Oxygen fugacity

CALLIOPE parameterises the redox state of the magma ocean through the oxygen fugacity \(f_{\mathrm{O}_2}\) at the surface, expressed in \(\log_{10}\) units relative to the iron-wüstite (IW) mineral buffer:

\[ \Delta\mathrm{IW} \equiv \log_{10} f_{\mathrm{O}_2} - \log_{10} f_{\mathrm{O}_2}^\mathrm{IW}(T). \]

Under the buffered-mode entry point equilibrium_atmosphere, the user supplies fO2_shift_IW = \(\Delta\mathrm{IW}\) as a scalar input and CALLIOPE computes the absolute \(\log_{10} f_{\mathrm{O}_2}\) at \(T = T_\mathrm{magma}\) by adding the buffer value to the shift. Under the authoritative-oxygen mode, \(\Delta\mathrm{IW}\) is instead a solver unknown that closes the system against a user-supplied total oxygen mass. Both modes share the parameterisations, conventions, and chemistry channels documented on this page; they differ only in whether \(\Delta\mathrm{IW}\) is an input or an output.

The IW mineral buffer

Iron in equilibrium with wüstite (FeO) in the presence of O\(_2\) obeys

\[ \mathrm{Fe} + \tfrac{1}{2}\,\mathrm{O_2} \rightleftharpoons \mathrm{FeO}, \]

which fixes a single curve \(\log_{10} f_{\mathrm{O}_2}^\mathrm{IW}(T)\) in \(T\)-\(f_{\mathrm{O}_2}\) space. CALLIOPE supports two parameterisations of this curve.

Fischer et al. (2011), `fischer` (default) ⁵

A simpler two-parameter fit of the 1-bar IW buffer. The fit reproduces the 1-bar curve in Fischer et al. (2011) ⁵ Fig. 6, which itself derives from Chase (1998) ³ NIST-JANAF tabulation. Fischer's own high-pressure measurements (\(\le\)200 GPa) extend the buffer to deep-mantle conditions but are not used by CALLIOPE.

\[ \log_{10} f_{\mathrm{O}_2}^\mathrm{IW}(T) = 6.94059 - \frac{28180.8}{T}. \]

Implemented as OxygenFugacity.fischer(T).

O'Neill & Eggins (2002), `oneill` (legacy) ¹⁰

A thermochemically-constrained fit derived from low-temperature equilibrium data, expressed as Bower et al. (2022) ¹ Equation (7):

\[ \log_{10} f_{\mathrm{O}_2}^\mathrm{IW}(T) = \frac{2\left[-244118 + 115.559\,T - 8.474\,T \ln T\right]}{\ln(10)\, R\, T}, \]

with \(R = 8.31441\) J K\(^{-1}\) mol\(^{-1}\). Bower et al. (2022) ¹ adopted this as the "IW buffer to which \(f_{\mathrm{O}_2}\) is referenced". CALLIOPE retains it under the model name oneill (function OxygenFugacity.oneill(T) in oxygen_fugacity.py); keep it as the choice when you need to reproduce results from the older literature line.

Choice of buffer

The two parameterisations cross near \(T \approx 1800\) K and diverge in opposite directions on either side: at \(T = 1500\) K Fischer is about \(0.4\) dex more reducing than O'Neill; at \(T = 3000\) K Fischer is about \(1.1\) dex more oxidising. The crossover means the difference is small (under \(0.05\) dex) near 1800 K but grows to several tenths of a dex by 2400 K and reaches roughly \(1\) dex at 3000 K. The choice matters at the few-tenths-of-a-dex level for inferred partial pressures across most of the magma-ocean range. CALLIOPE now defaults to Fischer 2011 because it sits within \(\sim\)0.2 dex of the Hirschmann composite used by atmodeller across the whole magma-ocean range and so produces cross-backend \(\Delta\mathrm{IW}\) values that agree to a few tenths of a dex rather than up to \(\sim 1\) dex with the older default. The legacy O'Neill choice remains available for reproducibility of pre-existing results; see Backend comparison for the quantitative comparison.

How \(\Delta\mathrm{IW}\) enters the chemistry

The shift \(\Delta\mathrm{IW}\) feeds the equilibrium chemistry through four channels:

1. The free \(\mathrm{O_2}\) partial pressure

In solve.get_partial_pressures:

fO2_model = OxygenFugacity()
p_d['O2'] = 10.0 ** fO2_model(ddict['T_magma'], fO2_shift)

This is what makes O\(_2\) a derived, not a solved, quantity. The mantle redox state pins \(p_\mathrm{O_2}\) at the surface, and through the speciation tree it pins the equilibrium between every oxidised/reduced couple.

2. The modified equilibrium constants

For every reaction \(A \to B + n_\mathrm{O_2}\,\mathrm{O_2}\), the equilibrium ratio \(G_\mathrm{eq} = p_B / p_A\) depends on \(f_{\mathrm{O}_2}\) as

\[ G_\mathrm{eq}(T, f_{\mathrm{O}_2}) = 10^{\,\log_{10} K_\mathrm{eq}(T) - n_\mathrm{O_2}\,\log_{10} f_{\mathrm{O}_2}}. \]

For the H\(_2\)O-H\(_2\) couple (\(n_\mathrm{O_2} = +0.5\)), reducing conditions (\(f_{\mathrm{O}_2}\) smaller, \(\Delta\mathrm{IW}\) more negative) drive \(G_\mathrm{eq}\) larger, which in turn drives more H\(_2\)O to dissociate into H\(_2\). This is the redox dependence visible in the redox-sweep tutorial, and it is the mechanism behind the H\(_2\)-dominated, long-lived magma-ocean atmospheres found in Nicholls et al. (2024) ⁸ Figure 6 at \(\Delta\mathrm{IW} \le -1\).

3. The S\(_2\) Gaillard solubility

The Gaillard et al. (2022) ⁷ sulfide solubility carries an explicit \(\ln f_{\mathrm{O}_2}\) term, so \(\Delta\mathrm{IW}\) enters the dissolved-S inventory directly. The implementation in solubility.SolubilityS2.gaillard calls back into OxygenFugacity() to compute the absolute \(f_{\mathrm{O}_2}\).

4. The N\(_2\) Dasgupta solubility

Similarly, the Dasgupta et al. (2022) ⁴ N\(_2\) solubility includes a \(-1.6\,\Delta\mathrm{IW}\) term in its exponent, so reducing conditions sharply increase the dissolved-N inventory. This is one mechanism by which planet-scale N partitioning is tied to mantle redox; see Nicholls et al. (2026) ⁹ for an application to L 98-59 d, where the inferred H\(_2\)-dominated atmosphere with photochemical SO\(_2\) implies \(\Delta\mathrm{IW}\) between IW-4 and IW-1.

Reference values for \(\Delta\mathrm{IW}\)

Reservoir	\(\Delta\mathrm{IW}\)	Source
Mercury surface	IW-2.8 to IW-5.4 (Fe-based: IW-2.8 to IW-4.5; sulphur-based: IW-5.4 via Namur et al. 2016)	Cartier & Wood (2019) ²
Mars upper mantle (shergottite source)	\(\approx\) IW (specifically IW-1.0 to IW-0.3 for QUE 94201)	Wadhwa (2001) ¹²
Mars shergottite parent melts	IW-1.0 to IW+1.9 (variation from crust assimilation)	Wadhwa (2001) ¹²
Iron-wüstite buffer	\(0\)	by definition
Earth's upper mantle (modern)	\(\approx\) IW+3.5	Sossi et al. (2020) ¹¹
Earth upper mantle (range)	FMQ\(\,\pm\,2\) (\(\approx\) IW+1.5 to IW+5.5)	Frost & McCammon (2008) ⁶
Earth mantle at \(\sim 8\) GPa	\(\approx\) FMQ\(-5\) (\(\approx\) IW-1.5)	Frost & McCammon (2008) ⁶
Earth transition zone (\(\sim\)14-23 GPa)	just below IW	Frost & McCammon (2008) ⁶
Earth lower mantle (\(>\)23 GPa)	metal-saturated (\(\sim\)1 wt% Fe\(^0\)); at or below IW	Frost & McCammon (2008) ⁶

CALLIOPE's PROTEUS-side default is fO2_shift_IW = 4.0, consistent with a near-surface terrestrial composition. Nicholls et al. (2024) ⁸ Table 2 explored \(\Delta\mathrm{IW} \in \{-5, -3, -1, 0, +1, +3, +5\}\) on a 7-point grid and demonstrated that the resulting atmospheric composition spans the full range from H\(_2\)-dominated reduced atmospheres (TRAPPIST-1 c-like) to H\(_2\)O/CO\(_2\)-dominated oxidised atmospheres (Earth-like).

Limitations

No \(f_{\mathrm{O}_2}\) evolution: \(\Delta\mathrm{IW}\) is a constant input, not a state variable. In reality the mantle \(f_{\mathrm{O}_2}\) should evolve with degree of crystallisation, fractional crystallisation depth, and atmospheric escape; CALLIOPE does not capture this and the user is responsible for choosing a representative value or sweeping over a grid.
No \(f_{\mathrm{O}_2}\) depth profile: the surface \(f_{\mathrm{O}_2}\) alone enters the chemistry. Bower et al. (2022) ¹ §2.3 and Sossi et al. (2020) ¹¹ discuss why the interface fugacity (rather than the deep-mantle value) is the relevant choice; this assumption is consistent with CALLIOPE's ideal-gas, single-temperature treatment but breaks down if a Fe-FeO equilibrium curve in the deep mantle differs by more than a few dex.
No solid-FeO buffering: when \(\Phi_\mathrm{global} \to 0\), there is no melt to buffer \(f_{\mathrm{O}_2}\) against the user-prescribed value. CALLIOPE keeps using the shift regardless of melt fraction, which is a reasonable bookkeeping choice but should not be over-interpreted physically.