High Pressure Melting Curve of Fe Determined by Inter‐Metallic Fast Diffusion Technique

The heat extracted from the core by the overlying mantle across the core‐mantle boundary controls the thermal evolution of the core. This in turn leads to the solidification of the inner core in association with the exsolution of light alloying elements into the liquid outer core. Although the temperature (T) at the inner core boundary (ICB) would be adjusted to account for the effects of the light elements, the melting T of Fe places an upper bound at the ICB and it is a vital point in the thermal profile of the core. Here, we determine the melting T of Fe in the multi‐anvil press by characterizing the interface of Fe‐W interaction. Our data place a tighter constraint on the melting curve of Fe between 8 and 21 GPa, that is directly applicable to small planetary bodies and serves as an anchor for melting curve of Fe at higher pressure.


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Lindemann's theory by Gilvarry (1956) proposed that melting occurs when the root-mean-square amplitude of thermal vibration reaches a critical value of the distance of nearest-neighbor atomic separation. Born (1939) also proposed that melting occurs when the shear modulus vanishes, and the system becomes thermodynamically unstable in the solid state. During melting, the input energy into the system is used exclusively in changing the phase of the substance by the action of bond breaking (Pedersen et al., 2016). Thus, it is an isothermal process that leads to a change in the physical and chemical transport properties.
Different methods have been adopted in detecting melting transition in materials. The integration of X-ray technology and spectroscopic methods in high-pressure studies has provided opportunity to detect melting through imaging or a change in spectra pattern. To detect melting, Boehler (1993) used the onset of convective motion while the change in the absorption spectra were adopted by Aquilanti et al. (2015) and Zhang et al. (2016). Likewise, Shen et al. (2004) and Anzellini et al. (2013) used the disappearance of X-ray diffraction peaks and the appearance of diffuse scattering as melting criteria. Since melting is accompanied by changes in transport properties of a material, relationships such as electrical resistivity-temperature (Ezenwa & Yoshino, 2021a;Ezenwa & Yoshino, 2021b;Geballe et al., 2021;Ohta et al., 2016;Pommier et al., 2021;Silber et al., 2018;Yin et al., 2022;Zhang et al., 2020), emissivity-temperature (e.g., Fischer & Campbell, 2010) as well as laser power-temperature (e.g., Lord et al., 2009) have been used as a melting criterion in static high-pressure studies. The lack of consensus in the melting curve of Fe (e.g., Boehler et al., 2002;Hou et al., 2021), possibly due to the uncertainties associated with the existing melting criteria necessitates the search for a new melting criterion. It is generally known that intermetallic diffusion processes are faster in liquid materials when compared with their respective solid phase(s) (e.g., Bartels et al., 1994), due to the higher degree of freedom in liquid. This suggests that a change in diffusion rate (diffusivity) could provide a unique diagnostic for solid-liquid transition. In this study, we precisely measured the melting temperature of Fe at various fixed pressures up to 21 GPa in LVP by characterizing the diffusivity of W in solid and liquid Fe at the melting transition.

Experimental Details
Experiments were performed in a 1000-ton Walker module type multi anvil LVP using 8/3 cell assembly, which comprises a precast MgO octahedron with pyrophyllite gasket. The pressure was calibrated using standards as fixed high temperature calibration points (Bertka & Fei, 1997;Hirose & Fei, 2002). For the quench experiments, we prepared consistent cell assembly parts to ensure reproducibility. The experiments along the melting were conducted in the temperature range of 1900-2100 K. We used the high-temperature pressure calibration curve which is close to the experimental temperatures. The estimated pressure change within 100 K is less than 0.2 GPa (Fei et al., 2004). Figure S1 in Supporting Information S1 shows the schematic of the 8/3 cell assembly. The starting samples W and Fe of purity 99.9wt% were commercially supplied by Alfa Aesar. Both W and Fe have a diameter of 0.5 mm with thicknesses of 0.2 and 1 mm, respectively. The samples were polished on both ends with a grain size of 0.5-micron polisher to minimize the roughness of the surface which could influence diffusion due to lattice imperfection. The cell assembly was stored in an oven for 24 hr at 1 atm and 393 K. The thermocouple (type C) wires (W5%Re and W26%Re) were placed in a four-hole cylindrical alumina forming a crossed junction at the base of the alumina. For close monitoring of the sample temperature, the W disc was placed in direct contact with the thermocouple junction and the wires were led out through the gasket ( Figure  S1 in Supporting Information S1). There was no correction for the effect of pressure on the thermocouple emf measurements. Using thermocouple type D (W3%Re and W25%Re) which is close in composition to type C used in our measurements Nishihara et al. (2020) demonstrated that pressure would influence the true thermocouple sensor reading by about −30 K at 16 GPa and 1173 K. Hence, the non-correction of the pressure influence on the thermocouple sensor in our study would not drastically change the interpretation of our results. The W and Fe samples were stacked and contained in an MgO sleeve. All ceramic parts were baked in an oven at 1273 K for 3 hr. The heat source was provided by passing a high alternating current through a cylindrical Re heater surrounding the MgO sample container while LaCrO 3 sleeve placed next to the heater served as thermal insulation.
Both pressure and temperature were automatically controlled through an in-house developed computer program. Experiments were performed at various fixed pressures up to a maximum pressure of 21 GPa. The samples were isothermally compressed at room temperature and then heated at a rate of about 100 K per minute and held constant for 2 min at the targeted temperature. The sample temperature was rapidly quenched by shutting off the heater power, followed by automatic decompression. The recovered post-mortem cell assembly was mounted in epoxy resin and wet ground along the long axis of the heater to expose the middle section of the W and Fe samples for chemical analysis. To characterize the Fe-W interaction and explore the change in the diffusion processes at the melting transition, Backscattered Electron (BSE) imaging, Energy Dispersive Spectroscopy (EDS) and Electron Probe Microanalysis (EPMA) were performed on each run sample. An accelerating voltage of 15 kV, a probe current of 30 nA, and a spot size (∼100 nm) beam were used for all analyses in this study. At each fixed pressure, several runs with different target temperatures were performed to determine the melting temperature.

Inter-Metallic Diffusion as a Melting Criterion
The use of inter-metallic diffusion as a melting criterion for a given metal say "B" involves the monitoring of the interface behavior of the metal with another metal assigned "A" across the melting transition of the metal B. The metal designated as A must have a higher melting temperature than metal B. Ideally, these two metals should be immiscible when B is in solid state. On melting of B, the two metals must react to form a uniform solution. Hence, this process can be used as a melting criterion.
The relative similarities in the chemical properties of metals (Benfey, 2007;Mendeleev, 2013) as well as crystallographic structures (Hume-Rothery & Raynor, 1962;Kornilov, 1953Kornilov, , 1954 govern the formation of solid solution in intermetallic alloys. In general, from the periodic table, elements in the same group can easily form solid solution with increasing temperature. However, across the period, the closer two metals are in their respective positions the more likely they would form solid solution with rising temperature. On the bases of Hume-Rothery rule, atomic diameter differences of ∼8%-10% between metals would lead to an unlimited solid solution, while diameter differences of ∼14%-16% and above would result in limited solid solution and immiscibility, respectively. Interestingly, W and Fe are of atomic diameter 2.82 Å and 2.54 Å respectively (Kornilov, 1953). From this, the percentage difference in their atomic diameter is about 9.9%. This implies that a pair of W and Fe can form an unlimited solution perhaps by direct substitution of atoms with increasing temperature. However, at 1 atm and at high temperature about 300-200 K below the melting temperature of Fe, experimental evidence by Reisner et al. (2019) shows that W slowly interacts with Fe forming a new stoichiometric interface boundary layer of FeW and Fe 2 W. Their study demonstrated that it would take about 48 hr for ∼3wt% W to diffuse into solid Fe at 1 atm and high temperature. The high metallic bonding in W (Brewer, 1968) which translates to a very high melting temperature (3695 K at 1 atm) and low thermal expansion coefficient (Wang and Reeber, 1998) could have caused a delay in the formation of FeW and Fe 2 W with increasing temperature.
Although, it is difficult to find an ideal candidate metal that would be immiscible with solid Fe at high temperature, the slow diffusion of solid W in solid Fe would enable the use of W-Fe interface behavior across Fe melting transition as a melting criterion if there is enough Fe material to accommodate the formation of the intermetallic boundary layer with pure Fe remaining for a given interaction time. Also, the formed solid FeW and Fe 2 W alloys must have higher melting temperature than Fe. This implies that the melting transition could be derived from the reaction at the interface between the intermetallic alloy of FeW/Fe 2 W compound and the pure Fe metal. At 1 atm, the phase diagram of Fe-W on the Fe rich side indicates that Fe with variable W composition up to 11 wt% has a minimum melting temperature value of 1802 K at ∼4.4 wt% (Goldbeck, 1982). This melting temperature is comparable to the melting temperature of 1809 K for pure Fe at 1 atm (see, Sterrett et al., 1965 and ref therein) within error. Since pressure enhances intermetallic diffusion, the melting temperature of the intermetallic alloy would increase relative to pure Fe with the promotion of W concentration via increasing pressure. On melting, it is expected that the loss of long-range order structure (Lennard-Jones and Devonshire, 1939) would lead to increase in the diffusion processes of W in liquid Fe due to the increased degree of freedom of the atomic displacement as well as increased vacancy formation in liquid. By employing shorter W and longer Fe sample wires in our cell design, the characterization of the change in FeW-Fe diffusive interaction across melting transition can be used as a criterion to detect melting in Fe. The difference in sample lengths was adopted in order to have enough pure Fe sample through the formation of the intermetallic boundary layer leading up to melting within a fixed run time.
We tested and established this technique at fixed pressure of ∼15 GPa. The duration of all experiments is 25 min. Figure 1 shows a comparison of the recovered samples before and after melting at 15 GPa. Shown in Figure 1a is the chemical analyses of the recovered run at 15 GPa and 2043 K. The BSE and EDS analyses show a formation of a thin intermetallic composition of FeW (gray color) at the W and Fe interface. EPMA analysis was also 4 of 9 performed on the sample as shown in Figure 1c. The results indicate that the intermetallic layer was formed by a solid-state diffusion process with a concentration gradient of ∼0.5 wt% per unit distance (Figure 2c). The overall percentage composition of W lies between 20 and 30 wt% in this boundary (∼30 μm wide). Judging from the phase diagram of Fe-W (Goldbeck, 1982), the intermetallic boundary would have a higher melting temperature than pure Fe. The quench texture and composition at the interface confirm the solid state at 15 GPa and 2043 K. With increasing temperature to 2073 K at 15 GPa, we show melting of the sample as indicated in Figure 1b. The BSE image and EDS analyses of the sample recovered show dendritic quench texture and distribution of W in the Fe sample region that are consistent with melting. The EPMA results shown in Figure 1d demonstrate an even dissolution of W in the range of 60-30 wt% in molten Fe with an average composition of 50 wt%. The variation of the measured W concentration is due to the spot analysis of the sample with dendritic texture (See Figure S2 in Supporting Information S1). By taking the average of the measured temperatures between the solid and melting runs, we determine the melting temperature of Fe to be 2058 K ± 15 K at 15 GPa. Table S1 lists the experimental conditions for each run, the identified phase, and the determined melting temperature at each given pressure. Using this procedure, a series of experiments at different temperatures were conducted at fixed pressures of 8, 10, 12, 14, 15, 18, and 21 GPa and the corresponding melting temperatures determined. Figure 2 illustrates the complete data set for all runs up to 21 GPa, forming a new precise melting curve of pure Fe between 8 and 21 GPa.

Melting Curve of Fe
The melting curve of Fe obtained in this study is shown in Figure 2. Based on the phase diagram of Fe (e.g., Swartzendruber, 1982), our experimental pressure range between 8 and 21 GPa indicates that our samples were melted from the face-centered cubic (fcc) − Fe crystal structure. Previous studies (e.g., Strong, 1959;Swartzendruber, 1982) have reported the (bcc) − − liquid triple point at around 5.2 GPa and 1991 K. It has been demonstrated that the melting curve of Fe changes its trend at this triple point (Silber et al., 2018;Strong et al., 1973) due to the structure preceding melting (Sanloup et al., 2000). The melting curve preceding the δ (bcc) phase was established from the experimentally determined melting temperatures below 5 GPa by previous studies (Liu and Basset., 1975;Silber et al., 2018;Strong, 1959). The trend of our determined melting curve compares well with the established triple point and is consistent with the data set of Silber et al. (2018) at lower pressure ( Figure 3). However, our curve is systematically lower by about 50 K when compared with measurements by Strong (1959) whose maximum investigated pressure was about 10 GPa and that of Silber et al., 2018 in the pressure range between 7 and 11 GPa. Both studies (Silber et al., 2018;Strong, 1959) employed change in electrical resistance as a melting criterion. Fe has a very small resistance change of about 9% at the melting transition, possibly because of assisted melting by residual magnetic scattering (Ezenwa & Secco, 2019). It is not clear what is the exact cause for the small systematic difference in the melting temperature, but such a difference could be attributed to factors such as pressure determination at high temperature or a weak signal in melting detection. Our melting curve is in fair agreement with the melting data recently reported by Yong et al. (2019) between 14 and 24 GPa. However, using our developed technique, we detect melting within 30 K. The error in our determined melting temperature is much smaller than that reported by Yong et al. (2019) (Figure 3).

Discussion and Implications
As previously highlighted, the change in the melting temperature of Fe as a function of pressure is of considerable theoretical and experimental interest in mineral physics since Fe is the most dominant component of the liquid outer and solid inner core. As such, Fe melting curve provides a constraint in core temperature as a function of depth, particularly at the ICB. Above 5 GPa, Fe would melt from a face-center cubic (fcc) crystal structure (γ-Fe). The slope of a melting curve alters when it passes through a triple point in a pressure-temperature phase diagram (Morard et al., 2018;Strong et al., 1973) due to the melting entropy difference between two solid polymorphs (Morard et al., 2018). It is not clear how much of this slope change will occur at the γ-ε-liquid boundary judging from the large uncertainties in the melting curve across this boundary from previous studies. Hence, we consider it reasonable to use our determined melting curve as a reference to establish the melting curve of γ-Fe up to the γ-ε-liquid boundary at about 80-90 GPa (Uchida et al., 2001;Yamazaki et al., 2012).
In DAC studies, the challenges associated with the melting detection criterion employed by various studies could have led to large discrepancy among  datasets, especially in the presence of a large temperature gradient associated with laser-heating (e.g., Anzellini et al., 2013;Hou et al., 2021). The use of our newly developed intermetallic fast diffusion as a melting criterion provides accurate measurements of the slope of the melting curve of γ-Fe, which can be used to assess the different melting curves established in DAC studies for the γ-Fe phase. The measurements in the large-volume pressure use a resistance heater to provide homogenous sample temperature, coupled with the use of a thermocouple sensor to directly measure the sample temperature. We extended the melting curve of γ-Fe up to 80 GPa based on the linear relationship between melting temperature and normalized isothermal volume contraction with increasing pressure (Kraut and Kennedy, 1966). To construct the melting curve, we first derived a metastable melting temperature from the thermodynamic data set of isothermal volume contraction of γ-Fe with increasing pressure up to 80 GPa at fixed 2000 K (Komabayashi & Fei, 2010). Since melting is an isothermal process, we then derived a normalized volume contraction along the melting boundary from our melting data set. By comparing the obtained linear graphs of temperature versus normalized volume contraction from our melting data and from the work of Komabayashi and Fei (2010), we obtained a melting temperature versus pressure data in the range of 21-80 GPa. As shown in Figure 4, the extended γ-Fe melting curve is in good agreement with the melting curve reported by Boehler (1993) using fluid flow and that by Shen et al. (1998) using the disappearance of X-ray diffraction peaks at lower pressure up to about 40 GPa. Above 40 GPa, it deviates from the above-mentioned studies but converges very closely with the melting curve determined using synchrotron Mössbauer spectroscope by Zhang et al. (2016) as well as high-pressure melting data by Jackson et al. (2013). Our melting curve is higher than the melting data by Sinmyo et al. (2019) which was determined by a change in resistance. In a sharp contrast, the present melting curve is much lower than that determined by Williams et al. (1987), Anzellini et al. (2013), andHou et al. (2021) through quenched texture, diffuse scattering ring in XRD and change in resistance as the melting criteria, respectively. It is interesting that different studies employing the same melting criterion such as a change in electrical resistance (Hou et al., 2021;Sinmyo et al., 2019) have reported large deviations in their results, suggesting that other factors such as the accuracy of the temperature sensor as well as inhomogeneity in sample temperature could affect the results. Anzellini et al. (2013) indicated that earlier DAC studies (e.g., Boehler, 1993) that identified melting transition at lower temperature may have identified the onset of fast recrystallization. Anzellini et al. (2013) determined melting temperature in the stability of γ-Fe phase seems to be too high to be consistent with our measurements at low pressure (Figure 4), unless the slope of the melting curve is about 300-500 K higher in the pressure range of 25-80 GPa.
Model studies (e.g., Hauck et al., 2013;Rivoldini and Van Hoolst, 2013) restrained by moment of inertia parameters have determined the radius of Mercury's core to be about 2, 020 km. The growth of planet's inner core enables its liquid outer core convection even when the heat flux through the CMB is less than the heat transported by conduction along its core adiabat. The inner core growth depends largely on the rate of core cooling by the mantle and the melting relationships of the core materials with pressure. Although the size of Mercury's inner core is not well known, studies (Dumberry & Rivoldini, 2015;Genova et al., 2019) have constrained the upper limit to about two-third the size of its entire core radius. The pressure at its CMB on the core side is approximately 5-7 GPa (e.g., Hauck et al., 2013). Assuming a silicon-bearing liquid outer core, the estimated pressure at its ICB is about 21 GPa if two-thirds of the core is solidified. Our melting data can be used to directly estimate the upper bound temperature at Mercury's ICB which should be about 2180 K.
Based on cosmochemical and seismological studies (e.g., Hirose et al., 2013;Li and Fei, 2003;McDonough and Sun, 1995), rocky planetary cores contain light alloying elements. Experiments have shown that the inclusion of light elements tends to decrease the melting temperature of Fe (e.g., Boehler, 1992;Chen et al., 2008;Fei & Bertka, 2005;Terasaki et al., 2011). Our newly developed technique could also provide opportunity for the investigation of the melting curve of systems such as the Fe-Si system which has not been systematically investigated in the pressure range of 5-25 GPa. In DAC studies, textural observations and X-ray diffraction studies have been the most effective methods in the identification of melting transition in the case of binary or ternary systems of Fe plus light element (e.g., Morard et al., 2011Morard et al., , 2017Terasaki et al., 2011). However, in the mega bar pressure range, it is challenging to recover samples for textural observation and in-situ melting detection from X-ray diffraction. This present technique would provide an alternative way to determine melting by monitoring an expected large change in unit cell parameter of the alloying metals before and after melting, verified by X-ray diffraction measurements and ex-situ characterization of the quenched samples. This can be readily extended to higher pressure which would provide a crosscheck between large-volume press and laser-heated diamond-anvil cell results.

Conclusion
We have tightly constrained the melting curve of Fe in LVP up to 21 GPa by characterization of the inter-metallic diffusive interaction between Fe-W at the melting transition of Fe. Our data set is in reasonable agreement with the multi-anvil study that reported the melting curve of Fe at high pressure between 14 and 24 GPa by monitoring the change in electrical resistivity upon melting, but ours provides much higher precision. The melting data are directly applicable to small planetary bodies. The estimated upper bound temperature at Mercury's ICB would be about 2180 K if two-thirds of the core were solidified. Using our established melting curve in LVP, we extended the melting curve of γ-Fe up to about 80 GPa by thermodynamic relations, which can be used to assess the existing melting curves by laser-heating DAC technique.

Data Availability Statement
The tabulated datasets were compiled and made available in Supporting Information S1. Raw EPMA data for Figure 1 and the datasets can be accessed in Mendeley Data Repository from the link https://data.mendeley.com/ datasets/6yt2wdrtsc/1.