Melting phase relations in the MgSiO₃–CaSiO₃ system at 24 GPa

Abstract

The Earth’s lower mantle is composed of bridgmanite, ferropericlase, and CaSiO₃-rich perovskite. The melting phase relations between each component are key to understanding the melting of the Earth’s lower mantle and the crystallization of the deep magma ocean. In this study, melting phase relations in the MgSiO₃–CaSiO₃ system were investigated at 24 GPa using a multi-anvil apparatus. The eutectic composition is (Mg,Ca)SiO₃ with 81–86 mol% MgSiO₃. The solidus temperature is 2600–2620 K. The solubility of CaSiO₃ component into bridgmanite increases with temperature, reaching a maximum of 3–6 mol% at the solidus, and then decreases with temperature. The same trend was observed for the solubility of MgSiO₃ component into CaSiO₃-rich perovskite, with a maximum of 14–16 mol% at the solidus. The asymmetric regular solutions between bridgmanite and CaSiO₃-rich perovskite and between MgSiO₃ and CaSiO₃ liquid components well reproduce the melting phase relations constrained experimentally.

Introduction

Determining melting phase relations in natural rock-forming, multi-component systems at high pressures is essential to understand the origin and consequences of the deep mantle melting (e.g., Williams and Garnero 1996) and the crystallization of the deep magma ocean of the early Earth (e.g., Ito et al. 2004). Although numerous melting experiments have been performed for these purposes, the range of parameter space (pressure (P), temperature (T), and compositions (X)) studied is limited for tracing the crystallization of the magma ocean and for a comprehensive investigation into the origin of the deep mantle melting. Indeed, it is not practical to cover the vast P–T–X space relevant to deep Earth events based on experiments alone. In contrast, experiments in simpler systems combined with thermodynamic models provide a powerful tool to understand the melting chemistry in the multi-component system. Such comprehensive datasets and thermodynamic models have been constructed for pressures up to those of the mantle transition zone (Gasparik 2003).

In addition, the occurrence of a certain degree of temperature gradient is inevitable in melting experiments using laser-heated diamond anvil cells or multi-anvil apparatuses, especially under the high pressures of the lower mantle. The temperature gradient obscures the melting chemistry in multi-component systems because phase segregation changes the bulk compositions of interest. In these cases, only limited information, such as phase equilibria and element partitioning between the phases in contact, can be extracted by assuming local chemical equilibrium. This problem could be overcome by experiments in simple binary systems.

The lower mantle is composed of bridgmanite, ferropericlase, and CaSiO₃-rich perovskite (Irifune 1994). Normal mid-ocean ridge basalt has a mineral assemblage of bridgmanite, CaSiO₃-rich perovskite, a SiO₂ phase, and an aluminous phase in the lower mantle (Hirose et al. 1999). Therefore, the binary melting phase relations between each mineral-constituting component are important in understanding the melting behaviors of the lower mantle. Previously, the MgO–MgSiO₃ system was investigated at the lower-mantle pressures using a multi-anvil press (Liebske and Frost 2012), but little is known about the other systems.

This paper is aimed at establishing the melting phase relations in the MgSiO₃–CaSiO₃ system at 24 GPa, avoiding ambiguous implications for Earth science that require other melting phase relations such as MgO–CaSiO₃. The eutectic composition, solidus temperature, and solubility between the two endmembers were investigated experimentally at 24 GPa. Although Gasparik (1996) has performed a melting experiment on the same system at 22.4 GPa, it is difficult to confirm melting from the backscattered electron image presented in his paper. In the present work, we re-investigate the system to find the interaction parameters between the MgSiO₃ and CaSiO₃ components by thermodynamic analysis in the framework for the endmembers (de Koker and Stixrude 2009; Stixrude and Lithgow-Bertelloni 2011; Bajgain et al. 2015) consistent with a previous work in the MgO-MgSiO₃ system (Liebske and Frost 2012).

Methods/experimental

Starting materials

We used three starting compositions, i.e., Mg_0.90Ca_0.10SiO₃ (Mg90), Mg_0.75Ca_0.25SiO₃ (Mg75), and Mg_0.50Ca_0.50SiO₃ (Mg50). Bubble-free, chemically homogenous glasses were synthesized for Mg_0.75Ca_0.25SiO₃ and Mg_0.50Ca_0.50SiO₃ in Pt crucibles at 2023 K using chemical mixtures of SiO₂ (99.9%, Kojundo Chemical Laboratory Co., Ltd.), MgO (> 99.99%, Kojundo Chemical Laboratory Co., Ltd.), and CaCO₃ (> 98.0%, Kanto Chemical Co., Inc.), after decarbonation at 1273 K for 16 h. The starting material with a bulk composition of Mg_0.90Ca_0.10SiO₃ was not vitrified completely, which is likely because of its high Mg content. We crushed and ground it to fine particles. The bulk composition of the latter starting material was confirmed via electron microprobe analysis after a subsolidus high-pressure and high-temperature experiment. Table 1 lists the bulk compositions of the starting materials. The starting materials were stored in a furnace at 383 K before use.

Table 1 Chemical compositions of starting materials

High-pressure melting experiments

High-pressure and high-temperature experiments were performed at 24 GPa and 2600–3000 K in a Kawai-type multi-anvil apparatus (Orange 2000, Geodynamics Research Center, Ehime University). A 10/4 (octahedron edge length/truncation edge length) cell assembly was chosen for pressure generation. The pressure of 24 GPa was calibrated based on the coexistence and dissociation of wadsleyite into bridgmanite and periclase at 2273 K in the Mg₂SiO₄ system (Fei et al. 2004), in addition to other phase transitions at room temperature, 1873 and 2273 K, as described in Zhou et al. (2017). The pressure calibration (Additional file 1: Figure S1) suggests that pressure does not vary with temperature above 1873 K. This is also evidenced by the recovery of bridgmanite from 2620 K (OS3099) since the pressure drop of less than 2 GPa is needed to be within the stability field of bridgmanite at that temperature (Gasparik 1996; Fei et al. 2004). A LaCrO₃ heater was chosen for temperature generation. The temperature was monitored with a W₉₇Re₃-W₇₅Re₂₅ thermocouple. No correction was applied for the pressure effect on the electromotive force of the thermocouple. The junction of the thermocouple was positioned at the center of the heater and between two Re capsules. The cell assembly used in this study is identical to that of Zhang et al. (1993) with respect to the octahedron dimension (edge length, 10 mm), the heater material (LaCrO₃), and the capsule material (Re). However, we used capsules with much smaller dimensions in both length and diameter and adopted a longer sleeve heater with a much smaller diameter to reduce the thermal gradient. Zhang et al. (1993) reported a temperature drop of ~ 50 K at a distance of ~ 0.5 mm from the hot spot axially. In this study, the center of the sample (~ 0.2 mm thick) is ~ 0.5 mm away axially from the thermocouple junction. Therefore, the uncertainty of the sample temperature caused by the temperature gradient is much less than 50 K. Indeed, we confirmed drastic changes of the melting phase relations within 100 K in a series of experiments. In addition, the obtained electron microprobe images of the recovered samples suggest that the thermal gradient in the axial direction was negligible and it was not the dominant factor of temperature uncertainty (i.e., no systematic error). Therefore, small, non-systematic changes in the geometry and temperature profile caused by the differences in the geometry of components (heater shape and size, the degree of cell deformation during heating, and so on) in each run lead to uncertainties in temperature. We adopted the variations in the heating power-temperature relationships (Additional file 1: Figure S2) as temperature uncertainties, although they represent the variations not only in the thermocouple-sample geometry and temperature profile but also in other factors not related to the temperature uncertainty but to the heating efficiency. Therefore, uncertainties would be overestimated. However, this is the best way to express the uncertainties in experimental temperatures (Table 2). The high-pressure samples were quenched by cutting off the power supply at the target temperatures after heating for 5–20 min. The experimental conditions and results are summarized in Table 2.

Table 2 Experimental summary

Phase identification and chemical analysis

The recovered samples were embedded in epoxy resin and finely polished for subsequent analyses. Phase identification was conducted with a microfocus X-ray diffraction (XRD) apparatus (Rigaku MicroMax-007HF) using Cu Kα radiation. The sample surface was coated with carbon and analyzed using a field emission-type scanning electron microscope (FE-SEM; JEOL JSM7000F). Chemical compositions were measured using an energy-dispersive X-ray spectrometer with a silicon-drift detector (EDS; Oxford Instruments X-Max^N) attached to the FE-SEM at 15 kV and 1 nA with collection times of ~ 30 s. Mg₂SiO₄ forsterite was used as the compositional standard for Mg and Si, while CaSiO₃ wollastonite was used for Ca. The grains of bridgmanite, CaSiO₃-rich perovskite were analyzed in a focused mode with a beam size of less than 1 μm. The quenched melt was analyzed in a scanning mode larger than the spatial scale of heterogeneity.

Thermodynamic modeling

Among many available thermodynamic approaches that can describe the melting behavior in a multi-component system, we followed that of Liebske and Frost (2012) to maintain consistency with the results for the MgO–MgSiO₃ system, which has been studied extensively at high pressures, including lower-mantle pressures. Liebske and Frost (2012) used the chemical potentials given by Lithgow-Bertelloni and Stixrude (2005) and Stixrude and Lithgow-Bertelloni (2011) for the solid phases and by de Koker and Stixrude (2009) for the liquid phases; these papers constructed an internally consistent thermodynamic model, although the equations, parameterization, and parameter set differ slightly from those of Liebske and Frost (2012). For solids, we used the thermodynamic descriptions and parameter sets of Lithgow-Bertelloni and Stixrude (2005) and Stixrude and Lithgow-Bertelloni (2011) based on usability and consistency with other components. For liquids, we followed Liebske and Frost (2012) to maintain consistency with the MgO–MgSiO₃ system. We describe all of these equations below for clarification.

Chemical equilibrium is attained when the chemical potentials (μ) of the components become equal between the liquid and solid phases:

$$ {\mu}_Y^{\mathrm{solid}}={\mu}_Y^{\mathrm{liquid}}, $$

(1)

where ${\mu}_Y^{\mathrm{phase}}$ represents the chemical potential of component Y, i.e., MgSiO₃ or CaSiO₃, in the assigned phase. The chemical potentials of solids and liquid are expressed as follows:

$$ {\displaystyle \begin{array}{l}{\mu}_Y^{\mathrm{phase}}={\mu}_Y^{o,\mathrm{phase}}+ RTln\left({a}_Y^{\mathrm{phase}}\right),\\ {}{\mu}_Y^{\mathrm{phase}}={\mu}_Y^{o,\mathrm{phase}}+ RTln\left({X}_Y^{\mathrm{phase}}\right)+ RTln\left({\gamma}_Y^{\mathrm{phase}}\right),\end{array}} $$

(2)

where ${\mu}_Y^o$, a _Y , X _Y , and γ _Y are the chemical potentials of the endmember, activity, molar concentration, and activity coefficient of component Y, respectively. The activity coefficients of solid and liquid are described by the asymmetric regular solution expressions

$$ RTln\left({\gamma}_{{\mathrm{MgSiO}}_3}^{\mathrm{phase}}\right)=\left[{A}_o-{A}_1\left(1-4{X}_{{\mathrm{MgSiO}}_3}^{\mathrm{phase}}\right)\right]{\left(1-{X}_{{\mathrm{MgSiO}}_3}^{\mathrm{phase}}\right)}^2, $$

(3)

$$ RTln\left({\gamma}_{{\mathrm{CaSiO}}_3}^{\mathrm{phase}}\right)=\left[{A}_o+{A}_1\left(1-4{X}_{{\mathrm{CaSiO}}_3}^{\mathrm{phase}}\right)\right]{\left(1-{X}_{{\mathrm{CaSiO}}_3}^{\mathrm{phase}}\right)}^2, $$

(4)

where A ₀ and A ₁ are expressed as follows:

$$ {A}_0=\left({W}_{{\mathrm{CaSiO}}_3\hbox{-} {\mathrm{MgSiO}}_3}^{\mathrm{phase}}+{W}_{{\mathrm{MgSiO}}_3\hbox{-} {\mathrm{CaSiO}}_3}^{\mathrm{phase}}\right)/2, $$

(5)

$$ {A}_1=\left({W}_{{\mathrm{CaSiO}}_3\hbox{-} {\mathrm{MgSiO}}_3}^{\mathrm{phase}}-{W}_{{\mathrm{MgSiO}}_3\hbox{-} {\mathrm{CaSiO}}_3}^{\mathrm{phase}}\right)/2, $$

(6)

where W _Y − Z is an interaction parameter between components Y and Z.

Once the chemical potentials of the liquid and solid endmembers are evaluated, we can determine the interaction parameter W by fitting the experimental data to the above equations.

The Gibbs free energy (G) of solid endmembers were determined from the Helmholtz free energy (F) as given by Lithgow-Bertelloni and Stixrude (2005) and Stixrude and Lithgow-Bertelloni (2011):

$$ G\left(P,T\right)=F\left(V,T\right)+ PV, $$

(7)

$$ F\left(V,T\right)={F}_0\left({V}_0,{T}_0\right)+{F}_{cmp}\left(V,{T}_0\right)+\left[{F}_{th}\left(V,T\right)-{F}_{th}\left(V,{T}_0\right)\right], $$

(8)

$$ P\left(V,T\right)={P}_{cmp}\left(V,{T}_0\right)+{P}_{th}\left(V,T\right), $$

(9)

where F is composed of a standard-state term (F ₀ ), a compression term (F _cmp ), and a thermal term (F _th ). Pressure (P) is composed of a compression term (P _cmp ) and a thermal term (P _th ). The compression term is expressed by the following third-order finite-strain (f) model:

$$ {F}_{cmp}\left(V,{T}_0\right)=\frac{9}{2}{V}_0{K}_{T0}{f}^2+\frac{9}{2}{V}_0{K}_{T0}\left({{K_T}_0}^{\prime }-4\right){f}^3 $$

(10)

$$ {P}_{cmp}\left(V,{T}_0\right)={\left(1+2f\right)}^{5/2}\left[3{K}_{T0}f+\frac{9}{2}{K}_{T0}\left({{K_T}_0}^{\prime }-4\right){f}^2\right], $$

(11)

$$ f=\frac{1}{2}\left[{\left(\frac{V_0}{V}\right)}^{2/3}-1\right], $$

(12)

where V ₀, K _T0, and K _T0 ^′ are the volume at a reference state, the bulk modulus, and the first derivative of the bulk modulus, respectively. The thermal term is derived from the Debye model:

$$ {F}_{th}\left(V,T\right)=9 nRTD\left(\frac{\theta }{T}\right), $$

(13)

$$ D(x)={x}^{-3}{\int}_0^x\ln \left(1-\exp \left(-t\right)\right){t}^2 dt, $$

(14)

$$ {P}_{th}\left(V,T\right)=\frac{\gamma }{V}\left({E}_{th}\left(V,T\right)-{E}_{th}\left(V,{T}_0\right)\right), $$

(15)

$$ {E}_{th}\left(V,T\right)=9 nRT{\left(\frac{\theta }{T}\right)}^{-3}{\int}_0^{\frac{\theta }{T}}\frac{t^3}{e^t-1}\kern0.5em dt, $$

(16)

$$ \theta ={\theta}_0{\left[1+6{\gamma}_0f+\frac{1}{2}\left(-12{\gamma}_0+36{\gamma_0}^2-18{q}_0{\gamma}_0\right){f}^2\right]}^{1/2}, $$

(17)

$$ \gamma ={\gamma}_0\frac{\left(1+2f\right)}{\left(1+6{\gamma}_0f\right)}, $$

(18)

where the parameters F(V ₀, T ₀), V ₀, K _T0, K _T0 ^′, θ ₀, γ ₀, and q ₀ were given by Stixrude and Lithgow-Bertelloni (2011).

For liquids, we followed Liebske and Frost (2012) to maintain consistency. The expressions of the solid compression terms F _cmp and P _cmp were also applied to liquids, F _cmp and P _cmp . The thermal part is expressed as follows:

$$ {F}_{th}\left(V,T\right)=-{S}_0\left(T-{T}_0\right)-{C}_v\left[T\ln \left(\frac{T}{T_0}\right)-\left(T-{T}_0\right)\right]-{C}_v\left(T-{T}_0\right)\left[\left({\gamma}_0-{\gamma}^{\prime}\right)\ln \left(\frac{V}{V_0}\right)+\left(\gamma -{\gamma}_0\right)\right], $$

(19)

$$ {P}_{th}\left(V,T\right)=\frac{\gamma }{V}{C}_v\left(T-{T}_0\right), $$

(20)

$$ \gamma ={\gamma}_0+{\gamma}^{\prime}\left(\frac{V}{V_0}-1\right), $$

(21)

where the parameters of MgSiO₃ liquid are from Liebske and Frost (2012), except for F ₀. For the liquid properties of the CaSiO₃ endmember, we used the result of ab initio calculations by Bajgain et al. (2015) by fitting their dataset to the above equations. The F ₀ and S ₀ of the CaSiO₃ liquid were determined by fitting the melting temperatures measured by Zerr et al. (1997), which are consistent with the results of molecular dynamics simulations by Liu et al. (2010) (Fig. 1). The thermodynamic parameters used in this study are summarized in Tables 3 and 4.

Fig. 1

Melting temperatures of CaSiO₃. The black line represents the melting temperatures calculated in this study, the crosses represent the experimental data by Gasparik et al. (1994), the blue circles by Shen and Lazor (1995), the orange squares by Zerr et al. (1997), and the dark gray dashed line represents the molecular dynamics simulation by Liu et al. (2010). The gray region is the stability region of CaSiO₃ perovskite at the melting temperatures (Gasparik et al. 1994)

Table 3 Thermodynamic properties of liquids

Table 4 Thermodynamic properties of solids

The previous study fitted F ₀ for standard melting points: MgO, 3070 K at 1 bar; MgSiO₃, 2900 K at 25 GPa. We found that the F ₀ fitted using the equations defined in the aforementioned methods is slightly different from those reported by previous studies. Nevertheless, we confirmed that we could reproduce the melting temperatures of MgO and MgSiO₃ and the MgO–MgSiO₃ melting phase relations using the parameters obtained in this study (Fig. 2, Additional file 1: Figures S3 and S4).

Fig. 2

Melting temperatures of MgSiO₃. The black line represents the melting temperatures calculated in this study, the crosses by Di Paola and Brodholt (2016), and the dashed gray line by de Koker and Stixrude (2009). The melting temperatures determined by high-pressure experiments are also plotted: orange squares, Zerr and Boehler (1993); blue circles, Shen and Lazor (1995)

Results and discussion

X-ray diffraction and phase identification

Additional file 1: Figure S5 shows the XRD patterns of the run products. Intense peaks from Re were unavoidably observed in all the patterns due to the small sample sizes in this study. Peaks from bridgmanite were evidently observed in OS3097 and OS3099, indicating that the experiments were performed outside of the garnet stability region, which is consistent with a previous study that reported the phase transformation of a MgSiO₃-rich composition from garnet to bridgmanite at 22.4 GPa (Gasparik 1996). Some weak peaks, presumably indexed to bridgmanite, were observed in OS3095 and OS3150 (e.g., Additional file 1: Figure S5e). The bridgmanite is interpreted as quench crystals from melt in the corresponding SEM images as shown later. CaSiO₃-rich perovskite was not identified in all the samples since it should be amorphized at the ambient pressure.

Electron microscopic observation and compositional analysis

Figures 3, 4, 5, and 6 show the backscattered electron images of the recovered samples. At 2600 K, the subsolidus phase assemblage of bridgmanite plus CaSiO₃-rich perovskite was obtained in both starting compositions (Fig. 3). Bridgmanite contains 5–6 mol% CaSiO₃ component, which is higher than that reported by Irifune et al. (2000) at 1773–2173 K, suggesting a positive temperature effect on solubility. We did not determine the solubility of MgSiO₃ component in CaSiO₃-rich perovskite because of its small grain size. At 2620 K, the sample OS3099-Mg90 had a quenched melt with quench textures surrounded by bridgmanite, which is interpreted as the liquidus phase, at the side part of the capsule (Fig. 4). At the central part, the subsolidus phase assemblage of bridgmanite plus CaSiO₃-rich perovskite was found. This observation suggests that the axial thermal gradient in the heater has been effectively reduced so that the radial thermal gradient mainly influenced the phase distribution in this sample. In the sample OS3099-Mg75, we observed a segregation of CaSiO₃-rich perovskite at the hot side part of the capsule (Fig. 4). This CaSiO₃-perovskite is interpreted as the liquidus phase, although we failed to find the quenched melt. At 2700 K, the samples were extensively molten, as clearly evidenced by the textures shown in Fig. 5. Both quenched melts exhibited textures similar to that of metal casting: fine grains attached to the mold wall; columnar grains arranged closely in parallel with the direction of cooling heat flow; and granular grains accumulated at the center. The sample OS3095-Mg90 had a bulk composition identical to that of the starting material, supporting a complete melting. In the sample OS3095-Mg75, CaSiO₃-rich perovskite, interpreted as the liquidus phase, was observed at the coldest part of the capsule. At 3000 K, the sample OS3150-Mg50 yielded a phase assemblage of CaSiO₃-rich perovskite plus melt (Fig. 6), identical to the sample OS3095-Mg75 (Fig. 5). The results of the chemical analyses are summarized in Table 5.

Fig. 3

Backscattered electron images for the samples recovered from high pressure and high temperature (Run No. OS3097). The subsolidus phase assemblage of bridgmanite (Bdg) plus CaSiO₃-rich perovskite (CaPv) was observed in both Mg90 and Mg75. T.C., thermocouple

Fig. 4

Backscattered electron images for the samples recovered from high pressure and high temperature (Run No. OS3099). In Mg90, a quenched melt was found at the hottest region surrounded by bridgmanite (Bdg), which is interpreted as the liquidus phase. In contrast, CaSiO₃-rich perovskite (CaPv) was found to be segregated in Mg75. We failed to find the melt. The subsolidus phase assemblage was observed at the central (cold) part of both samples

Fig. 5

Backscattered electron images for the samples recovered from high pressure and high temperature (Run No. OS3095). Quenched melts were found at most of the sample areas. In Mg75, the liquidus phase, CaSiO₃-rich perovskite (CaPv), was found at the coldest part

Fig. 6

Backscattered electron images for the sample recovered from high pressure and high temperature (Run No. OS3150-Mg50). A quenched melt was found at the hottest region surrounded by CaSiO₃-rich perovskite (CaPv), which is interpreted as the liquidus phase

Table 5 Chemical compositions of phases in run products

Melting phase relations in the MgSiO₃–CaSiO₃ system

The phase relations are summarized in Fig. 7. The solubility of MgSiO₃ component into CaSiO₃-rich perovskite is much higher than that predicted by theoretical calculations (Jung and Schmidt 2011). This is because Jung and Schmidt (2011) considered a tetragonal structure (I4/mcm) for CaSiO₃ perovskite, in addition to their use of oversimplified assumptions, as commented in their paper. Indeed, Komabayashi et al. (2007) showed that CaSiO₃ perovskite undergoes a phase transition from tetragonal to cubic above ~ 500 K at ~ 24 GPa. We determined the interaction parameter W ^solid by fitting our experimental data together with the dataset of Irifune et al. (2000) to the following equations:

$$ {W}_{{\mathrm{MgSiO}}_3\hbox{-} {\mathrm{CaSiO}}_3}^{\mathrm{solid}}\ \left[\mathrm{kJ}/\mathrm{mol}\right]=108(11)-0.024(5)\times \mathrm{T}\ \left[\mathrm{K}\right], $$

$$ {W}_{{\mathrm{CaSiO}}_3\hbox{-} {\mathrm{MgSiO}}_3}^{\mathrm{solid}}\ \left[\mathrm{kJ}/\mathrm{mol}\right]=52(8)+0.010(4)\times \mathrm{T}\ \left[\mathrm{K}\right]. $$

Fig. 7

Melting phase diagram in the MgSiO₃–CaSiO₃ system at 24 GPa. The filled red circle represents the composition of liquid above liquidus, the filled and open orange squares represent the liquid and coexisting bridgmanite (Bdg), the filled and open blue squares represent the liquid and coexisting CaSiO₃-rich perovskite (CaPv), the open diamonds represent the subsolidus phases, the cross symbols represent the subsolidus phases at 25 and 27 GPa reported in Irifune et al. (2000), and the dashed gray line represents the subsolidus phases at 25 GPa reported in Jung and Schmidt (2011)

The eutectic point was found to be located at 81–86 mol% MgSiO₃ and 2600–2620 K. The interaction parameter W ^liquid was determined by fitting our experimental data to the following equations:

$$ {W}_{{\mathrm{MgSiO}}_3\hbox{-} {\mathrm{CaSiO}}_3}^{\mathrm{liquid}}\ \left[\mathrm{kJ}/\mathrm{mol}\right]=1839(741)-0.59(26)\times \mathrm{T}\ \left[\mathrm{K}\right], $$

$$ {W}_{{\mathrm{CaSiO}}_3\hbox{-} {\mathrm{MgSiO}}_3}^{\mathrm{liquid}}\ \left[\mathrm{kJ}/\mathrm{mol}\right]=-1170(167)+0.39(6)\times \mathrm{T}\ \left[\mathrm{K}\right]. $$

The parameters were estimated using a linear least-squares method weighted by the compositional uncertainty. Therefore, the standard error of the fitting parameter, shown in parenthesis, does not include temperature uncertainty since the dataset from the literature did not describe how the temperature uncertainty was determined. The experimental data suggest that the interaction parameters approach to zero at high temperatures, suggesting an ideal behavior. These empirical (linear) fitting equations should not be extrapolated out from the experimental range. Figure 7 shows the resulting phase diagram from the thermodynamic calculations together with the experimental data. In prohibiting the extrapolation, the uncertainty of the thermodynamic model is described by the deviations from the experimental data (Figs. 1 and 7, Additional file 1: Figure S4).

Conclusions

Melting phase relations in the system MgSiO₃–CaSiO₃ were determined experimentally at 24 GPa. A thermodynamic model was constructed to reproduce the experimental data. The experimental dataset and the thermodynamic parameters derived in this study serve as the basis for understanding the complicated melting behaviors of the Earth’s lower mantle.

The advantage of establishing a thermodynamic model is that the degree of melting can be calculated at a certain P-T in a multi-component system. The seismic observations suggested a degree of melting of < 5–30% in ultra-low velocity zones at the base of the lower mantle (Williams and Garnero 1996). The estimation of the degree of melting depends on the wetting property and the elastic properties of the melt; conversely, these properties depend on the compositions of the melt; thus, they depend on the degree of melting. The degree of melting is also an important parameter in understanding the chemical differentiation during magma ocean crystallization. The melt-solid separation by rheological transition occurs at a crystal fraction of approximately 60% (e.g., Abe 1995). Therefore, it is of great importance to understand the phase relations together with the knowledge of the degree of melting at relevant pressures and temperatures. However, there are some difficulties in obtaining the degree of melting by high-pressure melting experiments under thermal gradients because the observed degree of melting arises from the thermal gradients, not from the phase equilibrium.

Figure 8 shows the eutectic temperatures of the MgO–MgSiO₃ system calculated using interaction parameters constrained from the chemical analyses of the recovered diamond anvil cell (DAC) samples (Ohnishi et al. 2017) and those measured directly by pyrometry in DAC experiments (Baron et al. 2017). As shown in Fig. 8, there is a deviation between these two studies with systematically lower temperatures in Baron et al. (2017). The deviation may have been caused by the inaccuracy of the temperature measurements by the pyrometer, the uncertain wavelength-dependent emissivity of the sample, and the effect of integrated radiation along the thermal gradients in an axial direction. The latter would be serious when transparent materials are used as a sample and pressure medium, resulting in an underestimation of the temperature. Indeed, the melting temperatures of endmembers show large discrepancies among DAC studies even if similar melting criteria were used. On the other hand, chemical analyses of the liquid and solid phases in contact (Ohnishi et al. 2017) present a much more reliable dataset because only an assumption of the local equilibrium is necessary without the information on temperature (it is noted that chemical equilibrium is assumed implicitly when melting temperatures are measured directly). Therefore, constructing the thermodynamic model using the Gibbs free energies of the endmembers by ab initio calculations and phase relations constrained by multi-anvil and DAC experiments combined with chemical analysis should be a promising approach to understand the mineralogy of deep mantle melting and comparisons with observations; however, this approach requires a step-by-step accumulation of relevant phase diagrams.

Fig. 8

Eutectic temperatures of the MgO–MgSiO₃ system calculated using interaction parameters constrained from chemical analyses of the recovered DAC samples (Ohnishi et al. 2017) and those measured directly by pyrometry in DAC experiments (Baron et al. 2017)