Low-Frequency Vibrations of Soft Colloidal Glasses

We conduct experiments on two-dimensional packings of colloidal thermosensitive hydrogel particles whose packing fraction can be tuned above the jamming transition by varying the temperature. By measuring displacement correlations between particles, we extract the vibrational properties of a corresponding “shadow” system with the same configuration and interactions, but for which the dynamics of the particles are undamped. The vibrational properties are very similar to those predicted for zero-temperature sphere packings and found in atomic and molecular glasses; there is a boson peak at low frequency that shifts to higher frequency as the system is compressed above the jamming transition.

Figure 1

The effective spring constant $k_{ij}$ between large particles, extracted from the stiffness matrix $K$ at $\varphi =0.885$ (squares), and $\varphi =0.840$ (diamonds). The results are fit to harmonic (solid), and Hertzian (dashed) repulsions, convoluted with the particle size distribution. Insets: Fitted interaction range $\sigma$ and energy $ϵ$ for the harmonic case as a function of $\varphi$. Solid line in upper inset: the particle diameter implied by the packing fraction.

Figure 2

$D(\omega )/\omega$ at $\varphi =0.859$ averaged over (a) different ratios $2N/T$ of the number of degrees of freedom to the number of frames, and (b) different values of the optical resolution relative to the average rms displacement, $ϵ/c$.

Figure 3

(a) The vibrational density of states relative to the Debye prediction, $D(\omega )/\omega$, for $\varphi =0.859$. The maximum in this plot, shown by the vertical dashed line, defines the boson peak frequency, ${\omega }^{*}$. (b) $D(\omega )$, and (c) the participation ratio $p(\omega )$. The dotted vertical lines in (b) and (c) indicate the frequencies of the modes shown in the remaining panels. (d)–(f) Eigenmodes at (d) low, (e) intermediate, and (f) high $\omega$. The size of each arrow scales with the displacement of the particle there, weighted by its mass: $|{\mathbf{u}}_i|\sqrt{m_i}$. Blue circles in (d) indicate high-displacement regions.

Figure 4

(a) $D(\omega )/\omega$ and (b) participation ratio vs $\omega$ as a function of packing fraction [as labeled in (b)]. Inset to (a): the boson peak frequency ${\omega }^{*}$ (circles, left) and inverse peak height ${\omega }^{*}/D({\omega }^{*})$ (triangles, right) vs $\varphi$.