First-principles study on the high-pressure physical properties of orthocarbonate Ca₂CO₄

Abstract

Orthorhombic Ca₂CO₄ is a recently discovered orthocarbonate whose high-pressure physical properties are critical for understanding the deep carbon cycle. Here, we study the structure, elastic and seismic properties of Ca₂CO₄-Pnma at 20–140 GPa using first-principles calculations, and compare them with the results of CaCO₃ polymorphs. The results show that the structural parameters of Ca₂CO₄-Pnma are in good agreement with the experimental results. It could be the potential host of carbon in the Earth's mantle subduction slab, and its low wave velocity and small anisotropy may be the reason why it cannot be detected in seismic observation. The thermodynamic properties of Ca₂CO₄-Pnma at high temperature and high pressure are obtained using the quasi-harmonic approximation method. This study is helpful in understanding the behavior of Ca-carbonate in the Earth’s lower mantle conditions.

Introduction

As the most important carbonate, CaCO₃ is transported to the deep mantle by subduction slab and plays a crucial role in the global long-term carbon cycle¹. It is also a mineral that plays a key role in biomineralization². However, CaCO₃ undergoes a series of phase transitions under high temperature and high pressure, forming various structures and polymorphs. So far, the predicted structures are mainly calcite, aragonite, post-aragonite, and pyroxene-like^{3,4,5,6,7,8,9}, and these structures and polymorphs have been experimentally verified^{3,4,6,9,10,11,12,13,14,15}. Some studies also considered the reaction of calcium carbonate with MgO, SiO₂, and MgSiO₃^7,10,16,17, while ignoring the reaction with CaO. Previously, Al-Shemali and Boldyrev¹⁸ mentioned the possible formation of calcium orthocarbonate Ca₂CO₄ in the CaCO₃ + CaO system under high pressure. Recently, using AIRSS¹⁹ and USPEX²⁰ crystal structure prediction methods, Sagatova et al.²¹ discovered a new structure of calcium orthocarbonate Ca₂CO₄ (space group Pnma) stable at 13–50 GPa and 2000 K, the carbon atoms in this phase are fourfold coordinated, and the structure is similar to high temperature and high pressure α'_H-Ca₂SiO₄ phase²². Afterward, they found that Ca₂CO₄-Pnma was stable in the pressure and temperature range of 20–100 GPa and 1000–2000 K using the density functional theory within quasi-harmonic approximation²³. Subsequently, Binck et al.²⁴ verified the results of Sagatova et al.²³ with single-crystal diffraction experiments. In addition, other alkaline earth orthocarbonates, Mg₂CO₄-Pnma²⁵, Mg₂CO₄-P2₁/c²⁵, Sr₂CO₄-Pnma²⁶, and Ba₂CO₄-Pnma²⁶ have also been predicted, of which Sr₂CO₄-Pnma²⁷ and Mg₂CO₄-P2₁/c²⁸ have been experimentally verified.

The elastic, seismic, and thermodynamic properties of Ca₂CO₄-Pnma under high pressure have not been investigated so far. Even the elastic constants of CaCO₃ polymorphs were only the experimental results of calcite^29,30,31,32 and aragonite^33,34 at ambient conditions. Using the first-principles method, Belkofsi et al. calculated the elastic constants of three calcite polymorphs(calcite-III, calcite-IIIb, calcite-VI)³⁵, and Huang et al. studied the elastic properties of aragonite, post-aragonite and P2₁/c³⁶. The thermal expansion coefficient^{37,38,39,40,41,42} and heat capacity^37,43,44,45 of calcite and aragonite were measured at ambient conditions, where there was a large difference between the fitted thermal expansion coefficient.

In this work, the structural properties, elastic properties, and seismic properties of Ca₂CO₄-Pnma at 20–140 GPa are studied using the first-principles calculations based on density functional theory and are compared with the results of CaCO₃ polymorphs. The thermodynamic properties of Ca₂CO₄-Pnma are obtained by quasi-harmonic approximation method.

Methods

First-principles calculations are done with using the VASP package^46,47 with projector-augmented wave⁴⁸. The exchange–correlation interactions adopt the Perdew-Burke-Ernzerhof functional within the generalized gradient approximation⁴⁹. The electronic configurations of the atoms are Ca: 3s²3p⁶4s², C: 2s²2p², O: 2s²2p⁴, respectively. The cutoff energy of the plane-wave basis is set to 900 eV. The k-point mesh generation and data processing are obtained by vaspkit program⁵⁰. The k-points mesh of Ca₂CO₄-Pnma, calcite, aragonite, P2₁/c-l, post-aragonite, P2₁/c-h and C222₁ are set to 5 × 7 × 4, 9 × 9 × 2, 7 × 4 × 6, 7 × 10 × 3, 8 × 7 × 8, 8 × 10 × 4, and 6 × 5 × 10 using the Monkhorst–Pack scheme⁵¹, respectively. The convergence criteria for energy and force are 1.0⨯10^–8 eV and 0.02 eV/Å, respectively. Based on the optimized lattice structure, the stress–strain method is used to obtain the elastic stiffness tensor. In order to ensure the accuracy of the elastic constants of Ca₂CO₄-Pnma, the elastic constants of calcite and aragonite are calculated and compared with the available experimental results^32,33. As shown in Table S1 (see Supplementary Material), the calculated results are in good agreement with the experimental results^32,33. The thermodynamic properties are calculated using the quasi-harmonic approximation method⁵² of the PHONOPY program^53,54, and the force constants are calculated using the density functional perturbation theory⁵⁵. The supercells of aragonite and Ca₂CO₄-Pnma adopt 2 × 2 × 2 and 2 × 2 × 1 unit cells, respectively. The convergence tests of the phonon spectrum calculations of aragonite and Ca₂CO₄-Pnma are shown in Tables S2, S3, and Figs. S1–S8 (see Supplementary Material).

Results and discussion

Structural properties

The lattice parameters and equations of state for Ca₂CO₄-Pnma are presented in Fig. 1. It is found that the calculated results are in good agreement with the available experimental²⁴ and previous theoretical results^21,24, indicating the validity of the structure. The sensitivity of the axis to compression is c > b > a. The unit-cell volume at 0 GPa is 303.38 ų and the bulk modulus and its first pressure derivative are K₀ = 113.40 GPa and K₀^′ = 4.00 by fitting the third-order Birch–Murnaghan equation, respectively, which are consistent with the results (V₀ = 302.0(3) ų, K₀ = 108(1) GPa, and K₀^′ = 4.43(3)) of Binck et al.²⁴.

Figure 1

Lattice parameters (a) and equation of state (b) for Ca₂CO₄-Pnma.

In order to better understand the elastic and seismic properties of Ca₂CO₄-Pnma, the candidate CaCO₃ structures (aragonite, P2₁/c-l, post-aragonite, P2₁/c-h, C222₁, ‘− l = low pressure’, ‘− h = high pressure’) in the Earth's mantle are considered. The relative stabilities of the CaCO₃ polymorphs considered in this work are evaluated from their enthalpies. According to Fig. S9 (see Supplementary Material), P2₁/c-l stabilizes above 30 GPa and retains its stability up to 46 GPa, while P2₁/c-h stabilizes above 75 GPa and retains its stability up to at least 140 GPa, which are consistent with the experimental and previous theoretical results^3,5. CaCO₃-C222₁ above 137 GPa is stable relative to post-aragonite, but this does not make any sense^5,56. Because in this interval, the modification P2₁/c-h is more favorable. For comparison with calcium orthocarbonate, four modifications of CaCO₃ must be considered, namely aragonite (20–35 GPa), P2₁/c-l (35–45 GPa), post-aragonite (45–75 GPa) and P2₁/c-h (75–140 GPa).

Elastic properties

The calculated elastic constants of Ca₂CO₄-Pnma are shown in Fig. 2 and Table 1. Within the studied pressure range, $c_{11} > c_{22} > c_{33}$, indicating that compression is easier along the c-axis than along the a- and b-axes. These results are consistent with those of Fig. 1, where the lattice parameter c decreases faster than the lattice parameters a and b with increasing pressure. The calculated elastic constants of CaCO₃ polymorphs are shown in Figs. S10–S13 and Tables S4–S7 (see Supplementary Material), respectively. Therefore, we believe that the calculated elastic constants are correct, but experimental verification is required.

Figure 2

Elastic constants of Ca₂CO₄-Pnma at 20–140 GPa.

Table 1 Calculated elastic constants (c_ij , in GPa), elastic modulus (B and G, in GPa), elastic anisotropy (A₁₀₀, A₀₁₀, and A₀₀₁), and wave velocities(V_P and V_S, in km/s), and seismic anisotropy (A_P and A_S, in %) of Ca₂CO₄-Pnma.

The bulk modulus (B) and shear modulus (G) of Ca₂CO₄-Pnma can be obtained by the Voigt⁵⁷-Reuss⁵⁸-Hill⁵⁹ scheme. As can be seen from Fig. 3 and Table 1, B is greater than G, indicating that with the change of volume, Ca₂CO₄-Pnma is more and more difficult to be compressed, and G is the main factor for the deformation of Ca₂CO₄-Pnma. The B and G of Ca₂CO₄-Pnma at < 75 GPa are larger than those of CaCO₃ polymorphs. The B of Ca₂CO₄-Pnma at 75–140 GPa is equal to that of P2₁/c-h, and the G is slightly larger and almost parallel.

Figure 3

Bulk modulus B and shear modulus G of Ca₂CO₄-Pnma and CaCO₃ polymorphs.

In order to evaluate the elastic anisotropy of Ca₂CO₄-Pnma, we adopt the scheme of Ravindran et al.⁶⁰. The shear anisotropic factors of A₁₀₀ in (100) plane, A₀₁₀ in (010) plane, and A₀₀₁ in (001) plane can be obtained from the following expression:

$$A_{100} = \frac{{4c_{44} }}{{c_{11} + c_{33} - 2c_{13} }}$$

(1)

$$A_{010} = \frac{{4c_{55} }}{{c_{22} + c_{33} - 2c_{23} }}$$

(2)

$$A_{001} = \frac{{4c_{66} }}{{c_{11} + c_{22} - 2c_{12} }}$$

(3)

The variation of shear anisotropic factors A₁₀₀, A₀₁₀ and A₀₀₁ of Ca₂CO₄-Pnma with pressure is displayed in Fig. 4 and Table 1. A₀₁₀ and A₀₀₁ gradually decrease with increasing pressure, A₁₀₀ first increases with the increase of pressure, and then gradually decreases at > 40 GPa. It can also be found that the elastic anisotropy of Ca₂CO₄-Pnma in the lower mantle conditions is very small, and the anisotropy of the (010) plane between [101] and [001] directions is the smallest.

Figure 4

Shear anisotropic factors of Ca₂CO₄-Pnma.

The compressional and shear wave velocities of minerals can be calculated from the elastic constants and densities. The compressional (V_P) and shear (V_S) wave velocities of Ca₂CO₄-Pnma and CaCO₃ polymorphs can be obtained from the Navier's equations⁶¹:

$$V_{P} = \sqrt {\frac{3B + 4G}{{3\rho }}} ,\quad V_{S} = \sqrt {\frac{G}{\rho }}$$

(4)

The densities and wave velocities of Ca₂CO₄-Pnma, CaCO₃ polymorphs and the Preliminary Reference Earth Model (PREM)⁶² are displayed in Fig. 5 and Table 1. From Fig. 5a, it is found that the densities of Ca₂CO₄-Pnma in the lower mantle is less than those of PREM, and greater than those of CaCO₃ polymorphs. As shown in Fig. 5b, the V_P and V_S of CaCO₄-Pnma and CaCO₃ polymorphs are lower than those of PREM, and the V_P and V_S of Ca₂CO₄-Pnma are greater than those of P2₁/c-l and post-aragonite, which are almost the same as those of P2₁/c-h. The wave velocities in various crystallographic directions can be obtained by solving the Christoffel equation $\left| {C_{ijkl} n_{j} n_{l} - \rho V^{2} \delta_{ik} } \right| = 0$⁶³. Figure 6 shows the wave velocities of Ca₂CO₄-Pnma along different crystallization directions at various pressures. The V_P of Ca₂CO₄-Pnma propagates the fastest in the [100] direction. The shear fast-wave velocity propagates the slowest in the [001] direction. With the increase of pressure, the propagation in the [100] and [010] directions become slower. The shear slow-wave velocity in [100] direction propagates more and more slowly as pressure increases.

Figure 5

Densities (a) and wave velocities (b) of Ca₂CO₄-Pnma, CaCO₃ polymorphs and PREM.

Figure 6

Wave velocities of Ca₂CO₄-Pnma along different crystallization directions at various pressures. Made using the AWESoMe program⁶⁴.

The anisotropy A_P of the compressional waves and the polarization anisotropy A_S of the shear waves are defined as⁶⁵:

$$A_{P} = \frac{{V_{P,\max } - V_{P,\min } }}{{V_{P,aggregate} }} \times 100\%$$

(5)

$$A_{S} = \frac{{\left| {V_{S1} - V_{S2} } \right|_{\max } }}{{V_{S,aggregate} }} \times 100\%$$

(6)

Figure 7 and Table 1 show the A_P and A_S of Ca₂CO₄-Pnma and CaCO₃ polymorphs. It can be seen that the seismic anisotropy A_P and A_S of Ca₂CO₄-Pnma are less than those of CaCO₃ polymorphs, and decrease with the increase of pressure, and gradually increase at > 45 GPa. The nonlinear dependence of seismic anisotropy on pressure can be attributed to the nonlinear pressure sensitivity of the wave velocity, which is caused by the nonlinear pressure dependence of its elastic modulus, especially the shear modulus.

Figure 7

Seismic anisotropy A_P (a) and A_S (b) of Ca₂CO₄-Pnma and CaCO₃ polymorphs.

The seismic properties of Ca₂CO₄-Pnma indicate that it could be the potential host of carbon in the subduction slab and coexists with CaCO₃ polymorphs, as suggested by Sagatova et al.^21,23. It was also verified by Binck et al.²⁴. The low wave velocity and small anisotropy of Ca₂CO₄-Pnma may be one of the reasons why it is impossible to detect the presence of carbonate in the lower mantle during the seismic observation of the subduction slab.

Thermodynamic properties

The thermodynamic parameters of minerals are a prerequisite for deriving the thermal state of the Earth's interior. In order to obtain the variation of thermodynamic parameters of Ca₂CO₄-Pnma with temperature and pressure, we first verify the constant pressure heat capacity C_P of aragonite at 0 GPa, and find that the calculated results are in good agreement with the experimental results⁴⁴(Fig. 8). On this basis, the predicted heat capacity and thermal expansion coefficient $\alpha$ of Ca₂CO₄-Pnma are shown in Figs. 9 and 10, respectively.

Figure 8

Constant pressure heat capacity C_P of aragonite at 0 GPa.

Figure 9

Constant volume heat capacity C_V (a) and constant pressure heat capacity C_P (b) of Ca₂CO₄-Pnma at various pressures.

Figure 10

Thermal expansion coefficient α of Ca₂CO₄-Pnma at various pressures.

Figure 9 shows that the constant capacity heat capacity C_V increases sharply with increasing temperature at low temperatures. Due to the suppression of non-harmonic effects under high pressure, the constant volume heat capacity C_V under high pressure and high temperature is very close to the Dulong Petit limit. The constant pressure heat capacity C_P is very close to the constant capacity heat capacity C_V. In addition, the effects of temperature and pressure on constant capacity heat capacity C_V and constant pressure heat capacity C_P are opposite, and the impact of temperature is more noteworthy.

It can be seen from Fig. 10 that thermal expansion coefficient α at low temperature increases rapidly with the increase of temperature and tends to flatten rapidly with the increase of temperature. With the increase of pressure, the thermal expansion coefficient $\alpha$ decreases rapidly, and the influence of temperature becomes less and less obvious, resulting in linear high temperature behavior.

Conclusions

On the basis of the determination of the stability for CaCO₃ polymorphs in the lower mantle conditions and the verification of the structural parameters of Ca₂CO₄-Pnma, we study the elastic, seismic and thermodynamic properties of Ca₂CO₄-Pnma, and compared the results with those of CaCO₃ polymorphs. The research shows that the densities of Ca₂CO₄-Pnma in the lower mantle are greater than those of CaCO₃ polymorphs, and the seismic anisotropies are less than those of CaCO₃ polymorphs. The wave velocities of Ca₂CO₄-Pnma and CaCO₃ polymorphs are relatively low, and the wave velocities of Ca₂CO₄-Pnma and CaCO₃-P2₁/c-h are almost the same. This means that the presence of carbonate in the lower mantle is unlikely to be detected by seismic observations of subducted slab. By verifying the constant pressure heat capacity of aragonite at 0 GPa, the thermodynamic properties of Ca₂CO₄-Pnma at high temperature and high pressure are calculated using the quasi-harmonic approximation method. The results of this study are helpful to better understand the behavior of calcium carbonate in the lower mantle conditions.