More accurate description of double layer interactions can be put forward on the primitive model. This model treats the electrostatic and hard-core interactions between all individual ions explicitly. However, it includes the solvent only in a "primitive" way, namely as a dielectric continuum. This model was studied in much detail in the theoretical community. Explicit expressions for the forces are mostly not available, but they are accessible with computer simulations, integral equations, or density functional theories.
The important finding from these studies is that the PB description represents only a mean-field approximation. This approximation is excellent in the so-called weak coupling regime, that is for monovalent electrolytes and weakly charged surfaces. However, this description breaks down in the strong coupling regime, which may be encountered for multivalent electrolytes, highly charged systems, or non-aqueous solvents. In the strong coupling regime, the ions are strongly correlated, meaning that each ion has an exclusion hole around itself. These correlations lead to strong ion adsorption to charged surfaces, which may lead to charge reversal and crystallization of these ions on the surface. These correlations may also induce attractive forces. The range of these forces is typically below 1 nm.