by Pashinkin et al. [158] , the process of vaporization of As 4 S 3 may occur according to the following reaction: Godovikov et al. [138] measured the saturation vapor pressure of As 4 S 3 at temperatures above its melting temperature, between 600 and 780 K. Using the Knudsen method, Pashinkin et al. [158] measured the vapor pressure of the metastable As 4 S 3 plastic phase between 400 and 430 K and the stable plastic phase between 434 and 481 K. Pashinkin et al. [158] also measured the vapor pressure of As 4 S 3ðoP28Þ þ As 4 S 3ðoÃÃÞ for temperatures between 402 and 429 K, and of As 4 S 3ðoP28Þ between 394 and 418 K using the Langmuir method. All measured data are presented in Fig. 5.

Thermodynamic Properties of the Gas in As-S Binary System
In addition to the complexity of the vaporization mechanism of As-S system, there is still a gap in the thermodynamic information available for the gaseous species. In fact, the thermodynamic properties for As 4 S 5ðgÞ and As 2 S 3ðgÞ are still unavailable in the literature. Mah [145] calculated the standard enthalpy of formation for the predominant species As 4 S 4ðgÞ and AsS ðgÞ but the values have significant uncertainties. A standard enthalpy of formation of 0.189 AE 0.038 kJ/mol for AsS ðgÞ was obtained based on the dissociation energy of 0.385 ± 0.038 kJ /mol from Sullivan. [160] For gaseous As 4 S 4ðgÞ , a standard enthalpy of formation of -2.092 AE 10.5 kJ /mol was obtained based on the sublimation studies of Munir et al. [151] and Gospodinov and Pashinkin. [133]

Experimental Determination of Ag-As-S Ternary Phase Diagram
Several minerals have compositions within the Ag-As-S ternary system, including proustite and xanthoconite ðAg 3 AsS 3 Þ which are very abundant, smithite and trechmannite (AgAsS 2 ) as well as a rare sulfosalt bearing called billingsleyite (Ag 7 AsS 6 ). Just like the system As-S, the Ag-As-S system tends to form as a glass under certain conditions. Measurements of phase diagram data of this system are presented in Table 12.

Ag 2 S-As 2 S 3 System
In the ternary Ag-As-S system, the quasi-binary Ag 2 S-As 2 S 3 is the most important experimentally and mineralogically. [165] Jaeger and van Klooster, [33] Elli and Giudici [161] and Kovaleva et al. [163] evaluated the Ag 2 S-As 2 S 3 phase diagram and reported two congruently melting compounds along this system: Ag 3 AsS 3 and AgAsS 2 : The measurements could not be performed for the entire composition range due to glass formation at high As 2 S 3 compositions. Solid solubility: No solid solubility has been reported in the Ag 2 S-As 2 S 3 system. Eutectic: According to Jaeger and van Klooster, [33] the eutectic reaction that forms Ag 2 S and Ag 3 AsS 3 takes place at a temperature of 742 K and a composition of 29 at.% As 2 S 3 . The eutectic reaction that forms Ag 3 AsS 3 and AgAsS 2 takes place at a temperature of 672 K and a composition around 58 at.% As 2 S 3 . This agrees with the eutectic measurements of Elli and Giudici [161] and Kovaleva et al. [163] Liquidus: The liquidus of Ag 2 S-Ag 3 AsS 3 measured by Kovaleva et al. [163] is slightly higher than the one measured by Jaeger and van Klooster. [33] The liquidus measurements of Ag 3 AsS 3 -AgAsS 2 determined by Jaeger and van Klooster [33] and Kovaleva et al. [163] are in good agreement and are higher than the liquidus measured by Wehmeier et al. [162] Gaudin and McGlashan [170] proposed a tentative Ag 2 S-As 2 S 3 phase diagram under non equilibrium conditions based on selective iridescent filming and reported an additional incongruent phase ''A''. Sommerlad [171] reported the synthesis of Ag 5 AsS 4 , and Hall [24] also reported an additional intermediate phase along this quasi-binary system. The phase has possibly a composition of Ag 5 AsS 4 by analogy with stephanite (Ag 5 SbS 4 ). [24] Roland [165] was not able to detect the formation of this phase.

Ag 3 AsS 3 Compound
Ag 3 AsS 3 Corresponds the mineral xanthoconite that undergoes a polymorphic transformation to form proustite as seen from Table 13. For this compound, Bryndzia and Kleppa [75] determined a standard heat of formation of 111:3 AE 3:4 kJ/mol using high-temperature calorimetry. This value is slightly lower than the one calculated by Gasanova et al. [172] from emf measurements of the following cell: ðÀÞPt Ag referenceelectrode ð Þ j j Ag 4 RbI 5 s ð Þ jðAg 2 S -As 2 S 3 -SÞjPtðþÞ. The measurements were made at temperatures from 300 to 380 K and the calculated enthalpy is 121:4 AE 2:8 kJ/mol. Using adiabatic calorimetry, Gorbunov et al. [173] and Gurevich et al. [174] measured the heat capacity of the compound in the 10 K to 350 K range and they are in very good agreement. The standard thermodynamic properties of Ag 3 AsS 3 are presented in Table 14.
Godovikov et al. [175] and Fedorova et al. [176] suggested that Ag 3 AsS 3 vaporizes incongruently according to the following reaction: Godovikov et al. [175] also calculated the vapor pressure as a function of temperature based on quartz manometer measurements for 846 K to 1013 K. Using the Knudsen method, Fedorova et al. [176] measured the vapor pressure for temperatures between 673 and 723 K. Godovikov et al. [138] also measured the vapor pressure for temperatures between 769 and 901 K using the isoteniscope method.

AgAsS 2 Compound
AgAsS 2 Is present in nature as the mineral trechmannite and it undergoes a polymorphic transformation to form smithite. The standard heat of formation of AgAsS 2 by Bryndzia and Kleppa [75] and Gasanova et al. [172] are in very good agreement. Using adiabatic calorimetry, Gurevich et al. [174] measured the heat capacity of the compound in the 10 K to 350 K range. The measured standard Table 12 Ag-As-S phase diagram data reported in the literature

References Experimental methods and results
Ag 2 S-As 2 S 3 [33] TA: Determination of the quasi-binary diagram up to a composition of 75 at.% As 2 S 3 [161] DTA and XRD: Determination of the quasi-binary diagram up to a composition of 58 at.% As 2 S 3 [162] DTA and XRD: Determination of the liquidus curves between compositions of 23 at.% and 67 at.% As 2 S 3 for annealed and annealed samples [163] DTA, XRD and microhardness: Determination of the quasi-binary diagram up to a composition of 70 at.% As 2 S 3 Ag 7 AsS 6 -Ag 3 AsS 3 [164] DTA, XRD and DSC: Prediction of the quasi-binary diagram for the entire composition range Ag 7 AsS 6 -Ag 2 S [164] DTA, XRD and DSC: Prediction of the quasi-binary diagram for the entire composition range AgAsS 2 -As 4 S 4 [165] Quenching experiments and XRD: Prediction of the quasi-binary diagram for the entire composition range in the presence of vapor Ag 3 AsS 3 -S [166] Quenching experiments and XRD: Determination of the cross section for the entire composition range in the presence of vapor Ag 3 AsS 3 -As [139] reproduced by [167] XRD: Determination of the eutectic temperature and the liquidus from 30 to 60% As Ag 0:5 As 0:5 -S [163] DTA, XRD and microhardness: Determination of the cross section for compositions from 30 to 58% S Ag 0:75 As 0:25 -S [163] DTA, XRD and microhardness: Determination of the cross section up to a composition of to 55% S Ag 0:5 As 0:5 -As [163] DTA, XRD and microhardness: Determination of the cross section up to a composition of to 45% As Ag 0:33 As 0:67 -As [163] DTA, XRD and microhardness: Determination of the cross section for compositions from 10 to 40% As Ag 2 S-As [139] reported by [168] Quenching experiments and XRD: Determination of the liquidus from 13 to 56% As [168] DTA and XRD: Determination of the cross section for compositions for the entire composition range Ag 3 AsS 3 -Ag [169] DTA and microhardness: Suggestion of the quasi-binary diagram up to a composition of 50% Ag thermodynamic properties and transformation temperatures are presented in Tables 14 and 15, respectively. Godovikov et al. [175] and Fedorova et al. [176] suggested that AgAsS 2 vaporizes incongruently according to the following reaction: AgAsS 2ðsÞ 1 3 Ag 3 AsS 3 ðsÞ þ 1 6 As 4 S 4ðgÞ þ 1 6 S 2ðgÞ ðEq 11Þ Using the Knudsen method, Fedorova et al. [176] measured the vapor pressure for temperatures between 575 and 623 K. Godovikov et al. [138] calculated the vapor pressure as a function of temperature based on quartz manometer measurements from 709 to 980 K. Using the same technique, Fedorova et al. [183] also calculated the vapor pressure for temperatures from 724 to 1013 K. The measurements of Godovikov et al. [138] and Fedorova et al. [183] are in good agreement.

Previous Thermodynamic Assessment of Ag-As-S Systems
The binary Ag-As system was previously assessed by Shi et al. [13] . The liquid phase was described by the substitutional solution model Ag; As ð Þ 1 , using the PARROT program of the Thermo-Calc software. [187] Shi et al. [13] also performed a thermodynamic assessment of the binary Ag-S system using the same program of the Thermo-Calc software. The liquid was modeled by the associate model Ag; Ag 2 S; S ð Þ 1 and Ag 2 S was considered as a stoichiometric compound. The sub-solidus phase equilibria in the As-S system were calculated by Emelina et al. [148] by constructing convex envelopes of primary thermodynamic surfaces using PhDi software. [188] The As-S system was also assessed by Prostakova et al. [14] with FactSage thermochemical software using the Modified Quasichemical Model (MQM) for the liquid phase similarly to the present work. However, different choices were made that distinguish the two optimizations. First, As 2 S 5 is obtained only at high pressures and Prostakova et al. [14] included it in their optimization. This compound is not considered in the present assessment. Also, different coordination numbers are set in this work. This choice is made because Liu et al. [189] investigated the structural stability of various As-S clusters based on DFT calculations and demonstrated that As 2 S 3 ð Þ n has the highest stability. The stability is due to the 4p-orbital in As atom binding with 3p-orbital of S atom, which decreases the energy level of the highest occupied molecular orbital. Additionally, Tsuchihashi et al. [190] studied the glasses in the As-S system, and the results show that the hardness, T g and the thermal expansion are maximum at 40 at.% As, which corresponds to the composition of As 2 S 3 . Hattori et al. [109] also reported that thermal conductivity of As-S glasses shows a maximum near the composition of As 2 S 3 . Since Tsuchihashi et al. [190] confirms that the SRO arrangement of glassy As 2 S 3 is similar to the crystalline structure, the maximum SRO is set in this work to occur at a composition of 40 at.% As. This selected SRO composition was also confirmed to be the most appropriate by studying the measurements made for the ternary Ag-As-S system as well as the favored oxidation state of arsenic. Thus, the values of the coordination numbers set in this work are Z As AsS =3 and Z S AsS =2, contrary to the values of Z As AsS =Z S AsS =3 used in the work of Prostakova et al. [14] . Finally, the most significant distinction between the two optimizations comes from the parameters of the excess Gibbs energy of the liquid solution. Linear terms to describe the liquid as used by Prostakova et al. [14] result in that the liquid is internally unstable at high temperatures and form inverted miscibility gaps, even if the real As-S liquid is not prone to separation. To correct the excess Gibbs energy of the liquid at high temperatures, CTln(T) terms were added in this work to avoid the formation of inverted liquid miscibility gaps in this system. TA 690 (417) … [162] DTA and XRD 686 (413) … [163] DTA and XRD 702 (429) … [184] Drop calorimetry 643 (370) 9.87 AE 0.84 This work … 696 (423) 7.32 Table 16 Enthalpies of formation of Ag-As-S ternary compounds predicted by DFT from the Open Quantum Materials Database [185] and the Materials Project [186] References Space group DH formation kJ mol -1 This led to an improvement compared with the previous assessment of this system.

Thermodynamic Modeling
All calculations and optimizations in this work were performed with the FactSage thermochemical software. [11,12] The properties of pure solid and liquid Ag, As and S are from SGTE. [191] The properties of the gaseous species S, S 2 , S 3 , S 4 , S 5 , S 6 , S 7 and S 8 are from JANAF Thermochemical Tables. [192] The properties of the gaseous species As, As 2 , As 3 and As 4 are from Degterov et al. [193] and other of their unpublished optimizations and evaluations.
The properties of the gaseous species AsS and As 4 S 4 are from Barin [194] based on the calculations of Mah. [145] The gaseous AgS is from Mills, [101] Ag is from Barin et al. [195] and Ag 2 is from Wagman et al. [196] 3.1 Thermodynamic Model for the Liquid Phase in the Binary Systems The Modified Quasichemical Model in the Pair Approximation suggested by Pelton et al. [8,9] was used to model the liquid phases to consider the short-range ordering (SRO). For a liquid binary A-B system, the following pair exchange reaction is considered between the A and B atoms on neighbouring lattice sites: Þ are the first nearest neighbor (FNN) atom pairs, and Dg AB is the non-configurational Gibbs energy change for the formation of 2 mol of A À B ð Þpairs. The pair and the overall mole (site) fractions are defined respectively as: n A , n B represent the number of moles of A and B, n AB represents the number of moles of pairs (A-B). If the coordination number of A is Z A , the ''coordination equivalent'' fraction of A is defined as: The ''coordination equivalent'' fraction of B is defined similarly. The Gibbs energy of the binary liquid is given by the following equation: Dg AB ðEq 16Þ g A and g B are the molar Gibbs energies of the pure liquid components, and DS configuration is the configurational entropy of mixing given by randomly distributing the (A-A), (B-B) and (A-B) pairs. The non-configurational Gibbs energy change for the formation of 2 mol of A À B ð Þpairs is expended in terms of the pair fractions: Dg AB , g i0 AB and g 0j AB are the temperature dependant model parameters that are optimized in this work for the Ag-As, Ag-S and As-S systems. The coordination numbers can be varied with composition to reproduce the SRO, as given by Eq 18 and 19. The composition of maximum SRO is determined by the ratio Z B / Z A : Z A AA and Z A AB are the coordination numbers when the nearest neighbours of an A atom are A atoms and B atoms respectively. Z B BB and Z B BA are defined similarly. In this work, Z Ag AgAg , Z As AsAs and Z S SS are set to be 6 for consistency with previous work.

Thermodynamic Model for the Liquid Phase in the Ternary System
Geometric models can be used to estimate the excess Gibbs energy of a ternary system from the parameters of the three binary subsystems. [197] For a given geometric model, the excess Gibbs energy of the ternary system at any composition p is calculated from the binary parameters at points a, b and c. The difference between the models is the placement of these three points which is very important to correctly interpolate the ternary system. After trying several extrapolation methods, the Kohler-Toop type extrapolation technique, with Ag as the asymmetric component, provided the most satisfactory description of the ternary system. It is then the selected technique in this work to optimize the ternary Ag-As-S liquid phase (Table 17).

Thermodynamic Model for Ag 2 S Solid Solutions
The low temperature monoclinic form is a semiconductor and has nearly an ideal stoichiometric composition. [43,198] The second bcc form has a high conductivity due to the increase of number of vacancies in the silver sublattice and the transfer of silver atoms to adjacent vacancy sites. Silver is distributed among the tetrahedrally and octahedrally coordinated positions. [199] At higher temperatures, the tetrahedral occupancy increases at the expense of the octahedral occupancy, and above 533 K the occupied sites are entirely tetrahedral. [199] The sulfur atoms arrangement stays unchanged. [199,200] Based on this information and by analogy to the model established by Waldner [201] for the Cu 2 S phase, a simplified sub-lattice model is used for Ag 2 S solid solutions and is presented in Table 7.

Results and Discussion
The optimization of the thermodynamic parameters was conducted using the FactSage thermochemical software. The Ag-As-S systems are assessed based on the literature data presented in the previous sections, and the optimized thermodynamic model parameters are presented in Table 18. The calculated Ag-As binary phase diagram is shown in Fig. 1, along with the experimental data. No information on the composition of maximum short-range ordering (SRO) of this system is reported. Nevertheless, analysis of the measured phase diagram data shows an increase of the slope of the tangent line of the liquidus curves around 20 at.% As. This indicates a strong maximum SRO around this composition. Thus, the coordination numbers Z Ag AgAs and Z As AgAs are set in this work to 1.5 and 6 respectively to reproduce SRO at 20 at.% As. There is also an inflection point on the liquidus of the arsenic rich side which is not uncommon. An inflection is considered traditionally as an indication of a tendency for an instable liquid phase at temperatures below the liquidus, meaning that it is prone to separation. It can also appear due to the properties of fusion of pure arsenic. [203] As seen from Fig. 1, the calculated eutectic and liquidus curves reproduce the measured data very well. However, silver solubilizes slightly less arsenic compared to the measured values below 625 K and above 900 K. The hcp phase (n) may not occur as a mineral according to Hall, [24] but it is still modeled in this work. The optimized Gibbs free energy of the phase is shown in Table 18. As explained in the previous section, there is a disagreement regarding the formation and the decomposition temperatures of the hcp phase. Elliot, [204] Baren [205] and Shi et al. [13] selected peritectic and eutectoid temperatures in accordance with the measurements of Eade and Hume-Rothery. [22] The same choice is made in this work since the measurements of Eade and Hume-Rothery [22] are considered more reliable than those of Heike and Leroux. [18] The activity data suggested by Lüdecke et al. [25] are very scattered and could Table 17 Crystallographic description of the phases in Ag-As-S system as reported by Schmid-Fetzer et al. [202] and Shi et al. [13] as well as the sublattice model of Ag 2 S selected in this work  not be reproduced in the work by Shi et al. [13] . After examining the experimental method and the inconsistency with phase diagram data, the activity data suggested by Lüdecke et al. [25] are considered doubtful and is not used in the present optimization. The calculated Ag-S binary phase diagram is shown in Fig. 2, and the different phases are identified with Pearson symbols. According to the integral molar quantities of the liquid phase determined by Fukatsu and Kozuka, [35] the SRO is maximum at a composition of Ag 2 S, which is expected. Considering the sharpness of the curves, the coordination numbers Z Ag AgS and Z S AgS are set in this work to 1.5 and 3 respectively to reproduce SRO at 33.33 at.% As. The solubility of sulfur in silver is very small as shown in Fig. 2. The most reliable solubility data obtained by Raub and Roschel [30] and Fueki et al. [31] are in very good agreement with the solubility calculated in this work. As reported in the literature, the low temperature form of Ag 2 S is considered stoichiometric, and the high temperature forms are considered non-stoichiometric. The homogeneity range, the non-symmetrical polymorphic temperatures, and the partial pressure of S 2 over the solid solutions are well reproduced. The optimized Gibbs free energies of the Ag 2 S phases are presented in Table 18. Table 3 shows that the selected heat of fusion of Ag 2 S is slightly higher than what was measured experimentally by Kelly [206] and nearly double of what was measured by Thompson and Flengas [52] and Koh and Itagaki. [62] This choice was necessary to adequately reproduce not only the binary Ag-S phase diagram, but also the Ag 2 S liquidus in the quasi-binary Ag 2 S-As 2 S 3 . As shown in Fig. 2(d), the sulphur contained in Ag 2 S completely vaporizes as observed by Piacente et al. [78] and becomes pure silver before this completely vaporizes too.
The calculated heat capacity of Ag 2 S as function of temperature is presented in Fig. 4(a). As was concluded by Sadonikov and Gusev, [66] the experimental data of Grønvold and Westrum [53] are considered the most reliable in the context of this work. Liu et al. [208] developed recently the zentropy theory to understand and predict anomalies of the physical properties of phases and this theory could be used to address the spikes of Fig. 4(a).
The optimization of the binary As-S system presents a great challenge. First, there are many contradictions and uncertainties in the measurements of this system as explained in the literature review. This is mainly due to the complexity of performing experiments of polymer structured samples. Above its melting temperature, sulfur is organized as ring shaped S 8 molecules, and at higher temperatures, it can form long polymer chains S n with high molecular weight. [209,210] The addition of arsenic to sulfur causes a significant increase in polymer chains and a reduction of S 8 molecules. This leads to glass formation and metastable tendencies for a wide range of compositions in the As-S system. [211,212] Also, the high vapor pressure in the arsenic rich region and its toxicity add to the difficulty of performing experiments to study this binary system. The calculated As-S binary phase diagram is shown in Fig. 5 and the optimized thermodynamic model parameters are presented in Table 18. Two calculations are superimposed on the figure, one with only the solids and the liquid phase, and the other including the gaseous species. As justified in Sect. 2.8, the coordination numbers set in this work are Z As AsS =3 and Z S AsS =2. By adding CTlnT terms to the model parameters of the liquid, no artifact miscibility gap is generated up to 6000 K. However, this also leads to the an unusual shape of the actual miscibility gap excisting in the arsenic rich region, hence it is represented by a dotted line. The extrapolation of this gap to the Ag-As-S ternary system is shown in the isothermal section of Fig. 7. As stated in Sect. 2.6, the thermodynamic information available in the literature for the gaseous species in the As-S system is incomplete. Despite this, the liquid was still optimized using the limited information available to reproduce the liquid-vapor equilibrium as best as possible. The azeotrope at the composition of As 2 S 3 measured by Jonker [81] is well reproduced. But this work suggests more than one azeotrope, as shown in Fig. 5(a), which was also predicted by Emelina. [148] Also, it is important to acknowledge that the MQM model used to optimize the liquid phase describes a monomeric melt and does not consider the complex structure of liquid As-S which tends to form polymer chains.
There is a large discrepancy in the measured standard enthalpies of formation of the three solid compounds in the As-S system, as presented in Tables 7, 9 and 11. In this work, low enthalpy values were selected for each compound. This was necessary to avoid stabilizing the liquid too much and consequently pushing the gas curve towards very high temperatures. With the few contradicting thermodynamic data available in the literature for As 4 S 3 , it is difficult to predict the behaviour of this compound very accurately. In this work, only the two orthorhombic crystalline phases were modeled, the plastic phase found by Blachnik et al. [87] and Chattopadhyay et al. [157] was not considered. The transformation and melting temperatures measured by Chattopadhyay et al. [157] are the most reliable in the context of this work and their values were selected in this optimization. However, it was not possible to optimize the system to have a congruent compound as found by Chattopadhyay et al. [157] , so As 4 S 3 is incongruent in this work as it was measured by Fedorova and Epova. [86] The calculated heat capacity of the three compounds in the As-S system is shown in Fig. 4. Blachnik et al. [87] performed heat capacity measurements, but the evaporation of their samples during the experiments caused uncertainties. The most reliable measurements are again those by Chattopadhyay et al. [157] . As seen in Fig. 4, the heat capacity of the liquid is high, and a possible explanation is its tendency to polymerize.
For the ternary Ag-As-S system, the thermodynamic properties of Ag 3 AsS 3 and AgAsS 2 are available in the

This work
Liquid Liquid1 + liquid2 (minor) Ag7AsS (HT) + liquid [164] x at. Ag (c) 6 Ag AsS (LT) + 7 6 Ag AsS (LT) 3 3 Ag2S-  literature and were optimized first to reproduce the vapor pressure measurements presented in Fig. 6. For Ag 7 AsS 6 , the enthalpy of formation and standard entropy were set to the values suggested by Gasanova et al. [172] since they are the only thermodynamic properties based on measurements available for this compound. The present calculations confirm the decomposition reactions 10 and 11 proposed by Godovikov et al. [175] and Fedorova et al. [176] . The liquid phase was then modeled and five ternary parameters, presented in Table 18, were used to reproduce more accurately the experimental data. Since three large positive terms are used to describe the three miscibility gaps in the Ag-S and As-S binary systems, three corrective ternary parameters have been added to balance these positive terms to avoid the formation of large ternary miscibility gaps and thus obtain a more precise interpolation. As there is no thermodynamic information available for Ag 2 AsS 2 and AgAsS and their decomposition temperatures are still uncertain, the Neumann-Kopp rule was used as a first approximation for these two compounds. Their Gibbs energies were then adjusted to reproduce as best as possible the available phase diagram data. A group of calculated phase diagrams together with the experimental data are shown in Fig. 7, 8, 9 and 10. Some phases are present at equilibrium in minor quantities and are identified in the figures as such. The quasi-binary Ag 2 S-As 2 S 3 is the most important in the Ag-As-S ternary system and it is well reproduced as seen from Fig. 7(b). The two quasi-binaries Ag 7 AsS 6 -Ag 3 AsS 3 and Ag 7 AsS 6 -Ag 2 S are also well reproduced as shown in Fig. 7(c) and (d). There is however a negligeable amount of a second liquid calculated, which was not detected by the measurements of Blachnik and Wickel. [164] The four cross sections measured by Kovaleva [163] along with the calculations from this work are presented in Fig. 8. Also, the predictions of Roland [139,[165][166][167] along with the calculations are presented in Fig. 9. There is a good agreement between the predictions and the calculations of Ag 3 AsS 3 -S and Ag 3 AsS 3 -As. However, a miscibility gap was predicted for AgAsS 2 -As 4 S 4 , but is not present in this work. Figure 10 shows the calculated Ag 2 S-As and Ag 3 AsS 3 -Ag along with the experimental data. Eutectic compositions and temperatures deviate slightly from measured values. Certain differences to what was measured for Ag 3 AsS 3 -Ag diagram were expected since Schmid-Fetzer et al. [202] already noted that in what was suggested by Chaus et al. [169] some phase diagram features were not considered. A summary of some invariant reactions involving four-phase intersection points with liquid in the ternary Ag-As-S system is presented in Table 19.

Conclusion
The increasing use of complex ores in smelters causes many problems due to their high arsenic content. To predict the behaviour of arsenic during the smelting process and help elaborate solutions, the development of thermodynamic models and their associated databases is essential. In this work, the thermodynamic information available in the literature was critically evaluated and was used to optimize the descriptions of the Ag-As, Ag-S and As-S binary systems. The MQM was applied to model the liquid phase while the CEF was utilized for solid solutions. The optimization of the binary As-S system presented a great challenge due the lack of complete thermodynamic information. The calculated phase diagrams for the three assessed binary systems provide an accurate description of all reliable experimental data. Using the Kohler-Toop interpolation, the Ag-As-S ternary system was assessed for the first time, and the calculations also provide a satisfactory description of the ternary data.
Acknowledgments The authors would like to thank the financial support from NSERC and CRC programs.
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