  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. the packing fraction increase by binary polydispersity equals 2f(1 -
f)X1(1 - X1)(u - 1)^2 + O((u - 1)^3), where f is the monosized packing
fraction, X1 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. 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 < u < 2 and beyond). The derived generic closed-form equation
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 shapes at
different compaction states. The explicit boundaries of this map appear to be
in good agreement with a broad collection of random close and random loose
packing data. Appendix A presents an overview of published monodisperse packing
fractions of (sphero)cylinders for aspect ratios from zero to infinity, and at
RLP and RCP packing configurations, and they are related to Onsager's theory.
Appendix B presents a derivation the binary packing fraction of disks in a
plane (R^2) and hyperspheres in R^D (D > 3) with small size difference.
