Temperature-dependent defect dynamics in the network glass SiO${}_2$

We investigate the long time dynamics of a strong glass former, SiO${}_2$, below the glass transition temperature by averaging single-particle trajectories over time windows which comprise roughly 100 particle oscillations. The structure on this coarse-grained time scale is very well defined in terms of coordination numbers, allowing us to identify ill-coordinated atoms, which are called defects in the following. The most numerous defects are O-O neighbors, whose lifetimes are comparable to the equilibration time at low temperature. On the other hand, SiO and OSi defects are very rare and short lived. The lifetime of defects is found to be strongly temperature dependent, consistent with activated processes. Single-particle jumps give rise to local structural rearrangements. We show that in SiO${}_2$ these structural rearrangements are coupled to the creation or annihilation of defects, giving rise to very strong correlations of jumping atoms and defects.

Figure 1

Examples for time-averaged single-particle (O-atom) trajectories for $T_{\mathrm{i}}=5000$ K at $T_{\mathrm{f}}=2500$ K (left figure) and at $T_{\mathrm{f}}=3250$ K (right figure).

Figure 2

Radial distribution function $g_{\alpha \beta }\left(r\right)$ as defined in Eq. (1) using different time averages $\Delta t_{\mathrm{av}}$ for the time average of ${\overline{\mathbf{r}}}_i\left(t\right)$. Here, for final temperatures, $T_{\mathrm{f}}=2500$ K quenched from $T_{\mathrm{i}}=5000$ K.

Figure 3

Radial distribution function as in Fig. 2, but here for final temperature $T_{\mathrm{f}}=3250$ K.

Figure 4

Radial distribution function $g_{\mathrm{SiO}}\left(r\right)$ for final temperatures $T_{\mathrm{f}}$. The left inset is an enlargement of the first peak and the right inset is an enlargement of farther peaks. The first peak height is increasing with decreasing temperature.

Figure 5

The radial distribution function, similar to Fig. 4, but here for $g_{\mathit{OO}}\left(r\right)$. The insets are enlargements of the first and second peaks.

Figure 6

Distribution of coordination number $P\left(z\right)$ for the number of Si neighbors of an O atom ($P_{\mathrm{OSi}}$ in left panel) and for the number of O neighbors of an Si atom ($P_{\mathrm{SiO}}$ in right panel). Thick lines are for $T_{\mathrm{f}}=2500$ K and thin dark lines are for $T_{\mathrm{f}}=3250$ K.

Figure 7

Distribution of coordination number $P\left(z\right)$ for the number of Si neighbors of an Si atom ($P_{\mathrm{SiSi}}$ in left panel) and for the number of O neighbors of an O atom ($P_{\mathit{OO}}$ in right panel). Thick lines are for $T_{\mathrm{f}}=2500$ K and thin dark lines are for $T_{\mathrm{f}}=3250$ K.

Figure 8

$M_{\alpha \beta }^{\mathrm{D}}$, the fraction of particles $i$ of type $\alpha \in \{$Si,O$\}$ which are defects (symbols) vs ${10}^4/T_{\mathrm{f}}$; also shown are Arrhenius fits $M_{\alpha \beta }^{\mathrm{D}}\left(T_{\mathrm{f}}\right)=C\exp \left(\frac{-E_{\mathrm{A}}}{k{T}_{\mathrm{f}}}\right)$ (lines), with $E_{\mathrm{A}}=1.5$$/2.0$$/1.7$$/0.36$ eV for SiSi/SiO/OSi/OO defects, respectively.

Figure 9

Normalized time correlation ${\tilde{C}}_{\mathrm{SiO}}^{\mathit{DD}}\left(t\right)$ as defined in Eqs. (5) and (6) for final temperatures $T_{\mathrm{f}}=2500$ K (top curve) to $T_{\mathrm{f}}=3250$ K (bottom curve).

Figure 10

Similar to Fig. 9, but here for ${\tilde{C}}_{\mathit{OO}}^{\mathit{DD}}$.

Figure 11

${\tau }_{\alpha \beta }^{\mathit{DD}}$ vs ${10}^4/T_{\mathrm{f}}$ for all $\alpha ,\beta \in \{$Si,O$\}$ (large open symbols) as compared to Arrhenius fits (lines) ${\tau }_{\alpha \beta }^{\mathit{DD}}=C\exp \left(\frac{E_{\mathrm{A}}}{k{T}_{\mathrm{f}}}\right)$ with $E_{\mathrm{A}}=3.76/2.92/3.37/4.39$ eV for SiSi/SiO/OSi/OO, respectively. For comparison, we show the time interval $\Delta t_{\mathrm{av}}$ over which we average (lower arrow), the time duration of jumps $\langle \Delta t_{\mathrm{d}}\rangle$ (upper arrow), and the equilibration time $t_{\mathrm{eq}}$ (stars).

Figure 12

Example of a time-averaged trajectory of an O atom at $T_{\mathrm{i}}=5000$ K and ${\chi }_i^{\mathrm{J}}\left(t\right)$, which indicates jump events as horizontal lines.

Figure 13

The fraction of jumping particles $M_{\alpha }^{\mathrm{J}}$ (symbols) vs ${10}^4/T_{\mathrm{f}}$ and fits with $M_{\alpha }^{\mathrm{J}}\left(T_{\mathrm{f}}\right)=C\exp \left(\frac{-E_{\mathrm{A}}}{k{T}_{\mathrm{f}}}\right)$ (lines) with $E_{\mathrm{A}}=2.89$ eV.

Figure 14

For comparison, we show for an O atom the defect functions ${\chi }_i^{\mathrm{D}}\left(t\right)$ of OO defects (top figure) and OSi defects (bottom figure) and the jump function ${\chi }_i^{\mathrm{J}}\left(t\right)$ (middle figure) at $T_{\mathrm{f}}=2500$ K.

Figure 15

Similar to Fig. 14, also at $T_{\mathrm{f}}=2500$ K, but for an Si atom we show the defect functions ${\chi }_i^{\mathrm{D}}\left(t\right)$ of SiO defects (top figure) and SiSi defects (bottom figure) and the jump function ${\chi }_i^{\mathrm{J}}\left(t\right)$ (middle figure).

Figure 16

Correlation of defects and jumps $A_{\alpha \beta }^{\mathit{DJ}}$ as defined in Eq. (10). The lines are a guide to the eye. The inset shows the same data in a semilogarithmic plot.

Figure 17

Similar to Fig. 14 for the same O atom, but here, for $T_{\mathrm{f}}=3250$ K, we show the defect functions ${\chi }_i^{\mathrm{D}}\left(t\right)$ of OO defects (top figure) and OSi defects (bottom figure) and the jump function ${\chi }_i^{\mathrm{J}}\left(t\right)$ (middle figure).