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

TL;DR

This work develops a Brownian dynamics framework for electrochemical capacitors with Thomas–Fermi electrodes, where electrode electrons respond adiabatically (Born–Oppenheimer) to ionic configurations. It yields an effective ion-only potential $V^{\rm eff}$ that incorporates applied voltage $\Delta\Psi$, global electroneutrality, and TF screening via a Green’s-function treatment, enabling computation of the average charge $\langle Q\rangle$ and differential capacitance from fluctuations. The approach is validated against analytical limits for isolated ions and against explicit-electrode benchmarks, achieving excellent agreement and substantial computational savings, and it demonstrates how the Thomas–Fermi length $l_{\rm TF}$ modulates ionic densities and capacitance in parallel-plate capacitors. The method enables larger-scale, longer-time simulations of Thomas–Fermi capacitors and provides a tractable route to relate electrode metallicity to electrochemical properties, with potential extensions to dynamic admittance and more detailed solvent descriptions.

Abstract

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.

Figures (7)

• Figure 1: Capacitor consisting of two Thomas-Fermi electrodes separated over a distance $L$ by an electrolyte solution, under an applied voltage $\Delta\Psi$. The electrolyte consists of explicit ions in an implicit solvent with permittivity $\varepsilon_0\varepsilon_{\rm s}$, while the electrodes are characterized by a permittivity $\varepsilon_0$ and a Thomas-Fermi screening-length $l_{\rm TF}$. Periodic boundary conditions in the $x$ and $y$ directions along the electrode-electrolyte interfaces, with box dimensions $L_x$ and $L_y$, are indicated by dashed lines. The dotted lines indicate the connexion with an electric circuit for $z\to\pm\infty$ that imposes the external voltage.
• Figure 2: Electrostatic force in the direction perpendicular to the implicit electrodes, as a function of the distance $d$ from them, for a system consisting of two "isolated" ions with charges $+e$ at $(0,0,-\frac{L}{2}+d)$ and $-e$ at $(0,0,+\frac{L}{2}-d)$ in vacuum (a) or in a solvent with relative permittivity $\varepsilon_{\rm s}=78$ (b), in the absence of voltage. The reported force is that on the positive charge; the force on the negative charge is simply the opposite. The numerical results with the present method (lines) for various Thomas-Fermi screening lengths $l_{\rm TF}$ (in atomic units, with $a_0\approx0.53$ Å the Bohr radius) indicated by colors are compared with the force corresponding to Eqs. \ref{['eq:UCoulRef:tot']} and \ref{['eq:UCoulRef:i']}, for independent ions next to an infinite interface between a dielectric medium and a Thomas-Fermi metal (symbols). The dotted line in panel b indicates the insulating limit $l_{\rm TF}\to\infty$ (see Eq. \ref{['eq:UCoulRef:i:ltfinfinity']}).
• Figure 3: Electrostatic force in the directions perpendicular (a) and parallel (b) to the implicit electrodes as a function of the distance $d$ from them, for a pair of ions with charges $+e$ at $(0,0,-\frac{L}{2}+d)$ and $-e$ at $(x,0,-\frac{L}{2}+d)$ with $x=0.714$ Å, in vacuum, for perfect metals ($l_{\rm TF}=0$). The reported force is that on the positive charge; the force on the negative charge is simply equal for the $z$ component and opposite for the $x$ one. The numerical results with the present method (lines) are compared with that obtained with the method of Ref. telles_efficient_2024 (symbols), which only applies to perfect metals.
• Figure 4: Equilibrium anion (red solid lines) and cation (blue dashed lines) density profiles from Brownian dynamics simulations of ions in an implicit solvent between perfect metals ($l_{\rm TF}=0$), with (a) the explicit electrode model from Ref. cats_capacitance_2022 (see text), (b) the present model of implicit electrodes and short-range interactions computed with the same atomic sites as in the previous panel and (c) the present fully implicit model where short-range interactions are computed via the Steele potential (see Eqs. \ref{['eq:Walls']}-\ref{['eq:Walls:Vrep']}). In all panels, the density profiles are shown for $\Delta\Psi=0$, $0.1$ and $0.2$ V from light to dark lines, respectively.
• Figure 5: Difference between the ion concentration profiles ($c_{\rm ions}=c_++c_-$) for various Thomas-Fermi screening lengths $l_{\rm TF}$ (in units of the Bohr radius $a_0$) and the concentration profiles for $l_{\rm TF}=0$, in the absence of voltage ($\Delta\Psi=0$). The system is identical to that of Fig. \ref{['fig:explicitimplicit']}, except for the change in $l_{\rm TF}$.