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 to consider larger systems, lower concentrations, and longer time scales concentrations than molecular simulations in order to predict the electrochemical properties of Thomas-Fermi capacitors and correlate them with the ion dynamics.

💡 Research Summary

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This paper introduces a computationally efficient description of metallic electrodes that incorporates the Thomas‑Fermi (TF) screening length (l_{\text{TF}}) as a tunable parameter, and integrates this description into Brownian dynamics (BD) simulations of electric double‑layer capacitors. The authors start by modeling the system as two parallel‑plate electrodes separated by an implicit‑solvent electrolyte. The electrodes consist of a uniform positive background charge density and a mobile electron density; the latter is treated within a Thomas‑Fermi kinetic‑energy functional. The electrolyte is represented by point ions embedded in a dielectric continuum of permittivity (\varepsilon_s).

A grand‑canonical treatment is adopted for the electrons: their densities can exchange particles with reservoirs that fix the electrochemical potentials (\mu_l) and (\mu_r). Global electroneutrality is enforced with a Lagrange multiplier, which also ensures that the imposed voltage (\Delta\Psi) equals the difference of the electrochemical potentials. The total free energy (\Omega) combines the Coulomb energy of all charges with the TF kinetic energy of the electrons.

The key methodological step is a Born‑Oppenheimer (BO) approximation for the electron density. By minimizing (\Omega) with respect to the electron densities while keeping the ionic configuration fixed, the authors obtain an Euler‑Lagrange equation that links the electron density fluctuation (\delta n(\mathbf r)) to the local electrostatic potential (\phi(\mathbf r)). Linearizing the TF kinetic functional around the bulk electron density yields a simple relation (\alpha,\delta n = e