Sub-Neptunes are the most common class of exoplanets in the Galaxy, yet their internal structure remains a matter of debate. Traditionally, it was thought that a sharp boundary lies beneath the hydrogen atmosphere, separating it from the silicate magma. However, recent calculations have shown that at the pressures and temperatures typical of sub-Neptune interiors, silicate and hydrogen can mix completely. This raises the challenge of describing the thermodynamic properties of such a supercritical fluid, without which it is impossible to build correct models of planetary interiors. This work fits into the line of research on phase equilibria and equations of state for silicate-hydrogen systems, which has been actively developing in recent years.
The authors performed first-principles molecular dynamics simulations within density functional theory using the VASP software package and the PBEsol exchange-correlation functional. They modeled a cell of 243 atoms (27 Mg, 27 Si, 81 O, 108 H), corresponding to the composition MgSiO3H4. First, crystalline MgSiO3 was melted, then hydrogen was added. The simulations were carried out at temperatures of 4000, 6000, and 8000 K and pressures up to ~600 GPa. The resulting pressure and internal energy data were fitted to a Helmholtz free energy, allowing the density, entropy, heat capacity, thermal expansion, bulk modulus, and Grüneisen parameter to be derived from a single thermodynamic surface.
The supercritical fluid MgSiO3H4 turned out to be less dense than the dry silicate melt under all conditions studied. The deviation from ideal mixing of MgSiO3 and H2 volumes reaches several percent and grows with pressure. Structural analysis showed that with increasing pressure, the Si–H coordination number increases, while the number of stable H–H bonds drops sharply: at 4000 K, the fraction of hydrogen atoms participating in long-lived H–H contacts falls below 0.1 at pressures around 340 GPa. This indicates a transition from molecular hydrogen to a more atomic form embedded in the silicate framework. Applying the equation of state to models of sub-Neptunes with a mass of 6 Earth masses and a total hydrogen inventory of 4% showed that depending on the pressure at the atmosphere-interior boundary (1 GPa vs. 5 GPa), the planet's radius changes by 5.7%.
This work provides, for the first time, a self-consistent equation of state for a supercritical silicate-hydrogen fluid, closing an important gap in sub-Neptune modeling. Unlike neighboring works, which mostly estimated phase equilibria or mixing at the level of thermodynamic models, here direct first-principles data on density and thermodynamic derivatives are obtained. This allows for a quantitative account of dissolved hydrogen in the interior and its influence on the radius and thermal evolution of planets.
A natural next step is to expand the composition to include iron, which can strongly affect density, redox conditions, and phase transitions. Experimental verification at high pressures and temperatures is also needed, though such experiments are still extremely rare. Joint efforts by theorists and experimentalists will allow the construction of equations of state for the full range of compositions, pressures, and temperatures relevant to sub-Neptunes.
The results will directly affect models of the internal structure and evolution of sub-Neptunes, as well as the interpretation of exoplanet mass and radius data.
Incorporate iron into the simulations and conduct experimental measurements of the density and compressibility of supercritical silicate-hydrogen fluids.
This work touches on a fundamental problem in planetary science: how to determine the composition and internal structure of exoplanets from limited observational data. It shows that ignoring the non-ideality of silicate-hydrogen mixing can lead to systematic errors in estimates of hydrogen content and the thermal state of the interior.
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What the work does
The authors performed first-principles molecular dynamics calculations of a supercritical MgSiO3H4 mixture at pressures up to 600 GPa and temperatures of 4000–8000 K, obtained an equation of state, and showed that the mixture's density is lower than that of dry silicate and that its behavior deviates from ideal mixing. Applying the equation of state to models of sub-Neptunes with a mass of 6 Earth masses, they found that the distribution of hydrogen between the atmosphere and the interior affects the planet's radius (a difference of 5.7% at boundary pressures of 1 and 5 GPa).
Where its strength is
The strength of the work lies in obtaining a self-consistent equation of state for a supercritical silicate-hydrogen fluid from first principles, which fills a gap in sub-Neptune modeling and allows accounting for the real distribution of hydrogen in the interior.
Where it could go next
Worth trying
Why this may matter
Accounting for supercritical mixing of silicate and hydrogen could change estimates of sub-Neptune radii by several percent, which is important for interpreting the observed radius–mass relation and revealing the internal structure of exoplanets.
Nearby works in our archive
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At a pressure of about 340 GPa and a temperature of 4000 K, only one in ten hydrogen atoms retains a bond with another hydrogen for longer than 100 femtoseconds; the rest are already incorporated into the silicate network.