Size-topology correlations in disk packings: Terminal bidispersity in order-disorder transitions

In random packings or tilings, the size distribution of individual elements (domains) and the statistics of numbers of neighbours of those domains are strongly correlated. In the case of circular disks forming a random packing in the plane, it has long been known empirically that a certain critical amount of bidispersity avoids crystallization of the packing. We demonstrate how the formalism of a simplified granocentric model allows for an analytical computation of the sizetopology correlation as a function of both size ratio and frequency of small disks. The results, obtained without free parameters, are in excellent agreement with the empirical findings of packing simulations concerning critical (terminal) bidispersity. It is also shown that, at equal size variance, the discrete (bidisperse) disk size distributions induce stronger disorder than continuously polydisperse disks.