Equation of State of Liquid Fe7C3 and Thermodynamic Modeling of the Liquidus Phase Relations in the Fe‐C System

We calculated the pressure‐volume relations of liquids Fe and Fe7C3 up to ∼360 GPa and 3000–8000 K based on the first‐principles molecular dynamics simulations. The results demonstrate that liquid Fe7C3 is similar in compressibility and thermal expansivity to liquid Fe at the Earth's core pressure range. We then obtained a thermodynamic model of the liquidus phase relations in the Fe‐C system at high pressures by using the thermal equation of state (EoS) of liquid Fe7C3 as well as that of liquid Fe3C estimated by interpolation between the volumes of Fe and Fe7C3 calculated in this study. The previously reported eutectic points in the Fe‐Fe3C system are reproduced with interaction parameters that are dependent on temperature but independent on pressure at >∼50 GPa. The melting curve of Fe7C3 should be close to that of Fe, which leads to a change in the eutectic system from Fe‐Fe3C to Fe‐Fe7C3 above 250 GPa as observed by previous experiments. The thermodynamic model also suggests that the solidus and liquidus of Fe3C are relatively low. The Fe‐Fe7C3 liquidus phase relations at the Earth's inner core boundary (ICB) pressure indicate the Fe7C3 inner core is unlikely. While the carbon content in the Earth's outer core may be small, the addition of 1 wt% C to liquid Fe drops the liquidus temperature to crystallize hcp‐Fe by ∼450 K, suggesting that the presence of carbon in the outer core is preferable for a relatively low core‐mantle boundary (CMB) temperature.


Introduction
Carbon is one of major candidates for the light elements included in the Earth's core (Hirose et al., 2021;Poirier, 1994).Its density deficit with respect to pure Fe is ∼4%-5% for the solid inner core and ∼7%-8% for the liquid outer core (Fei et al., 2016;Kuwayama et al., 2020).Carbon can be incorporated into both solid and liquid Fe alloys, affecting chemical and physical properties of both the inner and outer core.Especially, Fe 7 C 3 represents a high Poisson's ratio similar to that of the Earth's inner core and might thus be its major component (Chen et al., 2014;Prescher et al., 2015).Furthermore, the elastic properties of carbon-bearing hcp-Fe are in good agreement with inner core observations, supporting the need for carbon in the inner core (Li et al., 2018).On the other hand, the density and sound velocity profiles of the outer core are not compatible with those of liquid Fe-C, suggesting that carbon concentration should be minor in the liquid core (Badro et al., 2014;Umemoto & Hirose, 2020).
Liquidus phase relations, in particular the eutectic point in the Fe-C system are of great importance to understand the crystallization of carbon-bearing outer core liquid and have been examined under high pressures (Chabot et al., 2008;Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009;Wood, 1993) as well as at 1 bar (Chipman, 1972;Tanaka, 1967).Eutectic melting occurs between Fe and Fe 3 C up to 211-255 GPa (Chabot et al., 2008;Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009).Fe 7 C 3 and graphite (or diamond) also appear as liquidus phases for relatively C-rich (>5-6 wt%) liquids.Fe forms eutectic with Fe 7 C 3 instead of Fe 3 C at pressures higher than 211-255 GPa (Mashino et al., 2019).Melting experiments using a multi-anvil press revealed that carbon concentration in the Fe-C eutectic liquid decreases from 4.3 wt% at 1 bar to ∼3.8 wt% at 20 GPa (Chabot et al., 2008;Fei & Brosh, 2014).A further reduction was reported at higher pressures using a diamond-anvil cell (DAC) through in-situ measurements of liquid compositions by X-radiography to 44 GPa (Lord et al., 2009) and by Rietvelt analyses of X-ray diffraction patterns to 108 GPa (Morard, Andrault et al., 2017).Existing thermodynamic models also predicted the expansion of the liquidus field of Fe 3 C or Fe 7 C 3 with increasing pressure, resulting in the eutectic C content of <1 wt% (Wood, 1993) or ∼2 wt% at the ICB pressure (Fei & Brosh, 2014).On the other hand, recent DAC experiments combined with ex-situ textural and chemical characterizations of recovered samples demonstrated that the carbon content in the Fe-C eutectic liquid stays around ∼4 wt% in a wide pressure range from 23 to 255 GPa (Mashino et al., 2019).
The EoSs of both solid and liquid with an identical composition are key to thermodynamic modeling of the liquidus phase relations at high pressures.The compression curves of solid Fe 3 C and Fe 7 C 3 have been studied at room temperature to 187 GPa (Li et al., 2002;Ono & Mibe, 2010;Prescher et al., 2012;Sata et al., 2010;Scott et al., 2001) and to 158 GPa (Chen et al., 2012;Li et al., 2016;Prescher et al., 2015), respectively.Thermal EoSs were also constructed by experiments to 102 GPa for Fe 3 C (Litasov et al., 2013;McGuire et al., 2021) and to 79 GPa for Fe 7 C 3 (Lai et al., 2018;Nakajima et al., 2011) and by first-principles calculations at the core pressure range (Li et al., 2016;Vočadlo et al., 2002).In contrast, studies for the EoS of Fe-C liquid are limited to ∼10 GPa (Sanloup et al., 2011;Shimoyama et al., 2013Shimoyama et al., , 2016;;Terasaki et al., 2010) except Morard, Nakajima et al. (2017).Morard and others measured the volumes of liquids Fe-C to ∼60 GPa, but the liquid compositions were variable and thus constructing an EoS of liquid with a specific carbon content was difficult.
In this study, at first we report the volumes of liquids Fe and Fe 7 C 3 as functions of pressure and temperature based on first-principles molecular dynamics (FPMD) calculations and construct the thermal EoS for the latter.We also obtain the EoS of liquid Fe 3 C assuming ideal mixing between liquid Fe and Fe 7 C 3 .These EoSs are then used to have a thermodynamic model focusing on the liquidus phase relations in the Fe-C system between Fe and Fe 7 C 3 to the ICB pressure.We discuss the crystallization of C-bearing liquid Fe and the formation of the inner core based on the liquidus phase relations predicted by the present thermodynamic model at 330 GPa.

Methods
We performed FPMD calculations using pseudopotentials within the density-functional theory with 100 atoms for Fe 7 C 3 and 64 atoms for Fe in fixed cubic cells corresponding to those at ∼0-360 GPa.FPMD was performed at 10000 K for 2 picoseconds to generate a liquid structure.Radial distribution function and mean square displacements confirmed the liquid state.Then, the cell was quenched to a target temperature (3000-8000 K) and equilibrated in 2 picoseconds.Lastly, we performed FPMD for 10 picoseconds or longer to calculate pressure by averaging over the period.The temperature was controlled by velocity rescaling.Time steps were 2 femtoseconds at 10000 K for obtaining a liquid structure, and 1 femtosecond at target temperature for pressure calculations.The pseudopotentials for all atomic species were generated using Vanderbilt's method (Vanderbilt, 1990) with the partial core correction (Louie et al., 1982).A Perdew-Burke-Ernzerhof (PBE)-type generalized-gradient approximation (GGA) functional was used for exchange-correlation (XC) potential (Perdew et al., 1996).The electronic configurations for pseudopotential generation were 3s 2 3p 6 3d 6.5 4s 1 4p 0 for iron and 2s 2 2p 2 for carbon.The cutoff radii were 1.8 a.u.for iron and 1.3 a.u.for carbon with 1 a.u.= 0.529177 Å. Thermal excitation of electrons was considered by the Fermi-Dirac distribution.Γ point sampling was used.The cutoff energy for the plane-wave expansion was 30 Ry. Pressure was found to converge within 1 GPa, with respect to the number of atoms in a supercell, k-point sampling, and plane-wave cutoff energy.Uncertainty in pressure obtained in each simulation was <0.5 GPa, supporting the sufficient simulation time.Calculations have been carried out using the Quantum-ESPRESSO package (http://www.quantum-espresso.org) (Giannozzi et al., 2009).

Results
We obtained the P-V relations of liquids Fe and Fe 7 C 3 in a wide pressure range at 3000-8000 K and 3000-7000 K, respectively, with 1000 K intervals (Tables 1 and 2).The P-V relations of liquid Fe obtained in this study are within the variations in those reported by previous first-principles simulations at identical temperatures (Figure 1a) (de Koker et al., 2012;Ichikawa et al., 2014;Umemoto & Hirose, 2020;Vočadlo et al., 2003;Wagle & Steinle-Neumann, 2019).We applied pressure corrections to our calculations on liquid Fe such that they are consistent with earlier experimental results as often done in studies on Fe and Fe alloys using GGA-based simulations (Badro et al., 2014;Umemoto & Hirose, 2020).The experiments performed by Kuwayama et al. (2020) derived the Vinet EoS of liquid Fe at a reference temperature of 1811 K as: where V 0 is volume, K 0 is isothermal bulk modulus, and K′ is its pressure derivative at zero pressure.We obtained thermal expansivity by refitting the Anderson-Grüneisen (AG) model (Anderson et al., 1992) to Kuwayama et al.'s data: in which a is thermal expansivity, δ is AG parameter, κ is dimensionless parameter, and subscript zero denotes values at 1 bar.The pressures of the present theoretical calculations are compared to those given by the experimentally derived EoS (Equations 1 and 2) for identical temperatures and volumes (Figure 1b; Table 1).While these calculations and experiments show similar pressures, they match each other when we make pressure corrections to the former considering linear relations between them, below and above  2020) for identical temperatures and volumes of liquid Fe (see Table 3).a Pressures after corrections based on the results on liquid Fe (Figure 1b).b Pressures obtained in the FPMD calculations.
109 GPa separately (see Figure 1b for details).We applied the same pressure corrections to our calculations on liquid Fe 7 C 3 .Indeed, without pressure corrections, our thermodynamic model (see below) does not yield the reasonable melting curve of Fe 3 C, for example.Note that the calculations were performed only above 70 GPa since the spin transition is known to occur in solid Fe 7 C 3 at 53-70 GPa (Chen et al., 2012(Chen et al., , 2014;;Mookherjee et al., 2011;Prescher et al., 2015).
The obtained P-V-T relations of liquid Fe 7 C 3 (Table 2) were fitted by the Vinet EoS with a reference temperature of T 0 = 5000 K and the AG model (Equations 1 and 2) (Figure 2a, Table 3).We employed V 0 and a 0 from Jimbo and Cramb (1993) considering that their dV/dT at the reference temperature represents the a 0 value.Comparisons of isothermal compression curves given in Figure 2b show that while liquid Fe 7 C 3 is less compressible than liquid Fe at relatively low pressures, their compressibility becomes closer as pressure increases.In addition, the a 0 values are similar between liquids Fe and Fe 7 C 3 , indicating that their thermal expansivity is comparable to each other at relatively low pressure range.The thermal expansivity of Fe 7 C 3 is less pressuredependent than that of Fe, leading to less than ±15% difference at 330 GPa.
Subsequently we obtained the P-V-T relations of liquid Fe 3 C (Table 3, Figure 2b) from the thermal EoSs of liquids Fe and Fe 7 C 3 by considering the linear relation between the carbon content and the density and assuming the ideal mixing for volume (Badro et al., 2014;Umemoto & Hirose, 2020).V 0 and a 0 are again from the literature (Jimbo & Cramb, 1993).Our P-V-T EoS of liquid Fe 3 C yields ∼4% larger volume at 9.5 GPa than earlier experimental results (Terasaki et al., 2010).Other experimental studies on liquid Fe containing 5.7 wt% and 3.5 wt% C up to ∼10 GPa reported more compressible behaviors than the present modeling that assumes ideal mixing (Sanloup et al., 2011;Shimoyama et al., 2013Shimoyama et al., , 2016)).On the other hand, our modeling shows smaller volumes for Fe-C liquids than experiments conducted at 40-60 GPa (Morard, Nakajima et al., 2017).Such differences, however, are likely attributed to non-ideal mixing and the effect of magnetism below 70 GPa.

Thermodynamic Modeling of Liquidus Phase Relations in the Fe-C System
The melting phase relations of iron alloys can be thermodynamically modeled based on the EoSs of both solid and liquid phases and the melting curves for end members (see Figure 3 for procedures in the following sections).
The Gibbs free energy at a specific P-T condition is described as: (3) Melting of Fe, Fe 3 C, and Fe 7 C 3 is determined from the difference in the Gibbs free energy between solid and liquid: ∆G sol liq 1bar,T is described by a polynomial function which is often used to express G 1bar,T :  (Kuwayama et al., 2020) at 3000 K (pluses), 4000 K (crosses), 5000 K (circles), 6000 K (squares), 7000 K (diamonds), and 8000 K (triangles).A red line is the pressure correction applied to the calculations for liquid Fe 7 C 3 .A gray dashed line is a guide indicating P calc = P exp .
∆G sol liq 1bar,T = a + bT + cT ln T + dT 2 + eT 1 + f T 0.5 . (5) We only use a, b, and (when required) c parameters in our model for simplicity.Because we focus on the melting phase relations, we discuss based on ∆G sol liq P,T and do not determine G P,T explicitly in the following modeling, which reduces the number of parameters to be calculated.The Vinet EoS and the thermal expansivity from the AG model (Equations 1 and 2) are employed to calculate V T following Komabayashi (2014).Such formulation works well when it is extrapolated to high P-T or low-P/high-T conditions.

Thermodynamic Parameters for Solid and Liquid Fe
Although thermodynamic parameters for pure Fe were proposed in previous studies (e.g., Komabayashi, 2014), we employ those for liquid Fe based on the recent experimental data by Kuwayama et al. (2020) (see Section 2, Table 3).Those for solid fcc and hcp phases are adopted from Komabayashi (2014), which are consistent with recent experimental studies (Fei et al., 2016;Miozzi et al., 2020).The melting curve of Fe is required to calculate ∆G sol liq 1bar,T (Equation 4), but literature data published in the last decade show large variations, especially above 50 GPa (Anzellini et al., 2013;Aquilanti et al., 2015;Ezenwa & Fei., 2023;Hou et al., 2021;Jackson et al., 2013;Sinmyo et al., 2019;Zhang et al., 2016).Here we obtained a and b parameters in ∆G fcc liq 1bar,T and ∆G hcp liq 1bar,T for the melting curve of Fe to be in the middle of those reported by Anzellini et al. (2013) and Sinmyo et al. (2019), which represent relatively high and low estimations, respectively (Table 4, Figure 4a).The ∆G sol liq 1bar,T values agree with those in Komabayashi (2014) except for ∆G hcp liq 1bar,T below ∼2000 K that is too low to be relevant to melting phase relations (Figure 4b).

Melting Phase Relations in the Fe-Rich Side of the Eutectic
The Gibbs free energy of a liquid with an intermediate composition between Fe and FeC is given by: Figure 2. (a) Calculated pressure-volume relations of liquid Fe 7 C 3 at 3000 K (green), 4000 K (blue), 5000 K (black), 6000 K (red), and 7000 K (orange).Pressures were corrected for the results of calculations to be consistent with experimental data (Figure 1b).Lines represent the equation of state obtained by fitting.(b) Compression curves of liquid Fe 7 C 3 (blue), Fe 3 C (red), and Fe (black) at 3000 K (solid lines), 5000 K (dashed lines), and 7000 K (dotted lines).

Phase
V 0 (cm 3 /mol) K 0 (GPa) K' T 0 (K) a 0 (×10 where W Fe-FeC is Margules interaction parameter which depends on P and T, and R is gas constant.Considering carbon atoms are incorporated interstitially into liquid Fe, we define y C = x C /x Fe , where x i is mole fraction of atom i. Employing G′ liq P,T,y , which corresponds to the last two terms in Equation 6and indicates a difference between the Gibbs free energy obtained from Equation 6and that from the linear mixing of Fe and FeC liquids, chemical equilibrium between solid Fe and liquid with y C composition is described by: The right side of Equation 7 expresses the chemical potential of Fe in liquid Fe-C relative to that for pure liquid Fe at a given P-T condition.Indeed, carbon is partitioned into the solid fcc and hcp phases at high pressures as well as at 1 bar (Chabot et al., 2008;Chipman, 1972;Fei & Brosh, 2014;Mashino et al., 2019;Tanaka, 1967).To compensate the effect of carbon in solid phases on the solid-liquid equilibrium without increasing the number of fitting parameters, Equation 7 is modified as: in which D sol/liq C is partition coefficient defined as y sol C / y liq C .Such modification assumes similar curvature for coexisting solid and liquid in the C-poor portion of the G-y C diagram, making solid Fe and thus solid Fe-C alloy more stable when D sol/liq C is higher.The D sol/liq C has been determined by previous experimental studies (Chabot et al., 2008;Chipman, 1972;Fei & Brosh, 2014;Mashino et al., 2019;Tanaka, 1967) The temperature dependence of D sol/liq C has not been examined at high pressures and is therefore considered only at 1 bar.
Next the interaction parameter W Fe-FeC is obtained using Equation 8for each previous experimental datum reported up to 255 GPa on carbon concentration in liquid coexisting with solid Fe (Chabot et al., 2008;Chipman, 1972;Fei & Brosh, 2014;Mashino et al., 2019;Tanaka, 1967) (Figure 5a).The W Fe-FeC increases with increasing pressure to ∼50 GPa and decreases at higher pressures.Since excess volume is small under core pressures (Badro et al., 2014;Umemoto & Hirose, 2020) suggesting a small pressure dependence of W Fe-FeC , a reduction in W Fe-FeC at >136 GPa is likely to be attributed to a temperature effect.The variations in W Fe-FeC found in these earlier experiments are expressed as functions of P and T such that changes above 136 GPa are explained only by a temperature effect: where T is in K, and P is in GPa.When experimental temperatures are adjusted to a given temperature, for example, 3000 K, the W Fe-FeC values from experiments at ∼50 GPa are significantly higher than the fitting results The melting curve of fcc-Fe (dashed lines) and hcp-Fe (solid lines) calculated in this study (red) and a previous thermodynamic study (black) (Komabayashi, 2014).Experimental results are plotted as squares (Anzellini et al., 2013), triangles (Jackson et al., 2013), pentagons (Aquilanti et al., 2015), diamonds (Zhang et al., 2016), circles (Sinmyo et al., 2019), inverted triangles (Hou et al., 2021), and crosses (, Ezenwa & Fei, 2023).(b) ∆G sol liq 1bar,T of fcc-Fe (dashed lines) and hcp-Fe (solid lines) in this study (red) and a previous thermodynamic study (black) (Komabayashi, 2014).(Figure 5b), which may be because of a magnetic effect that is not included in our model (Chen et al., 2012;Mookherjee et al., 2011;Prescher et al., 2015;Vočadlo et al., 2002).

Melting of Fe 3 C and the Fe-Fe 3 C Eutectic Point
We calculated the ∆G sol liq P,T of Fe 3 C from the compositions of liquids coexisting with solid Fe 3 C (Chabot et al., 2008;Chipman, 1972;Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009;Tanaka, 1967) from the following equation: where G′ liq P,T,y is calculated with the interaction parameter W Fe-FeC from Equation 10 above.Then, ∆G sol liq 1bar,T (Fe 3 C) is obtained from each of these experimental data and expressed as the polynomial function using a, b, and c parameters (Equation 5) (Table 4, Figure 6a).The ∆G sol liq 1bar,T (Fe 3 C) values and the EoSs of solid and liquid Fe 3 C respectively from McGuire et al. (2021) and this study yield the melting curve of Fe 3 C (Figure 7).Indeed, Fe 3 C melts incongruently to liquid + solid Fe 7 C 3 (Fei & Brosh, 2014), and therefore the Fe 3 C melting curve is hypothetical and should be located between the solidus and liquidus temperatures.The present results are within the variations in experimentally measured solidus curve (Li et al., 2016;Lord et al., 2009;Nakajima et al., 2009;Takahashi et al., 2020) (Figure 7).
The eutectic temperature and composition are calculated from G′ liq P,T,y and the ∆G sol liq P,T of Fe and Fe 3 C (Figures 7 and 8).Our model shows that the carbon content in the fcc-Fe-Fe 3 C eutectic liquid decreases with increasing pressure, consistent with earlier experimental studies at 5-25 GPa (Fei & Brosh, 2014), while that The Margules interaction parameter W Fe-FeC calculated from compositions of liquids coexisting with solid Fe reported in previous experimental studies (Chabot et al., 2008;Chipman, 1972;Fei & Brosh, 2014;Mashino et al., 2019;Tanaka, 1967).Colors represent experimental temperatures.(b) W Fe-FeC adjusted to T = 3000 K using Equation 10.A dashed line indicates the pressure dependence at 3000 K.
in the hcp-Fe-Fe 3 C eutectic liquid is almost constant at 3.7-4.0wt% above 50 GPa in agreement with Mashino et al. (2019).The carbon content in the hcp-Fe-Fe 3 C eutectic liquid is higher than that in the fcc-Fe-Fe 3 C eutectic liquid around 20 GPa because hcp-Fe appears at liquidus at that pressure in our model.It may not be true, however, because ∆G sol liq 1bar,T (hcp-Fe) at low temperature relevant to such low pressure range is derived from linear extrapolation of ∆G sol liq 1bar,T (hcp-Fe) from high temperatures (Figure 4b).

Change From Fe-Fe 3 C Eutectic to Fe-Fe 7 C 3 Eutectic
The melting experiments performed by Mashino et al. (2019) found the coexistence of liquid + Fe + Fe 3 C at 211 GPa and liquid + Fe + Fe 7 C 3 at 255 GPa.While Liu et al. (2016) reported the decomposition of Fe 3 C into Fe + Fe 7 C 3 at lower pressures around 150 GPa, Fe 3 C was observed to ∼350 GPa at high temperatures by Tateno et al. (2010) and Takahashi et al. (2020).Here we adopt the melting phase relations reported by Mashino et al. (2019) and consider the change in the eutectic system from Fe-Fe 3 C to Fe-Fe 7 C 3 above 250 GPa.
The upper and lower bounds for ∆G sol liq 1bar,T (Fe 7 C 3 ) are tightly constrained by the following observations: (a) Fe 7 C 3 + liquid is stable rather than Fe 3 C at the melting temperature of Fe 3 C so that Fe 3 C melts incongruently, and (b) Fe 3 C + liquid is stable rather than Fe 7 C 3 + liquid at the Fe-Fe 3 C eutectic temperature below 250 GPa (Figure 6b).In addition, the liquidus field of Fe 3 C disappears at 250 GPa when the Fe 3 C solidus becomes identical with the eutectic temperature (Figure 7c).These constraints provide the polynomial equation for ∆G sol liq 1bar,T (Fe 7 C 3 ) (Equation 5, Table 4) to be consistent with experimental results from which the ∆G sol liq 1bar,T (Fe 7 C 3 ) value is calculated with Equation 12given below for each datum (Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009) (Figure 6b): The melting curve of Fe 7 C 3 calculated from ∆G sol liq 1bar,T (Fe 7 C 3 ) and its EoSs in the liquid and solid states (Table 3) is almost parallel to the melting curve of Fe (Figure 7a).We also calculated the solidus curve of Fe 3 C (melting of solid Fe 3 C to solid Fe 7 C 3 + liquid) from Equations 11 and 12 and the liquidus curve of Fe 3 C (crystallization of solid Fe 7 C 3 from liquid Fe 3 C) from Equation 12using W Fe-FeC , ∆G sol liq 1bar,T , and the EoSs of Fe 3 C (only for calculating the solidus) and Fe 7 C 3 (Figure 7).The solidus curve of Fe 3 C is within the variations in earlier experimental results and close to relatively low-temperature ones (Li et al., 2016;Lord et al., 2009;Nakajima et al., 2009;Takahashi et al., 2020).On the other hand, the liquidus curve of Fe 3 C and the melting curve of Fe 7 C 3 obtained here are lower by ∼500-1000 K than the Fe 3 C liquidus and the Fe 7 C 3 solidus, respectively, reported by previous experimental studies (Li et al., 2016;Lord et al., 2009;Takahashi et al., 2020) except that the experiments by Nakajima et al. (2009) determined the liquidus of Fe 3 C close to that of our model at ∼20-30 GPa.Such discrepancy might be due to experimental difficulties in determining melting temperatures in a diamond-anvil cell; previous results on the melting temperature of Fe have been inconsistent with each other by as much as ∼1000 K (Anzellini et al., 2013;Sinmyo et al., 2019).Since the melting temperature of Fe 7 C 3 is close to that of Fe, the liquidus field of Fe 7 C 3 does not expand even at core pressures unlike the thermodynamic calculations by Fei and Brosh (2014), resulting in the almost constant eutectic liquid composition above 250 GPa (Figure 8).1bar,T of Fe 3 C calculated from compositions of liquids coexisting with solid Fe 3 C reported by previous experiments (circles) (Chabot et al., 2008;Chipman, 1972;Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009;Tanaka, 1967).A red curve shows the results of fitting.(b) ∆G sol liq 1bar,T for Fe 7 C 3 from experimental studies (circles) (Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009).Gray inversed triangles are the upper bound based on the Fe 3 C melting temperature at 10, 20, 30, 50, 75, 100, 150, and 200 GPa calculated in this study.Gray triangles are the lower bound constrained from the Fe + Fe 3 C eutectic temperature at these pressures.A gray plus is a constraint to change the Fe-Fe 3 C eutectic to the Fe + Fe 7 C 3 eutectic system at 250 GPa.A red line is adopted in this study.(eutectic, black) calculated in this study.Filled red symbols show experimentally obtained solidus temperatures of Fe 3 C; circles (Lord et al., 2009), diamonds (Nakajima et al., 2009), squares (Li et al., 2016), and pentagons (Takahashi et al., 2020).Open red symbols indicate earlier experimental data for the Fe 3 C liquidus; diamonds (Nakajima et al., 2009), squares (Li et al., 2016), and pentagons (Takahashi et al., 2020).The solidus temperature of Fe 7 C 3 is given by blue filled circles (Lord et al., 2009).Black crosses indicate the eutectic temperature where a liquid coexisted with Fe and Fe 3 C (or Fe 7 C 3 ) (Mashino et al., 2019).Normal and inversed black triangles show the lower and upper bounds for such eutectic temperature, respectively (Chabot et al., 2008;Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009)

The Martian Core
The low density of the Martian core revealed by the InSight mission suggests that substantial amounts of light elements such as sulfur are included in the core of Mars (Irving et al., 2023;Khan et al., 2023;Samuel et al., 2023;Stähler et al., 2021).The tidal deformation as well as seismology indicates that the core is not completely solidified (Yoder et al., 2003).While the solid inner core was not detected by seismic waves transiting the upper part of the core (Irving et al., 2023), Fe and Fe-S alloys have been considered to be possible crystallizing solids in the Martian core from the liquidus phase relations in the Fe-S system (Helffrich, 2017;Stewart et al., 2007).
Carbon is one of major candidates for the Martian core light elements in addition to sulfur, and its estimated concentration ranges from less than 1.4 wt% to more than 3 wt% (Khan et al., 2022;Steenstra & van Westrenen, 2018).The liquidus phase relation at 40 GPa (close to the pressure at the center of the Martian core) obtained in the present modeling shows ∼3.5 wt% C in the Fe-Fe 3 C eutectic liquid (Figure 9a).According to the C-rich estimations (Khan et al., 2022(Khan et al., , 2023)), the carbon content in the core is close to the Fe-Fe 3 C eutectic composition, suggesting that Fe 3 C can be a crystallizing solid.The higher eutectic temperature of ∼2130 K in the Fe-Fe 3 C system than that of 1650 K in the Fe-Fe 3 S system (Mori et al., 2017) reinforces the possibility that Fe 3 C crystallizes earlier than Fe-S alloys as the core cools down.On the other hand, Fe 7 C 3 crystallizes from the liquid containing more than ∼5.4 wt% C (Figure 9a).Such carbon content is higher than the solubility of carbon in Fe-Ni-S liquid with >10 wt% S at ∼10 GPa (Tsuno et al., 2018).Therefore, the carbon content in the core of Mars is unlikely to be high enough to crystallize Fe 7 C 3 as a solid inner core.

The Earth's Core
Liquidus phase relations at 330 GPa constrain the composition and temperature of both the outer and inner core of the Earth.It has been argued that Fe 7 C 3 constitutes the inner core based on its high Poisson's ratio (compressional/ shear velocity ratio) (Chen et al., 2014;Prescher et al., 2015).The carbon content of the eutectic point in the Fe-Fe 7 C 3 system dictates the inner core phase if carbon is a predominant light element in the outer core.While carbon cannot be a single light element in the outer core to explain the observed density and sound velocity of the outer core, its concentration can be up to 3.9-4.2wt% C (Badro et al., 2014;Umemoto & Hirose, 2020).Such carbon content is slightly higher than the eutectic composition of Fe + 3.8 wt% C at 330 GPa predicted by the present thermodynamic model (Figure 9c).If Fe 7 C 3 is crystallizing at the solid inner core, the outer core liquid would be similar in carbon concentration to the Fe-Fe 7 C 3 eutectic liquid at 330 GPa.In this case, the ICB temperature is constrained to be close to the eutectic temperature of 3760 K that is substantially lower than previous estimates (Hirose et al., 2021).The 3760 K at the ICB gives 2760 K at the CMB when using Grüneisen parameter of 1.5 (Hirose et al., 2013;Vočadlo et al., 2003), which is lower than the Fe-Fe 3 C eutectic temperature at 136 GPa (Figure 9b) and therefore not feasible.Also such low CMB temperature suggests a minimal temperature difference across a thermal boundary layer in the lowermost mantle, which is also unlikely.Furthermore, the EoS of solid Fe 7 C 3 adopted in this study (Table 3) yields the lower density (∼12.2 g/cm 3 ) than that the PREM (∼12.8 g/ cm 3 ) (Dziewonski & Anderson, 1981) at the ICB even for ICB temperatures of less than 4000 K, consistent with Li et al. (2016) who obtained the density of Fe 7 C 3 at inner core conditions using ab initio molecular dynamics simulations.Earlier studies argued that the mixture of Fe 7 C 3 and hcp-Fe can explain the density of the inner core (Chen et al., 2012;Nakajima et al., 2011).Nevertheless, the Fe + Fe 7 C 3 cotectic temperatures in multi-component Fe alloy systems will be even lower than the eutectic temperature in the binary system.These suggest that Fe 7 C 3 is unlikely to be present in the Earth's inner core.
The density gap between the outer and inner core at the ICB suggests that the inner core is composed of hcp-Fe being depleted in light elements compared to the outer core liquid (Tateno et al., 2010).The upper limit of carbon concentration in the outer core is bounded by the amount of carbon in the Fe-Fe 7 C 3 eutectic liquid at 330 GPa.Indeed, the outer core carbon content has been considered to be minor compared to concentrations of other candidate light impurity elements (Hirose et al., 2021;Li et al., 2019).Despite its small amount, the amount of carbon in the outer core is important to constrain the ICB temperature.The liquidus temperature to crystallize Fe  (Chabot et al., 2008;Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009).drops by ∼450 K by the addition of 1 wt% C to Fe at 330 GPa (Figure 9c).Such effect of depressing the liquidus temperature is similar to that of 1 wt% S (Thompson et al., 2022) and larger than those by 1 wt% Si of less than 100 K (Edmund et al., 2022) and by 1 wt% O of ∼130 K (Oka et al., 2019).In addition, a large depression of the liquidus temperature of Fe by the presence of carbon at the CMB pressure (Figure 9b) is preferable for a relatively low CMB temperature of ∼3400-3600 K, which has been suggested by the solidus temperature of the pyrolitic lowermost mantle (Kim et al., 2020;Nomura et al., 2014).

Figure 3 .
Figure 3.A flowchart in the present thermodynamic modeling.See text for the definition of each parameter.For each item, the shape of the frame indicates how it was obtained.Black, Fe; red, Fe 3 C; blue, Fe 7 C 3 ; gray, liquid.

Figure 6 .
Figure 6.(a) ∆G sol liq1bar,T of Fe 3 C calculated from compositions of liquids coexisting with solid Fe 3 C reported by previous experiments (circles)(Chabot et al., 2008;Chipman, 1972;Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009;Tanaka, 1967).A red curve shows the results of fitting.(b) ∆G sol liq 1bar,T for Fe 7 C 3 from experimental studies (circles)(Fei & Brosh, 2014;Mashino et al., 2019;Nakajima et al., 2009).Gray inversed triangles are the upper bound based on the Fe 3 C melting temperature at10, 20,  30, 50, 75, 100, 150, and 200  GPa calculated in this study.Gray triangles are the lower bound constrained from the Fe + Fe 3 C eutectic temperature at these pressures.A gray plus is a constraint to change the Fe-Fe 3 C eutectic to the Fe + Fe 7 C 3 eutectic system at 250 GPa.A red line is adopted in this study.

Figure 9 .
Figure 9. Calculated liquidus phase relations at (a) 40 GPa, (b) 136 GPa, and (c) 330 GPa.Dotted phase boundaries such as the Fe/Fe + Fe 3 C boundary were not calculated in this study but drawn as guides.The liquidus on the Fe side in (a) was calculated using parameters for fcc-Fe.

Table 1
Results of First-Principles Calculations for Liquid Fe and Pressures to Match Experimental Data

Table 2
Results of First-Principles Calculations for Liquid Fe 7 C 3

Table 4
Parameters for Difference in Gibbs Free Energy Between Solid and Liquid at 1 Bar