Jean-Raymond Gavarri
email: gavarri.jr@univ-tln.fr

Claude Carel
email: c-carel@orange.fr

The complex nonstoichiometry of wüstite Fe 1-z O: review and comments

Keywords: wüstite, alpha-gamma transition, point defects, defect clusters, pseudo-phases, percolation

Thermodynamic properties and structural aspects of the nonstoichiometric wüstite Fe 1-z O, and its modificationsthe so-called pseudo-phases -as functions of departure z from stoichiometry and of equilibrium temperature are reviewed from 1960 to present. The complexity of the equilibrium phase diagram is described in some details. The first order transition W W' is specified on the iron / wüstite boundary near 1185 K. Transitions correlated to the modifications Wi at T(W)>1185 K and W'j at T(W') <1185 K (i and j =1,2,3) are re-examined. Structural determinations based on the characterization of point defects stabilization and of their clustering are reviewed. Additionally, the pseudo-phases are examined based on the transformation of defect clusters or of their mode of distribution (i.e., percolation or superstructure) with the inclusion of changes in electronic charge carriers.

Introduction

The nonstoichiometric iron monoxide Fe 1-z O is the main constituent of traditional blast furnace slags. Historically, this oxide was the subject of numerous studies in the general context of iron metallurgy improvement, and for its role in the fields of recycling of industrial waste, catalysis and nanoparticles.

Fe1-zO, also written equivalently as Fe y O or FeO x , is named after the German mineralogical appellation "wüstit" 1 . The wüstite is considered of particular interest by the geophysicists because it is a constituent of the lower mantel of earth. This oxide is in a steady state, between an invariant point near 600 °C and its melting temperature around 1400 °C, under an equilibrium oxygen pressure p' in the range 10 -25 to 10 -6 bar. Its large iron deficiency is expressed by the departure of stoichiometry z  [0.04 -0.18] around 1400 °C and z  0.065 near 600 °C. Because of so many papers published before, numerous reviews about the nonstoichiometric wüstite are available in the literature. So, the present authors make choices of some of them or equivalent works:

Goodenough [START_REF] Goodenough | Improved then translated into French by Casalot A: Les oxydes des métaux de transition[END_REF], Kofstad [START_REF] Per | Nonstoichiometry, diffusion, and electrical conductivity in Binary metal oxides[END_REF], Vallet [START_REF] Vallet | Collection 'Monographies de chimie minérale[END_REF], Burgmann [START_REF] Burgmann | The defect structure models for wüstite: A review[END_REF], Gokcen [START_REF] Gokcen | Thermodynamics[END_REF], Men' et al. [START_REF] Men | Compositiondefect-properties of solid phases. Method of cluster component[END_REF], Spencer, Kubaschewski [START_REF] Spencer | A thermodynamic assessment of the iron-oxygen system[END_REF], Bauer, Pianelli [START_REF] Bauer | A review of the lacunar structure of wüstite, and the necessity of new experimental researches[END_REF], Sorensen [START_REF] Cheetham | Thermodynamics and Defect Structure of Nonstoichiometric Oxides; section IX: Recommendations for Future Work[END_REF], Hazen, Jeanloz [START_REF] Hazen | Wüstite (Fe1-xO): a review of its defect structure and physical properties[END_REF], Gleitzer, Goodenough [START_REF] Gleitzer | Mixed-valence iron oxides in structure and bonding[END_REF], Gokcen [START_REF] Gokcen | Statistical thermodynamics of alloys[END_REF], Lykasov et al. [START_REF] Lykasov | Physics and chemistry properties of wüstite and its solutions[END_REF], Mrowec, Podgorecka [START_REF] Mrowec | Defect structure and transport properties of non-stoichiometric ferrous oxide[END_REF], Long, Grandjean [START_REF] Long | Mössbauer effect, magnetic and structural studies of wüstite[END_REF], Sundman [START_REF] Sundman | An assessment of the Fe-O system[END_REF], Wriedt [START_REF] Wriedt | The Fe-O (Iron-Oxygen) system. J. Phase Equilibria Section II: phase diagram evaluations[END_REF], Collongues [START_REF] Collongues | Nonstoichiometry in Oxides[END_REF], Gleitzer [START_REF] Gleitzer | Electrical properties of anhydrous iron oxides[END_REF], Smyth [START_REF] Smyth | The defect chemistry of metal oxides[END_REF], Desré, Hodaj [START_REF] Desré | Thermodynamique des matériaux[END_REF], Worral, Coley [START_REF] Worral | Defect structure of pseudo-phases of wüstite[END_REF].

Multiple studies have addressed the nature of the point defects (iron vacancies and interstitials) in wüstite, and the concept of defect clustering was systematically developed with a large variety of structural models. The NaCl face centered cubic structure of wüstite is characterized by a high rate of iron vacancies, and the existence of a certain proportion of iron ions occupying interstitial sites. Crystallographic determinations showed that these defects are not disordered in the lattice. Diverse models of clusters of iron vacancies and interstitials have been proposed.

Because it simplifies the structural description of defects, the oxygen site is assumed to be fully occupied (i.e., there is no significant amount of oxygen vacancies). Adopting the notation of Kröger and Vink [START_REF] Kröger | Solid State Physics[END_REF], the formula of this monoxide can be developed as t (V Fe q( ' ) ) ( z+t) O O [START_REF] Goodenough | Improved then translated into French by Casalot A: Les oxydes des métaux de transition[END_REF] The ionic species Fe 2+ and Fe 3+ , in octahedral sites, are noted Fe Fe and Fe Fe °. The Fe 3+ cations in tetrahedral (interstitial) sites are noted Fe i °°°. Iron vacancies noted V Fe q ( ' ) can be charged negatively corresponding to q=2 or q=1, or neutral corresponding to q=0. Electric neutrality would result from electrons jumping in the conduction band or interacting with holes in the valence band. It can be recalled that the band gap of wüstite was recently evaluated at about 1.0 eV at 25 °C, which means that a large population of electrons in the conduction band and/or holes in the valence band can be expected at high temperature, depending on the oxygen partial pressure [START_REF] Schrettle | Wüstite: electric, thermodynamic and optical properties of FeO[END_REF].

In this paper, we review historical (1960-2014) observations and concepts for wüstite at equilibrium and in quenched samples. Then, we show how the complex nonstoichiometry of wüstite can be better described using mixtures of point defects and their associations in clusters which fall into order progressively at increasing z.

The complex behavior of wüstite

Early work (1921-1961)

In 1921, G. Chaudron worked in H. Le Chatelier's lab. when he determined experimentally the external boundaries of the stable domain of ferrous oxide considered to be the daltonide 'FeO' [START_REF] Chaudron | Reversible reaction study of hydrogen and carbon oxide on metallic oxides[END_REF]. Point C is at the intersection of the two boundaries, where three solid phases (Fe, Fe3O4 and 'FeO') coexist at equilibrium at 570°C, temperature that has been revised particularly by Raccah, Vallet, Carel [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF][START_REF] Vallet | Contribution to the study of the solid nonstoichiometric iron monoxyde. Diagram T-P(O2)-X[END_REF][START_REF] Vallet | Evaluation of the molar thermodynamic properties of solid wüstites from their equilibrium thermodynamic study, Part I: formulas for partial and integral molar properties of the three Wi and three W'i[END_REF][START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF], and that is named Chaudron's point.

Jette and Foote demonstrated -measurements of the cell parameter and density-that wüstite was an iron deficient solid solution. They deducted that for the ionic compound, a missing Fe 2+ was necessarily compensated by the oxidation of two other Fe 2+ giving rise to electron holes Fe 3+ [START_REF] Jette | An X-ray study of the wüstite (FeO) solid solutions[END_REF].

Darken and Gurry studied in detail the phase diagram when determining the composition of wüstite chemically after slow quenching from  °C= 1100, 1200, 1300 and 1400. The boundaries with iron and magnetite were determined down to Chaudron's point. Additionally, the authors determined exhaustively the molar partial and integral thermodynamic properties of wüstite [START_REF] Darken | The System Iron-Oxygen. Part I. The Wüstite field and related equilibria[END_REF].

By comparison of calculated and experimental entropy values, Todd and Bonnickson, and Humphrey, King and Kelley deducted that the lacunary iron lattice envisaged by Jette and Foote was ordered [START_REF] Jette | An X-ray study of the wüstite (FeO) solid solutions[END_REF] [32] [START_REF] Humphrey | Some thermodynamic values for ferrous oxide[END_REF].

Brynestad and Flood observed that some (electron holes) Fe 3+ are necessarily located in tetrahedral sites written as [Fe III O 4 ] [START_REF] Brynestad | The redox equilibrium in wüstite and solid solutions of wüstite and magnesium oxide[END_REF].

Using neutron diffraction analyses, Roth determined the proportion of Fe 3+ occupying the tetrahedral sites of the fcc structure and concluded to the formation of clusters (2:1) constituted of two iron vacancies and one tetrahedral site occupied by a cation Fe 3+ [START_REF] Roth | Defects in the crystal and magnetic structures of ferrous oxide[END_REF]. Tarte et al. also showed directly by I.R. spectroscopy the existence of these Fe i °°°i ons [START_REF] Tarte | IR spectrum and presence of trivalent tetrahedral iron in the structure of iron protoxide Fe1-xO (Wüstite)[END_REF].

Using the formalism of the mass action law initially proposed by Carl Wagner et al. [START_REF] Wagner | Electrical conductivity of cobalt and iron oxides[END_REF] [38], Brynestad and Flood [START_REF] Brynestad | The redox equilibrium in wüstite and solid solutions of wüstite and magnesium oxide[END_REF], and subsequently Smyth [START_REF] Smyth | Deviations from Stoichiometry in MnO and FeO[END_REF] established a simplified expression linking the composition z to the equilibrium oxygen pressure as:

z  p' 1/s (2)
In a first attempt, Smyth characterized the defect structure of both Mn 1-z O and Fe 1-z O based on the experimental compositional data from Darken and Gurry [START_REF] Darken | The System Iron-Oxygen. Part I. The Wüstite field and related equilibria[END_REF]. His work often referenced has been continued by many authors in different areas of the defect chemistry, particularly the study of the clustering.

Modifications or pseudo-phases W1, W2, W3

Raccah-Vallet's team

During years 1962-65, three supposed « allotropic varieties » or « modifications » noted Wi (i=1,2,3) named later « subphases » and then « pseudo-phases » by other authors are discovered by Raccah (doctoral thesis [START_REF] Raccah | Study of the thermodynamic properties of iron protoxide[END_REF]), Vallet, and Kléman additionally [START_REF] Raccah | On thermodynamic properties of wüstite, a strictly regular solution of iron and oxygen[END_REF][START_REF] Raccah | New set of isotherms for the solid wüstite[END_REF][START_REF] Vallet | About new thermodynamic properties and a new phase diagram of solid wüstite[END_REF][START_REF] Vallet | Contribution to the study of the thermodynamic properties of the solid iron protoxide[END_REF][START_REF] Kléman | Thermodynamic properties of the solid iron protoxide. Experimental results applied to the drawing of the equilibrium diagram[END_REF] from their experimental measurements. An initial layout of the corresponding phase diagram is proposed, based on coordinates (Θ °C, x) [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF].

During their studies from 1961 to 1989, Vallet et al. establish correlations between mass variation, i.e., departure from stoichiometry expressed by x in FeO x , temperature T and oxygen partial pressure p' under equilibrium (see also [START_REF] Vallet | Collection 'Monographies de chimie minérale[END_REF]). They describe the phase diagram of wüstite FeO x above 911°C through the following three equations connecting log p' noted l' with the composition x and temperature T, and expressing the three thermodynamic behaviors or pseudo-phases W1, W2 and W3, successively [START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF]:

W1 :
l' = (46753.4T -1 -7.3781).x + (-78825.3T -1 + 16.0613) (3a) W2: l' = (-9568.9T -1 + 31.1728).x + (-18413.3T -1 -25.2569) (3b) W3: l' = (-33238.9T -1 + 48.3669).x + (6883.9T -1 -43.5669) (3c)

To better relate the nonstoichiometry and equilibrium pressure p' to the presence of defects (mainly cation vacancies and interstitials), we have expressed l' as a function of z instead of x and we have reported l' as a function of log z values. It can be recalled that these relationships derive from the general expression of equilibrium constants K(T) linking the composition z to the oxygen partial pressure p', following the basic setup:

½ O 2  O O + V Fe q ( ' ) + q h ° (4a) with [h °] = q.[V Fe q ( ' ) ] = q.z, K(T) = {[O O ].[V Fe q ( ' ) ].[h °]q } / (p') ½ = C.(z s / p' ) ½ (4b)  log z = (1/s).log(K(T)/C) 2 + (1/s) l' ( 4c
)
where C depends on q [START_REF] Smyth | Deviations from Stoichiometry in MnO and FeO[END_REF]. For iron vacancies expressed V Fe , exponent q( ' ) indicates a charge q times minus; h ° correspond to an electron hole. In the expression of K(T), we have neglected the activity coefficients, considering that they are slowly varying with z in a limited composition range. The activity [O O ] of oxygen on its site fcc is considered equal to 1 because no significant amount of vacancies can be present in the oxygen lattice. Finally, we have taken into account the relationshipq.[V Fe q(

' ) ]+[h °] = 0 (electroneutrality condition) with [V Fe q(

' ) ] = z.

The equilibrium constants depend on the values 2, 1, and 0 of q. The three hypothetical corresponding values of exponent s should be 6 (q=2), 4 (q=1), and 2 (q=0) (more likely close to these values, taking into account the various approximations made). In the remainder of this review, exponent s is taken as characteristic of the specific equilibrium equation connecting the fractions of point defects between them. Let us note that the relationships between activities and compositions z are not strictly linear. They assume the existence of three domains and allow a comparison with classical models of defect equilibria. In each domain, a unique model of defect equilibrium is supposed to be valid, which is not strictly exact because of the continuous evolution of interactions between defects, as composition and temperature vary in the phase diagram.

Fixed temperature and variable composition

To illustrate the different behaviors or transitions between one and another in Fig. 1 From the three segments observable on the curve l' vs log z in Fig. 1, we can derive three different values: s(W1) = 4.9, s(W2) = 6.7, and s(W3) = 8.9. These values are similar to the ones found by Toft Sørensen and El Sayed Ali, and Rekas and Mrowec [START_REF] Toft Sørensen | Defects on metal-deficient oxides: wüstite, Fe 1-y O[END_REF] [START_REF] Rekas | On defect clustering in the wustite phase[END_REF]. In reality, there is no reason for assuming linear relationships given the probable existence of mixtures of defects, and of their continuous evolution conditioned by the changes in charge of V Fe q(') (q=0,1,2).

Fixed composition and variable temperature

For a fixed value x = 1.130 (z = 0.115), we report in Fig. 2 the experimental values of l' versus 1/T using the data of Vallet and Carel [START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF]. Three quasi linear domains can be distinguished suggesting the existence of three Gibbs free energies of reaction in relation with the equilibrium constant K(T), with the classical relationship G(reaction) = H -TS = -RTlnK. The linear correlations are:

W1:

l' = -2.5994(10 From relation (7b), three values of enthalpies can be determined, linked to the changes in slopes and attributable to the pseudo-phases: W1:

H 1 /R = -25526 K  H 1 = -2.20 eV W2:
H 2 /R = -29322 K  H 2 = -2.52 eV W3: At the same time, a qualitative dilatometric study under CO2/CO indicates weak changes of slopes in curves l(T) with intersections located on the boundary W2/W3 [START_REF] Carel | Dilatometric study of diverse varieties of solid wüstite, and existence of a metastable point between the three varieties[END_REF]. In addition, several correlations between the cubic cell parameter obtained after quenching or in situ and the composition x in FeO x led correspond to the separation in three pseudo phases or subdomains of the equilibrium phase diagram [START_REF] Carel | Variation of the cell parameter for three wüstite varieties[END_REF] [START_REF] Carel | Introduction to description of phase diagram of solid wüstite. I: structural evidence of allotropic varieties[END_REF].

H 3 /R = -31008 K  H 3 = -2.67 eV
In the equations above, we have neglected the activity coefficients, assuming that the ionic interactions remained constant as z varies. Another point of view would be to consider that a significant evolution of the activity coefficients can be observed and that the variation is associated with molar fractions

[i] of point defects at equilibrium (the activity a i being proportional to the molar fraction: a i =  i .[i]). In the case of ionic solids, these activity coefficients  i are generally used to express deviations from ideal models of isolated defects at equilibrium.

Abnormal behavior near 1184/1185 K

A splitting of the phase diagram

Vallet proposes a separation of an upper subdomain W of the phase diagram from the lower subdomain W' on each part of the first order transformation -Fe  -Fe [START_REF] Vallet | On the thermodynamic properties of solid wüstite below 911°C[END_REF]. Each subdomain is constituted of three pseudo-phases W'j below and Wi above 1184 K. Because the properties of W' and W are distinct, the boundaries with or -iron of the stability domains are also distinct. That is the reason for the existence of points B and A, at the upper and lower end respectively of the corresponding curves in the phase diagram (see [START_REF] Vallet | Evaluation of the molar thermodynamic properties of solid wüstites from their equilibrium thermodynamic study, Part I: formulas for partial and integral molar properties of the three Wi and three W'i[END_REF], lines at 1184.15 K in Tables IV and V p. 723 and 724). The limit compositions on these boundaries y o(1) = 0.9427 or z o(1) = 0.0573, and y' o(1') = 0.9528 or z o(1') = 0.0472 of W1 and W'1 in equilibrium with Fe-O  and Fe-O ' are proposed at points A and B, respectively, both assumed at 1184 K, which cannot strictly occur as discussed below.

Thermodynamic inconsistency

In an extended and careful review published in 1991, Wriedt [START_REF] Wriedt | The Fe-O (Iron-Oxygen) system. J. Phase Equilibria Section II: phase diagram evaluations[END_REF] asks questions about the isotherm at 1184 K, and its assumed continuity, while it seems to correspond with transitions Wi/W'j, i.e., with five sub-boundaries for the indexes 1/1, 2/1, 2/2, 2/3 and 3/3. Are they linked to the - Fe transformation? These pending questions have no answer until the 1980s.

The basic equations governing the various equilibria between gases and solid phases (see equilibrium diagram in Fig. 3) are listed below. Particularly at the two boundaries involving-Fe cr or -Fe cr (subscript 'cr' for 'solid', i.e., 'crystallized'), equilibriacan be expressed:

1) on the boundaryFe/W'1 from point B to point C, (1-z o(1') ) -Fe cr + ½ O 2  Fe 1-zo(1 ' ) O  K   = exp (G o  /RT) (8a) -Fe cr  Fe gas  K (gas) = p  (Fe gas ) (8b) (1-z o(1') ) Fe gas + ½ O 2  Fe 1-zo(1') O  K (, gaz) = exp -(G o 1/RT) = 1 / (p  (Fe gas ). p'  ½ ) (8c)
2) similarly, on the boundary Fe/W1 from point A to melting, (1-z

o(1) ) -Fe cr + ½ O 2  Fe 1-zo(1') O  K   = exp (G o /RT) (8d) -Fe cr  Fe gas  K (gas) = p  (Fe gas ) (8e) (1-z o(1) ) Fe gas + ½ O 2  Fe 1-zo(1) O  K (, gaz) = exp -(G o 2/RT) = 1 / (p  (Fe gas ).p '  ½ ) (8f )
At the arbitrarily unique temperature of 1184 K, it is thus necessary to apply two different partial pressures p 1 and p 2 of Fe(gas), at equilibrium with the solid at points A and B. Obviously, at the unique Fe equilibrium temperature, it exists only one value of the pressure p(Fe gas ) = 3.005x10 -11 atm. Thus, the points A and B cannot be at the same temperature. Correlatively, in the case of the points A and B differing in temperatures T(A) and T(B), the compositions z o(W) (A) and z o(W') (B), the equilibrium partial pressures p'(A) and p 2 (A), p'(B) and p 1 (B) of oxygen and iron gas are necessarily different. Finally, a small difference in temperatures, probably a few degrees, are to be determined experimentally in order to adjust this set of parameters, which determines the phase diagram (Fig. II-3) close to the stable temperature of the -Fe transition. In this situation, the bottom row in Table IV should be changed to a value of T(A) higher than 1184/1185 K, and the header line of Table V to a value of T(B) also higher than 1185 K, but lower than T(A) in ( [START_REF] Vallet | Evaluation of the molar thermodynamic properties of solid wüstites from their equilibrium thermodynamic study, Part I: formulas for partial and integral molar properties of the three Wi and three W'i[END_REF] p. 723 and 724).

Note: If T(A) and T(B) differ from a typical value of 4 °C (see 2.3.4. Fe-O phase diagram close to 1185 K ), for example T(A) = 1191 K and T(B) = 1187 K, the partial pressures of iron are necessarily p(A)=p 1 =3.692 x10 -11 atm and p(B) = p 2 =3.220 x10 -11 atm following tabulated data [START_REF] Pankratz | Thermodynamic properties of elements and oxides[END_REF], [START_REF] Desai | Thermodynamic properties of iron[END_REF], [START_REF] Chase Jr | NIST-JANAF thermochemical tables[END_REF]. Thus, the differences between the two iron partial pressures is on the order of 0.4x10 -11

Phenomena observed close to the isotherm at 1185 K

Experimental confirmations of this hypothetical separation exist in the literature. Several of them are interesting to be examined. Löhberg and Stannek separate the pre-exponential term a and the energy term b = G/RT of an expression functionally similar to a Boltzmann distribution p H2O /p H2 = a.exp(b.z) expressed as a function of T -1 in two segments intersecting in the vicinity of the Fe transition temperature in ([55] Similarly, Janowski, Mrowec, Stokłosa separate in two segments the variation of the self-diffusion coefficient D* Fe of iron in wüstite as a function of T -1 from their own results and several sets from the literature in ([56] Fig. 7 p. 99). The intersection occurs for T -1 = 8.28 x 10 -4 K -1 , i.e., at 935°C, of the segments attributed to W and W'. Sundman, in an assessment of the entire Fe-O system, formulate a model for the 'pure iron corner' (solid solutions Fe-O  and Fe-O ' ) [START_REF] Sundman | An assessment of the Fe-O system[END_REF]. The assessment of Spencer and Kubaschewski is also relevant in the same context [START_REF] Spencer | A thermodynamic assessment of the iron-oxygen system[END_REF].

A departure from a monotonous variation of the electromotive force of galvanic cells, near the Fe transition, is evidenced from measurements at the iron/wüstite electrode, in several galvanic cells [57] [59]. In addition, authors [START_REF] Sjödén | On the Gibbs energy of formation of wustite[END_REF] publish their numerous determinations of the electromotive force (e.m.f.) between 866.7 and 1339.3 K. An anharmonic transitional zone between two separate continuous variations, not in continuation of each other, can be identified in the temperature range [1186-1194 K].

Consequences relevant to the Fe-O phase diagram

The  Fe first order transition in the 'pure iron corner' of the phase diagram Fe-O is not documented sufficiently. The left side in Fig. 3 displays literature data (relative to FeO x ) concerning the domain of this transition [START_REF] Spencer | A thermodynamic assessment of the iron-oxygen system[END_REF] [16] [START_REF] Swisher | Solubility, permeability and diffusivity of oxygen in solid iron[END_REF]. The temperature of the strictly pure FeFe transition is tabulated at 1184  3 K in the data base NIST-Janaf (1999-2011) [START_REF] Chase Jr | NIST-JANAF thermochemical tables[END_REF]. At present, it is generally taken to be 1185 K (point ø in Fig. 3). The  Fe transitional enthalpy is known with a twenty percent dispersion from assessed and experimental data, (see [START_REF] Guillermet | An assessment of the thermodynamic properties and the (p, T) phase diagram of iron[END_REF] Sundman takes up the molar fractions (  at point P, ' at point b in Fig. 3) of dissolved oxygen N O = x/(1+x) = 16 x 10 -6 and 6.8 x10 -6 in and -Fe respectively at 1185 K or 912 °C [START_REF] Sundman | An assessment of the Fe-O system[END_REF] (Fig. 3).

The separation of the boundary iron/wüstite into parts Fe-O  /W1 from point A and Fe-O ' /W'1 from point B arises distinctly in the temperature range characterized in [START_REF] Sjödén | On the Gibbs energy of formation of wustite[END_REF], following the suggested phase diagram in the right part of Fig. 3. Two types of two-phase domains ( + W'1) and (W1 + W'1) have been added hypothetically as well as an invariant point a' (W'1, Fe-O  , W1). As of now, the location of points A and B remains to be determined precisely. New experimental data is necessary to confirm the preceding suggestions.

For the subdomain of W' below 912 °C, the information equivalent to that of W above 912 °C is more difficult to access (see [START_REF] Vallet | Contribution to the study of the thermodynamic properties of the solid iron protoxide[END_REF] Figs. 6 and7 p. 12 and 13). An empirical relationship associated with the three W'j requires a curvature term. The analytical expression has been taken as follows:

l' = (a' j T -1 + b' j ) x 2 + (c' j T -1 + d' j ) x + (e' j T -1 + f' j ) (9)
The corresponding coefficients (a' j -f' j ) are available numerically in ([29] Table 1 p. 210). The sets of numerical relations detailed above for l' as functions of x and T -1 allow the determination of the transition lines or boundaries between domains of the pseudo-phases W1, W2 and W3, from -iron to magnetite between 912 and around 1400 °C, and W'1, W'2 and W'3, from -iron to magnetite between 912 and 592 °C, at point C, as drawn in Fig. 4 (see precedent drawings [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF] [27] [29]).

First evidences of phase modifications or pseudo-phases in the literature

Wagner jr's team

In the course of (thermodynamics -defect structure) approaches by Carl Wagner, Wagner jr et al. study the electrical conductivity  at equilibrium as function of log p O2 [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF]. With the help of their own thermogravimetric data l'(x) [START_REF] Swaroop | On the vacancy concentrations of wüstite (FeO x ) near the p to n transition[END_REF], they state that the composition of the break on their curves (x) well agree with that of the transitions Wi  Wi+1 determined in Raccah's work. So, the pseudo-phases are identified at equilibrium outside of Vallet's group for the first time (see [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF] Table IV p. 955; also Annex A Fig. A1 a-). The further studies of Wagner jr's group about wüstite were concerned with transport phenomena in the bulk and on the surface, in relation with the p to n transition.

Fender and Riley

From numerous accurate electromotive force measurements (e.m.f.) in galvanic cells, Fender and Riley [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF] brought the second experimental confirmation, outside of Vallet's team, of the likely existence of three pseudo-phases, and the location of the boundaries of the subdomains in the range 700-1350 °C. Their plot of the equilibrium diagram is in fairly good agreement with that of Vallet et al. (see Annex A, Fig. A2).

Partial molar enthalpies and entropies. Convergence point 

Fender and Riley, then later Vallet and Carel assessed the partial molar properties of an oxygen solid solution (H O  H O -½ H°O 2 , S O  S O -½ S°O 2 ) as functions of x  [1.058 -1.172] for the three pseudophases Wi (i=1,2,3) named also wI, wII and wIII (see [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF] Table I p. 796, and infra in Figs. 5 and6). The following relations were formulated in [START_REF] Vallet | Evaluation of the molar thermodynamic properties of solid wüstites from their equilibrium thermodynamic study, Part I: formulas for partial and integral molar properties of the three Wi and three W'i[END_REF] for the three Wi (i=1,2,3) at T >1184 K: and, for the three W'j (j=1,2,3) at T< 1184 K:

 H O = 19.14475(a i .x + c i ), ( 10h
(H O )' = 19.14475 (a' j .x 2 + c' j .

x + e' j ) (11h) (S O )' = -19.14475 (b' j .x 2 + d' j .x + f' j ) (11s)

Relations [START_REF] Hazen | Wüstite (Fe1-xO): a review of its defect structure and physical properties[END_REF] for the Wi, and (11) for the W'j are independent of T, in accordance with the property of regularity ([12] p. 81-85) of the solid solution wüstite identified in [START_REF] Raccah | On thermodynamic properties of wüstite, a strictly regular solution of iron and oxygen[END_REF] but contested in [START_REF] Marucco | Partial molar properties of oxygen mixing into the iron protoxyde at 1075°C. I.-Direct measurements of partial molar enthalpies by means of a high temperature microcalorimeter of the Tian-Calvet type[END_REF]. H O is represented in Fig. 5 from numerous available data sets. A convergence in the vicinity of the common intersection point  at x = 1.0713 (0.001) of the straight lines H O (x) from (10h) is globally noticeable.

Contrary to this contrived amalgamation process, the numerical adjustments presented next are related principally to the initial results from Raccah [START_REF] Raccah | Study of the thermodynamic properties of iron protoxide[END_REF] to [START_REF] Vallet | Contribution to the study of the thermodynamic properties of the solid iron protoxide[END_REF] which form a well populated and coherent data set at equilibrium, and for which it exists now a systematic interpretation by Vallet's group [START_REF] Vallet | Contribution to the study of the solid nonstoichiometric iron monoxyde. Diagram T-P(O2)-X[END_REF] [28] [START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF], and experimental confirmations [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF] [67] [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF].

The graphical representation of the relations (10h) and (11h), (10s) and (11s), in the case of the isotherms at 1000 and 820 °C, is given in Fig. 6. Along the isotherm at 1000 °C, the three Wi are represented by successive linear segments. For the two transitions Wi Wi+1 at decreasing iron content, a jump is observed toward the lower molar heat and entropy of solution which are exothermic and exo-entropic. For Above 1184 K, the exothermic H O for W1 increases with x and z corresponding to oxygen dissolution. For each O a vacancy and zero, one or two electronic holes are created simultaneously as the clustering is in progress (see equation (4a)). For W3, H O is decreasing with x, probably because of the formation of long-range ordering. For W2, H O varies in a manner intermediate between that for W1 and W3, which corresponds to the observation that W2 is a solid solution of W1 and W3 [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF]. Along the isotherm at 820 °C, the three W'i are represented by parabolic curves. Jumps (weaker in amplitude than the ones at 1000 °C) are observed toward higher values for the molar exothermic transitions W'jW'j+1, x 1'/2' = 1.071: (H O )' = +10.9 kJ, (S O )' = +6.3 J.K -1 and x 2'/3' = 1.1154: (H O )' = +2.9 kJ, (S O )' = +2.7 J.K -1 . A convergence of parabolas H O '(x) is observed for x = 1.0852 or y = 1-z = 0.9215 (0.004). This convergence occurs for a composition different (y =1.3 10 -2 ) from that of point C (y = 0.935), and is not defined as precisely for W' as it is for W.

The Gibbs-Duhem relation leads to the iron solution property H Fe from H O (Fig. 7). In the domain of W, exothermic transitional jumps are observed toward the higher values for the two transitions Wi  Wi+1 at equilibrium. At 1000 °C, the transition W1/W2 (x 1/2 =1.0782) is characterized by (H Fe ) 1/2 = 3.25 kJ.mol -1 , (S Fe ) 1/2 = 2.56 J. K -1 .mol -1 . The transition W2/W3 (x 2/3 =1.1164) is characterized by (H Fe ) 2/3 = 12.06 kJ.mol -1 , ( Fe ) 2/3 = 9.47 J.K -1 .mol -1 . In the domain of W', the equivalent jumps are exothermic as shown in Fig. 7 drawn for transitions at 820 °C.

As iron deficiency increases, the partial molar enthalpy of solution of iron H Fe decreases for W1 and increases for W3, variations that are opposite to that of H O . The five determinations of Asao et al. at 1000 °C are distributed on a curve noticeably parallel to the set of the successive segments concerning W1, W2 and W3 [START_REF] Asao | Thermodynamic properties of wüstite (FeO 1+y )[END_REF].

Equations x i/i+1 (T) of the lines -the so-called boundaries-separating the domain of stability of single pseudo-phases Wi and Wi+1 were adjusted following the analytical equations:

x i/i+1 = [(c i+1 -c i ) + T.(d i+1 -d i )] / [(a i -a i+1 ) + T.(b i -b i+1 )]
(see [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF] p. 3680, equations ( 6)). Numerically, they are adjusted now to

x 1/2 = (60412.0 -41.3182T) / (56322.3 -38.5509T) (12) x 2/3 = (25297.2 -18.310T) / (23670.0 -17.1941T) (13)
A metastable boundary can also be envisaged

x 1/3 = (85709.2 -59.6282T) / (79992.3 -55.7450T) (14)
The authors in [START_REF] Vallet | About new thermodynamic properties and a new phase diagram of solid wüstite[END_REF] remarked that the corresponding terms in the numerator and the denominator [of equations ( 12), ( 13) and ( 14)] are almost exactly proportional in a same ratio of 1.07, i.e., (c i+1 It is noticeable that a common « kinetic origin »  is characterized by x = 1.070  0.001 in W' (see Annex 2 « Re-analysis of kinetic data at 850 °C», derived from the experimental results [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF]).

-c i ) / (a i -a i+1 ) = (d i+1 -d i ) / (b i -b i+1 ) 1.

The triple point (W1, W2, W3) out of equilibrium

The point 8 (W1, W2, W3) was determined as the intersection of the extrapolated boundaries W1/W2 and W2/W3 at temperature 302 °C and composition y = 0.9322, directly below point  and point C (y C  0.935 0 ) (see [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF] Table III andFig Manenc et al. observed a disproportionation near 300 °C, in a miscibility gap, by inducing a slow drop in temperature [START_REF] Herai | Study of decomposition of the iron protoxide[END_REF][START_REF] Manenc | Structure of the iron protoxyde, recent results[END_REF]. Fender and Riley remarked that wII (W2) behaves [(G O vs 1-x)/T] in its specific region as a two-phase system above its critical temperature. Similarly to Vallet and Raccah [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF], they remarked that the extrapolation into the metastable domain below point C shows the phase separation into wI (W1) and wIII (W3) in the vicinity of 315 °C » [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF].

As early as 1977, Andersson and Sletnes studied conjointly the ordering and the preceding behavior below 300 °C, in reference with the spinodal decomposition using electron diffraction and microscopy (dark field images) [76] [77]. Typical patterns were attributed to the phase P' (periodicity of 2.6a o ), in the composition interval y (of Fe y O)  [0.95-0.92]. After heat treatment for 20 min at 300 °C, a tetragonal or orthorhombic symmetry was observed which can be compared to ordered P'-type regions. For compositions less than y = 0.91 in Fe 0.91 O (z ≥ 0.09), the more ordered structure identified as P" was observed even after quenching. Its parameters are a = b = c = 5a o .

These different observations indicate that W2 can be represented as a solid solution of W1 and W3, consistent with the determinations of [H Fe (x)] i e as determined below.

Trends to ordering at increasing z

The difference [H Fe (x)] i e between the partial molar enthalpy of iron in (FeO x ) i and FeO xo(i) on the boundary with Fe was envisaged as an excess type property in order to tentatively establish links between thermodynamic and defect structure properties [START_REF] Kléman | Thermodynamic properties of the solid iron protoxide. Experimental results applied to the drawing of the equilibrium diagram[END_REF]. The corresponding relations, using the previous formalism established for [

[ H O (x)] i e = [H O (x) -H O (x o )] i = 19.14475(a i /2(x -x o(i) )) (15h) [S O (x)] i e = [S O (x) -S O (x o )] i = -19.14475(b i /2(x -x o(i) )) (15s) [ H Fe (x)] i e = [H Fe (x)-H Fe (x o )] i = -19.14475(a i /4(x 2 -x o(i) 2 )) (16h) [ S Fe (x)] i e = [S Fe (x) -S Fe (x o )] i = 19.14475(b i /4(x 2 -x o(i) 2 )) (16s)
Substituting the coefficients a i , b i and x o(i) on the -Fe/W o(i) boundary by their numerical values yields Table 1. The sign of the term [H Fe (y)] i e as well as the sign of its derivative /x is the sign of -a i . This term corresponds to the energetic effect U on the iron lattice for the interaction between each additional O and a mole of FeO x into which it is dissolved (at constant iron content), a vacancy and zero, one or two electronic hole(s) being created (see supra relations 4a-4c).

] i e in J.K -1 .mol -1 .
The values of [H Fe ] i e in Table 1 show that:

1) for W1, the internal energy U m of the iron lattice (the term PV m is negligible compared to the contribution U m in H m ) decreases as z increases, which corresponds to the exothermic formation energy of clusters (m:n) that are thus favored (see Gokcen [START_REF] Gokcen | Thermodynamics[END_REF] p. 241-44, and [START_REF] Gokcen | Statistical thermodynamics of alloys[END_REF] p. 42-44, and Desré and Hodaj [START_REF] Desré | Thermodynamique des matériaux[END_REF] p. 91-96).

2) conversely, W3 is characterized by an increase in the lattice energy, which corresponds to the completion of the cluster percolation and the formation of the superstructure (2.55a x 2.55a x 2.55a) named P" by Manenc and characterized using HREM by Iijima [START_REF] Manenc | Structure of the iron protoxyde, recent results[END_REF] [78] [START_REF] Iijima | High resolution EM study of wüstite Proceed[END_REF].

3) likely because W2 is more than likely a solid solution of W1 and W3, the variation of U m is intermediate. Such trends were discussed previously by Fender and Riley when considering G O and H O in ( [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF] p. 797).

The positive and negative values of the term [S O ] i e (see Table I) correspond mainly to a vibrational entropy change in the oxygen sublattice [START_REF] Smyth | The defect chemistry of metal oxides[END_REF] [70] [START_REF] Lykasov | Thermodynamics of wustite[END_REF]. In the case of the initially disordered pseudo-phase W1, the excess entropy term weakly increasing with z shows that the disorder due to the formation of point defects is predominant. Reversely, in the case of the pseudo-phase W3, increasing short-and long-range orderings would be associated with increasing z values as indicated by a strong decrease of [  S Fe (x)] 3 e . These entropy variations are related to the increasing frequency of the Fe-O bond because of larger clusters and/or an increasing degree in their percolation until completion and superstructure. The pseudo-phase W2 can be thought to appear at the starting threshold of the percolation of clusters, when the lattice becomes more rigid.

Several approaches of investigation

Burgmann's review

In 1968, this author gathers some experimental results (electrical conductivity and transition p-to-n).

Because of numerous analogies with pyrrhotine Fe 1-z S [START_REF] Burgmann Jr | Contribution to the study of the iron-sulfur system, with limitation to the iron monosulfid (pyrrhotine)[END_REF], he describes tentatively the defect structure in terms of clusters (see [START_REF] Burgmann | The defect structure models for wüstite: A review[END_REF] Fig. 5 p. 173). He proposes a conduction model involving the formation of an acceptor band or of new states produced by the interaction of defects as z increases. The lattice contraction at increasing z reduces the gap between the valence and conduction bands, inducing a p-to-n transition for compositions z= 1/12 = 0.083 at 900 °C and z = 1/13 = 0.076 at 1300 °C ( [START_REF] Burgmann | The defect structure models for wüstite: A review[END_REF], Hillegas [82]). Such an overlap is also considered later by Molenda, Stoklosa and Znamirowski [START_REF] Molenda | The same authors. Electrical properties of manganese doped ferrous oxide at high temperatures[END_REF].

Manenc's phases P, P', P"

As the most important result of a seminal work, Manenc describes three types of « phases » named P, P' and P". He characterizes them after (rapid) quenching into water using X-ray diffraction. For an iron rich wüstite (z < 0.08), vacancies and ions Fe 3+ seem to be randomly distributed. This corresponds to the phase P which has a weakly incomplete NaCl-type structure alone. Other diffraction patterns of more iron deficient samples (0.08< z < 0.11) have additional reflections corresponding to a cubic cell with a repeating distance 2.6a o . Diffuse trails parallel to 100 directions connect these reflections. These features are characteristic of the phase P' alloyed with the phase P, and individually observed only in « oxygen rich » samples. Iron vacancies are ordered in the two phases [START_REF] Manenc | Structure of the iron protoxyde, recent results[END_REF][START_REF] Manenc | Existence of a superstructure in the iron protoxide[END_REF].

Diffraction for a heated single crystal such as z = 0.08 shows that a superstructure exists in the range 800-1000 °C for this composition. The phase P'' is observed in wüstite such as z  0.10 near 300 °C and when slowly cooled, at the threshold of disproportionation yielding Fe and Fe 3 O 4 . New superstructure lines are then characteristic of a commensurate 5X cubic cell [START_REF] Manenc | Structure of the iron protoxyde, recent results[END_REF][START_REF] Manenc | Existence of a superstructure in the iron protoxide[END_REF].

Mössbauer spectroscopy

Greenwood and Howe [START_REF] Greenwood | Mössbauer studies of Fe1-xO. I.-The defect structure of quenched samples[END_REF] interpret Mössbauer spectra in modeling the asymmetrical doublet as evidencing the differing defect structures of the pseudo-phases identified by Vallet and Raccah [START_REF] Vallet | Contribution to the study of the thermodynamic properties of the solid iron protoxide[END_REF], and Fender and Riley [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF] at high temperature. The quenching process produces nuclei leading to Manenc phases P, P', [(P+P') for z [0.05-0.08], (P') for z [0.08-0.10], and (P'') for z > 0.10] below point C. The phases are detectable from fine variations of quadrupole splitting resulting from cubic symmetry distortions. These distortions can be evaluated for the single cluster (4:1), and the Koch and Cohen cluster (13:4). From a sample such as z  0.05, the disproportionation gives rise to Fe 0.918 O (defect rich phase P') and Fe 0.976 O (defect poor phase P), because of a mixture of the (13:4) and (4:1) clusters.

Among numerous other Mössbauer studies reviewed by Long and Grandjean [START_REF] Long | Mössbauer effect, magnetic and structural studies of wüstite[END_REF], Romanov, Checherskaya and Tatsienko [START_REF] Romanov | Mössbauer spectra of non-stoichiometric and stoichiometric wustites[END_REF] [86], Hrynkiewicz et al. [START_REF] Hrynkiewicz | Mössbauer effect studies of ferrous oxides Fe 1-x O[END_REF], Pattek-Janczyk et al. [START_REF] Pattek-Janczyk | Double electron exchange in Fe 1-x O: a Mössbauer study[END_REF], this latter is of particular interest. The authors modeled the asymmetric doublet with a singlet attributed to Fe 3+ in tetrahedral sites, and two doublets attributed to octahedral Fe 2+ and Fe 3+ . They associated the maximum observed in the variation of the quadrupole splitting to the p-n transition described by [START_REF] Tannhauser | Conductivity in iron oxides[END_REF]. This transition is likely the same as the transition « metal  Mott insulator » identified by Molenda, tokłosa and Znamirowski [START_REF] Molenda | The same authors. Electrical properties of manganese doped ferrous oxide at high temperatures[END_REF].

Aldon and Jumas use the Mössbauer spectroscopy to evaluate precisely the reversible insertion of lithium and the effects on the local and long-range orderings of the wüstite lattice. They characterize the lithium as « an actor of electronic pressure » (see infra [START_REF] Aldon | Lithium-induced conversion reaction in wüstite Fe1-xO studied by 57 Fe Mössbauer[END_REF] in section 4.3. Nanocomposites. Epsilon-Fe).

First electronic-structural modeling

Goodenough's modeling [START_REF] Goodenough | Improved then translated into French by Casalot A: Les oxydes des métaux de transition[END_REF] is proposed as early as 1971. Considering the relations (2) associated with thermogravimetric analyses, and relation   p' 1/s (17) corresponding to the conductivity [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF], the result s=6 in the W1 subdomain above 1060 °C characterizes a statistical distribution of vacancies and electronic holes (Fe 3+ ), possibly corresponding to Manenc's phase P. Below 1060 °C, the changes in values of s correspond to an increasing association between vacancies and holes. In the W2 domain, triplets Fe 3+ -V Fe -Fe 3+ are partially ordered, improving Madelung energy. Numerous (4:1) clusters which form a superstructure according to Koch & Cohen (see infra section 7.2. The historical cluster (13:4) and [START_REF] Koch | The Defect Structure of Fe1-xO[END_REF]) correspond to W3. A mapping of the transfers of iron ions, and electronic exchanges is proposed as a mechanism, needing small polarons related to energy differences between the top of the valence band and an acceptor level [START_REF] Goodenough | Improved then translated into French by Casalot A: Les oxydes des métaux de transition[END_REF]. Later, this author with Gleitzer will consider the wüstite again from the point of view of its electronic properties and crystallographic structure. They will express some doubt concerning the role of the cluster (4:1) as only building unit of larger clusters (see [START_REF] Gleitzer | Mixed-valence iron oxides in structure and bonding[END_REF] p. 51-4).

Zvintchuk's team

In 1973, Tchiong Tki Khong in Zvintchuk's team, in collaboration with J. Manenc, studies samples of wüstite of composition z [0.065-0.108] quenched from 1100 °C [START_REF] Tki | About the structure of Fe1-xO][END_REF]. Electron diffraction reveals three forms I, II and III corresponding to the modifications wI, wII and wIII of the phase diagram by Fender and Riley [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF], and to the phases P, P', P''. A correspondence with the three Wi is specified. The authors describe a defect complex (16:6) of point symmetry m3m, formed of six tetrahedral corner sharing clusters (4:1) distributed on the six faces of an empty cube. This cluster is represented on Fig. 8.

Worral and Coley. Isotopic kinetic transitions

More than 40 years after Raccah, Vallet et al. [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF][START_REF] Raccah | On thermodynamic properties of wüstite, a strictly regular solution of iron and oxygen[END_REF][START_REF] Raccah | New set of isotherms for the solid wüstite[END_REF][START_REF] Vallet | About new thermodynamic properties and a new phase diagram of solid wüstite[END_REF][START_REF] Vallet | Contribution to the study of the thermodynamic properties of the solid iron protoxide[END_REF][START_REF] Carel | Dilatometric study of diverse varieties of solid wüstite, and existence of a metastable point between the three varieties[END_REF][START_REF] Carel | Variation of the cell parameter for three wüstite varieties[END_REF], Wagner jr et al. [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF] [66], and Fender and Riley [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF], Worral and Coley evidence kinetic transitions related evidently to the pseudo-phases [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF]. Following the process first used by Temkin et al. at the surface of iron oxides [START_REF] Nv | The Isotopic Exchange between CO and CO2 on various surfaces[END_REF][START_REF] Temkin | Kinetics and Mechanism of Isotopic exchange and Gas Reaction on Solid Surfaces[END_REF], and then applied by Grabke [START_REF] Grabke | Kinetics of oxygen exchange from CO2 to the surface of oxides[END_REF], Cramb and Belton [START_REF] Cramb | Studies of the interfacial kinetics of the reaction of CO2 with liquid iron by the 14 CO2-CO isotope exchange reaction[END_REF], Mori, Morita and Sano [START_REF] Mori | Determination of the rate of CO 2 dissociation on the surface of CaO-SiO 2 , CaO-Al 2 O 3 , CaO-SiO 2 -CaF 2 and CaO-SiO 2 -Fe x O Melts[END_REF], later by Zhang et al. [START_REF] Hu | Interfacial rate of CO2-CO reaction with the solid iron and iron oxide by isotope exchange technique at 1 273 K[END_REF] [98], Worral and Coley study the kinetics of the carbon exchange in the equilibrium W/(CO2/CO mixtures) in which a controlled proportion of 13 CO2 species was introduced. In such an environment, a specific equilibrium of the carbon isotopes is established following: 13 CO 2 + CO  13 CO + CO 2 which can be viewed as the sum of the two half-reactions: 13 as reflecting oxygen exchange on the surface of the sample and being conditioned by the electronic structure in the bulk. The apparent rate constant is expressed as a function of oxygen activity a O as follows:

k a = k 0 .a O -m or log k a = -m.log a O + logk 0 ( 18
)
The variation of log k a with the activity a O defined as being the ratio p(CO 2 )/p(CO), can be described by three successive linear segments at 995 °C, with m = 0.51 (w1), 0.66 (w2), 1.03 (w3) in ([73] Fig. 8 and equations [START_REF] Tarte | IR spectrum and presence of trivalent tetrahedral iron in the structure of iron protoxide Fe1-xO (Wüstite)[END_REF] [37] [START_REF] Wagner | Handbuch der Metallphysik Becker and Erler Kom[END_REF] p. 820-1).

The authors observe only one transition attributed to two pseudo phases w'1-2 and w'3 along the isotherm at 850 °C, no change being observed in the vicinity of the forecasted boundary w'1/w'2 by Vallet and Carel [START_REF] Vallet | Contribution to the study of the solid nonstoichiometric iron monoxyde. Diagram T-P(O2)-X[END_REF]. The only observed transition appears as a discontinuity at the location of the transition W'2  W'3 forecasted for x = 1.090 at equilibrium. Nevertheless, the authors state that three pseudo-phases w'1, w'2 and w'3 probably exist, but that two of them could not be separated, either because they were not detectable by electrical conductivity measurements as found before in [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF], or because they were possibly « obscured by the scatter ». Finally, Worral and Coley declared that, in their study, transitions at 850 °C and 995 °C were observed « correspond[ing] exactly with the proposed pseudo-phase boundaries in quite good agreement with the phase diagram by Vallet and Carel». Because of the identity of coefficients m in the correlations concerning Wi and W'j for i=j, they attributed in a following review [START_REF] Worral | Defect structure of pseudo-phases of wüstite[END_REF] three predominant clusters (7:2) type II to w1 and w'1, (12:4) type II to w2 and w'2 and (16:5) type I to w3 and w'3.

Note: In Annex B and Fig. B2, a graphical and numerical re-analysis of Fig. 9 by ([73] p. 821) shows a possible separation of the data in three sets at 850 °C, which correspond likely to the three W'j, provided that some data or groups of data are located in subdomains not proposed as stable. In Annex A, Fig. A1, the location of the observed transitions is represented by the set f-).

Takayama and Kimizuka's phase modifications

Using thermogravimetric analyses under equilibrium oxygen pressure p' defined by CO2/H2 mixtures, Takayama and Kimizuka observe significant modifications in curves representing l' vs log x in FeO x [START_REF] Takayama | Thermodynamic properties and subphases of wustite field determined by means of thermogravimetric method in the temperature range of 1100-1300°C[END_REF]. The authors conclude to the existence of a transition between pseudo-phases (see [START_REF] Takayama | Thermodynamic properties and subphases of wustite field determined by means of thermogravimetric method in the temperature range of 1100-1300°C[END_REF] Table I and Annex A Fig. A1 d-). Typical changes in slopes are clearly observed from their data.

The new representations for l' vs log z are done in Fig. 9. Two coefficients s can be defined as a first step, using the hypothesis of a transition implying two pseudo-phases proposed by the authors: s close to 4 and s close to 7.5. However, three correlations can be also defined as follows: The results of Takayama and Kimizuka [START_REF] Takayama | Thermodynamic properties and subphases of wustite field determined by means of thermogravimetric method in the temperature range of 1100-1300°C[END_REF] and those of Bransky and Hed [START_REF] Bransky | Thermogravimetric determination of the composition-oxygen partial pressure diagram of wustite (Fe1-yO)[END_REF] show a relative agreement in the same domain of temperature (see also Annex A Fig. A1 b-), but disagree with results of Fender, and those of Vallet because these authors did not observe any breaking point on the isothermal lines above 911 °C (see infra section 10. Percolation approach, Nota: Metastabilities).

Electrical properties. Transitions. Percolation

In an extensive review about electrical properties, Gleitzer points out that the modeling of the electrical conductivity of wüstite is relevant of the percolation concept [START_REF] Gleitzer | Electrical properties of anhydrous iron oxides[END_REF]. The superimposition of the two kinds of p-n and Wi  Wi+1 transitions, almost independent of each other, and both function of z, is already referenced in the Burgmann's review [START_REF] Burgmann | The defect structure models for wüstite: A review[END_REF].

Because of this common stress, it is not easy to correlate the p-n transition to the structure. Since the percolation process is characterized experimentally by a steep variation of the studied property at a threshold composition, the curves for the electrical conductivity vs composition can be interpreted in terms of upgradable clusters and long-range ordering in the subdomains of W1, W2 and W3 [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF] [100] [START_REF] Kozheurov | Electrical conductivity of wüstite[END_REF].

Later in the present review (see Section 10 Percolation approach, Fig. 20), a statistical distribution of clusters with two types of percolation modes is schematically represented through the phase diagram. This cluster distribution can justify structurally the singularities of the integral and partial thermodynamic properties (see above Figs. [START_REF] Goodenough | Improved then translated into French by Casalot A: Les oxydes des métaux de transition[END_REF][START_REF] Per | Nonstoichiometry, diffusion, and electrical conductivity in Binary metal oxides[END_REF][START_REF] Gokcen | Thermodynamics[END_REF][START_REF] Men | Compositiondefect-properties of solid phases. Method of cluster component[END_REF][START_REF] Spencer | A thermodynamic assessment of the iron-oxygen system[END_REF]. Also, it can justify the distinct rate constants observed in kinetic processes (see section 3.6. Kinetics transitions, and Annex B: Figs. B-1 and B-2).

The p to n transition

Tannhauser observe a thermo-electronic p to n transition around 1300 °C by thermal e.m.f. analyses of a highly nonstoichiometric wüstite [START_REF] Tannhauser | Conductivity in iron oxides[END_REF]. More precisely with Bransky [START_REF] Bransky | High-Temperature defect structure of ferrous oxide[END_REF], a change in sign of the Seebeck coefficient is observed for polycrystalline samples near O/Fe 1.09 or z  0.083 at T  [1010-1310 °C] and near z  0.066 at 910 °C (Fig. 10). The authors formalize the Seebeck coefficient , in the case of completely ionized vacancies, when the hopping process is predominant:

 = (k/e).(A/kT + ln (c o -c)/c) ( 22
)
where c = 'concentration' of charge carriers, c o = 'concentration' of available sites for the charge carriers (see also [START_REF] Heikes | A study of the transport properties of semiconductors with mixed valence[END_REF]).

The same formalism is used by Lafollet and Duquesnoy in the case of non-localized when the wüstite is considered as an intrinsic semiconductor, i.e., doped by its own impurities [START_REF] Lafollet | On the variations of the thermoelectric power coefficient of iron protoxide as a function of the partial pressure of oxygen at high temperature[END_REF].

They modeled  = 0 for z = 0.104 at 1100 °C, and z = 0.108 at 1000 °C (see [START_REF] Lafollet | On the variations of the thermoelectric power coefficient of iron protoxide as a function of the partial pressure of oxygen at high temperature[END_REF] in Fig. 10). Like in previous studies, Hodge and Bowen state that the experimental sign change of the thermal electromotive force, from positive to negative, occurs at increasing values of z [START_REF] Hodge | Measurement of low-temperature thermoelectric power for quenched wustite[END_REF]. Their model considers principally clusters (4:1) and electron holes trapped in octahedral sites adjacent to cluster vacancies. The thermally activated hopping of these holes permits continuous paths through the crystal because of the high point defect concentration between the zones separating the clusters.

In their studies on electrical conductivity, Ariya and Bratch, and Kozheurov and Mikhailov observe modifications to the relationship   (p') 1/n , closely related to the relationship z  (p') 1/n , where their coefficient n ( s) varied in such a way that three subdomains can be defined [START_REF] Ariya | Electrical conductivity of ferrous oxides at high temperatures[END_REF] [START_REF] Kozheurov | Electrical conductivity of wüstite[END_REF]. They find an increase of the conductivity in two composition ranges, from z  0.048 to 0.079 then from 0.079 to 0.097, in relation with the Fe/W and W/Fe3O4 boundaries.

Electrical conductivity

Neuschütz and Towidi establish a variation of the conductivity for T[700-1220 °C] from which they inferred that the defect electrons jumped between Fe 2+ and Fe 3+ , the activation energy of the process being 9.6 kJ.mol -1 [START_REF] Neuschütz | Electrical conductivity of wüstite[END_REF].

Following Molenda, Stokłosa and Znamirowski, the conductivity can be the sum of an « extrinsic conductivity σ y » related to the nonstoichiometry and of an « intrinsic component σ o » independent of the defect concentration, i.e., of p' [START_REF] Molenda | The same authors. Electrical properties of manganese doped ferrous oxide at high temperatures[END_REF]. Charge transport at high temperature would occur simultaneously in the valence and conduction bands, and the dopant acceptor band due to Fe 3+ in octahedral and tetrahedral sites. Near the composition z ≈ 0.08, the acceptor band would overlap the valence band giving rise to a broad metallic band close to the thermoelectric transition p to n for z ≈ 0.09 and corresponding to the stability limits of W1 and W2-3. Experimental details are given in ref. [START_REF] Molenda | The same authors. Electrical properties of manganese doped ferrous oxide at high temperatures[END_REF] (Fig. 2 p. 519 and Fig. 4, see also Annex A, Fig. A1 e-), where successive segments and breaks corresponding to the Raccah-Vallet's isotherms have been observed.

More recently, Toroker and Carter examined the possible improving of the conductivity of Fe 1-z O by doping the p-or n-conduction mode, with the aim of stabilizing the nanoscale fabrication of wüstite, and then adapting the nanoparticles to create efficient solar light conversion materials [START_REF] Toroker | Hole transport in nonstoichiometric and doped wüstite[END_REF].

They listed the procedures allowing the calculation of Marcus theory parameters for cluster models with hydrogen dopant and substitutional Cu + , Li + and Na + . They concluded that « iron vacancies drastically limit the hole conductivity » while Li, H and Na dopants improve it. So, it would be interesting to suppress the formation of vacancies by « alloying FeO », which would amplify the p-type conductivity.

Note: An extensive thermodynamic study of multiple substituted wüstites can be found in the work of Lykasov et al. which is reviewed partly in [START_REF] Lykasov | Physics and chemistry properties of wüstite and its solutions[END_REF] and [START_REF] Katkov | Wüstite solid solutions in Fe-Cu-O system[END_REF]. Structural aspects of calcio-and magnesio-wüstites are available infra (see Table II, and in Fig. 15)).

Nanocomposites. Epsilon-Fe

Aldon and Jumas examine the electrochemical redox phenomena at the negative electrode Fe/Fe 1-z0 O formed by wüstite of limit-composition z 0 (= 0.050 0.013) in Li-ion batteries [START_REF] Aldon | Lithium-induced conversion reaction in wüstite Fe1-xO studied by 57 Fe Mössbauer[END_REF]. The initial samples of hematite powder, reduced in H 2 at 800 °C, are quenched outside the furnace. The analyses by Mössbauer spectroscopy and X-ray diffraction, after the first discharge, yield the nature and concentration of the products of reduction of wüstite by Li. Nanoparticles of iron (2nm) embedded in a Li 2 O matrix are identified. A part of the iron atoms appearing in the disproportionation is made of the -Fe phase which is stressed in the matrix under a pressure evaluated at 6.7 GPa, and which is not ferromagnetic. It can be due to the extrusion from the W structure. At the same time, small clusters of lacunary hematite are formed. These phenomena have been still identified in the formation of « planetary materials, in the Earth's inner core ». See also geological observations by Hazen and Jeanloz [START_REF] Hazen | Wüstite (Fe1-xO): a review of its defect structure and physical properties[END_REF].

Other kinetic and diffusional works

Iron self-diffusion

In 1963, Desmarescaux and Lacombe study self-diffusion in using isotopes 55 Fe and 59 Fe [START_REF] Desmarescaux | Iron self-diffusion in the iron protoxide[END_REF]. They state experimentally (Fig. 11) that the self-diffusion coefficient D * Fe increases with z at a given temperature T. At increasing T, because the clusters are dissociated, the rate of free vacancies is increased, the selfdiffusion coefficient too. They deduce that self-diffusion proceeded through a vacancy driven mechanism, especially noticeable at high temperature. At lower temperatures, fewer free vacancies and more complex vacancy-electron holes exist. Diffusion is then slower. The number of vacancies associated to Fe 3+ cations [the clusters] increases with z, so a compensation with the free vacancies implies a quasi-constant D * Fe whatever z below 850 °C. These observations are interpreted in the Collongues's review [START_REF] Collongues | Nonstoichiometry in Oxides[END_REF] with the help of theoretical evaluations of the binding energies in clusters (see Sections 6.5. Cluster stability, and 6.6.

Diffusion and stability).

Isothermal kinetics

Landler and Komarek measure the loss of mass vs time M(t) in H2/H2O mixtures when reducing polycrystalline samples by thermogravimetry [START_REF] Landler | Reduction of wüstite within the wüstite phase in H2-H2O mixtures[END_REF]. In their calculations, the coefficients of iron self-and chemical diffusion decrease for increasing z at constant T[800 -1050 °C] because of « the increased freezing due to repulsive interaction of the vacancies ». That corresponds to activation energy independent of T. This observation is related primarily to the formation of clusters, then to the ordering of clusters in the bulk (see Section 2.7. Trends to ordering).

Considering the relationship log(M/M o ) = n.log t + k, in which M o is the total mass loss, reduction kinetics data from the authors at 950, 1000 and 1050 °C can be separated into three regimes II, III, and IV (see Annex B, Fig. B1). At 1000 °C, the calculated mean values of n are -0.72 (W3?), -0.49 (W2?) and -0.3 (W1?), respectively [START_REF] Carel | Reduction kinetics of solid wüstite in its stability domain[END_REF]. A brief anharmonic sequence is observed at the transitions between two reduction processes. The increase of 0.6 eV in the activation energies E III = 1.31 eV and E II = 1.89 eV, for the reduction sequences III then II, is observed a decreasing z, i.e., when the vacancy concentration decreases. It corresponds to the decrease of the ordering on the boundary W3W2. This variation is opposite to other results in the literature, particularly to ([112] Fig 5, see section 5.1. Self-diffusion). The three coefficients n are primarily the signature of the bulk modifications by Landler and Komarek, while the three kinetic coefficients m by Worral and Coley are primarily characteristic of surface phenomena [START_REF] Landler | Reduction of wüstite within the wüstite phase in H2-H2O mixtures[END_REF] [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF] (see [START_REF] Nowotny | Influence of the surface on the equilibration kinetics of non-stoichiometric oxides[END_REF]: Bulk and surface process in the course of the reduction).

Similarly, successive different reduction sequences using H2O/H2 gas mixtures that exhibit transitional anomalies can be observed by Rieke and Bohnenkamp [START_REF] Riecke | About the oxidation and reduction kinetics of wüstite in its existence domain[END_REF] (see Annex A, Fig. A1 c-).

In situ diffusion

Rickert and Weppner combine a solid galvanic cell for the ionic conduction by O 2-through the sample while with the other sample face in contact with a platinum foil [START_REF] Rickert | Electrochemical investigation of the chemical diffusion in wüstite by means of an oxidizing solid electrolyte[END_REF]. The diffusion is studied in situ during the relaxation from a composition to another, driven by controlled impulses of electrical potential. Their main result is that the chemical diffusion coefficient increases with increasing z (i.e., with the vacancy concentration), which disagrees with Landler and Komarek [START_REF] Landler | Reduction of wüstite within the wüstite phase in H2-H2O mixtures[END_REF] and not with any other authors. Transitional anomalies are clearly noticed (see [START_REF] Rickert | Electrochemical investigation of the chemical diffusion in wüstite by means of an oxidizing solid electrolyte[END_REF]

Relationships between defect structures and the phase diagram

We review key publications from authors who have attempted to solve the key challenge formulated by Carl Wagner in 1930 of linking more tightly defect structure and thermodynamics, such as partial molar properties or the equilibrium constant of formation of the clusters. Kinetics (diffusional) relations were also developed.

Per Kofstad and Zeev Hed

Theses authors remark that z increases at constant p' as T decreases, which is different than other oxides, and likely due to a complex defect structure [START_REF] Per | Nonstoichiometry, diffusion, and electrical conductivity in Binary metal oxides[END_REF] [118] (see section 5.1. Iron self-diffusion by authors [START_REF] Desmarescaux | Iron self-diffusion in the iron protoxide[END_REF]). They emphasize that there is some doubt that all the iron vacancies are doubly charged. The electron holes associated to a vacancy can jump into only one of the 12 next nearest octahedral neighbors (Fe Fe °) and one tetrahedral site (Fe i °°°)

. Most of octahedral vacancies are singly charged.

A model can thus be elaborated from the Roth complex [V Fe m( ' ) -Fe i n(°) -V Fe p(

' ) ] q( ' ) , n=1,3, (m,p) = 0,1,2. In the relationship   p' 1/s , the s value is larger than the value in the similar relationship connecting z to p' because vacancies would be neutral.

Toft Sørensen's works

In a book devoted to nonstoichiometric oxides, Toft Sørensen then with El Sayed Ali [9] [START_REF] Toft Sørensen | Defects on metal-deficient oxides: wüstite, Fe 1-y O[END_REF] tentatively identify the defect structure of the « subphases » in using thermogravimetric data [l'-z] at 1000 -1300 °C exclusively by Bransky and Hed [START_REF] Bransky | Thermogravimetric determination of the composition-oxygen partial pressure diagram of wustite (Fe1-yO)[END_REF]. They consider three main defects (doubly charged vacancy V Fe ", and tetrahedral complex defects (4:1) (m)-and (6:2)) as defined by Catlow et al. (see [START_REF] Cheetham | Thermodynamics and Defect Structure of Nonstoichiometric Oxides; section IX: Recommendations for Future Work[END_REF] chapter 2). They focus on the defect cluster differently ionized, which leads them to define a linear plot of l' vs log y (≡z) with minimal slopes 1/s in the relation z  (p') 1/s . A second criterion concerns the so-called exclusion envelope, i.e., the long-range order as a layer or a close packing structure. They localize in the wüstite equilibrium area T(z) the predominant defect (4:1) plus a layer structure, (4:1) plus a close packing, (6:2) plus a layer structure, according to T layers above 1300 °C, between 1300 and 1200 °C and below 1200°C, respectively. In addition, iron vacancies V Fe " are present on the right side of a line at composition z close to 0.09 (see [START_REF] Toft Sørensen | Defects on metal-deficient oxides: wüstite, Fe 1-y O[END_REF], Fig. 7 p. 20). So, a structural characterization is roughly proposed, and the phase diagram of Fender and Riley [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF] and Vallet and Raccah [START_REF] Vallet | Contribution to the study of the thermodynamic properties of the solid iron protoxide[END_REF] is approximately drawn again. In the course of their theoretical work about oxides properties, Men' et al. were concerned with the wüstite phase [START_REF] Men | Compositiondefect-properties of solid phases. Method of cluster component[END_REF] [START_REF] Men | Thermodynamic study of nonstoichiometric solid solutions with the MO type[END_REF]. Men' and Carel apply C.C.M. to model the wüstite solid solution [START_REF] Men | Investigation of wüstite in its homogeneity domain by means of the Cluster Component Method[END_REF] [121] [START_REF] Men | The cluster component method and ordering in nonstoichiometric monoxides with the rocksalt structure[END_REF]. The « cluster components » are defined from crystallographic positions forming unit blocks in the general spinel structure. For example, three cluster components ,  and , and their concentration as a function of z allow to build the lattice of the wüstite Fe 1-z O, and to model some of its physical properties, taking into account the pseudo-phases (Fig. 12).

The C(luster) C(omponent) M(ethod)

Other modeling approaches.

Mrowec et al. work on defect clustering in the wüstite phase using a collection of data (z, log p') from the literature [START_REF] Rekas | On defect clustering in the wustite phase[END_REF], [START_REF] Mrowec | Defect structure and transport properties of non-stoichiometric ferrous oxide[END_REF]. They envisage all the defect clusters suggested by Catlow and Fender [START_REF] Catlow | Calculations of the defect clustering in Fe1-xO[END_REF], i.e., (4:1), (6:2), (8:3), (10:4), (12:4) and (16:5) with several ionization degrees including the highest. Elaborate calculations of the equilibrium constants lead them to the assessment of the enthalpy and entropy of formation of all these ionized clusters, then to H O independently of z and T, and to S O for z  [0.055-0.145]. They conclude that the (4:1) cluster is the most likely to occur and that the ionization degree increases with temperature to reach the maximum value 5-at 1573 K. [START_REF] Nowotny | Defect structure and thermodynamic properties of the wustite phase (Fe1-yO)[END_REF] describe the defect structure and thermodynamic properties. They investigate the clustering of point defects (iron vacancies and electron holes) across the entire domain using the Debye-Hückel theory, which allows to use a product .[x i ] in place of [x i ], particularly for i=h °. With this approach, the activity coefficients are merged with the sole mean coefficient f  determined as a function of the ionic force of the solution (and not of its square root). They modeled the interactions during the formation of the cluster (4:1) taking into account activity instead of concentration. Sets of data (z, log p') and (H O , z) were gathered from the literature. They evaluated the enthalpy and entropy of formation of a cluster (4:1) 5-at -396.7 kJ.mol -1 and +267.4 J.K -1 .mol -1 , respectively. They concluded that this cluster (4:1) n-would be the constitutive module of more extended clusters. An expression of the equilibrium constant for the cluster formation is determined across the whole domain independently of the existence of pseudo-phase transitions.

Nowotny and Rekas

Cluster stability

Catlow and Fender try first to access the defect structure by means of large-scale computation of the binding energies following the Born model [START_REF] Catlow | Calculations of the defect clustering in Fe1-xO[END_REF]. They envisage in a theoretical manner the clusters unity (4:1), edge-sharing (6:2) and (8:3), corner sharing (16:5) precursor of Fe3O4. The location (zone [z, T] of the phase diagram) of these clusters is suggested qualitatively, but not in relation with the pseudo phases.

Later, Catlow et al. assessed the formation energy and the relative stability of the clusters by means of the Mott-Littleton method [START_REF] Catlow | The basic atomic processes of corrosion. II: Defect structures and cation transport in transition-metal oxides[END_REF]. The cluster (12:4) <110> type by Lebreton and Hobbs is considered as being the most stable extended cluster, after clusters (6:2) and (8:3) [START_REF] Lebreton | Defect structure of Fe 1-x O[END_REF]. In a summary-paper [START_REF] Catlow | Recent progress in defect studies of rock-salt transition metal oxides and magnetite[END_REF], four classes of clusters are envisaged: corner-sharing clusters (4:1), (7:2), and (16:5); edge-sharing clusters (6:2), (8:3) and (12:4); ZnS blende like cluster (10:4) [START_REF] Catlow | Recent progress in defect studies of rock-salt transition metal oxides and magnetite[END_REF] [128] [START_REF] Grimes | Defect clusters in nonstoichiometric 3d transition-metal monoxides[END_REF]; clusters (13:4) and (5:2) [130] [START_REF] Gartstein | Defect agglomeration in wüstite at high temperatures -I: The defect arrangement[END_REF]. The authors note that the (10:4) blende ZnS cluster envisaged by Grimes et al. [129] [132] is not the most likely because it was characterized without taking into account the geometric relaxation [START_REF] Catlow | Recent progress in defect studies of rock-salt transition metal oxides and magnetite[END_REF].

Tomlinson and Catlow attempt to relate the structural modeling of short-range order, and the macroscopic variation of z as a function of p'. The calculated binding energy per vacancy leads them to order the clusters as follows: [(8:3) +1 , (12:4), (6:2), (16:5) -1 , (12:5) -1 , (13:4) -2 , (4:1) -1 , (10:4) -4 , (5:2 M), (5:2 G)] from 2.24 to 0.71 eV, successively. Finally, they propose a mixture of clusters [(12:4), (6:2), (4:1)] for the incommensurate sub-phase P' [START_REF] Tomlinson | Computer simulation studies of Fe1-xO and Mn1-xO in Non-Stoichiometric Compounds Surfaces[END_REF].

Then, Grimes et al. consider the structure and stability of clusters with different coordination types [START_REF] Anderson | Defect clusters in wustite, Fe1-xO[END_REF] [START_REF] Minervini | Defect clustering in wüstite[END_REF]. They base their analysis on a generalized crystal field for the 3d electrons of Fe 3+ and Fe 2+ , including the energy of the orbitals 2s and 2p of oxygen. Their computation of the binding energy per vacancy favors the clusters <110> ZnS blende and <111> spinel type stacking. It rules out clusters (8:3) and (10:4), based on <100> stacking of (4:1) units, and the Koch and Cohen (13:4) cluster. Within the lower field of stoichiometry (P' phase), smaller clusters coexist including (4:1), (6:2), (7:2) <110>, (10:4) ZnS blende type, and (7:2) <111> spinel type. Larger clusters from the (12:4) <110> type, (18:6), (16:5) spinel <111> type, (16:5) <110> type and (16:7) <110> ZnS blende type clusters can be invoked at higher nonstoichiometry (P'' phase). Finally, they assert that the charge of the clusters should be zero in order to agree with experimental data.

Diffusion and stability

In a review about nonstoichiometric oxides, Collongues [START_REF] Collongues | Nonstoichiometry in Oxides[END_REF] envisages particularly the clustering as specified by Desmarescaux and Lacombe (see section 5.1. Self-diffusion of iron). The activation energy of the diffusion increases with increasing z, while the opposite effect is the principal rule. « Vacancy clusters » are the reason of this unusual behavior. Catlow et al.'s calculations of (Fe 3+ -vacancy) binding energy indicate that clusters, such as (6:2) or (8:3) of size lower than that of (16:5) (precursor of magnetite), are more stable than this latter cluster. This feature would indicate that even a cluster structurally far from magnetite is responsible of the reduced diffusion process. The existence of three domains in the phase diagram is envisaged.

Note: The classification of the clusters become more complicated because of the increasing number of proposals in the literature. Initially, only the criterion of corner or edge sharing of the basic (4:1) units was needed. Mixed cases were thought to be the exception. Lebreton and Hobbs [START_REF] Lebreton | Defect structure of Fe 1-x O[END_REF] introduced three way of clustering: edge sharing along <100> (Type I), corner sharing along <110> (Type II), and corner sharing along <111> (Type III). Mixed ways are also to be envisaged. Clusters based on composition alone are possible exceptions (e.g., cluster (10:3)). Labidi and Monty (see [START_REF] Labidi | P' and P" phase structures in Fe1-xO and Fe1-x-yCayO[END_REF] p. 100-101) resumed the situation differently with the types 1 (face shared) aligned in <100> direction, 2 (edge shared) aligned in <110> direction, and 3 (corner shared) aligned in <111> direction. Worral and Coley (see [START_REF] Worral | Defect structure of pseudo-phases of wüstite[END_REF] p. 24-26) classified the clusters in three types defined as either type 1 (corner shared as in magnetite), type 2 (edge shared sharing one octahedral vacancy), or type 3 (edge shared sharing two octahedral vacancies).

Structural approaches. Models for short-and long-range ordering

Roth model: the cluster (2:1)

Considering the experimental uncertainties inherent to X-ray diffraction on polycrystalline oxides, it is more suitable to use neutron diffraction experiments to determine correctly the site occupancies relative to iron atoms in octahedral and tetrahedral sites of the NaCl-like lattice. Initial results relative to quenched wüstite are obtained using neutron diffraction by W. L. Roth who determines the octahedral and tetrahedral site occupancies of vacancies and Fe 3+ cations, respectively represented by (z+t) and t [START_REF] Roth | Defects in the crystal and magnetic structures of ferrous oxide[END_REF]. The characterization of the ratio R = (z+t)/t was found close to R=2 for samples quenched from high temperatures. The neutron diffraction patterns obtained at 290 K and 4.2 K suggested the presence of defects constituted of two cation vacancies associated with one interstitial cation in the tetrahedral site. These two characteristics suggested a trend to the magnetite structure. In addition, the average magnetic moment per cation site was found to be much smaller than expected.

The historical Koch and Cohen cluster (13:4)

In 1969, Koch and Cohen publish a study by X-ray diffraction on a single crystal of wüstite Fe 0.902 O (z=0.098) obtained after quenching from 1000 °C, and « corresponding to the P' phase of Manenc » [START_REF] Koch | The Defect Structure of Fe1-xO[END_REF].

The authors proposed a new commensurate defect structure based on (13:4) large clusters (Fig. 13) distributed in the fcc lattice (cell parameter a) with a repetition distance (3a, 3a, 3a) noted as 3X. For the first time, a detailed crystallographic determination proposed a cluster corresponding to the corner sharing <110> type agglomeration of four basic (4:1) clusters. To perform their structural refinements, the authors fixed this cluster model assuming that the resulting superstructure should have a periodicity of 3X (the clusters being arranged only with regular distances of three cell parameters), in agreement with the known composition z=0.098, despite the fact that the additional peaks correspond to a superstructure 2.6X incompatible with their assumption.

Neutron diffraction

Using in situ neutron diffraction experiments, Cheetham, Fender and Taylor [START_REF] Cheetham | High temperature neutron diffraction study of Fe1-xO[END_REF] show that the ratio R = (z+t)/t ranges between 3 and 4, with a mean value of 3.2, which can be compatible with a specific (13:4) cluster (R = 3.25).

Using neutron diffraction at low and room temperature, coupled with magnetic scattering, Battle and Cheetham determine ratios R = 2.78, 2.82, and 3.01 and conclude that a cluster (8:3) can be consistent with their observations [START_REF] Battle | The magnetic structure of non-stoichiometric ferrous oxide[END_REF] and the preceding calculations of Catlow et al. (see [START_REF] Cheetham | Thermodynamics and Defect Structure of Nonstoichiometric Oxides; section IX: Recommendations for Future Work[END_REF] Chapter 8 p. 407-410: summary of these results by Cheetham). They couple neutron diffraction experiments to Mössbauer effect analyses and study the antiferromagnetic coupling of iron cations in octahedral and tetrahedral sites of the FeO lattice (see [START_REF] Battle | The magnetic structure of non-stoichiometric ferrous oxide[END_REF] Fig. 4 p. 343). For the first time, these authors show that atomic clustering observed in quenched samples (out of equilibrium) can be characterized from the study of magnetic properties of clusters in a lattice. This result will be used again by Saines et al. [START_REF] Saines | Coupling of the local defect and magnetic structure of wüstite Fe1-xO[END_REF].

The numerous results obtained by Radler, Cohen and Faber jr from neutron diffraction patterns of polycrystalline samples, under equilibrium, yield values of R > 3 corresponding to mixtures of clusters at high temperature, and R < 3 to clusters larger than (10:3) below 1173 K [START_REF] Radler | Point defect clusters in wüstite[END_REF]. This result could indicate a possible differentiation between the structures of W and W'. Schweika et al. find a ratio of vacancies to interstitials R = 4.0 (±0.5) obtained under in situ conditions by neutron diffuse scattering from a single crystal with composition z = 0.08, at T=1150 °C [START_REF] Schweika | In-situ study of the defect structure of wüstite Fe 1-x O by diffuse elastic neutron scattering[END_REF][START_REF] Schweika | Defect structure of ferrous oxide Fe 1-x O[END_REF]. Their analysis of the diffuse scattering patterns leads them to conclude that 30% of the vacancies are free, while a further 15% of the defects are bound in isolated clusters (4:1). In their analysis, the authors state that their results invalidate the predominance of large clusters at elevated z. For example, the (13:4) cluster cannot fit the diffuse scattering that they characterized near the origin.

Saines et al. in Cheetham's team study by neutron diffraction a polycrystalline sample of wüstite

Fe 0.902 O quenched from 900 °C (from the domain of W') [START_REF] Saines | Coupling of the local defect and magnetic structure of wüstite Fe1-xO[END_REF]. They derive nuclear and magnetic clusters from an elaborate Reverse Monte Carlo modeling technique. The nuclear structure is described by islands of V4T units connected along the <110> directions « into a Koch-Cohen arrangement ». The magnetic structure in the bulk between the clusters is described as a « non-collinear variant » of the antiferromagnetic structure along the [START_REF] Katkov | Wüstite solid solutions in Fe-Cu-O system[END_REF] axis as previously envisaged in [START_REF] Battle | The magnetic structure of non-stoichiometric ferrous oxide[END_REF]. Free vacancies are near the clusters. This structural approach can be compared with the paracrystalline description of the defect distribution by Welberry et al. [START_REF] Welberry | A paracrystalline description of defect distributions in wüstite, Fe 1-x O[END_REF] [143] [START_REF] Welberry | Local order in wüstite using a pair distribution function (PDF) approach[END_REF].

X-ray diffraction

Welberry et al. present a detailed study of X-ray diffuse scattering for a single crystal with composition z=0.057, obtained after quenching from a temperature which is not specified. Their sample corresponds either to the Manenc's phase P (see [START_REF] Manenc | Structure of the iron protoxyde, recent results[END_REF] [START_REF] Andersson | Decomposition and ordering in Fe 1-x O[END_REF]), or to the W1 domain proposed by Vallet et al. [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF]- [START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF]. The X-ray diffuse scattering analysis coupled with a paracrystalline modeling gives information on defect distribution, defect cluster size, number of interstitials, and lattice strain. The authors describe (statistically) a specific crystal constituted of ordered zones of clusters randomly distributed in a fcc lattice [START_REF] Welberry | A paracrystalline description of defect distributions in wüstite, Fe 1-x O[END_REF] [143] [START_REF] Welberry | Local order in wüstite using a pair distribution function (PDF) approach[END_REF].

Gavarri et alii 's works

In 1979, using in situ neutron diffraction experiments, Gavarri, Carel and Weigel develop an analysis of the structural evolution of wüstite in CO2/CO environments at high temperature [START_REF] Gavarri | Contribution to the structural study of solid wüstite at high temperature[END_REF]. A systematic determination of the ratio of the vacancy rate (z+t) divided by the interstitial rate t, R= (z+t)/t, is carried out. Quasi constant values of R close to 2.4 ± 0.4, at two equilibrium temperatures (985 and 1075 °C) and compositions z ranging between 0.058 and 0.120, are determined, with standard errors induced by the uncertainties on the separation between Bragg peaks and a complex diffuse scattering. Fig. 14 displays the values of R obtained by different authors. In the case of studies made in situ, most of the values of R range between 2.4 and 3.5. The authors observe two types of signals: (i) Bragg peaks from which the ratio R is determined, (ii) additional scattering corresponding to superstructure peaks, and thermal and static diffuse scattering associated with defect clustering, disorder and static distortions (see [START_REF] Gavarri | Contribution to the structural study of solid wüstite at high temperature[END_REF] and infra Fig. 15). The intensities I A (T, z) of this additional scattering increases linearly with composition z and decreases with temperature.

The ratios I A /I 220 , where I 220 is the intensity of the (220) Bragg peak, vary as follows:

• I A /I 220 (985 °C) = 0.19 + 4.05(z-0.062)

Therefore, these linear variations coupled with a small variation of the superstructure parameter k suggest a quasi-invariance of cluster size with clusters agglomerated in domains, and with the size of these domains increasing with z. The position of the centroid of this scattering allows the determination of an average superstructure corresponding to an irregular spacing of clusters quasi-constant with z or T. The average distance between clusters is evaluated to 2.7a 0 , a 0 being the cell parameter of ideal wüstite for a fixed temperature.

In later works by neutron diffraction on quenched samples of magnesio-and calcio-wüstites, using Rietveld refinements, Gavarri et al. [128] [146] determined R values close to 2.5 for pure wüstites or for magnesio-wüstites Fe (1-z-y) Mg y O. In the case of calcio-wüstites Fe (1-z-y) Ca y O, as the composition y increased, the ratio R increased strongly with R > 4 and the superstructure vanished. Under these conditions, the cluster sizes would be strongly modified, and a majority of free vacancies would appear likely. These studies demonstrate that the formation of clusters is stabilized by Mg doping, and destabilized by Ca doping, producing free vacancies coexisting with residual clusters.

Fig. 15 shows a neutron diffraction pattern obtained from a quenched polycrystalline calcio-wüstite (z= 0.065; y = 0.030; R = 3.3 ±0.5). In addition to Bragg peaks, the complex signal due to the superstructure and the structural disorder can be observed. It is to be noted that the diffuse scattering due to long-range ordering is similar in diffraction patterns of quenched and in situ samples. Consequently, only a slight difference would exist between high temperatures and room temperature for the nature of defects and their organization in the fcc lattice.

Table 2 lists the values of R corresponding to the quenched pure and substituted wüstites resulting from these studies (magnesio-and calcio-wüstites are noted WMg and WCa respectively). The values of the ratio R = (z+t)/t obtained from the literature on quenched, pure and Ca or Mg substituted, wüstites are displayed in Fig. 16. The utmost value R = 8.2 is obtained for a calcium rich calcio-wüstite (y=0.05).

6. T(ransmission) E(lectron) M(icroscopy) analyses

In the past, microstructural analyses using electron diffraction and HREM images were performed (see Andersson and Sletnes [START_REF] Andersson | Decomposition and ordering in Fe 1-x O[END_REF]). The images obtained by Iijima for a sample Fe 0.92 O « provided by Prof. Cohen » are often referenced [START_REF] Iijima | High resolution EM study of wüstite Proceed[END_REF]. The presence of the hypothetical cluster (10:4) <110> type with a ZnS-blende configuration has been particularly envisaged.

When re-interpreting the images obtained from electron microscopy experiments by Ishiguro and Nagakura [START_REF] Ishiguro | Structure of the commensurate phase P" of wüstite Fe 0.902 O studied by high resolution electron microscopy[END_REF], Gavarri, Carel, Weigel [START_REF] Gavarri | Re-examination of the cluster structure on quenched wüstites of P' and P" types[END_REF], then Nihoul, Gavarri, Carel [START_REF] Nihoul | The commensurate (10/4) cluster model in quenched wüstite P". New simulation of HREM direct images[END_REF] show that this cluster corresponding to the ideal ratio R = 2.5 is compatible with the experimental data available (at least for quenched wüstites). This cluster depicted in Fig. 17 is formed of a « FeO » cubic cell surrounded by an envelope of Fe 3+ cations with total 3D dimension (2a x 2a x 2a).

Fe 1-z O unit cell parameter

In the case of crystals quenched from1000 °C, and with a composition z = 0.10 [START_REF] Ishiguro | Structure of the commensurate phase P" of wüstite Fe 0.902 O studied by high resolution electron microscopy[END_REF], the formation of local ordered domains of these clusters can be at the origin of all observations corresponding to a mean ratio R close to 2.5, and a quasi-invariant superstructure. The authors propose a progressive ordering of clusters as z increased: for low z values, repetition distances 3a linkable to variety P by Manenc [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF], or W1, are observed. For intermediate z values, repetition distances 3a and (2.55a x 2.55a x 2.55a) are formed in local zones linkable to W3 or P".

A specific well-ordered superstructure can be based on a 3D supercell with parameters (5a x 5a x 5a). This superlattice can be associated with the limit evolution at the boundary W3/Fe3O4. Such a superstructure was depicted at equilibrium by Shaïovitch, Zvinchuk and Vergazov [START_REF] Yal | Lattice of an equilibrium superstructure of wüstite Fe 0.90 O. Fiz. Tverd. Tela[END_REF]. Finally, the distinction between « varieties of wüstite » or pseudo-phases can be due to both modifications of long-range order of clusters and the coexistence of different clusters that include free vacancies.

To define an isolated cluster correctly, particularly the ZnS blende defect cluster (10:4), it is necessary to take into account its envelope of Fe 2+ and Fe 3+ cations in octahedral sites. The charge of the cluster can be postulated as a first approximation by assuming the individual charges of vacancies and of iron atoms. In Fig. 18 is illustrated the possible arrangement of these defect clusters (10:4), in the fcc lattice. The minimum distance between clusters is [(2.5a) 2 + (0.5a) 2 ] ½ = 2.55a, and with allowed distances up to 3a. This model is based on a cluster core (10:4) with double envelope of octahedral iron cations.

If we assume that a cluster (m:n) has a global charge Q= 3n-mq (m vacancies with formal charges q (=2,1,0) and n interstitials Fe i °°° with charges +3), then the (10:4) cluster has a charge of Q= -8, +2, or +12

for q= 2, 1 or 0, respectively. The electroneutrality is ensured by envelopes with charges -Q. Using this type of defect structure, a model of ordered (10:4) clusters is proposed [START_REF] Gavarri | Re-examination of the cluster structure on quenched wüstites of P' and P" types[END_REF] [150], thus allowing simulation of HREM images in the two directions [001] and [START_REF] Kozheurov | Electrical conductivity of wüstite[END_REF] of the crystal. These hypothetical distributions of clusters (Fig. 19) are in agreement with the HREM images by Ishiguro and Nagakura [START_REF] Ishiguro | Structure of the commensurate phase P" of wüstite Fe 0.902 O studied by high resolution electron microscopy[END_REF].

Hypothesis of a mixture of defect clusters, and/or cluster zones

Mixture of clusters

Based on previous publications by Gavarri et al. [START_REF] Gavarri | Contribution to the structural study of solid wüstite at high temperature[END_REF] [151], a specific relationship can be established taking into account ordered zones of clusters and zones in which free vacancies are distributed. A mixture of different clusters (m:n) 1 and (m:n) 2 can coexist with additional free vacancies. As a first approximation, the authors consider a lattice based on N cells of Fe 1-z O, with a proportion V(z) of zones without defect. If N i is the number of cells where clusters (m:n) i are formed, with the characteristics m i and n i of a specific cluster, and the distance k i .a between two clusters, then the relation between cluster size and long-range order in the case of the fcc lattice of wüstite can be expressed as follows:

1-V(z) = 4z .  i {  i (k i ) 3 / (m i -n i ) } with  i = N i /N ( 25
)
For a unique cluster with continuous order (high z values), the relation [START_REF] Chaudron | Reversible reaction study of hydrogen and carbon oxide on metallic oxides[END_REF] yields:

(m-n) / (k) 3 = 4z  (m-n) = 4zk 3 (26)
Considering the experimental values of k available in the literature, ranging between 2.5 and 3, this parameter k is characteristic of zones in which clusters are ordered partially. This means that, for low values of z, zones without clusters would exist and that they would be occupied by isolated defects (i.e., free vacancies). The major part of the crystal lattice would be dominated by disordered point defects, and electrical conduction would be due to holes. For intermediate values of z, the cluster zones would be progressively joining together, forming percolation paths of ordered zones. A mixed conduction mode would be expected. For high values of z, the lattice would be occupied fully by ordered clusters, with likely small changes in the arrangement of clusters. Electrons in conduction band would mainly condition the electrical conduction.

This hypothetical description is in correspondence with the W1 disordered pseudo-phase and a conduction of p type, the intermediate pseudo-phase W2 as a mixture of cluster zones and free vacancy zones with mixed conduction, i.e., (holes +) and (electrons -) coexisting at short distance, and the ordered pseudophase W3 with conduction of n type from electrons in the conduction band.

Variation of parameter k

Numerous structural data can be found in the studies by Bauer, Pianelli et al. ([8] Part I and [START_REF] Bauer | II: [New structural examination of pure and substituted metastable wüstites[END_REF] Part II). These authors develop a series of experiments using X-ray diffraction on powders of quenched wüstites. They observe linear variations of the cell parameter as a function of composition z, and of the position of superstructure peaks. They characterize these peaks with parameter  = 1/k, where k designates the superstructure parameter discussed previously. Parameter  varies between 0.37 and 0.39, corresponding to values of k between 2.7 and 2.56. This variation observed in quenched samples is in good agreement with the model presented in (section 7.6. TEM analysis) with a minimum k value of 2.55. Authors [START_REF] Bauer | A review of the lacunar structure of wüstite, and the necessity of new experimental researches[END_REF][START_REF] Bauer | II: [New structural examination of pure and substituted metastable wüstites[END_REF] consider that their data did not permit to observe any phase transition. However, they suggest that wüstite can be described by an incommensurate lattice of clusters, with the presence of different clusters based on cluster (4:1).

Statistical analysis of diffuse scattering

Gartstein and Mason [START_REF] Gartstein | Reanalysis of wüstite electrical properties[END_REF] show that p conduction is observed in wüstite and that electrical conduction is based on a hopping mechanism between adjacent clusters. Later, Gartstein et al. publish a crystallographic analysis of a Fe 1-z O (z=0.07) single crystal using X-ray diffraction at 1173 K [START_REF] Gartstein | Defect agglomeration in wüstite at high temperatures -I: The defect arrangement[END_REF] and [START_REF] Gartstein | II: An electrical conduction model[END_REF]. They develop a detailed analysis of diffuse scattering and Bragg peaks, and conclude to the existence of a mixture of clusters (13:4) and asymmetric new clusters (5:2) constituted of 5 vacancies and 2 interstitials corresponding to the ideal ratio R = 2.5. They propose an ordering of these clusters with an average repetition distance of 2.5a, close to the hypothetic value of 2.55a proposed in [START_REF] Gavarri | Re-examination of the cluster structure on quenched wüstites of P' and P" types[END_REF] [START_REF] Nihoul | The commensurate (10/4) cluster model in quenched wüstite P". New simulation of HREM direct images[END_REF].

Summary

To conclude this section, it can be noted that there is an inherent difference between the concept of a mixture of disordered clusters with various forms, and the concept of zones with identical disordered clusters, coexisting with zones of free point defects. The concept of incommensurability, applied to specific compositions z, was also proposed as a mathematical approach of disorder (see Yamamoto [START_REF] Yamamoto | Modulated structure of wustite (Fe1-xO) (Three-dimensional modulation)[END_REF] and Weigel et al. [START_REF] Weigel | About symbols of the space group of a tri-incommensurate cubic wüstite, and Bravais groups of its crystal family in the 6-dimensional Euclidean space[END_REF]). Finally, the proposition of the irregular cluster (5:2), resulting from the analysis of diffuse scattering by Gartstein, Mason, Cohen [START_REF] Gartstein | Defect agglomeration in wüstite at high temperatures -I: The defect arrangement[END_REF] [153], seemed to be compatible with the quasi-constant value R=(z+t)/t = 2.4  0.4 [START_REF] Gavarri | Contribution to the structural study of solid wüstite at high temperature[END_REF].

Defect clustering and equilibrium equations

In this section, using basic equilibrium equations from the literature (see particularly [START_REF] Cheetham | Thermodynamics and Defect Structure of Nonstoichiometric Oxides; section IX: Recommendations for Future Work[END_REF] and [START_REF] Toft Sørensen | Defects on metal-deficient oxides: wüstite, Fe 1-y O[END_REF]), we propose a new presentation of defect equilibrium equations for wüstite.

It is important to remind that, in Fe 1-z O where only Fe 2+ (also noted Fe Fe ) and Fe 3+ cations (also noted Fe Fe ° or Fe Fe °°°, for cations in octahedral or tetrahedral sites, respectively) can coexist, cation vacancies can have one of three possible stable charges: doubly charged vacancies V Fe '', simply charged vacancies V Fe ', and neutral vacancies V Fe

x (noted V Fe ). Singly charged iron vacancies V Fe ' were proposed previously in place of doubly charged vacancies V Fe " by Kofstad and Hed [START_REF] Kofstad | Defect structure model for wustite[END_REF]. Vacancies of the oxygen sublattice are not considered because they occur in negligible numbers compared to the iron vacancies. In section 2.2. Modifications or pseudo-phases, the equilibrium constant (4b) of the hypothetical equilibria (4a), in which the activity of oxygen O O in the solid is taken equal to 1, can be developed with q=0, 1 or 2 following the general expression in which s =2(q+1):

log z = (1/s). l' + (1/s) log (K/C) 2 (with C = (q) 2q ) (27)
(identical to equation (4c)). The three hypothetical values of s corresponding to q = 2, 1, and 0 are s = 6, 4, and 2, respectively. The modifications of q can be described in terms of two equilibriums involving holes h ° in the valence band

V Fe "+ h °  V Fe ' and V Fe ' + h °  V Fe ( 28
)
z=0.10). The change in vacancy charge as z increases can be accompanied by injection of free electrons in the conduction band. As temperature T increases, the size of the cluster zones decreases, and the number of cluster zones increases. Correlatively, the population of free defects increases.

Taking into account all these considerations, the W1 pseudo-phase can be associated with a small number of clusters assembled in cluster zones, and a large population of free defects with numerous free V Fe '' vacancies. In this domain, holes can be the major electric charge carriers. The W3 pseudo-phase is constituted of large cluster zones, covering the entire lattice adjacent to the magnetite zone. The neutralization of vacancies (or decreasing q charges) shall be associated with a large number of electrons occupying the conduction band. As the W2 pseudo-phase was previously considered as a mix system of pseudo-phases W1 and W3 by several authors, properties can be expected to follow continuous evolutions. In other terms, an intermediate structural configuration can exist, corresponding to the formation of percolation paths allowing a mix conduction of holes and electrons.

Percolation approach

A primitive « composite picture » of a possible evolution of the defect structure needing only two clusters and distinct populations through the subdomains was proposed earlier [START_REF] Gavarri | Structure of wüstite at high temperature. Composite picture. Proceed. Round Table Meeting Iron and Manganese Oxides Sept. 8-9, 1980[END_REF]. Nearly at the same time, a similar simple model of the defect structure was proposed in terms of the so-called theoretical Cluster Component Method (see Section 6.3. The C. C. M., Fig. 12).

In Fig. 20, a representation of the distribution of cluster zones (CLZ) inside the lattice is drawn as a schematic percolation process. The size of cluster zones (CLZ) (represented by black squares) is conditioned by the equilibrium temperature. White squares represent zones with only disordered point defects (free vacancies, individual Fe 3+ in octahedral or tetrahedral sites).

-In the W1 domain, the CLZ's are isolated and immersed in a defect lattice. The transition line W1/W2 corresponds to a specific threshold z 1/2 value (for fixed T) where there is a starting percolation state.

-The W2 domain might be constituted of large black zones, with percolative paths through the lattice or both types of zones. The W2/W3 transition line might be reached when the residual white zones are no more percolative. -The W3 state is reached when black zones envelop white zones up to a limit state where all the lattice is covered by a continuum of clusters, i.e., at the boundary W3/Fe 3 O 4 .

The effect of temperature is to modify the sizes of clusters zones and the limits of composition for each boundary. As temperature increases, the cluster zones are destabilized, and the size of these zones decreases.

Because of the complexity of the system, it is obvious that no unique specific model can represent the various evolutions. Nevertheless, this approach shows that the modifications W1, W2, W3 may be justified simply from changes in electrical charges of point defects, so of clusters, and are strongly correlated with composition z, and the degree of ordering. This scheme implies nearly continuous transformations from Wi to W(i+1) at the level of the slight deviations from the main diffraction process. However, the discontinuities observed in the partial molar properties (see supra Fig. 6 and7), and in the kinetic reduction rates by [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF] at the compositions x i/j of the boundary of the subdomains correspond to abrupt and small-scale differences in internal energy and configurational entropy, due to the change of cluster type (or charge), and to the change in long range order of the clusters (see also [113] [114] in Fig. B1 of Annex B).

Note: Metastabilities encountered in most of the experimental studies can modify the location of the transitions ij, i.e., the characteristic z i/j values as determined by thermodynamic assessments, even suppressed them. As examples, the thermogravimetric isotherms (W area) in Raccah's study [START_REF] Raccah | Study of the thermodynamic properties of iron protoxide[END_REF] are grouped together considering a sole modification in certain temperature intervals, without break all along from -Fe/Wi to Wi/Fe3O4: W3 at 920, 950, 1000°C, W2 at 1067, 1122, 1140 °C, W1 at 1160, 1167,1200, 1250 °C.

Conclusions

Point defects: Up to now, the subdivision of the equilibrium phase diagram of wüstite appears to be complex with the existence of several domains corresponding to different thermodynamic behaviors and high order transitions (at least of second order). The departure from stoichiometry can be described by a complex system of cation vacancies noted V", V' or V x , depending on their electrical charge or neutrality, giving rise to Fe Fe ° (cations Fe 3+ ) coexisting with Fe Fe (cations Fe 2+ ) in octahedral sites, and Fe i °°° (cations Fe 3+ ) occupying the tetrahedral sites of the NaCl-type lattice. Changes in electrical charges must be taken into consideration, with electrons in the conduction band and holes in the valence band.

Clusters: Works from several different sources demonstrate the existence of clusters of defects constituted of iron vacancies coupled with cations Fe 3+ in tetrahedral interstitial sites, and mixture of Fe 2+ /Fe 3+ cations in octahedral sites of the basic fcc lattice. These clusters can coexist with free vacancies. However, at present, the exact form of clusters noted (m:n) (m vacancies V q( ' ) linked to n interstitials Fe i

°°°)

is far from being determined definitively.

The ratio R= (z+t)/t is obtained mainly from neutron diffraction experiments in the range 2 to 4 depending on the authors. Several authors observe quasi-constant R values in the range 2 to 3, while others observe a variable or constant value in the range 2.8 to 4. It can be noted that if the wüstite lattice is occupied solely by the (m:n) isolated cluster, then the ratio R is equal to (z+t)/t = m/n. The pseudo-phases: inside the initial phase diagram, some experimental evidences from different authors support the existence of distinct areas corresponding more than likely to pseudo-phases W1, W2, W3. The term « boundary » designates a threshold of percolation at the exact separation between two thermodynamic behaviors, corresponding to progressive evolutions of short-and long-range orderings. An additional first order transition W  W' is directly linked to the first order transition -Fe  -Fe at 912 °C.

Up to now, the lack of systematic studies close to the isotherm at 912 °C limits the correct definition of this part of the Fe-O phase diagram, despite the availability of some advanced thermodynamic studies. The interpretation of them remains to be developed.

Changes in cluster form:

With respect to the existence of clusters (m:n) and the variation of the ratio R observed by some authors, a first description of the pseudo-phases W1, W2, W3 is proposed in terms of structure changes of these clusters, with an increasing cluster size as z increased.

The ZnS-blende type (10:4) cluster can agree with neutron diffraction experiments at equilibrium and in quenched samples, and with electron microscopy images for some compositions z of quenched samples. The (12:4) <110> specific cluster, proposed by Lebreton and Hobbs, can also agree with a superstructure of P" type (5a x 5a x 10a) for a sufficiently high composition z.

The existence of separated ordered cluster zones can agree with the slowly variable k parameter. Let us note that significant changes in cluster form (e.g., (13:4) or (10:4) transformed into (4:1)) as z or T vary have never been demonstrated for the entire phase diagram. In addition, in studies of samples at equilibrium or after a quenching process, all observations of superstructure Bragg peaks or diffuse scattering due to disorder show that the k value (k close to 2.6-2.7) varies slowly with z, thus characterizing a quasi-constant repetition distance between clusters. If the form of clusters (m:n) would change strongly, the repetition distance k.a would also change strongly, which is not observed.

New possible description of the pseudo-phases: Pseudo-phase W1 (low z values and high T values) would be characterized likely by free vacancies coexisting with a weak proportion of small zones of clusters. Pseudo-phase W3 (high z values and low T values) would correspond to clusters covering the whole lattice with a minimum distance of 2.55a between clusters and to a 5X superstructure resulting from a zigzag ordering of clusters. Pseudo-phase W2 is identified as a solution of W1 and W3 of opposite energetic evolutions above its critical temperature near 300 °C. This pseudo-phase is likely constituted of zones of ordered clusters and zones of free vacancies, with a progressive build up from boundary z 1/2 to boundary z 2/3 at which the junction arises between cluster zones that percolate.

The p to n transition might be associated with a starting of this percolation. This transition could be located in the W2 domain in a non-exclusive manner and is likely connected with a given change in electrical charges of vacancies and clusters.

Finally, an additional feature can explain the observed electrical modifications. As z increases, the electrical charges of vacancies (V q( ' ) = V", V' or V x ) and clusters could change with q varying from q=2 (low z) to q=0 (high z), due to interactions between charges. This modification of charges q is associated with an increasing number of electrons populating the conduction band.

Transition close to 912 °C: Up to now, the three varieties W'1, W'2 and W'3 associated to -Fe below 912 °C are not strictly characterized from a structural point of view. According to Worral and Coley in their isotopic kinetic study, the defect structure identified on the surface of polycrystalline samples appears as being close to that identified for the pseudo-phases W1, W2, and W3.

______________ Annex B 1 -Separate diffusion sequences in the thermogravimetric study of reduction in H2/H2O gas mixtures

The first experimental confirmation of the three Wi out of Vallet's team is published by [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF] (see supra in Fig. A2). At the same time, the work of Landler and Komarek [START_REF] Landler | Reduction of wüstite within the wüstite phase in H2-H2O mixtures[END_REF] becomes available.

A numerical analysis makes be revealed three diffusion sequences distributed across the analyzed domain [START_REF] Carel | Reduction kinetics of solid wüstite in its stability domain[END_REF]. ) are statistically identical to those of equations [START_REF] Tarte | IR spectrum and presence of trivalent tetrahedral iron in the structure of iron protoxide Fe1-xO (Wüstite)[END_REF], [START_REF] Wagner | Electrical conductivity of cobalt and iron oxides[END_REF], [START_REF] Wagner | Handbuch der Metallphysik Becker and Erler Kom[END_REF] at 1268 K in ( [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF] p. 820). A discontinuity is observed at the location of the transition W'2  W'3 forecasted at equilibrium by [START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF]. It likely corresponds to a sharp process in the bulk and at the surface (nuclear or electronic change in cluster structure, change in percolation mode, germination of a superstructure (2.5a o ) x 3).

Fe 1 -

 1 z O = (Fe Fe ) (1-3z) (Fe Fe °)(2z-t) (Fe i °°°)

  below, we detail the relation log z vs l' (see relationship 4c) in the specific case of the Vallet et al. data, as obtained at 1000 °C (T=1273 K), using the relationships (3a, b, c) above, and transforming them into the following relationships: W1: log z = 0.2036 l' + 1.7511  s = 4.9 (±0.10) (5a) W2: log z = 0.1369 l' + 0.8352  s = 6.7 (±0.12) (5b) W3: log z = 0.1124 l' + 0.5246  s = 8.9 (± 0.15) (5c)

Fig. 1 .

 1 Fig. 1. The logarithm l' of oxygen partial pressure p' vs the logarithm of composition z derived from the data [29] at 1273 K.

Fig. 2 .

 2 Fig. 2. Variation of l' vs 1/T for a fixed value z = 0.115: three quasi linear correlations corresponding to the initial determinations in the three domains W1 (high T), W2 (intermediate T) and W3 (low T) give rise to three activation energies H i .

  atm. The corresponding partial pressures of oxygen are p'(A) = 4.829 x10 -17 atm and p'(B) = 4.229 x10 -17 atm, i.e. a difference of 0.6 x10 -17 atm. The compositions zo(A) = 0.0571 and zo(B) = 0.0468 correspond to a gap of 0.01 in composition.

Fig. 3 .

 3 Fig. 3. Equilibrium diagram (FeO x ) between W1 and W'1 and 'pure iron', with both limit points A and B of -Fe/W1 and -Fe/W'1 boundaries. Compositions N O = x/(1+x) are arbitrarily taken at 1184 K in paper [28].Dashed lines correspond with hypothetical limits.

Fig. 4 .

 4 Fig. 4. Phase diagram  °C (l', x) of solid wüstite FeO x under equilibrium following Raccah-Vallet-Carel; upper temperature 1250 °C, lower one 592 °C at point C; coordinate l' for log 10 p(O2). Points noted by a capital letter are tabulated in [27], except the point  in [28] and the point T in [62] [63] [64].

Fig. 5 .

 5 Fig. 5.  H O from some authors. ─ : derived for the Wi (i=1-3) i.e. at T [1193 -1523 K] from thermogravimetry + CO2/CO [28]; : w I,  -- : wII, and : wIII (two points only) by emf at T  [1023-1623 K] [67]; dash-dot line -.-.-: by calorimetry at 1348 K under vacuum (+ n O2 ) [68]; : by emf at T [873-1273 K] [69]; : by thermogravimetry + CO2/H2 at T  [1373-1673 K] [70]; : by emf at T  [823-1323 K] [71]; : by emf data at T  [873-1373 K] [72] [13].

  x 1/2 = 1.0782: (H O ) = -3.5 kJ, (S O ) = -2.3 J. K -1 , and x 2/3 = 1.1154: (H O ) = -10.5 kJ, (S O ) = -8.3 J.K -1 .

Fig. 6 .

 6 Fig. 6. Partial molar enthalpy ( H O -½H o O2 ) and entropy ( S O -½S°O 2 ) of solution of oxygen, through the stability field of wüstite, whatever the temperature. Transitional jumps are at 820 °C (W'i) and 1000 °C (Wi).Dashed lines are drawn in extrapolated or metastable zones.

Fig. 7 .

 7 Fig. 7. Variations of the partial molar enthalpy ( H Fe -H °Fe ) vs x for the isotherms at 1000 °C (Wi) and 820 °C (W'j). For comparison: -.-:  H Fe from e.m.f. measurements at 1000 °C by Asao et al. ([69]Table 3 p. 75); : Darken and Gurry determinations ([31] Table X p. 1408).

  07, also close to Chaudron's point composition. It results that the three pseudophases are under equilibrium at T (K) under the same oxygen pressure for a composition x  1.07 (e.g., p'  8 x10 -17 , 4 10 -15 or 8 x10 -11 atm at 1184, 1273, or 1573 K respectively), irrespective of index i. This result constitutes the reason why the property H O (= -274.8 kJ.mol -1 ) of equation (9h) is shared by the three Wi's with the same composition x 1.07 under the same equilibrium oxygen potential. The convergence at point  results of that fact. The iron partial molar properties converge also for H Fe = +87.8 kJ.mol -1 at 1000 °C. At invariant point C (-Fe, W'1, Fe3O4) of abscissa x C = 1.070 0 , these properties are merged (H O = -290.4 kJ.mol -1 and H Fe = + 21.4 kJ.mol -1 ).

  . p. 3681).

Fig. 8 .

 8 Fig. 8. The m3m cluster (16:6) of <110> type corresponding to (z+t)/t = 2.66 (from ref. [90]).

Fig. 9 .

 9 Fig. 9. Variation of l' vs log z for Fe 1-z O at 1250 °C. Three sets of data can be defined as three quasilinear relations.

l' = 4 .

 4 4539 log z -5.5685, l' = 6.1378 log z -3.6876, l' = 9.5059 log z -0.5046 (21) from which three values for s can be derived. A first domain for low z values yields a coefficient s = 4.5. So, it is possible to interpret the non-linear second domain in terms of two linear sequences with two coefficients s = 6.1 and s = 9.5: these two last coefficients are quite close to those obtained from the data of Vallet et al.

Fig. 10 :

 10 Fig. 10: Some determinations of the p-n transition by thermal e.m.f.. : Geiger et al. [66] not corrected for platinum ; : Bransky and Tannhauser [102]; : Lafollet and Duquesnoy [104] by modeling; : Hillegas jr and Wagner jr [105];  : Gartstein and Mason [106] by re-examination of Hillegas jr's thesis [82].

Fig. 11 :

 11 Fig. 11: Fe diffusion (from ref. [112], Fig.4) : log D * Fe vs 1/T. Fe % = 75.6  z= 0.112; 76.0  0.093; 76.2  0.083; 76.5  0.067.

  Fig. 6 and 7 p. 1856, and Fig. 8 p. 1857).

  unit blocks S.S. I, II, III: positions : (16d), : (16c), : (8b), : (8a), : (48f), : (32e); a and d are for iron, c and f for vacancies (see ref. (6), Table 7.1., p. 186). Short range order in terms of S.S and C.C. in the three subdomains; C.C. 1: I-6III, C.C. 2: II-6III, C.C. 3: I-6III-II. In the range [z o -z 1/2] , II and III increase, I decreases; in [z 1/2z 2/3 ], II and III decrease, 1 or 2 increases; in [z 2/3 -z 1 ], 1 and 2 decrease, 3 increases.

Fig. 12 :

 12 Fig. 12: C.C.M. applied to wüstite (from ref. [121]). The three pseudo phases from z o on the -Fe/W1 boundary to z 1 on the W3/Fe3O4 boundary described with the help of three cluster components , , .

Fig. 13 :

 13 Fig. 13: The Koch and Cohen cluster (13:4) associated with a 3X commensurate structure of wüstite Fe 0.902 O.

Fig. 14 :

 14 Fig. 14: R =(z+t)/t values vs z, determined under equilibrium, from the literature. : at 900°C [131]; : at 800°C and : at 925, 1050, 1125, 1180°C [136]; : at 1050°C and : at 850-1100°C [139]; : at 1150°C [141]; : at 1075°C and : at 985°C [145].

  • I A /I 220 (1075 °C) = 0.15 + 4.25 (z-0.058)

Fig. 15 .Fig. 16 .

 1516 Fig. 15. Neutron diffraction pattern ( = 1.9 Å) of a quenched polycrystalline calcio-wustite Fe 1-z-y Ca y O in room conditions [146]. Presence of a large diffuse scattering (I A ) due to the superstructure (k close to 2.7), and (110) small peak (36° 2) due to vacancies and interstitial Fe 3+ in the lattice. The intensity of I A decreases with the calcium fraction y. Cell parameter a(25 °C) = 4.3164 Å. R=(z+t)/t=3.3.

Fig. 17

 17 Fig. 17. The cluster (10:4) ZnS blende <110> type.

  Fig. 17. The cluster (10:4) ZnS blende <110> type. = Fe 3+ in tetrahedral site = Iron vacancy in octahedral site

Fig. 18 .Fig. 19 .

 1819 Fig. 18. Various orderings of clusters (10:4) in the fcc lattice of wüstite. The basic cluster can be the (13:4), the (10:4) or other possibilities with 4 interstitial sites occupied by cations Fe 3+ . The disorder can develop by statistical displacement in one, two or three directions. 2.55a could be the smaller inter-cluster distance (a = cell parameter of the "FeO" lattice).

Fig. 20 .

 20 Fig. 20. Three configurations for CLZ (in black) disordered distributions, typically at 1000 °C from Fe/W1 to W3/Fe 3 O 4 : W1 = majority of zones without defect clusters; W2 = intermediate configuration; W3 = majority of zones with defect clusters.

1 - 2 -

 12 Fig. A1. Transitional effects located in the diagram [ °C, x].

  Fig. B1. Typical runs of reduction (in H2/H2O gas mixtures under 1 atm) identified in ref. [113].

  4

  Jacobsson and Rosén, Guillermet and Per Gustafson, Sjödén, Seetharaman and Staffanson, Grønvold et al. publish new experimental thermodynamic data of highly improved accuracy about the molar thermodynamic properties of iron and wüstite [57] [58] [59] [60].

Table 3 p

 3 . 604). Very few experimental studies are available concerning the P(eritectoid) invariant point (Fe-O ' , W'1, Fe-O  ) close to the temperature of the pure -Fe  -Fe transition [61].

  H O (x)] i ,[H Fe (x)] i ,[S O (x)] i and [S Fe (x)] i leading to Figs. II-6 and II-7 above, are simply:

Table 1 -

 1

	FeO x /Fe 1-zO	x=1.0950 or z =0.0868	x=1.1420 or z =0.1243
	W1: a 1 = 46753.4 b 1 = -7.3781 x o(1) = 1.0572* z o(1) = 0.0541	[H O ] 1 e 16.90 -18.19 [H Fe ] 1 e ** [S O ] 1 e [S Fe ] 1 e ** 2.67 -2.87	[H O ] 1 e 37.94 [S O ] 1 e 5.99	e [H Fe ] 1 -41.72 e [S Fe ] 1 -6.58
	W2: a 2 = -9568.9 b 2 = 31.1728 x o(2) = 1.0522 z o(2) =0.0496	[H O ] 2 e -3.92 [S O ] 2 e -12.78	[H Fe ] 2 e 4.21 [S Fe ] 2 e 13.76	[H O ] 2 e -8.23 [S O ] 2 e -26.80	e [H Fe ] 2 9.03 e [S Fe ] 2 29.41
	W3: a 3 = -33238.9 b 3 = 48.3669 x o(3) =1.0482 z o(3) =0.0460	[H O ] 3 e -14.90 [S O ] 3 e -21.68	[H Fe ] 3 e 15.97 [S Fe ] 3 e 23.24	[H O ] 3 e -29.85 [S O ] 3 e -43.44	e [H Fe ] 3 32.69 e [S Fe ] 3 47.57

Variations at 1273 K of excess properties [H O ] i e and [S O ] i e . * For x= x o(i) , all terms are equal to zero. ** The terms [H O ] i e and [H Fe ] i e are expressed in kJ.mol -1 , and the terms [S O ] i e and [S Fe

  CO 2  13 CO + O ads and CO + O ads  CO 2

Table 2 -R = (z+t)/t values obtained from Rietveld analyses of neutron diffraction patterns of quenched substituted wüstites (from ref. [146]).

 2

	Fe 1-z-y Mg y O	Fraction y (Mg)	R = (z+t)/t (*)
	z=0.065	0.075	2.53(0.5)
	z=0.085	0.030	2.88(0.6)
	Fe 1-z-y Ca y O	Fraction y (Ca)	R = (z+t)/t
	z=0.0688	0.010	2.20(0.5)
	z=0.0649 (**)	0.030 (*)	3.31(0.5)
	z= 0.0534	0.050	8.22(4.0)
	z=0.085	0.010	2.37(0.5)
	z=0.0899	0.030	3.39(0.5)
	(*) experimental uncertainty (due to diffuse scattering); (**) See Fig. 16.

There is no equivalent available in the English language for the German phoneme 'ü'.

Acknowledgements:

The present publication can be considered as a Tribute to Professors Pierre Vallet (1906Vallet ( -1994)), Dominique Weigel and Aron Naumovitch Men, at Sverdlovsk (now Ekaterinburg), for being the first to announce the existence of "varieties of wüstite" in the phase diagram, and for the continuous impetus they provided to the work until the end of the 80ties. By the present work, two of their Associates present the culmination of 50 years of a research funded by the Universities of Rennes 1, Paris 13 and Toulon. We would like also to pay tribute to the memory of our regretted Polish colleagues, Professors Jan Janowski and Stanisława Jasienska of the AGH University of Krakow, who initiated a long collaboration with our laboratories, from the 1980s until this day.

____________________________________________________________________

Therefore, three domains corresponding to the three types of iron vacancies can exist in all systems presenting significant cation deficiency. In the case of Fe 1-z O, additional reactions occur since two valences (Fe 2+ and Fe 3+ ) exist:

Fe Fe + h °  Fe Fe ° (Fe 3+ in octahedral sites)

and Fe Fe °  Fe i °°° (Fe 3+ in tetrahedral sites) + V Fe" (additional cation vacancy) [START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF] As a first approximation, assuming that the sole point defects are iron vacancies, associated with Fe 3+ (Fe Fe ° )

and coexisting with Fe 2+ (Fe Fe ) in Fe 1-z O, the general relationship between composition z and oxygen partial pressure p' is as follows

In this expression, [Fe Fe ] ≈ 1-3z and [Fe Fe °] ≈ 2z.

As all molar fractions are related to the composition z, next relation can be derived:

If this simple model is correct, then the term (1-3z) should vary slowly with z in a reduced composition range, meaning that the activity of Fe Fe can be considered as being constant. Under this assumption, z is roughly proportional to (p') 1/s , with s = 2(q+1) being an integer characteristic of the nature of the defects. For q = 2 (doubly charged vacancies) we obtain s ≈ 6. For q = 1 (singly charged vacancies), s ≈ 4. For neutral vacancies, s ≈ 2. Now, let us consider complex clusters (m:n) Q with variable formal charge Q and successive reactions in which vacant tetrahedral sites V T are introduced:

The formation of stable cluster can be written:

with Q = 3n-qm (positive or negative charge). The cluster will be noted: CL = {n Fe i °°°, m V Fe q( ' ) } Q . The formal charge Q of a cluster is assumed to result from individual charges of vacancies with the invariant charge of interstitials Fe 3+ (Fe i °°°) as a first approximation. However, the Fe 3+ interstitial can be modified into Fe 2+ interstitial cation due to electron jumping, and as a result a more complete model should take into account various possible Q values of cluster charges due to the modification of interstitial cation charges.

Considering the formation of clusters CL coexisting with point defects Fe Fe ° and V Fe q( ' ) , a more elaborate equation (32a) can be expressed as follows:

with the hypothesis: Q = 3n-qm (positive or negative).

In equation (33a), the integer M  m is introduced arbitrarily to take into account the possible presence of additional free species Fe Fe ° and V Fe q( ' ) , associated with free electrons in the conduction band. The additional vacant tetrahedral sites are the vacant sites V T able to receive an interstitial cation Fe i °.

As a major fraction of these sites (8 sites for 4 FeO units in a fcc cell) is vacant, their activity can be considered as being quasi-constant, and does not play a significant role in the calculation of equilibrium constants. The resulting equilibrium constant is as follows, with the reduced notation

In this expression, the cluster concentration [CL] is considered as being proportional to z, at least as a first approximation. Applying the same approximation as in previous equilibrium equations, the equilibrium constant can be expressed as follows:

No activity coefficient is taken into consideration. Consequently, in a restricted range of composition z, no significant change of interactions occurs between point defects or clusters. To illustrate the possible values of parameter s corresponding to the three hypothetical q values, l' has been simulated as a function of log z using the expressions of K(T). The slopes of the curves (l' vs log z) were determined. The two specific clusters (4:1) and (10:4) have been chosen to calculate the various coefficients s depending on the charges q of vacancies.

The results are:

-for a cluster (4:1), the three coefficients s are equal to 5.7, 8.2 and 10.7 for q=2, 1 and 0, respectively; -for a cluster (10:4), the three coefficients s are equal to 5.4, 8.2 and 11 for q=2,1 and 0, respectively.

It can be noted that, for either the cluster (4:1) or the cluster (10:4), the coefficients s resulting from the calculations are quite similar for a given value of q. Given the approximations included in the model and expected experimental uncertainties, these simulated values (from s = 5 to 11) can be considered as being congruent with the experimental s values obtained in the literature and ranging between 4 and 10. In other terms, the existence of pseudo-phases might be reasonably thought to be associated with the variation of the electrical charges (q-) of cation vacancies, and, more generally, with the variation of the electrical charges Q of the clusters.

These changes of electrical charges are strongly correlated with the increasing density of them: in other terms, two neighboring vacancies V" with the same negative charge 2-can produce two singly charged V' or neutral V x vacancies, which will decrease the coulomb repulsion between them, and require additional free electrons migrating in the conduction band. Now, using this general approach delivering specific s values, it is possible to justify the various modifications observed by many authors including Vallet et al. [START_REF] Vallet | On the boundaries of the solid wüstite domain[END_REF][START_REF] Vallet | Contribution to the study of the solid nonstoichiometric iron monoxyde. Diagram T-P(O2)-X[END_REF][START_REF] Vallet | The Fe-O phase diagram in the range of the nonstoichiometric monoxide and magnetite at the Fe-rich limit, Bull. Alloy Phase Diagram (=>[END_REF], Geiger Levin and Wagner [START_REF] Geiger | Studies on the defect structure of wüstite using electrical conductivity and thermoelectric measurement[END_REF], Fender and Riley [START_REF] Fender | Thermodynamic Properties of Fe1-xO. Transitions in the Single-Phase Region[END_REF], Takayama and Kimizuka [START_REF] Takayama | Thermodynamic properties and subphases of wustite field determined by means of thermogravimetric method in the temperature range of 1100-1300°C[END_REF], Worral and Coley [START_REF] Worral | Kinetics of the reaction of CO 2 /CO gas mixtures with iron oxide[END_REF]. In particular, let us recall the values s(W1) = 4.6, s(W2) = 6.2, s(W3) = 9.2, extracted from the data by Vallet et al. , as well as values s(W1) = 4.8, s(W2) = 6, s(W3) = 9 extracted from the data by [START_REF] Takayama | Thermodynamic properties and subphases of wustite field determined by means of thermogravimetric method in the temperature range of 1100-1300°C[END_REF] (see section 3.7. Takayama and Kimizuka's phase modifications).

Considering the set of values s = 4.6, 6, 9 associated with W1, W2 and W3 respectively, for low values of z, the existence of doubly charged iron vacancies V Fe '' can be associated with a major population of holes corresponding to mode 'p' of conduction. For high values of z, the existence of singly charged or neutral vacancies can be associated with mode 'n' of conduction, i.e., to the injection of electrons in the conduction band. For intermediate values of z, an intermediate state would exist with a mix of holes in the valence band and electrons in the conduction band, following a rate varying from z 1/2 to z 2/3 .

These considerations can be supported by the p-n transition [START_REF] Tannhauser | Conductivity in iron oxides[END_REF] described many times in the literature (see Fig. 10). The authors provide a detailed interpretation for the existence of various defect configurations corresponding to pseudo-phases in the entire wüstite phase diagram.

For mean values of z, neutron diffraction data show that the clusters are assembled in limited ordered zones. For high values of z, the ordered cluster zones percolate (see [START_REF] Tki | About the structure of Fe1-xO][END_REF] HREM images for z=0.08 and