Percolation of spheres in continuum

The model of the continuum percolation of hard spheres with permeable shells, which describes phase transition sol-gel, has been investigate. Spheres have hard parts in radii $r$, which can't be blocked with each other, and permeable shells in width $d$, which can be blocked. Such spheres of the equal size have been randomly packing in the cub with linear size $L$. The probability of joining the spheres in a cluster is proportional to the volume of overlapping of permeable shells. Spheres belong to a cluster, if a communication between spheres arises. The percolation cluster is the cluster connecting bottom and top sides of the cube. The packing fraction, at which probability of occurrence of the percolation cluster is 0.5, is called as the percolation threshold. The percolation threshold corresponds to the gel point. The dependency of the percolation threshold of the hard spheres with permeable shells from a thickness of the shell has been obtained.