  Polymeric ionic liquids are emerging polyelectrolyte materials for modern
electrochemical applications. In this paper, we propose a self-consistent field
theory of the polymeric ionic liquid on a charged conductive electrode. Taking
into account the conformation entropy of rather long polymerized cations within
the Lifshitz theory and electrostatic and excluded volume interactions of ionic
species within the mean-field approximation, we obtain a system of
self-consistent field equations for the local electrostatic potential and
average concentrations of monomeric units and counterions. We solve these
equations in the linear approximation for the cases of a point-like charge and
a flat infinite uniformly charged electrode immersed in a polymeric ionic
liquid and derive analytical expressions for local ionic concentrations and
electrostatic potential, and derive an analytical expression for the linear
differential capacitance of the electric double layer. We also find a numerical
solution to the self-consistent field equations for two types of boundary
conditions for the local polymer concentration on the electrode, corresponding
to the cases of the specific adsorption absence (indifferent surface) and
strong short-range repulsion of the monomeric units near the charged surface
(hard wall case). For both cases, we investigate the behavior of differential
capacitance as a function of applied voltage for a pure polymeric ionic liquid
and a polymeric ionic liquid dissolved in a polar organic solvent. We observe
that the differential capacitance profile shape is strongly sensitive to the
adopted boundary condition for the local polymer concentration on the
electrode.

Soft Condensed Matter