Low-frequency vibrational mode anomalies and glass transition: Thermal stability, phonon scattering, and pressure effects

The relationship between the excess of low-frequency vibrational modes observed in glasses and the stability against thermal fluctuations is explored. Such study is performed by calculating the correlation of atomic displacements inside the glass. As a result, it is proved that thermal stability requires that modes present in the boson or floppy peak (due to the flexibility or rigidity of the glass atomic network) should be localized or strongly scattered. The glass transition is thus determined by the size of the quadratic mean displacement. Also, the 2/3 relationship between melting and glass transition temperature is shown to have its origins in the differences between the mean-free path of phonons due to scattering. The size of this scattering is estimated using the Boson peak frequency and sound velocity. Finally, the change in the glass transition temperature with pressure is obtained from the displacement of low-frequency modes.

Figure 1

(Color online) $\rho \left(E\right)/E^2$ for different glasses and their corresponding crystals. The data set was taken from Refs. 37, 41.

Figure 2

(Color online) The integral ${\int }_0^E\rho \left(E\right)/E^2$ for the glasses and crystals that appear in Fig. 1. The data set was also taken from Refs. 37, 41.

Figure 3

(Color online) Comparison between the ratio of the second inverse moments ${⟨{\omega }^{-2}⟩}_c/{⟨{\omega }^{-2}⟩}_g$ versus $T_g/T_m$. The dotted line is a guide to indicate the identity.

Figure 4

(Color online) A plot of the mean quadratic displacement as a function of $G$ for ${\lambda }_0=1$ and $k_c={10}^{-2}/L_0$ using arbitrary units as in Ref. 42.

Figure 5

(Color online) A plot of the mean quadratic displacement as a function of $L_0$ for $G=1$ and $k_c={10}^{-2}/L_0$ using arbitrary units as in Ref. 42.

Figure 6

(Color online) Normalized average mean quadratic as a function of the concentration of impurities with mass $m_A$ in a one-dimensional disordered chain for a fixed ratio between the masses $m_A/m_B=2$. The elastic constants are one, and the system has $N=100$ atoms. The solid line corresponds to the exact calculation using the Eq. (3), while the dotted line corresponds to the averaged mass approximation using Eq. (5).