Experimental investigation and thermodynamic
modelling of the Ag-Zr system

C. Antion, Marc Barrachin, P. Benigni, I. Crassous, Alexandre Decreton, E. Fischer, Sandra Gyasi, A. Janghorban, M. Lomello-Tafin, G. Mikaelian, et al.

► To cite this version:

C. Antion, Marc Barrachin, P. Benigni, I. Crassous, Alexandre Decreton, et al. Experimental investigation and thermodynamic modelling of the Ag-Zr system. Calphad, 2026, 92, pp.102922. 10.1016/j.calphad.2026.102922. hal-05524816

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# Experimental investigation and thermodynamic modelling of the Ag-Zr system

C. Antion<sup>a</sup>, M. Barrachin<sup>b,*</sup>, P. Benigni<sup>c</sup>, I. Crassous<sup>d</sup>, A. Decreton<sup>b</sup>, E. Fischer<sup>e</sup>, S. Gyasi<sup>b</sup>, A. Janghorban<sup>a</sup>, M. Lomello-Tafin<sup>a</sup>, G. Mikaelian<sup>c</sup>, I. Nuta<sup>f</sup>, J. Rogez<sup>c</sup>

## ABSTRACT

After reviewing existing studies and conducting a new experimental investigation on the Ag-Zr phase diagram, a new CALPHAD model for the Ag-Zr system is presented. These experiments include, differential thermal analyses over the whole composition range, silver activity measurements and a determination of the standard enthalpy of formation of AgZr by calorimetry. The discussion about how AgZr and AgZr₂ decompose is resolved, showing congruent melting and peritectic decomposition, respectively. The model accurately reproduces the phase diagram, but more experiments are needed for the liquidus and solidus of β(Zr), especially in zirconium-rich areas. It is noteworthy that sub-stoichiometric yttria crucibles have been successfully tested in this study with melts containing metallic zirconium. The model matches measured silver activities well, giving confidence in the estimated enthalpy of formation of compounds, which were previously thought to be not exothermic enough in other models. The calculated enthalpy of formation for AgZr fits within the experimental uncertainty range. Future experiments on the heat capacities of intermetallic compounds could further improve the Ag-Zr system model.

## 1. Introduction

Absorber rods in nuclear reactors serve to regulate core reactivity. Most pressurized water reactors (PWR) utilize rods made of silver-indium-cadmium alloy (80 wt% Ag, 15 wt% In, 5 wt% Cd), encased in a stainless steel (SS) (AISI 304 or 316) cylinder, which is then inserted into a Zircaloy-4 guide tube. During a low-pressure accident, the Ag-In-Cd alloy liquefies between 743 °C and 831 °C [1], but remains contained within its cladding without significant damage. If cooling is not restored, the core temperature continues to increase, causing a rise in the absorber rod's internal pressure (primarily due to helium gas and, to a lesser extent, volatile cadmium). This leads to cladding deformation until it contacts the guide tube. The released Ag-In-Cd melt can then interact with and chemically dissolve the Zircaloy-4 guide tube between 1000 °C and 1100 °C [2]. As temperatures rise, cadmium largely vaporizes, and the resulting Ag-In-Zr-(O-H) mixtures from the degraded absorber rod can move downwards in the core or partially vaporize. The vaporization of silver from these melts is crucial to assess, not only because it represents a significant mass of aerosol in the reactor coolant system but also due to its substantial impact on iodine chemistry, the main

contributor to short-to-mid-term gaseous environmental release.

Among the various chemical subsystems to be examined in modelling the thermodynamics of the complex Ag-In-Zr-O-(H) system, the Ag-Zr binary system, comprising the two main elements of the mixture, is of greatest importance. This system was initially modelled by Karakaya and Thompson [3] revised in Ref. [4], and later modelled by He et al. [5]. These studies reported that the system contains two intermetallic compounds, AgZr and AgZr₂, which undergo peritectic decomposition. However, more recent thermodynamic CALPHAD models [6,7] depict both AgZr and AgZr₂ intermetallic compounds as melting congruently. This highlights the ongoing discussion regarding the nature of AgZr and AgZr₂ transformations, as noted in the initial assessment by Karakaya and Thompson [3].

Despite this controversy and growing interest in this system for amorphous metallic glass formation, virtually no new experimental data have been produced since 1992. Recent preliminary findings based on metallographic examinations by Decreton et al. [8] suggest the probable decomposition of AgZr and AgZr₂, congruent and peritectic respectively, contradicting existing CALPHAD models of Ag-Zr. Furthermore, little attention has been given to measuring thermodynamic properties.

<footnote><sup>a</sup> Laboratoire SYMME, Polytech Annecy Chambéry-Université de Savoie, BP. 80439, Annecy-Le Vieux, Cedex, 74944, France</footnote>

<footnote><sup>b</sup> Autorité de Sûreté Nucléaire et Radioprotection, PSN-RES, SAM, BP, 13115, St Paul lez Durance, France</footnote>

<footnote><sup>c</sup> IM2NP, UMR7334, CNRS, Aix-Marseille Université, Campus de Saint Jérôme, Avenue Escadrille Normandie Niémen e Case 251, 13397, Marseille, Cedex, 20, France</footnote>

<footnote><sup>d</sup> FRAMATOME Centre de Recherche Composants, 60 avenue Paul Girod, CS 90 100, Ugine, Cedex, 73403, France</footnote>

<footnote><sup>e</sup> Univ. Grenoble Alpes, Grenoble INP, CMTC, 38000, Grenoble, France</footnote>

<footnote><sup>f</sup> Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000, Grenoble, France</footnote>

<footnote>* Corresponding author.
E-mail address: marc.barrachin@asnr.fr (M. Barrachin).</footnote>
Fitzner and Kleppa [9] reported direct calorimetric measurements of the standard enthalpy of formation for AgZr and AgZr₂, but these were later questioned by the same group [10]. Decreton et al. [8] attempted to determine the standard enthalpy of formation of AgZr and AgZr₂ using solution calorimetry in liquid aluminum at 723 °C, but were unsuccessful. Decreton et al. [11] also employed solution calorimetry in an acid bath (HNO₃+HF) at 25 °C to measure the standard enthalpy of formation for AgZr. However, the precision of their measurement was limited.

This paper integrates differential thermal analyses across the entire composition range with metallographic examinations. Its primary objective is to address the controversial nature of AgZr and AgZr₂ compositions. The study is supplemented by activity measurements and standard enthalpy measurements of AgZr by high-temperature calorimetry to develop a CALPHAD model of the Ag-Zr system that incorporates consistently the different experimental data. The paper is structured as follows: the next section will review existing literature. The third section will present experimental findings related to phase diagram characterization and activity measurements as well as calorimetric enthalpy determinations. The fourth section will introduce a new CALPHAD modeling approach for the Ag-Zr system.

## 2. Literature review

Many experimental investigations were reported on the phase diagram of the Ag-Zr binary system [12–18]. The system comprises several solid phases: Ag-rich fcc solution with extremely low Zr solubility, two Zr-rich solutions (hcp (αZr) and bcc (βZr)), and two tetragonal intermediate phases, AgZr and AgZr₂. Although [14] initially identified AgZr₃ using XRD, subsequent research by Betterton and Easton [15] determined this to be a misidentification of the AgZr₂ phase. Table 1 provides a summary of the crystal structures for the Ag-Zr phases.

The experimental investigations of the decomposition of AgZr and AgZr₂ yielded conflicting results. Betterton et al. [15] performed thermal, metallographic, and X-ray analyses on various samples containing 0–36 mol % Ag, concluding that AgZr₂ forms through a peritectic reaction involving the liquid phase and bcc phase (βZr). This finding was supported by Loboda et al. [17] using DTA studies. However, more recent DTA measurements by Zhang et al. [18] contradicted these results, suggesting that AgZr and AgZr₂ compounds melt congruently. Decreton et al. [8] later proposed that AgZr₂ likely forms via a peritectic reaction between the liquid and bcc (βZr) phase, based on metallographic observations of as-cast samples. In the Ag-rich region of the system, Raub et al. [12] identified an eutectic reaction between (Ag) and AgZr at 955 °C, which was subsequently confirmed by Loboda et al. [17] and Zhang et al. [18]. Table 2 provides a summary of the invariant reactions determined in various studies and those incorporated in different CALPHAD models.

Fitzner and Kleppa [9] employed high-temperature direct synthesis calorimetry to measure the standard enthalpies of formation for AgZr and AgZr₂ compounds at 1200 °C. However, X-ray powder diffraction, SEM, and EDX analysis revealed secondary phases in both samples, leading Kleppa to consider these results as indicative, as noted in the review by Guo and Kleppa [10]. Additionally, the enthalpies of

<table><caption>**Table 1**
Crystal structures of the stable and metastable (*) phases of the Ag-Zr system.</caption><thead><tr><th>Phase</th><th>Pearson Symbol</th><th>Space group</th><th>Strukturbericht</th><th>Prototype</th><th>Reference</th></tr></thead><tbody><tr><td>(Ag)</td><td>cF4</td><td>Fm 3 m</td><td>A1</td><td>Cu</td><td>[19]</td></tr><tr><td>Ag<sub>2</sub>Zr*</td><td></td><td>I4/mmm</td><td></td><td></td><td>[20]</td></tr><tr><td>AgZr</td><td>tP4</td><td>P4/nmm</td><td>B11</td><td>yCuTi</td><td>[14]</td></tr><tr><td>AgZr<sub>2</sub></td><td>tI6</td><td>I4/mmm</td><td>C11<sub>b</sub></td><td>MoSi<sub>2</sub></td><td>[21]</td></tr><tr><td>AgZr<sub>3</sub>*</td><td></td><td>I4/mmm</td><td>D<sub>022</sub>-Al<sub>3</sub>Ti</td><td></td><td>[20]</td></tr><tr><td>αZr</td><td>hP2</td><td>P6<sub>3</sub>/mmc</td><td>A3</td><td>Mg</td><td>[19]</td></tr><tr><td>βZr</td><td>cI2</td><td>Im 3 m</td><td>A2</td><td>W</td><td>[22]</td></tr></tbody></table>

formation for these compounds were predicted using the Miedema semi-empirical model [10] and Density Functional Theory (DFT), with results reported in various databases [23,20]. Table 3 presents whole values.

## 3. Experimental

### 3.1. Phase diagram determination

#### 3.1.1. Materials and methods

Arc melting under a purified argon atmosphere was used to elaborate samples of approximately 200 mg from Ag ingots (99.99 % pure) and cold-rolled Zr foils (99.95 % pure). Before alloy synthesis, pure Zr was melted to eliminate residual oxygen from the furnace. The alloys were remelted at least 12 times to ensure homogeneous composition. A weight loss of about 1.5 % was attributed to silver evaporation due to its higher vapor pressure (1.4 × 10⁻¹ atm.) [24] compared to Zr (2.97 × 10⁻⁸ atm.) [25] at Zr melting point 1854 °C). To achieve the desired post-synthesis alloy composition, an extra 20 mg of silver was added before each melting step. Scanning electron microscopy (SEM) was employed for metallographic examination. EDX without standard was used for quantitative element analysis in individual phases.

Phase transformations up to 1600 °C were detected using DTA with a SETARAM TGA92 apparatus. Various crucible materials (BN, W, Al₂O₃, and Y₂O₃-x) were tested, with substoichiometric yttrium crucibles chosen for their lower chemical reactivity to Zr-containing liquid alloys [26]. The sub-stoichiometric yttria Y₂O₃-x crucibles have been provided by Chatain et al. [26] from CEA who used them as a Knudsen effusion cell material in the Fe-Zr system. In the Fe-Zr study, the system exhibited low reactivity under specific experimental conditions, namely, temperatures ranging from 1230 to 1580 °C and compositions between 50 at.% and 95 at.% Zr. For our conditions (in particular temperatures lower than 1500 °C), we have confirmed limited reactivity for zirconium-rich Ag-Zr compositions (>50 at.% Zr) and observed more reduced contamination in silver-rich compositions (<50 at.% Zr). For compositions with a high silver content (~88 at.% Ag), the interaction between the sample and the crucible is not entirely absent. Detailed examination of the sample reveals minimal contamination, characterized by a few highly localized areas primarily composed of silver and yttrium, which exhibit an eutectic morphology consistent with the Y-Ag phase diagram [27].

Experiments were conducted under purified argon with a 5 °C. min⁻¹ heating/cooling rate. Due to supercooling phenomena, only thermal events during heating were considered for establishing the phase diagram.

#### 3.1.2. Results and discussion

Several samples from the silver-rich area of the diagram (11.8, 28.4, 40, 45, 50 mol%Zr), underwent DTA analysis (Table 4). The resulting DTA curves (example of the 40 % Zr composition in Fig. 1) typically exhibited two thermal events. The first event, occurring at 960 °C, corresponded to the invariant reaction involving the silver solid-solution (Ag) and AgZr. The second event was interpreted as the liquidus chosen at the end of the monovariant melting of AgZr. A rapid change in the ATD signal could be noticed in the endothermic direction between 1150 °C and the liquidus temperature (1161 °C), indicating a significant increase in the liquid phase fraction per unit temperature. This behaviour is consistent with the flat shape of the liquidus curve at the sample composition (40 at.% Zr).

The temperature of the first event is marginally lower than the melting point of pure silver (961.78 °C [28]). Microstructural examination failed to determine the nature of the invariant equilibrium due to the similarity between the liquid composition in the reaction and that of the solid silver solution. To confirm the eutectic nature of the heat effect, a bi-differential thermal analysis was performed on an alloy with a nominal composition of Ag₈₆Zr₁₄. This alloy was placed in the DTA
<table><caption>**Table 2**

Invariant reactions calculated in this work compared with experimental values.(*) = values read on author's figure. nd: not determined.</caption><thead><tr><th>Reaction</th><th>Type</th><th>T/K</th><th>T/°C</th><th>x<sub>Zr</sub> of the phases respectively in order of the phases indicated in the reaction</th><th>Reference</th><th>Method</th></tr></thead><tbody><tr><td rowspan="10">L↔(Ag) + AgZr</td><td rowspan="10">Eutectic</td><td>1228</td><td>955</td><td>0.025/nd./0.500</td><td>[12]</td><td rowspan="10">Experimental</td></tr><tr><td>1208</td><td>935 ± 10</td><td>(*)0.03/0/0.500</td><td>[17]</td></tr><tr><td>1226</td><td>953</td><td>(*)0.027/0/~0.500</td><td>[18]</td></tr><tr><td>1232 ± 2</td><td>959 ± 2</td><td>0.02/~0/0.500</td><td>This work</td></tr><tr><td>1213</td><td>940</td><td>0.025/0.001/0.485</td><td>[3]</td></tr><tr><td>1217</td><td>944</td><td>0.024/0.004/0.500</td><td>[4]</td></tr><tr><td>1214</td><td>941</td><td>0.056/~0.500</td><td>[5]</td></tr><tr><td>1229</td><td>956</td><td>0.069/~0.500</td><td>[6]</td></tr><tr><td>1227</td><td>954</td><td>0.049/0.033/0.500</td><td>[7]</td></tr><tr><td>1213</td><td>940</td><td>0.023/0.000/0.500</td><td>This work</td></tr><tr><td rowspan="6">L + AgZr<sub>2</sub>↔AgZr</td><td rowspan="6">Peritectic</td><td>1408</td><td>1135</td><td>(*)0.30/0.667/0.500</td><td>[17]</td><td rowspan="6">Experimental</td></tr><tr><td>1426</td><td>1153</td><td>(*)0.499/~0.667/~0.500</td><td>[18]</td></tr><tr><td>1409</td><td>1136</td><td>0.465/0.660/0.500</td><td>[3]</td></tr><tr><td>1420</td><td>1147</td><td>0.498/0.667/0.500</td><td>[4]</td></tr><tr><td>1396</td><td>1123</td><td>0.488/0.667/0.500</td><td>[5]</td></tr><tr><td>1437</td><td>1164</td><td>-/-/-</td><td>[15]</td></tr><tr><td rowspan="6">L↔AgZr + AgZr<sub>2</sub></td><td rowspan="6">Eutectic</td><td>1438 ± 2</td><td>1165 ± 2</td><td>0.585 ± 0.015/~0.500/~0.667</td><td>This work</td><td rowspan="6">Experimental</td></tr><tr><td>1427</td><td>1154</td><td>0.575/0.500/0.667</td><td>[6]</td></tr><tr><td>1426</td><td>1153</td><td>0.568/0.500/0.667</td><td>[7]</td></tr><tr><td>1439</td><td>1166</td><td>0.587/0.500/0.667</td><td>This work</td></tr><tr><td>1443</td><td>1170</td><td>(*)0.55/0.82/0.667</td><td>[17]</td></tr><tr><td>1464</td><td>1191</td><td>0.64/0.80/0.67</td><td>[15]</td></tr><tr><td rowspan="6">L+(βZr)↔AgZr<sub>2</sub></td><td rowspan="6">Peritectic</td><td>1459</td><td>1186</td><td>0.663 ± 0.004/0.80 ± 0.01/~0.67</td><td>This work</td><td rowspan="6">Experimental</td></tr><tr><td>1463</td><td>1190</td><td>0.635/0.802/0.667</td><td>[3]</td></tr><tr><td>1473</td><td>1200</td><td>0.653/0.785/0.667</td><td>[4]</td></tr><tr><td>1468</td><td>1195</td><td>0.623/~0.667</td><td>[5]</td></tr><tr><td>1459</td><td>1186</td><td>0.665/0.790/0.667</td><td>This work</td></tr><tr><td>1422</td><td>1149</td><td>0.705/0.804/0.667</td><td>[18]</td></tr><tr><td rowspan="3">L↔(βZr) + AgZr<sub>2</sub></td><td rowspan="3">Eutectic</td><td>1419</td><td>1146</td><td>0.742/nd./0.667</td><td>[6]</td><td rowspan="3">Experimental</td></tr><tr><td>1464</td><td>1191</td><td>0.683/0.814/0.667</td><td>[7]</td></tr><tr><td>1408</td><td>1135</td><td>0.500/0.500</td><td>[12]</td></tr><tr><td rowspan="4">L↔AgZr</td><td rowspan="4">Melting</td><td>>1456</td><td>>1183</td><td>0.500/0.500</td><td>This work</td><td rowspan="4">Experimental</td></tr><tr><td>1443</td><td>1170</td><td>0.500/0.500</td><td>[6]</td></tr><tr><td>1439</td><td>1166</td><td>0.500/0.500</td><td>[7]</td></tr><tr><td>1455</td><td>1182</td><td>0.500/0.500</td><td>This work</td></tr><tr><td rowspan="4">L↔AgZr<sub>2</sub></td><td rowspan="4">Melting</td><td>1439</td><td>1166</td><td>0.667/0.667</td><td>[18]</td><td rowspan="4">Experimental</td></tr><tr><td>1455</td><td>1182</td><td>0.667/0.667</td><td>[6]</td></tr><tr><td>1465</td><td>1192</td><td>0.667/0.667</td><td>[7]</td></tr><tr><td>1096</td><td>823</td><td>0.963/0.999/~0.667</td><td>[15]</td></tr><tr><td rowspan="6">(βZr)↔αZr + AgZr<sub>2</sub></td><td rowspan="6">Eutectoid</td><td>1093</td><td>820</td><td>0.961/0.978/0.670</td><td>[3]</td><td rowspan="6">Experimental</td></tr><tr><td>1093</td><td>820</td><td>0.966/0.985/0.667</td><td>[4]</td></tr><tr><td>1090</td><td>817</td><td>0.947/~0.667</td><td>[5]</td></tr><tr><td>1089</td><td>817</td><td>0.963/~0.667</td><td>[6]</td></tr><tr><td>1095</td><td>822</td><td>0.966/0.985/0.667</td><td>[7]</td></tr><tr><td>1098</td><td>825</td><td>0.969/0.987/0.667</td><td>This work</td></tr></tbody></table>

<table><caption>**Table 3**

Enthalpies of formation of the intermediate stable (and unstable) Ag-Zr compounds, determined by experiments and calculated (using CALPHAD method) at 298.15 K and by DFT at 0 K.</caption><thead><tr><th>Compounds</th><th>Δ<sub>f</sub> H/kJ/mol</th><th>Technique</th><th>References</th></tr></thead><tbody><tr><td rowspan="10">AgZr</td><td>-5.5</td><td>CALPHAD</td><td>[7]</td></tr><tr><td>-11</td><td>CALPHAD</td><td>[6]</td></tr><tr><td>-12.06</td><td>DFT</td><td>[20]</td></tr><tr><td>-12.45</td><td>DFT</td><td>[23]</td></tr><tr><td>-31</td><td>Miedema</td><td>[10]</td></tr><tr><td>-3.12 ± 3.13</td><td>Exp.</td><td>[9]</td></tr><tr><td>-22.3 ± 10.0</td><td>Exp.</td><td>[11]</td></tr><tr><td>-21.8 ± 5.7</td><td>Exp.</td><td>This work</td></tr><tr><td>-17.6</td><td>CALPHAD</td><td>This work</td></tr><tr><td rowspan="6">AgZr<sub>2</sub></td><td>-3.62</td><td>CALPHAD</td><td>[7]</td></tr><tr><td>-12</td><td>CALPHAD</td><td>[6]</td></tr><tr><td>-10.71</td><td>DFT</td><td>[20]</td></tr><tr><td>-10.5</td><td>DFT</td><td>[23]</td></tr><tr><td>-24</td><td>Miedema</td><td>[10]</td></tr><tr><td>-1.47 ± 2.89</td><td>Exp.</td><td>[9]</td></tr><tr><td>AgZr<sub>3</sub></td><td>-12.2</td><td>CALPHAD</td><td>This work</td></tr><tr><td rowspan="3">Ag<sub>2</sub>Zr</td><td>2.22</td><td>DFT</td><td>[23]</td></tr><tr><td>-4.92</td><td>DFT</td><td>[20]</td></tr><tr><td>-4.44</td><td>DFT</td><td>[23]</td></tr></tbody></table>

<table><caption>**Table 4**

Detected thermal events (in °C) on DTA curves for the different compositions.</caption><thead><tr><th>x<sub>Zr</sub> (mol%)</th><th>Heating</th><th>Liquidus</th><th>Liquidus</th></tr></thead><tbody><tr><td>11.8</td><td>Invariant temperature</td><td>1130</td><td>nd.<sup>(3)</sup></td></tr><tr><td>28.4</td><td>960</td><td>1155</td><td>nd.<sup>(3)</sup></td></tr><tr><td>40.0</td><td>960</td><td>1161</td><td>nd.<sup>(3)</sup></td></tr><tr><td>45.0</td><td>957</td><td>1170</td><td>nd.<sup>(3)</sup></td></tr><tr><td>&lt;50<sup>(1)</sup></td><td>959</td><td>1183</td><td>nd.<sup>(3)</sup></td></tr><tr><td>&gt;50<sup>(2)</sup></td><td>1164</td><td>1175</td><td>1134</td></tr><tr><td>54.8</td><td>uncertain</td><td>1184</td><td>1135</td></tr><tr><td>55.4</td><td>1165</td><td>1179</td><td>1146</td></tr><tr><td>60.0</td><td>1165</td><td>1184</td><td>1171</td></tr><tr><td>62.9</td><td>1165</td><td>1187</td><td>1175</td></tr><tr><td>65.0</td><td>1165</td><td>1192</td><td>1172</td></tr><tr><td>65.9</td><td>1162</td><td>1191</td><td>1173</td></tr><tr><td>69.7</td><td>1186</td><td>nd.<sup>(3)</sup></td><td>nd.<sup>(3)</sup></td></tr><tr><td>74.9</td><td>825 and 1186</td><td>1193</td><td>1174</td></tr><tr><td>85.0</td><td>825 and 1430<sup>(4)</sup></td><td>nd.<sup>(3)</sup></td><td>nd.<sup>(3)</sup></td></tr></tbody></table>

(1)The composition was slightly lower than 50 %.

(2)The composition was slightly higher than 50 %.

(3)nd.: not determined due to reaction between liquid phase and yttria crucible.

(4) Not invariant, but corresponds to the solidus temperature.
<figure><img src="image_3.png" /><figcaption>Fig. 1. DTA curve of an Ag₆₀Zr₄₀ sample with a heating rate of 5 °C.min⁻¹.</figcaption></figure>

sample cell, while a small piece of pure silver was positioned in the
reference cell. As illustrated in Fig. 2, the initial detected thermal event
aligns with the invariant plateau crossing at the previously reported
temperature at 960 °C, followed by the heat effect (at 962 °C) associated
with the melting of pure silver in the reference cell. Based on the analysis
of the DTA curves for compositions lower than 50 mol%Zr (Table 4) we
definitively suggest 959 ± 2 °C as the temperature for the eutectic
invariant point. A SEM-BSE image of as-cast sample of nominal
composition xZr = 15 mol.% is depicted in Fig. 3. The image reveals
darker zones of a primary phase identified as AgZr. Interspersed among
these zones is a silver solid solution (Ag) bordered by a thin layer of
material identical to the primary phase. This two-phase region repre-
sents the eutectic mixture of (Ag) and AgZr, which corresponded to an
average zirconium content of 2 mol%Zr.

Initially, Betterton et al. [15] proposed congruent melting of the equimolar AgZr compound, while Loboda et al. [17] and Zhang et al. [18] later suggested peritectic decomposition. However, our experimental work supports the congruent formation of this compound. Examination of a micrograph from an as-cast sample with Ag₄₃Zr₅₇ nominal composition and SEM-EDS composition measurements revealed primary solidification AgZr phase, followed by the solidification of a lamellar eutectic microstructure consisting of AgZr and AgZr₂ (Fig. 4). This observation rules out the peritectic formation of AgZr. The congruent melting point of AgZr was estimated from an extrapolation of liquidus temperatures obtained through DTA analysis of samples with compositions very close to AgZr. Our findings indicate that a temperature of 1183 °C can be considered as a lower bound of the melting

<figure><img src="image_4.png" /><figcaption>Fig. 3. SEM image (BSE mode, 2000x) of as-cast sample of composition Ag₈₅Zr₁₅</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. SEM image (BSE mode 1500X) of as-cast sample of composi-
tion Ag₄₃Zr₅₇.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. Bi-differential analysis of Ag₈₆Zr₁₄ placed in the sample cell vs. pure silver in the reference cell, with a heating rate of 5 °C.min⁻¹.</figcaption></figure>

temperature, which exceeds the previously reported experimental value
of 1135 °C by Raub [12].

Additional microstructural examinations support the previously
mentioned eutectic nature of the subsequent invariant, specifically L ↔
AgZr and AgZr₂. The microstructure of the as-cast Ag₄₀Zr₆₀ sample,
depicted in Fig. 5, exhibits the AgZr₂ primary phase surrounded by a
lamellar eutectic consisting of AgZr and AgZr₂. The eutectic point
composition is anticipated to fall between the two sample compositions:
57 mol % Zr (Figs. 4) and 60 mol % Zr (Fig. 5).

For zirconium compositions richer than 50 %, the determined
liquidus temperatures (Table 4) suggest a nearly flat liquidus line. Fig. 6
where the DTA signal is reported for the composition Ag₃₇,₁Zr₆₂,₉ shows
that the initial thermal event during heating corresponds to the eutectic
plateau, AgZr + AgZr₂ ↔ L, observed at 1165 ± 2 °C, which aligns well
with the value (1164 °C) reported by Betterton et al. [15]. At higher
temperature (1187 °C), an endothermic contribution from the
non-congruent melting of AgZr₂ was observed on the DTA curve. Similar
to the observations made for the composition of 40 at.% Zr, a rapid
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** SEM image (BSE mode 1000X) of as-cast sample of composition Ag₄₀Zr₆₀.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** DTA of a Ag₃₇.₁Zr₆₂.₉ sample performed in an yttria crucible under purified argon.</figcaption></figure>

change in the ATD signal for 62.9 at.% Zr is also detected between 1170 °C and the liquidus temperature (1187 °C). This indicates a significant increase in the fraction of the molten phase per unit temperature. This phenomenon is attributed to the shallow gradient of the liquidus curve at the sample composition (62.9 at.% Zr). The minimal temperature interval between the eutectic plateau and the liquidus curve corresponds with the observed thermal response.

Microscopic examination of an as-cast 66.7 mol% Zr composition sample (Fig. 7) showed that AgZr₂ was produced through a peritectic reaction involving liquid and (βZr). We noted the initial crystallization of (βZr), which is easily identifiable due to its eutectoid decomposition [15], surrounded by the peritectic formation of AgZr₂. The incomplete peritectic formation of AgZr₂ resulted in the remaining liquid ultimately solidifying into the previously mentioned eutectic mixture. Differential thermal analysis (DTA) was conducted on samples with zirconium content ranging from 65.9 to 85 mol%Zr, in conjunction with microstructure analysis. This allowed us to determine the peritectic temperature as 1186 °C. Our findings indicate that the invariant composition of the liquid falls between 65.9 and 66.7 mol%Zr, while that of (βZr) is closed to 80 mol%Zr.

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** SEM image (BSE mode 3500X) of as-cast sample of composition of Ag₃₃.₃Zr₆₆.₇⁻ (1) Primary βZr phase has turned into eutectoid αZr + AgZr₂ during cooling and is surrounded by grey AgZr₂ phase. (2) AgZr₂ phase (3) Eutectic AgZr + AgZr₂; AgZr in eutectic AgZr + AgZr₂ is the light grey phase.</figcaption></figure>

## 3.2. Activity measurements

Samples for activity measurements were synthesized at Framatome starting from known Ag and Zr. Three 800 g bulk alloys were produced using Zr chips (Zr D grade 99.87 % Framatome) and silver chips (Ag 99.9 % Goodfellow) to achieve targeted molar compositions of 10 %, 15 % and 17 % Ag. Framatome Analysis Centre gives Zr impurities (ppm by mass): Al 19; B < 0.4; C 46; Ca 0; Cd < 0.4; Co < 5; Cr 18; Cu < 10; Fe 127; Hf 51; Mg 0; Mn < 10; Mo < 10; N 11; Nb < 25; Ni < 10; O 850; Pb 0; S < 5; Si < 10; Sn < 30; Ta 0; Ti < 10; U 0; V < 10; W < 10; Zn 0.

To obtain the desired alloys, they are melted within an induction coil using levitation from a cold crucible. To ensure even metal distribution before melting, silver wire and zirconium chips were divided into 10 equal portions and layered in the compacting press. Initial layered mixtures were compressed using a compacting press at approximately 400 bar pressure (maximum capacity of 700 bar). This initial material distribution is essential due to the significant difference in melting temperatures as well as vapor pressures between zirconium (i.e. melting at 1850 °C) and silver (i.e. melting at 962 °C), which can cause difficulties in the mixing reaction and may lead to zirconium lump formation in the final ingot. This melting process was performed under an argon atmosphere at a pressure slightly lower than atmospheric pressure, which is approximately 900 mbar absolute pressure. All primary Zr-Ag samples were subjected to an identical heating protocol: the generator rapidly performed two intermediate power steps (to reach approximately 1/3 and 2/3 of the liquidus temperature) in order to reach the temperature required for the liquid state, according to the predictions of the phase diagram. After the second step, the first liquid droplets form at the base of the compact (at the bottom of the crucible). The third increase of the generator power allows the compact to be completely liquefied for good homogenisation of the skull. The maximum power of the generator is maintained until the operator observes the entire sample to be fully liquid. After this plateau (Table 5a), the generator was shut down, allowing the sample to cool to room temperature. Temperature measurements were obtained using a pyrometer calibrated with high-purity standard Zr.

It should be noted that a slight gas emission (reddish fumes) is visible throughout the melting process, even at the initial stage of electricity generation, when the compact is not yet liquid. Degassing diminishes in intensity due to silver activity decrease when mixing reaction proceeds.
<table><caption>**Table 5a**

Thermal preparation conditions for Ag-Zr alloy samples.</caption><thead><tr><th>Sample</th><th>Time along higher temperature plateau/s</th><th>Temperature of plateau and uncertainty /°C</th></tr></thead><tbody><tr><td>10 mol%Ag</td><td>260</td><td>1514 ± 7.5</td></tr><tr><td>15 mol%Ag</td><td>348</td><td>1452 ± 5</td></tr><tr><td>17 mol%Ag</td><td>260</td><td>1435 ± 7</td></tr></tbody></table>

Once the plateau of liquidus temperature is fully established, visible outgassing appears to cease. Each original sample had an initial weight of 800 ± 1 g, the melted ingot weighing 799 g, leading to a mass decrease of about 1 g during preparation. This loss can be attributed to the intense outgassing observed during melting, potentially due to silver vaporization, water desorption, or other volatiles impurities in the raw material. The resulting ingot was sectioned into cubes of 1 cm³ for use as samples loaded in Knudsen cells for activity measurements. To evaluate the composition and impurity levels of the solidified ingots, ICP-OES analyses were conducted at the Ugine Laboratory, focusing on Ag, O, and N contents (Table 5b).

According to literature, silver vapor consists of Ag(g) and Ag₂(g) [24]. However, the vapor pressure of Ag(g) is much higher than that of Ag₂(g). For instance, at 1127 °C, the vapor pressures of Ag and Ag₂ are 7.7 10⁻⁵ and 8.92 10⁻⁸ bar respectively. Due to this significant difference, the contribution of Ag₂(g) will be disregarded in the mass loss calculations presented thereafter since total uncertainty in the following measurements is at least more than 1 %.

The total vapor pressure of the gaseous phase, *p* (silver pressure in our case) was determined using the Knudsen effusion method, and derived from Equation (1):

$$p = \frac{\Delta m \sqrt{2\pi RT}}{sC \sqrt{M_{Ag}} \Delta t} \qquad (1)$$

In this equation, Δ*m* represents the mass loss over time Δ*t*, *T* is the experimental temperature, *R* is the gas constant, *M*<sub>Ag</sub> is the molar mass of silver, *s* is the effusion orifice section, *C* represents a geometrical coefficient called the Clausing factor [29].

For this study, we used a Knudsen multi-cell block comprising four boron nitride (type 7000) cells (crucible + lid) with an internal diameter of 0.0165 m corresponding to a sample vaporization section equal to *S* = 0.0002138 m². The four cells were loaded with pure Ag, 10 % Ag-Zr, 15 % Ag-Zr and 17 % Ag-Zr, respectively. The effusion orifice measured 0.002 m in both diameter (corresponding to a surface *s* = 3.1415 10⁻⁶ m²) and length. Given this configuration, the Clausing coefficient *C* was 0.514, resulting in a *sC/S* ratio of 0.00755.

Ag-Zr samples were heated to the desired temperature in an effusion furnace constructed in our laboratory [30,31]. A K-type thermocouple (chromel-alumel) inserted between two cells measured temperature which is recorded continuously by a 6100E series Eurotherm device. In a previous study on pure Ag [30], we showed that, in the multi-cell configuration, the temperature gradient between cells is less than 1 % up to the melting temperature. Samples weight were measured before and after experiments using a Mettler Toledo XS204 Analytical Balance (readability: 0.1 mg; repeatability (test weight): 0.07 mg (10 g scale)). Different temperature plateaus were chosen for each experiment

<table><caption>**Table 5b**

Compositions of the initial Ag-Zr alloys used in this study as measured by ICP OES (in mol% ± 0.05 mol% for Ag and Zr and ppm by mass for O, N impurities).</caption><thead><tr><th>Composition</th><th>Ag</th><th>Zr</th><th>O</th><th>N</th></tr></thead><tbody><tr><td>10 mol%Ag</td><td>10.90</td><td>88.98</td><td>635</td><td>14</td></tr><tr><td>15 mol%Ag</td><td>15.5</td><td>84.38</td><td>1061</td><td>111</td></tr><tr><td>17 mol%Ag</td><td>18.14</td><td>81.74</td><td>776</td><td>44</td></tr></tbody></table>

performed with the same sample composition. For each new experiment, a new sample is extracted from the same ingot.

In principle, Knudsen cell mass loss experiments are performed over a prolonged constant temperature plateau, which is of sufficient duration to disregard the temperature rise and fall steps. As a result, it is possible to determine an average temperature value at the plateau and assign an associated uncertainty related to accuracy of the temperature settings. If the temperature plateau cannot be maintained at a strictly constant temperature, additional uncertainty must be assigned, equivalent to the temperature interval over which *T* varies. Besides, pressure exhibits a strong dependence on temperature and therefore it is necessary to evaluate the impact of temperature drift on the plateau as well as for preceding and final temperature ramps. Along experiment each elemental mass loss during the *i*-th time interval of duration *δt*ᵢ, for which temperature is equal to *T*ᵢ, is known via Knudsen's formula:

$$\delta m_i = \frac{p_i \cdot sC \cdot \sqrt{M_{Ag}}}{\sqrt{2\pi R T_i}} \cdot \delta t_i \qquad (2)$$

where *p*ᵢ is the pressure of silver (Ag(g)) during the time interval *δt*ᵢ (=1 s in our experiments).

*p*ᵢ can be expressed for an alloy, using the Clausius-Clapeyron relation:

$$\ln p_i = \frac{A}{T_i} + B \qquad (3)$$

in which *A* = $-\frac{\Delta m_{exp}}{R} \overline{H}$ and *B* = $\frac{\Delta m_{exp}}{R} \overline{S}$ are partial thermodynamic functions that are constants over a wide temperature range for a given composition in a single-phase or two-phase regions.

By introducing Equation (3) into Equation (2), one obtains:

$$\delta m_i = \frac{sC \cdot \sqrt{M_{Ag}}}{\sqrt{2\pi R}} \left[ \exp^{\frac{A}{T_i} + B} \cdot \frac{\delta t_i}{\sqrt{T_i}} \right] \qquad (4)$$

For the entire experiment, Equation (4) has to be integrated over the time:

$$\Delta m = \sum_{i} \delta m_{i} = \frac{sC \cdot \sqrt{M_{Ag}}}{\sqrt{2\pi R}} \sum_{i} \left[ \exp^{\frac{A}{T_{i}+B}} \cdot \frac{\delta t_{i}}{\sqrt{T_{i}}} \right] \qquad (5)$$

The ratio between *δm*ᵢ and the total mass loss Δ*m* is then:

$$\frac{\delta m_i}{\Delta m} = \frac{\left[ \exp^{\frac{A}{T_i}} \cdot \frac{\delta t_i}{\sqrt{T_i}} \right]}{\sum_{i} \left[ \exp^{\frac{A}{T_i}} \cdot \frac{\delta t_i}{\sqrt{T_i}} \right]} \qquad (6)$$

Factor *A* is related to partial vaporization enthalpy of silver in the Ag-Zr mixture divided by *R*. This partial vaporization enthalpy is equal to the standard enthalpy of silver minus the partial mixing enthalpy of silver. In metallic alloys the mixing enthalpy is generally ten times lower than the standard enthalpy and equalizing *A* to standard enthalpy does not lead to significant errors. In the present case, by choosing a value for *A* taken from a preliminary CALPHAD optimization of Ag-Zr system (A = -33781.72 K), we can determine any elementary mass loss relative to the total loss mass.

As a consequence, the proportion of Δ*m* corresponding to the heating and cooling steps can be first obtained namely in a conventional Knudsen mass loss determination with a strictly constant temperature plateau. Subsequently, only the plateau at constant *T* was retained with Equation (1) to determine the pressure. The mass loss during the plateau (pl) can be easily calculated as:

$$\Delta m_{pl} = \Delta m - \sum_{begin_{exp}}^{begin_{pl}} \delta m_i - \sum_{end_{pl}}^{end_{exp}} \delta m_i \qquad (7)$$
where, *begin*<sub>*pl*</sub> and *end*<sub>*pl*</sub> are the time interval numbers corresponding to the beginning and the end of the plateau, and *begin*<sub>*exp*</sub>, and *end*<sub>*exp*</sub>, the initial and the last time interval numbers of the experiment.

Using Equation (6), it is also possible to choose any time interval, to determine the mass loss corresponding to this interval. Aiming at determining the pressure corresponding to the mean value of fluctuating temperatures during the plateau, only those time intervals in which the temperature differs from the mean temperature by less than 5K will be considered. Total mass loss over all selected time intervals is $\Delta m_{pl}^*$ (lower than $\Delta m_{pl}$ for the full plateau). A mass loss correction factor *D* can be defined by the following equation:

$$D = \frac{\Delta m_{pl}^*}{\Delta m} \qquad (8)$$

which gives an indication of the fraction of the total number of measurements that is taken into account for the pressure evaluation.

Knudsen effusion experimental data for the Ag and Ag-Zr samples are listed with their calculated pressure (Tables 6-9). The activity of silver in these alloys is calculated by dividing the Ag(g) vapor pressure over the alloys by the pure Ag(g) vapor pressure at the same temperature as measured in the multiple cell experiment.

At lower temperatures, studied compositions are in the same two-phases domain (βZr + AgZr<sub>2</sub>). For 83, 85 and 90 mol% Zr alloys, this domain extends up to 1150 °C (1523K), 1122 °C (1395K), to 1028 °C (1301K) respectively. Within this two-phase domain, the three compositions should correspond to the same activity (or pressure) for a given temperature. We can see that there is some activity variation between 83 and 85 and 90 mol% Zr as shown in Fig. 14. In the two-phase domain, for same temperature, we do not expect an activity variation for different compositions. In our case, by examining activity values in Tables 7-9 for the 3 measured compositions, we observe a small average deviation less than 0.01 at low temperature (at 1248 K and 1301 K). This average deviation increases with temperature since we calculate between activities at 83 and 85 %, 0.05 (at 1324 K and 1316 K), 0.06 (at 1370 K) and 0.09 (at 1384 K). This may be explained by a possible difficulty in the evaporation process. The effusion flow leads to a loss of Ag at the sample surface, which should be compensated by diffusion of Ag from the sample bulk towards the surface.

Indeed generally, the activation energies for diffusion are around 150 kJ mol<sup>-1</sup> for metallic alloys [32] meanwhile the vaporization enthalpy (assumed to be equal to the activation energy for vaporization in vacuum) for Ag at melting temperature (1234K) is 284 kJ mol<sup>-1</sup> [33, 34]. When temperature increases, evaporation flux increases more than diffusion flow. Thus, at high temperatures, the composition of the Ag surface decreases, and a depletion of the surface composition may occur. By contrast, measurements performed at low temperature in the biphasic domain (i.e. in our case, in experiments 3 and 4), are not affected by this competition.

For the calculation of partial free enthalpy, the following expression

is used:

$$\Delta \bar{G}_{Ag} = RT \ln a_{Ag} \qquad (9)$$

The associated uncertainty is calculated as:

$$d(\Delta \bar{G}_{Ag}) = R d T \ln a_{Ag} + RT \frac{d a_{Ag}}{a_{Ag}} \qquad (10)$$

From the experiment performed on pure silver, the melting temperature of silver is used to calibrate the temperature measurements by thermocouple with an uncertainty of ±10K. The value for $\frac{d a_{Ag}}{a_{Ag}}$ is taken equal to 0.05, corresponding to the largest mean deviation determined in the biphasic domain.

An additional source of uncertainty may arise from thermal gradients in the multiple cells. In Ref. [30], it has been shown with silver in four cells, that the pressure difference does not exceed 1 %.

Besides, the silver content of alloys was analyzed after each experiment and the composition evolution due to preferential loss by effusion of Ag did not exceed 1 %. This value can be retained as composition uncertainty.

### 3.3. Determination of the standard formation enthalpy of AgZr by drop solution calorimetry

#### 3.3.1. Materials and method

The standard formation enthalpy of AgZr, $\Delta_f H_{AgZr}$ is determined indirectly from the partial drop solution enthalpies of Zr, Ag and AgZr crystals at infinite dilutions in liquid Al at 1173 K according to:

$$\Delta_f H_{AgZr} = \Delta_{ds} H_{Ag.298K}^{\infty,1173K} + \Delta_{ds} H_{Zr.298K}^{\infty,1173K} - \Delta_{ds} H_{AgZr.298K}^{\infty,1173K} \qquad (11)$$

in which, the three drop-solution enthalpies are measured by Tian-Calvet isothermal calorimetry.

The partial drop solution enthalpies of Ag and Zr at infinite dilution in liquid Al at 1173 K were determined respectively in Refs. [35,36]. The selected values are $\Delta_{ds} H_{Ag.298K}^{\infty,1173K} = 42.2 \pm 0.2 \text{ kJ mol}^{-1}$ and $\Delta_{ds} H_{Zr.298K}^{\infty,1173K} = -130 \pm 9 \text{ kJ mol}^{-1}$. The drop solution of AgZr has been measured in the present work.

An AgZr sample was synthesized by arc melting as described in subsection 3.1.1. Next, the as-cast sample was wrapped in tantalum foil and inserted into a silica ampoule that was sealed under Ar. The role of the tantalum foil is to prevent a direct reaction between the AgZr sample and the silica ampoule during the annealing treatment. The sample was then annealed at 900 °C (1173K) for 40 days. Metallographic studies carried out in a previous study [11] showed that these annealing conditions make it possible to obtain a single-phase AgZr sample (Fig. 8). The calorimetric method has been described extensively in our previous works [8,35].

Some information is summarized here. The measurements were conducted under argon gas at a pressure slightly exceeding 1 bar,

<table><caption>**Table 6**
Knudsen effusion experimental data and obtained vapor pressure for pure silver.</caption><thead><tr><th></th><th>T/K</th><th>Temperature uncertainty/K (Standard deviation(±))</th><th>Measured total mass loss Δm/g</th><th>Mass correction factor D</th><th>Corrected mass loss Δm<sub>pl</sub>/g</th><th>p/Pa</th><th>Ag Activity</th></tr></thead><tbody><tr><td>n°1</td><td>1481.6</td><td>2.95</td><td>1.5295</td><td>0.77</td><td>1.1782</td><td>10.1040</td><td>1.00</td></tr><tr><td>n°2</td><td>1324.6</td><td>1.29</td><td>0.0564</td><td>0.55</td><td>0.0310</td><td>0.3130</td><td>1.00</td></tr><tr><td>n°3</td><td>1301.2</td><td>0.86</td><td>0.0269</td><td>0.39</td><td>0.0105</td><td>0.1045</td><td>1.00</td></tr><tr><td>n°4</td><td>1248.2</td><td>1.80</td><td>0.0071</td><td>0.51</td><td>0.0036</td><td>0.0242</td><td>1.00</td></tr><tr><td>n°5</td><td>1526.5</td><td>3.12</td><td>4.6474</td><td>0.96</td><td>4.4417</td><td>25.3963</td><td>1.00</td></tr><tr><td>n°6</td><td>1422.9</td><td>1.29</td><td>0.3866</td><td>0.42</td><td>0.1635</td><td>3.3104</td><td>1.00</td></tr><tr><td>n°7</td><td>1464.4</td><td>2.25</td><td>0.4569</td><td>0.91</td><td>0.4176</td><td>8.2084</td><td>1.00</td></tr><tr><td>n°8</td><td>1480.1</td><td>3.00</td><td>0.5717</td><td>0.81</td><td>0.4625</td><td>10.3221</td><td>1.00</td></tr><tr><td>n°9</td><td>1383.7</td><td>4.26</td><td>0.1466</td><td>0.73</td><td>0.1073</td><td>1.1578</td><td>1.00</td></tr><tr><td>n°10</td><td>1316.1</td><td>3.23</td><td>0.0259</td><td>0.90</td><td>0.0234</td><td>0.1815</td><td>1.00</td></tr><tr><td>n°11</td><td>1370.8</td><td>3.08</td><td>0.1685</td><td>0.75</td><td>0.1256</td><td>0.8179</td><td>1.00</td></tr></tbody></table>
<table><caption>**Table 7**

Knudsen effusion experimental data and obtained vapor pressure and Ag activity for 83 mol% Zr alloy</caption><thead><tr><th></th><th>T/K</th><th>Temperature uncertainty/K (Standard deviation (±))</th><th>Measured total mass loss Δm/g</th><th>Mass correction factor D</th><th>Corrected mass loss Δm<sup>2</sup><sub>pl</sub>/g</th><th>p/Pa</th><th>Ag activity a<sub>Ag</sub></th></tr></thead><tbody><tr><td>n°1</td><td>1481.6</td><td>2.95</td><td>0.3951</td><td>0.75</td><td>0.2961</td><td>2.2321</td><td>0.2209</td></tr><tr><td>n°2</td><td>1324.6</td><td>1.29</td><td>0.0091</td><td>0.55</td><td>0.0050</td><td>0.0440</td><td>0.1407</td></tr><tr><td>n°3</td><td>1301.2</td><td>0.86</td><td>0.0038</td><td>0.42</td><td>0.0016</td><td>0.0140</td><td>0.1335</td></tr><tr><td>n°4</td><td>1248.2</td><td>1.80</td><td>0.0007</td><td>0.54</td><td>0.0004</td><td>0.0022</td><td>0.0917</td></tr><tr><td>n°5</td><td>1526.5</td><td>3.12</td><td>1.3552</td><td>0.95</td><td>1.2807</td><td>6.4356</td><td>0.2534</td></tr><tr><td>n°6</td><td>1422.9</td><td>1.29</td><td>0.0857</td><td>0.42</td><td>0.0358</td><td>0.6367</td><td>0.1923</td></tr><tr><td>n°7</td><td>1464.4</td><td>2.25</td><td>0.1131</td><td>0.88</td><td>0.1000</td><td>1.7282</td><td>0.2105</td></tr><tr><td>n°8</td><td>1480.1</td><td>3.00</td><td>0.1499</td><td>0.80</td><td>0.1195</td><td>2.3431</td><td>0.2270</td></tr><tr><td>n°9</td><td>1383.7</td><td>4.26</td><td>0.029</td><td>0.74</td><td>0.0214</td><td>0.2034</td><td>0.1756</td></tr><tr><td>n°10</td><td>1316.1</td><td>3.23</td><td>0.0041</td><td>0.90</td><td>0.0037</td><td>0.0253</td><td>0.1392</td></tr><tr><td>n°11</td><td>1370.8</td><td>3.08</td><td>0.0311</td><td>0.77</td><td>0.0239</td><td>0.1369</td><td>0.1674</td></tr></tbody></table>

<table><caption>**Table 8**

Knudsen effusion experimental data and obtained vapor pressure and Ag activity for 85 mol% Zr alloy.</caption><thead><tr><th></th><th>T/K</th><th>Temperature uncertainty/K (Standard deviation (±))</th><th>Measured total mass loss Δm/g</th><th>Mass correction factor D</th><th>Corrected mass loss Δm<sup>2</sup><sub>pl</sub>/g</th><th>p/Pa</th><th>Ag activity a<sub>Ag</sub></th></tr></thead><tbody><tr><td>n°1</td><td>1481.6</td><td>2.95</td><td>0.2615</td><td>0.74</td><td>0.2615</td><td>1.9712</td><td>0.1951</td></tr><tr><td>n°2</td><td>1324.6</td><td>1.29</td><td>0.0046</td><td>0.55</td><td>0.0046</td><td>0.0411</td><td>0.1312</td></tr><tr><td>n°3</td><td>1301.2</td><td>0.86</td><td>0.0016</td><td>0.42</td><td>0.0016</td><td>0.0139</td><td>0.1333</td></tr><tr><td>n°4</td><td>1248.2</td><td>1.80</td><td>0.0004</td><td>0.54</td><td>0.0004</td><td>0.0022</td><td>0.0912</td></tr><tr><td>n°5</td><td>1526.5</td><td>3.12</td><td>1.1079</td><td>0.94</td><td>1.1079</td><td>5.5671</td><td>0.2192</td></tr><tr><td>n°6</td><td>1422.9</td><td>1.29</td><td>0.0320</td><td>0.42</td><td>0.0320</td><td>0.5691</td><td>0.1719</td></tr><tr><td>n°7</td><td>1464.4</td><td>2.25</td><td>0.0883</td><td>0.88</td><td>0.0883</td><td>1.5256</td><td>0.1859</td></tr><tr><td>n°8</td><td>1480.1</td><td>3.00</td><td>0.1062</td><td>0.79</td><td>0.1062</td><td>2.0830</td><td>0.2018</td></tr><tr><td>n°9</td><td>1383.7</td><td>4.26</td><td>0.0193</td><td>0.74</td><td>0.0193</td><td>0.1827</td><td>0.1578</td></tr><tr><td>n°10</td><td>1316.1</td><td>3.23</td><td>0.0034</td><td>0.90</td><td>0.0034</td><td>0.0234</td><td>0.1287</td></tr><tr><td>n°11</td><td>1370.8</td><td>3.08</td><td>0.0221</td><td>0.77</td><td>0.0221</td><td>0.1264</td><td>0.1545</td></tr></tbody></table>

<table><caption>**Table 9**

Knudsen effusion experimental data and obtained vapor pressure and Ag activity for 90 mol% Zr alloy.</caption><thead><tr><th></th><th>T/K</th><th>Temperature uncertainty/K (Standard deviation (±))</th><th>Measured total mass loss Δm/g</th><th>Mass correction factor D</th><th>Corrected mass loss Δm<sup>2</sup><sub>pl</sub>/g</th><th>p/Pa</th><th>Ag activity a<sub>Ag</sub></th></tr></thead><tbody><tr><td>n°1</td><td>1481.6</td><td>2.95</td><td>0.2532</td><td>0.75</td><td>0.2532</td><td>1.9083</td><td>0.1889</td></tr><tr><td>n°2</td><td>1324.6</td><td>1.29</td><td>0.0044</td><td>0.55</td><td>0.0044</td><td>0.0391</td><td>0.1250</td></tr><tr><td>n°3</td><td>1301.2</td><td>0.86</td><td>0.0015</td><td>0.42</td><td>0.0015</td><td>0.0128</td><td>0.1228</td></tr><tr><td>n°4</td><td>1248.2</td><td>1.80</td><td>0.0003</td><td>0.54</td><td>0.0003</td><td>0.0019</td><td>0.0781</td></tr><tr><td>n°5</td><td>1526.5</td><td>3.12</td><td>1.0349</td><td>0.94</td><td>1.0349</td><td>5.2003</td><td>0.2048</td></tr><tr><td>n°6</td><td>1422.9</td><td>1.29</td><td>0.0309</td><td>0.42</td><td>0.0309</td><td>0.5507</td><td>0.1664</td></tr><tr><td>n°7</td><td>1464.4</td><td>2.25</td><td>0.0853</td><td>0.88</td><td>0.0853</td><td>1.4738</td><td>0.1796</td></tr><tr><td>n°8</td><td>1480.1</td><td>3.00</td><td>0.1004</td><td>0.80</td><td>0.1004</td><td>1.9692</td><td>0.1908</td></tr><tr><td>n°9</td><td>1383.7</td><td>4.26</td><td>0.0188</td><td>0.74</td><td>0.0188</td><td>0.1780</td><td>0.1538</td></tr><tr><td>n°10</td><td>1316.1</td><td>3.23</td><td>0.0032</td><td>0.90</td><td>0.0032</td><td>0.0215</td><td>0.1186</td></tr><tr><td>n°11</td><td>1370.8</td><td>3.08</td><td>0.0210</td><td>0.77</td><td>0.0210</td><td>0.1201</td><td>0.1468</td></tr></tbody></table>

utilizing a Tian-Calvet high-temperature calorimeter. The temperatures of the reference and sample cells were monitored by a Pt10 %Rh–Pt thermocouple positioned between the cells within the isothermal alumina block. A sample of aluminum was placed in an alumina crucible, which was subsequently inserted into a vitreous silica tube. As the silica tube was gradually lowered into the calorimeter, maintained at the experimental temperature, the metallic load underwent melting. Following a transient thermal regime, the entire assembly achieved thermal equilibrium, resulting in a stable baseline. Subsequently, small fragments of AgZr, each weighing a few milligrams, were incrementally introduced. The masses were measured using a balance with an accuracy of $10^{-5}$ g. The differential signal of the calorimeter was continuously recorded throughout the process. After each addition, a thermal effect was observed before the signal returned to the baseline.

The annealed AgZr sample was cut into small pieces of 9–23 mg suitable for calorimetric experiments. Three experiments (runs 1–3) have been performed (Table 10) using all the available quantity of AgZr.

To check that the AgZr samples had dissolved properly in the liquid Al, post-calorimetry metallographic examinations were carried out using scanning electron microscopy (SEM) on the ingots solidified after runs 1 and 3. The two ingots were cut longitudinally, embedded in resin and polished. The chemical compositions of the individual phases were measured using EDX point analyses without standard.

### 3.3.2. Results and discussion

Parameters and values of enthalpy of drop solution of AgZr crystals are given in Table 10. The enthalpy values are plotted in Fig. 9. The enthalpies measured are widely dispersed and, in runs 1 and 2, become more exothermic as the mole fraction of AgZr increases. An explanation of these results is now proposed.

During drop solution calorimetric experiments of AgZr samples, the peaks observed (Fig. 10a, b and 10c) are composed of a brief initial endothermic contribution, associated with the heating and melting of the sample, followed by a strong exothermic contribution corresponding
<figure><img src="image_10.png" /><figcaption>Fig. 8. SEM images (BSE mode) of as-cast (left) and 40 days-annealed at 900 °C (right) samples of AgZr composition.</figcaption></figure>

<caption>Table 10</caption>

Partial drop-solution enthalpy of AgZr in liquid Al at 1173 K $\Delta_{ds}\overline{H}^{\infty}_{AgZr,298K}$ measured by Tian-Calvet isothermal calorimetry. The references states are crystalline AgZr at 298 K and liquid aluminum at 1173 K. For each run, $m_{Al}$ is the total mass of aluminum in the crucible before the first AgZr drop. It corresponds to the initial mass of the Al bath plus the masses of the 4 pure aluminum calibration samples. $m_{AgZr}$ values are the masses of the AgZr samples successively dropped in the Al bath. $x_{AgZr}$ is the molar fraction of AgZr in liquid Al corresponding to each measured drop solution enthalpy value.

<table><thead><tr><th>Run</th><th>m<sub>Al</sub> / mg</th><th>m<sub>AgZr</sub> / mg</th><th>x<sub>AgZr</sub> / %</th><th>Δ<sub>ds</sub> H<sup>∞</sup><sub>AgZr,298K</sub> / kJ mol<sup>-1</sup></th></tr></thead><tbody><tr><td rowspan="4">1</td><td rowspan="4">10570.84</td><td>15.15</td><td>0.09711</td><td>-60.7</td></tr><tr><td>22.89</td><td>0.34084</td><td>-103.5</td></tr><tr><td>10.48</td><td>0.55456</td><td>-125.4</td></tr><tr><td>10.06</td><td>0.68606</td><td>-119.0</td></tr><tr><td rowspan="4">2</td><td rowspan="4">10551.75</td><td>14.00</td><td>0.08990</td><td>-52.4</td></tr><tr><td>13.63</td><td>0.26727</td><td>-64.4</td></tr><tr><td>15.33</td><td>0.45311</td><td>-94.2</td></tr><tr><td>20.22</td><td>0.68115</td><td>-106.6</td></tr><tr><td rowspan="4">3</td><td rowspan="4">10597.80</td><td>10.09</td><td>0.06451</td><td>-43.5</td></tr><tr><td>9.66</td><td>0.19076</td><td>-47.2</td></tr><tr><td>8.80</td><td>0.30873</td><td>-41.7</td></tr><tr><td>10.55</td><td>0.43236</td><td>-14.7</td></tr></tbody></table>

to its dissolution in liquid aluminum accompanied by the formation of Al₃Zr. These first two observed contributions are quite frequently, but not systematically, followed by a third endothermic effect. When it occurs, this third effect is interpreted as the dissolution of Al₃Zr, a reaction whose endothermic character was highlighted in a previous study performed in our laboratory [36]. The lack of systematic observation of the dissolution of Al₃Zr is probably linked to a local saturation in Zr of the aluminum bath in the vicinity of the dissolving samples. This saturation is all the more probable as the number of samples already dissolved in the bath is high. Thus, it is often the first peaks that show a marked final endothermic contribution, this contribution tending to fade with the successive drops (e.g. Fig. 10a, run 1). Calorimetric signals of run 1 and 2 (Fig. 10a and b respectively) both showed this trend. This problem is exacerbated by the fact that the high density of the AgZr compound compared with that of liquid aluminum leads to an accumulation of samples towards the bottom of the crucible.

The complete dissolution of an AgZr sample in liquid aluminum cannot be guaranteed on the basis of peak shape analysis alone. Additional useful information on the dissolution of the samples is obtained by

<figure><img src="image_11.png" /><figcaption>Fig. 9. Enthalpy of drop solution of AgZr crystals from 298 K in liquid aluminum at 1173 K as a function of AgZr molar fraction in the aluminum bath for runs 1, 2 and 3.</figcaption></figure>

post-calorimetry metallographic examination using SEM.

In the AgZr ingot of run 1 (Fig. 11), white interdendritic precipitates and white crystals having a needle shape are observed. These phases are Al₃Zr D0.23 crystals as shown by Khvan et al. [37]. In addition, large clusters of Al₃Zr crystals, having a typical size of several hundreds of microns comparable to the size of the dropped AgZr fragments are also detected at the bottom of the run 1 ingot (Fig. 11) but have not been found in the run 3 ingot. The small black particles seen in Fig. 11 are aluminum oxides having roughly the stoichiometry of Al₂O₃ but containing a small amount of Ag (between 2 and 4 mol%). It was not possible to determine more precisely the nature of these minor precipitates by SEM.

Observation of the large Al₃Zr clusters indicates that dissolution of the AgZr samples was clearly not complete in run 1. As series 1 and 2 showed a similar evolution in the shape of the calorimetric signal with the mole fraction of AgZr, with quantitatively comparable values, it was decided to discard all the values from these two experiments.

In run 3, all the peaks show a clear final endothermic effect, but the last value is much less exothermic than the first three and this fourth value was also discarded. In fact, unlike the other samples, the fourth sample did not consist of a single piece; the total mass of 10.55 mg was
<figure><img src="image_12.png" /><figcaption>Fig. 10. Calorimetric signal recorded vs. time during the four successive drops of AgZr samples in a) run 1 2) run 2 and c) run 3. The baseline has been subtracted from the signal. In runs 1 and 2, the final endothermic signature is stronger for the first peak than for the second and disappears for the third and fourth peaks, whereas, in run 3, it is systematically observed for all the peaks.</figcaption></figure>

obtained by combining the last four small pieces of AgZr that remained. It is very likely that some of these small pieces floated on the aluminum bath, whose surface is slightly oxidized, but did not dissolve.

Finally, the partial drop solution enthalpy of AgZr at infinite dilution in liquid aluminum at 1173 K is calculated by taking the average of the three first values of run 3 giving $\Delta_{ds}\overline{H}_{AgZr,298K}^{\infty,1173K} = -44.1 \pm 7.0 \text{ kJ mol}^{-1}$. Using this value in Equation (11), the standard formation enthalpy of AgZr is calculated: $\Delta_{f}H_{AgZr,298K} = -21.8 \pm 5.7 \text{ kJ (mol at)}^{-1}$.

<figure><img src="image_13.png" /><figcaption>Fig. 11. Low magnification longitudinal section SEM observation of the bottom of the ingot from run 1. The Al₃Zr phase appears in white over the grey aluminum matrix. The micrograph shows needle-shaped Al₃Zr particles, interdendritic Al₃Zr and clusters of primary Al₃Zr precipitates.</figcaption></figure>

This determination makes it possible to estimate quantitatively the magnitude of the thermal effect associated with the drop solution of an AgZr particle which would lead to the formation of large clusters of Al₃Zr that do not ultimately dissolve. In this scenario, the chemical reaction reads as:

$$ \langle \text{AgZr} \rangle_{298K} + n(\text{Al})_{1173K} \rightarrow ((\text{Ag}, (n-3)\text{Al}))_{1173K} + \langle \text{Al}_3\text{Zr} \rangle_{1173K} \quad (12) $$

The enthalpy $\Delta H_{xx}$ of the above reaction can be calculated by the following Hess cycle:

$$ \Delta H_{xx} = - \Delta_{f}H_{AgZr,298K} + \Delta_{f}H_{Al_3Zr,298K} + \Delta H_{Al_3Zr,298K}^{1173K} + \Delta_{ds}\overline{H}_{Ag,298K}^{\infty,1173K} - 3\Delta H_{Al,298K}^{1173K} \quad 13 $$

The standard enthalpy of formation of the AgZr compound $\Delta_{f}H_{AgZr}$ has been determined above. Meschel and Kleppa [38] measured the enthalpy of formation of Al₃Zr by high-temperature direct synthesis calorimetry as:

$$ \Delta_{f}H_{Al_3Zr,298K}^{\infty} = -193.6 \pm 5.2 \text{ kJ mol}^{-1} $$

The heat increment of the aluminide from 298 K to the calorimeter temperature (1173 K) can be calculated using classical thermodynamic relations from the Gibbs energy function of Al₃Zr assessed by Fischer and Colinet [39]:

$$ \Delta H_{Al_3Zr,298K}^{1173K} = 91.2 \text{ kJ mol}^{-1} $$

The heat increment of pure aluminum from 298K to the calorimeter temperature (1173 K) is calculated using the SGTE data for pure elements [40]:

$$ \Delta H_{Al,298K}^{1173K} = 36.4 \text{ kJ mol}^{-1} $$

The uncertainties affecting these two last values are considered negligible towards the large uncertainties affecting the formation enthalpies $\Delta_{f}H_{AgZr,298K}$ and $\Delta_{f}H_{Al_3Zr,298K}$. The partial drop solution enthalpy of Ag in liquid Al at infinite dilution at 1173 K was reported by Gajavalli
[35] to be:

$$ \Delta_{ds} \overline{H}_{Ag,298K}^{\infty,1173K} = 42.2 \pm 0.2 \text{ kJ.mol}^{-1} $$

Considering all these values into Eq. (13), the enthalpy of the reaction is determined to be, $\Delta H_{xx} = -125.7 \pm 13 \text{ kJ mol}^{-1}$. This value is in remarkable quantitative agreement with the most exothermic values measured in runs 1 and 2 (Fig. 9). Thus, this calculation constitutes a quantitative consistency check which confirms the reliability of our experimental determination of $\Delta_f H_{AgZr,298K}$.

The AgZr enthalpy of formation obtained in this work is $\Delta_f H_{AgZr,298K} = -21.8 \pm 5.7 \text{ kJ (mol at)}^{-1}$. This value is in very good agreement with the result obtained formerly in our group by Decreton in his PhD thesis [11] using HF-HNO₃ solution calorimetry, albeit with greater uncertainty $\Delta_f H_{AgZr,298K} = -22.3 \pm 10.0 \text{ kJ (mol at)}^{-1}$. It should be noted that these two values are much more exothermic than the only other experimental value, obtained by Fitzner and Kleppa [9] by high temperature direct reaction calorimetry $\Delta_f H_{AgZr,298K} = -3.12 \pm 3.13 \text{ kJ (mol at)}^{-1}$.

# 4. Modelling of the Ag-Zr system

## 4.1. Thermodynamic models

Detailed crystallographic information about the different phases of the binary system Ag-Zr is given in Table 1. The thermodynamic models of the various phases are briefly introduced below. The Gibbs energy functions of all the phases are referred to the so-called “standard element reference” SER, defined using H°298.15K and S°0K for the stable state of the pure elements at 298.15 K and 1 bar. The temperature dependence of the molar Gibbs energy of the pure elements in either stable or metastable phases, referred to the SER state, is given by the expression:

$$ G - HSER = A + BT + CTLnT + DT^2 + ET^3 + \frac{F}{T} + GT^4 + HT^7 + IT^{-9} $$

The Gibbs energy functions of the elements are taken from the SGTE (Scientific Group Thermodata Europe) unary database [40].

The liquid, αZr and βZr solution phases are described by the substitution solution model, in which the Gibbs energy is expressed for the Φ phase, as:

$$ G^{\Phi} = x_{Ag}^{\Phi} G_{Ag}^{\Phi} + x_{Zr}^{\Phi} G_{Zr}^{\Phi} + RT(x_{Ag}^{\Phi} Ln x_{Ag}^{\Phi} + x_{Zr}^{\Phi} Ln x_{Zr}^{\Phi}) + exG^{\Phi} $$

where.

* $G_{Ag}^{\Phi}$ and $G_{Zr}^{\Phi}$ are the molar Gibbs energies of pure Ag and Zr in the Φ phase,

* $exG^{\Phi}$ is the excess Gibbs energy, expressed by a Redlich-Kister polynomial as follows:

$$ exG^{\Phi} = x_{Ag}^{\Phi}x_{Zr}^{\Phi} \sum_{\nu}^{\nu} L^{\Phi}_{Ag,Zr} [x_{Ag}^{\Phi} - x_{Zr}^{\Phi}]^{\nu} $$

where:

* R is the gas constant,

* $x_{Ag}^{\Phi}$ and $x_{Zr}^{\Phi}$, the mole fractions of the components Ag and Zr respectively,

* $L^{\Phi}_{Ag,Zr}$, a binary interaction parameter expressed as a linear function of temperature according to:

$$ L^{\Phi}_{Ag,Zr} = A_{\nu}^{\Phi} + B_{\nu}^{\Phi}T $$

The very small solubility of Zr in Ag(fcc_A1) is not considered in the present assessment. The AgZr and AgZr₂ compounds are modelled without a homogeneity domain, the microprobe measurements of Taguchi [41] on diffusion couples showing compositional gradients of the order of 0.2 mol% around the stoichiometry of the compounds, contradicting the existence of extended non-stoichiometric domains attributed to these compounds by Refs. [18,42].

The Gibbs energies for intermetallic compounds are expressed as following, assuming the Neumann-Kopp rule:

$$ G^{Ag,Zr_q}(T) = A + BT + p^{\circ} G^{fcc\_A1}_{Ag} + q^{\circ} G^{hcp\_A3}_{Zr} $$

in which.

* p and q, are the stoichiometric coefficients,

* $G_{Ag}^{fcc\_A1}$ and $G_{Zr}^{hcp\_A3}$, the reference terms corresponding to the Gibbs energies of pure silver and zirconium respectively,

* A and -B, the enthalpy of formation, $\Delta_f H_{298.15}^0$, and the entropy of formation, $\Delta_f S_{298.15}^0$, of the compound AgₚZrₚ, respectively. The parameters A and B have been evaluated during the optimization process.

The thermodynamic parameters given in Table A-1 have been optimized by using the optimization program BINGSS developed by Lukas et al. [43], which allows to take into account simultaneously all the selected available information, the phase diagram equilibria and the thermodynamic properties, each with a given weight.

## 4.2. CALPHAD modelling and discussion

We have reported in Table 2, our experimental results for the determination of the invariant reactions in the Ag-Zr system as well the results of our CALPHAD modelling for which we have considered the following experimental data.

* the eutectoid composition and temperature, 96.3 mol% Zr and 823 °C respectively, accurately determined by Betterton et al. [15].

* the eutectic transformation determined in this work at 959 ± 2 °C, L ↔ (Ag) + AgZr which is consistent with previous experimental works (955 °C [12], 953 °C [18]). The eutectic composition determined in our study is around 2 mol%Zr, which is in agreement with the value of Zhang et al. [18] 2.7 mol%Zr.

* the liquidus temperatures in the silver-rich composition range (xZr<50 mol%). The liquidus temperatures in the silver-rich part were previously determined by Raub et al. [12] by thermal analysis on cooling in an alumina crucible. They are likely imprecise due to the reactivity between alumina and liquid silver-zirconium alloys and the supercooling that can affect the determination of the liquidus. Their results, however, are consistent with the values measured in the present study in sub-stoichiometric yttria crucible during heating.

* the eutectic transformation L↔AgZr₂+AgZr at 1165 ± 2 °C, with an eutectic composition of 58.5 ± 1.5 mol% Zr, in agreement with Betterton et al. [15].

* the peritectic decomposition of AgZr₂, AgZr₂↔L+(βZr) at 1186 °C. The ended compositions of the peritectic plateau are 66.3 ± 0.4 et 80 ± 1 mol% Zr, respectively. These values are consistent with Betterton's data (Table 2).

For the modelling, our experimental work, as well as Betterton's work, were preferred to the data reported by Loboda et al. [17], and Zhang et al. [18]. The latter has not been selected due to the
experimental protocol in DTA.

* Loboda et al. reported a BeO crucible interaction, and consequently subjected the samples to a single temperature ramp. This method, which is similar to our own method for overcoming crucible reactivity, allowed to obtain qualitatively reliable thermograms. However, a high heating rate of 50 °C.min⁻¹ that they applied in their thermal analysis is a source of uncertainty in temperature measurements.

* Zhang et al. prepared samples in BN crucibles and performed DTA in the same crucibles. Decreton et al. [11] have shown that silver zirconium samples exhibit significant reactivity with BN. From this interaction, we can expect depleted zirconium compositions compared to the target ones, and as a consequence, larger uncertainties in the determination of transition temperatures. Furthermore, the experimental data reported by Zhang et al. are excluded from the optimization because, as highlighted by Okamoto [44], the phase diagram presents questionable features against the Okamoto criteria [45], such as an asymmetric liquidus around the AgZr₂ composition.

* Additionally, the results reported by Betterton et al. [15] concerning the zirconium-rich composition range were derived from metallographic observations of samples that were annealed and subsequently quenched, or analyzed following differential thermal analysis (DTA), all of which remained exclusively in the solid state. These results are not subject to biases arising from the reactivity of the samples, thereby ensuring their reliability.

Fig. 12 displays the computed phase diagram using the optimized parameters, with experimental data overlaid for comparison. The calculated invariant reactions were evaluated against those reported in the literature (Table 2). The results showed reasonable consistency with the experimental findings from this study, specifically the congruent melting of AgZr and the peritectic decomposition of AgZr₂. The calculated liquidus in equilibrium with AgZr exhibits a notably flat profile, in agreement with both current determinations and data reported by Raub [12]. It is important to note that optimized parameters did not produce an inverted miscibility gap in the liquid, even at elevated temperatures. In the Zr-rich region, the solvus identified by Ref. [15] is well reproduced. Additional experimental investigations should be necessary to verify the liquidus and solidus curves in Zr-rich areas.

Fig. 13 shows the calculated enthalpies of formation for the compounds, compared with data from existing literature. Up to now, the enthalpy of formation at 298 K for AgZr and AgZr₂ were only experimentally measured by Fitzner and Kleppa [9], though with a notable margin of error. Previous Calphad modeling [5,7] have depended on this dataset to refine the enthalpy of formation for both compounds. In a later publication, Guo and Kleppa themselves expressed doubts about their initial measurements, indicating that the reported values should be seen as merely indicative. Both DFT calculations and our experimental value for AgZr suggest more negative values. As a result, the computed enthalpies of formation for both compounds (Table 3) in our model are significantly more exothermic compared to earlier Calphad calculations [5,7]. Finally, the current optimized results fall within the wide error range of the experimental value for AgZr determined in this study and are slightly lower than the DFT data for AgZr₂.

The Ag activities displayed in Fig. 14, as calculated by modelling, do not quite match the experimental values. The three compositions studied belong to a common two-phase domain at low temperature and should have the same activity (pressure), as shown by the merging of the dotted, dashed and solid black curves into a single curve on the right-

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 12. Ag-Zr calculated phase diagram (top) and enlarged part compared with available experimental data (bottom).</figcaption></figure>

hand side of the Figure. However, the experimental results show different values for the three compositions in this part of the figure. The optimization without taking into account the Ag experimental activities as input data gives roughly the same modelling, this means that they are overall consistent with other input data, but not sufficiently precise to guide the optimization. We can see that the Zr-richest measured value cannot be realistic if the values at x<sub>Zr</sub> = 0.83 and 0.85 are correct. Values reported from He and al [5] and Hsiao et al. [7] are much higher than the measured ones. It is likely due to their higher enthalpies of formation of intermetallic compounds.

The calculated enthalpy of mixing of the liquid phase is consistent with the scattered experimental data of [9] (Fig. 15).
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Calculated enthalpies of formation of AgZr and AgZr₂ compounds at 298.15 K compared with values from literature (DFT, experiments and CALPHAD models).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Comparison between the calculated and experimental Ag activity in alloys (0.83, 0.85 and 0.90 mol fraction Zr), referred to Ag stable state and calculated literature data from Hsiao et al. [7] and He et al. [5].</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Enthalpy of mixing of the Ag-Zr liquid phase at 1200 °C calculated using our model, compared with experimental data from Fitzner et al. [9] and models from Hsiao et al. [7] and He et al. [5].</figcaption></figure>

**Table A 1**

Gibbs energy parameters for the Ag-Zr system optimized in this work, units are K, J mol·at⁻¹.

<table><tr><td>Liquid (Ag,Zr)</td><td><sup>0</sup>L<sub>Ag,Zr</sub> = -28952.70 + 17.06759 T<br/><sup>1</sup>L<sub>Ag,Zr</sub> = 22093.95-2.37621T<br/><sup>2</sup>L<sub>Ag,Zr</sub> = 3600</td></tr><tr><td>bcc_A2 (Ag,Zr)</td><td><sup>0</sup>L<sub>Ag,Zr</sub> = 8186.64-0.379T<br/><sup>1</sup>L<sub>Ag,Zr</sub> = 13555.37<br/><sup>2</sup>L<sub>Ag,Zr</sub> = 11201.11</td></tr><tr><td>hcp_A3 (Ag,Zr)</td><td><sup>0</sup>L<sub>Ag,Zr</sub> = 14111.42</td></tr><tr><td>AgZr</td><td>G<sub>AgZr</sub> = -17633.44 + 6.64124T + 0.5°G<sub>Ag</sub><sup>fcc</sup>A1 + 0.5°G<sub>Zr</sub><sup>hcp,A3</sup></td></tr><tr><td>AgZr<sub>2</sub></td><td>G<sub>AgZr2</sub> = -12201.51 + 3.33430T + 0.3333°G<sub>Ag</sub><sup>fcc</sup>A1 + 0.6667°G<sub>Zr</sub><sup>hcp,A3</sup></td></tr></table>

**5. Conclusion**

Following a thorough review of existing literature and a new experimental study of the Ag-Zr phase diagram, along with silver activity measurements and standard enthalpy of AgZr determination by calorimetry, a new thermodynamic model for the Ag-Zr binary system is introduced. The controversy about the nature of the decomposition of AgZr and AgZr₂ has been resolved, identifying congruent melting and peritectic decomposition, respectively. The phase diagram is accurately reproduced by the model. However, experimental validation is still needed for the liquidus and solidus of (βZr), despite the challenges posed by measurements for zirconium-rich compositions. The primary challenge in acquiring data within the zirconium-rich region at elevated temperatures is the prevention of melt contamination by the crucible. It is noteworthy that sub-stoichiometric yttria has been successfully tested in this study with melts containing metallic zirconium.

The optimization agrees well with the measured silver activities, providing confidence in the estimated enthalpy of formation of compounds, which were previously considered insufficiently exothermic in models like those of [5,7]. The enthalpy of formation for the AgZr compound, calculated using the optimized parameters matching phase diagram and thermodynamic properties, is within the experimental uncertainty range of the determination of the standard enthalpy of AgZr presently determined by calorimetry, while the AgZr₂ value agrees with the DFT calculation. Future experimental work on the heat capacities of intermetallic compounds could further improve the Ag-Zr system modelling.

**CRediT authorship contribution statement**

**C. Antion:** Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation. **M. Barrachin:** Writing – review & editing, Writing – original draft, Validation, Supervision, Project administration, Formal analysis. **P. Benigni:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation. **I. Crassous:** Writing – review & editing, Methodology, Investigation, Data curation. **A. Decreton:** Methodology, Investigation, Data curation. **E. Fischer:** Writing – review & editing, Writing – original draft, Validation, Software, Methodology, Investigation. **S. Gyasi:** Methodology, Investigation, Data curation. **A. Janghorban:** Writing – review & editing, Writing – original draft, Validation, Methodology, Investigation, Formal analysis, Data curation. **M. Lomello-Tafin:** Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Data curation. **G. Mikaelian:** Validation, Formal analysis, Data curation. **I. Nuta:** Writing – review & editing, Writing – original draft,
Validation, Methodology, Investigation, Formal analysis, Data curation.
**J. Rogez:** Writing – review & editing, Supervision, Methodology, Formal analysis.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors wish to thank Christian Chatillon for his helpful com-
ments on this work and Sylvie Chatain (CEA, France) for providing yttria
crucibles.

Data availability

Data will be made available on request.

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