## Abstract

Orthorhombic Ca_{2}CO_{4} 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 Ca_{2}CO_{4}-*Pnma* at 20–140 GPa using first-principles calculations, and compare them with the results of CaCO_{3} polymorphs. The results show that the structural parameters of Ca_{2}CO_{4}-*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 Ca_{2}CO_{4}-*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.

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## Introduction

As the most important carbonate, CaCO_{3} is transported to the deep mantle by subduction slab and plays a crucial role in the global long-term carbon cycle^{1}. It is also a mineral that plays a key role in biomineralization^{2}. However, CaCO_{3} 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, SiO_{2}, and MgSiO_{3}^{7,10,16,17}, while ignoring the reaction with CaO. Previously, Al-Shemali and Boldyrev^{18} mentioned the possible formation of calcium orthocarbonate Ca_{2}CO_{4} in the CaCO_{3} + CaO system under high pressure. Recently, using AIRSS^{19} and USPEX^{20} crystal structure prediction methods, Sagatova et al.^{21} discovered a new structure of calcium orthocarbonate Ca_{2}CO_{4} (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}-Ca_{2}SiO_{4} phase^{22}. Afterward, they found that Ca_{2}CO_{4}-*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 approximation^{23}. Subsequently, Binck et al.^{24} verified the results of Sagatova et al.^{23} with single-crystal diffraction experiments. In addition, other alkaline earth orthocarbonates, Mg_{2}CO_{4}-*Pnma*^{25}, Mg_{2}CO_{4}-*P*2_{1}/*c*^{25}, Sr_{2}CO_{4}-*Pnma*^{26}, and Ba_{2}CO_{4}-*Pnma*^{26} have also been predicted, of which Sr_{2}CO_{4}-*Pnma*^{27} and Mg_{2}CO_{4}-*P*2_{1}/*c*^{28} have been experimentally verified.

The elastic, seismic, and thermodynamic properties of Ca_{2}CO_{4}-*Pnma* under high pressure have not been investigated so far. Even the elastic constants of CaCO_{3} 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 *P*2_{1}/*c*^{36}. 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 Ca_{2}CO_{4}-*Pnma* at 20–140 GPa are studied using the first-principles calculations based on density functional theory and are compared with the results of CaCO_{3} polymorphs. The thermodynamic properties of Ca_{2}CO_{4}-*Pnma* are obtained by quasi-harmonic approximation method.

## Methods

First-principles calculations are done with using the VASP package^{46,47} with projector-augmented wave^{48}. The exchange–correlation interactions adopt the Perdew-Burke-Ernzerhof functional within the generalized gradient approximation^{49}. The electronic configurations of the atoms are Ca: 3*s*^{2}3*p*^{6}4*s*^{2}, C: 2*s*^{2}2*p*^{2}, O: 2*s*^{2}2*p*^{4}, 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 program^{50}. The *k*-points mesh of Ca_{2}CO_{4}-*Pnma*, calcite, aragonite, *P*2_{1}/*c*-l, post-aragonite, *P*2_{1}/*c*-h and *C*222_{1} 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 scheme^{51}, 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 Ca_{2}CO_{4}-*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 method^{52} of the PHONOPY program^{53,54}, and the force constants are calculated using the density functional perturbation theory^{55}. The supercells of aragonite and Ca_{2}CO_{4}-*Pnma* adopt 2 × 2 × 2 and 2 × 2 × 1 unit cells, respectively. The convergence tests of the phonon spectrum calculations of aragonite and Ca_{2}CO_{4}-*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 Ca_{2}CO_{4}-*Pnma* are presented in Fig. 1. It is found that the calculated results are in good agreement with the available experimental^{24} 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 Å^{3} and the bulk modulus and its first pressure derivative are *K*_{0} = 113.40 GPa and *K*_{0}^{′} = 4.00 by fitting the third-order Birch–Murnaghan equation, respectively, which are consistent with the results (*V*_{0} = 302.0(3) Å^{3}, *K*_{0} = 108(1) GPa, and *K*_{0}^{′} = 4.43(3)) of Binck et al.^{24}.

In order to better understand the elastic and seismic properties of Ca_{2}CO_{4}-*Pnma*, the candidate CaCO_{3} structures (aragonite, *P*2_{1}/*c*-l, post-aragonite, *P*2_{1}/*c*-h, *C*222_{1}, ‘− l = low pressure’, ‘− h = high pressure’) in the Earth's mantle are considered. The relative stabilities of the CaCO_{3} polymorphs considered in this work are evaluated from their enthalpies. According to Fig. S9 (see Supplementary Material), *P*2_{1}/*c*-l stabilizes above 30 GPa and retains its stability up to 46 GPa, while *P*2_{1}/*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}. CaCO_{3}-*C*222_{1} above 137 GPa is stable relative to post-aragonite, but this does not make any sense^{5,56}. Because in this interval, the modification *P*2_{1}/*c*-h is more favorable. For comparison with calcium orthocarbonate, four modifications of CaCO_{3} must be considered, namely aragonite (20–35 GPa), *P*2_{1}/c-l (35–45 GPa), post-aragonite (45–75 GPa) and *P*2_{1}/c-h (75–140 GPa).

### Elastic properties

The calculated elastic constants of Ca_{2}CO_{4}-*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 CaCO_{3} 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.

The bulk modulus (*B*) and shear modulus (*G*) of Ca_{2}CO_{4}-*Pnma* can be obtained by the Voigt^{57}-Reuss^{58}-Hill^{59} scheme. As can be seen from Fig. 3 and Table 1, *B* is greater than *G*, indicating that with the change of volume, Ca_{2}CO_{4}-*Pnma* is more and more difficult to be compressed, and *G* is the main factor for the deformation of Ca_{2}CO_{4}-*Pnma*. The *B* and *G* of Ca_{2}CO_{4}-*Pnma* at < 75 GPa are larger than those of CaCO_{3} polymorphs. The *B* of Ca_{2}CO_{4}-*Pnma* at 75–140 GPa is equal to that of *P*2_{1}/*c-h*, and the *G* is slightly larger and almost parallel.

In order to evaluate the elastic anisotropy of Ca_{2}CO_{4}-*Pnma*, we adopt the scheme of Ravindran et al.^{60}. The shear anisotropic factors of *A*_{100} in (100) plane, *A*_{010} in (010) plane, and *A*_{001} in (001) plane can be obtained from the following expression:

The variation of shear anisotropic factors *A*_{100}, *A*_{010} and *A*_{001} of Ca_{2}CO_{4}-*Pnma* with pressure is displayed in Fig. 4 and Table 1. *A*_{010} and *A*_{001} gradually decrease with increasing pressure, *A*_{100} first increases with the increase of pressure, and then gradually decreases at > 40 GPa. It can also be found that the elastic anisotropy of Ca_{2}CO_{4}-*Pnma* in the lower mantle conditions is very small, and the anisotropy of the (010) plane between [101] and [001] directions is the smallest.

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 Ca_{2}CO_{4}-*Pnma* and CaCO_{3} polymorphs can be obtained from the Navier's equations^{61}:

The densities and wave velocities of Ca_{2}CO_{4}-*Pnma*, CaCO_{3} 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 Ca_{2}CO_{4}-*Pnma* in the lower mantle is less than those of PREM, and greater than those of CaCO_{3} polymorphs. As shown in Fig. 5b, the *V*_{P} and *V*_{S} of CaCO_{4}-*Pnma* and CaCO_{3} polymorphs are lower than those of PREM, and the *V*_{P} and *V*_{S} of Ca_{2}CO_{4}-*Pnma* are greater than those of *P*2_{1}/*c-*l and post-aragonite, which are almost the same as those of *P*2_{1}/*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 Ca_{2}CO_{4}-*Pnma* along different crystallization directions at various pressures. The *V*_{P} of Ca_{2}CO_{4}-*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.

The anisotropy *A*_{P} of the compressional waves and the polarization anisotropy *A*_{S} of the shear waves are defined as^{65}:

Figure 7 and Table 1 show the *A*_{P} and *A*_{S} of Ca_{2}CO_{4}-*Pnma* and CaCO_{3} polymorphs. It can be seen that the seismic anisotropy *A*_{P} and *A*_{S} of Ca_{2}CO_{4}-*Pnma* are less than those of CaCO_{3} 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.

The seismic properties of Ca_{2}CO_{4}-*Pnma* indicate that it could be the potential host of carbon in the subduction slab and coexists with CaCO_{3} 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 Ca_{2}CO_{4}-*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 Ca_{2}CO_{4}-*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 results^{44}(Fig. 8). On this basis, the predicted heat capacity and thermal expansion coefficient \(\alpha\) of Ca_{2}CO_{4}-*Pnma* are shown in Figs. 9 and 10, respectively.

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 CaCO_{3} polymorphs in the lower mantle conditions and the verification of the structural parameters of Ca_{2}CO_{4}-*Pnma*, we study the elastic, seismic and thermodynamic properties of Ca_{2}CO_{4}-*Pnma*, and compared the results with those of CaCO_{3} polymorphs. The research shows that the densities of Ca_{2}CO_{4}-*Pnma* in the lower mantle are greater than those of CaCO_{3} polymorphs, and the seismic anisotropies are less than those of CaCO_{3} polymorphs. The wave velocities of Ca_{2}CO_{4}-*Pnma* and CaCO_{3} polymorphs are relatively low, and the wave velocities of Ca_{2}CO_{4}-*Pnma* and CaCO_{3}-*P*2_{1}/*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 Ca_{2}CO_{4}-*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.

## Data availability

The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

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## Acknowledgements

This work is supported by the Industrial Support and Guidance Project of Colleges and Universities of Gansu Province (No. 2022CYZC-37), the Key Natural Science Foundation of Gansu Province (No. 20JR5RA211) and the Talent Innovation and Entrepreneurship Project of Lanzhou City (No. 2020-RC-18).

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### Contributions

Z.-J.L. designed the calculations and wrote the manuscript. X.-W.S., C.-R.Z. and Z.-L.L analyzed the results. T.L., T.S. and X.-D.W. performed partial calculations. All authors reviewed the manuscript.

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Liu, ZJ., Li, T., Sun, XW. *et al.* First-principles study on the high-pressure physical properties of orthocarbonate Ca_{2}CO_{4}.
*Sci Rep* **13**, 11422 (2023). https://doi.org/10.1038/s41598-023-38604-w

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DOI: https://doi.org/10.1038/s41598-023-38604-w

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