  Classical theory of the electric double layer is based on the fundamental
assumption of a dilute solution of point ions. There are a number of situations
such as high applied voltages, high concentration of electrolytes, systems with
multivalent ions, or solvent-free ionic liquids where the classical theory is
often applied but the fundamental assumptions cannot be justified. Perhaps the
most basic assumption underlying continuum models in electrokinetics is the
mean-field approximation, that the electric field acting on each discrete ion
is self-consistently determined by the local mean charge density. This paper
considers situations where the mean-field approximation breaks down and
electrostatic correlations become important. A fourth-order modified Poisson
equation is developed that accounts for electrostatic correlations and captures
the essential features in a simple continuum framework. The theory is derived
variationally as a gradient approximation for non-local electrostatics, in
which the dielectric permittivity becomes a differential operator. The only new
parameter is a characteristic length scale for correlated ion pairs. The model
is able to capture subtle aspects of more detailed simulations based on Monte
Carlo, molecular dynamics, or density functional theory and allows for the
straightforward calculation of electrokinetic flows in correlated liquids, for
the first time. Departures from classical Helmholtz-Smoluchowski theory are
controlled by the dimensionless ratio of the correlation length to the Debye
screening length. Charge-density oscillations tend to reduce electro-osmotic
flow and streaming current, and over-screening of the surface charge can lead
to flow reversal. These effects also help to explain the apparent
charge-induced thickening of double layers in induced-charge electrokinetic
phenomena.
