Disorder-induced vibrational anomalies from crystalline to amorphous solids

The origin of the boson peak—an excess of the density of states over Debye's model in glassy solids—is still under intense debate, among which some theories and experiments suggest that the boson peak is related to a van-Hove singularity. Here we show that the boson peak and the van-Hove singularity are well separated entities by measuring the vibrational density of states of a two-dimensional granular system, where the packings are tuned gradually from a crystalline to polycrystalline structure and toward an amorphous material. We observe the coexistence of the boson peak and the van-Hove singularities being well separated in the polycrystals. The van-Hove singularities gradually shift to higher-frequency values while broadening their shape. They disappear completely when the structural disorder becomes sufficiently high. By analyzing the system at different degrees of disorder, we find that the boson peak is associated with spatially uncorrelated random flucutations of the shear modulus, whereas the smearing of the van-Hove singularities is associated with spatially correlated fluctuations of the shear modulus.

Figure 1

Images of the bond orientation parameter ${\psi }_6^i$ [(A)–(F)] and stresses [(a)–(f)] of packings of various degrees of disorder, showing force chains, i.e., those filament-like structures composed of particles carrying large stresses [49, 50]. Note that the same letters in upper and lower cases denote the same packing, and the color bar under panel (A) indicates the values of ${\psi }_6^i$. The ratio $R$ between the number of small and large disks and the disorder parameter $\eta$ are [(A) and (a)] $R=1.0$, $\eta =1.0$; [(B) and (b)] $R=6.25$, $\eta =0.74$; [(C) and (c)] $R=33.1$, $\eta =0.41$; [(D) and (d)] $R=178.92$, $\eta$ = 0.20; [(E) and (e)] (polycrystal): $R=1.0$, $\eta$ = 0.13; [(F) and (f)] (granular crystal): $R=1.0$, $\eta =0.0$.

Figure 2

The DOS's (a) and reduced DOS's (b) of packings of different disorder parameters $\eta$. Here each curve is ensemble averaged over 10 independent packings of the same $\eta$. Moreover, the frequency $\omega$ is normalized by the Debye frequency ${\omega }_D$. Inset of (b): The positions of the boson peak ${\omega }_b/{\omega }_D$ (red $\circ$), the first and second VHS – ${\omega }_{\mathrm{vH}1}/{\omega }_D$ (green $\square$) and ${\omega }_{\mathrm{vH}2}{\omega }_D$ (blue $▽$) vs $\eta$. The dashed line denotes the minimum $\eta$ beyond which the two VHS disappear. Here $p=26.5\pm 0.2$ N/m.

Figure 3

The DOS (a) and the reduced DOS (b) at different p collapse for the strongly disordered system ($\eta =1$). Only data points from four different values of the pressure $p$ are shown for clarity, results at other $p$ are similar. (c) The relative fluctuations of the shear modulus, $\delta G/\langle G\rangle$ vs $w/D$ at different pressure for the strongly disordered system ($\eta =1$), which is corresponding to (a) and (b). The results are independent of $p$. Note that here $w/D$ represents the rescaled course-grain size, where $D$ is the weighted average diameter of disks, i.e., $\frac{N_b{D}_b+N_s{D}_s}{N_b+N_s}$, here $N_b$ ($N_s$) indicates the number of the larger (smaller) disks for the whole system, and $D_b$ ($D_s$) indicates the diameter of the larger (smaller) disks.

Figure 4

The DOS (a) and the reduced DOS (b) rescaled by the Debye frequency ${\omega }_D$ at different pressure for the granular crystals ($\eta =0$). Each curve in (a) and (b) is ensemble averaged over 10 different runs at the same pressure. (c) With double $y$-axes: The left $y$ axis represents the frequencies of two VHS's [${\omega }_{\mathrm{vH}1}$ (red $\circ$, solid line) and ${\omega }_{\mathrm{vH}2}$ (red $\circ$, dashed line)] and the right $y$ axis represents the heights of two VHS's [$H_{\mathrm{vH}1}$ (blue $\square$, solid line) and $H_{\mathrm{vH}2}$ (blue $\square$, dashed line)] versus pressure. Note that the data of the second VHS at $p=5.45$ N/m are not shown here, due to the disappearance of the second VHS. (d) The relative fluctuations of shear modulus, $\delta G/\langle G\rangle$ vs $w/D$, at different pressure corresponding to (a) and (b). The results are dependent on $p$. Note that here $w/D$ represents the rescaled course-grain size, where $D$ is the weighted average diameter of disks. Each curve in (a)–(d) is ensemble averaged over 10 different runs at the same pressure.