# Research on Mechanical Properties of High-Pressure Anhydrite Based on First Principles

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

_{4}). The structure parameters of anhydrite crystal were obtained by means of first principles after structure optimization at 0~120 MPa. In comparison with previous experimental and theoretical calculation values, the results we obtained are strikingly similar to the previous data. The elastic constants and physical parameters of anhydrite crystal were also studied by the first-principles method. Based on this, we further studied the Young’s modulus and Poisson’s ratio of anhydrite crystal, the anisotropy factor, the speed of sound, the minimum thermal conductivity and the hardness of the material. It was shown that the bulk modulus and Poisson’s ratio of anhydrite crystal rose slowly with increasing pressure. The anisotropy characteristics of the Young’s modulus and shear modulus of anhydrite crystal were consistent under various pressure levels, while the difference in the anisotropy characteristics of the bulk modulus appeared. The acoustic velocities of anhydrite crystal tended to be stable with increasing pressure. The minimum thermal conductivity remained relatively unchanged with increasing pressure. However, the material hardness declined gradually with increasing pressure.

## 1. Introduction

_{3}samples to study the stability of its crystal structure [12]. With the development of high-speed computing, some software companies have developed software to calculate first principles, which has accelerated the progress in material research. At present, the popular programs on the market for first-principles calculation are VASP and CASTEP. Weck studied the physical properties of anhydrite crystal using VASP software, such as the heat capacity, entropy and free energy [13]. Benmakhlouf and Walker used CASTEP software to study the energy band and thermodynamic properties of MgSO

_{4}and carbonate [14,15]. In the early research, most scholars mainly used the first-principles method to study the physical and chemical properties of crystals, such as free energy, entropy and enthalpy. Later, Zhao et al. studied the elastic constant of CaCO

_{3}using first principles and obtained the shear modulus, bulk modulus, Poisson’s ratio and other mechanical parameters of CaCO

_{3}crystal [16]. DV Korabel’nikov also studied the physical properties of anhydrite rock using first principles, such as its structure parameter, elastic constants and vibrational energy states [17]. However, the specific research conditions—such as the pressure—were not specified in the research paper, nor is the specific elastic constant.

## 2. Theoretical Methods

#### 2.1. Research Methods

_{4}) crystal was first established. Because of the orthogonality of anhydrite crystals, there are nine independent elastic constants, which can be obtained through the following formula [18,19]:

#### 2.2. Specific Parameter Setting

^{−6}eV/atom was used for the electronic self-consistency loop. The sampling unit of k points in the Brillouin zone was 2 × 2 × 2. The optimization of the structure was calculated at six pressure points between 0~120 MPa. The values of the convergence thresholds in each set are given in Table 1.

## 3. Results and Discussion

#### 3.1. Properties of Unit Cell Under 0 MPa Pressure

_{4}) and shows the spatial distribution of anhydrite crystal atoms under a pressure of 0 MPa. Table 2 shows the fractional coordinates of each atom. There are 24 atoms in the unit cell.

#### 3.2. Relationship of Crystal Structure Parameters with Pressure Variation

^{3}at 0 MPa. In order to more clearly describe the changing relationship between the lattice constant and pressure, the lattice parameters and volume change ratio of the lattice under pressure are represented by a graph (as shown in Figure 2). The a0, b0 and c0 are the lattice parameters of anhydrite crystals under pressure of 0 MPa. It can be seen from the graph that the ratio of the lattice constant in each direction decreases with the increase of pressure.

#### 3.3. Pressure-Dependent Elastic Constants and Mechanical Properties

- ${C}_{ij}$—GPa.
- $P$—MPa.

_{R}and G

_{R}are, respectively, the bulk modulus and shear modulus of Reuss, and B

_{V}and G

_{V}are the bulk modulus and shear modulus of Voigt, respectively. In addition, Young’s modulus and Poisson’s ratio can also be obtained by the following formula:

#### 3.4. Pressure-Dependent Wave Velocity and Thermal Conductivity

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Peng, F.L.; Qiao, Y.K.; Zhao, J.W.; Liu, K.; Li, J.C. Planning and implementation of underground space in Chinese central business district (CBD): A case of Shanghai Hongqiao CBD. Tunn. Undergr. Space Technol.
**2020**, 95, 103176. [Google Scholar] [CrossRef] - Ning, Y.B.; Tang, H.M.; Zhang, B.C.; Shen, P.W.; Zhang, G.C.; Xia, D. Research on rock similar material proportioning test based on orthogonal design and its application in base friction physical model test. Rock Soil Mech.
**2020**, 6, 1–11. [Google Scholar] - Chikhaoui, M.; Belayachi, N.; Nechnech, A.; Hoxha, D. Experimental characterization of the hydromechanical properties of the gypsum soil of Sebkha of Oran. Period. Polytech Civ. Eng.
**2017**, 61, 706–717. [Google Scholar] [CrossRef][Green Version] - Marius, M.; Amalia, D.F.; Philippe, D.; Michael, A.; Dubois, P. Polylactide compositions. Part 1: Effect of filler content and size on mechanical properties of PLA/calcium sulfate composites. Polymer
**2008**, 48, 2613–2618. [Google Scholar] - Austad, T.; Shariatpanahi, S.F.; Strand, S.; Aksulu, H.; Puntervold, T. Low Salinity EOR Effects in Limestone Reservoir Cores Containing Anhydrite: A Discussion of the Chemical Mechanism. Energy Fuels
**2015**, 29, 6903–6911. [Google Scholar] [CrossRef] - Antao, S.M. Crystal-structure analysis of four mineral samples of anhydrite, CaSO
_{4}, using synchrotron high-resolution powder X-ray diffraction data. Powder Diffr.**2011**, 26, 326–330. [Google Scholar] [CrossRef] - Azimi, G.; Papangelakis, V.G. Mechanism and kinetics of gypsum–anhydrite transformation in aqueous electrolyte solutions. Hydrometallurgy
**2011**, 108, 122–129. [Google Scholar] [CrossRef] - Liu, C.J.; Zheng, H.F. A new approach to kinetics study of the anhydrite crystallization at 373 K using a diamond anvil cell with Raman spectroscopy. Rev. Sci. Instrum.
**2013**, 84, 044501. [Google Scholar] [CrossRef] - Tang, Y.; Gao, J.; Liu, C.; Chen, X.; Zhao, Y. Dehydration Pathways of Gypsum and the Rehydration Mechanism of Soluble Anhydrite γ-CaSO
_{4}. ACS Omega**2019**, 4, 7636–7642. [Google Scholar] [CrossRef][Green Version] - Aquilano, D.; Rubbo, M.; Catti, M.; Pavese, A.; Ugliengo, P. Theoretical equilibrium and growth morphology of anhydrite (CaSO
_{4}) crystal. J. Cryst. Growth**1992**, 125, 519–532. [Google Scholar] [CrossRef] - Rao, E.N.; Vaitheeswaran, G.; Reshak, A.H.; Auluck, S. Effect of lead and caesium on the mechanical, vibrational and thermodynamic properties of hexagonal fluorocarbonates: A comparative first principles study. RSC Adv.
**2016**, 6, 99885–99897. [Google Scholar] [CrossRef] - Etienne, B.; Marc, B.; Carlos, P.; Michele, L. First-principles modeling of sulfate incorporation and 34S/32S isotopic fractionation in different calcium carbonates. Chem. Geol.
**2014**, 374, 84–91. [Google Scholar] - Philippe, F.W.; Eunja, K.; Carlos, F.J.C.; David, C.S. First-principles study of anhydrite, polyhalite and carnallite. Chem. Phys. Lett.
**2014**, 594, 1–5. [Google Scholar] - Benmakhlouf, A.; Errandonea, D.; Bouchenafa, M.; Maabed, S.; Bouhemadou, A.; Bentabet, A. New pressure-induced polymorphic transitions of anhydrous magnesium sulfate. Dalton Trans.
**2017**, 46, 5058–5068. [Google Scholar] [CrossRef] [PubMed][Green Version] - Walker, S.M.; Becker, U. Uranyl (VI) and neptunyl (V) incorporation in carbonate and sulfate minerals: Insight from first-principles. Geochim. Cosmochim. Acta
**2015**, 161, 19–35. [Google Scholar] [CrossRef][Green Version] - Zhao, J.H.; Zhou, B.; Liu, B.G.; Guo, W.L. Elasticity of Single-Crystal Calcite by First-Principles Calculations. J. Comput. Theor. Nanosci.
**2009**, 6, 1181–1188. [Google Scholar] [CrossRef] - Korabel’nikov, D.V.; Zhuravlev, Y.N. Structural elastic electronic and vibrational properties of a series of sulfates from first principles calculations. J. Phys. Chem. Solids
**2018**, 119, 114–121. [Google Scholar] [CrossRef] - Myhill, R. The elastic solid solution model for minerals at high pressures and temperatures. Contrib. Mineral. Petrol.
**2018**, 173, 12. [Google Scholar] [CrossRef][Green Version] - Reshak, A.H.; Jamal, M. DFT calculation for elastic constants of orthorhombic structure within WIEN2K code: A new package (ortho-elastic). J. Alloys Compd.
**2012**, 543, 147–151. [Google Scholar] [CrossRef] - Xu, Y.Q.; Wu, S.Y.; Ding, C.C.; Wu, L.N.; Zhang, G.J. First-principles investigation on the mechanism of photocatalytic properties for cubic and orthorhombic KNbO
_{3}. Chem. Phys.**2018**, 504, 66–71. [Google Scholar] [CrossRef] - Chen, H.C.; Tian, W.Y. First-principles investigation of the physical properties of cubic and orthorhombic phase Na
_{3}UO_{4}. Phys. B**2017**, 524, 144–148. [Google Scholar] [CrossRef] - Pugh, S.F. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1954**, 45, 823–843. [Google Scholar] [CrossRef] - Frantsevich, I.N.; Voronov, F.F.; Bokuta, S.A. Elastic Constants and Elastic Moduli of Metals and Insulators Handbook; Frantsevich, I.N., Ed.; Naukova Dumka: Kiev, Ukraine, 1983; pp. 60–180. [Google Scholar]
- Jund, P.; Viennois, R.; Tao, X.M.; Niedziolka, K.; Tedenac, J.C. Physical properties of thermoelectric zinc antimonide using first-principles calculations. Phys. Rev. B
**2012**, 85, 224105. [Google Scholar] [CrossRef][Green Version] - Chen, X.Q.; Niu, H.; Li, D.; Li, Y. Intrinsic correlation between hardness and elasticity in polycrystalline materials and bulk metallic glasses. Intermetallics
**2011**, 19, 1275–1281. [Google Scholar] [CrossRef][Green Version] - Chung, D.H.; Buessem, W.R. The elastic anisotropy of crystals. J. Appl. Phys.
**1967**, 38, 2010–2012. [Google Scholar] [CrossRef] - Ranganathan, S.I.; Ostoja-Starzewski, M. Universal elastic anisotropy index. Phys. Rev. Lett.
**2008**, 101, 055504. [Google Scholar] [CrossRef][Green Version] - Nye, J.F. Physical Properties of Crystals; Oxford University Press: Oxford, UK, 1985. [Google Scholar]
- Bai, X. Study on Pure Mode Axes of Orthorhombic System Crystals. Master’s Thesis, Hebei University of Technology, Tianjin, China, 2016. [Google Scholar]
- Cahill, D.G.; Watson, S.K.; Pohl, R.O. Lower limit to the thermal conductivity of disordered crystals. Phys. Rev. B
**1992**, 46, 6131–6140. [Google Scholar] [CrossRef]

**Figure 2.**Relationship between lattice constants and volume ratio of anhydrite crystals and pressure.

**Figure 9.**Relationship between bulk modulus and pressure along the a and c axes under uniaxial compression.

**Figure 11.**Variation of longitudinal, transverse and average sound velocity of polycrystalline anhydrite with pressure.

**Figure 12.**Relationship between wave velocity and pressure in the [100], [001] and [110] directions of anhydrite crystals.

**Figure 13.**Relationship between the minimum thermal conductivity and pressure in the [100], [001] and [110] directions of anhydrite crystals.

Task | Geometry Optimization | Elastic Constants |
---|---|---|

Energy (eV/atom) | 1 × 10^{−5} | 2 × 10^{−7} |

Max. force (eV/Å) | 0.03 | 0.004 |

Max. stress (GPa) | 0.0008 | |

Max. displacement (Å) | 0.0001 | 0.00003 |

Element | Atom Number | Fractional Coordinates of Atoms | ||
---|---|---|---|---|

u | v | w | ||

O | 1 | 0.298 | 0.08 | 0 |

O | 2 | 0.798 | 0.08 | 0.5 |

O | 3 | −0.298 | 0.58 | 0 |

O | 4 | 0.202 | 0.58 | 0.5 |

O | 5 | 0.298 | 0.42 | 0 |

O | 6 | 0.798 | 0.42 | 0.5 |

O | 7 | −0.298 | −0.08 | 0 |

O | 8 | 0.202 | −0.08 | 0.5 |

O | 9 | 0.015 | 0.25 | 0.171 |

O | 10 | 0.515 | 0.25 | 0.671 |

O | 11 | −0.015 | 0.75 | −0.171 |

O | 12 | 0.485 | 0.75 | 0.329 |

O | 13 | 0.015 | 0.25 | −0.171 |

O | 14 | 0.515 | 0.25 | 0.329 |

O | 15 | −0.015 | −0.25 | 0.171 |

O | 16 | 0.485 | −0.25 | 0.671 |

S | 1 | 0.155 | 0.25 | 0 |

S | 2 | 0.655 | 0.25 | 0.5 |

S | 3 | −0.155 | 0.75 | 0 |

S | 4 | 0.345 | 0.75 | 0.5 |

Ca | 1 | 0.654 | 0.25 | 0 |

Ca | 2 | 1.154 | 0.25 | 0.5 |

Ca | 3 | −0.654 | 0.75 | 0 |

Ca | 4 | −0.154 | 0.75 | 0.5 |

Source | a | b | c | Source | a | b | c |
---|---|---|---|---|---|---|---|

Present | 6.292 | 7.087 | 7.102 | Exp2 [6] | 6.241 | 6.992 | 7 |

Exp [13] | 6.245 | 6.993 | 6.995 | Theo [17] | 6.355 | 6.982 | 7.013 |

Theo [13] | 6.28 | 7.08 | 7.09 |

P (Mpa) | A (Å) | B (Å) | C (Å) | b/a | c/a | V (Å^{3}) | Ρ (g/cm^{3}) |
---|---|---|---|---|---|---|---|

0 | 6.292317 | 7.086981 | 7.102321 | 1.126291 | 1.128729 | 316.7176 | 2.855024644 |

20 | 6.291652 | 7.086171 | 7.101235 | 1.126281 | 1.128676 | 316.5995 | 2.856089698 |

40 | 6.291209 | 7.085871 | 7.100233 | 1.126313 | 1.128596 | 316.5191 | 2.856814749 |

60 | 6.290695 | 7.084851 | 7.099219 | 1.126243 | 1.128527 | 316.4025 | 2.857867871 |

80 | 6.290117 | 7.08385 | 7.098251 | 1.126187 | 1.128477 | 316.2856 | 2.858923812 |

100 | 6.289599 | 7.083087 | 7.096716 | 1.126159 | 1.128326 | 316.1571 | 2.860085504 |

120 | 6.288812 | 7.08233 | 7.09611 | 1.126179 | 1.128371 | 316.0568 | 2.860993672 |

P | C11 | C12 | C13 | C22 | C23 | C33 | C44 | C55 | C66 |
---|---|---|---|---|---|---|---|---|---|

0 | 107.4598 | 33.0052 | 30.3394 | 135.3305 | 9.5501 | 107.1060 | 9.8914 | 19.0996 | 37.4988 |

20 | 106.4652 | 31.4676 | 28.6103 | 139.4924 | 9.3740 | 108.5523 | 8.5432 | 30.0487 | 37.6820 |

40 | 109.9706 | 31.1323 | 30.7150 | 137.5000 | 10.5535 | 108.1505 | 8.9690 | 7.7707 | 34.9036 |

60 | 110.2857 | 31.5445 | 32.2551 | 137.4208 | 11.4010 | 108.9114 | 9.3935 | 10.3985 | 36.6620 |

80 | 110.6258 | 31.8399 | 31.8132 | 139.4489 | 10.3298 | 110.0278 | 8.3825 | 28.1790 | 36.1350 |

100 | 109.9652 | 33.0277 | 31.9970 | 140.3028 | 10.8054 | 109.3896 | 8.2964 | 28.2512 | 37.3550 |

120 | 111.2584 | 33.7644 | 31.5085 | 139.7839 | 14.2967 | 109.0074 | 8.9327 | 11.0874 | 36.6704 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zeng, X.; You, S.; Li, L.; Lai, Z.; Hu, G.; Zhang, W.; Xie, Y. Research on Mechanical Properties of High-Pressure Anhydrite Based on First Principles. *Crystals* **2020**, *10*, 240.
https://doi.org/10.3390/cryst10040240

**AMA Style**

Zeng X, You S, Li L, Lai Z, Hu G, Zhang W, Xie Y. Research on Mechanical Properties of High-Pressure Anhydrite Based on First Principles. *Crystals*. 2020; 10(4):240.
https://doi.org/10.3390/cryst10040240

**Chicago/Turabian Style**

Zeng, Xianren, Shihui You, Linmei Li, Zhangli Lai, Guangyan Hu, Wenjuan Zhang, and Yuan Xie. 2020. "Research on Mechanical Properties of High-Pressure Anhydrite Based on First Principles" *Crystals* 10, no. 4: 240.
https://doi.org/10.3390/cryst10040240