Density of States of Colloidal Glasses

Glasses are structurally liquidlike, but mechanically solidlike. Most attempts to understand glasses start from liquid state theory. Here we take the opposite point of view, and use concepts from solid state physics. We determine the vibrational modes of a colloidal glass experimentally, and find soft low-frequency modes that are very different in nature from the usual acoustic vibrations of ordinary solids. These modes extend over surprisingly large length scales.

Figure 1

A two-dimensional image of the colloidal suspension at a volume fraction 0.60 acquired by confocal microscopy. The field of view is $100\times 100\mu \mathrm{m}$. The inset shows a few typical particle trajectories, which show that we obtain the DOS before large-scale rearrangements happen.

Figure 2

Left: Mean-square displacement versus time at different volume fractions $\varphi =0.54$, 0.57, 0.58, 0.59, 0.60 shows a plateau over which the soft modes in the system have been measured. The same plot in log-log scale is shown in the inset, where the plateau is not so evident. Right: pressure from the equation of state as a function of the plateau value of the mean-square displacement. The full line is a $P^{-3/2}$ fit.

Figure 3

Left: Density of states shown for the volume fractions, $\varphi =0.58$. The frequency axis is in the unit of $\mu {\mathrm{m}}^{-1}$. The dashed line shows the upper-limit of frequency ${\omega }_0$ until which the spectrum is reliable. Right: Closure of the gap: cumulative density of states $D^{\prime }(\omega )=\int D(\omega )d\omega$ at different volume fractions respectively $\varphi =0.54$, 0.57, 0.58, 0.59, 0.60. The frequencies along the horizontal axis are scaled by the pressure.

Figure 4

The two-dimensional eigenvector fields. The volume fraction is $\varphi =0.60$. Frequencies ($\mu {\mathrm{m}}^{-1}$) from left to right, top: 1.01, 3.40, bottom: 10.4, 18.5. The length scale associated with swirling structures (top left corner) is about 34 microns. Large displacements (big arrows) in the eigenvector field are mostly seen always at low frequencies and are often an isolated event.

Figure 5

Left: The average scalar product ${\overrightarrow{v}}_i•{\overrightarrow{v}}_j$, for a vol. fraction 0.60. This has been computed for eigenmodes in different parts of the spectrum, as shown in $D(\omega )$ in the inset. Lowest frequency modes have a long tail with a negative part characterizing the swirling motion. The shortest interparticle distance $\delta r_s=1.5\mu \mathrm{m}$ is indicated by the arrow. Right: The orientational correlation function for the lowest frequency modes for different packing fractions $\varphi =0.54$, 0.57, 0.58, 0.59, 0.60. The general shape of the modes seems only very weakly dependent on the packing fraction.