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Equations of State of Ca-Silicates and Phase Diagram of the Casio3 System Under Upper Mantle Conditions

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Equations of State of Ca-Silicates and Phase Diagram of the Casio3 System Under Upper Mantle Conditions minerals Article Equations of State of Ca-Silicates and Phase Diagram of the CaSiO3 System under Upper Mantle Conditions Tatiana S. Sokolova * and Peter I. Dorogokupets Institute of the Earth’s Crust, Siberian Branch of the Russian Academy of Sciences, 664033 Irkutsk, Russia; [email protected] * Correspondence: [email protected]; Tel.: +7-39-5251-1680 Abstract: The equations of state of different phases in the CaSiO3 system (wollastonite, pseudowol- lastonite, 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. Citation: Sokolova, T.S.; Dorogokupets, P.I. Equations of State Keywords: equation of state; the Helmholtz free energy; thermodynamic properties; CaSiO3; wollas- of Ca-Silicates and Phase Diagram of tonite; breyite; perovskite; phase transition; diamond; mantle the CaSiO3 System under Upper Mantle Conditions. Minerals 2021, 11, 322. https://doi.org/10.3390/ min11030322 1. Introduction Academic Editor: Fumagalli Patrizia The thermodynamic description and phase equilibria in the CaSiO3 system are im- portant for modern mineralogical research because Ca-silicates are common components Received: 29 January 2021 of the Earth’s crust and mantle. The low-pressure CaSiO3 polymorphs (wollastonite and Accepted: 16 March 2021 pseudowollastonite) are rock-forming minerals in the Earth’s crust and constitute cement Published: 19 March 2021 substances [1,2], while the high-pressure Ca-silicate phases (breyite (walstromite), larnite, titanite-structured CaSi2O5 and CaSiO3-perovskite) are stable in the mantle P-T conditions Publisher’s Note: MDPI stays neutral and detected as solid-phase inclusions in natural diamonds [3–7]. The presence of CaSiO3 with regard to jurisdictional claims in phase in diamonds indicates the possibility of transporting the substance from the lower published maps and institutional affil- mantle to the surface in an unchanged form and provides insights into the nature of dia- iations. mond 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 (CaSiO3) has a number of structural mod- ifications at relevant pressures and temperatures. Wollastonite structure is stable under Copyright: © 2021 by the authors. ambient conditions and transforms to pseudowollastonite phase at high temperatures Licensee MDPI, Basel, Switzerland. above ~1400 K [8]. The transition of wollastonite to walstromite-structured CaSiO3 is This article is an open access article detected at 3 GPa and 1173 K [9]. The name breyite for this phase is approved by IMA distributed under the terms and in 2018 [10], therefore it is used in the text below. The wollastonite–breyite equilibrium conditions of the Creative Commons was studied by different authors [1,11,12]. The subsequent decomposition of breyite to Attribution (CC BY) license (https:// association of larnite (Ca2SiO4) and titanite-structured CaSi2O5 was determined at higher creativecommons.org/licenses/by/ pressures by Kanazaki’s study [13]. The CaSiO3-perovskite is formed at pressure ~14 GPa 4.0/). Minerals 2021, 11, 322. https://doi.org/10.3390/min11030322 https://www.mdpi.com/journal/minerals Minerals 2021, 11, 322 2 of 17 from (larnite + CaSi2O5) assemblage [14] and retains its structure until lower mantle condi- tions. The phase diagram of the CaSiO3 is well studied at low pressures up to 3–4 GPa and high temperatures by experimental methods in [11,15]. In the present calculation, we also used the phase diagram of the CaSiO3 system, which was proposed in [1,11]. The thermodynamics of Ca-silicates can be calculated from their equations of state (EoS). The equations of state of wollastonite, pseudowollastonite, breyite, larnite (Ca2SiO4), titanite-structured CaSi2O5 and CaSiO3-perovskite can be constructed based on the thermo- dynamic model, which was successfully used in our previous studies for the MgSiO3–MgO system [16,17]. We use a thermodynamic model based on the Helmholtz free energy, which allows us to calculate a full set of thermodynamic properties (entropy, heat capac- ity, bulk moduli, thermal expansion, Gibbs energy, etc.) at given P-T or V-T parameters. The proposed thermodynamic model is based on optimization of different experimental measurements (dilatometric, X-ray diffraction, ultrasonic interferometry, etc.) and theoreti- cal data (calculations and reference) by analogy with our studies for metals [18–20] and compounds [21,22]. The proposed equations of state of studied Ca-silicates are constructed with a small number of fitting parameters. There are also other different forms of EoSes, which contain a small number of parameters and reliably describe the properties of metals and mixtures [23–26], but in this study, we will use our model that previously showed its versatility. 2. Thermodynamic Model The equations of state of wollastonite, pseudowollastonite, breyite, larnite (Ca2SiO4), titanite-structured CaSi2O5 and CaSiO3-perovskite are constructed using the thermody- namic model based on the Helmholtz free energy. According to the proposed thermo- dynamic model, the Helmholtz free energy can be represented in general form as the sum [27]: F(V, T) = U0 + FT0 (V) + Fth(V, T) + Fanh(V, T), (1) where U0 is the reference energy, FT0 is the potential part of the Helmholtz free energy at the reference isotherm T0 = 298.15 K, which depends only on volume; Fth is the thermal part of the Helmholtz free energy, which depends on volume and temperature; Fanh 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: −k PT0 (V) = 3K0X (1 − X) exp[h(1 − X)], (2) 1/3 0 where X = (V/V0) , h = 1.5K − k + 0.5, K0 is the isothermal bulk modulus under ambient −4 0 conditions (P0 = 1 bar = 10 GPa, T0 = 298.15 K), K = ¶K0/¶P0 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 (HO2)[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: ¶PT0 −k KT0(V) = −V = K0X exp[h(1 − X)][X + (1 − X)(hX + k)], (3) ¶V T Minerals 2021, 11, 322 3 of 17 ¶K 1 kX + 2X2h − X(1 + h) K0(V) = T0 = k + hX + . (4) ¶PT0 3 X + (1 − X)(hX + k) 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: −Qi Fth(V, T) = ∑ miRT ln 1 − exp , (5) i=1,2 T where mi is proportionality factor, which is calculated from the total number of atoms in the compound (i = 1, 2), m1 + m2 = 3n, n is the number of atoms in a chemical formula of the compound, Qi is the Einstein characteristic temperature, which depends only on volume, and R is the gas constant (R = 8.31451 Jmol−1K−1). The volume dependence of characteristic temperature in Equation (5) is determined by Al’tshuler equation [34]: g − g Q = Q x−g¥ exp 0 ¥ 1 − xb , (6) i 0i b where Q0i is the characteristic temperature under ambient conditions (i = 1, 2), x = V/V0, γ0 is the Grüneisen parameter at ambient conditions, γ¥ is the Grüneisen parameter at infinite compression, when x ! 0, and b 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: ¶ln Q b g = − = g¥ + (g0 − g¥)x , (7) ¶ ln V T where Q is the characteristic temperature in general case. We assume that the Grüneisen parameter will be the same for both characteristic tem- peratures (Q1 and Q2) in Equation (6). In addition, if γ¥ = 0, then Equation (7) transforms b to the classical form γ = γ0x .
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