Charge fluctuations from molecular simulations in the constant-potential ensemble

Abstract

We revisit the statistical mechanics of charge fluctuations in capacitors. In constant-potential classical molecular simulations, the atomic charge of electrode atoms are treated as additional degrees of freedom which evolve in time so as to satisfy the constraint of fixed electrostatic potential for each configuration of the electrolyte. The present work clarifies the role of the overall electroneutrality constraint, as well as the link between the averages computed within the Born-Oppenheimer approximation and that of the full constant-potential ensemble. This allows us in particular to derive a complete fluctuation-dissipation relation for the differential capacitance, that includes a contribution from the charge fluctuations (around the charges satisfying the constant-potential and electroneutrality constraints) also present in the absence of an electrolyte. We provide a simple expression for this contribution from the elements of the inverse of the matrix defining the quadratic form of the fluctuating charges in the energy. We then illustrate numerically the validity of our results, and recover the expected result for an empty capacitor with structureless electrodes at large inter-electrode distances. By considering a variety of liquids between graphite electrodes, we confirm that this contribution to the total differential capacitance is small compared to that induced by the thermal fluctuations of the electrolyte.

Capacitance and Structure of Electric Double Layers: Comparing Brownian Dynamics and Classical Density Functional Theory

“…16, is obtained by numerically differentiating the surface charge with respect to the potential difference, using the central difference formula. Alternatively, the same differential capacitance is extracted from the thermal fluctuations of the wall charge around the average, Q = Q − ⟨Q⟩ N, , over a single equilibrium simulation [53,54],…”

A semiclassical Thomas–Fermi model to tune the metallicity of electrodes in molecular simulations

“…As in the constant potential method neglecting the quantum nature of the electrons (corresponding to l TF = 0.0 Å), the charges are treated as dynamic variables, which are obtained at each time step of the simulation by enforcing the constant potential constraint ∂Etot/∂qi = 0. 5,15 Compared to this perfect metal case, the modifications of the algorithm are minimal and virtually do not add any computational cost.…”

“…4 These methods are based on the use of an extended Hamiltonian in which the electrode charges are additional degrees of freedom that obey a constant potential constraint at each simulation step. 5 They allowed to partly alleviate the main conceptual difficulty to represent the electrode-electrolyte interface at the molecular scale, which is the need to account for the electronic structure on the electrode side, while the electrolyte is usually better simulated using classical force fields because it requires a sampling of the configurational space beyond the reach of today's capabilities with ab initio calculations (see Ref. 6 for a recent review of classical molecular simulations of electrode-electrolyte interfaces).…”

MetalWalls: A classical molecular dynamics software dedicated to the simulation of electrochemical systems

“…The method was also discussed in the framework of Mass-Zero (MaZe) constrained molecular dynamics in (Coretti et al, 2020). An updated description of the statistical mechanics of the constant potential ensemble can be found in the reference (Scalfi et al, 2020).…”

“…The fulfillment of the electroneutrality constraint in this framework and its repercussions on the statistical mechanics of the constant-potential ensemble are discussed at length in (Scalfi et al, 2020).…”