The dielectric constant, although a simplified concept when considering atomic scales, enters many mean-field, electrochemical interface models and constant potential models as an important parameter. Here, we use ab initio and machine-learned molecular dynamics to scrutinize the behavior of the electronic contribution to ɛr(z) as a function of distance z from a Pt(111) surface. We show that the resulting dielectric profile can largely be explained as a sum of the metallic response and the density-scaled water response at the interface. A slight enhancement of the dielectric response close to the surface can be explained by elongated, strongly polarizable orbitals induced by metal/water bonding. In spite of this enhancement, our results suggest the presence of a region with a very low dielectric constant close to the surface (where the orientational dielectric response does not kick in yet), even for water in contact with hydrophilic metallic interfaces. This region will restrict the double layer capacitance to relatively low values even at potentials where dielectric saturation does not play a role yet. This finding has implications on possible interpretations of double layer capacitances, the dependence of surface electric fields on the ion size, and on electrochemical kinetics.
I. INTRODUCTION
The electric double layer (EDL) is one of the most fundamental building blocks in electrochemistry.1,2 The structure that solvent molecules and ions form at the EDL and the resulting potential profile at the interface have considerable influence on the energetics and kinetics of electrochemical reactions. The structure of the EDL and the interfacial potential profile have, hence, attracted great interest since the early stage of electrochemistry.
Often, it is mainly the structure of ions that is of interest. For example, the effect of ion type and concentration on electrochemical reactions is vividly discussed in the literature, in particular with respect to CO2 reduction,3–6 but also for hydrogen evolution.5–10 In other cases, the potential profile will be more relevant than the exact structure of the interface. This is the case, for example, when determining Frumkin corrections to reaction rates.11,12 A correct representation of the potential profile can also be expected to play a crucial role in constant potential methods, as developed in recent years to allow for ab initio calculations under applied bias in computational electrochemistry.13–16
Constant potential methods often make use of mean-field models of the electrolyte. These models are, however, highly sensitive to their parameterization.17,18 One of the most important factors determining the potential profile at the interface (and thus also the interfacial capacitance) is the dielectric constant, which describes the response of the solvent to the electric field at the interface.19–22
Determining the dielectric constant at the interface is challenging (in fact, it is even questionable whether the dielectric constant, which is a mean-field concept, is still a valid concept at the atomic scale in heterogeneous systems such as the solid/liquid interface23). Educated guesses and approximations are, therefore, inevitable, leading to interfacial mean-field models in which the dielectric constant differs drastically (see, e.g., Table 2 of Ref. 18).
In one of the most simplified descriptions, the dielectric constant ɛr is assumed to be constant and around six in the compact layer24–27 and takes the value of bulk water (i.e., ∼7828) for the remaining regions. A dielectric constant around six in the compact layer can be justified as follows:
When assuming a compact layer with a width of d ≈3 Å and ɛr = 6, one obtains a capacitance of C = ɛ0ɛr/d ≈ 20 μF/cm2—a value consistent with experimental results for many metal/electrolyte interfaces under conditions of high electrolyte concentrations and far from the potential of zero charge (pzc).2,29
ɛr = 6 corresponds to the dielectric response of water once the rotational degrees of freedom are frozen.30 Considering that molecule–surface interactions and dielectric saturation effects (i.e., all water molecules being preferentially aligned to the surface electric field already) likely play a role at the interface between a charged electrode and an aqueous electrolyte, a value of ɛr = 6 thus seems realistic.
Considering the strongly layered structure of the interface, a dielectric constant that is constant in the entire compact layer is, however, likely an oversimplification.
A better knowledge of the dielectric profile, i.e., how ɛr varies in space close to the interface, is needed, in particular as the profile of ɛr can influence the predictions and interpretations derived from mean-field models and experimental observations.
Currently, most studies targeting the dielectric profile at the interface are based on force field molecular dynamics,31–34 which suffer not only from errors stemming from force field parameterization35 (in particular at the interface, where good force fields have not been developed) but also from an approximate description of electron polarization at the interface.36 To overcome these issues, ab initio molecular dynamics (AIMD) simulations have recently been utilized to compute the dielectric profile.37 Despite the elegant implementation of a thermopotentiostat, this density functional theory (DFT)-based study suffers from two issues: First, the dielectric constant was extracted using a finite difference method. However, the cathode and anode show distinct dielectric profiles in these calculations, indicating conditions deviating from the pzc, without the exact potential being known. Second, the simulations were performed on Ne-like electrodes, which is little representative of realistic electrochemical interfaces.
In this work, we address the dielectric profile at the Pt(111)/water interface at the pzc. To do so, we make use of machine-learned potentials (MLPs) and extend the MD simulation timescale to a few nanoseconds. The dielectric response at the interface is further split into an electronic contribution and an orientational contribution. While the former is calculated by applying an electric field on the DFT level, the latter is approximated from the polarization fluctuation based on empirical atomic charges. Importantly, we find that even on a hydrophilic metallic surface such as Pt, the electronic dielectric response is low in the region outside the image charge plane, but below the first water region, suggesting the presence of a (narrow) “gap” region with a low dielectric constant as often suggested for graphite38 and in agreement with Ref. 39. Furthermore, we explain the electronic polarizability at the surface by combining a density scaled electronic polarizability of water with the electronic polarizability of the metal and accounting for a change in the polarizability of surface-near water due to a change in the shape (and polarizability) of molecular orbitals.
II. METHODS
A. Molecular dynamics simulations
The Pt(111)/H2O interface was modeled using a 3D periodic supercell containing two Pt(111)/water interfaces as shown in Fig. 1(a). A six atomic layer thick Pt(111) slab was used, with a surface unit cell of p(6 × 6) atoms. The Pt slabs were separated by 30 Å of water with 230 water molecules, leading to a water density of 1 ± 0.05 g/cm3 far away from the Pt surfaces (i.e., at z ≥ 9 Å, where z is the distance from the uppermost Pt layers) after equilibration.
Illustration of atomic models used. (a) 3D periodic supercell used in molecular dynamics simulations with machine-learned potentials and when computing the orientational dielectric constant. The red frame highlights atoms extracted when computing the electronic dielectric constant. (b) Setup used when computing the electronic dielectric constant. Dirichlet boundary conditions (DBCs) are applied in the z-direction as indicated by the charged plates and the external circuit. Red and blue lines: Hartree potential profiles V for two different DBCs. The slopes dV/dz in vacuum (highlighted by a black frame) provide .
Illustration of atomic models used. (a) 3D periodic supercell used in molecular dynamics simulations with machine-learned potentials and when computing the orientational dielectric constant. The red frame highlights atoms extracted when computing the electronic dielectric constant. (b) Setup used when computing the electronic dielectric constant. Dirichlet boundary conditions (DBCs) are applied in the z-direction as indicated by the charged plates and the external circuit. Red and blue lines: Hartree potential profiles V for two different DBCs. The slopes dV/dz in vacuum (highlighted by a black frame) provide .
Using a machine-learned potential (MLP; obtained as described in Sec. II B), molecular dynamics (MD) simulations were performed in the CP2K software package.40 The canonical (NVT) ensemble was approximated via the Nosé–Hoover thermostat with a chain length of 3. The temperature was set to 330 K, and the time step used in the propagation was set to 0.5 fs. During the MD simulations, the innermost two Pt layers are restrained by weak harmonic springs with a spring constant of 0.1 a.u. to ensure bulk properties.
B. Training and validation of the machine-learned potential
The MLP used in this work was trained with the DeePMD-kit software package.41 Training data for the MLP were obtained from density functional theory (DFT) calculations performed in the CP2K software package.40 The computational setup used in the DFT calculations is similar to that used in Ref. 42. In particular, PBE-D3 was used as exchange–correlation functional,43,44 the ionic cores were described by Goedecker–Teter–Hutter (GTH) pseudo-potentials with 1, 6, and 11 valence electrons for H, O, and Pt, respectively, and a double-ζ basis set with one set of polarization functions was employed. The plane wave energy cutoff was set to a high value of 1000 Ry to reduce the noise in the data and improve the fitting accuracy in the MLP training. All DFT calculations were based on calculations at the gamma point only.
A concurrent learning workflow DP-GEN was used to efficiently collect a training dataset that is fully representative for the configuration space. Each iteration of the concurrent learning workflow consists of a series of (i) training, (ii) exploration, and (iii) labeling.45
In the training step, a committee of four MLPs was trained based on the energies and forces. The Deep Potential Smooth Edition (DeepPot-SE) descriptor46 was chosen for the MLPs, and the cutoff was set to 6 Å. The number of neurons in each hidden layer of the embedding net and the fitting net was [25, 50, 100] and [240, 240, 240], respectively. The MLPs were trained in 2 × 105 steps, with each step using a batch size of 1. During learning, the learning rate was reduced exponentially from 5 × 10−3 to 1 × 10−8, where the learning rate was updated every 2000 steps. Both energies and atomic forces were included in the loss function for model optimization. The weight for energies in the loss function started from 0.02 and ended at 1, while that for atomic forces started from 1000 and ended at 1. In the first iteration, the training data were Pt(111)/water interfaces extracted from the AIMD trajectory from Ref. 42 (using a snapshot every 50 fs), Pt(111) slab and bulk water.
In the exploration step, new configurations were generated by performing MD simulations based on one of the MLPs trained in the previous step, using the atomic model of the Pt(111)/water interface described in Sec. II A. In each exploration step, MD simulations with various initial configurations were performed at various temperatures. Details on these MD simulations are given in the supplementary material, Sec. S2. Considering that MLPs are expected to be good at interpolating but poor at extrapolating, the deviations in the results predicted by the four MLPs trained in the previous step can be used to estimate in how far a given configuration is already represented by the existing training dataset. Configurations for which the deviations in atomic forces lay between 0.12 and 0.25 eV/Å were selected as candidates to form additional training data. In each iteration, a certain number of these candidate configurations (150 for the first 16 iterations and 100 for the remaining iterations) was chosen randomly from the candidate structures and added to the training dataset.
In the labeling step, DFT calculations were performed for the selected configurations from the exploration step with the computational setup specified above.
After 24 iterations of the concurrent learning workflow, the MLP trained on the collected dataset showed good accuracy and generalizability. A final training with 2 × 106 training steps (instead of 2 × 105 as used during training) was performed. Note that the initial training dataset obtained from AIMD trajectories was discarded after the first few training iterations. The MLP obtained in the final training step shows root mean square errors (RMSEs) of 0.43 meV/atom and 58.73 meV/Å for energies and atomic forces, respectively. Moreover, some important thermodynamic and dynamic properties of interfacial water were described well as detailed in the supplementary material, Sec. S1.
C. Electronic dielectric constant calculation
While the MLP used in this work enables the simulation up to nanoseconds, only energies and atomic forces can be predicted. To compute the electronic dielectric constant, we perform DFT calculations for several snapshots from the molecular dynamics trajectory to determine the density response to a small, perturbing electric field.23 By averaging over the density response (or equivalently the inverse dielectric constant) for several snapshots, a converged dielectric profile can be obtained, as described below.
Without loss of generality, we will formulate the calculation in the z direction, which is the direction parallel to the surface normal in our case.
For this approach to work, we still require well-defined boundary conditions for Eq. (3). These can be obtained either by using an approach based on Wannier centers and the Berry-phase polarization as sketched in Ref. 23, or by introducing a vacuum layer and setting P to zero in a region where ρ = 0 (i.e., in the vacuum region). Introducing a vacuum layer in the center of the Pt slab is no option, as the applied field would then be completely shielded in the water region by the two metal electrodes. Therefore, half-cell configurations with only one metal–water interface are extracted from various snapshots of the MD trajectory as follows: All Pt atoms and all water molecules residing at z ≤ 13 Å from the Pt surface are kept, and a vacuum layer of 30 Å is inserted, leading to cells as shown in Fig. 1(b). Based on the test shown in the supplementary material, Sec. S3, a water thickness of 13 Å is sufficient to eliminate the influence of the water–vacuum interface on the water–metal interface up to a few Angstrom away from the surface. In the resulting half-cell configurations, an external field (or ) can be applied indirectly in CP2K by using the implicit (generalized) Poisson solver and specifying Dirichlet boundary conditions (DBCs).47
Since we are interested in the electronic dielectric constant under conditions of zero charge, it may seem appropriate to set to zero. However, we deviate from this choice to account for the specific computational setup used here and to avoid unphysical band misalignment issues. The way in which this is done is explained and rationalized in detail in Sec. S4 of the supplementary material. Since it is possible to obtain a linear behavior in once the band alignment issues are accounted for [see Fig. 2(a) and Fig. S13 of the supplementary material], we believe the deviation from to be of little physical significance and the extracted electronic dielectric constant to represent the electronic dielectric constant at the pzc well.
(a) Charge density difference Δρ for a single snapshot obtained when changing the external field from to , where the perturbing external field is 0.09 V/Å. The strong independence of polarization charges Δρ on the value of suggests a linear response behavior. The blue solid and red dashed lines, denoting different values of applied, overlap. (b) Inverse electronic dielectric profile obtained from the polarization charges and averaging over several snapshots. z coordinates are given relative to the average position of the outermost Pt atoms at the metal–water interface.
(a) Charge density difference Δρ for a single snapshot obtained when changing the external field from to , where the perturbing external field is 0.09 V/Å. The strong independence of polarization charges Δρ on the value of suggests a linear response behavior. The blue solid and red dashed lines, denoting different values of applied, overlap. (b) Inverse electronic dielectric profile obtained from the polarization charges and averaging over several snapshots. z coordinates are given relative to the average position of the outermost Pt atoms at the metal–water interface.
III. RESULTS
A. Electronic dielectric response at the Pt(111)/water interface
To obtain the electronic polarizability of the Pt(111)/water interface, we proceed as detailed in Sec. II C. The polarization charge profile obtained for a representative snapshot is shown in Fig. 2(a). As expected, large polarization charges are observed at the metal/vacuum interface. These polarization charges shield the electric field inside the metal. The shielding does not occur at the center of the outermost Pt atoms but slightly beyond the edge, as expected due to the electron spillover at metal surfaces.48,49 At the metal/water interface, the shielding is partially lifted, as evidenced by a negative peak with an integrated total charge slightly smaller than that at the metal/vacuum interface. In the water region, the polarization charges fluctuate. These fluctuations reflect the layering of the water molecules and electronic polarization occurring within each layer. A negative peak in the polarization charges at the water/vacuum interface indicates that the polarization is lifted there.
The inverse electronic dielectric profile can be obtained from the polarization charges via Eq. (5). By averaging over 34 snapshots separated by 100 ps, a converged profile is obtained as shown in Fig. 2(b). Convergence tests can be found in Sec. S5 of the supplementary material. As expected, the inverse of the electronic dielectric constant is close to zero within the metal slab, indicating nearly perfect conductivity. Interestingly, a negative inverse dielectric constant is observed at the metal edges. Such a negative dielectric response has been observed earlier and can be rationalized as an indication of Le Châtelier’s principle: Positive polarization charges increase the electrostatic potential within metal with respect to vacuum. At the same time, the electron spillover will decrease, effectively moving the image charge plane inward as the surface charge increases (and outward as the surface charge becomes increasingly negative). This effect is captured in a negative polarizability in the region in which the image charge plane shifts.50
In the water region, the electronic dielectric constant rapidly decays to a value close to the experimental optical dielectric constant of water .30 This is expected as we consider only the electronic response here (the nuclei are fixed in each snapshot and cannot contribute to the dielectric constant as extracted from applying an external field).
(a) Atom density-derived electron density profile of water relative to the “bulk” value as approximated by the density between 5 and 10 Å (blue shaded region). (b) Electronic dielectric profile as obtained from the polarization charges and averaging over several snapshots for the metal/water (blue line) and metal/vacuum (red line) interfaces. Yellow line: dielectric profile expected for the structured water at the interface only [Eq. (6)]. Green line: idealized dielectric profile expected for a “non-interacting” metal/water interface [Eq. (7)]. The z-coordinates are relative to the average positions of the outermost Pt atoms at the metal–water interfaces.
(a) Atom density-derived electron density profile of water relative to the “bulk” value as approximated by the density between 5 and 10 Å (blue shaded region). (b) Electronic dielectric profile as obtained from the polarization charges and averaging over several snapshots for the metal/water (blue line) and metal/vacuum (red line) interfaces. Yellow line: dielectric profile expected for the structured water at the interface only [Eq. (6)]. Green line: idealized dielectric profile expected for a “non-interacting” metal/water interface [Eq. (7)]. The z-coordinates are relative to the average positions of the outermost Pt atoms at the metal–water interfaces.
As shown in Fig. 3(b), the density-scaled water electronic dielectric constant (yellow dashed line) captures the dielectric constant (blue solid line) well for z ≳ 3.7 Å. This suggests that the electronic dielectric response in this region is still dominated by the “pure” water response and is only modulated by the water structuring near the interface. As we move closer to the interface, we expect the dielectric response of the metal to play a role as well.
To visualize the effect of electron redistribution at the metal/water interface, we show in Fig. 4 the molecular orbitals calculated for water monomers in contact with Pt(111) slabs in typical “H-down” (physisorbed) and “O-down” (chemisorbed) configurations.51 Compared to their gas phase counterparts, the water molecules in contact with metal clearly show elongated molecular orbitals. This will not only enhance the electron density in the interfacial region, but the more diffuse orbitals can also be expected to enhance the polarizability of the molecule.52
Valence orbital of a “chemisorbed” (H-up) (a) and a “physisorbed” (H-down) (b) water molecule near the surface (blue) compared with the orbital that one would find for a molecule in the same geometry in gas phase (orange). The isosurfaces are illustrated with an isovalue of 0.01 e/bohr.3
Valence orbital of a “chemisorbed” (H-up) (a) and a “physisorbed” (H-down) (b) water molecule near the surface (blue) compared with the orbital that one would find for a molecule in the same geometry in gas phase (orange). The isosurfaces are illustrated with an isovalue of 0.01 e/bohr.3
Orbital polarizability as defined in Eq. (8) for single water molecules above the surface as a function of metal–water distances. “Chemisorbed” refers to water molecules in H-up geometry, and “physisorbed” refers to water molecules in H-down geometries. In both cases, the oxygen atoms are located at the top sites of metal slab.
Orbital polarizability as defined in Eq. (8) for single water molecules above the surface as a function of metal–water distances. “Chemisorbed” refers to water molecules in H-up geometry, and “physisorbed” refers to water molecules in H-down geometries. In both cases, the oxygen atoms are located at the top sites of metal slab.
Overall, the electronic response at the interface can thus be considered to be well described by the summation of effects from the idealized metal and idealized water, with deviations due to more diffuse and polarizable orbitals in the contact region between metal and water (i.e., 1 Å ≲ z ≲3.7 Å). The influence of more polarizable orbitals is relatively small though, enhancing the dielectric constant only by about 25%. Importantly, the electronic dielectric response ɛelec at the interface remains lower than ɛelec < 2.8 (i.e., only one point higher than in bulk solution) as soon as the response of the bare metal response has decayed for z ≳ 1 Å. This is interesting insofar as we do not expect the orientational and ionic dielectric response to contribute to the overall dielectric constant for z-values below 1.5 Å (see supporting information, Sec. S7, for more details). This suggests the presence of a “gap” region between the metal and the water in which the total dielectric constant is low (much lower than the average dielectric constant expected in the Helmholtz layer of 6), confining the majority of the potential drop to this region. To underline this statement, we also compute the orientational dielectric constant. As discussed in Sec. S6 of the supplementary material and shown in Fig. S14, the orientational dielectric response can indeed be expected to kick in only for z ≳ 1.5 Å from the surface, leaving a “gap” region between z ≈ 1 Å and z ≈ 1.5 Å in which the dielectric constant should not be much larger than 2.8. Note that we refrain from trying to add the orientational and electronic contributions. Adding dielectric constants (or polarizabilities, to be more exact) is not straightforward if the polarizabilities are not constant in space, as shown in Ref. 53. The authors are not aware of any method that would allow adding contributions if one of them (the electronic part in our case) contains regions with metallic behavior. The argument given above about the orientational dielectric response only kicking in at larger distances should, therefore, be seen as a qualitative statement.
IV. DISCUSSION
A. From dielectric profile to double layer capacitance
Average capacitance up to a certain distance z from the metal surface. Up to z = 1.5 Å (blue line), for C1(z), only the electronic response is taken into account. For C2, a larger dielectric constant is considered to account for the electronic, orientational, and ionic response. Red dashed line: using ɛr = 80 in C2; red dotted line: using ɛr = 6. The gray shade indicates the region between the metal surface and the image charge plane. The blue shade indicates the water density distribution in the z-direction.
Average capacitance up to a certain distance z from the metal surface. Up to z = 1.5 Å (blue line), for C1(z), only the electronic response is taken into account. For C2, a larger dielectric constant is considered to account for the electronic, orientational, and ionic response. Red dashed line: using ɛr = 80 in C2; red dotted line: using ɛr = 6. The gray shade indicates the region between the metal surface and the image charge plane. The blue shade indicates the water density distribution in the z-direction.
B. From dielectric response to electrocatalysis
To further emphasize the potential relevance of a region with a low dielectric response on electrocatalysis, we discuss two case studies: (i) the expected change in surface electric field with changing cation size and (ii) the influence of the potential profile on electrochemical kinetics. Care should be taken not to over-interpret the qualitative discussions as, likely, the presence of reactants at the interface will influence the dielectric profile and as a reaction may “feel” a local, non-time averaged dielectric response rather than the time-averaged response.60 Nevertheless, the qualitative picture sketched in the following should hold.
1. Ion size effect on the surface electric field
The surface electric field can play a crucial role in electrochemical reactions, as it can stabilize dipolar (adsorbed) species, such as adsorbed *CO2—a crucial intermediate in CO2 reduction.61 Ringe et al. invoked this idea for example to explain the dependence of the CO2 reduction rate on the ionic species in the electrolyte.3 The idea behind this ion size effect is that different ions approach the surface more or less closely, leading to differences in the double-layer capacitance and, hence, differences in the surface electric field at a given potential. The extent to which different ion sizes affect the double-layer capacitance, however, depends sensitively on the dielectric profile of the interface as illustrated in Fig. 7. When ɛr is constant at ɛr = 5, a change in (hydrated) ion radius (or ion position relative to the image charge plane) from r = 2.3 to r = 3 would change the capacitance by 27% from 20 to 15 μF cm2. Meanwhile, if ɛr changes from ɛr = 3 for z ∈ [1, 2] Å to ɛr = 11 thereafter, the same change in ion position would only cause a 13% decrease in the capacitance. As the surface electric field at constant potential is proportional to the capacitance, this suggests a much smaller (relative) change in the surface electric field in the latter case. Considering the low polarizability that we find in the region of 1 Å to ≥ 1.5 Å, one may thus expect the ion size to have a smaller effect on the interfacial electric field than naively expected.
Sketch of two interfaces, one with a constant ɛr (a) and the other with a region of low dielectric constant close to the interface and a higher ɛr thereafter (b). Lower panels: sketch of the resulting potential profile.
Sketch of two interfaces, one with a constant ɛr (a) and the other with a region of low dielectric constant close to the interface and a higher ɛr thereafter (b). Lower panels: sketch of the resulting potential profile.
2. Potential profile and electrochemical reaction kinetics
Electrochemical reaction kinetics can be expected to depend on the potential profile. This is captured, for example, in the Frumkin correction to the Butler–Volmer equation.11,12 Similarly, one may expect transition state energies of charge transfer reactions to depend on the potential present at the position of the center of charge of the transition state. This potential will, however, depend on the dielectric profile, as depicted in Fig. 7. When the double layer is marked by a “gap” region close to the interface, the potential increases rapidly to the applied potentials when moving away from the interface. For a constant ɛr, the potential continues to increase considerably over a much wider range of distances from the surface, leading to a different Frumkin correction.
Taken together, this showcases the immediate relevance of a region with a small dielectric constant above a Pt surface, as found in this work.
V. SUMMARY AND CONCLUSIONS
In summary, we have computed the electronic dielectric response at the Pt(111)/water interface. We find the interfacial dielectric response to be reasonably well captured by a combination of the response of the metal at a metal/vacuum interface and the electronic response of bulk water scaled by the surface-near (valence charge) density of the water. Deviations can be explained by the molecular orbitals of water molecules in contact with the Pt surface being more diffuse, leading to a somewhat increased polarizability of these orbitals. In spite of the resulting enhancement of the dielectric response at the interface, the electronic dielectric constant remains below 2.8 once the electronic response of the metal has decayed at ∼1 Å from the center of the outermost Pt layer. Since the first water molecules do not approach the surface any more closely than 2.0 Å, the low electronic dielectric constant at the interface suggests the presence of a “gap” region between the image charge plane of the metal and the first water layer in which the dielectric constant should be small. This conclusion has important implications on the double layer capacitance and the potential profile at the surface, which in turn can influence electrochemical reactions.
SUPPLEMENTARY MATERIAL
The following information can be found in the supplementary material: (1) validation of the machine-learned potential (energies and atomic forces, water structure, and water dynamics); (2) setup of MD sampling in the concurrent learning workflow; (3) water thickness convergence test; (4) details on the extraction of the electronic and orientational dielectric constant; (5) convergence tests of the electronic dielectric constant; (6) computational setup for maximal localized Wannier center calculation; and (7) orientational dielectric constant at the interface.
ACKNOWLEDGMENTS
J.-X.Z. acknowledges the Xiamen University and iChEM for a Ph.D. studentship and financial support by the Graduate School of Xiamen University for an internship at Leiden University. J.C. acknowledges funding from the National Science Fund for Distinguished Young Scholars (Grant No. 22225302), the National Natural Science Foundation of China (Grant Nos. 92161113, 21991151, 21991150, and 22021001), the Fundamental Research Funds for the Central Universities (Grant Nos. 20720220008, 20720220009, and 20720220010), Laboratory of AI for Electrochemistry (AI4EC), and IKKEM (Grant Nos. RD2023100101 and RD2022070501). The authors thank Harry Stern for discussions.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Jia-Xin Zhu: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Funding acquisition (supporting); Investigation (lead); Methodology (equal); Software (lead); Validation (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Jun Cheng: Conceptualization (supporting); Funding acquisition (lead); Methodology (supporting); Resources (lead); Supervision (supporting); Writing – review & editing (supporting). Katharina Doblhoff-Dier: Conceptualization (lead); Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Validation (supporting); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available in Zenodo repository at https://doi.org/10.5281/zenodo.13761448.