Time-dependent density functional theory for ion diffusion in electrochemical systems

Sparking potential over 1200 V by a falling water droplet

Hydrovoltaic technology has achieved notable breakthroughs in electric output via using the moving boundary of electric double layer, but the output voltage induced by droplets is saturated around 350 volts, and the underlying mechanism remains to be further clarified. Here, we show that falling water droplets can stably spark an unprecedented voltage up to 1200 volts within microseconds that they contact an electrode placed on top of an electret surface, approaching the theoretical upper limit. This sparking potential can be explained and described by a comprehensive model considering the water-electrode contact dynamics from both the macroscale droplet spreading and the microscale electric double layer formation, as well as the presence of a circuit capacitance. It is demonstrated that a droplet-induced electric spark is sufficient to directly ionize gas at atmospheric pressure and split water into hydrogen and oxygen, showing wide application potential in fields of green energy and intelligence.

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.

The Role of Energy Scales for the Structure of Ionic Liquids at Electrified Interfaces: A Theory-Based Approach

Ionic liquids offer unique bulk and interfacial characteristics as battery electrolytes. Our continuum approach naturally describes the electrolyte on a macroscale. An integral formulation for the molecular repulsion, which can be quantitatively determined by both experimental and theoretical methods, models the electrolyte on the nanoscale. In this article, we perform a systematic series expansion of this integral formulation, derive a description of chemical potentials in terms of higher-order concentration gradients, and rationalize the appearance of fourth-order derivative operators in modified Poisson equations, as recently proposed in this context. In this way, we formulate a rigorous multiscale methodology from atomistic quantum chemistry calculations to phenomenological continuum models. We apply our generalized framework to ionic liquids near electrified interfaces and perform analytical asymptotic analysis. Three energy scales describing electrostatic forces between ions, molecular repulsion, and thermal motion determine the shape and width of the long-ranging charged double layer. We classify the charge screening mechanisms dependent on the system parameters as dielectricity, ion size, interaction strength, and temperature. We find that the charge density of electrochemical double layers in ionic liquids either decays exponentially, for negligible molecular repulsion, or oscillates continuously. Charge ordering across several ion diameters occurs if the repulsion between molecules is comparable with thermal energy and Coulomb interactions. Eventually, phase separation of the bulk electrolyte into ionic layers emerges once the molecular repulsion becomes dominant. Our framework predicts the exact phase boundaries among these three phases as a function of temperature, dielectricity, and ion size.

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.

Relation between Charging Times and Storage Properties of Nanoporous Supercapacitors

An optimal combination of power and energy characteristics is beneficial for the further progress of supercapacitors-based technologies. We develop a nanoscale dynamic electrolyte model, which describes both static capacitance and the time-dependent charging process, including the initial square-root dependency and two subsequent exponential trends. The observed charging time corresponds to one of the relaxation times of the exponential regimes and significantly depends on the pore size. Additionally, we find analytical expressions providing relations of the time scales to the electrode’s parameters, applied potential, and the final state of the confined electrolyte. Our numerical results for the charging regimes agree with published computer simulations, and estimations of the charging times coincide with the experimental values.

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.

Effects of Kinetic Dielectric Decrement on Ion Diffusion and Capacitance in Electrochemical Systems

Diffusion of ionic components in electrolytes not only eliminates the gradients of ionic concentrations but also alters the local dielectric environment, and the coupling effect between kinetic dielectric decrement and ionic concentration gradient on the diffusion dynamics is not well understood. Herein, taking the charging process in electrical double layer systems as a case study, we conduct a multiscale investigation of ion diffusions in aqueous electrolytes by combining the dynamic density functional theory and an ion-concentration-dependent dielectric constant model. By properly considering the time evolutions of local dielectric constant coupled with ion density, we report an interesting phenomenon on the suppression of surface charge density that is not captured by conventional models. In addition, we show that the usage of aqueous electrolyte with small dielectric decrement coefficients promotes the capacitance, in quantitative agreement with experimental measurements.

Dynamic Adsorption of Ions into Like-Charged Nanospace: A Dynamic Density Functional Theory Study

The adsorption processes of ions into charged nanospace are associated with many practical applications. Whereas a large number of microporous materials have been prepared toward efficient adsorption of ions from solutions, theoretical models that allow for capturing the characteristics of ion dynamic adsorption into like-charged nanopores are still few. The difficulty originates from the overlapping of electric potentials inside the pores. Herein, a theoretical model is proposed by incorporating dynamic density functional theory with modified Poisson equation for investigating the dynamic adsorption of ions into like-charged nanoslits. This model is rationalized by comparing the theoretical predictions with corresponding simulation results. Afterward, by analyzing the adsorption dynamics, we show that the overlapping effect is associated with the pore size, ion bulk concentration, and surface charge density, and it plays a dominant role in the coupling between the total adsorption amount of ions and total adsorption time. Specifically, with weak overlapping effect, the total adsorption amount is intuitively proportional to the total adsorption time; however, when the overlapping effect is strong, the total adsorption amount may be inversely proportional to the total adsorption time, indicating that both high adsorption amount and short adsorption time can be achieved simultaneously. This work provides a meaningful insight toward the rational design and optimization of microporous materials for efficient ion adsorption.

Transient dynamics of electric double-layer capacitors: Exact expressions within the Debye-Falkenhagen approximation

We revisit a classical problem of theoretical electrochemistry: the response of an electric double-layer capacitor (EDLC) subject to a small, suddenly applied external potential. We solve the Debye-Falkenhagen equation to obtain exact expressions for key EDLC quantities: the ionic charge density, the ionic current density, and the electric field. In contrast to earlier works, our results are not restricted to the long-time asymptotics of those quantities. The solutions take the form of infinite sums whose successive terms all decay exponentially with increasingly short relaxation times. Importantly, this set of relaxation times is the same among all aforementioned EDLC quantities; this property is demanded on physical grounds but not generally achieved within approximation schemes. The scaling of the largest relaxation timescale τ_{1}, that determines the long-time decay, is in accordance with earlier results: Depending on the Debye length, λ_{D}, and the electrode separation, 2L, it amounts to τ_{1}≃λ_{D}L/D for L≫λ_{D} and τ_{1}≃4L^{2}/(π^{2}D) for L≪λ_{D}, respectively (with D being the ionic diffusivity).

Correlation of Ion Transport Hysteresis with the Nanogeometry and Surface Factors in Single Conical Nanopores

Better understanding in the dynamics of ion transport through nanopores or nanochannels is important for sensing, nucleic acid sequencing and energy technology. In this paper, the intriguing nonzero cross point, resolved from the pinched hysteresis current-potential (i-V) curves in conical nanopore electrokinetic measurements, is quantitatively correlated to the surface and geometric properties by simulation studies. The analytical descriptions of the conductance and potential at the cross point are developed: the cross-point conductance includes both the surface and volumetric conductance; the cross-point potential represent the overall/averaged surface potential difference across the nanopore. The impacts by individual parameter such as pore radius, half cone angle, and surface charges are systematically studied in the simulation that would be convoluted and challenging in experiments. The elucidated correlation is supported by and offer predictive guidance for experimental studies. The results also offer more quantitative and systematic insights in the physical origins of the concentration polarization dynamics in addition to ionic current rectification inside conical nanopores and other asymmetric nanostructures. Overall, the cross point serves as a simple yet informative analytical parameter to analyze the electrokinetic transport through broadly defined nanopore-type devices.

Computational Insights into Materials and Interfaces for Capacitive Energy Storage

Supercapacitors such as electric double-layer capacitors (EDLCs) and pseudocapacitors are becoming increasingly important in the field of electrical energy storage. Theoretical study of energy storage in EDLCs focuses on solving for the electric double-layer structure in different electrode geometries and electrolyte components, which can be achieved by molecular simulations such as classical molecular dynamics (MD), classical density functional theory (classical DFT), and Monte-Carlo (MC) methods. In recent years, combining first-principles and classical simulations to investigate the carbon-based EDLCs has shed light on the importance of quantum capacitance in graphene-like 2D systems. More recently, the development of joint density functional theory (JDFT) enables self-consistent electronic-structure calculation for an electrode being solvated by an electrolyte. In contrast with the large amount of theoretical and computational effort on EDLCs, theoretical understanding of pseudocapacitance is very limited. In this review, we first introduce popular modeling methods and then focus on several important aspects of EDLCs including nanoconfinement, quantum capacitance, dielectric screening, and novel 2D electrode design; we also briefly touch upon pseudocapactive mechanism in RuO2. We summarize and conclude with an outlook for the future of materials simulation and design for capacitive energy storage.

Characterization of the Electric Double Layer Formation Dynamics of a Metal/Ionic Liquid/Metal Structure

Although ionic liquids (ILs) have been used extensively in recent years as a high-capacitance "dielectric" in electric double layer transistors, the dynamics of the double layer formation have remained relatively unexplored. Better understanding of the dynamics and relaxation processes involved in electric double layer formation will guide device optimization, particularly with regard to switching speed. In this paper, we explore the dynamical characteristics of an IL in a metal/ionic liquid/metal (M/IL/M) capacitor. In particular, we examine a Au/IL/Au structure where the IL is 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl)trifluorophosphate. The experiments consist of frequency-dependent impedance measurements and time-dependent current vs voltage measurements for applied linear voltage ramps and abrupt voltage steps. The parameters of an equivalent circuit model are determined by fits to the impedance vs frequency data and subsequently verified by calculating the current vs voltage characteristics for the applied potential profiles. The data analysis indicates that the dynamics of the structure are characterized by a wide distribution of relaxation times spanning the range of less than microseconds to longer than seconds. Possible causes for these time scales are discussed.

Dynamics of Ion Transport in Ionic Liquids

A gap in understanding the link between continuum theories of ion transport in ionic liquids and the underlying microscopic dynamics has hindered the development of frameworks for transport phenomena in these concentrated electrolytes. Here, we construct a continuum theory for ion transport in ionic liquids by coarse graining a simple exclusion process of interacting particles on a lattice. The resulting dynamical equations can be written as a gradient flow with a mobility matrix that vanishes at high densities. This form of the mobility matrix gives rise to a charging behavior that is different to the one known for electrolytic solutions, but which agrees qualitatively with the phenomenology observed in experiments and simulations.

Kinetic Charging Inversion in Ionic Liquid Electric Double Layers

The charging kinetics of electric double layers (EDLs) has a pivotal role in the performance of a wide variety of nanostructured devices. Despite the prevalent use of ionic liquids as the electrolyte, relatively little is known on the charging behavior from a microscopic perspective. Here, we study the charging kinetics of ionic liquid EDLs using a classical time-dependent density functional theory that captures the molecular excluded volume effects and electrostatic correlations. By examining variations of the ionic density profiles and the charging density in response to an electrode voltage, we find that at certain conditions, the electrode charge shows a rapid surge in its initial response, rises quickly to the maximum, and then slowly decays toward equilibrium. The electrode charge and voltage may have opposite signs when the cell width is commensurate with the layer-by-layer ionic distributions. This unusual charging behavior can be explained in terms of the oscillatory structure of ionic liquids near the electrodes.