text: Interactions between various objects were studied within the DH and PB models by many researchers. Some of the relevant results are summarized in the following.
  Non-planar geometries: Objects of other than planar geometries can be treated within the Derjaguin approximation,  provided their size is substantially larger than the Debye length. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. The exponential nature of these repulsive forces and the fact that its range is given by the Debye length was confirmed experimentally by direct force measurements, including surface forces apparatus, colloidal probe technique, or optical tweezers.  The interaction free energy involving two spherical particles within the DH approximation follows the Yukawa or screened Coulomb potential
  where r is the center-to-center distance, Q is the particle charge, and a the particle radius. This expression is based on the superposition approximation and is only valid at large separations. This equation can be extended to more highly charged particles by reinterpreting the charge Q as an effective charge. To address the interactions in other situation, one must resort to numerical solutions of the DH or PB equation.
  Non-uniform or patchy charge distribution: Interaction between surfaces with non-uniform and periodic charge distribution has been analyzed within the DH approximation. Such surfaces are referred to have a mosaic or patch-charge distribution. One important conclusion from these studies is that there is an additional attractive electrostatic contribution, which also decays exponentially. When the non-uniformities are arranged in a quadratic lattice with spacing b, the decay length q of this additional attraction can be expressed as
  At high salt levels, this attraction is screened as the interaction between uniformly charged surfaces. At lower salt levels, however, the range of this attraction is related to the characteristic size of the surface charge heterogeneities.
  Three-body forces: The interactions between weakly charged objects are pair-wise additive due to the linear nature of the DH approximation. On the PB level, however, attractive three-body forces are present. The interaction free energy between three objects 1, 2, and 3 can be expressed as
  where F are the pair free energies and ΔF is the non-additive three-body contribution. These three-body contributions were found to be attractive on the PB level, meaning that three charged objects repel less strongly than what one would expect on the basis of pair-wise interactions alone.