Random close packing fraction of bidisperse discs: Theoretical derivation and exact bounds

A long-standing problem has been a theoretical prediction of the densest packing fraction of random packings, $\phi_{RCP}$, of same-size discs in $d=2$ and spheres in $3$. However, to minimize order, experiments and numerical simulations often use two-size discs and a prediction of the highest possible packing fraction, $\phi_{RCP}$, for these packings could be very useful. In such bidisperse packings, $\phi_{RCP}$ is a function of the sizes ratio, $D$, and concentrations, $p$, of the disc types. A disorder-guaranteeing theory is formulated here to derive the highest mathematically possible value of $\phi_{RCP}(p,D)$, using the concept of the cell order distribution. I also derive exact upper and lower bounds on this densest disordered packing fraction.