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

Significance

Impedance characterizes the macroscopic electric current response of electrochemical cells such as batteries and supercapacitors. While readily obtainable from experiments, interpreting impedance in terms of ion transport and solvent polarization at the molecular level has remained a major challenge. Here, we use a recently developed molecular simulation approach to compute the impedance of nanocapacitors comprising gold electrodes in contact with aqueous electrolyte solutions. We highlight an important coupling between solvent and ionic responses that is typically overlooked in continuum models. We also provide a simple interpretation of impedance in terms of interfacial capacitance and the electrolyte’s bulk conductivity. Our results further justify a simple equivalent circuit model allowing to extrapolate our findings to larger capacitors typically employed in electrochemical experiments.

Abstract

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

In recent decades, fueled by technological advances, the miniaturization of electrochemical devices has made tremendous progress. In this context, micro- and nanogap devices have demonstrated peculiar properties and promising performance in nanofluidics (1, 2), sensing (3–5), as well as energy storage and conversion applications (6, 7). Beyond their technological importance, nanoelectrochemical devices serve as valuable platforms for acquiring fundamental insights into solid–liquid interfaces (8, 9), highlighting how microscopic phenomena can impact macroscopic properties. For instance, nanofluidic experiments (10) [later corroborated by theoretical and computational studies (11, 12)] have shown how friction can be massively modified at water/carbon interfaces because of the coupling of electronic and dielectric fluctuations between the solid and the liquid. Recent studies have also focused on the dielectric properties of confined liquids between planar walls, revealing the influence of the interplane separation on the static dielectric response (13–15) and its anisotropy (16–19).

In electrochemical devices, and in particular for capacitors, the dynamical dielectric/electrical response is assessed through electrochemical impedance spectroscopy (EIS) experiments (20–22). Here, the impedance $Z(\omega )$ [or equivalently the admittance $Y(\omega )=1/Z(\omega )$] quantifies the system’s electric current response $I(\omega )$ to an oscillating voltage $\mathrm{\Delta }\mathrm{\Psi }(\omega )$, under linear response conditions. EIS results are commonly interpreted through equivalent circuit (EC) models (23, 24), which are fitted to the experimental data to extract parameters such as the electrical resistance $R$, capacitance $C$, and, more broadly, the distribution of relevant time scales governing polarization and charge transport (25). One of the most commonly used EC models is the transmission line model (TLM) (26–29), which is employed to model the charging dynamics of capacitors with porous electrodes. Despite their popularity, the physical interpretation of such representations is not always straightforward and has led to long-standing debates (see, e.g., ref. 30). For this reason, coupling experiments with theoretical and computational investigations is crucial to elucidate the physical mechanisms behind a given electrochemical signal.

Popular approaches to model the dynamical response of capacitors include analytic mean-field theories (31–35) and numerical simulations at the mesoscale (36, 37). The tractability of such approaches, however, relies on strong assumptions that often neglect important molecular details. In recent years, molecular simulations have proven a powerful tool to investigate the electrode-electrolyte interface that retain details about relevant molecular correlations (38–41). In particular, classical constant-potential molecular dynamics (MD) simulations have been extensively used to uncover the microscopic mechanisms that underlie macroscopic properties of electrochemical devices, such as supercapacitors (42–45). In this kind of simulation, the electrode charges fluctuate in response to changes in the electrolyte configuration, subject to the constraints of constant applied potential and global neutrality (46–48).

The dynamical behavior of the total charge $Q(t)$ of an electrode, which fluctuates in the constant-potential ensemble, provides a means to obtain important electrochemical information about the system. For example, following the classic approach of Johnson and Nyquist (49, 50), in fact anticipated in 1912 by De Haas-Lorentz in her PhD thesis (51), one can use the equilibrium charge or current fluctuations to compute the electrical response of an electrochemical system. Indeed, previous works have shown how the differential capacitance, which describes how the average electrode charge changes with applied potential $\mathrm{\Delta }\mathrm{\Psi }$, can be calculated from the charge fluctuations using a fluctuation–dissipation relation $C_{\mathrm{diff}}=\partial ⟨Q⟩/\partial \mathrm{\Delta }\mathrm{\Psi }=\beta ⟨\delta Q^2⟩$, where brackets denote an ensemble average, $\delta Q=Q-⟨Q⟩$ and $\beta =1/k_{\mathrm{B}}T$ with $k_{\mathrm{B}}$ the Boltzmann constant and $T$ the temperature (48, 52, 53). Furthermore, in recent work (54), this fluctuation–dissipation relation was extended to dynamical properties, in particular, to calculate the whole impedance spectrum from the total charge autocorrelation function $⟨\delta Q(0)\delta Q(t)⟩$ (QACF). This approach was illustrated for the case of nanocapacitors with pure water confined between electrodes.

In this work, we leverage this approach to investigate a far more challenging problem: to understand the electrochemical impedance of electrolyte solutions. Not only do the long time scales associated with ionic motion mean statistical sampling is more demanding than the pure water case, but, as we will see, differences in the underlying statistical mechanics of electrolytes vs dielectrics play an important role in interpreting the resulting impedance. For the simple aqueous electrolyte that we investigate, however, we show that a simple EC model comprising just bulk and interfacial elements is remarkably accurate in describing the simulation results. Even more surprisingly, we find that the interfacial contribution is purely capacitive, i.e., there is no measurable interfacial resistance. We also discuss the time scales of charge relaxation obtained in our MD simulations, and highlight the limitations of standard analytical theories. At the microscopic level, our simulations show that this arises from a coupling of solvent and ionic fluctuations, a feature that is only approximately taken into account in most implicit-solvent models of electrochemical response.

1. Results and Discussion

1.1. Electrode Charge Dynamics and Salt Concentration.

We consider NaCl(aq)-gold nanocapacitors at four different salt concentrations: 0.1, 0.5, 1.0, and 1.5 M. To emphasize the fundamental differences between electrolytes and dielectric media, we also compare to simulation results for pure water (54). A representative snapshot for the 1 M case is shown in Fig. 1A. For each concentration, the interelectrode distance $d\approx 5$ nm was obtained from the average of a constant pressure simulation at atmospheric pressure and $T=298$ K. Simulations at constant $\mathrm{\Delta }\mathrm{\Psi }$ (and constant volume) were performed with the fluctuating charge method, where the atomic charges on the electrode atoms are calculated self-consistently depending on the instantaneous electrolyte configuration at each simulation step (47). The consequences of such a Born–Oppenheimer (BO) approximation on charge fluctuations and electrochemical properties have been discussed in previous works (48, 54). In addition, recent works in the context of nanoscale flow have shown how dynamical coupling between electronic degrees of freedom in a solid and dielectric fluctuations in a fluid can impact interfacial friction (11, 12). Such effects are also neglected in the BO approximation; their potential impact on (high-frequency) impedance, which depends on the specific details of how they are described, lies beyond the scope of this work. We perform two types of simulations: i) we characterize the equilibrium fluctuations of the system with $\mathrm{\Delta }\mathrm{\Psi }=0$ V; ii) we interrogate the nonequilibrium response of the solution to a voltage step by instantaneously switching $\mathrm{\Delta }\mathrm{\Psi }=0\to 1$ V. More simulation details are provided in Materials and Methods and SI Appendix.

Fig. 1.

(A) Snapshot of the 1.0 M NaCl system, with the color of gold electrode atoms representing their instantaneous charge, as indicated by the color bar. Hydrogen, oxygen, sodium, and chloride atoms are represented as white, red, cyan, and green spheres, respectively. (B) Differential capacitance $C_{\mathrm{diff}}$ divided by the electrode area $A_{\mathrm{el}}$, as a function of NaCl concentration. (C) Electrode charge autocorrelation function divided by $A_{\mathrm{el}}$ for the five salt concentrations we considered. (D) Comparison of the responses to a voltage change from $\mathrm{\Delta }\mathrm{\Psi }=0$ to $1$ V (symbols) with the equilibrium charge ACFs $⟨\delta Q(0)\delta Q(t)⟩$, normalized as $(f(t)-f(\infty ))/(f(0)-f(\infty ))$, where $f(t)$ is either $Q(t)$ recorded during the charging process (symbols), or $⟨\delta Q(0)\delta Q(t)⟩$ at equilibrium (solid lines), for the 0.1 and 1 M systems. The uncertainty, calculated as 95% CI, is represented as error bars in panels (B) and (D), and with shaded areas in panel (C).

Fig. 1B shows the differential capacitance, $C_{\mathrm{diff}}$, estimated from the fluctuations of the total charge $Q$ on one of the two electrodes (48, 52) as a function of salt concentration. Note that these results do not account for the empty capacitor contribution $C_{\mathrm{diff}}^{\mathrm{empty}}$, which results from the fluctuations neglected by the BO approximation (48). This contribution depends in particular on the distance between the electrodes and typically amounts to $\sim$10% of the BO capacitance for the systems considered in the present work. The capacitance values for the pure water and 1 M cases are in excellent agreement with recent simulation results on the same setup (55) and compare favorably with simulations performed with different force-fields (56, 57). Our results show that the presence of ions introduces a positive contribution to capacitance. However, the capacitance does not depend on the salt concentration beyond 1 M in the considered range. A similar trend was found in simulations of supercapacitors (58) and in recent impedance experiments for NaCl solutions at planar electrodes, where $C_{\mathrm{diff}}$ was found to be almost constant for salt concentrations spanning six orders of magnitude (59). Interestingly, these results are in stark contrast with the cell capacitance values predicted by Debye–Hückel (DH) theory,

$$\begin{array}{c}\hfill C_{\mathrm{DH}}=\frac{1}{2}{A}_{\mathrm{el}}\frac{{\epsilon }_0{\epsilon }_{\mathrm{w}}}{{\lambda }_{\mathrm{D}}},\end{array}$$ [1]

where $A_{\mathrm{el}}$ is the surface area of each electrode, ${\epsilon }_0$ is the vacuum permittivity, ${\epsilon }_{\mathrm{w}}$ is the permittivity of the solvent, and the Debye screening length is given by

$$\begin{array}{c}\hfill {\lambda }_{\mathrm{D}}={\left(\frac{{\epsilon }_{\mathrm{w}}{\epsilon }_0{k}_{\mathrm{B}}T}{e^2{\sum }_{\alpha }{c}_{\alpha }{Z}_{\alpha }^2}\right)}^{\frac{1}{2}},\end{array}$$ [2]

where $e$ is the elementary charge and the sum runs over the species $\alpha$ with concentration $c_{\alpha }$ and valency $Z_{\alpha }$. Using ${\epsilon }_{\mathrm{w}}=70.7$ for SPC/E water (60), Eq. 1 predicts capacitances that are $\sim$25 to $\sim$100 times larger (at 0.1 M and 1.0 M, respectively) than the values obtained from simulations. This mismatch is not unexpected, given that DH theory implies a very simplified picture of the electrode/electrolyte interface. Furthermore, at sufficiently high salt concentrations (i.e., $\ge$0.5 M), ${\lambda }_{\mathrm{D}}$ becomes comparable to the ionic size, a regime where ionic correlations and molecular features of the solvent, which are neglected entirely in DH theory, are known to become important (38, 61). More sophisticated capacitance models could be used to fit MD results (58), but this falls beyond the scope of the present work.

Fig. 1C shows $⟨\delta Q(0)\delta Q(t)⟩$ for the different salt concentrations we consider. The QACFs in the presence of ions share some qualitative similarities with the pure water case. However, for finite salt concentrations, we observe an additional exponential-like slow mode with a decay rate that depends on concentration (discussed below, see also SI Appendix, Fig. S1). In particular, the relaxation rate of the slow mode is faster for higher salt concentrations. In the considered concentration range, it is known (and we will confirm later) that the ionic conductivity increases with salt concentration (62). The observed long-time behavior of the QACFs therefore suggests an intuitive connection with the ionic conductivity.

The impedance that we will derive from these QACFs describes the linear response between the current and voltage. This raises the issue of the voltage range over which the response is linear. We address it by performing nonequilibrium simulations where we subject the system to a step change $\mathrm{\Delta }\mathrm{\Psi }=0\to 1$ V, and measure $Q(t)$ as it relaxes to a new equilibrium. Results for the 0.1 and 1.0 M systems are presented in Fig. 1D, where we also compare to the predicted nonequlibrium response from the QACFs. The excellent agreement between the equilibrium and nonequilibrium results demonstrates that linear response is a valid approximation (SI Appendix), at least up to $\mathrm{\Delta }\mathrm{\Psi }=$ 1 V. This is consistent with previous results for water/gold nanocapacitors (54). Having established this fact, we now use these QACFs to obtain the impedance.

1.2. Impedance of Nanocapacitors as Bulk Impedance and Interfacial Capacitance.

1.2.1. Confined electrolytes.

Following the approach used in ref. 54 for pure water, we obtain $Z(\omega )$ from the QACF sampled at $\mathrm{\Delta }\mathrm{\Psi }=0$ V. As in this previous work, and for consistency with the BO contribution to the differential capacitance (see above), we limit our analysis to the impedance contributions within the BO approximation. Fig. 2 A and B report the real and imaginary parts of $Z(\omega )$, respectively. Here, to represent intensive quantities, $Z(\omega )$ is multiplied by $A_{\mathrm{el}}/d_{\mathrm{DDS}}$, where $d_{\mathrm{DDS}}=d-2w_{\mathrm{DDS}}$ is the effective width of the confined electrolyte based on the positions of the dielectric dividing surface at each electrode/electrolyte interface (16). Roughly speaking, $w_{\mathrm{DDS}}$ is the width of the vacuum layer between the first atomic plane of the electrode and the closest water molecules. In this work, we use $w_{\mathrm{DDS}}=1.27$ Å from ref. 54, which employed the same force field as this study (without salt).

Fig. 2.

(A) Real and (B) imaginary parts of the total impedance $Z(\omega )$ as calculated from confined simulations for all the salt concentrations we considered, as a function of frequency. (C) Real and (D) imaginary parts of interfacial impedance $Z_{\mathrm{int}}(\omega )$. The Inset in (C) shows the resistivity $\rho =\mathfrak{R}[Z(\omega \to 0)]A_{\mathrm{el}}/d_{\mathrm{DDS}}$, estimated as described below Eq. 9, and ${\rho }_{\mathrm{bulk}}=1/{\sigma }_0$, with the EH and Nernst–Einstein (NE) estimates of the conductivity. The Inset in (D) highlights the low-frequency behavior of $\mathfrak{I}[Z_{\mathrm{int}}(\omega )]$, as indicated by the black dashed box in the main panel. Solid lines indicate $\mathfrak{I}[Z_{\mathrm{int}}(\omega )]=-1/\omega C_{\mathrm{diff}}$, and the dotted line shows $\mathfrak{I}[Z_{\mathrm{int}}(\omega )]=-1/\omega C_{\mathrm{w},\mathrm{int}}$ given by Eq. 8. Symbols show results from our molecular simulations.

The impedance results display some distinctive features of capacitors at low frequencies (20). In particular, the real part $\mathfrak{R}[Z(\omega )]$ plateaus at $\omega \to 0$, corresponding to the total electrical resistance of each electrochemical system. For finite salt concentrations, resistance decreases with increasing salt concentration, suggesting an intuitive link with bulk conductivity, which will be further discussed below. Furthermore, the imaginary part $\mathfrak{I}[Z(\omega )]$ displays a clear ${\omega }^{-1}$ scaling at low frequencies, which is typical of capacitive systems. Above $\omega \gtrsim {10}^{11}$ rad/s, the impedance results do not depend on salt concentration and closely resemble that for pure water. This overlap suggests that, at high frequencies, the impedance reflects the individual motion of solvent molecules and ions, which, under the concentrations we consider, is dominated by water.

The global response of the confined systems encoded in $Z(\omega )$ includes all the molecular correlations that contribute to the total polarization or charge dynamics. Intuitively, we expect the global response to comprise “interfacial contributions” from electrolyte proximate to the electrodes, and a bulk-like contribution from the remaining electrolyte far from the electrodes. Experimentally, it can be challenging to rigorously disentangle such interfacial and bulk contributions to the charging dynamics (66). Here, we exploit the fact that molecular simulations allow us to characterize the bulk response, by performing simulations of the electrolyte in the absence of any explicit interfaces (SI Appendix). In particular, we use bulk simulations to estimate the electric conductivity $\sigma$ in two distinct ways: i) at finite frequencies [$\sigma (\omega )$] using a Green–Kubo (GK) approach (67); and ii) at zero frequency [${\sigma }_0\equiv \sigma (0)$] using the Einstein–Helfand (EH) approach (68). [Note that, while $\sigma (\omega )$ is a complex function, ${\sigma }_0$ is a real number.] More details on these calculations are provided in the Methods section.

1.2.2. Bulk electrolytes.

In Fig. 3, we compare our bulk simulations with experimental results obtained with different spectroscopic techniques (62–65). The experimental datasets were converted from permittivity to conductivity using the relation (67, 69) $\sigma (\omega )=i\omega {\epsilon }_0\left[\epsilon (\omega )-1\right]$. The simulation results for the 1 M system reported in Fig. 3A show that $\mathfrak{R}[\sigma (\omega )]$ reaches a global maximum at $\omega \approx 1.5\times {10}^{14}$ rad/s corresponding to the water librational band (70). At lower frequencies, we find a shoulder resembling the Debye-like modes of polar fluids and a plateau corresponding to ${\sigma }_0$. Comparing to experiments, we observe minor discrepancies; this is expected due to the limitations of the molecular model that we use. In particular, at high frequencies, the SPC/E model of water lacks any polarization modes resulting from intramolecular or electronic response (70). Nonetheless, overall agreement across the whole frequency range is fair. In Fig. 3B, we present $\mathfrak{R}[\sigma (\omega )]$ for all concentrations investigated. The dashed lines indicate ${\sigma }_0$ obtained by the EH method, which demonstrate the consistency between our two approaches to estimate the conductivity. (We note the large statistical errors for 0.1 M.) For concentrations other than 1 M, experimental data over the whole frequency range is scarcer, though we note that, similar to $Z(\omega )$, spectra overlap at high frequencies. Comparison to experimental data at low frequencies is shown in the Inset, where we observe reasonable agreement with our simulation results. Fig. 3C shows explicitly the comparison between ${\sigma }_0$ from simulation and $\mathfrak{R}[\sigma (\omega \to 0)]$ from experiment. While we find quantitative agreement at low concentrations, ${\sigma }_0$ is progressively underestimated at higher concentrations. Notwithstanding the limitations of the underlying model, the above analyses suggest that our bulk simulations faithfully capture the experimental conductivity of NaCl solutions.

Fig. 3.

(A) Conductivity calculated from bulk simulations (symbols), as a function of frequency, for the 1 M system compared with experimental optical measurements (solid lines) (62, 64, 65). (B) Conductivity from bulk simulations for all the salt concentrations considered calculated using the GK (solid lines) and EH (dashed lines) approaches. The Inset shows the low-frequency branches of conductivity from MD (symbols), compared with experimental measurements (solid lines) from ref. 62. (C) Conductivity at zero frequency ${\sigma }_0$ from bulk simulations with the EH approach (symbols) and from experimental measurements (62, 63) (dashed lines).

1.2.3. Interfacial impedance and low-frequency behavior.

To discern the interfacial and bulk-like contributions to the impedance, we adopt a subtraction technique similar to that used in experiments (66). Specifically, we define the bulk-like impedance as that which describes a hypothetical system in which the electrolyte retains bulk-like characteristics over the whole volume that it occupies,

$$\begin{array}{c}\hfill Z_{\mathrm{bulk}}(\omega )=\frac{d_{\mathrm{DDS}}}{A_{\mathrm{el}}}\frac{1}{\sigma (\omega )}.\end{array}$$ [3]

The interfacial contribution $Z_{\mathrm{int}}(\omega )$ is then defined as the excess contribution to the total impedance:

$$\begin{array}{c}\hfill Z_{\mathrm{int}}(\omega )=Z(\omega )-Z_{\mathrm{bulk}}(\omega ).\end{array}$$ [4]

We note that $Z_{\mathrm{int}}$ implicitly includes the two electrode-electrolyte interfaces; in our symmetric setup, the contribution of a single interface is simply $Z_{\mathrm{int}}(\omega )/2$. The results for the real and imaginary parts of $Z_{\mathrm{int}}(\omega )$ are shown in Fig. 2 C and D, respectively.

For finite ion concentration, it is remarkable that $\mathfrak{I}[Z_{\mathrm{int}}(\omega \to 0)]\approx -1/\omega C_{\mathrm{diff}}$ is well described by the static differential capacitance. This can be understood by acknowledging that, at low frequencies,

$$\begin{array}{cc}\hfill \mathfrak{R}[\sigma (\omega )]& ={\sigma }_0+{\sigma }_2{\omega }^2+{\sigma }_4{\omega }^4+\dots \hfill \end{array}$$ [5]

$$\begin{array}{cc}\hfill \mathfrak{I}[\sigma (\omega )]& ={\sigma }_1\omega +{\sigma }_3{\omega }^3+\dots \hfill \end{array}$$ [6]

where ${\sigma }_1={\epsilon }_0({\epsilon }_{\mathrm{eff}}-1)$, with ${\epsilon }_{\mathrm{eff}}$ an effective, frequency-independent, dielectric constant of the solution (69). As detailed in SI Appendix, both the real and imaginary parts of the bulk conductivity are well described by these expansions (SI Appendix, Figs. S4 and S5), and it is straightforward to show that

$$\begin{array}{c}\hfill \mathfrak{I}[Z_{\mathrm{int}}(\omega \to 0)]\approx -\frac{1}{\omega C_{\mathrm{diff}}}+\frac{{\sigma }_1\omega }{{\sigma }_0^2+(2{\sigma }_0{\sigma }_2+{\sigma }_1^2){\omega }^2}\frac{d_{\mathrm{DDS}}}{A_{\mathrm{el}}}.\end{array}$$ [7]

Provided that ${\sigma }_0>0$, which is true for finite ion concentration, the second term (which describes $\mathfrak{I}[Z_{\mathrm{bulk}}(\omega )]$ well, see SI Appendix, Fig. S7) vanishes as $\omega \to 0$. For dielectric systems such as pure water, however, this is no longer the case. As is apparent when setting ${\sigma }_0=0$ in Eq. 7, we instead expect $\mathfrak{I}[Z_{\mathrm{w},\mathrm{int}}(\omega )]\sim -1/\omega C_{\mathrm{w},\mathrm{int}}$ with an effective interfacial differential capacitance

$$\begin{array}{c}\hfill \frac{1}{C_{\mathrm{w},\mathrm{int}}}=\left[\frac{1}{C_{\mathrm{diff}}}-\frac{d_{\mathrm{DDS}}}{A_{\mathrm{el}}{\epsilon }_0({\epsilon }_{\mathrm{w}}-1)}\right],\end{array}$$ [8]

which, as seen in Fig. 2D, describes the simulation data well. It clear from Eq. 8 that $C_{\mathrm{diff}}$ contains both interfacial and bulk contributions. For the $d=5.07$ nm system that we have investigated, we estimate that $C_{\mathrm{w},\mathrm{int}}/A_{\mathrm{el}}\approx 2.80$ $\mathrm{\mu }$ F/cm², while ${\epsilon }_0({\epsilon }_{\mathrm{w}}-1)/d_{\mathrm{DDS}}\approx 13.17$ $\mathrm{\mu }$ F/cm². The total series capacitance $C_{\mathrm{diff}}/A_{\mathrm{el}}\approx 2.31$ $\mathrm{\mu }$ F/cm² is therefore dominated by the interfacial contribution. Evidently, as $d_{\mathrm{DDS}}$ increases, so too does the relative importance of the bulk contribution.

We now turn our attention to the real part of the interfacial impedance. Somewhat surprisingly, as shown in Fig. 2C, we find that $\mathfrak{R}[Z_{\mathrm{int}}]$ is negligible to within statistical uncertainty; this is the case for all concentrations we investigate, including pure water. This suggests that the real part of the cell impedance $\mathfrak{R}[Z]$ behaves at low frequency as the bulk contribution (SI Appendix):

$$\begin{array}{c}\hfill \mathfrak{R}[Z_{\mathrm{bulk}}(\omega \to 0)]\approx \frac{1}{{\sigma }_0}\left[1-\left(\frac{{\sigma }_0{\sigma }_2+{\sigma }_1^2}{{\sigma }_0^2}\right){\omega }^2\right]\frac{d_{\mathrm{DDS}}}{A_{\mathrm{el}}}.\end{array}$$ [9]

SI Appendix, Fig. S6 shows that this expansion, using the parameters fitted on the conductivity $\sigma (\omega )$, describes well the bulk impedance at low frequency. For the confined case, a quadratic fit (SI Appendix, Fig. S8) allows to us to obtain the cell resistance $R=\mathfrak{R}[Z(\omega \to 0)]$. These different estimates for the resistance, together with those obtained using the EH and Nernst-Einstein values for ${\sigma }_0$, are summarized in SI Appendix, Table S4. The Inset of Fig. 2C confirms the good agreement between the corresponding confined $R{A}_{\mathrm{el}}/d_{\mathrm{DDS}}$ and bulk ${\rho }_{\mathrm{bulk}}=1/{\sigma }_0$ resistivities, except for 0.1 M where $\rho \approx {\rho }_{\mathrm{bulk}}/3.6$. While it is possible that for this concentration the low-frequency regime might not be fully reached, we also note that the width of the EDL (${\lambda }_D\approx 9.1$ Å) is only five times smaller than the interelectrode distance $d$ and the separation between bulk and interfacial regions might be less straightforward (see also the discussion of timescales in Section 1.3).

For pure water, ${\sigma }_0$ vanishes, and we obtain a qualitatively different behavior (SI Appendix):

$$\begin{array}{cc}& \mathfrak{R}[Z_{\mathrm{w},\mathrm{bulk}}(\omega \to 0)]\hfill \\ \hfill & \approx \frac{{\sigma }_2}{{\sigma }_1^2}\left[1-\left(\frac{{\sigma }_2^2+2{\sigma }_1{\sigma }_3}{{\sigma }_1^2}-\frac{{\sigma }_4}{{\sigma }_2}\right){\omega }^2\right]\frac{d_{\mathrm{DDS}}}{A_{\mathrm{el}}},\hfill \end{array}$$ [10]

with ${\sigma }_1={\epsilon }_0({\epsilon }_{\mathrm{w}}-1)$. In this case, the overall resistance of the cell is $R\approx R_{\mathrm{w},\mathrm{bulk}}=d_{\mathrm{DDS}}{\sigma }_2/[A_{\mathrm{el}}{\epsilon }_0^2{({\epsilon }_{\mathrm{w}}-1)}^2$]. In SI Appendix, we further show that within the Debye relaxation model, ${\sigma }_2\approx {\epsilon }_0({\epsilon }_{\mathrm{w}}-1){\tau }_{\mathrm{w},\mathrm{bulk}}$, where ${\tau }_{\mathrm{w},\mathrm{bulk}}\approx 9$ ps is the Debye relaxation time of bulk water. By fitting the bulk conductivity at low frequencies, we obtain an estimate ${\sigma }_2/{\epsilon }_0({\epsilon }_{\mathrm{w}}-1)\approx 10$ ps consistent with this model.

1.2.4. EC model and effect of interelectrode distance.

From our analysis, we conclude that for finite salt concentrations, the interfacial contribution to the impedance is mostly capacitive and is well described by $Z_{\mathrm{int}}(\omega )\approx 1/i\omega C_{\mathrm{diff}}$. This is in stark contrast with recent findings on electrode-ionic liquid interfaces, where the resistance was dominated by interfacial contributions (71). Our observations mean that we can model the total impedance of our systems as an EC with a bulk component, parameterized on $\sigma (\omega )$ as in Eq. 3, in series with a purely capacitive element with $C_{\mathrm{diff}}=\beta ⟨\delta Q^2⟩$. Such EC models are already widely used by experimental groups to model macroscopic electrochemical cells with planar blocking electrodes (66). It is worth emphasizing, however, that this work brings direct evidence of this behavior at the molecular scale and confirms the validity of these EC models to represent nanocapacitors.

Finally, we note that, to the extent that $C_{\mathrm{diff}}$ is independent of $d$, which we might reasonably expect if ${\lambda }_{\mathrm{D}}\ll d$ (see, e.g., SI Appendix, Fig. S9), we can use this simple EC model to predict$Z(\omega )$ for different system sizes. We demonstrate this in Fig. 4, where we present results for $Z(\omega )$ for $2.5\lesssim d/\mathrm{nm}\lesssim 20$. The results from simulation compare favorably to those predicted by $Z(\omega )=1/i\omega C_{\mathrm{diff}}+d_{\mathrm{DDS}}/A_{\mathrm{el}}\sigma (\omega )$, where we have used $C_{\mathrm{diff}}$ obtained with $d=5.07$ nm. This suggests that, aside from the capacitive contribution arising from the EDL formation at the interface, the dielectric response of the confined electrolyte remains bulk-like. This observation is consistent with recent works that highlight how the dielectric response of nanoconfined water is essentially bulk-like, provided that one appropriately accounts for the boundary of the liquid (16, 54, 72–75). Our results extend this conclusion to systems with dissolved ions.

Fig. 4.

(A) Real and (B) imaginary parts of the impedance for different values of $d$ (as indicated in the legends), for the 1 M electrolyte. The solid lines show results obtained from molecular simulations, while the dashed lines indicate the prediction of the simple EC model in which the interfacial impedance is purely capacitive. Importantly, the simple EC model has been parameterized with $C_{\mathrm{diff}}$ from a single simulation with $d=5.07$ nm.

1.3. Effect of the Salt Concentration on Relaxation Times.

As shown in previous work, for pure water confined between the gold-electrodes, the relaxation of the QACF is well characterized by a single exponential with a characteristic time ${\tau }_{\mathrm{w}}={\tau }_{\mathrm{w},\infty }d/[d+({\epsilon }_{\mathrm{w}}-1)w_{\mathrm{DDS}}]$ (54), where ${\tau }_{\mathrm{w},\infty }\approx 8.6$ ps is close to the bulk value for the Debye relaxation mode of water. Upon introducing ions, the QACF is broadly characterized by two timescales: a fast relaxation that is comparable to ${\tau }_{\mathrm{w}}$, and a much slower relaxation ${\tau }_{\mathrm{slow}}$, which depends on salt concentration (Fig. 1C). A global timescale can be defined from the integral of the normalized QACF (54),

$$\begin{array}{c}\hfill {\displaystyle {\tau }_{\mathrm{Q}}={\int }_0^{\infty }\frac{⟨\delta Q(0)\delta Q(t)⟩}{⟨\delta Q^2⟩}\mathrm{d}t,}\end{array}$$ [11]

such that the admittance behaves at low frequency as $Y(\omega )\approx i\omega C_{\mathrm{diff}}/(1+i\omega {\tau }_{\mathrm{Q}})$. Consistent with our simple EC model, this translates in terms of impedance to $Z(\omega )\approx R+1/i\omega C_{\mathrm{diff}}$, with $R=\mathfrak{R}[Z(\omega \to 0)]$.

As shown in Fig. 5, beyond the initial jump from the pure water case (for which no ions from water autodissociation are captured by the present force field), ${\tau }_{\mathrm{Q}}$ decreases with increasing ion concentration, but remains on the order ${10}^{-11}$ s. It is compared with the “RC” times, using $C_{\mathrm{diff}}$ obtained from $⟨\delta Q^2⟩$ (Fig. 1B), and the total resistance estimated either as $R_{\mathrm{bulk}}=d_{\mathrm{DDS}}/A_{\mathrm{el}}{\sigma }_0$ or $R=\mathfrak{R}[Z(\omega \to 0)]$ (Fig. 2A). As expected, $R{C}_{\mathrm{diff}}$ agrees well with ${\tau }_{\mathrm{Q}}$ for the whole concentration range. The agreement obtained with $R_{\mathrm{bulk}}$ from bulk simulations is also good for the higher concentrations, but for the lowest concentration, 0.1 M, $R_{\mathrm{bulk}}{C}_{\mathrm{diff}}$ is approximately 3.6 times larger than ${\tau }_{\mathrm{Q}}$. As discussed above, this discrepancy between $R{C}_{\mathrm{diff}}$ and $R_{\mathrm{bulk}}{C}_{\mathrm{diff}}$ might be caused by the fact that we have not fully accessed the low frequency limit at 0.1 M, or simply due to the departure from the thin EDL limit (${\lambda }_D\ll d$) where the interfacial and bulk contributions can be decoupled.

Fig. 5.

Comparison of relevant characteristic times as a function of salt concentration from simulations (symbols) and analytical theories (dashed lines) (31, 33–35, 76). ${\tau }_Q$ is calculated as the integral of the normalized QACFs (Eq. 11). For ${\tau }_{\mathrm{ion}-\mathrm{tot}}$ (blue dots) and ${\tau }_{\mathrm{diff}}$ (green dots), the characteristic time is extracted by fitting the data (Fig. 6 B and C and SI Appendix, Fig. S1) with exponential functions. The $\mathit{RC}$ and $R_{\mathrm{bulk}}C$ characteristic times use the resistance computed from confined and bulk simulations, respectively, and $C=C_{\mathrm{diff}}$ from confined simulations. Note that $R_{\mathrm{bulk}}$ was calculated from the EH estimate of the bulk conductivity and using $d_{\mathrm{DDS}}$ instead of $d$, consistently with Eq. 3. Here, $\mathit{RC}$, $R_{\mathrm{bulk}}C$, ${\tau }_Q$, and ${\tau }_{\mathrm{diff}}$ were computed from equilibrium simulations, whereas ${\tau }_{\mathrm{ion}-\mathrm{tot}}$ was estimated from nonequilibrium simulations.

While the intermediate and high frequency range of $Z_{\mathrm{bulk}}(\omega )$ is dominated by the contribution of water, the behavior at low frequency significantly depends on the ions. We now examine three timescales frequently considered in the dynamics of ions in confined electrolytes obtained with analytical theories (31, 33–35, 37, 76). Such approaches are inherently approximate: they typically describe the solvent implicitly; use a common diffusion constant $D$ for anions and cations; rely on mean-field assumptions; and use a simplified representation of the electrode–electrolyte interface. Therefore, we limit ourselves to examining qualitative trends with salt concentration rather than how well these time scales compare quantitatively with our simulations. The first timescale is ${\tau }_d=d^2/4D$, which corresponds to ion diffusion over the distance between the electrodes. The second timescale is the ionic Debye relaxation time ${\tau }_{\mathrm{D}}={\lambda }_{\mathrm{D}}^2/D$, which corresponds to the relaxation of charge fluctuations in bulk electrolytes, with the diffusion of ions over the corresponding Debye screening length. The third timescale ${\tau }_{\mathrm{mix}}={\lambda }_{\mathrm{D}}(d-2{\lambda }_{\mathrm{D}})/2D$ mixes both ${\lambda }_{\mathrm{D}}$ and $d$ (31); it corresponds to the RC charging time assuming Nernst-Einstein conductivity and DH capacitance and reduces to ${\tau }_{\mathrm{mix}}\approx {\lambda }_{\mathrm{D}}d/2D$ in the limit of thin EDLs. The time scales introduced so far can be broadly interpreted as approximations or limiting cases of the more general solution obtained within the Debye–Falkenhagen approximation by Janssen and Bier (33).

We show these different time scales in Fig. 5 together with the simulation results for ${\tau }_{\mathrm{Q}}$. For the diffusion coefficient, we use $D=1.41\times {10}^{-9}$ m²/s, the average diffusion coefficient of Na⁺ and Cl⁻ ions computed from bulk simulations at 0.1 M. All the analytical timescales overestimate the relaxation time, but with different orders of magnitudes and trends with salt concentration. The worst estimate is ${\tau }_d$, by several orders of magnitude and no dependence on salt concentration; this result is not unsurprising as the build-up of EDLs does not require ions to diffuse over the whole electrochemical cell. While the order of magnitude of ${\tau }_{\mathrm{D}}$ is correct for the highest concentrations we consider, the agreement with ${\tau }_Q$ is worse for the lower concentrations, with ${\tau }_{\mathrm{D}}/{\tau }_Q\approx 10$ at 0.1 M. Naively, we would expect such a mean-field prediction to improve as concentration is decreased. This discrepancy suggests the observation that ${\tau }_{\mathrm{D}}\sim {\tau }_Q$ for the higher concentrations is largely coincidental. Finally, despite the wrong order of magnitude, only ${\tau }_{\mathrm{mix}}$ displays a trend with concentration similar to that of ${\tau }_{\mathrm{Q}}$. This observation suggests that ${\tau }_Q$, and more generally the system’s impedance, reflects the interplay between the dynamics of a bulk-like electrolyte and the interfacial capacitive behavior induced by the blocking electrodes. A more detailed discussion of the concentration dependence of ${\tau }_{\mathrm{D}}$, ${\tau }_{\mathrm{mix}}$, and ${\tau }_{\mathrm{Q}}$ is included in SI Appendix, Fig. S10. In a bid to rationalize the quantitative disagreement between theory and simulation, in the next section, we probe the relaxation process in terms of contributions from ions, water, and their cross-correlations.

1.4. Water and Ionic Contributions.

As shown in previous works, at $\mathrm{\Delta }\mathrm{\Psi }=0$ V and with global electroneutrality, the total electrode charge $Q$ is proportional to the total dipole moment of the solution $M_{\mathrm{tot}}$ in the direction perpendicular to the electrode surface (54, 77). For this reason, the behavior of $⟨\delta Q(0)\delta Q(t)⟩$ is determined entirely by $⟨M_{\mathrm{tot}}(0)M_{\mathrm{tot}}(t)⟩$. Clearly, in the case of an electrolyte solution, $M_{\mathrm{tot}}(t)=M_{\mathrm{wat}}(t)+M_{\mathrm{ion}}(t)$ has contributions from both the solvent water molecules and the dissolved ions. Molecular simulations offer the possibility of computing directly the ionic and solvent contributions, thus “dissecting” the global response to understand the source of specific features, as shown in previous works (78–80). We can therefore decompose the autocorrelation function of the total dipole moment:

$$\begin{array}{cc}\hfill ⟨\delta M_{\mathrm{tot}}(0)\delta M_{\mathrm{tot}}(t)⟩=& ⟨\delta M_{\mathrm{tot}}(0)\delta M_{\mathrm{ion}}(t)⟩\hfill \\ \hfill & +⟨\delta M_{\mathrm{tot}}(0)\delta M_{\mathrm{wat}}(t)⟩,\hfill \end{array}$$ [12]

with

$$\begin{array}{cc}\hfill ⟨\delta M_{\mathrm{tot}}(0)\delta M_{\mathrm{ion}}(t)⟩=& ⟨\delta M_{\mathrm{ion}}(0)\delta M_{\mathrm{ion}}(t)⟩\hfill \\ \hfill & +⟨\delta M_{\mathrm{wat}}(0)\delta M_{\mathrm{ion}}(t)⟩,\hfill \end{array}$$ [13]

and

$$\begin{array}{cc}\hfill ⟨\delta M_{\mathrm{tot}}(0)\delta M_{\mathrm{wat}}(t)⟩=& ⟨\delta M_{\mathrm{wat}}(0)\delta M_{\mathrm{wat}}(t)⟩\hfill \\ \hfill & +⟨\delta M_{\mathrm{wat}}(0)\delta M_{\mathrm{ion}}(t)⟩.\hfill \end{array}$$ [14]

The correlation functions on the right hand side of Eq. 12 rigorously relate the linear response of $M_{\mathrm{ion}}(t)$ and $M_{\mathrm{wat}}(t)$ to a change in voltage across the cell; consequently, they can be obtained by monitoring the time evolution of $M_{\mathrm{ion}}(t)$ and $M_{\mathrm{wat}}(t)$ upon the step change $\mathrm{\Delta }\mathrm{\Psi }=0\to 1$ V.

In Fig. 6A, we present the static limit of these normalized decomposed dipole correlation functions. For all ion concentrations, the total comprises positive contributions from $⟨\delta M_{\mathrm{ion}}^2⟩$ and $⟨\delta M_{\mathrm{wat}}^2⟩$ and a large negative contribution from the cross-correlation $⟨\delta M_{\mathrm{ion}}\delta M_{\mathrm{wat}}⟩$. This anticorrelation between the equilibrium fluctuations of the ionic and solvent contributions to the polarization was also found in bulk systems (78, 79, 81), and reflects the well-known Stillinger–Lovett (SL) conditions (82, 83). While ref. 81 provided a framework to understand the SL conditions for the total dipole moment in bulk systems in terms of “virtual electrodes” arising from the imposed boundary conditions, our results demonstrate that these SL conditions are also obeyed in the presence of explicit electrodes held at constant bias. See also ref. 84.

Fig. 6.

(A) Static correlations of the water and ionic contributions to the total dipole moment as a function salt concentration. (B and C) Normalized time correlation functions of the contributions to the total dipole moment for 0.1 and 1.0 M salt concentrations, as a function of time. In all panels, each color indicates the same combination of contributions $\alpha$ and $\beta$. In panels (B) and (C), we compare the results of equilibrium simulations (lines with error bars), those of nonequilibrium simulations (dotted lines), and exponential fits (dashed lines).

We now turn our attention to the nonequilibrium relaxation of $M_{\mathrm{ion}}$ and $M_{\mathrm{wat}}$, shown in Fig. 6 B and C for 0.1 M and 1.0 M, respectively. For both systems, we see an initial fast response of the solvent polarization (see dotted orange lines in Fig. 6 B and C). In fact, the negative values of $⟨M_{\mathrm{tot}}(0)M_{\mathrm{wat}}(t)⟩$ for $t\gtrsim {10}^{-12}$ s indicate that the solvent polarization overshoots its equilibrium value (SI Appendix, Eq. S7). On larger time scales, the ionic polarization begins to relax accompanied by further relaxation of the solvent polarization in an almost equal-and-opposite fashion. The relaxation of $M_{\mathrm{ion}}$ appears to follow a simple exponential time dependence, from which we can obtain an associated time scale ${\tau }_{\mathrm{ion}-\mathrm{tot}}$. In Fig. 5 we plot ${\tau }_{\mathrm{ion}-\mathrm{tot}}$ for all concentrations. Despite being roughly an order of magnitude smaller, we see that the dependence of ${\tau }_{\mathrm{ion}-\mathrm{tot}}$ on concentration closely follows that of ${\tau }_{\mathrm{mix}}$. The observed anticorrelations between the solvent and ionic polarization during the system’s nonequilibrium response are reminiscent of the SL conditions at equilibrium, and consistent with recent observations in bulk electrolyte solutions at finite wavevector and frequency (79). The impact of these anticorrelations appears to manifest as a much faster overall charging dynamics than predicted by ionic relaxation alone, as reflected by the order of magnitude discrepancy between ${\tau }_{\mathrm{ion}-\mathrm{tot}}$ and ${\tau }_{\mathrm{Q}}$. Also shown in Fig. 6 B and C are the correlation functions $⟨\delta M_{\mathrm{tot}}(0)\delta M_{\mathrm{ion}}(t)⟩$ and $⟨\delta M_{\mathrm{tot}}(0)\delta M_{\mathrm{wat}}(t)⟩$ obtained directly from the equilibrium fluctuations at $\mathrm{\Delta }\mathrm{\Psi }=0$ V, albeit at a lower time resolution (full lines). As expected, these results agree with those obtained from our nonequilibrium simulations.

In an attempt to quantify the impact of introducing ions, we have computed ${⟨\delta Q(0)\delta Q(t)⟩}_{\mathrm{diff}}$, by subtracting the pure water QACF from the QACF at finite salt concentration. We find that, for all investigated concentrations, ${⟨\delta Q(0)\delta Q(t)⟩}_{\mathrm{diff}}$ is well described by a simple exponential with characteristic time ${\tau }_{\mathrm{diff}}$ (SI Appendix, Fig. S1). The behavior of ${\tau }_{\mathrm{diff}}$ with concentration is shown in Fig. 5. For all concentrations, ${\tau }_{\mathrm{diff}}$ is roughly seven times larger than ${\tau }_Q$ and comparable to ${\tau }_{\mathrm{ion}-\mathrm{tot}}$. While it is tempting to attribute fast and slow timescales in the system into motions that involve ions and those that do not, such a decomposition is not straightforward. This can be seen in Fig. 6 B and C, where we further decompose the equilibrium correlation functions according to Eqs. 13 and 14. Not only do we see that $⟨\delta M_{\mathrm{ion}}(0)\delta M_{\mathrm{ion}}(t)⟩$ and $⟨\delta M_{\mathrm{ion}}(0)\delta M_{\mathrm{wat}}(t)⟩$ decay slowly, but so too does $⟨\delta M_{\mathrm{wat}}(0)\delta M_{\mathrm{wat}}(t)⟩$. In other words, fast and slow relaxation cannot be separated into purely solvent and ionic contributions.

To summarize, we find that analytical theories and implicit solvent descriptions can describe the qualitative scaling of the relaxation time scales with concentration. However, these predictions are far from quantitative agreement with simulation results. Our MD results highlight: i) ion and water polarization are strongly anticorrelated, and this effect is persistent in time; and ii) ion and water polarization relax with different time scales (single exponential with ${\tau }_{\mathrm{ion}-\mathrm{tot}}$, and a combination of ${\tau }_{\mathrm{w}}$ and ${\tau }_{\mathrm{ion}-\mathrm{tot}}$, respectively). These two features are absent in most analytical theories of electrolytes and present a significant challenge for implicit solvent models to faithfully capture relaxation in EDL capacitors.

2. Conclusions

In this work, we employed MD simulations to study electrode charge and polarization dynamics in gold/NaCl(aq) nanocapacitors. Using both equilibrium and nonequilibrium simulations, we investigated the impact of ion concentration and interelectrode distance on the charging dynamics. As expected, we found that ions induce slower dynamics compared to pure water systems, and more concentrated solutions result in faster charge relaxation. Using equilibrium charge autocorrelation functions, we obtained impedance spectra for the nanocapacitors that exhibit features typical for systems with blocking electrodes. Exploiting the fact that we can obtain the bulk conductivity directly in molecular simulations, we showed that the total impedance is well described by a simple sum of interfacial and bulk contributions. Remarkably, we found that the interfacial impedance for the considered systems is purely capacitive, i.e., interfacial resistance is negligible. We demonstrated that this simple EC model permits extrapolation to larger system sizes; while nanocapacitors are interesting in their own right, this observation will help to bridge the gap between molecular simulations and EIS experiments on larger length scales. We also showed that the electrolyte response, aside from the capacitive interfacial contribution, remains bulk-like even when the interelectrode separation is only a few nanometers.

Future work could focus on the influence of more complicated, possibly porous (42, 58) and heterogeneous electrode-electrolyte interfaces (85), as well as on the effect of screening within the electrode material via the Thomas-Fermi length (86, 87), many-body interactions at the surface [as done, e.g., for water on platinum electrodes (88, 89)] or even nonadiabatic effects (12). The same methodology could be applied and we expect the main conclusions to hold for a variety of nonreactive systems. If these more elaborate descriptions of the electrode material induce specific ion binding to the surface, this may require longer simulations to correctly sample the adsorption/desorption processes, but would in turn provide a direct link between the measured frequency-dependent impedance and the corresponding processes (possibly via more complex EC models, as often done to analyze experiments). Furthermore, the present theoretical framework to predict and analyze the impedance could be extended to include interfacial electron-transfer phenomena at a classical level (90), which would allow representing devices such as pseudocapacitors and batteries.

We also investigated the most relevant characteristic times of charge and polarization relaxation. While we found that analytical predictions fail to quantitatively describe the simulation results, they succeed in reproducing the qualitative behavior. By dissecting the global response into contributions from the ionic and solvent polarization, we showed that the failure of the analytic predictions originates from an anticorrelation between ions and water. In particular, we found that not only do the ions themselves relax slowly, but their presence also introduces a slow time scale in the solvent relaxation. Future coarse-grained descriptions of confined electrolytes could include the nonlocality and frequency-dependence of the solvent response (79, 91) beyond a uniform static permittivity (36, 92) to restore, at least partially, this fundamental physics.

3. Materials and Methods

3.1. Numerical Simulations.

Classical MD simulations were performed using MetalWalls (93, 94) and LAMMPS (95) [equipped with the ELECTRODE package (96)], which allows for the computation of custom properties on-the-fly. Equivalent simulations performed on the two software packages yielded consistent results, as already checked by other authors on similar systems (96). In this work, we simulated two different kinds of systems: bulk and confined simulations. The force field pameterization and simulation setups are similar to those used in previous works (86). Bulk simulations of the electrolyte, described by a simple point charge model [i.e., water with the SPC/E model (97) and NaCl ions modeled as Lennard–Jones particles with point charges (98)], were performed with 3D periodic boundary conditions (PBCs) with “tin-foil” boundary conditions. The confined system also included gold electrodes modeled as Lennard-Jones particles (99) with a Gaussian charge density fluctuating in magnitude (see below). Simulations performed with MetalWalls were carried out under 2D PBCs, while those with LAMMPS used 3D PBCs and a finite field to impose $\mathrm{\Delta }\mathrm{\Psi }$ (100, 101). In all the confined simulations, the spacing between the electrodes $d$ was equilibrated at constant atmospheric pressure and production runs were carried out at fixed separation. All the simulations are performed in the NVT ensemble and the temperature is fixed at $298$ K using a Nosé–Hoover chain thermostat (102), with five chained thermostats having all the same time constant of 1 ps. In confined simulations, we enforce the additional constraints of constant applied potential and global electroneutrality. The constant potential condition is ensured by the fluctuating charge method (38).

More simulation details and a list of all simulation runs can be found in SI Appendix.

3.2. Frequency-Dependent Impedance from Charge Fluctuations and Bulk Conductivity Spectrum.

The frequency-dependent electrical impedance and admittance were calculated from the dynamics of the equilibrium fluctuations of the electrode charge sampled from MD simulations, using the fluctuation–dissipation relation introduced in our previous work (54):

$$\begin{array}{c}\hfill Y(\omega )=\beta \left[i\omega ⟨\delta Q^2⟩+{\omega }^2{\int }_0^{\infty }⟨\delta Q(0)\delta Q(t)⟩e^{-i\omega t}\mathrm{d}t\right],\end{array}$$ [15]

where $Y=1/Z$ is the admittance, $Z$ is the impedance, $\beta =1/k_{\mathrm{B}}T$ with $k_{\mathrm{B}}$ the Boltzmann constant and $T$ the temperature, $\omega$ is the angular frequency, $Q$ is the electrode charge, and $\delta Q=Q-⟨Q⟩$, with $⟨·⟩$ denoting the canonical average. The numerical calculation of the Fourier–Laplace transform in Eq. 15 was performed as in ref. 54, using Filon–Lagrange integration and by applying a windowing procedure to $⟨\delta Q(0)\delta Q(t)⟩$ to suppress the numerical noise at large times. In this work, we used a normalized version of the windowing function, which allows preserving the magnitude of the original function at short times. A direct comparison between the raw and windowed autocorrelation functions is shown in SI Appendix, Figs. S10–S12.

In principle, the frequency-dependent conductivity in bulk systems can be estimated from the dynamics of the total dipole moment $\mathbf{M}$, calculated with unwrapped coordinates and including both salt and solvent species (67). In particular, the susceptibility $\chi (\omega )$ can be estimated from the total dipole moment (69) via

$$\begin{array}{c}\hfill \chi (\omega )=\frac{\beta }{3V}\left[⟨\delta {\mathbf{M}}^2⟩-i\omega {\int }_0^{\infty }⟨\delta \mathbf{M}(0)\delta \mathbf{M}(t)⟩e^{-i\omega t}\mathrm{d}t\right],\end{array}$$ [16]

where $V$ is the system volume. The frequency-dependent conductivity can be then calculated as

$$\begin{array}{c}\hfill \sigma (\omega )=i\omega \chi (\omega )=i\omega {\chi }^{\prime }(\omega )-\omega {\chi }^{″}(\omega ),\end{array}$$ [17]

where ${\chi }^{\prime }$ and ${\chi }^{″}$ denote the real and imaginary parts of the susceptibility. We used this approach to estimate the frequency-dependent conductivity from bulk water simulations. In the presence of free charge carriers (i.e., for finite salt concentrations), $⟨{\mathbf{M}}^2⟩$ becomes a divergent quantity, so the frequency-dependent conductivity has to be calculated using a different approach. The problem can be circumvented using electric currents $\mathbf{J}(t)=d\mathbf{M}/dt$ instead of the total polarization (67). The conductivity can be rewritten in the following form

$$\begin{array}{c}\hfill \sigma (\omega )=\frac{\beta }{3V}{\int }_0^{\infty }⟨\delta \mathbf{J}(0)\delta \mathbf{J}(t)⟩e^{-i\omega t}\mathrm{d}t.\end{array}$$ [18]

The Fourier–Laplace transform in Eqs. 16 and 18 was calculated numerically with the same windowing and integration scheme used to solve the transform in Eq. 15.

For finite salt concentrations, the DC conductivity [$\sigma (\omega \to 0)={\sigma }_0$] can be estimated from the behavior of $\mathfrak{R}[\sigma (\omega )]$ at low frequency (SI Appendix, Fig. S5). Alternatively, it can be computed using the Einstein–Helfand formalism which relates the total dipole moment of ions ${\mathbf{M}}_{\mathrm{ion}}$ to ${\sigma }_0$ (68):

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim}}⟨\mathrm{\Delta }{\mathbf{M}}_{\mathrm{ion}}^2(t)⟩=\frac{6V}{\beta }{\sigma }_{0}t+2⟨{\mathbf{M}}_{\mathrm{ion}}^2⟩,\end{array}$$ [19]

where $t$ is the time. In practice, we fit $⟨\mathrm{\Delta }{\mathbf{M}}_{\mathrm{ion}}^2(t)⟩$ with a line and extract ${\sigma }_0$ from the slope of the fitted line (SI Appendix, Fig. S2).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Input files and raw data for figures have been deposited on Zenodo (103).