Disorder Criterion and Explicit Solution for the Disc Random Packing Problem

Predicting the densest random disc packing fraction is an unsolved paradigm problem relevant to a number of disciplines and technologies. One difficulty is that it is ill defined without setting a criterion for the disorder. Another is that the density depends on the packing protocol and the multitude of possible protocol parameters has so far hindered a general solution. A new approach is proposed here. After formulating a well-posed form of the general protocol-independent problem for planar packings of discs, a systematic criterion is proposed to avoid crystalline hexagonal order as well as further topological order. The highest possible random packing fraction is then derived exactly: ${\varphi }_{\mathrm{RCP}}=0.852525\dots$. The solution is based on the cell order distribution that is shown to (i) yield directly the packing fraction; (ii) parametrize all possible packing protocols; (iii) make it possible to define and limit all topological disorder. The method is further useful for predicting the highest packing fraction in specific protocols, which is illustrated for a family of simply sheared packings that generate maximum-entropy cell order distributions.

Figure 1

The shape of a $k$-cell depends on $k-3$ internal angles, illustrated for a 4-cell in (a) and for a 5-cell in (b).

Figure 2

Increasing the disorder parameter $\alpha$, reduces $Q_5$ (right axis) and increases ${\varphi }_{\mathrm{RCP}}$ (left axis). At $\alpha =1$, ${\varphi }_{\mathrm{RCP}}=0.852525$. $Q_5$ tends to zero very slightly above $\alpha =1.0$ and well below $\alpha =1.1$ (dashed line). Beyond this point, packings contain only 3- and 4-cells and are therefore not truly disordered.