Molecular dynamics (MD) simulations have become a powerful tool for investigating electrical double layers (EDLs), which play a crucial role in various electrochemical devices. In this Review, we provide a comprehensive overview of the techniques used in MD simulations for EDL studies, with a particular focus on methods for describing electrode polarization, and examine the principle behind these methods and their varying applicability. The applications of these approaches in supercapacitors, capacitive deionization, batteries, and electric double-layer transistors are explored, highlighting recent advancements and insights in each field. Finally, we emphasize the challenges and potential directions for future developments in MD simulations of EDLs, such as considering movable electrodes, improving electrode property representation, incorporating chemical reactions, and enhancing computational efficiency to deepen our understanding of complex electrochemical processes and contribute to the progress in the field involving EDLs.

Electrical double layers (EDLs), solid–liquid interfaces between charged electrodes and electrolytes containing movable ions, are present in most electrochemical devices, such as batteries and supercapacitors, playing a crucial role in their performance.1–5 Modeling the EDL has become an important approach to understanding electrochemical phenomena, such as ionic structure, capacitance, and charging dynamics, further contributing to the design and optimization of these electrochemical devices.6–10 

In 1853, EDLs were initially defined by Helmholz as two flat planar carrying equal but opposite charges, representing the charged electrodes and the counterions in the electrolyte.11 Later, the Poisson–Boltzmann theory, developed by Gouy and Chapman, described the electrolyte side as an inhomogeneous ionic distribution in space.12,13 Stern further refined the theory by dividing the electrolyte into an inner layer (Stern layer) and a diffusion layer.14 Models based on Poisson–Boltzmann theory have been further developed to capture more physical properties, such as describing structural entropy by lattice gas models, thereby considering the effect of ion exclusion volumes.8,15,16 These theories provide valuable intuition for understanding a wide range of electrochemical phenomena and could serve as a guide for the optimization of electrochemical devices. However, with the advancement of electrochemical systems and especially the application of nanomaterials, these theories may struggle to characterize factors including nanoconfinement by electrodes, non-dilute electrolytes, and high working voltages (far exceeds the thermal voltage).9,10 Hence, simulations at the atomistic level are urgently needed to provide a precise microscopic picture of EDLs.

First-principles calculations based on quantum mechanics can capture the electron distribution of electrodes and electrolytes in electrochemical systems.10 Although some empirical models have been developed to address solvation and electrode polarization, the computational costs remain high, limiting the scope to systems involving a small number of atoms (usually within 1000).10,17 In contrast, molecular dynamics (MD) simulations offer sufficient sampling for systems containing complex electrodes and electrolytes at a reasonable computational expense. With the development of methods to describe electrode polarization (i.e., how to describe the voltage/current applied to electrodes), MD has become a powerful tool for exploring the thermodynamic and kinetic behavior of EDLs.7,9,10

In this Review, we aim to provide a comprehensive and up-to-date overview of the techniques used in classical MD simulations for EDL, with a particular focus on methods describing electrode polarization. We discuss the rationale behind these methods and examine the applicability of each to specific simulation systems. We then delve into the application of these methods across various fields, including supercapacitors, capacitive deionization, batteries, and electric double-layer transistors, highlighting recent advancements and insights. Finally, we discuss the current challenges and identify potential directions for future development, providing referable conclusions and recommendations for researchers in different fields.

Like conventional MD simulations, EDL modeling is based on interaction potential between atoms (i.e., force fields) to solve Newton’s equations of motion and obtain the trajectories of all atoms in the simulated system.7,9,10 Force fields describing electrolyte interactions are generally well-established and can be categorized into all-atom, united-atom, and coarse-grained models18,19 or polarizable and non-polarizable types depending on their ability to self-regulate according to local electric fields.20 Depending on research requirements, the electrode can be constructed as an ideal geometry (e.g., planar slab and slit pore) or near-realistic morphologies [e.g., conductive metal-organic frameworks (MOFs)], and the electrode atoms are usually set as immovable due to their negligible expansion and the computational cost. Non-electrostatic interactions between electrode and electrolyte atoms are well-established, such as the widely used Lennard-Jones potentials.7 As a result, the key to MD simulation of EDL lies in describing the electrostatic interactions associated with the electrodes, specifically, the representation of electrode polarization. We briefly summarize the methods for describing electrode polarization, as shown in Fig. 1, ranging from the straightforward constant-charge method (CCM) to several approaches with the constant potential method (CPM) that can self-consistently respond to the applied voltage/current and the motion of solvent and ions in electrolytes.

FIG. 1.

Schematics describing the method of electrode polarization. (a) Constant charge method applies uniform charge densities +σ (−σ) on flat electrode surfaces. (b) Green’s function method shows an array for Green’s function of single ith electrode grid points, the red point represents the boundary condition where G = 1 (j = i), and blue represents G = 0 (ji), and the system is controlled by Laplace equation ∇2Gi = 0. (c) ICC method calculates induced charge on surface elements on the dielectric boundary through the dot product of normal vector ni and local electric field Ei. (d) Fluctuating charge method allows the charge of electrode atoms to fluctuate in every time step to maintain electrode equipotential (atoms colored by their instantaneous charges).

FIG. 1.

Schematics describing the method of electrode polarization. (a) Constant charge method applies uniform charge densities +σ (−σ) on flat electrode surfaces. (b) Green’s function method shows an array for Green’s function of single ith electrode grid points, the red point represents the boundary condition where G = 1 (j = i), and blue represents G = 0 (ji), and the system is controlled by Laplace equation ∇2Gi = 0. (c) ICC method calculates induced charge on surface elements on the dielectric boundary through the dot product of normal vector ni and local electric field Ei. (d) Fluctuating charge method allows the charge of electrode atoms to fluctuate in every time step to maintain electrode equipotential (atoms colored by their instantaneous charges).

Close modal

The most straightforward and computationally inexpensive approach is CCM, which uniformly and explicitly distributes the net charge of an electrode discretely over its atoms or smoothly over its surface, as illustrated by Fig. 1(a). It is noteworthy that whether the charge distribution is discrete or smooth CCM is shown to have a non-negligible effect on the structure of EDL and the differential capacitance.21 In some simulations for slab systems containing a pair of planar electrodes, the electrodes do not carry any charge, and the applied voltage is represented by a uniform electric field parallel to the normal of the electrodes,7 which is theoretically equivalent to a CCM with smooth surface charges.

The CCM is often adopted to study the equilibrium properties of systems with open electrodes (defined as an electrode whose surface is in contact with bulk electrolyte22) that exhibit no confinement effects and can be reduced to one-dimensional structures, such as electrodes of graphene, carbon onions, and carbon nanotubes.23,24 In these simulations, electrolytes rearrange after applying electrode charges, and then, the information such as ionic structure can be obtained explicitly. However, the voltage between electrodes is not a priori and must be determined through post-processing based on Poisson’s equation with the applied electrode charge and charge distribution of electrolytes, allowing further analysis of potential-related information, such as differential capacitance.20 CCM can be extended to study dynamic charging and discharging processes in current mode by pre-setting the time-varying total charge on the electrode and uniformly distributing it among electrode atoms.25 The galvanostatic charge–discharge (GCD) processes of supercapacitors with electrolyte as ionic liquid (IL) and electrodes as graphene plates,25,26 slit-nanopore,25 and Mxenes27–29 were studied using this extended CCM.

The electrode atom charge in the CCM-based approach is pre-set and cannot dynamically respond to the movement of the electrolyte atoms, ignoring the fact that the electrode is equipotential. Hence, in principle, this approach is not accurate enough to describe the electrode polarization.20,25,30 In Secs. II B–II E, we introduce several methods describing electrode polarization that can ensure the equipotential state of electrodes and dynamically respond to changes in the electrolyte structure.

To make the electrodes equipotential in the MD system, the method of Green’s function can be employed.31,32 In an electrochemical system with a charge density distribution of ρe, Poisson’s equation 2ϕ=ρeε0 governs the distribution of electrical potential ϕ with boundary conditions ϕ = ϕ+ and ϕ = ϕ, where ϕ+(ϕ) is pre-setting electrical potential of the positive (negative) electrode. In the Green’s function method, ϕ can be divided into two parts as ϕ = ϕ′ + ϕc, where ϕ′ is the potential that considers the explicit charge in the simulated system (i.e., charges from electrolytes) and ϕc is the potential from the boundary conditions, which are implicitly caused by the induced charge of the electrodes. Hence, Poisson’s equation governing the electrical potential of the system can be divided into two parts as
2ϕ=ρeε0,
(1)
and
2ϕc=0,
(2)
with boundary conditions of
ϕc=ϕ+ϕforpositveelectrode,ϕϕfornegativeelectrode.
(3)
For ϕ′ in Eq. (1), classical electrostatic solution methods commonly used in MD, such as Ewald summation,33 can be adopted to solve it.
The Laplace equation with boundary conditions presented in Eqs. (2) and (3) can be numerically solved using Green’s function technique.32 In the technique, the discrete spatial array form of Green’s function Gi is employed to calculated ϕc. Gi is governed by ∇2Gi = 0, with the boundary condition of
Gji=1,j=i,0,ji,
(4)
where i is a single electrode grid point where unit potential will be applied and j represents other electrode grid points, as illustrated by Fig. 1(b). According to the linearity of the Laplace equation, potential field ϕc can be solved as
ϕc=iNeGiϕi,
(5)
where ϕi is the value of ϕc at the position of grid point i as given by Eq. (3) and Ne refers to the total number of electrode grids. The combination ϕc and ϕ′ yields ϕ, which eventually provides the electrostatic force of all charged particles in the electrolyte.
The equipotential electrodes can be described by the induced charge computation (ICC) method, which calculates the induced charge on polarized dielectric electrode–electrolyte boundaries, as illustrated in Fig. 1(c). These boundaries have varying permittivity at the interface between the electrode and electrolyte, and the electrical field controlled by the boundary conditions is34 
ϵoutEoutn=ϵinEinn,
(6)
EinEoutn=4πσϵout,
(7)
where ϵin (ϵout) and Ein (Eout) are the local permittivity and electric field inside (outside) the electrode media. σ is the induced charge density on the boundary, and n denotes the normal vector of the boundary surface. Equation (6) describes the continuity of polarization ɛE along the normal vector direction, while Eq. (7) elucidates the electric field mutation caused by induced charge σ.
To isolate the contribution of induced charge from Ein and Eout, the dielectric boundary is discretized into planar surface elements. For the surface element i, the induced charge will generate an electric field of 2πσiniϵout, and thus,
Eiin/out=Ei±2πσiniϵout,
(8)
where Ei is the electric field caused by the charge from electrolytes and induced charges on other surface elements. By combining Eqs. (6) and (8), the induced charge of element i can be expressed as
σi=ϵout2πϵinϵoutεin+εoutEini=fEini,
(9)
where coefficient f is defined as ϵout2πϵinϵoutεin+εout to simplify the formula. Since Ei is dependent on the induced charges of other surface elements, the direct calculation of σi takes the linear algebraic form of jδijfKijσj=fEielectrolyteni, where δij is Kronecker delta function. Kij is the electrical response between induced charges from element i to element j, only depending on the position of the elements.35  Eielectrolyte is the electric field from electrolyte charges.
Furthermore, Tyagi et al.34 developed an extension to this approach, known as ICC*. In this approach, Eq. (9) is solved by linear iteration as
σi(n+1)=ωfEini+1ωσin,
(10)
where ω is a relaxation coefficient determining the speed of convergence. At each iteration, σi is updated based on Ei, which is the external electric field calculated based on surface charges and electrolytes in the previous iteration. The charges that determine Ei are in explicit form, making this approach to allow for a scaleable and flexible algorithm that can accommodate different Coulomb solvers.

The ICC* method has undergone further development.36–38 The additional free charges, which exist in the absence of an electric field, are added to the induced charge in response to an external electric field [cf. Eq. (10)] as explicit charges at the electrode–electrolyte boundary.36 The free charge is obtained based on an identical surface shape but with the medium replaced by a vacuum on the electrolyte side and a conductor on the electrode side.36 

To accelerate the computation of ICC*, various methods have been adopted for the calculation of induced charge. One such approach adopted is the generalized minimal residual method, which is commonly used to solve large and sparse linear systems, select the optimal approximation of surface charges in the Krylov space, and enable convergence in fewer steps.36 Additionally, the pre-conditioning technique is employed to eliminate the ill-conditioned iteration matrix and improves the convergence of ICC*.37 Another technique to speed up the ICC* is to use the fast multipole method as the Coulomb solver.38 Notably, the fast multipole method offers significant advantages over O(NlogN) approaches, such as Particle Mesh Ewald (PME) and Particle–Particle Particle–Mesh (PPPM) by reducing time complexity to O(N) in extremely large systems.38 

We now discuss the method that allows the electrode atom charges to fluctuate at every time step, maintaining electrode equipotential, as illustrated by Fig. 1(d). Since the method was developed for simulating electrochemical cells by Reed et al.39,40 based on the work of Siepmann and Sprik,41 it has been further improved and advanced.25,42–51 Notably, this method has been integrated into software packages, such as MetalWalls,42,43 Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS),47–50 and GROMACS,25,51 for a widespread application in electrochemistry.

In this method, the charges of electrode atoms typically take the form of Gaussian charges with a given width 1η and fluctuating magnitude vector Q. The charge distribution of atom i positioned at Ri is expressed as
ρir=Qiη2π32exp[η2rRi2],
(11)
where Qi is the element of Q. The electrostatic energy Uele of the system is a function of charge magnitude Q and can be written in a quadratic form as
UeleQ=12QTAQQTB+C
(12)
In this equation, element Ai,j in A represents the electrostatic interaction coefficients describing the electrode atoms i and j, which depend on their positions. Element Bi in B is the potential at the position of electrode atom i caused by the charge from electrolytes. C is the electrostatic energy only considering the charge from electrolytes. The calculation of A and B depends on whether the system is periodic in 2D or 3D, as described in Refs. 39, 43, and 52. The potential of electrode atom i is ∂Uele/∂Qi, so the electrode equipotential conditions within an electrode are given by
UeleQi=UeleQj=ϕi,j,
(13)
where ϕ is the set potential for the electrode. To meet the equipotential constraint, Q can be solved by a technique of extended Lagrangian dynamics53,54 or, more commonly, by minimizing UT, which is given by
UT=UeleQTϕ
(14)
This minimization can be performed by numerical iteration.43,52 However, in most simulated systems, electrode atoms are fixed, and a more efficient approach is to pre-solve the inverse of matrix A before simulation and obtain electrode atom charges directly as
Q=A1B+ϕ.
(15)

This is the fundamental principle of the approach, and based on this foundation, the approach has undergone further development. For instance, the above form pertains to cases where the electrode potentials are set, while the calculated Q does not guarantee electrical neutrality in the system. For more practical scenarios, where the potential difference between the positive and negative electrodes is known and the system is electrically neutral, this can be achieved by modifying the matrix A or adding Lagrangian constraints.49,53,55 For the calculation of the potential acting on the electrode atoms from the electrolyte (vector B), the calculated speed can be improved by the meshed Ewald methods,56,57 implemented in widely used molecular dynamics software, such as GROMACS25,51 and LAMMPS.48,49 Simulation computation costs can also be reduced by employing a doubled-cell approach for systems with 2D periodicity.50 The Gaussian distribution of electrode charge can be replaced by a point charge form with an energy term proportional to the square of the electrode charge as the electrostatic form of point charges is directly present in the majority of molecular dynamics codes.58 When the force field describing the electrolyte is changed from non-polarized to polarized,20 the variables to be solved encompass not only the electrode charge but also the induced dipole of the electrolyte, which has recently been implemented in the software of MetalWall.43 

In addition to explicitly setting the potential difference, the potential difference between electrodes can also be achieved by the finite electric field method with a constant potential electrode with fluctuating charge.45,59 It should be noted that in the simulated system to which the method is applied, only one electrode is included (or rather two electrodes are next to each other) and the system is periodic in three dimensions, which saves computational costs.7,49 Similarly, by substituting the finite electric field with a finite electric displacement, this method can be extended to simulate the open-circuit state and the charging and discharging processes in the current control mode.44 For the simulation of charging and discharging processes in current control mode, it can also be achieved by setting the electrode potentials to be solved and imposing constraints on the sum of electrode charges.25 This method was applied to simulate galvanostatic charge–discharge processes for both planar and nanopore electrodes, capturing dynamics comparable to experimental results.25 

The discussion above has primarily focused on electrode atom charge fluctuations achieving pure electrostatic equipotentiality, considering electrodes as perfect conductors. However, this method can be easily extended to account for more quantum effects. One approach is to adjust the Gaussian width of the electrode charge to respond to the electrode’s metallicity, which is operationally the simplest method and allows simulations to obtain capacitance values close to experimental ones.60 Moreover, the energy determined by the fluctuating charge can include the band-structure energy from quantum calculations, thereby equilibrating the electrochemical potential of the electrode atoms. Then, the electrode charge is calculated by satisfying the constraint of equal electrochemical potential.61 The metallicity of the electrodes can also be considered using a semiclassical Thomas–Fermi theory, introducing the kinetic energy of electrons, where the degree of electrostatic screening of the electrodes can be described by the Thomas–Fermi length parameter.46 Recently, the fluctuating charge method has been developed to consider the metallicity of electrodes containing multiple atom types, where different types of atoms have distinct Gaussian widths and pre-determined chemical potentials.62 

Aside from the aforementioned approaches with CPM, the approach with explicit image charges63–68 and the core–shell methods69–71 can also enable electrode polarization and take image charge effects into account. Nevertheless, these two methods are limited to simple electrode structures or are not easily applicable to potential differences. Furthermore, they have been discussed in existing literature,7,9,10 and thus, will not be elaborated on in this Review.

The selection of methods for describing electrode polarization depends on various factors related to the system being simulated, such as the type of electrodes, electrolytes, voltage magnitude, and whether the simulation is in equilibrium or non-equilibrium. For CPM, methods based on image charges and core–shell are currently mainly applicable to systems with flat electrodes;63–71 the methods based on Green’s function have been applied to systems containing flat electrodes and slit nanopore,31,32,72 while CPM based on ICC* and fluctuating charge can be applied to flat, slit nanopore, and even disorder porous materials.7,9 On the other hand, CCM only requires changing the charges of electrode atoms in the force field, which enables applied to any system technically, but this approach may lead to incorrect results.21,30,47,73–78

CPM is more accurate and realistic than CCM,25,30,79 and it is also more complex to operate and requires significantly more computational resources. However, several studies have indicated that CCM can produce almost identical results to CPM in many systems.21,30,54,76 Therefore, determining the appropriate scenarios for employing the simpler and more cost-effective CCM, as well as situations must requiring CPM, is crucial to balance accuracy and computational efficiency.

The use of CCM for equilibrium simulations is influenced by factors such as electrode structure, electrolyte type, and voltage magnitude. For complex or nanoconfined electrodes, where electrode atom charges exhibit significant spatial inhomogeneity, the uniform charge application of CCM falls short when compared to CPM, rendering the latter indispensable.20,30,75 Therefore, CCM is generally more suitable for open electrode systems with ideal surfaces (e.g., graphene sheets, outer convex surfaces of carbon nanotubes, and carbon onions), although its applicability still depends on electrolyte type and voltage magnitude.

Simulations of systems containing ionic liquids (ILs) and flat electrodes have demonstrated that the consistency between CCM and CPM cannot be achieved at 9 V, a voltage far exceeding the electrochemical window of ILs.30 However, within the electrochemical window range, CCM can achieve EDL structures nearly identical to those produced by CPM.21,30,54,76 It has been reported that minor structural differences between CCM and CPM in IL systems result in differential capacitance disparities,73 possibly due to the structure’s heightened sensitivity to differential capacitance. As a result, CCM can be employed to obtain equilibrium EDL structures for systems containing open electrodes and ILs. The feasibility of using CCM for aqueous electrolytes remains uncertain, as some research has shown little difference between CCM and CPM,10 while others report marked discrepancies in the EDL structures obtained by the two methods.77 As for organic electrolytes, particularly those containing Li ions, CCM fails to capture accurate EDL structures, mainly because it neglects the pronounced image charge effect.47,80

In non-equilibrium simulations of voltage-driven charging and discharging processes, CCM will produce overly rapid relaxation times and non-physical heat generation, which makes CPM indispensable.30,54 However, for current-driven processes, our recent findings indicate that a CCM-based approach can be applied to the galvanostatic charge–discharge process simulations for planar electrode systems containing ILs, yielding EDL structures and GCD curves nearly identical to those generated by CPM.25 

Classical MD simulations combined with approaches describing electrode polarization can be applied to various fields involving EDLs. In this section, we will discuss the investigation of supercapacitors,1,81 capacitive deionization (CDI),82 batteries,4,83 and electrical double layer transistors (EDLTs),2,84 focusing on the key characteristics that can be obtained from MD simulations in these fields.

Supercapacitors, also known as electric double-layer capacitors, have emerged as an essential energy storage technology due to their unique ability to combine high power density, rapid charge/discharge rates, and long cycle life.1,9,81,85 Central to the performance of supercapacitors are the EDLs formed at the interfaces between the electrodes and the electrolyte. The characteristics of the EDL, such as its structure, charge distribution, and ion dynamics, significantly impact the energy density, power density, and overall performance of supercapacitors.6,9 Therefore, understanding the fundamental mechanisms governing EDL behavior is crucial for optimizing supercapacitor performance and enabling broader adoption. MD simulations have emerged as a powerful tool for investigating the complex phenomena occurring within EDLs and providing valuable insights into the structural, dynamic, and electrochemical properties of supercapacitor systems.7,9,10,22

1. Open electrode systems

In the majority of early simulations, supercapacitors were conceptualized as systems containing electrolytes and ideal open electrodes.8,86–91 From a technical standpoint, equilibrium simulations for these systems could be executed more cost-effectively using the CCM, as illustrated in Sec. II. The structures of EDLs and their relationships with capacitance have been extensively explored. These studies encompass a wide range of electrolytes, from aqueous32,92–94 and organic91,95 to ILs,8,86–90,96 and various electrode types, such as flat surfaces,72,97,98 convex carbon nanotube surfaces,24,99 and convex spherical surfaces of carbon onions.23 By utilizing statistical analysis from simulations, one-dimensional density distributions for various components, along with charge and potential distributions, can be obtained to determine the (differential) capacitance, effectively bridging the gap between the microscopic structure of the EDL and its macroscopic performance. The significant finding of these simulations involves the layered EDL structures in ILs,8,86–90,96,99,100 which transition from “over-screening” to “crowding.”16,101 Consequently, the differential capacitance deviates from the classical Gouy–Chapman–Stern model12–14 and instead exhibits a “camel shape” or “bell shape,” as predicted by lattice gas mean-field theory.16 Moreover, the dependence of differential capacitance and EDL structures on the type of ILs and temperature has been examined,87,89,99,102 and the effects caused by curvature23,99 and roughness100,102 of electrodes have also been evaluated.

With the advanced understanding of “EDL life,”92 the analysis of EDL structures is more refined. In a plane parallel to the electrode, the adsorbed ions in ILs have been found to undergo structural transitions between ordered and disordered phases.92,103 These transitions are profoundly related to charge fluctuations on the electrode, differential capacitance, and hysteresis in the kinetic processes.92,104,105

The refined description of the EDL structure was also represented in simulations exploring the impact of water impurities on the stability of ILs.72,97,98 It was revealed that even trace amounts of water present in the ILs could cause water to accumulate on the electrode surface, subsequently reducing the electrochemical window of the electrolyte.97 This reduction can be mitigated by employing hydrophilic ILs72 or incorporating salt into ILs [as depicted in Fig. 2(a)].98 In these studies, the location, orientation, hydrogen bonding, ion binding states, and infrared spectroscopy spectra of water molecules in the EDLs were meticulously characterized, with their voltage dependence also being comprehensively understood. For instance, it was observed that the addition of salts led to a significant blue shift [increase in wavenumber, Fig. 2(b)] in the O–H bond vibration of water molecules.98 This blue shift phenomenon increased the bond dissociation energy of the O–H bond, thereby decreasing water reactivity. Further analysis revealed that the blue shift phenomenon was caused by a reduction in water molecule clusters and the disruption of the hydrogen bond network as salts were added. This insight into the microscopic EDL structures of IL electrolytes containing trace amounts of water aids in understanding the macroscopic working voltage of these electrolytes. In conclusion, these studies emphasize that a more comprehensive understanding of EDL structures is necessary from both technical and mechanistic viewpoints, and it should not be confined to a one-dimensional description, even when the electrode has a smooth and flat surface.104,106

FIG. 2.

Applications of MD simulations in the investigation of supercapacitors. (a) Simulation system featuring an open electrode model of graphene and a humid IL electrolyte with added salt. (b) Computed infrared spectroscopy (IR) spectra of O–H bonds in pure water, humid ILs, and humid ILs with varying salt-to-water molar ratios. The gray lines denote the IR spectra, while the red, pink, and blue regions correspond to the first, second, and third fitted spectra, respectively. The cyan dashed lines represent the sum of the fitted spectra. (a) and (b) are reproduced with permission from Chen et al., Nat. Commun. 11, 5809 (2020). Copyright 2020 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (c) Simulation system featuring a slit nanopore electrode and an IL electrolyte. (d) Charging time constant (τc on the left y-axis) and capacitance (C on the right y-axis) as a function with pore size d. (c) and (d) are reproduced with permission from Mo et al., ACS Nano 14, 2395 (2020). Copyright 2020 American Chemical Society. (e) Simulation system featuring the realistic electrode models of Ni3(HITP)2 MOF and [Emim][BF4] electrolyte. (f) Two-demission charge density from electrolyte along the pore axis inside the MOF pores. (e) and (f) are reproduced with permission from Bi et al., Nat. Mater. 19, 552 (2020). Copyright 2020 Springer Nature.

FIG. 2.

Applications of MD simulations in the investigation of supercapacitors. (a) Simulation system featuring an open electrode model of graphene and a humid IL electrolyte with added salt. (b) Computed infrared spectroscopy (IR) spectra of O–H bonds in pure water, humid ILs, and humid ILs with varying salt-to-water molar ratios. The gray lines denote the IR spectra, while the red, pink, and blue regions correspond to the first, second, and third fitted spectra, respectively. The cyan dashed lines represent the sum of the fitted spectra. (a) and (b) are reproduced with permission from Chen et al., Nat. Commun. 11, 5809 (2020). Copyright 2020 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (c) Simulation system featuring a slit nanopore electrode and an IL electrolyte. (d) Charging time constant (τc on the left y-axis) and capacitance (C on the right y-axis) as a function with pore size d. (c) and (d) are reproduced with permission from Mo et al., ACS Nano 14, 2395 (2020). Copyright 2020 American Chemical Society. (e) Simulation system featuring the realistic electrode models of Ni3(HITP)2 MOF and [Emim][BF4] electrolyte. (f) Two-demission charge density from electrolyte along the pore axis inside the MOF pores. (e) and (f) are reproduced with permission from Bi et al., Nat. Mater. 19, 552 (2020). Copyright 2020 Springer Nature.

Close modal

2. Nanopore electrode systems

Following the experimental discovery of the anomalous increase in capacitance as pore size decreases,107–110 supercapacitor electrodes are primarily composed of nanoporous materials. To investigate the impact of nano-confinement, specifically the role of pore size on EDLs, electrodes are modeled as nanotubes111 or slit pores112–114 in MD simulations. Simulations based on CPM with Green’s function have been employed to explore the capacitance of slit-shaped nanopores with IL and found that capacitance exhibits a U-shaped scaling curve behavior with pore widths ranging from 0.75 to 1.26 nm, in which the left branch of the capacitance scaling curve corresponds to the anomalous capacitance increase.113 Concurrently, other MD simulations revealed a more complex oscillatory behavior in capacitance-pore size relationships, with maximum capacitance values observed at both 0.7 nm and a larger pore size of 1.4 nm.112 This oscillatory relationship was subsequently experimentally confirmed.115 The capacitance-pore size profile originates from changes in EDL structures under confinement, which can be explained as a superposition of EDLs on both electrode walls, analogous to wave interference. Vatamanu et al.114 further conducted simulations on several systems featuring various ILs and nanopore electrodes (slit pores, cylindrical pores, and slit pores with rough walls in different pore sizes) to investigate capacitance behaviors. They also observed the anomalous capacitance increase and found that electrode roughness enhances capacitance as well. A similar dependence of capacitance on pore size has also been investigated in aqueous and organic systems.116,117

The dynamic processes of EDLs under confinement, relating to the power density of supercapacitors, have been extensively investigated by MD simulations with nanotubes and slit nanopores. Kondrat and colleagues used MD based on CPM with Green’s function to study the charging dynamics of slit pores.118 For ionophilic pores (ions wetted in the pores at a non-polarized state), the charging process is a diffusion process accelerated by the collective effect. Moreover, when the voltage is large or close to zero, the diffusion within the pores is slower than in the bulk electrolyte. When the voltage is at an intermediate level, the diffusion is an order of magnitude larger than that in the bulk electrolyte. Conversely, when the electrodes are ionophobic (ions cannot enter the pores at a non-polarized state), the charging process can be accelerated due to the absence of overfilling or de-filling caused by ion congestion at the beginning of charging. The dependence of charging dynamics on the pore size of slit nanopore electrodes has been investigated by MD simulations based on CPM with fluctuating charge [Fig. 2(c)].119 It was found that, contrary to the conventional understanding that larger pores lead to faster charging, the charging time oscillates as the pore size increases [Fig. 2(d)]. Charging is accelerated at specific pore sizes, and these pores can simultaneously achieve higher energy density and larger power density, which are determined by the transition structures of ions.

Most of the dynamic processes, including the works discussed above, are driven by voltages in a jump-wise mode (voltage steps from 0 to a value). However, Breitsprecher and colleagues used CPM-MD simulations based on ICC* to study the charging process of slit pores and found that, compared to jump-wise voltages, applying linear or nonlinear ramp voltages could avoid congestion caused by counter-ions at the beginning of charging, resulting in faster charging rates.120,121 This finding has been experimentally confirmed.121 Similar acceleration due to ramp voltage has been observed in simulations of carbon nanotube charging.122 For discharging, since there are no congestion issues, directly stepping the voltage down to 0 V or even applying a reverse voltage can result in faster discharging rates.120,121 In addition, the dynamic processes of ion transport in slit nanopores driven by periodic triangular wave voltages, or square wave currents set by the current-mode CPM, have been investigated, which helps understand the cyclic voltammetry and GCD test in electrochemical measurements.25,123

3. Realistic electrode systems

To further explore energy storage mechanisms closer to real conditions, more complex MD simulation systems involving “realistic electrodes” should be employed. These account for the intricate structural and chemical properties of actual electrode materials. As discussed in Sec. II, employing CPM for conducting MD simulations with these electrode models is essential.

The most frequently adopted electrode material for supercapacitors is amorphous porous carbons.1,85 It is necessary to construct atomic structure models that can reflect the electrochemical features of these carbons before conducting MD simulations. Quenched molecular dynamics simulations were performed to generate a series of carbide-derived-carbon (CDC) models by tuning the thermal quench rate, which was compared with experimental pore size distributions, specific surface areas, and adsorption isotherms.124 These generated CDC models with ILs composed of supercapacitors were investigated using CPM with fluctuating charge,125–130 which get an experimentally matched specific mass capacitance,125 and demonstrated that a positive relationship exists between the local charge storage on the electrode and the degree of confinement.126 The charging dynamics of these systems, driven by jump-wise voltages, were also explored.127 It was found that the charging rate is dependent on the average pore size, with smaller pore sizes charging more slowly. The dynamic processes can be scaled up to a macroscopic level by fitting an equivalent resistor-capacitor circuit.127 Subsequent simulations involving these CDC electrodes showed that the capacitance and resistance in the equivalent circuit can be obtained directly from equilibrium simulations, without the need for fitting through dynamic process simulations.129 This multi-scale modeling opens the door for a quantitative comparison between molecular-scale simulations and macroscopic experiments.129 In addition to using quenched MD, the atomic structure model of amorphous carbon can also be constructed by adjusting the positions of the electrodes through Monte Carlo simulations to match in situ small-angle x-ray scattering experimental data.131 This constructed electrode model was adopted to investigate supercapacitors with an aqueous electrolyte and found the relation between the electrode’s atom charge and the degree of confinement similar to that found in ILs.126 

Materials with ordered structures, such as ordered carbons,132 MXenes,133,134 and MOFs,135,136 have shown potential as supercapacitor electrodes. The atomic models for these materials can be more easily obtained as their well-defined structures require less complex construction or modification compared to disordered materials. For ordered carbon materials, atomic structure models of zeolite-templated carbons137–139 and ordered mesoporous carbon140 have been adopted by MD simulations to study the mechanisms of these materials as supercapacitor electrodes. In zeolite-templated carbons, the capacitance was found to have a strong correlation with charge compensation per carbon atom, rather than a significant association with the electrode pore size.138 A smaller local surface curvature radius led to higher charge compensation per carbon atom and greater capacitance. However, charging rate was strongly correlated with pore size, with smaller sizes resulting in slower charging overall.138 Subsequently, the role of surface functional groups on the electrode of zeolite templated carbon was further analyzed in simulations.139 

Bi and colleagues performed CPM with fluctuating charges to investigate the capacitance and charging dynamics of supercapacitors composed of two-dimensional conductive MOFs and ILs, as illustrated in the simulation system in Fig. 2(e).51 The differential capacitance was found to exhibit a bell-shaped and double-humped pattern concerning pore size, and the relationship with the EDL structure inside the pores was clarified [Fig. 2(f)]. Moreover, by using a transmission line model, the microscopic simulation results were scaled up to the macroscopic scale, which was confirmed by experimental electrochemical measurements.51 Two-dimensional layered materials, such as MXenes and molybdenum disulfide, have also been studied as electrodes using MD simulations.27–29,62,141 For these materials, it is important to pay attention to whether the distance between electrode layers changes with alterations in the charging state. The GCD processes of supercapacitors composed of MXenes and ILs have been simulated, and, unlike most simulations where electrodes are fixed, the electrode volume was found to vary with charging, which is related to ion transportation.27,28

Desalination is a promising solution to address freshwater scarcity, which can be categorized into pressure-driven, thermal, and electrokinetic desalination.3 Pressure-driven desalination uses pressure to force water through a membrane, separating it from ions,142 while thermal desalination relies on evaporation and condensation to separate water from ions.143 Both methods are energy-intensive and not cost-effective for widespread application. Capacitive deionization (CDI), an emerging electrokinetic desalination technology, is a more cost-effective and energy-efficient water desalination process. CDI employs an electric field between a cathode and an anode to drive ions to the electrode, removing them from water [Fig. 3(a)].144 MD simulations can elucidate the microscopic mechanism of desalination [Fig. 3(b)] and improve the performance of CDI systems.145–153 

FIG. 3.

Application of MD simulations in CDI systems. (a) Schematic of CDI. Reproduced with permission from Sun et al., Adv. Funct. Mater. 33, 2213578 (2023). Copyright 2023 John Wiley and Sons. (b) MD simulation system of slit pore applying potential for desalination. Reproduced with permission from Bi et al., Sustainable Energy Fuels 4, 1285 (2020). Copyright 2020 The Royal Society of Chemistry. (c) The angle between the NO3 normal and longitude direction of CNT at different voltages. Reproduced with permission from Mao et al., Chem. Eng. J. 415, 128920 (2021). Copyright 2021 Elsevier. (d) Charge efficiency of the CDI system as a function of pore size. Reproduced with permission from Bi et al., Sustainable Energy Fuels 4, 1285 (2020). Copyright 2020 The Royal Society of Chemistry. (e) Calculated in-pore ion uptake and release concentration, salt concentration at the potential of zero charge, net charge concentration, and salt uptake concentration. Reproduced with permission from Zhang et al., Cell Rep. Phys. Sci. 3, 100689 (2022). Copyright 2022 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 3.

Application of MD simulations in CDI systems. (a) Schematic of CDI. Reproduced with permission from Sun et al., Adv. Funct. Mater. 33, 2213578 (2023). Copyright 2023 John Wiley and Sons. (b) MD simulation system of slit pore applying potential for desalination. Reproduced with permission from Bi et al., Sustainable Energy Fuels 4, 1285 (2020). Copyright 2020 The Royal Society of Chemistry. (c) The angle between the NO3 normal and longitude direction of CNT at different voltages. Reproduced with permission from Mao et al., Chem. Eng. J. 415, 128920 (2021). Copyright 2021 Elsevier. (d) Charge efficiency of the CDI system as a function of pore size. Reproduced with permission from Bi et al., Sustainable Energy Fuels 4, 1285 (2020). Copyright 2020 The Royal Society of Chemistry. (e) Calculated in-pore ion uptake and release concentration, salt concentration at the potential of zero charge, net charge concentration, and salt uptake concentration. Reproduced with permission from Zhang et al., Cell Rep. Phys. Sci. 3, 100689 (2022). Copyright 2022 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

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1. Ion adsorption capacity

Salt removal capacity is the primary parameter determining the effectiveness of a CDI system. To investigate the effect of voltage application mode on the salt removal capacity in a CDI system, MD simulations with the CCM were conducted.145 The simulations involved different CDI systems applying constant loading and pulse loading. The results indicate that the higher salt removal capacity of pulse loading is due to the intermittent load of the pulsed electric field. Using an appropriate load-unload frequency, the water dipole recovers from orientation polarization. In DC and extremely high-frequency voltage, the orientation polarization persists, preventing the electric field from driving ions to the electrode.

MD simulation is also used to explore the salt removal capacity of porous electrodes, such as carbon nanotubes (CNTs).146,147 Through MD simulations using CCM, it has been revealed that charged CNTs possess a hydrophilic nature, and water molecules penetrate ion pairs or clusters, which breaks the pair/cluster structure, making ion separation easier and eventually enhancing ion adsorption capacity and water purification. Moreover, the polarization of water molecules raises the dielectric constant and increases the availability of hydrogen bonding sites, making the dielectric response faster.148 This response significantly influences ion electrosorption capacity by affecting the Stern layer capacitance. Realistic electrode models are applied to investigate the ion adsorption capacity in amorphous porous carbons.149 In this work, the pore size is comparable to the ion size, bringing the huge potential for deionization. The molecular simulation results at high concentrations can be used to parameterize a modified Donnan model, which can then be used to predict specific ion adsorption at lower concentrations.149 

2. Selectivity of ions

Nanoporous electrodes with selective ion adsorption capabilities can expand the potential applications of CDI. MD simulation studies indicate that pore geometry and applied voltage affect selectivity, which is determined by ion properties, including their charge and geometry.150,151 Simulations with CCM show that the best selectivity between Cl and NO3 occurs when the pore size is 2.0 nm for uncharged electrodes.150 As voltage increases, the selectivity increases to a critical point before decreasing, with the maximum selectivity found at the critical point when the pore size is 1.5 nm. The mechanism of selectivity is examined from the perspective of molecular structure, such as hydrated radius and coordination number.150 The planar structure of NO3 makes it easier to dehydrate and enter the pore, thus being more competitive for adsorption near the critical point. The voltage can adjust the angle between the NO3 plane normal vector and the longitudinal direction of the CNT, thereby adjusting its entrance into the pore. Applying voltages to electrodes with 1.5 and 2.0 nm pores cause the normal of the NO3 plane and longitudinal direction to shift from being perpendicular to nearly parallel eventually [Fig. 3(c)]. This transition provides more strategies for NO3 to enter the pore, resulting in greater selectivity.150 

Besides the selectivity of anions, the selectivity of cations is also concerned and has been explored using MD simulations.151 The selectivity of Ca2+ over Na+ and its dependence on the applied voltage in slit pores has been analyzed using CCM. The competition between ion-pore electrostatic interactions and crowding effects is considered to be the controlling factor of selectivity. At higher voltages, the dehydration shell of Ca2+ is more difficult to break, while the solvation shell of Na+ is partially broken. As a consequence, Ca2+ demonstrates a larger volume and a decreased selectivity in comparison to Na+ under these conditions.151 

Beyond pore size, the shape of the electrode pores also significantly impacts ion transport processes,138,154,155 consequently playing a pivotal role in determining CDI selectivity. Through the application of density functional theory combined with a continuum solvation model, it has been observed that the cation binding energy in CNTs surpasses that in slit pores, especially for smaller ions, which makes CNTs more effective in separating Cs+/Li+, Cs+/Na+, and Cs+/K+ ions.156 However, a more thorough comprehension of this area warrants further exploration through future MD simulations.

3. Permselectivity improves efficiency

The efficiency of CDI in removing high molar strength saline water is often lower than expected due to the simultaneous counter-ion adsorption and co-ion desorption processes in nanoporous electrodes.157 MD simulations with CPM help address this problem by dynamically revealing underlying processes.152,153 It is found that in ionophobic pores (<0.6 nm), no ions are present inside the pore when not charged.152 When the potential is applied, only counter-ions are adsorbed, and no co-ion desorption occurs in these pores. The coordination number of water surrounding K+ ions in these pores is smaller than that in ionophilic pores (>1.0 nm), indicating a fractured hydration shell. In such ionophobic pores, ions must overcome a higher dehydration energy barrier to enter the pore, which explains the absence of ions when the electrode is not charged. The initial state and dehydration process in ionophobic pores exhibit complete permselectivity of counter-ions, improving charge efficiency [Fig. 3(d)], and is consistent with experiments.152 

MD simulations using the CPM have revealed the permselectivity properties near the transition stage between ionophobic and ionophilic pores.153 Ultra-micropores exhibit ionophobicity, where the ion removal process only involves counter-ion adsorption without co-ion desorption, leading to high charge efficiency. However, the dehydration energy barrier increases significantly in ultra-micropores, reducing the absolute number of ion adsorption. To achieve the best desalination performance, a balance between the ionophobic nature and dehydration cost must be considered through the net charge concentration adsorbed by pores. The simulations have shown that the maximum net charge concentration and salt uptake concentration occur at a pore size of 0.77 nm [Fig. 3(e)], resulting in the best desalination performance.153 

The battery is the most widely used electrochemical energy storage device, providing the foundation for modern society by powering mobile devices and electric vehicles, serving as a crucial component in future energy storage facilities.5 The increasing demand for large-scale energy storage, driven by the development of renewable energy sources, necessitates higher battery performance, including high energy density, safety, temperature endurance, and long lifetime.158 In a typical battery system, the interface between the electrode and electrolyte is the site of electron transfer, determining overall battery performance.159 The in-depth understanding of interfaces within batteries remains a challenge due to their in situ nature and atomic-scale presence,160 encouraging the application of in situ measurements161 and molecular simulations in battery research.4 

Using MD simulations, the atomic EDL structure in the interfacial region under various potentials can be precisely depicted. Crucial characteristics of this structure, such as the orientation of interfacial water,98 H-bond,162,163 and solvation structure,164 significantly affect the interfacial reaction properties.165 For instance, the orientation of interfacial water98 and the connectivity of H-bond networks162,163 have been shown to influence electron transfer from the electrode to water and proton transfer in the interfacial region. Thus, gaining detailed information on interfacial structures helps unveil the mechanisms underlying interface regulation, thereby facilitating the design of batteries with improved performance.166 

1. Understanding the interfacial structure in batteries

Mixed carbon-based electrolytes, which may include dimethyl carbonate (DMC), ethylene carbonate (EC), or 1,3-dioxolane (DOL) as solvent, are commonly employed in commercial batteries.167 Using ideal graphene sheets as electrodes, the interfacial structure of the EC:DMC(3:7)/LiPF6 electrolyte (with a solvent to LiPF6 molar ratio of about 2) under different potentials was simulated using CPM with fluctuating charge.168 For both positively and negatively charged electrodes, EC with higher polarity, despite being the minority solvent, replaces DMC as polarization increases [Fig. 4(a)].168 This is consistent with experimental observations of stronger adsorption of EC molecules on the LiCoO2 cathode than linear carbonates.169 Similarly, the interfacial structure of high concentration DMC/LiTFSI (with a solvent to LiPF6 molar ratio of about 1.2) was simulated.170 It was found that DMC is excluded from the positively charged electrode due to the distinct adsorption of the stable TFSI anion.170 This restrains the side reaction of the solvent, explaining the suppression of current collector corrosion in high concentration.171 

FIG. 4.

Application of MD simulations in the investigation of batteries. (a) The interfacial density of electrolyte components of the EC:DMC(3:7)/LiPF6 electrode under various potentials applied to carbon electrode. Reproduced with permission from Vatamanu et al., J. Phys. Chem. C 116, 1114 (2012). Copyright 2012 American Chemical Society. (b) and (c) Simulated probability (b) and electric field strength (c) of interfacial water for 21 m LiTFSI on an Au(111) electrode. Reproduced with permission from Li et al., Nat. Commun. 13, 5330 (2022). Copyright 2022 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (d) The comparison of 2D density distributions of interfacial water at the anode before (left sub-graphs) and after (sub-graphs) adding TEA+ under 0 V (upper sub-graphs) and −1.5 V (upper bottom sub-graphs). The white lines sketch the contours of TEA+ distribution. The unit of the color bar is # nm−3. Reproduced with permission from Zhou et al., Adv. Mater. 34, 2207040 (2022). Copyright 2023 John Wiley and Sons. (e) A scheme of the MD-DFT-data interactive model to investigate the effect of the EDL on electrolyte reduction and SEI formation. Reproduced with permission from Wu et al., J. Am. Chem. Soc. 145, 2473 (2023). Copyright 2023 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

FIG. 4.

Application of MD simulations in the investigation of batteries. (a) The interfacial density of electrolyte components of the EC:DMC(3:7)/LiPF6 electrode under various potentials applied to carbon electrode. Reproduced with permission from Vatamanu et al., J. Phys. Chem. C 116, 1114 (2012). Copyright 2012 American Chemical Society. (b) and (c) Simulated probability (b) and electric field strength (c) of interfacial water for 21 m LiTFSI on an Au(111) electrode. Reproduced with permission from Li et al., Nat. Commun. 13, 5330 (2022). Copyright 2022 Author(s), licensed under a Creative Commons Attribution (CC BY) license. (d) The comparison of 2D density distributions of interfacial water at the anode before (left sub-graphs) and after (sub-graphs) adding TEA+ under 0 V (upper sub-graphs) and −1.5 V (upper bottom sub-graphs). The white lines sketch the contours of TEA+ distribution. The unit of the color bar is # nm−3. Reproduced with permission from Zhou et al., Adv. Mater. 34, 2207040 (2022). Copyright 2023 John Wiley and Sons. (e) A scheme of the MD-DFT-data interactive model to investigate the effect of the EDL on electrolyte reduction and SEI formation. Reproduced with permission from Wu et al., J. Am. Chem. Soc. 145, 2473 (2023). Copyright 2023 Author(s), licensed under a Creative Commons Attribution (CC BY) license.

Close modal

In addition to organic electrolytes, aqueous electrolytes are competitive candidates for future energy storage systems due to their intrinsic safety.172 In recent years, the high-concentrated “water-in-salt” electrolyte (WISE), with a wide electrochemical window, enables the realization of practical aqueous batteries.173–175 To understand the interfacial structure of WISE, Vatamanu and Borodin176 conducted simulations of 28 m WISE comprising 21 m LiTFSI and 7 m LiOTF on polarized carbon electrodes using the CPM with fluctuating charge. It was revealed that TFSI anion shows stronger adsorption to the positively charged electrode than OTF anion. Simultaneously, water is largely excluded from the electrode surface, explaining the improved oxidation stability of WISE.176 By adding 21 m [MeEt3N][TFSI], an extreme concentration of 63 m can be achieved.177 As demonstrated by the MD simulations, the adsorbed water on the negatively charged electrode further decreases compared to 28 m WISE, achieving a wider electrochemical window.177 

The combination of in situ measurement and MD simulation can provide a more comprehensive and accurate depiction of the interfacial structure.79,170,178 By integrating shell-isolated nanoparticle-enhanced Raman spectroscopy (SHINERS) and MD simulations with CPM, an unconventional interfacial structure of 21 m LiTFSI at an Au(111) electrode was uncovered.79 Simulations revealed that most of the interfacial water is bound by Li+, with different H-bond numbers ranging from 0 to 2 [Fig. 4(b)], in good agreement with three O-H stretching bands observed by SHINERS.79 When polarization increases negatively, the adsorbed Li+ is inserted between the electrode and the first layer of water, weakening the electric field experienced by the water [Fig. 4(c)].79 The trends in the electric field as a function of potential correspond well with vibrational frequency shifts in SHINERS. The adsorbed Li+ also leads to unusual “H-up” interfacial water molecules, with the dipole pointing away from the electrode surface despite the negative electrode polarization.79 

2. Revealing the mechanisms of interface regulation to improve performance

Various strategies have been applied to regulate the interfacial structure for better battery performance. Using MD simulations, the mechanisms of interface regulation can be captured.165,179–182 For instance, in the case of a zinc-air battery, MD simulations based on CCM performed on the system with an ideal carbon electrode revealed that compared to SO42−, using OTF as the salt anion results in a reduced accumulation of water and increased presence of Zn2+ near the positively charged electrode, which suppresses the water-related oxygen reduction reaction and enhances reversibility.179 Additionally, the interfacial structure can be effectively regulated by introducing additives. Using CCM to simulate the system with an ideal carbon electrode, Ma et al.165 found that the addition of 4 m phosphonium cations effectively decreases the population of interfacial water. Zhou et al.180 found that adding 1 m TEA+ cation significantly improves the cathodic stability of a 13.8 m LiTFSI aqueous electrolyte with carbon-coated TiO2. By conducting CPM simulations for a carbon electrode, the hydrophobic TEA+ is found to occupy more space on the electrode surface as the potential of electrode becomes more negative, blocking water from touching the electrode surface [Fig. 4(d)] and successfully suppressing the decomposition of water.180 

The introduction of NO3 anion has been proven to effectively improve the oxidation stability of ether-based electrolytes on cathodes.183 Applying CCM on a carbon electrode, Zhang et al.181 found that the introduction of NO3 constructs a Li+-rich interface and forms unique polymer-like chain structures with high oxidation stability under positive potential. To consider the intrinsic charge of the cathode, Wang et al.182 coupled the atomic charge of the LiCoO2 electrode from DFT calculation to the CPM-MD simulation, revealing that NO3 is dramatically absorbed on the electrode surface under positive potential, effectively expelling the solvent from the interfacial region and preventing solvent oxidation. However, an increase in bound solvent was not observed even though Li+ is accumulated with the introduction of NO3 in the interfacial region.182 

The decomposition of electrolytes during battery cycling leads to the formation of solid–electrolyte interphase (SEI), which prevents side reactions and improves battery reversibility.184 MD simulations can help elucidate how the regulation of interfacial structure enhances SEI formation.181,185–187 For example, by conducting MD simulation on non-polarized Li metal, Huang et al.185 compared the interfacial structure of electrolytes with various solvents, revealing that the solvent tris(trimethylsilyl) borate has the strongest adsorption on the electrode surface, promoting PF6 and Li+ in the interfacial region, and leading to the formation of an inorganic-rich SEI. Using CPM, Li2DMC and Li2TFSI aggregates were found near the negatively charged carbon electrode when DMC was introduced into the LiTFSI aqueous electrolyte, and these aggregates were more easily reduced to form SEI, as predicted by quantum chemistry calculation.187 By conducting CPM simulations on a lithium anode, Wang et al.188 found that due to strong anion–cation association, the introduced NO3 can accumulate near the electrode even under negative polarization, facilitating the formation of nitride-enriched SEI.

Simulating the systems with a graphene electrode by CCM, Wu et al.189 explored the effect of fluoroethylene carbonate (FEC) on the interfacial structure of both carbonate and ether-based electrolytes. It is revealed that the Li-coordinated solvent is easier to be reduced on the negatively charged electrode than a free one.189 To complement the electron transfer process missing in MD simulations, reduction potentials of representative structures were calculated by quantum chemistry calculation to understand SEI formation [Fig. 4(e)].189 For the carbonate-based EC:EMC/LiPF6 electrolyte, the SEI is derived from both free and cation-coordinated solvents but not anions. With the introduction of FEC, the density of F atoms in the interfacial region increases, promoting the LiF component in the SEI.189 In contrast, for the ether-based DOL:DME/LiTFSI electrolyte, the SEI mainly originates from the Li+-coordinated solvents and anions, and the increase in F atoms due to FEC can only be observed at a low temperature of −40 °C,189 which is consistent with experimental observations.190 

The field-effect transistor is a critical component in modern semiconductor electronics. It comprises three terminals (source, gate, and drain) and controls the current flow by modulating the conductivity between the drain and source through changes in the gate-source voltage [Fig. 5(a)].2,191 Traditional field-effect transistors with inorganic solid-state materials as gate dielectrics suffer from limitations such as low capacitance and decreased carrier mobility.2,192,193 To overcome these issues, researchers have explored the use of EDLTs with liquid electrolytes instead. In EDLTs, the thinness of the EDL, which is only a few nanometers, results in high capacitance, leading to higher carrier accumulation, lower threshold voltage, and tunable electronic properties. Therefore, EDLTs have demonstrated promising performance and are considered a promising platform for next-generation electronic devices.2,84,194 The EDL between the channel and electrolyte is crucial in determining the performance of EDLTs, and using MD simulations to understand its microscopic characteristics is significant for understanding the working mechanism of EDLTs and optimizing their performance.195–199 

FIG. 5.

Application of MD simulations in the investigation of EDLTs. (a) Schematic of EDLTs. Reproduced with permission from Du et al., J. Mater. Sci. 50, 5641 (2015). Copyright 2015 Springer Nature. (b) Structures of the first adsorbed cation on the a-IGZO surface, and the voltages from top to bottom are 0, 0.4, and 1.0 V. Reproduced with permission from Black et al., ACS Appl. Mater. Interfaces 9, 40949 (2017). Copyright 2017 American Chemical Society. (c) 2D distribution surface charge carrier density at the a-IGZO surface. Reproduced with permission from ACS Zhao et al., Appl. Mater. Interfaces 10, 43211 (2018). Copyright 2018 American Chemical Society. (d) Charging dynamics of MoS2-based EDLTs for different concentrations of [Bmim][BF4] in acetonitrile. The top panel of (d) shows that a pulsed square-shaped gate voltage was applied to the systems studied, and the bottom panel exhibits the time evolution of the total charge of the channel surface. qe stands for the electric charge of an electron. Reproduced with permission from Zhao et al., ACS Appl. Mater. Interfaces 11, 13822 (2019). Copyright 2019 American Chemical Society.

FIG. 5.

Application of MD simulations in the investigation of EDLTs. (a) Schematic of EDLTs. Reproduced with permission from Du et al., J. Mater. Sci. 50, 5641 (2015). Copyright 2015 Springer Nature. (b) Structures of the first adsorbed cation on the a-IGZO surface, and the voltages from top to bottom are 0, 0.4, and 1.0 V. Reproduced with permission from Black et al., ACS Appl. Mater. Interfaces 9, 40949 (2017). Copyright 2017 American Chemical Society. (c) 2D distribution surface charge carrier density at the a-IGZO surface. Reproduced with permission from ACS Zhao et al., Appl. Mater. Interfaces 10, 43211 (2018). Copyright 2018 American Chemical Society. (d) Charging dynamics of MoS2-based EDLTs for different concentrations of [Bmim][BF4] in acetonitrile. The top panel of (d) shows that a pulsed square-shaped gate voltage was applied to the systems studied, and the bottom panel exhibits the time evolution of the total charge of the channel surface. qe stands for the electric charge of an electron. Reproduced with permission from Zhao et al., ACS Appl. Mater. Interfaces 11, 13822 (2019). Copyright 2019 American Chemical Society.

Close modal

1. Electrolyte structures

The effect of anion size on the interface structure in EDLTs with graphene and [Emim]+-based ILs has been investigated by MD simulations where electrodes are not polarized.195 It was found that the imidazolium ring of the cation tends to be parallel to the graphene surface and build up a positive charge density, the extent of which becomes weaker as the anion size increases. Based on the structures, the positive shift in the Dirac point voltage observed by experiments is attributed to the induced negative charge density close to the graphene.

MD simulations based on CPM have explored the EDL structures varying with gate-source voltage in EDLTs consisting of amorphous indium gallium zinc oxide (a-IGZO) and ILs, which achieve excellent transistor performances with a relatively high current on/off ratio above 107 and a threshold voltage less than 0.5 V.196,197 Simulations show that with the increase in the gate-source voltage, the number of cations in the first adsorption layer on the a-IGZO channel does not change significantly, while the orientation distribution of the imidazole ring dramatically changes as the imidazole ring is gradually distributed parallel to the channel surface [Fig. 5(b)].196 This orientational change significantly reduces the thickness of the EDL and consequently increases the capacitance, which is in agreement with the measurement of the atomic force microscope.196 The EDL structure transition shows a similar trend with the experimental conductivity of the a-IGZO channel, which exhibits a stepwise change with the gate-source voltage.196 

2. Distribution of charge carriers

In addition to investigating the EDL structures on the electrolyte side, MD simulations can also be used to explore the distribution of charge carriers on the solid side.196–198 In a study exploring the impact of ion size on the performance of EDLTs consisting of ILs and a-IGZO, MD simulations revealed that ILs with smaller ions can form a larger capacitance under the same gate-source voltage.197 Subsequently, the influence of EDL microstructures on the uniformity of carrier distribution on the channel surface was analyzed [Fig. 5(c)]. It was found that smaller cations can induce a more uniform distribution of charge carriers. However, [BF4], despite having the highest capacitance due to its small size, can induce the most non-uniform carrier distribution. Based on percolation theory analysis, it is concluded that the more uniform the carrier distribution, the higher the probability of forming conductive paths on the channel surface, potentially improving the conductivity.197 A method based on resistance network analysis was further developed to quantitatively obtain the surface conductivity of the channel from the surface charge carrier distribution. The results confirmed that the more uniform the carrier distribution, the higher the conductivity of the channel. The simulation results were also magnified at the nanoscale to quantitatively compare with the experimental measurements at the macroscopic scale based on the conductivity of the channel, proving the reliability of this analysis method.197 

3. Dynamics

Besides the static characteristics of EDLTs, their dynamic properties, especially the switching speed,2,84,198 are also crucial and can be studied using MD simulations.198,199 The dynamic performance of EDLTs mainly depends on the ion/molecule transport process in the EDLs, rather than the carrier mobility in the channel. To improve the dynamic performance of EDLTs with MoS2, an organic solvent of acetonitrile was added to the IL [Bmim][BF4] to enhance its diffusion performance, as investigated through MD simulations based on CPM.198 From the amount of accumulated charge on the channel surface after applying the gate-source voltage, it was found that the addition of the organic solvent significantly improved the dynamic performance of the device [Fig. 5(d)]. Through the analysis of the electrolyte conductivity at different concentrations, it was found that the electrolyte concentration corresponding to the highest conductivity can achieve the optimal dynamic performance of the device. Therefore, the dynamic performance of the EDLT is mainly determined by the diffusion performance of the electrolyte. The addition of an organic solvent weakens the interaction between the adsorbed ions and the channel surface, resulting in a more uniform distribution of charge carriers and ultimately enhancing the channel’s conductivity. Interestingly, the optimal electrolyte concentration for achieving the most uniform carrier distribution on the channel surface was also the same as the one that optimized the dynamic performance of the EDLTs, which suggests that the addition of an organic solvent can simultaneously improve the static and dynamic performance of EDLTs.198 Further simulations have shown that increasing the temperature can also enhance the dynamic performance of EDLTs, which is consistent with experimental findings.199 

MD simulation is a powerful tool for investigating EDLs at solid–liquid interfaces. Accurately modeling electrode polarization in these simulations remains a key challenge. We first provide an overview of several methods for describing electrode polarization in MD simulations, ranging from the straightforward CCM to more realistic CPM that maintains the equipotential of electrodes, including the approach with Green’s function, ICC (ICC*), and the widely used method with fluctuating charges. We discuss the scenarios in which these approaches are applicable and, in particular, analyze the cases in which it may be possible to use the simpler and more computationally efficient CCM.

Finally, we present the application of MD in exploring several areas involving EDLs, including supercapacitors, CDI, batteries, and EDLTs. By utilizing different approaches and models for electrodes and electrolytes, MD simulations provide valuable insights into the structures and dynamics of EDLs and their mechanisms on system performance. These insights are crucial for optimizing system performance and expanding the potential applications of these technologies.

In the future of MD simulations of EDLs, it will be crucial to improve CPM techniques to capture more accurate EDL behaviors, providing valuable insights for the optimization of electrochemical devices. One key area of improvement is considering the metallic properties of electrodes as many electrodes are not perfect conductors.7,46,62 For example, the materials of channels in EDLT and quantum capacitance effects in the field of supercapacitors require a more accurate representation of electrode properties in simulations. Another important aspect is the incorporation of chemical reactions into MD simulations.200,201 The development and the use of reactive force fields in CPM-MD may help integrate these reactions into simulations, leading to a more comprehensive understanding of the processes and mechanisms involved in pseudocapacitive behavior and batteries. Moreover, introducing movable electrodes in simulations can enable the investigation of electrode expansion27,28 and heat conduction at the solid–liquid interface.202,203 Finally, enhancing computational efficiency is essential. Efforts to increase the scale of studied systems and improve simulation times are necessary. Leveraging the power of GPUs and combining machine learning techniques can help accelerate simulations and enable more efficient analysis.204,205

Despite the promise and advancements in MD simulation techniques, it is crucial to recognize that these methods must be employed thoughtfully to avoid misleading results or the “garbage in, garbage out” phenomenon.206 In the pursuit of reliable and meaningful results, it is vital to adhere to best practices for simulations and calculations. This includes careful design of the system at the nanoscale, selecting the most appropriate method for the study, utilizing state-of-the-art parameters and accuracy, and ensuring the collection of sufficient statistical data.206 By addressing these considerations and following these guidelines, the full potential of MD simulations for EDLs can be harnessed, leading to a deeper understanding of the complex electrochemical processes underlying electrochemical device performance and contributing to ongoing progress in the electrochemical field.

The authors acknowledge the funding support from the National Natural Science Foundation of China (Grant Nos. T2325012, 52106090 and 52161135104) and the Program for HUST Academic Frontier Youth Team.

The authors have no conflicts to disclose.

Liang Zeng: Writing – original draft (lead); Writing – review & editing (lead). Jiaxing Peng: Writing – original draft (supporting). Jinkai Zhang: Writing – original draft (supporting). Xi Tan: Writing – original draft (supporting). Xiangyu Ji: Writing – original draft (supporting). Shiqi Li: Writing – original draft (supporting). Guang Feng: Conceptualization (lead); Project administration (lead); Supervision (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Perspectives for electrochemical capacitors and related devices
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