Hichem Khedim

Sandra Abdelouhab

Renaud Podor

Christophe Rapin

Michel Vilasi

Pierre-Jean Panteix

Michael Toplis

François Faure

Kinetic and equilibrium factors affecting saturation of chromium oxide in soda-silicate melts

Keywords: glass melts, chromium oxide, solubility, kinetic

Factors controlling the rate at which chromium oxide reaches saturation in Na2O-x SiO2 liquids have been studied as a function of melt composition and oxygen fugacity. Under an oxidising atmosphere, liquid Na2CrO4 or Cr2O3 crystals can be in equilibrium with soda-silicate melts, depending on the concentration of sodium in the studied system. Under reducing conditions, NaCrSi2O6 is stabilized in silica-rich melts when T is lower than 1160°C, while Cr2O3 is in equilibrium with the sodium-rich melts when T is above 1160°C. The chromium oxide (Cr2O3) to pyroxene (NaCrSi2O6) transformation is described in terms of the time required to attain chemical and textural equilibrium. Na2CrO4, NaCrSi2O6 and Cr2O3 phase stability domains are reported as well as the Na2O-SiO2-Cr2O3 phase diagram in the studied temperature and fO2 range.

Introduction

Most of the metallic alloys used in the glass making industry are chromia (Cr2O3)forming superalloys [1,2]. However, despite the fact that chromia exhibits low solubility in molten glass even at high temperature, there is a general lack of experimental data concerning the factors that affect chromia solubility in molten glasses. This state of art has stimulated specific studies of Cr2O3 solubility in the Na2O-xSiO2 system as a function of temperature (1000-1300°C), glass composition (x=1.5 to 4), partial pressure of oxygen (10 -18 atm < fO2 < 0.20 atm) and the addition of other oxides containing network-forming and/or network-modifying cations (B2O3, Al2O3 or CaO) [START_REF] Abdelouhab | Détermination de grandeurs physicochimiques dans les verres fondus -Relation avec le comportement du chrome et d'alliages chromine-formeurs[END_REF][START_REF] Abdelouhab | [END_REF][5][6][START_REF] Khedim | Vilasi -Solubility of chromium oxide in binary soda silicate meltssubmitted[END_REF]. One of the principal results of these recent studies is that the attainment of thermodynamic equilibrium involves a large number of reactions between numerous chromium-bearing species in mineral, liquid and gaseous phases, even in a compositionally simple system [5][6][START_REF] Khedim | Vilasi -Solubility of chromium oxide in binary soda silicate meltssubmitted[END_REF]. This complex situation is due to the fact that chromium can exhibit several oxidation states depending on temperature, oxygen fugacity fO2, and melt composition. For example, in air, Cr III and/or Cr VI species dominate, while under more reducing conditions Cr II may coexist with Cr III [5][6][START_REF] Khedim | Vilasi -Solubility of chromium oxide in binary soda silicate meltssubmitted[END_REF].

The work presented here considers the mechanisms leading to a thermodynamic equilibrium from a simple initial scenario of crystalline Cr2O3 placed in contact with liquids in the Na2O-xSiO2 system. In detail, the aim is to quantify the kinetics of individual reactions (under various fO2 conditions), to identify which of these processes may be rate-limiting, and to constrain the time-scale for attaining thermodynamic equilibrium. The reactions that may occur are presented in Figure 1 [5][6][START_REF] Khedim | Vilasi -Solubility of chromium oxide in binary soda silicate meltssubmitted[END_REF]. An essential step towards equilibrium involves the dissolution of crystalline Cr2O3, to form Cr III oxy-anions in the liquid (equation 1). Equation 1is written involving oxygen-species of the melt (O 2- (melt)) to highlight the fact that melt basicity influences the behaviour of chromium in this system.

½Cr2O3 (crystal) + xO 2- (melt)  Cr III Ox+3/2 2x- (melt)
Equation 1

In the simplest case these Cr III -bearing anions (Cr III Ox+3/2 2x- (melt)) may be stable.

Alternatively, depending on oxygen fugacity, reduction to Cr II -bearing species, or oxidation to Cr VI -bearing species may occur in the liquid (described by equations 2 and 3 respectively):

2Cr III Ox+3/2 2x- (melt) + (2y -2x -2)O 2- (melt)  2Cr II Oy (2y-2)- (melt) + 1/2O2 (g) Equation 2

2Cr III Ox+3/2 2x- (melt) + 3/2O2 (g) + (2y-2x -6)O 2- (melt) 2Cr VI Oy (2y-6)- (melt)

Equation 3

The reduction reaction (Eq 2) can be neglected in oxidising atmospheres such as air, but must be taken into account under more reducing conditions. On the contrary, the oxidation reaction described by Eq 3 can be neglected in a strongly reducing atmosphere. It is also worth of note that if oxygen fugacity conditions are highly reducing or highly oxidising, crystalline Cr2O3 may no longer be the stable phase in equilibrium with the liquid and other Cr-bearing phases may precipitate. Furthermore, under oxidising conditions the behaviour of chromium is complicated by the volatility of Cr VI (principally in the form of CrO3(g) [8]. This is particularly true for Na2Orich melts of the system Na2O-xSiO2 in air, for which Cr VI species are known to be stable [9,10].

Formation of CrO3(gas) may occur from Cr VI oxy-anions in the liquid (equation 4), or through direct oxidation/volatilisation of unreacted Cr2O3 crystals that are in contact with the surrounding atmosphere (equation 5). Both of these equilibria are considered in this study because CrO3 formation from Cr2O3 is known to occur in air when T>1000°C [11].

2CrOy (2y-6)- (melt) + 6O 2- (melt) = 2CrO3 (gas) + 2yO 2- (melt)

Equation 4

Cr2O3 (crystal) + 3/2O2 (gas) = 2CrO3 (gas) Equation 5

From a kinetic point of view, a constant concentration of chromium in the liquid phase will only be possible if the rate of Cr2O3 crystal dissolution (equation 1), and the rates of reduction or oxidation (equations 2 and 3), are rapid compared to the volatilisation of CrO3

(equations 4 and/or 5). Furthermore, if a phase different from Cr2O3 is stable in the liquidus then the crystallisation of this new phase may also affect achievement of the equilibrium state. In this study, the kinetics of individual dissolution and volatilisation reactions described above (equations 1 and 5) are quantified in the Na2O-xSiO2 system. The limits of the phase field of Cr2O3 are also considered, and alternative saturating phases and their stability fields are identified. The work is mainly experimental, but it also involves quantification of the liquid and solid compositions, as well as the relative proportions of the present phases, as detailed below.

I Experimental procedure

I.1. Synthesis of glasses

Several glasses were prepared for the Na2O-xSiO2 system with 1.5 ≤ x ≤ 4.0. 300g of each Cr-free glass were synthesised from mixtures of reagent Na2CO3 (Aldrich, 99.5%) and SiO2

(Aldrich, 99.9%). After weighing, the powders were mixed in a RETSCH® agate mortar for 15 minutes using a rotation speed of about 60 rpm. The subsequent mixture was then placed in a platinum crucible and held for 2 hours at 1200°C, then 2 hours at 1400°C and finally 24 hours at 1100°C. The time spent at 1400°C was kept relatively short in order to limit the loss of Na2O.

The chemical composition of some Cr-free glass was determined using Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES). These results (Table 1) show a small difference between the nominal Na / Si and those determined by ICP-AES. This offset does not exceed 3 relative % and can be attributed to a small extent of sodium volatilisation during the synthesis of the glass at high temperature. Chromium oxide was then added to each glass as crystalline Cr2O3 (Alfa Aesar, 99%). The mixtures were thoroughly mixed by grinding in an agate mortar.

I.2. Experimental methods

The experiments were performed in a 1750°C Nabertherm® muffle furnace. A Pt -10%Rh thermocouple was systematically introduced into the furnace near the sample in order to measure the temperature. Two different experimental procedures were employed.

For the first procedure, approximately 100 mg of the glass + Cr2O3 mixture was pressed onto platinum wire loops using polyvinyl alcohol as a binder and pre-heated for 2 minutes at T=1000°C. The diameters for the Pt-wire and the loops are approximately 0.2 mm and 2-3 mm respectively. Typically, six samples prepared in this way were hung together in the furnace, each sample being attached to an independent 1mm diameter Pt wire. The gas in the tube furnace is air, and individual wire loops were removed from the furnace after different durations.

The second procedure was designed for working under reducing conditions [12].

Approximately 100 mg of glass + Cr2O3 mixture was placed in a Pt-container, introduced into a silica tube which was sealed under secondary vacuum and then heated at the desired temperature in a muffle furnace. In order to limit Na2O volatilisation, a sample of the same glass composition was added in the reactor, buffering the partial pressure of Na2O. The oxygen fugacity in the reactor was controlled by the use of different oxide/metal solid buffers. After the required run duration, which is depending on temperature and glass composition, the silica tubes were removed from the furnace and quenched in cold water.

I.3. Sample preparation and analysis

Quenched samples were embedded in epoxy resin and then polished to 1 µm using anhydrous ethanol as a lubricant. The polished sections were then analysed using a fully automated CAMECA SX 100 electron microprobe with 5 Wavelength Dispersion Spectrometers (WDS). The elements analysed were Na, Si and Cr. The concentration of oxygen was calculated by stoichiometry assuming that the valence states of the cations are 1, 4 and 3 respectively. The standards used were NaAlSi3O8 for Na and Si, and Cr2O3 for Cr. The values of accelerating voltage and beam current were chosen depending on the distribution of Cr2O3 crystals in the glass. When the Cr2O3 crystals were smaller than 2 µm and the maximum distance between two crystals was less than 25 µm, the electron beam conditions were 10kV and 5nA. Because of the presence of abundant crystals of Cr2O3 which limited the size of glass pools, an incident-beam diameter of 5 µm was used. When the Cr2O3 crystals were larger than 2 µm and the maximum distance between two crystals was greater than 25 µm, the electron beam conditions are 25kV and 10nA. The incident beam diameter was 12 µm. The analysis conditions (accelerating voltage, beam current, counting times on peaks and backgrounds, average volume analysed) are reported in Table 2. In both cases, the three elements were analysed on three distinct wavelength dispersion spectrometers at the same time.

Comparative results of analyses performed on a Na2O -2SiO2 glass doped with 1 wt% Cr2O3 using both electron beam conditions are reported in Table 2 and illustrated in Figure 2.

These results indicate that the analyses performed with the second program are much more precise than those performed at lower acceleration voltage. Furthermore, the dispersion of the values around the mean is higher when the analyses are obtained using a low acceleration voltage. However, even in this case, electron microprobe measurements of the starting glasses are found to be systematically depleted in sodium relative to the values determined by ICP-AES (Table 1), implying some volatilisation of sodium under the electron beam at these conditions (the Na2O depletion is even more intense when the P1 program is used). However, the magnitude of this effect is limited and never exceeds 5 relative percent. Thus, a large beam size and a high voltage conditions (P2 program) have been used whenever possible. Unless otherwise stated, these are the analysis conditions for measurements shown in Table 2 such that all analyses are comparable.

I.4. Determination of the volume (and mass) fraction of Cr2O3 crystals in a glass pellet using image analysis

The quantification of the Cr loss by CrO3 volatilisation requires the determination of the bulk-Cr content of the samples after heat treatment. For this purpose, quantification of the relative proportions of residual Cr2O3 crystals and coexisting glass is required. To quantify the volume (or mass) fraction of Cr2O3 crystals, digital photographs of the polished sections surface of the sample were taken under reflected light using an Olympus Vanox AHMT optical microscope.

These images, in which glass is clearly distinguished from crystals, were treated using the AnalySIS software [START_REF]Software Analysis[END_REF], which allows the determination of the relative surface areas of glass and

Cr2O3. These values were then converted into relative mass proportions using the referenced density values () of glass and crystals [START_REF] Shermer | [END_REF]15]. The sample was then re-polished in order to remove approximately 100µm of thickness and another set of images were recorded and analysed. These procedures were repeated until the whole sample was investigated. The bulk Cr2O3 weight percentage of the samples was determined using the following equation:

) ( ) ( % ) ( ) ( % ) ( ) ( % ) ( % 1 1 3 2 3 2 1 3 2 3 2 3 2 = =                +           =    = = = n i n i n i glass e

II. Results

II.1 Volatilisation of Cr(VI) dissolved in the glass melt

Before considering how crystalline Cr2O3 dissolves in sodium-silicate melts (equations 1 to 3), the loss rates of Cr VI species are first considered (equations 4 and 5). In order to quantify the rate of the reaction 4, 10g of Na2O-2SiO2 glass doped with 1 wt% Cr2O3 was prepared from a mixture of Na2O-2SiO2 glass and Cr2O3 crystals, heated at 1100°C for 24h in air. This concentration of Cr2O3 being below the saturation level [5][6][START_REF] Khedim | Vilasi -Solubility of chromium oxide in binary soda silicate meltssubmitted[END_REF], no crystalline Cr2O3 should be present in the melt. The glass composition was determined by EPMA (Table 1), and the mass fraction of Cr2O3 in the glass was found to be 0.93  0.04. Observation under the optical microscope confirmed the absence of crystalline Cr2O3. This starting material was then used for subsequent time-series experiments. The obtained glass was yellow, indicating the presence of Cr VI [5][6][START_REF] Khedim | Vilasi -Solubility of chromium oxide in binary soda silicate meltssubmitted[END_REF].

150 mg beads of this glass were suspended on a Pt wire and heated at 997°C, 1107°C, 1207°C and 1310°C for run durations ranging from 10 to 120 minutes. The concentration of Cr2O3 after the heat treatment was quantified by EPMA from the core to the rim of each sample.

These measurements indicate that there is no chromium concentration gradient between the edge and the centre of the samples. On the other hand the average values of Cr2O3 are decreasing as time increases. This result is interpreted in terms of loss of Cr VI over time, while the lack of an internal concentration gradient implies that volatilisation is not limited by the diffusion of chromium in the glass.

In detail, the mass fraction of Cr2O3 remaining in the melt decreases with increasing temperature and run duration (Table 3; Figure 3). Assuming that loss of Cr VI occurs through a surface controlled process, it may be shown that the volatilisation rate at a constant temperature can be derived from the variation of C/C0 as a function of time, where C is the concentration of Cr in the liquid at the considered time, and Co is the initial concentration. The variation of C/Co is found to be linear as a function of time at constant temperature, allowing the definition of VR (the volatilisation rate of Cr VI ), at the measured temperature (Table 3). As expected, the volatilisation is faster at higher temperature. The kinetics of Cr volatilisation is totally different from the alkali-metal oxide (Na2O or K2O) volatilisation. Indeed, in the case of alkalis, volatilisation curves exhibit logarithmic variations with time exposure [16][17], while Cr volatilisation is linear with time.

Furthermore, when ln(VR) is plotted as a function of inverse temperature, an Arrhenian temperature dependence is found (Figure 4). From this correlation it is possible to quantify the activation energy for the volatilisation of Cr VI species dissolved in Na2O-2SiO2 liquid (equation 7):

       - = T R E C V R 1 ln Equation 7
where C is a constant, R is the ideal gas constant, T is the absolute temperature, and E is the activation energy. The value of E derived from this set of data is 108 kJ.mol -1 .

II.2 Volatilisation of CrO3 from Cr2O3 crystals

In the absence of melt, the formation and volatilisation of CrO3 from Cr2O3 crystals (equation 5), has been investigated by Stearns [18]. He showed that this reaction can be neglected when the temperature is lower than 1000°C, but in the temperature range of this work, this reaction must be taken into account. To confirm the influence of relation 5 and to quantify its effect, an additional experiment has been performed using Na2O-1.5SiO2 + 40wt% Cr2O3 mixture heated in air at 1300°C for 120 minutes. This amount of Cr2O3 is much higher than the equilibrium content of 8.15 wt% Cr2O3 for this composition at T=1300°C [START_REF] Abdelouhab | Détermination de grandeurs physicochimiques dans les verres fondus -Relation avec le comportement du chrome et d'alliages chromine-formeurs[END_REF]. The volatility of chromium in this experiment has been quantified by performing a mass balance calculation of the heat-treated sample. For this purpose the remaining Cr2O3 crystals has been quantified from 3D reconstruction of the sample (part I.4), and the liquid composition has been determined by EPMA. The mass of Cr2O3 lost from the sample may be determined using the following equation: Using these values, a loss of 14 wt% Cr2O3(volatilised) is obtained, resulting from both direct volatilisation of Cr2O3 crystal as well as volatilisation of Cr VI species dissolved in the liquid (the effects of reaction 4 and 5 cannot be decoupled). Assuming that the Cr VI volatilisation rate from the Na2O-1.5SiO2 liquid is approximately the same as that determined for the Na2O-2SiO2 melt, the predicted loss of Cr2O3 directly from the liquid after 2 hours is approximately 0.6 wt%. It can be concluded that the principal contribution to chromium loss during this experiment was the volatilisation of crystalline Cr2O3 (reaction 5) rather than loss from the liquid (reaction 4). This result implies that a sufficient excess of crystalline Cr2O3 must be added to the initial mixture when quantifying the equilibrium concentration of chromium in a Cr2O3 saturated liquid.

II.3 Dissolution kinetics of Cr2O3 under oxidising conditions

As noted above, in the Na2O-xSiO2 system, the dissolution of crystalline Cr2O3 can produce Cr(III) and/or Cr(VI) in the liquid when the system is in equilibrium with an oxidising atmosphere such as air (equations 1 and 3). In this section the experimental dissolution of crystalline Cr2O3 into sodium-silicate liquids is described as a function of temperature under such conditions. Liquids of Na2O-1.5SiO2, Na2O-2SiO2, Na2O-3.5SiO2 and Na2O-4SiO2 compositions, to which 5 to 20 wt% Cr2O3 have been added, were studied. Experiments were performed on glass beads placed directly in the furnace as described above. Identification of the presence of Cr(VI) species was simplified by the fact that such species impart a yellow colour to the glass.

Optical and SEM observations of heat treated samples show that for a short heat treatment, the external part of samples contains higher amounts of Cr(VI) and lower amounts of residual crystalline Cr2O3 compared to the central part of the sample (Figure 5). For this case, EPMA analyses show significant differences in total chromium content (corresponding to the sum of Cr(III) and Cr(VI)) between the core and rim, sample rims being systematically enriched in dissolved chromium in comparison to the core (Figure 6). With increasing time, the optically visible limit of the oxidised zone migrates towards the centre of the sample. In terms of liquid chemistry, the Cr-content of the sample rim is more or less constant, while the Cr-content of the core increases, finally reaching values observed at the sample rim.

In detail, the time required for core and rim compositions to converge depends on the liquid composition and temperature, as discussed below, while in certain cases chromium concentration decreases at the sample rim for long run duration.

Based upon these results the following sequence of events can be proposed:

-Upon heating above the glass transition, the dissolution of Cr2O3 crystals occurs througout the sample, producing Cr III in the liquid.

(equation 3) is limited by the availability of oxygen, initially introduced at the surface of the sample and then transported by diffusion to the bulk.

-During this stage preferential loss of CrO3 through volatilisation leads to more rapid elimination of residual Cr2O3 grains in parts of the liquid dominated by Cr VI species.

-Once the limit of the oxidised zone has moved through the whole sample, an approach to equilibrium may be reached, as illustrated by cases where the mole fractions of total chromium are equal at the rim and at the core of the sample (Figure 6).

-If the heat treatment of the sample is prolonged, the continuous loss of Cr VI species at the edge of the sample may lead to a decrease of the total chromium content of the liquid, by a combination of the mechanisms described in equations 4 and 5.

As noted above, the comparison of data for different liquids shows that the time to reach a state of equilibrium (convergence of core and rim compositions) depends on the liquid composition and temperature; For example, as illustrated in Figure 6, equilibrium is reached earlier in the Na2O-2SiO2 melt than in the Na2O-4SiO2 melt, even if the final total chromium content is lower in Na2O-4SiO2 compared to Na2O-2SiO2. This behaviour may be observed due to the rate of oxygen diffusion. For example, from a qualitative point of view, Na2O-4SiO2 liquid is more polymerised than Na2O-2SiO2, such that oxygen diffusion in the former composition is slower, thus requiring more time to reach an equilibrium state in terms of Cr concentration. This idea may be tested in a more quantitative way, given that the diffusion coefficent of oxygen is known to be proportional to shear viscosity [19,20]. Indeed, when the time required to reach equilibrium is considered as a function of the shear viscosity of the liquid (estimated from literature data for binary sodium silicates [21][22][23]), a very good correlation is observed (Figure 7), supporting the idea that the time required to reach equilibrium is indeed controlled by oxygen diffusion. This correlation seems to word as for a given melt composition at known temperature (i.e. a given liquid viscosity), the time to reach equilibrium may be determined from Figure 7 once a sample geometry correction has been made (all samples here are considered as glass pellets of 5mm diameter).

One final point of note is that electron micro-probe analyses of crystalline phases in these experiments showed that small drops of Na2CrO4 were observed on the platinum wire directly in contact with the silicate melt when x ≤ 2.5. This compound is a liquid at the experimental temperature, but is inferred to be the liquidus phase (rather than Cr2O3) for sodium-rich melts under oxidising conditions. For compositions with x>2.5, the stability of Cr2O3 depends on temperature, as detailed below.

II.4 Dissolution kinetics of Cr2O3 under reducing conditions

A similar procedure has been performed under reducing conditions in order to determine the effect of fO2 on the kinetics of dissolution. Under low fO2, oxidation of Cr III to Cr VI and associated volatility are not expected. The formation of Cr II species may be predominant (eq. 2).

For these experiments, the temperature is found to play a major role, and the results above and below 1160°C are distinguished.

II.4.1 T > 1160°C

When temperature is higher than 1160°C at low oxygen fugacity, the dissolution of Cr2O3 is often accompanied by a change of melt color from green to blue due to the presence of Cr II species. Crystalline textures and dissolved Cr content are reported in Figure 8 as a function of run duration. The dissolution process is found to be accompanied by an aggregation of Cr2O3 crystals and the formation of extensive zones enriched in dissolved Cr. In these conditions, the time necessary to reach equilibrium was established to be no longer than 16 hours. Similar behaviour is observed for all the compositions.

II.4.2 T<1160°C

In the temperature range from 1100 to 1160°C, under reducing conditions, a significant difference in behaviour is observed between silica-rich and silica-poor liquids. For silica-poor liquids (Na2O-1.5SiO2 and Na2O-2.5SiO2), Cr2O3 is the only crystalline product that is in equilibrium with the melt. In the case of Na2O-2SiO2 melt, under a Ni/NiO buffered atmosphere and T=1100°C (Figure 9), equilibrium is reached in less than 24 hours, and no significant changes are measured in the Na/Si ratio of the melt relative to the initial value.

On the other hand, when the melt composition is silica-rich (2.5 < x < 5), and the temperature less than 1160°C, a more complicated behaviour is observed. Several experiments have been performed as a function of temperature, with run durations up to 96 hours. The samples exhibit a variety of features as a function of run duration, which can be discussed as the following points:

1. For a short run durations, a crystal-free zone develops at the melt/gas interface, with extended zone where crystalline Cr2O3 is in equilibrium with the melt (Figure 10). During this step the melt contains a relatively high content of dissolved Cr.

2. After a few hours (the exact duration depends on temperature and melt composition, but is generally more than 16 hours) the Cr2O3 crystals initially present in the melt (1 to 5 µm in diameter) coalesce to form aggregates, typically 10 to 100µm in diameter. At this time, the precipitation of crystals of NaCrSi2O6 pyroxene (kosmochlor) is observed. The fact that the crystallisation of pyroxene does not take place immediately may imply that its nucleation is delayed. In this respect, the skeletal shape of pyroxene crystals is consistent with a rapid growth, potentially related to supersaturation. It is noted that the delayed nucleation and rapid growth are typical features of pyroxene crystallisation in both natural and experimental systems [24][25][26][27]. Immediately after the appearance of the first pyroxene crystals, two different zones of the samples which are enriched in crystalline Cr2O3 and NaCrSi2O6 can be distinguished respectively. The Cr concentration of the melt is observed to be significantly lower in the zones rich in pyroxene crystals.

3. With further increase in time, the complete dissolution of Cr2O3 and the precipitation of NaCrSi2O6 throughout the whole sample occur. The pyroxene crystals can be large enough to fall to the bottom of the melt drop. Some of these crystals are even big enough to fall out of the bead. Therefore, under these conditions, it would therefore appear that crystalline Cr2O3 is not stable, and that the chromium content of a liquid in equilibrium with pyroxene crystals is lower than that of the same liquid in equilibium with crystalline NaCrSi2O6 is the only product that is in equilibrium with the Na2O-3.25SiO2 melt after 50 hours run duration at this temperature.

II.5 Determination of the transition temperature from Cr2O3 to NaCrSi2O6

The total chromium solubilities determined in Na2O-3SiO2 melt are plotted in an Arrhenius diagram, for temperatures in the range from 1100 to 1250°C (Figure 12). The lower temperature domain corresponds to the liquids in equilibrium with NaCrSi2O6, while the higher temperature domain corresponds to the liquids in equilibrium with Cr2O3. Data from each domain are self-consistent, but there is a clear change in slope of the variation of Cr-content of the liquid between these two domains. The temperature at which the change in slope occurs is interpreted as the transition temperature from the domain in which the melt is in equilibrium with Cr2O3 to that in equilibrium with NaCrSi2O6. As illustrated in Figure 12, this temperature is estimated to be 1160(±15)°C for Na2O-3SiO2.

An interesting point to note is that for silica-rich melts at temperature lower than 1160°C, the melt may be in metastable equilibrium with Cr2O3 crystals for many hours before the appearance of pyroxene crystals. When the chromium content of these metastable liquid compositions is considered, an excellent agreement with equilibrium values extrapolated from higher temperature is observed (Figure 12). Finally, the reversibility of this reaction has been tested while dissolving pyrochlor crystals in soda-silicate melts at T=1200°C, in Na2O-3SiO2 melt. In this case, only Cr2O3 crystals are observed to be in equilibrium with the melt after 24 hours run duration.

II.6 Effect of fO2 and temperature on crystallisation

II.6.1 Decreasing the fO2

The study of the effect of decreasing the fO2 on Cr2O3 crystallisation in melt has been performed using Na2O-2SiO2 melt equilibrated with 3 wt% Cr2O3 in air (oxidising conditions), at T=1200°C. At this condition, Cr dissolved in the melt is dominantly in the form of Cr VI species [5][6][START_REF] Khedim | Vilasi -Solubility of chromium oxide in binary soda silicate meltssubmitted[END_REF]. The melt drop was placed in a graphite crucible and heated for 5 minutes at T=1200°C. In this reducing atmosphere, the Cr VI present at the edge of the melt drop is quickly reduced to the 3+ oxidation state. The Cr III content in this part of the melt becomes higher than the Cr2O3 solubility limit and results in a rapid precipitation of Cr2O3 crystals, as evidenced by the acicular morphology of these crystals (Figure 13). Moreover, Cr2O3 crystals display a preferential orientation that is typical of constrained growth under a compositional or thermal gradient [28].

II.6.2 Decreasing the melt temperature

The effect of a decrease in temperature has also been studied. The Cr III supersaturated melt drops used for reprecipitation experiments were prepared by completely dissolving Cr2O3 crystals in Na2O-3.25SiO2 melt at T=1250°C, in the presence of a Ni/NiO buffer. Samples were then heated at 1150°C for durations varying from 1 to 49 hours. The quantity and morphology of Cr2O3 and pyroxene crystals were systematically determined. The data collected are summarized in Figure 14 and 15. The Cr2O3 crystals nucleate very rapidly and the number of nucleus reaches a maximum after 9 hours (Figure 14b). Their platelet-like morphology (Figure 14a) is characteristic of crystals grown from a supersaturated liquid. Subsequently, the Cr2O3 crystal density slowly decreases and the mean size of the Cr2O3 crystals increases as a function of time, thus indicating grain ripening (Figure 14c).

The total number of NaCrSi2O6 crystals which have nucleated and grown in the melt (Figure 15a) are much more difficult to determine compared to the Cr2O3 grains. Indeed, the pyroxene crystals grow very rapidly (Figure 15b) and, as their density is higher than that of the melt, they sediment to the bottom of the melt drop. They are thus generally concentrated in this part of the sample and their distribution in the sample cross section is not as representative as that of Cr2O3. Nevertheless, the data reported in Figure 15b and 15c indicate that the size of these crystals is very large.

III Discussion: stability domains of chromium species and phases diagrams

III.1 Stability domains of Cr-containing solid phases in equilibrium with soda-silicate melts

The compositions of the solid phases that are in equilibrium with the silicate melt strongly depend on melt composition. Three distinct phases have been found to be in equilibrium with the liquid. Their respective stability domains are reported in Figure 16. Liquid Na2CrO4 is in equilibrium with Na2O-rich melt in the high fO2 domain. The phase stability domain of NaCrSi2O6 pyroxene covers the Na2O-poor part of our system, when temperature is lower than 1160°C, irrespective of fO2. Cr2O3 is the stable equilibrium phase in the rest of the (T, x, fO2) domain.

Data of the literature have only described the presence of Cr2O3 in equilibrium with sodasilicate melts [29][30][31][32], both during dissolution tests of chromium oxide, and studies of the corrosion of chromium-bearing alloys by molten silicates. In particular, the formation of pyroxene crystals is generally not reached during corrosion experiments performed using chromia-former alloys [33]. The formation of pyroxene crystals at the metal/melt interface was previously reported only once [START_REF] Abdelouhab | Détermination de grandeurs physicochimiques dans les verres fondus -Relation avec le comportement du chrome et d'alliages chromine-formeurs[END_REF]34], but was not explained in details.

III.2 Na2O-Cr2O3-SiO2 phase diagram

The Na2O-SiO2-Cr2O3 phase diagram can be established in the Cr2O3 and SiO2-rich domain, under reducing conditions. Two distinct phase diagrams corresponding to the temperature domains T<1160°C and T>1160°C are reported in Figure 17 The NaCrSi2O6 compound is not stable in the high-temperature part of the Na2O-SiO2-Cr2O3 phase diagram (Figure18). Only one-three phased domain, corresponding to the SiO2 -Cr2O3 -L3 equilibrium, still exists when T is higher than 1160°C. In this case, Cr2O3 or SiO2 crystals are in equilibrium with the Na2O-xSiO2 liquid compositions.

Concluding remarks

The objective of this work is to determine the dissolution kinetics of chromium oxide in sodasilicate melts, in order to determine the times necessary to attain the equilibrium state.

Equilibrium values and dissolution mechanisms have consequently been studied, requiring development of a variety of new experimental procedures:

❖ In order to limit sodium loss through volatilisation and thus maintain a constant glass composition, certain experiments were performed in a reactor, developed in our laboratory [5,12], in which temperature and oxygen fugacity may be independently controlled.

❖ The analysis conditions for the EPMA have been carefully studied and optimised in order to systematically control glass compositions, and the results of certain EPMA analyses have been compared to those obtained by ICP-AES measurements.

❖ A specific effort has been made to consider all possible reactions between Cr-bearing species. In detail, chromia dissolution in silicate melts can be described by The principle results obtained may be summarized as follows:

❖ The time to reach equilibrium mainly depends on temperature and melt composition (i.e.

viscosity and oxygen diffusivity). Indeed, at fixed temperature, a short equilibration time (a few minutes) can be expected when studying basic melts under oxidising atmospheres, while very long run durations are necessary to reach the thermodynamic equilibrium in silica-rich melts, under reducing conditions.

❖ The rate limiting effect of oxygen diffusion on the oxidation (or reduction) of chromium is clearly illustrated by the concentric zone of melt drops after short reaction times. In these cases, the differences between the chromium contents in the core and in the rim of the glass drops are important and are attributed to differences in the extent of the oxidation (or reduction) reaction.

❖ The main parameter that controls and limits the kinetics of chromium dissolution in soda silicate melts is the oxygen diffusion. This conclusion is supported by recent studies performed by Magnien et al. [35] and Roskosz et al. [36] on the kinetics of iron redox reactions in silicate melts. Magnien et al. [35] have determined that at superliquidus temperatures, the diffusion of oxygen represents the dominant mechanism in the kinetic of iron redox reactions in silicate melts. Roskosz et al. [36] described a more complex mechanism where the rate-limiting step is either diffusion of oxygen in the volume, or incorporation of oxygen at the sample surface that limits the bulk oxidation state of the liquid.

❖ The existence of the pyroxene phase has been clearly evidenced in the soda-silicate melts with the lowest Na contents and for temperatures lower than 1160°C. By taking the existence of this phase into account, the morphology of corroded chromium or chromiumalloys by molten glasses may be better understood.

  wt%Cr2O3 (volatilised) = wt%Cr2O3 (intial crystal) -(wt%Cr2O3 (melt) + wt%Cr2O3 (final crystal)) Equation8where: wt% Cr2O3(initial crystal) : Initial Cr2O3 content in the glass before the heat treatment = 40% wt% Cr2O3(melt) : Final Cr2O3 content dissolved in the glass after the heat treatment = 10% wt% Cr2O3(final crystal) : Final Cr2O3 content in the crystal after the heat treatment = 16%

  equilibrium at these conditions was checked by comparing the final Cr solubilities determined by dissolving Cr2O3 crystals in Na2O-3.25SiO2 melt (undersaturated experiment) or reprecipitating crystals from Cr2O3 supersaturated Na2O-3.25SiO2 melt (supersaturated experiment) under Ni/NiO buffered atmosphere, at T=1150°C. Final chromium contents are reported in Figure 11 as a function of run duration. Chromia dissolution and precipitation equilibria were reached after 25 hours run duration. No significant changes are measured in the final Na/Si ratio in the melt drops.

  and 18 respectively. The low-temperature phase diagram (Figure17) shows the presence of the NaCrSi2O6 compound, a Na2O-xSiO2 liquid phase that is enriched in Cr2O3 and three distinct phase domains consisting of: (a) SiO2 -Cr2O3 -L1 (b) L1 -NaCrSi2O6 -Cr2O3 (c) NaCrSi2O6 -Cr2O3 -L2

  5 equations: o an acid-base reaction (dissolution to produce Cr III species in the liquid) o two redox reactions (oxidation and reduction of dissolved Cr III into Cr VI and Cr II respectively) o two volatilisation reactions which produce CrO3 gas. This reaction can be due to (i) the volatilisation of dissolved Cr VI , and/or (ii) the oxidation of crystalline Cr2O3. The importance of these reactions has been quantified by specific experiments. It has been shown that they are of negligible importance for temperatures higher than 1000°C.

Figure 1 :

 1 Figure 1: Schematic representation of the equilibria involved in the dissolution of Cr2O3

Figure 2 :

 2 Figure 2: Distribution of the EPM analyses of a Na2O -2SiO2 glass doped with 1 wt% Cr2O3

Figure 3 :

 3 Figure 3: Variations of the Cr2O3 wt% dissolved in the initial Na2O-2SiO2 (+ 0.93% Cr2O3)

Figure 5 :

 5 Figure 5: SEM observations on Na2O-2SiO2 drops containing 15 Cr2O3 wt% heat treated at

Figure 6 :

 6 Figure 6: Dissolution kinetics of Cr2O3crystals at T=1230°C from supersaturated Na2O-2SiO2

Figure 7 :

 7 Figure 7: Time necessary to reach equilibrium reported as a function of melt viscosity

Figure 8 :

 8 Figure 8: Equilibration of Cr2O3 with Na2O-3SiO2 soda-silicate melts at T=1200°C

Figure 9 :Figure 10 :

 910 Figure 9: Kinetics of dissolution of chromium oxide in Na2O-2SiO2 at T=1150°C, fO2=2.3.10 -8 bars (Ni/NiO buffer)

Figure 11 :

 11 Figure 11: Dissolution kinetics of Cr2O3 in a Na2O-3.25SiO2 melt, T=1150°C, Ni/NiO fO2

Figure 12 :

 12 Figure 12: Arrhenius plot of Cr solubility in Na2O-3.25SiO2 melts in the studied temperature

Figure 13 :

 13 Figure 13: Morphology of Cr2O3 crystals precipated from a Cr VI -containing Na2O-2SiO2 melt

Figure 14 :

 14 Figure 14: Typical morphology of Cr2O3 nodules crystallised in N3.25S glass, T=1150°C,

Figure 15 :

 15 Figure 15: Typical morphology of pyroxene nodules crystallised in N3.25S glass, T=1150°C,

Figure 16 :

 16 Figure 16: Representation of the stability domains of the different phases that are in

Table 1 :

 1 Glass Compositions expressed in atomic%. EPMA = Electron Probe Micro Analyses

	; CA = Chemical Analyses
	Glass formulae	Theoretical	Theoretical	Na2O	SiO2	Na2O	SiO2
		Na2O	SiO2	(EPMA)	(EPMA)	(CA)	(CA)
	Na2O-1.5SiO2	40.00	60.00	39.08 (0.25)	58.25 (0.92)	39.45 (0.19)	59.09 (0.52)
	Na2O-2SiO2	33.33	66.67	32.20 (0.46)	66.73 (0.39)	33.07 (0.22)	65.59 (0.61)
	Na2O-2.5SiO2	28.57	71.43	28.83 (0.52)	71.17 (0.22)	-	-
	Na2O-3SiO2	25.00	75.00	24.21 (0.14)	77.44 (0.69)	-	-
	Na2O-3.5SiO2	22.22	77.78	21.44 (0.66)	77.16 (0.89)	22.38 (0.28)	76.78 (0.49)
	Na2O-4SiO2	20.00	80.00	20.09 (0.09)	81.73 (0.58)	-	-

Table 2 .

 2 EPM Analyses of a Na2O -2SiO2 glass doped with 1 wt% Cr2O3 using two different electron beam conditions (*average values and standard deviations calculated from 50 analyses).

	Program P1 Program P2

Table 3 :

 3 Volatilisation rates of dissolved Cr VI species in Na2O-2SiO2 (+ 0.93% Cr2O3) at

	different temperatures
	Temperature (°C)	993	1107	1207	1310
	Volatilisation rate of
	dissolved Cr VI	-1.52 (0.70)	-4.48 (1.44)	-5.80 (1.11)	-13.43 (2.73)
	(Cr2O3 wt% / s)*10 5

Figure 18: Na2O-SiO2-Cr2O3 phase diagrams under reducing conditions at T>1160°C.

Acknowledgements : The authors gratefully acknowledge J. Ravaux for performing electronprobe micro-analyses ("Service Commun d'Analyses par Microsonde Electronique" of Nancy).

This work was financially supported by the ANR through the Actimelt project.