Monte Carlo simulations of two-component Coulomb gases applied in surface electrodes

Abstract

In this work, we study the gapped Surface Electrode (SE), a planar system composed of two-conductor flat regions at different potentials with a gap G between both sheets. The computation of the electric field and the surface charge density requires solving Laplace’s equation subjected to Dirichlet conditions (on the electrodes) and Neumann Boundary Conditions over the gap. In this document, the GSE is modeled as a Two-Dimensional Classical Coulomb Gas having punctual charges +q and −q on the inner and outer electrodes, respectively, interacting with an inverse power law 1~r-potential. The coupling parameter Γ between particles inversely depends on temperature and is proportional to q2. Precisely, the density charge arises from the equilibrium states via Monte Carlo (MC) simulations. We focus on the coupling and the gap geometry effect. Mainly on the distribution of particles in the circular and the harmonically-deformed gapped SE. MC simulations differ from electrostatics in the strong coupling regime. The electrostatic approximation and the MC simulations agree in the weak coupling regime where the system behaves as two interacting ionic fluids. That means that temperature is crucial in finite-size versions of the gapped SE where the density charge cannot be assumed fully continuous as the coupling among particles increases. Numerical comparisons are addressed against analytical descriptions based on an electric vector potential approach, finding good agreement.

Comparative analysis of molecular dynamics and method of moments in two-dimensional concentric circular layers

“…Equilibrium states and charge density were derived from Monte Carlo (MC) simulations. In that study, interactions followed an inverse power law potential 1/r, where the coupling parameter Γ has been found inversely proportional to the system's temperature and proportional to q 2 . Furthermore, authors of [1] explored the electric vector potential and the Biot-Savart law analogy in electrostatics.…”

“…The plot in figure 8 shows the density profile on both sheets for ξ = −1, this is, inner particles having positive charge q 1 = q and outer ones carrying opposite charge q 1 = −q thus the system being globally neutral. The solid line in figure 8 corresponds to the MoM approximation calculated according to equation (2). MoM solution assumes that the density charge is continuous and it differs from the one in equation ( 8) (valid in the gapless limit) since MoM predicts a charge concentration in the R 3 boundary.…”

Special issue on soft matter research in Latin America

“…A more abstract problem connected with charged systems is analyzed by Salazar et al [21], who in their contribution compare an analytical electrostatic description of a gaped surface electrode, modeled in terms of a two-component Coulomb gas, with the corresponding Monte Carlo simulations, evidencing significant differences in the strong coupling regime.…”

Electric vector potential and the Biot-Savart like law in electrostatics

“…That strategy has been originally developed for the circular gaped SE in (Schmied, 2010). The same approach can be applied for arbitrary geometry gaped SE (Salazar et al, 2022). Therefore, one can understand the vector potential approach Θ here presented as a versatile way to find solutions in electrostatics.…”

“…For convenience, the potentials of the inner and outer sheets are V o and zero, respectively. This system is commonly referred to as gaped Surface Electrode (SE) Salazar et al, 2019Salazar et al, , 2022Schmied, 2010) and it plays an important role to model the static electric fields in SE ion traps where ion-trap networks including arrays of SE are also promising candidates in quantum processing Seidelin et al, 2006;Daniilidis et al, 2011;Mokhberi et al, 2017;Tao et al, 2018;Van Mourik et al, 2020) The surface electrode also has practical relevance in antenna theory. For instance, when considering a time-varying voltage between the sheets.…”