# Thermo-Elasticity of Materials from Quasi-Harmonic Calculations

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## Abstract

**:**

## 1. Introduction

_{2}SiO

_{4}: an end-member of the olivine solid solution series that is one of the most abundant silicates in the upper mantle of the Earth. The single-crystal thermo-elasticity of forsterite has been accurately determined experimentally at several temperatures from 300 K to 1700 K so that it represents an ideal system to validate and discuss our methodology [32].

## 2. Theoretical Aspects

#### 2.1. The Athermal Elastic Tensor

#### 2.1.1. Calculation from Analytical Forces

#### 2.1.2. Calculation from the Energy

#### 2.2. The Elastic Tensor at a Finite Temperature

- The determination of the equilibrium structure of the system at temperature T;
- The calculation of the second free-energy derivatives with respect to strain.

#### 2.2.1. The Equilibrium Structure at a Finite Temperature

_{2}O

_{3}[24], forsterite $\alpha $-Mg

_{2}SiO

_{4}[25], calcium stannate CaSnO

_{3}[45], and the molecular crystals of urea, purine and carbamazepine [43,46,47].

#### 2.2.2. Free Energy Derivatives with Respect to Strain

#### 2.2.3. Adiabatic versus Isothermal Elastic Moduli

#### 2.2.4. The Quasi-Static Approximation to Thermo-Elastic Moduli

## 3. The Implemented Algorithm

- A full structural relaxation of the system is performed (both atomic positions and lattice parameters are optimized). The static equilibrium structure, with volume ${V}_{0}$, is obtained.
- A space group symmetry-preserving QHA calculation is performed, which provides the thermal expansion of the system. A fully-automated algorithm is implemented in the Crystal program to perform this task [22,23], where four different volumes are explored (compressed and expanded with respect to ${V}_{0}$). For each volume, a volume-constrained, lattice symmetry-preserving structural relaxation is performed and phonon frequencies computed. By minimizing $F(V;T)$ at several temperatures, the $V\left(T\right)$ relation is determined. As a result, lattice parameters as a function of temperature are obtained.
- A value of temperature T is selected. Starting from the values of the lattice parameters at this temperature obtained at the end of the previous step, a volume-constrained, lattice symmetry-preserving structural relaxation is performed to get the equilibrium structure (also in terms of atomic positions) at the desired temperature.
- A given strain shape $\underline{\eta}$ is chosen, which will provide a linear combination of elastic stiffness constants according to Equation (16).
- The second free energy derivatives with respect to the strain are computed as discussed in Section 2.2.2. A fully-automated algorithm has been implemented in the Crystal program for this task. The starting point is represented by the optimized structure obtained at the end of step 3 above (i.e., the equilibrium structure at temperature T). The structure is deformed, in terms of the strain shape $\underline{\eta}$, into four strained configurations (two with positive and two with negative strain amplitude $\delta $). At each strained configuration, atomic positions are relaxed and phonon frequencies computed. The computed quasi-harmonic free energy as a function of strain amplitude is fitted to a second-order polynomial and the corresponding second-derivative determined.

## 4. Computational Parameters

^{−2}units) of the most diffuse $sp$ shells are 0.32 and 0.13 (Si), 0.59 and 0.25 (O), and 0.68 and 0.22 (Mg); the exponents of the d shells are 0.6 (Si), 0.5 (O), and 0.5 (Mg). The same basis set has already been successfully utilized in a couple of recent studies of vibrational and spectroscopic properties of forsterite [25,54].

**k**-points in the irreducible portion of the Brillouin zone. A pruned grid with 1454 radial and 99 angular points is used to calculate the DFT exchange-correlation contribution through numerical integration of the electron density over the unit cell volume. The self-consistent-field (SCF) convergence on energy was set to a value of 10

^{−10}hartree for all geometry optimizations and phonon frequency calculations. In the calculation of phonon frequencies, the Hessian matrix is computed numerically from finite differences of analytical forces computed at a set of displaced nuclear configurations: a step of 0.003 Å is used.

**k**-points (corresponding to a supercell containing 756 atoms). All quasi-harmonic properties are computed by working in terms of the primitive cell of the system (containing 4 formula units or equivalently 28 atoms), which already ensures convergence.

## 5. Results and Discussion

_{2}SiO

_{4}, that is characterized by nine independent elastic constants: ${C}_{11}$, ${C}_{22}$, ${C}_{33}$, ${C}_{44}$, ${C}_{55}$, ${C}_{66}$, ${C}_{12}$, ${C}_{13}$, ${C}_{23}$. The first step of our algorithm, as sketched in Section 3, consists of a full structural relaxation of the system (i.e., both atomic positions and lattice parameters are optimized by minimizing the static internal energy E of the system). The second step of the algorithm is a lattice symmetry-preserving quasi-harmonic calculation, where harmonic phonon frequencies are evaluated at four volumes (one compressed, one equilibrium and two expanded) according to the fully-automated scheme previously suggested and implemented in the Crystal program by some of the authors [22,23]. From the calculation of phonon harmonic frequencies at different volumes, the Helmholtz free energy of the system as a function of temperature and volume can be evaluated as in Equation (12). By minimizing this function with respect to volume at each temperature, the $V\left(T\right)$ relation is obtained that provides the thermal expansion of the system.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Volumetric thermal expansion of forsterite up to 2000 K. Experimental data are from Bouhifd et al. [55] (filled circles), Ye et al. [56] (empty circles) and Hazen [57] (empty squares). Lines correspond to quasi-harmonic computed values with different functionals of the DFT: LDA (blue), PBE (green), PBEsol (red), and PBE0 (yellow).

**Figure 2.**Thermal dependence of the isothermal ${K}^{\mathrm{T}}$ (left panel) and adiabatic ${K}^{\mathrm{S}}$ (right panel) bulk modulus of forsterite up to 2000 K. Experimental data (filled and empty black circles) are from Anderson et al. [58]. Lines correspond to quasi-harmonic computed values with different functionals of the DFT: LDA (blue), PBE (green), PBEsol (red), and PBE0 (yellow).

**Figure 3.**Adiabatic single-crystal elastic stiffness constants of forsterite as a function of temperature. Circles are experimental data by Isaak et al. [32] while lines correspond to fully quasi-harmonic computed values.

**Figure 4.**Effect of temperature on the Young modulus of forsterite.

**Left**: 3D representation of the spatial dependence of the Young modulus as a function of temperature from experimental elastic constants, quasi-harmonic computed constants, and quasi-static computed constants.

**Right**: 2D projections of the Young modulus in three different planes ($xy$, $xz$ and $yz$) as a function of temperature (blue for 250 K, green for 650 K, yellow for 1050 K and red for 1450 K).

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**MDPI and ACS Style**

Destefanis, M.; Ravoux, C.; Cossard, A.; Erba, A.
Thermo-Elasticity of Materials from Quasi-Harmonic Calculations. *Minerals* **2019**, *9*, 16.
https://doi.org/10.3390/min9010016

**AMA Style**

Destefanis M, Ravoux C, Cossard A, Erba A.
Thermo-Elasticity of Materials from Quasi-Harmonic Calculations. *Minerals*. 2019; 9(1):16.
https://doi.org/10.3390/min9010016

**Chicago/Turabian Style**

Destefanis, Maurizio, Corentin Ravoux, Alessandro Cossard, and Alessandro Erba.
2019. "Thermo-Elasticity of Materials from Quasi-Harmonic Calculations" *Minerals* 9, no. 1: 16.
https://doi.org/10.3390/min9010016