Experimental and theoretical thermal equations of state of MgSiO3 post-perovskite at multi-megabar pressures

The MgSiO3 post-perovskite phase is the most abundant silicate phase in a super-Earth’s mantle, although it only exists within the Earth’s lowermost mantle. In this study, we established the thermal equation of state (EoS) of the MgSiO3 post-perovskite phase, which were determined by using both laser-heated diamond anvil cell and density-functional theoretical techniques, within a multi-megabar pressure range, corresponding to the conditions of a super-Earth’s mantle. The Keane and AP2 EoS models were adopted for the first time to extract meaningful physical properties. The experimentally determined Grüneisen parameter, which is one of the thermal EoS parameters, and its volume dependence were found to be consistent with their theoretically obtained values. This reduced the previously reported discrepancy observed between experiment and theory. Both the experimental and theoretical EoS were also found to be in very good agreement for volumes at pressures and temperatures of up to 300 GPa and 5000 K, respectively. Our newly developed EoS should be applicable to a super-Earth’s mantle, as well as the Earth’s core-mantle boundary region.

These studies targeted the Earth's lowermost mantle. The maximum pressure is 136 GPa in the Earth's mantle. However, recently, one after another, super-Earths, which have masses of a few times that of the Earth, have been found in the extra solar system 14 . If super-Earths have a similar bulk composition with the Earth, MgSiO 3 PPv is believed to be an abundant silicate phase in the mantles of these huge terrestrial planets 15 . Although the pressure conditions of these super-Earths' mantles reach several hundreds of GPa, previously reported EoS of PPv, derived by LHDAC experiments, were limited up to approximately 150 GPa, and theoretical calculations were also limited up to 200 GPa. Moreover, PPv is also expected to exist in Uranus's and Neptune's rocky cores in which the pressure conditions reach approximately 700-800 GPa. The direct determination of the compression behavior of PPv at multi-megabar pressures is, therefore, important to the understanding of a super-Earth's deep interior, and Uranus's and Neptune's rocky cores.
In this study, we derived the thermal EoS of PPv using primary pressure scales (see Method), with consideration of the effects of uniaxial stress. We extended the experimental pressure up to 265 GPa and the calculations up to a pressure and temperature of 1.2 TPa and 5000 K, respectively. We determined the γ and its volume dependence, using the Al'tshuler et al.'s form 13 , both experimentally and theoretically. Moreover, the Keane and AP2 EoS models (All the EoS formulae are summarized in Supplementary Information A) were newly adopted to discuss the P-V-T relation and its derivatives that extract thermodynamic functions in the multi-megabar pressure range.

Results and Discussion
Thermal equation of state for the post-perovskite phase: Fitting the EoS to the experimental data. The results of powder X-ray diffraction experiments and the observed unit cell volumes of PPv phase are summarized in Supplementary Information B. The P-V-T data were fitted to the 3BM, Vinet, AP2, and Keane EoS with the MGD model. In the fitting of the experimental data, using the Keane EoS, Slater's formula for γ and the equation of β calculated by using γ 0 and γ ∞ were used to reduce the number of fitting parameters (see Supplementary Information A). In the EoS fitting, using the Keane model (fit 5), we successfully determined all of the parameters based on the present experimental data. The obtained γ ∞ and ′ ∞ K values satisfy the thermodynamic constraints 16 γ ∞ > 2/3 and ′ ∞ K > 5/3. The EoS parameters are summarized in Table 1 with the results of previous studies. The root-mean-square (RMS) of the fitting for volume and pressure were approximately 0.1 Å 3 and 0.8 GPa, respectively, for the experimental data (fits 5-7). However, the errors in the parameters are quite large for fit 5. These errors yield an unrealistically large volume (or pressure) error compared to the observed data. Alternatively, if we fix the V 0 and ′ ∞ K (and γ ∞ ) the errors become reasonably small, and it yields a comparative pressure and volume error to the RMS value (fit 6): 1σ P = 0.8 GPa and 1σ V = 0.11 Å 3 at 100 GPa and 300 K, respectively.
In fits 1-3 and 5 the V 0 was relatively small compared to those of the previous reports. Coincidentally, the bulk modulus (K 0 ) was large and its pressure derivative (K 0 ′ ) was small. It is well known that there is a trade-off relation between these parameters. The trade-off relations of V 0 and K 0 as a function of K 0 ′ are shown in Fig. 1a,b. The smaller V 0 and larger K 0 were obtained when K 0 ′ is small. This tendency has a large gradient, and thus it leads to fact that these parameters easily scatter, whereas γ was maintained almost constantly at approximately 1.5 in all of the fittings. Although both our and previous results are on the trade-off lines, the determination of the absolute value is the difficult problem.
In particular, the determination of the zero pressure volume is a difficult problem for the high pressure phases, such as the PPv phase, because these phases are not thermodynamically stable at ambient pressures. In these cases, the g-G plot 17 was used for the determination of V 0 18,19 , where g and G are the strain parameter and the normalized stress, respectively. The G includes a pressure term, and hence, its result depends on which pressure scale is applied. As a consequence, the g-G plot yielded a V 0 of 161.01 ± 0.38 Å 3 for the 3BM, 165.64 ± 0.45 Å 3 for the Vinet, 163.45 ± 0.42 Å 3 for the AP2, and 164.26 ± 0.42 Å 3 for the Keane model. Conversely, our theoretical calculations, from − 10 to 1200 GPa, enabled us to determine V 0 without any extrapolation. The theoretical data yields a V 0 of 164.22 ± 0.06 Å. The experiment and the calculation were done completely independently, and it was found that the experimental V 0 value derived using the Keane model and the theoretical V 0 value were remarkably consistent. The AP2 model also yielded similar V 0 values, and these values are consistent within the error of the result reported by Tsuchiya et al. 4 , whereas the 3BM and Vinet models yielded very different values. In fit 7, the V 0 was fixed to be 164.26 Å 3 , and γ ∞ and ′ ∞ K were also fixed to be the values obtained in fit 5. As a result, we obtained the parameters which agree with the present theoretical results, as discuss below. Since ′ ∞ K does not satisfy Keane's rule (K 0 ′ /2 < ′ ∞ K < K 0 ′ -1) there is a possibility that K 0 ′ and ′ ∞ K are slightly over-and underestimated, respectively, in fit 7. This fact indicates that the present experimental pressure is still not sufficient to determine ′ ∞ K precisely. Nevertheless, there was no significant discrepancy between the obtained parameters in fit 7 and the results from the ab initio data (fit 8).
The compression curve, based on the Keane model (fit 7), is shown in Fig. 2a, along with the experimental data at 300 K. Figure 2b shows the volume difference from the EoS, including both of the data at 300 K and at high temperature. Most of the data are reproduced by the EoS within 0.1% of the volume. The data at over 200 GPa were relatively scattered mainly due to the overlapping of the X-ray diffraction peaks from MgO and gold. If we use the same pressure scale and choose the quasi-hydrostatic data (St < 0.005, see Supplementary Information B), there is no discrepancy between the result of this study and that of Guignot et al. 10 . The differences between each EoS models are discussed below and in Supplementary Information C.
Fitting the EoS to the ab initio data. In the fitting of the EoS to the present ab initio data, RMS(P) for the 3BM, Vinet, AP2, and Keane models were 0.8, 1.0, 0.7, and 0.5 GPa, respectively, for all data, and 1.2, 4.2, 1.9, and 0.4 GPa, respectively, for the data at 300 K. The fitting of the data with the Keane EoS model showed the minimum RMS value. The Keane EoS parameters (fit 8) are shown in Table 1. The results for the other EoS models are summarized in Supplementary Information D, Table S4. Using the theoretical P-V-T data, with a wide P-T range, we determined all of the parameters to be free parameters, without using either Slater's formula for γ or the equation for β. The parameters obtained in fit 8 satisfy both the thermodynamic constraints and Keane's rule well, and these parameters agree with the experimental results (fit 7). Figure 3a shows the compression curves of the experimental Keane and AP2 EoS, and the theoretical Keane EoS. Here, it should be mentioned that the data, for example around 100 GPa and 4000 K, are fictitious, because the Pv-PPv phase boundary exist at around 119 GPa and 2400 K with a gradient of 11.5 MPa/K 3 . Since the melting temperature of PPv is 6200 K at 142 GPa 20 , the P-T condition shown in Fig. 3a are expected to be in the solid state. The valid and invalid quasi-harmonic approximation (QHA) regions are the same as in Tsuchiya et al. 5 .

The difference between the EoS models, and a comparison with previous studies.
The pressure differences with respect to fit 7 are shown in Fig. 3b. The difference between fits 6 and 7 was found to be less than 2 GPa in the pressure range from 100 to 300 GPa. This indicates that the difference in V 0 does not affect the calculated pressures in this pressure range. The difference in the 3BM, Vinet, and AP2 models were directly affected by the difference in the pressure scale. The ab initio calculations yielded a relatively larger pressure, approximately + 0.5-3.4%. In the case of high temperatures, the differences were reduced compared to those at 300 K. On the other hand, the volume differences (ΔV) at 100-140 GPa and 300-4000 K, with respect to fit 7, fell in only 0.1% of the AP2 EoS (fit 4) and the theoretical Keane EoS (fit 8), and in 0.4% of fit 4 and 0.8% of fit 8, up to 300 GPa and 5000 K. It should be noted that the experimental Keane EoS (fit 7) have their own errors: 1σ V = 0.1-0.4% and 1σ P = 1.2-1.9% for the ranges of 100-300 GPa and 300-5000 K, respectively. Figure 3c shows the comparison between fit 7 and the previous experimental works at 300 K. As shown in Fig. 2, when we use the same pressure scale, the results of this study (fit 7) are fairly consistent with the results obtained by Guignot et al. 10 . Therefore, the difference in Fig. 3c shows the difference in the pressure scales; i.e.,  Figure 3d shows the comparison between fit 7 and previous theoretical works at 300    Carlo (QMC) simulation 7 are in strong agreement with the present results, at around 100 GPa, although they show a slightly smaller pressure above 200 GPa. On the other hand, the EoS reported by Caracas and Cohen 6 yields much smaller pressures with respect to the present results.
These acceptable agreements found between the experiment and theory can be regarded as a highly reasonable result. The DFT, with LDA and QHA, is known to work satisfactorily well for the description of the thermal EoS of PPv, while the GGA with QHA predicts an approximately 10 GPa larger thermal pressure than the LDA result, as confirmed in the DFT studies of MgSiO 3 -PPv 2,4,5 . Lin et al. 7 recently attempted to verify the DFT, with LDA, and GGA formalism for electron many-body effects by applying the QMC technique, which is known as a more stringent treatment for the ground state properties. They then demonstrated that the correction for the static pressure, by the QMC simulation, is quite small for the LDA pressure. As a consequence, our LDA calculation, Oganov and Ono 2 , and the Lin et al. 7 's EoS are, thus, very similar to one another, as confirmed in Fig. 3d. We observed that the present theoretical prediction maintains its excellent agreement with the experiment, with an error of only a few percent, even at a much higher pressure, up to 300 GPa, reaching the conditions of the Earth's core.  We observed a peculiar discrepancy in the EoS between Caracas and Cohen 6 and the other prediction (Fig. 3d). The EoS of Caracas and Cohen 6 was obtained using both the LDA and GGA at the static conditions. Although the GGA result is found to be approximately 10% smaller pressure with respect to the earlier LDA predictions and present result (Fig. 3d), the LDA result (not depicted in Fig. 3d) is found to be a much smaller pressure (about 30% smaller at 100 GPa, and more at higher pressure). The temperature effect may change this situation. However, even if one considered the quantum correction to the free energy and evaluated the thermal pressure, it would be difficult to resolve the discrepancy among theoretical studies because the thermodynamic properties of Mg-PPv at such a low temperature of 300 K are almost the same as those calculated at the static condition 4 .
The thermal expansion and the Grüneisen parameter. The thermal expansion coefficient, at 130 GPa, is shown in Fig. 4. There are no significant differences between fits 6 and 7, despite the fact that their V 0 s are very different. The theoretical result (fit 8) is fairly consistent with the experimental Keane EoS (fit 7). The Vinet (fit 2) and AP2 (fit 4) EoS agree well, but 3BM (fit 1) shows slightly lower values. The previous theoretical report by Ono and Oganov 8 is also consistent. Although the difference between fit 7 and the results obtained by Guignot et al. 10 is not significant, their curvature is slightly different due to Guignot et al. 10 8 . On the other hand, the present theoretical calculation was performed over a wide P-T range, from − 10 GPa and 300 K to 1200 GPa and 5000 K, which yielded a value for β of approximately 2, without using the βγ 0 -γ ∞ relation. Consequently, the volume dependencies of γ obtained from the present experimental and theoretical works are quite consistent, as seen in Fig. 5.

Conclusions
We examined the P-V-T relationship of MgSiO 3 PPv at up to 265 GPa by using a LHDAC experiment, and up to 1200 GPa and 5000 K by the ab initio calculation within the DFT. The Keane and AP2 EoS models were newly adapted to this phase in order to establish the thermodynamic EoS at multi-megabar conditions, with the parameters set at infinite pressure. The parameters such as V 0 , γ 0 , γ ∞ , and ′ ∞ K were successfully determined by using data with a wide P-T range. The obtained parameters were found to be very consistent between those obtained in the experiment and those obtained by theoretical calculation. Furthermore, these parameters satisfy the rigorous thermodynamical constraints. The present result is the first report of the fully experimentally based Grüneisen parameter using LHDAC data. Its tendency with respect to the volume was fairly consistent with the theoretical prediction.
It has previously been predicted by ab initio computations that the PPv phase decomposes above approximately 900-1000 GPa 15,24 . The ab initio calculation in this study covered the pressure range of 0 to 1200 GPa. The experimental results are applicable up to approximately 300 GPa, which corresponds to the pressure at the core-mantle boundary of a super-Earth with a mass approximately 3 times that of the Earth 24 . The P-V-T data of present experimental EoS (fit7) and theoretical EoS (fit8) are summarized in Supplementary Information (Table  S5) for the convenience. When the experimental EoS were extrapolated to even higher pressures, the difference between the experimental model and the theoretical model was found to be not significant: ΔV < 1.6% up to 500 GPa and 5000 K (and doubled after an extrapolation to 1000 GPa and 5000 K). We strongly believe that the results of this study are considerably useful to discuss the physical properties of not only the Earth's lowermost mantle, but also a super-Earth's mantle.

Method
The choice of the pressure scale is the most important factor for the determination of the EoS. The MgO pressure scale, outlined by Speziale et al. 21 , was widely used; Guignot et al. 10 and Komabayashi et al. 11 also used this scale. However, Tange et al. 26 pointed out that this scale cannot reproduce the thermal expansion at 1 atm, and they proposed a primary scale which was based on the scale-free unified analysis (SFUA). Tange et al. 26 reported two sets of parameters for the 3 rd order Birch-Murnaghan (3BM) and Vinet EoS models. However, the pressure difference between the 3BM and Vinet EoS reached over 10 GPa above 200 GPa, due to the differences in their EoS formulae. Thus, a more physically applicable EoS model is required for ultra-high pressure conditions. In this study, the Keane 27,28 and AP2 29 EoS models were adopted as ultra-high pressure EoS models. All the EoS formulae  Table S3. The key points that should be noted with these EoS are, first, that they include the parameter at infinite pressure ( ′ ∞ K or a FG ). Second, these EoS yield an intermediate pressure that lies between 3BM and Vinet (Fig. S3). Third, the number of fitting parameters is same as for the case of 3BM and Vinet, with the MGD model, when the Slater's formula for the Grüneisen parameter γ = ′ − ∞ ∞ K ( / 2 1/6) is adopted 16 . The details of the experimental procedure and the ab initio calculation are summarized in Supplementary Information C.