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

In this article a closed-form equation is derived for the binary random packing fraction of similar particles with size ratios from unity to well over 2. To this end, the classic excluded volume model for spherocylinders and cylinders by Onsager [1] is revisited and employed to derive an asymptotically correct expression for these binary packings. From second-order perturbation it follows that the packing fraction increase by binary polydispersity equals 2f1(1 - f1)X1(1 - X1)(u - 1)^2 +O((u - 1)^3), with f1 as monosized packing fraction (which depends on particle shape and densification), X1 the number fraction of a component, and u the size ratio of the two particles. This equation has been found to be in excellent in agreement with a semi-empirical expression provided by Mangelsdorf and Washington [2] for random close packing of spheres, applicable to size ratios from unity to almost 2. The expression also agrees with computer simulations of binary random close sphere packings with size ratios up 2.5. Inserting the theoretical result based on Onsager's model, a general equation for binary packings is proposed, applicable to size ratios well over 2. This is expression is extensively compared with computer simulations of binary random loose sphere packings (1 < u < 2). The derived general closed-form equation, which contains monosized void fraction, size ratio and composition only, again appears to be in excellent agreement with this collection of computer-generated packing data.Comment: 29 pages, 4 figures, 3 table