Entropic insertion of a big sphere into a cylindrical vessel

We calculate the spatial distribution of the entropic potential between a big sphere and an even bigger vessel with cylindrical shape, which are immersed in small spheres, using the three-dimensional integral equation theory. The distribution is strongly dependent on relative magnitudes of the big-sphere diameter and the inner diameter of the vessel unless the latter is much larger than the former. For a fixed value of the inner diameter, a big sphere whose diameter lies in a specific range is spontaneously inserted into the vessel and strongly confined within a small space almost in the center of the vessel cavity.