Relation between Heterogeneous Frozen Regions in Supercooled Liquids and Non-Debye Spectrum in the Corresponding Glasses

Recent numerical studies on glassy systems provide evidence for a population of non-Goldstone modes (NGMs) in the low-frequency spectrum of the vibrational density of states $D(\omega )$. Similarly to Goldstone modes (GMs), i.e., phonons in solids, NGMs are soft low-energy excitations. However, differently from GMs, NGMs are localized excitations. Here we first show that the parental temperature $T^{*}$ modifies the GM/NGM ratio in $D(\omega )$. In particular, the phonon attenuation is reflected in a parental temperature dependency of the exponent $s(T^{*})$ in the low-frequency power law $D(\omega )\sim {\omega }^{s(T^{*})}$, with $2\le s(T^{*})\le 4$. Second, by comparing $s(T^{*})$ with $s(p)$, i.e., the same quantity obtained by pinning a $p$ particle fraction, we suggest that $s(T^{*})$ reflects the presence of dynamical heterogeneous regions of size ${\xi }^3\propto p$. Finally, we provide an estimate of $\xi$ as a function of $T^{*}$, finding a mild power law divergence, $\xi \sim (T^{*}-T_d{)}^{-\alpha /3}$, with $T_d$ the dynamical crossover temperature and $\alpha$ falling in the range $\alpha \in [0.8,1.0]$.

Figure 1

(a) Cumulative density of states $F(\omega )$ as a function of the parental temperature $T^{*}$ for $N={10}^3$. Temperatures decrease from yellow to blue, $T^{*}/T_d=1.87,1.56,1.50,1.44,1.38,1.31,1.25,1.19,1.13,1.06$. Inset: Inverse participation ratio $\mathcal{R}(\omega )$. (b) Probability distribution function $P(\mathrm{\Delta }r)$ of the total displacement $\mathrm{\Delta }r$ by varying temperature. (c) Cumulative function $F(\omega )$ as the fraction of frozen particles $p$ increases at high temperature $T^{*}/T_d=3.12$. $p$ increases from green to violet, $p=0.05,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9$. (d) $P(\mathrm{\Delta }r)$ at $T^{*}/T_d=3.12$ as $p$ increases from green to blue.

Figure 2

(a) Slope $s$ of the power law $F(\omega )\sim {\omega }^{s+1}$ as a function of the parental temperature $T^{*}$ for $N={10}^3$. (b) Variance of the distribution $P(\mathrm{\Delta }r)$ as a function of $T^{*}$. (c) Slope $s$ as a function of $p$ at $T^{*}=3.12$. (d) Variance of the distribution $P(\mathrm{\Delta }r)$ as a function of $p$ at $T^{*}=3.12$. Dashed lines are guides to the eye. In panels (a) and (c) the green dashed arrows sketch the mapping employed for measuring ${\xi }_{\mathrm{pin}}$.

Figure 3

(a) Correlation length ${\xi }_{\mathrm{pin}}^3$ defined in the main text as a function of temperature $T^{*}$ and estimated through different observables for $N={10}^3,12^3$. Diamonds refer to the $s(T^{*})=s(p)$ method, circles to ${\sigma }_2(T^{*})={\sigma }_2(p)$, and triangles using ${\sigma }_1$, i.e., fitting $P(\mathrm{\Delta }r)$ to $\gamma \mathrm{\Delta }{r}^2{e}^{-\mathrm{\Delta }{r}^2/{\sigma }_1^2}$ and thus considering ${\sigma }_1(T^{*})={\sigma }_1(p)$. The inset highlights the behavior of ${\xi }_{\mathrm{pin}}^3$ computed through the exponent $s$ for $N={10}^3,12^3$, cyan and red symbols, respectively. Dashed lines are fits to the power law $(T^{*}-T_d{)}^{-\alpha }$ with $\alpha =0.8$ (red) and $\alpha =1.0$ (cyan). (b) Comparison between the correlation length ${\xi }_{\mathrm{pin}}$ and the dynamic length ${\xi }_{\mathrm{dyn}}$. ${\xi }_{\infty }$ indicates the value of $\xi$ at high temperatures. The red-dashed line is the best fit $(T^{*}-T_d{)}^{-\alpha /3}$ with $\alpha \sim 0.8$.