Machine Learning Accelerated Finite-Field Simulations for Electrochemical Interfaces

Abstract

Electrochemical interfaces are of fundamental importance in electrocatalysis, batteries, and metal corrosion. Finite-field methods are one of the most reliable approaches for modeling electrochemical interfaces in complete cells under realistic constant-potential conditions. However, previous finite-field studies have been limited to either expensive ab initio molecular dynamics or less accurate classical descriptions of electrodes and electrolytes. To overcome these limitations, we present a machine learning-based finite-field approach that combines two neural network models: one predicts atomic forces under applied electric fields, while the other describes the corresponding charge response. Both models are trained entirely on first-principles data without employing any classical approximations. As a proof-of-concept demonstration in a prototypical Au(100)/NaCl­(aq) system, this approach accelerates fully first-principles finite-field simulations by roughly 4 orders of magnitude compared to ab initio molecular dynamics, allowing the extrapolation to cell potentials beyond the training range and accurate prediction of Helmholtz capacitance. Interestingly, we reveal a turnover of both density and orientation distributions of interfacial water molecules at the anode, arising from competing interactions between the positively charged anode and adsorbed Cl^– ions with water molecules as the applied potential increases. This novel computational scheme shows great promise in efficient first-principles modeling of large-scale electrochemical interfaces under potential control.

1. Introduction

Understanding electrochemical interfaces at the microscopic level is crucial for elucidating fundamental electrochemical processes in electrocatalysis, batteries, and corrosion. , When the electrode potential deviates from the potential of zero charge (PZC), excess charge accumulates on the electrode surface and is counterbalanced by an equal and opposite ionic charge in the electrolyte, leading to the formation of an electric double layer (EDL). , The structure of the EDL, including the distribution and orientation of ions and solvent molecules, determines the electrostatic potential (ESP) profile and the electric field at the interface, thereby significantly influencing the energetics and kinetics of electrochemical reactions. − Various electrochemical characterization techniques have allowed in situ investigations of the species and their compositions within EDLs, which, however, cannot provide more detailed atomistic structural information. Therefore, theoretical, particularly first-principles models are essential to obtain atomic-level insights into EDLs. −

A critical issue in molecular modeling of electrified interfaces is how to control the electrode potential, or equivalently, how to model a voltage across two electrodes. , In general, three different categories of approaches have been developed: grand-canonical, − counterions, ,− and finite-field methods. − The grand-canonical approach directly controls the Fermi energy level and electrode potential by varying the number of electrons. The implicit solvent is often used to maintain electroneutrality of the EDL or, in certain models, such as the constant inner potential method, to avoid polarization of water molecules next to the jellium by the counter charge. The counterions method charges the electrode by introducing hydrogen, alkali metal, or halogen atoms near the interface, but it, in principle, controls the surface charge density only rather than the electrode potential during simulations. Moreover, it is technically challenging to achieve a continuous variation in interfacial charge density because the number of ions can only be changed discretely, making it difficult to precisely tune the electrode potential to desired values. In contrast, finite-field methods, including both constant electric field (E) , and constant electric displacement (D) formulations, have been successfully applied to a wide range of electrode/electrolyte systems. ,,− A major advantage of finite-field methods is ensuring simulations of both electrode interfaces in a full cell at constant potentials. In addition, the charge response at the electrode surface arises from the polarization of free electrons within the metal electrode, naturally satisfying electroneutrality.

Applying finite-field methods to model the EDL of a realistic heterogeneous system needs to overcome the large temporal and spatial scales. For example, the ionic charge distribution within the electrolyte equilibrates in the nanosecond scale, which exceeds the capabilities of ab initio molecular dynamics (AIMD)-based finite-field simulations that typically are limited to 10–100 ps. Classical molecular mechanics (MM) approaches have been employed to characterize intermolecular interactions in bulk electrolytes; however, they are less accurate in describing the electronic structure of metal electrodes and metal–electrolyte interfaces in the presence of electric fields. , An alternative strategy is to employ hybrid quantum mechanical/molecular mechanics (QM/MM) approaches, e.g., by performing QM calculations on the electrode subsystem only. This scheme cannot, however, account for the charge transfer between the electrode and the electrolyte.

Machine learning potentials (MLPs) have been increasingly used in accelerating molecular dynamics (MD) simulations with first-principles accuracy. Several MLPs for simulating electrified interfaces have been combined with the classical Siepmann–Sprik model, in which electrodes follow Gaussian charge distributions with fluctuating magnitudes, while electrolytes are represented by point charge. For example, Zhang and co-workers − employed an atomistic ML approach (PiNN) to compute the charge response kernel and base charge, which were then used to generate response charges at electrode sites and to propagate classical MD simulations. Similarly, Grisafi et al. , developed the symmetry-adapted learning of three-dimensional electron densities (SALTED) approach to predict the electronic density of metal electrodes and charge response under finite-field perturbations so as to compute electrostatic forces that drive the dynamics of the metal–electrolyte interface. In contrast, Zhu and Cheng adopted the Deep Wannier model and classical Siepmann–Sprik model to describe the dielectric response in the electrolyte and metal electrode, respectively, allowing the estimation of long-range electrostatic energy. The remaining short-range energy was separately learned using another local atomic-descriptor-based MLP model. However, these studies retained the classical description of either electrodes or electrolytes.

In this work, we leverage two ML models to accelerate fully first-principles finite-field simulations of electrified interfaces and associated electrochemical properties. This ML-based finite-field approach includes a field-dependent MLP model for the metal–electrolyte interface to enable machine learning molecular dynamics (MLMD) simulations under constant potential and a ML electron density response (MLEDR) model to predict charge transfer both within the electrode and between the electrode and electrolyte at the electrochemical interface under an applied electric field. As a proof-of-concept, this approach is validated in a benchmark system, i.e., the Au(100)/NaCl­(aq) interface. Our ML models accurately capture not only potential-dependent interfacial structures but also the charge response and differential capacitance. This ML scheme offers first-principles consistency for the studied interface as AIMD simulations, while achieving a speedup of approximately 4 orders of magnitude. Furthermore, it eliminates the need for classical description of either the metal or the electrolytes, as both components are represented directly from first-principles data, making it a powerful tool for studying complex electrochemical systems with high efficiency and accuracy.

2. Methods

2.1. Machine Learning Models for Field-Dependent Potential Energy Surfaces and Electron Responses

To characterize the structure and dynamics of electrified interfaces under finite-field conditions, an accurate global potential energy surface (PES) capturing system-field interactions is essential. Although finite-field MD based on classical force fields or density functional theory (DFT) has been proposed, these methods suffer from limited accuracy and low computational efficiency, respectively. To this end, we adopted the field-induced recursively embedded atom neural network (FIREANN) approach, which employs the field-induced embedded atom density (FI-EAD) descriptor to characterize both atomic environments and system-field interactions. The FI-EAD descriptor comprises two componentsGaussian-type orbitals (GTOs) that depend on neighboring atoms and a field-dependent orbital. For a given central atom i, we define a combined orbital ${\psi }_{l_x{l}_y{l}_z}^{(m)}$ as a linear combination of the neighbor-atom GTOs and the field-dependent orbital:

$${\psi }_{l_x{l}_y{l}_z}^{(m)}=\sum _{j\ne i}^{N_{\mathrm{c}}}{c}_j{\phi }_{l_x{l}_y{l}_z}^{(m)}({\mathit{r⃗}}_{ij})+c_{\epsilon }{\phi }_{l_x{l}_y{l}_z}({\mathit{\epsilon ⃗}}_i)$$ 1

where j indexes each neighboring atom of the central atom, (m) denotes the mth GTO, and N _c is the number of neighbor atoms. c _j and c _ε are dimensionless combination coefficients that weigh the contribution of the GTOs and the field-dependent orbital, respectively. The GTO of the jth neighbor atom is defined as follows:

$${\phi }_{l_x{l}_y{l}_z}^{(m)}({\mathit{r⃗}}_{ij})={(x_{ij})}^{l_x}{(y_{ij})}^{l_y}{(z_{ij})}^{l_z}\exp [-\frac{(r_{ij}-r_{\mathrm{m}})}{2{\sigma }^2}]·f_{\mathrm{c}}(r_{ij})$$ 2

where ${\mathit{r⃗}}_{ij}$ , x _ij , y _ij , and z _ij are the Cartesian coordinate vector and its three components of the neighbor atom j relative to the corresponding central atom i, with r _ij being their distance; l = l _x + l _y + l _z specifies the orbital angular momentum; r _m and σ are hyperparameters to determine the center and the width of the radial Gaussian function; f _c(r _ij ) is a cosine-type cutoff function that makes the interaction smoothly decay to zero at the cutoff radius. In addition to the GTOs, we include the field-dependent orbital, which depends only on the components of the applied electric field $\mathit{\epsilon ⃗}=({\epsilon }_x,{\epsilon }_y,{\epsilon }_z)$ .

$${\phi }_{l_x{l}_y{l}_z}(\mathit{\epsilon ⃗})={({\epsilon }_x)}^{l_x}{({\epsilon }_y)}^{l_y}{({\epsilon }_z)}^{l_z}$$ 3

Each FI-EAD feature is formulated as the squared linear combination of the combined orbital ${\psi }_{l_x{l}_y{l}_z}^m$

$${\rho }_i^{(n)}=\sum _{l=0}^L\sum _{l_x,l_y,l_z}^{l_x+l_y+l_z=l}\frac{l!}{l_x!l_y!l_z!}{(\sum _{m=1}^{N_{\phi }}{d}_m^{(n)}{\psi }_{l_x{l}_y{l}_z}^{(m)})}^2$$ 4

Here, L is the maximum orbital angular momentum of primitive GTOs, d _m is the contraction coefficient of the mth combined orbital to generate the nth feature, and N _φ is the number of combined orbitals for a given orbital angular momentum l. By variation of these hyperparameters of GTOs, a vector of FI-EAD features is formed and serves as the input for each atomic neural network (NN) to predict atomic energy. A key advantage of the FI-EAD feature is its ability to implicitly capture three-body interactions with a computational scaling of O­(N _c). Besides, the combination coefficient c _j for the jth neighboring atom is produced by an atomic NN conditioned on the local environment of the jth atom, which can be used to generate new FI-EAD features and establish a message-passing-like iteration. At iteration t, we update the coefficient for neighbor atom j as

$${\mathit{c}}_j^{(t)}={\mathrm{N}\mathrm{N}}_j^{(t-1)}({\mathit{\rho }}_j^{(t-1)}({\mathit{c}}_j^{(t-1)},{\mathit{r⃗}}_j))$$ 5

where NN _j is the jth atomic NN module in the tth iteration and ρ _j is the FI-EAD descriptor of atom j generated in the t – 1st iteration. This form enables the incorporation of higher-order and nonlocal interactions, which are essential for accurately capturing electrostatic interactions at an electrified interface.

Another critical factor in characterizing electrified interfaces is the charge response of the metal electrode in the presence of electrolytes and an applied electric field, which can be predicted using our recently developed MLEDR model for representing the real-space electron density distribution. The architecture of the MLEDR model resembles the FIREANN model for PES but includes an additional component: ghost atoms. These ghost atoms are essentially grid points of electron density distribution obtained by DFT, whose local environments are characterized by the FI-EAD descriptor over surrounding real atoms and are mapped to the charge response value through an atomic NN.

To avoid redundancy, grid points were selected by our recently developed linearly independent feature-based grid sampling scheme. For clarity, we used ρ to denote the FI-EAD features and n to denote the charge density response. Specifically, the learning target of the MLEDR model is defined as

$$\mathrm{\Delta }n(\mathit{r};{\mathit{\epsilon ⃗}}_z)=n_{\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{e}}(\mathit{r};{\mathit{\epsilon ⃗}}_z)-n_{\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}}(\mathit{r})-n_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{e}}(\mathit{r})$$ 6

where $n_{\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{t}\mathrm{e}}(\mathit{r};{\mathit{\epsilon ⃗}}_z)$ is the electron density of the metal–electrolyte interface system at position r with an external field ${\mathit{\epsilon ⃗}}_z$ applied along the z-axis, n _metal( r ) and n _electrolyte( r ) are electron densities in the absence of any applied field of the metal electrode and electrolyte system, respectively. The learned charge response can be then used to calculate the charge density response at the electrode surface and subsequently the differential capacitance. More details about the model architecture, training setup, and hyperparameters are provided in Supporting Information and Table S1.

2.2. Finite-Field Methods, Data Sets, and Simulation Setup

In this work, we employ the finite-field approach introduced by Dufils et al. to set up electrochemical interfaces by imposing a voltage between two electrodes within three-dimensional periodic boundary conditions (PBC), without introducing a vacuum region. As illustrated in Figure , this method applies an electric field to a single electrode maintained at a constant potential and in contact with an aqueous ionic solution. The applied field induces interfacial charging, leading to the formation of two EDLs, one on each side of the electrode. The resulting electrostatic potential profile of the electrochemical interface system is shown in Figure S1. This setup has been successfully applied in simulations of electrochemical interfaces, encompassing both classical and first-principles MD. ,

1.

Schematic of a full cell setup to model a metal–electrolyte interface with the finite-field method, where the Au(100) electrodes are separated by an aqueous NaCl solution of 5.5 M. The applied cell potential is along the z axis given by $\mathrm{\Delta }\mathrm{\Psi }={\mathrm{\epsilon ⃗}}_z·L_z$ , with L _z being the corresponding box length. ρ _i and ρ_g denote the descriptor of central atom i and grid point used in the PES and MLEDR model, respectively.

To train the FIREANN PES and MLEDR models, we constructed a comprehensive data set of finite-field MD configurations for an Au(100)/aqueous NaCl interface. The electrochemical interface system comprised a five-layer Au(100) periodic slab represented by a 4 × 4 supercell containing 80 Au atoms and an adjacent electrolyte region of 25 Å, which included 100 water molecules and 10 NaCl ions pairs (approximately 5.5 M). The initial atomic positions of the electrolyte were randomized using the Packmol package. The system was then equilibrated using a classical constant-potential MD scheme under three applied electric fields: 0.0, 0.029, and 0.057 V/Å along the surface-normal (z-axis), corresponding to cell potentials of 0, 1, and 2 V, respectively. For each electric field, 10 independent AIMD simulations were carried out for 20 ps. Configurations were extracted every 200 fs from each trajectory and added to the initial training set, resulting in 3000 configurations for PES training. An active learning workflow, based on the uncertainty of predicted atomic forces, was employed to progressively sample the configuration space. Specifically, MLMD simulations were run under different conditions to generate many configuration candidates. An uncertainty metric, denoted as F _msd, is defined as the maximum standard deviation of the predictions for all atomic forces by four trial models of a given configuration. A newly sampled configuration was accepted if its F _msd was between 0.2 and 0.4 eV/Å. The lower and upper bounds here were used to excluding redundant configurations already learned and unphysical configurations, respectively. After each iteration of active learning, four trial models were updated with the augmented data set from different initial NN weights and biases. The active learning procedure was terminated until almost no additional configurations could be accepted. This generated additional 9139 configurations, finally yielding 12,139 configurations for training the FIREANN PES and MLEDR models. For all data sets and subsequent descriptor calculations, Å is used for atomic coordinates and interatomic distance, V/Å for electric field strength, e/ų for charge density response, and eV/Å for atomic forces, while meV/Å is used for model prediction error metrics. During all training processes, 90% of the configurations under each applied electric field condition were randomly selected as the training set, while the remaining configurations were held out as the validation set to assess the model’s generalization performance. For training the field-dependent PES with FIREANN, only atomic forces were used. This is because, under periodic boundary conditions with an applied field, the polarization of a periodic system is multivalued and defined only modulo a polarization quantum. Consequently, absolute energies of different configurations can differ by arbitrary constants depending on the polarization branch. Such inconsistencies render total energies unsuitable as training targets for a single-valued PES model. In contrast, atomic forces are uniquely determined by the local environment and are invariant to the chosen polarization branch. The force-trained PES is continuous with respect to atomic forces and suitable for MD simulations. Simultaneously, the field-induced charge response defined in eq was computed for each configuration and used to train the MLEDR model. This dual training approach enabled our ML framework to accurately reproduce both atomic forces and charge transfer behavior under applied electric fields. To further demonstrate the computational efficiency of the FIREANN model, we built a larger electrochemical interface system comprising five layers of Au(100) slabs with an 8 × 8 supercell and an electrolyte containing 800 water molecules and 80 NaCl ion pairs (approximately 5.5 M), totaling 2880 atoms. For the MLEDR model, electron density responses (Δn) were obtained from CP2K in CUBE format on a uniform 3D grid with a spacing of ∼0.16 Å. For the MLEDR model, electron density response was outputted on a three-dimensional grid evenly spaced by 0.16 Å. The data set was split into 90% for training and 10% for validation.

All DFT calculations were carried out using the open source CP2k/Quickstep package with the Perdew–Burke–Ernzerhof (PBE) functional, Goedecker–Teter–Hutter (GTH) pseudopotentials, , and double-ζ basis sets with one set of polarization functions (DZVP). DFT calculations and AIMD simulations were performed using a single Γ-point sampling in the Brillouin zone for computational efficiency since we used a sufficiently large supercell. A plane-wave energy cutoff of 800 Ry was employed, and Grimme’s D3 dispersion correction was applied to capture dispersion interactions. All MLMD simulations based on the FIREANN PES were conducted with large-scale atomic/molecular massively parallel simulator (LAMMPS) in the canonical (NVT) ensemble using a time step of 0.5 fs. Two separate thermostats were applied to the electrode and electrolyte regions. Specifically, we set the electrolyte region to a target temperature of 330 K and the electrode region to 220 K. This approach ensures that both the electrode and the electrolyte maintain the proper temperature distribution throughout the thermalization and production runs, as demonstrated in Figure S2. In each condition, the interfacial system was equilibrated for 1 ns to ensure relaxation of the ion distribution within the electrolyte under the applied finite-field, followed by 5 ns of sampling.

3. Results and Discussion

3.1. Validation of MLPs and MLEDR for Electrochemistry Interface

The accuracy of the FIREANN PES is first assessed by evaluating the root-mean-square error (RMSE) of atomic forces predicted by FIREANN against DFT calculations under finite electric fields. The total energy and the polarization of the periodic system under an external field are not well-defined targets for ML, as the polarization is a multivalued property. Even so, as shown in Figure S3, it can reliably predict energy differences since forces are the gradients of the energy, and the energy difference and relative stability of configurations are still captured accurately, while the absolute values of energy may shift by a constant. As listed in Table S2 of the Supporting Information, the PES model achieves an RMSE for atomic forces of approximately 43 meV/Å, which is comparable to values reported in previous studies, , and the validation data set shows similar RMSEs as the training data set. Further tests indicate that the FIREANN PES model exhibits excellent extrapolation ability in the applied electric field, supercell size, and composition for the Au(100)/NaCl­(aq) interface (Figures S4 and S5). As shown in Figure S6, the MLEDR model reproduces the charge response with low RMSE values of 0.2369, 0.3232, and 0.4210 meV/ų at cell potentials of 0, 1, and 2 V, respectively, demonstrating high fidelity of the predicted charge response. To further assess the generalizability of the model beyond the validation data set, we monitor the electrolyte polarization varying with the cell potential during MD simulations. As shown in Figure , as the cell potential varies from 0 to 4 V, to −4 V, and back to 0 V, by 0.5 V per 0.5 ns, the total electrolyte polarization follows accordingly and sufficiently. The electrolyte polarization consists of two componentsthe orientational relaxation of solvent water molecules and the directional transport of solute ions. Here the latter dominates the polarization response due to the high electrolyte concentration. Importantly, although the training data set includes data points with three discrete cell potentials only, the model accurately covers the electrolyte polarization in the entire range from −4 V to +4 V. This demonstrates both interpolation and extrapolation capabilities of the FIREANN PES, allowing continuous variation of the cell potential in atomistic simulations.

2.

Variation of electrolyte polarization under the varying cell potential for the Au(100)/NaCl­(aq) interface system. The gray area in panel (a) indicates the range of cell potentials included in the training data set of FIREANN model. Panel (b) illustrates two contributions to electrolyte polarization: ion transfer (red) and water molecular orientation (yellow). Panel (c) compares the variation of two components of electrolyte polarizations obtained by FIREANN-based MD and AIMD simulations. FIREANN results were repeated three times to estimate the standard deviations.

Figure c compares the electrolyte polarizations at different potentials obtained from AIMD and MLMD simulations. It should be noted that the typical time scale of AIMD simulations (e.g., 10–100 ps) is insufficient for the electrolyte to fully relax, as shown in Figure S7 of Supporting Information, all these results were obtained by first performing equilibrated MLMD simulations for 1 ns with the FIREANN PES for a given cell potential (in place of the classical MD simulations in literature), followed by 5 ns MLMD simulations and 30 ps AIMD simulations. The total electrolyte polarization and individual contributions from water and ions with MLMD simulations agree well with AIMD results, demonstrating the accuracy of our FIREANN PES. Additionally, the computational efficiency of the FIREANN PES model is evaluated by comparing the performance of MLMD to AIMD simulations. Specifically, for this electrochemical interface system with 400 atoms, MLMD achieved a speed of 7.9 × 10^–4 s/atom/CPU per MD step in comparison with 8.5 s/atom/CPU per MD step for AIMD. This represents a speedup of approximately 4 orders of magnitude by the MLMD approach over AIMD.

3.2. Differential Helmholtz Capacitance

The surface excess charge density on the electrode directly determines the strength of the interfacial electric field, the orientation of interfacial water molecules, and the hydration states of the ions. In finite-field methods, the surface excess charge density is induced by the applied electric field and depends on the electrolyte configuration, while the average charge response correlates with the cell potential. The Helmholtz capacitance, which characterizes the surface charge response of an EDL system as a function of the cell potential, can be computed using the following formula

$$C_{\mathrm{H}}=2·\left(\frac{\mathrm{d}{\sigma }_{\mathrm{m}}}{\mathrm{d}\mathrm{\Delta }\mathrm{\Psi }}\right)$$ 7

where ΔΨ is the sum of the potential differences across both sides of the metal electrode and σ_m is the surface charge response density imposed on the metal electrode surface. The factor of 2 in eq arises because ΔΨ represents the total potential difference across both double layers at two metal–electrolyte interfacesi.e., the potential difference between the two Helmholtz layers on either side of the metal electrode. A total of 2000 configurations were extracted from MLMD trajectories with 10 ns for each cell potential. The corresponding charge responses were predicted by using the MLEDR model. Afterward, the interfacial charge response density is obtained by integrating the charge response from the central layer of the metal slab toward the left or right electrolyte region and averaging over an ensemble of configurations in the MLMD simulation for each cell potential. The capacitance is then obtained by differentiating the surface charge response density with respect to the cell potential. As shown in Figure a, charge transfer from the electrolyte to the electrode occurs at the cell potential of 0 V. Further charge analysis reveals that this transfer originates from chloride ions (Cl^–) specific adsorbed on the electrode surface. As the applied electric field increases, charge transfer occurs within the electrode, resulting in the formation of two electrified interfaces. As illustrated in Figure b, the Helmholtz capacitance is calculated with a maximum value of approximately 20.8 μF/cm² at the cell potential of 0 V. Our result is consistent with the prediction of Guo et al. in similar Au(100)/electrolyte systems based on an ML accelerated counterions approach, with comparable capacitances of 17–19 μF/cm². However, additional DFT single-point calculations are required for postprocessing to estimate averaged work function and surface charge density in that work. The computed capacitance here is within the same order of magnitude but lower by a factor of 2–3 compared to the experimental values that were typically reported for Au(100), along with other electrolytes like NaClO₄, which range from 40–65 μF/cm² under comparable surface charge densities. This discrepancy may be attributed to the higher electrolyte concentration (5.5 M NaCl) in our simulation than in experiments and the different electrolyte used in experiments, as well as potential errors of the PBE-D3 functional used for generating data set. It is worth noting that changing the sign of the cell potential reflects the change of applied field direction, which changes the sides of the anode and cathode but would not change the overall dynamics. This feature is fully captured by our FIREANN PES. Hence, only positive cell potentials are discussed hereafter unless otherwise stated.

3.

(a) Response of charge density averaged over the xy plane and (b) the cell potential-dependent differential capacitance of the Au(100)/NaCl­(aq) interface with the salt concentration of 5.5 M. The gray area in all panel indicates the Au electrode.

3.3. Structure of Au(100)/NaCl­(aq) Interface

The structure and composition of EDLs play a critical role in determining the activity and selectivity of catalytic reactions, including nitrogen reduction and oxidation, , hydrogen/oxygen evolution reaction, − water oxidation, and electro/photochemical CO and CO₂ reduction reactions (CO₍₂₎RR). − Atomic structures of the electrified Au(100)/NaCl­(aq) interface obtained from 0 to 4.0 V were analyzed from above MLMD trajectories during 5 ns. This extended timescale enhances the statistical reliability and enables the calculation of the smooth potential of mean force (PMF).

Figure a,b shows the equilibrium ion concentration profiles varying with the applied potential. As the cell potential increases, corresponding to an increase in the excess surface charge density on the electrode, oppositely charged ions accumulate near the electrode surface. This behavior is consistent with the expected ion–electrode Coulomb interactions. An interesting phenomenon is that the peak of the Na⁺ concentration near the anodic surface hardly moves as the cell potential increases. This counterintuitive trend can be attributed to the rapid adsorption of Cl^– onto the anodic surface, resulting in local charge inversion, in agreement with previous findings. Furthermore, the preferential adsorption of Cl^– over Na⁺ on the electrode surface can be rationalized by the weaker solvation of larger Cl^– ions by water molecules, making them more prone to direct surface adsorption. In contrast, Na⁺ ions, being smaller and more strongly solvated, experience greater repulsive solvation forces. The concentration profile of Na⁺ exhibits multiple peaks on the cathodic surface when the cell potential is 3 or 4 V. This is likely due to the strong electrostatic interactions at high cell potentials, disrupting the Na⁺ hydration layer, thereby allowing a fraction of Na⁺ ions to approach the cathodic surface more closely. Additionally, the concentration distributions obtained from MLMD simulations are much less oscillatory than those from AIMD in Figure S8, owing to an over 100 times longer simulation time. Indeed, AIMD with insufficient relaxation time leads to some artificial peaks of the concentration profiles, particularly in the bulk region, underscoring the advantage of MLMD simulations in analyzing atomic structures of electrode–electrolyte interfaces.

4.

Concentration distributions and PMFs of cations (Na⁺) and anions (Cl^–) as a function of the distance to the electrode under various cell potentials obtained from FIREANN simulations. The gray area in each panel represents the Au electrode being positively (anode) or negatively (cathode) charged on the respective side.

The PMFs for the approach of ions to the electrode surface are directly related to the equilibrium ion concentration profiles in Figure a,b through the following relation

$$\mathrm{P}\mathrm{M}\mathrm{F}(z)=-RT\ln \frac{c(z)}{c(z_0)}$$ 8

where PMF­(z) denotes the free energy of ions at position z relative to that at the bulk (defined by the midpoint of the electrolyte layer at z ₀) along the direction normal to the interface. As shown in Figure c,d, the electrostatic repulsion drives Cl^– ions away from the negatively charged electrode, while Na⁺ ions are increasingly attracted toward it with increasing cell potential. A notable feature is the presence of an energy barrier for Cl^– adsorption on the anode with a potential-dependent height of approximately 10 kJ/mol. This barrier arises from the high surface coverage of Cl^–, which effectively screens the interaction between the positively charged electrode and Na⁺ ions. As a result, the free energy profiles of Na⁺ in the anodic region remain relatively unchanged across different cell potentials.

Next, we analyze the structural distributions of water molecules at the electrified Au(100)/NaCl­(aq) interface. Figure a presents the water density distributions at different cell potentials, which consistently exhibit a pronounced peak approximately 3.1 Å from the electrode surface. This indicates a well-defined first solvation layer forms, defined as the set of water molecules within the first pronounced oxygen density peak adjacent to the metal electrode, extending up to the first minimum in the density profile. Figure b illustrates the dipole orientation profiles, defined as ${\rho }_{{\mathrm{H}}_2\mathrm{O}}·\cos \theta$ , where θ is the angle between the bisector of ∠HOH and the positive z-axis. With zero cell potential, most interfacial water molecules adapt themselves so that the O–H bond points toward the electrode surface. This is because of the hydration of Cl^– ions that are directly adsorbed onto the electrode. As the cell potential increases, the orientation of the water dipole becomes progressively more ordered, following the direction of the electric field.

5.

(a) Density and (b) dipole orientation distributions of water molecular along the surface normal, within the Au electrode (gray plates) cathode and anode. Normalized orientational distributions (θ) for water molecules in the cathodic (c) and anodic (d) interface obtained from FIREANN simulations, where the insets show the dominant water orientation at different peaks.

More specifically, Figure c,d displays the angular distributions for interfacial water molecules located within 4 Å of the cathode and anode surface, respectively. Interestingly, water molecules near the cathodic interface exhibit a pronounced peak at approximately 45°, corresponding to an orientation with one O–H bond pointing toward the electrode, while the other O–H bond lies nearly parallel to the interface, as shown in the inset of Figure c. This configuration allows the parallel O–H bond to participate in hydrogen bonding with surrounding water molecules in the same interfacial layer and facilitating the formation of an extended hydrogen-bonding network, thereby enhancing interfacial structural stability. , Furthermore, this effect strengthens with increasing cell potential, leading to an increasingly sharper angular distribution. In contrast, charge inversion occurs in the anodic interface due to the excess adsorption of Cl^– ions on the positively charged electrode. These anions tend to attract one of the hydrogen atoms in water molecules toward the electrode, yielding dominant angular distributions around ∼135°, consistent with the previous finding. At higher cell potentials (e.g., 3–4 V), the stronger charge response in the electrode renders more positive charge concentrated on the anodic surface, manifesting stronger interactions with interfacial water molecules, while the Cl^– coverage is limited by its chemical potential and cannot increase proportionally, as seen in Figure b and representative snapshots in Figure S9. As a result, the water molecule tends to reorient and allow its negatively charged center (i.e., oxygen atom) toward the anode, resulting in the significant shift of the angular distribution to smaller angles. More detailed tests show in Figure S10 that this shift starts at a cell potential of approximately 2.5 V.

AIMD simulations yield similar distributions as discussed above (Figure S8) but with much higher computational costs. Moreover, the well-trained FIREANN potential allows us to study the structure and dynamics of the Au(100)/NaCl­(aq) interface with a more extended cell. To this end, we performed MLMD simulations using a cell setup with 5 metallic layers and 64 atoms in each layer of the electrode, plus a thicker electrolyte layer with 50 Å, consisting of 2880 atoms. The corresponding distributions of ions and water molecules under various cell potentials are presented in Figure S11, which compare well with those obtained in the smaller cell for training. These results clearly demonstrate the scalability of our FIREANN model to more complex systems than those trained.

4. Conclusions

In this work, we propose an efficient strategy to accelerate finite-field simulations of electrochemical interfaces by leveraging machine learning techniques. A key feature of this strategy is using a field-induced (nonlocal) message-passing neural network framework to learn both atomic forces and charge responses of the metal–electrolyte system computed by density functional theory in the presence of external electric fields, thus allowing highly efficient machine learning molecular dynamics simulations and the evaluation of Helmholtz capacitance under various constant cell potentials. We validate this approach in a benchmark Au(100)/NaCl­(aq) interfacial system, demonstrating its exceptional accuracy and superior computational efficiencyboth in time and in system size scalabilitycompared to conventional finite-field ab initio molecular dynamics simulations. Furthermore, this approach exhibits excellent extrapolation capabilities beyond the range of cell potentials included in the training data, allowing us to identify a turnover in the orientation distribution of interfacial water molecules at the anodic interface at high cell potentials. This reorientation is because the positively charged anode interacts more strongly with water molecules than adsorbed Cl^– anions as the applied potential increases. The proposed machine learning strategy offers broad potential for efficient modeling large-scale electrochemical interfaces from first-principles under applied voltage control. It is particularly well suited for investigating metal–electrolyte interfacial structures and electrocatalytic reactions, especially when combined with enhanced sampling techniques to explore the associated free energy landscapes.

The original FIREANN package is available in the main branch of the FIREANN repository at https://github.com/bjiangch/FIREANN. The FIREANN code for electron response is a branch of the original package in the “FIREANN for Density and Response” branch at https://github.com/bjiangch/FIREANN/tree/FIREANN-for-Density-and-Response. A README file indicating a clear entry, along with a “quick start” example and step-by-step instructions, as well as the atomic forces data set for PES model training, are provided in the “FIREANN for Density and Response” branch. Due to the overly large size of the data set for charge response, the CP2K input parameters necessary for generating the charge response for the electrochemical interface systems are provided.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.5c00792.

• Additional information for the performance of the FIREANN model under different cell potentials; variation of electrolyte polarization from MLMD and AIMD; snapshot in the finite-field MLMD trajectories under varying cell potential; detailed results on the dipole orientation of anodic interface water under varying cell potentials; concentration distributions of ions and density and dipole orientation distributions of water molecules from AIMD; and concentration profiles of ions, density, and dipole orientation distributions of water molecules in the more extended cell (PDF)

The authors declare no competing financial interest.