# Ab Initio Thermoelasticity of Liquid Iron-Nickel-Light Element Alloys

^{*}

^{†}

## Abstract

**:**

_{3}-dominant mantle. But the H-rich core likely causes a distinct Fe depletion for the bulk Earth composition.

## 1. Introduction

_{T}) and shear (K

_{S}) moduli of iron and iron-LE alloys are key to interpreting seismological observations and then constructing a compositional model of the core [5,8]. However, those of the liquid states at the OC pressure (P) and temperature (T) (from ~136 to ~329 GPa and from ~4000 to ~6000 K) are still limitedly clarified experimentally. So far, static experiments have been performed up to less than 100 GPa [9,10,11]. Higher-P behavior of liquid iron was on the other hand investigated by shock wave experiments in multi-Mbar condition [2,12,13,14,15,16]. The temperature, however, changes along the principal Hugoniot and dramatically increases with increasing pressure to more than 8000 K at the pressure of the inner core (IC)-OC boundary (P

_{ICB}) of ~329 GPa, which is far higher than the expected actual ICB temperature (T

_{ICB}) of ~5000–6000 K [17,18,19,20,21,22]. Experimental determination of thermoelasticity of liquid iron alloys in the whole P, T condition of the earth’s OC thus remains technically impractical.

_{P}) of liquid iron and iron-LE alloys at the OC conditions in order to constrain the OC composition by interpreting seismological observations. These parameters for the Fe-O, Fe-Si, Fe-S, Fe-C, Fe-Ni, and Fe-Si-O system were calculated [23,24,25]. However, the data points in these studies were limited; two particular compositions of Fe

_{0.82}Si

_{0.10}O

_{0.08}and Fe

_{0.79}Si

_{0.08}O

_{0.13}only were considered [24] and two particular pressures of the core-mantle boundary (CMB) and ICB only were considered [23]. In particular, in the latter, empirical pressure corrections of 10 GPa and 8 GPa were adopted at the CMB and ICB respectively, though the optimized OC compositions are essentially sensitive to these corrections. Meanwhile, some studies have been performed throughout the whole OC P, T conditions for pure Fe [7,26], Fe-S [27], and Fe-H [28]. However, different formulations were employed to model their thermal equations of state, making a quantitative comparison of the reported thermoelasticity not easy.

_{ICB}and discuss the possible OC composition.

## 2. Results and Discussion

#### 2.1. Effects of LE on the Thermoelasticity of Liquid Iron

_{P}(Table 1), but trends are different depending on the type of LE. It is found that incorporations of larger Si and S atoms have only marginal effects on the volume (volume per atom), then the EoS is nearly unchanged (Figure S1). In contrast, incorporations of smaller O, C, and in particular H atoms reduce the volume considerably in the whole OC P range. These are related to the fact that the Fe-Si and S alloys are so-called substitutional-type, while the Fe-O, C, and H alloys are interstitial-type as recognized generally in lower P condition.

_{P}, but the systematics is less pronounced since the effects of LEs on ρ and K

_{S}are partially cancelled. Perturbation ratios ($\partial \mathrm{ln}{V}_{\mathrm{P}}/\partial \mathrm{ln}\rho $) are sometimes referred to discuss the chemical heterogeneity in Earth’s deep interior [30,31]. In the present cases, absolute values of this ratio are always smaller than 1, indicating that the effects of LE incorporations are always much larger in ρ than in V

_{P}.

#### 2.2. Optimized Compositions

_{P}between the Fe-Ni-X liquid alloys and the preliminary reference earth model (PREM) [32] are then evaluated as $\sum}[{\left(\frac{\rho -{\rho}_{PREM}}{{\rho}_{PREM}}\right)}^{2}+{\left(\frac{{V}_{\mathrm{P}}-{V}_{{\mathrm{P}}_{PREM}}}{{V}_{{\mathrm{P}}_{PREM}}}\right)}^{2}]$ (Figure 1), where the summation is taken over the whole OC pressure range. It is clearly demonstrated that the misfits are sensitive to the LE concentration and temperature but not so sensitive to the Ni concentration. The best-fit compositions along two adiabats (T

_{ICB}= 5000 K and 6500 K) with three different Ni/(Fe + Ni) ratios, which can be defined by the minima of misfits, are listed in Table 2 with misfits and the ρ and V

_{P}of best-fit compositions along two adiabats are shown in Figure 2.

_{P}are insufficient to determine the OC composition uniquely. Therefore, some other information, e.g., melting phase relations, partitioning behavior between solids and liquids, the bulk earth (BE) compositional property and so on, are quite helpful to place further constraints on the LE composition, but all of these are not well understood at the moment.

_{ICB}since the amount of LEs required to reproduce the PREM decreases for higher T (Table 2). However, the misfit values are insensitive to temperature without any systematic variations, meaning that it is difficult to constrain T

_{ICB}through this optimization. Based on the calculated results at two different T

_{ICB}, the best-fit compositions are represented as a function of T

_{ICB}within the first-order as follows:

_{ICB}should correspond to the freezing temperature of the OC liquid, melting phase relations of the Fe-LEs systems at the P

_{ICB}are quite important to place further constraints on the OC composition. There are however almost no available data with enough quality at the moment. Some experiments, though all were conducted at substantially lower pressures than P

_{ICB}, suggest that the eutectic temperature of Fe-FeS system is more than 1000 K lower than the melting temperature of pure Fe [19,34], while solidus or eutectic temperatures are not quite different (within a few 100 K) in the Fe-FeSi [35] and Fe-O systems [36]. A large drop in the melting temperature might also be expected in the Fe-H system [37]. The T

_{CMB}is usually thought to be ~4000 K [38,39] and its adiabatic extrapolation leads to ~5200 K for the T

_{ICB}[7]. This might be ~1000 K lower than the T

_{M}of pure Fe at the P

_{ICB}, suggesting S and H as the potential LEs in the OC. But even larger temperature drops could exist in the ternary or quaternary systems, so it is hard to exclude Si and O from the LE candidates based on this discussion.

^{41}. Partitioning of LEs between solid and liquid phases is therefore thought to be required, namely LEs dissolving in the OC should strongly prefer liquid to solid. Again, nothing can be conclusive before the melting phase relations of the Fe-LEs systems are clarified at P

_{ICB}, but extrapolations of experimental knowledge obtained at lower pressures suggest that the strong partitioning occurs in the Fe-O system [36] but not in the Fe-S [19,34], Fe-Si [35,40], and Fe-H systems [37].

#### 2.3. Bulk Earth Composition

_{3}-dominant, the Mg/Si ratios of the BE expected with the best-fit composition for the core decreases by ~0.2. Then, the Mg/Si and Mg/Fe ratios of all the best-fit compositions except for Si and H-bearing cases match the ratios of CI chondrite and OCCAM model. In this case, Si-rich and H-rich OC with MgSiO

_{3}-dominant lower mantle lead to a too Si-rich and Fe-poor BE composition, respectively.

## 3. Conclusions

_{P}of pure Fe, but the effects are counterintuitively larger for the Si and S incorporations than for the O, C, and H incorporations. Any best-fit alloy composition except the C-rich case can reproduce the ρ and V

_{P}of the actual OC in the comparable level, so that the information of ρ and V

_{P}only are insufficient to determine the OC composition uniquely. Melting phase relations and LE partitioning in the Fe-Ni-LE systems at the P

_{ICB}are therefore essential to place a tighter constraint on the OC chemistry. The Si-rich best-fit composition for the core with an assumption of the pyrolytic mantle predicts the CI chondritic BE composition, but the O and S-rich best-fit compositions for the core with the MgSiO

_{3}-dominant mantle also lead to the similar chemistry for the BE. The H-rich best-fit composition however causes a distinct deficit of Fe for the BE. In future studies, it might be important to investigate correlations between LEs in higher-order multicomponent systems, which are ignored in this study.

## 4. Methods

#### 4.1. Ab Initio Molecular Dynamics Simulations

_{1−X}O

_{X}; (Fe-Ni)

_{1−X}Si

_{X}; (Fe-Ni)

_{1−X}S

_{X}; (Fe-Ni)

_{1−X}C

_{X}; (Fe-Ni)

_{1−X}H

_{X}, at different atomic fractions (${X}_{\mathrm{O}}\le 0.5,{X}_{\mathrm{Si}}\le 0.3,{X}_{\mathrm{S}}\le 0.3,{X}_{\mathrm{C}}\le 0.3,{X}_{\mathrm{H}}\le 0.4$). Three Ni/(Fe + Ni) ratios of 0, 0.06 (consistent with the geochemically modeled value) [43], and 0.12 are examined.

^{−15}s) for the Fe-LE systems, which is the same for previous studies [7,17,33,49], and 0.5 fs for the Fe-H system. Some results (pure Fe and Fe

_{1−X}O

_{X}systems) are, in part, already reported in the previous studies [7,49]. MD cells basically contain 50 atoms as in our previous study [7] but 100 atoms for the optimized compositions, and T is controlled by the kinetic energy scaling method. The validity of the cell size with 50–100 atoms for liquid iron can be seen in previous calculations [17,33], where a minor variation of the melting temperature of iron (~100 K) was found with changing the cell size from 67 to 980 atoms. Thermodynamic properties of liquid iron were also found to be sufficiently converged for this cell size.

^{2}3p

^{6}3d

^{6.5}4s

^{1}4p

^{0}is pseudized with a sufficiently small core radius of 2.0 a.u. for Fe; 2s

^{2}2p

^{4}, with a core radius of 1.5 a.u. for O; 3s

^{2}3p

^{4}, with a core radius of 1.7 a.u. for S; 3s

^{2}3p

^{2}3d

^{0}, with a core radius of 1.4 a.u. for Si; 2s

^{2}2p

^{2}, with a core radius of 1.1 a.u. for C; 1s

^{1}, with a core radius of 0.8 a.u. for H; and 3s

^{2}3p

^{6}3d

^{8}4s

^{2}4p

^{0}with a core radius of 2.0 a.u. for Ni by the Vanderbilt scheme [53] with non-linear core corrections [54]. We apply a kinetic energy cutoff of 50 Ry and spin polarization is not taken into account. These conditions are already well tested in our previous calculations [7,49,55] and are fairly similar to those in calculations by other groups [24]. Liquids in principle have no periodic structure, thus the Γ point only is sampled in our simulations. All the MD simulations are conducted in P,T condition from ~80 to ~500 GPa and from 4000 to 8000 K (Figure S1), which covers the whole P,T range of the core. Standard deviations in calculated T and P are found to be typically ~50 K and ~3 GPa at 5000 K and ~130 GPa and ~120 K and ~6 GPa at 8000 K and ~400 GPa, respectively.

#### 4.2. EoS Analysis of Liquid Iron Alloys

_{0}, we used the Vinet (Morse-Rydberg) Equation,

_{0}. The internal thermal energy is represented by a second-order polynomial of temperature with a volume dependent second-order coefficient,

_{0}in order to constrain the reference isotherm as tightly as possible within the broad pressure range. Although in the previous study [7], the following functional form for $\gamma \left(V\right)$

_{0}, ${K}_{{T}_{0}}$, ${K}_{{T}_{0}}^{\u2019}$, $\gamma $, e

_{0}, and g) to calculate P at a given V, T. These parameters are determined by least squares analyses on the datasets obtained from the AIMD calculations. The derived EoS parameters for best-fit compositions are summarized in Table S1.

_{T}and α are obtained based on the thermodynamic definitions as (∂P/∂V)

_{T}= −K

_{T}/V and (∂P/∂T)

_{V}= αK

_{T}, respectively. K

_{T}is then converted to K

_{S}as

_{P}is then calculated for each composition (Figure S1) as ${V}_{\mathrm{P}}=\sqrt{\frac{{K}_{\mathrm{S}}}{\rho}}$ along two different adiabats explained below.

#### 4.3. Adiabats

_{P}are calculated along the adiabats anchored by two possible ICB temperatures: T

_{ICB}= 5000 K and T

_{ICB}= 6500 K. The former T

_{ICB}is found to give ~3700 K at 136 GPa, which is close to a proposed core-mantle boundary temperature [38,39]. The latter corresponds to the melting temperature (T

_{M}) of pure iron at 329 GPa [17,18], which would be close to the upper bound of ICB temperature since T

_{M}of iron-LE alloys are expected in general to be lower than the T

_{M}of pure Fe.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The misfit from the value of the preliminary reference earth model (PREM) as a function of atomic fraction of LEs (X

_{LE}). Three different Ni fractions, 0 (

**a**,

**d**), 0.06 (

**b**,

**e**), and 0.12 (

**c**,

**f**) and two different T

_{ICB}of 5000 K (

**a**–

**c**) and 6500 K (

**d**–

**f**) are examined. Filled circles are the results of molecular dynamics (MD) and dashed lines are the cubic spline interpolations. The atomic fractions at the minima correspond to the best-fit LE concentrations. Open circles indicate the misfits obtained from MD with the best-fit compositions, which are in good agreement with the minima of spline interpolations. Errors in the misfits originated in the fitting procedures are comparable to the size of symbols.

**Figure 2.**ρ and V

_{P}for the best-fit compositions along two adiabats with T

_{ICB}= 5000 K and 6500 K (red, Fe-Ni-O; blue, Fe-Ni-Si; purple, Fe-Ni-S; yellow, Fe-Ni-C; green, Fe-Ni-H). Solid lines correspond to the results determined from raw thermoelasticity data and shaded regions correspond to the uncertainties in pressure with +10 GPa [33]. Dashed lines indicate the PREM values [32]. The ρ and V

_{P}of almost all alloys overlap, indicating that these data only are insufficient to determine the OC composition uniquely.

**Table 1.**Effects of light element (LE) incorporation on ${V}_{\mathrm{P}}$ and $\rho $ of liquid Fe calculated at the P

_{CMB}and 4000 K and at the P

_{ICB}and 5300 K. X

_{LE}represents the fraction of LE in atom%.

P, T Condition | O | Si | S | C | H | |
---|---|---|---|---|---|---|

P = P_{CMB}T = 4000 K | $\frac{\partial \mathrm{ln}{V}_{\mathrm{P}}}{\partial {X}_{\mathrm{LE}}}$ | 0.05(1) | 0.13(1) | 0.06(1) | 0.16(1) | 0.02(1) |

$\frac{\partial \mathrm{ln}\rho}{\partial {X}_{\mathrm{LE}}}$ | −0.34(1) | −0.51(1) | −0.41(1) | −0.30(1) | −0.24(1) | |

$\frac{\partial \mathrm{ln}{V}_{\mathrm{P}}}{\partial \mathrm{ln}\rho}$ | −0.14(1) | −0.26(1) | −0.16(1) | −0.54(1) | −0.10(1) | |

P = P_{ICB}T = 5300 K | $\frac{\partial \mathrm{ln}{V}_{\mathrm{P}}}{\partial {X}_{LE}}$ | 0.09(1) | 0.21(1) | 0.16(1) | 0.20(1) | 0.07(1) |

$\frac{\partial \mathrm{ln}\rho}{\partial {X}_{\mathrm{LE}}}$ | −0.31(1) | −0.48(1) | −0.38(1) | −0.31(1) | −0.21(1) | |

$\frac{\partial \mathrm{ln}{V}_{\mathrm{P}}}{\partial \mathrm{ln}\rho}$ | −0.29(1) | −0.44(1) | −0.42(1) | −0.63(1) | −0.35(1) |

**Table 2.**Best-fit compositions of binary and ternary alloys at T

_{ICB}= 5000 K and 6500 K. Misfit, Mg/Si, and Mg/Fe represent a misfit in V

_{P}and ρ from the PREM, Mg/Si, and Mg/Fe ratios expected to the bulk earth with the pyrolytic mantle, respectively. Errors from the fitting procedures are represented in parentheses.

T_{ICB} | Best-Fit Composition | $\mathbf{Misfit}(\times {10}^{-2})$ | Mg/Si | Mg/Fe |
---|---|---|---|---|

$5000$ K | Fe_{0.78}O_{0.22} | 1.8(1) | 1.25(1) | 1.03(1) |

Fe_{0.85}Si_{0.15} | 2.7(1) | 1.06(1) | 1.04(1) | |

Fe_{0.81}S_{0.19} | 1.6(1) | 1.25(1) | 1.08(1) | |

Fe_{0.80}C_{0.20} | 11.2(1) | 1.25(1) | 1.01(1) | |

Fe_{0.70}H_{0.30} | 1.9(1) | 1.25(1) | 0.97(1) | |

Fe_{0.73}Ni_{0.05}O_{0.22} | 0.8(1) | 1.25(1) | 1.10(1) | |

Fe_{0.80}Ni_{0.05}Si_{0.15} | 1.3(1) | 1.06(1) | 1.10(1) | |

Fe_{0.76}Ni_{0.05}S_{0.19} | 0.6(1) | 1.25(1) | 1.14(1) | |

Fe_{0.75}Ni_{0.05}C_{0.20} | 11.8(1) | 1.25(1) | 1.07(1) | |

Fe_{0.64}Ni_{0.04}H_{0.32} | 1.1(1) | 1.25(1) | 1.03(1) | |

Fe_{0.69}Ni_{0.09}O_{0.22} | 0.7(1) | 1.25(1) | 1.15(1) | |

Fe_{0.74}Ni_{0.10}Si_{0.16} | 1.4(1) | 1.05(1) | 1.17(1) | |

Fe_{0.71}Ni_{0.10}S_{0.19} | 1.2(1) | 1.25(1) | 1.21(1) | |

Fe_{0.7}Ni_{0.09}C_{0.21} | 10.4(1) | 1.25(1) | 1.13(1) | |

Fe_{0.6}Ni_{0.08}H_{0.32} | 0.9(1) | 1.25(1) | 1.09(1) | |

$6500$ K | Fe_{0.82}O_{0.18} | 4.8(1) | 1.25(1) | 1.02(1) |

Fe_{0.88}Si_{0.12} | 1.7(1) | 1.09(1) | 1.02(1) | |

Fe_{0.85}S_{0.15} | 0.8(1) | 1.25(1) | 1.05(1) | |

Fe_{0.84}C_{0.16} | 7.7(1) | 1.25(1) | 1.00(1) | |

Fe_{0.74}H_{0.26} | 0.1(1) | 1.25(1) | 0.97(1) | |

Fe_{0.77}Ni_{0.05}O_{0.18} | 0.9(1) | 1.25(1) | 1.08(1) | |

Fe_{0.82}Ni_{0.05}Si_{0.13} | 1.8(1) | 1.08(1) | 1.08(1) | |

Fe_{0.79}Ni_{0.05}S_{0.16} | 0.2(1) | 1.25(1) | 1.12(1) | |

Fe_{0.78}Ni_{0.05}C_{0.17} | 7.5(1) | 1.25(1) | 1.06(1) | |

Fe_{0.69}Ni_{0.04}H_{0.27} | 2.3(1) | 1.25(1) | 1.02(1) | |

Fe_{0.71}Ni_{0.1O0.19} | 2.2(1) | 1.25(1) | 1.15(1) | |

Fe_{0.77}Ni_{0.1}Si_{0.13} | 1.3(1) | 1.08(1) | 1.15(1) | |

Fe_{0.74}Ni_{0.1}S_{0.16} | 0.6(1) | 1.25(1) | 1.18(1) | |

Fe_{0.72}Ni_{0.1C0.18} | 7.7(1) | 1.25(1) | 1.13(1) | |

Fe_{0.64}Ni_{0.09H0.27} | 1.0(1) | 1.25(1) | 1.09(1) |

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## Share and Cite

**MDPI and ACS Style**

Ichikawa, H.; Tsuchiya, T.
Ab Initio Thermoelasticity of Liquid Iron-Nickel-Light Element Alloys. *Minerals* **2020**, *10*, 59.
https://doi.org/10.3390/min10010059

**AMA Style**

Ichikawa H, Tsuchiya T.
Ab Initio Thermoelasticity of Liquid Iron-Nickel-Light Element Alloys. *Minerals*. 2020; 10(1):59.
https://doi.org/10.3390/min10010059

**Chicago/Turabian Style**

Ichikawa, Hiroki, and Taku Tsuchiya.
2020. "Ab Initio Thermoelasticity of Liquid Iron-Nickel-Light Element Alloys" *Minerals* 10, no. 1: 59.
https://doi.org/10.3390/min10010059