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

Abstract

A long-standing problem has been a theoretical prediction of the densest packing fraction of random packings, $φ_{RCP}$, 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, $φ_{RCP}$, for these packings could be very useful. In such bidisperse packings, $φ_{RCP}$ 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 $φ_{RCP}(p,D)$, using the concept of the cell order distribution. I also derive exact upper and lower bounds on this densest disordered packing fraction.

Figures (6)

• Figure 1: The four possible $3$-cell configurations and their areas, $S_{3i}$, $i=a,b,c,d$ (shaded). The areas the discs occupy within the shaded cell areas are, respectively, $S^{disc}_{3i}$.
• Figure 2: The dependence of $u$ on $p$, described by eq. (\ref{['puRelation']}).
• Figure 3: The packing fraction, $\phi_3(p,D)$, for $1.0\leq D\leq6.0<D_{max}=\sqrt{3}/(2-\sqrt{3})$. The blue line follows the highest packing fraction for each mixture. $D=1.0$ recovers the fully trigonal crystal, $\phi_3=\pi/\left(2\sqrt{3}\right)$.
• Figure 4: For any choice of $D$, a 3-cells system would be disordered when $u$ is between $u_{min}(D)$ (purple curve) and $u_{max}(D)$ (green curve). Converting from $u$ to $p$, using eq. (\ref{['puRelation']}), the lowest curve extends from $p_{min}(D=1)=0.175$ to $p_{min}(D=6.4)=0.312$ and the upper curve from $p_{max}(D=1)=0.825$ to $p_{max}(D=6.4)=0.939$. The figure shows that, when $p$ and $D$ are chosen independently, disorder is assured for any value of $1<D<D_{max}$ within a wide region, demarcated by the two dashed lines. Using again eq. (\ref{['puRelation']}), this range corresponds to $p_{low}\equiv0.312<p<0.825\equiv p_{high}$. Also shown is the curve corresponding to the common choice of both disc types occupying the same area (blue curve), in which $p$ depends on $D$. The sharp drop of the upper bound in these packings, when $D\lesssim3$, limits the range of $p$, for which disorder is assured.
• Figure 5: A typical example of $\phi_{RCP}$ of bidisperse planar disc packing as a function of $p$ for $D=3.0$. It is bounded above by $\phi_3$ and below by $\phi_{low}$.