Large-scale atomistic simulations of magnesium oxide exsolution driven by machine learning potentials: Implications for the early geodynamo

The precipitation of magnesium oxide (MgO) from the Earth’s core has been proposed as a potential energy source to power the geodynamo prior to the inner core solidiﬁcation. Yet, the stable phase and exact amount of MgO exsolution remain elusive. Here we utilize an iterative learning scheme to develop a uniﬁed deep learning interatomic potential for the Mg-Fe-O system valid over a wide pressure-temperature range. This potential enables direct, large-scale simulations of MgO exsolution processes at the Earth’s core-mantle boundary. Our results suggest that Mg exsolves in the form of crystalline Fe-poor ferropericlase as opposed to a liquid MgO component presumed previously. The solubility of Mg in the core is limited, and the present-day core is nearly Mg-free. The resulting exsolution rate is small yet nonnegligible, suggesting that MgO exsolution can provide a potentially important energy source, although it alone may be diﬃcult to drive an early geodynamo.


Introduction
Chemical buoyancy due to the crystallization of the inner core is believed to have supplied energy to power the geodynamo in the last 0.5-1 billion years (Nimmo, 2015).Paleomagnetic records suggest the existence of a very early (3.4 Ga) magnetic field in the Earth's history prior to the inner core crystallization (Tarduno et al., 2010).The energy source of this early geodynamo is enigmatic.Radiogenic heat production in the core may not be sufficient to sustain an early dynamo (Frost et al., 2022).The basal magma ocean may be electrically conductive (Stixrude et al., 2020), but the scale and longevity of a convective basal magma ocean are uncertain.
Recent studies propose that exsolution of oxides from the core upon cooling, such as MgO (O'Rourke and Stevenson, 2016) or SiO2 (Hirose et al., 2017), may be a viable mechanism to power an early dynamo.
Experimental studies on metal-silicate partitioning suggest that the solubility of Mg is highly sensitive to temperature (Badro et al., 2016;Du et al., 2017).The high-temperature equilibration between the metallic and silicate melts during the core formation process may result in a few wt% of MgO dissolved in the core.
Upon cooling, Mg is expected to precipitate out of the core as its solubility drops.However, the efficiency of this mechanism, especially for MgO oxide, remains controversial (Badro et al., 2016;Du et al., 2017).
The precipitation rate of MgO has been widely estimated using Mg partitioning behaviors in the metalsilicate system (Badro et al., 2018;Du et al., 2019;Liu et al., 2019).This estimation, strictly speaking, is unjustified.In contrast to the core formation process, where metallic and silicate melts equilibrate, the precipitation process involves the equilibration between the metallic melt and exsolution, where the exact phase and chemistry of exsolution depend on bulk compositions and thermodynamic conditions (Helffrich et al., 2020).Previous estimates, however, implicitly assume that exsolved MgO is a component of liquid silicate (Badro et al., 2018;Du et al., 2019;Liu et al., 2019).This assumption is questionable, as MgO is more refractory than SiO2 which may exsolve out of the core in solid-sate (Hirose et al., 2017).Therefore, a careful examination of the Mg exsolution process is necessary.
In this study, we combine enhanced sampling, feature selection, and deep learning to develop a unified machine learning potential (MLP) for the Mg-Fe-O system.This MLP is used to perform large-scale molecular dynamics simulations to study the exsolution of Mg from core fluids.Unlike previous computational studies based on free energy calculations (Davies et al., 2018;Wahl and Militzer, 2015;Wilson et al., 2023), this method does not prescribe the state of the exsolved phase (Sun et al., 2022).The results inform the stable state of MgO precipitation, Mg and O partitioning between core fluid and exsolution, and the efficiency of MgO exsolution in powering an early geodynamo.

Development of Machine Learning Potential
A machine learning potential (MLP) is a non-parametric model that approximates the Born-Oppenheimer potential energy surface.We follow the same approaches outlined in our previous work on Mg-Si-O (Deng et al., 2023) and Mg-Si-O-H (Peng and Deng, 2024) where details of the machine learning process can be found.To briefly summarize our approach, the MLP is trained on a set of configurations drawn from multithermal and multibaric (MTMP) simulations (Piaggi and Parrinello, 2019), which are used to efficiently sample the multi-phase configuration space.We use the structure factor of B1 MgO as the collective variable to drive the sampling, and an iterative learning scheme as described by (Deng et al., 2023) to efficiently select distinct samples from molecular dynamics trajectories.High-accuracy ab initio calculations are performed on the selected sample configurations to derive the corresponding energies, atomic forces, and stresses.The DeePMD approach is employed to train an MLP which takes a configuration (a structure of a given atomic arrangement) and predicts its energy, atomic forces, and stresses without iterating through the time-consuming self-consistent field calculation (Wang et al., 2018;Zhang et al., 2018).The details of DeePMD approach and density functional theory (DFT) calculations can be found at Supplementary Information (Text S1, Figure S1).Our MLP explores a wide compositional space, trained on Mg-Fe-O systems of varying Mg:Fe:O ratios, including the pure endmembers, Fe and MgO, as well as intermediate compositions denoted by (MgO)aFebOc , where a=0-64, b = 0-64, c = 0-16 with 2a+b≥ 64.The final training set consists of 4466 configurations generated at pressure up to 200 GPa and temperature up to 8000 K.

Two-phase molecular dynamics simulation
Two-phase simulations are performed on a pure MgO system to determine the melting point of B1 MgO.Alfè (2005) found that systems of 432 atoms are sufficient to yield converged melting points as those larger systems.Here, supercells of 432 atoms are constructed and then relaxed for 1000 steps at desired pressure and temperature conditions in the NPT ensemble.The relaxed cell is then used to perform NVT simulations at high temperatures far exceeding the melting temperatures, with the atoms of half the cell fixed and the force applied to these atoms set to 0. The resulting structure is half-molten and half-crystalline.We relax this structure again at the target pressure and temperature for 1000 steps to obtain the initial configuration for two-phase simulations.Simulations on the two-phase supercell of solid-liquid coexistence were then performed.If the whole cell is molten (or crystallized) at the end, the simulation temperature is above (or below) the melting point.The state of the system can be determined by analyzing the radial distribution functions, allowing us to pinpoint the upper and lower bounds of the melting point.

Exsolution simulation
We construct systems of various Mg:Fe:O ratios by substituting/removing Mg and/or O atoms of supercells of B1 MgO.Initial configurations are melted at 8000 K and 140 GPa under the NPT ensemble for ~10 ps.
We inspect trajectories and radial distribution functions to ensure systems are fully molten and well relaxed.
The resulting configurations are further used to perform simulations at 140 GPa and target temperatures under the NPT ensemble for up to several nanoseconds to simulate the exsolution process.

Gibbs dividing surface
To determine the composition of the two coexisting phases, for every frame we locate the Gibbs dividing surface (GDS) that separates the whole cell into an oxide region, a metallic region, and two interfaces in between (Sega et al., 2018;Willard and Chandler, 2010).For every snapshot, we calculate the coarsegrained instantaneous density field at point r by where  $ is the position of the ith atom; s is the coarse-graining length, and is set as 2.5 Å here (Willard and Chandler, 2010).The instantaneous surface s(t) is defined as a contour surface of the instantaneous coarse-grained density.The proximity of ith atom to this surface is where (t) is the surface normal in the direction of the density gradient at that point.This instantaneous density field is then projected onto each atom and associated with the corresponding proximity.We find that ρ() follows the expected form where roxide and rmetal are the density of the oxide phase and metal phase, respectively; a0 and a1 are the positions of the Gibbs dividing surfaces; w is the thickness of the interface.Fitting ρ() to Eq. ( 3) yields the location of the Gibbs dividing surfaces, as well as the density of each phase.The metal and oxide phases are defined by  $ > 2 and  $ < −2, respectively.We count the number of atoms of metal and oxide phases of each frame.A long trajectory after the system is equilibrated and used to determine the average concentrations in each phase and associated standard deviations.For more details, the reader is referred to (Deng and Du, 2023;Sega et al., 2018;Willard and Chandler, 2010;Xiao and Stixrude, 2018).

Benchmarks of the Machine Learning Potential
We compare the energies, atomic forces, and stresses from the MLP to those from DFT calculations for 15078 configurations that are not included in the training set (Figure S2).The root-mean-square errors of the energies, atomic forces, and stresses are 6.34 meV/atom −1 , 0.27 eV/Å −1 , and 0.48 GPa, respectively.
We perform two additional tests to further examine the reliability of the MLP.First, we perform MD simulations with supercells of B1 MgO solid, MgO liquid, and a Mg-Fe-O liquid mixture, respectively.
These supercells are larger than the training configurations.The root-mean-square error of energy prediction by the MLP with respect to the DFT calculations is similar to the error in the testing sets (Figure S3).This verification test further proved the accuracy of energy prediction and also demonstrated the transferability of the MLP to structures larger than the train/test sets.Second, we calculate the melting point of B1 MgO at 140 GPa using the solid-liquid two-phase coexistence method with a supercell of 432 atoms.
For both DFT and MLP, the system crystalizes at 7700 K and melts at 7800 K, suggesting a melting point of 7750±50 K at 140 GPa and further validating the robustness of the MLP.The melting temperature is also consistent with previous studies (Alfè, 2005;Du and Lee, 2014).

System convergence of the exsolution simulation
The robust MLP of Mg-Fe-O system allows for large-scale exsolution simulations.We first examine the convergence of Fe liquid composition with respect to the simulation cell size by performing exsolution simulations at 5000 K and 140 GPa with five Mg-Fe-O liquid mixtures, where ratios of Mg, O and Fe atoms are fixed as 2:2:3, i.e., Mg64O64Fe96, Mg512O512Fe768, Mg1728O1728Fe2592, Mg2304O2304Fe3456, Mg3136O3136Fe4704.
For all simulation, the system quickly demixes to form MgO-rich and Fe-rich region, and subsequently, MgO-rich region spontaneously crystallizes to form ferropericlase while metallic phase remains liquid.The resulting atomic fraction of Mg, O in the metallic phase converges when system size reaches 2000 atoms (Figure S4).Large systems also yield better statistics and thus the smaller uncertainties in the atomic fraction.Based on this test, all the partitioning results reported here are derived from simulations performed with systems of more than 2000 atoms to ensure convergence and robust statistics.

Exsolution process
In all simulations considered, exsolution spontaneously occurs within a few picoseconds at 4000 K to a few nanoseconds at 5500 K. Exsolutions are all solid ferropericlase with small amounts of FeO.The interfaces between exsolution and Fe liquid are typically irregular as they form spontaneously without interference.
Taking the exsolution simulation of Mg2088Fe3456O2520 liquid at 140 GPa and 5500 K as an example (Figure 1; Supplementary movie 1).It starts with a homogeneous liquid (Figure 1a), and quickly demixes to form patches of MgO-rich liquid and Fe-rich liquid with a continuous drop of potential energy.Within around 250 ps, MgO-rich patches and Fe-rich patches conglomerate, respectively, dividing the whole cell into two regions: one enriched in MgO and the other Fe.MgO-rich region remains liquid for another 750 ps until a sudden crystallization occurs to form ferropericlase (Figure 1b).The crystallization is a rapid process accompanied by a significant drop in potential energy.Quickly after ferropericlase crystallizes, the potential energy plateaus.The element exchange between ferropericlase and residual Fe liquid continues within the interface region.We analyze the trajectories at this stage, calculate the Gibbs dividing surface, and determine the average composition of each phase for the last 100 ps.The chemical compositions of both phases are shown in Figure 1c.The metallic liquid is oxygen rich and magnesium poor.The exsolved ferropericlase is of B1 structure and is nearly stochiometric (Mg0.974Fe0.026)O.Similar analyses have been applied to all other exsolution simulations, and the compositions of simulation products are summarized in Table S1.with the separation of ferropericlase and metallic liquid.The sudden drop of internal energy corresponds to the crystallization of ferropericlase.

Element partitioning
We further analyze the element partitioning between the exsolved ferropericlase and Fe liquid considering two dissociation reactions: MgO ox = Mg met + O met and FeO ox = Fe met + O met , where superscripts ox and met indicate oxide and metal, respectively.We also consider Mg exchange reaction with MgO ox + Fe met = FeO ox + Mg met , but the fit is poor, as also reported in other studies (Badro et al., 2018;Liu et al., 2019) standard thermodynamic model with the non-ideality described by the epsilon formalism of (Ma, 2001).
The phase relation between ferropericlase and Fe liquid has been studied mostly at low pressures and low temperatures (Asahara et al., 2007;Frost et al., 2010;Ozawa et al., 2008).Unfortunately, Mg contents in metallic melts were not reported and only  2 7 were reported in these studies.Thus, we only include  2 7 of these experiments in the fitting (Texts S2, S3; Tables S2).
Our calculated  2 34 is slightly larger than that reported by a recent ab initio calculation (Wilson et al., 2023) where the pure B1 MgO is assumed as the exsolved phase, while our exsolution simulations show that precipitates contain small amounts of FeO (Figure 1, Table S1).The incorporation of FeO in the exsolved phase likely changes the free energy of the system, leading to different in  2 34 .Wahl and Militzer (2015) also perform ab initio simulations on the Mg-Fe-O system but focus on high temperatures close to the solvus closure and do not report Mg partitioning results at the conditions overlapping this study, which precludes a direct comparison.Compared with the previously determined  2 34 between Fe liquid and silicate melt (Badro et al., 2018;Du et al., 2019;Liu et al., 2019),  2 34 between Fe liquid and solid ferropericlase shows similar temperature dependence but is overall approximately one order of magnitude lower (Figure 2a), indicating a low Mg content in Fe liquid when equilibrated with ferropericlase.This is expected as MgO preferentially enters ferropericlase when silicate melt crystalizes (Boukaré et al., 2015).
Oxygen partitioning between ferropericlase and liquid Fe is strongly controlled by temperature, in agreement with previous experiments (Asahara et al., 2007;Frost et al., 2010;Ozawa et al., 2008) and calculations (Davies et al., 2018). 2 7 between Fe liquid and silicate melt derived by (Liu et al., 2019) generally aligns with  2 7 between Fe liquid and solid ferropericlase, especially at high temperatures.Our  2 7 can be well fitted with previous experimental data to a unified thermodynamic model (Asahara et al., 2007;Ozawa et al., 2008), except for the four data points reported by (Frost et al., 2010) (Text S3; Table S3).At around 30-70 GPa and with similar oxygen contents in the liquid Fe,  2 7 of (Frost et al., 2010) are around half log unit higher than those of (Ozawa et al., 2008) and our extrapolated results.The source of this discrepancy is unknown but may arise from the carbon contamination. 2 7 reported by an early DFT calculation (Davies et al., 2018) is around 0.3-0.6 log unit higher than those of (Ozawa et al., 2008) and our results at similar conditions.We note that Davies et al. (2018) calculated the chemical potential of FeO for defect-free (Mg,Fe)O.Yet, both our simulations and previous studies (Karki and Khanduja, 2006;Van Orman et al., 2003) support the existence of defects in ferropericlase at high temperatures, which may lower the free energy of the host mineral and enrich FeO in ferropericlase, leading to a reduced  2 7 .(Badro et al., 2018), (Liu et al., 2019), and (Du et al., 2019), respectively.(b) Experimental studies include (Ozawa et al., 2008) (O08, downward triangle), (Asahara et al., 2007) (A07, diamond), (Frost et al., 2010) (F10, upward triangle).DFT study includes (Davies et al., 2018) (D18, square).All previous results are normalized to 140 GPa for a direct comparison using the best-fit pressure dependence, , where P/T are experimental/calculation pressure/temperature, and  7 is a fitted constant (Table S3).L19 (dashed line) indicates the O exchange coefficient of silicate-melt calibrated by (Liu et al., 2019).Uncertainties of the exchange coefficients of this study are roughly represented by the symbol size.

Exsolution rate and geodynamo
Earth's accretion and differentiation in its early history likely resulted in a core much hotter than it is today.
The precipitation of light elements due to the secular cooling of the core may have provided a vital energy source to drive the geodynamo.The energetics of the exsolution-powered dynamo hinge on the cooling rate and the exsolution rate.Here, we adopt a core thermal evolution model proposed by O'Rourke et al. (2017) where the CMB temperature (TCMB) drops from around 5000 K to around 4100 K over the first ~3.8 billion years (Gyr) with a cooling rate of ~230 K Gyr -1 .Given this thermal history of the core, the phase of exsolution and the associated exsolution rate can be further determined using the element partitioning models, along with knowledge of the initial core composition.All previous modeling of Mg exsolution from the core assume that MgO exsolve as a component of silicate melts.However, our simulations show that MgO should exsolve as a component of crystalline ferropericlase, at least when light elements other than Mg and O are absent (Badro et al., 2018;Du et al., 2019;Liu et al., 2019;Mittal et al., 2020).Here, we first examine the Mg exsolution and its potential to drive the early geodynamo for an Mg-and O-bearing core, and then we discuss the effects of additional light elements.
To model Mg exsolution from a core fluid with only Mg and O as light elements, we first determine the saturation conditions under which Mg precipitates.Previous N-body simulations and metal-silicate equilibrium experiments suggest that the Earth's core following its formation may contain 1.6-5 wt% O (Fischer et al., 2017;Liu et al., 2019;Rubie et al., 2015).The corresponding saturation magnesium concentration in the core is 0.04-0.19wt% at 140 GPa, as determined using the  2 34 between metal and ferropericlase, which is significantly lower than that determined by  2 34 between metal and silicate melt (Liu et al., 2019) (Figure 3a).This difference is expected, as the former  2 34 value is about one order of magnitude smaller than the latter (Figure 2a).Hence, our work implies that a substantial amount of Mg may have already been exsolved by the time the core cools to a TCMB of 5000 K. Further cooling reduces the Mg solubility in the core, with concentrations approaching 0.02-0.003wt% at a TCMB of 4000 K.This suggests a diminishingly small amount of Mg in the present-day outer core.We use these saturation magnesium conditions as the initial core composition in our exsolution modeling.
Despite the contrasting Mg solubilities in the core, the compositions of the exsolutions are similar and exhibit a comparable trend with temperature.Specifically, the exsolved phase in both models becomes increasingly FeO-rich with cooling.At 4000 K, the exsolution contains up to 20 wt% FeO (Figure 3b).
Throughout the thermal history, TCMB is lower than the solidus of exsolved ferropericlase (Deng et al., 2019), indicating that exsolutions remain solid.
The resulting exsolution rates decrease with temperature, with the values dropping from 1.4-5.6 ×10 -6 K -1 at 5000 K to 0.2-1.0×10 -6 K -1 at 4000 K. Exsolution rates of ferropericlase are approximately one order of magnitude smaller than those predicted for silicate melt exsolution (Figure 3c).Exsolutions are depleted in iron and enriched in Mg.As a result, they are lighter than the outer core fluid and thus provide the buoyancy flux that may sustain an exsolution-driven dynamo (O' Rourke and Stevenson, 2016).Converting the exsolution rate to the magnetic field intensity is model dependent, however.The upper bound of the exsolution rates (1-5.6 ×10 -6 K -1 ) derived here are similar to the previous reports (Badro et al., 2016;Du et al., 2017).While Du et al. (2019) conclude that this exsolution rate is not sufficient to power the early geodynamo alone, Badro et al. (2018) use a scaling law that relates the exsolution rate to dipolar magnetic field intensity ( DEFGHI6 J;K8L6 ) and argue that MgO exsolution can well produce the dipolar magnetic field intensity at Earth's surface consistent with observations.We follow (Badro et al., 2018) to convert the exsolution rate to  DEFGHI6 J;K8L6 (Figure 3d).The results show that  DEFGHI6 J;K8L6 generated by the upper bound exsolution rate is broadly consistent with the paleo-intensities records dating back to 3.4 Gyr (Tarduno et al., 2010), and that generated by the lower bound rate is overall smaller than the observations and thus may not be sufficient (Tarduno et al., 2015).Overall, we find that MgO exsolution alone may be difficult to power the early geodynamo, but it is nevertheless an important energy source.
While the exact composition of the core remains unknown, it may contain other light elements, such as S, Si, C, and H (Hirose et al., 2021).As the core cools, the solubility of these light elements tends to decrease, leading to their exsolution.For example, in a core composed solely of Si, O, and Fe, the exsolved phase would likely be solid SiO2 (Hirose et al., 2017;Zhang et al., 2022).The study by Helffrich et al. (2020) on the joint solubility of Mg, O, and Si in liquid Fe suggests that the presence of Si enhances the retention of Mg in metal, thereby reducing the extent of MgO exsolution.It is crucial to note, however, that their thermodynamic model is based on data from the silicate melt-Fe system and the SiO2-Fe system without considering ferropericlase.As a result, in their model, MgO is implicitly treated as a component of liquid rather than as solid ferropericlase.Adding further complexity, instead of precipitating separate MgO and SiO2 solids, a Mg-Fe-Si-O system may yield exsolutions of MgSiO3 bridgmanite or post-perovskite.Indeed, bridgmanite and post-perovskite with low iron content are quite refractory, with melting temperatures exceeding the TCMB assumed here and thus may form stable exsolution phases (Deng et al., 2023;Zerr and Boehler, 1993).Whether bridgmanite, post-perovskite, solid SiO2, B1 MgO, or liquid is the stable exsolution phase depends on their free energies and is still open to question.Consequently, a comprehensive re-evaluation of the phase relations in the Mg-Si-O-Fe system and more broadly, in the Mg-Si-O-C-H-S-Fe system, which considers exsolutions as solids, is warranted.This study marks a first attempt to demonstrate the significance of solid exsolutions and the substantially different behaviors they exhibit during exsolution.element partitioning models from this study with crystalline ferropericlase as the exsolved phase (solid lines) and those from a recent study with silicate melts as the exsolved phase (dashed lines) (Liu et al., 2019), respectively.Red and green denote initial oxygen concentration in the core of 5 wt% and 1.6 wt% at 5000 K, respectively.

Conclusion
We developed a machine learning potential of ab initio quality for Mg-Fe-O system using the iterative training scheme, which enables large-scale atomistic simulations of Mg exsolution processes at 4000-5500 K and 140 GPa without any ad hoc assumptions regarding the stable exsolution phase.The exsolved phase is solid Fe-poor ferropericlase across all the thermodynamic conditions considered.Using the Gibbs dividing surface method, we analyze simulation trajectories, obtain the chemical composition of exsolved phases and liquid phases, and determine Mg and O exchange coefficients.The results show that partitioning of Mg into the exsolved phase is significantly enhanced when compared to scenarios where the exsolved phase is assumed to be liquid, as in previous studies (Badro et al., 2018;Du et al., 2019;Liu et al., 2019;Mittal et al., 2020).The resulting small Mg exchange coefficients suggest a reduced Mg solubility in the core.Assuming a reasonable initial core composition with 1.6-5 wt% oxygen, the MgO exsolution rate may be insufficient to generate the dipolar magnetic field at the Earth's surface with intensities that align with the paleomagnetic record.
Though not the focus of this study, it is noteworthy that our oxygen exchange coefficients are smaller than the previous ab initio results, indicating a reduced transport of FeO from ferropericlase into the core fluid (Davies et al., 2018), with implications for the dynamics of long-term core-mantle interaction.Moreover, solid exsolution may encapsulate distinctive core-characteristic signatures and transport them into the certain regions of the overlaying mantle (Helffrich et al., 2018), offering a valuable window to probe the core-mantle interaction (Deng and Du, 2023)

Introduction
Chemical buoyancy due to the crystallization of the inner core is believed to have supplied energy to power the geodynamo in the last 0.5-1 billion years (Nimmo, 2015).Paleomagnetic records suggest the existence of a very early (3.4 Ga) magnetic field in the Earth's history prior to the inner core crystallization (Tarduno et al., 2010).The energy source of this early geodynamo is enigmatic.Radiogenic heat production in the core may not be sufficient to sustain an early dynamo (Frost et al., 2022).The basal magma ocean may be electrically conductive (Stixrude et al., 2020), but the scale and longevity of a convective basal magma ocean are uncertain.
Recent studies propose that exsolution of oxides from the core upon cooling, such as MgO (O'Rourke and Stevenson, 2016) or SiO2 (Hirose et al., 2017), may be a viable mechanism to power an early dynamo.
Experimental studies on metal-silicate partitioning suggest that the solubility of Mg is highly sensitive to temperature (Badro et al., 2016;Du et al., 2017).The high-temperature equilibration between the metallic and silicate melts during the core formation process may result in a few wt% of MgO dissolved in the core.
Upon cooling, Mg is expected to precipitate out of the core as its solubility drops.However, the efficiency of this mechanism, especially for MgO oxide, remains controversial (Badro et al., 2016;Du et al., 2017).
The precipitation rate of MgO has been widely estimated using Mg partitioning behaviors in the metalsilicate system (Badro et al., 2018;Du et al., 2019;Liu et al., 2019).This estimation, strictly speaking, is unjustified.In contrast to the core formation process, where metallic and silicate melts equilibrate, the precipitation process involves the equilibration between the metallic melt and exsolution, where the exact phase and chemistry of exsolution depend on bulk compositions and thermodynamic conditions (Helffrich et al., 2020).Previous estimates, however, implicitly assume that exsolved MgO is a component of liquid silicate (Badro et al., 2018;Du et al., 2019;Liu et al., 2019).This assumption is questionable, as MgO is more refractory than SiO2 which may exsolve out of the core in solid-sate (Hirose et al., 2017).Therefore, a careful examination of the Mg exsolution process is necessary.
In this study, we combine enhanced sampling, feature selection, and deep learning to develop a unified machine learning potential (MLP) for the Mg-Fe-O system.This MLP is used to perform large-scale molecular dynamics simulations to study the exsolution of Mg from core fluids.Unlike previous computational studies based on free energy calculations (Davies et al., 2018;Wahl and Militzer, 2015;Wilson et al., 2023), this method does not prescribe the state of the exsolved phase (Sun et al., 2022).The results inform the stable state of MgO precipitation, Mg and O partitioning between core fluid and exsolution, and the efficiency of MgO exsolution in powering an early geodynamo.

Development of Machine Learning Potential
A machine learning potential (MLP) is a non-parametric model that approximates the Born-Oppenheimer potential energy surface.We follow the same approaches outlined in our previous work on Mg-Si-O (Deng et al., 2023) and Mg-Si-O-H (Peng and Deng, 2024) where details of the machine learning process can be found.To briefly summarize our approach, the MLP is trained on a set of configurations drawn from multithermal and multibaric (MTMP) simulations (Piaggi and Parrinello, 2019), which are used to efficiently sample the multi-phase configuration space.We use the structure factor of B1 MgO as the collective variable to drive the sampling, and an iterative learning scheme as described by (Deng et al., 2023) to efficiently select distinct samples from molecular dynamics trajectories.High-accuracy ab initio calculations are performed on the selected sample configurations to derive the corresponding energies, atomic forces, and stresses.The DeePMD approach is employed to train an MLP which takes a configuration (a structure of a given atomic arrangement) and predicts its energy, atomic forces, and stresses without iterating through the time-consuming self-consistent field calculation (Wang et al., 2018;Zhang et al., 2018).The details of DeePMD approach and density functional theory (DFT) calculations can be found at Supplementary Information (Text S1, Figure S1).

Two-phase molecular dynamics simulation
Two-phase simulations are performed on a pure MgO system to determine the melting point of B1 MgO.Alfè (2005) found that systems of 432 atoms are sufficient to yield converged melting points as those larger systems.Here, supercells of 432 atoms are constructed and then relaxed for 1000 steps at desired pressure and temperature conditions in the NPT ensemble.The relaxed cell is then used to perform NVT simulations at high temperatures far exceeding the melting temperatures, with the atoms of half the cell fixed and the force applied to these atoms set to 0. The resulting structure is half-molten and half-crystalline.We relax this structure again at the target pressure and temperature for 1000 steps to obtain the initial configuration for two-phase simulations.Simulations on the two-phase supercell of solid-liquid coexistence were then performed.If the whole cell is molten (or crystallized) at the end, the simulation temperature is above (or below) the melting point.The state of the system can be determined by analyzing the radial distribution functions, allowing us to pinpoint the upper and lower bounds of the melting point.

Exsolution simulation
We construct systems of various Mg:Fe:O ratios by substituting/removing Mg and/or O atoms of supercells of B1 MgO.Initial configurations are melted at 8000 K and 140 GPa under the NPT ensemble for ~10 ps.
We inspect trajectories and radial distribution functions to ensure systems are fully molten and well relaxed.
The resulting configurations are further used to perform simulations at 140 GPa and target temperatures under the NPT ensemble for up to several nanoseconds to simulate the exsolution process.

Gibbs dividing surface
To determine the composition of the two coexisting phases, for every frame we locate the Gibbs dividing surface (GDS) that separates the whole cell into an oxide region, a metallic region, and two interfaces in between (Sega et al., 2018;Willard and Chandler, 2010).For every snapshot, we calculate the coarsegrained instantaneous density field at point r by where  $ is the position of the ith atom; s is the coarse-graining length, and is set as 2.5 Å here (Willard and Chandler, 2010).The instantaneous surface s(t) is defined as a contour surface of the instantaneous coarse-grained density.The proximity of ith atom to this surface is where (t) is the surface normal in the direction of the density gradient at that point.This instantaneous density field is then projected onto each atom and associated with the corresponding proximity.We find that ρ() follows the expected form where roxide and rmetal are the density of the oxide phase and metal phase, respectively; a0 and a1 are the positions of the Gibbs dividing surfaces; w is the thickness of the interface.Fitting ρ() to Eq. ( 3) yields the location of the Gibbs dividing surfaces, as well as the density of each phase.The metal and oxide phases are defined by  $ > 2 and  $ < −2, respectively.We count the number of atoms of metal and oxide phases of each frame.A long trajectory after the system is equilibrated and used to determine the average concentrations in each phase and associated standard deviations.For more details, the reader is referred to (Deng and Du, 2023;Sega et al., 2018;Willard and Chandler, 2010;Xiao and Stixrude, 2018).

Benchmarks of the Machine Learning Potential
We compare the energies, atomic forces, and stresses from the MLP to those from DFT calculations for 15078 configurations that are not included in the training set (Figure S2).The root-mean-square errors of the energies, atomic forces, and stresses are 6.34 meV/atom −1 , 0.27 eV/Å −1 , and 0.48 GPa, respectively.
We perform two additional tests to further examine the reliability of the MLP.First, we perform MD simulations with supercells of B1 MgO solid, MgO liquid, and a Mg-Fe-O liquid mixture, respectively.
These supercells are larger than the training configurations.The root-mean-square error of energy prediction by the MLP with respect to the DFT calculations is similar to the error in the testing sets (Figure S3).This verification test further proved the accuracy of energy prediction and also demonstrated the transferability of the MLP to structures larger than the train/test sets.Second, we calculate the melting point of B1 MgO at 140 GPa using the solid-liquid two-phase coexistence method with a supercell of 432 atoms.
For both DFT and MLP, the system crystalizes at 7700 K and melts at 7800 K, suggesting a melting point of 7750±50 K at 140 GPa and further validating the robustness of the MLP.The melting temperature is also consistent with previous studies (Alfè, 2005;Du and Lee, 2014).

System convergence of the exsolution simulation
The robust MLP of Mg-Fe-O system allows for large-scale exsolution simulations.We first examine the convergence of Fe liquid composition with respect to the simulation cell size by performing exsolution simulations at 5000 K and 140 GPa with five Mg-Fe-O liquid mixtures, where ratios of Mg, O and Fe atoms are fixed as 2:2:3, i.e., Mg64O64Fe96, Mg512O512Fe768, Mg1728O1728Fe2592, Mg2304O2304Fe3456, Mg3136O3136Fe4704.
For all simulation, the system quickly demixes to form MgO-rich and Fe-rich region, and subsequently, MgO-rich region spontaneously crystallizes to form ferropericlase while metallic phase remains liquid.The resulting atomic fraction of Mg, O in the metallic phase converges when system size reaches 2000 atoms (Figure S4).Large systems also yield better statistics and thus the smaller uncertainties in the atomic fraction.Based on this test, all the partitioning results reported here are derived from simulations performed with systems of more than 2000 atoms to ensure convergence and robust statistics.

Exsolution process
In all simulations considered, exsolution spontaneously occurs within a few picoseconds at 4000 K to a few nanoseconds at 5500 K. Exsolutions are all solid ferropericlase with small amounts of FeO.The interfaces between exsolution and Fe liquid are typically irregular as they form spontaneously without interference.
Taking the exsolution simulation of Mg2088Fe3456O2520 liquid at 140 GPa and 5500 K as an example (Figure 1; Supplementary movie 1).It starts with a homogeneous liquid (Figure 1a), and quickly demixes to form patches of MgO-rich liquid and Fe-rich liquid with a continuous drop of potential energy.Within around 250 ps, MgO-rich patches and Fe-rich patches conglomerate, respectively, dividing the whole cell into two regions: one enriched in MgO and the other Fe.MgO-rich region remains liquid for another 750 ps until a sudden crystallization occurs to form ferropericlase (Figure 1b).The crystallization is a rapid process accompanied by a significant drop in potential energy.Quickly after ferropericlase crystallizes, the potential energy plateaus.The element exchange between ferropericlase and residual Fe liquid continues within the interface region.We analyze the trajectories at this stage, calculate the Gibbs dividing surface, and determine the average composition of each phase for the last 100 ps.The chemical compositions of both phases are shown in Figure 1c.The metallic liquid is oxygen rich and magnesium poor.The exsolved ferropericlase is of B1 structure and is nearly stochiometric (Mg0.974Fe0.026)O.Similar analyses have been applied to all other exsolution simulations, and the compositions of simulation products are summarized in Table S1.with the separation of ferropericlase and metallic liquid.The sudden drop of internal energy corresponds to the crystallization of ferropericlase.

Element partitioning
We further analyze the element partitioning between the exsolved ferropericlase and Fe liquid considering two dissociation reactions: MgO ox = Mg met + O met and FeO ox = Fe met + O met , where superscripts ox and met indicate oxide and metal, respectively.We also consider Mg exchange reaction with MgO ox + Fe met = FeO ox + Mg met , but the fit is poor, as also reported in other studies (Badro et al., 2018;Liu et al., 2019) standard thermodynamic model with the non-ideality described by the epsilon formalism of (Ma, 2001).
The phase relation between ferropericlase and Fe liquid has been studied mostly at low pressures and low temperatures (Asahara et al., 2007;Frost et al., 2010;Ozawa et al., 2008).Unfortunately, Mg contents in metallic melts were not reported and only  2 7 were reported in these studies.Thus, we only include  2 7 of these experiments in the fitting (Texts S2, S3; Tables S2).
Our calculated  2 34 is slightly larger than that reported by a recent ab initio calculation (Wilson et al., 2023) where the pure B1 MgO is assumed as the exsolved phase, while our exsolution simulations show that precipitates contain small amounts of FeO (Figure 1, Table S1).The incorporation of FeO in the exsolved phase likely changes the free energy of the system, leading to different in  2 34 .Wahl and Militzer (2015) also perform ab initio simulations on the Mg-Fe-O system but focus on high temperatures close to the solvus closure and do not report Mg partitioning results at the conditions overlapping this study, which precludes a direct comparison.Compared with the previously determined  2 34 between Fe liquid and silicate melt (Badro et al., 2018;Du et al., 2019;Liu et al., 2019),  2 34 between Fe liquid and solid ferropericlase shows similar temperature dependence but is overall approximately one order of magnitude lower (Figure 2a), indicating a low Mg content in Fe liquid when equilibrated with ferropericlase.This is expected as MgO preferentially enters ferropericlase when silicate melt crystalizes (Boukaré et al., 2015).
Oxygen partitioning between ferropericlase and liquid Fe is strongly controlled by temperature, in agreement with previous experiments (Asahara et al., 2007;Frost et al., 2010;Ozawa et al., 2008) and calculations (Davies et al., 2018). 2 7 between Fe liquid and silicate melt derived by (Liu et al., 2019) generally aligns with  2 7 between Fe liquid and solid ferropericlase, especially at high temperatures.Our  2 7 can be well fitted with previous experimental data to a unified thermodynamic model (Asahara et al., 2007;Ozawa et al., 2008), except for the four data points reported by (Frost et al., 2010) (Text S3; Table S3).At around 30-70 GPa and with similar oxygen contents in the liquid Fe,  2 7 of (Frost et al., 2010) are around half log unit higher than those of (Ozawa et al., 2008) and our extrapolated results.The source of this discrepancy is unknown but may arise from the carbon contamination. 2 7 reported by an early DFT calculation (Davies et al., 2018) is around 0.3-0.6 log unit higher than those of (Ozawa et al., 2008) and our results at similar conditions.We note that Davies et al. (2018) calculated the chemical potential of FeO for defect-free (Mg,Fe)O.Yet, both our simulations and previous studies (Karki and Khanduja, 2006;Van Orman et al., 2003) support the existence of defects in ferropericlase at high temperatures, which may lower the free energy of the host mineral and enrich FeO in ferropericlase, leading to a reduced  2 7 .(Badro et al., 2018), (Liu et al., 2019), and (Du et al., 2019), respectively.(b) Experimental studies include (Ozawa et al., 2008) (O08, downward triangle), (Asahara et al., 2007) (A07, diamond), (Frost et al., 2010) (F10, upward triangle).DFT study includes (Davies et al., 2018) (D18, square).All previous results are normalized to 140 GPa for a direct comparison using the best-fit pressure dependence, , where P/T are experimental/calculation pressure/temperature, and  7 is a fitted constant (Table S3).L19 (dashed line) indicates the O exchange coefficient of silicate-melt calibrated by (Liu et al., 2019).Uncertainties of the exchange coefficients of this study are roughly represented by the symbol size.

Exsolution rate and geodynamo
Earth's accretion and differentiation in its early history likely resulted in a core much hotter than it is today.
The precipitation of light elements due to the secular cooling of the core may have provided a vital energy source to drive the geodynamo.The energetics of the exsolution-powered dynamo hinge on the cooling rate and the exsolution rate.Here, we adopt a core thermal evolution model proposed by O'Rourke et al. (2017) where the CMB temperature (TCMB) drops from around 5000 K to around 4100 K over the first ~3.8 billion years (Gyr) with a cooling rate of ~230 K Gyr -1 .Given this thermal history of the core, the phase of exsolution and the associated exsolution rate can be further determined using the element partitioning models, along with knowledge of the initial core composition.All previous modeling of Mg exsolution from the core assume that MgO exsolve as a component of silicate melts.However, our simulations show that MgO should exsolve as a component of crystalline ferropericlase, at least when light elements other than Mg and O are absent (Badro et al., 2018;Du et al., 2019;Liu et al., 2019;Mittal et al., 2020).Here, we first examine the Mg exsolution and its potential to drive the early geodynamo for an Mg-and O-bearing core, and then we discuss the effects of additional light elements.
To model Mg exsolution from a core fluid with only Mg and O as light elements, we first determine the saturation conditions under which Mg precipitates.Previous N-body simulations and metal-silicate equilibrium experiments suggest that the Earth's core following its formation may contain 1.6-5 wt% O (Fischer et al., 2017;Liu et al., 2019;Rubie et al., 2015).The corresponding saturation magnesium concentration in the core is 0.04-0.19wt% at 140 GPa, as determined using the  2 34 between metal and ferropericlase, which is significantly lower than that determined by  2 34 between metal and silicate melt (Liu et al., 2019) (Figure 3a).This difference is expected, as the former  2 34 value is about one order of magnitude smaller than the latter (Figure 2a).Hence, our work implies that a substantial amount of Mg may have already been exsolved by the time the core cools to a TCMB of 5000 K. Further cooling reduces the Mg solubility in the core, with concentrations approaching 0.02-0.003wt% at a TCMB of 4000 K.This suggests a diminishingly small amount of Mg in the present-day outer core.We use these saturation magnesium conditions as the initial core composition in our exsolution modeling.
Despite the contrasting Mg solubilities in the core, the compositions of the exsolutions are similar and exhibit a comparable trend with temperature.Specifically, the exsolved phase in both models becomes increasingly FeO-rich with cooling.At 4000 K, the exsolution contains up to 20 wt% FeO (Figure 3b).
Throughout the thermal history, TCMB is lower than the solidus of exsolved ferropericlase (Deng et al., 2019), indicating that exsolutions remain solid.
The resulting exsolution rates decrease with temperature, with the values dropping from 1.4-5.6 ×10 -6 K -1 at 5000 K to 0.2-1.0×10 -6 K -1 at 4000 K. Exsolution rates of ferropericlase are approximately one order of magnitude smaller than those predicted for silicate melt exsolution (Figure 3c).Exsolutions are depleted in iron and enriched in Mg.As a result, they are lighter than the outer core fluid and thus provide the buoyancy flux that may sustain an exsolution-driven dynamo (O'Rourke and Stevenson, 2016).Converting the exsolution rate to the magnetic field intensity is model dependent, however.The upper bound of the exsolution rates (1-5.6 ×10 -6 K -1 ) derived here are similar to the previous reports (Badro et al., 2016;Du et al., 2017).While Du et al. (2019) conclude that this exsolution rate is not sufficient to power the early geodynamo alone, Badro et al. (2018) use a scaling law that relates the exsolution rate to dipolar magnetic field intensity ( DEFGHI6 J;K8L6 ) and argue that MgO exsolution can well produce the dipolar magnetic field intensity at Earth's surface consistent with observations.We follow (Badro et al., 2018) to convert the exsolution rate to  DEFGHI6 J;K8L6 (Figure 3d).The results show that  DEFGHI6 J;K8L6 generated by the upper bound exsolution rate is broadly consistent with the paleo-intensities records dating back to 3.4 Gyr (Tarduno et al., 2010), and that generated by the lower bound rate is overall smaller than the observations and thus may not be sufficient (Tarduno et al., 2015).Overall, we find that MgO exsolution alone may be difficult to power the early geodynamo, but it is nevertheless an important energy source.
While the exact composition of the core remains unknown, it may contain other light elements, such as S, Si, C, and H (Hirose et al., 2021).As the core cools, the solubility of these light elements tends to decrease, leading to their exsolution.For example, in a core composed solely of Si, O, and Fe, the exsolved phase would likely be solid SiO2 (Hirose et al., 2017;Zhang et al., 2022).The study by Helffrich et al. (2020) on the joint solubility of Mg, O, and Si in liquid Fe suggests that the presence of Si enhances the retention of Mg in metal, thereby reducing the extent of MgO exsolution.It is crucial to note, however, that their thermodynamic model is based on data from the silicate melt-Fe system and the SiO2-Fe system without considering ferropericlase.As a result, in their model, MgO is implicitly treated as a component of liquid rather than as solid ferropericlase.Adding further complexity, instead of precipitating separate MgO and SiO2 solids, a Mg-Fe-Si-O system may yield exsolutions of MgSiO3 bridgmanite or post-perovskite.Indeed, bridgmanite and post-perovskite with low iron content are quite refractory, with melting temperatures exceeding the TCMB assumed here and thus may form stable exsolution phases (Deng et al., 2023;Zerr and Boehler, 1993).Whether bridgmanite, post-perovskite, solid SiO2, B1 MgO, or liquid is the stable exsolution phase depends on their free energies and is still open to question.Consequently, a comprehensive re-evaluation of the phase relations in the Mg-Si-O-Fe system and more broadly, in the Mg-Si-O-C-H-S-Fe system, which considers exsolutions as solids, is warranted.This study marks a first attempt to demonstrate the significance of solid exsolutions and the substantially different behaviors they exhibit during exsolution.element partitioning models from this study with crystalline ferropericlase as the exsolved phase (solid lines) and those from a recent study with silicate melts as the exsolved phase (dashed lines) (Liu et al., 2019), respectively.Red and green denote initial oxygen concentration in the core of 5 wt% and 1.6 wt% at 5000 K, respectively.

Conclusion
We developed a machine learning potential of ab initio quality for Mg-Fe-O system using the iterative training scheme, which enables large-scale atomistic simulations of Mg exsolution processes at 4000-5500 K and 140 GPa without any ad hoc assumptions regarding the stable exsolution phase.The exsolved phase is solid Fe-poor ferropericlase across all the thermodynamic conditions considered.Using the Gibbs dividing surface method, we analyze simulation trajectories, obtain the chemical composition of exsolved phases and liquid phases, and determine Mg and O exchange coefficients.The results show that partitioning of Mg into the exsolved phase is significantly enhanced when compared to scenarios where the exsolved phase is assumed to be liquid, as in previous studies (Badro et al., 2018;Du et al., 2019;Liu et al., 2019;Mittal et al., 2020).The resulting small Mg exchange coefficients suggest a reduced Mg solubility in the core.Assuming a reasonable initial core composition with 1.6-5 wt% oxygen, the MgO exsolution rate may be insufficient to generate the dipolar magnetic field at the Earth's surface with intensities that align with the paleomagnetic record.
Though not the focus of this study, it is noteworthy that our oxygen exchange coefficients are smaller than the previous ab initio results, indicating a reduced transport of FeO from ferropericlase into the core fluid (Davies et al., 2018), with implications for the dynamics of long-term core-mantle interaction.Moreover, solid exsolution may encapsulate distinctive core-characteristic signatures and transport them into the certain regions of the overlaying mantle (Helffrich et al., 2018), offering a valuable window to probe the core-mantle interaction (Deng and Du, 2023) increased from the Gamma-point only to a 2×2×2 Monkhorst-Pack mesh.We found this high precision recalculation to be important for optimizing the robustness of the MLP (Deng and Stixrude, 2021b) .atoms in metallic phase as a function of system size (i.e., number of atoms in the system) at 5000 K and 140 GPa.The ratio of the numbers of Mg, O, and Fe are 2:2:3 for all the systems.

Text S3 Regression
We fit the exsolution simulation results and previous experiments on ferropericlase-Fe partitioning simultaneously to resolve the thermodynamic parameters.We exclude the data with reported carbon contamination (Du et al., 2019b).Early experiments do not report Mg contents in the liquid Fe and are therefore only used to constrain O partitioning.We first explore fitting the oxygen partitioning independently using data from this study and experimental results from (Asahara et al., 2007;Frost et al., 2010;Ozawa et al., 2008).The goodness of fit (measured by R 2 ) dramatically drops from 0.93 to 0.74 as long as the four data points reported by (Frost et al., 2010) are included.This is largely due to the conflicting results between (Frost et al., 2010) and (Ozawa et al., 2008) at around 30-70 GPa.We therefore reject the data of (Frost et al., 2010) for fitting.
We then fit both  & '( and  & * simultaneously using the data of this study and (Asahara et al., 2007;Ozawa et al., 2008).Based on F-test, the model with or without  '( and  '( '( fit the data equally well.As such,  '( and  '( '( are set to be 0. The negligible roles of  '( and  '( '( on partitioning are consistent with previous studies on metal-silicate systems (Badro et al., 2018;Du et al., 2019a;Liu et al., 2019).The resulting fitted parameters are listed in Table S3.Only data of Asahara et al., 2007 andOzawa et al., 2008 are used for fitting.

Figure 1 .
Figure 1.Molecular dynamics simulation of spontaneous ferropericlase exsolution from a homogeneous Mg2088Fe3456O2520 liquid at 140 GPa and 5500 K (NPT ensemble).(a) The initial configuration at 1 fs with a homogeneous distribution of Mg (green), Fe (yellow), and O (red) atoms.(b) The final configuration at 1.5 ns.Dark red planes are the Gibbs diving surfaces that separates the whole systems into crystalline ferropericlase, interface, and metallic liquid.The cell dimension is 50.9Å × 39.9 Å × 35.3 Å initially (a) and becomes 47.9 Å × 37.6 Å × 33.3 Å at the end of the simulation (b).(c) Evolution of number of atoms in liquid (liq) and solid exsolution (sol) in the last 100 ps.(d) Evolution of potential energy.Energy drops

Figure 2 .
Figure 2. Mg (a) and O (b) exchange coefficients as a function of oxygen content in the iron (XO) and temperature at 140 GPa.Solid circles are the results from this study, and solid lines are the best-fit curves (Supplementary Texts S2, S3).The color of curves and symbols represents the value of XO.Previous calculations and experiments are also shown for comparison.(a) W23 denotes the DFT calculation result by (Wilson et al., 2023) (upward triangle).B18 (dotted line), L19 (dashed line), and D19 (dotted-dashed line) represents the Mg exchange coefficient between silicate melt and Fe liquid calibrated by(Badro et al.,

Figure 3 .
Figure 3. MgO solubility (a), chemical composition of the exsolution (b), exsolution rate (c), intensity of the dipolar magnetic field at Earth's surface produced by exsolution-driven dynamo (d) based on the Our MLP explores a wide compositional space, trained on Mg-Fe-O systems of varying Mg:Fe:O ratios, including the pure endmembers, Fe and MgO, as well as intermediate compositions denoted by (MgO)aFebOc , where a=0-64, b = 0-64, c = 0-16 with 2a+b≥ 64.The final training set consists of 4466 configurations generated at pressure up to 200 GPa and temperature up to 8000 K.

Figure 1 .
Figure 1.Molecular dynamics simulation of spontaneous ferropericlase exsolution from a homogeneous Mg2088Fe3456O2520 liquid at 140 GPa and 5500 K (NPT ensemble).(a) The initial configuration at 1 fs with a homogeneous distribution of Mg (green), Fe (yellow), and O (red) atoms.(b) The final configuration at 1.5 ns.Dark red planes are the Gibbs diving surfaces that separates the whole systems into crystalline ferropericlase, interface, and metallic liquid.The cell dimension is 50.9Å × 39.9 Å × 35.3 Å initially (a) and becomes 47.9 Å × 37.6 Å × 33.3 Å at the end of the simulation (b).(c) Evolution of number of atoms in liquid (liq) and solid exsolution (sol) in the last 100 ps.(d) Evolution of potential energy.Energy drops

Figure 2 .
Figure 2. Mg (a) and O (b) exchange coefficients as a function of oxygen content in the iron (XO) and temperature at 140 GPa.Solid circles are the results from this study, and solid lines are the best-fit curves (Supplementary Texts S2, S3).The color of curves and symbols represents the value of XO.Previous calculations and experiments are also shown for comparison.(a) W23 denotes the DFT calculation result by (Wilson et al., 2023) (upward triangle).B18 (dotted line), L19 (dashed line), and D19 (dotted-dashed line) represents the Mg exchange coefficient between silicate melt and Fe liquid calibrated by(Badro et al.,

Figure 3 .
Figure 3. MgO solubility (a), chemical composition of the exsolution (b), exsolution rate (c), intensity of the dipolar magnetic field at Earth's surface produced by exsolution-driven dynamo (d) based on the

Figure S1 .
Figure S1.Convergence tests of total energy (a) and pressure (b) with varying energy cutoffs (ENCUT flag in VASP) for a mixture of Mg64O64Fe64 at the static condition.An energy cutoff of 800 eV is sufficient to obtain converged results for both energy and pressure.

Figure S2 .
Figure S2.Comparisons of energies (a), atomic forces (b), and stresses (c) between DFT and the machine learning potential (MLP) for all the test data at temperatures up to 8000 K and pressures up to ~200 GPa.15078 energies, 5988786 force components, and 135702 stress components are included in these comparisons.The red dashed lines are guides for perfect matches. .

Table S1 Summary of exsolution simulations at 140 GPa. NMg , NFe , and NO are the numbers of 129 Mg, Fe, O atoms, respectively.
The difference between the composition of the bulk system and the sum 130 of those of ferropericlase and metallic liquid yields the composition of the corresponding interface.

Table S2 Summary of previous experimental results on ferropericlase-Fe element partitioning. Experiments with reported carbon and sulfur contamination are excluded
. (see the content in a separate supplementary file)

Table S2 .
Summary of previous experimental results on ferropericlase-Fe element partitioning.Experiments with reported carbon and sulfur contamination are excluded.XFeO, XFe, XO are the FeO content in ferropericlase, Fe content in metallic liquid, O content in metallic liquid, respectively.