Linear screening of the electrostatic potential around spherical particles with non-spherical charge patterns

We derive analytical expressions for the screened electrostatic potential caused by a charged spherical colloid that contains point charges distributed in an arbitrary manner in its interior. We consider both the cases of uniform and discontinuous dielectric constants. The solution is based on an expansion of the electrostatic potentials on the various regions of space in spherical harmonics involving spherical Bessel functions of the third kind. Tetrahedral charge arrangements as well as a random charge distribution inside the confining sphere are considered explicitly as representative examples.