Brownian dynamics simulations of electric double-layer capacitors with tunable metallicity

We introduce an efficient description of electrodes, characterized by their Thomas-Fermi screening length lTF inside the metal, for Brownian dynamics (BD) simulations of capacitors. Within a Born-Oppenheimer approximation for the electron charge density inside the electrodes, we derive the effective many-body potential for ions in an implicit solvent between Thomas-Fermi electrodes, taking into account the constraints of applied voltage and of global electro-neutrality of the system, as well as the 2D periodic boundary conditions along the electrode surfaces. We derive the average charge and the fluctuation-dissipation relation for the differential capacitance, highlighting the contribution of the fluctuations of the net ionic dipole moment, as well as those from the solvent polarization and of the electron density, whose fluctuations are suppressed within the Born-Oppenheimer description. We demonstrate the relevance of this model by validating its predictions against known results for the force on ions as a function of the ion-surface distance in simple geometries. The equilibrium ionic density profiles from BD simulations are in excellent agreement with those from an explicit electrode model for perfect metals and are obtained at a significantly lower computational cost. Finally, we discuss with the present model the effect of the Thomas-Fermi screening length on the equilibrium ionic density profiles and the capacitance. While limited to parallel plate capacitors, the present simulation method allows us to consider larger systems, lower concentrations, and longer time scales than molecular simulations in order to predict the electrochemical properties of Thomas-Fermi capacitors and correlate them with the ion dynamics.