Bulk modulus along jamming transition lines of bidisperse granular packings

We present three-dimensional discrete element method simulations of bidisperse granular packings to in-vestigate their jamming densities phi J and dimensionless bulk moduli K as functions of the size ratio 8 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 8 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, phi mono J approximate to 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, phi similar to mono J approximate to 0.61, due to its smooth nature. At size ratios 8 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 8 = 0.15, the discontinuous jump in K as a function of XS is obtained at XS* approximate to 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.