Comparative study on high-pressure physical properties of monoclinic MgCO₃ and Mg₂CO₄

Abstract

The physical properties of Mg-carbonate at high temperature and pressure are crucial for understanding the deep carbon cycle. Here, we use first-principles calculations to study the physical properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c under high pressure. The research shows that the structure and equation of state of MgCO₃-C2/m are in good agreement with the experimental results, and the phase transition pressure of Mg₂CO₄ from pnma to P2₁/c structure is 44.66 GPa. By comparing the elastic properties, seismic properties and anisotropy of MgCO₃-C2/m and Mg₂CO₄-P2₁/c, it is found that the elastic modulus and sound velocity of Mg₂CO₄-P2₁/c are smaller than those of MgCO₃-C2/m, while the anisotropy is larger than that of MgCO₃-C2/m. These results indicate that Mg₂CO₄-P2₁/c exists in the deep mantle and may be the main reason why carbonate cannot be detected. The minimum thermal conductivity of MgCO₃-C2/m and Mg₂CO₄-P2₁/c is the largest in the [010] direction and the smallest in the [001] direction. The thermodynamic properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are predicted using the quasi-harmonic approximation (QHA) method.

Introduction

Magnesite (space group $R\overline{3} c$) is subducted into the deep mantle as the primary carbon carrier, and its high-temperature and high-pressure physical properties are critical for understanding the deep carbon cycle^1,2. Previous studies have mainly focused on the structural phase transition of MgCO₃-$R\overline{3} c$ under high-temperature and high-pressure conditions, transforming to monoclinic MgCO₃ (space group C2/m) at around 80 GPa. Oganov et al. used the USPEX method to predict for the first time that MgCO₃-C2/m is most stable in the lower mantle greater than 82.4 GPa³. MgCO₃-C2/m has 30 atoms, in which adjacent oxygen atoms form tetrahedra around carbon atoms, and Mg atoms are in octet and tenfold coordination. Subsequently, the structure of MgCO₃-C2/m was verified experimentally^4,5,6 and theoretically^{7,8,9,10,11,12,13}. Recently, Gavryushkin et al. used USPEX and AIRSS methods to find that MgCO₃ reacts with MgO to form Mg₂CO₄, which has two structures, orthorhombic (space group Pnma) and monoclinic (space group P2₁/c), and its structure is P2₁/c when the pressure is higher than 50 GPa¹⁴. Their experiments later confirmed the existence of Mg₂CO₄-P2₁/c at the temperature and pressure of the Earth's lower mantle¹⁵. Mg₂CO₄-P2₁/c has 28 atoms, it is isostructural to β-Ca₂SiO₄¹⁶, and the atoms at the two positions Mg(1) and Mg(2) are six-coordinated, with octahedral coordination polyhedra. Earlier reports^17,18,19,20, although the structure of Mg₂CO₄ was not determined, believed that it is stable at high pressure.

To understand the carbon cycle in the deep earth, it is crucial to study the structure, phase transition, equations of state, elasticity, thermodynamics, and thermal conductivity of MgCO₃-C2/m and Mg₂CO₄-P2₁/c under high pressure. Recently, Maeda et al. measured the equation of state of MgCO₃-C2/m at high pressure⁶. Since it is extremely difficult to measure the elastic constants, thermodynamic parameters, and thermal conductivity of minerals experimentally, the properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c have not been reported experimentally. Even the elastic constant of MgCO₃-$R\overline{3} c$ can only be measured to 13.7 GPa²¹, and its thermodynamic properties are only at low pressure, and the results at high pressure are extrapolated^22,23,24.

In the present work, the structures, phase transitions, equations of state, elastic properties, seismic properties, and minimum thermal conductivity of MgCO₃-C2/m and Mg₂CO₄-P2₁/c at high pressure are investigated using first-principles calculations and compared with the available experimental and theoretical results. The QHA method is adopted to research the thermodynamic properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c.

Methods

First-principles calculations are used to investigate the high-pressure physical properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c using projector-augmented wave (PAW)²⁵ as implemented in VASP^26,27. The electronic configurations: 2p⁶3s² for Mg, 2s²2p² for C and 2s²2p⁴ for O are considered for the valence electrons. The Perdew–Burke–Ernzerhof modified solid (PBEsol) in the generalized gradient approximation (GGA)²⁸ describes the exchange and correlation potentials. The generation of k-point mesh and the calculation of elastic properties utilize the vaspkit program²⁹. The cutoff energy for the plane wave is set to 900 eV. The k-points mesh of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are set to 4 × 5 × 5 and 13 × 9 × 7 using the Monkhorst–Pack scheme³⁰, respectively. The thermodynamic properties are calculated by the QHA method³¹.

Results and discussion

Structural parameters, phase transition and equation of state

The crystal structures of MgCO₃-C2/m and Mg₂CO₄-P2₁/c in the unit cell are shown in Fig. 1. The optimized lattice parameters are summarized in Table 1 and compared with available experimental and previously calculated results. At 110 GPa, the calculated results of MgCO₃-C2/m are consistent with the experimental results⁵. The results for Mg₂CO₄-P2₁/c at 100 GPa agree well with the previous calculations¹⁴.

Figure 1

Crystal structures of MgCO₃-C2/m (a) and Mg₂CO₄-P2₁/c (b) in unit cell. The crystal structures are drawn by VESTA³².

Table 1 Lattice parameters of MgCO₃-C2/m and Mg₂CO₄-P2₁/c at 110 GPa and 100 GPa, respectively, compared with experimental and previous calculations.

As shown in Fig. 2, the present calculated transition pressure from Mg₂CO₄-Pnma to Mg₂CO₄-P2₁/c is 44.66 GPa, while Gavryushkin et al. calculated 52 GPa¹⁴. This error may be caused by the use of different exchange correction functions, PBEsol is used in the present work, while PBE was used by Gavryushkin et al.¹⁴. The accuracy of using PBEsol to calculate the properties of Mg-carbonate has been examined in the previous studies¹³. In the previous study, MgCO₃-C2/m was stable in the lower mantle above 80 GPa^{3,5,6,7,8,9,10,11,12,13,33}. Therefore, in order to facilitate comparison, the high-pressure properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c in the pressure range of 50–140 GPa are investigated in this work.

Figure 2

Enthalpy difference between Mg₂CO₄-P2₁/c and Mg₂CO₄-Pnma.

The equations of state for MgCO₃-C2/m and Mg₂CO₄-P2₁/c at 50 to 140 GPa are shown in Fig. 3. It is found that the equation of state of MgCO₃-C2/m is in good agreement with available experimental data⁶. The equation of state of Mg₂CO₄-P2₁/c is smaller than that of MgCO₃-C2/m, and is almost parallel. The formula unit volume of MgCO₃-C2/m is smaller than that of Mg₂CO₄-P2₁/c, which is in line with their molecular formula composition relationship.

Figure 3

Equation of state for MgCO₃-C2/m and Mg₂CO₄-P2₁/c.

Elastic properties

For monoclinic MgCO₃-C2/m and Mg₂CO₄-P2₁/c, there are 13 independent elastic constants ($c_{11}$, $c_{12}$, $c_{13}$, $c_{15}$, $c_{22}$, $c_{23}$, $c_{23}$, $c_{25}$, $c_{33}$, $c_{35}$, $c_{44}$, $c_{46}$, $c_{55}$ and $c_{66}$). The elastic constants are calculated using the stress–strain method²⁹. In this work, all calculated elastic stiffness constants c_ij are checked using the mechanical stability criteria³⁴ and find that they all meet the mechanical stability criteria in the studied pressure range, thus they are mechanically stable.

The elastic constants of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are plotted in Figs. 4 and 5, respectively. It is found from Fig. 4 that at 50–110 GPa, $c_{22} > c_{33} > c_{11}$, indicating that the a-axis of MgCO₃-C2/m is the most easily compressed, and the b-axis is the least compressed. At > 110 GPa, $c_{33} > c_{22} > c_{11}$, the c-axis is least likely to be compressed. From Fig. 5, it can be seen that $c_{22} > c_{11} > c_{33}$ in the studied pressure range, indicating that the c-axis of Mg₂CO₄-P2₁/c is the most easily compressed, and the b-axis is the least compressible. In the previous study¹³, the elastic constants of MgCO₃-$R\overline{3} c$ are consistent with the experimental results²¹. Therefore, the predicted elastic constants of MgCO₃-C2/m and Mg₂CO₄-P2₁/c in this work should be correct, but it needs to be further verified by experiments.

Figure 4

Elastic constants of MgCO₃-C2/m.

Figure 5

Elastic constants of Mg₂CO₄-P2₁/c.

Based on the calculated elastic constants, the bulk modulus B and shear modulus G of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are calculated using the Voigt-Reuss-Hill method^35,36,37. As shown in Fig. 6, the bulk modulus B and shear modulus G of MgCO₃-C2/m and Mg₂CO₄-P2₁/c increase with increasing pressure, and the bulk modulus B and shear modulus G of MgCO₃-C2/m are larger than those of Mg₂CO₄-P2₁/c at 50–140 GPa.

Figure 6

Bulk modulus B and shear modulus G of MgCO₃-C2/m and Mg₂CO₄-P2₁/c.

Seismic properties

Based on the calculated bulk and shear moduli and density, the compression (V_P) and shear (V_S) velocities of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are given by the following expressions³⁸:

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

(1)

$$V_{S} = \sqrt {\frac{G}{\rho }}$$

(2)

As shown in Fig. 7, the density of Mg₂CO₄-P2₁/c is larger than that of MgCO₃-C2/m, and the difference between their bulk modulus and shear modulus is smaller, respectively. Therefore, the V_P and V_S of MgCO₃-C2/m are larger than those of Mg₂CO₄-P2₁/c in the studied pressure range, and their V_P and V_S tend to be parallel, respectively (See Fig. 8).

Figure 7

Density of MgCO₃-C2/m and Mg₂CO₄-P2₁/c.

Figure 8

Compression (V_P) and shear (V_S) velocities of MgCO₃-C2/m and Mg₂CO₄-P2₁/c.

The V_P and V_S of MgCO₃-C2/m and Mg₂CO₄-P2₁/c along different directions can be obtained by solving the Christoffel equation³⁹ $\left| {C_{ijkl} n_{j} n_{l} - \rho V^{2} \delta_{ik} } \right| = 0$. In order to visualize the propagation wave velocities of MgCO₃-C2/m and Mg₂CO₄-P2₁/c along different directions, the AWESoMe program^40,41 is used to plot their 3D representations of V_P and V_S and shear wave splitting and polarization vectors at various pressures (Figs. 9 and 10).

Figure 9

3D representation of the V_P and V_S and the shear wave splitting and polarization vectors of MgCO₃-C2/m at various pressures.

Figure 10

3D representation of the V_P and V_S and the shear wave splitting and polarization vectors of Mg₂CO₄-P2₁/c at various pressures.

The anisotropy A_P of the V_P for MgCO₃-C2/m and Mg₂CO₄-P2₁/c is defined as⁴²:

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

(3)

The polarization anisotropy A_S of the V_S is defined as

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

(4)

The seismic anisotropies of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are shown in Fig. 11. The seismic anisotropy of MgCO₃-C2/m at 75 GPa is found to be in good agreement with the previous theoretical calculations¹⁰. The anisotropy of Mg₂CO₄-P2₁/c is obviously larger than that of MgCO₃-C2/m. The seismic anisotropy of MgCO₃-C2/m and Mg₂CO₄-P2₁/c showed obvious nonlinear dependence on pressure. This is mainly due to the nonlinear pressure of wave velocity caused by the nonlinear pressure dependence of the elastic moduli of MgCO₃-C2/m and Mg₂CO₄-P2₁/c, especially their shear moduli.

Figure 11

Seismic anisotropy of MgCO₃-C2/m and Mg₂CO₄-P2₁/c.

Although the previous experimental^4,5,6 and theoretical^{7,8,9,10,11,12,13} studies obtained MgCO₃-C2/m at high temperature and pressure, they did not consider the reaction with MgO, the main mineral of the Earth's lower mantle. The theoretical¹⁴ and experimental¹⁵ results of Gavryushkin et al. show that MgCO₃ reacts with MgO to form Mg₂CO₄-P2₁/c orthocarbonate under the temperature and pressure conditions of the lower mantle. By comparing the high-pressure physical properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c, it is found that their seismic anisotropy is quite different, while the equation of state, elastic modulus, density and wave velocity have similar relationship with pressure. The low wave velocities of Mg₂CO₄-P2₁/c are more suitable to explain the existence of low-velocity zone near the subducting slab. Therefore, we believe that Mg₂CO₄-P2₁/c may exist in the deep mantle, providing strong evidence for carbon storage in carbonate minerals, which may be the main reason why carbonate rocks cannot be detected in the lower mantle.

Minimum thermal conductivity

The thermal conductivity of minerals is critical to understanding the Earth's thermal balance and history⁴³. The minimum thermal conductivity of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are calculated using Cahill’s model:

$$K_{\min } = ({{k_{B} } \mathord{\left/ {\vphantom {{k_{B} } {2.48}}} \right. \kern-\nulldelimiterspace} {2.48}})n^{2/3} (v_{P} + 2v_{S} )$$

(5)

The anisotropy of the minimum thermal conductivity can be calculated by changing Eq. (5) into the following form:

$$K_{min} = ({{k_{B} } \mathord{\left/ {\vphantom {{k_{B} } {2.48}}} \right. \kern-\nulldelimiterspace} {2.48}})n^{2/3} (v_{P} + v_{S1} + v_{S2} )$$

(6)

where k_B is Boltzmann’s constant, n is the atomic number density per unit volume. The minimum thermal conductivities of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are shown in Fig. 12, and it is found that their minimum thermal conductivities increase with the increase of pressure, and that of MgCO₃-C2/m is larger than that of Mg₂CO₄-P2₁/c. In the studied pressure range, K_min[010] > K_min[100] > K_min[001], indicating that the thermal conductivity in the [010] direction is the largest and the thermal conductivity in the [001] direction is the smallest.

Figure 12

Minimum thermal conductivity of MgCO₃-C2/m (a) and Mg₂CO₄-P2₁/c (b).

Thermodynamic properties

Thermodynamic parameters are the preconditions for deriving the thermal state of the Earth's interior. Therefore, the thermodynamic properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c are crucial for studying the thermal state of the lower mantle. The constant volume heat capacity (C_V) and the thermal expansion coefficient ($\alpha$) of MgCO₃-C2/m and Mg₂CO₄-P2₁/c at various pressures are depicted in Figs. 13 and 14, respectively. The C_V and $\alpha$ of MgCO₃-C2/m are larger than those of Mg₂CO₄-P2₁/c under the same pressure.

Figure 13

Constant volume heat capacity C_V of MgCO₃-C2/m (a) and Mg₂CO₄-P2₁/c (b).

Figure 14

Thermal expansion coefficient $\alpha$ of MgCO₃-C2/m (a) and Mg₂CO₄-P2₁/c (b).

Conclusions

On the basis of verifying the structure and equation of state of MgCO₃-C2/m, the phase transition pressure of Mg₂CO₄-P2₁/c is determined. The high-pressure physical properties of MgCO₃-C2/m and Mg₂CO₄-P2₁/c at 50–140 GPa are investigated by first-principles calculations. By comparison, it is found that the elastic modulus and wave velocity of Mg₂CO₄-P2₁/c are smaller than those of MgCO₃-C2/m, and the density and seismic anisotropy are larger than those of MgCO₃-C2/m. The low wave velocity of Mg₂CO₄-P2₁/c may be more suitable to explain the existence of the low-velocity zone near the subducting slab. Therefore, it is believed that Mg₂CO₄-P2₁/c may exist in the deep mantle, providing strong evidence for carbon storage in carbonates and the reason why it cannot be detected in the lower mantle. The minimum thermal conductivity of MgCO₃-C2/m is greater than that of Mg₂CO₄-P2₁/c, and their minimum thermal conductivity is the largest in the [010] direction and the smallest in the [001] direction. The constant volume heat capacity C_V and thermal expansion coefficient $\alpha$ of MgCO₃-C2/m are larger than those of Mg₂CO₄-P2₁/c. Unfortunately, there are no experimental data on the elastic constants, thermodynamic parameters, and minimum thermal conductivity of MgCO₃-C2/m and Mg₂CO₄-P2₁/c, so further verification is required.