Equations of State of Ca-Silicates and Phase Diagram of the CaSiO3 System under Upper Mantle Conditions

The equations of state of different phases in the CaSiO3 system (wollastonite, pseudowollastonite, breyite (walstromite), larnite (Ca2SiO4), titanite-structured CaSi2O5 and CaSiO3-perovskite) are constructed using a thermodynamic model based on the Helmholtz free energy. We used known experimental measurements of heat capacity, enthalpy, and thermal expansion at zero pressure and high temperatures, and volume measurements at different pressures and temperatures for calculation of parameters of equations of state of studied Ca-silicates. The used thermodynamic model has allowed us to calculate a full set of thermodynamic properties (entropy, heat capacity, bulk moduli, thermal expansion, Gibbs energy, etc.) of Ca-silicates in a wide range of pressures and temperatures. The phase diagram of the CaSiO3 system is constructed at pressures up to 20 GPa and temperatures up to 2000 K and clarifies the phase boundaries of Ca-silicates under upper mantle conditions. The calculated wollastonite–breyite equilibrium line corresponds to equation P(GPa) = −4.7 × T(K) + 3.14. The calculated density and adiabatic bulk modulus of CaSiO3-perovskite is compared with the PREM model. The calcium content in the perovskite composition will increase the density of mineral and it good agree with the density according to the PREM model at the lower mantle region.


Introduction
The thermodynamic description and phase equilibria in the CaSiO 3 system are important for modern mineralogical research because Ca-silicates are common components of the Earth's crust and mantle. The low-pressure CaSiO 3 polymorphs (wollastonite and pseudowollastonite) are rock-forming minerals in the Earth's crust and constitute cement substances [1,2], while the high-pressure Ca-silicate phases (breyite (walstromite), larnite, titanite-structured CaSi 2 O 5 and CaSiO 3 -perovskite) are stable in the mantle P-T conditions and detected as solid-phase inclusions in natural diamonds [3][4][5][6][7]. The presence of CaSiO 3 phase in diamonds indicates the possibility of transporting the substance from the lower mantle to the surface in an unchanged form and provides insights into the nature of diamond formation. Thus, the present study will expand the possibility for thermodynamic modeling of physicochemical processes in the Earth's deep mantle.
It is well known that calcium metasilicate (CaSiO 3 ) has a number of structural modifications at relevant pressures and temperatures. Wollastonite structure is stable under ambient conditions and transforms to pseudowollastonite phase at high temperatures above~1400 K [8]. The transition of wollastonite to walstromite-structured CaSiO 3 is detected at 3 GPa and 1173 K [9]. The name breyite for this phase is approved by IMA in 2018 [10], therefore it is used in the text below. The wollastonite-breyite equilibrium was studied by different authors [1,11,12]. The subsequent decomposition of breyite to association of larnite (Ca 2 SiO 4 ) and titanite-structured CaSi 2 O 5 was determined at higher pressures by Kanazaki's study [13]. The CaSiO 3 -perovskite is formed at pressure~14 GPa

Thermodynamic Model
The equations of state of wollastonite, pseudowollastonite, breyite, larnite (Ca 2 SiO 4 ), titanite-structured CaSi 2 O 5 and CaSiO 3 -perovskite are constructed using the thermodynamic model based on the Helmholtz free energy. According to the proposed thermodynamic model, the Helmholtz free energy can be represented in general form as the sum [27]: where U 0 is the reference energy, F T0 is the potential part of the Helmholtz free energy at the reference isotherm T 0 = 298.15 K, which depends only on volume; F th is the thermal part of the Helmholtz free energy, which depends on volume and temperature; F anh is an additional contribution to the Helmholtz free energy, which connects with intrinsic anharmonicity, as a function of temperature and volume. The potential part of the Helmholtz free energy from Equation (1) is determined by the integration of the equation of pressure at the reference isotherm. Many different equations are known to express the equation of pressure at the reference isotherm, each of which has its advantages and limitations (review in [17]). We use the Kunc equation [28], which is flexible in calculations because it contains parameter k: where X = (V/V 0 ) 1/3 , η = 1.5K − k + 0.5, K 0 is the isothermal bulk modulus under ambient conditions (P 0 = 1 bar = 10 −4 GPa, T 0 = 298.15 K), K = ∂K 0 /∂P 0 and k is an additional parameter. When k = 2, the Kunc equation corresponds to the Vinet equation [29], which is often used in calculation of equations of state of metals in solid state physics. The values of k = 5 and k > 5 transform the Kunc equation into one of the forms of the Holzapfel equation (HO 2 ) [30] and into the Birch-Murnaghan equation [31], respectively. We use k = 5 for all Ca-silicate phases in the present study.
Differentiating Equation (2) with respect to volume at constant temperature, we determine the isothermal bulk modulus at the reference isotherm and calculate of its derivation by pressure: The thermal part of the Helmholtz free energy from Equation (1) can be calculated by Debye, Einstein, Bose-Einstein models, or their combinations [32,33]. A more simple approach is to use the Einstein model with two characteristic temperatures, which makes it possible to good approximate of the heat capacity in the temperature range from~100 K to the melting point of the substance: where m i is proportionality factor, which is calculated from the total number of atoms in the compound (i = 1, 2), m 1 + m 2 = 3n, n is the number of atoms in a chemical formula of the compound, Θ i is the Einstein characteristic temperature, which depends only on volume, and R is the gas constant (R = 8.31451 Jmol −1 K −1 ). The volume dependence of characteristic temperature in Equation (5) is determined by Al'tshuler equation [34]: where Θ 0i is the characteristic temperature under ambient conditions (i = 1, 2), x = V/V 0 , γ 0 is the Grüneisen parameter at ambient conditions, γ ∞ is the Grüneisen parameter at infinite compression, when x → 0, and β is an additional fitting parameter. The advantage of Equation (6) is that it can be explicitly differentiated in the simple form to determine the Grüneisen parameter: where Θ is the characteristic temperature in general case. We assume that the Grüneisen parameter will be the same for both characteristic temperatures (Θ 1 and Θ 2 ) in Equation (6). In addition, if γ ∞ = 0, then Equation (7) transforms to the classical form γ = γ 0 x β .
Differentiating Equation (5) with respect to temperature at constant volume, we determine entropy and then calculate internal energy: Differentiating Equation (5) with respect to volume at constant temperature, we obtain the thermal part of pressure: Differentiating Equation (9) with respect to temperature at constant volume and Equation (10) with respect to volume at constant temperature, we obtain the isochoric heat capacity and thermal part of isothermal bulk modulus: where q is additional parameter, which is calculated from Equation (7) according to [34]. Differentiating Equation (10) with respect to temperature at constant volume, we determine the slope: The additional contribution of the intrinsic anharmonicity to the Helmholtz free energy in Equation (1) can be expressed by classical equation from [27]: The contribution of the intrinsic anharmonicity to entropy, internal energy, thermal part of pressure, isochoric heat capacity, thermal part of isothermal bulk modulus and slope can be obtained by analogy with Equations (8)- (13), respectively: where a = a 0 x m , a 0 is the intrinsic anharmonicity parameter and m is the anharmonic analogue of the Grüneisen parameter. The Equation (10) for the thermal part of pressure and equation for anharmonicity part of pressure (P anh in Equation (15)), as can be seen, are very similar. The Grüneisen parameter in Equation (10) is the coefficient of proportionality between pressure and internal energy, just like the parameter m in Equation (15). Therefore, we use this name for the parameter m. The parameters a 0 and m are fitting parameters in the general cause, but we use the fixed value for m (m = 1) in the present study.
The full equations of pressure and isothermal bulk modulus are calculated as the sum of potential, thermal, and anharmonicity components: P = P T0 (V) + P th (V,T) + P anh (V,T) and K T = K T0 (V) + K Tth (V,T) + K Tanh (V,T). Further, it is easy to calculate the coefficient of thermal expansion α = (∂P th /∂T) V / K T , the heat capacity at constant pressure C P = C V + α 2 TVK T , the adiabatic bulk modulus K S = K T + VT(αK T ) 2 /C V , and the thermodynamic Grüneisen parameter γ th = αVK T /C V = αVK S /C P . The enthalpy and Gibbs energy are determined from the linear relations H = E th + PV − P 0 V 0 and G = F(V,T) + PV − P 0 V 0 , respectively. The parameter U 0 in Equation (1) define the Gibbs energy under ambient conditions, which also can be found from the simple relation: 298 and S 298 are enthalpy and entropy at ambient conditions, respectively.
Thus, Equations (1)-(15) contain a group of fixed parameters (V 0 , k, m i ) and a group of fitting parameters (K 0 , K', Θ 0i , γ 0 , γ ∞ , β, a 0 , m) derived by least squares method in Excel worksheet. Using the proposed approach, it is easy to calculate a full set of thermodynamic functions of Ca-silicates that depend on temperature, pressure, and volume.

Results
The equations of state of studied Ca-silicates were constructed using thermodynamic model described above. Table 1 shows parameters of equations of state for wollastonite (Wol), pseudowollastonite (PsWol), breyite (Brt), larnite (β-Lrn), titanite-structured CaSi 2 O 5 (Ca-Tit) and CaSiO 3 -perovskite (Ca-Pv) obtained by simultaneous optimization of known thermochemical measurements of the heat capacity, enthalpy, thermal expansion, and numerous P-V-T data including modern X-ray diffraction measurements for the Ca-silicate minerals. The optimization procedure is described in detail in our previous works [16,20,33]. In the present study, we also offer the open working MS Excel spreadsheets for each of the studied Ca-silicates (Files S1-S6 in Supplementary Material). Detailed instructions and descriptions of worksheets are provided in [17,19]. Therefore, below we will focus on the features of calculating of equation of state for each of the phases.

Results
The equations of state of studied Ca-silicates were constructed using thermodynamic model described above. Table 1 shows parameters of equations of state for wollastonite (Wol), pseudowollastonite (PsWol), breyite (Brt), larnite (β-Lrn), titanite-structured CaSi2O5 (Ca-Tit) and CaSiO3-perovskite (Ca-Pv) obtained by simultaneous optimization of known thermochemical measurements of the heat capacity, enthalpy, thermal expansion, and numerous P-V-T data including modern X-ray diffraction measurements for the Ca-silicate minerals. The optimization procedure is described in detail in our previous works [16,20,33]. In the present study, we also offer the open working MS Excel spreadsheets for each of the studied Ca-silicates (Files S1-S6 in Supplementary Material). Detailed instructions and descriptions of worksheets are provided in [17,19]. Therefore, below we will focus on the features of calculating of equation of state for each of the phases. Wollastonite (Wol). A review of the heat capacity measurements of wollastonite and other CaSiO3 phases can be found in [35]. The most reliable measurements we found of molar volume of wollastonite as a function of temperature are published in [36]. The only known measurement of the volume of wollastonite as a function of pressure in [37] shows the following values of K0 = 54.7 GPa and ∂K/∂P = 23.5. However, at these values, our equation of state of wollastonite becomes unstable at temperatures above 700 K (this is easily verified by substituting the parameters to the present Excel file for wollastonite).
Wollastonite (Wol). A review of the heat capacity measurements of wollastonite and other CaSiO 3 phases can be found in [35]. The most reliable measurements we found of molar volume of wollastonite as a function of temperature are published in [36]. The only known measurement of the volume of wollastonite as a function of pressure in [37] shows the following values of K 0 = 54.7 GPa and ∂K/∂P = 23.5. However, at these values, our equation of state of wollastonite becomes unstable at temperatures above 700 K (this is easily verified by substituting the parameters to the present Excel file for wollastonite). Therefore, we use measurements of volume, depending on pressure for pseudowollastonite from [38]. The simultaneous optimization of known experimental data for wollastonite at γ 0 = 1, β = 1 and γ ∞ = 0 determine the obtained parameters of its equation of state (Table 1). Figure 1a shows a comparison of the calculated isobaric (C P ) and isochoric (C V ) heat capacity of wollastonite with measurements from [39][40][41][42]; Figure 1c shows calculated molar volume at reference pressure (P 0 = 1 bar = 10 −4 GPa) with Swamy's data [36]; and Figure 1e shows calculated compressibility as a function of pressure with measurements from [37]. It can be seen that calculated data are in good agreement with different reference and experimental data.
The calculated thermodynamic functions of wollastonite at different temperatures and pressures (0.0001 and 3 GPa) are presented in Table 2. The last column shows the Gibbs energy at given P-T parameters. The parameter U 0 is calculated from the enthalpy at ambient conditions ∆H f,298 = −1633.750 kJmol −1 from [43] and entropy S 298 = 81.358 Jmol −1 K −1 at ambient conditions from Table 2, then U 0 = ∆G f,298 = −1,633,750 − 81.358 × 298.15 = −1,658,007 Jmol −1 . The same scheme was used for calculation of the Gibbs energy for Mgsilicates in the Mg 2 SiO 4 system [16]. We have shown calculation thermodynamic functions for wollastonite in the Table 2. Further tabulated values of calculated thermodynamic functions for other Ca-silicates are presented in the Supplementary Material (Tables S1-S5). Table 2. Thermodynamic functions of wollastonite at different pressures and temperatures.  Figure 1. The calculated heat capacity (a,b), molar volume at reference pressure (c,d) and compressibility as a function of pressure (e,f) of wollastonite (Wol) and pseudowollastonite (PsWol) in comparison with different reference and experimental data from [36,37,[39][40][41][42] for Wol and from [38,42,[44][45][46][47] for PsWol. Table 2. Thermodynamic functions of wollastonite at different pressures and temperatures.

Pseudowollastonite (PsWol).
As follows from the review in [2], wollastonite at ambient conditions has a triclinic structure (space group P1). The crystal structure of a synthetic two-layer polytype of pseudowollastonite was studied using single-crystal X-ray diffraction data at pressures up to 10 GPa by Yang and Prewitt [38]. It is determined that such a polytype has monoclinic structure (space group C2/c) and molar volume at ambient condition (V 0 = 796.9 A 3 = 39.99 cm 3 mol −1 ) coincides with the volume of wollastonite ( Table 2). The molar volume of pseudowollastonite was studied at temperature range 307-1793 K by Richet et al. [44]. There are two-, four-, and six-layer polytypes of CaSiO 3 -pseudowollastonite in this temperature range. The parameters of K 0 and K' for our calculation were extracted from [38]; parameters of Θ 01 and Θ 02 are calculated from the heat capacity and relative enthalpy measurements from [42,[45][46][47]. The parameter U 0 is calculated from the condition of equality of the Gibbs energy of wollastonite and pseudowollastonite at T = 1398 K and P = 0.0001 GPa.
The calculated thermodynamic functions of pseudowollastonite are compared with different reference and experimental data [38,44] by analogy with wollastonite on Figure 1b,d,f. The tabulated values of calculated thermodynamic functions of pseudowollastonite as a function of temperature at different pressures (0.0001 and 3 GPa) is presented in the Supplementary Material (Table S1).
Breyite (Brt). Regarding of breyite, in the literature, we can find the different names of its structural isomorphs (Ca-walstromite, wollastonite-II, walstromite-like CaSiO 3 (III)). The crystal structures of breyite (or walstromite), CaSiO 3 -walstromite, and wollastonite-II are very similar. This issue is discussed in detail by Joswig et al. [49]. Breyite probably crystallizes in the triclinic structure (space group P1) and it is stable in the pressure range from~3 GPa and above, since the remaining pressure of investigated inclusions of breyite in natural diamonds [3,4] was also estimated to be 3 GPa. The direct P-V-T measurements of breyite are carried out in [5,50]. The parameters of V 0 , K 0 and K' are taken from Anzolini et al. [5]. The parameter U 0 is calculated from the phase transition of wollastonite-breyite. The calculated volume of breyite as a function of temperature and pressure is compared with experimental data on Figure 2a (Table S2).
The calculated thermodynamic functions of pseudowollastonite are compared with different reference and experimental data [38,44] by analogy with wollastonite on Figure  1b,d,f. The tabulated values of calculated thermodynamic functions of pseudowollastonite as a function of temperature at different pressures (0.0001 and 3 GPa) is presented in the Supplementary Material (Table S1).
Breyite (Brt). Regarding of breyite, in the literature, we can find the different names of its structural isomorphs (Ca-walstromite, wollastonite-II, walstromite-like CaSiO3(III)). The crystal structures of breyite (or walstromite), CaSiO3-walstromite, and wollastonite-II are very similar. This issue is discussed in detail by Joswig et al. [49]. Breyite probably crystallizes in the triclinic structure (space group P1) and it is stable in the pressure range from ~3 GPa and above, since the remaining pressure of investigated inclusions of breyite in natural diamonds [3,4] was also estimated to be 3 GPa. The direct P-V-T measurements of breyite are carried out in [5,50]. The parameters of V0, K0 and K' are taken from Anzolini et al. [5]. The parameter U0 is calculated from the phase transition of wollastonite-breyite. The calculated volume of breyite as a function of temperature and pressure is compared with experimental data on Figure 2a (Table S2).  Larnite (β-Lrn). As temperature increases, the composition of Ca 2 SiO 4 can form a series of polymorphs (γ-, β-, α L -, α' H -and α). The review of crystal structures of Ca 2 SiO 4 is presented for example in [51,52]. The β-larnite is formed as metastable monoclinic phase (space group P2 1 /n11) in the stability field of γ-Ca 2 SiO 4 by its cooling. We considered here only the beta modification, which is typical for natural larnite [53]. Larnite (β phase) has a wide stability field at high pressures and temperatures; therefore, this phase is used in the present study. The parameter V 0 is taken from Remy et al. [51], and other parameters (K 0 , K', Θ 0i , γ 0 , γ ∞ and β) are calculated from optimization of experimental measurements of heat capacity and P-V-T data from [51,[54][55][56][57]. The parameter U 0 is calculated from the enthalpy at ambient conditions ∆H f,298 = −2307.04 kJmol −1 from [43] and entropy at ambient conditions S 298 = 124.047 Jmol −1 K −1 from The calculated volume of larnite as a function of pressure and temperature is compared with experimental data on Figure 3a,b.
1 Figure 3. The calculated volume of larnite (β phase) as a function of temperature (a) and pressure (b) is compared with experimental data from [51,56,57]. [3][4][5], but thermochemical data for this phase are not available. Therefore, the present equation of state of titanite-structured CaSi 2 O 5 (space group C2/c) is constructed on the values of parameters of V 0 , K 0 and K' from [58], and other parameters (Θ 0i , γ 0 , γ ∞ and β) are calculated from optimization of experimental data from [2,58] (Table S4).

Titanite-structured CaSi 2 O 5 (Ca-Tit). The titanite-structured CaSi 2 O 5 phase is identified by in situ XRD analyses in natural diamonds from Guinea and Brazil
Perovskite (Ca-Pv). CaSiO 3 -perovskite is considered as the one of the main Cabearing phase in the Earth's lower mantle [59]. Therefore, findings of Ca-perovskite in natural diamond [6] may explain the mechanisms of diamond formation in general. The question of nature of "super-deep" diamond is discussed widely; however, the aim of this study is to reliably calculate the thermodynamic functions of CaSiO 3 -perovskite in a wide range of pressures and temperatures. CaSiO 3 -perovskite is stabilized in the cubic structure (space group Pm3m) at temperatures above 1000 K and pressures above 11-16 GPa, according to [1]. At low temperatures up to 500 K and high pressures up to 156 GPa, the CaSiO 3 -perovskite has a distorted cubic structure, which is identified in [60,61] as tetragonal modification (space group I4/mcm). The direct measurements of heat capacity and bulk modulus as a function of temperature at ambient pressure for cubic Ca-perovskite are not available [62]; therefore, all parameters (K 0 , K', Θ 0i , γ 0 , γ ∞ and β) for proposed equation of state were calculated by authors. However, numerous experimental measurements of molar volume of Ca-perovskite at different pressures and temperatures are known ( Table 3). The proposed equation of state of CaSiO 3 -perovskite is constructed on the modern P-V-T measurement from [60,61,63] and simultaneous sound velocities and density measurements at pressures up to 23 GPa and temperatures up to 1700 K from [64]. The pressure measurements in [63] were recalculated based on the equation of state of Pt from [65]; the pressure from [60] was calculated using a self-consistent pressure scale of Pt from [66]; and pressure from [64] was monitored during the experiment by the unit-cell volume of NaCl [67]. The differences in measurements of volume in Table 3 are significant, so the proposed equation of state of Ca-perovskite here is important. Figure 4a,b shows the deviations of the experimental measured pressure in [60,63] and calculated pressure from the proposed equation of state. It can be seen that the most differences are observed at pressure over 100 GPa. Figure 5a,b shows a difference between calculated pressure and adiabatic bulk modulus and experimental data from [64]. The tabulated values of calculated thermodynamic functions of cubic CaSiO 3 -perovskite as a function of temperature at different pressures (0.0001, 10, 15 and 100 GPa) are presented in the Supplementary Material (Table S5). cation (space group I4/mcm). The direct measurements of heat capacity and bulk modulus as a function of temperature at ambient pressure for cubic Ca-perovskite are not available [62]; therefore, all parameters (K0, K', Θ0i, γ0, γ∞ and β) for proposed equation of state were calculated by authors. However, numerous experimental measurements of molar volume of Ca-perovskite at different pressures and temperatures are known ( Table 3). The proposed equation of state of CaSiO3-perovskite is constructed on the modern P-V-T measurement from [60,61,63] and simultaneous sound velocities and density measurements at pressures up to 23 GPa and temperatures up to 1700 K from [64]. The pressure measurements in [63] were recalculated based on the equation of state of Pt from [65]; the pressure from [60] was calculated using a self-consistent pressure scale of Pt from [66]; and pressure from [64] was monitored during the experiment by the unit-cell volume of NaCl [67]. The differences in measurements of volume in Table 3 are significant, so the proposed equation of state of Ca-perovskite here is important. Figure 4a,b shows the deviations of the experimental measured pressure in [60,63] and calculated pressure from the proposed equation of state. It can be seen that the most differences are observed at pressure over 100 GPa. Figure 5a,b shows a difference between calculated pressure and adiabatic bulk modulus and experimental data from [64]. The tabulated values of calculated thermodynamic functions of cubic CaSiO3-perovskite as a function of temperature at different pressures (0.0001, 10, 15 and 100 GPa) are presented in the Supplementary Material (Table S5).  [63] (a) and in [60] (b) and calculated pressure from the proposed equation of state of Ca-perovskite at different isotherms. The pressure in [63] is recalculated from the equation of state of Pt [65]; pressure in [60] is determined using the equation of state of Pt [66].

Figure 4.
Difference between experimental measured pressure in [63] (a) and in [60] (b) and calculated pressure from the proposed equation of state of Ca-perovskite at different isotherms. The pressure in [63] is recalculated from the equation of state of Pt [65]; pressure in [60] is determined using the equation of state of Pt [66].

Discussion
The proposed equations of state of wollastonite, pseudowollastonite, breyite (walstromite), larnite, titanite-structured CaSi2O5 and CaSiO3-perovskite allow us to calculate the phase diagram of the CaSiO3 system ( Figure 6). The wollastonite-pseudowollastonite equilibrium corresponds to the data from [11,48], which were used for calibration of the Gibbs energy of pseudowollastonite. At a pressure of about 3 GPa and temperature of Figure 5. Difference between experimental measured pressure (a) and adiabatic bulk modulus (b) from [64] and calculated pressure and adiabatic bulk modulus from the proposed equation of state of Ca-perovskite. The pressure in [64] is calculated from the equation of state of NaCl [67].

Discussion
The proposed equations of state of wollastonite, pseudowollastonite, breyite (walstromite), larnite, titanite-structured CaSi 2 O 5 and CaSiO 3 -perovskite allow us to calculate the phase diagram of the CaSiO 3 system (Figure 6). The wollastonite-pseudowollastonite equilibrium corresponds to the data from [11,48], which were used for calibration of the Gibbs energy of pseudowollastonite. At a pressure of about 3 GPa and temperature of about 1100 K wollastonite transforms to breyite (walstromite). The Gibbs energy of breyite is calibrated based on measurements from [11,15]. The calculated wollastonite-breyite equilibrium line coincides with the line P(GPa) = −4.7 × T(K) + 3.14, that was recommended by Essene [11] (Figure 6). At higher pressures, breyite transforms to the two-phase assembly of larnite (Ca 2 SiO 4 ) and CaSi 2 O 5 [1]. Based on this phase transition, the sum of the Gibbs energy of both phases is calculated. The phase boundary of breyite-(larnite + CaSi 2 O 5 ) is calculated from equation 3Brt = β-Lrn + CaSi 2 O 5 [12,57], using equilibrium of the Gibbs energy of breyite, larnite and Ca-titanite. After that, we can estimate the Gibbs energy of Ca-perovskite and calculate the line of phase transition from equation β-Lrn + CaSi 2 O 5 = 3Ca-Pv [12,57]. It should be noted that the position of the calculated equilibrium lines (breyite-(larnite + CaSi 2 O 5 ) and (larnite + CaSi 2 O 5 )-Ca-perovskite) depends entirely on the choice of the primary experimental data, which have a very significant scatter (see Figure 6). We performed a calibration of the Gibbs energy of Ca-perovskite based on the thermochemical data from [12], because we consider these data to be the most reliable. The stability field of high-pressure Ca-silicates were studied by [71,72] and significant deviations were determined. Thus, the obtained phase diagram of the CaSiO 3 system is based on modern P-V-T measurements for Ca-silicates and clarifies the phase boundaries under upper mantle conditions at the depth up to~600 km. The phase equilibria in the CaSiO3 system were studied in [1,12,15,62] and stability field of mineral association (Lrn + CaSi2O5) and Ca-perovskite in [71,72].
There are many first-principles calculations based on density functional theory (DFT) for CaSiO3-perovskite [73][74][75][76], however some of these were made at zero temperature [75,76]. The equation of state of CaSiO3-perovskite based on ab initio molecular dynamic (AIMD) simulations was constructed in [74]. The authors believe that tetragonal structure of Ca-perovskite is stable at temperature range 1000-4000 K and pressure range 15-130 GPa. However, this assumption does not agree well with the melting line calculations in Figure 6. Calculated phase diagram of the CaSiO 3 system under upper mantle P-T conditions. Thick black lines are our calculations, symbols show the different experimental studies. The phase equilibria in the CaSiO 3 system were studied in [1,12,15,62] and stability field of mineral association (Lrn + CaSi 2 O 5 ) and Ca-perovskite in [71,72].
The equation of state of CaSiO 3 -perovskite based on ab initio molecular dynamic (AIMD) simulations was constructed in [74]. The authors believe that tetragonal structure of Caperovskite is stable at temperature range 1000-4000 K and pressure range 15-130 GPa. However, this assumption does not agree well with the melting line calculations in the CaSiO 3 system (see Figure 7). First-principle molecular dynamics calculations to investigate the structure of Ca-perovskite at high temperatures and pressures were made in [77]. The cubic structure of CaSiO 3 -perovskite was found to be stable at temperatures higher 1000 K and at all pressures according to [77]. The phase boundaries in Ca-perovskite based on LDA and GGA approximations were calculated by Stixrude et al. [73]. The location of the phase transition from tetragonal to cubic structure in Ca-perovskite is shown in Figure 7. The gray area indicates the stability field of the tetragonal structure of Ca-perovskite according to [73,77]. The melting line in the CaSiO3 system was studied experimentally at pressure up to 58 GPa in [1,78,79] and based on first-principles calculations in [80,81]. Figure 7 shows the comparison of the experimental data and different calculations. As can be seen, the experimental data from [79] are in good agreement with calculations [80,81] at the temperature range 2750-3500 K and at the pressure range 15-30 GPa. The melting line from [78] is lower than other estimates and difference increases at high pressures. The melting of Ca-SiO3-perovskite is predicted in the range of very high temperatures and pressures up to 6400 K and 300 GPa, respectively, according to the recent study [81]. The melting temperature of CaSiO3-perovskite significantly increases with pressure and is higher than that of MgSiO3-perovskite.
The proposed equation of state of Ca-perovskite is of great interest for deep mineralogy. The thermodynamic parameters of the Earth's lower mantle can be estimated using the equation of state of Ca-perovskite. Table 4 shows the parameters of the Earth's lower mantle according to the PREM model (Preliminary Reference Earth Model) [82], where temperature is calculated from [83]. We calculated the density and adiabatic bulk modulus of CaSiO3-perovskite at these P-T parameters (from Table 4) and compared it with the PREM model and with the density and adiabatic bulk modulus of MgSiO3-perovskite from our previous study [16] (Figure 8). It can be seen that the density of Ca-perovskite is  [1,[78][79][80][81]; different dotted lines represent first-principle calculations of the phase boundaries in Ca-perovskite [73,77]. The gray area indicates the stability field of the tetragonal structure of Ca-perovskite.
The melting line in the CaSiO 3 system was studied experimentally at pressure up to 58 GPa in [1,78,79] and based on first-principles calculations in [80,81]. Figure 7 shows the comparison of the experimental data and different calculations. As can be seen, the experimental data from [79] are in good agreement with calculations [80,81] at the temperature range 2750-3500 K and at the pressure range 15-30 GPa. The melting line from [78] is lower than other estimates and difference increases at high pressures. The melting of CaSiO 3 -perovskite is predicted in the range of very high temperatures and pressures up to 6400 K and 300 GPa, respectively, according to the recent study [81]. The melting temperature of CaSiO 3 -perovskite significantly increases with pressure and is higher than that of MgSiO 3 -perovskite.
The proposed equation of state of Ca-perovskite is of great interest for deep mineralogy. The thermodynamic parameters of the Earth's lower mantle can be estimated using the equation of state of Ca-perovskite. Table 4 shows the parameters of the Earth's lower mantle according to the PREM model (Preliminary Reference Earth Model) [82], where temperature is calculated from [83]. We calculated the density and adiabatic bulk modulus of CaSiO 3 -perovskite at these P-T parameters (from Table 4) and compared it with the PREM model and with the density and adiabatic bulk modulus of MgSiO 3 -perovskite from our previous study [16] (Figure 8). It can be seen that the density of Ca-perovskite is higher than the density of Mg-perovskite by about 0.1-0.15 gcm −3 and almost coincides with the density according to the PREM model (Figure 8a). The adiabatic bulk moduli of Ca-perovskite and Mg-perovskite differs slightly. However, with increasing pressure, adiabatic bulk modulus of CaSiO 3 -perovskite exceeds the PREM model by 40 GPa at pressures of 120-130 GPa (Figure 8b). Moreover, we added the first-principle calculated data for the density and adiabatic bulk modulus of Ca-perovskite according to Li et al. [74]. The calculated density from [74] and calculated density from our equation of state for CaSiO 3 -perovskite are very similar, especially in the high pressure range above 80 GPa (Figure 8a). The calculated values of the adiabatic bulk modulus are in less agreement (Figure 8b). ( Figure 8a). The calculated values of the adiabatic bulk modulus are in less agreement (Figure 8b). Thus, the obtained results of the present study can be used to interpret of the phase and seismic boundaries in the Earth's upper mantle. Since calcium is one of the main components of the Earth's mantle, it has its influence on the location of the observed seismic boundaries. At a depth of about ~350 km, CaSiO3 phase is contained as a component of clinopyroxene minerals, whereas below 350 km, it is concentrated as a component of complex garnet solid solutions (majorite garnet) [9,59]. The phase of CaSiO3-perovskite is sta-  [16] and CaSiO 3 -perovskite from the present equation of state are calculated from the mantle geotherm in comparison with the density and adiabatic bulk modulus according to the PREM model [82] and calculation from [74].
Thus, the obtained results of the present study can be used to interpret of the phase and seismic boundaries in the Earth's upper mantle. Since calcium is one of the main components of the Earth's mantle, it has its influence on the location of the observed seismic boundaries. At a depth of about~350 km, CaSiO 3 phase is contained as a component of clinopyroxene minerals, whereas below 350 km, it is concentrated as a component of complex garnet solid solutions (majorite garnet) [9,59]. The phase of CaSiO 3 -perovskite is stabilized at increasing pressure and is considered one of the main minerals of the lower mantle, comprising up to 7 wt.% of peridotitic mantle, according to the modern geophysical data [84]. There is the view that the transition of the CaSiO 3 component in the garnet solid solution to form denser of Ca-perovskite is associated with seismic discontinuity at a depth of 520 km [59]. The phase transition of wadsleyite-ringwoodite in the Mg 2 SiO 4 system also occurs at this depth at pressure 17-18 GPa, but it is characterized by a slight increase in density and adiabatic bulk modulus as was calculated and shown in our earlier study [16]. Accordingly, the phase transition in olivine cannot fully explain the seismic boundary at 520 km. Moreover, the olivine-wadsleyite transition at the depth 410 km at 14-15 GPa calculated in [16] corresponded to a temperature of~1700 K [16], but according to the estimations in [85], the temperature in this depth is 1790 to 1830 K. The calculated phase transition of (larnite + CaSi 2 O 5 )-Ca-perovskite occurs at a depth of 410 km at temperatures 1700-1800 K ( Figure 6). Although the phase transitions in the CaSiO 3 system cannot be directly responsible for the seismic boundaries in the upper mantle, they have an effect on the location of discontinuities and point to the role of calcium-containing minerals in the upper mantle.
It is well known that magnesium is the most abundant element in mantle minerals, but the ratios of main elements Mg, Fe, Ca, and Al remain an open question. The calculated density of MgSiO 3 -perovskite from the mantle geotherm [83] is too low with respect to the PREM model. However, the calculated density of CaSiO 3 -perovskite is in good agreement with the density by the PREM model (Figure 8a). The calcium content in the perovskite composition will increase the density of mineral, as well as the presence of iron. Such a composition will obviously be close to the real composition of the mantle material. The obtained thermodynamic data in the present study can be used in the calculation of silicate mixtures of different compositions, as was done for example in [86]. Thus, the possibilities of thermodynamic modeling for future calculations and investigation of deep mineralogy in the area that is not available for direct experimentation are opened.

Conclusions
We have constructed the equations of state of wollastonite, pseudowollastonite, breyite, larnite (Ca 2 SiO 4 ), titanite-structured CaSi 2 O 5 and CaSiO 3 -perovskite using a thermodynamic model based on the Helmholtz free energy. The proposed equations of state are developed based on optimization of different experimental measurements at ambient conditions and to a high temperatures and pressures. The full set of thermodynamic properties (volume, thermal expansion, entropy, isobaric and isochoric heat capacity, bulk moduli, Gibbs energy, etc.) of studied Ca-silicates is calculated at given pressures and temperatures and is presented in the form of Tables to the present article. The calculated properties are compared with reference and experimental data. It is shown that the proposed equations of state reliably describe properties of studied Ca-silicates in a wide range of pressures and temperatures.
The phase diagram of the CaSiO 3 system is constructed at pressures up to 20 GPa and temperatures up to 2000 K. The calculated phase diagram are compared with experimental data for different phases and clarifies the phase boundaries in the CaSiO 3 system under upper mantle conditions. The phase transition of majorite garnet, which contains CaSiO 3 phase in its composition, to Ca-perovskite is associated with seismic boundary at the depth 520 km; therefore, it points to the role of calcium-containing minerals in the upper mantle region. The calculated density of CaSiO 3 -perovskite is in good agreement with the density according to the PREM model at the lower-mantle region. The calculated thermodynamic data for Ca-perovskite open the possibilities of thermodynamic modeling at high pressures and temperatures.