Hydrogen-helium immiscibility boundary in planets

Abstract

The location of the hydrogen-helium (H/He) immiscibility boundary controls whether and where helium rain occurs in giant planets, yet it remains uncertain because high-pressure experiments are challenging and ab initio simulations are limited in system size and simulation time. We map this boundary by computing composition-dependent chemical potentials from large-scale molecular dynamics driven by machine learning potentials trained on three density functional approximations (PBE, vdW-DF, and the hybrid HSE). The three functionals yield consistent immiscibility boundaries, and the demixing temperatures are typically ~2000 K lower than previous ab initio simulations using small system sizes across the pressure range of 100-1000 GPa. Fitting the H/He mixing free energy to a Redlich-Kister regular solution model rationalizes the thermodynamic driving force for phase separation and provides a predictive representation of the boundary. Comparison with current planetary interior profiles indicates that helium rain is plausible in Saturn but unlikely in the warmer interior of Jupiter. Our results narrow the uncertainty in the H/He immiscibility boundary and provide inputs for planetary models that couple demixing, heat transport, and composition gradients in gas giants.

Figures (3)

• Figure 1: Thermodynamic behaviors of H/He mixtures predicted by different machine learning potentials (MLPs). (a-c) Pressure as a function of density for five H/He mixtures ($x_{\rm He}$ = 0, 0.25, 0.5, 0.75, and 1.0) at (a) 2000 K, (b) 6000 K, and (c) 10000 K. Results are from different vdW-DF MLPs: CACE (circles), MACE (triangles), N2P2 (crosses), and a $\Delta$-learning vdW-DF model (pluses) built upon the PBE MLP baseline. All results are compared to reference vdW-DF data (dashed lines with diamonds) from AIMD calculations schottler2018ab. (d, h) Snapshots from MLP MD simulations showing: (d) homogeneous mixture and (h) phase separation. White and red particles indicate hydrogen and helium atoms, respectively. (e, i) Derivative of the helium chemical potential $({\partial \mu_{\rm He}}/{\partial \ln x_{\rm He}})_{T, P}$ computed from the S0 method cheng2022computing. The MD simulations were performed using MLPs trained on vdW-DF. Blue, red, and yellow symbols indicate MLP architectures employing CACE, N2P2, and MACE, respectively. The vertical dashed black lines represent the equilibrium helium fractions ($x_{\rm He}$) of the He-poor ($x_{\rm He} = 0.19$) and He-rich ($x_{\rm He} = 0.77$) phases during demixing at 7000 K and 800 GPa. (f, j) Computed chemical potentials of H (triangles) and He (pentagons), compared with ideal solution behaviors (dotted light gray lines). (g, k) Mixing free energy per atom for H/He mixtures ($\Delta G^{\rm mix}$) as a function of helium fractions. Pink diamonds denote the $\Delta G^{\rm mix}$ at the vdW-DF DFT level, promoted from the vdW-DF N2P2 MLP to DFT using free energy perturbation (FEP). The schematic dashed green line in (k) represents the hypothetical $\Delta G^{\rm mix}$ curve for a homogeneous H/He mixture inside the demixing region.
• Figure 2: Immiscibility diagrams of H/He mixtures predicted using MLPs trained on PBE, vdW-DF, and HSE. (a) Schematic immiscibility diagram of the H/He mixtures with two projections. The blue surface denotes the upper bounds of the two-phase coexistence region. (b) A sketch of the H/He immiscibility and helium rain in the interior models of gas giants. (c) Pressure as a function of density for $x_{\rm He}=0.25$ at 6000 K. (d, e, f) Composition-dependent immiscibility boundaries of H/He mixtures using MLPs trained on three XC functionals: (d) PBE, (e) vdW-DF, and (f) HSE. The dotted purple and green lines mark the estimated molecular–atomic transition boundaries for pure hydrogen and for a mixture with $x_{\rm He}=0.05$ based on classical MD simulations, respectively. In (e), the cyan stars indicate the transition temperatures obtained from PIMD simulations driven by vdW-DF MLP for $x_{\rm He}=0.05$ at 150 and 300 GPa. Planetary isentropes for Jupiter militzer2013abnettelmann2008abschottler2018ab (red band) and Saturn nettelmann2013saturn (solid brown line) are indicated. The dashed lines denote the present-day thermal profiles of Jupiter and Saturn based on the HMH24 model howard2024evolution, while the dotted brown line denotes another thermal profile of Saturn from the MF20 model mankovich2020evidence. (g) H/He immiscibility boundaries at $x_{\rm He}=0.089$. For reference, previous studies at a similar composition ($x_{\rm He} = 0.085-0.11$) are included as follows: PBE AIMD results analyzed by assuming ideal entropy of mixing lorenzen2011metallization, PBE AIMD results analyzed with non-ideal entropy morales2010evidencemorales2013hydrogen, vdW-DF AIMD results analyzed with non-ideal entropy schottler2018ab, vdW-DF MLP chang2024theoretical, SCAN+rVV10 MLP chang2024theoretical, and laser-shock experiments brygoo2021evidence.
• Figure 3: Redlich-Kister (R-K) model analyses of the mixing free energy ($\Delta G^{\rm mix}$) and the resulting immiscibility boundary in the H/He system. (a) Mixing free energy per atom of H/He mixtures computed from 150 GPa to 1000 GPa and 2000 K to 12000 K, using the vdW-DF MLP. The solid lines denote the third-order R-K model fitted individually at each $P$-$T$. The dashed lines indicate the modeled $\Delta G^{\rm mix}$ for assumed homogeneous H/He mixtures in the phase separation regime, derived from the individual fits. The dotted lines show $\Delta G^{\rm mix}$ derived from a global R-K fit. (b) Pressure and temperature dependence of the R-K parameters ($\omega_i$). The dashed lines, with shaded uncertainty bands, represent the polynomial fits for the $P$-$T$ dependence of the $\omega_1(T, P)$ and $\omega_2(T, P)$ parameters derived from the MD simulation data. (c) Comparison of H/He immiscibility boundaries from different methods at a helium fraction of $x_{\rm He}=0.089$. The blue, red, and yellow solid lines correspond to direct MD simulation results based on PBE, vdW-DF, and HSE MLPs, respectively. The shaded areas represent the associated error estimates from Monte Carlo sampling by propagating the uncertainty of fitted $\omega_i$. The pink stars represent the immiscibility boundaries derived with the methodology of individual $P$-$T$ fit in earlier AIMD studies morales2013hydrogenschottler2018ab for a vdW-DF subset at 150 GPa and 200 GPa. Previous studies at similar compositions ($x_{\rm He} = 0.08-0.09$) are included: PBE AIMD results analyzed by assuming ideal entropy of mixing lorenzen2011metallization (blue crosses), PBE AIMD results analyzed with non-ideal entropy morales2010evidencemorales2013hydrogen (blue hexagons), vdW-DF AIMD results with non-ideal entropy schottler2018ab (purple diamonds).