# Anharmonic Effects on the Thermodynamic Properties of Quartz from First Principles Calculations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) and structure (framework of SiO

_{4}tetrahedra), has several high-pressure and high-temperature polymorphs. In particular, in temperature (at ambient pressure) quartz shows a displacive phase transition at about 847 K [3] from the α phase (trigonal) to the β phase (hexagonal) which makes quite difficult to model its behavior as a function of temperature through and above the phase transition itself, whereas in pressure (at room temperature) it is stable up to 2.5 GPa (quartz-coesite phase transition, see [5,17]). Recently, reliable pressure-volume-temperature equations of state (PVT EoS) for quartz have been determined accounting for the continuous phase transition in temperature [8]. A reliable determination of the bulk modulus (K) as a function of temperature has been determined from the PVT experimental data, directly using its definition (involving the partial derivative of the pressure with respect to the volume, at any given temperature) instead of some EoS fit, due to its strong correlation with the K’ parameter (essentially, the first derivative of K with respect to P in the expression K = K

_{0}+ K’P) and the significant dependence of K’ on temperature (e.g., [8]).

## 2. Methods and Computational Details

#### 2.1. Strategy of Computation

#### 2.2. Computational Details

^{3}volume range. Phonon dispersion correction was accounted for by computing the vibrational frequencies with the supercell approach: the 2 × 2 × 2, 3 × 3 × 3, 4 × 1 × 1 (and the symmetry equivalent 1 × 4 × 1 supercell) and 1 × 1 × 4 supercells were employed; the resulting phonon density of state (PDOS) is shown in Figure 1.

_{1}vibrational mode (the soft-mode), the E mode and the mode of symmetry (2) at the K-point (0 0 1/2) in the Brillouin zone. The E (Q) energy profile along each anharmonic mode was computed by means of the SCANMODE keyword in the CRYSTAL17 input file [20]. Energy data from the SCANMODE’S were processed by means of the anharm module of the QM-thermodynamic program [26]. Some details of the implementation are provided in the form of a tutorial on the program website (https://qm-thermodynamics.readthedocs.io/en/main/_static/anharm.html (accessed on 1 September 2021).

## 3. Results and Discussion

_{p}= C

_{v}+ VTKα

^{2}. On the other hand, at low temperatures (from 300 K down to 0 K) the computed specific heat perfectly follows the trend of the experimental data by [27] (the curve from [5] is the result of a power series fitting of experimental data from the high temperature region down to 300 K, whereas those below 300 K have been extrapolated by the same power series, and therefore they should not be considered).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Calculated bulk modulus including the anharmonic correction (1), within the harmonic approximation (2) and excluding the anharmonic modes (3).

**Figure 3.**Calculated (this study) and experimental [8] quartz bulk modulus up to 750 K.

**Figure 4.**Comparison between the calculated thermal expansion values (this study) and those derived from the experimental data by [3].

**Figure 6.**Calculated specific heat (C

_{v}) as a function of temperature and a constant volume. The solid line is the C

_{v}calculated including the phonon dispersion in a supercell 2 × 2 × 2 with the calculated frequencies over a range of unit-cell volumes; the dashed line reports the C

_{v}obtained from a calculation of a supercell 2 × 2 × 2; dotted-dashed line shows the C

_{v}calculated from a calculation of a supercell 3 × 3 × 3; the dotted line is the C

_{v}calculated including only the vibrational modes at the Brillouin zone center.

**Figure 8.**Quartz-coesite phase boundary. Comparison between the calculated data (solid line) and the experimental ones from the thermodynamic database by [5].

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Murri, M.; Prencipe, M.
Anharmonic Effects on the Thermodynamic Properties of Quartz from First Principles Calculations. *Entropy* **2021**, *23*, 1366.
https://doi.org/10.3390/e23101366

**AMA Style**

Murri M, Prencipe M.
Anharmonic Effects on the Thermodynamic Properties of Quartz from First Principles Calculations. *Entropy*. 2021; 23(10):1366.
https://doi.org/10.3390/e23101366

**Chicago/Turabian Style**

Murri, Mara, and Mauro Prencipe.
2021. "Anharmonic Effects on the Thermodynamic Properties of Quartz from First Principles Calculations" *Entropy* 23, no. 10: 1366.
https://doi.org/10.3390/e23101366