Realistic atomic model for charge storage and charging dynamics of amorphous porous carbons

Abstract

Amorphous porous carbons have been widely used as electrodes for energy storage. However, due to their structural heterogeneity and complex pore topology, the absence of reliable atomic carbon models hinders understanding energy storage mechanisms through molecular simulations. Here, we develop a modeling approach capturing the rich experimental information of small angle X-ray scattering, gas adsorption, and material density, integrating three-dimensional morphology construction with atomic structure generation. Using the built atomic model, constant-potential molecular simulations are performed to investigate the capacitive behavior of amorphous porous carbons, well-validated by experimental measurements of capacitance and impedance. Voronoi sphere analyses uncover the interplay between pore structure and charge storage: ultramicropores (< 0.7 nm) are found to enhance the capacitance through ion exchange, while large micropores (> 0.7 nm) exhibit lower capacitance with negligible ion number change. The charging dynamics are quantified by the proposed multi-scale impedance model, bridging microscopic simulations and macroscopic experiments. This modeling framework paves the way toward molecular understanding of energy storage in amorphous porous materials.

Introduction

Porous carbon materials, featuring superior specific surface area, excellent conductivity, great abundance, and low cost^1,2, have been widely used as electrodes for supercapacitors³ and batteries⁴. Structures of nanoporous carbons, including pore size, pore shape, pore connectivity, and specific surface area (SSA), are of great significance to energy storage^5,6,7. To examine the effects of the pore structure on energy storage, experiments of porous carbon supercapacitors uncovered that the pore with size close to the ionic diameter could enhance the capacitance^8,9, and pore walls with smaller areas of graphene-like domains exhibited higher capacitance^10,11.

To understand the capacitive behavior of nanoporous electrodes at the atomic level, molecular simulations have been playing a pivotal role⁶. Using ideal pore models, like nanoslits and nanotubes, to epitomize porous carbons¹², molecular simulations could effectively explore the influence of pore size on charge storage. Monte Carlo simulations of nanoslits revealed that the electrode with a wider pore size distribution (PSD) has a lower capacitance, ignoring the pore connectivity of porous carbons¹³. Adopting quenched molecular dynamics (QMD) to obtain amorphous nanoporous carbon models¹⁴, the pore with larger curvature, exerting higher confinement on electrolyte ions, is found to store charge more efficiently¹⁵. Nevertheless, QMD-generated models rely on the simulation cooling rate and material density¹⁶, and none of the other experimental data is taken into account¹⁷. Notably, small angle x-ray scattering (SAXS) can capture three-dimensional (3D) structures of porous carbons¹⁸, which could be used by the Gaussian random field (GRF) method to construct nanoporous models displaying the morphology of electrode matrix¹⁹. These 3D electrode matrix models demonstrated that the concave surface of porous carbons helps to improve charge storage^19,20. However, the van der Waals (vdW) interactions between the electrode and electrolyte are neglected due to the absence of carbon atoms in 3D electrode matrix models, which is key to modeling electrode-electrolyte interfaces^21,22. Therefore, realistic atomic models, based on more experimental information^23,24, are required to investigate the capacitive behavior of amorphous porous carbons, especially the charging dynamics related to the ion transport inside electrode pores¹².

Herein, we developed a realistic amorphous electrode model with the atomic structure, built upon the SAXS, gas adsorption, and density of nanoporous carbons. With this carbon model, we conducted constant-potential molecular dynamics (MD) simulations of carbon electrodes with the ionic liquid electrolyte to study their capacitance and charging process. The MD-obtained differential capacitance is in good agreement with experimental measurements of the same electrode and electrolyte, with regard to the magnitude and trend. To dissect the energy storage mechanism, the Voronoi sphere method was proposed to analyze the in-pore ion structure and transport across different-sized pores under polarizations. Analyses demonstrate that ultramicropores (< 0.7 nm) dominate charge storage through the ion exchange process, while large micropores (> 0.7 nm) make minor contributions due to the weakly changed numbers of in-pore cations and anions. Particularly, ultramicropores render different capacitances, depending on how they connect with large micropores. Meanwhile, multi-scale models were developed not only to characterize impacts of the pore size on the charge storage and charging dynamics, but also to bridge molecular simulations and macroscopic measurements.

Results

Atomic model of amorphous porous carbons

The 3D structure of the selected porous carbon (YP-50F, density of 0.86 g cm⁻³) is obtained by the SAXS measurements, and the SAXS profile for nanopores was extracted with the scattering vector $q$ ranging from 1 to 6 nm⁻¹ (Fig. 1a). Using the classical GRF (cGRF) method²⁵, the SAXS profile can be characterized by a pair of geometry statistical parameters: the domain length $l$ refers to the periodical structure of the pore void and pore wall, and the correlation length $\xi$ accounts for the association in the pore size, direction, and shape between neighboring pores²⁶. These parameters originate from the Teubner-Strey model²⁶, which describes SAXS profiles with scattering peaks induced by periodic arrangements of pore walls. In this model, the scattering intensity $I$ is a function of scattering vector $q$ as$:$ $I\left(q\right)={I}_{0}/\left(1+{c}_{1}{q}^{2}+{c}_{2}{q}^{4}\right)$, where ${I}_{0}$ is the scattering intensity at $q=0$, and ${c}_{1}$, ${c}_{2}$ are fitting coefficients for the model (details see Supplementary Section 1). The two GRF parameters could be obtained by $\xi=1/\scriptstyle\sqrt{\scriptstyle\sqrt{\frac{1}{4{c}_{2}}}+\frac{{c}_{1}}{{4c}_{2}}}$ and $l=2\pi /\scriptstyle\sqrt{\scriptstyle\sqrt{\frac{1}{4{c}_{2}}}-\frac{{c}_{1}}{4{c}_{2}}}$, which are adopted to access the position of scattering peak ${q}_{\max }$:

$${q}_{\max }={\left[{\left(\frac{2\pi }{l}\right)}^{2}-\frac{1}{{\xi }^{2}}\right]}^{\frac{1}{2}}$$

(1)

Fig. 1: Realistic atomic model of amorphous nanoporous carbons.

a SAXS intensity versus scattering vector $q$ of the porous carbon sample in experiments (gray dots), fitting curves by cGRF method (blue) and the mGRF method (red), plotted in log-log scale. b 3D electrode matrix structure generated by mGRF model (top panel) and the atomic structure obtained by H3D-RMD (bottom panel). c PSD from the experimental gas adsorption (black) and GCMC simulation of the atomic carbon model (red). Source data are provided as a Source Data file.

However, fitting the experimental SAXS curve by cGRF yields l = 4.46 nm and ξ = 0.19 nm, giving $\frac{2\pi }{l} < \frac{1}{\xi }$, suggesting that the scattering peak vanishes and a ‘Guinier knee’ occurs²⁷ (detailed in Supplementary Section 1). This conflicts with the fact that the SAXS profile exhibits two scattering peaks around 3.4 and 4.9 nm⁻¹ (Fig. 1a). To account for multiple scattering peaks, the GRF method was developed with multiple sets of statistical geometry parameters, named multiple scattering peaks GRF (mGRF) method, and its autocorrelation function is formulated as:

$${g}_{{mGRF}}\left(r\right)=\mathop{\sum }\limits_{i=1}^{N}{a}_{i}\frac{1}{\cosh \left(\frac{r}{{\xi }_{i}}\right)}\frac{\sin \left(\frac{2\pi r}{{l}_{i}}\right)}{\frac{2\pi r}{{l}_{i}}}$$

(2)

where the weight ${a}_{i}$ is related to the fraction of the surface area, and N is determined based on the number of scattering peaks (Supplementary Fig. 2).

Using the mGRF method, the two scattering peaks of SAXS are captured, situated at 3.4 and 4.7 nm^-1, very close to the experimental observation (Fig. 1a). Based on mGRF fitting of the SAXS data (Supplementary Section 2), thousands of 3D electrode matrix models for amorphous nanoporous carbons were generated, incorporating the pore void fraction from gas adsorption measurements, and then filtered by the experimental SAXS. To validate the SSA of the 3D electrode matrix model, the Monte Carlo integration method²⁸ was taken to calculate the SSA from the pore structure (defined as ${\rho }_{{matrix}}\left({{\bf{x}}}\right)$, see Supplementary Section 1). The filtered 3D electrode matrix model was found to have an SSA of 1573 m2 g^-1 (top panel of Fig. 1b), close to the experimental value (1517 m² g^-1) evaluated by the quenched solid density functional theory (QSDFT) model²⁹. As a comparison, the 3D electrode matrix model obtained by the cGRF method gave an SSA of 1194 m² g^-1, further from the experiment.

To take vdW interactions between the electrode and electrolyte into account, we proposed the hybrid 3D reverse molecular dynamics (H3D-RMD) method to construct the amorphous porous carbon model with the atomic structure, combining the 3D-RMD with the reactive force field MD. In 3D-RMD simulations, the pore structure of the atomic model is expressed by the spatial occupation of atoms ${\rho }_{{atom}}\left({{\bf{x}}}\right)$, and each atom occupies a sphere with a vdW radius ${r}_{{vdW}}$:

$${\rho }_{{atom}}\left({{\bf{x}}}\right)=H\left({r}_{{vdW}}- \mathop{\min }\limits_{{{{\bf{R}}}}_{i}}\left\{{||}{{{\bf{R}}}}_{i}-{{\bf{x}}}{||}\right\}\right)$$

(3)

where $H$ is the Heaviside step function, and $\left\{{{{\bf{R}}}}_{i},{i}=1,\ldots,{N}_{{atom}}\right\}$ denote the atom positions. Here, ${\rho }_{{atom}}\left({{\bf{x}}}\right)=1$ represents the pore wall, and ${\rho }_{{atom}}\left({{\bf{x}}}\right)=0$ means the pore void. Therefore, the difference in pore structure between the atomic model and the mGRF-generated 3D electrode matrix model is expressed as:

$${\chi }^{2}=\frac{1}{V}{\int }_{V}{\left[{\rho }_{{atom}}\left({{\bf{x}}}\right)-{\rho }_{{matrix}}\left({{\bf{x}}}\right)\right]}^{2}{{\rm{d}}}{{\bf{x}}}$$

(4)

where V is the box volume. Accordingly, the 3D-RMD simulations minimize ${\chi }^{2}$ to guarantee that the pore structure of the resulting atomic model matches that of the 3D electrode matrix model. The reactive force field MD ensures the stability of the chemical structure (details see Supplementary Section 3). During the electrode construction by H3D-RMD, the density of porous carbon was taken to determine the total atom number in the model, and the annealing process was employed to produce the atomic structure of the amorphous porous carbon electrode²⁴ (bottom panel of Fig. 1b, the selection of the reactive force field is discussed in the Methods and Supplementary Section 3). The SSA of the atomic model is calculated as 1694 m² g^-1, matching the experiment. To further validate the atomic model, the PSD was evaluated through grand canonical Monte Carlo (GCMC) simulations (see Supplementary Fig. 6 and Methods), which aligns with the gas adsorption measurement (Fig. 1c), confirming the reliability and accuracy of the atomic model constructed by mGRF and H3D-RMD methods. Meanwhile, the simulated TEM image of the atomic model was found to be similar to the experimental one (Supplementary Fig. 4).

To verify the generality of the approach to constructing realistic atomic models, we applied the same modeling process, including the mGRF and H3D-RMD, to two additional amorphous porous carbons (YP-80F and ACC-5092-20, Supplementary Fig. 5). The resulting atomic models were validated by gas adsorption and cyclic voltammetry measurements. The simulated adsorption isotherms, PSDs and capacitances all agree with the experiments (Supplementary Figs. 6-7 and Supplementary Table 1), demonstrating that the methodology consisting of mGRF and H3D-RMD could be a generalized tool for realistically modeling amorphous porous carbons.

Capacitance and ion packing of nanoporous carbon electrodes

Using the amorphous atomic model in Fig. 1b, constant-potential MD simulations³⁰ were performed to explore the charge storage of supercapacitors with nanoporous carbon electrodes and ionic liquid [EMIM][BF₄] electrolyte (Fig. 2a). With a series of simulations at equilibrium under different applied voltages, the gravimetric differential capacitance ${C}_{d}$ was acquired (see Methods), consistent with electrochemical measurements of the same nanoporous carbon and electrolyte in both the magnitude and curve shape (Fig. 2b). This consistency is hardly achieved by mimicking the porous carbon with the widely-used nanoslit model (Fig. 2b). The dependence of C_d on the electrode potential could be understood by the accumulated ions within electrode pores (Fig. 2c). Specifically, at the potential of zero charge (PZC), the electrolyte wets the electrode²⁰, with anions exhibiting a higher accumulation than cations. This leads to a positive PZC value (0.14 V), close to the experiment (0.16 V). Under polarization, the ratio of ion number variation to potential difference increases for both negative and positive electrodes (inset of Fig. 2c), corresponding to the U-shaped ${C}_{d}$ curve. Furthermore, ${C}_{d}$ of the positive electrode is higher than the negative electrode, both in the simulation and experiment. This could be attributed to the smaller size of anions, making them more likely to approach pore walls and display stronger image charge effects³¹.

Fig. 2: Capacitance of nanoporous carbon supercapacitor and its origin.

a Configuration of MD simulation system with the porous carbon electrode and ionic liquid electrolyte ([EMIM][BF₄]). b Differential capacitance of experiments (black), atomic model (red) and nanoslit model (blue) for porous carbons. The nanoslit model uses the average pore size of the porous carbons. Dashed lines represent the PZC values. c Total ion number as a function of the electrode potential. The subfigure represents the rate of ion number change with the electrode potential. d Schematic of the Voronoi sphere method. Red dots represent in-pore Voronoi nodes; the pore size d is defined as the diameter of the Voronoi sphere (in dark grey) with the Voronoi node as the center. e Number of ions in different-sized pores at the PZC. Pore sizes are classified into ultramicropores ( < 0.7 nm) and large micropores ( > 0.7 nm). f, g Change of ion number in different-sized pores, compared to the PZC, for negative (f) and positive (g) electrodes. h–j Local electrode charge (LEC) of ions in different-sized pores at the PZC (h), under negative (i) and positive (j) polarizations. Error bars in b, c, e–j indicate one standard deviation of four independent simulations. Source data are provided as a Source Data file.

The number of ions in the whole porous electrode gives a limited understanding of the charge storage, because the impacts of local structures are averaged out¹⁵. To assess how the ion packing inside different-sized pores affects the capacitance, the Voronoi sphere method was proposed to characterize local pore structures and quantify in-pore ion distributions. The pore size is defined as the diameter of the Voronoi sphere (Fig. 2d and Supplementary Section 5), and then pores are categorized into two types: ultramicropores (< 0.7 nm) and large micropores (> 0.7 nm). The number of ions in different-sized pores is calculated at the PZC and under the applied voltage of 2 V (Fig. 2e-g, calculation details see Supplementary Section 6). At the PZC, the ultramicropores tend to adsorb more ions (both cations and anions) than large micropores. Specifically, anions predominantly occupy the 0.42 nm pore and cations prefer the 0.49 nm pore, demonstrating that anions, with smaller size and stronger image charge effects, can enter smaller pores more effectively. Under polarization, the ion response presents two distinct trends depending on pore sizes. In ultramicropores, both negative and positive electrodes show increased counter-ions and decreased co-ions, indicating an ion exchange process (Fig. 2f-g). In stark contrast, large micropores have negligible changes in both cation and anion numbers, suggesting that the electrolyte in these pores behaves like ion reservoirs.

To evaluate the charge stored by different electrode pores, the local electrode charge (LEC) is calculated by taking the concept of ‘counter charge’¹⁵, which quantifies the charge of electrode atoms surrounding adsorbed ions (Fig. 2h-j). At the PZC (Fig. 2h), the LEC gets enlarged as the pore size decreases, and the LEC near anions is higher than near cations, due to their smaller size allowing them to get closer to the pore wall. Under polarization, the LEC is significantly larger in ultramicropores than in large micropores of both negative (Fig. 2i) and positive (Fig. 2j) electrodes. This is attributed to the increased separation and decoupling of cation-anion pairs in ultramicropores (Supplementary Fig. 11), and the ion could induce more charge than an electrically neutral cation-anion pair^32,33. The higher LEC in ultramicropores means more efficient screening of adsorbed counter-ions, leading to an enhanced capacitance of ultramicropores¹⁹.

To scrutinize the in-pore ion packing, the radial ion number density is quantified for different-sized pores (Fig. 3a-c and Supplementary Fig. 12, details see Supplementary Section 5). At the PZC, steric effects in ultramicropores enforce a monolayer of ions to stay at the pore center (Fig. 3a). As the pore size increases, ions nearly deplete in the pore center, with both cations and anions adsorbing on the pore wall. Further enlarging pore size introduces a second layer of ions in the pore center, leading to a peak-valley-peak density distribution. Under polarization, the changes in the number density of ultramicropores align with the ion exchange process (Figs. 3b-c vs. 2f-g). For large micropores, counter-ions migrate from the pore center to the pore wall (Fig. 3b), while the depletion regions at the PZC are enriched with co-ions desorbed from pore walls, forming another ion layer (Supplementary Fig. 13, schematics see Supplementary Fig. 14). During such a process, some ions may leave large micropores but are almost commensurably replenished, resulting in little change of the ion number within these pores.

Fig. 3: Impact of pore size and connectivity on ion packing and charge storage.

a Radial number density of cations for different-sized pores at the PZC. r = 0 denotes the pore center. b, c Change of radial number density of cations in negative (b) and positive (c) electrodes, compared with the PZC. d Schematic of ultramicropores with different connections to large micropores. Ultramicropores, showing Voronoi sphere overlap with large micropores, are defined as facial ultramicropores, and the others are deep ultramicropores. e LEC of ions in deep and facial ultramicropores. f Change of ions in deep and facial ultramicropores, compared with the PZC. Error bars in (e) and (f) indicate one standard deviation of four independent simulations. Source data are provided as a Source Data file.

Another key factor affecting charge storage is how the ultramicropores connect with large micropores. Based on their connections to large micropores, ultramicropores are classified into deep and facial ones (Fig. 3d, details see Supplementary Section 7). Figure 3e shows that deep ultramicropores have a larger LEC than facial ones in both negative and positive electrodes, leading to higher capacitance. This is because cation-anion pairs are more separated (Supplementary Fig. 16a-b), and ion pairs get more decoupled in deep ultramicropores (Supplementary Fig. 16c-d). As a result, deep ultramicropores display stronger ion exchange than the facial ones in both negative and positive electrodes (Fig. 3f). Meanwhile, deep ultramicropores in the positive electrode tend to store more charge than negative electrode due to stronger ion exchange (Fig. 3f), which renders molecular insight into why the capacitance of the positive electrode is higher than the negative one (Fig. 2b).

Charging dynamics and multi-scale impedance model

We then focus on the charging dynamics of the nanoporous carbon supercapacitor. To recognize how the charging process responds to the applied voltage (Fig. 4a), a fractal equivalent circuit (FEC) model is developed for nanoporous carbon electrodes possessing the geometry of the atomic model in Fig. 1b. In the FEC model, the charging processes of nanoporous electrodes are characterized at two levels (Fig. 4b): level I represents ion transport in ultramicropores and level II accounts for ion diffusion in large micropores. The impedance of the FEC ${Z}_{{FEC}}$ is derived (details in Supplementary Section 8):

$$\begin{array}{c}{Z}_{{FEC}}\left(\omega \right)=\sqrt{{R}_{{{\rm{II}}}}\frac{{L}_{{{\rm{II}}}}\sqrt{{R}_{{{\rm{I}}}}}\coth \left[{L}_{{{\rm{I}}}}\sqrt{{R}_{{{\rm{I}}}}{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{I}}}}2\pi {r}_{{{\rm{I}}}}}\right]}{{n}_{{{\rm{I}}}}\sqrt{{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{I}}}}2\pi {r}_{{{\rm{I}}}}}+\sqrt{{R}_{{{\rm{I}}}}}{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{II}}}}2\pi {r}_{{{\rm{II}}}}{L}_{{{\rm{II}}}}\coth \left[{L}_{{{\rm{I}}}}\sqrt{{R}_{{{\rm{I}}}}{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{I}}}}2\pi {r}_{{{\rm{I}}}}}\right]}}\\ \coth \left[{L}_{{{\rm{II}}}}\sqrt{{R}_{{{\rm{II}}}}\frac{{n}_{{{\rm{I}}}}\sqrt{{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{I}}}}2\pi {r}_{{{\rm{I}}}}}+\sqrt{{R}_{{{\rm{I}}}}}{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{II}}}}2\pi {r}_{{{\rm{II}}}}{L}_{{{\rm{II}}}}\coth \left[{L}_{{{\rm{I}}}}\sqrt{{R}_{{{\rm{I}}}}{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{I}}}}2\pi {r}_{{{\rm{I}}}}}\right]}{{L}_{{{\rm{II}}}}\sqrt{{R}_{{{\rm{I}}}}}\coth \left[{L}_{{{\rm{I}}}}\sqrt{{R}_{{{\rm{I}}}}{\left(j\omega \right)}^{\alpha }{C}_{{{\rm{I}}}}2\pi {r}_{{{\rm{I}}}}}\right]}}\right]\end{array}$$

(5)

where ${R}_{{{\rm{I}}}}$, ${R}_{{{\rm{II}}}}$ represent resistances per unit length along the pore axis; ${r}_{{{\rm{I}}}}$, ${r}_{{{\rm{II}}}}$ denote the average half-pore sizes, and ${L}_{{{\rm{I}}}}$, ${L}_{{{\rm{II}}}}$ are pore lengths (defined as the length of ion path in Fig. 4b and Supplementary video); ${n}_{{{\rm{I}}}}$ is the number of ultramicropores branching from large micropores (Supplementary Table 2). The capacitance (C_I and C_II) and in-pore ionic conductivity (${\sigma }_{{{\rm{I}}}}=1/{R}_{{{\rm{I}}}}\pi {r}_{{{\rm{I}}}}^{2}$ and ${\sigma }_{{{\rm{II}}}}=1/{R}_{{{\rm{II}}}}\pi {r}_{{{\rm{II}}}}^{2}$) of ultramicropores and large micropores were then obtained, using the FEC model to fit the charging process of the supercapacitor system modeled by constant-potential MD simulations (Fig. 4a and Supplementary Table 3). The R² of the FEC fitting reaches 0.998, demonstrating that the FEC is capable of describing the MD-obtained charging process of the amorphous nanoporous carbon electrode. The FEC fitting renders higher capacitance of ultramicropores than large micropores (6.01 vs. 4.44 μF cm⁻²), supporting the aforementioned observation that ultramicropores show higher LEC (Fig. 2i-j). Meanwhile, the ionic conductivity of electrolytes in ultramicropores is much smaller than in large micropores (3.90 × 10⁻² vs. 2.33 × 10⁻¹ S m⁻¹), indicating that ultramicropores would tremendously slow down the ion transport. Based on the capacitance and in-pore ionic conductivity, the time constants of charging process are calculated for different pores, using $\tau=C{L}_{{eff}}^{2}/2\sigma r$, where ${L}_{{eff}}$ is the effective pore length accounting for the ion path. For ultramicropores, L_eff = 2.85 nm, and then τ_I = 25.06 ns; for large micropores, L_eff = 3.73 nm and τ_II = 1.77 ns. Therefore, the ultramicropores display a much slower charging process (Fig. 4c), suggesting that the charging could be bottlenecked by ultramicropores.

Fig. 4: Charging dynamics of porous carbon electrodes.

a MD-obtained charging process of the nanoporous carbon electrode under 2 V voltage (blue), fitted by the FEC model (red). b Schematic of the FEC model describing the charging process of MD simulation. Levels I and II account for the ion transport in ultramicropores and large micropores, respectively. An ion path is given in the schematic (the right panel). c Charging processes of ultramicropores and large micropores obtained by the FEC model. The charges have been normalized by their equilibrium values. d, e LEC of in-pore ions as a function of pore size for negative (d) and positive (e) electrodes. f Schematic of a real electrode consisting of packed carbon particles. g Nyquist plot of EIS from the electrochemical measurement and the prediction by the MIM. Source data are provided as a Source Data file.

To unveil the influence of the pore size on charging dynamics, time-dependent changes in LECs were analyzed for different-sized pores (Fig. 4d-e). Under polarization, the LECs in ultramicropores stabilize about 100 ns; whereas in large micropores, the changes of LECs become negligible after 10 ns, implying that the large micropores reach equilibrium around 10 ns (close to 5τ_II = 8.85 ns). The slower charging in ultramicropores arises from the more frequent collisions between ions and pore walls, compared to large micropores (Supplementary Fig. 17), manifested by the more confined ion motion paths in ultramicropores (Supplementary Fig. 18a-b vs. Supplementary Fig. 18c-d).

Considering that the real electrode consists of closely packed carbon particles on the current collector (Fig. 4f), the multi-scale impedance model (MIM) is constructed for the carbon supercapacitor cell, by scaling up porous carbon electrodes modeled in MD simulations to real carbon particles via the FEC model (details see Supplementary Section 10). Using the MIM, the electrochemical impedance spectroscopy (EIS) of a supercapacitor cell is assessed and found to quantitatively match experimental measurements (Fig. 4g). This illustrates that the MIM incorporating FEC is able to predict the charging process of nanoporous carbon supercapacitors at the macroscopic level, based on molecular simulations with the atomic electrode model developed in this work.

Discussion

We have developed a comprehensive modeling framework for investigating the energy storage of amorphous porous carbon electrodes. The electrode model was built on the SAXS, pore void fraction, SSA, PSD, and density of porous carbons. Capturing the multiple scattering peaks observed in SAXS measurements, the mGRF method was developed to generate the 3D electrode matrix model, incorporating the pore void fraction. The atomic structure was subsequently constructed for the amorphous porous carbon model, constrained by the material density, via the proposed H3D-RMD method, which helps to properly account for vdW interactions between the electrode and electrolyte. The atomic electrode model exhibits PSD and SSA, matching well with experimental measurements and thus confirming its reliability.

The atomic electrode model was utilized to explore the capacitive behavior of nanoporous carbon supercapacitors through constant-potential MD simulations, and the predicted differential capacitance was validated by experimental measurements in terms of both the magnitude and trend. To uncover the relationship between local pore structures (e.g., pore size and pore connectivity) and charge storage, the Voronoi sphere method was proposed to analyze the ion packing in different amorphous carbon pores. Ultramicropores (< 0.7 nm) exhibit ion exchange processes and higher capacitance, owing to more efficiently separated and decoupled cation-anion pairs; while large micropores (> 0.7 nm) show negligible ion number change and lower capacitance, since ions transport between the pore wall and the pore center, with cations and anions remaining paired. Notably, how ultramicropores connect with large micropores remarkably impacts the capacitance: deep ultramicropores could store more charge than facial ultramicropores, due to stronger image charge effects and ion exchange.

The charging process of amorphous nanoporous carbon electrodes was delineated by the proposed FEC model, which reveals that ultramicropores exhibit higher capacitance but smaller ionic conductivity than large micropores. The slower charging dynamics of ultramicropores originate from more frequent collisions between ions and pore walls. A multi-scale model was developed to predict the impedance of a porous carbon supercapacitor cell, based on MD simulations, and achieve quantitative agreement with electrochemical measurements, thereby bridging the computational microscopy and experimental macroscopy on the charging dynamics of nanoporous carbon electrodes.

This work provides a strategy for employing MD simulations to gain molecular insights into the charge storage and charging dynamics of amorphous porous carbon electrodes. The methods developed for modeling and analyzing amorphous porous carbons could be adopted for other fields where porous media play a vital role, such as catalyst³⁴, electrochemical desalination³⁵, hydrogen storage³⁶, carbon capture³⁷, gas separation³⁸, and oil recovery³⁹. Future work could take into account the flexibility of electrodes and polarizability of electrolyte force fields to explore their influence on the gas adsorption and energy storage in porous materials^40,41,42.

Methods

Material preparation and experimental measurements

The YP-50F and YP-80F were purchased from Kuraray Chemical, Japan; the ACC-5092-20 was purchased from Kynol, Germany; the polytetrafluoroethylene (PTFE) binder (60 wt% dispersion in water) was purchased from Aladdin; the electrolyte ([EMIM][BF₄]) was obtained from Lanzhou Yulu Fine Chemical Co., Ltd. and purified via the Schlenk line at 85 °C for 48 h before further use.

The PSD and SSA of the porous carbon were measured with a Quantachrome Autosorb IQ system. Specifically, porous carbon samples (~ 60 mg) were activated under vacuum for 24 h at 200 °C. Afterwards, liquid nitrogen baths were used to measure nitrogen adsorption isotherms at 77 K. The PSD and SSA were evaluated by adsorption isotherm curves, based on the QSDFT model²⁹. SAXS experiments were performed on the SAXS/WAXS SYSTEM (XENOCS, France) using a monochromatic X-ray source (λ = 0.15148 nm) and a Pilatus 100 K detector for data collection.

A porous carbon electrode film was prepared by mixing carbon powder (90 wt%) with PTFE binder (10 wt%) in ethanol. Thoroughly stirred by the grinding rod, the solvent was removed to yield a dough-like solid mixture. The resulting solid was transferred to the rolling press (MSK-2150, Shenzhen Kejing Star Technology Co., Ltd.) and rolled to give a carbon film of approximately 150 µm in thickness with the mass loading of ~10 mg cm⁻². Finally, the as-prepared film was dried under vacuum at 80 °C for 2 days. An oversized porous carbon film was used as a counter electrode, and an Ag wire (99.99% pure, Gaoss Union Optoelectronic Technology Co., Ltd) was employed as a pseudo-reference electrode. The above electrodes were fabricated into the Swagelok cell with the glassy carbon electrodes used as collectors and the Whatman GF/A membrane (Cytiva, Sweden, thickness 260 µm, pore size 1.6 µm) used as the separator. Before any electrochemical measurements, the Ag wire electrode is cleaned by sonication in ethanol. The as-prepared cells were left for over 24 h to ensure the open-circuit potential stable. All measurements were carried out under dry and oxygen-free conditions in an argon-filled glove box (H₂O < 0.01 ppm, O₂ < 0.01 ppm, temperature 25 °C ± 2 °C), using a BioLogic VMP-3e potentiostat. EIS measurements were performed to measure the differential capacitance ${C}_{d}$, using a multi-sinusoidal signal with an amplitude of 5 mV at each potential and the drift correction implemented in the BioLogic potentiostat. The ${C}_{d}$ is then calculated from the lowest frequency (f = 10 mHz) of the EIS curve by ${C}_{d}=-1/2\pi f{{\rm{Im}}}\left(Z\right)$, where ${{\rm{Im}}}\left(Z\right)$ is the imaginary part of the impedance. The cyclic voltammetry experiments were performed at a scan rate of 1 mV s^-1 with the potential difference ranging from 0 V to 1 V or 2 V for 5 cycles to ensure stable capacitance values obtained. More than 5 cells were tested for a single electrochemical measurement. The specific capacitances and specific currents were calculated based on the mass of the active material.

Construction of the atomic model for porous carbon

To build a 3D atomic structure of the nanoporous carbon, the mGRF model and the H3D-RMD method were proposed, in which the atomic model was built based on the experimental SAXS, pore void fraction, PSD, SSA, and density of nanoporous carbons. The mGRF model was developed to capture the structural information of the SAXS data and generate a 3D electrode matrix model with a grid resolution of 0.1 nm to resolve the spatial occupation of the pore wall (details see Supplementary Sections 1 and 2). The SAXS of the model is produced by the Fourier transform¹⁹, and the SSA of the model was calculated via the Monte Carlo integration approach, computing the surface areas of the accessible pore volume²⁸. The simulated TEM image of the atomic model was generated by the Prismatic software⁴³.

To overcome the limitation that the atomic structure is absent in the 3D electrode matrix model, the H3D-RMD method was developed (Supplementary Section 3), in which the annealing process is conducted to construct the atomic electrode model, with the temperature gradually decreasing from 15,000 K to 298 K at a rate of 20 K ps⁻¹. The Tersoff force field⁴⁴ was applied to govern the formation and breakage of interatomic bonds, ensuring the chemical stability of the atomic structure. The atomic structure obtained from the annealing process is relaxed via MD simulation of reactive force field ReaxFF⁴⁵, in which the temperature gradually increases to 1500 K and then decreases to 298 K. Alternative force fields for the annealing and relaxation processes would lead to different structures^46,47, as discussed in Supplementary Fig. 3. In this work, the usage of the Tersoff force field in the annealing process and the ReaxFF force field in the relaxation process could provide the best stability of the final structure. The relaxed atomic model was validated by comparing its PSD obtained from the adsorption isotherm of the GCMC simulation performed by LAMMPS⁴⁸.

MD simulation

The MD simulation system consists of two porous carbon electrodes immersed in the electrolyte (Fig. 2a), taking the Amber ff94 for electrode atoms⁴⁹ and the coarse-grained model for [EMIM][BF₄]⁵⁰. Simulations were performed in the NVT ensemble using the customized constant-potential MD program based on GROMACS⁵¹. The voltage between the two electrodes was maintained by the constant-potential method³⁰, which keeps the potential constant by the fluctuating charges on electrode atoms in each step. A GPU-accelerated mix-precision implementation of the constant potential method in the MD simulation was developed to enhance the calculation efficiency and accuracy. The system was maintained at 298 K using the Nosé-Hoover thermostat. The electrostatic interactions of the whole system, including electrode-electrode, electrode-electrolyte and electrolyte-electrolyte interactions, were calculated based on the particle mesh Ewald method, with the grid spacing of 0.1 nm for fast Fourier transform and the B-spline interpolation in order 4 for charge spreading in a real space grid. The vdW interactions were calculated by Lennard-Jones potentials under Lorentz-Berthelot combination rules. A cutoff length of 1.2 nm was set for both non-electrostatic and real-space electrostatic interactions. The system was initially annealed from 500 to 298 K for 10 ns, followed by another 20 ns at the PZC to reach equilibrium. The analyses were conducted based on the equilibrium system after performing simulations under polarization for 150 ns. The differential capacitance from the simulation was calculated by the partial derivative of the equilibrium charge on electrode atoms to the electrode potential⁵². The initial and final atomic coordinates from MD simulations are provided in Supplementary Data 1.