# Bulk Modulus along Jamming Transition Lines of Bidisperse Granular Packings

We present 3D DEM simulations of bidisperse granular packings to investigate their jamming densities, $\phi_J$, and dimensionless bulk moduli, $K$, as a function of the size ratio, $\delta$, and the concentration of small particles, $X_{\mathrm S}$. We determine the partial and total bulk moduli for each packing and report the jamming transition diagram, i.e., the density or volume fraction marking both the first and second transitions of the system. At a large enough size difference, e.g., $\delta \le 0.22$, $X^{*}_{\mathrm S}$ divides the diagram with most small particles either non-jammed or jammed jointly with large ones. We find that the bulk modulus $K$ jumps at $X^{*}_{\mathrm S}(\delta = 0.15) \approx 0.21$, at the maximum jamming density, where both particle species mix most efficiently, while for $X_{\mathrm S} < X^{*}_{\mathrm S}$ $K$ is decoupled in two scenarios as a result of the first and second jamming transition. Along the second transition, $K$ rises relative to the values found at the first transition, however, is still small compared to $K$ at $X^{*}_{\mathrm S}$. While the first transition is sharp, the second is smooth, carried by small-large interactions, while the small-small contacts display a transition. This demonstrates that for low enough $\delta$ and $X_{\mathrm S}$, the jamming of small particles indeed impacts the internal resistance of the system. Our new results will allow tuning 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.

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