The distribution of forces affects vibrational properties in hard sphere glasses

Abstract

We study theoretically and numerically the elastic properties of hard sphere glasses, and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero-temperature, we argue that the presence of certain pairs of particles interacting with a small force $f$ soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting $P(f)\sim f^{{\theta }_e}$ the force distribution of such pairs and ${\varphi }_c$ the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale ${\omega }^{\ast }$, rising up to it as $D(\omega )\sim {\omega }^{2+a}$, and decaying above ${\omega }^{\ast }$ as $D(\omega )\sim {\omega }^{-a}$ where $a=(1-{\theta }_e)/(3+{\theta }_e)$ and $\omega$ is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with $⟨\delta R^2⟩\sim 1/\mu \sim ({\varphi }_c-\varphi {)}^{\kappa }$ where $\kappa =2-2/(3+{\theta }_e)$, and (iii) continuum elasticity breaks down on a scale ${\ell }_c\sim 1/\sqrt{\delta z}\sim ({\varphi }_c-\varphi {)}^{-b}$ where $b=(1+{\theta }_e)/(6+2{\theta }_e)$ and $\delta z=z-2d$, where $z$ is the coordination and $d$ the spatial dimension. We numerically test (i) and provide data supporting that ${\theta }_e\approx 0.41$ in our bi-disperse system, independently of system preparation in two and three dimensions, leading to $\kappa \approx 1.41$, $a\approx 0.17$, and $b\approx 0.21$. Our results for the mean-square displacement are consistent with a recent exact replica computation for $d=\infty$, whereas some observations differ, as rationalized by the present approach.

5 pages + 4 pages supplementary information

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