ab Initio DFT and MD Simulations Serving as an Anchor for Correcting Melting Curves Reported by DAC and SW Experiments—Some Transition Metals as Illustrative Examples

Abstract

The pressure–temperature scales in DAC and shock wave (SW) experiments should be corrected by taking into account the thermal pressure shifts. In the present contribution, it is further claimed that first-principle ab initio DFT and MD simulations should serve as an anchor for correcting the pressures and temperatures reported by DAC and SW experiments. It was concluded that upon deriving the actual pressure sensed by the explored sample, the thermal pressure and the temperature shifts must be taken into account when constructing melting curves. Therefore, melting curves measured by diamond anvil cells for 3d elements do not contribute to a better understanding of the geophysical Earth’s inner core. In addition, the advantage of the Lindemann–Gilvarry vs. Simon–Glatzel fitting procedure of melting curves is shown.

1. Introduction

During the last six decades, diamond anvil cells (DACs) have been frequently used to determine the equations of state (EOSs) and melting curves of elemental metals [1]. However, melting curves derived using DAC experiments have never matched the shock wave (SW) experimental results nor the theoretical density functional theory (DFT) simulations [1,2,3,4], where chromium metal is an exception (see the Section 4). It is a well-known consensus that the pressure-transmitting medium (PTM) and the packing procedure of the investigated sample, namely, the packing environment and the gasket type, play a role in the inconsistencies of reported melting curves. This gives rise to the question of whether DAC measurements contribute to a better understanding of high-pressure physics.

Many experiments showed that the melting curves of the same element measured using a laser-heated DAC (LH-DAC), as reported by different experiments, reveal different melting points using different PTMs [5,6]. The pressure medium is expected to distribute the pressure homogeneously within the pressure chamber, preventing non-hydrostatic effects, such as pressure gradients, shear stress, or inhomogeneous pressure [1]. Upon increasing the temperature at each pressure, the examined sample and the PTM are both subject to increases in their volumes. Nevertheless, the volume expansion is suppressed by the chamber’s finite volume, provoking an increase in the thermal pressure over the whole system. In some cases, the PTM melts and remains liquid during the entire experiment [7]. Therefore, the pressure reported does not necessarily present the actual pressure experienced by the sample. The effects of the PTM on the measured sample, and the PTM’s response to P and T changes throughout the experiment must be taken into account. Therefore, the pressure scale of the reported melting curves and the equation of state (EOS) isotherms should be corrected according to the actual pressure in the cell; illustrative examples are shown below.

In a recent publication, Zhang, Y. et al. [8] took into account the radiation absorption by a LiF window, thus correcting the reported SW data for elemental vanadium. This approach is strongly supported by the first-principle DFT-Z method, verifying that this procedure is indeed reasonable. In addition, Zhang, Y. et al. [8] suggested that this procedure should also be applied to all the d-electron transition metals. Zhang’s results clearly confirm the linear increase in the thermal pressure shifts upon elevating the applied temperature (see Figures 1–4), as predicted by first-principle theories [9].

In a previous paper, it was argued that if isochoric conditions exist in the DAC chambers [4], increasing the temperature causes additional thermal pressure. In practice, a mechanical pressure gauge to measure this thermal pressure (P_th) does not exist. Therefore, the actual pressure must be estimated, either from the pressure shift of each melting point relative to the initial ambient pressure (see Figures 1–4) or from first-principle calculations of the lattice component in the P–V–T equation of state [9]. The direct determination of the thermal pressure and temperature at the melt from experimental results is not yet available. In the present contribution, the linear rise of P_th vs. the temperature, as predicted by first-principle assumptions [9], was confirmed. This allowed for the extrapolation of the thermal pressure and the thermal temperature up to the melt, thus correcting the pressure scales and the melting temperatures of the transition metals Fe, V, Ir, and Pt.

2. Analysis of the Published Data

All the presented data in the present paper were downloaded and reanalyzed using B. Tummers’ datathief [10]. The obtained melting curves were fitted by applying the constraint that the fitting volume parameter V of the experimental equation of state (EOS) at ambient temperature must simultaneously fit the experimental melting and EOS data. Note that the Grüneisen parameter γ is a fitting parameter assigned in the manuscript “combined approach” (see the Appendix A). The corroboration of this fitting procedure was confirmed using the ab initio DFT-Z methodology [3].

3. Ilustrative Examples

3.1. Melting Curve of Elemental Vanadium

The melting curve of vanadium reported by Zhang, Y. et al. [8] (therein Figure 4) was determined using in situ X-ray diffraction in a LH-DAC and is depicted in Figure 1. The blue solid line presents the melting curve of V fitted with the combined approach constraint, which was confirmed using first-principle simulations (red solid line, DFT-Z method [3]). In Figure 1a, the colored asterisks represent the thermal pressure shifts as a function of the applied temperature. The dashed black lines represent the applied pressures at 300 K. The angle (difference) between the dashed lines and the asterisk points indicates that the thermal pressure (P_th) was caused by increasing the temperature, thus confirming the assumption that isochoric conditions exist in the DAC chamber when using KCl PTM. The horizontal colored double arrow lines indicate the thermal pressure shifts (P_th) at the melt. In Figure 1b, the corrected melting curve of V as derived from Figure 1a is presented.

The contribution of Zhang, Y. et al. [8] clearly demonstrates that all the experimental SW and DAC data reported to date need pressure and temperature scale corrections. As shown, there is no discrepancy between the static DAC and the dynamic SW measurements of V if the radiation absorption by the LiF window is correctly taken into account. The proposed approach was indeed confirmed by first principles using the ab initio DFT-Z method simulations [3].

3.2. ε-Iron

The isochoric behavior of ɛ-iron was reported by Sinmyo, R. et al. (2019) [11] and is depicted in Figure 2a. This result totally contradicts the quasi-isobaric behavior of ɛ-iron in laser-heated DAC claimed by Anzellini et al. [12]. The reason for this discrepancy comes from the packing of the iron foil with Al₂O₃ adding Ar PTM compared with KCl PTM in Anzellini’s experiments.

Four melting points of ε-phase iron (out of 10) reported by Sinmyo, R. et al. [11] are displayed in Figure 2a, corroborating the melting points claimed by Murphy, C. et al. (2011), where the thermally corrected melting points (marked X) were derived using inelastic X-ray scattering and phonon density of states [13]. The green squares in Figure 2a are the melting data reported by Anzellini et al. [12] (therein Figure 2). The red open circles are DAC melting results derived by Sinmyo et al. and fitted using the combined approach constraint (green solid line). The purple open circle is the triple point liquid–fcc–hcp structures. The colored asterisks represent pressure–temperature thermal shifts, confirming the isochoric condition in the DAC chamber. The horizontal colored double arrow lines indicate the thermal pressure shifts (P_th) at the melt based on the theoretical ab initio melting curve. The black circles are the experimental results obtained by Tateno et al. [14] and indicate that iron at pressures above the ICB is in a solid state.

The pressure–temperature-corrected scale based on the theoretical ab initio anchor deduced from Figure 2a is the main result for the melting curve of ε-iron metal. Note that the red asterisks in Figure 2b show the P_th shifts at the melt, which were fitted using the combined approach with Vinet fit parameters 163.4/5.55/1.6 (blue solid line, see the Appendix A). A possible explanation for the discrepancy between Anzellini et al.’s [12] and Sinmyo et al.’s [11] experiments is given in the discussion.

Figure 2. Melting curve of ε-iron. (a) The magenta solid line represents the theoretical molecular dynamics (MD) of the ab initio calculated melting curve from Alfe et al. [15]. The double-arrow lines indicate the thermal shifts relative to the theoretical anchor and the experimental melting curve (dashed). The green squares present the experimental DAC melting points reported by Anzellini et al. [12]. The red open circles are DAC melting results derived by Sinmyo et al. [11] and were fitted by adopting the combined approach constraint (green solid line). The purple open circle is the triple point. The blue X points are Murphy, C. et al.’s [13] thermally corrected melting points, which were derived using inelastic X-ray scattering The colored asterisks represent pressure–temperature thermal shifts, as reported by Sinmyo, R. et al. [11]. The black circles present Tateno et al.’s experimental results [14]. (b) Pressure–temperature corrected scale based on the theoretical ab initio anchor (deduced from Figure 2a and marked by the red asterisks), which was fitted using the combined approach with Vinet fit parameters 163.4/5.55/1.6 (blue solid line, see Appendix A).

Figure 2. Melting curve of ε-iron. (a) The magenta solid line represents the theoretical molecular dynamics (MD) of the ab initio calculated melting curve from Alfe et al. [15]. The double-arrow lines indicate the thermal shifts relative to the theoretical anchor and the experimental melting curve (dashed). The green squares present the experimental DAC melting points reported by Anzellini et al. [12]. The red open circles are DAC melting results derived by Sinmyo et al. [11] and were fitted by adopting the combined approach constraint (green solid line). The purple open circle is the triple point. The blue X points are Murphy, C. et al.’s [13] thermally corrected melting points, which were derived using inelastic X-ray scattering The colored asterisks represent pressure–temperature thermal shifts, as reported by Sinmyo, R. et al. [11]. The black circles present Tateno et al.’s experimental results [14]. (b) Pressure–temperature corrected scale based on the theoretical ab initio anchor (deduced from Figure 2a and marked by the red asterisks), which was fitted using the combined approach with Vinet fit parameters 163.4/5.55/1.6 (blue solid line, see Appendix A).

3.3. Iridium

Iridium attracted considerable interest in the scientific community due to its outstanding elastic and thermal properties. It is one of the most incompressible metals in nature, exhibiting a face-centered cubic (fcc) structure up to 78 GPa with a measured bulk modulus B_o = 341(3) GPa according to the combined approach calculation. Based on ab initio simulations, above ~78 GPa up to 600 GPa, the r-hcp structure was proposed [3]. At room temperature (RT), Ir exhibits a density of 22.56 g/cc and a high shear modulus of G_o = 210 GPa. Thus, thanks to its shear modulus, chemical inertness, refractory nature, and phase stability, Ir is used for crucibles and thermocouples. The experimental pressure–temperature data at 26 and 36 GPa of the fcc phase derived by Anzellini et al. [16] is depicted in Figure 3a. The thermally corrected scale is depicted in Figure 3b. In addition, as shown in Figure 3c, a triple point is expected at ~80 GPa (500 K) [3], predicting a phase transition from fcc to a disordered hexagonal close-packed phase (r-hcp). The extrapolation of the combined approach up to 600 GPa (blue solid line) with the same fitting parameters 341/4.4/2.55 was corroborated via a DFT simulation. The dashed black lines present the ambient applied pressures. The angle between the dashed black lines and the colored round points are due to the thermal pressure (P_th) shifts provoked by the increase in temperature. The horizontal black double arrows in Figure 3c indicate the thermal pressure shifts (P_th).

3.4. Platinum

Platinum is interesting, as it exhibits an fcc to r-hcp phase transition when experiencing increasing pressure, as shown in Figure 4a.

Recently reported thermal shifts by Anzellini et al. [17] (therein Figure 4) show a difference in the pressure shifts between the fcc and hcp phase. The corrected pressure scale at the melt, which is derived from Figure 4a (colored double arrows), is proposed in Figure 4b. The pressure–temperature melting points (blue diamonds) at the r-hcp region are shown in the figure. The corrected melting points are fitted with the combined approach (red solid line), which was corroborated using the first-principle Z method (green hexagrams) [3]. By utilizing the EOS parameters according to Zha et al. [18] (red solid line), the corrected melting points were fitted with BM 273/4.2/2.63 parameters, where γ_o is a free parameter.

4. Discussion

It is assumed that the temperatures and the pressures of the present transition metals are indeed correct, and the derived temperatures, either from a thermocouple or using an optical pyrometer (black body spectra), within the errors, are reasonable [1]. The advantages and disadvantages of these methods were beyond the scope of the present contribution. However, the fact that the data, within the errors, of V DAC experiments embedded in NaCl or KCl match the corrected SW data lead to the conclusion that the reported initial pressures and temperatures are reliable.

In many published articles, isochoric or quasi-isobaric conditions in the DAC chambers were reported [7,11,12,14]. Quasi-isobaric behavior means that due to a temperature rise, the sample expansion could not be suppressed by the PTM, leading to melting curves lower than those predicted by the ab initio simulations. Nevertheless, in the case of the isochoric condition in the DAC, this means that the PTM prevents thermal volume expansion, thus provoking an increase in the thermal pressure.

In conclusion, the pressure-transmitting medium (PTM) and packing procedure of the examined sample explain the discrepancies between the experimental reported melting points. In addition, in the case of V, by taking into account the absorption of the LiF window in the SW experiments, no discrepancy between the SW and DAC results exists [8]. This led to the conclusion that the first-principle DFT-Z methodology [3] should serve as an anchor for the pressure and temperature scale corrections.

The actual pressure on the V sample in the LH-DAC chamber can be estimated from first-principle calculations utilizing the P–V–T equation of state [9]. However, the use of experimental data to directly determine the actual pressure and temperature at the melt in LH-DAC experiments has not been possible up to now. In the present contribution, a method to directly derive the thermal pressure and the temperature at the melt is proposed. The corrected scale (actual pressure) of V melting points and the proposed melting curve are depicted in Figure 1b. Note that the experiment proposed by Zhang et al. was performed with KCl PTM [8], while the experiment by D. Errandonea et al. was performed with NaCl PTM [6]. The corrected SW data, which was confirmed using the DFT-Z method [3], provide evidence that the correction proposed by Y. Zhang et al. is reasonable and can be applied to other transition metals, like Fe, Ir, or Pt, as follows.

In the case of ɛ-iron metal, the colored asterisks in Figure 2a represent the pressure–temperature thermal shifts, as expected in the isochoric condition in a DAC chamber. This isochoric behavior relates to the packing proposed by R. Sinmyo et al. (Al₃O₃ + Ar). Again, the linear increase in the temperature, as theoretically predicted by the P–V–T equation of state [9], was experimentally confirmed. The discrepancy between Sinmyo et al. [11] and Anzellini et al. [12] melting curves is explained by the different PTMs.

In the case of iridium, the extrapolation of the combined approach up to 600 GPa (Figure 3c, red solid line) with the same fitting parameters for low and extreme pressures (341/4.4/2.55) corroborated the assumption of L. Burakovsky et al. that a second-order smooth transition from fcc to hcp exists [3]. The colored open circles in Figure 4c present the Z method calculations of the EOS (P–V) for the fcc and r-hcp phases. The dashed black lines present the angle between the applied pressures and the calculated thermal shift (P_th), similar to V and Fe. Note that L. Burakovsky et al. related the phase transition from fcc to r-hcp to the 5d electron rearrangement. The loss of the fcc stability drives the hex structure via a second-order phase transition, suggesting a random layer stacking.

The elastic properties of platinum show high density and low strength, which is usually used for crucibles at low temperatures. In addition, platinum serves as a pressure standard in DAC X-ray experiments, as well as in high-pressure shock experiments. There were earlier reports of experimental melting curves for Pt [19,20]. The difference in the thermal pressure shifts between the fcc and hcp is clearly observed and is demonstrated in Figure 4a, where the corrected melting scale at the melt (r-hcp structure) is depicted in Figure 4b. Similar to ε-iron, the experimental results are below the first-principle calculated melting curve. We attributed this behavior of the Pt metal to the KCL PTM in both experiments.

Nevertheless, a large uncertainty in the estimated temperature, which was fitted with the Planck function in the grey-body approximation and Wien pyrometry, could lead to a large span in the errors of the reported temperatures from the DAC measurements.

The phenomenon of isobaric behavior of vanadium recrystallization [6] different from the isochoric behavior claimed by Zhang, Y. et al. [8] must be related to the elastic properties of NaCl PTM.

In Figure 5, the high-pressure studies of chromium metal utilizing KBr PTM are depicted. Cr metal is the only case where the theoretical approach using ab initio quantum molecular dynamics (QMD) simulations based on the Z methodology matches the DAC experiments [21,22]. Note the advantage of the combined Lindemann–Gilvarry approach vs. Simon–Glatzel fitting curve procedure, revealing Bo, Bo′, and γ_o simultaneously.

5. Conclusions

The pressure and temperature scales reported in the DAC experiment do not represent the actual pressure experienced by the sample in the cell. The different response of the PTM’s to P and T changes is the reason for the variety in melting curves reported in the literature. Therefore, melting curves derived from diamond anvil cells for 3d elements do not contribute to a better understanding of the geophysical Earth’s inner core. It is suggested that the melting curves derived using first-principle DFT-Z methodology or quantum molecular dynamics (QMD) should serve as an anchor for the pressures and temperatures corrections.

Upon deriving the actual pressure sensed by the explored sample, the thermal pressure and the temperature shifts must be taken into account when constructing melting curves.

The corrected pressure scales for metallic Fe (3d), V (4d), Ir, and Pt (5d) transition metals are proposed.