Ioffe-Regel criterion and viscoelastic properties of amorphous solids

We show that viscoelastic effects play a crucial role in the damping of vibrational modes in harmonic amorphous solids. The relaxation of a given plane elastic wave is described by a memory function of a semi-infinite one-dimensional mass-spring chain. The initial vibrational energy spreads from the first site of the chain to infinity. In the beginning of the chain, there is a barrier, which significantly reduces the decay of vibrational energy below the Ioffe-Regel frequency. To obtain the parameters of the chain, we present a numerically stable method, based on the Chebyshev expansion of the local vibrational density of states.

Figure 1

(a) Damped harmonic oscillator with the mass $m_{\mathbf{q}}$ (the ball), the stiffness $k_{\mathbf{q}}^{\mathrm{inst}}$ (the spring), and the damping ${\eta }_{\mathbf{q}}$ (the dashpot). (b) Harmonic oscillator with a general viscoelastic element defined by the memory function $K_{\mathbf{q}}\left(t\right)$ (the rectangle).

Figure 2

(a) General case of the semi-infinite one-dimensional mass-spring model. The displacements of masses $m_{\mathbf{q}},m_{1\mathbf{q}},m_{2\mathbf{q}},\cdots$ are denoted by $u_{\mathbf{q}},u_{1\mathbf{q}},u_{2\mathbf{q}},\cdots ,$ respectively. Each spring is denoted by its stiffness. (b) Finite representation of the same chain with three sites. The tail of the chain is replaced by a general viscoelastic element defined by the memory function $K_{3\mathbf{q}}\left(t\right)$ and denoted by the rectangle.

Figure 3

The LVDOS ${\mathcal{F}}_n\left(\omega \right)$ shown as a ratio ${\mathcal{F}}_n\left(\omega \right)/{\mathcal{F}}_{\infty }\left(\omega \right)$ for $\varkappa =1$ for different values of $n$ (up to $n=100$).

Figure 4

Parameters of the chain as a function of scaled site number $n{\omega }_c/{\omega }_{\max }$ obtained using the RMT for different values of $\varkappa$. For small values of $\varkappa$, dots merge into lines. (a) Normalized stiffness of vertical springs $k_n/m_n{\omega }_c^2$. (b) Normalized stiffness of horizontal springs $b_n/b_{\infty }$. (c) Normalized mass $m_n/m_{\infty }$. In (a) the dashed line marks the Ioffe-Regel criterion $k_n/m_n={\omega }_c^2$.

Figure 5

Regular one-dimensional mass-spring model, which corresponds to the low-frequency approximation.

Figure 6

Relaxation of elastic plane waves in the framework of the RMT for (a) $\varkappa =0.01$ and (b) $\varkappa =0$ for the same set of wave numbers $q$. Solid lines show the precise result, which corresponds to the general chain (Fig. 2). Dashed lines represent a relaxation in the low-frequency approximation, which corresponds to the regular chain (Fig. 5).

Figure 7

Function ${\mathcal{F}}_{\mathbf{q}}\left(\omega \right)$ calculated using the KPM for a numerical random matrix with $N={100}^3$ atoms, the parameter $\varkappa =1$, and the wave number $q=1$. Averaging over 100 realizations was applied. The vertical dashed line shows the chosen position of ${\omega }_{\max }$.

Figure 8

Parameters of the chain as a function of scaled site number $\varkappa n/{\omega }_{\max }$ obtained using the numerical random matrix model (68) for different values of the parameter $\varkappa$ and the wave number $q$.