Experimental study and thermodynamic calculations of phase relations in the Fe–C system at high pressure

Experimental study and thermodynamic calculations of phase relations in the Fe–C system at high pressure

Earth and Planetary Science Letters 408 (2014) 155–162 Contents lists available at ScienceDirect Earth and Planetary Science Letters www.elsevier.co...

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Earth and Planetary Science Letters 408 (2014) 155–162

Contents lists available at ScienceDirect

Earth and Planetary Science Letters www.elsevier.com/locate/epsl

Experimental study and thermodynamic calculations of phase relations in the Fe–C system at high pressure Yingwei Fei a,∗ , Eli Brosh b a b

Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road, N.W., Washington, DC 20015, USA NRCN, P.O. Box 9001, Beer-Sheva 84190, Israel

a r t i c l e

i n f o

Article history: Received 13 June 2014 Received in revised form 16 September 2014 Accepted 26 September 2014 Available online xxxx Editor: J. Brodholt Keywords: high-pressure phase relation thermodynamic model iron–carbon system Earth’s core Fe–C melting

a b s t r a c t We have conducted a series of melting experiments in the Fe–C system at pressures up to 25 GPa in the temperature range of 1473–2073 K. The results define the phase relations at several pressures, including the eutectic temperature and composition as a function of pressure, carbon partitioning between solid iron and liquid, and change of melting relations involving iron carbides. In order to interpolate and extrapolate the phase relations over a wide pressure and temperature range, we have established a comprehensive thermodynamic model in the Fe–C binary system. The calculated phase diagrams at pressures of 5, 10, and 20 GPa reproduce the experimental data, including the solubility of carbon in solid iron and the effect of pressure on the eutectic temperature and composition. The formation of Fe7 C3 at pressures above 5 GPa is correctly modeled and the change of phase relations in the Fe–C system between 5 and 10 GPa is captured in the model. The model provides predictions of the phase relations at 136 GPa and 330 GPa, based on existing knowledge of the thermochemistry of the system at lower pressure. The calculated phase relations can be used to understand the role of carbon during inner core crystallization, predicting carbon distribution between the inner and outer cores and mineralogy of the solid inner core. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The thermochemistry and high-pressure phase relations in the Fe–C system are fundamental for understanding the role of carbon in planetary cores as well as in industrial diamond synthesis processes and the steel technology. The thermochemistry of Fe–C alloys at ambient pressure has been critically reviewed by Chipman (1972), and the phase equilibrium and lattice parameter data were summarized by Okamoto (1992). The thermodynamic evaluation of the system by Gustafson (1985) is in general agreement with the review by Chipman (1972). This model is widely accepted as a basis for extending the phase equilibrium calculations to the multicomponent systems, including systems relevant to geophysics such as the Fe–C–Si system (Lacaze and Sundman, 1991), and the Fe– C–Ni system at ambient pressure (Gabriel et al., 1987) and at high pressure (Kocherzhinskii et al., 1993). There are limited experimental studies of the phase relations in the Fe–C system at high pressure. Earlier high-pressure experiments (e.g., Strong and Chrenko, 1971; Tsuzuki et al., 1984;

*

Corresponding author. E-mail address: [email protected] (Y. Fei).

http://dx.doi.org/10.1016/j.epsl.2014.09.044 0012-821X/© 2014 Elsevier B.V. All rights reserved.

Shterenberg et al., 1975; Kocherzhinskii et al., 1992) were conducted to understand the conditions of iron carbide formation and carbon solubility in the melt up to 8 GPa, in an effort to refine the diamond synthesis process. Hirayama et al. (1993) investigated the melting relations in the Fe–C system up to 12 GPa in a multianvil apparatus. Specifically, they examined the eutectic melting and observed a small increase of eutectic temperature with increasing pressure (∼7 ◦ C/GPa), but no resolvable change in eutectic composition. Recently, Chabot et al. (2008) determined the phase relations in the Fe–Fe3 C system at 5 GPa, indicating a eutectic temperature between 1473 and 1498 K, and a eutectic composition of about 4.7 wt.% C. The melting relations in the more carbon-rich region (> eutectic carbon content) have been recently examined at 5, 10, and 14 GPa (Nakajima et al., 2009). Their experiments focused on the carbon solubility in the melts coexisting with either graphite/carbon or iron carbides. In this study, we have systematically investigated the phase relations in the Fe–Fe3 C system at 5, 10, and 20 GPa and determined the pressure effect on the eutectic temperature up to 25 GPa. We have further developed a thermodynamic model of the Fe–C system by assessing the existing thermochemical and thermophysical data. The model reproduces the high-pressure phase equilibrium data reasonably well and provides

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Fig. 1. Representative SEM images of quenched textures of the recovered samples in different phase stability fields: (a) coexisting solid Fe and Fe3 C phases at 20 GPa and 1823 K; (b) solid Fe coexisting with Fe–C melt at 20 GPa and 1923 K; (c) solid Fe7 C3 coexisting with Fe–C melt at 10 GPa and 1743 K; and (d) solid Fe3 C coexisting with Fe–C melt at 10 GPa and 1668 K.

the calculated phase relations at pressures pertinent to the Earth’s core. 2. Experimental procedure We conducted melting experiments in the Fe–C system up to 25 GPa, using the multi-anvil presses at the Geophysical Laboratory of the Carnegie Institution of Washington, described by Bertka and Fei (1997). Starting materials were prepared using mixtures of fine powder of pure iron and graphite with different carbon contents (1 to 7 wt.% C). The starting materials were then packed into MgO capsules dried by a handheld torch. Three well-calibrated highpressure assemblies (18/11, 10/5, and 8/3) with different sample sizes were used to achieve different pressure ranges. All assemblies were dried at 1000 ◦ C and stored in a dry oven afterward. The 18/11 (18-mm octahedron edge length and 11-mm truncated edge length of the tungsten carbide cube) assembly is identical to that used by Corgne et al. (2007), and consists of a cast MgO octahedron (Aremco 584OS) with finned gaskets, ZrO2 sleeve thermal insulator, and graphite heater. All experiments at 5 GPa were performed using this assembly. Pressure calibration was described in Bertka and Fei (1997). The experiments at 10 GPa were performed using the 10/5 assembly, which consists of a precast MgO octahedron with pyrophyllite gaskets, ZrO2 sleeve thermal insulator, and rhenium cylindrical heater. A similar configuration was used for the 8/3 assembly with the exception that the ZrO2 sleeve was replaced by a LaCrO3 sleeve for more stable thermal insulation at higher pressure. The 8/3 assembly has been extensively calibrated

(Hirose and Fei, 2002) and was used for experiments at 20 and 25 GPa in this study. For each pressure, we carried out a series of experiments over a range of temperatures to determine melting points. Each set of melting experiments was conducted using a consistent experimental configuration to ensure high reproducibility of the experiments at a constant pressure. Temperatures were measured with a W5%Re–W26%Re thermocouple which was inserted axially into the sample chamber through a 4-bore alumina rod. Samples were heated at a rate of 100 ◦ C/min to the set temperature value of the experiments. The duration for each experiment varied from 1 to 5 h, depending on the final set temperatures. The recovered samples were mounted in epoxy resin and polished for quench texture and chemical composition analyses with a JEOL JXA-8900 electron microprobe at the Geophysical Laboratory. We obtained backscattered electron images of the recovered samples which show unambiguous melting quench textures of the molten samples (cf., Fig. 1). Thin ferropericlases rims against the MgO capsule walls were observed in the recovered charge. It is more difficult to obtain precise carbon contents of the phases in the quenched samples (e.g., Chabot et al., 2008; Dasgupta and Walker, 2008). We used a procedure similar to that described by Chabot et al. (2008) and Deng et al. (2013) to analyze carbon contents. All samples were not carbon-coated because the Fe–C samples are conductive. In order to produce reliable and selfconsistent carbon measurements, we first established a calibration curve using standards including pure Fe, NIST Fe–C alloy standards with 0.215 and 0.969 wt.% C, and synthetic Fe3 C (6.69 wt.% C) (Deng et al., 2013). We re-polished the standards for every new

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Table 1 Experimental run conditions, coexisting phases, and phase compositions. Run no.

P (GPa)

T (K)

Phases

C (wt.%)

Fe (wt.%)

Fe Fe Fe melt Fe melt Fe melt melt Fe Fe3 C melt Fe3 C Fe Fe3 C melt Fe3 C melt melt melt melt Fe melt Fe melt Fe Fe3 C melt Fe7 C3 melt Fe3 C melt Fe melt Fe Fe3 C Fe Fe3 C melt Fe melt Fe3 C Fe7 C3 Fe3 C Fe7 C3 melt Fe7 C3 melt + C Fe melt melt Fe Fe3 C Fe melt melt melt Fe3 C melt Fe3 C melt Fe3 C

1.32 1.35 1.53 4.21 1.08 3.31 0.72 2.43 3.70 1.39 5.56 3.97 6.05 – 5.75 4.09 6.10 4.30 5.91 5.09 3.71 1.23 3.70 0.70 3.53 1.55 6.26 5.70 8.27 4.22 5.99 5.50 0.78 2.97 1.07 6.50 0.77 6.56 4.41 0.96 3.26 6.70 8.36 6.77 8.37 6.08 8.34 7.67 0.53 1.81 2.87 0.67 6.44 1.26 3.15 2.85 3.32 6.18 3.24 6.09 4.71 6.42

99.20 98.75 99.14 96.75 98.18 97.17 99.12 96.46 96.45 98.11 93.73 95.63 93.13 – 94.69 95.20 93.54 94.14 94.11 94.48 95.99 98.76 96.47 98.59 97.05 97.57 92.43 94.83 91.84 96.04 93.55 93.73 98.67 97.26 98.31 93.45 99.27 93.75 96.46 99.16 96.97 93.95 92.36 94.15 92.47 93.67 91.57 92.17 98.98 97.91 96.95 99.54 92.95 99.21 96.69 96.19 96.19 92.96 95.40 92.30 94.60 92.52

PR393 PR401 PR404

5 5 5

1498 1523 1543

PR599

5

1673

PR617

5

1823

PL234 PR426

5 5

1513 1473

PR392

5

1523

PR445

5

1513

PR429

5

1543

PR433 PR437 PR406 M972 M978

5 5 5 10 10

1583 1598 1613 1618 1643

M967

10

1668

M973

10

1618

M976

10

1743

M980

10

1668

M1104 M1108

10 10

1823 1823

PL210

20

1723

PL211

20

1823

PL212 PL199

20 20

1873 1923

PL219

20

1823

PL226

20

1873

PL222

20

1903

PL447 PL452

20 20

2073 2073

PL452b LO588

20 25

2073 1913w

LO589

25

1933w

LO590 LO604

25 25

2003 2023w

LO600

25

2023w

LO601

25

2023

(0.10) (0.10) (0.33) (0.10) (0.04) (0.24) (0.07) (0.60) (0.10) (0.10) (0.06) (0.18) (0.15) (0.10) (0.31) (0.19) (0.16) (0.72) (0.11) (0.15) (0.10) (0.18) (0.30) (0.14) (0.20) (0.29) (0.33) (0.05) (0.11) (0.08) (0.18) (0.09) (0.13) (0.12) (0.20) (0.13) (0.23) (0.19) (0.14) (0.13) (0.04) (0.08) (0.26) (0.20) (0.22) (0.24) (0.16) (0.14) (0.15) (0.14) (0.22) (0.32) (0.68) (0.17) (0.19) (0.60) (0.10) (0.77) (0.85) (0.35) (0.09)

Total

(0.29) (0.40) (0.22) (0.80) (0.50) (0.36) (0.20) (0.48) (0.24) (0.63) (0.71) (0.31) (0.02) (0.10) (0.35) (0.25) (0.16) (0.27) (0.16) (0.20) (0.35) (0.21) (0.18) (0.30) (0.61) (0.26) (0.49) (0.49) (0.28) (0.25) (0.69) (0.22) (0.31) (0.19) (0.57) (0.15) (0.20) (0.20) (0.39) (0.17) (0.28) (0.17) (0.32) (0.29) (0.35) (0.55) (0.34) (0.44) (0.14) (0.28) (0.32) (0.25) (0.42) (0.21) (0.45) (0.89) (0.27) (0.80) (0.99) (0.51) (0.08)

100.52 100.10 100.67 100.96 99.26 100.47 99.84 98.89 100.16 99.50 99.29 99.60 99.18 – 100.44 99.29 99.64 98.44 100.01 99.57 99.69 99.99 100.17 99.29 100.58 99.12 98.69 100.53 100.11 100.26 99.54 99.22 99.45 100.23 99.39 99.94 100.04 100.30 100.86 100.12 100.23 100.65 100.74 100.92 100.84 99.75 99.90 99.84 99.51 99.72 99.82 100.21 99.39 100.47 99.84 99.03 99.51 99.14 98.64 98.39 99.31 98.94

(0.31) (0.37) (0.45) (0.85) (0.50) (0.19) (0.20) (0.17) (0.21) (0.53) (0.72) (0.37) (0.13) (0.22) (0.15) (0.41) (0.21) (0.58) (0.24) (0.26) (0.34) (0.12) (0.10) (0.28) (0.42) (0.18) (0.45) (0.54) (0.30) (0.19) (0.65) (0.15) (0.21) (0.16) (0.58) (0.16) (0.20) (0.14) (0.34) (0.20) (0.31) (0.17) (0.45) (0.37) (0.28) (0.77) (0.49) (0.31) (0.08) (0.34) (0.28) (0.24) (0.38) (0.22) (0.35) (0.31) (0.30) (0.28) (0.16) (0.18) (0.09)

“w” indicates the temperature estimated from the power–temperature relationship.

probe session to remove any contamination from the previous sessions. We always placed the standards next to the unknown and frequently checked the established calibration curve. The carbon content is derived from the linear relationship between the carbon Kα peak counts and known carbon concentrations of the standards. There is measurable, consistent instrument carbon background based on measurements of the pure Fe standard that was corrected through the calibration regression line.

3. Experimental results We conducted a series of melting experiments in the Fe-rich region of the Fe–C system at constant pressures (5, 10, 20, and 25 GPa) up to 2023 K. Table 1 lists experimental conditions and coexisting phases in the quenched samples. The chemical compositions for each phase were determined with electron microprobe, using procedure described above.

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Fig. 2. Phase relations in the Fe–C system at (a) 5 GPa, (b) 10 GPa, and (c) 20 GPa. The solid lines represent the calculated results. The experimental data are plotted using different symbols. Solid circles (blue) represent compositions of solid iron coexisting with either melt or Fe3 C. Open circles (blue) represent compositions of melt coexisting with solid iron. Solid squares (blue) represent compositions of iron carbides (Fe3 C or Fe7 C3 ). Open squares (blue) represent compositions of melt coexisting with iron carbides. Open diamonds (blue) indicate only melt observed. The melt compositions (open red squares with cross) coexisting with either graphite or diamond from Nakajima et al. (2009) and experimental data (green: solid circle – solid iron, solid square – carbide, and open symbols – coexisting liquid) at 5 GPa from Chabot et al. (2008) are also plotted for comparison. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3.1. Phase relations at 5 GPa At 5 GPa, we observed the onset of melting at a temperature between 1473 K and 1523 K. The eutectic temperature must be very close to 1513 K because experiments PL234 and PR445 at this temperature showed melt and solid, respectively. Such an observation only occurs near the melting temperature due to the small temperature uncertainty (±10 K) for each run. The result is consistent with that of Chabot et al. (2008) who found the eutectic temperature at 5 GPa between 1473 K and 1523 K. Our result on eutectic temperature (1513 ± 10 K) is about 100 K lower than that of Strong and Chrenko (1971) and about 50 K higher than that of Hirayama et al. (1993). In the Fe-rich region, solid metallic C-bearing face-centered cubic (FCC) iron coexists with Fe–C melt above the eutectic temperature (Fig. 2a). Carbon solubility in the solid iron coexisting with liquid or iron carbide is a function of temperature, decreasing with increasing temperature above the eutectic temperature. The experiments at 1543 K (PR404), 1673 K (PR599), and 1823 K (PR617) define the Fe + melt two-phase loop as a function of temperature (Fig. 2a). On the Fe3 C side of the eutectic point, we observed Fe3 C coexisting with Fe–C melt, but the coexisting field occurs only in a small temperature interval (<150 K). At temperatures higher than 1650 K, the melt coexists with graphite (Nakajima et al., 2009), indicating that Fe3 C melts incongruently. By determining the compositions of Fe–C melts coexisting with either solid iron or Fe3 C, we determined a eutectic composition of 4.0 ± 0.3 wt.% C at 5 GPa, which is slightly lower than the value at 1 atm (4.3 wt.% C). The eutectic point was further confirmed

by experiment PL234, which indicated complete melting at 1513 K using a starting material with 4 wt.% C. We used synthetic Fe3 C as the standard for carbon concentration analyses (Chabot et al., 2008; Deng et al., 2013). However, the measured composition of the iron carbide phase, coexisting with either solid iron or Fe–C melt, generally has lower carbon content than that of Fe3 C, indicating that the Fe3 C phase is nonstoichiometric, i.e., it may contain carbon defects. This is in a general agreement with the work of Walker et al. (2013) on the nonstoichiometry of iron carbides at high pressure. At the eutectic temperature, the carbon solubility in solid iron is the highest, about 1.5 wt.% C. The carbon atoms are likely entering the octahedral interstitial sites of FCC iron, as is seen in the austenite phase under ambient pressure conditions (Laneri et al., 2002). 3.2. Phase relations at 10, 20, and 25 GPa At 10 GPa, we accurately determined the eutectic point at 1618 K and 3.7 ± 0.3 wt.% C (Table 1). The quenched Fe3 C phase also shows evidence for C defects compared to the stoichiometric Fe3 C phase. Phase relations of the iron carbide side are significantly different from those at 5 GPa because of the formation of a new iron carbide compound, Fe7 C3 . At 1743 K, the Fe–C melt coexists with Fe7 C3 , instead of Fe3 C, indicating a melting reaction Fe3 C = Fe7 C3 + melt (Fig. 2b). The formation of Fe7 C3 at 10 GPa is consistent with the previous observation of Fe7 C3 at pressures above 6 GPa in the excess carbon system (Tsuzuki et al., 1984), and a recent study of Nakajima et al. (2009) in the Fe–C system

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up to 14 GPa. The observed phase relations are also in a general agreement with the inferred phase diagram by Lord et al. (2009). At subsolidus temperatures, we observed the coexisting field of Fe + Fe3 C at low bulk carbon content and the field of Fe3 C + Fe7 C3 at high bulk carbon content (>6 wt.% C) at 20 GPa (Fig. 2c). The noticeable change of the phase relation in the Fe–C between 10 and 20 GPa is the increase of eutectic temperature with increasing pressure. Limited experimental data at 25 GPa show similar phase relations with increasing eutectic temperature (Table 1). We have also observed subtle changes, including slight decrease of the maximum C solubility in FCC-Fe with increasing pressure, in an agreement with the results of Walker et al. (2013), and apparent shift of the eutectic composition towards Fe-rich side. 4. Thermodynamic modeling We have developed a comprehensive thermodynamic model to reproduce and calculate phase relations in the Fe–C system at high pressure and temperature. The modeling was performed according to the CALPHAD (Calculation of Phase Diagrams) method (Saunders and Miodownik, 1998; Lukas et al., 2007). The Gibbs energy of each phase was expressed as a function of temperature, pressure and composition. These expressions of the Gibbs energy contain adjustable parameters that are fitted to available experimental data. The calculations were done by means of the ThermoCalc software (Andersson et al., 2002). In this study, we used the CALPHAD model of Fe–C at ambient pressure by Gustafson (1985) with recent modifications by Hallstedt et al. (2010) and Djurovic et al. (2011) as a basis for the high-pressure modeling. While there are several other carbides identified in the Fe–C system (Okamoto, 1992), the only carbides modeled in the present work are Fe3 C (cementite) and Fe7 C3 . The ambient pressure model was extended to high pressures by the following conventional scheme:

P G ( T , P , x) = G ( T , P 0 , x) +





V T , P , x d P 

(1)

P0

where P 0 is the ambient pressure, x is the composition and V ( T , P  , x) is a composition-dependent equation of state (EOS). There are several commonly used equations of state for modeling the volume as a function of pressure and temperature, which include the Murnaghan EOS (e.g., Fabrichnaya et al., 2004; Holland and Powell, 1998), Birch–Murnaghan EOS with temperature-dependent parameters (Saxena et al., 1993), and Tait EOS (Holland and Powell, 2011). However, these models often lead to inconsistencies in the predictions for the thermophysical properties at high pressure (Hammerschmidt et al., 2014). Instead, we use the formulation by Brosh et al. (2007), which enables us to express V ( T , P  , x) by means of an interpolation between the ambient pressure Gibbs energy and the Gibbs energy at extreme pressures, derived from the quasiharmonic model of solids. This approach allows us to avoid many of the inconsistencies of previous works while preserving the Gibbs energy formalism (1) and using the existing models for the ambient-pressure thermochemistry. The details of the EOS formulation and the thermodynamic descriptions (TDB file, zipped) are given in the online appendix. 4.1. Elements: iron (Fe) and carbon (C) Following the CALPHAD methodology, we first developed the model for the end-members. The model for iron is based on the one described by Brosh et al. (2007). The parameters were slightly changed here (Table S1) to fit the new melting data of iron by Anzellini et al. (2013) and the HCP–FCC equilibrium boundary by

159

Komabayashi et al. (2009). We also reproduced the sound velocity of liquid iron (Blairs, 2007) at near-ambient pressure more closely. The changes did not affect other calculated features of the thermophysics of iron, such as the Hugoniot curve and the bulk sound velocity along the Hugoniot, as shown in Brosh et al. (2007). Fig. S1 shows the calculated phase diagram of iron up to 350 GPa. The thermodynamic properties of the graphite and diamond phases of pure carbon (C) were reviewed by Day (2012). The properties of liquid carbon and the high-pressure phase diagram were reviewed by Savvatimskiy (2005) and by Bundy et al. (1996), respectively. Fig. S2 shows the calculated phase diagram of carbon. The current version of the carbon phase diagram shows the melting curve of diamond has a positive slope at high pressure in agreement with the phase diagram evaluation by Bundy et al. (1996), based on interpretation of the shockwave data by Shaner et al. (1984). Complete thermodynamic models for pure carbon were developed by Gustafson (1986), Khishchenko et al. (2005), van Thiel and Ree (1989), and Fried and Howard (2000). At ambient pressure, our model for pure carbon is identical to that of Gustafson (1986). The EOS parameters of graphite and diamond were fitted to existing data on the thermophysical properties, reviewed by Day (2012). The known graphite–diamond equilibrium boundary was also used to constrain the EOS parameters. The calculated thermophysical properties of graphite and diamond and the corresponding experimental data are shown in Fig. S3. At pressures above 10 GPa, the model EOS of liquid carbon was constrained by the rising melting curve of diamond and by the assumption that the high-pressure carbon liquid should be diamondlike, characterized by a molar volume similar to diamond and a very high bulk modulus (Fried and Howard, 2000). At low pressures, the model EOS of liquid carbon is constrained by the initial positive slope of the melting curve, which implies much higher molar volume at ambient pressure. Also, the strong curvature of the melting curve implies a low bulk modulus for the liquid below 10 GPa. Sekine (1993) also derived a very low ambient-pressure bulk modulus for liquid carbon, based on the analysis of shockwave data. Fig. S4 shows the calculated bulk modulus of liquid carbon as a function of pressure, which follows to the aforementioned constraints. The calculated melting curve of carbon is shown in Fig. S2. While the shape of the melting curve is reproduced, the calculated melting temperature is some 100 K higher than the experimental results of Togaya (1997). This inconsistency can only be remedied by changing the underlying ambient-pressure model, at the cost of hampering the extension of our effort to higher order systems. For example, melting curve could be adjusted by changing the heat capacity of liquid carbon slight, but the effect is relatively minor and the change would not be justified considering relatively large uncertainty in melting temperature determination. 4.2. The Fe–C binary system Several previous attempts have been made to model the Fe–C binary system at high pressure. Some models primarily targeted pressures up to 8 GPa in relation to diamond synthesis (e.g., Turkevich, 1992; Muncke, 1979), whereas others targeted higher pressures for geophysical applications (Wood, 1993). The models by Turkevich (1992) and Wood (1993) predict a negative slope for the melting of diamond at high pressures. The model by Rouquette et al. (2008) did not include the formation of Fe7 C3 carbide. Our high-pressure model follows the thermodynamic description by Gustafson (1985). Specific sets of EOS parameters were attributed to the solution end-members assigned in his model. For example, the variation of the molar volume of interstitial FCC_A1 solid solutions was expressed through the EOS parameters

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Fig. 3. (a) Calculated melting temperatures (red – Fe3 C eutectic, blue – Fe3 C peritectic, and black – Fe7 C3 peritectic) as a function of pressure (solid curves), compared with experimental data (dashed curves from Lord et al., 2009). The red circles represent experimental data on the Fe3 C eutectic melting from this study (solid circle – solid and open – melt). Other Fe3 C eutectic melting data (red symbols) from Hirayama et al. (1993) (solid triangles), Strong and Chrenko (1971) (open triangle), Kocherzhinskii et al. (1992) (crosses), and Nakajima et al. (2009) (open diamond). Selected Fe7 C3 peritectic melting data (black symbols) are from Kocherzhinskii et al. (1992) (crosses), and Nakajima et al. (2009) (open diamond). (b) Calculated eutectic compositions as a function of pressure, compared with experimental data (diamonds from Lord et al., 2009; circles from this study). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

of carbon in the Fe:C end-member, which is an imaginary compound where the FCC_A1 lattice sites are filled by Fe and the octahedral sites are filled with carbon. The liquid is modeled as a substitutional solution and the variation of its molar volume is expressed via the pressure-dependence of an interaction parameter. The details of the model and the EOS parameters are given in the online appendix. For the compounds Fe3 C and Fe7 C3 , the effect of ferromagnetism and its disappearance at high pressure on the molar volume were modeled by a pressure-dependent Curie temperature in the Hillert–Jarl term for the magnetic Gibbs energy, commonly used in CALPHAD. Since the models of Gustafson (1985), Hallstedt et al. (2010), and Djurovic et al. (2011) included a separate magnetic term only for Fe3 C but not for Fe7 C3 , we modified the ambient-pressure description of Fe7 C3 in a manner that enabled such separation but maintained consistency with the model of Hallstedt et al. (2010) and Djurovic et al. (2011) at high temperature. While this is an improvement over previous models, our model still suffers from the deficiency of treating the carbides as stoichiometric compounds instead of having a range of compositions as revealed by Walker et al. (2013) and confirmed in this study. In the fitting of EOS parameters, both direct measurements of thermophysical properties and phase equilibria were used for optimization. Fig. S5 shows some calculations of the thermophysical properties, compared with the corresponding experimental results. There is general good agreement between the calculations and the experimental data. The complex behavior of the volume thermal expansion coefficient of Fe3 C (cementite) is reproduced by including the pressure-dependence of the Curie temperature (Fig. S5c) in the magnetic Gibbs energy term. The model parameters (Table S1) are adjusted by a global optimization of the available experimental data. In general, the current model represents the best fit to a majority of data within the experimental uncertainties, but certain compromises are necessary to interpret various experimental datasets as discussed below. At 1 bar, our model is identical to that of Gustafson (1985), indicating that both cementite and Fe7 C3 carbides are metastable. Hence, the eutectic equilibrium is between liquid, graphite and FCC-Fe at ambient pressure. The liquid + cementite + FCC-Fe eutectic is calculated to become stable at pressures above 0.124 GPa in our model. The calculated phase diagrams at pressures of 5, 10, and 20 GPa are shown in Fig. 2, compared with the experimental results from the multi-anvil experiments. The calculations reproduce the

general phase relations in the binary system well. The effect of pressure on the eutectic temperature up to 25 GPa is precisely determined in this study, which is closely reproduced in the model without major adjustment of the thermophysical properties (Fig. 3a). The calculated eutectic temperatures at higher pressures are also in a good agreement with the measured eutectic temperatures in the laser-heated diamond-anvil cell (LHDAC) up to 70 GPa (Lord et al., 2009). However, with the existing thermophysical data, we cannot reproduce the measured melting curves of Fe3 C and Fe7 C3 by Lord et al. (2009) which showed significantly higher melting temperatures than the measurements of Nakajima et al. (2009) and Kocherzhinskii et al. (1992). Our model, with constraints posed by the existing thermophysical data, does not allow increasing the peritectic melting points of Fe3 C and Fe7 C3 without significantly changing the calculated eutectic temperatures which are in a good agreement with several experimental measurements. Furthermore, all quench experiments (Nakajima et al., 2009; Kocherzhinskii et al., 1992; this study) showed small temperature interval between eutectic and Fe3 C or Fe7 C3 peritectic temperatures. Such a rapid increase in the peritectic melting point of Fe7 C3 will also result in significant changes to the melting relations when extrapolating to higher pressure as discussed below. The formation of Fe7 C3 at pressures above 5 GPa is modeled according to the results of Kocherzhinskii et al. (1992) who identified the onset of Fe7 C3 peritectic reaction at around 5.8 GPa. This is a lower pressure than the ∼9 GPa found in earlier experiments by Shterenberg et al. (1975). Its stability extends to the entire pressure range of interest. The change of phase relations in the Fe–C system between 5 and 10 GPa is captured in the model, although tighter constraints on the peritectic melting points of Fe3 C and Fe7 C3 would help to define the phase relations in the C-rich regions at high pressures. The current experiments showed a slight decrease in carbon content at the eutectic point with increasing pressure, from 4.3 wt.% C at ambient pressure to about 3.5 wt.% C at 20 GPa. The model reproduces the effect of pressure on the eutectic composition well. It predicts an almost linear decrease in the eutectic carbon content with increasing pressure (Fig. 3b) which is contrary to the abrupt decrease above 20 GPa reported by Lord et al. (2009). There is no mechanism to produce such a drastic change in the thermodynamic modeling without some fundamental changes in the system. The change could be related to the high-to-low spin transition at about 20 GPa, but further experimental confirmation is needed.

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The maximum solubility of carbon in FCC-iron is 2.08 wt.% at ambient pressure. Carbon atoms occupy the octahedral interstitial sites in metallic iron. We modeled the decrease in molar volume of the C-bearing with increasing carbon content (Fig. S5a). The calculations predict that the maximum carbon solubility in FCCiron decreases slightly with increasing pressure, consistent with our observations (Fig. 2) as well as with that of Walker et al. (2013). At higher pressures, the stable iron phase is hexagonal close-packed (HCP) iron. We assumed the same effect of the dissolved carbon in HCP-iron on the molar volume as that in the FCC-iron. The calculated maximum carbon solubility in HCP-iron shows an increase across the FCC–HCP transition, but this increase is not constrained by any experimental measurements. Further understanding of carbon solubility in HCP-iron and its substitution mechanism is needed to model the interaction between solid carbon and iron at high pressure. Accurate measurements of the molar volumes as a function of composition for C-bearing iron and iron carbides at high pressure and temperature are also crucial for improving the model, especially addressing the nonstoichiometry of carbides at high pressure. 5. Discussions and geophysical implications Although the experimental data in the Fe–C system are limited, our current thermodynamic model provides a basis for assessing the available data, identifying gaps in the experimental data, and exploring the phase relations at conditions where experimental data are not yet available. Calculations and experimental measurements up to 25 GPa show good agreement in the eutectic temperature and composition, and the carbon solubility in solid iron. The calculated phase diagrams in the Fe–C binary system as a function of pressure reproduce the experimentally determined phase relations including the formation of Fe7 C3 at high pressure, although the peritectic melting temperatures of the iron carbides are less well constrained. The model is directly applicable for understanding the role of carbon in the cores of the smaller terrestrial planets such as Mars and Mercury. There are almost no equilibrium data in the Fe–C binary system at pressures above 25 GPa. The goal of this study is also to predict phase relations at higher pressures using an optimized thermodynamic model. We have calculated phase relations at 136 GPa and 330 GPa (Fig. 4), corresponding to pressures at the Earth’s core–mantle boundary (CMB) and inner core boundary (ICB), respectively. It is seen that the general shape of the phase diagram does not change with pressure significantly. There are a couple of noticeable changes including the disappearance of the peritectic point of Fe7 C3 because Fe7 C3 melts congruently at high pressure. We also see the appearance of a eutectic point between Fe7 C3 and diamond around 30 GPa, which needs to be verified in future experiments. The appearance of congruent melting of Fe7 C3 and of another eutectic point between Fe7 C3 and C seems to be closely related to the melting curve of Fe7 C3 at high pressure. The reason for developing a eutectic point is that the temperature at which the liquid with the composition of Fe7 C3 in equilibrium with diamond becomes lower than the congruent melting point of the Fe7 C3 carbide. It is possible that these features are artificial because of the steep melting curve of Fe7 C3 at high pressure. At 136 GPa, the phase relations between the two carbides of iron are basically the same as at 20 GPa. Namely, both are stable and the Fe3 C melts peritectically, decomposing to liquid + Fe7 C3 . At 330 GPa, our model predicts that Fe3 C is no longer stable below liquidus temperature. This result is based on long range extrapolations and is very sensitive to specific model parameters, particularly the thermal equations of state of iron and iron carbides. However, it is in line with the theoretical predictions of Bazhanova et al. (2012) regarding the relative stability of iron-

Fig. 4. Calculated phase diagrams in the Fe–C system at 20 GPa (black dotted lines), 136 GPa (blue dotted lines) and at 330 GPa (solid red lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

carbides at high pressure. Like our model, their calculation also predicts that at 330 GPa, Fe7 C3 should be more stable than Fe3 C. Unlike Bazhanova et al. (2012), we did not model the Fe2 C, because of the lack of suitable experimental data. The present model, as well as the ab-initio calculations of Bazhanova et al. (2012), also does not include any solution models to deal with nonstoichiometry of the carbides, which may have influence on the stability field of carbides at very high pressure and temperature. For applications to the Earth’s core, we generally consider compositions only in the Fe-rich region (e.g., <5 wt.% C). With solidification of the inner core, the mineralogy (C-bearing iron or iron carbide) of the inner core is controlled by the bulk carbon content of the core and the eutectic composition of the system at the ICB pressure. The calculated eutectic composition and temperature at 330 GPa are 2.24 wt.% C and 5100 K, respectively. If the bulk carbon content in the core is less than 2.24 wt.%, a metallic iron core with some dissolved carbon is expected. On the other hand, higher bulk carbon content would lead to an iron carbide inner core. Our model predicts that the carbide should be Fe7 C3 . The possibility of different carbides becoming stable at high pressures exists (e.g., Bazhanova et al., 2012). However, if this is indeed the case, it may be conjectured that the shape of the phase diagram should be essentially the same as predicted by the present model, with the only change being the stoichiometry of the highest-melting carbide. Carbon solubility in solid iron is an important factor in modeling the density contrast between the inner and outer core. At ICB conditions, HCP-Fe is the stable phase of iron. There are no experimental data on the carbon solubility in HCP-iron and the effect of carbon on the density of a Fe–C alloy. The current model adopts the behavior of FCC-iron in the Fe–C system for the HCP phase, which is likely not an adequate approximation. Future experiments

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should investigate the carbon substitution mechanism in the HCP phase and its effect on density at high pressure and temperature. Additional experimental priorities include accurate determination of the peritectic melting temperatures of the iron carbides, extending the measurements of the eutectic composition to pressures above 20 GPa, and verifying the change of the melting relation from incongruent to congruent melting in Fe7 C3 . For Earth’s core mineralogy application, it is important to determine which iron carbide is stable at the liquidus. Acknowledgements We thank Yu Wang for conducting the experiments as part of the pre-doctoral training program. We also thank Liwei Deng for experimental assistance, Caitlin Murthy and Neil Bennett for useful comments, and Dave Walker for critical review of the manuscript. This work was supported by NASA and NSF grants (to Y.F.) and by the Carnegie Institution of Washington. Appendix A. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.epsl.2014.09.044. References Andersson, J.O., Helander, T., Höglund, L., Shi, P.F., Sundman, B., 2002. Thermo-Calc & DICTRA, computational tools for materials science. Calphad 26, 273–312. Anzellini, S., Dewaele, A., Mezouar, M., Loubeyre, P., Morard, G., 2013. Melting of iron at Earth’s inner core boundary based on fast X-ray diffraction. Science 340, 464–466. Bazhanova, Z.G., Oganov, A.R., Gianola, O., 2012. Fe–C and Fe–H systems at pressures of the Earth’s inner core. Phys. Usp. 55, 489–497. Bertka, C.M., Fei, Y., 1997. Mineralogy of the Martian interior up to core–mantle boundary pressures. J. Geophys. Res. 102 (B3), 5251–5264. Blairs, S., 2007. Review of data for velocity of sound in pure liquid metals and metalloids. Int. Mater. Rev. 52, 322–344. Brosh, E., Makov, G., Shneck, R.Z., 2007. Application of CALPHAD to high pressures. Calphad 31, 173–185. Bundy, F.P., Bassett, W.A., Weathers, M.S., Hemley, R.J., Mao, H.K., Goncharov, A.F., 1996. The pressure–temperature phase and transformation diagram for carbon; updated through 1994. Carbon 34, 141–153. Chabot, N.L., et al., 2008. The Fe–C system at 5 GPa and implications for Earth’s core. Geochim. Cosmochim. Acta 72, 4146–4158. Chipman, J., 1972. Thermodynamics and phase diagram of the Fe–C system. Metall. Trans. 3, 55–64. Corgne, A., Keshav, S., Fei, Y., McDonough, W.F., 2007. How much potassium is in the Earth’s core? New insights from partitioning experiments. Earth Planet. Sci. Lett. 256, 567–576. Dasgupta, R., Walker, D., 2008. Carbon solubility in core melts in a shallow magma ocean environment and distribution of carbon between the Earth’s core and the mantle. Geochim. Cosmochim. Acta 72, 4627–4641. Day, H.W., 2012. A revised diamond–graphite transition curve. Am. Mineral. 97, 52–62. Deng, L., Fei, Y., Liu, X., Gong, Z., Shahar, A., 2013. Effect of carbon, sulfur and silicon on iron melting at high pressure: implications for composition and evolution of the planetary terrestrial cores. Geochim. Cosmochim. Acta 114, 220–233. Djurovic, D., Hallstedt, B., von Appen, J., Dronskowski, R., 2011. Thermodynamic assessment of the Fe–Mn–C system. Calphad 35, 479–491. Fabrichnaya, O., Saxena, S.K., Richet, P., Westrum, E.F., 2004. Thermodynamic Data, Models and Phase Diagrams in Multicomponent Oxide Systems. Springer. Fried, L.E., Howard, W.M., 2000. Explicit Gibbs free energy equation of state applied to the carbon phase diagram. Phys. Rev. B 61, 8734–8743. Gabriel, A., Gustafson, P., Ansara, I., 1987. A thermodynamic evaluation of the C–Fe– Ni system. Calphad 11, 203–218. Gustafson, P., 1985. A thermodynamic evaluation of the Fe–C system. Scand. J. Metal. 14, 259–267.

Gustafson, P., 1986. An evaluation of the thermodynamic properties and the P , T phase diagram of carbon. Carbon 24, 169–176. Hallstedt, B., et al., 2010. Thermodynamic properties of cementite (Fe3 C). Calphad 34, 129–133. Hammerschmidt, T., et al., 2014. Including the effects of pressure and stress in thermodynamic functions. Phys. Status Solidi B 251, 81–96. Hirayama, Y., Fujii, T., Kurita, K., 1993. The melting relation of the system, iron and carbon at high-pressure and its bearing on the early-stage of the Earth. Geophys. Res. Lett. 20, 2095–2098. Hirose, K., Fei, Y., 2002. Subsolidus and melting phase relations of basaltic composition in the uppermost lower mantle. Geochim. Cosmochim. Acta 66, 2099–2108. Holland, T.J.B., Powell, R., 1998. An internally-consistent thermodynamic dataset for phases of petrological interest. J. Metamorph. Geol. 16, 309–344. Holland, T.J.B., Powell, R., 2011. An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids. J. Metamorph. Geol. 29, 333–383. Khishchenko, K.V., Fortov, V.E., Lomonosov, I.V., 2005. Multiphase equation of state for carbon over wide range of temperatures and pressures. Int. J. Thermophys. 26, 479–491. Kocherzhinskii, Y.A., Kulik, O.G., Turkevich, V.Z., Ivanchenko, S.A., 1992. Phase equilibria in the Fe–C system. Sverkhtverdye Materialy 6, 3–9 (in Russian). Kocherzhinskii, Y.A., Kulik, O.G., Turkevich, V.Z., 1993. Phase equilibria in the Fe– Ni–C and Fe–Co–C systems under high temperatures and high pressures. High Temp., High Press. 25, 113–116. Komabayashi, T., Fei, Y., Meng, Y., Prakapenka, V., 2009. In-situ X-ray diffraction measurements of the g–e transition boundary of iron in an internally-heated diamond anvil cell. Earth Planet. Sci. Lett. 282, 252–257. Lacaze, J., Sundman, B., 1991. An assessment of the Fe–C–Si system. Metall. Trans. A 22, 2211–2223. Laneri, K.F., Desimoni, J., Zarragoicoechea, G.J., Fernández-Guillermet, A., 2002. Distribution of interstitials in fcc iron–carbon austenite: Monte Carlo simulations versus Mössbauer analysis. Phys. Rev. B 66, 134201. Lord, O.T., Walter, M.J., Dasgupta, R., Walker, D., Clark, S.M., 2009. Melting in the Fe–C system to 70 GPa. Earth Planet. Sci. Lett. 284, 157–167. Lukas, H.L., Fries, S.G., Sundman, B., 2007. Computational Thermodynamics, the CALPHAD Method. Cambridge University Press, Cambridge. Muncke, G., 1979. Physics of diamond growth. In: Field, J.E. (Ed.), The Properties of Diamond. Academic Press, pp. 473–501. Nakajima, Y., Takahashi, E., Suzuki, T., Funakoshi, K., 2009. Carbon in the core revisited. Phys. Earth Planet. Inter. 174, 202–211. Okamoto, H., 1992. The C–Fe (carbon–iron) system. J. Phase Equilib. 13, 543–565. Rouquette, J., Dolejš, D., Kantor, I.Yu., McCammon, C.A., Frost, D.J., Prakapenka, V.B., Dubrovinsky, L.S., 2008. Iron–carbon interactions at high temperatures and pressures. Appl. Phys. Lett. 92, 121912. Saunders, N., Miodownik, A.P., 1998. CALPHAD, A Comprehensive Guide. Pergamon Press, London. Savvatimskiy, A.I., 2005. Measurements of the melting point of graphite and the properties of liquid carbon (a review for 1963–2003). Carbon 43, 1115–1142. Saxena, S.K., Chatterjee, N., Fei, Y., Shen, G., 1993. Thermodynamic Data on Oxides and Silicates. Springer, New York. Sekine, T., 1993. An evaluation of the equation of state of liquid carbon at very high pressure. Carbon 31, 227–233. Shaner, J.W., Brown, J.M., Swenson, C.A., McQueen, R.G., 1984. Sound velocity of carbon at high pressures. J. Phys., Colloq. 8, C8–235. Shterenberg, L.E., Slesarev, V.N., Korsunskaya, I.A., Kamenetskaya, D.S., 1975. The experimental study of the interaction between melt, carbides and diamond in the iron–carbon system at high pressures. High Temp., High Press. 7, 517–522. Strong, H.M., Chrenko, R.M., 1971. Further studies on diamond growth rates and physical properties of laboratory-made diamonds. J. Phys. Chem. 75, 1838–1843. Togaya, M., 1997. Pressure dependences of the melting temperature of graphite and the electrical resistivity of liquid carbon. Phys. Rev. Lett. 79, 2474–2477. Tsuzuki, A., Sago, S., Hirano, S., Naka, S., 1984. High temperature and pressure preparation and properties of iron-carbides Fe7 C3 and Fe3 C. J. Mater. Sci. 19, 2513–2518. Turkevich, V.Z., 1992. Thermodynamic calculation of the phase diagram of the Fe–C system up to 8 GPa in the range of equilibria with the liquid phase. In: Anthology “New Developments in Superhard Materials”. The Academy of Science of the Ukrainian Soviet Republic, Kiev, pp. 4–8 (in Russian). van Thiel, M., Ree, F.H., 1989. Theoretical description of the graphite, diamond, and liquid phases of carbon. Int. J. Thermophys. 10, 227–236. Walker, D., Dasgupta, R., Li, J., Buono, A., 2013. Nonstoichiometry and growth of some Fe carbides. Contrib. Mineral. Petrol. 166, 935–957. Wood, B.J., 1993. Carbon in the core. Earth Planet. Sci. Lett. 117, 593–607.