Random packing fraction of binary similar particles: Onsager's excluded volume model revisited

In this paper, the binary random packing fraction of similar particles with size ratios ranging from unity to well over 2 is studied. The classic excluded volume model for spherocylinders and cylinders proposed by Onsager [1] is revisited to derive an asymptotically correct expression for these binary packings. From a Taylor series expansion, it follows that the packing fraction increase by binary polydispersity equals $2f(1-f) X_1(1-X_1) (u-1)^2+O ((u-1)^3)$, where $ f$ is the monosized packing fraction, $X_1$ is the number fraction of a component, and $u$ is the size ratio of the two particles. This equation is in excellent agreement with the semi-empirical expression provided by Mangelsdorf and Washington [2] for random close packing (RCP) of spheres. Combining both approaches, a generic explicit equation for the bidisperse packing fraction is proposed, which is applicable to size ratios well above 2. This expression is extensively compared with computer simulations of the random close packing of binary spherocylinder packings, spheres included, and random loose sphere packings $(1\le u\le 2)$. The derived generic closed-form and parameter-free equation, which contains a monosized packing fraction, size ratio, and composition of particle mix, appears to be in excellent agreement with the collection of computer-generated packing data using four different computer algorithms and RCP and random loose packing (RLP) compaction states. Furthermore, the present analysis yields a monodisperse packing fraction map of a wide collection of particle types in various compaction states. The explicit RCP–RLP boundaries of this map appear to be in good agreement with all reviewed data. Appendix A presents a review of published monodisperse packing fractions of (sphero)cylinders for aspect ratios $ l/d$ from zero to infinity and in RLP and RCP packing configurations, and they are related to Onsager's model. Appendix B presents a derivation of the binary packing fraction of disks in a plane $(R^2)$ and hyperspheres in $R^{D}(D$ > 3) with a small size difference, again using this model.