# High-Pressure Melting Experiments of Fe3C and a Thermodynamic Model of Fe-C Liquids for the Earth's Core

T. Komabayashi, C. Mcguire, S. Thompson, G. D. Bromiley, A. Bravenec, A. Pakhomova

► To cite this version:

T. Komabayashi, C. Mcguire, S. Thompson, G. D. Bromiley, A. Bravenec, et al., High-Pressure Melting Experiments of Fe3C and a Thermodynamic Model of Fe-C Liquids for the Earth's Core. Journal of Geophysical Research: Solid Earth, 2024, 129 (9), pp.e2024JB029641. 10.1029/2024JB029641. hal-05098756

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# JGR Solid Earth

## RESEARCH ARTICLE

10.1029/2024JB029641

**Key Points:**

* Melting experiments of Fe₃C were conducted to 85 GPa in diamond anvil cells with in situ X-ray diffraction and textural observation

* A thermodynamic model for melting of the system Fe-C which is applicable to the center of the Earth was established

* The Earth's core cannot be approximated by the system Fe-C and should include another light element

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

T. Komabayashi
tetsuya.komabayashi@ed.ac.uk

**Citation:**

Komabayashi, T., McGuire, C., Thompson, S., Bromiley, G. D., Bravenec, A., & Pakhomova, A. (2024). High-pressure melting experiments of Fe₃C and a thermodynamic model of Fe-C liquids for the Earth's core. *Journal of Geophysical Research: Solid Earth*, 129, e2024JB029641. https://doi.org/10.1029/2024JB029641

Received 3 JUN 2024
Accepted 10 SEP 2024
Corrected 17 DEC 2024

This article was corrected on 17 DEC 2024. See the end of the full text for details.

**Author Contributions:**

**Conceptualization:** T. Komabayashi

**Data curation:** T. Komabayashi, C. McGuire

**Formal analysis:** T. Komabayashi, C. McGuire

**Funding acquisition:** T. Komabayashi, C. McGuire

**Investigation:** T. Komabayashi, C. McGuire, S. Thompson, G. D. Bromiley, A. Bravenec, A. Pakhomova

**Methodology:** T. Komabayashi, C. McGuire

© 2024. The Author(s). This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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## High-Pressure Melting Experiments of Fe₃C and a Thermodynamic Model of Fe-C Liquids for the Earth's Core

T. Komabayashi¹ Ⓓ, C. McGuire¹,² Ⓓ, S. Thompson¹, G. D. Bromiley¹, A. Bravenec¹,³ Ⓓ, and A. Pakhomova⁴,⁵

¹School of GeoSciences and Centre for Science at Extreme Conditions, University of Edinburgh, Edinburgh, UK, ²Now at Lawrence Livermore National Laboratory, Livermore, CA, USA, ³Now at Department of Earth and Space Sciences, University of Washington, Seattle, WA, USA, ⁴Deutsches Elektronen-Synchrotron (DESY), Hamburg, Germany, ⁵Now at European Synchrotron Radiation Facility, Grenoble, France

**Abstract** Melting experiments of Fe₃C were conducted to 85 GPa in laser-heated diamond anvil cells with in situ X-ray diffraction and post-experiment textural observation. From the determined pressure-temperature conditions of the melting curve for Fe₃C, together with literature data on the melting point of diamond and eutectic point of the system Fe-Fe₃C/Fe₇C₃ under high pressures, we established a self-consistent thermodynamic model for high-pressure melting of the system Fe-C including the mixing parameters for liquids. The results show that mixing of Fe and C liquids is negatively nonideal from 1 bar to the pressure at the center of the Earth. The departure from ideal mixing becomes progressively larger with increasing pressure, which leads to greatly stabilized liquids under core pressures. The modeled carbon content in eutectic melts under core pressures is 3.3–4.4 wt%. From the Gibbs free energy, we derived an internally consistent parameters for Fe-C outer cores which included the crystallizing points at their bottoms, isentropic thermal profiles, and densities and longitudinal seismic wave speeds ($Vp$). While the addition of carbon in excess of the eutectic melt composition effectively reduces the density of iron liquid, the $Vp$ of iron liquid is not greatly changed. Therefore, the low density and high $Vp$ of PREM relative to pure iron cannot be reconciled by an Fe-C liquid. Therefore, the Earth's core cannot be approximated by the system Fe-C and should include another light element.

**Plain Language Summary** The presence or absence of carbon in the Earth's central core has long been discussed in order to account for the observed core density lower than iron because the identification of such light elements in the core will provide critical information about the origin and evolution of the solid Earth. The relevance of carbon in the Earth's core is testable by comparing the density and seismic wave velocities of candidate Fe-C materials with observations. Here we present a new experiment-based thermodynamic model for the system Fe-C. We first determined the melting points of Fe₃C to 85 GPa. We then assessed thermodynamic parameters of Fe-C liquids, from the constrained pressure-temperature conditions of the melting curve for Fe₃C, together with literature data. From the established model, we consistently computed the properties of hypothetical Fe-C liquid outer cores including the density and longitudinal seismic wave velocity. While a carbon concentration greater than 4 wt% effectively reduces the density of iron liquid, the seismic velocity is not greatly changed, and therefore carbon cannot reconcile the low density and high velocity nature of the Earth's outer core. Therefore, the Earth's core cannot be approximated by the system Fe-C and should include another light element.

### 1. Introduction

The presence of carbon in the Earth's central core has long been discussed because of its high cosmic abundance and chemical affinity to iron (Benz & Elliott, 1961), and as part of global carbon cycle. At the same time, carbon is one of the possible impurity elements in the core among conventionally considered five candidates: Si, S, O, C, and H (Hirose et al., 2013; Komabayashi, 2021; Poirier, 1994) which result in low densities compared with iron (Ogino et al., 1984), in order to account for observed core density deficits: 3%–4% for the inner core and 7%–8% for the outer core (Fei et al., 2016; Komabayashi, 2014; Kuwayama et al., 2020). The identification of light elements in the core has been considered to provide critical information about the origin and evolution of the solid Earth and the origin of magnetic fields. Despite a research history of more than 70 years (Birch, 1952), the presence or absence of carbon in the core is yet to be determined.
**Project administration:** T. Komabayashi

**Resources:** T. Komabayashi

**Supervision:** T. Komabayashi

**Writing – original draft:** T. Komabayashi

**Writing – review & editing:** C. McGuire,
G. D. Bromiley, A. Pakhomova

Element partitioning studies indicated that the core should have incorporated oxygen and silicon as dominant light elements during core formation, if partitioning of elements between metal and silicate in the magma ocean occurred under equilibrium conditions (Rubie et al., 2011; Siebert et al., 2013; Wade & Wood, 2005). Sulfur might also have been partitioned to the core as an equilibrium core formation process, with an amount of no more than 2 wt% (e.g., Boujibar et al., 2014). Similarly, the carbon content in the core has been proposed to be on the order of 0.1–0.4 wt% from metal-silicate partitioning studies (Dasgupta & Walker, 2008; Dasgupta et al., 2013). Also the observed carbon isotopic composition of silicate Earth can be explained by core formation which fractionated carbon isotopes and as a results, the Earth's core would contain ~1 wt% carbon (Wood et al., 2013). This implies that a significant amount (>>1 wt%) of carbon in the core is not expected from the geochemical point of view. On the other hand, comparing physical properties of Fe-C alloys between laboratory measurements/computer simulations and geodetic observations, the presence of carbon in the core has been repeatedly suggested. Mashino et al. (2019) determined eutectic relations of the system Fe-Fe₃C in diamond anvil cells (DAC) to 250 GPa and reported that solid Fe₃C is stable up to 250 GPa but Fe₇C₃ is the liquidus phase for liquid Fe₃C above that pressure, so the crystallizing phase in the Fe-C core at the outer core-inner core boundary (ICB, 330 GPa) would be Fe₇C₃ if the bulk composition of the outer core is more carbon-rich than the eutectic composition. Nakajima et al. (2011) constructed a thermal equation of state (EoS) for solid Fe₇C₃ and suggested that it can account for the inner core density when the core temperature is as low as 3600 ± 2000 K. Prescher et al. (2015) reported that the high Poisson's ratio of the Earth's inner core can be explained by orthorhombic Fe₇C₃ among different possible Fe-light element alloys. As such, a carbon-rich core would satisfy some of the observed properties of the inner core.

In this study, the relevance of the system Fe-C to the Earth's core is tested by comparing geophysically observable parameters, which include density and seismic wave velocity of candidate materials. As the liquid outer core dominates the core volume, constraining the outer core composition would be the key to successful identification of the light elements in the core. Because laboratory experiments on liquid iron alloys under high pressure, particularly core pressures, is very challenging, first-principles calculations have led the field (Badro et al., 2014; Ichikawa & Tsuchiya, 2020; Umemoto & Hirose, 2020; Yokoo et al., 2024). Direct experimental examination of the longitudinal seismic wave speed ($Vp$) of Fe-C liquids under core pressures in DAC were reported by Nakajima et al. (2015). These previous theoretically or experimentally constrained $Vp$ data were, however, not necessarily guaranteed to be consistent with phase relations. $Vp$ of a liquid phase can be calculated from the adiabatic bulk modulus and density which are second and first derivatives respectively, of the Gibbs free energy which controls the stability of the phase. Therefore, the phase stability such as the crystallizing point and its seismic velocity should be consistently presented for discussions regarding the outer core. In order to apply a hypothetical core model to the Earth, a self-consistent set of thermodynamic parameters is needed for liquids and solids of each candidate system, which includes phase equilibria and EoS of constituent phases. As such, establishing a thermodynamic model for each system is a very promising approach.

The thermodynamic model of the system Fe-C at 1 bar has been repeatedly updated (Chipman, 1972; Gustafson, 1985). Liquids show negatively nonideal mixing behavior at 1 bar, which stabilizes a liquid field in the temperature-composition (T-X) phase diagram more than ideal mixing (Gustafson, 1985). The mixing properties at high pressure should be investigated to derive a reliable thermodynamic model for Fe-C liquids under Earth's core conditions. Wood (1993) made pioneering work on phase relations of the system Fe-C under high pressures by constructing a thermodynamic model using his own experimental data to 5 GPa and available physical properties of phases at that time. His work highlighted the importance of the stability of Fe₃C and Fe₇C₃, and nonideality of Fe-C liquids in order to address core processes. Since then, the system Fe-C under high pressure has been extensively studied by both experiment and theory (Chabot et al., 2008; Dasgupta & Walker, 2008; Dasgupta et al., 2013; Fei & Brosh, 2014; Li et al., 2002; Litasov et al., 2013; Lord et al., 2009; Mashino et al., 2019; McGuire et al., 2021; Mookherjee et al., 2011; Morard et al., 2017; Nakajima et al., 2009, 2011; Rohrbach et al., 2014; Yokoo et al., 2024). Fei and Brosh (2014) produced an updated thermodynamic model based on new experimental data to 25 GPa, and also highlighted the importance of the stability of Fe₃C and Fe₇C₃. On the other hand, from first-principles calculation, Yokoo et al. (2024) examined pressure-volume-temperature (P-V-T) relations of Fe₇C₃ liquid at 60–360 GPa and 3000–7000 K, which were then used for modeling liquidus phase relations of the subsystem Fe-FeC to core pressures. As such, a further extended experiment-based Fe-C liquids model is needed which can be reliably applied to P-T conditions greater than the previous models (Fei & Brosh, 2014; Wood, 1993). Moreover, all these existing thermodynamic models were focused on producing melting phase diagrams rather than providing the physical properties of liquids under core conditions, although P-
<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) Experimental and calculated P-T conditions on the melting of Fe₃C of this study. The light blue, black solid, half-filled, and open symbols denote stability of Fe₃C + C, Fe₃C + Fe₇C₃, Fe₃C + Fe₇C₃ + liquid, and Fe₇C₃ + liquid, respectively. Existing experimental data by Liu et al. (2016) are shown. The newly calculated boundary for the reaction Fe₃C + C = Fe₇C₃ in this study is also plotted. (b) The eutectic reaction of Fe + Fe₃C = liquid from the thermodynamic model of this study. Existing experimental data are also plotted (Fei & Brosh, 2014; Mashino et al., 2019; Morard et al., 2017). The black and purple solid lines are from (a).</figcaption></figure>

V-T EoS of Fe-C liquids were included (Fei & Brosh, 2014; Wood, 1993; Yokoo et al., 2024). We have been establishing thermodynamic models for Fe-alloys which produce consistent sets of properties of the systems such as phase relations, isentrope of a convecting fluid, and density and Vp of liquids (Komabayashi, 2014, 2020; Thompson et al., 2022).

For modeling Fe-C liquids, melting relations of carbides under high pressure need to be determined by experiment. Figure 1 shows a summary of recent experimental data on the following two carbide-bearing reactions:

$$ \text{The eutectic reaction: } Fe + Fe_3C = \text{liquid} \quad (1) $$

and

$$ \text{The solidus for Fe}_3\text{C: Fe}_3\text{C} = \text{Fe}_7\text{C}_3 + \text{liquid} \quad (2) $$

Thermodynamics requires that the eutectic reaction should take place at a lower temperature than the solidus of Fe₃C. Recent laser-heated DAC experiments reported Reaction 1 (Mashino et al., 2019; Morard et al., 2017) at temperatures higher than Reaction 2 (Liu et al., 2016) up to 85 GPa, although temperature uncertainty in experiment may resolve this thermodynamic inconsistency. Two independent groups determined the P-T conditions of Reaction 1 from textural observation of recovered samples, and reported consistent results (Figure 1) (Mashino et al., 2019; Morard et al., 2017). In contrast, Reaction 2 was determined by an in situ X-ray diffraction (XRD) study only (Liu et al., 2016). Molten textures need to be confirmed by post-mortem textural observation, as was done on Reaction 1 (Mashino et al., 2019; Morard et al., 2017), for a precisely consistent phase diagram. Moreover, the crystal structure of solid Fe₇C₃ is contentious. On Reaction 2, Liu et al. (2016) observed solid Fe₇C₃ with an orthorhombic symmetry, consistent with Prescher et al. (2015) who claimed a structure with the orthorhombic symmetry from a single crystal XRD from ambient to 205 GPa and above 3700 K, while other experiments by Nakajima et al. (2011) confirmed the stability of a hexagonal structure up to 71.5 GPa and 1973 K by in situ powder XRD. The hexagonal structure was further supported by recent first-principles calculations at pressures greater than 150 GPa (Das et al., 2017; Raza et al., 2015). As mentioned above, Prescher et al. (2015) reported that the high Poisson's ratio of orthorhombic Fe₇C₃ can account for the observation of the inner core. Therefore, the crystal structure of Fe₇C₃ needs to be carefully considered when a thermodynamic model is established. Furthermore, the low P-T stability of solid Fe₇C₃ have not been defined. At 1 bar, Fe₃C, if stable, would coexist with a carbon phase in the binary diagram which means that Fe₇C₃ is not stable (e.g., Gustafson, 1985). Previous experiments reported that solid Fe₇C₃ became stable at 10 GPa but not at 5 GPa (Nakajima et al., 2009). As such there is likely an equilibrium reaction:

$$ Fe_3C + C (\text{diamond}) = Fe_7C_3 \quad (3) $$

Since the expected temperature conditions for Reaction 3 are low, chemical equilibrium may not be achieved within the experimental timescale. One can place constraints on the P-T locations for the boundary once the thermodynamic parameters for the phases have been established based on high temperature data.

Here we present a new experiment-based thermodynamic model for the system Fe-C, with the inclusion of recent updates on the physical properties of relevant phases. We first determined the melting point of Fe₃C (Reaction 2) under high pressures in DAC, based on both in situ X-ray diffraction and post-mortem textural observations. We
then modeled both thermochemical and thermophysical properties of Fe-C liquids, including melting of diamond and the mixing properties of Fe-C liquids, so that calculated melting points of Fe₃C under high pressures are consistent with our experimental data. We compute the crystallizing points of hypothetical Fe-C outer cores, their isentropes, and geophysically observable properties (the density and *Vp*), all of which are derived from the Gibbs free energy calculations. Then we compare the computed density and *Vp* with those measured by seismology and discuss the relevance of the system Fe-C to the core.

## 2. Methods

### 2.1. Experimental Procedure

High-pressure-temperature experiments were conducted in laser-heated DAC. High pressures were generated with a pair of opposed diamond anvils and rhenium gasket. The culet size of the diamond was 300 µm. The starting material was a solid Fe₃C phase (cementite) synthesized in a piston-cylinder press at the University of Edinburgh, which was also used in McGuire et al. (2021). The powder mixture of Fe and C for the piston-cylinder synthesis run was slightly enriched in C, so the starting material in this study was Fe₃C + C. During laser heating, a tiny amount of an Fe₇C₃ phase was recognized in the XRD pattern at relatively low temperatures, which can be explained by Reaction 3 as discussed later. The presence of Fe₇C₃ ensures the observed melting reaction is Fe₃C = Fe₇C₃ + liquid (Reaction 2), rather than the eutectic melting (Fe + Fe₃C = liquid, Reaction 1). The sample was ground and made into pellets with 10 µm thickness and was sandwiched between layers of compressed pellets of Al₂O₃. High temperatures were generated with a double-sided laser-heating system at P02.2, PETRA III, DESY (Konopkova et al., 2021). The diameter of the laser beam was about 20 µm. Temperatures were measured by the spectroradiometric method. See Konopkova et al. (2021) for details of the laser-heating setup at the beamline. Heating was ramped from the first observable emission by pyrometry in laser power steps of ~1 W (0.5 W per side) until approaching 90% of the homologous temperature for the reported Fe-Fe₃C eutectic point (Morard et al., 2017) at a given pressure. Each recorded temperature and pressure point was held for several seconds. At the end of the heating cycle, we increased the laser power in a larger step change (several Watts) to melt the sample and simultaneously recorded a diffraction pattern and pyrometry before immediately quenching. We took the average of the temperatures measured at both sides of the sample along the laser beam path. An uncertainty of 5% which is typical to the modern laser-heating systems (e.g., Morard et al., 2011; Thompson et al., 2022) and the temperature difference between the two sides of the sample were adopted as the overall uncertainty in temperature.

In-situ XRD experiments were performed at the beamline P02.2. The details of the experimental set up at the beamline including the method of aligning the paths of laser beam, X-rays, and temperature measurement were given in Konôpková et al. (2021). Monochromatic X-rays of 0.290 Å were focused on sample position with areas of 2 µm * 2 µm. XRD data were collected on a flat panel detector (Perkin Elmer XRD 1621), with an exposure time of 1 s. The two-dimensional diffraction data were converted to conventional one-dimension patterns using the software package Dioptas (Prescher & Prakapenka, 2015).

We first compressed the samples to target pressures at room temperature using the Raman shift of the diamond anvils as a pressure gauge (Akahama & Kawamura, 2004). The samples were then heated by laser for melting and then quenched by turning the laser power off. The pressure determination under high temperatures was made based on the unit-cell volumes and the EoS of both solid Fe₃C (McGuire et al., 2021) and Fe₇C₃ where possible. For consistency, the pressure values throughout this study are based on Fe₃C. When Fe₃C is not available (i.e., on or above melting point), we use the pressure values from Fe₇C₃. The pressure uncertainty was obtained from the uncertainties in the unit-cell volume of the sample and in the temperature uncertainty through employing its EoS. The EoS for solid Fe₇C₃ was newly assessed in this study by fitting the *P-V-T* data of Nakajima et al. (2011). The assessment was primarily made in order to maintain the consistency with Fe-C liquids under core *P-T* conditions as discussed later. Nevertheless, the experimental conditions of the present study are mostly covered by Nakajima et al. (2011)'s data set, hence the calculated pressures are consistent with Nakajima et al. (2011)'s original EoS. The *P-V-T* EoS for solid Fe₃C and Fe₇C₃ in this study are consistent with the *P-V-T* EoS of hcp Fe proposed by Dewaele et al. (2006), based on which Komabayashi (2014, 2020) and Thompson et al. (2022) constructed thermodynamic models for Fe, Fe-FeO, Fe-Si, and Fe-Fe₃S. Therefore the experimental data and thermodynamic model in this study are consistent with those earlier data sets.
Textural observations of recovered samples were made with a field-emission gun scanning electron microscope (SEM) attached to a Focused Ion Beam system (Zeiss Crossbeam 550 FIB-SEM), installed at the University of Edinburgh. Recovered samples were placed in the FIB unit and milled with an in situ milling system using Ga ions. They were milled with a trapezoidal geometry to expose the laser heated region. Initial millings were made using relatively high beam currents (30–65 nA), followed by polishing the surface for clearer visibility of textures with smaller beam currents (7–3 nA). Backscattered electron imaging was used for textural analysis of the samples, operating at either 3 or 15 kV.

## 2.2. Thermodynamics

The above obtained experimental data together with existing literature experimental data were used to model melting relations in the system Fe-C under high pressure. A thermodynamic potential, the Gibbs free energy, of phases was parameterized as a function of P, T, and X. The molar Gibbs free energy (G) of a phase at a given P-T condition can be calculated via:

$$G = G_T^\circ + \int_{1\text{bar}}^{P} V_T \, dP \quad (4)$$

where $G_T^\circ$ and $V_T$ are the molar Gibbs free energy at 1 bar and T of interest and the molar volume at T. $G_T^\circ$ is expressed with a polynomial function as:

$$G_T^\circ = a + bT + cT \ln T + dT^2 + eT^{-1} + fT^{0.5}. \quad (5)$$

The volume integration term in Equation 4 is calculated from a P-V-T EoS of the phase. As in our previous works on different systems, we adopted the formulation of the Vinet-Anderson-Grüneisen (Vinet-A-G) EoS. The Vinet EoS is used for the pressure at compression.

$$P_{298} = 3K_0x^{-2}(1-x) \exp\left[\frac{3}{2}(K' - 1) * (1-x)\right] \quad (6)$$

where $x \equiv (V/V_0)^{1/3}$ and, $P_{298}$, $K_0$, $K'$, and $V_0$ are the pressure at 298 K, the isothermal bulk modulus, its pressure derivative, and the molar volume at 1 bar and 298 K, respectively.

The thermal expansion coefficient (thermal expansivity) at compression was modeled using the A-G parameter (Anderson et al., 1992; Komabayashi, 2014; Thompson et al., 2022).

$$\frac{\alpha}{\alpha_0} = \exp\left[-\frac{\delta_0}{\kappa}(1 - \eta^\kappa)\right] \quad (7)$$

where $\alpha$ is the thermal expansivity at compression, $\alpha_0$ is the thermal expansivity at 1 bar, $\eta \equiv V/V_0$, $\delta_0$ is the A-G parameter at $\eta = 1$ (i.e., 1 bar), and $\kappa$ is the dimensionless parameter.

T-X phase diagrams for eutectic relations in the system Fe-C were calculated using the thermodynamic data for the end-members and mixing properties of liquids (e.g., Nakamura & Banno, 1991). When an Fe phase is at equilibrium with a liquid phase (i.e., partial melting),

$$\mu_{\text{Fe}}^{\text{Liquid}} = \mu_{\text{Fe}}^{\text{Solid}} \quad (8)$$

where, for instance, $\mu_{\text{Fe}}^{\text{Liquid}}$ is the chemical potential of Fe for the liquid phase, which can also be expressed as,

$$\mu_{\text{Fe}}^{\text{Liquid}} = G_{\text{Fe}}^{\text{Liquid}} + RT \ln a_{\text{Fe}}^{\text{Liquid}} \quad (9)$$

where $G_{\text{Fe}}^{\text{Liquid}}$, R, and $a_{\text{Fe}}^{\text{Liquid}}$ is the molar Gibbs free energy of the liquid Fe phase (i.e., pure Fe), the gas constant, and the thermodynamic activity of Fe in the liquid phase, respectively. In the same way, for $\mu_{\text{Fe}}^{\text{Solid}}$, we can write,
$$ \mu_{\text{Fe}}^{\text{Solid}} = G_{\text{Fe}}^{\text{Solid}} + RT \ln a_{\text{Fe}}^{\text{Solid}}. \qquad (10) $$

Note that Fe phases are not pure Fe but have some solubility of C when coexisting with a carbon-bearing phase (e.g., Fei & Brosh, 2014) while the solid carbon phase (i.e., diamond) does not accommodate Fe at all, which means

$$ \mu_{C}^{\text{Solid}} = G_{C}^{\text{Solid}}. \qquad (11) $$

Using Equations 8–10, one obtains,

$$ \ln \frac{a_{\text{Fe}}^{\text{Liquid}}}{a_{\text{Fe}}^{\text{Solid}}} = -(G_{\text{Fe}}^{\text{Liquid}} - G_{\text{Fe}}^{\text{Solid}})/RT. \qquad (12) $$

The activity is converted from the composition of the phase. For example, $a_{\text{Fe}}^{\text{Liquid}}$ is described as,

$$ a_{\text{Fe}}^{\text{Liquid}} = \Gamma_{\text{Fe}}^{\text{Liquid}} X_{\text{Fe}}^{\text{Liquid}} \qquad (13) $$

where $\Gamma_{\text{Fe}}^{\text{Liquid}}$ is the activity coefficient and $X_{\text{Fe}}^{\text{Liquid}}$ is the mole fraction of Fe in the liquid phase with,

$$ X_{\text{Fe}}^{\text{Liquid}} + X_{C}^{\text{Liquid}} = 1. \qquad (14) $$

Equation 14 requires that the properties of carbon-baring liquids, for example, Fe₃C, need to be for Fe₀.75C₀.25, namely 0.75*Fe + 0.25*C, not 3*Fe + 1*C. The nonideal interaction between Fe and C atoms can be described by introducing the activity coefficient. The Redlich and Kister model (Guggenheim, 1937; Redlich & Kister, 1948) applies to express $\Gamma_{\text{Fe}}^{\text{Liquid}}$ via,

$$ RT \ln \Gamma_{\text{Fe}}^{\text{Liquid}} = (X_{C}^{\text{Liquid}})^{2} * L_{\text{Fe,C}} \qquad (15) $$

$$ RT \ln \Gamma_{C}^{\text{Liquid}} = (1 - X_{C}^{\text{Liquid}})^{2} * L_{\text{Fe,C}} \qquad (16) $$

where $L_{\text{Fe,C}}$ represents the interaction parameter. The mixing properties of Fe and C liquids (i.e., $L_{\text{Fe,C}}$) under high pressures were assessed in this study.

# 3. Results

## 3.1. High-P-T Experiments

We conducted three sets of laser-heating experiments with in situ XRD at about 50, 70, and 85 GPa at P02.2, PETRA-III. The experimental P-T conditions and results are summarized in Figure 1a and Table 1. As demonstrated in Table 1, the pressure values calculated from Fe₃C and Fe₇C₃ are consistent throughout the experimental P-T conditions. This is important as we construct a self-consistent thermodynamic model for the system Fe-C. A series of XRD patterns are shown in Figure 2 (3rd run) and Figure S1 (1st and 2nd runs).

In all the runs, we confirmed Reaction 2 Fe₃C = Fe₇C₃ + liquid and Reaction 3 Fe₃C + C = Fe₇C₃. In addition, the followings were observed in common. The samples at target pressures showed broad peaks from Fe₃C phase and the Al₂O₃ pressure medium due to the stress which was developed during compression. Then laser heating to 1500–1800 K released the stress and peak sharpening was observed. Diffraction peaks from a minor amount of Fe₇C₃ phase were clearly observed in the first few high-temperature XRD patterns which should have formed by Reaction 3 (Table 1; Figure 2). We increased the laser power significantly at the end of each heating cycle to melt the sample. We then quenched the samples by turning the laser power off. During the in situ runs, only the hexagonal structure was observed for solid Fe₇C₃. The recovered samples were then processed with the FIB and observed by SEM at the University of Edinburgh. We did not observe diffuse scatterings with in situ XRD. A diffuse scattering may not be observed every time one has a melt in a sample chamber. One of the key factors to making it observed is
<table><caption>**Table 1**

Experimental Conditions and Results</caption><thead><tr><th rowspan="2"></th><th colspan="3">Pressure (GPa)</th><th rowspan="2">Observed solid phases by in situ XRD</th><th colspan="2">Unit-cell volume (Å³)</th><th rowspan="2">Remarks</th></tr><tr><th>Fe₃C</th><th>Fe₇C₃</th><th>Temperature (K)</th><th>Fe₃C</th><th>Fe₇C₃</th></tr></thead><tbody><tr><td rowspan="6">Run 1</td><td>36.9 (5.0)</td><td></td><td>300</td><td>Fe₃C</td><td>134.7 (1.7)</td><td></td><td rowspan="6">Broad XRD peaks</td></tr><tr><td>50.3 (1.1)</td><td>52.7 (0.9)</td><td>1570 (90)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>133.8 (0.3)</td><td>162.3 (0.2)</td></tr><tr><td>48.9 (1.0)</td><td>52.1 (1.0)</td><td>1530 (90)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>134.1 (0.3)</td><td>162.4 (0.2)</td></tr><tr><td>48.6 (1.0)</td><td>51.2 (1.2)</td><td>1570 (90)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>134.3 (0.2)</td><td>162.9 (0.3)</td></tr><tr><td>50.3 (1.0)</td><td>52.3 (1.3)</td><td>1950 (100)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>134.7 (0.3)</td><td>163.9 (0.4)</td></tr><tr><td>49.5 (2.0)</td><td>53.4 (1.2)</td><td>2430 (120)</td><td>Fe₃C (minor) + Fe₇C₃</td><td>136.2 (0.7)</td><td>165.2 (0.2)</td></tr><tr><td rowspan="4">Run 2</td><td>55.2 (6.5)</td><td></td><td>300</td><td>Fe₃C</td><td>129.2 (1.8)</td><td></td><td rowspan="4">Broad XRD peaks</td></tr><tr><td>71.3 (5.9)</td><td></td><td>1720 (130)</td><td>Fe₃C + Fe₇C₃ (120 only)</td><td>127.7 (1.3)</td><td></td></tr><tr><td>71.4 (3.6)</td><td></td><td>1910 (150)</td><td>Fe₃C + Fe₇C₃ (120 only)</td><td>128.0 (0.8)</td><td></td></tr><tr><td>71.3 (4.3)</td><td></td><td>1900 (150)</td><td>Fe₃C + Fe₇C₃ (120 only)</td><td>128.1 (1.0)</td><td></td></tr><tr><td rowspan="10">Run 3</td><td></td><td>73.9 (2.1)</td><td>2520 (240)</td><td>Fe₃C (211, 102 only) + Fe₇C₃</td><td></td><td>157.3 (0.1)</td><td></td></tr><tr><td>86.8 (1.7)</td><td></td><td>300</td><td>Fe₃C</td><td>124.3 (0.4)</td><td></td><td>Broad XRD peaks</td></tr><tr><td>85.9 (1.3)</td><td></td><td>1770 (100)</td><td>Fe₃C</td><td>124.2 (0.3)</td><td></td><td>Broad XRD peaks</td></tr><tr><td>86.4 (1.9)</td><td>85.5 (5.6)</td><td>1910 (130)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>124.3 (0.4)</td><td>151.7 (1.7)</td><td></td></tr><tr><td>84.3 (2.0)</td><td>84.9 (5.7)</td><td>1880 (100)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>124.7 (0.4)</td><td>151.8 (1.8)</td><td></td></tr><tr><td>83.9 (1.8)</td><td>85.3 (5.9)</td><td>1860 (100)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>124.8 (0.4)</td><td>151.6 (1.8)</td><td></td></tr><tr><td>83.4 (1.9)</td><td>85.3 (5.0)</td><td>1870 (90)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>124.9 (0.4)</td><td>151.6 (1.5)</td><td></td></tr><tr><td>82.6 (2.6)</td><td>83.7 (4.1)</td><td>2180 (110)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>125.6 (0.6)</td><td>153.0 (1.3)</td><td></td></tr><tr><td>83.6 (2.5)</td><td>83.9 (5.1)</td><td>2180 (120)</td><td>Fe₃C + Fe₇C₃ (minor)</td><td>125.4 (0.6)</td><td>152.9 (1.6)</td><td></td></tr><tr><td></td><td>81.9 (1.7)</td><td>2930 (160)</td><td>Fe₇C₃</td><td></td><td>155.7 (0.3)</td><td></td></tr></tbody></table>

enough amount of melt in the sample chamber (Sinmyo et al., 2019). One would need a liquid layer as thick as 4–5 µm to observe diffuse scatterings (Morard et al., 2017; Thompson et al., 2022). As can be seen in the SEM images (Figure 3), the thicknesses of the liquid portions in the samples in this study were about 2 µm or less.

In Run 1, the sample was first compressed to 37 GPa at room temperature and then laser heated. The pressure on heating increased to 50 GPa due to thermal pressure. On a temperature increase from 1950 to 2430 K, we observed significant reductions in peak intensity of Fe₃C, and instead Fe₇C₃ became the dominant phase (Figure S1). As such, the solid phase assemblage at 2430 K was Fe₇C₃ + Fe₃C(minor) which represents the solidus assemblage for the Fe₃C bulk composition, and therefore these P-T conditions should be at/close to the solidus for Fe₃C. SEM images showed melted textures at the heated spot, as evidenced by migration of part of the sample (i.e., liquid) into the pressure media (Figure 3a).

In Run 2, the sample was heated at 71 GPa (Table 1). At relatively low temperatures to 1910 K, XRD peaks from solid Fe₃C were observed together with only one minor peak of (120) for Fe₇C₃ (Figure S1). On a significant increase in temperature to 2520 K, only two peaks of (211) and (102) were recognized for Fe₃C, and instead, Fe₇C₃ became the dominant phase with many new peaks. The solid phase assemblage at 2520 K is Fe₃C (minor) + Fe₇C₃, which represents the solidus assemblage for Fe₃C. SEM images showed clear textural evidence for sample melting, as seen in Run 1 (Figure 3b).

Figure 2 shows a series of XRD patterns of the highest pressure run 3 at about 85 GPa. The temperature increased markedly from 2180 to 2930 K at the end of the run. At that temperature, only peaks from Fe₇C₃ were recognized, which means that the temperature was above the solidus. SEM images clearly demonstrated a melted texture (Figure 3c).

From changes in XRD patterns and SEM observations, it is most likely that the significant reductions of peak intensity of Fe₃C observed at the highest temperature of each run indicate that the reaction of Fe₃C = Fe₇C₃ + liquid took place. The 1st and 2nd runs show the solidus assemblage (solid Fe₃C + Fe₇C₃) while the 3rd run shows the liquidus assemblage (solid Fe₇C₃). This is consistent with the SEM images in Figure 3. The
<figure><img src="image_4.png" /><figcaption>**Figure 2.** XRD patterns collected during in situ laser-heating run on Fe₃C at about 85 GPa (3rd run) with Al₂O₃ pressure medium.</figcaption></figure>

recovered samples from the 1st and 2nd runs retain plate-shaped geometries, while the 3rd run shows an offset of the sample at the heated spot, which implies that a greater amount of melt was generated. Thus we placed constraints on the melting points of Fe₃C under pressure.

The determined melting points of Fe₃C are compared with reported melting temperatures of the system Fe-C (Liu et al., 2016; Mashino et al., 2019; Morard et al., 2017) (Figure 1). The solidus temperature of Fe₃C constrained in each run is slightly greater or similar to the reported eutectic point at the same pressure (Mashino et al., 2019; Morard et al., 2017) (Figure 1b). Since the observed reaction (Fe₃C = Fe₇C₃ + liquid) must be located at temperatures higher than the eutectic temperature, the present experimental data place tight constraints on the *P*-*T* conditions of the reaction.

## 3.2. Thermodynamics of Melting of the System Fe-C

We have constructed a thermodynamic model for high-pressure eutectic melting of the system Fe-C using the newly obtained and existing experimental data. The parameters for $G°_T$ and EoS of the phases are listed in Tables 2 and 3 respectively. On the Fe end-member, an experiment-based thermodynamic model established by Komabayashi (2014) was adopted. The model is applicable to 360 GPa and 7000 K and he derived the EoS parameters for liquid iron by using *P*-*T* conditions of experimentally constrained melting points to 200 GPa. The EoS and thermodynamics for Fe liquid by Komabayashi (2014) were tested against theoretical EoS (Li et al., 2022; Wagle & Steinle-Neumann, 2019) and in petrological applications (Armstrong et al., 2019; Hirschmann, 2021). Since the core density deficit and its reconciliation by the addition of light elements are discussed based on the properties of pure iron, maintaining consistency with the previously established liquid Fe properties (Komabayashi, 2014) is an important guiding principle on exploring the nature of the system Fe-C.

We also adopted the thermodynamic database for solid Fe₃C from Hallstedt et al. (2010) and McGuire et al. (2021) (Tables 2 and 3). The $G°_T$ and EoS parameters for solid Fe₇C₃ have been reevaluated in this study using the *P*-*V*-*T* data set by Nakajima et al. (2011) so that calculated phase relations reproduce low-pressure melting relations (Fei & Brosh, 2014; Nakajima

et al., 2011) and high-pressure invariant point at 255 ± 25 GPa and 3570 ± 180 K where Fe₃C, Fe₇C₃, Fe and liquid are stable (Mashino et al., 2019). Modeling the eutectic system Fe-C includes modeling of carbon phases under high pressures, namely diamond and its counterpart liquid phase (Figure 4). $G°_T$ for diamond was taken from Dinsdale (1991). We assessed the room-temperature EoS for diamond from data reported by Angel et al. (2015). The thermal part was also based on Angel et al. (2015), but we assume $\delta_0 = 0$ as the $\alpha_0$ is already small at 1 bar (0.4 * 10⁻⁵/K) and its volume (i.e., pressure) dependence is very difficult to assess for core conditions. $G°_T$ for liquid carbon (diamond-like) was from Fried and Howard (2000) with adjustments to make the melting points higher than their model which predicted low melting points under high pressures, particularly at *P* > 300 GPa (Figure 4b). The calculated melting points are close to experimental data on graphite melting up to 10 GPa (Togaya, 1997) (Figure 4a); note that the melting points of graphite at *P* < 10 GPa should be close to that of diamond (Fried & Howard, 2000). The EoS parameters for diamond-like liquid carbon were assessed so that the calculated melting line is consistent with shock wave data at *P* > 550 GPa (Brygoo et al., 2007; Eggert et al., 2010). As a consequence, our modeled melting curve of diamond is generally consistent with available experimental data at low pressure and very high pressures.

We modeled the solubility of carbon in solid Fe phases, which slightly affects the calculated eutectic composition described below. This was modeled by ideally mixing Fe and C infinitely diluted in Fe. The model was fitted to the data of Fei and Brosh (2014) and Mashino et al. (2019). This approach is validated by the successful fitting to
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 3.** Backscattered electron images of recovered samples: (a) 1st run, (b) 2nd run, and (c) 3rd run. (a) Infiltration of the sample metal (i.e., liquid) along the grain boundaries of the pressure medium (indicated by arrows) was observed, which serves as a piece of evidence of melting. (b) A thinning of the sample and infiltration of the liquid sample into the pressure medium were observed at the heated portion. (c) Migration/offset of the sample at the heated spot was observed which indicates melting of the sample during laser heating. Infiltration of the sample melt into the pressure medium is also recognized.</figcaption></figure>

the experimental data of small carbon content in solid Fe phases (<~1 wt%) at a wide pressure range of 60–255 GPa (Figure 5).

We assessed the mixing properties of Fe and C liquids (i.e., $L_{Fe,C}$) under high pressures by reproducing experimental data on eutectic temperature at 10–255 GPa by thermodynamic calculation (Fei & Brosh, 2014; Lord et al., 2009; Mashino et al., 2019; Morard et al., 2017; Nakajima et al., 2009) (Figures 6 and 7a). The optimized $L_{Fe,C}$ as a function of pressure is:

$$L_{Fe,C} = -818.1 * P - 115360 \ (P \text{ in GPa}) \qquad (17)$$
<caption>**Table 2**</caption>

Gibbs Energies of End-Member Phases at 1 bar, T (G°<sub>T</sub>) (J mol<sup>-1</sup>) = a + bT + cT ln T + dT<sup>2</sup> + eT<sup>-1</sup> + ext

<table><thead><tr><th>Phase</th><th>a</th><th>b</th><th>c</th><th>d</th><th>e</th><th>ext.</th></tr></thead><tbody><tr><td>fcc iron<sup>a</sup></td><td>16,300.921</td><td>381.47162</td><td>-52.2754</td><td>0.000177578</td><td>-395355.4</td><td>-2,476.28*T<sup>0.5</sup></td></tr><tr><td>hcp iron<sup>b</sup></td><td>12,460.921</td><td>386.99162</td><td>-52.2754</td><td>0.000177578</td><td>-395355.4</td><td>-2,476.28*T<sup>0.5</sup></td></tr><tr><td>Liquid iron<sup>c</sup></td><td>-9,007.3402</td><td>290.29866</td><td>-46</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Solid Fe<sub>3</sub>C<sup>d</sup></td><td>-10195.86075</td><td>690.9498876</td><td>-118.47637</td><td>-0.0007</td><td>590,527</td><td>0</td></tr><tr><td>Liquid Fe<sub>3</sub>C<sup>e</sup></td><td>-7,444.9</td><td>1,033.2308</td><td>-166.22896</td><td>-0.000101651</td><td>1,627,524</td><td>0</td></tr><tr><td>Solid Fe<sub>7</sub>C<sub>3</sub><sup>f</sup></td><td>44,825.37935</td><td>1,727.374719</td><td>-299</td><td>-0.00175</td><td>1,476,317.5</td><td>0</td></tr><tr><td>Liquid Fe<sub>7</sub>C<sub>3</sub><sup>g</sup></td><td>918.3</td><td>253.937</td><td>-40.84945</td><td>-0.000007899</td><td>579,190</td><td>0</td></tr><tr><td>Solid C (diamond)<sup>h</sup></td><td>-16359.441</td><td>175.61</td><td>-24.31</td><td>-0.0004723</td><td>2,698,000</td><td>-2.61*10<sup>8</sup>*T<sup>-2</sup> + 1.11*10<sup>10</sup>*T<sup>-3</sup></td></tr><tr><td>Solid C in iron (fcc and hcp)<sup>i</sup></td><td>163,100</td><td>156.953</td><td>-28.7039</td><td>-0.00004</td><td>1,873,436</td><td>0</td></tr><tr><td>Liquid C (diamond-like)<sup>j</sup></td><td>105,100</td><td>184.953</td><td>-28.7039</td><td>-0.00004</td><td>1,873,436</td><td>0</td></tr></tbody></table>

Calculated eutectic relations at several different pressures are shown in Figure 6, and the comparison of calculated and experimentally determined Reactions 1 and 2 is shown in Figure 1. Our solidus for Fe<sub>3</sub>C (Fe<sub>3</sub>C = Fe<sub>7</sub>C<sub>3</sub> + liquid) is calculated consistently with our experiments and is placed at a greater temperature than the eutectic data from multianvil experiments (Fei & Brosh, 2014). Considering the uncertainty in temperature in DAC experiments (Liu et al., 2016; Mashino et al., 2019; Morard et al., 2017; this study), all these data are consistent with our thermodynamic model in Figure 1.

In Figure 6a, the calculated diamond-liquidus curves (Fei & Brosh, 2014; this study) do not seem to match the experimental data by Nakajima et al. (2009) (triangles) at 10 GPa; the experimental data are more temperature

<caption>**Table 3**</caption>

Thermophysical Parameters in the Vinet-Anderson-Grüneisen Equation

<table><thead><tr><th>Phase</th><th>V<sub>0</sub> (cm<sup>3</sup> mol<sup>-1</sup>)</th><th>K<sub>0</sub> (GPa)</th><th>K'</th><th>α<sub>0</sub> (*10<sup>-5</sup> K<sup>-1</sup>)</th><th>δ<sub>0</sub></th><th>κ</th></tr></thead><tbody><tr><td>fcc iron<sup>a</sup></td><td>6.82</td><td>163.4</td><td>5.38</td><td>7</td><td>5.5</td><td>1.4</td></tr><tr><td>hcp iron<sup>b</sup></td><td>6.753</td><td>163.4</td><td>5.38</td><td>5.8</td><td>5.1</td><td>1.4</td></tr><tr><td>Solid Fe<sub>3</sub>S<sup>c</sup></td><td>28.379</td><td>120.0</td><td>5.7</td><td>3.3</td><td>6.1</td><td>1.4</td></tr><tr><td>Solid Fe<sub>3</sub>C<sup>d</sup></td><td>22.823</td><td>232.0</td><td>5.09</td><td>5.6</td><td>7.4</td><td>0.2</td></tr><tr><td>Solid Fe<sub>7</sub>C<sub>3</sub><sup>e</sup></td><td>55.37229</td><td>250</td><td>4.05</td><td>4.9</td><td>5.2</td><td>1.2</td></tr><tr><td>Solid C (diamond)<sup>f</sup></td><td>3.414625</td><td>444</td><td>4</td><td>0.4</td><td>0</td><td>1.4</td></tr><tr><td>Solid C in iron (fcc and hcp)<sup>g</sup></td><td>3.45</td><td>300</td><td>4</td><td>2</td><td>5.5</td><td>1.4</td></tr><tr><td>Liquid iron<sup>h</sup></td><td>6.880</td><td>148.0</td><td>5.8</td><td>9.0</td><td>5.1</td><td>0.56</td></tr><tr><td>Liquid FeO<sup>i</sup></td><td>13.16</td><td>128.0</td><td>3.85</td><td>4.7</td><td>4.5</td><td>1.4</td></tr><tr><td>Liquid Fe<sub>0.9235</sub>Si<sub>0.0765</sub> (Fe-4 wt%Si)<sup>j</sup></td><td>6.909</td><td>150.3</td><td>5.81</td><td>9.0</td><td>5.1</td><td>0.56</td></tr><tr><td>Liquid Fe<sub>3</sub>S (Fe-16 wt%S)<sup>k</sup></td><td>30.90</td><td>70.0</td><td>6.5</td><td>5.0</td><td>4.1</td><td>0.56</td></tr><tr><td>Liquid Fe<sub>3</sub>C (Fe-6.7 wt%C)<sup>l</sup></td><td>24.22</td><td>153.7</td><td>5.35</td><td>8.0</td><td>5.35</td><td>0.92</td></tr><tr><td>Liquid Fe<sub>7</sub>C<sub>3</sub> (Fe-8.4 wt%C)<sup>m</sup></td><td>59.023</td><td>155.19</td><td>5.27</td><td>7.79</td><td>5.42</td><td>1</td></tr><tr><td>Liquid C (diamond-like)<sup>n</sup></td><td>4.2</td><td>200</td><td>4.6</td><td>1</td><td>5.00</td><td>1.4</td></tr></tbody></table>

<footnote><sup>a</sup>fcc iron: Komabayashi (2014). <sup>b</sup>hcp iron: Komabayashi (2014). <sup>c</sup>Liquid iron: Komabayashi (2014). <sup>d</sup>Solid Fe<sub>3</sub>C: Hallstedt et al. (2010). <sup>e</sup>Liquid Fe<sub>3</sub>C: This study. <sup>f</sup>Solid Fe<sub>7</sub>C<sub>3</sub>: McGuire et al. (2021); This study. <sup>g</sup>Liquid Fe<sub>7</sub>C<sub>3</sub>: This study. <sup>h</sup>Solid C (diamond): Dinsdale (1991). <sup>i</sup>Solid C in iron (fcc and hcp): This study. <sup>j</sup>Liquid C (diamond-like): Fried and Howard (2000); This study.</footnote>

<footnote><sup>a</sup>fcc iron: Komabayashi (2014). <sup>b</sup>hcp iron: Refitted to the data of Dewaele et al. (2006) by Komabayashi (2014). <sup>c</sup>Solid Fe<sub>3</sub>S: Thompson et al. (2020, 2022). <sup>d</sup>Solid Fe<sub>3</sub>C: McGuire et al. (2021). <sup>e</sup>Solid Fe<sub>7</sub>C<sub>3</sub>: Nakajima et al. (2011); This study. <sup>f</sup>Solid C (diamond): Angel et al. (2015); This study. <sup>g</sup>Solid C in iron (fcc and hcp): This study. <sup>h</sup>Liquid iron: Komabayashi (2014). <sup>i</sup>Liquid FeO: Komabayashi (2014). <sup>j</sup>Liquid Fe<sub>0.9235</sub>Si<sub>0.0765</sub> (Fe-4 wt%Si): Komabayashi (2020). <sup>k</sup>Liquid Fe<sub>3</sub>S (Fe-16 wt%S): Thompson et al. (2022). <sup>l</sup>Liquid Fe<sub>3</sub>C (Fe-6.7 wt%C): This study. <sup>m</sup>Liquid Fe<sub>7</sub>C<sub>3</sub> (Fe-8.4 wt%C): This study. <sup>n</sup>Liquid C (diamond-like): This study.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Melting curves of diamond to (a) 400 GPa and (b) 1,000 GPa from shock experiments (Brygoo et al., 2007; Eggert et al., 2010), first-principles calculations (Benedict et al., 2014; Correa et al., 2006; Wang et al., 2005), and thermodynamic modeling (Fried & Howard, 2000; this study). In (a), the uncertainties in each data point of Correa et al. (2006) and Wang et al. (2005) are similar to the size of each symbol. Also plotted are experimental measurements on the melting point of graphite (Togaya, 1997), which could serve as the maximum for the melting point of diamond at the same pressure. The same reference applies for the same symbol/line in both figures. A high-pressure transition of diamond to the BC8 structure is from Benedict et al. (2014).</figcaption></figure>

insensitive. For the graphite-liquidus at lower pressures, experimental studies also reported temperature insensitive data up to 2273 K at 2 GPa (Dasgupta & Walker, 2008) and 5 GPa (Nakajima et al., 2009), whereas a thermodynamic calculation showed a temperature dependent curve at 5 GPa (Wood, 1993). As such, the thermodynamic models show temperature dependent diamond/graphite-liquidus curves (Fei & Brosh, 2014; Wood, 1993; this study) and in contrast, experimental studies reported temperature insensitive curves (Dasgupta & Walker, 2008; Nakajima et al., 2009). While the mixing models of the thermodynamic studies may not fully capture the interaction between Fe and C liquids at those pressures (Fei & Brosh, 2014; Wood, 1993; this study),
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Carbon contents in solid iron phases at or near eutectic temperature (Fei & Brosh, 2014; Mashino et al., 2019; Rohrbach et al., 2014; Walker et al., 2013).</figcaption></figure>

the experimental data should also include large uncertainties because they relied on interpretation of recovered sample texture which showed many crystals precipitated upon quenching. Interestingly, the graphite-liquidus curve at 1 bar is temperature dependent (Chipman, 1972) similar to Wood (1993)'s calculation at 5 GPa. Our calculation reinforces that the thermodynamic calculations produce a temperature-dependent diamond-liquidus curve.

Thus we have established the thermodynamic model for the eutectic system. The modeled eutectic compositions and temperatures at selected pressures are shown in Table 4. In Figure 7, we also compare the modeled eutectic points with existing experimental data (Chabot et al., 2008; Fei & Brosh, 2014; Lord et al., 2009; Mashino et al., 2019; Morard et al., 2017) and thermodynamic models (Fei & Brosh, 2014; Wood, 1993; Yokoo et al., 2024). The eutectic temperature increases monotonously with increasing pressure and the experimental measurements on eutectic temperature are well reproduced by our model (Figure 7a). In contrast, the experimental data on eutectic composition are largely scattered (Figure 7b), which clearly indicates that the determination of eutectic temperature is more reliable than eutectic compo-

sition in DAC. Our model shows a decrease in carbon content in eutectic melts to 2.8 wt% at 70 GPa, followed by an increase at greater pressures. At the ICB, the eutectic melt contains 4.4 wt% C. These observations are generally consistent with the data by Fei and Brosh (2014) and Morard et al. (2017). The most recent experimental data at 200 GPa (Mashino et al., 2019) are also consistent with our calculated eutectic points. Although our results are not in excellent agreement with the low pressure data by Mashino et al. (2019), the overall difference is about 1 wt% C, which is likely within experimental uncertainty.

Existing thermodynamic models (Fei & Brosh, 2014; Wood, 1993; Yokoo et al., 2024) show similar trends with increasing pressure, namely the carbon content in eutectic melt decreases to 40–100 GPa and levels off at greater pressures, possibly with a slight increase (Figure 7b). Note that (a) Wood (1993)'s model shows much lower carbon contents above 100 GPa, but its pressure dependence is qualitatively consistent with ours, (b) Yokoo et al. (2024) separately calculated the eutectic reactions for fcc iron + Fe₃C and hcp iron + Fe₃C for practical reasons, which are supposed to be connected on the fcc-hcp transition in Fe, and therefore implies that the carbon content in eutectic melt increases from 40 GPa in their model. As such, the pressure dependence of the eutectic composition is qualitatively agreed by all the thermodynamic models. The increase in carbon content above 70 GPa in our study is because of the enhanced melting point of Fe compared to Fe₃C.

The complete P-T phase diagram for the system Fe-C except for graphite stability is presented in Figure 8. We newly included calculated results of Reaction 3 which defines the low P-T stability limit of Fe₇C₃. The stability of Fe₇C₃ is strongly temperature dependent at low pressures. Reaction 3 occurs at about 1600 K below 100 GPa (Figure 8), which is consistent with our experiments (Figure 1a). Absence of Fe₇C₃ to 5 GPa in experiments (Nakajima et al., 2009) can also be explained by Reaction 3; Reaction 3 theoretically takes place at 1600 K, where however diamond + liquid is the stable assemblage and therefore one cannot observe Fe₇C₃ at any temperature. With increasing pressure, Reaction 3 becomes a stable reaction, namely it occurs at a lower temperature than diamond + liquid due to the increased melting points (e.g., 10 GPa in Figure 6a). On the other hand, Reaction 3 becomes pressure-temperature dependent above 100 GPa and solid Fe₇C₃ is stable above 150 GPa at 300 K.

# 4. Discussions

## 4.1. Melting of Fe₃C in DAC Experiments

We have constrained the monomineralic melting points (i.e., solidus) of Fe₃C via Fe₃C = liquid + Fe₇C₃ (Reaction 2) to 85 GPa, which are at similar temperatures to the reported eutectic points for Fe + Fe₃C (Reaction 1) (Mashino et al., 2019; Morard et al., 2017). Since the solidus of Fe₃C should be at higher temperatures than the eutectic reaction, our observations place tight constraints on the solidus of Fe₃C. Our thermodynamic model shows that the difference in temperature between the two reactions is 100 and 230 K at 20 and 70 GPa respectively (Figure 1b).
<figure><img src="image_10.png" /><figcaption>**Figure 6.** Eutectic relations in the Fe-C system at (a) 10 GPa, (b) 136 GPa, (c) 250 GPa, and (d) 330 GPa. The red, green, black and purple lines are calculated by Fei and Brosh (2014), Wood (1993), this study, and Yokoo et al. (2024), respectively. In (a), the symbols are experimental data: blue (Fei & Brosh, 2014); black (Nakajima et al., 2009). The solid circles denote the compositions of solid iron coexisting with melt. The open circles denote the compositions of melt coexisting with solid iron. The solid squares represent the compositions of iron carbides (Fe₃C or Fe₇C₃). The open squares represent the compositions of melt coexisting with iron carbides. The open diamond shows that only melt was observed. The open triangles show the melt compositions coexisting with diamond. In (b), the right-pointing triangle denotes the liquid composition coexisting with solid Fe in an experiment (Mashino et al., 2019). In (c), the circle symbol denotes the experimental condition where an eutectic assemblage (solid Fe + solid Fe₇C₃ + liquid) was observed (Mashino et al., 2019).</figcaption></figure>

The observed melting points of Fe₃C are consistent with, but slightly higher than, those of Liu et al. (2016). This small discrepancy could be explained by the difference in the way the reaction was detected. In their in situ XRD experiments, Liu et al. (2016) judged Reaction 2 based on the initial rise in a gradual increase in the intensity of diffuse scatterings with increasing temperature, which they interpreted as the appearance of a liquid. However, as Reaction 2 is an incongruent melting reaction, the amount of liquid produced should suddenly, rather than gradually, increase. Indeed, the data point that Liu et al. (2016) adopted as the onset of melting did not show a significant increase, considering the uncertainty, in diffuse signal intensity. Hence the temperatures for Reaction 2 by Liu et al. (2016) could have been underestimated, although the discrepancy can still be within experimental uncertainty.
<figure><img src="image_11.png" /><figcaption>**Figure 7.** (a) Eutectic temperatures of the Fe-Fe₃C/Fe₇C₃ system (Fei & Brosh, 2014; Lord et al., 2009; Mashino et al., 2019; Morard et al., 2017). Calculated eutectic temperatures (Fei & Brosh, 2014; this study) and subsolidus reaction (McGuire et al., 2021; this study) are also shown. (b) Carbon contents in eutectic melt in the Fe-Fe₃C/Fe₇C₃ system. Experimental data are plotted (Chabot et al., 2008; Fei & Brosh, 2014; Lord et al., 2009; Mashino et al., 2019; Morard et al., 2017). Results of thermodynamic calculations by Wood (1993), Fei and Brosh (2014), Yokoo et al. (2024), and this study are also shown. CMB, core-mantle boundary; ICB, inner core boundary.</figcaption></figure>

Liu et al. (2016) observed Fe₇C₃ with an orthorhombic symmetry as a product of Reaction 2 (Figure 1) while we observed hexagonal Fe₇C₃ throughout the experiments (Figure 2, Figure S1). Reaction 2 occurred in both studies at similar temperatures (Figure 1) as discussed above, and therefore, the Gibbs free energy difference between the two structures would be small. In this study, we modeled the system with hexagonal Fe₇C₃ because it is the stable phase under core pressures (Das et al., 2017; Raza et al., 2015) and its *P*-*V*-*T* data were experimentally collected to 70 GPa (Nakajima et al., 2011).
<table><caption>**Table 4**
Melting Points of Fe, Fe₃C, and Fe₇C₃ and Eutectic Points of the System Fe-Fe₃C/Fe₇C₃ Under High Pressures</caption><thead><tr><th>P (GPa)</th><th>Fe (fcc or hcp) (K)</th><th>Fe₃C (via Fe₃C = Liq + Fe₇C₃) (K)</th><th>Fe₇C₃ (K)</th><th>Eutectic (K)</th><th>C content in eutectic point (wt%)</th></tr></thead><tbody><tr><td>10</td><td>2070 (fcc)</td><td>1710</td><td>1880</td><td>1600 (Fe-Fe₃C)</td><td>3.5</td></tr><tr><td>20</td><td>2289 (fcc)</td><td>1910</td><td>2230</td><td>1810 (Fe-Fe₃C)</td><td>3.2</td></tr><tr><td>40</td><td>2636 (fcc)</td><td>2250</td><td>2680</td><td>2150 (Fe-Fe₃C)</td><td>3</td></tr><tr><td>70</td><td>3032 (fcc)</td><td>2700</td><td>3080</td><td>2470 (Fe-Fe₃C)</td><td>2.8</td></tr><tr><td>100</td><td>3342 (fcc)</td><td>3045</td><td>3340</td><td>2790 (Fe-Fe₃C)</td><td>3</td></tr><tr><td>136</td><td>3901 (hcp)</td><td>3410</td><td>3580</td><td>3100 (Fe-Fe₃C)</td><td>3.3</td></tr><tr><td>250</td><td>5418 (hcp)</td><td>3800</td><td>4190</td><td>3650 (Fe-Fe₃C)</td><td>4.3</td></tr><tr><td>330</td><td>6382 (hcp)</td><td>-</td><td>4630</td><td>4000 (Fe-Fe₇C₃)</td><td>4.4</td></tr></tbody></table>

## 4.2. Melting Point of Fe₇C₃ Under High Pressures

Here we compare results of thermodynamic models (Fei & Brosh, 2014; Yokoo et al., 2024; this study) and experimental measurements (Liu et al., 2016; Lord et al., 2009) on the melting points of Fe₇C₃ and the liquidus of Fe₃C via Fe₇C₃ + liquid = liquid (Figure 9). Note that Yokoo et al. (2024) and this study calculated the melting of point of Fe₇C₃ (solid Fe₇C₃ = liquid Fe₇C₃). Fei and Brosh (2014) calculated the peritectic reaction of solid Fe₇C₃ = diamond + liquid, and Lord et al. (2009) did not confirm the reaction products and therefore the reaction they observed might be either solid Fe₇C₃ = liquid Fe₇C₃ or solid Fe₇C₃ = diamond + liquid, depending on the pressure.

The experimental studies showed high melting temperatures of Fe₇C₃ (Lord et al., 2009) and liquidus of Fe₃C (Liu et al., 2016) (Figure 9). The melting points of Fe₇C₃ by Yokoo et al. (2024)'s model are the lowest among the thermodynamic studies up to 100 GPa. Our new calculations show similar melting points of Fe₇C₃ to the peritectic point by Fei and Brosh (2014)'s model to 30 GPa, but the reaction curve becomes less steep which gives low melting temperatures above 110 GPa. Liu et al. (2016) reported the liquidus of Fe₃C via Fe₇C₃ + liquid = liquid by in situ XRD. Determination of liquidus by in situ XRD is very challenging because the bulk composition probed by X-ray may not necessarily be the same as the starting material nearer the completion of melting due to the mobility of partial melts (as evidenced here in Figure 3).

Yokoo et al. (2024) calculated the *P-V-T* relationships of Fe₇C₃ liquid above 70 GPa using first-principles calculations and applied corrections in pressure to mitigate the problem of underestimation of volume, which is often found in their method. They then developed a thermodynamic model of melting of Fe₇C₃ by combining the EoS of liquid and solid Fe₇C₃. They did not discuss the structure of solid Fe₇C₃ but reported its thermal EoS parameters from a combination of different literature data. Our EoS of solid Fe₇C₃ are based on Nakajima et al. (2011)'s *P-V-T* data set. Moreover, the experimental pressures calculated from the unit-cell volumes and EoS of Fe₃C and of Fe₇C₃ are consistent in this study (Table 1), which indicates that the melting relations of these two carbides were consistently calculated in our model. Self-consistency is important for calculating properties of the system precisely as discussed later.

## 4.3. Eutectic Point Under High Pressures

The eutectic points of the Fe rich portion of the system Fe-C to core pressures were calculated as intersections of the Fe-liquidus and Fe₃C/Fe₇C₃-liquidus curves and therefore influenced by the thermodynamics of the end-members including the entropy of fusion and the interaction parameter for liquids. The interaction parameter *L*<sub>Fe,C</sub> was assessed to 255 GPa by reproducing existing experimental data (Table 4 and Figure 7a). The eutectic melt becomes carbon-poor at pressures to 70 GPa and then reverses to carbon-rich of 3.3–4.4 wt%C

<figure><img src="image_12.png" /><figcaption>**Figure 8.** A pressure-temperature phase diagram of the system Fe-C. All the reaction boundaries were calculated from the thermodynamic models developed by Komabayashi (2014) for iron and this study for the Fe-C phases. CMB, core-mantle boundary; ICB, inner core boundary.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 9.** Comparison of experimental measurements (Liu et al., 2016; Lord et al., 2009) and thermodynamic models (Fei & Brosh, 2014; Yokoo et al., 2024; this study) on the *P*-*T* locations of reaction boundaries in the system Fe-C. The blue lines show the Fe + Fe₃C eutectic reaction, the red lines denote Fe₃C = liquid + Fe₇C₃, and the green lines show melting of Fe₇C₃; Yokoo et al. (2024) and this study calculated the reaction of solid Fe₇C₃ = liquid Fe₇C₃, and Fei and Brosh (2014) calculated the peritectic reaction of solid Fe₇C₃ = diamond + liquid. Two thick lines are experimental measurements (purple, the liquidus of Fe₃C via Fe₇C₃ + liquid = liquid in a bulk composition of Fe₃C (Liu et al., 2016)); green, solid Fe₇C₃ = liquid Fe₇C₃ or solid Fe₇C₃ = diamond + liquid depending on the pressure (Lord et al., 2009). The gray curves are phase relations in iron (Komabayashi, 2014).</figcaption></figure>

at the core pressures. The eutectic compositions calculated in this study are generally consistent with experimental values (Fei & Brosh, 2014; Morard et al., 2017). While it seems that our eutectic compositions are not consistent with data of Mashino et al. (2019) to 200 GPa, the difference is on the order of 1 wt% C. Considering the precision of chemical analysis in the DAC experimental studies, this may not be a significant disagreement. On the other hand, Mashino et al. (2019) reported contamination by oxygen in the samples which could have affected the carbon content in eutectic liquid.

## 4.4. Mixing Properties of Fe and C Liquids Under Core Conditions

The mixing of Fe and C liquids at 1 bar shows strongly negative nonideality which stabilizes liquids relative to ideal mixtures (Chipman, 1972; Gustafson, 1985) (Figure 10). We discuss their mixing behavior based on Gibbs free energy curves (i.e., *G*-X) at core pressures from our modeling which reproduces experimentally determined eutectic points. The *G*-X curves that we fit are compared with hypothetical ideal mixing lines of Fe and C liquids (black) in Figure 10. The difference in the free energy between the two mixing models becomes larger with increasing pressure; they are -42 and -72 kJ/mol at Xc = 0.25 (Fe3C), respectively at the top and bottom of the outer core (Figure 10). The effect of these difference can be found in Figure 11 where phase relations with ideal and optimized mixing models were calculated at 136 GPa. With ideal mixing assumed, the calculated melting point of Fe₇C₃ at 136 GPa is greater than 6800 K, which does not place the eutectic point consistently with experiments. As such, negatively nonideal mixing is implicit for Fe-C liquids at any depth in the outer core, as well as at 1 bar.

The enhanced negative departure from ideal mixing of Fe-C liquids under high pressures is caused by a negative volume of mixing (*V*<sub>mix</sub>). The *V*<sub>mix</sub> for Fe₃C and Fe₇C₃ liquids under core conditions are -0.15 and -0.17 cm³/mol respectively (Table 5). At 1 bar, Jimbo and Cramb (1993) measured the effect of carbon on the density of Fe liquid up to 4 wt% C. Adopting the parameters for a graphite-like carbon liquid at 1 bar by Fried and Howard (2000), the *V*<sub>mix</sub> for Fe₃C at 1 bar and 1800 K is calculated to be -0.69 cm³/mol, while the diamond-like liquid modeled in this study yields -0.16 cm³/mol at the same *P*-*T* conditions. This implies that the *V*<sub>mix</sub> for iron and diamond-like carbon liquids may be constant from 1 bar to 330 GPa (Figure 12a) and the interaction between Fe and C atoms becomes reduced on the transition from graphite-like to diamond-like at about 5 GPa (Fried & Howard, 2000).

The negative *V*<sub>mix</sub> means the volumes of mixtures are smaller than the linear mixing products. Because of this shrinkage, the addition of carbon to liquid iron only causes a mild reduction of density. This will have pronounced effects on their *P*-wave velocity under the outer core conditions as discussed later.

## 4.5. The Equation of State for the Liquid Phases

The molar volume or density of Fe-C liquids can be calculated by differentiating the Gibbs free energy after mixing Fe and C liquids (Callen, 1960). For readers' interest, fitted EoS parameters for liquid Fe₃C and Fe₇C₃ are listed in Table 3. Also calculated physical properties of solid and liquid Fe₃C and diamond are given in Table S1.

Table 3 compares the EoS parameters of various Fe-light element alloys (Komabayashi, 2014, 2020; Thompson et al., 2022) and Table 5 shows calculated physical properties of these alloy liquids under core conditions. Fe₃C and Fe₇C₃ liquids show similar or slightly smaller K₀ values to that of iron (Figure 12a) although they are mixtures of iron and diamond-like carbon liquids. This is because of nonideality as discussed in Section 4.4. Due to the negative *V*<sub>mix</sub> which is constant with respect to pressure as discussed above, an intermediate Fe-C liquids behave as a softer material than the end-member liquids, and therefore have lower bulk moduli.

Figure 12b compares the compression curves for solid and liquid Fe₇C₃ between Yokoo et al. (2024) and this study at 5000 K. The curves for solid differ particularly at 50–100 GPa, which means Yokoo et al. (2024)'s EoS is
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 10.** Gibbs free energies of liquids in the system Fe-C at (a) 1 bar and 1800 K and conditions relevant to the (b) top and (c) bottom of the outer core. The black curves represent the Gibbs free energies modeled assuming ideal mixing between Fe and C liquids. The red lines are based on the nonideal mixing models fitted to the phase relations (Gustafson, 1985; This study).</figcaption></figure>

not consistent with an experimental P-V-T data set by Nakajima et al. (2011) to 71.5 GPa and 1973 K, as the EoS of this study was calibrated against it. The curves for liquid also differ above 50 GPa; Yokoo et al. (2024)'s model produces greater molar volumes. At 330.4 GPa and 5000 K, the models of Yokoo et al. (2024) and this study yield the density of Fe₇C₃ liquid to be 11.90 and 12.11 g/cm³ respectively; this difference does not significantly affect discussions on the density of the outer core, but will be significant when integrated with pressure in Gibbs free energy calculations (Equation 4).

## 4.6. Carbon in the Core

In order to discuss the relevance of carbon to the Earth's outer core, one needs to obtain the profiles of geophysically observable properties, which include density and *Vp* over the outer core depth range. In order to calculate these profiles, one first needs to constrain the temperature profile. Many of the previous works assumed core isentropes based on the Grüneisen parameter for pure Fe. However, as demonstrated by our previous works, the isentrope depends on the composition. In addition, the crystallization temperature of a liquid core also depends on the composition, so the core temperature profile strongly relies on the composition. Having established the thermodynamic model, we can consistently calculate the temperature, density and *Vp* for a hypothetical liquid core. Below we calculate profiles for a core with a composition of Fe₃C.

We first calculate the crystallizing point (i.e., liquidus) at the ICB (Figure 6). Then the isentrope for the liquid core is calculated (Figure 13) using,

$$dT/dP = T * \alpha * V/Cp \quad (18)$$

where *α*, *V*, and *Cp* are the thermal expansivity, molar volume, and heat capacity at constant pressure, which is equivalent to using the Grüneisen parameter. All of *α*, *V*, and *Cp* are consistently derived from the Gibbs free energy. A mantle geotherm (Brown & Shankland, 1981) and the solidus for the mantle at the core-mantle boundary (CMB, 136 GPa) (Fiquet et al., 2010) can be used to place geodynamic constraints on the core composition (Figure 13). Morard et al. (2017) considered some amount of carbon or sulfur present in the Earth's core in order to lower the crystallizing point at the ICB compared to pure iron, which is needed to avoid extensive melting at the bottom of the mantle. They requested a 2 wt% carbon or sulfur which would make the core temperature below the mantle solidus at the CMB (ca. 4180 K) (Fiquet et al., 2010). This discussion was however based on assumed core isentropes. Our newly calculated core isentrope revised the needed amount of carbon in the core to be >ca 2.5 wt% if there is no sulfur.

In Figure 14, we calculate the density and *Vp* profiles for an Fe-6.7 wt%C (Fe₃C) liquid core along its isentrope in Figure 13. For comparison, results of first-principles calculations (Badro et al., 2014; Ichikawa & Tsuchiya, 2020) and of static experiments (Nakajima et al., 2015) and the profiles of a seismological model (PREM, Dziewonski & Anderson, 1981) are plotted. The density of Fe-6.7 wt%C liquid starting at *T*<sub>ICB</sub> = 4530 K in this study is comparable to that of Fe-3.9 wt%C in the first-principles works starting at *T*<sub>ICB</sub> = 6300–6500 K (Badro et al., 2014; Ichikawa & Tsuchiya, 2020). In contrast, our model shows only a slight increase in *Vp* by the addition of 6.7 wt%C which does not match the PREM profile while both ab-initio studies reported enhanced *Vp* for Fe-3.9 wt%C liquids. Nakajima et al. (2015) conducted inelastic X-ray scattering experiments in laser-heated DAC and obtained *Vp* data of Fe-3.9 wt%C liquid. They constructed a thermal EoS for the liquid from *Vp* data to calculate the density. Their profiles, in Figure 14, show *Vp* and density that are too high and too low, respectively, compared to the other studies. This can
<figure><img src="image_17.png" /><figcaption>**Figure 11.** Effects of nonideality on calculated phase relations at 136 GPa. The black lines represent phase relations with the optimized nonideal mixing parameters for Fe-C liquids in this study. The green lines were calculated assuming ideal mixing for the liquids. The blue symbol is same as in Figure 6b. The nonideal mixing model greatly stabilizes liquids to make the calculated solidus consistent with the experimental measurements.</figcaption></figure>

be explained by the possible presence of a positive dispersion (Scopigno et al., 2005) in their measurements with a minimum momentum transfer of about 3 nm⁻¹.

The core profiles depend on the crystallizing point of the outer core and its isentrope, which requires a set of consistent phase diagrams and the Grüneisen parameter. As discussed above, our isentrope, density and *Vp* profiles are self-consistent, in contrast to the existing reports. With such a self-consistent approach, this study newly provides *Vp* for a liquid Fe₃C core (Figure 14). The *Vp* of liquid Fe₃C is slightly greater than pure iron (Figure 15). The addition of carbon effectively reduces the density of liquid iron, while the *Vp* is not effectively enhanced, which makes it difficult to reconcile the PREM profile. In addition, the density of the precipitating phase, Fe₇C₃, at the ICB is the same as the coexisting liquid (12.41 g/cm³) which is much less dense than the PREM inner core profile (Figure 14). Therefore, a core model with carbon as the sole light element can be rejected.

Figure 15 shows the effects of carbon concentration on density, adiabatic bulk modulus, and *Vp* of our experiment-based thermodynamic model and first-principles calculations (Badro et al., 2014; Ichikawa & Tsuchiya, 2020). Results are qualitatively in reasonable agreement (i.e., positive or negative). However, the density and *Vp* profiles show clear departure from linear trends in this study, which indicates the strong nonideal behavior, contrary to first-principles results. In addition to the fact that the crystallizing temperature at the ICB varies with carbon content, the nonideality of Fe-C liquids makes it complicated to assess the effects of carbon on the density and *Vp*. Nevertheless, in this paper, we have demonstrated that carbon cannot be a major light element in the core.

<table><caption>**Table 5**

Calculated Physical Properties of Fe-Alloy Liquids</caption><thead><tr><th><i>P</i> (GPa)</th><th><i>T</i> (K)</th><th>Composition<sup>a</sup></th><th><i>V</i> (cm<sup>3</sup> mol<sup>-1</sup>)</th><th><i>K</i><sub>T</sub> (GPa)</th><th>α*(10<sup>-5</sup> K<sup>-1</sup>)</th><th><i>S</i> (J K<sup>-1</sup> mol<sup>-1</sup>)</th><th>γ<sup>b</sup></th><th><i>Vp</i> (km s<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="10">136</td><td rowspan="10">4000</td><td>Fe</td><td>5.182</td><td>587</td><td>1.78</td><td>104.3</td><td>1.52</td><td>7.77</td></tr><tr><td>Fe<sub>0.82</sub>O<sub>0.18</sub> (Fe-5.9 wt%O)</td><td>4.871</td><td>551</td><td>1.54</td><td>108.7</td><td>1.07</td><td>7.67</td></tr><tr><td>Fe<sub>0.9235</sub>Si<sub>0.0765</sub> (Fe-4 wt%Si)</td><td>5.224</td><td>589</td><td>1.81</td><td>104.4</td><td>1.63</td><td>8.00</td></tr><tr><td>Fe<sub>0.75</sub>S<sub>0.25</sub> (Fe-16 wt%S)</td><td>5.010</td><td>614</td><td>0.93</td><td>124.0</td><td>0.71</td><td>7.95</td></tr><tr><td>Fe<sub>0.9</sub>S<sub>0.1</sub> (Fe-6 wt%S)</td><td>5.113</td><td>597</td><td>1.45</td><td>117.8</td><td>1.17</td><td>7.81</td></tr><tr><td>Fe<sub>0.75</sub>C<sub>0.25</sub> (Fe-6.7 wt%C)</td><td>4.496</td><td>581</td><td>1.59</td><td>102.9</td><td>1.12</td><td>7.90</td></tr><tr><td>Fe<sub>0.7</sub>C<sub>0.3</sub> (Fe-8.4 wt%C)</td><td>4.371</td><td>581</td><td>1.54</td><td>102.1</td><td>1.07</td><td>7.96</td></tr><tr><td>C</td><td>3.052</td><td>680</td><td>0.27</td><td>80.0</td><td>0.19</td><td>13.17</td></tr><tr><td>Fe<sup>b</sup></td><td>4.250</td><td>1,224</td><td>0.86</td><td>108.4</td><td>1.39</td><td>9.99</td></tr><tr><td rowspan="10">330</td><td rowspan="10">6000</td><td>Fe<sub>0.82</sub>O<sub>0.18</sub> (Fe-5.9 wt%O)</td><td>3.914</td><td>1,122</td><td>0.79</td><td>116.5</td><td>0.86</td><td>9.69</td></tr><tr><td>Fe<sub>0.9235</sub>Si<sub>0.0765</sub> (Fe-4 wt%Si)<sup>c</sup></td><td>4.288</td><td>1,227</td><td>0.88</td><td>107.5</td><td>1.51</td><td>10.28</td></tr><tr><td>Fe<sub>0.75</sub>S<sub>0.25</sub> (Fe-16 wt%S)</td><td>4.108</td><td>1,287</td><td>0.52</td><td>134.5</td><td>0.70</td><td>10.40</td></tr><tr><td>Fe<sub>0.9</sub>S<sub>0.1</sub> (Fe-6 wt%S)</td><td>4.193</td><td>1,248</td><td>0.73</td><td>124.4</td><td>1.09</td><td>10.12</td></tr><tr><td>Fe<sub>0.75</sub>C<sub>0.25</sub> (Fe-6.7 wt%C)</td><td>3.659</td><td>1,181</td><td>0.78</td><td>108.6</td><td>1.02</td><td>10.04</td></tr><tr><td>Fe<sub>0.7</sub>C<sub>0.3</sub> (Fe-8.4 wt%C)</td><td>3.553</td><td>1,175</td><td>0.76</td><td>108.1</td><td>0.96</td><td>10.10</td></tr><tr><td>C</td><td>2.497</td><td>1,263</td><td>0.15</td><td>90.6</td><td>0.17</td><td>16.22</td></tr></tbody></table>

<footnote><sup>a</sup>Fe (Komabayashi, 2014); Fe<sub>0.82</sub>O<sub>0.18</sub> (Komabayashi, 2014); Fe<sub>0.9235</sub>Si<sub>0.0765</sub> (Komabayashi, 2020); Fe<sub>0.75</sub>S<sub>0.25</sub> (Thompson et al., 2022); Fe<sub>0.9</sub>S<sub>0.1</sub> (Thompson et al., 2022); Fe<sub>0.75</sub>C<sub>0.25</sub> (This study); Fe<sub>0.7</sub>C<sub>0.3</sub> (This study). <sup>b</sup>γ, Grüneisen parameter which is given by γ = αVK<sub>T</sub>/C<sub>v</sub>, where C<sub>v</sub> is the heat capacity at a constant volume. <sup>c</sup>6000 K at 330 GPa is below the melting points for Fe and Fe-4 wt%Si.</footnote>
<figure><img src="image_18.png" /><figcaption>**Figure 12.** (a) Compression behaviors of Fe-C liquids from thermodynamic models of this study. The Fe-C liquids were calculated by mixing Fe and C liquids using the nonideal mixing parameters. The dashed red curve is for a hypothetical liquid Fe₃C assuming ideal mixing of Fe and C liquids. (b) Comparison of Fe₇C₃ phases between this study and Yokoo et al. (2024).</figcaption></figure>

The above results do not deny the existing geochemical arguments that a small amount of carbon would be in the core from metal-silicate partitioning studies (Dasgupta & Walker, 2008; Dasgupta et al., 2013) and from the consideration on carbon isotopic composition of silicate Earth (Wood et al., 2013). Recently Hirose et al. (2021) proposed a core composition model by integrating mineral physics, geochemistry, and cosmochemistry data, which is Fe + 5% Ni + 1.7% S + 0%–4.0% Si + 0.8%–5.3% O + 0.2% C + 0%–0.26% H (by weight) for the outer core. Another recent model proposed a composition of Fe + 5% Ni + 4.1(1.4)% Si + 0.47(0.16)% H (by weight) for the outer core based on first-principles molecular dynamic simulations and neural network methods, which was fitted to the seismological observations (Liu & Jing, 2024). They reported that adding carbon increased the misfit and therefore did not consider it as a major element in the core. Nevertheless the possibility of the presence of a small amount of carbon in the core was not excluded (Liu & Jing, 2024). Our results shows that a small amount (e.g., 0.2 wt% from Hirose et al. (2021)'s model) of carbon can be present in the core regardless the amounts of other light elements since that level of concentration of carbon will not have pronounced effects on the density and *Vp* of liquid iron (Figure 14).

As mentioned in Introduction, it is important to develop a core model from a self-consistent thermodynamic database for a better understanding of the thermal and chemical state of the Earth's core. Komabayashi (2020) developed a thermodynamic model and demonstrated that the density and *Vp* for a liquid core with a composition
<figure><img src="image_19.png" /><figcaption>**Figure 13.** Isentropic temperature profiles of Fe (Komabayashi, 2014), Fe-4 wt%Si (Komabayashi, 2020) and Fe-5.9 wt%O (Komabayashi, 2014), Fe-16 wt%S (Thompson et al., 2022), Fe-6 wt%S (Thompson et al., 2022), and Fe-6.7 wt%C (this study) outer cores. The mantle profile (Brown & Shankland, 1981) and mantle solidus at the CMB (Fiquet et al., 2010) are shown.</figcaption></figure>

of Fe-5.8(0.6) wt%Si–0.8(0.6) wt%O match the PREM values for the outer core. As a next logical step, a new model for an Fe-Si-O-C core with a small amount of carbon needs to be established with assessing the mixing properties for the multicomponent system.

Lastly, the depletion of carbon in the core might also be consistent with inferred systematics of C and S in iron meteorite bodies (e.g., Hirschmann et al., 2024), which imply degassing and preferential C loss in parent bodies of iron meteorites. If the Earth accreted from a significant proportion of differentiated bodies, as evidenced from iron

<figure><img src="image_20.png" /><figcaption>**Figure 14.** Density and *V*<sub>p</sub> of Fe-C liquids over the outer core conditions. The profiles are along each isentrope starting at a temperature of the inner core boundary (e.g., Figure 13 for this study). Results from the first-principles calculations (Badro et al., 2014; Ichikawa & Tsuchiya, 2020) and experiments (Nakajima et al., 2015) are shown. For reference, isentropic profiles of liquid Fe (gray dashed line, Komabayashi, 2014) and the seismologically constrained profiles (PREM, Dziewonski & Anderson, 1981) are also plotted. The open hexagon denotes the density of Fe<sub>7</sub>C<sub>3</sub> which precipitates from the Fe<sub>3</sub>C outer core.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Figure 15.** Effects of carbon on the *Vp*, adiabatic bulk modulus (*Ks*), and density (*ρ*) of liquid iron at 330 GPa. Data from the first-principles calculations (Badro et al., 2014; Ichikawa & Tsuchiya, 2020) and thermodynamic modeling (this study) are compared. Note that the parameters are normalized to those for pure iron.</figcaption></figure>

meteorites, its composition might already be preferentially C-depleted relative to other light elements. As such, a C-rich core would also not be expected.

**5. Summary and Conclusions**

We have conducted high-pressure experiments on melting of Fe₃C to 85 GPa. We then assessed the thermo-dynamics of Fe-C liquids from experimental data including our new experiments and literature data on diamond melting. We developed a thermodynamic model of melting of the system Fe-C under core pressures. The mixing of Fe and C liquids are nonideal from 1 bar to the pressure at the center of the Earth. The calculated carbon content in eutectic point under the core pressures is 3.3–4.4 wt%. From the Gibbs free energy, we derived an internally consistent parameters for Fe-C outer cores which included the crystallizing points at their bottoms, isentropic thermal profiles, and densities and longitudinal seismic wave speeds. While the addition of carbon in excess of the eutectic melt composition effectively reduces the density of iron liquid, the *Vp* of iron liquid is not greatly changed. Therefore, the low density and high *Vp* of PREM relative to pure iron cannot be reconciled by an Fe-C liquid. Therefore, the Earth's core cannot be approximated by the system Fe-C and should include another light element.

**Data Availability Statement**

The data files used in this paper are available in Komabayashi et al. (2024).

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**Acknowledgments**

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research were carried out at PETRA III (proposal no. I-20190593 EC). The FIB-SEM at the University of Edinburgh (EPSRC Grant (EP/P030564/1)) was used for the textural analysis in this study. The FIB-SEM operator Tom Glen helped us analyze samples. Two anonymous reviewers are thanked for their constructive comments which have helped us improve the manuscript. This research was supported by the European Research Council (ERC) Consolidator Grant to TK (647723) and by the Natural Environment Research Council (NERC) to TK (NE/ W005832/1). This work was performed in part under the auspices of the US Department of Energy at Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (LLNL- JRNL-819849).
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Erratum

The supporting information of the originally published version of this article contained some errors in Table S1.
The errors have been corrected, and this may be considered the authoritative version of record.