Structural and mechanical characteristics of sphere packings near the jamming transition: From fully amorphous to quasi-ordered structures

Mechanically stable sphere packings are generated in three-dimensional space using the discrete element method, which span a wide range in structural order, ranging from fully amorphous to quasi-ordered structures, as characterized by the bond orientational order parameter. As the packing pressure, $p$, varies from the marginally rigid limit at the jamming transition ($p \approx 0$) to that of more robust systems ($p \gg 0$), the coordination number, $z$, follows a familiar scaling relation with pressure, namely, $\Delta z = z - z_c \sim p^{1/2}$, where $z_c = 2d = 6$ ($d=3$ is the spatial dimension). While it has previously been noted that $\Delta z$ does indeed remain the control parameter for determining the packing properties, here we show how the packing structure plays an influential role on the mechanical properties of the packings. Specifically, we find that the elastic (bulk $K$ and shear $G$) moduli, generically referred to as $M$, become functions of both $\Delta z$ and the structure, to the extent that $M-M_c \sim \Delta z$. Here, $M_c$ are values of the elastic moduli at the jamming transition, which depend on the structure of the packings. In particular, the zero shear modulus, $G_c=0$, is a special feature of fully amorphous packings, whereas more ordered packings take larger, positive values, $G_c > 0$.