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:

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

The user supplies fO2_shift_IW = \(\Delta\mathrm{IW}\) as a scalar input; CALLIOPE computes the absolute \(\log_{10} f_{\mathrm{O}_2}\) at \(T = T_\mathrm{magma}\) by adding the buffer value to the shift. This page documents the buffer parameterisations, the physical meaning of \(\Delta\mathrm{IW}\), and how \(f_{\mathrm{O}_2}\) enters the chemistry.

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.

O'Neill & Eggins (2002), `oneill` (default)

A thermochemically-constrained fit derived from low-temperature equilibrium data, expressed as Bower et al. (2022) 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) adopted this as the "IW buffer to which \(f_{\mathrm{O}_2}\) is referenced". This is the function OxygenFugacity.oneill(T) in oxygen_fugacity.py.

Fischer et al. (2011), `fischer`

A simpler two-parameter fit calibrated against high-pressure (\(\sim\)25 GPa) experimental data:

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

Implemented as OxygenFugacity.fischer(T).

Choice of buffer

The two parameterisations agree to within \(\sim\)0.3 dex near \(T \approx 2000\) K but diverge with increasing temperature, reaching \(\sim\)0.7 dex by \(T = 2500\) K and growing further at higher \(T\). The disagreement is comparable to or smaller than the typical uncertainty in \(\Delta\mathrm{IW}\) inferred from petrological observations (Sossi et al. 2020 give \(\Delta\mathrm{IW} = +3.5 \pm 0.5\) for Earth's modern surface mantle), so the choice between buffers rarely changes inferred partial pressures meaningfully. CALLIOPE defaults to O'Neill & Eggins because Bower et al. (2022) used it and it was validated end-to-end against PROTEUS coupled runs.

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

The shift \(\Delta\mathrm{IW}\) feeds the equilibrium chemistry through two 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) Figure 6 at \(\Delta\mathrm{IW} \le -2\).

3. The S\(_2\) Gaillard solubility

The Gaillard et al. (2022) 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) 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) Section 4 for an application to L 98-59 d, where the inferred SO\(_2\)/H\(_2\) atmosphere implies \(\Delta\mathrm{IW} \approx -1\).

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

Reservoir	\(\Delta\mathrm{IW}\)	Source
Mercury surface	\(\sim -5\)	Cartier & Wood (2019)
Asteroidal material	\(\sim -2\)	Doyle et al. (2019)
Mars mantle	\(-3\) to \(0\)	Wadhwa (2001)
Iron-wüstite buffer	\(0\)	by definition
Earth's upper mantle \(f_{\mathrm{O}_2}\)	\(+3.5\)	Sossi et al. (2020)
Earth upper mantle	\(+1\) to \(+5\) (i.e. FMQ\(\,\pm\,2\))	Frost & McCammon (2008)
Earth deep (lower) mantle	\(\sim -2\) to \(-3\) (\(\sim 5\) log units below FMQ at \(\sim 8\) GPa)	Frost & McCammon (2008)

CALLIOPE's PROTEUS-side default is fO2_shift_IW = 4.0, consistent with a near-surface terrestrial composition. Nicholls et al. (2024) 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) §2.3 and Sossi et al. (2020) 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.