The Effects of Non-ideal Mixing in Planetary Magma Oceans and Atmospheres

TL;DR

This study implements a fully global non-ideal thermodynamic framework for magma-ocean planets by extending a prior equilibrium model to include fugacity corrections for gas species and activity coefficients for silicate melts and metallic liquids. By solving a network of 18 independent reactions among 25 phase components under mass balance, the authors quantify how non-ideality alters atmosphere-magma-ocean exchange (AMOI) pressures, hydrogen partitioning, and water inventories for both a proto-Earth analog and sub-Neptune analogs. They find that non-ideality yields modest shifts in the AMOI state for embryos (typically <20% and often negligible) but can be more pronounced for hotter, higher-pressure sub-Neptunes, where simultaneous gas and melt/metal non-ideality can raise or lower interior and atmospheric hydrogen budgets by comparable magnitudes. The work demonstrates the necessity of a self-consistent global treatment of non-ideality to interpret atmospheric spectra and interior structures, and it suggests that the so-called fugacity crisis may be mitigated when melt activities are properly accounted.

Abstract

Sub-Neptunes with hydrogen-rich envelopes are expected to sustain long-lived magma oceans that continuously exchange volatiles with their overlying atmospheres. Capturing these interactions is key to understanding the chemical evolution and present-day diversity of sub-Neptunes, super-Earths, and terrestrial planets, particularly in light of new JWST observations and upcoming missions. Recent advances in both geochemistry and astrophysics now allow the integration of experimental constraints and thermodynamic models across melt, metal, and gas phases. Here we extend a global chemical equilibrium model to include non-ideal behavior in all three phases. Our framework combines fugacity corrections for gas species with activity coefficients for silicate and metal species, enabling a fully coupled description of volatile partitioning. We show that for planetary embryos (0.5 M$_\oplus$ at 2350 K), non-ideality introduces only modest corrections to atmosphere-magma ocean interface (AMOI) pressures, volatile inventories, and interior compositions. In contrast, for sub-Neptunes with higher temperatures ($\approx$ 3000 K) and pressures, non-ideal effects are more pronounced, though still modest in absolute terms$-$typically within 20% and at most a factor of two. Including activity and fugacity coefficients simultaneously increases the AMOI pressure, enhances water retention in the mantle and the envelope. Our results demonstrate that non-ideality must be treated globally: applying corrections to only one phase leads to incomplete or even misleading trends. These findings highlight the importance of self-consistent global thermodynamic treatments for interpreting atmospheric spectra and interior structures of sub-Neptunes and super-Earths.

Figures (10)

• Figure 1: Contour plots of the fugacity coefficients ($\phi_i$) for H2O, CH4, CO2, CO, H2, and for the ratio $\phi_{\ce{H2}}/\phi_{\ce{H2O}}$, shown as a function of pressure and temperature. A value of $\phi_i=1$ corresponds to ideal behavior, while deviations indicate non-ideality in the gas phase. For direct comparison, the color scale is shared across CH4, CO2, and CO, and separately for H2O and H2. Fugacity coefficients become large at high pressures, but remain relatively small over the pressure range relevant for sub-Neptunes, which typically have surface pressures below 10 GPa young_phase_2024gilmore_core-envelope_2026
• Figure 2: Activity coefficient $\gamma$ and activities $a$ for H2 in silicate melt as a function of the mole fraction of H2 in the silicate phase. Curves are shown for different pressures and temperatures. Both the activities and the activity coefficient are strongly temperature dependent. At lower temperatures, the dependence on pressure becomes significant.
• Figure 3: Gibbs free energy of H2 dissolution in silicate melt. The blue curve shows the original fit based on the solubility experiments of hirschmann_solubility_2012. The red curve shows the revised model, which adopts the free energy of mixing from gilmore_core-envelope_2026 anchored to the experimental free energy of reaction from hirschmann_solubility_2012. The revised relation yields more negative Gibbs free energies at high temperatures, implying greater H2 solubility in silicates at those conditions.
• Figure 4: Plots of pressure corrections for the intra-melt reaction $\rm MgSiO_3 \rightleftharpoons MgO + SiO_2$ (R2) at temperatures ranging from 3800 K to 6000 K. Results using the simpler approach for estimating the effects of thermal pressure (Equation \ref{['eqn:Pthermal_simple']}) are shown in the left panel, and results using the integrated approach for estimating thermal pressure effects (Equation \ref{['eqn:Pthermal_integrated']}) are shown at right. A fiducial adiabat through a molten interior of a sub-Neptune with a total mass of 6 M$_\oplus$ and 3 weight percent hydrogen is shown by the open circles. See text.
• Figure 5: Plot of the log of the equilibrium constant for the reaction $\rm MgO_\text{silicate} \rightleftharpoons Mg_\text{gas} + 1/2O_{2\text{,gas}}$ as a function of pressure and temperature with and without pressure correction of the 1-bar standard-state Gibbs free energy of formation for the melt species MgO, $\hat{G}^{o}_{f,{\rm MgO, melt}}$. Note the rapid initial decrease with $P$ is the same for both the corrected and uncorrected cases.