Vibrations in Jammed Solids: Beyond Linear Response


Corey O'Hern (Yale University)


We propose a novel `phase diagram' for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize three distinct classes of behavior as a function of the input kinetic energy $T_0$ and packing fraction deviation from jamming onset $\Delta \phi=\phi - \phi_J$ using molecular dynamics simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) can only occur above jamming onset for $\Delta \phi > \Delta \phi_c(T_0)$; they possess an average instantaneous coordination number that is greater than the isostatic value required for mechanically stable packings. ICS display harmonic vibrational response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of $2N-2$ sharp peaks at the eigenfrequencies associated with the dynamical matrix. Hypo-coordinated solids (HCS) occur both above and below jamming onset. For $\Delta \phi_c(T_0) > \Delta \phi > \Delta \phi_{cb}(T_0)$ (for $\Delta \phi >0$) and $|\Delta \phi| < \Delta \phi_{cb}(T_0)$ (for $\Delta \phi <0$), the network of interparticle contacts fluctuates and the average coordination number is below the isostatic value, but cage-breaking particle rearrangements do not occur. The vibrational response of HCS is strongly nonharmonic. The density of vibrational modes $D(\omega)$ for HCS is not a collection of sharp peaks, and its precise form depends on the measurement method. For $\Delta \phi < \Delta \phi_{cb}(T_0)$ ($\Delta \phi>0$) and $|\Delta \phi| > \Delta \phi_{cb}(T_0)$ ($\Delta \phi<0$), particle rearrangements occur, the structural relaxation time is finite, and the density of vibrational modes resembles that for dense liquids with nonzero intercept $D(0)$ and slope $D'(0) <0$ at zero frequency.

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