The partition function can be expressed as a functional integral over all possible simple-ion distributions
Equation (5) differs from the exact action by terms accounting for higher-order correlations among simple ions. Dropping these terms, as we have in Eq. (5), yields a tractable but thermodynamically inconsistent theory . Minimizing to implement the mean field approximation yields the Poisson-Boltzmann equation
By considering only one ionic distribution, the mean field approximation neglects fluctuations and higher-order correlations among the simple ions. Even this simplified formulation has no analytic solution except for the simplest geometries. Derjaguin, Landau , Verwey and Overbeek  (DLVO) invoked the Debye-Hückel approximation to linearize the Poisson-Boltzmann equation. Solving for the potential outside a sphere of radius carrying charge yields 
Although the Debye-Hückel approximation cannot be valid near the surface of a highly charged sphere, nonlinear effects should be confined to a thin surface layer. Viewed at longer length scales, nonlinear screening should only renormalize and [alexander84,lowen92,lowen93,gisler94,belloni98].
In this approximation, we obtain the effective pair potential by integrating Eq. (7) over the surface of a second sphere separated from the first by a center-to-center distance . This integration is facilitated by assuming the second sphere's presence does not disrupt the first sphere's ion cloud. The resulting superposition approximation yields a screened Coulomb repulsion for the effective inter-sphere interaction,
The full DLVO potential includes a term accounting for dispersion interactions. These are negligibly weak for well-separated spheres [19,20], however, and are omitted from .