Redox systematics of a magma ocean with variable pressure-temperature gradients and composition

Abstract

Oxygen fugacity in metal-bearing systems controls some fundamental aspects of the geochemistry of the early Earth, such as the FeO and siderophile trace element content of the mantle, volatile species that influence atmospheric composition, and conditions for organic compounds synthesis. Redox and metal-silicate equilibria in the early Earth are sensitive to oxygen fugacity (fO₂), yet are poorly constrained in modeling and experimentation. High pressure and temperature experimentation and modeling in metal-silicate systems usually employs an approximation approach for estimating fO₂ that is based on the ratio of Fe and FeO [called “ΔIW (ratio)” hereafter]. We present a new approach that utilizes free energy and activity modeling of the equilibrium: Fe + SiO₂ + O₂ = Fe₂SiO₄ to calculate absolute fO₂ and relative to the iron-wüstite (IW) buffer at pressure and temperature [ΔIW (P,T)]. This equilibrium is considered across a wide range of pressures and temperatures, including up to the liquidus temperature of peridotite (4,000 K at 50 GPa). Application of ΔIW (ratio) to metal-silicate experiments can be three or four orders of magnitude different from ΔIW (P,T) values calculated using free energy and activity modeling. We will also use this approach to consider the variation in oxygen fugacity in a magma ocean scenario for various thermal structures for the early Earth: hot liquidus gradient, 100 °C below the liquidus, hot and cool adiabatic gradients, and a cool subsolidus adiabat. The results are used to assess the effect of increasing P and T, changing silicate composition during accretion, and related to current models for accretion and core formation in the Earth. The fO₂ in a deep magma ocean scenario may become lower relative to the IW buffer at hotter and deeper conditions, which could include metal entrainment scenarios. Therefore, fO₂ may evolve from high to low fO₂ during Earth (and other differentiated bodies) accretion. Any modeling of core formation and metal-silicate equilibrium should take these effects into account.

The terrestrial planets are thought to have undergone a hot early thermal history in which much of their mantles were molten (1 and 2). During accretion of the planets, conditions were so reducing that metal was stable, and typically oxygen fugacity (fO₂) was several log fO₂ units below the iron-wustite buffer (IW).

Knowing fO₂ helps understand at least four major aspects of the geochemistry of the early Earth. First, fO₂ has a strong control on the conditions of core formation and metal-silicate equilibrium. The fO₂ has a strong effect on the partitioning of siderophile elements that depends upon the valence of the element in the silicate melt (SI Appendix, Fig. S1). Because the partition coefficients depend on fO₂—a 2+ cation will have a log D dependence of 0.5 and a 4+ cation will have a logD dependence of 1—the effect on D can be large. Some recent models have argued that the siderophile element concentrations of the Earth’s primitive upper mantle (PUM) can only be explained if the Earth was very reduced at the beginning of accretion (IW-4), but then became more oxidized in the later stages of accretion (IW-2) (3, 4). On the other hand, some models for Earth accretion can explain as many as 12 siderophile elements by metal-silicate equilibrium at high pressure and temperature and a fixed relative fO₂ near IW-1 or IW-1.5 (5). The differences in these two approaches reflect the different way of calculating fO₂ for the siderophile element partitioning expressions; the former approach uses a relative fO₂ approximation, whereas the latter approach uses absolute fO₂. Clearly, the way fO₂ is calculated at high temperatures and pressures has an influence on the outcome of partitioning and thermodynamics models. Second, during Earth’s earliest history, the molten mantle (magma ocean) may have extended to depths near 1,000 km, with convective overturn providing an efficient stirring mechanism for mobilization of interior volatiles. Therefore the deeper parts of the silicate mantle may have had a profound effect on the chemical composition and density of the Earth’s early atmosphere and in particular whether the atmosphere was reduced or oxidized (6). Third, there is some debate about whether Earth’s early biochemical building blocks were reduced species such as methane and ammonia or oxidized species such as CO₂ and H₂O (7). The stability of these two extreme end members cases is ultimately related to the conditions of oxygen fugacity established near the surface but in the magma ocean stage of the Earth. Understanding the baseline of fO₂ established in Earth’s early mantle is critical to the subsequent evolution of conditions suitable for complex biochemical syntheses. And fourth, in addition to simple sources of hydrocarbons required for complex organic molecule formation, biochemical equilibria require many trace metals for various processes or energetic cycles. For example, Mn, W, and Mo are all involved in the reactions near hydrothermal vents and hyperthermophile bacteria (8, 9), P is an essential part of ATP and ADP (10), Cu is in the equivalent of hemoglobin (called hemocyanin) for cuttlefish blood (11), V is an integral part of the cellular functions of tunicates (sea squirts) (12), Mn has a critical role in the Krebs cycle and photosynthesis (13), Cr is an efficient participant in Fischer-Tropsch synthesis (14), and Ni and Co are known catalysts in many biochemical processes (15). An understanding of how these elements were established in the early mantle and crust thus has direct relevance to the origin and development of life and biochemical systems, as well as constraining the nature of early differentiation.

Experiments at high pressures and temperatures are typically solid media experiments that do not have the flexibility of fixing oxygen fugacity with a gas mixture or even a solid oxygen buffer. However, many elemental solubilities and partitioning equilibria are dependent upon fO₂. Although fO₂ can be circumvented in some cases by consideration of Fe-M distribution coefficient (where M is a siderophile trace element such as Ni or Co), these do not eliminate the effects of alloy or silicate liquid compositional variation or the specific chemical effects of S in the silicate liquid, for example. Calculation of oxygen fugacity in high pressure and temperature experiments in metal-silicate systems is usually approximated relative to the IW buffer by considering the ratio of Fe in the metal and FeO in the silicate melt [e.g., ref. 16; called here “ΔIW (ratio)”—see below]. The relative fO₂ approach (ΔIW) was originally developed by ref. 17 and supposed to be temperature independent but only across 300 °C; it is not likely to be valid across thousands of degrees such as that considered in core formation modeling. Because ΔIW (ratio) is a quick and easy calculation to make, it has been applied to a huge variety of metallic (Fe-Ni-S-C-O-Si systems) and silicate liquids (SiO₂, Al₂O₃, TiO₂, FeO, MgO, CaO, Na₂O, K₂O systems). This approach has surely led to values that have little meaning, yet are applied with great confidence. For example, it is this approach that has led to many researchers proposing that the PUM is at “IW-2.4” (e.g., ref. 18). This approach also leads to the conclusion that an experiment at 1 GPa and 1,800 °C and one at 25 GPa, 2,400 °C, for example, with the same metal and silicate compositions, will have the same relative fO₂! This is unlikely because it assumes the volumetric properties of both the metallic and silicate liquids are the same across this large pressure and temperature (PT) range.

The goal of this work is twofold: first, to determine the actual value of fO₂ in any given metal-silicate experiment, and second to model the evolution of fO₂ in an accreting body. We utilize equations of state (EOS) and solution models for metals and silicate melts, and new data for the IW reference buffer at high pressure and temperature. We have calculated fO₂ for many experiments reported in the literature, using the equilibrium 2Fe(metal) + SiO₂(liq) + O₂ = Fe₂SiO₄(liq). With these calculations we demonstrate the differences between the ΔIW (ratio) approximation and the calculation of absolute fO₂ (ΔIW (P,T), and discuss the implications for models of core formation and differentiation.

Model

Relative fO₂: The Approximation Approach [ΔIW (ratio)].

Calculation of oxygen fugacity in high pressure and temperature experiments in metal-silicate systems is usually approximated by the ratio of Fe in the metal and FeO in the silicate melt:

[1]

According to the equilibrium 2Fe + O₂ = 2FeO, and where IW is the iron-wüstite reference oxygen buffer, XFe is the mole fraction of Fe in the metal, and XFeO is the mole fraction of FeO in the silicate melt. As one can see this approach contains no information about pressure or temperature.

Absolute fO₂: High PT Activity Models and EOS Approach [ΔIW (P,T)].

The simple equilibrium 2Fe + O₂ = 2FeO might initially seem like the best way to constrain fO₂, but because the solution model of ref. 19 does not have an FeO component we instead have chosen the equilibrium involving Fe₂SiO₄ and SiO₂:

[2]

which can be applied to any metal-silicate pairs that contain Fe metal (or FeNiS system) and FeO-bearing silicate melt. fO₂ can be calculated from the Gibbs free energy of equilibrium 2 and activities of Fe₂SiO₄, SiO₂, and Fe in the silicate and metal, using the relation:

[3]

The activities and Gibbs free energy were calculated for each experiment discussed herein, as described below, and the value of fO₂ at the IW buffer at high pressure and temperature was calculated as described below.

Modeling this equilibria at high PT conditions requires satisfactory activity-composition relations and thermodynamic or EOS data.

Activities.

We utilize activities of Fe in the FeNi alloy using the thermodynamic adapted power series concept for calculating activities for the alloy (20, 21), and Fe in the Fe-Ni-S-O system using an associated regular solution model (22). Activities of SiO₂ and Fe₂SiO₄ in the silicate melt are calculated at reference pressure (19). In utilizing the latter model we have assumed that all Fe in silicate melt was FeO, which is a reasonable assumption because at IW 99% of total Fe should be Fe²⁺ (23), and the experiments being considered are all at fO₂ lower than IW. The models of refs. 20 and 21 are internally consistent with the silicate liquid model of ref. 19. The latter has been calibrated against olivine-liquid and pyroxene-liquid equilibria over a broad range of composition and temperature, which demonstrates that the model accurately predicts both the activity of SiO₂ and Fe₂SiO₄ in multicomponent silicate liquids.

Equations of state.

The EOS for FeNi alloy is based on X-ray diffraction data from diamond anvil experiments up to 30 GPa and fit to the Murnaghan equation (24). This expression is calibrated for FeNi alloys up to 10% Ni. We have used the FeNi alloy as a proxy for the FeNiS system, utilizing the 1 bar volume data for the liquid alloy Fe end member. The EOS for silicate melt up to 40 GPa is used and is based on a fourth order Taylor expansion of the volume about the reference pressure (25–28). The model has been tested against available sink/float density data, shock compression data (fayalite), mineral fusion curves (silica polymorphs), and molecular dynamics simulations and is the most comprehensive formalism for multicomponent silicate liquids at high pressure. The model of refs. 25–28 is adopted rather than the internally consistent formulation in ref. 19 because the latter is only applicable to pressures up to 3 GPa.

Subsolidus equilibria.

Ref. 29 presents a thermodynamic database for the system Fe-Mg-Si-O across a wide range of pressures, including the wadsleyite and ringwoodite polymorphs of Fe₂SiO₄. These thermodynamic data are used for Fe metal, O₂, SiO₂, and Fe₂SiO₄ to calculate fO₂ at subsolidus conditions.

Reference buffers.

Volumes of Fe, FeO, Ni, and NiO have been recently measured (30) and are used to calculate the position of the IW and nickel–nickel oxide buffers as high PT reference points. The Campbell et al. calculations (30) are based on measurements between 3 and 90 GPa using in situ X-ray techniques with multianvil and diamond anvil assemblies. Precision is the best available because the measurements were acquired on both metal and oxide phases simultaneously.

Online version available.

Researchers wishing to use this log fO₂ calculation for their own work can access it at ctserver.ofm-research.org/KR_logfo2/. We caution the users to only include experiments that fall within the FeNiS system, FeO-bearing multicomponent silicate melts that are representative of terrestrial mantle compositions, and up to 50 GPa and 4,000 K.

Results and Discussion

Calculations for Specific Experiments.

Calculations have been carried out for a few specific experiments from the literature chosen to represent certain extremes of composition or PT conditions (SI Appendix, Table S1). For example, high T experiments on peridotitic silicate melts from refs. 31 and 32 are chosen to investigate the effect of very high T (2,873–3,000 K at 25–26 GPa). Several high PT experiments from refs. 33 and 34 were chosen for additional comparisons. In general, the calculated values of ΔIW(P,T) are 1 to 2 log fO₂ units higher than those obtained using the ΔIW (ratio) approximation (Fig. 1). Experiments with sulfur-rich metallic liquids (35, 36) were chosen at very low and very high pressures (0.8 vs. 25 GPa), respectively. The difference between ΔIW(ratio) and ΔIW(P,T) is larger for both experiments than all of the sulfur-free experiments considered above. In addition, the low pressure experiment with S-bearing metallic liquid yields a calculated ΔIW (P,T) value nearly 5 log fO₂ units higher than the ΔIW(ratio) value (Fig. 1). This difference is due primarily to lowered Fe activity in the metal phase but also due to the volume integral, as discussed below.

Fig. 1.

Calculated ΔIW (P,T) (solid circles) compared to ΔIW (ratio) (open circles). Individual experiments are from the following references: experiments of [ PM93 (35) and LA01 (36)] involve FeNiS melts, the high PT experiments of [WW05 (32)], utilize MgO-rich melts and FeNi metal, and several other experiments at higher temperatures are from [I98 (31)], [WW01 (33)], or low pressures, [S88 (34)]. Note that all examples yield higher ΔIW (P,T) values than ΔIW (ratio).

The cause of the differences between ΔIW (ratio) and ΔIW (P,T) calculations is the volume integral for FeO liquid [i.e., (Fe₂SiO₄ - SiO₂)/2] relative to FeO solid. The volume of FeO (liquid) is much larger than the volume of FeO solid, so relative to the IW buffer, the integral of that volume difference is what drives the increase in fO₂. These results suggest that calculated relative fO₂ from the ratio approximation, ΔIW (ratio), could be too low. As a result ΔIW (P,T) should be considered in detailed modeling efforts for a complete understanding of polybaric metal-silicate equilibria.

Calculations for the Early Earth: Magma Ocean Isentropes, Liquidus, Subliquidus, and Subsolidus Gradients.

The dominant locations and mechanisms for equilibration between metal and silicate in a magma ocean are at the base of the magma ocean, or between small metallic liquid droplets and silicate melt (37, 38). The PT conditions at the base of a magma ocean will be determined by phase equilibria of peridotite, and the locations of the liquidus and solidus. However, because settling velocities of small iron droplets (approximately 0.5 ms^-1) are much lower than typical convection velocities (approximately 10 ms^-1), iron droplets may remain entrained in convective cells for a significant period of time in the magma ocean. Accumulation through sedimentation at the base of the ocean will be a slow and gradual process, compared to the timescale required for chemical equilibration (e.g., refs. 39 and 40). Furthermore, ref. 41 suggest that droplets settle out in approximately the Stokes’ settling time but show that strong density currents develop in the droplet bearing region of the magma ocean due to density perturbations. The velocities of these density currents range up to 50 ms^-1, which is much greater than thermal convection velocities. In addition, the gravitational energy of sinking droplets is converted into heat that raises the temperature at the base of the magma ocean by at least several hundred degrees. These observations all indicate that the temperature at the metal-silicate interface may be above the liquidus, and along an isentropic or adiabatic gradient within a convective cell (Fig. 2).

Fig. 2.

Entrainment of metallic droplets in a magma ocean setting may occur either along the base of the ocean (A), or along the boundary between a falling metallic body and silicate melt (B) (39 and 40). In a magma ocean setting, this interaction and equilibration between metal and silicate melt may take place at temperatures above the liquidus, because entrainment would likely occur along an isentrope (shaded circular area on right side of diagram).

The thermal conditions for the early Earth are a topic of active research, with the liquidus and solidus of the primitive mantle being studied by several groups (42–44). At pressures between 3 and 25 GPa, the multianvil experiments of (42, 45–47) constrain the liquidus and solidus location for primitive mantle composition such as KLB-1 peridotite. At higher pressures there is more discrepancy: ref. 43 propose a liquidus that reaches 4,000 K at 50 GPa, with a wide melting interval of close to 1,000 K between the liquidus and the solidus, whereas ref. 44 proposes a liquidus that reaches 3,200 K at 50 GPa, with a narrower melting interval of 400 K. Because of the large difference in liquidus location between these two studies we will present the results of calculations using both constraints to illustrate the differences. For the calculations below, we will consider (a) a magma ocean gradient based on the ref. 43 phase diagram, (b) a magma ocean gradient based on the cooler ref. 44 phase diagram, (c) a liquidus based on the ref. 43 phase diagram with a subliquidus gradient that is approximately 100 K below the liquidus (composition in SI Appendix, Table S2), and (d) a subsolidus gradient that represents the period of time after the magma ocean has cooled (Fig. 3). In addition, we will consider the effects of silicate compositional changes, with basalt, primitive peridotite, and a mantle composition that changes from reduced (0.5 wt% FeO) to oxidized (9.0 wt% FeO) as the Earth grows (or as PT conditions increase to 50 to 60 GPa; SI Appendix, Table S2).

Fig. 3.

Summary of phase relations for peridotite at high pressure and temperature conditions. Phase relations at pressures lower than 30 GPa are taken from (42, 45, 46). Higher pressure phase relations are from (43, 44). Also shown are the various PT gradients used in the modeling in this paper.

Basalt vs. peridotite.

To illustrate the potential differences between basaltic and peridotite melts, we can calculate fO₂ for two melt compositions along the 2,600 K adiabat of ref. 44 (11 K/GPa gradient) between 1 bar and 50 GPa (SI Appendix, Table S3). The difference between ΔIW (ratio) and ΔIW (P,T) becomes larger at high pressures (> 25 GPa) for the basalt and peridotite (Fig. 4). The peridotite, which would have a ΔIW (ratio) value of -2.4, starts with a Δ IW (P,T) = -1.52 at 1 bar and becomes as reduced as ΔIW = -5.53 at 50 GPa (a 4.01 log unit shift; Fig. 4). Similarly, the basalt has a value of ΔIW(ratio) = -1.9 regardless of PT conditions, whereas the ΔIW (P,T) = -1.28 at 1 bar, and -2.18 at 50 GPa, which is an overall shift of approximately 0.9 log fO₂ units with pressure. The ΔIW (P,T) of basalt and peridotite stay similar up to approximately 15 GPa, but then the peridotite becomes more reduced at high pressures, presumably due to effects of alkalis in the basaltic liquid stabilizing the presence of ferric iron (23). In the peridotite, a relative paucity of alkalis and the larger intrinsic volume of Fe₂O₃ (and presumed higher compressibility as inferred from 1-bar measurements) converts oxidized iron to its reduced form and lowers the intrinsic fO₂.

Fig. 4.

A comparison of ΔIW(P.T) calculated for basalt and peridotite (SI Appendix, Table S3) along relatively cool adiabat based the Andrault et al. phase diagram (44). Note that both compositions start slightly oxidized, and then become more reduced at high pressures. The largest error contribution to log fO₂ is from the FeO volume as a function of pressure: Error in FeO at 1 bar is approximately 0.5%, but the error is approximately 3% at 10 GPa, which corresponds to 0.042 J/ bar or 0.1 log ₁₀ units at 2,000 K and 10 GPa. The error at 50 GPa is approximately 0.5 log ₁₀fO₂ units. The error contribution from melt and metal activities for these compositions is insignificant relative to the volume integral. ΔIW (ratio) values for these two compositions are -1.9 (basalt) and -2.4 (peridotite).

Peridotite with different PT gradients.

If we focus on peridotite and consider different gradients, it becomes clear that fO₂ does not change considerably along various gradients. For example, use of the hotter ref. 43 liquidus PT gradient results in progressively more reduced conditions with ΔIW (P,T) value of -6.0 at 50 GPa (Fig. 5). This reduction is not specific to the peridotite composition or to the liquidus gradient—it is also produced for a subliquidus composition and along a hot isentrope (Fig. 6). This reduction at higher temperatures is likely due to the dominant role of thermal expansion (dV/dT) over compression (dV/dT) as pressure increases. Liquids become less compressible at high pressures, the effect of higher temperatures dominates, causing reduction.

Fig. 5.

ΔIW (P,T) calculated for peridotite using various PT gradients. Also included are the results for a subsolidus scenario. The hottest gradient produces very low ΔIW (P,T) values at high pressures.

Fig. 6.

ΔIW (P,T) calculated for scenarios in which the mantle composition starts with low FeO content (reduced), and then changes during accretion to high FeO content (oxidized). Two PT gradients are considered—the hot adiabat based on the Fiquet et al. phase diagram (43), and the relatively cool adiabat based on the Andrault et al. phase diagram (44). Note the reducing and oxidizing results caused by these scenarios, respectively.

Oxygen fugacity was also calculated for the subsolidus equilibrium:

[4]

using the thermodynamic database of ref. 29 and using the polymorphs of Fe₂SiO₄. The results show that the mantle could be as reduced at IW-3 to IW-5 at these conditions, similar to the results for equilibria involving liquid silicates (Fig. 5).

Reduced to oxidized with various gradients.

Some accretion models (3, 4, 48) allow fO₂ to evolve, or really the mantle FeO content to increase, during accretion. We can model the effect of changing magma ocean composition during accretion to test if these models actually do predict oxidation. For example, as FeO is added to a peridotite, and pressure and temperature are held fixed at 20 GPa and 2,500 °C, ΔIW(P,T) calculated using our approach varies from -3 to -1.8 (SI Appendix, Fig. S2). Perhaps a more realistic scenario, however, is where Earth is growing (i.e., P and T are gradually increasing) while FeO increases. In such a scenario, the two thermal PT gradients (43, 44) yield similar results up to 50 GPa (Fig. 6). Again, the dominance of the thermal effects in a high pressure magma ocean is responsible for this change. Differences between hot or cool magma ocean adiabats are very minor, as all calculations lead to values of ΔIW (P,T) of approximately -5.5 to -6.0 at 50 GPa.

Implications for Accretion Models and Early Differentiation.

These results offer new insights into several significant problems in planetary geochemistry and accretion, core-mantle partitioning, and the secular evolution of Earth’s oxidized mantle.

Evolution of fO₂ during accretion of the Earth.

Some modeling approaches have used fixed fO₂ during accretion of the Earth (e.g., refs. 5 and 49). On the other hand, variable fO₂ is necessary in some siderophile element modeling (3, 4), but it is not clear if such oxidation is possible or how it happens. The heterogeneous accretion models typically assume a starting relative fO₂ of IW-4 and a final relative fO₂ of IW-2 (e.g., ref. 3; or from -3.5 to -2 in model of ref. 4). But the results presented here—that even a mantle changing composition from low to high FeO as Earth grew becomes increasingly reduced at depth—indicate that fO₂ variation might be significant. The change in relative fO₂ in previous models (3, 4) is the opposite of the change we calculate for a growing magma ocean that typically results in reduction to values close to IW-6. Although the manner in which oxygen fugacity is calculated varies from study to study, it is clear that significant reduction at high PT conditions must be considered in future modeling.

Evolution of planetary mantle fO₂.

Pressure-dependent metal-silicate equilibria exacerbate the problem and long-puzzling observation that the FeO-poor (and hence “reduced”) terrestrial mantle is actually the most oxidized, compared to that estimated for Mars (equilibrated with metal at approximately 14 GPa; (50), to the much lower pressure Moon [4.5 GPa; (51)] and asteroid 4 Vesta [0.1 MPa; (52, 53)].

A long standing conundrum has been that Earth’s core formed under reduced conditions (ΔIW = -2), whereas the oldest record of mantle oxygen fugacity indicates no significant difference from the current mantle (close to fayalite-magnetite-quartz buffer; e.g., refs. 54–56). These observations have suggested either that the mantle was oxidized early (32, 57), or perhaps gradually between 4.5 and 3.8 Ga due to plate tectonics and recycling (e.g., refs. 58 and 59). Our results indicate that the terrestrial mantle may have been even more reduced during the accretion and magma ocean stage. The role of the Mg-perovskite pump oxidation mechanism (e.g., refs. 32 and 57) would have to be even greater if simple metal-silicate equilibria produced early mantle reduction.

Conclusions

As a result of our calculations the following conclusions can be drawn:

• The ratio approach to calculating ΔIW underestimates this value for metal-silicate experiments from the literature.

• Entrainment of metallic droplets during the early metal-silicate equilibria can lead to reduction due to the effects of high temperature. Superliquidus conditions of several hundred degrees and near-liquidus conditions can lead to significant reduction in the pressure range between 20 and 50 GPa.

• Increasing the FeO content of Earth’s mantle during accretion does not lead to oxidation, but rather to significant reduction relative to the IW buffer. This is the opposite of what is assumed in many accretion models.

• Magma oceans that have active zones of metal-silicate equilibration can be significantly zoned in oxygen fugacity with most scenarios indicating shallow oxidized portions and deep reduced portions. None of these become more oxidized than the IW buffer.