The tail effect on the shape of an electrical double layer differential capacitance curve

Understanding the Electric Double-Layer Structure, Capacitance, and Charging Dynamics

Significant progress has been made in recent years in theoretical modeling of the electric double layer (EDL), a key concept in electrochemistry important for energy storage, electrocatalysis, and multitudes of other technological applications. However, major challenges remain in understanding the microscopic details of the electrochemical interface and charging mechanisms under realistic conditions. This review delves into theoretical methods to describe the equilibrium and dynamic responses of the EDL structure and capacitance for electrochemical systems commonly deployed for capacitive energy storage. Special emphasis is given to recent advances that intend to capture the nonclassical EDL behavior such as oscillatory ion distributions, polarization of nonmetallic electrodes, charge transfer, and various forms of phase transitions in the micropores of electrodes interfacing with an organic electrolyte or ionic liquid. This comprehensive analysis highlights theoretical insights into predictable relationships between materials characteristics and electrochemical performance and offers a perspective on opportunities for further development toward rational design and optimization of electrochemical systems.

Double-Edged Sword of Ion-Size Asymmetry in Energy Storage of Supercapacitors

The advanced supercapacitor is of great significance for renewable energy storage. Achieving its high energy and high power densities remains a huge challenge. Herein, the contribution of ion-size asymmetry to the charging behavior of a supercapacitor is systematically studied using time-dependent density functional theory (TDDFT). We track the time evolution of the ionic microstructure inside the porous electrode and its reservoir and reveal a kinetic charge inversion in the asymmetrical ion-size cases. Compared with the symmetrical ion-size case, we find that the ion-size asymmetry has a double-edged sword effect on the energy storage of a supercapacitor: it accelerates the charging process yet reduces the differential capacitance. Additionally, the energy density and power density can simultaneously increase in the asymmetrical cases, which provides important insights toward the experimental design of supercapacitors with high energy and high power densities.

Capacitive energy storage in single-file pores: Exactly solvable models and simulations

Understanding charge storage in low-dimensional electrodes is crucial for developing novel ecologically friendly devices for capacitive energy storage and conversion and water desalination. Exactly solvable models allow in-depth analyses and essential physical insights into the charging mechanisms. So far, however, such analytical approaches have been mainly limited to lattice models. Herein, we develop a versatile, exactly solvable, one-dimensional off-lattice model for charging single-file pores. Unlike the lattice model, this model shows an excellent quantitative agreement with three-dimensional Monte Carlo simulations. With analytical calculations and simulations, we show that the differential capacitance can be bell-shaped (one peak), camel-shaped (two peaks), or have four peaks. Transformations between these capacitance shapes can be induced by changing pore ionophilicity, by changing cation-anion size asymmetry, or by adding solvent. We find that the camel-shaped capacitance, characteristic of dilute electrolytes, appears for strongly ionophilic pores with high ion densities, which we relate to charging mechanisms specific to narrow pores. We also derive a large-voltage asymptotic expression for the capacitance, showing that the capacitance decays to zero as the inverse square of the voltage, C ∼ u^-2. This dependence follows from hard-core interactions and is not captured by the lattice model.

Stabilizing the Nanosurface of LiNiO2 Electrodes by Varying the Electrolyte Concentration: Correlation with Initial Electrochemical Behaviors for Use in Aqueous Li-Ion Batteries

This study attempted to stabilize the nanosurface of LiNiO₂ (LNO) electrodes by varying the electrolyte concentration, significantly influencing its initial electrochemical behaviors for use in aqueous lithium-ion batteries. The charge/discharge capacities, reversibility, and cyclability of LNO were improved during initial cycles with an increase in the concentration of lithium bis(trifluoromethanesulfonyl)imide (LiTFSI). As determined by the galvanostatic intermittent titration technique, the superior diffusivity of Li⁺ ions in the LNO electrode is also obtained in the concentrated electrolyte. Nanoscale observation of the LNO surface revealed that its morphology is maintained relatively well in the concentrated electrolyte while it is destroyed in dilute electrolytes after the initial electrochemical cycles. These results are considered to be attributable to the variation of the interface condition in the electrical double layer with an increase in the electrolyte concentration, thus stabilizing the nanosurface of LNO by suppressing the dissolution of Ni ions from the surface. Additionally, in situ X-ray diffraction analysis demonstrated that LNO shows more stable phase transitions and volume changes as the electrolyte concentration increases, indicating that its structural changes in bulk can be directly related to the state of the nanosurface, which has a positive impact on the initial electrochemical behaviors in this system.

Capillary Ionization and Jumps of Capacitive Energy Stored in Mesopores

We study ionic liquid-solvent mixtures in slit-shaped nanopores wider than a few ion diameters. Using a continuum theory and generic thermodynamic reasoning, we reveal that such systems can undergo a capillary ionization transition. At this transition, the pores spontaneously ionize or deionize upon infinitesimal changes of temperature, slit width, or voltage. Our calculations show that a voltage applied to a pore may induce a capillary ionization, which-counterintuitively-is followed by a re-entrant deionization as the voltage increases. We find that such ionization transitions produce sharp jumps in the accumulated charge and stored energy, which may find useful applications in energy storage and heat-to-energy conversion.

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

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.

Temperature dependence of differential capacitance in the electric double layer.Symmetric valency 1:1 electrolytes

The differential capacitance of an electric double layer formed by an aqueous solution of KNO₃ on a glassy carbon electrode is measured by impedance analysis at constant frequency. Results are obtained at electrolyte concentrations of 0.1 mol/dm³, 0.5 mol/dm³, and 1.0 mol/dm³, and at a series of temperatures, viz., 288 K, 298 K, 308 K, 318 K, and 328 K. The differential capacitance envelopes reveal a rich, complex pattern of maxima, minima, and local minima, whose magnitude and position change with a change in solution concentration. At the two lower concentrations, the temperature dependence of the capacitance, for example, at zero electrode potential, shows an alternating positive-negative behavior, while at the highest concentration of 1.0 mol/dm³, the slope of the differential capacitance-electrode potential curve is always positive. The experimental results are supplemented by a numerical grand canonical Monte Carlo simulation study of a restricted primitive model double layer but with an off-center cationic charge achieved by displacing the charge center from the ion sphere center toward its surface. The simulations, performed at the electrolyte concentration of 1.0 mol/dm³ and constant cation charge center displacement, and at varying electrode potentials and temperatures, show, in general, a negative temperature dependence of the differential capacitance. However, this temperature dependence can also be positive for a negative electrode charge and for a sufficiently large gradient of the cation charge center displacement with temperature. This feature is seen to be associated with an increase in the entropy of formation of the double layer.

Asymmetric double-layer charging in a cylindrical nanopore under closed confinement

This article presents a physical-mathematical treatment and numerical simulations of electric double layer charging in a closed, finite, and cylindrical nanopore of circular cross section, embedded in a polymeric host with charged walls and sealed at both ends by metal electrodes under an external voltage bias. Modified Poisson-Nernst-Planck equations were used to account for finite ion sizes, subject to an electroneutrality condition. The time evolution of the formation and relaxation of the double layers was explored. Moreover, equilibrium ion distributions and differential capacitance curves were investigated as functions of the pore surface charge density, electrolyte concentration, ion sizes, and pore size. Asymmetric properties of the differential capacitance curves reveal that the structure of the double layer near each electrode is controlled by the charge concentration along the pore surface and by charge asymmetry in the electrolyte. These results carry implications for accurately simulating cylindrical capacitors and electroactuators.

Theoretical study of the effect of π^{+}-π^{+} association in imidazolium ionic liquids at charged interfaces

We develop an extended classical density-functional theory to describe clustering of imidazolium-based cations into linear chains, driven by π-π stacking. We find that the associating system displays a similar short-ranged structure to the completely dissociated fluid. We also construct a restricted primitive model for associating ionic species in an RTIL+solvent mixture. The double-layer formed in these systems exhibits strong overscreening by the cation chains, as expected. Significantly enhanced capacitance is also observed for the case where counterions are the associating species. The established density-functional method can be also used to describe polydisperse polyelectrolyte models.

Kinetic-Dominated Charging Mechanism within Representative Aqueous Electrolyte-based Electric Double-Layer Capacitors

The chemical nature of electrolytes has been demonstrated to play a pivotal role in the charge storage of electric double-layer capacitors (EDLCs), whereas primary mechanisms are still partially resolved but controversial. In this work, a systematic exploration into EDL structures and kinetics of representative aqueous electrolytes is performed with numerical simulation and experimental research. Unusually, a novel charging mechanism exclusively predominated by kinetics is recognized, going beyond traditional views of manipulating capacitances preferentially via interfacial structural variations. Specifically, strikingly distinctive EDL structures stimulated by diverse ion sizes, valences, and mixtures manifest a virtually identical EDL capacitance, where the dielectric nature of solvents attenuates ionic effects on electrolyte redistributions, in stark contradiction with solvent-free counterpart and traditional Helmholtz theory. Meanwhile, corresponding kinetics evolve conspicuously with ionic species, intimately correlated with ion-solvent interactions. The achieved mechanisms are subsequently illuminated by electrochemical measurements, highlighting the crucial interplay between ions and solvents in regulating EDLC performances.

Structure of an electric double layer containing a 2:2 valency dimer electrolyte

The structure of a planar electric double layer formed by a 2:2 valency dimer electrolyte in the vicinity of a uniformly charged planar hard electrode is investigated using density functional theory and Monte Carlo simulations. The dimer electrolyte consists of a mixture of charged divalent dimers and charged divalent monomers in a dielectric continuum. A dimer is constructed by two tangentially tethered rigid spheres, one of which is divalent and positively charged and the other neutral, whereas the monomer is a divalent and negatively charged rigid sphere. The density functional theory reproduces well the simulation results for (i) the singlet distributions of the various ion species with respect to the electrode, and (ii) the mean electrostatic potential. Comparison with earlier results for a 2:1/1:2 dimer electrolyte shows that the double layer structure is similar when the counterion has the same valency.

Non-Faradaic Energy Storage by Room Temperature Ionic Liquids in Nanoporous Electrodes

The enhancement of non-Faradaic charge and energy density stored by ionic electrolytes in nanostructured electrodes is an intriguing issue of great practical importance for energy storage in electric double layer capacitors. On the basis of extensive molecular dynamics simulations of various carbon-based nanoporous electrodes and room temperature ionic liquid (RTIL) electrolytes, we identify atomistic mechanisms and correlations between electrode/electrolyte structures that lead to capacitance enhancement. In the symmetric electrode setup with nanopores having atomically smooth walls, most RTILs showed up to 50% capacitance increase compared to infinitely wide pore. Extensive simulations using asymmetric electrodes and pores with atomically rough surfaces demonstrated that tuning of electrode nanostructure could lead to further substantial capacitance enhancement. Therefore, the capacitance in nanoporous electrodes can be increased due to a combination of two effects: (i) the screening of ionic interactions by nanopore walls upon electrolyte nanoconfinement, and (ii) the optimization of nanopore structure (volume, surface roughness) to take into account the asymmetry between cation and anion chemical structures.