Internal Friction and Vulnerability of Mixed Alkali Glasses

Based on a hopping model we show how the mixed alkali effect in glasses can be understood if only a small fraction $c_V$ of the available sites for the mobile ions is vacant. In particular, we reproduce the peculiar behavior of the internal friction and the steep fall (“vulnerability”) of the mobility of the majority ion upon small replacements by the minority ion. The single and mixed alkali internal friction peaks are caused by ion-vacancy and ion-ion exchange processes. If $c_V$ is small, they can become comparable in height even at small mixing ratios. The large vulnerability is explained by a trapping of vacancies induced by the minority ions. Reasonable choices of model parameters yield typical behaviors found in experiments.

Figure 1

(a) Normalized diffusion coefficient $D_A(x)/D_A(0)$ of $A$ ions for 5 different reduced temperatures $\theta =k_{B}T/{ϵ}_{\mathrm{mis}}$. The inset shows $D_A(x)$ for small $x$ at $\theta =1/10$ (open symbols) in comparison with the results in the absence of a correlation induced trapping effect (full symbols). (b) Measured data (symbols) for $(1-x){\mathrm{Na}}_2\mathrm{O}\mathrm{\text{−}}x{\mathrm{Rb}}_2\mathrm{O}\mathrm{\text{−}}4{\mathrm{B}}_2{\mathrm{O}}_3$ glasses at 652 K (redrawn from 19) and fit by the model (lines) with $\theta =1/9$ and asymmetric mismatch barriers $u_{\mathrm{mis}}^B=2u_{\mathrm{mis}}^A={ϵ}_{\mathrm{mis}}$.

Figure 2

(a) Internal friction $Q^{-1}$ and spectral components [cf. Eq. (2)] as a function of $\omega$ for $x=0.3$ and $\theta =1/10$. (b) Change of the internal friction $Q^{-1}$ with $x$ for $\theta =1/10$.

Figure 3

(a) $H_{\mathrm{m}}$ in dependence of ${\theta }^{-1}={ϵ}_{\mathrm{mis}}/k_{B}T$ at $x=0.5$. The inset shows the decrease of $c_{AB}$ with increasing ${\theta }^{-1}\propto {ϵ}_{\mathrm{mis}}$ (from analytical calculation). (b) $H_{\mathrm{m}}$ as a function of ion radii ratio of various mixed alkali pairs for $x=0.5$ and $\omega /2\pi =0.4\mathrm{Hz}$ fixed.

Figure 4

(a) Activation energies of the diffusion coefficients shown in Fig. 1, and of the spectral components of the internal friction. (b) $E_{\mathrm{Na}}$, $E_{\mathrm{Rb}}$, and $E_{\mathrm{m}}$ in $(1-x){\mathrm{Na}}_2\mathrm{O}\mathrm{\text{−}}x{\mathrm{Rb}}_2\mathrm{O}\mathrm{\text{−}}3{\mathrm{SiO}}_2$ glasses (redrawn from [18]). (c) $Q^{-1}$ calculated from Fig. 2 after applying time-temperature scaling for ${ϵ}_{\mathrm{mis}}=2u_0=1\mathrm{eV}$ and $\omega /\nu ={10}^{12}$ (see text). (d) $Q^{-1}$ in $(1-x){\mathrm{Na}}_2\mathrm{O}\mathrm{\text{−}}x{\mathrm{Rb}}_2\mathrm{O}\mathrm{\text{−}}3{\mathrm{SiO}}_2$ glasses for $\omega /2\pi =0.4\mathrm{Hz}$ (redrawn from [18]).