Equation of state for random sphere packing with arbitrary adhesion and friction

We systematically generate a large set of random micro-particle packings over a wide range of adhesion and friction by means of adhesive contact dynamics simulation. The ensemble of generated packings covers a range of volume fraction $\varphi$ from $0.135\pm 0.007$ to $0.639\pm 0.004$, and of coordination number $Z$ from $2.11\pm 0.03$ to $6.40\pm 0.06$. We determine $\varphi$ and $Z$ at four limits (random close packing, random loose packing, adhesive close packing, and adhesive loose packing), and find a universal equation of state $\varphi (Z)$ to describe packings with arbitrary adhesion and friction. From a mechanical equilibrium analysis, we determine a critical friction coefficient ${\mu }_{\mathrm{f},\mathrm{c}}$: when the friction coefficient $\mu_{\rm f}$isbelow$\mu_{\rm f, c}$, particles’ rearrangements are dominated by sliding, otherwise, they are dominated by rolling. Because of this reason, both $\varphi ({\mu }_{\mathrm{f}})$ and $Z({\mu }_{\mathrm{f}})$ change sharply across $\mu_{\rm f, c}$. Finally, we generalize the Maxwell counting argument to micro-particle packings, and show that the loosest packing, i.e., adhesive loose packing, satisfies the isostatic condition at $Z=2$.