Bulk modulus along jamming transition lines of bidisperse granular packings

We present three-dimensional discrete element method simulations of bidisperse granular packings to investigate their jamming densities φJ and dimensionless bulk moduli K as functions of the size ratio δ and the concentration of small particles XS. We determine the partial and total bulk moduli for packings near their jamming densities, including a second transition that occurs for sufficiently small δ and XS when the system is compressed beyond its first jamming transition. While the first transition is sharp, exclusively with large-large contacts, the second is rather smooth, carried by small-large interactions at densities much higher than the monodisperse random packing baseline, φJmono≈0.64. When only nonrattlers are considered, all the effective transition densities are reduced, and the density of the second transition emerges rather close to the reduced baseline, φJmono≈0.61, due to its smooth nature. At size ratios δ≤0.22 a concentration XS∗ divides the diagram - either with most small particles nonjammed or jammed jointly with large ones. For XS<XS∗, the modulus K displays different behaviors at first and second jamming transitions. Along the second transition, K rises relative to the values found at the first transition; however, is still small compared to K at XS∗. Explicitly, for our smallest δ=0.15, the discontinuous jump in K as a function of XS is obtained at XS∗≈0.21 and coincides with the maximum jamming density where both particle species mix most efficiently. Our results will allow tuning or switching the bulk modulus K or other properties, such as the wave speed, by choosing specific sizes and concentrations based on a better understanding of whether small particles contribute to the jammed structure or not, and how the micromechanical structure behaves at either transition.