Electrochemistry meets polymer physics: polymerized ionic liquids on an electrified electrode

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. We derive an analytical expression for the linear differential capacitance of the electric double layer. Solving numerically the self-consistent field equations, we establish a range of the potential drop, when the linear theory is valid and analyze the differential capacitance as a function of applied voltage for a pure polymeric ionic liquid and a polymeric ionic liquid dissolved in a highly polar organic solvent. Our findings can be used in different electrochemical applications, such as supercapacitors, batteries, fuel cells, electrodeposition, etc.