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Phonon Spectra, Nearest Neighbors, and Mechanical Stability of Disordered Colloidal Clusters with Attractive Interactions

We investigate the influence of morphology and size on the vibrational properties of disordered clusters of colloidal particles with attractive interactions. From measurements of displacement correlations between particles in each cluster, we extract vibrational properties of the corresponding "shadow" glassy cluster, with the same geometric configuration and interactions as the "source" cluster but without damping. Spectral features of the vibrational modes are found to depend strongly on the average number of nearest neighbors, $\bar{NN}$, but only weakly on the number of particles in each glassy cluster. In particular, the median phonon frequency, $ω_{med}$, is essentially constant for $\bar{NN}$ $<2$ and then grows linearly with $\bar{NN}$ for $\bar{NN}$ $>2$. This behavior parallels concurrent observations about local isostatic structures, which are absent in clusters with $\bar{NN}$ $<2$ and then grow linearly in number for $\bar{NN}$$>2$. Thus, cluster vibrational properties appear to be strongly connected to cluster mechanical stability (i.e., fraction of locally isostatic regions), and the scaling of $ω_{med}$ with $\bar{NN}$ is reminiscent of the behavior of packings of spheres with repulsive interactions at the jamming transition. Simulations of random networks of springs corroborate observations and suggest that connections between phonon spectra and nearest neighbor number are generic to disordered networks.