Jamming of Bidisperse Frictional Spheres.

By generalizing a geometric argument for frictionless spheres, a model is proposed for the jamming density $\phi_J$ of mechanically stable packings of bidisperse, frictional spheres. The monodisperse, friction-dependent jamming density $\phi_J^{\mathrm{mono}}(\mu_s)$ is the only input required in the model, where $\mu_s$ is the coefficient of friction. The predictions of the model are validated by robust estimates of $\phi_J$ obtained from computer simulations of up to $10^7$ particles for a wide range of $\mu_s$, and size ratios up to 40:1. For the largest size ratio, the frictional $\phi_J$ below $f^{s}_{\mathrm{max}}$ is slightly greater than the model prediction, but is still bounded by the frictionless model. Although $\phi_J$ varies nonmonotonically with the volume fraction of small spheres $f^s$ for all $\mu_s$, its maximum value $\phi_{J,\mathrm{max}}$ at $f^{s}_{\mathrm{max}}$ are both $\mu_s$-dependent. The optimal $f^{s}_{\mathrm{max}}$ is characterized by a sharp transition in the fraction of small rattler particles.