Computational Amperometry of Nanoscale Capacitors in Molecular Simulations

Impedance of nanocapacitors from molecular simulations to understand the dynamics of confined electrolytes

Nanoelectrochemical devices have become a promising candidate technology across various applications, including sensing and energy storage, and provide new platforms for studying fundamental properties of electrode/electrolyte interfaces. In this work, we employ constant-potential molecular dynamics simulations to investigate the impedance of gold-aqueous electrolyte nanocapacitors, exploiting a recently introduced fluctuation-dissipation relation. In particular, we relate the frequency-dependent impedance of these nanocapacitors to the complex conductivity of the bulk electrolyte in different regimes, and use this connection to design simple but accurate equivalent circuit models. We show that the electrode/electrolyte interfacial contribution is essentially capacitive and that the electrolyte response is bulk-like even when the interelectrode distance is only a few nanometers, provided that the latter is sufficiently large compared to the Debye screening length. We extensively compare our simulation results with spectroscopy experiments and predictions from analytical theories. In contrast to experiments, direct access in simulations to the ionic and solvent contributions to the polarization allows us to highlight their significant and persistent anticorrelation and to investigate the microscopic origin of the timescales observed in the impedance spectrum. This work opens avenues for the molecular interpretation of impedance measurements, and offers valuable contributions for future developments of accurate coarse-grained representations of confined electrolytes.

Computational dielectric spectroscopy on solid-solution interface by time-dependent voltage applied molecular dynamics simulation

A frequency-dependent dielectric constant characterizes the dielectric response of a medium and also represents the time scale of system's collective dynamics. Although it is valuable not only academically but also practically for developing advanced devices, getting the value of a solution at the interface with a solid or electrode surface is challenging both experimentally and computationally. Here, we propose a computational method that imitates the dielectric spectroscopy and AC impedance measurement. It combines a time-dependent voltage applied molecular dynamics simulation with an equivalent circuit representation of a system composed of a solution confined between two identical electrodes. It gives the frequency-dependent dielectric constants of the bulk solution and the interface simultaneously. Unlike the conventional method, it does not require computation of a dipole autocorrelation function and its Fourier transformation. Application of the method on a system of water confined between polarizable Pt electrodes gives the static dielectric constant and the relaxation time of the bulk water in good agreement with previous simulation results and experimental values. In addition, it gives a much smaller static dielectric constant at the interface, consistent with previous observations. The outline of the dielectric dispersion curve of the interface seems similar to that of the bulk, but the relaxation time is several times faster.

Charging and discharging a supercapacitor in molecular simulations

As the world moves more toward unpredictable renewable energy sources, better energy storage devices are required. Supercapacitors are a promising technology to meet the demand for short-term, high-power energy storage. Clearly, understanding their charging and discharging behaviors is essential to improving the technology. Molecular Dynamics (MD) simulations provide microscopic insights into the complex interplay between the dynamics of the ions in the electrolyte and the evolution of the charge distributions on the electrodes. Traditional MD simulations of (dis)charging supercapacitors impose a pre-determined evolving voltage difference between the electrodes, using the Constant Potential Method (CPM). Here, we present an alternative method that explicitly simulates the charge flow to and from the electrodes. For a disconnected capacitor, i.e., an open circuit, the charges are allowed to redistribute within each electrode while the sum charges on both electrodes remain constant. We demonstrate, for a model capacitor containing an aqueous salt solution, that this method recovers the charge-potential curve of CPM simulations. The equilibrium voltage fluctuations are related to the differential capacitance. We next simulate a closed circuit by introducing equations of motion for the sum charges, by explicitly accounting for the external circuit element(s). Charging and discharging of the model supercapacitor via a resistance proceed by double exponential processes, supplementing the usual time scale set by the electrolyte dynamics with a novel time scale set by the external circuit. Finally, we propose a simple equivalent circuit that reproduces the main characteristics of this supercapacitor.

Molecular Understanding of Charging Dynamics in Supercapacitors with Porous Electrodes and Ionic Liquids

Porous electrodes and ionic liquids could significantly enhance the energy storage of supercapacitors. However, they may reduce the charging dynamics and power density due to the nanoconfinement of porous electrodes and the high viscosity of ionic liquids. A comprehensive understanding of the charging mechanism in porous supercapacitors with ionic liquids provides a crucial theoretical foundation for their design optimization. Here, we review the progress of molecular simulations of the charging dynamics in supercapacitors consisting of porous electrodes and ionic liquids. We highlight and delve into the breakthroughs in the ion transport and charging mechanism for electrodes with subnanometer pores and realistic porous structures. We also discuss future directions for the charging dynamics of supercapacitors.

Theory and Simulations of Ionic Liquids in Nanoconfinement

Room-temperature ionic liquids (RTILs) have exciting properties such as nonvolatility, large electrochemical windows, and remarkable variety, drawing much interest in energy storage, gating, electrocatalysis, tunable lubrication, and other applications. Confined RTILs appear in various situations, for instance, in pores of nanostructured electrodes of supercapacitors and batteries, as such electrodes increase the contact area with RTILs and enhance the total capacitance and stored energy, between crossed cylinders in surface force balance experiments, between a tip and a sample in atomic force microscopy, and between sliding surfaces in tribology experiments, where RTILs act as lubricants. The properties and functioning of RTILs in confinement, especially nanoconfinement, result in fascinating structural and dynamic phenomena, including layering, overscreening and crowding, nanoscale capillary freezing, quantized and electrotunable friction, and superionic state. This review offers a comprehensive analysis of the fundamental physical phenomena controlling the properties of such systems and the current state-of-the-art theoretical and simulation approaches developed for their description. We discuss these approaches sequentially by increasing atomistic complexity, paying particular attention to new physical phenomena emerging in nanoscale confinement. This review covers theoretical models, most of which are based on mapping the problems on pertinent statistical mechanics models with exact analytical solutions, allowing systematic analysis and new physical insights to develop more easily. We also describe a classical density functional theory, which offers a reliable and computationally inexpensive tool to account for some microscopic details and correlations that simplified models often fail to consider. Molecular simulations play a vital role in studying confined ionic liquids, enabling deep microscopic insights otherwise unavailable to researchers. We describe the basics of various simulation approaches and discuss their challenges and applicability to specific problems, focusing on RTIL structure in cylindrical and slit confinement and how it relates to friction and capacitive and dynamic properties of confined ions.

Frequency-Dependent Impedance of Nanocapacitors from Electrode Charge Fluctuations as a Probe of Electrolyte Dynamics

The frequency-dependent impedance is a fundamental property of electrical components. We show that it can be determined from the equilibrium dynamical fluctuations of the electrode charge in constant-potential molecular simulations, extending in particular a fluctuation-dissipation relation for the capacitance recovered in the low-frequency limit and provide an illustration on water-gold nanocapacitors. This Letter opens the way to the interpretation of electrochemical impedance measurements in terms of microscopic mechanisms, directly from the dynamics of the electrolyte, or indirectly via equivalent circuit models as in experiments.

A theory for the stabilization of polar crystal surfaces by a liquid environment

Polar crystal surfaces play an important role in the functionality of many materials and have been studied extensively over many decades. In this article, a theoretical framework is presented that extends existing theories by placing the surrounding solution environment on an equal footing with the crystal itself; this is advantageous, e.g., when considering processes such as crystal growth from solution. By considering the polar crystal as a stack of parallel plate capacitors immersed in a solution environment, the equilibrium adsorbed surface charge density is derived by minimizing the free energy of the system. In analogy to the well-known diverging surface energy of a polar crystal surface at zero temperature, for a crystal in solution it is shown that the "polar catastrophe" manifests as a diverging free energy cost to perturb the system from equilibrium. Going further than existing theories, the present formulation predicts that fluctuations in the adsorbed surface charge density become increasingly suppressed with increasing crystal thickness. We also show how, in the slab geometry often employed in both theoretical and computational studies of interfaces, an electric displacement field emerges as an electrostatic boundary condition, the origins of which are rooted in the slab geometry itself, rather than the use of periodic boundary conditions. This aspect of the work provides a firmer theoretical basis for the recent observation that standard "slab corrections" fail to correctly describe, even qualitatively, polar crystal surfaces in solution.

ELECTRODE: An electrochemistry package for atomistic simulations

Constant potential methods (CPM) enable computationally efficient simulations of the solid-liquid interface at conducting electrodes in molecular dynamics (MD). They have been successfully used, for example, to realistically model the behavior of ionic liquids or water-in-salt electrolytes in supercapacitors and batteries. The CPM models conductive electrodes by updating charges of individual electrode atoms according to the applied electric potential and the (time-dependent) local electrolyte structure. Here we present a feature-rich CPM implementation, called ELECTRODE, for the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), which includes a constrained charge method and a thermo-potentiostat. The ELECTRODE package also contains a finite-field approach, multiple corrections for non-periodic boundary conditions of the particle-particle particle-mesh solver, and a Thomas-Fermi model for using non-ideal metals as electrodes. We demonstrate the capabilities of this implementation for a parallel-plate electrical double-layer capacitor, for which we have investigated the charging times with the different implemented methods and found an interesting relationship between water and ionic dipole relaxations. To prove the validity of the one-dimensional correction for the long-range electrostatics, we estimated the vacuum capacitance of two co-axial carbon nanotubes and compared it to structureless cylinders, for which an analytical expression exists. In summary, the ELECTRODE package enables efficient electrochemical simulations using state-of-the-art methods, allowing one to simulate even heterogeneous electrodes. Moreover, it allows unveiling more rigorously how electrode curvature affects the capacitance with the one-dimensional correction.

Unified polarizable electrode models for open and closed circuits: Revisiting the effects of electrode polarization and different circuit conditions on electrode-electrolyte interfaces

A precise understanding of the interfacial structure and dynamics is essential for the optimal design of various electrochemical devices. Herein, we propose a method for classical molecular dynamics simulations to deal with electrochemical interfaces with polarizable electrodes under the open circuit condition. Less attention has been paid to electrochemical circuit conditions in computation despite being often essential for a proper assessment, especially comparison between different models. The present method is based on the chemical potential equalization principle, as is a method developed previously to deal with systems under the closed circuit condition. These two methods can be interconverted through the Legendre transformation, so that the difference in the circuit conditions can be compared on the same footing. Furthermore, the electrode polarization effect can be correctly studied by comparing the present method with the conventional simulations with the electrodes represented by fixed charges, since both of the methods describe systems under the open circuit condition. The method is applied to a parallel-plate capacitor composed of platinum electrodes and an aqueous electrolyte solution. The electrode polarization effects have an impact on the interfacial structure of the electrolyte solution. We found that the difference in the circuit conditions significantly affects the dynamics of the electrolyte solution. The electric field at the charged electrode surface is poorly screened by the nonequilibrium solution structure in the open circuit condition, which accelerates the motion of the electrolyte solution.

Fully periodic, computationally efficient constant potential molecular dynamics simulations of ionic liquid supercapacitors

Molecular dynamics (MD) simulations of complex electrochemical systems, such as ionic liquid supercapacitors, are increasingly including the constant potential method (CPM) to model conductive electrodes at a specified potential difference, but the inclusion of CPM can be computationally expensive. We demonstrate the computational savings available in CPM MD simulations of ionic liquid supercapacitors when the usual non-periodic slab geometry is replaced with fully periodic boundary conditions. We show how a doubled cell approach, previously used in non-CPM MD simulations of charged interfaces, can be used to enable fully periodic CPM MD simulations. Using either a doubled cell approach or a finite field approach previously reported by others, fully periodic CPM MD simulations produce comparable results to the traditional slab geometry simulations with a nearly double speedup in computational time. Indeed, these savings can offset the additional cost of the CPM algorithm, resulting in periodic CPM MD simulations that are computationally competitive with the non-periodic, fixed charge equivalent simulations for the ionic liquid supercapacitors studied here.

Microscopic Simulations of Electrochemical Double-Layer Capacitors

Electrochemical double-layer capacitors (EDLCs) are devices allowing the storage or production of electricity. They function through the adsorption of ions from an electrolyte on high-surface-area electrodes and are characterized by short charging/discharging times and long cycle-life compared to batteries. Microscopic simulations are now widely used to characterize the structural, dynamical, and adsorption properties of these devices, complementing electrochemical experiments and in situ spectroscopic analyses. In this review, we discuss the main families of simulation methods that have been developed and their application to the main family of EDLCs, which include nanoporous carbon electrodes. We focus on the adsorption of organic ions for electricity storage applications as well as aqueous systems in the context of blue energy harvesting and desalination. We finally provide perspectives for further improvement of the predictive power of simulations, in particular for future devices with complex electrode compositions.

Modeling galvanostatic charge-discharge of nanoporous supercapacitors

Molecular modeling has been considered indispensable in studying the energy storage of supercapacitors at the atomistic level. The constant potential method (CPM) allows the electric potential to be kept uniform in the electrode, which is essential for a realistic description of the charge repartition and dynamics process in supercapacitors. However, previous CPM studies have been limited to the potentiostatic mode. Although widely adopted in experiments, the galvanostatic mode has rarely been investigated in CPM simulations because of a lack of effective methods. Here we develop a modeling approach to simulating the galvanostatic charge-discharge process of supercapacitors under constant potential. We show that, for nanoporous electrodes, this modeling approach can capture experimentally consistent dynamics in supercapacitors. It can also delineate, at the molecular scale, the hysteresis in ion adsorption-desorption dynamics during charging and discharging. This approach thus enables the further accurate modeling of the physics and electrochemistry in supercapacitor dynamics.

Constant potential simulations on a mesh

Molecular dynamics simulations in a constant potential ensemble are an increasingly important tool to investigate charging mechanisms in next-generation energy storage devices. We present a highly efficient approach to compute electrostatic interactions in simulations employing a constant potential method (CPM) by introducing a particle-particle particle-mesh solver specifically designed for treating long-range interactions in a CPM. Moreover, we present evidence that a dipole correction term-commonly used in simulations with a slab-like geometry-must be used with caution if it is also to be used within a CPM. It is demonstrated that artifacts arising from the usage of the dipole correction term can be circumvented by enforcing a charge neutrality condition in the evaluation of the electrode charges at a given external potential.