Figure 1

The structures and MSD of four typical samples. (a) From $A$ to $D$, the samples vary from amorphous to polycrystalline with compression. The total dimensions are $146\times 146\mu {\mathrm{m}}^2$. Different colors correspond to different ${\Psi }_{6i}$ values. (b) MSD curves for the disordered particles (open symbols) and ordered particles (solid symbols) in $A$ and $D$. Higher MSD of disordered particles demonstrates their larger motions. All curves are below the dashed line of free diffusion, as the result of caging. $B$ and $C$ are shown in the inset.Reuse & Permissions

Figure 2

The density of states and participation ratio of the normal modes. (a) The density of states $D(\omega )$ of the four samples. In each curve, we can identify a plateau, which shrinks with compression from $A$ to $D$, consistent with the jamming prediction. The inset shows the $D(\omega )/\omega$ spectrum. (b) The participation ratio $P$ at different frequencies. The solid curves are the smoothed results of data. At small $\omega$, the modes are quasilocalized with small $P$ values, while after the characteristic frequency ${\omega }^{*}$, $P$ reaches a stable value. ${\omega }^{*}$ increases with compression.Reuse & Permissions

Figure 3

Vibration distribution in real and Fourier spaces. (a) The polarization vectors of three typical modes, ${\omega }_1<{\omega }_2<{\omega }_3$, in sample $A$. The low-frequency mode (${\omega }_1$) has large-scale collective correlations, while the other two (${\omega }_2$ and ${\omega }_3$) look random. (b) The transverse and longitudinal Fourier components of the three modes within the first Brillouin zone. At ${\omega }_1$, the long-wavelength transverse component dominates. (c) The ratio of the mean squared vibration amplitude between disordered particles and ordered particles, $\overline{A_{\mathrm{dis}}^2}/\overline{A_{\mathrm{ord}}^2}$. It is initially larger than 1 but drops to unity after the transition frequency ${\omega }^{†}$. (d) The average magnitude of transverse and longitudinal components of all samples. The average is taken over the small $q$ area [$-\frac{\pi }{2\sigma }<q_x,q_y<\frac{\pi }{2\sigma }$]. The transverse curves (solid) dominate the longitudinal ones (dashed) at low frequencies. Each transverse curve contains a plateau with the onset frequency ${\omega }^{\oplus }$, as illustrated by the arrows. (e) The agreement of ${\omega }^{*}$, ${\omega }^{†}$, and ${\omega }^{\oplus }$.Reuse & Permissions

Figure 4

Simulations at various temperatures and volume fractions. (a) Different conditions represented by different shapes (for $T$) and colors (for $\varphi$). The solid and open symbols represent systems with and without a plateau in $D(\omega )$, respectively. (b) The comparison of ${\omega }^{†}$ and ${\omega }^{*}$ determined by $\overline{A_{\mathrm{dis}}^2}/\overline{A_{\mathrm{ord}}^2}$ and $P$, respectively. Reasonable agreement appears in all samples, suggesting that the correspondence between the low-frequency quasilocalization and large vibrations at disordered sites is universal.Reuse & Permissions