Hard spheres cannot overlap owing to infinite repulsive energies between particles at contact. The volume that a hard-sphere excludes other hard spheres to occupy is identical to a sphere at the center of a hard-sphere with a diameter equal to the center-to-center distance of two contacting hard spheres. This distance is the sum of the radii of two hard spheres at contact . The phase space that a particle forbids the other counterpart particle to occupy is called excluded volume and schematically demonstrated in Fig. .