Heating through the glass transition: A rigidity approach to the boson peak

Using molecular dynamics, we study the relationship between the excess of low-frequency vibrational modes (Boson peak, BP) and the glass transition for a bidispersive glass interacting through a truncated Lennard-Jones potential. The evolution of the BP with increasing temperature is correlated with the average coordination, as predicted by rigidity theory. This is due to a lack of atomic “contacts,” as is confirmed by taking a crystal with broken bonds. We show how the quadratic mean displacement $\left(⟨u^2⟩\right)$ is enhanced by the BP. When $⟨u^2⟩$ is obtained on short time scales or measured on inherent structures, the glass transition temperature $T_g$ is determined by the position and height of the BP. Between the melting temperature $T_m$ and $T_g$, the nature of the relaxation processes exhibit phase separation, where the backbone increases its rigidity while the smaller atoms diffuse away to form separate crystals.

Figure 1

(Color online) Total internal energy $⟨U⟩=⟨U_{AA}⟩+⟨U_{BB}⟩+⟨U_{AB}⟩$ versus temperature $T$. Insets: (a) energy of $A\text{−}A$ bonds, (b) $B\text{−}B$ bonds, and (c) $A$ and $B$ bonds.

Figure 2

(Color online) Diffusion constant $D$ as a function of the temperature. Triangles are particles $A$ $\left(D_A\right)$, circles are particles $B$ $\left(D_B\right)$, and squares are the whole mixture $D_{AB}$.

Figure 3

(Color online) Nearest neighbors $⟨Z⟩$ of (a) type $A$ around $A$, (b) type $B$ around $B$, and (c) type $B$ around $A$. We can see an increase in $⟨Z_{AA}⟩$ above $T_g$. A similar behavior is shown in (b), however, $B$ spheres have more mobility, like in a fluid. A decrease is observed in $⟨Z_{AB}⟩$ indicating phase separation.

Figure 4

(Color online) Reduced density of vibrational modes $g\left(\omega \right)/{\omega }^2$ for glasses at different temperatures. At $T=1.3>T_g$, there is a reduction in the Boson peak with respect to temperatures below $T_g$. ${\omega }_0$ is a characteristic frequency. For comparison, the result obtained for a crystal at low $T$ is presented. Inset (a): evolution of the $⟨{\omega }^{-2}⟩$ at low frequencies as a function of the temperature $T$. Inset (b): the same quantity but now in function of the coordination number $⟨Z_{AA}⟩$.

Figure 5

(Color online) Reduced density of vibrational modes $g\left(\omega \right)/{\omega }^2$ for a fcc lattice of $A$ atoms with a concentration $x$ of $B$ atoms with a reduced diameter. The number of $B$ particles is indicated in the inset for each curve. Inset (a): number of $B$ atoms as a function of the average coordination. Inset (b): evolution of the $⟨{\omega }^{-2}⟩$ at low frequencies as a function of the coordination number $⟨Z_{AA}⟩$.

Figure 6

(Color online) Square mean displacement. Green squares represent short scales of time while green circles show $⟨u^2⟩$ on the inherent structures. On the same plot, with red diamonds and downward triangles, we present the same quantity calculated from $g\left(\omega \right)$ and $g_A\left(\omega \right)$, using Eq. (3).