Carl Schreck

This thesis describes comprehensive computational and theoretical studies of the mechanical and vibrational properties of athermal particulate systems, such as granular media, foams, and emulsions, modeled as frictionless particles with spherical or ellipsoidal shapes. First, we investigate the mechanical properties of static packings of ellipsoidal particles. While amorphous packings of spherical particles are isostatic at jamming onset and possess the minimum contact number $z=z_{\rm iso}$ required for mechanical stability, packings of ellipsoidal particles are generally hypostatic with $z < z_{\rm iso}$. To investigate the deviation in contact number from the isostatic value, we measured the dynamical matrix of static packings of ellipsoidal particles. At jamming onset, we find that the harmonic response of these systems vanishes, and the potential energy scales as $\delta^4$ for perturbations $\delta$ along special directions in configuration space called `quartic modes’. Quartic modes give rise to novel power-law scaling of the static shear modulus and their number matches the deviation in the contact number from the isostatic value. Second, we investigated the dependence of the structural and mechanical properties of static bidisperse disk packings to two packing generation protocols. The first involved thermally quenching liquid configurations to zero temperature followed by compression to reach packing fractions $\phi_J$ at jamming onset. The second involved breaking the system up into a square grid and initializing alternating squares with small and large particles followed by compression to $\phi_J$. We find that {\it isostatic} packings exist over a range of packing fractions for large systems, in contrast to the view in the community that amorphous mechanically stable disk packings exist at a single packing fraction. We also compare the structural and mechanical properties of isostatic and hyperstatic packings. In addition, we also study the vibrational response of mechancially stable disk packings to thermal fluctuations. We find that, in contrast to atomic and molecular systems, systems that interact via repulsive contact forces possess no harmonic regime in the large system limit for all values of overcompression studied, and at jamming onset for all system sizes. To show this, we performed fixed energy simulations following perturbations with amplitude $\delta$ along all eigendirections of the dynamical matrix. The fluctuations abruptly spread to all modes for extremely small perturbation amplitudes, $\delta > \delta_c$, where $\delta_c$ is the amplitude at which a single contact breaks, and to a continuous frequency band for $\delta>\Delta\phi$. We also show that the density of vibrational modes deviates strongly from that predicted from the dynamical matrix in the nonharmonic regime.