H Khedim

T Katrina

R Podor

P J Panteix

C Rapin

M Vilasi

Solubility of Cr2O3 and speciation of chromium in soda-limesilicate melts

Chromium oxide solubility was measured in soda-lime-silicate melts using a thermochemical reactor allowing an independent control of temperature, glass composition and oxygen fugacity in a closed system. This investigation performed 1 in the ternary Na2O-CaO-xSiO2 (NCS) system showed that the physicochemical and thermochemical models used for chromium chemistry description in binary melts are applicable in the ternary system. The presence of Cr(II), Cr(III) and Cr(VI) has been evidenced and their respective contributions in the total chromium content dissolved in the melt were determined. A new approach based on optical basicity allowed the quantification of the basicity of the melt and consequently the free oxygen ions activity (aO 2-). The treatment of the obtained results led to the determination of thermodynamical properties of the system. The standard enthalpies and entropies corresponding to the oxidation and reduction reactions were determined.

Introduction

The dissolution processes of minerals and refractory materials in molten silicates are of major interest for geologists (relict grains, xenoliths, magma assimilation) and chemists (corrosion of ceramics by molten glasses). It has been shown that the solubility of elements in silicate melts strongly depends on glass basicity (alkaline oxides content), temperature and oxygen fugacity [START_REF] Khedim | Redox-control solubility of chromium oxide in soda-silicate melts[END_REF][START_REF] Thiemsorn | Redox ratio and optical absorption of polyvalent ions in industrial glasses[END_REF][START_REF] Calas | Crystal field spectroscopy of Cr3+ in glasses: Compositional dependence and thermal site expansion[END_REF][START_REF] Berry | The effect of composition on Cr 2+ /Cr 3+ in silicate melts[END_REF] . This latter parameter influences the redox properties of the glass and yields to the stabilization of high or low oxidation states for the multivalent elements in the melt. Therefore, effects related to the action of both O 2-(basic effect) and O2 (oxidizing effect) rule together the formation of the oxide polyhedron, the well known tetrahedra or octahedra, which stabilize the metallic or semi-metallic solute in the molten silicate liquid.

Oxygen is one of the principal reactants in redox equilibria which occur in molten glasses.

The equilibrium state of the glass strongly depends on its composition, which conditions the O 2-activity (i.e. the basicity). Thus, for glasses of fixed O 2-activity, the measurement of dissolved oxygen can be used to control some processes, such as the control of the properties of the melt (viscosity, quality of refining) or the final state of the material (colour, ageing…) 5- 8 .

Melts containing one or more multivalent elements are in equilibrium when all the components taking part in the redox reactions assume a common value of the potential [START_REF] Claes | Redox chemistry in glass melts out of equilibrium with their gaseous atmosphere[END_REF] .

Reactions involving oxygen which are known to be generally slow around 25°C, are very fast at temperature higher than 1000°C. In this case, free oxygen ions O 2-, which are imposed by the glass composition [START_REF] Claes | Redox chemistry in glass melts out of equilibrium with their gaseous atmosphere[END_REF][START_REF] Johnston | Oxidation-Reduction Equilibria in Molten Na2O-2SiO2 Glass in Contact with Metallic Copper and Silver[END_REF][START_REF] Krol | Oxidation-reduction behavior of antimony silicate glasses prepared from raw materials and cullet[END_REF] , take part in redox reactions.

The redox reactions between oxygen and multivalent elements in the glass melt are often described as followed:

- - - + + +  + +  2 2 ) ( ) ( 2 4 O n ne O n ne M M n m m - + + + +  + 2 ) ( 2
) ( 2 4 O n M O n M n m m
Eq. 1 This reaction, in satisfying the equilibrium constant, allows the prediction of the influence of the state variables on this equilibrium:

• For constant temperature and glass composition (aO 2-

), an increase in oxygen fugacity should increase the oxidized species (M (m+n)+ ) content.

• For constant temperature and oxygen fugacity (fO2), an increase in the O 2activity should increase the concentration of the reduced species (M (m)+ ).

Several studies confirm the first prediction 12-14 , but second was rejected by many others 15-18 . In this case, increasing basicity leads to an increase of the oxidized species concentration.

This disagreement is explained by taking account of the oxyacid character of the cations which interact with the free oxygen ions. This leads to the formation of oxy-complexes which present different stabilities, charges, symmetries and coordination numbers 19 .

By taking into account the combined effect of the redox reaction and the oxy-acidbasicity, equilibrium described by Eq. 1 can be expressed by the equation:

- - - - - -        - - + + ) 2 ( 2 2 ) 2 ( 2 4 n m y y m x x MO O n x y O n MO
Eq. 2

In this equilibrium, the oxidized and reduced species are complexed by free oxygen ions, whereby the x and y indices are the coordination numbers of resulting oxo-complex species. If the value of the stoichiometric coefficient (y-x-n/2) related to (O 2-) is positive, an increase the basicity stabilizes the highest oxidation state. In the contrary case, the reverse occurs. The equilibrium constant of the reaction can be given by:

( ) ( ) ( ) 4 2 ) 2 ( ) 2 ( 2 ) 2 ( ) .( . ) ( n m x x n y x n m y y fO MO a O a MO a T K - - + - - - - - = Eq. 3
where ai is the activity of the i element and fO2 is the oxygen fugacity.

By replacing activities by the expression (ai=iCi), the equilibrium constant can be written as:

  ( )  ( ) 4 2 ) 2 ( ) 2 ( 2 ) 2 ( . . . . ) ( ) ( ) ( n m x x M n y x n m y y M fO MO O a MO T K m n m - - + - - - - - + + + =   Eq. 4
with :

i : activity coefficient of the i component in the melt

[i] : concentration of the i component in the melt

Since the concentrations of the multivalent ions in melts are generally low, applying Henry's law, their activities can be equated to their concentrations 5,19 . Consequently, the equilibrium constant K(T) expression can be evaluated, using only the concentrations of the multivalent elements, the oxygen fugacity and the activity of the free oxygen ions:

  ( )  ( ) 4 2 ) 2 ( ) 2 ( 2 ) 2 ( . . ) ( n m x x n y x n m y y fO MO O a MO T K - - + - - - - - =
Eq. 5

Consequently:

    ( )     4 2 ) 2 ( 2 ) 2 ( ) 2 ( . ). ( n n y x m x x n m y y fO O a T K MO MO - - - - - - - - = = 
Eq. 6

with  as the redox ratio Chromium can exist in various oxidation states. It is often used in metallurgy for its properties of hardness and corrosion resistance. Much of the equipments used in the glass industry involves superalloys containing up to 30% chromium. The ability of these alloys to resist corrosion is related to the chemical durability of the chromium oxide (Cr2O3) layer formed at the alloy/glass interface [START_REF] Abdelouhab | Electrochemical Study of Chromium Corrosion in Na2O-xSiO2 Melts[END_REF][START_REF] Carton | Corrosion of chromium in glass melts[END_REF] . In the ambient environments, chromium is primarily in the Cr(III) form, but also in the Cr(VI) form in some oxidizing environments close to the surface [START_REF] Burns | Mineralogy of chromium[END_REF] . Nevertheless, several studies showed the existence of Cr(II) in silicates melted under reducing conditions [START_REF] Berry | A XANES determination of the oxidation state of chromium in silicate glasses[END_REF][START_REF] Barnes | The distribution of chromium among orthopyroxene, spinel and silicate liquid at atmospheric pressure[END_REF][START_REF] Roeder | Crystallization of Chromite and Chromium Solubility in Basaltic Melts[END_REF][START_REF] Hanson | The systematics of Cr(III) and Cr(II) partitioning between olivine and liquid in the presence of spinel[END_REF][START_REF] Poustovetov | The distribution of Cr between basaltic melt and chromian spinel as an oxygen geobarometer[END_REF] . The behavior of chromium in the silicate melts is of interest not only in glass manufacture [START_REF] Duffy | A review of optical basicity and its applications to oxidic systems[END_REF] , but also in many other areas such as nuclear waste immobilization [START_REF] Schreiber | Redox Chemistry in Candidate Glasses for Nuclear Waste Immobilization[END_REF] , slag recycling 28 and geochemistry [START_REF] Schreiber | Mutual interactions of redox couples via electron exchange in silicate melts: Models for geochemical melt systems[END_REF] . However, a good description of the chemistry of chromium in silicate melts passes inevitably by a good comprehension of the chemical equilibria (acid-bases, redox) taking place between chromium and the melt.

In a previous work [START_REF] Khedim | Redox-control solubility of chromium oxide in soda-silicate melts[END_REF] , a chemical model describing chromium chemistry in the Na2O-SiO2 binary melts was developed. A description of the reactions occurring when chromium oxide is in contact with the glass melt is given in Figure 1. The first reaction (Eq. 7) is the direct dissolution of Cr2O3 into the melt as a Cr(III) oxo-complex. This is an acid-base reaction which is independent of the fO2.

) ( 2 3 ) ( ) ( ) ( 2 1 2 ) 3 2 ( 3 2 melt O x melt O III Cr crystal O Cr x x - - -       - + 
Eq. 7

Cr(III) species can be oxidized to Cr(VI) species (Eq. 8), under oxidizing conditions, or reduced to Cr(II) species (Eq. 9), under reducing conditions

) ( 2 3 ) ( ) ( ) ( 4
3 ) ( ) ( 2 ) 6 2 ( 2 ) 3 2 ( melt O y x melt O VI Cr gas O melt O III Cr y y x x - - - - -       + - +  +
Eq. 8

) ( 4 3 ) ( ) ( 2 1 ) ( ) ( 2 ) 6 2 ( 2 ) 3 2 ( gas O melt O VI Cr O x z melt O III Cr y y x x +        + - + - - - - -
Eq. 9

The equilibrium constants for the dissolution, oxidation, and reduction reactions can be written as:

    ) ( . 2 3 2 III Cr a O a K x d       - - = Eq. 10       4 3 2 2 3 2 . ) ( . ) ( fO III Cr a O a VI Cr a K y x ox       + - - = Eq. 11       ) 2 1 ( 2 4 1 2 . ) ( . ) ( + - - = x z red O a III Cr a fO II Cr a K
Eq. 12 where ai = i..i and ai is the activity, i the activity coefficient and i the mole fraction of species i.

In glass melts, polyvalent ion contents are generally very low, and in conformity with Henry's law, the activity can be equated to the concentration, so that:

  ( ) -       - + = 2 2 3 log ) ( log O a x K III Cr diss Eq. 13       ox K O a y x fO III Cr VI Cr log log 2 3 log 4 3 ) ( ) ( log 2 2 +       + - - = - Eq. 14       red K O a x z fO III Cr II Cr log log 2 1 log 4 1 ) ( ) ( log 2 2 +       + - + - = - Eq. 15
The model developed in our previous works 1, 31 allows the determination of the redox ratio (Cr(II)/Cr(III) and Cr(VI)/Cr(III)). In this work, this model has been extended to a ternary system in order to study Cr2O3 solubility in the Na2O-CaO-SiO2 system.

Experimental

Sodium silicate and soda lime silicate glasses with the general composition of Na2O-CaO-xSiO2 (x=3, 4 and 7) were prepared from reagent-grade SiO2 (Chempur 99.9 %), Na2CO3

(Chempur 99.5%) and CaCaO3 (Chempur 99%). Appropriate amounts of the reagent powders were weighed and mixed, introduced into a Pt-Au (95wt% of platinum and 5wt% of gold) crucible and heated in a muffle furnace. This type of crucible presents the advantage of a low wetability with the glass, compared to a pure platinum crucible. In these conditions the glass does not adhere to the crucible walls.

To minimize the sodium losses by volatilization [START_REF] Walter | Vapor pressure and vapor fractionation of silicate melts of tektite composition[END_REF][START_REF] Konta | Volatility of oxides from silicates melts and the origin of moldavites[END_REF] , not less than 200 grams of glass were melted and the following thermal cycle was used: 1200°C for 2h, 1400°C for 2h and then 1100°C for 24h. The melts were then quenched and finely crushed. Glass compositions were systematically checked by electron probe microanalysis: the intended and attained glass compositions are reported in Table 1. Ten grams of each resulting glass composition was mixed with 5 wt.% Cr2O3, finely grounded and melted at 1200°C for 5 minutes in a platinum crucible. The process was performed three times to ensure a regular distribution of the Cr2O3 grains in the glass network. A mixed Cr2O3-glass quantity (100 mg) was taken from the mixture, put in a carbon crucible, heated at 1200 °C for 1 minute and removed from the furnace to be quenched in air. Due to the poor wetability between the melt and the carbon crucible, the sample forms a glass ball.

The Cr2O3 solubility limit in the investigated melts was performed using the reactor described in our previous works [START_REF] Khedim | Redox-control solubility of chromium oxide in soda-silicate melts[END_REF][START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF][START_REF] Mathieu | Control of alkali-metal oxide activity in molten silicates[END_REF] . It allows the independent control of oxygen fugacity, or in an alumina crucible in order to prevent reactions with the silica tube and to avoid alloying between the solid buffer and the platinum crucible. The silica tube is then evacuated and directly sealed under secondary vacuum. Direct measurements show that the residual pressures inside the thermochemical cell never exceed Ptotal < 10 -6 bar at room temperature.

When prepared, the silica tubes are introduced in a muffle furnace directly at the experimental temperature on an alumina support for a vertical maintain of the cell. After the heat treatment, the silica tubes are removed from the furnace and directly quenched into cold water. The presence of both metal and oxide phases in the solid buffer after each run is systematically checked by optical microscopy observation and X-ray diffraction.

Determinations of the total amount of dissolved chromium and of the final glass composition were performed using a CAMECA SX100 Electron Microprobe. The analyses conditions were normalized to an acceleration voltage of 25kV and a 6nA beam current. The electron beam was defocalized to a 15 µm diameter, to perform the analysis in a relatively large volume of the material, and to limit Na volatilization.

Results and discussions

To understand the effect of the partial substitution of alkaline oxide (Na2O) by the oxide (CaO) on the chromia solubility, this one will be studied in the Na2O-CaO-xSiO2 system (noted NCS) for the three following melt compositions: x=3, 4 and 7, as a function of temperature and oxygen fugacity. The liquidus temperatures in the NCS system are higher than in the Na2O-SiO2 system; therefore, the selected temperature range lay between 1200 and 1300°C.

Figure 2 shows three characteristic curves of chromium oxide solubility in the NCS system. These three curves show the variation of the dissolved total chromium content with the oxygen fugacity, and illustrate the influence of the three experimental parameters: oxygen fugacity, temperature and liquid composition. The obtained results show that:

-Regardless of oxygen fugacity, chromium oxide solubility increases with the temperature for a given liquid composition.

-Under oxidizing atmosphere, the solubility increases with the basicity (alkaline oxide content) of the glass.

-Under a reducing atmosphere, the solubility decreases when the basicity increases.

-Independently of the temperature or composition of the glass, the general shapes of the curves remain unchanged. They always comprise three distinct fields of solubility which are characteristic of the presence of the three chromium valences: Cr(II), Cr(III)

and Cr(VI).

The model developed in our previous works [START_REF] Khedim | Redox-control solubility of chromium oxide in soda-silicate melts[END_REF][START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF] was applied to the experimental results obtained for the NCS system to determine the speciation of the chromium species as well as the redox parameters in the ternary system. When equilibrium is reached, the total chromium dissolved in the glass (Cr(tot)) is the sum of all chromium species : Cr(tot) = Cr(II) + Cr(III) + Cr(VI) Eq. 16

Using the same assumption as in the Na2O-SiO2 system, we consider that the Cr(III) content in the melt remains constant as a function of (fO2) for a given temperature [START_REF] Khedim | Redox-control solubility of chromium oxide in soda-silicate melts[END_REF][START_REF] Roeder | Crystallization of Chromite and Chromium Solubility in Basaltic Melts[END_REF] . The minimum of chromium content in the glass corresponds to the Cr(III) contribution. Therefore, under oxidizing conditions, Cr(II) content can be neglected compared to Cr(III) and Cr(VI). In the same way, under reducing conditions, Cr(VI) can be neglected compared to Cr(III) and Cr(II). With these considerations, it can be written:

Under oxidizing conditions Cr(tot) = Cr(III) + Cr(VI) Eq. 17

Under reducing conditions

Cr(tot) = Cr(II) + Cr(III) Eq. 18

Using this formalism, log(Cr(II)/Cr(III)) and log(Cr(VI)/Cr(III)) were plotted as a function of log(fO2) for the Na2O-CaO-3SiO2 system and reported on Figure 3. The obtained results

show clearly that the two parameters are linearly dependent. The slope values obtained under reducing and oxidizing conditions are respectively -0.24 and 0.74. These are in good agreement with the theoretical values of the expected slopes (Eq. 14 and Eq. 15).

Variations of log(Cr(II)/Cr(III)) and log(Cr(VI)/Cr(III)) are reported as a function of log(fO2) at T=1250°C for three different glass compositions (NC3S, NC4S and NC7S) in Figure 4. The results show that log(Cr(II)/Cr(III)) decreases with increasing basicity of the melt (Figure 4A), although this variation is weak. On the contrary, log(Cr(VI)/Cr(III))

increases with increasing melt basicity. Therefore, an increase in the basicity of the glass (alkaline oxides content) leads to stabilize the oxidized forms. This behavior is identical to that described in the Na2O-SiO2 system 1 . These results are consistent with those obtained by other authors for various systems and various multivalent elements [START_REF] Thiemsorn | Redox ratio and optical absorption of polyvalent ions in industrial glasses[END_REF][START_REF] Pretorius | Activity-Composition Relations of Chromium Oxide in Silicate Melts at 1500°C under Strongly Reducing Conditions[END_REF][START_REF] Tilquin | Effect of basicity on redox equilibria in sodium silicate melts: An in situ electrochemical investigation[END_REF] .

The influence of the temperature on the redox parameters was studied for the NC3S liquid composition. Values of log(Cr(II)/Cr(III)) and log(Cr(VI)/Cr(III)) are plotted as a function of log(fO2) in Figure 5 for three different temperatures (1200, 1250 and 1300°C). The results reveal that whatever the imposed oxygen fugacity is, the redox ratio increases with increasing melt temperature. This same behavior was also observed in the Na2O-SiO2 system 1 and in several studies treating of the solubility of the multivalent species in silicate melts [START_REF] Abdelouhab | Electrochemical Study of Chromium Corrosion in Na2O-xSiO2 Melts[END_REF][START_REF] Kido | High temperature spectroscopy of manganese and chromium doped glasses with the basic composition 16Na2O [middle dot] 10CaO [middle dot] 74SiO2[END_REF][START_REF] Hrma | Chromium phase behavior in a multi-component borosilicate glass melt[END_REF] .

The treatment of the whole set of data should lead to the determination of formulas for chromium complexes formed in the NCS system, as well as the thermodynamic properties related to dissolution, oxidation and reduction processes of chromia. However, the determination of these properties cannot be realized without knowing the values of the aO 2-in the given liquid. This parameter is essential for a complete and consistent exploitation of the results. The following development permits a determination of the activity of the oxide ions aO 2-.

In glass networks, the aO 2-was introduced as a basicity indicator [START_REF] Lux | Säuren und Basen im Schmelzflu: Die Bestimmung der Sauerstoffionenkonzentration[END_REF][START_REF] Dietzel | Die Kationenfeldstärken und ihre Beziehungen zu Entglasungsvorgägen, zur Verbindungsbildung und zu den Schmelzpunkten von Silikaten[END_REF] . Nevertheless, no standard state can be defined for this ion. Moreover, this basicity depends on the bond strength between the alkali and oxygen ion. When the size of the alkali ion increases, the bond strength decreases. As a consequence the mobility of the oxide ions is higher [START_REF] Paul | Chemistry of Glasses[END_REF][START_REF] Zarzycki | Les Verres et L'Etat Vitreux[END_REF] . Furthermore, it is generally agreed, in binary melts, to equate the activity of the oxide ions (aO 2-) to the activity of the sodium oxide (aNa2O).

In the case of binary melts (Na2O-SiO2 in this study), aO 2-is equated to the Na2O activity [START_REF] Stegmaier | The significance of basicity of glass melts and the investigation of its measurements. I[END_REF][START_REF] Toop | Some new ionic concepts of silicate slags[END_REF][START_REF] Perander | Acidity and ionic structure of molten alkali silicates[END_REF] . In the case of ternary glasses, the situation is more complex. One of the key points is to establish where aO 2-can still be equated to Na2O activity when adding another modifying oxide (CaO) to the Na2O-SiO2 network. In this case, there is competition between the two basic species Na2O and CaO. From this, two assumptions can be proposed.

(i) If Na2O is much more basic than CaO, the aO 2-could be equated to the Na2O activity.

Then the contribution of calcium oxide on the acid-base properties can be neglected. Several studies have determined the Na2O activity in binary Na2O-xSiO2 and ternary NCS system [START_REF] Abdelouhab | Determination of Na2O activities in silicate melts by EMF measurements[END_REF][START_REF] Kim | The activity of sodium oxide in molten float glass by EMF measurements[END_REF][START_REF] Tsukihashi | Measurement of the activity of sodium oxide in sodium oxidesilicon dioxide melts by chemical equilibration method[END_REF][START_REF] Yamaguchi | Measurement of activity of Na2O in sodium oxide-silicon dioxide-phosphorus pentoxide ternary melt using sodium beta-alumina as a solid electrolyte[END_REF][START_REF] Neudorf | Thermodynamic Properties of Na2O-SiO2-CaO Melts at 1000°C to 1100°C[END_REF][START_REF] Rego | Thermodynamic activity of Na2O in Na2O-CaO-SiO2, Na2O-MgO-SiO2, and Na2O-CaO-SiO2-Al2O3 melts at 1400°C[END_REF] .

The values applied in this work are those measured by Neudorf et al. [START_REF] Neudorf | Thermodynamic Properties of Na2O-SiO2-CaO Melts at 1000°C to 1100°C[END_REF] and Abdelouhab et al. [START_REF] Abdelouhab | Determination of Na2O activities in silicate melts by EMF measurements[END_REF] (Table 2). In this case, basicity is given by the value of log[aNa2O]. By comparing the two series of data, aNa2O values in NC3S and NC4S are the same as in the Na2O-2.5SiO2 and Na2O-3.5SiO2 melts, respectively (Table 2). Thus, these glasses have equivalent basicities.

However, the NC7S composition does not have an equivalent in the binary melt.

(ii) If Na2O and CaO basicities are close, aO 2-cannot be equated to the aNa2O. Initially, we propose, in the binary system, to equate the O 2-activity (aO 2-=aNa2O in binary system) to the calculated optical basicity. Optical basicity values (Table 3) were calculated according to the Duffy and Ingram model [START_REF] Duffy | Interpretation od Glass Chemistry in terms of the Optical Concept[END_REF][START_REF] Duffy | Behaviour of Basicity Indicator Ions in relation to the Ideal Optical Basicity of Glasses[END_REF][START_REF] Duffy | Establishment of an optical scale for Lewis basicity in inorganic oxyacids, molten salts, and glasses[END_REF] . The dependance of log[aO 2-] at T=1250°C on the optical basicity is plotted in Figure 6.

For comparison, the optical basicity values of the ternary melts (Table 3) are reported in Figure 6. This interpolation highlights that O 2-activities in ternary Na2O-CaO-x(3 and 4)SiO2

glasses are comparable to O 2-activities in binary Na2O-x(1.7 and 2.15)SiO2 glasses, respectively.

The oxide ions activities aO 2-obtained from the optical basicities interpolation are inconsistent with the Na2O activities measured by Abdelouhad et al. [START_REF] Abdelouhab | Determination of Na2O activities in silicate melts by EMF measurements[END_REF] (Table 2). To validate one of these two assumptions, we compare results concerning chromia solubility in the two vitreous systems Na2O-SiO2 31 and NCS. According to the equations Eq. 13, Eq. 14 and Eq.

15, redox ratios are expressed as a function of oxygen fugacity and O 2-activity. In these equations, for a given temperature and oxygen fugacity, only aO 2-has an effect on redox parameters. This means that for a given temperature and oxygen fugacity, the redox ratios should be the same for two glasses with the same aO 2-.

The comparison of log (Cr(VI)/Cr(III))) vs log(fO2) of the ternary glass NC3S with the redox ratio of various binary systems [START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF] is plotted in Figure 7. This plot clearly shows that for given fO2 and temperature, the redox ratio obtained in the ternary glass NC3S is closest to the characteristic redox ratio of N2.5S. In accordance with Eq. 6, it can thus be assumed that the basicities (i.e. O 2-activity) of NC3S and N2.5S are equivalent. In the same way, Figure 8 exhibits that the NC4S ternary glass presents a basicity which is close to the basicity of N3.5S and N3S binary glasses. It although seems to be a bit closer to the basicity of N3.5S glass.

Regarding the results reported on Table 4, it can be considered that aNa2O = aO 2-.

The determination of O 2-activities permit the characterization of the physico-chemical properties of chromium in the NCS system:

-the identification of chromium complexes formed in NCS system, -the determination of the thermodynamic properties for dissolution, oxidation and reduction of the chromium species in NCS melts, according to the physico-chemical model suggested by our previous works [START_REF] Khedim | Redox-control solubility of chromium oxide in soda-silicate melts[END_REF][START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF] .

Determination of the chromium complexes formed in NCS systems at 1250 °C was achieved by plotting the logarithms of (Cr(III), Cr(II)/Cr(III) and Cr(VI)/Cr(III)) as a function of O 2-activities in the system (Figure 9). According to Eq. 13, Eq. 14 and Eq. 15, the slopes of the straight lines are directly correlated to the (x-3/2), (x-y+3/2) and (z-x+1/2) parameters for a given temperature and f(O2) concerning log[Cr(VI)/Cr(III)] can be exploited. The slope of the obtained straight line is equal to 0,514 (R²=0,99). This allows a calculation of the (x-y) value which is equal to 2.

Results
Some of the most stable Cr(VI) formulas are dichromate or chromate 37 , (Cr2O7) 2-or (CrO4) 2- respectively. Considering these two possible Cr(VI) formulas, Cr(III) formulas would be Cr2O3 or (CrO2) -. However, the Cr2O3 formula for the Cr(III) complex cannot be considered:

we assume that the Cr2O3 species is a solid precipitate in the melt.

At this stage, it is only possible to formulate assumptions concerning the nature of the chromium complexes formed in the Na2O-CaO-SiO2 system. The whole of the experimental data related to this system must be supplemented by complementary solubility measurements, covering a broader range of glass compositions.

The thermochemical model, developed in previous works [START_REF] Khedim | Redox-control solubility of chromium oxide in soda-silicate melts[END_REF][START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF] , is based on the knowledge of the nature of the Cr(II), Cr(III) and Cr(VI) complexes through the determination of the values of the x, y and z parameters. As these latter parameters have not been obtained for this system, only an approximate calculation of the thermodynamic properties can be proposed.

The development consists in simplified equations which do not take into account the nature of the chromium complexes [START_REF] Kido | High temperature spectroscopy of manganese and chromium doped glasses with the basic composition 16Na2O [middle dot] 10CaO [middle dot] 74SiO2[END_REF]55 :

-+ +  The related equilibrium constants are written as follows: As mentioned previously, the O 2-activity is related to the glass composition. The concentration of the multivalent element is considered to be relatively low compared to the concentration of the free oxygen ions, so that for a given temperature and glass composition, O 2-activity is not affected during a redox reaction and remains constant [START_REF] Johnston | Oxidation-Reduction Equilibria in Molten Na2O-2SiO2 Glass in Contact with Metallic Copper and Silver[END_REF][START_REF] Krol | Oxidation-reduction behavior of antimony silicate glasses prepared from raw materials and cullet[END_REF] . It is thus advantageous to define equilibrium constants KT' freed from the O 2-activity term as follows:

+ = 3 ' aCr K Tdiss Eq. 22 T K   + = log
Eq. 31

The slopes and the Y-axis-intercepts of each straight line are related to the standard enthalpies and entropies for each reaction, in accordance with the following equation:

R S T R H K 3 . 2 1 3 . 2 log 0 0  +        - =
Eq. 32

By identification, the general formula expressing the standard entropies S 0 and standard enthalpies H 0 of the reactions can be written as:

 R S 3 . 2 0 =  Eq. 33  R H diss 3 . 2 0 - =  Eq. 34
The values concerning the standard enthalpies and entropies of dissolution, oxidation and reduction of the chromia in NC3S glass system are given in Table 5.

Conclusion

Several important points of the chemistry of chromium in the silicate melts have been highlighted in this study:

-The model previously developed for binary Na2O-xSiO2 1,31 has been successfully applied to the ternary Na2O-CaO-xSiO2 system.

-The solubility of Cr2O3 has been determined as function of temperature, glass composition and oxygen fugacity.

-Cr2O3 is dissolved as Cr(II), Cr(III) and Cr(VI) species. The speciation of chromium ions has been determined as a function of oxygen fugacity and glass composition. The log[Cr(II)/Cr(III)] and log [Cr(VI)/Cr(III)] ratios vary with -1/4log(fO2) and 3/4log(fO2) respectively, independently of the melt composition and temperature.

-The (H°diss, S°diss), (H°red, S°red) and (H°ox, S°ox) have been estimated

for Cr(III) → Cr(VI) and Cr(III) → Cr(II) redox reactions. The obtained Cr(III) species can be oxidized to Cr(VI) (Eq. 8)or reduced to Cr(II) complexes (Eq. 9).

  temperature and alkali volatilization in the silicate melt. The cell is constituted by a sealed silica tube of around 25-30 cm 3 (Ø ext = 22 mm, h  120 mm) containing several components that impose the thermochemical parameters of the system. The Na2O partial vapor pressure is imposed by around 5 g of Na2O-CaO-xSiO2 source melt (with 1< x <7) and located in a platinum crucible (Ø ext = 18 mm, h = 20 mm), shut by a platinum lid. The studied glass balls (samples) are suspended in the reactor from the lid. The Pt crucible hosting the source and the sample constitute the reactor of the cell. The oxygen fugacity in the device is controlled by a solid M/MxOy buffer (M = Metallic element; Fe/FeO, FeO/Fe3O4, Co/CoO, Ni/NiO and Fe3O4/Fe2O3) incorporated below the reference reservoir either at the bottom of the silica tube

  concerning log[Cr(III)] and log[Cr(II)/Cr(III)] variations, as a function aO 2-, cannot be directly exploited to calculate the x, y and z parameters necessary for the identification of the formulas for the chromium complexes. Indeed, the variation of log[Cr(III)] as well as log[Cr(II)/Cr(III)] present a very low variation with aO 2-. This makes impossible the determination of the (3/2-x) and (z-x+1/2) parameters. Nevertheless, results

Figure 1 :

 1 Figure 1: Chemical reactions occurring when Cr2O3 crystals are in contact with the melt. The

Figure 2 :Figure 4 :Figure 5 :

 245 Figure 2 : Variation of Cr (at.%) content as function of log fO2 :

Figure 6 :

 6 Figure 6 : log a(O 2 ) -in binary melts at 1250°C as a function of calculated optical basicity. Dotted lines correspond to the extrapolation of a(O 2-) in ternary melts from their optical

Figure 7 :

 7 Figure 7: log[Cr(VI)/Cr(III)] as a function of log(fO2) at T=1250°C, in the ternary Na2O-

Figure 8 :

 8 Figure 8: log[Cr(VI)/Cr(III)] as a function of log(fO2) at T=1250°C, in the ternary Na2O-

Figure 9 :

 9 Figure 9 : Variation as a function of log[a(O 2-)] at T=1250°C of:

Table Captions Table 1 :

 Captions1 Composition of glasses used as started materials

	Aimed glass Composition	Na2CO3 wt.%	SiO2 wt.%	CaCO3 wt.%	Na (at.%) EPMA	Si (at.%) EPMA	Ca (at.%) EPMA	x
	Na2O-1.5SiO2	69.68	59.25	-	25.95 (0.40)	20.17 (0.19)	-	1.56 (0.04)
	Na2O-2SiO2	58.19	65.97	-	21.91 (0.44)	22.38 (0.22)	-	2.04 (0.06)
	Na2O-2.5SiO2	49.95	70.79	-	19.22 (0.49)	23.72 (0.25)	-	2.47 (0.09)
	Na2O-3SiO2	43.75	74.41	-	16.34 (0.22)	25.16 (0.11)	-	3.08 (0.06)
	Na2O-3.5SiO2	38.93	77.24	-	15.00 (0.33)	25.83 (0.17)	-	3.466 (0.10)
	Na2O-CaO-3SiO2	40.89	60.42	35.21	14.04 (0.25)	21.32 (0.15)	7.02 (0.09)	3.03 (0.06)
	Na2O-CaO-4SiO2	34.04	67.06	29.31	10.98 (0.16)	23.64 (0.12)	5.85 (0.10)	4.30 (0.08)
	Na2O-CaO-7SiO2	22.64	78.08	19.50	7.57 (0.27)	26.82 (0.14)	3.70 (0.08)	7.09 (0.17)

Table 3 :

 3 Optical basicity values calculated for the binary end ternary systems according toDuffy and

	Ingram model 52-54
	x (Na2O-xSiO2)	1.5	2	2.5	3
	Optical Basicity	0.748	0.703	0.663	0.648
	x(NC-xS)	3	4	7
	Optical Basicity	0.718	0.678	0.612

Table 5 :

 5 Thermodynamic data related to chromium oxide reactions in Na2O -CaO -3SiO2H° (kJ.mol -1 ) S° (J.mo1 l-.K -1 ) H° (kJ.mol -1 ) S° (J.mo1 l-.K -1 ) H° (kJ.mol -1 ) S° (J.mol -l .K -1 )

	melt composition
	Dissolution		Oxidation		Reduction
	33 ± 11	26 ± 9	50 ± 15	54 ± 16	172 ± 37	61 ± 16

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

Acknowledgements

The authors are grateful to J. Ravaux and S. Mathieu for performing EPMA (Service Commun d'Analyses par Microsonde Electronique de Nancy). This work was financially supported by the ANR through the Actimelt project.

The results show that logK'T is linearly correlated to (1/T), according to the following expression:

Table 4 : log[aO 2-] as well as the composition of the binary equivalent with the ternary Na2O -CaO -xSiO2 for each studied case

Melt composition

Na2O-CaO-3SiO2 Na2O-CaO-4SiO2 Na2O-CaO-7SiO2


H Khedim

R Podor

P J Panteix

C Rapin

M Vilasi

Solubility of chromium oxide in binary soda silicate melts

Keywords: 82, 60, Hc

The total chromium oxide (Cr2O3) solubility is measured in the liquids belonging to the Na2O-SiO2 system using a thermochemical reactor which allows the independent control of temperature (1125°C<T<1250°C), glass composition (1.5<x<3) and oxygen fugacity (-15<log fO2<-0.61). The contribution of each chromium species (Cr(II), Cr(III) and Cr(VI)) and the total Cr content dissolved in the melt are determined using a model describing the chromium chemistry in melts involving dissolution, redox and oxo-complex reactions. A manipulation of the obtained results allows the determination of physicochemical properties of chromium species in melts (chromium complexes formulas) as well as thermodynamic properties (entropies and enthalpies) related to the dissolution, reduction and oxidation processes of chromium oxide in binary melts.

Introduction

Chromium can exist under various oxidation states depending on environmental conditions. It is often used in metallurgy as an alloying element due to its hardening and corrosion resistance properties. In the glass industry, most of the equipments are made up of superalloys containing up to 30% of chromium. The lifetime of metallic parts, which are in contact with molten glasses, is mainly controlled by high temperature corrosion, and the degradation of the alloys or superalloys immersed in the molten glasses is governed by dissolution and oxido-reduction reactions [1][2][3][4]. The corrosion resistance is conditioned by the formation of a protective layer of chromium oxide at the metal/glass interface. As a consequence, the knowledge of chromium oxide dissolution in simple silicate melts will provide useful information concerning the understanding of the ability of these alloys to resist against glass melts corrosion.

Several studies have been dedicated to the chromium oxide dissolution process in silicate melts [4][START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF][START_REF] Khedim | [END_REF][7][8][9][10][11]. The first type of studies deal with corrosion reaction of chromium containing alloys [4,[START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF]. Other studies report the dissolution of chromium oxides in glass melts [START_REF] Khedim | [END_REF][7][8][9][10][11].

The authors have demonstrated that the solubility of chromium oxide strongly depends on glass basicity (melt composition), temperature and oxygen fugacity [START_REF] Khedim | [END_REF][7][8]12]. This latter parameter influences the redox properties of the glass and yields to the stabilization of high or low oxidation states of the multivalent elements in the melt. Therefore the effects related to the action of both O 2-(basicity effect) and O2 (redox effect) rule together the chromium chemistry in silicate melts, as described by Tilquin et al [10]. Oxygen is one of the principal reactants in redox equilibria occuring in molten glasses [13,14]. The equilibrium state of the medium strongly depends on its composition, which fixes the O 2-activity (i.e. for a given basicity). Thus, for glasses of fixed O 2-activity, the measurement of dissolved oxygen can be used to control some processes, such as the melt properties (viscosity, quality of refining) or the final state of the material (colour, ageing…) [START_REF] Paul | Chemistry of Glasses 2nd Ed[END_REF][START_REF] Rawson | Properties and Applications of Glass[END_REF][START_REF] Claes | [END_REF][START_REF] Doremus | Glass Science[END_REF].

Regarding the behavior of Cr2O3 in molten glasses, the various interactions occurring between the silicate melt and chromia are presented in previous works [START_REF] Khedim | [END_REF][8][9][10][11] and the related chemical reactions are described hereafter:

) ( 2 3 ) ( ) ( ) ( 2 1 2 ) 3 2 ( 3 2 melt O x melt O III Cr crystal O Cr x x - - -       - + 
Eq. 1

Cr(III) species can be oxidized into Cr(VI) species (Eq. 2), under oxidizing conditions, or reduced to Cr(II) species (Eq. 3), under reducing conditions:

) ( 2 3 ) ( ) ( ) ( 4
3 ) ( ) ( 2 ) 6 2 ( 2 ) 3 2 ( melt O y x melt O VI Cr gas O melt O III Cr y y x x - - - - -       + - +  + Eq. 2 ( ) gas O melt O II Cr melt O x z melt O III Cr z z x x 2 ) 2 2 ( 2 ) 3 2 ( 4 1 ) ( ) ( ) ( 2
1 ) ( ) ( +        + - + - - - - -

Eq. 3

The equilibrium constants for the dissolution, oxidation, and reduction reactions can be expressed as:

)] ( [ . ] [ 2 3 2 III Cr a O a K x diss       - - = Eq. 4 ) ( log 2 3 log )] ( log[ 2-       - + = O a x K III Cr diss Eq. 4 3 2 2 3 2 )]. ( [ ] [ )]. ( [ fO III Cr a O a VI Cr a K y x ox       + - - = Eq. 6 ox K O a y x fO III Cr VI Cr log ] [ log 2 3 log 4 3 )] ( [ )] ( [ log 2 2 +       + - - = - Eq. ) 2 1 ( 2 4 1 2 ] [ )]. ( [ )]. ( [ + - - = x z red O a III Cr a fO II Cr a K Eq. 8 red K O a x z fO III Cr II Cr log ] [ log 2 1 log 4 1 )] ( [ )] ( [ log 2 2 +       + - + - = - Eq.
Where ai = i..i With ai is the activity, i the activity coefficient and i the molar fraction of species i.

In glass melts, polyvalent ions contents are generally very low. Consequently, their concentrations are also very low. Therefore, in conformity with Henry's law, the activity can be assimilated to the concentration.

The model developed in our previous works [START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF][START_REF] Khedim | [END_REF] allowed the determination of the redox ratio (Cr(II)/Cr(III) and Cr(VI)/Cr(III)), the chromium complexes formulas and the thermodynamic data concerning the reactions occurring in the Na2O-3SiO2 melt composition.

The aim of this work is to determine the chromium speciation (Cr(II), Cr(III) and Cr(VI) contents and their ratios) in the Na2O-SiO2 system by taking into account the model developed in a previous work [START_REF] Khedim | [END_REF]. The expected results should allow the determination of the chromium complexes formulas (x, y and z parameters) and thermodynamic data related to the dissolution, oxidation and reductions processes.

Experimental

Soda-silicate glasses with the general composition of Na2O-xSiO2 (x=1.5, 2, 2.5 and 3) were synthetised from reagent grade SiO2 (Chempur 99.9 %) and Na2CO3 (Chempur 99.5%).

Appropriate amounts of the reagent powders were weighed and mixed, then introduced in a Pt95Au5 crucible and heated in a muffle furnace. In order to minimise the sodium loss by volatilisation [START_REF] Walter | [END_REF]20], not less than 200 grams of glass were melted and the thermal cycle was defined as follows: 1200°C for 2h, 1400°C for 2h and then 1100°C for 24h. The melts were then quenched and finally crushed. Glass compositions were systematically checked by electron probe microanalysis: the aimed and obtained glass compositions are reported in Table 1.

Ten grams of each resulting glass composition were mixed to 5 wt.% Cr2O3, finely ground and melted at 1200°C for 5 minutes in a platinum crucible. The process was performed three times in order to ensure a regular distribution of the Cr2O3 grains in the glass network. The required quantity of glass mixed with Cr2O3 (about 100 mg) was taken from the mixture, put in a carbon crucible, heated at 1200 °C for 1 minute and finally quenched in air. Due to the high surface tension between the melt and the carbon crucible, the sample forms a perfect glass ball.

Alkali elements volatilise easily from silicate melts at high temperature [21]. The rate of sodium volatilisation increases with increasing temperature and alkali content, and decreases with increasing O2 and sample size [22]. This can lead to drastic changes in the melt composition during long term experiments. In order to avoid volatilisation, the experiments were performed in a closed system using a method developed in the laboratory [START_REF] Khedim | [END_REF]23]. It allows the independent control of oxygen fugacity, temperature and alkali activity in the silicate melt.

The experiment principle is as described in previous works [START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF][START_REF] Khedim | [END_REF]23]: the cell is constituted by a sealed silica tube of around 25-30 cm 3 (Ø ext = 22 mm, h  120 mm) containing several components that impose the thermochemical parameters of the system. The studied glass balls (samples) were mounted on a platinum lid by using a platinum wire. In order to limit the Na2O volatilisation from the sample, the Na partial vapour pressure was imposed to the sample by adding a platinum crucible (Ø ext = 18 mm, h = 20 mm) containing 5 g of glass (source) which has the same composition with the sample. Assuming that the reference melt quantity in the reactor is large enough as compared to the molten samples, this device will ensure that, at thermodynamic equilibrium, the aNaO1/2 in the sample will be the same than in the reference melt according to the following reactions: NaO1/2 (reference melt) → Na(g) + 1/4 O2 Eq. 10 Na(g) + 1/4 O2 → NaO1/2 (sample) Eq. 11 aNaO1/2 (sample) = aNaO1/2 (reference melt) Eq. 12

The oxygen fugacity in the device is controlled by a solid M/MxOy buffer (M = Metallic element; Fe/FeO, FeO/Fe3O4, Co/CoO, Ni/NiO and Fe3O4/Fe2O3) incorporated below the reference reservoir either at the bottom of the silica tube or in an alumina crucible in order to prevent reactions with the silica tube and to avoid alloying between the solid buffer and the platinum crucible. The silica tube was then evacuated and directly sealed under secondary vacuum. Direct measurements show that the residual pressures inside the thermochemical cell never exceed 10 -6 bar at room temperature.

The silica tubes were introduced in a muffle furnace directly at the experimental temperature on an alumina support for a vertical maintain of the cell.

After the adequate heat treatment [START_REF] Khedim | Kinetic Study of the Chromium Solubility in Soda-Silicate Melts[END_REF], the silica tubes were removed from the furnace and directly quenched into water. The cooling rate is estimated to be 50°C/s for the first 10s of the quenching. Each part of the reactor was weighed before and after experiment in order to look for potential leaks. Special attention, by careful observation and analyses, was paid onto the Na2O-xSiO2 melt reservoir and the melt drop holder. Finally, the presence of both metal and oxide phases in the solid buffer after each run was checked by optical microscope and X-ray diffraction.

The glass compositions as well as the total amount of dissolved chromium in melts were determined using a CAMECA SX100 electron microprobe at the Université Henri Poincaré, Nancy-Université (France). The main difficulty, when analysing glasses with high Na contents is to determine the sodium content with accuracy and with limited sodium volatilisation during measurement. The optimised conditions for the analysis of homogeneous glasses are an acceleration voltage of 25 kV, a beam intensity of 8 nA, an electron beam size of 10 m. The counting times on peaks and backgrounds are (6 s, 3 s), (10 s, 5 s) and (30s, 15) respectively for Na, Si and Cr. Each element was analysed with a specific spectrometer allowing the quantification of the three elements at once. However, this also allows the limitation of both analysis duration as well as Na2O volatilisation. Each glass composition and total chromium content were determined by a mean of 15-20 individual analyses. The standards which were used in these analyses are Cr2O3 for Cr and NaAlSi3O8 for Na and Si.

The oxygen content was determined by stoechiometry associating the oxidation numbers I, IV and III for Na, Si and Cr, respectively.

The presence of excess chromia crystallites in the sample was systematically checked by electron backscattered imaging (Figure 1). In order to measure the dissolved chromium species with accuracy, the analysed zone was chosen in the vicinity of remaining Cr2O3 crystallites. Not less than 25 to 30 microprobe analyses were performed on each sample. The results were averaged and the standard deviation was calculated: the extreme data, which are too far from the mean value, were rejected and the procedure was repeated. The data used in this work are the mean values determined from 10 measurements performed by following this procedure. The standard deviations have been considered as the error bars. They are always in the range of 5% from the mean value. This proves that there is no local segregation of Cr2O3 crystals and that the analysed samples have reached their equilibrium state.

Results

Only equilibrium values were taken into account. The minimum run duration necessary to reach the equilibrium state was determined by performing a preliminary kinetic investigation [START_REF] Khedim | Kinetic Study of the Chromium Solubility in Soda-Silicate Melts[END_REF]: in most cases, it was about 2 hours. The chromium solubility in molten silicates was studied as a function of temperature, oxygen fugacity and glass composition. All the obtained results are summarised in Table 2.

The total chromium content in the melt was plotted as a function of oxygen fugacity for different glass compositions (Figure 2). It can be concluded that the chromium content variations with melt composition follow the same phenomenon for any values of temperature:

-Under an oxidizing atmosphere, the solubility increases with the basicity (alkaline oxides content) of glass.

-Under a reducing atmosphere, the solubility decreases when the basicity increases.

The total chromium content was also plotted as a function of oxygen fugacity for different temperatures for a given glass composition (N2S) in Figure 3. It can be concluded that the total chromium content in the melt increases with temperature for any values of oxygen fugacity.

Independently of the temperature or the glass composition, the general shape of the obtained curves remains unchanged. It always comprises three distinct fields of solubility which are characteristic of the presence of the three chromium valences (Cr(II), Cr(III) and Cr(VI)) [START_REF] Khedim | [END_REF].

Discussion

The speciation of the chromium species has been determined using the same model as that proposed in previous works [START_REF] Khedim | Etude de la solubilité de la chromine dans les silicates fondus -Détermination de Grandeurs physicochimiques et Thermodynamiques[END_REF][START_REF] Khedim | [END_REF] : when the equilibrium is reached, the total chromium dissolved in the glass Cr(tot) is considered as the sum of all chromium species: Cr(tot) = Cr(II) + Cr(III) + Cr(VI) Eq. 13

Using the same assumption, for a given temperature, we consider that the Cr(III) species dissolved in the melt which is in equilibrium with solid Cr2O3 crystallites, remains constant for any value of oxygen fugacity [START_REF] Khedim | [END_REF]11]. This assumption is convinced by the fact that chromium oxide dissolution (Eq. 1) is an acid-base reaction which does not depend on oxygen fugacity. The minimum value of chromium content in the glass corresponds to the Cr(III) contribution. Therefore, under oxidizing conditions, Cr(II) content can be neglected as compared to Cr(III) and Cr(VI). In the same way, under reducing conditions, Cr(VI) can be neglected as compared to Cr(III) and Cr(II). With these considerations, it can be simplified as the following equations:

Under oxidizing conditions Cr(tot) = Cr(III) + Cr(VI) Eq. 14

Under reducing conditions

Cr(tot) = Cr(II) + Cr(III) Eq. 15

Chromium speciation could be determined by considering these assumptions. Thus the redox ratios, defined as Rox=Cr(VI)/Cr(III) and Rred =Cr(II)/Cr(III), can be calculated. In order to confirm or reinforce the proposed model, the plots of the redox ratio values as a function of logfO2 are performed. In fact, Eq. 7 and Eq. 9 show that, for a given melt composition and temperature, the logarithm of the redox ratio varies linearly with oxygen fugacity and the slope value is equal to -0.25 under reducing conditions and 0.75 under oxidizing conditions.

Logarithms of the obtained redox ratios in the Na2O-2SiO2 glass composition have been plotted as a function of logfO2 for different temperatures (Figure 4): the correlation between both parameters is linear with slope values thus in perfect agreement with the theoretical values. This behavior is observed for each temperature of the studied temperature range.

Redox ratios values being given and confirmed and the complex formulas of the various chromium species formed in molten glass can be determined by plotting the logarithms of Cr(III), Rox and Rred values, for a given temperature and oxygen fugacity, as a function of a(O 2-) of the melt. The variations should be linear. The x, y and z parameters relative to each chromium complex (Cr(II)Oz (2z-2)-, Cr(III)Ox (2x-3)-and Cr(VI)Oy (2y-6)-) can be thus deduced from the slope values of the obtained lines (x-1.5, y-x-1.5 and z-x+0.5) in agreement with Eq. 5, 7 and 9 respectively.

First of all, a reliable data set concerning the O 2-activities should be available. In the case of the Na2O-SiO2 binary melts, the O 2-activity can be assimilated to the Na2O activity [25][26][27]. The values of Na2O activities (in the Na2O-SiO2 system) that are in the literature are strongly dispersed. Therefore, we decided to select the most recent data set that was published by Abdelouhab et al. [28,[START_REF] Abdelouhab | Détermination de grandeurs physico-chimiques dans les verres fondus -relation avec le comportement à la corrosion du chrome et d'alliages chromine-formeurs[END_REF]. Logarithms of Cr(III), Rox and Rred values were plotted as a function of log[a(O 2-)] for different temperatures, and for a given fO2 : examples of these kinds of plots are given in Figure 5. As expected, the variations are linear. The obtained slope values, the calculated x, y and z parameters as well as the chromium complexes formulas are summarized in Table 3. The slope values presented here are the average values of the slopes determined experimentally for the different fO2. The error bars correspond to the standard deviations of the experimental slopes.

These parameters are extrapolated after multiple processing of experimental data, and it should be noted that the x, y and z values remain constant with the temperature (Figure 6), even if the slope values differ from one temperature to another. Consequently, important information has to be highlighted: only one complex, for each oxidation state of chromium, can exist in the studied range of temperature. At this stage, the compositions suggested for the chromium species are Cr2O 2+ , Cr3O4 + and Cr2O7 2-for Cr(II), Cr(III) and Cr(VI) respectively.

Tilquin et al. [10], who studied chromium oxide dissolution in soda silicate melts, have determined that (y-x) is equal to 2,88 ( 3) and suggested that the composition of the Cr(VI) complex is CrO4 2-. Consequently, the composition suggested for the chromium complex containing Cr(III) is CrO + . This is not in agreement with the compositions of the complexes obtained in this work. This difference can be explained by the fact that Tilquin et al. [10] have performed their study at T=1000°C while the temperature domain explored during this work rises from 1125 to 1350°C. Indeed, the chromium complex compositions can drastically change with melting temperature.

As it has just been shown, x, y and z parameters are constant in the range of temperatures studied here (Figure 6). Consequently, it is possible to carry out calculations allowing the determination of thermodynamic data (standard enthalpies and entropies) relative to the dissolution, oxidation and reduction reactions [START_REF] Khedim | [END_REF] of chromia for the various Na2O-SiO2 melt compositions. The thermodynamic development is as follows:

For given oxygen fugacity and glass composition, Eq. 5, 7 and 9 can be written:

A III Cr = )] ( log[
Eq. 16

B fO III Cr VI Cr + = 2 log 4 3 )] ( [ )] ( [ log
Eq. 17

C fO III Cr II Cr + - = 2 log 4 1 )] ( [ )] ( [ log Eq. 18 with ] [ log 2 3 log 2-       - + = O a x K A diss Eq. 19 ] [ log 2 3 log 2-       + - - = O a y x K B ox Eq. 20 ] [ log 2 1 log 2-       + - + = O a x z K C red Eq. 21
By plotting log[Cr(III)] as a function of (1/T) for a given glass composition, we notice that the behavior follows an Arrhenius law (Figure 7) and can be written:

T A A III Cr ' ' ' )] ( log[ + =
Eq. 22

Equilibrium constant, as well as O 2-, activity depends on temperature as shown by the following equations:

R S T R H K 3 . 2 1 3 . 2 log 0 0  +        - = Eq. 23 T O a   + = -)] ( log[ 2 Eq. 24
As it has been mentioned before, O 2-activity is comparable to Na2O activity.  and  parameters are well known for the Na2O-SiO2 system [28]. By replacing the Kdiss and aO 2- with their respective expressions, log[Cr(III)] as a function of (1/T) can be divided into entropic and enthalpic terms:

                  - -  - +             - -  = T x R H x R S III Cr   2 3 3 . 2 2 3 3 . 2 )] ( log[ 0 0 Eq. 25 With ' 2 3 3 . 2 0 A x R S =             - -   and ' ' 2 3 3 . 2 0 A x R H =             - -  - 
At this step, standard enthalpy and entropy of chromium dissolution process can be expressed, for a glass composition, as a function of the determined parameters:

            - + =   2 3 ' 3 . 2 0 x A R S diss
Eq. 26

            - - - =   2 3 ' ' 3 . 2 0 x A R H diss
Eq. 27

B and C parameters (Eq. 16 and Eq. 17) correspond to the Y-axis intercept when plotting log[Cr(VI)/Cr(III)] and log [Cr(II)/Cr(III)] respectively as a function of logfO2 (Figure 4). The evolution of this two parameters (B and C) as a function of reverse temperature (1/T), for a given glass composition, is linear (Figure 8) and follows an Arrhenius law as :

T B B B ' ' '+ = Eq. 28 T C C C ' ' '+ = Eq. 29
Considering B and C expressions (Eq. 17 and Eq. 18), and by replacing logKox, logKred and log aO 2-(Eq. 20 and Eq. 21) by their respective expressions, B and C parameters can be written as:

T R H x y R S x y B ox ox 3 . 2 2 3 3 . 2 2 3 0 0  -       - - +        +       - - =   Eq. 30 T R H x z R S x z C red red 3 . 2 2 1 3 . 2 2 1 0 0  -       + - +        +       + - =   Eq. 31
Standard enthalpies and entropies of oxidation and reduction reactions can thus be determined by identification and expressed as follows:

            - - - =   2 3 ' 3 . 2 0 x y B R S ox
Eq. 32

      -       - - =  ' ' 2 3 3 . 2 0 B x y R H ox  Eq. 33             + - - =   2 1 ' 3 . 2 0 x z C R S red Eq. 34       -       + - =  ' ' 2 1 3 . 2 0 C x z R H red  Eq. 35
Standard entropies and enthalpies for chromium dissolution, reduction and oxidation in the Na2O-SiO2 system, obtained by the thermochemical calculations, are reported in Table 4.

Considering the set of data related to the dissolution process, H°diss and S°diss values, given for the whole Na2O-SiO2 composition range, are homogeneous around a median value of 85 ± 5 kJ.mol -1 and 22 ± 5 J.mol -1 .K -1 respectively. Moreover, if we compare H°diss values, calculated in this study, with those measured by Kim et al. [START_REF] Kim | [END_REF] in borosilicate systems it comes that all these values are coherent. This justifies a posteriori the chosen approach.

Conclusion

Several important features of the chromium chemistry in silicate melts were drawn up in this study. Among those, it is advisable to recall that:

-The development of an original experimental device allowed the determination of the Cr2O3 solubility limits in the Na2O-xSiO2 system as a function of experimental parameters.

-The chromium species speciation and thus the redox ratios could be determined by using a relatively simple model.

-The influence of the experimental parameters (temperature, oxygen fugacity and glass composition) on the redox ratio has been evidenced. In fact, it has been shown that an increase in the temperature leads to an increase in the redox ratio and that basicity of the melt stabilises the oxidised forms.

-The unicity of the complex compositions, associated to the various chromium valences, in the studied range of temperature and glass composition (Na2O-SiO2 system) have been proved.

-Chromium complexes compositions Cr2O7 2-, Cr3O4 + and Cr2O 2+ are proposed for Cr(II), Cr(III) and Cr(VI) species respectively.

-The thermodynamic data related to the dissolution, the reduction and the oxidation processes of chromium oxide in the soda-silicate system could be determined.

A complete set of data, concerning the chromium solubility limits in the Na2O-SiO2 system is proposed as a function of temperature, glass composition and oxygen fugacity. A recent work [31] presents a similar approach in calcium silicate base melts, with a phenomenology in good agreement with ours. Nevertheless, further studies have been performed here: oxo-complexes formulas, thermodynamic data and evolution of redox with melt basicity have been proposed.

Figure 2 :

 2 Figure 2: Total chromium contents reported as a function of oxygen fugacity for different

Figure 3 :

 3 Figure 3: Plots of total chromium contents in the Na2O-2SiO2 melt as a function of oxygen

Figure 4 :

 4 Figure 4: Plots of log(Rred) (a) and log(Rox) (b) determined in the Na2O-2SiO2 system as a

Figure 5 :

 5 Figure 5: Logarithmic plot of Cr(III) (a), Rox (b) and Rred (c) as a function of aO 2-.

Figure 6 :

 6 Figure 6: Stoichiometric coefficients x, y and z of the chromium oxo-complexes as a function

Figure 7 :Figure 8 :

 78 Figure 7: log[Cr(III)] reported as a function of 1/T for 4 compositions of the Na2O-SiO2

Table 1 :

 1 Composition of glasses used as starting materials

	Aimed glass Composition	Theoretical Na (at%)	Theoretical Si (at%)	Na (at.%) WDS	Si (at.%) WDS	X
	Na2O-1.5SiO2	26.67	20.00	25.95 (0.40)	20.17 (0.19)	1.56 (0.04)
	Na2O-2SiO2	22.22	22.22	21.91 (0.44)	22.38 (0.22)	2.04 (0.06)
	Na2O-2.5SiO2	19.04	23,81	19.22 (0.49)	23.72 (0.25)	2.47 (0.09)
	Na2O-3SiO2	16.67	25.00	16.34 (0.22)	25.16 (0.11)	3.08 (0.06)
	Na2O-3.5SiO2	14.81	25.92	14.94 (0.19)	25.86 (0.09)	3.46 (0.06)

Table 2 :

 2 Total content of chromium dissolved in the Na2O-SiO2 melts as a function of temperature, glass composition and oxygen fugacity.

	T=1250°C

Table 3 :

 3 x, y and z values and chromium complex formulae determined for the different investigated temperatures in the Na2O-SiO2 melts

		T=1250°C	T=1200°C	T=1175°C	T=1125°C	Average values
	x-3/2	-0.18 (0.03)	-0.17 (0.04)	-0.15 (0.03)	-0.15 (0.03)
	y-x-3/2	0.73 (0.13)	0.66 (0.14)	0.42 (0.08)	0.40 (0.08)
	z-x+1/2	-0.30 (0.05)	-0.26 (0.05)	-0.24 (0.04)	-0.36 (0.07)
	x	1.35 (0.03)	1.33 (0.04)	1.35 (0.03)	1.35 (0.03)	1.345
	y	3.50 (0.16)	3.49 (0.18)	3.27 (0.10)	3.24 (0.11)	3.375
	z	0.55 (0.08)	0.57 (0.09)	0.61 (0.08)	0.49 (0.10)	0.555
	Cr(III)	Cr3O4 +	Cr3O4 +	Cr3O4 +	Cr3O4 +
	Cr(VI)	Cr2O7 2-	Cr2O7 2-	Cr2O7 2-	Cr2O7 2-
	Cr(II)	Cr2O 2+	Cr2O 2+	Cr2O 2+	Cr2O 2+

Table 4 :

 4 Standard entropies and enthalpies of chromium dissolution, reduction and oxidation reactions in the Na2O-SiO2 system

		H°diss	S°diss	H°oxy	S°oxy	H°red	S°red
		(kJmol -1 )	(JK -1 mol -1 )	(kJ mol -1 )	(JK -1 mol -1 )	(kJmol -1 )	(JK -1 mol -1 )
	Na2O-1.5SiO2	85 ± 4	18 ± 5	-105 ± 8	35 ± 5	218 ±	49 ± 8
	Na2O-2SiO2	88 ± 1	24 ± 4	-63 ± 14	67 ± 10	185 ±	21 ± 7
	Na2O-2.5SiO2	79 ± 11	20 ± 6	-120 ± 6	30 ± 6	228 ±	48 ± 7
	Na2O-3SiO2	84 ± 15	26 ± 5	-115 ± 22	30 ± 6	246 ±	61 ± 12

Acknowledgements

The authors are grateful to J. Ravaux and S. Mathieu for performing EPMA (Service Commun d'Analyses par Microsonde Electronique de Nancy). This work was financially supported by the ANR BLAN06-3_134633 through the Actimelt project.