# First-Principles Study of the Elastic Properties of Nickel Sulfide Minerals under High Pressure

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{3}S

_{2}), and polydymite (Ni

_{3}S

_{4}) under high pressure are investigated. Our calculated results show that the crystal structures of these Ni sulfides are well predicted. These Ni sulfides are mechanically stable under the high pressure of the upper mantle. The elastic constants show different changing trends with increasing pressure. The bulk modulus of these Ni sulfides increases linearly with pressure, whereas shear modulus is less sensitive to pressure. The universal elastic anisotropic index A

^{U}also shows different changing trends with pressure. Furthermore, the elastic wave velocities of Ni sulfides are much lower than those of olivine and enstatite.

## 1. Introduction

_{3}S

_{2}), and polydymite (Ni

_{3}S

_{4}) are the main Ni sulfide minerals, which are thermodynamically stable and widespread in mantle peridotite [4,5].

_{3}S

_{2}), and polydymite (Ni

_{3}S

_{4}), which are the main Ni sulfides in the upper mantle, under high pressure. The results of this study can provide key parameters for understanding the structural stability of sulfide minerals in the deep earth, and are of great significance for interpreting the wave velocity structure and understanding the dynamic properties of the Earth’s interior.

## 2. Methods

_{3}S

_{2}), and polydymite (Ni

_{3}S

_{4}) were 8 × 8 × 8, 9 × 9 × 9, and 6 × 6 × 6, respectively. During geometry optimization, the convergence threshold for the maximum stress was 0.01 GPa. In the elastic constants calculation, the convergence thresholds for energy change, maximum force, and maximum displacement were 1 × 10

^{−6}eV/atom, 0.002 eV/Å, and 1 × 10

^{−4}Å, respectively. Six distorted structures were generated for each strain pattern. The maximum strain amplitude was 0.003 to make sure that the crystal deformation was within the range of linear elasticity. In order to obtain the elastic constants with high accuracy, the elastic constants were calculated three times under each pressure, and the one with the smallest error was taken.

_{ij}is the elastic compliance, and the subscripts V and R in these equations denote the Voigt and Reuss moduli, respectively.

## 3. Results

#### 3.1. Crystal Structure

_{3}S

_{2}) contains five atoms in the rhombohedral primitive cell with space group R32 [29]. Polydymite (Ni

_{3}S

_{4}) has a cubic Fd-3m structure with 56 atoms per unit cell [30]. The primitive cell containing 14 atoms was used in the elastic calculation. The crystal structures of millerite, heazlewoodite, and polydymite calculated under 0 GPa are given in Table 1, together with the measured and previously calculated results. The atom positions in their structures under 0 GPa are given in Table S1. It can be found that our calculated results are in good agreement with the measured data, with errors no more than 0.4%, and it is also in good agreement with the previously calculated results.

_{3}S

_{2}, with the pressure increasing, the lattice constant a and axial angle α show different trends of variation. The lattice constant a decreases with pressure, whereas the axial angle α increases with pressure. For Ni

_{3}S

_{4}, the lattice constant a also has a negative correlation with pressure.

#### 3.2. Equation of State

_{0}, B

_{0}and B’

_{0}represent the unit cell volume, bulk modulus, and its pressure derivative under 0 GPa, respectively, which are the key parameters in the equation of state and can be obtained by fitting the third-order Birch–Murnaghan equation. The calculated equation of state parameters of these Ni sulfides are listed in Table 2. It can be found that there are some differences between our calculated parameters and the previous data [33]. Such differences arise due to the fact that our parameters were calculated at 0 K while the previous data were calculated at room temperature.

_{0}, which decreases with increasing pressure, is defined as the volume anticompressibility. Figure 2 shows the relative volume of millerite, heazlewoodite, and polydymite as a function of pressure. As shown in Figure 2, Ni

_{3}S

_{4}has the lowest volume anticompressibility, because it has the largest relative volume decrease.

#### 3.3. Elastic Constant

_{11}, C

_{12}, C

_{13}, C

_{14}, C

_{33}, and C

_{44}). Polydymite belongs to the cubic system, which has three independent elastic constants (i.e., C

_{11}, C

_{12}, and C

_{44}). The variation of the elastic constants of millerite, heazlewoodite, and polydymite with pressure is presented in Figure 3. From Figure 3, we can find that the elastic constants have different change trends with pressure.

_{14}has always been the smallest of the six components in the pressure range of ~0–24 GPa, and it is very insensitive to pressure change. C

_{44}is the second smallest of these components, and it increases slowly with the increase of pressure. The values of C

_{11}and C

_{33}are relatively large; both of them increase linearly with pressure. However, the difference between C

_{11}and C

_{33}increases gradually with increasing pressure, indicating that the difference of the compressibility in [001] and [100] directions increases gradually with increasing pressure. The difference between C

_{12}and C

_{13}is almost the same in the pressure range.

_{14}has been the smallest component in the pressure range studied, and it is insensitive to pressure change. Such phenomenon is very similar to millerite. C

_{11}and C

_{33}have always been the two largest components, and the difference between them is very small in the pressure range, implying that the difference of the compressibility in those directions is small. The second smallest component, C

_{44}, is also insensitive to pressure change. C

_{13}and C

_{12}increase linearly with increasing pressure, but differ in increments.

_{11}, which represents the uniaxial deformation, keeps the maximum all the time. C

_{44}, which represents the pure shear deformation, is very insensitive to pressure, whereas C

_{12}increases linearly with pressure.

#### 3.4. Elasticity Modulus

_{11}= C

_{22}, C

_{13}= C

_{23}, C

_{24}= −C

_{14}, C

_{55}= C

_{44}, C

_{66}= (C

_{11}− C

_{12})/2, S

_{11}= S

_{22}, S

_{13}= S

_{23}, S

_{24}= −S

_{14}, S

_{55}= S

_{44}, and S

_{66}= (S

_{11}− S

_{12})/2. Taking these conditions into account, Equations (3)–(6) can be simplified as follows:

_{11}= C

_{22}= C

_{33}, C

_{12}= C

_{13}= C

_{23}, C

_{44}= C

_{55}= C

_{66}, and S

_{11}= S

_{22}= S

_{33}, S

_{12}= S

_{13}= S

_{23}, S

_{44}= S

_{55}= S

_{66}. The bulk modulus (B

_{V}, B

_{R}) and shear modulus (G

_{V}, G

_{R}) can be simplified as follows:

_{3}S

_{2}, and Ni

_{3}S

_{4}increase with pressure, and the bulk modulus of Ni

_{3}S

_{2}is the largest in the pressure range of ~0–24 GPa. When the pressure is larger than 23 GPa, the bulk modulus of Ni

_{3}S

_{4}displays a much greater increasing range, corresponding to the greater increase in C

_{11}and C

_{12}of Ni

_{3}S

_{4}. Nevertheless, for the shear modulus, the three of them are all insensitive to pressure change. With the increase of pressure, NiS has a moderate increase. Ni

_{3}S

_{2}increases at a slower rate, whereas Ni

_{3}S

_{4}remains almost invariant.

## 4. Discussion

#### 4.1. Mechanical Stability of Ni Sulfides under High Pressure

_{ij}satisfy the Born stability criteria [34,36]. For a trigonal system, the mechanical stability conditions are as follows:

_{ij}of NiS and Ni

_{3}S

_{2}satisfy the criteria in the pressure range of ~0–24 GPa, indicating that millerite and heazlewoodite are mechanically stable under the pressure of the upper mantle. According to Equation (19), the elastic constants C

_{ij}of Ni

_{3}S

_{4}also satisfy the criteria. Thus, polydymite is also mechanically stable under the pressure of the upper mantle.

#### 4.2. Elastic Anisotropy of Ni Sulfides

^{U}), which is the most effective method to evaluate the elastic anisotropy of minerals due to it considering both bulk and shear moduli [40], is used to evaluate the elastic anisotropy of Ni sulfides. A

^{U}was defined as follows:

^{U}is identically zero. The deviation of A

^{U}from zero corresponds to the extent of elastic anisotropy. The pressure dependence of the A

^{U}for these Ni sulfides is shown in Figure 5. According to the elastic anisotropy diagram (EAD) proposed by Ranganathan and Ostojastarzewski [40], the elastic anisotropy of these Ni sulfides is small. On the whole, with the increase of pressure, the A

^{U}of Ni

_{3}S

_{2}increases, the A

^{U}of Ni

_{3}S

_{4}is almost unchanged, and the A

^{U}of NiS decreases. Under low pressure, the elastic anisotropy of NiS is stronger than that of Ni

_{3}S

_{2}and Ni

_{3}S

_{4}. With pressure increasing to greater than 5 GPa, the A

^{U}of Ni

_{3}S

_{2}becomes the largest, indicating that the anisotropy of Ni

_{3}S

_{2}is the strongest. In the pressure range of ~0–24 GPa, the anisotropy of Ni

_{3}S

_{4}is the weakest.

#### 4.3. Elastic Wave Velocity of Ni Sulfides

_{P}and shear wave velocity vs. can be calculated according to the bulk modulus, shear modulus, and density:

_{P}and shear wave velocity vs. of these three Ni sulfides are compared with those of olivine (i.e., Fe

_{2}SiO

_{4}-Fa and (Mg

_{0.875}Fe

_{0.125})SiO

_{4}-Fa

_{12.5}) [42] and enstatite (unpublished), which can represent the major constituents of the upper mantle in Figure 6. It can be seen that the V

_{P}of NiS, Ni

_{3}S

_{2}, and Ni

_{3}S

_{4}increases with increasing pressure. The V

_{P}decreases in the order of NiS > Ni

_{3}S

_{4}> Ni

_{3}S

_{2}. Compared with the V

_{P}of olivine and enstatite, the V

_{P}of NiS is about 22–26% lower. However, the vs. of NiS, Ni

_{3}S

_{2}, and Ni

_{3}S

_{4}is less affected by pressure, especially for Ni

_{3}S

_{2}and Ni

_{3}S

_{4}. The vs. decreases in the order of NiS > Ni

_{3}S

_{2}> Ni

_{3}S

_{4}. Similarly, compared with the vs. of Fo, Fa

_{12.5}, and enstatite, the vs. of NiS is about 25–35% lower. Previous studies found that magmatic processes can concentrate sulfide in the upper mantle, and sulfide melt can cause velocity anomalies [14,17]. Furthermore, Padilha et al. ([43] and references therein) show that sulfide minerals can be concentrated and cause high conductivity anomalies in the lithosphere. According to the big difference in seismic wave velocity between these sulfides and silicates, Ni sulfides may be one potential reason for velocity anomalies in the upper mantle.

## 5. Conclusions

_{3}S

_{2}increases, and the elastic constants show different changing trends. The bulk moduli of these Ni sulfides decrease in the order of Ni

_{3}S

_{2}> NiS > Ni

_{3}S

_{4}, and there is a positive correlation between them and the pressure, whereas the shear modulus decreases in the order of NiS > Ni

_{3}S

_{2}> Ni

_{3}S

_{4}, and it is insensitive to pressure. These Ni sulfides are mechanically stable in the studied pressure. The universal elastic anisotropic index A

^{U}also shows different changing trends with pressure. Under low pressure, the elastic anisotropy of NiS is the strongest, but the anisotropy of Ni

_{3}S

_{2}is the strongest when the pressure is larger than 5 GPa. Ni sulfides may be one potential reason for velocity anomalies in the upper mantle.

## Supplementary Materials

_{3}S

_{2}). Table S4: Pressure dependence of parameters for polydymite (Ni

_{3}S

_{4}).

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Elastic constants of Ni sulfides as a function of pressure. (

**a**) millerite; (

**b**) heazlewoodite; (

**c**) polydymite.

**Figure 5.**The universal elastic anisotropic index (A

^{U}) of millerite, heazlewoodite, and polydymite as a function of pressure.

**Figure 6.**Comparison between the elastic wave velocity of Ni sulfides and that of olivine and enstatite.

**Table 1.**Comparison of the calculated and measured crystal structures of millerite (NiS), heazlewoodite (Ni

_{3}S

_{2}), and polydymite (Ni

_{3}S

_{4}) under 0 GPa.

Millerite | This Study | Calc. [31] | Calc. [32] | Expt. [28] | Error (%) |
---|---|---|---|---|---|

a (Å) | 9.5955 | 9.5917 | 9.62 | 9.6190 | 0.15 |

c (Å) | 3.1498 | 3.1434 | 3.15 | 3.1499 | 0.06 |

V (Å^{3}) | 251.161 | 250.450 | 252.399 | 0.10 | |

Heazlewoodite | This Study | Calc. [31] | Calc. [32] | Expt. [29] | Error (%) |

a (Å) | 4.0820 | 4.0765 | 4.09 | 4.0821 | 0.1 |

α (°) | 89.3177 | 89.367 | 89.4 | 89.475 | 0 |

V (Å^{3}) | 68.0029 | 67.730 | 68.014 | 0.4 | |

Polydymite | This Study | Calc. [31] | Calc. [32] | Expt. [30] | Error (%) |

a (Å) | 9.4757 | 9.4702 | 9.49 | 9.457 | 0.1 |

V (Å^{3}) | 849.813 | 849.332 | 845.785 | 0.4 |

Mineral | V_{0} (Å^{3}) | B_{0} (GPa) | B’_{0} |
---|---|---|---|

NiS | 251.15 | 123.79 | 5.00 |

Ni_{3}S_{2} | 67.99 | 127.93 | 4.83 |

Ni_{3}S_{2} [34] | 68.74 | 119.2 | 4.7 |

Ni_{3}S_{4} | 850.83 | 116.65 | 4.74 |

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**MDPI and ACS Style**

Zhang, Q.; Tian, Y.; Liu, S.; Yang, P.; Li, Y.
First-Principles Study of the Elastic Properties of Nickel Sulfide Minerals under High Pressure. *Minerals* **2020**, *10*, 737.
https://doi.org/10.3390/min10090737

**AMA Style**

Zhang Q, Tian Y, Liu S, Yang P, Li Y.
First-Principles Study of the Elastic Properties of Nickel Sulfide Minerals under High Pressure. *Minerals*. 2020; 10(9):737.
https://doi.org/10.3390/min10090737

**Chicago/Turabian Style**

Zhang, Qiuyuan, Ye Tian, Shanqi Liu, Peipei Yang, and Yongbing Li.
2020. "First-Principles Study of the Elastic Properties of Nickel Sulfide Minerals under High Pressure" *Minerals* 10, no. 9: 737.
https://doi.org/10.3390/min10090737