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

Domain: neuroscience
Published: 2026-02-06
Editorial outcome: minor_revision

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 cost1,2, 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 storage5–7. 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 capacitance8,9, and pore walls with smaller areas of graphene-like domains exhibited higher capacitance10,11.

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 storage19,20. 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 interfaces21,22. Therefore, realistic atomic models, based on more experimental information23,24, 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 ξ 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(q)=I0= 1+c1q2+c2q4, where I0 is the scattering intensity at q=0, and c1, c2 are fitting coefficients for the model (details see Supplementary Section 1). The two GRF parameters could be obtained by ξ=1= \sqrt{1 +c1} and l=2\pi= \sqrt{1-c1}, which are adopted to access the position of scattering peak qmax:
qmax = 2\pi \sqrt{2 \xi^2 - l^2 \over l^2 \xi^2} (1)

However, fitting the experimental SAXS curve by cGRF yields l=4.46 nm and ξ=0.19 nm, giving 2\pi < 1, 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(r)= \sum_{i=1}^{N} ai \exp(-r/\xi i) cos(2\pi r/li) (2)
where the weight ai 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 ρmatrix(x), 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 ρatom(x), and each atom occupies a sphere with a vdW radius rvdW: 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. The number of atoms in the model is expressed as:
ρatom(x)=H(rvdW - min ||Ri - x||) (3)
where H is the Heaviside step function, and Ri, i=1,...,Natom denote the atom positions. Here, ρatom(x)=1 represents the pore wall, and ρatom(x)=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= 1/V \int_V [ρatom(x)-ρmatrix(x)]^2 dx (4)
where V is the box volume. Accordingly, the 3D-RMD simulations minimize χ2 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 the gas adsorption isotherms. The simulated gas adsorption isotherms, PSD 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 Cd 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 Cd 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 Cd curve. Furthermore, Cd 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.

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 pair32,33. 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.

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 ZFEC is derived (details in Supplementary Section 8):
ZFEC(ω)= RII \sqrt{LI/RII} coth \sqrt{LI RI (jω)α CI 2πrI} + RII (jω)α CI 2πrII LI coth \sqrt{LI RI (jω)α CI 2πrI} / (coth \sqrt{LI RI (jω)α CI 2πrI} + \sqrt{RI (jω)α CI 2πrI} LII coth \sqrt{LI RI (jω)α CI 2πrI} RI (jω)α CI 2πrI) (5)
where RI, RII represent resistances per unit length along the pore axis; rI, rII denote the average half-pore sizes, and LI, LII are pore lengths (defined as the length of ion path in Fig. 4b and Supplementary video); nI is the number of ultramicropores branching from large micropores (Supplementary Table 2). The capacitance (CI and CII) and in-pore ionic conductivity (σI =1=RIπrI2 and σII =1=RII πrII2) 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 τeff=CL2 =2σr, where Leff is the effective pore length accounting for the ion path. For ultramicropores, Leff=2.85 nm, and then τI =25.06 ns; for large micropores, Leff=3.73 nm and τII =1.77 ns. Therefore, the ultramicropores display a much slower charging process (Fig. 4c), suggesting that the charging could be bottlenecked by ultramicropores.

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 5τII =8.85 ns). The slower charging in ultramicropores, compared to large micropores (Supplementary Fig. 17), is 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.

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 100K 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 Cd, using a multi-sinusoidal signal with an amplitude of 5 mV at each potential and the drift correction implemented in the BioLogic potentiostat. The Cd is then calculated from the lowest frequency (f=10 mHz) of the EIS curve by Cd = -1=2πfIm(Z), where Im(Z) 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 structures46,47, 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.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This paper reports a good quality work. However, authors study only one activated carbon of YP-50F. Authors mention that activated carbon has a complex structure. This is correct. Therefore, authors must study at least three kinds of activated carbon having different porosities. YP-50F is well-known but the purpose of science is to find general view and understanding. The nitrogen adsorption isotherm of YP-50F has a hysteresis, indicating the presence of slit-shaped mesopores. Authors may use activated carbon without mesopores and also heat-treated activated carbon having different crystallinity. Then, authors could discuss whether their approach is adequate or not. In particular, authors do not use the original experimental approaches. Then, a general view on activated carbon with the relevance to supercapacitor science is necessary for publication on Nature Communications. This is the main concern.\nI have following comments additionally.\n1. Specific surface area: Authors may describe the method of surface area evaluation. This is because many authors use BET method, even though the BET method gives rise to 10-30% overestimation. The correct evaluation is indispensable for comparison with the theoretical value. Discussion on page 4 is not correct.\n2. Recently, carbon science has rapid progresses in carbon structure modeling. Authors may refer more recent papers except for ref. 40.\n3. SAXS is a really good method to get the open and close porosity. Is there detailed explanation on the presence of two peaks.\n4. Is there any description on how authors determined the specific surface area, which is associated with the comment 1. Authors may show the adsorption isotherm. The surface area depends on the evaluation model, but the adsorption isotherm gives the truth.\n5. I wonder if the Voronoi sphere model is fit for description of the carbon pore structure.\nAuthors may measure XRD. The XRD of YP-50F indicates the presence of graphitic like structure. The nanoscale structure of activated carbon consists of the slit-shaped unit, even though the units form a 3D curve structure.\n6. Authors show two kinds of ultramicropores in Fig.3.e. Authors may apply high resolution TEM method to compare with the model. The carbon model consists of random packing of carbon atoms. Activated carbon consists of planar -electron conjugated unit, leading to relatively high electrical conductivity. I do not agree with this model. Authors introduce the term of facial ultramicropores. However, the model indicates the defect on the pore wall. As to the deep ultramicropore in the figure is not realistic. Activation reaction removes the unstable structure shown in the model.\nAuthors can discuss using pore size distribution. Also, authors can partially fill the ultramicropores with the ionic liquid. Then, authors can get a more reliable discussion.", "reviewer_2": "The manuscript presents the development of a molecular simulation framework to study charge storage mechanisms in realistic porous carbon electrodes. The authors employ a commonly used ionic liquid as the electrolyte and demonstrate how a realistic 3D porous structure can be generated by integrating experimental data such as SAXS and gas adsorption measurements. The study also includes analysis of charge storage and ion dynamics within these non-ideal porous environments.\nOverall, this is a compelling and timely contribution. However, I have reservations about its suitability for Nature Communications in its current form.\nMajor Comments:\nEmphasis on Methodology Buried in Supplementary Information: The most novel and impactful aspect of the manuscript is the methodology for constructing 3D porous structures that are consistent with experimental data. Unfortunately, this methodology is largely relegated to the Supplementary Information. I strongly recommend bringing this material into the main text. A clearer and more detailed exposition of this approach would significantly enhance the manuscript’s appeal and originality.\nLimited New Insights on Charge Storage Mechanisms: While the manuscript offers a solid analysis of ion dynamics and charge storage, the conclusions, such as the enhanced charge separation in ultramicropores (<0.7 nm), have already been reported in previous studies using idealized pore geometries. The introduction rightly notes that earlier studies often rely on slit or cylindrical pores. Therefore, it would be very valuable to explicitly compare the performance of the realistic structure to that of ideal geometries with equivalent average pore size and surface area. For instance, in Figure 2b, including data for slit or cylindrical pores as a baseline would allow readers to assess the added value of modeling realistic structures.\nValidation and Model Limitations: Figure 2b shows qualitative agreement between simulation and experiment, but notable quantitative discrepancies, particularly at positive voltages where differences exceed 50%. While the model captures the position of the potential of zero charge (PZC) very well, the overall mismatch raises important questions. Is the 3D structure generated truly representative of the real electrode? Or are the discrepancies due to limitations in the ionic liquid force field? Including a comparison with simulations on flat surfaces using the same ionic liquid could help disentangle these effects.\nGCMC Undersampling of Larger Pores: Figure 1c shows that the GCMC modeling underpredicts the occurrence of pores larger than 1.5 nm. Could this be a limitation of the small simulation volume (6 nm box size)? Does the H3D-RMD method inherently suppress the formation of larger pores? It would be helpful to see a discussion of the variability of the resulting structures, including: How sensitive are the final structures to the initial configuration? Would using a larger simulation box allow better sampling of mesopores? How reproducible is the SAXS agreement across different generated structures?\nSummary:\nThis manuscript has the potential to make a significant contribution, particularly through its integration of experimental data into realistic molecular modeling of porous carbon structures. However, its current focus underemphasizes the most novel aspects of the work, and does not yet fully demonstrate the advantages of realistic pore modeling over idealized geometries. Strengthening these aspects and addressing the discrepancies between simulations and experiments would, in my view, make the manuscript more suitable for a high-impact venue such as Nature Communications.", "reviewer_3": "The authors report the results of atomistic simulations aimed at modeling elechtrochemical storage devices in the form of supercapacitors, composed of amorphous carbon electrodes and an ionic liquid electrolyte. The focus is on the generation of realistic models for amorphous carbon electrodes, with a modelling approach aimed at reproducing experimental observables (namely a 3D structure obtained from SAXS measurements and a method adapted from the Gaussian random field method). The 3D structure is then discretized using a newly-introduced approach (\"H3D-RMD\") focusing on matching densities.\nThe generated 3D model is shown to match the experimental SAXS profile. Then, the discretized, atomistic model of an amorphous carbon electrode is shown to match to a satisfying accuracy the experimentally-measured pore size distribution. From constant-potential molecular dynamics simulations, the authors extract interesting insights regarding the charging mechanisms with an unexpected contribution from pores smaller than 0.7 nm. Finally, atomistic simulation results are extrapolated using an equivalent circuit to the macroscopic limite, allowing to compare the charging dynamics to experimental data (in particular impedance measurements).\nI don't have much to say regarding this work. The manuscript is well written, the results are clearly presented, and there are convincing comparisons to experimental data. The results presented here should appeal to a broad audience.\nThere is just one point that I'd like to hear from the authors before I can definitely recommend the work for publication. In the H3-RMD method and in the constant potential simulations, there are interatomic potentials involved. In particular, the H3-RMD method relies on the Tersoff and ReaxFF potentials (the precise parameterizations are cited by the authors), and the constant potential MD uses the Amber ff94 potential for the electrode atoms.\nThese models are known to be relatively imprecised when describing pure carbon structures. For instance, the Tersoff potential in its original parameterization for carbon struggles at reproducing the vibrational properties of some allotropes (see 10.1103/PhysRevB.81.205441). The accuracy of the potential might not be critical to the two steps (obtaining realistic electrode models and modeling the charging of the device), but I believe it should be tested.\nCan the authors perform additional simulations and swapping the models (or perhaps changing some of the parameters), to see how that affects the results, for the H3D-RMD step?\nFor the constant potential MD, recent results have shown that allowing for the electrodes to relax significantly affects the observed charging dynamics (10.1021/acsnano.5c07490). The influence of the interatomic potential seems important in this case. Since I suppose these are expensive simulations and the authors perhaps cannot perform a set of additional ones, could they discuss this point, and perhaps propose a strategy to test for the influence of the potential itself?\nFinally, in the constant potential MD simulations, what is the potential used for electrode-electrolyte interactions (it is not mentioned)? And are electrode-electrode electrostatic interactions included or not?"}, "author_response": "We have studied all the comments carefully and made a point-by-point response to each comment. Specifically, reviewers’ comments are copied in blue, and each comment is followed by our response in black. The manuscript has been revised, accordingly, in red. After each response to a comment, a brief summary is provided of what has been changed or added and where they are positioned in the revised manuscript and corresponding Supplementary Information.\nTo carefully make responses to all the comments, we have accomplished a series of necessary modeling and experimental work, performed the corresponding analysis, and reflected it all in the new, majorly revised version of the manuscript.\nFor your information, we briefly summarized the work done for this revision:\n1) Atomic models of two additional porous carbons were generated by the mGRF and the H3D-RMD methods developed here, and were validated by gas adsorption and cyclic voltammetry measurements. The simulated adsorption isotherms, PSDs and capacitances all match the experiments, indicating the generality of the modeling method.\n2) An extended comparison between the pore size of slit-shaped pores and the pore sizes measured by the Voronoi sphere was performed, further confirming that the Voronoi sphere method is suitable for describing the pore size of porous carbons.\n3) The details of methodology, including the theories about the peak positions of the SAXS curve and the H3D-RMD procedure, were moved from the Supplementary Information to the main text, further highlighting the novelty of the article.\n4) Comparison between the capacitance of the atomic model and the nanoslit model indicated that the atomic model for the porous carbon electrode is much more proper for exploring energy storage mechanisms.\n5) Additional simulations were performed to explore the impact of force fields when constructing atomic models using the H3D-RMD method.\n6) Other detailed information about modeling and experiment methods was provided: (1) surface area evaluation and (2) the simulation settings for CPM MD.\n7) New figures/data and discussions were added: (1) deep and facial ultramicropores in the atomic model and (2) discussions and outlook with reference to some literature."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Authors revised their manuscript considering my comments. The revised manuscript is quite nice.\nI recommend the acceptance of the present manuscript.", "reviewer_2": "I believe the authors did an excellent job addressing reviewers comments. They conducted additional simulations and analysis requested by reviewers. I believe the revised manuscript is ready for publication and will be of great interest to a broad community of researchers interested in nano science.", "reviewer_3": "The authors have satisfyingly addressed my comments."}, "author_response": ""}]

Decision letter