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Voltage E and oxygen fugacity (fO₂) are both measures of oxidation state. The relation between fO₂ and E for a given electrolytic medium can be established by the anode reaction where oxygen is produced. In the case of aqueous solutions, conversion is provided by half‐reaction 1.15 and Equation 1.18. We can then replace E‐pH diagrams with analogous logfO₂‐pH diagrams. In this treatment, the actual speciation state of solutions is still the key to investigate the system, but half‐reactions are not considered, and the equilibrium values of overall reactions are used, same as for activity plots. As for E‐pH plots, boundaries will shift by varying the total amount of soluble elements in the electrolytic solution, hence the activity of dissolved ionic species or the corresponding gas fugacity (e.g. when carbonates or sulfides and sulfates are present).

Figure 1.4 shows that in logfO₂‐pH diagrams phase, boundaries that in E‐pH diagrams were dependent on pH only (and not on the concentration of dissolved ions) are horizontal. Moreover, a quick comparison between figures 2 (Eh‐pH diagram) and 4 (logfO₂‐pH diagram) also shows that on the logfO₂ vs. pH representation the pyrite–magnetite boundary, which appears only for pH > 7, maintains the positive slope. At the T conditions of Figure 1.4 this boundary is represented by the overall reaction:

(1.42)

Besides, at 145°C (water saturated conditions) the pyrite–pyrrhotine boundary is defined at pH > 7 and is positive because:

(1.43)

whereas for pH < 7 the boundary is horizontal, given by the proton‐free reaction:

(1.44)

As we have already seen, when considering high‐temperature non‐aqueous (oxide) systems in the inner Earth geospheres, there is no acid–base framework and anchoring fO₂ or E to pH makes no sense in absence of the solvent liquid water.

To measure the chemical potential of redox exchanges in higher temperature geological systems, geoscientists turned their attention to molecular oxygen transfer among molecular components such as oxides or mineral‐like macromolecular entities.

The practice between geoscientists becomes to assess criteria for fO₂ (or aO₂) estimations disconnected from the formal description of the acid–base character of magmas. In particular, techniques were established involving mineral phases coexisting in igneous rock to establish thermodynamic or empiric laws and trends from quenched glasses via indirect measurements, most often of spectroscopic nature (e.g., Neuville et al., 2020). This change of perspective reflects the obvious consideration that geoscientists deal with samples (solidified rocks) made accessible at Earth’s surface and which represent the final snapshots at the end of a long thermal and chemical evolution, whose a posteriori reconstruction is the objective of the geochemical (lato sensu) investigation.

We may then say that for practical reasons geoscientists remained anchored to the original Lavoisier‐like definition of oxidation occurring in combustion processes, related to the exchange of oxygen molecules. The fact that most of the chemical analyses were from techniques in which oxygen was not directly determined but allowed to give oxides has also further favored these approaches.

In this framework, a mutual exchange of knowledge has always characterized the field of geochemistry and petrology on one side and that of metal extraction in metallurgy in the other. Relations of the type

(1.45)

with v the charge (positive) of the cation of the metal M in the corresponding oxide. Reaction 1.45 is the main target of extractive metallurgy (see also Reaction 1.34), but also sketches the ensemble of processes that occurred since early Earth’s evolution to segregate the metallic core.

Ellingham diagrams (Ellingham 1944; Figure 1.5) are used in metallurgy to evaluate the ease of reduction of metal oxides, as well as chlorides, sulfides, and sulfates. The diagram shows the variation of the standard Gibbs free energy of formation, ΔG⁰, with temperature for selected oxides and is used to predict the equilibrium temperature for reactions of the type of Reaction 1.45 and particularly the oxygen fugacity under which ore will be reduced to its metal. The standard Gibbs energy change of formation of a compound (the Gibbs energy change when one mole of a compound is formed from elements at P = 1 bar) is given by:

(1.46)

with R the universal gas constant and A and B constants.

To compare the relative stabilities of the various oxides, the Ellingham diagram is prepared for oxidation reactions involving one mole of oxygen. For the oxidation of a metal, ΔG⁰ represents the chemical affinity of the metal for oxygen. When the magnitude of ΔG⁰ is negative, the oxide phase is stable over the metal and oxygen gas. Furthermore, the more negative the value, the more stable the oxide is. The Ellingham diagram also indicates which element will reduce which metal oxide. The similarity between the electromotive force series (E⁰) and the Ellingham diagram, which rates the tendency of metals to oxidize, should be easily recognized.

When both Me and Me^ν+_2/νO in Reaction 1.45 are in their standard states, the equilibrium constant, K₄₅, corresponding to this reaction can be expressed as:

(1.47)

where fO₂⁰ is the pure gas component gas fugacity at standard state (in this case 1 bar and T of interest). If, at any temperature, the acting oxygen fugacity is greater than the calculated value from Equation 1.47, spontaneous oxidation of metal M occurs, while oxide Me^ν+_2/νO_(s) decomposes to metal Me and gaseous oxygen at the oxygen partial pressure less than the equilibrium value. In other words, an element is unstable, and its oxide is stable at higher oxygen potentials than its ΔG_f⁰–T line on the Ellingham diagram. Therefore, the larger negative value for ΔG_f⁰ an oxide has, the more stable it is. In the Ellingham diagram of Figure 1.4, it can be seen, for example, that the reduction of Cr₂O₃ by carbon is possible (from the thermodynamic standpoint) at temperature above 1250°C and at each reported temperature by aluminum. It is worth noting that the Ellingham line for the formation of carbon monoxide (CO) has a negative slope, while those of all other oxides have positive slopes. As a result, at sufficiently high temperatures, carbon will reduce even the most stable oxides.

Figure 1.5 (a) Ellingham diagram for the main components of the melt/slag (solid lines) and possible reducing agents (dotted lines). The slopes of the lines representing ΔG_f⁰–T relations change at the temperature in which the phase transformations of reactants or products occur. Modified from Zhang et al. (2014). (b) Full Ellingham diagram for some relevant oxides including CO₂ equilibria and normographic scales for oxygen fugacity and related quantities via CO/CO₂ and H₂/H₂O ratios at P_tot = 1 bar. The scale of /_O ratio is used in the same manner as that of P_CO/P_CO2 ratio, except that the point H on the ordinate at T= 0 K is used instead of point C.

Modified from Hasegawa (2014). All reactions’ components are considered in their pure stable phase t 1 bar and T of interest.

Reaction 1.45 illustrates in fact how pairs of metals and their oxides, both having unitary activity, can be used as redox buffers, such that fO₂ values can be easily fixed at any temperature. Even in the presence of a third phase, such as silicate melts or any other liquid, gas–solid assemblages allow a straightforward application of Equation 1.47 to impose fO₂, unless solid phases are not refractory, and dissolve other components exchanges with the coexisting liquid. Reaction 1.1, involving metal iron and wustite, is a typical example (so‐called IW buffer) of one of these gas–solid equilibria fixing fO₂.

In order to illustrate the effect of the fluid phase, refined versions of the Ellingham diagrams included normographic scales, which are designed so that the equilibrium oxygen partial pressure or the corresponding P_H2/P_H2O or P_CO/P_CO2 ratios between metal and its oxide can be read off directly at a given temperature by drawing the line connecting point O for P_O2, C for P_CO/P_CO2, or H for P_H2/P_H2O in Figure 1.5 and the condition of interest (Reaction 1.45 at T of interest) and then extending it to the corresponding normographic scale.

It is now quite obvious to see how both metallurgists and petrologists could then develop techniques to constrain fO₂ to investigate melting and sub‐solidus conditions of oxides and silicates. After the seminal studies of Bowen and Schairer (1932, 1935) on FeO–SiO₂ and MgO–FeO–SiO₂ systems and in which the authors used iron crucibles in an inert (O₂‐free) atmosphere to equilibrate the phases with metallic iron at very low but also unknown P_O2, early fO₂ control techniques were applied by Darken and Gurry (1945, 1946), who detailed the Fe–O system based on the use of CO and CO₂, or CO₂ and H₂, conveyed in a gas‐mixer supplying a continuous mixture in definite constant volume proportions.

Later Eugster, in his early experiments to determine the phase relations of annite had to prevent its oxidation and formation of magnetite (Eugster 1957, 1959; Eugster and Wones 1962). He then developed the double capsule technique, with a Pt capsule containing the starting material, surrounded by a larger gold capsule. A metal–oxide or oxide–oxide pair plus H₂O was then placed between the two metal containers. In these experiments, reaction involving OH‐bearing minerals and H₂O provides a fixed and known hydrogen fugacity (Eugster 1977), so through the dissociation Reaction 1.14 measurement of fH₂ allows calculating both fO₂ given the fH₂O at the experimental pressure (aH₂O = 1)of the internal capsule.

Since Eugster, many metal–oxide and oxide–oxide assemblages have been used in experimental petrology that have been called “redox buffers.” These buffers have contributed to our understanding of the role of fO₂ on melt phase equilibria and mineral composition on Earth, also under volatile saturated conditions due to the possibility of evaluating fluid phase speciation at the experimental P and T conditions (e.g., Pichavant et al., 2007; Frost and McCammon, 2008. Mallmann and O’Neill, 2009; Feig et al., 2010). The results of experimental petrology made it possible to systematically collect glasses (quenched melts) to measure ratios of metals in their different oxidation states, particularly Fe^II/Fe^III, and relate such ratios to experimental P, T, and fO₂, glass composition, or other spectroscopic observations about glass/melt structure and the local coordination of targeted metals (Neuville, 2020 and references therein).

Obviously on Earth iron is the most abundant multivalent element and because of its speciation behaviour it gives rise to many reactions involving minerals and liquid (i.e., natural melts, particularly silicate ones). Reactions such as Reaction 1.1 are then useful for providing a scale with geological significance for fO₂ conditions that are recorded in rock and minerals formed in past and present Earth environments, from the core (in which Fe⁰ dominates) up to shallow crust and through all igneous environments where it exists, and Fe^II and Fe^III. To provide systematics for the aO₂ conditions the following gas–solid reactions, other than Reaction 1.1, were then assessed (Figure 1.6):

(1.48)

(1.49)

(1.50)

(1.51)

All these equilibria have the interesting feature of displaying unitary activities for oxide component appearing as pure phases, such that their equilibrium constants simply describe the variations of O₂ activity (aO₂) with temperature:

(1.52)

or, by defining fugacity, with temperature and pressure:

(1.53)

with A, B, and C constants. The pressure term, C, allows computing directly fO₂ rather than aO₂ at the pressure of interest. Mineral assemblages making up Reactions 1.1 and 1.44 to 1.47 are not necessarily occurring in deep Earth and igneous environments, whereas the actual multi‐component space of phases fix the aO₂ via multiple equilibria in which solid solutions and/or the presence of relatively mobile liquid (silicate melts) and/or supercritical fluids play a fundamental role.

Figure 1.6 Common solid oxygen buffers used in petrology and geochemistry. The lines represent the fugacity‐temperature conditions where the phases coexist stably. FMQ: fayalite, magnetite, quartz; IW: iron–wüstite; WM: wüstite–magnetite; HM magnetite–hematite. Oxygen fugacity values were computed for a total pressure of 1 bar. Also reported is the value of logfO₂ in air (PO₂ ~0.21 bar).

However, when one of these mineral “buffers” is selected as a reference, the acting logfO₂ can be given as a relative value, without temperature:

(1.54)

A relative fO₂ scale embodying temperature effects bears just a practical implication (tracking fO₂ variations with respect to a reference) but not a real meaning about logfO₂ evolution in igneous systems and related environments (e.g., Moretti and Steffansson, 2020). In particular, a common misconception was that in a system a given value of Δbuffer (e.g., ΔQFM = 0.5) could represent some kind of “magic number” characteristic of the whole “rock system” throughout its thermal and chemical evolution. In natural environments oxygen activity (hence fugacity) in fact varies to accommodate the compositional variations and the speciation state of the mineral/melt/fluid phases, and also when highly mobile volatile components are involved, such that fO₂ can thus be fixed by factors that are external to the system object of the thermodynamic description.

Similarly, the rock system evolution cannot be approximated by a unique Fe^II/Fe^III ratio, that the system had when completely molten. Indeed, two rocks that have ideally crystallized along the QFM buffer may have different proportions of fayalite ad magnetite because of the crystallization style but different Fe^II/Fe^III bulk ratios (Frost, 1991). However, in some systems, it is possible that the melt fixed the redox potential of the system via iron oxidation state, with the Fe^II/Fe^III ratio approaching unity (e.g. Moretti et al., 2013).

It is also important to recall that fO₂ is just a thermodynamic parameter used to conveniently report the oxidation state of a system, particularly when O₂ is not an existing gaseous species that could be detected if the system were accessible to measurements. This is clearly proved by the very low values reported in ordinates in Figure 1.6. Therefore, fO₂ turns out to be by‐product of thermodynamic calculations applied to the analyses from natural samples, in which the true redox observables are the oxidation states of iron and other elements in minerals and liquids. The common practice is then to measure the concentration ratio of redox couples of multiple valence elements in melts (Fe^II/Fe^III, but also S^‐II/S^VI, V^III/V^V, etc.) or in gases (e.g., H₂/H₂O, CO/CO₂, H₂S/SO₂) and relate them to fO₂ via thermodynamic calculations using appropriate standard state thermochemical data. As fO₂ is provided, its value is then anchored via Equation 1.54 to a given gas‐solid buffer of the Reaction type 1.1 or 1.48 to 1.51. This is quite easy for gases, in which governing equilibria are directly solved if the gas analysis is provided (e.g., Giggenbach, 1980, 1987; Aiuppa et al., 2011 and references therein) and also for solid–solid equilibria, such as in case of coexisting iron–titanium oxide solid solutions titanomagnetite (Fe₃O₄‐Fe₂TiO₄) and hemo‐ilmenite (Fe₂O₃–FeTiO₃) (Buddington and Lindsley, 1964) or for peridotite assemblages in the mantle (e.g., Mattioli and Wood, 1988; Gudmundsson and Wood, 1995), in which the good thermodynamic characterization of solid solutions allows quite accurately treating component activities, based on mineral analyses.

On the contrary, it is much less straightforward for fO₂ estimates by oxidation states of Fe and/or S measured in glasses, as the oxybarometers derived by the study of synthetic quenched melts still suffer from too many empiric approaches. Because of their polymerized nature, silicate melts do not allow a precise distinction between solute and solvent like in aqueous solutions, where complexes and solvation shells can be easily defined in which covalence forces exhaust (see also Moretti et al., 2014). In fact, melt composition largely affects the ligand constitution and then the speciation state of redox‐sensitive elements. In the case of the Fe^II/Fe^III ratio, the most common redox indicator for melts/glasses, the choice of components in reaction:

(1.55)

over a large compositional range (e.g., from mafic to silicic) does not offer the possibility to find accurate and internally consistent expressions for the activity coefficients of oxide components γFeO and γFeO_1.5 (with FeO_1.5 conveniently replacing Fe₂O₃) that solve the reaction equilibrium constant:

(1.56)

in which the term within integral is the difference of partial molar volumes of iron components. Expansion of excess contribution to its Gibbs free energy of mixing is used to define γFeO and γFeO_1.5 in melt mixtures. However, when these are adopted to solve Equation 1.56 for measured Fe^II/Fe^III values, they show success only over limited compositional datasets (Moretti, 2020 and references therein). Armstrong et al. (2019) calibrated volumes and interaction parameters of activity coefficients entering equation (1.56) for an andesitic melt (easily quenchable as a glass) with fO₂ buffered by the Ru‐Ru₂O assemblage in the T‐range 1673K to 2473K and for pressures up 23 GPa. Data fitting showed that volume term in Equation 1.56 turns from negative to positive for P > 10 GPa, which yielding iron oxidation with increasing pressure. The calibrated Equation 1.56 was then used by the authors to demonstrate how the mantle oxidized after the Earth’s core started to form by a deep magma ocean with initial Fe^III/Fe_tot = 0.04 from which FeO disproportionated to Fe₂O₃ plus metallic iron at high temperature. The separation of Fe⁰ to the core raised the oxidation state of the upper mantle and of exsolved gases that were forming the atmosphere (Armstrong et al., 2019).

The search for one general formulation for all melt compositions of interest in petrology and geochemistry led to empirical expressions, in which adjustable parameters are introduced without the formal rigor requested by Equation 1.56 (e.g. Kress and Carmichael, 1991). These formulations furnish quite accurate fO₂ values from measured Fe^II/Fe^III ratios within the compositional domain in which they have been calibrated. Besides, they often violate reaction stoichiometry and do not ensure internal consistency: if used to calculate activities they fail the application of the Gibbs‐Duhem principle relating all component activities within the same phase (e.g., Lewis and Randall, 1961). Such expressions then treat fO₂ as a Maxwell’s demon, doing what we need it to do to fit the calibration data and with the consequence that outside the calibration domain, all the unpredictable non‐idealities are discharged on the fO₂ terms, resulting in biased calculations of fluid speciation, or other phase equilibria constraints.

As extensively treated in Moretti (2020), unpredictable non‐idealities reflect counterintuitive behaviors that cannot be accounted for by activity coefficients used in the equilibrium constant of Reaction 1.55. It is well known that depending on melt composition alkali addition (i.e. decreasing pO^2–) can either oxidize or reduce iron in the melt. This occurs because of the change of speciation due to the amphoteric behavior of Fe^III, which depending on composition and then pO^2– can behave as either network former or modifier (see the conceptualization provided by Ottonello et al., 2001; Moretti, 2005, 2020; Le Losq et al., 2020; see also Reaction 1.41). Models that define the melt (oxo)acidity (Reaction 1.27) hence pO^2– (e.g; polymeric models based on the Toop and Samis mixing of bridging, non‐bridging and free oxygens; see Moretti, 2020, and references therein) allows solving speciation and set activity‐composition relations of ionic and molecular species, just as for aqueous solutions and molten salts.

The problem of unsolved compositional behaviors due to speciation, that are not accounted for by typical oxide‐based approaches to mixtures, is exacerbated when dealing with the mutual exchanges involving iron and another redox‐sensitive elements, such as sulfur. Sulfur‐bearing melt species play a special role since the oxidation of sulfide to sulfate involves eight electrons: for any increment of the Fe^III/Fe^II redox ratio, there is an eight‐fold increment for sulfur species (S^–II/S^VI; e.g., Moretti and Ottonello, 2003; Nash et al., 2019; Cicconi et al., 2020b; Moretti and Stefansson, 2020). Sulfur in magmas partitions between different phases (gas, solids such as pyrrhotite and anhydrite, and liquid as well, such as immiscible Fe–O–S liquids; Baker and Moretti, 2011 and references therein). The large electron transfer makes S^–II/S^VI a highly sensitive indicator to fO₂ changes in a narrow range (typically around QFM and NNO buffers in magmatic melts; Moretti, 2020 and references therein), whereas its effectiveness as a buffer of the redox potential is limited by the abundance of sulfur in magma, significantly lower than iron.

The modelling of joint Fe and S redox exchanges is still a major challenge which sees contrasting approaches (see Moretti, 2020). Formulations exist with various degrees of empiricism, but even those displaying better performances in exploring the fO₂ – fS₂ space of natural silicate melts (Moretti and Baker, 2008) should be carefully tested in reproducing phase diagrams involving multiple phases, including coexisting Fe–O–S melt, FeS_(s), and Fe₃O_4(s). Introduction of sulfur equilibria in petrogenetic grids would be a major step forward for modelling in igneous petrology. Besides, it would provide the liaison with processes occurring in late‐ to post‐magmatic stages, prior to further cooling down to real hydrothermal conditions dominated by condensed water (see Figure 1.4). For late‐ to post‐magmatic stages, such as in the case of porphyry‐copper ore formation, logfO₂‐logfS₂ diagrams. Figure 7 reliably summarizes phase relations in the Fe–S–O system, in a way similar to Figure 1.2 and 1.4. It is worth noting that the diagram in Figure 1.7 can also be seen as resulting from reactivity of a sub‐solidus mixed iron molten oxide–sulfide, in which the two main ligands are O^2– and S^2– (half‐reactions 1.6, 1.11, and 1.12).

Figure 1.7 Two‐redox potential fO₂‐fS₂ diagram. The conformation of stability fields in the Fe–O–S space is essentially the same also for large fO₂ and fS₂ variations with temperature

(redrawn from Nadoll et al., 2011).

A natural assemblage of pyrite + magnetite + pyrrhotite corresponds then to the triple point marked by a star in Figure 1.7, which at a given T is invariant for fO₂ and fS₂ values given by the simultaneous occurrence of Reaction 1.10 and:

(1.57)

that allow identifying the stable phase as a function of temperature and fugacities (or activities) of reference gas species. It is worth noting that in absence of water (no H in the system represented in Figure 1.7) the boundary between FeS₂ and FeS is a function of fS₂ only (see Reaction 1.10) but not of fO₂, as instead reported in Figure 1.4.