Excluded volume geometry and packing fraction in binary convex hyperparticle mixtures

In this paper the excluded volume of binary similar hyperparticles with small size difference in D-dimensional Euclidean spaces R^{2}, R^{3}, and R^{4} is studied using two different statistical geometry approaches. These geometric approaches, concerning orientation geometry and integral geometry, yield the excluded volume of particle pairs. The excluded volume of rectangles, based on orientation geometry, in Euclidean space R^{2} is used to derive an explicit equation for the bidisperse packing fraction, which is compatible with the expression published previously. Next, the excluded volumes of pairs of convex particles in D=2,3, and 4, resulting from integral geometry, are presented. These excluded volumes are identical with the specific ones for circles and rectangles (D = 2) and (sphero)cylinders (D = 3), derived by orientation geometry. Furthermore, these orientation geometry-based excluded volumes contain geometric measures: particle volume, surface area, mean curvature, and the second quermassintegral. They allow for the derivation of closed-form expressions for the random packing fraction of binary convex similar hyperparticles in Euclidean spaces R^{2}, R^{3}, and R^{4}.