Surface Charge Density in Electrical Double Layer Capacitors with Nanoscale Cathode–Anode Separation

Abstract

Using a dynamic density functional theory, we study the charging dynamics, the final equilibrium structure, and the energy storage in an electrical double layer capacitor with nanoscale cathode–anode separation in a slit geometry. We derive a simple expression for the surface charge density that naturally separates the effects of the charge polarization due to the ions from those due to the polarization of the dielectric medium and allows a more intuitive understanding of how the ion distribution within the cell affects the surface charge density. We find that charge neutrality in the half-cell does not hold during the dynamic charging process for any cathode–anode separation, and also does not hold at the final equilibrium state for small separations. Therefore, the charge accumulation in the half-cell in general does not equal the surface charge density. The relationships between the surface charge density and the charge accumulation within the half-cell are systematically investigated by tuning the electrolyte concentration, cathode–anode separation, and applied voltage. For high electrolyte concentrations, we observe charge inversion at which the charge accumulation exceeds the surface charge at special values of the separation. In addition, we find that the energy density has a maximum at intermediate electrolyte concentrations for a high applied voltage.

Impedance response of ionic liquids in long slit pores

“…´" 0.3. This is close to the typical value for the ionic liquids which inspired the above parameter sets, and has been considered in other works [Qing et al, 2021]. It follows from the limit model (and the expressions for the complex impedance) that changing the pore length results in a simple rescaling of the impedance response, and so we fix H " 100 µm for our numerical results.…”

“…There have been a number of DDFT studies for ionic liquids; the majority of these consider a nanoscale parallel plate capacitor set-up, such as Jiang et al [2014], Lian et al [2016b] and Qing et al [2021], who studied charging dynamics. Babel et al [2018] applied a sinusoidal voltage on the walls of a nanoscale parallel-plate capacitor to calculate the impedance response, performing numerical simulations both with and without mean-field electrostatics and hard-sphere effects, finding that the latter had the most significant influence on the result.…”

Structure–Capacitance Relationships of Graphene/Ionic Liquid Electrolyte Double Layers

“…Kornyshev’s theory/model for double layers is often employed to rationalize experimental IL capacitance profiles. , However, while this model can predict a minimum in capacitance at the PZC, the minimum is only observed for low ion concentrations that do not resemble ionic liquids; rather, for high ion concentrations, the theory predicts a single maximum (at the PZC). Application of low-concentration limit of the model to ionic liquids has sometimes been justified based on ion-pairing and/or ion association arguments, yet this rationale is inconsistent with conductivity data of pure ionic liquids and also electrostatic screening analysis. , Other theoretical models proposed by Oldham, Bazant et al, Limmer, and Wang and co-workers , similarly predict that capacitance profiles of ionic liquids and/or concentrated electrolytes would exhibit a single maximum at the PZC, and decrease with voltage as C ∼ Δ V –1/2 . The C ∼ Δ V –1/2 behavior predicted by these models is a saturation effect, and qualitatively explains the high voltage tails of the capacitance profiles in Figure ; however, as emphasized, the minima and camel shape of the profile are not explained by the theories/models for high ion content.…”

A coarse-grained model of room-temperature ionic liquids between metal electrodes: a molecular dynamics study

“…First, the local charge density profile ρ q ( z ) is given by〈 ρ q ( z )〉 = q (〈 ρ + ( z )〉 − 〈 ρ − ( z )〉),where the angular brackets denote the ensemble average and ρ − ( z ) and ρ + ( z ) are the anion and cation density profiles, respectively, which are spatially averaged over the x - and y -directions and computed by binning the real systems in the z -direction into 100 intervals, counting the number of relevant ions, and dividing the counts by the volume of the bins. Then, as shown by Hautman et al 46 and derived by Qing et al , 48 the surface charge density σ q is obtained usingwhere Δ V is the constant potential difference between the two electrodes. The static term σ s q accounts for the direct response to the applied potential in the dielectric medium with relative permittivity ε r , while the polarization term σ p q accounts for the charge separation.…”

“…First, the local charge density profile r q (z) is given by hr q (z)i = q(hr + (z)i À hr À (z)i), (7) where the angular brackets denote the ensemble average and r À (z) and r + (z) are the anion and cation density profiles, respectively, which are spatially averaged over the x-and y-directions and computed by binning the real systems in the z-direction into 100 intervals, counting the number of relevant ions, and dividing the counts by the volume of the bins. Then, as shown by Hautman et al 46 and derived by Qing et al, 48 the surface charge density s q is obtained using…”