Thermodynamic Properties of Liquid Fe-Mg Alloys Under Outer-Core Conditions Using First-Principles Molecular Dynamics

Abstract

Magnesium (Mg) partitioning behavior between solid and liquid iron (Fe) shows that Mg slightly favors liquid Fe under conditions at the Earth’s core. This means that the Earth’s outer core may contain more Mg than previously thought. However, geophysical properties such as density (ρ) and sound velocity (V_P) of liquid Fe-Mg alloys under outer-core conditions have yet to be explored. Considering that the liquid outer core includes approximately 10 wt.% light elements, here we established an equation of state (EoS) of liquid Fe-Mg alloys with Mg less than 10 wt.% to study the thermodynamic properties under outer-core conditions using first-principles molecular dynamics. The results show that adding Mg to liquid Fe will clearly reduce the ρ, isothermal bulk modulus (K_T), and adiabatic bulk modulus (K_S). Meanwhile, it will increase the V_P. In order to access the Mg content in the outer core, the ρ and V_P of liquid Fe-Mg alloys along the geotherm are compared with the preliminary reference Earth model. Assuming Mg is the only light element, the maximum content of Mg required is approximately 2.9–6.1 wt.% due to the temperature uncertainty at the inner core boundary (ICB). Further, considering the geochemical constraints (the partition coefficient between liquid and solid Fe, molten Fe-alloy, and silicate melts), the content of Mg is further constrained to below 0.5 wt.%.

1. Introduction

The early Earth was a magma ocean. With the Earth cooling, siderophile and lithophile elements began to separate, and Earth’s solidification led to the formation of the core and mantle until metal–silicate equilibrium was achieved [1]. The liquid outer core, which is the intermediate layer between the Earth’s inner core and the mantle, is primarily composed of liquid iron (Fe) [2,3]. However, seismic observations indicate that the ρ and V_P of pure Fe are approximately 10% higher and 5% lower than those of the outer core, respectively [4]. Therefore, the liquid outer core may contain some impurities—light elements, such as sulfur (S), silicon (Si), oxygen (O), hydrogen (H), carbon (C), and nitrogen (N)—which reduce ρ and increase V_P [5,6,7,8,9]. However, as one of the abundant elements in Earth, the role of Mg in the composition and dynamics of the outer core remains understudied.

Emerging evidence indicates that early precipitation of Mg-bearing minerals from the core could have supplied the energy required to sustain the geodynamo [10,11]. As the Earth crystallized and solidified, the majority of Mg concentrated in the lower mantle [12]. Theoretical investigations of Mg partitioning between solid and liquid Fe nevertheless reveal a modest preference for liquid Fe, yet a non-negligible fraction of Mg still enters the solid Fe state. This implies that the outer core could host approximately ~6 wt.% in earlier times and ~1 wt.% by now, with a higher Mg inventory than previously thought [13]. But the experimental studies about the partitioning of Mg between molten Fe-alloys and silicate melts at high pressure and temperature indicate that Mg incorporates less than 0.5 wt.% in the core formation process [14]. In addition, the lowermost mantle may exist in a partially molten state, which could facilitate the transfer of Mg to the outer core [15,16] to enrich the Mg content over geological time.

Meanwhile, Mg has received considerable attention as a potential light element in the Earth’s inner core [17,18]. Mg has never been seriously considered for inclusion in the inner core before, since under environmental conditions, Mg does not form an alloy with Fe at all. However, high pressure and high temperature can break this phenomenon. As long as the pressure increases to 45 GPa, Mg can be more siderophile with increasing temperature [19]. So the solid solubility of Mg in Fe alloy increases rapidly [20]. This miracle can be attributed to electronegativity differences, as Mg is dramatically changed under high pressure, which also led to the discovery of unexpected Fe-Mg compounds (FeMg₄, FeMg₃, FeMg₂, Fe₂Mg₃, FeMg, and Fe₂Mg) under inner core pressures [21]. A recent study shows that oxygen can reduce the enthalpy of Fe-Mg alloy and promote its stability under high pressure [22]. Given that the inner core appeared approximately one billion years ago and grew with the crystallization of the outer core [23], it is reasonable to deduce that the outer core contains the Mg element.

Although the amount of Mg in the outer core is small, it is likely to contribute an important light-element effect to liquid Fe alloy under the Earth’s outer-core conditions. Therefore, assuming Mg to be the sole light element in the outer core, we explored the effect of Mg on the physical properties of liquid Fe-Mg alloys (with Mg less than 10 wt.%) by first-principles molecular dynamics (FP-MD) simulations. This approach was adopted to disentangle the intrinsic effects of Mg concentration on thermodynamic properties and constrain Mg content in the outer core from a geophysical aspect. We firstly established an EoS of liquid Fe-Mg alloys and then investigated the effect of Mg on ρ, the coefficient of thermal expansivity (α), the Grüneisen parameter (γ), K_T and K_S, and V_P under the Earth’s outer-core conditions. Consequently, compared with ρ and V_P from the PREM, we constrained the maximum concentration of Mg in the Earth’s outer core.

2. Methods

2.1. Computational Method

We simulated the liquid Fe-Mg alloys by FP-MD based on the density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP) [24,25]. All FP-MD simulations are constrained by the canonical ensemble with a fixed number of atoms (N = 108 atoms), constant volume (V), and constant temperature (T), and constrained by a Nosé–Hoover thermostat [26,27]. The electronic configurations considered were fourteen valence electrons (3p⁶3d⁷4s¹) for Fe and eight valence electrons (2p⁶3s²) for Mg. The cutoff radii were 2.2 a.u. for Fe and 1.7 a.u. for Mg. To ensure the accuracy of the calculation results, we set the cutoff energy to 450 eV after the cutoff energy test (see Figure S1 in Supplementary Materials). The finite temperature effects in the electronic structure and force calculations were incorporated using the Fermi–Dirac smearing method [28]. Gamma point sampling was used.

The initial liquid Fe structure was obtained by melting face-centered, cubic phase Fe at 10,000 K for 5 ps with a time step of 0.5 fs. In order to explore the effect of Mg on molten Fe alloys, we studied five compositions: pure Fe, Fe₁₀₃Mg₅ with 2.06 wt.% Mg, Fe₉₈Mg₁₀ with 4.24 wt.% Mg, Fe₉₃Mg₁₅ with 6.96 wt.% Mg, and Fe₈₈Mg₂₀ with 8.98 wt.% Mg. The liquid Fe-Mg alloy structures were obtained by randomly replacing Fe atoms with Mg atoms to a fixed Mg component. Each Fe-Mg alloy was quenched to a target temperature of 4000, 5000, or 6000 K. The target pressures were realized by varying the simulation cell box. Here, the time step set was 1 fs. Each simulation run lasted for 10 ps. The simulation duration was long enough to ensure system stabilization (see Figure S2 in Supplementary Materials). The pressure was the statistical average of the latter 7 ps with the first 3 ps to reach equilibrium. We monitored the liquid state by mean square displacement (MSD) and radial distribution function (RDF). Finally, we collected the P-V-T data of the liquid Fe-Mg alloys to fit an EoS. The P-V-T data are listed in the Zenodo reservoir [29].

2.2. Pressure Correction

Considering the pressure underestimated from DFT calculations, we systematically corrected the raw simulated pressure according to experimental data using the DAC approach [30]. During the correction process, it was the high-pressure and high-temperature experimental data of pure Fe reported by Kuwayama et al. [30] that were used as a benchmark. The correction method employed was proposed by French et al. [31], with pressure deviation and volume satisfying the following relationship:

$$∆P(\mathrm{V})={∆P}_0{\left(\frac{V_{0c}}{V}\right)}^{\chi +1}\mathrm{e}\mathrm{x}\mathrm{p}\left[{\displaystyle \frac{\chi +1}{\chi }}\left(1-{\left({\displaystyle \frac{V_{0\mathrm{c}}}{V}}\right)}^{\chi }\right)\right]$$

(1)

where $\mathsf{\Delta }{P}_0$, $V_{0c}$, and $\chi$ are parameters, which were obtained through least-squares fitting. We have presented the experimentally determined pressures (P_exp), the first-principles calculated pressures (P_cal), and the pressure deviations (∆P) for the pure Fe under various conditions in Table S1 of the Supplementary Materials. The parameters $\mathsf{\Delta }{P}_0$, $V_{0c}$, and $\chi$ were fitted using ∆P and V, yielding results of 19.13 GPa, 96.22 × 10⁻³cm³/g, and −0.5, respectively. Applying the correction function, we conducted a systematic correction of all theoretically pressures of liquid Fe-Mg alloys at different temperatures in this study. After that, the corrected pressure data (P_corr) were used to construct an EoS. The necessity of pressure correction is explicitly stated in the Supplementary Materials.

2.3. Equation of State of Liquid Fe-Mg Alloy

Based on the corrected P-V-T data of liquid Fe-Mg alloys, we constructed their thermal EoS. The EoS applied here is according to the Murnaghan [32,33] and Mie–Grüneisen–Debye (MGD) EoS [34]. The detailed formulation of the EoS refers to previous work [35]. The parameters in the EoS were fitted according to the P-V-T data of liquid Fe and liquid Fe-Mg alloys from FP-MD simulations by least-squares algorithm. The EoS parameters are listed in

Table 1. EoS parameters of liquid Fe-Mg.

Parameter a	Value
$K_{T_0-\mathrm{F}\mathrm{e}}$(GPa)	80.24
$K_{T_0-\mathrm{M}\mathrm{g}}$(GPa)	222.39
$K_{T_0-\mathrm{F}\mathrm{e}}^{\prime }$	3.38
$K_{T_0-\mathrm{M}\mathrm{g}}^{\prime }$	0.21
$V_{0-\mathrm{F}\mathrm{e}}$ (10−3 cm3/g)	172.31
$V_{0-\mathrm{M}\mathrm{g}}$ (10−3 cm3/g)	392.95
e0 (10−7/K)	2.19
$\mathrm{g}$	$-$4.76
γ0	1.43

^a Subscript zero means the values at 0 GPa and 6000 K.

. Due to the content of Mg being lower than 10 wt.%, we dealt with Fe-Mg by ideal mixing as used by Xie et al [35]. The bulk modulus $K_{T_0}$, the derivative of bulk modulus $K_{T_0}^{\prime }$, and volume V₀ at reference temperature T₀ are associated with the concentrations of Fe (X_Fe) and Mg (X_Mg). They are expressed as

$$K_{T_0}=K_{T_0-\mathrm{F}\mathrm{e}}{X}_{\mathrm{F}\mathrm{e}}+K_{T_0-\mathrm{M}\mathrm{g}}{X}_{\mathrm{M}\mathrm{g}}$$

(2)

$$K_{T_0}^{\prime }=K_{T_{0-\mathrm{F}\mathrm{e}}}^{\prime }{X}_{\mathrm{F}\mathrm{e}}+K_{T_{0-\mathrm{M}\mathrm{g}}}^{\prime }{X}_{\mathrm{M}\mathrm{g}}$$

(3)

and

$$V_0=V_{0-\mathrm{F}\mathrm{e}}{X}_{\mathrm{F}\mathrm{e}}+V_{0-\mathrm{M}\mathrm{g}}{X}_{\mathrm{M}\mathrm{g}}$$

(4)

To assess the accuracy of the established EoS of Fe-Mg alloys, we compared the computational isothermal ρ and V_P of liquid Fe with previous studies [30,34,35,36], considering the scarcity of available references about Fe-Mg alloys under outer-core conditions. Further details are discussed in Section 3.1.

3. Results

3.1. Thermodynamic Properties of Liquid Fe with Pressure Correction

In order to verify the accuracy of the EoS, we derived the ρ, K_S, and V_P of liquid pure Fe from the EoS obtained by pressure correction and compared them with the DAC [30] and shock-wave compression approaches [36] (see Figure 1). The comparison of uncorrected results with experimental and theoretical data is displayed in the Supplementary Materials. The ρ at 4000 and 5000 K from the EoS is consistent with the values from the DAC approaches [30]. At 5000 K, the ρ difference between the DAC and DFT approaches at the ICB is approximately 0.2 g/cm³. Our calculated ρ differs from the experimental value of Kuwayama et al. by only 0.138 g/cm³ (1.15% relative error), and the V_P discrepancy is 0.04 km/s (0.44%). With the temperature increase, the ρ decreases, which is also in line with DAC results. As for the K_S, it shows excellent agreement with DAC results [30] as well as V_P. In addition, the consistency with DAC data ensures the reliability of the EoS. The results also illustrate that pressure is a primary factor affecting ρ and V_P, while the influence of temperature on V_P is quite limited.

3.2. The Effects of Mg on Liquid Fe-Mg Alloys

After obtaining the EoS, we explored the impact of Mg concentration on the thermodynamic properties of liquid Fe-Mg alloys, including K_T, α, γ, and K_S, as shown in Figure 2. Evidently, its effect on K_T and K_S cannot be overlooked. The addition of Mg can decrease both the K_T and K_S. Taking Fe-8.98 wt.% Mg as an example, its shortages of K_T and K_S compared with liquid pure Fe are −7.31% (Figure 2a) and −7.22% (Figure 2d) at 330 GPa. This implies that it is necessary to consider the impact of light element concentrations on K_T and K_S and include them as constraints to study the composition of the outer core. However, the concentration of Mg has a minimal impact on α (Figure 2b) and γ (Figure 2c). With the increase of Mg content in liquid Fe-Mg alloys, the α and γ increase slightly. The 8.98 wt.% Mg only results in a +2.38% increase for α. The concentration of Mg exhibits negligible influence on γ along the pressure range of the outer core.

In the following, we calculated the ρ and V_P under the isothermal conditions at 4000, 5000, and 6000 K to assess the approximate range of Mg concentration in the outer core. As shown in Figure 3, the addition of Mg decreases the ρ of liquid Fe-Mg alloys notably, as well as with temperature. Since the atomic mass of Mg is much smaller than Fe, the ρ of Fe-Mg alloys is inversely proportional to the concentration of Mg. As for the impact on the isothermal V_P, the increasing Mg content can indeed increase the V_P at temperatures of 4000 to 6000 K. By combining the constraints of ρ and V_P, we predict that the outer core requires approximately 2.06~6.96 wt.% Mg to match the PREM. Above all, the impact of Mg on ρ and V_P is apparent. It can decrease the ρ of liquid Fe alloys and increase the V_P, which indicates that it is probably a candidate light element in the outer core.

4. Discussions

It is well-known that the temperature in the outer core increases along a well-defined adiabatic gradient from the core mantle boundary (CMB) to the ICB. The adiabatic temperature profile, also known as the geotherm profile, is not well determined due to the challenges of extremely high pressure and temperature in the outer core. Here, the adiabatic temperature profile along the geotherm in the outer core is derived by $T=T_{\mathrm{ICB}}{\left({\displaystyle \frac{\rho }{{\rho }_{\mathrm{ICB}}}}\right)}^{\gamma }$, where T_ICB is the temperature at the ICB, namely the anchor temperature, and ρ_ICB is the ρ at the ICB. The ρ is taken from the PREM, and γ is calculated by the EoS.

There has been no work reported about the geotherm of the Earth’s outer core being predicted by Fe-Mg alloys. Hence, we first used liquid Fe to reproduce the geotherm profile and compared it with previous work to estimate the EoS used in this work (see Figure 4a). We took the two anchoring temperatures of 5000 and 5400 K at the ICB, as used in previous work [30,32,37,38]. With the anchoring temperature T_ICB = 5400 K, the T_CMB predicted is 4229 K, in comparison with the reported 4005 K [30] and 3981 K [39]. If the temperature at the ICB is reduced to T_ICB = 5000 K, the temperature at the CMB is estimated to be 3916 K. It is also in agreement with previous T_CMB estimates ranging from 3654 to 3829 K [30,34,40]. The coincidences here suggest that the geotherm of the Earth’s outer core predicted by the EoS is reliable.

Consequently, we considered a wider anchoring temperature range (4850–7100 K), which is widely accepted, to study the Mg effect on the prediction of the geotherm profile. The sampled four anchoring temperatures at the ICB were 7100 K [41,42], 6350 K [43,44], 5400 K [45,46], and 4850 K [47]. The temperature profiles are plotted in Figure 4b. Assuming the temperature at the ICB is 5400 K, it corresponds to T_CMB = 4229 K and T_CMB = 4268 K for Fe and Fe-8.98 wt.% Mg, respectively. The addition of 8.98 wt.% Mg raises the temperature at the CMB by 39 K. Therefore, if Mg is a possible light element in the outer core, the temperature profile in the outer core is higher than that predicted with pure Fe. It is evident that the impact of Mg on the temperature of the outer core is comparable to that of O [35], H [48], and C [49]. The temperature profile rises with the addition of Mg.

In order to constrain the Mg content in the outer core, the ρ and V_P along the geotherm of the outer core were calculated. The ρ-P and V_P-P profiles are displayed in Figure 5. Assuming the temperature at the ICB is 5400 K, it is Fe-4.24 wt.% Mg that generally reproduces the PREM density profile (see Figure 5a). In comparison with the PREM, liquid Fe–4.24 wt.% Mg alloy yields absolute deviations of 0.119 g/cm³ in ρ and 0.238 km/s in V_P, corresponding to discrepancies of 1.06% and 2.53%, respectively. However, the slope of ρ-P is higher than that of the PREM. In terms of the V_P, the addition of Mg indeed increases the V_P. Similarly, the slope of the V_P-P curve for the Fe-Mg alloys is less than that of the PREM. To match the V_P profiles of the PREM, the outer core would need to contain more than 10 wt.% Mg. However, this is inconsistent with geochemical constraints (see Figure 5b). So, we conclude that a single Mg concentration cannot match the entire ρ-P or V_P-P of the PREM along the geotherm. It implies that Mg is not the sole light element in the outer core. Hence, a multiple content model is required to align the ρ and V_P curve of the PREM. In addition, when establishing such a model, consideration should be given to those with a lower ρ and higher V_P gradient than that of the PREM. Suggested light elements such as S [50] and C [49] are the possible light elements alloyed with Fe in the outer core, assuming that Mg is a candidate light element.

We further explored the Mg concentration at the boundaries of the outer core, namely the CMB and ICB. We found a linear relationship for both the ρ and V_P of the Fe-Mg liquid alloy, such that ${\left({\displaystyle \frac{\partial \mathsf{\rho }}{\partial \mathrm{X}}}\right)}_{\mathrm{T}}<0$ and ${\left({\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{p}}}{\partial \mathrm{X}}}\right)}_{\mathrm{T}}>0$ (X = wt.% of Mg) at the CMB and ICB (see Figure 6). The magnitude of the ${-\left({\displaystyle \frac{\partial \mathsf{\rho }}{\partial \mathrm{X}}}\right)}_{\mathrm{T}}$ contributed by Mg is 5.2–7.7 times larger than the ${\left({\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{p}}}{\partial \mathrm{X}}}\right)}_{\mathrm{T}}$ (see Table S2 in Supplementary Materials), a disparity that markedly exceeds the signatures of other candidate light elements. For C, N, and Si, the ${-\left({\displaystyle \frac{\partial \mathsf{\rho }}{\partial \mathrm{X}}}\right)}_{\mathrm{T}}$ is 2.5–3.0 times the corresponding ${\left({\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{p}}}{\partial \mathrm{X}}}\right)}_{\mathrm{T}}$, whereas for H, O, and S, the ratio rises to 4.5–5.0 [40]. The results indicate that it is difficult to simultaneously satisfy the ρ at the CMB and ICB with liquid Fe-Mg alloy (see Figure 6a). Considering the uncertainty of the temperature at the ICB, the ρ constraints suggest that the concentration of Mg at the CMB is approximately 2.9–5.4 wt.%, while at the ICB, it is around 3.8~6.1 wt.%. And the Mg concentration uncertainty is 0.6 wt.% at 4850 K and 0.8 wt.% at 7100 K. Additionally, the Mg content at the ICB is higher than that at the CMB. This aligns with reports that light elements are stratified within the outer core and that their concentration increases with depth [51]. However, matching the V_P of the PREM at both the top and the bottom of the outer core would demand Mg contents exceeding 10 wt.%, a value already above the total light elements content inferred for the outer core and thus implausible. Even so, Figure 6b indicates that the ICB requires a higher Mg concentration than the CMB. This discrepancy can serve as one piece of evidence supporting the notion that Mg is not the sole light element in the outer core.

However, in a multi-light-element system, the interactions among these elements cannot be neglected. For example, increasing the O concentration alters the partition coefficient of H, driving additional H into the inner core [8]. Therefore, it is necessary to discuss the composition of the outer core by considering a multi-component system composed of Mg and other candidate light elements (such as an Fe-Mg-S system or Fe-Mg-O [22,49]) to match the ρ-P or V_P-P profile of the PREM. Here, we propose that the presence of Mg in the outer core is likely due to its retention after the solidification of the magma ocean. Additionally, the growth and crystallization of the inner core may have resulted in a higher Mg content at the boundary of the inner core, contributing to the higher Mg content at the ICB than that at the CMB observed here.

5. Conclusions

We employed FP-MD simulations to model the Fe-Mg alloys under outer-core conditions. Based on the corrected P-V-T data, we fit an EoS to calculate the thermodynamic properties of liquid Fe-Mg alloys. As a result, the addition of Mg will reduce ρ, K_T, and K_S, and increase the V_P of liquid Fe-Mg alloys. Further, ρ and V_P along the geotherm of the outer core are calculated to constrain the Mg content in the outer core. The addition of Mg can indeed reduce the ρ and increase the V_P. Therefore, Mg is probably a candidate light element in the outer core. With T_ICB = 5400 K, it is Fe-4.24 wt.% Mg that could match the ρ of the PREM along the geotherm. Due to its inability to simultaneously satisfy ρ and V_P, it cannot be a dominant light element. Under the assumption that Mg is the sole light element, we derived an upper limit of 2.9–6.1 wt.% Mg that could satisfy the ρ of the PREM at the ICB and CMB. The uncertainty of temperature at the CMB will result in a deviation of approximately 2.5 wt.% Mg and 2.3 wt.% Mg. But cosmochemical and geochemical constraints suggest that the actual Mg content is likely significantly lower, approximately 0.5 wt.%, which further narrows the Mg concentration in the outer core.