(2) |

where

is the total Coulomb energy, the prefactor results from integrals over momenta, and is the thermal energy at temperature . All charged species in the system, including the fixed macroions, contribute to . The macroions also exclude simple ions from their interiors, so their volumes are excluded from volume of integration . Equation (3) implicitly adopts the primitive model, approximating the solvent's influence through its dielectric constant, . We will reconsider the solvent's role in Sec. 3 when we examine hydrodynamic and electrohydrodynamic coupling.

The partition function can be expressed as a functional integral over all possible simple-ion distributions

where is one particular simple-ion distribution with action

is the volume of the system and the prime on the integral in Eq. (4) indicates that the simple ions' number is conserved: .

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 [10]. Minimizing to implement the mean field approximation yields the Poisson-Boltzmann equation

where the subscript is the concentration of simple ions of type far from charged surfaces.

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 [11], Verwey and Overbeek [12] (DLVO) invoked the Debye-Hückel approximation to linearize the Poisson-Boltzmann equation. Solving for the potential outside a sphere of radius carrying charge yields [13]

The monotonic decay of correlations within the simple ion distribution is described by the Debye-Hückel screening length, , given by

(8) |

We will consider only monovalent simple ions with .

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 .