Pressure Effects on the Thermodynamic Properties of MgSiO₃ Akimotoite

Abstract

The thermodynamic properties of MgSiO₃ akimotoite at high temperatures and high pressures are important for investigating the phase equilibria of the Earth’s transition zone and the upper part of the lower mantle. In this paper, we present the self-consistent unit-cell volume, elastic properties, and in particular, thermodynamic properties including thermal expansion, heat capacity, entropy, and Grüneisen parameter of MgSiO₃ akimotoite at pressures up to 30 GPa and temperatures to 2000 K using an iterative numerical method and available experimental data, which are consistent with the previous studies. The results show that the determined thermal expansion, heat capacity, entropy, and Grüneisen parameter exhibit a nonlinear and negative relationship with increasing pressure. Additionally, the pressure derivatives of these thermodynamic parameters along with the temperature are also presented.

1. Introduction

Akimotoite (Ak) is one of the high-pressure polymorphs of MgSiO₃ with the ilmenite-type structure. It is believed to be stable between 18 and 25 GPa, and from 900 to 2200 K, and occurs mainly in cold stagnant slabs at the bottom of the mantle transition zone or in the uppermost region of the lower mantle [1,2,3,4]. Recent studies suggest that akimotoite is stable in the harzburgite layers of subducting slabs rather than in the mid-ocean ridge basalt or pyrolite [4,5]. Akimotoite is stable at the low pressure of bridgmanite (Brg) with the perovskite-type structure and at the low temperature of majorite (Mj) with the garnet-type structure in the pure MgSiO₃ system, which is expected to contribute to the structure of the 660 km discontinuity [6,7]. Therefore, the elastic properties of akimotoite have been studied for decades by numerous experimental measurements [3,4,8,9,10], and theoretical calculations [11,12,13,14,15], providing a wealth of data for understanding the seismic discontinuities and density jumps caused by akimotoite-related phase transitions.

On the other hand, knowledge of the thermodynamic properties of MgSiO₃ akimotoite is essential for the study of the geotherms and phase equilibria. However, compared with the sufficient elastic moduli and volume data of MgSiO₃ akimotoite, the thermodynamic properties of MgSiO₃ akimotoite, especially at high-pressure conditions are still lacking. At ambient pressure conditions, the thermal expansion of MgSiO₃ akimotoite has been determined from the unit-cell volume by using X-ray diffraction experiments [10,16,17], while the heat capacity and entropy in a wide range of temperatures have been presented from calorimetric measurements [16,17,18]. However, thermodynamic properties such as thermal expansion, heat capacity, entropy, and Grüneisen parameter at simultaneous high temperatures and high pressures have only been obtained by First-principles calculations [11,19].

Decades ago, Davis and Gordon introduced a numerical iterative procedure by which the density, the elastic moduli, and especially the thermal expansion, and heat capacity of the metal melt, at high temperatures and high pressures, could be obtained from adiabatic elastic wave velocities [20]. Recently, the feasibility of this method has been demonstrated for solid minerals [21]. Therefore, self-consistent thermodynamic properties at high-pressure conditions can be conveniently determined from the numerous published experimental data on minerals.

In this paper, we evaluated the unit-cell volume, the elastic properties, and most importantly, the thermodynamic properties of MgSiO₃ akimotoite at high temperature and high-pressure, with the combination of the available experimental data and a numerical iterative procedure. Comparisons with previous work have been made to verify the accuracy of our results. In addition, the pressure effects on the thermal expansion, heat capacity, entropy, and Grüneisen parameter at different temperature conditions were presented.

2. Methods

2.1. Calculation Procedure

The basic calculation procedure, which is described in our recent studies [21,22], is presented in the following. Note that the uncertainties of the derived parameters at different temperature and pressure conditions are derived from the propagation of the experimental measurement errors by means of a Taylor series expansion.

The thermal expansion (α) at certain pressure can be calculated with the volume (V) as a function of temperature through Equation (1).

$$\alpha \left(T\right)={\displaystyle \frac{1}{V}}\left({\displaystyle \frac{\partial V}{\partial T}}\right)$$

(1)

And the integration of Equation (1) can be expressed as Equation (2).

$$V\left(T\right)=V_{0}exp\left[{\int }_{T_0}^T\alpha \left(T\right)dT\right]$$

(2)

here V₀ refers to the volume at ambient conditions. Also, the isothermal pressure derivative of volume can be expressed as Equation (3),

$${\left({\displaystyle \frac{\partial V}{\partial P}}\right)}_T=-V^2\left({\displaystyle \frac{1}{v_{\Phi }^2}}+{\displaystyle \frac{T{\alpha }^2}{C_P}}\right)$$

(3)

where C_P is the heat capacity. And v_Φ is the bulk sound velocity, which can be determined with the P-wave velocity (v_P) and the S-wave velocity (v_S) via Equation (4). Equation (5) can be used to evaluate the isothermal derivative of the heat capacity with respect to pressure.

$$v_{\Phi }={\left(v_p^2-{\displaystyle \frac{4}{3}}{v}_s^2\right)}^{{\displaystyle \frac{1}{2}}}$$

(4)

$${\left({\displaystyle \frac{\partial C_P}{\partial P}}\right)}_T=-VT\left[{\alpha }^2+{\left({\displaystyle \frac{\partial \alpha }{\partial T}}\right)}_P\right]$$

(5)

The calculation process starts by using the experimentally measured volume of MgSiO₃ akimotoite at ambient pressure conditions (V(P₀,T)) to determine the thermal expansion as a function of temperature (α(P₀,T)) by Equation (1). Meanwhile, the heat capacity of MgSiO₃ akimotoite as a function of temperature (C_P(P₀,T)) can be fitted from published experimental data with empirical equations. Thus, with the obtained α(P₀,T) and C_P(P₀,T), the approximate volume at an arbitrary reference pressure (V(P₀+ΔP,T)) can be estimated using Equation (3). Also, the bulk velocity of MgSiO₃ akimotoite as a function of temperature and pressure (v_Φ(P,T)) can be obtained using the P-wave velocity and S-wave velocity at high temperatures and high pressures (v_P(P,T) and v_S(P,T)) via Equation (4). Therefore, the resulting V(P₀+ΔP,T) can be used to update the values of α(P₀+ΔP,T) and C_P(P₀+ΔP,T) via Equations (1) and (5), respectively. Finally, iteration of this loop results in converged volume, thermal expansion, and heat capacity of MgSiO₃ akimotoite as a function of temperature and pressure.

Also, with the combinations of V(P,T), v_P(P,T), v_S(P,T), the adiabatic bulk modulus (K_S), isothermal bulk modulus (K_T), and shear modulus (G) of MgSiO₃ akimotoite at simultaneous high temperatures and high pressures can be determined through Equations (6)–(8), respectively.

$$K_S={\displaystyle \frac{1}{V}}\left(v_p^2-{\displaystyle \frac{4}{3}}{v}_s^2\right)$$

(6)

$$K_T={\displaystyle \frac{K_S}{1+\alpha \gamma T}}$$

(7)

$$G={\displaystyle \frac{1}{V}}{v}_S^2$$

(8)

Itis worth noting that the parameter γ in Equation (7) stands for the Grüneisen parameter, which can be derived from Equation (9). Furthermore, with the entropy measured at ambient conditions, the entropy at high temperature and high-pressure conditions can also be calculated via Equations (10) and (11), respectively [23].

$$\gamma ={\displaystyle \frac{\alpha K_{S}V}{C_P}}$$

(9)

$${S\left(T\right)}_P={S\left(T_0\right)}_P+{\int }_{T_0}^T\left({\displaystyle \frac{C_P}{T}}\right)dT$$

(10)

$${S\left(P\right)}_T={S\left(P_0\right)}_T-{\int }_{P_0}^P\alpha VdP$$

(11)

2.2. Thermoelastic Data of MgSiO₃ Akimotoite

To date, there are only a few experimental measurements on the sound velocity of MgSiO₃ akimotoite. The sound velocity of MgSiO₃ akimotoite at ambient pressure conditions was determined by Weidner and Ito [8] using Brillouin spectroscopy, and the high temperature and high-pressure data were determined only by Zhou et al. [3] using ultrasonic interferometry techniques in the mantle transition zone environment.

In the paper by Zhou et al. [3], they presented a linear pressure dependence and a nonlinear temperature dependence of the sound velocity for MgSiO₃ akimotoite, which is in good agreement with the result by Weidner and Ito [8]. Therefore, in this study, we used the proposed fitting equations for v_P and v_S by Zhou et al. [3] (

Table 1. Sound velocity, unit-cell volume, heat capacity and entropy data of MgSiO₃ akimotoite used in this study.

Data type	Temperature Range	Pressure Range	Method	References
	K	GPa
Sound velocity	~1500	~25.7	Ultrasonic interferometry	Zhou et al. (2014) [3]
Unit-cell volume	298–876	Ambient	X-ray diffraction	Ashida et al. (1988) [16]
	298–773	Ambient	X-ray diffraction	Kojitani et al. (2022) [17]
Heat capacity	170–700	Ambient	Calorimetry	Ashida et al. (1988) [16]
	1.90–302.43	Ambient	Thermal relaxation method	Akaogi et al. (2008) [18]
	300–820	Ambient	Calorimetry	Kojitani et al. (2022) [17]
Standard entropy	298.15	Ambient	Calorimetry	Akaogi et al. (2008) [18]

), which are shown as Equations (12) and (13), respectively.

$$v_P\left(P,T\right)=10211\left(8\right)+55.8\left(4\right)\times P-0.352\left(12\right)\times \left(T-300\right)-1.22\left(11\right)\times {10}^{-4}{\times \left(T-300\right)}^2\hspace{1em}\mathrm{m}/\mathrm{s}$$

(12)

$$v_S\left(P,T\right)=5862\left(6\right)+21.9\left(3\right)\times P-0.253\left(9\right)\times \left(T-300\right)-1.39\left(8\right)\times {10}^{-4}\times {\left(T-300\right)}^2 \mathrm{m}/\mathrm{s}$$

(13)

In general, the temperature variation of the experimentally measured isobaric volume is needed to derive the thermal expansion (Equation (1)) to proceed with the calculation. The high-temperature unit-cell volume of MgSiO₃ akimotoite at ambient pressure conditions was reported to be 876 K by Ashida et al. [16], and the data were recently remeasured by Kojitani et al. [17]. Thus, the available volume data from these papers are analyzed with the EoSFit7 Program [24], and then fitted with the empirical equation proposed by Fei [25]. The obtained equation for the thermal expansion of MgSiO₃ akimotoite as a function of temperature is shown in Equation (14). The calculated results via Equation (2) and Equation (14) are shown in Figure 1, which are highly consistent with the experimentally determined data, with the largest difference less than 0.2%.

$$\alpha \left(T\right)=2.61\left(7\right)\times {10}^{-5}+0.54\left(14\right)\times {10}^{-8}T-1.28\left(14\right)\times T^{-2} {\mathrm{K}}^{-1}$$

(14)

In the studies by Ashida et al. [16] and Kojitani et al. [17], they also reported the high-temperature heat capacity of MgSiO₃ akimotoite with calorimetric measurements to ~800 K, and the results were expressed by the Maier-Kelley equation [26] and a polynomial formula shown as Equation (15), respectively. The low-temperature data were accurately determined by Akaogi et al. [18]. These heat capacity values are fitted with Equation (15), as well as Equations (16)–(18), which are three classical empirical formulas proposed by Hass and Fisher [27], Berman and Brown [28], and Richet and Fiquet [29], respectively. The determined fitting coefficients, reduced χ², and R² are listed in

Table 2. Fitting coefficients for the heat capacity of MgSiO₃ akimotoite using different empirical equations.

Fitting Equations	Reduced χ2	R2	References of the Empirical Equations
$C_P=c_0+c_{1}T+c_2{T}^{-0.5}+c_3{T}^{-1}+c_4{T}^{-2}+c_5{T}^{-3}$ $c_0=-247\left(433\right)\times {10}^2$ $c_1=-8\left(9\right)\times {10}^{-2}$ $c_2=2\left(2\right)\times {10}^4$ $c_3=-2\left(2\right)\times {10}^5$ $c_4=1\left(2\right)\times {10}^7$ $c_5=-3\left(9\right)\times {10}^8$	1.19	0.996	Kojitani et al. (2022) [17]
$C_P=c_0+c_{1}T+c_3{T}^2+c_4{T}^{-0.5}+c_5{T}^{-2}$ $c_0=494\left(54\right)$ $c_1=-2.8\left(6\right)\times {10}^{-1}$ $c_2=1.2\left(3\right)\times {10}^{-4}$ $c_3=-6.4\left(7\right)\times {10}^3$ $c_4=2.4\left(4\right)\times {10}^6$	1.21	0.996	Hass and Fisher (1976) [27]
$C_P=c_0+c_1{T}^{-0.5}+c_2{T}^{-2}+c_3{T}^{-3}$ $c_0=146\left(4\right)$ $c_1=-6\left(1\right)\times {10}^2$ $c_2=-5.3\left(4\right)\times {10}^6$ $c_3=6.1\left(5\right)\times {10}^8$	1.18	0.996	Berman and Brown (1985) [28]
$C_P=c_0+c_{1}ln\left(T\right)+c_2{T}^{-1}+c_3{T}^{-2}+c_4{T}^{-3}$ $c_0=155\left(133\right)$ $c_1=-5\left(39\right)$ $c_2=-1\left(2\right)\times {10}^4$ $c_3=-4\left(3\right)\times {10}^6$ $c_4=5\left(2\right)\times {10}^7$	1.19	0.996	Richet and Fiquet (1991) [29]

. Furthermore, the comparisons between the calculated heat capacity of MgSiO₃ akimotoite using Equations (15)–(18) and the available experimental results [16,17,18] are also shown in Figure 2.

The heat capacity of MgSiO₃ akimotoite at 2000 K was estimated to be 129.512 J/mol∙K by Hofmeister and Ito (Model C) [30], and 135.83 J/mol∙K by Kojitani et al. [17]. Therefore, after considering the reduced χ² and R² of the obtained fitting equations, as well as the previously estimated values, we chose Equation (17) in this study. The largest difference between the value calculated using Equation (17) and the results by Ashida et al. [16], Akaogi et al. [18], and Kojitani et al. [17] are 3.7%, 2.3%, and 2.5% above room temperature, respectively.

Finally, the standard entropy used in the present study was from Akaogi et al. [18] (Table 1), which was determined to be S°₂₉₈ = 53.7(4) J/mol∙K.

$$C_P=c_0+c_{1}T+c_2{T}^{-0.5}+c_3{T}^{-1}+c_4{T}^{-2}+c_5{T}^{-3}\hspace{1em}\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K}$$

(15)

$$C_P=c_0+c_{1}T+c_2{T}^2+c_3{T}^{-0.5}+c_4{T}^{-2} \mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K}$$

(16)

$$C_P=c_0+c_1{T}^{-0.5}+c_2{T}^{-2}+c_3{T}^{-3} \mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K}$$

(17)

$$C_P=c_0+c_{1}ln\left(T\right)+c_2{T}^{-1}+c_3{T}^{-2}+c_4{T}^{-3} \mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K}$$

(18)

3. Results

3.1. High Temperature and High Pressure Unit-Cell Volume of MgSiO₃ Akimotoite

Our determined unit-cell volume of MgSiO₃ akimotoite variation with pressure at different temperatures is shown in Figure 3, together with the results carried out from X-ray diffraction experiments [3,9,10], and First-principle calculations [11]. At room temperature, our result shows good consistency with the X-ray diffraction measurements by Reynard et al. [9], and the difference is less than 0.55%. The excellent agreement is also maintained at high temperature conditions, and the largest differences are 0.43% and 0.53% compared with those of Wang et al. [10] and Zhou et al. [3], respectively. Meanwhile, our obtained unit-cell volume is generally lower than the theoretical result by Karki and Wentzcovitch [11], and the largest differences are 0.87%, 0.98%, and 1.0% at 300 K, 1000 K, and 2000 K, respectively.

3.2. High Temperature and High-Pressure Elastic Properties of MgSiO₃ Akimotoite

Our derived elastic moduli of MgSiO₃ akimotoite as a function of pressure and temperature are shown in Figure 4. In Figure 4b,c, the calculated adiabatic bulk modulus and shear modulus are in good agreement with the experimental results under the relevant conditions [3], and the largest differences are 2.1%, 1.1%, and 0.4% at 300 K, 700 K, and 1500 K, respectively. Compared with the results obtained from using first-principles calculations [11,14], our adiabatic bulk modulus and shear modulus are generally larger than those of Hao et al. [14], and Karki and Wentzcovitch [11] (shown in Figure 4b,d, respectively). While our shear modulus at 300 K is larger than that by Hao et al. [14], then with increasing temperature, the values become smaller than those by Hao et al. [14] at 1500 K and 2000 K.

As shown in Figure 4, all the elastic moduli generally display a linear dependence on pressure and a nonlinear temperature dependence within the uncertainties. Therefore, we adopt the data to the equation of M = M₀ + (∂M/∂P) × P + (∂M/∂T) × (T − 300) + (∂²M/∂T²) × (T − 300)², where M refers to the elastic moduli, M₀ refers to the elastic moduli at ambient conditions, ∂M/∂P is the first pressure derivative at fixed temperature, and ∂M/∂T, ∂²M/∂T² are the first and second temperature derivatives of M at fixed pressure, respectively. The fitting parameters are listed in

Table 3. Isothermal bulk modulus, adiabatic bulk modulus, and shear modulus of MgSiO₃ akimotoite at ambient conditions, and their pressure and temperature derivatives.

Temperature	KT	KS	G	References
K	GPa	GPa	GPa
		M0 (GPa)
300	--	212	132	Weidner and Ito (1985) [8]
298	212	--	--	Reynard et al. (1996) [9]
300	201	--	--	Karki & Wentzcovitch (2002) [11]
298	210	--	--	Wang et al. (2004) [10]
300	221	226	136	Zhang et al. (2005) [12]
2000	158.1 (6)	--	85.7	Li et al. (2009) [13]
300	207 (3)	218.9 (6)	131.8 (3)	Zhou et al. (2014) [3]
300	202	204	127	Hao et al. (2019) [14]
300	205 (1)	209 (2)	--	Siersch et al. (2021) [4]
300	221.5 (11)	222.4 (13)	130.8 (5)	This study
		∂M/∂P
300	4.64	--	--	Karki & Wentzcovitch (2002) [11]
298	5.6 (8)	--	--	Wang et al. (2004) [10]
300	3.94	3.85	1.04	Zhang et al. (2005) [12]
2000	3.7 (2)	--	4.5 (3)	Li et al. (2009) [13]
300	4.6	4.62 (3)	1.64 (1)	Zhou et al. (2014) [3]
300	4.40	4.39	1.64	Hao et al. (2019) [14]
300	--	4.4	--	Siersch et al. (2021) [4]
300	4.49 (1)	4.39 (1)	1.54 (1)	This study
	∂M/∂T (×10−2 GPa/K)
300	−2.5	--	--	Karki & Wentzcovitch (2002) [11]
298	−4.0 (1)	--	--	Wang et al. (2004) [10]
300	−3.0	−2.16	−1.79	Zhang et al. (2005) [12]
300	--	−1.99(9)	−1.58(4)	Zhou et al. (2014) [3]
300	−2.3	−1.719	−1.242	Hao et al. (2019) [14]
300	−2.943 (1)	−1.937 (1)	−1.645 (1)	This study
	∂2M/∂T2 (×10−6 GPa/K2)
300	--	−2.9 (8)	−6.7 (4)	Zhou et al. (2014) [3]
300	--	−1.26	−1.94	Hao et al. (2019) [14]
300	−0.847 (6)	−2.060 (8)	−5.351 (4)	This study

, along with the published results for comparison.

The isothermal bulk modulus, adiabatic bulk modulus, and shear modulus at ambient conditions are determined as K_T0 = 221.5 (11) GPa, K_S0 = 222.4 (13) GPa, and G₀ = 130.8 (5) GPa, respectively. At ambient conditions, the pressure derivatives of the elastic moduli are determined to be ∂K_T/∂P = 4.49 (1), ∂K_S/∂P = 4.39 (1), and ∂G/∂P = 1.54 (1), and the first temperature derivatives of the elastic moduli are determined as ∂K_T/∂T = −2.943 (1) × 10⁻² GPa/K, ∂K_S/∂T = −1.937 (1) × 10⁻² GPa/K, and ∂G/∂T = −1.645 (1) × 10⁻² GPa/K, the second temperature derivatives of the elastic moduli are determined as ∂²K_T/∂T² = −0.847 (6) × 10⁻⁶ GPa/K², ∂²K_S/∂T² = −2.060 (8) × 10⁻⁶ GPa/K², and ∂²G/∂T² = −5.351 (4) × 10⁻⁶ GPa/K². These values are generally within the range of the available values listed in Table 3 [3,4,8,9,10,11,12,13,14].

3.3. High Temperature and High-Pressure Thermodynamic Properties of MgSiO₃ Akimotoite

The calculated thermal expansion of MgSiO₃ akimotoite as a function of temperature is presented in Figure 5. Using the unit-cell volume data presented by Ashida et al. [16], Saxena et al. [31] obtained the thermal expansion of MgSiO₃ akimotoite at ambient pressure conditions. Also, the thermal expansion data has been obtained from experimentally measured entropy by Chopelas [32], as well as from First-principles calculations by Karki and Wentzcovitch [11]. Recently, Kojitani et al. [17] also presented the thermal expansion of MgSiO₃ akimotoite as a function of temperature by fitting the X-ray diffraction determined unit-cell volumes. At 0 GPa, our calculated thermal expansion is determined to be α₀ = 1.35 (4) × 10⁻⁵/K at 300 K, which is slightly lower than the previous results [11,17,31,32]. However, in the temperature range of ~700–2000 K, our data is in great agreement with the results by Karki and Wentzcovitch [11] and Kojitani et al. [17], within 2.9% and 3.4%, respectively, and ~5.7% and ~6.8% larger than those of Saxena et al. [31] and Chopelas [32], respectively. The good consistency also holds at high-pressure conditions and above 700 K. The average differences are ~0.7%, ~2.9%, and ~7.3% at 10 GPa, 20 GPa, and 30 GPa, respectively, compared to the theoretical results [11].

Figure 6 shows the temperature dependence of the calculated heat capacity at 0 GPa, 10 GPa, 20 GPa, and 30 GPa, along with previous results at ambient pressure conditions [16,17,31], and at high-pressure conditions [11]. The heat capacity of MgSiO₃ akimotoite was measured by differential scanning calorimetry at ambient pressure conditions [16,17], and by First-principles calculations at high-pressure conditions [11]. Also, Saxena et al. [31] presented the fitting equation for the heat capacity as a function of temperature using the data from Watanabe [33]. As mentioned above, since the heat capacity data at high temperature and ambient pressure conditions used in this study were from the investigations by Ashida et al. [16], Akaogi et al. [18], and Kojitani et al. [17], our calculated heat capacity of MgSiO₃ akimotoite at 0 GPa shows good agreement with these published results [16,17]. Our calculated heat capacity is also in good agreement with the results obtained by Saxena et al. [31] and Karki and Wentzcovitch [11], with the largest differences being less than 2.5% and 1.5%, respectively. Compared to the first calculation result at high-pressure conditions [11], our result is ~0.9% lower than that at 10 GPa, while ~1.3%, and ~1.4% higher than that at 20 GPa, and 30 GPa, respectively.

The entropy of minerals at high temperatures and high-pressures is a valuable thermodynamic parameter in determining reliable phase diagrams [34], and its value can be obtained from heat capacity measurements using adiabatic calorimetry. The entropy of MgSiO₃ akimotoite at ambient pressure conditions has been reported by Ashida et al. [16], and Saxena et al. [31], and currently, high-pressure values have only been determined by First-principles calculations [11]. Our calculated entropy of MgSiO₃ akimotoite at high temperatures and high pressures is shown in Figure 7, which is generally, in good agreement with the previous investigations.

The Grüneisen parameter is of considerable importance in geosciences, as it often appears in equations describing the thermoelastic behavior of minerals at high-pressures and temperatures [35]. With the combination of the obtained thermodynamic and elastic data, our calculated Grüneisen parameter of MgSiO₃ akimotoite at high temperatures and high-pressures is illustrated in Figure 8, along with the available results at ambient pressure conditions by Saxena et al. [31], and at high-pressure conditions by Karki and Wentzcovitch [11].

At ambient conditions, the Grüneisen parameter of MgSiO₃ akimotoite is determined to be γ₀ = 1.03 (25), which is lower than those by Saxena et al. [31] (γ₀ = 1.72), and Karki and Wentzcovitch [11] (γ₀ = 1.29). As the temperature increases to ~900 K, our Grüneisen parameter increases rapidly, and then the value increases slightly with a very small temperature derivative as 0.00007/K, and the values are generally higher than those presented by Saxena et al. [31], as well as Karki and Wentzcovitch [11]. Also, the differences in the temperature dependencies below ~500 K are probably caused by the different temperature derivatives of the thermal expansion and heat capacity under the low-temperature conditions (Figure 5 and Figure 6). Under high-pressure conditions, the calculated Grüneisen parameter shows a temperature derivative as 0.00002/K at 10 GPa, −0.00002/K at 20 GPa, and −0.00005/K at 30 GPa, which shows a similar pattern to the result of Karki and Wentzcovitch [11].

4. Discussions

The pressure effects on the thermodynamic properties of minerals are prerequisites for deriving the thermal state of the Earth’s interior. Therefore, the calculated thermal expansion, heat capacity, entropy, and Grüneisen parameter of MgSiO₃ akimotoite as a function of pressure at various temperatures are illustrated in Figure 9a–d, respectively.

As shown in Figure 9a–d, it is obvious that all these thermodynamic parameters tend to be negative and nonlinear with increasing pressure. Thus, we fitted our calculated results to a polynomial equation of N = N₀ + (∂N/∂P)_T × P + (∂²N/∂P²)_T × P² at fixed temperatures, where N refers to the thermodynamic properties, N₀ refers to M at ambient pressure conditions, (∂N/∂P)_T and (∂²N/∂P²)_T refer to the first and second pressure derivatives of N, respectively. The fitting coefficients are listed in

Table 4. Thermal expansion, heat capacity, entropy, and Grüneisen parameter of MgSiO₃ akimotoite at ambient conditions, and their first and second pressure derivatives at different temperatures.

T K	α 10−5/K	CP J/mol∙K	S J/mol∙K	γ
	N0
	10−5/K	J/mol∙K	J/mol∙K
300	1.35 (4)	77 (17)	54(1)	1.03 (25)
700	1.44 (3)	116 (9)	138 (10)	1.35 (2)
1000	3.03 (20)	124 (8)	181 (14)	1.37 (2)
1500	3.37 (27)	130 (7)	233 (16)	1.41 (7)
2000	3.66 (35)	133 (6)	271 (18)	1.44 (12)
	∂N/∂P
	10−7/K∙GPa	J/mol∙K∙GPa	J/mol∙K∙GPa	10−2/GPa
300	−5.84 (2)	−0.8066 (6)	−0.315 (2)	−2.200 (7)
700	−5.79 (2)	−0.2356 (6)	−0.681 (2)	−0.981 (3)
1000	−6.64 (3)	−0.2078 (11)	−0.758 (2)	−1.096 (4)
1500	−8.01 (4)	−0.2472 (20)	−0.848 (3)	−1.377 (5)
2000	−9.42 (5)	−0.3178 (28)	−0.929 (4)	−1.643 (6)
	∂2N/∂P2
	10−9/K∙GPa2	10−3J/mol∙K∙GPa2	10−3J/mol∙K∙GPa2	10−4/GPa2
300	6.40 (6)	0.53 (2)	5.14 (6)	−1.210 (2)
700	6.29 (6)	1.60 (2)	5.99 (6)	0.762 (10)
1000	7.13 (10)	2.36 (3)	6.69 (7)	0.945 (13)
1500	9.12 (14)	3.72 (6)	7.90 (9)	1.243 (18)
2000	11.19 (18)	5.20 (9)	9.17 (11)	1.482 (21)

. Additionally, the first and second pressure derivatives of thermal expansion, heat capacity, entropy, and Grüneisen parameter of MgSiO₃ akimotoite as a function of temperature are shown in Figure 10a–d, respectively.

In Table 4 and Figure 10a, the first pressure derivative of thermal expansion is ∂α/∂P = −5.84 (2) × 10⁻⁷/K∙GPa at 300 K, then the value increases to ~−5.53 × 10⁻⁷/K∙GPa at ~450 K, and decreases with the temperature to −9.42 (5) × 10⁻⁷/K∙GPa at 2000 K. The value of ∂C_P/∂P is calculated to be −0.8066 (6) J/mol∙K∙GPa at 300 K, and the value increases rapidly to ~−0.2077 J/mol∙K∙GPa at ~1000 K, and then decreases slowly to −0.3178 (28) J/mol∙K∙GPa at 2000 K (Table 4, Figure 10b). The pressure effect on the entropy of MgSiO₃ akimotoite is shown in Table 4 and Figure 10c, which generally increases with the temperature, and ∂S/∂P decreases from −0.315 (2) J/mol∙K∙GPa to −0.929 (4) J/mol∙K∙GPa in the temperature range of 300–2000 K. In addition, ∂γ/∂P is determined to be −2.200 (7) × 10⁻²/GPa at 300 K, then, similar to the heat capacity, the value tends to increase to a maximum of ~−0.979 × 10⁻²/GPa at ~650 K, and finally decreases to −1.643 (6) × 10⁻²/GPa at 2000 K (Table 4, Figure 10d). Also, with the combination of the temperature variation of ∂α/∂P and ∂C_P/∂P (Figure 10a,b, respectively), it seems that the pressure derivative of the Grüneisen parameter is largely dependent on the pressure effects on the thermal expansion and heat capacity.

The phase transition boundaries and the Clapeyron slopes of the MgSiO₃ Majorite-akimotoite-bridgmanite system are of great geophysical interest as they are thought to be one of the causes of the boundary between the Earth’s transition zone and lower mantle. Although plenty of work has been done to determine the phase transition diagram of this MgSiO₃ system, the results are still controversial [19,36,37,38]. In the meantime, the phase transition boundaries in the MgSiO₃ system are usually derived using Equation (19):

$$∆G\left(P,T\right)=∆H°\left(T\right)-T∆S°\left(T\right)+\underset{1 atm}{\overset{P}{\int }}∆V\left(P,T\right)dP=0$$

(19)

where ΔG(P,T) and ΔV(P,T) are the Gibbs free energy and the volume changes of two coexisting phases, respectively, ΔH°(T) and ΔS°(T) are the enthalpy and the entropy changes of two coexisting phases at ambient pressure conditions (1 atm) and temperature T, respectively. This equation provides a convenient way to estimate the phase transition boundaries, avoiding the use of thermodynamic properties at high-pressure conditions, which have been employed by a majority of previous investigations [19,36,37].

On the other hand, the Clausius-Clapeyron equation (Equation (20)) is another classical equation to derive the phase equilibrium:

$${\displaystyle \frac{dP}{dT}}={\displaystyle \frac{∆S\left(P,T\right)}{∆V\left(P,T\right)}}$$

(20)

here dP/dT is the Clapeyron slope, and ΔS(P,T) and ΔV(P,T) are the entropy and volume changes between the two different phases, respectively. Therefore, the Clapeyron slope of the MgSiO₃ system can be determined from the entropy data of related minerals at high temperatures and high-pressures [38]. However, similar to akimotoite, the experimentally measured thermodynamic properties of majorite and bridgmanite at high temperatures and high-pressures have only been presented under ambient pressure conditions. Therefore, more data is needed for further studies.

5. Conclusions

To summarize, we used a numerical iterative method and published experimental data to determine the self-consistent thermodynamic parameters of MgSiO₃ akimotoite in a wide temperature and pressure range. All the obtained thermodynamic parameters, including thermal expansion, heat capacity, entropy, and Grüneisen parameter show a nonlinear and negative behavior with the increasing pressure. The pressure derivatives of the thermal expansion, heat capacity, and Grüneisen parameter generally increase with the temperature to ~450 K, ~1000 K, and ~650 K, respectively, and then decrease with the temperature. The pressure effect on entropy, on the other hand, generally increases with the temperature in the temperature range studied. This work provides a basic understanding of the thermodynamic properties of MgSiO₃ akimotoite under high temperature and high-pressure conditions. Further work is needed to be done to obtain more accurate thermodynamic properties when considering the phase equilibria associated with MgSiO₃ akimotoite.