  We study ionic liquids interacting with electrified interfaces. The ionic
fluid is modeled as a Coulomb lattice gas. We compare the ionic density
profiles calculated using a popular modified Poisson-Boltzmann equation with
the explicit Monte Carlo simulations. The modified Poisson-Boltzmann theory
fails to capture the structural features of the double layer and is also unable
to correctly predict the ionic density at the electrified interface. The
lattice Monte Carlo simulations qualitatively capture the coarse-grained
structure of the double layer in the continuum. We propose a convolution
relation that semiquantitatively relates the ionic density profiles of a
continuum ionic liquid and its lattice counterpart near an electrified
interface.
