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Silicon Self-Diffusion in Stishovite: Calculations of Point Defect Parameters Based on the cBΩ Thermodynamic Model^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

^{act}= cBΩ, where B is the isothermal bulk modulus, Ω is the mean atomic volume, and c is a dimensionless constant. In this way, other important point defect parameters, such as the activation volume ${v}^{act}$, the activation entropy ${s}^{act}$, and the activation enthalpy ${h}^{act}$ may be estimated if the thermoelastic properties of the material are known over a wide temperature and pressure range. Our calculations are based on previously reported self-diffusion coefficients in stishovite single crystals measured at 14 GPa and at temperatures from 1400 to 1800 °C, in the [110] and [001] directions, by Shatskiy et al. (Am. Mineral. 2010, 95, 135–43). Furthermore, the EOS of stishovite, proposed by Wang et al. (J. Geophys. Res. 2012, 117, B06209) has been used for the accurate implementation of the cBΩ model. Our results suggest that the aforementioned point defect parameters exhibit considerable temperature dependence over the studied temperature range (1000–2000 °C). The estimated activation volumes (4.4–5.3 cm

^{3}/mol, in the range of 1400–1800 °C) are in agreement with reported experimental results. Our study confirms the potential of the cBΩ model for the theoretical investigation of diffusion processes in minerals, in order to overcome the experimental difficulties and the lack of experimental diffusion data in mantle conditions.

## 1. Introduction

_{2}, with the tetragonal rutile structure (P4

_{2}/mnm, space group 136). Stishovite is hardly found on Earth’s surface but is the predominant form of silica in the lower mantle [1]. The key role of stishovite as a potential mineral that stores water in its crystal structure and transports it into the deep mantle has been recently clarified [2]. Furthermore, the appearance of stishovite in SiO

_{2}-rich fragments has been proposed to explain the presence of seismic reflectors in the mid-mantle region [3]. These seismic reflectors would be affected by the rheological properties of the related materials undergoing plastic deformation under mantle conditions. Plastic deformation occurs either by diffusion or dislocation creep which are both controlled by atomic diffusion [3,4]. Silicon self-diffusion in stishovite was studied for the first time by Shatskiy et al. at the temperature range of 1400–1800 °C and at 14 GPa, after their successful synthesis of large single crystals [5,6]. Their results suggest a weak anisotropy of diffusion along the [001] and [110] directions. Based on these experimental diffusion data, in the present work, we apply the so-called cBΩ thermodynamic model [7] to further estimate important point defect parameters of Si self-diffusion in stishovite. According to this model, proposed by Varotsos and Alexopoulos [8,9], the diffusion coefficients of point defects can be calculated if the thermoelastic properties of the host material are known. A concise description of the model is given in the next section.

_{1−x}Ge

_{x}), and minerals [10,11,12,13,14,15,16,17,18,19,20]. However, diffusion in minerals has been less studied in the framework of the cBΩ model, as compared to other materials, probably due to the lack of complete sets of thermoelastic properties necessary to implement the model at temperatures and pressures similar to that of the Earth’s interior. Systematic studies of diffusion in minerals in the framework of the cBΩ model have been carried out by Zhang et al. [12,14] who have studied oxygen self-diffusion in silicate and oxide minerals, and H, Na, and K diffusion in plagioclase feldspar. Notably, the model has also been used to explain the emission of seismic electrical signals (SES) as precursors of earthquake events and to describe the thermodynamical and rheological properties of Earth’s mantle [21,22].

## 2. Methodology

#### 2.1. The cBΩ Thermodynamic Model

^{act}is a dimensionless constant, independent of temperature and pressure, under certain conditions [7]. As a consequence, in the case of a single self- or hetero-diffusion process, the diffusion coefficients D are expressed via the well-known Arrhenius equation, as follows:

#### 2.2. Bulk Properties of Stishovite

## 3. Results and Discussion

_{B}T are shown in Figure 2. The linear dependence of logD versus BΩ/k

_{B}T implies the validity of the cBΩ model, according to Equation (6). From the linear fittings of the data, the parameter ${c}^{act}$ has been extracted for the two crystallographic directions (refer to Table 1). Obviously, these values are very close to each other, due to the similar slopes of the two lines, as shown in Figure 2. Similar values of ${c}^{act}$ have been reported for He diffusion in forsterite and olivine, i.e., 0.160–0.189, depending on the crystallographic direction of diffusion [13].

^{3}/mol) in the range of 1400–1800 °C are comparable to the reported value of (6.0 ± 1.0) cm

^{3}/mol, taking into account the experimental uncertainties. The above calculated values of ${v}^{act}$ are close to the mean atomic volume Ω

_{ο}, calculated at 1000 °C (8.53 × 10

^{−30}m

^{3}). Specifically, ${v}^{act}$ varies from 0.89 to 1.04 Ω

_{ο}, in the range of 1400–1800 °C, where the diffusion experiments in stishovite were carried out by Shatskiy et al. [5]. It is worth mentioning that the sign and value of activation volume is indicative of the involved type of diffusion mechanism, namely vacant or interstitial sites. Values of activation volume comparable to the mean atomic volume probably suggests vacancy-mediated self-diffusion in stishovite [15,19].

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Isothermal bulk modulus B, mean atomic volume Ω, and volume thermal expansivity β, at 14 GPa, in the temperature range of 1000–2000 °C, calculated on the basis of the P-V-T EOS of stishovite reported by Wang et al. [23].

**Figure 2.**Experimental diffusion coefficients of Si in stishovite [5,6] for the two crystallographic directions, [110] and [001], as a function of the dimensionless quantity BΩ/k

_{B}T. From the linear fit of the lines, the parameter ${c}^{act}$ is derived in each case. The correlation factors, R

^{2}are also given in parentheses.

**Figure 3.**Calculated point defect parameters of Si self-diffusion across the [110] and [001] crystallographic directions in stishovite (

**a**) activation Gibbs free energy, ${g}^{act}$; (

**b**) activation entropy, ${s}^{act}$; (

**c**) activation enthalpy, ${h}^{act}$; (

**d**) activation volume, ${v}^{act}$. Reported experimental values of activation enthalpy and activation volume have also been included in the plots.

**Table 1.**Calculated values of ${c}^{act}$, activation enthalpy, activation entropy, activation Gibbs free energy, and activation volume for Si diffusion in stishovite. Reported experimental values of activation enthalpy and activation volume are also included.

Direction | ${\mathit{c}}^{\mathit{a}\mathit{c}\mathit{t}}$ | ${\mathit{h}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}}^{\mathit{a}\mathit{c}\mathit{t}}(\mathbf{kJ}/\mathbf{mol})$ | ${\mathit{h}}_{\mathit{e}\mathit{x}\mathit{p}}^{\mathit{a}\mathit{c}\mathit{t}}(\mathbf{kJ}/\mathbf{mol})$ | ${\mathit{s}}^{\mathit{a}\mathit{c}\mathit{t}}$ $({\mathit{k}}_{\mathbf{B}}\mathbf{units})$ | ${\mathit{g}}^{\mathit{a}\mathit{c}\mathit{t}}(\mathbf{kJ}/\mathbf{mol})$ | ${\mathit{v}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}}^{\mathit{a}\mathit{c}\mathit{t}}({\mathbf{cm}}^{3}/\mathbf{mol})$ | ${\mathit{v}}_{\mathit{e}\mathit{x}\mathit{p}}^{\mathit{a}\mathit{c}\mathit{t}}({\mathbf{cm}}^{3}/\mathbf{mol})$ |
---|---|---|---|---|---|---|---|

[110] | 0.182 ± 0.016 | (350–429) ± 19 | 322 ± 28 ^{1} | (5.7–8.6) ± 0.3 | (289–267) ± 17 | (3.9–5.6) ± 0.6 | 6.0 ± 1.0 ^{2} |

[001] | 0.188 ± 0.021 | (359–438) ± 21 | 334 ± 39 ^{1} | (5.9–8.9) ± 0.3 | (299–276) ± 17 | (4.0–5.8) ± 0.6 | - |

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**MDPI and ACS Style**

Saltas, V.; Vallianatos, F.
Silicon Self-Diffusion in Stishovite: Calculations of Point Defect Parameters Based on the *cBΩ* Thermodynamic Model. *Environ. Sci. Proc.* **2021**, *6*, 6.
https://doi.org/10.3390/iecms2021-09341

**AMA Style**

Saltas V, Vallianatos F.
Silicon Self-Diffusion in Stishovite: Calculations of Point Defect Parameters Based on the *cBΩ* Thermodynamic Model. *Environmental Sciences Proceedings*. 2021; 6(1):6.
https://doi.org/10.3390/iecms2021-09341

**Chicago/Turabian Style**

Saltas, Vassilios, and Filippos Vallianatos.
2021. "Silicon Self-Diffusion in Stishovite: Calculations of Point Defect Parameters Based on the *cBΩ* Thermodynamic Model" *Environmental Sciences Proceedings* 6, no. 1: 6.
https://doi.org/10.3390/iecms2021-09341